No-tension Material Research Articles

Simple Nonlinear Numerical Modeling for Unreinforced and FRP-Reinforced Masonry Domes

This paper presents a new method to model the nonlinear behavior of double-curvature masonry structures, possibly reinforced by composite materials, by means of conventional elasto-plastic analyses. The method is meant to be used in professional design, especially for assessment and retrofitting purposes, based on the exploitation of the simplest nonlinear finite elements available in commercial software, namely, trusses with elasto-fragile and elasto-ductile behavior (Cutoff Bars, according for instance to the definition provided by Strand7 R3.1.3a). Numerical static nonlinear analyses are carried out by considering elastic hexahedral elements for bricks and by lumping nonlinearities on joints. These are assumed, in turn, to be elastic–brittle and elastic–plastic by using 1D elements, namely, Point Contacts, under the No-Tension Material hypothesis, and Cutoff Bars, respectively, assigning a small tensile resistance to the material. The reinforcement, realized with FRP hooping strips, is successfully modeled in a similar fashion, i.e., by applying perfectly bonded elastic–plastic Cutoff Bars at the extrados of the dome, where debonding is accounted for in a conventional way, limiting the tensile strength according to Italian Standards’ indications. The procedure is validated against benchmark models with the same geometry, using experimental data and more refined structural model results for comparison. After an in-depth analysis of the obtained results, in terms of capacity curves, the robustness and accuracy of the proposed approach are assessed.

On the Buckling of the No-tension Material Masonry Column

Masonry columns, subjected to eccentric compression, crack due to tension if the eccentricity is larger than the size of the core of the section. Previous studies have assumed that the cracks have so small spacing that the cracked tension side can be neglected during the analysis.The critical load can be determined using this assumption. However, experimental experience has shown that the cracks have large spacing, approximately equal to one and a half times the cross-section height. Therefore, the crack-free parts between the cracks influence the lateral deflection and the critical load. Considering the above-mentioned phenomenon, we determined the elastic critical buckling load of the cracked masonry column.

Seismic vulnerability prediction of masonry aggregates: Iterative Finite element Upper Bound limit analysis approximating no tensile resistance

A novel 3D Finite Element Upper Bound limit analysis approach for the evaluation of the seismic vulnerability of masonry aggregates when the collapse may occur both locally and globally is presented.The model relies on a 3D discretization of the aggregate by means of infinitely resistant hexahedron elements and rigid plastic quadrilateral interfaces, where all plastic dissipation occurs.The interface behavior is assumed isotropic, with friction, cohesion, and tension cutoff. The collapse horizontal acceleration is obtained in the discretized model solving large-scale linear programming problems with standard numerical routines. According to a consolidated literature, the vulnerability prediction should be done with pre-assigned failure mechanisms formed by large macro-blocks mutually roto-translating, and assuming a material unable to withstand tensile stresses without mutual sliding between macro-blocks. However, such assumptions may be responsible for an overestimation of the load-carrying capacity and the determination of a realistic mechanism is neither obvious nor possible in any case. Discretizing the aggregate into 3D elements and interfaces, the failure mechanism is automatically found in the solution of the limit analysis problem, but enforcing the no-tension material hypothesis may be responsible for the activation of parasite sliding between contiguous elements or stalling issues of the numerical algorithm. In the first case, the failure mechanism changes considerably and the computational evaluation of the horizontal acceleration at collapse can be affected by unsafe inaccuracy.The present approach iteratively approximates the no-tension material model at low computational cost, progressively shrinking the failure surface and evaluating (i) collapse accelerations, (ii) failure mechanisms active, (iii) power dissipated on interfaces and (iv) power expended by loads not dependent on the collapse multiplier. By analyzing the failure mechanisms obtained, it is possible to select where spurious sliding is not dominant. Enforcing to zero the power dissipated on interfaces it is possible to estimate the collapse multiplier corresponding to the twofold assumption that (i) parasite sliding is negligible and (ii) the material approximates in a technically acceptable way the no-tension one.The model is first benchmarked on a masonry portal, where a reference solution is available, and then applied to two historical aggregates located in Italy.

