Dissertation2002 Melis

8/12/2019 Dissertation2002 Melis

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Istituto Universitario

di Studi Superiori di Pavia

Universit degli Studi

di Pavia

EUROPEAN SCHOOL OF ADVANCED STUDIES IN

REDUCTION OF SEISMIC RISK

ROSE SCHOOL

DISPLACEMENT-BASED SEISMIC ANALYSIS FOR

OUT OF PLANE BENDING OF

UNREINFORCED MASONRY WALLS

A Dissertation Submitted in Partial

Fulfilment of the Requirements for the Master Degree in

EARTHQUAKE ENGINEERING

Master Thesis by: Supervisors:

Giammichele Melis Guido Magenes

Michael C. Griffith

April, 2002


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ii

The dissertation entitled Dispalcement-based seismic analysis for out of plane bending of

unreinforced masonry walls, by Giammichele Melis, has been approved in partial fulfilment of the

requirements for the Master Degree in Earthquake Engineering.

Guido Magenes_____________________________________

M.J. Nigel Priestley__________________________________


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Abstract

iii

ABSTRACT

The importance of assessing the seismic resistance of existing masonry structures has drawn strong

and growing interest in the recent years.

While conservative rules may be acceptable for the design of new masonry structures implying only a

minor and negligible economic penalty, on the other hand in the case of existing buildings the same

degree of conservatism may hold the balance between the necessity of strengthening or not the

structure under investigation, with hence a huge difference in the economic balance.

This study has investigated on the applicability of a displacement-based procedure to predict the

response of unreinforced masonry walls when dynamically loaded, taking into account their reserve

capacity due to rocking.

It has been found that the procedure proposed is reasonably accurate and at the same time fairly

conservative, with a more acceptable degree of conservatism with respect to traditional methods of

assessing seismic performance of unreinforced masonry walls based on elastic stress calculations

which lead to excessively conservative results.


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Acknoledgements

iv

ACKNOWLEDGEMENTS


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Index

vi

LIST OF TABLES

2.1 Empirically derived tri-linear F-defining displacements.

4.1 Tests walls data.

4.2 Analysed walls.

4.3 Peak responses.

5.1 Equivalent aspect ratio and thickness.

5.2 First set of 22 walls.

5.3 Refined set of 8 walls selected for THA analysis.

5.4 Earthquake ground motions used in the sensitivity study.

5.5 T1and T2values.

5.6 Mean value and standard deviation of the error.

5.7 Mean value and standard deviation of the error for max=U.

6.1 Mean value and standard deviation of the error for 2/U=0.250.50.


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Index

vii

LIST OF FIGURES

2.1 Deflection of Rocking Walls.

2.2 Bi-linear F-relationship.

2.3 Simply supported wall at incipient rocking and at point of instability.

2.4 Parapet wall at incipient rocking and at point of instability.

2.5 Bi-linear and real Semi-rigid F-relationship.

2.6 Tri-linear, Bi-linear and real Semi-rigid F-relationship.

3.1 SDOF Idealisation.

3.2 50mm Wall Non-linear vs. Frequency.

3.3 Non-load bearing 110mm Wall Non-linear vs. Frequency.

4.1 Experimental vs. analytical predictions of wall displacements.

4.2 Experimental vs. analytical predictions of wall acceleration.

4.3 F.E.A.P. model vs. ROWMANRY analytical predictions of wall displacements.

4.4 F.E.A.P. model vs. ROWMANRY analytical predictions of wall acceleration.

5.1 Unreinforced masonry wall support configurations.

5.2 Rigid Force Deformation relationship Set 1.

5.3 Rigid Force Deformation relationship Set 2.

5.4 Generic tri-linear F-relationship for URM walls.

5.5 Earthquake records: accelerograms.5.6 Earthquake records: displacement spectra.


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Index

viii

5.7 Earthquake records: acceleration spectra.

5.8.a Wall 1 F-relationship for 1/U=0.050.20.

5.8.b Wall 1 F-relationship for 2/U=0.250.50.

5.9 Definition of the secant stiffness values K1, K2and KS.

5.10 max/Uvs. period (T1, T2or TS) for Wall 1, Artificial earthquake.

5.11 Variation in Sd/maxvalues vs. max/Ufor all earthquakes: Wall 1.

5.11.bisVariation in Sd/maxvalues vs. max/Ufor all earthquakes: Wall 1 (resized).

5.12 Variation in Sd/maxvalues vs. max/Ufor all earthquakes: Wall 4a.

5.12.bisVariation in Sd/maxvalues vs. max/Ufor all earthquakes: Wall 4a (resized).

5.13 Comparison between Sd and analytical max values vs. max/U for the Artificial earthquake

record.

5.13.bisComparison between Sd and analytical max values vs. max/U for the Artificial earthquake

record (resized).

5.14 Comparison between Sdand analytical maxvalues vs. max/Ufor the san Salvador earthquake

record.

5.14.bisComparison between Sdand analytical maxvalues vs. max/Ufor the San Salvador earthquake

record (resized).

6.1 Sd(T2)/maxvs. max/Ufor 2/U=0.25.







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Chapter1 Introduction

1

1. INTRODUCTION

The importance of assessing the seismic resistance of existing masonry structures has drawn strong

and growing interest in the recent years.

While conservative rules may be acceptable for the design of new masonry structures implying only a

minor and negligible economic penalty, on the other hand in the case of existing buildings the same

degree of conservatism may hold the balance between the necessity of strengthening or not the

structure under investigation, with hence a huge difference in the economic balance.

In particular it has been shown [1] that traditional methods of assessing seismic performance of

unreinforced masonry (URM) walls based on elastic stress calculations lead to excessively

conservative results when compared with more realistic methods of assessment based on stability and

energy considerations.

With this starting point the purpose of this research is first to address the problem of assessing the out-

of-plane seismic resistance of rocking unreinforced masonry (URM) walls when their reserve

capacity is considered and then to investigate the applicability of a linearized displacement-based

(DB) procedure to predict the maximum displacement (and eventually the failure) of URM walls when

dynamically loaded.

One of the main reference for the research which is here presented has been the analysis procedure

proposed at the University of Adelaide and Melbourne by K. Doherty, M.C. Griffith et al. to model the

reserve capacity of rocking brick masonry walls subjected to out-of-plane bending [2]. Such a

procedure rests on a linearized displacement based (DB) approach capable of predicting the response

of URM walls subjected to a dynamic excitation.

In order to lead a series of time history analyses capable of validate the effectiveness of the proposed

DB procedure and to analyse the influence that the characteristic parameters of the walls have on the


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Chapter1 Introduction

2

application of such a procedure, it has been necessary to characterise the real force-displacement

relationship of the rocking walls.

According to the approach based on the experimental results of the shaking table tests led at the

University of Adelaide [3], the real non-linear force-displacement relationship of rocking unreinforced

brick masonry walls has been modelled with a tri-linear relationship described with further details in

the following section.

The analysis procedure which is here presented is not completely new compared with other methods

that like the one proposed are opposing to the traditional methods based on elastic stress calculations

for evaluating the seismic performance of unreinforced masonry structures. With respect to other

methods, like the one proposed by M.J.N. Priestley and founded on the equal energy observation [1],

the DB approach herein examined and validated has the main advantage of being rather

straightforward both in the definition of the equivalent structure and in the evaluation of the maximum

response.

Nevertheless it should be remarked that the approach proposed by M.J.N. Priestley has the big merit to

have surpassed the traditional one in favour of an approach which takes into account the reserve

capacity of the URM walls. Hence it constitutes the starting point of all the following researches, this

one included, which assess the seismic resistance of URM walls based on stability mechanisms rather

than static strength.


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Chapter2 Tri-linear model

2. TRI-LINEAR MODEL

The study presented in this paper is focused on a specific class of URM walls which comprises parapet

walls (cantilever walls) and simply-supported walls vertically spanning between supports at ceiling

and floor levels where the support motions can be assumed to move simultaneously.

For dynamic analysis the URM walls which agree with the description given above, can be idealised

as single-degree-of-freedom (SDOF) systems, resting on the fundamental modal deflection of the

rocking walls (schematically depicted in Figure2.1) for parapet walls and simply supported walls.

Parapet Wall Simply Supported Wall

Figure 2.1 Deflection of Rocking Walls

In order to define the parameters characterising the equivalent structure (or substitute structure

according to the definition given by Shibata and Sozen [4]) to be used in the DB approach, an

extensive series of time history analyses has been performed.

It is well known that non linear time-history analysis (THA) based on time step integration is a

representative and reliable method of accounting for the time dependent nature of URM wall response,

provided that the non-linear Force-Displacement (F-) relationship and the damping properties of the

wall are accurately represented in the analytical model.


