Failure Limits of Shallow Grid Shells: The physics behind the concept of equivalent thickness and numerical validation.

The present paper is concerned with long span, synclastic grid shells with a relatively small rise. The shell is subjected to symmetrically distributed loading. Such shells are commonly used in the building industry due to their considerable high strength to weight ratio and ability to span large areas. There are three possible failure modes of this type of structure: buckling, snap-through and yielding. The present analysis shows that buckling always occurs before snap-through or yielding. The buckling problem is solved first analytically by considering an equivalent continuous shell of the same weight. The numerical solution is presented next using beam elements in the commercial finite element package ADINA. Comprehensive parametric study is performed by varying the key parameters characterizing the geometry of the shell. This includes the span to rise ratio, panel size, and grid topology. The failure envelope is constructed by changing one parameter at a time, and determining the dominant failure mode in a given range of parameters. In addition, a thorough analysis of a 2D arch was preformed which proves that the solution to the eigenvalue problem gives a solution to the one of nonlinear prebuckling equilibrium. Thus, the former solution method is used for the present 3D grid shell problem. It can be concluded that a relatively simple analytical solution with a suitably defined equivalent thickness of the shell provides sufficient accuracy on the failure load which could be used in preliminary weight efficient design. In addition, the present paper offers a much needed in insight into the interplay between bending and membrane response of lattice structures.