Analytical Micro-Modeling of Masonry Periodic Unit Cells Elastic Properties

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1 *Revised Manscript (Unmarked) Click here to view linked References Analytical Micro-Modeling of Masonry Periodic Unit Cells Elastic Properties Anastasios Drogkas 1, Pere Roca, Climent Molins Departament d Enginyeria de la Constrcció, Universitat Politècnica de Catalnya, Camps Diagonal Nord, Bilding C1, Jordi Girona 1-3 UPC, Barcelona, Spain Highlights The mechanical properties of masonry composites are derived from PUC calclations Several different masonry wall and pillar typologies are investigated The orthotropic elastic characteristics are determined with very low comptational cost The nmerical reslts are compared to experimental data A parametric investigation is performed to determine the accracy of the model compared to FE calclations Keywords Micro-modeling; analytical modeling; masonry; elastic modli 1. Introdction 1.1 State of the Art Masonry strctres crrently constitte a very large part of the existing bilt environment. They inclde a large amont of common bt still-in-se bildings as well as otstanding cltral heritage strctres. Masonry bildings are comprised of strctral members sch as load bearing walls, pillars and valts composed of single 1 Corresponding Athor addresses: anastasios.drogkas@pc.ed (Anastasios Drogkas), pere.roca.fabregat@pc.ed (Pere Roca), climent.molins@pc.ed (Climent Molins) 1

2 or mltiple brick or stone masonry leaves with possible interior rbble infills. An accrate strctral analysis of sch bildings is important as a way to verify their capacity against gravity loads and horizontal actions sch as wind and earthqake. However, the strctral analysis of masonry bildings faces significant challenges de to the complex behavior of masonry which stems from its composite character, the inflence of the geometric arrangement of nits and the fact that most tilized mortars only provide a weak bond between nits. One of the conseqences of the composite natre of masonry is fond in its anisotropic mechanical response. Another emerging conseqence is the difficlty of predicting the average masonry mechanical properties, sch as, in particlar, the masonry compressive strength and the Yong s modls, from the properties of the constitent materials. Even when the elastic properties of the constitent materials are known, it is difficlt to derive the elastic properties of the composite withot resorting to sophisticated analysis tools. Frthermore, reslts for one masonry typology may not be readily sitable for the evalation of the properties of another. Therefore, the development of simple tools for the derivation of the orthotropic elastic properties of a variety of masonry typologies based on the elastic properties of their constitent materials, while maintaining a single analysis methodology throghot, constittes a sond framework for the frther development of analysis tools based on the detailed micro-modeling approach. Masonry strctres composed of periodically repeating patterns may be simplified in order to facilitate their analysis. In this sense, it is possible to calclate the elastic and strength properties of masonry strctres throgh the analysis of a geometrically repeating part. This part may be frther simplified by taking into accont symmetry arising from geometrical and loading conditions. A nmber of analytical and nmerical models has been proposed for the analysis of masonry periodic nit cells sing a variety of methods. These have been employed for the derivation of the elastic and inelastic properties of masonry composites as a reslt of the interaction of their two phases: nits and mortar. The cells are analyzed considering appropriate bondary conditions, kinematic compatibilities and stress eqilibrim in order to derive the strains and stress state in the cell for different loading conditions. Nonlinear properties may be derived from the iterative soltion of the problem nder the assmptions of plasticity or damage models for the behavior of the materials in the cell. 2

3 Throgh these comptations in the micro- or meso-scale it is possible to calclate the orthotropic behavior of the masonry composite in the macro-scale, arising from the geometrical arrangement of the two phases in the cell. Elastic [1 4] and strength properties [5 8] of the composite for in-plane and ot-of-plane load conditions have been calclated sing sch techniqes. Research has extended from the stdy of periodic masonries to nonperiodic masonries [7,9]. Methods for the analysis of masonry cells sing analytical methods have been proposed, mostly focsing on the analysis of stack bond and rnning bond masonry wall typologies [4,10,11]. Finite element representations of masonry nit cells have also been proposed [12] in which the interaction of the two phases, the reslting stress and strains in the cell and the nonlinear behavior of the materials may be accrately represented. This approach has also been adopted for the verification of the accracy of analytical models, sch as the ones already described, the prodction of in-plane strength domain crves for masonry membranes and the exection of two-scale analyses. However, FE calclations may reqire time for the creation of the models and high comptational effort. A comparison of analytical model and FE model reslts may be fond in [3]. A micromechanical approach for the analysis of periodically reinforced composites, according to which a repeating cell of the composite is discretized into sb-cells with different properties and arranged in a reglar grid, has been proposed in the past [13]. Eqilibrim and compatibility conditions are assembled in a set of closed form expressions and can be solved in a single analysis step for the derivation of, for example, the average elastic properties of the composite. The benefit of this approach is its comptational efficiency and relative simplicity. Masonry periodic nit cells, seen as reglar arrangements of sqare or cbic sb-cells with different material properties, can be analyzed sing the same approach. Adaptations of this method, capable of providing closed form expressions for the elastic properties of stack and rnning bond masonry, have been proposed [14,15]. Masonry analyzed in this manner is sally idealized as an infinitely thin or infinitely thick membrane, these assmptions being accordingly eqivalent to a plane stress and plane strain approach. However, the existence of transversal joints, gaps or other discontinities, which reslt in non-constant geometric strctre along the depth of the masonry, render these two approaches fndamentally not accrate. Analysis of the nit 3

4 cell taking into accont these discontinities mst consider the actal finite thickness and actal geometry of the masonry strctre. Comptations on cells where the actal finite thickness of the masonry is considered allow for a more accrate representation of ot-of-plane stresses, which, while only marginally affecting the initial elastic stiffness, may strongly inflence the compressive strength of the composite [16]. However, since it is intended to apply the models proposed here in nonlinear analysis in a following paper, it is deemed necessary to inclde a realistic representation of the ot-off-plane stresses sing three-dimensional models for linear elastic analysis as well. With this type of models, the tre thickness of the masonry may be easily taken into accont in finite element calclations. Throgh comptations on periodic nit cells it is possible to make fairly accrate predictions of the compressive and tensile strength and the elastic modli of masonry. Sch techniqes may be, therefore, sed for two-scale modeling of masonry walls [17 21]. 1.2 Objectives The prpose of this paper is to present a simple analytical calclation method for the derivation of the elastic characteristics of masonry strctres. The method is based on the analysis of masonry periodic nit cells, the smallest repeating geometrical entity representative of the overall masonry geometric pattern, in an attempt to derive masonry composite orthotropic macro-properties from material and geometrical micro-properties. The micro properties considered are those of the nits, the mortar and the infill. Strctral members allowing this type of modelling inclde single- and mlti-leaf walls and pillar-like strctres. These properties are to be determined for normal and shear loading. The models are formlated based on three-dimensional elasticity in order to inclde the inflence of ot-of-plane stresses on the response. These macro-properties may be sed for the analysis of large walls and other strctres in fll mlti-scale models or may be sed to provide the information needed for analysis with orthotropic material models, sch as the Hill, Rankine-Hill or the Hoffmann criteria. The comptational cost associated with FE analysis of cells in two-scale analyses is still relatively high, especially when the analyses are carried ot on cells in a threedimensional configration. Analytical models for the analysis of the cells present an advantageos, in terms of 4

