Modelling the cracks produced by settlements in masonry structures

The present work is concerned with the prediction of the crack pattern produced by given kinematical data (settlements/distortions) in masonry constructions. By adopting the simplified model of Heyman, extending it to masonry structures treated as continuous bodies, we restrict the analysis to the search of displacement fields which are piecewise rigid. Restricting to small strains and displacements we look for the solution of the kinematical problem by minimizing the potential energy of the structure. A variational approximation of the minimum problem is obtained by considering a fixed finite element subdivision of the structure into rigid blocks. Two case studies are presented to illustrate the way in which a particular fracture pattern can be identified as the one associated to the minimum of the energy in this restricted class of piecewise rigid displacements.     

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.     

Structural rigidity optimization with frictionless unilateral contact

The problem of maximization the global rigidity (measured by the compliance) of an elastic structure with frictionless unilateral contact is considered in the framework of topology optimization. The frictionless unilateral contact is introduced in the continuous formulation of the elastic problem (under the assumption of small strains and small displacements) in the regularized form of an interface with an asymmetric behavior law relating the normal component of the stress vector transmitted through the contact surface to the normal displacement (in the case of contact with a rigid foundation) or the jump of normal displacement (in the case of internal contact of two surfaces of the elastic medium). Using the concept of homogeneous thermodynamical potentials, we extend a convergent and numerically efficient optimization algorithm introduced in the framework of linear elasticity to this nonlinear case of an elastic structure with unilateral contact. Numerical examples in two-dimensional elasticity are presented.     

Explicit solutions for depressed masonry arches loaded until collapse—Part I: a one-dimensional nonlinear elastic model

In this paper the equilibrium problem for masonry arches is formulated in terms of a suitable set of nonlinear ordinary differential equations. We show that by making a small number of simple hypotheses it is possible to find the explicit expressions for the displacements and rotations of the cross-sections of an in-plane loaded masonry arch. To this end, the masonry arch is schematised as a curved, one-dimensional nonlinear elastic beam made of a material that is by hypothesis incapable of withstanding significant tensile stresses. In this first part of the two-part paper, the one-dimensional model and the explicit expressions for the displacements and rotations, obtained by integrating the set of differential equations, are presented. In particular, the formal expressions for displacement, stress and strain fields are illustrated in full detail for an explicit, albeit approximate, solution for a statically determinate depressed arch subjected to a uniform vertical load.     

Green's function of the displacement boundary value problem for a heat source in an infinite plane with an arbitrary shaped rigid inclusion

Summary Green's functions of the displacement boundary value problem are derived within two-dimensional thermoelasticity for a heat source in an infinite plane with an arbitrary shaped rigid inclusion. The following two cases are considered: either rigid-body displacements and rigid-body rotations of the inclusion are allowed or no rigid-body displacements and no rigid-body rotations of the inclusion are possible. To solve these problems, fundamental solutions are developed for a point heat source, for rigid-body rotations of the inclusion, and for concentrated loads acting on the inclusion. Complex stress functions, temperature function, a rational mapping function and the thermal dislocation method are used for the analysis. In analytical examples, distributions of stresses are developed for an infinite plane with a rectangular rigid inclusion. Received 5 August 1998; accepted for publication 1 December 1998     

Equilibrium and collapse analysis of masonry bodies

This paper gives a general formulation of the statics of the masonry continuum within the conceptual framework set up by J. Heyman in his fundamental and pioneering studies of masonry arches and vaults. Here the masonry body will be represented by an assemblage of rigid particles of stones, held together only by compressive forces, and liable to crack as soon as tensile stresses begin to develop. The very small size of the stones, compared to the overall dimensions of the body, permits a treatment in terms of a continuum.The admissible mechanism displacement and stress fields of the masonry body are analysed, and an appropriate formulation of the virtual work equation is given. A variational inequality, involving the sign of the work of external loads along the mechanisms-necessary and sufficient for the existence of the admissible equilibrium states-is then proved. The collapse of the body is then properly formulated and, finally, new versions of the kinematical and statical theorems of failure are proved.

