Finite Element Analysis
March 8, 2017 | Author: Omer Anwaar | Category: N/A
Short Description
Finite Element Analysis...
Description
IW-CAAD 2004 Understanding and Using
Finite Element Analysis July 19-21, 2004 Moratuwa, Sri Lanka Asian Center for Engineering Computations and Software
Asian Institute of Technology, Thailand
Understanding and Using
Finite Element Analysis
Buddhi S. Shrama
The Objective
Finite Element Analysis
• To understand the fundamentals of the Finite Element Method and the Finite Element Analysis • To apply the Finite Element Analysis Tools for Modeling and Analysis of Structures • Use SAP2000 as Tool for Finite Element Modeling and Analysis of Structures
ACECOMS, AIT
Finite Element Analysis
The Program • • • • • • • • •
What is FEM and Why it is needed Fundamental concepts in FEM and FEA Concept of Stiffness Finite Elements and their Usage Constructing Finite Element Models Applying Loads to FE Models Interpreting FE Results Modeling Different Types of Structures using FE Intro to Non-linear and Dynamic Analysis
ACECOMS, AIT
What is Finite Element Analysis and Why do We Need It!
The Structural System
Finite Element Analysis
STRUCTURE EXCITATION Loads Vibrations Settlements Thermal Changes
pv
RESPONSES Displacements Strains Stresses Stress Resultants
ACECOMS, AIT
The Need For Analysis
Finite Element Analysis
We need to determine the Response of the Structure to Excitations
Analysis
so that: We can ensure that the structure can sustain the excitation with an acceptable level of response
Design
ACECOMS, AIT
Analysis of Structures
Finite Element Analysis
xx yy zz pvx 0 x y z Real Structure is governed by “Partial Differential Equations” of various order pv
Direct solution is only possible for: • Simple geometry • Simple Boundary • Simple Loading. ACECOMS, AIT
The Need for Structural Model STRUCTURE
RESPONSES
Finite Element Analysis
EXCITATION Loads Vibrations Settlements Thermal Changes
pv
Displacements Strains Stress Stress Resultants Structural Model
ACECOMS, AIT
The Need for Modeling
Finite Element Analysis
A - Real Structure cannot be Analyzed: It can only be “Load Tested” to determine response
B - We can only analyze a “Model” of the Structure C - We therefore need tools to Model the Structure and to Analyze the Model ACECOMS, AIT
Finite Element Method and FEA • Finite Element Analysis (FEA)
Finite Element Analysis
“A discretized solution to a continuum problem using FEM”
• Finite Element Method (FEM) “A numerical procedure for solving (partial) differential equations associated with field problems, with an accuracy acceptable to engineers”
ACECOMS, AIT
From Classical to FEM
Equilibrium
Finite Element Analysis
Actual Structure
xx yy zz pvx 0 x y z “Partial Differential Equations”
FEM
Assumptions
Classical
Structural Model
Kr R
Stress-Strain Law
Compatibility
t
_
_
“Algebraic Equations” _
dV p u dV p u ds t v
t s
v
(Principle of Virtual Work)
K = Stiffness r = Response R = Loads ACECOMS, AIT
Simplified Structural System Deformations (u)
Loads (F)
Finite Element Analysis
Fv
u
K
(Stiffness)
F Equilibrium Equation
F=Ku ACECOMS, AIT
The Total Structural System STRUCTURE
RESPONSES
Finite Element Analysis
EXCITATION pv
• Static • Dynamic
• Elastic • Inelastic
• Linear • Nonlinear
Eight types of equilibrium equations are possible! ACECOMS, AIT
Finite Element Analysis
The Main Equilibrium Equations 1. Linear-Static Ku F
Elastic
2. Linear-Dynamic
Elastic
Mu(t ) Cu(t ) Ku(t ) F (t )
3. Nonlinear - Static
Elastic OR Inelastic
Ku FNL F
4. Nonlinear-Dynamic
Elastic OR Inelastic
Mu(t ) Cu(t ) Ku(t ) F (t ) NL F (t ) ACECOMS, AIT
Finite Element Analysis
The Basic Analysis Types Excitation
Structure
Response
Basic Analysis Type
Static
Elastic
Linear
Linear-Elastic-Static Analysis
Static
Elastic
Nonlinear
Nonlinear-Elastic-Static Analysis
Static
Inelastic
Linear
Linear-Inelastic-Static Analysis
Static
Inelastic
Nonlinear
Nonlinear-Inelastic-Static Analysis
Dynamic
Elastic
Linear
Linear-Elastic-Dynamic Analysis
Dynamic
Elastic
Nonlinear
Nonlinear-Elastic-Dynamic Analysis
Dynamic
Inelastic
Linear
Linear-Inelastic-Dynamic Analysis
Dynamic
Inelastic
Nonlinear
Nonlinear-Inelastic-Dynamic Analysis
ACECOMS, AIT
Special Analysis Types
Finite Element Analysis
• Non-linear Analysis – – – – –
P-Delta Analysis Buckling Analysis Static Pushover Analysis Fast Non-Linear Analysis (FNA) Large Displacement Analysis
• Dynamic Analysis – Free Vibration and Modal Analysis – Response Spectrum Analysis – Steady State Dynamic Analysis
ACECOMS, AIT
The Finite Element Analysis Process Engineer
Evaluate Real Structure
Finite Element Analysis
Create Structural Model Discretize Model in FE Software
Solve FE Model Interpret FEA Results
Engineer
Physical significance of Results
ACECOMS, AIT
The Fundamentals In Finite Element Method
Finite Element Analysis
From Continuum to Structure From Structure To Structural Model
ACECOMS, AIT
Solid – Structure - Model
Finite Element Analysis
3D SOLIDS
Discretization
Simplification (geometric)
3D-CONTINUM MODEL (Governed by partial differential equations)
CONTINUOUS MODEL OF STRUCTURE
(Governed by either partial or total differential equations)
DISCRETE MODEL OF STRUCTURE (Governed by algebraic equations) ACECOMS, AIT
Equilibrium
Finite Element Analysis
Actual Structure
xx yy zz pvx 0 x y z “Partial Differential Equations”
Structure
Assumptions
Continuum
Structural Model
Kr R
Stress-Strain Law
Compatibility
t
_
_
“Algebraic Equations” _
dV p u dV p u ds t v
t s
v
(Principle of Virtual Work)
K = Stiffness r = Response R = Loads ACECOMS, AIT
Continuum Vs Structure
Finite Element Analysis
• A continuum extends in all direction, has infinite particles, with continuous variation of material properties, deformation characteristics and stress state • A Structure is of finite size and is made up of an assemblage of substructures, components and members
ACECOMS, AIT
Finite Element Analysis
Physical Categorization of Structures • Structures can be categorized in many ways. • For modeling and analysis purposes, the overall physical behavior can be used as basis of categorization – – – – –
Cable or Tension Structures Skeletal or Framed Structures Surface or Spatial Structures Solid Structures Mixed Structures
ACECOMS, AIT
Structure, Member, Element • Structure can be considered as an assemblage of “Physical Components” called Members
Finite Element Analysis
– Slabs, Beams, Columns, Footings, etc.
• Physical Members can be modeled by using one or more “Conceptual Components” called Elements – 1D elements, 2D element, 3D elements – Frame element, plate element, shell element, solid element, etc.
ACECOMS, AIT
Structural Members Continuum
Regular Solid (3D)
Finite Element Analysis
y Plate/Shell (2D) x z t(b,h) h
t
z x
L b
Dimensional Hierarchy of Structural Members
ACECOMS, AIT
Finite Element Analysis
The Reference System • To convert continuum to structures, the first step is to define a finite number of reference dimensions • The Four Dimensional Reference System: – Three Space Dimensions, x, y, z – One Time Dimension, t
• The Entire Structural System is a function of Space and Time – S (x, y, z, t)
ACECOMS, AIT
Global Axis and Local Axis
Finite Element Analysis
• Global Axis used to reference the overall structure and to locate its components: Also called the Structure Axis
Z
Y
• Local Axis used to reference the quantities on part of a structure or a member or an element: Also called the Member Axis or Element Axis
X
ACECOMS, AIT
Finite Element Analysis
The General Global Coordinate System • The global coordinate system is a threedimensional, right-handed, rectangular coordinate system. • The three axes, denoted X, Y, and Z, are mutually perpendicular and satisfy the right-hand rule. • The location and orientation of the global system are arbitrary. The Z direction is normally upward, but this is not required. • All other coordinates systems are converted or mapped back and forth to General Coordinate System
ACECOMS, AIT
Polar Coordinate Systems
Finite Element Analysis
• Polar coordinates include – Cylindrical CR-CA-CZ coordinates – Spherical SB-SA-SR coordinates.
• Polar coordinate systems are always defined with respect to a rectangular XY-Z system.