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4

Hence the correct definition of the two mentioned above parameters, the Force-Displacement

relationship and the Damping properties, of the examined URM walls, is the first essential point to be

defined.

For what concerns the Force-Displacement relationship, recalling that the attention in this is study

focused on assessing the seismic resistance of existing masonry structures, it has first of all to be

underlined that solely the Post-Cracked F-relationship has been here considered, taking into account

that many of the existing buildings are already cracked due to low strength mortar, settlement of

footings or temperature and moisture changes and that past earthquakes or accidental damage may also

have previously induced cracking.

The first simplification which can be done to infer the F-relationship is to consider the URM walls

as rigid bodies thus having an infinite material stiffness so that the corresponding theoretical Force-

Displacement relationship is the bilinear relation depicted in Figure 2.2 with initial rigid resistance F 0

related to the geometry (i.e. height and thickness), to the self weight, to the overburden weight and to

the boundary conditions of the walls.

An expression of F0can be inferred from moment equilibrium at the point of instability.

Figure 2.2 - Bi-linear F- relationship

For a rocking simply supported wall it is assumed a triangular-shaped relative displacement profile.

The accuracy of this assumption has been verified with shaking table tests and THA as described in

[3]. Under this assumption moment equilibrium, at the point of incipient rocking, of the upper half of

the wall about the pivot point in the cracked cross section at the mid-height of the wall (refer to Figure

2.3), leads to:

62222

0 hFhRtW

= 62222

0 hFhRtMg

=


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5

where R, the horizontal reaction at the top of the wall, can be obtained by considering rotational

equilibrium of the simply supported wall as a whole about the pivot point at the base, and can be

written as

h

tgMF

R

= 220

and the rigid threshold resistance:

h

tgMF = 30

Considering that for walls simply supported at their top and bottom and with uniformly distributed

mass, the effective mass has been calculated to be three-fourths of the total mass (Me= M), based

on standard calculation techniques [2], the expression of the rigid threshold resistance can be hence re-

written as:

h

tgMF e = 40

Figure 2.3 - Simply supported wall at Incipient Rocking and at Point of Instability

For a parapet wall, under the assumption of a triangular-shaped relative displacement profile (refer to

Figure 2.4), moment equilibrium with respect to the pivot point at the base of the wall leads to the

following expression:

hFt

W3

2

20= hF

tMg

3

2

20=

and considering that also in this case the effective mass is Me = M, the expression of the rigid

threshold resistance can be written as follows:

h

tgM

h

tgMF e ==

3

40


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6

Figure 2.4 - Parapet wall at Incipient Rocking and at Point of Instability

Referring to the scheme illustrated in Figure 2.3 of a simply supported wall at Incipient Rocking and at

Point of Instability, when the lateral force (i.e. the resultant of the inertial forces) exceeds F0 (the

Rigid threshold resistance), a mid-height displacement occurs, permitted by the rigid rotation at the

three joints (pivot points) formed at the cracked section: at the top, at the mid height and at the base of

the wall. P-effects then reduce the gravity restoring moments as the resultant of the vertical forces

above the mid-height crack moves toward the walls back compressive face. With further

displacement P- effects continue to reduce the lateral resistance of the wall creating a negative

stiffness arm, K0, of the bi-linear F-relationship.

When the vertical force resultant above the mid-height crack moves outside of the wall thickness the

resistance to overturning is reduced to zero. The displacement at which this occurs is termed the rigid

instability displacement, instability(rigid).

For a real URM wall the individual blocks of the wall can deform significantly when subjected to high

compressive forces with two main consequences.

The first one is that the pivot points possess finite dimensions rather than being infinitesimal like

supposed in the rigid-body model so that the resistance to rocking is associated with a lever arm

significantly less than half the wall thickness (as for a rigid wall) with hence a displacement at thepoint of instability (U=instability) which is less than the one theoretically obtainable when a rigid-body

model is considered. The second consequence is that the wall possess finite lateral stiffness (rather

than being rigid) prior to incipient rocking.

The real semi-rigid F- relationship as it can be observed during the tests [3] deviates significantly

from the bilinear relationship for the reasons explained above and assumes a curvilinear profile. Such

a profile is represented, together with the Bi-linear F- relationship, in Figure 2.5 where it can be

observed that the peak semi-rigid Resistance force is much less than the rigid threshold resistance

previously defined and that the displacement at the point of instability is lower than in the rigid case.


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7

Figure 2.5 - Bi-linear and real Semi-rigid F- relationship

The real curvilinear relationship can be idealised with a tri-linear model defined by three displacement

parameters 1, 2, uand the force parameter F0and represented in Figure 2.6.

Figure 2.6 - Tri-linear, Bi-linear and real Semi-rigid F- relationship

To construct the tri-linear model, the bilinear model is first constructed in accordance with F 0and K0.

The amplitude of the force plateau is, therefore, controlled by the ratio 2/u. For displacements in the

range exceeding 2, the tri-linear and the bi-linear models coincide. For displacements between 1and


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8

2the force is constant. The initial slope of the tri-linear model is governed by the force amplitude of

the plateau and by the value of 1.

The ratios 1/u and 2/u are related to the material properties and the state of degradation of the

mortar joints at the pivot points.

It has been proposed [2] that for walls in new, moderately degraded and severely degraded

condition the ratios of 1/uand 2/upossess the values indicated in Table 2.1.

Table 2.1 - Empirically derived tri-linear F- defining displacements.State of degradation at cracked joint 1/u 2/u

New 6% 28%

Moderate 13% 40%

Severe 20% 50%

Recognising that the definition of the parameters given above is debatable because it is based on a

qualitative, and hence subjective, measure of the state of the walls, the influence of selecting an

incorrect or an inappropriate value for the ratios 1/uand/or 2/uhas been investigated with further

details. The results are presented in Section 5.


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Chapter3 SDOF Idealisation

3. SDOF IDEALIZATION

As mentioned in the previous Section, for dynamic analysis the URM walls considered in this study

can be idealised as SDOF systems.

The equation of motion governing the rocking behaviour of the cracked URM walls, also considering

what has been defined about the F-relationship of the walls, can be written as the equation of motion

of a SDOF non-linear system, that is:

)())(()()( taMtftvCtaM gSDOFSSDOFSDOF =++ (3.1a)

where

a(t) and v(t) and (t) are the SDOF system acceleration, velocity and displacement

respectively;

fS((t)) is the resisting force of the SDOF system, which is a function of the displacement (t);

MSDOFis the mass of the SDOF;

CSDOFis the damping coefficient equal to 2MSDOF

and diving byMSDOFequation 3.1acan be re-written as follows:

)())((

)()( taM

tftv

M

Cta g

SDOF

S

SDOF

SDOF =

++ (3.1b)


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10

The resisting force considering the F-relationship of a rocking URM wall can be written as follows:

( )( )t

Ff U

U

S

=

1

20 for (t) 1

( )20 = UUS

Ff for 1< (t) 2

( )( )tF

f UU

S

= 0 for (t) > 2

and takes into account of the non-linearity of the system.

Equation 3.1a (and 3.1b) applies if the effective mass of the system is considered (i.e. of the total

mass for a non load-bearing simply supported wall) and if a(t), v(t) and (t) are respectively the

acceleration, the velocity and the displacement of the considered (single) degree of freedom.

Because it can be more suitable for a simply supported rocking wall to refer to the mid-height wall

response (which is the point of maximum response), equation 3.1 can be re-written putting in evidence

the wall mid-height displacement, velocity and acceleration.

Referring to the SDOF idealisation depicted in Figure 3.1 and considering that, based on the

assumption of a triangular shaped relative displacement profile, the displacement of the considered

degree of freedom is two thirds of the displacement at the mid-height of the wall :

SDOF=2/3 m

2/3

2/3

1/2

1/2

Figure 3.1 SDOF Idealisation.

and similarly:

aSDOF=2/3 am

vSDOF=2/3 vm

where

amand vmand mare the wall mid-height acceleration, velocity and displacement respectively;

Equation 3.1 can be rearranged as follows:

( )( )g

S

mm aM

tf

vM

C

a 2

3

2

3

=

+

+ (3.2)


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11

where

M is the total mass of the wall;

C is the damping coefficient equal to 2M (notice that CSDOF= 2MSDOF=2/3 C).

The circular frequency which appears in the expression of the damping coefficient C is herein

evaluated with reference to the initial stiffness of the tri-linear F-relationship, which is:

( )

1

20

= U

U

initial

FK

thereforeM

K

M

K initial

SDOF

initial ==2

3

The correct definition of the damping coefficient should take into account the change in the circular

frequency of the system as the stiffness changes along the tri-linear curve.

