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Analysis of masonry structures: review of and recent trends in homogenization t

Analysis of masonry structures: review of and recent trends in homogenization techniques1 Paulo B. Lourenc ¸o, Gabriele Milani, Antonio Tralli, and Alberto Zucchini Abstract: The mechanics of masonry structures have been underdeveloped for a long time in comparison with other fields of knowledge. Presently, nonlinear analysis is a popular field in masonry research and homogenization techniques play a major role despite the mathematical and conceptual difficulties inherent to this approach. This paper addresses different homogenization techniques available in published literature, aiming at defining a first catalogue and at discussing the ad- vantages and disadvantages of the different approaches. Finally, special attention is given to a micromechanical based model and a model based on a polynomial expansion of the microstress field. These seem promising and accurate strat- egies for advanced structural analysis. Key words: masonry, homogenization, limit analysis, finite elements, in-plane loads, out-of-plane loads. Re ´sume ´ : L’aspect me ´canique des structures en mac ¸onnerie s’est peu de ´veloppe ´ par rapport a ` d’autres domaines techni- ques. L’analyse non line ´aire est pre ´sentement un domaine populaire de la recherche en mac ¸onnerie et les techniques d’ho- moge ´ne ´isation y jouent un ro ˆle important, malgre ´ les difficulte ´s mathe ´matiques et conceptuelles associe ´es a ` cette approche. Le pre ´sent article traite des diffe ´rentes techniques d’homoge ´ne ´isation disponibles dans la litte ´rature, vise a ` de ´fi- nir un premier catalogue et discute des avantages et des inconve ´nients des diffe ´rentes approches. Finalement, une attention particulie `re est accorde ´e a ` un mode `le microme ´canique et a ` un mode `le base ´ sur une extension d’un polyno ˆme du domaine des microcontraintes. Ce sont des strate ´gies qui semblent prometteuses et pre ´cises pour des analyses structurales pousse ´es. Mots-cle ´s : mac ¸onnerie, homoge ´ne ´isation, analyse a ` la limite, e ´le ´ments finis, charge dans le plan, charge hors plan. [Traduit par la Re ´daction] 1. Introduction Masonry is a heterogeneous material that consists of units and joints. Units can be bricks, blocks, ashlars, adobes, irregular stones, and others. Mortar can be clay, bitumen, chalk, lime- or cement-based, glue, or others. The huge number of possible combinations generated by the geometry, nature, and arrangement of units as well as the character- istics of mortars raises doubts about the accuracy of the term ‘‘masonry.’’ Still, much information can be gained from the study of regular masonry structures, in which a periodic repetition of the microstructure occurs due to a con- stant arrangement of the units (or constant bond). The difficulties in performing advanced testing of these types of structures are quite large due to the numerous var- iations of masonry, the large scatter of in situ material prop- erties, and the impossibility of reproducing it all in a specimen. Therefore, most of the advanced experimental re- search carried out in the last few decades concentrated in brick–block masonry and its relevance for design. Accurate modelling requires a comprehensive experimental descrip- tion of the material, which seems mostly available at the present state of knowledge (CUR 1997; Lourenc ¸o 1998). The global field of structural analysis of masonry struc- tures encompasses several different approaches and a com- prehensive review is given in Lourenc ¸o (2002). The present paper focuses exclusively on the analysis of unreinforced masonry structures making use of homogenization tech- niques, which has been receiving a growing interest from the scientific community. A comprehensive review of the current state of the art, aiming at a classification of approaches, and a discussion on advantages and disadvantages of the different techniques is included. To overcome approximations and limitations identified, two powerful approaches are addressed in detail: (i) a micromechanical homogenization approach that consid- ers additional internal deformation mechanisms and (ii) a polynomial expansion of the microstress field inside an ele- mentary masonry cell. 2. Homogenization theory: basic assumptions This section