Acta Polytechnica CTU Proceedings doi:10.14311/APP.2020.26.0001 Acta Polytechnica CTU Proceedings 26:1–6, 2020 © Czech Technical University in Prague, 2020 available online at http://ojs.cvut.cz/ojs/index.php/app HOMOGENIZATION OF UNREINFORCED OLD MASONRY WALL COMPARISON OF SCALAR ISOTROPIC AND ORTHOTROPIC DAMAGE MODELS Vasco Bernardo, Tomáš Krejčí, Tomáš Koudelka, Michal Šejnoha∗ Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mechanics, Thákurova 7, 166 29 Prague 6, Czech Republic ∗ corresponding author: sejnom@fsv.cvut.cz Abstract. Masonry is a heterogeneous composite material made of bricks bonded by a mortar matrix. Modeling such a material on macroscale typically calls for homogenization adopting a suitable constitutive model capable of capturing its quasi-brittle behavior. The present contribution concentrates on comparison and potential application of classical isotropic and orthotropic damage models in the framework of strain based first order numerical homogenization. As an illustrative example, a representative volume element in terms of a periodic unit cell is constructed to address the response of an unreinforced masonry (URM) structure typical of “Placa” buildings (mixed masonry-reinforced concrete buildings) built in Portugal. The performance of the two models is examined on the basis of macroscopic stress-strain curves constructed for both tensile and compressive loading. The selected geometrical constraints clearly identify the differences in the predictive capabilities of the two models. Keywords: Homogenization of unreinforced masonry wall, periodic unit cell, scalar isotropic damage model, orthotropic damage model, “Placa” buildings. 1. Introduction The Portuguese housing stock results from more than eight centuries of history expansion. The building area consists mostly of residential buildings and it is esti- mated that half of the existing building stock in Lisbon is represented by old masonry buildings [1]. Four ty- pologies of masonry buildings are typically recognized in the Lisbon Metropolitan Area: buildings built be- fore 1755, “Pombalino” buildings built after the 1755 Earthquake, “Gaioleiro” buildings built between 1870 and 1930 and “Placa” buildings 1930 to 1960 [2]. Figure 1. Scheme of "Placa" building typology. The “Placa” buildings, see Fig. 1, now enjoy a partic- ular interest as they represent 32 % of the Portuguese housing stock [1]. As seen in Fig. 1, the structure consists of a concrete slab simple supported by ma- sonry walls. The exterior walls of these buildings are mainly composed of resistant stone masonry walls or by solid clay bricks with thickness greater than 0.50 m. The interior walls are usually made of solid or hollow clay bricks with thickness up to 0.30 m. The slabs are made of reinforced concrete typically with a total thickness of about 0.10 m and frequently with a single reinforcement layer and poor concrete. Such building experience high vulnerability to damage when subject to seismic loads. This is attributed to in- sufficient strength or deformation capacity of masonry walls. It is also worth pointing out that no impact of earthquake has been considered in their design as the first Portuguese seismic design regulation appeared not until 1958. Predicting the bearing capacity of such buildings is thus of paramount importance, par- ticularly in areas with a moderate to high seismic risk. A reliable prediction of the evolution of potential failure due to earthquake then strongly depends on the selected constitutive law for both concrete and masonry. Herein, this issue is addressed in the light of a typically used isotropic scalar damage model [3] and more advanced orthotropic damage model al- lowing for an independent evolution of damage in three principal strain directions [4]. While concrete is usually assumed isotropic the masonry is only lo- cally isotropic at the level of individual constituents whereas on macroscale it possesses an orthotropic ma- terial symmetry. To allow for treating masonry as a homogeneous anisotropic medium thus requires ho- 1 http://dx.doi.org/10.14311/APP.2020.26.0001 