For simplicity a constant damping coefficient value has been adopted and the suitability of such an

assumption has been checked a posteriori.

To perform the required time history analyses a specific non linear spring element to be used with the

finite element program F.E.A.P. [5] has been developed coding the tri-linear model described above.

In the time history analyses a damping coefficient corresponding to a damping ratio = 5 % was

assumed, equal to the lower bound of the damping ratio values obtained from the experimental

observations (i.e. based on the results of the tests led at the university of Adelaide by K.T. Doherty et

al.).

Figures 3.2 and 3.3 present the correlation between the frequency of each response cycle versus the

non-linear damping ratio for the 50 mm wall specimens at various levels of applied overburden and

for non-loadbearing 110 mm wall specimens. For both sets of walls a lower bound estimate of of

around 5% was observed.

The selection of such a constant value, = 5 %, appears to be a conservative one. As illustrated in the

results of the time history analyses reported in the following the use of a lower bound for the dampingratio value leads to a slight increase in the displacement and in the acceleration response (with a bigger

relevance in the second case) which contains an acceptable and favourable degree of conservatism

when that value is used in the prediction of the wall maximum displacement.


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12

0.0%

2.5%

5.0%

7.5%

10.0%

12.5%

15.0%

17.5%

20.0%

22.5%

25.0%

0 1 2 3 4 5 6 7 8 9

FREQUENCY (Hz)

PERCENTCRITICALDAMPING(PSI)

NO PRECOMP

0.075MPa

0.15MPa

Figure 3.2 - 50mm Wall - Non-linear vs. Frequency (From [8]).

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

0 1 2 3 4 5 6FREQUENCY (Hz)

PERCENTCRITICALDAMPING(PSI)

Figure 3.3 - Non-loadbearing 110mm Wall - Non-linear vs. Frequency (From [8]).

What has been described above for simply supported walls can be extended to the case of a

cantilevered wall.

It has to be underlined that in the latter case equation 3.2 is applicable with amand vm representing

respectively the acceleration and the velocity at two third of the wall height. The remaining quantities

have the same meaning than in the case of a simply supported wall.


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Chapter4 Tests

4. TESTS

Using the previously mentioned non-linear model a parametric study was carried out to investigate

which are the parameters that primarily characterise the tri-linear model in terms of wall response

when a time history analysis is performed and how do these parameters affect the displacement based

assessment procedure.

Before starting the parametric study a comparison between the analytical and the experimental results

coming from the tests performed at the Adelaide University [8] has been done. This was made both to

have a support of the theoretical assumptions (i.e. the Tri-linearF-approximation and the parameters

assumed in the SDOF approxiamtion) and also to test the reliability of the F.E.A.P. model coded for

the specific purpose of this parametric study.

Doherty et al. [3] and [8] conducted at the Adelaide University an extensive set of numerical analyses

in order to verify the ability of the tri-linear model to accurately simulate the dynamic response of

URM walls subjected to earthquake support motion. Several of the walls tested by Doherty were

numerically re-analysed using as input in the THA the actual shaking table acceleration.

In following some of the results obtained from these comparisons are presented.

The analysed walls have the characteristics summarised in Table 4.1 and were tested with three

different input motions: 50% El Centro, 66% El Centro and four times the Nahanni earthquake

accelerogram record.

Table 4.1 Tests wall data.

Wall Data

L [m] 1.00 Length

h [m] 1.50 Height

t [mm] 110 Thickness

[N/m3] 18000 Specific Weight


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Chapter4 Tests

14

In Table 4.2 the characteristics of the SDOF idealizations used as input values in the numerical

analyses are reported. It has to be remarked that the differences between the three walls considered in

the analyses lie in the 1 and 2 values as a consequence of the different state of degradation of the

tested walls.

Table 4.2 Analysed walls.

Wall a

P [N] 0 Vertical Load

M [kg] 302.9 Total Mass

F0[N] 871.20 Force at Incipient Rocking

F'0[N] 1306.80 3/2 Force at Incipient Rocking

u[m] 0.1100 Displacement at Point of Instability

1[m] 0.0070 Characteristic displacement 1


C [kg/sec] 629.1 Viscous Damping Factor

Wall b

P [N] 0 Vertical Load








Wall cP [N] 0 Vertical Load









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Chapter4 Tests

15

In Figure 4.1 and 4.2 the time histories in terms of displacements and accelerations for the three walls

are reported.

Midheight Displacement Response

-0,020

-0,015

-0,010

-0,005

0,000

0,005

0,010

0,015

0,020

0 1 2 3 4 5 6 7 8 910

11

12

13

14

15

16

17

18

19

20

21

22

Time [sec]

Midhe

ightResponse

[m]

ExperimentalNumerical

a) Wall a


-0,150

-0,100

-0,050

0,000

0,050

0,100

0,150

0 1 2 3 4 5 6 7 8 910

11

12

13

14

15

16

17

18

19

20

21

22

Time [sec]

Midhe

ightResponse

[m]

Experimental

Numerical

b) Wall b


-0,060

-0,050

-0,040

-0,030

-0,020

-0,010

0,000

0,010

0,020

0,030

0,040

0 1 2 3 4 5 6 7 8 910

11

12

Time [sec]

Midhe

ightResponse

[m

]

Experimental

Numerical

c) Wall cFigure 4.1 Experimental versus analytical predictions of wall displacements.


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Chapter4 Tests

16

Midheight Acceleration Response

-5,00

-4,00

-3,00

-2,00

-1,00

0,00

1,00

2,00

3,00

4,00

5,00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Time [sec]

MidheightRespons

e[m/s2]

Experimental

Numerical

a) Wall a


-10,00

-8,00

-6,00

-4,00

-2,00

0,00

2,00

4,00

6,00

8,00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Time [sec]

MidheightResponse[m/sec2]

Experimental

Numerical

b) Wall b


-6,00

-4,00

-2,00

0,00

2,00

4,00

6,00

0 1 2 3 4 5 6 7 8 9 10 11 12

Time [sec]

Midhe

ightResponse

[m/s2]

Experimental

Numerical

c) Wall c

Figure 4.2 Experimental versus analytical predictions of wall accelerations.


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Chapter4 Tests

17

In Figure 4.1 a good correlation between the experimental and analytical displacement response can be

noticed.

From Figure 4.2 it can be observed that the accelerations predicted with the analytical model are

bigger than the actual ones (i.e. the ones recorded with the shaking-table tests) as a consequence that a

lower bound for the damping ratio value has been used in the numerical model. For the same reason,

in the corresponding displacement plots, it can be observed that the experimental response appears to

drop off (damp out) much more rapidly than the analytical response. This is consistent with an under

estimate of damping in the numerical model. It is worth noting that in the plots reported in Figure 4.1

(Experimental vs. analytical predictions of wall displacements) the amplitude of the first peak

following each significant impulse in the ground motion is reasonably well predicted by the analysis

(this is because the amplitude of the first peak is much less sensitive to damping as are the subsequent

peaks).

The assumption of a lower bound for the damping ratio value results hence in being a conservative

one, still within the range of an acceptable approximation of the exact solution (i.e. the real

acceleration of the wall specimen in the tests).

The good, but non-perfect, correspondence between the numerical and the experimental results (for

both the displacement and the acceleration responses) is due in part to the approximation of the real

curvilinear F- relationship of the rocking wall with a tri-linear one and in part to the high non-

linearity of the real walls rocking response, even at small displacements.

The real rocking wall is highly non-linear for all the displacements beyond the cracking displacement

which is only a few millimetres, much less than the value assumed for the displacement 1.

The main reason of having assumed for Wall_a, Wall_b and Wall_c the characteristics summarised in

Table 4.2 is that, as a countercheck for the reliability of the model, its results (i.e. the results reported

above for Wall_a, Wall_b and Wall_c) have been compared with the theoretical results obtained by

Doherty at the Adelaide University with the program ROWMANRY.

The good correspondence between the time history responses of the two programs can be observed in

Figure 4.3 and 4.4 even if it has to be underlined that the two responses are not perfectly coincident.

This is due to the fact that ROWMANRY uses an event-based time history routine instead of the

traditional step-by-step non-linear time history analysis used in F.E.A.P..

The event-based model effectively checks after each half-cycle of response (each zero crossing) to see

whether the damping used for the analysis of that half-cycle of response was correct for the amplitude

of displacement for that half-cycle of response. If not, the damping was adjusted and the analysis

repeated for that half-cycle of response.


