Shear failure is the dominant mode of failure observed in manymasonry buildings subjected to lateral loading due to earthquakes,wind (in tall and slender structures), support settlements, or unsym-metrical vertical loading. Lateral loading can produce both diago-nal cracking failures and shear failures of the horizontal joints. Jointresistance is of particular concern in the analysis of the load-bearingunreinforced masonry structures that are rather common amongolder buildings in many countries in the world. The shear generallyacts in combination with compression caused by the self-weightand floor loads. Confinement by, for instance, structural framesto in-fill walls may also lead to shear compression.
The present state of knowledge concerning shear strength andshear load-displacement behavior of masonry is far less advancedthan that concerning masonry behavior in compression, eventhough shear failure is an important, often governing mode of fail-ure in many masonry buildings (Van Zijl 2004). This lack of under-standing is reflected by the low values of shear resistance allowedby present U.S. building codes [ASCE 31-03 (ASCE 2003)].Information on the post-peak behavior and on the deformations as-sociated with pre-peak and post-peak responses are also lacking.Only recently, the terms softening and dilatancy were introducedin the research community (Lourenço et al. 1998; Van Zijl2004). Knowledge of such behavior is essential if adequate analyti-cal models are to be developed to describe the in-plane behavior of
masonry walls. Most of the research conducted to date regardingthe masonry shear behavior has been limited to determining thepeak shear stress and its affecting parameters.
A variety of experimental approaches (Fig. 1) has been adoptedin the last two decades to determine the shear behavior of joints ofunreinforced masonry. A widely used approach is the compressiveloading of a prismatic masonry specimen that contains a single jointat an angle, θ, to the applied load as illustrated in Fig. 1(a) (Nusset al. 1978; Hamid and Drysdale 1980). The nature of this force-controlled test makes it impossible to obtain data in the post-peakrange, as the specimen collapses in an unstable manner after attain-ing its strength. Studies using this approach have, however, haveprovided valuable information concerning the factors (includingmortar type) that influence the peak shear stress.
Van der Pluijm (1993) presents the most complete characteriza-tion of the masonry shear behavior for solid clay and calcium-silicate units. The test setup shown in Fig. 1(b) allows to apply aconstant confining pressure upon shearing. The confining (compres-sive) stresses were applied at three different levels, namely, 0.1, 0.5,and 1.0 MPa. Thereby, the specimen edges could translate in the di-rection normal to the shearing deformation. The uplift or displace-ment normal to the shear joint, which is known as dilatancy, was alsomeasured. Armaanidis (1998)measured a dilatation angle from 23.5to 34.5° for limestone using a direct shear test. He proposed that theshear strength at the weak discontinuities of limestone be a com-bined effect of both the internal friction angle (ϕ) and the dilatancyangle (φ) and, proposed the following expression:
τu ¼ cþ σn tanðϕþ φÞ ð1Þ
Hansen (1999), Gottfredsen (1997), and Chaimoon and Attard(2009) also used the same experimental technique in their study.
Many researchers (Yokel and Fattal 1975; Calvi et al. 1985;Gabor et al. 2006) have used the test configuration shown inFig. 1(c) to study the shear strength of masonry subjected todiagonal compression. The concentrated diagonal load creates
1Civil Engineer, Ph.D. Research Student, Graduate School of Engineer-ing, Hokkaido Univ., Kita-ku, Sapporo 060-8628, Japan (correspondingauthor). E-mail: [email protected]
2Professor, Graduate School of Engineering, Hokkaido Univ., Kita-ku,Sapporo 060-8628, Japan. E-mail: [email protected]
in-plane shear stress along the joints of the specimen. The distri-bution of normal and shear stresses along any given joint is stronglynonuniform, with the result that shear strength determined from thistest represents an average value of progressive failure events, be-cause of the stress redistribution during the failure process ratherthan reflecting a true material property. The post-peak behavior anddeformations cannot be obtained realistically by means of this ex-perimental configuration.
The triplet test configuration shown in Fig. 1(d) was adopted byLourenço et al. (2004) as recommended by European Standard EN1052-4 [European Committee for Standardization (CEN) 2000].This test was conducted to verify the Mohr-Coulomb criterion witha cohesion value of the order of 1.4 MPa, and the initial frictioncoefficient (tanϕ) of 1.03. Copeland and Saxer (1964) used thesame specimen configuration to identify the parameters affectingthe shear bond between brick and mortar.
