Finite Element Modelling of Tuff Block Masonry Panels subject to Compressive Loading

Transcription

1 Paper 54 Civil-Comp Press, 2015 Proceedings of the Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing, J. Kruis, Y. Tsompanakis and B.H.V. Topping, (Editors), Civil-Comp Press, Stirlingshire, Scotland Finite Element Modelling of Tuff Block Masonry Panels subject to Compressive Loading A. Garofano 1 and A. Ielardi 2 1 Applied Computing and Mechanics Laboratory (IMAC) École Polytechnique Fédérale de Lausanne (EPFL), Switzerland 2 Department of Engineering, University of Sannio, Benevento, Italy Abstract Tuff block masonry is a widely used material for masonry constructions, including many historical buildings, in southern Italy and other Mediterranean regions. The necessity to achieve a more detailed knowledge of the behaviour of this kind of masonry, characterized by high variability in terms of physical and mechanical properties, represents a key issue particularly in the non-linear field. This paper illustrates the results of the finite element modelling and calibration of the constitutive behaviour of yellow tuff block masonry panels subjected to uniaxial loading. The satisfying performance of the proposed models is shown in comparison with the numerical results of available experimental tests. A detailed three-dimensional model of the masonry panels was firstly developed using a micro-modelling technique. For tuff units and mortar joints, individually described in the model, different sets of mechanical characteristics were assumed. The non-linear behaviour of masonry components was described through a smeared cracking method following a multi-directional crack approach. The mechanical properties and the constitutive models of the single materials, with particular attention to the softening behaviour, were based on the experimental data. Afterwards, an equivalent simplified bi-dimensional macro-model was proposed and a homogeneous material defined. Finally, recommended values for the mechanical parameters, useful for micro or macro-modelling approaches of tuff masonry structures, are provided. Keywords: tuff masonry, mortar joints, compressive behaviour, non-linear material, multi-scale modelling, finite element. 1 Introduction Tuff masonry structures built over the last centuries in the Mediterranean area represent a consistent part of the existing buildings and require special prevention 1

2 measures, especially since they are often located in highly seismic areas. This type of masonry, often assembled according to many different arrangements, is widely used also in other areas of the world, like Turkey and Japan, thanks to the abundant availability of raw materials. Tuff rock is generally characterized by a high variability in terms of physical and mechanical properties, linked to the process of formation and consolidation. In addition to the difficulties associated to the large scatter in the mechanical and geometrical properties of the masonry constituents and due to the construction procedure, a higher level of uncertainties are related particularly to the non-linear behaviour of such materials. Considering the costs of experimental characterization through laboratory testing, the high time consuming, the expensive resources, the specific testing machines and measurement devices, the large scatter attained on the construction process, it is clear the convenience of developing proper knowledge of the material through numerical modelling based on available data from tests carried out on this type of masonry. For this purpose, data were collected from available works where tests on the tuff block masonry panels [1], and on individual materials, tuff stones and mortar [2, 3], have been carried out. Extensive studies on panels made with Neapolitan yellow tuff and hydraulic pozzolana-based mortar have been performed in (Augenti and Parisi, 2011) [1], where the uniaxial monotonic compressive behaviour along the directions parallel and orthogonal to the mortar bed joints was studied and analytical stressstrain relationships were defined. In (Lignola et al., 2009) [4] a Finite Element model of masonry panels subjected to diagonal loading was constructed. The model was employed to evaluate the in-plane response of tuff masonry walls strengthened with an innovative cementitious matrix composite grid (CMG) system. A micromodelling approach and parametric analyses were adopted to understand the contribution of basic materials (mortar, tuff blocks and CMG strengthening) and the effect of the workmanship defects. Issues concerning Finite Element procedures have been addressed in (Miccoli et al., 2014) [5] by modelling earth block masonry panels subjected to uniaxial compression and diagonal compression tests. The experimental behaviour was described through non-linear models capable of describing cracking phenomena. The reliability of different numerical strategies was assessed. Both macro and micro-modelling techniques were used to simulate the isotropic or orthotropic behaviour of masonry. The model calibration was carried out on the basis of characterization tests and by sensibility analysis of the input parameters to understand their influence on the compressive and shear behaviour of masonry. The present paper illustrates the results of the Finite Element modelling and calibration of the constitutive behaviour of yellow tuff block masonry panels subjected to uniaxial loading. As mentioned before, experimental tests, found in literature, performed on similar materials were reviewed and considered for the comparison with the results obtained through the numerical simulations. The experimental tests evidenced that yellow tuff masonry presents a typical non-linear behaviour, due to cracking and loss of stiffness starting at early stages of loading. The study was not limited to the elastic properties but the whole non-linear range was investigated and characterized as well. Different Finite Element modelling 2

