1

Review of Modelling of Masonry Shear

M. Dhansekar

Professor, Faculty of Built Environment & Engineering, Queensland University of Technology, Brisbane, Australia.

Ph. +61 7 3138 6666

Fax. + 61 7 3137 1170

Email: m.dhanasekar@qut.edu.au

URL: http://staff.qut.edu.au/staff/dhanasek/

This article is prepared on invitation from the editors Professors Muralikrishnan

and Ramamurthy, IIT Madras, India.

Abstract:

Masonry is extensively used in buildings as structural and cladding elements. With the increase in

demand for understanding the performance of all building elements, experts from various fields

are seeking to include masonry as a sub-system of their respective major models. It is imperative

that these models account for the true behaviour of masonry dictated by the presence of mortar

joints acting as planes of weakness, the interaction between the grout and masonry shell as well as

the grouted cores and unreinforced masonry in various forms of masonry design. The purpose of

this paper is, therefore, to provide a state-of-the-art review on the modelling of masonry in the

context of the fundamental behaviour of various forms of masonry to loading, with particular

attention to shear loading. It is shown that the modelling methods and the parameters must be

carefully chosen as one method cannot be claimed suitable for all circumstances.

Keywords: Structural Masonry, Micro Modelling, Macro Modelling, Interface

Modelling, Reinforced Masonry

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1. Introduction

Masonry is widely used in buildings either as load bearing walls or cladding/ infill

walls. Whilst masonry in buildings do not exhibit measurable distress under

gravity loading, its failure often is widely reported even under moderate lateral

loading; such failures occur both in the load bearing and in the cladding masonry

elements causing significant loss of lives and damages to properties. With the

realisation of the more frequent occurrence of extreme events, loading codes of

most nations demand adequate detailing of the structural and nonstructural

(including fittings) elements against the lateral loading. To appropriately detail

masonry elements, design engineers require thorough understanding of masonry

behaviour and modelling methods. The purpose of this paper is to raise the

awareness of the proper understanding of the performance of masonry under

loading, with particular attention to shear loading, to minimise the vulnerability of

masonry (FEMA307 [37]).

2. Theory of Structural Masonry

Masonry is strong in compression but weak in tension; it is also characterised by

mortar joints acting as planes of weakness. Under loading (vertical, lateral or

combination of vertical and lateral) unreinforced masonry behaves in a complex

manner exhibiting modes of failure classed as brittle. Where fully reinforced,

masonry is said to behave similar to reinforced concrete and fails in a more ductile

manner [36, 67, 69]. Failure of mixed constructions involving reinforced

elements and unreinforced masonry can be defined anywhere within the broader

spectrum of highly brittle to moderately ductile range.

Modelling of masonry without full understanding of their behaviour, especially

the effect of mortar joints and/ or the manner in which the grout and masonry

shells interact, can lead to erroneous results.

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2.1 Unreinforced Masonry

Masonry is predominantly unreinforced and is vulnerable to seismic and high

wind actions. Masonry derives stability and increased out-of-plane flexural and

in-plane shear capacities from imposed vertical loads; therefore, it is prudent to

use them as structural members with the slabs directly resting on them. In the

modern masonry buildings, walls running on two orthogonal vertical planes are

connected to each other transmitting loads from the slabs that form cellular system

capable of distributing the lateral loading efficiently across the walls in their

stiffer direction, thus requiring much thinner walls. With the reduction in

thickness, these walls become more sensitive to stress levels, especially where the

walls are ungrouted and only face shell bedded. As at least two materials with

vastly differing elastic properties form masonry, appropriate material modelling,

structural idealisation and choice of modelling method must be carefully

identified to obtain the desired results commensurate to the expected levels of

accuracy.

2.1.1 Shear in unreinforced masonry

Shear in unreinforced masonry is largely resisted due to deformation in mortar

bed joints and is defined conveniently by Mohr-Coulomb failure envelop that

relates the shear strength u to the shear bond strength 0 , an apparent friction

coefficient and vertical stress vf in the joint as shown in equation (1); this linear

equation is usually applicable to a small range of vf , typically 2MPa or lower

reflecting the significant nonlinearity the vertical stress can cause to masonry.

0u vf (1)

Where any insert is provided in masonry, such as flashing or damp proof course,

0 will vanish and the shear capacity of such elements is solely dependent on

and vf [17, 21]. The value of for cyclic loading is critical in earthquake design

of partition or cladding walls as may not have been mobilised and is usually

quite low compared to its static counterpart.

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In spite of its simplicity, equation (1) provides a reasonable conservative estimate

of shear capacity of URM walls and hence is adopted in the European (BS5628

[8], BS-EN1996 [10]) and the Australian standards (AS3700 [2]). The American

(MSJC [56]) and Canadian standards (CSA-S304.1 [11]) relate the shear strength

of masonry to the square-root of its characteristic compressive strength 'mf ; the

proportionality constants vary between the standards.

