Rigid impacts of three-dimensional rocking structures

Studies of rocking motion aim to explain the remarkable earthquake resistance of rocking structures. State-of-the-art assessment methods are mostly based on planar models, despite ongoing efforts to understand the significance of three-dimensionality. Impacts are essential components of rocking motion. We present experimental measurements of free-rocking blocks on a rigid surface, focusing on extreme sensitivity of impacts to geometric imperfections, unpredictability, and the emergence of three-dimensional motion via spontaneous symmetry breaking. These results inspire the development of new impact models of three-dimensional facet and edge impacts of polyhedral objects. Our model is a natural generalization of existing planar models based on the seminal work of George W. Housner. Model parameters are estimated empirically for rectangular blocks. Finally, new perspectives in earthquake assessment of rocking structures are discussed.

     

Limit analysis by linear programming of 3D masonry structures with associative friction laws and torsion interaction effects

In this paper, a formulation for limit analysis of three-dimensional masonry structures discretized as rigid block assemblages interacting through no-tension and frictional contact interfaces is developed. Linear and piecewise linearized yield functions are used for rocking, sliding and torsion failure. A simple yield condition has been defined to take into account interaction effects of shear force with torsion and bending moment. Associative flow rules are considered for strain rates. On the basis of the developed governing equations, the limit analysis problem has been formulated as a nonlinear mathematical program. An iterative solution procedure based on linear programming is used to solve the limit analysis problem and to take into account nonlinearities due to the influence of bending moments and shear stresses on torsion strength. The results of experimental investigations on out-of-plane masonry walls constrained at one edge and different examples from literature were considered for validation. Comparison with existing formulations is carried out.     

Limit equilibrium analysis and the minimum thickness of circular masonry arches to withstand lateral inertial loading

In this paper, we compute the location of the imminent hinges and the minimum thickness, t, of a circular masonry arch with mid-thickness radius, R, and embracing angle, β, which can just sustain its own weight together with a given level of a horizontal ground acceleration, ε g. Motivated from the recent growing interest in identifying the limit equilibrium states of historic structures in earthquake prone areas, this paper shows that the value of the minimum horizontal acceleration that is needed to convert an arch with slenderness (t/R, β) into a four-hinge mechanism depends on the direction of the rupture at the imminent hinge locations. This result is obtained with a variational formulation and the application of the principle of stationary potential energy, and it is shown that a circular arch becomes a mechanism with vertical ruptures when subjected to a horizontal ground acceleration that is slightly lower than the horizontal acceleration needed to create a mechanism with radial ruptures. The paper explains that the multiplicity on the solution for the minimum uplift acceleration is a direct consequence of the multiple possible ways that a masonry arch with finite thickness may rupture at a given location. The paper further confirms that the results obtained with commercially available distinct element software are in very good agreement with the rigorous solution.     

Rocking of rigid block on nonlinear flexible foundation

The two prime models used currently to describe rocking of rigid bodies, the Housner’s model and the Winkler foundation model, can capture some of the salient features of the physics of this important problem. These two models involve either null or linear interaction between the block and the foundation.Hopefully, some additional aspects of the problem can be captured by an enhanced nonlinear model for the base-foundation interaction. In this regard, what it is adopted in this paper is the Hunt and Crossley’s nonlinear impact force model in which the impact/contact force is represented by springs in parallel with nonlinear dampers. In this regard, a proper mathematical formulation is developed accounting for the possibility of uplifting in the case of strong excitation. Further, an averaging procedure has been developed to expeditiously derive the steady state response amplitude in case of harmonic base excitation. The analytical study is supplemented by experimental tests developed in the Laboratory of Experimental Dynamics at the University of Palermo, Italy. In this context, because of the obvious relevance for historical monuments, free-rocking tests are presented for several marble-block geometries on both rigid and flexible foundations. Numerical vis-à-vis experimental data are examined, showing that the proposed nonlinear model is sufficiently versatile to capture additional aspects of the physics of the problem even for quite soft foundation materials.     