briefly recalls the basis of the theory of homogenization applied to masonry structures, with particu- lar emphasis on running bond texture. Consider a masonry wall consisting of the periodic arrangements of masonry Received 24 September 2006. Revision accepted 27 June 2007. Published on the NRC Research Press Web site at cjce.nrc.ca on 19 December 2007. P.B. Lourenc ¸o.2 Associate Professor, Department of Civil Engineering, University of Minho, Azure ´m, 4800-058 Guimara ˜es, Portugal. G. Milani and A. Tralli. University of Ferrara, Via Saragat 1, 44100 Ferrara, Italy. A. Zucchini. ENEA, FIS.MET, v. Don Fiammelli, 2, 40129 Bologna, Italy. Written discussion of this article is welcomed and will be received by the Editor until 31 March 2008. 1This article is one of a selection of papers published in this Special Issue on Masonry. 2Corresponding author (e-mail: pbl@civil.uminho.pt). 1443 Can. J. Civ. Eng. 34: 1443–1457 (2007) doi:10.1139/L07-097 # 2007 NRC Canada units and mortar joints as shown in Fig.1. The periodicity allows one to regard as the repetition of a representative element of volume (REV or elementary cell) Y. Let x ¼ ½x1; x2 be a frame of reference for the global des- cription of (macroscopic scale) and y ¼ ½y1; y2; y3 be a frame of reference for Y. The Y module is defined as Y ¼ ! t=2; t=2½, where Y 2 <3 is the elementary cell and ! 2 <2 represents the middle plane of the plate (Caill- erie 1984). The @Y boundary surface of the elementary cell (see Fig. 1) is defined as @Y ¼ @Yl [ @Yþ 3 [ @Y 3 , where @Y 3 ¼ ! ½t=2. The basic idea of homogenization consists in introducing averaged quantities representing the macroscopic stress and strain tensors (denoted here, respectively, as E and S), as follows: ½1 E ¼< " " " " " >¼ 1 V Z Y " " " " "ðuÞ dY ¼< >¼ 1 V Z Y dY where V stands for the volume of the elementary cell, e and stand for the local quantities (stresses and strains, respec- tively), and <*> is the average operator. According to Anthoine (1995) and Cecchi et al. (2005), the homogenization problem in the linear elastic range in the presence of coupled membranal and flexural loads, under the assumption of the Kirchhoff-Love plate theory, can be written as follows: ½2a div ¼ 0 ½2b ¼ að yÞ" ½2c " " " " " ¼ E þ y3 þ symðgrad uperÞ ½2d e3 ¼ 0 on @Yþ 3 and @Y 3 ½2e n antiperiodic on @Yl ½2f uper periodic on @Yl where is the microscopic stress tensor (microstress), a(y) represents a !-periodic linear elastic constitutive law for the components (masonry units and mortar), E is the macro- scopic in-plane strain tensor, is the out-of-plane strain tensor (curvature tensor), uper is a o-periodic displacement field, e3 is the unit vector normal to the masonry middle plane, and n is the unit vector normal to the internal cell boundary. Equation [2a] represents the microequilibrium for the elementary cell with zero body forces, usually neglected in the framework of homogenization. Furthermore, in eq. [2c], the microstrain tensor " " " " " is ob- tained as a linear combination among macroscopic E and tensors and a periodic strain field. E and tensors are re- lated to the macroscopic displacement field components U1ðx1; x2Þ, U2ðx1; x2Þ, and U3ðx1; x2Þ by means of the classic relations E ¼ ðU; þ U;Þ=2, with Ei3 ¼ 0, and ¼ U3; with i3 ¼ 0, ; ¼ 1; 2, and i = 1,2,3. Macroscopic homogenized membrane and bending con- stants, M and N, respectively, can be obtained solving the elastostatic problem [2a–f] and making use of the classic relations: ½3 N ¼ h i ¼ AE þ B M ¼ hy3 i ¼ BTE þ D where A, B, and D are the constitutive homogenized plate tensors. Usually, the elementary cell has a central sym- metry, hence B = 0. As a rule, a solution for the problem given by eqs. [2a–f] can be obtained using standard finite element (FE) packages, as suggested for the in-plane case by Anthoine (1995). The governing equations in the non- linear case are formally identical to eq. [3], provided that a non linear stress–strain law for the constituent materials is assumed. 2.1. Closed-form solutions in the linear elastic range This section briefly recalls some of the most popular sim- plified approaches that have appeared in the past in tech- nical literature for obtaining homogenized elastic moduli for masonry. As the elastostatic problem given in eq. [3] cannot be uploads/Ingenierie_Lourd/ lourenco-recent-developments-in-homogenization-techniques-pdf 1 .pdf