Implementation of an elastic no-tension material model in a sequentially linear analysis framework

Implementation of an elastic no-tension material model in a sequentially linear analysis framework

Energy-based modelling of in-plane fragility curves for the 2D ultimate capacity of Italian masonry buildings

The high seismic hazard of the Italian territory and the vulnerability of its historic masonry heritage require the development of fragility curves that must be increasingly reliable and robustly correlated to exposure. To date, national-scale seismic risk analyses mainly use empirical curves derived from the statistical analysis of damage induced by past events. These curves have shown good reliability, but they correlate only with a few typological-structural characteristics of the building, such as the number of floors, the vertical structure typology or the construction period. The present research paper aims to overcome this limitation with a hybrid approach that provides a better exposure characterisation. Specifically, the proposed strategy integrates the SAVE and Piecewise Rigid Displacement (PRD) methods. SAVE is an empirical approach based on the damage assessment due to past seismic events used to identify a seismic behaviour of a structure, while the PRD method is a numerical approach that solves the boundary value problem for normal, rigid, no-tension material. It can model different structural typologies, and as a result, it also provides the value of the horizontal static multiplier that drives the masonry construction to collapse. An extended numerical campaign is carried out considering a sample of 750 masonry buildings distributed throughout the Italian territory and extracted from the PLINIVS typological database. Looking at each construction, first, a PRD analysis is conducted to define its seismic capacity, paying special attention to modelling construction details. After that, the SAVE method is used to classify the construction in a specific seismic vulnerability class, i.e., from A to C, with decreasing vulnerability. All the buildings belonging to the same class are then collected, and three fragility curves representative of the collapse state (one for each vulnerability class) are derived and validated against empirical and analytical ones commonly adopted in the Literature. The integrated methodology shows a good agreement between simulations and observations, confirming the viability of the proposed hybrid methodology for the large-scale assessment of masonry buildings, providing an effective strategy to plan mitigation and rehabilitation interventions.

An analysis of masonry arches reinforced with CFRP

In this paper a numerical and experimental investigation of masonry arches is developed with an analysis of the effects of strengthening by means of advanced composite materials. More precisely, the focus of the analysis concerns the effects on the strength and stiffness improvement of damaged solid clay brick masonry arches by strengthening with carbon fibre reinforced plastic (CFRP). The masonry arch is modelled by assuming a no-tension material law. Moreover, the interface between mortar and bricks are modelled by assuming a unilateral contact behaviour. A 3D finite element model of the structure is developed accounting for the effective geometry of the brick-mortar components of the masonry arch and of the reinforcement layer. An experimental campaign was developed aimed to study the behaviour of damaged solid clay brick masonry arches strengthened with carbon fibre reinforced plastic.

Simple lower bound limit analysis model for masonry double curvature structures

A new Lower Bound LB plate and shell limit analysis Finite Element FE model for the analysis at collapse of masonry double curvature structures is presented. The discretization relies into hexahedrons assumed infinitely resistant and quadrilateral interfaces where all plastic dissipation occurs. On such interfaces, the flexural behavior is ruled by the interaction between bending moment and axial load, whereas the shear and torsional behavior are modeled by means of an in-plane tangential force, out-of-plane shear and a plate torque. The resultant limit analysis problem obtained by such a formulation is particularly straightforward and the number of variables to deal with very limited. Equilibrium is indeed imposed only on hexahedrons and admissibility on interfaces between adjoining elements. Masonry can be modeled obeying a classic no-tension material or with more complex linearized failure surfaces, for example derived from suitable homogenization techniques. A simple Linear Programming LP problem is so derived, where the actual thickness of the structure is accurately accounted for, a key feature to assess the stability of a vault in case of the no-tension material assumption. The numerical model is validated by means of several meaningful structural examples. A detailed comparison with numerical data available in the literature, obtained for the same examples with alternative numerical approaches shows the accuracy of the method proposed and its usefulness for a fast and reliable prediction of the load carrying capacity of masonry double curvature structures.

