1 introduction - qut · th94 international masonry conference, guimarães 2014 ... finally the...

Nazir, Shahid and Dhanasekar, Manicka

9th

International Masonry Conference, Guimarães 2014 4

1 INTRODUCTION

Masonry is used in engineered structures despite its complexities as an anisotropic, low tensile

and highly variable material. Even under simple stress states, masonry exhibits complex failure

modes. When subjected to the lateral loading due to earthquake or wind, the non-linear behaviour of

mortar joints plays a crucial role in the response of masonry structures. In the masonry systems

containing strong units – weak interface combination, failure of units is virtually eliminated and the

failure is limited to mortar joints that act as the planes of weakness. Epoxy or polymer-cement mortars

that exhibit high adhesion and interfacial tensile bond strength (closer to the modulus of rupture of the

units), can modify this behaviour to the one involving a combined mechanism of interfacial and block

failure. To replicate such failure, the blocks/ bricks cannot be modelled as rigid or elastic as in

conventional masonry; therefore, damage of units is included in the model presented in this paper.

A number of modelling methods are developed in recent times depending upon the level of the

accuracy required and the simplicity desired. ”Micro” (or material scale) modelling methods employ

individual properties of the constituent materials and their interfaces whilst the “macro” modelling

approaches use homogenisation of the blocks and the mortar. Homogenised macro method is

efficient for the modelling of large scale walls or full buildings as they require less computational effort

compared to the micro modelling. On the other hand, the micro modelling can provide an insight into

the localisation of the block – mortar interfaces depending on the levels of detail required, which the

macro modelling cannot offer unless multi scale approach is employed.

Micro modelling approach based on contact or interface models are reported in the literature. The

modelling of interface in masonry is commonly handled through zero thickness elements by various

researchers [2-5]. The interface can exhibit damages due to tensile, shear and compressive stress,

which are modelled using different strategies; for example, Nazir and Dhanasekar [6] have used a

surface contact model with damage plasticity for thin layer mortared masonry, Mahaboonpachai et al.

[7] used damage failure criterion; Spada et al. [8] and Haach et al. [2] have used non associated

plasticity theory for shear tension regime with a tension cut-off and Mohr Coulomb type failure criteria

Finite Element Modelling Method for Conventional and Thin Layer Mortared Masonry Systems

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International Masonry Conference Dresden 2010 5

for shear-compression regime. In the micro models, the clay brick or the concrete block is kept elastic

in most publications [2, 6, 7] or even rigid [3].

The modelling method reported in this paper considers the concrete blocks with nonlinear solid

element formulated using damage plasticity model in Lee at el [9] and Lublinar et el [10]. The method

considers the body of the polymer mortar layer and the associated unit-mortar interfaces as a zero

thickness element defining shear-tension failure using an elliptical failure envelope and shear-

compression using Mohr-Coulomb failure criterion. The main novelty in this model is the nonlinear

post-failure plastic displacements is determined explicitly from the derivatives of the constitutive

equations . This approach is used to update the plastic displacement vectors, interfacial stresses

tensors and the Jacobian matrices. This approach can provide very large nonlinear plastic

deformations which can be problematic in the predictor-corrector type integration schemes due to

convergence difficulties especially where deformation accumulates with negligible increase in forces.

Furthermore, the explicit integration scheme is easy to formulate as detailed in Section 3 of this

paper. Another novelty is the ability of this model to reproduce the failure of this masonry through

joints as well as the masonry units as described in Section 9. The model has been calibrated through

data sets from compression, flexural tension and shear tests on thin layer mortared masonry prisms,

beams and triplets respectively [11]. The orthotropic behaviour of thin layer mortared masonry was

examined under uniaxial tensile and compressive loading applied parallel and perpendicular to bed

joints. Finally the behaviour of a thin layer mortared masonry shear wall tested by da Porta [xx] was

successfully reproduced using this model.

