nonlinear finite element analysis of grouted and ungrouted hollow

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Nonlinear finite element analysis of grouted and ungrouted hollowinterlocking mortarless block masonry system

Waleed A. Thanoona,∗, Ahmed H. Alwathafb, Jamaloddin Noorzaeic, Mohd. Saleh Jaafarc,Mohd. Razali Abdulkadirc

a College of Engineering and Architecture, University of Nizwa, Nizwa, Omanb Civil Engineering Department, Faculty of Engineering, Sana’a University, Sana’a, Yemen

c Civil Engineering Department, Faculty of Engineering, University Putra Malaysia, 43400 UPM-Serdang, Malaysia

Received 1 November 2006; received in revised form 23 October 2007; accepted 26 October 2007

Abstract

The main feature of the interlocking hollow block masonry is the replacement of mortar layers commonly used in bonded masonry withinterlocking keys (protrusions and grooves). This study covers the modelling and the analysis of interlocking mortarless ungrouted (hollow) andgrouted concrete block system subjected to axial compression loads using FEM. The main features simulated in the developed finite element codeare the mechanical characteristics of the interlocking dry joints including the geometric imperfection of the shell beds of the blocks, the interactionbetween block units, the progressive debonding between the block and grout and material nonlinearity. The applicability of the proposed FE modelis investigated by demonstrating the nonlinear structural response and failure mechanism of individual block, ungrouted and grouted interlockingmortarless prisms. The results found show good agreement with the experimental test results.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: Finite element method; Nonlinear analysis; Constitutive relations; Interlocking masonry block; Modelling of dry joint; Geometric imperfection;Block–grout bond

1. Introduction

Numerous analytical models have been developed tosimulate the behaviour of different types of structural masonrysystems using Finite element technique. Two main approacheshave been employed in the masonry modelling depending onthe type of the problem and the level of accuracy. The macro-modelling approach intentionally makes no distinction betweenunits and joints but smears the effect of joints presence throughthe formulation of a fictitious homogeneous and continuousmaterial equivalent to the actual one which is discrete andcomposite [1–3]. The alternative micro-modelling approachanalyzes the masonry material as a discontinuous assembly ofblocks, connected to each other by joints at their actual position,the latter being simulated by appropriate constitutive models ofinterface [4–6]. An extensive critical review for the analyticalmodels of different masonry system can be found in Ref. [7].

∗ Corresponding author. Tel.: +968 25466401; fax: +968 25443629.E-mail address: [email protected] (W.A. Thanoon).

Interlocking mortarless load bearing hollow block systemis different from conventional mortared masonry systems inwhich the mortar layers are eliminated and instead the blockunits are interconnected through interlocking protrusions andgrooves [8,9]. It was experimentally found that the behaviour ofthis system is highly affected by the behaviour of the mortarless(dry) joint in both elastic and inelastic stages of loading. Thegeometric imperfection of the block bed is an important factorinfluences the structural behaviour of the mortarless masonrysystem [11,12]. Hence, the modelling of mortarless joint playsa significant role in the overall simulation of the system.Moreover, in order to accurately estimate the system strengthand failure mechanism, it is necessary to have proper model forthe stress–strain behaviour of its different masonry materials ina micro-model framework.

The complex interaction between block units, dry joint andgrouting material (if any) has to be well understood underdifferent stages of loading; i.e. elastic, inelastic and failure.For interlocking mortarless masonry system, very limited

0141-0296/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2007.10.014

Please cite this article in press as: Thanoon WA, et al. Nonlinear finite element analysis of grouted and ungrouted hollow interlocking mortarless block masonrysystem. Engineering Structures (2007), doi:10.1016/j.engstruct.2007.10.014

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ARTICLE IN PRESS2 W.A. Thanoon et al. / Engineering Structures ( ) –

Notations

f ′c, σo Uniaxial compressive strength of the material

(block unit or grout)εo Uniaxial strain corresponding to the ultimate

strength, f ′c .

