15 Materials and Structures Chimeneas

ORIGINAL ARTICLE

A comparison of different failure criteria in a numericalseismic assessment of an industrial brickwork chimney

F. J. Pallares Æ A. Aguero Æ S. Ivorra

Received: 19 January 2007 / Accepted: 14 April 2008 / Published online: 24 April 2008

� RILEM 2008

Abstract A theoretical analysis using three well-

known masonry analysis constitutive models is

performed on a masonry structure to simulate the

response of the structure to specific seismic forces.

The results of the three numerical approaches are

compared and a discussion is presented, mainly

intended for professionals, concerning the suitability

of the three models and the limitations of each

numerical approach. The aim of the study is to

evaluate the relative accuracy of the three different

models and their suitability for determining the

failure mode of the masonry chimney. The models

studied are: a linear elastic constitutive model, an

elastic-plastic Drucker-Prager’s type model and a

model including cracking and/or crushing in the

material using Willam-Warnke’s criterion. A macro-

modelling approach is used because of the great

number of elements forming the structure and the

computational demand. Seismic actions are

synthetically generated and scaled until chimney

failure, in accordance with the present regulations on

seismic-proof constructions in Europe and Spain.

Conclusions for researchers and professionals are

obtained to determine the suitability of each model

according to the results required and the available

calculation capacity.

Keywords Accelerogram � Cracking �Failure criteria � Masonry chimney �Seismic analysis

1 Introduction

Industrial brickwork chimneys built during the indus-

trial revolution towards the end of the 19th and early

20th century are common in many European countries.

They were built to get rid of smoke and to create the

necessary draught for industrial processes as, e.g., in

textile or paper manufacturing. Very few of them

remain in use, since they became obsolete when new

energy generation systems made their appearance in

the 20th century. In many cities these chimneys form a

characteristic landscape and are often protected by law

as part of the cultural heritage. Figure 1 shows one of

the many industrial chimneys that can be observed in

the city of Valencia (Spain).

In the existing scientific literature, there are few

references to this type of construction. In [1] the

F. J. Pallares (&)

Department of Applied Physics, Polytechnic University of

Valencia, c/Camino de Vera s/n, Valencia 46022, Spain

e-mail: [email protected]

A. Aguero

Department of Continuous Medium and Theory of

Structures, Polytechnic University of Valencia, c/Camino

de Vera s/n, Valencia 46022, Spain

S. Ivorra

Department of Construction Engineering, University of

Alicante, Apartado 99, Alicante 03090, Spain

Materials and Structures (2009) 42:213–226

DOI 10.1617/s11527-008-9379-5

authors analyse the typology and structure of indus-

trial chimneys built between 1870 and the first

decades of the 20th century in the Italian regions of

Piedemont and Veneto, and the problems associated

with their restoration.

In [2] the authors study the behaviour of three

typical chimneys in these areas using the finite

element method with a linear analysis, taking into

account the chimney self-weight, wind, temperature

differences and earthquakes as acting loads. Pallares

et al. [3] considers the behaviour of a chimney when

a seismic action is introduced, while Aoki and Sabia

[4] studies the structural characterization of a brick

chimney using experimental tests and model

updating.

This paper continues the investigation of this type

of construction. The aim is to evaluate the relative

accuracy of the different models and their suitability

for determining failure modes and crack patterns in a

seismic analysis. The different models are based on

the use of three different masonry failure criteria

usually proposed in the literature, and the study

focuses on the comparison of the criteria. The study

of the seismic vulnerability of these chimneys and

similar masonry structures nowadays involves a

considerable number of professionals, many of whom

apply simplified analysis, equivalent static analyses

or linear elastic dynamic analyses, since the use of

more refined models is normally reserved for research

purposes due to their complexity. However, the

currently available commercial software with com-

monly used masonry failure criteria for both

professionals and researchers can lead to more

realistic approaches.