5 comptational cost, alternative. Finally, the low comptational cost of sch analytical models makes the prodction of large databases of reslts ndemanding. The verification of the model is performed throgh a comparison against a FE reslt benchmark following a parametric investigation. A comparison is also carried ot with a range of reslts from the existing inventory of experimental data. This stdy incldes the analysis of varios masonry wall geometrical typologies: single leaf stack bond, single-leaf rnning bond, doble-leaf Flemish bond and three-leaf walls with a rnning bond oter leaf. In the last case, the infill constittes a third material phase, with different mechanical properties from the nits and mortar normally considered in sch analyses. These masonry typologies, and especially the last three, represent a significant portion of existing and new strctres, with the available research focsing primarily, if not entirely, on rnning rather than Flemish bond strctres. Three-leaf walls with rbble infill are also commonly encontered both in large strctres and common bildings. Stack bond pillars are commonly sed as specimens for experimental investigation, therefore a large body of experimental work exists dealing with their elasticity and strength properties. English bond prisms are similarly sed for experimental investigation prposes as well as in strctral practice. The stdy focses on masonries characterized by nits with an elastic stiffness higher than that of the mortar. However, the applicability of the model is examined throgh an analysis of experimental case stdies with a wide range of properties. It has been noted that there is a lack of experimental reslts concerning the horizontal and transversal Yong s modli, the shear modli and the Poisson s ratios of masonry in experimental cases which inclde also a detailed mechanical characterization of the nits and mortar. This lack of experimental reslts has been compensated by sing finite element reslts as benchmark cases. 2. Overview of the Models 2.1 Derivation of Wall Periodic Unit Cells The derivation of the periodic nit cell is accomplished throgh the identification of the repeating pattern in the strctre and its sbseqent simplification de to symmetry conditions. The for masonry typologies stdied, in order of complexity, are stack bond, rnning bond, Flemish bond and three-leaf masonry with a 5

6 rnning bond oter leaf and infill. The derivation patterns for the masonry wall typologies considered in this work are shown in Figre 1. Stack bond and rnning bond walls are single leaf strctres with no variation of the geometry along the thickness. Flemish bond walls inclde a thin transversal joint and the nits are oriented as either header or stretcher blocks, meaning that for the largest portion of the wall the geometry changes along the thickness. Three-leaf walls are composed of oter leaves of any type and an interior leaf whose thickness may be larger than that of the oter leaves. Increase in the geometrical complexity of the composite reslts in an increase in the size of the repeating pattern and, therefore, of the nit cell. In stack bond and rnning bond masonry the layot of the masonry does not change throgh the thickness and all nits are identically oriented, while in Flemish bond masonry the midthickness is characterized by a transversal mortar joint and intersecting header nits. The anisotropic behavior of the masonry composite is the reslt of the geometrical arrangement of components, which are here considered isotropic, althogh the orthotropic behavior of bricks may be implemented with only minor alterations in the model. The infill is also considered macroscopically isotropic and is normally more deformable than the oter leaf. 2.2 Derivation of Pillar Periodic Unit cells The periodic nit cells for pillar-like strctres are derived similarly to the wall typologies. Two different strctral types are considered: stack bond prisms and English bond pillars. The derivation of the cells is shown in Figre Discretization of the Derived Cells The analytical model for the analysis of masonry periodic nit cells is based on the discretization of the cell in strctral parts, the modeling of their interaction in terms of deformation and stress distribtion and eqilibrim nder external loads. The eqilibrim problem will be primarily solved for three normal stress components, σ xx, σ yy and σ zz and six shear stress components, σ xy, σ xz, σ yx, σ yz, σ zx and σ zy in order to obtain the elastic properties of the cells in the three orthogonal directions dictated by the geometric arrangement of the masonry. 6

7 The discretization of the periodic nit cells is done as illstrated in Figre 3. It reslts in each cell being discretized into cboid parts of nits, bed joints, head joints, transversal joints, cross joints and infill. Specifically: the stack bond cell is discretized into one stretcher nit, one bed joint, one head joint and one cross joint part; the rnning bond cell is discretized into for stretcher nit, one bed, two head and two cross joint parts; the Flemish bond cell is discretized into for header nit, six stretcher nit, three bed joint, for head joint, seven cross joint and six transversal joint parts; the three-leaf cell is discretized similarly to the way the rnning bond cell with the addition of nine transversal joint parts representing the infill. The stack bond pillar is discretized into a nit part and a mortar part arranged in a layered pattern and the English bond pillar is discretized into for nit parts, one bed joint part two head joint parts and for cross joint parts. The orthogonal discretization carried ot separates the cell into parts allocated in an orthogonal grid, with each part belonging in one horizontal, one vertical and one transversal strip of cboid parts. As a reslt, the stack bond cell is divided into two horizontal, two vertical and two transversal strips, the rnning bond cell into three horizontal, three vertical and six transversal strips and the Flemish bond cell into six horizontal, ten vertical and fifteen transversal strips. The type of the loading applied on the cell and its orientation compared to the orthogonal grid dictates the assmptions made for the elastic analysis of the cell, which are based on the strains to which each strip is sbjected. The proposed discretization of mortar joints and nits allows for simplified assmptions in the formlations of the model. The final discretization scheme is show in Figre Development of the Analytical Model 3.1 General Assmptions and Eqations The deformation of the cell faces, and, therefore, the total strain of the cell, depends on the loading applied. Application of normal stress reslts only in normal global cell strains and the application of shear stress reslts only in shear global cell strains. This means that nder normal stress the total deformation of all strips, or the average strain of each one, is eqal in the three principal directions. Under the application of normal stress, either only normal or both normal and shear stresses may arise in the cboid parts. In trn, nder the application of shear stress, only shear stresses of the same geometrical orientation arise in the cboid parts. All stresses and strains are assmed constant in the cboid parts. For linear 7

