Reinforced Concrete Frame Element with Bond Interfaces. I: Displacement-Based, Force-Based, and Mixed Formulations

This is the first of two papers that presents the theory and applications of three different formulations of reinforced concrete frame elements with bond slip in the reinforcing bars. This paper presents the governing differential equations of the problem (strong form) and the three different element formulations (weak forms). The first is the displacement-based formulation, which is derived from the principle of stationary potential energy. The second is the force-based formulation, which is derived from the principle of stationary complementary energy. The third is the two-field mixed formulation, which is derived from the principle of stationary Hellinger-Reissner potential. The selection of the displacement and force shape functions for the different formulations is discussed. Tonti’s diagrams are used to conveniently represent the equations that govern both the strong and the weak forms of the problem. This paper derives the general matrix equations of the three formulations. Implementation of the formulations in a general-purpose nonlinear structural analysis software and a set of applications are discussed in the companion paper.

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