Nonlinear finite elements/Euler Bernoulli beams

Euler-Bernoulli BeamEdit

 
Euler-Bernoulli beam

DisplacementsEdit

 

StrainsEdit

 
 

Strain-Displacement RelationsEdit

 
 

The displacements

 

The derivatives

 

von Karman strainsEdit

The von Karman strains

 

Equilibrium EquationsEdit

Balance of forcesEdit

 

Stress ResultantsEdit

 

Constitutive RelationsEdit

Stress-Strain equationEdit

 

Stress Resultant - Displacement relationsEdit

 

Extensional/Bending StiffnessEdit

 

If   is constant, and  -axis passes through centroid

 

Weak FormsEdit

Axial EquationEdit

 

where

 

Bending EquationEdit

 

where

 

Finite Element ModelEdit

 
Finite element model for Euler Bernoulli beam
 

where  .

Hermite Cubic Shape FunctionsEdit

 
Hermite shape functions for beam

Finite Element EquationsEdit

 

where

 
 

Symmetric Stiffness MatrixEdit

 

Load VectorEdit

 

Newton-Raphson SolutionEdit

 

where

 

The residual is

 

For Newton iterations, we use the algorithm

 

where the tangent stiffness matrix is given by

 

Tangent Stiffness MatrixEdit

 

Load StepsEdit

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 LockingEdit

Recall

 

where

 
 
Mebrane locking in Euler-Bernoulli beam

For Hinged-HingedEdit

Membrane strain:

 

or

 

Hence, shape functions should be such that

 

  linear,   cubic   Element Locks! Too stiff.

Selective Reduced IntegrationEdit

  • 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 integrationEdit

 

Assume   = constant.