Diagonal compressive strength of masonry samples—experimental and numerical approach

ORIGINAL ARTICLE

Diagonal compressive strength of masonry samples—experimental and numerical approach

Rui Sousa • Hipolito Sousa • Joao Guedes

Received: 25 October 2011 / Accepted: 16 August 2012 / Published online: 28 August 2012

� RILEM 2012

Abstract Masonry is a structural material that

presents a quite complex behaviour that depends on

the mechanical and geometrical characteristics of the

units, the mortar and the link between these two

elements. In particular, the characterization of the

shear behaviour of masonry elements involves proper

experimental campaigns that make these analyses

particularly expensive. The main objective of this

paper is to present a case study on the characterization

of the shear behaviour of masonry through a method-

ology that merges a small number of laboratory tests

with computer simulations. The methodology is

applied to a new masonry system that has recently

been developed in Portugal, and involves a FEM

numerical approach based on micro3D modelling of

masonry samples using nonlinear behaviour models

that are calibrated through a small number of labora-

tory tests. As a result, the characterization of the

masonry shear behaviour trough this methodology

allowed simulating, with reasonably accuracy, a large

set of expensive laboratory tests using numerical tools

calibrated with small experimental resources.

Keywords Masonry � Lightweight concrete units �Diagonal compression � Laboratory tests �Computer simulations � Sensitivity analysis

List of symbols

An Net area of masonry sample

B Height of the masonry sample

D Isotropic scalar degradation variable

dc Compressive damage variable

dmax Maximum aggregate size

Del0

Initial (undamaged) elastic stiffness

dt Tensile damage variable

e Thickness of the mortar parallel joints

E0 Modulus of elasticity

F Compression load

fl Tensile flexural strength

Fmax Maximum compression load

Fmax Maximum compression load

Gp Potential plastic flow

g Full width of the mortar strips

G Shear modulus

GF Fracture energy

GFo Base value of the fracture energy

H Length of the masonry sample

hs Depth of a sample

Kc Ratio between the tensile and compressive

stress invariants at initial yield

L Distance between measurement points of

Dv and Dh

N Percentage of gross area of the unit that is

solid

�p Effective hydrostatic pressure

�q Von Mises equivalent effective stress

R2 Coefficient of determination

R. Sousa (&) � H. Sousa � J. Guedes

GEQUALTEC, Faculty of Engineering, University of

Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

e-mail: [email protected]

Materials and Structures (2013) 46:765–786

DOI 10.1617/s11527-012-9933-z

sc Weight factor to control the recovery of the

compressive stiffness

st Weight factor to control the recovery of the

tensile stiffness

t Total thickness of the wall

ai, bi, ci Adimensional parameters

c Shear strain

cmax Shear strain for the smax

Dh Horizontal extensions

Dv Vertical shortening

e Total strain

ecu Strain for rcu or ultimate strain

ec,limit Limit compressive strain

eel Elastic strain

ec Compression strain

emax Strain of masonry for the Fmax

epl Plastic strain

~epl Multi-axial equivalent plastic strain

~eplc

Compressive equivalent plastic strain

~eplt

Tensile plastic strain

et Tensile strain

m Poisson coefficient

r Cauchy stress

�r Effective stress

rb0 Initial equi-biaxial compressive yield stress

rc Uniaxial compression stress

�rc Compressive effective stresses

rc0 Initial uniaxial compressive yield stress

rcu Compressive strength (maximum

compression stress)

�r_

maxMaximum principal effective stress

(algebraic value)

rt Uniaxial tensile stress

�rt Tensile effective stresses

rto Uniaxial tensile strength

s Shear stress

smax Shear strength or maximum shear stress

w Dilation angle

[ Parameter that defines the rate at which Gp

approaches the asymptote

1 Introduction

Masonry is a material that presents a quite complex

behaviour that depends on the mechanical and

geometrical characteristics of the units, the mortar

and the link between these two elements. In particular,

this complexity arises from the large scatter observed

in the mechanical characteristics of the materials

involved and of the interface between units and

mortar. In fact, experimental results are essential to

characterize the units and the mortar as separated

elements, but also the masonry through samples where

both elements and their interactions are taken into

account. The results and the knowledge got from these

campaigns have been essential to define the empirical

based rules which have been adopted in most design

codes for masonry and, moreover, to develop and

calibrate numerical tools to simulate masonry

structures.

However, this involves expensive resources, even

when small size samples are used. In general, labo-

ratory tests are difficult to prepare and require specific

instrumentation and careful execution. On the other

hand, the development of computer tools and numer-

ical models has made possible to reduce the difficulties

and costs of experimental characterization by means

of computer-simulated tests. Nevertheless, the accu-

racy and efficiency of these tools depends on the

available experimental information necessary to cal-

ibrate the numerical models. After being properly

verified and calibrated through experimental data,

numerical tools can be a very powerful tool to simulate

and extrapolate experimental results to other condi-

tions with lower time consuming and financial costs.

Many behaviour models have been developed

during the last decades to simulate concrete type

materials (with tensile brittle behaviour) and, in

association to this, masonry type materials. In this

last case, the simulations can be done either by using

refined models, where units, mortar and interfaces are

represented separately through proper behaviour laws

calibrated for the different elements, or by using more

global models, where masonry is considered to be a

heterogeneous material ruled by a unique behaviour

law that integrates its global behaviour.

This paper describes the characterization of the

shear behaviour of a new masonry system made with

lightweight concrete units that has been recently

developed in Portugal. This characterization was

made through the application of an integrated method

where numerical analyses are used together with a

limited number of experiments. The results allowed

characterizing the shear behaviour of the masonry

system for a set of pre-defined characteristics, but also

for controlled variations on some of the geometrical

766 Materials and Structures (2013) 46:765–786

and mechanical properties of the masonry, following

an extensive sensitivity analysis.