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Sommario Il presente articolo fornisce una formulazione generale della statica del continuo murario, seguendo l'indirizzo concettuale espresso da Heyman nel suo fondamentale e pionieristico studio su archi e volte in muratura.Nel presente lavoro il solido murario viene schematizzato come un assemblaggio di elementi rigidi di pietra, tenuti insieme da forze di compressione e soggetti a frattura non appena si inneschino trazioni. Le dimensioni ridotte dei conci, in relazione alle dimensioni globali del corpo, consentono di trattare il problema nel continuo.Vengono quindi analizzati i possibili meccanismi, ed i campi di tensione e spostamenti associati, espressi secondo una appropriata formulazione del Principio dei lavori virtuali. Viene poi mostrato come una opportuna diseguaglianza variazionale-che coinvolge il segno del lavoro delle forze esterne lungo i meccanismi-costituisca condizione necessaria e sufficiente per l'esistenza degli stati ammissibili di equilibrio. Utilizzando tale condizione viene quindi fornita una nuova versione dei teoremi statico e cinematico di collasso.

     

含铰接杆系结构几何非线性分析子结构方法

将细长杆系结构按长度方向划分为多个子结构,由于在子结构坐标系下的节点位移均是小位移,可以将子结构内部自由度凝聚到边界. 考虑到子结构端面在变形过程中保持为刚性截面,将端面节点自由度进一步凝聚到端面形心点,这样每一个子结构就减缩成形式上只有两个节点的广义梁单元,大大减缩了自由度. 大位移大转动是细长杆系结构产生几何非线性效应的一个重要原因,基于共旋坐标法,建立了随单元一起运动的随动坐标系,推导了子结构单元的节点力平衡方程及其切线刚度阵. 同时,考虑到工程机械中细长杆系结构含有相互铰接的刚体加强块,给出了非独立自由度节点力转换到独立参数下的广义节点力及其导数. 最后,通过履带式起重机的副臂工况算例,给出了其在不同载荷下的臂架结构位移,验证了方法的正确性.      

A fringe-compensation technique for stress analysis by reflection holographic interferometry

The use of holographic interferometry for stress analysis of nontransparent objects is limited by rigid-body displacements of the object. These displacements can alter the fringe patterns and often cause the fringes to disappear completely. A technique of compensation for this deterioration of the fringe pattern forreal-time holographic interferometry is described in this paper. It is especially designed to permit the accurate measurement of the out-of-plane component of strain near regions of stress concentration in plates that are subjected to in-plane loading. It is first shown that the fringes caused by a pure rigid-body displacement can be eliminated almost completely by translations of the hologram and rotation of the illumination wave. This procedure is first described when the displacement is known; then when it is unknown. A method to estimate the error made in the correction is presented. In actual stress-analysis problems, the object is both rigidly displaced and strained. Assuming the rigid displacement is known and corrected as previously, the analysis is developed to relate the fringe pattern to the strain-related displacement. This analysis takes into account the optical modifications of the system that are necessary to achieve the rigid-body-displacement correction. When the rigid-body displacement is unknown, the method is shown still to be workable through the use of various symmetries and boundary conditions. Two sample interferograms are presented as illustrations. Quantitative treatment of data from one of these are presented in a companion paper.     

On contact interaction between a moving massive body and a linear elastic half space

We study a class of problems involving the motion of a linear elastic body in frictional contact with a linear elastic half space. The dynamic effects considered are the inertial properties of the body regarded as rigid. We study only those regimes of contact interaction for which the slip velocity with the body taken as absolutely rigid and the time rate of change of the elastic displacements of points of the body and the half space that are on the contact surface are of the same order of magnitude. This work generalizes previous work on similar problems in that we simultaneously consider inertia forces of the body and the convective term in the slip-velocity due to the rigid-body velocity of the slider/indentor. Thus regimes of contact interaction investigated include rolling/sliding and shift-torsion type. We propose a variational formulation of the following two problems: (a) finite contact area and shift-torsion type of contact kinematics, (b) local contact area and general kinematics at the contact surface. Results for an elastic cylinder contacting an elastic half-plane are also given.     