ACECOMS, AIT
Finite Element Analysis
Local Coordinate Systems • Each part (joint, element, or constraint) of the structural model has its own local co-ordinate system used to define the properties, loads, and response for that part. • In general, the local co-ordinate systems may vary from joint to joint, element to element, and constraint to constraint
ACECOMS, AIT
Finite Element Analysis
Local Axis and Natural Axis • The elements and variation of fields can often be described best in terms “Natural Coordinates” • Natural coordinates may be linear or curvilinear • Shape functions can are used to associate the local system and natural system
ACECOMS, AIT
Finite Element Analysis
Primary Relationships
ACECOMS, AIT
Finite Element Analysis
The Basic Structural Quantities • • • • • •
Loads Actions Deformations Strains Stresses Stress Resultants
The main focus of Structural Mechanics is to develop relationships between these quantities
The main focus of FEM is solve these relationships numerically
ACECOMS, AIT
Mechanics Relationships
Finite Element Analysis
Load
Action
Stress Resultant
Deformation
Stress
Strain
ACECOMS, AIT
Finite Element Analysis
Primary Relationships • • • • • •
Load – Action Relationship Action – Deformation Relationship Deformation – Strain Relationship Strain – Stress Relationship Stress – Stress Resultant Relationship Stress Resultant – Action Relationship
• Most of these relationships can defined mathematically, numerically and by testing
ACECOMS, AIT
Action - Deformation Relationship • This involves two types of relationships
M
Finite Element Analysis
– Deformations produced due to given Actions PL • Example:
M
AE
– Actions needed to produce or restrain certain Deformation • Example: EA P L
• Moment-Curvatures, Load-Deflection Curves are samples of this relationship • The represents to “Element Stiffness”
M
f
P P
d
ACECOMS, AIT
Simplified Examples of Action-Deformation L3 v 6 EI
V
3M 2V L
V M
Finite Element Analysis
M
v
P
PL AE
P
2
L 2 EI
V
2M V L
M
ACECOMS, AIT
Deformation – Strain Relationship
Finite Element Analysis
• In general, strain is the first derivative of deformation
• Basic Deformation and Corresponding Strains are: – – – –
Shortening Curvature Shearing Twisting
Axial Strain Axial Strain Shear Strain Shear Strain + Axial Strain
• Total Strain is summation of strains from different deformations ACECOMS, AIT
Strain – Stress Relationship
Finite Element Analysis
• The resistance of the material to strain, derived from the stiffness of the material particles • For a general Isotropic Material 1 v v v 0 0 0 v 1 v v 0 0 0 x x v v 1 v 0 0 0 y y z 1 2 v 0 0 0 E 0 0 z 2 xy 1 v 1 2v xy 1 2v 0 0 0 0 0 yz yz 2 zx 1 2v zx 0 0 0 0 0 2
kfc
fy
• For 2D, Isotropic Material, V=0
xx E x
xy G xy ACECOMS, AIT
Finite Element Analysis
The Stress Strain Components • The Hook's law is simplified form of Stress-Strain relationship • Ultimately the six stress and strain components can be represented by 3 principal summations
yy
y x
yz
z
xy
zy zz
yx
zx
xz
xx
At any point in a continuum, or solid, the stress state can be completely defined in terms of six stress components and six corresponding strains. ACECOMS, AIT
Secondary Relationships • Global Axis - Local Axis
Finite Element Analysis
– Geometric Transformations Matrices
• Local Axis - Natural Axis – Shape Functions – Jacobian Matrix
ACECOMS, AIT
What are Shape Functions
Finite Element Analysis
• Shape Functions or Interpolation Functions provide a means of computing value of any quantity (field) at some point based on the value specified at specific locations • Shape Functions are used in FEM to relate the values ate Nodes to those within the Element – Nodal Displacements to Element Deformation – Nodal Stresses to Stresses within the Element
• Shape Functions can be in 1D, 2D or in 3D • Shape Functions can be Liner or Polynomials
ACECOMS, AIT
Finite Element Analysis
One Dimensional Shape Functions
N1 (s) 0.5 s (1 s) S =-1 S=0
S=0
S =+1
N1 (s) (1 s)
S=1
N 2 (s) (1 s)(1 s) N3 (s) 0.5 s (1 s)
N1 (s) s S is the “Natural Coordinate System”
w( s) N1 ( s) w1 N 2 ( s) w2 N 3 ( s) w3 3
w( s) N i wi i 1
ACECOMS, AIT
The Jacobian Matrix
Finite Element Analysis
• Jacobian Matrix relates the derivative of Nodal Displacement, directly with Element Strains • The Strain is Derivative of Displacement • Displacements are specified on nodes, in Element Local Axis • For computing K. strains are needed in element in “Natural Coordinates” • Shape Functions relate Nodal Displacements with Element Displacements
N 3 N 2 w N1 J w1 w2 w3 s s s s J N i , s wi ACECOMS, AIT
Finite Element Analysis
The Concept of DOF
ACECOMS, AIT
Finite Element Analysis
The Concept of DOF • In a continuum, each point can move in infinite ways • In Structure, movement of each point is represented or resolved in limited number of ways, called Degrees Of Freedom (DOF) • The DOF of range from 1 to 7 depending on type and level of structural model and the element being considered • Global and Local DOF have different meaning and significance ACECOMS, AIT
The Basic Six DOF • Three Translations along the reference axis
Finite Element Analysis
– Dx, Dy, Dz
• Three Rotations about the reference axis – Rx, Ry, Rz
ACECOMS, AIT
Finite Element Analysis
The Seven Degrees of Freedom • The General Beam Element may have 7 degrees of freedom • The seventh degree is Warping • Warping is out-of plane distortion of the beam crosssection
ry uy y
u x rx x z uz rz wz
Each section on a beam member can have seven Degrees Of Freedom (DOF) with respect to its local axis.
ACECOMS, AIT
Finite Element Analysis
Actions and DOF
ACECOMS, AIT
Finite Element Analysis
The Complete DOF Picture § § § § § § §
uz Axial deformation Axial strain Axial stress ux Shear deformation Shear strain Shear stress uy Shear deformation Shear strain Shear stress rz Torsion Shear strain Shear stress r y Curvature Axial strain Axial stress rx Curvature Axial strain Axial stress wz Warping Axial strain Axial stress
ACECOMS, AIT
Global Structural DOF
Finite Element Analysis
• Only 3 DOF are really needed at Global Level • The deformation of the structure can be defined completely in terms of 3 translations of points with respect to Global Axis
• Rotations may be defined arbitrarily at various locations for convenience of modeling and interpretation
ACECOMS, AIT
Local DOF and Natural DOF
Finite Element Analysis
• DOF can be defined for local movements of joints and elements in 3 Orthogonal reference system
• Natural DOF can be defined in terms of Natural Coordinates System of the element which may be orthogonal or curvilinear • Relationship between Global, Local and Natural DOF is established through Transformation Matrices ACECOMS, AIT
Types of DOF in SAP2000 • Active – the displacement is computed during the analysis
Finite Element Analysis
• Restrained – the displacement is specified, and the corresponding reaction is computed during the analysis
• Constrained – the displacement is determined from the displacements at other degrees of freedom
• Null – the displacement does not affect the structure and is ignored by the analysis
• Unavailable – The displacement has been explicitly excluded from the analysis ACECOMS, AIT
Constraints and Restraints • Restraints:
Finite Element Analysis
– Direct limits on the DOF – External Boundary Conditions – Fixed Support , Support Settlement
• Constraints – Linked or dependent limits on DOF – Internal linkages within the structure, in addition to or in place of normal connections – Rigid Diaphragm, Master-Slave DOF
ACECOMS, AIT
Body Constraints
Finite Element Analysis
• A Body Constraint causes all of its constrained joints to move together as a three-dimensional rigid body. • All constrained joints are connected to each other by rigid links and cannot displace relative to each other. • This Constraint can be used to: – Model rigid connections, such as where several beams and/or columns frame together – Connect together different parts of the structural model that were defined using separate meshes – Connect Frame elements that are acting as eccentric stiffeners to Shell elements ACECOMS, AIT
Finite Element Analysis
Constraints in SAP2000 • A constraint is a set of two or more constrained joints. • The displacements of each pair of joints in the constraint are related by constraint equations. • The types of behavior that can be enforced by constraints are: – Rigid-body behavior – Equal-displacement behavior – Symmetry and anti-symmetry conditions
ACECOMS, AIT
Constraints in SAP2000
Finite Element Analysis
• Rigid-body behavior – Rigid Body: fully rigid for all displacements – Rigid Diaphragm: rigid for membrane behavior in a plane – Rigid Plate: rigid for plate bending in a plane – Rigid Rod: rigid for extension along an axis – Rigid Beam: rigid for beam bending on an axis
ACECOMS, AIT
The Concept of Stiffness
Finite Element Analysis
What is Stiffness ? • In structural terms, stiffness may be defined as “Resistance to Deformation” • So for each type of deformation, there is a corresponding stiffness • Stiffness can be considered or evaluated at various levels • Stiffness is also the “constant” in the ActionDeformation Relationship
For Linear Response
uF Ku F F K u
ACECOMS, AIT
The Structure Stiffness Stress/Strain
Material Stiffness
Cross-section Geometry
Finite Element Analysis
EA, EI
Section Stiffness
Member Geometry
EA/L
Member Stiffness
Structure Geometry Structure Stiffness
ACECOMS, AIT
Finite Element Analysis
Structure Stiffness • The overall resistance of the structures to over all loads, called the Global Structure Stiffness. • Derived from the sum of stiffness of its members, their connectivity and the boundary or the restraining conditions. ACECOMS, AIT
Finite Element Analysis
Member and Element Stiffness • The resistance of each Element to local actions called the Element Stiffness This is derived from the cross-section stiffness and the geometry of the Element. • In FEM, the Member Stiffness can be derived from stiffness of Elements used to model the Member
ACECOMS, AIT
Beam Element Cross-section Stiffness •
Finite Element Analysis
•
The resistance of the cross-section to unit strains. This is derived from the cross-section geometry and the stiffness of the materials from which it is made. For each of degree of freedom, there is a corresponding stiffness, and a corresponding cross-section property
§ § § § § § §
uz Cross-section area, Ax ux Shear Area along x, SAx uy Shear Area along y, SAy rz Torsional Constant, J rx Moment of Inertia, Ixx r y Moment of Inertia, Iyy wz Warping Constant, Wzz or Cw
ACECOMS, AIT
Finite Element Analysis
Computing Element Stiffness • Assume Nodal Displacements (Deformations) • Determine Deformations within the element using “Shape Functions” • Determine the Strains within the element using Strain-Displacement Relationship • Determine Stress within the element using Stress-Strain Relationship • Use the principle of Virtual Work and integrate the product of stress and strain over the volume of the element to obtain the Stiffness
ACECOMS, AIT
Deriving the Basic Stiffness Equation Internal Work
I .W dv
External Work
E.W F
V
I .W dv
Finite Element Analysis
T
Stress-Strain
D Strain-Disp.
B
V
I .W T D dv V
I .W T B T D B dv V
T I .W B D B dv V T
Equilibrium
E.W I .W T F B D B dv V T
T
T F B D B dv V F K
ACECOMS, AIT
Stiffness Equation: An Example E
D B
Finite Element Analysis
F
DE 1 B L
1 L
L K V
K B D B dv T
V
K
1 1 E dv L L
E L2
dv
V
E K 2 AL L EA K L
EA
ACECOMS, AIT
The Matrices in FEM Global Nodal Deformations T-Matrix Global-Local Cords.