2 Literature review

2.1 New masonry systems

Considerable changes have been reported on masonry

wall construction during the last decades with the

introduction or extended use of lightweight materials

and new types of units [19]. In more recent years, new

concepts for masonry systems have been developed in

some Southern European countries. These systems are

considered innovative, since they are able to accom-

plish several functionality aspects (e.g., structural and

fire resistance, thermal and acoustic insulation, water

tightness) without the need for complementary mea-

sures. Many of these systems consist on one-leaf walls

made of lightweight materials and large units with

high percentage of voids. These new units are lighter,

improving block handling, but keeping structural

resistance [10]. The need to understand and to properly

characterize the mechanical behaviour of these inno-

vative systems has motivated experimental and

numerical studies, although the majority of these

studies are related to the behaviour of clay masonry

(e.g. [8, 26, 29]).

2.2 Masonry analysis

Masonry, in general, is a structural brittle material

made of units jointed together using, usually, hydrau-

lic mortar that makes the units work together. The

different construction methods, and the large geomet-

rical and mechanical variations that both units and

mortar may exhibit, make the characterization of

masonry a complex issue, in particular if no experi-

mental studies exist to sustain these analyses. Calde-

rini et al. [5], for instance, based on observations of

seismic damage on complex masonry walls and on

experimental laboratory tests, report that masonry

subjected to in-plane loading has two typical types of

behaviour which are associated to different failure

modes:

• flexural behaviour (rocking, crushing);

• shear behaviour (sliding shear failure, diagonal

cracking).

However, a mixed shear/flexural behaviour may

also occur, depending on the intensity of the axial

loading. The behaviour depends on parameters such

as: the geometry of the wall (units and wall aspect ratio

and cross-section), the mechanical characteristics of

the masonry materials (mortar joints, units and

interfaces), the boundary conditions and the level of

axial load. In fact, most of the available studies on

masonry refer to experimental campaigns involving

masonry samples using different units, mortar and

types of joints. Many non-linear models have been

developed or modified to fit the masonry behaviour. In

general, the models are based on a Finite Element (FE)

formulation where non-linear damage constitutive

models and friction models are used (e.g. [3, 4, 38].

However, models based on Discrete Element (DE)

formulations have also been reported with success

[24].

The simulation of masonry structures through a FE

formulation can follow models with different detailing

approaches. The less detailed approach represents

masonry as an equivalent continuum material with an

appropriate constitutive global homogeneous model

(e.g. [28], while the more detailed represents the

masonry units, the joints and the interfaces explicitly,

i.e. using different FE elements and material models

(e.g. [7, 25]. The first approach is more suitable for

modelling the global response of large structures due

to its lower time consuming and computational cost, in

opposition to the second approach, which use is only

justifiable when more refined analysis are requested in

smaller samples. Nevertheless, the more detailed

analyses have also been used to determine the

necessary parameters to calibrate more global models,

or as a complement, or an alternative to laboratory

experiments [3].

Concerning the constitutive models, recent studies

suggest the use of plastic damage constitutive models

for FE analysis of masonry elements (e.g. [3, 32]). In

general, these constitutive models were first developed

to simulate other brittle materials, such as concrete or

rocks (e.g. [15, 27, 37]), allowing the consideration of

their main failure mechanisms, i.e. cracking and

crushing through the use of two scalar variables which

monitor local damage under tension and compression.

Many studies are found involving the use of

experimental data to calibrate and validate constitutive

models that are used in the numerical analyses.

However, only a few involved the prediction of

Materials and Structures (2013) 46:765–786 767

masonry behaviour, being mostly oriented to the in-

plane compression (e.g. [2, 40] and in-plane shear

behaviour (e.g. [16, 17]) of masonry samples.

Regarding shear behaviour, Fouchal et al. [16]

developed a numerical model based on the adhesion

intensity to simulate the mechanical behaviour of the

interfaces in masonry elements. The mechanical

characteristics were determined experimentally using

samples made of solids bricks and mortar. The

numerical simulation of masonry shear tests and the

comparison to available experimental data (triplets

tested according to European standard EN 1052-3 and

diagonal compression tests based on American stan-

dard ASTM E519) gave good results and provided the

coefficients required to model the interfaces: stiffness

parameters and viscosity and friction coefficients.

Gabor et al. [17] present a numerical and an

experimental analysis of the in-plane shear behaviour

of masonry panels made with clay hollow units. A two

dimensional FE model was used adopting an elastic

behaviour for the units and an elastic-perfectly plastic

formulation (Drucker–Prager) for the mortar, i.e.

considering the overall non-linear behaviour of the

masonry concentrated on the mortar that was consid-

ered perfectly bonded to the units. The material

parameters were determined through compressive

tests on masonry and mortar samples and direct shear

tests on masonry triplets (European standard EN

1052-3). The numerical analysis consisted on the

simulation of diagonal compression tests on masonry

panels made in laboratory according to RILEM

recommendations. The model was able to estimate,

with good accuracy, the ultimate load and strain, the

plastic strain evolution and the failure mode.

Regardless of the more or less success of the

different modelling approaches to simulate the in-

plane behaviour of masonry, the actual investigations

are still looking for a more accurate and generalized

model to simulate the different aspects that can

influence the masonry mechanical behaviour.

3 Work methodology

An integrated method was used to predict the in-plane

shear behaviour of a new masonry system that has

recently been developed through a national co-funded

research project in Portugal. In particular it aims

analysing the influence of the geometrical and

mechanical properties of the constituent materials in

the behaviour of the masonry. The method involves a

FE numerical approach based on a detailed 3D

modelling of diagonal compressive tests (standard

tests) on masonry samples, using nonlinear behaviour

models that are calibrated through a small number of

laboratory tests, reducing time consuming and finan-

cial cost. The method consists of the following steps:

• Characterization of the mechanical properties of

the masonry constituents, units and mortar,

through laboratory tests and available literature;

• Characterization of the shear behaviour of

masonry through laboratory tests performed on

small samples according to American standard

ASTM E [1].

