Compatible model for herringbone bond masonry: Linear elastic homogenization,failure surfaces and structural implementation

A simplified kinematic procedure at a cell level is proposed to obtain in-plane elastic moduli and macroscopic masonry strength domains in the case of herringbone masonry. The model is constituted by two central bricks interacting with their neighbors by means of either elastic or rigid-plastic interfaces with friction, representing mortar joints. The herringbone pattern is geometrically described and the internal law of composition of the periodic cell is defined.A sub-class of possible elementary deformations is a-priori chosen to describe joints cracking under in-plane loads. Suitable internal macroscopic actions are applied on the Representative Element of Volume (REV) and the power expended within the 3D bricks assemblage is equated to that expended in the macroscopic 2D Cauchy continuum. The elastic and limit analysis problem at a cell level are solved by means of a quadratic and linear programming approach, respectively.To assess elastic results, a standard FEM homogenization is also performed and a sensitivity analysis regarding two different orientations of the pattern, the thickness of the mortar joints and the ratio between block and mortar Young moduli is conducted. In this way, the reliability of the numerical model is critically evaluated under service loads.When dealing with the limit analysis approach, several computations are performed investigating the role played by (1) the direction of the load with respect to herringbone bond orientation, (2) masonry texture and (3) mechanical properties adopted for joints.At a structural level, a FE homogenized limit analysis is performed on a masonry dome built in herringbone bond. In order to assess limit analysis results, additional non-linear FE analyses are performed, including a full 3D numerical expensive heterogeneous approach and models where masonry is substituted with an equivalent macroscopic material with orthotropic behavior and possible softening. Reliable predictions of collapse loads and failure mechanisms are obtained, meaning that the approach proposed may be used by practitioners for a fast evaluation of the effectiveness of herringbone bond orientation.     

Discrete and continuous models for in plane loaded random elastic brickwork

The aim of this paper is to study non-periodic masonries – typical of historical buildings – by means of a perturbation approach and to evaluate the effect of a random perturbation on the elastic response of a periodic masonry wall. The random masonry is obtained starting from a periodic running bond pattern. A random perturbation on the horizontal positions of the vertical interfaces between the blocks which form the masonry wall is introduced. In this way, the height of the blocks is uniform, while their width in the horizontal direction is random. The perturbation is limited such as each block has still exactly 6 neighboring blocks. In a first discrete model, the blocks are modeled as rigid bodies connected by elastic interfaces (mortar thin joints). In other words, masonry is seen as a “skeleton” in which the interactions between the rigid blocks are represented by forces and moments which depend on their relative displacements and rotations. A second continuous model is based on the homogenization of the discrete model. Explicit upper and lower bounds on the effective elastic moduli of the homogenized continuous model are obtained and compared to the well-known effective elastic moduli of the regular periodic masonry. It is found that the effective moduli are not very sensitive to the random perturbation (less than 10%). At the end, the Monte Carlo simulation method is used to compare the discrete random model and the continuous model at the structural level (a panel undergoing in plane actions). The randomness of the geometry requires the generation of several samples of size L of the discrete masonry. For a sample of size L, the structural discrete problem is solved using the same numerical procedure adopted in [Cecchi, A., Sab, K., 2004. A comparison between a 3D discrete model and two homogenized plate models for periodic elastic brickwork, International Journal of Solids Structures 41 (9–10), 2259–2276] and the average solution over the samples gives an estimation which depends on L. As L increases, an asymptotic limit is reached. One issue is to find the minimum size for L and to compare the asymptotic average solution to the one obtained from the continuous homogenized model.     

