Stiffness Matrix for Elastic‐Softening Beams

Abstract

Softening, the decrease of bending moment capacity of a section at advanced curvature, may occur at locations within a statically indeterminate reinforced concrete structure prior to the attainment of maximum load. The stiffness matrix for an elastic flexural member with a softening portion is derived, employing the necessary assumption of a finite softening or discontinuity length. For positive values of the softening parameter the stiffness values are as for a beam with a prismatic haunch. For negative values of the parameter stiffness coefficients become unstable at the critical softening parameter for a fixed-end beam. The derived stiffness coefficients allow use of standard frame analysis programs without special treatment for softening. An example of softening at the base of one column of a portal frame is given, showing the effect of softening on sway displacement. The frame demonstrates instability at the critical softening parameter for the particular position and length of the softening hinge.