σ1p & σ2p Tensile and compressive stress respectivelyf ′t Tensile strength of block or grout

Eo Initial tangent modulus of elasticityfeq Equivalent tensile strength

FE analyses have been reported in the literature [13,14].The existing FE analyses are simplified and hence showinaccurate prediction for the structural response of the masonrysystem compared to actual behaviour of the system foundexperimentally. Furthermore, the existing models ignore theinteraction between masonry block units, mortarless jointand grout as well as are incapable of simulating the failuremechanism of the masonry system.

In this study, a finite element model is proposed andan incremental-iterative program is developed to predictthe behaviour and failure mechanism of the system undercompression load. The nonlinear progressive contact behaviour(closing-up) of dry joint that takes into account the geometricimperfection of the block bed interfaces is included based onthe experiment testing data. The developed contact relations fordry joint within specified bounds can be used for any mortarlessmasonry system efficiently with less computational effort. Thebond between block and grout is considered to simulate thedebonding and slipping of the block–grout interface.

Furthermore, the stress–strain behaviour of masonry blocksand grout materials under compression for uniaxial andbiaxial stress state was modelled. Material nonlinearity inthe compressive stress field is considered for the masonryconstituents (block and grout) in the orthogonal directions andthe effect of micro-cracking confinement and softening on thestress–strain relationship under biaxial stresses are includedemploying the equivalent uniaxial strain concept. The modelallows for the progressive local failure of the masonry materials(crushing and cracking). After cracking, a smeared crack modelis adopted and the compressive strength reduction in thecracked block is considered. Both material and joint modelsare proposed based on the comprehensive experimental testscarried out by the authors and published elsewhere [11,12]

The interlocking block system investigated in this study iscalled Putra Block. It is a load bearing block system consistingof three types of blocks namely the stretcher block, cornerblock and half block as shown in Fig. 1. The blocks weredeveloped at Universiti Putra Malaysia [9,10]. The stretcherblock is the main unit used in the construction of the wall;the corner block unit is used to fit the junctions and at theend of the walls while the half block unit is to be used tocomplete the courses of the wall so that the vertical joint willbe staggered. The interlocking mechanism of the block relies onthe insertion of the protrusions of the block to that of the nextcourse. In addition, the assembled blocks provide continuous

Fig. 1. Putra interlocking block units [9].

hollow voids in the wall that allow the casting of reinforcedconcrete (grout) stiffeners in vertical and horizontal directionsto enhance stability and integrity of the wall. The structuralbehaviour of full scale mortarless interlocking walls subjectedto eccentric compressive loads was investigated experimentallyby Thanoon et al. [15].

Previous studies on hollow block masonry under compres-sion showed that the differences in ultimate strength using 2D(plane stress and plane strain) and 3D FE analysis were notmore than 5% [4,16]. The accuracy can be increased if the FEmesh is used in the critical section where the failure is initiated.Also using 3D discretization involves high computational cost.Therefore, plane stress 2D continuum is adopted in this study.Fig. 2 shows the hollow and grouted prisms modelled usingEight-nodded isoparametric element to simulate the masonryconstituents and three-nodded isoparametric interface elementof zero thickness located between material elements to modelthe interface characteristics of the dry joint and bond betweenblock and grout. In hollow prism, the cross webs of the unitare not in contact because the webs are not aligned verticallyin successive courses. Hence, there is no element representingthe webs between joint elements in FE modelling. However, inthe grouted prism, different elements thicknesses were used tomodel the grout and block web.

2. Modelling of mortarless joint

The authors have investigated the characteristics ofmortarless joints experimentally [11,12]. The experimentaltests focus on the contact behaviour in mortarless jointsdue to geometric imperfection of the block beds. The testswere performed on small wall panels and the deformationcharacteristic of the interlocking dry joint at different locationsalong the wall was investigated considering different sourcesof block bed imperfection [11]. Modified triplet tests onmortarless interlocking panels with different levels of axialcompression were carried out to evolve the deformation andshear strength characteristics of the system at the joint interface.Failure criterion and stiffness of the interlocked bed joints undercombined normal-shear load was proposed [12]. Figs. 3 and 4show the joint response under axial compression load and shear


ARTICLE IN PRESSW.A. Thanoon et al. / Engineering Structures ( ) – 3

(a) Hollow prism. (b) Grouted prism. (c) Different types of finite elements.

Fig. 2. Ungrouted and grouted prisms and elements used in FE analysis.

