An analytical approach to determining crossing and veering phenomena

The aim of this paper is to give formal analytical proof of the crossing phenomena in the case of a planar frame structure that consists of two Euler-Bernoulli beams. The criteria for distinguishing veering from crossing are presented. It is supposed that frame elements are made of homogeneous materials and with circular cross section. An analytical approach to solving the problem is applied with respect to the Euler-Bernoulli beam theory, thus no discretization techniques were used. The variable system parameter is the diameter of the cross sections. For different values of the diameter eigenvalue loci may cross or abruptly diverge-veer apart. Based on the matrix transformation known from linear algebra the analytical solution for the presented problem is proposed and illustrated by an example.