ECIV 720 A Advanced Structural Mechanics and Analysis

1
ECIV 720 A Advanced Structural Mechanics and
Analysis

• Lecture 16 17
• Higher Order Elements (review)
• 3-D Volume Elements
• Convergence Requirements
• Element Quality

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Higher Order Elements
Quadrilateral Elements
Recall the 4-node
4 generalized displacements ai
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Higher Order Elements
Quadrilateral Elements
Assume Complete Quadratic Polynomial
9 generalized displacements ai
9 BC for admissible displacements
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9-node quadrilateral
9-nodes x 2dof/node 18 dof
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9-node element Shape Functions
Following the standard procedure the shape
functions are derived as
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N1,2,3,4 Graphical Representation
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N5,6,7,8 Graphical Representation
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N9 Graphical Representation
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Polynomials the Pascal Triangle
Degree
Pascal Triangle
1
.
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Polynomials the Pascal Triangle
To construct a complete polynomial
etc
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Incomplete Polynomials
3-node triangular
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Incomplete Polynomials
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8-node quadrilateral
Assume interpolation
8 coefficients to determine for admissible displ.
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8-node quadrilateral
8-nodes x 2dof/node 16 dof
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8-node element Shape Functions
Following the standard procedure the shape
functions are derived as
h
x
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N1,2,3,4 Graphical Representation
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N5,6,7,8 Graphical Representation
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Incomplete Polynomials
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6-node Triangular
Assume interpolation
6 coefficients to determine for admissible displ.
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6-node triangular
6-nodes x 2dof/node 12 dof
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6-node element Shape Functions
Following the standard procedure the shape
functions are derived as
LiArea coordinates
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Other Higher Order Elements
12-node quad
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Other Higher Order Elements
16-node quad
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3-D Stress state
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3-D Stress State
Assumption Small Deformations
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Strain Displacement Relationships
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3-D Finite Element Analysis
Solution Domain is VOLUME
Simplest Element (Lowest Order) Tetrahedral
Element
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3-D Tetrahedral Element
Can be thought of an extension of the 2D CST
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3-D Tetrahedral
Shape Functions
Volume Coordinates
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Geometry Isoparametric Formulation
In view of shape functions
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Jacobian of Transformation
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Strain-Displacement Matrix
B is CONSTANT
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Stiffness Matrix
Element Strain Energy
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Force Terms
Body Forces
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Element Forces
Surface Traction
Applied on FACE of element
eg on face 123
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Stress Calculations
Stress Invariants
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Stress Calculations
Principal Stresses
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Other Low Order Elements
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Degenerate Elements
Still has 24 dof
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Degenerate
Still has 24 dof
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Higher Order Elements 10-node 4-hedral
Z
2

Y
X

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15-node 5-hedral
Z
Y
X
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15-node 5-hedral Shape Functions
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20-node 6-hedral
z
24
23
8
16
15
22
5
20
h
7
x
13
14
6
17
4
19
12
11
18
1
3
Z
9
10
2
Y
X
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20-node 6-hedral Shape Functions
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Convergence Considerations
For monotonic convergence of solution
Requirements
Elements (mesh) must be compatible
Elements must be complete
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Monotonic Convergence
No of Elements
For monotonic convergence the elements must
be complete and the mesh must be compatible
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Mixed Order Elements
Consider the following Mesh
4-node
8-node
Incompatible Elements
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Mixed Order Elements
We can derive a mixed order element for grading
4-node
8-node
7-node
By blending shape functions appropriately
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Convergence Considerations
For monotonic convergence of solution
Requirements
Elements (mesh) must be compatible
Elements must be complete
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Element Completeness
For an element to be complete
Assumption for displacement field
must accommodate

• RIGID BODY MOTION
• CONSTANT STRAIN STATE

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Element Completeness
Consider
This is not a complete polynomial
However,
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Element Completeness
Assume displacement field
Test for ELEMENT completeness
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Element Completeness
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Element Completeness
In order for the computed displacements to be
the assumed ones we must satisfy
Condition for element completeness
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Effects of Element Distortion
Loss of predictive capability of isoparametric
element
No Distortion
1
x2y
xy2
Behavior accurately predicted
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Effects of Element Distortion
Angular Distortion
1
x2y
xy2
Predictability is lost for all quadratic terms
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Effects of Element Distortion
Quadratic Curved Edge Distortion
1
x2y
xy2
Predictability is lost for all quadratic terms
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Effects of Element Distortion
The advantage (reduced of dof) of using 8-node
higher order element based on an incomplete
polynomial is lost when high element distortions
are present
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Effects of Element Distortion
Loss of predictive capability of isoparametric
element
No Distortion
9-node
1
x2y
xy2
x2y2
Behavior accurately predicted
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Effects of Element Distortion
9-node
Angular Distortion
Behavior predicted better than 8-node
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Effects of Element Distortion
Quadratic Curved Edge Distortion
9-node
Predictability is lost for high order terms
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Effects of Element Distortion
The advantage (reduced of dof) of using higher
order element based on an incomplete polynomial
is lost when high element distortions are present
For angular distortion 9-node element shows
better behavior
For Curved edge distortion all elements give low
order prediction
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Polynomial Element Predictability
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Tests of Element Quality
Eigenvalue Test Identify Element
Deficiencies Patch Test Convergence of Solutions
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Eigenvalue Test
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Eigenvalue Test
Eigenproblem
As many eigenvalues l as dof
For each l there is a solution for d
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Displacement Modes Stiffness Matrix
For all eigenvalues and modes
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Eigenvalue Test
Scale d so that
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Eigenvalue Test
Rigid Body Motion gt System is not strained gt U0
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Rigid Body Motion
Rigid Body Modes
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Displacement Modes Stiffness Matrix
Consider the 2-node axial element
Identify all possible modes of displacement
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Displacement Modes Stiffness Matrix
Consider the 4-node plane stress element
1
t1 E1 v0.3
1
Solve Eigenproblem
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Displacement Modes Stiffness Matrix
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Displacement Modes Stiffness Matrix
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Displacement Modes Stiffness Matrix
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Displacement Modes Stiffness Matrix
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Displacement Modes Stiffness Matrix
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Displacement Modes Stiffness Matrix
The eigenvalues of the stiffness matrix display
directly how stiff the element is in the
corresponding displacement mode
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Patch Test
Objective
Examine solution convergence for displacements,
stresses and strains in a particular element type
with mesh refinement
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Patch Test - Procedure
Build a simple FE model
Consists of a Patch of Elements
At least one internal node
Load by nodal equivalent forces consistent with
state of constant stress
Internal Node is unloaded and unsupported
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Patch Test - Procedure
Compute results of FE patch
If (computed sx) (assumed sx) test passed