Force-based higher-order beam element with flexural–shear–torsional interaction in 3D frames. Part I: Theory

An innovative higher-order beam theory, capable of accurately taking into account flexural–shear–torsional interaction, is originally combined with a force-based formulation to derive the corresponding finite element. The selected set of higher-order deformation modes leads to an explicit and direct interaction between three-dimensional shear and normal stresses. Namely, cross-sectional displacement and strain fields are composed of independent and orthogonal modes, which results in unambiguously defined generalised cross-sectional stress-resultants and in a minimisation of the coupling of equilibrium equations. On the basis of work-equivalency to three-dimensional continuum theory, dual one-dimensional higher-order equilibrium and compatibility equations are derived. The former, which govern an advanced form of beam equilibrium, are strictly satisfied via stress fields arising from the solution of the corresponding systems of coupled differential equations. The formulation, which is numerically validated in a companion paper for both linear and nonlinear material response, inherently avoids shear-locking and accurately accounts for span loads. Finally, the superiority of force-based approaches over displacement-based ones, well established for inelastic behaviour, is also demonstrated for the linear elastic case.