CE 72.32 (January 2016 Semester) Lecture 6 - Overview of Finite Element Analysis

Dr. Naveed Anwar
Executive Director, AIT Consulting
Affiliated Faculty, Structural Engineering
Director, ACECOMS
Design of Tall Buildings
Hybrid Learning System

Dr. Naveed Anwar
Director, ACECOMS
Lecture 6: Overview of Finite
Element Analysis
Design of Tall Buildings

Design of Tall Building: Hybrid Learning System, Dr. Naveed Anwar
• Some of the material presented in these notes are based on following
sources:
– Edward L. Wilson, “Three-Dimensional Static and Dynamic Analysis of
Structures”
– ETABS User and Technical Manuals
– Graham H. Powell, “Modeling for Structural Analysis”
– Lecture Notes from Prof. Worsak Kanok-Nukulchai
– Notes from various workshops conducted by Dr. Naveed Anwar
– Seminar notes from Computers and Structures Incorporated, USA
– SAP2000 User and Technical Manuals
– Other Papers and References
Acknowledgements
3

Fundamental Principles of FE
4

The Structural System
EXCITATION
Loads
Vibrations
Settlements
Thermal Changes
RESPONSES
Displacements
Strains
Stresses
Stress Resultants
STRUCTURE
pv
5

Simplified Structural System
Loads (F) Deformations (u)
Fv
F = K u
F
K (Stiffness)u
Equilibrium Equation
6

The Total Structural System
EXCITATION RESPONSES
STRUCTURE
pv
• Static
• Dynamic
• Elastic
• Inelastic
• Linear
• Nonlinear
Eight types of equilibrium equations are possible!
7

The Need For Analysis
• We need to determine the response of the
structure to excitations
so that
• We can ensure that the structure can sustain
the excitation with an acceptable level of
response
Analysis
Design
8

Analysis of Structures
pv






xx yy zz
vx
x y z
p    0
Real Structure is governed by “Partial Differential
Equations” of various order
Direct solution is only possible for:
• Simple Geometry
• Simple Boundary
• Simple Loading
Equilibrium Equation: The Sum of Body Forces and
Surface Tractions is equal to Zero
9

The Need for Structural Model
Structural
Model
EXCITATION
Loads
Vibrations
Settlements
Thermal Changes
RESPONSES
Displacements
Strains
Stress
Stress Resultants
STRUCTURE
pv
10

• A - Real Structure cannot be analyzed: It can only be a “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
The Need for Modeling
11

CONTINUOUS MODEL
OF STRUCTURE
(Governed by either
partial or total differential
equations)
DISCRETE MODEL OF
STRUCTURE
(Governed by algebraic
equations)
3D-CONTINUUM
MODEL
(Governed by partial
differential equations)
Solid – Structure Model
Simplification
(geometric)
Discretization
3D SOLIDS
12

13
Discretization
Object Elements
Nodes

Finite Element Modeling
14
Examples
2D 3D

Finite Element Method: The Analysis Tool
• Finite Element Analysis (FEA)
– “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”
15

From Classical to FEM Solution






xx yy zz
vx
x y z
p    0
 t
v
t
s
t
v
dV p udV p uds
_ _ _
  
Assumptions
Equilibrium
Compatibility
Stress-Strain Law
(Principle of Virtual Work)
“Partial Differential
Equations”
Classical
Actual Structure
RKr 
“Algebraic
Equations”
K = Stiffness
r = Response
R = Loads
FEM
Structural Model
16

FEA Overall Process
17
• Prepare the FE Model
– Discretize (mesh) the structure
– Prescribe loads
– Prescribe supports
• Perform calculations (solve)
– Generate stiffness matric (k) for
each element
– Connect elements (assemble K)
– Assemble loads (into load vector R)
– Impose supports conditions
– Solve equations (KD = R) for
displacements
• Post-process

The Finite Element Analysis Process
Evaluate Real Structure
Create Structural Model
Discretize Model in FE
Solve FE Model
Interpret FEA Results
Physical Significance of Results
Engineer
Software
Engineer
18

Design of Tall Building: Hybrid Learning System, Dr. Naveed Anwar 19

Brief History of FEM
20

Archimedes FEM
Archimedes’ problem (circa 250 B.C.): rectification of the circle as limit of
inscribed regular polygons

• Grew out of aerospace industry
• Post-WW II jets, missiles, space flight
• Need for light weight structures
• Required accurate stress analysis
• Paralleled growth of computers
Brief History