砌体结构破坏倒塌过程的离散单元模拟

基于传统颗粒离散单元模型（DEM）,开发了一种绑定式离散单元模型（BTDEM）用于砌体结构崩塌机制的研究,并对一个典型砌体结构的破坏倒塌过程进行数值模拟。结果证实BTDEM方法能够详细地模拟出砌体建筑结构的崩塌过程,同时还可以模拟出砌块的破碎,是一种普适性较强的模拟砌体结构建筑的高效数值方法。BTDEM方法具有计算速度...     

燃气爆炸作用下蒸压加气混凝土砌体墙的加固性能

为研究燃气爆炸作用下蒸压加气混凝土砌体墙的加固性能，基于有限元软件LS-DYNA，建立了砌体墙简化数值模型，分析了GB 50779-2012 石油化工控制室抗爆设计规范中建议的荷载作用下砌体墙高度和厚度的影响，对比了玄武岩纤维(basalt fiber reinforced plastic, BFRP)布与喷涂式聚脲对蒸压加气混凝土单向砌体墙的加固效果，并以防止砌体墙倒塌为设计目标，给出了加固建议。研究表明，本文中建立的简化数值模型能较好地模拟燃气爆炸作用下蒸压加气混凝土砌体墙的变形和破坏模式，计算结果与试验吻合良好；《规范》建议荷载作用下，未加固砌体墙以弯曲破坏为主，随着墙体高度增加，破坏模式由弯曲破坏向剪切破坏转变；BFRP布条加固可以有效提高墙体抗弯刚度和压拱效应，而聚脲涂层加固对抗弯刚度提高有限但墙体拉拱效应明显，二者均能显著提高墙体抗爆性能；加固墙体均发生弯曲破坏，BFRP布条材料的断裂一般发生在墙体位移最大处，而聚脲涂层材料的断裂发生在跨端边界处。     

Application of the NSCD method to analyse the dynamic behaviour of stone arched structures

In this paper, a masonry arch is simulated in order to assess both its structural and seismic vulnerability. The non-smooth contact dynamics (NSCD) computational method is used to simulate this type of structure as a collection of bodies under the hypothesis of unilateral constraints and frictional contact, with or without cohesion. Sinusoidal oscillations in three dimensions and real earthquake data have been applied to the supporting base element of the arch model.The primary aim of this study is to better understand the dynamic behaviour of the masonry arch, a typical component of historic unreinforced masonry (URM) structures. This study also assesses the influences of the input parameters on the mechanical and dynamic behaviour of the arch structure. Its collapse mechanism is studied for both cohesive and non-cohesive contact.In addition, we examine the behaviour under seismic loading of the Arles aqueduct, a real historical arched structure located in the south-east of France. Significant information can be obtained from the comparison of the results of advanced numerical analysis, taking into account the precise geometry of the model, the mechanical characteristics of the materials and the observation of the in situ monuments after their collapse.     

In plane loaded masonry walls: DEM and FEM/DEM models. A critical review

This work is dedicated to the assessment of the nonlinear behaviour of masonry panels with regular texture and subject to in-plane loads, by means of numerical pushover analysis and an analytical homogenized model. Two numerical models are considered and adopted for performing a set of numerical tests: a discrete model developed by authors and a discrete/finite element model frequently adopted in rock mechanics field and effectively extended to masonry structures. In both models the hypotheses of rigid blocks and elastic–plastic joints following a Mohr–Coulomb yield criterion are adopted. The aim of this work is twofold: (1) a comparison and a calibration of the numerical models, evaluating their effectiveness in determining ultimate loads and collapse mechanisms of masonry panels, by assuming a nonlinear homogenized model for regular masonry as reference solution; (2) the evaluation of sensitivity of masonry behaviour and numerical models to panel dimension ratio and to varying masonry texture. In a first case study, sliding collapse mechanisms changing to overturning collapse mechanisms for increasing panel and block height-to-width ratio are obtained and the results given by the numerical models turn out to be in good agreement. Furthermore, a second case study, dedicated to square panels supported at base ends and vertically loaded, shows different ‘arch mechanisms’ depending on block height-to-width ratio.     