2 DESCRIPTION OF INTERFACE MODEL

Conventional masonry, encompassing stronger units and weaker joints fail due to a combination of

mortar body cracking and delamination of block-mortar interfaces whilst the unit remains undamaged

especially under tension and shear dominated loading. To represent such failure it is customery to

use zero thickness joint elements to represnet both the mortar-block interfaces and the mortar joints

with the units modelled as rigid or elastic. In thin layer mortared masonry, due to the usage of higher


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adhesive mortar, concrete block/ clay brick can also fail under tension/ shear. Therefore, in this paper,

concrete blocks are considered deformable capable of damaging plastically whilst a specifically

formulated zero thickness joint element is used to represent the mortar and the interfaces. This

modelling concept is shown in Fig. 1(a).

Yield surface for the zero thickness joint element is shown in Fig. 1(b). The yield surface is divided

into two regimes; i) shear-tension and ii) shear-compression. The yield surfaces are treated differently

in different publications that deals with the interface under shear-tension regime; for example, Spada

et al. [8] has used an elliptical criterion for shear-tension, whilst Lourenco [4] has used tension cut-off.

Shear-compression is however mostly modelled using Mohr Coulomb type failure criterion. In this

study, elliptical criterion is used for shear-tension regime and Mohr Coulomb criterion for shear-

compression regime as shown in Fig. 1(b).

Normal stiffness ( nk ) and shear stiffness ( sk ) of the interface are defined as functions of the

Young’s modulus of masonry block and the masonry instead of block and mortar as normally found in

the literature [3, 5]. It is because the behaviour of 2mm thick mortar is very hard to physically

determine experimentally; simple mortar cylinder or similar tests cannot be considered representative

of the mortar layer in masonry. The Young’s modulus of masonry and concrete blocks can be

determined from standard tests [11, 12]. Therefore normal stiffness ( nk ) and shear stiffness ( sk ) was

defined using masonry and concrete block properties and the formulation by Pluijm [13] is modified for

thin layer mortared masonry as given in (1);

( )

mun

u ju m u

E E

E h tk

E h

2(1 )

ns

kk

(1)

Once the analysis converges to shear-tension regime, possibilities of failure due to i) dominant

tension and ii) dominant shear should be checked. These two modes are separated by equivalent

displacement ratios as given in Eq. (2).

1n

t

no

u

u 2

st

so

u

u (2)


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In shear-tension regime, when failure is dominated by tensile stresses (1t >

2t ) the initial failure will

occur due to tensile strength exceedance; therefore,for unloading the slope of the tension softening

curve will be considered first at the start, explicitly, by taking derivative of the constitituve equations

below, and reprsented by Figure 2(a).

(3)

(4)

(5)

In the shear-tension regime, when failure is dominated by shear stresses (1t <

2t ) the initial failure

will occur due to shear bond strength exceedance; therefore, unloading will be determined as given in

Fig. 2(b) with the stated equations.

(6)

(a): Modelling of wall

(b): Initial yield surface

Figure 1: Wall modelling and yield surface

2

1 1 1

2

1( , ) ( ) ( ) tf p r p f

1

11 ) e(

t

ft

f p

G

tr p f

1

1

1

2

e

t

ft

t

f p

Gt

f

f

G

rh

p

2

2 2 2

2

2( , ) ( ) ( ) of p r p c


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(7)

(8)

When the analysis converges to shear-compression regime, two possible modes could occur: i)

dominant shear and ii) dominant compression. These two modes are again separated by the

equivalent displacement ratios as given in Eq. (9).

1n

c

no

u

u 2

sc

so

u

u (9)

If 1c is less than or equal to

2c then the failure will occur due to shear stress; else if 1c is greater

than2c , then the failure will occur due to the dominant compressive stresses. Under shear-

compression regime, when failure is dominated by shear stresses (1c <

2c ) the failure condition is

defined as given in Eq. (10).

2 2 2 2( , ) tan ( ) of p r p c (10)

Plastic shear behaviour in shear-compression will be the same as given in Fig. 2(b); however, due

to Mohr-Coulomb failure criteria, the slope of the shear softening curve is dependent on sf

G , so for

shear under compression of the modified shear energy sf

G will be calculated as in (11):

( ) 0.13s sf m fG G (11)

In shear-compression regime, when failure is dominated by compression stress (1c >

2c ) the failure

condition and softening will be as given in Fig. 2(c).

Due to the simplicity of the model, plastic corrections have been computed explicitly for each type

of failure. At each step of loading, the flow direction N and the slope of the softening curve ( h ) will

be substituted into Eq. (13) to determine d , which will be used to calculate the normal and the shear

plastic displacements (Eq. (14)).