8 elastic analysis, the nits and the mortar are modeled as three-dimensional isotropic contina. Perfect bond is considered at the nit-mortar interface for the linear elastic comptations performed here, so that all deformation of the cell is acconted for in the nits, mortar and infill. Isotropic linear elasticity stress-strain relations in three dimensions apply for every cboid component nder normal and shear stress. This incldes the nits, the mortar and the infill. These relations are expressed as: ii, n ii, n n n jj, n kk, n n E E (1) ij, n ij, n 1 n En (2) where the sb-index n refers to the identifier of the cboid, σ ii and ε ii are the applied normal stress and calclated normal strain along axis i, σ ij and ε ij are the shear stress and strain in plane ij, E n is the Yong s modls and ν n is the Poisson s ratio. When a cboid with dimensions deformation of the cboid D i and D j is sbjected to a shear stress ij the contribtion of the shear ij d i to its normal deformation in the direction i is taken as being half the displacement at its top de to the shear stress. Therefore, it is defined as d ij i Dj (3) ij, n 2 The three Yong s modli for each cell configration are calclated by se of the eqation E (4) c, i ii ii The Poisson s ratios are determined by the eqation (5) c, ij jj ii The three shear modli for each cell configration are calclated by se of the eqation: 8 c, ij ij 2 ij G (6)

9 The x, y and z axes in the model correspond to the horizontal, vertical and transversal directions. The transversal direction is normal to the face of the walls and the vertical direction is parallel to the axis of the pillars. The total strain in each of the three directions and the three reslting planes depends on the type of cell. The cboid parts are designated by a set of initials. Throghot all the cases these are: for nits in general, s and d for stretcher and header nits respectively where both are present (sch as the Flemish bond case), h for head joints, c for cross joints, b for bed joints and t for transversal joints. Infill is designated as i. Dimensions are designated according to their orientation: l corresponds to a horizontal length, h to a vertical height and t to a transversal thickness. The dimension symbols are finally sffixed, m or i for nits, mortar and infill respectively, meaning that h is the nit height, h m is the thickness of the bed joint, l is the nit length, l m is the thickness of the head joint, t is the width of the nit, t m is the thickness of the transversal joint and t i is the transversal thickness of the infill. Figre 4 illstrates the discretization of the single leaf wall cells along with the naming convention for each component in detail. The discretization of the mlti leaf walls is shown in Figre 5 and that of the pillars in Figre 6. Normal stress eqilibrim eqations are formed according to ii, n A n ii A0 (7) n while shear stress eqilibrim eqations are formed according to ij, n A n ij A0 (8) n i. where A is the total cross sectional area of the cell and A n is the cross sectional area of cboid n in direction All normal and shear stress eqilibrims at the faces or cross sections of the cell assre global eqilibrim of internal and external stress. External stresses are averaged over the srface of the cell, meaning that there may exist a mismatch between the external average stress and the stress of an individal cboid. 9

10 3.2 Stack Bond Wall Cell Sbjected to Normal Stress De to the simple geometrical layot of the cell, which does not inclde complex geometrical interlocking of mortar joints and nits, shear stress components are disregarded for applied normal stress. Normal strain conformity is assmed as follows: zz, h zz, zz, c zz, b xx, h xx, c xx, xx, b (9) yy, h yy, yy, c yy, b The above strain conformity relations assme eqal ot-of-plane strains for all cboids and eqal in-plane strains for parallel cboids in a given direction. Horizontal normal stress eqilibrim conditions at the right and left face of the cell are as follows:,, h 2 h 2 h 2 h 2 xx m xx xx b m (10),, h 2 h 2 h 2 h 2 xx m xx h xx c m Vertical normal stress eqilibrim conditions at the top and bottom faces of the cell are as follows:,, l 2 l 2 l 2 l 2 (11) yy m yy h m yy,, l 2 l 2 l 2 l 2 yy m yy c m yy b Transversal normal stress eqilibrim at the front face of the cell is as follows: 10

11 l 2 l 2h 2 h 2 zz m m l 2h 2 l 2h 2 l 2h 2 l 2h 2 zz, h m zz, zz, c m m zz, b m (12) The normal strains of the entire cell along the three axes can be calclated as, l 2, l 2 l 2 l 2 xx xx h m xx m, h 2, h 2 h 2 h 2 (13) yy yy h yy c m m zz zz, h Cell Sbjected to Shear Stress For xy shear it is assmed that (14) xy, h xy, xy, c xy, b The shear stress eqilibrim conditions at the top and right faces of the cell are as follows:,, l 2 l 2 l 2 l 2 xy m xy h m xy,, h 2 h 2 h 2 h 2 (15) xy m xy xy b m The cell shear strain can be calclated as, h 2, h 2 h 2 h 2 (16) xy xy c m xy h m For xz shear it is assmed that xz, xz, b (17) xz, c xz, h xz 11

12 Shear stress eqilibrim at the right face of the cell is as follows:,, h 2 h 2 h 2 h 2 (18) xz m xz b m xz The cell shear strain is eqal to, l 2, l 2 l 2 l 2 (19) xz xz xz h m m For yz strain it is assmed that yz, h yz, (20) yz, c yz, b Shear stress eqilibrim at the front and top faces of the cell is as follows: l 2 l 2h 2 h 2 yz m m l 2h 2 l 2h 2 l 2h 2 l 2h 2 yz, h m yz, yz, c m m yz, b m (21),, l 2 l 2 l 2 l 2 yz m yz h m yz The cell shear strain is eqal to:, h 2, h 2 h 2 h 2 (22) yz yz yz b m m 3.3 Rnning Bond Wall Cell Sbjected to Normal Stress The geometrical interlocking of the nits and the mortar reqires that the in plane shear stresses be taken into accont for normal stress loading conditions. Therefore, apart from the three components of normal stress for each cboid component, the xy shear stress and strains also have been inclded for the formlation of the system of eqations. In trn, the compatibility conditions take into accont the shear deformation of the cboids, primarily those of the bed and cross joints, for applied loads in the horizontal direction. The reslting shear stresses are negligible for loading in the vertical and transversal directions. 12

13 The horizontal normal stress eqilibrim conditions at the left face and at a cross section across the middle of the cell are expressed as follows:, 2,, h h h 2 h h 2 (23) xx m xx xx c m xx h, 1,, 1 h h h 2 h h 2 xx m xx xx b m xx The vertical normal stress eqilibrim conditions at the top face and at a cross section across the middle of the cell are as follows: l 2 l 2 l 2 l 2 l 2 l 2 (24) yy m yy, h m yy, 1 m yy, 2 m l 2 l 2 l 2 l 2 l 2 l 2 yy m yy, c m yy, b m yy, c m The transversal normal stress eqilibrim condition at the front face of the cell is as follows: h h l 2 l 2, 2, 1 2 2, 2 2 2, 2, 2 2 zz m m h l l l l h l l l zz h m zz m zz m m zz c m zz b m (25) The shear stress eqilibrim conditions at the top face and at a cross section across the middle of the cell are as follows: xy, h m xy, 1 m xy, 2 m l 2 l 2 l 2 l 2 0 (26) l 2 l 2 l 2 l 2 0 xy, c m xy, b m xy, c m Deformation compatibility of the cboids, considering both normal and shear deformation along the horizontal axis and only normal deformation along the vertical and transversal axis, is as follows: zz, h zz, 1 zz, 2 zz, c zz, b yy, h yy, 1 yy, 2 13

