Displacement‐based finite difference approximations of derivatives of the tangent stiffness matrix with respect to the load parameter

The vehicle to investigate to which extent energy‐based categorization of buckling can be linked up with spherical geometry is the so‐called consistently linearized eigenproblem. This investigation requires computation of the first and the second derivative of the tangent stiffness matrix $\tilde{\bf{K}}_T$ with respect to a dimensionless load parameter λ in the frame of the Finite Element Method (FEM). A finite‐difference approximation of the first derivative of $\tilde{\bf{K}}_T$ , redefined as a directional derivative, has proved to meet the requirements of computational efficiency and sufficient accuracy. It represents a displacement‐based finite‐difference approximation, abbreviated as DBFDA. The present work is devoted to the computation of a DBFDA of the second derivative of ˜ KT with respect to λ. For the special case of a two‐dimensional co‐rotational beam element, an analytical solution of this derivative is presented. A circular arch, subjected to a vertical point load on its apex, serves as an example for numerically assessing the usefulness of the computed DBFDAs of the first and the second derivative of $\tilde{\bf{K}}_T$ with respect to λ. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)