Symmetry of tangent stiffness matrices of 3D elastic frames
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 tangent stiffness matrices derived by some researchers are tenuously attributed to the use of Lagrangian formulations, while the asymmetry of corotational tangent stiffness matrices is commonly attributed to the noncommutativity of spatial rotations. In this paper, the inconsistency regarding the symmetry 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 invariably asymmetric. A “correction matrix” that enforces the proper rotational behavior of nodal moments into the conventional geometric stiffness matrix of an Updated Lagrangian spatial beam element is presented. It is demonstrated through a numerical example that adoption of this correction matrix is necessary 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 analysis of a space frame is also explained.
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