15th International Brick and Block Masonry Conference
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A SHEAR RESPONSE SURFACE FOR THE CHARACTERIZATION OF UNIT-MORTAR INTERFACES
Parisi, Fulvio1; Augenti, Nicola2 1 PhD, Post-Doc, University of Naples Federico II, Department of Structural Engineering, [email protected] 2 Professor, University of Naples Federico II, Department of Structural Engineering, [email protected]
Shear behaviour of unit-mortar interfaces is typically characterized through the Mohr- Coulomb failure model and shear stress versus shear strain diagrams. In porous stone masonry types such as tuff masonry, dilatancy plays also a key role and shear strength of unit-mortar interfaces at zero confining normal stress is non-zero due to the slip surface’s roughness. To characterize nonlinear shear behaviour for tuff masonry assemblages, direct shear tests were carried out under different pre-compression levels. This paper summarises the experimental program discussing the main results. Empirical formulas are presented to define shear failure at both peak and residual stress levels. Shear deformation capacity, strength degradation, mode II fracture energy, and dilatancy coefficient were computed. Multiple regression analysis was applied to derive a shear response surface including both stress-strain diagrams and the frictional strength model. Constraints on the continuity of both the shear response surface and its first partial derivatives were imposed to nonlinear regression analysis, in order to represent shear softening behaviour in the inelastic range. The surface was defined in a dimensionless space to be used, in principle, for other stone masonry interfaces. This empirical model allows to simulate the shear behaviour over the whole range of allowable strains, and hence the stress-strain diagram at any confining stress level. The experimental results and the proposed empirical models could be employed in both micro-modelling numerical strategies and simplified nonlinear analysis methods based on the macro-element idealisation of masonry walls with openings.
Keywords: Unit-mortar interface, direct shear tests, dilatancy, mode II fracture energy, shear softening, shear response surface
INTRODUCTION Unit-mortar interfaces considerably affect the overall behaviour of masonry under any loading condition. Two basic failure modes can be identified for such interfaces: tensile failure (mode I) and shear failure (mode II). The former is the separation of the interface normal to the joints, the latter may consist of a shear failure of the mortar joint or sliding mechanism of the masonry units (i.e., bricks, blocks, stones, etc.). Since unit-mortar interfaces act as planes of weakness, mechanical properties of masonry significantly change with the loading direction. Therefore, the mechanical behaviour of masonry along those material discontinuities has been deeply investigated for a number of masonry classes (Atkinson et al. 1989; van der Pluijm 1993; Binda et al. 1994; Lourenço & Ramos 2004). Past studies have highlighted the potential for strain softening in both dry and mortar joints with rough surfaces. Furthermore, a significant dilatational behaviour of the masonry joint under shearing deformation has been 15th International Brick and Block Masonry Conference
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detected in most cases, that is, a transverse expansion resulting in a volume growth of mortar. This means a non-isochoric plasticity, as opposed to metals and plastics (van Zijl 2004), and hence a pressure build-up under the normal uplift if the volume increase of the masonry joint is prevented or resisted by confining boundary conditions. As a result, shear strength of the unit-mortar interface may significantly increase with the normal compressive stress even though shearing dilatancy is arrested at high pressure levels and at large shear strains.
Shear strength of unit-mortar interfaces gives a major contribution to both shear and compressive strengths of masonry (Atkinson et al. 1982; McNary & Abrams 1985). Nevertheless, this issue has not yet been investigated in the case of tuff masonry, which until now has been widely used around the world even in earthquake-prone regions. This lack of knowledge inspired the authors to carry out a series of direct shear tests with the aim of (1) developing empirical models to be used in nonlinear analysis of masonry structures, and (2) assessing mechanical properties employed in both numerical and analytical models. Data processing was first addressed at evaluating shear modulus, shear strength at zero confining stress (i.e., cohesion), and friction coefficient in the range of small strains. Fracture energy and dilatancy angle were also estimated to be used in finite element (FE) analyses. Secondly, the Mohr-Coulomb failure criterion was characterized at both peak and residual shear stress levels and could be combined with a cap model in compression (Lourenço & Rots 1997) and a cut-off in tension (Binda et al. 1994). Several regression analysis techniques were employed (1) to define τ−γ constitutive laws at different pre-compression levels, and (2) to obtain a shear response surface τ(σ,γ) including shear stress-strain diagrams and the Mohr- Coulomb failure model.
EXPERIMENTAL PROGRAM A series of monotonic direct shear tests on double-layer masonry specimens made of tuff stones and mortar joints were carried out. Those experimental tests were preferred over couplet and triplet tests, although the latter have been adopted as the European standard method (CEN 2002). In fact, triplet tests are more difficult to be controlled in the post-peak range of the stress-strain response because two joints are tested all together, while couplet tests are affected by parasite bending effects inducing a non-uniform distribution of stresses at the unit-mortar interface.
