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15th International Brick and Block Masonry Conference

Florianópolis – Brazil – 2012

NUMERICAL SIMULATIONS OF MASONRY LABORATORY TESTS: A SENSITIVITY ANALYSIS OF THE COMPRESSIVE BEHAVIOUR

Sousa, Rui 1; Sousa, Hipólito2 1 Msc, Researcher, GEQUALTEC, Faculty of Engineering, University of Porto, Civil Engineering Department,

[email protected] 2 PhD, Professor, GEQUALTEC, Faculty of Engineering, University of Porto, Civil Engineering Department,

[email protected]

The main objective of this paper is to demonstrate how computer simulations of laboratory tests can be useful for studying the compressive behaviour of masonry systems, made with materials with different properties, without consuming too many laboratory resources. This paper describes a sensitivity analysis applied successfully to a case study, namely in a shell bedded masonry system made with lightweight concrete units that was developed in Portugal for single leaf walls. The sensitivity analysis consisted in computer simulations of simple compressive tests on masonry through the use of 3D Finite Element Method and non-linear constitutive model, which was calibrated through a small number of laboratory tests carried on masonry samples and masonry materials. In the analysis some variations on the properties of the masonry materials (units and joints) was considered, and the results obtained are presented and discussed.

Keywords: Lightweight concrete masonry, Simple compression, Laboratory tests, Computer simulations, Sensitivity analysis INTRODUCTION The mechanical behaviour of masonry is complex and experimental characterization through laboratory testing is a key factor to understanding it. However, masonry laboratory tests require high time consuming and expensive resources. This situation is aggravated when further experimental studies on the influence of some aspects on the mechanical behaviour of masonry are needed (e.g. changes in geometry and strength of masonry materials), given the significant increase of samples needed to achieve representative results. Meanwhile, the evolution of computers and nonlinear numerical models for masonry has reduced these difficulties by allowing more accurate simulations of laboratory tests. However, studies based on the use of computer simulations of laboratory tests, in particular to predict the masonry mechanical behaviour due to changes of the constituent materials, aren’t easily found, although some studies about the prediction of the compressive strength of masonry have been recently reported, e.g. Barbosa et al. (2010) or Vyas&Reddy (2010). The contribution of this work is to demonstrate the potential use of computer simulations of laboratory tests to predict the compressive behaviour of masonry systems, with low time consuming and financial costs. To that end, a masonry system was used as a case study, in particular a shell bedded masonry system made with lightweight concrete units that has been developed in the scope of a research project.



Starting from a numerical model calibrated and validated by means of experimental tests made to the masonry system and materials (units and mortar), the sensitivity analysis consisted in the simulations of compressive tests made to masonry samples, whose mechanical and geometric properties of the materials have been modified. The work consisted in the following steps:

• characterization of the mechanical properties of the masonry constituents, units and mortar, through laboratory tests and available literature;

• characterization of the compressive behaviour of masonry through laboratory tests performed on small samples according to European standards;

• construction of a 3D micro-model of masonry sample using a commercial FEM software (ABAQUS) and a plastic damage constitutive model to simulate the units and mortar behaviour;

• calibration and validation of the numerical model through the experimental data obtained from laboratory tests;

• determination of the main properties of the masonry system for a set of variations on some of the geometrical and mechanical properties of the masonry materials.

DESCRIPTION OF THE MASONRY SYSTEM The masonry system used as case study was developed under a co-funded national research project called OTMAPS, i.e. “Thermal and mechanical optimization of single leaf masonry” (in English). The research project involved numerical and experimental studies and some results have already been published, with particular reference to the mechanical behaviour (Sousa&Sousa 2010; Sousa&Sousa 2011). The masonry system is made with lightweight materials, namely vertically perforated concrete units made with light expanded clay aggregates (Figure 1a and Figure 1b) and factory-made lightweight mortar for the joints. The bed joints have two mortar strips (shell bedded masonry) with 10mm of thickness and the perpend joints are filled to the full height, forming 40% of the width of the unit (Figure 1c). The main properties of the masonry materials (unit and mortar joints) were determined through standard laboratory tests and some results are presented in Table 1.

