Evaluation of a Statistically Equivalent Periodic Unit Cell for a quasi-periodic masonry

The paper presents a method to estimate the Statistically Equivalent Periodic Unit Cell (SEPUC) corresponding to a masonry with quasi-periodic texture. The identification of the texture and the constituent phases (unit blocks and mortar joints) is achieved by means of digital image processing techniques applied to color image of the masonry wall. A statistical analysis of geometrical parameters (width and height of blocks, thickness and length of mortar joints) allows to estimate their probability distribution and to identify the typology of the texture. Subsequently a Monte Carlo analysis is performed using several tentative SEPUCs generated with different dimensions of blocks and joints according to the estimated distributions. A criterion was eventually proposed to identify, among the numerically generated ones, the SEPUC which is more suitable to model the behavior of masonry wall. The SEPUC is analyzed with techniques available for periodic texture, applying periodic boundary conditions, in order to estimate the equivalent elastic stiffness. The proposed method is validated comparing the results in the elastic range obtained with SEPUC and those obtained imposing essential and natural boundary conditions on the original texture.     

The implementation of normal and/or tangential boundary conditions in finite element codes for incompressible fluid flow

Various techniques for implementing normal and/or tangential boundary conditions in finite element codes are reviewed. The principle of global conservation of mass is used to define a unique direction for the outward pointing normal vector at any node on an irregular boundary of a domain containing an incompressible fluid. This information permits the consistent and unambiguous application of essential or natural boundary conditions (or any combination thereof) on the domain boundary regardless of boundary shape or orientation with respect to the co-ordinate directions in both two and three dimensions. Several numerical examples are presented which demonstrate the effectiveness of the recommended technique.     

Limit states for brick masonry based on homogenization approach

In this paper, a procedure is developed for assessing the strength of brick masonry based on homogenization theory. The approach invokes a lower bound analysis whereby plastically admissible stress fields are constructed in the constituents involved, subject to periodic boundary conditions and static equilibrium requirements. The critical load is obtained by solving a constrained optimization problem. The analysis employs a set of specific loading histories such as axial tension, pure shear and biaxial tension–compression at different orientations of bed joints. The performance of this approach is verified against numerical solutions based on finite element analysis. In the second part of this paper, a methodology is outlined for identification of a macroscopic failure criterion that incorporates a critical plane approach. A quantitative verification of this criterion is carried out for different loading conditions and the results are compared with the experimental data available in the literature.     

混凝土动态双轴拉压破坏准则细观数值模拟研究

正常服役期内的混凝土结构往往处于复杂应力状态,并且不可避免地会受到偶发动力载荷作用.对于复杂载荷作用下的混凝土力学性能研究,破坏准则是基础.受试验设备等条件限制,现有的动态双轴拉压破坏准则形式复杂、缺乏更高应变率和侧应力比范围且尚未综合考虑应变率和侧应力比的耦合作用.为进一步提出适用范围更高且更准确的混凝土动态双轴拉压...     

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.     

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.     

Dynamic response of rocking cracked masonry walls

The behaviour of masonry constructions results to be very far from the one characterizing ductile structures. In masonry constructions, the seismic action activates a rocking motion rather than a dissipating mechanism. A strength resource of masonry structures, properly reinforced in order to avoid early local failures, consists in exhibiting rocking behaviour, until a failure condition is attained. Aim of the paper is to investigate the dynamic behaviour of masonry single storey walls, according to Housner’s studies and innovatively introducing the effect of diagonal cracks developing from the toes of the piers and shown by typical post-earthquake cracking patterns. The proposed procedure can be easily applied to the case of multi-storey regular masonry walls with openings representing the main resistant structural components of a masonry building. Starting from the evaluation of the incipient rocking acceleration of the system, the free and forced motions of the wall are examined. In the paper, according to the classical Housner’s approach, the energy dissipation occurring during the impact is modelled. Finally, a numerical application, considering a simple constant horizontal acceleration impulse of given duration has been carried out.

     

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 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.     

