Analysis of Straight and Skewed Box Girder Bridge by Finite Strip Method
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International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN ( ISSN 2250-2459, 2250-2459, Volume 2, Issue 11, November 2012)
Analysis of Straight and Skewed box Girder Gird er Bridge by Finite Strip Method Prof. Dr. S.A.Halkude S.A.Halkude1, Prof. Akim C.Y. 2 Principal, Walchand Institute of Technology, Sholapur Asst. Professor in Civil Engg.Dept., Deogiri I nstitute of Engineering and Management Studies, Aurangabad II. FINITE STRIP MODELING OF BRIDGES
Abstract — Generally a box girder is a beam and hence bending and shear actions exist in longitudinal direction. Due to the thin walled and widespread cross-section, the shear is not uniform even along the width of horizontal plates. When subjected to eccentric loading, the section exhibits torsion. An exact close form solution for such a continuum with a complex behavior is almost impossible. Some approximate analytical methods with simplifying assumptions for loading and geometry have been used for analysis. The objective of the research reported in this paper is to obtain analysis of box Girder Bridge by Finite Strip Method considering various values of skew angles considering their effect on each nodal line at specific distance interval along the span of girder by considering self weight and point load.
Selection of the best approach and the most suitable type of strip is based mainly on the bridge geometry, namely the shape and support conditions. The load conditions should also be considered. For a right or skew slab of box Girder Bridge with a constant cross-section and simply supported ends, the finite strip method is the most efficient method. It reduces to half bandwidth of the stiffness matrix significantly. Therefore the time consumption in forming this matrix is reduced Generating a finite strip model Depending upon the objective of analysis and the loading conditions, the suitable finite strip model can be chosen. In using the finite strip method for the overall analysis of a single span skew box girder bridge, the following modeling will provide adequate accuracy.
Keywords — Finite Strip Method, Box Girder, Skew angle, Nodal lines, Strips, Load Vector, Stiffness matrix, Strain energy, Finite Element Method.
I. I NTRODUCTION
For a structure with constant cross-section and end boundary conditions that do not change transversely, st ress analysis can be performed using finite strip instead of finite elements. In each strip, the displacement component at any point is expressed e xpressed in terms of the displacement parameters of nodal lines by means of simple polynomials in the transverse direction and continuously differentiable smooth series in the longitudinal direction. With the stipulation that such series should satisfy the boundary conditions at the ends of the strip. Using Strain-displacement relationships, the strain energy of the structure and the potential energy of external loads can be expressed in terms of the displacement parameters should make the total potential energy of the structure become minimum. This yields a set of linear algebraic equations with the displacement parameters, the displacement and stress components at any point in the structure can be obtained.
The flanges and webs are idealized as the orthotropic plates with equivalent elastic properties. Each web and flange is divided into minimum three strips. If only the deflection and longitudinal stress are required, few strips are are sufficient. If more more accurate accurate results are desired more strips should be used considering load act on a nodal line. In regions with a higher stress gradient, narrower strips should be used. The width of strip should change gradually from one strip to another. If the cross section and loads are both symmetrical about the centerline, only half of the bridge needs to be analyzed. If the bridge is subjected to uniform load only, five to ten symmetrical harmonies are adequate for the analysis
Numbering nodal lines and strips In order to minimize the half bandwidth of a stiffness matrix in a finite strip model, the nodal lines should be numbered so as to keep the difference between the numbers of all the nodal lines within each strip at a minimum.
191
International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 11, November 2012)
M(y) = - EI
=0
3.1
The deflection of the beam by a half – sine wave is, w(y) = δ sin 3.2 In order to obtain the best approximation the total potential energy developed in the system become a minimum. П=U+W
П= ∫
3.3
The first derivative of П with respect to δ must be zero
Modelling of box girder
П =0 δ
The numbering of nodal lines and strips for the proposed box girder model is done as shown below. The Fig.2.2 shows the Finite Strip Model of box girder whereas the Fig.2.3 shows the Finite Element Model of box girder.
П = П=
3.4
δ
∫
δ
δ
δ
3.6
Then the value of
3.5
is obtained by solving eq. 3.4
3.7
The deflection function proposed in Equation 1.2 becomes w(y) =
sin
3.8
From which the bending moment is obtained as III. THEORETICAL FORMULATION Consider a simply supported beam carrying a central point load at centre and having span L.