Design of out-of-plane shear response is non-controversial and can be performed

using equation 1 for URM walls. Design of masonry walls for inplane shear is,

however, more complex; the provisions in various standards can at best be

described as reflecting the current poor state-of-the-art. Walls under inplane shear

loading are said to exhibit three modes of failure, namely, (1) diagonal shear (2)

sliding shear and (3) flexural rocking. It is claimed that the diagonal shear is more

brittle, although several test results on walls that exhibit diagonal shear failure

capable of being laterally drifted beyond 1%. For example although URM shear

walls that exhibit pinched cyclic load displacement relationship typically

associated with poor hysteretic energy dissipation capacity due primarily to the

rocking mechanism are considered not suitable for seismic zones, from a detailed

dynamic analyses and statistical studies of unreinforced masonry shear walls

exhibiting rocking mechanism and using the input of 164 recorded European

earthquake events, Lestuzzi et al. [50] concluded that the limited hysteresis does

not lead to excessive displacement demand. This is an important finding as it

contradicts the widely held perception on such masonry shear walls.

Generally the reinforced and unreinforced masonry shear walls exhibit mixed

modes of failure. Often another potentially brittle mode of failure, corner

crushing, is not considered in the design. Some detailed examination of the

strength properties of masonry (tensile bond, shear bond and compressive) by the

author [13] has shown the tensile and shear bond strengths increase significantly

(up to 175%) whilst the compressive strength remained relatively unaffected. It

was concluded that it’s not so much of the age, but the effect of drying of well-

cured specimens increase their two bond strengths (tensile & shear). The

compressive strength, whose failure depends on the crushing / cracking of the

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units, has not shown any significant effect of drying / age. The masonry failure

surface, defined by these three strength properties of masonry, usually consider 28

day moist strength properties; to date no literature is found on the effect of non-

proportional changes in the strength properties of masonry, reflecting the effect of

drying, to the behaviour of masonry shear walls. When the shear and tensile bond

strength values are increased by 175% whilst the compressive strength are

retained at original level, it was possible to predict the localised compression

failure at the corners as the mid zone of the wall was able to resist higher lateral

load due to increased shear bond strength [13].

Another aspect that requires careful consideration for the evaluation of the

behaviour of masonry shear walls is dilatancy that is defined as the vertical

translation of masonry units during sliding, causing volume change, which in turn

can generate or increase confining pressure if the masonry is restrained in some

way. Ignoring dilatancy in the calculation of shear capacity can lead to very

conservative predictions, whereas inappropriate use of dilatancy (especially in

‘confined’ or infill masonry [39]) leads to highly unconservative predictions. Van

Zijl [72] has shown that dilatancy coefficients should be made pressure dependent

in order to make safe predictions.

For the determination of the lower bound capacities of diagonally compressed

URM wall panels, Roca [63] provided two simple expressions, one for deep

beams subjected to concentrated vertical loading and the other for deep beams

subjected to uniformly distributed vertical loading. These analytical expressions

can help calibrating FE models for problems with no experimental data.

2.2 Reinforced Masonry

Where the seismic/ wind forces are high, masonry is invariably reinforced. There

are many forms of reinforced masonry. In some designs all cores of walls are

grouted and reinforced and in others only a few selected hollow cores are grouted

and reinforced. In the later, the grouted and reinforced cores can be regarded as

concealed RM columns/ mullions [14, 42] – figure 1 shows a photo; where, it can

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be seen that owing to the staggered web shell in hollow blocks, the grouted

mullion/ concealed column can never have straight edges.

Figure 1 Concealed mullion in wide spaced reinforced masonry

2.2.1 Compression behaviour

RM has outer masonry shells and inner grout containing reinforcing steel. The

outer shells consist of solid bricks or hollow blocks bonded together by mortar

bed joints. The inner grout is usually made from lean concrete and is poured into

the hollow cores containing steel reinforcement such that it forms a composite

vertical member by binding the outer shells and the reinforcing bars. Owing to

the limited space within the hollow cores of the masonry shells, it is difficult to

detail the lateral ties or spirals, which leaves the reinforcement laterally untied and

the grout remain unconfined laterally. Where the Poisson’s ratio of the grout is

much larger than that of the masonry shell, due to incompatibility of lateral

expansion of these two materials, the shells are subjected to lateral grout pressure

[47, 68]. This phenomenon leads to premature spalling of the shells. Similar

spalling of (highly stressed compression) shell also occurs where masonry is

subjected to eccentric loading or lateral bending [35, 60]. Figure 2 shows spalling

of web shells of grouted prisms under axial compression and hollow prism

(spalling circled) under eccentric compression.

Figure 2 Web shell spalling in axially loaded grouted prisms and eccentrically

loaded hollow (dry-stack) prisms

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Although it is well known that the compressive strength of brittle materials

increase with lateral confinement, only a few researchers have examined methods

of improving the behaviour of masonry subjected to axial compression using

some form of confinement of the grout. Typically round bars, flat bars and plates

were used as joint reinforcement to confine the grout. Hatzinikolas et al. [43]

concluded that such joint reinforcement system adversely affected the behaviour

of the grouted masonry and reduced the strength due to stress concentration within

the joints. Drysdale and Hamid [35, 36] found that the round bars enhanced the

compressive strength of masonry by a small amount of 2% whilst the plate

reinforcement system provided 15% increase. A similar level of increase in

compressive strength of grouted concrete masonry was reported by Priestly and

Elder [58] based on tests carried out on prisms containing stainless steel plates in

the mortar joints to overcome the potential corrosion problem triggered by lack of

cover. With a view to minimizing lateral expansion of grout, fibre reinforcement

is sometimes used; more details on micro (fibre) and macro reinforcement effects

on masonry can be found in [7].

