Analytical plane-stress recovery of non-prismatic beams under partial cross-sectional loads and surface forces

High levels of strength- and stiffness-to-mass ratio can be achieved in slender structures by lengthwise tailoring of their cross-sectional areas. During use, a non-prismatic beam element can be subject to surface forces or loads acting on only a part of their cross-section. Practical examples involve tapered aircraft wings, wind turbine and helicopter rotor blades under fluid pressure and shear forces; arched beams in bridges subject to vehicular traction forces and tensile stresses in tendons of prestressed concrete. Presently, beam theories generalise the external loads to the entire cross-sectional area. However, this technique does not accurately describe surface-load boundary conditions and beams under partial cross-sectional loads. Hence, an efficient analytical plane-stress recovery methodology is introduced in the present study that generalises the external load to a specific sub-area of the cross-section of homogeneous non-prismatic beams with one plane of symmetry. As a result, the transverse stress components become piecewise distributions, i.e. non-smooth but continuous in the thickness direction. Additional novelties include the boundary equilibrium recovered considering applied surface loads and terms up to second-order derivatives of the internal forces to define the transverse stress field. Closed-form solutions for the specific case of non-prismatic beams with a rectangular cross-section loaded both on top and bottom surfaces are presented. For validation purposes, different numerical examples are modelled with results compared to solid-like finite element analyses as well with relevant analytical theories. The results show that the developed formulation predicts the stress field in non-prismatic beams under surface forces and non-uniform loads applied to a part of the cross-sectional area with goods levels of accuracy. The error associated with the proposed method escalates with the taper angle, such that a 10◦ taper angle could result in a 6% error at the surfaces and reduced values for interior zones, while the analytical state-of-the-art models were not able to predict the transverse stresses correctly.