This thesis develops a Joint Shell Punching Model (JSPME) for modelling punching failure at edge and corner columns of flat slabs without and with shear reinforcement. Significantly, the JSPME requires significantly less computational time than nonlinear analysis with 3-D solid elements. This makes the JSPME suitable for modelling full floor plates of multi-storey buildings. In the JSPME, 3-D joint elements are combined with 2-D shell elements to simulate the separation of the slab from the column at punching failure. The punching resistance of the joint elements is calculated in terms of the relative slab-column rotation using the so-called Critical Shear Crack theory (CSCT) of Muttoni (2008). The developed JSPME is shown to accurately predict punching resistance of laboratory tested edge and corner column specimens as well as virtual specimens analysed with NLFEA using 3-D solid elements. This thesis examines the influence of column cross-section aspect ratio, and biaxial loading eccentricity, on punching resistance at edge columns seemingly for the first time. Due to lack of experimental data, it was found necessary to calibrate the JSPME for rectangular columns and biaxial loading against the predictions of NLFEA with 3-D solid elements. A refined shear field method is developed for accounting for loading eccentricity in the CSCT. The thesis also develops load enhancement expressions to account for loading eccentricity in the closed form version of the CSCT which is adopted in the next generation of Eurocode 2. The final part of this thesis extends the JSPMES to model full-size flat-slab specimens. The punching resistances of edge and corner column connections in full floor plates are shown to be typically greater or equal to the strength of comparable isolated sub-assemblies thereby justifying the development of design rules from tests on sub-assemblies.