A computational model for the limit analysis of three-dimensional masonry structures

This paper extends previous work on the limit analysis of ductile frames and plane masonry arches to the limit analysis of three-dimensional masonry structures. A lower-bound approach is developed which can handle three-dimensional collapse mechanisms involving any combination of sliding, twisting and hingeing at the block interfaces. A computer program for determining the collapse load of such structures is used to study (a) the equilibrium limits of a block with four contact points resting on an inclined plane and (b) the collapse of a semicircular arch of four blocks. The paper also describes experimental and computational work on a radially symmetric model dome of 380 blocks subject to foundation settlement.

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Sommario Il presentre contributo estende al campo delle structture tridimensionali in muratura un precedente lavoro sull'analisi limite di telai duttili ed archi in muratura piani. Si e' sviluppato un approccio statico che analizza meccanismi di collasso tridimensionale ottenuti per combinazione dei meccanismi semplici di scorrimento e rotazione nel piano e fuori dal piano delle superfici di interfaccia tra i blocchi. Si descrivono (a) i limiti di equilibrio di un blocco con 4 punti di contatto su base inclinata, (b) le condizioni di collasso di un arco semicircolare costituito da quattro blocchi, applicando un programma di calcolo redatto per l'analisi e la definizione del carico di collasso di tali strutture. La terza parte dell'articolo presenta il lavoro sperimentale e di calcolo sviluppato su un modello di cupola a simmetria radiale costituita da 380 blocchi soggetta a cedimenti fondali.

     

A numerical model for non-linear dynamic analysis of slender masonry structures

A numerical model is presented to enable performing non-linear dynamic analysis of slender masonry structures and elements, such as towers and columns or masonry walls in out-of-plane flexure. Such structures are represented via a continuous one-dimensional model. The main mechanical characteristics of the material in all sections along the height of such structures are taken into account by means of a non-linear elastic constitutive law formulated in terms of generalized stress and strain, under the assumption that the material has no resistance to tension and limited compressive strength. The relations defined herein for the general case of hollow rectangular cross-sections are also aimed at enabling study of towers, bell-towers and similar slender structures.     

A micro-mechanical model for the homogenisation of masonry

Masonry is a composite material made of units (brick, blocks, etc.) and mortar. For periodic arrangements of the units, the homogenisation techniques represent a powerful tool for structural analysis. The main problem pending is the errors introduced in the homogenisation process when large difference in stiffness are expected for the two components. This issue is obvious in the case of non-linear analysis, where the tangent stiffness of one component or the tangent stiffness of the two components tends to zero with increasing inelastic behaviour.The paper itself does not concentrate on the issue of non-linear homogenisation. But as the accuracy of the model is assessed for an increasing ratio between the stiffness of the two components, the benefits of adopting the proposed method for non-linear analysis are demonstrated. Therefore, the proposed model represents a major step in the application of homogenisation techniques for masonry structures.The micro-mechanical model presented has been derived from the actual deformations of the basic cell and includes additional internal deformation modes, with regard to the standard two-step homogenisation procedure. These mechanisms, which result from the staggered alignment of the units in the composite, are of capital importance for the global response. For the proposed model, it is shown that, up to a stiffness ratio of one thousand, the maximum error in the calculation of the homogenised Young's moduli is lower than five percent. It is also shown that the anisotropic failure surface obtained from the homogenised model seems to represent well experimental results available in the literature.     

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.     

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 simple and effective nonlinear elastic one-dimensional model for the structural analysis of masonry arches

In this paper the static response of a masonry arch is studied by way of a one-dimensional nonlinear elastic model in which masonry is regarded as a material with bounded tensile and compressive strengths. By following an approach analogous to that followed in the theory of bending of elastic beams, the equilibrium problem for the arch leads to a free-boundary, nonlinear differential problem. An approximate solution to such problem can be pursued by means of an ad hoc iterative procedure, illustrated in detail herein. The results obtained in three case studies are compared with some numerical and experimental results available in the literature. In addition, the case of an actual arch undergoing spreading of the springings is considered, and the distribution and possible evolution of the cracking pattern discussed.     

