Tri-Diagonal Matrix Method for Complex Structures

If the structure can be arranged in sections so that members from each section attach only in the section or to the two adjacent sections, then it is possible to generate the stiffness matrix directly in tri-diagonal submatrix form from the generalized force displacement equation for each member and the equilibrium equations for each joint of the structure. The solution for the generalized displacements is accomplished by using the submatrices of the stiffness matrix. The largest matrix that has to be inverted has a size equal to the number of generalized displacements in the largest section. The equations are set up for six generalized forces in each member, six generalized displacements at each joint, variable area and moment of inertia of members, variable temperature in members, large deflections, inelastic effects in axial load and bending, preloads or residual loads, local loads between joints, and various end conditions on the members. An example compares the forces and deflections in a simple two-dimensional redundant truss for pinned joints and for rigid joints.