The idea of placing web reinforcement within the grouted cores of the vertically

reinforced masonry was first developed by Kumar and reported in [16, 47, 48,

49]. By placing triangular shaped web reinforcement within each reinforced cores

of clay block masonry, he achieved 32% increase in compressive strength of clay

block masonry over the grouted, unconfined clay block masonry strength.

2.2.2 Shear behaviour

Shear in RM is largely resisted due to aggregate interlock in the grout, dowel

action of the flexural reinforcement (perpendicular to the direction of shear) and

the tensile action of shear reinforcement (parallel to the direction of shear) with no

significant effect of mortar joint [67, 69]. Therefore the fully grouted reinforced

(in both horizontal and vertical directions) masonry could be regarded analogues

to reinforced concrete; unfortunately this analogy does not bring much comfort as

in spite of significant research, shear in reinforced concrete still remains as a

subject requiring further development [73, 74].

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Walls subjected to vertical out-of-plane bending will invariably be provided with

vertical steel; horizontal shear in such walls would only derive marginal benefit of

the vertical steel but will derive excellent benefit from horizontal reinforcement if

provided in zones of high shear (mid high levels). If the same wall is subjected to

vertical shear, the vertical reinforcement can prove to be effective. Shear

behaviour of reinforced masonry can be inferred from the theory of reinforced

concrete as long as the masonry is fully grouted and reinforced - for example,

reinforced masonry beams [15, 17, 25, 31]. For the out-of-plane design, the

Australian [2], European [10] and British [9] Standards provide a singly

reinforced formula by either limiting the maximum effective tensile reinforcement

or by specifying the neutral axis parameter be less than a conservative limit far

below the balanced limit (for example, 0.40 in AS3700 [2]). Masonry is seldom

designed as doubly reinforced as restraining compression steel is always a

challenge.

The North American standards define the nominal shear strength of reinforced

masonry as a function of moment-to-shear ratio '0.16 2 m

Mf

Vd

. The CSA-

S304 [11] perhaps is the only standard that includes the size effect in reinforced

masonry beams; the nominal shear strength is defined for fully grouted and

reinforced masonry beam is defined as '4000.16 1

2000 m

df

.

Whilst fully reinforced masonry can be regarded behaving similar to reinforced

concrete under shear (both inplane and out-of-plane), behaviour under inplane

shear is not well understood. Designers may conceive the partially grouted

masonry walls as ‘mixed’ system consisting of reinforced core elements and

unreinforced masonry panels spanning between the cores by naming them as

partially grouted reinforced masonry. The wall may also be conceived as a

reinforced masonry in which the spacing of walls is spaced wider than the fully

reinforced masonry (for example, spacing of vertical reinforcement up to 2m) and

are called as wide spaced reinforced masonry [30, 42].

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The design standards reflect the poor state-of-the-art of the shear wall failure

mechanisms as they provide widely differing design clauses with extremely

unconservative/ unsafe provisions in many places and too conservative/

uneconomical clauses in others. Through a comparative study of various masonry

standards using 75 test data on fully reinforced masonry shear walls reported in

the literature, Dickey and Lissel [34] determined that the North American codes

outperform the Australian, European and British codes in terms of economic

design – whilst all standards were conservative. The situation becomes rather

more complex when partially grouted/ wide spaced reinforced masonry walls are

considered. With limited data on 12 full-scale wide spaced reinforced masonry

shear walls, Dhanasekar et al [30] have shown that the provisions in CSA-S04

[11] and AS3700 [2] are unsafe.

It is therefore clear that whilst unreinforced and fully reinforced masonry walls

can be designed safely using the provisions in most design standards, it appears

safe design may not be achieved through standard provisions for the partially

reinforced/ wide spaced reinforced masonry walls. Further research is urgently

needed to examine the inplane shear behaviour of partially reinforced masonry

walls; this is especially so because these walls are quite economical and can prove

to be an adequate structural system for those countries that experience moderate,

intra-plate type earthquakes and/ or cyclonic wind. These walls can be analogised

to the much celebrated confined masonry system that has large unreinforced

masonry panels surrounded by tension columns and beams (that confine the

masonry) [39].

3. Modelling Strategies

Modelling of masonry is either for fundamental understanding of its material

behaviour or for the determination of its structural behaviour. For material

characterisation, detailed micro modelling in two or three dimension is necessary.

For structural characterisation, depending on the need, simplified or more

elaborate modelling can be performed.

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Masonry cracks under serviceability loading conditions; cracking can also be an

important issue under concentrated loading. Fracture mechanics approach might

well be required to fully understand the masonry response to cracking;

unfortunately, it’s not easy to translate the classical theories of fracture to

masonry because masonry is a composite consisting of several brittle materials;

furthermore, masonry cracks at ultimate under compression in a direction parallel

to the direction of loading. Some limited research is reported in this area mainly

by Shrive and co-workers [61] and Rots [60] but the area remains largely

unexplored, leaving serviceability crack limit calculations wide open in all

standards for unreinforced masonry. This is despite the fact that such cracking is

quite common in masonry.

3.1 Geometrical Modelling

Masonry is generally used in walling applications, although columns/ piers, arches

and massive dams/ abutments are also built of masonry. The ultimate behaviour

characterised by the complete load deflection response of masonry is perhaps the

most important aspect for the design, which is mostly studied experimentally. It’s

seldom possible to carry out experiment on all common masonry types; therefore,

sound numerical modelling techniques are required for appropriate understanding.

