Fixed-End Force Corrections for Beam Elements in Linear Static Analysis

In a structure containing beams, some of the beam members may be subjected to distributed loads. Consider the problem of a three-member beam structure subjected to a point load P and a distributed load Q.

In practical engineering analysis, it is of interest to obtain element forces and moments at the joints (nodes) and along the length of each individual member. If only the nodal deformations and the element stiffnesses are used to calculate the element nodal forces and moments, as is standard in the finite element procedure, the calculation yields only approximate results (as would be expected in a finite element solution). This can be unsatisfactory in linear static analysis and a "fixed-end force correction" is best applied to improve the results.

ADINA provides an option for this fixed-end force correction.

Shown below is the solution to the problem illustrated above. Three beam elements are used to model the frame. The shear force and moment diagrams are shown. The fixed-end force correction option is employed and seven section points are used per beam member to plot the respective quantities.

In the following figure, solutions to the problem are shown where the fixed-end force correction is not used. In the case of one element per member, the obtained shear force and moment diagrams differ markedly from those obtained previously. However, the use of additional elements gives improved results and convergence is of course obtained as the number of elements is increased.

We should note that this fixed-end force correction is calculated using linear static beam theory and hence is not applicable to general nonlinear and dynamic analyses.