A Simplifies Mechanics-based Procedure for the Seismic Risk Assessment of Unreinforced Masonry Buildings

I.U.S.S. Istituto Universitario

Università degli Studi

di Studi Superiori di Pavia

di Pavia

EUROPEAN SCHOOL OF ADVANCED STUDIES IN REDUCTION OF SEISMIC RISK

ROSE SCHOOL

A SIMPLIFIED MECHANICS – BASED PROCEDURE FOR THE SEISMIC RISK ASSESSMENT OF UNREINFORCED MASONRY BUILDINGS

An Individual Study Submitted in Partial Fulfilment of the Requirements for the PhD Degree in

EARTHQUAKE ENGINEERING

By LUIS FERNANDO RESTREPO VELEZ Supervisor: Prof. GUIDO MAGENES December, 2003

The dissertation entitled “A Simplified Mechanics-Based Procedure for the Seismic Risk Assessment of Unreinforced Masonry Buildings”, by Luis Fernando Restrepo Vélez, has been approved as second dissertation, in partial fulfilment of the requirements for the PhD Degree in Earthquake Engineering.

Guido Magenes

Paolo Emilio Pinto

Abstract

ABSTRACT

A simplified mechanics-based procedure for the seismic risk assessment of unreinforced masonry buildings is developed, based on some of the ideas presented in the original procedure proposed by Calvi [1999]. This new procedure is based on the application of a displacement-based approach for vulnerability evaluation of classes of buildings, but relevant changes have been carried out, in order to correct the main drawbacks identified. One drawback is related to the out-of-plane behaviour that has not been originally included. Another drawback is related to the definition of the joint probability density function JPDF of displacement ∆ and period T; the real JPDF must consider the fact that ∆ and T are not two independent variables. For the in-plane mechanisms the demand is represented by the displacement response spectrum which is obtained from the regional probabilistic seismic hazard study, and must be defined by the median or mean value for each spectral period and also by the standard deviation. For the out-of-plane mechanisms the demand is represented according with the formulation presented in Eurocode 8 to compute the acceleration for non-structural components. Appropriate definition of the JPDF would be necessary to be included in the methodology; however, two main problems arise: the first is the definition of the JPDF, which should include the effect of all the related sources of uncertainty and that cannot be estimated from the sample with enough accuracy, but rather have to be possibly chosen by means of a heuristic approaching. The second problem is related with the integration of the JPDF over the failure domain, in order to compute the probability of failure. An alternative path has been chosen to overcome these problems, by estimating the CDF of ∆ and T using first order reliability methods (FORM). Considering all the sources of uncertainty, the strategy used in the procedure can treat the uncertainty coming from the material properties and structural response, and the uncertainty coming from the statistics of the population of buildings. The former is directly included in the computation of the probability of failure for a single building or class of building, whilst the latter is considered in a second stage. As a complement of the proposed methodology a survey data form has been designed to gather the data field required for the application of the methodology in real cases.

i

Acknowledgments

ACKNOWLEDGEMENTS

Being far is not an easy thing, neither for those who stay there, or for us living here. My deepest gratitude to my family, for all the help and support during this period that I have spent far from them. Thank you very much for your understanding and unconditional love. A very special mention to the ROSE School for all the academic and financial opportunities that they have given me so far, and specially to Professor Guido Magenes for his clear explanations on the masonry issues, for the excellent guidance, for the opportunity of working as his research assistant and for his generosity on providing financial support. A true feeling of gratitude to Prof. Paolo E. Pinto and to Dr. Paolo Franchin for all the invaluable explanations and help concerning the probability and reliability issues. Hopefully, now I have got the idea. The discussion sessions with my colleagues Ms. Helen Crowley and Ms. Rosamaria Iaccino were always very fruitful; thanks to them that helped me to improve my work. The most important place in this acknowledgements if for my wife Lina, for her endless patience and for putting my priorities always first than her priorities. My infinite love is for you.

ii

Acknowledgments

AGRADECIMIENTOS

Estar lejos de la familia no es algo fácil, tanto para los que están allá como para los que estamos acá. Mis más profundos agradecimientos para mi familia, por toda la ayuda y el respaldo que me han brindado durante todo este tiempo que he estado lejano. Muchas gracias por su comprensión y amor incondicional. Una mención muy especial para la ROSE School por todas las oportunidades de carácter académico y financiero que me han dado hasta ahora, y muy especialmente al Profesor Guido Magenes por sus claras explicaciones relacionadas con los temas de mampostería, por ser un excelente guía, por la oportunidad de poder trabajar como su asistente de investigación y por su gran generosidad al proporcionarme patrocinio. Un verdadero sentimiento de gratitud para el Profesor Paolo E. Pinto y para el Dr. Paolo Franchin por toda las invaluables explicaciones relacionadas con los temas de probabilidad y confiabilidad. Espero haber finalmente captado la idea. Las sesiones de discusión con mis colegas Helen Crowley y Rosamaria Iaccino fueron siempre muy provechosas y contribuyeron a mejorar la calidad de mi trabajo. El sitio más importante en estos agradecimientos es para mi esposa Lina, por su paciencia sin límites y por poner mis prioridades siempre por delante de las suyas. Mi infinito amor es para ti.

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Index

A SIMPLIFIED MECHANICS – BASED PROCEDURE FOR THE SEISMIC RISK ASSESSMENT OF UNREINFORCED MASONRY BUILDINGS

INDEX

page ABSTRACT …………………………………………………………………………………….

i

ACKNOWLEDGEMENTS …………………………………………………………………….

ii

AGRADECIMIENTOS …………………………………………………………………………

iii

INDEX …………………………………………………………………………………………..

iv

LIST OF TABLES ………………………………………………………………………………

vi

LIST OF FIGURES ……………………………………………………………………………..

vii

1.

INTRODUCTION ………………………………………………………………………..

1

2.

VULNERABILITY AND LOSS ESTIMATION PROCEDURES ……………………..

3

2.1

DAMAGE PROBABILITY MATRIX ………………………..…………………...

4

2.2

GNDT I AND II LEVEL APPROACHES ..………………………….…………….

6

2.3

VULNUS ..………….………………………………………………………………

8

2.4

HAZUS …………………………………………………………………………….

10

2.5

FaMIVE ….……….………………….…….………………………………………

14

iv

Index

page 2.6 2.7 3.

15

GENERAL REMARKS …….……………………………………………………...

18

MeBaSe PROCEDURE ...……………………………………………………………….

19

3.1

PROCEDURE FOR IN-PLANE FAILURE MECHANISM ....................................

19

3.1.1 3.1.2 3.1.3

Demand …………………………................................................................. Limit States …………………....................................................................... Simplified Structural Model ………………….............................................

19 20 20

PROCEDURE FOR OUT-OF-PLANE FAILURE MECHANISM ……………….

29

3.2.1 3.2.2 3.2.3

Demand …………………………................................................................. Limit States …………………....................................................................... Simplified Structural Model ………………….............................................

30 33 34

PROBABILISTIC CONSIDERATIONS ...………………….……………………..

36

3.3.1 3.3.2 3.3.3 3.3.4 3.3.5

37 40 41 42 48

3.2

3.3

3.4 4.

SIMPLIFIED DISPLACEMENT – BASED PROCEDURE FOR CLASESS OF BUILDINGS …….…………………………………………………………………

Uncertainties …………..………………....................................................... Definition of the JPDF for structural capacity ……………………………. Definition of the Seismic Demand ………………………………………... FORM Procedure ………..……………….................................................... Probability of failure for In-plane and Out-of-plane mechanisms ................

DAMAGE PROBABILITY AT URBAN OR REGIONAL SCALE ……………

APPLICATION EXAMPLES ……………………………………………………………. 4.1

50 52

PROBABILITY OF FAILURE FOR IN-PLANE RESPONSE OF A 4-STORIES UNREINFORCED MASONRY BUILDING ……………………………………...

52

PROBABILITY OF FAILURE FOR OUT-OF-PLANE RESPONSE OF SINGLE WALLS ……………………………………………………………………………..

59

5.

SURVEY DATA FORM …………………………………………………………………

63

6.

CONCLUSIONS ………………………………………………………………….………

68

6.1

SUMMARY …………………………………….......................................................

68

6.2

FINAL REMARKS ……………………………………...........................................

70

6.3

FUTURE RESEARCH ……………………………………………………………..

70

4.2

7.

BIBLIOGRAPHY ………………………………………………………………………..

ANNEX 1.

FORTRAN Code for the MeBaSe Procedure

ANNEX 2.

Survey data form v

72

Index

LIST OF TABLES

2.1

2.7 2.8 3.1 3.2 3.3 4.1 4.2 4.3

Classification of vulnerability assessment methods, according to input, methodology and output [Dolce et al., 1994]. European Macroseismic Scale EMS [Gruntal, 1998]. DPM Irpinia [i.e. Braga et al.; Giovinazzi and Lagomarsino, 2001]. Scores and relative weights to compute Iv. Classification and corresponding values of the vulnerability factors [Bernardini, Gori and Modena, 1990]. Vulnerability factors related to qualitative judgements and their corresponding weights [Bernardini, Gori and Modena, 1990]. Linguistic relationship between a and I3 [Bernardini, Gori and Modena, 1990]. Limit states for masonry structures, Calvi [1999]. κ1 and κ2 values, according to the number of stories. Displacement ratios for the tri-linear model. ∆LS according to the boundary conditions. Computation of the seismic coefficient C. Computation of median values for ∆LS and TLS.. Parameters for computing FTLS and F∆ LS using FORM procedure.

4.4 4.5 4.6 4.7 4.8

Median and standard deviation values for vector X. Probability of failure for each given Limit State. Probability of failure for each given Limit State for the same seismic demand. Median and standard deviation values for vector X. Probability of failure for different demand levels.

2.2 2.3 2.4 2.5 2.6

vi

Index

LIST OF FIGURES

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 4.1 4.2

Differentiation of structures (buildings) into vulnerability classes [Gruntal, 1998]. Acceleration versus damage ratio tri-linear curves for masonry buildings proposed in the GNDT II level approach [after GNDT, 1993]. Definition of the safety criterion [Bernardini, 2000]. Example of building capacity curve and demand spectrum [FEMA, 1999]. Example of fragility curves for slight, moderate, extensive and complete damage states [FEMA, 1999], for a specific class of buildings. Example of building damage estimation process [FEMA, 1999]. Out-of-plane mechanisms taken into account in the FaMIVE procedure [D’Ayala and Speranza, 2002]. Displacement demand and capacity for LS4 and masonry building class of building (ground acceleration 0.3g), Calvi, [1999]. Acceleration and Demand Response Spectra according to Eurocode 8, for ag = 0.3g, S = 1.0 and k = 1.0. Simplified equivalent SDOF model. Deformed shapes for different limit states and in-plane failure modes. Simplified model for the definition of κ2. Definition of κ2 according to the storey at failure. Elastic-Perfectly plastic response for two different structures. Force distribution, storey shears, storey strengths and seismic coefficients. Boundary conditions considered. Seismic response of an unreinforced masonry building [Paulay and Priestley, 1992]. Flexible floor response to ground excitation [Paulay and Priestley, 1992]. Example of acceleration and displacement response spectra computed with Eurocode 8. Tri-linear simplified model. Comparison of the limiting displacement ∆2 and the displacement response spectrum DRS. Sources of uncertainty. Components for different uncertainty models. Sources of uncertainty on capacity response for in-plane walls. Sources of uncertainty on capacity response for out-of-plane walls. Seismic hazard curve of Peak Ground Acceleration for Gaziantep, Turkey, adapted from Restrepo-Vélez and Bommer [2003]. Approximation of the limit state function g in the normalized standard space. Subdivision of the hypothetic region in zones. Plan view and façade for the in-plane assessment example. Comparison of median values of capacity and demand for each given limit state. vii

Index

4.3

4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 5.1 5.2 5.3 5.4

Cumulative density functions CDF for demand FD. Cumulative density functions CDF for TLS, and ∆LS conditional to TLS = 1.0 sec. Comparison of median values of capacity and demand for each given limit state, for performance level 2 – PGA 0.15g. Variation of the probability of failure Pf with the coefficient of correlation ρ’ij, for LS4. Variation of the sensitivity factors αsen with the coefficient of correlation ρ’ij, for LS4. Comparison of median values of capacity and demand. Cumulative density functions CDF for demand FD. Cumulative density functions CDF for TLS, and ∆LS conditional to TLS = 1.5 sec. Variation of the sensitivity factors αsen with the coefficient of correlation ρ’ij, for LS4. Main screen for the website used to manipulate the database. Partial view of the screen used to enter the write the information to the database. Partial view of the screen used to control and to modify the information written in the database. Output view of the screen used to write the input file required for the MeBaSe procedure.

viii

Chapter 1. Introduction

1.

INTRODUCTION

A big research effort has been made on the prediction of earthquakes in the last decades, and in fact the exploration of new techniques aiming to foresee the occurrence of seismic events is in a continuous progress. However, the possibility of knowing in advance the occurrence of a major earthquake is still far from being a reality, and the preparedness to face an eventual emergency has to be made from the point of view of the prevention rather than the prediction. Several analytical tools have been developed around the world in order to estimate, with different degrees of accuracy, the vulnerability of buildings and the probable loss of lives and economic resources, due to the occurrence of an earthquake. Those tools are intended to be used by government agencies, and even by insurance companies, as a mean for planning of emergency preparedness procedures and response strategies, and also for the reconstruction phases. Nonetheless, most of the current available tools require a large amount of resources, in terms of money, time and computational effort, in order to be properly implemented and effectively used. Additionally, large portions of buildings in non-developed countries, and an important portion of buildings in developed countries, especially old constructions, are built with unreinforced masonry, under several forms. In most modern design codes the use of unreinforced masonry has been banned for the construction of new buildings in moderate to high seismic regions. However, in low seismic regions or in those countries were the use of an official building code has not been yet implemented, the use of unreinforced masonry is still one of the most preferable choices for new low-rise buildings, considering its relatively low cost and easy construction procedures. From the arguments exposed in the previous paragraphs, it is possible to say that the development of a simplified methodology for the seismic vulnerability, or better seismic risk assessment of unreinforced masonry buildings, would have a considerable relevance, especially taking into account the social 1

Chapter 1. Introduction

benefits that some new developments in this field will bring to poor communities in developing countries, providing useful tools for governments and other agencies. The simplified methodology presented in this dissertation includes, apart from the considerations for the in-plane failure mechanism, a formulation for the out-of-plane behaviour. Out-of-plane collapse mechanism has received less research compared to the in-plane behaviour, and the aspects related to the dynamic behaviour and vulnerability assessment still offer a lot of space for new developments. This dissertation is organized into four chapters, covering from the theoretical development of the methodology to the discussion of some numerical examples. Chapter 2 is a general review of some of the methodologies most extensively used in Europe and America, for the assessment of the vulnerability and loss estimation.

This review includes the

Damage Probability Matrix DPM, the GNDT first and second level approaches, the VULNUS , the FaMIVE and the HAZUS procedures. Chapter 3 is dedicated to the development of a new simplified methodology for the seismic risk assessment of unreinforced masonry buildings. The methodology includes the procedures for the evaluation of in-plane and out-of-plane failure mechanisms, and also the complete analysis and development of the probabilistic framework, necessary in order to considerer the main sources of uncertainties existing in both the methodology and the physical phenomena. The procedure included in this dissertation is presented as a first step of a major development that is being carried out as a part of a PhD research project. In Chapter 4 some numerical examples are presented with the purpose of illustrating in a detailed way the steps of the procedure, including the probabilistic considerations.

Finally, the survey form used to gather the required information for the proposed procedure is presented in Chapter 5.

2

Chapter 2. Vulnerability and Loss Estimation Procedures

2.

VULNERABILITY AND LOSS ESTIMATION PROCEDURES

The purpose of this chapter, rather than doing a full description of some of the most used vulnerability and loss estimation procedures, is to explain the basic ideas of those procedures, showing some of their advantages and limitations, aiming to highlight the interest and potential of developing rational mechanics – based methodologies. Several methods for vulnerability assessment have been developed in recent years, considering different approaches for in the input data and for the output. Corsanego and Petrini [1990] proposed the following classification for the methods, according to the type of building: •

Direct: these methods are subdivided in two groups. The typological group is based on data gathered after the occurrence of real earthquakes that are statistically manipulated to obtain damage probability matrices for a limited number of classes of buildings. The mechanical group is based on numerical models and the output is usually the level of damage for a given building of a given class.

•

Indirect: these techniques use both the data obtained after the occurrence of real earthquakes and the data obtained through numerical models.

•

Conventional: these correspond to methodologies based on judgements given by experts. The output is a vulnerability index for a given class of building but this vulnerability is not directly correlated with any specific level of damage.

A different classification was proposed by Dolce et al. [1994], based on the input, the methodology and the output, considering the options shown in Table 2.1. 3


Table 2.1. Classification of vulnerability assessment methods, according to input, methodology and output [Dolce et al., 1994] Input Method Output 1. Damage data 1. Statistical methods 1. Absolute vulnerability 2. Geometric and qualitative 2. Mechanical methods 2. Relative vulnerability features 3. Methods based on expert 3. Mechanic features judgements 4. Seismic demand features 5. Geological and geotechnical data

Different input-method-output combinations are possible, resulting in methods with different levels of applicability and accuracy. Nonetheless, Dolce et al. [1994] recognized that for masonry buildings just hybrid expert-statistical methods are available, and that no method based on mechanical features exists. Absolute quantification of the vulnerability is not available neither. The selection of the adequate methodology must be based on the following criteria [Dolce, 1997]: •

Purpose and regional scale of the damage scenario.

•

Accuracy of the earthquake scenario used for the damage scenario.

•

Available resources in terms of time and funds to prepare the building catalogue.

•

Quantity and accuracy required in the information according to the different types of feasible methodologies.

•

Existing information, either from government bodies, databases, libraries, etc.

From the available literature and considering the vast collection of methodologies, the following six procedures are explained in more detail, taking into account that they have been used or proposed to be used for the assessment of unreinforced masonry buildings.

2.1

DAMAGE PROBABILITY MATRIX

The damage probability matrix method or DPM is based on the idea that a set of buildings having a common structural typology would have the same behaviour under the action of an earthquake, and as a consequence, the level of damage would be the same for the set of buildings. The damage is characterized by some level of uncertainty described by a damage probability matrix [i.e. Whitman, 1973; Di Pasquale et al., 2001]. Each element of the matrix is expressed according to Eq.(2.1): 4


DPM (DV , I , T ) = P(DV | I , T )

(2.1)

where DV corresponds to a given level of damage, T is an specific structural typology and I is the earthquake intensity, normally described by some macroseismic scale, for example the European Macroseismic Scale EMS , given in Table 2.2 [Gruntal, 1998]. Table 2.2. European Macroseismic Scale EMS [Gruntal, 1998] Scale Description Not felt I Scarcely felt II Weak III Largely observed IV Strong V Slightly damaging VI Damaging VII Heavily damaging VIII Destructive IX Very destructive X Devastating XI Completely devastating XII

This methodology is considered as a direct method because it allows estimating the vulnerability in one single step, considering the building as a member within a specific class. The level or class of vulnerability according to the structural typology of masonry buildings is shown in Figure 2.1, according to the proposal given in the EMS-98, being A the highest vulnerability and F the lowest.

Figure 2.1. Differentiation of structures (buildings) into vulnerability classes [Gruntal, 1998]

According to Eq.(2.1), the DPM would be a matrix having the probability of reaching some level of damage for a given earthquake intensity. An example of a DPM is shown in Table 2.3 prepared after the Irpinia earthquake in Italy [i.e. Braga et al., 1982; Giovinazzi and Lagomarsino, 2001].

5


Table 2.3. DPM Irpinia [i.e. Braga et al, 1982; Giovinazzi and Lagomarsino, 2001] EMS Intensity Class A Class B Class C

V 0.020 0.010 0.005

VI 0.284 0.185 0.065

VII 0.423 0.284 0.167

VIII 0.726 0.501 0.334

IX 0.860 0.700 0.500

X 0.923 0.850 0.700

One of the main disadvantages of the DPM method is the use of a discrete measure of the earthquake intensity through the use of a macroseismic intensity scale, rather than using a different definition of intensity, for example a continuous parameter like acceleration or displacement, with a better correlation with the level of damage. A key issue here is that the input is described by the effects of the ground – motion on structures, rather than other parameters measuring directly the input ground – motion, which is somehow like using the responses to compute the response. Another disadvantage is that the DPM does not consider the uncertainty on the demand, which one of the components of uncertainty that are required to be included in a complete vulnerability assessment, as discussed in Section 3.3.1. Finally, the DPM method does not allow the estimation of the vulnerability for a single building, but just the evaluation as a part of a class of buildings; thus, it not possible to individuate all the features of each specific building.