• Construction of 3D models of masonry samples

using a FEM program and a plastic damage type

model to simulate the units and mortar behaviour;

• Calibration and validation of the numerical model

through experimental results obtained from labo-

ratory tests;

• Characterization of the shear behaviour of the

masonry samples through computer-simulated

shear tests, considering different geometrical and

mechanical properties for the masonry

constituents.

During the calibration and validation procedures,

potential numerical fracture patterns were determined

and compared to the fracture patterns observed in the

laboratory tests.

4 Masonry system: general characteristics

The masonry system used as case study was developed

under a co-funded national research project, called

‘‘Thermal and mechanical optimization of single leaf

masonry (OTMAPS)’’, which aimed conceiving a

single leaf masonry system with optimal thermal

performance and enough mechanical strength to be

used as structural masonry.

This project involved numerical and experimental

studies and some results have already been published

[34–36].

The masonry system was developed for single leaf

walls with units made of lightweight concrete with

light expanded clay aggregates (Leca�). The units

measure 350 9 350 9 190 mm3 and have a blind


surface for laying mortar (Fig. 1a, b). Factory-made

lightweight mortar is used in all masonry joints.

The bed joints consist on two 10 mm thick mortar

strips (shell bedded masonry), 120 mm wide each,

positioned along the sides of the units. The perpend

joints are represented by three hollow columns defined

by the irregular lateral surfaces of the units in contact,

filled in with mortar, forming 40 % of the width of the

unit (Fig. 2).

The shape and the material of the units and, in

particular, the dimensions and distribution of the

cavities was decided within the research project to full

fill thermal requirements, but also handling capabilities.

The main properties of the masonry materials

(concrete units and mortar joints) are summarized in

Table 1. These results were determined through

standard laboratory tests in the scope of the referred

research project [34].

5 Determination of the masonry properties

5.1 Materials (units and mortar)

A set of laboratory tests was performed to determine

the necessary parameters to define the constitutive

laws of the masonry materials, namely the uniaxial

stress–strain behaviour curves of the units and the

joints mortar. These parameters are essential, not only

to characterize the masonry, but also to give input data

to the numerical model.

Given the difficulties associated with the dimen-

sional representation of test samples (e.g. the concrete

samples were extracted from the units webs and shells,

meaning small samples) and the carrying out of the

tests (e.g. difficulties to adjust the loading rate and the

displacement measuring capabilities of the test device

to the brittle nature of the masonry constituents), as

Fig. 1 Base unit: a scheme

with unit dimensions (in

mm), b blind surface for

laying mortar and c opposite

side


imposed by the standards, the properties of the units

concrete and the joints mortar were estimated through

laboratory tests done in this study, but also by using

expressions in regulations/codes for concrete struc-

tures, i.e. Model Code 90 [6], and experimental data

from other studies [22, 30, 39].

5.1.1 Mortar

To characterize the tensile behaviour of the mortar, the

direct tensile strength (rto) and the fracture energy

(GF) were estimated from the mathematical expres-

sions of Model Code 90:

rto ¼ fl1:5 0:01hsð Þ0:7

1þ 1:5 0:01hsð Þ0:7ð1Þ

GF ¼ GFo

rcu

10

� �0:7

ð2Þ

and by using the mean flexural and compressive

strength obtained from the flexural and compressive

tests performed in this study on the mortar (following

the European standard EN 1015-11 [12]): a tensile

flexural strength (fl) of 2.45 N/mm2 and a compressive

strength (rcu) of 11.45 N/mm2.

The base values of fracture energy (GFo) are defined

in MC90 as a function of the maximum aggregate size

(dmax). A regression technique was used to obtain GFo

as function of dmax.

The compressive constitutive law of the mortar

(before and after the compressive peak stress) was also

estimated from the Model Code 90 mathematical

expressions:

• for compressive strains (ec) lower or equal to than

ec,limit;

rc ¼ �� E0

rcuec � ec

ecu

� �2

1þ E0

rcu=ecu� 2

� �ec

ecu

rcu ð3Þ

• for compressive strains (ec) higher than ec,limit;

rc ¼�"

1

ec;limit=ecu

u� 2

ec;limit=ecu

� �2

!ec

ecu

� �2

þ 4

ec;limit=ecu

� u

� �ec

ecu

#�1

rcu

ð4ÞThe limit compressive strain (ec,limit) has no

significance other than limiting the applicability of

expression (3). The expressions that define the

parameter u, as well as ec,limit, can be found in

Model Code 90.

In this case, the mean compressive strength (rcu),

the ultimate strain (ecu) and the modulus of elasticity

(E0) had to be previously determined or estimated. The

compressive strength was obtained from the compres-

sive tests on the mortar samples. The modulus of

elasticity was estimated from test data reported in

Fig. 2 Masonry assembly details

Table 1 Properties of the masonry materials (unit and mortar

joints)

Properties Average value Standard test

references

Unit dimensions

(length 9 width 9 height)

350 9 350 9 190 mm EN 772-16

Total volume of holes in unit 26 % EN 772-9

Unit weight 18.5 kg EN 772-13

Unit compressive strength

(gross area)

2.6 N/mm2 EN 772-1

Mortar compressive strength 11.5 N/mm2 EN 1015-11

fl—mortar flexural strength 2.5 N/mm2 EN 1015-11

Mortar dry density 1,300 kg/m3‘ EN 1015-10


other experimental studies carried out on similar

mortars [39], i.e. the experimental results of the

compressive strength and the modulus of elasticity of

several cement/sand based mortars (12 types of

general purpose mortars) were used to obtain a

probable function to calculate E0 (in N/mm2), through

a regression technique:

E0 ¼ 2; 766r0:6305cu ð5Þ

Finally, the default value indicated in Model Code

90 for the ecu was adopted (around of -2 9 10-3 mm/

mm).