Meli (1973) used the test configuration shown in Fig. 1(e) toinvestigate bond and friction of joints with different unit types.A linear variation of the shear strength with confining pressurewas observed. Bond strength was found to vary with the mortarand unit types. Hamid and Drysdale (1980) also used the test con-figuration shown in Fig. 1(e) to study the shear response of bothgrouted and ungrouted concrete masonry. Their results showed thatthe coefficient of friction decreased with an increase in the confin-ing stress and that grouted specimens yielded friction coefficientsthat were considerably higher than ungrouted specimens. Data con-cerning the deformation in the direction of the shear load showedthat ungrouted masonry has a considerably higher initial shear stiff-ness in comparison to grouted concrete masonry. With the increasein normal stresses both the shear strength and the shear stiffnessincrease. The post-peak frictional response under shear loading thatwas applied in the same direction as the initial shear force was alsodetermined.
Abdou et al. (2006), El-Sakhawy et al. (2002), and Atkinsonet al. (1989) conducted direct shear tests on masonry couplets as
shown in Fig. 1(f). They used a servo-controlled system to measurethe shear load-displacement characteristics for different types ofbrick and mortar. Abdou et al. (2006) tested both hollow and solidbricks, and found that the shear stiffness of masonry with hollowbricks is higher than that of masonry with solid bricks, because ofthe mortar that entered inside the holes and acts as an abutment,thus giving more shear resistance than the solid brick. He alsofound that the ultimate shear strength and residual friction are in-dependent of the brick types. Only one type of mortar was used(20 MPa), so no direct correlation between mortar grade and shearstrength could be established. The joint failure could be well rep-resented by the Mohr-Coulomb criterion when a shear load is ap-plied together with compression. Atkinson et al. (1989) conducteda series of tests on various types of brick and mortar for both staticand cyclic loading. Some of the results along with their test meth-ods are given in Table 1 and Fig. 2.
Previous studies on joint shear behavior while providing insightinto some of the parameters influencing the shear strength generallydo not provide detailed information related to the constitutivebehavior that would be required to set up analytical models for sim-ulating the structural response under different loading conditions.Such a model will require definitions of (1) shear stiffness for initialloading states; (2) peak and residual stresses; and (3) the effect ofmaterials properties and normal loads on shear strength/stiffnessand dilatancy. As in the case of rock joints (Goodman 1976;Armaanidis 1998), dilatancy is the normal expansion or contractionupon shearing. A complex relationship exists among joint normalstiffness, normal displacement, and shear displacement (Saeb andAmadei 1988; Van Zijl 2004). Dilatancy can produce an increase inthe normal load resulting in increased shear strength, when the nor-mal boundary condition is displacement controlled. This applies,for example, to in-fill panels where the stiffness of the frameenclosing the panel affects its normal displacements. An evaluationof the dilatancy requires the measurement of both normal and sheardisplacements prior to and after the peak shear stress. It was also
θ
(a) (b) (c)
(d) (e) (f)
Fig. 1. Different types of shear test specimens: (a) Nuss shear test (1978); (b) Van der Pluijm test (1993); (c) diagonal tension test; (d) triplet test;(e) Meli test (1973); (f) direct shear
observed that the effect of dilatancy under high compression is mar-ginally small and can be neglected (Gabor et al. 2006).
This paper examines the shear failure mode occurring in hori-zontal joints and the shear stress-slip behavior of unreinforced brickmasonry under static loading. Nonstandard tests were conducted onfour series of masonry samples; triplet shear specimens [Fig. 1(d)]were used, in order to study the effect of mortar strength on jointshear behavior. At the outset, the paper describes the experimentalapparatus and the sample preparation procedures. Then the exper-imental program is outlined. This is followed by a description ofthe experimental results. An analytical approach has been taken tocorrelate the shear strength with confining pressure and mortarstrength. Finally, a macro mechanical model for shear stress slip isproposed with numerical examples. Finally, some exhaustive con-clusions are drawn on the basis of the observed results.
