Chaimoon 2009 Engineering-Structures

Engineering Structures 31 (2009) 103–112

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Engineering Structures

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Experimental and numerical investigation of masonry under three-pointbending (in-plane)Krit Chaimoon, Mario M. Attard ∗School of Civil and Environmental Engineering, University of New South Wales, Sydney, NSW, 2052, Australia

a r t i c l e i n f o

Article history:Received 27 June 2007Received in revised form30 July 2008Accepted 30 July 2008Available online 3 September 2008

Keywords:Unreinforced masonryThree-point bendingFractureSoftening

a b s t r a c t

The aim of this study is to investigate the failure and post-peak behaviour of masonry panels with lowbond strength mortar under three-point bending (TPB) both experimentally and numerically. Full-scalemasonry panels with two different mortar strengths were tested under TPB. The material parameterswere obtained from compression, TPB and shear tests on bricks and brick–mortar interfaces. Theexperimental results are detailed. Adopted for the numerical study was a micro-model finite elementformulation in which masonry is modelled using expanded brick units with zero thickness brick–mortarinterfaces. The simulation of the masonry TPB tests compared well with the experimental results.Investigations were the crack propagation through the masonry and the effects of dilatancy on the postpeak response.

Crown Copyright© 2008 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Unreinforced masonry is a composite of bricks, usually madefrom clay, and mortar joints. The fracture behaviour of masonrystructures is influenced by several factors, e.g. material propertiesof brick and mortar, geometry of bricks, joint thickness, propertiesof brick–mortar bond, etc. In plain concrete, the notched three-point bending test is used to determine the mode I fractureproperties such as the flexural tensile strength and the modeI fracture energy. Masonry panels were tested in three-pointbending by Guinea et al. [15]. The mortar and brick units used byGuinea et al. [15] had similar strengths with the mortar having atensile strength of approximately 73% of the brick units. The failuremode therefore consisted of a single vertical crack penetrating thepanel through the mid-span region above the notch. The fracturewas therefore primarily mode I. Masonry used in practice can havea variety of mortar to brick strengths and hence the failure inmasonry panels will often consist of both mode I and mode IIfracture. This is especially true for masonry panels with low bondstrength as compared to the strength of the brick units. The aimof this study is to investigate the behaviour both experimentallyand numerically of three-point bending masonry panels withrelatively low strength mortar. The basic elastic and inelasticfracture material properties for the masonry components: bricks,mortar and brick–mortar interfaces, were obtained from several

∗ Corresponding author. Tel.: +61 2 9385 5075; fax: +61 2 9385 6139.E-mail addresses: [email protected] (K. Chaimoon),

[email protected] (M.M. Attard).

0141-0296/$ – see front matter Crown Copyright© 2008 Published by Elsevier Ltd. Aldoi:10.1016/j.engstruct.2008.07.018

standard tests including compression, shear and TPB. Full-scalemasonry panels with two different mortar strengths were alsotested under TPB. The failuremode of themasonry panels, involveda zig-zagging crack pattern through the bed andheadmortar joints,and incorporated both tensile and shear type fracture. Thematerialparameters and the full load versus deflection and CMOD testresults are presented.For numerical purpose, in general, there are two main ap-

proaches adopted for masonry modelling: macro-modelling andmicro-modelling. In the macro-modelling approach, homogenisa-tion techniques and continuum-based theories are usually adoptedfor large and practice-oriented analysis. The interaction betweenunits and mortar is generally ignored for the global structuralbehaviour and a relation is established between average strainsand average stresses. The material parameters must be obtainedfrom masonry tests of sufficiently large size under homogeneousstates of stress. Macro-modelling approaches have been adoptedby many researchers such as Grande et al. [14], Wu and Hao [30],Betti and Vignoli [4], Cavicchi and Gambarotta [7], ElGawady et al.[12], Massart et al. [20], Milani et al. [22], Carpinteri et al. [6] andLourenco et al. [18]. In the micro-modelling approach, a repre-sentation of the unit, mortar and the unit/mortar interface mustbe included. Cohesive crack models are usually adopted in themicro-modelling formulation through the interface elements. Thisapproach is suitable for small structural elements with particularinterest in strongly heterogeneous states of stress and strain. Theprimary aim of micro-modelling is to closely represent masonryfrom the knowledge of the properties of each constituent and theinterface. The experimental data must be obtained from labora-tory tests on masonry constituents and small masonry samples.