http://ojs.cvut.cz/ojs/index.php/app V. Bernardo, T. Krejčí, T. Koudelka, M. Šejnoha Acta Polytechnica CTU Proceedings mogenization. This issue is partially addressed here with reference to one specific topology of a masonry wall linked to the “Placa" building. This choice is supported by available experimental data provided by the National Laboratory for Civil Engineering in Portugal (LNEC) [5, 6]. The paper is organized as follows. Following this introductory part, the two constitutive models are briefly reviewed in Section 2. The basic grounds of the theory of homogenization are introduced next in Section 3. The core of the present work appears in Section 4 devoted to the numerical analysis of unreinforced masonry wall via homogenization. The principal findings are finally summarized in Section 5. 2. Damage models Generally, damage models distinguish three states of the material - virgin material, damaged material and pseudo-undamaged states. In a virgin state, the mate- rial contains no defects and behaves as linearly elastic. At points where the strength limit is exceeded the material experiences damage. In the most simple case of a scalar isotropic damage model, the damage evolu- tion is governed by a dimensionless damage parameter ω written for one-dimensional case as ω = Ad A , (1) where Ad is the part of the cross section with evolved defects, i.e with the material in a damaged state and A is the total cross section area. The corresponding stress-strain relation reads σ = (1 −ω)Deε, (2) where σ is the stress vector, De is the elastic stiffness matrix and ε is the vector of strain components. The scalar damage parameter ω ∈ [0; 1] character- izes the material state, where ω = 0 represents an undamaged virgin state, while ω = 1 corresponds to fully evolved defects. The transition states from 0 to 1 are described by the damage evolution law. The present formulation builds on one-dimensional traction-separation law given by [7] σ = ft exp ( − wcr uf ) , (3) where ft is the tensile strength, wcr is the crack open- ing and uf is the parameter controlling the slope of the softening branch. To partially avoid mesh depen- dency of the results typical of materials with softening the crack opening is smeared over the element as [7] κ−κe = ωκ = wcr lch , (4) where the Mazars equivalent strain κ is provided by κ = √√√√ 3∑ α=1 H(εα)2, (5) where H(·) is the Heaviside function, κe is the elastic part of κ and lch is the element characteristic length. Combining a 1D format of Eq. (2) with Eqs. (4) and (3) and replacing the strain ε with the maximum equiva- lent strain in the loading history κ̄ yields the resulting nonlinear equation to be solved for the damage pa- rameter ω (1 −ω)Eκ̄ = ft exp ( − ωlchκ̄ uf ) , (6) for states where κ̄ exceeds the elastic threshold ε0. The scalar isotropic damage model can be used successfully for the description of 1D stress states, e.g., 1D tension or bending. A clear drawback is evident from a general multidimensional format of stress-strain relation (2) suggesting the reduction of the entire stiffness matrix even in cases when dam- age evolves in one direction only. The remedy can be provided by anisotropic [4] or orthotropic dam- age models. Introducing two independent damage parameters Dtα and Dcα for tension and compression, respectively, gives an equivalent format of Eq. (6) now written for individual principal strain directions as ( 1 −Dβα ) E|εβα| = fβ exp ( − Dβαlch|εβα| u β f ) , (7) where β ∈ [t,c] identifies either tensile or compressive failure. Further details regarding both models can be found in [8] and [9]. 