3 strategies are available for masonry, depending on the scale of observation, each one useful to understand and reproduce a specific behaviour of a masonry structure. The well-known macro and micro-modelling techniques, characterized by different advantages and fields of application, have been both considered in the present study. The micro-modelling is usually applied to small elements requiring a more detailed representation, allowing the investigation of local phenomena, while the macromodelling is used for the global modelling of the entire structure, where the elements are large enough to assume the stress distribution essentially as uniform, and resulting in a reduced computational cost. A detailed three-dimensional model of the masonry panels was firstly developed in this study using a micro-modelling technique. The mechanical properties and the constitutive models of the single materials, with particular attention to the softening behaviour, have been calibrated with respect to the experimental data. Afterwards, the tested panels were described through a bidimensional macro-model, assuming a unique equivalent homogeneous material. The numerical results provided by both the detailed and the simplified models have been found in good agreement with the experimental tests and the suggested mechanical parameters can be useful for future applications. 2 Experimental data The experimental tests available in literature considered in this study were carried out by Augenti and Parisi [1] with the aim to define an analytical model able to approximate the non-linear behaviour of masonry panels made by Neapolitan yellow tuff subjected to uniaxial compression. The specimens were tested under compression both in the orthogonal and parallel directions to the mortar bed joints. The dimensions of tuff blocks employed in the construction of the specimens were 300 x 150 x 100 mm 3, with a unit weight of kn/m 3. A hydraulic mortar with a content of pozzolana with average to low mechanical properties was used. Preliminary tests on both materials composing the masonry specimens were carried out in order to evaluate their mechanical properties [2, 3]. The mean compressive strength of tuff blocks and mortar were, respectively, f bm = 4.13 N/mm 2 and f mm = 2.5 N/mm 2. Masonry panels were built with overall dimensions of 650 x 610 x 150 mm 3. Eight monotonic compressive tests were performed along the direction orthogonal to mortar bed joints, while seven monotonic compressive tests were performed in the direction parallel to mortar bed joints. The set-up was prepared according the ASTM C b and ASTM E11-04 standards, and the tests were carry out under load control up to half of the expected strength and under displacement control in the non-linear field [1]. In order to read the deformations during the tests, two LVDT transducers were placed on both sides of the specimens; the first one orthogonal and the second one parallel to the loading direction. The test set-up and a damaged specimen after the test are illustrated in Figure 1. 3

4 (a) (b) Figure 1: Masonry panel before (a) and after (b) the uniaxial compression test [1] In order to elaborate the experimental results, the compressive stress was evaluated as the ratio between the compressive force applied on top of the wall and the area of the horizontal cross-section of the panel, while the axial deformation was calculated as the average displacement given by the transducers and their length. These parameters allowed to obtain the experimental σ ε diagrams in the direction orthogonal to bed joints, reported in Figure 2. The tests showed that the elastic part of the curve is quite well defined and characterized by a low dispersion of data. A higher dispersion of experimental values is found regarding the peak stress. The results concerning the post-elastic behaviour were also quite dispersed, due to the higher variability related to the fracture process of the material. As a result, higher uncertainties affect the estimation of the value of the material s fracture energy. σ A1 A2 A3 A4 A5 A6 A7 A ε ( ) Figure 2: Results of the compressive tests on tuff masonry panels in the direction orthogonal to bed joints 4