5.1.2 Units

To characterize the tensile behaviour of the light-

weight concrete of the units, the direct tensile strength

(rto) and the fracture energy (GF), the expressions (1)

and (2) were also used.

However, the average values of the concrete

properties needed in the two previous expressions

were obtained from laboratory tests performed in the

units, namely from flexural tests on samples extracted

directly from the units webs and shells (giving

fl = 2.0 N/mm2), Fig. 3, and from compressive tests

performed in the complete units (giving E0 and rcu, as

it will be referred to afterwards).

This procedure allowed using more reliable con-

crete samples and, therefore, getting more consistent

data to include in the numerical model, since the use of

larger samples (more in agreement to the standards)

would not be able to represent, in such a realistic way,

the very particular physical rearrangement of the

constituents of the concrete as they exist within the

units. Moreover, the use of test samples from masonry

units takes also into account the influence of the

manufacturing process (vibro-compression moulding)

on the mechanical properties of the concrete. Taking

this into consideration, the uniaxial compressive

behaviour of the concrete, in particular the compres-

sive behaviour before the peak stress (or compressive

strength, rcu), was also measured directly from the

units, namely trough laboratory tests performed in two

loading directions (perpendicular and parallel to the

bed joints), according to the procedures set out in

standard EN 772-1 [14]. Six units were tested, three

for each loading direction. Two displacement trans-

ducers were set on the units to measure the vertical

shortening and the horizontal extension (Dv and Dh)

(Fig. 4).

The results of these tests enabled to determine the

constitutive compression laws of the units in both

loading directions, up to the peak stress. The stresses

were calculated for the net area of the unit’s (surface

perpendicular to the loading direction). The similarity

between the results for the two loading directions

allowed estimating, using a regression technique, a

single polynomial function to represent the units

compressive law on both directions. This function,

that presents a high correlation to the experimental

results (R2 = 0.96), was adopted in the numerical

model to simulate the compressive behaviour law

before the compressive strength, rcu (Fig. 5).

To estimate the behaviour of the concrete after

the peak stress, the mathematical expressions (3) and

(4) were also used. In these expressions the prop-

erties of the concrete obtained from the compressive

behaviour law were inserted (modulus of elasticity,

E0, compressive strength, rcu, and ultimate strain,

ecu). Other experimental data on lightweight concrete

available in the literature were also considered [30],

in particular on the validation of the geometrical

shape of the compressive stress–strain diagrams after

the peak stress (a constant residual stress for high

values of strain is expected for this kind of

concrete).

Notice that the need to characterize the compres-

sive behaviour of both the concrete and the mortar

after the peak stress, i.e. the descending branch of the

diagram for high values of strain, is to prevent

convergence problems during the non-linear numeri-

cal simulations.

Figure 6 represents the constitutive compressive

laws estimated for the mortar and the concrete, and

Table 2 gives the results estimated for the mechanical

characteristics.

It is stressed that the constitutive law of the concrete

used in the units represent a simplified and integrated

way of considering the influence of various factors,

apart from the material itself, on the behaviour and

fracture mechanism of the units. Among those factors,

some are particularly complex and difficult to imple-

ment in standard non-linear analysis, such as the

manufacturing process, already referred to, and the

potential geometrical instability of the webs and shells

within the units.


5.2 Masonry system

Diagonal tension (shear) tests were carried out

according to ASTM E [1]. This test was chosen

because of its versatility on the application to different

types of masonry and of its capability to provide

representative values of the shear mechanical

parameters.

The data obtained through these experiments are

essential to understand the mechanical behaviour of

the masonry system. This campaign complements the

information got from the previous tests and allows

calibrating the masonry as a whole. Three types of

masonry assemblies were built in laboratory condi-

tions, thus:

• 4 samples with filled perpend joints—assembly A

(reference);

• 4 samples with unfilled perpend joints—assembly

B;

• 4 samples built with surface finishing made of a

20 mm thick layer of unreinforced rendering

mortar (same mortar as used in the joints) and

with filled perpend joints—assembly C.

The dimensions of the masonry samples were

adjusted to the laboratory equipment: 800 9 800 9

350 mm3 (length 9 height 9 width).

The vertical shortening (Dv) and the horizontal

extensions (Dh) were measured by four displacement

transducers, two for each side of the samples. A

compression load (F) was applied at a constant speed

(0.1 N/mm2/min) to achieve the maximum strength

in an interval of 15–20 min (Figs. 7, 8). All

measurements were taken up to the maximum load

(Fmax).

The shear stress and strain diagrams (s–c) obtained

from numerical regression techniques applied to the

experimental results of each of the three tested

assemblies are represented in Fig. 9. The coefficients

of determination (R2) ranged between 0.95 and 0.98.

The main shear characteristics determined through

these regression curves are:

• the maximum compression load (Fmax);

• the maximum shear stress (smax);

• the shear strain for the maximum shear stress

(cmax);

• the shear modulus (G).

Fig. 3 Examples of

samples prepared from webs

and shells removed from the

units, and the flexural test

set-up


Fig. 4 Simple compression

tests on masonry units:

a perpend and b parallel to

the bed joints

Fig. 5 Compressive behaviour of the concrete used in the

masonry units (up to rcu)

Fig. 6 Constitutive compressive laws estimated for the mortar

and concrete


These characteristics were calculated by means of

the following expressions defined in the American

standard ASTM E [1]:

s ¼ 0:707F

An

ð6Þ

An ¼ðbþ hÞ

2t � n ð7Þ

G ¼ sc

ð8Þ

c ¼ Dhþ Dv

Lð9Þ

The shear modulus (G) was calculated for strains

corresponding to 1/3 of the maximum shear stress. The

main shear characteristics of the masonry assemblies

are given in Table 3.

