Matrix Solution of Force Relaxation Method

An analysis of space trusses is presented based on the bar-force relaxation method. Bar-force distribution matrixes are formulated from bar and joint stiffnesses. Bar forces are calculated from the joint forces and distribution matrixes. Over-distribution matrixes are developed for single bars and triangular truss panels. These balance the residuals between joints or in panels after one relaxation cycle. The structure is considered as a group of triangular panels. The residual forces in the panels are liquidated in a single cycle, and therefore, cyclic relaxation operations are required only between panels. Because of the small ratio of joint displacements to bar lengths, the first relaxation cycle between panels nearly balances all panels to equilibrium, and convergence is rapid. All matrixes are of second order for two-dimensional trusses and third order for three-dimensional problems. The only mathematical operations involved are matrix addition, multiplication, and inversion.