2.2

GNDT I AND II LEVEL APPROACHES

The Gruppo Nazionale per la Difesa dai Terremoti – GNDT is the Italian government research body in charge of the seismic risk evaluation and definition of the required measures to reduce it. Two, somehow complementary, approaches have been published by this group, with the aim of being applied in the assessment of the seismic risk in the Italian territory. The GNDT I level approach is nothing more than a DPM method, having three classes of vulnerability, from A to C, each of these having a DPM. For this approach the demand is considered through the use of the EMS-98 intensity scale and the damage is described by means of a qualitative description, according to the level of damaged reached by the building. Further description of this methodology can be found elsewhere [e.g. GNDT, 1993]. The GNDT II level approach is based on a survey form designed to gather information regarding the typology and constructive features for each single building, that are combined afterwards to get a vulnerability index IV. Eleven parameters are combined with different scores and relative weights, depending on four classes of vulnerability, as shown in Table 2.4. IV is an absolute measure from 0 to 382.5 but eventually can be normalized from 0 to 100, being 0 the best vulnerability condition and 100 the worst.

6


Table 2.4. Scores and relative weights to compute Iv Parameter

Class

Type and layout of resistant system Quality of resistant system Conventional resistant Location of building and foundations Horizontal elements (floor, diaphragm) Configuration in plan Configuration in height Max. distance between walls Roof Non-structural elements State of conservation

A 0 0 0 0 0 0 0 0 0 0 0

B 5 5 5 5 5 5 5 5 15 0 5

Weight C 20 25 25 25 15 25 25 25 25 25 25

D 45 45 45 45 45 45 45 45 45 45 45

1.00 0.25 1.50 0.75 variable 0.50 variable 0.25 variable 0.25 1.00

For each vulnerability index there is a corresponding curve correlating the damage ratio and the demand represented by the PGA, by means of a tri-linear curve resembling somehow the so called “fragility curves”, which are better explained in Section 2.4. The damage is computed in terms of economical loss, correlated as a function of the peak ground acceleration PGA [Giovinazzi and Lagomarsino, 2001]. Figure 2.2 shows the acceleration versus damage ratio tri-linear curves for masonry buildings proposed in the GNDT II level approach.

1.0 Iv = 100

Damage ratio

0.8

0.6

0.4 Iv = 0

0.2

0.0 0.0

0.1

0.2

PGA [g]

0.3

0.4

0.5

Figure 2.2. Acceleration versus damage ratio tri-linear curves for masonry buildings proposed in the GNDT II level approach [after GNDT, 1993]

Some important drawbacks of the I and II level approaches is that their intervals of confidence are very large, both for the arbitrary way used to defined the points and weight for the vulnerability index, and for lack of correlation between PGA and the level of damage of a given building [Giovinazzi and Lagomarsino, 2001].

7


2.3

VULNUS

The VULNUS procedure was developed in the second half of the ‘80s at the University of Padova, with the purpose of evaluating the seismic vulnerability of a single building or group of buildings [Bernardini, Gori and Modena, 1990]. The methodology is based on the evaluation of the geometrical and mechanical characteristics of each building, which is combined with the evaluation of some other important factors controlling the response of the structure, that are handled through qualitative judgments. The whole procedure is developed under the fuzzy set theory that is used for the definition of the safety criterion. This method could be considered inside the mechanical group because it makes use of the so called collapse multipliers. The geometrical and mechanical characteristics are described with two indices or collapse multipliers, according to Eqs (2.2) and (2.3):

I1 =

min (V x , V y )

(2.2)

W

(

I 2 = min I 2' + I 2'' i

)

(2.3)

i

I1 is the collapse multiplier for in-plane behaviour considering shear failure at ground floor, being W the total weight of the building and Vx and Vy the strength at mid-storey height of the ground floor, according to Eq.(2.4):

  {Vx ,V y } = {Fx , Fy } f t 1 + ( W ) 1.5ω  f t Fx + Fy 

{

1

2

(2.4)

}

where Fx , Fy are the total areas of the walls in the x and y direction respectively, ft is the tensile strength of masonry [Ministero dei Lavori Pubblici, 1981] and ω is a factor to include the effects of plan regularity. In this expression it has been assumed that the walls are rigidly jointed to the slabs and are subjected to uniform vertical compression. I2 is the collapse multiplier for the out-of-plane behaviour, considering each single wall i and several failure modes, namely: overturning, flexural tension, arch crushing, shoulders overturning and tension. A detailed description of the computation procedure for this collapse multiplier can be found in Bernardini, Gori and Modena [1990].

8


Once the in-plane I1 and out-of-plane I2 indices have been computed, a safety criterion is chosen to estimate the vulnerability of a given building. The definition of the safety criterion is shown in Figure 2.3, where c1 = 0.5, c2 = 1.0, c3 = 0.1 and c4 = 1 are the values typically assumed for buildings in Italy. The parameter u is defined with Eq.(2.5), being A the maximum base shear divided by the total weight of the building W.

c + c − c + [(I 1 / A − c1 )(I 2 / A − c1 )] u= 3 1 2 2c3 + ac 4

1

2

(2.5)

The parameter a depends on the qualitative judgments that are performed taken into account large databases gathered in Italy by means of several surveys performed in the past. An important feature of this parameter is that it can be updated after the continuous improving of the database [Bernardini, 2000]. The qualitative judgments are expressed as a combination of seven vulnerability factors Si and their corresponding weights Wi, and is evaluated with Eq.(2.6).

Figure 2.3. Definition of the safety criterion [Bernardini, 2000]

I3 = ∑ i

Wi S i 45 × 3.15

(2.6)

Tables 2.5 and 2.6 shows the proposed values for the size and the weight of each vulnerability factor, respectively.

9


Table 2.5. Classification and corresponding values of the vulnerability factors [Bernardini, Gori and Modena, 1990] Class Size S 1 Good or corresponding to code 0 2 Almost good 15 3 Almost poor 30 4 Poor or unsafe 45

Table 2.6. Vulnerability factors related to qualitative judgements and their corresponding weights [Bernardini, Gori and Modena, 1990] Vulnerability factors Weight W 1 Walls system quality 0.15 2 Soil and foundations interaction 0.75 3 Floors interaction 0.50 4 Elevation regularity 0.50 5 Roof interaction 0.50 6 Interaction of non-structural elements 0.25 7 General maintenance conditions 0.50 Total 3.15

Once I3 has been obtained, a linguistic relationship between a and I3 is established, according to Table 2.7. Table 2.7. Linguistic relationship between a and I3 [Bernardini, Gori and Modena, 1990] j=1 If I3 is very large then a is very large j=2 If I3 is large then a is large j=3 If I3 is medium then a is medium j=4 If I3 is small then a is small j=5 If I3 is very small then a is very small

After the three parameters I1, I2 and I3 have been computed and the linguistic relationship has been established, fuzzy set theory is applied in order to compute the vulnerability value. Details of the corresponding fuzzy set theory are not reported here but can be found in Bernardini, Gori and Modena [1990]. The main advantages of this method are that it allows to compute an absolute value for the vulnerability with respect to the intensity of the given ground motion, and also allows to classify the surveyed buildings in an orderly way with respect to the vulnerability measure.

2.4

HAZUS

HAZUS is an acronym for “HAZard in the U.S.” and corresponds to a methodology developed by the Federal Emergency Management Agency [Whitman et al., 1997; FEMA, 1999]. The methodology is based on three fundamental concepts: capacity curve, design point and fragility curve. 10


Capacity curve is the relationship between the lateral load resistance of a given structure and its characteristic lateral displacement [Kircher et al., 1997], and is typically obtained by means of a static pushover analysis. The capacity curve is then converted to spectral acceleration and roof displacement in order to be compared with the demand spectrum. Figure 2.4 shows an example of the capacity curve and the demand spectrum for a given building. It can be seen that the capacity curve is controlled by the yield capacity, or restoring force, and the ultimate capacity, being possible to represent with this single curve the corresponding strength at a certain displacement limit state, which is turns can be directly correlated with some damage limit state. The damage limit states considered by HAZUS are: slight, moderate, extensive and complete; their description can be found in Kircher et al . [1997]. For a given building class, defined by the structural system, the building type and the occupancy class, a specific capacity curve is defined, for which the design point (Sa, Sd) is obtained. The design point corresponds to the intersection of capacity and demand curves.

Figure 2.4. Example of building capacity curve and demand spectrum [FEMA, 1999]

The fragility curve concept represents the cumulative density function CDF for the probability of reaching or exceeding a specific damage limit state for a given peak response to a given ground motion demand. In HAZUS the fragility curves are assumed to be represented by lognormal functions, hence they can be fully described by the median and the standard deviation, according to Eq.(2.7):

  1  Sd ln _ P[ds S d ] = Φ   β ds   S d ,ds 

11

   

(2.7)

Chapter 2. Vulnerability and Loss Estimation Procedures _

where: S d ,ds is the median spectral displacement for which the building reaches the damage limit state ds, β ds is the standard deviation of the natural logarithm of spectral displacement for the damage limit state and Φ is the standard normal cumulative density function. The variability of a given damage state βds is obtained with Eq.(2.8), where βD is the variability of the ground-motion demand, βC is the variability of the capacity response and βT,ds is the variability of the damage state threshold.

(CONV [β C , β D ])2 + (β T ,ds )2

β ds =

(2.8)

In Eq.(2.8) CONV represents the convolution process applied to the variability of the ground-motion demand and the capacity response that is necessary to carry out, taken into account that demand spectrum is dependent on building capacity. The convolution process is explained in Kircher et al . [1997].

Nonetheless, when the variability of the ground-motion demand has already been

incorporated into the seismic hazard assessment, which is the normal practice, the variability of a given damage state must be computed with Eq.(2.9) in order to avoid a double counting of βD.

β ds =

(β C )2 + (β T ,ds )2

(2.9)

_

In HAZUS, typical median values S d , ds of spectral displacement and standard deviation values β ds of the natural logarithm of spectral displacement are given for 36 different classes of buildings, based on experimental tests, experience and judgment. Those parameters are strictly valid for buildings within the United States. Figure 2.5 shows an example of fragility curves for slight, moderate, extensive and complete damage states obtained as explained above. Once the design point and the fragility curve for a specific class of building have been computed, the probability of reaching or exceeding a specific damage limit state is obtained. The process of building damage estimation is schematically shown in Figure 2.6. It is worth noting that in HAZUS fragility curves for non-structural drift sensitive components and non-structural acceleration sensitive components are also provided.

12


Figure 2.5. Example of fragility curves for slight, moderate, extensive and complete damage states [FEMA, 1999], for a specific class of buildings

Figure 2.6. Example of building damage estimation process [FEMA, 1999]

An important advantage of HAZUS is its ability to estimate the damage under a given earthquake scenario, considering damages not just in buildings but also in lifelines, transportation systems, utility systems, essential and high potential loss facilities. In fact, with HAZUS is possible to compute the damage due to earthquake hazard, inundation, fire and hazardous materials. 13


The main disadvantage stems from its complexity. HAZUS is so complete and powerful that is takes a lot of resources to be implemented in a real application for a small to medium community. In HAZUS’s users manual [FEMA, 1999] it is stated that the methodology allows for different levels of funding which means different levels of inventory collection. However, when low levels of funding are available, the information that is missing has to be assumed as similar to the one provided in the HAZUS database, which has been validated to be used in the United States.

2.5

FaMIVE

The Failure Mechanism Identification and Vulnerability Evaluation method, FaMIVE [D’Ayala and Speranza, 2002], has been developed in recent years aiming to the assessment of the seismic vulnerability of historic buildings in town centres. The procedure makes use of an electronic survey form that helps to gather the information on the field and allows also to store it in a database. Limit state and collapse mechanism analyses are in the core of the procedure, taking into account in the process the friction coefficient of masonry, a detailed modeling of the restriction provided by the lateral supporting walls and also the structural retrofit carried out in the buildings. FaMIVE uses a predefined set of rules in the definition of the most feasible collapse mechanism, for which a load factor or collapse multiplier is associated by means of a static equivalent procedure. This load factor is further manipulated to lead to the definition of some vulnerability value. D’Ayala and Speranza claim that this procedure can be applied to “medium size samples of buildings” without sacrificing the benefit of a detailed knowledge of the mechanical and geometrical features of each individual building. The application of the procedure starts with the identification of all the structural typologies existing in the zone being studied. Afterwards, a detailed survey of the relevant geometrical parameters (height, length, thickness, number of stories, strengthening devices) for every façade wall on each building is carried out.

The surveyor must associate each building with one of the structural typologies

previously defined for the zone; then, according with existing information and the type of building, the equivalent shear capacity ESC is computed for each façade wall, varying the number of stories participating in the mechanism. The eight collapse mechanisms considered in the procedure are presented in Figure 2.7. The selected collapse mechanisms are then ordered according to their ESC 14


and finally a judgment is made, considering the danger posed by each mechanism. The final result is the vulnerability of the specific building. The advantages of this procedure are the use of mechanical criteria for the definition of the vulnerability and the inclusion of the most common out-of-plane failure mechanisms.

Other

advantages are the use of the friction coefficient in the definition of the failure mechanisms and also the possibility of applying the procedure to specific buildings. The main drawbacks are not to consider the displacement rather than the acceleration demand and also to ignore the inherent uncertainty existing in the seismic demand; being these two aspects quite relevant, as will be discussed in Chapter 3.

Figure 2.7. Out-of-plane mechanisms taken into account in the FaMIVE procedure [D’Ayala and Speranza, 2002]

2.6

SIMPLIFIED DISPLACEMENT – BASED PROCEDURE FOR CLASSES OF BUILDINGS

A simplified procedure for the evaluation of the vulnerability of masonry buildings, subject to in-plane failure mechanisms, was developed by Calvi [1999], based on the application of the displacementbased approach for vulnerability evaluation of classes of buildings. That procedure considers the energy dissipation and displacement capacity of the existing buildings, and through a very simplified probabilistic approach, computes the probability of occurrence of a specific damage state limit for a given earthquake motion.

15


In general, the steps suggested in the procedure are: •

Definition of the demand. For a given site or area being studied, the demand is defined in terms of an elastic displacement response spectra DRS, based on the selection of an specific return period, or better probability of exceedance. The selected response spectra must include the effects of soil condition and geographical location. Response spectra taken from seismic design codes could be a suitable choice to define the demand.

•

Definition of performance levels or limit states. The limit states are defined in terms of interstorey drift. Table 2.8 shows the limit states for masonry structures used by Calvi [1999] and based on a work by Magenes and Calvi [1997]. The first two levels of limit state corresponding to Undamaged Buildings and Slightly Damaged-Usable Buildings have been mixed into one single limit state, taking into account that masonry buildings do not exhibit a clear definition of the elastic limit, because cracking develops at very early loading stages. No coefficient of variation for drift is shown for LS1-LS2 and LS3; however, from experimental results it can be stated that the scatter for ultimate limit state is much more reduced than the scatter for initial and intermediate limit states, [Magenes and Calvi, 1997].

Limit State LS1 – LS2 LS3 LS4

•

Table 2.8. Limit states for masonry structures, Calvi [1999] Coefficient of Median Drift δLS (%) ξ (%) Variation for Drift (%) 0.1 --2 0.3 --10 0.5 10.9 15

µLS 1 1 + 3/n 1 + 6/n

Definition of simplified structural models. The simplified models are defined considering a linear displacement profile for LS1-LS2 and a soft-storey displacement profile with failure at the ground floor for the LS3 and LS4. The corresponding displacement for any given limit state is computed with Eq. (2.10), for which hs is the typical storey height and n is the number of stories.

∆ LS = 0.67hs nδ y + (δ LS − δ y )hs

(2.10)

The period of vibration is computed with Eq. (2.11), using the same ductility factors µLS shown in Table 2.8.

TLS = 0.04(nhs )

16

3

4

µ LS

(2.11)


•

Definition of a joint probability density function JPDF in the displacement ∆ − period T space, for which a uniform distribution is assumed, limited by the computed minimum and maximum expected displacement capacities, and by the minimum and maximum expected period of vibration.

•

Computation of the probability that the demand exceeds the capacity. The probability is computed by integrating the JPDF of ∆-T in the areas above the DRS. Figure 2.8 shows the displacement demand and the capacity for masonry building class and a ground acceleration of 0.3g.

Figure 2.8. Displacement demand and capacity for LS4 and masonry building class of building (ground acceleration 0.3g), Calvi, [1999]

The main advantage of this methodology is the possibility to evaluate the seismic vulnerability of a given class of buildings, considering different levels of quality in the data and in the refinement in modelling and analysis [Calvi, 1999]. Here it is worth stressing the importance of this methodology in proposing a mechanical formulation for the solution of the problem. Two main drawbacks can be identified in this methodology: First, the out-of-plane behaviour has not been included arguing that out-of-plane collapse mechanism is only feasible when ties or transversal walls are absent. Out-of-plane behaviour has received little attention in the literature, and has been considered as a marginal, or at least, a local structural problem. However, the experience in recent earthquakes and numerous works available in the literature [e.g. Giuffrè, 1999; D’Ayala et al., 1997] have shown that the outward out-of-plane collapse of façade walls is the most probable failure mode. They also have shown that the out-of-plane collapse occurs before the activation of some in-plane 17


mechanism. Besides, it would be possible that some out-of-plane collapse occurs, even in buildings with ties or good connection of the out-of-plane walls, especially considering that these walls are sensitive to displacement rather than to acceleration, according to what has been shown by Doherty et al [2002].

The second drawback is related to the definition of the JPDF of ∆ and T. To treat the JPDF f ∆ ,T (∆, T ) as a uniform distribution for the ∆−T space, considering both variables as independent, was useful for the description of the original methodology, but the real JPDF must take into account the fact that ∆ and T are not two independent variables.

2.7

GENERAL REMARKS

The purpose of the revision reported in this chapter was to identify possible features that ideally would be considered also in the new procedure, and problems that would be either solved or avoided in order to have a sounder methodology. The features that have been identified to be ideally included in the new procedure are: •

The use of displacement or drift as an indicator of the demand level, which has a better correlation with the level of damage.

•

To consider the uncertainty on the demand, recognized to be one of the components of uncertainty that are required to be included in a complete seismic risk assessment.

•

The use of a mechanics – based approach for the definition of the structural response.

Conversely, the problems identified to be overcome are: •

The use of a discrete measure of the earthquake intensity through the use of a macroseismic intensity scale.

•

The uncertainties in the methodology originated in the arbitrary definition of factors and weights.

•

Complexity in the application of the procedure in real cases. 18

Chapter 3. MeBaSe Procedure

3.

MeBaSe PROCEDURE

In this Chapter a new proposal for the seismic risk assessment of unreinforced masonry buildings is presented. The methodology is based on the general ideas proposed by Calvi [1999], but relevant changes have been carried out, in order to correct the main drawbacks identified in Section 2.6.

3.1

PROCEDURE FOR IN-PLANE FAILURE MECHANISM

3.1.1

Demand

The definition of the demand representing the ground motion was originally proposed to be represented by the elastic displacement response spectrum DRS (ξ = 5%) for the area of interest, given Tr, source, path and site-effects. Being the demand represented by a displacement response spectrum, then displacement demand is a function of period, as shown in Eq.(3.1).

The importance of

underlining such an obvious concept will be clear in the following paragraphs.

∆ demand = f (T )

(3.1)

The demand can be obtained from regional probabilistic seismic hazard studies or from available seismic codes. In order to apply the probabilistic procedure presented in Section 3.3, the demand must be defined by the median or mean value for each spectral period and also by the standard deviation of the corresponding demand parameter. Figure 3.1 shows an example of the acceleration response spectra ARS and the displacement response spectra DRS for ag = 0.3g, S = 1.0 and k = 1.0, defined according to Eurocode 8 [CEN, 2001] with 5% of critical damping ratio, and illustrating the random nature of the demand.

19


0.8

0.16

0.7

0.14 ARS

0.6

0.12

Sa ('g)

DRS

0.5

0.10

0.4

0.08

0.3

0.06

0.2

0.04

0.1

0.02

0.0

0.00 0.0

1.0

2.0

3.0

4.0

Figure 3.1. Acceleration and Demand Response Spectra according to Eurocode 8, for ag = 0.3g, S = 1.0 and k = 1.0

Given the fact that additionally to acceleration or displacement, some other ground-motion parameters could influence the response of masonry structures [Penazzo, 2001], the demand could also include duration/energy-related parameters. Nonetheless this issue is not explored in the current development.

3.1.2

Limit States

Appropriate median values for the drift limit, corresponding to each given limit state, must be used. Brick masonry, stone masonry, tuff masonry and other types of masonry have different drift limits that have to be obtained from testing or from the available literature. In case they need to be assumed, a higher level of uncertainty must be set accordingly. Alternatively, the values presented in Table 2.8, proposed by Calvi [1999], could be used.

