THEORETICAL BACKGROUND OF MASONRY MODEL

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2. Homogenization Techniques in Masonry Structures Masonry structures can be numerically analyzed if an accurate stress-strain relationship is employed for each constituent material and each constituent material is then separated individually. However, a threedimension-analysis of a masonry structure involving even a very simple geometry would require a large number of elements and the nonlinear analysis of the structure would certainly be intractable. To overcome this computational difficulty, the orthotropic material properties proposed by Pande et al.5,6 can be introduced to model the masonry structure in the sense of an equivalent homogenized material. The equivalent material properties introduced in Pande et al. are based on a strain energy concept. The details of the procedure to obtain equivalent elastic parameters based on the homogenization technique are given in the following. The basic assumptions made to derive the equivalent material properties through the strain energy considerations are: 1. Brick and mortar are perfectly bonded 2. Head or bed mortar joints are assumed to be continuous The second assumption is necessary in the homogenization procedure and it has been shown7 that the assumption of continuous head joints instead of staggered joints, as they appear in practice, does not have any significant effect on the stress states of the constituent materials. Let the orthotropic material properties of the masonry panel be denoted by E x , E y , E z , ν xy , ν xz , ν yz , G xy , by G yz , G xz , Fig. 1. The stress/strain relationship of the homogenized masonry material is represented σ = ⎡ ⎣ D⎤ ⎦ ε (1) or 5 G. N. Pande, B. Kralj, and J. Middleton. Analysis of the compressive strength of masonry given by the equation α ( ′) ( ) β k = b m . The Structural Engineer, 71:7-12, 1994. f K f f 6 G. N. Pande, J. X. Liang, and J. Middleton. Equivalent elastic moduli for brick masonry. Comp. & Geotech., 8:243-265, 1989. 7 R. Luciano and E. Sacco. A damage model for masonry structures. Eur. J. Mech., A/Solids, 17:285-303,1998. THEORETICAL BACKGROUND OF MASONRY MODEL page 2 / 20
ε = ⎡ ⎣ C⎤ ⎦ σ (2) where, σ = T { σxxσ yyσzzτxyτ yzτxz} { xx, yy, zz, xy, yz, xz} ε = ε ε ε γ γ γ T (3) are the vectors of stresses and strains in the Cartesian coordinate system. ⎡ 1 ν xy ν ⎤ xz ⎢ − − 0 0 0 ⎥ ⎢ Ex Ex Ex ⎥ ⎢ ν yx 1 ν ⎥ yz ⎢− − 0 0 0 ⎥ ⎢ Ey Ey Ey ⎥ ⎢ ν ⎥ ⎢ ν zx zy 1 − − 0 0 0 ⎥ ⎢ Ez Ez E ⎥ z ⎡ ⎣C⎤ ⎦ = ⎢ ⎥ ⎢ 1 0 0 0 0 0 ⎥ ⎢ G ⎥ xy ⎢ ⎥ ⎢ 1 0 0 0 0 0 ⎥ ⎢ G ⎥ yz ⎢ ⎥ ⎢ 1 ⎥ ⎢ 0 0 0 0 0 ⎥ ⎣ Gxz ⎦ (4) The details of the derivation of orthotropic elastic material properties of masonry in terms of the properties of the constituents are given in Appendix I. In the mathematical theory of homogenization, there has been an issue relating to the sequence of homogenization, if there are more than two constituents. For example, if we homogenize bricks and mortar in head joints first and then homogenize the resulting material with bed joints at the second stage, then result may not be the same if we had followed a different sequence. However, it has been shown in the case of masonry, the sequence of homogenization does not have any significant influence. Here we present in Appendix I equations for equivalent properties if bricks and bed joints are homogenized first. It is noted that, in Pande et al., the equivalent material properties were derived with the brick and the head mortar joint being homogenized first. The equivalent orthotropic material properties derived from the homogenization procedure are used to construct the stiffness matrix in the finite element analysis procedure and, from this equivalent stress/strains are then calculated. The stresses/strains in the constituent materials can be evaluated through structural relationships, i.e., THEORETICAL BACKGROUND OF MASONRY MODEL page 3 / 20
Page 1: THEORETICAL BACKGROUND OF MASONRY M
Page 5 and 6: 3. Criteria for Failure for Constit
Page 7 and 8: easons. Fig.4 Example of in-plane d
Page 9 and 10: Fig.6 Cracked and deformed shape at
Page 11 and 12: APPENDIX I ORTHOTROPIC PROPERTIES O
Page 13 and 14: ε ε ε γ γ γ xxi xx yyi yy yyi
Page 15 and 16: where, µν b b χb = 1 − ν 2 2
Page 17 and 18: eq ( + ) µ eq ν yx ν yzνzx λeq
Page 19 and 20: S S S hj xy hj zx hj,21 2 1 ν ⎜

THEORETICAL BACKGROUND OF MASONRY MODEL

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