The materials adopted for the specimens were: (1) yellow tuff stones from Naples, Italy, 300×150×100 mm in size, with uniaxial compressive strength fb = 4.13 MPa, Young’s modulus Eb = 1540 MPa, and shear modulus Gb = 544 MPa; and (2) premixed hydraulic mortar based on natural sand and a special binder with pozzolana-like reactive aggregates (water/sand ratio by weight 1:6.25, i.e., 4 L of water per 25 kg of sand). The mortar had low- medium mechanical characteristics and consistency to reproduce the actual conditions of tuff masonry in ancient buildings; indeed, it had an uniaxial compressive strength fm = 2.5 MPa, Young’s modulus Em = 1520 MPa, and shear modulus Gm = 659 MPa. Both strengths and elastic moduli were characterized through compressive tests according to European standards.
As shown in Figure 1, each specimen consisted of a single-leaf, double-layer tuff masonry assemblage with a mortar bed joint. The gross dimensions of the specimen were 852×210×150 mm, while both head and bed joints had a thickness of 10 mm. The layers were shifted to each other in order to apply and measure the horizontal forces via simple contact 15th International Brick and Block Masonry Conference
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between devices and specimen. Two linear variable differential transformers (LVDTs) were placed on each side of the specimen to measure horizontal and vertical relative displacements induced by the shearing deformation and dilatancy. The vertical and horizontal LVDTs had a stroke of 20 and 50 mm, respectively. The latter were fixed to both masonry layers.
765 300 10 145 10 300 tuff stones LVDT #1 LVDT #2 71 100 45 10 72 210 210 100 bed 68
300 10 145 10 300 joint 150 263 261 241 765 765 Figure 1: Specimen geometry (dimensions in mm) and arrangement of LVDTs
The experimental program consisted of three series of deformation-controlled tests performed at different pre-compression load levels. Each experimental test was carried out by subjecting the masonry specimen to increasing shear deformation along the mortar bed joint. The test set-up was slightly different from that used by Atkinson et al. (1989) for clay brick masonry, even if the ability to induce a rather uniform distribution of shear and normal stresses along the bed joint was preserved (Fig. 2). Both the applied and resisting lateral forces were applied at a distance from the bed joint which was defined on the basis of the size of the hydraulic jack and the height of the masonry layers. The masonry specimen was placed onto a L-shaped steel I-beam which was enabled to slip over the rigid base of an universal testing machine by means of two Teflon layers and two lateral unequal angles. A double-effect hydraulic jack with load capacity of 500 kN was positioned against the beam and a reaction frame. A further L-shaped I-beam was installed over the specimen and was forced against a load cell (with load capacity of 100 kN) anchored to another reaction frame. After two further Teflon layers were placed over the upper metallic beam, the hydraulic actuator of the universal testing machine (with load capacity of 3000 kN in compression) was pushed against the specimen. The loading plate of the universal machine was pinned to minimise bending effects.
actuator reaction hydraulic jack LVDT #1 LVDT #2 load cell reaction frame frame
two Teflon layers
HEA160
M18 bolts Unequal angles
two Teflon layers Figure 2: Experimental set-up for direct shear tests 15th International Brick and Block Masonry Conference
Florianópolis – Brazil – 2012
Three sets of direct shear tests were carried out at different pre-compression loads corresponding to about 5%, 10% and 15% of the ultimate axial force resisted by the effective (horizontal) cross-section of the specimen. Namely, three direct shear tests were carried out under a pre-compression load of 25 kN corresponding to a vertical confining stress σ = 0.25 MPa (specimens C1 to C3). A second group of four tests were performed under a pre-compression load of 50 kN corresponding to σ = 0.50 MPa (specimens C4 to C7). A third set of two tests were conducted under a pre-compression load of 75 kN corresponding to σ = 0.75 MPa (specimens C8 and C9).
The loading protocol of each direct shear test consisted of two stages. In the former a vertical confining pressure was imposed to the specimen by means of the hydraulic actuator and was kept constant during the complete test duration. At that stage the upper beam allowed to get a rather uniform distribution of the vertical pressure over the specimen. In the second stage, the lower masonry layer was subjected to a horizontal load with a magnitude calibrated by a computer program in a way to get a target increasing stroke of the hydraulic jack (and hence an increasing shearing deformation of the bed joint) under a given normal pressure level. The shear load was applied at a displacement rate of 10 μm/s, so that nonlinear response of the bed joint was fully measured during the test until a target displacement (corresponding to a shear strain of about 10%) was reached. The horizontal force applied to the specimen was transmitted to the lower masonry layer resulting in a relative movement with respect to the upper layer. The load cell at the opposite side with respect to the hydraulic jack enabled the estimation of the actual shear force resisted by the bed joint. The testing system was able to avoid uncontrolled actions potentially caused by unbalanced forces; no in-plane and out-of- plane rotations were detected.
