NUMERICAL MODELLING OF THE IN-PLANE BEHAVIOUR OF …Numerical modelling of the in-plane behavior of rubble stone masonry 5 Figure 4: Rubble masonry wall studied by [13]. Regarding

SAHC2014 – 9th International Conference on Structural Analysis of Historical Constructions

F. Peña & M. Chávez (eds.) Mexico City, Mexico, 14–17 October 2014

NUMERICAL MODELLING OF THE IN-PLANE BEHAVIOUR OF RUBBLE STONE MASONRY

Nicola Tarque1, Andrea Benedetti2, Guido Camata3 and Enrico Spacone3

1 Civil Engineering Division, Pontificia Universidad Católica del Perú Av. Universitaria 1801, Lima, Peru

[email protected]

2 Department of Civil, Chemical, Environmental, and Materials Engineering, University of Bologna Via Zamboni 33, Bologna, Italy

[email protected]

3 Department of Engineering and Geology, University G. d’Annunzio Viale Pindaro 44, Pescara, Italy

[email protected] [email protected]

Keywords: stone masonry, numerical modelling, in-plane behaviour, FEM

Abstract. It is well known that unreinforced stone masonry has a poor behaviour under hori-zontal loads. Since the material is brittle, this type of masonry is very vulnerable and in the case of earthquakes damage, collapses and causalities are very likely to occur. In order to better understand its behaviour and to propose efficiency reinforcement system, it is im-portant to numerical evaluate the seismic response of stone masonry. In this work a simplified methodology to reproduce a shear-compression test of rubble stone masonry is proposed within the finite element method (FEM). The methodology represents the stone units as rigid bodies and the mortar as plastic material with compression and tension degradation, both using plane stress elements. It is known that the stone shapes through the wall thickness may-be even irregular, but here those are assumed to fit a plane model. The experimental tests gathered from a literature review showed that the tensile and compressive strengths of the mortar are the parameter that affects the most the seismic behaviour of the masonry. For this reason the elastic and inelastic part of the constitutive laws used to numerically represent the mortar are calibrated with attention. The results of the numerical models show that the ap-proach is able to reproduce fairly well the experimental test analysed (monotonic test) in terms of capacity curve and failure pattern. The advantage of this simplified methodology is the use of a limited number of degrees of freedom which allows the reduction of the computa-tion time, leaving the possibility to analyse complete walls.

Nicola Tarque, Andrea Benedetti, Guido Camata and Enrico Spacone

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1 INTRODUCTION Stone masonry is one of the oldest material used for construction around the world. For ex-

ample, early European populations as Greeks and Romans used stone as decorative and struc-tural elements. Later, in the middle age the work done by Gothic builders was remarkable considering more complex shape constructions as abbeys and castles.

Stone masonry could be general classified according to the unit shapes: rubble masonry

and ashlar masonry. As it is specified by [1], a vast part of historical stone constructions are built considering dry-stone masonry without bonding mortar, irregular stone masonry with bonding mortar, rubble masonry with irregular bonding mortar thickness, and with a combina-tion of the three techniques. Looking at the wall thickness the stone wall can be composed on (Figure 1): Single leaf: stone bricks are bound together using mortar. The bricks may be character-

ized by irregular shapes and the mortar joints are normally thick and horizontally or sub-horizontally disposed.

Double-leaf: two different layers can be identified on the cross section. The leaves can be completely separated by a vertical joint, which can be filled by mortar, or the leaves can be interlocked (courses are slightly overlapped in the transversal direction).

Three-leaf: this is characterized by two load-bearing external leaves and an inner leaf, this last is composed with a very heterogeneous material consisted on mortar (sometimes mud), small rubble stones and other loose materials. Thick mortar bed joints are horizon-tally disposed on external layers. Sometimes there is a transversal connection.

a) Single leaf b) double-leaf without and with interlocking c) three-leaf

Figure 1: Classification of different cross sections of stone masonry, modified from [2].

As part of the seismic risk mitigation and conservation of the heritage, some researchers have been studied how to numerically approach the pseudo static and dynamic behaviour of stone walls and its possible reinforcements. For this, finite element (FEM) and discrete ele-ment (DEM) methods have been used. However, due to the few experimental data and due to the diversity of the mechanical properties of stone walls constituents (especially the binder material, mortar), the numerical analyses still remains an open topic for the scientific and en-gineering community.