14 (27) yy, c yy, b xx, h xx, c xx, 1 xx, b l 2 l 2 l 2 l 2 h 4 xx, h m xx, 1 m xx, 2 m xy, h xy, 1 xy, 2 2 l 2 l 2 l 2 2 h 2 xx, c m xx, b m xy, c xy, b m The following assmptions are made abot the normal and shear stress distribtion in the cell: (28) xx, 1 xx, 2 xy, 1 xy, 2 xy, h These assmptions, copled with the global shear stress eqations for the cell reslts in zero shear stresses for the nits and head joint. They also reslt in identical horizontal effective stresses for the 1 and 2 cboids, which represent the nit. This invalidates the partition of the nit into two parts, bt the distinction is meaningfl for non-linear analysis, where the two nit parts may sstain different degrees of damage. The normal strains of the entire cell along the three axes are eqal to, l 2, 1 l 2 l 2, 2 l 2,, 1, 2 h 4 l 2 l 2 xx xx h m xx m xx m xy h xy xy m, h 2, h, 2 h 2 h h (29) yy yy h yy c m yy m zz zz, h Cell Sbjected to Shear Stress For xy shear it is assmed that (30) xy, h xy, 1 xy, 2 14

15 xy, c xy, b The shear stress eqilibrim conditions at a horizontal cross section and a vertical cross section at the middle of the cell are as follows: l 2 l 2 l 2 l 2 l 2 l 2 (31) xy m xy, c m xy, b m xy, c, m, 1,, 1 h h h 2 h h 2 xy m xy xy b m xy The cell shear strain is eqal to, h 2, h, 2 h 2 h h (32) xy xy h xy c m xy m For xz shear it is assmed that xz, 1 xz, b (33) xz, 1 xz, 2 xz, c xz, 2 The stress eqilibrim conditions at the front and left face and at a cross section at the center of the cell are as follows: h h l 2 l 2, 2, 1 2 2, 2 2 2, 2, 2 2 xz m m h l l l l h l l l xz h m xz m xz m m xz c m xz b m,,, 2 h h h 2 h h 2 xz m xz h xz c m xz (34), 1,, 1 h h h 2 h h 2 xz m xz xz b m xz The cell shear strain is eqal to, l 2, 1 l 2 l 2, 2 l 2 l 2 l 2 (35) xz xy h m xy m xy m m 15

16 For yz shear it is assmed that (36) yz, h yz, 1 yz, 2 yz, c yz, b Shear stress eqilibrims at the top and front faces of the cell are as follows: l 2 l 2 l 2 l 2 l 2 l 2 (37) yz m yz, h m yz, 1 m yz, 2 m h h l 2 l 2, 2, 1 2 2, 2 2 2, 2, 2 2 yz m m h l l l l h l l l yz h m yz m yz m m yz c m yz b m The cell shear strain is eqal to, h 2, h, 2 h 2 h h (38) yz yz h yz c m yz m 3.4 Flemish Bond Wall Cell Sbjected to Normal Stress As in the rnning bond cell model, the in-plane shear deformation of the bed joint is taken into accont for normal stress loading. The transfer of stress across the transversal joint is also taken into accont withot, however, considering the effects of shear stress in other planes. Also, as in the case of the rnning bond cell, shear stresses are negligible for vertical and transversal applied normal stress. The horizontal normal stress eqilibrim conditions at the left face and at two cross sections along the length of the cell are as follows: h h t t 2 xx m m, 1, 1, 3, 2, 2, 3 h / 2 h h / 2 t h / 2 h h / 2 t 2 xx d xx b m xx s xx d xx c m xx t m h h t t 2 xx m m, 1, 1, 2, 2, 3, 2 h / 2 h h / 2 t h / 2 h h / 2 t 2 xx h xx c m xx s xx h xx c m xx t m (39) 16

17 h h t t 2 xx m m, 1, 2, 1, 1, 4, 1 h / 2 h h / 2 t h / 2 h h / 2 t 2 xx s xx b m xx s xx t xx c m xx t m The vertical normal stress eqilibrim conditions at the top face and at a cross section across the middle of the cell are as follows: t t 2t 2 l l 2 yy m m t 2 l l 2 l t 2 l t 2 t yy, d1 yy, h1 m yy, s1 m yy, s2 m yy, s3 t 2 l l 2 l t 2 l t 2 t 2 yy, d 2 yy, h2 m yy, t1 m yy, t 2 m yy, t3 m (40) t t 2t 2 l l 2 yy m m t 2 l l 2 l t 2 l t 2 t yy, b1 yy, c1 m yy, b2 m yy, c1 m yy, b1 t 2 l l 2 l t 2 l t 2 t 2 yy, c2 yy, c3 m yy, c4 m yy, c3 m yy, c2 m The transversal normal stress eqilibrim conditions at the front face and at a cross section across the middle of the cell are as follows: h h t 2 l l 2 zz m m t 2 l l 2 l t 2 l t 2 h zz, d1 zz, h1 m zz, s1 m zz, s2 m zz, s3 t 2 l l 2 l t 2 h zz, b1 zz, c1 m zz, b2 m m (41) h h t 2 l l 2 zz m m t 2 l l 2 l t 2 l t 2 h zz, d 2 zz, h2 m zz, t1 m zz, t 2 m zz, t3 t 2 l l 2 l t 2 h zz, c2 zz, c3 m zz, c4 m m The shear stress eqilibrim conditions at the top face and at a vertical cross section across the middle of the cell are as follows: t 2 l l 2 l t 2 l t 2 t xy, d1 xy, h1 m xy, s1 m xy, s2 m xy, s3 t 2 l l 2 l t 2 l t 2 t 2 0 xy, d 2 xy, h2 m xy, t1 m xy, t 2 m xy, t3 m (42) t 2 l l 2 l t 2 l t 2 t xy, b1 xy, c1 m xy, b2 m xy, c1 m xy, b1 t 2 l l 2 l t 2 l t 2 t 2 0 xy, c2 xy, c3 m xy, c4 m xy, c3 m xy, c2 m 17