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104 K. Chaimoon, M.M. Attard / Engineering Structures 31 (2009) 103–112

Fig. 1. (a) Original brick. (b) Typical brick specimen for TPB tests. (c) Typical brick specimen for shear testing. Dimensions are in mm.

Two levels of refinement arewidely used in the literature includingdetailed micro-modelling and simplified micro-modelling. In thedetailed micro-modelling, the units, mortar and the unit/mortarinterfaces are all modelled separately. In the simplified micro-modelling, the properties of the mortar and the unit/mortar inter-face are lumped into a common element,while expanded elementsare used to present the brick units. Some accuracy is obviously lost,however the reduction in computational effort results in a modelwhich would be applicable to a wider range of structures. Severalresearchers have adopted micro-models to study the complex be-haviour of masonry structures, see [26,21,11,5,13,25,17,16].Since the best insight into the behaviour of masonry structures

can be obtained from the use of a micro-model, the micro-modelling approach is used in this study. The micro-modelpurposed by Chaimoon and Attard [10] for the simulation offracture in masonry was utilised for comparison with the testresults. The finite element formulation is based on a triangularunit, constructed from constant strain triangles, with nodes alongits sides but not at the vertex or the center of the unit. Fractureis modelled through a constitutive softening-fracture law at theboundary nodes. The constitutive law is a single branch softeninglaw. The material within the triangular unit remains linear elastic.Triangular units are grouped into rectangular zones which mimicbrick units and mortar joints. The path-dependent softeningbehaviour is solved using a linear complementarity problem(LCP) formulation, in non-holonomic rate form within a quasi-prescribed displacement approach. The inelastic failure surface ismodelled using a Mohr–Coulomb failure surface with a tensioncut-off. The similarity between the experimental and numericalpredictions was good indicating that the numerical formulation of[10] provided a robust tool for the simulation of masonry fractureunder shear and tension.Section 2 describes the materials used in this work and the

tests performed. Section 3 presents the micro-model and thecomparison between the experimental and numerical results.Finally, conclusions are made in Section 4.

2. Materials and tests

2.1. Materials and specimens

The brick units usedwere solid clay bricks of 230×110×76mm(Austral Bowral Brown dry pressed brick). There is a frog at thetop bed as shown in Fig. 1(a). For compression testing, six of thebrick units (three for testing in the direction perpendicular to thebed face and three for testing in the direction parallel to the bedface) were cut into dimensions of 230×110×50mm to eliminatethe effects of the frog. For the TPB brick tests, the frog was filledwith plaster to reduce the effects of roughness and lack of a planesurface on the loaded face and a central notch 10mm in depth wascut. Fig. 1(b) shows a typical brick TPB specimen. In addition, brickspecimens of 230 × 110 × 50 mm were cut for shear testing as

Table 1Compressive strength and mix proportions of the mortar

Mortar type Compressive strength(MPa)

Cement:lime:sand:water(weight ratio)

W 7.26± 0.07 1:0.00:6.05:1.58S 16.79± 0.20 1:0.12:3.50:1.10

Fig. 2. (a) Typical masonry specimen for compression tests. (b) Typical masonryspecimen for shear testing. Dimensions are in mm.