基于巴西劈裂法的饱水煤样能量耗散特征研究

为研究饱水煤样拉伸破坏过程中的能量耗散规律,利用RMT-150B岩石力学实验系统对赵固二矿二1煤层煤样在自然和饱水状态下进行巴西圆盘劈裂实验,探讨了自然与饱水状态下煤样的应力、强度及能量变化。结果表明:巴西劈裂实验时,加载过程应力-时间关系与实验控制方式有关。采用载荷控制方式时,峰值前拉应力与时间呈良好线性关系,达到峰值时应力瞬间跌落为零,表现出明显脆性破坏特征;采用变形控制方式时峰值前拉应力与时间呈非线性关系,峰值后出现分次破坏特征;采用两种控制方式在巴西圆盘劈裂实验破坏前没有本质区别;两种状态下不同层位煤样抗拉强度和峰值能率存在差异,自然状态下层位煤样抗拉强度较高(1.80MPa),中层位煤样抗拉强度较低(1.23MPa),上层位煤样抗拉强度介于两者之间(1.52MPa),饱水后上中下层位煤样的抗拉强度和峰值能率均有不同程度降低,抗拉强度平均软化系数分别为0.65,0.61和0.61,平均峰值能率降幅为36.0%,56.1%和40.6%;两种状态下煤样抗拉强度与峰值能率大致呈线性关系,表明抗拉强度越高的煤样,能量耗散越少,煤样抗拉强度越低,能量耗散越多。     

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.     

无网格局部强弱法求解不规则域问题

无网格局部彼得洛夫-伽辽金(meshless local Petrov-Galerkin,MLPG)法是一种具有代表性的无网格方法,在计算力学领域得到广泛应用.然而,这种方法在边界上需执行积分运算,通常很难处理不规则求解域问题.为了克服MLPG法的这种局限性,提出了无网格局部强弱(meshless local strong-weak,MLSW)法.MLSW法采用MLPG法离散内部求解域,采用无网格介点(meshless intervention-point,MIP)法施加自然边界条件,并采用配点法施加本质边界条件,避免执行边界积分运算,可适用于求解各类复杂的不规则域问题.从理论上讲,这种结合式方法,既保持了MLPG法稳定而精确计算的优势,同时兼备配点型方法在处理复杂结构问题时简洁而灵活的优势,实现了弱式法和强式法的优势互补.此外,MLSW法采用移动最小二乘核(moving least squares core,MLSc)近似法来构造形函数,是对传统移动最小二乘(moving least squares,MLS)近似法的一种改进.MLSc使用核基函数代替通常的基函数,有利于数值求解的精确性和稳定性,而且其导数近似计算变得更为简单.数值算例结果初步表明:这种新方法实施简单,求解稳定、精确,表现出适合工程运用的潜力.     

直角结合异材界面端应力强度系数的经验公式

由不同材料结合而成的材料(简称异材或双材料)的力学性能及其可靠性评价是工程中亟待解决的问题。表征界面端奇异应力场大小的应力强度系数是结合异材强度评价的依据，本文针对工程中最常见的直角结合异材，通过对大量不同材料组合的异材的边界元数值分析，提出了界面端应力强度系数的近似计算公式，无量纲化后的应力强度系数的值只与异材Dundurs参数a，卢有关，该公式具有较高的精度，可以作为一般工程上的应力强度系数的计算以及异材结构设计的依据。     

Computations of a laminar backward‐facing step flow at Re=800 with a spectral domain decomposition method

The two‐dimensional laminar incompressible flow over a backward‐facing step is computed using a spectral domain decomposition approach. A minimum number of subdomains (two) is used; high resolution being achieved by increasing the order of the basis Chebyshev polynomial. Results for the case of a Reynolds number of 800 are presented and compared in detail with benchmark computations. Stable accurate steady flow solutions were obtained using substantially fewer nodes than in previously reported simulations. In addition, the problem of outflow boundary conditions was examined on a shortened domain. Because of their more global nature, spectral methods are particularly sensitive to imposed boundary conditions, which may be exploited in examining the effect of artificial (non‐physical) outflow boundary conditions. Two widely used set of conditions were tested: pseudo stress‐free conditions and zero normal gradient conditions. Contrary to previous results using the finite volume approach, the latter is found to yield a qualitatively erroneous yet stable flow‐field. Copyright © 1999 John Wiley & Sons, Ltd.     

On the equilibrium and admissibility coupling in NT vaults of general shape

In the paper a approach for dealing with masonry vault analysis is outlined, based on the selection of membrane stress surfaces obeying equilibrium and masonry material admissibility at the meanwhile.Thereafter the problem is expressed in function of a suitably defined stress function allowing some simplification and the search of the solution is, then, set up by an energetic approach.In general, assuming the stress function as a generic function of the membrane surface, it is demonstrated that, under gravitational loads, the equilibrium of the vault implies its admissibility. This result is very significant since coupling of equilibrium and strength explains why, anciently, it was so easy to build up masonry vaults, by simply hypothesizing a resistant shape under the assigned loads and, at the same time, resulting also in the satisfaction of the material requirements as regards to stress admissibility.A “direct stress” approach is outlined, as well, for obtaining analytical solutions for each vault shape, and, as an example, its application to the case of the barrel vaults is reported.     