18 Deformation compatibility of the cboids, considering both normal and shear deformation along the horizontal axis and only normal deformation along the vertical and transversal axes, is as follows: t 2 l l 2 l t 2 l t 2 xx, d1 xx, h1 m xx, s1 m xx, s2 m xx, s3 h 4 xy, d1 xy, h1 xy, s1 xy, s2 xy, s3 t 2 l l 2 l t 2 l t 2 xx, d 2 xx, h2 m xx, t1 m xx, t 2 m xx, t3 xy, d 2 xy, h2 xy, t1 xy, t 2 xy, t3 h 4 (43) t 2 l l 2 l t 2 l t 2 xx, d1 xx, h1 m xx, s1 m xx, s2 m xx, s3 h 4 xy, d1 xy, h1 xy, s1 xy, s2 xy, s3 t 2 l l 2 l t 2 l t 2 xx, b1 xx, c1 m xx, b2 m xx, c1 m xx, b1 xy, b1 xy, c1 xy, b2 xy, c1 xy, b1 m h 2 t 2 l l 2 l t 2 l t 2 xx, d1 xx, h1 m xx, s1 m xx, s2 m xx, s3 h 4 xy, d1 xy, h1 xy, s1 xy, s2 xy, s3 t 2 l l 2 l t 2 l t 2 xx, c2 xx, c3 m xx, c4 m xx, c3 m xx, c2 xy, c2 xy, c3 xy, c4 xy, c3 xy, c2 m h 2 t t 2 t t 2 zz, d1 zz, d 2 m zz, b1 zz, c2 m t t 2 t t 2 zz, d1 zz, d 2 m zz, s3 zz, t3 m t t 2 t t 2 zz, d1 zz, d 2 m zz, h1 zz, h2 m t t 2 t t 2 zz, d1 zz, d 2 m zz, c1 zz, c3 m t t 2 t t 2 zz, d1 zz, d 2 m zz, s2 zz, t 2 m t t 2 t t 2 zz, d1 zz, d 2 m zz, s1 zz, t1 m t t 2 t t 2 zz, d1 zz, d 2 m zz, b2 zz, c4 m xx, b1 xx, c2 xx, d1 18

19 xx, s3 xx, t3 xx, c1 xx, c3 xx, h1 xx, h2 xx, s2 xx, t 2 xx, s1 xx, t1 xx, b2 xx, b2 xx, c4 yy, d1 yy, h1 yy, s1 yy, s2 yy, s3 yy, d 2 yy, h2 yy, t1 yy, t2 yy, t3 yy, b1 yy, c1 yy, b2 yy, c2 yy, c3 yy, c4 zz, s1 zz, s2 zz, s3 zz, d1 zz, h1 xy, d1 xy, d 2 xy, h1 xy, h2 xy, s1 xy, t1 xy, s2 xy, t 2 xy, s3 xy, t3 xy, b1 xy, c2 xy, c1 xy, c3 19

20 The following assmptions are made abot the normal stress distribtion in the cell: zz, c1 zz, c3 zz, b2 zz, c4 (44) zz, d1 zz, d 2 zz, s1 zz, t1 zz, s2 zz, t 2 zz, s3 zz, t3 The above set of eqations assmes that the transversal normal stress is eqal for the cboids of the two leaves of masonry, be they nit or mortar. They also, again, lead to eqal effective transversal stresses in the different cboids comprising a single header brick. Damaged stresses in non-linear analysis, however, are different. For the shear stresses it is assmed that (45) xy, s1 xy, s2 xy, s3 xy, h1 xy, d1 xy, b1 xy, b2 The normal strains of the entire cell along the three axes are eqal to: xx xx, d1 t 2 xx, h1 lm xx, s1 l 2 lm t 2 xx, s2 lm xx, s3 t 2 l 2lm t 2 xy, d1 xy, h1 xy, s1 xy, s2 xy, s3 h 4, 1 h 2, 1 h, 3 h 2 h h (46) yy yy d yy b m yy s m, 1 t, 2 t 2 t t 2 zz zz d zz d m m 20

21 3.4.2 Cell Sbjected to Shear Stress For xy shear it is assmed that: (47) xy, d1 xy, h1 xy, s1 xy, s2 xy, s3 xy, d 2 xy, h2 xy, t1 xy, t2 xy, t3 xy, b1 xy, c1 xy, b2 xy, c2 xy, c3 xy, c4 The shear stress eqilibrim condition at the left face and at a cross-section along the mid height of the cell are as follows: t t /2h h xy m m, 1, 1, 3, 2, 2, 3 h 2 h h 2 t h 2 h h 2 t 2 xy d xy b m xy s xy d xy c m xy t m t t 2l 2 l t 2 xy m m t 2 l l 2 l t 2 l t 2 t xy, b1 xy, c1 m xy, b2 m xy, c1 m xy, b1 t 2 l l 2 l t 2 l t 2 t 2 xy, c2 xy, c3 m xy, c4 m xy, c3 m xy, c2 m (48) The cell shear strain is eqal to, 1 h 2, 1 h, 3 h 2 h h (49) xy xy d xy b m xy s m For xz shear it is assmed that (50) xz, d1 xz, s1 xz, s2 xz, s3 xz, b1 xz, c1 xz, b2 xz,h2 xz, t1 xz, t2 xz, t3 xz, c2 xz, c3 xz, c4 The shear stress eqilibrim conditions at the front face, transversal section at the middle, the left face and a vertical section along the middle of the cell are as follows: l 2 l t 2h 2 hm xz m t 2 l l 2 l t 2 l t 2 h xz, d1 xz, h1 m xz, s1 m xz, s2 m xz, s3 t l l l t 2 l t 2 h xz, b1 xz, c1 m xz, b2 m xz, c1 m xz, b1 m

22 l 2 l t 2h 2 hm 2 2 xz m t 2 l l 2 l t 2 l t 2 h xz, d 2 xz, h2 m xz, t1 m xz, t 2 m xz, t3 t l l 2 l t 2 l t 2 h 2 xz, c2 xz,c3 m xz, c4 m xz,c3 m xz,c2 m 2 (51) t t 2h h xz m m, 1, 1, 3, 2, 2, 3 h 2 h h 2 t h 2 h h 2 t 2 xz d xz b m xz s xz d xz c m xz t m t t 2h h xz m m, 1, 2, 1, 1, 4, 1 h 2 h h 2 t h 2 h h 2 t 2 xz s xz b m xz s xz t xz c m xz t m The cell shear strain is eqal to:, 1 t 2, 1 l, 1 l 2 l t 2,s2 l,s3 t 2 l 2 l t 2 (52) xz xz d xz h m xz s m xz m xz m For yz shear it is assmed that: (53) yz, d1 yz, h1 yz, s1 yz, s2 yz, s3 yz, d 2 yz, h2 yz, t1 yz, t2 yz, t3 yz, b1 yz, c1 yz, b2 yz, c2 yz, c3 yz, c4 The shear stress eqilibrim conditions at the front face and a horizontal cross section at the middle of the cell follows: l 2 l t 2h 2 hm 2 yz m t 2 l l 2 l t 2 l t 2 h yz, d1 yz, h1 m yz, s1 m yz, s2 m yz, s3 t 2 l l 2 l t 2 l t 2 h yz, b1 yz, c1 m yz, b2 m yz, c1 m yz, b1 m 2 (54) 2 t t 2t 2 l l 2 yz m m t 2 l l 2 l t 2 l t 2 t yz, b1 yz, c1 m yz, b2 m yz, c1 m yz, b1 t 2 l l 2 l t 2 l t 2 t 2 yz, c2 yz, c3 m yz, c4 m yz, c3 m yz, c2 m The cell shear strain is eqal to 22