Experimental Program
The materials used in the preparation of the triplet shear test spec-imens included one type of wire-cut clay brick and four types ofmortar with different proportions and strengths indicated as E, M,S, and N, respectively. The brick used here had an actual dimensionof 250 × 120 × 70 mm. Bricks were immerged in water the daybefore the construction of the tested specimen assembly. They werethen dried in normal laboratory conditions for at least 1 day prior tobuilding the specimens to ensure saturation degree of 80%. The
brick compressive strength was 17 MPa. The three types of ordi-nary mortar (M, S, N) used were prepared following the provisionsof ASTM C270 (ASTM 2007a) for the construction of masonrywalls. The type E mortar was prepared as high strength mortar withcomparatively low water-cement ratio. The reason for choosingthese mortar types was to study the effect of mortar types andstrength on the shear strength. The workability of the mortar wasmonitored using the flow test per ASTM C 1437 (ASTM 2007b).For each batch of mortar type used in the construction of thespecimens 10 cylinders (∅ ¼ 50 mm) were cast to determine thecompressive and splitting strength of the mortar. Table 2 summa-rizes the results for the four types of mortar used in the presentinvestigation.
Specimen Preparation
The specimens were built with two full bricks, one 3=4 brick and1=4 brick bonded together by a 10 mm-thick mortar joint as shownin Fig. 3(a). To ensure the correct dimension of the mortar joints, atimber block that was thicker than the bricks by 10 mm was placedover the first brick. More than the needed amount of mortar wasthen placed on the top face of the brick with a trowel. The one3=4 brick (170 mm) was then placed in such a way that a smallportion of it (60 mm) exceeded the bottom brick and rested onthe timber block. Another quarter brick was then placed 30 mmapart from the 3=4 brick. The second brick was then placed on thesetwo cut bricks, tapped with a wooden mallet, and leveled in two
Table 1. Various Experimental Results on Interface Shear Stress-Slip Test
Author’s name and test method f 0cm (MPa) Em (GPa) f 0
cb (MPa) σn (MPa) τu (MPa) τ r (MPa) c (MPa) ϕi (Deg) ϕr (Deg) tm (mm)
Hansen (1999) for solid clay brickswith couplet specimen [Fig. 1(b)]
Note: fcm, f 0cb = uniaxial compressive strength of mortar and brick respectively; Em = Young’s modulus of mortar; σn = normal precompression; τu = ultimate
shear strength; τ r = residual shear stress; c = interface cohesion; ϕi = initial friction angle; ϕr = residual friction angle; tm = thickness of mortar.
Fig. 2. Shear-slip relationship of different experimental data from (a) Van der Pluijm (1993); (b) Hansen (1999); (c) Chaimoon and Attard (2009) forsolid clay bricks and (d) Hansen (1999); (e) Lourenco et al. (2004); (f) Abdou et al. (2006) for perforated clay bricks
Table 2. Specification of Mortar Used in Triplet Shear Test Specimen
directions with a sprite level to create a 10 mm-thick mortar joint.The excess mortar, which squeezed to the sides, was removed witha trowel, and the sides of the mortar joint were flattened at the samelevel of bricks on all sides. The timber block was then removed, andthe specimen was left in place for 5 days to allow the mortar todevelop sufficient strength. During these 5 days, the specimenswere covered with thin plastic sheeting for curing. After the 5 daysof initial curing, the plastic sheet was removed. The specimenswere then left for an additional 23 days to cure under ambient con-ditions in the laboratory before testing, which began 28 days fromconstruction.
Instrumentation and Test Setup
Five specimens for each mortar type were built and cured for28 days. Before testing, the length of the mortar joint was measured.Two steel plates were attached on both sides of the specimen andkept in position with four bolts. A uniform confining pressurewas exerted on the specimen using a manually controlled hydraulicjack having a load gauge. When the expected level of pressure wasreached, the specimen was ready for the shear test. Four linear dis-placement (LVDT) gauges were attached on the top of the 3=4 brickthat will be load for shear [Fig. 3(b)] on opposite sides of joints torecord the shear displacement. The specimen was designed in such away that the applied load be transferred through the upper 3=4 brickas shear and the confining pressure be carried out by both the top andbottom bricks. The area resisting to shear and compression was cal-culated accordingly. Fig. 4 shows the loading and support arrange-ments used for testing of the specimens. Synthetic elastomers wereused to ensure a uniform load distribution over the area and supports.