In practical applications, masonry buildings consisting of shear walls are treated

as an assembly of springs for the determination of push-over drift. Recently,

Tena-Colunga and Cano-Licona [70] extended such models for partial and full

damage status of the walls using the definition of a shear area factor. The idea is

to adjust the shear area depending on the stiffness (or height to length ratio) of the

masonry shear walls. Such analytical method is useful to approximate

conservative analysis of large scale buildings.

Where out-of-plane flexure of wall or column is desired, one dimensional line

element modelling may be efficient. An efficient 1D layered finite element was

formulated by the author and coworkers [3, 4, 5] for the flexural analysis of RM

columns including bond-slip. The new 1D finite element was formulated using

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mixed approach that satisfied the equilibrium and compatibility conditions

simultaneously. A special purpose computer program was developed using the

layered, 1D line element for the analysis of RM columns subjected to monotonic

axial and lateral loading. The 1D model prediction was validated using the lateral

load-deflection response of RM columns tested at the University of Calgary,

carried out as part of a collaborative research project. The model can also account

for bond-slip that is quite critical in wide spaced reinforced masonry applications

as there is only limited surface area of steel is available. Furthermore the grout in

masonry may not be compacted well, causing potential poor bond with the steel

surface.

As wall structures can generally be regarded as plane stress bodies, idealisation of

the walls on their mid thickness vertical plane is commonly adopted and meshed

using 2D plane stress continuum elements – especially for solid masonry.

However the hollow block masonry that is ungrouted and face shell bedded do not

strictly comply with the plane stress assumptions as the face shells (being thin

members) exhibit tendency for out-of-plane buckling under compression and the

web shells with no mortar joints beneath or above them acts like diaphragms

connecting the two face shells. The face shell buckling has the potential to induce

cracking in the web shells; in such situations, the effective heights of the face

shells will increase dramatically leading to pre-mature collapse that is often brittle

(many standards provide quite conservative capacity reduction factors to avoid

catastrophic failure). Shell elements can only predict such failure – thus

warranting expensive 3D modelling [66]. With a view to addressing the

computational costs, the author and his co-workers have developed a family of

quasi 3D elements (figure 3) that require only 2D meshing, saving enormous time

and cost [32, 33, 38, 77].

Figure 3 A Quasi 3D Finite Element

x

y

z

2h1

2h2

x

y

i j

k

l

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Figure 3 shows a typical hybrid stress element formulated from the 3D Hellinger-

Reissner principle with the capability of predicting 3D states of the stresses and

strains. In this element is treated as a function of z to represent the traction

free condition on the lateral surfaces. According to the equilibrium equation in the

z-direction, and could not be zero everywhere but functions of (x, y, z).

3.2 Material Modelling

Where micro modelling is used, the properties of the units, mortar, grout and

reinforcement as well as the interface properties of interacting surfaces between

units and mortar/ units and grout/ mortar and grout/ and grout and reinforcement

are required. These properties are often determined from uniaxial tests on

standard sized test specimens subjected to uniform state of stress. Important

parameter is the interface properties as masonry behaviour can be significantly

affected by these properties. With the reduction in thickness of the bed joint,

significant strength gain of masonry can be obtained [18]. Higher bond strengths

of masonry can also increase the strength of masonry [65] as well as enhancing

the ductility of the masonry material [62]. Further details of brick-mortar

interaction can be seen in [75].

True behaviour of masonry can only be assessed through biaxial testing of panels,

which is quite an involved and expensive proposition. Although not attempted,

mesoscopic properties of masonry for its characterisation under biaxial loading

can be carried out using micro modelling; multi-scale modelling [54, 52] method

can be exploited for this purpose. Recent studies on multi-scale modelling

application to masonry exhibit the feasibility for such studies. The down side is

that multi-scale method is computationally expensive; it will, however, be less

expensive than the other alternate, biaxial testing. To date comprehensive study

on the biaxial behaviour of masonry is only determined experimentally for only

solid clay brick masonry [22, 26, 28]; these data are widely cited over the past

three decades. The test data are typically presented as a failure surface as shown

in figure 4. For hollow and grouted concrete masonry, some data are reported in

recent times [23, 24, 29].

z

yz zx

13

Figure 4: Failure surface for masonry [13]

All the failure surfaces are 3D shaped consisting of intersecting convex envelopes

of conical/ truncated conical/ cap shapes. They illustrate the profound influence of

shear stress on the normal stresses, which implicitly describe the effect of bed

joints. Flow theories of plasticity can be applied to these surfaces by considering

them as initial ‘yield’ surfaces as the flow vectors can be normal to these convex

surfaces containing no dips, gaps or overlaps. These material models are suitable

for implementation into commercial FE codes (eg. ABAQUS [1]).

These failure/ yield surfaces can only be implemented into modelling of masonry

where smeared cracking is considered. Unless care is taken, mesh size can

seriously affect the results of the analysis – known as mesh pathology that

adversely affects the uniqueness of unloading curves. These problems are widely

studied in reinforced concrete modelling, where it is shown that by limiting the

size of the elements to not less than three times the size of the coarse aggregates

and by explicitly defining the Mode I and Mode II energies in input data, mesh

pathology in reinforced concrete structures can be overcome by determining the

isotropic characteristic length h A where = 2 for linear elements and

A is the area of the largest element and incorporating h as a material property.