• 1940s
– Hrennikoff (1941) - Lattice of 1D bars
– McHenry (1943) - Model 3D solids
– Courant (1943) - Variational form
– Levy (1947, 1953) - Flexibility and Stiffness
• 1950-60s
– Argryis and Kelsey (1954) - Energy Principle for Matrix Methods
– Turner, Clough, Martin and Topp (1956) - 2D elements
– Clough (1960) - Term “Finite Elements”
• 1980s - Wide applications due to:
– Integration of CA/CAE – automated mesh generation and graphical display of
analysis results
– Powerful and low cost computers
• 2000s – FEA in CAD; Design Optimization in FEA; Nonlinear FEA; Better CAD/CAE
Integration
Developments

• Mechanical/Aerospace/Civil/Automotive Engineering
• Structural/Stress Analysis
– Static/Dynamic
– Linear/Nonlinear
• Fluid Flow
• Heat Transfer
• Electromagnetic Fields
• Soil Mechanics
• Acoustics
• Biomechanics
Current Applications

Prerequisite Concepts
A Brief Review of
Theory of Elasticity
25

Method of Virtual Forces
Method of Virtual Displacements
Conservation of Linear
Momentum
26

Potential Energy and Kinetic Energy
Oscillation of Pendulum
Stationary Energy Principle
Energy as a Function of a Typical
Displacement
27

Stress Equilibrium Equations
0
0
0



b
zyzxz
b
yzyxy
b
xzxyx
Z
zyx
Y
zyx
X
zyx


















The Sum of Body Forces and Surface Tractions is equal to Zero
28

Strain Displacement Relationships
z
v
y
w
γ
z
w
ε
x
w
z
u
γ
y
v
ε
x
v
y
u
γ
x
u
ε
yzz
xzy
xyx

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
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





ntdisplacemeofcomponents
zandyx,thearew)v,(u,
29

3D Stress-Strain Relationships
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30

3D Material Matrix
 
  
 

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E
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:Note
31

Plane Stress and Plane Strain Approximations
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
2
1
00
01
01
1 

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E
D
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  
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Plane Stress
Plane Strain
32

• Cramer’s Rule
• Inverse Method
• Gaussian Elimination
• Gauss-Seidel Iteration
Solution of Systems of Linear Algebraic Equations
33

Modeling of Materials
34

• Local coordinate system
• Each material has its own material local coordinate system, which is used to
define the elastic and thermal properties.
• The axes of the material local coordinate system are denoted as 1, 2, and 3.
• The material coordinate system is aligned with the local coordinate system for
each element.
• Significant only for orthotropic and anisotropic materials
Material Properties
35

Stresses and Strains – General Notation
Definition of Stress Components in the Material Local Coordinate System
36

• The elastic mechanical properties relate the behavior of the stresses and strains
within the material
• The stresses are defined as forces per unit area acting on an elemental cube
aligned with the material axes
• Not all stress components exist in every element type.
Stresses and Strains
37

xx
yy
zz
xy
zx
yx
zy
xz
yz
x
y
z
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.
xx
yy
zz
xy
zx
yx
zy
xz
yz
x
y
z
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.
• The Hook's law is a simplified form
of Stress-Strain relationship
• Ultimately, the six stress and strain
components can be represented by
3 principal summations
Stress and Strains Components
38

Stress-strains Relationship
• For a general Isotropic Material
• For 2D, Isotropic Material, V=0
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z
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v
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vvv
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vv
E
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xxx E  xyxy G 
39

• The behavior of an isotropic material is independent of the direction of loading
or the orientation of the material
• Shearing behavior is uncoupled from ex-tensional behavior and is not affected
by temperature change
• The isotropic mechanical and thermal properties relate strain to stress and
temperature
Isotropic Materials
40

• The behavior of an orthotropic material can be different in each of the three
local coordinate directions.
• Shearing behavior is un-coupled from extensional behavior and is not affected
by temperature change.
• The orthotropic mechanical and thermal properties relate strain to stress and
temperature change as shown on the next slide.
Orthotropic Materials
41

• The behavior of an anisotropic material can be different in each of the three
local coordinate directions
• Shearing behavior can be fully coupled with extensional behavior and can be
affected by temperature change
• The anisotropic mechanical and thermal properties relate strain to stress and
temperature change as shown on the next slide.
Anisotropic Materials
42