2

2 2 e( )

o

fs

c p

G

or p c

2

2

2

2

e

o

fs

s

c p

Go

f

c

G

rh

p


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2 2

en

s

e

NN

N

(12)

.

.

N kdud

N k N h

(13)

2

pnn

pss

Nud

Nu

(14)

From these plastic displacement increments, the elastic displacement increments can be

determined and hence the stress. Finally, the Jacobian is determined as in Eq. (15):

:

.

pk N k N

kN k N h

(15)

(a): Tension softening and failure criteria (b): Shear softening and failure criteria

(c): Compression model and failure criterial

For AB;

3

2

2

3 3

2

22

( )2

np np

c o

np np

p

u uh f

p p

u u

For BC; 3

22( )

( )

np

r c

nr np

p uh f

u u

For CD; 3

32 exp 2r c r c nr

nr np nr np r e

f f p uh p

u u u u

3 3 3 3( , ) ( ) cf p r p f

Figure 2: Failure models and softening behaviour for Interface element


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The complete algorithm of the analytical interface model was coded in FORTRAN into a user

subroutine (UMAT) and was incorporated into ABAQUS [14] finite element package, which was used

as a solver. The flowchart of the algorithm is shown in Fig. 6. Masonry block was represented by four-

noded continuum plane stress (CPS4) elements in this study. Equivalent mortar joint interface was

represented by zero thickness four-noded (COH4D) cohesive elements, which were assigned the

properties coded in UMAT.

3 CALIBRATION

The interface element was calibrated using three experimental results (1) masonry prisms tested

under axial compression, (2) triplet specimens tested under shear and (3) masonry beams tested

under four-point bending; only thin layer mortared masonry under monotonic deformation controlled

loading was considered. Not all datasets required for the modelling were available from these tests,

especially Gtf, G

sf, , and . The unknown data were adjusted until the results from the FE model

compare well with the experimental outputs. It is important to note that the calibrated datasets, given

in Table 2, were used for all other numerical examples without any further modification. The

compressive strength of the concrete masonry unit was kept equal 18MPa and the tensile strength

(modulus of rupture) of the block was equal to 2MPa and the material behaviour of model was defined

using damage plasticity model presented by Lee at el [9] and Lublinar et el [10].

Each concrete block was 390mm long and × 90mm thick. The thickness of mortar joint was 2mm.

Therefore the size of four-high stack bonded prism for thin layer mortared masonry was 390mm ×

368mm. Due to symmetry; quarter of the prism was modelled for numerical testing. Loading,

boundary conditions and deformed shape is given in Fig. 3(a). Deformation controlled loading,c , was

monotonically increased at a rate of 2000 steps/mm. Each half block was divided into 32 elements

and 195mm long mortar joint was divided into 8 interface elements along 195mm length. Fig 3(b)

shows excellent match between the numerical output and the experimental stress-displacement

curve.


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A 7 unit-long masonry beam, 1346mm long ╳ 390mm high ╳90mm thick, was tested for flexural

tension. Out of plane thickness of mortar joint was equal to 60mm. Due to symmetry, half of the

(a): Compression model

(b): Comparison of experimental and numerical results

(c): Flexural beam model (d): Flexural beam model

(e): Flexural beam model (f): Flexural beam model

Figure 3: Calibration of model


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flexural beam was modelled for numerical testing. Loading, boundary conditions and deformed shape

are shown in Fig. 3(c). The failure mechanism produced by the model is same as reported by

experimental observations. Furthermore, the numerical output in terms stress-displacement curve

overlaps experimental curve as shown in Fig. 3(d). It can be seen that there is sudden drop in stress

after attaining the peak load but that is not replicated by the model.

Table 1: Parameters for the Interface

Sr.

No.