A homogenized Love–Kirchhoff model for out-of-plane loaded random 2D lattices: Application to “quasi-periodic” brickwork panels

A homogenization procedure for finding the bending stiffness of a 2D regular lattice with random local interactions is proposed. The kinematic and static methods are used to provide explicit upper and lower bounds for the homogenized moduli. The proposed homogenization procedure is applied to a masonry obtained by a random perturbation of the periodic running bond masonry [Cecchi, A., Sab, K., 2009. Discrete and continuous models for in plane loaded random elastic brickwork. Eur. J. Mech. A 28, 610–625].A numerical evaluation of the scatter between the discrete models and the 2D Love–Kirchhoff model is performed on a test case, for various values of the random perturbation parameter and of the parameter that characterizes the heterogeneity of the wall. As expected, when the number of heterogeneities in the structure is large enough, the average response of the random discrete model converges to an asymptotic response. It is shown that this asymptotic response is very close to that of the periodic discrete model which is in turn very close to the response of the deterministic homogenized model. Similarly to the conclusion of Cecchi and Sab [Cecchi A., Sab K., 2009. Discrete and continuous models for in plane loaded random elastic brickwork. Eur. J. Mech. A. 28, 610–625.] dedicated to in-plane loading, the present results concerning out-of-plane loading show (both by means of a discrete model and a homogenized model) that the running bond pattern may be used successfully to analyze historical masonries with blocks having irregular widths in the horizontal direction.     

A homogenized viscoelastic model for masonry structures

A linear viscous model for evaluating the stresses and strains produced in masonry structures over time is presented. The model is based on rigorous homogenization procedures and the following two assumptions: that the structure is composed of either rigid or elastic blocks, and that the mortar is viscoelastic. The hypothesis of rigid block is particularly suitable for historical masonry, in which stone blocks may be assumed as rigid bodies, while the hypothesis of elastic blocks may be assumed for newly constructed brickwork structures. The hypothesis of viscoelastic mortar is based on the observation that non-linear phenomena may be concentrated in mortar joints. Under these assumptions, constitutive homogenized viscous functions are obtained in an analytical form.Some meaningful cases are discussed: masonry columns subject to minor and major eccentricity, and a masonry panel subject to both horizontal and vertical loads. The major eccentricity case is analysed taking into account both the effect of viscosity and the no-tension hypothesis, whereas the bi-dimensional loading case is analysed to verify the sensitivity of masonry behaviour to viscous function. In the masonry wall considered, the principal stresses are both of compression, and the no-tension assumption may therefore be discounted.     

A homogenized Reissner–Mindlin model for orthotropic periodic plates: Application to brickwork panels

This paper describes a new procedure for the homogenization of orthotropic 3D periodic plates. The theory of Caillerie [Caillerie, D., 1984. Thin elastic and periodic plates. Math. Method Appl. Sci., 6, 159–191.] – which leads to a homogeneous Love–Kirchhoff model – is extended in order to take into account the shear effects for thick plates. A homogenized Reissner–Mindlin plate model is proposed. Hence, the determination of the shear constants requires the resolution of an auxiliary 3D boundary value problem on the unit cell that generates the periodic plate. This homogenization procedure is then applied to periodic brickwork panels.A Love–Kirchhoff plate model for linear elastic periodic brickwork has been already proposed by Cecchi and Sab [Cecchi, A., Sab, K., 2002b. Out-of-plane model for heterogeneous periodic materials: the case of masonry. Eur. J. Mech. A-Solids 21, 249–268 ; Cecchi, A., Sab, K., 2006. Corrigendum to A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork [Int. J. Solids Struct., vol. 41/9–10, pp. 2259–2276], Int. J. Solids Struct., vol. 43/2, pp. 390–392.]. The identification of a Reissner–Mindlin homogenized plate model for infinitely rigid blocks connected by elastic interfaces (the mortar thin joints) has been also developed by the authors Cecchi and Sab [Cecchi A., Sab K., 2004. A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork. Int. J. Solids Struct. 41/9–10, 2259–2276.]. In that case, the identification between the 3D block discrete model and the 2D plate model is based on an identification at the order 1 in the rigid body displacement and at the order 0 in the rigid body rotation.In the present paper, the new identification procedure is implemented taking into account the shear effect when the blocks are deformable bodies. It is proved that the proposed procedure is consistent with the one already used by the authors for rigid blocks. Besides, an analytical approximation for the homogenized shear constants is derived. A finite elements model is then used to evaluate the exact shear homogenized constants and to compare them with the approximated one. Excellent agreement is found. Finally, a structural experimentation is carried out in the case of masonry panel under cylindrical bending conditions. Here, the full 3D finite elements heterogeneous model is compared to the corresponding 2D Reissner–Mindlin and Love–Kirchhoff plate models so as to study the discrepancy between these three models as a function of the length-to-thickness ratio (slenderness) of the panel. It is shown that the proposed Reissner–Mindlin model best fits with the finite elements model.     

