Instructors Solutions Manual
January 21, 2017 | Author: Tamara Knox | Category: N/A
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Description
Instructor's Solutions Manual
M. Asghar Bhatti
Advanced Topics in Finite Element Analysis of Structures with Computations Using Mathematica and Matlab
John Wiley & Sons, Inc. 2006
CHAPTER ONE
Essential Background
1.1 An engineering analysis problem is formulated in terms of the following second order boundary value problem -u x4 - u£ + u££ = x; 0 < x < 1
uH0L = 4 and u£ H1L = 1
Derive a suitable weak form for use with the Galerkin method. Clearly indicate how the boundary conditions will be handled.
2
Essential Background
With uHxL as an assumed solution the residual is eHxL = -uHxL x4 - x - u£ HxL + u££HxL
Multiplying by wHxL and writing integral over the given limits, the Galerkin weighted residual is Ÿ 0 H-u w x4 - w x - w u£ + w u££L dx = 0 1
Using integration by parts, the order of derivative in w u££ can be reduced to 1 as follows. Ÿ 0 Hw u££L dx = wH1L u£ H1L - wH0L u£ H0L + Ÿ 0 H-u£ w £ L dx 1
1
Combining all terms, the weighted residual now is as follows.
wH1L u£ H1L - wH0L u£ H0L + Ÿ 0 H-w Iu x4 + x + u£ M - u£ w £ L dx = 0 1
Consider the boundary terms wH1L u£ H1L - wH0L u£ H0L
Each one of these terms gives rise to two possibilities -wH0L u£ H0L
wH1L u H1L £
Either -u£ H0L is known or wH0L = 0
Either u£ H1L is known or wH1L = 0
From these requirements the possible boundary conditions are as follows: -u H0L is given
NBC 1 2
£
u£ H1L is given
EBC or
wH0L = 0 ï Must satisfy uH0L boundary condition
or
wH1L = 0 ï Must satisfy uH1L boundary condition
Given NBC for the problem: u£ H1L - 1 = 0
Rearranging: H u£ H1L Ø 1 L
Given EBC for the problem: uH0L - 4 = 0
therefore with admissible solutions Hthose satisfying EBCL: H w H0L Ø 0 L Thus the boundary terms in the weak form reduce to: wH1L Assuming admissible solutions the final weak form is as follows. wH1L + Ÿ 0 H-w Iu x4 + x + u£ M - u£ w £ L dx = 0 1
3
1.2 An engineering analysis problem is formulated in terms of the following ordinary differential equation 2
d u du ÅÅÅÅ ÅÅÅÅÅÅ - x ÅÅÅÅ ÅÅÅÅ = u; 0 < x < 1 dx dx2 duH0L duH1L uH0L = ÅÅÅÅÅÅÅÅ ÅÅÅÅÅ - 2; ÅÅÅÅÅÅÅÅ ÅÅÅÅÅ = 1 dx dx
Obtain a suitable weak form for the problem. What is the order of the differential equation? Is the boundary condition at x = 0 a natural or an essential boundary condition? Is the boundary condition at x = 1 a natural or an essential boundary condition?
(i) Second-order (ii) Natural (iii) Natural (iv)
4
Essential Background
With uHxL as an assumed solution the residual is eHxL = -uHxL - x u£ HxL + u££HxL
Multiplying by wi HxL and writing integral over the given limits, the Galerkin weighted residual is Ÿ 0 H-u wi - x u£ wi + u££ wi L dx = 0 1
Using integration by parts, the order of derivative in wi u££ can be reduced to 1 as follows. Ÿ 0 Hwi u££ L dx = wi H1L u£ H1L - wi H0L u£ H0L + Ÿ 0 H-u£ wi £ L dx 1
1
Combining all terms, the weighted residual now is as follows.
wi H1L u£ H1L - wi H0L u£ H0L + Ÿ 0 H-u wi - u£ Hx wi + wi £ LL dx = 0 1
Consider the boundary terms
wi H1L u£ H1L - wi H0L u£ H0L
Each one of these terms gives rise to two possibilities -wi H0L u£ H0L
wi H1L u H1L £
Either -u£ H0L is known or wi H0L = 0
Either u£ H1L is known or wi H1L = 0
From these requirements the possible boundary conditions are as follows: -u H0L is given
wi H0L = 0 ï Must satisfy uH0L boundary condition
NBC 1 2
£
u£ H1L is given
EBC
wi H1L = 0 ï Must satisfy uH1L boundary condition
or or
Given NBC for the problem: uH0L - u£ H0L + 2 = 0 u£ H1L - 1 = 0
£ i u H0L Ø u H0L + 2 yz Rearranging: jj z u£ H1L Ø 1 k {
wi H1L - HuH0L + 2L wi H0L
Thus the boundary terms in the weak form reduce to:
Assuming admissible solutions the final weak form is as follows. wi H1L - HuH0L + 2L wi H0L + Ÿ 0 H-u wi - u£ Hx wi + wi £ LL dx = 0 1
(v)
5
Linear solution Starting assumed solution: uHxL = a0 + x a1 Weighting functions Ø 81, x<
Substitute into the weak form and perform integrations to get: Weight
Equation
1
-2 a0 - a1 - 1 = 0
x
a0 5 a1 ÅÅ - ÅÅÅÅ Å3ÅÅÅÅÅ + 1 = 0 - ÅÅÅÅ 2
Solving these equations we get 16 15 :a0 Ø - ÅÅÅÅÅÅÅÅÅ , a1 Ø ÅÅÅÅÅÅÅÅÅ > 17 17 Substituting into the admissible solution we get the following solution of the problem. 1 uHxL = ÅÅÅÅÅÅÅÅÅÅ H15 x - 16L 17
1.3 Steady state heat flow through long hollow circular cylinders can be described by the following ordinary differential equation. dTHrL d ÅÅÅÅ ÅÅ Ik A ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ M + A Q = 0; ri < r < ro dr dr
THri L = Ti ; THro L = T0
where r is the radial coordinate, THrL is the temperature, k is the thermal conductivity, Q is the heat generation per unit area, A = 2 p r L the surface area, L is the length of the cylinder, ri is the inner radius, and ro is the outer radius. The boundary conditions specify the temperature on the inside and outside of the cylinder respectively. Derive finite element equations for a typical two node linear element for the problem with nodes at r1 and r2 . Assume k and Q are constant over the element. Note that A is a function of r and is not constant over the element.
Derivation of element equations
Element nodes: 8r1 , r2 <
r2 -r ÅÅÅÅÅÅ Interpolation functions, NT = I ÅÅÅÅÅÅÅÅ r2 -r1
r1 -r ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ r1 -r2 M
6
Essential Background
1 ÅÅÅÅÅÅ B T = dNT êdx = I ÅÅÅÅÅÅÅÅ r -r 1
2
kHrL = 2 k L p r
1 ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ M r -r 2
1
pHrL = 0
qHrL = 2 L p Q r
jij 2 k k jj ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅ{Å jj r2 Hr1 -r2 L2 T j k k = Ÿ r1 H2 k L p r B B L „r = jj 2 2 jj jj 2 k L p ijj ÅÅÅÅr22ÅÅÅ - ÅÅÅÅr21ÅÅÅ yzz j ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ k ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅ{Å k Hr1 -r2 L Hr2 -r1 L i r2 r2 y L p jj ÅÅÅÅ22ÅÅÅ - ÅÅÅÅ21ÅÅÅ zz
zyz zz zz zz z ij r22 r12 yz z zz 2 k L p j ÅÅÅÅ2ÅÅÅ - ÅÅÅÅ2ÅÅÅ z z k ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅ{ÅÅ z Hr2 -r1 L2 { i r2 r2 y 2 k L p jj ÅÅÅÅ22ÅÅÅ - ÅÅÅÅ21ÅÅÅ zz k ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅ{ÅÅ Hr1 -r2 L Hr2 -r1 L
r2 1 1 r Tq = Ÿ r1 H2 L p Q r NL „r = :- ÅÅÅÅÅ L p Q Hr1 - r2 L H2 r1 + r2 L, - ÅÅÅÅÅ L p Q Hr1 - r2 L Hr1 + 2 r2 L> 3 3
The complete element equations are as follows. ij 2 k L p jij ÅÅÅÅr22ÅÅÅ - ÅÅÅÅr21ÅÅÅ zyz jj ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ k ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅ{Å jj Hr1 -r2 L2 jj jj 2 2 jj jj 2 k L p ijj ÅÅÅÅr22ÅÅÅ - ÅÅÅÅr21ÅÅÅ yzz j ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ { kÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ k Hr1 -r2 L Hr2 -r1ÅÅÅÅLÅÅ 2
2
yz zz zz T zz ij 1 zz j 2 2 ij r2 r1 yz z zz k T2 2 k L p j ÅÅÅÅ2ÅÅÅ - ÅÅÅÅ2ÅÅÅ z z {Å z k ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅ Å Hr2 -r1 L2 { i r2 r2 y 2 k L p jj ÅÅÅÅ22ÅÅÅ - ÅÅÅÅ21ÅÅÅ zz k ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅ{Å Hr1 -r2 L Hr2 -r1 L
1 yz ijjj - ÅÅÅÅ3Å L p Q Hr1 - r2 L H2 r1 + r2 L yzzz z = jj z { k - ÅÅÅÅ1Å L p Q Hr1 - r2 L Hr1 + 2 r2 L z{ 3
1.4 Consider solution of the following second order boundary value problem using two node linear elements. 2
d u du ÅÅÅÅ ÅÅÅÅÅÅ = ÅÅÅÅ ÅÅÅÅ ; 0 < x < 100 dx dx2
uH0L = 50; uH100L = 10 (a) Show that the following is an appropriate weak form for a typical linear element with nodes at arbitrary locations x1 and x2 Ÿ x1 Hu£ Hwi + wi £ LL dx = 0 x2
where wi HxL are suitable weighting functions. (b) Using the weak form given in (a), and the assumed solution written in terms of following interpolation functions i u1 y i u1 y uHxL = H N1 N2 L jj zz; u£ HxL = H N£1 N£2 L jj zz k u2 { k u2 {
show that the element equations for a two node linear element for this problem are as follows.
7
ij Ÿx 2 HN1 + N£1 L N£1 dx Ÿx 2 HN1 + N£1 L N£2 dx yz u 1 jj 1 zz ij 1 yz ij 0 yz jj x zj z = j z j 2 HN + N£ L N£ dx x2 HN + N£ L N£ dx zz k u2 { k 0 { Ÿ Ÿ 2 2 2 1 2 2 x1 k x1 { x
x
(c) Carrying out integrations, the element equations in (b) can be expressed as follows. -L + 2 L - 2 y i u1 y i 0 y 1 i zz jj zz = jj zz ÅÅÅÅ ÅÅÅÅ jj 2L -L - 2 L + 2 { k u2 { k 0 { k
where L = x2 - x1 , the element length. Using three of these elements, with nodes located at 0, 60, 90, and 100 determine an approximate solution of the problem.
8
Essential Background
With uHxL as an assumed solution the residual is eHxL = u£ HxL - u££ HxL
Multiplying by wi HxL and writing integral over the given limits, the Galerkin weighted residual is Ÿ x1 Hwi u£ - wi u££ L dx = 0 x2
Using integration by parts, the order of derivative in -wi u££ can be reduced to 1 as follows. Ÿ x1 H-wi u££ L dx = wi Hx1 L u£ Hx1 L - wi Hx2 L u£ Hx2 L + Ÿ x1 Hu£ wi £ L dx x2
x2
Combining all terms, the weighted residual now is as follows. wi Hx1 L u£ Hx1 L - wi Hx2 L u£ Hx2 L + Ÿ x1 Hu£ Hwi + wi £ LL dx = 0 x2
Consider the boundary terms
wi Hx1 L u£ Hx1 L - wi Hx2 L u£ Hx2 L
Each one of these terms gives rise to two possibilities wi Hx1 L u£ Hx1 L
-wi Hx2 L u Hx2 L £
Either u£ Hx1 L is known or wi Hx1 L = 0
Either -u£ Hx2 L is known or wi Hx2 L = 0
From these requirements the possible boundary conditions are as follows: u£ Hx1 L is given
NBC 1 2
-u Hx2 L is given £
wi Hx1 L = 0 ï Must satisfy uHx1 L boundary condition
EBC or or
wi Hx2 L = 0 ï Must satisfy uHx2 L boundary condition
Given EBC for the problem: uHx1 L - u1 = 0 uHx2 L - u2 = 0
i wi Hx1 L Ø 0 yz z therefore with admissible solutions Hthose satisfying EBCL: jj k wi Hx2 L Ø 0 { All boundary terms vanish. Assuming admissible solutions the final weak form is as follows. Ÿ x1 Hu£ Hwi + wi £ LL dx = 0 x2
Assumed solution
9
i u1 y uHxL = H N1 N2 L jj zz ª NT d k u2 {
i u1 y u£ HxL = H N£1 N£2 L jj zz ª BT d k u2 { Weighting functions wi = Ni Weak form
2 £ £ Ÿx1 HNi + Ni L u dx = 0
x
Two equations
2 2 £ £ £ £ Ÿx1 HN1 + N1 L u dx = 0; Ÿx1 HN2 + N2 L u dx = 0
x
x
Writing together in a matrix form ‡
x2
x1
ij N1 + N1 yz jj z N£ N£2 £ zH 1 N + N 2 2 k { £
i0y i u1 y L jj zz dx = jj zz k0{ k u2 {
ij Ÿx 2 HN1 + N£1 L N£1 dx Ÿx 2 HN1 + N£1 L N£2 dx yz u 1 jj 1 zz ij 1 yz ij 0 yz jj x zj z = j z j 2 HN + N£ L N£ dx x2 HN + N£ L N£ dx zz k u2 { k 0 { Ÿ Ÿ 2 2 2 1 2 2 x x k 1 { 1 x
x
Linear assumed solution x-x2 ÅÅÅÅÅÅ uHxL = I ÅÅÅÅÅÅÅÅ x1 -x2
iu y x- x1 ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ j 1 zz x2 -x1 M j k u2 {
8n1, n2< = 8Hx - x2L ê Hx1 - x2L, Hx - x1L ê Hx2 - x1L< ê. x2 Ø Hx1 + LL; 8b1, b2< = 8D@n1, xD, D@n2, xD 2 2 yz ij ÅÅÅÅ12 Hs - 1L s zz jj z jj jj -Hs - 1L Hs + 1L zzz zz jj zz jj 1 zz jj ÅÅÅÅ s Hs + 1L zz jj 2 zz; Nu = jjjj 0 zz zzz jjj zz jj zzz jjj 0 zz jj zzz jjj 0 zz jj { k0
2 I ÅÅÅÅÅ2ÅÅÅÅÅÅ + ÅÅ2ÅÅ M jij ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅ L jj jj jj 4 s jj - ÅÅÅÅLÅÅÅÅ jjj jj 2 I ÅÅsÅÅ + ÅÅÅÅs+1 ÅÅÅÅÅÅÅ M 2 jjj ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅ2ÅÅÅÅÅÅ L Bu = jjjj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
Stress & strain interpolation:
2 T Ÿx1 Bu A Ns
s -1
s
zyz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz z {
ij 0 yz jj zz jj 0 zz jj zz jj zz jjj 0 zzz jj zz Ns = jjjj 1 zzzz; jj zz jj s zz jj zz jjj 0 zzz jj zz jj zz k0{
ij 0 yz jj zz jj 0 zz jj zz jj zz jjj 0 zzz jj zz Ne = jjjj 0 zzzz jj zz jj 0 zz jj zz jjj 1 zzz jj zz jj zz ks {
x
After carrying out integrations we get
k1 = ‡
x2
x1
k2 = ‡
x2
x1
ij 0 jj jj L Bu A NTs dx = ‡ Bu A NTs ÅÅÅÅÅÅ ds = jjjj 0 jj 2 -1 jj k0
0 0 -A
2A ÅÅÅÅ ÅÅ ÅÅ 3
0 0 0
4A ÅÅ ÅÅ - ÅÅÅÅ 3
0 0 A
2A ÅÅÅÅ ÅÅ ÅÅ 3
1
1 i0 0 0 0 L Ns NTe dx = ‡ Ns NTe ÅÅÅÅÅ ds = jjjj 2 -1 k0 0 0 0
0 L 0 y zz z 0 0 ÅÅÅÅL3Å z{
0 0 yz zz zz 0 0 zzzz zz z 0 0 z{
7
k3 = ‡
x2
k4 = ‡
x2
k5 = ‡
x2
rq = ‡
x2
x1
x1
x1
x1
1 i -1 0 L Ns BuT dx = ‡ Ns BuT ÅÅÅÅÅÅ ds = jjjj 2 4 2 -1 k ÅÅÅÅ3Å - ÅÅÅÅ3Å
1
0
ÅÅÅÅ32Å
0
1 i0 0 0 L Ne E NTe dx = ‡ Ne E NTe ÅÅÅÅÅÅ ds = jjjj 2 -1 k0 0 0
0 0 0y zz z 0 0 0 z{
0 0 LE
0
0 0 0
LE ÅÅÅÅ ÅÅ Å 3
1 i0 0 0 L 0 L Ne NTs dx = ‡ Ne NTs ÅÅÅÅÅ ds = jjjj L 2 -1 k 0 0 0 0 ÅÅÅÅ3Å
yz zz z {
0 0y zz z 0 0 z{
Lq ij ÅÅÅÅ ÅÅÅÅÅ y jj 6 zzz jj 2 L q zz L Nu q dx = ‡ Nu q ÅÅÅÅÅÅ ds = jjjj ÅÅÅÅÅÅÅÅ ÅÅÅÅ zzz jj 3 zzz 2 -1 jj L q zz k ÅÅÅÅ6ÅÅÅÅÅ { 1
Combining these terms we get the followign element equations ij 0 jj jj jj 0 jj jj jj jj 0 jj jj jj 1 jj jj jj - ÅÅÅÅ2Å jj 3 jjj jj 0 jjj jj k0
0 0
0 0
-A
2A ÅÅÅÅ ÅÅÅÅÅ 3
0
0
4A - ÅÅÅÅ ÅÅÅÅÅ 3
0 0
0
0
A
2A ÅÅÅÅ ÅÅÅÅÅ 3
0
-1
0
0
L
ÅÅÅÅ43Å
- ÅÅÅÅ23Å
0
0
0
0
0
-L
0
LE
0
- ÅÅÅÅL3Å
0
0
0
yz Lq i ÅÅÅÅ ÅÅÅÅÅ + FL zz u zz ji 1 zy jjjj 6 z z j j 2Lq 0 zzz jj u zz jj ÅÅÅÅÅÅÅÅ zz jjj 2 zzz jjj 3 ÅÅÅÅ zz jj zz jj Lq 0 zzz jjj u3 zzz jj ÅÅÅÅ zz jj zz jjj 6ÅÅÅÅÅ + FR zz jj a zz = jj 0 zz jj 1 zz jj 0 zz jj zz jj zj z j ÅÅÅÅL3Å zzzz jjjj a2 zzzz jjj 0 zz jj b zz jjj zj z 0 zzz jjj 1 zzz jjjj 0 zz k b { j 2 LE z k0 ÅÅÅÅ Å3ÅÅÅÅ z{ 0
yz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz {
5.2 Show that the following five parameter assumed stress field is suitable for a 4 node plane stress element. sx = b1 + b2 - b5 s
8
Multifield Formulations
sy = b1 - b2 - b4 t txy = b3 + b4 s + b5 t
Initial matrix of stress modes 0 0 ij 1 1 j P T = jjjj 1 -1 0 -t jj 1 s k0 0
-s y zz 0 zzzz zz t {
Step 1: Identify work stress modes t-1 ij ÅÅÅÅ ÅÅÅÅÅÅ jj 4 j j B T = jjjj 0 jjj s -1 j ÅÅÅÅ k 4ÅÅÅÅÅÅ
P 1 = 81, 1, 0<
0
1-t ÅÅÅÅ ÅÅ ÅÅÅ 4
s -1 ÅÅÅÅ ÅÅÅÅÅÅ 4
0
t-1 ÅÅÅÅ ÅÅÅÅÅ 4
ÅÅÅÅ14Å
H-s - 1L
0
ÅÅ41ÅÅ H-s - 1L 1-t ÅÅÅÅ ÅÅÅÅÅ 4
0
0
s +1 ÅÅÅÅ ÅÅÅÅÅÅ 4
0
s +1 ÅÅÅÅ ÅÅÅÅÅÅ 4
t+1 ÅÅÅÅ ÅÅÅÅÅÅ 4
1-s ÅÅÅÅ ÅÅÅÅÅÅ 4
Ÿ Ÿ P 1 B T dsdt = 8-1, -1, 1, -1, 1, 1, -1, 1<
w1 = 1 ï Work mode
P 2 = 81, -1, 0<
Ÿ Ÿ P 2 B T dsdt = 8-1, 1, 1, 1, 1, -1, -1, -1<
w2 = 1 ï Work mode
P 3 = 80, 0, 1<
Ÿ Ÿ P 3 B T dsdt = 8-1, -1, -1, 1, 1, 1, 1, -1<
w3 = 1 ï Work mode
P 4 = 80, -t, s <
1 1 1 1 Ÿ Ÿ P 4 B T dsdt = : ÅÅÅÅÅ , 0, - ÅÅÅÅÅ , 0, ÅÅÅÅÅ , 0, - ÅÅÅÅÅ , 0> 3 3 3 3 1 w4 = ÅÅÅÅÅ ï Work mode 3
P 5 = 8-s , 0, t<
yz zz zz 1-s zz ÅÅÅÅ Å4ÅÅÅÅÅ zz zz z 1 ÅÅÅÅ4Å H-t - 1L z{
ÅÅÅÅ41Å H-t - 1L 0
t+1 ÅÅÅÅ ÅÅÅÅÅÅ 4
9
1 1 1 1 Ÿ Ÿ P 5 B T dsdt = :0, ÅÅÅÅÅ , 0, - ÅÅÅÅÅ , 0, ÅÅÅÅÅ , 0, - ÅÅÅÅÅ > 3 3 3 3 1 w5 = ÅÅÅÅÅ ï Work mode 3 Stress matrix satisfying work criteria 0 0 ij 1 1 j P T = jjjj 1 -1 0 -t jj 1 s k0 0
-s y zz 0 zzzz zz t {
Step 2: Perform eigenvalue test to determine optimum set ij 1.0989 j C = jjjj 0.32967 jj k0
0
1.0989
0
0
1. -0.3 jij j j D = jj -0.3 1. jj 0. k 0.
t-1 jij ÅÅÅÅ4ÅÅÅÅÅÅ jj j B T = jjjj 0 jj jj s -1 k ÅÅÅÅ4ÅÅÅÅÅÅ
yz zz zz zz z 0.384615 {
0.32967
0 s -1 ÅÅÅÅ ÅÅÅÅÅÅ 4 t-1 ÅÅÅÅ ÅÅÅÅÅ 4
0. y zz 0. zzzz zz 2.6 { 1-t ÅÅÅÅ ÅÅ ÅÅÅ 4
0
0
ÅÅ41ÅÅ
ÅÅÅÅ14Å H-s - 1L
H-s - 1L
1-t ÅÅÅÅ ÅÅÅÅÅ 4
zyz zz zz zz zz zz 1 ÅÅÅÅ4Å H-t - 1L z{
ÅÅÅÅ41Å H-t - 1L 0
t+1 ÅÅÅÅ ÅÅÅÅÅÅ 4
0
0
s +1 ÅÅÅÅ ÅÅÅÅÅÅ 4
0
s +1 ÅÅÅÅ ÅÅÅÅÅÅ 4
t+1 ÅÅÅÅ ÅÅÅÅÅÅ 4
1-s ÅÅÅÅ ÅÅÅÅÅÅ 4
1-s ÅÅÅÅ ÅÅÅÅÅÅ 4
Stiffness matrix for a displacement based element 0.494505 jij jj 0.178571 jj jj jj -0.302198 jj jj jj -0.0137363 j k = jjj jjj -0.247253 jj jjj -0.178571 jj jj jj 0.0549451 jj k 0.0137363
0.178571
-0.302198
-0.0137363 -0.247253
-0.178571
0.0549451
0.494505
0.0137363
0.0549451
0.0137363
0.494505
-0.178571
-0.178571
-0.247253
-0.0137363
0.0549451
-0.0137363 -0.247253
0.0549451
-0.178571
0.494505
0.0137363
-0.302198
-0.178571
0.0549451
0.0137363
0.494505
0.178571
-0.302198
-0.247253
-0.0137363 -0.302198
0.178571
0.494505
0.0137363
0.0137363
-0.0137363 -0.247253
0.178571
-0.302198
-0.302198
-0.247253
-0.0137363 0.0549451
0.178571
Eigenvalues of displacement based stiffness matrix
81.42857, 0.769231, 0.769231, 0.494505, 0.494505, 0, 0, 0<
Examine stress matrix: {{1, 1, 0, 0, -s}, {1, -1, 0, -t, 0}, {0, 0, 1, s, t}}
0.178571
0.494505 -0.178571
10
Multifield Formulations
-1 jij jj -1 jj jj jj k a = jjjj -1 jj 1 jj ÅÅ3ÅÅ jj jj k0
-1 1
-1
1
1
1
1
1
-1 -1
1
-1 -1
1
1
1
1
0
- ÅÅÅÅ13Å
0
ÅÅ31ÅÅ
0
- ÅÅÅÅ13Å
ÅÅ13ÅÅ
0
- ÅÅÅÅ13Å
0
ÅÅÅÅ13Å
0
jij 5.6 jj jj 8.88178 µ 10-16 jj j k b = jjjj 0. jj jj 0 jj jj k0
k -1 b
-1
8.88178 µ 10-16
0.
0 0
10.4
0.
0.
10.4 0
0
0
4.8
0
0
0.
jij 0.178571 jj jj -1.52503 µ 10-17 jj j = jjjj 0. jj jj 0. jj jj k 0.
0.394027 jij jj 0.178571 jj jj jj -0.20172 jj jj jj -0.0137363 j k = jjj jjj -0.347731 jj jjj -0.178571 jj jj jj 0.155423 jj k 0.0137363
zyz -1 zzzz zz -1 zzzz zz z 0 zzz zz zz - ÅÅÅÅ31Å { 1
-1.52503 µ 10-17 0.0961538 0. 0. 0.
zyz zz 0 zzzz zz 0 zzz zz 0. zzzz zz 4.8 { 0
zyz zz zz 0. 0. 0. zz zz zz 0.0961538 0. 0. zz zz zz 0. 0.208333 0. zz z 0. 0. 0.208333 {
0.
0.
0.
0.178571
-0.20172
-0.0137363 -0.347731
-0.178571
0.155423
0.394027
0.0137363
0.155423
0.0137363
0.394027
-0.178571
-0.178571
-0.347731
-0.0137363
0.155423
-0.0137363 -0.347731
0.155423
-0.178571
0.394027
0.0137363
-0.20172
0.178571
-0.178571
0.155423
0.0137363
0.394027
0.178571
-0.20172
-0.347731
-0.0137363 -0.20172
0.178571
0.394027
0.0137363
-0.0137363 -0.347731
0.178571
-0.20172
0.0137363
0.394027
-0.20172
-0.347731
-0.0137363 0.155423
0.178571
-0.178571
Eigenvalues:81.42857, 0.769231, 0.769231, 0.0925926, 0.0925926, 0, 0, 0< There are 5 nonzero eigenvalues. Therefore the stress matrix is OK.
5.3 Check to see if the following stress interpolation matrix is a reasonable choice for a 4 node plane stress element. ij 1 s j P = jjjj 0 1 jj k0 0 T
0 0y zz s t 0 zzzz zz 1 s t { t
11
Initial matrix of stress modes
0 0y zz 0 zzzz zz 1 s t {
ij 1 s j P T = jjjj 0 1 jj k0 0
t
s
t
Step 1: Identify work stress modes t-1 ij ÅÅÅÅ ÅÅÅÅÅÅ jj 4 j j B T = jjjj 0 jjj s -1 j ÅÅÅÅ k 4ÅÅÅÅÅÅ
P 1 = 81, 0, 0<
0
1-t ÅÅÅÅ ÅÅ ÅÅÅ 4
s -1 ÅÅÅÅ ÅÅÅÅÅÅ 4
0
t-1 ÅÅÅÅ ÅÅÅÅÅ 4
ÅÅÅÅ14Å
H-s - 1L
0
ÅÅ41ÅÅ H-s - 1L 1-t ÅÅÅÅ ÅÅÅÅÅ 4
0
0
s +1 ÅÅÅÅ ÅÅÅÅÅÅ 4
0
s +1 ÅÅÅÅ ÅÅÅÅÅÅ 4
t+1 ÅÅÅÅ ÅÅÅÅÅÅ 4
1-s ÅÅÅÅ ÅÅÅÅÅÅ 4
Ÿ Ÿ P 1 B T dsdt = 8-1, 0, 1, 0, 1, 0, -1, 0<
w1 = 1 ï Work mode
P 2 = 8s , 1, 0<
Ÿ Ÿ P 2 B T dsdt = 80, -1, 0, -1, 0, 1, 0, 1<
w2 = 1 ï Work mode
P 3 = 8t, s , 1<
2 2 4 2 4 4 2 4 Ÿ Ÿ P 3 B T dsdt = :- ÅÅÅÅÅ , - ÅÅÅÅÅ , - ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ , - ÅÅÅÅÅ > 3 3 3 3 3 3 3 3 4 w3 = ÅÅÅÅÅ ï Work mode 3
P 4 = 80, t, s <
1 1 1 1 Ÿ Ÿ P 4 B T dsdt = : ÅÅÅÅÅ , 0, - ÅÅÅÅÅ , 0, ÅÅÅÅÅ , 0, - ÅÅÅÅÅ , 0> 3 3 3 3 1 w4 = ÅÅÅÅÅ ï Work mode 3
P 5 = 80, 0, t<
1 1 1 1 Ÿ Ÿ P 5 B T dsdt = :0, ÅÅÅÅÅ , 0, - ÅÅÅÅÅ , 0, ÅÅÅÅÅ , 0, - ÅÅÅÅÅ > 3 3 3 3 1 w5 = ÅÅÅÅÅ ï Work mode 3
Stress matrix satisfying work criteria
yz zz zz 1-s zz ÅÅÅÅ Å4ÅÅÅÅÅ zz zz z 1 ÅÅÅÅ4Å H-t - 1L z{
ÅÅÅÅ41Å H-t - 1L 0
t+1 ÅÅÅÅ ÅÅÅÅÅÅ 4
12
Multifield Formulations
0 0y zz t 0 zzzz zz s t {
1 s t jij j j P = jj 0 1 s jj k0 0 1 T
Step 2: Perform eigenvalue test to determine optimum set ij 1.0989 j C = jjjj 0.32967 jj k0
0
1.0989
0
0
-0.3 ij 1. j D = jjjj -0.3 1. jj 0. k 0.
t-1 ij ÅÅÅÅ ÅÅÅÅÅÅ jj 4 j j B T = jjjj 0 jj jj s -1 k ÅÅÅÅ4ÅÅÅÅÅÅ
yz zz zz zz z 0.384615 {
0.32967 0. y zz 0. zzzz zz 2.6 {
0
1-t ÅÅÅÅ ÅÅ ÅÅÅ 4
s -1 ÅÅÅÅ ÅÅÅÅÅÅ 4
0
t-1 ÅÅÅÅ ÅÅÅÅÅ 4
ÅÅÅÅ14Å
0
H-s - 1L
ÅÅ41ÅÅ H-s - 1L 1-t ÅÅÅÅ ÅÅÅÅÅ 4
yz zz zz 1-s zz ÅÅÅÅ Å4ÅÅÅÅÅ zz zz z 1 ÅÅÅÅ4Å H-t - 1L z{
ÅÅÅÅ41Å H-t - 1L 0
t+1 ÅÅÅÅ ÅÅÅÅÅÅ 4
0
0
s +1 ÅÅÅÅ ÅÅÅÅÅÅ 4
0
s +1 ÅÅÅÅ ÅÅÅÅÅÅ 4
t+1 ÅÅÅÅ ÅÅÅÅÅÅ 4
1-s ÅÅÅÅ ÅÅÅÅÅÅ 4
Stiffness matrix for a displacement based element ij 0.494505 jj jj 0.178571 jj jj jj -0.302198 jj jj jj -0.0137363 k = jjj jj -0.247253 jj jj jjj -0.178571 jj jj 0.0549451 jj j k 0.0137363
0.178571
-0.302198
-0.0137363 -0.247253
-0.178571
0.0549451
0.494505
0.0137363
0.0549451
-0.178571
-0.247253
-0.0137363
0.0137363
0.494505
-0.178571
0.0549451
-0.0137363 -0.247253
0.0549451
-0.178571
0.494505
0.0137363
-0.302198
-0.178571
0.0549451
0.0137363
0.494505
0.178571
-0.302198
-0.247253
-0.0137363 -0.302198
0.178571
0.494505
0.0137363
0.0137363
0.494505
-0.0137363 -0.247253
0.178571
-0.302198
-0.302198
-0.247253
-0.0137363 0.0549451
0.178571
Eigenvalues of displacement based stiffness matrix
81.42857, 0.769231, 0.769231, 0.494505, 0.494505, 0, 0, 0<
Stress matrix satisfying work criteria 1 s t jij j j P = jj 0 1 s jj k0 0 1 T
0 0y zz t 0 zzzz zz s t {
There are 1 possible submatrices with 5 stress modes in each.
Examine stress matrix: {{1, s, t, 0, 0}, {0, 1, s, t, 0}, {0, 0, 1, s, t}}
0.178571
-0.178571
13
-1 jij jj 0 jj jj jj 2 j k a = jjj - ÅÅ3ÅÅ jj jj ÅÅ1ÅÅ jj 3 jj jj k0
4. jij jj -1.2 jj jj k b = jjjj 0. jj jj 0 jj j k0
k -1 b
zyz zz zz zz z - ÅÅÅÅ43Å zzzz zz z 0 zzzz zz zz - ÅÅÅÅ31Å {
0
1
0
1
0
-1
0
-1
0
-1
0
1
0
1
- ÅÅÅÅ23Å
- ÅÅÅÅ43Å
ÅÅ32ÅÅ
ÅÅÅÅ43Å
ÅÅÅÅ43Å
ÅÅÅÅ23Å
0
- ÅÅÅÅ31Å
0
ÅÅÅÅ13Å
0
- ÅÅÅÅ13Å
ÅÅÅÅ13Å
0
- ÅÅ31ÅÅ
0
ÅÅÅÅ31Å
0
-1.2
0.
5.33333
-0.4
-0.4
13.0667
0.
-0.4
0
0.
0.268141 jij jj 0.060471 jj jj = jjjj 0.00185589 jj jj 0.000154657 jj j k 0.
0.325056 jij jj 0.0984212 jj jj jj -0.223063 jj jj jj 0.0304255 j k = jjj jjj -0.315591 jj jjj -0.132419 jj jj jj 0.213598 jj k 0.00357239
zyz zz zz zz zz -0.4 0. zz zz zz 4.8 0. zz z 0. 3.46667 { 0
0
0.
0
zyz zz zz zz zz 0. zz zz zz 0. zz z 0.288462 {
0.060471
0.00185589 0.000154657 0.
0.20157
0.00618629 0.000515524 0.
0.00618629
0.0769162
0.00640968
0.000515524 0.00640968 0.208867 0.
0.0984212
0.
0.
-0.223063
0.0304255
-0.315591
-0.132419
0.213598
0.
0.276055
0.0165063
0.135334
-0.139923
-0.250261
0.0249953
-
0.0165063
0.428734
-0.120608
0.102496
-0.0850634 -0.308168
0.
0.135334
-0.120608
0.259558
0.00214038 -0.169375
-0.139923 0.102496
0.00214038 0.438838
-0.250261 -0.0850634 -0.169375 0.0249953
-0.308168
-0.161127 0.189165
0.210954
0.088042
-
-0.225744
-
0.210954
0.386858
0.00652806 0.
0.088042
-0.225744
0.00652806
0.320314
-
-0.225516
-0.073172
0.0327787
-0.119565
0.
Eigenvalues:81.2171, 0.715005, 0.650081, 0.121892, 0.085197, 0, 0, 0< There are 5 nonzero eigenvalues. Therefore the stress matrix is OK.
5.4 Check to see if the following stress interpolation matrix is a reasonable choice for a 4 node plane stress element. ij s 2 1 0 0 -s yz jj zz P = jjjj 0 -1 0 -t 0 zzzz jj zz t { k 0 0 1 s T
14
Multifield Formulations
Initial matrix of stress modes ij s 2 jj P = jjjj 0 jj k0 T
1
0 0
-1 0 -t 0
1 s
-s yz zz 0 zzzz zz t {
Step 1: Identify work stress modes t-1 ij ÅÅÅÅ ÅÅÅÅÅÅ jj 4 j jj T B = jjj 0 jj jj s -1 k ÅÅÅÅ4ÅÅÅÅÅÅ
0 s -1 ÅÅÅÅ ÅÅÅÅÅÅ 4 t-1 ÅÅÅÅ ÅÅÅÅÅ 4
P 1 = 9s 2 , 0, 0=
1-t ÅÅÅÅ ÅÅ ÅÅÅ 4
0
0
ÅÅ41ÅÅ
ÅÅÅÅ14Å H-s - 1L
H-s - 1L
1-t ÅÅÅÅ ÅÅÅÅÅ 4
0
0
s +1 ÅÅÅÅ ÅÅÅÅÅÅ 4
0
s +1 ÅÅÅÅ ÅÅÅÅÅÅ 4
t+1 ÅÅÅÅ ÅÅÅÅÅÅ 4
1-s ÅÅÅÅ ÅÅÅÅÅÅ 4
1 1 1 1 Ÿ Ÿ P 1 B T dsdt = :- ÅÅÅÅÅ , 0, ÅÅÅÅÅ , 0, ÅÅÅÅÅ , 0, - ÅÅÅÅÅ , 0> 3 3 3 3
1 w1 = ÅÅÅÅÅ ï Work mode 3
P 2 = 81, -1, 0<
Ÿ Ÿ P 2 B T dsdt = 8-1, 1, 1, 1, 1, -1, -1, -1<
w2 = 1 ï Work mode
P 3 = 80, 0, 1<
Ÿ Ÿ P 3 B T dsdt = 8-1, -1, -1, 1, 1, 1, 1, -1<
w3 = 1 ï Work mode
P 4 = 80, -t, s <
1 1 1 1 Ÿ Ÿ P 4 B T dsdt = : ÅÅÅÅÅ , 0, - ÅÅÅÅÅ , 0, ÅÅÅÅÅ , 0, - ÅÅÅÅÅ , 0> 3 3 3 3 1 w4 = ÅÅÅÅÅ ï Work mode 3
P 5 = 8-s , 0, t<
1 1 1 1 Ÿ Ÿ P 5 B T dsdt = :0, ÅÅÅÅÅ , 0, - ÅÅÅÅÅ , 0, ÅÅÅÅÅ , 0, - ÅÅÅÅÅ > 3 3 3 3 1 w5 = ÅÅÅÅÅ ï Work mode 3
Stress matrix satisfying work criteria
yz zz zz zz zz zz z 1 ÅÅÅÅ4Å H-t - 1L z{
ÅÅÅÅ41Å H-t - 1L 0
t+1 ÅÅÅÅ ÅÅÅÅÅÅ 4
1-s ÅÅÅÅ ÅÅÅÅÅÅ 4
15
2 jij s jj P = jjj 0 jj k0 T
1
0
0
-1 0
-t
0
s
1
-s zy zz 0 zzzz zz t {
Step 2: Perform eigenvalue test to determine optimum set ij 1.0989 j C = jjjj 0.32967 jj k0
0
1.0989
0
0
1. -0.3 jij j j D = jj -0.3 1. jj 0. k 0.
t-1 jij ÅÅÅÅ4ÅÅÅÅÅÅ jj j B T = jjjj 0 jj jj s -1 k ÅÅÅÅ4ÅÅÅÅÅÅ
yz zz zz zz z 0.384615 {
0.32967
0 s -1 ÅÅÅÅ ÅÅÅÅÅÅ 4 t-1 ÅÅÅÅ ÅÅÅÅÅ 4
0. y zz 0. zzzz zz 2.6 { 1-t ÅÅÅÅ ÅÅ ÅÅÅ 4
0
0
ÅÅ41ÅÅ
ÅÅÅÅ14Å H-s - 1L
H-s - 1L
1-t ÅÅÅÅ ÅÅÅÅÅ 4
zyz zz zz zz zz zz 1 ÅÅÅÅ4Å H-t - 1L z{
ÅÅÅÅ41Å H-t - 1L 0
t+1 ÅÅÅÅ ÅÅÅÅÅÅ 4
0
0
s +1 ÅÅÅÅ ÅÅÅÅÅÅ 4
0
s +1 ÅÅÅÅ ÅÅÅÅÅÅ 4
t+1 ÅÅÅÅ ÅÅÅÅÅÅ 4
1-s ÅÅÅÅ ÅÅÅÅÅÅ 4
1-s ÅÅÅÅ ÅÅÅÅÅÅ 4
Stiffness matrix for a displacement based element 0.494505 jij jj 0.178571 jj jj jj -0.302198 jj jj jj -0.0137363 j k = jjj jj -0.247253 jj jj jj -0.178571 jj jj jj 0.0549451 jj k 0.0137363
0.178571
-0.302198
-0.0137363 -0.247253
0.494505
0.0137363
0.0549451
0.0137363
0.494505
-0.178571
0
-0.0137363
-0.178571
0.494505
0.0137363
-0.302198
0.0137363
0.494505
0.178571
-0.302198
-0.247253
-0.0137363 -0.302198
0.178571
0.494505
0.0137363
0.0137363
-0.0137363 -0.247253
0.178571
-0.302198
-0.302198
-0.247253
-0.0137363 0.0549451
0
-1 0
-t
0
s
1
-0.247253
-0.0137363 -0.247253
0.0549451
0.178571
Stress matrix satisfying work criteria 1
-0.178571 0.0549451
0.0549451
81.42857, 0.769231, 0.769231, 0.494505, 0.494505, 0, 0, 0<
ij s 2 jj P = jjjj 0 jj k0
0.0549451
-0.178571
Eigenvalues of displacement based stiffness matrix
T
-0.178571
-s yz zz 0 zzzz zz t {
There are 1 possible submatrices with 5 stress modes in each.
Examine element eigenvalues with stress matrix
0.178571
0.494505 -0.178571
16
Multifield Formulations
2 jij s jj P = jjj 0 jj k0 T
1
0 0
-1 0 -t 0
1 s
ij - ÅÅ31ÅÅ jj jj jj -1 jj jj k a = jjj -1 jj jj ÅÅ1ÅÅ jj 3 jj jj k0
0
ÅÅÅÅ13Å
0
ÅÅÅÅ13Å
0
1
1
1
1
-1 -1
-1 -1
- ÅÅÅÅ31Å
1
1
1
1
0
- ÅÅÅÅ13Å
0
ÅÅÅÅ13Å
0
- ÅÅÅÅ31Å
ÅÅÅÅ13Å
0
- ÅÅÅÅ31Å
0
ÅÅÅÅ31Å
0
ij 0.8 jj jj 1.73333 jj j k b = jjjj 0. jj jj 0 jj j k0
k -1 b
-s zy zz 0 zzzz zz t {
yz zz zz zz z 0 zzzz zz 0. zzzz z 4.8 {
1.73333
0.
0
0
10.4
0.
0
0
0.
10.4 0
0
0
4.8
0
0
0.
ij 1.95652 jj jj -0.326087 jj j = jjjj 0. jj jj 0. jj j k 0.
ij 0.269804 jj jj 0.0543478 jj jj jjj -0.077496 jj jj -0.13796 j k = jjj jj -0.223507 jj jj jj -0.0543478 jj jj jj 0.0311997 jj k 0.13796
yz zz z -1 zzzz zz -1 zzz zz z 0 zzzz zz zz - ÅÅ31ÅÅ { 0
yz zz zz zz zz zz 0.0961538 0. 0. zz zz zz 0. 0.208333 0. zz 0. 0. 0.208333 {
-0.326087 0.
0.
0.
0.150502
0.
0.
0. 0. 0.
0.
0.0543478
-0.077496
-0.13796
-0.223507
0.269804
0.13796
0.0311997
-0.0543478 -0.223507
-0.0543478 0.0311997
0.13796
0.269804
-0.0543478 0.0311997
0.0311997
-0.0543478 0.269804
-0.13796
-0.13796
-0.223507
0.13796
-0.077496
0.0543478
-0.0543478 0.0311997
0.13796
0.269804
0.0543478
-0.077496
-0.223507
-0.13796
-0.077496
0.0543478
0.269804
0.13796
-0.13796
-0.223507
0.0543478
-0.077496
0.13796
0.269804
-0.077496
0.0543478
-0.223507
-0.13796
0.0311997
-0.0543478
Eigenvalues:80.769231, 0.769231, 0.434783, 0.0925926, 0.0925926, 0, 0, 0< There are 5 nonzero eigenvalues. Therefore the stress matrix is OK.
5.5 Using one 4/1 element determine stresses in the cantilever plate due to a temperature rise of 100 °C. The plate is 5 mm thick, 50 mm long, 20 mm wide at the base and 10 mm wide at the tip. Use the following material properties. E = 70 GPa; n = 0.49; a = 23 µ 10-6 ë °C
17
Report displacements and effective stresses at element centroids.
y 20 10 0 0
50
x
Figure 5.10.
Global equations at start of the element assembly process 0 jij jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj k0
0
0 0 0 0
0
0 0 0 0
0
0 0 0 0
0
0 0 0 0
0
0 0 0 0
0
0 0 0 0
0
0 0 0 0
0
0 0 0 0
0 0 y i u1 zz jj 0 0 zzzz jjjj v1 zz jj 0 0 zzzz jjjj u2 zz jj 0 0 zzzz jjjj v2 zz jj 0 0 zzz jjj u3 zz jj 0 0 zzzz jjjj v3 zz jj 0 0 zzzz jjjj u4 zj 0 0 { k v4
ij 1.16978 µ 107 jj Plane strain C = jjjj 1.16511 µ 107 jj k0
ij 46697.8 jj jj 0 C d = jjjj jjj 0 j k0
0
0 zyz jij zyz zz jj 0 zz zz jj zz zz jj zz zz jj 0 zz zz jj zz zz jj zz zz jj 0 zz zz jj zz zz = jj zz zzz jjj 0 zzz zz jj zz zzz jjj 0 zzz zz jj zz zzz jjj 0 zzz zz jj zz z j z { k0{
1.16511 µ 107 7
1.16978 µ 10 0
yz zz zz zz zz zz 46697.8 0 zz 0 23348.9 {
0
0
46697.8 0
0
0 0
k = 1.16667 µ 107
G = 23348.9
Initial strain vector due to temperature change 23 ÅÅÅÅÅÅÅÅÅÅÅÅ a = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 1000000
yz zz zz zz 0 zz 23348.9 { 0
DT = 100
Interpolation functions and their derivatives
e T0 = H 0.0034477 0.0034477 0 L
18
Multifield Formulations
1 1 1 1 NT = : ÅÅÅÅÅ Hs - 1L Ht - 1L, - ÅÅÅÅÅ Hs + 1L Ht - 1L, ÅÅÅÅÅ Hs + 1L Ht + 1L, - ÅÅÅÅÅ Hs - 1L Ht + 1L> 4 4 4 4
t-1 1-t t+1 1 ∑NT ê∑s = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Å , ÅÅÅÅÅ H-t - 1L> 4 4 4 4
s -1 1 s +1 1-s ∑NT ê∑t = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-s - 1L, ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ > 4 4 4 4 P T = 81<
Mapping to the master element xHs,tL = 25 s + 25 5ts 5s 15 t 25 yHs,tL = - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅ ÅÅÅ 2 2 2 2
i 25 J = jjjj 5 5t k ÅÅ2ÅÅ - ÅÅÅÅ2ÅÅ
yz z 15 5s z ÅÅÅÅ Å2 Å - ÅÅÅÅ Å2ÅÅÅ z{
0
375 125 s detJ = ÅÅÅÅÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 2 2
Gauss quadrature points and weights Point
Weight
1
s Ø -0.57735 t Ø -0.57735
1.
2
s Ø -0.57735 t Ø 0.57735
1.
3
s Ø 0.57735 t Ø -0.57735
1.
4
s Ø 0.57735 t Ø 0.57735
1.
Computation of element matrices at 8-0.57735, -0.57735< with weight = 1. 0 i 25 zyz J = jj k 3.94338 8.94338 {
detJ = 223.584
NT = H 0.622008 0.166667 0.0446582 0.166667 L
∑NT ê∑s = H -0.394338 0.394338 0.105662 -0.105662 L
∑NT ê∑t = H -0.394338 -0.105662 0.105662 0.394338 L -0.00881854 jij jj 0 j B T = jjjj jj 0 jj k -0.0440927
0.0176371
0
-0.0440927
0
-0.0118146 0
0.0118146
0
0
0
0
0
0
0
0.0118146
0.00236292 0.0440927
-0.00881854 -0.0118146 0.0176371
0.00236292 0
-0.011181
0
19
B Tv =
8-0.00881854, -0.0440927, 0.0176371, -0.0118146, 0.00236292, 0.0118146, -0.0111815, 0.0440927< -0.00587903 jij jj 0.00293951 j B Td = jjjj jj 0.00293951 jj k -0.0440927 P =H1 L
0.0039382
0.0117581
-0.0293951
-0.00587903 -0.0078764 -0.00078764 0.0078764
0.0146976
-0.00587903 0.0039382
-0.00881854 -0.0118146
0.0176371
0.00157528
0.00372
-0.00078764 -0.0039382 0.00372 0.0118146
0.00236292
T
k a = H -1.97169 -9.85844 3.94338 -2.64156 0.528312 2.64156
k b = H 0.0000191644 L 10690.7 jij jj 676.628 jj jj jj 1636.92 jj jj jj -4422.37 j k c = jjj jjj -2864.57 jj jj -181.302 jj jj jj -9463.08 jj j k 3927.05
-0.0039382 -0.007
0.0146976
0.04409
9.85844 L
-2.5
zyz -181.302 -3734.82 -3745.74 -13017.8 zzzz zz 3250.42 2893.91 -362.604 -438.613 -870.947 -4092.22 -2016.87 zzzz zz 2814.09 -362.604 2595.5 1184.97 -754.032 3600. -4655.55 zzzz zz -181.302 -438.613 1184.97 767.559 48.5797 2535.62 -1052.25 zzz zz -3734.82 -870.947 -754.032 48.5797 1000.74 1003.67 3488.11 zzzz zz -3745.74 -4092.22 3600. 2535.62 1003.67 11019.7 -857.93 zzzz z -13017.8 -2016.87 -4655.55 -1052.25 3488.11 -857.93 14185.2 { 676.628
1636.92
-4422.37 -2864.57 -181.302 -9463.08 3927.05
13938.5
3250.42
2814.09
r Te = H -158721. -793604. 317442. -212646. 42529.1 212646. -201250. 793604. L
Computation of element matrices at 8-0.57735, 0.57735< with weight = 1. 0 i 25 zyz J = jj k 1.05662 8.94338 {
detJ = 223.584
NT = H 0.166667 0.0446582 0.166667 0.622008 L
∑NT ê∑s = H -0.105662 0.105662 0.394338 -0.394338 L
∑NT ê∑t = H -0.394338 -0.105662 0.105662 0.394338 L ij -0.00236292 jj jj 0 T B = jjjj jj 0 jj k -0.0440927
B Tv =
0
0.00472584
0
0.0152742 0
-0.0176371
-0.0440927
0
-0.0118146 0
0.0118146 0
0
0
0
0
-0.00236292 -0.0118146 0.00472584
0
0
0.0118146 0.0152742 0.0440927
8-0.00236292, -0.0440927, 0.00472584, -0.0118146, 0.0152742, 0.0118146, -0.0176371, 0.0440927<
20
Multifield Formulations
-0.00157528 jij jj 0.00078764 j B Td = jjjj jj 0.00078764 jj k -0.0440927
0.0039382
0.00315056
-0.0293951
-0.00157528 -0.0078764 -0.00509139 0.0078764
0.0146976
-0.00157528 0.0039382
-0.00236292 -0.0118146
PT = H 1 L
0.0101828
k b = H 0.0000191644 L
0.00587
-0.00509139 -0.0039382 0.00587
0.00472584
k a = H -0.528312 -9.85844 1.05662 -2.64156 3.41506
ij 10188.3 jj jj 181.302 jj jj jjj 2641.8 jj jj -1184.97 j k c = jjj jj -2970.75 jj jj jj -3418.7 jj jj jj -9859.34 jj k 4422.37
-0.0039382 -0.011
0.0146976
0.0118146
2.64156
0.0152742
0.04409
-3.94338 9.85844 L
-9859.34 4422.37 y zz -3814.45 -3250.42 -13315. zzzz zz 870.947 884.151 -97.1594 -226.257 -1136.39 -3299.7 362.604 zzzz zz 3567.74 -97.1594 1088.19 919.527 -594.765 362.604 -4061.16 zzzz zz 2198.17 -226.257 919.527 2352.6 314.024 844.402 -3431.72 zzz zz -3814.45 -1136.39 -594.765 314.024 2189.52 4241.07 2219.69 zzzz zz -3250.42 -3299.7 362.604 844.402 4241.07 12314.6 -1353.26 zzzz z -13315. 362.604 -4061.16 -3431.72 2219.69 -1353.26 15156.5 { 181.302
2641.8
-1184.97 -2970.75 -3418.7
13561.7
870.947
3567.74
r Te = H -42529.1 -793604. 85058.3
2198.17
-212646. 274913. 212646. -317442. 793604. L
Computation of element matrices at 80.57735, -0.57735< with weight = 1. 0 i 25 yz J = jj z 3.94338 6.05662 k {
detJ = 151.416
NT = H 0.166667 0.622008 0.166667 0.0446582 L
∑NT ê∑s = H -0.394338 0.394338 0.105662 -0.105662 L
∑NT ê∑t = H -0.105662 -0.394338 0.394338 0.105662 L -0.0130217 jij jj 0 j B T = jjjj jj 0 jj k -0.0174458 B Tv
0
0.0260434
-0.00604339 0
0
-0.00697
-0.0174458 0
-0.0651085 0
0.0651085
0
0
0
0
0
0
0.0651085
-0.00604339 0.017445
0
-0.0130217 -0.0651085 0.0260434
8-0.0130217, -0.0174458, 0.0260434, -0.0651085, -0.00604339, 0.0651085, -0.00697831, 0.0174458< =
ij -0.00868113 jj jj 0.00434056 B Td = jjjj jjj 0.00434056 j k -0.0174458 PT = H 1 L
0.00581525
0.0173623
0.0217028
-0.00402893 -0.0217028
-0.0116305 -0.00868113 -0.0434056 0.00201446 0.00581525
-0.00868113 0.0217028
-0.0130217 -0.0651085
0.0260434
-0.004
0.0434056
0.00232
0.00201446
-0.0217028
0.00232
0.0651085
-0.00604339 0.01744
21
k a = H -1.97169 -2.64156 3.94338 -9.85844 -0.915064 9.85844
k b = H 0.0000129785 L ij 1875.31 jj jj 267.715 jj jj jj 2417.13 jj jj jj -3604.55 k c = jjj jjj -3644.77 jj jj 2371. jj jj jj -647.667 jj j k 965.835
-1.05662 2.64156 L
yz z -3245.88 -5076.09 -1090.08 -1113.42 zzzz zz 4068.24 18184.1 -1998.25 -15728.8 -2605.42 -4872.42 535.431 zzzz zz 4155.35 -1998.25 22380.5 5067.37 -20539. 535.431 -5996.82 zzzz zz -3245.88 -15728.8 5067.37 15159.1 -463.697 4214.53 -1357.8 zzz zz zz -5076.09 -2605.42 -20539. -463.697 20111.7 698.12 5503.4 zz zz -1090.08 -4872.42 535.431 4214.53 698.12 1305.56 -143.468 zzzz z -1113.42 535.431 -5996.82 -1357.8 5503.4 -143.468 1606.84 { 267.715
2417.13
-3604.55 -3644.77 2371.
2034.16
4068.24
4155.35
-647.667 965.835
r Te = H -158721. -212646. 317442. -793604. -73662.6 793604. -85058.3 212646. L
Computation of element matrices at 80.57735, 0.57735< with weight = 1. 0 i 25 yz J = jj z k 1.05662 6.05662 {
detJ = 151.416
NT = H 0.0446582 0.166667 0.622008 0.166667 L
∑NT ê∑s = H -0.105662 0.105662 0.394338 -0.394338 L
∑NT ê∑t = H -0.105662 -0.394338 0.394338 0.105662 L ij -0.00348915 jj jj 0 B T = jjjj jj 0 jjj k -0.0174458
-0.0165108
0
0.00697831
0
-0.0174458
0
-0.0651085 0
0.0130217 0
0.0651085 0
0
0
0
0
0
-0.00348915 -0.0651085 0.00697831
0
0.0651085 0.0130217 0.0174458
B Tv =
8-0.00348915, -0.0174458, 0.00697831, -0.0651085, 0.0130217, 0.0651085, -0.0165108, 0.0174458< ij -0.0023261 jj jj 0.00116305 T B d = jjjj jj 0.00116305 jj k -0.0174458 PT = H 1 L
0.0217028
0.0046522
-0.0023261 -0.0434056 -0.00434056 0.0434056
0.00581525
-0.0023261 0.0217028
-0.00348915 -0.0651085 0.00697831
k a = H -0.528312 -2.64156 1.05662 -9.85844 1.97169
k b = H 0.0000129785 L
0.00868113
-0.0217028 -0.01100
0.00581525 -0.0116305
0.005503
-0.00434056 -0.0217028 0.005503 0.0651085
9.85844
-2.5
0.0130217 2.64156 L
0.017445
22
Multifield Formulations
1133.4 jij jj 71.7341 jj jj jj 3900.96 jj jj jj -965.835 j k c = jjj jj -4229.9 jj jj jj -267.715 jj jj jj -804.452 jj k 1161.82
zyz -267.715 -5514.94 -894.101 -1231.01 zzzz zz -14558.6 -4068.24 -4558.85 3513.59 zzzz zz 3604.55 -19661.3 -2103.28 -5761.65 zzzz zz 15786.2 999.127 3002.26 -4335.96 zzz zz zz 999.127 20582. 3336.83 4594.2 zz zz 3002.26 3336.83 2361.05 -339.449 zzzz z -4335.96 4594.2 -339.449 2398.46 {
71.7341
3900.96
-965.835 -4229.9
1477.72
1090.08
5268.23
1090.08
15216.5
-535.431
5268.23
-535.431 20154.7
-267.715 -14558.6 3604.55 -5514.94 -4068.24 -19661.3 -894.101 -4558.85 -2103.28 -1231.01 3513.59
r Te = H -42529.1 -212646. 85058.3
-5761.65
-267.715 -804.452 1161.82
-793604. 158721. 793604. -201250. 212646. L
After summing contributions from all points, the element equations as follows: k a = H -5. -25.
10.
k b = H 0.0000642857 L ij 23887.7 jj jj 1197.38 jj jj jjj 10596.8 jj jj -10177.7 j k c = jjj jj -13710. jj jj jj -1496.72 jj jj jj -20774.5 jj k 10477.1
ij 412777. jj jj 6 jjj 1.94564 µ 10 jj jj jj -767181. jj jj jj 1.93427 µ 106 k = jjjj jj -402599. jj jj jj -1.94594 µ 106 jj jj jj jj 757003. jj j 6 k -1.93397 µ 10
1197.38
25. L
-10.
10596.8
-10177.7 -13710.
-1496.72 -20774.5 10477.1
-1496.72 -18140.3 -8980.35 -28677.2
31012.1
9279.69
15805.4
9279.69
37178.6
-2993.45 -30952.3 -8681.
15805.4
-2993.45 46218.8
10776.4
-41549.1 2394.76
-1496.72 -30952.3 10776.4
34065.4
898.035
10596.8
-10177.7
43884.
9279.69
15805.4
9279.69
27000.9
-2694.1
-20475.2 -10177.7 15805.4
-2694.1
33347.
-18140.3 -8681.
-41549.1 898.035
-8980.35 -16823.2 2394.76 -28677.2 2394.76
1.94564 µ 106
6
9.75323 µ 10
-3.87961 µ 10 6
-3.87961 µ 10 6
-1.94594 µ 10
6
-9.74036 µ 10 3.87991 µ 10
6
-9.7509 µ 10
1.59273 µ 10
6
9.73803 µ 10 9.76844 µ 10
6
3.88021 µ 10
-9.76377 µ 10 6
-1.57238 µ 10 6
3.89128 µ 10
6
3.89128 µ 10
6
-9.7427 µ 10
805000. -2.0125 µ 106
-9.76377 µ 10 1.94534 µ 10
6
1.94534 µ 10
9.76611 µ 10
1.93427 µ 10
6
9.73803 µ 10
3.
3. -
-3.87961 µ 10 6
75
-
6 6
-767181.
yz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz z {
6
6
422954.
402500. 2.0125 µ 106
-9.74036 µ 10 3.88021 µ 10
746826. -1.93367 µ 10
-1.93367 µ 10
6
6
6
6
6
Complete element equations for element 1
-1.94594 µ 10
6
-20475.2
-1.94594 µ 106 6
-3.89188 µ 10
746826.
-16823.2 2394.76
-402599. 6
-3.89188 µ 10 6
6
6 6
9.73803 µ 10
10596.8
1.93427 µ 106
-767181.
6
r = H -402500. -2.0125 µ 106 T
-25. 5. 25.
1. -
-805000. 2.0125 µ 106 L
23
jij 412777. jj jj 1.94564 µ 106 jjj jj jj -767181. jj jj jj 1.93427 µ 106 jj jj jj jj -402599. jj jj jj -1.94594 µ 106 jj jjj jj 757003. jj j 6 k -1.93397 µ 10
1.94564 µ 106
1.93427 µ 106
-767181.
6
6
-1.94594 µ 106
-402599.
6
6
6
75700
9.75323 µ 10
-3.87961 µ 10
9.73803 µ 10
-1.94594 µ 10
-9.74036 µ 10
3.8799
-3.87961 µ 106
1.59273 µ 106
-3.89188 µ 106
746826.
3.88021 µ 106
-1.572
6
6
9.73803 µ 10
-3.89188 µ 10 6
-1.94594 µ 10
6
-9.74036 µ 10 6
6
-1.93367 µ 10 6
3.88021 µ 10
-9.76377 µ 10
6
-9.7509 µ 10
9.76844 µ 10
-1.93367 µ 10
746826. -1.57238 µ 10
6
6
6
6
3.87991 µ 10
6
6
3.89128 µ 10
3.89128 µ 10
-9.7427 µ 10
6
1.94534 µ 10
@1, 2D
@1, 3D
@1, 4D
@1, 5D
@1, 6D
@3, 3D
@3, 4D
@3, 5D
@3, 6D
@5, 3D
@5, 4D
@5, 5D
@5, 6D
@2, 2D
@2, 3D
@4, 2D
@4, 3D
@3, 2D
@5, 2D
@6, 2D
@6, 3D
@7, 2D
@7, 3D
@8, 2D
@8, 3D
@2, 4D
@4, 4D
@6, 4D
@7, 4D
@8, 4D
9.76611 µ 10
@2, 5D
@4, 5D
@6, 5D
@7, 5D
@8, 5D
@2, 6D
@4, 6D
@6, 6D
@7, 6D
@8, 6D
-3.879 6
-3.87961 µ 10 6
3.8912 -7671
6
6
1.93427 µ 10
1 jij zyz jj 2 zz jj zz jj zz jj 3 zz jj zz jj zz jj 4 zz j z Locations for element contributions to a global vector: jjj zzz jjj 5 zzz jj zz jjj 6 zzz jj zz jjj 7 zzz jj zz j z k8{
ij @1, 1D jj jj @2, 1D jj jj jj @3, 1D jj jj jj @4, 1D and to a global matrix: jjj jj @5, 1D jj jj jjj @6, 1D jj jj @7, 1D jj j k @8, 1D
1.94534 µ 10
422954.
The element contributes to 81, 2, 3, 4, 5, 6, 7, 8< global degrees of freedom.
-9.76377 µ 10 6
-767181.
6
6
9.73803 µ 10
1.5825 -3.891
@1, 7D
@1, 8D y zz @2, 8D zzzz zz @3, 7D @3, 8D zzzz zz @4, 7D @4, 8D zzzz zz @5, 7D @5, 8D zzz zz @6, 7D @6, 8D zzzz zz @7, 7D @7, 8D zzzz z @8, 7D @8, 8D {
@2, 7D
Adding element equations into appropriate locations we have ij 412777. jj jj jj 1.94564 µ 106 jj jj jj -767181. jj jj jj 1.93427 µ 106 jjj jj jj -402599. jj jj jj -1.94594 µ 106 jj jjj jj 757003. jj jj 6 k -1.93397 µ 10
1.94564 µ 106 6
6
9.75323 µ 10
-3.87961 µ 10 6
-3.87961 µ 10 6
-1.94594 µ 10
6
-9.74036 µ 10 6
-9.7509 µ 10
1.59273 µ 10
6
6
9.73803 µ 10
-1.94594 µ 10
-3.89188 µ 10 6
9.76844 µ 10
6
3.88021 µ 10
-9.76377 µ 10 6
-1.57238 µ 10 6
3.89128 µ 10
6
3.89128 µ 10
6
-9.7427 µ 10
-9.74036 µ 10 3.88021 µ 10
746826. -1.93367 µ 10
-1.93367 µ 10 6
6
6
6
6
746826.
-1.94594 µ 106
-402599. 6
-3.89188 µ 10 6
3.87991 µ 10
6 6
9.73803 µ 10
6
1.93427 µ 106
-767181.
-9.76377 µ 10 1.94534 µ 10
6
1.94534 µ 10
9.76611 µ 10
-3.879
-3.87961 µ 10 6
1.93427 µ 10
6
9.73803 µ 10
3.8912 -7671
6 6
-767181.
3.8799 -1.572
6
6
422954.
75700
1.5825 -3.891
24
Multifield Formulations
Essential boundary conditions Node
dof
Value
1
u1 v1
0 0
4
u4 v4
0 0
Remove 81, 2, 7, 8< rows and columns. After adjusting for essential boundary conditions we have ij 1.59273 µ 106 jj jj 6 jjj -3.89188 µ 10 jj jj jj 746826. jj j 6 k 3.88021 µ 10
-3.89188 µ 106 6
6
9.76844 µ 10
-1.93367 µ 10 6
422954.
6
1.94534 µ 106
-1.93367 µ 10 -9.76377 µ 10
yz i u2 zz jj z -9.76377 µ 10 zzzz jjjj v2 zz jjj z 1.94534 µ 106 zzz jjj u3 zz j z 9.76611 µ 106 { k v3 3.88021 µ 106
746826.
6
yz ijj 805000. zz jj zz jj -2.0125 µ 106 zz = jj zz jj zz jj 402500. zz jj j { k 2.0125 µ 106
yz zz zz zz zz zz zz zz {
Solving the final system of global equations we get
8u2 = 0.202609, v2 = -0.0323944, u3 = 0.205101, v3 = 0.0523294<
Complete table of nodal values u
v
1
0
0
2
0.202609
-0.0323944
3
0.205101
0.0523294
4
0
0
Computation of reactions
Equation numbers of dof with specified values: 81, 2, 7, 8<
Extracting equations 81, 2, 7, 8< from the global system we have
ij 412777. jj jj jj 1.94564 µ 106 jj jjj jj 757003. jj j 6 k -1.93397 µ 10
1.94564 µ 106 6
9.75323 µ 10
6
3.87991 µ 10
6
-9.7509 µ 10
1.93427 µ 106
-767181. 6
-3.87961 µ 10
6
-1.57238 µ 10 6
3.89128 µ 10
Substituting the nodal values and re-arranging
6
9.73803 µ 10
6
3.89128 µ 10
6
-9.7427 µ 10
-402599. 6
-1.94594 µ 10 -767181.
-1.94594 µ 106
757003.
6
3.87991
6
1.58256
-9.74036 µ 10 -3.87961 µ 10
6
1.93427 µ 10
6
9.73803 µ 10
-3.8915
25
ij R1 yz ijjj 412777. jj z jj R2 zzz jjjj 1.94564 µ 106 jj zz = jj jj zz j jjj R3 zzz jjjj 757003. j z jj k R4 { jk -1.93397 µ 106
1.94564 µ 106
-767181.
1.93427 µ 106
-402599.
-1.94594 µ 106
9.75323 µ 106
-3.87961 µ 106
9.73803 µ 106
-1.94594 µ 106
-9.74036 µ 106
3.87991 µ 106
-1.57238 µ 106
3.89128 µ 106
-767181.
-3.87961 µ 106
-9.7509 µ 106
3.89128 µ 106
-9.7427 µ 106
1.93427 µ 106
9.73803 µ 106
Carrying out computations, the reactions are as follows. Label
dof
Reaction
R1
u1
-1.74623 µ 10-10
R2
v1
2179.61
R3
u4
1.16415 µ 10-10
R4
v4
-2179.61
Sum of Reactions dof: u
0
dof: v
2.32831 µ 10-10
Solution for element 1
Nodal displacements = H 0 0 0.202609 -0.0323944 0.205101 0.0523294 0 0 L
7 jij 1.16978 µ 10 jj Plane strain C = jjj 1.16511 µ 107 jj k0
e T0 = H 0.0034477 0.0034477 0 L
1.16511 µ 107 7
1.16978 µ 10 0
zyz zz zz 0 zz z 23348.9 { 0
25 Solution at 8s, t< = 80, 0< ï 8x, y< = :25, ÅÅÅÅÅÅÅÅÅÅ > 2
1 1 1 1 Interpolation functions = : ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ > 4 4 4 4 u = 0.101927
v = 0.00498375
e T = H 0.00406879 0.00282413 0 1.68106 µ 10-16 0 0 L
s T = H -6.36646 µ 10-12 -58.1229 -190.003 3.92508 µ 10-12 Principal stresses = H -6.36646 µ 10-12
Effective stress Hvon MisesL = 168.63
-58.1229 -190.003 L
0 0L
26
Multifield Formulations
Solution at 8s, t< = 8-1, -1< ï 8x, y< = 80, 0<
Interpolation functions = 81, 0, 0, 0<
u = 0.
v = 0.
e T = H 0.00405217 0. 0 -0.000647888 0 0 L
s T = H -33098.5 -33287.7 -33287.7 -15.1275 0 0 L
Principal stresses = H -33097.3 -33287.7 -33288.9 L
Effective stress Hvon MisesL = 191.033
Solution at 8s, t< = 8-1, 1< ï 8x, y< = 80, 20<
Interpolation functions = 80, 0, 0, 1<
u = 0.
v = 0.
e T = H 0.00410201 0. 0 0.00104659 0 0 L
s T = H -32515.5 -32707.1 -32707.1 24.4367 0 0 L
Principal stresses = H -32512.4 -32707.1 -32710.1 L
Effective stress Hvon MisesL = 196.175
Solution at 8s, t< = 81, -1< ï 8x, y< = 850, 10< Interpolation functions = 80, 1, 0, 0<
u = 0.202609
v = -0.0323944
e T = H 0.00400234 0.00847238 0 -0.00209318 0 0 L s T = H 65031. 65239.8 64844.1 -48.8733 0 0 L
Principal stresses = H 65250.6
65020.1 64844.1 L
Effective stress Hvon MisesL = 353.106
Solution at 8s, t< = 81, 1< ï 8x, y< = 850, 20<
Interpolation functions = 80, 0, 1, 0<
u = 0.205101
v = 0.0523294
e = H 0.00410201 0.00847238 0 0.00129578 0 0 L T
s T = H 66197. 66401.1 66005.4 30.2549 0 0 L
Principal stresses = H 66405.5
66192.6
Effective stress Hvon MisesL = 346.676
66005.4 L
27
Solution summary Nodal solution x
y
u
v
1
0
0
0
0
2
50
10
0.202609
-0.0323944
3
50
20
0.205101
0.0523294
4
0
20
0
0
Solution at element centroids Coord
Disp
Stresses
Principal stresses
Effective Stress
-6.36646 µ 10-12 -58.1229 -190.003
168.63
-12
25 25 ÅÅÅÅ ÅÅ 2
1
-6.36646 µ 10 -58.1229 -190.003 3.92508 µ 10-12 0 0
0.101927 0.00498375
Support reactions Node
dof
Reaction
1
1
-1.74623 µ 10-10
1
2
2179.61
4
1
1.16415 µ 10-10
4
2
-2179.61
Sum of applied loads Ø H 2.32831 µ 10-10
0L
Sum of support reactions Ø H 0 2.32831 µ 10-10 L
Ansys solution
PRINT U
NODAL SOLUTION PER NODE
***** POST1 NODAL DEGREE OF FREEDOM LISTING *****
LOAD STEP= TIME=
1
1.0000
SUBSTEP=
1
LOAD CASE=
0
28
Multifield Formulations
THE FOLLOWING DEGREE OF FREEDOM RESULTS ARE IN GLOBAL COORDINATES
NODE
UX
1
0.0000
2
0.19728
3
0.19728
4
0.0000
UY
UZ
0.0000
USUM
0.0000
0.0000
-0.29478E-01
0.0000
0.19947
0.58957E-01
0.0000
0.20590
0.0000
0.0000
0.0000
MAXIMUM ABSOLUTE VALUES NODE
2
VALUE
3
0.19728
PRINT S
0.58957E-01
0
3
0.0000
0.20590
PRIN ELEMENT SOLUTION PER ELEMENT
***** POST1 ELEMENT NODAL STRESS LISTING *****
LOAD STEP= TIME=
1
1.0000
SUBSTEP=
1
LOAD CASE=
0
THE FOLLOWING X,Y,Z VALUES ARE IN GLOBAL COORDINATES
ELEMENT= NODE
1 S1
PLANE182 S2
S3
SINT
SEQV
1
45.014
-91.606
-184.25
229.26
199.76
2
26.497
-73.089
-184.25
210.75
182.60
3
21.007
-67.599
-184.25
205.26
178.31
4
46.726
-93.318
-184.25
230.98
201.53
29
AnsysFiles\Chap05\Prb5-5Data.txt
!* Problem 5-5 !* UP Formulation. êPREP7 !* ET,1,PLANE182 !* !* KEYOPT,1,1,0 KEYOPT,1,3,2 KEYOPT,1,6,1 KEYOPT,1,10,0 !* !* !* MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,EX,1,,70000 MPDATA,PRXY,1,,.499 UIMP,1,REFT,,, MPDATA,CTEX,1,,23*10**H-6L K,1,,,, K,2,50,10,, K,3,50,20,, K,4,0,20,, A,1,2,3,4 esize,100 MSHKEY,1 AMESH,ALL DL,4, ,ALL, êSOL
FINISH TUNIF,100, êSTATUS,SOLU ERESX,NO
30
Multifield Formulations
SOLVE êPOST1
FINISH PRNSOL,U,Y
5.6 Analyze Problem 5.5 using one 9/3 quadrilateral element.
y 20
6
3
8
15 10
5 7
2 4
5 0
9
1 x 0
10
20
30
40
Global equations at start of the element assembly process
50
31
0 jij jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0 0 0
0 0 0 0
7 jij 1.16978 µ 10 jj Plane strain C = jjj 1.16511 µ 107 jj k0
46697.8 jij jj 0 j C d = jjjj jj 0 jjj k0
0
1.16511 µ 107 7
1.16978 µ 10 0
zyz zz zz zz zz 46697.8 0 zz zz 0 23348.9 {
0
0
46697.8 0
0
0 0 7
k = 1.16667 µ 10
zyz zz zz 0 zz z 23348.9 {
0 zyz jij zyz zz jj 0 zz zz jj zz zz jj zz zz jj 0 zz zz jj zz zz jj zz zz jj 0 zz zz jj zz zz jj zz zz jj 0 zz zz jj zz zz jj zz zz jj 0 zz zz jj zz zz jj zz zz jj 0 zz zz jj zz zz jj zz zz jj 0 zz zz jj zz zz jj 0 zz zz jj zz zz = jj zz zz jj 0 zz zzz jjj zzz zz jj zz zzz jjj 0 zzz zz jj zz zzz jjj 0 zzz zz jj zz zzz jjj 0 zzz zz jj zz zzz jjj 0 zzz zz jj zz zzz jjj 0 zzz zz jj zz zzz jjj zzz zz jj 0 zz zzz jjj zzz zz jj 0 zz zz jj zz z j z { k0{
0
G = 23348.9
Initial strain vector due to temperature change 23 a = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 1000000
0 0 0 y i u1 zz jj 0 0 0 zzzz jjjj v1 zz jj 0 0 0 zzzz jjjj u2 zz jj 0 0 0 zzzz jjjj v2 zz jj 0 0 0 zzz jjj u3 zz jj 0 0 0 zzzz jjjj v3 zz jj 0 0 0 zzzz jjjj u4 zz jj 0 0 0 zzz jjj v4 zz jj 0 0 0 zzzz jjjj u5 zz jj 0 0 0 zzzz jjjj v5 zz jj 0 0 0 zzz jjj u6 zz jj 0 0 0 zzzz jjjj v6 zz jj 0 0 0 zzzz jjjj u7 zz jj 0 0 0 zzz jjj v7 zz jj 0 0 0 zzzz jjjj u8 zz jj 0 0 0 zzzz jjjj v8 zz jj 0 0 0 zzzz jjjj u9 zj 0 0 0 { k v9
DT = 100
e T0 = H 0.0034477 0.0034477 0 L
Interpolation functions and their derivatives 1 1 1 NT = : ÅÅÅÅÅ Hs - 1L s Ht - 1L t, - ÅÅÅÅÅÅ Hs - 1L Hs + 1L Ht - 1L t, ÅÅÅÅÅ s Hs + 1L Ht - 1L t, 4 2 4 1 1 1 - ÅÅÅÅÅÅ s Hs + 1L Ht - 1L Ht + 1L, ÅÅÅÅÅ s Hs + 1L t Ht + 1L, - ÅÅÅÅÅÅ Hs - 1L Hs + 1L t Ht + 1L, 2 4 2 1 1 ÅÅÅÅÅ Hs - 1L s t Ht + 1L, - ÅÅÅÅÅÅ Hs - 1L s Ht - 1L Ht + 1L, Hs - 1L Hs + 1L Ht - 1L Ht + 1L> 4 2
32
Multifield Formulations
1 1 1 ∑NT ê∑s = : ÅÅÅÅÅ H2 s - 1L Ht - 1L t, -s Ht - 1L t, ÅÅÅÅÅ H2 s + 1L Ht - 1L t, - ÅÅÅÅÅÅ H2 s + 1L It2 - 1M, 4 4 2 1 1 1 ÅÅÅÅÅ H2 s + 1L t Ht + 1L, -s t Ht + 1L, ÅÅÅÅÅ H2 s - 1L t Ht + 1L, - ÅÅÅÅÅÅ H2 s - 1L It2 - 1M, 2 s It2 - 1M> 4 4 2
1 1 1 ∑NT ê∑t = : ÅÅÅÅÅ Hs - 1L s H2 t - 1L, - ÅÅÅÅÅ Is 2 - 1M H2 t - 1L, ÅÅÅÅÅ s Hs + 1L H2 t - 1L, -s Hs + 1L t, 4 2 4 1 1 1 ÅÅÅÅÅ s Hs + 1L H2 t + 1L, - ÅÅÅÅÅÅ Is 2 - 1M H2 t + 1L, ÅÅÅÅÅ Hs - 1L s H2 t + 1L, -Hs - 1L s t, 2 Is 2 - 1M t> 4 2 4
P T = 81, s, t<
Mapping to the master element xHs,tL = 25 s + 25 5ts 5s 15 t 25 yHs,tL = - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅ ÅÅÅ 2 2 2 2
i 25 J = jjjj 5 5t k ÅÅ2ÅÅ - ÅÅÅÅ2ÅÅ
0 15 ÅÅÅÅ ÅÅ 2
-
5s ÅÅÅÅ ÅÅÅÅ 2
yz zz z {
375 125 s detJ = ÅÅÅÅÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 2 2
Gauss quadrature points and weights Point
Weight
1
s Ø -0.774597 t Ø -0.774597
0.308642
2
s Ø -0.774597 t Ø 0.
