Finite element approach for reinforced concrete beam cracking

The study aimed to construct a finite element model of bending reinforced concrete beams taking into account crack formation. An incremental orthotropic model was used for modeling the elastic-plastic deformation of concrete. The finite element stiffness matrix was formulated based on a complementary energy functional. By employing Lagrange multipliers, the functional was augmented with nodal equilibrium equations derived via the principle of virtual displacements. Crack formation was incorporated by introducing additional unknown nodal displacements and using independent stress approximation functions within the element subdomains separated by the crack. The stiffness matrix of the rectangular finite element was derived by approximating the stress fields within the element domain using linear functions. To solve the non-linear problem, an incremental-iterative algorithm was proposed to track the progressive sequence of crack initiation within the finite elements. The Euclidean force norm was used to evaluate the solution residual. The number of required iterations is determined chiefly by the total number of initiated cracks. Once no new crack initiation occurs during a load step, the residual norm approaches zero within just two iterations. To validate the proposed model, a numerical analysis of a reinforced concrete beam subjected to a uniformly distributed load was performed. The numerical results derived via the proposed method were validated against the LIRA-SAPR simulation data.