A kinematic FE limit analysis model for thick English bond masonry walls

Two-wythes masonry walls arranged in English bond texture were often used in the past as bearing panels in seismic area. On the other hand, earthquake surveys have demonstrated that masonry strength under horizontal actions is usually insufficient, causing premature collapses of masonry buildings, often ascribed to out-of-plane actions. Furthermore, many codes of practice impose for new brickwork walls a minimal slenderness, which for instance is fixed by the Italian O.P.C.M. 3431 equal to 12 for artificial bricks and 10 for natural blocks masonry.For the above reasons, the analysis at failure of English bond brickwork walls under out-of-plane actions is a topic that deserves consideration, despite the fact that almost the totality of the studies of masonry at failure is devoted to running bond arrangements. Furthermore, it must be noted that an approach based on the analysis of running bond texture – in comparison with English bond pattern – is not suitable for the investigation of the behavior of bearing panels.In this framework, in the present paper, a Reissner–Mindlin kinematic limit analysis approach is presented for the derivation of the macroscopic failure surfaces of two-wythes masonry arranged in English bond texture. In particular, the behavior of a 3D system constituted by infinitely resistant bricks connected by joints reduced to interfaces with frictional behavior and limited tensile/compressive strength is identified with a 2D Reissner–Mindlin plate. In this way, assuming both an associated flow rule for the constituent materials and a finite subclass of possible deformation modes, an upper bound approximation of macroscopic English bond masonry failure surfaces is obtained as a function of macroscopic bending moments, torsion and shear forces.Several examples of technical relevance are treated both at a cell level and at a structural level, addressing the differences in terms of collapse loads and failure surfaces due to different textures and constituent laws for joints. Finally, two meaningful structural examples consisting of a panel in cylindrical flexion and a masonry slab constrained at three edges and out-of-plane loaded are discussed. A detailed comparison in terms of deformed shapes at collapse and failure loads between a 2D FE Reissner–Mindlin limit analysis approach and a full 3D heterogeneous FE model shows the reliability of the results obtained using the kinematic identification approach proposed.     

Homogenization of masonry walls with a computational oriented procedure. Rigid or elastic block?

The idea behind this paper is to present a numerical procedure for the analysis of masonry walls, based on the application of an asymptotic homogenization method. In this paper, a masonry wall, obtained by the regular repetition of blocks between which mortar is laid, is modelled as a periodic body in the two plane directions. The local problem is formulated for a base cell tied to the geometry of the body and in a position to generate it entirely through some law of its internal composition. Two homogenized models are formulated: the first envisages that both phases, block and mortar, behave in linear elastic fashion; the other envisages that the mortar behaves in linear elastic fashion, while the block is infinitely stiff. The two models are described theoretically and the construction of the model according to the characteristic module is numerically defined. In the case where the infinitely stiff (rigid) block is assumed, not only is the formulation of the model made extremely simple, but any numerical problems tied to great differences in the numerical values characterizing the constitutive modules of the two phases are overcome. In this regard, the domain of applicability of this model is sought both by comparing the homogenized constitutive functions, while varying the ratios of the elastic coefficients of the mortar and the block, with the rigid solution, and by analysing the structural behaviour that derives from the application, or not, of the rigid model, this being done for two sample problems. It should be underlined that the rigid-block model furnishes qualitatively sound structural answers even for very low ratios between the elastic moduli of the two phases composing the wall, and furnishes answers that are quantitatively sound as well for ratios of the order of 30:1, a realistic ratio in the case of ancient walls. The results obtained can be extended to heterogeneous materials in general, that is, to many of the innovatory materials, the composites, where the constituent phases have stiffness characteristics that are rather different and the condition of regularity of alternation of the phases is adequately plausible.     