0.493827
3
s Ø -0.774597 t Ø 0.774597
0.308642
4
s Ø 0. t Ø -0.774597
0.493827
5
s Ø 0. t Ø 0.
0.790123
6
s Ø 0. t Ø 0.774597
0.493827
7
s Ø 0.774597 t Ø -0.774597
0.308642
8
s Ø 0.774597 t Ø 0.
0.493827
9
s Ø 0.774597 t Ø 0.774597
0.308642
Computation of element matrices at 8-0.774597, -0.774597< with weight = 0.308642
33
i 25 J = jj k 4.43649
yz z 9.43649 {
0
detJ = 235.912
NT =
H 0.472379 0.274919 -0.06
-0.0349193 0.007621 -0.0349193 -0.06
∑N ê∑s = H -0.876028 1.06476 T
-0.18873 -0.109839 0.0239718 -0.135242 0.11127
∑N ê∑t = H -0.876028 -0.509839 0.11127 T
ij -0.0185668 jj jj 0 T B = jjjj jj 0 jj k -0.0928341
0.274919 0.16 L
0
0.0521782
-0.135242 0.0239718 -0.109839 -0.18873 1.0647 0
-0.0096417 0
-0.0928341 0
-0.0540284 0
0
0
0
-0.50983
-0.0185668 -0.0540284 0.0521782
B Tv = 8-0.0185668, -0.0928341, 0.0521782, -0.0540284,
-0.0018502
0.0117915
0
0
0
0
0.0117915
-0.0096417 -0.0143318
-0.0096417, 0.0117915, -0.00185023, -0.0143318, 0.000508067, 0.00254033, -0.00334409, -0.0116398, 0.008, -0.02, -0.040417, 0.112834, 0.0131336, 0.0656682< ij -0.0123779 jj jj 0.00618894 B Td = jjjj jjj 0.00618894 j k -0.0928341 PT = H 1
0.0309447
0.0347855
0.0180095
-0.0064278 -0.00393049 -0.001233
-0.0618894 -0.0173927 -0.0360189 0.0032139
0.00786098
0.0006167
-0.0173927 0.0180095
0.0032139
-0.00393049 0.0006167
-0.0185668 -0.0540284 0.0521782
0.0117915
-0.0096417
0.0309447
-0.774597 -0.774597 L
-0.014331
-3.93394 -0.702036 0.858566 -0.13472 -1.04353 0. ij -1.3519 -6.75948 3.79922 jj j -2.94286 3.04722 0.543795 -0.665042 0.104353 0.808318 k a = jj 1.04717 5.23587 jj 1.04717 5.23587 -2.94286 3.04722 0.543795 -0.665042 0.104353 0.808318 k ij 6.24107 µ 10-6 jj j k b = jjj -4.83431 µ 10-6 jj j -6 k -4.83431 µ 10
-4.83431 µ 10-6 3.74464 µ 10-6 3.74464 µ 10-6
-4.83431 µ 10-6 yz zz z 3.74464 µ 10-6 zzzz zz 3.74464 µ 10-6 {
34
Multifield Formulations
15433.1 jij jj 976.778 jj jj jj 6331.08 jj jj jj -9372.04 jj jj jj -1455.21 jj jj jj 1769.85 jj jj jj 2339.81 jj jj jj -9.57679 jj jj -422.314 j k c = jjjj jj -26.7288 jj jj jjj 1977.81 jj jj 282.843 jj jj jj 2819.83 jj jj jj -1683.48 jj jj -16107.2 jj jj jj 8753.3 jj jj jj -10916.9 jj j k -690.945
976.778
6331.08
-9372.04 -1455.21 1769.85
20121.6
7195.48
9722.46
7195.48
11134.2
-1597.58 -2223.47 188.292
9722.46
-1597.58 11245.5
-1386.68 -2223.47 455.58 -2177.
188.292
257.711 3074.32
-1386.68 -2177.
-9.57679 -4
257.711
3074.32
-2
1097.58
1017.51
-1
455.58
-2299.41 -1384.64 1591.1
447.105
-64.4277 -246.866 -193.707 39
25
-2299.41 -64.4277 473.216
259.651
-352.743 -4
1097.58
-1384.64 -246.866 259.651
356.96
15.0272
-6
1017.51
1591.1
471.42
0.2
0.262061
11
-26.7288 -173.245 256.459
-193.707 -352.743 15.0272 39.8208
-48.4305 -64.027
-550.613 -196.899 -266.048 37.9454
59.5719
-7.05206 -84.1264 0.7
235.488
297.633
15.5558
673.622
-1237.31 -160.25
2554.98
995.523
1128.89
1473.05
2783.28
-1284.27 -575.777 220.92
3956.18
447.943
3159.08
-7814.22 -15144.6 7534.29 -22468.5 -2960.4
-194.235 -256.301 57.0709
3145.28
-17404.2 422.81
-690.945 -4478.43 6629.51
1029.38
-14233.5 -5089.88 -6877.39 980.896
yz z 9.43649 {
-7
192.86
-7 46
-1309.4
-2579.73 -1011.44 44
3678.42
1221.39
-3538.53 -2
-1251.94 -1655.11 6.77435
29
-182.297 -2174.69 18
1539.95
Computation of element matrices at 8-0.774597, 0.< with weight = 0.493827 0
453.755
-17.7064 -5 388.662
-58.1843 -665.709 -236.864 624.579
r Te = H -108828. -544138. 305837. -316682. -56513.9 69114.6
i 25 J = jj k 2.5
2339.81
-10844.9 -84004.5 2977.9
detJ = 235.912
NT = H 0. 0. 0. -0.0872983 0. 0. 0. 0.687298 0.4 L
∑NT ê∑s = H 0. 0. 0. -0.274597 0. 0. 0. -1.2746 1.54919 L ∑NT ê∑t = H -0.343649 -0.2 0.0436492 0. -0.0436492 0.2
ij 0.0036417 jj jj 0 B T = jjjj jjj 0 jj k -0.036417
0
0.00211943
0
-0.000462557 0
-0.036417 0
-0.0211943 0
0
0
0
0.0036417
-0.0211943 0.00211943
B Tv = 80.0036417, -0.036417, 0.00211943, -0.0211943,
0.343649 0. 0. L -0.01098
0.00462557
0
0
0
0
0.00462557
-0.000462557 0
-0.000462557, 0.00462557, -0.0109839, 0, 0.000462557, -0.00462557, -0.00211943, 0.0211943, -0.0036417, 0.036417, -0.0509839, 0, 0.0619677, 0<
35
0.0024278 jij jj -0.0012139 j B Td = jjjj jj -0.0012139 jj k -0.036417
PT = H 1
0.012139
0.00141295
0.00706477
-0.000308371 -0.00154186
-0.024278 -0.000706477 -0.0141295 0.000154186
-0.00
0.00308371
0.0036
0.012139
-0.000706477 0.00706477
0.000154186
-0.00154186
0.0036
0.0036417
-0.0211943
0.00462557
-0.000462557 0
-0.774597 0. L
0.00211943
-4.24258 0.246914 -2.46914 -0.0538879 0.538879 -1.27962 0 0.053 ij 0.424258 j k a = jjjj -0.328629 3.28629 -0.191258 1.91258 0.0417414 -0.417414 0.991189 0 -0.04 jj 0 0 0 0 0 0 0 0 k0
ij 9.98571 µ 10-6 jj k b = jjjj -7.73489 µ 10-6 jj k0 ij 3655.56 jj jj -120.249 jj jj jj 2127.49 jj jj jj -69.9834 jj jj jjj -464.317 jj jjj 15.2736 jj jj -145.074 jj jj jj 1088.06 jj jj jj 464.317 k c = jjjj jjj -15.2736 jj jj -2127.49 jj jj jj 69.9834 jj jj jj -3655.56 jj jj jj 120.249 jj jj -673.392 jj jj jj 5050.44 jj jj jj 818.467 jj j k -6138.5
r Te
-7.73489 µ 10-6 -6
5.99142 µ 10 0
-120.249 2127.49 4846.02
0 yz zz z 0 zzzz z 0{
-69.9834 -464.317 15.2736
-69.9834 2820.33
-69.9834 1238.18
15.2736
-40.7295 -270.227 8.88905
2820.33
-40.7295 1641.4
8.88905
15.2736
-270.227 8.88905
58.976
-615.525 8.88905
-358.229 -1.94
-725.372 -84.4317 -422.158 18.4269
-145.074 1088.06 -84.4317 633.238
27
-358.229 -422.158 -63.3238 -8 -1.94
18.4269
-138.201 -5
78.182
92.1343
13.8201
1.9
92.1343
437.564
0
-1
-108.806 633.238
-63.3238 -138.201 13.8201
0
328.173
13
-15.2736 270.227
-8.88905 -58.976
58
1.94
-18.4269 138.201
615.525
-8.88905 358.229
1.94
-78.182
-92.1343 -13.8201 -1
69.9834
-1238.18 40.7295
270.227
-8.88905 84.4317
-2820.33 40.7295 120.249
-1641.4
-2127.49 69.9834
-4846.02 69.9834
-8.88905 358.229 464.317
422.158
-15.2736 145.074
-2820.33 -15.2736 615.525
-633.238 -2 63.3238
8.8
-1088.06 -4
725.372
108.806
15
-3366.96 -391.907 -1959.53 85.532
427.66
2031.04
0
-8
64.149
0
1523.28
64
0
10
-505.044 2939.3
-293.93
-641.49
4092.33
476.339
2381.69
-103.959 -519.795 -2468.6
613.85
-3572.54 357.254
= H 34152.8 -341528. 19876.5
779.692
-77.9692 0
-198765. -4337.97 43379.7
-1851.45 -7
-103009. 0 4337.97
Computation of element matrices at 8-0.774597, 0.774597< with weight = 0.308642 0 i 25 yz z J = jj k 0.563508 9.43649 {
detJ = 235.912
NT =
H -0.06
46
-615.525 -725.372 -108.806 -1
-0.0349193 0.007621 -0.0349193 -0.06
0.274919 0.472379 0.274919 0.16 L
-4337
36
Multifield Formulations
∑NT ê∑s = H 0.11127 ∑N ê∑t = T
H 0.18873
-0.876028 -0.50983
0.109839 -0.0239718 0.135242 -0.11127 0.509839 0.876028 -1.06476 -0.619677 L
0.004 jij jj 0 j B T = jjjj jj 0 jj k 0.02 B Tv
-0.135242 0.0239718 -0.109839 -0.18873 1.06476
0
-0.00567204 0
0.00101613
0
-0.00471659 0
0.02
0
0.0116398
0
-0.00254033 0
0.014
0
0
0
0
0
0
0
0.004
0.0116398
-0.00567204 -0.00254033 0.00101613
0.0143318
-0.0
0.000846778
-0.0
= 80.004, 0.02, -0.00567204, 0.0116398, 0.00101613,
-0.00254033, -0.00471659, 0.0143318, -0.00728341, -0.0117915, 0.0413725, 0.0540284, -0.0371336, 0.0928341, -0.0178502, -0.112834, 0.0262673, -0.0656682< ij 0.00266667 jj jj -0.00133333 T B d = jjjj jj -0.00133333 jjj k 0.02 P =H1 T
-0.00666667 -0.00378136 -0.00387993 0.000677422 0.0133333
0.00189068
0.00775985
-0.000338711 -0.00169356 0.00
-0.00666667 0.00189068
-0.00387993 -0.000338711 0.000846778
0.00
0.004
-0.00567204 -0.00254033
0.01
0.0116398
-0.774597 0.774597 L
0.00101613
1.45625 -0.412995 0.847521 0.0739871 -0.184968 -0.343426 1.0435 ij 0.29125 j -0.656487 -0.0573102 0.143275 0.266017 -0.808 k a = jjjj -0.225601 -1.12801 0.319905 jj 1.12801 -0.319905 0.656487 0.0573102 -0.143275 -0.266017 0.8083 k 0.225601 ij 6.24107 µ 10-6 jj j k b = jjj -4.83431 µ 10-6 jj j -6 k 4.83431 µ 10
-4.83431 µ 10-6 -6
3.74464 µ 10
-6
-3.74464 µ 10
4.83431 µ 10-6 -6
-3.74464 µ 10
3.74464 µ 10-6
yz zz zz zz zz z {
37
716.305 jij jj 45.3357 jj jj jj 344.344 jj jj jj -245.63 jj jj jj -77.1625 jj jj jj 46.0672 jj jj jj 444.541 jj jj jj -225.347 jj jj -466.971 j k c = jjjj jj -194.192 jj jj jjj 2212.19 jj jj 1161.8 jj jj jj 2819.83 jj jj jj -1683.48 jj jj -3998.41 jj jj jj -95.3983 jj jj jj -1994.67 jj j k 1190.85 r Te = H 23445.6
45.3357
344.344
-245.63
-77.1625 46.0672
933.916
207.728
489.126
-40.3088 -108.258 204.377
207.728
303.263
-37.4141 -63.3346 3.77699
489.126
-37.4141 361.81
444.541 344.25
-225.347 -4 617.669
11.0911
-76.825
-75.9781 423.625
20
-1.46282 -72.7601 3.86435
34
-40.3088 -63.3346 11.0911
13.3117
-108.258 3.77699
-76.825
-1.46282 16.3836
11.1785
-90.6762 -4
204.377
344.25
-75.9781 -72.7601 11.1785
399.627
-38.3072 -2
617.669
-1.20076 423.625
3.86435
84.9131
-139.693 209.791
34.1487
-584.105 -219.932 -240.883 45.0355
-90.6762 -38.3072 503.42
21
-41.3404 -209.433 212.86
35
55.3177
-240.497 -324.669 48
-570.414 537.211
-1066.8
-138.042 212.455
2730.77
1166.04
1026.58
-240.903 -239.645 1296.88
1473.05
2314.5
-405.314 -486.464 53.4575
3956.18
-138.027 2807.5
53.4575
-362.686 -2003.33 1323.55 -5236.81 -1078.6
-1041.99 -1637.21 286.708 -2798.49 97.6362
446.192
-2804.99 207.04
yz z 7.5 {
0
-141.219 -1
-246.317 -2558.4
25
1194.73
-1038.11 -3522.52 -5
-37.8143 -1880.87 99.8943 288.966
-2344.
-14889.9 -27645.8 84004.5
88 -1
-42691.
detJ = 187.5
NT = H 0. 0.687298 0. 0. 0. -0.0872983 0. 0. 0.4 L
∑NT ê∑s = H -0.343649 0. 0.343649 0.2
0. L
-0.0436492 0. 0.0436492 -0.2
∑NT ê∑t = H 0. -1.2746 0. 0. 0. -0.274597 0. 0. 1.54919 L
ij -0.013746 jj jj 0 B T = jjjj jjj 0 jj k0
-3 15
618.907
344.11
5955.96
2658.95
-1105.27 -1 1423.48
-598.724 -408.507 3313.68
-1985.94 -37.8143 423.521
117228. -33246.1 68225.4
874.088
Computation of element matrices at 80., -0.774597< with weight = 0.493827 i 25 J = jj k 4.43649
84
-1.20076 -1
0
0.0301586
0
0.008
0
-0.001
0
0
-0.169946 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.013746 0
0.008
0
0.013746 0
-0.013746 -0.169946 0.0301586
B Tv = 8-0.013746, 0, 0.0301586, -0.169946, 0.013746, 0, 0.008, 0,
-0.00174597, 0, 0.00649731, -0.0366129, 0.00174597, 0, -0.008, 0, -0.0366559, 0.206559< ij -0.00916398 jj jj 0.00458199 B Td = jjjj jj 0.00458199 jj k0
0
0.0201057
0
-0.0100529 -0.113297 -0.00458199 0
-0.00266667
0
-0.0100529 0.0566487
-0.00266667
-0.013746 -0.169946
0.0566487
0.0301586
0.00916398
0
-0.00458199 0 0
0.00533333
0.013746 0
38
Multifield Formulations
PT = H 1
0. -0.774597 L
-15.7358 1.27277 0 ij -1.27277 0 2.79246 j 0 0 0 0 0 k a = jjjj 0 jj -0.985887 0 k 0.985887 0 -2.16303 12.1889
ij 7.93651 µ 10-6 jj k b = jjjj 0 jj -6 k -6.14759 µ 10 544.668 jij jj 0 jj jj jj -1195. jj jj jj -3366.96 jj jj jj -544.668 jj jj jj 0 jj jj jj -316.991 jj jj jj 0 jj jj 69.1819 j k c = jjjj jj 0 jj jj jjj -257.448 jj jj -725.372 jj jj jjj -69.1819 jj jj 0 jj jj jj 316.991 jj jj jj 0 jj jj jj 1452.45 jj k 4092.33
0 -6.14759 µ 10-6 yz zz zz 0 0 zz zz -6 0 4.7619 µ 10 {
0
-1195.
408.501
5050.44
-896.25
5050.44
65062.2
-3693.55 1195.
0
yz z 7.5 {
0 0
-0.573775 0 0.125224
0 -0.466
-316.991 0
69
-408.501 0
-237.743 0
-5050.44 695.477
-2939.3
-1
-896.25
-3693.55 85220.2
3366.96
896.25
1959.53
521.607
-4
0
1195.
544.668
0
316.991
0
-6 0
3366.96
-408.501 -5050.44 896.25 0
0
408.501
0
237.743
695.477
1959.53
316.991
0
184.485
0
-4
-237.743 -2939.3
521.607
0
237.743
0
138.364
0
0
-151.785 -427.66
51.8864
641.49
-113.839 0
-69.1819 0
1088.06
14016.9
-795.732 257.448
-40.2631 0
-51.8864 0
8.7
-30.1973 0
-1088.06 149.832
-633.238 -3
-193.086 -795.732 18359.7
725.372
193.086
422.158
112.374
0
427.66
69.1819
0
40.2631
0
-8
113.839
0
51.8864
0
30.1973
0
151.785
-51.8864 -641.49 0
-695.477 -1959.53 -316.991 0
237.743
2939.3
-521.607 0
-6138.5
-79079.1 4489.28
1089.34
4489.28
-1452.45 6138.5
102458. 0
detJ = 187.5
NT = H 0. 0. 0. 0. 0. 0. 0. 0. 1. L
∑NT ê∑s = H 0. 0. 0. 0.5
0. 0. 0. -0.5 0. L
∑NT ê∑t = H 0. -0.5 0. 0. 0. 0.5
-184.485 0
-237.743 0
-9
40
-138.364 0
-845.309 3572.54
18
-103580. -4092.33 -1089.34 -2381.69 -633.982 51
Computation of element matrices at 80., 0.< with weight = 0.790123 0
0 -0.161664 0 0.601603 0 0
-3366.96 -544.668 0
r Te = H -102458. 0 224793. -1.26673 µ 106
i 25 J = jj k 2.5
0.740741 0
0. 0. 0. L
59629.6
0 -13013.9 0 48429. -2
39
0 jij jj 0 j B T = jjjj jj 0 jj k0
0
0.00666667
0
0
-0.0666667 0 0
0
0
0
0
0 0
0
0
-0.0666667 0.00666667
0 0
0
0.02 0 0 0.0666667
ij 0 jj jj 0 T B d = jjjj jj 0 jj k0
0
0.00444444
0
0
0 -0.00444444 -0.0222
0
-0.00222222 -0.0444444 0
0 -0.00666667 0
0
0 0.00222222
0.044444
0
-0.00222222 0.0222222
0
0 -0.00666667 0
0
0 0.00222222
-0.0222
0
-0.0666667
0
0 0
0 0.0666667
-0.0066
0
0 0
0 0 -0.00666667 0
0
0
0 0 0
0.0666667
0
0
0 0 0
0
0
0.02 0
-0.00666667 0
B Tv = 80, 0, 0.00666667, -0.0666667, 0, 0, 0.02, 0, 0, 0, -0.00666667, 0.0666667, 0, 0, -0.02, 0, 0, 0<
PT = H 1
0.0222222
0. 0. L
ij 0 0 j k a = jjjj 0 0 jj k0 0
0.00666667
r Te = H 0
0 0.0133333
0.02 0
0.987654 -9.87654 0 0 2.96296
0
0 0 -0.987654 9.87654
0 0
-2.9629
0
0
0 0 0
0
0 0 0
0
0 0
0
0
0
0 0 0
0
0 0 0
0
0 0
0
0
0
0
0
0 0
0
0 0 0
0
0 0
0
0
0
0
0 0
0
0 0 0
0
0 0
0
15578.7
-512.459 0
0 614.95
-4612.13 0 0 -15578.7 512.459
0.0000126984 0 0 y zz jij j j k b = jj 0 0 0 zzzz zz jj 0 0{ k0 0 jij jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj 0 j k c = jjjj jj 0 jj jj jj 0 jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj k0
0
0 0
0
-512.459 20652.1
0
0 3074.75
461.213
0 0 512.459
-20652.1 0 0
0
0
0
0
0 0
0
0 0 0
0
0 0
0
0
0
0
0 0
0
0 0 0
0
0 0
0
614.95
3074.75
0
0 1844.85
0
0 0 -614.95
-3074.75 0 0
0
-4612.13 461.213
0
0 0
1383.64
0 0 4612.13
-461.213 0 0
0
0
0
0
0 0
0
0 0 0
0
0 0
0
0
0
0
0 0
0
0 0 0
0
0 0
0
0 -614.95
4612.13
0 0 15578.7
-512.459 0 0
0
-15578.7 512.459
0
512.459
-20652.1 0
0 -3074.75 -461.213 0 0 -512.459 20652.1
0 0
0
0
0
0
0 0
0
0 0 0
0
0 0
0
0
0
0
0 0
0
0 0 0
0
0 0
0 0 614.95
0
-614.95
-3074.75 0
0 -1844.85 0
3074.75
0 0
0
4612.13
-461.213 0
0 0
-1383.64 0 0 -4612.13 461.213
0 0
0
0
0
0
0 0
0
0 0 0
0
0 0
0
0
0
0
0 0
0
0 0 0
0
0 0
0 79506.2 -795062. 0
0 238519. 0 0 0 -79506.2 795062. 0 0 -238519.
Computation of element matrices at 80., 0.774597< with weight = 0.493827
40
Multifield Formulations
0 i 25 J = jj k 0.563508 7.5
yz z {
detJ = 187.5
NT = H 0. -0.0872983 0. 0. 0. 0.687298 0. 0. 0.4 L
∑NT ê∑s = H 0.0436492 0. -0.0436492 0.2
0.343649 0. -0.343649 -0.2
∑NT ê∑t = H 0. 0.274597 0. 0. 0. 1.2746 0. 0. -1.54919 L 0.00174597 jij jj 0 j B T = jjjj jj 0 jj k0 B Tv
-0.000825267 0
0
0. L
-0.00174597 0
0.008
0
0
0.0366129
0
0
0
0
0
0
0
0
0
-0.000825267 0
0.00174597 0.0366129
= 80.00174597, 0, -0.000825267, 0.0366129, -0.00174597, 0, 0.008, 0,
-0.00174597 0
0.013746, 0, -0.00383064, 0.169946, -0.013746, 0, -0.008, 0, 0.00465591, -0.206559< 0.00116398 jij jj -0.000581989 j B Td = jjjj jj -0.000581989 jj k0 P =H1 T
0
-0.000550178 -0.0122043
-0.00116398 0
0
0.000275089
0.0244086
0.000581989
0
-0
0
0.000275089
-0.0122043
0.000581989
0
-0
0.00174597 0.0366129
0. 0.774597 L
ij 0.161664 0 -0.0764136 3.39008 j 0 0 0 k a = jjjj 0 jj k 0.125224 0 -0.0591897 2.62595
ij 7.93651 µ 10 jj k b = jjjj 0 jj -6 k 6.14759 µ 10 -6
-6
0 6.14759 µ 10 0 0
0 4.7619 µ 10-6
yz zz zz zz zz {
-0.000825267 0
0.0
-0.00174597 0
-0.161664 0 0.740741 0 1.27277
0 -0.354689
0
0 0
0 0
0 0
-0.125224 0 0.573775 0 0.985887 0 -0.274741
41
8.78726 jij jj 0 jj jj jj -4.15348 jj jj jj -92.1343 jj jj jj -8.78726 jj jj jj 0 jj jj jj 40.2631 jj jj jj 0 jj jj 69.1819 j k c = jjjj jj 0 jj jj jjj -19.2792 jj jj -427.66 jj jj jj -69.1819 jj jj jj 0 jj jj -40.2631 jj jj jj 0 jj jj jj 23.4327 jj j k 519.795 r Te = H 13013.9
0
-4.15348 -92.1343 -8.78726 0
6.59044
138.201
-3.11511 0
-6.59044 0
138.201
2900.05
-21.7746 4.15348
-138.201 -19.0312 633.238
40.2631
0 -3
3.11511
0
8.78726
0
-40.2631 0
0
6.59044
0
-30.1973 0
184.485
0
31
138.364
0
4.15348
92.1343
-6.59044 -138.201 3.11511 0
-19.0312 -422.158 -40.2631 0
30.1973
633.238
-14.2734 0
-422.158 -14.2734 -7
-30.1973 0
0
-32.7002 -725.372 -69.1819 0
51.8864
1088.06
-24.5252 0
641.49
13461.2
-101.071 19.2792
316.991
-51.8864 0
-6
0
54
237.743
0
-641.49
-88.3371 2939.3
-14.4594 -101.071 17942.9
427.66
14.4594
-1959.53 -66.2528 -3
0
69.1819
0
-316.991 0
-51.8864 -1088.06 24.5252
0
51.8864
0
0
32.7002
725.372
-1 -5
-237.743 0
40.2631
0
-184.485 0
-30.1973 -633.238 14.2734
0
30.1973
0
-138.364 0
-779.692 -16361.2 122.846
-23.4327 779.692
107.368
-3572.54 18
19.0312
17.5745 0
122.846
422.158
-21808.5 -519.795 -17.5745 2381.69
-6151.29 272902. -13013.9 0 59629.6
yz z 5.56351 {
0
0
102458. 0
80.5262
-3
40
-28552.5 1.26673
detJ = 139.088
∑N ê∑s =
H 0.18873
30.1973
92.1343
NT = H -0.06 0.274919 0.472379 0.274919 -0.06 T
69
-3.11511 -21.7746 3865.58
Computation of element matrices at 80.774597, -0.774597< with weight = 0.308642 i 25 J = jj k 4.43649
0
-0.0349193 0.007621 -0.0349193 0.16 L
-1.06476 0.876028 0.509839 -0.11127 0.135242 -0.0239718 0.109839 -0.619677 L
∑NT ê∑t = H 0.11127
ij 0.004 jj jj 0 B T = jjjj jjj 0 j k 0.02
-0.509839 -0.876028 1.06476
-0.18873 -0.109839 0.0239718 -0.13524
0
-0.026328
0
0.02
0
-0.0916398 0
-0.15746
0
0.191382
0
0
0
0
0
0
0
0.004
-0.0916398 -0.026328
-0.15746
0.0629839 0.191382
0.0629839 0
B Tv = 80.004, 0.02, -0.026328, -0.0916398, 0.0629839,
-0.0135691 0
-0.15746, -0.0135691, 0.191382, 0.00156912, -0.0339228, 0.00891321, -0.0197427, -0.0017235, 0.00430876, 0.00870737, -0.0243088, -0.044553, 0.111382<
-0.0135691
42
Multifield Formulations
0.00266667 jij jj -0.00133333 j B Td = jjjj jj -0.00133333 jj k 0.02
PT = H 1
-0.00666667 -0.017552 0.0133333
0.00877599
-0.00666667 0.00877599 0.004
0.774597 -0.774597 L
0.0305466
0.0419892
0.0524866
-0.009046
-0.0610932 -0.0209946 -0.104973 0.00452304 0.0305466
-0.0916398 -0.026328
-0.0209946 0.0524866
0.00452304
-0.15746
0.191382
0.0629839
0.858566 -1.13021 -3.93394 2.70379 -6.75948 -0.582499 8.21573 ij 0.171713 j k a = jjjj 0.133008 0.665042 -0.87546 -3.04722 2.09435 -5.23587 -0.451202 6.36387 jj 3.04722 -2.09435 5.23587 0.451202 -6.36387 k -0.133008 -0.665042 0.87546
ij 3.67957 µ 10-6 jj j k b = jjj 2.85018 µ 10-6 jj j -6 k -2.85018 µ 10 ij 422.314 jj jj 26.7288 jj jj jj -1977.81 jj jj jj -282.843 jj jj jj -2819.83 jj jj 1683.48 jj jj jj 3764.03 jj jj jj -783.556 jj jj jj -671.648 k c = jjjj jj 122.127 jj jj jj -348.126 jj jj 231.449 jj jj jj 77.1625 jj jj jj -46.0672 jj jj jj -440.76 jj jj jj 239.527 jj jj jj 1994.67 jj k -1190.85 r Te = H 13822.9
2.85018 µ 10-6 2.20774 µ 10-6 -2.20774 µ 10-6
-2.85018 µ 10-6 yz zz z -2.20774 µ 10-6 zzzz zz 2.20774 µ 10-6 {
26.7288
-1977.81 -282.843 -2819.83 1683.48
550.613
-15.5558 -2554.98 -1473.05 -3956.18 948.656
5061.01
-1
806.102
12247.
-8555.43 -17101.6 4613.33
30
-2554.98 806.102
11918.
8012.09
17622.1
-5881.35 -23080.7 99
-1473.05 12247.
8012.09
30152.9
-3313.5
-31347.4 -5913.15 54
-3956.18 -8555.43 17622.1
-3313.5
37111.2
10654.4
-41130.2 -1
948.656
-17101.6 -5881.35 -31347.4 10654.4
36958.6
-867.647 -6
5061.01
4613.33
-156.978 3060.71
-23080.7 -5913.15 -41130.2 -867.647 49134.6
26
991.284
5485.99
-1976.47 -6535.8
-900.424 -740.928 4113.15
1180.06
7237.61
-198.274 1499.81
3866.18
-308.541 -3948.84 -871.355 68
1066.8
-491.962 -1166.04 2182.69
-575.828 4717.25
260.707
11
-6.58102 -8697.81 -1 1530.8
-5170.83 -2
857.796
161.809
40.3088
-335.131 -219.245 -825.11
108.258
234.113
-482.216 90.6715
-1015.52 -291.549 1125.5
-213.83
1926.46
1174.69
-618.457 -4821.01 -782.929 84
-614.833 -1227.46 2747.33
4569.5
90.6715
-1 54
-351.169 5665.12
1449.91
-6335.9
-2
1041.99
-8663.21 -5667.52 -21329.3 2343.88
22174.2
4182.8
-3
2798.49
6051.87
29094.3
13
69114.6
yz z 5.56351 {
0
-783.556 -6
-15.5558 9343.77
-12465.4 2343.88
-26251.4 -7536.6
-90982.3 -316682. 217655. -544138. -46891.2 661366. 5422.47
Computation of element matrices at 80.774597, 0.< with weight = 0.493827 i 25 J = jj k 2.5
3764.03
detJ = 139.088
NT = H 0. 0. 0. 0.687298 0. 0. 0. -0.0872983 0.4 L
∑NT ê∑s = H 0. 0. 0. 1.2746 0. 0. 0. 0.274597 -1.54919 L
43
∑NT ê∑t = H 0.0436492 -0.2
ij -0.000784562 jj jj 0 B T = jjjj jjj 0 j k 0.00784562
-0.343649 0. 0.343649 0.2
-0.0436492 0. 0. L
0
0.00359485
0
0.00784562
0
-0.0359485 0
-0.0617684 0
0
0
0
0
-0.000784562 -0.0359485 0.00359485
B Tv = 8-0.000784562, 0.00784562, 0.00359485,
0.00617684 0
0
0.05098 0
-0.0617684 0.00617684
0
-0.0359485, 0.00617684, -0.0617684, 0.0509839, 0, -0.00617684, 0.0617684, -0.00359485, 0.0359485, 0.000784562, -0.00784562, 0.0109839, 0, -0.0619677, 0< -0.000523041 jij jj 0.000261521 j B Td = jjjj jj 0.000261521 jj k 0.00784562 P =H1 T
-0.00261521
0.00239657
0.0205895
0.033
0.00523041
-0.00119828 -0.0239657 -0.00205895 -0.041179
-0.01
-0.00261521
-0.00119828 0.0119828
-0.01
-0.000784562 -0.0359485
0.774597 0. L
0.0119828
0.00359485
0.0041179
-0.00205895 0.0205895 -0.0617684
0.00617684 0
ij -0.0538879 0.538879 0.246914 -2.46914 0.424258 -4.24258 3.50184 j k a = jjjj -0.0417414 0.417414 0.191258 -1.91258 0.328629 -3.28629 2.71251 jj 0 0 0 0 0 0 k0
ij 5.88731 µ 10 jj k b = jjjj 4.56029 µ 10-6 jj k0 -6
-6
4.56029 µ 10
-6
3.53239 µ 10 0
0 yz zz z 0 zzzz z 0{
0 -0.424258 0 -0.328629 0 0
44
Multifield Formulations
100.032 jij jj -3.29051 jj jj jj -458.343 jj jj jj 15.0771 jj jj jj -787.547 jj jj jj 25.9061 jj jj jj -85.532 jj jj jj 641.49 jj jj 787.547 j k c = jjjj jj -25.9061 jj jj jjj 458.343 jj jj -15.0771 jj jj jj -100.032 jj jj jj 3.29051 jj jj -18.4269 jj jj jj 138.201 jj jj jj 103.959 jj j k -779.692
-3.29051 -458.343 15.0771
-787.547 25.9061
-85.532
641.49
78
132.608
15.0771
-607.606 25.9061
-1044.02 -427.66
-64.149
-2
15.0771
2100.12
-69.083
3608.53
-118.702 391.907
-2939.3
-3
-607.606 -69.083
2784.05
-118.702 4783.68
293.93
11
25.9061
-118.702 6200.34
3608.53
1959.53
-203.959 673.392
-5050.44 -6
-1044.02 -118.702 4783.68
-203.959 8219.53
3366.96
505.044
20
-427.66
391.907
1959.53
673.392
5558.2
0
-6
-64.149
-2939.3
293.93
-5050.44 505.044
0
4168.65
50
-6200.34 203.959
-673.392 5050.44
62
-25.9061 -3608.53 118.702 1044.02
-4783.68 203.959
118.702
-15.0771 -2100.12 69.083
3366.96
-8219.53 -3366.96 -505.044 -2
-3608.53 118.702
-391.907 2939.3
36
607.606
69.083
-2784.05 118.702
-4783.68 -1959.53 -293.93
-1
3.29051
458.343
-15.0771 787.547
-25.9061 85.532
-641.49
-7
-132.608 -15.0771 607.606
-25.9061 1044.02
427.66
64.149
25
-92.1343 84.4317
145.074
1197.45
0
-1
0
898.086
10
422.158
-13.8201 -633.238 63.3238
725.372
-1088.06 108.806
519.795
-476.339 -2381.69 -818.467 -4092.33 -6755.65 0
77.9692
3572.54
r Te = H -4337.97 43379.7
19876.5
-357.254 6138.5
-613.85
81
-5066.74 -6
0
-198765. 34152.8 -341528. 281898. 0 -34152.8 34152
Computation of element matrices at 80.774597, 0.774597< with weight = 0.308642 0 i 25 yz J = jj z k 0.563508 5.56351 {
detJ = 139.088
NT =
H 0.007621 -0.0349193 -0.06 ∑NT ê∑s =
0.274919 0.472379 0.274919 -0.06
-0.0349193 0.16 L
H -0.0239718 0.135242 -0.11127 0.509839 0.876028 -1.06476 0.18873 ∑N ê∑t = T
H -0.0239718 0.109839 0.18873 -0.000861752 jij jj 0 j B T = jjjj jj 0 jj k -0.00430876
0.109839 -0.619677 L
-1.06476 0.876028 0.509839 -0.11127 0.135242 -0.619677 L
0
0.00496467 0
-0.00521544 0
0.0247
-0.00430876
0
0.0197427
0
0.0339228
0
0
0
0
0
0
0
-0.000861752 0.0197427
0.00496467 0.0339228
B Tv = 8-0.000861752, -0.00430876, 0.00496467, 0.0197427,
-0.00521544, 0.0339228, 0.0247074, -0.191382, 0.0314919, 0.15746, -0.0446559, 0.0916398, 0.008, -0.02, 0.00384562, 0.0243088, -0.0222765, -0.111382<
-0.00521544 -0.19
45
-0.000574502 jij jj 0.000287251 j B Td = jjjj jj 0.000287251 jj k -0.00430876 PT = H 1
-0.0065809 -0.00347696 -0.0113076
0.00143625
0.00330978
-0.00287251
-0.00165489 0.0131618
0.00143625
-0.00165489 -0.0065809 0.00173848
-0.000861752 0.0197427
0.774597 0.774597 L
0.00173848
0.00496467
0.0339228
ij -0.0369936 -0.184968 0.213125 0.847521 -0.22389 1.45625 j k a = jjjj -0.0286551 -0.143275 0.165086 0.656487 -0.173424 1.12801 jj k -0.0286551 -0.143275 0.165086 0.656487 -0.173424 1.12801 ij 3.67957 µ 10-6 jj j k b = jjj 2.85018 µ 10-6 jj j -6 k 2.85018 µ 10
ij 19.6011 jj jj 1.24058 jj jj jj -90.9824 jj jj jj -10.0728 jj jj jj -140.499 jj jj 42.0585 jj jj jj 798.087 jj jj jj -216.912 jj jj jj -716.305 k c = jjjj jj -45.3357 jj jj jj -344.344 jj jj 245.63 jj jj jj 77.1625 jj jj jj -46.0672 jj jj jj -109.413 jj jj jj -2.6105 jj jj jj 506.693 jj k 32.0692
2.85018 µ 10-6 2.20774 µ 10-6 2.20774 µ 10-6
2.85018 µ 10-6 yz zz z 2.20774 µ 10-6 zzzz zz 2.20774 µ 10-6 {
0.01
0.0226152
-0.0
-0.0113076
-0.0
-0.00521544 -0.1
-8.21573 1
1.06065
0.821573 -6.36387 1 0.821573 -6.36387 1
1.24058
-90.9824 -10.0728 -140.499 42.0585
25.5559
-2.75866 -117.975 -44.3174 -190.836 236.446
-216.912 -7
798.087
1080.71
-4
-2.75866 423.622
32.7481
636.683
-215.745 -3623.27 1123.84
33
-117.975 32.7481
545.614
237.612
869.097
-44.3174 636.683
237.612
1189.79
-59.1114 -6679.56 173.117
-190.836 -215.745 869.097
-1278.31 -4926.66 36 51
-59.1114 1565.18
440.404
-8805.63 -1
236.446
-3623.27 -1278.31 -6679.56 440.404
37528.4
-1579.86 -2
1080.71
1123.84
-4926.66 173.117
-8805.63 -1579.86 49561.9
79
-45.3357 3324.87
368.1
5134.41
-1536.99 -29165.3 7926.83
26
-933.916 100.813
4311.27
1619.54
6973.92
-207.728 1517.14
1045.14
3427.17
533.2
-489.126 -1187.7
2195.69
-1199.01 4388.
-8640.69 -39493.7 16 -19053.6 -3441.39 12
40.3088
-342.694 -205.064 -735.797 -76.7913 4100.72
108.258
224.659
-9.92462 506.554
527.785
-2
-948.535 -1204.43 5313.54
16
799.737
-214.248 -4536.12 1093.8
39
215.475
1081.95
-487.888 202.313 70.233
-143.301 -4.54436 660.521
-1139.03 -6122.24 95
32.0692
-2351.92 -260.384 -3631.94 1087.22
660.626
-71.3121 -3049.67 -1145.62 -4933.15 6112.18
r Te = H -2977.98 -14889.9 17156.6
68225.4
-24544.7 -8
7053.29
-5607.22 -1
20630.7
27936.7
-1
-18023.1 117228. 85382. -661366. 108828. 5
After summing contributions from all points, the element equations as follows:
-33.3333 3.33333 -8.33333 6.66667 ij -1.66667 -8.33333 6.66667 j -3.33333 0 2.77778 -8.33333 4.44444 k a = jjjj 0.555556 8.33333 jj 2.22222 5.55556 -4.44444 22.2222 -2.77778 5.55556 1.11111 k
0
1.66
0
1.11
-11.1111 1.66
46
Multifield Formulations
jij 0.0000642857 j k b = jjjj -7.14286 µ 10-6 jj k0
20900.4 jij jj 926.544 jj jj jj 5076.64 jj jj jj -13424.6 jj jj jj -6298.03 jj jj jj 3582.64 jj jj jj 6839.13 jj jj jj 494.157 jj jj -887.011 j k c = jjjj jj -185.309 jj jj jjj 1551.65 jj jj 823.594 jj jj jj 1900.03 jj jj jj -3335.56 jj jj -21070.8 jj jj jj 14083.5 jj jj jj -8011.93 jj j k -2964.94
-7.14286 µ 10-6 0.0000214286 0
zyz zz zz 0 zz z 0.0000214286 { 0
926.544
5076.64
-13424.6 -6298.03 3582.64
6839.13
494.157
-8
27025.4
12518.6
8851.99
12518.6
108084.
-5133.74 15134.4
-2903.17 -8506.91 494.157
9453.22
-1
8851.99
-5133.74 138234.
-13877.6 -17684.2 -2470.78 25
12065.7
21439.8
-2470.78 -24793.5 78
12065.7
38615.9
-3644.4
-37378.
-11118.5 42
-8506.91 -13877.6 21439.8
-2903.17 15134.4
-3644.4
47878.8
14824.7
-49652.9 -3
494.157
-17684.2 -2470.78 -37378.
14824.7
83453.2
-2470.78 -3
9453.22
-2470.78 -24793.5 -11118.5 -49652.9 -2470.78 105829.
13
-185.309 2549.85
782.415
4296.69
-3397.33 -36389.7 13589.3
-1205.74 782.415
3353.68
3088.48
5924.94
823.594
12788.8
-466.703 3933.49
1884.38
-466.703 17758.9
3150.25
2935.29
3046.48
-905.954 2922.11
247.078
-1465.17 -988.313 10342.7
33
-11859.8 -16313.8 3953.25
8914.59
-1235.39 -13480.6 494.157
73
-28774.7 3953.25
-905.954 -1232.92 247.078
-18000.5 -1235.39 11009.8
-2964.94 -112571. 5600.44 -11774.1 5600.44
-947.133 -22691.7 3788.53
-947.133 4391.4
-25986.
-149767. 4447.41
4447.41
34
-12353.9 -48911.6 14 14
3788.53
-28549.1 -1
8025.1
-988.313 -6
494.157
-18758.2 74
29306.8
-1317.75 -1
-31019.9 -1317.75 45040.7
-3
47
jij 301211. jj jj 926061. jjj jj jj -675202. jj jj jj 3.08882 µ 106 jj jj jj jj -324278. jj jj jj 629572. jj jj jj 42293.4 jj jj jj -1.15177 µ 106 jjj jj jj 152009. j k = jjj jj 526089. jj jjj jj -58277.4 jj jj jj 1.50763 µ 106 jj jj jj jj -224121. jj jj jj 223793. jj jj jj 572788. jj jj jj -1.13818 µ 106 jj jj jj jj 213577. jj j 6 k -4.61202 µ 10
-675202.
926061. 6
6
3.08882 µ 106
-324278.
629572.
42
6
1.
5.20667 µ 10
-2.93462 µ 10
8.76162 µ 10
-25062.1
-811767.
-2.93462 µ 106
2.04699 µ 106
-7.60564 µ 106
541409.
-866996.
6
6
8.76162 µ 10
-7.60564 µ 10
-25062.1 1.28571 µ 10
-3.23767 µ 10
2.30206 µ 10
681201.
-333158.
7
6
3.68271 µ 10
-1.45063 µ 10 6
-1.19576 µ 10 6
431592.
2.77175 µ 10
-1.61807 µ 10
235819. -2.29594 µ 10 6
6
6
4.66367 µ 10
-2.56016 µ 10 6
-2.90943 µ 10
6
-3.63703 µ 10
7
-1.15344 µ 10
6
2.30848 µ 10
-1.15406 µ 10
9.22371 µ 10
r T = H -134167. -670833. 536667. -2.68333 µ 106
1.49478 µ 10
-588158.
-217198.
-4.62403 µ 10
6
3.68245 µ 10
1.43909 µ 10
6
5.76576 µ 10
-
6
52
6
-
-1.30861 µ 10
-2.86965 µ 10
-2.63631 µ 10 7
5.
-1.26281 µ 10
6
3. -
6
6
7.00782 µ 10 6
54
-504614.
303628. 7
6
640832.
Complete element equations for element 1
6.74026 µ 10
2.
6
6
-1.41792 µ 10
6
-
-3.78879 µ 10
540552.
95
6
6
6
2.77279 µ 10
641702.
6.72325 µ 10
-369019.
1.12765 µ 10
-2.06168 µ 10
6
-2.93031 µ 10
6
6
115946. 6
72435.3
5.11431 µ 10
-
-2.88799 µ 10 6
632311.
6
1.17657 µ 10
-2.56854 µ 10
955341. 1.42921 µ 10
597824. 6
1.05183 µ 10
-1.15474 µ 10 -218235.
7 6
-2.56854 µ 10 6
6
6 6
1.17657 µ 10
-2.87121 µ 10 6
-5.16204 µ 10
7
-208251. 6
4.11322 µ 10
-5.16204 µ 10
-866996. 6
6
6
541409.
-811767.
7
6
7.13962 µ 10
7
-1.15537 µ 10
-
268333. -670833. 536667. 0 134167. 670833.
48
Multifield Formulations
jij 301211. jj jj 926061. jjj jj jj -675202. jj jj jj 3.08882 µ 106 jj jj jj jj -324278. jj jj jj 629572. jj jj jj 42293.4 jj jj jj -1.15177 µ 106 jjj jj jj 152009. jj jj jj 526089. jj jj jj jj -58277.4 jj jj 6 jjj 1.50763 µ 10 jj jj jj -224121. jj jj jj 223793. jj jj jj 572788. jj jj jj -1.13818 µ 106 jj jj jj jj 213577. jj j 6 k -4.61202 µ 10
-675202.
926061. 6
6
3.08882 µ 106
-324278.
629572.
6
42293
5.20667 µ 10
-2.93462 µ 10
8.76162 µ 10
-25062.1
-811767.
1.2857
-2.93462 µ 106
2.04699 µ 106
-7.60564 µ 106
541409.
-866996.
-2082
8.76162 µ 106
-7.60564 µ 106
4.11322 µ 107
-5.16204 µ 106
1.17657 µ 107
-25062.1
6
-5.16204 µ 10
541409.
-811767.
7
-866996. 6
1.28571 µ 10
-3.23767 µ 10 6
-2.87121 µ 10
2.30206 µ 10
681201.
-333158. 6
7
6
3.68271 µ 10
-1.45063 µ 10 6
-1.19576 µ 10 6
431592.
2.77175 µ 10
-1.61807 µ 10
235819. -2.29594 µ 10 6
-2.90943 µ 10
6
-3.63703 µ 10
7
-1.15344 µ 10
5.11431 µ 10
6
2.30848 µ 10
6.74026 µ 10
-1.15406 µ 10 6
6
9.22371 µ 10
1.49478 µ 10
-588158.
-217198.
-4.62403 µ 10
6
3.68245 µ 10
1.43909 µ 10
6
5.76576 µ 10
-7100 52276
6
-5756
-2.86965 µ 10 6
7.13962 µ 10
-3.090 7
-1.15537 µ 10
81, 2, 7, 8, 13, 14, 15, 16, 17, 18, 11, 12, 5, 6, 3, 4, 9, 10< global degrees of freedom.
The element contributes to
-4515
6
-1.30861 µ 10
-2.63631 µ 10 7
5.5435
-1.26281 µ 10
6
3.4666 -8602
6
6
7.00782 µ 10
640832.
54860
-504614.
303628. 7
-5786
6
-1.41792 µ 10
6
2.1353
6
-3.78879 µ 10
6
2.77279 µ 10
641702.
-2.887
6
540552.
95534
6
-369019.
1.12765 µ 10 6
6.72325 µ 10
-2.93031 µ 10
6
6
-2.56016 µ 10 6
72435.3
-2.06168 µ 10 6
4.66367 µ 10
632311. -218235.
6
-2.88799 µ 10 6
1.42921 µ 10
6
115946. 6
-2.56854 µ 10
955341.
-1.15474 µ 10
597824. 6
1.05183 µ 10
-2.56854 µ 10 6
-3.237 6
6
1.17657 µ 10
-208251. 6
6
-2.305
49
1 jij jj 2 jj jj jj 7 jj jj jj 8 jj jj jj 13 jj jj jj 14 jj jj jj 15 jj jj jj 16 jj jj 17 j Locations for element contributions to a global vector: jjjj jj 18 jj jj jjj 11 jj jjj 12 jj jjj 5 jj jjj 6 jj jjj 3 jj jjj jj 4 jj jj jj 9 jj k 10
@1, 1D jij jj @2, 1D jj jj jj @7, 1D jj jj jj @8, 1D jj jj jj @13, 1D jj jj jj @14, 1D jj jj jj @15, 1D jj jj jj @16, 1D jj jj @17, 1D j and to a global matrix: jjjj jj @18, 1D jj jj jjj @11, 1D jj jj @12, 1D jj jj jj @5, 1D jj jj jj @6, 1D jj jj @3, 1D jj jj jj @4, 1D jj jj jj @9, 1D jj j k @10, 1D
@1, 2D
@1, 7D
@1, 8D
zyz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz z {
@1, 13D
@1, 14D
@1, 15D
@1, 16D
@
@7, 16D
@
@2, 2D
@2, 7D
@2, 8D
@2, 13D
@2, 14D
@2, 15D
@2, 16D
@8, 2D
@8, 7D
@8, 8D
@8, 13D
@8, 14D
@8, 15D
@8, 16D
@14, 14D
@14, 15D
@14, 16D
@7, 2D
@7, 7D
@7, 8D
@7, 13D
@13, 2D @13, 7D @13, 8D @13, 13D @14, 2D @14, 7D @14, 8D @14, 13D @15, 2D @15, 7D @15, 8D @15, 13D @16, 2D @16, 7D @16, 8D @16, 13D @17, 2D @17, 7D @17, 8D @17, 13D
@18, 2D @18, 7D @18, 8D @18, 13D
@7, 14D @13, 14D @15, 14D @16, 14D @17, 14D
@6, 14D
@6, 15D
@6, 16D
@5, 13D
@5, 14D
@3, 2D
@3, 7D
@3, 8D
@3, 13D
@9, 2D
@4, 7D @9, 7D
@4, 8D @9, 8D
@4, 13D @9, 13D
@10, 2D @10, 7D @10, 8D @10, 13D
Adding element equations into appropriate locations we have
@17, 16D
@12, 16D
@5, 8D
@4, 2D
@16, 16D
@12, 15D
@5, 7D
@6, 13D
@17, 15D
@15, 16D
@12, 14D
@5, 2D
@6, 8D
@16, 15D
@18, 16D
@11, 14D
@6, 7D
@15, 15D
@13, 16D
@18, 15D
@11, 13D
@6, 2D
@13, 15D
@18, 14D
@11, 2D @11, 7D @11, 8D
@12, 2D @12, 7D @12, 8D @12, 13D
@7, 15D
@3, 14D @4, 14D @9, 14D
@10, 14D
@11, 15D @5, 15D @3, 15D @4, 15D @9, 15D
@10, 15D
@11, 16D
@5, 16D @3, 16D @4, 16D @9, 16D
@10, 16D
@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
50
Multifield Formulations
jij 301211. jj jj 926061. jjj jj jj 572788. jj jj jj -1.13818 µ 106 jj jj jj jj -224121. jj jj jj 223793. jj jj jj -675202. jj jj jj 3.08882 µ 106 jjj jj jj 213577. jj jj jj -4.61202 µ 106 jj jj jj jj -58277.4 jj jj 6 jjj 1.50763 µ 10 jj jj jj -324278. jj jj jj 629572. jj jj jj 42293.4 jj jj jj -1.15177 µ 106 jj jj jj jj 152009. jj j k 526089.
926061.
-1.13818 µ 106
572788. 6
6
6
-224121.
-6752
223793. 6
5.20667 µ 10
4.66367 µ 10
-2.90943 µ 10
235819.
-2.29594 µ 10
-2.934
4.66367 µ 106
5.54303 µ 106
-577614.
1.16281 µ 106
-4.08606 µ 106
-2.560
6
6
2.3084
6
11594
6
-2.90943 µ 10
6
-577614.
5.83295 µ 10 6
1.16281 µ 10
235819. 6
-2.29594 µ 10
6
-2.93462 µ 10 6
1.71579 µ 10 6
-4.08606 µ 10
6
-2.56016 µ 10 6
8.76162 µ 10
-3.63703 µ 10
-4.55118 µ 10
7
-2.30519 µ 10
6
6
-1.15344 µ 10 -1.19576 µ 10 6
6
-1.2071 µ 10
-4.43508 µ 10
-25062.1 1.28571 µ 10
6
-2.30519 µ 10 7
-1.96329 µ 10
5.20856 µ 10
64170
115946.
641702.
2.0469
6
-2.06168 µ 10
6
-1.90122 µ 10 6
2.30603 µ 10
5.94221 µ 10
-4.83995 µ 10
8.76142 µ 10
-1.41792 µ 10 6
522765.
-575638.
-45156.3
-710073.
681201.
213411. 1.88425 µ 10
dof
Value
1
u1 v1
0 0
2
u2 v2
0 0
3
u3 v3
0 0
6
-2.87083 µ 10
Essential boundary conditions Node
6
-863456. 6
Remove 81, 2, 3, 4, 5, 6< rows and columns. After adjusting for essential boundary conditions we have
-1.618 6
6
9.2237 43159
6
3.68245 µ 10
5.74256 µ 10
64083
1.18349 µ 10 6
-1.26281 µ 10
6
-7.605
6
6
540552. 6
2.77279 µ 10
-1.15384 µ 10
-294332.
-1.15363 µ 10
6
7
6.9172 µ 10 7
6
-2.86965 µ 10
-575638.
3.68271 µ 10
-1.96329 µ 10 6
-1.30861 µ 10
-2.87121 µ 10 6
6
1.43909 µ 10 6
6
2.30848 µ 10
6
303628.
-811767.
6
-3294.38 6
2.77175 µ 10
-2.90899 µ 10 -1.15406 µ 10
6
-2.90899 µ 10
880970. 6
7
6.74026 µ 10 6
1.71579 µ 10
6
54140 -8669 -2082
6
1.72741 µ 10
-2.87032 µ 10
2.3020
-235445.
186147.
-3331
-468487.
-811619.
-1.450
51
6 jij 2.04699 µ 10 jj jj -7.60564 µ 106 jjj jj jj 640832. jj jj jj 9.22371 µ 106 jj jj jj jj 431592. jj jj jj -1.61807 µ 106 jj jj jj 541409. jj jj jj -866996. jj jj jj jj -208251. jj jj jj 2.30206 µ 106 jj jj jj -333158. jj jj 6 k -1.45063 µ 10
-7.60564 µ 106 7
9.22371 µ 106
640832. 6
-1.61807 µ 106
431592. 7
6
4.11322 µ 10
7.00782 µ 10
-4.62403 µ 10
597824.
5.11431 µ 10
7.00782 µ 106
9.79256 µ 106
-9.22557 µ 106
2.04426 µ 106
2.21776 µ 106
7
-4.62403 µ 10
6
-9.22557 µ 10 6
2.04426 µ 10
597824. 6
7
2.21776 µ 10 6
-5.16204 µ 10 7
-4.62381 µ 10 6
-2.63631 µ 10 6
1.17657 µ 10
-3.23767 µ 10
7
-1.15474 µ 10 632311. -218235.
6
5.76576 µ 10
-1.15537 µ 10 6
-3.09067 µ 10
6
-2.30584 µ 10
6
-2.30584 µ 10 7
2.30903 µ 10
-511852.
-3.45992 µ 10 6
-4.83377 µ 10
7
-1.15491 µ 10
-4.62381 µ 10
5.76576
643559.
-599691.
-50461 7
-599691. 6
1.49478 µ 10
4.113 µ 10
-58815
-588158.
1.05183
-217198.
-860299.
5.54352 µ 10
-1.15512 µ 10
-1.483 µ 10
v
0
0
2
0
0
3
0
0
4
0.101861
-0.0284674
5
0.104682
0.00699846
6
0.101556
0.0428647
7
0.191474
0.0198994
8
0.180643
0.017815
9
0.190381
0.0143666
Computation of reactions
Equation numbers of dof with specified values: 81, 2, 3, 4, 5, 6<
Extracting equations 81, 2, 3, 4, 5, 6< from the global system we have
72435.3
7
1.12765
1.17653 µ 10
u7 = 0.191474, v7 = 0.0198994, u8 = 0.180643, v8 = 0.017815, u9 = 0.190381, v9 = 0.0143666<
1.42921
6
2.81204 µ 10 6
8u4 = 0.101861, v4 = -0.0284674, u5 = 0.104682, v5 = 0.00699846, u6 = 0.101556, v6 = 0.0428647,
u
955341. 7
-194794.
Complete table of nodal values
-2.5685 6
Solving the final system of global equations we get
1
-2.6363
1866.81
3788.53 6
-5.1620 7
-504614. 7
7.13962 µ 10 6
9.25033 µ 10 1866.81
6
5.11431 µ 10
7
541409.
52
Multifield Formulations
jij 301211. jj jj 926061. jj jj jj jj 572788. jj jj jj -1.13818 µ 106 jjj jj jjj -224121. jj k 223793.
926061.
-1.13818 µ 106
572788. 6
6
5.20667 µ 10
6
4.66367 µ 10
6
-2.90943 µ 10
6
4.66367 µ 10
5.54303 µ 10 6
-2.90943 µ 10
1.16281 µ 10
235819. 6
-2.29594 µ 10
1.71579 µ 10
6
1.71579 µ 10 6
-4.08606 µ 10
-2.90899 µ 10
-2.560
6
2.3084
6
11594
-2.90899 µ 10
-1.96329 µ 10
880970. 6
-2.934
6
-4.08606 µ 10
6
5.83295 µ 10 6
6
-2.29594 µ 10
235819. 1.16281 µ 10
6
-6752
223793. 6
-577614.
-577614.
-224121.
6
-1.96329 µ 10
6
5.20856 µ 10
64170
Substituting the nodal values and re-arranging
i 301211. ij R1 yz jjjj zz jj jj jj R2 zz jj 926061. zz j jjj jj R zzz jjjj 572788. jj 3 zz jj z jj jj R4 zzz = jjjj zz jj -1.13818 µ 106 jj z jj jj R5 zzz jjjj zz jj -224121. jj z j j k R6 { jj k 223793.
926061.
-1.13818 µ 106
572788. 6
5.20667 µ 10
6
6
4.66367 µ 10
6
6
-2.90943 µ 10
-224121.
223793. -2.29594 µ 106
235819. 6
4.66367 µ 10
5.54303 µ 10
-577614.
1.16281 µ 10
-4.08606 µ 106
-2.90943 µ 106
-577614.
5.83295 µ 106
1.71579 µ 106
-2.90899 µ 106
235819.
1.16281 µ 106
1.71579 µ 106
880970.
-1.96329 µ 106
-2.29594 µ 106
-4.08606 µ 106
-2.90899 µ 106
-1.96329 µ 106
5.20856 µ 106
53
Carrying out computations, the reactions are as follows. Label
dof
Reaction
R1
u1
301.583
R2
v1
1158.08
R3
u2
-603.166
R4
v2
52.7036
R5
u3
301.583
R6
v3
-1210.78
Sum of Reactions dof: u
-3.20142 µ 10-10
dof: v
-1.45519 µ 10-10
Solution for element 1
Nodal displacements = H 0 0 0.101861 -0.0284674 0.191474 0.0198994 0.180643 0.017815 0.1903 7 jij 1.16978 µ 10 jj Plane strain C = jjj 1.16511 µ 107 jj k0
e T0 = H 0.0034477 0.0034477 0 L
1.16511 µ 107 7
1.16978 µ 10 0
zyz zz zz 0 zz z 23348.9 { 0
25 Solution at 8s, t< = 80, 0< ï 8x, y< = :25, ÅÅÅÅÅÅÅÅÅÅ > 2
Interpolation functions = 80, 0, 0, 0, 0, 0, 0, 0, 1<
u = 0.104682
v = 0.00699846
e T = H 0.00361488 0.00475547 0 -0.00013954 0 0 L s T = H 17192.7 17246. 17023.9 -3.2581 0 0 L
Principal stresses = H 17246.2 17192.5 Effective stress Hvon MisesL = 200.888
17023.9 L
Solution at 8s, t< = 8-1, -1< ï 8x, y< = 80, 0<
Interpolation functions = 81, 0, 0, 0, 0, 0, 0, 0, 0<
u = 0.
v = 0.
e T = H 0.00431938 0. 0 -0.00267538 0 0 L
s T = H -29972.8 -30174.5 -30174.5 -62.4673 0 0 L
54
Multifield Formulations
Principal stresses = H -29955. -30174.5 -30192.3 L
Effective stress Hvon MisesL = 228.892
Solution at 8s, t< = 8-1, 1< ï 8x, y< = 80, 20<
Interpolation functions = 80, 0, 0, 0, 0, 0, 1, 0, 0<
u = 0.
v = 0.
e T = H 0.0043169 0. 0 0.00314184 0 0 L
s T = H -30001.8 -30203.4 -30203.4 73.3586 0 0 L
Principal stresses = H -29977.9 -30203.4 -30227.3 L
Effective stress Hvon MisesL = 238.291
Solution at 8s, t< = 81, -1< ï 8x, y< = 850, 10<
Interpolation functions = 80, 0, 1, 0, 0, 0, 0, 0, 0<
u = 0.191474
v = 0.0198994
e = H 0.00418423 -0.000280483 0 -0.00069576 0 0 L T
s T = H -34821.6 -35030.1 -35017. -16.2452 0 0 L
Principal stresses = H -34820.3 -35017. -35031.4 L
Effective stress Hvon MisesL = 204.21
Solution at 8s, t< = 81, 1< ï 8x, y< = 850, 20<
Interpolation functions = 80, 0, 0, 0, 1, 0, 0, 0, 0<
u = 0.190381
v = 0.0143666
e T = H 0.00329833 -0.000826072 0 0.00143732 0 0 L s T = H -51541.5 -51734.1 -51695.5 33.5598 0 0 L
Principal stresses = H -51535.8 -51695.5 -51739.8 L
Effective stress Hvon MisesL = 185.828
Solution summary Nodal solution
55
x
y
u
v
1
0
0
0
0
2
0
10
0
0
3
0
20
0
0
4
25
5
0.101861
-0.0284674
5
25
25 ÅÅÅÅ ÅÅ 2
0.104682
0.00699846
6
25
20
0.101556
0.0428647
7
50
10
0.191474
0.0198994
8
50
15
0.180643
0.017815
9
50
20
0.190381
0.0143666
Solution at element centroids Coord
25 25 ÅÅÅÅ ÅÅ 2
1
Disp
Stresses
Principal stresses
Effective Stress
0.104682 0.00699846
17192.7 17246. 17023.9 -3.2581 0 0
17246.2 17192.5 17023.9
200.888
Support reactions Node
dof
Reaction
1
1
301.583
1
2
1158.08
2
1
-603.166
2
2
52.7036
3
1
301.583
3
2
-1210.78
Sum of applied loads Ø H -2.91038 µ 10-10
1.16415 µ 10-10 L
Sum of support reactions Ø H -3.20142 µ 10-10
-1.45519 µ 10-10 L
5.7 The formulation for the 4/1 element discussed in the text can handle plane stress and plane strain problems. Extend this formulation to analyze axisymmetric problems. Using only one element based on this formulation compute stresses in a thick cylinder subjected to an internal pressure of 1 MPa. The inner and outer diameters of the cylinder are 20 mm and 40 mm respectively. The cylinder is 30 mm long. One end of the cylinder is fixed and the other is free. Assume E = 200 GPa and n = 0.499.
56
Multifield Formulations
Quadrilateral Elements for Axisymmetric Problems Axisymmetric problems are formulated in terms of cylindrical coordinate system; r-z-q, where r is the radial direction, z is the axial symmetry direction, and q is the circumferential direction. Because of axial symmetry the displacement in the q direction is zero. The radial displacement (u) and the axial displacement HwL are independent of q and therefore v = 0 grq = gqz = 0 trq = tqz = 0 Thus an axisymmetric problem is described in terms of the following displacement, stress, and strain terms. T Displacement vector: uHr, zL = H u w L
Assumed solution over element ij u yz ij N1 0 N2 0 j z=j k w { k 0 N1 0 N2
ij u1 jj jj w1 … y jjjj zz jj u2 … { jjj jjj w2 jj k ª
yz zz zz zz zz zz = NT d zz zz zz zz {
p = PT b
T Stress vector: s = H sr sq sz trz L
Strain vector: e = H er eq ez grz LT
Strain-displacement relationships (assuming small displacements) ∑u ∑w ∑u ∑w er = ÅÅÅÅ ÅÅÅ eq = ÅÅÅÅurÅ ez = ÅÅÅÅ ÅÅÅÅ grz = ÅÅÅÅ ÅÅÅ + ÅÅÅÅ ÅÅÅÅ ∑r ∑z ∑z ∑r
The strain-displacement relationships give
57
∑u i ÅÅÅÅ ÅÅÅ ∑r ij er yz jjjj jj zz jj ∑w jj ez zz jj ÅÅÅÅ∑zÅÅÅÅ jj zz = jj jj zz jj jj eq zz jj ÅÅÅÅur Å jj zz jj j ∑u ∑w k grz { j ÅÅÅÅ k ∑zÅÅÅ + ÅÅÅÅ∑rÅÅÅÅ
∑N1 zyz jij ÅÅÅÅ∑rÅÅÅÅÅÅ zz jj zz jj zz jj 0 zz = jj zz jj zz jj N1 ê r zz jj zz jj ∑N1 { k ÅÅÅÅ∑zÅÅÅÅÅÅ
… zy ij u1 yz z zz jj zz jj w1 zzz ∑N1 ∑N2 zz j z ÅÅÅÅ∑zÅÅÅÅÅÅ 0 ÅÅÅÅ∑zÅÅÅÅÅÅ … zzz jjj zz T zzz jjj u2 zzz = B d zz 0 N1 ê r 0 … zzz jjj zz jj w2 zzz z zj ∑N1 ∑N2 ∑N2 ÅÅÅÅ∑rÅÅÅÅÅÅ ÅÅÅÅ∑zÅÅÅÅÅÅ ÅÅÅÅ∑rÅÅÅÅÅÅ … { jk ª z{ 0
∑N2 ÅÅÅÅ ÅÅÅÅÅÅ ∑r
0
The derivatives of the interpolation functions with respect to r and z are computed using the mapping. The mapping is r = H N1 N2 ∑r ij ÅÅÅÅ ÅÅ ∑s j j J = jj ∑z ÅÅ k ÅÅÅÅ ∑s
ij r1 yz ij z1 yz jj zz j z … L jjj r2 zzz and z = H N1 N2 … L jjjj z2 zzzz jj zz jj zz k ª { k ª {
∑r ÅÅÅÅ ÅÅ y i J11 J12 ∑t z zz = jj z k J21 J22 ∑z z ÅÅÅÅ Å Å ∑t {
yz ∑r ∑z ∑r ∑z z detJ = » ÅÅÅÅ ÅÅ ÅÅÅÅÅÅ - ÅÅÅÅ ÅÅ ÅÅÅÅÅÅ » ∑s ∑t ∑t ∑s {
∑Ni ∑Ni ∑Ni 1 ÅÅÅÅ ÅÅÅÅÅ = ÅÅÅÅÅÅÅÅ ÅÅ IJ22 ÅÅÅÅ ÅÅÅÅÅ - J21 ÅÅÅÅ ÅÅÅÅÅ M ∑r detJ ∑s ∑t
∑Ni ∑Ni ∑Ni 1 ÅÅÅÅ ÅÅÅÅÅ = ÅÅÅÅÅÅÅÅ ÅÅ I-J12 ÅÅÅÅ ÅÅÅÅÅ + J11 ÅÅÅÅ ÅÅÅÅÅ M ∑z detJ ∑s ∑t
Knowing the B matrix the deviatoric and volumetric strain-displacement matrices can then be evaluated as follows. BvT = mT BT and BdT = HI - ÅÅÅÅ13 m mT L BT
where mT = H 1 1 1 0 L and I is a 4 µ 4 identity matrix. The element equations are obtained using numerical integration. With a constant thickness h, the volume integration reduces to area integration and the surface integration to line integrals. Element stiffness matrix ka = 2 p Ÿ Ÿ P BvT r dA = 2 p Ÿ-1 Ÿ-1 P BvT r detJ ds dt = 2 p ‚ 1
1
A
kb = 2 p Ÿ Ÿ P ÅÅÅÅ1k P T r dA = 1 1 2 p Ÿ-1 Ÿ-1 A
P ÅÅÅÅ1k P T r detJ ds dt = 2 p ‚
m i=1
m i=1
T ‚j=1 wi wj PHsi , tj L Bv Hsi , tj L rHsi , tj L detJHsi , tj L n
1 ‚j=1 wi wj PHsi , tj L ÅÅÅÅk P T Hsi , tj L rHsi , tj L detJHsi , tj L n
58
Multifield Formulations
kc = 2 p Ÿ Ÿ Bd Cd BdT r dA = 1 1 2 p Ÿ-1 Ÿ-1 A
Bd Cd BdT r detJ ds dt = 2 p ‚
k = kc + kaT kb-1 ka
0 ij 2 G 0 jj jj 0 2 G 0 Cd = jjjj jj 0 0 2G jj 0 0 k 0
m i=1
T ‚j=1 wi wj Bd Hsi , tj L Cd Bd Hsi , tj L rHsi , tj L detJHsi , tj L n
0y zz 0 zzzz zz G = ÅÅÅÅÅÅÅÅEÅÅÅÅÅÅÅÅÅ k = ÅÅÅÅÅÅÅÅÅEÅÅÅÅÅÅÅÅÅÅÅÅ 2 H1+nL 3 H1-2 nL 0 zzz zz G{
Equivalent load vector due to body forces:
rb =
i br 2 p Ÿ Ÿ N jj k bz A
1 1 i br yz z r dA = 2 p ‡ ‡ N jj k bz { -1 -1
i br yz z r detJ ds dt = 2 p „ „ wi wj NHsi , tj L jj k bz { i=1 j=1 m
n
Equivalent load vector due to distributed loads: i qr rq = 2 p ‡ Nc jj k qz C
1 yz i qr z rc dc = 2 p ‡ Nc jj { k qz -1
i qr yz z rc Jc da = 2 p „ wi Nc Hai L jj k qz { i=1 n
yz z rHsi , tj L detJHsi , tj L {
yz z rc Hai L Jc Hai L {
4/1 — U/P Quadrilateral Element The simplest element is a 4 node quadrilateral element as shown in Figure 5.11. The interpolation functions for mapping and the displacements are as follows. ij jj jj jj jj Interpolation functions Ø jjj jj jj jj jj k
ÅÅÅÅ14Å H1 - s L H1 - tL yz zz zz ÅÅÅÅ14Å Hs + 1L H1 - tL zzzz zz z ÅÅÅÅ14Å Hs + 1L Ht + 1L zzzz zz zz ÅÅÅÅ14Å H1 - s L Ht + 1L {
A constant pressure is assumed over the element. Thus the stress interpolation matrix is a 1 µ 1 matrix with the value1. That is
59
PT = 881 4 4 4 4
t-1 1-t t+1 1 ∑NT ê∑s = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Å , ÅÅÅÅÅ H-t - 1L> 4 4 4 4
s -1 1 s +1 1-s ∑NT ê∑t = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-s - 1L, ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ > 4 4 4 4 P T = 81<
Mapping to the master element rHs,tL = 5 s + 15 zHs,tL = 15 t + 15
i 5 0 yz J = jj z k 0 15 {
detJ = 75
Gauss quadrature points and weights Point
Weight
1
s Ø -0.57735 t Ø -0.57735
1.