A simple meso-macro model based on SQP for the non-linear analysis of masonry double curvature structures

A 3D model for the evaluation of the non-linear behavior of masonry double curvature structures is presented. In the model, the heterogeneous assemblage of blocks is substituted with a macroscopically equivalent homogeneous non-linear material. At the meso-scale, a curved running bond representative element of volume (REV) constituted by a central block interconnected with its six neighbors is discretized through of a few six-noded rigid wedge elements and rectangular interfaces. Non linearity is concentrated exclusively on joints reduced to interface, exhibiting a frictional behavior with limited tensile and compressive strength with softening. The macroscopic homogenous masonry behavior is then evaluated on the REV imposing separately increasing internal actions (in-plane membrane actions, meridian and parallel bending, torsion and out-of-plane shear). This simplified approach allows to estimate heuristically the macroscopic stress–strain behavior of masonry at the meso-scale. The non-linear behavior so obtained is then implemented at a structural level in a novel FE non-linear code, relying on an assemblage of rigid infinitely resistant six-noded wedge elements and non-linear interfaces, exhibiting deterioration of the mechanical properties.Several numerical examples are analyzed, consisting of two different typologies of masonry arches (a parabolic vault and an arch in a so-called “skew” disposition), a ribbed cross vault, a hemispherical dome and a cloister vault. To fully assess numerical results, additional non-linear FE analyses are presented. In particular, a simplified model is proposed, which relies in performing at a structural level a preliminary limit analysis – which allows to identify the failure mechanism – and subsequently in modeling masonry through elastic elements and non-linear interfaces placed only in correspondence or near the failure mechanism provided by limit analysis. Simulations performed through an equivalent macroscopic material with orthotropic behavior and possible softening are also presented, along with existing experimental evidences (where available), in order to have a full insight into the capabilities and limitations of the approach proposed.     

A Reissner–Mindlin limit analysis model for out-of-plane loaded running bond masonry walls

Earthquake surveys have demonstrated that the lack of out-of-plane strength is a primary cause of failure in many traditional forms of masonry. Moreover, bearing walls are relatively thick and, as a matter of fact, many codes of practice impose a minimal slenderness for them, as for instance the recent Italian O.P.C.M. 3431 [2005. Ulteriori modifiche ed integrazioni all’OPCM 3274/03 (in Italian) and O.P.C.M. 3274, 20/03/2003, Primi elementi in materia di criteri generali per la classificazione sismica del territorio nazionale e di normative tecniche per le costruzioni in zona sismica (in Italian)], in which the upper bound slenderness is fixed respectively equal to 12 for artificial bricks and 10 for natural blocks masonry. In this context, a formulation at failure for regular assemblages of bricks based both on homogenization and Reissner–Mindlin theory seems particularly attractive. In this paper a kinematic limit analysis approach under the hypotheses of the thick plate theory is developed for the derivation of the macroscopic failure surfaces of masonry out-of-plane loaded. The behavior of a 3D system of blocks connected by interfaces is identified with a 2D Reissner–Mindlin plate. Infinitely resistant blocks connected by interfaces (joints) with a Mohr–Coulomb failure criterion with tension cut-off and compressive cap are considered. Finally, an associated flow rule for joints is adopted. In this way, the macroscopic masonry failure surface is obtained as a function of the macroscopic bending moments, torsional moments and shear forces by means of a linear programming problem in which the internal power dissipated is minimized, once that a subclass of possible deformation modes is a priori chosen. Several examples of technical relevance are presented and comparisons with previously developed Kirchhoff–Love static [Milani, G., Lourenço, P.B., Tralli, A., 2006b. A homogenization approach for the limit analysis of out-of-plane loaded masonry walls. J. Struct. Eng. ASCE (in press)] and kinematic [Sab, K., 2003.Yield design of thin periodic plates by a homogenisation technique and an application to masonry walls. C.R. Mech. 331, 641–646] failure surfaces are provided. Finally, two meaningful structural examples are reported, the first concerning a masonry wall under cylindrical flexion, the second consisting of a rectangular plate with a central opening out-of-plane loaded. For both cases, the influence of the shear strength on the collapse load is estimated.     

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.     

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。     

A macro-block model for in-plane loaded masonry walls with non-associative Coulomb friction

This paper presents a macro-block model and a simplified procedure for the prediction of the collapse load and the failure mechanism of in-plane loaded masonry walls, with non-associative frictional contact interfaces. The macro-block modeling is based on the assumption that the failure involves a single crack where all the possible relative motions among micro-blocks are concentrated. Two limiting conditions for the ultimate load factor are kinematically computed in closed form, by use of minimization routines of limit analysis. The results are compared against micro-block modeling and other macro-block models through a few illustrative examples and a parametric analysis. The influence of the main mechanical and geometrical parameters, such as unit aspect, scale effect, overload and friction is also shown.     

