Shell structures made of reinforced cementitious composites are often curved and slender, making their failure process complex due to the strong interaction of material and geometric nonlinearities. A new lattice approach is proposed for modelling this failure process. The model is an extension of a 3D frame element formulation based on the rigid-body spring concept. The shell geometry is discretised into a lattice of elements, with cross-sections of the elements selected so that the shell volume is fully represented. Geometric nonlinearity is modelled by considering large rotations. Material nonlinearity is incorporated by evaluating the vectorial constitutive response at multiple points of the cross-sections of the elements with a damage-plasticity constitutive model relating generalised internal forces to normalised displacement jumps. Interaction of the cementitious matrix and the reinforcement is modelled by link elements. The model is applied to multiple benchmarks for the elastic response of shells. It is then used to study the failure response of reinforced cementitious shells.