Nonlinear finite elements/Euler Bernoulli beams

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Euler-Bernoulli Beam[edit]

Euler-Bernoulli beam

Displacements[edit]

Strains[edit]

Strain-Displacement Relations[edit]

The displacements

The derivatives

von Karman strains[edit]

The von Karman strains

Equilibrium Equations[edit]

Balance of forces[edit]

Stress Resultants[edit]

Constitutive Relations[edit]

Stress-Strain equation[edit]

Stress Resultant - Displacement relations[edit]

Extensional/Bending Stiffness[edit]

If is constant, and -axis passes through centroid

Weak Forms[edit]

Axial Equation[edit]

where

Bending Equation[edit]

where

Finite Element Model[edit]

Finite element model for Euler Bernoulli beam

where .

Hermite Cubic Shape Functions[edit]

Hermite shape functions for beam

Finite Element Equations[edit]

where

Symmetric Stiffness Matrix[edit]

Load Vector[edit]

Newton-Raphson Solution[edit]

where

The residual is

For Newton iterations, we use the algorithm

where the tangent stiffness matrix is given by

Tangent Stiffness Matrix[edit]

Load Steps[edit]

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[edit]

Recall

where

Mebrane locking in Euler-Bernoulli beam

For Hinged-Hinged[edit]

Membrane strain:

or

Hence, shape functions should be such that

linear, cubic Element Locks! Too stiff.

Selective Reduced Integration[edit]

  • 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[edit]

Assume = constant.