M(y) =
EI
d 2w
=
dy 2
sin
3.9
To improve the accuracy using a series of sine function r
w(y) =
m sin
m y l
m1
3.10
Applying the energy approach again yields. r
1
m
w(y) =
m 1
r
M(y) =
m m 1
The boundary conditions at both ends y = 0 and y = l are,
1 2
4
sin
sin
sin
sin
3.11
This will give resulting deflection and moment at mid span
w(y) = 0 192
International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 11, November 2012)
Proposed work for analysis of bridge is descritized into no. of strips.
After solving the above equation for
a0 , a1 , a2 , a3 ......
the displacement function 3.12 can be written as
Where
N1 ( x) (1 3 X 2 +2 X 3 ) N2 ( x) x(1 2 X + X 2 ) N3 ( x) ( 3 X 2 2 X 3 ) N4 ( x) x( X 2 X )
In each strip the displacement components at any point is
In a matrix form, r
fm( x)sin
w(x,y) =
m y l
m 1 r
(a
=
a1 x a2 x a3 x ....) sin 2
0
3
m 1
w(x,y) =
m y l
3.12
wim r im m y [ N1 , N2 ,N 3, N 4 ] sin w l m 1 jm jm
3.16
Where a0 , a1 , a2 , a3 ...... are the unknown displacement
More concisely
parameters.
r
[ N ]
Consider deflection amplitudes for a plate strip. As shown above In this there are four displacement parameters
w(x,y) =
wim , w jm , im and
Energy Formulation
r
wi =
jm i.e.
m y
wim sin
r
w
jm
sin
i =
j =
To
(
(
dx
dw dx
l
r
)i =
im
sin
3.13
………… 3.17
sin jm
m y
m1
the
wim , w jm , im and
four
jm
The Equation 3.17 can be written in the following matrix form.
2w 2 x 2 w 2 dxdy y 2w 2 xy
l
r
include
l
m y
m 1
) j =
m y
The strain energy of a plate strip is given by
m y
m 1
dw
sin
l
m 1
w j =
m 1
m
l displacement
3.14
U=
3.14
2
parameters
a third order polynomial
1
is
required in equation. 2 3 fm( x) = a0 a1 x a2 x a3 x
U=
1 2
193
dxdy
International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 11, November 2012)
U
W
1
2 m1
k m m
m
U t
T
r
m 1
pm
k2
k3
13lb 70
lb3 210
1
S
k Dy
12l
k Dy
4lb
4 m
4 m
5b
15
k 2
k Dxy 2 m
k Dxy 2 m
k5
k 1
k6
k3
6l
k D1 2 m
5b
2lb 15
6l
k D1 2 m
2l b
S
k Im Im
m1
Im
r
T
I 1 m1
pIm Im
Now the Total potential energy of the entire plate is now expressed as
t Ut W t
Dx
b3
T
Wt
k2
symmetrical
r
2 I 1
m
k 1 Kii b K ij k b 3 k m Kij K jj k4 b b k5 k1
The strain energy and the potential energy of the entire plate is,
T
r
In order to obtain the best approximation the total potential energy developed in the system become a minimum.
t 0 , wim
Dx
t 0 im
This yields a set of linear equation, after solving it gives an unknown displacements and bending moments at any point inside the structure
11lb2
3l 3l l km4 Dy km2 Dxy km2 D1 2 Dx 420 5 5 b
IV. VALIDATION OF PROGRAM
k4
k5 k6
9lb 140
km4 Dy
12l 5b
km2 Dxy
6l 5b
km2 D1
6l b3
Dx
13lb2
l l 3l km4 Dy km2 Dxy km2 D1 2 Dx 840 5 10 b
lb3 280
km4 Dy
Load vector
lb 15
km2 Dxy
lb 30
km2 D1
l Dx b
pm for point load P 0 at ( x0 , y0 ) is,
N1 ( x0 ) N ( x ) pm 2 0 p0 sin km y0 N3 ( x0 ) N 4 ( x0 )
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International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 11, November 2012) Table No. 4.1 Node no. 01 to 08 and skew angle 15 0 displacement for distance along beam
Comments:
Bending Moment is obviously zero at both the supports and nature of graph observed to be sinusoidal in nature. Variation of Bending Moment (Mx, My and Mxy) observed at 3.0 m, 6.0 m and 9.0 m from the left to right support shown by the graphs is plus or minus 5 % while comparing program to staad-pro results.
