Buckling of columns with an intermediate elastic restraint

This paper presents a comprehensive set of exact stability criteria for Euler columns with an intermediate elastic restraint. A subset of this class of problem is the buckling problem of columns with an intermediate rigid support where the elastic restraint takes on an infinite stiffness. Also, this study reiterates the existence of a critical elastic restraint stiffness in which the buckled mode switches to a higher-buckling mode of the corresponding column without an intermediate support. It is clear that this critical stiffness value exists only when the restraint is placed at the node of the higher-buckling mode and the buckling load associated with this critical stiffness value is the maximum achievable value that can be attained with an intermediate elastic restraint