Fig. 3. Close-up deformation versus the compressive stress of dry joint [11].

Fig. 4. Shear slip under different precompressive stresses in dry joint [12].

load respectively [11,12]. The load–deformations relations thatrequired in the FE modelling of the joint is derived based onthese experimental results.

The proposed mathematical model that can describe thenonlinear compressive stress in terms of the joint closure is:

σn = adbn + cdn (1)

whereσn normal compressive stress (N/mm2)

dn close-up deformation (mm)a, b, c constants determined from data analysis of the test

results.Fig. 3 shows the upper and lower bounds of Eq. (1).

Although high variation in close-up deformations was

measured at dry joints at different location along the wall joint,the average deformation of all specified joints in a wall yieldssimilar deformation [11]. Hence, Eq. (1) yields an averagecompressive stress on the contacted area of the dry joint fora family curves obtained at different location along the dryjoint in the tested specimen. These variations in closing-upbehaviour are due to block bed irregularity and variation ofthe blocks height. The constants a, b, and c were evaluatedusing nonlinear regression analysis. Although the modelling ofdry joint (geometric imperfection) of the block bed interfacesunder compressive load is based on the experimental dataobtained by testing interlocking Putra Block system, it mightalso be applicable to any other dry stack block system. Onlythe constants (a, b and c) need to be evaluated experimentallyfor other mortarless block systems.

Fig. 5 showed the failure criteria and shear-slip model ofthe joint under combined shear-normal forces. These modelswere available in the developed computer code in order toinvestigate the structural response of the interlocking systemunder eccentric/shear load. However, this study is limited toinvestigate the behaviour of the joint under concentric axialcompressive load in which the contact behaviour of the jointis more dominant and will be discussed in more detail.

The normal tangent stiffness of the joint, kn , developed bydifferentiating of Eq. (1) as follows:

kn = Ad Bn + c (2)

where A = a.b and B = b − 1. At zero deformation (at theorigin) kn = kni = c, where kni is the initial normal stiffnessand Eq. (2) can be written as

kn = kni + Ad Bn . (3)

The relative displacement between the corresponding twopoints lying on the opposite faces of the continuum elements(Fig. 1) can be expressed as:

{d} = [B] j {δ} (4)

where{d} relative displacement vector at a point (tangential (ds)

and normal (dn) displacement, i.e. shear slip and close-updisplacement).

[B] j strain–displacement matrix of the zero thicknessinterface element.



Fig. 5. Shear strength envelope and shear load–slip model of dry joint [12].

{δ} nodal displacement of the top and bottom nodes of theinterface.

The stress–deformation constitutive relation for the dry jointis expressed as in the standard form:

{σ } = [D j ]{d}

or{τsσn

}=

[ks 00 kn

] {dsdn

}(5)

wherekn normal tangent stiffness of the joint determined from

Eq. (3).ks elastic shear stiffness (estimated from shear load–slip

curves of the bed joints from the shear test [12], =103 N/mm3).ds , dn relative shear and normal movement of the contacted

interfaces.The dry joint is assumed to lose all its stiffness when σn

reaches the masonry block compressive strength ( f ′c).

3. Masonry materials modelling

3.1. Stress–strain relation

The experimental data of stress–strain tests at different loadlevel were used to model the stress–strain behaviour of masonryblock and grout materials [27]. The stress–strain relation underuniaxial compression load is:

σ =p(ε/εo)σo

p − 1 + (ε/εo)p (6)

where,σ , ε instantaneous values of the stress and the strain

respectivelyσo, εo the ultimate stress (peak) and the corresponding strain

respectivelyp material parameter constant depends on the shape of the

stress–strain diagrams.The material parameter (p) was calculated in Ref. [17] based

on the initial tangent modulus. To avoid the inaccuracy incalculating the initial tangent modulus, in this study, nonlinearregression analysis was used for more accurate evaluation ofthe material parameter (p). Fig. 6 shows the experimental

Fig. 6. Comparison of test data and the best fit relation.

test data [27] of the block and grout materials as well as thesuggested best fit curve. A comparison with the well-knownformula suggested by Saenz [18], which is commonly used forthe simulation of compressive stress–strain curves of masonryand concrete under uniaxial stress state [2,19], is also shown inthe same figure.