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Chapter4 Tests

18

Getting back to the correlation with the experimental results, neither ROWMANRY or the F.E.A.P.

model gives perfect correlation with the tests results (the real curvilinear F- relationship is

approximated with a tri-linear one), but it has again to be underlined the good correlation between the

experimental and the numerical results in terms of peak displacement response.

Table 4.3 Peak responses.

Maximum absolute displacement [m]

Shaking Table F.E.A.P.

Wall a 0.017 0.015

Wall b 0.104 0.110 = U

Wall c 0.032 0.047

It has to be observed that even when the numerical analysis leads to estimate for the wall a failure that

doesnt occur in the test (Wall_b), such a prediction is still within the range of an acceptable degree of

conservatism: the maximum experimental displacement (i.e. the displacement recorded in the test) was

only the 95% of the ultimate displacement U.


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Chapter4 Tests

19


-0,020

-0,015

-0,010

-0,005

0,000

0,005

0,010

0,015

0 1 2 3 4 5 6 7 8 910

11

12

13

14

15

16

17

18

19

20

21

22

Time [sec]

MidheightRespo

nse[m]

ROWMANRY

FEAP

a) Wall 1


-0.150

-0.100

-0.050

0.000

0.050

0.100

0.150

0 1 2 3 4 5 6 7 8 910

11

12

13

14

15

16

17

18

19

20

21

22

Time [sec]

MidheightResponse[m]

ROWMANRY

FEAP

b) Wall 2


-0.060

-0.050

-0.040

-0.030

-0.020-0.010

0.000

0.010

0.020

0.030

0.040

0 1 2 3 4 5 6 7 8 910

11

12

Time [sec]

Midheight

Response[m]

ROWMANRY

FEAP

c) Wall 3

Figure 4.3 F.E.A.P. model vs. ROWMANRY analytical predictions

of wall displacements.


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Chapter4 Tests

20


-5,00

-4,00

-3,00

-2,00

-1,00

0,00

1,00

2,00

3,00

4,00

5,00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Time [sec]

MidheightResponse[m/s2]

ROWMANRY

FEAP

a) Wall 1


-5.00

-4.00

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

5.00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Time [sec]

MidheightResponse[m/sec2]

ROWMANRY

FEAP

b) Wall 2


-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

0 1 2 3 4 5 6 7 8 9 10 11 12

Time [sec]

MidheightResponse[m/s2]

ROWMANRY

FEAP

c) Wall 3

Figure 4.4 F.E.A.P. model vs. ROWMANRY analytical predictions

of wall accelerations.


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Chapter5 Sensitivity analysis

5. SENSITIVITY ANALYSIS

In order to select for the analyses a set of walls which was not excessively large but at the same time

capable of representing the full range of interest (i.e. a realistic range for design and assessment

purposes in terms of geometric characteristics of the walls and applied load), a normalised force-

displacement relationship has been considered.

The selection of a normalised F-relationship allows to avoid the repetition of analyses where walls

with different geometric characteristics and boundary conditions would anyway lead to the same F-

relationship simply scaled in proportion to the ratio of the thickness versus the height of the wall. Such

a condition would imply that the same results would be obtained if the accelerogram used as input

motion is scaled in proportion.

For what concerns the boundary conditions, it has to be underlined that the response of a non-

loadbearing simply supported wall is the same as the response of a parapet wall with the same

thickness and one-quarter of the aspect ratio (h/t), as it has been shown in Section 2 when the

expression of the rigid threshold resistance has been derived from moment equilibrium in the two

cases. Proceeding in a similar way it can be shown [8] that when an overburden pressure is applied the

effect can be modelled by further reducing the aspect ratio of the equivalent parapet wall. These

equivalencies in terms of aspect ratio take to an equivalent thickness such that the ratio between the

equivalent and the actual thickness varies between one and three quarter as summarised in the

following Table 5.1.


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22

Table 5.1. Equivalent aspect ratio and thickness.

Support Type (h/t)eq/(h/t)actual teq/tactual

Rigid parapet 1 1

Rigid non-loadbearing simply supported wall with base

reaction at the leeward face

1/4 1

Rigid loadbearing simply supported wall with top and

base reaction at the leeward face

1/[4(1+)] 1

Rigid loadbearing simply supported wall with top reaction

at the centreline and base reaction at the leeward face

1/[4(1+)] (1+3/4)/(1+)

varies between

and 1

The symbol in the previous table defines the ratio of overburden weight and self weight of the upper

half of the wall above mid-height:

2W

P=

With reference to a simply supported wall, two boundary conditions have been considered along the

top edge of the URM walls: the first corresponding to a Slabboundary condition (which places the

vertical load at the edge of the wall face) and the second corresponding to a Timber Bearerboundary

condition (which maintains the vertical load along the wall centreline). These two conditions are

depicted in Figure 5.1 (respectively 5.1c and 5.1d) together with the configurations corresponding to a

parapet wall (5.1a) and to a simply supported non-loadbearing wall (5.1b).

Figure 5.1 Unreinforced masonry wall support configurations.


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23

The two boundary conditions have been taken into account in the ratio F0/W (W being the weight of

the wall) and in the values of the ratio u/tthat have been calculated respectively as:

F0/W = 4 t/h (1+) ; u/t= 1 in the first boundary condition, and

F0/W = 4 t/h (1+3/4) ;u/t= (1+3/4)/(1+) 0.8 for what concerns the second

boundary condition described above.

On the basis of the considerations reported above the following range of properties has been selected:

t [mm] 110 500 Thickness

h [m] 3.0 5.0 Height

p [MPa] 0.00 0.50 Axial compression

Being the axial compression p defined aslt

Pp

= , with P the applied vertical load and l the length of

the wall (always considered unitary).

All the possible combination (with t/h between 1/30 and 1/5) have been considered taking into account

the following values

t = 110mm, 200mm, 300mm, 400mm500mm;

h = 3.0m, 3.3m(only for the 110mmthickness wall), 4.0m, 5.0m;

p = 0.00MPa, 0.10MPa, 0.20MPa, 0.30MPa, 0.40MPa, 0.50MPa.

A total number of 168 walls resulted from such a combination.

Among these walls only those having an aspect ratio between 1/30 and 1/5 and a F0/W ratio less than

one have been selected. Their characteristics are presented in Table 5.2 referring to a 1mwide strip of

a wall with a specific weight of 18000 N/m3.


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24

Table 5.2. First set of 22 walls.

Wall

No.

t

[mm]

h

[m]

p

[Mpa]

h/t Fo

[N]

W

[N])

F0/W u

[mm]

u/t

1 200 3.0 0.1 15.0 10908 10800 1.01 160 0.80

1a 110 3.0 0.2 27.3 5702 5940 0.96 86 0.78

1b 200 4.0 0.2 20.0 14832 14400 1.03 158 0.79

1c 200 5.0 0.3 25.0 17280 18000 0.96 156 0.78

1d 300 4.0 0.1 13.3 20088 21600 0.93 246 0.82

1e 300 5.0 0.2 16.7 28080 27000 1.04 240 0.80

1f 500 5.0 0.1 10.0 48150 45000 1.07 415 0.83

2 200 5.0 0.2 25.0 12600 18000 0.70 160 0.80

2a 300 5.0 0.1 16.7 17280 27000 0.64 249 0.83

3 200 5.0 0.1 25.0 7740 18000 0.43 166 0.83

3a 110 3.3 0.1 30.0 3071 6534 0.47 89 0.81

4 110 3.3 0.0 30.0 849 6534 0.13 110 1.00

4a 200 5.0 0.0 25.0 2880 18000 0.16 200 1.00

5 300 3.0 0.0 10.0 6480 16200 0.40 300 1.00

5a 400 4.0 0.0 10.0 11520 28800 0.40 400 1.00

5b 500 5.0 0.0 10.0 18000 45000 0.40 500 1.00

6 500 3.0 0.0 6.0 18090 27000 0.67 500 1.006a 110 3.0 0.1 27.3 4099 5940 0.69 110 1.00

6b 200 4.0 0.1 20.0 10944 14400 0.76 200 1.00

6c 300 5.0 0.1 16.7 20790 27000 0.77 300 1.00

7 400 5.0 0.1 12.5 37080 36000 1.03 400 1.00

8 200 2.0 0.0 10.0 720 7200 0.10 200 1.00

In Figure 5.2 the normalised Force-Displacement relationship is depicted for the walls included in

Table 5.2 and considered as rigid blocks.

Something has to be underlined about the procedure followed in the selection of the parameters of the

walls to be analysed.