Kinematic FE limit analysis homogenization model for masonry walls reinforced with continuous FRP grids

A homogenization model for periodic masonry structures reinforced with continuous FRP grids is presented. Starting from the observation that a continuous grid preserves the periodicity of the internal masonry layer, rigid-plastic homogenization is applied directly on a multi-layer heterogeneous representative element of volume (REV) constituted by bricks, finite thickness mortar joints and external FRP grids. In particular, reinforced masonry homogenized failure surfaces are obtained by means of a compatible identification procedure, where each brick is supposed interacting with its six neighbors by means of finite thickness mortar joints and the FRP grid is applied on the external surfaces of the REV. In the framework of the kinematic theorem of limit analysis, a simple constrained minimization problem is obtained on the unit cell, suitable to estimate – with a very limited computational effort – reinforced masonry homogenized failure surfaces.A FE strategy is adopted at a cell level, modeling joints and bricks with six-noded wedge shaped elements and the FRP grid through rigid infinitely resistant truss elements connected node by node with bricks and mortar. A possible jump of velocities is assumed at the interfaces between contiguous wedge and truss elements, where plastic dissipation occurs. For mortar and bricks interfaces, a frictional behavior with possible limited tensile and compressive strength is assumed, whereas for FRP bars some formulas available in the literature are adopted to reproduce the delamination of the truss from the support.Two meaningful structural examples are considered to show the capabilities of the procedure proposed, namely a reinforced masonry deep beam (0°/90° continuous reinforcement) and a masonry beam in simple flexion for which experimental data are available. Good agreement is found between present model and alternative numerical approaches.     

A multi-scale computational model of crystal plasticity at submicron-to-nanometer scales

Plastic flow in crystal at submicron-to-nanometer scales involves many new interesting problems. In this paper, a unified computational model which directly combines 3D discrete dislocation dynamics (DDD) and continuum mechanics is developed to investigate the plastic behaviors at these scales. In this model, the discrete dislocation plasticity in a finite crystal is solved under a completed continuum mechanics framework: (1) an initial internal stress field is introduced to represent the preexisting stationary dislocations in the crystal; (2) the external boundary condition is handled by finite element method spontaneously; and (3) the constitutive relationship is based on the finite deformation theory of crystal plasticity, but the discrete plastic strains induced by the slip of the newly nucleated or propagating dislocations are calculated by dislocation dynamics methodology instead of phenomenological evolution equations used in conventional crystal plasticity. These discrete plastic strains are then localized to the continuum material points by a Burgers vector density function proposed by us. Various processes, such as loop dislocation evolution, dislocation junction formation etc., are simulated to verify the reliability of this computational model. Specifically, a uniaxial compression test for micro-pillars of Cu is simulated by this model to investigate the ‘dislocation starvation hardening’ observed in the recent experiment.     

Dynamic analysis for axially moving viscoelastic panels

In this study, stability and dynamic behaviour of axially moving viscoelastic panels are investigated with the help of the classical modal analysis. We use the flat panel theory combined with the Kelvin–Voigt viscoelastic constitutive model, and we include the material derivative in the viscoelastic relations. Complex eigenvalues for the moving viscoelastic panel are studied with respect to the panel velocity, and the corresponding eigenfunctions are found using central finite differences. The governing equation for the transverse displacement of the panel is of fifth order in space, and thus five boundary conditions are set for the problem. The fifth condition is derived and set at the in-flow end for clamped–clamped and clamped-simply supported panels. The numerical results suggest that the moving viscoelastic panel undergoes divergence instability for low values of viscosity. They also show that the critical panel velocity increases when viscosity is increased and that the viscoelastic panel does not experience instability with a sufficiently high viscosity coefficient. For the cases with low viscosity, the modes and velocities corresponding to divergence instability are found numerically. We also report that the value of bending rigidity (bending stiffness) affects the distance between the divergence velocity and the flutter velocity: the higher the bending rigidity, the larger the distance.     

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

Meccanica - 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...     