To include the biaxial stresses effect on the uniaxialstress–strain relation given by Eq. (6), the procedure inRef. [27] was used and is summarized below.

Rewriting Eq. (1) in terms of equivalent uniaxial strain, weobtain (for i = 1, 2):

σi =p(εiu/εi p)σi p

p − 1 + (εiu/εi p)p (7)

whereσi p maximum (peak) compressive principal stress in

direction iεi p equivalent uniaxial strain corresponding to maximum

(peak) compressive principle stress σi p

εiu the equivalent uniaxial strain.The equivalent uniaxial strain εiu eliminates Poisson’s

effect; whereas the strengthening due to the micro-crackingconfinement in biaxial compression stress and softening incompression–tension stress fields are incorporated in σi p andεi p respectively [20–22]. Thus a single relation (Eq. (7)) canrepresent infinite variety of monotonic biaxial loading curves.



Fig. 7. Stress–Strain curves of masonry material for different biaxial stressratio.

Fig. 8. Masonry envelope for different stress states.

The tangent moduli, E1t and E2t for a given principal stressdirections are found as the slopes of the σ1 versus ε1u and theσ2 versus ε2u curves for the current ε1u and ε2u as follows:

Ei t =pEs

[p − 1 + (εiu/εi p)

p− p(εiu/εi p)

p][

p − 1 + (εiu/εi p)p]2 (8)

whereEs , the secant modulus at the peak (maximum stress)

σi p/εi p.Fig. 7 depicts the equivalent uniaxial stress–strain curves

for masonry element loaded with different biaxial stress ratio,α, (α = σ1/σ2) in the compression and tension fields. In thetension field, the relation is linear and the slope is equal tothe initial tangent modulus (Eo) at the origin. The strength isreduced when α < 0.0 whereas for α > 0.0, the strength isenhanced due to micro-cracking confinement. The maximumstress (peak), σi p, and the corresponding strain εi p will be foundfrom the biaxial failure criteria.

3.2. Failure criteria

The proposed failure envelope implements the biaxialstrength envelope shown in Fig. 8 for all stress states andsummarized in Table 1 [28].

Table 1Failure criterion under different stress states

Stress state Criteria

Compression–compressionstate (Viccho [23])

Kc1 = 1 + 0.92(σ2/ f ′c) − 0.76(σ2/ f ′

c)2

Kc2 = 1 + 0.92(σ1/ f ′c) − 0.76(σ1/ f ′

c)2

σ1p = Kc1 f ′c ε1p = Kc1εo

σ2p = Kc2 f ′c ε2p = Kc2εo

Tension–compression(Cerioni et al. [2])

σ1p = feq =

√1 − (σ2/ f ′

c)2 f ′

tσ2p = σ1p/α ≤ 0.65 f ′

cε1p = σ1p/Eo

ε2p = εo[−1.6q3+ 2.25q2

+ 0.35q]

q = σ2p/ f ′c

Tension–tension (Kupferet al. [24])

σ1p = f ′t ≥ σ2p

ε1p = f ′t /Ec ≥ ε2p = σ2p/Eo

An incremental relationship is assumed between strainsand stresses, which in differential form and in the principaldirections can be written for undamaged masonry (see Box I).

The tangent moduli of elasticity, E1 and E2, along theprinciple stress directions are evaluated in the compressive fieldfrom a nonlinear equivalent uniaxial stress–strain relation basedon Eq. (8) and from linear relation in the tensile field as shownin Fig. 7.

After cracking, the tangential elasticity modulus andPoisson’s ratio are reduced to zero in the directionperpendicular to the crack direction. Instead of zeros, verysmall values are substituted in the program to avoid singularityin stiffens matrices. The masonry along cracks is stillresisting compressive stress after cracking. It was found thatthe compressive strength of concrete after cracking can besignificantly reduced by the tensile strain in the transversedirection [25,26]. To account for this effect, the followingformula is adopted to obtain σ2p after cracking:

σ2p

f ′c

=1

0.8 + 0.34(ε1/εo)≤ 1.0 (9)

where εo is the compressive strain relative to the uniaxialcompressive strength, f ′

c , and ε1 is the tensile strain normal tothe crack direction.