Considering at this stage the model of the cracked unreinforced masonry walls as rigid blocks has the

benefit that it simplifies the problem, taking also into account that the definition of the parameters 1

and 2is so far not influent.


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25

Rigid Force-Deflection Relationship

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 u/t

Fo/W

1

1a

1b

1c

1d

1e

1f2

2a

3

3a

4

4a

5

5a

5b

6

6a

6b

6c

7

8

Figure 5.2 Rigid Force-Deformation Relationship Set 1.

The adaptation to the model of the cracked unreinforced masonry walls as deformable (semi-rigid)

blocks, according to what explained in Section 2, can be done only in a next step (i.e. when the Time

History Analyses have to be performed).

A more refined selection has been hence done in order to reject walls with very similar properties,

getting to the set of walls presented in ascending order of F0/W values in Table 5.3 (and highlighted in

the previous Table 5.2) which corresponds to the set of eight walls analysed in the time history

analyses and whose normalised Force-Displacement relationships are represented in Figure 5.3.

Table 5.3. Refined set of 8 walls selected for the THA.

Wall

No.

t

[mm]

H

[m]

P

[Mpa]

h/t Fo

[N]

W

[N])

F0/W u

[mm]

u/t

1 200 3.0 0.1 15.0 10908 10800 1.01 160 0.80

1a 110 3.0 0.2 27.3 5702 5940 0.96 86 0.78

6 500 3.0 0.0 6.0 18090 27000 0.67 500 1.00

2a 300 5.0 0.1 16.7 17280 27000 0.64 249 0.83

3a 110 3.3 0.1 30.0 3071 6534 0.47 89 0.81

5 300 3.0 0.0 10.0 6480 16200 0.40 300 1.00

4a 200 5.0 0.0 25.0 2880 18000 0.16 200 1.00

4 110 3.3 0.0 30.0 849 6534 0.13 110 1.00


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26

Rigid Force-Deflection Relationship

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20

u/t

Fo/W

1

1a

2a

3a

4

4a

5

6

Figure 5.3 Rigid Force-Deformation Relationship Set 2.

The generic normalised load versus deflection characteristics for all eight walls are shown in Figure

5.4 where it can be seen that the tri-linear F-relationship used in the numerical model is contained

within the triangularF- relationship for a rigid-body analysis of a wall with a horizontal crack at its

mid-height (its base for a cantilever wall) and no consideration of deformations (tensile or

compressive) in the mortar.

Figure 5.4. Generic tri-linear F- relationship for URM walls.


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27

Six different earthquake records were used to study in which way different characteristics of the

earthquake ground motions affect the behaviour of URM walls.

The earthquake ground motions used for this purpose are listed in Table 5.4 and presented in Figure

5.5. Their respective displacement and acceleration response spectra are given in Figure 5.6 and 5.7.

Table 5.4 Earthquake ground motions used in the sensitivity study.

Earthquake Description

El Centro Recorded at El Centro during the Imperial Valley, California earthquake, 18thMay

1940. Magnitude 6.6, Epicentral Distance 8km. Accelerogram component NS.

Taft Recorded at Kern Country, Taft Lincoln School Tunnel, California, 21stJuly 1952.

Accelerogram component S69E. Epicenter 35 00 00N

119 02 00W. Seismograph station 35 09 00N 119 27 00W.

San Salvador Recorded at San Salvador, El Salvador, 10th October 1986. Magnitude 5.4,

Epicenter 13 67 00N 89 19 00W.

Gemona Recorded at Gemona, Friuli, Italy, 15th

September 1976. Magnitude 6.1,

Accelerogram component NS.

Sturno Recorded at sturno, Irpinia, Italy, 23rd

November1980. Magnitude 6.9,

Accelerogram component EW.

Artificial Code Compatible earthquake.


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28

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.0 10.0 20.0 30.0 40.0 50.0 60.0

t [sec]

a[g]

(a) El Centro

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.0 10.0 20.0 30.0 40.0 50.0 60.0

t [sec]

a[g]

(b) Taft

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.0 5.0 10.0 15.0 20.0 25.0

t [sec]

a[g]

(c) San Salvador

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

0.0 2.0 4.0 6.0 8.0 10.0 12.0

t [sec]

a[g]

(d) Gemona

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0

t [sec]

a[g]

(e) Sturno

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0

t [sec]

a[g]

(f) Code Compatible

Figure 5.5 Earthquake records: accelerograms.


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29

ElCentro 5%damping

0.00

0.10

0.20

0.30

0.40

0.0 1.0 2.0 3.0 4.0 5.0T [sec]

S

d[m]

(a) El Centro

Taft 5%damping

0.00

0.05

0.10

0.15

0.20

0.0 1.0 2.0 3.0 4.0 5.0

T [sec]

S

d[m]

(b) Taft

SanSalvador 5%damping

0.00

0.10

0.20

0.30

0.40

0.0 1.0 2.0 3.0 4.0 5.0T [sec]

Sd[m]

(c) San Salvador

Gemona 5%damping

0.00

0.10

0.20

0.30

0.40

0.0 1.0 2.0 3.0 4.0 5.0

T [sec]

Sd[m]

(d) Gemona

Sturno 5%damping

0.00

0.20

0.40

0.60

0.80

0.0 1.0 2.0 3.0 4.0 5.0

T [sec]

Sd[m]

(e) Sturno

Code Compatible_EQ 5%damping

0.00

0.02

0.04

0.06

0.08

0.0 1.0 2.0 3.0 4.0 5.0T [sec]

Sd[m]

(f) Code Compatible

Figure 5.6 Earthquake records: displacement spectra, 5% damping.

The input motions (i.e. the accelerograms described in Table 5.4) used for the Time History analyses

have been scaled in order to lead the wall under consideration into a significative range of

displacements 1


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30

ElCentro 5%damping

0.0

2.0

4.0

6.0

8.0

10.0

0.0 1.0 2.0 3.0 4.0 5.0

T [sec]

Sa[m/sec

2]

(a) El Centro

Taft 5%damping

0.0

2.0

4.0

6.0

8.0

0.0 1.0 2.0 3.0 4.0 5.0

T [sec]

Sa[m/sec

2]

(b) Taft

SanSalvador 5%damping

0.00

5.00

10.00

15.00

20.00

0.0 1.0 2.0 3.0 4.0 5.0T [sec]

Sa[m/sec

2]

(c) San Salvador

Gemona 5%damping

0.00

5.00

10.00

15.00

20.00

0.0 1.0 2.0 3.0 4.0 5.0

T [sec]

Sa[m/sec

2]

(d) Gemona

Sturno 5%damping

0.00

4.00

8.00

12.00

16.00

0.0 1.0 2.0 3.0 4.0 5.0

T [sec]

Sa[m/sec

2]

(e) Sturno

Code Compatible_EQ 5%damping

0.00

1.00

2.00

3.00

4.00

0.0 1.0 2.0 3.0 4.0 5.0T [sec]

Sa[m/sec

2]

(f) Code Compatible

Figure 5.7 Earthquake records: acceleration spectra, 5% damping.

It is worth noting that according to the selection of the walls described in the first part of this Section

the set of eight walls considered in this study encompasses:

wall slenderness ratios (h/t) ranging between 6 and 30;

wall heights (h) ranging between 3mand 5m;

wall thickness (t) ranging between 110mmand 500mm;

axial compression (p) ranging between 0 and 0.2MPa.

The walls were modelled as equivalent single-degree-of-freedom (SDOF) systems as previously

described in Section 2.


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31

Having thus confirmed the suitability of the tri-linear model to model the out-of-plane response of

simply-supported or cantilever URM walls, a more detailed parametric study has been conducted

to study the effect of the walls initial stiffness and strength on the post-cracking dynamic response of

these walls.

These parameters, the stiffness and the strength of the walls, are strictly related to the selection of the

parameters 1 and 2 that define the tri-linear F- relationship together with the values of the

displacement at point of static instability Uand the force at incipient rockingF0.

From the Shaking-table tests performed at the Adelaide University [8] it appears that the values of 1

and 2mainly depend on the material properties and on the state of degradation at the cracked joints.

Since an exact quantification of such parameters in a real existing masonry wall can be problematic if

not impossible, it was considered essential to determine how the dynamic response of a wall is

affected by variations in

1 and

2 (and therefore in initial stiffness and static strength Fu) once thegeometric characteristics, the boundary conditions and the vertical loading are given.

The values of 1/Uand 2/Uhave been varied in the ranges considered of practical interest, that is

1/U= 0.05 0.20 and 2/U= 0.25 0.50.