23 , 3 h 2, 1 h, 1 h 2 h h (55) yz yz s yz b m yz d m 3.5 Three Leaf Wall Cell Sbjected to Normal Stress The three leaf-model retains the set of assmptions made for the rnning bond model, which still apply to the oter masonry leaf. A nmber of frther assmptions are made for the infill and the stress eqilibrim conditions are modified to accommodate the new geometrical entities. The horizontal normal stress eqilibrim conditions at the left face and at a cross section across the middle of the cell are as follows: h h t t 2 xx m i, 2,,, 3, 4, 1 h 2 h h 2 t h 2 h h 2 t 2 xx xx c m xx h xx i xx i m xx i i (56) h h t t 2 xx m i, 1,, 1, 2, 5, 2 h 2 h h 2 t h 2 h h 2 t 2 xx xx b m xx xx i xx i m xx i i The vertical normal stress eqilibrim conditions at the top face and at a cross section across the middle of the cell are as follows: yy lm 2 l 2t ti 2 l 2 l 2 l 2 l 2 t l 2 l 2 l 2 l 2 t 2 yy, h m yy, 1 m yy, 2 m yy, i1 m yy, i2 m yy, i3 m i (57) yy lm 2 l 2t ti 2 l 2 l 2 l 2 l 2 t l 2 l 2 l 2 l 2 t 2 yy,c m yy,b m yy,c m yy, i4 m yy, i5 m yy, i4 m i The transversal normal stress eqilibrim conditions at the front and back faces of the cell are as follows: h h l 2 l 2, 2, 1 2 2, 2 2 2, 2, 2 2 zz m m h l l l l h l l l zz h m zz m zz m m zz c m zz b m (58) h h l 2 l 2, 1 2, 2 2 2, 3 2 2, 4 2, zz m m h l l l l h l l l zz i m zz i m zz i m m zz i m zz i m 23

24 The shear eqilibrim conditions at the left face, a vertical cross section of the cell and at the top face of the cell read:, 2,,, 3, 4, 1 h 2 h h 2 t h 2 h h 2 t 2 0 xy xy c m xy h xy i xy i m xy i i, 1,,, 2, 5, 2 h 2 h h 2 t h 2 h h 2 t 2 0 (59) xy xy b m xy c xy i xy i m xy i i Constant normal stress is assmed in the infill, so that xx, i1 xx, i2 xx, i3 xx, i4 xx, i5 (60) yy, i1 yy, i2 yy, i3 yy, i4 yy, i5 zz, i1 zz, i2 zz, i3 zz, i4 zz, i5 Zero in-plane shear is assmed in the infill, leading to (61) xy, i1 xy, i2 xy, i3 xy, i4 xy, i5 0 Two additional displacement conformity eqations are introdced to accommodate the deformation of the infill. They read: l 2 l 2 l 2 l 2 h 4 xx, h m xx, 1 m xx, 2 m xy, h xy, 1 xy, 2 l 2 l 2 l 2 l 2 h 2 xx, i1 m xx, i2 m xx, i3 m xy, i1 xy, i2 xy, i3 m (62) l 2 l 2 l 2 l 2 h 4 xx, h m xx, 1 m xx, 2 m xy, h xy, 1 xy, 2 2 l 2 l 2 l 2 2 h 2 xx, c m xx, b m xy, c xy, b m The total cell strain in the horizontal and vertical direction remains the same. The strain in the transversal direction is now defined as: 24

25 , t, 2 t 2 t t 2 (63) zz zz h zz i i i Cell Sbjected to Shear Stress For xy shear the same assmptions as for the rnning bond cell apply, althogh several additional assmptions are needed. Based on these assmptions the eqilibrim eqations are again adjsted. Constant shear strain is assmed in the infill, so that (64) xy, i1 xy, i2 xy, i3 xy, i4 xy, i5 The shear stress eqilibrim conditions at the top face and at a horizontal cross-section across the middle of the cell are expressed as xy lm 2 l 2t ti 2 l 2 l 2 l 2 l 2 t l 2 l 2 l 2 l 2 t 2 xy, h m xy, 1 m xy, 2 m xy, i1 m xy, i2 m xy, i3 m i (65) xy lm 2 l 2t ti 2 l 2 l 2 l 2 l 2 t l 2 l 2 l 2 l 2 t 2 xy,c m xy,b m xy,c m xy, i4 m xy, i5 m xy, i4 m i Deformation conformity is assmed between the leaves, leading to h 2 h h 2 h 2 h h 2 (66) xy, h1 xy, c1 m xy, 2 xy, i1 xy, i4 m xy, i3 For xz shear the deformation of the cell is dominated by the deformation of the infill. Therefore, the system of eqations and the assmptions need to be adjsted. The shear stress eqilibrim conditions at the back of the cell and at a vertical cross-section at the middle of the cell are as follows: h h l 2 l 2, 1 2, 2 2 2, 3 2 2, 4 2, xz m m h l l l l h l l l xz i m xz i m xz i m m xz i m xz i m h h t t 2 xz m i, 1,, 1, 2, 5, 2 h 2 h h 2 t h 2 h h 2 t 2 xz xz b m xz xz i xz i m xz i i 25 (67)

26 The stress assmptions read: (68) xz, i1 xz, i2 xz, i3 xz, i4 xz, i5 The strain assmptions read: (69) xz, h xz, 1 xz, 2 xz, c xz,b The total strain of the cell is defined as:, t, 1 t 2 t t 2 (70) xz xz h xz i i i For yz shear the deformation of the cell is again dominated by the deformation of the infill when its elastic modls is low. However, the deformation profile changes for higher vales of the Yong s modls of the infill since in this case the deformation of the oter leaf becomes more significant. Overall, the deformation is still dominated by the different stiffness between the oter and inner leaves. The stress eqilibrim conditions at the top, a horizontal cross section at mid height and at the front of the cell are as follows: yz lm 2 l 2t ti 2 l 2 l 2 l 2 l 2 t l 2 l 2 l 2 l 2 t 2 yz, h m yz, 1 m yz, 2 m yz, i1 m yz, i2 m yz, i3 m i yz lm 2 l 2t ti 2 l 2 l 2 l 2 l 2 t l 2 l 2 l 2 l 2 t 2 yz, c m yz, b m yz, c m yz, i4 m yz, i5 m yz, i4 m i (71) h h l 2 l 2, 2, 1 2 2, 2 2 2, 2, 2 2 yz m m h l l l l h l l l yz h m yz m yz m m yz c m yz b m The stress assmptions read: yz, i1 yz, h 26

