Complete Stiffness Matrices for Buckling Analysis of Frames

The present paper attempts to explain the significance of the various terms that must be included in an element stiffness matrix of a beam finite element to yield correct results in buckling analyses of spatial frames. The complete stiffness matrices for the buckling and post‐buckling analysis of three‐dimensional elastic framed structures are derived for the $C^0‐\mathrm{two}‐\mathrm{noded}$ straight prismatic beam element with doubly symmetric cross section. The assumed small‐strain hypothesis permitted closed‐form expressions to be arrived at. Derivations of the elastic and geometric matrices of a generic beam element are first reviewed. All stress resultants are included in the geometric stiffness matrix. The contribution of the conservative external surface loads having moment resultants to the stiffness matrix is then formulated. The so‐called energy method of stiffness derivation used commonly by engineers is argued to yield correct results when relevant load‐stiffness terms are included. The resulting expressions are easier to understand than their counterparts known from the papers advocating semitangential rotations and moments. Finally, the actual way of application of the external moments is included in the system stiffness matrix and it is shown that the tangential operator of the assembled system is symmetric. The present element stiffness matrices can be easily incorporated into finite‐element codes by only correcting those already implemented.