Similar approach is used for unreinforced masonry [6] although masonry is

distinctly orthotropic.

Theoretically there are two factors that affect the softening part of the load

displacement curves: (1) appropriate energy release criteria during crack opening

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due to tension (Mode I fracture) and crack sliding due to shear (Mode II fracture)

and (2) inclusion of characteristic length in the material model to eliminate the

mesh size dependency problems.

3.2.1 Energy release criteria

Measuring fracture energies under tension and shear require careful preparation of

specimen as well as high-precision displacement control equipment. The work

carried out at Delft University of Technology, Netherlands [64] is considered

classical for the purpose; a typical result is provided in figure 5 for tension. A

similar result is provided for shear in [52]. As masonry exhibits significant

nonlinearity under compression, a complete stress-strain curve for compression,

including softening is essential for its appropriate modelling.

Figure 5 Crack Opening Mode I Fracture for Masonry [52]

The model shown in Figure 5 is useful when the structure is subjected to

monotonically increasing load. Where the load path involves cyclic deformation

of varying magnitude, much work is essential. Hypothetical cyclic strain – stress

models are provided in [40]. Experimentally measured cyclic stress-strain curves

for unconfined and confined masonry are reported in [23, 24].

3.2.2 Characteristic length

The second factor is the characteristic length – a mesh dependent parameter. The

maximum isotropic characteristic length 2 x yh h h (for elements with linear

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shape function) where &x yh h are the length of the element in the 1 and 2

directions, should satisfy the energy criteria as shown in equation 2:

2

fi i

i

G Eh

f (2)

in which fG is the fracture energy (measured as area under curves shown in

figure 5), E and tf are the stiffness and strength in the direction of measurement

of fG respectively (subscript i stands for the direction of measurement). In other

words, it could be inferred that one should provide an upper limit of element size

during meshing if they have reliable experimental data on E, fG and tf ;

alternatively the model should be adjusted for h and G as usually E and f for

materials are reliably estimated.

However, for distinctly orthotropic materials like masonry, it appears that

different characteristic lengths for the two orthotropic directions are necessary.

Discrepancy between the values used for the characteristic lengths in the x and y

directions has been found in the analysis of URM walls reported in Lourenco [52].

For example, in the analysis of the ETH Zurich shear walls, the length and the

height of 4-noded plane stress elements used by Lourenco [52] were 150mm and

133.3mm respectively; the characteristic length of these elements, therefore, is

2 150 133.3 200h mm if isotropy is assumed. However this was not used

as inferred from the following calculation: the tensile strength of masonry parallel

to the x and y directions was reported equal to 0.28 MPa and 0.05 MPa

respectively; Young’s modulus of masonry parallel xE and perpendicular yE

to the bed joints was equal to 2460 MPa and 5460 MPa respectively. Fracture

energy for the x and y directions used in the analysis was 0.02 Nmm/mm2. The

characteristic length (h) of elements was thus determined as 628 mm and 43,680

mm respectively (compared to the isotropic characteristic length of 200 mm). This

shows that the value of the characteristic lengths of elements actually used in the

analysis by Lourenco [52] for the x and y directions were different; This

discrepancy from the established idea of characteristic length for isotropic

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materials must be properly understood in modelling – especially where fracture

energy across the two orthotropic directions are not available. Based on the above

observation, two characteristic lengths, one for the normal (y) direction and the

other for the parallel (x) direction to masonry bed joint, were also adopted in [20].

Gambarotta and Lagomarsino [41] have also expressed the similar concern on the

characteristic length of elements for the modelling of URM shear walls to cyclic

loading in such a way that the localisation based mesh pathology can be avoided.

Whilst they specified the minimum height of smeared cracking masonry element

should be the related to the width of crack band (or mortar thickness), they agreed

that no guideline could be provided to the width of the element based on the

current knowledge. This leaves the analysts to come out their own interpretation

of characteristic length for masonry as it stands until some fundamental research

is carried out. Mosler and Meschke [55] have strongly recommended usage of a

tensorial characteristic length to deal with the mesh pathological problems of the

failure of anisotropic materials.

3.2.3 Application of damage mechanics

Recent modellers of unreinforced masonry walls tend to derive the properties of

masonry from repetitive unit volume elements of masonry using the theory of

homogenisation with a view to incorporating them into macro modelling. Whilst

elastic properties are determined accurately using homogenisation, its application

to nonlinear material properties is far from complete [52]. Therefore, an elastic

damage mechanics approach is tried with some limited success. In this method, a

simple positive scalar D that ranges from zero (no damage) to one (fully damaged)

is used in the theory of isotropic continuum damage mechanics. Equation 3 relates

the damaged state modulus and stress &E to their undamaged state counterparts

&E :

1

1

E D E

D

(3)

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For masonry a damage tensor ijklD is more appropriate than the scalar D , because

the material is anisotropic consisting of interfaces. The weakening process of the

materials and the interfaces is represented by the ijklD .

