Abstract
This paper presents a parametric stability study of groin, or cross vaults, a structural element widely used in old masonry construction, particularly in Gothic architecture. The vaults’ stability is measured using the geometric safety factor (GSF), computed by evaluating the structure’s minimum thickness through a thrust network analysis (TNA). This minimum thickness is obtained by formulating and solving a specific constrained nonlinear optimisation problem. The constraints of this optimisation enforce the limit analysis’s admissibility criteria, and the equilibrium is calculated using independent force densities on a fixed horizontal projection of the thrust network. The parametric description of the vault’s geometry is defined with respect to the radius of curvature of the vault and its springing angle. This detailed parametric study allows identifying optimal parameters which improve the vaults’ stability, and a comprehensive comparison of these results was performed with known results available for two-dimensional pointed arches. Moreover, an investigation of different force flows represented by different form diagrams was performed, providing a better understanding of the vaults’ structural behaviour, and possible collapse mechanisms were studied by observing the points where the thrust network touches the structural envelope in the limit states. Beyond evaluating the GSF, the groin vault’s stability domain was described to give additional insights into the structural robustness. Finally, this paper shows how advances in equilibrium methods can be useful to understand and assess masonry groin vaults.
Highlights
Given the lack of literature about the minimum thickness problem in groin vaults, this paper aims to fill this gap using the thrust network analysis (TNA)-based optimisation framework described in [25]
The implementation of this paper considers the flow of forces, i.e., the topology, and geometry of the thrust network, fixed in its horizontal projection, as in [21]
The structure of the paper is as follows: a parametric description of the vaults is presented in Section 2; the numerical optimisation problem needed to find the minimum thickness is illustrated in Section 3; the assumptions of the force flow are described in Section 4, where three different topologies are considered; Section 5 presents the results of minimum thickness of the parametric groin vaults obtained; Section 6 presents a stability-domain analysis of selected vaults from the parametric analysis; and Section 7 reports the conclusions
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. To give additional information about the robustness of the structure, in the present paper stability domains are defined This domain measures the size of the space of admissible stress states by computing the vault’s minimum and maximum thrusts and observing how these evolve towards the limit state. The structure of the paper is as follows: a parametric description of the vaults is presented in Section 2; the numerical optimisation problem needed to find the minimum thickness is illustrated in Section 3; the assumptions of the force flow are described, where three different topologies are considered; Section 5 presents the results of minimum thickness of the parametric groin vaults obtained; Section 6 presents a stability-domain analysis of selected vaults from the parametric analysis; and Section 7 reports the conclusions The structure of the paper is as follows: a parametric description of the vaults is presented in Section 2; the numerical optimisation problem needed to find the minimum thickness is illustrated in Section 3; the assumptions of the force flow are described in Section 4, where three different topologies are considered; Section 5 presents the results of minimum thickness of the parametric groin vaults obtained; Section 6 presents a stability-domain analysis of selected vaults from the parametric analysis; and Section 7 reports the conclusions
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