Testing and Measurements
For each type of mortar, five specimens were tested with a constantconfining pressure of 0.25, 0.50, 1.0, 1.25, and 1.0 MPa, respec-tively, resulting in a total of 20 specimens. At the beginning, therequired confining pressure is applied through the hydraulic jack.Then specimen was transferred under the actuator of a universaltesting machine to apply the shear load as compression. The maxi-mum loading capacity of the vertical actuator is 1,000 kN. Theshear load was applied at a rate of 0.05 mm, and the correspondingshear displacement was measured by means of four LVDTsattached to two opposite sides of the specimen [Fig. 3(b)] andrecorded through a data logger. The confining pressure was kept
almost constant throughout the entire loading process. Fig. 5 givesa confining pressure as a function of time for each specimen tested;the plots show that there is little fluctuation of the confining pres-sure. This is due to the fact that when two rough surfaces of brickand mortar slide over each other dilatancy takes place, whichcauses an increase in volume, and thus pressure on the steel plate.This excess pressure somehow contributes to the overestimation ofthe shear strength at the interface, but for the sake of simplicity ofthe analysis, the dilatancy effect is neglected in the numerical mod-eling. A more detailed explanation of this exclusion is given in thesubsequent paragraph.
Test Results and Discussion
It is quite obvious that the ultimate shear strength increases withincreasing confining pressure normal to the shearing surface.
Load
Reaction plate
(a) (b)
Steel bolt
Steel plate
Hydraulic jack
LVDT
Load cell
Fig. 3. (a) A typical triplet shear specimen; (b) schematic diagram of instrumentation
Fig. 4. Instrumentation and test setup of shear specimen
However, this is not the only governing factor that influences theshear strength of the brick-mortar interface. The other factors are(1) characteristics of bond between mortar and brick; (2) character-istics of brick and mortar; (3) coefficient of friction between the twosliding surfaces; and (4) the overall quality of the joint. Since dur-ing the fabrication of the specimens an overall uniformity was dif-ficult to be attained, some inconsistencies are inevitable. Figs. 6(aand b) show the nominal shear stress as a function of the sheardisplacement for some of the tested specimens. According to thefigures, it is easy to see that just before the peak shear stressthe stiffness is very high with very little shear deformation. Theinterlocking between the grains of the brick and the mortar underconfining pressure is the main reason for the high stiffness of theshear load-displacement relationship. There is a barely detectablehardening phase but just for a short range before the peak load. Asthe imposed shear displacement overcomes the interlock betweenbrick and mortar, a phenomenon of volume increase (dilatancy)takes place and gives rise to a much higher strength than expected.In the present study, the effect of dilatancy was not considered dueto the fact that at a confining pressure higher than 0.5 MPa thisdilatancy becomes marginally small, the effect of internal frictiondominates over dilatancy (Armaanidis 1998), and therefore makesit possible to evaluate the shear strength by means of the Mohr-Coulomb criterion. Moreover, the high confining pressure restrictsthe upward dilatant deformation of the specimen and turns it intodeformation of brick and mortar by squeezing them laterally atconstant volume.
As previously mentioned, the interface behaves like a quasi-brit-tle material and exhibits a very small hardening branch that appearsbetween the elastic limit and the peak stress. The post-peak damageand release of strain energy is quite evident as the stress dropsgradually. After the initial damage, the shearing surface readjustsand relocates its position for new sliding resistance after losing thecohesive bond at the brick-mortar interface. This stage is calledresidual stress and depends mainly on the interface static frictionand confining pressure; in the following, it will be indicated asresidual shear strength. After reaching the residual shear strength,the relative movement between the two sliding surfaces turns into arigid body movement with very little (or no) relative shear defor-mation. This stage can be considered as a complete failure stage,and the whole phenomenon can be indicated as dynamic friction,something that is beyond static equilibrium and static analysis.
In this study, two important parameters, confining pressure andmortar strength, were noticed as major factors contributing to theshear capacity at the brick-mortar interface. The increase in shearstrength with increasing confining pressure for different mortarstrengths (N, S, M, and E) can be seen in Fig. 7(a). The experimen-tal parameters found from the triplet shear test are given in Table 3.It is quite evident from Fig. 7(a) that the shear strength does in-crease with increasing confining pressure in a rather nonlinear fash-ion for different mortar strengths, which is incompatible withMohr-Coulomb criterion but consistent with the Mohr-Coulombfailure envelop or rupture line. For a specific material, the ruptureline may be a curve as shown in Fig. 8.
The Fig. 7(a) also indicate that the strength of mortar have asignificant role on the peak shear stress. The other two parameters,namely, cohesion (c) and internal friction angle (ϕ), which are theinherent properties of the interface between brick and mortar, alsovary with the mortar strength and confining pressure. The cohesionc is independent of normal stress and only increases a little withincreasing mortar strength whereas the friction angle ϕ is depen-dent on normal stress but cannot be verified independently at thispresent stage of knowledge.