Finite Element Analysis
Element Nodal Deformations N-Matrix Shape Functions
Deformation in Element Space B-Matrix Strain-Deforrmation
Strain In Element Space D-Matrix Stress-Strain
Stress in Element Space
ACECOMS, AIT
What is Stiffness Matrix
Finite Element Analysis
• The actions and deformations of different DOF in an element are not independent – One action may produce more than one deformations – One Deformation may be caused by more than one Action
• A Stiffness Matrix relates various Deformation and actions within an Element • A Stiffness Matrix is generalized expression of overall element stiffness
ACECOMS, AIT
Element Stiffness Matrix r5
r2
r3
r6
r1
r4
Finite Element Analysis
Node1
Node2
R1
K11
K12
K13
K14
K15
K16
r1
R2
K21
K22
K23
K24
K25
K26
r2
R
K
K
K
K
K
K
r
K
K
K
K
K
K
r
R5
K51
K52
K53
K54
K55
K56
r5
R6
K61
K62
K63
K64
K65
K66
r6
3
R
4
=
31 41
32 42
33 43
34 44
35 45
36 46
3 4
ACECOMS, AIT
A 2D Frame Element Stiffness U2
U2
U3
U3
E ,A ,I ,L U1
U1
Finite Element Analysis
Node1
Node2
(P1)1
EA/L
0
0
-EA/L
0
0
(U1)1
(P2)1
0
12EI/L3
6EI/L2
0
-12EI/L3
6EI/L2
(U2)1
(P3)1
0
6EI/L2
4EI/L
0
-6EI/L2
2EI/L
(U3)1
(P1)2 =
-EA/L
0
0
EA/L
0
0
(U1)2
(P2)2
0
-12EI/L3
-6EI/L2
0
12EI/L3
-6EI/L2
(U2)2
(P3)2
0
6EI/L2
2EI/L
0
-6EI/L2
4EI/L
(U3)2
( U1)1
(U2)1
(U3)1
(U1)2
(U2)2
(U3)3
ACECOMS, AIT
Direct Stiffness Method and FEM
Finite Element Analysis
• Basically there is no conceptual difference between DSM and FEM. DSM is a special case of the general FEM • Direct Stiffness Method (DSM) – The terms of the element stiffness matrix are defined explicitly and in close form (formulae) – It is mostly applicable to 1D Elements (beam, truss)
• Finite Element Method – The element stiffness matrix terms are computed by numerical integration of the general stiffness equation
ACECOMS, AIT
Isoparametric Elements
Finite Element Analysis
Introduction • In real world, the problem domains are such that they have no proper shape • It is difficult to find the exact solution of the real problems • Isoparametric elements are used to discretize a complex shape problem domain into a number of geometrical shapes • Analysis is carried out on the simple discretized shapes and then the result is integrated over the actual problem domain to get the approximate numerical solution ACECOMS, AIT
Finite Element Analysis
1D Isoparametric Shape • Consider the example of a bar element • For simplification, let the bar lie in x-axis • First, relate the Global coordinate X to natural coordinate system with variable r, Y
x2 x1
U1
U2
Z
X, U r = -1
r
r = +1
1 r 1 ACECOMS, AIT
1D Isoparametric Shape Transformation is given by: 1 1 X (1 r ) X 1 (1 r ) X 2 2 2
Finite Element Analysis
h1
Y
x2 x1
U1
U2
Z
X, U r = -1
r
r = +1
h2
1 1 are interpolation of h1 (1 r ) and h2 (1 r ) shape functions 2 2
The bar global displacements are shown by: 2
U hiU i i 1
ACECOMS, AIT
1D Isoparametric Shape
Finite Element Analysis
Element Strains can be calculated by: dU dr dr dX dU U 2 U1 dr 2 dX X 2 X 1 L and dr 2 2
Where L is the length of the bar
ACECOMS, AIT
1D Isoparametric Shape Therefore, we have
Finite Element Analysis
U 2 U1 L
Buˆ So, Strain displacement transformation matrix can be shown as:
1 B 1 1 L ACECOMS, AIT
1D Isoparametric Shape The Stiffness Matrix is given by:
K BT EB dV
Finite Element Analysis
v
Where E is the Elasticity constant Therefore, we have
AE K 2 L
1 1 1 1 1 Jdr 1
Where, • A = area of the bar • J = Jacobian relating an element length in the global coordinate system to an element length in the natural coordinate system
dX J dr L so J 2 ACECOMS, AIT
1D Isoparametric Shape Therefore, K is evaluated as
Finite Element Analysis
Substituting the value of r from
AE 1 1 K L 1 1 And put in
2
1 1 X (1 r ) X 1 (1 r ) X 2 2 2
To get
rX
U hiU i i 1
( X 1 X 2) / 2 L/2
ACECOMS, AIT
Finite Element Analysis
Example 01 Derive • Interpolation Matrix H • Strain Displacement Interpolation Matrix B • Jacobian Operator J for the three-node element as shown in figure 1
3
2 X, U
r = -1 x1
r=0 L/2
r = +1 L/2
ACECOMS, AIT
Example 01
Finite Element Analysis
Finding the interpolation functions of the given element
r h1 (1 r ) 2
+1
r = -1
r=0
+1 r = -1
r=0
r h2 (1 r ) 2
r = +1
+1
r = -1
h3 1 r
r = +1
r=0
r = +1
2
ACECOMS, AIT
Example 01
Finite Element Analysis
So,
H h1 h2 h3
The strain displacement matrix B is obtained by
dH dr 1 1 B J 1 ( r ) ( r ) 2r 2 2 B J 1
ACECOMS, AIT
Example 01 For Jacobian Operator x h1 x1 h2 x2 h3 x3 r r L (1 r ) x1 (1 r )( x1 L) (1 r 2 )( x1 ) 2 2 2 L L x x1 r 2 2 dx J dr L J 2 L 2 J 1 ; det J 2 L
Finite Element Analysis
x
ACECOMS, AIT
2D Isoparametric Element
Finite Element Analysis
• Linear and quadratic two-dimensional isoparametric finite elements use the same shape function for specification of the element shape and interpolation of the displacement field 3
5
2
1
4
2
1
6
5
7
4
3
4
1 3 2
ACECOMS, AIT
Finite Element Analysis
2D Isoparametric Element • Shape functions Ni are defined in local coordinates
, (1 , 1)
• The same shape functions are used for interpolations of displacements of coordinates
u N i ui ; v N i vi
x N i xi ; y N i yi
ACECOMS, AIT
2D Isoparametric Element
Finite Element Analysis
• Shape functions for linear quadratic twodimensional isoparametric elements are shown here • Linear Elements 4-node:
1 N i (1 o )(1 o ) 4
ACECOMS, AIT
2D Isoparametric Element
Finite Element Analysis
Quadratic Elements 8-nodes 1 1 N i (1 o )(1 o ) (1 2 )(1 o ) 4 4 1 (1 o )(1 2 ) i 1, 3, 5, 7 4 1 N i (1 2 )(1 o ) i 2, 6 2 1 N i (1 o )(1 2 ) i 4, 8 2
where
o i ;o i ACECOMS, AIT
Finite Element Analysis
Example 02 • Derive the expressions needed for the calculation of Stiffness Matrix of the isoparametric 4-node finite element shown in the figure. Assume plane stress or plane strain conditions
y, v
or s
1
2
or r y4
3 4
x4
x, u
ACECOMS, AIT
Example 02
Finite Element Analysis
• The four interpolation functions for the linear quadratic isoparametric element are 1 h1 (1 r )(1 s ) 4 1 h 2 (1 r )(1 s ) 4 1 h 3 (1 r )(1 s ) 4 1 h 4 (1 r )(1 s ) 4
y, v
or s
1
2
or r y4
3 4
x4
x, u
ACECOMS, AIT
Example 02 The coordinate interpolations for the element is given by 4
4
i 1
i 1
Finite Element Analysis
x hi xi ; y hi yi Using the interpolation functions, the coordinate interpolations for this element are 1 1 1 1 x (1 r )(1 s) x1 (1 r )(1 s) x2 (1 r )(1 s) x3 (1 r )(1 s) x4 4 4 4 4 1 1 1 1 y (1 r )(1 s) y1 (1 r )(1 s) y2 (1 r )(1 s) y3 (1 r )(1 s) y4 4 4 4 4
ACECOMS, AIT
Example 02 The displacement interpolations for the element is given by 4
4
i 1
11
Finite Element Analysis
u hi ui ; v hi vi Using the interpolation functions, the coordinate interpolations for this element are 1 1 1 1 u (1 r )(1 s)u1 (1 r )(1 s)u2 (1 r )(1 s)u3 (1 r )(1 s)u4 4 4 4 4 1 1 1 1 v (1 r )(1 s)v1 (1 r )(1 s)v2 (1 r )(1 s)v3 (1 r )(1 s)v4 4 4 4 4
ACECOMS, AIT
Example 02 The element strains are given by
T xx yy xy
Finite Element Analysis
xx
u v u v ; yy ; xy x y y x
To evaluate the displacement derivatives, we need to evaluate
x r r x s s
y r x y s y
or
J r x
ACECOMS, AIT
Example 02
Finite Element Analysis
where x 1 1 1 1 (1 s ) x1 (1 s ) x2 (1 s ) x3 (1 s ) x4 r 4 4 4 4 x 1 1 1 1 (1 r ) x1 (1 r ) x2 (1 r ) x3 (1 s ) x4 s 4 4 4 4 y 1 1 1 1 (1 s ) y1 (1 s ) y2 (1 s ) y3 (1 s ) y4 r 4 4 4 4 y 1 1 1 1 (1 r ) y1 (1 r ) y2 (1 r ) y3 (1 r ) y4 s 4 4 4 4
ACECOMS, AIT
Example 02 For any value of r and s
Finite Element Analysis
1 r 1 and 1 s 1 We can form the Jacobian matrix. Assuming we evaluate J at r r and s s i
j
x 1 r J y s at r ri and s s j ACECOMS, AIT
Example 02
Finite Element Analysis
To evaluate the element strains, we use u 1 1 1 1 (1 s )u1 (1 s)u2 (1 s )u3 (1 s )u4 r 4 4 4 4 u 1 1 1 1 (1 r )u1 (1 r )u2 (1 r )u3 (1 s )u4 s 4 4 4 4 v 1 1 1 1 (1 s )v1 (1 s)v2 (1 s)v3 (1 s )v4 r 4 4 4 4 v 1 1 1 1 (1 r )v1 (1 r )v2 (1 r )v3 (1 r )v4 s 4 4 4 4
ACECOMS, AIT
Example 02
Finite Element Analysis
Simplifying the above relations, we get u x 1 s 0 (1 s ) 0 (1 s ) 0 1 s 0 1 J 1 uˆ u 1 r 0 1 r 0 ( 1 r ) 0 ( 1 s ) 0 4 y and v x 0 1 s 0 (1 s ) 0 (1 s ) 0 1 s 1 J 1 uˆ v 0 1 r 0 1 r 0 ( 1 r ) 0 ( 1 s ) 4 y
Where
uˆ T u1 v1 u2 v2 u3 v3 u4 v4 where r ri and s s j ACECOMS, AIT
Example 02 Strain-displacement transformation is given by
Finite Element Analysis
ij Bij uˆ So, we can get
0 (1 s) 0 1 s 0 1 s 0 (1 s) 1 Bij 0 1 r 0 1 r 0 (1 r ) 0 (1 r ) 4 1 r 1 s 1 r (1 s) (1 r ) (1 s ) (1 r ) 1 s
ACECOMS, AIT
Example 02 Stiffness Matrix K is given by K tij ij Fij
Finite Element Analysis
i, j
where Fij BijT CBij det J ij
In the above expressions, C is the material property matrix, t is the thickness of the element at the sampling point (r,s)
ACECOMS, AIT
Finite Element Analysis
Example 03 • Calculate the deflection uA of the structural model shown
Z U4
A
U3
U1
Bar with xsectional area = 1cm2
6 cm U6 U5
U2
0.1cm U8
Y
U7= uA
0.5 cm2 each
E= 30 x 106 N/cm2
6 cm
0.3 0.1cm Section AA A 8 cm
ACECOMS, AIT
Example 03
Finite Element Analysis
By symmetry and boundary conditions, we only need to evaluate the stiffness coefficient corresponding to uA We know that
Z U4
A
U3
U6
x r J x s
y r y s
U1
Bar with xsectional area = 1cm2
6 cm
U5
U2
U8 Y
U7= uA E= 30 x 106 N/cm2
6 cm
0.3
A 8 cm
ACECOMS, AIT
Example 03 So, we have
Finite Element Analysis
4 0 J 0 3 Now, calculating B
3(1 s ) 1 B ... 0 48 4(1 r )
...