PROCESSING OF EXPERIMENTAL DATA Force and displacement readings were carefully processed accounting for the effective shear force transferred by the unit-mortar interface, as well as the actual gage length of the bridged horizontal LVDT. The effective shear force resisted by the bed joint was not equal to that applied by the hydraulic jack, nor that measured by the load cell located on the reaction frame. Indeed, a rate of the applied shear force (measured by a further load cell placed between the hydraulic jack and the lower beam) was dissipated through the interface between the lower layer and the base of the universal machine and was absorbed as axial deformation of stone units and head joints. Conversely, the force measured by the opposite load cell was lower than the effective shear force resisted by the bed joint, since the former was affected by frictional dissipation along the upper beam-actuator interface. Therefore, the effective shear force was computed as the sum of the load cell reading and the frictional force dissipated along the beam-actuator interface. This force was assumed as proportional to the ratio between the contact area of the actuator plate and the one of the lower masonry layer. To plot the experimental shear stress-strain diagrams (Figs. 3(a)–(c)), the computed shear forces were divided by the effective bed joint area and the relative horizontal displacement readings were divided by the bed joint thickness. Those computations were limited to the strain range [0,0.1] which was divided into a number of sub-ranges to associate shear stresses falling in a given sub-range with its central shear strain. A large shear strain ductility was identified for all tests generated by a high frictional dissipation capacity of the bed joint. Accordingly to what has been reported in the literature (Homand et al. 2001; Misra 2002), rough fracture surfaces induced strength degradation under 15th International Brick and Block Masonry Conference
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increasing inelastic displacement. As the shear strain increases, the joint roughness reduces ranging from the mobilization of the asperities in the pre-peak region to their gradual smoothening in the post-peak region. Conversely, no softening was recorded for masonry assemblages with dry joints according to Vasconcelos & Lourenço (2009).
0.4 0.4 0.4 C1 0.3 C2 0.3 0.3 C3 0.2 mean 0.2 0.2 [MPa] [MPa]
C4 C5 [MPa] C8 τ τ 0.1 0.1 C6 C7 τ 0.1 C9 mean mean 0 0 0 0 0.05 0.1 0 0.05 0.1 00.050.1 γ γ γ (a) (b) (c) Figure 3: Experimental and mean stress-strain curves for: (a) σ = 0.25 MPa; (b) σ = 0.50 MPa; and (c) σ = 0.75 MPa
The mean values of peak shear strength, residual strength, strength degradation ratio (SDR = τr/τp), and peak strain are reported in Table 1. Both peak and residual shear strengths increase with the confining stress acting normally to the bed joint. It is worth noting that the scatter of the peak shear strength was very limited, being the coefficient of variation (CoV) equal to 2.11%, 4.95%, and 5.72% for a pre-compression level of 0.25, 0.50, and 0.75 MPa, respectively. The uncertainty in the residual strength estimation is higher, with a CoV ranging between 9.96% and 16.97%. Shear softening plays a relevant role given that SDR ranges from 0.63 (for σ = 0.25 MPa) to 0.80 (for σ = 0.50 MPa). For triplets made of solid mud bricks and hydraulic lime mortar with joint thickness of 10 mm, Mirabella Roberti et al. (1997) found a SDR between 0.51 and 0.69 for confining stress ranging between 0.12 and 1.25 MPa. The shear strain at the peak stress increases with the confining normal stress.
Table 1: Mean strength and strain parameters
Test series σ [MPa] τp [MPa] τr [MPa] SDR γp [%] 1 0.25 0.22 0.14 0.63 0.52 2 0.50 0.29 0.23 0.80 1.10 3 0.75 0.36 0.26 0.72 1.50
In simplified seismic analysis methods based on the macro-element idealisation of masonry walls with openings, the lateral stiffness of each macro-element depends on both shear and flexural contributions (Parisi 2010). Although the shear modulus reduces as the lateral displacement increases, the secant shear modulus at one-third of the peak shear stress is widely used in the abovementioned methods. Such a parameter was found to be 93, 72, and 63 MPa for a pre-compression level of 0.25, 0.50, and 0.75 MPa, respectively. The uncertainty in shear modulus is quite limited, since CoV did not exceed 15%.