• Local Coordinate System
• Each material has its own material local coordinate system used to define the
elastic and thermal properties.
• Significant only for orthotropic and anisotropic materials.
• The axes of the material local coordinate system are denoted as 1, 2, and 3.
• The material coordinate system is aligned with the local coordinate system for
each element.
Material Properties
43

• These properties are given at a series of specified material temperatures, t.
• Properties at other temperatures are obtained by linear interpolation between
the two nearest specified temperatures.
• Properties at temperatures outside the specified range use the properties at the
nearest specified temperature.
Temperature Dependent Properties
44

Temperature Dependent Properties
Determination of Property Ematt at Temperature Tmatt from Function E(T)
45

Modeling of Geometry
46

CONTINUOUS MODEL
OF STRUCTURE
(Governed by either
partial or total differential
equations)
DISCRETE MODEL OF
STRUCTURE
(Governed by algebraic
equations)
3D-CONTINUUM
MODEL
(Governed by partial
differential equations)
Simplification
(geometric)
Discretization
3D SOLIDS
47

• A continuum extends in all directions, 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.
Continuum vs. Structure
48

• A structure can be considered as an assemblage of “Physical Components” called
Members
– 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.
Structure, Member, Element
49

Structural Members
Dimensional Hierarchy of Structural Members
Continuum
Regular Solid
(3D)
Beam (1D)
b h
L>>(b,h)

b
h
t
z
Plate/Shell (2D)
x z
t<<(x,z)
 x
z
y
x L
50
Dimensional Hierarchy of Structural Members

• 1D Elements (Beam type)
– Only one dimension is actually modeled as a line, the other two dimensions
are represented by stiffness properties
– Can be used in 1D, 2D and 2D
• 2D Elements (Plate type)
– Only two dimensions are actually modeled as a surface, the third dimension
is represented by stiffness properties
– Can be used in 2D and 3D Model
• 3D Elements (Brick type)
– All three dimensions are modeled as a solid
– Can be used in 3D Model
Basic Categories of Finite Elements
51

Physical Categorization of Structures
• Structures can be categorized in many ways.
• For modeling and analysis purposes, the overall physical behavior can be used as
the basis of categorization
– Cable or Tension Structures
– Skeletal or Framed Structures
– Surface or Spatial Structures
– Solid Structures
– Mixed Structures
52

Structure Types
• Cable Structures
• Cable Nets
• Cable Stayed
• Bar Structures
• 2D/3D Trusses
• 2D/3D Frames, Grids
• Surface Structures
• Plate, Shell
• In-Plane, Plane Stress
• Solid Structures
53

Global Modeling or “Macro Model”
• A model of the whole structure
• The objective is to get an overall structural
response
• Results in the form of member forces and stress
patterns
• Global modeling is the 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
54

Global Modeling of Structural Geometry
Various Ways to Model a Real Structure
55

Local Model or “Micro Model”
• Model of a single member or part of a
member
• Model of the cross-section, opening, joints,
connection
• The objective is to determine the local
stress concentration, cross-section
behavior, modeling of cracking, bond,
anchorage, etc.
• Needs finite element modeling, often using
very fine mesh, advance element features
and non-linear analysis
• Mostly suitable for research, simulation,
experiment verification and theoretical
studies
56

Element-Based Modeling
• The structure is directly divided into
specific Finite Elements such as:
– Truss Element
– Beam Element
– Plate Element
– Shell Element
– Solid/Brick Element
• The engineer has to select and
define each element individually to
represent structure geometry
57

• The Finite Elements are discretized representation of the continuous structure
• Generally, they correspond to the physical structural components but dummy or
idealized elements may also be used at certain times.
• Elements behavior is completely defined within its boundaries and is not
directly related to other elements.
• Nodes are imaginary points which serve to provide connectivity across element
boundaries and are used to describe arbitrary quantities.
Nodes and Finite Elements
58

• Identify basic issue and need for modeling
• See how materials are represented and modeled
• Explore the types and properties of various elements
• Select appropriate elements for representing different structural members
The Selection of Elements

Some Finite Elements
Truss and Beam Elements (1D,2D,3D)
Plane Stress, Plane Strain, Axisymmetric, Plate and Shell Elements (2D,3D)
Brick Elements
60

• In several softwares, the graphic objects representing the structural members
are automatically divided into finite elements for analysis
• This involves
– Object-based Modeling
– Auto Meshing
– Auto Load Computation
– Auto Load Transfer
Current Modeling Trend
61