A micro-mechanical homogenisation model for masonry: Application to shear walls

An improved micro-mechanical model for masonry homogenisation in the non-linear domain, is proposed and validated by comparison with experimental and numerical results available in the literature. Suitably chosen deformation mechanisms, coupled with damage and plasticity models, can simulate the behaviour of a basic periodic cell up to complete degradation and failure. The micro-mechanical model can be implemented in any standard finite element program as a user supplied subroutine defining the mechanical behaviour of an equivalent homogenised material. This work shows that, with the proposed model, it is possible to capture and reproduce the fundamental features of a masonry shear wall up to collapse with a coarse finite element mesh. The main advantage of such homogenisation approach is obviously the possibility to simulate real complex structures while taking into consideration the arrangement of units and mortar, which would otherwise require impractical amount of finite elements and computer resources.     

Non-linear micropolar and classical continua for anisotropic discontinuous materials

Non-linear Cosserat and Cauchy anisotropic continua equivalent to masonry-like materials, like brick/block masonry, jointed rocks, granular materials or matrix/particle composites, are presented.An integral procedure of equivalence in terms of mechanical power has been adopted to identify the effective elastic moduli of the two continuous models starting from a Lagrangian system of interacting rigid elements. Non-linear constitutive functions for the interactions in the Lagrangian system are defined in order to take into account both the low capability to carry tension and the friction at the interfaces between elements. The non-linear problem is solved through a finite element procedure based on the iterative adjustment of the continuum constitutive tensor due to the occurrence of some limit situation involving the contact actions of the discrete model.Differences between the classical and the micropolar model are investigated with the aid of numerical analyses carried out on masonry walls made of blocks of different size. The capability of the micropolar continuum to discern, unlike the classical continuum, the behaviour of systems made of elements of different size is pointed out. It is also shown that for anisotropic materials, even in the elastic case, the micropolar solution in general does not tend to the classical solution when the size of the elements vanishes.     

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...     

Cosserat model for periodic masonry deduced by nonlinear homogenization

The paper deals with the problem of the determination of the in-plane behavior of periodic masonry material. The macromechanical equivalent Cosserat medium, which naturally accounts for the absolute size of the constituents, is derived by a rational homogenization procedure based on the Transformation Field Analysis. The micromechanical analysis is developed considering a Cauchy model for masonry components. In particular, a linear elastic constitutive relationship is considered for the blocks, while a nonlinear constitutive law is adopted for the mortar joints, accounting for the damage and friction phenomena occurring during the loading history. Some numerical applications are performed on a Representative Volume Element characterized by a selected commonly used texture, without performing at this stage structural analyses. A comparison between the results obtained adopting the proposed procedure and a nonlinear micromechanical Finite Element Analysis is presented. Moreover, the substantial differences in the nonlinear behavior of the homogenized Cosserat material model with respect to the classical Cauchy one, are illustrated.     

Towards a multiscale analysis of periodic masonry brickwork: A FEM algorithm with damage and friction

The aim of this paper is to present an effort towards a multiscale model of the inelastic behaviour of masonry brick panels and the relative solution algorithm. The essential features of the inelastic behaviour, such as the damage development within the bed joints and the frictional dissipation over cracks’ faces, are taken into account. Micromechanical solutions are adopted in order to trace the guidelines for modelling the mentioned phenomena, and FEM analyses of large-scale panels are shown.     

