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Nonlinear finite elements/Euler Bernoulli beams

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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.