2
s Ø -0.57735 t Ø 0.57735
1.
3
s Ø 0.57735 t Ø -0.57735
1.
4
s Ø 0.57735 t Ø 0.57735
1.
Computation of element matrices at 8-0.57735, -0.57735< with weight = 1. i 5 0 yz J = jj z k 0 15 {
detJ = 75
NT = H 0.622008 0.166667 0.0446582 0.166667 L
∑NT ê∑s = H -0.394338 0.394338 0.105662 -0.105662 L
∑NT ê∑t = H -0.394338 -0.105662 0.105662 0.394338 L
62
Multifield Formulations
-0.0788675 jij jj 0 j B T = jjjj jj 0.0513494 jj k -0.0262892
0
0.0788675
0
0.0211325
-0.0262892 0
-0.00704416 0
0
0
0.013759
-0.0788675 -0.00704416 0.0788675
-0.021132
0
0.00704416 0
0.00368672 0
0.013759
0.00704416 0.0211325
0.0262892
B Tv =
8-0.0275181, -0.0262892, 0.0926266, -0.00704416, 0.0248192, 0.00704416, -0.00737345, 0.0262892< ij -0.0696948 jj jj 0.00917269 B Td = jjjj jjj 0.0605221 j k -0.0262892 PT = H 1 L
0.00876306
0.047992
-0.0175261 -0.0308755 0.00876306
-0.0171165
0.00234805 0.00234805
-0.0788675 -0.00704416 0.0788675
k a = H -157.08 -150.065 528.734
-40.2097 141.674
k b = H 0.000171247 L
ij 6.81637 µ 106 jj jj jj 605886. jj jj jj -3.48156 µ 106 jj jj jj -838752. k c = jjjj jj jj -1.02229 µ 106 jj jj -162347. jj jj jj jj 1.49274 µ 106 jj k 395213.
-3.48156 µ 106
605886. 6
2.71953 µ 10
829746. 2.72222 µ 10
6
0.0211325
0.02628
-42.0894 150.065 L
40.2097
-1.02229 µ 106
-162347.
-2.2746 µ 10
-45913.9
-728697.
-45913.9
705457.
-222330.
-838752.
6
-2.2746 µ 10
-45913.9
2.39382 µ 10
255941.
609477.
-45913.9
705457.
255941.
212985.
12302.6
-728697.
-222330.
609477.
12302.6
195254.
6
-838752.
-1.02229 µ 10
776356.
-184509.
224743.
283764.
-561502.
-728697.
-222330.
-76034.3
Computation of element matrices at 8-0.57735, 0.57735< with weight = 1. i 5 0 yz J = jj z k 0 15 {
0.00245
-0.00458635 -0.00234805 0.01621 0.00704416
6
6
829746.
-0.00234805 -0.018
0.0128594
-0.00469611 -0.00827307 0.00469611
detJ = 75
NT = H 0.166667 0.0446582 0.166667 0.622008 L
∑NT ê∑s = H -0.105662 0.105662 0.394338 -0.394338 L
∑NT ê∑t = H -0.394338 -0.105662 0.105662 0.394338 L -0.0211325 jij jj 0 j B T = jjjj jj 0.013759 jj k -0.0262892 B Tv
0
0.0211325
0
0.0788675
0
-0.078867
-0.0262892 0
-0.00704416 0
0.00704416 0
0
0
0
0.00368672
-0.0211325 -0.00704416 0.0211325
0.013759
0.00704416 0.0788675
0.0513494 0.0262892
8-0.00737345, -0.0262892, 0.0248192, -0.00704416, 0.0926266, 0.00704416, -0.0275181, 0.0262892< =
63
-0.0186747 jij jj 0.00245782 j B Td = jjjj jj 0.0162169 jj k -0.0262892 PT = H 1 L
0.00876306
0.0128594
0.00234805
-0.00234805 -0.0696
0.047992
-0.0175261 -0.00827307 -0.00469611 -0.0308755 0.00469611 0.00876306
-0.00458635 0.00234805
-0.0211325 -0.00704416 0.0211325
k a = H -42.0894 -150.065 141.674
k b = H 0.000171247 L
ij 733678. jj jj 162347. jj jj jj jj -184509. jj jj -224743. jj k c = jjjj jj -1.02229 µ 106 jj jj jjj -776356. jj jj jj 1.49274 µ 106 jj j k 838752.
-40.2097 528.734
0.00704416
40.2097
0.0788675
0.026289
-157.08 150.065 L
162347.
-184509.
-224743. -1.02229 µ 106
-776356.
1.492
520967.
222330.
-76034.3 561502.
-728697.
-395
222330.
212985.
-12302.6 705457.
-255941.
-1.0
195254.
609477.
1623
-76034.3 -12302.6 561502.
705457.
222330.
-728697. -255941.
609477. 6
-395213. -1.02229 µ 10 283764.
162347.
222330. 6
2.72222 µ 10
6
2.39382 µ 10
45913.9 6
-3.48156 µ 10
838752.
8387 6.816
6
-728697. -829746.
45913.9
-3.4
45913.9
-2.2746 µ 10
Computation of element matrices at 80.57735, -0.57735< with weight = 1. i 5 0 yz J = jj z k 0 15 {
0.009172
-0.0171165 -0.00234805 0.060522
-605
detJ = 75
NT = H 0.166667 0.622008 0.166667 0.0446582 L
∑NT ê∑s = H -0.394338 0.394338 0.105662 -0.105662 L
∑NT ê∑t = H -0.105662 -0.394338 0.394338 0.105662 L -0.0788675 jij jj 0 j B T = jjjj jj 0.00931788 jjj k -0.00704416 B Tv
0
0.0788675
0
0.0211325
-0.00704416 0
-0.0262892 0
0
0.0347748
0
-0.0788675
-0.0262892 0.0788675
0
0.0262892 0
0.00931788 0 0.0262892
-0.0211325 0.00249672
0.0211325 0.00704416
8-0.0695496, -0.00704416, 0.113642, -0.0262892, 0.0304504, 0.0262892, -0.0186358, 0.00704416< =
-0.0556843 jij jj 0.0231832 j B Td = jjjj jj 0.0325011 jj k -0.00704416 P =H1 L T
0.00234805
0.0409867
-0.00469611 -0.0378808
0.00876306
0.0109824
-0.0175261 -0.0101501
-0.00876306 -0.014 0.0175261
0.0062
0.00234805
-0.00310596 0.00876306
-0.00083224 -0.00876306 0.0087
-0.0788675
-0.0262892
0.0262892
0.0788675
0.0211325
0.0070
64
Multifield Formulations
k a = H -586.229 -59.3748 957.884
-221.59 256.664
k b = H 0.000252868 L
ij 5.3074 µ 106 jj jj jjj 128735. jj jj jj -3.56374 µ 106 jj jj jj -997803. k c = jjjj jj -1.08693 µ 106 jj jj jj 601707. jj jj jj jj 1.38674 µ 106 jj j k 267360.
-3.56374 µ 106
128735. 6
3.53478 µ 10
1.46595 µ 10
6
-3.35874 µ 10
6
-3.35874 µ 10
-1.08545 µ 10
-1.07601
-45913.9
552914.
-1.43234
-1.08545 µ 10
6
-1.07601 µ 10 -361601.
-45913.9
6
1.46595 µ 10
419013.
6
1.46595 µ 10
640903.
12302.6
6
419013.
12302.6
769274.
6
128735.
-159211.
267360.
-392799.
-112274.
552914. -1.43234 µ 10
6
-1.07601 µ 10
12302.6
Computation of element matrices at 80.57735, 0.57735< with weight = 1. i 5 0 yz J = jj z k 0 15 {
6
4.01574 µ 10
-1.08693 µ 10
899971.
6
601707.
6
3.90247 µ 10 6
-1.08693 µ 106
-997803.
6
1.46595 µ 10
6
221.59 -157.08 59.3748 L
detJ = 75
NT = H 0.0446582 0.166667 0.622008 0.166667 L
∑NT ê∑s = H -0.105662 0.105662 0.394338 -0.394338 L
∑NT ê∑t = H -0.105662 -0.394338 0.394338 0.105662 L ij -0.0211325 jj jj 0 B T = jjjj jjj 0.00249672 j k -0.00704416
0
0.0211325
0
-0.0788675
0.0788675 0
-0.00704416 0
-0.0262892 0
0
0.00931788
0
-0.0211325
-0.0262892 0.0211325
0.0262892 0
0.0347748 0
0.00931788
0.0262892 0.0788675 0.00704416
B Tv =
8-0.0186358, -0.00704416, 0.0304504, -0.0262892, 0.113642, 0.0262892, -0.0695496, 0.00704416< ij -0.0149206 jj jj 0.00621192 B Td = jjjj jj 0.00870864 jj k -0.00704416 PT = H 1 L
0.00234805
0.00876306
0.0409867
-0.0175261 -0.0378808
-0.00876306 -0.055 0.0175261
0.0231
0.00234805
-0.00083224 0.00876306
-0.00310596 -0.00876306 0.0325
-0.0211325
-0.0262892
0.0262892
k a = H -157.08 -59.3748 256.664
k b = H 0.000252868 L
0.0109824
-0.00469611 -0.0101501
0.0211325
-221.59 957.884
0.0788675
221.59 -586.229 59.3748 L
0.0070
65
jij 406952. jj jj 34494.6 jjj jj jj -159211. jj jj jj -267360. j k c = jjjj jj -1.08693 µ 106 jj jj jj -128735. jj jj jj jj 1.38674 µ 106 jj j k 361601.
-159211.
34494.6
-267360.
392799.
-112274.
-12302.6
640903.
-12302.6
552914.
-112274.
-12302.6
-12302.6
6
-1.07601 µ 10
6
-1.46595 µ 10
6
-267360.
-1.08693 µ 10 6
419013.
45913.9
-601707.
6
7.0636 µ 10
2.91082 µ 10
7.47858 µ 10 6
-5.82165 µ 10
-4.21845 µ 106
-465732.
-582165.
-3.60942 6
-3.37655
6
2.05698 µ
6
116433.
2.51674 µ 10 6
-116433.
7.37408 µ 10
3.37655 µ 10
6
-582165.
2.51674 µ 10
3.37655 µ 10
7.47858 µ 10
-3.60942 µ 106
-3.37655 µ 106
2.05698 µ 106
116433.
6
6
-1.86293 µ 10 6
2.36747 µ 10
-4.21845 µ 10 582165.
-3.35874
-1.46595 µ 10
7.37408 µ 6
2.32866 µ
6
-5.82165
-7.38901 µ 10
465732. 6
-3.60942 µ 10
-2.91082 µ 10
4.66353 µ 108
-2.10178 µ 109
5.79448 µ 108
-2.09861 µ 109
-5.82242 µ 108
1.
8
8
8
8
8
4.
-9.27931 µ 10 8
-9.27931 µ 10 8
-9.31424 µ 10
8
-2.62177 µ 10 8
4.19627 µ 10
9
4.19131 µ 10
-2.04486 µ 10
9.31424 µ 10
1-a ÅÅÅÅ ÅÅÅÅÅÅ M 2
3.30583 µ 10
-3.21152 µ 10 8
4.19131 µ 10
1.16018 µ 10
9
4.19627 µ 10
3.30583 µ 10
-2.62177 µ 10
9.27931 µ 10
8
-5.79448 µ 10 8
2.52746 µ 10
5. -
8
1.16367 µ 10
8
-3.21152 µ 10 1.16367 µ 10
9
-2.10178 µ 10 8
8
9
9
5.82242 µ 10
-2.62177 µ 10 9
-1.16018 µ 10 8
1.16018 µ 10
9 9
-1.16018 µ 10
9
8
8 9
-2.09861 µ 10 8
-9.31424 µ 10
-1.16367 µ 10
9
4.63558 µ 10
2.52746 µ 10
9
-1.16367 µ 10 8
0 0 0L
9 9
2.52746 µ 10
NBC on side 4 with q = 8-1, 0< 0 0
-116433.
6
Computation of element matrices resulting from NBC a+1 ÅÅÅÅÅÅ NTc = I ÅÅÅÅ 2
-2.32866 µ 106 -5.82165 µ 10
6
997803.
6
-942.478 418.879 L
523.599
6
2.91082 µ 10 6
2.13917 µ 10
0 0 0 0
6
4.01574 µ 6
-3.56374 µ 10
-1.07601 µ 10
-7.38901 µ 106
45913.9
3.90247 µ 10
-523.599 1884.96
931463.
419013.
6
1.43234 µ 10
6
1.08545 µ 10
899971.
-1.46595 6
1.43234 µ 10
769274.
552914. 6
k b = H 0.00084823 L
rT = H 0
-1.07601
288317.
k a = H -942.478 -418.879 1884.96
ij 1.06046 µ 109 jj jj 8 jjj 4.66353 µ 10 jj jj jj -2.10178 µ 109 jj jj jj 5.79448 µ 108 k = jjjj jj -2.09861 µ 109 jj jj jj -5.82242 µ 108 jj jj jj jj 1.05296 µ 109 jj j 8 k -4.63558 µ 10
-128735.
392799.
After summing contributions from all points, the element equations as follows:
7 jij 1.32644 µ 10 jj jj 931463. jj jj jj jj -7.38901 µ 106 jj jj jj -2.32866 µ 106 k c = jjjj jj -4.21845 µ 106 jj jj jj -465732. jj jj jj jj 5.75895 µ 106 jj j 6 k 1.86293 µ 10
-1.08693 µ 106
1. -
66
Multifield Formulations
rHaL = 10
zHaL = 15 - 15 a
drêda = 0
dzêda = -15
J c = 15
Gauss point = -0.57735
Weight = 1.
r Tq = H 199.169
743.309
NTc = H 0.211325 0 0 0.788675 L 0
0 0 0 0
Gauss point = 0.57735 NTc
0
0L J c = 15
Weight = 1.
= H 0.788675 0 0 0.211325 L
r Tq = H 743.309
J c = 15
0 0 0 0
199.169
0L
Summing contributions from all Gauss points r Tq = H 942.478 0 0
0 0 0 942.478 0 L
Complete element equations for element 1 ij 1.06046 µ 109 jj jj 8 jjj 4.66353 µ 10 jj jj jj -2.10178 µ 109 jj jj jj 5.79448 µ 108 jj jj jj -2.09861 µ 109 jj jj jj -5.82242 µ 108 jj jj jj 9 jjj 1.05296 µ 10 jj 8 k -4.63558 µ 10
4.66353 µ 108
-2.10178 µ 109
5.79448 µ 108
-2.09861 µ 109
-5.82242 µ 108
1.0529
8
8
8
8
8
4.6355
2.13917 µ 10
-9.27931 µ 10 8
-9.27931 µ 10 8
9
4.19627 µ 10
-1.16367 µ 10
-1.16367 µ 10 8
-9.31424 µ 10
8
-2.62177 µ 10 8
9
4.19131 µ 10
-2.04486 µ 10
8
9.31424 µ 10
3.30583 µ 10
-3.21152 µ 10 8
Node
dof
Value
1
u1 w1
0 0
2
u2 w2
0 0
Remove 81, 2, 3, 4< rows and columns. After adjusting for essential boundary conditions we have
1.16018 µ 10
9
4.19627 µ 10
3.30583 µ 10
-5.794 8
-5.79448 µ 10 8
2.52746 µ 10
5.8224 -2.101
8
1.16367 µ 10 9.27931 µ 10
-3.21152 µ 10 1.16367 µ 10
9
8
-2.098 8
9
-2.10178 µ 10 8
-2.62177 µ 10
Essential boundary conditions
4.19131 µ 10
9
5.82242 µ 10
-2.62177 µ 10 9
-1.16018 µ 10 8
1.16018 µ 10
9 9
-1.16018 µ 10
9
-2.09861 µ 10 8
8 9
9
4.63558 µ 10
-9.31424 µ 10 9
9
2.52746 µ 10
2.52746 µ 10
1.0604 -4.663
67
9 jij 4.19627 µ 10 jj jj 1.16367 µ 109 jjj jj jj -2.10178 µ 109 jj jj 8 k 9.27931 µ 10
zyz ji u3 zz j 2.52746 µ 10 zzzz jjjj w3 zz jjj z -4.66353 µ 108 zzz jjjj u4 zz z 2.13917 µ 108 { k w4
1.16367 µ 109
-2.10178 µ 109
9.27931 µ 108
8
8
8
3.30583 µ 10
-5.79448 µ 10
-5.79448 µ 108
1.06046 µ 109
2.52746 µ 108
-4.66353 µ 108
yz ji 0 zyz zz jj zz zz jj 0 zz zz = jj zz zz jj z zzz jjj 942.478 zzz zz jj zz { k0 {
Solving the final system of global equations we get
9u3 = 0.0000972836, w3 = -0.0000706801, u4 = 0.000150769, w4 = -9.80215 µ 10-6 =
Complete table of nodal values u
w
1
0
0
2
0
0
3
0.0000972836
-0.0000706801
4
0.000150769
-9.80215 µ 10-6
Solution for element 1
Nodal displacements = H 0 0 0 0 0.0000972836 -0.0000706801 0.000150769 -9.80215 µ 10-6 L
ij 3.34223 µ 107 jj jj jj 3.32889 µ 107 C = jjj jj jj 3.32889 µ 107 jj k0
3.32889 µ 107
3.32889 µ 107
7
7
3.32889 µ 10
7
3.32889 µ 10
3.34223 µ 107
0
0
3.34223 µ 10
Solution at 8s, t< = 80, 0< ï 8x, y< = 815, 15<
yz zz zz zz 0 zz zz zz 0 zz z 66711.1 { 0
1 1 1 1 Interpolation functions = : ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ > 4 4 4 4
u = 0.0000620132
e T = H -2.67428 µ 10-6
v = -0.0000201206 -1.34137 µ 10-6
4.13421µ 10-6
s T = H 3.59008 3.76792 4.49849 0.0727363 0 0 L
Principal stresses = H 4.49849
3.79388 3.56412 L
Effective stress Hvon MisesL = 0.843295
Solution summary Nodal solution
1.09032 µ 10-6
0 0L
68
Multifield Formulations
r
z
u
w
1
10
0
0
0
2
20
0
0
0
3
20
30
0.0000972836
-0.0000706801
4
10
30
0.000150769
-9.80215 µ 10-6
Solution at selected points on the elements r
1
15
z
15
Disp
Stresses
Principal stresses
Effective Stress
0.0000620132 -0.0000201206
3.59008 3.76792 4.49849 0.0727363 0 0
4.49849 3.79388 3.56412
0.843295
5.8 Programming Project Implement either 4/1 or 9/3 U/P formulation in fe2Quad program discussed in Chapter 2.
CHAPTER SIX
Plates and Shells
6.1 An a µ b simply supported thin plate is subjected to a load qHx, yL = q0 sinHp x ê aL sinHp y ê bL
where q0 is a constant. (i) Show that a solution of the same form as the load satisfies the governing differential equation and the simple support boundary conditions. (ii) Determine the maximum deflection and the maximum stresses sx and sy for a plate with E = 106 , n = 0.3, h = 0.2, q0 = 10, a = 10, and b = 5.
2
Plates and Shells
y a
b
x
Figure 6.22.
Try a solution of the following form px py wHx,yL = c sinJ ÅÅÅÅÅÅÅÅÅÅ N sinJ ÅÅÅÅÅÅÅÅÅÅÅ N a b
where c is a constant. By differentiating and substituting into appropriate expressions, the moments Mx and My are as follows. py px ÅÅ Å M sinI ÅÅÅÅ ÅÅÅÅÅ M D c p2 Hn a 2 + b 2 L sinI ÅÅÅÅ a b ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ ÅÅÅÅ Mx = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ a2 b2 py px ÅÅÅÅ M sinI ÅÅÅÅ ÅÅ Å M D c p2 Ha 2 + b 2 nL sinI ÅÅÅÅ a b ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ ÅÅÅÅ My = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 2 2 a b
The simple support boundary conditions are as follows. w = 0 along all four sides Mx = 0 along x = 0 and x = a
3
My = 0 along y = 0 and y = b Check for zero displacement along the sides 80, 0, 0, 0<
Check for zero moments along the sides Along along x = 0 and x = a 80, 0< 80, 0<
The constant c is determined by substituting the solution into the governing differential equation. py px ÅÅÅÅ M sinI ÅÅÅÅ ÅÅÅÅÅ M Ha 2 + b 2 L c p4 sinI ÅÅÅÅ px py a b ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ ÅÅÅÅ ÅÅÅÅÅÅ L - sinJ ÅÅÅÅÅÅÅÅÅÅ N sinJ ÅÅÅÅÅÅÅÅÅÅÅ N q0 = 0 DH ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 4 4 a b a b 2
Using the given numerical data
D = 732.601
c = 0.0560523 px py w = 0.0560523 sinJ ÅÅÅÅÅÅÅÅÅÅ N sinJ ÅÅÅÅÅÅÅÅÅÅÅ N 10 5
The maximum deflection clearly occurs at the center.
Maximum deflection = 0.0560523
ï
a 4 b 4 q0 c = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ 2 D Ha 2 + b 2 L p4
4
Plates and Shells
Maximum Mx = 8.91626
Maximum sx = 1337.44
Maximum My = 17.4272
Maximum sy = 2614.09
6.2 Determine deflection and stresses at the center of a simply supported rectangular plate subjected to a uniformly distributed load. Use the following numerical data. 10 µ 5 plate h = 0.2 E = 106 n = 0.3 q = 10 Taking advantage of symmetry and model a 1/4 of the plate using only one element. From the available literature on plates determine the exact solution and compare the finite element solution with the exact solution.
2.5
0
y4
3
1
2
0
5
x
Figure 6.23.
(i) Finite element solution The element sides 1-2 and 1-4 are simply supported, and the other two sides have symmetry boundary condition. Thus the following essential boundary conditions are imposed.
5
Side1 - 4 : w = 0 qx = 0 Side1 - 2 : w = 0 qy = 0 Side 2 - 3 : qy = 0 Side 3 - 4 : qx = 0
Equations for element 1
i 0 y i 5 y i5y 2 Ø jj zz 3 Ø jjjj 5 zzzz 4 Ø jjjj 5 zzzz k0{ k ÅÅ2ÅÅ { k ÅÅ2ÅÅ {
i0y Element nodes: 1 Ø jj zz k0{ 5 a = ÅÅÅÅÅ 2
5 b = ÅÅÅÅÅ 4
h = 0.2
E = 1000000
n = 0.3
D = 732.601
q = 10.
1236.63 -275.458 260.22 521.612 -187.546 -348.132 565.568 -32.2 ij 1146.37 jj jj 1236.63 2021.98 -219.78 521.612 908.425 0 -565.568 505.495 0 jj jj jj -275.458 -219.78 761.905 187.546 0 175.824 32.2344 0 190.4 jj jj jj 260.22 521.612 187.546 1146.37 1236.63 275.458 -1058.46 1192.67 -55.6 jj jj 908.425 0 1236.63 2021.98 219.78 -1192.67 959.707 0 jj 521.612 jj jj -187.546 0 175.824 275.458 219.78 761.905 -55.6777 0 -29.3 jj jj jj -348.132 -565.568 32.2344 -1058.46 -1192.67 -55.6777 1146.37 -1236.63 275.4 jj jj jj 565.568 505.495 0 1192.67 959.707 0 -1236.63 2021.98 -219.7 jj jj jj -32.2344 0 190.476 -55.6777 0 -29.304 275.458 -219.78 761.9 jj jj jj -1058.46 -1192.67 55.6777 -348.132 -565.568 -32.2344 260.22 -521.612 -187.5 jj jj 959.707 0 565.568 505.495 0 -521.612 908.425 0 jjj 1192.67 j 55.6777 0 -29.304 32.2344 0 190.476 187.546 0 175.8 k
Essential BC Ø
Node
dof
Value
1
1 2 3
0 0 0
2
1
0
2
3
0
3
2
0
3
3
0
4
1
0
4
2
0
6
Plates and Shells
Global matrices after incorporating EBC
0 yz ij 2021.98 -1192.67 z j K = jjjj -1192.67 1146.37 187.546 zzzz zz jj 0 187.546 761.905 { k
RT = H 13.0208 31.25 -26.0417 L
Solution of global equations Ø 80.0738563, 0.114294, -0.0623135<
Solution for element 1
2 5 Local coordinates, s = ÅÅÅÅÅ ijx - ÅÅÅÅÅ yz 5 k 2{
4 5 t = ÅÅÅÅÅ ijy - ÅÅÅÅÅ yz 5 k 4{
1 1 Interpolation functions = :- ÅÅÅÅÅÅÅÅÅÅÅÅÅ Hx - 5L H2 y - 5L I2 x2 - 5 x + 8 y2 - 10 y - 25M, - ÅÅÅÅÅÅÅÅÅÅÅÅÅ Hx - 5L H5 - 2 yL2 y, 625 125 1 1 1 ÅÅÅÅÅÅÅÅÅÅÅÅÅ Hx - 5L2 x H2 y - 5L, ÅÅÅÅÅÅÅÅÅÅÅÅÅ x H2 y - 5L I2 x2 - 15 x + 2 y H4 y - 5LM, ÅÅÅÅÅÅÅÅÅÅÅÅÅ x H5 - 2 yL2 y, 125 625 125 2 2 2 1 ÅÅÅÅÅÅÅÅÅÅÅÅÅ Hx - 5L x2 H2 y - 5L, - ÅÅÅÅÅÅÅÅÅÅÅÅÅ x y I2 x2 - 15 x + 8 y2 - 30 y + 25M, ÅÅÅÅÅÅÅÅÅÅÅÅÅ x y2 H2 y - 5L, - ÅÅÅÅÅÅÅÅÅÅÅÅÅ Hx - 5L x2 y, 625 125 125 125 2 2 2 ÅÅÅÅÅÅÅÅÅÅÅÅÅ Hx - 5L y I2 x2 - 5 x + 2 y H4 y - 15LM, - ÅÅÅÅÅÅÅÅÅÅÅÅÅ Hx - 5L y2 H2 y - 5L, - ÅÅÅÅÅÅÅÅÅÅÅÅÅ Hx - 5L2 x y> 625 125 125 Nodal dof = H 0 0 0 0 0.0738563 0 0.114294 0 0 0 0 -0.0623135 L
wHx,yL = 0.000265537 y x3 - 0.00448407 y x2 - 0.000562514 y3 x - 0.000844824 y2 x + 0.0305532 y x 0.00159322 x y - 0.00896813 y ij yz jj zz zz -0.00337508 y x - 0.00168965 x zz jj zz 2 2 k 0.00159322 x - 0.0179363 x - 0.00337508 y - 0.0033793 y + 0.0611063 {
y = jjjj
732.601 219.78 0 y zz jij j j 0 zzzz D C = jj 219.78 732.601 zz jj 0 256.41 { k 0
-0.42542 y x + 0.371351 x + 6.57006 y yz ij Mx yz ij z j zz jj zz jj My zzz = jjj 2.12243 y x + 1.23784 x + 1.97102 y zz jj zz jj zz jj zz jj 2 2 k Mxy { k -0.408519 x + 4.59904 x + 0.865406 y + 0.866486 y - 15.6683 {
ij Vx yz ji 1.30539 y + 1.23784 zy z=j j z k Vy { k 1.30539 x + 6.57006 {
-63.813 y x + 55.7027 x + 985.509 y ij sx yz ij yz jj zz jj zz j z zz j s 318.364 y x + 185.676 x + 295.653 y Maximum in-plane stresses, jjj y zzz = jjj zz jj zz jj zz 2 2 t -61.2778 x + 689.857 x + 129.811 y + 129.973 y 2350.24 k xy { k {
7
i tzx zy ji 9.79043 y + 9.28378 zy z=j z Maximum transverse shear stresses, jj k tzy { k 9.79043 x + 49.2755 { 5 5 Solution at x = ÅÅÅÅÅ and y = ÅÅÅÅÅ 2 4 w = 0.0595864
Moments = 87.81152, 12.191, -4.28862<
Maximum in-plane stresses = 81171.73, 1828.64, -643.292<
Shears = 82.86958, 9.83354<
Maximum transverse shear stresses = 821.5218, 73.7516<
Solution summary Nodal solution x-coord
y-coord
w
qx
qy
1
0
0
0
0
0
2
5
0
0
0.0738563
0
3
5
ÅÅ52ÅÅ
0.114294
0
0
4
0
ÅÅ52ÅÅ
0
0
-0.0623135
Element solution
1
Coord
Disp
Max in-plane stresses
Max transverse shear stresses
ÅÅ52ÅÅ
0.0595864
1171.73 1828.64 -643.292
21.5218 73.7516
ÅÅ54ÅÅ
(ii) Exact solution 16 q a b w = ÅÅÅÅÅÅÅÅ ÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ ÅÅÅÅ m, n = 1, 3, 5, … m 2 n 2 2 p6 D „ „ mpx npy sinH ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ L sinH ÅÅÅÅÅÅÅÅ ÅÅÅÅ L
m
n m n IH ÅÅÅÅaÅÅ L +H ÅÅbÅÅ L M
The maximum occurs at the mid-span. Using the numerical data given, and three terms in the sum, we get
Maximum deflection = 0.0864965
8
Plates and Shells
x-coord
y-coord
w
qx
qy
1
0
0
0
0
0
2
5
0
0
0.055621
0
3
5
ÅÅÅÅ52Å
0.0864965
0
0
4
0
ÅÅÅÅ52Å
0
0
-0.0332987
(iii) Ansys solution
PRINT U
NODAL SOLUTION PER NODE
***** POST1 NODAL DEGREE OF FREEDOM LISTING *****
LOAD STEP= TIME=
1
1.0000
SUBSTEP=
1
LOAD CASE=
0
THE FOLLOWING DEGREE OF FREEDOM RESULTS ARE IN GLOBAL COORDINATES
NODE
UX
UY
UZ
USUM
1
0.0000
0.0000
0.0000
0.0000
2
0.0000
0.0000
0.0000
0.0000
3
0.0000
0.0000
4
0.0000
0.0000
0.99239E-01 0.99239E-01 0.0000
0.0000
MAXIMUM ABSOLUTE VALUES NODE VALUE
PRINT S
0
0
0.0000
0.0000
3
3
0.99239E-01 0.99239E-01
ELEMENT SOLUTION PER ELEMENT
9
***** POST1 ELEMENT NODAL STRESS LISTING *****
LOAD STEP= TIME=
1
1.0000
SUBSTEP=
1
LOAD CASE=
0
SHELL RESULTS FOR TOP/BOTTOM ALSO MID WHERE APPROPRIATE
THE FOLLOWING X,Y,Z VALUES ARE IN GLOBAL COORDINATES
ELEMENT= NODE
1 SX
SHELL63 SY
SZ
SXY
SYZ
SXZ
1
-368.51
65.586
0.0000
-1467.8
0.0000
0.0000
2
158.72
-764.88
0.0000
-731.76
0.0000
0.0000
3
2610.8
6043.4
0.0000
246.42
0.0000
0.0000
4
1165.7
173.58
0.0000
-489.64
0.0000
0.0000
1
368.51
-65.586
-10.000
1467.8
0.0000
0.0000
2
-158.72
764.88
-10.000
731.76
0.0000
0.0000
3
-2610.8
-6043.4
-10.000
-246.42
0.0000
0.0000
4
-1165.7
-173.58
-10.000
489.64
0.0000
0.0000
AnsysFiles\Chap06\Prb6-2Data.txt
!* Problem 6.2 !* 1 plate element model êPREP7 !* ET,1,SHELL63 !* KEYOPT,1,1,2
10
Plates and Shells
KEYOPT,1,2,0 KEYOPT,1,3,1 KEYOPT,1,5,0 KEYOPT,1,6,2 KEYOPT,1,7,0 KEYOPT,1,8,0 KEYOPT,1,9,0 KEYOPT,1,11,0 !* R,1,0.2, , , , , , RMORE, , , , RMORE RMORE, , !* !* MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,EX,1,,1000000 MPDATA,PRXY,1,,0.3 N,1,,,,,,, N,2,5,0,,,,, N,3,5,2.5,,,,, N,4,0,2.5,,,,, e,1,2,3,4 D,1, , , , , ,UZ,ROTX,ROTY, , , D,2, , , , , ,UZ,ROTY, , , , D,3, , , , , ,ROTX,ROTY, , , , D,4, , , , , ,UZ,ROTX,, , , SFE,1,1,PRES, ,10, , , êSOL
FINISH êSTATUS,SOLU SOLVE êPOST1
FINISH !* PRNSOL,U,Z
11
6.3 Determine deflection and stresses at the centroid of a triangular plate subjected to a uniformly distributed. The top side of the plate is simply supported and the left side is clamped. The third side is free. Use the following numerical data. 10 µ 5 plate h = 0.2 E = 106 n = 0.3 q = 10 Use only one finite element to model the plate using only one element.
y 5
0
3
2
1 x 0
10
Figure 6.24.
The element side 1-3 is clamped and 2-3 is simply supported. Thus the following essential boundary conditions are imposed. Side1 - 3 : w = 0 qx = 0 qy = 0 Side 2 - 3 : w = 0 qy = 0
Equations for element 1
i0y Element nodes: 1 Ø jj zz k0{ h = 0.2
i 10 zy i0y 2 Ø jj z 3 Ø jj zz k 5 { k5{
E = 1000000
n = 0.3
D = 732.601
q = 10
12
Plates and Shells
ij 219.78 jj jj 466.422 jj jj jj -112.332 jj jj jjj -21.978 jj jj 90.3541 jj jj jj -100.122 jj jj jj -197.802 jj jj jj 542.125 jj k -7.32601
466.422 -112.332
-21.978
90.3541
133.089
-262.515
695.971 2637.36
473.748
-976.801
133.089
109.89
-202.686
1691.09
695.971 473.748
-262.515 -976.801 -202.686
573.871 -1086.69
-100.122 -197.802 799.756 -599.512 2515.26
-361.416
-40.293
58.6081
140.415
36.63
3101.34
-466.422
-213.675
48.84
285.714
-501.832
-51.2821
566.545
112.332
-466.422
903.541 -280.83
-40.293
140.415
-213.675 -501.832
Node
dof
Value
1
1 2 3
0 0 0
2
1
0
2
3
0
3
1 2 3
0 0 0
Essential BC Ø
-415.14
112.332
-87.9121
36.63
-164.835
-87.9121
566.545
799.756 2515.26
58.6081
903.541 -280.83
-7.32601
-1086.69
-599.512 -361.416 -164.835 -415.14
542.125
48.84
-51.2821
1666.67 91.5751
91.5751 952.381
Global matrices after incorporating EBC K = H 573.871 L
RT = H -62.5 L
Solution of global equations Ø 8-0.10891<
Solution for element 1 1 1 Interpolation functions = : ÅÅÅÅÅÅÅÅÅÅÅÅÅ Hy - 5L Ix2 - 2 y x + 4 y2 - 10 y - 50M, ÅÅÅÅÅÅÅÅÅÅÅÅÅ Hy - 5L Ix2 - 2 y x + 4 Hy - 5L yM, 250 100 1 1 1 1 - ÅÅÅÅÅÅÅÅÅ x Hy - 5L2 , ÅÅÅÅÅÅÅÅÅÅÅÅÅ x I-x2 + H2 y + 5L x - 4 Hy - 5L yM, ÅÅÅÅÅÅÅÅÅ x Hy - 5L y, - ÅÅÅÅÅÅÅÅÅÅÅÅÅ x Ix2 - 2 y x + 4 Hy - 5L yM, 500 50 100 25 1 1 1 ÅÅÅÅÅÅÅÅÅÅÅÅÅ Ix3 + H5 - 4 yL x2 + 8 Hy - 5L y x + 4 H15 - 2 yL y2 M, ÅÅÅÅÅÅÅÅÅÅÅÅÅ Hx - 2 yL2 Hy - 5L, - ÅÅÅÅÅÅÅÅÅÅÅÅÅ Hx - 10L x Hx - 2 yL> 500 100 100
Nodal dof = H 0 0 0 0 -0.10891 0 0 0 0 L wHx,yL = 0.010891 x y - 0.00217819 x y2 0 jij zyz j zz j y = jj -0.00435638 x zz jj zz k 0.0217819 - 0.00871277 y {
13
732.601 219.78 0 y jij zz j j D C = jj 219.78 732.601 0 zzzz jj zz 0 256.41 { k 0
M 0.957447 x yz jij x zyz ijj zz jj M zz jj zz jj y zz = jj 3.19149 x zz zzz jj jjj z k Mxy { k 2.23404 y - 5.58511 {
ij Vx yz ij 3.19149 yz z=j j z 0 { k Vy { k
143.617 x ij sx yz ji zyz jj zz jj zz j z j s j z Maximum in-plane stresses, jj y zz = jj 478.723 x zz zz jj zz jj t 335.106 y 837.766 k { xy k {
i tzx yz ij 23.9362 yz z=j z Maximum transverse shear stresses, jj 0 { k tzy { k 10 10 Solution at x = ÅÅÅÅÅÅÅÅÅÅ and y = ÅÅÅÅÅÅÅÅÅ 3 3 w = 0.0403369
Moments = 83.19149, 10.6383, 1.8617<
Maximum in-plane stresses = 8478.723, 1595.74, 279.255<
Shears = 83.19149, 0<
Maximum transverse shear stresses = 823.9362, 0<
Solution summary Nodal solution qx
qy
x-coord
y-coord
w
1
0
0
0
0
0
2
10
5
0
-0.10891
0
3
0
5
0
0
0
Element solution Coord 1
(iii) Ansys solution
10 ÅÅÅÅ ÅÅ 3 10 ÅÅÅÅ ÅÅ 3
Disp
Max in-plane stresses
Max transverse shear stresses
0.0403369
478.723 1595.74 279.255
23.9362 0
14
Plates and Shells
PRINT ROT
NODAL SOLUTION PER NODE
***** POST1 NODAL DEGREE OF FREEDOM LISTING *****
LOAD STEP= TIME=
1
1.0000
SUBSTEP=
1
LOAD CASE=
0
THE FOLLOWING DEGREE OF FREEDOM RESULTS ARE IN GLOBAL COORDINATES
NODE 1
ROTX
ROTY
ROTZ
RSUM
0.0000
0.0000
0.0000
0.0000
2 -0.19599
0.0000
0.0000
0.19599
3
0.0000
0.0000
0.0000
0.0000
MAXIMUM ABSOLUTE VALUES NODE
0
0
2
-0.19599
0.0000
0.0000
0.19599
VALUE
2
PRINT S
ELEMENT SOLUTION PER ELEMENT
***** POST1 ELEMENT NODAL STRESS LISTING *****
LOAD STEP= TIME=
1
1.0000
SUBSTEP=
1
LOAD CASE=
0
SHELL RESULTS FOR TOP/BOTTOM ALSO MID WHERE APPROPRIATE
15
THE FOLLOWING X,Y,Z VALUES ARE IN GLOBAL COORDINATES
ELEMENT= NODE
1 SX
SHELL63 SY
SZ
SXY
SYZ
SXZ
1
-2584.4
-775.33
0.0000
301.52
0.0000
0.0000
2
775.33
2584.4
0.0000
2562.9
0.0000
0.0000
3
0.17053E-12-0.56843E-13
0.0000
753.79
0.0000
0.0000
3
0.17053E-12-0.56843E-13
0.0000
753.79
0.0000
0.0000
1
2584.4
775.33
-10.000
-301.52
0.0000
0.0000
2
-775.33
-2584.4
-10.000
-2562.9
0.0000
0.0000
3 -0.17053E-12 0.56843E-13 -10.000
-753.79
0.0000
0.0000
3 -0.17053E-12 0.56843E-13 -10.000
-753.79
0.0000
0.0000
AnsysFiles\Chap06\Prb6-3Data.txt
!* Problem 6.3 !* 1 plate element model êPREP7 !* ET,1,SHELL63 !* KEYOPT,1,1,2 KEYOPT,1,2,0 KEYOPT,1,3,1 KEYOPT,1,5,0 KEYOPT,1,6,2 KEYOPT,1,7,0 KEYOPT,1,8,0 KEYOPT,1,9,0 KEYOPT,1,11,0
16
Plates and Shells
!* R,1,0.2, , , , , , RMORE, , , , RMORE RMORE, , !* !* MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,EX,1,,1000000 MPDATA,PRXY,1,,0.3 N,1,,,,,,, N,2,10,5,,,,, N,3,0,5,,,,, e,1,2,3,3 D,1, , , , , ,UZ,ROTX,ROTY, , , D,2, , , , , ,UZ,ROTY, , , , D,3, , , , , ,UZ,ROTX,ROTY, , , , SFE,1,1,PRES, ,10, , , êSOL
FINISH êSTATUS,SOLU SOLVE êPOST1
FINISH !* PRNSOL,U,Z
6.4 Use four node quadrilateral Mindlin plate element to determine deflection and stresses at the center of a simply supported rectangular plate subjected to a uniformly distributed load. Use the following numerical data. 10 µ 5 plate h = 0.2 E = 106 n = 0.3 q = 10 Taking advantage of symmetry and model a 1/4 of the plate using only one element. Compare the solution with that obtained in Problem 6-2.
17
y 10
5
x
Figure 6.25.
The element sides 1-2 and 1-4 are simply supported, and the other two sides have symmetry boundary condition. Thus the following essential boundary conditions are imposed. Side1 - 4 : w = 0 qx = 0 Side1 - 2 : w = 0 qy = 0 Side 2 - 3 : qy = 0 Side 3 - 4 : qx = 0
2.5
0
y4
3
1
2
0
5
x
18
Plates and Shells
Global equations at start of the element assembly process D = 732.601
G = 384615.
1 jij j j C = jj 0.3 jj k0
zyz zz zz zz 0.35 {
0.3
0
1
0
0
Mapping to master element 1 1 1 1 NT = : ÅÅÅÅÅ Hs - 1L Ht - 1L, - ÅÅÅÅÅ Hs + 1L Ht - 1L, ÅÅÅÅÅ Hs + 1L Ht + 1L, - ÅÅÅÅÅ Hs - 1L Ht + 1L> 4 4 4 4 5s 5 xHs,tL = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 2 2 5t 5 yHs,tL = ÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4 5 ji ÅÅ2ÅÅ J = jjj j0 k
0 zy zz zz ÅÅÅÅ54Å {
25 detJ = ÅÅÅÅÅÅÅÅÅÅ 8
Interpolation functions 1 1 1 1 NT = : ÅÅÅÅÅ Hs - 1L Ht - 1L, - ÅÅÅÅÅ Hs + 1L Ht - 1L, ÅÅÅÅÅ Hs + 1L Ht + 1L, - ÅÅÅÅÅ Hs - 1L Ht + 1L> 4 4 4 4
t-1 1-t t+1 1 ∑NT ê∑s = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-t - 1L> 4 4 4 4
s -1 1 s +1 1-s ∑NT ê∑t = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-s - 1L, ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ > 4 4 4 4
Gauss quadrature points and weights for bending stiffness Point
Weight
1
s Ø -0.57735 t Ø -0.57735
1.
2
s Ø -0.57735 t Ø 0.57735
1.
3
s Ø 0.57735 t Ø -0.57735
1.
4
s Ø 0.57735 t Ø 0.57735
1.
Gauss quadrature points and weights for shear stiffness
1
Point
Weight
s Ø 0. t Ø 0.
4.
19
Computation of element matrices at 8-0.57735, -0.57735< with weight = 1. ij ÅÅÅÅ52Å J = jjjj k0
0 yz zz zz ÅÅÅÅ54Å {
25 detJ = ÅÅÅÅÅÅÅÅÅÅ 8
NT = H 0.622008 0.166667 0.0446582 0.166667 L
∑NT ê∑s = H -0.394338 0.394338 0.105662 -0.105662 L
∑NT ê∑t = H -0.394338 -0.105662 0.105662 0.394338 L
0.157735 0 0 -0.157735 0 0 -0.042265 0 ij 0 0 j B T = jjjj 0 -0.31547 0 0 -0.0845299 0 0 0.0845299 0 0 jj 0.0845299 0 0.042265 -0.0845299 0 k 0 -0.157735 0.31547 0 0.157735 0 jij jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 k = jjjj jj 0 jj jj jj 0 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
0
0
0 0
247.778
-74.0486 0 41.1139
0
0 0
0
23.4925
0 -66.3919 19.8413
0 0
0 0.31 -0.0 0
0 -222.5
30
-74.0486 136.705
0 30.7148
-35.593 0 19.8413
-36.63
0 23.4925
-
0
0
0 0
0
0 0
0
0 0
0
41.1139
30.7148
0 36.2945
19.8413
0 -11.0164 -8.23001 0 -66.3919 -
23.4925
-35.593
0 19.8413
62.6859
0 -6.2948
9.53711
0 -37.039
-
0
0
0 0
0
0 0
0
0 0
0
-66.3919 19.8413
0 -11.0164 -6.2948 0 17.7897
-5.31645 0 59.6187
-
19.8413
-36.63
0 -8.23001 9.53711
0 -5.31645 9.81499
0 -6.2948
17
0
0
0 0
0 0
0
0 0
0
-222.5
23.4925
0 -66.3919 -37.039 0 59.6187
-6.2948
0 229.273
19
30.7148
-64.4822 0 -42.3261 -36.63
0 -8.23001 17.2779
0 19.8413
83
ij 19.4378 yz zz jj zz jj 0 zzz jjj zz jj 0 zzz jjj zz jj jjj 5.20833 zzz zz jj zzz jjj 0 zz jj zzz jjj 0 zz j j r q = jj z jj 1.39557 zzz zz jj zz jj zz jj 0 zz jj zz jj 0 zz jj z jj jj 5.20833 zzz zz jj zz jj zz jj 0 zz jj z j { k0
0
Computation of element matrices at 8-0.57735, 0.57735< with weight = 1.
20
Plates and Shells
5 ji ÅÅÅÅ2Å J = jjj j0 k
0 zy zz zz ÅÅÅÅ54Å {
25 detJ = ÅÅÅÅÅÅÅÅÅÅ 8
NT = H 0.166667 0.0446582 0.166667 0.622008 L
∑NT ê∑s = H -0.105662 0.105662 0.394338 -0.394338 L
∑NT ê∑t = H -0.394338 -0.105662 0.105662 0.394338 L
0 0 0.042265 0 0 -0.042265 0 0 -0.157735 0 jij j j 0 -0.0845299 0 0 0.0845299 0 0 B = jj 0 -0.31547 0 jj 0.0845299 0 0.157735 -0.0845299 0 k 0 -0.042265 0.31547 0 0.042265 T
ij 0 jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj 0 j k = jjjj jj 0 jj jj jj 0 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
0
0
229.273
-19.8413 0 59.6187
-19.8413 83.8343
0 0 0 8.23001
0
0 0
0
0 0.31 -0.1
0 0
0
0 -222.5
-
6.2948
0 -66.3919 37.039
17.2779
0 42.3261
-36.63
0 -30.7148 -
0
0 0
0
0
0 0
0
0 0
59.6187
8.23001
0 17.7897
5.31645
0 -11.0164 6.2948
0 -66.3919 -
6.2948
17.2779
0 5.31645
9.81499
0 8.23001
9.53711
0 -19.8413 -
0
0
0 0
0
0 0
0
0 0
-19.8413 0 41.1139
-66.3919 42.3261
0
0
0 -11.0164 8.23001
0 36.2945
-36.63
0 6.2948
9.53711
0 -19.8413 62.6859
0 -23.4925 -
0
0
0 0
0
0 0
0 0
-222.5
-30.7148 0 -66.3919 -19.8413 0 41.1139
37.039
-23.4925 -64.4822 0 -19.8413 -36.63
ij 5.20833 yz zz jj zz jj 0 zz jj zz jj zz jj 0 zz jjj jj 1.39557 zzz zzz jjj zz jj zzz jjj 0 zz jj zzz jjj 0 zz r q = jjj jj 5.20833 zzz zz jj zzz jjj 0 zz jj zzz jjj 0 zz jj zz jjj jj 19.4378 zzz zz jj zz jj zz jj 0 zz jj { k0
0
-23.4925 0 247.778
0 -30.7148 -35.593
Computation of element matrices at 80.57735, -0.57735< with weight = 1.
0 74.0486
0 7 1
21
5 ji ÅÅÅÅ2Å J = jjj j0 k
0 zy zz zz ÅÅÅÅ54Å {
25 detJ = ÅÅÅÅÅÅÅÅÅÅ 8
NT = H 0.166667 0.622008 0.166667 0.0446582 L
∑NT ê∑s = H -0.394338 0.394338 0.105662 -0.105662 L
∑NT ê∑t = H -0.105662 -0.394338 0.394338 0.105662 L
0 0 0.157735 0 0 -0.157735 0 0 -0.042265 0 0 jij j j 0 -0.31547 0 0 0.31547 0 0 0.0845 B = jj 0 -0.0845299 0 jj 0 0.042265 -0.31547 0 -0.042 k 0 -0.157735 0.0845299 0 0.157735 0.31547 T
ij 0 jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj 0 j k = jjjj jj 0 jj jj jj 0 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
0
0
0 0
36.2945
-19.8413 0 41.1139
0
0 0
0
-30.7148 0 -66.3919 42.3261
0 0
0
0 -11.0164 8
-19.8413 62.6859
0 -23.4925 -35.593
0 37.039
-36.63
0 6.2948
9
0
0
0 0
0
0 0
0
0 0
0
41.1139
-23.4925 0 247.778
74.0486
0 -222.5
-30.7148 0 -66.3919 -
-30.7148 -35.593
0 74.0486
136.705
0 -23.4925 -64.4822 0 -19.8413 -
0
0 0
0
0 0
0 -222.5
-23.4925 0 229.273
0
-66.3919 37.039 42.3261
-36.63
0
0
0
0 0
-19.8413 0 59.6187
0 6
0 -30.7148 -64.4822 0 -19.8413 83.8343
0 8.23001
1
0 0
0
0 0
0
-11.0164 6.2948
0 -66.3919 -19.8413 0 59.6187
8.23001
0 17.7897
5
8.23001
0 -19.8413 -36.63
17.2779
0 5.31645
9
ij 5.20833 yz zz jj zz jj 0 zz jj zz jj zz jj 0 zz jjj jj 19.4378 zzz zzz jjj zz jj zzz jjj 0 zz jj zzz jjj 0 zz r q = jjj jj 5.20833 zzz zz jj zzz jjj 0 zz jj zzz jjj 0 zz jj zz jjj jj 1.39557 zzz zz jj zz jj zz jj 0 zz jj { k0
9.53711
0
0 0 0 6.2948
Computation of element matrices at 80.57735, 0.57735< with weight = 1.
22
Plates and Shells
5 ji ÅÅÅÅ2Å J = jjj j0 k
0 zy zz zz ÅÅÅÅ54Å {
25 detJ = ÅÅÅÅÅÅÅÅÅÅ 8
NT = H 0.0446582 0.166667 0.622008 0.166667 L
∑NT ê∑s = H -0.105662 0.105662 0.394338 -0.394338 L
∑NT ê∑t = H -0.105662 -0.394338 0.394338 0.105662 L
0 0 0.042265 0 0 -0.042265 0 0 -0.157735 0 0 jij j j 0 -0.31547 0 0 0.31547 0 0 0.0845 B = jj 0 -0.0845299 0 jj 0 0.157735 -0.31547 0 -0.157 k 0 -0.042265 0.0845299 0 0.042265 0.31547 T
ij 0 jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj 0 j k = jjjj jj 0 jj jj jj 0 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
0
0
0 0
17.7897
-5.31645 0 59.6187
0
0 0
0
-8.23001 0 -66.3919 19.8413
0 0
0
0 -11.0164 -
-5.31645 9.81499
0 -6.2948
17.2779
0 19.8413
-36.63
0 -8.23001 9
0
0
0 0
0
0 0
0
0 0
59.6187
-6.2948
0 229.273
19.8413
0 -222.5
23.4925
0 -66.3919 -
-8.23001 17.2779
0 19.8413
83.8343
0 30.7148
-64.4822 0 -42.3261 -
0
0 0
0
0 0
0
0 247.778
-74.0486 0 41.1139
0
-66.3919 19.8413
0 0
0
0
0 -222.5
30.7148
19.8413
-36.63
0 23.4925
-64.4822 0 -74.0486 136.705
0 30.7148
-
0
0
0 0
0
0
0 0
0
-11.0164 -8.23001 0 -66.3919 -42.3261 0 41.1139
30.7148
0 36.2945
1
-6.2948
-35.593
0 19.8413
6
ij 1.39557 yz zz jj zz jj 0 zz jj zz jj zz jj 0 zz jjj jj 5.20833 zzz zzz jjj zz jj zzz jjj 0 zz jj zzz jjj 0 zz r q = jjj jj 19.4378 zzz zz jj zzz jjj 0 zz jj zzz jjj 0 zz jj zz jjj jj 5.20833 zzz zz jj zz jj zz jj 0 zz jj { k0
9.53711
0 -37.039
-36.63
Computation of element matrices at 80., 0.< with weight = 4.
0 0
2
0 23.4925
23
5 ji ÅÅÅÅ2Å J = jjj j0 k
0 zy zz zz ÅÅÅÅ54Å {
25 detJ = ÅÅÅÅÅÅÅÅÅÅ 8
NT = H 0.25 0.25 0.25 0.25 L ∑NT ê∑s = H -0.25
0.25 0.25 -0.25 L
∑NT ê∑t = H -0.25
-0.25
i -0.1 0 B T = jj k -0.2 -0.25
ij 40064.1 jj jj 40064.1 jj jj jjj -20032.1 jj jj 24038.5 jj jj jj 40064.1 jj jj jj -20032.1 k = jjjj jj -40064.1 jj jj jj 40064.1 jj jj -20032.1 jj jj jj -24038.5 jj jj jj 40064.1 jj j k -20032.1
0.25 0.25 L
0.25 0.1
0
-0.2 -0.25
0
0.25 0.1
0
0.25 -0.1 0
0
-0.25
0
0.2
0.2
-0.25
0.25 y zz 0 {
40064.1
-20032.1 24038.5
40064.1
-20032.1 -40064.1 40064.1
50080.1
0
40064.1
50080.1
0
-40064.1 50080.1
0
0
50080.1
20032.1
0
50080.1
20032.1
500
40064.1
20032.1
40064.1
40064.1
20032.1
-24038.5 40064.1
200
50080.1
0
40064.1
50080.1
0
-40064.1 50080.1
0
0
50080.1
20032.1
0
50080.1
20032.1
0
500
-40064.1 20032.1
-24038.5 -40064.1 20032.1
40064.1
-40064.1 200
50080.1
0
40064.1
50080.1
0
-40064.1 50080.1
0
0
50080.1
20032.1
0
50080.1
20032.1
500
-40064.1 -20032.1 -40064.1 -40064.1 -20032.1 24038.5
0
0
-20
-40064.1 -20
50080.1
0
40064.1
50080.1
0
-40064.1 50080.1
0
0
50080.1
20032.1
0
50080.1
20032.1
500
0
After summing contributions from all points, the element equations as follows: ij 40064.1 jj jj 40064.1 jj jj jj -20032.1 jj jj jj 24038.5 jj jj jj 40064.1 jj jj -20032.1 j k = jjjj jj -40064.1 jj jj jj 40064.1 jj jj -20032.1 jj jj jj -24038.5 jj jj jj 40064.1 jj j k -20032.1
r = H 31.25 T
40064.1
-20032.1 24038.5
40064.1
-20032.1 -40064.1 40064.1
-20032.1
50611.3
-119.048 40064.1
50281.6
-9.15751 -40064.1 49814.6
119.048
-119.048 50373.2
20032.1
9.15751
50043.5
20032.1
119.048
49933.6
40064.1
20032.1
40064.1
40064.1
20032.1
-24038.5 40064.1
20032.1
50281.6
9.15751
40064.1
50611.3
119.048
-40064.1 49613.1
-9.15751
-9.15751 50043.5
20032.1
119.048
50373.2
20032.1
9.15751
49970.2
-40064.1 20032.1
-24038.5 -40064.1 20032.1
40064.1
-40064.1 20032.1
49814.6
119.048
40064.1
49613.1
119.048
49933.6
20032.1
-9.15751 49970.2
9.15751
-40064.1 50611.3 20032.1
-40064.1 -20032.1 -40064.1 -40064.1 -20032.1 24038.5 49613.1
-9.15751 40064.1
49814.6
9.15751
49970.2
-119.048 49933.6
0 0 31.25
0
20032.1
0 31.25
0 0 31.25
Complete element equations for element 1
-40064.1 -20032.1
-119.048 -40064.1 50281.6
0 0L
20032.1
-119.048
-119.048 50373.2 9.15751
-9.15751 50043.5
24
Plates and Shells
ij 40064.1 jj jj 40064.1 jj jj jjj -20032.1 jj jj 24038.5 jj jj jj 40064.1 jj jj jj -20032.1 jj jj jj -40064.1 jj jj jj 40064.1 jj jj -20032.1 jj jj jj -24038.5 jj jj jj 40064.1 jj j k -20032.1
40064.1
-20032.1 24038.5
50611.3
-119.048 40064.1
40064.1
-20032.1 -40064.1 40064.1
-20032.1 -24
50281.6
-9.15751 -40064.1 49814.6
119.048
-40
-119.048 50373.2
20032.1
9.15751
50043.5
20032.1
119.048
49933.6
-20
40064.1
20032.1
40064.1
40064.1
20032.1
-24038.5 40064.1
20032.1
-40
50281.6
9.15751
40064.1
50611.3
119.048
-40064.1 49613.1
-9.15751 -40
-9.15751 50043.5
20032.1
119.048
50373.2
20032.1
9.15751
49970.2
-40064.1 20032.1
-24038.5 -40064.1 20032.1
40064.1
-40064.1 20032.1
49814.6
119.048
40064.1
49613.1
-40064.1 50611.3
119.048
49933.6
20032.1
-9.15751 49970.2
9.15751
20032.1
-40064.1 -20032.1 -40064.1 -40064.1 -20032.1 24038.5 49613.1
-9.15751 40064.1
49814.6
9.15751
49970.2
-119.048 49933.6
20032.1
2403
-119.048 -40
-119.048 50373.2
-20
-40064.1 -20032.1 4006
-119.048 -40064.1 50281.6 20032.1
-20
9.15751
-40
-9.15751 50043.5
-20
The element contributes to 81, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12< global degrees of freedom. Adding element equations into appropriate locations we have ij 40064.1 jj jj 40064.1 jj jj jj -20032.1 jj jj jj 24038.5 jj jj 40064.1 jjj jj jjj -20032.1 jj jj -40064.1 jj jj jj 40064.1 jj jj jj -20032.1 jj jj jj -24038.5 jj jj jj 40064.1 jj k -20032.1
40064.1
-20032.1 24038.5
40064.1
-20032.1 -40064.1 40064.1
-20032.1 -24
50611.3
-119.048 40064.1
50281.6
-9.15751 -40064.1 49814.6
119.048
-40
-119.048 50373.2
20032.1
9.15751
50043.5
20032.1
119.048
49933.6
-20
40064.1
20032.1
40064.1
40064.1
20032.1
-24038.5 40064.1
20032.1
-40
50281.6
9.15751
40064.1
50611.3
119.048
-40064.1 49613.1
-9.15751 -40
-9.15751 50043.5
20032.1
119.048
50373.2
20032.1
9.15751
49970.2
-40064.1 20032.1
-24038.5 -40064.1 20032.1
40064.1
-40064.1 20032.1
49814.6
119.048
40064.1
49613.1
-40064.1 50611.3
119.048
49933.6
20032.1
-9.15751 49970.2
9.15751
20032.1
-40064.1 -20032.1 -40064.1 -40064.1 -20032.1 24038.5 49613.1
-9.15751 40064.1
49814.6
9.15751
49970.2
-119.048 49933.6
Essential boundary conditions
20032.1
2403
-119.048 -40
-119.048 50373.2
-20
-40064.1 -20032.1 4006
-119.048 -40064.1 50281.6 20032.1
-20
9.15751
-40
-9.15751 50043.5
-20
25
Node 1 2 2 3 3 4 4
dof
Value
w1 Hqx L1 Hqy L1
0 0 0
Hqy L2 w2
0
Hqx L3
0
Hqy L3
0
Hqx L4
0
0
w4
0
Remove 81, 2, 3, 4, 6, 8, 9, 10, 11< rows and columns. After adjusting for essential boundary conditions we have -40064.1 -119.048 y ij Hqx L2 ij 50611.3 zz jj jj jj -40064.1 40064.1 20032.1 zzzz jjjj w3 jj zz jj j 50373.2 { k Hqy L4 k -119.048 20032.1
yz ij 0 yz zz jj zz = jj 31.25 zzzz zz zzz jjj z z { { k0
Solving the final system of global equations we get
8Hqx L2 = 0.0539835, w3 = 0.0682754, Hqy L4 = -0.0270237<
Complete table of nodal values w
qx
qy
1
0
0
0
2
0
0.0539835
0
3
0.0682754
0
0
4
0
0
-0.0270237
Computation of reactions
Equation numbers of dof with specified values: 81, 2, 3, 4, 6, 8, 9, 10, 11<
Extracting equations 81, 2, 3, 4, 6, 8, 9, 10, 11< from the global system we have
26
Plates and Shells
40064.1 jij jj 40064.1 jj jj jj -20032.1 jj jj jj 24038.5 jj jj jj -20032.1 jj jj jj 40064.1 jj jj jj -20032.1 jj jj jj -24038.5 jj k 40064.1
40064.1
-20032.1 24038.5
50611.3
-119.048 40064.1
-20032.1 -40064.1 40064.1
40064.1
-20032.1 -24
50281.6
-9.15751 -40064.1 49814.6
119.048
-40
-119.048 50373.2
20032.1
9.15751
50043.5
49933.6
-20
40064.1
20032.1
119.048
20032.1
40064.1
40064.1
20032.1
-24038.5 40064.1
20032.1
-40
-9.15751 50043.5
20032.1
119.048
50373.2
20032.1
49970.2
-20
49814.6
119.048
40064.1
49613.1
9.15751
119.048
49933.6
20032.1
-9.15751 49970.2
-40064.1 50611.3 20032.1
-40064.1 -20032.1 -40064.1 -40064.1 -20032.1 24038.5 -9.15751 40064.1
49613.1
9.15751
-20
-40064.1 -20032.1 4006
-119.048 -40064.1 50281.6
49814.6
-119.048 -40
-119.048 50373.2 9.15751
-40
Substituting the nodal values and re-arranging R 40064.1 jij 1 zyz jij jj R zz jj 40064.1 jj 2 zz jj jj z j jj R zzz jjj -20032.1 jjj 3 zzz jjj jj zz jj jjj R4 zzz jjj 24038.5 jj zz jj jjj R5 zzz = jjj -20032.1 jj zz jj jjj R6 zzz jjj 40064.1 jj zz jj jjj R zzz jjj -20032.1 jj 7 zz jj jjj zz jj jj R8 zzz jjj -24038.5 jj zz jj k R9 { k 40064.1
40064.1
-20032.1 24038.5
50611.3
-119.048 40064.1
40064.1 50281.6
-9.15751 -40064.1 49814.6
119.048
20032.1
9.15751
50043.5
119.048
49933.6
40064.1
20032.1
40064.1
40064.1
20032.1
-24038.5 40064.1
20032.1
20032.1
119.048
50373.2
20032.1
9.15751
49970.2
49814.6
119.048
40064.1
49613.1
9.15751
-40064.1 50611.3
-119.04
119.048
49933.6
20032.1
-9.15751 49970.2
20032.1
-119.048 50373.2
-40064.1 -20032.1 -40064.1 -40064.1 -20032.1 24038.5
-40064.1 -20032
49613.1
-9.15751 40064.1
dof
Reaction
R1
w1
Hqx L1
-62.5
R2
Hqy L1
-21.2613
R3 R4
w2
Hqy L2
-51.0235
Hqx L3
24.7318
Hqy L3
-56.8543
R7 R8
w4
Hqx L4
-11.4765
R9
20032.1
-9.15751 50043.5
Label
R6
-20032
-119.048 50373.2
49814.6
Carrying out computations, the reactions are as follows.
R5
-20032.1 -40064.1 40064.1
17.8096
14.8412 -49.4431
-119.048 -40064.1 50281.6
9.15751
27
Sum of Reactions dof: w
-125.
dof: qx
-127.559
dof: qy
57.3826
Solution for element 1
Nodal solution = H 0 0 0 0
0.0539835 0 0.0682754 0 0 0
5 Solution at 8s, t< = 8-1, 1< ï 8x, y< = :0, ÅÅÅÅÅ > 2 Interpolation functions = 80, 0, 0, 1< w = 0.
bx = 0.0270237
by = 0.
Moments = 83.95952, 1.18785, -2.77166<
Maximum in-plane stresses = 8593.927, 178.178, -415.749<
Shears = 8-856.963, 0.<
Maximum transverse shear stresses = 8-6427.22, 0.<
Solution at 8s, t< = 8-1, -1< ï 8x, y< = 80, 0<
Interpolation functions = 81, 0, 0, 0< w = 0.
bx = 0.
Moments = 80., 0., -5.54005<
by = 0.
Maximum in-plane stresses = 80., 0., -831.007<
Shears = 80., 0.<
Maximum transverse shear stresses = 80., 0.<
5 Solution at 8s, t< = 8-1, 1< ï 8x, y< = :0, ÅÅÅÅÅ > 2 Interpolation functions = 80, 0, 0, 1< w = 0.
bx = 0.0270237
by = 0.
Moments = 83.95952, 1.18785, -2.77166<
Maximum in-plane stresses = 8593.927, 178.178, -415.749<
Shears = 8-856.963, 0.<
Maximum transverse shear stresses = 8-6427.22, 0.<
0 -0.0270237 L
28
Plates and Shells
Solution at 8s, t< = 81, -1< ï 8x, y< = 85, 0<
Interpolation functions = 80, 1, 0, 0< w = 0.
bx = 0.
by = 0.0539835
Moments = 84.7458, 15.8193, -2.76839<
Maximum in-plane stresses = 8711.871, 2372.9, -415.258<
Shears = 80., -1709.83<
Maximum transverse shear stresses = 80., -12823.7<
5 Solution at 8s, t< = 81, 1< ï 8x, y< = :5, ÅÅÅÅÅ > 2
Interpolation functions = 80, 0, 1, 0< w = 0.0682754
bx = 0.
by = 0.
Moments = 88.70532, 17.0072, 0.<
Maximum in-plane stresses = 81305.8, 2551.08, 0.<
Shears = 8875.325, 1750.65<
Maximum transverse shear stresses = 86564.94, 13129.9<
Solution summary Nodal solution x-coord
y-coord
w
qx
qy
1
0
0
0
0
0
2
5
0
0
0.0539835
0
5
ÅÅ52ÅÅ
0.0682754
0
0
0
ÅÅ52ÅÅ
0
0
-0.0270237
3 4
Element solution
1
Coord
Disp
Max in-plane stresses
0 ÅÅ52ÅÅ
0. 0.0270237 0.
593.927 178.178 -415.749
Support reactions
Max transverse shear stresses -6427.22 0.
M 3.959 1.187 -2.7
29
Node
dof
Reaction
1
1
-62.5
1
2
-21.2613
1
3
17.8096
2
1
-51.0235
2
3
24.7318
3
2
-56.8543
3
3
14.8412
4
1
-11.4765
4
2
-49.4431
Sum of applied loads Ø H 125. 0 0 L
Sum of support reactions Ø H -125. -127.559 57.3826 L
(iii) Ansys solution
PRINT U
NODAL SOLUTION PER NODE
***** POST1 NODAL DEGREE OF FREEDOM LISTING *****
LOAD STEP= TIME=
1
1.0000
SUBSTEP=
1
LOAD CASE=
0
THE FOLLOWING DEGREE OF FREEDOM RESULTS ARE IN GLOBAL COORDINATES
NODE
UX
UY
UZ
USUM
1
0.0000
0.0000
0.0000
0.0000
2
0.0000
0.0000
0.0000
0.0000
3
0.0000
0.0000
4
0.0000
0.0000
0.68593E-01 0.68593E-01 0.0000
0.0000
30
Plates and Shells
MAXIMUM ABSOLUTE VALUES NODE VALUE
0
0
0.0000
0.0000
PRINT ROT
3
3
0.68593E-01 0.68593E-01
NODAL SOLUTION PER NODE
***** POST1 NODAL DEGREE OF FREEDOM LISTING *****
LOAD STEP= TIME=
1
SUBSTEP=
1.0000
1
LOAD CASE=
0
THE FOLLOWING DEGREE OF FREEDOM RESULTS ARE IN GLOBAL COORDINATES
NODE 1 2
ROTX
ROTY
0.0000 0.53979E-01
3
0.0000
4
0.0000
ROTZ
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
-0.27034E-01
0.0000
RSUM 0.0000 0.53979E-01 0.0000 0.27034E-01
MAXIMUM ABSOLUTE VALUES NODE VALUE
PRINT S
2
4
0.53979E-01-0.27034E-01
0 0.0000
2 0.53979E-01
ELEMENT SOLUTION PER ELEMENT
***** POST1 ELEMENT NODAL STRESS LISTING *****
31
LOAD STEP= TIME=
1
1.0000
SUBSTEP=
1
LOAD CASE=
0
SHELL RESULTS FOR TOP/BOTTOM ALSO MID WHERE APPROPRIATE
THE FOLLOWING X,Y,Z VALUES ARE IN GLOBAL COORDINATES
ELEMENT= NODE 1
1 SX
SHELL143 SY
0.28990E-10 0.56616E-10
SZ
SXY
0.0000
-831.13 -415.22
2
711.81
2372.7
0.0000
3
1306.0
2550.9
0.0000
4
594.16
178.25
0.0000
-415.91
1 -0.28990E-10-0.56616E-10 -10.000
831.13
2
-711.81
-2372.7
-10.000
415.22
3
-1306.0
-2550.9
-10.000
4
-594.16
-178.25
-10.000
-0.18360E-10
0.18360E-10 415.91
SYZ
SXZ
0.71054E-13 0.88818E-13 76.532 76.532 0.15632E-12
0.88818E-13 34.436 34.436
0.19895E-12 0.88818E-13 76.532 76.532 0.11369E-12
0.88818E-13 34.436 34.436
AnsysFiles\Chap06\Prb6-4Data.txt
6.5 Prepare a short report on different plate and shell elements available in Ansys or any other commercial finite element software available. Look through the documentation and determine the theory on which each element is based.
32
Plates and Shells
6.6 Computational project The so-called flat-slab system consists of a reinforced concrete slab supported directly by columns. Drop panels (areas with increased slab thickness) are provided over the columns to avoid punching shear failure. For the typical symmetric system shown in Figure 6.25, the columns are spaced with L = 4 m and w = 5 m in the y direction. The slab thickness is t = 0.13 m. The columns are 0.4 m µ 0.4 m square. The drop panels are 0.1 m thick. In each direction the drop panels extend to a distance of spacing ê 6. The factored design load on the slab (including its own weight) is 15.85 kN ê m 2 . The goal of the analysis is to find stresses and deflections in the slab. For concrete assume E = 22.5 GPa and n = 0.17.
Slab w Drop panel
w
Center Lines L
L
Figure 6.25 Plan view and section of a flat slab supported by columns with drop panels
Column
33
(a) Approximate analysis using an equivalent frame A common design office technique to analyze flat slab systems is to use the equivalent frame method. The equivalent frame is defined by considering half the slab width from each side of a column line as a beam supported by columns of the floor below and above as shown in Figure 6.26. Use the plane frame element to analyse this equivalent frame. Assuming story height to be 3m
h
C.L.
Beam with width = L and Thickness = t h
w
w
wê2
Figure 6.26 Equivalent frame (b) Finite element analysis using plate element The equivalent frame method obviously gives only an estimate of the deflections and stresses in the slab. For a more complete picture of stresses create a finite element model using plate elements. The slab at the column locations can be considered fixed. Account for the increased thickness in the drop panel zones by changing the element thickness in these regions. Compare finite element results with those from the equivalent frame and comment of the suitability of the approximate method.
