Virtual Principle for Determination Initial Displacements of Reinforced Concrete and Prestressed Concrete (Overtop) Members

Theoretical approach with analytical and numerical procedure for determination initial displacement of a reinforced and prestressed concrete members, simple and cantilever beams, loaded by axial forces and bending moments is proposed. It is based on the principle of minimum potential energy with equality of internal and external forces. The equations for strain internal energy have been derived, including compressed and tensile concrete and reinforcement. The energy equations of the external forces with axial flexural displacement effects have been derived from the assumed sinusoidal curve. The trapezoid rule is applied to integrate the segment strain energy. The proposed method uses a non-linear stress-strain curve for the concrete and bilinear elastic-plastic relationship for reinforcement; equilibrium conditions at a sectional level to generate the strain energies along the beam. At the end of this article are shown three specific numerical examples with comparative, experimental (two tests) results with the excellent agreement and one calculation result with a great disagreement, by obtaining results of virtual principle method. With this method is avoiding the adoption of an unsure (EJ), as in the case of underestimating or overestimate initial flexural rigidity.