an orthotropic continuum model for the analysis of masonry structures

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26 1995 TNO-95-NM-R0712⎧⎪D −1 (σ⎪ n+1 − σ trial ∂g) 1⎪∂ f 2⎪+∆λ t, n+1 ⎪ +∆λ c, n+1 ⎪ = 0∂σ∂σ⎪⎪ n+1 ⎪ n+1⎨ f 1 = ( 1 / 2 ξ n+1P T t ξ n+1 ) 1 / 2+ 1 / 2 π T ξ n+1 = 0 .⎪⎪ f⎪ 2 = ( 1 / 2 σ n+1P T c, n+1 σ n+1 ) 1 / 2− σ c, n+1 = 0⎩This system of non-linear equations is solved with a regular Newton-Raphson method and the Jacobiannecessary for this procedure reads (note that the subscript n+1 is dropped in the derivativesand matrices for convenience)(68)⎡⎢D −1 ∂ 2 g 1+∆λ t, n+1∂σ⎢2 +∆λ ∂ 2 f 2c, n+1∂σ 2−−−−−−−−−−−−−−−−⎢TJ = ⎢⎛ ∂ f 1 ⎞⎢⎝ ∂σ ⎠⎢ −−−−−−−−−−−−−−−−⎢T⎢⎛ ∂ f 2 ⎞⎣⎝ ∂σ ⎠|+|+|∂g 1∂σ +∆λ ∂ 2 g 1t, n+1∂σ ∂κ t−−−−−−−−−−∂ f 1∂κ t−−−−−−−−−−0|+|+|∂ f 2∂σ +∆λ ∂ 2 f 2c, n+1∂σ ∂κ c−−−−−−−−−−⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦0 . (69)−−−−−−−−−−∂ f 2∂κ c4.2 Consistent tangent operator - Corner regimeDifferentiation of the update equations and the consistency conditions (d f 1, n+1 = 0 and(d f 2, n+1 = 0) results inJ⎧dσ⎫ ⎧⎪ n+1 dε⎫⎪⎬⎪⎭ ⎪ n+1⎪⎨dλ t, n+1 = ⎨ 0 ⎬⎪dλ c, n+1⎪ 0 ⎪⎩⎩ ⎭. (70)Then, the consistent tangent operator is given byD ep = ∂σ ⎪⎪ = (J −1 ) 4×4∂ε⎪ n+1, (71)in which (J −1 ) 4x4 is the top-left 4×4 submatrix of the inverse of J. The consistent tangent operatorcan also be written in other fashion by means of a condensation of the matrix J. The derivation ofan expression equivalent to eqs. (45,65) is given in Louren•o (1994).An investigation on the performance of the numerical implementation is given in Appendix A.
TNO-95-NM-R0712 1995 275. EXAMPLESIn the present section the masonry model proposed in this report is used to analyse different structuresfound in the literature.5.1 TU Eindhoven tests on shear wallsSeveral tests on masonry shear walls were carried out within the scope of the STRUCTURALMASONRY program from the Netherlands, see Raijmakers and Vermeltfoort (1992) and Vermeltfoortand Raijmakers (1993). In Louren•o (1994) some of the specimens are analysed with a micromodeland very good agreement is found with the experimental results. The parameters necessaryto characterize the micro-material model are available from micro-experiments but almost noexperimental data are available on the behaviour of the composite.The purpose of this report is not a sharp reproduction of the experimental results and the main concernof the author is to reproduce the behaviour observed in the experiments. For this purpose theexact knowledge of the macro-material parameters is not capital (and, in any case, it is not knownfor this particular structure).The Þrst shear wall analysed by Louren•o (1994) is used for comparison between the micro- andmacro-model. The specimen consists of a pier with a width/height ratio of one (990 × 1000 mm 2 ),built up with 18 layers (16 layers unrestrained) of Joosten solid clay bricks (dimensions 204 x 98 x50 mm 3 ) and 10 mm thick mortar (1:2:9, cement:lime:sand by volume). The pier was subjected toa vertical precompression force P = 30 kN before a horizontal load was monotonically increasedunder top displacement control d until failure (see Fig. 23). The cracking patterns are depicted inFig. 24 for the two tests carried out.Pda) Phase 1 - Vertical loading b) Phase 2 - Horizontal loadingFig. 23 - Loads for Vermeltfoort shear wall
Page 1 and 2: Delft University of TechnologyFacul
Page 3 and 4: TNO-95-NM-R0712 1995 11. INTRODUCTI
Page 5 and 6: TNO-95-NM-R0712 1995 3surfaces sugg
Page 7 and 8: TNO-95-NM-R0712 1995 52. TENSION -
Page 9 and 10: TNO-95-NM-R0712 1995 7τxyτ uσxft
Page 11 and 12: TNO-95-NM-R0712 1995 9whereúκ t =
Page 13 and 14: TNO-95-NM-R0712 1995 11σ n+1 = η
Page 15 and 16: TNO-95-NM-R0712 1995 132.3 Features
Page 17 and 18: TNO-95-NM-R0712 1995 15Ñ The norma
Page 19 and 20: TNO-95-NM-R0712 1995 173. COMPRESSI
Page 21 and 22: TNO-95-NM-R0712 1995 19σ cσ pσ m
Page 23 and 24: TNO-95-NM-R0712 1995 21[MPa]-10.0-8
Page 25 and 26: TNO-95-NM-R0712 1995 238.0/ f myf m
Page 27: TNO-95-NM-R0712 1995 254. A COMPOSI
Page 31 and 32: TNO-95-NM-R0712 1995 29Horizontal f
Page 33 and 34: TNO-95-NM-R0712 1995 31a) Total def
Page 35 and 36: TNO-95-NM-R0712 1995 33The explanat
Page 37 and 38: TNO-95-NM-R0712 1995 355.2 ETH Zuri
Page 39 and 40: TNO-95-NM-R0712 1995 37For the nume
Page 41 and 42: TNO-95-NM-R0712 1995 39a) Total def
Page 43 and 44: TNO-95-NM-R0712 1995 415.2.2 Wall W
Page 45 and 46: TNO-95-NM-R0712 1995 43c) Cracks d)
Page 47 and 48: TNO-95-NM-R0712 1995 455.2.3 Compar
Page 49 and 50: TNO-95-NM-R0712 1995 477. REFERENCE
Page 51 and 52: TNO-95-NM-R0712 1995 49APPENDIX APE
Page 53 and 54: TNO-95-NM-R0712 1995 51Plastic step
Page 55: TNO-95-NM-R0712 1995 53-10.0-8.0yxd

an orthotropic continuum model for the analysis of masonry structures

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