(d) Figure 1: (a) Scheme of the unit with dimensions (in mm), (b) unit blind side for laying

mortar joints, (c) unit opposite side and (d) masonry assembly details



Table 1: Main properties of the materials (unit and mortar joints)

Properties Average Values Standard test references Unit Dimensions (Length x Width x Height) 350x350x190 mm EN 772-16

Total volume of holes in unit 26% EN 772-9 Unit weight 18,5 kg EN 772-13

Mortar dry density 1300 kg/m3 EN 1015-10 MASONRY MATERIALS (UNITS AND MORTAR PROPERTIES) Laboratory tests were made to determine certain specific properties needed to define the constitutive model parameters of the masonry materials, such as, for example, the stress-strain conditions before and after the maximum tensile and compressive strength. Given the difficulties associated with the dimensional representation of test samples and the available testing equipment, some properties were estimated through simple standard laboratory tests carried out on the units and mortar joints, mathematical expressions from model codes for concrete structures, i.e., Model Code 90, and other test data available from empirical studies (Neville 1995; Veiga 1997). In short, the mortar properties, such as the direct tensile strength (σto), the fracture energy (GF) and the compressive constitutive law (before and after the compressive strength, σcu), were estimated from Model Code 90 mathematical expressions (CEB-FIB 1990), the experimental results of the flexural and compressive tests performed according to the European standard EN 1015-11 (1999), and other test data reported in studies carried out on similar mortars (Veiga 1997). As for the concrete properties used in the units, the direct tensile strength (σto) and fracture energy (GF) were estimated with a similar approach. However, in this case the samples were prepared directly from webs and shells removed from the units. The compressive behaviour of concrete units was determined in two load directions: perpendicular and parallel to the bed joints. The tests were conducted according to procedures established in the European standard EN 772-1 (2000). Six units were tested, three for each direction of loading. Displacement transducers were used to measure the vertical and horizontal deformations, ∆v and ∆h (Figure 2a and Figure 2b). These tests and the use of numerical regression techniques enabled the determination of a unique constitutive law, up to the compressive strength, σcu, since an acceptable scatter in the experimental results was obtained (R2=0,96), Figure 2c. As for the behaviour after σcu, this was estimated trough the mathematical expressions of the Model Code 90 and other experimental data available for lightweight concrete (Neville 1995).



(a)

(b)

(c)

Figure 2: Simple compression tests in masonry units and results - (a) load perpend and (b) parallel to the bed joints, and (c) compressive behaviour of the concrete

Moreover, some of the mechanical properties of the lightweight concrete were determined through this constitutive law, namely the Poisson coefficient (ν) and the modulus of elasticity (E0) for 40% of the compressive strength (σcu), Table 2. Stresses were calculated in the net area of the units (perpendicular to the direction of loading). This data was implemented in the constitutive model used in the numerical simulations. It is stressed that the constitutive law of the concrete used in the units represent a simplified and integrated way of considering the influence of various factors, apart from the material itself, on the behaviour and fracture mechanism of the units. Table 2: Mechanical characteristics of the concrete and mortar

Material Modulus of

elasticity Poisson

Coefficient Compressive

strength Strain for

σcu Tensile

strength Fracture energy

(tensile) E0 (N/mm2) ν σcu (N/mm2) εcu (mm/m) σt0 (N/mm2) GF (Nmm/mm2)

Concrete 5427 0.16 3.12 0.91 1.27(*) 0.0085(*) Mortar 12866(*) 0.20(*) 11.45 2.15 (*) 1.57(*) 0.0175(*)