3.1.3

Simplified Structural Model

3.1.3.1. Definition of the Displacement Limit State ∆LS. A simplified equivalent SDOF system, based on the well known substitute – structure method [Shibata and Sozen, 1976], suitable to model the in-plane behaviour of masonry buildings must be used here. Figure 3.2 shows the general features of the SDOF, in which a multi degree of freedom system MDOF with masses mfi located at each storey height hi and subjected to a system of lateral forces Fi is represented by a SDOF with effective mass meff and effective stiffness Keff, given by the ratio between the yielding force of the system Fy and the displacement at the effective height heff for a given displacement demand or limit state ∆LS.

20

Chapter 3. MeBaSe Procedure ∆i

mfi

Fi

meff j

F Fy

∆eff

hT hi

Keff

heff

∆y

∆LS

∆

Figure 3.2. Simplified equivalent SDOF model

Being presented the simplified SDOF model now in Figure 3.3 four possible displacement profiles for different limit states and in-plane failure modes are shown. The profile (a) corresponds to the limit states LS1 and LS2 for which none of the members of the structure has reached the yield displacement ∆y (i.e. an appreciable extension of cracking). The yield displacement at the effective height he, assuming a triangular displacement shape, can be computed with Eq.(3.2). The triangular shape is justified for low-rise buildings dominated by a shear deformation failure mode, as is the case for most old-unreinforced masonry buildings.

∆ y = κ 1 hT δ y

(3.2)

The profiles (b), (c) and (d) correspond to the limit states LS3 and LS4, in which either a soft storey mechanism or a weak beam mechanism has been developed as the most probable failure mode. Profile (b) is the most typical situation for low rise unreinforced masonry buildings, but profile (c) could be also possible, either at top or intermediate stories, depending on the relative strengths of the stories; or also profile (d), depending on the relative strengths of the spandrel beams and the piers.

(a)

(b)

(c)

(d)

Figure 3.3. Deformed shapes for different limit states and in-plane failure modes

21


The maximum displacement for a given limit state ∆LS, can be computed as the summation of the yield displacement ∆y Eq.(3.2), and the plastic displacement ∆p Eq.(3.3), being hsp the effective height of the piers going into the inelastic range, when openings are present, or the storey height when there are no openings.

∆ p = κ 2 (δ LS − δ y )hsp

(3.3)

∆ LS = ∆ y + ∆ p

(3.4)

∆ LS = κ 1 hT δ y + κ 2 (δ LS − δ y )hsp

(3.5)

When the building has a regular distribution of masses and a uniform storey height κ1 is equal to 0.667. However, when the buildings is better represented by a distributed mass corresponding to the mass of the masonry walls plus lumped masses corresponding to the mass at each floor, the κ1 has to be computed in the way described below. In Eqs. (3.3) and (3.5) a key issue is the definition of κ1 and κ2 that have been defined by carrying out the following analysis: •

A simplified model like the one shown in Figure 3.4 was used, for which lumped masses mf were considered at each floor, and the masonry is assumed to have a distributed mass mm per unit length. The definition of some other relevant parameter is shown in the figure.

x

mfi

∆yi

∆pi

mmdx

hT -h1

dx hT

j hsp

hi x

h1

Figure 3.4. Simplified model for the definition of κ2

22


•

The effective displacement was computed according to Eq.(3.6), assuming that for any given storey i, the yield displacement ∆ y i is obtained with Eq.(3.7) and the plastic displacement ∆ p i with Eq.(3.8).

∆e =

(

2

∑ mi ∆ yi + ∆ p i

)

∑ m (∆

)

n

i =1 n

i =1

i

+ ∆ pi

yi

(3.6)

∆ y i = hi δ y

(3.7)

∆ p i = (δ LS − δ y )hsp

(3.8)

Eq.(3.6) can be written in an explicit way, including the terms corresponding to the distributed and the lumped masses, as appears in Eq.(3.9), being Mm computed with Eq.(3.10) and Nm computed with Eq.(3.11).

∆e =

n

n

n

δ y2 ∑ hi2 m f + 2δ y δ p hsp ∑ hi m f + δ p2 hsp2 ∑ m f + M m i

i =1

i

i= j

i

i= j

(3.9)

n

n

δ y ∑ hi m f + δ p hsp ∑ m f + N m i

i =1

h1

i= j

hT

i

M m = ∫ mm (xδ y ) dx + ∫ mm (xδ y + δ p hsp ) dx 2

2

h1

hT

0

h1

N m = ∫ mm xδ y dx + ∫ mm (xδ y + δ p hsp )dx

•

(3.10)

h1

0

(3.11)

From Eq.(3.9) and assuming a ductility factor µ = 1, which means δp = 0, it is possible to compute ∆e for the yielding state. Then equalling ∆e to ∆LS in Eq.(3.5) it can be computed κ1.

•

Now, with Eq.(3.9) and for different ductility factors, an effective displacement ∆e is computed, which is then equated to ∆LS in Eq.(3.5), with the κ1 previously defined. By rearranging Eq.(3.5),

κ2 is obtained with Eq.(3.12):

23


κ 2 = (∆ LS − κ 1 hT δ y ) / (δ LS − δ y )hsp

(3.12)

As an example, Figure 3.5 shows the variation of κ2 for different ductility factors µ, and considering failure at different heights, for a five storey building with a mass ratio mf / mm equal to 0.3. It can be noticed that κ2 is almost insensitive to the variation of the ductility level. From the analysis of a set of buildings, ranging from 1 to 6 stories and all the buildings with the same mass ratio mf / mm equal to 0.3, it has been computed the following values of κ2 according to the number of stories n, which are presented in Table 3.1. These values are considered valid when the weak storey is formed in the two lower thirds of the building. Finally, with κ1 and κ2 from Table 3.1, the displacement for any given limit state ∆LS can be computed with Eq.(3.5).

1.4 1.2 1.0 0.8

κ2

µ =1

0.6

µ =2 µ =3

0.4

µ =4

0.2

µ =5

0.0 1

2

3 4 Story at failure

5

Figure 3.5. Definition of κ2 according to the storey at failure

Table 3.1. κ1 and κ2 values, according to the number of stories Number of Stories κ1 κ2 n 1 0.790 0.967 2 0.718 0.950 3 0.698 0.918 4 0.689 0.916 5 0.684 0.900 6 0.681 0.881

24

6


3.1.3.2. Definition of the Ductility for a Given Limit State ∆LS. From Eqs.(3.2) and (3.5), the ductility for a given limit state µLS, is obtained with Eqs.(3.13) and (3.14) for the case when there are openings, or with Eq.(3.15) for the case when there are not openings and the height hsp is the same for all the stories.

µ LS =

∆p ∆ LS ∆ y + ∆ p = = 1+ ∆y ∆y ∆y

µ LS = 1 +

(δ LS − δ y )hsp

µ LS = 1 +

κhT δ y (δ LS − δ y )

κnδ y

(3.13)

(3.14)

(3.15)

3.1.3.3. Definition of the Capacity for Given Limit State ∆LS. Unlike Eq.(3.1), in this case the displacement capacity for a given limit state computed with Eq.(3.5) does not depend on the period of the structure, but rather on its material and geometrical properties. For two buildings with similar materials and geometrical properties, this relationship will depend just on the total height hT as shown in Eq.(3.16).

Thus, it is necessary to find a relationship between structural period and total

height hT = g (T ) , in order to be able to compare demand and capacity.

∆ capacity = f (hT )

(3.16)

∆ capacity = f (g (T ))

(3.17)

Let us consider for example two structures with the same given height and with one and two walls, corresponding to curves 1 and 2 in Figure 3.6, respectively. These structures will reach the yielding displacement ∆y when the applied lateral force reaches the strength of each wall Fy, assuming a simple bilinear elasto – plastic idealisation. Beyond this point, no additional force is necessary to take the structures to some displacement limit state ∆LS. The effective period of these structures will be a function of the secant stiffness to the displacement capacity for the given limit state.

25


F Fy,2 KLS,2 Fy,1 KLS,1

∆y

∆LS ∆

Figure 3.6. Elastic-Perfectly plastic response for two different structures

From Figure 3.6, Eqs.(3.18) and (3.19) can be stated, relating the stiffness and displacement at yield, with the equivalent stiffness and displacement for a given limit state.

Fy = K y ∆ y = K LS ∆ LS

K LS =

K y∆ y ∆ LS

=

Ky

µ LS

(3.18)

(3.19)

Two different approaches can be followed beyond this stage. The first one would be to consider the elastic period of the structure Ty as a function of total height hT, according to Eq.(3.20), which is equivalent to those type of formulations used in seismic codes. In this case a fundamental assumption is made: two buildings with the same height have the same or approximately the same elastic period, regardless their yielding strength.

T y = αhTβ

(3.20)

From Eq.(3.19), the relationship shown in Eq.(3.21) can be established between effective period for a given limit state and elastic period, including the effect of ductility, and combining the standard-code form shown in Eq.(3.20), to relate elastic period with total height of the structure, with Eq.(3.21), a general expression for the effective period in terms of total height and ductility is obtained, according to Eq.(3.22).

TLS = T y µ LS 26

(3.21)


TLS = αhTβ µ LS

(3.22)

Now, by using Eqs.(3.2), (3.13) and (3.22), a simple expression relating displacement and effective period for a given limit state is found, recalling the assumption already explained:

∆ LS =

κδ y 2 β −1 T

α h 2

TLS2

(3.23)

Eq.(3.23) must be carefully understood, because some misleading interpretation could be drawn at first sight. Through the entire derivation of this equation it was already shown that the dependency of ∆LS on TLS is just a mathematical manipulation made in order to be able to compare the capacity given by this equation, with the demand given by Eq.(3.2). In fact, this relationship must not be read as if the effective period of a structure is changed, its capacity will increase; the true meaning of this equation is that for a given displacement capacity an effective period is obtained. In the first approach described above, just the height and some ductility measure of the building was considered, and no reference to the strength was made, which leads to inconsistencies in the procedure, as shown in Figure 3.6. For the second approach let us consider two structures, the first with one single wall and strength Fy,1 and the second with two identical walls and strength Fy,2 that will be the double than the first one, but the yielding and limit state displacement capacities will be about the same. For the two structures the secant stiffness and the effective period will not be the same. Thus, it is necessary to find an expression to relate the effective period to both ductility and strength. A second and more rigorous alternative, which is the one followed in this dissertation, considers that two buildings of the same height can have different strength and hence different effective stiffness. Thus, it is required that the maximum displacement at given limit state ∆LS, to be expressed as a function of the strength Fy and the effective period TLS, through Eqs.(3.24) and (3.25).

M∆ LS Fy

TLS = 2π

∆ LS =

Fy 4π M 2

27

TLS2

(3.24)

(3.25)


Having found an expression for the relationship defined in Eq. (3.17), an additional step is yet necessary to define the strength of the structure Fy, in order to do a proper use of the modified methodology. The strength Fy can be computed in terms of the seismic coefficient C with Eq.(3.26).

Fy = CW = CMg

(3.26)

Combining Eqs.(3.25) and (3.26), a relationship between displacement ∆LS and effective period TLS for a given limit state, Eq.(3.27) can be defined, being C computed by Eq. (3.28). Figure 3.7 shows the meaning of C for every storey m. For a given force distribution (a) the storey shears can be computed (left b), and then considering the strength distribution (right b), the ratio between the strength and the shear is obtained. This ratio is assumed to be the implicit seismic coefficient for every storey, and the minimum Cm value will be the one controlling the overall response of the structure.

Cg 2 TLS 4π 2

(3.27)

C = min{C m }

(3.28)

∆ LS =

Cm can be computed with Eq. (3.29) [Benedetti & Petrini, 1984], where: Am is the normal area of all the resistant elements in direction of minimal strength, τkm the reference shear strength for masonry at the level m, n the number of floors, and γm the ratio between Am and the maximum area Bm of the longitudinal or transversal walls.

This expression is similar to Eq.(2.3) used in the VULNUS

methodology. 1

n  2 Wi ∑   1 i=m   + Cm = A τ 1 m km n hiWi 1.5τ km Am (1 + γ m )   Wt ∑ n   i=m ∑ h jW j j =1

28

(3.29)


Fi

Vm

Fym

(a)

(b)

Cm (c)

Figure 3.7. Force distribution, storey shears, storey strengths and seismic coefficients

It has to be recognized that Eq.(3.29) is rather simplistic because it considers the in-plane shear strength as the only mechanism to withstand the seismic action. Besides, in this equation it has not been included any influence of three-dimensional effects, like torsion and distribution of strength in plan, for the computation of the seismic coefficient. A slight modification of Eq.(3.28) is proposed here under the form of Eq.(3.30), in which φc is a correction coefficient included to cope with the possible three-dimensional effects and other sources of strength not included in the original formulation. Research is currently being carried out on the development of some analytical expression to allow for the computation of φc based on some basic information.

C=

1

φc

min{C m }

(3.30)

In order to apply all the steps proposed in the methodology by Calvi [1997], two additional steps are still missing: the definition of a joint probability density function JPDF in the ∆-T space, and finally, the computation of the probability that the demand exceeds the capacity. Those two steps are treated in detail in Section 3.3.

3.2

PROCEDURE FOR OUT-OF-PLANE FAILURE MECHANISM

A similar procedure to that proposed for the in-plane failure mechanism for unreinforced masonry walls has to be developed for the out-of-plane failure mechanism, in order to have a complete model to

29


be used on the seismic assessment of unreinforced masonry buildings, considering both modes of failure. The general procedure is intended to follow the same steps outlined in the original procedure by Calvi [1999], but new approaches are necessary to define both the demand and the response of the structural model to considerer the features of the out-of-plane behaviour. The procedure developed so far and explained below is restricted to parapets and simple supported walls, either with loadbearing or non-loadbearing walls, according to the boundary conditions shown in Figure 3.8.

Timber bearer boundary condition

W

W

(c)

(d)

Slab boundary condition

t

Slip joint boundary condition

h

(a)

(b)

Parapet wall

Simply-supported Non-loadbearing wall parapet wall

Simply-supported Loadbearing wall

Figure 3.8. Boundary conditions considered

3.2.1

Demand

In the same way as the demand defined for the in-plane failure mechanism, the objective for the outof-plane failure mechanism is also to define the demand as a function of period, as shown in Eq.(3.1). Figure 3.9 shows the path of seismic energy through an unreinforced masonry building [Paulay and Priestley, 1992]. The input acceleration and displacement at each level for the out-of-plane walls is due to the combined effect of the end in-plane-walls response and the floor diaphragm response. Paulay and Priestley proposed a procedure to estimate the input acceleration for the out-of-plane walls, considering the effects already described, as shown in Figure 3.10.

30


Figure 3.9. Seismic response of an unreinforced masonry building [Paulay and Priestley, 1992]

Figure 3.10. Flexible floor response to ground excitation [Paulay and Priestley, 1992]

A similar procedure is adopted in Eurocode 8 [CEN, 2001] to compute the acceleration for nonstructural elements, according to Eq.(3.31):

S ane

 h  3α 1 + ne  hT   = 2  Tne   1 + 1 − T1  

(3.31)

where: α is equal to the peak ground acceleration of the given ground – motion, hne is the height at which the non – structural element is placed, hT is the total height of the building, Tne is the period of vibration of the non-structural element and T1 is the period of the first mode of vibration of the building.

31


The numerator in Eq.(3.31) considers the amplification of the acceleration due to the level of the nonstructural element with respect to the ground level; the higher the element is placed, the larger the amplification is, being maximum for an element placed at the roof level. The denominator in the same equation represents the dynamic interaction described by the ratio between the period of vibration of the non-structural element and the period of the first mode of vibration of the building. When both periods of vibration are the same there is no effect on the acceleration of the non-structural element, when the period of the non-structural element goes to zero the de – amplification effect of the period ratio is null, and the period tends to infinite the denominator tends to infinite and the acceleration goes to zero. Also a de – amplification takes place when Tne is larger than T1. Little experimental evidence is available to compute the effect of location in height and period ratio on the displacement for the non-structural element, and apparently no definitive conclusions have been drawn so far. In the New Zealand code for assessment [NZSEE, 2002] it has been suggested that the displacement for non-structural elements can be computed with Eq.(3.32), but being this a cautious assumption.

S d ne = S ane

Tne2 4π 2

(3.32)

As an example Figure 3.11 shows the acceleration and displacement response spectra for a nonstructural element placed at 12 meters from the ground level, in a 15 m high building with T1 = 0.40 sec, for PGA = 0.30g.

Demand Response Spectra

1.0

0.10

Sa

0.08

Sd Sa ('g)

0.8

0.06

0.6 0.04

0.4

Sd (m)

1.2

0.02

0.2

0.00

0.0 0.0

0.5

1.0

1.5

2.0 2.5 T ne (sec)

3.0

3.5

4.0

Figure 3.11. Example of acceleration and displacement response spectra computed with Eurocode 8

32


It can be assumed that the previous equations developed for non-structural elements can also be used for out-of-plane walls.

3.2.2

Limit States

To define proper limit states for the assessment of out-of-plane unreinforced masonry walls it is useful to refer to the work done by Doherty et al. [2002] and Griffith et al. [2003], in which the semi-rigid nonlinear behaviour of a bending wall is described by means of a tri-linear simplified model, as shown in Figure 3.12. Fo corresponds to the force at incipient rocking and is also called the “Rigid threshold resistant”, ∆u is the ultimate displacement or the displacement at the point of static instability. The straight line defined by these two parameters means that the wall is cracked before the motion starts, and that the wall is behaving as a set of rigid bodies. The real behaviour of the semi-rigid wall, which is a consequence of the stiffness of the wall before cracking and also of the finite rather than infinitesimal dimensions of the pivot points, is represented by the continuous curve. Doherty et al [2002] and Griffith et al. [2003] have shown that the nonlinear nature of the masonry walls, subjected to out-of-plane motion, can be described by the tri-linear model, defined by the parameters ∆1 and ∆2, being these parameters estimated from the material properties and the state of degradation of the mortar at the pivot points, as a proportion of the ultimate displacement.

Table 3.2 shows the ratios proposed by Doherty et al [2002] for three different

conditions of the cracked joint.

F Fo

F

K2 ∆1

∆2

∆u

Figure 3.12. Tri-linear simplified model

33

∆


Table 3.2. Displacement ratios for the tri-linear model State of Degradation at Cracked Joint New Moderate Severe

∆1/∆u (%) 6 13 20

∆2/∆u (%) 28 40 50

Griffith et al. [2003] have shown that the displacement ∆2 can be used as a valid parameter, when the main objective is to determine whether a wall will collapse or not, and in fact they stated: •

The inaccuracy in the evaluation of ∆2 has no relevant effect in the application of the simplified procedure.

•

The maximum strength defined by the force plateau limited by ∆2 is a rough estimate of the real maximum strength.

•

The maximum wall strength estimated is relatively insensitive to the mechanical material properties like the elastic modulus of elasticity and the compressive strength.

It is important to underline that for the current definitions of limit state it is possible just to identify whether a wall would collapse or not, thus “stable” or “collapsed” and the only possible limit state at this stage.

3.2.3

Simplified structural models

The ultimate displacement or the displacement at the point of instability is a function of the thickness of the wall, the boundary conditions and Ψ, which is the ratio between the “overburden weight” (axial force applied at the top the wall) and the self-weight of the upper-half of the wall above mid-height.

∆ u = f (t , B.C., Ψ )

(3.33)

Fo is the force required to initiate the rocking motion, and is a function of the overburden ratio Ψ, the effective mass and the geometry of the wall, and can be expressed according to Eq.(3.34):

Fo =

4(1 + Ψ )M e gt h

34

(3.34)


From Figure 3.12 and once the parameters Fo, ∆2 and ∆u have been computed or estimated, it is possible to compute the force corresponding to the plateau, by using equation (3.35), and the stiffness to the point of limiting displacement is determined with equations (3.36) and (3.37).

 ∆  F = Fo 1 − 2   ∆u 

K2 =

K2 =

Fo ∆2

(3.35)

F ∆2

(3.36)

 ∆2   1 −  ∆u 

(3.37)

Now, considering the relationship between stiffness and period, Eq.(3.38), and Eqs. (3.34) and (3.37), it is possible to get an expression to link the period T2 and the limiting displacement ∆2, as is presented in Eq.(3.39).

K2 =

4π 2 M e T22

      π 2∆2h T2 =    1 − ∆ 2 (1 + Ψ )gt   ∆   u   

(3.38)

1

2

(3.39)

For a given limiting displacement than can be conservatively assumed as ∆LS = φ∆u, being φ a factor between 0.8 and 1.0, and knowing the ratio ρ2 = ∆2/∆u, it is now possible to compute the effective period, with the use of Eq. (3.40), and this value has to be compared with the demand, defined in Section 3.2.1, to evaluate whether for the computed period TLS, the limiting displacement ∆LS would be greater or lower than the demand defined by the displacement response spectrum DRS, as shown in Figure 3.13.