5 Mean values,, standard deviation, s, and the coefficient of variation, CoV, were evaluated for the following values: σ p : compressive strength; σ r : residual stress (defined along the softening branch as the normal stress corresponding to an inelastic strain ε i = 1.5%); ε p : strain corresponding to the compressive strength; ε u : ultimate strain; μ ε = ε u / ε p : ductility; E 1/3 : secant modulus corresponding to ⅓ of the compressive strength; E 1/2 : secant modulus corresponding to ½ of the compressive strength; ν 1/3 : Poisson s coefficient corresponding to ⅓ of the compressive strength; ν 1/2 : Poisson s coefficient corresponding to ½ of the compressive strength; G 1/3 : shear modulus corresponding to ⅓ of the compressive strength; G 1/2 : shear modulus corresponding to ½ of the compressive strength. The values of such parameters are summarized in Table 1. σ p σ r σ r / σ p (%) ε p (%) ε u (%) μ ε ( ) s CoV (%) E 1/3 E 1/2 ν 1/3 ( ) ν 1/2 ( ) G 1/3 G 1/ s CoV (%) Table 1: Results from compressive tests along the direction orthogonal to bed joints 3 Three-dimensional micro-modelling of masonry panels The three-dimensional Finite Element model of the masonry panels subjected to compressive loading in the direction orthogonal to bed joints has been built implementing a micro-modelling technique performed through the code DIANA [6]. Mortar joints and units have been described on the basis of their respective geometry and through different sets of mechanical properties defining the linear, non-linear and cracking behaviour. At the contact between units and mortar joints perfect bond conditions were assumed. Possible approaches for modelling and numerical simulation of the non-linear behaviour of brittle or quasi-brittle materials through smeared cracking approaches are basically the following two: a total strain based approach, fixed crack concept or 5

6 rotating crack concept, and a multi-directional crack approach based on strain decomposition; the latter one was used for the present work. 3.1 Adopted finite elements For the three-dimensional model, solid brick elements, illustrated in Figure 3, have been used. In this case, both the masonry blocks and the mortar joints composing the structure of the wall were modelled with 20-noded elements (indicated as CHX60 in DIANA [6]). The employed elements are isoparametric elements characterized by a quadratic interpolation of geometry and displacements and Gauss integration in the 20 nodes of the element. Typically, a rectangular brick element approximates the following strain and stress distributions over the element s volume as follows: the strain ε xx and stress σ xx vary linearly in x direction and quadratically in y and z direction; the strain ε yy and stress σ yy vary linearly in y direction and quadratically in x and z direction; the strain ε zz and stress σ zz vary linearly in z direction and quadratically in and y direction. A 3x3x3 integration scheme has been applied in the code, obtaining optimal stress points. Figure 3: Finite Element (CHX60) used for 3-D model of masonry panels [6] 3.2 Adopted non-linear material model The non-linear model adopted for description of both materials composing the masonry panel is a multi-directional fixed crack model. In such model, cracking phenomena occurring in quasi-brittle materials during loading are smeared into continuum and are described through a combination of tension cut-off, tension softening and shear retention. In particular, the criterion showed in Figure 4(a) has been employed for the modelling of the material behaviour of both joints and units. For the tension side, a cut-off based on constant value of strength has been considered, while for compression a Von Mises criterion has been chosen. Regarding the shear behaviour, the shear stiffness of the material is usually reduced due to the cracking. This reduction is generally known as shear retention. In case of full shear retention the elastic shear modulus, G, is not reduced. In case of a reduced shear stiffness, the shear retention factor, β, is less or equal to one, but greater than zero. In the present case a constant shear retention has been adopted, according to Figure 4(b). 6

7 (a) (b) Figure 4: (a) Material model adopted for mortar and blocks based on Rankine tensile cut-off criterion and Von Mises yield condition; (b) Constant shear retention [6] (a) (b) Figure 5: (a) Linear tensile softening; (b) Parabolic compressive behaviour [6] Furthermore, a linear tension softening has been employed, Figure 5(a), resulting in a crack stress equal to zero at the ultimate crack strain. In case of linear tension softening, the formulation of the crack stress is given by: 1 0, 0,, (1) with a ultimate crack strain equal to:, 2 (2) The non-linear behaviour of materials in compression is described by a parabolic function (see Figure 5(b)) defined by three characteristic values: the strain, α c/3, at which one third of the maximum compressive strength, f c, is reached; the strain, α c, at which the maximum compressive strength is reached; and the ultimate strain, α u, at which the material is completely softened in compression. These values can be determined as follows: 7

8 / 1 3 ; / / ; 3 (3) 2 The parabolic compression curve is finally described by the following formulation: (4) 3.3 Geometry definition and meshing The masonry wall has been described via a three-dimensional model, with dimensions of 650 x 610 x 150 mm 3, taking into account the actual geometry of masonry units, bed joints and head joints. The geometry of the panel was then discretized through a three-dimensional mesh, composed by the solid elements previously described. In particular, the mesh is composed by 1748 elements, with a total of 8900 nodes, as it can be seen in Figure 6(a). In addition, different material properties have been assigned for masonry units and mortar joints (see Figure 6(b)). (a) (b) Figure 6: Three-dimensional mesh division (a) and material assignment (b) for the masonry panel 8