(*) Obtained indirectly through empirical expressions from the literature MASONRY SYSTEM Compressive strength tests were performed according to European standard EN 1052-1 (1998). Four samples measuring 800x525x350mm3 (height x length x wide) were built in laboratory conditions. The vertical shortening (∆v) was measured in the direction of the compression load, F, through four displacement transducers, two mounted on each side of the samples (Figure 3a and Figure 3b). For practical reasons all measurements were taken up to the maximum load, Fmax. The main mechanical properties were determined from the experimental curve, using the formulae given in EN 1052-1 (Table 3). The modulus of elasticity (E) was determined by considering the vertical shortening (∆v) corresponding to one third of the maximum compression load (Fmax). Figure 3c gives the load-strain compression



diagrams (F-ε) for the samples and their representative curve, calculated through numerical regression (designated as the “experimental”). Figure 4 shows a representative example of the fracture pattern observed in the test samples.

(a) (b) (c)

Figure 3: Examples of masonry compression tests - (a) Plan of a test sample (in mm), (b) test set-up and (c) load-strain diagrams for the tested samples

Figure 4: Typical fracture pattern of masonry samples - (a) vertical cracking of units

near the parallel joints and (b) horizontal crushing of units near the loading area Table 3 - Mechanical properties of masonry determined from the experimental curve

Fmax (kN) εmax (mm/m) E (N/mm2) R2 326.8 0.66 3185 0.97

These experimental results were used for the calibration and validation of the numerical model chosen for the masonry. NUMERICAL SIMULATIONS The model used in the numerical simulations was based on the following aspects:

• 3D Finite Element Method with micro-modelling of the masonry test sample used in laboratory tests, i.e., discretization of units and joints linked together through contact points generated in the interfaces;

(a)

(a)

(b)



• application of a plastic damage model developed developed by Lubliner et al.(1989), later improved by Lee et al. (1998), to simulate the non-linear the behaviour of the masonry materials (mortar and concrete);

• simplification in modelling the behaviour of the unit-joint interfaces by considering perfect contact conditions.

a) Contact between masonry materials A simplification was undertaken to facilitate the process of numerical convergence, since certain problems appeared by using friction models (e.g., Mohr-Coulomb) in the simulation of the contact between units and mortar joints. It was decided to use perfect contact conditions, i.e., a model that considers total transmission of loads (normal and tangential) through contact points generated between the units and the mortar joints. b) Constitutive model for Masonry Materials The main concepts used in this model will be briefly described to contextualize the work carried out in this study, namely the parameters involved in the calibration of the constitute model. More details about the model can be found in the given references. The model is able to simulate the non-linear behaviour of concrete and mortar, such as tensile cracking and compressive crushing. This can be implemented in the model through the uniaxial constitutive laws, usually represented by tensile/compression stress-strain relations. As an alternative to the tensile constitutive laws, the concept of brittle fracture, proposed by Hilleborg (1976), can be used trough a value for fracture energy (GF).The model uses the concepts of classical plastic theory, in particular strain decomposition, elasticity, and plastic flow. The main features of the model are:

• The hardening variables simulate the failure mechanisms (tensile cracking and compressive crushing) and control the evolution of the yield or failure surface, i.e. the tensile equivalent plastic strain, ε�t

pl, and compressive equivalent plastic strain, ε�cpl;

• The development of the plastic strains is governed by a potential plastic flow (G), through a Drucker-Prager hyperbolic function:

(4)

• Yield function is defined as a function of the hardening variables and the effective stresses in the following form:

(5)

(6)

on the condition 0 ≤ α ≤ 0,5 (7)

on the condition 0,5 ≤ Kc ≤ 1 (8)

𝐺𝐺 = �(ϵ 𝜎𝜎𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡(𝜓𝜓))2 + 𝑞𝑞� – �̅�𝑝 𝑡𝑡𝑡𝑡𝑡𝑡(𝜓𝜓)

Y�σ,� ε�pl� =1

1 − α�q� − 3αp� + β�ε�pl�⟨σ��max ⟩ − γ⟨−σ��max ⟩�–σ � c � ε�c

pl�

β�ε�pl� =σ � c � ε�c

pl�

σ � t � ε�tpl�

(1 − α) − (1 + α)

α =σb0 σc0� − 1

2 σb0 σc0� − 1

γ = 3 (1 − Kc)2 Kc − 1



IMPLEMENTATION OF THE NUMERICAL MODEL A masonry sample with equal geometrical characteristics as the one tested in the laboratory conditions was simulated by micro 3D modelling in the FE commercial software ABAQUS (Figure 5). This software had implemented the constitutive model presented in 4.3.