The relationship between confining pressure and residual shearstrength is shown in Fig. 7(b). Once the interface cohesion is lost,the ratio of residual shear strength to confining pressure (or residualfriction coefficient) increases to an almost constant rate that is in-dependent of the mortar strength.
The failure modes of the shear test specimens were predomi-nantly interface failures. In all cases, the mortar separated from ei-ther the inner brick or the outer, or both. No substantial damageswere seen on the brick surfaces; rather, some small mortar piecesappeared to remain attached to the brick surfaces (Fig. 9),
Fig. 5. Time history of the vertical load for each specimen
(a) (b)
Fig. 6. Experimental results of shear stress versus shear deformation for (a) 0.5 MPa; (b) 1.0 MPa of confining pressure
something that indicates that if the brick strength is higher than themortar strength, damage takes place within the mortar. During theincrease of the shear load, a minor crack was observed, propagatingalmost halfway into the depth of the outer bricks at the peak shearload. This crack can be regarded as a flexural crack. Since the crackdoes not reach the interface, its effect on the average shear strengthat the interface is not very significant.
Shear Strength and Shear Stress-Slip Relationship
The shear capacity of masonry joints with moderate confining pres-sure can be predicted by the Mohr-Coulomb criterion (Lourençoet al. 2004), which establishes a linear relationship between the
(a) (b)
Fig. 7. Effect of normal stress: (a) variation of shear strength with normal stress; (b) variation of residual shear strength with normal stress
Table 3. Shear Strength Parameters from Triplet Shear Test
shear strength τu and the normal compressive stress σn (see Table 3)by τu ¼ cþ σn tanϕ. Here, c represents the cohesion between thebrick-mortar interface and tanϕ is the tangent of the friction angleof the interface. This relationship can be observed in other exper-imental findings as shown in Table 1 in which the Mohr-Coulombcriteria obtained from the test results of each mortar strength casesis shown. It can be observed in Fig. 7(a) that the shear strengthincreases with both confining pressure and mortar strength in anonlinear fashion. The value of interface cohesion c representsthe quality of bond between the brick-mortar interfaces. The in-crease in mortar strength will increase the interface cohesion butthis is not the only factor that controls the interface cohesion. Thereare other influential factors like the type of brick, the surface rough-ness of the brick, the absorption of the brick, and the brick strengththat also affects the quality of the interface cohesion. Since both thisstudy and the previous studies did not deal with those parametersexplicitly, a clear idea about their influence on the interface cohe-sion is still open to investigate.
The internal frictional angle ϕ at the interface is a function of thematerial properties and surface properties of the two sliding surfa-ces. In static conditions, by definition it should be equal to a con-stant that is called the coefficient of static friction. During theevolution of sliding under compression, however, both of the slid-ing surfaces undergo substantial deformation and after the loss ofinterface cohesion, the value of the frictional angle is different fromthe static one. Until (and unless) an accurate analysis of the bricksurface texture, abrasion characteristics of both brick and mortar,and pore structure of the brick are explicitly carried out, conclusiveremarks cannot be drawn on the relationship between the static anddynamic coefficient of internal friction as a function of brick andmortar characteristics. The residual shear strength is defined as theconstant value after the shear strength from which the residualfriction angle can be obtained. Both the initial and the residualfrictional angles are somehow independent of mortar strength.The latter lies around 45° as in the case of the experiments in thisstudy [Fig. 7(b) and Table 3] with some scattering though. Fig. 6(a)shows some of the shear stress-slip relationships. The pre-peakstiffness does not change noticeably with the increase of confiningpressure; rather, a change in stiffness is observed with the increaseof mortar strength. High strength mortar offers strong interface co-hesion, which in turn increases the interface stiffness. On the otherhand, higher confining pressure levels merely increase the shearcapacity of the interface with the stiffness almost constant. Never-theless, it cannot be concluded that the confining pressure has noeffect on interface stiffness unless more tests are carried out.
Numerical Modeling
The failure analysis of masonry structures has been based on mod-eling techniques developed in modern concrete mechanics. Forfully grouted reinforced masonry, where the influence of mortarjoints is marginal, the smeared crack approach can be applied tothe analysis of such masonry structures (Lotfi and Shing 1991).On the other hand, the behavior of unreinforced masonry maynot be modeled accurately by the smeared crack approach as un-reinforced concrete behavior cannot. Although intact brick unitsmay be assumed homogeneous and isotropic, the presence ofmortar joints makes unreinforced masonry composite both hetero-geneous and anisotropic and shows distinctive directional proper-ties at the time of load-reaction interaction. In the finite-elementanalysis of unreinforced masonry structures, the effect of mortarjoints as the major source of weakness and material nonlinearityhas been accounted for with different levels of refinement.