The results presented here are obtained within the

framework of a theoretical study and show good

agreement with real failure modes observed in actual

earthquakes. The numerical failure modes obtained in

[5], which use the same criteria presented here, are in

accordance with the real failure modes observed

in actual earthquakes in [6], collapsing the chimneys

at the base or in the top third of the structure, mainly

depending on the height of the chimney. The practical

modelling guidelines and conclusions outlined here

may be of use to both professionals and those

involved in research.

2 Description

In Europe and America many of these chimneys were

built in seismic areas, so it is important to know their

behaviour in case of a low to moderate intensity

earthquake and also what criteria can lead to accurate

results. This study focuses on the Spanish Mediter-

ranean area using the European code [7] and the

Spanish standard [8].

To achieve the aforementioned goals, the finite

element method (e.g. [9]) is used as a numerical tool to

study the response of the structure when subjected to

seismic tremors. There are many examples in which

masonry has been studied through this numerical

method in recent years. Advanced models for small

structures have been used to micro-model both bricks

and mortar, involving considerable computational

cost, unaffordable for large structures. Examples of

this micromodelling approach, usually employed to

compare with experimental results obtained in the

laboratory, can be found in [10] or [11].

Large structures are analysed through macromod-

elling approaches in which average stress states are

considered in the material. Examples of this kind of

modelling technique can be found in [12, 13].

Fig. 1 Industrial masonry chimney

214 Materials and Structures (2009) 42:213–226

However, few works have been published relating

to the specific dynamic behaviour of masonry in

earthquakes, although researchers have recently

started focusing on this field, [14].

Different models can provide a fair approximation

of the response of the structure to earth tremors,

permitting the investigation of various aspects,

according to the objective of the study, and this is

one of the points of this paper.

In the present study, numerical analyses were

carried out in order to: (a) compare the three different

criteria usually employed when performing dynamic

analyses on masonry structures, (b) determine the

important aspects to be taken into account in seismic

analysis, (c) show the suitability of each criterion

according to the results required and the calculation

capacity available.

The criteria studied are commonly used in calcu-

lations for geo-materials such as masonry. These

criteria, used to separate the linear from the non-

linear behaviour of the material, are the Drucker-

Prager criterion and the Willam-Warncke criterion.

The third analysis carried out is a linear analysis

without any plastic or failure criterion. The results of

this elastic analysis were compared with the results

from the non-linear analyses and to evaluate the

quality of the selected mesh and the results obtained.

Examples and justification for the use of these models

is provided below in each case by references.

None of the models provides the stiffness degra-

dation of masonry caused by successive plastic

deformations resulting from cyclic behaviour [15,

16], apart from consideration of the cracking/plastic

process in masonry due to a seismic action and its

progression in time.

The study is set out as follows:

(1) Description of the problem, establishing the

geometric definition of the structure, specifying

height, base section, thickness, etc.

(2) Discretization of the continuum through the

finite element method in order to reduce the

degrees of freedom to a discrete number,

assigning macromodel properties. Boundary

conditions and initial values are then fixed to

set up the model for the seismic action.

(3) Generation of the artificial accelerograms to be

used in the calculations from the response

spectrum proposed in the standards.

(4) Setting up yield criteria: with the model and the

seismic action defined, a yield/crack criterion is

needed to register the inelastic behaviour of

masonry during the loading process. Three

different criteria are studied, resulting in three

analyses: linear elastic analysis, elastic-plastic

analysis (Drucker-Prager) and a linear elastic

analysis until cracking or crushing occurs in

masonry (Willam-Warncke).

(5) Calculation and results of the analyses.

(6) Conclusions derived from the analyses.

3 Properties of the chimney

The structure studied in the present work is an

industrial chimney made of masonry. One particular

chimney was chosen whose dimensions and circular

section are representative of the great number of these

chimneys in Mediterranean coastal areas and Europe.

