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Updated Element Axes

In a geometrically nonlinear analysis the axes of the element must deform with the element. This has to resolve the difference between the original (undeformed) configuration and the current (deformed) configuration.

For 1D elements the deformed direction cosines can be represented by a new x vector based on the deformed positions of the ends of the element and the average rotation of the element about its x-axis.

For 2D elements the undeformed configuration can be represented by direction cosines based on the (r0,s0,t0)\left( r_{0},s_{0},t_{0} \right) axes.

D0=[r0s0t0]D_{0} = \left\lbrack r_{0}|s_{0}|t_{0} \right\rbrack

The deformed configuration at iican be represented by direction cosines based on the deformed (ri,si,ti)\left( r_{i},s_{i},t_{i} \right) axes.

Di=[risiti]D_{i} = \left\lbrack r_{i}|s_{i}|t_{i} \right\rbrack

The base direction cosines De0D_{e0} can then be updated using

Dei=De0D0TDiD_{ei} = D_{e0}{D_{0}}^{T}D_{i}