Object-Based Modeling
• Level 1
– The nodes are defined first by
coordinates and then elements are
defined that connect the nodes
• Level 2
– The elements are defined directly,
either numerically or graphically and
the nodes are created automatically
• Level 3
– The structure is represented by
generic objects and the elements and
nodes are created automatically
62

Joint Elements
63

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
applied on 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.
64

• Joints, also known as nodal points or nodes, are fundamental parts of every
structural model
• A joint is defined by specifying its label, j, and three spatial coordinates, x, y, z,
that locate the joint in space.
– 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
– 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
Joint Elements
65

• 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
Joint Elements
66

• The joint local coordinate axes may be further modified by the use of three joint
coordinate angles, denoted as a, b, and c.
• The joint coordinate angles specify rotations of the local coordinate system
about its own current axes.
• The resulting orientation of the joint local coordinate system is obtained
according to the following procedures:
1. The local system is first rotated about its +3 axis by angle a
2. The local system is next rotated about its resulting +2 axis by angle b
3. The local system is lastly rotated about its resulting +1 axis by angle c
Joint Coordinate Angles
67

Joint Coordinate Angles
68

Degrees of Freedom
• Every joint of the structural model may have up to six displacement components
or degrees of freedom
– The joint may translate along its three local axes. These translations are
denoted U1, U2, and U3.
– The joint may rotate about its three local axes. These rotations are denoted
R1, R2, and R3.
69

Degrees of Freedom
• Each degree of freedom in the structural model must be one of the following
types:
– Active
• the displacement is computed during the 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
70

• By default, all six degrees of freedom are available to every joint. This default
should generally be used for all three-dimensional structures.
• The degrees of freedom that are not specified as being available are called
unavailable degrees of freedom.
• Any stiffness, loads, mass, restraints, or constraints that are applied to the
unavailable degrees of freedom are ignored by the analysis.
Degrees of Freedom - Availability
71

• If the displacement of a joint along any one of its available degrees of freedom
is known, such as at a support point, that degree of freedom is restrained.
• The known value of the displacement may be zero or non-zero, and may be
different in different Load Cases
• The force along the restrained degree of freedom that is required to impose the
specified restraint displacement is called the reaction, and is determined by the
analysis
• Unavailable degrees of freedom are essentially restrained
Restrained Degrees of Freedom
72

• Any joint that is part of a constraint or weld may have one or more of its
available degrees of freedom constrained.
• The program automatically creates a master joint to govern the behavior of each
constraint.
• The displacement of a constrained degree of freedom is then computed as a
linear combination of the displacements along the degrees of freedom at the
corresponding master joint
• A degree of freedom may not be both constrained and restrained
Constrained Degrees of Freedom
73

Simple Spring Restraints
• Independent springs stiffness in each DOF
74

Coupled Spring Restraints
• General Spring Connection
• Global and skewed springs
• Coupled 6x6 user-defined spring stiffness
option (for foundation modeling)
75

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
76

One Dimensional Elements
77

• Simple Frame Elements for
– Beam, Column
– Truss, Bracing, etc.
• Non-Linear Link Element for
– Hook, Gap, Damper
– Base Isolators
– Friction
• Plastic Hinge Element
The Frame Elements
78

DOF for 1D Elements
Dx
Dy
DxDz
Dy
Dx
Dy
Rz
Dy
RxRz DxDz
Dy
Rx
Rz
Ry
2D Truss 2D Beam 3D Truss
2D Frame 2D Grid 3D Frame
Dy
Rz
79

Joint Connectivity
• A frame element is represented by a
straight line connecting two joints, i and j.
• The two joints must not share the same
location in space.
• The two ends of the element are denoted
end i and end j, respectively.
80

Local Coordinate System
81

82

The Frame Elements
Frame Elements in FE Model
Element Forces
Element Forces
Frame Elements in
FE Model
83

• A frame section is a set of material and geometric properties that describes the
cross-section of one or more frame elements.
• Sections are defined independently of the frame elements. and are assigned to
the elements.
• Section properties are of two basic types:
– Prismatic — all properties are constant along the full element length
– Non-prismatic — the properties may vary along the element length
Section Properties
84

Isoparametric Formulation of 1D Element
A Simple Example of an Isoparametric Element
85

A Tapered Bar Example
Displacement Transformation
86

An Arbitrary, Two-Dimensional Frame
Element
Three-dimensional Frame Element
Member Forces in Local Reference Systems
87