ACECOMS, AIT
Example 03 Stiffness K for an Area is,
K BT EB t det J dr ds 3(1 s ) E 1 3 (1 s ) (0.1)(12)dr ds 3 ( 1 s ) 0 4 ( 1 r ) 1 48 1 2 2(1 )(1 r )
1 1
Finite Element Analysis
K
1
2
K 1336996.34 N / cm The stiffness of the truss is AE/L, or
(1)(30 X 106 ) k 3750000 N / cm 8
ACECOMS, AIT
Example 03 Hence, Ktotal = 6.424 x 106 N/cm
Finite Element Analysis
Now, since P = Ku Therefore, u = P/K
6000 4 u 9 . 34 X 10 cm 6 6.424 X 10
u 9.34 X 104 cm ACECOMS, AIT
Shell Element
Finite Element Analysis
• A Shell element is used to model shell, membrane, and plate behavior in planar and three-dimensional structures • The membrane behavior uses an isoparametric formulation that includes translational in-plane stiffness components and a rotational stiffness component in the direction normal to the plane of the element.
ACECOMS, AIT
Shell Element Axis 3
Finite Element Analysis
Axis 2
Face 3
J3
Axis 1
Face 2
J2 J4
Face 4
Face5 Bottom Face6 Top
J1
Face 1
ACECOMS, AIT
Finite Element Analysis
Shell Elements • A simple quadrilateral Shell Element • Two dimensional plate bending and membrane elements are combined to form a four-node shell element y
y z
y
uy
x
x
z
ux
+
z
x
uy uz
ux
=
z
Plate Bending Element
Membrane Element
Shell Element
ACECOMS, AIT
Shell Elements
Finite Element Analysis
• A simple quadrilateral Shell Element • A thin-plate (Kirchhoff) formulation is normally used that neglects transverse shearing deformation
• A thick plate (Mindlin/Reissner) formulation can also be chosen which includes the effects of transverse shearing deformation
ACECOMS, AIT
What are The Finite Elements (in SAP2000)
Finite Element Analysis
Nodes and Finite Elements • The Finite Elements are discretized representation of the continuous structure • Generally they correspond to the physical structural components but sometimes dummy or idealized elements my also be used • Elements behavior is completely defined within its boundaries and is not directly related to other elements • Nodes are imaginary points used describe arbitrary quantities and serve to provide connectivity across element boundaries
ACECOMS, AIT
Basic Categories of Finite Elements • 1 D Elements (Beam type) – Only one dimension is actually modeled as a line, other two dimensions are represented by stiffness properties
Finite Element Analysis
– Can be used in 1D, 2D and 2D
• 2 D Elements (Plate type) – Only two dimensions are actually modeled as a surface, third dimension is represented by stiffness properties – Can be used in 2D and 3D Model
• 3 D Elements (Brick type) – All three dimensions are modeled as a solid – Can be used in 3D Model ACECOMS, AIT
Finite Element Analysis
The Joint or Node
ACECOMS, AIT
Finite Element Analysis
Basic Properties of Joints • All elements are connected to the structure at the joints • The structure is supported at the joints using Restraints and/or Springs • Rigid-body behavior and symmetry conditions can be specified using Constraints that apply to the joints • Concentrated loads may be applied at the joints • Lumped masses and rotational inertia may be placed at the joints • Loads and masses applied to the elements are transferred to the joints • Joints are the primary locations in the structure at which the displacements are known (the supports) or are to be determined ACECOMS, AIT
Joint Local Coordinates
Finite Element Analysis
• By default, the joint local 1-2-3 coordinate system is identical to the global X-Y-Z coordinate system • It may be necessary to use different local coordinate systems at some or all joints in the following cases: – Skewed Restraints (supports) are present – Constraints are used to impose rotational symmetry – Constraints are used to impose symmetry about a plane that is not parallel to a global coordinate plane – The principal axes for the joint mass (translational or rotational) are not aligned with the global axes – Joint displacement and force output is desired in another coordinate system
ACECOMS, AIT
Finite Element Analysis
Joint Local Coordinates
ACECOMS, AIT
Finite Element Analysis
Spring Restraints on Joints • Any of the six degrees of freedom at any of the joints in the structure can have translation or rotational spring support conditions. • Springs elastically connect the joint to the ground. • The spring forces that act on a joint are related to the displacements of that joint by a 6x6 symmetric matrix of spring stiffness coefficients. – Simple Springs – Coupled Springs
ACECOMS, AIT
Simple Spring Restraints
Finite Element Analysis
• Independent spring stiffness in each DOF
ACECOMS, AIT
Finite Element Analysis
Coupled Spring Restraints • General Spring Connection • Global and skewed springs • Coupled 6x6 user-defined spring stiffness option (for foundation modeling)
ACECOMS, AIT
Finite Element Analysis
Stiffness Matrix for Spring Element
where u1 ,u2 ,u3 ,r1 ,r2 and r3 are the joint displacements and rotations, and the terms u1, u1u2, u2, ... are the specified spring stiffness coefficients.
ACECOMS, AIT
Some Sample Finite Elements
Finite Element Analysis
Truss and Beam Elements (1D,2D,3D)
Plane Stress, Plane Strain, Axisymmetric, Plate and Shell Elements (2D,3D)
Brick Elements
ACECOMS, AIT
Finite Element Analysis
One Dimensional Elements
ACECOMS, AIT
DOF for 1D Elements Dy
Dx
2D Truss
Finite Element Analysis
Dy
Dy Rz
Dz
Dx
3D Truss
2D Beam
Ry Dy Rz
Dy Dx
Rz
Dy Dz
Rx
Dx
Rx
Rz
2D Frame
2D Grid
3D Frame
ACECOMS, AIT
Variation of 1D Elements
Finite Element Analysis
• Based on DOF – – – – –
2D Truss 3D Truss 2D Beam 3D Beam 2D Grid
• Based on Behavior – Thick Beam/ Thin Beam – Liner/ Isoperimetric
• Non-Linear Elements – – – – – – –
NL Link Gap Element Tension Only Compression Only Friction Cable Damper
ACECOMS, AIT
Finite Element Analysis
Usage of 1D Elements
3D Frame
2D Grid
2D Frame ACECOMS, AIT
Finite Element Analysis
Nonlinear Link Element in SAP2000
ACECOMS, AIT
Finite Element Analysis
Two Dimensional Elements
ACECOMS, AIT
DOF for 2D Elements Ry ?
Ry ? Dy Rz
Rx
Dx
Finite Element Analysis
Dy
Dy
Membrane
Plate
Dz
Dx
Rx
Rz
Shell
ACECOMS, AIT
Membrane Element General
Finite Element Analysis
• • • •
Total DOF per Node = 3 (or 2) Total Displacements per Node = 2 Total Rotations per Node = 1 (or 0) Membranes are modeled for flat surfaces
R3
U2
U2 Node 4
Node 3
U1 3
U1 2
1
Application • For Modeling surface elements carrying in-plane loads
R3
U2
Node 1
R3
U2
Node 2
U1
U1
Membrane
ACECOMS, AIT
Variation of Membrane Elements Plain-Strain Assumptions
x 1 unit
Finite Element Analysis
x2 x1
x3 3D Problem
x
2D Problem
Plane Strain Problem
Plane Stress Problem
ACECOMS, AIT
Finite Element Analysis
Plate Element General • Total DOF per Node = 3 • Total Displacements per Node = 1 • Total Rotations per Node = 2 • Plates are for flat surfaces
U3
U3
R2
Node 3
R2
Node 4
R1 3
R1 2
1
Application • For Modeling surface elements carrying out of plane loads
U3
R2
Node 1
U3
R2
Node 2
R1
R1
Plate
ACECOMS, AIT
Shell Element General
Finite Element Analysis
• • • •
Total DOF per Node = 6 (or 5) Total Displacements per Node = 3 Total Rotations per Node = 3 Used for curved surfaces
U3, R3
U3, R3 U2, R2
U2, R2
Node 3
Node 4
U1, R1 3
Application • For Modeling surface elements carrying general loads
U1, R1
2
U3, R3 1
U3, R3
U2, R2
Node 1
U2, R2 Node 2
U1, R1
U1, R1
Shell
ACECOMS, AIT
Finite Element Analysis
Variations of Plate Elements – Based on Behavior – 2D Plane Stress – 2D Plane Strain – Axisymetric Solid – Plate – Shell
– Based on Number of Nodes – 3 Node, 6 Node – 4 Node, 8 Node, (9 Node)
– Based on Material Model – Rubber – Soil – Laminates – Isotropic/ Orthotropic
ACECOMS, AIT
Finite Element Analysis
Shell Elements in SAP2000
ACECOMS, AIT
Finite Element Analysis
Shell Elements in SAP2000
ACECOMS, AIT
Finite Element Analysis
Local Cords for Shell Element • Each Shell element has its own local coordinate system used to define Material properties, loads and output. • The axes of this local system are denoted 1, 2 and 3. The first two axes lie in the plane of the element the third axis is normal
ACECOMS, AIT
Finite Element Analysis
Three Dimensional Elements
ACECOMS, AIT
DOF for 3D Elements Dy
Finite Element Analysis
Dz
Dx
Solid/ Brick
ACECOMS, AIT
Finite Element Analysis
Brick Element in SAP2000 • 8-Node Brick • Bricks can be added by using Text Generation in V7. New version 8 will have graphical interface for Bricks
ACECOMS, AIT
Finite Element Analysis
Connecting Dissimilar Elements
ACECOMS, AIT
Connecting Different Types of Elements Truss Truss
Finite Element Analysis
Frame
Shell
Membrane
Plate
Shell
Solid
OK
OK
Dz
OK
OK
OK
Rx, Ry, Rz
OK
Rx, Ry, Rz, Dz
Rx ? Dx, Dy
Rx ?