In general, the fracture pattern consisted of cracks

opening in a direction more or less parallel to the

applied compression load (F). This pattern was

particularly evident in assembly ‘‘C’’ (with mortar

rendering—Fig. 10) where the main fracture line

follows an almost straight diagonal line from the

loading corners. As for assemblies A and B, although

the fracture pattern tends to follow the same diagonal,

the cracks tag along the joints, near the joint-unit

interfaces, contouring the units (Figs. 11, 12). This

behaviour was more evident in the masonry samples

with unfilled perpend joints (assembly B), which

exhibited a more brittle behaviour when compared to

the other assemblies (Fig. 11).

Local damage in the units near the loading point

was observed in all the assemblies.

The results from this set of tests, together with the

results referred in Sect. 5.1, gave the necessary

information to verify and calibrate the numerical

models, providing the necessary confidence to perform

the other experimental tests on a numerical, but reliable

basis. More detailed information and conclusions

about this experimental work can be found in [35].

6 Numerical simulations

6.1 Implemented model: general description

The numerical simulations were performed using a

micro 3D FE modelling of masonry samples. In

particular, a non-linear constitutive model was used

for the mortar and the concrete, as it will be explained

in 6.3. For simplification purposes, no friction models

were adopted at the unit-joint interfaces (e.g., Mohr–

Coulomb), but rigid links instead (Fig. 13), as it will

be explained in the next section.

Fig. 7 Illustration of a diagonal tension (shear) sample used in

laboratory tests (dimensions in mm)

Table 2 Estimated mechanical characteristics of the concrete and mortar

Material Elasticity modulus,

E0 (N/mm2)

Poisson

coefficient, mCompression

strength, rcu

(N/mm2)

Strain for rcu,

ecu (N/mm2)

Tensile strength

(direct), rt0 (N/mm2)

Fracture energy

(tensile),

GF (Nmm/mm2)

Concrete (units) 5,427 0.16 3.12 0.91 0.63a 0.0085a

Mortar (joints) 12,866a 0.20a 11.45 2.15a 1.57a 0.0182a

a Obtained indirectly through empirical expressions from the literature


6.2 Modelling the unit-joint interfaces

The links between the units and the mortar joints were

considered rigid, i.e. a full transmission of tangential

and vertical displacements was adopted between the two

elements contact interfaces to overcome convergence

problems detected when using friction models (e.g.,

Mohr–Coulomb) during non-linear regimes. Although

being a simplification, this was sustained by observations

and measurements obtained from the laboratory tests.

During the application of loading, no evident fracture

pattern was observed on the interfaces. Only when the

maximum load was reached the fracture pattern was

clearly visible and the maximum displacement measured

parallel to the bed joints was lower than 0,5 mm.

In the case of the perpend joints, it was not considered

any head-to-head contact between units. It’s highlighted

that the perpend joints have gaps of about 3–5 mm (see

Figs. 8a, 12), which are unlikely to be filled by the unit’s

deformability, at least until the maximum load is reached,

as it was observed during the laboratory tests in the

masonry assemblies A (reference sample) and B (sample

with unfilled perpend joints). Therefore, for the perpend

joints no contact or friction models were considered in

assembly B and a perfect local contact was introduced in

the interfaces between the concrete units and the mortar

joints in the case of assemblies A and C (Fig. 14).

6.3 Modelling the masonry materials

The tensile and compressive behaviour of the concrete

and mortar were simulated with a plastic damage

Fig. 8 Diagonal tension test set-up in the laboratory: a A and b C assemblies

Fig. 9 Shear behaviour of the three masonry assemblies tested

in laboratory (regression analysis curves)

Table 3 Shear characteristics of the masonry assemblies

Assembly R2 Fmax

(kN)

smax

(N/mm2)

cmax

(mm/m)

G(N/mm2)

A 0.98 173.8 0.59 0.80 1,504

B 0.95 92.2 0.31 0.41 1,726

C 0.96 182.7 0.61 0.69 1,851


model developed by Lubliner et al. [27], later

improved by Lee and Fenves [23]. A full description

of the development and principles of this constitutive

model can be found in the relevant references.

However, to contextualize and understand the work

carried out in this study, the main concepts used in this

model are briefly described.

This model simulates the non-linear behaviour of

brittle materials (e.g., concrete, mortar, natural rocks)

subjected to monotonic or cyclic loading. For the

concrete and the mortar the main nonlinearities are on

the tensile and compression behaviour curves, which

may induce cracking and crushing failures mecha-

nisms. These nonlinearities are simulated in the model

through uniaxial constitutive laws, usually represented

by stress–strain behaviour curves in tensile (rt–et) or

compression (rc–ec) determined experimentally

(Fig. 15).

The uniaxial tensile constitutive laws use the

concept of fracture energy (GF), proposed by Hille-

borg et al. [20]. The values of fracture energy, tensions

and extensions can be determined experimentally or,

in the case of materials such as concrete or mortar,

estimated from specific literature (e.g. Model Code

Fig. 10 Example of the fracture pattern observed in masonry assembly C (with mortar rendering)

Fig. 11 Example of the fracture pattern in assembly B (unfilled perpend joints)


90). The fracture energy can be provided directly to

the model and, if properly used, it may help solving

convergence problems in the post-peak tensile regime,

in particular when dealing with concrete elements with

little or no reinforcement at all.

This model uses the concepts of classical plastic

theory, in particular strain decomposition, elasticity,

and plastic flow.

(a) Strain variables

Strain is decomposed into elastic and plastic

deformations:

e ¼ eel þ epl ð10Þ

(b) Hardening variables

The failure mechanisms are tensile cracking and

compressive crushing that are characterized by two

hardening variables:

~eplt —tensile equivalent plastic strain

~eplc —compressive equivalent plastic strain

The evolution of the yield (or failure) surface is

controlled by these hardening variables.