shown in Fig. 1(c). Notches were cut around each of the brick shearspecimen to produce a uniform plane of weakness.Two proportions of mortar, see Table 1, were mixed to provide

two different strength mortars. Both types of mortar (types W andS) were separately cast in moulds at the same time of constructionas all the masonry specimens. The mortar type notations ‘‘W’’ and‘‘S’’ indicate a weak and strong mortar, respectively. Three mortarcylinders, 100 mm in diameter and 200 mm in height, were usedfor compression testing for each mortar type. The compressivestrengths at 28 days were determined according to the Australianstandard AS1012.9:1999 and are listed in Table 1. Shear and TPBtests on mortar specimens were also undertaken and the resultswere detailed in [8]. The shear and TPB mortar results are notlisted here as they were found not to provide representativematerial properties of the mortar in the masonry panels becauseof the significant influence of the brick’s water absorption on theproperties of the masonry mortar joints.Stacked bond specimens consisting of five bricks with four

joints were constructed for masonry compression tests, seeFig. 2(a). To determine the shear properties of the brick–mortarinterface, specimens made of two bricks and a mortar joint wereprepared as shown in Fig. 2(b). One side of the mortar jointcontained the brick frog. Prior to testing, both sides of each of thetwo-brick specimens were cut symmetrically in order to eliminatethe outer brick frogs on the exposed faces.The masonry panels with the two different types of mortar

were laid by a masonry mason. The panels were four coursesin height. The length of the panels was equivalent to six brickswith five header joints. The mortar thickness excluding frog wasapproximately 10 mm.

K. Chaimoon, M.M. Attard / Engineering Structures 31 (2009) 103–112 105

Table 2Compression and shear test results of brick and masonry

Material Compressive strength (MPa) Cohesion (MPa) Initial friction coefficienttanφ

Residual frictioncoefficient

Mode II fracture energy (N/mm)

Brick (31.31± 6.56)a(21.68± 0.54)b

0.62 4.20 1.74 0.564–2.436

Masonry (Mortar type W) 18.97± 0.86 – – – –Masonry (Mortar type S) 18.71± 1.04 – – – –Masonry joint(Mortar type W)

– 0.43 0.70 0.74 0.03–0.12

Masonry joint(Mortar type S)

– 0.18 0.89 0.74 0.03–0.12

a Compressive strength in the direction perpendicular to bed face.b Compressive strength in the direction parallel to bed face.

Fig. 3. Geometry and dimensions of the masonry panels.

Fig. 3 gives the dimensions of the TPB masonry panels. Prior totesting, the frog in the brick at the position of themid-span loadingpoint was filled with plaster to allow loading and a central notch3 mm wide was sawn up to a specific depth shown as dimension‘‘a’’ in Fig. 3.

2.2. Tests

Compression tests for the brick and masonry specimens wereperformed according to the Australian standards AS4456:2003 andAS3700, respectively. For all the shear tests, a modified version ofthe shear test set-up introduced by Van Der Pluijm [28] was used.Van Der Pluijm’s shear test rig [28] was designed to provide anapproximately uniform state of stress along the tested shear plane.Details of themodified shear test rig can be found in [8,9]. The sheartest rig is shown schematically in Fig. 4. Four displacements, twovertical and two horizontal, were measured with linear variabledisplacement transducers (LVDTs) which were installed after theapplication of the initial confining pressure and before applyingthe shear loading. The positions of the LVDTs are shown inFig. 4(a). Load was applied continuously under a controlled rateuntil the peak load was reached and then for some specimensunloading–reloading was applied at various stages along the post-peak path by manual control of strain rate. All data was recordeduntil well into the residual load stage. Table 2 summarises thecompression and shear test results for the brick and masonryspecimens. The brick friction coefficients aremuch higher than thejoint friction coefficients. The high value of 4.2 for the brick frictioncoefficient is not yet clear. Report on the shear properties of brickis lacking in the literature. Further research should clarify this.The shear test results were also used to examine the un-

loading–reloading characteristics and the dilatancy behaviour.Fig. 5(a), (b) illustrate the typical unloading–reloading behaviouralong the residual stage of the load path, of the bricks and thebrick–mortar interfaces. The bricks exhibited unloading–reloading

Fig. 4. (a) Modified shear test rig. (b) Loading on the specimen.

