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Computing the collapse load and identifying the associated mechanism of block assemblage structures is an important task in historical restoration works. The research in this thesis is to contribute to the modelling and analysis of rigid block structures in dry frictional contact. The models can be applied to various structures such as masonry arches) vaults) domes, walls or piles in heritage buildings. Estimates of the collapse load are made by solving the underlying mathematical programming problems. 2-D and 3-D models are formulated and different contact assumptions (concavity and convexity) are investigated. Both associated and nonassociated flow rules are accounted for. The associative problem can be robustly solved by the bound theorems of classical plasticity theory. However, the result is often not valid. The nonassociated rule which shows up as a complementarity relationship in the governing system is the major challenge of the project. The 3-D model is more difficult to deal with as compared to the 2-D case. Proposed methods are presented to treat the nonlinear 3-D problem (Lorentz cone) by formulating it as a second-order cone problem or by piecewise linearizing the cone as a polyhedral. Various computational methods are proposed to obtain the best upper bound collapse load for the nonassociative model. One method formulates some extended complementarity problems that are able to restrict the domain of the collapse load variable to search for better solutions. The best method uses nonlinear programming) specifically mathematical programming with equilibrium constraints problems) to attempt to directly minimize the collapse load. Some enumerative schemes are also attempted to map out all the nonassociative solutions but proved to be computationally expensive. Various 2-D and 3-D examples are demonstrated for several different types of structures.