To simulate the shear response of masonry in the plastic range, strain softening has to be characterized for representing the progressive internal crack growth, due to the presence of micro-cracks and material heterogeneity. Softening behaviour of tuff masonry is believed to be increased by the presence of some pumice inclusions within the stone units, which result in higher concentrations of cracking and acceleration of crack formation over non-porous masonry units (e.g., clay bricks). Shear softening of the unit-mortar interface may 15th International Brick and Block Masonry Conference
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conveniently be described via the mode II fracture energy, defined as the integral of the shear stress versus relative horizontal displacement diagram. The fracture energy increment considerably reduced under increasing shear strain and the mean mode II fracture energy was found to be 0.12, 0.13, and 0.16 Nmm/mm2 for a pre-compression level of 0.25, 0.50, and 0.75 MPa, respectively. Since van Zijl (2004) found that the mode II fracture energy increases linearly with the confining stress, a linear regression analysis was carried out providing the following relationship (with a coefficient of determination R2 = 0.92):
II =+ G f 0.10 0.07σ (1) so the mode II fracture energy at zero-confining stress was 0.1 Nmm/mm2. This value is in agreement with that obtained by van der Pluijm (1993) in the case of masonry specimens with II 2 solid clay or calcium-silicate units, for which Gf was between 0.01 and 0.25 Nmm/mm . Dilatancy was also investigated by means of normal displacement versus shear displacement curves (i.e., δv-δh diagrams). In the context of the Mohr-Coulomb failure criterion, one can assume that the effective friction angle of the bed joint is the sum of an inherent angle due to adhesion and a dilatancy angle defined as ψ = arctan(δv/δh). The dilatational behaviour under shearing deformation was consistent with that observed by other researchers (e.g., Misra 2002) and consisted of three different trends. In the first one, the normal displacements were negative (i.e., the masonry layers tended to approach). Afterwards, vertical displacements became positive and increased with the shear displacements. Finally, as the shearing deformation increased, the vertical displacements tended to a constant steady-state value. The following relation between dilatancy coefficient and confining stress was found (R2 = 0.29):
tanψ =−0.34 0.42σ (2)
CHARACTERIZATION OF THE MOHR-COULOMB FAILURE MODEL The shear strength of the unit-mortar interface was characterized through linear regression analysis, assuming the Mohr-Coulomb failure criterion for predicting both peak and residual shear strengths. This strength criterion establishes a linear relationship between the shear stress τ and the normal stress σ as follows:
τμσ=+⋅c (3) where c and μ are respectively the cohesion and the friction coefficient (i.e., the tangent of the friction angle φ) of the contact surface, assumed to be strength parameters at peak and residual stress levels. Their characterization led to the following relations:
=+ τ p 0.15 0.29σ (4)
=+ τr 0.08 0.26σ (5) with R2 equal to 0.96 and 0.78 respectively. Such models point out a friction angle of 16° and 15° for the peak and residual stress states, respectively. It is worth noting that the residual cohesion of the bed joint was not set to zero within regression analysis, because it is believed to be associated with a non-zero frictional resistance of the slip surface. This evidence was 15th International Brick and Block Masonry Conference
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observed by van der Pluijm (1993), while no residual cohesion was detected for dry masonry joints by Lourenço & Ramos (2004) through cyclic couplet tests. The cohesive bond strength and the friction coefficient reduced of 47% and 10%, respectively, from the peak to the residual state, whilst results of linear regression analysis by Binda et al. (1994) showed that cohesion and friction coefficient reduce respectively of 61% and 12% for sandstone masonry triplets, and of 43% and 26% for calcareous stone masonry.
SHEAR RESPONSE SURFACE A nonlinear stress-strain model was first fitted to the experimental data associated with each pre-compression level (Fig. 4(a)) and then it was suitably regularised for further processing (Fig. 4(b)). The aim of those computations was to obtain a dimensionless shear response surface of the unit-mortar interface by merging the stress-strain diagrams and the Mohr- Coulomb failure criterion together. Such a surface is to be distinguished from the classical failure surface (which does not include the constitutive law) and was defined to get a unique function τ(σ,γ) describing the shear stress under varying shear strain and confining stress.
The symbol τ denotes the shear stress τ normalized to the peak strength τp, σ represents the given confining stress σ normalized to a reference normal stress σref, and γ stands for the given shear strain γ normalized to the shear strain γp corresponding to the peak strength. It is underlined that this strain was identified for each experimental stress-strain curve and γp does not represents the maximum shear strain reached during the tests. It was employed in this study just to emphasize the relationship between the shear strain and the peak shear strength.