Symbol Conventional Masonry Thin Layer Mortar Masonry

Value Units Reference Value Units Reference

1 uE 16700 MPa Formula 16700 MPa Formula

2 juE 4687 MPa [11] 8217 MPa [11]

3 tf 0.30 MPa [13] 1.0 MPa [15]

4 oc 0.375 MPa [12] 1.25 MPa [12]

5 cuf 18.0 MPa [11] 18.0 MPa [11]

6 Gtf 0.012 N/mm [4] 0.18 N/mm Calibrated

7 Gsf 0.0125 N/mm [4] 0.225 N/mm Calibrated

8 uh 190 mm Standard 190 Mm Standard

9 mt 10.0 mm [11] 2.0 Mm [11]

10 0.30 - Assumed 0.30 - Assumed

11 μ

0.44 mm [1] 0.64 mm Calibrated

The size of triplet was 390mm high ╳ 270mm wide ╳ 90mm thick, as shown in Fig. 3(e). Loading,

boundary conditions and deformed shape are shown in Fig. 3(e). The failure mechanism produced by


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the model is same as reported by experimental observations. Furthermore, the numerical output in

terms stress-displacement curve overlaps experimental curve as shown in Fig. 3(f).

Based on the detailed calibration, the FE model has been tuned such that its predictions match

well with that of the experimental outputs as shown in Fig. 3. The calibrated materials data,

considered as representative of the thin layer mortared masonry, are presented in Table 1.

5. Numerical Examples

In this section, the behaviour of thin layer mortared masonry investigated for (i); a wallette to find

its biaxial loading and (ii) a shear wall under in-plane shear load are presented.

5.1. Biaxial behaviour of thin layer mortared masonry

Thin layer mortared masonry was tested under uniaxial and biaxial deformation controlled loading

applied parallel and perpendicular to bed joints. A thin layer mortared masonry wallette was

considered for this purpose, as shown in Figure 4(a) and due to symmetry 1/4th of this wallette was

modelled as shown in Figure 4(b).

In Case I, when deformation controlled load applied normal to bed joint, tn , the wallette behaviour

was elastic initially and failure occurred at tensile stress of 1MPa and displacement of 0.041mm as

shown in Figure 4(c). After initial failure, the tension softening occurred until a residual stress of

0.012MPa and the total displacement was 1.216mm. The tensile failure caused the delamination of

masonry blocks at joint locations as shown in Fig. 4(d).

In Case II, when deformation controlled compression load was applied normal to bed joint,cn , the

wallette behaviour was elastic initially up to almost half of compressive strength and then there was

compression hardening as shown in Fig. 4(e). This compression hardening was found up to a

compressive strength of -9.24MPa. After this peak stress, there was compression softening which

continued up to a residual stress of -4.81MPa, as shown in Fig. 4(f).


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In Case III, when deformation controlled tensile load was applied normal to bed joint,tp , the

wallette behaviour was elastic initially up to its tensile strength. Instead of failure and softening, there

(b); Part modelled

(a); Wallette considered

(c); Tensile test normal to bed joint

(e); Compression test normal to bed joint

Figure 4: Biaxial testing of wallette

(d); Tension Failure normal to bed joint

(f); Compression failure normal to bed joint


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was hardening observed up to 1.45MPa followed by softening as in Fig. 5(a). This might have been

due to the horizontal joints being under shear although the vertical joints were under tension. Under

tensile loading, initially both the vertical and the horizontal joints remained elastic. As tensile loading

parallel to bed joint keep increased, the vertical joints under tension failed as shown in Fig. 5(b). Once

the applied tensile forces caused shear stress at horizontal joints more than its shear strength

(1.25MPa), failure occurred accompanied with softening.

When observing the failure mechanism of wallette under tension parallel to bed joint, it is observed

that the vertical joints exhibit delamination and bed joints show shear failure as shown in Fig. 4(e).

(a); Tensile test parallel to bed joint

(c); Compression test parallel to bed joint

(b); Tensile failure parallel to bed joint

(d); Tensile failure parallel to bed joint

Figure 5; Testing of wallette


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In Case IV, under the deformation controlled compressive load applied parallel to bed joint,cp , the

wallette behaviour was elastic initially followed by failure due to compressive stress just equal to half

of compressive strength of thin layer mortared masonry. After initial failure, it was expected to have

compressive hardening but analysis exited due to instability of the individual columns of the

delaminated masonry courses. This instability is reported for conventional masonry by Dhanasekar

[16]. When compression deformation loading applied parallel to bed joint. In case of thin layer

mortared masonry this instability still occurred but with a higher compression of -6.12MPa as shown in

Fig. 4(f).