(c) Stress–strain relations

The model considers the reduction of material stiff-

ness through a unique and isotropic scalar degradation

variable (d). This scalar variable is governed by the

hardening variables and the Cauchy stresses in the

following generic form:

r ¼ 1� dð ÞDel0 e� epl� �

ð11Þ

1� dð Þ ¼ 1� scdtð Þ 1� stdcð Þ ð12ÞIf no damage variables are introduced, the model

assumes a plastic behaviour. The plastic formulation is

performed in terms of effective stress, calculated in the


Fig. 12 Example of the fracture pattern in assembly A (filledperpend joints)

Fig. 13 Behaviour adopted for the masonry materials and unit-

joint interfaces (vertical cut)

Fig. 14 Contact model used for the perpend joints for

assemblies A and C (horizontal cut)


�r ¼ Del0 e� epl� �

ð13Þ

(d) Plastic flow

The development of the plastic strains is governed by a

potential plastic flow (Gp), established according to

following law:

epl ¼ koGp �rð Þ

o�rð14Þ

The flow potential used is the Drucker–Prager

hyperbolic function:

Gp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 rto tanðwÞð Þ2þ�q

q� �p tanðwÞ ð15Þ

(e) Yield function

The yield surface defined as a function of the

hardening variables and the effective stresses in the

following form:

Y �r; ~epl� �

� 0 ð16Þ

Y �r; ~epl� �

¼ 1

1� ai�q� 3ai�pjbi ~epl

� ��r_

max � ci � �r_

max

� �

� �rc ~eplc

� �ð17Þ

The adimensional parameters are related to the

material properties, thus allowing the calibration of

the constitutive model, and are calculated in the


bi ~epl� �

¼�rc ~epl

c

� �

�rt ~eplt

� � 1� aið Þ � ð1þ aiÞ ð18Þ

ai ¼rb0=rc0 � 1

2rb0=rc0 � 1on the condition 0 � ai� 0:5

ð19Þ

ci ¼3ð1� KcÞ2Kc � 1

on the condition 0:5�Kc� 1

ð20ÞSome studies indicate that this constitutive model is

quite capable of simulating the facture mechanisms in

simple and in reinforced concrete structures [21].

6.4 Implementation of the numerical model

The constitutive model presented in Sect. 6.3 is

implemented in the FE software that was used in this

study. The mechanical behaviour of the masonry was

simulated by micro three-dimensional modelling of

the masonry samples tested in the laboratory (Figs. 16,

17).

Given the complex shape of the masonry units, 4-

node tetrahedral FEs were used to facilitate the

meshing process. The masonry samples were simu-

lated with total restraint of displacements at the

loading corners, therefore with a higher local confine-

ment effect when compared to the physical samples.

The Newton–Raphson iterative base method was

chosen, although the program can automatically select

other methods, depending on the convergence process.

The loading was simulated by imposing, in one of

the supporting corners, a linear smooth variation of

displacements in the direction of the compression load

applied to the samples tested in the laboratory. The

corresponding compression load (F) was determined,

afterwards, through the sum of the nodal reactions.

Fig. 15 Typical stress–strain diagrams for concrete (uniaxial

constitutive laws): a tensile and b compressive curves


The results of the calculations, namely the com-

pression load and the shear strain diagrams (F–c),

were used to determine the mechanical characteristics

of the masonry through the expressions 1–4.

7 Calibration and validation of the numerical

model

7.1 General aspects

The calibration and validation of the numerical model

involved the simulation of the experimental tests

performed on the units and on the masonry (referred in

Sect. 5) to verify the capability of the model to

properly represent the masonry system tested in

laboratory conditions. This process involves the

comparison between the test results determined

experimentally in the laboratory and the results

obtained in the numerical simulations, in particular

the comparison between the response curves and the

mechanical properties of masonry and units. More-

over, this procedure was split in two phases. The first

consisted on simulating the compression tests per-

formed in laboratory on the concrete units, and the

second the diagonal shear tests on the masonry

assemblies. These simulations allowed verifying, first

the units behaviour, i.e. the concrete behaviour law

alone when integrated in the units, and then the

masonry system as an assembly of units, joints and

interfaces.

7.2 Calibration of the units’ constitutive

model—phase 1

The calibration of the units compressive behaviour

was made by simulating the compressive tests per-

formed in laboratory. A micro 3D FE modelling of a

unit was constructed and displacements were imposed

to simulate the loading. The other possible displace-

ments of the loading surfaces nodes were restrained;

thus a higher confinement of the unit was expected in

the numerical simulations. The verification and cali-

bration was then conducted with a sensitivity analysis

performed through the variation of some of the

parameters of the constitutive model. The parameters

related to the material properties that were obtained

directly from the experimental tests, or indirectly by

using experimental values from the experimental

campaign and the expressions from the literature,

namely: the elastic modulus (E0), the Poisson’s ratio

(m), the uniaxial tensile and compressive strength (rto

and rcu), the ultimate compressive strain (ecu) and the

fracture energy (GF), were set constant and equal to the

values established in Table 2. As for the other model

Fig. 16 Geometric model used for masonry test sample with

and without filled perpend joints (assemblies A and B)

Fig. 17 Geometric model used for masonry test sample with

rendering mortar (assembly C)


parameters, a range of values was selected according

to following limits:

• Dw ¼ 15�; 45�½ �;• D 2¼ 0:1; 1½ �;• DKc ¼ 2=3; 1½ �;• Drb0=rc0 ¼ 1; 1:3½ �:

During the simulations, only one parameter was

changed at a time, keeping constant the others

parameters for a set of predefined values.

Through direct comparison to the experimental

results, it was obtained the best possible approxima-

tion between the numerical and the experimental

compressive stress and strain diagrams. In this case,

the best fit gave a small difference, between 2 and 3 %,

for ecu and rcu (Fig. 18). Moreover, it was found that

Kc and w were the most influential variable parame-

ters. Table 4 presents the values adopted for the model

parameters after the calibration of the units.

