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

Abstract

a b s t r a c tAn 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 higherorder 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.

Force-based higher-order beam element with flexural-shear-torsional interaction in 3D frames. Part II: Applications

“…The formulation by Heyliger and Reddy [2] is a displacement-based approach-using Lagrange and Hermite interpolation functions-which stands out as perhaps the most fundamental difference in relation to the proposed theory. Other advantageous features of the latter, such as the definition of the generalised forces and relevant orthogonality properties, can be found in the companion paper [1]. The following comparisons with the present beam theory are obviously restricted to the scope of applicability of the above equation, which is planar loading; the full three-dimensional capabilities of the proposed force-based HOBT will be illustrated in Section 5 of this document.…”

“…They are intrinsically related to the boundary conditions (BCs) assumed, which for this model consist of a combination of imposed generalised nodal displacements and/or forces, in a total of 76 (38 at each extremity) as derived by Correia et al [1]. The special nature of such effects-when applied together with a flexibility formulation-naturally asks for a careful analysis in order to expose this yet undisclosed behaviour.…”

“…For the current loading case the behaviour of the element is governed by the differential equilibrium equations shown in system (3) of Appendix A of Correia et al [1]. Hence the generalised stress-resultants therein included are expected to play a role in the response.…”

“…The transfer of the advantages of force-based formulations to higher-order beam theories (HOBTs), which to the authors' knowledge was proposed for the first time in the companion paper [1], is herein investigated from the application viewpoint. The theoretical background previously introduced indicated that the novel element satisfies an advanced form of equilibrium, in-between that of the classical Timoshenko beam theory and the complete threedimensional equilibrium, which is herein confirmed with numerical examples.…”

“…Modes 1 through 8 are the first-order ones, being composed of the six modes corresponding to classical Timoshenko beam theory, followed by two related to warping and distortion. Out of the remaining 30 higher-order displacement modes, defined in the companion paper based on normalised Legendre polynomials [1], seven are also depicted: modes 9 and 11 are the first describing a nonlinear displacement variation throughout the cross-section, parabolic along the in-plane axes; modes 14 and 15 are exemplificative of the way in which significant physical phenomena associated with cross-sectional in-plane deformation (e.g., Poisson's effect or confinement in reinforced concrete beams) may be accounted for in the present formulation; modes 21 and 31 complement modes 1, 5 and 11 as cross-sectional representations of the initial five Legendre polynomials; finally, mode 36 is a sample of those modes featuring a product of polynomials of different degree along the two orthogonal cross-sectional coordinate axes.…”