Upper bound limit analysis of masonry retaining walls using PIV velocity fields

Masonry is the building of structures from individual units, with or without the use of mortar as a bonding component. Stone masonry structures are frequent in the regions where stones are ubiquitous. In Europe, it was massively used until the nineteenth century to build earth retaining walls. For instance, they represent 85% of the retaining walls in Great Britain and 60% in France. Most of these masonry walls are currently perfectly safe, showing an average durability of more than a century. The best economic interest is to maintain the stock in good order by identifying and repairing the structures at risk of collapse and to preserve the currently satisfactory structures at the lowest investment level. Unfortunately, there is not sufficient scientific knowledge to do so reliably. This work aims to develop a design method suitable for existing structures. The analytical model is based upon the yield design theory which provides a rigorous framework and has proven to be effective for this kind of structures. The strength domain of stone masonry is determined using the homogenization theory. The modelisation is carried out in 2D. An experimental campaign was carried out in order to verify the proposed model. The experimental setup is in 2D thanks to the use of Schneebeli rods and is considered to be a physical model and is not scaled. The theoretical failing load and the theoretical kinematics of the failure are compared with the experimental failing load and the actual kinematics of the failure. The discrepancies and differences between the two sets are then discussed.     

基于各向异性损伤本构模型的砌体结构抗震分析

进一步完善了应力张量的线性变换方法,并将其应用于砌体材料本构模型的开发。在整体式有限元模型中,因不区分砌块和砂浆而将其视为匀质的连续材料,难以用受拉和受压两个损伤变量准确描述灰缝的II型滑移破坏。为解决这一问题,提出应再引入一个针对II型滑移破坏的损伤变量。基于以上研究工作,对砌体结构的振动台试验进行了模拟。模拟结果进一步验证了本文提出的弹塑性损伤本构模型的有效性,以及在进行结构非线性分析方面的优越性。由于本文提出的本构模型能够较为真实地描述砌体墙的破坏模式,因此使用该模型进行结构非线性分析,除了能够获取结构的位移和应力等反应外,还能较为准确地实时提供结构中的损伤分布状态信息,找出结构的薄弱部位,并据此合理地设计结构或进行相应的结构修复。     

Masonry walls strengthened with composite materials – dynamic out-of-plane behavior

The dynamic behavior of masonry walls strengthened with composite materials and subjected to dynamic out-of-plane loading is analytically investigated. The analytical model derived in the paper focuses on one-way action through the height of the wall and it is based on dynamic equilibrium and the compatibility conditions between the structural components (masonry units, mortar joints, FRP reinforcement, and adhesive layers). The cracking and the nonlinear and dissipative behavior of the mortar material, the breathing of cracks, the rocking phenomenon, the development of arching forces, the interaction between the existing wall and the composite system, and the formation of debonded zones near the cracked mortar joints are considered in the nonlinear dynamic analysis. A numerical study that examines the capabilities of the model, quantifies the response of the strengthened wall to dynamic loads such as free and forced vibrations and seismic base excitation, and compares it to the response of the unstrengthened wall is presented. A summary and conclusions close the paper.     

Localized stress percolation through dry masonry walls. Part I – Experiments

Transmission photoelasticity on scale models is shown to disclose the stress distribution within dry masonry walls. This distribution is found to be complicated by unilateral joints between elements, where ‘randomness constrained within a geometrical scheme’ of contact points occurs, so that stress percolation results highly localized, evidencing ‘unloading islands’ in a ‘stress stream’. These findings are theoretically explained in Part II of this paper from both micromechanical and continuous modelling perspectives.     

