Analytic Solution for Two Dimensional Beam Problems: Pure Stress Boundary Conditions

Abstract

© The Author(s), under exclusive licence to Springer Nature India Private Limited 2026.Closed-form analytical solutions for displacement, strain, and stress serve as essential benchmarks for validating numerical methods. This work constructs complete function sets that generate two classes of strong solutions to the governing partial differential equations of two-dimensional beams. The boundary conditions are of Neumann type, prescribing displacement derivatives (strains) or equivalently stresses along all surfaces. The proposed framework is demonstrated on orthotropic and isotropic beams, effectively reproducing continuous and discontinuous surface stresses. The results provide a versatile set of benchmark solutions for assessing beam mechanics models.

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