Dynamics of a spacecraft with large flexible appendage constrained by multi-strut passive damper

This paper is concerned with the dynamics of a spacecraft with multi-strut passive damper for large flexible appendage.The damper platform is connected to the spacecraft by a spheric hinge,multiple damping struts and a rigid strut.The damping struts provide damping forces while the rigid strut produces a motion constraint of the multibody system.The exact nonlinear dynamical equations in reducedorder form are firstly derived by using Kane’s equation in matrix form.Based on the assumptions of small velocity and small displacement,the nonlinear equations are reduced to a set of linear second-order differential equations in terms of independent generalized displacements with constant stiffness matrix and damping matrix related to the damping strut parameters.Numerical simulation results demonstrate the damping effectiveness of the damper for both the motion of the spacecraft and the vibration of the flexible appendage,and verify the accuracy of the linear equations against the exact nonlinear ones.     

Limit analysis of masonry structures with free discontinuities

We formulate a novel procedure for the limit analysis of two-dimensional masonry structures subject to arbitrary loading conditions. The proposed approach works in the framework of free discontinuity methods, on examining collapse mechanisms that exhibit free crack opening discontinuities. The load bearing capacity and the collapse mechanism of the structure are obtained through a fully variational approach, by minimizing a kinetic functional that admits the collapse crack pattern as a variable. Numerical examples illustrate the practical application of the proposed procedure to the limit analysis of a variety of masonry walls and arches subject to foundation settlements, vertical and horizontal forces.     

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.     

Variation of Small Amplitude Vibration Dynamic Properties with Displacement in Reinforced Concrete Structures

Correlating damage level and changes in dynamic characteristics of a structure forms the basis for damage detection techniques in structural health monitoring. In reinforced concrete building structures such correlation is not well established. A damage detection technique capable of identifying the structural condition of the system based on its small amplitude vibration response is desirable because such response is easier to obtain. It is a common practice in engineering applications to estimate dynamic parameters from small-amplitude vibrations assuming a linear behavior of the structure. This simplification causes inaccurate estimation of the dynamic properties in reinforced concrete structures due to the presence of nonlinear elastic behavior. In this study no such assumption is made and a linear model is only used for sets of data corresponding to the same displacement amplitude of a nonlinear elastic system. The trends found between small-amplitude vibration dynamic properties and past levels of maximum displacement in various reinforced concrete structures are reported. In addition to analytical and numerical studies, results from a series of laboratory tests are reported to demonstrate the use of the approach. One full-scale three-story reinforced concrete flat-plate building and six small-scale reinforced concrete beams were examined. In this study, small displacements are defined as displacements below an overall drift ratio of 0.03%. The displacement dependence of the dynamic properties is considered explicitly. It was found that while fundamental frequencies of the examined reinforced concrete specimens were found to decrease uniformly as past peak displacement level increased, the equivalent viscous damping ratio was found to increase until the past peak displacement reached the neighborhood of nominal yield displacement and then observed to decrease when the specimens are pushed beyond the nominal yield displacement level, which has not been reported in literature before. Recommendations are provided as to how small amplitude vibration tests should be set up to avoid misleading observations due to nonlinear response at small amplitude response, observations that could lead to erroneous conclusions regarding the damage state of a structure.     

Identification of the representative crack length evolution in a multi-level interface model for quasi-brittle masonry

It is proposed here to identify the law of crack length evolution with a small number of parameters governing a recently presented model (Rekik and Lebon, submitted for publication) describing the interface behavior in damaged masonry. Studies on non-confined medium- and large-sized masonry structures have shown that it is necessary to obtain a linear increasing crack in the post-peak part of the “stress–strain or –displacement” diagram. In confined masonry structures showing softening and sliding parts, the results obtained with this crack evolution failed to match the experimental data. The crack lengths identified in the post-peak part at several points on the experimental “stress–displacement” diagram show that the representative crack length is a bilinear or trilinear function describing the increase in the crack length with respect to the decrease in the shear stress. Numerical studies on medium- and large-sized masonry structures consisting of the same materials subjected to various loads were performed to determine the ultimate crack length, and the results are relatively insensitive to the size of the masonry and the type of the load applied. The numerical local fields determined in the elementary and full-scale structures investigated were used to test the validity of the present model at the local scale, as well as to obtain an additional unilateral condition in the case of compressed masonry structures in order to prevent overlapping between the masonry components.     