Rx, Ry, Rz
OK
OK
OK
Dx, Dy
OK
OK
Rx, Rz
OK
Rx, Rz
OK
OK
Rx, Rz
Rx, Ry, Rz
OK
Rx, Ry, Rz, Dz
Dx, Dz
OK
Rx, Rz
OK
OK
Dz
Dx, Dz
OK
OK
Membrane
Plate
Frame
Solid
Orphan Degrees Of Freedom: 0
1
2
3
4
ACECOMS, AIT
Connecting Dissimilar Elements
Finite Element Analysis
• When elements with different degree of freedom at ends connect with each other, special measures may need to be taken to provide proper connectivity depending on Software Capability
Beams to Plates
Beam to Brick
Plates to Brick ACECOMS, AIT
Connecting Dissimilar Elements
Finite Element Analysis
• When members with mesh of different size or configuration need to be connected we may have to: – – – –
Use special connecting elements Use special Constraints Use mesh grading and subdivision Use in-compatible elements (Zipper Elements in ETABS) – Automatic “Node” detection and internal meshing by the Software
ACECOMS, AIT
Finite Element Analysis
Connecting Beams with Membrane
Modeling Shear-Walls using Panels only
Modeling Shear-Walls using Panels, Beams, Columns
(No Moment continuity with Beams and Columns unless 6 DOF Shell is used)
(Full Moment continuity with Beams and Columns is restored by using additional beams)
ACECOMS, AIT
Meshing Slabs and Walls
Finite Element Analysis
“Zipper”
In general the mesh in the slab should match with mesh in the wall to establish connection
Some software automatically establishes connectivity by using constraints or “Zipper” elements
ACECOMS, AIT
How to Apply Loads to Finite Element Model
Loads To Design Actions • Loads
Finite Element Analysis
• Load Cases
• Load Combinations • Design Envelopes • Design Actions
ACECOMS, AIT
Load Cases • Load cases are defined by the user and used for analysis purpose only
Finite Element Analysis
• Static Load Cases – Dead Load – Live Load – Wind Load
• Earthquake Load Cases – Response Spectrum Load Cases – Time History Load Cases
• Static Non-Linear Load Cases
ACECOMS, AIT
Load Combinations • The Load Combinations may be created by the program, user defined or a combination of both.
Finite Element Analysis
• Some Examples: [Created by the program] – – – – – – – – – –
1.4ΣDL 1.4ΣDL + 1.7(ΣLL + ΣRLL) 0.75[1.4ΣDL + 1.7(ΣLL + ΣRLL) + 1.7WL] 0.75[1.4ΣDL + 1.7(ΣLL + ΣRLL) - 1.7WL] 0.9ΣDL + 1.3WL 0.9ΣDL - 1.3WL 1.1 [1.2ΣDL + 0.5(ΣLL + ΣRLL) + 1.0E] 1.1 [1.2ΣDL + 0.5(ΣLL + ΣRLL) - 1.0E] 1.1 (0.9ΣDL + 1.0E) 1.1 (0.9ΣDL - 1.0E)
ACECOMS, AIT
Applying Gravity Loads
Finite Element Analysis
• All gravity loads are basically “Volume Loads” generated due to mass contained in a volume • Mechanism and path must be found to transfer these loads to the “Supports” through a Medium • All type of Gravity Loads can be represented as: – Point Loads – Line Loads – Area Loads – Volume Loads
ACECOMS, AIT
Load Transfer Path
Finite Element Analysis
• The Load is transferred through a medium which may be: – – – – –
A Point A Line An Area A Volume A system consisting of combination of several mediums
• The supports may be represented as: – – – –
Point Supports Line Supports Area Supports Volume Supports
ACECOMS, AIT
Finite Element Analysis
Graphic Object Representation Object
Load
Geometry Medium
Support Boundary
Point
Point Load Concentrated Load
Node
Point Support Column Support
Line
Beam Load Wall Load Slab Load
Beam / Truss Connection Element Spring Element
Line Support Wall Support Beam Support
Area
Slab Load Wind Load
Plate Element Shell Element Panel/ Plane
Soil Support
Volume
Seismic Load Liquid Load
Solid Element
Soil Support
ETABS and SAP200 uses graphic object modeling concept ACECOMS, AIT
Load Transfer Path is difficult to Determine Load
• Complexity of Load Transfer Mechanism depend on:
Vol.
Finite Element Analysis
Area
– Complexity of Load – Complexity of Medium – Complexity of Boundary
Line
Point Line Line
Area
Volume
Medium
Area Volume
Boundary
ACECOMS, AIT
Finite Element Analysis
Load Transfer Path is difficult to Determine
Line
Area
Volume
Transfer of a Point Load to Point Supports Through Various Mediums ACECOMS, AIT
Finite Element Analysis
Simplified Load Transfer
To Lines
To Points
To Lines and Points
Transfer of Area Load ACECOMS, AIT
Applying Wind Loads
Finite Element Analysis
• At least 3 basic Wind Load Cases should be considered – Along X-Direction – Along Y Direction – Along Diagonal
• Each Basic Wind Load Case should be entered separately into load combinations twice, once with (+ve) and once with (-ve) sign • Total of 6 Wind Load Cases should considered in Combinations, but only 3 Load Cases need to be defined and analyzed ACECOMS, AIT
Applying Wind Loads
Finite Element Analysis
At least 3 Basic Load Case for Wind Load should be considered
Wx
Diagonal wind load may be critical for special types and layouts of buildings
Wy
Wxy ACECOMS, AIT
Wind Load Combinations
Finite Element Analysis
Comb1
Comb2
Comb3
Comb4
Comb5
Comb6
Wx
+f
-f
0
0
0
0
Wy
0
0
+f
-f
0
0
Wxy
0
0
0
0
+f
-f
(f) Is the load factor specified for Wind in Example: Comb = 0.75(1.4D + 1.7W) will need Six the design codes Actual Combinations
Six Additional Load Combinations are required where ever “Wind” is mentioned in the basic Load Combinations
Comb1= 0.75(1.4D + 1.7Wx) Comb2 = 0.75(1.4D - 1.7Wx) Comb3 = 0.75(1.4D + 1.7Wy) Comb4 = 0.75(1.4D - 1.7Wy) Comb5 = 0.75(1.4D + 1.7Wxy) Comb6 = 0.75(1.4D - 1.7Wxy) ACECOMS, AIT
Finite Element Analysis
Nature of Dynamic Loads • • • •
Free Vibration Forced Vibration Random Vibration Seismic Excitation • Response Spectrum • Time History • Steady-State Harmonic Load • Impact • Blast
ACECOMS, AIT
Getting and Interpreting Finite Element Results
What Results Can We Get ? Finite Element Analysis
(in SAP2000)
ACECOMS, AIT
Finite Element Analysis
At Joints • • • • •
Joint Displacements Spring Reactions Restrained Reactions Constrained Forces Results Available For: – For all Available DOF – Given on the “Local Joint Coordinates” – Given for all Load Case, Mode Shapes,Response Spectrums, Time Histories, Moving Loads, and Load Combinations
ACECOMS, AIT
For Frame Elements • The Actions Corresponding to Six DOF at Both Ends, in Local Coordinate System 2
2
1
Finite Element Analysis
1
+V2
+M2 +P 2
2
3
3
+V3
3
+V3 +P
+V2
+T
+M3
3
+M3
+T
+M2 ACECOMS, AIT
Finite Element Analysis
For Shell Element • The Shell element internal forces (also called stress resultants) are the forces and moments that result from integrating the stresses over the element thickness. • The results include the “Membrane Results” (in-plane forces) and “Plate Bending Results” • The results are given for Element Local Axis • It is very important to note that these stress resultants are forces and moments per unit of in-plane length
ACECOMS, AIT
Finite Element Analysis
Shell Stress Resultants
ACECOMS, AIT
Finite Element Analysis
Membrane Results
ACECOMS, AIT
Finite Element Analysis
Plate Bending Results
ACECOMS, AIT
Finite Element Analysis
Obtaining Design Actions From Basic Results
ACECOMS, AIT
Obtaining Envelop Results Comb1
Comb2
Comb3 Comb N
Finite Element Analysis
Load Case -1
Load Case - 2 Load Case - 3
Envelop Results
Load Case - M
Total
P1
P2
P3
PN
Max, P Min, P
ACECOMS, AIT
Can Envelop Results be Used for Design ?