5.2. Thin layer mortared masonry shear walls

Finally this model was validated by analysing a thin layer mortared masonry shear wall (Fig. 7(a))

reported in Da Porto [1].. The shear wall was constructed using 240mm long ╳ 250mm high ╳ 300mm

thick concrete masonry units. The shear wall was constructed of glue mortar with joint thickness equal

to 1 to 1.4mm thickness; therefore, the wall size was equal to 984mm long ╳ 1250mm high ╳ 300mm

thick. It is reported that the concrete masonry unit had 43% perforations. Therefore, for analytical

purposes the thickness of wall was considered equal to 300mm ╳ (1.0 - 0.43) = 171mm. On top of

wall, a reinforced concrete beam was provided with dimension of 984mm long ╳ 250mm high ╳

300mm thick. The masonry shear wall constructed on a reinforced concrete footing of 1500mm long ╳

250mm high ╳ 450mm thick.

Top and bottom concrete beams were kept elastic. The modulus of elasticity for top and bottom

concrete beams was defined equal to 30GPa and poison’s ratio equal to 0.3. The modulus of

elasticity of unit reported by Da Porto is 9328MPa normal to bed joint but there is no data for elasticity

modulus parallel to bed joint and hence was assumed as 6370MPa. For masonry units Poison’s Ratio

equal to 0.22, the compressive strength of concrete masonry units was 8.71 MPa and tensile strength

was 0.434MPa.

The combined or homogenised mortar joint and unit-mortar (zero thick element) was assigned ae

shear bond strength of 0.6 MPa which is usually 1.25 times of tensile bond strength, therefore, tensile


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bond strength of thin layer glue mortared masonry interface was taken equal to 0.5 MPa. The Poisons

ratio for interface was considered equal to 0.21. The tensile and shear energies were equal to 0.8

Nmm/mm2 and 1.0 Nmm/mm2, respectively. The friction coefficient was kept equal to 0.64 and mortar

thickness equal to 2mm.

The each concrete masonry unit, 240mm long x 250mm, high was divided into 6 x 6 elements. The

length of each interface element was kept at least equal to 41mm. Each element size of concrete top

and bottom beam was less equal than 50mm x 50mm. The bottom floor beam was restrained for any

translational and rotational movement. At the top surface of bond beam of wall, a uniformly distribute

stress equal 0.62MPa applied in first step which causes pre-compression equal 1.16MPa (0.17% of

masonry compressive strength) on mortared bedded area of the shear wall. In last step, the

displacement controlled in-plane horizontal deformation applied monotonically on left end of top

concrete beam. The meshing loading and boundary conditions are shown in Fig. 7(b).

The numerical model is shown to predict the experimental results quite well; load-displacement

curves and the collapse mechanism are shown in Fig. 7. Considering the collapse mechanism, it is

found that the failure mechanism can be reproduced by the numerical model. Comparing with

experiments, there was tensile failure in the bottom most concrete masonry layer as well as the

smeared cracking along the diagonal from one corner to other corner with crushing at compression

corner as shown in Fig.7(c). Failure occurred mainly through the concrete masonry units with some

cracks through the joints and comparable to experimental results shown in Fig. 7(e).


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(a): Experimental setup [1]

Pre-comp (σp) = 0.62MPa In-Plane

Displacement (δH)

Fix bottom

(b): Boundary conditions

(c) : Failure mechanism of thin mortared masonry shear walls

(d) : Load-displacement for shear wall (e) : Failure mechanism

Figure 7: Thin layer mortared masonry shear wall


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6 CONCLUSIONS

Based on this investigation, following conclusions can be made;

Nonlinear modelling of thin layer mortared masonry joint can be carried out using explicit

integration scheme within a implicit finite element framework for zero thickness four node interface

element. In interface element modelling the plastic behaviour of masonry joint can be determined

explicitly from constitutive equations and used to update the stresses, displacement and Jacobian.

This interface element can replicate the behaviour of meso level masonry wallettes as well as

masonry shear walls when failure is through mortar joints or combination of masonry joints and

masonry units.

