THEORETICAL BACKGROUND OF MASONRY MODEL

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σ = σ σ b [ S ] b ⎡S bj = ⎣ bj ⎦ ⎡S σ ⎤σ ⎤σ hj = ⎣ hj ⎦ (5) where subscripts b, bj and hj represent brick, bed joint ad head joint, respectively. The structural relationships for strains can similarly be established. The structural matrices S are listed in Appendix II. From the results listed in Pande et al., it can shown that the orthotropic material properties are functions of Fig.1 Coordinate System used in Masonry Panel 1. Dimensions of the brick, length, height and width 2. Young’s modulus and Poisson’s ratio of the brick material 3. Young’s modulus and Poisson’s ratio of the mortar in the head and bed joints 4. Thickness of the head and bed mortar joints It must be noted that the geometry of masonry has to be modeled in the reference of the above figure. This is because the homogenization is performed based on the coordinate system. THEORETICAL BACKGROUND OF MASONRY MODEL page 4 / 20
3. Criteria for Failure for Constituents Failure of masonry can be based on the micromechanical behaviour. At every loading step, once the equivalent stresses/strains in the masonry structure are calculated, stresses/strains of the constituent materials can be derived based on the structural relationship in eq. (5). The maximum principal stress is calculated in each constituent level (i.e., Brick, Bed joint, and Head joint) and is compared to the tensile strength defined by user. If the maximum principal stress exceeds the tensile strength at current step, the stiffness contribution of the constituent to the whole element is forced to be negligible. For the nonlinear stress-strain relation of constituents, even the elastio-perfectly plastic relation could be simulated. This can be numerically implemented by substituting the stiffness of the constituent with very small value as Ei zero (where the subscript ‘i’ could be brick, bed joint, or head joint). If users set the ‘Stiffness Reduction Factor’ as very small value, the masonry model will behave with nonlinearity. In the same reason, if the ‘Stiffness Reduction Factor’ is set to be unit value, the masonry model will behave elastically (refer to the figure 2 below). Fig.2 Stress-Strain of a constituent of masonry model In this way, the local failure mode can be evaluated. For better understanding this kind of equivalent nonlinear stress-strain relationship theory, see Lee et al.(1996). Once cracking occurs in any constituent material, the effect is smeared onto the neighboring equivalent orthotropic material through another homogenization. Although there are a number of criteria for the masonry model such as Mohr-Coulomb and so on, the masonry model in MIDAS currently determines the tensile failure referring only to the user-input tensile strength. More advanced failure criteria are developed in the near future based on the abundant research. After the tensile cracks occur, the crack positions can be traced by post processor of solid stresses. THEORETICAL BACKGROUND OF MASONRY MODEL page 5 / 20
Page 1 and 2: THEORETICAL BACKGROUND OF MASONRY M
Page 3: ε = ⎡ ⎣ C⎤ ⎦ σ (2) where,
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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