Limit Analysis Of Masonry Structures Research Articles

Limit analysis of masonry domes with oculus and lantern: A comparison between different approaches

This paper aims to illustrate and compare different approaches for the limit analysis of masonry domes of revolution with an oculus at the top. It specifically focuses on the influence of the limited compressive and tensile strength of masonry on the bearing capacity of the dome. The first assessment is conducted through a semi-analytical formulation that revisits Durand-Claye’s method within the framework of the lower bound theorem of limit analysis. The second investigation is performed using a kinematic approach, which allows for closed-form solutions by exploiting the upper bound theorem of limit analysis and the virtual work theorem. These approaches are applied to a case study that was experimentally tested, involving a hemispherical dome with an oculus. The dome is subjected to its self-weight and a vertical load at the crown. Parametric investigations are presented to compare the results obtained through the proposed methods with those derived from the classical limit analysis of masonry structures, based on Heyman’s hypotheses.

Technical reports and theoretical studies about the structural behaviour of masonry domes in the 18th century

The development of the scientific theory on the stability of vaulted structures was driven mainly during the 18th century based on applying the laws of Mechanics and the knowledge of the properties of materials. In the specific case of structures such as domes, the development of the theory is closely related to the theory of arches and vaults. Still, the main contributions are made in technical reports on the construction, stability, and repair of domes, such as the dome of Saint Peter's in Rome and Sainte-Geneviève or Pantheon in Paris. This article, therefore, analyses the relationship between theory and practice and the development achieved in the specific subject of masonry domes during the 18th century from the point of view of the Limit Analysis of Masonry Structures. In a more general sense, the objective of this research is to study in greater depth the theory of Limit Analysis itself, that is, to learn from the work carried out in the 18th century by mathematicians, architects, and engineers, who, although unaware of the existence of the theoretical framework of Limit Analysis as such, moved within it.

ArchLab: a MATLAB tool for the Thrust Line Analysis of masonry arches

Abstract According to Heyman’s safe theorem of the limit analysis of masonry structures, the safety of masonry arches can be verified by finding at least one line of thrust entirely laying within the masonry and in equilibrium with external loads. If such a solution does exist, two extreme configurations of the thrust line can be determined, respectively referred to as solutions of minimum and maximum thrust. In this paper it is presented a numerical procedure for determining both these solutions with reference to masonry arches of general shape, subjected to both vertical and horizontal loads. The algorithm takes advantage of a simplification of the equations underlying the Thrust Network Analysis. Actually, for the case of planar lines of thrust, the horizontal components of the reference thrusts can be computed in closed form at each iteration and for any arbitrary loading condition. The heights of the points of the thrust line are then computed by solving a constrained linear optimization problem by means of the Dual-Simplex algorithm. The MATLAB implementation of presented algorithm is described in detail and made freely available to interested users (https://bit.ly/3krlVxH). Two numerical examples regarding a pointed and a lowered circular arch are presented in order to show the performance of the method.

Masonry structures in the presence of foundation settlements and unilateral contact problems

This paper aims at presenting a new perspective on the modeling of the influence of foundation settlements in masonry structures. We show that it is possible to model in a consistent way the crack pattern and the associated mechanism induced by applied settlements starting from a proper treatment of unilateral contact constraints between contiguous blocks, in which the structure is partitioned after the onset of settlements. We extend two classic variational formulations for the Signorini problem, namely the minimum of the total potential energy and a complementary boundary formulation, to the case of contact between multiple no-tension rigid bodies satisfying Heyman's hypotheses for the limit analysis of masonry structures. The discretization of the proposed variational formulations directly entails two dual linear programming problems. The proposed formulations allow to correctly evaluate the effects of foundation settlements and their influence on the ultimate load bearing capacity of masonry structures.

Thrust network analysis of masonry helical staircases

ABSTRACTWe exploit a recent generalization of the Thrust Network Analysis (TNA) to analyse masonry structures of extremely complicated and audacious geometry such as helicoidal staircases. Actually, differently from available methods, based on ad-hoc solutions holding for each particular structure, the TNA allows for studying the equilibrium of structures of arbitrary shape by invoking the principles of limit analysis of masonry structures dictated by Heyman. Four paradigmatic monumental staircases, with increased degree of complexity, have been selected as case studies in order to show the versatility and accuracy of the proposed approach. The results entailed from the generalized TNA are also successfully compared, for the first time in the literature, with those related to the adoption of available analytical methods for the analysis of no-tension membranes.

Limit analysis of masonry structures with free discontinuities

We formulate a novel procedure for the limit analysis of two-dimensional masonry structures subject to arbitrary loading conditions. The proposed approach works in the framework of free discontinuity methods, on examining collapse mechanisms that exhibit free crack opening discontinuities. The load bearing capacity and the collapse mechanism of the structure are obtained through a fully variational approach, by minimizing a kinetic functional that admits the collapse crack pattern as a variable. Numerical examples illustrate the practical application of the proposed procedure to the limit analysis of a variety of masonry walls and arches subject to foundation settlements, vertical and horizontal forces.