In figure 5.8a and 5.8b are depicted respectively the influence that the change in the value of 1/U(in

Figure 5.8a) and of 2/U(in Figure 5.8b) have on theF-relationship for a given wall.

In particular the F- relationships represented in Figure 5.8 refer to wall 1, characterised by the

following parameters

l [m] 1.0 Length

h [m] 3.0 Height

t [mm] 200 Thickness

[N/m3] 18000 Specific Weight

P [N] 20000 Vertical load

and with 1/

Uand

2/

U respectively varying in the ranges 0.05

0.20 and 0.25

0.50.


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32

Force-Displacement Relationship

0

2500

5000

7500

10000

12500

15000

0.00 0.05 0.10 0.15 0.20

Displacement [m]

Force[N]

D1/Du = 0.050

D1/Du = 0.075

D1/Du = 0.100

D1/Du = 0.125

D1/Du = 0.150

D1/Du = 0.175

D1/Du = 0.200

Figure 5.8a Wall 1 F- relationship for 1/U= 0.050.20

Force-Displacement Relationship

0

2500

5000

7500

10000

12500

15000

0.00 0.05 0.10 0.15 0.20

Displacement [m]

Force[N]

D2/Du = 0.25

D2/Du = 0.30

D2/Du = 0.35

D2/Du = 0.40

D2/Du = 0.45

D2/Du = 0.50

Figure 5.8b Wall 1 F- relationship for 2/U= 0.250.50

An example of the results obtained from these analyses is shown in Figure 5.10 for Wall 1 where the

maximum wall displacement MAX (normalised by the theoretical ultimate wall displacement, U) is


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33

plotted versus the effective natural period of the wall for the ranges of ratios for 1/U and 2/U

mentioned above.

Three different methods of natural period calculation were used for the abscissa of each value of

MAX/Uin this figure: T1, T2, and TS. The values of T1were computed using the stiffness of the initial

elastic segment of the tri-linear F- relationship used in each analysis, i.e. K1=FU/1 (being FU the

Semi-rigid threshold resistance) as indicated in Figure 5.9. The values of T2were calculated using

the secant stiffness going through the point on the tri-linear F-curve at the point where = 2, i.e.

K2=FU/2 depicted in Figure 5.9. The values of TSwere calculated using the secant stiffness at the

point = max, i.e.KS= F(=max)/max (see Figure 5.9).

Figure 5.9 Definition of the secant stiffness valuesK1,K2andKs

The natural periods T1and T2, respectively based on the stiffness of the initial elastic segment and on

the secant stiffness at =2, depend hence on the values of 1and 2(or, which is the same, on the

values of the ratios 1/uand 2/u).

The variation in the values of T1and T2for all the examined walls with changing the values of 1/u

and 2/uare reported in Table 5.5.


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34

Wall

No.

Fo

(N)

W

(N)

u

(mm)

1/u 2/u T1(sec)

2(sec)

Wall

No.

Fo

(N)

W

(N)

u

(mm)

1/u 2/u T1(sec)

2(sec)

0.05 0.35 0.18 0.48 0.05 0.35 0.20 0.52

0.08 0.35 0.23 0.48 0.08 0.35 0.25 0.52

0.10 0.25 0.24 0.38 0.10 0.25 0.26 0.41

0.10 0.30 0.25 0.43 0.10 0.30 0.27 0.47

0.10 0.35 0.26 0.48 0.10 0.35 0.28 0.52

0.10 0.40 0.27 0.53 0.10 0.40 0.29 0.58

0.10 0.45 0.28 0.59 0.10 0.45 0.30 0.65

0.10 0.50 0.29 0.65 0.10 0.50 0.32 0.71

0.13 0.35 0.29 0.48 0.13 0.35 0.32 0.52

0.15 0.35 0.31 0.48 0.15 0.35 0.34 0.52

0.18 0.35 0.34 0.48 0.18 0.35 0.38 0.52

1 10908 10800 160

0.20 0.35 0.36 0.48

3a 3071

6534

89

0.20 0.35 0.40 0.52

0.05 0.35 0.14 0.36 0.05 0.35 0.39 1.04

0.08 0.35 0.17 0.36 0.08 0.35 0.50 1.04

0.10 0.25 0.18 0.28 0.10 0.25 0.52 0.82

0.10 0.30 0.19 0.32 0.10 0.30 0.54 0.93

0.10 0.35 0.19 0.36 0.10 0.35 0.56 1.04

0.10 0.40 0.20 0.40 0.10 0.40 0.58 1.16

0.10 0.45 0.21 0.44 0.10 0.45 0.60 1.28

0.10 0.50 0.22 0.49 0.10 0.50 0.63 1.42

0.13 0.35 0.22 0.36 0.13 0.35 0.63 1.04

0.15 0.35 0.24 0.36 0.15 0.35 0.68 1.04

0.18 0.35 0.26 0.36 0.18 0.35 0.75 1.04

1a 5702 5940 86

0.20 0.35 0.27 0.36

5 6480

16200

300

0.20 0.35 0.79 1.04

0.05 0.35 0.39 1.04 0.05 0.35 0.51 1.34

0.08 0.35 0.50 1.04 0.08 0.35 0.64 1.34

0.10 0.25 0.52 0.82 0.10 0.25 0.67 1.06

0.10 0.30 0.53 0.93 0.10 0.30 0.69 1.20

0.10 0.35 0.56 1.04 0.10 0.35 0.72 1.34

0.10 0.40 0.58 1.16 0.10 0.40 0.75 1.50

0.10 0.45 0.60 1.28 0.10 0.45 0.78 1.66

0.10 0.50 0.63 1.42 0.10 0.50 0.82 1.83

0.13 0.35 0.63 1.04 0.13 0.35 0.82 1.34

0.15 0.35 0.68 1.04 0.15 0.35 0.88 1.34

0.18 0.35 0.74 1.04 0.18 0.35 0.96 1.34

6 18090 27000 500

0.20 0.35 0.79 1.04

4a 2880

18000

200

0.20 0.35 1.02 1.34

0.05 0.35 0.28 0.75 0.05 0.35 0.42 1.11

0.08 0.35 0.36 0.75 0.08 0.35 0.53 1.11

0.10 0.25 0.37 0.59 0.10 0.25 0.55 0.87

0.10 0.30 0.39 0.67 0.10 0.30 0.57 0.99

0.10 0.35 0.40 0.75 0.10 0.35 0.59 1.11

0.10 0.40 0.42 0.83 0.10 0.40 0.62 1.23

0.10 0.45 0.44 0.92 0.10 0.45 0.64 1.36

0.10 0.50 0.46 1.02 0.10 0.50 0.67 1.51

0.13 0.35 0.46 0.75 0.13 0.35 0.67 1.11

0.15 0.35 0.49 0.75 0.15 0.35 0.72 1.11

0.18 0.35 0.54 0.75 0.18 0.35 0.79 1.11

2a 17280 27000 249

0.20 0.35 0.57 0.75

4 849 6534

110

0.20 0.35 0.84 1.11

Table 5.5 T1and T2values.


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35

Wall-1 2/u=0.350

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

T [sec]

max/u

T1 [sec]

TS [sec]

Sd/Du

Wall-1 1/u=0.100

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

T [sec]

max/u

T2 [sec]

TS [sec]

Sd/Du

(a) Variation in max/Uwith changes in 21 and for max/U< 0.54.0 Artificial eq.

Wall-1 2/u=0.350

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

1.600

1.800

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

T [sec]

max/

u

T1 [sec]

Sd/Du

Wall-1 1/u=0.100

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

1.600

1.800

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

T [sec]

max/u

T2 [sec]

Sd/Du

(b) Variation in max/Uwith changes in 21 and for max/U> 0.58.0 Artificial eq.

Figure 5.10 max/Uversus period (T1, T2or Ts) for Wall 1, Artificial earthquake

For the two plots on the left-hand side of Figure 5.10, the data points (from left to right) correspond to

the case where 2/U= 0.35 and 1/U=0.05, 0.075, 0.10, 0.125, 0.15, 0.175 and 0.20, respectively.

For the two plots on the right-hand side of Figure 5.10, the data points (from left to right) correspond

to the case where 1/U= 0.10 and 2/U= 0.20, 0.25, 0.30, 0.35, 0.40, 0.45 and 0.50, respectively.

It is of interest to note how sensitive the maximum wall response (max/U) is to changes in initial

stiffness (1/U) and strength (2/U).

Plots of the same kind of those of Figure 5.10 are reported in an Appendix for all the examined cases,

that is the walls reported in Table 5.3 subjected to the earthquake input motions presented in Table 5.4.