Fig. 5. Typical unloading–reloading behaviour: (a) Bricks and (b) Brick–mortarinterfaces.

behaviour similar to the behaviour of plain concrete as the unload-ing stiffness was less than the initial tangent stiffness because of


Fig. 6. Dilatancy behaviour: (a) Bricks (b) Brick–mortar interfaces.

Fig. 7. Evolution of tanψ with shear displacement for brick–mortar interfaces.

damage (Gb2,Gb3,Gb4,Gb5 < Gb1). The unloading stiffness of thebrick–mortar interfaces at the residual stage was very similar tothe initial tangent stiffness (Gj1 ≈ Gj2 ≈ Gj3 ≈ Gj4 ≈ Gj5).Fig. 6(a), (b) show the typical dilatancy behaviour found in the

brick units and the brick–mortar interfaces, respectively. The di-latancy coefficient, defined as the tangent to the inelastic openingversus inelastic shear displacement curve, of the brick–mortar in-terfaces is plotted against the shear/slip displacement in Fig. 7. Thedilatancy coefficient decreases with increasing slip displacement.At large slip displacement, the dilatancy coefficient tends to zeroas the opening displacement is arrested with almost pure frictionslip. Fig. 8 presents the evolution of total confining pressure withshear displacement for brick–mortar interfaces.All TPB tests were carried out under crack mouth opening

displacement (CMOD) control. During the test, the load-pointdisplacement and CMOD were continuously recorded. For thebrick TPB tests, a CMOD rate of 0.032 mm/min (3200 µε/min),was used so that the maximum load was reached within about30–60 s after the start to comply with the RILEM TC-50 FMC Draft

Fig. 8. Evolution of total confining pressure with shear displacement forbrick–mortar interfaces.

Table 3TPB test results for the brick units

Flexural strength (MPa) Mode I fracture energy (N/mm)

10.09± 0.42 0.312± 0.01

Fig. 9. Load–CMOD curves obtained from the brick TPB tests.

Fig. 10. TPB test set-up scheme for the masonry panel.

Recommendation [24]. For the masonry panels it was necessary touse a CMOD rate of 0.09 mm/min (9000 µε/min) for panels withmortar typeWand a CMOD rate of 0.12mm/min (12 000 µε/min)for panels with mortar type S. Table 3 summarises the flexuralstrength and mode I fracture energy obtained from the brick TPBtests. Fig. 9 presents the brick TPB load–CMOD curves. Fig. 10shows the test set-up for the TPB tests on the masonry panels. Thetest results for the masonry panels will be discussed in Section 3.3.


(a) Basic triangular unit. (b) Single masonry unit.

Fig. 11. Modelling for masonry units.

3. Micro-model and numerical simulation

3.1. Masonry modelling

In order to simulate the fracture of the masonry beams,the micro-model proposed by Chaimoon and Attard [10] wasused. The mortar thickness and the brick–mortar interfaces arelumped into a zero-thickness interface while the dimensions ofthe brick units are expanded to keep the geometry of a masonrystructure unchanged. Each masonry unit is further subdividedinto interior brick elements which have borders representing themortar interfaces or internal brick interfaces. The masonry unitis subdivided using triangular units formed by assembling nineconstant strain finite element triangles and condensing out thefreedoms at the vertices and the centre (see Fig. 11(a)). There aretwo nodes on each of the three sides of the triangular unit. Thetriangular unitwas first developed byAttard and TinLoi [2] andwasused for simulating fracture in quasi-brittle materials, see [3,1].Fracture is captured through a constitutive softening-fracture lawat the boundary nodes along the sides of the triangular unit. Thematerial within the triangular unit remains linear elastic. Differentinelastic constitutive properties are assigned to the brick–mortarinterface around the perimeter of the masonry unit and to theinterior brick interfaces. Fig. 11(b) shows the simplest model fora single masonry unit using four triangular units. In this study,fracture is restricted to the horizontal and vertical brick–mortarinterfaces as there was no brick failure in the TPB masonry panelstested.The basic triangular finite element unit has generalised

interface displacements which correspond to the outward normaland tangential (anti-clockwise defined as positive) displacementsat the interface nodes. The conjugate generalized forces are theoutward normal Qn and shear force Qs at the interface nodes (seeFig. 11(a)).