0.5 0.5 experimental σ = 0 0.4 empirical 0.4 σ = 0.25 MPa σ = 0.75 MPa σ = 0.50 MPa 0.3 0.3 σ = 0.75 MPa σ = 1.00 MPa σ = 0.50 MPa [MPa] 0.2 [MPa] 0.2 τ τ σ 0.1 = 0.25 MPa 0.1
0 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 γ γ
(a) (b) Figure 4: Stress-strain diagrams: (a) empirical models fitting experimental data for different pre-compression levels and (b) regularised curves
Nonlinear regression analysis on stress-strain data was carried out for two strain ranges, separately, in order to simulate the softening behaviour. Therefore, for each given confining γ stress level, the authors identified a ‘crossing’ shear strain c corresponding to the counterflexure point of the constitutive model, so as to get the best fit to experimental data and to attain the continuity of both constitutive function and its first derivative. The stress- strain diagrams at σ = 0 and σ = 1.00 MPa were derived by extrapolating those directly associated with experimental data. Both cohesive and frictional terms were assumed to be dependent on the shear strain, in order to define the shear stress at the interface as a linear function of the confining stress as follows:
15th International Brick and Block Masonry Conference
Florianópolis – Brazil – 2012
τσ(), γγμγσ=+⋅c() () (6) where σ represents the confining stress σ normalized to a reference normal stress σref assumed to be 0.18 times the uniaxial compressive strength of masonry. The adoption of a linear relationship between the shear stress and the normal stress allowed to split the multiple regression analysis in two simple nonlinear regression analyses carried out on the shear strain. To describe the variation of the crossing shear strain versus the normal stress, the authors derived the following equation:
=+ γc 1 0.275σ (7)
The latter equation was obtained to divide the (σ,γ) plane in two regions (denoted by Ω1 and
Ω2) and, thus, to split the response surface for simulating shear softening (Fig. 5). Given a σ γ normal stress i , one can estimate the shear strain c,i at which corresponds the counterflexure point of the stress-strain diagram.
Figure 5: Construction procedure of the shear response surface
Figure 6 illustrates the shear response surface derived for the unit-mortar interface (R2 = 0.96) where both cohesive and frictional terms are response functions defined as: c(γ)=− 2.719 γ 4.644 γ 234 +2.289 γ −0.6 γ (8)
μ (γ)=− 0.082 γ 1.695 γ 23 −+1.54 γ 0.371γ 4 (9) for γ ∈+[0,(1 0.275σ)]; and
15th International Brick and Block Masonry Conference
Florianópolis – Brazil – 2012
c(γ)=+ 0.464 0.146 γ +0.034 γ 23344 −⋅+⋅3.8 10−−γ 1.5 10 γ (10)
μ (γ)=+ 0.64 0.002 γ −0.001γ 24354 +⋅1.5 10−−γ −4.474 ⋅ 10 γ (11) for γ ∈+[(1 0.275σ),5] . It is underlined that the shear response surface defines the shearing behaviour of the unit-mortar interface from the elastic range to the inelastic range, including strength degradation due to strain softening. The response parameters c( γ ) and μ( γ ) should not be confused with the classical strength parameters of the Mohr-Coulomb failure criterion (denoted as c and μ in this study), because they depend on the shear strain and are used to characterize the shear response over the whole range of allowable shear strains. Therefore, cohesion and friction coefficient could be used in linear equivalent models, while the shear response surface could be employed within nonlinear models.
Figure 6: Shear response surface of the unit-mortar interface
CONCLUSIONS The shear response of stone-mortar interface was characterized for tuff masonry through deformation-controlled direct shear tests performed at different confining stress levels. Strength and deformation parameters of the stone-mortar interface were first computed to be used for both design and assessment purposes. The mode II fracture energy was also evaluated for refined FE analyses. The Mohr-Coulomb failure model was characterized and then merged with stress-strain diagrams fitted to experimental data. Nonlinear multiple regression analysis allowed to develop a shear response surface. In such an interface model the shear stress is a linear function of the confining normal stress, whereas cohesive and frictional response functions of the shear strain are used instead of the classical strength parameters of the failure model. The surface is dimensionless and describe the shear response of the stone-mortar interface over the whole range of allowable strains. Furthermore, stress- strain diagrams at any confining stress level can be derived.
ACKNOWLEDGEMENTS This work was carried out in the framework of the ReLUIS-DPC 2010-2013 project (Line AT1-1.1 - ‘Evaluation of the Vulnerability of Masonry Buildings, Historical Centres and Cultural Heritage’) funded by the Italian Department of Civil Protection. 15th International Brick and Block Masonry Conference
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