It is 30 m high without reinforcement, as can be seen in

Fig. 2a. The dimensions shown were established by

rules of thumb taken from different handbooks such as

[17] or [18] and compared with the actual dimensions

measured from other 30-m chimneys. Variability in

dimensions is not a parameter studied here.

The structure can be clearly divided into three

parts, [19]:

(1) Base: Its task is to distribute stresses on the

foundation. It is not always necessary and

sometimes does not exist. Shapes can vary

widely: square, hexahedron, octagonal, etc.

(2) Shaft: This is the chimney strictly speaking. Its

tasks were to lead the smoke to a great height to

avoid environmental contamination and to cre-

ate the necessary draught to facilitate

combustion. It is formed by a tube with varied

cross-sections: circular, square, hexagon, and

octagon. It is variable in height with prismatic

or helicoidal shapes.

(3) Crown: The upper part of the chimney with an

aesthetic purpose.

As previously stated, the discretization of the

continuum is made through 3D solid elements with

plastic and cracking capabilities, depending on the

failure criteria used. Eight node bricks with three

degrees of freedom per node are used.

Materials and Structures (2009) 42:213–226 215

Figure 2b shows three tested meshes of the finite

element discretization used for the chimney. The

central mesh was chosen for the calculations, since

accurate enough results were obtained from it com-

pared with the finer mesh, having a total of 20,320

nodes with 60,800 active d.o.f. and 15,280 elements.

4 Seismic action

Since time-dependent transient responses were

looked for, accelerograms were used as input for

the seismic action. In this way a comparison could be

made between the models of the time (instant of the

seismic motion) in which the initial crack appeared

and how the cracking pattern spread. Each accelero-

gram was synthetically generated [20] compatible

with the seismic spectrum proposed in the current

European code [7] and in accordance with the Spanish

norms for the Mediterranean coast (Valencia), [8].

A total of five accelerograms were used to compare the

different criteria.

The initial synthetic accelerograms generated from

the standards were scaled to produce cracks

(a) (b) Fig. 2 (a) Longitudinal

section of the industrial

brickwork chimney studied

in the present work.

(b) Discretization using 3D

solid elements. Isometric

view. Tested meshes


(or plastic deformation) and the collapse of the

structure in order to study the failure process. The

peak associated ground acceleration was 0.06 g.

Figure 3a shows the response spectrum for the

city of Valencia from the Spanish norms, while

Fig. 3b shows one of the artificial accelerograms

generated.

5 Calculation values

As stated above, a macro-structural approach was

adopted in order to model the whole structure, due

to the great number of elements forming the

chimney. More detailed models would become

extremely demanding from a computational point

of view, and the heterogeneities and uncertainties

usually present in masonry structures make it

advisable to use models with a small number of

mechanical parameters.

The industrial chimneys were built of masonry due

to its natural properties, which were suited to the use

for which the chimneys were conceived: it was easy

to handle and had good thermal and mechanical

properties.

The number of unavoidable uncertainties is some-

times high when professionals carry out structural

assessments, and discrepancies between calculated

and experimental values may occur. In the present

paper, rough but representative values for the

masonry mechanical parameters were used to per-

form the calculations.

For the masonry, the values used in the calcula-

tions were:

Fig. 3 (a) Response spectrum. (b) Artificial accelerogram generated; time (s) versus acceleration (a/g)

Uniaxial compressive strength: fc ¼ 6,500,000 N/m2

Uniaxial tensile strength: ft ¼ 200,000 N/m2

Elasticity modulus: E ¼ 6e8 N/m2

Poisson ratio: m ¼ 0:2

q ¼ 1,600 kg=m3

ð1Þ

Due to the lack of laboratory data, these values

were chosen from references such as [17, 21] or [22].

These provide valuable information on the strengths

of masonry used for chimneys and are typical values

for masonry structures from the end of the 19th until

the middle of the 20th century. The low tensile

strength was chosen in order to consider the lime

mortar used in the first chimneys built in the 19th

century, being a typical mortar used prior to 1950, or

chimneys constructed with cement mortar whose

mechanical properties are deteriorated due to climatic

conditions. This is a key parameter in the seismic

behaviour of chimneys and must be estimated

carefully in each case.