Masonry Spiral Stairs: A Comparison between Analytical and Numerical Approaches

The present paper applies the Linear Arch Static Analysis (LASA), which models the masonry material as unilateral, i.e., No-Tension material in the sense of Heyman, and the Safe Theorem of the Limit Analysis to the study of masonry spiral stairs. A comparison is made with a refined FE analysis of the same problem, obtained by means of the ANSYS Parametric Design Language (APDL). The objective is to prove that LASA can be a valid alternative to other more complex numerical methods, such as FE, especially when the modeling parameters, such as the boundary conditions, cannot be exactly defined. The case study of a small spiral staircase placed in the tower of Nisida, a small island close to Naples, Italy is taken into consideration. The results show that the LASA analysis provides results that fall within two limit FE cases in terms of stress and overall thrust, providing at the same time a meaningful insight into the equilibrium state of the structure.

Analysis of Linear Elastic Masonry-Like Solids Subjected to Settlements

A linear elastic no-tension material model is implemented in this contribution to cope with the analysis of masonry-like solids in case of either elastic or inelastic settlements. Instead of implementing an incremental non-linear approach, an energy-based method is adopted to address the elastic no-tension equilibrium. Under a prescribed set of compatible loads, and possible enforced displacements, a solution is found by distributing an equivalent orthotropic material having negligible stiffness in tension, such that the overall strain energy is minimized and the stress tensor is negative semi-definite all over the domain. A preliminary implementation of the proposed method is given by adopting a heuristic approach to turn the multi-constrained minimization problem into an unconstrained one. Numerical simulations focus on a wall with an opening subjected to either inelastic settlement or standing on elastic soil.

NONLINEAR FE ANALYSIS OF A MASONRY SPIRAL STAIRCASE IN NISIDA: A REFINED NUMERICAL CASE STUDY

Following previous work by some of the present authors based on the linear arch static analysis (LASA), which models the masonry material as a no-tension material according to Heyman (Heyman, J., The Stone Skeleton, Int. J. Solids Struct., vol. 2, no. 2, pp. 249-279, 1966) and on the safe theorem of the limit analysis (LA), an in-depth numerical study of a case study based on a masonry spiral staircase in Nisida, near Naples, is here presented by means of an accurate finite element (FE) model. The nonlinear FE model has been obtained by the use of the ANSYS Parametric Design Language (APDL), and a precise representation of all the material involved and of the boundary conditions, has been obtained. The results confirm that LASA can be an alternative to much more complex numerical analyses, such as FE, but it cannot account for the main cause of collapse or stress redistribution in these type of structures, that is sagging and subsidences. The results are presented and discussed in some detail.

On blocks detection in unilateral masonry-like structures: a rigid-elastic displacement approach

In the present paper, a rigid block model for detection and identification of the cracks' pattern in a masonry-like structure subjected to dead loads and finite displacements is presented. The equilibrium problem is formulated by means of a suitable variational formulation for the boundary value problem of normal elastic no-tension material. Specifically, static equilibrium states of masonry-like structures are obtained through the linear conic optimisation of total potential energy. The independent variables of the optimisation problem are elastic strains and displacements, both assumed to be continuous. Indeed, the displacement and the strain fields are described in terms of nodal values through a FEM-like formulation and both of them are treated as independent variables of the optimisation problem. The study provides the basis for future developments in the field of mechanics and mechanisms of fracture. Two simple case studies are also discussed in order to tap the promising potential of the proposed approach.