Thin layer mortared masonry is orthotropic in nature. It has 1.45times higher tensile strength

parallel to bed joint as compare to tensile strength normal to bed joint. But compressive strength

normal to bed joint is 1.51times higher than the compressive strength parallel to bed joints. The use of

high adhesive, smaller thickness mortar joints increases the strength under biaxial compression of

masonry parallel to bed joint more prominently as compared to normal to bed joint. Further increase in

adhesion of masonry can causes the failure through block which could completely eliminate the

orthotropy, which is problematic for modelling of masonry.

Model is successfully validated through a thin layer mortared masonry shear wall experiment.

Results compared well with the load-displacement curve and failure patternThere is an ongoing effort

in developing the model and future effort includes but not limited to; (1) enhancement of interface

element model to handle cyclic loading, (2) enhancement of interface element model for reinforced

masonry, (3) study of model to simulate the bond between composites like FRP and concrete and (4)

3-dimensional analysis.

ACKNOWLEDGEMENTS

The work was funded through the Australian Research Council’s ARC LP0990514


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The support of the concrete masonry association of Australia (CMAA), AdBri Masonry Pty Ltd and

Rockcote Australia is gratefully acknowledged.

Reference

[1] F. Da Porto, "In Plane Cyclic Behaviour of Thin Layer Joint Masonry," PhD, University of Trento,Italy, 2005.

[2] V. G. Haach, G. Vasconcelos, and P. B. Lourenço, "Parametrical study of masonry walls subjected to in-plane loading through numerical modeling," Engineering Structures, vol. 33, pp. 1377-1389, 2011.

[3] K. M. Dolatshahi and A. J. Aref, "Two-dimensional computational framework of meso-scale rigid and line interface elements for masonry structures," Engineering Structures, vol. 33, pp. 3657-3667, 2011.

[4] Lourenço and Rots, "Multisurface Interface Model for Analysis of Masonry Structures," Journal of Engineering Mechanics, vol. 123, pp. 660-668, 1997.

[5] P. B. Lourenco, "Computational strategies for masonry structures," PhD Thesis, Delft University, Netherlands, 1996.

[6] S. Nazir and M. Dhanasekar, "Modelling the failure of thin layered mortar joints in masonry," Engineering Structures, vol. 49, pp. 615-627, 2013.

[7] T. Mahaboonpachai, T. Matsumoto, and Y. Inaba, "Investigation of interfacial fracture toughness between concrete and adhesive mortar in an external wall tile structure," International Journal of Adhesion and Adhesives, vol. 30, pp. 1-9, 2010.

[8] A. Spada, G. Giambanco, and P. Rizzo, "Damage and plasticity at the interfaces in composite materials and structures," Computer Methods in Applied Mechanics and Engineering, vol. 198, pp. 3884-3901, 2009.

[9] J. Lee, and G. L. Fenves, , "Plastic-Damage Model for Cyclic Loading of Concrete Structures," Journal of Engineering Mechanics, vol. 124, pp. 892–900, 1998.

[10] J. Lubliner, J. Oliver, S. Oller, and E. Oñate, "A Plastic-Damage Model for Concrete," International Journal of Solids and Structures, vol. 25, pp. 299–329, 1989.

[11] J. A. Thamboo, M. Dhanasekar, and C. Yan, "EFFECTS OF JOINT THICKNESS, ADHESION AND WEB SHELLS TO THE FACE SHELL BEDDED CONCRETE MASONRY LOADED IN COMPRESSION," Australian Journal of Engineering Structures, 2013 (Accepted).

[12] S. A. International, "Australian Standards for Masonry Structures," in AS3700, ed, 2011. [13] V. Pluijm, "Shear behaviour of bed joints," Drexel University, Philadelphia, Pennsylvania, USA,

1993, pp. 125–36. [14] ABAQUS, "ABAQUS user subroutine’s reference manual," ed: Hibbit, Karlson and Sorenson,

Inc, 2010, pp. Version 6.9-1. [15] J. A. Thamboo, M. Dhanasekar, and C. Yan, "Characterisation of Flexural Tensile Bond

Strength in Thin Bed Masonry," in International Brick and Block Masonry Conference (IB2MaC), Florianopolis, Brazil 2012.

[16] M. Dhanasekar, "The performance of brick masonry subjected to in-plane loading," PhD Thesis, University of Newcastle, Australia, 1985.

1 introduction - qut · th94 international masonry conference, guimarães 2014 ... finally the...

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