Finite Element Analysis
• Actions Interact with each other, effecting the stresses • For Column Design: • For Beam Design: • For Slabs:
P
P, Mx, My Mx, Vy, Tz
Mx, My, Mxy – At least 3 Actions from each combination must be considered together as set
Mx My
• Therefore, Envelop Results Can Not be Used • Every Load Combinations must be used for design with complete “Action Set”
ACECOMS, AIT
• For static loads, Design Actions are obtained as the cumulative result from each load combination, as set for all interacting actions • The final or critical results from design of all load combinations are adopted
Combinations Load Cases
Finite Element Analysis
Design Actions For Static Loads
Design Actions Obtained as set from all Combinations
ACECOMS, AIT
Static, Dynamic and Nonlinear Results For a Single Action:
Finite Element Analysis
Static Load Case Response Spectrum Load Case
1
+ 1 for each Time Step
Time History Load Case
Static Non-linear Load Case
OR 1 for envelop
Load Combination Table
1 for each Load Step OR 1 for Envelop
ACECOMS, AIT
Response Spectrum Case – All response spectrum cases are assumed to be earthquake load cases
Finite Element Analysis
– The output from a response spectrum is all positive. – Design load combination that includes a response spectrum load case is checked for all possible combinations of signs (+, -) on the response spectrum values
– A 3D element will have eight possible combinations of P, M2 and M3 and eight combinations for M3, V, T
ACECOMS, AIT
Response Spectrum Results for Action Set
Maximum Results obtained by: SRSS, CQC, etc.
P, Mx, My>
+P, +Mx, +My +P, +Mx, -My +P, -Mx, +My +P, -Mx, -My
-P, +Mx, +My -P, +Mx, -My -P, -Mx, +My -P, -Mx, -My
Load Combination Table
Finite Element Analysis
Design Actions needed for Columns:
ACECOMS, AIT
Time History Analysis Results Option – 2: Design For All Values (At each time step)
Finite Element Analysis
Max Val
T (sec)
Option – 1: Envelope Design
Min Val
Response Curve for One Action
ACECOMS, AIT
Time-History Results – The default design load combinations do not include any time history results
Finite Element Analysis
– Define the load combination, to include time history forces in a design load combination – Can perform design for each step of Time History or design for envelops for those results – For envelope design, the design is for the maximum of each response quantity (axial load, moment, etc.) as if they occurred simultaneously. – Designing for each step of a time history gives correct correspondence between different response quantities ACECOMS, AIT
Time History Results
Finite Element Analysis
– The program gets a maximum and a minimum value for each response quantity from the envelope results for a time history – For a design load combination any load combination that includes a time history load case in it is checked for all possible combinations of maximum and minimum time history design values. – If a single design load combination has more than one time history case in it, that design load combination is designed for the envelopes of the time histories, regardless of what is specified for the Time History Design item in the preferences.
ACECOMS, AIT
Static Non Linear Results
Finite Element Analysis
– The default design load combinations do not include any Static Nonlinear results
– Define the load combination, to include Static Nonlinear Results in a design load combination – For a single static nonlinear load case the design is performed for each step of the static nonlinear analysis.
ACECOMS, AIT
Finite Element Analysis
Obtaining Reinforcement From Actions
ACECOMS, AIT
Finite Element Analysis
Computing Rebars For Beam Elements • For Beam type elements (1D elements) design actions like Axial force, moments, and shear force are output directly. • These actions can be used directly for design purposes • Generally, design is carried out in two parts
• Axial- Flexural: • Shear Torsion:
y
My Vy Nx x
z
Vz
Mz
Tx
3D Beam Column y
My
P, Mx, My T, Vx, Vy
z
Mz
• Beam Design: • Column Design:
Mx, Vy, T Mx, My, P
x
Nx
Biaxial & Load
ACECOMS, AIT
Computing Rebars For Beam Elements Ast : To resist tension due to My
Asc + Al/4: To resist compression due to moment Mx (doubly reinforced beams) and tension due to Torsion
Asc : To resist compression
Finite Element Analysis
due to My (may not be needed)
Asvt + Asv/2: To resist shear due to Torsion. Must be closed hoops on sides of the section
Asw + Al/4 : To resist secondary tension in deep beams due to moment and due to Torsion
Ast + Al/4 : To resist main tension due to moment and tension due to Torsion ACECOMS, AIT
Computing Rebars For Plate Elements
Finite Element Analysis
• Moment output for plate type elements in Finite Element Analysis is reported in moment per unit width along the local axis of the plate element. These need to be converted to moments along x and y for design purposes.
• The following procedure can be used: The portion of a plate element bounded by a crack is shown in the Adjoining figure. The moment about an axis dx =k dy parallel to the crack may be given as: mc ds mx dy mxy kdycos my kdy mxy dy sin 2
dy mc mx k 2 m y 2kmxy dx
dy
ds
Crack
ACECOMS, AIT
Computing Rebars For Plate Elements mxy kdy
•
Finite Element Analysis
•
The plate needs to be reinforced with bars in the x and y direction The corresponding moment capacity at the assumed crack is 2
dy mrc mrx k 2 mry dx
my kdy mxy dy mx dy
ms ds
mry kdy
mrx dy
•
Where mrc must equal or exceed mc solving for the minimum we get 1 mry m y mxy k
mrc ds
mrx , mry Positive moment capacities per unit width ACECOMS, AIT
Computing Rebars For Plate Elements
Finite Element Analysis
•
•
The reinforcement at the bottom of the slab in each direction is designed to provide resistance for the positive moment
The reinforcement at the top of the slab in each direction is designed to provide resistance for the negative moment
mry m y mxy mrx mx mxy
mry and mrx are set to zero if they yield a negative value
mry m y mxy mrx mx mxy
mry and mrx are set to zero if they yield a positive value
ACECOMS, AIT
Finite Element Analysis
Computing Rebars For Brick Elements • For Brick elements the FEA results in the nodal stresses and strains. • The stresses on the brick elements need to be integrated along x and y direction to obtain forces. • Stress variation in both the directions may be considered and integrated. • These forces are then used to find the moment about the two orthogonal axes and the net axial force. Similar approach is used to obtain shear forces in two directions • After the axial forces, moments and shear forces are obtained then the section can be designed as a rectangular beam
ACECOMS, AIT
Computing Rebars For Brick Elements Sample Calculations for P and M
Finite Element Analysis
Following equations are based on the assumption that there is no stress variation in the transverse direction Pi C1 C2 T ........ n
P Pi i 1
C1
M i C1 x1 C2 x2 Tx3 .......
x1 x3
n
M Mi i 1
CL
T
x2
C2
ACECOMS, AIT
Modeling Structures Using FEM
Global Modeling or ”Macro Model”
Finite Element Analysis
• A model of the Whole Structure • Objective is to get Overall Structural Response • Results in the form of member forces and stress patterns
• Global Modeling is same for nearly all Materials • Material distinction is made by using specific material properties • Global Model may be a simple 2D beam/ frame model or a sophisticated full 3D finite element model • Generally adequate for design of usual structures
ACECOMS, AIT
Local Model or “Micro Model”
Finite Element Analysis
• Model of Single Member or part of a Member • Model of the Cross-section, Opening, Joints, connection
• Objective: To determine local stress concentration, cross-section behavior, modeling of cracking, bond, anchorage etc. • Needs finite element modeling, often using very fine mesh, advance element features, non-linear analysis • Mostly suitable for research, simulation, experiment verification and theoretical studies ACECOMS, AIT
Global Modeling of Structural Geometry
Finite Element Analysis
(a) Real Structure
(b) Solid Model
(c) 3D Plate-Frame
(e) 2D Fram e
(d) 3D Fram e
(f) Grid-Plate
Fig. 1 Various Ways to Model a Real Struture
ACECOMS, AIT
The Basic Issues • Which Model to be used ?
Finite Element Analysis
– 3D or 2D – Frame or Grid – Plate, Membrane, Shell, Solid
• Which Elements to be used ? – Beam, Plate, Brick – Size and number of elements
• Which Solution to be used ? – – – –
Linear or Nonlinear Static or Dynamic Linear static or Nonlinear dynamic Linear dynamic or Nonlinear static ACECOMS, AIT
Finite Element Analysis
Overall Procedure – Linear Static • • • • • • • • •
Setup the Units to be used Define Basic Material Properties Define Cross-sections to be used Draw, generate Nodes and Elements Assign XSections, Restraints, Constraints etc. Apply Loads to Nodes and Elements Run the Analysis Check Basic Equilibrium and Deformations Interpret and use the Results ACECOMS, AIT
What Type of Analysis Should be Carried out!
Finite Element Analysis
The type of Analysis to be carried out depends on the Structural System – The Type of Excitation (Loads) – The Type Structure (Material and Geometry) – The Type Response
ACECOMS, AIT
Finite Element Analysis
• • • • • • • •
P-Delta Analysis Buckling Analysis Static Pushover Analysis Response Spectrum Analysis Fast Non-Linear Analysis (FNA) Steady State Dynamic Analysis Free Vibration and Modal Analysis Large Displacement Analysis
ACECOMS, AIT
• Static Excitation – When the Excitation (Load) does not vary rapidly with Time – When the Load can be assumed to be applied “Slowly”
Finite Element Analysis
• Dynamic Excitation – When the Excitation varies rapidly with Time – When the “Inertial Force” becomes significant
• Most Real Excitation are Dynamic but are considered “Quasi Static” • Most Dynamic Excitation can be converted to “Equivalent Static Loads” ACECOMS, AIT
Static
Dynamic
Self Load
Normal Operation
At lifting/ placement
Superimposed Dead Load
Normal Operation
At placement
Live Load
Normal Operation
Depends on type
Highway Traffic
Quasi Static
Impact
Water/ Liquid
Normal Operation
Filling, Sloshing
Creep, Shrinkage
Static
No Dynamic Component
Wind
Equivalent Static
Random Vibration
Seismic Excitation
Equivalent Static
Response Spectrum, Time History
Vibratory Machines
Equivalent Static
Impulse At Startup ACECOMS, AIT Steady State at
Finite Element Analysis
Excitation/ Load
• Elastic Material
Finite Element Analysis
– Follows the same path during loading and unloading and returns to initial state of deformation, stress, strain etc. after removal of load/ excitation
• Inelastic Material – Does not follow the same path during loading and unloading and may not returns to initial state of deformation, stress, strain etc. after removal of load/ excitation
• Most materials exhibit both, elastic and inelastic behavior depending upon level of loading. ACECOMS, AIT
Finite Element Analysis
Creating Finite Element Models
ACECOMS, AIT
Finite Element Analysis
Model Creation Tools • • • • • • • • •
Defining Individual Nodes and Elements Using Graphical Modeling Tools Using Numerical Generation Using Mathematical Generation Using Copy and Replication Using Subdivision and Meshing Using Geometric Extrusions Using Parametric Structures
ACECOMS, AIT
Graphic Object Modeling
Finite Element Analysis
• Use basic Geometric Entities to create FE Models • Simple Graphic Objects – – – –
Point Object Line Object Area Object Brick Object
Represents Node Represents 1D Elements Represents 2D Elements Represents 3D Elements
• Graphic Objects can be used to represent geometry, boundary and loads • SAP2000, ETABS and SAFE use the concept of Graphic Objects
ACECOMS, AIT
Finite Element Analysis
Modeling Objects and Finite Elements • Structural Members are representation of actual structural components • Finite Elements are discretized representation of Structural Members • The concept of Graphic Objects can be used to represent both, the Structural Members as well as Finite Elements • In ETABS, the Graphic Objects representing the Structural Members are automatically divided into Finite Elements for analysis and then back to structural members for result interpretation
ACECOMS, AIT
Finite Element Analysis
Unstable Structures
ACECOMS, AIT
When is Structure Unstable in FEM Solution • When the Global Stiffness Matrix is Singular
Finite Element Analysis
– The determinant of matrix is zero – Any diagonal element in the matrix is zero
• When the Global Stiffness Matrix is IllConditioned – The numerical values in various matrix cells are of grossly different order – Numerical values are either too small or too large
ACECOMS, AIT
Why are the FEM Models Unstable • Restraint Instability – Not enough Boundary Restraints
Finite Element Analysis
• Geometric Instability – – – –
Not enough Elements Not enough stiffness of Elements Elements not connected properly Presence of Orphan Degrees Of Freedom
• Material Instability – Not enough Material Stiffness, (E, G) – Not enough Cross-section Stiffness (A, I, J, ..)