A continuous energy-based numerical approach to predict fracture mechanisms in masonry structures: CDF method

In the present paper, we propose the Continuous Displacement for Fracture (CDF) method, a continuous energy-based numerical approach to find mechanisms and crack patterns exhibited by 2D masonry structures subjected to given loads and settlements. The structure is modelled through the normal, rigid, no-tension material, and the equilibrium problem is solved as the minimum of the total potential energy (TPE). With the CDF method the solution is sought in the space of continuous functions. The CDF performances are compared and illustrated against the PRD approach that finds the TPE minimum in the space of small, piecewise-rigid displacements.The CDF method is displacement-based approach, allowing for a direct control of the effects of foundation settlements. Some problems are proposed to benchmark the methodology against both PRD and analytical solutions to also clearly illustrate its peculiarities. Finally, its use and potentials are benchmarked and compared on a case study. CDF provides results in good agreement with both the PRD approach and another more sophisticated model. The main outcome is that, although more computationally cumbersome, CDF is mesh independent and perfectly captures a clear subdivision of the structural domain into macro-regions behaving as rigid or quasi-rigid bodies.

Topology optimization of load-bearing capacity

The present work addresses the problem of maximizing a structure load-bearing capacity subject to given material strength properties and a material volume constraint. This problem can be viewed as an extension to limit analysis problems which consist in finding the maximum load capacity for a fixed geometry. We show that it is also closely linked to the problem of minimizing the total volume under the constraint of carrying a fixed loading. Formulating these topology optimization problems using a continuous field representing a fictitious material density yields convex optimization problems which can be solved efficiently using state-of-the-art solvers used for limit analysis problems. We further analyze these problems by discussing the choice of the material strength criterion, especially when considering materials with asymmetric tensile/compressive strengths. In particular, we advocate the use of a L1-Rankine criterion which tends to promote uniaxial stress fields as in truss-like structures. We show that the considered problem is equivalent to a constrained Michell truss problem. Finally, following the idea of the SIMP method, the obtained continuous topology is post-processed by an iterative procedure penalizing intermediate densities. Benchmark examples are first considered to illustrate the method overall efficiency while final examples focus more particularly on no-tension materials, illustrating how the method is able to reproduce known structural patterns of masonry-like structures. This paper is accompanied by a Python package based on the FEniCS finite-element software library.

BUCKLING DELAMINATION IN COMPRESSED NO-TENSION HOMOGENEOUS BRITTLE BEAM-COLUMNS REINFORCED WITH FRP

The main issue of this paper is the instability of no-tension structural members reinforced with FRP. This study concerns the instability of FRP reinforcement. The primary instability problem of a compressed element involves the partialization of the inflex section. In particular, in the case of a compressed slender element reinforced on both tense and compressed side FRP delamination phaenomenon could occur on the latter.
 This entails the loss of the reinforcement effectiveness in the compressed area for nominal load values much lower than material effective strength. Therefore, structural elements or portions thereof which absorb axial components in the direction of the reinforcement may exhibit relatively modest performance with respect to the unreinforced configuration.
 By employing a no-tension material linear in compression, an analytical solution for FRP buckling delamination length is provided.
 The main objective of this paper is to provide a simplified tool that allows to evaluate the critical load of the reinforced beam-column and to predict the tension at which delamination and the loss of effectiveness of reinforcement in the compressed area could occur.

Out-of-plane local mechanism analysis with finite element method

Abstract The stability check of masonry structures is a debated problem in Italy that poses serious problems for its extensive use. Indeed, the danger of out of plane collapse of masonry walls, which is one of the more challenging to evaluate, is traditionally addressed not using finite element models (FEM). The power of FEM is not properly used and some simplified method are preferred. In this paper the use of the thrust surface is suggested. This concept allows to to evaluate the eccentricity of the membrane stresses using the FEM method. For this purpose a sophisticated, layered, finite element with a no-tension material is used. To model a no-tension material we used the smeared crack method as it is not mesh-dependent and it is well known since the early ’80 in an ASCE Report [1]. The described element has been implemented by the author in the program Nòlian by Softing.