27 yz, i2 yz, 1 (72) yz, i3 yz, 2 yz, i4 yz, c yz, i5 yz, b Deformation conformity conditions read: t t 2 t t 2 xz, h xz, t1 i xz, 2 xz, t3 i h 2 h h 2 h 2 h h 2 (73) xz, h xz, c m xz, 2 xz, 1 xz, b m xz, 1 Since deformation conformity is not rigidly imposed, the total strain of the cell is defined according to the geometrical average of the cboids participating in the total deformation of the cell according to yz lm yz, ht yz, i1 ti 2 2 yz, ct yz, i4 ti 2lm l lm h hm yz, 1 t yz, i2 ti 2 2 m 2 2 m 2 i l l l h h t t lm,, yz bt yz i ti l 2 m yz, 2 t yz, i3 ti 2 2 (74) 3.6 Stack Bond Pillar For the stack bond pillar model the system of eqations developed by Haller for vertical normal stress is sed [22]. According to Haller [20], the horizontal and transversal deformation eqality of the two components reads: and zz, zz, b (75) xx, xx, b The horizontal and transversal stress eqilibrim reads: 27

28 h h, h, h and,, xx m xx xx b m h h h h (76) zz m zz zz b m The vertical stress eqilibrim demands that both components develop vertical stress eqal to the external load according to (77) yy, yy, b yy The total vertical strain of the cell is, h 2, h 2 h 2 h 2 (78) yy yy yy b m m 3.7 English Bond Pillar Shear stresses in the components are disregarded for applied normal stress. Normal strain conformity in the cell is achieved by assming xx, 1 xx, b1 xx, 3 xx, h2 xx, c2 xx, 4 and xx, 2 xx, c1 xx, h1 xx, c4 xx, c3 xx, c5 yy, 3 yy, h1 yy, 4 yy, c5, yy, b1 yy, c1 yy, c2 yy, c3 and yy, 2 yy, c4 yy, 1 yy, h2 (79) zz, 2 zz, c1 zz, h1 zz, 1 zz, b1 zz, 3 and zz, c4 zz, c3 zz, c5 zz, h2 zz, c2 zz, 4 Normal stress eqilibrim in the horizontal, vertical and transversal directions is expressed as h h l 2 xx m l 2 l 2 t 2 h 2 l 2 l 2 t 2 h l 2 l 2 t / 2 h 2 xx, h1 m xx, c5 m xx, c1 m xx, c3 m m xx, 2 m xx, c4 m h h l 2 xx m l 2 l 2 t 2 h 2 l 2 l 2 t 2 h l 2 l 2 t / 2 h 2 xx, 3 m xx, 4 m xx, b1 m xx, c2 m m xx, 1 m xx, h2 m, 3, 1, 5, 4 l 2 l 2 l 2 l 2 t / 2 l 2 l 2 l 2 l 2 l 2 t 2 yy yy m yy h m m yy c m yy m m, 1, 1, 3, 2 yy l 2 l 2 yy b l 2 lm 2 yy c tm / 2 l 2 lm 2 yy c lm 2 yy c l 2 lm 2 tm 2 (80) 28

29 , 1, 2, 4, 2 l 2 l 2 l 2 l 2 t / 2 l 2 l 2 l 2 l 2 l 2 t 2 yy yy m yy m m yy c m yy h m m ( h h )( l 2) zz m l 2 l 2 l 2 h 2 l 2 l 2 l 2 h l 2 l 2 l 2 h 2 zz, 3 m zz, h1 m zz, b1 m zz, c1 m m zz, 1 m zz, 2 m ( h h )( l 2) zz m l 2 l 2 l 2 h 2 l 2 l 2 l 2 h l 2 l 2 l 2 h 2 zz, 4 m zz, c5 m zz, c2 m zz, c3 m m zz, h2 m zz, c4 m The total vertical strain of the cell is, 3 h 2, 1 h, 1 h 2 h 2 h h 2 (81) yy yy yy b m yy m 3.8 Calclation For linear elastic analysis the soltion from which the elastic modli of the cell are derived can be accomplished in a single analysis step by solving the linear system of eqations derived from the above expressions. For the application of normal stress, the systems need to be solved for the nknown normal stresses and strains and shear stresses and strains, where the latter two are considered. For the application of shear stress, the systems need to be solved for the arising shear stresses and strains. For normal stress loading, the nknowns inclde three normal stresses, three normal strains, one shear stress and one shear strain vale for each cboid. However, in the stack bond wall, the stack bond pillar and the English bond pillar models shear stress and strain nknowns are not inclded. In smmary, for normal stress loading, the stack bond wall system, consisting of for cboids, is solved for 24 nknowns, the rnning bond wall for 40 nknowns for its five cboids, the Flemish bond wall for 128 nknowns for its sixteen cboids and the three-leaf wall is solved for 80 nknowns arising from ten cboids. The stack bond pillar has two cboids and its system has 12 nknowns and the English bond pillar has twelve cboids reslting in a system of 72 nknowns. For shear stress loading each cboid has one shear stress and one shear strain nknown components. Therefore, the nmber of nknowns is: 8 for the stack bond wall, 10 for the rnning bond wall, 32 for the Flemish bond wall and 20 for the three-leaf wall model. 29

30 Overall, obtaining closed form expressions for the stresses and strains in the cboids, as well as for the elastic modli of the cells, nder conditions of applied normal stress is extremely difficlt even for the simple case of stack bond masonry. Expressions for the shear modli of the cells are far easier to be obtained de to the mch smaller nmber of nknowns. However, the linear systems of eqations can be solved with very little effort sing simple linear algebra or any basic symbolic math software. 4. Verification of the Model The capacity of the model to predict the elastic properties of masonry cells nder normal and shear loading is initially evalated throgh a comparison with the reslts obtained from FE analyses. The meshes sed for this verification are shown in Figre 7. Apart from the comparison between the reslts obtained from the proposed model and the FE analyses, it is desirable to compare the two approaches in terms of comptational cost. The masonry nit cell is spposed to represent a volme of masonry inside an extended composite continm. Therefore, bondary conditions assring displacement conformity and periodicity at the cell faces need to be applied. Stress distribtion in the constitents of the composite is not niform bt rather depends on the relative elasticity parameters of the involved components. The imposed bondary conditions ensre that the distribtion of stress in the components is based on strain and deformation compatibility nder normal and shear loading. Periodicity conditions for a cell shown in Figre 8a are imposed by tying the displacements of nodes in opposite faces of the FE model. The tyings can be described as keeping the distance between two pairs of nodes eqal, specifying a nmber of controlling nodes: p1, p2, p3, p4 and p5. These controlling nodes also serve to flly describe the deformations necessary for the derivation of the elastic properties of the cell. Therefore the displacement of a node in face i2 or j2 for the application of a normal stress in the i direction or a shear stress in the ij plane can be derived from the displacement of the node in the opposite face i1 or j1 and the displacement of node p1, p2 and p3 as follows: d d d d (82) i2 i1 p2 p1 d d d d (83) j2 j1 p3 p1 30