Singular stress fields in masonry structures: Derand was right

The idea of a no-tension (NT) material underlies the design of masonry structures since antiquity. Based on the NT model, the safety of the structure is a problem of geometry rather than of strength materials, in the same spirit of the “rules of proportion” of the medieval building tradition. The use of singular stress fields for equilibrium problems of NT materials in 2d, has been recently proposed by Lucchesi et al. to produce statically admissible stress fields; here we introduce a simple way to construct singular stresses, based on the Airy’s stress formulation. We interpret the singular part of such stress fields as axial contact forces acting on ideal 1d structures arising inside the body, in the same spirit of Strut and Tie methods. A number of simple problems of equilibrium concerning typical walls, arches and portals, is solved in terms of stress fields having regular and singular parts, by adopting the direct and the stress function formulation. The validity of the rules of proportion described by Derand and Gil is also verified.     

On the minimal mass reinforcement of masonry structures with arbitrary shapes

This paper presents a tensegrity approach to the minimal mass design of tensile reinforcements of masonry structures with arbitrary shapes. The proposed strengthening methodology allows for the design of minimal mass resisting mechanisms of systems formed by a network of masonry rods, mainly working in compression, and grids of tensile reinforcements. Assuming a perfectly plastic response by each member, the existence of such resisting mechanisms ensures that the reinforced structure is stable under the examined loading conditions, due to the safe theorem of the limit analysis of elastic-plastic bodies. The approach proposed in this paper includes an explicit determination of the state of prestress to be applied to tensile reinforcements, in order that they are effective under pre-existing loading conditions. Several benchmark examples illustrate the potential of this approach when dealing with minimal mass reinforcements of 3D models of masonry walls under in-plane and out-of-plane forces, and a structural complex formed by a cloister vault resting on supporting walls.     

Stress analysis of masonry vaults and static efficacy of FRP repairs

In the paper, the basic principles for the analysis of structures made by Not-Resisting-Tension (NRT) material are introduced; the theory is then applied for investigating the static behaviour of a NRT masonry arch model and to test the effect of reinforcements made by FRP strips of variable length. A wide experimental campaign is developed and numerical/experimental comparison is provided in order to evaluate the skill of the adopted model in capturing the real behaviour of the structure with or without reinforcement.     

Predictive model for the collapse load of masonry assemblage with two piers joined by a spandrel

The masonry assemblage composed of two piers connected by a spandrel can be considered a repetitive unit in large masonry walls with openings, occurring in masonry buildings. In this work, the collapse load of the above-mentioned masonry assemblage is predicted by solving a system of nonlinear equations, where the normal force in the spandrel is a root of an equilibrium equation of fourth degree. Piers and spandrel are assumed rigid and nonlinearity (crushing and no tensile strength) is concentrated at the pier-foundation and pier–spandrel interfaces. The model also takes into account the effect of a timber lintel supporting the spandrel and anchored into the two adjacent piers. This approach valid for assemblages with one spandrel can be extended for the evaluation of the collapse load of structures composed of N piers connected by N ? 1 spandrels. The system of nonlinear equations is easily solved with an iterative method and the collapse load provided by the solution agrees well with the experimental result.     

3D macro and micro-block models for limit analysis of out-of-plane loaded masonry walls with non-associative Coulomb friction