The objective in this work is to demonstrate the capability of a simplified methodology to

study the in-plane behaviour of rubble stone single leaf walls. Although the stone walls may

Numerical modelling of the in-plane behavior of rubble stone masonry

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vary in thickness, it is assumed that it has a constant thickness. Here the stones are treated as rigid body elements with three degree of freedom (DOFs) while the mortar is treated within the plasticity theory, in particular using a plastic-damage model. This model characterizes the mortar by its compressional and tensional constitutive law and some damage parameters for reproducing degradation specially due to reversal loads. The simplified methodology allows to save computational time with great accuracy in the numerical response.

2 FINITE ELEMENT MODELS Finite element method (FEM) using calibrated stress-strain relationships and modelling

techniques represents a suitable numerical approach to reproduce the masonry behaviour. Previous research results have shown that the response of masonry structures up to failure can be successfully modelled using techniques applied to concrete mechanics because both are considered brittle materials [3][4][5]. The numerical modelling of masonry walls can follow either the micro-modelling of each of its components (discontinuous or discrete approach) or the macro-modelling of the wall (continuum approach), thus assuming that the complete ma-sonry wall is homogeneous.

In the discontinuous/discrete approach the failure zones are placed in pre-assigned weak

zones, such as the mortar joints or brick middle. Besides, physical separation can be expected between bricks. However, this approach is computationally expensive for the analysis of large masonry structures. The continuum approach performs well in cases where the damage zones are spread over the wall, and not limited to few bricks and mortar joints [6] [7]. Each of these approaches (represented in Figure 2) have advantages and disadvantages related to the degree of accuracy and computation time.

a) Discrete approach b) Continuum approach

Figure 2: Representation of cracking within a discrete and continuum approach win FEM, modified from [6].

In the present paper, a sort of combination of both techniques is used for the simplified methodology. The mortar (elastic and plastic behaviour) is represented by the continuum ap-proach and the stone units as rigid bodies with 3 degree of freedom (DOF). This allows to have just cracking and crushing at the mortar zones simulating the discrete approach. The re-duction of DOFs for the stones helps to save computational time when analysing other walls.

2.1 Damage model for quasi brittle materials The plastic-damage model used here to represent the mortar is the one developed by [8]

[9] and implemented in the FEM software Abaqus/Standard [10]. This model is a continuum, plastic-based, damage model for concrete, where the two main failure mechanisms are tensile


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cracking and compressive crushing of the material. This model assumes that failure of a qua-si-brittle material (mortar in this case) could be effectively modelled using its uniaxial tension, uniaxial compression and plasticity characteristics. The stiffness degradation in the damage-plastic based model is represented by the damage factors (dt, dc) that reduce the elasticity module in tension and compression for reversal loads (3). The tension and compression con-stitutive laws are the principal input data, as well as the fracture energy for each of them. The fracture energy represents the inelastic area below the stress-strain diagram divided by the el-ement characteristic length, h. This last value is used to avoid mesh dependency in the results (e.g. [11]), and is the length of a line across an element for a first-order element; or half of the same typical length for a second-order element.

a) Tensional behaviour b) Compressional behaviour

Figure 3: Response of concrete under tension and compression loads implemented in Abaqus for the concrete damaged plasticity model, modified from [12].

3 EXPERIMENTAL TEST NUMERICALLY REPRODUCED HERE

To prove the capability of the proposed methodology, some shear-compression tests per-formed by [13] has been reproduced here. They performed a complete research programme at the University of Minho to experimentally evaluate the in-plane seismic performance and failure pattern of ancient stone masonry without and with bonding mortar, and considering different pre-compressional loads. Three masonry panel typologies were studied: dry-stone masonry (WS), irregular stone mansonry (WI) and rubble stone masonry (WR), in this work just the WR wall’s behaviour is of interest. A total of 10 WS, 7 WI and 7 WR with dimen-sions of 1000 x 1200 x 200 mm were built by [13]. Figure 4 shows an sketch of a rubble ma-sonry wall reproduced in this paper.

