Nonlinear finite elements/Buckling of beams

Nonlinear post-buckling

Buckling of beams.

Newton-Raphson

Standard Newton-Raphon methods perform poorly for bucking problems.

File:BucklingNR.png 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 $\mathrm{\Delta }u$ in each Newton iteration.

Assume $F$ = independent of geometry. Then

$F=\lambda \overline{F}$

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

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

File:BucklingArc.png Arc-length method

The load increment is computed using

$\lambda =\pm \sqrt{\mathrm{\Delta }{s}^2-\mathrm{\Delta }{u}_n^2}$

The reference arc length

$\mathrm{\Delta }{s}_0=\frac{F}{n_{\text{loadstep}}}$

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