Nonlinear finite elements/Euler Bernoulli beams

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Euler-Bernoulli Beam[edit | edit source]

Euler-Bernoulli beam

Displacements[edit | edit source]

Strains[edit | edit source]

Strain-Displacement Relations[edit | edit source]

The displacements

The derivatives

von Karman strains[edit | edit source]

The von Karman strains

Equilibrium Equations[edit | edit source]

Balance of forces[edit | edit source]

Stress Resultants[edit | edit source]

Constitutive Relations[edit | edit source]

Stress-Strain equation[edit | edit source]

Stress Resultant - Displacement relations[edit | edit source]

Extensional/Bending Stiffness[edit | edit source]

If is constant, and -axis passes through centroid

Weak Forms[edit | edit source]

Axial Equation[edit | edit source]


Bending Equation[edit | edit source]


Finite Element Model[edit | edit source]

Finite element model for Euler Bernoulli beam

where .

Hermite Cubic Shape Functions[edit | edit source]

Hermite shape functions for beam

Finite Element Equations[edit | edit source]


Symmetric Stiffness Matrix[edit | edit source]

Load Vector[edit | edit source]

Newton-Raphson Solution[edit | edit source]


The residual is

For Newton iterations, we use the algorithm

where the tangent stiffness matrix is given by

Tangent Stiffness Matrix[edit | edit source]

Load Steps[edit | edit source]


  • 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 | edit source]



Mebrane locking in Euler-Bernoulli beam

For Hinged-Hinged[edit | edit source]

Membrane strain:


Hence, shape functions should be such that

linear, cubic Element Locks! Too stiff.

Selective Reduced Integration[edit | edit source]

  • 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 | edit source]

Assume = constant.