7.3 Validation of the masonry

behaviour—phase 2

After calibrating the constitutive model of the units, it

were performed the numerical simulations of the shear

diagonal tests done in the laboratory. The three types

of masonry assemblies (A–C) were simulated, sepa-

rately, using the model in Sect. 6.1 and the material

properties listed in Tables 2 and 4.

This validation was done by comparing the exper-

imental to the numerical results for each type of

masonry assembly, in particular the compression load

and the shear versus strain diagrams (F–c), and the

masonry mechanical characteristics related to the

shear behaviour (Fmax, c and G).

The first calculations gave good results when

compared to the experimental results, in particular a

good estimation of Fmax and cmax, with variations

below 9 %. The major difference was found in the

shear modulus with variations between 10 and 23 %,

especially in the samples with unfilled joints (assem-

bly B). In spite of this, it was decided not to make any

changes on the model parameters, since the masonry

assemblies B were also those that presented the

highest dispersion in the experimental results

(R2 = 0.95) and, thus, the numerical results were

considered to be within an acceptable range of values.

The experimental and numerical diagrams (F–c)

comparing the responses for each assembly type are

presented in Fig. 19. The mechanical characteristics

and the comparative ratios obtained are shown in

Table 5. These ratios have the experimental results as

common denominator.

A detailed analysis of the stresses and the defor-

mations, in particular of the principal stresses distri-

bution, was undertaken to identify potential fracture/

cracking patterns and, therefore, to further validate the

Fig. 18 Comparison between the experimental and the numer-

ical response curves after calibration (concrete units in simple

compression)

Table 4 Variable model parameters adopted for the concrete

units after calibration (best fit)

w (�) [ rbo/rco Kc

45 0.10 1.16 1.00

Fig. 19 Comparative diagrams of the experimental and numer-

ical results obtained for the three masonry assemblies after

calibrating the units behaviour laws


numerical model. It is assumed that the fracture

patterns, or crack directions coincide with the align-

ments of the minimum principal stresses, which can

then be compared to the patterns observed in the

laboratory tests.

The minimum principal stresses distribution on the

assemblies and the corresponding global deformation

(magnified about 250 times) are represented in

Figs. 20, 21 and 22 for the maximum compression

load (breaking point).

The main aspects to be noted in the results of the

numerical tests on the masonry assemblies that are

consistent to the experimental results from the labo-

ratory tests are:

• the compressive stresses are concentrated in an

area more or less coincident with the direction of

the compression load (except for assembly C, the

higher stress concentration areas define a ‘‘zig-

zag’’ line around the mortar joints);

• there is a higher concentration of stresses along the

bed joints on areas closer to the perpend joints,

especially when these are unfilled (assembly B);

• there is a high concentration of stresses and

deformation in the units located near the load

application points (corners of the masonry

samples).

8 Masonry tests using numerical analysis

8.1 General aspects

Starting from the calibrated and validated model, it is

now possible to perform other tests on the same

Table 5 Mechanical proprieties and ratios obtained after the calibration process—numerical (num.) versus experimental (exp.)

Assembly Mechanical properties Ratios

Fmax num–exp (kN) cnum–exp (mm/m)

Gnum–exp (N/mm2)

Fnum/Fexp cnum/cexp Gnum/Gexp

A 172.2–173.8 0.75–0.80 1,354–1,504 0.99 0.94 0.90

B 96.4–92.2 0.37–0.41 1,330–1,726 1.05 0.91 0.77

C 189.3–182.7 0.68–0.69 1,607–1,851 1.04 0.99 0.87

Fig. 20 Minimum

principal stresses and global

deformation at breaking

point—assembly A (filled

perpend joints)


masonry assemblies, or on other samples using the

same masonry, but different dimensions or shapes,

with low time consuming and financial cost. In this

particular case, the same masonry assemblies were

tested taking into account the influence of the

mechanical properties of the materials and of the

geometrical shape of the joints on the mechanical

behaviour of the masonry system. The results were

Fig. 21 Minimum


deformations at breaking

point—assembly B (unfilled

perpend joints)

Fig. 22 Minimum


deformation at breaking

point—assembly C

(rendering mortar and filled

perpend joints)


compared to the reference assembly (A) properties

listed in Table 5. The parameters involved in this

sensitivity analysis were:

• the compressive strength of the mortar joints; it

was adopted the same initial thickness (10 mm),

but different compressive strengths (0.5 9 rcu and

2.0 9 rcu);

• the compressive strength of the rendering mortar

used in assembly C; it was still considered

unreinforced mortar with the same thickness

(20 mm), but with a higher compressive strength

(2.0 9 rcu);

• the thickness of the parallel joints (e); it was

adopted a mortar with the characteristics of the

reference situation (assembly A), but with differ-

ent thicknesses (0.5 9 e and 2.0 9 e);

• the geometrical configuration of the parallel joints;

it was adopted a mortar with the characteristics of

the reference situation (assembly A), but with

discontinuous joints (g/t = 0.685, where ‘‘g’’ is

the full width of the mortar strips and ‘‘t’’ the total

thickness of the wall) were replaced by full joints

(g/t = 1.0).

In order to consistently consider the variation of the

mortar compressive strength (rcu) on the global

material characteristics, the mechanical properties

(E0 and GF) and constitutive laws of the mortar

considered in the simulations with half or twice the

mortar compressive strength (0.5 9 rcu and 2 9 rcu)

were estimated as described in Sect. 5.1 (results are

presented in Table 6). In the case of the uniaxial

tensile strength (rt0), the value was estimated through

Table 6 Mechanical properties estimated for the mortar used

in the sensitivity analysis

Ref. E0 (N/

mm2)

m rcu (N/

mm2)

rt0 (N/

mm2)

GF (Nmm/

mm2)

0.5 9 rcu 8,311 0.2 5.73 0.97 0.011

2 9 rcu 19,917 0.2 22.9 2.43 0.029

Table 7 Mechanical properties and ratios for the masonry determined in the sensitivity analysis

Parameters Fmax (kN) c (mm/m) G (N/mm2) Fvar/Fref cvar/cref Gvar/Gref

Ref.