TLS

  π 2 ρ 2φ∆ u h  =   φ (1 − ρ 2 )(1 + Ψ )gt 

35

1

2

(3.40)


Now, the two additional steps: definition of a joint probability density function JPDF in the ∆-T space, and the computation of the probability that the demand exceeds the capacity, can be applied, according to the details shown in the next section.

Sd Safe Fails

∆LS ∆2

T2

TLS

T

Figure 3.13. Comparison of the limiting displacement ∆2 and the displacement response spectrum DRS

It was already said that the current development for out-of-plane mechanism includes just the configurations shown in Figure 3.8. Extension of this ideas to more complex failure mechanisms is currently under way, assuming that all the mechanism being studied can be represented by a tri – linear model similar to the one shown in Figure 3.12.

3.3

PROBABILISTIC CONSIDERATIONS

In Sections 3.1 and 3.2 the simplified methodologies for the vulnerability assessment of in-plane and out-of-plane unreinforced masonry walls were presented. These new methodologies are based on two basic concepts: a mechanical-based approach for the formulation of the response of the structure and the use of a joint probability density function JPDF for the computation of the probabilities of exceedance of a given limit state. In this Section all the steps required for the probabilistic considerations of the methodologies are developed, from the identification of the sources of uncertainty, to the definition of the JPDF and the integration procedures to compute the probability of exceedance.

36


3.3.1

Uncertainties

The problem of estimation of the seismic risk of a structure is a very cumbersome matter, especially when a full description of the uncertainty of all the parameters involved is required. In any seismic risk assessment procedure the uncertainties come from both the demand and the response sides. The uncertainty is recognized as originated in three different sources, namely: ground-motion demand, damage state threshold and capacity response, according to the definition used in the HAZUS procedure [Whitman et al., 1997; FEMA, 1999] as shown in Figure 3.14. Ground motion demand Sources of uncertainty

Damage state threshold Capacity response

Figure 3.14. Sources of uncertainty

3.3.1.1. Uncertainties on ground-motion demand.

Despite the fact that there is a generalized

consensus about the sources of the uncertainty, there are different approaches and methodologies used to consider the uncertainty in a seismic risk and loss estimation analysis, as is explained below. In a general sense, the uncertainties in ground-motion demand are classified in two different types [Kennedy et al., 1980]: randomness βR and uncertainty βU components. Abrahamson [2000] proposed a similar model for the variability, as formed by the aleatory and epistemic components.

The

uncertainty known as aleatory is related to the randomness of the earthquake generating process and there is no way to reduce it. The second type is the epistemic uncertainty, which is originated in the lack of knowledge of the phenomenon and it is possible to reduce it with additional research and a better understanding of the different parameters that affect the phenomenon. Abrahamson [2000] and Bommer [2002] have proposed that the two types of uncertainty can be treated in different ways to compute the ground-motion demand in seismic hazard assessment: the aleatory uncertainty is customarily represented by the standard deviation of the PDF for the prediction equation, and the epistemic uncertainty is represented with the formulation of a logic tree. Some researchers [e.g. Restrepo-Vélez and Bommer, 2003] have indicated that apart from the scatter in the probability density function of the prediction equations, the uncertainty in seismic hazard assessment has some other sources, for example the definition of the geometric features of the seismic sources, the definition of activity features of the seismic sources such as maximum magnitude, 37


recurrence model and parameters for the recurrence model, and also the selection of the prediction equation; these sources of uncertainty belong to the group of epistemic uncertainty [Abrahamson, 2000]. Whitman et al. [1997] considered that it is not practical to separate these two components of variability, and instead they proposed to compute composite “best-estimate” variability. In HAZUS, the uncertainty related with the ground-motions demand is represented by the spatial and the temporal variability components. The spatial variability is implicitly included in the variability of damage probability matrices or fragility curves, while the uncertainty due to the temporal variability could be evaluated and/or chosen by the user through the selection of the magnitude M or the recurrence rate λ of the earthquake selected for the earthquake scenario [FEMA, 1999]. Figure 3.15 shows the different components for the ground-motion uncertainty models related above. It is worth underlining that the temporal component proposed in HAZUS, rather than a component of uncertainty is a deterministic variable, because once a recurrence rate λ or magnitude M is fixed, the uncertainty about the temporal component, defined according to HAZUS, does not exist any longer. Uncertainty / Variability Kennedy et al. , [1980]

Randomness

Uncertainty

Aleatory

Epistemic Logic tree

Abrahamson, [2000]

σ of PDF of prediction eq. Spatial Whitman et al. , [1997]

Temporal

Spatial

Temporal

Spatial Variability in fragility curves

Temporal Selection of λ

Figure 3.15. Components for different uncertainty models

In any case, regardless the definition of uncertainty in ground-motion used, all the components of variability related to source, path, site and temporal features have to be considered, either in the seismic hazard assessment and/or directly in the seismic risk assessment, in order to include the full effect of this uncertainty. Increasing the level of complexity of this component of uncertainty, the demand could be expressed also as combination of several peak ground-motion parameters, given the fact that additionally to acceleration, velocity or displacement, some other peak ground-motion parameters could have influence in the response of masonry structures [Penazzo, 2001].

Bazzurro and Cornell [2002]

proposed a new methodology to compute the ground-motion demand for several peak ground-motion parameters at the same time, with a vector-valued probabilistic seismic hazard assessment or VPSHA. 38


This type of analysis is required when several peak-ground motion parameters are going to be included in the hazard assessment, because it is not correct to assume that peak ground acceleration, peak ground displacement, duration, spectral ordinates, etc, are all simultaneously certain number of standard deviation above the median value [Restrepo-Vélez and Bommer, 2003]. 3.3.1.2. Uncertainties on damage state threshold. Tables 2.8 and 3.2 show the parameters needed for defining the damage state thresholds for in-plane and out-of-plane failure mechanisms, respectively. Those can be considered as the median values for the threshold, and the uncertainty has to be represented by the standard deviation or the coefficient of variation of every one of the damage state threshold. 3.3.1.3. Uncertainties on capacity response. The uncertainties on capacity response come from the propagation of the uncertainty on each one of the parameters affecting the response. Figures 3.16 and 3.17 show the sources of uncertainty on capacity response for in-plane and out-of-planes, respectively. Those sources have been listed, according to the methodologies proposed in Sections 3.1 and 3.2.

Geometrical

Sources of uncertainty on capacity response for in-plane walls

Mixed Geometrical/ Material Properties

Material Properties

Others

Height Area of resistant walls Mass distribution Irregularities Type of slabs and roof Masonry specific weight Quality of masonry Age (state of conservation, maintenance, design code) Use (loads) Pre-existing damages

Figure 3.16. Sources of uncertainty on capacity response for in-plane walls

Geometrical

Sources of uncertainty on capacity response for out-of-plane walls

Mixed Geometrical/ Material Properties

Height of building and walls Boundary conditions Thickness of the walls Type of slabs and roof Presence of chains or other restrainers Masonry specific weight Texture of masonry (double leaf, single leaf, staggering ratio, shape factor, etc) Quality of masonry

Material Properties

Others

Pre-existing damages

Figure 3.17. Sources of uncertainty on capacity response for out-of-plane walls

39


3.3.2

Definition of the JPDF for Structural Capacity

It has to be recognized that Eqs. (3.27) and (3.39) are not proper descriptions of the dependency of ∆ on T, and this dependency must rather be expressed with Eq. (3.41), by including the uncertainty in the relationship.

∆ = g (T , ε )

(3.41)

According to the original proposal by Calvi, an appropriate definition of the JPDF f ∆ ,T (∆, T ) would be necessary to be included in the methodology at this stage.

However, two main problems might

arise: the first is the definition of the JPDF, which should include the effect of all the sources of uncertainty related in Section 3.3.1 and that cannot be estimated from the sample with enough accuracy, but rather would be needed to be possibly chosen by means of a heuristic approaching [Pinto and Giannini, 2001]. The second problem is related with the integration of the JPDF over the failure domain, in order to compute the probability of failure. In most of the cases the JPDF has a very complicated mathematical form, and in addition the integration must be performed in a multidimensional space, being the integral difficult to compute. To avoid complications from the mathematical point of view, Monte Carlo methods could be applied in order to compute the probability that DRS exceed the capacity. Some special Monte Carlo techniques, like Importance Sampling or Latin Hypercube Sampling [McKay, Beckman and Conover, 1979] would be useful to reduce the computing time. An alternative path, and somehow an easier one, can be followed to overcome the mathematical problems related above. Let us consider the classical reliability formulation represented in Eq. (3.42), for which many references can be found [e.g. Pinto and Franchin, 2003], in which FD(α) represents the CDF of the demand being less or equal to some value α, and fC(α) is the PDF of the structural capacity SC: ∞

Pf = ∫ [1 − FD (α )] f SC (α )dα

(3.42)

0

In the particular formulation representing the capacity for a given limit state LS in terms of the displacement capacity for that given limit state ∆LS and the corresponding effective period TLS, Eq.(3.42) takes the form of Eq. (3.43), where Pf is the probability of failure, FD is the CDF of the

40


displacement demand ∆LS = x conditional on period TLS = y and f ∆ LS TLS is the JPDF for the displacement limit state ∆LS and effective period at the same limit state TLS.

Pf = ∫ ∫ [1 − FD (∆ LS = x TLS = y )] f ∆ LS TLS (x, y )dxdy

(3.43)

y x

The JPDF f ∆ LS TLS can be expressed as the product of the PDF of ∆LS conditional to some TLS times the PDF of TLS, according to Eq. (3.44), leading to the computation of the probability of failure Pf with Eq.(3.45):

f ∆ LS TLS ( x, y ) = f ∆ LS

TLS

(x y ) f T ( y )

Pf = ∫ ∫ [1 − FD (∆ LS = x TLS = y )] f ∆ LS

(3.44)

LS

TLS

(x y ) f T ( y )dxdy LS

(3.45)

y x

Extensively known reliability methods like first order (FORM) or second order (SORM) reliability methods can be used to compute, in an approximate way, the CDF of a complicated function of random variables [i.e. Pinto and Giannini, 2001], from which the corresponding PDF can be afterwards numerically evaluated. This is the approach that is proposed here to compute the required

f ∆ LS

TLS

(x y ) and

f TLS ( y ) included in Eq.(3.45).

In order to compute the CDF of a random variable using FORM analysis, for example TLS, a G limit state function must be created, as shown in Eq.(3.46):

G = α − TLS

(3.46)

By assigning parametric values to α in a suitable range, for example from 0.01 sec to 4.0 sec, and running FORM analysis for each discrete value of α, the result will be the probability of TLS being less or equal than α, i.e. the cumulative density function CDF of TLS. As a final step, the CDF obtained is differentiated numerically to obtain f TLS ( y ) .

In a similar fashion, f ∆ LS

TLS

(x y ) can be obtained, taking due care of the fact that this is a conditional

distribution. This problem can be solved by using Eq.(3.27) and fixing the period to some specific value Ti, considering as random variables just those required to compute the seismic coefficient C, and

41


then following the procedure suggested above. The resulting PDF would be the distribution of ∆LS conditional to Ti, f ∆ LS

3.3.3

Ti

(x Ti ) .

The same procedure must be followed for every single value of T.

Definition of the Seismic Demand

In the classical reliability formulation, the probability that the demand exceeds some given value α,

[1 − FD (α )], is represented as the product of the hazard function H(I) of the intensity parameter of interest I times some factor to take into account the non-linear response of the structure, including also the uncertainty on the structural response. The hazard function is obtained through a probabilistic seismic hazard analysis of the site of interest, and in general has the shape shown as an example in Figure 3.18.

Annual frequencies of exceedance

1.0.E-02

1.0.E-03

1.0.E-04

1.0.E-05 0.00

0.10

0.20

0.30 0.40 PGA [g]

0.50

0.60

0.70

Figure 3.18. Seismic hazard curve of Peak Ground Acceleration for Gaziantep, Turkey, adapted from Restrepo-Vélez and Bommer [2003]

For example, in a typical application of the reliability theory known as the SAC/FEMA method, which is intended for the seismic design and assessment of moment resisting frames [Cornell et al., 2002], the probability that the seismic demand D exceeds a given value d, [1 − FD ( d )] , is given by Eq. (3.47): −

k

1 k2

β 2 d  b P (D > d ) = k o   e 2 b a

2

D

(3.47)

where the first factor is the probability that the seismic hazard exceeds the demand D required to produce the response d, and the factor with the exponential function accounts for the uncertainty on 42


the structural response. It is worth noting that the probability obtained with Eq.(3.42) corresponds to the total annual probability of failure. Now that the concept of demand in the classical seismic reliability has been clarified, it is appropriate to explain the definition of the seismic demand proposed to be used in this dissertation. One of the main objectives of the methodology being developed is its application in the evaluation of damage distribution under a given earthquake scenario. In this case it is assumed that the level of seismic hazard is chosen from the start, when the earthquake scenario is defined, for example: a seismic event for an annual frequency of exceedance N equal to 0.0021 (equivalent to a return period of 475 years), corresponds to the 10% probability of being exceeded q in a lifetime L of 50 years, assuming the extensively used Poissonian model in Eq. (3.48):

q = 1 − e − LN

(3.48)

When such a selection is made, it is implicit that all the aleatory uncertainties on the seismic hazard analysis given by the scatter on the prediction equation have been already considered, through the integration process across the scatter, being left just to consider the epistemic uncertainties inherent to the definition of the earthquake scenario or the computation of the seismic hazard itself. In this dissertation it is proposed that for a scenario event the CDF of demand will be computed from the median values of the displacement response spectrum given by the probabilistic seismic hazard analysis, and the standard deviation representing the epistemic uncertainty in the definition of the spectrum. For the computation of this CDF it will be used the widely accepted assumption of lognormality. The aleatory component of the uncertainty on the demand it is assumed to be included when the probabilistic seismic hazard assessment is carried out, thus the uncertainty represented by the standard deviation of the prediction equation is considered when the integration across the scatter is performed. To include the aleatory component also as a part of the uncertainty in the definition of the spectrum would be a double counting of this uncertainty, being an overestimation of the uncertainty and a mistake from the conceptual point of view.

43


3.3.4

FORM Procedure

Having discussed the goal of using FORM, let us describe in a brief way the general procedure using as an example Eq.(3.24). In short, the purpose of FORM is to approximate, in the normalized standard space, the limit state g function by means of a linear function tangent to the g function at the design point which is the most likely failure point, as shown in Figure 3.19. β is the distance from the origin to the design point and α is a unitary vector on the direction of β. The components of α are called sensitivity factors and they represent the influence of each random variable ui in the norm of β, which can be assigned with a probabilistic meaning.

Details regarding FORM analysis can be found

elsewhere [e.g. Pinto and Giannini, 2001]. Eq.(3.24) can be written in a more explicit way in Eq.(3.49), hence the G limit state function will be expressed with Eq.(3.50):

TLS = 2π

κ 1 hT δ y + κ 2 (δ LS − δ y )hsp  K 1 K 1τ km 1 + 2 φc  τ km

1

 2  g 

+ ε TLS

(3.49)

G = α − TLS (τ km , δ y , δ LS , hsp , hT )

U2

Safe domain

(3.50)

g(U) = 0

FORM approximation

Failure domain Design point

u2*

β α2

SORM approximation

α α1

u1*

U1

Figure 3.19. Approximation of the limit state function g in the normalized standard space

44


For the purpose of this dissertation, ε TLS has been assumed equal to zero, and φ c ,τkm, δy, δLS, hsp and hT and considered as the only random variables on which TLS depends. In upcoming research that is being currently carried out, a more complete and proper dependence of TLS on a higher number of random variables is been used, and appropriate values for φ c and ε TLS are proposed. Before applying the procedure, the following statements have to be done: •

Due to the nonlinear nature of the limit state function G, as can be seen from Eqs.(3.46) and (3.49), it is necessary to use the Hasofer-Lind index, which is invariant with respect to the form of the limit state function.

•

Considering that the random variables X: { φ c , hT, hsp, δy, δLS, τkm,} are not Gaussian, a transformation is needed. The variables in X are assumed to be non Gaussian because is not possible to have negative values for none of them. Besides, in order to use the Hasofer-Lind index, all the random variables must be transformed into standard Gaussian random variables. Hereafter it is assumed that all the variables on vector X are lognormally distributed. Nonetheless, the FORM procedure can be applied to any type of marginal distribution of the individual random variables, provided that a proper transformation is carried out.

•

Taken into account that JPDF of X is unknown, the Rosemblatt transformation is not applicable and the Nataf transformation has to be used [Liu and Der Kiureghian, 1986].

The steps required to found the so called “design point” or “most likely failure point” are the following: 1. To create the correlation matrix P with the coefficient of correlation between variables xij. Correlation between random variables can be estimated from the statistical data gathered during in the survey.

[ ]

P = ρ ij

(3.51)

2. To uncorrelate the random variables by computing the “corrected” correlation matrix P’ by using the Nataf transformation.

45


3. To factorize matrix Cy = P’ by using the Cholesky decomposition. Cy = [σiσj ρ ij' ] = P’ because are the covariance matrices of standard normal variables, hence σi = σj = 1.0: P’ = AAT

(3.52)

4. To compute the inverse A-1. 5. To select starting point x1 (could be the mean values of X, µi). 6. To transform random variables X to correlated standard normal variables Y, with Eqs.(3.53) to (3.55) for the case of lognormally distributed variables, where µ correspond to the mean and σ is the standard deviation for each random variable on vector X:

 σ σ y = ln  x  µ x 

2    + 1  

µ y = ln µ x − 0.5σ y2

y=

ln x − µ y

σy

(3.53)

(3.54)

(3.55)

A transformation from the X space to the Y space of the standard normal variables is always possible, regardless the type of distribution assumed for any of the random variables. The key point is just to equate the probability of failure, in order to find the corresponding standard normal value y corresponding to x. For example, assuming that x is distributed according to the Beta distribution, with parameters p, q, a and b, the corresponding transformation can be performed with Eq.(3.56), being Φ−1 the inverse standard normal cumulative density function:

P[x ≤ x ] = FBeta ( p ,q ,a ,b ) ( x )

⇒

y = Φ −1 (FBeta ( p ,q ,a ,b ) ( x ))

(3.56)

7. To transform correlated standard normal variables Y to independent standard normal variables U, with Eq.(3.57): U = A-1 Y 46

(3.57)


8. To compute the Jacobian J of the transformation and the inverse J-1 .

J=

∂ (u1 ,..., u n ) ∂ (x1 ,..., x n )

(3.58)

9. To compute the gradient of the limit state function G: ∇ G(x). 10. To compute the gradient of the limit state function in the standard normal space g with Eq.(3.59):

∇ g(u) = ∇ G(x) J-1

(3.59)

11. To compute u(2) with Eq.(3.60):

(2)

u =

∇g (u (1) )u (1) − g (u (1) ) ∇g (u

)

(1) 2

∇g (u (1) )

(3.60)

12. To compute x(2) with Eq.(3.61): x(2) = x(1) + J-1 (u(2) - u(1))

(3.61)

13. To compare x(1) and x(2) and iterate until convergence is achieved. 14. Once the procedure has converged, to compute the vector of sensitivity factors αsen

with

Eq.(3.62):

αsen = −

∇g (u )

∇g (u )

2

(3.62)

15. To compute the Hasofer-Lind reliability index βHL, with Eq.(3.61):

β HL = αsen u

(3.63)

16. To compute the CDF for TLS with Eq.(3.64). It has already been said that FORM is a procedure to get the approximate CDF of a random variable, based on the CDF of the random variables of which the first is a function. The accuracy of the approximation will depend on the shape of the g 47


limit state and specially on the flatness or the curvature of the g function in the neighbourhood around the design point.

FTLS ≅ Φ(− β HL )

(3.64)

Once this procedure has been coded, it can readily be used to compute the approximate CDF of basically any random variable related to some others random variables through a complex non-linear function. The FORM procedure described above has been coded into a FORTRAN program which is presented in Annex 1. As an alternative another computer program is freely available and can be adapted to be used in this assessment procedure [Haukaas, 2001].

3.3.5

Probability of failure for In-plane and Out-of-plane mechanisms

With all the tools required to compute the probability of failure Pf, already available, the procedure is nearly complete. The details regarding each specific failure mechanism are explained next. 3.3.5.1. Probability of failure for In-plane mechanism. According to Eq.(3.45), apart from the CDF of the demand FD, it is required to compute the PDFs f TLS ( y ) and f ∆ LS

TLS

(x y ) .

