# Tracking multi-directional intersecting cracks in numerical modelling of masonry shear walls under cyclic loading

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## Abstract

In-plane cyclic loading of masonry walls induces a complex failure pattern composed of multiple diagonal shear cracks, as well as flexural cracks. The realistic modelling of such induced localized cracking necessitates the use of costly direct numerical simulations with detailed information on both the properties and geometry of masonry components. On the contrary, computationally efficient macro-models using standard smeared-crack approaches often result in a poor representation of fracture in the simulated material, not properly localized and biased by the finite element mesh orientation. This work proposes a possible remedy to these drawbacks of macro-models through the use of a crack-tracking algorithm. The macro-modelling approach results in an affordable computational cost, while the tracking algorithm aids the mesh-bias independent and localized representation of cracking. A novel methodology is presented that allows the simulation of intersecting and multi-directional cracks using tracking algorithms. This development extends the use of localized crack approaches using tracking algorithms to a wider field of applications exhibiting multiple, arbitrary and interacting cracking. The paper presents also a novel formulation including into an orthotropic damage model the description of irreversible deformations under shear loading. The proposed approach is calibrated through the comparison with an experimental test on a masonry shear wall against in-plane cyclic loading.

## Keywords

Continuum damage mechanics Crack-tracking Cyclic shear loading Intersecting cracks Masonry walls## Notes

