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

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

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Nonlinear finite elements

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