Frame model
AnsysFiles\Chap06\Prb6-6FlatSlab\frameModel.tif
34
Plates and Shells
Input data file
AnsysFiles\Chap06\Prb6-6FlatSlab\frameModel.txt
êTITLE,Frame
!*
êPREP7
CMSEL,S,_Y1
!*
CMSEL,S,_Y
ET,1,BEAM3
CMDELE,_Y
*set,L,4
CMDELE,_Y1
*set,w,5
LESIZE,ALL,wê5, , , ,1, , ,1,
*set,h,3
FLST,2,9,4,ORDE,2
*set,t,0.13
FITEM,2,1
*set,d,0.1
FITEM,2,-9
*set,c,0.4
LMESH,P51X
*set, ta, Ht*L*w + d*HLê3L*Hwê3LLêHL*wL
LATT,1,2,1, , , ,
EPLOT
35
!* R,1,L*ta,HL*ta**3Lê12,ta,0,0,0,
FLST,2,13,2,ORDE,6
R,1,L*t,HL*t**3Lê12,t,0,0,0,
FITEM,2,7
R,2,c*c,Hc**4Lê12, c, , , ,
FITEM,2,-11
!*
FITEM,2,18
!*
FITEM,2,-22
MPTEMP,,,,,,,,
FITEM,2,29
MPTEMP,1,0
FITEM,2,-31
MPDATA,EX,1,,22.5E6
SFBEAM,P51X,1,PRES,15.850*4, , , , , ,
MPDATA,PRXY,1,,0.17
!*
K,1,0,0,,
FLST,2,6,3,ORDE,6
K,2,0,h,,
FITEM,2,1
K,3,0,2*h,,
FITEM,2,3
K,4,w,0,,
FITEM,2,-4
K,5,w,h,,
FITEM,2,6
K,6,w,2*h,,
FITEM,2,-7
K,7,2*w,0,,
FITEM,2,9
K,8,2*w,h,,
!*
êGO
K,9,2*w, 2*h,, K,10,2.5*w,h,,
DK,P51X, , , ,0,ALL, , , , , ,
LSTR,
3,
2
FLST,2,1,3,ORDE,1
LSTR,
2,
1
FITEM,2,10
LSTR,
2,
5
!*
LSTR,
5,
4
êGO
LSTR,
5,
6
DK,P51X, , , ,0,UX,ROTZ, , , , ,
LSTR,
5,
8
FINISH
LSTR,
8,
9
LSTR,
8,
7
êSTATUS,SOLU
LSTR,
8,
10
SOLVE
êSOL
FITEM,5,1
êPOST1
FITEM,5,-2
PLDISP,1
FITEM,5,4
ETABLE,smaxi,NMISC, 1
FITEM,5,-5
ETABLE,smini,NMISC, 2
FITEM,5,7
ETABLE,smaxj,NMISC, 3
FITEM,5,-8
ETABLE,sminj,NMISC, 4
CM,_Y,LINE
ETABLE,mzi,SMISC, 6
LSEL, , , ,P51X
ETABLE,mzj,SMISC, 12
CM,_Y1,LINE
SABS,1
CMSEL,S,_Y
SMAX,maxij,SMAXI,SMAXJ,1,1,
FLST,5,6,4,ORDE,6
FINISH
36
Plates and Shells
!*
PLETAB,MAXIJ,NOAV
!*
!* !*
Deformed shape
AnsysFiles\Chap06\Prb6-6FlatSlab\defFrame.tif
Stresses
AnsysFiles\Chap06\Prb6-6FlatSlab\stressFrame.tif
37
Compute the total volume of one panel and divide it by the panel area to get the average slab thickness. L = 4; w = 5; t = .13; d = .1; ta = Ht * L * w + d * HL ê 3L * Hw ê 3LL ê HL * wL 0.141111
Deformed shape
AnsysFiles\Chap06\Prb6-6FlatSlab\defFrame2.tif
38
Plates and Shells
Stresses
AnsysFiles\Chap06\Prb6-6FlatSlab\stressFrame2.tif
39
Plate model
AnsysFiles\Chap06\Prb6-6FlatSlab\plateModel.tif
40
Plates and Shells
Input data file
AnsysFiles\Chap06\Prb6-6FlatSlab\Shell63Model.txt êFILNAME,Plates,0 êPREP7 !* ET,1,SHELL63 !* *set,L,4 *set,w,5 *set,h,3 *set,t,0.13 *set,d,0.1 *set,c,0.4 R,1,t+d, , , , , , R,2,t, , , , , ,
CMSEL,S,_Y CMDELE,_Y CMDELE,_Y1 !* FLST,2,9,5,ORDE,3 FITEM,2,1 FITEM,2,-8 FITEM,2,20 ESIZE,wê10,0, AESIZE,P51X,wê20, MSHKEY,0 FLST,5,10,5,ORDE,4 FITEM,5,1
41
R,2,t, , , , , ,
FITEM,5,-8
MPTEMP,,,,,,,,
FITEM,5,11
MPTEMP,1,0 MPDATA,EX,1,,22.5E6 MPDATA,PRXY,1,,0.17 RECTNG,0,2.5*L,0,2.5*w, BLC5,L,wê2,Lê3,wê3 BLC5,2*L,wê2,Lê3,wê3 BLC5,L,1.5*w,Lê3,wê3 BLC5,2*L,1.5*w,Lê3,wê3 RECTNG,0,Lê6,.5*w-wê6,.5*w+wê6 RECTNG,0,Lê6,1.5*w-wê6,1.5*w+wê6 RECTNG,0,Lê6,2.5*w-wê6,2.5*w RECTNG,L-Lê6,L+Lê6,2.5*w-wê6,2.5*w RECTNG,2*L-Lê6,2*L+Lê6,2.5*w-wê6,2.5*w FLST,3,9,5,ORDE,2 FITEM,3,2 1,P51X,,delete,keep
RECTNG,0,c,wê2-cê2,wê2+cê2 RECTNG,0,c,1.5*w-cê2,1.5*w+cê2 RECTNG,0,c,2.5*w-c,2.5*w RECTNG,L-cê2,L+cê2,wê2-cê2,wê2+cê2 RECTNG,L-cê2,L+cê2,1.5*w-cê2,1.5*w+cê2 RECTNG,L-cê2,L+cê2,2.5*w-c,2.5*w RECTNG,2*L-cê2,2*L+cê2,wê2-cê2,wê2+cê2 RECTNG,2*L-cê2,2*L+cê2,1.5*w-cê2,1.5*w+cê2 RECTNG,2*L-cê2,2*L+cê2,2.5*w-c,2.5*w ASBA,
6,
1
ASBA,
7,
12
ASBA,
8,
13
ASBA,
9,
16
ASBA,
3,
17
ASBA,
5,
18
ASBA,
2,
14
ASBA,
4,
15
ASBA,
10,
19
FLST,2,10,5,ORDE,4 FITEM,2,1
CM,_Y,AREA ASEL, , , ,P51X CM,_Y1,AREA CHKMSH,'AREA' CMSEL,S,_Y !* AMESH,_Y1 !* CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !* EPLOT êSOL
FINISH
FITEM,3,-10 ASBA,
FITEM,5,20
FLST,2,30,4,ORDE,12 FITEM,2,3 FITEM,2,-4 FITEM,2,47 FITEM,2,49 FITEM,2,-51 FITEM,2,53 FITEM,2,-54 FITEM,2,57 FITEM,2,-66 FITEM,2,68 FITEM,2,-78 FITEM,2,80 !*
êGO DL,P51X, ,ALL, FLST,2,2,4,ORDE,2 FITEM,2,1 FITEM,2,-2 DL,P51X, ,SYMM FLST,2,10,5,ORDE,4
42
Plates and Shells
FLST,2,10,5,ORDE,4
FITEM,2,-8
FITEM,2,1
FITEM,2,11
FITEM,2,-8
FITEM,2,20
FITEM,2,11
AGLUE,P51X
êGO
FITEM,2,20
CM,_Y,AREA ASEL, , , ,
11
!*
CM,_Y1,AREA
êSTATUS,SOLU
SFA,P51X,1,PRES,15.85
CMSEL,S,_Y !*
SOLVE
CMSEL,S,_Y1 AATT,
1,
2, 1,
0,
CMSEL,S,_Y
Deformed shape
AnsysFiles\Chap06\Prb6-6FlatSlab\defPlate.tif
êPOST1
FINISH PLDISP,1
43
von Mises Stresses
AnsysFiles\Chap06\Prb6-6FlatSlab\vonMisesPlate.tif
Comparison Frame model: use only the slab thickness for computing beam section properties Maximum displacement: 0.00539 m;
Maximum stress = 12,204 kN ê m2
Frame model: account for drop panel by assigning an average slab thickness Maximum displacement: 0.005284 m; Finite element analysis using plate element Maximum displacement: 0.004 m;
Maximum stress = 10,447 kN ê m2 Maximum stress = 9,546 kN ê m 2
CHAPTER SEVEN
Introduction to Nonlinear Problems
7.1 Consider the following nonlinear differential equation 2
d u - ÅÅÅÅ ÅÅÅÅÅÅ + u3 = 0; 0 < x < 2 dx2
with the boundary conditions
uH0L = 1 ê 27; u£ H2L = 5 ê 3
Obtain a suitable weak form. Starting with a quadratic polynomial uHxL = a0 + a1 x + a2 x2 obtain an approximate solution of the problem.
Weak form
2
≥ 3 Ÿ0 H-u + u L wi dx = 0 2
Integration by parts
@-u£ wi Dx=2 - @-u£ wi Dx=0 + Ÿ0 Hu£ wi£ + u3 wi L dx = 0 2
EBC at x = 0 ï wi H0L = 0
@-u£ wi Dx=2 + Ÿ0 Hu£ wi£ + u3 wi L dx = 0 2
NBC at x = 2. ï u£ H2L = 5 ê 3
@-5 ê 3 wi Dx=2 + Ÿ0 Hu£ wi£ + u3 wi L dx = 0 2
ClearAll@WeakFormD; WeakForm@w_, u_D := -H5 ê 3 w ê. x Ø 2L + IntegrateA D@u, xD D@w, xD + u3 w, 8x, 0, 2 4 4 4 4
t-1 1-t t+1 1 ∑NT ê∑s = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-t - 1L> 4 4 4 4
s -1 1 s +1 1-s ∑NT ê∑t = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-s - 1L, ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ > 4 4 4 4
Mapping to the master element
75 jij ÅÅÅÅ2ÅÅ jj jj 85 jj ÅÅÅÅ2ÅÅ j Initial coordinates = jjj jj 0 jj jj jj k0
zyz zz z 0 zzzz zz 85 z ÅÅÅÅ ÅÅÅ zzz 2 z zz z 75 z ÅÅÅÅ Å2 ÅÅ { 0
5ts 5s x0 Hs,tL = - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ - 20 t + 20 4 4 5ts 5s y0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ + 20 t + 20 4 4
5t ij ÅÅ54ÅÅ - ÅÅÅÅ ÅÅ 4 J = jjj 5t 5 j ÅÅÅÅ k 4ÅÅÅ + ÅÅ4ÅÅ
Current configuration
5s - ÅÅÅÅ ÅÅ Å - 20 yz 4 zz z 5s ÅÅÅÅ4ÅÅÅÅ + 20 z{
25 s detJ = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + 50 8
101
75 jij ÅÅÅÅ2ÅÅÅ jj jj 85 j ÅÅÅÅÅÅÅ Updated coordinates = jjjj 2 jj 0 jj jj k0
zyz zz zz zz 0 zz z 19.4243 zzzz zz 14.0888 { 0
5ts 5s xHs,tL = - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ - 20 t + 20 4 4 yHs,tL = 1.33387 t s + 1.33387 s + 8.37826 t + 8.37826
25 Hs+16L i ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ 8 Deformation gradient, F p = 1êdetJ jjjj k 16.2046 t + 16.2046
zyz zz 3.33468 s + 16.2046 t + 37.1503 { 0
Gauss quadrature points and weights Point
Weight
1
s Ø -0.57735 t Ø -0.57735
1.
2
s Ø -0.57735 t Ø 0.57735
1.
3
s Ø 0.57735 t Ø -0.57735
1.
4
s Ø 0.57735 t Ø 0.57735
1.
Computation of element matrices at 8-0.57735, -0.57735< with weight = 1. i 1.97169 -19.2783 yz J = jj z k 0.528312 19.2783 {
i 0.4 J -T = jj k 0.4
detJ = 48.1958
-0.0109618 y zz 0.04091 {
0 i1 yz Deformation gradient, F p = jj z; k 0.142105 0.536754 {
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
yz ij 1 0 0.142105 0 z j 0.536754 zzzz F = jjjj 0 0 0 zz jj k 0 1 0.536754 0.142105 { _
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.153412, 0.158893, 0.0411067, -0.0465876<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.173867, 0.153412, 0.0465876, -0.0261326<
102
Geometric Nonlinearity
-0.153412 -0.0218007 0.158893 0.0225796 0.0411067 0.00584149 -0.0465876 -0.00 jij j j G = jj 0 -0.0933239 0 0.0823446 0 0.0250061 0 -0.01 jj k -0.173867 -0.107052 0.153412 0.107087 0.0465876 0.0286845 -0.0261326 -0.02 T
-165.924 jij jj 149.828 j S = jjjj jj 0 jj k0
149.828
zyz zz zz zz -165.924 149.828 zzzz z 149.828 -1603.97 { 0
0
-1603.97 0
0
0 0
T
B =
ij -0.153412 jj jj -0.173867 jj jj jj 0 jj k0
yz zz zz zz z -0.0465876 zzzz z -0.0261326 {
0
0.158893 0
0.0411067 0
-0.0465876 0
0
0.153412 0
0.0465876 0
-0.0261326 0
-0.153412 0
0.158893 0
0.0411067 0
-0.173867 0
0.153412 0
0.0465876 0
3713.19 jij jj 1619.28 jj jj jj -3462.88 jj jj jj -1636.66 j k c = jjj jjj -994.946 jj jj -433.886 jj jj jj 744.637 jj j k 451.266
ij -2139.89 jj jj 0 jj jj jj 1887.47 jj jj jj 0 k s = jjj jj 573.383 jj jj jjj 0 jj jj -320.955 jj j k0 1573.3 jij jj 1619.28 jj jj jj -1575.41 jj jj jj -1636.66 j k = jjj jjj -421.563 jj jjj -433.886 jj jj jj 423.681 jj k 451.266
1619.28
-3462.88 -1636.66 -994.946 -433.886 744.637
451.266
8061.1
-1575.71 -7181.65 -433.886 -2159.97 390.316
1280.51
-1575.71 3260.95
1573.76
927.876
422.211
-725.943 -420.26
-7181.65 1573.76
6412.66
438.543
1924.32
-375.642 -1155.32
-433.886 927.876
438.543
266.595
116.259
-199.525 -120.916
-2159.97 422.211
1924.32
116.259
578.761
-104.585 -343.113
390.316
-725.943 -375.642 -199.525 -104.585 180.831
89.9108
1280.51
-420.26
217.922
0
1887.47
-1155.32 -120.916 -343.113 89.9108
zyz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz {
yz z -320.955 zzzz zz zz 0 -1669.24 0 -505.745 0 287.522 0 zz zz 1887.47 0 -1669.24 0 -505.745 0 287.522 zzzz zz zz 0 -505.745 0 -153.637 0 85.9997 0 zz z 573.383 0 -505.745 0 -153.637 0 85.9997 zzzz zz zz 0 287.522 0 85.9997 0 -52.5661 0 zz z -320.955 0 287.522 0 85.9997 0 -52.5661 { -2139.89 0
0
573.383
0
-320.955 0
1887.47
0
573.383
0
1619.28
-1575.41 -1636.66 -421.563 -433.886 423.681
451.266
5921.21
-1575.71 -5294.18 -433.886 -1586.58 390.316
959.559
-1575.71 1591.7
1573.76
422.131
422.211
-438.421 -420.26
-5294.18 1573.76
4743.41
438.543
1418.57
-375.642 -867.802
-433.886 422.131
438.543
112.958
116.259
-113.525 -120.916
-1586.58 422.211
1418.57
116.259
425.124
-104.585 -257.113
390.316
-438.421 -375.642 -113.525 -104.585 128.265
89.9108
959.559
-420.26
165.356
-867.802 -120.916 -257.113 89.9108
zyz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz {
103
28.6966 y z jij jj -6615.67 zzz zz jj zz jjj jj 162.841 zzz zz jj z jj jj 5772.9 zzz zz r i = jjj jj -7.68924 zzz zz jj z jj jj 1772.66 zzz zz jj z jj jj -183.848 zzz zz jj k -929.89 {
Computation of element matrices at 8-0.57735, 0.57735< with weight = 1. i 0.528312 -19.2783 zy z J = jj k 1.97169 19.2783 { i 0.4 J -T = jj k 0.4
detJ = 48.1958
-0.04091 y zz 0.0109618 {
0 yz i1 z; Deformation gradient, F p = jj k 0.530345 0.924993 {
8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
ij 1 0 0.530345 0 yz _ j z F = jjjj 0 0 0 0.924993 zzzz jj zz k 0 1 0.924993 0.530345 {
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0261326, 0.0465876, 0.153412, -0.173867<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.0465876, 0.0411067, 0.158893, -0.153412< GT =
ij -0.0261326 -0.0138593 0.0465876 0.0247075 0.153412 0.0813614 -0.173867 -0.0922096 yz jj z jj 0 -0.0430932 0 0.0380234 0 0.146975 0 -0.141905 zzzz jj zz j 0.0411067 0.0648939 0.158893 0.226174 -0.153412 -0.242187 { k -0.0465876 -0.04888 ij -20.7918 jj jj 241.261 S = jjjj jj 0 jj k0
yz zz zz zz z -20.7918 241.261 zzzz z 241.261 -230.129 { 0
0
-230.129 0
241.261
0
0 0
BT =
ij -0.0261326 jj jj -0.0465876 jj jj jjj 0 j k0
yz zz zz zz z -0.173867 zzzz zz -0.153412 {
0
0.0465876 0
0.153412 0
-0.173867 0
0
0.0411067 0
0.158893 0
-0.153412 0
-0.0261326 0
0.0465876 0
0.153412 0
-0.0465876 0
0.0411067 0
0.158893 0
104
Geometric Nonlinearity
51.4647 jij jj -1.15235 jj jj jj -44.5957 jj jj jj -23.0169 j k c = jjj jj -173.302 jj jj jj -61.7309 jj jj jj 166.433 jj k 85.9001
ij 3.55572 jj jj 0 jj jj jj -15.2673 jj jj jj 0 k s = jjj jjj -45.2669 jj jj 0 jj jj jj 56.9785 jj j k0 ij 55.0204 jj jj -1.15235 jj jj jjj -59.8631 jj jj -23.0169 j k = jjj jj -218.569 jj jj jj -61.7309 jj jj jj 223.412 jj k 85.9001
ij 515.523 yz z jj jj 76.5185 zzz zz jj z jj jj -431.295 zzz zz jj zz jj jjj -308.085 zzz zz r i = jj jjj -1693.84 zzz zz jj jjj -918.221 zzz zz jj jjj 1609.61 zzz zz jj z j k 1149.79 {
-1.15235 -44.5957 -23.0169 -173.302 -61.7309 166.433
85.9001
-21.8293 -133.664 -58.4865 -516.977 81.4682
498.842
151.799
-21.8293 71.7595
40.4673
240.646
132.388
-267.81
-133.664 40.4673
128.923
133.576
485.89
-151.026 -481.149
-58.4865 240.646
133.576
830.761
423.422
-898.105 -498.511
-516.977 132.388
485.89
423.422
1844.45
-494.079 -1813.36
81.4682
-267.81
-151.026 -898.105 -494.079 999.481
563.637
498.842
-151.026 -481.149 -498.511 -1813.36 563.637
1795.67
yz zz zz zz zz zz 0 23.6193 0 79.7965 0 -88.1484 0 zz z -15.2673 0 23.6193 0 79.7965 0 -88.1484 zzzz zz zz 0 79.7965 0 263.275 0 -297.805 0 zz z -45.2669 0 79.7965 0 263.275 0 -297.805 zzzz zz zz 0 -88.1484 0 -297.805 0 328.974 0 zz z 56.9785 0 -88.1484 0 -297.805 0 328.974 { 0
-15.2673 0
3.55572
0
-45.2669 0
-15.2673 0
56.9785
-45.2669 0
-1.15235 -59.8631 -23.0169 -218.569 -61.7309 223.412 155.355
-21.8293 -148.932 -58.4865 -562.243 81.4682
0
56.9785
85.9001 555.82
-21.8293 95.3788
40.4673
320.443
132.388
-355.959 -151.026
-148.932 40.4673
152.543
133.576
565.686
-151.026 -569.297
-58.4865 320.443
133.576
1094.04
423.422
-1195.91 -498.511
-562.243 132.388
565.686
423.422
2107.73
-494.079 -2111.17
81.4682
-355.959 -151.026 -1195.91 -494.079 1328.46
563.637
555.82
-151.026 -569.297 -498.511 -2111.17 563.637
2124.65
Computation of element matrices at 80.57735, -0.57735< with weight = 1. i 1.97169 -20.7217 yz z J = jj k 0.528312 20.7217 {
i 0.4 J -T = jj k 0.4
-151.026
zyz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz {
-0.0101982 y zz 0.0380604 {
detJ = 51.8042
yz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz z {
105
0 i1 yz Deformation gradient, F p = jj z; k 0.132207 0.573695 {
1 0 0.132207 0 jij zyz j j F = jj 0 0 0 0.573695 zzzz jj zz k 0 1 0.573695 0.132207 { _
8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.156657, 0.161757, 0.0382434, -0.0433425<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.161757, 0.142726, 0.0572736, -0.0382434<
ij -0.156657 -0.0207112 0.161757 0.0213854 0.0382434 0.00505605 -0.0433425 -0.00 j -0.0927989 0 0.0818814 0 0.0328576 0 -0.02 G = jjjj 0 jj -0.161757 -0.111259 0.142726 0.111668 0.0572736 0.029512 -0.0382434 -0.02 k T
ij -148.175 jj jj 126.326 S = jjjj jj 0 jjj k0
126.326
yz zz zz zz z -148.175 126.326 zzzz zz 126.326 -1294.66 { 0
0
-1294.66 0
0
0 0
BT =
ij -0.156657 jj jj -0.161757 jj jj jjj 0 jj k0
yz zz zz zz z -0.0433425 zzzz zz -0.0382434 {
0
0.161757 0
0.0382434 0
-0.0433425 0
0
0.142726 0
0.0572736 0
-0.0382434 0
-0.156657 0
0.161757 0
0.0382434 0
-0.161757 0
0.142726 0
0.0572736 0
ij 3362.36 jj jj 1653.37 jj jj jj -3184.53 jj jj jj -1647.25 k c = jjj jj -1031.12 jj jj jjj -447.496 jj jj 853.292 jj j k 441.38
1653.37
-3184.53 -1647.25 -1031.12 -447.496 853.292
441.38
6567.27
-1596.33 -5874.42 -484.776 -2266.9
1574.05
427.735
-1596.33 3044.92
1574.76
955.493
443.525
-815.885 -421.955
-5874.42 1574.76
5269.42
494.45
2016.94
-421.955 -1411.94
-484.776 955.493
494.45
331.652
122.813
-256.024 -132.487
2016.94
122.813
790.396
-118.842 -540.437
-2266.9
443.525
427.735
-815.885 -421.955 -256.024 -118.842 218.616
113.063
1574.05
-421.955 -1411.94 -132.487 -540.437 113.063
378.327
yz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz z {
106
Geometric Nonlinearity
-1611.59 jij jj 0 jj jj jj 1425.38 jj jj jj 0 j k s = jjj jj 568.14 jj jj jj 0 jj jj jj -381.929 jj k0
ij 1750.77 jj jj 1653.37 jj jj jj -1759.15 jj jj jj -1647.25 k = jjj jjj -462.981 jj jjj -447.496 jj jj jj 471.364 jj k 441.38
ij -143.948 yz z jj jj -5654.8 zzz zz jj zz jj jjj 307.628 zzz zz jj jjj 4925.07 zzz zz j r i = jj jj -81.2507 zzz zz jj jjj 2049.4 zzz zz jj zz jjj jj -82.4286 zzz zz jj k -1319.67 {
zyz -381.929 zzzz zz zz 0 -1264.92 0 -499.388 0 338.935 0 zz zz 1425.38 0 -1264.92 0 -499.388 0 338.935 zzzz zz zz 0 -499.388 0 -202.563 0 133.811 0 zz z 568.14 0 -499.388 0 -202.563 0 133.811 zzzz zz zz 0 338.935 0 133.811 0 -90.8174 0 zz z -381.929 0 338.935 0 133.811 0 -90.8174 { 0
1425.38
-1611.59 0
1653.37 4955.68 -1596.33 -4449.04 -484.776 -1698.76 427.735 1192.12
0
568.14
0
-381.929 0
1425.38
0
568.14
0
yz zz zz zz z 1780. 1574.76 456.104 443.525 -476.949 -421.955 zzzz zz 1574.76 4004.49 494.45 1517.55 -421.955 -1073. zzzz zz 456.104 494.45 129.089 122.813 -122.213 -132.487 zzz zz 443.525 1517.55 122.813 587.834 -118.842 -406.626 zzzz zz -476.949 -421.955 -122.213 -118.842 127.798 113.063 zzzz z -421.955 -1073. -132.487 -406.626 113.063 287.51 { -1759.15 -1647.25 -462.981 -447.496 471.364
441.38
-1596.33 -4449.04 -484.776 -1698.76 427.735
1192.12
Computation of element matrices at 80.57735, 0.57735< with weight = 1. i 0.528312 -20.7217 zy z J = jj k 1.97169 20.7217 {
i 0.4 J -T = jj k 0.4
detJ = 51.8042
-0.0380604 y zz 0.0101982 {
0 i1 zyz; Deformation gradient, F p = jj k 0.493403 0.934891 {
8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
yz ij 1 0 0.493403 0 _ z j 0.934891 zzzz F = jjjj 0 0 0 zz jj k 0 1 0.934891 0.493403 {
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
107
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0382434, 0.0572736, 0.142726, -0.161757<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.0433425, 0.0382434, 0.161757, -0.156657< GT =
ij -0.0382434 -0.0188694 0.0572736 0.028259 0.142726 0.0704217 -0.161757 -0.0798112 yz jj z jj 0 -0.0405206 0 0.0357534 0 0.151225 0 -0.146458 zzzz jj zz j k -0.0433425 -0.0571388 0.0382434 0.072414 0.161757 0.213245 -0.156657 -0.22852 { ij -17.9534 jj jj 220.581 S = jjjj jj 0 jj k0
yz zz zz zz z -17.9534 220.581 zzzz z 220.581 -194.611 { 0
0
-194.611 0
220.581
0
0 0
BT =
-0.0382434 jij jj -0.0433425 jj jj jj 0 jj j k0
zyz zz zz zz -0.161757 zzzz z -0.156657 {
0
0.0572736 0
0.142726 0
-0.161757 0
0
0.0382434 0
0.161757 0
-0.156657 0
-0.0382434 0
0.0572736 0
0.142726 0
-0.0433425 0
0.0382434 0
0.161757 0
ij 61.6597 jj jj 30.1915 jj jj jj -77.5582 jj jj jj -50.2583 k c = jjj jj -230.117 jj jj jjj -112.676 jj jj 246.016 jj j k 132.743
17.5827 jij jj 0 jj jj jj -26.3308 jj jj jj 0 j k s = jjj jj -65.6197 jj jj jj 0 jj jj jj 74.3678 jj k0
30.1915
-77.5582 -50.2583 -230.117 -112.676 246.016
132.743
142.726
-49.2438 -133.765 -112.676 -532.663 131.729
523.701
-49.2438 124.309
64.2905
289.451
183.78
-336.202 -198.827
-133.765 64.2905
136.923
187.567
499.217
-201.599 -502.376
-112.676 289.451
187.567
858.809
420.513
-918.143 -495.404
-532.663 183.78
499.217
420.513
1987.92
-491.618 -1954.48
131.729
-336.202 -201.599 -918.143 -491.618 1008.33
561.488
523.701
-198.827 -502.376 -495.404 -1954.48 561.488
1933.15
0
-26.3308 0
0
17.5827
0
-26.3308 0
0
32.2623
0
-26.3308 0
32.2623
0
0
98.2681
-65.6197 0
98.2681
0
-104.2
0
74.3678
0
-104.2
yz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz z {
zyz zz zz zz zz 98.2681 0 -104.2 0 zz zz 0 98.2681 0 -104.2 zzzz zz zz 244.896 0 -277.544 0 zz z 0 244.896 0 -277.544 zzzz zz zz -277.544 0 307.376 0 zz z 0 -277.544 0 307.376 { -65.6197 0
74.3678
-65.6197 0
74.3678
108
Geometric Nonlinearity
79.2425 jij jj 30.1915 jj jj jj -103.889 jj jj jj -50.2583 j k = jjj jj -295.737 jj jj jj -112.676 jj jj jj 320.384 jj k 132.743
ij 459.709 yz z jj jj 226.863 zzz zz jj z jj jj -383.741 zzz zz jj zz jj jjj -440.741 zzz zz r i = jj jjj -1715.66 zzz zz jj jjj -846.664 zzz zz jj z jj jj 1639.69 zzz zz jj k 1060.54 {
30.1915
-103.889 -50.2583 -295.737 -112.676 320.384
132.743
160.309
-49.2438 -160.096 -112.676 -598.282 131.729
598.069
-49.2438 156.571
64.2905
387.719
183.78
-440.402 -198.827
-160.096 64.2905
169.186
187.567
597.485
-201.599 -606.575
-112.676 387.719
187.567
1103.71
420.513
-1195.69 -495.404
-598.282 183.78
597.485
420.513
2232.82
-491.618 -2232.02
131.729
-440.402 -201.599 -1195.69 -491.618 1315.71
561.488
598.069
-198.827 -606.575 -495.404 -2232.02 561.488
2240.53
After summing contributions from all points, the element equations as follows: 3458.33 jij jj 3301.69 jj jj jj -3498.32 jj jj jj -3357.19 j k = jjj jjj -1398.85 jj jjj -1055.79 jj jj jj 1438.84 jj k 1111.29
rT = H 0
3301.69
-3498.32 -3357.19 -1398.85 -1055.79 1438.84
1111.29
11192.6
-3243.11 -10052.3 -1089.82 -4445.87 1031.25
3305.57
-3243.11 3623.65
3253.28
1586.4
-1711.73 -1192.07
1181.9
-10052.3 3253.28
9069.64
1254.14
4099.29
-1150.22 -3116.68
-1089.82 1586.4
1254.14
2439.79
1083.01
-2627.33 -1247.32
-4445.87 1181.9
4099.29
1083.01
5353.5
-1209.12 -5006.93
1031.25
-1711.73 -1150.22 -2627.33 -1209.12 2900.22
1328.1
3305.57
-1192.07 -3116.68 -1247.32 -5006.93 1328.1
4818.04
0 0 0 0
0 0 0L
Complete element equations for element 1 3458.33 jij jj 3301.69 jj jj jj -3498.32 jj jj jj -3357.19 jj jj jj -1398.85 jj jj jj -1055.79 jj jj jj 1438.84 jj k 1111.29
3301.69
-3498.32 -3357.19 -1398.85 -1055.79 1438.84
1111.29
11192.6
-3243.11 -10052.3 -1089.82 -4445.87 1031.25
3305.57
-3243.11 3623.65
3253.28
1586.4
1181.9
-1711.73 -1192.07
-10052.3 3253.28
9069.64
1254.14
4099.29
-1150.22 -3116.68
-1089.82 1586.4
1254.14
2439.79
1083.01
-2627.33 -1247.32
-4445.87 1181.9
4099.29
1083.01
5353.5
-1209.12 -5006.93
1031.25
-1711.73 -1150.22 -2627.33 -1209.12 2900.22
1328.1
3305.57
-1192.07 -3116.68 -1247.32 -5006.93 1328.1
4818.04
zyz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz {
zyz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz {
0. Du zyz jij 1 zyz jij zyz zz jj Dv zz jj 0. zz zz jj 1 zz jj zz zz jj z j z zz jj Du zzz jjj 0. zzz zzz jjj 2 zzz jjj zzz zz jj zz jj zz zzz jjj Dv2 zzz jjj 0. zzz zz jj zz = jj zz zzz jjj Du3 zzz jjj 0. zzz zz jj zz jj zz zzz jjj Dv3 zzz jjj 0. zzz zz jj zz jj zz zzz jjj Du zzz jjj 0. zzz zz jj 4 zz jj zz zj z j z { k Dv4 { k 0. {
109
The element contributes to 81, 2, 3, 4, 5, 6, 7, 8< global degrees of freedom. ij 1 yz jj zz jj 2 zz jj zz jj zz jj 3 zz jjj zzz jj 4 zz j z Locations for element contributions to a global vector: jjj zzz jj 5 zz jj zz jj zz jjj 6 zzz jj zz jjj 7 zzz jj zz k8{
@1, 1D jij jj @2, 1D jj jj jj @3, 1D jj jj jj @4, 1D j and to a global matrix: jjj jjj @5, 1D jj jjj @6, 1D jj jjj @7, 1D jj j k @8, 1D
@1, 2D
@1, 3D
@1, 4D
@1, 5D
@1, 6D
@3, 3D
@3, 4D
@3, 5D
@3, 6D
@5, 3D
@5, 4D
@5, 5D
@5, 6D
@2, 2D
@2, 3D
@4, 2D
@4, 3D
@3, 2D
@5, 2D
@6, 2D
@7, 2D
@8, 2D
@6, 3D
@7, 3D
@8, 3D
@2, 4D
@4, 4D
@6, 4D
@7, 4D
@8, 4D
@2, 5D
@4, 5D
@6, 5D
@7, 5D
@8, 5D
@2, 6D
@4, 6D
@6, 6D
@7, 6D
@8, 6D
@1, 7D
@1, 8D y zz @2, 7D @2, 8D zzzz zz @3, 7D @3, 8D zzzz zz @4, 7D @4, 8D zzzz z @5, 7D @5, 8D zzzz zz @6, 7D @6, 8D zzzz zz @7, 7D @7, 8D zzzz z @8, 7D @8, 8D {
Adding element equations into appropriate locations we have ij 3458.33 jj jj 3301.69 jj jj jj -3498.32 jj jj jj -3357.19 jj jj jj -1398.85 jj jj -1055.79 jj jj jj 1438.84 jj j k 1111.29
3301.69
-3498.32 -3357.19 -1398.85 -1055.79 1438.84
1111.29
11192.6
-3243.11 -10052.3 -1089.82 -4445.87 1031.25
3305.57
-3243.11 3623.65
3253.28
1586.4
1181.9
-1711.73 -1192.07
-10052.3 3253.28
9069.64
1254.14
4099.29
-1150.22 -3116.68
-1089.82 1586.4
1254.14
2439.79
1083.01
-2627.33 -1247.32
-4445.87 1181.9
4099.29
1083.01
5353.5
-1209.12 -5006.93
1031.25
-1711.73 -1150.22 -2627.33 -1209.12 2900.22
1328.1
3305.57
-1192.07 -3116.68 -1247.32 -5006.93 1328.1
4818.04
ij Du1 yz ij 0 yz z j jj zz jj Dv1 zzz jjj 0 zz zz zzz jjj jjj zz jj Du zz jj 0 zz jjj 2 zzz jjj zz zz jj jj zz zz jjj Dv2 zzz jjj 0 zz zz = jj jj zz jjj Du3 zzz jjj 0 zz zz jj jj zz jjj Dv3 zzz jjj -2000. zzz zz jj jj zz jjj Du zzz jjj 0 zzz jj 4 zz jj zz j z j z k Dv4 { k 0 { After assembly of all elements the global matrices are as follows.
yz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz z {
110
Geometric Nonlinearity
3458.33 jij jj 3301.69 jj jj jj -3498.32 jj jj jj -3357.19 j K T = jjj jj -1398.85 jj jj jj -1055.79 jj jj jj 1438.84 jj k 1111.29
yz ij 0. zz jj zz jj 0. zz jj zz jj zz jj 0. zz jj zz jj zz jj 0. zz; RE = jjj zz zz jjj 0. z jj jj -2000. zzz zz jj zz jj zzz jjj 0. zz jj { k 0.
3301.69
-3498.32 -3357.19 -1398.85 -1055.79 1438.84
1111.29
11192.6
-3243.11 -10052.3 -1089.82 -4445.87 1031.25
3305.57
-3243.11 3623.65
3253.28
1586.4
1181.9
-1711.73 -1192.07
-10052.3 3253.28
9069.64
1254.14
4099.29
-1150.22 -3116.68
-1089.82 1586.4
1254.14
2439.79
1083.01
-2627.33 -1247.32
-4445.87 1181.9
4099.29
1083.01
5353.5
-1209.12 -5006.93
1031.25
-1711.73 -1150.22 -2627.33 -1209.12 2900.22
1328.1
3305.57
-1192.07 -3116.68 -1247.32 -5006.93 1328.1
4818.04
ij -859.98 yz z jj jj 11967.1 zzz zz jj z jj jj 344.568 zzz zz jj z jj jj -9949.14 zzz zz; RI = jjj z jjj 3498.44 zzz zz jj jjj -2057.18 zzz zz jj z jj jj -2983.03 zzz zz jj k 39.229 {
ij 859.98 yz z jj jj -11967.1 zzz zz jjj jj -344.568 zzz zzz jjj zz jj jjj 9949.14 zzz zz R = RE - RI = jj jjj -3498.44 zzz zz jj jjj 57.1788 zzz zz jj z jj jj 2983.03 zzz zz jj k -39.229 {
System of equations ij 3458.33 jj jj 3301.69 jj jj jj -3498.32 jj jj jj -3357.19 jj jj jj -1398.85 jj jj -1055.79 jj jj jj 1438.84 jj j k 1111.29
3301.69
-3498.32 -3357.19 -1398.85 -1055.79 1438.84
1111.29
11192.6
-3243.11 -10052.3 -1089.82 -4445.87 1031.25
3305.57
-3243.11 3623.65
1586.4
1181.9
-1711.73 -1192.07
-10052.3 3253.28
9069.64
1254.14
4099.29
-1150.22 -3116.68
-1089.82 1586.4
1254.14
2439.79
1083.01
-2627.33 -1247.32
-4445.87 1181.9
4099.29
1083.01
5353.5
-1209.12 -5006.93
1031.25
-1711.73 -1150.22 -2627.33 -1209.12 2900.22
1328.1
3305.57
-1192.07 -3116.68 -1247.32 -5006.93 1328.1
4818.04
ij Du1 yz ij 859.98 yz z j jj z jj Dv1 zzz jjj -11967.1 zzz zz zzz jjj jjj jj Du zz jj -344.568 zzz zz jjj 2 zzz jjj zz zz jj jj z jjj Dv2 zzz jjj 9949.14 zzz zz = jj jj zz jjj Du3 zzz jjj -3498.44 zzz zz jj jj zz jjj Dv3 zzz jjj 57.1788 zzz zz jj jj zz jjj Du zzz jjj 2983.03 zzz jj 4 zz jj zz z j j z k Dv4 { k -39.229 {
Essential boundary conditions
3253.28
zyz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz {
zyz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz {
111
Node
dof
Value
1
Du1 Dv1
0 0
2
Du2 Dv2
0 0
3
Du3
0
4
Du4
0
Remove 81, 2, 3, 4, 5, 7< rows and columns. After adjusting for essential boundary conditions we have -5006.93 y i Dv3 ij 5353.5 zz jj j k -5006.93 4818.04 { k Dv4
yz ij 57.1788 yz z=j z { k -39.229 {
Solving the final system of global equations we get 8Dv3 = 0.10921, Dv4 = 0.10535<
Complete table of nodal values Du
Dv
1
0
0
2
0
0
3
0
0.10921
4
0
0.10535
Total increments since the start of this load step Du
Dv
1
0
0
2
0
0
3
0
-9.0586
4
0
-9.2673
Total nodal values u
v
1
0
0
2
0
0
3
0
-22.9665
4
0
-23.3059
Solution for element 1 Initial configuration
112
Geometric Nonlinearity
zyz zz z 0 zzzz zz 85 z ÅÅÅÅ ÅÅ zzz 2 z zz z 75 z ÅÅÅÅ Å2 Å {
75 jij ÅÅÅÅ2ÅÅ jj jj 85 jj ÅÅÅÅ2ÅÅ j Nodal coordinates = jjj jj 0 jj jj jj k0
0
5ts 5s x0 Hs,tL = - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ - 20 t + 20 4 4 5ts 5s y0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ + 20 t + 20 4 4
5t ij ÅÅ54ÅÅ - ÅÅÅÅ ÅÅ 4 J = jjj 5t 5 j ÅÅÅÅ k 4ÅÅÅ + ÅÅ4ÅÅ
5s - ÅÅÅÅ ÅÅ Å - 20 yz 4 zz z 5s ÅÅÅÅ4ÅÅÅÅ + 20 z{
25 s detJ = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + 50 8
Current configuration II
0 jij jj 0 j Nodal displacements = jjjj jj 0 jj k0
zyz zz zz zz -22.9665 zzzz z -23.3059 {
0
0
75 ij ÅÅÅÅ ÅÅÅ jj 2 jj 85 jj ÅÅÅÅÅÅÅ Updated coordinates = jjjj 2 jj 0 jj jj k0
yz zz zz zz 0 zz zz 19.5335 zzzz zz 14.1941 { 0
5ts 5s xII Hs,tL = - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ - 20 t + 20 4 4
yII Hs,tL = 1.33484 t s + 1.33484 s + 8.4319 t + 8.4319
5t i ÅÅÅÅ5Å - ÅÅÅÅ ÅÅÅ 4 J II = jjjj 4 k 1.33484 t + 1.33484
zyz zz 1.33484 s + 8.4319 { 5s - ÅÅÅÅ ÅÅÅÅ - 20 4
detJ II = 3.33709 s + 16.1569 t + 37.2366
25 Hs+16L i ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ 8 Deformation gradient, F IIp = H1êdetJ L jjjj k 16.1569 t + 16.1569
yz zz z 3.33709 s + 16.1569 t + 37.2366 { 0
Solution at 8s, t< = 8-0.57735, -0.57735< ï 8x, y< = 830.4087, 3.23802< i 1.97169 -19.2783 zy J II = jj z k 0.564169 7.66123 {
detJ II = 25.9818
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
113
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.107715, 0.118572, 0.0288623, -0.0397193<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.322521, 0.284577, 0.0864193, -0.0484756< B TL =
-0.107715 0 0.118572 0 0.0288623 0 -0.0397193 0 jij zyz jj 0 -0.322521 0 0.284577 0 0.0864193 0 -0.0484756 zzzz jj jj zz k -0.322521 -0.107715 0.284577 0.118572 0.0864193 0.0288623 -0.0484756 -0.0397193 { 0 i1 yz z; F IIp = jj k 0.141687 0.539088 {
Det@F IIp D = 0.539088
Element thickness, hII = 1.2287
0.141687 y i 1. zz Left Cauchy-Green tensor = jj 0.141687 0.310691 { k
i 0.0100376 0.0381908 yz Green-Lagrange strain tensor, e = jj z k 0.0381908 -0.354692 { i -248.75 85.5623 zy Cauchy stress tensor, s = jj z k 85.5623 -665.012 {
Principal stresses = H -681.914 -231.849 L
Effective stress Hvon MisesL = 600.549
i -164.767 148.436 yz Second PK stresses = jj z -1582.36 { k 148.436 r Ti = H -25.5844 6552.83
-164.274 -5717.64 6.85532 -1755.83 183.003
Solution at 8s, t< = 8-0.57735, 0.57735< ï 8x, y< = 88.14797, 12.0844< i 0.528312 -19.2783 yz J II = jj z k 2.10551 7.66123 {
920.634 L
detJ II = 44.6381
8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 80.000465431, 0.0231187, 0.0626961, -0.0862803<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.0503006, 0.0443829, 0.171557, -0.165639<
114
Geometric Nonlinearity
B TL
0.000465431 0 0.0231187 0 0.0626961 0 -0.0862803 0 jij j j = jj 0 -0.0503006 0 0.0443829 0 0.171557 0 -0. jj k -0.0503006 0.000465431 0.0443829 0.0231187 0.171557 0.0626961 -0.165639 -0.
0 i1 zyz; F IIp = jj k 0.528782 0.926184 {
Det@F IIp D = 0.926184
Element thickness, hII = 1.02589
0.528782 y i 1. zz Left Cauchy-Green tensor = jj k 0.528782 1.13743 {
i 0.139805 0.244875 yz Green-Lagrange strain tensor, e = jj z k 0.244875 -0.071092 { i -21.5213 222.607 zy Cauchy stress tensor, s = jj z 36.3325 { k 222.607
Principal stresses = H 231.884
-217.073 L
Effective stress Hvon MisesL = 388.879
i -20.4488 240.045 yz Second PK stresses = jj z -227.186 { k 240.045
r Ti = H -513.225 -78.9458 429.656
309.517
1687.07
924.563
Solution at 8s, t< = 80.57735, -0.57735< ï 8x, y< = 832.6854, 3.88946< i 1.97169 -20.7217 yz J II = jj z k 0.564169 9.20257 {
-1603.5 -1155.13 L
detJ II = 29.8351
8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.119634, 0.129089, 0.0251346, -0.0345893<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.280866, 0.247823, 0.099447, -0.066404< B TL =
0.129089 0 0.0251346 0 -0.0345893 0 ij -0.119634 0 yz jj zz jj 0 zz -0.280866 0 0.247823 0 0.099447 0 -0.066404 jj zz j z k -0.280866 -0.119634 0.247823 0.129089 0.099447 0.0251346 -0.066404 -0.0345893 { 0 i1 yz z; F IIp = jj 0.131817 0.575921 k {
Element thickness, hII = 1.20193
Det@F IIp D = 0.575921
115
0.131817 yz i 1. Left Cauchy-Green tensor = jj z k 0.131817 0.349061 {
i 0.00868792 0.0379582 yz Green-Lagrange strain tensor, e = jj z k 0.0379582 -0.334158 { i -212.568 76.1713 yz Cauchy stress tensor, s = jj z -588.715 { k 76.1713
Principal stresses = H -603.555 -197.728 L
Effective stress Hvon MisesL = 532.95
i -147.143 125.231 yz Second PK stresses = jj z -1278.25 { k 125.231 r Ti = H 144.747
5602.63
-307.073 -4879.23 80.0465
Solution at 8s, t< = 80.57735, 0.57735< ï 8x, y< = 88.75802, 14.5157< i 0.528312 -20.7217 yz J II = jj z k 2.10551 9.20257 {
-2030.79 82.2801 1307.39 L
detJ II = 48.4915
8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.0154644, 0.0371745, 0.057714, -0.0794241< 9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.0463035, 0.0408561, 0.172807, -0.16736<
B TL =
-0.0154644 0 0.0371745 0 0.057714 0 -0.0794241 0 jij zyz jj 0 -0.0463035 0 0.0408561 0 0.172807 0 -0.16736 zzzz jj jj zz -0.0794241 { k -0.0463035 -0.0154644 0.0408561 0.0371745 0.172807 0.057714 -0.16736 0 i1 zyz; F IIp = jj k 0.491949 0.936053 {
Det@F IIp D = 0.936053
Element thickness, hII = 1.02227
0.491949 yz i 1. z Left Cauchy-Green tensor = jj k 0.491949 1.11821 {
i 0.121007 0.230245 yz Green-Lagrange strain tensor, e = jj z k 0.230245 -0.0619026 { i -18.416 205.643 yz Cauchy stress tensor, s = jj z k 205.643 30.9974 {
116
Geometric Nonlinearity
Principal stresses = H 213.412 -200.831 L
Effective stress Hvon MisesL = 358.8
i -17.6222 219.484 yz Second PK stresses = jj z -191.983 { k 219.484
r Ti = H -457.902 -228.794 382.551 441.737
1708.91 853.871 -1633.56 -1066.81 L
After summing contributions from all points the internal load vector is as follows: r Ti = H -851.964 11847.7
340.859
Global internal load vector RTI = H -851.964 11847.7
340.859
-9845.61 3482.88 -2008.18 -2971.78 6.07127 L
-9845.61 3482.88 -2008.18 -2971.78 6.07127 L
After assembling all element internal force vectors, the global internal force and the external load vectors are as follows. -851.964 y z jij jj 11847.7 zzz zz jj z jj jj 340.859 zzz zz jj z jj jj -9845.61 zzz zz j zz; RI = jjj jjj 3482.88 zzz zz jj jjj -2008.18 zzz zz jj jjj -2971.78 zzz zz jj z j 6.07127 { k
0. zyz jij zz jj 0. zz jj zz jj zz jj 0. zz jj zz jj zz jj 0. zz j zz RE = jjj zz jjj 0. zz jj z jjj -2000. zzz zz jj zz jjj 0. zz jj zz j 0. { k
Corresponding to unrestrained dof i -2008.18 yz i -2000. yz z; RE = jj z; RI = jj { k 6.07127 { k 0. »»RE »» = 2000.;
»»R »» = 10.1868
Convergence parameter = 0.0000259427 Solution converged to the desired tolerance.
i 8.1799 yz R = RE - RI = jj z k -6.07127 {
117
Load factor 1 0.8 0.6 0.4 0.2 v3 5
10
15
20
118
Geometric Nonlinearity
PRINT U
NODAL SOLUTION PER NODE
***** POST1 NODAL DEGREE OF FREEDOM LISTING *****
LOAD STEP= TIME=
2
2.0000
SUBSTEP=
2
LOAD CASE=
0
THE FOLLOWING DEGREE OF FREEDOM RESULTS ARE IN GLOBAL COORDINATES
NODE
UX
UY
UZ
USUM
1
0.0000
0.0000
0.0000
0.0000
2
0.0000
0.0000
0.0000
0.0000
3
0.0000
-22.699
0.0000
22.699
4
0.0000
-23.119
0.0000
23.119
MAXIMUM ABSOLUTE VALUES NODE VALUE
PRINT S
0
4
0
4
0.0000
-23.119
0.0000
23.119
ELEMENT SOLUTION PER ELEMENT
***** POST1 ELEMENT NODAL STRESS LISTING *****
LOAD STEP= TIME=
2
2.0000
SUBSTEP=
2
LOAD CASE=
0
119
THE FOLLOWING X,Y,Z VALUES ARE IN ROTATED GLOBAL COORDINATES, WHICH INCLUDE RIGID BODY ROTATION EFFECTS
ELEMENT= NODE
1
PLANE182
SX
SY
SZ
SXY
SYZ
SXZ
2
-96.862
-673.71
-0.21259E-10
120.56
0.0000
0.0000
3
-37.615
24.078
0.22737E-12
202.24
0.0000
0.0000
4
-44.693
25.377
218.44
0.0000
0.0000
1
-82.677
-780.81
147.72
0.0000
0.0000
AnsysFiles\Chap09\Prb9-3Data.txt
!*Problem 9.3 !* One element model !* Large displacement analysis !* Hyper elastic material êPREP7
!* Element type !* ET,1,PLANE182 KEYOPT,1,1,0 KEYOPT,1,3,3 KEYOPT,1,6,0 KEYOPT,1,10,0 !* R,1,1 !* Material property *set, e, 1000 *set, nu, 0.25 *set, mu, eêH2*H1+nuLL *set, k, eêH3*H1-2*nuLL *set, d, 2êk
0.0000 -0.22737E-12
120
Geometric Nonlinearity
*set, P, 2000 !* Ansys neo-Hookean model MPTEMP,,,,,,,, MPTEMP,1,0 TB,HYPE,1,1,2,NEO TBTEMP,0 TBDATA,,mu,d,,,, !* k,1,37.5,0 k,2,42.5,0 k,3,0,42.5 k,4,0,37.5 A,1,2,3,4 ESIZE,100 AMESH,1 DL,1, ,ALL êSOLU
DL,3, ,UX ANTYPE,0 NLGEOM,1 ARCLEN,1,1,0.0001 NCNV,2,0,0,0,0 RESCONTRL,DEFINE,ALL,1,1 ERESX,NO OUTRES,ERASE OUTRES,ALL,1 AUTOTS,-1.0 !* First load step !* No applied load !* Used for initialization of !* the arc-length controls LSWRITE,1, !* Specify applied forces FK,3,FY,-P LSWRITE,2, LSSOLVE,1,2,1 FINISH !* Postprocessing
121
êPOST1 SET,LAST PRNSOL,UX
9.4 Taking large displacements into consideration determine stresses in an Aluminum bracket shown in Figure 9.17. A load of 10,000 lb is applied near the tip. The other end can be considered fixed. Assume a compressible neo-Hookean material with E = 10.6 µ 106 psi and n = 0.35.
1000 lb
5 4 3 Thickness = 1.25 in
2 1 0 0
Figure 9.17.
2
4
6
8
10
Aluminum bracket
(a) Use only two elements and obtain a solution using Mathematica/Matlab. (b) Obtain solution using an available finite element computer program. For a more realistic analysis apply the load over 1 inch length and assume a 1/4 in fillet at the inside corner of the bracket.
y 4 3 2 1 0
4
6 5
2
3 1 1 0
2 2
4
6
8
10
x
122
Geometric Nonlinearity
Specified nodal loads Node
dof
Value
6
Dv6
-1
(i) Load factor = 1 Set the state of all elements to their initial values.
Iteration = 1 Global equations at start of the element assembly process ij 0 jj jj 0 jj jj jjj 0 jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
E = 10600.;
n = 0.35;
Plane stress analysis.
0 y i Du1 y i 0 y zz jj zz zz jj 0 zzzz jjjj Dv1 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzzz jjjj Du2 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzzz jjjj Dv2 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzz jjj Du3 zzz jjj 0 zzz zz jj zz zz jj 0 zzzz jjjj Dv3 zzzz jjjj 0 zzzz zz zz jj zz = jj 0 zzzz jjjj Du4 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzz jjj Dv4 zzz jjj 0 zzz zz jj zz zz jj 0 zzzz jjjj Du5 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzzz jjjj Dv5 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzz jjj Du6 zzz jjj 0 zzz zz jj zz zz jj 0 { k Dv6 { k -1 {
Initial thickness = 1.25 g = 0.461538
Interpolation functions and their derivatives 1 1 1 1 NT = : ÅÅÅÅÅ Hs - 1L Ht - 1L, - ÅÅÅÅÅ Hs + 1L Ht - 1L, ÅÅÅÅÅ Hs + 1L Ht + 1L, - ÅÅÅÅÅ Hs - 1L Ht + 1L> 4 4 4 4
t-1 1-t t+1 1 ∑NT ê∑s = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-t - 1L> 4 4 4 4
s -1 1 s +1 1-s ∑NT ê∑t = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-s - 1L, ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ > 4 4 4 4
Mapping to the master element
123
0 jij jj 5 jj ÅÅÅÅÅ j 2 Initial coordinates = jjjj jj ÅÅÅÅ5Å jj 2 jj k0 5s 5 x0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4
0y zz z 0 zzzz zz zz 3 zzz zz z 4{
ts s 7t 7 y0 Hs,tL = - ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4 4 4 ij ÅÅ54ÅÅ J = jjj j - ÅÅtÅÅ - ÅÅÅÅ1Å k 4 4
0 ÅÅÅÅ74Å
-
ÅÅ4sÅÅ
yz zz zz {
35 5s detJ = ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ 16 16
Current configuration
0y zz z 0 zzzz zz zz 3 zzz zz z 4{
0 jij jj 5 jj ÅÅÅÅÅ j 2 Updated coordinates = jjjj jj ÅÅÅÅ5Å jj 2 jj k0 5s 5 xHs,tL = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4 ts s 7t 7 yHs,tL = - ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4 4 4
5 ij - ÅÅÅÅ ÅÅ Hs - 7L 16 Deformation gradient, F p = 1êdetJ jjjj k0
yz zz zz 5 ÅÅ Hs - 7L { - ÅÅÅÅ 16 0
Gauss quadrature points and weights Point
Weight
1
s Ø -0.57735 t Ø -0.57735
1.
2
s Ø -0.57735 t Ø 0.57735
1.
3
s Ø 0.57735 t Ø -0.57735
1.
4
s Ø 0.57735 t Ø 0.57735
1.
Computation of element matrices at 8-0.57735, -0.57735< with weight = 1.
124
Geometric Nonlinearity
i ÅÅÅÅ5Å 0 J = jjjj 4 k -0.105662 1.89434
i 0.8 J -T = jj k0
0.0446224 y zz 0.527889 {
zyz zz {
detJ = 2.36792
i1 Deformation gradient, F p = jj k0
0y zz; 1{
ij 1 0 0 0 yz j z F = jjjj 0 0 0 1 zzzz jj zz k0 1 1 0 { _
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.333066, 0.310755, 0.0892449, -0.0669336<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.208166, -0.055778, 0.055778, 0.208166< GT =
-0.333066 0 0.310755 0 0.0892449 0 -0.0669336 0 jij zyz jj 0 -0.208166 0 -0.055778 0 0.055778 0 0.208166 zzzz jj jj zz 0.055778 0.0892449 0.208166 -0.0669336 { k -0.208166 -0.333066 -0.055778 0.310755 ij 0 jj jj 0 S = jjjj jj 0 jj k0
0 0 0 0
0 0y zz 0 0 zzzz zz 0 0 zzzz z 0 0{
T
B =
ij -0.333066 jj jj -0.208166 jj jj jj 0 jj k0
0
0.310755
0
-0.055778 0
0
-0.0669336 0
0
0.208166
0
0.0892449 0
-0.333066 0
0.310755
-0.208166 0
-0.055778 0
ij 6089.58 jj jj 2685.59 jj jj jj -5076.91 jj jj jj -247.986 k c = jjj jjj -1631.7 jj jj -719.602 jj jj jj 619.031 jj j k -1718.
yz zz zz zz z -0.0669336 zzzz z 0.208166 {
0.0892449 0 0.055778
0.055778
0
0
yz z -618.055 -719.602 -930.085 -427.886 -1922.99 zzzz zz -1538.1 4898.86 -671.398 1360.35 412.133 -1182.31 1797.37 zzzz zz -618.055 -671.398 1278.83 66.4476 165.607 852.936 -826.38 zzzz zz -719.602 1360.35 66.4476 437.212 192.817 -165.869 460.337 zzz zz -930.085 412.133 165.607 192.817 249.216 114.652 515.262 zzzz zz -427.886 -1182.31 852.936 -165.869 114.652 729.144 -539.701 zzzz z -1922.99 1797.37 -826.38 460.337 515.262 -539.701 2234.1 { 2685.59
-5076.91 -247.986 -1631.7
3471.13
-1538.1
-719.602 619.031
-1718.
125
0 jij jj 0 jj jj jj 0 jj jj jj 0 j k s = jjj jj 0 jj jj jj 0 jj jj jj 0 jj k0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
ij 6089.58 jj jj 2685.59 jj jj jj -5076.91 jj jj jj -247.986 k = jjj jjj -1631.7 jj jjj -719.602 jj jj jj 619.031 jj k -1718.
ij 0 yz jj zz jj 0 zz jj zz jj zz jjj 0 zzz jj zz jjj 0 zzz r i = jjj zzz jj 0 zz jj zz jjj 0 zzz jj zz jjj zzz jj 0 zz jj zz k0{
0y zz 0 zzzz zz 0 zzzz zz 0 zzzz zz 0 zzz zz 0 zzzz zz 0 zzzz z 0{
-719.602 619.031 -1718. y zz -618.055 -719.602 -930.085 -427.886 -1922.99 zzzz zz -1538.1 4898.86 -671.398 1360.35 412.133 -1182.31 1797.37 zzzz zz -618.055 -671.398 1278.83 66.4476 165.607 852.936 -826.38 zzzz zz -719.602 1360.35 66.4476 437.212 192.817 -165.869 460.337 zzz zz -930.085 412.133 165.607 192.817 249.216 114.652 515.262 zzzz zz -427.886 -1182.31 852.936 -165.869 114.652 729.144 -539.701 zzzz z -1922.99 1797.37 -826.38 460.337 515.262 -539.701 2234.1 { 2685.59
-5076.91 -247.986 -1631.7
3471.13
-1538.1
Computation of element matrices at 8-0.57735, 0.57735< with weight = 1. i ÅÅÅÅ5Å 0 J = jjjj 4 k -0.394338 1.89434 i 0.8 J -T = jj k0
0.166533 y zz 0.527889 {
zyz zz {
i1 Deformation gradient, F p = jj k0
detJ = 2.36792
0y zz; 1{
ij 1 0 0 0 yz _ j z F = jjjj 0 0 0 1 zzzz jj zz k0 1 1 0 {
8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
126
Geometric Nonlinearity
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.1502, 0.0669336, 0.333066, -0.2498<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.208166, -0.055778, 0.055778, 0.208166< GT =
0 0.0669336 0 0.333066 0 -0.2498 0 ij -0.1502 yz jj zz jj 0 zz -0.208166 0 -0.055778 0 0.055778 0 0.208166 jj zz j z -0.055778 0.0669336 0.055778 0.333066 0.208166 -0.2498 { k -0.208166 -0.1502 ij 0 jj jj 0 S = jjjj jj 0 jj k0
0 0 0 0
0 0y zz 0 0 zzzz zz 0 0 zzzz z 0 0{
BT =
-0.1502 jij jj -0.208166 jj jj jj 0 jj j k0
0.0669336
0
0.333066 0
-0.2498
0
-0.055778 0
0.055778 0
0.208166 0
-0.1502
0
0
0.333066 0
-0.055778 0
0.055778 0
0.0669336
-0.208166 0
ij 1639.56 jj jj 1211.1 jj jj jj -371.315 jj jj jj 65.2487 k c = jjj jj -2654.01 jj jj jjj -1032.84 jj jj 1385.77 jj j k -243.511
0 jij jj 0 jj jj jj 0 jj jj jj 0 j k s = jjj jj 0 jj jj jj 0 jj jj jj 0 jj k0
zyz zz zz zz -0.2498 zzzz z 0.208166 {
0
0
-243.511 y zz -1746.05 zzzz zz -280.437 261.749 -144.613 1086.43 -114.652 -976.859 539.701 zzzz zz 467.852 -144.613 208.724 -460.337 102.393 539.701 -778.969 zzzz zz zz -1977.27 1086.43 -460.337 5622.18 719.602 -4054.6 1718. zz z -1166. -114.652 102.393 719.602 1445.75 427.886 -382.136 zzzz zz 1046.61 -976.859 539.701 -4054.6 427.886 3645.69 -2014.19 zzzz z -1746.05 539.701 -778.969 1718. -382.136 -2014.19 2907.15 { 1211.1
-371.315 65.2487
-2654.01 -1032.84 1385.77
2444.2
-280.437 467.852
-1977.27 -1166.
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0y zz 0 zzzz zz 0 zzzz zz 0 zzzz zz 0 zzz zz 0 zzzz zz 0 zzzz z 0{
1046.61
127
1639.56 jij jj 1211.1 jj jj jj -371.315 jj jj jj 65.2487 j k = jjj jj -2654.01 jj jj jj -1032.84 jj jj jj 1385.77 jj k -243.511
ij 0 yz jj zz jj 0 zz jjj zzz jj 0 zz jjj zzz jj zz jj 0 zz r i = jjj zzz jjj 0 zzz jj zz jjj 0 zzz jj zz jj zz jj 0 zz jj zz k0{
-243.511 y zz 2444.2 -280.437 467.852 -1977.27 -1166. 1046.61 -1746.05 zzzz zz -280.437 261.749 -144.613 1086.43 -114.652 -976.859 539.701 zzzz zz 467.852 -144.613 208.724 -460.337 102.393 539.701 -778.969 zzzz zz zz -1977.27 1086.43 -460.337 5622.18 719.602 -4054.6 1718. zz z -1166. -114.652 102.393 719.602 1445.75 427.886 -382.136 zzzz zz 1046.61 -976.859 539.701 -4054.6 427.886 3645.69 -2014.19 zzzz z -1746.05 539.701 -778.969 1718. -382.136 -2014.19 2907.15 { -371.315 65.2487
1211.1
-2654.01 -1032.84 1385.77
Computation of element matrices at 80.57735, -0.57735< with weight = 1. i ÅÅÅÅ5Å 0 J = jjjj 4 k -0.105662 1.60566 i 0.8 J -T = jj k0
0.0526449 y zz 0.622796 {
yz zz z {
i1 Deformation gradient, F p = jj k0
detJ = 2.00708
0y zz; 1{
1 0 0 0 jij j j F = jj 0 0 0 1 jj k0 1 1 0 _
8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
zyz zz zz zz {
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.321033, 0.29471, 0.10529, -0.0789674<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.0658061, -0.245592, 0.245592, 0.0658061< GT =
0.29471 0 0.10529 0 -0.0789674 0 ij -0.321033 0 yz jj zz jj 0 zz -0.0658061 0 -0.245592 0 0.245592 0 0.0658061 jj zz j z 0.245592 0.10529 0.0658061 -0.0789674 { k -0.0658061 -0.321033 -0.245592 0.29471
128
Geometric Nonlinearity
0 jij jj 0 j S = jjjj jj 0 jj k0
0 0y zz 0 0 zzzz zz 0 0 zzzz z 0 0{
0 0 0 0
BT =
ij -0.321033 jj jj -0.0658061 jj jj jj 0 jj k0
0.29471
0
-0.245592 0
0.245592 0
0.0658061
-0.321033
0
0
0
ij 0 jj jj 0 jj jj jj 0 jj jj jj 0 k s = jjj jjj 0 jj jj 0 jj jj jj 0 jj j k0
693.603
0.10529
-0.245592 0 -3878.97 1620.97
0
0.245592 0 -1601.88 -1880.24 1039.37
1199.94
330.856
-242.089 -935.806 -1022.72 -88.6525
330.856
4301.13
-2376.32 730.323
1408.73
-1152.48
-242.089 -2376.32 3429.82
118.612
-935.806 730.323
118.612
1067.24
848.976
-195.69
-1022.72 1408.73
-2268.71 848.976
2683.53
-377.468
-88.6525 -1152.48 636.732 64.8675
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
ij 4441.48 jj jj 693.603 jj jj jjj -3878.97 jj jj 1620.97 j k = jjj jj -1601.88 jj jj jj -1880.24 jj jj jj 1039.37 jj k -434.338
0.10529
0.29471
-0.0658061 0
4441.48 jij jj 693.603 jj jj jj -3878.97 jj jj jj 1620.97 j k c = jjj jj -1601.88 jj jj jj -1880.24 jj jj jj 1039.37 jj k -434.338
0
636.732 0y zz 0 zzzz zz 0 zzzz zz 0 zzzz zz 0 zzz zz 0 zzzz zz 0 zzzz z 0{
-195.69
yz zz zz zz z -0.0789674 zzzz z 0.0658061 {
-0.0789674 0
0
-2268.71 636.732
-377.468 308.807
-919.016 -31.7819 607.899
-170.612
0
-434.338 y zz 64.8675 zzzz zz 636.732 zzzz zz -919.016 zzzz zz -31.7819 zzz zz 607.899 zzzz zz -170.612 zzzz z 246.25 {
-434.338 y zz zz zz zz 330.856 4301.13 -2376.32 730.323 1408.73 -1152.48 636.732 zzzz zz -242.089 -2376.32 3429.82 118.612 -2268.71 636.732 -919.016 zzzz zz -935.806 730.323 118.612 1067.24 848.976 -195.69 -31.7819 zzz zz -1022.72 1408.73 -2268.71 848.976 2683.53 -377.468 607.899 zzzz zz -88.6525 -1152.48 636.732 -195.69 -377.468 308.807 -170.612 zzzz z 64.8675 636.732 -919.016 -31.7819 607.899 -170.612 246.25 { 693.603
-3878.97 1620.97
1199.94
330.856
-1601.88 -1880.24 1039.37
-242.089 -935.806 -1022.72 -88.6525 64.8675
129
0 jij zyz jj 0 zz jj zz jjj zzz jj 0 zz jj zz jj zz jj 0 zz r i = jjj zzz jj 0 zz jj zz jj zz jj 0 zz jj zz jj zz jj 0 zz jj zz k0{
Computation of element matrices at 80.57735, 0.57735< with weight = 1. i ÅÅÅÅ5Å 0 J = jjjj 4 k -0.394338 1.60566 i 0.8 J -T = jj k0
0.196473 y zz 0.622796 {
zyz zz {
i1 Deformation gradient, F p = jj k0
detJ = 2.00708
0y zz; 1{
ij 1 0 0 0 _ j F = jjjj 0 0 0 1 jj k0 1 1 0
8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
yz zz zz zz z {
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.10529, 0.00705308, 0.392947, -0.29471<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.0658061, -0.245592, 0.245592, 0.0658061< GT =
0 0.00705308 0 0.392947 0 -0.29471 0 ij -0.10529 yz jj zz jj 0 zz -0.0658061 0 -0.245592 0 0.245592 0 0.0658061 jj zz j z -0.245592 0.00705308 0.245592 0.392947 0.0658061 -0.29471 { k -0.0658061 -0.10529 ij 0 jj jj 0 S = jjjj jj 0 jj k0
0 0 0 0
0 0y zz 0 0 zzzz zz 0 0 zzzz z 0 0{
130
Geometric Nonlinearity
BT =
ij -0.10529 jj jj -0.0658061 jj jj jj 0 jj k0
0.392947 0
-0.29471
-0.245592
0.245592 0
0.0658061 0
-0.10529
0
-0.0658061 0
0
0
0
227.482
127.487
589.712
-1925.05 -848.976 1281.75
31.7819
294.021
244.026
682.478
-848.976 -1097.3
120.802
244.026
596.202
-56.8706 -475.788 -910.717 -247.901 723.562
682.478
-56.8706 2574.83
377.468
-2200.83 -2547.04 1667.99
-710.266
-848.976 -475.788 -2200.83 7184.4
3168.42
-4783.55 -118.612
4095.18
-1408.73 -450.839
-1097.3
-910.717 -2547.04 3168.42
377.468
-247.901 1667.99
120.802
723.562
-4783.55 -1408.73 3749.71
-636.732
-710.266 -118.612 -450.839 -636.732 1040.3
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0y zz 0 zzzz zz 0 zzzz zz 0 zzzz zz 0 zzz zz 0 zzzz zz 0 zzzz z 0{
227.482
127.487
589.712
-1925.05 -848.976 1281.75
31.7819
294.021
244.026
682.478
-848.976 -1097.3
120.802
244.026
596.202
-56.8706 -475.788 -910.717 -247.901 723.562
682.478
-56.8706 2574.83
ij 515.816 jj jj 227.482 jj jj jj 127.487 jj jj jj 589.712 k = jjj jjj -1925.05 jj jjj -848.976 jj jj jj 1281.75 jj k 31.7819
ij 0 yz jj zz jj 0 zz jj zz jj zz jj 0 zz jjj zzz jj 0 zz j z r i = jjj zzz jj 0 zz jj zz jj zz jjj 0 zzz jj zz jjj 0 zzz jj zz k0{
0.392947 0 0.245592 0
0.00705308 0
ij 515.816 jj jj 227.482 jj jj jj 127.487 jj jj jj 589.712 k c = jjj jjj -1925.05 jj jj -848.976 jj jj jj 1281.75 jj j k 31.7819
ij 0 jj jj 0 jj jj jjj 0 jj jj 0 j k s = jjj jj 0 jj jj jj 0 jj jj jj 0 jj k0
0.00705308 0 -0.245592
yz zz zz zz z -0.29471 zzzz z 0.0658061 {
0 0
377.468
-2200.83 -2547.04 1667.99
-710.266
-848.976 -475.788 -2200.83 7184.4
3168.42
-4783.55 -118.612
4095.18
-1408.73 -450.839
-1097.3
-910.717 -2547.04 3168.42
377.468
-247.901 1667.99
120.802
723.562
-4783.55 -1408.73 3749.71
-636.732
-710.266 -118.612 -450.839 -636.732 1040.3
yz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz z {
yz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz z {
131
After summing contributions from all points, the element equations as follows: ij 12686.4 jj jj 4817.77 jj jj jj -9199.71 jj jj jj 2027.95 k = jjj jjj -7812.64 jj jjj -4481.65 jj jj jj 4325.91 jj k -2364.07
r =H0 T
4817.77
-9199.71 2027.95
7409.29
-1243.66 290.187
-1243.66 10057.9 290.187
-3249.2
-7812.64 -4481.65 4325.91 -4481.65 -4216.11 907.534
-3249.2
2701.31
7492.2
-2476.11 -4547.75 3697.36
795.493
-3559.55
-4481.65 2701.31
-2476.11 14311.
4929.82
-9199.71
-4216.11 795.493
-4547.75 4929.82
8473.68
-1243.66
907.534
-3559.55 3697.36
-3483.36 3697.36
0 0 0 0
0 0 0L
-9199.71 -1243.66 8433.35
-3234.63 2027.95
290.187
-3361.24
Complete element equations for element 1 ij 12686.4 jj jj 4817.77 jj jj jj -9199.71 jj jj jj 2027.95 jj jj jj -7812.64 jj jj -4481.65 jj jj jj 4325.91 jj j k -2364.07
4817.77
-9199.71 2027.95
7409.29
-1243.66 290.187
-1243.66 10057.9 290.187
-3249.2
-7812.64 -4481.65 4325.91 -4481.65 -4216.11 907.534
-3249.2
2701.31
7492.2
-2476.11 -4547.75 3697.36
795.493
-3559.55
-4481.65 2701.31
-2476.11 14311.
4929.82
-9199.71
-4216.11 795.493
-4547.75 4929.82
8473.68
-1243.66
907.534
-3559.55 3697.36
-3483.36 3697.36
-9199.71 -1243.66 8433.35
-3234.63 2027.95
290.187
-3361.24
The element contributes to 81, 2, 3, 4, 5, 6, 7, 8< global degrees of freedom. ij 1 yz jj zz jj 2 zz jjj zzz jj 3 zz jjj zzz jj zz jj 4 zz Locations for element contributions to a global vector: jjj zzz jjj 5 zzz jj zz jjj 6 zzz jj zz jj zz jj 7 zz jj zz k8{
-2364.07 y zz -3483.36 zzzz zz 3697.36 zzzz zz -3234.63 zzzz zz 2027.95 zzz zz 290.187 zzzz zz -3361.24 zzzz z 6427.81 {
-2364.07 y i Du1 y i 0. y zz jj zz jj zz -3483.36 zzzz jjjj Dv1 zzzz jjjj 0. zzzz zz jj zz jj zz 3697.36 zzzz jjjj Du2 zzzz jjjj 0. zzzz zz jj zz jj zz -3234.63 zzzz jjjj Dv2 zzzz jjjj 0. zzzz zz jj zz = jj zz 2027.95 zzz jjj Du3 zzz jjj 0. zzz zz jj zz jj zz 290.187 zzzz jjjj Dv3 zzzz jjjj 0. zzzz zz jj zz jj zz -3361.24 zzzz jjjj Du4 zzzz jjjj 0. zzzz zj z j z 6427.81 { k Dv4 { k 0. {
132
Geometric Nonlinearity
@1, 1D jij jj @2, 1D jj jj jj @3, 1D jj jj jj @4, 1D j and to a global matrix: jjj jj @5, 1D jj jj jj @6, 1D jj jj jj @7, 1D jj k @8, 1D
@1, 2D
@1, 3D
@1, 4D
@1, 5D
@1, 6D
@3, 3D
@3, 4D
@3, 5D
@3, 6D
@5, 3D
@5, 4D
@5, 5D
@5, 6D
@2, 2D
@2, 3D
@4, 2D
@4, 3D
@3, 2D
@5, 2D
@6, 2D
@7, 2D
@8, 2D
@6, 3D
@7, 3D
@8, 3D
@2, 4D
@4, 4D
@6, 4D
@7, 4D
@8, 4D
@2, 5D
@4, 5D
@6, 5D
@7, 5D
@8, 5D
@2, 6D
@4, 6D
@6, 6D
@7, 6D
@8, 6D
@1, 7D
@1, 8D y zz @2, 7D @2, 8D zzzz zz @3, 7D @3, 8D zzzz zz @4, 7D @4, 8D zzzz zz @5, 7D @5, 8D zzz zz @6, 7D @6, 8D zzzz zz @7, 7D @7, 8D zzzz z @8, 7D @8, 8D {
Adding element equations into appropriate locations we have 12686.4 jij jj 4817.77 jj jj jj -9199.71 jj jj jj 2027.95 jj jj jj -7812.64 jj jj jj -4481.65 jj jj jj 4325.91 jj jj jj -2364.07 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
4817.77
-9199.71 2027.95
7409.29
-1243.66 290.187
-1243.66 10057.9 290.187
-3249.2
-7812.64 -4481.65 4325.91
-2364.07 0
-4481.65 -4216.11 907.534
-3483.36 0
-3249.2
2701.31
7492.2
-2476.11 -4547.75 3697.36
795.493
0
-3234.63 0
-4481.65 2701.31
-2476.11 14311.