For solid masonry in running bond, assuming damage occurs only in the mortar

matrix, Luciano and Saco [53] proposed a damage representation of a unit cell of

masonry using a six digit (000000 for undamaged to 111111 for fully damaged)

representation for monotonic loading condition. Gambarotta and Lagomarsino

[40, 41] proposed damage models for mortar only and then extended the damage

model using homogenisation for the masonry and applied the model for shear

walls subjected to cyclic lateral loading. These models together with simple

isotropic failure criterion for units and mortar were implemented into a FE code as

a user material subroutine and applied for 3D analysis of larger masonry

buildings.

4. Modelling Methods

Finite element method (FEM) is largely used in the modelling of masonry, similar

to other engineering problems. It is well known that FE methods only provide

solution to equilibrium problems and where modelling of instability is the primary

focus, they inherently fail. Although powerful contact modelling methods are

emerging to address such weaknesses, FE methods are overburdened with the

additional level of (contact) nonlinearity in addition to the material nonlinearities

that are integral to masonry modelling.

Contrary to achieving equilibrium through nodal connectivity in the FE methods,

nodal or surface contact is used to achieving equilibrium of assembled bodies in

distinct (or discrete) element method (DEM) [79] and spring connectivity between

nodes are used in applied element method (AEM) [46]. The bodies themselves

may be rigid or deformable but they are not formulated through shape functions as

the finite elements are. Large masonry buildings can be modeled using these

techniques (DEM/ AEM) with some simplified assumption that the nonlinearities

and failure are localised along the contacting planes.

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4.1 Finite Element Methods

Finite element analysis of masonry walls in general, although have provided much

insight into the behaviour of masonry [27, 52, 57, 80], are generally regarded as

too cumbersome in terms of the requirement of large amount of input data,

especially related to nonlinear biaxial properties, as well as the time taken for the

analysis including post processing of the innumerable results affected by various

material interfaces.

4.1.1 FE Solution Techniques: Implicit/ Explicit

Common solution technique used in most finite element models reported in the

literature is related to the conventional implicit technique that requires the fully

assembled structural stiffness matrix for the solution of equilibrium equations. As

the masonry cracks, this stiffness matrix tend to become ill-conditioned posing

severe convergence problem, especially during the softening stages of the wall

behaviour. Through some highly powerful methods such as the arc-length method

[44], ill-posed equations are solved in commercial packages using double

precision arithmetic – a process that requires significant computational cost.

In contrast, explicit solution technique never assembles the full system stiffness

matrix; rather it solves for the internal variables using the theory of dynamic wave

propagation in solids. It can handle softening stages of the nonlinear response of

walls quite adequately and hence can be regarded computationally efficient.

Although explicit technique is more suitable for high dynamic events such as

impact, static load tests could also be simulated if due care is taken to minimise

the kinetic energy due to rapid cracking/ load shedding. As iterations are not

performed, much smaller increments of the applied load are required for the

explicit technique. This technique provides solution when the global energy

equilibrium is ensured; as such it is suitable where the global structural behaviour

is of prime importance; its application to stress analysis where accuracy is

primarily desired can be a problem. Unlike the implicit method, the explicit

solution technique is only conditionally stable. Choosing an appropriate time step

19

is vitally important to obtain stable solutions. The time step should be kept lower

than the maximum period maxt of the structure as given in equation 4.

2max

max

21t

(4)

In which max is the highest Eigen value of the system and is the

corresponding critical damping.

Explicit methods are shown to provide realistic results of the global behaviour of

masonry shear walls in recent times – by the author and coworker [19, 20] for

wide spaced reinforced masonry subjected to monotonic loading and recently by

Karapita et al. [45] for unreinforced masonry subjected to cyclic loading. These

early successes prompt further studies for comprehensive examination of this

method.

4.1.2 Material Nonlinearity in FE Formulation

Irrespective of the method of solving (implicit/ explicit), the FE models must

handle the material nonlinearity that includes plasticity, damage and cracking/

fracture. Nonlinearities due to interface bond failure/ slip in masonry structures

could be classed under material nonlinearity also. Nonlinear solution algorithms

have attained high level of perfection these days and many commercial packages

provide very good routines to take care of nonlinearities. Normally they use

incremental, iterative techniques that are automated and auto-scaled. Correct

choice of data and solution strategies are more important in the current scenario

rather than trying to improve the efficiency of the solution schemes.

In all nonlinear methods, the stiffness varies as a function of the accumulated

strains. For nonlinear analyses, displacement control techniques provide better

solution; therefore, predefined displacement history data are provided to infer the

structural response. The nonlinear algorithms could be regarded as “predictor –

corrector” schemes in which first elastic displacements are obtained (“predictive”)

for a given increment of external loading. The excess (or residual) load that

20

violates a yield criterion is then dissipated into the structure by allowing the

structure to deform (“corrective”) further through a process of iteration. This

scheme continues until the excess/ residual load becomes negligibly small. At

this stage it is regarded that nonlinear equilibrium is achieved.

Material nonlinearity is usually defined in two facets: First definition of a yield

criterion and second a flow rule. There are many classical yield criteria and flow

rules available in the theory of plasticity. For example, concrete/ rock type brittle

materials are modelled using Drucker-Prager yield criterion. For masonry, the

failure criteria reported in Section 3.2 of this paper are the best available to date.

There is a tendency to treat fully reinforced masonry as reinforced concrete by

disregarding the planes of weakness (bed joints); this proposition requires further

thorough examination prior to adoption as a standard process. The author

believes that there is absolutely no justification to assume isotropy of the partially

reinforced or confined masonry.

