Nonlinear finite elements/Buckling of beams

Standard Newton-Raphon methods perform poorly for bucking problems.

• Also called Modified Riks Method.
• Control the size of the load step using a parameter ${\displaystyle \lambda }$.
• Solve for both ${\displaystyle \lambda }$ and ${\displaystyle \Delta u}$ in each Newton iteration.

Assume ${\displaystyle F}$ = independent of geometry. Then

${\displaystyle F=\lambda ~{\bar {F}}}$

${\displaystyle \lambda }$ can be thought of as a normalized load parameter.

${\displaystyle {\text{Residual}}~=r(u,\lambda )=K(u)~u-\lambda ~{\bar {F}}}$

The load increment is computed using

${\displaystyle \lambda =\pm {\sqrt {\Delta s^{2}-\Delta u_{n}^{2}}}}$

The reference arc length

${\displaystyle \Delta s_{0}={\cfrac {F}{n_{\text{loadstep}}}}}$