Mortar joints (rcu = 11.45 N/mm2)

Without rendering 172.2 0.75 1,354 – – –

g/t = 0.69

e = 10 mm

Var.

Mortar joints (0.5 9 rcu) 169.4 0.79 1,315 0.98 1.06 0.97

Mortar joints (2.0 9 rcu) 175.8 0.74 1,364 1.02 0.99 1.01

Rendering (rcu) 189.3 0.68 1,607 1.10 0.91 1.19

Rendering (2.0 9 rcu) 216.3 0.56 1,893 1.26 0.74 1.40

g/t = 1 194.8 0.74 1,513 1.13 0.98 1.12

e = 0.5 9 e 167.7 0.72 1,332 0.97 0.96 0.98

e = 2.0 9 e 177.5 0.78 1,361 1.03 1.03 1.01

Fig. 23 Behaviour of the masonry samples with joints made of

mortars with different mechanical strength


the compressive strength (0.5 9 rcu and 2 9 rcu) and

using the Model Code 90 expression:

rto ¼ 1:4rcu

10

� �2=3

ð21Þ

In the analysis, only one parameter was changed at

a time, being the other parameters constant and equal

to those of the reference sample (assembly A).

8.2 Results

The mechanical behaviour of the new masonry

samples with updated parameters (var.) was compared

to that of the masonry assembly A (ref.) through the

final diagrams and the ratios between the mechanical

properties determined in both numerical simulations

(var./ref.), namely the maximum compression load

(Fmax) and the corresponding shear strain (cmax) and

shear modulus (G) (Table 7). Figures 23, 24, 25 and

26 show the set of final diagrams (F–c) that correspond

to the considered variations in relation to the reference

sample.

The results show that the use of higher strength

rendering mortar and continuous parallel joints are the

factors that mostly improve the shear behaviour of

masonry (in plan), namely its strength and stiffness.

These findings seem consistent with known experi-

mental studies and code specifications. Oliveira and

Hanai [31] reported an increase of about 55 % in the

shear strength (diagonal compression) of the masonry

when using unreinforced mortar (with compressive

strength of 4 N/mm2) and an increase of about 30 %

when using higher compressive strength mortar (23 N/

mm2). Moreover, although reinforced mortar was used

in the experiments of Hamid et al. [18] and Drysdale

et al. [11], these authors refer an overall increase of the

shear strength (diagonal compression) of about 200 %

and of the stiffness of about 75 % in clay masonry

samples when using rendering mortar with higher

strength capability.

In the case of continuous joints, the increase of the

shear capacity of masonry is explained by the increase

of the bond area between the units and the mortar

Fig. 24 Behaviour of the masonry samples with rendering

mortars with different mechanical strength

Fig. 25 Behaviour of the masonry samples with continuous

(g/t = 1) and discontinuous parallel mortar joints (g/t = 0.69)

Fig. 26 Behaviour of the masonry samples with joints of

different thicknesses (e)


joints. This increase in strength is evident in the

formulae of Eurocode 6 [13], where the shear strength

of the masonry depends on the bond strength (desig-

nated as the initial shear strength) and on the

coefficient g/t.

On the other hand, increasing the thickness of the

parallel joints and (or) the mechanical strength of the

mortar joints have no significant influence on the shear

behaviour of this masonry system. No experimental

studies about these aspects in the shear behaviour were

accessed or found yet by the authors in order to better

sustain these numerical findings.

9 Conclusions

The present paper describes the study of the shear

behaviour of a masonry system by using an approach

that involves the merging of experimental and numer-

ical results. In particular, it describes the steps

followed in the analysis of the shear behaviour of a

new masonry system made of lightweight concrete

units that were recently developed in Portugal. The

shear behaviour was studied by loading masonry test

assemblies in compression along their diagonal, as

defined in the American standard test ASTM E [1]. A

limited number of experimental tests were performed

to produce experimental data to verify and calibrate a

numerical model that was used to simulate a series of

new conditions and possibilities for the new masonry

system.

The following conclusions can be drawn from this

study:

• It was possible to characterize the mechanical

properties of the masonry using, not only exper-

imental results from standard tests on the masonry

system, but also simplified tests and other data

from technical literature, standards and codes

available for the materials;

• Despite the simplifications made to model the

interfaces between units and joints, which were

considered rigid, the adopted numerical model

characterized fairly well the behaviour of the

masonry system in shear (through diagonal com-

pression) and consistently represented the main

aspects of the failure mechanism observed in the

laboratory tests, at least until the maximum load

(breaking point);

• However, the simplification made on modelling

the interfaces may result in important deviations to

the experimental results in the post-peak shear

behaviour of the masonry;

• This last aspect can be clarified later by performing

laboratory tests after the breaking point and by

using friction models in the joints, although it is

likely that the use of these models can significantly

increase the time and computational resources

required for numerical convergence.

The results of this study show that is possible to

accurately characterize the shear behaviour of

masonry through computer simulations of laboratory

tests, in particular the American standard test defined

in ASTM E [1], by using simple models that can be

calibrated using a small number of laboratory tests,

either on the masonry units and materials, or on

masonry samples. This practical study has shown the

potential of using computer simulations of laboratory

tests, reducing the need for the involvement of

important laboratory resources and costs in the

research and development of masonry systems.

Acknowledgments The authors gratefully acknowledged

ADI-Portuguese Innovation Agency and the Company Maxit-

Portugal for the help provided in the OTMAPS research project.

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Diagonal compressive strength of masonry samples—experimental and numerical approach

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