Matrix stiffness method

The matrix stiffness method is how almost all structural analysis software works. It treats a structure as an assembly of members (elements) connected at joints (nodes). The unknowns are the displacements and rotations at the nodes. For each element, a stiffness matrix relates the forces at its ends to the displacements at its ends. These are transformed from local member axes to common global axes and added into one structure stiffness matrix K. Then K d = F, where F holds the loads, is solved for the displacements d, and member forces follow.

A two-dimensional truss member has two degrees of freedom per node, giving a 4 × 4 matrix based on A E / L. A plane frame member has three per node (two translations and one rotation), giving a 6 × 6 matrix with axial terms A E / L and bending terms 12 E I / L³, 6 E I / L² and 4 E I / L or 2 E I / L, the same coefficients seen in the slope-deflection method. Supports are applied by removing restrained degrees of freedom from the system. Loads between nodes are converted to equivalent nodal loads using fixed-end forces.

The method is systematic, which makes it perfect for computers, but it can also produce nonsense quickly if the model is wrong. Engineers check that reactions balance the loads, that deflected shapes look sensible and that a few key results agree with hand methods. The same framework extends to finite elements, second-order effects and dynamics.