Stress recovery of laminated non‑prismatic beams under layerwise traction and body forces

Emerging manufacturing technologies, including 3D printing and additive layer manufacturing, ofer scope for making slender heterogeneous structures with complex geometry. Modern applications include tapered sandwich beams employed in the aeronautical industry, wind turbine blades and concrete beams used in construction. It is noteworthy that state-of-the-art closed form solutions for stresses are often excessively simple to be representative of real laminated tapered beams. For example, centroidal variation with respect to the neutral axis is neglected, and the transverse direct stress component is disregarded. Also, non-classical terms arise due to  interactions between stifness and external load distributions. Another drawback is that the external load is assumed to react uniformly through the cross-section in classical beam formulations, which is an inaccurate assumption for slender structures loaded on only a sub-section of the entire cross-section. To address these limitations, a simple and efcient yet accurate analytical stress recovery method is presented for laminated non-prismatic beams with arbitrary cross-sectional shapes under layerwise body forces and traction loads. Moreover, closed-form solutions are deduced for rectangular cross-sections. The proposed method invokes Cauchy stress equilibrium followed by implementing appropriate interfacial boundary conditions. The main novelties comprise the 2D transverse stress feld recovery considering centroidal variation with respect to the neutral axis, application of layerwise external loads, and consideration of efects where stifness and external load distributions difer. A state of plane stress under small linear-elastic strains is assumed, for cases where beam thickness taper is restricted to 15◦. The model is validated by comparison with fnite element analysis and relevant analytical formulations.