Experimentally each panel was subjected to a pre-compressional vertical load followed by

an horizontal top cyclic displacement. The pre-compression values were 0.50, 0.875 and 1.25 MPa, corresponding to 100 kN, 175 kN and 250 Kn, respectively; and allowed to study the influence of axial vertical loads. The mortar compression strength for WI and WR was regis-tered around 3.0 MPa, a low value to simulate deteriorated stone walls typically found in his-torical stone masonry [14]. The adopted dimensions for the walls and stone units were about 1:3 scale for single leaf walls found in the northern region of Portugal.


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Figure 4: Rubble masonry wall studied by [13].

Regarding the experimental behaviour of the WR walls, it was shown that level of pre-compressional loads greatly influenced the in-plane behaviour. Walls with a vertical load of 100 kN had a flexural failure (rocking at the base) with a large rotation at the top wall; while the other walls -175 and 250 kN- shown a mixed of flexural and shear failure for the former and shear failure for the latter. Later, [1] numerically reproduced the in-plane behaviour of these walls by a simplified micro-modelling approach, which is out of the scope of this paper. In the following sections a comparison of the experimental Force vs Displacement curves of these walls are shown.

4 NUMERICAL MODEL

4.1 Description of the model In this work just the rubble stone masonry (WR) was reproduced in Abaqus/Standard con-

sidering the 3 different pre-compressional loads. The mortar was represented by the plastic-damage model implemented in Abaqus and the stones were assumed as rigid bodies. The stone and mortar arrangements were preliminary drawn in a CAD software, specifying planar surfaces, and then exported into the FEM software as one part. To have a rigid body behav-iour for the stones, first an elastic material with modulus of elasticity (E=20200 MPa) and Poisson’s module (v= 0.20) was specified for all stones, then a Rigid Body constrain was as-signed to each of the them. The stone’ specific weight was 26.0x10-06 N/mm3.

Just the wall texture of Figure 4a was used for different analysis, it should be noticed that

in the experimental tests different stone arrangement and mortar thickness could be found for each of the 7 WR. Regarding the boundary conditions and loads, here the wall base was fully fixed to the foundation and the vertical and the horizontal load were placed at the top stone unit, as seen in Figure 5a. For each stone unit a reference point, RP, was assigned automati-cally coincident to the centre of gravity of the stone. For the mesh, 4-node rectangular shell elements and 3-node triangular elements were used to represent the stone and mortar, respec-tively (Figure 5b). The characteristic element length h was kept close to 20 mm for the mortar.

The material properties used for the stone and mortar were based on the values reported by

[1][13]. However, not all the required information was available, for example the fracture en-ergy in compression Gfc and tension Gft and the modulus of elasticity E for the mortar to be used with the plastic-damage model. The tensional strength was kept as 0.05 MPa as reported by [1]. The compressional strength was close to the value reported by [13], 3 MPa. The elas-ticity of module was initially computed as the joint stiffness (kn= 2 MPa/mm, [1]) times the


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thickness of the mortar. The tensional softening curve for the mortar followed an exponential shape (Figure 6a) and the compressional hardening/softening curve followed a parabolic shape (Figure 6b). A summary of the material properties used are specified in the following tables. The calibration process was performed varying each of the previous parameters (E mortar and Table 2) one by one and running the analyses until reach an acceptable agreement in terms of numerical and experimental Force vs Displacement curve and failure pattern of the wall. Since the experimental test considered a cyclic displacement, here just the envelope of the cyclic response was used. The calibration process was done based on the results of the wall with a pre-compression load of 175 kN and an horizontal load applied from left to right (Figure 5a).

a) sketch of the test b) Numerical mesh

Figure 5: Representation of the shear-compression test.

Table 1: Elastic material properties for the mortar.

Mortar E (MPa) v γm (N/mm3)

115 0.25 20.0 E-06

Table 2: Mortar material properties used for the plastic-damage model.

Tension (softening) Compression (hardening/softening) ft (N/mm2) Gft (Nmm/mm2) fc (N/mm2) fci (N/mm2) Gfc (Nmm/mm2) εp (mm/mm)

0.05 0.01 2.50 2.30 4.90 0.0175

a) Tensional behaviour b) Compressional behaviour

Figure 6: Constitutive laws used for the representation of the rubble stone walls.