6 Cracking and plastic criteria

The plastic and cracking criteria used in the calcu-

lations presented here are the well-established

Drucker-Prager and Willam-Warncke criteria applied

in geo-materials, adapted to the case under study

(masonry).

It has been said that none of the models used

provides or takes into account either the stiffness

degradation in masonry caused by successive plastic

deformations or cracks resulting from low cycle

fatigue, or material fatigue. However, the purpose of

this paper is to compare both frequently used criteria


in masonry and to understand the cracking progres-

sion in the material, since cracking is the main

phenomenon that governs the response and loads

applied to the chimney do not lead to crushing or

plastic deformation due to excessive compression, as

will be shown in the results presented. Nevertheless, a

reduced elasticity modulus is considered to take into

account the degradation process [23].

Both the Drucker-Prager and Willam-Warncke

criteria are frequently used in masonry structures to

determine the end of the elastic behaviour and the

beginning of the non-linear range. The Drucker-Prager

model has been employed in a brittle material such as

masonry in different situations to simulate initial

cracks at the onset of plastic deformation [13]. Addessi

et al. [24] combines both damage and a Drucker-

Prager plastic criterion to reproduce the behaviour of

masonry to cyclic loading, while Cerioni et al. [25]

applies Drucker-Prager criterion to the case of Parma

Cathedral Bell-Tower, subjected to the El Centro

earthquake comparing different finite elements. The

Willam-Warncke criterion showed good results with

masonry in [26], obtaining a 2D failure surface quite

similar to that presented in [27]. Litewka and Szojda

[28] uses a Willam-Warnke limit surface at material

failure for brittle material, considering brick masonry

in the numerical model proposed, with good agreement

to experimental results. The WW criterion is consid-

ered in this paper as the reference criterion because its

use in seismic analysis of masonry chimneys [5] gives

results that agree with actually observed failure modes,

as presented in [6], and it makes suitable allowance for

the cracking phenomenon. This is why it will also be

considered as the reference criterion, theoretically

estimating the appearance of the first crack, although

no experimental results can be provided to support this

piece of information.

Their capacity for reproducing seismic behaviour

in masonry chimneys will be shown in the next

section. These criteria can be stated as:

• Drucker-Prager criterion [29]

This plastic criterion, included in many non-

linear software packages, states the beginning of

plastic strains if:

b � I1 þffiffiffiffiffi

J2

p� ry ¼ 0 ð2Þ

where b, parameter related to the internal friction

angle; I1, first stress invariant; ry, parameter

related to cohesion and J2, deviatoric stress

invariant.

Both parameters are fitted in order to obtain the ftand fc strengths fixed in (1).

• Willam-Warncke criterion [30]

This cracking criterion states initial crack if an

equation such as (3) is satisfied:

f ðrm; sm; hÞ ¼ffiffiffi

3p sm

qðrm; smÞ� 1 ¼ 0 ð3Þ

where

h is the angle of similarity given by: cos h ¼ 2r1�r2�r3

2ffiffi

3pffiffiffi

J2

p ;J2 second invariant of stress deviator tensor; r1; r2; r3

are the principal stresses; rm is the mean normal stress:

rm ¼ r1þr2þr3

3; qc is the deviatoric length for h = 60�,

and qt is the deviatoric length for h = 0�; sm is the

mean shear strength: sm ¼ffiffiffiffiffiffiffi

25

J2

q

:

More information about these criteria can be found

in [31].

Parameters to define the failure surface are those

given in (1).

7 Results

The calculations were carried out in a Pentium-IV

computer, 2.8 GHz processor, 1 GB RAM and about

50 h were needed to complete one seismic motion.