转换矩阵在非线性材料杆系分析中的应用

证明了在杆系中,力的转换矩阵与位移的转换矩阵互为转置矩阵,当静不定非线性杆系静力平衡方程确定,而变形协调条件难以确定时,利用转置矩阵可以方便求得静不定非线性杆系的内力及有关节点位移.非线性材料杆系应力-应变关系 σ=Bε1/n中的幂n=2时,非线性材料静不定桁架有可能存在两个解;而采用常规方法求解静不定非线性杆系内力时...     

非线性材料杆系结构的内力求解

随着科技的发展,已研发出了许多非线性材料,这些非线性材料组成的结构已在工程实际中得到了广泛应用.本文研究了静不定非线性材料结构杆件内力的求解这一关键问题.利用非线性材料结构余能及静力平衡条件,构造了新泛函数,对新泛函数进行一阶求导,即可方便求得静不定非线性材料结构杆件的内力.同时,指出了非线性材料杆系结构的内力求解时存在漏解的原因:结构材料的应力与应变本构关系为非线性幂函数,把结构余能对杆系的内力求一阶偏导,令所得到的位移方程等于零,该方程为一元二次方程,应有两个实数解,而有关文献却遗漏了一个实数解.     

Virtual displacements principle,existence and uniqueness for elastic no tension bodies

Aim of this paper is the formulation of the virtual displacement principle, and some its applications, for bodies made of masonry like–material, i.e. the elastic no tension material (ENT). In place of the common adoption of continuously diffused smeared cracks, used to include both cracks and fracture strains, the paper considers the direct presence of discontinuities of the displacement fields, i.e. the cracks occurring in masonry bodies. The paper makes use of the notion of the new boundary of the body which includes the fractures, following the approach of Volpert and Hudjaev (Analysis in classes of discontinuous functions and equations of mathematical physics, Nijhoff, Dordrecht, 1985). In this framework the paper extends to the ENT bodies a previous formulation of the principle of virtual displacements, given by the Author, for the simpler case of the rigid no tension bodies. The consequent study of the existence and uniqueness of the admissible equilibrium concludes the paper.     

Finite-Element Analysis of Shells of Revolution by Two Doubly Curved Quadrilateral Elements

Two finite-element models, one derived from the principle of minimum total potential energy and the other derived from a modification of the same, are considered for linear analysis of shells of revolution. The first is a compatible displacement model in which an element interior displacement field that generates compalible interelement boundary displacements and rotations is assumed, but no rigid-body displacement modes are included in the element displacement field; the second is a hybrid displacement model in which rigid-body modes are included explicitly in the displacement field in the interior of the element and a Lagrangian multiplier technique is needed to satisfy the     

葵花型索穹顶结构等效平面桁架静力计算方法

选取n次轴对称葵花型索穹顶结构的1/n,将空间轴对称结构等效为一次超静定平面桁架结构.给出了葵花型索穹顶结构撑杆不在同一径向上的内力和位移处理方法,推导了力法求解约束力的计算公式,提出了小变形时各点位移计算的位移传递矩阵和计算公式.最后,通过一个算例介绍了该方法的计算过程并与有限元计算结果进行了比较,内力和位移误差均在工程设计要求内,验证了方法的正确性和准确性.     

Displacement and strain measurement with automated grid methods

An image-processing-based automated grid method is investigated to determine the method's displacement and strain accuracy limits, and how these limits are influenced by the choice of camera-calibration models. A CCD camera and a PC-based frame grabber are used to record grid spot motion, then ordering and centroiding are used to identify each spot and calculate their individual displacements. The displacements are fitted with a moving biquadratic surface, and the strains are obtained by analytical differentiation of that surface. Camera-calibration models which are considered include various combinations of image-perspective transformation, image stretching, and elliptical-lens distortion. The strain and displacement accuracy are explored through rigid-body motion and uniaxial tension tests. In the process, sensitivity to in-plane and out-of-plane rigid-body translation, and extreme sensitivity to in-plane rigid-body rotation (for non-synchronized frame grabbers) are confirmed. It is found that under the best conditions the displacement accuracy is 015 pixels and that the strain accuracy is 120 microstrain. Finally, the automated grid method is used to investigate the strains developed in an aluminum perforated strip subjected to uniaxial tension.