3.2. Constitutive law and solution algorithm

At the level of the interface nodes, the inelastic failure surface isa function of the normal and shear interface forces. Fig. 12 showsthe adopted failure surface at each of the interface nodes for thebrick and brick–mortar joints. The failure surface consists of aMohr–Coulomb linear inelastic surface and a tension cut-off. Theangle φ is the friction angle and ψ is the dilatancy angle. Thefailure surface involves interface forces as opposed to conventionalcontinuum models, which are based on stresses. No compressioncap is introduced and a constant dilatancy angle is assumed here.The flow rule is non-associated when the friction and dilatancyangles are different. A piece-wise linear assemblage of all theinelastic failure surfaces within the complete structural model isadopted following the procedures in [19].Each of the failure planes is associated with an irreversible de-

formation vector called the interface multiplier vector (analogousto plastic multipliers used in classical plasticity). For the nodal in-terface inelastic failure surface shown in Fig. 12, λi collects the

Fig. 12. Mohr–Coulomb with tension cut-off failure surface.

interface multipliers into a vector given by

λi ={λt λs1 λs2

}T (1)

where λt is the multiplier associated with the tension cut-off; λs1and λs2 are the multipliers associated with the Mohr–Coulombportions of interface inelastic failure surface. In Fig. 12, Qirepresents the resultant force vector at an interface node at anystage of the analysis, Qty is the tensile inelastic failure force, andQsy is the shear inelastic failure force. The inelastic failure forcesare grouped into the initial inelastic failure vector ri, defined by

ri ={Qty Qsy Qsy

}T. (2)

The interface failure surface is defined by the inelastic failurevector ri and the orientations of the normals to each of the failureplanes. The interface normalitymatrixNi contains the orientationsof the normal to each failure plane. The dilatancy matrix Vi definesthe flow rule for the interface irreversible deformationmultipliers.The dilatancy angle is taken as independent of the interfaceinelastic multipliers. The normality and dilatancy matrices aredefined by

Ni =

[0 cosφ − cosφ1 sinφ sinφ

](3)

Vi =

[0 cosψ − cosψ1 sinψ sinψ

]. (4)

The tensile inelastic failure force Qty is estimated from the productof the material tensile strength stress ft obtained from a puretension test, and half the interface length Li (see Fig. 11(a)) and thespecimen thickness ti. That is,

Qty =ftLiti2. (5)

Similarly, Qsy is defined by

Qsy =c cosφLiti2

(6)


Fig. 13. Single branch softening law.

where c is the cohesion. The softening constitutive law for theinterface forces is a single-branch softening curve (see Fig. 13)with λnc denoting the critical opening or sliding displacement. Themultiplier λi can only have positive values along the descendingbranch. Elastic unloading is allowed from the descending softeningbranch. Once the critical displacement is reached, the multiplierλi is free to go either forwards or backwards. The criticaldisplacements are approximated here by

λtc =2GIfft, λsc =

2GIIfc cosψ

(7)

where GIf ,GIIf are the mode I and mode II fracture energies,

respectively.The evolution of the inelastic failure surface is represented by

an interface hardening (softening)matrixHi. If themultipliers havenot exceeded their critical values (given in Eq. (7)) then the generalform of the matrix Hi is defined by

Hi =

−Qtyλtc

−Qtyλscβ −

Qtyλscβ

−Qsyλtcβ −

Qsyλsc

0

−Qsyλtcβ 0 −

Qsyλsc

(8)

where β is an interaction parameter. The off-diagonal terms inthe matrix above represent interaction between the tension andshear failure planes. The evolution of the interface inelastic failuresurface can be represented by the vector ξi given by

