Pile caps are critical structural members that transfer loads from columns and bridge piers to the pile foundation. They are large concrete blocks with significant disturbed (D-) regions, low shear span-to-depth ratio, low longitudinal reinforcement ratio, and typically no vertical ties which makes them vulnerable to shear failure. The current major concrete codes of practice suggest both the sectional approach, and the Strut-and-Tie Method (STM) for the design of pile caps. Neither of the approaches accurately capture the ultimate capacity, force-deformation behaviour, complex strain variation, or the final mode of failure of pile caps. Compatibility Strut-and-Tie Method (C-STM) is an efficient, minimalist, non-linear modelling approach that satisfies equilibrium, compatibility, and constitutive material relationships and effectively simulates the behaviour of shear-critical reinforced concrete members. A displacement based three-dimensional (3D) C-STM is proposed to simulate the behaviour of four-pile caps. Appropriate geometry and material constitutive relationships, including tension stiffening and softening effects, are incorporated into the model. The effects of concrete softening due to transverse tensile strains are also incorporated in the model. To establish the generality of the model, the load and deformation corresponding to cracking, yield, and failure of 33 four-pile cap specimens with varying shapes, sizes and material properties are considered. The model satisfactorily simulates the overall force-deformation behaviour, internal strain behaviour, and sequentially predicts the non-linear events that lead to the failure of the pile caps. The results from the 3D C-STM are in good agreement with the experimental results and observations of four-pile cap specimens. © 2019 Elsevier Ltd