Structural analysis of non-prismatic beams: Critical issues, accurate stress recovery, and analytical definition of the Finite Element (FE) stiffness matrix

Non-prismatic beams are widely employed in strategic structures like bridges and sport arenas, requiring accurate analyses for a reliable and effective design. Unfortunately, features of non-prismatic beams lead their modeling to be a non-trivial task: (i) variations of both cross-section area and second moment of area impede an easy computation of analytical solutions compelling to use approximated methods; (ii) stress distributions in prismatic and non-prismatic beams are substantially different, as proved by analytical results available since the beginning of the past century; and (iii) the peculiar stress distribution in non-prismatic beams entails complicated constitutive relations, as highlighted in recent publications. Usually, commercial software does not properly account for all the features of non-prismatic beams, leading to inconsistent structural analyses, erroneous estimations of the stress distribution, and -consequently- coarse predictions of the structural element strength. The present paper proposes a strategy to effectively overcome the above-mentioned problems. We derive an accurate analytical model for 2D non-prismatic beams, able to handle the non-trivial stress distribution and the complicated constitutive relations. Thereafter, we compute both homogeneous and particular solutions using the symbolic calculus software MAPLE and we analytically define the Finite Element (FE) stiffness matrix for a planar, symmetric, linearly-tapered beam. Finally, we compare the proposed FE and SAP2000 solutions, considering several beams with different geometries, loads, and constraints. Numerical results highlight the reliability of the proposed modeling strategy, since the resulting FE consistently handles all the critical issues of non-prismatic beams with an extremely low computational cost. Conversely, SAP2000 solution remarks the need of ad hoc analysis tools and modeling strategies to be used for the design of non-prismatic structural elements.