Figure 5: Geometric model of a masonry sample used for numerical simulations

Given the complex shape of the masonry units, a 4-node tetrahedral finite element was used to facilitate the meshing process and the Newton-Raphson iterative method was used. The sample support conditions were simulated with total restraint of displacements. The loading was simulated by imposing vertical displacements in one of the supports, in the direction of the compression load, with a linear variation and uniform growth of displacements. The compression load (F) was determined by the sum of the nodal reactions. CALIBRATION AND VALIDATION OF THE NUMERICAL MODEL The numerical model was calibrated in two stages. The first consisted of calibrating the concrete constitutive model through computer simulations of the laboratory compressive tests performed on the units. The second stage consisted of the validation and, if necessary, the calibration of the masonry numerical model through computer simulations of the laboratory compressive tests carried out on the masonry samples. a) Calibration of the constitutive model - Stage 1 The material parameters that had a direct determination (from tests) or indirect (from literature) were set constant and equal to the values established in Table 2. For the model parameters that couldn’t be determined, a range of values was set according to following: ∆ψ = [15º; 45º]; ∆ϵ = [0.1; 1]; ∆σb0/σc0 = [1; 1.3]; ∆Kc = [0.67; 1]. In the best possible approximation between the numerical simulations and the experimental results, a small difference between 2 and 3 % was obtained for εcu and σcu. Table 4 presents the values these parameters after the calibration of the units. It should be noted that the properties defined in Table 2 and in Table 4 for the concrete and mortar were used in the numerical simulations of the masonry tests. Table 4: Variable model parameters adopted for the concrete after calibration (best fit)

ψ (degrees) 𝛜 σbo/σco Kc

45º 0.10 1.16 1.00



b) Validation of the masonry numerical model - Stage 2 A good approximation on the first attempt was found between the numerical and experimental results. Small differences between the mechanical proprieties were found (5 to 12%) and the load-strain diagrams (F- ε) were very similar (Figure 5).

Figure 5: Diagram between experimental and numerical results (after the calibration)

Table 5: Mechanical proprieties for masonry: numerical (num.) VS experimental (exp.)

Reference E (N/mm2) Fmax (kN) εmax (mm/m) Ratios between the mechanical proprieties

Enum /Eexp Fnum /Fexp εnum /εexp Experimental 3185 327 0.66 1.05 0.98 1.12 Numerical 3345 322 0.74 A detailed analysis of stresses and deformations was performed to identify potential fracture patterns, which were then compared with the patterns observed in the laboratory tests. An example of the minimum and maximum principal stresses and global deformation for the maximum compression load (breaking point) is given in Figure 6 and certain stress concentration areas are pointed out to enable some conclusions to be drawn.

(a) (b)

Figure 6: Stresses and global deformation at the maximum load (breaking point) - (a) Minimum principal (compressive stresses) and Maximum principal (tensile stresses)



The numerical results seem consistent with the experimental results, since a high concentration of tensile stresses in the units near the discontinuity of the parallel joints, Figure 6b, can induce vertical cracking in those units. Moreover, a high concentration of compressive stresses in the units near the loading area, Figure 6a, can induce horizontal crushing of those units. Therefore, theses concentrations of stresses can create fracture patterns that are similar to the ones observed in laboratory tests (see Figure 4). According to this analysis, it was considered that the numerical model used for the masonry satisfactorily reproduces the main failure mechanisms observed in the laboratory tests and these results were used as reference in the sensitivity analysis (reference sample). SENSITIVITY ANALYSIS Based on the calibrated model for the masonry, a sensitivity analysis about the influence of the masonry materials on the compressive behaviour of masonry was carried out. Some possible variations in masonry materials used in the reference situation were considered, in particular in the following parameters:

• replacing the discontinuous joints, g/t=0.69, with continuous joints, g/t = 1; • mortar joints with half or twice the initial compressive strength (0.5σcu and 2σcu); • concrete units with twice the initial compressive strength (2σcu).