The second facet of a flow rule is typically well established in the theory of

plasticity. The flow rules may be “associated” (plastic strain tensor direction

remains outward normal to the yield surface) or “non-associated” (plastic strain

tensor oblique to the yield surface). It should be noted that the flow rule provides

only the direction of strain tensor. As stresses cannot exceed yield, any excess

stress (calculated based on elastic predictor) must be reduced back to the yield

surface. The direction predicted by the flow rule helps in reducing the excess

stresses to the yield surface. Furthermore, the yield surface itself might change its

shape/ size or position depending on the material data/ load history. When the

loading is simple monotonic, isotropic hardening rule is considered sufficient; for

cyclic loading, kinematic hardening rule allowing for the translation of the yield

surface away from the origin to account for the accumulation of residual stresses

(plastic strains) must be accounted for in detail.

Lofti and Shing [51] provided an appraisal of smeared crack modelling techniques

and used the approach for masonry and steel reinforcement for the analysis of

reinforced masonry walls. It was assumed that the uncracked masonry followed a

plasticity model, the cracked masonry an orthotropic material model and the steel

21

reinforcement an elastic hardening-plastic material. This approach predicted

flexure dominant behaviour but couldn’t capture the brittle shear behaviour of RM

walls due to the unrealistic kinematic constraint introduced by the smeared crack

assumption. This problem can be overcome with the use of discrete crack

approach [12, 52, 55] to model the major diagonal crack in a shear wall. Although

this approach predicts the behaviour well, it requires the location of the major

diagonal crack as a priori.

By modelling the shear walls using plasticity-based interface elements for mortar

joints and smeared crack elements for masonry units and by treating the

reinforcing steel as an elastic-hardening plastic material by means of a smeared

overlay on top of the smeared crack elements, RM shear walls are modelled. The

compressive and tensile behaviour of masonry units can be governed by Von

Mises failure surface with a Rankine type tension cut-off. A similar failure

criterion was formulated for masonry by Lourenco [52], who presented a macro

model capable of accurately predicting the behaviour of the URM shear walls

until the walls exhibited large lateral drift associated with wide crack.

Zhuge [80] developed a two-dimensional plane stress element model for the non-

linear analysis of URM shear walls. This model was developed using a

homogeneous material model to predict the detailed load-deflection characteristics

and critical limit states of URM walls under inplane earthquake ground

acceleration.

4.1.3 New Generation Finite Elements

A four noded 2D isoparametric hybrid stress element with six degrees of freedom

per node termed as a quasi 3D element with 24 stress parameters (Q3D24) is

reported in Section 3.1 of this paper. In spite of exhibiting convergence with h-

refinement, the element has remained computationally more expensive than the

standard plane element, but unlike the plane element it could provide out-of-plane

deformation and stress fields just from quasi 2D analysis. With reference to this

22

capability, it can be argued that the Q3D24 element is less expensive than the 3D

counterpart in an overall sense.

Trefftz Finite Element (TFE) [59] technique is preferred where there exist

significant nonlinearities due to contact or moving boundaries or discontinuities.

Since 1926 when Trefftz published the first paper on this formulation, a number

of contributions that examine problems of some elastic and non elastic fields

using Trefftz approach have emerged. The advantage is that TFE is formulated

using two shape functions (as against a single shape function that formulates

conventional finite elements); first is a domain shape function that accounts for

intra element nonlinearities and the second is a boundary (or, frame) shape

function that allows for inter-element compatibility. The separation of the intra

and inter element shape functions possess the advantage of not imposing explicit

relation between nodal degrees of freedom and the intra element deformation

point-by-point. Trefftz approach can be referred to as the boundary-type solution

procedure employing regular T-complete functions satisfying the governing

equations.

Masonry containing cracks, voids or other defects can be analysed using very

complex, fine mesh especially around the defects in the conventional finite

element analysis. To effectively describe singularity of stress fields at the crack

tip, commercial finite element programs provide singular elements formulated by

collapsing rectangular elements into triangular shape for 2D analysis and

rectangular prism elements into triangular pyramid shape for 3D analysis.

Depending on the anticipated state of stress singularity around the crack tip, mid

side nodes in the collapsed elements could either be moved to the quarter point

position closer to the crack tip or left unchanged in their original location.

With the objective of making the meshing simpler especially around the crack or

other defects, using the variational principle, complex variable technique, and

mapping function Tong et al [71] formulated several super-elements based on

hybrid element technique. As the stress singularity at the crack tip has been

included in the formulation of these elements, the stress intensity factors were

23

reported directly from the nodal displacements, thus eliminating the use of any

weak forms such as the J-integrals for this purpose.

The author and co-workers presented two Trefftz boundary element formulations

based on the Galerkin technique and the collocation point technique with

particular attention to modelling mode III fracture problems [76]. First, original

formulations and the solution of the Mode III crack on elastic problem are

deduced based on the Trefftz functions satisfying the Laplace governing equation.

Then stiffness matrix and equivalent nodal flow vector are formed from the

approximate solution satisfying the boundary condition using the Galerkin method

and the collocation point method [78]. Near crack tips results are improved

through an auxiliary function.

Trefftz finite element (TFE) formulations appear more suitable for dealing with

cracking in masonry; they do not appear to have significant advantage in dealing

with material plasticity and damage although some contributions have emerged in

the literature in recent times for plasticity.