Although several earthquakes were simulated, the

following figures present different instants for only

one of the earthquake motions tested, in order to

compare the criteria appropriately. In the figures,

qðrm; hÞ ¼2qcðq2

c � q2t Þ cos hþ qcð2qt � qcÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4ðq2c � q2

t Þ cos2 hþ 5q2t � 4qtqc

p

4ðq2c � q2

t Þ cos2 hþ ðqc � 2qtÞ2


stresses, plastic deformations and cracks are plotted

according to each criterion. Results for the three

criteria are presented below:

7.1 Linear elastic analysis

This elastic model has many weaknesses, but it is

widely used in practical masonry analysis and

conclusions can be obtained. Results for this linear

elastic model with no cracking or plastic criteria are

in agreement with those obtained from the non-linear

analyses until the non-linearities are involved in the

chimney response, which occurs when the equation

for each non-linear criterion is satisfied.

The high level of tensile stresses in the bottom and

middle sections of the chimney suggests the need to

use non-linear models to detect possible cracks, as the

tensile strength is exceeded (see Fig. 4a). The

masonry compressive strength seems not to have a

significant effect on the response of the chimney due

to the low gravity stresses experienced. Furthermore,

vertical stresses under self-weight are used to check

that the weight of the model has been properly

introduced, comparing it with simplified calculations

(Fig. 4b). It is confirmed that the chimney remains in

the elastic range as stresses are below the elastic

limit, as expected.

In order to evaluate the quality of the mesh

adopted in the subsequent calculations, three meshes

were tested in this linear elastic model, as can be

observed in Fig. 2b. Results were compared for the

three meshes, to determine the mesh that provided the

best results with the lowest computational effort. No

significant differences could be obtained using more

refined meshes.

The model developed for this elastic study will be

used in further work, currently under development, to

obtain numerical modal frequencies and vibration

modes to compare with those obtained from experi-

mental studies, validating or updating the parameters

of the numerical model.

7.2 Drucker-Prager criterion (D-P)

Figure 5a, b show longitudinal stresses and plastic

strains given by the Drucker-Prager criterion at

t = 5.34 s from the beginning of the seismic

action, this being the time in which plastic

deformation is initiated, while Fig. 6a and b show,

in the same way, at instant t = 5.62 s, longitudinal

stresses and plastic strains when the plastic strains

have progressed and extended to both sides of the

base of the chimney.

Time t = 5.70 s is the instant when plastic strains

reach the shaft, seen in Fig. 7a, where plastifications

at the base and middle sections are shown, while in

Fig. 7b, at t = 5.72 s, plastic strains and stresses are

displayed, the latter remaining constant when the

plastic criterion is satisfied according to the perfect

plasticity criterion used.

Figures 8a, b; 9a, b; 10a, b show the growth of the

plastic strains in the shaft in the right border at

t = 6 s and the final state of the plastic areas when

the seismic action has finished.

Finally, Fig. 11 displays the evolution of plastic

strains with time belonging to the two edge nodes on

both sides of the chimney base as an example of the

different results that can be obtained from the

analysis.Fig. 4 (a) Stresses higher than tensile strength at the bottom.

(b) Vertical stresses under self-weight


7.3 Willam-Warncke criterion (W-W)

In order to compare both criteria appropriately, results

are displayed showing the time of the appearance of the

first crack at different points of the chimney and the

final crack pattern for the same seismic action.

Figure 12 shows the instant of the appearance of

the first crack at the base of the chimney. This instant

Fig. 5 t = 5.34 s.

(a) Longitudinal stresses

(vertical axis) throughout

the chimney. Isometric

view, (N/m2); (b) Plastic

strains at chimney base.

Isometric view (zoom).

Maxima at the base of the

chimney

Fig. 6 t = 5.62 s.

(a) Longitudinal stresses

(vertical axis) throughout

the chimney. Frontal view,

(N/m2); (b) Plastic strains

on both sides of chimney

base. Frontal view (zoom)


coincides with the moment at which the beginning of

plastic strains using the D-P criterion is achieved, as

can be checked.

