ECIV 720 A Advanced Structural Mechanics and Analysis

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ECIV 720 A Advanced Structural Mechanics and
Analysis

• Non-Linear Problems in Solid and Structural
  Mechanics
• Special Topics

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Introduction
Nonlinear Behavior Response is not directly
proportional to the action that produces it.
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Introduction
Recall Assumptions

• Small Deformations
• Linear Elastic Behavior

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Introduction
Linear Behavior
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Introduction
A. Small Displacements
Inegrations over undeformed volume
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Introduction
B. Linear Elastic Material
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Introduction
C. Boundary Conditions do not change (Implied
Assumption)
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Introduction
If any of the assumptions is NOT satisfied
NONLINEARITIES
Geometric Assumption A or C not satisfied
Material Assumption B not satisfied
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Classification of Nonlinear Analysis
Small Displacements, small rotations Nonlinear
stress-strain relation
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Classification of Nonlinear Analysis
Large Displacements, large rotations and small
strains Linear or nonlinear material behavior
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Classification of Nonlinear Analysis
Large Displacements, large rotations and large
strains Linear or nonlinear material behavior
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Classification of Nonlinear Analysis
Change in Boundary Condition
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Classification of Nonlinear Analysis
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Nonlinear Analysis
Cannot immediately solve for d
Iterative Process Required to obtain d so that
equilibrium is satisfied
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Solution Methods
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Newton-Raphson
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Newton Raphson
With initial conditions
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Modified Newton-Raphson
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SPECIAL TOPICS

• Boundary Conditions
• Elimination Approach
• Penalty Approach
• Special Type Elements

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Boundary Conditions Elimination Approach
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Boundary Conditions Elimination Approach
Boundary Conditions
u1a
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Boundary Conditions Elimination Approach
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Boundary Conditions Elimination Approach

KffufPf Kfsus
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Boundary Conditions Elimination Approach
uf
Pf
Kff
Kfs
Ksf
Kss
us
Ps
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Boundary Conditions Elimination Approach
Ksfuf KssusPs
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Boundary Conditions Penalty Approach
Boundary Conditions
u1a
kC large stiffness
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Boundary Conditions Penalty Approach
Contributes to P
Consequently, for Equilibrium
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Boundary Conditions Penalty Approach
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Choice of C
Rule of Thumb
Penalty approach is easy to implement
Error is always introduced and it depends on C
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Changing Directions of Restraints
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Changing Directions of Restraints
4
2
3
1
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Changing Directions of Restraints
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Changing Directions of Restraints
4
2
Introduce Transformation
3
1
In stiffness matrix
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Connecting Dissimilar ElementsSimple Cases
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Connecting Dissimilar ElementsSimple Cases
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Connecting Dissimilar ElementsSimple Cases
Beam
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Connecting Dissimilar ElementsSimple Cases
Beam
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Connecting Dissimilar ElementsEccentric
Stiffeners
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Connecting Dissimilar ElementsEccentric
Stiffeners
1
3
2
4
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Connecting Dissimilar ElementsEccentric
Stiffeners
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Connecting Dissimilar ElementsEccentric
Stiffeners
3,4 Slave
1,2 Master
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Connecting Dissimilar ElementsEccentric
Stiffeners
The assembly displays the correct stiffness in
states of pure stretching and pure bending
The assembly is too flexible when curvature
varies Use finer mesh
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Connecting Dissimilar ElementsRigid Elements
Generalization of Eccentric Stiffeners
Multipoint Constraints
Rigid element is of any shape and size
Use it to enforce a relation among two or more dof
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Connecting Dissimilar ElementsRigid Elements
e.g.
1-2-3 Perfectly Rigid
Rigid Body Motion described by u1, v1, u2
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Connecting Dissimilar ElementsRigid Elements
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Elastic Foundations
RECALL
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Elastic Foundations
RECALL
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Elastic Foundations
RECALL
Additional stiffness Due to Elastic Support
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Elastic Foundations
RECALL
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Elastic Foundations General Cases
Foundation
z
y
x
Soil
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Elastic Foundations General Cases
Winkler Foundation Stiffness Matrix

• s is the foundation modulus
• H are the Shape functions of the
• attached element

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Winkler Foundations

• Resists displacements normal to surface only
• Deflects only where load is applied
• Adequate for many problems

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Other Foundations

• Resists displacements normal to surface only
• They entire foundation surface deflects
• More complicated by far than Winkler
• Yields full matrices

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Elastic Foundations General Cases
z
y
x
Soil
Infinite
Infinite
Infinite
Infinite
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Infinite Elements
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Infinite Elements
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Infinite Elements
Use Shape Functions that force the field variable
to approach the far-field value at infinity but
retain finite size of element
or
Use conventional Shape Functions for field
variable Use shape functions for geometry that
place one boundary at infinity
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Shape functions for infinite geometry
Mapped Element
Element in Physical Space
Reasonable approximations
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Shape functions for infinite geometry
Node 3 need not be explicitly present