4929.82
-9199.71 2027.95
0
-4216.11 795.493
-4547.75 4929.82
8473.68
-1243.66 290.187
0
907.534
-3559.55 3697.36
-9199.71 -1243.66 8433.35
-3361.24 0
-3483.36 3697.36
-3234.63 2027.95
290.187
-3361.24 6427.81
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
ij Du1 yz ij 0 yz z j jj z jj Dv1 zzz jjj 0 zzz zz jjj zzz jjj jj Du zz jj 0 zzz jj 2 zz jj zz jj z j z jj Dv zzz jjj 0 zzz 2 zz jjj zzz jjj zz jj zz jj z jjj Du3 zzz jjj 0 zzz jj zz jj zz jjj Dv3 zzz jjj 0 zzz jj zz = jj zz jjj Du4 zzz jjj 0 zzz zz jj zz jj jjj Dv4 zzz jjj 0 zzz zz jj zz jj jjj Du zzz jjj 0 zzz zz jj 5 zz jj zz jjj zz jj jj Dv5 zzz jjj 0 zzz zz jjj zzz jjj z jjj Du6 zzz jjj 0 zzz zz jj zz jj k Dv6 { k -1 { E = 10600.;
-3559.55 3697.36
n = 0.35;
Initial thickness = 1.25
0 0 0y zz 0 0 0 zzzz zz 0 0 0 zzzz zz 0 0 0 zzzz z 0 0 0 zzzz zz 0 0 0 zzzz zz 0 0 0 zzzz zz 0 0 0 zzz zz 0 0 0 zzzz zz 0 0 0 zzzz zz 0 0 0 zzz zz 0 0 0{
133
g = 0.461538
Plane stress analysis.
Interpolation functions and their derivatives 1 1 1 1 NT = : ÅÅÅÅÅ Hs - 1L Ht - 1L, - ÅÅÅÅÅ Hs + 1L Ht - 1L, ÅÅÅÅÅ Hs + 1L Ht + 1L, - ÅÅÅÅÅ Hs - 1L Ht + 1L> 4 4 4 4
t-1 1-t t+1 1 ∑NT ê∑s = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-t - 1L> 4 4 4 4
s -1 1 s +1 1-s ∑NT ê∑t = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-s - 1L, ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ > 4 4 4 4
Mapping to the master element
ij ÅÅÅÅ52Å jj jj j 10 Initial coordinates = jjjj jj 10 jj j k0
3 yz zz z 3 zzzz zz 4 zzz zz 4{
5ts 35 s 5t 45 x0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 8 8 8 8
t 7 y0 Hs,tL = ÅÅÅÅÅÅ + ÅÅÅÅÅ 2 2 5t 35 ji ÅÅÅÅ8ÅÅÅ + ÅÅÅÅ8ÅÅ J = jjj j0 k
5s ÅÅÅÅ ÅÅÅÅ - ÅÅÅÅ58Å zy 8 zz zz ÅÅ12ÅÅ {
Current configuration
5 jij ÅÅÅÅ2Å jj jj 10 Updated coordinates = jjjj jj 10 jj j k0
5t 35 detJ = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 16 16
3 zy zz z 3 zzzz zz 4 zzz zz 4{
5ts 35 s 5t 45 xHs,tL = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 8 8 8 8 t 7 yHs,tL = ÅÅÅÅÅÅ + ÅÅÅÅÅ 2 2
5 Ht+7L ij ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ 16 Deformation gradient, F p = 1êdetJ jjjj k0
Gauss quadrature points and weights
yz zz z 5 Ht+7L z ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ { 16
0
134
Geometric Nonlinearity
Point
Weight
1
s Ø -0.57735 t Ø -0.57735
1.
2
s Ø -0.57735 t Ø 0.57735
1.
3
s Ø 0.57735 t Ø -0.57735
1.
4
s Ø 0.57735 t Ø 0.57735
1.
Computation of element matrices at 8-0.57735, -0.57735< with weight = 1. i 4.01416 J = jjjj k0
-0.985844 y zz zz ÅÅ12ÅÅ {
i 0.249118 0 yz z J -T = jj k 0.491184 2. {
i1 Deformation gradient, F p = jj k0
detJ = 2.00708
0y zz; 1{
1 0 0 0 jij j j F = jj 0 0 0 1 jj k0 1 1 0 _
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
zyz zz zz zz {
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0982367, 0.0982367, 0.0263225, -0.0263225<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.982367, -0.0176327, 0.263225, 0.736775<
0.0982367 0 0.0263225 0 -0.0263225 ij -0.0982367 0 j -0.982367 0 -0.0176327 0 0.263225 0 GT = jjjj 0 jj 0.263225 0.0263225 0.736775 k -0.982367 -0.0982367 -0.0176327 0.0982367
ij 0 jj jj 0 S = jjjj jj 0 jj k0
0 0 0 0
0 0y zz 0 0 zzzz zz 0 0 zzzz z 0 0{
ij -0.0982367 jj jj -0.982367 T B = jjjj jj 0 jj k0
0
0.0982367
0
0
-0.0176327 0
0.0263225 0
-0.0263225 0
0.263225
0
0.736775
0
0.0263225 0
-0.0982367 0
0.0982367
-0.982367
-0.0176327 0
0
0.263225
0
0 0
135
9917.16 jij jj 3168.42 jj jj jj -241.283 jj jj jj -910.717 j k c = jjj jj -2657.29 jj jj jj -848.976 jj jj jj -7018.58 jj k -1408.73
ij 0 jj jj 0 jj jj jj 0 jj jj jj 0 k s = jjj jjj 0 jj jj 0 jj jj jj 0 jj j k0
3168.42 41284.5 -2200.83 644.265 -848.976 -11062.2 -118.612 -30866.6
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
ij 9917.16 jj jj 3168.42 jj jj jjj -241.283 jj jj -910.717 j k = jjj jj -2657.29 jj jj jj -848.976 jj jj jj -7018.58 jj k -1408.73
ij 0 yz jj zz jj 0 zz jjj zzz jj 0 zz jjj zzz jj zz jj 0 zz r i = jjj zzz jjj 0 zzz jj zz jjj 0 zzz jj zz jjj 0 zzz jj zz j z k0{
3168.42 41284.5 -2200.83 644.265 -848.976 -11062.2 -118.612 -30866.6
-241.283 -910.717 -2657.29 -848.976 -7018.58 -1408.73 y zz -2200.83 644.265 -848.976 -11062.2 -118.612 -30866.6 zzzz zz 414.957 -56.8706 64.6516 589.712 -238.326 1667.99 zzzz zz -56.8706 108.323 244.026 -172.63 723.562 -579.958 zzzz zz 64.6516 244.026 712.02 227.482 1880.62 377.468 zzz zz 589.712 -172.63 227.482 2964.1 31.7819 8270.69 zzzz zz -238.326 723.562 1880.62 31.7819 5376.28 -636.732 zzzz z 1667.99 -579.958 377.468 8270.69 -636.732 23175.9 { 0y zz 0 zzzz zz 0 zzzz zz 0 zzzz zz 0 zzz zz 0 zzzz zz 0 zzzz z 0{
-241.283 -910.717 -2657.29 -848.976 -7018.58 -1408.73 y zz -2200.83 644.265 -848.976 -11062.2 -118.612 -30866.6 zzzz zz 414.957 -56.8706 64.6516 589.712 -238.326 1667.99 zzzz zz -56.8706 108.323 244.026 -172.63 723.562 -579.958 zzzz zz 64.6516 244.026 712.02 227.482 1880.62 377.468 zzz zz 589.712 -172.63 227.482 2964.1 31.7819 8270.69 zzzz zz -238.326 723.562 1880.62 31.7819 5376.28 -636.732 zzzz z 1667.99 -579.958 377.468 8270.69 -636.732 23175.9 {
Computation of element matrices at 8-0.57735, 0.57735< with weight = 1. i 4.73584 J = jjjj k0
-0.985844 y zz zz ÅÅ12ÅÅ {
detJ = 2.36792
136
Geometric Nonlinearity
i 0.211156 0 yz J -T = jj z k 0.416333 2. {
i1 Deformation gradient, F p = jj k0
ij 1 0 0 0 _ j F = jjjj 0 0 0 1 jj k0 1 1 0
0y zz; 1{
8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
yz zz zz zz z {
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0223112, 0.0223112, 0.0832666, -0.0832666<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.832666, -0.167334, 0.375501, 0.624499<
0.0223112 0 0.0832666 0 -0.0832666 0 -0.0223112 0 jij j j G = jj 0 -0.832666 0 -0.167334 0 0.375501 0 0.6 jj -0. k -0.832666 -0.0223112 -0.167334 0.0223112 0.375501 0.0832666 0.624499 T
0 jij jj 0 j S = jjjj jj 0 jjj k0
0 0 0 0
0 0y zz 0 0 zzzz zz 0 0 zzzz zz 0 0{
ij -0.0223112 jj jj -0.832666 B T = jjjj jjj 0 jj k0 ij 8081.84 jj jj 719.602 jj jj jj 1594.04 jj jj jj -114.652 k c = jjj jjj -3726.85 jj jj -1032.84 jj jj jj -5949.03 jj j k 427.886
0
0.0223112
0
-0.167334 0
0
0.0832666 0
-0.0832666 0
0.375501
0
0.624499
0
0.0832666 0
-0.
0.375501
0.62
-0.0223112 0
0.0223112
-0.832666
-0.167334 0
0
0
0
yz z -26162.9 zzzz zz -460.337 350.444 -144.613 -636.606 65.2487 -1307.88 539.701 zzzz zz 7010.33 -144.613 1415.76 -280.437 -3142.41 539.701 -5283.68 zzzz z -1977.27 -636.606 -280.437 1987.61 1211.1 2375.85 1046.61 zzzz zz -15765.9 65.2487 -3142.41 1211.1 7180.64 -243.511 11727.6 zzzz zz 1718. -1307.88 539.701 2375.85 -243.511 4881.06 -2014.19 zzzz z -26162.9 539.701 -5283.68 1046.61 11727.6 -2014.19 19718.9 { -114.652 -3726.85 -1032.84 -5949.03 427.886
719.602
1594.04
34918.5
-460.337 7010.33
-1977.27 -15765.9 1718.
137
0 jij jj 0 jj jj jj 0 jj jj jj 0 j k s = jjj jj 0 jj jj jj 0 jj jj jj 0 jj k0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0y zz 0 zzzz zz 0 zzzz zz 0 zzzz zz 0 zzz zz 0 zzzz zz 0 zzzz z 0{
719.602
1594.04
34918.5
-460.337 7010.33
ij 8081.84 jj jj 719.602 jj jj jj 1594.04 jj jj jj -114.652 k = jjj jjj -3726.85 jj jjj -1032.84 jj jj jj -5949.03 jj k 427.886
ij 0 yz jj zz jj 0 zz jj zz jj zz jjj 0 zzz jj zz jjj 0 zzz r i = jjj zzz jj 0 zz jj zz jjj 0 zzz jj zz jjj zzz jj 0 zz jj zz k0{
yz z -26162.9 zzzz zz -460.337 350.444 -144.613 -636.606 65.2487 -1307.88 539.701 zzzz zz 7010.33 -144.613 1415.76 -280.437 -3142.41 539.701 -5283.68 zzzz zz -1977.27 -636.606 -280.437 1987.61 1211.1 2375.85 1046.61 zzz zz -15765.9 65.2487 -3142.41 1211.1 7180.64 -243.511 11727.6 zzzz zz 1718. -1307.88 539.701 2375.85 -243.511 4881.06 -2014.19 zzzz z -26162.9 539.701 -5283.68 1046.61 11727.6 -2014.19 19718.9 { -114.652 -3726.85 -1032.84 -5949.03 427.886 -1977.27 -15765.9 1718.
Computation of element matrices at 80.57735, -0.57735< with weight = 1. i 4.01416 J = jjjj k0
-0.264156 y zz zz ÅÅ12ÅÅ {
i 0.249118 0 yz z J -T = jj k 0.131612 2. {
i1 Deformation gradient, F p = jj k0
detJ = 2.00708
0y zz; 1{
ij 1 0 0 0 yz _ j z F = jjjj 0 0 0 1 zzzz jj zz k0 1 1 0 {
8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
138
Geometric Nonlinearity
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0982367, 0.0982367, 0.0263225, -0.0263225<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.263225, -0.736775, 0.802582, 0.197418<
0.0982367 0 0.0263225 0 -0.0263225 0 -0.0982367 0 jij j j G = jj 0 -0.263225 0 -0.736775 0 0.802582 0 0.1 jj -0. k -0.263225 -0.0982367 -0.736775 0.0982367 0.802582 0.0263225 0.197418 T
0 jij jj 0 j S = jjjj jj 0 jj k0
0 0 0 0
0 0y zz 0 0 zzzz zz 0 0 zzzz z 0 0{
ij -0.0982367 jj jj -0.263225 B T = jjjj jjj 0 jj k0 ij 1094.34 jj jj 848.976 jj jj jj 1498.3 jj jj jj 1408.73 k c = jjj jj -2191.17 jj jj jjj -1880.24 jj jj -401.469 jj j k -377.468
ij 0 jj jj 0 jj jj jjj 0 jj jj 0 j k s = jjj jj 0 jj jj jj 0 jj jj jj 0 jj k0
0
0.0982367
0
-0.736775 0
0
0.0263225 0
-0.0263225 0
0.802582
0
0.197418
0
0.0263225 0
-0.
0.802582
0.19
-0.0982367 0
0.0982367
-0.263225
-0.736775 0
0
0
0
-2191.17 -1880.24 -401.469 -377.468 y zz -31.7819 -2192.48 zzzz zz 118.612 5758.61 -2376.32 -5713.89 1620.97 -1543.01 636.732 zzzz zz 8182.46 -2376.32 23264.1 330.856 -25213. 636.732 -6233.61 zzzz zz -935.806 -5713.89 330.856 6374.03 693.603 1531.03 -88.6525 zzz zz zz -9042.3 1620.97 -25213. 693.603 27499.5 -434.338 6755.8 zz zz -31.7819 -1543.01 636.732 1531.03 -434.338 413.449 -170.612 zzzz z -2192.48 636.732 -6233.61 -88.6525 6755.8 -170.612 1670.29 { 848.976
1498.3
1408.73
3052.32
118.612
8182.46
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0y zz 0 zzzz zz 0 zzzz zz 0 zzzz zz 0 zzz zz 0 zzzz zz 0 zzzz z 0{
-935.806 -9042.3
139
1094.34 jij jj 848.976 jj jj jj 1498.3 jj jj jj 1408.73 j k = jjj jj -2191.17 jj jj jj -1880.24 jj jj jj -401.469 jj k -377.468
ij 0 yz jj zz jj 0 zz jjj zzz jj 0 zz jjj zzz jj zz jj 0 zz r i = jjj zzz jjj 0 zzz jj zz jjj 0 zzz jj zz jj zz jj 0 zz jj zz k0{
-2191.17 -1880.24 -401.469 -377.468 y zz 3052.32 118.612 8182.46 -935.806 -9042.3 -31.7819 -2192.48 zzzz zz 118.612 5758.61 -2376.32 -5713.89 1620.97 -1543.01 636.732 zzzz zz 8182.46 -2376.32 23264.1 330.856 -25213. 636.732 -6233.61 zzzz zz -935.806 -5713.89 330.856 6374.03 693.603 1531.03 -88.6525 zzz zz zz -9042.3 1620.97 -25213. 693.603 27499.5 -434.338 6755.8 zz zz -31.7819 -1543.01 636.732 1531.03 -434.338 413.449 -170.612 zzzz z -2192.48 636.732 -6233.61 -88.6525 6755.8 -170.612 1670.29 { 848.976
1498.3
1408.73
Computation of element matrices at 80.57735, 0.57735< with weight = 1. i 4.73584 J = jjjj k0
-0.264156 y zz zz ÅÅ12ÅÅ {
i 0.211156 0 yz J -T = jj z k 0.111556 2. {
i1 Deformation gradient, F p = jj k0
detJ = 2.36792
0y zz; 1{
1 0 0 0 jij j j F = jj 0 0 0 1 jj k0 1 1 0 _
8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
zyz zz zz zz {
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0223112, 0.0223112, 0.0832666, -0.0832666<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.223112, -0.776888, 0.832666, 0.167334<
0.0223112 0 0.0832666 0 -0.0832666 0 ij -0.0223112 0 jj j G = jj 0 -0.223112 0 -0.776888 0 0.832666 0 0.1 jj -0.223112 -0.0223112 -0.776888 0.0223112 0.832666 0.0832666 0.167334 -0. k T
140
Geometric Nonlinearity
0 jij jj 0 j S = jjjj jj 0 jj k0
0 0 0 0
0 0y zz 0 0 zzzz zz 0 0 zzzz z 0 0{
ij -0.0223112 jj jj -0.223112 T B = jjjj jj 0 jj k0 603.516 jij jj 192.817 jj jj jj 1989.13 jj jj jj 412.133 j k c = jjj jjj -2252.35 jj jj -719.602 jj jj jj -340.289 jj j k 114.652
ij 0 jj jj 0 jj jj jj 0 jj jj jj 0 k s = jjj jj 0 jj jj jjj 0 jj jj 0 jj j k0
0
0.0223112
0
-0.776888 0
0
0.0832666 0
-0.0832666 0
0.832666
0
0.167334
0
0.0832666 0
-0.
0.832666
0.16
-0.0223112 0
0.0223112
-0.223112
-0.776888 0
0
0
zyz -1858.37 zzzz zz 66.4476 7038.59 -671.398 -7423.52 -247.986 -1604.19 852.936 zzzz zz 8722.38 -671.398 30397.7 -1538.1 -32552.4 1797.37 -6567.71 zzzz zz -719.602 -7423.52 -1538.1 8405.9 2685.59 1269.98 -427.886 zzz zz -9376.41 -247.986 -32552.4 2685.59 34993.2 -1718. 6935.55 zzzz zz 460.337 -1604.19 1797.37 1269.98 -1718. 674.505 -539.701 zzzz z -1858.37 852.936 -6567.71 -427.886 6935.55 -539.701 1490.54 { 192.817
1989.13
412.133
-2252.35 -719.602 -340.289 114.652
2512.4
66.4476
8722.38
-719.602 -9376.41 460.337
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0y zz 0 zzzz zz 0 zzzz zz 0 zzzz zz 0 zzz zz 0 zzzz zz 0 zzzz z 0{
192.817
1989.13
412.133
-2252.35 -719.602 -340.289 114.652
2512.4
66.4476
8722.38
-719.602 -9376.41 460.337
603.516 jij jj 192.817 jj jj jj 1989.13 jj jj jj 412.133 j k = jjj jjj -2252.35 jj jjj -719.602 jj jj jj -340.289 jj k 114.652
0
zyz -1858.37 zzzz zz 66.4476 7038.59 -671.398 -7423.52 -247.986 -1604.19 852.936 zzzz zz 8722.38 -671.398 30397.7 -1538.1 -32552.4 1797.37 -6567.71 zzzz zz -719.602 -7423.52 -1538.1 8405.9 2685.59 1269.98 -427.886 zzz zz -9376.41 -247.986 -32552.4 2685.59 34993.2 -1718. 6935.55 zzzz zz 460.337 -1604.19 1797.37 1269.98 -1718. 674.505 -539.701 zzzz z -1858.37 852.936 -6567.71 -427.886 6935.55 -539.701 1490.54 {
141
0 jij zyz jj 0 zz jj zz jjj zzz jj 0 zz jj zz jj zz jj 0 zz r i = jjj zzz jj 0 zz jj zz jj zz jj 0 zz jj zz jj zz jj 0 zz jj zz k0{
After summing contributions from all points, the element equations as follows: ij 19696.9 jj jj 4929.82 jj jj jj 4840.18 jj jj jj 795.493 k = jjj jj -10827.7 jj jj jjj -4481.65 jj jj -13709.4 jj j k -1243.66
rT = H 0
-10827.7 -4481.65 -13709.4 -1243.66 y zz -61080.4 zzzz zz -2476.11 13562.6 -3249.2 -13709.4 2027.95 -4693.41 3697.36 zzzz zz 24559.4 -3249.2 55185.9 -1243.66 -61080.4 3697.36 -18665. zzzz zz -4481.65 -13709.4 -1243.66 17479.6 4817.77 7057.48 907.534 zzz zz -45246.7 2027.95 -61080.4 4817.77 72637.5 -2364.07 33689.7 zzzz zz 2027.95 -4693.41 3697.36 7057.48 -2364.07 11345.3 -3361.24 zzzz z -61080.4 3697.36 -18665. 907.534 33689.7 -3361.24 46055.7 { 4929.82
4840.18
81767.7
-2476.11 24559.4
0 0 0 0
795.493
-4481.65 -45246.7 2027.95
0 0 0L
Complete element equations for element 2 ij 19696.9 jj jj 4929.82 jj jj jj 4840.18 jj jj jj 795.493 jj jj jjj -10827.7 jj jjj -4481.65 jj jj -13709.4 jj j k -1243.66
-10827.7 -4481.65 -13709.4 -1243.66 y i Du3 y i 0. y zz jj zz jj zz -61080.4 zzzz jjjj Dv3 zzzz jjjj 0. zzzz zz jj zz jj zz -2476.11 13562.6 -3249.2 -13709.4 2027.95 -4693.41 3697.36 zzzz jjjj Du5 zzzz jjjj 0. zzzz zz jj zz jj zz 24559.4 -3249.2 55185.9 -1243.66 -61080.4 3697.36 -18665. zzzz jjjj Dv5 zzzz jjjj 0. zzzz zz jj zz = jj zz -4481.65 -13709.4 -1243.66 17479.6 4817.77 7057.48 907.534 zzz jjj Du6 zzz jjj 0. zzz zz jj zz jj zz -45246.7 2027.95 -61080.4 4817.77 72637.5 -2364.07 33689.7 zzzz jjjj Dv6 zzzz jjjj 0. zzzz zz jj zz jj zz 2027.95 -4693.41 3697.36 7057.48 -2364.07 11345.3 -3361.24 zzzz jjjj Du4 zzzz jjjj 0. zzzz zj z j z -61080.4 3697.36 -18665. 907.534 33689.7 -3361.24 46055.7 { k Dv4 { k 0. { 4929.82
4840.18
81767.7
-2476.11 24559.4
795.493
-4481.65 -45246.7 2027.95
The element contributes to 85, 6, 9, 10, 11, 12, 7, 8< global degrees of freedom.
142
Geometric Nonlinearity
5 jij jj 6 jj jj jj 9 jj jj jj 10 j Locations for element contributions to a global vector: jjj jj 11 jj jj jj 12 jj jj jj 7 jj k8
ij @5, 5D jj jj @6, 5D jj jj jj @9, 5D jj jj jj @10, 5D and to a global matrix: jjj jjj @11, 5D jj jjj @12, 5D jj jj jj @7, 5D jj k @8, 5D
@5, 6D @6, 6D
@9, 6D
@5, 9D
@5, 10D
@9, 9D
@9, 10D
@6, 9D
@6, 10D
@10, 6D @10, 9D @10, 10D
@11, 6D @11, 9D @11, 10D
zyz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz {
@5, 11D
@5, 12D
@9, 11D
@9, 12D
@6, 11D
@10, 11D
@11, 12D
@7, 11D
@7, 12D
@12, 11D
@8, 6D
@8, 11D
@7, 9D @8, 9D
@7, 10D
@8, 10D
@10, 12D
@11, 11D
@12, 6D @12, 9D @12, 10D @7, 6D
@6, 12D
@12, 12D @8, 12D
@5, 7D
@5, 8D y zz @6, 8D zzzz zz @9, 7D @9, 8D zzzz zz @10, 7D @10, 8D zzzz zz @11, 7D @11, 8D zzz zz @12, 7D @12, 8D zzzz zz @7, 7D @7, 8D zzzz z @8, 7D @8, 8D {
@6, 7D
Adding element equations into appropriate locations we have ij 12686.4 jj jj 4817.77 jj jj jj -9199.71 jj jj jj 2027.95 jj jj jjj -7812.64 jj jjj -4481.65 jj jj 4325.91 jj jj jj -2364.07 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj k0
4817.77
-9199.71 2027.95
-7812.64 -4481.65 4325.91
-2364.07 0
0
7409.29
-1243.66 290.187
-4481.65 -4216.11 907.534
-3483.36 0
0
-1243.66 10057.9 290.187
-3249.2
-3249.2
2701.31
7492.2
-2476.11 -4547.75 3697.36
795.493
-3559.55 3697.36
0
0
-3234.63 0
0
-4481.65 2701.31
-2476.11 34007.9
9859.63
-22909.1 784.289
-4216.11 795.493
-4547.75 9859.63
90241.4
784.289
-60790.2 -2476.11 2455
-22909.1 784.289
19778.6
-6722.48 -4693.41 3697
907.534
-3559.55 3697.36
4840.18
795.
-3483.36 3697.36
-3234.63 784.289
-60790.2 -6722.48 52483.5
3697.36
-18
0
0
0
4840.18
-2476.11 -4693.41 3697.36
13562.6
-32
0
0
0
795.493
24559.4
3697.36
-18665.
-3249.2
5518
0
0
0
-10827.7 -4481.65 7057.48
907.534
-13709.4 -12
0
0
0
-4481.65 -45246.7 -2364.07 33689.7
After assembly of all elements the global matrices are as follows.
2027.95
-61
143
12686.4 jij jj 4817.77 jj jj jj -9199.71 jj jj jj 2027.95 jj jj jj -7812.64 jj jj jj -4481.65 K T = jjjj jj 4325.91 jj jj jj -2364.07 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0 ij 0. yz z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz z jj jj 0. zzz zz j zz; RE = jjjj jj 0. zzz zz jj z jj jj 0. zzz z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z j k -1. {
4817.77
-9199.71 2027.95
-7812.64 -4481.65 4325.91
-2364.07 0
7409.29
-1243.66 290.187
-4481.65 -4216.11 907.534
-3483.36 0
-1243.66 10057.9 290.187
-3249.2
-3249.2
2701.31
7492.2
-2476.11 -4547.75 3697.36
795.493
-3559.55 3697.36
0
-3234.63 0
-4481.65 2701.31
-2476.11 34007.9
9859.63
-22909.1 784.289
-4216.11 795.493
-4547.75 9859.63
90241.4
784.289
-60790.2 -
-22909.1 784.289
19778.6
-6722.48 -
907.534
-3559.55 3697.36
-3483.36 3697.36
-3234.63 784.289
0
0
0
0
0
0
0
0
0
0
ij 0 yz jj zz jj 0 zz jjj zzz jj 0 zz jjj zzz jj zz jjj 0 zzz jj zz jjj 0 zzz jj zz jjj 0 zzz RI = jjjj zzzz; jj 0 zz jj zz jj zz jj 0 zz jjj zzz jj 0 zz jjj zzz jj 0 zz jj zz jj zz jj 0 zz jj zz j z k0{
4
-60790.2 -6722.48 52483.5
3
4840.18
-2476.11 -4693.41 3697.36
1
795.493
24559.4
3697.36
-18665.
-
0
-10827.7 -4481.65 7057.48
907.534
-
0
-4481.65 -45246.7 -2364.07 33689.7
2
ij 0. yz z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz z jj jj 0. zzz zz j zz R = RE - RI = jjjj jj 0. zzz zz jj z jj jj 0. zzz zz jjj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj k -1. {
144
Geometric Nonlinearity
System of equations 12686.4 jij jj 4817.77 jj jj jj -9199.71 jj jj jj 2027.95 jj jj jj -7812.64 jj jj jj -4481.65 jj jj jj 4325.91 jj jj jj -2364.07 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
4817.77
-9199.71 2027.95
7409.29
-1243.66 290.187
-1243.66 10057.9 290.187
-3249.2
-7812.64 -4481.65 4325.91
-2364.07 0
-4481.65 -4216.11 907.534
-3483.36 0
-3249.2
2701.31
7492.2
-2476.11 -4547.75 3697.36
795.493
-3559.55 3697.36
0
-3234.63 0
-4481.65 2701.31
-2476.11 34007.9
9859.63
-22909.1 784.289
-4216.11 795.493
-4547.75 9859.63
90241.4
784.289
-60790.2 -2476.
-22909.1 784.289
19778.6
-6722.48 -4693.
907.534
-3559.55 3697.36
4840.18
-3483.36 3697.36
-3234.63 784.289
-60790.2 -6722.48 52483.5
0
0
0
4840.18
-2476.11 -4693.41 3697.36
0
0
0
795.493
24559.4
3697.36
-18665.
-3249.
0
0
0
-10827.7 -4481.65 7057.48
907.534
-13709
0
0
0
-4481.65 -45246.7 -2364.07 33689.7
2027.95
Essential boundary conditions Node
dof
Value
1
Du1 Dv1
0 0
2
Du2 Dv2
0 0
Remove 81, 2, 3, 4< rows and columns. After adjusting for essential boundary conditions we have
3697.36 13562.6
145
34007.9 jij jj 9859.63 jj jj jj -22909.1 jj jj jj 784.289 jj jj jj 4840.18 jj jj jj 795.493 jj jj jj -10827.7 jj k -4481.65
-10827.7 -4481.65 y zz 90241.4 784.289 -60790.2 -2476.11 24559.4 -4481.65 -45246.7 zzzz zz 784.289 19778.6 -6722.48 -4693.41 3697.36 7057.48 -2364.07 zzzz zz -60790.2 -6722.48 52483.5 3697.36 -18665. 907.534 33689.7 zzzz zz -2476.11 -4693.41 3697.36 13562.6 -3249.2 -13709.4 2027.95 zzz zz 24559.4 3697.36 -18665. -3249.2 55185.9 -1243.66 -61080.4 zzzz zz -4481.65 7057.48 907.534 -13709.4 -1243.66 17479.6 4817.77 zzzz z -45246.7 -2364.07 33689.7 2027.95 -61080.4 4817.77 72637.5 { 9859.63
-22909.1 784.289
4840.18
795.493
Du 0 y jij 3 zyz jij z jj Dv zz jj 0 zzz jj 3 zz jj zz z j jj z jj Du zzz jjj 0 zzz jj 4 zz jj zz z j jj z jj Dv4 zzz jjj 0 zzz zz jj jj zz zz = jj jj z jj Du5 zz jj 0 zzz zz jj jj zz z j jj z jj Dv5 zzz jjj 0 zzz zz jj jj zz jjj Du zzz jjj 0 zzz jj 6 zz jj zz j z j z Dv -1. k 6{ k {
Solving the final system of global equations we get
8Du3 = 0.00106552, Dv3 = -0.000827007, Du4 = 0.00156468, Dv4 = 0.000898983, Du5 = 0.00128101, Dv5 = -0.00627425, Du6 = 0.00205808, Dv6 = -0.00627746<
Complete table of nodal values Du
Dv
1
0
0
2
0
0
3
0.00106552
-0.000827007
4
0.00156468
0.000898983
5
0.00128101
-0.00627425
6
0.00205808
-0.00627746
Total increments since the start of this load step
146
Geometric Nonlinearity
Du
Dv
1
0
0
2
0
0
3
0.00106552
-0.000827007
4
0.00156468
0.000898983
5
0.00128101
-0.00627425
6
0.00205808
-0.00627746
Total nodal values u
v
1
0
0
2
0
0
3
0.00106552
-0.000827007
4
0.00156468
0.000898983
5
0.00128101
-0.00627425
6
0.00205808
-0.00627746
Solution for element 1 Initial configuration
0 jij jj 5 jj ÅÅÅÅÅ j 2 Nodal coordinates = jjjj jj ÅÅÅÅ5Å jj 2 jj k0 5s 5 x0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4
0y zz z 0 zzzz zz zz 3 zzz zz z 4{
ts s 7t 7 y0 Hs,tL = - ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4 4 4 ij ÅÅ54ÅÅ J = jjjj t 1 k - ÅÅ4ÅÅ - ÅÅÅÅ4Å
Current configuration II
yz zz z 7 s z ÅÅÅÅ4Å - ÅÅ4ÅÅ {
0
35 5s detJ = ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ 16 16
ij 0 jj jj 0 Nodal displacements = jjjj jj 0.00106552 jj k 0.00156468
yz zz zz zz z -0.000827007 zzzz zz 0.000898983 {
0
0
147
0 jij jj 5 jj ÅÅÅÅÅ Updated coordinates = jjjj 2 jj 2.50107 jj j k 0.00156468
zyz zz zz zz zz 2.99917 zzz zz 4.0009 { 0 0
xII Hs,tL = -0.000124791t s + 1.24988 s + 0.000657549 t + 1.25066 yII Hs,tL = -0.250431 t s - 0.250431 s + 1.75002 t + 1.75002
i 1.24988 - 0.000124791t 0.000657549 - 0.000124791s yz J II = jj z k -0.250431 t - 0.250431 1.75002 - 0.250431 s { detJ II = -0.313039 s - 0.0000537159 t + 2.18747
i -0.3125 s - 0.0000539974 t + 2.18745 0.000821937 - 0.000155989 s Deformation gradient, F IIp = H1êdetJ L jj 2.18752 - 0.313039 s k -0.000750622 t - 0.000750622
Solution at 8s, t< = 8-0.57735, -0.57735< ï 8x, y< = 80.528621, 0.800754< 0.000729598 y i 1.24995 zz J II = jj { k -0.105845 1.8946
yz z {
detJ II = 2.36823
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.333097, 0.310751, 0.0892532, -0.0669064<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.208009, -0.0558898, 0.0557358, 0.208163< B TL
0.310751 0 0.0892532 0 -0.0669064 0 -0.333097 0 jij j j -0.208009 0 -0.0558898 0 0.0557358 0 0.20 = jj 0 jj -0.208009 -0.333097 -0.0558898 0.310751 0.0557358 0.0892532 0.208163 -0. k
0.000385147 y i 0.99999 zz; F IIp = jj { k -0.000133978 1.00014
Det@F IIp D = 1.00013
Element thickness, hII = 1.24991
0.000251224 y i 0.999981 zz Left Cauchy-Green tensor = jj 0.000251224 1.00028 k {
i -9.62896 µ 10-6 Green-Lagrange strain tensor, e = jjj k 0.000125573
i 0.480481 0.986226 yz z Cauchy stress tensor, s = jj k 0.986226 1.66284 { Principal stresses = H 2.2215 -0.0781818 L
0.000125573 yz zz 0.000141093 {
148
Geometric Nonlinearity
Effective stress Hvon MisesL = 2.2616
0.98558 y zz 1.66273 {
i 0.47976 Second PK stresses = jj k 0.98558
r Ti = H -1.08099 -1.99626 0.27881 0.632081 0.289652 0.534897 0.512534 0.829284 L
Solution at 8s, t< = 8-0.57735, 0.57735< ï 8x, y< = 80.529463, 2.98846< 0.000729598 yz i 1.2498 J II = jj z -0.395018 1.8946 k {
detJ II = 2.36817
8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.15031, 0.0669082, 0.333106, -0.249705<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.208079, -0.0557959, 0.0556419, 0.208233< B TL =
-0.15031 0 0.0669082 0 0.333106 0 -0.249705 0 jij zyz jj 0 -0.208079 0 -0.0557959 0 0.0556419 0 0.208233 zzzz jj jj zz 0.0556419 0.333106 0.208233 -0.249705 { k -0.208079 -0.15031 -0.0557959 0.0669082 0.000385147 y i 0.999964 zz; F IIp = jj k -0.000500014 1.00014 {
Det@F IIp D = 1.00011
Element thickness, hII = 1.24993
-0.000114795 y i 0.999928 zz Left Cauchy-Green tensor = jj -0.000114795 1.00028 { k
i -0.0000358438 -0.0000574759 yz Green-Lagrange strain tensor, e = jj z k -0.0000574759 0.000141093 {
-0.450655 y i 0.163023 zz Cauchy stress tensor, s = jj k -0.450655 1.55304 { Principal stresses = H 1.68636
0.0297033 L
Effective stress Hvon MisesL = 1.6717
-0.451146 y i 0.16339 Second PK stresses = jj zz k -0.451146 1.55222 {
r Ti = H 0.205037 -0.756047 0.106716 -0.34575 0.0865179 -0.188561 -0.398271 1.29036 L
Solution at 8s, t< = 80.57735, -0.57735< ï 8x, y< = 81.97194, 0.678535<
149
0.000585501 yz i 1.24995 J II = jj z k -0.105845 1.60543 {
detJ II = 2.00677
8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.321047, 0.294675, 0.10533, -0.0789578<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.0656985, -0.245735, 0.245589, 0.0658444<
B TL =
-0.321047 0 0.294675 0 0.10533 0 -0.0789578 0 jij jj 0 -0.0656985 0 -0.245735 0 0.245589 0 0.0658444 jj jj -0.0656985 -0.321047 -0.245735 0.294675 0.245589 0.10533 0.0658444 -0.0789578 k 0.000364648 y i 0.999989 zz; F IIp = jj { k -0.000158066 0.999856
Det@F IIp D = 0.999845
Element thickness, hII = 1.2501
0.000206531 y i 0.999977 zz Left Cauchy-Green tensor = jj { k 0.000206531 0.999712
yz i -0.0000113582 0.0001033 Green-Lagrange strain tensor, e = jj z -0.000143871 { k 0.0001033
i -0.745286 0.810884 yz Cauchy stress tensor, s = jj z -1.78668 { k 0.810884 Principal stresses = H -2.22965 -0.302313 L
Effective stress Hvon MisesL = 2.09492
i -0.745841 0.811486 yz Second PK stresses = jj z -1.78681 { k 0.811486
r Ti = H 0.466606 -0.358613 -1.05083 1.70087
0.302654 -0.886506 0.281569 -0.455746 L
Solution at 8s, t< = 80.57735, 0.57735< ï 8x, y< = 81.97261, 2.53233< 0.000585501 yz i 1.2498 J II = jj z k -0.395018 1.60543 {
detJ II = 2.0067
8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
yz zz zz zz z {
150
Geometric Nonlinearity
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.105333, 0.00690847, 0.393108, -0.294684<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.0657772, -0.24563, 0.245484, 0.0659231< B TL =
0.00690847 0 0.393108 0 -0.294684 0 ij -0.105333 0 yz jj zz jj 0 zz -0.0657772 0 -0.24563 0 0.245484 0 0.0659231 jj zz j z 0.00690847 0.245484 0.393108 0.0659231 -0.294684 { k -0.0657772 -0.105333 -0.24563 0.000364648 y i 0.999958 zz; F IIp = jj k -0.000589909 0.999856 {
Det@F IIp D = 0.999814
Element thickness, hII = 1.25013
-0.000225289 y i 0.999915 zz Left Cauchy-Green tensor = jj k -0.000225289 0.999712 {
i -0.0000422613 -0.000112596 yz Green-Lagrange strain tensor, e = jj z k -0.000112596 -0.000143871 {
i -1.11992 -0.884544 yz Cauchy stress tensor, s = jj z k -0.884544 -1.91613 {
Principal stresses = H -2.48803 -0.548023 L
Effective stress Hvon MisesL = 2.26431
i -1.11928 -0.884595 yz Second PK stresses = jj z k -0.884595 -1.91756 { r Ti = H 0.44189
0.549915 0.525642 1.16538 -1.64916 -2.05231 0.681624 0.33702 L
After summing contributions from all points the internal load vector is as follows: r Ti = H 0.0325383 -2.56101 -0.139661 3.15257
-0.970333 -2.59248 1.07746
2.00092 L
Global internal load vector
RTI = H 0.0325383 -2.56101 -0.139661 3.15257
Solution for element 2 Initial configuration
ij ÅÅÅÅ52Å jj jj j 10 Nodal coordinates = jjjj jj 10 jj j k0
3 yz zz z 3 zzzz zz 4 zzz zz 4{
-0.970333 -2.59248 1.07746
2.00092 0 0 0 0
151
5ts 35 s 5t 45 x0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 8 8 8 8 t 7 y0 Hs,tL = ÅÅÅÅÅÅ + ÅÅÅÅÅ 2 2 5t 35 ij ÅÅÅÅ ÅÅÅ + ÅÅÅÅ ÅÅ 8 8 j j J = jj k0
5s ÅÅÅÅ ÅÅÅÅ - ÅÅÅÅ58Å yz 8 zz zz ÅÅ12ÅÅ {
5t 35 detJ = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 16 16
Current configuration II
ij 0.00106552 jj jj 0.00128101 Nodal displacements = jjjj jj 0.00205808 jjj k 0.00156468
ij 2.50107 jj jj 10.0013 Updated coordinates = jjjj jj 10.0021 jj k 0.00156468
-0.000827007 y zz -0.00627425 zzzz zz -0.00627746 zzzz zz 0.000898983 {
2.99917 y zz 2.99373 zzzz zz 3.99372 zzzz z 4.0009 {
xII Hs,tL = 0.625069 t s + 4.37518 s - 0.624681 t + 5.62649
yII Hs,tL = -0.000432299 t s - 0.00315592 s + 0.500431 t + 3.49688
0.625069 s - 0.624681 i 0.625069 t + 4.37518 J II = jj k -0.000432299 t - 0.00315592 0.500431 - 0.000432299 s
yz z {
detJ II = 0.0000812872 s + 0.312534 t + 2.1875
Deformation gradient, F IIp = H1êdetJ L
0.000193195 s + 0.000242835 t + 0.00150665 y ij 0.312535 t + 2.18759 zz j { k -0.000216149 t - 0.00157796 0.0000811446 s + 0.312499 t + 2.18741
Solution at 8s, t< = 8-0.57735, -0.57735< ï 8x, y< = 83.6695, 3.20963< -0.985565 y i 4.01429 zz J II = jj -0.00290633 0.50068 k {
detJ II = 2.00701
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.0989446, 0.0982206, 0.0265121, -0.0257881<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.982371, -0.0176953, 0.263226, 0.736841<
152
Geometric Nonlinearity
B TL
0.0982206 0 0.0265121 0 -0.0257881 0 -0.0989446 0 jij j j = jj 0 -0.982371 0 -0.0176953 0 0.263226 0 0 jj 0.263226 0.0265121 0.736841 k -0.982371 -0.0989446 -0.0176953 0.0982206
0.00062524 yz i 1.00003 z; F IIp = jj k -0.000724021 0.999933 {
Det@F IIp D = 0.999968
Element thickness, hII = 1.25002
-0.0000988481 y i 1.00007 zz Left Cauchy-Green tensor = jj k -0.0000988481 0.999867 {
-0.0000493558 y i 0.0000344195 zz Green-Lagrange strain tensor, e = jj k -0.0000493558 -0.0000667784 { -0.388076 y i 0.13288 zz Cauchy stress tensor, s = jj k -0.388076 -0.660676 {
Principal stresses = H -0.818908 0.291112 L Effective stress Hvon MisesL = 0.99687
-0.387573 y i 0.133354 zz Second PK stresses = jj k -0.387573 -0.661316 { r Ti =
H 0.923461 1.72462 0.0499721 -0.0662984 -0.247441 -0.462111 -0.725992 -1.19621 L
Solution at 8s, t< = 8-0.57735, 0.57735< ï 8x, y< = 82.53147, 3.78777< -0.985565 y i 4.73606 zz J II = jj -0.00340551 0.50068 k {
detJ II = 2.3679
8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.022909, 0.0221899, 0.0835328, -0.0828137<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.832699, -0.167358, 0.375468, 0.624589< B TL =
-0.022909 0 0.0221899 0 0.0835328 0 -0.0828137 0 jij zyz jj 0 -0.832699 0 -0.167358 0 0.375468 0 0.624589 zzzz jj jj zz -0.0828137 { k -0.832699 -0.022909 -0.167358 0.0221899 0.375468 0.0835328 0.624589 0.000648377 y i 1.00005 zz; F IIp = jj k -0.000719093 0.999943 {
Det@F IIp D = 0.999989
153
Element thickness, hII = 1.25001
-0.0000707857 y i 1.00009 zz Left Cauchy-Green tensor = jj -0.0000707857 0.999886 k {
-0.0000353223 y i 0.0000461511 Green-Lagrange strain tensor, e = jj zz k -0.0000353223 -0.0000570463 { -0.277901 y i 0.315896 zz Cauchy stress tensor, s = jj k -0.277901 -0.493639 {
Principal stresses = H -0.579856 0.402113 L Effective stress Hvon MisesL = 0.855041
-0.277355 y i 0.316226 zz Second PK stresses = jj k -0.277355 -0.494092 {
r Ti = H 0.66352 1.23551 0.158409 0.226277 -0.230738 -0.617312 -0.591191 -0.844478 L
Solution at 8s, t< = 80.57735, -0.57735< ï 8x, y< = 88.30481, 3.20628< -0.263797 y i 4.01429 J II = jj zz k -0.00290633 0.500181 {
detJ II = 2.00711
8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.0984239, 0.0976999, 0.0269026, -0.0261786<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.263157, -0.736862, 0.802578, 0.197442<
0.0976999 0 0.0269026 0 -0.0261786 0 ij -0.0984239 0 j -0.263157 0 -0.736862 0 0.802578 0 0.19 B TL = jjjj 0 jj -0. k -0.263157 -0.0984239 -0.736862 0.0976999 0.802578 0.0269026 0.197442 0.000736388 y i 1.00003 zz; F IIp = jj k -0.000724021 0.99998 {
Det@F IIp D = 1.00001
Element thickness, hII = 1.24999
0.0000123265 y i 1.00007 zz Left Cauchy-Green tensor = jj k 0.0000123265 0.99996 {
i 0.0000344195 Green-Lagrange strain tensor, e = jjjj -6 k 6.20301 µ 10
6.20301 µ 10-6 yz zz z -0.0000200211 {
154
Geometric Nonlinearity
i 0.331193 0.0483926 yz Cauchy stress tensor, s = jj z k 0.0483926 -0.096405 {
Principal stresses = H 0.336602 -0.101813 L Effective stress Hvon MisesL = 0.397413
i 0.331101 0.048703 zyz Second PK stresses = jj k 0.048703 -0.0963392 { r Ti =
H -0.113732 0.0516994 -0.00828217 0.190085 0.119795 -0.190851 0.0022192 -0.050933 L
Solution at 8s, t< = 80.57735, 0.57735< ï 8x, y< = 88.0002, 3.78384< -0.263797 y i 4.73606 zz J II = jj k -0.00340551 0.500181 {
detJ II = 2.36799
8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.0224706, 0.0217515, 0.0838615, -0.0831424<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.223099, -0.776918, 0.832618, 0.167399<
0.0217515 0 0.0838615 0 -0.0831424 0 ij -0.0224706 0 j B TL = jjjj 0 -0.223099 0 -0.776918 0 0.832618 0 0.16 jj -0. k -0.223099 -0.0224706 -0.776918 0.0217515 0.832618 0.0838615 0.167399 0.000742587 y i 1.00005 zz; F IIp = jj -0.000719093 0.999982 k {
Det@F IIp D = 1.00003
Element thickness, hII = 1.24998
0.0000234484 yz i 1.00009 Left Cauchy-Green tensor = jj z k 0.0000234484 0.999965 {
i 0.0000461511 0.0000117707 yz Green-Lagrange strain tensor, e = jj z k 0.0000117707 -0.0000174127 { i 0.483992 0.0920555 Cauchy stress tensor, s = jj k 0.0920555 -0.0153638
Principal stresses = H 0.500422 -0.0317935 L Effective stress Hvon MisesL = 0.517053
i 0.483817 0.0924134 Second PK stresses = jj k 0.0924134 -0.0152319
yz z {
yz z {
155
r Ti =
H -0.092981 0.00402289 -0.180533 0.0412579 0.34701 -0.0150136 -0.0734963 -0.0302672 L After summing contributions from all points the internal load vector is as follows: r Ti = H 1.38027
3.01586
0.0195665 0.391321 -0.0113738 -1.28529 -1.38846 -2.12189 L
Global internal load vector
RTI = H 0.0325383 -2.56101 -0.139661 3.15257
0.409935 0.423378 -0.311005 -0.120975 0.0195665
After assembling all element internal force vectors, the global internal force and the external load vectors are as follows. ij 0.0325383 yz z jj jj -2.56101 zzz zz jj z jj jj -0.139661 zzz zz jj zz jj zz jj 3.15257 zz jj z jj jj 0.409935 zzz z jj jj 0.423378 zzz zz jj zz; j RI = jj jj -0.311005 zzz zz jj z jj jj -0.120975 zzz z jj jj 0.0195665 zzz zz jj z jj jj 0.391321 zzz zz jj z jj jj -0.0113738 zzz zz jj z j k -1.28529 {
ij 0. yz z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz z jj jj 0. zzz zz jj zz j RE = jj jj 0. zzz zz jj z jj jj 0. zzz z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z j k -1. {
Corresponding to unrestrained dof 0.409935 y jij z jj 0.423378 zzz jj zz jj z jj -0.311005 zzz jj zz jj z jj -0.120975 zzz j zz zz; RI = jjj jjj 0.0195665 zzz jj zz jjj 0.391321 zzz jj zz jj z jj -0.0113738 zzz jj zz k -1.28529 {
»»RE »» = 1.;
0. y jij z jj 0. zzz jj zz jj z jj 0. zzz jj zz jj z jj 0. zzz j zz zz; RE = jjj jjj 0. zzz jj z jj 0. zzz jj zz jj z jj 0. zzz jj zz j z -1. k {
»»R »» = 0.83288
Convergence parameter = 0.346844
Iteration = 2
-0.409935 y jij z jj -0.423378 zzz jj zz jj z jj 0.311005 zzz jj zz jj z jj 0.120975 zzz j zz zz R = RE - RI = jjj jjj -0.0195665 zzz jj zz jjj -0.391321 zzz jj zz jjj 0.0113738 zzz jj zz j z 0.285288 k {
156
Geometric Nonlinearity
Global equations at start of the element assembly process 0 jij jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
E = 10600.;
n = 0.35;
0 y i Du1 y i 0 y zz jj zz zz jj 0 zzzz jjjj Dv1 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzzz jjjj Du2 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzzz jjjj Dv2 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzz jjj Du3 zzz jjj 0 zzz zz jj zz zz jj 0 zzzz jjjj Dv3 zzzz jjjj 0 zzzz zz jj zz zz = jj 0 zzzz jjjj Du4 zzzz jjjj 0 zzzz zz zz jj zz jj 0 zzz jjj Dv4 zzz jjj 0 zzz zz zzz jjj zzz jjj 0 zzz jjj Du5 zzz jjj 0 zzzz zz jj zz zz jj 0 zzzz jjjj Dv5 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzz jjj Du6 zzz jjj 0 zzz zz jj zz zz jj 0 { k Dv6 { k -1 {
Initial thickness = 1.25 g = 0.461538
Plane stress analysis.
Interpolation functions and their derivatives 1 1 1 1 NT = : ÅÅÅÅÅ Hs - 1L Ht - 1L, - ÅÅÅÅÅ Hs + 1L Ht - 1L, ÅÅÅÅÅ Hs + 1L Ht + 1L, - ÅÅÅÅÅ Hs - 1L Ht + 1L> 4 4 4 4
t-1 1-t t+1 1 ∑NT ê∑s = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-t - 1L> 4 4 4 4
s -1 1 s +1 1-s ∑NT ê∑t = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-s - 1L, ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ > 4 4 4 4
Mapping to the master element
0 jij jj 5 jj ÅÅÅÅÅ j 2 Initial coordinates = jjjj jj ÅÅÅÅ5Å jj 2 jj k0 5s 5 x0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4
0y zz z 0 zzzz zz zz 3 zzz zz z 4{
ts s 7t 7 y0 Hs,tL = - ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4 4 4 ij ÅÅ54ÅÅ J = jjj j - ÅÅtÅÅ - ÅÅÅÅ1Å k 4 4
0 ÅÅÅÅ74Å
-
ÅÅ4sÅÅ
yz zz zz {
35 5s detJ = ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ 16 16
157
Current configuration
0 jij jj 5 jj ÅÅÅÅÅ Updated coordinates = jjjj 2 jjj 2.50107 jj k 0.00156468
zyz zz zz zz zz 2.99917 zzz zz 4.0009 { 0 0
xHs,tL = -0.000124791t s + 1.24988 s + 0.000657549 t + 1.25066 yHs,tL = -0.250431 t s - 0.250431 s + 1.75002 t + 1.75002
i -0.3125 s - 0.0000539974 t + 2.18745 0.000821937 - 0.000155989 s yz z Deformation gradient, F p = 1êdetJ jj 2.18752 - 0.313039 s k -0.000750622 t - 0.000750622 {
Gauss quadrature points and weights Point
Weight
1
s Ø -0.57735 t Ø -0.57735
1.
2
s Ø -0.57735 t Ø 0.57735
1.
3
s Ø 0.57735 t Ø -0.57735
1.
4
s Ø 0.57735 t Ø 0.57735
1.
Computation of element matrices at 8-0.57735, -0.57735< with weight = 1. i ÅÅÅÅ5Å 0 J = jjjj 4 k -0.105662 1.89434
i 0.8 J -T = jj k0
0.0446224 y zz 0.527889 {
zyz zz {
detJ = 2.36792
0.000385147 y i 0.99999 zz; Deformation gradient, F p = jj k -0.000133978 1.00014 {
0 -0.000133978 0 ij 0.99999 yz jj zz j zz 0.000385147 0 1.00014 F = jj 0 zz jj z 1.00014 -0.000133978 { k 0.000385147 0.99999 _
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.333066, 0.310755, 0.0892449, -0.0669336<
158
Geometric Nonlinearity
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.208166, -0.055778, 0.055778, 0.208166<
0.0000446237 0.310752 -0.0000416345 0.089244 -0.0000119 jij -0.333063 j GT = jjjj -0.0000801746 -0.208196 -0.0000214827 -0.0557859 0.0000214827 0.0557859 jj -0.333085 -0.0556578 0.310806 0.0558119 0.08925 k -0.208293
ij 0.47976 jj jj 0.98558 S = jjjj jj 0 jj k0
yz zz zz zz z 0.98558 zzzz z 1.66273 {
0.98558
0
0
1.66273
0
0
0
0.47976
0
0.98558
BT =
-0.333066 jij jj -0.208166 jj jj jj 0 jj j k0
0
0.310755
0
0
-0.055778 0
-0.0669336 0
0.055778
0
0.208166
0
0.0892449 0
-0.333066 0
0.310755
-0.208166 0
-0.055778 0
ij 6089.42 jj jj 2683.69 jj jj jj -5076.83 jj jj jj -246.243 k c = jjj jj -1631.66 jj jj jjj -719.093 jj jj 619.066 jj j k -1718.36
2683.69 3467.74 -1536.33 -617.289 -719.093 -929.178 -428.273 -1921.27
zyz zz zz zz -0.0669336 zzzz z 0.208166 {
0.0892449 0
0.055778
0
0
-1718.36 y zz -1536.33 -617.289 -719.093 -929.178 -428.273 -1921.27 zzzz zz 4898.54 -672.705 1360.33 411.658 -1182.04 1797.37 zzzz zz -672.705 1279.25 65.9805 165.402 852.967 -827.367 zzzz zz 1360.33 65.9805 437.201 192.681 -165.878 460.432 zzz zz 411.658 165.402 192.681 248.972 114.755 514.803 zzzz zz -1182.04 852.967 -165.878 114.755 728.857 -539.449 zzzz z 1797.37 -827.367 460.432 514.803 -539.449 2233.84 { -5076.83 -246.243 -1631.66 -719.093 619.066
ks = ij 0.775317 jj jj 0 jj jj jj -0.224349 jj jj jj 0 jj jj jj -0.207745 jj jj 0 jj jj jj -0.343222 jj j k0
0
-0.224349 0
0.775317
0
0
0.0513134
-0.207745 0
-0.224349 0 0
0.0601142
0
0.112922
-0.224349 0
0.0513134
0
0.0601142
0
0
0
0.0556652
0
0.0919661
-0.207745 0
0.0601142
0
0.0556652
0
0
0
0.0919661
0
0.138334
0.112922
0
0.0919661
0
0.0601142 0.112922
-0.343222 0
yz z -0.343222 zzzz zz zz 0 zz zz 0.112922 zzzz zz zz 0 zz z 0.0919661 zzzz zz zz 0 zz z 0.138334 {
-0.343222 0
-0.207745 0
159
6090.2 jij jj 2683.69 jj jj jj -5077.06 jj jj jj -246.243 j k = jjj jj -1631.86 jj jj jj -719.093 jj jj jj 618.723 jj k -1718.36
2683.69
-5077.06 -246.243 -1631.86 -719.093 618.723
3468.51
-1536.33 -617.514 -719.093 -929.385 -428.273
-1536.33 4898.59
-672.705 1360.39
-617.514 -672.705 1279.31
65.9805
411.658
-1181.93
165.462
852.967
-719.093 1360.39
65.9805
437.256
192.681
-165.786
-929.385 411.658
165.462
192.681
249.028
114.755
-428.273 -1181.93 852.967 -1921.61 1797.37
ij 1.08099 yz z jj jj 1.99626 zzz zz jj z jj jj -0.27881 zzz zz jj z jj jj -0.632081 zzz zz r i = jjj z jjj -0.289652 zzz zz jj jjj -0.534897 zzz zz jj z jj jj -0.512534 zzz zz jj k -0.829284 {
-165.786 114.755
-827.254 460.432
514.895
728.995 -539.449
-1718.36 y zz -1921.61 zzzz zz 1797.37 zzzz zz -827.254 zzzz zz 460.432 zzz zz 514.895 zzzz zz -539.449 zzzz z 2233.97 {
Computation of element matrices at 8-0.57735, 0.57735< with weight = 1. i ÅÅÅÅ5Å 0 J = jjjj 4 k -0.394338 1.89434 i 0.8 J -T = jj k0
0.166533 y zz 0.527889 {
yz zz z {
detJ = 2.36792
0.000385147 y i 0.999964 zz; Deformation gradient, F p = jj k -0.000500014 1.00014 {
0 -0.000500014 0 ij 0.999964 yz _ j zz zz F = jjjj 0 0.000385147 0 1.00014 zz jj z 1.00014 -0.000500014 { k 0.000385147 0.999964 8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.1502, 0.0669336, 0.333066, -0.2498<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.208166, -0.055778, 0.055778, 0.208166<
0.0000751022 0.0669312 -0.0000334678 0.333054 -0.0001665 ij -0.150195 j GT = jjjj -0.0000801746 -0.208196 -0.0000214827 -0.0557859 0.0000214827 0.0557859 jj -0.150117 -0.0557503 0.066971 0.0559043 0.333085 k -0.208217
160
Geometric Nonlinearity
0.16339 jij jj -0.451146 j S = jjjj jj 0 jj k0
zyz zz zz zz 0.16339 -0.451146 zzzz z -0.451146 1.55222 {
-0.451146 0
0
1.55222
0
0
0 0
BT =
ij -0.1502 jj jj -0.208166 jj jj jj 0 jj k0
yz zz zz zz z -0.2498 zzzz z 0.208166 {
0
0.0669336
0
0.333066 0
-0.2498
0
-0.055778 0
0.055778 0
0.208166 0
-0.1502
0
0
0.333066 0
-0.055778 0
0.055778 0
0.0669336
-0.208166 0
1640.68 jij jj 1211.43 jj jj jj -371.493 jj jj jj 65.635 j k c = jjj jj -2655.62 jj jj jj -1032.11 jj jj jj 1386.43 jj k -244.953
ij 0.126499 jj jj 0 jj jj jj 0.055903 jj jj jj 0 k s = jjj jjj 0.0262311 jj jj 0 jj jj jj -0.208633 jj j k0 ij 1640.8 jj jj 1211.43 jj jj jjj -371.437 jj jj 65.635 j k = jjj jj -2655.59 jj jj jj -1032.11 jj jj jj 1386.22 jj k -244.953
0
-244.953 y zz 2442.61 -280.044 467.736 -1976.53 -1164.73 1045.14 -1745.62 zzzz zz -280.044 261.584 -144.599 1086.15 -115.008 -976.246 539.651 zzzz zz 467.736 -144.599 208.771 -460.687 102.636 539.651 -779.144 zzzz zz -1976.53 1086.15 -460.687 5623.04 717.907 -4053.58 1719.31 zzz zz -1164.73 -115.008 102.636 717.907 1445.14 429.215 -383.044 zzzz zz 1045.14 -976.246 539.651 -4053.58 429.215 3643.4 -2014. zzzz z -1745.62 539.651 -779.144 1719.31 -383.044 -2014. 2907.8 { -371.493 65.635
1211.43
-2655.62 -1032.11 1386.43
0
0.055903
0
0.0262311
0
-0.208633
0.126499
0
0.055903
0
0.0262311
0
0
0.0264317
0
0.0163097
0
-0.0986443
0.055903
0
0.0264317
0
0.0163097
0
0
0.0163097
0
0.0183278
0
-0.0608686
0.0262311
0
0.0163097
0
0.0183278
0
0
-0.0986443 0
-0.208633 0
-0.0608686 0
-0.0986443 0
0.368146
-0.0608686 0
-244.953 y zz -1745.82 zzzz zz -280.044 261.611 -144.599 1086.17 -115.008 -976.344 539.651 zzzz zz 467.792 -144.599 208.797 -460.687 102.653 539.651 -779.242 zzzz zz -1976.53 1086.17 -460.687 5623.06 717.907 -4053.64 1719.31 zzz zz -1164.7 -115.008 102.653 717.907 1445.16 429.215 -383.105 zzzz zz 1045.14 -976.344 539.651 -4053.64 429.215 3643.77 -2014. zzzz z -1745.82 539.651 -779.242 1719.31 -383.105 -2014. 2908.17 { 1211.43
-371.437 65.635
-2655.59 -1032.11 1386.22
2442.74
-280.044 467.792
-1976.53 -1164.7
1045.14
161
-0.205037 y z jij jj 0.756047 zzz zz jj z jj jj -0.106716 zzz zz jj zz jj zz jj 0.34575 zz j z r i = jjj jj -0.0865179 zzz zz jj z jj jj 0.188561 zzz zz jj z jj jj 0.398271 zzz zz jj k -1.29036 {
Computation of element matrices at 80.57735, -0.57735< with weight = 1. i ÅÅÅÅ5Å 0 J = jjjj 4 k -0.105662 1.60566 i 0.8 J -T = jj k0
0.0526449 y zz 0.622796 {
zyz zz {
detJ = 2.00708
0.000364648 y i 0.999989 Deformation gradient, F p = jj zz; { k -0.000158066 0.999856
0.999989 0 -0.000158066 0 jij zyz j zz j F = jj 0 0.000364648 0 0.999856 zz jj zz 0.999856 -0.000158066 { k 0.000364648 0.999989 _
8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.321033, 0.29471, 0.10529, -0.0789674<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.0658061, -0.245592, 0.245592, 0.0658061<
0.0000507443 0.294707 -0.0000465836 0.105289 -0.0000166 ij -0.321029 j -0.0000895545 -0.245556 0.0000895545 0.245556 GT = jjjj -0.0000239961 -0.0657967 jj -0.320976 -0.245482 0.294707 0.245627 0.105236 k -0.0659224
ij -0.745841 jj jj 0.811486 S = jjjj jj 0 jj k0
0.811486
yz zz zz zz z -0.745841 0.811486 zzzz z 0.811486 -1.78681 { 0
0
-1.78681 0
0
0 0
162
Geometric Nonlinearity
BT =
ij -0.321033 jj jj -0.0658061 jj jj jj 0 jj k0
0
0.29471
-0.245592 0
0.245592 0
0.0658061
-0.321033
0
0
0
0.29471
-0.0658061 0
ij 4442.07 jj jj 692.533 jj jj jj -3879.11 jj jj jj 1622.41 k c = jjj jjj -1602.36 jj jj -1880.22 jj jj jj 1039.41 jj j k -434.723
0 0.10529
-0.245592 0 -3879.11 1622.41
692.533
0.245592 0 -1602.36 -1880.22 1039.41
1199.61
332.254
-242.847 -935.76
-1021.84 -89.0271
332.254
4301.32
-2377.53 730.332
1408.22
-1152.53
-242.847 -2377.53 3432.95
118.065
-2270.24 637.057
-935.76
1067.72
849.331
-195.692
-2270.24 849.331
2683.77
-377.331
730.332
-1021.84 1408.22
118.065
-89.0271 -1152.53 637.057
ij -0.126242 jj jj 0 jj jj jjj 0.225621 jj jj 0 j k s = jjj jj -0.0389239 jj jj jj 0 jj jj jj -0.0604549 jj k0 ij 4441.95 jj jj 692.533 jj jj jj -3878.89 jj jj jj 1622.41 k = jjj jjj -1602.4 jj jjj -1880.22 jj jj jj 1039.34 jj k -434.723
0.10529
65.0707
637.057
-195.692 -377.331 308.821
-919.855 -31.6353 608.31
-170.699
0
-434.723 y zz 65.0707 zzzz zz 637.057 zzzz zz -919.855 zzzz zz -31.6353 zzz zz zz 608.31 zz zz -170.699 zzzz z 246.475 {
0
0.225621
0
-0.0389239 0
-0.126242
0
0.225621
0
-0.0389239 0
0.30703
0
0.194964
0.30703
0
0
-0.727614 0
0.225621
0
0
0.30703
-0.727614 0
-0.0389239 0 0
0.194964
-0.0604549 0
692.533
-0.0604549
0
-0.185838
0
-0.0822684
0.30703
0
-0.185838
0
0
-0.0822684 0
0.194964
0
-3878.89 1622.41
-1602.4
-0.0522404
-0.0822684 0 -1880.22 1039.34
1199.49
332.254
-242.622 -935.76
-1021.87 -89.0271
332.254
4300.59
-2377.53 730.639
1408.22
-1152.34
-242.622 -2377.53 3432.22
118.065
-2269.94 637.057
-935.76
1067.54
849.331
-195.774
-2269.94 849.331
2683.58
-377.331
730.639
-1021.87 1408.22
118.065
-89.0271 -1152.34 637.057 65.0103
ij -0.466606 yz z jj jj 0.358613 zzz zz jj z jj jj 1.05083 zzz zz jjj jj -1.70087 zzz j zzz j r i = jj jj -0.302654 zzz zz jj zz jj jjj 0.886506 zzz zz jj jjj -0.281569 zzz zz jj k 0.455746 {
yz zz zz zz z -0.0789674 zzzz z 0.0658061 {
-0.0789674 0
0 0
637.057
-919.66
-195.774 -377.331 308.768 -31.6353 608.228
-170.699
-434.723 y zz 65.0103 zzzz zz 637.057 zzzz zz -919.66 zzzz zz -31.6353 zzz zz 608.228 zzzz zz -170.699 zzzz z 246.422 {
163
Computation of element matrices at 80.57735, 0.57735< with weight = 1. i ÅÅÅÅ5Å 0 J = jjjj 4 k -0.394338 1.60566 0.196473 y zz 0.622796 {
i 0.8 J -T = jj k0
yz zz z {
detJ = 2.00708
0.000364648 y i 0.999958 zz; Deformation gradient, F p = jj k -0.000589909 0.999856 {
0 -0.000589909 0 ij 0.999958 yz _ j zz zz 0.000364648 0 0.999856 F = jjjj 0 zz jj z 0.999856 -0.000589909 { k 0.000364648 0.999958 8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.10529, 0.00705308, 0.392947, -0.29471<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.0658061, -0.245592, 0.245592, 0.0658061< 0.0000621114 0.00705278 -4.16068 µ 10-6 jij -0.105285 jj G = jjj -0.0000239961 -0.0657967 -0.0000895545 -0.245556 jj -0.105236 -0.245579 0.00719694 k -0.0658417
ij -1.11928 jj jj -0.884595 S = jjjj jj 0 jjj k0
0
0
0 0
0.0000895545 0.245556 0.245725
yz zz zz zz z -1.11928 -0.884595 zzzz zz -0.884595 -1.91756 {
-0.884595 0 -1.91756
0
BT =
-0.10529 jij jj -0.0658061 jj jj jj 0 jj j k0
-0.000231
0.39293
T
0.392745
zyz zz zz zz -0.29471 zzzz z 0.0658061 {
0
0.00705308 0
0.392947 0
-0.29471
0
-0.245592
0
0.245592 0
0.0658061 0
-0.10529
0
0.00705308 0
0.392947 0
-0.245592
0.245592 0
-0.0658061 0
0
0
164
Geometric Nonlinearity
516.22 jij jj 227.489 jj jj jj 128.108 jj jj jj 590.142 j k c = jjj jj -1926.56 jj jj jj -849.002 jj jj jj 1282.23 jj k 31.3705
ij -0.0827175 jj jj 0 jj jj jj -0.132023 jj jj jj 0 k s = jjj jjj 0.308706 jj jj 0 jj jj jj -0.0939658 jj j k0 ij 516.138 jj jj 227.489 jj jj jjj 127.976 jj jj 590.142 j k = jjj jj -1926.25 jj jj jj -849.002 jj jj jj 1282.14 jj k 31.3705
227.489
128.108
590.142
-1926.56 -849.002 1282.23
31.3705
293.987
244.444
682.491
-849.002 -1097.18 377.069
120.697
244.444
596.419
-55.7166 -478.104 -912.278 -246.422 723.551
682.491
-55.7166 2575.84
-2202.44 -2547.09 1668.02
-711.245
-849.002 -478.104 -2202.44 7190.02
3168.52
-4785.36 -117.076
-1097.18 -912.278 -2547.09 3168.52
4094.71
-1407.24 -450.448
377.069
-246.422 1668.02
120.697
723.551
-0.132023
0
-4785.36 -1407.24 3749.55
-637.845
-711.245 -117.076 -450.448 -637.845 1041.
-0.0827175 0
0
0.308706
0
-0.0939658 0
-0.132023
0
0.308706
0
-0
0
-0.28262
0
0.492715
0
-0.0780731 0
-0.132023
0
-0.28262
0
0.492715
0
-0
0
0.492715
0
-1.15211 0
0.350685
0
0.308706
0
0.492715
0
-1.15211 0
0
-0.0780731 0
0.350685
0
-0.178646
0
0.350685
0
-0
-0.0939658 0
-0.0780731 0
0.3
227.489
127.976
590.142
-1926.25 -849.002 1282.14
31.3705
293.905
244.444
682.359
-849.002 -1096.87 377.069
120.603
244.444
596.136
-55.7166 -477.612 -912.278 -246.5
682.359
-55.7166 2575.56
-2202.44 -2546.6
723.551
1668.02
-711.323 -117.076
-849.002 -477.612 -2202.44 7188.87
3168.52
-4785.
-1096.87 -912.278 -2546.6
3168.52
4093.56
-1407.24 -450.098
377.069
-246.5
1668.02
-4785.
-1407.24 3749.37
120.603
723.551
-711.323 -117.076 -450.098 -637.845 1040.82
ij -0.44189 yz z jj jj -0.549915 zzz zz jjj jj -0.525642 zzz zzz jjj zz jj jjj -1.16538 zzz zz r i = jj jjj 1.64916 zzz zz jj jjj 2.05231 zzz zz jj jjj -0.681624 zzz zz jj z j k -0.33702 {
zyz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz {
After summing contributions from all points, the element equations as follows:
-637.845
yz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz z {
165
12689.1 jij jj 4815.15 jj jj jj -9199.41 jj jj jj 2031.94 j k = jjj jj -7816.11 jj jj jj -4480.43 jj jj jj 4326.43 jj k -2366.66
rT = H 0
-2366.66 y zz 7404.64 -1239.67 290.016 -4480.38 -4212.83 904.906 -3481.83 zzzz zz -1239.67 10056.9 -3250.55 2699.59 792.591 -3557.12 3697.63 zzzz zz 290.016 -3250.55 7495.88 -2479.08 -4548.42 3697.69 -3237.48 zzzz zz -4480.38 2699.59 -2479.08 14316.7 4928.44 -9200.21 2031.03 zzz zz zz -4212.83 792.591 -4548.42 4928.44 8471.33 -1240.6 289.92 zz zz 904.906 -3557.12 3697.69 -9200.21 -1240.6 8430.9 -3362. zzzz z -3481.83 3697.63 -3237.48 2031.03 289.92 -3362. 6429.39 { 4815.15
0 0 0 0
-9199.41 2031.94
-7816.11 -4480.43 4326.43
0 0 0L
Complete element equations for element 1 12689.1 jij jj 4815.15 jj jj jj -9199.41 jj jj jj 2031.94 jj jj jj -7816.11 jj jj jj -4480.43 jj jj jj 4326.43 jj k -2366.66
-2366.66 y i Du1 y i 0. y zz jj zz jj zz 7404.64 -1239.67 290.016 -4480.38 -4212.83 904.906 -3481.83 zzzz jjjj Dv1 zzzz jjjj 0. zzzz zz jj zz jj zz -1239.67 10056.9 -3250.55 2699.59 792.591 -3557.12 3697.63 zzzz jjjj Du2 zzzz jjjj 0. zzzz zz jj zz jj zz 290.016 -3250.55 7495.88 -2479.08 -4548.42 3697.69 -3237.48 zzzz jjjj Dv2 zzzz jjjj 0. zzzz zz jj zz = jj zz -4480.38 2699.59 -2479.08 14316.7 4928.44 -9200.21 2031.03 zzz jjj Du3 zzz jjj 0. zzz zz jj z j z zz jj Dv zzz jjj 0. zzz -4212.83 792.591 -4548.42 4928.44 8471.33 -1240.6 289.92 zz jj 3 zz jj zz zz jj zz jj zz 904.906 -3557.12 3697.69 -9200.21 -1240.6 8430.9 -3362. zzzz jjjj Du4 zzzz jjjj 0. zzzz zj z j z -3481.83 3697.63 -3237.48 2031.03 289.92 -3362. 6429.39 { k Dv4 { k 0. { 4815.15
-9199.41 2031.94
-7816.11 -4480.43 4326.43
The element contributes to 81, 2, 3, 4, 5, 6, 7, 8< global degrees of freedom. 1 jij zyz jj 2 zz jj zz jj zz jj 3 zz jj zz jj zz jj 4 zz j z Locations for element contributions to a global vector: jjj zzz jjj 5 zzz jj zz jjj 6 zzz jj zz jjj 7 zzz jj zz j z k8{
ij @1, 1D jj jj @2, 1D jj jj jj @3, 1D jj jj jj @4, 1D and to a global matrix: jjj jj @5, 1D jj jj jjj @6, 1D jj jj @7, 1D jj j k @8, 1D
@1, 2D
@1, 3D
@1, 4D
@1, 5D
@1, 6D
@3, 3D
@3, 4D
@3, 5D
@3, 6D
@5, 3D
@5, 4D
@5, 5D
@5, 6D
@2, 2D
@2, 3D
@4, 2D
@4, 3D
@3, 2D
@5, 2D
@6, 2D
@7, 2D
@8, 2D
@6, 3D
@7, 3D
@8, 3D
@2, 4D
@4, 4D
@6, 4D
@7, 4D
@8, 4D
@2, 5D
@4, 5D
@6, 5D
@7, 5D
@8, 5D
Adding element equations into appropriate locations we have
@2, 6D
@4, 6D
@6, 6D
@7, 6D
@8, 6D
@1, 7D
@1, 8D y zz @2, 8D zzzz zz @3, 7D @3, 8D zzzz zz @4, 7D @4, 8D zzzz zz @5, 7D @5, 8D zzz zz @6, 7D @6, 8D zzzz zz @7, 7D @7, 8D zzzz z @8, 7D @8, 8D {
@2, 7D
166
Geometric Nonlinearity
12689.1 jij jj 4815.15 jj jj jj -9199.41 jj jj jj 2031.94 jj jj jj -7816.11 jj jj jj -4480.43 jj jj jj 4326.43 jj jj jj -2366.66 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
4815.15
-9199.41 2031.94
-7816.11 -4480.43 4326.43
-2366.66 0
7404.64
-1239.67 290.016
-4480.38 -4212.83 904.906
-3481.83 0
-1239.67 10056.9 290.016
-3250.55 2699.59
-3250.55 7495.88
792.591
-3557.12 3697.63
-2479.08 -4548.42 3697.69
-4480.38 2699.59
-2479.08 14316.7
4928.44
-9200.21 2031.03
0
-4212.83 792.591
-4548.42 4928.44
8471.33
-1240.6
0
8430.9
-3362.
