Nonlinear finite elements/Buckling of beams

Newton-Raphson
Standard Newton-Raphon methods perform poorly for bucking problems.

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} $$

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}}} $$