Finite element analysis (FEA) is used by industrial designers and biomechanicists to estimate the performance of engineered structures or human skeletal and soft tissues subjected to varying regimes of stress and strain. FEA is rarely applied to problems of biomechanical design in animals, despite its potential to inform structure-function analysis. Non-invasive techniques such as computed tomography scans can be used to generate accurate three-dimensional images of structures, such as skulls, which can form the basis of an accurate finite element model. Here we have applied this technique to the long skull of the large carnivorous theropod dinosaur Allosaurus fragilis. We have generated the most geometrically complete and complex FEA model of the skull of any extinct or extant organism and used this to test its mechanical properties and examine, in a quantitative way, long-held hypotheses concerning overall shape and function. The combination of a weak muscle-driven bite force, a very `light' and `open' skull architecture and unusually high cranial strength, suggests a very specific feeding behaviour for this animal. These results demonstrate simply the inherent potential of FEA for testing mechanical behaviour in fossils in ways that, until now, have been impossible.