From Eqs.(3.24), (3.26), (3.30) and (3.46), the limit state function for TLS is expressed according to Eq.(3.65):

G = 2π

κ 1 hT δ y + κ 2 (δ LS − δ y )hsp  K K 1τ km 1 + 2 φc  τ km 1

1

 2  g 

−α

(3.65)

where K1 and K2 are constant for a given building, and are computed with Eqs. (3.66) and (3.67):

K1 =

1 Wt ∑ i=m

n

∑h W j =1

48

Am

hiWi

n

j

j

(3.66)

Chapter 3. MeBaSe Procedure n

K2 =

∑W

i

i=m

(3.67)

1.5 Am (1 + γ m )

Now, by applying the FORM procedure described in Section 3.3.3 the CDF of TLS is computed. The corresponding PDF f TLS is obtained with Eq.(3.68):

f TLS =

dFTLS

(3.68)

dTLS

For ∆LS an equivalent computation is carried out, considering the fact that in this case we are interested in computing a conditional PDF. Firstly, from Eqs. (3.25), (3.29), (3.30), (3.66) and (3.67) the displacement ∆LS is expressed as a function of period TLS and the other random variables. Secondly, the period is set to some deterministic value TLSi , and afterwards the FORM procedure is applied to Finally, the conditional PDF f ∆ LS

the limit state function G shown in Eq.(3.70).

TLS i

(x y )

is

computed with Eq.(3.71):

∆ LS

 T2 1 K = LS2 K1τ km 1 + 2 4π φ c  τ km

TLS2 i 1  K G= K 1τ km 1 + 2 2 4π φ c  τ km

f∆

LS T LS i

=

dF∆ LS

TLS i

d∆ LS

1

 2  g 

(3.69)

1

 2  g − α 

(3.70)

(3.71)

3.3.5.2. Probability of failure for Out-of-plane mechanism. The limit state function G for TLS, corresponding to the out-of-plane failure mechanism described in Section 3.2, is computed with Eq.(3.72), being ρ2, ∆LS, h, φ, Ψ and t the random variables on which G depends, and ∆LS computed according to Table 3.3 depending on the boundary conditions. The PDF f TLS is also obtained with Eq.(3.68).

49


  π 2 ρ 2 ∆ LS h  G =   φ (1 − ρ 2 )(1 + Ψ )gt 

1

2

−α

(3.72)

In a similar fashion to the in-plane mechanism, the CDF of ∆LS would be conditional to period TLSi , because both ∆LS and TLS depends on some common random variables, namely φ and Ψ, and as a consequence they are correlated. From Eq.(3.40) the limit state function for ∆LS will be Eq.(3.73):

G = TLS2 i

φ (1 − ρ 2 )(1 + Ψ )gt −α π 2 ρ2h

(x y ) depends on the boundary conditions, as shown in Table 3.3.

Additionally, the CDF F∆ LS

TLS i

corresponding PDF f ∆

( x y ) is also computed with Eq.(3.71).

LS T LS i

(3.73)

The

Table 3.3. ∆LS according to the boundary conditions Boundary conditions Simply-supported wall with vertical load at wall’s top centerline Simply-supported wall with vertical load at wall’s top edge Parapet wall with no vertical load

3.4

∆LS = φ∆u

(1 + 0.75Ψ ) t φ (1 + Ψ ) φ t

φ t

DAMAGE PROBABILITY AT URBAN OR REGIONAL SCALE

The main goal of the proposed methodology is the loss estimation at urban or regional scale; thus, a general framework is still required in order to apply the procedures developed for in-plane and out-ofplane failure mechanisms, going from the simple computation of the probability of failure of a specific class of buildings, to a more general loss estimation procedure, including several classes of buildings for the building stock within a geographical region. This final step in the MeBaSe methodology is presented in the following. Figure 3.20 shows a hypothetic region that has been subdivided in n1 zones, each of them corresponding to a subregion, a census tract, a mail-code zone, a neighbourhood, a block within the city, or any other subdivision considered suitable for the purpose of the loss assessment process. The building stock comprised within the particular zonei (i = 1 to n1) is subdivided in building classes

50


CBj(mt,#s,st) (j = 1 to n2), according to the material (masonry type, age, quality) mt, number of stories #s and possibly, soil type st. Once the region of interested has been subdivided in zones and the corresponding building classes have been identified, the probability of failure Pf ij , which is the probability of reaching or exceeding the specified limit state of interes, for each CBj(mt,#s,st) in each zonei is computed, as explained in sections 3.3.4.1 for in-plane failure mechanism or 3.3.4.2 for out-of-plane failure mechanism.

3

1

5

2

4

i 7 6

Zonei: {CB1, CB2, CB3, …, CBj, …, CBn2} 8 n1

Figure 3.20. Subdivision of the hypothetic region in zones

The probability of failure in a given zone for a specific class of buildings CBj(mt,#s,st) would be computed with Eq.(3.74), where Pf

CB ( mt , # s , st )

is the probability of failure for a given class of buildings,

Pf ij is the probability of failure for a single building of the class CBj(mt,#s,st) within the zonei, computed with Eq.(3.45), and Pmt , # s , st is the joint probability that a building within the population has a given material mt, certain number of stories #s and an specific soil type st, which can be statistically inferred from the sample gathered during the data survey:

Pf

CB j ( mt , # s , st )

= Pf ij Pmt , # s , st

(3.74)

The total probability of failure in the whole region for the same class of buildings CBj(mt,#s,st) would be computed with Eq.(3.75):

Pf

n1

CB j ( mt , # s , st )T

51

= ∑ Pf i =1

CB j ( mt , # s , st )i

(3.75)

Chapter 4. Application Examples

4.

APPLICATION EXAMPLES

4.1

PROBABILITY OF FAILURE FOR IN-PLANE RESPONSE OF A 4-STOREY UNREINFORCED MASONRY BUILDING

The first application example corresponds to a hypothetical 4-story building, built with unreinforced brick masonry. The plan view of the typical storey and the façade are shown in Figure 4.1. For the material properties it has been assumed a specific weight pm equal to 2.0 MN/m3, and a reference shear strength τkm equal to 0.20 MPa.

Figure 4.1. Plan view and façade for the in-plane assessment example

The computation of the seismic coefficient C, according to Eqs.(3.28) and (3.29) is shown in Table 4.1, being C = 0.27. Here it has been assumed the correction factor φc = 1.0. The weights for the slabs and applied load have been assumed as typical for reinforced concrete slabs and residential use.

52


Table 4.1. Computation of the seismic coefficient C Story Ground 1 2 3

hs j [m] 3.15 3.15 3.15 3.15

hj [m] 3.15 6.30 9.45 12.60

Vj pm W walls [m3] [MN/m3] [MN] 38.21 0.02 0.76 38.21 0.02 0.76 38.21 0.02 0.76 38.21 0.02 0.76

W slab

W load

W j-tot

% W walls

[MN] 0.45 0.45 0.45 0.90

[MN] 0.31 0.31 0.31 0.31

[MN] 1.53 1.53 1.53 1.60

[%] 50 50 50 48

h t = 12.6 h sp = 2.0 Story

mj 2 [MN-s /m] Ground 0.156 1 0.156 2 0.156 3 0.163

Wj

τ km

[MN] 1.53 1.53 1.53 1.60

[MPa] 0.20 0.20 0.20 0.20

Am 2 [m ] 5.24 5.24 5.24 5.24

Bm 2 [m ] 6.89 6.89 6.89 6.89

γm

hjWj

Σh i W i

ΣW i

Cm

1.31 1.31 1.31 1.31

[MN-m] 4.82 9.64 14.46 20.16

[MN-m] 49.08 44.26 34.62 20.16

[MN] 5.43 3.90 2.37 0.80

0.27 0.27 0.31 0.46

Σh j W j = 49.08

W t = 6.19

C = 0.27

If a deterministic assessment was to be performed, just the median values of the drift for a given limit state would be required. By using median drift values and Eqs. (3.2), (3.3), (3.5), (3.24) and (3.26) it would be possible to compute the corresponding displacement and effective period for each given limit state, as shown in Table 4.2. In Figure 4.2 the comparison of median values of capacity and demand for three different limit states and performance levels (levels of seismic input) is shown. According to these results it could be concluded that for an expected performance level 1 the capacity of the building is higher than the demand, hence for this case the building is safe. For the performance level 2 the building is marginally unsafe but for the performance level 3 the building is clearly unsafe. The key question would be: how safe or unsafe is the building, considering the uncertainty in demand, capacity and the definition of the limit state thresholds? Now, let us apply the proposed MeBaSe methodology to find an answer this question. Table 4.2. Computation of median values for ∆LS and TLS LS

δ LS

1-2 3 4

[%] 0.10 0.30 0.50

∆y [m] 0.009 0.009 0.009

∆p [m] 0.000 0.004 0.007

∆LS [m] 0.009 0.012 0.016

T LS [sec] 0.36 0.43 0.49

First, the definition of the displacement demand is required. In this example it has been used the same DRS shown in Figure 3.1, for ag = 0.05g, 0.15g and 0.3g, for performance levels 1, 2 and 3 respectively; S = 1.0 and k = 1.0, defined according to Eurocode 8 [CEN, 2001]. For each limit state, damping ratios of 5%, 10% and 15% have been used, respectively. The variability of the ground-

53


motion demand βD has been assumed to be equal to 0.5, which means that the standard deviation is 0.65 times the mean value.

0.04 Perf. Level 1 Perf. Level 2

Sd (m)

0.03

Perf. Level 3

0.02 LS 4 LS 3

0.01

LS 1-LS 2

0.00 0.0

0.2

0.4

T (sec)

0.6

0.8

1.0

Figure 4.2. Comparison of median values of capacity and demand for each given limit state

Figure 4.3 shows the cumulative density functions CDF for the demand FD(∆LS) given that TLS = y. The surface shown in this figure is not the JPDF of TLS-∆LS; referred in Sections 2.5 and 3.3.2, but rather it is the CDF for the demand required in Eq.(3.45). The next step is the definition of the CDF for TLS and ∆LS conditional to TLSi, by using the FORM procedure. In Tables 4.3 and 4.4 the required parameters, and median and standard deviation values for the X vector are presented. The standard deviations have been assumed to be representative and have been used just with the purpose of carrying out this example. Proper values must be selected from the literature and/or computed from the survey data. Figure 4.4 shows the cumulative density function CDF for TLS and, as an example, the CDF for ∆LS conditional to TLS = 1.0 sec, that correspond to the limit state LS4, computed according to the FORM procedure presented in Section 3.3.3. In this figure different values for the coefficient of correlation ρ ij' have been used to consider the correlation between all variables in the X vector, in order to show the influence on the estimation of the CDF.

54

FD(∆LS=x|TLS=y)


T LS [sec] ∆ LS [m]

Figure 4.3. Cumulative density functions CDF for demand FD

Table 4.3. Parameters for computing FTLS and F∆ LS using FORM procedure Parameters 0.00 0.689 0.916 K1 0.87 K2 0.39

α κ1 κ2

Table 4.4. Median and standard deviation values for vector X Vector X

τkm [MPa] hT [m]

δy δLS

hsp [m]

φc

µx

σx

0.20 12.60 0.001 0.005 2.00 1.00

0.05 2.00 0.001 0.001 0.50 0.20

55


1.0

1.0

0.9

0.9

0.8

0.8 0.7

0.6

LS)

ρ = 0.0 ρ = 0.1

0.5

CDF (

CDF (TLS )

0.7

ρ = 0.5 ρ = 0.8

0.4

0.6 0.5 0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0 0.0

0.2

0.4

0.6 T LS [sec]

0.8

1.0

ρ = 0.0 ρ = 0.1 ρ = 0.5 ρ = 0.8

0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 ∆ LS [m]

1.2

Figure 4.4. Cumulative density functions CDF for TLS, and ∆LS conditional to TLS = 1.0 sec

Finally, by computing the integral in Eq.(3.39), the probability of failure for each given limit state is obtained, as shown in Table 4.5. For these results a coefficient of correlation ρ ij' = 0.5 has been used. The selection of the same coefficient of correlation between all the possible pairs of random variables is rather arbitrary. In fact, intuitively it could be said that a high correlation is expected for example between the ht and τkm, whilst hsp could be poorly correlated with all the rest of variables. In a formal application, an appropriate estimation of the coefficients of correlation can be done. For each limit state a different level on the demand has been used, corresponding the performance levels 1, 2 and 3 shown in Figure 4.2. Table 4.5. Probability of failure for each given Limit State Limit State LS1-LS2 LS3 LS4

Probability of Failure 0.002 0.278 0.819

When the same seismic demand is used to compute the probability of failure for each limit state, the probability decreases when going to more demanding limit states, as expected. Table 4.6 shows the probability of failure for each limit state under performance level 2. Table 4.6. Probability of failure for each given Limit State for the same seismic demand Limit State LS1-LS2 LS3 LS4


56


In this specific example the differences in the probabilities of failure are not as large as would be expected, due to the similitude in the displacement response spectra in the range of effective period corresponding to this building, scaled to the respective damping values, as can be seen in Figure 4.5.

0.04 Perf. Level 1 Perf. Level 2

Sd (m)

0.03

Perf. Level 3

0.02 LS 4 LS 3

0.01

LS 1-LS 2

0.00 0.0

0.2

0.4

T (sec)

0.6

0.8

1.0

Figure 4.5. Comparison of median values of capacity and demand for each given limit state, for performace level 2 – PGA 0.15g

The shear resistant areas on each direction, Am and Bm, have been assumed as deterministic variables due to the fact that this a specific building, whose dimensions are known. In case that instead of this building, a set of 4-storey brick masonry buildings were to be analyzed, with no specific information on single buildings, Am and Bm could be treated both as random variables for the computation of FTLS . In Figure 4.4 are also presented the CDFs corresponding to coefficients of correlation equal to 0.0, 0.1 and 0.8. It can be seen that the selection of the coefficient of correlation has a quite limited effect in the computation of FTLS but has some notable influence in F∆ LS . In the same way, the computed probability of failure varies, as shown in Figure 4.6 for the limit state LS4, being conservative to use a low or even null coefficient of correlation. The influence of each random variable in the computation of FTLS and F∆ LS is presented in Figure 4.7, showing the variation of the sensitivity factors αsen with respect to the variation of the coefficient of correlation ρ’ij. The sensitivity factors vary for each design point, this is for each point on the CDF. The results showed in Figure 4.7 correspond to the sensitivity for the median values of TLS and ∆LS conditional to TLS = 1.0 sec

57


0.26 0.25 0.24

Pf

0.23 0.22 0.21 0.20 0.19 0.18 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ρ ' ij

Figure 4.6. Variation of the probability of failure Pf with the coefficient of correlation ρ’ij, for LS4 1.2

0.6 ht

0.4

1.0

δy δ LS

0.2

0.8 0.6

h sp

0.0

τ km φc

α sen

α sen

-0.2

τ km

0.4

-0.4

φc

0.2 0.0

-0.2

-0.6

-0.4

-0.8

-0.6

-1.0

-0.8 0.0

0.1

0.2

0.3

0.4

ρ ' ij

0.5

0.6

0.7

0.8

0.0

0.1

0.2

0.3

0.4

ρ ' ij

0.5

0.6

0.7

0.8

Figure 4.7. Variation of the sensitivity factors αsen with the coefficient of correlation ρ’ij, for LS4

In this figure it can be seen that for low coefficients of correlation all the random variables used in the computation of CDF for TLS have influence in the uncertainty, whilst for high coefficients of correlation the biggest influence in the uncertainty comes from the referential shear strength τkm, the total height hT and the yield drift limit δy and the correction factor φc. In the case of the conditional CDF for ∆LS, both random variables have a relatively equal influence for low coefficients of correlation, and for high coefficients of correlation the biggest influence in the uncertainty comes from the correction factor φc, having τkm an almost negligible influence.

58


4.2

PROBABILITY OF FAILURE FOR OUT-OF-PLANE RESPONSE OF SINGLE WALLS

The second application example is related with the assessment of the out-of-plane behaviour of the façade wall at the fourth storey of the building used for the first example. The building has reinforced concrete slabs and the façade wall at each storey is bearing on the floor slab; hence it can be assumed that each wall at intermediate levels will behave as simply-supported load-bearing walls, according to Figure 3.7(c). It has been selected a median value for the overburden weight ratio Ψ equal to 0.50, a φ factor equal to 0.8, and ρ2 equal to 0.40. With this median values the corresponding median values for displacement limit state ∆LS and effective period TLS were computed and then compared with the demand. Figure 4.8 shows the displacement response spectra corresponding to Eurocode 8, for PGA equal to 0.3g, 0.5g and 1.0g, along with the mean capacity computed for the out-of-plane wall used in this example. It can be seen that for PGA equal to 0.3g and 0.5g the mean capacity of the wall exceeds the demand, and the wall would not collapse. When PGA is increased to 1.0g, the capacity of the wall is below the spectral demand, which means that the wall would be unstable under such demand. 0.40 0.35

Sa 0.3g

0.30

Sa 0.5g Sa 1.0g

Sd (m)

0.25 0.20

Capacity of the out-of-plane wall

0.15 0.10 0.05 0.00 0.0

0.5

1.0

1.5

2.0 2.5 T ne (sec)

3.0

3.5

4.0

Figure 4.8. Comparison of median values of capacity and demand

As the first step in the application of the MeBaSe procedure it has been used the DRS for PGA 1.0g shown in Figure 4.8. The variability of the ground-motion demand βD was again assumed to be equal to 0.5. Figure 4.9 shows the corresponding cumulative density functions CDF for the demand FD(∆LS) given that TLS = y.

59


Then, to obtain the CDF for TLS, and ∆LS conditional to TLSi by using the FORM procedure, the median and standard deviation values for the X vector presented in Table 4.7 were used. In this example a coefficient of correlation ρ ij' = 0.50 was also used for the correlation between all variables in the X vector. Figure 4.10 shows the cumulative density functions CDF for TLS and the CDF for ∆LS

FD(∆LS = x|TLS = y)

conditional to TLS = 1.5 sec, calculated with the FORM already described.

T LS [sec] ∆ LS [m ]

Figure 4.9. Cumulative density functions CDF for demand FD

Table 4.7. Median and standard deviation values for vector X Vector X

ρ2 h [m]

φ Ψ t [m]

µx

σx

0.40 3.00 0.80 0.50 0.30

0.10 0.50 0.20 0.20 0.05

60


1.0

1.0

0.9

0.9

0.8

0.8 0.7

0.6 0.5 0.4

LS)

ρ = 0.0 ρ = 0.1 ρ = 0.5 ρ = 0.8

CDF (

CDF (TLS )

0.7

0.6 0.5 0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0 0.0

0.5

1.0

1.5

2.0 2.5 T LS [sec]

3.0

3.5

ρ = 0.0 ρ = 0.1 ρ = 0.5 ρ = 0.8

0.0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 ∆ LS [m]

4.0

Figure 4.10. Cumulative density functions CDF for TLS, and ∆LS conditional to TLS = 1.5 sec

The probability of failure for each given limit state, obtained with the integral shown in Eq.(3.39), are presented in Table 4.8. Table 4.8. Probability of failure for different demand levels PGA 0.3g 0.5g 1.0g


In the same way as for the in-plane mechanism, in Figure 4.10 are also presented the CDFs corresponding to coefficients of correlation equal to 0.0, 0.1 and 0.8. In this example the selection of the coefficient of correlation has an almost negligible effect in the computation of FTLS and a moderate effect for F∆ LS . Moreover, the influence of each random variable in the computation of FTLS and

F∆ LS has also been analyzed and is presented in Figure 4.11, showing the variation of the sensitivity factors for the median values of TLS and ∆LS. For low coefficients of correlation used in the computation of CDF for TLS the biggest influence comes from the displacement ratio ρ2 and height of the wall h, and the lowest influence comes from the thickness of the wall t, being in fact feasible to consider it as a parameter rather than a random variable, taking into account its null influence.

For high coefficients of correlation the biggest

influence in the uncertainty comes again from the displacement ratio ρ2 whilst the rest of the random variables have low to null influence. In the case of the conditional CDF for ∆LS, the biggest influence comes again from the displacement ratio ρ2, and the other random variable have a medium to low influence, for all the coefficients of correlation.

61


0.6

1.0

0.4

0.8 0.6

0.2

0.4

α sen

-0.2

ρ2

0.2

h

0.0

α sen

0.0

-0.6

h

-0.2

φ

Ψ

-0.4

Ψ

t

-0.6

t

φ

-0.4

ρ2

-0.8

-0.8

-1.0

-1.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0

ρ ' ij

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ρ ' ij

Figure 4.11. Variation of the sensitivity factors αsen with the coefficient of correlation ρ’ij, for LS4

62

Chapter 5. Survey Data Form

5.

SURVEY DATA FORM

The MeBaSe procedure presented in Chapter 3 is complemented with a survey data form, developed with the purpose of being use on field surveys. The new survey form resembles the appearance of the GNDT first and second level forms [GNDT, 1993], but includes new parameters required for the application of the new procedure. A version of the survey data form with the corresponding User’s Manual written in Italian is presented in Annex 2. The survey form is divided in eight sections, each one being described as follows:

•

Data related to the survey form

Basic information like date, number of survey form and surveyor is included in this section and is intended to be used for identification and reference purposes.