The linear elastic displacement response spectra (for 5% damping) are also plotted in Figure 5.10 to

enable comparisons between the displacement spectra and the calculated values. In Figure 5.10a, it can

be seen that for support motion that generates displacement response onto the plateau of the Fcurve (12, refer to Figure 5.4), it appears that good estimates of the wall response can be


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36

obtained using the linear elastic response spectra and wall natural period calculated based on the initial

stiffness, 1T . For displacement response greater than 2 (max/U >0.5, Figure 5.10b), however, it

appears that better estimates of the maximum wall response are given using the elastic spectrum with

values of period, T2. Note that in Figure 5.10b, plots of max/U versus TSwere omitted because at

large displacements, the secant stiffness is so small that the corresponding periods become extremely

large, giving extremely poor correlation with the linear elastic displacement spectrum.

To confirm that these trends were consistent, Sd(T)/max was plotted versus max/U for all three

methods (i.e. using T1, T2and Ts) for several combinations of the walls listed in Table 5.3 under the

effect of the earthquake motions presented in Table 5.4 and whose response spectra are given in

Figure 5.6 and 5.7.

Two walls characterised by opposite strength conditions, i.e. wall 1 and wall 4a, have been analysed

under the effect of all the input motion presented in Table 5.4.

Furthermore all the walls have been analysed under the effect of two specific earthquakes, which is the

Artificialand the San Salvador earthquakes.

The results of these analyses are presented in the following Figures in terms of S d/maxvs. max/U.

First the full scale has been used and then all the plots have been resized to the same range of Sd/max

in order to make the comparison more clear.


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38

0.00

1.00

2.00

3.00

4.00

5.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(T1)/M

AX

El Centro

Taft

San Salvador

Gemona

Sturno

Artificial

(a) Response spectrum predictions using T1values for period.

0.00

1.00

2.00

3.00

4.00

5.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(T2)/M

AX

El Centro

Taft

San Salvador

Gemona

Sturno

Artificial

(b) Response spectrum predictions using T2values for period.

0.00

1.00

2.00

3.00

4.00

5.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(TS)/M

AX

El Centro

Taft

San Salvador

Gemona

Sturno

Artificial

(c) Response spectrum predictions using TSvalues for period.

Figure 5.11_bis Variation inSd/maxvalues versus max/Ufor all earthquakes: Wall 1 (resized).


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39

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(T1)/M

AX

El Centro

Taft

San Salvador

Gemona

Sturno

Artificial


0.00

1.00

2.00

3.00

4.00

5.00

6.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(T2)/M

AX

El Centro

Taft

San Salvador

Gemona

Sturno

Artificial


0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(TS)/M

AX

El Centro

Taft

San Salvador

Gemona

Sturno

Artificial


Figure 5.12 Variation inSd/maxvalues versus max/Ufor all earthquakes: Wall4a.


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40

0.00

1.00

2.00

3.00

4.00

5.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(T1)/M

AX

El Centro

Taft

San Salvador

Gemona

Sturno

Artificial


0.00

1.00

2.00

3.00

4.00

5.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(T2)/M

AX

El Centro

Taft

San Salvador

Gemona

Sturno

Artificial


0.00

1.00

2.00

3.00

4.00

5.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(TS)/M

AX

El Centro

Taft

San Salvador

Gemona

Sturno

Artificial


Figure 5.12_bis Variation inSd/maxvalues versus max/Ufor all earthquakes: Wall4a (resized).


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41

0.00

0.50

1.00

1.50

2.00

2.50

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(T1)/M

AX

Wall 1

Wall 1a

Wall 2a

Wall 3a

Wall 4

Wall 4a

Wall 5

Wall 6


0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(T2)/M

AX

Wall 1

Wall 1a

Wall 2a

Wall 3a

Wall 4

Wall 4a

Wall 5

Wall 6


0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(TS)/M

AX

Wall 1

Wall 1a

Wall 2a

Wall 3a

Wall 4

Wall 4a

Wall 5Wall 6

(c) Response spectrum predictions using Tsvalues for period.

Figure 5.13 Comparison betweenSdand analytical maxvalues versus max/Ufor the Artificialearthquake record.


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42

0.00

1.00

2.00

3.00

4.00

5.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(T1)/M

AX

Wall 1

Wall 1a

Wall 2a

Wall 3a

Wall 4

Wall 4a

Wall 5

Wall 6


0.00

1.00

2.00

3.00

4.00

5.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(T2)/M

AX

Wall 1

Wall 1a

Wall 2a

Wall 3a

Wall 4

Wall 4a

Wall 5

Wall 6


0.00

1.00

2.00

3.00

4.00

5.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(TS)/M

AX

Wall 1

Wall 1a

Wall 2a

Wall 3a

Wall 4

Wall 4a

Wall 5Wall 6


Figure 5.13_bis Comparison betweenSdand analytical maxvalues versus max/Ufor theArtificial earthquake record (resized).


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43

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(T1)/M

AX

Wall 1

Wall 1a

Wall 2a

Wall 3a

Wall 4

Wall 4a

Wall 5

Wall 6


0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(T2)/M

AX

Wall 1

Wall 1a

Wall 2a

Wall 3a

Wall 4

Wall 4a

Wall 5

Wall 6


0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(TS)/M

AX

Wall 1

Wall 1a

Wall 2a

Wall 3a

Wall 4

Wall 4a

Wall 5Wall 6


Figure 5.14 Comparison betweenSdand analytical maxvalues versus max/Ufor San SalvadorEarthquake.


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44

0.00

1.00

2.00

3.00

4.00

5.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(T1)/M

AX

Wall 1

Wall 1a

Wall 2a

Wall 3a

Wall 4

Wall 4a

Wall 5

Wall 6


0.00

1.00

2.00

3.00

4.00

5.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(T2)/M

AX

Wall 1

Wall 1a

Wall 2a

Wall 3a

Wall 4

Wall 4a

Wall 5

Wall 6


0.00

1.00

2.00

3.00

4.00

5.00

0.0 0.2 0.4 0.6 0.8 1.0

MAX/ u

Sd(TS)/M

AX

Wall 1

Wall 1a

Wall 2a

Wall 3a

Wall 4

Wall 4a

Wall 5Wall 6


Figure 5.14_bis Comparison betweenSdand analytical maxvalues versus max/Ufor SanSalvador Earthquake (resized).


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45

The wall displacement evaluated from a THA has been considered as the exact one.

The reliability of the simplified elastic spectrum approach has been evaluated from the error in

percentage of the displacements computed from the displacement response spectra following the

method described above (i.e. using T1, T2or TS).

Several ranges of maximum response have been defined to work out whether the use of T 1rather than

T2can be considered more reliable or vice versa.

Those ranges have been selected on the basis of the plots shown in Figures from 5.11 to 5.14.

The mean value and the standard deviation of the error (error defined as the difference in percentage

from the exact value) have been considered as the parameters more representative of the

approximation of the method and are given in the Table 5.6 and 5.7 for all the examined cases divided

in different ranges of interest.

In particular it can be observed that the mean error done in predicting maxusing Sd(T1)appears to be at

the very most equal to 24% when the maximum displacements are less than 50% of U.The standard

deviation of the error in such conditions varies between 17% and 27%. However, the spectral

estimates are consistently less than the calculated (so-called actual) displacements so while this

approach appears to be in average reasonably accurate, it is also on the non-conservative side of the

likely real maximum values. This is especially true in the region of max > 0.5Uwhere the spectral

estimates using T1are well below the calculated values of max.

Furthermore, the mean error done in predicting maxwith the simplified elastic spectrum approach and

the natural period T2 (in other words using Sd(T2)) is at the very most equal to 13% when the

maximum displacements are greater than 50%U and equal to a mean error of 12% when the

maximum displacements are greater than 70%U.

The standard deviation of the error in these last conditions (when the maximum displacements are

greater than 70%U), varies between 9% and 24%. The degree and gravity of uncertainty that can be

inferred from the mean value and the standard deviation of the error seems hence to be acceptable

compared with the natural uncertainties of the problem that lie in the accurate definition of the realF-

relationship and its idealisation.

It appears that this method, while not giving very good predictions of maximum displacements in the

small amplitude range (max < 0.5U), seems to work very well in the large amplitude (max > 0.7U),

displacement region, hence suitable for predicting whether a wall will collapse or not (see Table 5.6).