The analysis considered the variation of one parameter at a time, keeping the other parameters constant and equal to the reference sample. In order to consistently consider the variation of the mortar compressive strength on the global material characteristics, the mechanical properties of the mortar and concrete were estimated as described in section 3.1 (Table 6). Table 6: Mechanical properties estimated for the mortar and concrete used in the sensitivity analysis

Ref. E0 (N/mm2) υ σcu (N/mm2) σt0 (N/mm2) GF (Nmm/mm2) 0.5σcu (mortar) 8311 0.20 5.73 0.96 0.010 2σcu (mortar) 19917 0.20 22.90 2.42 0.029

2σcu (concrete) 6682 0.16 6.24 2.02 0.014 RESULTS AND DISCUSSION The mechanical behaviour of masonry with different material parameters (var.) was compared with the mechanical behaviour of the reference masonry sample (ref.) through ratios between the mechanical properties determined in the numerical simulations (var./ref.). Table 7 shows the mechanical properties of the masonry and its comparative ratios, while Figure 7 gives the comparative diagrams (F-ε) with the variations considered in relation to the reference sample. Table 7: Mechanical properties of the masonry (sensitivity analysis)

Sample Parameters E (N/mm2)

Fmax (kN)

εmax (mm/m)

Ratios Evar/Eref Fmax,var/max,ref εmax,var/εmax,ref

ref. Mortar (σcu=11.45 N/mm2) Concrete (σcu=3.12 N/mm2) Parallel joints (g/t=0.69)

3345 322.0 0.74 - - -

var.

Mortar (0.5σcu) 3286 316.3 0.77 0.98 0.98 1.04 Mortar (2σcu) 3418 329.0 0.73 1.02 1.02 0.98 Concrete (2σcu) 4286 592.3 1.14 1.28 1.84 1.53 Parallel joints (g/t=1) 3874 406.2 0.79 1.16 1.26 1.06



(a)

(b)

(c)

Figure 7: Compressive behaviour of masonry made with (a) discontinuous parallel joints (g/t=0.69), (b) units and (c) parallel joints with different mechanical strength

According to the results obtained in the simulations the most influential parameters for the compressive strength of masonry were the use of continuous joints and the use of more resistant units, since this resulted in a substantial increase of stiffness and load capacity of the masonry. However, the mortar strength had no significant influence. These findings are consistent with several scientific studies (e.g. Drysdale et al. 1994) and masonry code specifications, Eurocode 6 (EN 1996-1-1 2005). For example, the greater influence of the unit properties compared with the mortar joints can be deduced in the formulae provided in Eurocode 6 for masonry in simple compression. Moreover, in the worst case admitted in Eurocode 6 for shell bedded masonry (i.e. joints with g/t=0.4), the compressive strength can be reduced to half of the value for masonry made with continuous joints. CONCLUSIONS The numerical model has characterized the compressive behaviour of masonry rather well, at least until the maximum load is reached (breaking point). The simplification introduced for modelling the interfaces between masonry materials may result in some inaccuracy after the breaking point, but this simplification can save computational resources in the event of difficulties with the numerical convergence. The results indicate that it should be possible to use of numerical simulation of laboratory tests by means of a numerical model calibrated with a very small number of tests performed on masonry and on masonry materials. NOMENCLATURE General Symbols Greek letters A loading area E modulus of elasticity of masonry E0 modulus of elasticity of the masonry materials F compression load Fmax maximum compression load G potential plastic flow g full width of the mortar strips GF fracture energy Kc ratio between the tensile and compressive