### Acknowledgements

This research has received the financial support from the MINECO (Ministerio de Economia y Competitividad of the Spanish Government) and the ERDF (European Regional Development Fund) through the MULTIMAS project (Multiscale techniques for the experimental and numerical analysis of the reliability of masonry structures, ref. num. BIA2015-63882-P).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- 1.Huerta S (2008) The analysis of masonry architecture: a historical approach. Archit Sci Rev 51(4):297–328CrossRefGoogle Scholar
- 2.Roca P, Cervera M, Gariup G, Pelà L (2010) Structural analysis of masonry historical constructions. Classical and advanced approaches. Arch Comput Methods Eng 17:299–325CrossRefMATHGoogle Scholar
- 3.Theodossopoulos D, Sinha B (2013) A review of analytical methods in the current design processes and assessment of performance of masonry structures. Constr Build Mater 41:990–1001CrossRefGoogle Scholar
- 4.Addessi D, Marfia S, Sacco E, Toti J (2014) Modeling approaches for masonry structures. Open Civ Eng J 2:288–300CrossRefGoogle Scholar
- 5.Orduña A, Lourenço PB (2003) Cap model for limit analysis and strengthening of masonry structures. J Struct Eng 129(10):1367–1375CrossRefGoogle Scholar
- 6.Block P, Ciblac T, Ochsendorf J (2006) Real-time limit analysis of vaulted masonry buildings. Comput Struct 84:1841–1852CrossRefGoogle Scholar
- 7.Gilbert M, Casapulla C, Ahmed H (2006) Limit analysis of masonry block structures with non-associative frictional joints using linear programming. Comput Struct 84:873–887CrossRefGoogle Scholar
- 8.Milani G, Lourenço P, Tralli A (2006) Homogenised limit analysis of masonry walls, part I: failure surfaces. Comput Struct 84:166–180CrossRefGoogle Scholar
- 9.Roca P, Molins C, Marí AR (2005) Strength capacity of masonry wall structures by the equivalent frame method. J Struct Eng 131(10):1601–1610CrossRefGoogle Scholar
- 10.Lagomarsino S, Penna A, Galasco A, Cattari S (2013) TREMURI program: an equivalent frame model for the nonlinear seismic analysis of masonry buildings. Eng Struct 56:1787–1799CrossRefGoogle Scholar
- 11.Siano R, Sepe V, Camata G, Spacone E, Roca P, Pelà L (2017) Analysis of the performance in the linear field of equivalent-frame models for regular and irregular masonry walls. Eng Struct 145:190–2010CrossRefGoogle Scholar
- 12.Endo Y, Pelà L, Roca P, Porto F, Modena C (2015) Comparison of seismic analysis methods applied to a historical church struck by 2009 L ’ Aquila earthquake. Bull Earthq Eng 13:3749–3778CrossRefGoogle Scholar
- 13.Saloustros S, Pelà L, Roca P, Portal J (2014) Numerical analysis of structural damage in the church of the Poblet monastery. Eng Fail Anal 48:41–61CrossRefGoogle Scholar
- 14.Pelà L, Cervera M, Roca P (2013) An orthotropic damage model for the analysis of masonry structures. Constr Build Mater 41:957–967CrossRefGoogle Scholar
- 15.Lourenço PB, Rots JG (1997) Multisurface interface model for analysis of masonry structures. J Eng Mech 123(7):660–668CrossRefGoogle Scholar
- 16.Macorini L, Izzuddin BA (2011) A non-linear interface element for 3D mesoscale analysis of brick-masonry structures. Int J Numer Methods Eng 85:1584–1608CrossRefMATHGoogle Scholar
- 17.Massart TJ, Peerlings RHJ, Geers MGD (2007) An enhanced multi-scale approach for masonry wall computations with localization of damage. Int J Numer Methods Eng 69:1022–1059CrossRefMATHGoogle Scholar
- 18.Addessi D, Sacco E (2012) A multi-scale enriched model for the analysis of masonry panels. Int J Solids Struct 49(6):865–880CrossRefGoogle Scholar
- 19.Petracca M, Pelà L, Rossi R, Oller S, Camata G, Spacone E (2017) Multiscale computational first order homogenization of thick shells for the analysis of out-of-plane loaded masonry walls. Comput Methods Appl Mech Eng 315:273–301ADSMathSciNetCrossRefMATHGoogle Scholar
- 20.Clemente R (2006) Análysis estructural de edificios históricos mediante modelos localizados de fisuración. Ph.d. thesis, Universitat Politècnica de CatalunyaGoogle Scholar
- 21.Pelà L (2009) Continuum damage model for nonlinear analysis of masonry structures. Ph.d. thesis, Universitat Politècnica de CatalunyaGoogle Scholar
- 22.Cervera M, Pelà L, Clemente R, Roca P (2010) A crack-tracking technique for localized damage in quasi-brittle materials. Eng Fract Mech 77(13):2431–2450CrossRefGoogle Scholar
- 23.Roca P, Cervera M, Pelà L, Clemente R, Chiumenti M (2013) Continuum FE models for the analysis of Mallorca Cathedral. Eng Struct 46:653–670CrossRefGoogle Scholar
- 24.Pelà L, Cervera M, Oller S, Chiumenti M (2014) A localized mapped damage model for orthotropic materials. Eng Fract Mech 124–125:196–216CrossRefGoogle Scholar
- 25.Cervera M, Chiumenti M (2006) Smeared crack approach: back to the original track. Int J Numer Anal Methods Geomech 30(12):1173–1199CrossRefMATHGoogle Scholar
- 26.Slobbe A, Hendriks M, Rots J (2014) Smoothing the propagation of smeared cracks. Eng Fract Mech 132:147–168CrossRefGoogle Scholar
