A new interpretation of Saint-Venant compatibility conditions

Traditional approaches, which are based on displacements, often introduce errors in stress computations due to indirect treatment of boundary conditions and stress recovery processes. This becomes a significant limitation when analyzing anisotropic structures where precise evaluation of stress fields is critical. To address this issue, the authors propose a new system of Beltrami-Mitchell-type equations for orthotropic bodies formulated directly in terms of stresses. These equations are derived from the Saint-Venant strain compatibility conditions and the generalized Hooke's law for orthotropic bodies, and in contrast to the well-known Beltrami-Michell equation, without using equilibrium equations. The resulting formulation is mathematically consistent and well-suited for numerical implementation. Finite-difference schemes are developed, and several benchmark problems are solved, including orthotropic rectangular plates under parabolic and uniform edge loads, and a cantilever beam. Iterative numerical procedures are employed to study the stress distribution. The results demonstrate high accuracy, improved satisfaction of physical conditions, and close agreement with classical solutions. This is explained by the formulation boundary value problems of elasticity theory directly for stresses in contract to displacement-based formulation. This approach can be generalized to solve thermal and plastic boundary value problems for isotropic and anisotropic bodies. The proposed approach is applicable to practical simulations in structural analysis of composite and orthotropic materials in aerospace, mechanical, and civil engineering