短柱型复合材料加筋壁板轴向压缩破坏分析方法研究

复合材料加筋结构可作为航空结构中的承力部件,其损伤与破坏对航空器的结构安全和服役性能至关重要.本文通过试验和数值仿真手段研究了短柱型复合材料结构压缩失效机理和极限承载力.通过短柱型单加筋板的轴向压缩破坏试验,分析梳理出界面脱粘和材料压溃两种典型失效形式;分别建立加筋板壳单元模型和实体单元模型,引入内聚力模型和Hashi...     

Masonry as structured continuum

A structured continuum model is formulated to describe the behaviour of block masonry modelled as distinct rigid body systems with elastic interfaces. A correspondence between the two motions is obtained by postulating a relationship between the displacement fields of the continuum and the discrete models. The constitutive functions for the dynamic actions of the continuum are derived by equating the power of the two models.A preliminary version was presented at AIMETA '92, congress held at Trento, Italy, on 28 September – 2 October 1992.     

A multi-scale enriched model for the analysis of masonry panels

A multi-scale model for the structural analysis of the in-plane response of masonry panels, characterized by periodic arrangement of bricks and mortar, is presented. The model is based on the use of two scales: at the macroscopic level the Cosserat micropolar continuum is adopted, while at the microscopic scale the classical Cauchy medium is employed. A nonlinear constitutive law is introduced at the microscopic level, which includes damage, friction, crushing and unilateral contact effects for the mortar joints. The nonlinear homogenization is performed employing the Transformation Field Analysis (TFA) technique, properly extended to the macroscopic Cosserat continuum. A numerical procedure is developed and implemented in a Finite Element (FE) code in order to analyze some interesting structural problems. In particular, four numerical applications are presented: the first one analyzes the response of the masonry Representative Volume Element (RVE) subjected to a cyclic loading history; in the other three applications, a comparison between the numerically evaluated response and the micromechanical or experimental one is performed for some masonry panels.     

Numerical simulation of the failure process of unreinforced masonry walls due to concentrated static and dynamic loading

Finite element analyses of brick masonry subjected to in-plane concentrated static and dynamic loads are carried out to study crack initiation and propagation during the failure process of unreinforced masonry walls. The numerical model is firstly validated by the experimental tests by using the same materials parameters and loading conditions. Then, the static and dynamic concentrated loads are applied to the mortar joints and brick, respectively, and numerical simulations are used to compare the fracture characteristics for these loads. In addition, a comparison of fracture mechanisms for the concentrated loads on the mortar joint and brick is also given. Finally, the effect of dynamic pressure (P_max) on the failure mechanism of brick masonry is considered.     

砌体填充墙的抗爆性能

为了揭示砌体填充墙的抗爆破坏机理,在野外实验中,测得了爆炸条件下砌体填充墙上的爆炸荷载及位移,得到了墙的抗爆性能、破坏模式以及碎片的飞散和分布情况。实验结果表明,墙体的破坏模式与荷载的大小有关,其破坏主要由灰缝的破坏引起。结合实验现象,采用分离式建模的精细化数值模拟方法,得到了不同荷载条件下裂缝的发展过程、墙体的边界条件对墙体的破坏模式的影响,确定了墙体不同破坏等级时的药量,进一步说明本文中数值模拟方法的合理性。     

屋面光伏板风荷载特性数值分析

风荷载在屋面光伏阵列结构体系设计中起控制作用。采用计算风工程的方法分析讨论了屋面光伏板的风荷载特性。数值算法采用分离涡模拟方法。数值计算结果与现有风洞实验数据的比较,验证了本文方法的正确性。考虑影响光伏板风荷载的因素主要有光伏板在屋面上的安装位置、安装倾角、光伏阵列之间的距离和风向等。计算结果表明,屋面处脱落的涡对安装在不同位置的光伏阵列风荷载的影响较明显。当倾角由15°增加到45°时,电池板受到的风荷载随着倾角的增加而增大。在一定阵列间距范围内,光伏板风荷载主要表现为前排对下游光伏板的遮挡影响。本文方法与结果能为屋面光伏建筑结构设计提供重要参考。     

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.