9 In order to reproduce the actual constraint conditions during the experimental test of the panels, high attention was paid to the modelling of constraints imposed onto the model. The base of the panel has been considered fully supported upon the application of the compressive loading. For all the nodes belonging to the top face of the wall, a rigid constraint in the same direction of the applied vertical load has been defined, in order to assure a uniform vertical displacement to the top of the wall during the analysis, as imposed by the presence of the rigid beam. Since the top rigid beam is a C steel profile, the displacements of the nodes on the top surface of the wall have been constrained to have the same displacement in X direction, while it has been chosen to leave free the possible displacements in the Y direction. The difference in the boundary conditions assumed in the modelling for the base and the top of the panel depends on the fact that during the experiments the constraining conditions between the two sides were not the same due to a different contact between the faces of the specimen and the steel profiles. However, it is remarked that such effects are quite difficult to assess and correctly quantify [7, 8]. Due to the high non-linearity of the material, it was chosen to apply the load in terms of control of displacement on the top surface of the panel. A three-dimensional view of the model in terms of geometry, mesh and materials assignment is reported in Figure 7. (a) (b) (c) Figure 7: Three-dimensional model of the masonry panel: geometry definition (a), meshing (b) and assigned materials (c) 4 Calibration and validation of the numerical model The mechanical characteristics of materials employed for the masonry panels have been obtained from the work by Augenti and Parisi [1] and from the results of the preliminary characterization tests carried out by Augenti and Romano [2, 3] for the determination of the mechanical properties of the component materials of the panels. It is recalled that the masonry blocks, made by Neapolitan yellow tuff, have a unit weight ρ = kn/m 3, the mean compressive strength is f bm = 4.13 N/mm 2, the Young s modulus is E b = 1540 N/mm 2, while the shear modulus is G b = 444 N/mm 2 [1]. The mean compressive strength of the pozzolana-based mortar is f mm = 2.5 9

10 N/mm 2. The Young s modulus of the mortar is E m = 1520 N/mm 2, while the shear modulus is G m = 659 N/mm 2 [1, 9]. The values of the Young s modulus for the masonry blocks are rather low for this kind of stones, as admitted by the same authors, as also confirmed looking at the resulting values of the Young s modulus obtained by the tests on the complete panels. This can be probably associated to some bias in the evaluation of the Young s modulus due to the fact that in [2] also the stroke measurement of displacements during the characterization tests was included in the calculations. Reconsidering the data from the characterization tests reported in [2] and reevaluating the Young s modulus on the basis of the values of the displacements measured by LVDTs, a more accurate value could be found for the elastic modulus of tuff blocks. The same considerations and checks have been carried out for the determination of the Young s modulus of the mortar employed for the joints. Therefore, the analysis of the original experimental characterization data led to the estimation of Young s modulus E b = 2230 N/mm 2 for the tuff blocks and E m = 1980 N/mm 2 for the mortar. These values have been found to be also in agreement with typical ranges of values for yellow tuff stone and for similar types of hydraulic mortars made with sand and pozzolana aggregates [10 14]. The mechanical properties of tuff blocks and mortar joints have been summarized in Table 2. The tensile fracture energies of tuff units and mortar joints were assigned on the basis of typical values found in literature for materials with similar characteristics and strength properties [4, 15] and calibrated assuming variations within a range of ±20%. Furthermore, different values have been considered for the compressive fracture energies, according to available recommendations [16]. In this case, for materials with compressive strength values below 12 N/mm 2, a value of the ductility index d u (defined as the ratio between the fracture energy and the strength) equal to 1.6 mm is provided. Therefore, values of compressive fracture energy of 6.6 N/mm and 4 N/mm, can be obtained respectively for tuff units and mortar joints. These values were assumed as an upper bound for the compressive fracture energies of both materials, while ⅓ of them was assumed as a lower bound. The values reported in Table 2 result from a calibration process, varying in the numerical analyses a single parameter at a time and assuming different combinations. Young s modulus Poisson s coefficient Shear retention factor Tensile strength Tensile fracture energy Compressive strength Compressive fracture energy Material Tuff unit Mortar joint E ν ( ) β ( ) f t G ft (N/mm) f c G fc (N/mm) Table 2: Mechanical properties for tuff units and mortar joints assumed in the model 10

