Closed-form analytical solution procedure for element design in D regions

This paper presents a novel procedure for solving the equations system of the rotating crack model used for reinforced concrete. It is implemented in the programme NonOPt where it is used to optimise the reinforcement design of D regions. The procedure is based on solving explicit closed-form relations without the need to incrementally increase the applied loads. The solution procedure is based on a secant modulus approach and is developed initially on the basis that the stress-strain response of the steel and concrete is linearly elastic. Subsequently the effect of material nonlinearities is included and the solution procedure is adapted accordingly. A reinforcement design procedure for membrane elements is described along with some case studies. The design procedure minimises the amount of reinforcement required to satisfy predefined design constraints. Material nonlinearities are taken into account, stress and strain compatibilities are satisfied and the design considers both the ultimate and serviceability limit states through the application of appropriate design constraints.