Rigid End Offsets
• Rigid End connections to model large joints
• Automated end offset evaluation and
assignment
88

End Releases
• Easily model non-fixed connections
by general “End-Release”
– Axial
– Shear
– Torsion
– Moment
89

Non-prismatic Frame Elements
• Multiple non-prismatic segments over element length to model beams of variable
sections
90

Usage of 1D Elements
3D Frame
2D Grid
2D Frame
91

Two Dimensional Elements
92

Plate ShellMembrane
DOF for 2D Elements
Dx
Dy
Dy
Ry ?
Rx
Dz
Dy
Rx
Rz
Ry ?
Dx
93

Two-dimensional Elements
Four-to Nine-Node Two-Dimensional Isoparametric Elements
94

DOF for 2D Elements
95

DOF for 2D Elements
96

2D Elements in FE Model
2D Elements in FE Model
Stress Results
97

• General
– 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
– Pure membrane behavior; only the in-plane forces and the normal (drilling)
moment can be supported
• Application
– For Modeling surface elements carrying in-plane loads
– Walls, Deep Beams, Domes, Thin Shells
The Membrane Element

The Membrane Element
99

Variation of Membrane Elements
Plane Stress ProblemPlane Strain Problem
100

• General
– Total DOF per Node = 3
– Total Rotations per Node = 2
– Plates are for flat surfaces
• Application
– For Modeling surface elements carrying out of plane loads
– Slabs, Footings, Mats
Plate Element

• Plate
Plate Element

• General
– Total DOF per Node = 6 (or 5)
– Total Rotations per Node = 3
– Used for curved surfaces
• Application
– For Modeling surface elements carrying general loads
– Every surface type member
Shell Element

• Shell
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 and the third axis is normal.
Local Cords for Shell Element

Local Cords for Shell Element

• Four-node Quadrilateral Shell Element
Shell Elements in SAP2000/ETABS

• Three-node Triangular Shell Element
Shell Elements in SAP2000/ETABS

• A Shell Section is a set of material and geometric properties describing the
cross-section of one or more Shell elements.
• Sections are defined independently of the Shell elements, and are referenced
during the definition of the elements.
• Section Type
– Membrane, Plate, Shell
• Thickness Formulation
– THICK: A thick-plate (Mindlin-Reissner) formulation is used which includes
the effects of transverse shear deformation
– THIN: A thin-plate (Kirchhoff) formulation is used that neglects transverse
shearing deformation
Section Properties

• A Zero Energy Hourglass Displacement Mode
Eight- and Five-Point Integration Rules
Numerical Integration in 2D-Hourglass Modes

• Quadrilateral Plate Bending Element
The Quadrilateral Elements

Typical Element Side
Positive Displacements in Plate Bending Element
Node Point
Transverse Shears
112

Cantilever Beam Modeled using
One Plate Element
Use of Plate Element to Model
Torsion in Beams
113

Typical Side of Quadrilateral Element
Zero Energy Displacement Mode
114

Formation of Flat Shell Element
115
A Simple Quadrilateral Shell Element

Modeling Curved Shells with Flat Elements
Use of Flat Elements to Model Arbitrary Shells
116

Use of Solid Elements for Shell Analysis
• Cross-Section of Thick Shell
Structure Modeled with Solid
Elements
Scordelis-Lo Barrel Vault Example
Hemispherical Shell Example
117

A Solid Element
118

• A solid element is three- to nine-node element used to model axisymmetric solids
under axisymmetric loading.
• The element models a representative two-dimensional cross section of the three-
dimensional axisymmetric solid.
• The axis of symmetry must be one of the global axes, and the element must exist
in one of the global principal planes.
• The geometry, loading, displacements, stresses, and strains are assumed not to
vary in the circumferential direction.
• Any displacements that occur in the circumferential direction do not affect the
element.
A Solid Element

Three-Node Triangular Element
Four-to-Nine Node Quadrilateral Element
A Solid Element
120

• Each solid element has its own element local coordinate system used to
define material properties, loads and output.
• The axes of this local system are denoted 1, 2 and 3.
• These axes are always parallel to the axes of the global coordinate system
with the same positive sense.
• Local axis 1 is parallel to the radial direction, axis 2 is parallel to the axis of
symmetry, and axis 3 is tangent to the circumferential direction of the
axisymmetric solid.

