A finite difference method for the static limit analysis of masonry domes under seismic loads

The static limit analysis of axially symmetric masonry domes subject to pseudo-static seismic forces is addressed. The stress state in the dome is represented by the shell stress resultants (normal-force tensor, bending-moment tensor, and shear-force vector) on the dome mid-surface. The classical differential equilibrium equations of shells are resorted to for imposing the equilibrium of the dome. Heyman’s assumptions of infinite compressive and vanishing tensile strength, alongside with cohesive-frictional shear response, are adopted for imposing the admissibility of the stress state. A finite difference method is proposed for the numerical discretization of the problem, based on the use of two staggered rectangular grids in the parameter space generating the dome mid-surface. The resulting discrete static limit analysis problem results to be a second-order cone programming problem, to be effectively solved by available convex optimization softwares. In addition to a convergence analysis, numerical simulations are presented, dealing with the parametric analysis of the collapse capacity under seismic forces of spherical and ogival domes with parameterized geometry. In particular, the influence that the shear response of masonry material and the distribution of horizontal forces along the height of the dome have on the collapse capacity is explored. The obtained results, that are new in the literature, show the computational merit of the proposed method, and quantitatively shed light on the seismic resistance of masonry domes.

Graphical Modelling of Hoop Force Distribution for Equilibrium Analysis of Masonry Domes

The behaviour of axisymmetric masonry shells can be simulated by a system of forces constituted by meridian forces acting in the vertical planes, and by hoop forces acting circumferentially. A crucial component for the assessment of these structures using the Modified Thrust Line Method (MTLM) is the determination of hoop forces, whose computation is strenuous, limiting the practical application of MTLM. Working around this limitation, the current research introduces a strategy to manipulate the hoop forces by graphically implementing a function describing their distribution. The adaptiveness of this distribution function not only allows the application of MTLM for the analysis of a range of geometries, but also enables the simulation of membrane behaviour, arch behaviour and their combination, for considering partially cracked structures. Taking this into account, the approach is applied in the case studies illustrated within the current research.

Masonry Dome Behavior under Gravity Loads Based on the Support Condition by Considering Variable Curves and Thicknesses

It is necessary to recognize masonry domes’ behavior under gravity loads in order to strengthen, restore, and conserve them. The neutral hoop plays a crucial role in identifying the masonry dome’s behavior to distinguish between its tensile and compressive regions. When it comes to determining the neutral hoop position in a dome with the same brick material, in addition to determining the dome’s curve and thickness, the support condition located on the boundary line is a significant parameter that has received less attention in the past. Therefore, this research aims to comprehensively define masonry dome behaviors based on the support condition’s effect on the masonry dome’s behavior, in addition to thickness and curve parameters, by determining neutral hoop(s). The method is a graphical and numerical analysis to define the sign-changing positioning in the first principal stress (hoop stress), based on the shell theory and extracted from a finite element method (FEM) Karamba3D analysis of a macro-model. The case studies are in four types of supports: condition fixed, free in the X- and Y-axes, free in all axes (domes placed on a drum), and free in all axes (domes placed on a pendentive and a drum). For each support condition, twelve curves and four varied thicknesses for each curve are considered. Results based on the dome’s variables show that, in general, four types of masonry domes behavior can be identified: single-masonry dome behavior with no neutral hoop; double-masonry dome behavior where all hoops are compressive with a single neutral hoop; double-masonry dome behavior where hoops are compressive and tensile with a single neutral hoop; and treble-masonry dome behavior with double neutral hoops.

Brunelleschi’s Dome: A New Estimate of the Thrust and Stresses in the Underlying Piers

The paper deals with the insurgence of the thrust, together with its valuation, in masonry domes, giving special attention to the Brunelleschi’s Dome in Florence. After a recalling of the kinematical approach in the context of the Heyman masonry model, the thrust of Brunelleschi’s Dome is estimated as the maximum of the set of all the kinematical ones. Comparisons are made with other valuations made by the usual, but less accurate, statical approach. The knowledge of the thrust allows an evaluation of the stresses acting in the supporting piers: their base sections are all compressed, with level stresses sufficiently low. This result shows the extraordinary conception of Filippo Brunelleschi’s Dome and the favorable design of the pillar sections and of the drum positioning, due, perhaps, to Arnolfo di Cambio or to the succeeding Masters.