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46

Table 5.6

MAX(Sd- MAX)/MAX

(T1)

(Sd- MAX)/MAX

(T2)

(Sd- MAX)/MAX

(TS)Error

Wall 1

-16% 209% 61% Mean Value50%U> MAX> 0

27% 191% 44% Standard Deviation

-59% 9% 116% Mean ValueMAX> 50%U


-67% -5% 131% Mean ValueMAX> 70%U


Wall 4a

-12% 48% 27% Mean Value50%U> MAX> 0


-35% -6% 72% Mean ValueMAX> 50%U


-34% -12% 61% Mean ValueMAX> 70%U


San Salvador

-24% 184% 45% Mean Value50%U> MAX> 0


-52% 3% 9% Mean ValueMAX> 50%U


-58% -4% -20% Mean ValueMAX> 70%U


Artificial

-10% 85% 59% Mean Value50%U> MAX> 0


-41% 13% 116% Mean ValueMAX> 50%U


-47% -1% 117% Mean ValueMAX> 70%U



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Table 5.7

MAX(Sd- MAX)/MAX

(T1)

(Sd- MAX)/MAX

(T2)

(Sd- MAX)/MAX

(TS)Error

Wall 1

-68% -7% - Mean ValueMAX= U

11% 9% - Standard Deviation

Wall 4a



San Salvador



Artificial




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Chapter6 Conclusions

6. CONCLUSIONS

This study has shown the applicability of a simplified linearized displacement-based procedure to

predict the response of URM walls subjected to specific earthquake ground motion. In particular when

the DB approach is applied to assess the possibility of failure of existing walls, it has been shown that

the results appear to be reasonably accurate and fairly conservative.

The main advantage of the method is that it is straightforward in its application and at the same time it

takes into account of the real behaviour of the URM walls (i.e. their reserve capacity due to rocking)

when dynamically loaded.

The natural period T2, which is the effective natural period to be used in the DB approach when the

objective is to predict whether or not the wall will fail, is a function of the maximum force plateau (the

so-called rigid threshold resistance) and of the displacement 2, hence it is a function of the tri-linear

F-relationship for the wall under investigation.

The key point is hence the definition of the characteristic displacement 2.

It is important to underline that the reliability of the method doesnt seem to be affected by the

position of the value 2 with respect to the ultimate displacement U (for a given wall). In other

words the method doesnt seem to work better or worst depending on the absolute value of the natural

period T2(given that the value of T2assumed for the examined wall approximates accurately well the

real T2 value of the wall). This is shown in Figures 6.1 to 6.6 where S d(T2)/max vs. max/U is

reported for the six value of 2/U considered for each examined wall and for maximum

displacements greater than 50%U.


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2/U=0.25

0.00

0.50

1.00

1.50

2.00

2.50

0.5 0.6 0.7 0.8 0.9 1.0

MAX/ u

Sd(T2)/M

AX

Figure 6.1 Sd(T2)/maxvs. max/Ufor 2/U=0.25.

2/U=0.30

0.00

0.50

1.00

1.50

2.00

2.50

0.5 0.6 0.7 0.8 0.9 1.0

MAX/ u

Sd(T2)/M

AX


2/U=0.35

0.00

0.50

1.00

1.50

2.00

2.50

0.5 0.6 0.7 0.8 0.9 1.0

MAX/ u

Sd(T2)/M

AX



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50

2/U=0.40

0.00

0.50

1.00

1.50

2.00

2.50

0.5 0.6 0.7 0.8 0.9 1.0

MAX/ u

Sd(T2)/M

AX


2/U=0.45

0.00

0.50

1.00

1.50

2.00

2.50

0.5 0.6 0.7 0.8 0.9 1.0

MAX/ u

Sd(T2)/M

AX


2/U=0.50

0.00

0.50

1.00

1.50

2.00

2.50

0.5 0.6 0.7 0.8 0.9 1.0

MAX/ u

Sd(T2)/M

AX



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52

For maximum displacements greater than 70%U, which is the range of interest for the values of max

when the objective is to assess the failure of the wall, the mean error varies between 1% and 11% and

the standard deviation of the error varies between 13% and 26%. It can be observed that these ranges

are essentially the same inferred for the general case and reported in Table 5.6 (i.e. between 1% and

12% for the mean error and between 11% and 24% for the standard deviation of the error) revealing

that the method proposed possess the same degree of reliability in the full range of 2/Uwhich has

been considered.

Furthermore the time history analyses performed for this specific purpose have shown that large

changes in the value of the ratio 1/Udo not imply big changes in the value of max/U. This suggests

that an inaccuracy in the evaluation of 1 doesnt invalidate the results of the DB procedure and

emphasise the importance of correctly define the displacement 2.It has to be considered that the point on the tri-linearF-curve where =2determines the maximum

force plateau, corresponding roughly to the maximum strength of the wall. A study recently carried

out at the University of Pavia by L.Picchi [10] showed that the maximum wall strength calculated

according to the currently available theories is relatively insensitive to the wall material properties

such as the elastic modulus and the compressive strength.

In particular in the study mentioned above it has been shown that for a likely range of material

properties (i.e. varying the elastic modulus of the masonry from 500Mpa up to 10000Mpa), the

ultimate wall strength varies by only as much as 10%.

With this result, it is remarkable the possibility of establishing a value of 2 for a wall without the

need for detailed material property data. Once calculated an estimate of Fu (for instance with the

method proposed by Priestley [6]) the value of 2is easily determined on the triangular rigid-bodyF-

envelope. With this point established, it is possible to calculate the secant stiffness, K2, and go on to

assess whether or not collapse will occur using the corresponding period, T2, and the 5% damped

elastic displacement spectra for the earthquake motion being considered.

The accuracy of the method doesnt seem to be conditioned by the characteristics of the earthquakes

considered as input motions. It has anyway to be underlined that even if the selection of the six

earthquakes described in Table 5.4 has been done trying to select seismic excitation with different

characteristics, a complete analysis with all six earthquakes has been done only for the two walls at the

boundaries, in terms of strength, of the considered set of walls (i.e. wall 1and wall 4a). The entire set

of wall has been analysed under the effect of the artificial and San Salvador earthquakes, not revealing

any particular influence of the characteristic of the two earthquakes (quite different one from the other

as it can be noticed in the spectra of Figures 5.5 and 5.6) in the response of the walls.

A more complete set of earthquakes should be considered in order to confirm this trend.


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Still to be confirmed is whether other more complex out-of-plane failure mechanisms can be modelled

in a similar fashion. A study on this issue constitutes an important future development of this work.

There appears to be no reason, in principle, that it can not be done if a comparable rigid body

mechanism exists which has a force-displacement response that can be described with a tri-linear

model.

Another issue to be investigated is whether this approach will work when the hysteretic model is not

symmetric or when friction is accounted for in the model (i.e. when the non-linear F- relationship

becomes inelasticinstead of elastic).


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Chapter7 References

7. REFERENCES

[1] M.J.N.Priestley Seismic behaviour of unreinforced masonry walls Bulletin of the New Zeland

National Society for Earthquake Engineering 1985 18(2)

[2] K.T.Doherty, M.C.Griffith, N.Lam and J.Wilson Displacement-based seismic analysis for out-of-

plane bending of unreinforced masonry wall- Earthquake Engineering and Structural Dynamics 2002

[3] K.T.Doherty, B.Rodolico, N.Lam, J.Wilson and M.C.Griffith The modelling of earthquake

induced collapse of unreinforced masonry walls combining force and displacement principals. 12

World Conference of Earthquake Engineering

[4] A.Shibata, M.A.Sozen Substitute-structure method for seismic design of R/C Journal of

Structural Division, Proceedings of the American Society of Civil Engineering 1976 102, ST1.

[5] FEAP A Finite Element Analysis Program Robert L. Taylor Department of Civil and

Environmental Engineering University of California at Berkeley

[6] T.Paulay, M.J.N.Priestley Seismic Design of Reinforced Concrete and Masonry Buildings. John

Wiley and Sons, Inc. 1992

[7] Anil K. Chopra Dynamics of Structures. Prentice Hall. 2001

[8] Kevin Thomas Doherty An investigation of the weak links in the seismic load path of unreinforced

masonry buildings.Thesis submitted to the Faculty of Engineering at The University of Adelaide for

the Degree of Doctor of Philosophy.

[9] M.J.N.Priestley, R.J.Evison, A.J.Carr Seismic Response of Structures free to rock on their

foundations. Bulletin of the New Zeland National Society for Earthquake Engineering 1978 11(3)

[10] Luigino Picchi Risposta sismica per azioni fuori dal piano di pareti murarie - Tesi di Laurea in

Ingegneria Civile Relatore: Prof. Ing. Guido Magenes Universit di Pavia

[11] George W. Housner The behaviour of inverted pendulum structures during earthquakes.

Bulletin of the Seismological Society of America 1963 53(2)

Dissertation2002 Melis

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