ACECOMS, AIT
Structure Types • Cable Structures • Cable Nets • Cable Stayed
Finite Element Analysis
• Bar Structures • 2D/3D Trusses • 2D/3D Frames, Grids
• Surface Structures • Plate, Shell • In-Plane, Plane Stress
• Solid Structures ACECOMS, AIT
How to Model the Foundations
Soil-Structure Interaction • Simple Supports
Finite Element Analysis
• Fix, Pin, Roller etc. • Support Settlement
• Elastic Supports • Spring to represent soil • Using Modulus of Sub-grade reaction
• Full Structure-Soil Model • Use 2D plane stress elements • Use 3D Solid Elements
ACECOMS, AIT
Modeling of Foundations and Mats
Soil
Finite Element Analysis
Modeling of Mat Beam
Plate
Brick
Constraint
Yes
Yes
Yes
Spring
Yes
Yes
Yes
Brick
No
Yes
Yes
ACECOMS, AIT
Finite Element Analysis
Computing Soil Spring • A = Spacing of Springs in X • B = Spacing of Springs in Y • Ks = Modulus of sub-grade reaction (t/cu m etc.) • K = Spring constant (t/m etc)
B B
A
A
K= ks*A*B ACECOMS, AIT
Finite Element Analysis
Raft as Beam-Grid, Soil as Spring • The raft is represented as a grillage of beams representing slab strips in both directions • The soil is represented by spring • This approach is approximate and does not consider the Mxy or the torsional rigidity of the mat
ACECOMS, AIT
Finite Element Analysis
Raft as Plate, Soil as Spring • The raft is modeled using Plate (or Shell) elements • At least 9-16 elements should be used in one panel • Soil springs may be located or every node or at alternate nodes • Not suitable fro very thick rafts like thick pile caps etc
ACECOMS, AIT
Finite Element Analysis
Raft as Brick, Soil as Spring • The raft is represented by brick elements, soil as springs • More than one layer of brick elements should be used along thickness (usually 3-5) unless higher order elements are used • Suitable for very thick mats and pile caps etc. • Difficult to determine rebars from brick results
ACECOMS, AIT
Finite Element Analysis
Raft as Plate, Soil as Brick • The raft is represented by plate elements, soil as bricks • Soil around the mat should also be modeled (min 2 times width)
ACECOMS, AIT
Raft as Brick, Soil as Brick •
Finite Element Analysis
•
• • •
The raft is represented by brick elements, soil as bricks also More than one layer of brick elements should be used along thickness (usually 3-5) unless higher order elements are used Soil around the mat should also be modeled (min 2 times width) Suitable for very thick mats and pile caps etc. Difficult to determine rebars from brick results
ACECOMS, AIT
Finite Element Analysis
Modeling of Cellular Mats • The top slab, the walls and the bottom slab should be modeled using plate elements • More than one plate element layer should be used in the walls • The soil may be represented by springs or by bricks
ACECOMS, AIT
Finite Element Analysis
Modeling of Piles • For analysis and design of individual Pile, it can be modeled as beam element and soil around it as series of lateral and vertical springs • For analysis of super structure, entire pile can be represented by a single a set of springs
ACECOMS, AIT
Finite Element Analysis
Using Nonlinear Springs to Model Soil • The springs used to represent may be either linear or non linear • The non-linear response of the soil can be obtained from actual tests • The non-linear response can then be used to determine “K” for various levels of load or deformation • Nonlinear springs are especially useful for vertical as well as lateral response of piles and pile groups ACECOMS, AIT
Finite Element Analysis
Modeling of Shear Walls
ACECOMS, AIT
Finite Element Analysis
Modeling of Planner Walls
Using Truss
Using Beam and Column
Using Panels, Plates and Beams ACECOMS, AIT
Frame Model for Planer Walls
Finite Element Analysis
H
t B
Rigid Zones
• Specially Suitable when H/B is more than 5 • The shear wall is represented by a column of section “B x t” • The beam up to the edge of the wall is modeled as normal beam • The “column” is connected to beam by rigid zones or very large cross-section
ACECOMS, AIT
Using Plates to Model Walls
Finite Element Analysis
Multiple elements greater accuracy in determination of stress distribution and allow easy modeling of openings
Using Plate Elements only (No Moment continuity with Beams and Columns unless 6 DOF Shell is used)
Using Plate Elements with Beams, Columns (Full Moment continuity with Beams and Columns) ACECOMS, AIT
Truss Model for Planner Walls •
Finite Element Analysis
•
txt
• •
C t x 2t B
t
•
For the purpose of analysis, assume the main truss layout based on wall width and floor levels Initial member sizes can be estimated as t x 2t for main axial members and t x t for diagonal members Use frame elements to model the truss. It is not necessary to use truss elements Generally single diagonal is sufficient for modeling but double diagonal may be used for easier interpretation of results The floor beams and slabs can be connected directly to truss elements
ACECOMS, AIT
Finite Element Analysis
Modeling of Cellular Shear Walls
Uniaxial
Biaxial ACECOMS, AIT
Finite Element Analysis
Modeling Walls With Openings
Plate-Shell Model
Rigid Frame Model
Truss Model
ACECOMS, AIT
Introduction To Dynamic Analysis
What is Seismic Analysis
Finite Element Analysis
Determination of Structural Response due to Seismic Excitation • The Seismic Excitation is Dynamic in nature • So the Response is governed by
“The Dynamic Equilibrium Equation” • The question is how to solve this equation?
ACECOMS, AIT
Finite Element Analysis
The General
Dynamic Analysis
ACECOMS, AIT
Why Dynamic Analysis – In General • Capture the Realistic Behavior of Structures
Finite Element Analysis
• No Conservative Approximations in Analysis • Puts Check on Structural Irregularities • Identifies Ductility Demands
• Lower Base Shears • Required by Code
ACECOMS, AIT
Basic Dynamic Equilibrium – No Damping Static Elastic Only: Displacement (U)=Force (P) /Stiffness(K) P(u,a)
M
Finite Element Analysis
U = P/K or K u = P K Inertia Only : Acceleration (a)=Force (P) / Mass(M) a = P/M or Ma = P
BOTH :
Ma+Ku=P ACECOMS, AIT
Basic Dynamic Equilibrium – With Damping F
Finite Element Analysis
FI + F D + FS = F
F(t)I + F(t)D + F(t)S = F(t) M a(t) + C v(t) + K u(t) = F(t) M u’’(t) +C u’(t) + K u(t) = F(t) (Second order differential equation for linear structural behavior)
F = External Force FS = Internal Forces FD = Energy Dissipation Forces FI= Inertial Force (t) = Varies with time u’’ = Acceleration (a) u’ = Velocity (v) u = Displacement M = Mass C = Damping K = Stiffness
ACECOMS, AIT
Basics of Structure Dynamics
Finite Element Analysis
• Idealization for a Single Floor – Mass less Column, Entire mass is concentrated on the roof – Rigid roof, Rigid ground – Column is flexible in lateral direction but rigid in vertical direction
Roof
Column
Ground
ACECOMS, AIT
What is Dynamic Response ? • If the roof is displaced laterally by a distance uo and then released the structure will oscillate around its equilibrium position.