stress invariants at initial yield L distance between measurement points of ∆v

𝛼, β, 𝛾 adimensional parameters; ∆h, ∆v horizontal vertical displacements 𝜀 total strain εc uniaxial compressive strain εcu strain for σcu εmax strain of masonry for Fmax 𝜀̃𝑝𝑙 multi-axial equivalent plastic strain 𝜀�̃�𝑝𝑙 compressive equivalent plastic strain 𝜀�̃�𝑝𝑙 tensile equivalent plastic strain

ν Poisson coefficient



or ∆h �̅�𝑝 effective hydrostatic pressure 𝑞𝑞� Von Mises equivalent effective stress t total thickness of the wall

𝜎𝜎� effective stress 𝜎𝜎𝑏0 initial equi-biaxial compressive yield stress σc uniaxial compressive stress 𝜎𝜎�𝑐 compressive effective stress 𝜎𝜎�𝑡 Tensile effective stress 𝜎𝜎𝑐0 initial uniaxial compressive yield stress σcu uniaxial compressive strength 𝜎𝜎��𝑚𝑎𝑥 maximum principal effective stress (algebraic

value) σto uniaxial tensile strength 𝜓𝜓 dilation angle ϵ “eccentricity” parameter

ACKNOWLEDGEMENTS The authors gratefully acknowledged ADI- Innovation Agency and the company Maxit-Portugal for the help provided in the OTMAPS research project. REFERENCES Barbosa, C.S., Lourenço, P.B., Hanai, J.B. On the compressive strength prediction for concrete masonry prisms. Materials and Structures, 43 (2010):331-344. CEB-FIP. Model Code 90 - Design Code. Lausanne: Thomas Telford Ed., 1990. Drysdale, R., Hamid A., Baker, L. Masonry structures. Behavior and design. New Jersey: Prentice Hall&Englewood Cliffs, 1994. EN 1015-11. Methods of test for mortar masonry - Part 11: determination of flexural and compressive strength of hardened mortar. CEN, Brussels; 1999. EN 1052-1. Methods of test for masonry - Part 1: Determination of compressive strength. CEN, Bruxels,1998. EN 1996-1-1. Design of masonry structures - Part 1-1: general rules for reinforced and unreinforced masonry. CEN, Brussels; 2005. EN 772-1. Methods of test for masonry units. Part 1: determination of compressive strength. CEN, Brussels, Belgium, 2000. Hilleborg, A., Modeer, M., Petersson, P. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res, 6 (1976): 773-782. Lee, J., Fenves, G. Plastic-damage model for cyclic loading of concrete structures. J Eng Mech, 124 (8) (1998): 892-900. Lubliner, J., Oliver, J., Oller, S., Oñate, E. A plastic-damage model for concrete. Int J Solid Struct, 25 (3) (1989): 299-326. Neville, A. Properties of concrete (4th ed.). Harlow: Longman Scientific & Technical, 1995. Sousa R., Sousa H. Influence of head joints and unreinforced rendering on shear behaviour of lightweight concrete masonry. In:Proceedings of the 9th Australasian Masonry Conference, Queenstown (NZ), 2011, pp 515-522. Sousa, R., Sousa, H. Experimental evaluation of some mechanical properties of large lightweight concrete and clay masonry and comparison with EC6 expressions. In: Proceedings of the 8th International masonry conference Dresden (DE), 2010, pp 545-554. Uday Vyas, Ch. V., Reddy, B.V.V. Prediction of solid block masonry prism compressive strength using FE model. Materials and Structures, 43 (2010):719-735. Veiga, M. Comportamento de argamassas de revestimento de paredes. Contribuição para o estudo da sua resistência à fendilhação (in portuguese). Ph.D. dissertation, University of Porto, 1997.

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