- 27.Oliver J, Huespe AE (2004) Continuum approach to material failure in strong discontinuity settings. Comput Methods Appl Mech Eng 193(30–32):3195–3220ADSMathSciNetCrossRefMATHGoogle Scholar
- 28.Meschke G, Dumstorff P (2007) Energy-based modeling of cohesive and cohesionless cracks via X-FEM. Comput Methods Appl Mech Eng 196(21–24):2338–2357ADSCrossRefMATHGoogle Scholar
- 29.Dias-Da-Costa D, Alfaiate J, Sluys LJ, Júlio E (2010) A comparative study on the modelling of discontinuous fracture by means of enriched nodal and element techniques and interface elements. Int J Fract 161(1):97–119CrossRefMATHGoogle Scholar
- 30.Zhang Y, Lackner R, Zeiml M, Mang H a (2015) Strong discontinuity embedded approach with standard SOS formulation: element formulation, energy-based crack-tracking strategy, and validations. Comput Methods Appl Mech Eng 287:335–366ADSMathSciNetCrossRefGoogle Scholar
- 31.Saloustros S, Pelà L, Cervera M (2015) A crack-tracking technique for localized cohesive-frictional damage. Eng Fract Mech 150:96–114CrossRefGoogle Scholar
- 32.Saloustros S, Pelà L, Cervera M, Roca P (2016) Finite element modelling of internal and multiple localized cracks. Comput Mech 59(2):299–316CrossRefMATHGoogle Scholar
- 33.Faria R, Oliver J (1993) A rate dependent plastic-damage constitutive model for large scale computations in concrete structures. CIMNEGoogle Scholar
- 34.Cervera M, Oliver J, Faria R (1995) Seismic evaluation of concrete dams via continuum damage models. Earthq Eng Struct Dyn 24(9):1225–1245CrossRefGoogle Scholar
- 35.Faria R, Oliver J, Cervera M (2004) Modeling material failure in concrete structures under cyclic actions. J Struct Eng 130:1997–2005CrossRefGoogle Scholar
- 36.Roca P, Cervera M, Pelà L, Clemente R, Chiumenti M (2012) Viscoelasticity and damage model for creep behavior of historical masonry Structures. Open Civ Eng J 6:188–199CrossRefGoogle Scholar
- 37.Pelà L, Bourgeois J, Roca P, Cervera M, Chiumenti M (2016) Analysis of the effect of provisional ties on the construction and current deformation of Mallorca Cathedral. Int J Archit Herit 10:418–437CrossRefGoogle Scholar
- 38.Faria R, Oliver J, Cervera M (1998) A strain-based plastic viscous-damage model for massive concrete structures. Int J Solids Struct 35:1533–1558CrossRefMATHGoogle Scholar
- 39.Petracca M, Pelà L, Rossi R, Oller S, Camata G, Spacone E (2016) Regularization of first order computational homogenization for multiscale analysis of masonry structures. Comput Mech 57:257–276MathSciNetCrossRefMATHGoogle Scholar
- 40.Lubliner J, Oliver J, Oller S, Oñate E (1989) A plastic-damage model for concrete. Int J Solids Struct 25(3):299–326CrossRefGoogle Scholar
- 41.Petracca M, Pelà L, Rossi R, Zaghi S, Camata G, Spacone E (2017) Micro-scale continuous and discrete numerical models for nonlinear analysis of masonry shear walls. Constr Build Mater, (in press)Google Scholar
- 42.Oliver J, Cervera M, Oller S, Lubliner J (1990) Isotropic damage models and smeared crack analysis of concrete. In: Proceedings of SCI-C computer aided analysis and design of concrete structures, pp. 945–957Google Scholar
- 43.Bazant Z, Oh B (1983) Crack band theory for fracture of concrete. Mater Struct 16:155–177Google Scholar
- 44.Cervera M, Chiumenti M (2006) Mesh objective tensile cracking via a local continuum damage model and a crack tracking technique. Comput Methods Appl Mech Eng 196(1–3):304–320ADSCrossRefMATHGoogle Scholar
- 45.Cervera M (2003) Viscoelasticity and Rate-dependent Continuum Damage Models. Monography N-79. tech. rep., BarcelonaGoogle Scholar
- 46.Rabczuk T, Bordas S, Zi G (2010) On three-dimensional modelling of crack growth using partition of unity methods. Comput Struct 88(23–24):1391–1411CrossRefGoogle Scholar
- 47.Song J-H, Belytschko T (2009) Cracking node method for dynamic fracture with finite elements. Int J Numer Methods Eng 77:360–385MathSciNetCrossRefMATHGoogle Scholar
- 48.de Borst R, Nauta P (1985) Non-orthogonal cracks in a smeared finite element model. Eng Comput 2(1):35–46CrossRefGoogle Scholar
- 49.Anthoine A, Magenes G, Magonette G (1994) Shear compression tensting and analysis of brick masonry walls. In: 10th European conference on earthquake engineering, (Vienna), pp. 1657–1662Google Scholar
- 50.Magenes G, Calvi GM (1997) In-plane seismic response of brick masonry walls. Earthq Eng Struct Dyn 26:1091–1112CrossRefGoogle Scholar
- 51.Binda L, Mirabella Roberti G, Tiraboschi C, Abbaneo S (1994) Measuring masonry material properties. U.S.-Italy Workshop on Guidelines for Seismic Evaluation and Rehabilitation of Unreinforced Masonry Buildings, pp. 326–347Google Scholar
- 52.COMET (2016) Coupled mechanical and thermal analysis. http://www.cimne.com/comet/
- 53.GiD (2016) The personal pre and post-processor. http://www.gidhome.com/
- 54.Saloustros S, Pelà L, Cervera M, Roca P (2017) An enhanced finite element macro-model for the realistic simulation of localized cracks in masonry structures: a large-scale application. Int J Archit Herit. doi: 10.1080/15583058.2017.1323245 MATHGoogle Scholar