Nonlinear finite elements/Timoshenko beams

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Timoshenko Beam[edit]

Timoshenko beam.

Displacements[edit]

Strains[edit]

Principle of Virtual Work[edit]

where

= shear correction factor

Taking Variations[edit]

Take variation

Take variation

Take variation

Internal Virtual Work[edit]

Integrate by Parts[edit]

Get rid of derivatives of the variations.

Collect terms[edit]

Euler-Lagrange Equations[edit]

Constitutive Relations[edit]

Then,

where

Equilibrium Equations[edit]

Weak Form[edit]

Finite element model[edit]

Trial Solution[edit]

Element Stiffness Matrix[edit]

Choice of Approximate Solutions[edit]

Choice 1[edit]

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

Nearly singular stiffness matrix ().

Choice 2[edit]

= 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 3[edit]

= 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 Locking[edit]

Example Case[edit]

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 1[edit]

Linear , Linear , Linear .

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

Example Case 2[edit]

Linear , Quadratic , Linear .

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

Example Case 3[edit]

Quadratic , Quadratic , Linear .

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

Example Case 4[edit]

Cubic , Quadratic , Linear .

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

Overcoming Shear Locking[edit]

Option 1[edit]

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

Option 2[edit]

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