•

Location of the building

In this section the information required for the complete identification of the building within a data base is collected.

63


•

Type of usage

Residential, commercial or public usage is considered in this section, including the frequency of use and the type of activity in case that the main use is commercial. This information is used in the estimation of the storey masses. This type of data is fundamental in the application of the procedure.

•

Site morphology

Site features like the position of the building within the block, slopes and the historic origin condition of the building is considered in this section. This information is presently not used in the procedure but it has been included for eventual application of other procedures used for calibration or comparison purposes.

•

Age and maintenance condition

The age of construction, age and type of eventual retrofit interventions, maintenance condition and state of degradation of the mortar joints are surveyed in this part.

The last two features are

fundamental in the application of the procedure, especially for the selection of the displacement ratio

ρ2.

•

Structural typology

The structural typology of the vertical elements, the slabs and roof is considered here, using the same classification proposed by the GNDT [1993]. This information is fundamental in the application of the procedure, particularly for the definition of the reference shear strength for masonry and the identification of the possible out-of-plane collapse modes.

•

General features of the building

The information gathered in this section is related to the plan and height regularity, the direction of the joists in the floor systems, demolitions, attachments, restrictions, etc. The use of this information has not been included yet in the procedure, but could have relevance in the determination of the

64


probability of failure for certain types of buildings, thus it is included for eventual updates or refinements of the procedure. Other information compiled in this section refers to the total area, the storey height, the shear resistant areas of walls, and the direction, thickness, length and other features of the walls in the perimeter of the building. These parameters are fundamental for the evaluation of the in-plane and out-of-plane probabilities of failure.

•

Overturning mechanisms associated to existing damages

The last section on the survey form is devoted to the compilation of the possible overturning mechanisms associated to existing damages, for which the more general out-of-plane collapse modes have been included in the form. Finally, as part of the survey form, a website has been developed in order to write the information into a database. Figure 5.1. shows the main screen of the website.

Figure 5.1. Main screen for the website used to manipulate the database

The website has three different modules, namely: Module 1 which is used to write the information to the database, Module 2 that helps to read the information stored in the database and allows to modify

65


the information when some correction is required, and Module 3 that creates a text file with the relevant information required for the application of the MeBaSe procedure. Figures 5.2 to 5.4. shows some partial views of the three modules.

Figure 5.2. Partial view of the screen used to enter the write the information to the database

Figure 5.3. Partial view of the screen used to control and to modify the information written in the database

66


Figure 5.4. Output view of the screen used to write the input file required for the MeBaSe procedure

67

Chapter 6. Conclusions

6.

CONCLUSIONS

6.1

SUMMARY

A simplified mechanics-based procedure for the seismic risk assessment of unreinforced masonry buildings (MeBaSe) has been developed, based on the concept of displacement-based approach for vulnerability evaluation of classes of buildings, that was originally proposed by Calvi [1999]. The procedure developed can be classified as a direct method because allows the estimation of the probability of failure of a given building or class of building by using numerical models (mechanical group), but also makes use of data collected from surveys on populations of buildings from which relevant statistical parameters used in the analysis are obtained (typological group). As a first step, a revision of the state-of-the-art procedures for vulnerability assessment of buildings, in particular unreinforced masonry buildings, was carried out, from which advantages and disadvantages of the current procedures were identified. Considering this, it can be stated that the MeBaSe procedure complies with the advantages and solves, at least partially, the disadvantages identified in existing procedures, because it is a procedure based on displacement capacity and demand, includes the uncertainty in the demand response spectra, and allows the computation of the probability of reaching or exceeding certain limit state for a building or class of buildings, which is the direct measure of the seismic risk being estimated. Besides, the procedure uses a continuous definition of the earthquake intensity and can be readily applied in real cases. Additionally, the procedure proposed here includes the explicit computation of the probability of failure for out-of-plane behaviour, and includes also the treatment of the uncertainty on the ∆ and T space, avoiding a complicated definition of the JPDF

f ∆ ,T (∆, T ) .

68


Considering the in-plane mechanisms, the demand represented by the displacement response spectrum is to be obtained from the regional probabilistic seismic hazard study, and must be defined by the median or mean value for each spectral period and also by the standard deviation. Concerning the limit states, appropriate median values corresponding to each given limit state must be used. Brick masonry, stone masonry, tuff masonry and other types of masonry have different drift limits that have to be obtained from testing or from the available literature. In case they need to be assumed, a higher level of uncertainty must be set accordingly. Statistical evaluation of available experimental data is currently under way. For the out-of-plane mechanism, the procedure developed in this dissertation is based on the response of parapets and simple supported walls, either with loadbearing or non-loadbearing walls. Equivalent procedures for other wall configuration and failure modes are currently under development. In the same way as the demand defined for the in-plane failure mechanism, the demand is expressed as a function of period, which is taken from Eurocode 8 to compute the acceleration for out-of-plane walls. To define proper limit states for the assessment of out-of-plane unreinforced masonry walls it has been used a semi-rigid nonlinear behaviour of a bending wall described by means of a tri-linear simplified model. This model is defined by the parameters ∆1 and ∆2, estimated from the material properties and the state of degradation of the mortar. Regarding the probabilistic aspects of the new procedure, it is accepted that the uncertainty is originated in three different sources, namely: ground-motion demand, damage state threshold and capacity response, and that all these are necessary to be considered for a complete description of the uncertainty in a vulnerability assessment analysis. In order to overcome the numerical problems that might arise in the definition of the JPDF for TLS and ∆LS an alternative path has been chosen, by using FORM reliability method to compute in an approximate way the cumulative density function CDF of ∆LS and TLS. Taking into account all the possible sources of uncertainty, the strategy used here can treat the uncertainty coming from the material properties and structural response, and the uncertainty coming from the statistics of the population of buildings. The former is directly included in the computation of probability of failure of a single building or class of buildings, whilst the latter is consider in a second stage. Finally, as a complement of the proposed methodology, a survey data form has been designed to gather the data field required for the application of the methodology at urban scale in real cases. This survey form allows gathering the information required for the application of the MeBaSe procedure and also additional information useful in the application of other procedures that could be eventually used for calibration or comparison purposes.

69


6.2

FINAL REMARKS

In the definition of the κ1 and κ2 parameters for the computation of the displacement for any given limit state, presented in Section 3.1.3.1, a simplified model considering the distributed mass of the masonry walls and the lumped masses at each storey has been proposed. For this definition a mass ratio mf / mm equal to 0.3 was used, showing that the parameter κ2 is almost insensitive to the variation of the ductility level. The values proposed for κ1 and κ2 are considered valid when the weak storey is formed in the two lower thirds of the building. Nonetheless, a complete and formal computation of the displacement for the given limit state could also be performed by means of Eq.(3.9). From the results obtained in the numerical examples presented in Chapter 4 it was stated that the selection of the coefficient of correlation has null or very limited effect in the computation of the cumulative density function for the effective period TLS and the displacement limit state ∆LS, and that the use of a null or low coefficient of correlation between the random variables is conservative, for the computation of the probability of failure. The relative influence of each random variable used in the estimation of the cumulative density function for TLS and ∆LS was also performed, showing a variation of the influence on the total uncertainty of each variable, according to the level of correlation considered between the random variables.

6.3

FUTURE RESEARCH

The mechanical models used in the definition for the in-plane and out-of-plane response are simple and do not include factors to consider, for example, irregularities in plan and height for the case of inplane failure, or bidirectional bending for out-of-plane failure. More refined mechanical models could be developed in order to consider more complex behaviour for both failure mechanisms. The MeBaSe procedure is flexible enough to allow the inclusion of new mechanical models with more parameters; this task would not complicate in excess the application of the procedure. The demand model used for the out-of-plane behaviour, taken from Eurocode 8, is still has to be validated as an adequate representation of the displacement demand for out-of-plane walls, specially considering the presence of flexible slabs and the possible out-of-phase movement between the floor and ceiling of a given storey. Regarding also the out-of-plane failure mechanism, the displacement ratios ∆1/∆u and ∆2/∆u proposed by Doherty et al [2002] for three different conditions of the cracked joint, must be confirmed or

70


modified according to the type and quality of the masonry customarily used in the region where the methodology is to be applied. Besides, the definition of other limit states for out-of-plane failure modes, apart from “stable” or “collapsed” would allow a better correlation with the level of damage. Finally, considering that unreinforced masonry is usually a material showing highly non-linear degradation properties, the effect of duration as a component of the demand also needs to be addressed.

71

Chapter 7. Bibliography

7.

BIBLIOGRAPHY

1. Abrahamson, N.A. [2000]. “State of the Practice of Seismic Hazard Evaluation”, Proceed. Earthquake Engineering, GeoEng2000 Conference, Melbourne, Australia, November 2000. 2. Bazzurro, P. and Cornell, C.A. [2002]. "Vector-Valued Probabilistic Seismic Hazard Analysis (VPSHA)", Proceed. 7th U.S. National Conference on Earthquake Engineering, Boston, MA, July. 3. Braga, F., Dolce, M. and Liberatore, D. [1982]. “Southern Italy November 23, 1980 Earthquake: Sect. 5 A Satistical Study on Damaged Buildings and a Ensuing Review of the MSK-76 Scale”. Proceed. 7th European Conference on Earthquake Engineering, Athens, September. 4. Benedetti, D. and Petrini, V. [1984]. “Sulla Vulnerabilità Sismica di Edifici in Muratura: Proposta di un Metodo di Valutazione. L’industria delle Costruzioni No.149, marzo. (In Italian). 5. Bernardini, A. [2000]. “La Vulnerabilità degli Edifici: Valutazione a Scala Nazionale della Vulnerabilità Sismica degli Edifici Ordinari”. (A cura di A. Bernardini) CNR-GNDT, Roma. (In Italian). 6. Bernardini, A., Gori, R. and Modena, C. [1990]. “An Application of Coupled Analytical Models and Experiential Knowledge for Seismic Vulnerability Analyses of Masonry Buildings”. In A. Koridze (ed) Engineering Aspects of Earthquake Phenomena, Vol. 3: 161-180. Oxon: Omega Scientific. 7. Bommer, J.J. [2002]. “Deterministic vs. Probabilistic Seismic Hazard Assessment: An Exaggerated and Obstructive Dichotomy”. Journal on Earthquake Engineering, Vol. 6, Special Issue No. 1, 43-73. 8. Calvi, G.M. [1999]. “A Displacement-Based Approach for Vulnerability Evaluation of Classes of Buildings”. Journal on Earthquake Engineering, Vol 3, N.3, 411-438. 9. CEN. [2001]. “Eurocode 8: Design of Structures for Earthquake Resistance. Part 1: General Rules, Seismic Actions and Rules for Buildings”. Draft 4, Final Project Team Draft (Stage 34), December 2001. 10. Cornell, A. C., Jalayer, F., Hamburger, R., and Fouth, D. [2002]. “The Probabilistic Basis for the 2000 SAC/FEMA Steel Moment Frame Guidelines”. Journal of Engineering Mechanics – ASCE, 128(4):526-533. 72


11. Corsanego, A. and Petrini, V. [1990]. “Seismic Vulnerability of Buildings - Work in Progress”. Proceed- SEISMED 3. Trieste, Italy. 12. D’Ayala, D., Spence, R., Oliveira C. and Pomonis, A. [1997]. “Earthquake Loss Estimation for Europe’s Historic Town Centres”. Earthquake Spectra, 13(4), 773-794. 13. D’Ayala, D. and Speranza, E. [2002]. “An Integrated Procedure for the Assessment of Seismic Vulnerability of Historic Buildings”. Proceed. 12th European Conference on Earthquake Engineering, London. Paper No.561. 14. Di Pasquale, G., Goretti, A., Dolce, M. and Martinelli, A. [2001]. “Confronto fra Differenti Modelli di Vulnerabilità degli Edifici”. L’Ingegneria Sismica in Italia, Atti X Convegno Nazionale ANIDIS, Potenza-Matera. (In Italian). 15. Doherty, K.T., Griffith, M.C., Lam, N. and Wilson, J. [2002]. “Displacement-based Seismic Analysis for Out-of-plane Bending of Unreinforced Masonry Walls ”. Earthquake Engineering and Structural Dynamics, 31: 833-850. 16. Dolce, M. [1997]. “La Valutazione della Vulnerabilità per le Analisi di Rischio e gli Scenari di Danno”. L’Ingegneria Sismica in Italia, Atti 8° Convegno Nazionale ANIDIS, Taormina. (In Italian). 17. Dolce, M., Kappos, A., Zuccaro, G., and Coburn, A.W. [1994]. “State of the Art Report of W.G. 3 – Seismic Risk and Vulnerability”. Proceed. 10th European Conference on Earthquake Engineering”, Wien. 18. FEMA [1999]. “HAZUS 1999. Earthquake Loss Estimation Methodology. Technical Manual”. Federal Emergency Management Agency, Washington D.C. 19. Giovinazzi, S., Lagomarsino, S. [2001]. “Una Metodologia per l’analisi di Vulnerabilità Sismica del Costruito”. L’Ingegneria Sismica in Italia, Atti X Convegno Nazionale ANIDIS, PotenzaMatera. (In Italian). 20. Giuffrè, A. [1999]. “Sicurezza e Conservazione del Centri Storici, Il Caso Ortigia”. (A cura di A. Giuffrè) CNR-GNDT, Ed. Laterza, 2nd ed, Bari. (In Italian). 21. GNDT. [1993]. “Rischio Sismico di edifici Pubblici – Parte I Aspetti Metodologici”. Consiglio Nazionale delle Richerche - Gruppo Nazionale per la Difesa dai Terremoti, Roma. (In Italian). 22. Griffith, M.C., Magenes, G., Melis, G. and Picchi, L. [2003] “Evaluation of Out-of-Plane Stability of Unreinforced Masonry Walls Subjected to Seismic Excitation”. Journal on Earthquake Engineering, Vol. 7, Special Issue No. 1, 141-169. 23. Gruntal, G. [1998]. “European Macroseismic Scale 1998”. Chaiers dur Centre Européen du Géodynamique et de Séismologie, Volume 15, Luxembourg. 24. Haukass, T. [2001]. “FERUM, Finite Element http://www.ce.berkeley.edu/~haukaas/FERUM/ferum.html.

Reliability

Using

Matlab”.

25. Kennedy, R.P., Cornell, C.A., Campbell, R.L., Kaplan, S., and Perla, H.F. [1980]. “Probabilistic Seismic Safety on an Existing Nuclear Power Plant”. Nuclear Engineering and Design, Vol 59(2): 315-338 26. Kircher, C., Nassar, A., Kustu, O. and Holmes, W. [1997]. “Development of Building Damage Functions for Earthquake Loss Estimation”. Earthquake Spectra, 13(4): 663-682. 73


27. Liu, P and Der Kiureghian, A. [1986]. “Multivariate Distribution Models with Prescribed Marginals and Covariances”. Probabilistic Engineering Mechanics, Vol 1, N°2. 28. Magenes, G. and Calvi, G.M. [1997]. “In-Plane Seismic Response of Brick Masonry Walls”. Earthquake Engineering and Structural Dynamics, Vol 26, 1091-1112. 29. McKay, Beckman and Conover. [1979] Technometrics, 21(2):239-245. 30. Ministero dei Lavori Pubblici. [1981]. “Istruzioni per la Compilazione della Scheda di Rilevamento, Esposizione e Vulnerabilità Sismica degli Edifici”. GNDT, Regioni EmiliaRomagna e Toscana, Firenze. (In Italian). 31. NZSEE. [2002]. “Assessment and Improvement of the Structural Performance of Buildings in Earthquake”. Recommendations of an NZSEE Study Group on Earthquake Risk Buildings Prepared for the Building Industry Authority. 32. Paulay, T., and Priestley, M.J.N. [1992]. “Seismic Design of Reinforced Concrete and Masonry Structures”, John Wiley & Sons, New York. 33. Penazzo, P. [2001]. “Influence of strong-motion characteristics on inelastic demand in masonry structures”. MSc Dissertation. London: Imperial College. 34. Pinto, P.E. and Giannini, R. [2001]. “Methods for Seismic Reliability Analysis”. Notes for a course at the European School of Advanced Studies in Reduction of Seismic Risk-ROSE School. 35. Pinto, P.E. and Franchin, P. [2003]. “Probabilistic Seismic Risk Assessment of Structures”. Notes for a Short Course at the European School of Advanced Studies in Reduction of Seismic Risk-ROSE School. 36. Restrepo-Vélez, L.F., and Bommer, J.J. [2003] “An Exploration of the Nature of the Scatter in Ground-Motion Prediction Equations and the Implications for Seismic Hazard Assessment.”. Journal on Earthquake Engineering, Vol. 7, Special Issue No. 1, 1-29. 37. Shibata, A. and Sozen, M.A. [1976]. “Substitute-Structure Method for Seismic Design in R/C”. Journal of the Structural Division – ASCE, 102:1-18. 38. Sudret, B. and Der Kiureghian, A. [2000]. “Stochastic Finite Element Methods and Reliability, A State – of – the – Art Report”. Department of Civil and Environmental Engineering , University of California at Berkeley, Report UCB/SEMM-2000/08. 39. Whitman, R.V. [1973]. “Damage Probability Matrices for Prototype Buildings”. Massachusetts Institute of Technology, Department of Civil Engineering Research, Report R73-57, Cambridge, Massachusetts. 40. Whitman, R.V., Anagnos, T., Kircher, C., Lagorio, H.C., Lawson, R.S., and Schenider, P. [1997]. “Development of a National Earthquake Loss Methodology”. Earthquake Spectra, 13(4): 643661.

74

Annex 1

FORTRAN Code for the MeBaSe Procedure c**********************************************************************] c c MeBaSe c MECHANICAL BASED PROCEDURE FOR THE SEISMIC RISK ESTIMATION OF c UNREINFORCED MASONRY BUILDINGS c c MAIN PROGRAM: main.for c Coded: Luis F. Restrepo V. c January 2002 c Ver 1.0 c c INPUT VARIABLES : c c OUTPUT VARIABLES : c c**********************************************************************] program main implicit none integer io, n1, n2, n3, n4, n5 c----------------------------------------------------------------------] c Reads input file: input.dat c----------------------------------------------------------------------] write(*,*) '

1. Reading input file'

call readfirstline(io,n1,n2,n3,n4,n5) call mebase(io,n1,n2,n3,n4,n5) end

A1.1

Annex 1 c**********************************************************************] c c MeBaSe c MECHANICAL BASED PROCEDURE FOR THE SEISMIC RISK ESTIMATION OF c UNREINFORCED MASONRY BUILDINGS c c SUBROUTINE: mebase.for c Coded: Luis F. Restrepo V. c November 2002 c Ver 1.0 c c INPUT VARIABLES : c c io : Flag 1: In-plane 2:Out-of-plane c n1 : Number of constants c n2 : Number of random variables for T LS c n3 : Number of random variables for Delta LS c n4 : Number of points defining DRS c n5 : Number of discretization points c c OUTPUT VARIABLES : c c pf : Probability of failure c c**********************************************************************] subroutine mebase(io,n1,n2,n3,n4,n5) integer io, n1, n2, n3, n4, n5, i, j, eqcn, naux double precision xd(n1+1), m(n2), sd(n2), x(n2), p(n2,n2), d1, d2 double precision prob, tls(n5), cdftls(n5), deltals(n5) double precision cdfdeltals(n5,n5), pdftls(n5), pdfdeltals(n5,n5) double precision pf, drs(n4,2), drsint(n5), sddrs, sdy, my, y double precision cdfdemand(n5,n5), xaux(n2) c----------------------------------------------------------------------] c Reads input file: input.dat c----------------------------------------------------------------------] call readdata(n1,n2,d1,d2,xd,m,sd,x,p,drs,sddrs) c----------------------------------------------------------------------] c Computes the CDF of Tls c----------------------------------------------------------------------] open(6,file = 'output.dat') write(*,*) ' 2. Computing CDF of Tls' write(6,1000) xd(1) = 0.0d0 xd(n1+1) = 0.0d0 eqcn = 1 naux = n2 prob = 0.0d0 do i = 1,n2 xaux(i) = x(i) end do do i = 1,n5 xd(1) = xd(1)+d1 call form(io,eqcn,n1,n2,naux,xd,m,sd,xaux,p,prob) tls(i) = xd(1) cdftls(i) = prob write(6,1100) tls(i), cdftls(i) end do c----------------------------------------------------------------------] c Computes the PDF of Tls c----------------------------------------------------------------------] write(*,*) ' 3. Computing PDF of Tls' call pdf(n5,tls,cdftls,pdftls) c----------------------------------------------------------------------] c Computes the CDF of Deltals conditional on Tls(i) c----------------------------------------------------------------------] open(6,file = 'output.dat') write(*,*) ' 4. Computing CDF of Deltals' write(6,1200) do i = 1,n5 write(*,*) 'tls(',i,')' write(6,*) 'tls(',i,')' xd(1) = 0.0d0 xd(n1+1) = tls(i) eqcn = 2 naux = n3