Roof
Finite Element Analysis
Column
Ground
One Cycle uo
1
uo
-uo
2
3
4
5 ACECOMS, AIT
Dynamic Response Displacement
uo
1
5
Amplitude
Finite Element Analysis
2
4
time 3
-uo
• The oscillation will continue forever with the same amplitude uo and the structure will never come to rest. • Actual structure will oscillate with decreasing amplitude and will eventually come to rest.
uo
1
uo
-uo
2
3
4
5 ACECOMS, AIT
Damped Dynamic Response
Finite Element Analysis
Mass m
Stiffness K Damping C
Idealized One storey Building
• To incorporate damping or dying out of dynamic response feature into the idealized structure, an energy absorbing element should be introduced. • Viscous damper is the most commonly used energy absorbing element in the dynamic modeling of structures ACECOMS, AIT
Finite Element Analysis
Displacement, Velocity, Acceleration • • • • •
Displacement Velocity Acceleration Time Period Frequency
Change in Location Rate of Change of Displacement wrt Time Rate of Change of Velocity wrt Time The time taken to complete one cycle The no. of cycles per second
u du v u dt d 2u a v u 2 dt ACECOMS, AIT
Finite Element Analysis
Free Vibration Analysis
ACECOMS, AIT
Free Vibration Analysis • Definition
Finite Element Analysis
•
– Natural vibration of a structure released from initial condition and subjected to no external load or damping Main governing equation -Eigen Value Problem
M
u c u K ut Pt t t
• Solution gives – Natural Frequencies – Associated mode shapes – An insight into the dynamic behavior and response of the structure ACECOMS, AIT
Free Vibration
Finite Element Analysis
M u’’(t) +C u’(t) + K u(t) = F(t) M u’’(t) + K u(t) = 0 Which leads to eigenvalue problem
K n w n2 M n
K w M 0 det K w M 0 2 n
n
2 n
Solution of above equation yields a polynomial of order n for w , which in turn gives n mode shapes
• •
No external force is applied No damping of the system
w = natural frequencies F = Mode shape
A mode shape is set of relative (not absolute) nodal displacement for a particular mode of free vibration for a specific natural frequency
ACECOMS, AIT
Modal Analysis
Finite Element Analysis
• Determination of natural frequencies and mode shapes. • No external load or excitation is applied to the structure. • Obtained from eigenvalue analysis. • There are as many modes as there are DOF in the system
ACECOMS, AIT
Finite Element Analysis
Analysis for
Ground Motion
ACECOMS, AIT
Basic Dynamic for Ground Motion
Finite Element Analysis
mu cu ku F F mu mg mug k w ; c 2w m m mu cu ku mug mu 2w mu mw 2u mug u 2w u w 2u ug • The unknown is displacement and its derivatives ( velocity, acceleration) • Variables are ground acceleration, damping ratio and circular frequency
ACECOMS, AIT
Ground Motion Input and Displacement Output
u 2w u w u ug Finite Element Analysis
2
ACECOMS, AIT
• Determination the total dynamic response of structure as the sum of response of all mode shapes using the ground acceleration at each time step
+ Damping Ratio for each mode
0.15
Acceleration (a/g)
Finite Element Analysis
Response History Analysis
0.1 0.05 0 -0.05 0
5
10
15
20
25
30
35
-0.1 -0.15 Time (Second)
ACECOMS, AIT
Finite Element Analysis
Modal Displacements for Ground Motion
ACECOMS, AIT
Finite Element Analysis
Response Spectrum Analysis
ACECOMS, AIT
What are Response Spectra
u 2w u w u ug Finite Element Analysis
2
• For a ground acceleration at particular time, for a given time period and damping ratio, a single value of displacement, velocity and acceleration can be obtained • Output of the above (u, v, a) equation are the dynamic response to the ground motion for a structure considered as a single DOF • A plot of the “maximum” response for different ground motion history, different time period and damping ratio give the “Spectrum of Response” ACECOMS, AIT
Finite Element Analysis
Response Spectrum Generation
ACECOMS, AIT
Finite Element Analysis
Response Spectrum Generation
ACECOMS, AIT
Spectral Parameters S v wS d
Finite Element Analysis
• Spectral Displacement • Pseudo Spectral Velocity • Pseudo Spectral Acceleration
Sd Sv Sa
S a wSv w 2 S d
u v u
du dt
d 2u a v u 2 dt
ACECOMS, AIT
Finite Element Analysis
Spectra For Different Soils
ACECOMS, AIT
Finite Element Analysis
How to Use Response Spectra • For each mode of free vibration, corresponding Time Period is obtained. • For each Time Period and specified damping ratio, the specified Response Spectrum is read to obtain the corresponding Acceleration • For each Spectral Acceleration, corresponding velocity and displacements response for the particular degree of freedom is obtained • The displacement response is then used to obtain the corresponding stress resultants • The stress resultants for each mode are then added using some combination rule to obtain the final response envelop ACECOMS, AIT
Modal combination Rules • ABS SUM Rule
Finite Element Analysis
• Add the absolute maximum value from each mode. Not so popular and not ro used in practice
r
• SRSS • Square Root of Sum of Squares of the peak response from each mode. Suitable for well separated natural frequencies.
• CQC • Complete Quadric Combination is applicable to large range of structural response and gives better results than SRSS.
ro
ro
N
n0
n 1
N
2 r n0 n 1
N
N
i 1 n 1
r r
in i 0 n 0
ACECOMS, AIT
Response Spectrum Analysis
• Uses modal combination rules to determine total peak response from all modes
Spectral Acceleartion
Finite Element Analysis
• Determination of peak response of the structure based on a design or specified response spectrum and the specified mode shapes 1.4 1.2 1 0.8 0.6 0.4 0.2 0
0% 2% 5%
0
1
2
3
4
5
Time Period (Sec)
ACECOMS, AIT
Introduction to Non-linear Analysis
Basic Sources of Non-Linearity • Geometric Non-Linearity
M
Finite Element Analysis
M
• Material Non-Linearity • • Compound Non-Linearity
M
f
P P
• Large Displacements d
ACECOMS, AIT
Geometric Non Linearity
Finite Element Analysis
• The deformations change the basic relationships in the stiffness evaluation • Example: Axial Load Changes Bending Stiffness
• The deformation produce additional actions, not present at initial conditions • Example: Axial load causes additional moments
ACECOMS, AIT
Material Non-Linearity
Finite Element Analysis
• The basic material “constants” (E, G, v) etc. change with level of strain
kfc
• Example: Stress-Strain curve is non-linear
• The cross-section properties change with level of strain • Example: Cracking in reinforced concrete reduces A, I etc
b
Kd N.A
d
yt
As
ACECOMS, AIT
Finite Element Analysis
Material Non-linearity Moment –Curvature curve generated for a rectangular column with circular core. The outer portion is modeled by stressstrain curve for low strength unconfined concrete where as the core is modeled by lightly confined concrete. Observe the drop in moment capacity as the outer concrete fails.
Semi-confined, High Strength Concrete
Rectangular Whitney Curve
ACECOMS, AIT
Types of Non-Linearity • Smooth , Continuous
Finite Element Analysis
– Softening – Hardening
• • • • • •
Discontinuous Snap-through Bifurcation Elastic Buckling In-Elastic Buckling P-Delta ACECOMS, AIT
Non-linear Analysis in SAP2000
Finite Element Analysis
• The non-linear analysis is “Always” carried out together with Time History Dynamic analysis • Non-linear behavior can be modeled by: – NL Link Element – For Dynamic –Nonlinear • Elastic Stiffness for Linear Analysis • Gap, Hook, Damper, Isolator for Nonlinear – Hinge Element – For Static Pushover • Material Non linearity • Load-Deflection Curves
ACECOMS, AIT
Introduction to Push-over Analysis
Why Pushover Analysis
Finite Element Analysis
• Buildings do not respond as linearly elastic systems during strong ground shaking • Improve Understanding of Building Behavior - More accurate prediction of global displacement - More realistic prediction of earthquake demand on individual components and elements - More reliable identification of “bad actors”
• Reduce Impact and Cost of Seismic Retrofit - Less conservative acceptance criteria - Less extensive construction
• Advance the State of the Practice ACECOMS, AIT
Performance Based Design - Basics • Design is based not on Ultimate Strength but rather on Expected Performance
Finite Element Analysis
– Basic Ultimate Strength does not tell us what will be performance of the structure at Ultimate Capacity
• Performance Based Design Levels – – – – –
Fully Operational Operational Life Safe Near Collapse Collapse ACECOMS, AIT
Finite Element Analysis
Pushover Spectrum
ACECOMS, AIT
Finite Element Analysis
Pushover Demand Curves
ACECOMS, AIT
Finite Element Analysis
Earthquake Push on Building
ACECOMS, AIT
Finite Element Analysis
The Pushover Curve
ACECOMS, AIT
Finite Element Analysis
Pushover Capacity Curves
ACECOMS, AIT
Finite Element Analysis
Demand Vs Capacity
ACECOMS, AIT
Non-linearity Considered in Pushover
Finite Element Analysis
• Material nonlinearity at discrete, user-defined hinges in frame/line elements. 1. Material nonlinearity in the link elements. • Gap (compression only), hook (tension only), uniaxial plasticity base isolators (biaxial plasticity and biaxial friction/pendulum)..
2. Geometric nonlinearity in all elements. • Only P-delta effects • P-delta effects plus large displacements
3. Staged (sequential) construction. • Members can be added or removed in a sequence of stages during each analysis case.
ACECOMS, AIT
Finite Element Analysis
Important Considerations • • • • •
Nonlinear analysis takes time and patience Each nonlinear problem is different Start simple and build up gradually. Run linear static loads and modal analysis first Add hinges gradually beginning with the areas where you expect the most non-linearity. • Perform initial analyses without geometric nonlinearity. Add P-delta effects, and large deformations, much later.
ACECOMS, AIT
Finite Element Analysis
Important Considerations • Mathematically, static nonlinear analysis does not always guarantee a unique solution. • Small changes in properties or loading can cause large changes in nonlinear response. • It is Important to consider many different loading cases, and sensitivity studies on the effect of varying the properties of the structure • Nonlinear analysis takes time and patience. Don’t Rush it or Push to Hard
ACECOMS, AIT
Finite Element Analysis
Procedure for Static Pushover Analysis 1. Create a model just like for any other analysis. 2. Define the static load cases, if any, needed for use in the static nonlinear analysis (Define > Static Load Cases). 3. Define any other static and dynamic analysis cases that may be needed for steel or concrete design of frame elements. 4. Define hinge properties, if any (Define > Frame Nonlinear Hinge Properties). 5. Assign hinge properties, if any, to frame/line elements (Assign > Frame/Line > Frame Nonlinear Hinges). 6. Define nonlinear link properties, if any (Define > Link Properties).
ACECOMS, AIT
Finite Element Analysis
Procedure for Static Pushover Analysis 7. Assign link properties, if any, to frame/line elements (Assign > Frame/Line > Link Properties). 8. Run the basic linear and dynamic analyses (Analyze > Run). 9. Perform concrete design/steel design so that reinforcing steel/ section is determined for concrete/steel hinge if properties are based on default values to be computed by the program. 10. For staged construction, define groups that represent the various completed stages of construction. 11. Define the static nonlinear load cases (Define > Static Nonlinear/Pushover Cases).
ACECOMS, AIT
Finite Element Analysis
Procedure for Static Pushover Analysis 12. Run the static nonlinear analysis (Analyze > Run Static Nonlinear Analysis). 13. Review the static nonlinear results (Display > Show Static Pushover Curve), (Display > Show Deformed Shape), (Display > Show Member Forces/Stress Diagram), and (File > Print Tables > Analysis Output). 14. Perform any design checks that utilize static nonlinear cases. 15. Revise the model as necessary and repeat. ACECOMS, AIT
View more...
Comments