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4.2 Numerical analyses For the analysis the sequence of loading was equal as in the experimental test: a) boundary

conditions were specified at the wall base, b) gravity loads were imposed, c) the pre-compressional load was added to the previous step and kept constant, d) the last step consid-ered an application of an horizontal load at the wall top until failure. For all cases the relative vertical deformation was kept constant after applying the pre-compressional load. Besides, juts for the walls subjected to 175 kN and 250 kN as vertical load, the rotation of the upper stone units was restricted to avoid large rotations. All the analysis considered a full Newton-Raphson iterative procedure and an automatic stabilization was selected for the convergence criterion, with a specified dissipated energy fraction of 0.0002 and an adaptive stabilization with maximum ratio of stabilization to strain energy of 0.05 for gravity and pre-compressional loads; for the horizontal loads just a specified dissipated energy fraction of 0.001 was im-posed.

4.3 Results The first round of analyses was performed for the wall with pre-compressional axial load

of 175 kN and monotonic horizontal load applied from left to right. After reached a good agreement in terms of numerical results, the material property values were fixed (values shown in previous tables) and the other analyses considering variation of the vertical load and direction of the horizontal load were performed.

The experimental tests consisted on a cyclic top horizontal displacement. Here just a

monotonic analysis was performed, so for a comparison of the Force vs Displacement curves just the envelope of the experimental cyclic curves were taken into account as in [1]. [13] re-ported that the major difference in terms of failure pattern was found for the walls under 100 kN of axial load, this was due to the flexional failure ending in a rocking behaviour. The other axial loads allowed a mixed of flexional and shear failure with crushing at toe zones and di-agonal cracking at the centre wall.

Figure 7 shows the comparison of all numerical and experimental curves (these last named

with a prefix WR) considering variation of axial load and direction of horizontal loads. Be-sides the numerical results reported by [1] are plotted and named S&L 2009. In all cases an acceptable agreement is seen, especially for the wall with 175 kN as axial load. When the failure is major controlled by shear, as it is the case for Figure 7b and 7c, 3 zones in the Force vs Displacement curve could be identified. The first, which is the elastic, is controlled by the elastic modulus of the mortar. Then, the mortar tensile strength controlled the slope change of the F-D curve and the tensile fracture energy allowed the stress re-distribution. Finally, the compressional stress concentration at the wall toe and the crack thickness defines the maxi-mum lateral strength until the wall collapse.

Some differences in terms of post-peak behaviour, seen in Figure 7c, can be due to the dif-

ferent stone arrangement used in the experimental test and in the numerical models. Besides, since the stone arrangement of rubble stone walls is not symmetric, an slightly different be-haviour it is expected to see when horizontal loads change direction.


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loading from left to right loading from right to left

a) Rubble masonry, pre-compression load 100 kN


b) Rubble masonry, pre-compression load 175 kN


c) Rubble masonry, pre-compression load 250 kN Figure 7: Comparison of numerical and experimental Force vs Displacement curves.

The numerical failure pattern seen for walls under 100 kN as axial behaviour was con-trolled by the mortar tensional failure at bottom wall zone (see red rectangle in Figure 8a and Figure 9a), which allowed to have rocking. Some diagonal cracking were initiated at the cen-tre wall; however, the maximum strength of 35 to 40 kN was governed by the flexional failure.

For the walls with 175 kN as axial load some horizontal mortar tensional failure was ob-

served at the bottom and top wall zones (see Figure 8b and Figure 9b) but with more diagonal cracking at the middle wall. The loss of stiffness in the numerical curve (Figure 7b) was due to the gradually cracking initiation, while the maximum strength was strongly influenced by the compression at the toe wall. The post peak behaviour of the curves was controlled by a mix of the fracture energy in tension and compression.


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The wall under 250 kN as axial load had an increment of the maximum lateral strength, but the failure was registered for less displacement values, which indicates a more brittle failure. Almost null horizontal cracking was registered at the top and bottom wall edges (see Figure 8c and Figure 9c) due to the high level of pre-compression at the wall. Figure 9 shows the failure pattern of the walls but considering horizontal loads from right to left, here the same failure tendency described before was observed.

a) Axial load 100 kN b) Axial load 175 kN c) Axial load 250 kN

Figure 7: Numerical failure pattern of the walls at the last step. Horizontal load from left to right.

a) Axial load 100 kN b) Axial load 175 kN c) Axial load 250 kN

Figure 9: Numerical failure pattern of the walls at the last step. Horizontal load from right to left.