ξi = ri + Hiλi. (9)The path-dependent softening behaviour in finite incrementalform will need to be solved [2,3]. The non-holonomic relationshipbetween the structure generalised forces Q and the structuralvector of plastic multipliers λ takes the linear complementarityproblem (LCP) form:

ϕ = NTQ− Hλ− R

ϕ ≤ 0 λ ≥ 0 ϕTλ = 0(10)

with N being the structure normality matrix, R the vector of ini-tial failure values and H the structure hardening matrix. The for-mulation is then cast in terms of a quasi-prescribed displacementformat which maintains the load vector. Complementarity is en-forced between the potential function vector ϕ and the incrementof the inelastic multiplier vector. The problem is solved in incre-mental steps as a series of linear complementarity problems. Ateach event, a set of active inelastic multipliers is maintained andupdated, with unloading inelastic points removed from the activeset. Once the critical either crack opening or sliding displacement isreached, the rows and columns of the hardening matrix associatedwith the corresponding inelastic multiplier is made zero. Solutionsto the LCP are obtained using the algorithm in [27]. When mul-tiple solutions are detected, the equilibrium solution which pro-vides the minimum increment in external work is taken as thecritical solution.

Fig. 14. Finite element modelling for the masonry panels under TPB.

Table 4Parameters used for the mortar interfaces

Mortar type Tension Shearft (MPa) GIf (N/mm) c (MPa) tanφ tanψ GIIf (N/mm)

W 0.086 0.002 0.344 0.74 0.0 0.0370S 0.128 0.003 0.216 0.89 0.0 0.0125

3.3. Simulation of masonry panels under TPB

Fig. 14 presents the finite element model employed for themasonry beams with mortar types W and S. In order to reflectthe fact that the area where the joint and unit were bondedtogether was smaller than the area of the bed face of brick, thereare only two elements along brick length. Many researchers alsoused a mesh with two elements along brick length and obtainedpromising results, see [23,25,17]. Although there was a centralnotch in each panel with mortar type S, the notch was neglectedin the modelling because the notch depth was very small incomparison with the panel depth. The elastic modulus of 3360MPa obtained from a fit of the initial linear part of the test resultswas employed and a Poisson’s ratio of 0.2 was assumed in thesimulations. The parameters used for the mortar interfaces arelisted in Table 4. The shear parameters were within 20% of theaverage experimental values and within the scatter of the testresults. The model employed a constant dilatancy angle of zero asdilatancy was thought not to be important for the TPB test. Furtherdiscussion about this assumption is made later in this paper. Thetensile bond strengths were estimated based on the values of thecohesion. Van Der Pluijm [28] carried out an extensive testingprogram and obtained a ratio of the cohesion to the tensile bondstrength that ranged between 1.3 and 6.5. In this study the ratiosof 4 and 1.7 were employed for the interfaces withmortar typesWand S, respectively. Using the estimated tensile bond strengths, thevalues of the mode I fracture energy were then estimated.The experimental and numerical results are compared in

Figs. 15 and 16 for the masonry beams with mortar types W andS, respectively. A reasonable comparison was achieved for eachmortar type in terms of the load–displacement and load–CMODcurves. The finite elementmodelwas able to capture the post-peakbehaviour reasonably well.The crack patterns at failure for both mortar types are

depicted and compared with the deformations obtained from thesimulations (note the deformations were scaled to enhance thevisualization). Fig. 17 illustrates the comparison for the masonrypanel with mortar type W, while Fig. 18 shows the comparisonfor mortar type S. The cracks zig-zagged through the head andbed joints without any crack penetrating the bricks. The crackpatterns clearly show that the failure of the masonry beams withthe relatively low bond strength mortar was governed by bothtensile and shear fracture (mode I and mode II fracture) of themortar joints. For the panel with mortar type W, the predictionwas slightly different from the experimental crack patternwhereasthe crack pattern of the panel with mortar type S provided a much


Fig. 15. Comparisons of the experimental and numerical results of the masonrybeamwith mortar type W: (a) load–displacement curve and (b) load–CMOD curve.

