Exact Solutions for Torsion and Warping of Axial-Loaded Beam-Columns Based on Matrix Stiffness Method

Abstract

The typically-used element torsional stiffness GJ/L (where G is the shear modulus, J the St. Venant torsion constant, and L the element length) may severely underestimate the torsional stiffness of thin-walled nanostructural members, due to neglecting element warping deformations. In order to investigate the exact element torsional stiffness considering warping deformations, this paper presents a matrix stiffness method for the torsion and warping analysis of beam-columns. The equilibrium analysis of an axial-loaded torsion member is conducted, and the torsion-warping problem is solved based on a general solution of the established governing differential equation for the angle of twist. A dimensionless factor is defined to consider the effect of axial force and St. Venant torsion. The exact element stiffness matrix governing the relationship between the element-end torsion/warping deformations (angle and rate of twist) and the corresponding stress resultants (torque and bimoment) is derived based on a matrix formulation. Based on the matrix stiffness method, the exact element torsional stiffness considering the interaction of torsion and warping is derived for three typical element-end warping conditions. Then, the exact element second-order stiffness matrix of three-dimensional beam-columns is further assembled. Some classical torsion-warping problems are analyzed to demonstrate the established matrix stiffness method.

Cross-sectional Analysis of Beams Subjected to Saint-Venant Torsion Using the Green’s Theorem and the Finite Difference Method

“…This means that the stress function (x,y,z) is constant along the boundary of the cross-section. Observing that sin = -dy/ds and cos = dz/ds, the shearing stress component t at the boundary directed along the tangent is Likewise, the shearing stress component n at the boundary directed along the outer normal to the boundary is Observing Equation (8), the shearing stress component n is zero; this is in agreement with the condition of stress free outer surfaces. Therefore, the resultant shearing stress at the boundary is t.…”

“…Pluzsik et al [7] presented a theory for thin-walled, closed section, orthotropic beams which takes into account the shear deformation in restrained warping induced torque; the analytical ("exact") solution of simply supported beams subjected to a sinusoidal load was developed for this purpose. Pan et al [8] presented a matrix stiffness method for the torsion and warping analysis of beam-columns in order to investigate the exact element torsional stiffness considering warping deformations; the equilibrium analysis of an axial-loaded torsion member was conducted, and the torsion-warping problem was solved based on a general solution of the established governing differential equation for the angle of twist. Pavazza et al [9] presented a novel theory of torsion of thin walled beams ("shear deformable beams") of arbitrary open cross-sections with influence of shear; the theory is based on the classical Vlasov's theory of thin-walled beams of open cross-section, as well on the Timoshenko's beam bending theory.…”

Cross-sectional Analysis of Beams Subjected to Saint-Venant Torsion Using the Green’s Theorem and the Finite Difference Method

“…Pan et al [7] presented a matrix stiffness method for the torsion and warping analysis of beam-columns in order to investigate the exact element torsional stiffness considering warping deformations; the equilibrium analysis of an axialloaded torsion member was conducted, and the torsion-warping problem was solved based on a general solution of the established governing differential equation for the angle of twist. Pavazza et al [8] presented a novel theory of torsion of thin walled beams ("shear deformable beams") of arbitrary open cross-sections with influence of shear; the theory is based on the classical Vlasov's theory of thin-walled beams of open cross-section, as well on the Timoshenko's beam bending theory.…”

“…In the case of cross-sections without openings and recalling that the stress function has a constant value  B at the boundary, the Green's theorem applied to the second term at the right-hand side of Equation ( 12c) is given by where A B is the area enclosed by the outer boundary of the cross-section. Let us express the stress function as follows (12e) with the function  * being zero along the boundary and satisfying Equation (7). Substituting Equations (12d) and ( 12e)…”

Nanomechanics and Plasticity

“…Considering the coupled loading scenarios, Lu et al [ 20 ] investigated the deformation mechanisms of nanowire under coupled tension-torsion loads. Pan et al [ 21 ] derived the exact solutions for torsion and warping of axial-loaded beam-columns in order to avoid the underestimation of the torsional stiffness of thin-walled nanostructures. With the advancement of fabrication technology, more complex nanostructures have been reported, such as heterostructures, Janus structures, helical/spiral structures [ 22 ], nanoscrolls [ 23 ], and three-dimensional networks [ 24 ].…”