Nonlinear finite elements/Euler Bernoulli beams

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




Bending Equation edit




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




Symmetric Stiffness Matrix edit


Load Vector edit


Newton-Raphson Solution edit




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


  • 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




Mebrane locking in Euler-Bernoulli beam

For Hinged-Hinged edit

Membrane strain:




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.