Nonlinear finite elements/Timoshenko beams

Timoshenko BeamEdit

 
Timoshenko beam.

DisplacementsEdit

 

StrainsEdit

 
 

Principle of Virtual WorkEdit

 

where

 
 

  = shear correction factor

Taking VariationsEdit

 

Take variation

 
 

Take variation

 
 

Take variation

 

Internal Virtual WorkEdit

 
 

Integrate by PartsEdit

Get rid of derivatives of the variations.

 

Collect termsEdit

 

Euler-Lagrange EquationsEdit

 
 

Constitutive RelationsEdit

 

Then,

 

where

 

Equilibrium EquationsEdit

 

Weak FormEdit

 

Finite element modelEdit

Trial SolutionEdit

 

Element Stiffness MatrixEdit

 

Choice of Approximate SolutionsEdit

Choice 1Edit

  = linear ( )
  = linear ( )
  = linear ( ).

Nearly singular stiffness matrix ( ).

Choice 2Edit

  = linear ( )
  = quadratic ( )
  = linear ( ).

The stiffness matrix is ( ). We can statically condense out the interior degree of freedom and get a ( ) matrix. The element behaves well.

Choice 3Edit

  = linear ( )
  = cubic ( )
  = quadratic ( )

The stiffness matrix is ( ). We can statically condense out the interior degrees of freedom and get a ( ) matrix. If the shear and bending stiffnesses are element-wise constant, this element gives exact results.

Shear LockingEdit

Example CaseEdit

Linear  , Linear  , Linear  .

 

But, for thin beams,

 

If constant  

 

Also

  1.   Non-zero transverse shear.
  2.   Zero bending energy.

Result: Zero displacements and rotations   Shear Locking!

Recall

 

or,

 

If   and   constant

 

If there is only bending but no stretching,

 

Hence,

 

Also recall:

 

or,

 

If   and   constant, and no membrane strains

 

Hence,

 

Shape functions need to satisfy:

 

Example Case 1Edit

Linear  , Linear  , Linear  .

  • First condition   constant   constant. Passes! No Membrane Locking.
  • Second condition   linear   constant. Fails! Shear Locking.

Example Case 2Edit

Linear  , Quadratic  , Linear  .

  • First condition   constant   quadratic. Fails! Membrane Locking.
  • Second condition   linear   linear. Passes! No Shear Locking.

Example Case 3Edit

Quadratic  , Quadratic  , Linear  .

  • First condition   linear   quadratic. Fails! Membrane Locking.
  • Second condition   linear   linear. Passes! No Shear Locking.

Example Case 4Edit

Cubic  , Quadratic  , Linear  .

  • First condition   quadratic   quadratic. Passes! No Membrane Locking.
  • Second condition   linear   linear. Passes! No Shear Locking.

Overcoming Shear LockingEdit

Option 1Edit

  • Linear  , linear  , linear  .
  • Equal interpolation for both   and  .
  • Reduced integration for terms containing   - treat as constant.

Option 2Edit

  • Cubic  , quadratic  , linear  .
  • Stiffness matrix is  .
  • Hard to implement.