11 In the analysis cases reported in the following it is shown the effect of the compressive fracture energy of tuff units and mortar joints on the uniaxial σ ε curve of the masonry panel. In all the cases, the parameters reported in Table 2 were kept constant, while the values of the fracture energies were varied. Some combinations are reported for reference. The calibration led to a satisfactory fitting of the numerical non-linear curve with the experimental ones. A summary of the cases reported in the following is given in Table 3. Figure 8 (a) 8 (b) 9 (a) Case Tuff unit Mortar joint G fc,b (N/mm) G fc,j (N/mm) Table 3: Compressive fracture energy for tuff units and mortar joints assumed in the analysis cases In Figure 8(a) the results of two analysis cases are reported. In particular, the compressive fracture energy of tuff blocks, G fc,b, was set to a value of 6.0 N/mm, while two different values, 0.8 and 1.5 N/mm, have been considered for the compressive fracture energy of mortar joints, G fc,j. In Figure 8(b) the results of two more cases are illustrated. For these cases, the compressive fracture energy of tuff blocks, Gfc,b, was set to a value of 3.2 N/mm, lower than the previous cases, while the same two different values, 0.8 and 1.5 N/mm, of the previous case for the compressive fracture energy of mortar joints, Gfc,j, have been considered. It can be noticed that the stiffness of the stress-strain curve given by the model is in good agreement with the average stiffness of the experimental curves. Also in terms of strength the model gives a satisfactory result, being the peak value of the numerical curve very similar to the value of the regression curve and to the strength measured from the experiments. If the compressive fracture energy of the blocks is kept constant, the influence of the compressive fracture energy of the mortar joints can be seen in the post-peak behaviour of the models, characterized by different a softening branch. 11

12 σ σ 5.0 Experimental curves 4.5 Gfc,j = 1.5 N/mm; Gfc,b = 6.0 N/mm 4.0 Gfc,j = 0.8 N/mm; Gfc,b = 6.0 N/mm ε ( ) Experimental curves Gfc,j = 1.5 N/mm; Gfc,b = 3.2 N/mm Gfc,j = 0.8 N/mm; Gfc,b = 3.2 N/mm ε ( ) (b) Figure 8: Effect of mortar joint s compressive fracture energy: curves with tuff unit s compressive fracture energy G fc,b = 6.0 N/mm (a) and G fc,b = 3.2 N/mm (b) In Figure 9(a) the curves obtained considering a compressive fracture energy of mortar joints, Gfc,j, of 1.5 N/mm, and a compressive fracture energy of tuff blocks, Gfc,b, of 3.2, 4.5 and 6.0 N/mm, are reported. A higher value of compressive fracture energy of the blocks influences the softening behaviour for higher strain levels, and allows the non-linear curve to keep a higher residual stress value. The curve corresponding to the case with Gfc,j = 1.5 N/mm and Gfc,b = 4.5 N/mm is the one providing the best fitting with the experimental curves in terms of stiffness, compressive strength and softening behaviour, and is compared with the experimental curves in Figure 9(b). As it is shown, a good agreement with the experimental curves is obtained both in the pre and post-peak phase of the compressive behaviour of the masonry. The initial stiffness of the curve fits well the experimental curves range, while the softening behaviour is a good approximation of the experimental one. The numerical model provides a maximum compressive stress of 3.89 N/mm 2, corresponding to only 1.8% less than the average experimental compressive stress, equal to 3.96 N/mm 2. (a) 12

