This work establishes a solution to the problem of assessing the capacity of multi-object assemblies to withstand external forces without becoming unstable. Our physically-grounded approach handles arbitrary structures made from rigid objects of any shape and mass distribution without relying on heuristics or approximations. The result is a method that provides a foundation for autonomous robot decision-making when interacting with objects in frictional contact. Our strategy relies on a contact interface graph representation to reason about instabilities and makes use of object shape information to decouple sub-problems and improve efficiency. Our algorithm can be used by motion planners to produce safe assembly transportation plans, and by object placement planners to select better poses. Compared to prior work, our approach is more generally applicable than commonly used heuristics and more efficient than dynamics simulations.