A1.2

Annex 1 prob = 0.0d0 do j = 1,n2 xaux(j) = x(j) end do

c c c

c 5 c c c c c

do j = 1,n5 xd(1) = xd(1)+d2 call form(io,eqcn,n1,n2,naux,xd,m,sd,xaux,p,prob) deltals(j) = xd(1) cdfdeltals(j,i) = prob ******************** These lines are to avoid ******************** ********* numerical problems when CDF is already > 0.999 ********* *************** when displacement is extremely small ************* if(prob.ge.0.999) then xd(1) = xd(1)-d2 do k = j,n5 xd(1) = xd(1)+d2 deltals(k) = xd(1) cdfdeltals(k,i) = prob write(6,1100) deltals(k), cdfdeltals(k,i) end do goto 5 else end if ****************************************************************** write(6,1100) deltals(j), cdfdeltals(j,i) end do continue end do do i = 1,n5 do j = 1,n5 write(6,1100) deltals(j), cdfdeltals(i,j) end do end do

c----------------------------------------------------------------------] c Computes the PDF of Deltals conditional on Tls(i) c----------------------------------------------------------------------] write(*,*) '

5. Computing PDF of Deltals'

do i = 1,n5 write(6,*) 'tls(',i,')' call pdf(n5,deltals,cdfdeltals(:,i),pdfdeltals(:,i)) do j = 1,n5 write(6,1100) deltals(j), pdfdeltals(j,i) end do end do c----------------------------------------------------------------------] c Interpolates the DRS to the same Tls of the PDF c----------------------------------------------------------------------] write(*,*) '

10

6. Interpolating the Demand Response Spectrum'

do i = 1,n5 j = 0 j = j+1 if(tls(i).eq.drs(j,1)) then drsint(i) = drs(j,2) else if(tls(i).lt.drs(j+1,1)) then drsint(i) = (drs(j+1,2)-drs(j,2))*(tls(i)-drs(j,1))/ 1 (drs(j+1,1)-drs(j,1))+drs(j,2) else goto 10 end if end if end do

c----------------------------------------------------------------------] c Computes the CDF of the demand for a given Tls c----------------------------------------------------------------------] write(*,*) ' 7. Computing CDF of the demand for a given Tls' call probdemand(n5,sddrs,drsint,deltals,cdfdemand) c----------------------------------------------------------------------] c Computes the probability of failure Pf c----------------------------------------------------------------------] write(*,*) ' pf = 0.0d0

8. Computing the probability of failure Pf'

do i = 1,n5

A1.3

Annex 1 do j = 1,n5 pf = pf+(1-cdfdemand(i,j))*pdfdeltals(j,i)*pdftls(i) end do end do write(*,1500) pf write(6,1500) pf c----------------------------------------------------------------------] c Formats c----------------------------------------------------------------------] 1000 1100 1200 1300 1400 1500

format(/, 10x, 'Tls(i) CDF of Tls(i)', /) format(5x, F10.4,3x, F12.10) format(/, 6x, 'Deltals(i) CDF of Deltals(i)', /) format(/, 10x, 'Tls(i) PDF of Tls(i)', /) format(/, 6x, 'Deltals(i) PDF of Deltals(i)', /) format(2x, 'PROBABILITY OF FAILURE Pf = ', F5.3) return end

A1.4

Annex 1 c**********************************************************************] c c MeBaSe c MECHANICAL BASED PROCEDURE FOR THE SEISMIC RISK ESTIMATION OF c UNREINFORCED MASONRY BUILDINGS c c SUBROUTINE: readfirstline.for c Coded: Luis F. Restrepo V. c November 2002 c Ver 1.0 c c INPUT VARIABLES : c c OUTPUT VARIABLES : c c io : Flag 1: In-plane 2:Out-of-plane c n1 : Number of constants c n2 : Number of random variables for T LS c n3 : Number of random variables for Delta LS c n4 : Number of points defining DRS c n5 : Number of discretization points c c**********************************************************************] subroutine readfirstline(io,n1,n2,n3,n4,n5) implicit none integer io, n1, n2, n3, n4, n5 c----------------------------------------------------------------------] c Reads input file: input.dat c----------------------------------------------------------------------] open(5,file = 'input.dat') read(5,*) io, n1, n2, n3, n4, n5 return end

A1.5

Annex 1 c**********************************************************************] c c MeBaSe c MECHANICAL BASED PROCEDURE FOR THE SEISMIC RISK ESTIMATION OF c UNREINFORCED MASONRY BUILDINGS c c SUBROUTINE: readdata.for c Coded: Luis F. Restrepo V. c November 2002 c Ver 1.0 c c INPUT VARIABLES : c c n1 : Number of constants c n2 : Number of random variables for T LS c c OUTPUT VARIABLES : c c d1 : Step increment for T LS c d2 : Step increment for Delta LS c xd : Vector of constants c m : Vector of mean values of X c sd : Vector of standard deviation values of X c x : Vector of initial values of X in FORM analysis c p : Matrix of correlation coefficients c drs(i,1) : Periods of the Demand Response Spectra c drs(i,2) : Ordinates of the Demand Response Spectra c sddrs : Standard deviation of thd Demand Response Spectra c c**********************************************************************] subroutine readdata(n1,n2,d1,d2,xd,m,sd,x,p,drs,sddrs) implicit none integer i, j, n1, n2 double precision d1, d2 double precision xd(n1), m(n2), sd(n2), x(n2), p(n2,n2), drs(20,2) double precision sddrs c----------------------------------------------------------------------] c Reads input file: input.dat c----------------------------------------------------------------------] read (5,*) d1, d2 do i = 1,n1 read(5,*) xd(i) end do do i = 1,n2 read(5,*) m(i), sd(i), x(i) end do read (5,*) ((p(i,j), j = 1,n2), i = 1,n2) read (5,*) ((drs(i,j), j = 1,2), i = 1,20) read(5,*) sddrs return end

A1.6

Annex 1 c**********************************************************************] c c MeBaSe c MECHANICAL BASED PROCEDURE FOR THE SEISMIC RISK ESTIMATION OF c UNREINFORCED MASONRY BUILDINGS c c SUBROUTINE: form.for c Coded: Luis F. Restrepo V. - Convergence criteria taken from FERUM c November 2002 c Ver 1.0 c c This subroutine performs the FORM (First Order Reliability Method) c analysis using the Nataf transformation c c INPUT VARIABLES : c c io : Flag 1: In-plane 2:Out-of-plane c eqcn : Equation type (1. Tls, 2. Deltals) c n1 : Number of constants c n2 : Number of random variables c naux : Number of 'efective' random variables c xd : Vector of constants c m : Vector of mean values of X c sd : Vector of standard deviation values of X c x : Vector of initial values of X c p : Correlation matrix P c c OUTPUT VARIABLES : c c prob : Probability of failure c c**********************************************************************] subroutine form(io,eqcn,n1,n2,naux,xd,m,sd,x,p,prob) implicit none integer i, j, k, n1, n2, naux, io, eqcn, flag, iter, maxiter double precision m(n2), sd(n2), xd(n1+1), x(n2), p(n2,n2), sdy, my double precision maux, sdaux, paux, alpha, pp, low, ilow, y, u, ux double precision xaux, dja, ja, ija, gradgcap, gradg, delta, xx double precision gcap0, gcap, gcapf, g, tol1, tol2, err double precision normg, od, normd, d(n2), beta, prob allocatable :: maux(:), sdaux(:), paux(:,:), alpha(:), pp(:,:) allocatable :: low(:,:), ilow(:,:), y(:), u(:), ux(:), xaux(:) allocatable :: dja(:,:), ja(:,:), ija(:,:), gradgcap(:), gradg(:) allocatable :: xx(:) c----------------------------------------------------------------------] c Form vectors y matrices for maux, sdaux, xaux and paux c----------------------------------------------------------------------] allocate(maux(naux),sdaux(naux),paux(naux,naux),alpha(naux)) allocate(pp(naux,naux),low(naux,naux),ilow(naux,naux),y(naux)) allocate(u(naux),ux(naux),xaux(naux),dja(naux,naux),ja(naux,naux)) allocate(ija(naux,naux),gradgcap(naux),gradg(naux),xx(naux)) do i = 1,naux maux(i) = m(i+n2-naux) sdaux(i) = sd(i+n2-naux) xaux(i) = x(i+n2-naux) do j = 1,naux paux(i,j) = p(i+n2-naux,j+n2-naux) end do end do c----------------------------------------------------------------------] c Computes correlation matrix P' by using the Nataf transformation c----------------------------------------------------------------------] call nataf(naux,maux,sdaux,paux,pp) c----------------------------------------------------------------------] c Performs the Cholesky decomposition of P' = A * Atrans c----------------------------------------------------------------------] call cholesky(naux,pp,low) c----------------------------------------------------------------------] c Performs the inversion of A c----------------------------------------------------------------------] call inversion(naux,low,ilow) c----------------------------------------------------------------------] c Transforms r.v.'s X to correlated standard normal r.v.'s Y c----------------------------------------------------------------------]

A1.7

Annex 1

do i = 1,naux sdy = sqrt(log((sdaux(i)/maux(i))**2+1)) my = log(maux(i))-0.5d0*sdy**2 y(i) = (log(xaux(i))-my)/sdy end do c----------------------------------------------------------------------] c Transforms r.v.'s Y to uncorrelated standard normal r.v.'s U c---------------------------------------------------------------------] do i = 1,naux u(i) = 0.0d0 end do

c c c

c

do i = 1,naux do j = 1,naux u(i) = u(i)+ilow(i,j)*y(j) ******************** These lines are to avoid ******************** ********* numerical problems when the transformed r.v.' ********** *********************** are extremely large ********************** if(ux(i).gt.1000000) then prob = 0.999999 goto 5 else end if ****************************************************************** end do end do

c----------------------------------------------------------------------] c Starts iteration to find the 'design point' c----------------------------------------------------------------------] flag = 0 tol1 = 0.001 tol2 = 0.001 iter = 1 maxiter = 100 call feval(io,eqcn,n1,naux,xd,xaux,gcap0) do while(flag.eq.0) c----------------------------------------------------------------------] c Transforms r.v.'s U to the original space of X c----------------------------------------------------------------------] do i = 1,naux ux(i) = 0.0d0 do j = 1,naux ux(i) = ux(i)+low(i,j)*u(j) end do end do do i = 1,naux sdy = sqrt(log((sdaux(i)/maux(i))**2+1)) my = log(maux(i))-0.5d0*sdy**2 xaux(i) = exp(ux(i)*sdy+my) end do c----------------------------------------------------------------------] c Computes Jacobian of the transformation J and the inverse of J c----------------------------------------------------------------------] do i = 1,naux do j = 1,naux dja(i,j) = 0.0d0 ja(i,j) = 0.0d0 ija(i,j) = 0.0d0 end do end do do i = 1,naux do j = 1,naux sdy = sqrt(log((sdaux(i)/maux(i))**2+1)) dja(i,i) = 1/sdy/xaux(i) end do end do do i = 1,naux do j = 1,naux do k = 1,naux ja(i,j) = ja(i,j)+ilow(i,k)*dja(k,j) end do end do end do

A1.8

Annex 1

call inversion(naux,ja,ija) c----------------------------------------------------------------------] c Computes gradient of the G limite-state function c----------------------------------------------------------------------] call feval(io,eqcn,n1,naux,xd,xaux,gcap) do i = 1,naux xx(i) = 0.0d0 gradgcap(i) = 0.0d0 end do do i = 1,naux delta = sdaux(i)/500.0d0 do j = 1,naux xx(j) = xaux(j) end do xx(i) = xaux(i) + delta call feval(io,eqcn,n1,naux,xd,xx,gcapf) gradgcap(i) = (gcapf-gcap)/delta end do c----------------------------------------------------------------------] c Computes gradient of the transformed g limite-state function c----------------------------------------------------------------------] do i = 1,naux gradg(i) = 0.0d0 end do do i = 1,naux do j= 1,naux gradg(i) = gradg(i)+gradgcap(j)*ija(j,i) end do end do

c----------------------------------------------------------------------] c Checks convergence c----------------------------------------------------------------------] normg = 0.0d0 od = 0.0d0 normd = 0.0d0 do i = 1,naux normg = normg+gradg(i)**2 alpha(i) = 0.0d0 xx(i) = 0.0d0 end do normg = sqrt(normg) do i = 1,naux alpha(i) = -gradg(i)/normg end do do i = 1,naux od = od+u(i)*alpha(i) end do do i = 1,naux xx(i) = u(i)-od*alpha(i) end do do i = 1,naux normd = normd+xx(i)**2 end do normd = sqrt(normd) od=abs(gcap/gcap0) if((abs(gcap/gcap0)
A1.9

Annex 1 end if c c c

c c

c

******************** These lines are to avoid ******************** ************ numerical problems when the probability ************* ************************ is extremely low ************************ if(iter.eq.(maxiter-1)) then prob = 0.0d0 goto 5 end if ****************************************************************** *********** It seems that these lines are not required *********** if(iter.eq.maxiter) then write(*,*) 'Maximum number of iterations exceeded' stop else end if ****************************************************************** end do

c----------------------------------------------------------------------] c Computes Hasofer-Lind reliability index and probability of failure c----------------------------------------------------------------------] beta = 0.0d0 do i = 1,naux beta = beta+alpha(i)*u(i) end do call normalcdf(-beta,prob) 5

return end

A1.10

Annex 1 c**********************************************************************] c c MeBaSe c MECHANICAL BASED PROCEDURE FOR THE SEISMIC RISK ESTIMATION OF c UNREINFORCED MASONRY BUILDINGS c c SUBROUTINE: nataf.for c Coded: Luis F. Restrepo V. c November 2002 c Ver 1.0 c c This subroutine computes the correlation matrix P' by using the c Nataf transformation (Liu and Der Kiureghian, 1986). Is valid just c for variables with lognormal distribution c c INPUT VARIABLES : c c n : Number of random variables c m : Vector of mean values of X c sd : Vector of standard deviation values of X c p : Correlation matrix P c c OUTPUT VARIABLES : c c pp : Correlation matrix P' c c**********************************************************************] subroutine nataf(n,m,sd,p,pp) integer i, j, n double precision m(n), sd(n), p(n,n), pp(n,n), f, cvi, cvj

1

do i = 1,n cvi = sd(i)/m(i) do j = 1,n cvj = sd(j)/m(j) f = (log(1+p(i,j)*cvi*cvj))/(p(i,j)*sqrt((log(1+cvi**2))* (log(1+cvj**2)))) pp(i,j) = p(i,j)*f end do end do return end

A1.11

Annex 1 c**********************************************************************] c c MeBaSe c MECHANICAL BASED PROCEDURE FOR THE SEISMIC RISK ESTIMATION OF c UNREINFORCED MASONRY BUILDINGS c c SUBROUTINE: cholesky.for c Coded: Luis F. Restrepo V. - Taken from NUMERICAL RECIPES IN FORTRAN c November 2002 c Ver 1.0 c c This subroutine performs the decomposition of A = L * Ltransp c c INPUT VARIABLES : c c n : Dimension of matrix 'A' c x : Positive-definite symmetric matrix 'A' to decompose c c OUTPUT VARIABLES : c c lo : Lower triangular matrix L c c**********************************************************************] subroutine cholesky(n,x,lo) integer i, j, k, n double precision x(n,n), a(n,n), lo(n,n), d(n), sum do i = 1,n do j = 1,n a(i,j) = x(i,j) end do end do do i = 1,n do j = i,n sum = a(i,j) do k = i-1,1,-1 sum = sum-a(i,k)*a(j,k) end do if(i.eq.j)then if(sum.le.0.)pause 'cholesky failed' d(i) = sqrt(sum) else a(j,i) = sum/d(i) endif end do end do do i = 1,n do j = 1,n lo(i,j) = a(i,j) end do end do do i = 1,n do j = 1,n if(j.gt.i) then lo(i,j) = 0.0d0 else if(j.eq.i) then lo(i,j) = d(i) else continue end if end if end do end do return end

A1.12

Annex 1 c**********************************************************************] c c MeBaSe c MECHANICAL BASED PROCEDURE FOR THE SEISMIC RISK ESTIMATION OF c UNREINFORCED MASONRY BUILDINGS c c SUBROUTINE: inversion.for c Coded: Luis F. Restrepo V. - Taken from www.nauticom.net/www/jdtaft c November 2002 c Ver 1.0 c c This subroutine performs the inversion of A c c INPUT VARIABLES : c c n : Dimension of matrix 'A' c a : Positive-definite symmetric matrix 'A' to invert c c OUTPUT VARIABLES : c c ia : Inverse matrix of 'A' c c**********************************************************************] subroutine inversion(n,a,ia) integer i, j, k, n, n2 double precision a(n,n), ia(n,n), d(n,2*n), alpha, beta c----------------------------------------------------------------------] c Creates the reduction matrix c----------------------------------------------------------------------] n2 = 2*n do i = 1,n do j = 1,n d(i,j) = a(i,j) d(i,n+j) = 0.0d0 end do d(i,n+i) = 1.0d0 end do c----------------------------------------------------------------------] c Performs the reduction c----------------------------------------------------------------------]

5

do i = 1,n alpha = d(i,i) if(alpha.eq.0.0d0) go to 10 do j = 1,n2 d(i,j) = d(i,j)/alpha end do do k = 1,n if((k-i).eq.0.0d0) go to 5 beta = d(k,i) do j = 1,n2 d(k,j) = d(k,j) - beta*d(i,j) end do end do end do

c----------------------------------------------------------------------] c Copy results into output matrix c----------------------------------------------------------------------] do i = 1,n do j = 1,n ia(i,j) = d(i,j+n) end do end do 10

return print *,'*** ERROR: Trying to invert a singular matrix ***' end

A1.13

Annex 1 c**********************************************************************] c c MeBaSe c MECHANICAL BASED PROCEDURE FOR THE SEISMIC RISK ESTIMATION OF c UNREINFORCED MASONRY BUILDINGS c c SUBROUTINE: feval.for c Coded: Luis F. Restrepo V. c November 2002 c Ver 1.0 c c This subroutine evaluates the G limit state function c c INPUT VARIABLES : c c io : Flag 1: In-plane 2:Out-of-plane c eqcn : Equation type (1. Tls, 2. Deltals) c n1 : Number of constants c n2 : Number of random variables c xd : Vector of constants c x : Vector of random variables c c OUTPUT VARIABLES : c c g : function value c c**********************************************************************] subroutine feval(io,eqcn,n1,n2,xd,x,g) integer n1, n2, io, eqcn double precision xd(n1+1), x(n2), g, num, den, grav, pi grav = 9.81d0 pi = 3.14159265358979 if(io.eq.0) then if(eqcn.eq.1) then num = xd(2)*x(1)*x(2)+xd(3)*(x(3)-x(2))*x(4) den = grav/x(6)*xd(4)*x(5)*(1+xd(5)/x(5))**0.5 g = 2.0d0*pi*sqrt(num/den)-xd(1) else if(eqcn.eq.2) then num = grav*xd(4)*x(1)/x(2)*(1+xd(5)/x(1))**0.5 g = num/4/(pi**2)*(xd(6)**2)-xd(1) else write(*,*) 'Wrong Equation number!' stop end if end if else if(eqcn.eq.1) then num = pi**2*x(1)*x(3)*x(5)*x(2) den = x(3)*(1-x(1))*(1+x(4))*grav*x(5) g = sqrt(num/den)-xd(1) else if(eqcn.eq.2) then num = x(3)*(1-x(1))*(1+x(4))*grav*x(5) den = pi**2*x(1)*x(2) g = (xd(2)**2)*num/den-xd(1) else write(*,*) 'Wrong Equation number!' stop end if end if end if return end

A1.14

Annex 1 c**********************************************************************] c c MeBaSe c MECHANICAL BASED PROCEDURE FOR THE SEISMIC RISK ESTIMATION OF c UNREINFORCED MASONRY BUILDINGS c c SUBROUTINE: searchcriteria.for c Coded: Luis F. Restrepo V. - Adapted from FERUM c November 2002 c Ver 1.0 c c This subroutine computes the search direction c c INPUT VARIABLES : c c gcap : G limite-state function c gradgcap : Gradient of the G limite-state function c u : Random variables in the U space c c OUTPUT VARIABLES : c c d : Search direction c c**********************************************************************] subroutine searchcriteria(n2,gcap,gradg,u,d) implicit none integer i, n2 double precision gcap, gradg(n2), u(n2), d(n2), alpha(n2),normg double precision daux normg = 0.0d0 daux = 0.0d0 do i = 1,n2 normg = normg+gradg(i)**2 alpha(i) = 0.0d0 d(i) = 0.0d0 end do normg = sqrt(normg) daux = gcap/normg do i = 1,n2 alpha(i) = -gradg(i)/normg end do do i = 1,n2 daux = daux+alpha(i)*u(i) end do do i = 1,n2 d(i) = daux*alpha(i)-u(i) end do return end