5 CONCLUSIONS This papers shows an alternative methodology to fast reproduce the in-plane behaviour of

stone walls with a uniform wall thickness. Within the finite element method (FEM), the pro-posed methodology represents the mortar with a continuum approach and the stone units as rigid bodies. This last allows to reduce the number of DOFs in the analysis, consequently sav-ing computation time. A plastic-damage material with stiffness degradation due to cracking and crushing is calibrated to reproduce the material behaviour of the mortar. Some shear-compression tests performed by [13] has been numerically reproduced here. The good agree-ment in terms of Force vs Displacement curves and failure pattern with the experimental tests validates the assumption that stone units are rigid enough while all the damage is concentrated to the mortar material. Besides, it has been seen that the axial load is an important parameter that influences on the type of failure, low axial loads allow a flexural failure, while high axial


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loads allow shear failure. Since this methodology is efficient computationally, complete stone walls could be modelled to evaluate their in-plane seismic response. Further developments may include the calibration of damage parameters to reproduce the cyclic in-plane behavior of stone walls.

REFERENCES [1] R. Senthivel, P.B. Lourenço, Finite element modelling of deformation characteristics of

historical stone masonry shear walls. Engineering Structures, 31, 1930-1943, 2009. doi:10.1016/j.engstruct.2009.02.046

[2] L. Binda, G. Baronio, D. Penazzi, M. Palma, C. Tiraboschi, Characterization of stone masonry walls in seismic areas: data-base on the masonry sections and materials inves-tigations. Proceedings of the 9th National Conference, Italian Seismic Engineering, on CD-ROM (only available in Italian), Turin, Italy, 1999.

[3] H.R. Lotfi, P.B. Shing, Interface Model Applied to fracture of masonry Structures. ASCE, 120 (1), 63-80, 1994.

[4] N. Tarque, G. Camata, E. Spacone, H. Varum, M. Blondet, Non-linear dynamic anal-yses of a full scale unreinforced adobe module. Earthquake Spectra Journal, early online publication, http://dx.doi.org/10.1193/022512EQS053M, 2013.

[5] L. Pelà, M. Cervera, M., P. Roca, An orthotropic damage model for the analysis of ma-sonry structures. Construction and Building Materials, 41, pp. 57-967, 2013.

[6] Midas FEA: Advanced nonlinear and detail analysis program, Analysis and Algorithm, Total strain crack, chapter 13. 2011.

[7] C. Sui, M.Y. Rafiq, Laterally loaded masonry wall panels: a review of numerical mod-els. Journal of the International Masonry Society, 22(2), 47-52, 2009.

[8] J. Lubliner, J. Oliver, S. Oller, E. Oñate, A plastic-damage model for concrete. Interna-tional Journal of Solids and Structures, 25(3), 299-326, 1989.

[9] J. Lee, G. Fenves, Plastic-Damage Model for Cyclic Loading of Concrete Structures. Journal of Engineering Mechanics, 124(8), 892-900, 1998.

[10] Abaqus 6.9 SIMULIA, Abaqus/CAE Extended Funcionality EF2, Manual. Dassault Systemss Corporation, Providence, RI, USA, 2009.

[11] P.B. Lourenço, Computational strategies for masonry structures. Ph.D. thesis, Delft University, Delft, The Netherlands, 1996.

[12] A. Wawrzynek, A. Cincio, Plastic-damage macro-model for non-linear masonry struc-tures subjected to cyclic or dynamic loads. Proceedings of the Conf. Analytical Models and New Concepts in Concrete and Masonry Structures (AMCM’2005), Gliwice, Po-land, 2005.

[13] G. Vasconcelos, Experimental investigations on the mechanics of stone masonry: Char-acterization of granites and behaviour of ancient masonry shear walls. Ph.D. thesis, University of Minho, Guimaraes, Portugal, 2005.

[14] G. Magenes, A. Penna, A. Galasco, M. Rota, Experimental characterization of stone masonry mechanical properties. Proceedings of the 8th International Masonry Confer-ence, Dresden, Germany, 2010.

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NUMERICAL MODELLING OF THE IN-PLANE BEHAVIOUR OF …Numerical modelling of the in-plane behavior of rubble stone masonry 5 Figure 4: Rubble masonry wall studied by [13]. Regarding