A boundary integral method for two-dimensional (non)-Newtonian drops in slow viscous flow

A boundary integral method for the simulation of the time-dependent deformation of Newtonian or non-Newtonian drops suspended in a Newtonian fluid is developed. The boundary integral formulation for Stokes flow is used and the non-Newtonian stress is treated as a source term which yields an extra integral over the domain of the drop. The implementation of the boundary conditions is facilitated by rewriting the domain integral by means of the Gauss divergence theorem. To apply the divergence theorem smoothness assumptions are made concerning the non-Newtonian stress tensor. The correctness of these assumptions in actual simulations is checked with a numerical validation procedure. The method appears mathematically correct and the numerical algorithm is second order accurate. Besides this validation we present simulation results for a Newtonian drop and a drop consisting of an Oldroyd-B fluid. The results for Newtonian and non-Newtonian drops in two dimensions indicate that the steady state deformation is quite independent of the drop-fluid. The deformation process, however, appears to be strongly dependent on the drop-fluid. For the non-Newtonian drop a mechanical model is developed to describe the time-dependent deformation of the cylinder for small capillary numbers.     

A formal finite element approach for open boundaries in transport and diffusion ground-water problems

In the application of the finite element method to diffusion and convection-dispersion equations over a ground-water domain, the Galerkin technique was used to incorporate Neumann (or second-type) and Cauchy (or third-type) boundary conditions. While mass movement through open boundaries is a priori unknown, these boundaries are usually treated as a zero Neumann condition at some far distance from the domain of interest. Nevertheless, cheaper and better solutions can be obtained if these unknown conditions are adequately incorporated in the weak formulation and in the transient solution schemes (open boundary condition). Theoretical and numerical proofs are given of the equivalences between this approach and a ‘well-posed’ problem in a semi-infinite domain with a zero Neumann condition at a boundary placed at infinity. Transport and diffusion equations were applied in one dimension to show the numerical performances and limitations of this procedure for some linear and non-linear problems. No a priori limitations are foreseen in order to find similar solutions in two or three dimensions. Thus the spatial discretization in the proximity of open boundaries could be drastically reduced to the domain of interest.     

轴向运动弦线横向振动的频域分析

轴向运动弦线是多种工程系统的模型。为明确轴向运动横向振动的频域特性，及探索频域方法的应用特点．本文用频域方法分析轴向运动弦线的横向振动。基于轴向运动弦线横向振动方程和边界条件．通过Laplace变换导出频率域中的控制方程，并将该控制方程和边界条件用状态变量表示。由状态空间中的控制方程导出特征方程，从而求出固有频率。由轴向运动弦线的矩阵函数计算得到系统的传递函数，然后用留数定理计算传递函数的Laplace逆变换．这样就可以得到时域响应。最后分析了轴向运动弦线的横向共振，若简谐外激励的频率与系统固有频率相同，系统响应将随时间无限增加。     

On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory： equilibrium, governing equation and static deflection

This paper has successfully addressed three critical but overlooked issues in nonlocal elastic stress field theory for nanobeams： （i） why does the presence of increasing nonlocal effects induce reduced nanostructural stiffness in many, but not consistently for all, cases of study, i.e., increasing static deflection, decreasing natural frequency and decreasing buckling load, although physical intuition according to the nonlocal elasticity field theory first established by Eringen tells otherwise？ （ii） the intriguing conclusion that nanoscale effects are missing in the solutions in many exemplary cases of study, e.g., bending deflection of a cantilever nanobeam with a point load at its tip; and （iii） the non-existence of additional higher-order boundary conditions for a higher-order governing differential equation. Applying the nonlocal elasticity field theory in nanomechanics and an exact variational principal approach, we derive the new equilibrium conditions, do- main governing differential equation and boundary conditions for bending of nanobeams. These equations and conditions involve essential higher-order differential terms which are opposite in sign with respect to the previously studies in the statics and dynamics of nonlocal nano-structures. The difference in higher-order terms results in reverse trends of nanoscale effects with respect to the conclusion of this paper. Effectively, this paper reports new equilibrium conditions, governing differential equation and boundary condi- tions and the true basic static responses for bending of nanobeams. It is also concluded that the widely accepted equilibrium conditions of nonlocal nanostructures are in fact not in equilibrium, but they can be made perfect should the nonlocal bending moment be replaced by an effective nonlocal bending moment. These conclusions are substantiated, in a general sense, by other approaches in nanostructural models such as strain gradient theory, modified couple stress models and experiments.