Homogenized rigid-plastic model for masonry walls subjected to impact

A simple rigid-plastic homogenization model for the analysis of masonry structures subjected to out-of-plane impact loads is presented. The objective is to propose a model characterized by a few material parameters, numerically inexpensive and very stable. Bricks and mortar joints are assumed rigid perfectly plastic and obeying an associated flow rule. In order to take into account the effect of brickwork texture, out-of-plane anisotropic masonry failure surfaces are obtained by means of a limit analysis approach, in which the unit cell is sub-divided into a fixed number of sub-domains and layers along the thickness. A polynomial representation of micro-stress tensor components is utilized inside each sub-domain, assuring both stress tensor admissibility on a regular grid of points and continuity of the stress vector at the interfaces between contiguous sub-domains. Limited strength (frictional failure with compressive cap and tension cut-off) of brick-mortar interfaces is also considered in the model, thus allowing the reproduction of elementary cell failures due to the possible insufficient resistance of the bond between units and joints.Triangular Kirchhoff-Love elements with linear interpolation of the displacement field and constant moment within each element are used at a structural level. In this framework, a simple quadratic programming problem is obtained to analyze entire walls subjected to impacts.In order to test the capabilities of the approach proposed, two examples of technical interest are discussed, namely a running bond masonry wall constrained at three edges and subjected to a point impact load and a masonry square plate constrained at four edges and subjected to a distributed dynamic pressure simulating an air-blast. Only for the first example, numerical and experimental data are available, whereas for the second example insufficient information is at disposal from the literature. Comparisons with standard elastic–plastic procedures conducted by means of commercial FE codes are also provided. Despite the obvious approximations and limitations connected to the utilization of a rigid-plastic model for masonry, the approach proposed seems able to provide results in agreement with alternative expensive numerical elasto-plastic approaches, but requiring only negligible processing time. Therefore, the proposed simple tool can be used (in addition to more sophisticated but expensive non-linear procedures) by practitioners to have a fast estimation of masonry behavior subjected to impact.     

砖石古塔模型振动台试验的离散元模拟分析

砖石古塔块体间黏结强度较低,受地震作用易产生裂缝后发生破坏,为研究砖石古塔在地震作用下开裂及裂缝发展机制,以玄奘塔1/8缩尺模型结构为对象,建立离散元模型,计算了地震波激励下结构的加速度及位移反应,与振动台试验结果进行对比,分析了塔体开裂破坏全过程。结果表明,数值计算所得结构的加速度及位移反应与振动台试验结果一致,当地震烈度较低时,两者顶层位移变化曲线基本一致;烈度提高后,塔体开裂导致结构动力响应的计算值与试验结果出现差异,但两者的变化规律相同。在地震波激励下,塔体首层先开裂,随地震烈度的提高延伸至中部塔层,块体第2层出现阶梯状错动,模型损伤过程与试验裂缝发展状况基本吻合。研究结果为砖石古塔地震损伤及破坏过程分析提供了参考。     

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.     

用于多种夹层板等效的有限元分析法

针对夹层板力学性能解析法难于计算复杂结构的夹层板且通用性差的问题,本文采用有限元分析法研究了夹层板性能的等效方法。对夹层板的代表体单元模型施加位移约束,模拟弯曲变形时线性独立的应变分量和弯曲内力;根据夹层板内力与应变的本构关系,求出刚度矩阵;最后由刚度矩阵得出宏观等效弹性常数,从而把夹层板等效成连续材料的单层板单元。将该方法与解析法计算结果进行比较得到的夹层板单元四个主要弹性常数误差在0.2%以内,验证了该方法的有效性;另外采用该方法等效三种典型结构夹层板,比较实际模型和等效模型的弯曲响应,得到的误差均在1.4%以内,表明该方法在不考虑复杂多变的夹芯结构时具有通用性。     

A multi-scale architecture for multi-scale simulation and its application to gas-solid flows

A multi-scale hardware and software architecture implementing the EMMS （energy-minimization multi-scale） paradigm is proven to be effective in the simulation of a two-dimensional gas-solid suspension. General purpose CPUs are employed for macro-scale control and optimization, and many integrated cores （MlCs） operating in multiple-instruction multiple-data mode are used for a molecular dynamics simulation of the solid particles at the meso-scale. Many cores operating in single-instruction multiple- data mode, such as general purpose graphics processing units （GPGPUs）, are employed for direct numerical simulation of the fluid flow at the micro-scale using the lattice Boltzmann method. This architecture is also expected to be efficient for the multi-scale simulation of other comolex systems.