3. Numerical homogenization In the present study, we limit our attention to a strain based first order homogenization only. To this end, consider a heterogeneous body loaded on its outer boundary by displacement field compatible with a macroscopically uniform strain E in an equivalent ho- mogeneous medium. This allows us in the framework 1st order homogenization to split the local displace- ment and strain as [10] ∆u(x) = ∆E x + ∆u∗(x), (8) ∆ε(x) = ∆E + ∆ε∗(x), (9) where u∗(x) is the fluctuation part of the local dis- placement field u(x), ε∗(x) and ε(x) are the corre- sponding strains, and x is the spatial coordinate. Next, suppose that the computational model is de- fined in terms of a periodic cell (PUC) taking the actual brick layout into account. For “Placa” building the assumed PUC appears in Fig. 2. The wall thick- ness is 250 mm, constituted by bricks with nominal dimensions of 250×120×70 mm and 10 mm for the mortar layer. Because the present formulation assumes the macro- scopic strain increment ∆E in Eq. (8) be prescribed, the solution is searched in terms of the fluctuation rather than total displacements. The stepping stone in deriving ∆u∗ is the Hill lemma written as〈 δεT(x)∆σ(x) 〉 = 0, (10) 2 vol. 26/2020 Homogenization of unreinforced old masonry wall Figure 2. Periodic RVE for the masonry wall. where the left hand side represents the volume average of virtual work done by local stress and strain fields. In the framework of finite element discretization, we approximate ∆u∗(x) in terms of their nodal values ∆r to write ∆u∗(x) = N(x)∆r, ∆ε∗(x) = B(x)∆r. (11) Substituting from Eq. (11)2 into (10) yields the final system of algebraic equations in the form K∆r = ∆f, (12) where K = 1 Ω ∫ Ω BT(x)L(x)B(x) dΩ, (13) ∆f = 1 Ω ∫ Ω BT(x)L(x)∆E dΩ, (14) where L is the material stiffness matrix. It is clear that in order to satisfy 〈∆ε(x)〉 = ∆E, (15) the volume average of the fluctuation part of the dis- placement field must disappear. This is achieved by adopting the periodic boundary conditions, which in this particular example of a rectangular PUC amount to the same displacements u∗ on the opposite bound- aries of PUC. The rigid body motion is then con- strained by simply fixing the corner displacements. Note that Eq. (12) can be sought as part of the solution of more complex computational framework based on multiscale analysis illustrated in Fig. 3. In such a case, the actual masonry wall is replaced by an equivalent homogeneous one, so there is no need to consider all the geometrical details on macroscale. Such a distinction between bricks and mortar is made on the mesoscale only being represented by a specific PUC. In case of a fully uncoupled analysis, the the- ory of homogenization then serves to provide all the necessary data needed in the macroscopic constitutive model. The unit cell simulations then represent a virtual tester to substitute for complex or even unfea- sible large scale laboratory experiments. This model- ing concept is adopted henceforth providing both the effective elastic properties and various macroscopic stress-strain curves for the selected loading scenarios to compare performance of the two selected models. Figure 3. Framework for the homogenization of masonry wall. 4. Numerical study The present section is devoted to the derivation of macroscopic response for various loading scenarios limiting attention to prescribed macroscopic uniform strains, as suggested in Section 3. Figure 4. Masonry PUC finite element model. The geometrical model displayed in Fig. 2 is ex- ploited. The corresponding, relatively fine, finite ele- ment mesh is plotted in Fig. 4. This mesh consisting of four node quadrilateral elements is used first in the derivation of effective elastic properties and then in the prediction of a nonlinear response when loading the PUC beyond the material elastic limits. 3 V. Bernardo, T. Krejčí, T. Koudelka, M. Šejnoha Acta Polytechnica CTU Proceedings Material E [GPa] ν [-] ft [MPa] Gft [N/m] fc [MPa] ρ [kN/m3] Brick 13.0 0.2 2.0 58.0 40.0 18.0 Mortar 0.7 0.2 0.1 10.0 1.30 17.5 Table 1. Material parameters of brick and mortar. 