13 σ σ Experimental curves Gfc,j = 1.5 N/mm; Gfc,b = 6.0 N/mm Gfc,j = 1.5 N/mm; Gfc,b = 4.5 N/mm Gfc,j = 1.5 N/mm; Gfc,b = 3.2 N/mm ε ( ) Experimental curves Gfc,j = 1.5 N/mm; Gfc,b = 4.5 N/mm ε ( ) (a) (b) Figure 9: (a) Effect of tuff unit s compressive fracture energy (curves with tuff unit s compressive fracture energy G fc,j = 1.5 N/mm); (b) Comparison between the best fitting numerical curve and experimental results The results of the calibrated model (Figure 9(b)) were analysed also in terms of deformations and stresses distribution. It was possible to identify and compare the simulated cracking distribution of the panel with the experimental one. As an example, Figure 10 reports the principal stress distribution, considered as an indication of the failure pattern and localization of possible cracks, at the step corresponding to the maximum compressive stress. In particular, from Figure 10(a), a concentration of compressive stress, leading to partial crushing, is found in the blocks at the contact areas with the bed joints, which, instead, failed in compression. Furthermore, the maximum principal stress distribution illustrated in Figure 10(b) allows to identify the areas of the blocks interested by cracking, where tensile stresses higher than the tuff tensile strength could be read. The stress analysis showed that the model was able to satisfactorily simulate the failure pattern observed in the tested panels. The difference in the constraint conditions at the top and bottom of the panel can be found in the model 13

14 too (see Figure 10(a)). The partial crushing of the lower tuff blocks, starting at the external corners and progressing during the analysis, could be also seen in the experiments where higher damage was found in the same zones (see Figure 1(b)). (a) (b) Figure 10: Stress distribution (in N/mm 2 ) over the masonry panel corresponding to: (a) minimum (predominantly compressive) principal stresses and (b) maximum (predominantly tensile) stresses 5 Equivalent bidimensional macro-modelling of masonry panels A bidimensional macro-model of the tested panels was constructed and a homogenized equivalent material was calibrated also considering the results obtained from the analyses on the three-dimensional detailed model. The reduced computational cost of this simplified model is useful in case of future applications on more extended models. The bidimensional model of the masonry panel was built using 2-D flat plane stress elements (membrane elements). The element used in the bidimensional model is illustrated in Figure 11 and is an eight-node quadrilateral isoparametric plane stress element (indicated as CQ16M in DIANA [6]). It is based on quadratic interpolation and Gauss integration. For this element, the strain ε xx varies linearly in x direction and quadratically in y direction; the strain ε yy varies linearly in y direction and quadratically in x direction; the shear strain γ xy varies quadratically in both directions. A 2x2 integration scheme was adopted in the code. Figure 11: Finite Element (CQ16M) used for 2-D model of masonry panels [6] 14

15 The geometry of the bidimensional panel is illustrated in Figure 12(a), and was made by surfaces with an overall dimension of 650 x 610 mm. The thickness of 150 mm of the wall was given as an attached property to the elements. As illustrated in Figure 12(b), the mesh of the panel is composed by 440 elements, with a total of 1405 nodes. The dimensions (X, Y) of the elements is x mm 2. (a) (b) Figure 12: Two-dimensional geometry definition (a) and meshing (b) for the masonry panel The actual constraint conditions of the panel during the experiments have been assigned to the model: the base of the panel has been considered fixed, while all the nodes belonging to the top line of the model have been constrained by means of a rigid beam in the direction of the applied vertical load. The possible displacements in the X direction have been left free. Regarding the material used in the bidimensional modelling of the panels, the same non-linear model considered in the detailed three-dimensional model has been adopted. The cracking model is again a multi-directional fixed crack model, defined by a tension cut-off according to Rankine, a Von Mises criterion for compression, and a constant shear retention. The homogeneous material employed in the analysis has a Young s modulus E = 2200 N/mm 2 and a Poisson s coefficient ν = The Young s modulus was assessed through a theoretical procedure for a masonry composed of regular blocks and mortar joints, according to the following formula [17]: where: E: Young s modulus of masonry; E m : Young s modulus of mortar joints; E b : Young s modulus of tuff units; t m : mortar joint thickness; t b : tuff unit thickness. 1 / (5) / 15

16 It was possible, therefore, to take into account not only the brick and mortar Young s moduli, E b and E m, but also the ratio t m /t b between the joint and brick thicknesses. It is also noticed that the values provided by the previous formulation provided a good estimation of the Young s modulus of the tuff masonry, in agreement with the average value obtained from the experiments. The tensile strength assumed for the homogeneous material corresponds to the lowest value of tensile strength between two materials constituting the masonry wall, mortar and tuff, and it is equal to f t = 0.23 N/mm 2, corresponding to tuff units. The tensile fracture energy was estimated on the basis of the lowest value of the ultimate tensile strain between the two components of the masonry, resulting in a tensile fracture energy G ft = N/mm for the homogeneous material, while the compressive fracture energy was calibrated on the basis of the experimental results and the numerical results from the three-dimensional model. The mechanical properties employed for the definition of the homogenous material are finally summarized in Table 4. Young s modulus E Poisson s coefficient ν ( ) Shear retention factor β ( ) Tensile strength f t Tensile fracture energy G ft (N/mm) Compressive strength f c Compressive fracture energy G fc (N/mm) Table 4: Mechanical properties for the homogeneous material assumed in the model σ Experimental curves Three-dimensional model Bidimensional model ε ( ) Figure 13: Comparison between bidimensional homogeneous model and detailed three-dimensional model 16

