Buckling Modes at Coincident Singularities of Stiffness Matrix

When the exact differential equation is used to obtain the stiffness coefficients for an axially loaded beam element, the entries of the stiffness matrix of a frame with such elements are complex functions of the load parameter. A simple sign-counting procedure can be employed to find critical loads for a given frame and loading. The load parameter becomes critical when some of the matrix eigenvalues pass through zero and become negative. When the axial force in an element approaches the critical load for the element as a clamped beam, one of the negative eigenvalues approaches minus infinity and, after this load is exceeded, returns from plus infinity. This type of singularity is called a pole. It is possible that a critical point and a pole of some multiplicities occur at the same value of the load parameter. Two situations, different from a modal point of view, are discussed with most attention drawn to the case of buckling with a zero displacement vector. The mechanism of forming such buckling modes is considered.