FEA-Lecture 4B-Non-linear PDF

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Lecture 4B - Finite Element Analysis for Non-linear analysis...

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6/03/2019

Nonlinear FEA A/Prof. Faiz Shaikh

Introduction: 

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We use linear approximation because: 

Linear solutions are easier to compute.

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The computational cost is lower.

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Solutions can be superposed on each other.

However, linear analysis is not adequate and nonlinear analysis is necessary when: 

Designing high performance components.

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Establishing the causes of failure.

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Simulating true material behaviour.

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Gain better understanding of physical phenomena.

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Non-linear FE analysis 

In linear FEA the stiffness matrix is not a function of displacements.

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This assumptions is what defines linear analysis.

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The stiffness matrix is created once and it remain unchanged throughout the entire deformation process.

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In certain problems the deformation significantly changes the structure stiffness, making it necessary to update the stiffness matrix during deformation and thus require nonlinear analysis.

Non-linear FE analysis 

Solution of non linear static and dynamic problems is complex.

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Solutions require more than a single step.

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Involves load step and time step increment.

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Most commercials programs include routines to optimize the stepping.

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Materials, geometry or both (materials and geometry)

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Geometric non-linearity 

Deformation changes the stiffness of the structure.

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Stiffness does not remain constant throughout the process of deformation.

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Large displacement  When deformation of a structure is large the original stiffness matrix no longer adequately represents the structure.

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Larger displacement problems are divided into two types: 

Those results in small strain.

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Those results in large finite strains. Simple Truss Undeformed Configuration

Under Large Deformation Truss Has a Different Geometry Thus Implying a New Stiffness Response

Geometric non-linearity (cont’d) 

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In small strain condition, materials remain elastic and structure returns to its original configuration when load is removed. In large strain condition, elements undergo permanent deformations.  Requires non-linear material properties.  combined material and geometric nonlinearities.

Due to large displacement, geometry changes and its stiffness matrix needs to be adjusted.  Recalculates the stiffness matrices after adjusting the nodal coordinates with calculated displacements. 

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Example of a problem require non linear geometry analysis 

Flat membrane under pressure.

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The only mechanism available to resist the load is bending stiffness due to bending stress.

Linear analysis accounts only the initial (before deformation takes place) stiffness produced by bending stress.

Nonlinear analysis is required to account for membrane stiffness that develops during the deformation process.

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During deformation process, the membrane acquires membrane stiffness in addition to bending stiffness.

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Neglecting to account for the nonlinear effect will cause a 230% error in the displacement results.

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Material non-linearity 

Linear material is defined by two parameters: E and 

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A nonlinear material model doesn’t follow linear relation between stress and strain.

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Does not have constant modulus of elasticity (E).

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Using such material, the model stiffness matrix changes during loading process.

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Stiffness matrix needs to be recalculated during solution process.

Static Failure 

Under static loading conditions, material failure can occur when a structure is stressed beyond the elastic limit.

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Two types of static failures: ductile failure and brittle failure Brittle failure

Ductile failure

Strain (ε󰇜

Ductile fracture after necking

Brittle fracture

Figure: Stress-strain curves for brittle and ductile materials. 10

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Material non-linearity 

Non linear analysis require more complex definition and depends on the entire stress-strain curve.

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In reality material exhibits plasticity and creep.

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Non-linear analysis Idealized stress-strain curves supplied to FE.

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Approximated in a bilinear or multi linear way.

Nonlinear material model for steel 

Elastic perfectly plastic

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Elastic-plastic with strain hardening.

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Yield criteria biaxial state of stress.

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Nonlinear material model for concrete 

The concrete can be modeled using the following characteristics:

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Compressive stress-strain relationship.- Multi-linear stress-strain curve

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Yield condition - Bi-axial state of stress

Yield criterion of ductile material 

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Yield criteria for brittle materials 

Maximum Normal Stress Criterion

Also known as the Coulomb, or Rankine criterion, is often used to predict the failure of brittle materials.  Failure occurs when the principal stresses exceed the ultimate tensile strength t, or the ultimate compressive strength c, 

-c < {1, 2} < t Where, 1 and 2 are the principal stresses for 2D stress. Graphically, this criterion requires that the two principal stresses lie within the green zone.

Yield criteria for brittle materials(cont’d)

TheMohrTheoryofFailure, alsoknownastheCoulomb‐Mohrcriterion Mohr'stheoryisoftenusedinpredicting thefailureofbrittlematerials,andisapplied tocasesof2Dstress. Mohr'stheoryrequiresthatthetwoprincipalstresses liewithinthegreenzone.

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