17 In Figure 13 the result in terms of σ ε curve obtained from the analysis on the bidimensional model is shown. The constitutive curve is also compared with the σ ε curve obtained from the detailed three-dimensional model, showing good agreement. The choice of mechanical parameter previously discussed and the parametric analyses evidenced also the negligible effect of the tensile properties and the important role of the compressive behaviour in the constitutive curve of the wall. 6 Mechanical properties for multi-scale modelling of tuff masonry The choice of the parameters for masonry constitutive models is a difficult task in particular when detailed modelling techniques are followed, but also in case of the adoption of more simplified modelling approaches. Moreover, for structures built with this type of material, it is often possible to achieve only a poor knowledge of the masonry behaviour, limited to the definition of few mechanical properties. In addition, the application of experimental testing procedures to obtain an exhaustive set of mechanical parameters is not always possible due to time or economical restrains and limitations. Nevertheless the experimental results can be affected by bias due to the method of testing, scale problems or restraining effects of boundary conditions, which may affect the correct measurement of a given parameter [7, 8]. Particularly significant in case of masonry elements is also the effect of the workmanship on the overall mechanical behaviour which is difficult to be caught by models based on single components or small scale tests and not always can be easily described by numerical models [4]. Micro-modelling parameters Tuff blocks Mortar joints E ν ( ) E ν ( ) * * Compressive behaviour (parabolic constitutive law) f c G fc (N/mm) f c G fc (N/mm) * ( ) f c * ( ) f c Tensile behaviour (linear softening law) f t ε t,ult ( ) G ft (N/mm) f t ε t,ult ( ) G ft (N/mm) ( ) f c (5 10) f t / E ( ) f t ( ) f c ( ) f t / E ( ) f t * From tests Shear behaviour (constant shear retention) β ( ) β ( ) Table 5: Values of mechanical properties for modelling of tuff masonry with the micro-modelling approach 17

18 Macro-modelling parameters Equivalent homogeneous material E ν ( ) * Compressive behaviour (parabolic constitutive law) f c G fc (N/mm) * ( ) f c Tensile behaviour (linear softening law) f t ε t,ult ( ) G ft (N/mm) ( ) f c (5 10) f t / E ( ) f t Shear behaviour (constant shear retention) * From tests β ( ) 0.2 Table 6: Values of mechanical properties for the definition of the equivalent tuff masonry in macro-modelling approach On the basis of the results from experimental testing of similar types of masonry, and with respect to the results obtained from the numerical models described in the previous sections, a list of recommended values for mechanical parameters for tuff masonry models is given in the Tables 5 and 6. In the case of a detailed modelling of masonry structures, the mechanical parameters are provided with respect to the two components materials of tuff masonry, tuff blocks and mortar joints, as summarized in Table 5. Furthermore, the mechanical properties of the equivalent homogeneous tuff masonry material are reported in Table 6, in case a macro-modelling approach is chosen. 7 Conclusions Tuff masonry is a construction material widely employed in many existing structures and historical buildings in Italy and other Mediterranean regions. The analysis of such type of buildings is often limited by the incomplete knowledge of the mechanical behaviour of the material composing the structure. In addition, the variability of mechanical properties of tuff masonry and of its single components, blocks and mortar, add more sources of uncertainties in the definition of parameters to be used in the models, particularly when the non-linear behaviour needs to be determined. This work was focused on the study of the behaviour of tuff block masonry panels, in order to achieve a more accurate and comprehensive characterization of this type of material. Detailed Finite Element modelling procedures have been employed, allowing to take into account different material characteristics. A detailed 18