better comparison. The simulated crack patterns for both mortartypes was reasonable.The simulation of the masonry beam with mortar type S was

used for further parameter investigation to help provide a betterunderstanding of how the cracks propagate through the bed andhead joints. The sequence and pattern of regions at the interfacenodes of the finite element model where inelastic failure orunloading was active are detailed for several points along theload–displacement curve in Fig. 19. The considered points (A, B,and C) on the load–displacement curve are as shown in Fig. 16(a).The symbols used in Fig. 19 denote either an activated inelasticfailure surface, unloading from a failure surface or a criticaldisplacement either opening or sliding being activated. Initiallytensile failure took place in the bottomhead joints in a symmetricalpattern around the midspan region. When the critical openingdisplacement was reached at the central bottom head joint, theother two bottom head joints where tensile inelastic failure wasactive then elastically unloaded. The crack propagated up thecentral head joint. The pattern of inelastic failure points at the peakload, point A, are depicted in Fig. 19(a). At this stage the criticalcrack opening displacement had only been reached in the verticalbottom central brick–mortar joint. Shortly after the peak load, thecracks propagated further through the central head joint undertension. Inelastic failure associated with shear was also activatedalong the central bed joint. Cracking in the second layer head jointsfollowed and led to a snap back in the load displacement responsecorresponding to point B and with the failure points illustrated inFig. 19(b). The failure pointswhen the crack propagated in the thirdlayer head joint are presented in Fig. 19(c). Finally, the collapse

Fig. 16. Comparisons of the experimental and numerical results of the masonrybeam with mortar type S: (a) load–displacement curve and (b) load–CMOD curve.

(a) Experimental.

(b) Numerical.

Fig. 17. Comparison of the experimental and numerical crack patterns at failurefor the masonry panel with mortar type W under TPB.

mechanism formedwith an asymmetric stepped crack through thehead and bed joints as shown in Fig. 19(d).The assumption of zero dilatancy in the TPB test will

be discussed here. Note the present formulation is limitedto a constant dilatancy angle which does not degrade withincreasing inelastic shear displacement. Van Zijl [29] detailedthe role of dilatancy in shear-compression. Any restraint on thebrick–mortar interface opening caused by dilatancy was shownto result in large compressive stresses on the interfaces. VanZijl incorporated a variable dilatancy coefficient to reproduce


(a) Experimental.

(b) Numerical.

Fig. 18. Comparison of the experimental and numerical crack patterns at failurefor the masonry panel with mortar type S under TPB.

(a) The failure points at the peak load, point A.

(b) The failure points when the snap back occurs, point B.

(c) The failure points when the cracks propagate through three headjoints, point C.

(d) The failure points at failure.

Fig. 19. The simulation results — Progression of the failure points at severallocations along the load–displacement curve of the masonry beam with mortartype S. © Critical displacement N Inelastic tensile failure Inelastic shear failure• Unloading.

Fig. 20. Effects of dilatancy angle on the simulation of masonry beam with mortartype W: (a) on load–displacement curve and (b) on load–CMOD curve.

experimentalmeasurements of brick normal uplift during shearingalong a brick–mortar interface. Small masonry shear experimentsin which the brick mortar interface opening was prevented,were used to verify the formulation. In the TPB experiment,there is minimal restraint against uplift during shearing of thebrick–mortar interfaces associated with dilatancy. Three differentdilatancy coefficients (tanψ) of 0, 0.1 and 0.2, were used andthe numerical responses are compared in Figs. 20 and 21. Thesevalues were chosen from an inspection of the initial experimentaldilatancy coefficients which varied from 0.034 to 0.438 formasonry joints with mortar type W and from 0.086 to 0.266 formortar type S.Figs. 20 and 21 show the load–displacement and load–CMOD