0
-3237.48 2031.03
289.92
-3362.
6429.39
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
904.906
-3557.12 3697.69
-9200.21 -1240.6
289.92
-3481.83 3697.63
ij Du1 yz ij 0 yz z j jj z jj Dv1 zzz jjj 0 zzz zz jjj zzz jjj jj Du zz jj 0 zzz zz jjj 2 zzz jjj zz jj zz jj z jjj Dv2 zzz jjj 0 zzz jj zz jj zz jjj Du3 zzz jjj 0 zzz jj zz jj zz jjj Dv3 zzz jjj 0 zzz jj zz = jj zz jjj Du zzz jjj 0 zzz jj 4 zz jj zz z jj z j jj Dv4 zzz jjj 0 zzz zz jj zz jj zz jj zz jj jjj Du5 zzz jjj 0 zzz zz jj zz jj jjj Dv5 zzz jjj 0 zzz zz jj zz jj jjj Du6 zzz jjj 0 zzz zz jj zz jj z j z j k Dv6 { k -1 { E = 10600.;
0
-3237.48 0
n = 0.35;
Plane stress analysis.
Initial thickness = 1.25 g = 0.461538
Interpolation functions and their derivatives 1 1 1 1 NT = : ÅÅÅÅÅ Hs - 1L Ht - 1L, - ÅÅÅÅÅ Hs + 1L Ht - 1L, ÅÅÅÅÅ Hs + 1L Ht + 1L, - ÅÅÅÅÅ Hs - 1L Ht + 1L> 4 4 4 4
t-1 1-t t+1 1 ∑NT ê∑s = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-t - 1L> 4 4 4 4
s -1 1 s +1 1-s ∑NT ê∑t = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-s - 1L, ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ > 4 4 4 4
0 0 0y zz 0 0 0 zzzz zz 0 0 0 zzzz zz 0 0 0 zzzz zz 0 0 0 zzz zz 0 0 0 zzzz zz 0 0 0 zzzz zz 0 0 0 zzz zz 0 0 0 zzzz zz 0 0 0 zzzz zz 0 0 0 zzz zz 0 0 0{
167
Mapping to the master element
5 jij ÅÅÅÅ2Å jj jj 10 Initial coordinates = jjjj jjj 10 jj k0
3 zy zz z 3 zzzz zz 4 zzz zz 4{
5ts 35 s 5t 45 x0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 8 8 8 8
t 7 y0 Hs,tL = ÅÅÅÅÅÅ + ÅÅÅÅÅ 2 2 5t 35 ij ÅÅÅÅ ÅÅÅ + ÅÅÅÅ ÅÅ 8 8 J = jjjj k0
5s ÅÅÅÅ ÅÅÅÅ - ÅÅÅÅ58Å yz 8 zz zz 1 ÅÅ2ÅÅ {
5t 35 detJ = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 16 16
Current configuration
ij 2.50107 jj jj 10.0013 Updated coordinates = jjjj jj 10.0021 jj k 0.00156468
2.99917 y zz 2.99373 zzzz zz 3.99372 zzzz z 4.0009 {
xHs,tL = 0.625069 t s + 4.37518 s - 0.624681 t + 5.62649 yHs,tL = -0.000432299 t s - 0.00315592 s + 0.500431 t + 3.49688 Deformation gradient, F p = 1êdet
0.000193195 s + 0.000242835 t + 0.00150665 y i 0.312535 t + 2.18759 zz J jj k -0.000216149 t - 0.00157796 0.0000811446 s + 0.312499 t + 2.18741 {
Gauss quadrature points and weights Point
Weight
1
s Ø -0.57735 t Ø -0.57735
1.
2
s Ø -0.57735 t Ø 0.57735
1.
3
s Ø 0.57735 t Ø -0.57735
1.
4
s Ø 0.57735 t Ø 0.57735
1.
Computation of element matrices at 8-0.57735, -0.57735< with weight = 1.
168
Geometric Nonlinearity
i 4.01416 J = jjjj k0
-0.985844 y zz zz ÅÅ12ÅÅ {
detJ = 2.00708
i 0.249118 0 yz z J -T = jj k 0.491184 2. {
0.00062524 y i 1.00003 zz; Deformation gradient, F p = jj k -0.000724021 0.999933 {
0 -0.000724021 0 ij 1.00003 yz jj zz j zz 0 0.00062524 0 0.999933 F = jj zz jj z 0.999933 -0.000724021 { k 0.00062524 1.00003 _
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0982367, 0.0982367, 0.0263225, -0.0263225<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.982367, -0.0176327, 0.263225, 0.736775<
0.0000711255 0.0982401 -0.0000711255 0.0263234 -0.000019058 ij -0.0982401 jj j G = jj -0.000614215 -0.982302 -0.0000110247 -0.0176315 0.000164578 0.263207 jj -0.982462 -0.0975189 -0.0175719 0.0982429 0.26325 0.0261301 k T
ij 0.133354 jj jj -0.387573 S = jjjj jj 0 jjj k0
0
0
0 0
-0.0982367 jij jj -0.982367 j B T = jjjj jj 0 jj k0 ij 9923.53 jj jj 3191.3 jj jj jj -243.577 jj jj jj -910.168 k c = jjj jj -2659. jj jj jjj -855.106 jj jj -7020.95 jj j k -1426.02
yz zz zz zz z 0.133354 -0.387573 zzzz zz -0.387573 -0.661316 {
-0.387573 0 -0.661316 0
0
0.0982367
0
0
-0.0176327 0
0.0263225 0
-0.0263225 0
0.263225
0
0.736775
0
0.0263225 0
-0.0982367 0
0.0982367
-0.982367
-0.0176327 0
0
3191.3
-243.577 -910.168 -2659.
41286.9
-2200.29 646.232
-2200.29 414.85 646.232
243.879
-855.106 65.2663
243.879
-11062.8 589.567
-173.157 229.125
-135.9
-236.539 723.353
-30870.3 1667.79
712.477 1881.26
-581.464 382.102
0
-855.106 -7020.95 -1426.02 y zz -30870.3 zzzz zz 589.567 -236.539 1667.79 zzzz zz -173.157 723.353 -581.464 zzzz zz 229.125 1881.26 382.102 zzz zz 2964.27 36.4142 8271.68 zzzz zz 36.4142 5376.23 -623.868 zzzz z 8271.68 -623.868 23180.1 {
-855.106 -11062.8 -135.9
-57.0637 65.2663
-57.0637 108.389
0.263225
0 0
169
-1.78559 jij jj 0 jj jj jj 0.0601851 jj jj jj 0 j k s = jjj jj 0.478448 jj jj jj 0 jj jj jj 1.24696 jj k0 ij 9921.74 jj jj 3191.3 jj jj jj -243.517 jj jj jj -910.168 k = jjj jjj -2658.52 jj jjj -855.106 jj jj jj -7019.7 jj k -1426.02
0
0.0601851
0
0.478448
0
1.24696
-1.78559
0
0.0601851
0
0.478448
0
1.2
0
0.00608146
0
-0.0161266 0
-0.05014
0
0.00608146
0
0.0601851 0 0
-0.0161266 0
0.478448
0
-0.0161266 0
-0.1282
-0.0161266 0
-0
0
-0.334122 0
-0.1282
0
-0
0
-0.05014
0
-0.334122
0
-0.862697 0
1.24696
0
-0.05014
0
-0.334122
0
-1426.02 y zz -30869.1 zzzz zz 589.567 -236.589 1667.79 zzzz zz -173.174 723.353 -581.514 zzzz zz 229.125 1880.92 382.102 zzz zz 2964.14 36.4142 8271.35 zzzz zz 36.4142 5375.37 -623.868 zzzz z 8271.35 -623.868 23179.3 {
3191.3
-243.517 -910.168 -2658.52 -855.106 -7019.7
41285.1
-2200.29 646.292
-2200.29 414.856 646.292
-855.106 -11062.3 -135.9
-57.0637 65.2502
-57.0637 108.396
243.879
-855.106 65.2502
243.879
-11062.3 589.567
-173.174 229.125
-135.9
-236.589 723.353
-30869.1 1667.79
ij -0.923461 yz z jj jj -1.72462 zzz zz jj zz jj jjj -0.0499721 zzz z jj jj 0.0662984 zzz j zzz j r i = jj jj 0.247441 zzz zz jj z jj jj 0.462111 zzz zz jj z jj jj 0.725992 zzz zz jj { k 1.19621
712.349 1880.92
-581.514 382.102
Computation of element matrices at 8-0.57735, 0.57735< with weight = 1. i 4.73584 J = jjjj k0
0
-0.985844 y zz zz ÅÅ12ÅÅ {
detJ = 2.36792
i 0.211156 0 yz z J -T = jj k 0.416333 2. {
0.000648377 y i 1.00005 Deformation gradient, F p = jj zz; k -0.000719093 0.999943 {
1.00005 0 -0.000719093 0 jij zyz j zz j 0.000648377 0 0.999943 F = jj 0 zz jj zz 0.999943 -0.000719093 { k 0.000648377 1.00005 _
8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
-0
170
Geometric Nonlinearity
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338< 8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0223112, 0.0223112, 0.0832666, -0.0832666<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.832666, -0.167334, 0.375501, 0.624499<
0.0000160438 0.0223122 -0.0000160438 0.0832704 -0.0000598764 ij -0.0223122 j -0.000108496 -0.167325 0.000243466 0.375479 GT = jjjj -0.000539881 -0.832618 jj -0.0217112 -0.167327 0.0224303 0.375572 0.0829918 k -0.832719
ij 0.316226 jj jj -0.277355 S = jjjj jj 0 jj k0
0
0
0 0
ij -0.0223112 jj jj -0.832666 T B = jjjj jj 0 jj k0 ij 8083.93 jj jj 738.912 jj jj jjj 1593.82 jj jj -110.762 j k c = jjj jj -3729.53 jj jj jj -1041.52 jj jj jj -5948.22 jj k 413.368
yz zz zz zz z 0.316226 -0.277355 zzzz z -0.277355 -0.494092 {
-0.277355 0 -0.494092 0
0
0.0223112
0
0
-0.167334 0
0.0832666 0
-0.0832666 0
0.375501
0
0.624499
0
0.0832666 0
-0.
0.375501
0.62
-0.0223112 0
0.0223112
-0.832666
-0.167334 0
0
0
0
yz z -26167.4 zzzz zz -456.448 350.27 -143.847 -636.867 63.4507 -1307.22 536.844 zzzz zz 7011.53 -143.847 1416.11 -282.236 -3142.66 536.844 -5284.98 zzzz zz -1985.95 -636.867 -282.236 1989.58 1214.87 2376.82 1053.32 zzz zz -15765.9 63.4507 -3142.66 1214.87 7179.96 -236.801 11728.6 zzzz zz 1703.49 -1307.22 536.844 2376.82 -236.801 4878.63 -2003.53 zzzz z -26167.4 536.844 -5284.98 1053.32 11728.6 -2003.53 19723.8 { -110.762 -3729.53 -1041.52 -5948.22 413.368
738.912
1593.82
34921.7
-456.448 7011.53
-1985.95 -15765.9 1703.49
ks = ij -1.04401 jj jj 0 jj jj jj -0.19205 jj jj jj 0 jj jj jjj 0.519321 jj jjj 0 jj jj 0.716739 jj j k0
0 yz z 0.716739 zzzz zz zz 0 -0.0343542 0 0.0981922 0 0.128212 0 zz zz -0.19205 0 -0.0343542 0 0.0981922 0 0.128212 zzzz zz zz 0 0.0981922 0 -0.251055 0 -0.366458 0 zz z 0.519321 0 0.0981922 0 -0.251055 0 -0.366458 zzzz zz zz 0 0.128212 0 -0.366458 0 -0.478493 0 zz z 0.716739 0 0.128212 0 -0.366458 0 -0.478493 { 0
-0.19205
-1.04401 0
0
0.519321
0
0.716739
-0.19205
0
0.519321
0
171
8082.88 jij jj 738.912 jj jj jj 1593.63 jj jj jj -110.762 j k = jjj jj -3729.01 jj jj jj -1041.52 jj jj jj -5947.5 jj k 413.368
zyz -26166.7 zzzz zz -456.448 350.235 -143.847 -636.768 63.4507 -1307.1 536.844 zzzz zz 7011.34 -143.847 1416.07 -282.236 -3142.56 536.844 -5284.85 zzzz zz -1985.95 -636.768 -282.236 1989.33 1214.87 2376.45 1053.32 zzz zz -15765.3 63.4507 -3142.56 1214.87 7179.71 -236.801 11728.2 zzzz zz 1703.49 -1307.1 536.844 2376.45 -236.801 4878.15 -2003.53 zzzz z -26166.7 536.844 -5284.85 1053.32 11728.2 -2003.53 19723.3 { -110.762 -3729.01 -1041.52 -5947.5
738.912
1593.63
34920.7
-456.448 7011.34
413.368
-1985.95 -15765.3 1703.49
ij -0.66352 yz z jj jj -1.23551 zzz zz jj z jj jj -0.158409 zzz zz jj zz jj jjj -0.226277 zzz zz r i = jj jjj 0.230738 zzz zz jj jjj 0.617312 zzz zz jj z jj jj 0.591191 zzz zz jj k 0.844478 {
Computation of element matrices at 80.57735, -0.57735< with weight = 1. i 4.01416 J = jjjj k0
-0.264156 y zz zz ÅÅ12ÅÅ {
detJ = 2.00708
i 0.249118 0 yz J -T = jj z k 0.131612 2. {
0.000736388 y i 1.00003 zz; Deformation gradient, F p = jj k -0.000724021 0.99998 {
0 -0.000724021 0 ij 1.00003 yz _ j zz zz F = jjjj 0 0.000736388 0 0.99998 zz jj z 0.99998 -0.000724021 { k 0.000736388 1.00003 8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0982367, 0.0982367, 0.0263225, -0.0263225<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.263225, -0.736775, 0.802582, 0.197418<
0.0000711255 0.0982401 -0.0000711255 0.0263234 -0.000019058 ij -0.0982401 j GT = jjjj -0.000193835 -0.263219 -0.000542552 -0.736761 0.000591011 0.802565 jj -0.0980442 -0.736728 0.0987682 0.802628 0.0257408 k -0.263306
172
Geometric Nonlinearity
0.331101 jij jj 0.048703 j S = jjjj jj 0 jj k0
0
0
-0.0963392 0
0
0 0
ij -0.0982367 jj jj -0.263225 T B = jjjj jj 0 jj k0 1095.55 jij jj 850.373 jj jj jj 1499.48 jj jj jj 1413.55 j k c = jjj jjj -2193.25 jj jj -1885.17 jj jj jj -401.785 jj j k -378.761
0
0.0982367
0
-0.736775 0
0
0.0263225 0
-0.0263225 0
0.802582
0
0.197418
0
0.0263225 0
-0.
0.802582
0
0.19
-0.0982367 0
0.0982367
-0.263225
-0.736775 0
0
0
-2193.25 -1885.17 -401.785 -378.761 y zz 3051.16 123.441 8181.61 -940.738 -9040.51 -33.076 -2192.26 zzzz zz 123.441 5755.29 -2363.6 -5712.65 1606.83 -1542.13 633.324 zzzz zz 8181.61 -2363.6 23268.4 316.719 -25215.3 633.324 -6234.75 zzzz zz -940.738 -5712.65 316.719 6375.2 708.884 1530.7 -84.8646 zzz zz -9040.51 1606.83 -25215.3 708.884 27499.4 -430.549 6756.41 zzzz zz -33.076 -1542.13 633.324 1530.7 -430.549 413.211 -169.699 zzzz z -2192.26 633.324 -6234.75 -84.8646 6756.41 -169.699 1670.6 { 850.373
ij -0.0024111 jj jj 0 jj jj jj -0.049207 jj jj jj 0 k s = jjj jj 0.0384331 jj jj jjj 0 jj jj 0.013185 jj j k0 1095.55 jij jj 850.373 jj jj jj 1499.43 jj jj jj 1413.55 j k = jjj jjj -2193.21 jj jjj -1885.17 jj jj jj -401.772 jj k -378.761
zyz zz zz zz 0.331101 0.048703 zzzz z 0.048703 -0.0963392 {
0.048703
1499.48
1413.55
-0.049207 0
0
-0.0024111 0
0.0384331
-0.049207 0
0
0.013185
0
0.0384331
0
0.01
0
0.0377474
0
0.152335
0
0.03
0
-0.140875 0
-0.049207
0
-0.140875 0
0
0.152335
0
-0.14995
0
-0.040818
0
0.0384331
0
0.152335
0
-0.14995
0
-0.
0
0.0377474
0
-0.040818 0
0.013185
0
0.0377474
0
0.152335
-0.0101144 0
-0.040818 0
-2193.21 -1885.17 -401.772 -378.761 y zz 3051.15 123.441 8181.56 -940.738 -9040.47 -33.076 -2192.24 zzzz zz 123.441 5755.15 -2363.6 -5712.5 1606.83 -1542.09 633.324 zzzz zz 8181.56 -2363.6 23268.3 316.719 -25215.1 633.324 -6234.71 zzzz zz -940.738 -5712.5 316.719 6375.05 708.884 1530.66 -84.8646 zzz zz -9040.47 1606.83 -25215.1 708.884 27499.2 -430.549 6756.37 zzzz zz -33.076 -1542.09 633.324 1530.66 -430.549 413.201 -169.699 zzzz z -2192.24 633.324 -6234.71 -84.8646 6756.37 -169.699 1670.59 { 850.373
1499.43
1413.55
-0.
173
0.113732 y z jij jj -0.0516994 zzz zz jj zz jjj jj 0.00828217 zzz zz jj z jj jj -0.190085 zzz zz r i = jjj jj -0.119795 zzz zz jj z jj jj 0.190851 zzz zz jj z jj jj -0.0022192 zzz zz jj k 0.050933 {
Computation of element matrices at 80.57735, 0.57735< with weight = 1. i 4.73584 J = jjjj k0
-0.264156 y zz zz ÅÅ12ÅÅ {
detJ = 2.36792
i 0.211156 0 yz z J -T = jj k 0.111556 2. {
0.000742587 y i 1.00005 Deformation gradient, F p = jj zz; { k -0.000719093 0.999982
1.00005 0 -0.000719093 0 jij zyz j zz j F = jj 0 0.000742587 0 0.999982 zz jj zz 0.999982 -0.000719093 { k 0.000742587 1.00005 _
8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0223112, 0.0223112, 0.0832666, -0.0832666<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.223112, -0.776888, 0.832666, 0.167334<
-0.0000160438 0.0832704 -0.0000598764 ij -0.0223122 0.0000160438 0.0223122 j -0.000576907 -0.776874 0.000618327 0.832651 GT = jjjj -0.00016568 -0.223108 jj -0.0221504 -0.776907 0.0228695 0.832766 0.0826664 k -0.223139
ij 0.483817 jj jj 0.0924134 S = jjjj jj 0 jj k0
0.0924134
yz zz zz zz z 0.483817 0.0924134 zzzz z 0.0924134 -0.0152319 {
0
0
-0.0152319 0
0
0 0
ij -0.0223112 jj jj -0.223112 T B = jjjj jj 0 jj k0
0
0.0223112
0
0
-0.776888 0
0.0832666 0
-0.0832666 0
0.832666
0
0.167334
0
0.0832666 0
-0.
0.832666
0.16
-0.0223112 0
0.0223112
-0.223112
-0.776888 0
0
0
0
174
Geometric Nonlinearity
603.791 jij jj 194.181 jj jj jj 1989.48 jj jj jj 416.965 j k c = jjj jj -2253.38 jj jj jj -724.695 jj jj jj -339.892 jj k 113.548
194.181
1989.48
416.965
2512.16
71.2814
8722.2
71.2814
7037.67
-654.575
8722.2
-654.575 30399.3
-724.695 -9375.52 459.232
-724.695 -7424.85 -1556.14 -9375.52 -266.026 -32551.7 459.232
-1602.3
-1858.85 849.319
ij 0.00119184 jj jj 0 jj jj jj -0.00514793 jj jj jj 0 k s = jjj jjj -0.00444799 jj jj 0 jj jj jj 0.00840409 jj j k0 ij 603.793 jj jj 194.181 jj jj jjj 1989.48 jj jj 416.965 j k = jjj jj -2253.38 jj jj jj -724.695 jj jj jj -339.884 jj k 113.548
zyz -1858.85 zzzz zz -7424.85 -266.026 -1602.3 849.319 zzzz zz -1556.14 -32551.7 1793.74 -6569.83 zzzz zz 8409.73 2704.6 1268.5 -423.768 zzz zz 2704.6 34989.9 -1713.88 6937.31 zzzz zz 1268.5 -1713.88 673.701 -539.1 zzzz z -423.768 6937.31 -539.1 1491.37 { -2253.38 -724.695 -339.892 113.548
1793.74 -6569.83
0
-0.00514793 0
0.00119184
0
-0.00514793 0
-0.00444799 0
0
-0.0359809
0
0
0.021
0.0192124
0.008
-0.00514793 0
-0.0359809
0
0.0192124
0
0
0
0.0166001
0
-0.03
0.0192124
-0.00444799 0
0.0192124
0
0.0166001
0
0
0.0219164
0
-0.0313645
0
0.001
0.00840409
0
0.0219164
0
-0.0313645
0
194.181
1989.48
416.965
2512.16
71.2814
8722.2
71.2814
7037.63
-654.575
8722.2
-654.575 30399.3
-724.695 -7424.83 -1556.14 -9375.52 -266.026 -32551.7 459.232
-1602.28 1793.74
-1858.84 849.319
yz ij 0.092981 z jj jj -0.00402289 zzz zzz jjj zz jj 0.180533 zzz jjj zz jj jjj -0.0412579 zzz zz r i = jj zzz jjj -0.34701 zz jj jjj 0.0150136 zzz zz jj jjj 0.0734963 zzz zz jj z j k 0.0302672 {
-0.00444799 0
-6569.81
yz z -1858.84 zzzz zz -7424.83 -266.026 -1602.28 849.319 zzzz zz -1556.14 -32551.7 1793.74 -6569.81 zzzz zz 8409.75 2704.6 1268.46 -423.768 zzz zz 2704.6 34989.9 -1713.88 6937.28 zzzz zz 1268.46 -1713.88 673.702 -539.1 zzzz z -423.768 6937.28 -539.1 1491.37 { -2253.38 -724.695 -339.884 113.548
-724.695 -9375.52 459.232
After summing contributions from all points, the element equations as follows:
175
19704. jij jj 4974.76 jj jj jj 4839.02 jj jj jj 809.59 j k = jjj jj -10834.1 jj jj jj -4506.49 jj jj jj -13708.9 jj k -1277.87
rT = H 0
4974.76
4839.02
81769.1
-2462.02 24561.4
-2462.02 13557.9 24561.4
809.59 -3219.08
-3219.08 55192.
-4506.49 -13708.8 -1277.77 -45243.6 1993.82 1993.74
-4688.06 3687.27
-61086.8 3687.28
0 0 0 0
-61082.5
0 0 0L
-18670.9
Complete element equations for element 2 19704. jij jj 4974.76 jj jj jj 4839.02 jj jj jj 809.59 jj jj jj -10834.1 jj jj jj -4506.49 jj jj jj -13708.9 jj k -1277.87
4974.76
4839.02
81769.1
-2462.02 24561.4
-2462.02 13557.9 24561.4
809.59 -3219.08
-3219.08 55192.
-4506.49 -13708.8 -1277.77 -45243.6 1993.82 1993.74
-61082.5
-4688.06 3687.27
-61086.8 3687.28
-18670.9
-10834.1 -4506.49 -13708.9 -1277.87 y zz -4506.49 -45243.6 1993.74 -61086.8 zzzz zz -13708.8 1993.82 -4688.06 3687.28 zzzz zz -1277.77 -61082.5 3687.27 -18670.9 zzzz zz 17486.5 4857.47 7056.5 926.788 zzz zz 4857.47 72633. -2344.81 33693.2 zzzz zz 7056.5 -2344.81 11340.4 -3336.2 zzzz z 926.788 33693.2 -3336.2 46064.5 {
-10834.1 -4506.49 -13708.9 -1277.87 y i Du3 y i 0. y zz jj zz jj zz -4506.49 -45243.6 1993.74 -61086.8 zzzz jjjj Dv3 zzzz jjjj 0. zzzz zz jj zz jj zz -13708.8 1993.82 -4688.06 3687.28 zzzz jjjj Du5 zzzz jjjj 0. zzzz zz jj zz jj zz -1277.77 -61082.5 3687.27 -18670.9 zzzz jjjj Dv5 zzzz jjjj 0. zzzz zz jj zz = jj zz 17486.5 4857.47 7056.5 926.788 zzz jjj Du6 zzz jjj 0. zzz zz jj zz jj zz 4857.47 72633. -2344.81 33693.2 zzzz jjjj Dv6 zzzz jjjj 0. zzzz zz jj zz jj zz 7056.5 -2344.81 11340.4 -3336.2 zzzz jjjj Du4 zzzz jjjj 0. zzzz zj z j z 926.788 33693.2 -3336.2 46064.5 { k Dv4 { k 0. {
The element contributes to 85, 6, 9, 10, 11, 12, 7, 8< global degrees of freedom. 5 jij jj 6 jj jj jj 9 jj jj jj 10 j Locations for element contributions to a global vector: jjj jjj 11 jj jjj 12 jj jjj 7 jj j k8
ij @5, 5D jj jj @6, 5D jj jj jj @9, 5D jj jj jj @10, 5D and to a global matrix: jjj jj @11, 5D jj jj jjj @12, 5D jj jj @7, 5D jj j k @8, 5D
@5, 6D @6, 6D
@9, 6D
@5, 9D
@5, 10D
@9, 9D
@9, 10D
@6, 9D
@6, 10D
@10, 6D @10, 9D @10, 10D
@11, 6D @11, 9D @11, 10D
zyz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz {
@5, 11D
@5, 12D
@9, 11D
@9, 12D
@6, 11D
@10, 11D
@11, 12D
@7, 11D
@7, 12D
@12, 11D
@8, 6D
@8, 11D
@7, 9D @8, 9D
@7, 10D
@8, 10D
Adding element equations into appropriate locations we have
@10, 12D
@11, 11D
@12, 6D @12, 9D @12, 10D @7, 6D
@6, 12D
@12, 12D @8, 12D
@5, 7D
@5, 8D y zz @6, 8D zzzz zz @9, 7D @9, 8D zzzz zz @10, 7D @10, 8D zzzz zz @11, 7D @11, 8D zzz zz @12, 7D @12, 8D zzzz zz @7, 7D @7, 8D zzzz z @8, 7D @8, 8D {
@6, 7D
176
Geometric Nonlinearity
12689.1 jij jj 4815.15 jj jj jj -9199.41 jj jj jj 2031.94 jj jj jj -7816.11 jj jj jj -4480.43 jj jj jj 4326.43 jj jj jj -2366.66 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
4815.15
-9199.41 2031.94
-7816.11 -4480.43 4326.43
-2366.66 0
7404.64
-1239.67 290.016
-4480.38 -4212.83 904.906
-3481.83 0
-1239.67 10056.9 290.016
-3250.55 2699.59
-3250.55 7495.88
792.591
-3557.12 3697.63
-2479.08 -4548.42 3697.69
0 0
0
0
-3237.48 0
0
-4480.38 2699.59
-2479.08 34020.7
9903.2
-22909.1 753.157
-4212.83 792.591
-4548.42 9903.2
90240.4
753.144
-60796.9 -2462.02 2456
-22909.1 753.144
19771.3
-6698.19 -4688.06 3687
904.906
-3557.12 3697.69
-3481.83 3697.63
-3237.48 753.157
0
0
0
0
0
0
0
0
0
-10834.1 -4506.49 7056.5
0
0
0
-4506.49 -45243.6 -2344.81 33693.2
4839.02
809.
-60796.9 -6698.19 52493.9
3687.28
-18
4839.02
-2462.02 -4688.06 3687.28
13557.9
-32
809.59
24561.4
3687.27
-18670.9 -3219.08 5519 926.788
-13708.8 -12 1993.82
-61
After assembly of all elements the global matrices are as follows. ij 12689.1 jj jj 4815.15 jj jj jjj -9199.41 jj jj 2031.94 jj jj jj -7816.11 jj jj jj -4480.43 K T = jjjj jj 4326.43 jj jj jj -2366.66 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0 ij 0. yz z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz z jj jj 0. zzz zz j zz; RE = jjjj jj 0. zzz zz jj z jj jj 0. zzz z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z j k -1. {
4815.15
-9199.41 2031.94
-7816.11 -4480.43 4326.43
-2366.66 0
7404.64
-1239.67 290.016
-4480.38 -4212.83 904.906
-3481.83 0
-1239.67 10056.9 290.016
-3250.55 2699.59
-3250.55 7495.88
792.591
-3557.12 3697.63
-2479.08 -4548.42 3697.69
0
-3237.48 0
-4480.38 2699.59
-2479.08 34020.7
9903.2
-22909.1 753.157
-4212.83 792.591
-4548.42 9903.2
90240.4
753.144
-60796.9 -
-22909.1 753.144
19771.3
-6698.19 -
904.906
-3557.12 3697.69
4
-3481.83 3697.63
-3237.48 753.157
0
0
0
0
0
0
0
0
0
-10834.1 -4506.49 7056.5
926.788
-
0
0
0
-4506.49 -45243.6 -2344.81 33693.2
1
ij 0.0325383 yz z jj jj -2.56101 zzz zz jj z jj jj -0.139661 zzz zz jj zz jj zz jj 3.15257 zz jj z jj jj 0.409935 zzz z jj jj 0.423378 zzz zz j zz; RI = jjjj jj -0.311005 zzz zz jj z jj jj -0.120975 zzz z jj jj 0.0195665 zzz zz jj z jj jj 0.391321 zzz zz jj z jj jj -0.0113738 zzz zz jj z j k -1.28529 {
-60796.9 -6698.19 52493.9
3
4839.02
-2462.02 -4688.06 3687.28
1
809.59
24561.4
3687.27
ij -0.0325383 yz zz jj zz jj 2.56101 zz jjj jj 0.139661 zzz zzz jjj zz jj jjj -3.15257 zzz zz jj jjj -0.409935 zzz zz jj jjj -0.423378 zzz zz R = RE - RI = jjjj z jj 0.311005 zzz zz jj z jj jj 0.120975 zzz zz jjj jj -0.0195665 zzz zz jj z jj jj -0.391321 zzz zz jjj jj 0.0113738 zzz zz jj z j k 0.285288 {
-18670.9 -
177
System of equations 12689.1 jij jj 4815.15 jj jj jj -9199.41 jj jj jj 2031.94 jj jj jj -7816.11 jj jj jj -4480.43 jj jj jj 4326.43 jj jj jj -2366.66 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
4815.15
-9199.41 2031.94
-7816.11 -4480.43 4326.43
-2366.66 0
7404.64
-1239.67 290.016
-4480.38 -4212.83 904.906
-3481.83 0
-1239.67 10056.9 290.016
-3250.55 2699.59
-3250.55 7495.88
792.591
-3557.12 3697.63
-2479.08 -4548.42 3697.69
0
-3237.48 0
-4480.38 2699.59
-2479.08 34020.7
9903.2
-22909.1 753.157
-4212.83 792.591
-4548.42 9903.2
90240.4
753.144
-60796.9 -2462.
-22909.1 753.144
19771.3
-6698.19 -4688.
904.906
-3557.12 3697.69
4839.02
-3481.83 3697.63
-3237.48 753.157
-60796.9 -6698.19 52493.9
3687.28
0
0
0
4839.02
-2462.02 -4688.06 3687.28
13557.9
0
0
0
809.59
24561.4
0
0
0
-10834.1 -4506.49 7056.5
926.788
-13708
0
0
0
-4506.49 -45243.6 -2344.81 33693.2
1993.82
Essential boundary conditions Node
dof
Value
1
Du1 Dv1
0 0
2
Du2 Dv2
0 0
Remove 81, 2, 3, 4< rows and columns. After adjusting for essential boundary conditions we have
3687.27
-18670.9 -3219.
178
Geometric Nonlinearity
34020.7 jij jj 9903.2 jj jj jj -22909.1 jj jj jj 753.157 jj jj jj 4839.02 jj jj jj 809.59 jj jj jj -10834.1 jj k -4506.49
9903.2
-22909.1 753.157
4839.02
809.59
90240.4
753.144
-60796.9 -2462.02 24561.4
753.144
19771.3
-6698.19 -4688.06 3687.27
-60796.9 -6698.19 52493.9
3687.28
-18670.9
-2462.02 -4688.06 3687.28
13557.9
-3219.08
24561.4
3687.27
-4506.49 7056.5
-18670.9 -3219.08 55192. 926.788
-45243.6 -2344.81 33693.2
-13708.8 -1277.77 1993.82
-61082.5
-10834.1 -4506.49 y zz -4506.49 -45243.6 zzzz zz 7056.5 -2344.81 zzzz zz 926.788 33693.2 zzzz zz -13708.8 1993.82 zzz zz -1277.77 -61082.5 zzzz zz 17486.5 4857.47 zzzz z 4857.47 72633. {
Du -0.409935 y jij 3 zyz jij z jj Dv zz jj -0.423378 zzz jj 3 zz jj zz z j jj z jj Du zzz jjj 0.311005 zzz jj 4 zz jj zz z j jj z jj Dv4 zzz jjj 0.120975 zzz zz jj jj zz zz = jj jj z jj Du5 zz jj -0.0195665 zzz zz jj jj zz z j jj z jj Dv5 zzz jjj -0.391321 zzz zz jj jj zz jjj Du zzz jjj 0.0113738 zzz jj 6 zz jj zz j z j z Dv 0.285288 k 6{ k {
Solving the final system of global equations we get
8Du3 = 0.000182584, Dv3 = -0.00016137, Du4 = 0.000296708, Dv4 = 0.000163753, Du5 = 0.000189277, Dv5 = -0.00118062, Du6 = 0.00033142, Dv6 = -0.00117187<
Complete table of nodal values Du
Dv
1
0
0
2
0
0
3
0.000182584
-0.00016137
4
0.000296708
0.000163753
5
0.000189277
-0.00118062
6
0.00033142
-0.00117187
Total increments since the start of this load step
179
Du
Dv
1
0
0
2
0
0
3
0.0012481
-0.000988377
4
0.00186139
0.00106274
5
0.00147029
-0.00745487
6
0.0023895
-0.00744933
Total nodal values u
v
1
0
0
2
0
0
3
0.0012481
-0.000988377
4
0.00186139
0.00106274
5
0.00147029
-0.00745487
6
0.0023895
-0.00744933
Solution for element 1 Initial configuration
0 jij jj 5 jj ÅÅÅÅÅ j 2 Nodal coordinates = jjjj jj ÅÅÅÅ5Å jj 2 jj k0 5s 5 x0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4
0y zz z 0 zzzz zz zz 3 zzz zz z 4{
ts s 7t 7 y0 Hs,tL = - ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4 4 4 ij ÅÅ54ÅÅ J = jjjj t 1 k - ÅÅ4ÅÅ - ÅÅÅÅ4Å
Current configuration II
yz zz z 7 s z ÅÅÅÅ4Å - ÅÅ4ÅÅ {
0
35 5s detJ = ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ 16 16
ij 0 jj jj 0 Nodal displacements = jjjj jj 0.0012481 jj k 0.00186139
yz zz zz zz z -0.000988377 zzzz zz 0.00106274 {
0
0
180
Geometric Nonlinearity
0 jij jj 5 jj ÅÅÅÅÅ Updated coordinates = jjjj 2 jj 2.50125 jj j k 0.00186139
zyz zz zz zz zz 2.99901 zzz zz 4.00106 { 0 0
xII Hs,tL = -0.000153322 t s + 1.24985 s + 0.000777372 t + 1.25078 yII Hs,tL = -0.250513 t s - 0.250513 s + 1.75002 t + 1.75002
i 1.24985 - 0.000153322 t 0.000777372 - 0.000153322 s yz J II = jj z k -0.250513 t - 0.250513 1.75002 - 0.250513 s { detJ II = -0.313141 s - 0.0000735751t + 2.18745
i -0.3125 s - 0.0000739709 t + 2.18743 0.000971716 - 0.000191653 s Deformation gradient, F IIp = H1êdetJ L jj 2.18752 - 0.313141 s k -0.000892714 t - 0.000892714
Solution at 8s, t< = 8-0.57735, -0.57735< ï 8x, y< = 80.528678, 0.800774< 0.000865893 y i 1.24994 zz J II = jj { k -0.105879 1.89465
yz z {
detJ II = 2.36828
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.333104, 0.31075, 0.0892549, -0.0669014<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.20798, -0.0559108, 0.055728, 0.208162< B TL
0.31075 0 0.0892549 0 -0.0669014 0 -0.333104 0 jij j j -0.20798 0 -0.0559108 0 0.055728 0 0.20 = jj 0 jj -0.20798 -0.333104 -0.0559108 0.31075 0.055728 0.0892549 0.208162 -0. k
0.000457095 y i 0.999987 zz; F IIp = jj { k -0.00015934 1.00017
Det@F IIp D = 1.00015
Element thickness, hII = 1.2499
0.000297833 y i 0.999974 zz Left Cauchy-Green tensor = jj 0.000297833 1.00033 k {
i -0.0000131903 0.000148861 yz Green-Lagrange strain tensor, e = jj z 0.000166215 { k 0.000148861 i 0.543784 1.16919 yz Cauchy stress tensor, s = jj z k 1.16919 1.9509 { Principal stresses = H 2.61189
-0.117206 L
181
Effective stress Hvon MisesL = 2.67243
i 0.542768 1.16829 yz z Second PK stresses = jj k 1.16829 1.95077 {
r Ti = H -1.25599 -2.35391 0.3067 0.752606 0.336541 0.630728 0.612747 0.970575 L
Solution at 8s, t< = 8-0.57735, 0.57735< ï 8x, y< = 80.529678, 2.98853< 0.000865893 yz i 1.24976 J II = jj z -0.395146 1.89465 k {
detJ II = 2.3682
8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.150331, 0.0669038, 0.333116, -0.249688<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.208063, -0.0557994, 0.0556165, 0.208246< B TL =
-0.150331 0 0.0669038 0 0.333116 0 -0.249688 0 jij zyz jj 0 -0.208063 0 -0.0557994 0 0.0556165 0 0.208246 zzzz jj jj zz 0.0556165 0.333116 0.208246 -0.249688 { k -0.208063 -0.150331 -0.0557994 0.0669038 0.000457095 y i 0.999951 zz; F IIp = jj k -0.000594666 1.00017 {
Det@F IIp D = 1.00012
Element thickness, hII = 1.24992
-0.000137466 y i 0.999902 zz Left Cauchy-Green tensor = jj -0.000137466 1.00033 { k
i -0.0000490964 -0.000068846 yz Green-Lagrange strain tensor, e = jj z k -0.000068846 0.000166215 {
i 0.108928 -0.53965 yz Cauchy stress tensor, s = jj z k -0.53965 1.80056 { Principal stresses = H 1.95805
-0.0485642 L
Effective stress Hvon MisesL = 1.98278
-0.540374 y i 0.109438 Second PK stresses = jj zz k -0.540374 1.79942 {
r Ti = H 0.283888 -0.868791 0.110706 -0.40427 0.0185657 -0.235695 -0.41316 1.50876 L
Solution at 8s, t< = 80.57735, -0.57735< ï 8x, y< = 81.97198, 0.678516<
182
Geometric Nonlinearity
0.000688852 yz i 1.24994 J II = jj z k -0.105879 1.60538 {
detJ II = 2.0067
8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.32105, 0.294669, 0.105338, -0.0789562<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.0656797, -0.245761, 0.245589, 0.0658514< B TL =
-0.32105 0 0.294669 0 0.105338 0 -0.0789562 0 jij jj 0 -0.0656797 0 -0.245761 0 0.245589 0 0.0658514 jj jj -0.0656797 -0.32105 -0.245761 0.294669 0.245589 0.105338 0.0658514 -0.0789562 k 0.000429014 y i 0.999984 zz; F IIp = jj { k -0.000187987 0.999827
Det@F IIp D = 0.999812
Element thickness, hII = 1.25013
0.000240955 y i 0.999969 zz Left Cauchy-Green tensor = jj { k 0.000240955 0.999654
i -0.000015559 0.000120526 Green-Lagrange strain tensor, e = jj -0.000172696 k 0.000120526
i -0.917838 0.946055 yz Cauchy stress tensor, s = jj z -2.15293 { k 0.946055
yz z {
Principal stresses = H -2.66515 -0.405612 L
Effective stress Hvon MisesL = 2.48728
i -0.918599 0.946902 yz Second PK stresses = jj z -2.15313 { k 0.946902
r Ti = H 0.583345 -0.407219 -1.26174 2.02667
0.340316 -1.0764 0.338083 -0.543044 L
Solution at 8s, t< = 80.57735, 0.57735< ï 8x, y< = 81.97277, 2.53225< 0.000688852 yz i 1.24976 J II = jj z k -0.395146 1.60538 {
detJ II = 2.00662
8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
yz zz zz zz z {
183
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.105342, 0.00688115, 0.393142, -0.294681<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.0657723, -0.245637, 0.245466, 0.0659439< B TL =
0.00688115 0 0.393142 0 -0.294681 0 ij -0.105342 0 yz jj zz jj 0 zz -0.0657723 0 -0.245637 0 0.245466 0 0.0659439 jj zz j z k -0.0657723 -0.105342 -0.245637 0.00688115 0.245466 0.393142 0.0659439 -0.294681 { 0.000429014 y i 0.999942 zz; F IIp = jj k -0.000701579 0.999827 {
Det@F IIp D = 0.999769
Element thickness, hII = 1.25016
-0.000272598 y i 0.999884 zz Left Cauchy-Green tensor = jj k -0.000272598 0.999655 {
i -0.0000578855 -0.000136234 yz Green-Lagrange strain tensor, e = jj z k -0.000136234 -0.000172696 {
i -1.43105 -1.07031 yz Cauchy stress tensor, s = jj z k -1.07031 -2.3302 { Principal stresses = H -3.04152 -0.719721 L
Effective stress Hvon MisesL = 2.75314
i -1.43014 -1.07045 yz Second PK stresses = jj z k -1.07045 -2.33225 {
r Ti = H 0.554764 0.667311 0.634825 1.41739
-2.07041 -2.49044 0.880817 0.405735 L
After summing contributions from all points the internal load vector is as follows: r Ti = H 0.16601 -2.96261 -0.209513 3.79239
-1.37498 -3.17181 1.41849
Global internal load vector
RTI = H 0.16601 -2.96261 -0.209513 3.79239
Solution for element 2 Initial configuration
ij ÅÅÅÅ52Å jj jj j 10 Nodal coordinates = jjjj jj 10 jj j k0
3 yz zz z 3 zzzz zz 4 zzz zz 4{
-1.37498 -3.17181 1.41849
2.34202 L
2.34202 0 0 0 0 L
184
Geometric Nonlinearity
5ts 35 s 5t 45 x0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 8 8 8 8 t 7 y0 Hs,tL = ÅÅÅÅÅÅ + ÅÅÅÅÅ 2 2 5t 35 ij ÅÅÅÅ ÅÅÅ + ÅÅÅÅ ÅÅ 8 8 j j J = jj k0
5s ÅÅÅÅ ÅÅÅÅ - ÅÅÅÅ58Å yz 8 zz zz ÅÅ12ÅÅ {
5t 35 detJ = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 16 16
Current configuration II
ij 0.0012481 jj jj 0.00147029 Nodal displacements = jjjj jj 0.0023895 jjj k 0.00186139
ij 2.50125 jj jj 10.0015 Updated coordinates = jjjj jj 10.0024 jj k 0.00186139
-0.000988377 y zz -0.00745487 zzzz zz -0.00744933 zzzz zz 0.00106274 {
2.99901 y zz 2.99255 zzzz zz 3.99255 zzzz z 4.00106 {
xII Hs,tL = 0.625076 t s + 4.37519 s - 0.624617 t + 5.62674
yII Hs,tL = -0.000511394 t s - 0.00374464 s + 0.500514 t + 3.49629
0.625076 s - 0.624617 i 0.625076 t + 4.37519 J II = jj k -0.000511394 t - 0.00374464 0.500514 - 0.000511394 s
yz z {
detJ II = 0.000103241s + 0.31254 t + 2.1875
Deformation gradient, F IIp = H1êdetJ L
0.000217372 s + 0.000287254 t + 0.00179341 y ij 0.312538 t + 2.18759 zz j { k -0.000255697 t - 0.00187232 0.000103051s + 0.312502 t + 2.18741
Solution at 8s, t< = 8-0.57735, -0.57735< ï 8x, y< = 83.66971, 3.20931< -0.985505 y i 4.0143 zz J II = jj -0.00344939 0.500809 k {
detJ II = 2.007
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.0990773, 0.098218, 0.0265477, -0.0256884<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.982367, -0.0177075, 0.263225, 0.73685<
185
B TL
0.098218 0 0.0265477 0 -0.0256884 0 -0.0990773 0 jij j j = jj 0 -0.982367 0 -0.0177075 0 0.263225 0 0 jj 0.263225 0.0265477 0.73685 k -0.982367 -0.0990773 -0.0177075 0.098218
0.000748383 zy i 1.00004 z; F IIp = jj k -0.000859306 0.999925 {
Det@F IIp D = 0.999961
Element thickness, hII = 1.25003
-0.00011101 y i 1.00007 zz Left Cauchy-Green tensor = jj k -0.00011101 0.99985 {
-0.0000554155 y i 0.0000360978 zz Green-Lagrange strain tensor, e = jj k -0.0000554155 -0.000075168 { -0.435824 y i 0.117495 zz Cauchy stress tensor, s = jj k -0.435824 -0.754763 {
Principal stresses = H -0.935198 0.29793 L Effective stress Hvon MisesL = 1.11444
-0.435167 y i 0.118136 zz Second PK stresses = jj k -0.435167 -0.755611 { r Ti =
H 1.04491 1.96849
0.0483133 -0.0738611 -0.279983 -0.527456 -0.813242 -1.36718 L
Solution at 8s, t< = 8-0.57735, 0.57735< ï 8x, y< = 82.53175, 3.7876< -0.985505 y i 4.73608 zz J II = jj -0.00403989 0.500809 k {
detJ II = 2.36789
8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.0230204, 0.0221674, 0.0835828, -0.0827297<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.832701, -0.167362, 0.37546, 0.624603<
0.0221674 0 0.0835828 0 -0.0827297 0 ij -0.0230204 0 j -0.832701 0 -0.167362 0 0.37546 0 0.62 B TL = jjjj 0 jj 0.0835828 0.624603 -0. k -0.832701 -0.0230204 -0.167362 0.0221674 0.37546 0.000774416 y i 1.00005 zz; F IIp = jj -0.000853046 0.999937 { k Element thickness, hII = 1.25001
Det@F IIp D = 0.999986
186
Geometric Nonlinearity
-0.0000787213 yz i 1.0001 Left Cauchy-Green tensor = jj z k -0.0000787213 0.999875 {
-0.0000392695 y i 0.0000492962 zz Green-Lagrange strain tensor, e = jj k -0.0000392695 -0.0000628075 { -0.309056 y i 0.329401 zz Cauchy stress tensor, s = jj k -0.309056 -0.549821 {
Principal stresses = H -0.647586 0.427167 L Effective stress Hvon MisesL = 0.937266
-0.308351 y i 0.329845 Second PK stresses = jj zz k -0.308351 -0.550414 {
r Ti = H 0.739284 1.3762 0.17471 0.252088 -0.261966 -0.687484 -0.652028 -0.940805 L
Solution at 8s, t< = 80.57735, -0.57735< ï 8x, y< = 88.30502, 3.20533< -0.263729 y i 4.0143 zz J II = jj k -0.00344939 0.500219 {
detJ II = 2.00712
8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.0984593, 0.0976, 0.0270111, -0.0261519<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.263143, -0.736873, 0.802571, 0.197444<
0.0976 0 0.0270111 0 -0.0261519 0 ij -0.0984593 0 j B TL = jjjj 0 -0.263143 0 -0.736873 0 0.802571 0 0.19 jj 0.802571 0.0270111 0.197444 -0. k -0.263143 -0.0984593 -0.736873 0.0976 0.00087344 y i 1.00004 zz; F IIp = jj -0.000859306 0.999984 { k
Det@F IIp D = 1.00002
Element thickness, hII = 1.24999
0.0000140894 y i 1.00007 zz Left Cauchy-Green tensor = jj k 0.0000140894 0.999968 {
i 0.0000360978 Green-Lagrange strain tensor, e = jjjj -6 k 7.08968 µ 10
i 0.369411 0.0553136 Cauchy stress tensor, s = jj k 0.0553136 -0.0381346
yz z {
7.08968 µ 10-6 zy zz z -0.0000157826 {
187
Principal stresses = H 0.376785 -0.0455085 L Effective stress Hvon MisesL = 0.401478
i 0.369291 0.0556636 yz Second PK stresses = jj z k 0.0556636 -0.0380408 { r Ti = H -0.12777 0.0115125 -0.011803 0.0840445 0.136411 -0.0730373 0.00316262 -0.0225197 L
Solution at 8s, t< = 80.57735, 0.57735< ï 8x, y< = 88.00049, 3.78293< -0.263729 y i 4.73608 zz J II = jj k -0.00403989 0.500219 {
detJ II = 2.36801
8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.0225004, 0.0216474, 0.0839727, -0.0831197<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.223095, -0.776917, 0.832603, 0.167409< B TL
0.0216474 0 0.0839727 0 -0.0831197 0 ij -0.0225004 0 jj j -0.223095 0 -0.776917 0 0.832603 0 0.16 = jj 0 jj -0. k -0.223095 -0.0225004 -0.776917 0.0216474 0.832603 0.0839727 0.167409
0.000880415 y i 1.00005 zz; F IIp = jj k -0.000853046 0.999987 {
Det@F IIp D = 1.00004
Element thickness, hII = 1.24998
0.000027316 y i 1.0001 zz Left Cauchy-Green tensor = jj { k 0.000027316 0.999975
i 0.0000492962 0.0000137115 yz Green-Lagrange strain tensor, e = jj z k 0.0000137115 -0.0000124696 { i 0.542931 0.107239 yz Cauchy stress tensor, s = jj z k 0.107239 0.0575902 { Principal stresses = H 0.56557
0.0349512 L
Effective stress Hvon MisesL = 0.548929
i 0.542697 0.107649 yz Second PK stresses = jj z k 0.107649 0.0577759 { r Ti =
H -0.106975 -0.0451719 -0.211822 -0.125565 0.399234 0.168584 -0.0804381 0.00215335 L
188
Geometric Nonlinearity
After summing contributions from all points the internal load vector is as follows: r Ti =
H 1.54945
3.31103
-0.00060097 0.136706 -0.00630497 -1.11939 -1.54255 -2.32835 L
Global internal load vector
RTI = H 0.16601 -2.96261 -0.209513 3.79239
0.174468 0.139225 -0.124059 0.0136769 -0.00060097
After assembling all element internal force vectors, the global internal force and the external load vectors are as follows. yz ij 0.16601 zz jj zz jj -2.96261 zz jj z jj jj -0.209513 zzz zz jj zz jj zz jj 3.79239 zz jj zz jj zz jjj 0.174468 zz jj zz zz jjj 0.139225 zz; j RI = jj jj -0.124059 zzz zz jj z jj jj 0.0136769 zzz z jj jj -0.00060097 zzz zz jj zzz jjj zz jj 0.136706 zz jjj jj -0.00630497 zzz zzz jjj { k -1.11939
ij 0. yz z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj zz jj jjj 0. zzz zz jj jjj 0. zzz zz j RE = jj jj 0. zzz zz jj z jj jj 0. zzz z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z j k -1. {
Corresponding to unrestrained dof 0.174468 jij zyz jj 0.139225 zz jj zz jj z jj -0.124059 zzz jj zz jj z jj 0.0136769 zzz jj zz zz; RI = jj jjj -0.00060097 zzz jj zz jjj 0.136706 zzz jj zz jjj -0.00630497 zzz jj zz j z k -1.11939 {
»»RE »» = 1.;
0. y jij z jj 0. zzz jj zz jj z jj 0. zzz jj zz jj z jj 0. zzz jj zz zz; RE = jj jjj 0. zzz jj z jj 0. zzz jj zz jj z jj 0. zzz jj zz j z k -1. {
»»R »» = 0.313662
Convergence parameter = 0.049192
Iteration = 3 Global equations at start of the element assembly process
-0.174468 y jij z jj -0.139225 zzz jj zz jj z jj 0.124059 zzz jj zz jj z jj -0.0136769 zzz jj zz zz R = RE - RI = jj jjj 0.00060097 zzz jj zz jjj -0.136706 zzz jj zz jjj 0.00630497 zzz jj zz j z k 0.119394 {
189
0 jij jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
E = 10600.;
n = 0.35;
0 y i Du1 y i 0 y zz jj zz zz jj 0 zzzz jjjj Dv1 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzzz jjjj Du2 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzzz jjjj Dv2 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzz jjj Du3 zzz jjj 0 zzz zz jj zz zz jj 0 zzzz jjjj Dv3 zzzz jjjj 0 zzzz zz jj zz zz = jj 0 zzzz jjjj Du4 zzzz jjjj 0 zzzz zz zz jj zz jj 0 zzz jjj Dv4 zzz jjj 0 zzz zz zzz jjj zzz jjj 0 zzz jjj Du5 zzz jjj 0 zzzz zz jj zz zz jj 0 zzzz jjjj Dv5 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzz jjj Du6 zzz jjj 0 zzz zz jj zz zz jj 0 { k Dv6 { k -1 {
Initial thickness = 1.25 g = 0.461538
Plane stress analysis.
Interpolation functions and their derivatives 1 1 1 1 NT = : ÅÅÅÅÅ Hs - 1L Ht - 1L, - ÅÅÅÅÅ Hs + 1L Ht - 1L, ÅÅÅÅÅ Hs + 1L Ht + 1L, - ÅÅÅÅÅ Hs - 1L Ht + 1L> 4 4 4 4
t-1 1-t t+1 1 ∑NT ê∑s = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-t - 1L> 4 4 4 4
s -1 1 s +1 1-s ∑NT ê∑t = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-s - 1L, ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ > 4 4 4 4
Mapping to the master element
ij 0 jj jj ÅÅÅÅ5Å jj 2 Initial coordinates = jjjj jj ÅÅÅÅ5Å jj 2 jj k0 5s 5 x0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4
0y zz z 0 zzzz zz zz 3 zzz zz z 4{
ts s 7t 7 y0 Hs,tL = - ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4 4 4 5 ji ÅÅ4ÅÅ J = jjj j - ÅÅtÅÅ - ÅÅÅÅ1Å k 4 4
Current configuration
0 ÅÅÅÅ74Å
-
ÅÅ4sÅÅ
zyz zz z {
35 5s detJ = ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ 16 16
190
Geometric Nonlinearity
0 jij jj 5 jj ÅÅÅÅÅ Updated coordinates = jjjj 2 jj 2.50125 jj j k 0.00186139
zyz zz zz zz zz 2.99901 zzz zz 4.00106 { 0 0
xHs,tL = -0.000153322 t s + 1.24985 s + 0.000777372 t + 1.25078 yHs,tL = -0.250513 t s - 0.250513 s + 1.75002 t + 1.75002
i -0.3125 s - 0.0000739709 t + 2.18743 0.000971716 - 0.000191653 s Deformation gradient, F p = 1êdetJ jj 2.18752 - 0.313141 s k -0.000892714 t - 0.000892714
Gauss quadrature points and weights Point
Weight
1
s Ø -0.57735 t Ø -0.57735
1.
2
s Ø -0.57735 t Ø 0.57735
1.
3
s Ø 0.57735 t Ø -0.57735
1.
4
s Ø 0.57735 t Ø 0.57735
1.
Computation of element matrices at 8-0.57735, -0.57735< with weight = 1. i ÅÅÅÅ5Å 0 J = jjjj 4 k -0.105662 1.89434
i 0.8 J -T = jj k0
0.0446224 y zz 0.527889 {
yz zz z {
detJ = 2.36792
0.000457095 y i 0.999987 zz; Deformation gradient, F p = jj k -0.00015934 1.00017 {
0 -0.00015934 0 ij 0.999987 yz _ j zz zz F = jjjj 0 0.000457095 0 1.00017 zz jj z 1.00017 -0.00015934 { k 0.000457095 0.999987
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.333066, 0.310755, 0.0892449, -0.0669336<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.208166, -0.055778, 0.055778, 0.208166<
yz z {
191
0.0000530709 0.310751 -0.0000495158 0.0892437 -0.00001422 -0.333062 jij j j G = jj -0.000095152 -0.208201 -0.0000254959 -0.0557873 0.0000254959 0.0557873 jj -0.333089 -0.0556353 0.310816 0.0558181 0.0892508 k -0.208316 T
0.542768 jij jj 1.16829 j S = jjjj jj 0 jj k0
zyz zz zz zz 0.542768 1.16829 zzzz z 1.16829 1.95077 {
1.16829
0
0
1.95077
0
0
0 0
T
B =
ij -0.333066 jj jj -0.208166 jj jj jj 0 jj k0
0
0.310755
0
-0.055778 0
0
-0.0669336 0
0
0.208166
0
0.0892449 0
-0.333066 0
0.310755
-0.208166 0
-0.055778 0
6089.42 jij jj 2683.35 jj jj jj -5076.84 jj jj jj -245.92 j k c = jjj jjj -1631.66 jj jj -719.002 jj jj jj 619.075 jj j k -1718.43
yz zz zz zz z -0.0669336 zzzz z 0.208166 {
0.0892449 0 0.055778
0.055778
0
0
-1718.43 y zz 3467.12 -1536. -617.152 -719.002 -929.013 -428.35 -1920.96 zzzz zz -1536. 4898.51 -672.952 1360.34 411.57 -1182. 1797.38 zzzz zz -617.152 -672.952 1279.34 65.8941 165.365 852.978 -827.555 zzzz zz -719.002 1360.34 65.8941 437.201 192.656 -165.881 460.452 zzz zz -929.013 411.57 165.365 192.656 248.928 114.776 514.719 zzzz zz -428.35 -1182. 852.978 -165.881 114.776 728.807 -539.404 zzzz z -1920.96 1797.38 -827.555 460.452 514.719 -539.404 2233.79 { 2683.35
-5076.84 -245.92
-1631.66 -719.002 619.075
ks = ij 0.907939 jj jj 0 jj jj jjj -0.25869 jj jj 0 jj jj jj -0.243282 jj jj jj 0 jj jj jj -0.405968 jj k0
0
-0.25869
0 -0.25869
0.907939
0
0
0.0532282 0
-0.25869
0
0
0.0693156 0
-0.243282 0 0
0.136146
-0.405968 0
-0.243282 0 0
-0.243282 0
0.0693156
0
0.136146
0.0693156
0
0.0532282 0 0.0651871
0.0693156 0
yz z -0.405968 zzzz zz zz 0 zz zz 0.136146 zzzz zz zz 0 zz z 0.108779 zzzz zz zz 0 zz z 0.161044 {
-0.405968 0
0
0.108779
0.0651871
0
0
0.108779
0
0.161044
0.136146
0
0.108779
0
192
Geometric Nonlinearity
6090.33 jij jj 2683.35 jj jj jj -5077.1 jj jj jj -245.92 j k = jjj jj -1631.9 jj jj jj -719.002 jj jj jj 618.669 jj k -1718.43
-1718.43 y zz 3468.03 -1536. -617.411 -719.002 -929.256 -428.35 -1921.36 zzzz zz -1536. 4898.56 -672.952 1360.41 411.57 -1181.86 1797.38 zzzz zz -617.411 -672.952 1279.4 65.8941 165.435 852.978 -827.419 zzzz zz -719.002 1360.41 65.8941 437.266 192.656 -165.772 460.452 zzz zz -929.256 411.57 165.435 192.656 248.993 114.776 514.828 zzzz zz -428.35 -1181.86 852.978 -165.772 114.776 728.968 -539.404 zzzz z -1921.36 1797.38 -827.419 460.452 514.828 -539.404 2233.96 { -5077.1
2683.35
-245.92
-1631.9
-719.002 618.669
ij 1.25599 yz z jj jj 2.35391 zzz zz jj z jj jj -0.3067 zzz zz jj z jj jj -0.752606 zzz zz r i = jjj z jjj -0.336541 zzz zz jj jjj -0.630728 zzz zz jj z jj jj -0.612747 zzz zz jj k -0.970575 {
Computation of element matrices at 8-0.57735, 0.57735< with weight = 1. i ÅÅÅÅ5Å 0 J = jjjj 4 k -0.394338 1.89434 i 0.8 J -T = jj k0
0.166533 y zz 0.527889 {
yz zz z {
detJ = 2.36792
0.000457095 y i 0.999951 zz; Deformation gradient, F p = jj k -0.000594666 1.00017 {
0 -0.000594666 0 ij 0.999951 yz _ j zz zz F = jjjj 0 0.000457095 0 1.00017 zz jj z 1.00017 -0.000594666 { k 0.000457095 0.999951 8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.1502, 0.0669336, 0.333066, -0.2498<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.208166, -0.055778, 0.055778, 0.208166<
0.000089319 0.0669303 -0.0000398032 0.33305 -0.000198063 ij -0.150193 j GT = jjjj -0.000095152 -0.208201 -0.0000254959 -0.0557873 0.0000254959 0.0557873 jj -0.150101 -0.0557447 0.0669779 0.0559275 0.333089 k -0.208225
193
0.109438 jij jj -0.540374 j S = jjjj jj 0 jj k0
zyz zz zz zz 0.109438 -0.540374 zzzz z -0.540374 1.79942 {
-0.540374 0
0
1.79942
0
0
0 0
BT =
ij -0.1502 jj jj -0.208166 jj jj jj 0 jj k0
yz zz zz zz z -0.2498 zzzz z 0.208166 {
0
0.0669336
0
0.333066 0
-0.2498
0
-0.055778 0
0.055778 0
0.208166 0
-0.1502
0
0
0.333066 0
-0.055778 0
0.055778 0
0.0669336
-0.208166 0
1640.92 jij jj 1211.51 jj jj jj -371.534 jj jj jj 65.7069 j k c = jjj jj -2655.96 jj jj jj -1031.99 jj jj jj 1386.58 jj k -245.221
ij 0.138086 jj jj 0 jj jj jj 0.0674712 jj jj jj 0 k s = jjj jjj 0.0462485 jj jj 0 jj jj jj -0.251806 jj j k0 ij 1641.05 jj jj 1211.51 jj jj jjj -371.466 jj jj 65.7069 j k = jjj jj -2655.92 jj jj jj -1031.99 jj jj jj 1386.33 jj k -245.221
0
-245.221 y zz 2442.33 -279.971 467.716 -1976.4 -1164.51 1044.87 -1745.54 zzzz zz -279.971 261.557 -144.598 1086.12 -115.077 -976.145 539.646 zzzz zz 467.716 -144.598 208.782 -460.755 102.686 539.646 -779.184 zzzz zz -1976.4 1086.12 -460.755 5623.3 717.598 -4053.46 1719.56 zzz zz -1164.51 -115.077 102.686 717.598 1445.05 429.474 -383.23 zzzz zz 1044.87 -976.145 539.646 -4053.46 429.474 3643.02 -2013.99 zzzz z -1745.54 539.646 -779.184 1719.56 -383.23 -2013.99 2907.95 { -371.534 65.7069
1211.51
-2655.96 -1031.99 1386.58
0
-0.251806
0
0
-
0
-0.11183
0
0.0143938
0
-
0.0674712 0 0
0
0.0299646 0
0.0674712
0
0
0.0143938 0
0.0462485
0
0.0143938 0
-0.00692401 0
-
0
-0.11183
0
-0.0537182
0
0.417354
0
-0.11183
0
-0.0537182
0
0.
-0.251806 0
0.0462485
0.0462485
0 0.138086
0.0674712 0 0.0143938
0.0299646 0
-0.00692401 0
-0.0537182 0
-245.221 y zz -1745.79 zzzz zz -279.971 261.587 -144.598 1086.14 -115.077 -976.257 539.646 zzzz zz 467.783 -144.598 208.812 -460.755 102.701 539.646 -779.296 zzzz zz -1976.4 1086.14 -460.755 5623.3 717.598 -4053.51 1719.56 zzz zz -1164.46 -115.077 102.701 717.598 1445.05 429.474 -383.284 zzzz zz 1044.87 -976.257 539.646 -4053.51 429.474 3643.44 -2013.99 zzzz z -1745.79 539.646 -779.296 1719.56 -383.284 -2013.99 2908.37 { 1211.51
-371.466 65.7069
-2655.92 -1031.99 1386.33
2442.47
-279.971 467.783
-1976.4
-1164.46 1044.87
194
Geometric Nonlinearity
-0.283888 y z jij jj 0.868791 zzz zz jj z jj jj -0.110706 zzz zz jj zz jj zz jj 0.40427 zz j z r i = jjj jj -0.0185657 zzz zz jj z jj jj 0.235695 zzz zz jj zz jj zz jj 0.41316 zz jj -1.50876 { k
Computation of element matrices at 80.57735, -0.57735< with weight = 1. i ÅÅÅÅ5Å 0 J = jjjj 4 k -0.105662 1.60566 i 0.8 J -T = jj k0
0.0526449 y zz 0.622796 {
zyz zz {
detJ = 2.00708
0.000429014 y i 0.999984 Deformation gradient, F p = jj zz; { k -0.000187987 0.999827
0.999984 0 -0.000187987 0 jij zyz j zz j F = jj 0 0.000429014 0 0.999827 zz jj zz 0.999827 -0.000187987 { k 0.000429014 0.999984 _
8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.321033, 0.29471, 0.10529, -0.0789674<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.0658061, -0.245592, 0.245592, 0.0658061<
0.0000603501 0.294706 -0.0000554018 0.105288 -0.000019793 ij -0.321028 j -0.000105362 -0.245549 0.000105362 0.245549 GT = jjjj -0.0000282318 -0.0657948 jj -0.320965 -0.245462 0.294705 0.245633 0.105225 k -0.0659428
ij -0.918599 jj jj 0.946902 S = jjjj jj 0 jj k0
0.946902
yz zz zz zz z -0.918599 0.946902 zzzz z 0.946902 -2.15313 { 0
0
-2.15313 0
0
0 0
195
BT =
ij -0.321033 jj jj -0.0658061 jj jj jj 0 jj k0
0
0.29471
-0.245592 0
0.245592 0
0.0658061
-0.321033
0
0
0
0.29471
-0.0658061 0
ij 4442.21 jj jj 692.35 jj jj jj -3879.16 jj jj jj 1622.67 k c = jjj jjj -1602.47 jj jj -1880.23 jj jj jj 1039.42 jj j k -434.794
0 0.10529
-0.245592 0 -3879.16 1622.67
692.35
0.245592 0 -1602.47 -1880.23 1039.42
1199.57
332.508
-242.983 -935.763 -1021.69 -89.0953
332.508
4301.37
-2377.76 730.339
-1152.55
1408.13
-242.983 -2377.76 3433.53
117.969
-2270.54 637.119
-935.763 730.339
117.969
1067.82
849.404
-195.694
-1021.69 1408.13
-2270.54 849.404
2683.84
-377.308
-89.0953 -1152.55 637.119
ij -0.160537 jj jj 0 jj jj jjj 0.271972 jj jj 0 j k s = jjj jj -0.0385604 jj jj jj 0 jj jj jj -0.0728747 jj k0 ij 4442.05 jj jj 692.35 jj jj jj -3878.89 jj jj jj 1622.67 k = jjj jjj -1602.51 jj jjj -1880.23 jj jj jj 1039.35 jj k -434.794
0.10529
65.1072
637.119
-195.694 -377.308 308.824
-920.012 -31.6097 608.388
-170.715
0
-434.794 y zz 65.1072 zzzz zz 637.119 zzzz zz -920.012 zzzz zz -31.6097 zzz zz 608.388 zzzz zz -170.715 zzzz z 246.516 {
0
0.271972
0
-0.0385604 0
-0.160537
0
0.271972
0
-0.0385604 0
0.364818
0
0.233081
0.364818
0
0
-0.869871 0
0.271972
0
0
0.364818
-0.869871 0
-0.0385604 0 0
0.233081
-0.0728747 0
692.35
-0.0728747
0
-0.228505
0
-0.0977527
0.364818
0
-0.228505
0
0
-0.0977527 0
0.233081
0
-3878.89 1622.67
-0.0624539
-0.0977527 0
-1602.51 -1880.23 1039.35
1199.41
332.508
-242.711 -935.763 -1021.73 -89.0953
332.508
4300.5
-2377.76 730.704
1408.13
-1152.32
-242.711 -2377.76 3432.66
117.969
-2270.17 637.119
-935.763 730.704
117.969
1067.6
849.404
-195.792
-1021.73 1408.13
-2270.17 849.404
2683.61
-377.308
-89.0953 -1152.32 637.119 65.0343
ij -0.583345 yz z jj jj 0.407219 zzz zz jj z jj jj 1.26174 zzz zz jjj jj -2.02667 zzz j zzz j r i = jj jj -0.340316 zzz zz jj zz jj zzz jjj 1.0764 zz jj jjj -0.338083 zzz zz jj k 0.543044 {
yz zz zz zz z -0.0789674 zzzz z 0.0658061 {
-0.0789674 0
0 0
637.119
-195.792 -377.308 308.762
-919.779 -31.6097 608.29
-170.715
-434.794 y zz 65.0343 zzzz zz 637.119 zzzz zz -919.779 zzzz zz -31.6097 zzz zz zz 608.29 zz zz -170.715 zzzz z 246.454 {
196
Geometric Nonlinearity
Computation of element matrices at 80.57735, 0.57735< with weight = 1. i ÅÅÅÅ5Å 0 J = jjjj 4 k -0.394338 1.60566 0.196473 y zz 0.622796 {
i 0.8 J -T = jj k0
yz zz z {
detJ = 2.00708
0.000429014 y i 0.999942 zz; Deformation gradient, F p = jj k -0.000701579 0.999827 {
0 -0.000701579 0 ij 0.999942 yz _ j zz zz 0.000429014 0 0.999827 F = jjjj 0 zz jj z 0.999827 -0.000701579 { k 0.000429014 0.999942 8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.10529, 0.00705308, 0.392947, -0.29471<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.0658061, -0.245592, 0.245592, 0.0658061< 0.0000738691 0.00705267 -4.94829 µ 10-6 jij -0.105284 jj G = jjj -0.0000282318 -0.0657948 -0.000105362 -0.245549 jj -0.105225 -0.245575 0.00722416 k -0.0658475 T
ij -1.43014 jj jj -1.07045 S = jjjj jj 0 jjj k0
yz zz zz zz z -1.43014 -1.07045 zzzz zz -1.07045 -2.33225 {
-1.07045 0
0
-2.33225 0
0
0 0
0.245746
BT =
-0.10529 jij jj -0.0658061 jj jj jj 0 jj j k0
-0.00027568
0.392924
0.000105362 0.245549 0.392707
zyz zz zz zz -0.29471 zzzz z 0.0658061 {
0
0.00705308 0
0.392947 0
-0.29471
0
-0.245592
0
0.245592 0
0.0658061 0
-0.10529
0
0.00705308 0
0.392947 0
-0.245592
0.245592 0
-0.0658061 0
0
0
197
516.307 jij jj 227.495 jj jj jj 128.228 jj jj jj 590.229 j k c = jjj jj -1926.89 jj jj jj -849.024 jj jj jj 1282.35 jj k 31.2995
ij -0.102331 jj jj 0 jj jj jj -0.160099 jj jj jj 0 k s = jjj jjj 0.381903 jj jj 0 jj jj jj -0.119473 jj j k0 ij 516.205 jj jj 227.495 jj jj jjj 128.068 jj jj 590.229 j k = jjj jj -1926.5 jj jj jj -849.024 jj jj jj 1282.23 jj k 31.2995
227.495
128.228
590.229
-1926.89 -849.024 1282.35
31.2995
293.988
244.528
682.503
-849.024 -1097.18 377.001
120.686
244.528
596.467
-55.4987 -478.553 -912.591 -246.142 723.562
682.503
-55.4987 2576.06
-2202.77 -2547.14 1668.03
-711.422
-849.024 -478.553 -2202.77 7191.23
3168.6
-4785.79 -116.811
-1097.18 -912.591 -2547.14 3168.6
4094.72
-1406.99 -450.407
377.001
-246.142 1668.03
120.686
723.562
-4785.79 -1406.99 3749.59
-638.05
-711.422 -116.811 -450.407 -638.05
1041.14
0
0.381903
0
-0.119473
0
-0.160099
0
0.381903
0
-0.11
0
0.597498
0
-0.0936026 0
-0.160099 0
-0.343796
0
0.597498
0
-0.09
0
0.597498
0
-1.42528 0
0.44588
0
0.381903
0
0.597498
0
-1.42528 0
0
-0.0936026 0
0.44588
0
-0.232804
0
0.44588
0
-0.23
0
-0.160099
-0.102331 0 0
-0.343796
-0.119473 0
-0.0936026 0
-849.024 1282.23
0.445
227.495
128.068
590.229
-1926.5
293.885
244.528
682.343
-849.024 -1096.8
244.528
596.124
-55.4987 -477.956 -912.591 -246.236 723.562
682.343
-55.4987 2575.71
377.001
-2202.77 -2546.54 1668.03
31.2995 120.567 -711.516
-849.024 -477.956 -2202.77 7189.81
3168.6
-4785.35 -116.811
-1096.8
4093.3
-1406.99 -449.961
-912.591 -2546.54 3168.6
377.001
-246.236 1668.03
120.567
723.562
ij -0.554764 yz z jj jj -0.667311 zzz zz jjj jj -0.634825 zzz zzz jjj zz jj jjj -1.41739 zzz zz r i = jj jjj 2.07041 zzz zz jj jjj 2.49044 zzz zz jj jjj -0.880817 zzz zz jj z j k -0.405735 {
zyz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz {
-4785.35 -1406.99 3749.35
-638.05
-711.516 -116.811 -449.961 -638.05
1040.91
After summing contributions from all points, the element equations as follows:
yz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz z {
198
Geometric Nonlinearity
12689.6 jij jj 4814.7 jj jj jj -9199.39 jj jj jj 2032.69 j k = jjj jj -7816.83 jj jj jj -4480.25 jj jj jj 4326.57 jj k -2367.14
rT = H 0
4814.7
-9199.39 2032.69
-7816.83 -4480.25 4326.57
7403.79
-1238.93 290.004
-4480.19 -4212.24 904.422
-1238.93 10056.8 290.004
-3250.81 2699.29
-3250.81 7496.58
792.033
-3556.67
-2479.66 -4548.58 3697.78
-4480.19 2699.29
-2479.66 14318.
