Consistent derivation of a beam model from the Saint Venant’s solid model

• Published in: International journal of solids and structures Vol. 159; pp. 90 - 110
• Main Authors: Paradiso, Massimo, Marmo, Francesco, Rosati, Luciano
• Format: Journal Article
• Language: English
• Published: New York Elsevier Ltd 01.03.2019; Elsevier BV
• Subjects: Beam theory; Cantilever beams; Center of gravity; Center of shear; Center of twist; Cross-sections; Derivation; Displacement; Equivalence; Finite element method; Saint Venant; Shear flexibility; Shear forces; Three dimensional models; Center of twist; Shear flexibility; Saint Venant; Beam theory; Center of shear
• ISSN: 0020-7683; 1879-2146
• DOI: 10.1016/j.ijsolstr.2018.09.021

We illustrate the consistent derivation of a one-dimensional (1D) beam model from the Saint Venant three-dimensional (3D) solution of a beam-like solid that aims at the formulation of a 1D finite element able to reproduce the 3D elastic solution. The derivation is achieved by enforcing the equivalence between the 3D solid model and the 1D beam both in terms of elastic energy and displacements of the beam axis, i.e. the axis joining the center of gravity of the cross sections of the solid model. The result is obtained by proving the equivalence of two flexibility operators, one providing the elastic energy of the solid and the other yielding the kinematic parameters of the free end section of the beam-like solid assumed to be a cantilever. To this end an intrinsic expression of the displacement field of the solid due to a shear force acting at an arbitrary point of the end section is derived by expressing the constraint condition at the root section as recently proposed in the literature. The arbitrariness of the point of application of the shear force naturally qualifies the center of twist as the only point that is needed to quantify the amount of elastic energy and displacements associated with tangential stresses. In other words the center of twist represents the only point of the cross section that is actually required both in the derivation of the beam model and in the derivation of the relevant finite element.