curves with different values of dilatancy angle of the masonryTPB beams with mortar types W and S, respectively. Dilatancywould only be a factorwhen the crack path propagates through thehorizontal bed joint under predominately-inelastic shear failureactivated by the Mohr–Coulomb failure surface. As can be seen,dilatancy has an effect after the first peak load is reached byincreasing the peak load slightly and increasing the second peakload particularly in the case of panel with mortar type W. Thisimplies that a constant zero dilatancy provides a conservativeestimate of the load deformation response for the TPB specimenstested as compared to results with other values of constantdilatancy. Note that the predicted response is in some cases higherthan the observed experimental resistance because average valuesare used for all parameters.The progression of inelastic failure/cracking at three points

along the load path identified as X, Y and Z in Fig. 21(a), arepresented in Fig. 22(a)–(c). The simulation of masonry beam with


Fig. 21. Effects of dilatancy angle on the simulation of masonry beam with mortartype S: (a) on load–displacement curve and (b) on load–CMOD curve.

mortar type S was still utilised for this purpose. Point X representsthe stage at which shear inelastic failure is first initiated in themortar bed joint, as seen in Fig. 22(a). Dilatancy would thereforehave an effect only after this stage of loading. As stated earlier,the load–displacement curve was unaffected by dilatancy untilafter the peak load (see Fig. 19(a)) and only after the tensile crackbegun to propagate the second layer of head joint (point Y, seeFig. 21(a)). The dilatancy effect then became more dominant withsome interface nodes attaining their critical shear displacements,see Fig. 22(b). The dilatancy angle should have a value of zerowhenthe critical shear displacement is reached as can be observed fromthe experimental results in Fig. 6(b). The present formulation doesnot allow a degradation of the dilatancy angle and hence this leadsto an overestimation of the load carrying capacity. Presented inFig. 22(c) is the inelastic failure/cracking distribution at load pointZ well down the post-peak load path. Beyond load point Z, thethree values of the dilatancy angle have very little effect on theload deformation path as there is no further restraint to openingof the horizontal bed joints.

4. Conclusions

In this study, the experimental failure behaviour of masonrypanels with low bond strength under three-point bending (TPB)has been investigated. Full-scale masonry panels with twodifferent strengths of mortar were tested under TPB. Fracture testson bricks, mortar, and brick–mortar interfaces were performed.The basic material parameters were obtained from compression,TPB and shear tests on bricks and brick–mortar interfaces. Theexperimental crack propagation of the masonry panels involved

(a) The failure points when the dilatancy first occurs, at point X.

(b) The failure points at point Y.

(c) The failure points at point Z.

Fig. 22. The simulation results of the masonry beam with mortar type S — thefailure points at relevant locations on the load–displacement curve to reflect thedilatancy effects.© Critical displacementN Inelastic tensile failure Inelastic shearfailure • Unloading.

both tensile and shear fracture (mode I and mode II fracture)of the mortar joints rather than just mode I fracture alone. Thecracks zigzagged through the head and bed mortar joints withoutbrick failure. The TPB test as a standard test to extract the modeI fracture energy, therefore has limited use for masonry panels oflow strength mortar joints.The micro-model purposed by Chaimoon and Attard [10]

was used to determine numerical predictions of the test resultsincluding the load–displacement and CMOD curves and the crackpatterns. The micro-model relies on subdividing a masonry panelinto expanded brick/masonry units and zero-thickness mortarjoints. Fracture is captured through a constitutive softening-fracture law at the finite element interface nodes. Differentinelastic constitutive properties are assigned to the brick–mortarinterface around the perimeter of the masonry unit and tothe interior brick interfaces. The path-dependent non-holonomicsoftening behaviour is solved in incremental steps as a series oflinear complementarity problems. The numerical results provideda good match to the experimental results even through thenumerical formulation assumed a zero dilatancy. The effects ofincluding some dilatancy were studied. As expected, using azero constant dilatancy provided a good reproduction of the loadresponse for the TPB specimens.

Acknowledgement

Mr. Krit Chaimoon is supported financially by a Thai govern-ment research scholarship.


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Documents

Chaimoon 2009 Engineering-Structures