• All joints for a given element must lie in one of the principal global planes.
• The radial coordinate must not be negative for any of the element’s joints.
• Solid elements should not be connected to joints that are connected to
other types of elements unless special provisions are made to enforce
axisymmetric conditions upon these joints.
Joint Connectivity

123

• The solid element activates the three translational degrees of freedom at
each of its connected joints.
• Rotational degrees of freedom are not activated.
• Contributes stiffness only to the degrees of freedom in the plane of the
element.
• It is necessary to provide restraints or other supports for the translational
degrees of freedom that are normal to this plane
Degrees of Freedom

Three Dimensional Elements
125

• The Solid element is an eight-node element used to model three-
dimensional solid structures.
• The incompatible bending modes significantly improve the bending
behavior of the element if the element geometry is of a rectangular form.
• The local coordinate system for each Solid element is identical to the
global system.
The Solid Element

The Solid Element
Solid Element Joint Connectivity and Face Definitions
127

Six-Node Plane Triangle and Ten-Node
Solid Tetrahedral Elements
Eight- to 27- Node Solid Element
128

• Each Solid element has six quadrilateral faces, with a joint located at each of the
eight corners
• The relative position of the eight joints: the paths j1-j2-j3 and j5-j6-j7 should
appear counter-clockwise when viewed along the direction from j5 to j1.
Joint Connectivity

• The Solid element activates the three translational degrees of freedom at each
of its connected joints.
• Rotational degrees of freedom are not activated.
• This element contributes stiffness to all of these translational degrees of
freedom.
Degrees of Freedom

• Each Solid element has its own element local coordinate system used to define
Material properties, loads and output.
• The axes of this local system are denoted 1, 2 and 3.
• These axes always correspond with the global coordinate axes X, Y and Z,
respectively, regardless of the orientation of the element.

Stress in Solid ElementsSolid Elements in FE Model
Solid Elements in FE Model
132

Modeling of
Boundary Conditions, Supports
133

• Any of the six degrees of freedom at any of the joints in the structure can have
translational 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.
Springs

Springs
• In a joint local coordinate system, the spring forces and moments F1, F2, F 3, M1,
M2 and M3 at a joint are given by:
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.
135

Joints in FE Model
Spring Support
Hinge Support
Joint Load
136

Joint Results
Reaction Forces
Joint Displacement
137

Using Constraints in
Structural Model
138

Rigid Diaphragm Approximation
Utilization of Displacement Constraints in
Portal Frame Analysis
139

Floor Diaphragm Constraints
Column Connected Between Horizontal Diaphragms
140

Rigid Constraints
Rigid Body Constraints
141

Use of Constraints in Beam-shell Analysis
Connection of Beam to Slab by Constraints
142

Use of Constraints in Shear Wall Analysis
Beam-Column Model of Shear Wall
143

Use of Constraints for Mesh Transitions
Use of Constraints to Merge Different Finite Element Meshes
144

Connecting Dissimilar Elements
145

Connecting Different Types of Elements
Truss Frame Membrane Plate Shell Solid
Truss
OK OK Dz OK OK OK
Frame
Rx, Ry, Rz OK
Rx, Ry, Rz,
Dz
Rx ?
Dx, Dy
Rx ? Rx, Ry, Rz
Membrane
OK OK OK Dx, Dy OK OK
Plate
Rx, Rz OK Rx, Rz OK OK Rx, Rz
Shell
Rx, Ry, Rz OK
Rx, Ry, Rz,
Dz
Dx, Dz OK Rx, Rz
Solid
OK OK Dz Dx, Dz OK OK
0
Orphan Degrees of Freedom
1 2 3 4
146

Connecting Dissimilar Elements
• 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 the Software Capability
Beams to Plates Beam to Brick Plates to Brick
147

Meshing Slabs and Walls
In general, the mesh in the slab
should match with the mesh in
the wall to establish connection
Some software automatically
establishes connectivity by using
constraints or “Zipper” elements
“Zipper”
148

Modeling Non-linear Behavior
149

Nonlinear Link Element
150

• These model concentrated nonlinearity
The Nonlinear Hinges

• These model distributed NonLinearity
Nonlinear Multilayer Shell

Dr. Naveed Anwar
Director, ACECOMS
Thank You

CE 72.32 (January 2016 Semester) Lecture 6 - Overview of Finite Element Analysis

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CE 72.32 (January 2016 Semester) Lecture 6 - Overview of Finite Element Analysis