In general, the approach towards its numerical representation canfocus on the micromodeling of the individual components, viz. unit(e.g., brick, block) and mortar, or the macromodeling of masonry asa composite (Rots 1991). Depending on the level of accuracy andthe simplicity desired, the modeling technique can be categorizedthree possible ways (Fig. 10):1. Detailed micromodeling: Units and mortar in the joints are
represented by continuum elements whereas the unit-mortarinterface is represented by interface elements.
2. Simplified micromodeling: Expanded units are represented bycontinuum elements whereas the behavior of the mortar jointsand unit-mortar interface is lumped into interface elements.
3. Macromodeling: Units, mortar, and the unit-mortar interfaceare smeared out in the continuum elements. In the macro ana-lysis, masonry is considered as a single material (also knownas a homogenized material), which inherently includes theeffect of mortar joints.
In the simplified micromodeling approach, masonry units aremodeled with continuum elements, while mortar joints are modeledby means of interface elements. Each joint, consisting of mortarand the two unit-mortar interfaces, is lumped into an average inter-face while the units are expanded in order to keep the geometryunchanged. Masonry is thus considered as a set of elastic blocksbonded by potential fracture/slip lines at the joints. Accuracy is lostsince Poisson’s effect of the mortar is not included. Early attemptswith this approach were made by Arya and Hegemier (1978), Page(1978), and more recently, Rots (1991). Obviously, the approachwith this level of refinement is computationally intensive for theanalysis of large masonry structures, but it is certainly a valuableresearch tool and also a viable alternative to the costly and oftentime-consuming laboratory experiments. From a modeling pointof view, the aforementioned approach is similar to the discreteelement method, which was originally proposed by Cundall (1971)in the area of rock mechanics.
In this current study, the simplified micromodeling approach isadopted for simulating the behavior and failure mechanisms of ma-sonry assemblages based on the behavior of the basic constituents.A simple but general model for shear cracking in the masonry inter-face is proposed. It is defined in terms of shear stress on the averageplane of the crack and the corresponding relative shear displace-ment. In the following sections, the formulation of the interfacemodel is explained, and the applicability of the interface modelto mortar joints is validated by experimental results. The proposedmodel can be implemented directly as the constitutive law of theinterface element in the context of discrete crack analysis.
A simplified micromodeling approach is proposed here to modelthe masonry interface. The mortar thickness and the brick-mortarinterfaces are lumped into zero-thickness 8-node interface elementswhile the dimensions of the brick unit are expanded to keep thegeometry of a masonry structure unchanged and modeled as a20-node solid element. At the level of the interface Gauss integra-tion point, the inelastic failure surface is a function of the normaland interface shear stresses. Fig. 11 shows the adopted failuresurface at each of the interface Gauss integration points for brick-mortar joints. The failure surface consists of a Mohr-Coulomblinear inelastic surface and a tension cutoff. The limit of theMohr-Coulomb surface is assigned by adopting a compressioncap line when the normal compression at the respective Gauss pointgoes beyond the compressive strength of masonry and high enoughto crush the mortar or bricks, whichever is weaker in strength. Themaximum shear capacity at the interface at a particular level of nor-mal stress is given by the Mohr-Coulomb criterion in Eq. (2)
τu ¼ cþ σn tanϕ ð2Þ
The local bond stress-slip (τ -δ) relation up to the peak can begiven by Eq. (3) (Fig. 12) (Dai et al. 2005)
τ ¼ 2BGf½expð−BδÞ − expð−2BδÞ� ð3Þ
In this equation only two parameters, interface fracture energyGf and interface initial shear stiffness B, are needed to define thebond stress-slip relationship. The interface initial shear stiffness Bis a function of mortar thickness tm, Young’s modulus of brick andmortar Eb, Em, normal precompression σn and mortar compressivestrength fcm. In Eq. (4), B reads as
B ¼ 2.75logðEbEmtmÞ
exp½4.039 expð−0.155f 0cmÞσ0.525
n � ð4Þ
The fracture energy parameter Gf can be given by Eq. (5) oncethe peak shear stress τu and shear stiffness B are known, whichreads as
Gf ¼ τu=0.5B ð5Þ
δu ¼ ln
�2
B
�¼ 0.693=B ð6Þ
Here δu is the shear deformation corresponding to peak shearstress τu. The post-peak regime of the bond stress-slip (τ -δ) rela-tionship in Fig. 12 can be given by the following equation:
τ ¼ τu
�δuδ
�n
ð7Þ
where n = function of fracture energy Gf and can be given byEq. (8)
n ¼ 1.01 expð−2.288GfÞ ð8Þ
