Abstract A line-segment approximation for a nonlinear elastic force has been developed which makes it possible to obtain analytical solutions to a number of transient-load problems which formerly could be solved only by graphical or numerical methods. It is shown that for many problems involving even large nonlinearities, two line segments only are sufficient to give satisfactory accuracy. For some problems where the classical analytical approximate methods of Lindstedt or of Kryloff and Bogoliuboff can be applied, it is shown that the bilinear method leads to an improved accuracy.