Eigenvalues for Beam‐Columns on Elastic Supports

The eigenvalues associated with beam‐columns, which rest on elastic supports, are determined by an initial‐value numerical method. These eigenvalues can be related to the buckling loads, critical foundation constants, or natural frequencies of vibration. Interrelationships between these eigenvalues are examined and three‐way dimensionless critical diagrams display the results of this study for single‐span structures with various boundary conditions and for multi‐span structures. These show that, for the cases presented, the square of the natural frequency varies linearly with respect to both the axial load and the elastic support constant. For all cases studied, there are different, but similar, interactive relationships for each mode shape. These critical diagrams enable the analyst to determine the axial load‐support constant‐natural frequency combination for a critical condition to exist—that is, an eigenvalue solution with a unique mode shape. Several example problems are included to illustrate the use of the dimensionless critical diagrams.