After the complete loss of cohesion and permanent deformationat the interface, only a fraction of shear stress can be transferredthrough the joint, which is residual shear stress τ res. Eq. (9) givesthe magnitude of this residual shear stress, which depends on thelevel of available compression pressure and the tangent of theresidual friction angle ϕres. A maximum slip δmax corresponds tothe point on the curve where shear stress τ is equal to the residualshear stress τ res in Eq. (7) (Fig. 12)
τ res ¼ σres tanϕres ð9Þ
The relaxation of shear stress can happen at any stage of loadingand at any location, which causes an unloading at a particular Gausspoint during the evolution of loading. For compatibility andcompleteness of the constitutive relationship, it is necessary to in-troduce an unloading and reloading path along which the unloadingand reloading can take place. In this model a suitable unloading andreloading path is introduced for pre-peak and post-peak regimes.Abdou et al. (2006) and Chaimoon and Attard (2009) observedquite similar loading and unloading path during their experimentalprocedure for the shear test (Fig. 13). The unloading stiffness isless than the initial tangent stiffness Geff because of the partial lossof interface cohesion especially at the pre-peak regime. In the
Mohr-circleτ
τ
Fig. 11. Failure surface of masonry joint with compression cap pro-posed by Chaimoon and Attard (2007)
Fig. 12. Shear stress-slip model for the interface
Fig. 13. Typical unloading-reloading behavior of brick-mortarinterfaces
post-peak regime, the cohesion is completely lost and the over-whelming damage occurs on aggregate interlocking between brickand mortar grains. Once, the shear force due to aggregate interlock-ing is lost, the remaining shear resistance comes only from the in-terface friction in the order of residual stress where a state ofcomplete damage prevails. Subsequent equations for unloadingand reloading are postulated considering this phenomenologicalobservation from the shear test on the brick-mortar coupletassemblage.
Loading: δ > δpmaxFor pre-peak regime:δ ≤ δu
τ ¼ 2BGf½expð−BδÞ − expð−2BδÞ� ð10Þ
For post-peak regime 1: δu < δ ≤ δmax
τ ¼ τu
�δuδ
�n
ð11Þ
(a) (b)
(c) (d)
(e) (f)
Fig. 14. Comparison of simulation with experimental data from (a) and (b) Hansen (1999); (c) Van der Pluijm (1993); (d) Chaimoon and Attard(2007); (e) and (f) from this study
Here K0 = fracture parameter; α = stress reduction factor; δp =plastic deformation; δe = elastic deformation; Gi = initial shearmodulus of the interface at zero pre-compression, and Geff = effec-tive shear modulus at any level of precompression.
For post-peak regime 1 and 2: δ > δuSame as Eq. (25)Here Gunld = unloading shear modulus; τpmax = stress corre-
sponds to δpmax; and D0 = damage parameter.
Model Implementation and Validation
The proposed analytical model needs only the Young’s modulus ofboth brick and mortar Eb and Em, the thickness of the mortar tm andthe overburden pressure σn. In most cases, experimental resultsgiven in Table 1 show good agreement (Fig. 14) with that ofthe analytical curves. In some cases, the pre-peak stiffness andthe peak shear stress were simulated quite closely whereas infew cases the post-peak regimes show a little difference with theexperimental one. This can partly be explained because of the ex-perimental shear stress gives an average stress over the entire inter-face under investigation where a single softening constitutive law isnot valid because of the variation of interface properties and cohe-sion over the sliding surface due to the lack of uniformity at thetime of specimen fabrication involving human error, whereas innumerical models all of this variability is ignored and a singlebranch softening constitutive law is provided; hence, the softeningbehavior appears to be of the same nature for all cases. Moreover, inan experimental procedure the damage is gradual and the locationwhere a complete state of interface damage prevails is rather diffi-cult to locate; hence the exact point from where the residual stress isgoing to initiate is also difficult to determine. For numerical mod-els, it is done when the softening shear stress meets the residualshear stress criterion, so the junction of these two is not smoothbut an abrupt change in direction and slope that is evident in someexperimental results also. In addition, the post-peak behavior dur-ing the testing procedure is so delicate and abrupt that it is quiteimpossible to obtain solicited data from the test unless one has verysophisticated and well controlled experimental facilities. In this ex-perimental procedure the authors did not have that level of controlover the post-peak regime and that is the reason for the straight linesoftening and inconsistent variation in the post-peak regime formost of the cases. This may also be true for other experimentalresults such as Lourenço et al. (2004) in Fig. 15. Nevertheless, themodel curve can predict the ultimate shear strength and residualshear stress quite correctly, and this is the merit of this model overfew others limitations. If an accurate estimation of the empiricalparameter B is ensured, the initial stiffness and shear displacementat peak shear stress will be very close to that of the experimentalresults.