4928.26
-9200.43
-4212.24 792.033
-4548.58 4928.26
8470.95
-1240.04
904.422
-3556.67 3697.78
-3481.55 3697.71
0 0 0 0
0 0 0L
-9200.43 -1240.04 8430.52
-3238.01 2031.59
289.873
-3362.16
Complete element equations for element 1 12689.6 jij jj 4814.7 jj jj jj -9199.39 jj jj jj 2032.69 jj jj jj -7816.83 jj jj jj -4480.25 jj jj jj 4326.57 jj k -2367.14
4814.7
-9199.39 2032.69
-7816.83 -4480.25 4326.57
7403.79
-1238.93 290.004
-4480.19 -4212.24 904.422
-1238.93 10056.8 290.004
-3250.81 2699.29
-3250.81 7496.58
792.033
-3556.67
-2479.66 -4548.58 3697.78
-4480.19 2699.29
-2479.66 14318.
4928.26
-9200.43
-4212.24 792.033
-4548.58 4928.26
8470.95
-1240.04
904.422
-3556.67 3697.78
-3481.55 3697.71
-9200.43 -1240.04 8430.52
-3238.01 2031.59
289.873
-3362.16
The element contributes to 81, 2, 3, 4, 5, 6, 7, 8< global degrees of freedom. 1 jij zyz jj 2 zz jj zz jj zz jj 3 zz jj zz jj zz jj 4 zz j z Locations for element contributions to a global vector: jjj zzz jjj 5 zzz jj zz jjj 6 zzz jj zz jjj 7 zzz jj zz j z k8{
ij @1, 1D jj jj @2, 1D jj jj jj @3, 1D jj jj jj @4, 1D and to a global matrix: jjj jj @5, 1D jj jj jjj @6, 1D jj jj @7, 1D jj j k @8, 1D
@1, 2D
@1, 3D
@1, 4D
@1, 5D
@1, 6D
@3, 3D
@3, 4D
@3, 5D
@3, 6D
@5, 3D
@5, 4D
@5, 5D
@5, 6D
@2, 2D
@2, 3D
@4, 2D
@4, 3D
@3, 2D
@5, 2D
@6, 2D
@7, 2D
@8, 2D
@6, 3D
@7, 3D
@8, 3D
@2, 4D
@4, 4D
@6, 4D
@7, 4D
@8, 4D
@2, 5D
@4, 5D
@6, 5D
@7, 5D
@8, 5D
Adding element equations into appropriate locations we have
@2, 6D
@4, 6D
@6, 6D
@7, 6D
@8, 6D
-2367.14 y zz -3481.55 zzzz zz 3697.71 zzzz zz -3238.01 zzzz zz 2031.59 zzz zz 289.873 zzzz zz -3362.16 zzzz z 6429.69 {
-2367.14 y i Du1 y i 0. y zz jj zz jj zz -3481.55 zzzz jjjj Dv1 zzzz jjjj 0. zzzz zz jj zz jj zz 3697.71 zzzz jjjj Du2 zzzz jjjj 0. zzzz zz jj zz jj zz -3238.01 zzzz jjjj Dv2 zzzz jjjj 0. zzzz zz jj zz = jj zz 2031.59 zzz jjj Du3 zzz jjj 0. zzz zz jj zz jj zz 289.873 zzzz jjjj Dv3 zzzz jjjj 0. zzzz zz jj zz jj zz -3362.16 zzzz jjjj Du4 zzzz jjjj 0. zzzz zj z j z 6429.69 { k Dv4 { k 0. {
@1, 7D
@1, 8D y zz @2, 8D zzzz zz @3, 7D @3, 8D zzzz zz @4, 7D @4, 8D zzzz zz @5, 7D @5, 8D zzz zz @6, 7D @6, 8D zzzz zz @7, 7D @7, 8D zzzz z @8, 7D @8, 8D {
@2, 7D
199
12689.6 jij jj 4814.7 jj jj jj -9199.39 jj jj jj 2032.69 jj jj jj -7816.83 jj jj jj -4480.25 jj jj jj 4326.57 jj jj jj -2367.14 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
4814.7
-9199.39 2032.69
-7816.83 -4480.25 4326.57
-2367.14 0
7403.79
-1238.93 290.004
-4480.19 -4212.24 904.422
-3481.55 0
-1238.93 10056.8 290.004
-3250.81 2699.29
-3250.81 7496.58
792.033
-3556.67 3697.71
-2479.66 -4548.58 3697.78
-4480.19 2699.29
-2479.66 14318.
4928.26
-9200.43 2031.59
0
-4212.24 792.033
-4548.58 4928.26
8470.95
-1240.04 289.873
0
904.422
-3556.67 3697.78
-9200.43 -1240.04 8430.52
-3362.16 0
-3481.55 3697.71
-3238.01 2031.59
289.873
-3362.16 6429.69
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
ij Du1 yz ij 0 yz z j jj z jj Dv1 zzz jjj 0 zzz zz jjj zzz jjj jj Du zz jj 0 zzz zz jjj 2 zzz jjj zz jj zz jj z jjj Dv2 zzz jjj 0 zzz jj zz jj zz jjj Du3 zzz jjj 0 zzz jj zz jj zz jjj Dv3 zzz jjj 0 zzz jj zz = jj zz jjj Du zzz jjj 0 zzz jj 4 zz jj zz z jj z j jj Dv4 zzz jjj 0 zzz zz jj zz jj zz jj zz jj jjj Du5 zzz jjj 0 zzz zz jj zz jj jjj Dv5 zzz jjj 0 zzz zz jj zz jj jjj Du6 zzz jjj 0 zzz zz jj zz jj z j z j k Dv6 { k -1 { E = 10600.;
0
-3238.01 0
n = 0.35;
Plane stress analysis.
Initial thickness = 1.25 g = 0.461538
Interpolation functions and their derivatives 1 1 1 1 NT = : ÅÅÅÅÅ Hs - 1L Ht - 1L, - ÅÅÅÅÅ Hs + 1L Ht - 1L, ÅÅÅÅÅ Hs + 1L Ht + 1L, - ÅÅÅÅÅ Hs - 1L Ht + 1L> 4 4 4 4
t-1 1-t t+1 1 ∑NT ê∑s = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-t - 1L> 4 4 4 4
s -1 1 s +1 1-s ∑NT ê∑t = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-s - 1L, ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ > 4 4 4 4
0 0 0y zz 0 0 0 zzzz zz 0 0 0 zzzz zz 0 0 0 zzzz zz 0 0 0 zzz zz 0 0 0 zzzz zz 0 0 0 zzzz zz 0 0 0 zzz zz 0 0 0 zzzz zz 0 0 0 zzzz zz 0 0 0 zzz zz 0 0 0{
200
Geometric Nonlinearity
Mapping to the master element
5 jij ÅÅÅÅ2Å jj jj 10 Initial coordinates = jjjj jjj 10 jj k0
3 zy zz z 3 zzzz zz 4 zzz zz 4{
5ts 35 s 5t 45 x0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 8 8 8 8
t 7 y0 Hs,tL = ÅÅÅÅÅÅ + ÅÅÅÅÅ 2 2 5t 35 ij ÅÅÅÅ ÅÅÅ + ÅÅÅÅ ÅÅ 8 8 J = jjjj k0
5s ÅÅÅÅ ÅÅÅÅ - ÅÅÅÅ58Å yz 8 zz zz 1 ÅÅ2ÅÅ {
5t 35 detJ = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 16 16
Current configuration
ij 2.50125 jj jj 10.0015 Updated coordinates = jjjj jj 10.0024 jj k 0.00186139
2.99901 y zz 2.99255 zzzz zz 3.99255 zzzz z 4.00106 {
xHs,tL = 0.625076 t s + 4.37519 s - 0.624617 t + 5.62674 yHs,tL = -0.000511394 t s - 0.00374464 s + 0.500514 t + 3.49629 Deformation gradient, F p = 1êdet
0.000217372 s + 0.000287254 t + 0.00179341 y i 0.312538 t + 2.18759 zz J jj k -0.000255697 t - 0.00187232 0.000103051s + 0.312502 t + 2.18741 {
Gauss quadrature points and weights Point
Weight
1
s Ø -0.57735 t Ø -0.57735
1.
2
s Ø -0.57735 t Ø 0.57735
1.
3
s Ø 0.57735 t Ø -0.57735
1.
4
s Ø 0.57735 t Ø 0.57735
1.
Computation of element matrices at 8-0.57735, -0.57735< with weight = 1.
201
i 4.01416 J = jjjj k0
-0.985844 y zz zz ÅÅ12ÅÅ {
detJ = 2.00708
i 0.249118 0 yz z J -T = jj k 0.491184 2. {
0.000748383 y i 1.00004 zz; Deformation gradient, F p = jj k -0.000859306 0.999925 {
0 -0.000859306 0 ij 1.00004 yz jj zz j zz 0 0.000748383 0 0.999925 F = jj zz jj z 0.999925 -0.000859306 { k 0.000748383 1.00004 _
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0982367, 0.0982367, 0.0263225, -0.0263225<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.982367, -0.0176327, 0.263225, 0.736775<
0.0000844154 0.0982402 -0.0000844154 0.0263234 -0.000022619 ij -0.0982402 jj j G = jj -0.000735187 -0.982293 -0.000013196 -0.0176314 0.000196993 0.263205 jj -0.982476 -0.0973852 -0.0175598 0.0982445 0.263254 0.0260943 k T
ij 0.118136 jj jj -0.435167 S = jjjj jj 0 jjj k0
0
0
0 0
-0.0982367 jij jj -0.982367 j B T = jjjj jj 0 jj k0 ij 9924.65 jj jj 3195.57 jj jj jj -244.005 jj jj jj -910.065 k c = jjj jj -2659.3 jj jj jjj -856.251 jj jj -7021.34 jj j k -1429.26
yz zz zz zz z 0.118136 -0.435167 zzzz zz -0.435167 -0.755611 {
-0.435167 0 -0.755611 0
0
0.0982367
0
0
-0.0176327 0
0.0263225 0
-0.0263225 0
0.263225
0
0.736775
0
0.0263225 0
-0.0982367 0
0.0982367
-0.982367
-0.0176327 0
0
0.263225
0
-856.251 -7021.34 -1429.26 y zz -856.251 -11062.8 -139.131 -30870.8 zzzz zz -2200.19 414.834 -57.1018 65.381 589.539 -236.21 1667.75 zzzz zz 646.618 -57.1018 108.404 243.851 -173.261 723.315 -581.761 zzzz zz -856.251 65.381 243.851 712.557 229.432 1881.36 382.968 zzz zz zz -11062.8 589.539 -173.261 229.432 2964.27 37.2801 8271.8 zz zz -139.131 -236.21 723.315 1881.36 37.2801 5376.19 -621.464 zzzz z -30870.8 1667.75 -581.761 382.968 8271.8 -621.464 23180.7 { 3195.57
-244.005 -910.065 -2659.3
41287.
-2200.19 646.618
0 0
202
Geometric Nonlinearity
-2.03731 jij jj 0 jj jj jj 0.067772 jj jj jj 0 j k s = jjj jj 0.545896 jj jj jj 0 jj jj jj 1.42364 jj k0
ij 9922.61 jj jj 3195.57 jj jj jj -243.937 jj jj jj -910.065 k = jjj jjj -2658.75 jj jjj -856.251 jj jj jj -7019.92 jj k -1429.26
0
0.067772
0
0.545896
0
1.42364
0
-2.03731 0
0.067772
0
0.545896
0
1.4
0
0.00605314
0
-0.0181595 0
0.067772
0
0.00605314
0
0
-0.0181595 0
0.545896
0
0
-0.0556657 0
1.42364
0
3195.57 41284.9 -2200.19 646.686 -856.251 -11062.3 -139.131 -30869.4
ij -1.04491 yz z jj jj -1.96849 zzz zz jj zz jj jjj -0.0483133 zzz z jj jj 0.0738611 zzz j zzz j r i = jj jj 0.279983 zzz zz jj z jj jj 0.527456 zzz zz jj z jj jj 0.813242 zzz zz jj { k 1.36718
-0.146272
-0.0181595 0 -0.381464
-0.0556657 0
-0.985844 y zz zz ÅÅ12ÅÅ {
-0.0181595 0
detJ = 2.36792
i 0.211156 0 yz z J -T = jj k 0.416333 2. {
0.000774416 y i 1.00005 Deformation gradient, F p = jj zz; k -0.000853046 0.999937 {
1.00005 0 -0.000853046 0 jij zyz j zz j 0.000774416 0 0.999937 F = jj 0 zz jj zz 0.999937 -0.000853046 { k 0.000774416 1.00005 _
8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
-
0
-0.381464
0
-0.146272
0
-
0
-0.986513
0
-0.381464
0
-
-243.937 -910.065 -2658.75 -856.251 -7019.92 -1429.26 y zz -2200.19 646.686 -856.251 -11062.3 -139.131 -30869.4 zzzz zz 414.84 -57.1018 65.3628 589.539 -236.266 1667.75 zzzz zz -57.1018 108.41 243.851 -173.279 723.315 -581.817 zzzz zz 65.3628 243.851 712.411 229.432 1880.98 382.968 zzz zz 589.539 -173.279 229.432 2964.12 37.2801 8271.42 zzzz zz -236.266 723.315 1880.98 37.2801 5375.2 -621.464 zzzz z 1667.75 -581.817 382.968 8271.42 -621.464 23179.7 {
Computation of element matrices at 8-0.57735, 0.57735< with weight = 1. i 4.73584 J = jjjj k0
-0.0556657 0
203
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338< 8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0223112, 0.0223112, 0.0832666, -0.0832666<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.832666, -0.167334, 0.375501, 0.624499<
-0.0000190325 0.0832707 -0.0000710303 ij -0.0223123 0.0000190325 0.0223123 j -0.000129586 -0.167324 0.000290794 0.375477 GT = jjjj -0.00064483 -0.832613 jj -0.0215995 -0.167325 0.0224526 0.375583 0.082941 k -0.832724
ij 0.329845 jj jj -0.308351 S = jjjj jj 0 jj k0
0
0
0 0
ij -0.0223112 jj jj -0.832666 T B = jjjj jj 0 jj k0 ij 8084.25 jj jj 742.51 jj jj jjj 1593.77 jj jj -110.037 j k c = jjj jj -3729.99 jj jj jj -1043.14 jj jj jj -5948.02 jj k 410.663
yz zz zz zz z 0.329845 -0.308351 zzzz z -0.308351 -0.550414 {
-0.308351 0 -0.550414 0
0
0.0223112
0
0
-0.167334 0
0.0832666 0
-0.0832666 0
0.375501
0
0.624499
0
0.0832666 0
-0.
0.375501
0.62
-0.0223112 0
0.0223112
-0.832666
-0.167334 0
0
0
yz z -26168. zzzz zz -455.724 350.235 -143.705 -636.907 63.1153 -1307.09 536.313 zzzz zz 7011.69 -143.705 1416.16 -282.571 -3142.67 536.313 -5285.18 zzzz zz -1987.57 -636.907 -282.571 1989.93 1215.57 2376.97 1054.57 zzz zz -15765.7 63.1153 -3142.67 1215.57 7179.75 -235.549 11728.6 zzzz zz 1700.78 -1307.09 536.313 2376.97 -235.549 4878.14 -2001.55 zzzz z -26168. 536.313 -5285.18 1054.57 11728.6 -2001.55 19724.6 { -110.037 -3729.99 -1043.14 -5948.02 410.663
742.51
1593.77
34922.
-455.724 7011.69
-1987.57 -15765.7 1700.78
ks = ij -1.16298 jj jj 0 jj jj jj -0.213936 jj jj jj 0 jj jj jjj 0.578499 jj jjj 0 jj jj 0.798418 jj j k0
0
yz z 0.798418 zzzz zz zz 0 -0.038317 0 0.109252 0 0.143001 0 zz zz -0.213936 0 -0.038317 0 0.109252 0 0.143001 zzzz zz zz 0 0.109252 0 -0.280018 0 -0.407732 0 zz z 0.578499 0 0.109252 0 -0.280018 0 -0.407732 zzzz zz zz 0 0.143001 0 -0.407732 0 -0.533687 0 zz z 0.798418 0 0.143001 0 -0.407732 0 -0.533687 { 0
-0.213936 0
-1.16298
0
0.578499
-0.213936 0
0
0.798418
0.578499
0
0
204
Geometric Nonlinearity
8083.08 jij jj 742.51 jj jj jj 1593.55 jj jj jj -110.037 j k = jjj jj -3729.41 jj jj jj -1043.14 jj jj jj -5947.22 jj k 410.663
zyz -26167.2 zzzz zz -455.724 350.197 -143.705 -636.798 63.1153 -1306.95 536.313 zzzz zz 7011.48 -143.705 1416.12 -282.571 -3142.56 536.313 -5285.04 zzzz zz -1987.57 -636.798 -282.571 1989.65 1215.57 2376.56 1054.57 zzz zz -15765.1 63.1153 -3142.56 1215.57 7179.47 -235.549 11728.2 zzzz zz 1700.78 -1306.95 536.313 2376.56 -235.549 4877.61 -2001.55 zzzz z -26167.2 536.313 -5285.04 1054.57 11728.2 -2001.55 19724. { -110.037 -3729.41 -1043.14 -5947.22 410.663
742.51
1593.55
34920.8
-455.724 7011.48
-1987.57 -15765.1 1700.78
ij -0.739284 yz z jj jj -1.3762 zzz zz jjj jj -0.17471 zzz zzz jjj zz jj jjj -0.252088 zzz zz r i = jj jjj 0.261966 zzz zz jj jjj 0.687484 zzz zz jj z jj jj 0.652028 zzz zz jj k 0.940805 {
Computation of element matrices at 80.57735, -0.57735< with weight = 1. i 4.01416 J = jjjj k0
-0.264156 y zz zz ÅÅ12ÅÅ {
detJ = 2.00708
i 0.249118 0 yz J -T = jj z k 0.131612 2. {
0.00087344 y i 1.00004 zz; Deformation gradient, F p = jj k -0.000859306 0.999984 {
0 -0.000859306 0 ij 1.00004 _ j F = jjjj 0 0.00087344 0 0.999984 jj 0.999984 -0.000859306 k 0.00087344 1.00004
yz zz zz zz z {
8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0982367, 0.0982367, 0.0263225, -0.0263225<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.263225, -0.736775, 0.802582, 0.197418<
0.0000844154 0.0982402 -0.0000844154 0.0263234 -0.000022619 ij -0.0982402 j GT = jjjj -0.000229911 -0.26322 -0.000643529 -0.736764 0.000701007 0.802569 jj -0.098009 -0.736716 0.0988683 0.802633 0.0256324 k -0.26332
205
0.369291 jij jj 0.0556636 j S = jjjj jj 0 jj k0
0
0
-0.0380408 0
0
0 0
ij -0.0982367 jj jj -0.263225 T B = jjjj jj 0 jj k0 1095.77 jij jj 850.63 jj jj jj 1499.66 jj jj jj 1414.45 j k c = jjj jjj -2193.6 jj jj -1886.08 jj jj jj -401.834 jj j k -379.001
0
0.0982367
0
-0.736775 0
0
0.0263225 0
-0.0263225 0
0.802582
0
0.197418
0
0.0263225 0
-0.
0.802582
0.19
-0.0982367 0
0.0982367
-0.263225
-0.736775 0
0
0
0
-1886.08 -401.834 -379.001 y zz 3050.89 124.341 8181.3 -941.653 -9040.02 -33.317 -2192.17 zzzz zz 124.341 5754.57 -2361.21 -5712.3 1604.19 -1541.93 632.684 zzzz zz 8181.3 -2361.21 23268.8 314.074 -25215.2 632.684 -6234.85 zzzz zz -941.653 -5712.3 314.074 6375.3 711.736 1530.61 -84.1558 zzz zz zz -9040.02 1604.19 -25215.2 711.736 27498.8 -429.84 6756.4 zz zz -33.317 -1541.93 632.684 1530.61 -429.84 413.16 -169.527 zzzz z -2192.17 632.684 -6234.85 -84.1558 6756.4 -169.527 1670.62 { 850.63
ij 0.00955074 jj jj 0 jj jj jj -0.0209536 jj jj jj 0 k s = jjj jj 0.00578836 jj jj jjj 0 jj jj 0.0056145 jj j k0 1095.78 jij jj 850.63 jj jj jj 1499.64 jj jj jj 1414.45 j k = jjj jjj -2193.59 jj jjj -1886.08 jj jj jj -401.828 jj k -379.001
zyz zz zz zz 0.369291 0.0556636 zzzz z 0.0556636 -0.0380408 {
0.0556636
1499.66
1414.45
0
-0.0209536 0
0.00955074
0
0
-0.0630821 0
-2193.6
0.00578836
-0.0209536 0 0.0671329
0
0.0056145
0.00578836
0
0
0.0169028
0.0671329
0
-0.054933
0
-0.0179882
0
-0.054933
0
-0.0209536 0
-0.0630821 0
0
0.0671329
0
0.00578836
0
0.0671329
0
0.0169028
0
-0.0179882 0
0.0056145
0
0.0169028
0
-0.0045290
-0.0179882 0
-2193.59 -1886.08 -401.828 -379.001 y zz 3050.9 124.341 8181.28 -941.653 -9040.01 -33.317 -2192.17 zzzz zz 124.341 5754.51 -2361.21 -5712.24 1604.19 -1541.92 632.684 zzzz zz 8181.28 -2361.21 23268.7 314.074 -25215.2 632.684 -6234.83 zzzz zz -941.653 -5712.24 314.074 6375.24 711.736 1530.59 -84.1558 zzz zz -9040.01 1604.19 -25215.2 711.736 27498.8 -429.84 6756.38 zzzz zz -33.317 -1541.92 632.684 1530.59 -429.84 413.155 -169.527 zzzz z -2192.17 632.684 -6234.83 -84.1558 6756.38 -169.527 1670.62 { 850.63
1499.64
1414.45
206
Geometric Nonlinearity
0.12777 zy jij jj -0.0115125 zzz zz jj zz jj zz jj 0.011803 zz jj z jj jj -0.0840445 zzz zz j z r i = jjj jj -0.136411 zzz zz jj z jj jj 0.0730373 zzz zz jj z jj jj -0.00316262 zzz zz jj k 0.0225197 {
Computation of element matrices at 80.57735, 0.57735< with weight = 1. i 4.73584 J = jjjj k0
-0.264156 y zz zz ÅÅ12ÅÅ {
detJ = 2.36792
i 0.211156 0 yz z J -T = jj k 0.111556 2. {
0.000880415 y i 1.00005 Deformation gradient, F p = jj zz; { k -0.000853046 0.999987
1.00005 0 -0.000853046 0 jij zyz j zz j F = jj 0 0.000880415 0 0.999987 zz jj zz 0.999987 -0.000853046 { k 0.000880415 1.00005 _
8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0223112, 0.0223112, 0.0832666, -0.0832666<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.223112, -0.776888, 0.832666, 0.167334<
0.0000190325 0.0223123 -0.0000190325 0.0832707 -0.0000710303 ij -0.0223123 j -0.000683984 -0.776878 0.000733092 0.832655 GT = jjjj -0.000196431 -0.223109 jj -0.0221206 -0.776906 0.0229736 0.83278 0.0825552 k -0.223143
ij 0.542697 jj jj 0.107649 S = jjjj jj 0 jj k0
0.107649
yz zz zz zz z 0.542697 0.107649 zzzz z 0.107649 0.0577759 {
0
0
0.0577759 0
0
0 0
ij -0.0223112 jj jj -0.223112 T B = jjjj jj 0 jj k0
0
0.0223112
0
0
-0.776888 0
0.0832666 0
-0.0832666 0
0.832666
0
0.167334
0
0.0832666 0
-0.
0.832666
0.16
-0.0223112 0
0.0223112
-0.223112
-0.776888 0
0
0
0
207
603.832 jij jj 194.435 jj jj jj 1989.51 jj jj jj 417.864 j k c = jjj jj -2253.53 jj jj jj -725.641 jj jj jj -339.809 jj k 113.342
ij 0.0124846 jj jj 0 jj jj jj 0.0327791 jj jj jj 0 k s = jjj jjj -0.0465932 jj jj 0 jj jj jj 0.00132949 jj j k0 ij 603.845 jj jj 194.435 jj jj jjj 1989.54 jj jj 417.864 j k = jjj jj -2253.58 jj jj jj -725.641 jj jj jj -339.808 jj k 113.342
zyz -1858.9 zzzz zz 72.1809 7037.36 -651.439 -7424.95 -269.383 -1601.92 848.641 zzzz zz 8722.01 -651.439 30399.1 -1559.49 -32551. 1793.06 -6570.11 zzzz zz -725.641 -7424.95 -1559.49 8410.3 2708.13 1268.19 -422.997 zzz zz -9375.19 -269.383 -32551. 2708.13 34988.7 -1713.1 6937.51 zzzz zz 459.025 -1601.92 1793.06 1268.19 -1713.1 673.548 -538.985 zzzz z -1858.9 848.641 -6570.11 -422.997 6937.51 -538.985 1491.5 { 194.435
1989.51
417.864
-2253.53 -725.641 -339.809 113.342
2512.07
72.1809
8722.01
-725.641 -9375.19 459.025
0
0.0327791
0
-0.0465932
0
0.00132
0.0124846
0
0.0327791
0
-0.0465932
0
0
0.0929684
0
-0.122333
0
-0.0034
0.0327791
0
0.0929684
0
-0.122333
0
0
-0.122333
0
0.173888
0
-0.0049
-0.122333
0
0.173888
0
-0.0465932 0 0
-0.00341425 0
0.00132949
0
-0.00341425 0
194.435
1989.54
417.864
2512.09
72.1809
8722.05
72.1809
7037.46
-651.439
8722.05
-651.439 30399.2
-725.641 -7425.07 -1559.49 -9375.23 -269.383 -32551.1 459.025
-1601.93 1793.06
-1858.9
848.641
yz ij 0.106975 z jj jj 0.0451719 zzz zz jj zz jj zz jj 0.211822 zz jj zz jj zz jj 0.125565 zz r i = jjj z jjj -0.399234 zzz zz jj jjj -0.168584 zzz zz jj jjj 0.0804381 zzz zz jj z j k -0.00215335 {
-0.00496172 0
-6570.11
0.00704
-0.00496172 0
yz z -1858.9 zzzz zz -7425.07 -269.383 -1601.93 848.641 zzzz zz -1559.49 -32551.1 1793.06 -6570.11 zzzz zz 8410.47 2708.13 1268.18 -422.997 zzz zz 2708.13 34988.8 -1713.1 6937.51 zzzz zz 1268.18 -1713.1 673.555 -538.985 zzzz z -422.997 6937.51 -538.985 1491.5 { -2253.58 -725.641 -339.808 113.342
-725.641 -9375.23 459.025
After summing contributions from all points, the element equations as follows:
208
Geometric Nonlinearity
19705.3 jij jj 4983.15 jj jj jj 4838.8 jj jj jj 812.215 j k = jjj jj -10835.3 jj jj jj -4511.11 jj jj jj -13708.8 jj k -1284.25
rT = H 0
4983.15
4838.8
812.215
81768.7
-2459.39 24561.5
-2459.39 13557. 24561.5
-3213.46
-3213.46 55192.4
-4511.12 -13708.7 -1284.14 -45242.6 1987.46 1987.36
-4687.06 3685.38
-61087.6 3685.39
0 0 0 0
-61082.1
0 0 0L
-18671.8
Complete element equations for element 2 19705.3 jij jj 4983.15 jj jj jj 4838.8 jj jj jj 812.215 jj jj jj -10835.3 jj jj jj -4511.11 jj jj jj -13708.8 jj k -1284.25
4983.15
4838.8
81768.7
-2459.39 24561.5
-2459.39 13557. 24561.5
812.215 -3213.46
-3213.46 55192.4
-4511.12 -13708.7 -1284.14 -45242.6 1987.46 1987.36
-61082.1
-4687.06 3685.38
-61087.6 3685.39
-18671.8
-10835.3 -4511.11 -13708.8 -1284.25 y zz -4511.12 -45242.6 1987.36 -61087.6 zzzz zz -13708.7 1987.46 -4687.06 3685.39 zzzz zz -1284.14 -61082.1 3685.38 -18671.8 zzzz zz 17487.8 4864.87 7056.31 930.387 zzz zz 4864.87 72631.2 -2341.21 33693.5 zzzz zz 7056.31 -2341.21 11339.5 -3331.52 zzzz z 930.387 33693.5 -3331.52 46065.9 {
-10835.3 -4511.11 -13708.8 -1284.25 y i Du3 y i 0. y zz jj zz jj zz -4511.12 -45242.6 1987.36 -61087.6 zzzz jjjj Dv3 zzzz jjjj 0. zzzz zz jj zz jj zz -13708.7 1987.46 -4687.06 3685.39 zzzz jjjj Du5 zzzz jjjj 0. zzzz zz jj zz jj zz -1284.14 -61082.1 3685.38 -18671.8 zzzz jjjj Dv5 zzzz jjjj 0. zzzz zz jj zz = jj zz 17487.8 4864.87 7056.31 930.387 zzz jjj Du6 zzz jjj 0. zzz zz jj zz jj zz 4864.87 72631.2 -2341.21 33693.5 zzzz jjjj Dv6 zzzz jjjj 0. zzzz zz jj zz jj zz 7056.31 -2341.21 11339.5 -3331.52 zzzz jjjj Du4 zzzz jjjj 0. zzzz zj z j z 930.387 33693.5 -3331.52 46065.9 { k Dv4 { k 0. {
The element contributes to 85, 6, 9, 10, 11, 12, 7, 8< global degrees of freedom. 5 jij jj 6 jj jj jj 9 jj jj jj 10 j Locations for element contributions to a global vector: jjj jjj 11 jj jjj 12 jj jjj 7 jj j k8
ij @5, 5D jj jj @6, 5D jj jj jj @9, 5D jj jj jj @10, 5D and to a global matrix: jjj jj @11, 5D jj jj jjj @12, 5D jj jj @7, 5D jj j k @8, 5D
@5, 6D @6, 6D
@9, 6D
@5, 9D
@5, 10D
@9, 9D
@9, 10D
@6, 9D
@6, 10D
@10, 6D @10, 9D @10, 10D
@11, 6D @11, 9D @11, 10D
zyz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz {
@5, 11D
@5, 12D
@9, 11D
@9, 12D
@6, 11D
@10, 11D
@11, 12D
@7, 11D
@7, 12D
@12, 11D
@8, 6D
@8, 11D
@7, 9D @8, 9D
@7, 10D
@8, 10D
Adding element equations into appropriate locations we have
@10, 12D
@11, 11D
@12, 6D @12, 9D @12, 10D @7, 6D
@6, 12D
@12, 12D @8, 12D
@5, 7D
@5, 8D y zz @6, 8D zzzz zz @9, 7D @9, 8D zzzz zz @10, 7D @10, 8D zzzz zz @11, 7D @11, 8D zzz zz @12, 7D @12, 8D zzzz zz @7, 7D @7, 8D zzzz z @8, 7D @8, 8D {
@6, 7D
209
12689.6 jij jj 4814.7 jj jj jj -9199.39 jj jj jj 2032.69 jj jj jj -7816.83 jj jj jj -4480.25 jj jj jj 4326.57 jj jj jj -2367.14 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
4814.7
-9199.39 2032.69
-7816.83 -4480.25 4326.57
-2367.14 0
7403.79
-1238.93 290.004
-4480.19 -4212.24 904.422
-3481.55 0
-1238.93 10056.8 290.004
-3250.81 2699.29
-3250.81 7496.58
792.033
-3556.67 3697.71
-2479.66 -4548.58 3697.78
0 0
0
0
-3238.01 0
0
-4480.19 2699.29
-2479.66 34023.3
9911.41
-22909.2 747.338
-4212.24 792.033
-4548.58 9911.41
90239.7
747.317
-60797.7 -2459.39 2456
19770.
-6693.68 -4687.06 3685
904.422
-3556.67 3697.78
-22909.2 747.317
-3481.55 3697.71
-3238.01 747.338
0
0
0
0
0
0
0
0
0
-10835.3 -4511.12 7056.31
0
0
0
-4511.11 -45242.6 -2341.21 33693.5
4838.8
812.
-60797.7 -6693.68 52495.6
3685.39
-18
4838.8
-2459.39 -4687.06 3685.39
13557.
-32
812.215
24561.5
3685.38
-18671.8 -3213.46 5519 930.387
-13708.7 -12 1987.46
-61
After assembly of all elements the global matrices are as follows. ij 12689.6 jj jj 4814.7 jj jj jjj -9199.39 jj jj 2032.69 jj jj jj -7816.83 jj jj jj -4480.25 K T = jjjj jj 4326.57 jj jj jj -2367.14 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0 ij 0. yz z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz z jj jj 0. zzz zz j zz; RE = jjjj jj 0. zzz zz jj z jj jj 0. zzz z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z j k -1. {
4814.7
-9199.39 2032.69
-7816.83 -4480.25 4326.57
-2367.14 0
7403.79
-1238.93 290.004
-4480.19 -4212.24 904.422
-3481.55 0
-1238.93 10056.8 290.004
-3250.81 2699.29
-3250.81 7496.58
792.033
-3556.67 3697.71
-2479.66 -4548.58 3697.78
0
-3238.01 0
-4480.19 2699.29
-2479.66 34023.3
9911.41
-22909.2 747.338
-4212.24 792.033
-4548.58 9911.41
90239.7
747.317
-60797.7 -
19770.
-6693.68 -
904.422
-3556.67 3697.78
-22909.2 747.317
4
-3481.55 3697.71
-3238.01 747.338
0
0
0
0
0
0
0
0
0
-10835.3 -4511.12 7056.31
930.387
-
0
0
0
-4511.11 -45242.6 -2341.21 33693.5
1
yz ij 0.16601 zz jj zz jj -2.96261 zz jj z jj jj -0.209513 zzz zz jj zz jj zz jj 3.79239 zz jj zz jj zz jj 0.174468 zz jj zz jj 0.139225 zz j zz; RI = jjjj jj -0.124059 zzz zz jj z jj jj 0.0136769 zzz z jj jj -0.00060097 zzz zz jj zzz jjj zz jj 0.136706 zz jjj jj -0.00630497 zzz zzz jjj { k -1.11939
-60797.7 -6693.68 52495.6
3
4838.8
-2459.39 -4687.06 3685.39
1
812.215
24561.5
3685.38
ij -0.16601 yz zz jj zz jj 2.96261 zz jj z jj jj 0.209513 zzz zz jj z jj jj -3.79239 zzz zz jj z jj jj -0.174468 zzz z jj jj -0.139225 zzz zz j zz R = RE - RI = jjjj jj 0.124059 zzz zz jj z jj jj -0.0136769 zzz zz jjj jj 0.00060097 zzz zz jj z jj jj -0.136706 zzz zz jjj jj 0.00630497 zzz zzz jjj k 0.119394 {
-18671.8 -
210
Geometric Nonlinearity
System of equations 12689.6 jij jj 4814.7 jj jj jj -9199.39 jj jj jj 2032.69 jj jj jj -7816.83 jj jj jj -4480.25 jj jj jj 4326.57 jj jj jj -2367.14 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
4814.7
-9199.39 2032.69
-7816.83 -4480.25 4326.57
-2367.14 0
7403.79
-1238.93 290.004
-4480.19 -4212.24 904.422
-3481.55 0
-1238.93 10056.8 290.004
-3250.81 2699.29
-3250.81 7496.58
792.033
-3556.67 3697.71
-2479.66 -4548.58 3697.78
0
-3238.01 0
-4480.19 2699.29
-2479.66 34023.3
9911.41
-22909.2 747.338
-4212.24 792.033
-4548.58 9911.41
90239.7
747.317
-60797.7 -2459.
19770.
-6693.68 -4687.
904.422
-3556.67 3697.78
-22909.2 747.317
4838.8
-3481.55 3697.71
-3238.01 747.338
-60797.7 -6693.68 52495.6
3685.39
0
0
0
4838.8
-2459.39 -4687.06 3685.39
13557.
0
0
0
812.215
24561.5
0
0
0
-10835.3 -4511.12 7056.31
930.387
-13708
0
0
0
-4511.11 -45242.6 -2341.21 33693.5
1987.46
Essential boundary conditions Node
dof
Value
1
Du1 Dv1
0 0
2
Du2 Dv2
0 0
Remove 81, 2, 3, 4< rows and columns. After adjusting for essential boundary conditions we have
3685.38
-18671.8 -3213.
211
34023.3 jij jj 9911.41 jj jj jj -22909.2 jj jj jj 747.338 jj jj jj 4838.8 jj jj jj 812.215 jj jj jj -10835.3 jj k -4511.11
9911.41
-22909.2 747.338
4838.8
812.215
90239.7
747.317
-60797.7 -2459.39 24561.5
747.317
19770.
-6693.68 -4687.06 3685.38
-60797.7 -6693.68 52495.6
3685.39
-18671.8
-2459.39 -4687.06 3685.39
13557.
-3213.46
24561.5
3685.38
-4511.12 7056.31
-18671.8 -3213.46 55192.4 930.387
-45242.6 -2341.21 33693.5
-13708.7 -1284.14 1987.46
-61082.1
-10835.3 -4511.11 y zz -4511.12 -45242.6 zzzz zz 7056.31 -2341.21 zzzz zz 930.387 33693.5 zzzz zz -13708.7 1987.46 zzz zz -1284.14 -61082.1 zzzz zz 17487.8 4864.87 zzzz z 4864.87 72631.2 {
Du -0.174468 y jij 3 zyz jij z jj Dv zz jj -0.139225 zzz jj 3 zz jj zz z j jj z jj Du zzz jjj 0.124059 zzz jj 4 zz jj zz z j jj z jj Dv4 zzz jjj -0.0136769 zzz zz jj jj zz zz = jj jj z jj Du5 zz jj 0.00060097 zzz zz jj jj zz z j jj z jj Dv5 zzz jjj -0.136706 zzz zz jj jj zz jjj Du zzz jjj 0.00630497 zzz jj 6 zz jj zz j z j z Dv 0.119394 k 6{ k {
Solving the final system of global equations we get
8Du3 = 0.0000332154, Dv3 = -0.0000360654, Du4 = 0.000057925, Dv4 = 0.0000269595, Du5 = 0.0000350328, Dv5 = -0.000236237, Du6 = 0.0000618104, Dv6 = -0.00023317<
Complete table of nodal values Du
Dv
1
0
0
2
0
0
3
0.0000332154
-0.0000360654
4
0.000057925
0.0000269595
5
0.0000350328
-0.000236237
6
0.0000618104
-0.00023317
Total increments since the start of this load step
212
Geometric Nonlinearity
Du
Dv
1
0
0
2
0
0
3
0.00128132
-0.00102444
4
0.00191931
0.0010897
5
0.00150532
-0.00769111
6
0.00245131
-0.0076825
Total nodal values u
v
1
0
0
2
0
0
3
0.00128132
-0.00102444
4
0.00191931
0.0010897
5
0.00150532
-0.00769111
6
0.00245131
-0.0076825
Solution for element 1 Initial configuration
0 jij jj 5 jj ÅÅÅÅÅ j 2 Nodal coordinates = jjjj jj ÅÅÅÅ5Å jj 2 jj k0 5s 5 x0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4
0y zz z 0 zzzz zz zz 3 zzz zz z 4{
ts s 7t 7 y0 Hs,tL = - ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4 4 4 ij ÅÅ54ÅÅ J = jjjj t 1 k - ÅÅ4ÅÅ - ÅÅÅÅ4Å
Current configuration II
yz zz z 7 s z ÅÅÅÅ4Å - ÅÅ4ÅÅ {
0
35 5s detJ = ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ 16 16
ij 0 jj jj 0 Nodal displacements = jjjj jj 0.00128132 jj k 0.00191931
yz zz zz zz z -0.00102444 zzzz zz 0.0010897 {
0
0
213
0 jij jj 5 jj ÅÅÅÅÅ Updated coordinates = jjjj 2 jj 2.50128 jj j k 0.00191931
zyz zz zz zz zz 2.99898 zzz zz 4.00109 { 0 0
xII Hs,tL = -0.0001595 t s + 1.24984 s + 0.000800158 t + 1.2508 yII Hs,tL = -0.250529 t s - 0.250529 s + 1.75002 t + 1.75002
0.000800158 - 0.0001595 s y i 1.24984 - 0.0001595 t zz J II = jj k -0.250529 t - 0.250529 1.75002 - 0.250529 s { detJ II = -0.313161 s - 0.0000786647 t + 2.18744
i -0.3125 s - 0.000079085 t + 2.18742 Deformation gradient, F IIp = H1êdetJ L jj k -0.000920857 t - 0.000920857
Solution at 8s, t< = 8-0.57735, -0.57735< ï 8x, y< = 80.528689, 0.800777< 0.000892245 y i 1.24993 zz J II = jj { k -0.105886 1.89466
0.0010002 - 0.000199375 s y zz {
2.18752 - 0.313161 s
detJ II = 2.36829
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.333105, 0.31075, 0.0892553, -0.0669004<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.207974, -0.0559149, 0.0557265, 0.208163< B TL
0.31075 0 0.0892553 0 -0.0669004 0 -0.333105 0 jij j j -0.207974 0 -0.0559149 0 0.0557265 0 0.20 = jj 0 jj -0.207974 -0.333105 -0.0559149 0.31075 0.0557265 0.0892553 0.208163 -0. k
0.000471006 y i 0.999986 zz; F IIp = jj { k -0.000164363 1.00017
Det@F IIp D = 1.00016
Element thickness, hII = 1.2499
0.000306725 y i 0.999972 zz Left Cauchy-Green tensor = jj 0.000306725 1.00034 k {
i -0.0000141023 0.000153304 yz Green-Lagrange strain tensor, e = jj z 0.000169822 { k 0.000153304 i 0.548046 1.20409 yz Cauchy stress tensor, s = jj z k 1.20409 1.99056 { Principal stresses = H 2.67289
-0.134282 L
214
Geometric Nonlinearity
Effective stress Hvon MisesL = 2.7425
i 0.546967 1.20314 yz z Second PK stresses = jj k 1.20314 1.99042 {
r Ti = H -1.28166 -2.41272 0.304829 0.778127 0.343421 0.646485 0.633413 0.988104 L
Solution at 8s, t< = 8-0.57735, 0.57735< ï 8x, y< = 80.52972, 2.98854< 0.000892245 yz i 1.24975 J II = jj z -0.395171 1.89466 k {
detJ II = 2.3682
8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.150336, 0.0669029, 0.333118, -0.249685<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.20806, -0.0558001, 0.0556117, 0.208249< B TL =
-0.150336 0 0.0669029 0 0.333118 0 -0.249685 0 jij zyz jj 0 -0.20806 0 -0.0558001 0 0.0556117 0 0.208249 zzzz jj jj zz 0.0556117 0.333118 0.208249 -0.249685 { k -0.20806 -0.150336 -0.0558001 0.0669029 0.000471006 y i 0.999947 zz; F IIp = jj k -0.000613413 1.00017 {
Det@F IIp D = 1.00012
Element thickness, hII = 1.24992
-0.000142294 y i 0.999895 zz Left Cauchy-Green tensor = jj -0.000142294 1.00034 { k
i -0.0000524916 -0.0000712678 yz Green-Lagrange strain tensor, e = jj z k -0.0000712678 0.000169822 {
i 0.0831202 -0.558607 yz Cauchy stress tensor, s = jj z k -0.558607 1.82981 { Principal stresses = H 1.99318 -0.0802474 L
Effective stress Hvon MisesL = 2.03449
i 0.0836601 -0.559382 yz Second PK stresses = jj z k -0.559382 1.8286 { r Ti =
H 0.307042 -0.878348 0.108727 -0.412858 -0.0099939 -0.249602 -0.405775 1.54081 L
215
Solution at 8s, t< = 80.57735, -0.57735< ï 8x, y< = 81.97199, 0.678511< 0.00070807 y i 1.24993 zz J II = jj { k -0.105886 1.60537
detJ II = 2.00668
8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.321051, 0.294667, 0.105339, -0.0789559<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.0656764, -0.245766, 0.24559, 0.0658528< B TL =
0.294667 0 0.105339 0 -0.0789559 0 ij -0.321051 0 jj jj 0 -0.0656764 0 -0.245766 0 0.24559 0 0.0658528 jj j 0.24559 0.105339 0.0658528 -0.0789559 k -0.0656764 -0.321051 -0.245766 0.294667 0.000440983 y i 0.999983 zz; F IIp = jj k -0.000193914 0.99982 {
Det@F IIp D = 0.999804
Element thickness, hII = 1.25013
0.000246994 y i 0.999967 zz Left Cauchy-Green tensor = jj { k 0.000246994 0.99964
i -0.0000166348 0.000123549 yz Green-Lagrange strain tensor, e = jj z -0.000179773 { k 0.000123549
i -0.960746 0.969766 zy Cauchy stress tensor, s = jj z -2.24303 { k 0.969766 Principal stresses = H -2.76443 -0.439343 L
Effective stress Hvon MisesL = 2.57305
i -0.961547 0.970672 yz Second PK stresses = jj z -2.24325 { k 0.970672
r Ti = H 0.614004 -0.41149 -1.30809 2.09976
0.343581 -1.12564 0.3505 -0.562629 L
Solution at 8s, t< = 80.57735, 0.57735< ï 8x, y< = 81.9728, 2.53224< 0.00070807 y i 1.24975 J II = jj zz k -0.395171 1.60537 {
detJ II = 2.00659
8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
yz zz zz zz z {
216
Geometric Nonlinearity
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.105344, 0.00687574, 0.393149, -0.294681<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.0657715, -0.245639, 0.245463, 0.0659479< B TL =
-0.105344 0 0.00687574 0 0.393149 0 -0.294681 0 jij zyz jj 0 -0.0657715 0 -0.245639 0 0.245463 0 0.0659479 zzzz jj jj zz k -0.0657715 -0.105344 -0.245639 0.00687574 0.245463 0.393149 0.0659479 -0.294681 { 0.000440983 y i 0.999938 zz; F IIp = jj k -0.000723696 0.99982 {
Det@F IIp D = 0.999758
Element thickness, hII = 1.25016
-0.000282747 y i 0.999876 zz Left Cauchy-Green tensor = jj -0.000282747 0.999641 k {
i -0.0000618886 -0.000141305 yz Green-Lagrange strain tensor, e = jj z k -0.000141305 -0.000179773 {
i -1.50945 -1.11017 zy Cauchy stress tensor, s = jj z k -1.11017 -2.43257 { Principal stresses = H -3.1733 -0.768716 L
Effective stress Hvon MisesL = 2.8673
i -1.50849 -1.11033 yz Second PK stresses = jj z k -1.11033 -2.43478 {
r Ti = H 0.582059 0.694731 0.658052 1.47981 -2.17227 -2.59277 0.932164 0.418233 L
After summing contributions from all points the internal load vector is as follows: r Ti = H 0.221441 -3.00782 -0.236477 3.94484
Global internal load vector
RTI = H 0.221441 -3.00782 -0.236477 3.94484
Solution for element 2 Initial configuration
-1.49527 -3.32153 1.5103 2.38451 L
-1.49527 -3.32153 1.5103 2.38451 0 0 0 0 L
217
5 jij ÅÅÅÅ2Å jj jj 10 Nodal coordinates = jjjj jj 10 jj j k0
3 zy zz z 3 zzzz zz 4 zzz zz 4{
5ts 35 s 5t 45 x0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 8 8 8 8
t 7 y0 Hs,tL = ÅÅÅÅÅÅ + ÅÅÅÅÅ 2 2 5t 35 ij ÅÅÅÅ ÅÅÅ + ÅÅÅÅ ÅÅ 8 8 J = jjj j0 k
5s ÅÅÅÅ ÅÅÅÅ - ÅÅÅÅ58Å yz 8 zz zz ÅÅ12ÅÅ {
5t 35 detJ = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 16 16
Current configuration II
ij 0.00128132 jj jj 0.00150532 Nodal displacements = jjjj jj 0.00245131 jj k 0.00191931 ij 2.50128 jj jj 10.0015 Updated coordinates = jjjj jjj 10.0025 jj k 0.00191931
-0.00102444 y zz -0.00769111 zzzz zz -0.0076825 zzzz z 0.0010897 {
2.99898 y zz 2.99231 zzzz zz 3.99232 zzzz zz 4.00109 {
xII Hs,tL = 0.625077 t s + 4.37519 s - 0.624604 t + 5.62679
yII Hs,tL = -0.000526383 t s - 0.00385972 s + 0.500531 t + 3.49617
0.625077 s - 0.624604 i 0.625077 t + 4.37519 zyz J II = jj k -0.000526383 t - 0.00385972 0.500531 - 0.000526383 s { detJ II = 0.000109593 s + 0.312541 t + 2.18751
Deformation gradient, F IIp = H1êdetJ L
0.000218742 s + 0.000295622 t + 0.00185061 zy jij 0.312538 t + 2.18759 z k -0.000263192 t - 0.00192986 0.000109395 s + 0.312503 t + 2.18741 {
Solution at 8s, t< = 8-0.57735, -0.57735< ï 8x, y< = 83.66975, 3.20924< -0.985492 y i 4.0143 zz J II = jj k -0.00355581 0.500835 {
detJ II = 2.007
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
218
Geometric Nonlinearity
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.0991034, 0.0982175, 0.0265547, -0.0256688<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.982367, -0.0177101, 0.263224, 0.736852<
0.0982175 0 0.0265547 0 -0.0256688 0 ij -0.0991034 0 j -0.982367 0 -0.0177101 0 0.263224 0 0 B TL = jjjj 0 jj 0.263224 0.0265547 0.736852 k -0.982367 -0.0991034 -0.0177101 0.0982175 0.000774083 y i 1.00004 zz; F IIp = jj -0.000885817 0.999923 k {
Det@F IIp D = 0.999959
Element thickness, hII = 1.25003
-0.000111826 y i 1.00007 Left Cauchy-Green tensor = jj zz k -0.000111826 0.999846 {
-0.0000558188 y i 0.0000364017 zz Green-Lagrange strain tensor, e = jj k -0.0000558188 -0.0000770663 {
-0.439028 y i 0.113111 zz Cauchy stress tensor, s = jj -0.439028 -0.776383 k {
Principal stresses = H -0.956572 0.2933 L Effective stress Hvon MisesL = 1.13209
-0.438336 y i 0.11378 zz Second PK stresses = jj k -0.438336 -0.777265 {
r Ti = H 1.05389
2.0226 0.047378 -0.0736844 -0.282388 -0.541954 -0.818878 -1.40696 L
Solution at 8s, t< = 8-0.57735, 0.57735< ï 8x, y< = 82.5318, 3.78756< -0.985492 y i 4.73608 zz J II = jj k -0.00416362 0.500835 {
detJ II = 2.36789
8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.0230422, 0.022163, 0.0835926, -0.0827134<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.832701, -0.167363, 0.375458, 0.624606< B TL
0.022163 0 0.0835926 0 -0.0827134 0 -0.0230422 0 jij j j -0.832701 0 -0.167363 0 0.375458 0 0.62 = jj 0 jj 0.375458 0.0835926 0.624606 -0. k -0.832701 -0.0230422 -0.167363 0.022163
219
0.00080028 yz i 1.00005 F IIp = jj z; k -0.000879172 0.999936 {
Det@F IIp D = 0.999986
Element thickness, hII = 1.25001
-0.0000789872 y i 1.0001 zz Left Cauchy-Green tensor = jj k -0.0000789872 0.999872 {
-0.0000393982 y i 0.0000496829 zz Green-Lagrange strain tensor, e = jj k -0.0000393982 -0.0000639453 { i 0.329244 -0.3101 yz Cauchy stress tensor, s = jj z k -0.3101 -0.561914 { Principal stresses = H -0.6592 0.42653 L Effective stress Hvon MisesL = 0.947439
-0.309363 y i 0.329705 zz Second PK stresses = jj k -0.309363 -0.562527 {
r Ti = H 0.741847 1.4061 0.175214 0.258015 -0.263155 -0.701188 -0.653906 -0.962925 L
Solution at 8s, t< = 80.57735, -0.57735< ï 8x, y< = 88.30506, 3.20514< -0.263716 y i 4.0143 zz J II = jj k -0.00355581 0.500227 {
detJ II = 2.00712
8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.0984663, 0.0975805, 0.0270324, -0.0261466<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.26314, -0.736874, 0.802569, 0.197445< B TL
0.0975805 0 0.0270324 0 -0.0261466 0 -0.0984663 0 jij j j = jj 0 -0.26314 0 -0.736874 0 0.802569 0 0.19 jj -0.26314 -0.0984663 -0.736874 0.0975805 0.802569 0.0270324 0.197445 -0. k
0.000899929 y i 1.00004 zz; F IIp = jj k -0.000885817 0.999986 { Element thickness, hII = 1.24998
Det@F IIp D = 1.00002
0.0000140668 y i 1.00007 zz Left Cauchy-Green tensor = jj 0.0000140668 0.999972 k {
220
Geometric Nonlinearity
i 0.0000364017 Green-Lagrange strain tensor, e = jjjj -6 k 7.07842 µ 10
i 0.380506 0.0552245 Cauchy stress tensor, s = jj k 0.0552245 -0.0156503
Principal stresses = H 0.38806
-0.0232046 L
Effective stress Hvon MisesL = 0.400167
i 0.380382 0.0555749 Second PK stresses = jj k 0.0555749 -0.0155527
yz z {
7.07842 µ 10-6 zy zz z -0.0000140274 {
yz z {
r Ti = H -0.130458 -0.00331056 -0.00894053 0.042453 0.137003 -0.0277672 0.00239561 -0.
Solution at 8s, t< = 80.57735, 0.57735< ï 8x, y< = 88.00055, 3.78275< -0.263716 y i 4.73608 zz J II = jj k -0.00416362 0.500227 {
detJ II = 2.36801
8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.0225062, 0.0216271, 0.0839944, -0.0831153<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.223094, -0.776916, 0.832599, 0.167411< B TL
0.0216271 0 0.0839944 0 -0.0831153 0 ij -0.0225062 0 jj j -0.223094 0 -0.776916 0 0.832599 0 0.16 = jj 0 jj -0. k -0.223094 -0.0225062 -0.776916 0.0216271 0.832599 0.0839944 0.167411
0.000906948 y i 1.00005 zz; F IIp = jj k -0.000879172 0.999989 {
Det@F IIp D = 1.00004
Element thickness, hII = 1.24997
0.0000277225 y i 1.0001 Left Cauchy-Green tensor = jj zz k 0.0000277225 0.999979 {
i 0.0000496829 0.000013915 yz Green-Lagrange strain tensor, e = jj z k 0.000013915 -0.0000105107 { i 0.555892 0.108835 yz Cauchy stress tensor, s = jj z k 0.108835 0.0828801 { Principal stresses = H 0.579732 0.0590401 L Effective stress Hvon MisesL = 0.552583
221
i 0.555649 0.109246 yz Second PK stresses = jj z k 0.109246 0.0830751 { r Ti =
H -0.108901 -0.0619801 -0.214694 -0.183627 0.406424 0.231313 -0.0828285 0.0142944 L After summing contributions from all points the internal load vector is as follows:
r Ti = H 1.55638 3.36341 -0.0010432 0.0431563 -0.002116 -1.0396 -1.55322 -2.36697 L
Global internal load vector
RTI = H 0.221441 -3.00782 -0.236477 3.94484
0.0611095 0.0418766 -0.0429142 0.0175473 -0.0010
After assembling all element internal force vectors, the global internal force and the external load vectors are as follows. 0.221441 y z jij jj -3.00782 zzz zz jj z jj jj -0.236477 zzz zz jj zz jj zz jj 3.94484 zz jj z jj jj 0.0611095 zzz zz jj z jj jj 0.0418766 zzz zz; j RI = jjj z jj -0.0429142 zzz zz jj z jj jj 0.0175473 zzz zz jjj jj -0.0010432 zzz zz jjj jj 0.0431563 zzz zzz jjj zz jj jjj -0.002116 zzz z j { k -1.0396
0. y z jij jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz j RE = jjj z jj 0. zzz zz jj z jj jj 0. zzz z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z j -1. { k
Corresponding to unrestrained dof ij 0.0611095 yz jj z jj 0.0418766 zzz jj zz jj z jj -0.0429142 zzz jjj zz jj 0.0175473 zzz j j zzz; RI = jj jj -0.0010432 zzz jj zz jj zz jjj 0.0431563 zzz jj zz jjj -0.002116 zzz jj zz k -1.0396 {
»»RE »» = 1.;
ij 0. yz jj z jj 0. zzz jj zz jj z jj 0. zzz jj zz jj z jj 0. zzz j zz; RE = jj jj 0. zzz jj zz jj zz jjj 0. zzz jj z jj 0. zzz jj zz j z k -1. {
»»R »» = 0.10523
Convergence parameter = 0.00553672
ij -0.0611095 yz jj z jj -0.0418766 zzz jj zz jj z jj 0.0429142 zzz jjj zz jj -0.0175473 zzz j j zzz R = RE - RI = jj jj 0.0010432 zzz jj zz jj zz jjj -0.0431563 zzz jj zz jjj 0.002116 zzz jj zz k 0.0395959 {
222
Geometric Nonlinearity
Iteration = 4 Global equations at start of the element assembly process ij 0 jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj jj jj 0 jj k0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
0
0 0 0 0
0 0 0 0 0
E = 10600.;
n = 0.35;
0 y i Du1 y i 0 y zz jj zz zz jj 0 zzzz jjjj Dv1 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzzz jjjj Du2 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzzz jjjj Dv2 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzz jjj Du3 zzz jjj 0 zzz zz jj zz zz jj 0 zzzz jjjj Dv3 zzzz jjjj 0 zzzz zz jj zz zz = jj 0 zzzz jjjj Du4 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzz jjj Dv4 zzz jjj 0 zzz zz jj zz zz jj 0 zzzz jjjj Du5 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzzz jjjj Dv5 zzzz jjjj 0 zzzz zz jj zz zz jj 0 zzz jjj Du6 zzz jjj 0 zzz zz jj zz zz jj 0 { k Dv6 { k -1 {
Initial thickness = 1.25 g = 0.461538
Plane stress analysis.
Interpolation functions and their derivatives 1 1 1 1 NT = : ÅÅÅÅÅ Hs - 1L Ht - 1L, - ÅÅÅÅÅ Hs + 1L Ht - 1L, ÅÅÅÅÅ Hs + 1L Ht + 1L, - ÅÅÅÅÅ Hs - 1L Ht + 1L> 4 4 4 4
t-1 1-t t+1 1 ∑NT ê∑s = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-t - 1L> 4 4 4 4
s -1 1 s +1 1-s ∑NT ê∑t = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-s - 1L, ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ > 4 4 4 4
Mapping to the master element
ij 0 jj jj ÅÅÅÅ5Å jj 2 Initial coordinates = jjjj jj ÅÅÅÅ5Å jj 2 jj k0 5s 5 x0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4
0y zz z 0 zzzz zz zz 3 zzz zz z 4{
ts s 7t 7 y0 Hs,tL = - ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4 4 4
223
5 ji ÅÅ4ÅÅ J = jjj j - ÅÅtÅÅ - ÅÅÅÅ1Å k 4 4
0 ÅÅÅÅ74Å
-
ÅÅ4sÅÅ
zyz zz z {
35 5s detJ = ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ 16 16
Current configuration
ij 0 jj jj ÅÅÅÅ5Å j Updated coordinates = jjjj 2 jj 2.50128 jj j k 0.00191931
yz zz zz 0 zz zz z 2.99898 zzz zz 4.00109 { 0
xHs,tL = -0.0001595 t s + 1.24984 s + 0.000800158 t + 1.2508 yHs,tL = -0.250529 t s - 0.250529 s + 1.75002 t + 1.75002
i -0.3125 s - 0.000079085 t + 2.18742 Deformation gradient, F p = 1êdetJ jj k -0.000920857 t - 0.000920857
0.0010002 - 0.000199375 s y zz 2.18752 - 0.313161 s {
Gauss quadrature points and weights Point
Weight
1
s Ø -0.57735 t Ø -0.57735
1.
2
s Ø -0.57735 t Ø 0.57735
1.
3
s Ø 0.57735 t Ø -0.57735
1.
4
s Ø 0.57735 t Ø 0.57735
1.
Computation of element matrices at 8-0.57735, -0.57735< with weight = 1. i ÅÅÅÅ5Å 0 J = jjjj 4 k -0.105662 1.89434
i 0.8 J -T = jj k0
0.0446224 y zz 0.527889 {
yz zz z {
detJ = 2.36792
0.000471006 y i 0.999986 zz; Deformation gradient, F p = jj k -0.000164363 1.00017 {
0 -0.000164363 0 ij 0.999986 yz _ j zz zz F = jjjj 0 0.000471006 0 1.00017 zz jj z 1.00017 -0.000164363 { k 0.000471006 0.999986 8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
224
Geometric Nonlinearity
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.333066, 0.310755, 0.0892449, -0.0669336<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.208166, -0.055778, 0.055778, 0.208166<
0.0000547439 0.310751 -0.0000510768 0.0892436 -0.0000146 ij -0.333062 j -0.0000262718 -0.0557875 0.0000262718 0.0557875 GT = jjjj -0.0000980477 -0.208202 jj -0.333089 -0.0556309 0.310817 0.0558193 0.0892508 k -0.20832
ij 0.546967 jj jj 1.20314 S = jjjj jjj 0 j k0
yz zz zz zz z 0.546967 1.20314 zzzz z 1.20314 1.99042 {
1.20314
0
0
1.99042
0
0
0 0
BT =
ij -0.333066 jj jj -0.208166 jj jj jj 0 jjj k0
0
0.310755
0
0
-0.055778 0
-0.0669336 0
0.055778
0
0.208166
0
0.0892449 0
-0.333066 0
0.310755
-0.208166 0
-0.055778 0
ij 6089.43 jj jj 2683.29 jj jj jjj -5076.85 jj jj -245.859 j k c = jjj jj -1631.66 jj jj jj -718.985 jj jj jj 619.075 jj k -1718.44
yz zz zz zz z -0.0669336 zzzz zz 0.208166 {
0.0892449 0
0.055778
0
0
2683.29
-5076.85 -245.859 -1631.66 -718.985 619.075
3467.01
-1535.94 -617.127 -718.985 -928.984 -428.367
-1535.94 4898.51
-673.001 1360.34
-617.127 -673.001 1279.36
65.8778
411.553
-1181.99
165.359
852.982
-718.985 1360.34
65.8778
437.202
192.652
-165.881
-928.984 411.553
165.359
192.652
248.92
114.781
-428.367 -1181.99 852.982 -1920.9
1797.39
-165.881 114.781
-827.594 460.456
514.705
728.799 -539.396
-1718.44 y zz -1920.9 zzzz zz 1797.39 zzzz zz -827.594 zzzz zz 460.456 zzz zz 514.705 zzzz zz -539.396 zzzz z 2233.79 {
ks = ij 0.928711 jj jj 0 jj jj jjj -0.26337 jj jj 0 jj jj jj -0.248847 jj jj jjj 0 jj jj -0.416494 jj j k0
0
-0.26337
0
-0.248847 0
0.928711
0
-0.26337
0
0
0.0512168 0
-0.26337
0
0
0.0705699 0
-0.248847 0 0
0.141584
-0.416494 0
0.0705699
0.0512168 0 0.0666785
0.0705699 0
yz z -0.416494 zzzz zz zz 0 zz zz 0.141584 zzzz zz zz 0 zz z 0.111599 zzzz zz zz 0 zz z 0.163311 {
-0.416494 0
-0.248847 0 0
0.141584
0.0705699
0
0
0.111599
0.0666785
0
0
0.111599
0
0.163311
0.141584
0
0.111599
0
225
6090.36 jij jj 2683.29 jj jj jj -5077.11 jj jj jj -245.859 j k = jjj jj -1631.91 jj jj jj -718.985 jj jj jj 618.659 jj k -1718.44
2683.29
-5077.11 -245.859 -1631.91 -718.985 618.659
3467.94
-1535.94 -617.39
-1535.94 4898.56 -617.39
-718.985 -929.233 -428.367
-673.001 1360.41
-673.001 1279.41
65.8778
411.553
-1181.85
165.429
852.982
-718.985 1360.41
65.8778
437.268
192.652
-165.769
-929.233 411.553
165.429
192.652
248.987
114.781
-428.367 -1181.85 852.982 -1921.32 1797.39
ij 1.28166 yz z jj jj 2.41272 zzz zz jj z jj jj -0.304829 zzz zz jj zz jj jjj -0.778127 zzz zz r i = jj jjj -0.343421 zzz zz jj jjj -0.646485 zzz zz jj z jj jj -0.633413 zzz zz jj k -0.988104 {
-165.769 114.781
-827.452 460.456
514.816
728.963 -539.396
-1718.44 y zz -1921.32 zzzz zz 1797.39 zzzz zz -827.452 zzzz zz 460.456 zzz zz 514.816 zzzz zz -539.396 zzzz z 2233.96 {
Computation of element matrices at 8-0.57735, 0.57735< with weight = 1. i ÅÅÅÅ5Å 0 J = jjjj 4 k -0.394338 1.89434 i 0.8 J -T = jj k0
0.166533 y zz 0.527889 {
yz zz z {
detJ = 2.36792
0.000471006 y i 0.999947 zz; Deformation gradient, F p = jj k -0.000613413 1.00017 {
0 -0.000613413 0 ij 0.999947 yz _ j zz zz F = jjjj 0 0.000471006 0 1.00017 zz jj z 1.00017 -0.000613413 { k 0.000471006 0.999947 8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.1502, 0.0669336, 0.333066, -0.2498<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.208166, -0.055778, 0.055778, 0.208166<
0.0000921348 0.0669301 -0.000041058 0.333049 -0.00020430 ij -0.150192 j GT = jjjj -0.0000980477 -0.208202 -0.0000262718 -0.0557875 0.0000262718 0.0557875 jj -0.150098 -0.0557436 0.0669792 0.055932 0.333089 k -0.208226
226
Geometric Nonlinearity
0.0836601 jij jj -0.559382 j S = jjjj jj 0 jj k0
zyz zz zz zz 0.0836601 -0.559382 zzzz z -0.559382 1.8286 {
-0.559382 0
0
1.8286
0
0
0 0
BT =
ij -0.1502 jj jj -0.208166 jj jj jj 0 jj k0
yz zz zz zz z -0.2498 zzzz z 0.208166 {
0
0.0669336
0
0.333066 0
-0.2498
0
-0.055778 0
0.055778 0
0.208166 0
-0.1502
0
0
0.333066 0
-0.055778 0
0.055778 0
0.0669336
-0.208166 0
1640.97 jij jj 1211.53 jj jj jj -371.542 jj jj jj 65.7207 j k c = jjj jj -2656.04 jj jj jj -1031.98 jj jj jj 1386.61 jj k -245.273
ij 0.13659 jj jj 0 jj jj jj 0.0695538 jj jj jj 0 k s = jjj jjj 0.0534347 jj jj 0 jj jj jj -0.259578 jj j k0 ij 1641.1 jj jj 1211.53 jj jj jjj -371.473 jj jj 65.7207 j k = jjj jj -2655.99 jj jj jj -1031.98 jj jj jj 1386.36 jj k -245.273
0
-245.273 y zz 2442.29 -279.957 467.714 -1976.39 -1164.47 1044.81 -1745.53 zzzz zz -279.957 261.553 -144.598 1086.12 -115.091 -976.128 539.647 zzzz zz 467.714 -144.598 208.784 -460.769 102.696 539.647 -779.194 zzzz zz -1976.39 1086.12 -460.769 5623.37 717.54 -4053.45 1719.61 zzz zz -1164.47 -115.091 102.696 717.54 1445.04 429.527 -383.268 zzzz zz 1044.81 -976.128 539.647 -4053.45 429.527 3642.96 -2013.99 zzzz z -1745.53 539.647 -779.194 1719.61 -383.268 -2013.99 2907.99 { -371.542 65.7207
1211.53
-2656.04 -1031.98 1386.61
0
0.0695538
0
0.0534347
0
-0.259578
0
0.13659
0
0.0695538
0
0.0534347
0
-0
0
0.0303117
0
0.0132592
0
-0.113125
0
0.0695538
0
0.0303117
0
0.0132592
0
-0
0
0.0132592
0
-0.0172099 0
0.0534347
0
0.0132592
0
0
-0.113125 0
-0.259578 0
-0.0172099 0
-0.0494839 0
-0.113125 0
-0.0494839 0 0.422187
-0.0494839 0
-245.273 y zz -1745.79 zzzz zz -279.957 261.583 -144.598 1086.13 -115.091 -976.241 539.647 zzzz zz 467.783 -144.598 208.815 -460.769 102.71 539.647 -779.307 zzzz zz -1976.39 1086.13 -460.769 5623.35 717.54 -4053.5 1719.61 zzz zz -1164.42 -115.091 102.71 717.54 1445.02 429.527 -383.317 zzzz zz 1044.81 -976.241 539.647 -4053.5 429.527 3643.38 -2013.99 zzzz z -1745.79 539.647 -779.307 1719.61 -383.317 -2013.99 2908.41 { 1211.53
-371.473 65.7207
-2655.99 -1031.98 1386.36
2442.42
-279.957 467.783
-1976.39 -1164.42 1044.81
-0 0 0.4
227
-0.307042 y z jij jj 0.878348 zzz zz jj zz jjj jj -0.108727 zzz zz jj z jj jj 0.412858 zzz zz r i = jjj jj 0.0099939 zzz zz jj z jj jj 0.249602 zzz zz jj z jj jj 0.405775 zzz zz jj k -1.54081 {
Computation of element matrices at 80.57735, -0.57735< with weight = 1. i ÅÅÅÅ5Å 0 J = jjjj 4 k -0.105662 1.60566 i 0.8 J -T = jj k0
0.0526449 y zz 0.622796 {
zyz zz {
detJ = 2.00708
0.000440983 y i 0.999983 Deformation gradient, F p = jj zz; { k -0.000193914 0.99982
0.999983 0 -0.000193914 0 jij zyz j zz j F = jj 0 0.000440983 0 0.99982 zz jj zz 0.99982 -0.000193914 { k 0.000440983 0.999983 _
8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.321033, 0.29471, 0.10529, -0.0789674<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.0658061, -0.245592, 0.245592, 0.0658061<
0.0000622526 0.294705 -0.0000571483 0.105288 -0.000020417 ij -0.321027 j -0.000108302 -0.245548 0.000108302 0.245548 GT = jjjj -0.0000290194 -0.0657943 jj -0.320962 -0.245458 0.294705 0.245634 0.105223 k -0.0659466
ij -0.961547 jj jj 0.970672 S = jjjj jj 0 jj k0
0.970672
yz zz zz zz z -0.961547 0.970672 zzzz z 0.970672 -2.24325 { 0
0
-2.24325 0
0
0 0
228
Geometric Nonlinearity
BT =
ij -0.321033 jj jj -0.0658061 jj jj jj 0 jj k0
0
0.29471
-0.245592 0
0.245592 0
0.0658061
-0.321033
0
0
0
0.29471
-0.0658061 0
ij 4442.25 jj jj 692.318 jj jj jj -3879.17 jj jj jj 1622.72 k c = jjj jjj -1602.49 jj jj -1880.23 jj jj jj 1039.42 jj j k -434.807
-3879.17 1622.72
0 0.10529
-0.245592 0
692.318
0.245592 0 -1602.49 -1880.23 1039.42
1199.56
332.558
-243.008 -935.767 -1021.67 -89.1086
4301.39
-2377.81 730.34
1408.12
-1152.55
-243.008 -2377.81 3433.65
117.952
-2270.6
637.131
-935.767 730.34
117.952
1067.85
849.42
-195.694
-1021.67 1408.12
-2270.6
849.42
2683.86
-377.304
65.1138
637.131
-195.694 -377.304 308.826
-920.045 -31.6052 608.406
0.282057 0 0
0.282057 0
0
-0.9015
0
0.282057
0
-0.9015
0
0.377887 0
-0.0363789 0 0
0.241556 0 0
-0.075577
0.377887
0
0.241556
0
0
0.377887
0
0.24
-0.240253 -0.101254
0
-0.101254
0
-0.240253
0
-0.
0
-0.0647248 0
-0.101254
0
-1602.53 -1880.23 1039.35
1199.39
332.558
-242.726 -935.767 -1021.7
-89.1086
332.558
4300.49
-2377.81 730.718
-1152.31
1408.12
-242.726 -2377.81 3432.75
117.952
-2270.22 637.131
-935.767 730.718
117.952
1067.61
849.42
-195.795
-1021.7
-2270.22 849.42
2683.62
-377.304
1408.12
-89.1086 -1152.31 637.131 65.0382
637.131
0 -0.
0.241556 0
-3878.89 1622.72
-434.807 y zz 65.1138 zzzz zz 637.131 zzzz zz -920.045 zzzz zz -31.6052 zzz zz 608.406 zzzz zz -170.719 zzzz z 246.525 {
-0.0363789 0
0.377887 0
-0.075577
-170.719
-0.0363789 0
0 -0.170101
692.318
ij -0.614004 yz z jj jj 0.41149 zzz zz jj z jj jj 1.30809 zzz zz jjj jj -2.09976 zzz j zzz j r i = jj jj -0.343581 zzz zz jj zz jj jjj 1.12564 zzz zz jj jjj -0.3505 zzz zz jj k 0.562629 {
yz zz zz zz z -0.0789674 zzzz z 0.0658061 { 0
332.558
-89.1086 -1152.55 637.131
ij -0.170101 jj jj 0 jj jj jjj 0.282057 jj jj 0 j k s = jjj jj -0.0363789 jj jj jj 0 jj jj jj -0.075577 jj k0 ij 4442.08 jj jj 692.318 jj jj jj -3878.89 jj jj jj 1622.72 k = jjj jjj -1602.53 jj jjj -1880.23 jj jj jj 1039.35 jj k -434.807
0.10529
-0.0789674 0
0 0
-195.795 -377.304 308.761
-919.803 -31.6052 608.304
-170.719
-434.807 y zz 65.0382 zzzz zz 637.131 zzzz zz -919.803 zzzz zz -31.6052 zzz zz 608.304 zzzz zz -170.719 zzzz z 246.46 {
-0.