Figure 13 shows stresses for the instant at which

cracks appear in the left edge of the chimney base and

how cracking progresses in depth and extends to both

sides of the base due to the seismic action. Here, as

before, the agreement with the previous criterion is

clear.

Finally, Fig. 14 presents the instant t = 5.72 s at

which cracks appear in the shaft of the chimney.

Once more, agreement can be found with regard to

the D-P criterion. However, an important item is

shown in Fig. 15, where displacements at the base of

the chimney (fitting with the seismic action imposed

at the base) and amplifications at the crown can be

observed until t = 5.94 s, when the chimney fails.

This failure is not recorded using D-P criterion.

The results displayed for both criteria have been

presented for one synthetically generated seismic

action, but the study was performed for different

seismic actions, for which the conclusions are

outlined in the following section.

8 Conclusions

Several seismic movements were used to study the

seismic behaviour of an industrial masonry chimney in

order to compare widely used masonry failure criteria.

The results of the three numerical approaches were

compared and a discussion presented, mainly intended

for use by practicing engineers, concerning the

suitability and limitations of each numerical approach,

evaluating the seismic response and failure mode in the

type of structure shown. The main characteristics

compared between the criteria, usually required in

seismic analysis, were: appearance of the first crack in

the different parts of the chimney and failure mode.

Fig. 7 (a) t = 5.70 s. Plastic strains along the right chimney border. Initial plastic strains at a height of 18 m. (b) Longitudinal

stresses (vertical axis) along the right chimney border, t = 5.72 s

Fig. 8 t = 6 s. (a) Longitudinal stresses (vertical axis)

throughout the chimney. Isometric view, (N/m2). Maximum

located in the shaft of the chimney. (b) Plastic strains in the

shaft. Isometric view


As a result of the study, the following conclusions

are drawn and comments are made for each criterion,

with an outline of the general conclusions and a

summary of the whole work:

8.1 Linear elastic model

Elastic linear analysis does not capture the failure

mechanism but indicates qualitatively possible failure

modes and cracking areas, fully resisting the seismic

motion undergone by the chimney. It provides insight

into the response of the structure as a first approx-

imation, and is the fastest and easiest analysis to

estimate the appearance of cracks and will indicate

the need for a more refined (non-linear) analysis.

8.2 Drucker-Prager criterion

This criterion is capable of representing the appear-

ance of the first cracks, producing similar predictions

of initial cracks to the W-W criterion, as has been

shown. Following the first cracks, however, failure

mode and subsequent cracks can be wrongly pre-

dicted by the D-P criterion (as compared to W-W

criterion), since tensile strength still remains after

D-P surface is reached, leading to cases (crack

Fig. 9 (a) Growth of the plastic strains throughout the chimney, t = 6 s (right border). (b) Enlargement of the plastic strains area at

the base, t = 12 s. Isometric view

Fig. 10 Final state of plastic strains on the left border (a) and the right border (b) of the chimney


patterns and failure modes) not in accordance with

W-W predictions. The residual tensile strength

remaining in the plastic areas can lead to a chimney

fully withstanding some earthquakes when W-W

predicts collapse.

Taking this into account, when as a consequence of a

seismic motion a part of the chimney plastifies (cracks),

e.g. the bottom left part of the base, if the next cracks (or

tensile demanding area) are on the opposite side or in a

different part of the chimney (e.g., shaft), so that tensile

strength is not being demanded from a previous plastic

area, these cracks will be in accordance with the W-W

criterion. However, if the next cracks involve previous

plastic areas through tensile stresses, the subsequent

cracks and failure mode could be erroneous.

In general, this criterion consumes less computer

time than the Willam-Warncke criterion and can give

a good understanding of the seismic behavior, taking

into account the limitations commented.

