Beam Displacements and Rotations

Displacements reported are element displacements; element end values may differ from nodal displacements when the element has end releases, is offset, or shear deformation is included in the analysis due to its section property having a shear area. Such differences are reported in the Beam End Relative Displacement output, contours, and diagrams if the element has an end release.

Element rotations are calculated by interpolation from the element end rotations. The end rotations for an element are those of the node unless the element end has degrees of freedom that are released. In these situations the rotation at the end of the element has to be calculated.

The basic equation relating force and displacement for the element if

$$\left\{ f \right\} = \left[ K \right] \left\{ u \right\}$$

This can be partitioned depending on whether a degree of freedom is released on not.

$$\left\{\begin{array}{l} f_{r} \\ f_{f} \end{array}\right\}=\left[\begin{array}{ll} K_{rr} & K_{r f} \\ K_{f r} & K_{f f} \end{array}\right]\left\{\begin{array}{l} u_{r} \\ u_{f} \end{array}\right\}$$

where the subscripts r and f refer to released and fixed respectively. Knowing the forces that result from the loads on the element and the displacements at the fixed degrees of freedom we can extract the following equation for the displacements at the released degrees of freedom.

$$\left\{u_{r}\right\}=[K]^{-1}\left(\left\{f_{r}\right\}-\left[K_{r f}\right]\left\{u_{f}\right\}\right)$$