Limit analysis of masonry walls by rigid block modelling with cracking units and cohesive joints using linear programming

In this paper a rigid block model with cracking units and cohesive joints is developed for limit analysis of masonry structures using linear programming. The model is applicable to in-plane loaded unreinforced masonry block walls with regular textures and provides as output failure loads and collapse mechanisms. A simplified micro-modelling approach is adopted, based on the discretization of masonry units into triangular blocks separated by contact interfaces. Tensionless contact interfaces are used to model dry joints and cohesive contacts are introduced for mortar joints and internal unit interfaces. Failure modes at contact interfaces involve cracking, crushing and sliding. An iterative solution procedure is used to model non-associative flow rule in sliding and to take into account no-tension behaviour of cohesive joints in case of cracking. The modelling approach is validated and applied to different structural examples from the literature, including unconfined and confined masonry panels and a masonry beam. To show the accuracy of the proposed model and the improvement of the predicted response, the obtained results are compared with experimental tests and with the outcomes of standard rigid block models using a single block per masonry unit and tensionless interfaces.

Homogenized limit analysis of masonry structures with random input properties: polynomial Response Surface approximation and Monte Carlo simulations

The uncertainty often observed in experimental strengths of masonry constituents makes critical the selection of the appropriate inputs in finite element analysis of complex masonry buildings, as well as requires modelling the building ultimate load as a random variable. On the other hand, the utilization of expensive Monte Carlo simulations to estimate collapse load probability distributions may become computationally impractical when a single analysis of a complex building requires hours of computer calculations. To reduce the computational cost of Monte Carlo simulations, direct computer calculations can be replaced with inexpensive Response Surface (RS) models. This work investigates the use of RS models in Monte Carlo analysis of complex masonry buildings with random input parameters. The accuracy of the estimated RS models, as well as the good estimations of the collapse load cumulative distributions obtained via polynomial RS models, show how the proposed approach could be a useful tool in problems of technical interest.

The Analysis of Masonry Architecture: A Historical Approach

Traditional unreinforced masonry architecture has disappeared from new building activity in the western world. Nevertheless, the architectural heritage of masonry must be preserved, and this involves structural analysis. The classical theory of structures does not apply well to such heterogeneous structures with unknown boundary conditions. Nevertheless, there exists a theory of masonry structures based in simple assumptions about the material: good compressive strength, almost no tensile strength and a constructive care to avoid sliding failure. The theory was born at the end of the 17th Century, developed during the 18th and was applied in the 19th Century. It was abandoned and eventually forgotten at the beginning of the 20th Century. After half a century, in the 1960s, Heyman incorporated the old theory within the frame of modern limit analysis with its implicit treasure of critical observation and experience. The safe theorem permits using equilibrium equations and simple material statements cited. No affirmation about boundary conditions, impossible to know and essentially changing, is made (other than the usual about strength and small displacements). In the first part of the paper, an outline of the old theory is summarised and discussed. In the second part, the main ideas and concepts of limit analysis of masonry structures are discussed.

Load carrying capacity of 2D FRP/strengthened masonry structures

An adaptive discontinuous finite element model is formulated for limit analysis of masonry structures strengthened by fiber composites. The model is able to predict the effects of fracture damage and delamination on the load carrying capacity of the reinforced structures. A numerical investigation on the collapse mechanisms of masonry structures under plain strain/stress is presented, accounting for different mechanical properties of FRP–masonry interface and different placements of the reinforcement in the masonry structures.

Collapse of Masonry Buttresses

This paper investigates the collapse of masonry buttresses under concentrated lateral loads. A fracture forms at the collapse state, significantly decreasing the resistance to overturning. Conventional analysis assumes that a masonry buttress acts monolithically to resist lateral loads. The current paper demonstrates that this approach is clearly unsafe, and the possibility of a fracture at the collapse state must be considered in the design and assessment of masonry buttresses. By treating the masonry as a continuum, infinitely strong in compression, with no resistance to tension and no possibility for sliding, the writers demonstrate the form of the fracture and determine the critical failure load for typical buttress forms. This approach follows in the tradition of limit analysis of masonry structures as developed by Heyman. General methods are proposed for the overturning analysis of masonry buttresses, and calculation examples are provided. Finally, methods for evaluating the safety of existing buttresses are presented and discussed.

A numerical method for the limit analysis of masonry structures

The paper presents a numerical method for the limit analysis of structures made of a rigid no-tension material. Firstly, we formulate the constrained minimum problem resulting from the application of the kinematic theorem, which characterizes the collapse multiplier as the minimum of all kinematically admissible multipliers. Subsequently, by using the finite element method, we derive the corresponding discrete minimum problem in which the objective function is linear and the inequality constraints are linear as well as quadratic. The method is then applied to some examples for which the collapse multiplier and a collapse mechanism are explicitly known. Lastly, the solution to the minimum problem calculated via numerical codes for quadratic programming problems, is compared to the exact solution.