If the modulus of mortar is not readily available, Eq. (26) is rec-ommended by European Committee for Standardization (CEN)(2001) for normal weight concrete and can be used for mortaras well
Em ¼ 22,000
�fmc10
�1=3
ð26Þ
The interface cohesion c and the initial friction angle ϕ are thetwo inherent properties of mortar and brick, the materials used inthis experiment. One is independent of stress and the other is astress-dependent parameter. For numerical analysis of masonrywall for shear, any reasonable value of these two will produce con-sistent result. However, for normal strength mortar and brick arange from 0.15 to 0.25 for c and 50 to 65° for ϕ will yield a goodapproximation of experimental results. For the residual friction an-gle τ res, a reasonable approximate value from 45 to 55° can be usedsafely for model implementation.
Conclusions
In this study, some results obtained on mortar joints in brick ma-sonry under shear and compression, are compared with other sim-ilar results from the literature. First, various experimental results are
compared on the basis of two major affecting parameters, namely,confining pressure and mortar strength. From these comparisonsthe following conclusions can be drawn:1. The shear capacity of the joints for both solid and hollow
bricks will definitely rise with an increase in confining pres-sure acting normal to the joint. The resulting relationshipshows a nonlinear tendency, something that is consistent withthe Mohr-Coulomb failure envelope. Once cohesion is lost (orthe shear stress reduces to the residual shear strength), the con-fining pressure plays a significant role also for the residualshear capacity.
2. The shear strength does increase with the increase of mortarstrength, as the interface cohesion increases with the increaseof mortar strength, but a definitive relationship cannot beestablished at this current stage of knowledge. Strong mortarwith weak brick and weak mortar with strong brick will be-have differently, and as a result, the shear strength will be veryscattered in nature.
3. The shear stress-slip relationship can best be described as pre-peak and post-peak regime. In the pre-peak stage, the stiffnessis somehow constant throughout the loading process, andshows little hardening phase near the peak shear stress. So,the pre-peak behavior can be said to be elastoplastic. In thepost-peak regime, the damage is rather gradual and shear stressreaches to a constant value after the loss of interface cohesion;this stage is called residual shear strength.
4. Brick type, its surface roughness, and mortar texture definitelyhave an effect on interface friction. Until and unless some vig-orous investigation is carried out to know what these effectsare, an explicit correlation between friction coefficient andmortar strength cannot be established in this study. Merelya variation of friction coefficient can be shown with respectto mortar strengths and the strength of bricks.
5. The so-called dilatancy, which causes an upward displacementof the brick units upon sliding, has some marginal effect on theoverall deformation of brick-mortar assemblages but this de-formation can be neglected on the grounds that the deforma-tion of the brick and mortar themselves are large enough incomparison with the dilatant deformation.
6. The proposed model equations have uniqueness in their simpli-city and can predict the peak shear stress as well as the initialstiffness and the softening behavior of the interface quite wellfor the experimental results examined in this paper. The neces-sary information for the proposed model is only the Young’smodulus of brick and mortar, the mortar thickness, and the nor-mal compressive stress acting on the interface. Any reasonablevalue for c and ϕ will produce good numerical approximation.
Acknowledgments
The authors gratefully acknowledge the financial and technicalsupport for this experimental work from Hokkaido University.Mr. Kimura Tsutomu of Hokkaido University and Mr. Dillon Lunnof North Carolina State University are appreciated for their assis-tance during the experimental work. The MEXT is also greatly ac-knowledged for providing a scholarship to the first author.
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