Dynamic Stiffness Matrix for a Beam Element with Shear Deformation

Abstract

A method for calculating the dynamic transfer and stiffness matrices for a straight Timoshenko shear beam is presented. The method is applicable to beams with arbitrarily shaped cross sections and places no restrictions on the orientation of the element coordinate system axes in the plane of the cross section. These new matrices are needed because, for a Timoshenko beam with an arbitrarily shaped cross section, deflections due to shear in the two perpendicular planes are coupled even when the coordinate axes are chosen to be parallel to the principal axes of inertia.

Local coordinate systems-based method to analyze high-order modes of n-step Timoshenko beam

“…The frequency equation derived from the reformatted governing differential equations is significantly simplified, such that it allows determination of high-order modes. The accuracy, capacity, and efficiency of the proposed methods are sufficiently corroborated by the EDS scheme, with the W-W algorithm capturing modal frequencies reliably (Williams and Wittrick, 1970; Wittrick and Williams, 1971; Williams and Wittrick, 1983; Williams and Wittrick, 1983; Williams and Kennedy, 2010; Howson and Williams, 1973; Pilkey and Kitis, 1995; Banerjee and Williams, 1996; Banerjee, 1997, 2001, 2003; Yuan et al., 2007; Li et al., 2008; Yu and Roesset, 2001; Greco and Pau, 2012).…”

“…The accuracy, capacity, and efficiency of the LCS method in acquiring high-order modes of a stepped Timoshenko beam are assessed using the well-known EDS scheme involving the W-W algorithm as a reference. As widely demonstrated (Howson and Williams, 1973; Pilkey and Kitis, 1995; Banerjee and Williams, 1996; Banerjee, 1997, 2001, 2003; Yuan et al., 2007; Li et al., 2008; Yu and Roesset, 2001; Greco and Pau, 2012), the EDS scheme is a valid method to achieve high-order modes of a stepped Timoshenko beam, with no need for solving the governing differential equations of the beam.…”

“…EDS scheme: the EDS method, underpinned by the sophisticated Wittrick-Williams (W-W) algorithm (Williams and Wittrick, 1970;Wittrick and Williams, 1971;Williams and Wittrick, 1983;Williams and Kennedy, 2010), is a robust scheme for obtaining solutions of high-order modes of discontinuous or stepped Timoshenko beams (Howson and Williams, 1973;Pilkey and Kitis, 1995;Banerjee and Williams, 1996;Banerjee, 1997Banerjee, , 2001Banerjee, , 2003Yuan et al, 2007;Li et al, 2008;Yu and Roesset, 2001;Greco and Pau, 2012) with the EDS method, the EDS matrix of a beam member is formulated using frequency-dependent exact dynamic shape functions derived from general wave solutions of the Timoshenko beam's equations for the beam member. As the exact dynamic shape functions can readily capture all necessary high-frequency wave modes of interest, extremely highly accurate solutions can be characterized, without the need for refined meshes for a beam member (Howson and Williams, 1973;Yuan et al, 2007).…”

“…The EDS matrices of beam members are assembled to form the global EDS matrix for the stepped beam. With the W-W algorithm, the global EDS matrix is processed by counting the number of modal frequencies exceeded by a given frequency, rather than directly calculating modal frequencies from the global EDS matrix, to give modal frequencies (Williams and Wittrick, 1970;Wittrick and Williams, 1971;Williams and Wittrick, 1983;Williams and Kennedy, 2010;Howson and Williams, 1973;Pilkey and Kitis, 1995;Banerjee and Williams, 1996;Banerjee, 1997Banerjee, , 2001Banerjee, , 2003Yuan et al, 2007;Li et al, 2008;Yu and Roesset, 2001;Greco and Pau, 2012). In principle, the EDS method involves some steps similar to those of the FE method: formation of an element stiffness matrix, assembly to form a global stiffness matrix, introduction of boundary conditions, evaluation of node displacement variables, etc.…”