Piecewise rigid displacement (PRD) method: a limit analysis-based approach to detect mechanisms and internal forces through two dual energy criteria

This paper proposes an extension of the Piecewise Rigid Displacement (PRD) method based on a new dual linear programming problem that minimises the complementary energy. Before, the PRD method had been applied to solve the kinematical problem for masonry-like structures composed of normal, rigid, no-tension (NRNT) material minimising the total potential energy. Specifically, the PRD method frames this minimum-energy search as a linear programming problem whose solutions are displacements and singular strain fields (cracks).Here, we show that the corresponding dual linear programming problem discretises the minimum of the complementary energy and returns, as solutions, admissible internal stress states compatible with the crack pattern obtained by solving the primal problem. Thus, these two minimum-energy criteria are dually connected, and their combined use allows coupling mechanisms and internal forces with settlements or homogeneous boundary displacements. This allows addressing different mechanical problems: equilibrium and stability of the reference configuration, effects of settlements, and mechanisms due to overloading (e.g. horizontal forces). Since the NRNT material represents the extension to continuum media of Heyman's material model, the PRD method offers an extremely fast, limit analysis-based, displacement approach that allows simultaneously finding mechanisms and compatible internal forces for any boundary condition, loads and geometry.

Assessment of 3D Linear Elastic Masonry-Like Vaulted Structures

A novel approach is adopted to assess the static behavior of vaulted structures, such as cantilevered masonry stairs, assuming a linear elastic no-tension material model. Masonry is substituted by an equivalent orthotropic material whose elastic properties vary locally and with a negligible stiffness where tensile strain occurs. In order to recover a tension-free state of stress, an energy-based minimization procedure is carried out to establish the distribution and the orientation of the equivalent material for a given compatible load. The capability of the approach in defining purely compressive stress solutions in masonry walls under dead load and both in-plane and out-of-plane live loads has already been assessed. A meaningful application to a cantilevered masonry stair is here presented; the results are in good agreement with those available in the technical literature on historical masonry constructions.

Catenary arch of finite thickness as the optimal arch shape

Based on the analogy with the load path of a homogeneous hanging chain, catenary has been for over three centuries considered to be an ideal masonry arch’s shape. In the present research, after the thrust line theory, complete insight into the equilibrium analysis of catenary arch of finite thickness under its own weight is provided. It is concluded that, in accordance with the common criterion—the existence of only normal (axial) forces—the ideality has previously been unjustifiably attached to such an arch, since there are shear forces present as well. However, with respect to so-called Couplet-Heyman’s assumptions (unilateral rigid no-tension material), it is shown that any catenary arch is stable under its own weight.

Experimental investigation of progressive instability and collapse of no-tension brickwork pillars

The progressive instability behaviour of compressed dry-stone rectangular pillars loaded with an eccentric load is assessed experimentally and compared with the theory. Photoelastic compression tests were designed and executed on polymethyl-methacrylate brickwork pillars to reveal, i) the load-bearing capacity of the structure and the load-lateral displacement relation, ii) the effect of the eccentricity in the stress distribution along the structure, iii) the collapse mode of the system at high eccentricity. By employing a no-tension material model with linear behaviour in compression, new analytical, closed-form expressions for deformed shape of the structure, location of the neutral axis in a generic cross section and axial displacement are provided. The photoelastic stress analysis outcome fully confirms the analytical predictions for both low and high eccentricity loadings.

Collapse load of a masonry arch after actual displacements of the supports

In this work, the collapse load of a masonry arch subjected to an actual horizontal displacement of the supports is assessed, in the deformed configuration of the structure, by means of limit analysis theory. Masonry is modelled as a rigid in compression, no-tension material and avoiding the occurrence of sliding failures. A numerical tool, based on the approach of the kinematical theorem of the collapse state, is proposed for the collapse computation of circular arches subjected to dead loads and to incremental concentrated load applied at their crown. A parametric study has been carried out in order to develop a deeper understanding of the influence of the involved parameters. It is shown that the existence of a kinematically admissible collapse mechanism is correlated to a thickness/mean radius ratio dependent on the value of the actual horizontal displacement of the supports. In the paper, a relationship between these two last parameters is proposed.