3.3. Modelling of block–grout interface

The significant parameters that govern the bond between theblock and grouts are the tensile bond strength and shear bondstrength of the interface. Debonding (separation) occurs whenthe normal force across the interface is tensile and its valueexceeds the tensile bond strength (σt ′). This means that thenodes located at both sides of the interfaces (Fig. 1) are freeto move separately.

Shear failure is initiated along the block–grout interfacewhen the shear stress is more than or equal to the shear strength(τu) that shown in Fig. 9. The envelope in this figure is dividedinto three regions depending on the value of the normal stressσn (tension is positive) and given by the following equations:

for σn ≥ σt ′ τu = 0.0 (10a)

for σt ′ > σn ≥ 0.0 τu = |τo| − (|τo|/σt ′)σn (10b)



{dσ } = [Dc]{dε}dσ1dσ2dτ12

=1

1 − ν2

E1 ν√

E1 E2 0ν√

E1 E2 E2 00 0 0.25(E1 + E2 − 2ν

√E1 E2)

dε1

dε2dγ12

where v =

√v1v2 and v1 = v2 = 0.2

Box I.

Fig. 9. shear strength envelope of block–grout interface.

for σn ≤ 0.0 τu = |τo| − µσn (10c)

whereσt ′ tensile bond strength on the interface (N/mm2)

τu shear strength (mm)τo shear bond on the interface or cohesion strength (shear

strength at zero normal stress) (N/mm2)

µ friction coefficient of the interface.The stress–deformation relation of the block–grout interface

is governed by the constitutive relation in Eq. (4). In this case,the interface is assumed to lose the shear stiffness when theshear stress is more than the shear strength defined by theEq. (10). The interface is assumed to lose all its stiffnesswhen σn reaches the compressive strength ( f ′

c) of the weakermaterial (block or grout). Moreover, when the normal stress,σn , is tension and more than tensile bond strength (σt ′) theinterface will lose all its stiffness at that point. In the undamagedblock–grout interface, high values (1010 N/mm3) are assignedto ks and kn to diminish relative displacement before theinterface failure.

4. Finite element analysis

Incremental-iterative finite element program has beendeveloped to implement the proposed mortarless masonrymodel. Nonlinear analysis is started from the beginning of theload application because of the nonlinearity characteristic of thejoint contact behaviour. Fig. 10 shows the adopted calculationto predict the complete nonlinear response of the masonrysystem for i th iteration for any incremental load.

Table 2Masonry block properties

Typeofmaterial

Eo(N/mm2)

f ′c

(N/mm2)εo Material

parameter,p

f ′t

(N/mm2)εcr

Block 10 932 −22.0 −0.0023 8 1.98 0.000181Grout 9 562.5 −20.4 −0.0028 4.2 2.03 0.000212

Table 3Dry joint element properties (JE)

Joint kni (N/mm3) ks (N/mm3) Coeff. A Coeff. B

1 & 4 5.0 103 14 004.20 3.282 & 3 5.0 103 984.90 6.46

Table 4Block–grout interface element properties (BE)

kn (N/mm3) ks (N/mm3) M τo (N/mm2) σt ′ (N/mm2)

1010 1010 0.6 0.3a 0.36b

a Ref. [16].b Ref. [4].

The results of FE analysis were compared with thosefound experimentally; a unit block [27], interlocking mortarlessungrouted and grouted prisms [11]. All the properties of theblock and grout as well as dry joint and block–grout bondinterfaces are shown in Tables 2–4.

4.1. Block unit

The compressive stress versus axial strain of the finiteelement model and the results of the block unit tested underaxial compression are compared in Fig. 11. Experimentally,under axial compression, block specimen shows conical shear-compression type of failure at the face-shell of the block unit asshown in Fig. 12(a). This mode of failure is well predicted bythe FE model as shown in Fig. 12(b), which shows the crushedarea bounded by dotted lines. The close matching between theexperimental and theoretical analysis indicates that the adoptedmasonry material modelling and failure criteria are capable ofpredicting both the nonlinear stress–strain response and thefailure mode of the masonry.

