Influence of constitutive relation on stress eigenvectors of homogeneous deep beam subjected to point loads

This paper aims to conduct a numerical study on a two-dimensional homogenous deep beam subjected to equally spaced four-point bending. The existing works study the behaviour of deep beams for linearised elasticity-based concrete models wherein Cauchy stress and linearised strain are explicitly related. The strut angle in the strut-and-tie method is used to analyse and design deep beams, ideally representing the stress eigenvectors. Further, the literature suggests that the constitutive relation used for modeling plain concrete has no bearing on strut angle. Recently, a non-linear, non-dissipative constitutive model has been proposed for plain concrete having linearised strain as a non-invertible function of Cauchy stress. Additionally, the analytical solution of the deep beam boundary value problem for the non-linear elastic plain concrete model is infeasible. Therefore, an existing numerical scheme with multifield formulation and appropriate element-free basis functions is employed to study deep beams. The existing multifield formulation assumes divergence-free Airy’s stress potential function and displacement field as primary unknowns. The Airy’s function is written as a linear combination of Hermite-type moving least square approximants. The displacement field is assumed using a linear combination of Lagrange Moving least squares approximants. The Cauchy stress is obtained from Airy’s function, and linearised strain is computed from the assumed displacement field satisfying strain-displacement relationship point wise. The constitutive relation between stress and linearised strain is satisfied in a weak integral sense such that the assumed stress and displacement field satisfy traction and displacement boundary conditions, respectively. The unknown coefficients in the assumed stress function and displacement field are obtained by casting the boundary value problem into a minimisation problem with the Frobenius norm of the constitutive relation as the objective function, and the traction and displacement boundary conditions are enforced as constraints. Finally, the numerically computed stress eigenvectors are obtained and compared for linear elastic Hooke’s model and non-linear elastic constitutive model obtained from an implicit constitutive theory framework.