4.1. Effective elastic properties from homogenization The basic prerequisite in the application of homoge- nization is the knowledge of material data of individual phases as also intimated in Fig. 3. In case of “Placa” masonry, the computational model consists of clay bricks and mortar produced from the cement, lime and sand mixture in the proportions of 1:3. To col- lect the required properties, the specimens extracted during rehabilitation works from a building located in the center of Lisbon were examined in the National Laboratory for Civil Engineering in Portugal (LNEC), see Fig. 5. The resulting material properties of both phases are listed in Table 1. The same material prop- erties for bed and head joint are assumed. Further details regarding the actual experimental program are available in [5, 6, 11]. (a) (b) Figure 5. Axial compression tests: a) Solid clay bricks, b) Mortar [5]. The effective elastic plane-stress stiffness matrix is found first from four independent elasticity solutions when the unit cell is loaded in turn by each of the components of the macroscopic strain E equal to one while the remaining components vanish. The volume averages of the local stress then furnish individual columns of the homogenized effective stiffness matrix, see [12] for further details. The extracted Young’s moduli and Poisson’s ratios appear in Table 2 clearly identifying the orthotropic material symmetry the “Placa” masonry. E11 E22 G12 ν12 ν21 [GPa] [GPa] [GPa] [−] [−] 6.68 4.03 0.67 0.089 0.147 Table 2. Equivalent elastic properties for RVE. 4.2. Nonlinear response of “Placa” masonry This section compares the macroscopic nonlinear re- sponse of “Placa” masonry due to simple tension and compression applied independently in the X and Y directions as plotted in Fig. 6 also identifying the pre- scribed components of the macroscopic strain E. This is clearly a constrained problem as only one strain component in the principal material direction is as- sumed nonzero. However, because of heterogeneity all local strain and stress components become active to give rise damage evolution due to tensile failure even in case of compressive loading. (a) (b) Figure 6. Applied loading: a) ET = {±Exx, 0, 0}, b) ET = {0,±Eyy , 0}. The resulting macroscopic stress strain curves (Σ– E) are plotted in Fig. 7. The corresponding damage pattern associated with the last converged step for individual loading conditions appears in Figs. 8 - 11. It is interesting to point out clear differences be- tween the predictions provided by the two constitutive models observed in Figs. 7(a-c). Particularly the dif- ference in the predicted strength is attributed to the fact that unlike in scalar isotropic damage model the orthotropic damage model reduces the stiffness inde- pendently in the principal strain direction according to Eq. (7) providing the condition |εβα| > ε0 either for tension or compression is met. This is also sup- ported by the evolution of stress component in the direction normal to the applied load. In this con- 4 vol. 26/2020 Homogenization of unreinforced old masonry wall -0.002 -0.0015 -0.001 -0.0005 0 E xx [-] 0 -1.5 -1 -0.5 Σ [ M P a ]Σ xx - orthotropic model Σ yy - othotropic model Σ xx - scalar isotropic model Σ yy - scalar isotropic model (a) -0.002 -0.0015 -0.001 -0.0005 0 E yy [-] 0 -1.5 -1 -0.5 Σ [ M P a ] Σ xx - orthotropic model Σ yy - othotropic model Σ xx - scalar isotropic model Σ yy - scalar isotropic model (b) 0 0.0005 0.001 0.0015 0.002 E xx [-] -0.2 -0.1 0 0.1 0.2 0.3 0.4 Σ [ M P a ] Σ xx - orthotropic model Σ yy - othotropic model Σ xx - scalar isotropic model Σ yy - scalar isotropic model (c) 0 0.0005 0.001 0.0015 0.002 E yy [-] 0 0.05 0.1 0.15 Σ [ M P a ] Σ xx - orthotropic model Σ yy - othotropic model Σ xx - scalar isotropic model Σ yy - scalar isotropic model (d) Figure 7. Macroscopic stress-strain curves: a) com- pression (Exx < 0) applied in X direction, b) com- pression (Eyy < 0) applied in Y direction, c) tension (Exx > 0) applied in X direction, d) tension (Eyy > 0) applied in Y direction. (a) (b) (c) Figure 8. Damage pattern due to Exx < 0 (com- pression): a) isotropic model, b) orthodam-11, c) orthodam-22. (a) (b) (c) Figure 9. Damage pattern due to Eyy < 0 (com- pression): a) isotropic model, b) orthodam-11, c) orthodam-22. (a) (b) (c) Figure 10. Damage pattern due to Exx > 0 (tension): a) isotropic model, b) orthodam-11, c) orthodam-22. (a) (b) (c) Figure 11. Damage pattern due to Eyy > 0 (tension): a) isotropic model, b) orthodam-11, c) orthodam-22. 