4.2. Ungrouted (hollow) prism

Fig. 13(a) shows a half of the three courses height prismthat used in the analysis. 2D finite element discretization of



Fig. 10. Nonlinear solution procedure.

Fig. 11. Stress–strain curves of block unit.

the prism in x–y plane is adopted as shown in Fig. 13(b).The web and face-shell elements are modelled with different

thicknesses. The compressive load was applied incrementallyand the boundary condition at the top and bottom of the prismwere simulated as reported in the experimental test [11].

Fig. 14(a) shows the deformed mesh of the prism andFig. 14(b) shows the average axial deformation of the gaugelength between the Demec Points located as shown in thesame figure on both prism sides. The results obtained from theprogram and the test results [11] are shown together in thisfigure for comparison purposes. A good agreement betweenthe program results and the experimental work is distinguished.It can be shown that the dry joints affect predominantlythe hollow prism deformation especially at the lower andmiddle load levels (45% of the ultimate load). This behaviouris due to the nonlinear gradual closure of the contactedinterfaces (or seating) of the dry bed joints. Beyond thisload, the joints will be mostly in maximum contact conditionand the prism deformation is affected by the masonry block



(a) Block crushing from the test [26]. (b) FE model crushing.

Fig. 12. Mode of failure of block unit.

(a) Half of hollow prism. (b) Finite element mesh.

Fig. 13. Hollow prism and FE mesh.

(a) Deformed mesh. (b) Axial deformation versus compressive load.

Fig. 14. Mode of deformation of hollow prism.

shortening only; till the sudden failure of the specimen. Thecompressive strength provided by the FE model is 13.2 N/mm2

(11.2 N/mm2 from the test), which is 15.0% higher than theaverage strength of the tested prisms.



(a) Maximum principal stress, S1. (b) Minimum principal stress, S2.

Fig. 15. Principal stresses distribution of hollow prism.

Maximum and Minimum principal stresses (S1, S2)obtained by the FE analysis at a compressive load of 48kN (average compressive stress is 2.0 N/mm2) are shownin Fig. 15(a) and (b) respectively. The small unsymmetricaldistribution of the stresses in the prism is due to the block bedimperfection. As predicted by the program, high tensile stressis induced in the webs as shown in Fig. 15(a) and compressivestress is higher at the face-shell as shown in Fig. 15(b).

Cracks initiate when the principal tensile stress reachesthe equivalent tensile strength. Fig. 16(a) shows the predictedcracks pattern near the failure and Fig. 16(b) shows theobserved cracks in the experimental test [11]. A single line inFig. 16(a) means single crack and two cross lines means doublecracks in the orthogonal directions which occurred at differentloading stages. As can be seen, a relatively good agreementbetween the FE mode results and the experimental test.

4.3. Grouted prism

The FE mesh of the grouted prism was constructed as in thehollow prism with additional bond element located betweenthe block unit and the grouted core as shown in Fig. 17.Bond element (BE) properties of the block–grout interface areavailable in Table 4. An equivalent thickness for the grout andblock web is calculated to take into consideration the differencein the material properties of the web block and grout in themiddle region of the prism. The calculated equivalent thicknessis based on the ratio of the modulus of elasticity of block andgrout.

(a) FE model result. (b) Test result [10].

Fig. 16. Cracks pattern in hollow prism.

The deformed mesh and the axial deformation are shown inFig. 18(a) and (b) respectively. The results of grouted prismshows different mode of deformation compared to ungroutedprism. Unlike the hollow prism, the high initial deformationat the lower load levels is not exists in the grouted prism dueto grout which eliminates the effect of joint seating observedin hollow prism. However noticeable deformation occurs atthe higher load levels, mainly, because of the progressivefailure of the bond between block and grout. FE results as



(a) Half of grouted prism. (b) Finite element mesh.

Fig. 17. Grouted prism and FE mesh.

(a) FE deformed mesh. (b) Axial deformation versus compressive load.

Fig. 18. Mode of deformation of grouted prism.

well as experimental results show linear variation of load withdeformation within the elastic range of loading, followed bynonlinear style of relation reflecting the degradation of stiffnessdue to the nonlinearity of material and gradual debonding ofblock–grout interface.

