AbstractThe direct integration method for structural systems having force and/or stiffness discontinuities is known to present considerable numerical difficulties. A non‐linear one‐degree‐of‐freedom test problem having these characteristics is developed and its exact oscillation period determined. The effects on amplitude and phase of small perturbations of the switching point, such as are caused by fixed‐step integration, have been studied. Artificial energy changes introduced at discontinuities by fixed‐step explicit and implicit methods are investigated. For a number of commonly used algorithms orders of convergence in fixed‐step integration of the test problem degrade from design value before, toO(h) after, the first discontinuity traversal.O(h) convergence is maintained for any number of discontinuity traversals thereafter, but design order can be recovered by locating the switching point sufficien