5 V. Bernardo, T. Krejčí, T. Koudelka, M. Šejnoha Acta Polytechnica CTU Proceedings text, the results plotted in Fig. 7(d) should not be surprising once realizing the geometrical arrangement of the PUC. Clearly, if tension is applied along the Y axis, the weak mortar behaves as the weakest link. So, in this case, the stiffness along the X axis, which essentially keeps the original value in case of the or- thotropic model, provides no support resulting in a similar behavior predicted by both models. Although not exactly coincident with the axes of material orthotropy the damage parameter developed in the (11) principal direction (e.g., direction (11) is close to the X direction for the Exx strain component prescribed) is significantly greater in comparison to the one in the second (22) principal direction, which appears negligible. It should also be mentioned that even though the two macroscopic normal stresses (Σxx, Σyy) Figs. 7(a,b) are both compressive due to the applied constraints, e.g. Exx 6= 0, while Eyy = 0, the local failure is in tension because of positive strains devel- oped as a result of material discontinuity and stress concentration at vertices of the brick and mortar in- terface. This causes the damage pattern in Figs. 8 and 10 and Figs. 9 and 11 be similar. However, be- cause of the geometrical arrangement, the two loading conditions predict different strength limits. But as already mentioned, the maximum compressive stress reached should not be associated with the compressive strength of the masonry material. Given the proper- ties in Table 1, it is not surprising that the damage localizes solely in the mortar phase for all loading scenarios. 5. Conclusions The present paper was concerned with numerical mod- eling of masonry structure associated with the “Placa” building in the framework of 1st order homogenization. Apart from the derivation of effective properties of the selected masonry topology the research objectives focused on the prediction of macroscopic response em- ploying two different constitutive models. Limiting attention to tensile failure, the performance of a simple isotropic scalar damage model was compared to more advanced orthotropic damage model. The latter one allows for an independent evolution of damage in three principal directions of the local strain, thus leading in general to more stiff response as illustrated in this for both compression and tension loading condition. In this regard, the loading was introduced by pre- scribing the rate of macroscopic strain. Only the load- ing in principal directions of material orthotropy was considered thus introducing some constraints to the macroscopic response. While such loading conditions appear useful in highlighting the essential differences in the behavior of the two damage models, they can- not be used in the estimation of effective parameters of the macroscopic model such as the tensile strength or the fracture energy in tension. Prediction of such parameters would require combining the stress driven homogenization up to the onset of failure with a spe- cific format of the strain-driven procedure to tract the descending part of the macroscopic stress-strain law consistent with a unidirectional stress state. This study will be the subject of our future research. Acknowledgements The financial support supported by the FCT project No.PD/BD/135325/2017 and by the GAČR grant No. 18-24867S is gratefully acknowledged. References [1] F. Tiago, J. Milosevic, R. Bento. Seismic vulnerability assessment of a mixed masonry rc building aggregate by linear and nonlinear analyses. Bulletin of Earthquake Engineering 14:1765–2327, 2016. [2] M. Monteiro, R. Bento. Characterization of ”placa” buildings. ICIST DTC, Portugal 2012. [3] G. Mlani, C. Valente, C. Alessandri. The narthex of the church of the Nativity in Bethlehem: A non-linear finite eelement approach to predict the structural damage. 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WIT Press, Southampton, Boston, 2013. 6 Acta Polytechnica CTU Proceedings 26:1–6, 2020 1 Introduction 2 Damage models 3 Numerical homogenization 4 Numerical study 4.1 Effective elastic properties from homogenization 4.2 Nonlinear response of ``Placa'' masonry 5 Conclusions Acknowledgements References