Symmetry of tangent stiffness matrices of 3D elastic frame

In the literature, the symmetry of the element tangent stiffness matrix of a spatial elastic beam has been a subject of debate. The symmetry of the ta ngent stiffness matrices derived by some researchers are tenuously attribut ed to the use of Lagrangian formulations, while the asymmetry of corotation al tangent stiffness matrices is commonly attributed to the noncommutativit y of spatial rotations. In this paper, the inconsistency regarding the symm etry of element tangent stiffness matrices formulated in the Lagrangian and the corotational frameworks is resolved. It is shown that, irrespective of the formulation framework, the element tangent stiffness matrix is invaria bly asymmetric. A "correction matrix" that enforces the proper rotational b ehavior of nodal moments into the conventional geometric stiffness matrix o f an Updated Lagrangian spatial beam element is presented. It is demonstrat ed through a numerical example that adoption of this correction matrix is n ecessary for the detection of the lowest buckling mode of a space dome. The fact that the structure tangent stiffness matrix becomes symmetric at the converged solution of each increment in the geometrically nonlinear analysi s of a space frame is also explained.