Nonlinear finite elements/Buckling of beams

Nonlinear Post-Buckling

Buckling of beams.

Newton-Raphson

Standard Newton-Raphon methods perform poorly for bucking problems.

Predicting buckling with Newton_Raphson.

Arc Length Method

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

Assume $F$ = independent of geometry. Then

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

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

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

Arc-length method

The load increment is computed using

$\lambda = \pm\sqrt{\Delta s^2 - \Delta u_n^2}$

The reference arc length

$\Delta s_0 = \cfrac{F}{n_{\text{loadstep}}}$