31 where d p1, d p2 and d p3 are the displacement vectors of nodes p1, p2 and p3 in the ij plane and d i1, d i2, d j1 and d j2 are the displacement vectors of a node in faces i1, i2, j1 and j2 in the ij plane. Eqal displacements in the k direction are imposed at the nodes of each of the external faces parallel to the ij plane for both applied normal and shear stress. The elastic properties of the composite cell may be derived by registered displacements in the above designated controlling nodes, where d pn, i is eqal to the displacement of node n in direction i. The total dimension of the cell in each principal direction i (horizontal, vertical and transversal) is designated as D i. The Yong s modli and the Poisson s ratios of the cell may be calclated from E d d d i p2, i p1, i c, i ii ii ii ii Di Di (84) and d j d d p3, j d i p1, j d p2, i d p1, i c, ij jj ii D j D i D j Di d d d d d d k i p5, k p4, k p2, i p1, i c, ik kk ii Dk D i D j Di (85) where d i, d j and d k are the relative displacements between two opposing faces in the i, j or k direction respectively, which can be expressed by the difference in displacement of opposing controlling nodes in a given direction. The shear modls of the cell is eqal to G d d d d Dj Di c, ij ij 2ij ij ij ij p4, i p1, i p4, j p1, j (86) The deformation profile of the cell nder normal and shear applied stress is shown in Figre 8b and c. 31

32 A parametric investigation is condcted in order to verify the accracy of the models for a wide range of ratios of nit-to-mortar and mortar-to-infill Yong s modls, as shown in Table 1. These properties and dimensions have been applied to all typologies of masonry nder stdy. The parametric investigation of the three-leaf wall is condcted by assming a ratio of 30 for the nit-to-mortar Yong s modls and altering the Yong s modls of the infill ranging from highly deformable to stiffer than the oter leaf. The wall models have been tested in all normal and shear directions and the pillar models have been tested for vertical loading. Finally, a comparison is made between the reslts obtained by the model and experimental reslts drawn from the existing literatre. 5. Reslts 5.1 Parametric Investigation Overview The analytical approach prodced elastic reslts nearly identical to those obtained from FE analysis for all models and for almost the entire range of material properties. The Yong s modli and the Poisson s ratios of the masonry composites were calclated with great accracy, with the differences being restricted to a nmber of shear modli in ot-off-plane loading. The comptational cost of the analytical models is negligible, the reslts being prodced practically instantaneosly on an ordinary compter, even for the largest systems of eqations. The FE models reqired a comptational time ranging from several seconds to a few mintes for the prodction of the reslts. The reslts of the parametric investigation and their comparison with FEM calclations are illstrated in Figre 9 throgh Figre 13. The Yong s modli and Poisson s ratios for the stack bond case (Figre 9) are very well calclated. The shear modli in the xy and xz planes are very similar de to the existence of the continos bed mortar joint. The in-plane shear modls is well approximated, while the two ot-off-plane modli are slightly overestimated by the model. 32

33 An interesting phenomenon is the tendency of the horizontal and vertical Yong s modli to be drastically redced for a redction of the Yong s modls of the mortar with the transversal modls being redced only slightly. For high vales of the mortar modls the three elastic modli are of the same order of magnitde while for low vales the transversal modls is mch higher. Certain conclsions may be drawn from this observation concerning the applicability of plane analysis methods for stack bond masonry strctres. According to the relative deformability in the in-plane and ot-of-plane directions, plane stress analysis may provide reasonable reslts for very rigid mortars and plane strain for very deformable mortars. The vertical and transversal Yong s modls for rnning bond is almost exactly the same as for stack bond, bt rnning bond has a higher horizontal modls de to the staggered non-continos layot of the horizontal joints, as illstrated in Figre 10. The model slightly nderestimates the shear modls in the xz plane for highly deformable mortars. Overall, the reslts are not drastically different from the ones obtained for stack bond masonry, althogh the accracy of the model is slightly higher in the stack bond case. Similar observations as those made for the relative deformability of the stack bond masonry in the in-plane and ot-of-plane directions can be made for rnning bond masonry as well. The fact that both typologies are single leaf strctres with constant geometry across the thickness of the cell indicate that this trend of the transversal Yong s modls of masonry to become mch higher than the two in-plane ones may be shared by other singe leaf typologies. This assmption cold be checked by frther FEM or analytical calclations. At this point a comparison may be here made between the proposed model and the model proposed by Zcchini & Lorenço [15] for rnning bond masonry. The models are nearly eqally accrate for the prediction of the Yong s modls in all three orthogonal directions and the ν xy, ν xz and ν yz Poisson s ratios: the average absolte error is 1.4% for the proposed model, with the maximm absolte error being 6%, while the Zcchini & Lorenço model has a 1.3% average and 6% maximm absolte error. For the prediction of the shear modls the proposed model has an average absolte error of 3% and a maximm absolte error of 18%, which is registered for the G xz modls for E /E m = The Zcchini & Lorenço model has a maximm error of only 5%, again for the G xz modls, which is registered for the G xz modls for E /E m = 10. The reason for the redced accracy of the proposed model is the inclsion of shear stresses and strains only in the plane of the 33

34 applied external stress, whereas the Zcchini & Lorenço model incldes shear stresses and strains in the yz plane in the bed joint. In the Flemish bond wall case (Figre 11), de to the existence of the transversal joint, decreasing the Yong s modls of the mortar reslts in a decrease in the transversal Yong s modls of the cell. The discrepancy between the nmerical and analytical reslts for the shear modls in xz shear are similar to those in the previos cases, bt all the remaining elastic properties are very well calclated. The transversal joint eliminates the distinct higher rigidity in the transversal direction for highly deformable mortars noted in the stack bond and rnning bond typologies. However, the complex geometry across the thickness of the strctre wold render the plane stress and plane strain methods of analysis inappropriate by definition. The model for the three-leaf wall case, whose reslts are shown in Figre 12, is capable of approximating the FEM reslts very well for the entire range of vales. Of interest is the shift in the deformation profile of the cell nder yz shear as the Yong s modls of the infill increases. For low vales the obtained shear modls is eqal to the modls in the xz plane and for higher vales it is eqal to the modls in the xy plane. This is de to the deformability of the infill dominating the global response when highly deformable. The horizontal and vertical Yong s modls and the in-plane shear modls are only slightly affected by a redction of the Yong s modls of the infill. The rest of the elastic modli are greatly redced. The infill inflences the behavior of the masonry in a similar way to a continos vertical or horizontal joint bt in a transversal orientation. The analytical models for the stack bond and English bond pillar, shown in Figre 13, prodced reslts identical to the ones obtained from FE analyses. The deformation of the bed joint nder vertical loading dominates the response in terms of global elastic stiffness and, as sch, the existence of head and cross joints in the English bond pillar case has very small inflence on the final reslts. 34