Funicularity in elastic domes: Coupled effects of shape and thickness

The present work aims to analyse the behaviour of thin or shallow shells in order to find a simple way for defining quantitatively the concept of Funicularity. The first mention in the scientific literature about the condition of Funicularity was due to Dermot O’Dwyer, who describe the funicular analysis of masonry vaults.He started by studying arches in which the stress state can be represented with a line of thrust defined by a funicular polygon. This calculation procedure becomes more difficult when we deal with 3-D structures like shells. He proposes a method to evaluate the surface of thrust and to determinate the ultimate load of collapse This method consists of seven different steps that allow to find the force network by imposing two sets of constraints: the first is related to the height of the structural nodes that should lie into the thickness of the element; the second set of constraints concerns the vertical equilibrium of the forces in each node. Then, O’Dwyer’s idea has been improved by John Ochsendorf and his student Philippe Block. They designed a new type of analysis, the Thrust Network Analysis,starting from Maxwell’s concept of reciprocal figures. There will be two figures, the primal grid Γ and the dual grid Γ*,which can be considered the reciprocal of the former. As a matter of fact, the equilibrium of a node in Γ can be represented by a closed polygon in Γ* and vice versa. Another approach to mention is the one proposed by Francesco Marmo and Luciano Rosati. Their method is analogous to TNA, but they tried to simplify the process by considering only the primal grid and not the dual one. Consequently, they deleted some geometric hypothesis. Furthermore, they studied the problem of shells loaded by horizontal loads,in order to apply their method also in seismic areas. It is worth noting that in TNA,all the loads have to be vertical ones. The abovementioned approaches are complex,employing a large number of equations to be solved.Therefore, the aim of the present thesis is to propose a simple and effective method to describe the surface of thrust governing the static behaviour of shell structures. Our studies have been implemented by “SAP2000” FEM software: at first,it has been necessary to define the radius R,the height H,the number of divisions along the vertical axes Z,and the angular divisions of the shell mesh;then,the element type has been adopted by choosing “thick shell”, which has been characterized by Concrete C28/35 material with E=32308 MPa and ν=0.2. Thus, the parametric analysis has been performed by varying the shell relative thickness, t/2R,and the shallowness ratio, H/R,as in the following: H/R=5/5;4/5;3/5;2/5;1/5 t/2R=1/15;1/25;1/40;1/70;1/100;1/200;1/500 Finally,35 finite element models have been obtained from the combination of these two dimensionless parameters. From the SAP models,in which the shell structures are subjected to dead load only,we have obtained the bending moments and axial forces Mx,My,Nx,Ny for each shell node, from which it is possible to derive the eccentricities of the surface of thrust: ex=Mx/Nx ey=My/Ny eav=(ex+ey)/2 From the analysis of the obtained surfaces of thrust,it is possible to observe that,by decreasing the shell relative thickness,the surface of thrust tends to overlap with the geometrical axis of the structure particularly in the boundary constraint region:it means that, for each shallowness ratio, the bending moment acting at the shell supports tends to vanish.