The importance of using bond element in the block–groutinterface can be recognized from the finite element analysisresults with and without bond element as shown in Fig. 18(b).In the analysis results without bond element, premature failurein the grouted core occurred because the grout compressivestrength is lower than the block and the program stoppedat 181.7 kN which equal to average compressive stress of5.3 N/mm2. However, the results with bond element show thatthe prism sustains load more than that observed in ignoring thebond element because part of the load was allocated to break theblock–grout interface bond and thus gradual debonding occursbefore the overall collapse. The predicted debonding failure of

the block face-shell from the grouted core is nearly similarto the observed failure in the experimental test as shown inFig. 19. The compressive strength predicted by the model is12.5 N/mm2 (11.5 N/mm2 from test) which is 8.0% higherthan the average strength of the tested prism (deformationreading was stopped at stress of 9.36 N/mm2).

The variation of the stress distribution in the grouted prismis relatively less compared to the hollow prism as shown inthe principal stresses (S1, S2) counter in Fig. 20 which plottedat a compressive load of 69.0 kN (average compressive stressis 2.0 N/mm2). In terms of load, the grouted prism carriedhigher load compared to ungrouted prism. However, as thecross sectional area of the grouted prism is higher than theungrouted prism, the stresses becomes lower. Furthermore, thefull capacity of grout strength was not achieved due to theweb-shell splitting failure. This phenomenon was also observedin experimental testing [15]. It is attributed to the similar



(a) FE model result. (b) Test results [10].

Fig. 19. Block–grout debonding of grouted prism.

(a) Maximum principal stress, S1. (b) Minimum principal stress, S2.

Fig. 20. Principal stresses distribution of grouted prism.

failure modes observed in grouted and hollow prisms. Theaxial compressive load in the grout produces bilateral expansion

of the grout which is confined by the web-shell faces of theblock. This in turn will create additional tensile stresses at the



web-shell interface leading to a splitting crack at this junctionbefore material crushing failure or the full utilization of thegrout strength.

Maximum tensile stress is detected in the grout as shown inFig. 20(a). However cracks were not induced in the subsequentincremental loads because the tensile stress value still much lessthan the grout tensile strength and higher compressive stress ispredicted in the grout as shown in Fig. 20(b).

5. Summary and conclusion

1. Finite element model of mortarless block masonry systemhas been developed. The model is at micro-level whichincludes the modelling of masonry materials, mortarlessdry joint and block–grout interface behaviour. Moreover,the proposed model is capable of predicting the nonlinearresponse of the mortarless masonry system as well as itsfailure mode.

2. The contact behaviour of mortarless joint taking intoconsideration the geometric imperfection of the block bedinterfaces is simulated depending on the results obtainedexperimentally. The developed contact relations for dryjoint within specified bounds describe accurately thevariation of the contact properties and can be used forany mortarless masonry system efficiently in the nonlinearsolution procedure of the system.

3. The presented stress–strain relationship of masonry blockand grout material based on the fitting experimental datais capable of simulating the material nonlinear stress–strainbehaviour for different masonry constituents effectively.Furthermore, it can be incorporated successfully into themodel to include the biaxial state of stress and the materialnonlinearity in the orthogonal directions using the equivalentuniaxial strain concept. The effect of micro-crackingconfinement and softening on the masonry behaviour underbiaxial stresses can be included efficiently also by thepresented stress–strain relation.

4. The bond between block and grout interfaces has beenmodelled as well considering the debonding and slippingfailure of block–grout interface.

5. The FE model is capable of predicting accurately thedeformation of the mortarless ungrouted and grouted prismunder compression. In the ungrouted prism, the dry jointinfluences the prism deformation predominantly from theinitial loading stage up to 45% of the ultimate loadsreflecting the seating effect of dry joint due to the blockbed imperfection. In this range of loading, the stiffness ofungrouted prism increases due to increasing of the contactedparticles in the dry joint interface.

6. Unlike the ungrouted prism, the high initial deformation atlower load levels disappears in the grouted prism due to thepresence of grout, which dominates the overall behaviourof grouted prism. However, noticeable deformation occursat the higher load levels due to the debonding between theblock and grout.

7. The developed FE model can predict the cracking patternof ungrouted prism and block–grout debonding of groutedprism as well as the failure load with acceptable accuracy.

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