Results and discussion Comparison of displacement and bending moment (transverse, longitudinal and twisting) values as obtained by the computer program and Staad-Pro at 3.0 m, 6.0 m and 9.0 m from the left to right support shown by the graphs. It is observed from these graphs that values of downward displacement and bending moment obtained by the computer program and that obtained by Staad-Pro is comparable.
Comments:
Displacement is obviously zero at both the supports and nature of graph observed to be parabolic in nature.
V. PARAMETRIC WORK
The variation of displacement observed at 3.0 m, 6.0 m and 9.0 m from the left to right support shown by the graphs is plus or minus 5 % while comparing program to staad-pro results.
The close agreement of the results of the computer program based on the finite strip method with those given by the Staad-Pro confirms that the computer program developed is accurate and reliable.
195
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On basis of this the parametric work is done by 0 0 0 considering the skew angles15 ,30 &45 The results obtained are tabulated in the form of tables and graphs.
Comments:
Table No. 5.1 Node no. 1 to 8 and skew angle 15 0, 300&450 displacement for distance along beam
At support their downward displacement is obviously zero. However in the one third portion near the either support as we go towards the centre displacement decreases with increase in skew angle i.e. about 5 to 10% In the central one third portion it is seen that displacement is reducing
Comments :
Bending moment obviously zero at both the support.
For given skew angle Mx varies in a sinusoidal form
The maximum bending moment observe at centre.
196
With increase in skew angle Mx decreases up to skew 0 angle 30 then it again increases
International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 11, November 2012)
VI. CONCLUSION
Comments:
Bending moment obviously zero at both the support. For given skew angle Mx varies in a sinusoidal form The maximum bending moment observe at centre. With increase in skew angle bending moment increases and its nature reverses.
The computer time and effort required for the analysis of skew box girder using newly developed program is less as compared to Finite Element Analysis using Staad-Pro. Finite Strip Method require less input data because of the smaller number of mesh lines involved due to the reduction in dimensional analysis but in case of Finite Element Method it is somewhat tedious and difficult. Finite Strip Method involves less number of equations and matrix with narrow bandwidth, consequently much less computing time for solution, where as Finite Element Method involves more no. of equations and matrix so it is time consumable. In Finite Strip Method it is easy to specify only those locations at which displacement and bending moments are required and then accordingly the computations In an analysis of box Girder Bridge by Finite Strip Method considering loadings i.e. point load at centre, the variation of nodal displacement and nodal bending moments as obtained by computer program is almost same. It is observed that the longitudinal bending moment My decreases and the twisting moments M xy increases with an increase in the skew angle. VII. NOTATIONS
P ……….. load w ………. Displacement M ………. Bending Moment ∏ ………. Total potential energy W ………. Potential energy U ………. Strain energy r ……….. No. of series terms [N] ……… Matrix of transverse shape function Mx ……… Transverse bending moment My ……… Longitudinal bending moment
Comments: Bending moment obviously zero at both the support. For given skew angle Mxy varies in a sinusoidal form The maximum bending moment observe at centre. With increase in skew angle bending moment increases and its nature reverse
197
Mxy …….
Twisting moment
{M} …….
Moment vector
Dx ………
Flexural rigidity
Dxy ……..
Torsional rigidity
D1 ………
Coupling rigidity
International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 11, November 2012)
{K} ……..
[11 ] Matrix, Finite Element, Computer and Structural analysis By Madhujit Mukhopadhyay.
Stiffness matrix
{P} ……… Load vector
[12 ] Matrix methods of structural analysis by Madhu B. Kanchi.
{ᵟ} ………. Curvature vector
[13 ] Combined boundary element / Finite strip analysis of bridges. Journal of Structural Engineering ASCE, 120,716-727.
s ………… no. of strips t …………
[14 ] Analysis of Curved box girder bridges. Structural Division, ASCE, 99, 799-819
is subscript representing the whole structure.
[15 ] Application of the Finite strip method in the analysis of concrete box bridges ICJ, Jan 1994, 47-52
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