To sum up, the use of plastic criteria such as the

DP criterion employed here, and used in many

software packages, provides suitable approximations

of the stress–strain fields until the appearance of theFig. 11 Final state of plastic strains on the left edge (a) and

the right edge (b) of the base

Fig. 12 (a) Longitudinal

stresses (vertical axis)

throughout the chimney.

Isometric view, (N/m2);

(b, c, d) Crack pattern.

Frontal view (b-global,

c-base zoom) and oblique

view (d-base zoom). Cracks

at the base and the lower

part of the chimney


first cracks, even in the case of brittle material such as

masonry, but can fail to predict the crack pattern at

failure.

8.3 Willam-Warncke criterion

This criterion provides more realistic information

than the DP criterion on the progression of the crack

pattern during the earthquake if a time-dependent

analysis is desired. This is because it makes proper

allowance for the cracking phenomenon reducing the

stiffness matrix of the cracked elements and intro-

ducing crack planes.

One of its drawbacks is that it requires the largest

computational effort, but results are in accordance

with observations of the failure modes of real

chimneys in earthquakes, showing reasonable results

for both crack pattern and the first crack appearance.

8.4 General conclusions and comments

– After the study of the different accelerograms, the

chimney studied presents a failure mode by

toppling over the base due to the extent of the

cracks, based on the results from the W-W

criterion. This is not adequately recorded by the

D-P criterion in any of the cases studied or by the

elastic criterion.

– Failure proceeds from cracking and not from

crushing of masonry in brickwork chimneys, as a

result of the low tensile strength exhibited by the

masonry and the type of action. The masonry

compressive strength does not have a significant

effect on the response of this type of structure,

due to the low gravity stresses experienced, the

low tensile strength being the key parameter.

– The W-W criterion is considered as the reference

criterion in this study, since it makes proper

Fig. 13 Instant of

appearance of cracks on the

left border. (a) Longitudinal

stresses (vertical axis) along

the chimney. Isometric

view, (N/m2); (b, c) Crack

pattern. Frontal view (b-

base zoom) and oblique

view (c-base zoom)


allowance for cracking phenomenon, as previously

explained, modifying the stiffness matrix when

cracks are produced, and the calculated failure mode

matches well with the failure modes observed in real

chimneys that have experienced earthquakes. Sim-

ilar results have been reached in [13].

– It is really important to take the masonry tensile

strength into account in order to adequately

estimate the failure mode. The results shown

here agree with the observations made in the cited

literature, even though they are based on numer-

ical simulations.

In conclusion, the low tensile strength exhibited by

masonry in this type of construction results in cracking,

which is the main contributing factor to non-linear

behaviour. It is therefore important to include precise

information on this factor in the model. The stress

redistribution and increase of the plastic area influence

the crack pattern and, consequently, the failure mode

obtained, so it is therefore possible to conclude that the

results obtained, mainly with the W-W cracking

criterion, are useful for evaluating the maximum

seismic action that a chimney can withstand and the

failure mode in the case of excessive cracking, while

D-P can provide fair approximations to initial cracks

and crack patterns.

The numerical models shown are reasonably able

to predict the response to the seismic action, accord-

ing to the degree of accuracy, the results looked for,

and the available computer capacity, giving valuable

information on possible crack pattern distributions in

earthquakes.

These conclusions and comments may provide

useful information on the repair and retrofit applica-

tions to this type of structure carried out by practising

engineers or researchers, and some guidelines to

assess the reliability of measures taken to preserve

these or similar constructions from earthquakes.

All the approaches have their limitations, and

uncertainties of masonry properties in this type of

construction should induce professionals to act with

caution.

The conclusions and comments stated could be

extrapolated to similar masonry structures such as

towers, bell towers, minarets or lighthouses.

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Fig. 14 t = 5.72 s. Cracks

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Frontal view

Fig. 15 Displacements of the base and crown of the chimney.

The displacement of the base node (2) corresponds to the

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15 Materials and Structures Chimeneas