229
Computation of element matrices at 80.57735, 0.57735< with weight = 1. i ÅÅÅÅ5Å 0 J = jjjj 4 k -0.394338 1.60566 0.196473 y zz 0.622796 {
i 0.8 J -T = jj k0
yz zz z {
detJ = 2.00708
0.000440983 y i 0.999938 zz; Deformation gradient, F p = jj k -0.000723696 0.99982 {
0 -0.000723696 0 ij 0.999938 yz _ j zz zz 0.000440983 0 0.99982 F = jjjj 0 zz jj z 0.99982 -0.000723696 { k 0.000440983 0.999938 8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.10529, 0.00705308, 0.392947, -0.29471<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.0658061, -0.245592, 0.245592, 0.0658061< 0.0000761978 0.00705264 -5.10428 µ 10-6 jij -0.105283 jj G = jjj -0.0000290194 -0.0657943 -0.000108302 -0.245548 jj -0.105223 -0.245573 0.00722954 k -0.0658485 T
ij -1.50849 jj jj -1.11033 S = jjjj jj 0 jjj k0
yz zz zz zz z -1.50849 -1.11033 zzzz zz -1.11033 -2.43478 {
-1.11033 0
0
-2.43478 0
0
0 0
0.24575
BT =
-0.10529 jij jj -0.0658061 jj jj jj 0 jj j k0
-0.00028437
0.392922
0.000108302 0.245548 0.392699
zyz zz zz zz -0.29471 zzzz z 0.0658061 {
0
0.00705308 0
0.392947 0
-0.29471
0
-0.245592
0
0.245592 0
0.0658061 0
-0.10529
0
0.00705308 0
0.392947 0
-0.245592
0.245592 0
-0.0658061 0
0
0
230
Geometric Nonlinearity
516.326 jij jj 227.497 jj jj jj 128.253 jj jj jj 590.248 j k c = jjj jj -1926.96 jj jj jj -849.032 jj jj jj 1282.38 jj k 31.2868
ij -0.10701 jj jj 0 jj jj jj -0.166651 jj jj jj 0 k s = jjj jjj 0.399366 jj jj 0 jj jj jj -0.125706 jj j k0 ij 516.219 jj jj 227.497 jj jj jjj 128.086 jj jj 590.248 j k = jjj jj -1926.56 jj jj jj -849.032 jj jj jj 1282.25 jj k 31.2868
227.497
128.253
590.248
-1926.96 -849.032 1282.38
31.2868
293.99
244.546
682.509
-849.032 -1097.18 376.989
120.685
244.546
596.48
-55.4557 -478.645 -912.658 -246.087 723.567
682.509
-55.4557 2576.11
-2202.83 -2547.16 1668.04
-711.457
-849.032 -478.645 -2202.83 7191.5
3168.63
-4785.9
-1097.18 -912.658 -2547.16 3168.63
4094.75
-1406.94 -450.403
376.989
-246.087 1668.04
120.685
723.567
-4785.9
-116.764
-1406.94 3749.61
-638.09
-711.457 -116.764 -450.403 -638.09
1041.17
0
-0.166651
0
0.399366
0
-0.125706
0
-0.10701
0
-0.166651
0
0.399366
0
-0.12
0
-0.358974
0
0.62195
0
-0.0963249 0
-0.166651 0
-0.358974
0
0.62195
0
-0.09
0
0.62195
0
-1.49046 0
0.46914
0
0.399366
0
0.62195
0
-1.49046 0
0
-0.0963249 0
0.46914
0
-0.247109
0
0.46914
0
-0.24
-0.125706 0
-0.0963249 0
0.469
227.497
128.086
590.248
-1926.56 -849.032 1282.25
31.2868
293.883
244.546
682.343
-849.032 -1096.78 376.989
120.559
244.546
596.121
-55.4557 -478.023 -912.658 -246.184 723.567
682.343
-55.4557 2575.75
-2202.83 -2546.54 1668.04
-711.554
-849.032 -478.023 -2202.83 7190.01
3168.63
-4785.43 -116.764
-1096.78 -912.658 -2546.54 3168.63
4093.26
-1406.94 -449.934
376.989
-246.184 1668.04
120.559
723.567
ij -0.582059 yz z jj jj -0.694731 zzz zz jjj jj -0.658052 zzz zzz jjj zz jj jjj -1.47981 zzz zz r i = jj jjj 2.17227 zzz zz jj jjj 2.59277 zzz zz jj jjj -0.932164 zzz zz jj z j k -0.418233 {
zyz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz {
-4785.43 -1406.94 3749.36
-638.09
-711.554 -116.764 -449.934 -638.09
1040.93
After summing contributions from all points, the element equations as follows:
yz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz z {
231
12689.8 jij jj 4814.63 jj jj jj -9199.39 jj jj jj 2032.83 j k = jjj jj -7816.98 jj jj jj -4480.23 jj jj jj 4326.61 jj k -2367.24
rT = H 0
4814.63
-9199.39 2032.83
-7816.98 -4480.23 4326.61
7403.64
-1238.79 290.01
-4480.17 -4212.14 904.328
-1238.79 10056.7
-3250.86 2699.23
-3250.86 7496.73
290.01
791.922
-3556.59
-2479.77 -4548.62 3697.8
-4480.17 2699.23
-2479.77 14318.2
4928.24
-9200.49
-4212.14 791.922
-4548.62 4928.24
8470.89
-1239.94
904.328
-3556.59 3697.8
-3481.51 3697.73
0 0 0 0
0 0 0L
-9200.49 -1239.94 8430.46
-3238.12 2031.7
289.87
-3362.19
Complete element equations for element 1 12689.8 jij jj 4814.63 jj jj jj -9199.39 jj jj jj 2032.83 jj jj jj -7816.98 jj jj jj -4480.23 jj jj jj 4326.61 jj k -2367.24
4814.63
-9199.39 2032.83
-7816.98 -4480.23 4326.61
7403.64
-1238.79 290.01
-4480.17 -4212.14 904.328
-1238.79 10056.7 290.01
-3250.86 2699.23
-3250.86 7496.73
791.922
-3556.59
-2479.77 -4548.62 3697.8
-4480.17 2699.23
-2479.77 14318.2
4928.24
-9200.49
-4212.14 791.922
-4548.62 4928.24
8470.89
-1239.94
904.328
-3556.59 3697.8
-3481.51 3697.73
-9200.49 -1239.94 8430.46
-3238.12 2031.7
289.87
-3362.19
The element contributes to 81, 2, 3, 4, 5, 6, 7, 8< global degrees of freedom. 1 jij zyz jj 2 zz jj zz jj zz jj 3 zz jj zz jj zz jj 4 zz j z Locations for element contributions to a global vector: jjj zzz jjj 5 zzz jj zz jjj 6 zzz jj zz jjj 7 zzz jj zz j z k8{
ij @1, 1D jj jj @2, 1D jj jj jj @3, 1D jj jj jj @4, 1D and to a global matrix: jjj jj @5, 1D jj jj jjj @6, 1D jj jj @7, 1D jj j k @8, 1D
@1, 2D
@1, 3D
@1, 4D
@1, 5D
@1, 6D
@3, 3D
@3, 4D
@3, 5D
@3, 6D
@5, 3D
@5, 4D
@5, 5D
@5, 6D
@2, 2D
@2, 3D
@4, 2D
@4, 3D
@3, 2D
@5, 2D
@6, 2D
@7, 2D
@8, 2D
@6, 3D
@7, 3D
@8, 3D
@2, 4D
@4, 4D
@6, 4D
@7, 4D
@8, 4D
@2, 5D
@4, 5D
@6, 5D
@7, 5D
@8, 5D
Adding element equations into appropriate locations we have
@2, 6D
@4, 6D
@6, 6D
@7, 6D
@8, 6D
-2367.24 y zz -3481.51 zzzz zz 3697.73 zzzz zz -3238.12 zzzz zz zz 2031.7 zz zz zz 289.87 zz z -3362.19 zzzz z 6429.76 {
-2367.24 y i Du1 y i 0. y zz jj zz jj zz -3481.51 zzzz jjjj Dv1 zzzz jjjj 0. zzzz zz jj zz jj zz 3697.73 zzzz jjjj Du2 zzzz jjjj 0. zzzz zz jj zz jj zz -3238.12 zzzz jjjj Dv2 zzzz jjjj 0. zzzz zz jj z=j z zz jj Du3 zzz jjj 0. zzz 2031.7 zz jj zz jj zz zz jj z j z zz jj Dv3 zzz jjj 0. zzz 289.87 zz jj zz jj zz zz jj z j z -3362.19 zzz jjj Du4 zzzz jjjj 0. zzzz zj z j z 6429.76 { k Dv4 { k 0. {
@1, 7D
@1, 8D y zz @2, 8D zzzz zz @3, 7D @3, 8D zzzz zz @4, 7D @4, 8D zzzz zz @5, 7D @5, 8D zzz zz @6, 7D @6, 8D zzzz zz @7, 7D @7, 8D zzzz z @8, 7D @8, 8D {
@2, 7D
232
Geometric Nonlinearity
12689.8 jij jj 4814.63 jj jj jj -9199.39 jj jj jj 2032.83 jj jj jj -7816.98 jj jj jj -4480.23 jj jj jj 4326.61 jj jj jj -2367.24 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
4814.63
-9199.39 2032.83
-7816.98 -4480.23 4326.61
-2367.24 0
7403.64
-1238.79 290.01
-4480.17 -4212.14 904.328
-3481.51 0
-1238.79 10056.7
-3250.86 2699.23
-3250.86 7496.73
290.01
791.922
-3556.59 3697.73
-2479.77 -4548.62 3697.8
-4480.17 2699.23
-2479.77 14318.2
4928.24
-9200.49 2031.7
0
-4212.14 791.922
-4548.62 4928.24
8470.89
-1239.94 289.87
0
904.328
-3556.59 3697.8
-9200.49 -1239.94 8430.46
-3362.19 0
-3481.51 3697.73
-3238.12 2031.7
289.87
-3362.19 6429.76
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
ij Du1 yz ij 0 yz z j jj z jj Dv1 zzz jjj 0 zzz zz jjj zzz jjj jj Du zz jj 0 zzz zz jjj 2 zzz jjj zz jj zz jj z jjj Dv2 zzz jjj 0 zzz jj zz jj zz jjj Du3 zzz jjj 0 zzz jj zz jj zz jjj Dv3 zzz jjj 0 zzz jj zz = jj zz jjj Du zzz jjj 0 zzz jj 4 zz jj zz z jj z j jj Dv4 zzz jjj 0 zzz zz jj zz jj zz jj zz jj jjj Du5 zzz jjj 0 zzz zz jj zz jj jjj Dv5 zzz jjj 0 zzz zz jj zz jj jjj Du6 zzz jjj 0 zzz zz jj zz jj z j z j k Dv6 { k -1 { E = 10600.;
0
-3238.12 0
n = 0.35;
Plane stress analysis.
Initial thickness = 1.25 g = 0.461538
Interpolation functions and their derivatives 1 1 1 1 NT = : ÅÅÅÅÅ Hs - 1L Ht - 1L, - ÅÅÅÅÅ Hs + 1L Ht - 1L, ÅÅÅÅÅ Hs + 1L Ht + 1L, - ÅÅÅÅÅ Hs - 1L Ht + 1L> 4 4 4 4
t-1 1-t t+1 1 ∑NT ê∑s = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-t - 1L> 4 4 4 4
s -1 1 s +1 1-s ∑NT ê∑t = : ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ H-s - 1L, ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ > 4 4 4 4
0 0 0y zz 0 0 0 zzzz zz 0 0 0 zzzz zz 0 0 0 zzzz zz 0 0 0 zzz zz 0 0 0 zzzz zz 0 0 0 zzzz zz 0 0 0 zzz zz 0 0 0 zzzz zz 0 0 0 zzzz zz 0 0 0 zzz zz 0 0 0{
233
Mapping to the master element
5 jij ÅÅÅÅ2Å jj jj 10 Initial coordinates = jjjj jjj 10 jj k0
3 zy zz z 3 zzzz zz 4 zzz zz 4{
5ts 35 s 5t 45 x0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 8 8 8 8
t 7 y0 Hs,tL = ÅÅÅÅÅÅ + ÅÅÅÅÅ 2 2 5t 35 ij ÅÅÅÅ ÅÅÅ + ÅÅÅÅ ÅÅ 8 8 J = jjjj k0
5s ÅÅÅÅ ÅÅÅÅ - ÅÅÅÅ58Å yz 8 zz zz 1 ÅÅ2ÅÅ {
5t 35 detJ = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 16 16
Current configuration
ij 2.50128 jj jj 10.0015 Updated coordinates = jjjj jj 10.0025 jj k 0.00191931
2.99898 y zz 2.99231 zzzz zz 3.99232 zzzz z 4.00109 {
xHs,tL = 0.625077 t s + 4.37519 s - 0.624604 t + 5.62679 yHs,tL = -0.000526383 t s - 0.00385972 s + 0.500531 t + 3.49617 Deformation gradient, F p = 1êdet
0.000218742 s + 0.000295622 t + 0.00185061 y i 0.312538 t + 2.18759 zz J jj k -0.000263192 t - 0.00192986 0.000109395 s + 0.312503 t + 2.18741 {
Gauss quadrature points and weights Point
Weight
1
s Ø -0.57735 t Ø -0.57735
1.
2
s Ø -0.57735 t Ø 0.57735
1.
3
s Ø 0.57735 t Ø -0.57735
1.
4
s Ø 0.57735 t Ø 0.57735
1.
Computation of element matrices at 8-0.57735, -0.57735< with weight = 1.
234
Geometric Nonlinearity
i 4.01416 J = jjjj k0
-0.985844 y zz zz ÅÅ12ÅÅ {
detJ = 2.00708
i 0.249118 0 yz z J -T = jj k 0.491184 2. {
0.000774083 y i 1.00004 zz; Deformation gradient, F p = jj k -0.000885817 0.999923 {
0 -0.000885817 0 ij 1.00004 yz jj zz j zz 0 0.000774083 0 0.999923 F = jj zz jj z 0.999923 -0.000885817 { k 0.000774083 1.00004 _
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0982367, 0.0982367, 0.0263225, -0.0263225<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.982367, -0.0176327, 0.263225, 0.736775<
0.0000870198 0.0982403 -0.0000870198 0.0263234 -0.000023316 ij -0.0982403 jj j G = jj -0.000760434 -0.982291 -0.0000136492 -0.0176313 0.000203758 0.263204 jj -0.982479 -0.0973589 -0.0175573 0.0982447 0.263254 0.0260872 k T
ij 0.11378 jj jj -0.438336 S = jjjj jj 0 jjj k0
0
0
0 0
-0.0982367 jij jj -0.982367 j B T = jjjj jj 0 jj k0 ij 9924.87 jj jj 3196.41 jj jj jj -244.087 jj jj jj -910.044 k c = jjj jj -2659.36 jj jj jjj -856.476 jj jj -7021.42 jj j k -1429.89
yz zz zz zz z 0.11378 -0.438336 zzzz zz -0.438336 -0.777265 {
-0.438336 0 -0.777265 0
0
0.0982367
0
0
-0.0176327 0
0.0263225 0
-0.0263225 0
0.263225
0
0.736775
0
0.0263225 0
-0.0982367 0
0.0982367
-0.982367
-0.0176327 0
3196.41 41287. -2200.17 646.701 -856.476 -11062.8 -139.765 -30870.9
0
0.263225
0
-244.087 -910.044 -2659.36 -856.476 -7021.42 -1429.89 y zz -2200.17 646.701 -856.476 -11062.8 -139.765 -30870.9 zzzz zz 414.831 -57.1098 65.403 589.534 -236.147 1667.75 zzzz zz -57.1098 108.407 243.846 -173.283 723.309 -581.825 zzzz zz 65.403 243.846 712.574 229.492 1881.38 383.138 zzz zz 589.534 -173.283 229.492 2964.27 37.4499 8271.83 zzzz zz -236.147 723.309 1881.38 37.4499 5376.18 -620.994 zzzz z 1667.75 -581.825 383.138 8271.83 -620.994 23180.9 {
0 0
235
-2.09138 jij jj 0 jj jj jj 0.0676898 jj jj jj 0 j k s = jjj jj 0.560383 jj jj jj 0 jj jj jj 1.46331 jj k0 ij 9922.78 jj jj 3196.41 jj jj jj -244.02 jj jj jj -910.044 k = jjj jjj -2658.8 jj jjj -856.476 jj jj jj -7019.96 jj k -1429.89
0
0.0676898
0
0.560383
0
1.46331
0
-2.09138
0
0.0676898
0
0.560383
0
1
0
0.00595831
0
-0.0181374 0
0.00595831
0
0.0676898 0 0
-0.0181374 0
0.560383
0
-0.150154
-0.0181374 0
0
-0.0555107 0
1.46331
0
-0.392091
-0.0555107 0
-0.0181374 0
-244.02
-2200.17 646.769
-910.044 -2658.8
Computation of element matrices at 8-0.57735, 0.57735< with weight = 1. -0.985844 y zz zz ÅÅ12ÅÅ {
detJ = 2.36792
i 0.211156 0 yz z J -T = jj k 0.416333 2. {
0.00080028 y i 1.00005 Deformation gradient, F p = jj zz; k -0.000879172 0.999936 {
1.00005 0 -0.000879172 0 jij zyz j zz j 0.00080028 0 0.999936 F = jj 0 zz jj zz 0.999936 -0.000879172 { k 0.00080028 1.00005 _
8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
-
0
-0.392091
0
-0.150154
0
-
0
-1.0157
0
-0.392091
0
-
-856.476 -7019.96 -1429.89 y zz -856.476 -11062.3 -139.765 -30869.4 zzzz zz -2200.17 414.837 -57.1098 65.3849 589.534 -236.202 1667.75 zzzz zz 646.769 -57.1098 108.413 243.846 -173.301 723.309 -581.88 zzzz zz -856.476 65.3849 243.846 712.423 229.492 1880.99 383.138 zzz zz -11062.3 589.534 -173.301 229.492 2964.12 37.4499 8271.43 zzzz zz -139.765 -236.202 723.309 1880.99 37.4499 5375.17 -620.994 zzzz z -30869.4 1667.75 -581.88 383.138 8271.43 -620.994 23179.9 { 3196.41
41284.9
ij -1.05389 yz z jj jj -2.0226 zzz zz jj zz jj jjj -0.047378 zzz z jj jj 0.0736844 zzz j zzz j r i = jj jj 0.282388 zzz zz jj z jj jj 0.541954 zzz zz jj z jj jj 0.818878 zzz zz jj k 1.40696 { i 4.73584 J = jjjj k0
-0.0555107 0
236
Geometric Nonlinearity
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338< 8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0223112, 0.0223112, 0.0832666, -0.0832666<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.832666, -0.167334, 0.375501, 0.624499<
0.0000196154 0.0223123 -0.0000196154 0.0832707 -0.0000732057 ij -0.0223123 j -0.000133914 -0.167323 0.000300506 0.375476 GT = jjjj -0.000666366 -0.832612 jj -0.0215777 -0.167325 0.0224569 0.375586 0.0829311 k -0.832725
ij 0.329705 jj jj -0.309363 S = jjjj jj 0 jj k0
0
0
0 0
ij -0.0223112 jj jj -0.832666 T B = jjjj jj 0 jj k0 ij 8084.31 jj jj 743.212 jj jj jjj 1593.76 jj jj -109.896 j k c = jjj jj -3730.08 jj jj jj -1043.45 jj jj jj -5947.98 jj k 410.136
yz zz zz zz z 0.329705 -0.309363 zzzz z -0.309363 -0.562527 {
-0.309363 0 -0.562527 0
0
0.0223112
0
0
-0.167334 0
0.0832666 0
-0.0832666 0
0.375501
0
0.624499
0
0.0832666 0
-0.
0.375501
0.62
-0.0223112 0
0.0223112
-0.832666
-0.167334 0
0
0
0
yz z -26168.1 zzzz zz -455.582 350.228 -143.677 -636.915 63.0497 -1307.07 536.209 zzzz zz 7011.72 -143.677 1416.17 -282.637 -3142.67 536.209 -5285.22 zzzz zz -1987.89 -636.915 -282.637 1990. 1215.71 2377. 1054.82 zzz zz -15765.7 63.0497 -3142.67 1215.71 7179.71 -235.305 11728.6 zzzz zz 1700.26 -1307.07 536.209 2377. -235.305 4878.05 -2001.16 zzzz z -26168.1 536.209 -5285.22 1054.82 11728.6 -2001.16 19724.7 { -109.896 -3730.08 -1043.45 -5947.98 410.136
743.212
1593.76
34922.
-455.582 7011.72
-1987.89 -15765.7 1700.26
ks = ij -1.18795 jj jj 0 jj jj jj -0.218887 jj jj jj 0 jj jj jjj 0.589943 jj jjj 0 jj jj 0.816896 jj j k0
0 yz z 0.816896 zzzz zz zz 0 -0.0392988 0 0.11152 0 0.146665 0 zz zz -0.218887 0 -0.0392988 0 0.11152 0 0.146665 zzzz zz zz 0 0.11152 0 -0.285264 0 -0.416199 0 zz z 0.589943 0 0.11152 0 -0.285264 0 -0.416199 zzzz zz zz 0 0.146665 0 -0.416199 0 -0.547362 0 zz z 0.816896 0 0.146665 0 -0.416199 0 -0.547362 { 0
-0.218887
0
0.589943
0
0.816896
-1.18795
0
-0.218887
0
0.589943
0
237
8083.12 jij jj 743.212 jj jj jj 1593.54 jj jj jj -109.896 j k = jjj jj -3729.49 jj jj jj -1043.45 jj jj jj -5947.17 jj k 410.136
zyz -26167.3 zzzz zz -455.582 350.189 -143.677 -636.804 63.0497 -1306.92 536.209 zzzz zz 7011.5 -143.677 1416.13 -282.637 -3142.56 536.209 -5285.07 zzzz zz -1987.89 -636.804 -282.637 1989.71 1215.71 2376.58 1054.82 zzz zz -15765.1 63.0497 -3142.56 1215.71 7179.42 -235.305 11728.2 zzzz zz 1700.26 -1306.92 536.209 2376.58 -235.305 4877.51 -2001.16 zzzz z -26167.3 536.209 -5285.07 1054.82 11728.2 -2001.16 19724.2 { -109.896 -3729.49 -1043.45 -5947.17 410.136
743.212
1593.54
34920.8
-455.582 7011.5
-1987.89 -15765.1 1700.26
ij -0.741847 yz z jj jj -1.4061 zzz zz jjj jj -0.175214 zzz zzz jjj zz jj jjj -0.258015 zzz zz r i = jj jjj 0.263155 zzz zz jj jjj 0.701188 zzz zz jj z jj jj 0.653906 zzz zz jj k 0.962925 {
Computation of element matrices at 80.57735, -0.57735< with weight = 1. i 4.01416 J = jjjj k0
-0.264156 y zz zz ÅÅ12ÅÅ {
detJ = 2.00708
i 0.249118 0 yz J -T = jj z k 0.131612 2. {
0.000899929 y i 1.00004 zz; Deformation gradient, F p = jj k -0.000885817 0.999986 {
0 -0.000885817 0 ij 1.00004 yz _ j zz zz F = jjjj 0 0.000899929 0 0.999986 zz jj z 0.999986 -0.000885817 { k 0.000899929 1.00004 8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0982367, 0.0982367, 0.0263225, -0.0263225<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.263225, -0.736775, 0.802582, 0.197418<
0.0000870198 0.0982403 -0.0000870198 0.0263234 -0.0000233169 ij -0.0982403 j GT = jjjj -0.000236883 -0.263221 -0.000663045 -0.736765 0.000722266 0.80257 jj -0.0980021 -0.736714 0.098888 0.802634 0.0256111 k -0.263322
238
Geometric Nonlinearity
0.380382 jij jj 0.0555749 j S = jjjj jj 0 jj k0
0
0
-0.0155527 0
0
0 0
ij -0.0982367 jj jj -0.263225 T B = jjjj jj 0 jj k0 1095.81 jij jj 850.679 jj jj jj 1499.69 jj jj jj 1414.63 j k c = jjj jjj -2193.66 jj jj -1886.26 jj jj jj -401.842 jj j k -379.048
0
0.0982367
0
-0.736775 0
0
0.0263225 0
-0.0263225 0
0.802582
0
0.197418
0
0.0263225 0
-0.
0.802582
0.19
-0.0982367 0
0.0982367
-0.263225
-0.736775 0
0
0
0
-2193.66 -1886.26 -401.842 -379.048 y zz 3050.83 124.516 8181.23 -941.831 -9039.9 -33.3639 -2192.15 zzzz zz 124.516 5754.42 -2360.74 -5712.22 1603.67 -1541.89 632.558 zzzz zz 8181.23 -2360.74 23268.8 313.554 -25215.2 632.558 -6234.86 zzzz zz -941.831 -5712.22 313.554 6375.3 712.294 1530.58 -84.0166 zzz zz -9039.9 1603.67 -25215.2 712.294 27498.7 -429.701 6756.38 zzzz zz -33.3639 -1541.89 632.558 1530.58 -429.701 413.149 -169.493 zzzz z -2192.15 632.558 -6234.86 -84.0166 6756.38 -169.493 1670.62 { 850.679
ij 0.0137169 jj jj 0 jj jj jj -0.0102907 jj jj jj 0 k s = jjj jj -0.00618355 jj jj jjj 0 jj jj 0.00275739 jj j k0 1095.82 jij jj 850.679 jj jj jj 1499.68 jj jj jj 1414.63 j k = jjj jjj -2193.67 jj jjj -1886.26 jj jj jj -401.839 jj k -379.048
zyz zz zz zz 0.380382 0.0555749 zzzz z 0.0555749 -0.0155527 {
0.0555749
1499.69
1414.63
0
-0.0102907 0
0.0137169
0
-0.00618355 0
0
-0.0321549 0
-0.0102907
0
-0.0321549 0
0
0.0338297
-0.0102907 0 0.0338297
0.00275
-0.00618355 0 0
0.00861
0.0338297
0
0
-0.0185815
0
-0.0090
-0.00618355 0
0.0338297
0
-0.0185815
0
0
0.00861588
0
-0.00906465 0
0.00275739
0
0.00861588
0
-0.0023
-0.00906465 0
-2193.67 -1886.26 -401.839 -379.048 y zz 3050.84 124.516 8181.22 -941.831 -9039.91 -33.3639 -2192.15 zzzz zz 124.516 5754.38 -2360.74 -5712.19 1603.67 -1541.88 632.558 zzzz zz 8181.22 -2360.74 23268.8 313.554 -25215.1 632.558 -6234.85 zzzz zz -941.831 -5712.19 313.554 6375.28 712.294 1530.58 -84.0166 zzz zz -9039.91 1603.67 -25215.1 712.294 27498.7 -429.701 6756.38 zzzz zz -33.3639 -1541.88 632.558 1530.58 -429.701 413.146 -169.493 zzzz z -2192.15 632.558 -6234.85 -84.0166 6756.38 -169.493 1670.62 { 850.679
1499.68
1414.63
239
0.130458 zy jij jj 0.00331056 zzz zz jj z jj jj 0.00894053 zzz zz jj z jj jj -0.042453 zzz zz j z r i = jjj jj -0.137003 zzz zz jj z jj jj 0.0277672 zzz zz jj z jj jj -0.00239561 zzz zz jj k 0.0113752 {
Computation of element matrices at 80.57735, 0.57735< with weight = 1. i 4.73584 J = jjjj k0
-0.264156 y zz zz ÅÅ12ÅÅ {
detJ = 2.36792
i 0.211156 0 yz z J -T = jj k 0.111556 2. {
0.000906948 y i 1.00005 Deformation gradient, F p = jj zz; { k -0.000879172 0.999989
1.00005 0 -0.000879172 0 jij zyz j zz j F = jj 0 0.000906948 0 0.999989 zz jj zz 0.999989 -0.000879172 { k 0.000906948 1.00005 _
8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑x0 , ∑N2 ê∑x0 , ∑N3 ê∑x0 , ∑N4 ê∑x0 = = 8-0.0223112, 0.0223112, 0.0832666, -0.0832666<
9∑N1 ê∑y0 , ∑N2 ê∑y0 , ∑N3 ê∑y0 , ∑N4 ê∑y0 = = 8-0.223112, -0.776888, 0.832666, 0.167334<
0.0000196154 0.0223123 -0.0000196154 0.0832707 -0.0000732057 ij -0.0223123 j -0.000704597 -0.776879 0.000755185 0.832657 GT = jjjj -0.000202351 -0.22311 jj -0.0221148 -0.776906 0.022994 0.832782 0.0825336 k -0.223143
ij 0.555649 jj jj 0.109246 S = jjjj jj 0 jj k0
0.109246
yz zz zz zz z 0.555649 0.109246 zzzz z 0.109246 0.0830751 {
0
0
0.0830751 0
0
0 0
ij -0.0223112 jj jj -0.223112 T B = jjjj jj 0 jj k0
0
0.0223112
0
0
-0.776888 0
0.0832666 0
-0.0832666 0
0.832666
0
0.167334
0
0.0832666 0
-0.
0.832666
0.16
-0.0223112 0
0.0223112
-0.223112
-0.776888 0
0
0
0
240
Geometric Nonlinearity
603.838 jij jj 194.484 jj jj jj 1989.51 jj jj jj 418.039 j k c = jjj jj -2253.56 jj jj jj -725.824 jj jj jj -339.792 jj k 113.301
1989.51
418.039
2512.05
72.3561
8721.96
72.3561
7037.29
-650.826
8721.96
-650.826 30399.
-725.824 -7424.95 -1560.14 -270.037 -32550.8
458.984
-1601.85 1792.93
ij 0.0162783 jj jj 0 jj jj jj 0.0457981 jj jj jj 0 k s = jjj jjj -0.0607516 jj jj 0 jj jj jj -0.00132483 jj j k0 ij 603.855 jj jj 194.484 jj jj jjj 1989.55 jj jj 418.039 j k = jjj jj -2253.62 jj jj jj -725.824 jj jj jj -339.793 jj k 113.301
-725.824 -9375.1
-9375.1
-1858.91 848.507
zyz -1858.91 zzzz zz -7424.95 -270.037 -1601.85 848.507 zzzz zz -1560.14 -32550.8 1792.93 -6570.14 zzzz zz 8410.38 2708.81 1268.12 -422.845 zzz zz 2708.81 34988.4 -1712.95 6937.53 zzzz zz 1268.12 -1712.95 673.517 -538.963 zzzz z -422.845 6937.53 -538.963 1491.52 { -2253.56 -725.824 -339.792 113.301
194.484
-6570.14
458.984
0
0.0457981
0
-0.0607516 0
0.0162783
0
0.0457981
0
-0.0607516 0
-0.00132
0
0.13802
0
-0.170921
0
-0.01289
0.0457981
0
0.13802
0
-0.170921
0
0
-0.170921
0
0.226728
0
0.0049443
-0.0607516
0
-0.170921
0
0.226728
0
0
-0.0128969 0
0.00494435
0
0.0092773
0.00494435
0
-0.00132483 0
-0.0128969 0
yz z -1858.91 zzzz zz 72.3561 7037.42 -650.826 -7425.12 -270.037 -1601.86 848.507 zzzz zz 8722. -650.826 30399.1 -1560.14 -32551. 1792.93 -6570.16 zzzz zz -725.824 -7425.12 -1560.14 8410.61 2708.81 1268.13 -422.845 zzz zz -9375.16 -270.037 -32551. 2708.81 34988.6 -1712.95 6937.54 zzzz zz 458.984 -1601.86 1792.93 1268.13 -1712.95 673.527 -538.963 zzzz z -1858.91 848.507 -6570.16 -422.845 6937.54 -538.963 1491.53 { 194.484
1989.55
418.039
-2253.62 -725.824 -339.793 113.301
2512.07
72.3561
8722.
-725.824 -9375.16 458.984
ij 0.108901 yz z jj jj 0.0619801 zzz zz jj z jj jj 0.214694 zzz zz jj z jj jj 0.183627 zzz zz r i = jjj z jjj -0.406424 zzz zz jj jjj -0.231313 zzz zz jj jjj 0.0828285 zzz zz jj z j k -0.0142944 {
After summing contributions from all points, the element equations as follows:
241
19705.6 jij jj 4984.79 jj jj jj 4838.76 jj jj jj 812.726 j k = jjj jj -10835.6 jj jj jj -4512.01 jj jj jj -13708.8 jj k -1285.5
rT = H 0
-10835.6 -4512.01 -13708.8 -1285.5 y zz 81768.7 -2458.88 24561.5 -4512.02 -45242.4 1986.11 -61087.8 zzzz zz -2458.88 13556.8 -3212.35 -13708.7 1986.21 -4686.87 3685.02 zzzz zz 24561.5 -3212.35 55192.4 -1285.38 -61082. 3685.01 -18672. zzzz zz -4512.02 -13708.7 -1285.38 17488. 4866.31 7056.28 931.092 zzz zz -45242.4 1986.21 -61082. 4866.31 72630.8 -2340.51 33693.5 zzzz zz 1986.11 -4686.87 3685.01 7056.28 -2340.51 11339.3 -3330.61 zzzz z -61087.8 3685.02 -18672. 931.092 33693.5 -3330.61 46066.2 { 4984.79
0 0 0 0
4838.76
812.726
0 0 0L
Complete element equations for element 2 19705.6 jij jj 4984.79 jj jj jj 4838.76 jj jj jj 812.726 jj jj jj -10835.6 jj jj jj -4512.01 jj jj jj -13708.8 jj k -1285.5
-10835.6 -4512.01 -13708.8 -1285.5 y i Du3 y i 0. y zz jj zz jj zz 81768.7 -2458.88 24561.5 -4512.02 -45242.4 1986.11 -61087.8 zzzz jjjj Dv3 zzzz jjjj 0. zzzz zz jj zz jj zz -2458.88 13556.8 -3212.35 -13708.7 1986.21 -4686.87 3685.02 zzzz jjjj Du5 zzzz jjjj 0. zzzz zz jj zz jj zz 24561.5 -3212.35 55192.4 -1285.38 -61082. 3685.01 -18672. zzzz jjjj Dv5 zzzz jjjj 0. zzzz zz jj zz = jj zz -4512.02 -13708.7 -1285.38 17488. 4866.31 7056.28 931.092 zzz jjj Du6 zzz jjj 0. zzz zz jj zz jj zz -45242.4 1986.21 -61082. 4866.31 72630.8 -2340.51 33693.5 zzzz jjjj Dv6 zzzz jjjj 0. zzzz zz jj zz jj zz 1986.11 -4686.87 3685.01 7056.28 -2340.51 11339.3 -3330.61 zzzz jjjj Du4 zzzz jjjj 0. zzzz zj z j z -61087.8 3685.02 -18672. 931.092 33693.5 -3330.61 46066.2 { k Dv4 { k 0. { 4984.79
4838.76
812.726
The element contributes to 85, 6, 9, 10, 11, 12, 7, 8< global degrees of freedom. 5 jij jj 6 jj jj jj 9 jj jj jj 10 j Locations for element contributions to a global vector: jjj jjj 11 jj jjj 12 jj jjj 7 jj j k8
ij @5, 5D jj jj @6, 5D jj jj jj @9, 5D jj jj jj @10, 5D and to a global matrix: jjj jj @11, 5D jj jj jjj @12, 5D jj jj @7, 5D jj j k @8, 5D
@5, 6D @6, 6D
@9, 6D
@5, 9D
@5, 10D
@9, 9D
@9, 10D
@6, 9D
@6, 10D
@10, 6D @10, 9D @10, 10D
@11, 6D @11, 9D @11, 10D
zyz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz zz {
@5, 11D
@5, 12D
@9, 11D
@9, 12D
@6, 11D
@10, 11D
@11, 12D
@7, 11D
@7, 12D
@12, 11D
@8, 6D
@8, 11D
@7, 9D @8, 9D
@7, 10D
@8, 10D
Adding element equations into appropriate locations we have
@10, 12D
@11, 11D
@12, 6D @12, 9D @12, 10D @7, 6D
@6, 12D
@12, 12D @8, 12D
@5, 7D
@5, 8D y zz @6, 8D zzzz zz @9, 7D @9, 8D zzzz zz @10, 7D @10, 8D zzzz zz @11, 7D @11, 8D zzz zz @12, 7D @12, 8D zzzz zz @7, 7D @7, 8D zzzz z @8, 7D @8, 8D {
@6, 7D
242
Geometric Nonlinearity
12689.8 jij jj 4814.63 jj jj jj -9199.39 jj jj jj 2032.83 jj jj jj -7816.98 jj jj jj -4480.23 jj jj jj 4326.61 jj jj jj -2367.24 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
4814.63
-9199.39 2032.83
-7816.98 -4480.23 4326.61
-2367.24 0
7403.64
-1238.79 290.01
-4480.17 -4212.14 904.328
-3481.51 0
-1238.79 10056.7 290.01
-3250.86 2699.23
-3250.86 7496.73
791.922
-3556.59 3697.73
-2479.77 -4548.62 3697.8
0 0
0
0
-3238.12 0
0
-4480.17 2699.23
-2479.77 34023.8
9913.03
-22909.2 746.199
-4212.14 791.922
-4548.62 9913.03
90239.5
746.175
-22909.2 746.175
19769.8
-6692.8
-4686.87 3685
-60797.9 -6692.8
52495.9
3685.02
-18
13556.8
-32
904.328
-3556.59 3697.8
4838.76
812.
-60797.9 -2458.88 2456
-3481.51 3697.73
-3238.12 746.199
0
0
0
4838.76
-2458.88 -4686.87 3685.02
0
0
0
812.726
24561.5
3685.01
-18672.
-3212.35 5519
0
0
0
-10835.6 -4512.02 7056.28
931.092
-13708.7 -12
0
0
0
-4512.01 -45242.4 -2340.51 33693.5
1986.21
-61
After assembly of all elements the global matrices are as follows. ij 12689.8 jj jj 4814.63 jj jj jjj -9199.39 jj jj 2032.83 jj jj jj -7816.98 jj jj jj -4480.23 K T = jjjj jj 4326.61 jj jj jj -2367.24 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0 ij 0. yz z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz z jj jj 0. zzz zz j zz; RE = jjjj jj 0. zzz zz jj z jj jj 0. zzz z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z j k -1. {
4814.63
-9199.39 2032.83
-7816.98 -4480.23 4326.61
-2367.24 0
7403.64
-1238.79 290.01
-4480.17 -4212.14 904.328
-3481.51 0
-1238.79 10056.7 290.01
-3250.86 2699.23
-3250.86 7496.73
791.922
-3556.59 3697.73
-2479.77 -4548.62 3697.8
0
-3238.12 0
-4480.17 2699.23
-2479.77 34023.8
9913.03
-22909.2 746.199
-4212.14 791.922
-4548.62 9913.03
90239.5
746.175
-22909.2 746.175
19769.8
-6692.8
-
-60797.9 -6692.8
52495.9
3
904.328
-3556.59 3697.8
4
-60797.9 -
-3481.51 3697.73
-3238.12 746.199
0
0
0
4838.76
-2458.88 -4686.87 3685.02
1
0
0
0
812.726
24561.5
3685.01
-18672.
-
0
0
0
-10835.6 -4512.02 7056.28
931.092
-
0
0
0
-4512.01 -45242.4 -2340.51 33693.5
1
ij 0.221441 yz z jj jj -3.00782 zzz zz jj z jj jj -0.236477 zzz zz jj zz jj zz jj 3.94484 zz jj z jj jj 0.0611095 zzz z jj jj 0.0418766 zzz zz j zz; RI = jjjj jj -0.0429142 zzz zz jj z jj jj 0.0175473 zzz zz jjj jj -0.0010432 zzz zz jj z jj jj 0.0431563 zzz zz jjj jj -0.002116 zzz zzz jjj { k -1.0396
ij -0.221441 yz zz jj zz jj 3.00782 zz jj z jj jj 0.236477 zzz zz jj z jj jj -3.94484 zzz zz jj z jj jj -0.0611095 zzz zz jj jjj -0.0418766 zzz zz R = RE - RI = jjjj z jj 0.0429142 zzz zz jj z jj jj -0.0175473 zzz zz jjj jj 0.0010432 zzz zz jj z jj jj -0.0431563 zzz zz jjj jj 0.002116 zzz zzz jjj k 0.0395959 {
243
System of equations 12689.8 jij jj 4814.63 jj jj jj -9199.39 jj jj jj 2032.83 jj jj jj -7816.98 jj jj jj -4480.23 jj jj jj 4326.61 jj jj jj -2367.24 jj jj 0 jj jj jj 0 jj jj jj 0 jj j k0
4814.63
-9199.39 2032.83
-7816.98 -4480.23 4326.61
-2367.24 0
7403.64
-1238.79 290.01
-4480.17 -4212.14 904.328
-3481.51 0
-1238.79 10056.7 290.01
-3250.86 2699.23
-3250.86 7496.73
791.922
-3556.59 3697.73
-2479.77 -4548.62 3697.8
-3238.12 0
-4480.17 2699.23
-2479.77 34023.8
9913.03
-22909.2 746.199
-4212.14 791.922
-4548.62 9913.03
90239.5
746.175
-22909.2 746.175
4838.76
-60797.9 -2458.
19769.8
-6692.8
-4686.
-3481.51 3697.73
-3238.12 746.199
-60797.9 -6692.8
52495.9
3685.02
0
0
0
4838.76
-2458.88 -4686.87 3685.02
0
0
0
812.726
24561.5
3685.01
-18672.
-3212.
0
0
0
-10835.6 -4512.02 7056.28
931.092
-13708
0
0
0
-4512.01 -45242.4 -2340.51 33693.5
1986.21
904.328
-3556.59 3697.8
0
Essential boundary conditions Node
dof
Value
1
Du1 Dv1
0 0
2
Du2 Dv2
0 0
Remove 81, 2, 3, 4< rows and columns. After adjusting for essential boundary conditions we have
13556.8
244
Geometric Nonlinearity
34023.8 jij jj 9913.03 jj jj jj -22909.2 jj jj jj 746.199 jj jj jj 4838.76 jj jj jj 812.726 jj jj jj -10835.6 jj k -4512.01
-10835.6 -4512.01 y zz 90239.5 746.175 -60797.9 -2458.88 24561.5 -4512.02 -45242.4 zzzz zz 746.175 19769.8 -6692.8 -4686.87 3685.01 7056.28 -2340.51 zzzz zz -60797.9 -6692.8 52495.9 3685.02 -18672. 931.092 33693.5 zzzz zz -2458.88 -4686.87 3685.02 13556.8 -3212.35 -13708.7 1986.21 zzz zz 24561.5 3685.01 -18672. -3212.35 55192.4 -1285.38 -61082. zzzz zz -4512.02 7056.28 931.092 -13708.7 -1285.38 17488. 4866.31 zzzz z -45242.4 -2340.51 33693.5 1986.21 -61082. 4866.31 72630.8 { 9913.03
-22909.2 746.199
4838.76
812.726
Du -0.0611095 y jij 3 zyz jij z jj Dv zz jj -0.0418766 zzz jj 3 zz jj zz z j jj z jj Du zzz jjj 0.0429142 zzz jj 4 zz jj zz z j jj z jj Dv4 zzz jjj -0.0175473 zzz zz jj jj zz zz = jj jj z jj Du5 zz jj 0.0010432 zzz zz jj jj zz z j jj z jj Dv5 zzz jjj -0.0431563 zzz zz jj jj zz jjj Du zzz jjj 0.002116 zzz jj 6 zz jj zz j z j z Dv 0.0395959 k 6{ k {
Solving the final system of global equations we get
9Du3 = 6.18995 µ 10-6 , Dv3 = -8.82096 µ 10-6 , Du4 = 0.0000118247, Dv4 = 3.81438 µ 10-6 , Du5 = 6.62093 µ 10-6 , Dv5 = -0.0000498326, Du6 = 0.0000118227, Dv6 = -0.0000488354=
Complete table of nodal values Du
Dv
1
0
0
2
0
0 -6
3
6.18995 µ 10
-8.82096 µ 10-6
4
0.0000118247
3.81438 µ 10-6
5
6.62093 µ 10-6
-0.0000498326
6
0.0000118227
-0.0000488354
Total increments since the start of this load step
245
Du
Dv
1
0
0
2
0
0
3
0.00128751
-0.00103326
4
0.00193114
0.00109351
5
0.00151194
-0.00774094
6
0.00246313
-0.00773134
Total nodal values u
v
1
0
0
2
0
0
3
0.00128751
-0.00103326
4
0.00193114
0.00109351
5
0.00151194
-0.00774094
6
0.00246313
-0.00773134
Solution for element 1 Initial configuration
0 jij jj 5 jj ÅÅÅÅÅ j 2 Nodal coordinates = jjjj jj ÅÅÅÅ5Å jj 2 jj k0 5s 5 x0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4
0y zz z 0 zzzz zz zz 3 zzz zz z 4{
ts s 7t 7 y0 Hs,tL = - ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ 4 4 4 4 ij ÅÅ54ÅÅ J = jjjj t 1 k - ÅÅ4ÅÅ - ÅÅÅÅ4Å
Current configuration II
yz zz z 7 s z ÅÅÅÅ4Å - ÅÅ4ÅÅ {
0
35 5s detJ = ÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ 16 16
ij 0 jj jj 0 Nodal displacements = jjjj jj 0.00128751 jj k 0.00193114
yz zz zz zz z -0.00103326 zzzz zz 0.00109351 {
0
0
246
Geometric Nonlinearity
0 jij jj 5 jj ÅÅÅÅÅ Updated coordinates = jjjj 2 jj 2.50129 jj j k 0.00193114
zyz zz zz zz zz 2.99897 zzz zz 4.00109 { 0 0
xII Hs,tL = -0.000160908 t s + 1.24984 s + 0.000804661t + 1.2508 yII Hs,tL = -0.250532 t s - 0.250532 s + 1.75002 t + 1.75002
i 1.24984 - 0.000160908 t 0.000804661 - 0.000160908 s yz J II = jj z k -0.250532 t - 0.250532 1.75002 - 0.250532 s { detJ II = -0.313165 s - 0.0000799989 t + 2.18744
i -0.3125 s - 0.0000804243 t + 2.18742 0.00100583 - 0.000201135 s Deformation gradient, F IIp = H1êdetJ L jj 2.18752 - 0.313165 s k -0.000926698 t - 0.000926698
Solution at 8s, t< = 8-0.57735, -0.57735< ï 8x, y< = 80.528692, 0.800777< 0.000897562 y i 1.24993 zz J II = jj { k -0.105887 1.89466
yz z {
detJ II = 2.36829
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.333106, 0.31075, 0.0892554, -0.0669002<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.207973, -0.0559158, 0.0557263, 0.208163< B TL
0.31075 0 0.0892554 0 -0.0669002 0 -0.333106 0 jij j j -0.207973 0 -0.0559158 0 0.0557263 0 0.20 = jj 0 jj -0.207973 -0.333106 -0.0559158 0.31075 0.0557263 0.0892554 0.208163 -0. k
0.000473813 y i 0.999986 zz; F IIp = jj { k -0.000165406 1.00017
Det@F IIp D = 1.00016
Element thickness, hII = 1.2499
0.00030849 y i 0.999972 zz Left Cauchy-Green tensor = jj 0.00030849 1.00034 k {
i -0.0000143411 0.000154186 yz Green-Lagrange strain tensor, e = jj z 0.000170125 { k 0.000154186 i 0.54645 Cauchy stress tensor, s = jj k 1.21102 Principal stresses = H 2.68044
1.21102 y zz 1.9932 {
-0.140793 L
247
Effective stress Hvon MisesL = 2.75354
i 0.545358 1.21007 yz z Second PK stresses = jj k 1.21007 1.99307 {
r Ti = H -1.28435 -2.42117 0.302211 0.784057 0.344141 0.64875
Solution at 8s, t< = 8-0.57735, 0.57735< ï 8x, y< = 80.529728, 2.98854< 0.000897562 yz i 1.24975 J II = jj z -0.395176 1.89466 k {
0.638
0.988362 L
detJ II = 2.3682
8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.150337, 0.0669028, 0.333119, -0.249685<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.20806, -0.0558003, 0.0556107, 0.208249< B TL =
-0.150337 0 0.0669028 0 0.333119 0 -0.249685 0 jij zyz jj 0 -0.20806 0 -0.0558003 0 0.0556107 0 0.208249 zzzz jj jj zz 0.0556107 0.333119 0.208249 -0.249685 { k -0.20806 -0.150337 -0.0558003 0.0669028 0.000473813 y i 0.999946 zz; F IIp = jj k -0.000617304 1.00017 {
Det@F IIp D = 1.00012
Element thickness, hII = 1.24992
-0.000143377 y i 0.999893 zz Left Cauchy-Green tensor = jj -0.000143377 1.00034 { k
i -0.0000533813 -0.0000718105 yz Green-Lagrange strain tensor, e = jj z k -0.0000718105 0.000170125 {
i 0.0736461 -0.562858 yz Cauchy stress tensor, s = jj z k -0.562858 1.82972 { Principal stresses = H 1.99464
-0.0912729 L
Effective stress Hvon MisesL = 2.04181
i 0.0741917 -0.563643 yz Second PK stresses = jj z k -0.563643 1.8285 { r Ti =
H 0.313874 -0.876397 0.107553 -0.413686 -0.0200339 -0.253813 -0.401393 1.5439 L
248
Geometric Nonlinearity
Solution at 8s, t< = 80.57735, -0.57735< ï 8x, y< = 81.97199, 0.678509< 0.000711761 y i 1.24993 zz J II = jj { k -0.105887 1.60537
detJ II = 2.00668
8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.321051, 0.294667, 0.10534, -0.0789558<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.0656758, -0.245767, 0.24559, 0.0658531< B TL =
0.294667 0 0.10534 ij -0.321051 0 jj jj 0 -0.0656758 0 -0.245767 0 jj j 0.24559 k -0.0656758 -0.321051 -0.245767 0.294667 0.000443282 y i 0.999983 zz; F IIp = jj k -0.000195144 0.999818 {
0 0.24559
0
0.10534
0.0658531
Det@F IIp D = 0.999801
yz z 0.0658531 zzzz zz -0.0789558 {
-0.0789558 0
Element thickness, hII = 1.25013
0.000248061 y i 0.999966 zz Left Cauchy-Green tensor = jj { k 0.000248061 0.999636
i -0.0000169165 0.000124083 yz Green-Lagrange strain tensor, e = jj z -0.000181687 { k 0.000124083
i -0.972242 0.973957 zy Cauchy stress tensor, s = jj z -2.26735 { k 0.973957 Principal stresses = H -2.78938 -0.450216 L
Effective stress Hvon MisesL = 2.59374
i -0.97305 0.974877 yz Second PK stresses = jj z k 0.974877 -2.26759 {
r Ti = H 0.622573 -0.410861 -1.31917 2.11786
0.343125 -1.13952 0.35347
Solution at 8s, t< = 80.57735, 0.57735< ï 8x, y< = 81.97281, 2.53223< 0.000711761 y i 1.24975 J II = jj zz k -0.395176 1.60537 {
detJ II = 2.00659
8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
-0.567479 L
249
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.105344, 0.00687462, 0.393151, -0.294681<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.0657714, -0.24564, 0.245462, 0.0659487< B TL =
-0.105344 0 0.00687462 0 0.393151 0 -0.294681 0 jij zyz jj 0 -0.0657714 0 -0.24564 0 0.245462 0 0.0659487 zzzz jj jj zz 0.00687462 0.245462 0.393151 0.0659487 -0.294681 { k -0.0657714 -0.105344 -0.24564 0.000443282 y i 0.999937 zz; F IIp = jj k -0.000728286 0.999818 {
Det@F IIp D = 0.999755
Element thickness, hII = 1.25016
-0.000285039 y i 0.999874 zz Left Cauchy-Green tensor = jj -0.000285039 0.999637 k {
i -0.0000629378 -0.00014245 yz Green-Lagrange strain tensor, e = jj z -0.000181687 { k -0.00014245
i -1.53024 -1.11917 zy Cauchy stress tensor, s = jj z k -1.11917 -2.46012 { Principal stresses = H -3.20708 -0.78328 L
Effective stress Hvon MisesL = 2.89601
i -1.52927 -1.11934 yz Second PK stresses = jj z k -1.11934 -2.46237 { r Ti = H 0.58904
0.701655 0.663245 1.49663
-2.19833 -2.61861 0.946043 0.420323 L
After summing contributions from all points the internal load vector is as follows: r Ti = H 0.241136 -3.00677 -0.24616 3.98487
-1.5311 -3.3632 1.53612 2.3851 L
Global internal load vector
RTI = H 0.241136 -3.00677 -0.24616 3.98487
Solution for element 2 Initial configuration
-1.5311 -3.3632 1.53612 2.3851 0 0
0 0L
250
Geometric Nonlinearity
5 jij ÅÅÅÅ2Å jj jj 10 Nodal coordinates = jjjj jj 10 jj j k0
3 zy zz z 3 zzzz zz 4 zzz zz 4{
5ts 35 s 5t 45 x0 Hs,tL = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 8 8 8 8
t 7 y0 Hs,tL = ÅÅÅÅÅÅ + ÅÅÅÅÅ 2 2 5t 35 ij ÅÅÅÅ ÅÅÅ + ÅÅÅÅ ÅÅ 8 8 J = jjj j0 k
5s ÅÅÅÅ ÅÅÅÅ - ÅÅÅÅ58Å yz 8 zz zz ÅÅ12ÅÅ {
5t 35 detJ = ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅ 16 16
Current configuration II
ij 0.00128751 jj jj 0.00151194 Nodal displacements = jjjj jj 0.00246313 jj k 0.00193114 ij 2.50129 jj jj 10.0015 Updated coordinates = jjjj jjj 10.0025 jj k 0.00193114
-0.00103326 y zz -0.00774094 zzzz zz -0.00773134 zzzz z 0.00109351 {
2.99897 y zz 2.99226 zzzz zz 3.99227 zzzz zz 4.00109 {
xII Hs,tL = 0.625077 t s + 4.37519 s - 0.624601 t + 5.6268
yII Hs,tL = -0.000529293 t s - 0.00388313 s + 0.500534 t + 3.49615
0.625077 s - 0.624601 i 0.625077 t + 4.37519 zyz J II = jj k -0.000529293 t - 0.00388313 0.500534 - 0.000529293 s { detJ II = 0.000111499 s + 0.312542 t + 2.18751
Deformation gradient, F IIp = H1êdetJ L
0.000218202 s + 0.000297248 t + 0.00186253 zy jij 0.312538 t + 2.18759 z k -0.000264647 t - 0.00194157 0.0001113 s + 0.312503 t + 2.18741 {
Solution at 8s, t< = 8-0.57735, -0.57735< ï 8x, y< = 83.66975, 3.20923< -0.98549 y i 4.0143 zz J II = jj k -0.00357754 0.50084 {
detJ II = 2.007
8N1 , N2 , N3 , N4 < = 80.622008, 0.166667, 0.0446582, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
251
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.0991087, 0.0982174, 0.0265561, -0.0256648<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.982367, -0.0177107, 0.263224, 0.736853<
0.0982174 0 0.0265561 0 -0.0256648 0 ij -0.0991087 0 j -0.982367 0 -0.0177107 0 0.263224 0 0 B TL = jjjj 0 jj 0.263224 0.0265561 0.736853 k -0.982367 -0.0991087 -0.0177107 0.0982174
0.00077971 y i 1.00004 zz; F IIp = jj -0.000891232 0.999922 k {
Det@F IIp D = 0.999959
Element thickness, hII = 1.25003
-0.000111615 y i 1.00007 Left Cauchy-Green tensor = jj zz k -0.000111615 0.999845 {
-0.0000557122 y i 0.0000364488 zz Green-Lagrange strain tensor, e = jj k -0.0000557122 -0.0000775621 {
i 0.11158 -0.4382 yz Cauchy stress tensor, s = jj z k -0.4382 -0.782171 {
Principal stresses = H -0.961168 0.290578 L
Effective stress Hvon MisesL = 1.13471
i 0.112253 -0.4375 zy Second PK stresses = jj z k -0.4375 -0.783057 { r Ti =
H 1.05223
2.03666
0.0469646 -0.0732221 -0.281943 -0.545722 -0.817248 -1.41772 L
Solution at 8s, t< = 8-0.57735, 0.57735< ï 8x, y< = 82.53181, 3.78755< -0.98549 y i 4.73608 zz J II = jj k -0.00418872 0.50084 {
detJ II = 2.36789
8N1 , N2 , N3 , N4 < = 80.166667, 0.0446582, 0.166667, 0.622008<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.394338, -0.105662, 0.105662, 0.394338<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.0230466, 0.0221621, 0.0835945, -0.0827101<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.832701, -0.167363, 0.375457, 0.624606<
252
Geometric Nonlinearity
B TL
0.0221621 0 0.0835945 0 -0.0827101 0 -0.0230466 0 jij j j = jj 0 -0.832701 0 -0.167363 0 0.375457 0 0.62 jj -0. k -0.832701 -0.0230466 -0.167363 0.0221621 0.375457 0.0835945 0.624606
0.000805842 yz i 1.00005 z; F IIp = jj k -0.000884471 0.999935 {
Det@F IIp D = 0.999985
Element thickness, hII = 1.25001
-0.0000787251 y i 1.0001 zz Left Cauchy-Green tensor = jj k -0.0000787251 0.999872 {
-0.0000392663 y i 0.000049697 zz Green-Lagrange strain tensor, e = jj k -0.0000392663 -0.0000642129 { -0.309071 y i 0.328281 zz Cauchy stress tensor, s = jj k -0.309071 -0.565085 {
Principal stresses = H -0.661588 0.424784 L Effective stress Hvon MisesL = 0.948247
-0.308327 y i 0.328744 zz Second PK stresses = jj k -0.308327 -0.5657 {
r Ti = H 0.739373 1.41385
0.17464
0.259654 -0.262247 -0.704458 -0.651766 -0.969043 L
Solution at 8s, t< = 80.57735, -0.57735< ï 8x, y< = 88.30507, 3.2051< -0.263713 y i 4.0143 J II = jj zz k -0.00357754 0.500229 {
detJ II = 2.00712
8N1 , N2 , N3 , N4 < = 80.166667, 0.622008, 0.166667, 0.0446582<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.394338, 0.394338, 0.105662, -0.105662<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.0984677, 0.0975765, 0.0270368, -0.0261455<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.263139, -0.736874, 0.802568, 0.197445< B TL
0.0975765 0 0.0270368 0 -0.0261455 0 ij -0.0984677 0 jj j = jj 0 -0.263139 0 -0.736874 0 0.802568 0 0.19 jj -0.263139 -0.0984677 -0.736874 0.0975765 0.802568 0.0270368 0.197445 -0. k
0.000905244 y i 1.00004 zz; F IIp = jj { k -0.000891232 0.999986 Element thickness, hII = 1.24998
Det@F IIp D = 1.00002
253
0.0000139679 yz i 1.00007 Left Cauchy-Green tensor = jj z k 0.0000139679 0.999973 {
i 0.0000364488 Green-Lagrange strain tensor, e = jjjj -6 k 7.02876 µ 10
7.02876 µ 10-6 yz zz z -0.0000134267 {
i 0.383614 0.0548364 yz z Cauchy stress tensor, s = jj k 0.0548364 -0.00819444 {
Principal stresses = H 0.391144 -0.0157245 L Effective stress Hvon MisesL = 0.399239
i 0.383491 0.0551849 yz Second PK stresses = jj z k 0.0551849 -0.00809669 {
r Ti = H -0.130971 -0.00813712 -0.00746583 0.0285736 0.136437 -0.0127802 0.00200046 -
Solution at 8s, t< = 80.57735, 0.57735< ï 8x, y< = 88.00056, 3.78271< -0.263713 y i 4.73608 zz J II = jj k -0.00418872 0.500229 {
detJ II = 2.36802
8N1 , N2 , N3 , N4 < = 80.0446582, 0.166667, 0.622008, 0.166667<
8∑N1 ê∑s, ∑N2 ê∑s, ∑N3 ê∑s, ∑N4 ê∑s< = 8-0.105662, 0.105662, 0.394338, -0.394338<
8∑N1 ê∑t, ∑N2 ê∑t, ∑N3 ê∑t, ∑N4 ê∑t< = 8-0.105662, -0.394338, 0.394338, 0.105662<
9∑N1 ê∑xII , ∑N2 ê∑xII , ∑N3 ê∑xII , ∑N4 ê∑xII = = 8-0.0225074, 0.021623, 0.0839989, -0.0831144<
9∑N1 ê∑yII , ∑N2 ê∑yII , ∑N3 ê∑yII , ∑N4 ê∑yII = = 8-0.223094, -0.776916, 0.832598, 0.167412<
0.021623 0 0.0839989 0 -0.0831144 0 ij -0.0225074 0 j B TL = jjjj 0 -0.223094 0 -0.776916 0 0.832598 0 0.16 jj 0.832598 0.0839989 0.167412 -0. k -0.223094 -0.0225074 -0.776916 0.021623 0.000912247 y i 1.00005 zz; F IIp = jj -0.000884471 0.99999 k {
Det@F IIp D = 1.00004
Element thickness, hII = 1.24997
0.0000277221 y i 1.0001 zz Left Cauchy-Green tensor = jj { k 0.0000277221 0.99998
i 0.000049697 0.0000139146 Green-Lagrange strain tensor, e = jjj -6 k 0.0000139146 -9.84878 µ 10 i 0.558862 0.108833 yz z Cauchy stress tensor, s = jj k 0.108833 0.0909341 {
zyz z {
254
Geometric Nonlinearity
Principal stresses = H 0.582937 0.0668598 L Effective stress Hvon MisesL = 0.552549
i 0.558618 0.109242 yz Second PK stresses = jj z k 0.109242 0.0911304 { r Ti = H -0.1091 -0.0672987 -0.214507 -0.20215 0.407165 0.251162 -0.0835585 0.0182862 L After summing contributions from all points the internal load vector is as follows: r Ti =
H 1.55153
3.37507
-0.000368144 0.0128562 -0.000588578 -1.0118 -1.55057 -2.37613 L
Global internal load vector
RTI = H 0.241136 -3.00677 -0.24616 3.98487
0.0204331 0.011878 -0.0144529 0.00896964 -0.0003
After assembling all element internal force vectors, the global internal force and the external load vectors are as follows. yz ij 0.241136 zz jj zz jj -3.00677 zz jj zz jj zz jjj -0.24616 zz jj zz jj 3.98487 zz jj zz jj zz jj 0.0204331 zz jj zz jj zz jj 0.011878 zz j zz; RI = jjj jj -0.0144529 zzz zz jj z jj jj 0.00896964 zzz z jj jj -0.000368144 zzz zz jj zz jj zz jj 0.0128562 zzz jjj zz jj jjj -0.000588578 zzz z j { k -1.0118
ij 0. yz z jj jj 0. zzz zz jj zz jj jjj 0. zzz z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz j RE = jjj z jj 0. zzz zz jj z jj jj 0. zzz z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z jj jj 0. zzz zz jj z j k -1. {
Corresponding to unrestrained dof ij 0.0204331 yz jj zz jj 0.011878 zz jj zz jj z jj -0.0144529 zzz jj zz jj z jj 0.00896964 zzz j zz; RI = jj jj -0.000368144 zzz jj zz jj zz jjj 0.0128562 zzz jj zz jjj -0.000588578 zzz jj zz k -1.0118 {
ij 0. yz jj z jj 0. zzz jj zz jj z jj 0. zzz jj zz jj z jj 0. zzz j zz; RE = jj jj 0. zzz jj zz jj zz jjj 0. zzz jj z jj 0. zzz jj zz j z k -1. {
ij -0.0204331 yz jj z jj -0.011878 zzz jj zz jj z jj 0.0144529 zzz jj zz jj z jj -0.00896964 zzz j zz R = RE - RI = jj jj 0.000368144 zzz jj zz jj zz jjj -0.0128562 zzz jj zz jjj 0.000588578 zzz jj zz k 0.0117983 {
255
»»RE »» = 1.;
»»R »» = 0.0339544
Convergence parameter = 0.000576451 Solution converged to the desired tolerance.
PRINT U
NODAL SOLUTION PER NODE
***** POST1 NODAL DEGREE OF FREEDOM LISTING *****
LOAD STEP= TIME=
2
2.0000
SUBSTEP=
2
LOAD CASE=
0
THE FOLLOWING DEGREE OF FREEDOM RESULTS ARE IN GLOBAL COORDINATES
NODE
UX
UY
UZ
USUM
1
0.0000
0.0000
0.0000
0.0000
2
0.0000
0.0000
0.0000
0.0000
3
0.12891E-02-0.10366E-02
0.0000
0.16541E-02
4
0.19347E-02 0.10938E-02
0.0000
0.22225E-02
5
0.15135E-02-0.77565E-02
0.0000
0.79028E-02
6
0.24662E-02-0.77464E-02
0.0000
0.81295E-02
MAXIMUM ABSOLUTE VALUES NODE VALUE
PRINT S
6
5
0.24662E-02-0.77565E-02
0 0.0000
ELEMENT SOLUTION PER ELEMENT
6 0.81295E-02
256
Geometric Nonlinearity
***** POST1 ELEMENT NODAL STRESS LISTING *****
LOAD STEP= TIME=
2
SUBSTEP=
2.0000
2
LOAD CASE=
0
THE FOLLOWING X,Y,Z VALUES ARE IN ROTATED GLOBAL COORDINATES, WHICH INCLUDE RIGID BODY ROTATION EFFECTS
ELEMENT= NODE 1
1
PLANE182
SX
SY
SZ
0.54467
1.9912
2 -0.97708 3
-1.5389
4
0.68717E-01
ELEMENT= NODE
SYZ
SXZ
1.2130
0.0000
0.0000
-2.2776
-0.90949E-12 0.97497
0.0000
0.0000
-2.4717
-0.18190E-11 -1.1223
0.0000
0.0000
1.8266
-0.18190E-11-0.56424
0.0000
0.0000
2
-0.27285E-11
SXY
PLANE182
SX
SY
SZ
SXY
SYZ
SXZ
4
0.32729
-0.56670
-0.13642E-11-0.30821
0.0000
0.0000
3
0.11054
-0.78491
0.27285E-11-0.43730
0.0000
0.0000
5
0.38467
-0.49327E-02 0.90949E-12 0.54555E-01
0.0000
0.0000
6
0.55965
0.0000
0.0000
0.94391E-01-0.45475E-12 0.10869
AnsysFiles\Chap09\Prb9-4Data2Elem.txt
!*Problem 9.4 !* Two element model !* Large displacement analysis !* Hyper elastic material
257
êPREP7 !* Element type !* ET,1,PLANE182 KEYOPT,1,1,0 KEYOPT,1,3,3 KEYOPT,1,6,0 KEYOPT,1,10,0 !* R,1,1.25 !* Material property *set, e, 10600 *set, nu, 0.35 *set, mu, eêH2*H1+nuLL *set, k, eêH3*H1-2*nuLL *set, d, 2êk *set, P, 1 !* Ansys neo-Hookean model MPTEMP,,,,,,,, MPTEMP,1,0 TB,HYPE,1,1,2,NEO TBTEMP,0 TBDATA,,mu,d,,,, !* k,1,0,0 k,2,5ê2,0 k,3,5ê2, 3 k,4,10, 3 k,5,10, 4 k,6,0, 4 A,1,2,3,6 A,3,4,5,6 ESIZE,100 AMESH,ALL êSOLU
DL,1, ,ALL ANTYPE,0 NLGEOM,1
258
Geometric Nonlinearity
ARCLEN,1,1,0.0001 NCNV,2,0,0,0,0 RESCONTRL,DEFINE,ALL,1,1 ERESX,NO OUTRES,ERASE OUTRES,ALL,1 AUTOTS,-1.0 !* First load step !* No applied load !* Used for initialization of !* the arc-length controls LSWRITE,1, !* Specify applied forces FK,5,FY,-P LSWRITE,2, LSSOLVE,1,2,1 FINISH !* Postprocessing êPOST1
SET,LAST PRNSOL,UX
Deformed shape for load =1000 lb
259
vonMises stresses for load =1000 lb
260
Deformed shape for load =10,000 lb
Geometric Nonlinearity
261
vonMises stresses for load =10,000 lb
262
Geometric Nonlinearity
AnsysFiles\Chap09\Prb9-4DataMesh.txt
!*Problem 9.4 !* Large displacement analysis !* Hyper elastic material êPREP7
!* Element type !* ET,1,PLANE182 KEYOPT,1,1,0 KEYOPT,1,3,3 KEYOPT,1,6,0 KEYOPT,1,10,0 !* R,1,1.25 !* Material property *set, e, 10600 *set, nu, 0.35 *set, mu, eêH2*H1+nuLL *set, k, eêH3*H1-2*nuLL *set, d, 2êk *set, P, 1 !* Ansys neo-Hookean model MPTEMP,,,,,,,, MPTEMP,1,0 TB,HYPE,1,1,2,NEO TBTEMP,0 TBDATA,,mu,d,,,, !* k,1,0,0 k,2,5ê2,0 k,3,5ê2, 3 k,4,10, 3 k,5,10, 4 k,6,0, 4 k,7,9, 4
263
LSTR,1,2 LSTR,2,3 LSTR,3,4 LFILLT,2,3,1ê4, , LSTR,4,5 LSTR,5,7 LSTR,7,6 LSTR,6,1 AL,1,2,4,3,5,6,7,8 ESIZE,1ê8 MSHKEY,0 AMESH,ALL êSOLU
DL,1, ,ALL ANTYPE,0 NLGEOM,1 ARCLEN,1,1,0.0001 NCNV,2,0,0,0,0 RESCONTRL,DEFINE,ALL,1,1 ERESX,NO OUTRES,ERASE OUTRES,ALL,1 AUTOTS,-1.0 !* First load step !* No applied load !* Used for initialization of !* the arc-length controls LSWRITE,1, !* Specify applied forces SFL,6,PRES,Pê1.25, LSWRITE,2, !* Apply large load !* to see the large disp effects SFL,6,PRES,10*Pê1.25, LSWRITE,3, LSSOLVE,1,3,1 FINISH !* Postprocessing
264
Geometric Nonlinearity
êPOST1 SET,LAST PRNSOL,UX
9.5 Side view of a pry bar is shown in Figure 9.18. The cross section of the bar is rectangular with thickness = 1/2 in and width (perpendicular to the plane of paper) = 0.75 in. The other dimensions (in inches) are shown in the figure. Assume a compressible neo-Hookean material with E = 29 µ 106 psi and n = 0.3. A load of P = 200 lb is applied at the center of the handle.
P
30°
15.3564
6 5.40192 r=3 120°
Figure 9.18.
Pry bar
(a) Taking large displacements into consideration and using a frame model (for example BEAM3 element in Ansys) determine maximum deflection and stress in the bar. (b) Using a plane stress model determine maximum deflection and von Mises stress in the bar. To avoid stress concentration assume the load to be uniformly distributed over a 3 in length.
Deformed shape
265
Maximum stress
AnsysFiles\Chap09\Prb9-5BeamModel.txt
266
Geometric Nonlinearity
!* Problem 9-5 !* Pry bar - Beam Model !* Use k-in units êPREP7
ET,1,BEAM3 R,1,0.75*.5,0.75*H.5**3Lê12,0.5,0,0,0, MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,EX,1,,29000 *set,t,1ê2 *set,a,6 *set,r,3+tê2 *set,b,16 *set,p,3.14159ê6 *set,xc, 15.3564 *set,yc,5.40192-tê2 k,1,-a, 0 k,2,-aê2, 0 k,3,0,0 k,4,xc-r*sinHpL, -yc-r*cosHpL k,5,xc+r,-yc k,6,xc+r,0 k,7, xc, -yc L,1,2 L,2,3 L,3,4 LARC,4,5,7,r, L,5,6 ESIZE,1,0, LMESH,ALL DK,6, , , ,0,UX,UY, , , , , DK,5, , , ,0,UX, , , , , , FK,2,FY,-0.2 êSOLU
FINISH NLGEOM,1 SOLVE FINISH
267
êPOST1 ETABLE,smaxI,NMISC, 1 ETABLE,smaxJ,SMISC,3 PRETAB,SMAXI,SMAXJ
Deformed shape
vonMises stresses
268
Geometric Nonlinearity
The simple beam element model results compare very well with those from the plane stress model. With the small displacement assumption we get the following deformed shape.
269
AnsysFiles\Chap09\Prb9-5PlaneStressModel.txt
!* Problem 9-5 !* Pry bar !* Use k-in units
êPREP7 *set,a,6 *set,r,3
*set,b,16 *set,t,0.5 *set,p,H22ê7Lê6 *set,xc, b*cosHpL + r*sinHpL *set,yc,-b*sinHpL + r*cosHpL k,1,-a, 0 k,2,0, 0 k,3,b*cosHpL, -b*sinHpL k,4, xc+r, yc
270
Geometric Nonlinearity
k,5,xc+r,0 k,6,-a, -t k,7,0, -t
k,8,xc - Hr + tL*sinHpL, yc - Hr + tL*cosHpL k,9,xc+r+t, yc k,10,xc+r+t,0 k,11,-1.5,0 k,12,-4.5,0 k,13,xc,yc l,6,7 l,7,8 LARC,8,9,13,r+t, l,9,10 l,10,5 l,5,4 LARC,3,4,13,r l,3,2 l,2,11 l,11,12 l,12,1 l,1,6 FLST,2,12,4 FITEM,2,1 FITEM,2,2 FITEM,2,3 FITEM,2,4 FITEM,2,5 FITEM,2,6 FITEM,2,7 FITEM,2,8 FITEM,2,9 FITEM,2,10 FITEM,2,11 FITEM,2,12 AL,P51X ET,1,PLANE42 KEYOPT,1,3,3 R,1,0.75,
271
MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,EX,1,,29000 MPDATA,PRXY,1,,0.3 ESIZE,tê2 LESIZE,3,tê4, , , , , , ,1 LESIZE,7,tê4, , , , , , ,1 AMESH,1 êSOLU
FINISH DK,10, , , ,0,ALL, , , , , , DK,9, , , ,0,UX, , , , , , SFL,10,PRES,0.200êH3*0.75L, NLGEOM,1 SOLVE êPOST1
FINISH PLESOL,S,EQV,0,1 NSORT,U,X *GET,maxUx,SORT, ,MAX NSORT,U,Y *GET,maxUy,SORT, ,MAX NSORT,S,EQV *GET,maxSEQV,SORT, ,MAX
9.6 Compute linearized buckling load for the three bar truss shown in Figure 9.19. All members have the same cross-sectional area and are of the same material, A = 0.001 m2 and E = 200 GPa. The dimensions in meters are shown in the figure.
272
Geometric Nonlinearity
P 3
0
-4
0
4
HmL
Figure 9.19.
e = 200 10^3; A = 0.001 * 10002 ; P = 1000.; n1 = 80, 3
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