Analytical displacement solutions for statically determinate beams based on a trilinear moment–curvature model

A standard approach is presented to obtain analytical solutions for deflection field of determinate beams subjected to conventional loading patterns. The solutions are based on a trilinear moment–curvature response using a deflection hardening behavior characterized by flexural crack initiation, inelastic response due to crack extension, and full plastic hinge formation. Methodology for full span deflection and rotation distributions are presented for multiple cases that include three- and four-point bending, uniform load, concentrated moment, and cantilever beams. The proposed approach provides analytical expressions for the curvature, rotation, and deflection at any point along the beam, and correlated to stress or strain distribution. The procedure can therefore be integrated into a serviceability-based design approach. A parametric study of the effects of model parameters on the stages of the response is addressed. Several case studies involving steel fiber reinforced concrete (SFRC), textile reinforced concrete (TRC) and ultra-high performance concrete (UHPC) are conducted and the simulated load-deflection responses are verified against the experimental data from several published experiments. Size effect on the serviceability limits of beams with spans ranging from 0.22 to 8.6 m is studied by tracking the full-range moment–curvature and load-deflection responses.