ECIV 720 A Advanced Structural Mechanics and Analysis

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ECIV 720 A Advanced Structural Mechanics and
Analysis

• Lecture 12
• Isoparametric CST
• Area Coordinates
• Shape Functions
• Strain-Displacement Matrix
• Rayleigh-Ritz Formulation
• Galerkin Formulation

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FEM Solution Area Triangulation
Area is Discretized into Triangular Shapes
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FEM Solution Area Triangulation
One Source of Approximation
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FEM Solution Nodes and Elements
Points where corners of triangles meet Define
NODES
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Non-Acceptable Triangulation
Nodes should be defined on corners of ALL
adjacent triangles
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FEM Solution Nodes and Elements
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FEM Solution Objective

• Use Finite Elements to Compute Approximate
  Solution At Nodes

• Interpolate u and v at any point from Nodal
  values q1,q2,q6

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Intrinsic Coordinate System
2 (0,1)
3 (0,0)
1 (1,0)
Parent
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Area Coordinates
Location of P can be defined uniquely
Area Coordinates
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Area Coordinates and Shape Functions
Area Coordinates are linear functions of x and h
Are equal to 1 at nodes they correspond to
Are equal to 0 at all other nodes
Natural Choice for Shape Functions
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Shape Functions
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Geometry from Nodal Values
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Intrinsic Coordinate System
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Displacement Field from Nodal Values
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Strain Tensor from Nodal Values of Displacements
Need Derivatives
Strain Tensor
u and v functions of x and h
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Jacobian of Transformation
J
Jacobian of Transformation
J
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Jacobian of Transformation Physical Significance
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Jacobian of Transformation Physical Significance
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Jacobian of Transformation Physical Significance
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Jacobian of Transformation
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Strain Tensor from Nodal Values of Displacements
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Strain Tensor from Nodal Values of Displacements
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Strain Tensor from Nodal Values of Displacements
e B q
Looks Familiar?
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Strain-Displacement Matrix
Is constant within each element - CST
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Stresses
e B q
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Element Stiffness Matrix ke
e B q
s D B qe
ke
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Formulation of Stiffness Equations
t
Assume Plane Stress
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Total Potential Approach
Total Potential
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Total Potential Approach
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Total Potential Approach
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Total Potential Approach
Work Potential of Body Forces
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WP of Body Forces
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WP of Body Forces
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WP of Body Forces
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WP of Body Forces
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Total Potential Approach
Work Potential of Tractions
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WP of Traction
Distributed Load acting on EDGE of element
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WP of Traction
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WP of Traction
Components
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WP of Traction
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WP of Traction
2
3
1
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WP of Traction
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Total Potential Approach
Work Potential of Concentrated Loads
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WP of Concentrated Loads
Indicates that at location of point loads a node
must be defined
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In Summary
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After Superposition
Minimizing wrt Q
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Galerkin Approach
Galerkin
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Galerkin Approach
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Galerkin Approach
Introduce Virtual Displacement Field f
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Galerkin Approach
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Element Stiffness Matrix ke
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Galerkin Approach
Virtual Work Potential of Body Forces
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dWP of Body Forces
As we have already seen
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dWP of Body Forces
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Galerkin Approach
Virtual Work Potential of Traction
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dWP of Traction
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dWP of Traction
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Galerkin Approach
Virtual Work Potential of Point Loads
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dWP of Concentrated Loads
Indicates that at location of point loads a node
must be defined
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In Summary
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After Superposition
Y is arbitrary and thus
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Stress Calculations
ee Be qe
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Stress Calculations
Constant
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Summary of Procedure
Tt (force/length)
Nodes should be placed at
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Summary of Procedure
For Every Element Compute

• Strain-Displacement Matrix B

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Summary of Procedure
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Summary of Procedure

• Node Equivalent Traction Vector

For ALL loaded sides
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Summary of Procedure
Collect ALL Point Loads in Nodal Load Vector
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Summary of Procedure
Form Stiffness Equations
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Summary of Procedure
Apply Boundary Conditions
For Every Element Compute Stress
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Example
Tt200 lb/in
(0,2)
(3,2)
fx0
fy60 lb/in2
(0,0)
(3,0)
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ANSYS Solution Coarse Mesh
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2-D Constant Stress Triangle
Comments

• First Element for Stress Analysis
• Does not work very well
• When in Bending under-predicts displacements
• Slow convergence for fine mesh
• For in plane strain conditions Mesh Locks
• No Deformations

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Element Defects
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Element Defects
Constant Stress Triangles
Y-Deflection X-Stress about ¼ of actual
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Element Defects
x10, y10 x2a, y20 x30, y3a
?
Spurious Shear Strain Absorbs Energy Larger
Force Required
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Element Defects
Rubber Like Material n0.5
Mesh Lock