NURBS solid modeling for the three-dimensional limit analysis of curved rigid block structures

In this paper, we present a novel computational approach for the generalized three-dimensional upper bound limit analysis of curved rigid block structures. This approach allows to discretize structures with complex geometries into a finite number of curved blocks, while retaining an overall exact description of the structural geometry and improving the accuracy of the solution in terms of both collapse load multiplier and failure mechanism. To this aim, the discretization of a given spatial curved structure is obtained by using NURBS solids, where NURBS stands for Non-Uniform Rational B-Spline. A NURBS solid is a closed region of space delimited by NURBS boundary surfaces: no information is provided about the space within the volume, except for the mathematical expression of the boundary surfaces. NURBS solids allow to discretize a complex spatial structure into a finite number of solid elements while maintaining unaffected the initial geometry, independently on the number of elements used. Each NURBS solid element is considered as a rigid and infinitely resistant block and an upper bound limit analysis is applied to the obtained assembly. The external power is evaluated on the original geometry: the resultants of surface and volume forces, and the coordinates of the corresponding loaded points are evaluated through a new boundary integration formula, which allows avoiding any additional discretization based on standard regular elements (voxel or tetrahedral). On the other hand, the internal power is evaluated according to an associated flow rule on the common boundary NURBS surfaces between adjacent solid elements, thus representing zero-thickness interfaces. The upper-bound formulation stems into a linear programming problem, from which the kinematic load multiplier, the generalized velocities of each block, and contact forces at interfaces are obtained. Finally, the Gothic arches with curved cross-sections located within the Carmo Convent (Lisbon, Portugal) and a double curvature groin vault from the St John Hospital (Jerusalem) are studied to prove the effectiveness of the proposed NURBS solid-based approach for the three-dimensional limit analysis of generally shaped rigid block structures.