Symmetry of Tangent Stiffness Matrices of 3D Elastic Frame

Spatial Rotation Kinematics and Flexural–Torsional Buckling

“…However, Teh & Clarke (1999b) have demonstrated through a heuristic example that in the direct stiffness method of matrix structural analysis, the asymmetric part of the correction stiffness matrix (16) vanishes at equilibrium when the element tangent stiffness matrix is assembled to the structure tangent stiffness matrix. This demonstration is in agreement with the more rigorous theoretical treatment by Simo & Vu-Quoc (1986) and Nour-Omid & Rankin (1991).…”

“…The symmetric part of the correction stiffness matrix (16) coincides with the additional stiffness matrix (14), and hence the semi-tangential adjustment matrix, for When the modified Newton-Raphson iteration method is used, in which case the structure tangent stiffness matrix is determined at the start of each new increment only following the convergence of the last increment, the correction stiffness matrix (16) reduces to the additional stiffness matrix (14). A symmetrised tangent stiffness matrix can therefore be employed in conjunction with the modified Newton-Raphson iteration method (Teh & Clarke 1999b), resulting in much efficiency. This is also true for linear buckling analysis as the computed element forces will always be in equilibrium with each other.…”

“…Under conservative loadings, a tangent stiffness matrix T k (which includes the geometric stiffness matrix) derived directly from an energy principle is invariably symmetric (Oran 1973, Teh & Clarke 1999b, since…”

Second order moments in torsion members

“…An elastic buckling solution for the concentrated end torque M o acting on the simply supported member shown in The solutions obtained using the finite element method of Teh and Clarke [13] for the flexural boundary conditions illustrated in Fig. 2 are given in Table 1 by the values of the buckling factor k in…”

Conceptual issues in geometrically nonlinear analysis of 3D framed structures

“…The geometrically nonlinear analysis of 3D framed structures has received considerable attention by numerous researchers [1][2][3][4][5][6][7][8][9], particularly focussing on the treatment of the difficulties associated with finite nodal rotations in 3D space. These difficulties arise mainly from the non-commutativity of finite rotations about fixed axes and the dual issue of nonconservative moments about fixed axes.…”