Strain-Displacement Relations[edit | edit source]
The von Karman strains
Stress Resultant - Displacement relations[edit | edit source]
Extensional/Bending Stiffness[edit | edit source]
If is constant, and -axis passes through centroid
Finite element model for Euler Bernoulli beam
Hermite Cubic Shape Functions[edit | edit source]
Hermite shape functions for beam
Symmetric Stiffness Matrix[edit | edit source]
The residual is
For Newton iterations, we use the algorithm
where the tangent stiffness matrix is given by
- 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.
Mebrane locking in Euler-Bernoulli beam
Hence, shape functions should be such that
linear, cubic Element Locks!
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
Assume = constant.