4.2 Other Numerical Methods

As stated earlier, AEM and DEM are emerging as popular methods for modelling

masonry – especially larger building assemblages. The fundamental idea of these

two methods is defining contact between boundaries of blocks. Contact problems

have attracted much attention due to their inherent complexity and frequent

occurrence in engineering practice. A variety of numerical approaches for

calculating contact zones, and dividing them into stick and slip subzones, as well

as determining the contact stresses, are available in the literature. Amongst these

approaches the interface element (IE) methodology, due to its ease of numerical

implementation, has been investigated by many researchers. The kinematic

inconsistency associated with the classical joint element of Goodman [13, 27]

usually results in spurious oscillations of tangential traction. To circumvent this

difficulty, the Newton-Cotes integration scheme was used by several authors

instead of the widely adopted Gaussian quadrature [5,20].

24

As TFE approach assimilates the merits of the conventional FE and boundary

element (BE) methods and, moreover, discards some of their drawbacks: Firstly,

the element formulation calls for integration along the element boundaries only,

and secondly, some problems with singular or local effects can be treated easily if

exact local solution functions are available. The author and coworkers have

developed a TFE model specifically for elastic contact problems with the help of

Kaliakin-Li (KL) interface elements [76]. The model consisted of a 4-node hybrid

Trefftz element used in the discretisation of both contacting bodies; 4-node KL

elements have been embedded for simulating the potential contact zone. To assess

the reliability of the KL elements for predicting the contact behaviour, an

interfacial constitutive relation, namely penalized normal contact and tangential

friction law, has been adopted. The TFE-interface model has been implemented

into a commercial FE package, ABAQUS, via user element subroutine (UEL).

Two benchmark examples have been presented to illustrate the applicability of the

overall model to contact problems. Good agreements have been achieved between

the results from the present approach and analytical as well as the conventional

FE solutions obtained from ABAQUS software [1].

From static contact analysis using finite elements, one could discover that the

uniform bearing stress assumption at the interface is inappropriate. From the

results reported, it could be inferred that although the bulk of the interface is

under an “average” stress, at the point of singularity (where there is a sudden

discontinuity) the stress was found to peak significantly (approximately four times

average stress – sufficient to cause local damage). Contact interface failure in

masonry is so important and the TFE method with KL contact appears to be an

effective approach for solving the problem.

5. Summary and Conclusions

This paper has provided a review of the fundamental behaviour of masonry under

loading with a view to identifying appropriate modelling method; both the

unreinforced and reinforced masonry as well as the wide space reinforced

25

masonry are considered. The following conclusions have emerged from the

review:

Unreinforced and fully reinforced masonry behaviour under out-of-plane

shear is reasonably well understood.

Inplane shear behaviour of masonry in general is not fully well

understood; all masonry walls are thought to fail due to diagonal shear/

sliding shear/ rocking flexure, of which the diagonal shear is believed to

be more brittle than the other two modes of failure – although

contradicting results are reported in the literature. Modelling inplane shear

behaviour of masonry is currently the most sought topic of research.

Some inplane shear design provisions in AS3700 [2] and CSA-S304 [11]

are unsafe. As this form of construction is quite economical, there is a

pressing need to urgently revise the current provisions so that safer designs

can be achieved.

Shear and tensile bond strengths can increase dramatically when masonry

is bone-dry whilst its compressive strength will remain relatively

unaffected. This can cause highly brittle corner crushing mode of failure –

especially if masonry walls are designed to transfer concentrated lateral

loading at corner rather than distributed by the rigid diaphragm.

Some special elements are formulated specially for simplified geometric

modelling of masonry; they are:

o 1D element incorporating grouting and reinforcement including

bond-slip;

o 2D element capable of providing three dimensional state of stress.

Masonry can be modelled using micro modelling approach in FE; however

careful interface modelling is essential. Masonry can also be analysed

using macro modelling; such models require special failure surface for

masonry under biaxial stress state.

Smeared crack modelling of masonry is commonly adopted as it is

economical; energy based characteristic length of the element must be

included in such models to avoid mesh pathology. As modern masonry is

distinctly orthotropic, research should be directed to evaluating

characteristic lengths for each of the axis of orthotropy.

26

Explicit solution technique is emerging as a viable approach for modelling

statically loaded masonry walls subjected to nonlinearity and cracking.

Trefftz finite element method, that incorporates two shape functions – one

for deformation of interior of the element and the other for the boundary

deformation related to inter-element compatibility – is proving to be

efficient for dealing cracking and contact problems; the method can be

tried for efficient masonry modelling.

Applied element method (AEM) and distinct (or discrete) element method

(DEM) are emerging as powerful methods in modelling masonry as these

methods largely rely on the interface contact formulation between large

blocks that may be rigid or deformable. Progressive failure of structure,

that cannot be easily modelled using FE methods can be readily modelled

in AEM/ DEM.

Acknowledgements

The author acknowledges a range of undergraduate and research students who have contributed to

the development of the ideas/ data reported in the paper. Recent deliberations with Professors Page

(Newcastle, NSW), Shrive (Calgary, Canada) and Drysdale (McMaster, Canada) as well as

industry partners Alan Pearson (CMAA) and Wayne Holt (AdBri Masonry) are also fondly

acknowledged.

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