an orthotropic continuum model for the analysis of masonry structures

More documents

Recommendations

Info

24 1995 TNO-95-NM-R0712since the original work of Baºant and Oh (1983). As stated in Section 2., in the present work h isassumed to be related to the area of an element, cf. eq. (13). However, this approach is generallyused in engineering practice only for modelling tensile behaviour with linear elastic pre-peakbehaviour followed by inelastic softening until total degradation of strength.The constitutive relation shown in Fig. 16 features pre-peak hardening and a residual plateau.Clearly, the hardening branch of the constitutive relation is stable and should not be adjusted as afunction h but also the residual plateau is constant and independent of the h value. To demonstratethe veracity of the deÞnition of a mesh independent release of energy upon mesh reÞnement anexample of a simple bar loaded in uniaxial is given. The problem is similar to the well-known problemof a simple bar loaded in tension proposed by CrisÞeld (1982).Consider the bar shown in Fig. 20 which is divided in n elements with n = 10, 20 and 40 elements.The length of the bar is 50 mm and the tranversal section of the bar has unit dimensions(1. 0 mm 2 × 1. 0 mm 2 ). The compressive fracture energies are assumed to equal G fcx = 10. 0 N/mmand G fcy = 5. 0 N/mm. For the rest of the material properties the values used in the previous sectionare assumed. One element is slightly imperfect (10%) to trigger the localization: f cx = 9. 0 N/mm 2 ,f cy = 4. 5 N/mm 2 ,G fcx = 9. 0 N/mm and G fcy = 4. 5 N/mm. The other material parameters remainthe same.Imperfect elementFig. 20 - Simple bar with imperfect element loaded in compressionThe load-displacement response of the bar is depicted in Fig. 21a for the energy-based regularizationmethod (note that ÒdisplacementÓ is understood as the relative displacement between the endsof the bar). It can be observed that the response is totally independent from the number of elements.The response of the bar with a constitutive model which has not been modiÞed by the size of theÞnite element mesh, see Fig. 21b, shows a dramatic mesh-dependent behaviour in the post-peakresponse. The brittleness of the response increases with an increasing number of elements.10.010.0Load [MN]8.06.04.0n = 10, 20 and 40Load [MN]8.06.04.0n = 20n = 102.02.0n = 400.00.0 10.0 20.0 30.0 40.0 50.0-3Displacement [10 mm]0.00.0 10.0 20.0 30.0 40.0 50.0-3Displacement [10 mm]a) Energy-based regularization b) No regularizationFig. 21 - Load-displacement diagram for simple bar with imperfect element
TNO-95-NM-R0712 1995 254. A COMPOSITE YIELD SURFACE FOR MASONRYThe two yield surfaces detailed in the previous section are now combined in a composite yield surfaceas illustrated in Fig. 22.σyσxσy0.1 0.2 0.3σ xyσxFig. 22 - Proposed composite yield surface with iso-shear stress linesf tx = 1. 0, f ty = 0. 5, f mx = 10. 0, f my = 5. 0 [N/mm 2 ] − α = 1. 0, β =−1. 0, γ = 3. 0Contour spacing: 0.1 f mxA full description of the compressive and tensile parts of the composite yield surface are given inthe previous sections and will not be repeated here. Only the aspects relative to the corner regimewill be addressed in the present section.One of the most important issues of multi-surface plasticity is to deÞne the number of active yieldsurfaces. Simo et al. (1988) have proposed an algorithm in which the assumption is made that thenumber of active yield surfaces in the Þnal stress state is less than or equal to the number of activeyield surfaces in the trial stress state. This implies that it is not possible that a yield surface, whichis inactive in the trial state, becomes active during the return mapping. This is not valid for the proposedyield surface. Due to the small number of yield surfaces (two), a trial and error iterative procedureis used, in which the return mapping process is restarted if a non-admissible solution isfound, see Louren•o (1994) for a complete description. From the experience of the author, thisleads to a robust and always convergent algorithm. The disadvantage is that the return mappingalgorithm might have to be restarted before the correct solution is obtained.4.1 Return mapping algorithm - Corner regimeThe return mapping algorithm in the frame of a implicit Euler backward integration scheme isgiven in Section 2. for single surface plasticity. For multisurface plasticity, the most importantassumption is KoiterÕs (1953) generalisation of the plastic strain rate asúεúε p = ú∂g 1 λ t∂σ+ ú∂ f 2 λ c∂σ . (67)Note that no coupling is assumed between the compressive and tensile regimes.Upon algebraic manipulation, the return mapping algorithm for the corner regime results in the followingset of six equations containing 6 unknowns (σ n+1 , ∆κ t, n+1 =∆λ t, n+1 and ∆κ c, n+1 =∆λ c, n+1 ),cf. eqs. (29,61),
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: TNO-95-NM-R0712 1995 238.0/ f myf m
Page 29 and 30: TNO-95-NM-R0712 1995 275. EXAMPLESI
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

Create successful ePaper yourself

Delete template?

Save as template?