In this paper, a masonry system composed of a façade wall connected with two sidewalls and subjected to out-of-plane loading is investigated within the framework of three-dimensional limit analysis. Two different modeling approaches, namely macro and micro-block models, are adopted. A rigid-perfectly plastic model with dry contact interfaces governed by Coulomb failure criterion is assumed for masonry walls with regular units and staggering (non-standard limit analysis). Three classes of failure modes are investigated, involving rocking, sliding, twisting failure and combinations of them. The macro-block model is based on the assumption that the failure involves a number of cracks which separate the structure into a few macro-blocks and all the possible relative motions among micro-blocks are concentrated along the cracks. Two limiting conditions for the ultimate load factor are kinematically computed by use of minimization routines. The micro-block model is based on a concave contact formulation in which contact points are located at the corners of interfaces, allowing failure modes involving opening and sliding to be simulated. An iterative solution procedure is used to solve the non-associative friction problem, with second order cone programming (SOCP) used to allow the conic yield function to be solved directly. Both models are validated against experimental outcomes from the literature. A parametric analysis is carried out in order to highlight the influence of each geometrical and mechanical parameter on the prevalence of a mechanism over the other. The presence of an unrestrained horizontal floor system with different orientations is also analyzed.     

A mixed lumped stress–displacement approach to the elastic problem of masonry walls

We present a novel approach to the elastic problem of masonry walls, which generalizes the lumped stress method presented in [Fraternali, 2001], [Fraternali, 2007] and [Fraternali, 2010] and Fraternali et al. (2002). The generalization consists of a mixed lumped stress-displacement approach to the elastic problem of a wall that incorporates no-tension elements. Such an approach depends on the nodal values of the Airy stress function and the displacements of selected (“pivot”) nodes. The latter coincide with inter-element and boundary nodes. The mixed lumped stress-displacement method can be conveniently coupled with standard finite element and boundary element approaches. Numerical applications dealing with recurrent structural elements are given, showing that such a method is able to capture some essential features of the actual response of masonry constructions.     

Experimental investigations of the joint-mortar behaviour

An important failure mode of the masonry walls is the shearing process in joint mortar. In order to understand better this phenomenon, an experimental study is carried out on half brick couplet specimen. Load/Unload shear tests are performed to assess the type of the shear behaviour of the joint mortar. The cohesion and the internal friction angle are then derived from linear regression while assuming Mohr–Coulomb criterion. In particular, the influence of holes on the joint behaviour is studied by comparing results obtained with both solid and hollow bricks. In both cases, the experimental results show that there is not any stiffness degradation even in the softening regime. Actually, the shear modulus remains constant. Hence, the joint behaviour is considered to be elastoplastic, independently on the brick type. However, it seems that the presence of holes increases the stiffness but does not affect the internal friction angle of the joint mortar.     

Masonry structures made of monolithic blocks with an application to spiral stairs

The broad topic of the present work is Statics and Kinematics of masonry structures, made of monolithic blocks, modelled a la Heyman, that is rigid bodies loaded by external forces, submitted to unilateral constraints, and undergoing small displacements, under the simplifying assumption that sliding on rough interfaces is prevented. Specifically, in this work, we study the effect, in terms of internal forces, of specified loads, by using given settlements/eigenstrains to trigger special regimes of the internal forces. Although our scope here is the analysis of masonry structures composed by monolithic pieces, and whose blocks are not likely to break at their inside, the method we propose can also be applied to generic masonry structures, such as those made of bricks or small stones. Heyman’s assumptions translate, for unilateral continua, into a normality assumption which allows to employ the two theorems of Limit Analysis. These continuous structures may actually fracture everywhere at their inside, forming rigid blocks in relative displacement among each other. Such piecewise rigid-body displacements in masonry are physiological, and rather than the result of over-loading, are most likely the direct product of small changes of the displacement type boundary conditions. However, when in a part of the structure a specific piecewise rigid-body displacement nucleates, that part of the structure exhibits a one degree of freedom mechanism, and becomes statically determined. Therefore, the internal forces can be computed, despite the original uncracked structure being abundantly overdetermined, and then admitting infinite many statically admissible stress regimes. With these assumptions, in the present paper we study the equilibrium and the effect of settlements in a masonry structure made of monolithic blocks. In particular, the triple helical stair of the convent of San Domingos de Bonaval, located in the Bonaval district of Santiago de Compostela, is considered as case study.