19 three-dimensional model of the masonry panels was firstly developed based on a micro-modelling technique. The masonry panel was, therefore, represented through a separate geometrical discretization of units and joints, for which non-linear cracking models with distinct characteristics were employed. At the contact between units and mortar joints with perfect bond conditions were assumed. The non-linear behaviour of tuff blocks and mortar joints have been opportunely calibrated and the comparison with available experimental data resulted in a satisfactory performance of the global model. Secondly, the tuff masonry panels were also modelled through a macro-model based on a simplified bidimensional approach. The aim was the definition of an equivalent homogeneous material representative of this type of masonry. The numerical results provided by the simplified model have been found in good agreement with the experimental tests and the reduced computational effort is useful for future analytical applications. Finally, recommended values for mechanical properties for tuff masonry and for single components, tuff and mortar, have been suggested on the basis of the results provided by the proposed FE models and the available experimental data. References [1] N. Augenti, F. Parisi, Constitutive models for tuff masonry under uniaxial compression, Journal of Materials in Civil Engineering, 22(11), [2] N. Augenti, A. Romano, Preliminary experimental results for advanced modelling of tuff masonry structures, 6 th International Conference on Structural Analysis of Historical Constructions (SAHC), Bath, United Kingdom, 2-4 July [3] N. Augenti, A. Romano, Qualification tests for micro and macro-modeling of tuff masonry structures. 14 th International Brick and Block Masonry Conference, Sydney, February [4] G. Lignola, A. Prota, G. Manfredi, Nonlinear analyses of tuff masonry walls strengthened with cementitious matrix-grid composites, Journal of Composite for Construction, 13(4), , [5] L. Miccoli, A. Garofano, P. Fontana, U. Müller, Static behaviour of earth block masonry: experimental testing and Finite Element Modelling, 9 th International Masonry Conference (IMC), International Masonry Society (IMS), Guimarães, Portugal, 7-9 July [6] TNO DIANA Finite Elements Analysis, User s Manual Release , Element Library, Material Library, Analysis Procedures. TNO Building and Construction Research, Department of Computational Mechanics: A.A. Delft, The Netherlands, [7] A.T. Vermeltfoort, Effects of the width and boundary conditions on the mechanical properties of masonry prisms under compression, Proceedings of the 11 th International Brick and Block Masonry Conference, Shanghai, P.R. China, pp , October

20 [8] L. Barbosa, A. Lima, A. Santos, Study of the influence of compressive strength and thickness of capping mortar on compressive strength of prisms of structural clay blocks, 15 th International Brick and Block Masonry Conference, Florianópolis, Brazil, 3-6 June [9] N. Augenti, F. Parisi, Constitutive modelling of tuff masonry in direct shear, Construction and Building Materials 25, p , [10] G. Marcari, G. Fabbrocino, P.B. Lourenço, Investigation into compressive behaviour of tuff masonry panels, Wondermasonry 2, Workshop on Design for Rehabilitation of Masonry Structures, Lacco Ameno, Ischia Italy, [11] G. Marcari, G. Fabbrocino, P.B. Lourenço, Mechanical properties of tuff and calcarenite stone masonry panels under compression, 8 th International Masonry Conference (IMC), International Masonry Society (IMS), Dresden, Germany, 4-7 July, [12] B. Calderoni, E.A. Cordasco, L. Guerriero, P. Lenza, G. Manfredi, Mechanical behaviour of post-medieval tuff masonry of the Naples area, Masonry International, Journal of the International Masonry Society, vol. 21, no. 3, [13] B. Calderoni, G. Cecere, E.A. Cordasco, L. Guerriero, P. Lenza, G. Manfredi, Metrological definition and evaluation of some mechanical properties of post-medieval Neapolitan yellow tuff masonry, Journal of Cultural Heritage, 11(2): , [14] A. Prota, G. Marcari, G. Fabbrocino, G. Manfredi, C. Aldea, Experimental In-Plane Behavior of Tuff Masonry Strengthened with Cementitious Matrix Grid Composites, Journal of Composites for Construction, Vol. 10, No. 3, June 1, [15] P.B. Lourenço, Computational strategies for masonry structures, Doctoral Thesis, Delft University of Technology, Delft University Press, [16] P.B. Lourenço, Recent advances in masonry structures: micromodelling and homogenization, Galvanetto U., Aliabadi M.F.H., Editors, London: Imperial College Press, , [17] M. Como, Statics of historic masonry constructions, Springer Series in Solid and Structural Mechanics 1, M. Frémond and F. Maceri Eds. DOI /