Nonlinear finite elements/Euler Bernoulli beams

Euler-Bernoulli Beam

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Euler-Bernoulli beam

Displacements

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Strains

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Strain-Displacement Relations

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The displacements

 

The derivatives

 

von Karman strains

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The von Karman strains

 

Equilibrium Equations

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Balance of forces

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Stress Resultants

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Constitutive Relations

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Stress-Strain equation

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Stress Resultant - Displacement relations

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Extensional/Bending Stiffness

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If   is constant, and  -axis passes through centroid

 

Weak Forms

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Axial Equation

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where

 

Bending Equation

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where

 

Finite Element Model

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Finite element model for Euler Bernoulli beam
 

where  .

Hermite Cubic Shape Functions

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Hermite shape functions for beam

Finite Element Equations

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where

 
 

Symmetric Stiffness Matrix

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Load Vector

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Newton-Raphson Solution

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where

 

The residual is

 

For Newton iterations, we use the algorithm

 

where the tangent stiffness matrix is given by

 

Tangent Stiffness Matrix

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Load Steps

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Recall

 
  • Divide load into small increments.
 
  • Compute   and   for first load step,
 
 
Stiffness of Euler-Bernoulli beam.
  • Compute   and   for second load step,
 
  • Continue until F is reached.

Membrane Locking

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Recall

 

where

 
 
Mebrane locking in Euler-Bernoulli beam

For Hinged-Hinged

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Membrane strain:

 

or

 

Hence, shape functions should be such that

 

  linear,   cubic   Element Locks! Too stiff.

Selective Reduced Integration

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  • Assume   is linear ;~~   is cubic.
  • Then   is constant, and   is quadratic.
  • Try to keep   constant.
 
  •   integrand is constant,   integrand is fourth-order ,   integrand is eighth-order

Full integration

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Assume   = constant.