A1.15

Annex 1 c**********************************************************************] c c MeBaSe c MECHANICAL BASED PROCEDURE FOR THE SEISMIC RISK ESTIMATION OF c UNREINFORCED MASONRY BUILDINGS c c SUBROUTINE: normalcdf.for c Coded: Luis F. Restrepo V. - some parts taken from NUMERICAL RECIPES IN FORTRAN c November 2002 c Ver 1.0 c c This subroutine computes the normal cdf of Beta c c INPUT VARIABLES : c c beta : Normal standard variable c c OUTPUT VARIABLES : c c prob : Probability of being less or equal than beta c c**********************************************************************] subroutine normalcdf(beta,prob) integer i, it, n double precision beta, prob, betahl, pi, delta, x, sum pi = 3.14159265358979 n = 20 prob = 0.0d0 if(beta.ge.0.0d0)then betahl = beta it = 2**(n-2) delta = (betahl)/it x = 0.5*delta sum = 0.0d0 do i = 1,it sum = sum + (1/sqrt(2.0d0*pi)*exp(-0.5d0*(x)**2)) x = x + delta end do prob = 0.5+sum*delta else betahl = -1*beta it = 2**(n-2) delta = (betahl)/it x = 0.5*delta sum = 0.0d0 do i = 1,it sum = sum + (1/sqrt(2.0d0*pi)*exp(-0.5d0*(x)**2)) x = x + delta end do prob = 1-(0.5+sum*delta) end if if(prob.lt.0) then prob = 0.0d0 else end if return end

A1.16

Annex 1 c**********************************************************************] c c MeBaSe c MECHANICAL BASED PROCEDURE FOR THE SEISMIC RISK ESTIMATION OF c UNREINFORCED MASONRY BUILDINGS c c SUBROUTINE: pdf.for c Coded: Luis F. Restrepo V. c December 2002 c Ver 1.0 c c This subroutine computes the PDF of a given CDF c c INPUT VARIABLES : c c n : size of vector x c x : vector of variable x c cdfx : CDF of variable x c c OUTPUT VARIABLES : c c pdfx : PDF of variable x c c**********************************************************************] subroutine pdf(n,x,cdfx,pdfx) integer i, n double precision x(n), cdfx(n), pdfx(n), sum sum = 0.0d0 pdfx(1) = 0.0d0 do i = 2,n pdfx(i) = cdfx(i)-cdfx(i-1) end do return end

A1.17

Annex 1 c**********************************************************************] c c MeBaSe c MECHANICAL BASED PROCEDURE FOR THE SEISMIC RISK ESTIMATION OF c UNREINFORCED MASONRY BUILDINGS c c SUBROUTINE: probdemand.for c Coded: Luis F. Restrepo V. - based on the normalcdf.for routine c November 2002 c Ver 1.0 c c This subroutine computes the normal cdf of the demand for a given period c c INPUT VARIABLES : c c sddrs : Standard deviation of the demand c drsint : Ordinates of the demand response spectra c deltals : Vector of periods c c OUTPUT VARIABLES : c c cdfdemand : Matrix with the CDF of demand given Tls c c**********************************************************************] subroutine probdemand(sddrs,drsint,deltals,cdfdemand) integer i, j, k, double precision double precision double precision

it, n, m sddrs, sddrs1, drsint(100), deltals(100) cdfdemand(100,100), sdy, my, funca, funcb beta1, beta2, prob, betahl, pi, delta, x, sum

pi = 3.14159265358979 n = 20 m = 6 do i = 1,100 prob = 0.0d0 sddrs1=drsint(i)*(1-sddrs) sdy = log((sddrs1/drsint(i))**2+1) my = drsint(i)*exp(-0.5*sdy) do j = 1, 100 beta1 = (log(deltals(j)/my))/sqrt(sdy) if(beta1.ge.0.0d0)then if(j.eq.1) then betahl = beta1 it = 2**(n-2) delta = (betahl)/it x = 0.5*delta sum = 0.0d0 do k = 1,it sum = sum + (1/sqrt(2.0d0*pi)*exp(-0.5d0*(x)**2)) x = x + delta end do prob = 0.5+sum*delta else beta2 = (log(deltals(j-1)/my))/sdy betahl = beta1 - beta2 it = 2**(m-2) delta = betahl/it x = beta2 + 0.5*delta sum = 0.0d0 do k = 1,it sum = sum+1/sqrt(2.0d0*pi)*exp(-0.5d0*(x)**2) x = x + delta end do prob = prob+sum*delta end if else if(j.eq.1) then betahl = -1*beta1 it = 2**(n-2) delta = betahl/it x = 0.5*delta sum = 0.0d0 do k = 1,it sum = sum + (1/sqrt(2.0d0*pi)*exp(-0.5d0*(x)**2)) x = x + delta end do prob = 1-(0.5+sum*delta) else beta2 = (log(deltals(j-1)/my))/sdy betahl = beta1 - beta2 it = 2**(m-2) delta = betahl/it x = beta2 + 0.5*delta sum = 0.0d0

A1.18

Annex 1 do k = 1,it sum = sum+1/sqrt(2.0d0*pi)*exp(-0.5d0*(x)**2) x = x + delta end do prob = prob+sum*delta end if end if if(prob.lt.0) then prob = 0.0d0 else end if cdfdemand(i,j) = prob end do end do return end

A1.19

Annex 1

Input file for In-plane example 0 5 6 2 20 100 0.012 0.0015 0.00 0.689 0.916 0.87 0.39 12.60 2.00 12.60 0.001 0.001 0.001 0.005 0.001 0.005 2.00 0.50 2.00 0.20 0.05 0.20 1.00 0.20 1.00 1.000 0.001 0.001 0.001 0.001 0.001 1.000 0.001 0.001 0.001 0.001 0.001 1.000 0.001 0.001 0.001 0.001 0.001 1.000 0.001 0.001 0.001 0.001 0.001 1.000 0.001 0.001 0.001 0.001 0.001 0.00 0.0000000 0.01 0.0000040 0.02 0.0000172 0.03 0.0000413 0.04 0.0000780 0.05 0.0001290 0.06 0.0001960 0.08 0.0003851 0.10 0.0006589 0.15 0.0014825 0.20 0.0026356 0.25 0.0041182 0.30 0.0059302 0.35 0.0080716 0.40 0.0105425 0.45 0.0167731 0.50 0.0186368 0.60 0.0223641 2.00 0.0745471 4.00 0.0745471 1.65

0.001 0.001 0.001 0.001 0.001 1.000

A1.20

Annex 1

Input file for Out-of-plane example 1 0.04 0.00

1

5 0.0250

5

20

0.40 0.10 3.00 0.50 0.80 0.20 0.50 0.20 0.30 0.05 1.000 0.800 0.800 0.800 0.800 1.000 0.800 0.800 0.800 0.800 1.000 0.800 0.800 0.800 0.800 1.000 0.800 0.800 0.800 0.800 0.001 0.00000 0.010 0.00003 0.020 0.00010 0.030 0.00024 0.040 0.00044 0.050 0.00070 0.060 0.00103 0.080 0.00192 0.100 0.00314 0.150 0.00791 0.300 0.04135 0.500 0.11990 0.700 0.16859 0.900 0.17456 1.000 0.17096 1.400 0.15115 2.000 0.13120 2.400 0.12326 3.000 0.11547 4.001 0.10796 1.65

100 0.40 3.00 0.80 0.50 0.30 0.800 0.800 0.800 0.800 1.000

A1.21

Annex 2

ISTRUZIONI PER LA COMPILAZIONE DELLA SCHEDA DI RILIEVO DEGLI EDIFICI IN MURATURA

INDICE

pag. INDICE ……………………………………………………………………………………….....

2

1.

NOTE ESPLICATIVE GENERALI ………………………………………………….......

3

2.

SEZIONE 1: DATI RELATIVI ALLA SCHEDA ……………………..............................

4

3.

SEZIONE 2: LOCALIZZAZIONE EDIFICIO ..………………………………………….

4

4.

SEZIONE 3: CONDIZIONI D’USO ……………………………………………………..

5

5.

SEZIONE 4: MORFOLOGIA DEL SITO ………………………………………………..

5

6.

SEZIONE 5: ETA E STATO DI MANUTENZIONE ………………………….………...

6

7.

SEZIONE 6: TIPOLOGIA STRUTTURALE ………………………………………….....

6

8.

SEZIONE 7: CARATTERISTICHE GENERALI DELL’EDIFICIO ………….………...

8

9.

SEZIONE 8: MECCANISMI DI RIBALTAMENTO ASSOCIATI A DANNO ESISTENTI .........................................................................................................................

10.

SEZIONE 9: SEZIONE DI CONTROLLO ………….………...........................................

A2.1

10 12

Annex 2

1. 1.1

NOTE ESPLICATIVE GENERALI ISTRUZIONI GENERALI

Per la compilazione della scheda dovrà essere utilizzata sempre la matita in modo da poter facilmente correggere eventuali errori. Gli zeri vanno scritti barrati. I dati numerici vanno riportati allineati con il margine di destra. Il punto, nelle caselle in cui è presente, separa la parte intera de quella decimale; ove non presente, i numeri sono interi. Le caselle definiscono il numero massimo di caratteri che si devono riportare. Qualora il valore numerico superasse lo spazio massimo previsto, si inseriscono delle lettere sulla sinistra per le cifre eccedenti (Z = 0, A = 1, B = 2, ...) escludendo per approssimazione le ultime cifre sulla destra. Ad esempio, nelle caselle 488 a 491, si riporta: B 3 4 1 Se la superficie media coperta è di 23.414 mq A C 2 8 Se la superficie media coperta è di 132.751 mq A C 2 7 Se la superficie media coperta è di 132.749 mq Le tabelle presente nella scheda verranno compilate rilevando l’edificio a partire dai piani inferiori. Nell’impossibilità di reperire un dato lasciare in bianco la casella o le caselle corrispondenti. I manuali di istruzioni per la compilazione delle schede di I e II livelli, publicati per il CNR – GNDT, si intendono come complemento di queste manuale. 1.2

QUALITÀ DELL’INFORMAZIONE

E – Qualità elevata: Informazioni prevalentemente dirette (misure effettuate in sito, letture di elaborati grafici affidabili, visione diretta degli elementi di informazione) con un grado di attendibilità vicino alla certezza. M – Qualità media: Informazioni prevalentemente dedotte (letture indirette quali quelle desunte da fotografie, misure desunte da elaborati non esecutivi, saggi non distruttivi di scarsa attendibilità, letture dirette su situazioni analoghe, informazioni orali di persone di fiducia del rilevatore) con un grado di attendibilità intermedio fra il precedente (E) ed il seguente (B). B – Qualità bassa: Informazioni prevalentemente presunte (misure dedotte da ragionevoli ipotesi conoscitive quali quelle sulle usuali modalità e sulle più frequenti scelte progettuali, informazioni orali diverse dalle precedenti con un grado di attendibilità di poco superiore ad una scelta puramente casuale della classe.

A – Informazione assente: Con un grado di attendibilità intorno ai limiti di una scelta casuale. In questi casi la valutazione del rilevatore ha valore puramente indicativo. A2.2

Annex 2

2.

SEZIONE 1: DATI RELATIVI ALLA SCHEDA

Data (caselle 1 a 6): Giorno, mese ed anno della data del rilievo. Scheda # (caselle 7 a 10): Numero progressivo che identifica la scheda tra tutte quelle compilate da una stessa squadra: tale numero viene confermato sulla scheda nel momento in cui questa viene consegnata, compilata, dal caposquadra al coordinatore. È chiaro che ogni squadra deve proporre

una sua numerazione progressiva per le corrispondenti schede compilate. Rilevatore (caselle 11 a 26): Nome e cognome del capo della squadra di rilievo. Si inseriscono tutte le lettere che entrano nelle caselle senza lasciare caselle bianche fra parole distinte.

3.

SEZIONE 2: LOCALIZZAZIONE EDIFICIO

Comune (caselle 27 a 41): Nome del Comune per esteso. Si inseriscono tutte le lettere che entrano nelle caselle senza lasciare caselle bianche fra parole distinte. Foglio (caselle 42 a 44): Si riporta il numero del foglio della cartografia di rilevazione in cui si trova l’edificio. Mappale (caselle 45 a 47): Si riportano, ove possibile, i dati relativi all’edificio. Particella (caselle 48 a 51): Si riportano, ove possibile, i dati relativi all’edificio. Provincia (caselle 52 a 53): Riportare il numero di codice ISTAT della Provincia. Via (caselle 54 a 65): Riportare il nome della via, viale, ecc. tenendo presenti le seguenti convenzioni: • • • • • •

Si inseriscono tutte le lettere che entrano nelle caselle. Non si lasciano spazi bianchi tra parole distinte. Si omettono tutte le preposizioni e congiunzioni. I numeri vanno sempre riportati in fondo in cifre arabe. Per i nomi di persona si riporta l’iniziale del nome e del cognomem, ove questo è presente; in caso contrario, il nome per esteso. I nome dei Santi, “Santo”, “San”, ecc. si abbreviano con “S”seguita immediatamente dal nome.

Esempi: Via della Battaglia di San Martino: BATTAGLIASMA Piazza Giacomo Leopardi: GLEOPARDI Corso Vittorio Emanuele II: VITTORIOEMA2 Piazza XXV Aprile: APRILE25 # Civico (caselle 66 a 68): Riportare il numero più recente del portone principale (o presunto tale); ove presenti (e ove possibile) riportare anche le lettere. Il portone di cui è indicato il numero civico deve essere sempre individuato nella cartografia di rilevazione con una freccia. Altro # civico (caselle 69 a 71): Ove questo è presente oppure ove un secondo portone o accesso principale esiste, riportare il numero ed anche indicarlo nella cartografia di rilevazione con una freccia. A2.3

Annex 2

Denominazione storica del bene (caselle 72 a 84): Riportare il nome oppure la denominazione storica del palazzo; tenendo conto delle stesse convenzioni usati per la Via (caselle 54 a 65). # Aggregato (caselle 85 a 87): Si indica in queste caselle il numero di aggregato strutturale provvisorio assegnato dal coordinatore. # Edificio (caselle 88 a 90): Numero d’ordine provvisorio dell’edificio. Queste caselle vengono riempite sono nel caso in cui l’aggregato strutturale sia costituito da due o più edifici. In questo caso il numero d’ordine dell’aggregato strutturale da mettere in scheda rimane costante, e gli edifici dello stesso isolato vengono numerati progressivamente a partire da 1.

4.

SEZIONE 3: CONDIZIONI D’USO

Uso piano terra (casella 91): Si riportano le condizioni d’uso del piano terra. Uso piani superiori (casella 92): Si riportano le condizioni d’uso dei piani superiori. Se i piani superiori hanno diversi condizioni d’uso si riporta quella prevalente per i piani superiori. # Appartamenti (caselle 93 a 94): Si riporta il numero di appartamenti nell’edificio per cui le condizioni d’uso sono residenziale. Attività commerciale (caselle 95 a 109): Si riporta la attività commerciale prevalente d’accordo con le seguenti categorie: • • • • • • •

Produzione Commercializzazione Deposito, stoccaggio Uffici Misto, servizi pubblici Locali amministrazione Palestre

Condizioni d’uso (casella 110): Si intende riferirsi soltanto alle considerazioni temporali. Per abbandonato si intende quando la mancata utilizzazione funzionale ha comportato anche uno stato di degrado fisico.

5.

SEZIONE 4: MORFOLOGIA DEL SITO

Morfologia del sito (casella 111): Si riporta la morfologia del sito. Fondazioni in roccia (casella 112): L’opzione roccia dovrà essere indicata in presenza di roccia affiorante, anche se sovrastata da depositi superficiali parzialmente disgregati. Pendenza percentuale del terreno (caselle 114 a 115): Si vuole rilevare la situazione altimetrica media del terreno su cui insiste il fabbricato, valutata nella direzione ortogonale alla isoipse. Posizione nell’aggregato (casella 117): Si riporta la posizione del’edificio nell’aggregato.

A2.4

Annex 2

Inquadramento storico nell’aggregato (casella 118): Si riporta l’inquadramento storico del’edificio nell’aggregato.

6.

SEZIONE 5: ETÀ E STATO DI MANUTENZIONE

Età di costruzione (casella 119): Si riporta il codice che distingue l’epoca della prima costruzione. Nei casi di difficoltà nell’individuare l’età della costruzione, si può cercare di interpretare induttivamente sulla base della “tipologia e tecnologia edilizia” (nei casi dubbi tra due classi contigue scegliere la precedente). Evitare in ogni caso di omettere il dato. Età ultimo intervento (casella 124): Si riporta il codice relativo all’epoca dell’ultimo intervento più significativo. Se nell’edificio non è stato effettuato alcun intervento la colonna dovrà essere sbarrata Tipo ultimo intervento (casella 129): Si riporta il codice presente in tabella, relativo al tipo dell’ultimo intervento più significativo (si riporta 0 in caso di nessun intervento). Stati di fatto (casella 130): Si tiene conto con questa voce dello stato di conservazione degli edifici, secondo le seguente definizione: •

•

• •

P – Pessimo: Edifici che presentano pareti fuori piombo e/o lesioni gravi anche se non diffuse. Edifici caratterizati da grave deterioramento dei materiali. Edifici che, pur non presentando lesioni, sono caratterizati da uno stato di conservazione delle murature tale da determinare una grave diminuzione di resistenza. S – Scadente: Edifici con lesioni di media entità (ampiezza della lesione: 2-3 mm) o con lesioni capillari di origine sismica. Edifici che, pur non presentando lesioni, sono caratterizati da uno stato di conservazione delle murature tale da determinare una significativa diminuzione di resistenza D – Discreto: Edifici che presentano lesioni capillari non diffuse, ad eccezione del caso in cui queste siano state prodotte da terremoti. B – Buono: Murature in buone condizioni senza lesioni visibili.

Condizione dei giunti (casella 132): Si riporta il grado di deterioramento dei giunti tenendo conto la seguente classificazione: • • •

7.

N – Nuovo: Se la malta dei giunti è in perfetto stato di conservazione, senza fessure e senza nessun grado di deterioramento. M – Moderato: Se la malta esibisce fessure di qualsiasi lunghezza ma con una profondità minore al 90% dello spessore della parete. In questa categoria è possibile avere qualche grado di deterioramento. S – Severo: Se la malta esibisce fessure di una profondità superiore al 90% dello spessore della parete.

SEZIONE 6: TIPOLOGIA STRUTTURALE

Tipologia strutturale (caselle 134 a 145): Individua la tipologia dominante secondo le codifiche indicate nella scheda. Riportare per ciascun elemento strutturale (verticali, orinzzontali e coperture) la tipologia corrispondente ed il numero di piani per i quali si ripete. La scheda prevede la descrizione di un massimo di quattro tipologie differenti. Qualora queste siano più numerose occorre accorpare con A2.5

Annex 2

gli stessi criteri usati per i dati metrici. La tipologia relativa alle coperture deve essere presente solo nell’ultima riga riguardante l’ultimo piano, per cui in caso di coesistenza al penultimo piano di parziale copertura e struttura orizzontale si indicherà solo la tipologia di quest’ultima, ponendo invece sull’ultima riga la tipologia della copertura più elevata (ovviamente solo se riguarda un piano preso in considerazione e non una superfetazione irrilevante). Di seguito si riportano, con qualche maggior dettaglio, le definizioni già contenute nella scheda. Strutture verticali • • • • • • • • • • • • •

A : Muratura a sacco o a due paramenti formata da pietre di pezzature molto varie, male intessuta e priva di collegamento tra i due fogli. B : Muratura a sacco formata da pietre di pezzature più regolare, bene intessuta e priva di collegamento tra i due fogli, oppure come sopra con spigoli, mazzette e/o ricorsi in pietra squadrata o mattoni pieni. C : Muratura di pietra sbozzata di cattiva qualità in presenza di irregolarità. D : Muratura di pietra sbozzata con spigoli, mazzette e/o ricorsi in mattoni pieni e/o pietra squadrata. E : Muratura di pietra arrotondata o ciottoli di fiume di pezzatura varia senza mazzette e/o ricorsi in pietra squadrata e/o mattoni pieni. F : Come sopra con spigoli, mazzette e/o ricorsi in pietra squadrata e/o mattoni pieni. G : Muratura in blocchetti di tufo o pietra da taglio di dimensioni costanti. H : Muratura in blocchetti di calcestruzzo prefabbicati, con inerti ordinari. I : Muratura in blocchetti di calcestruzzo prefabbicati con inerti leggeri (argilla espansa, ecc.) omogenei in tutta l’estensione. L : Muratura in laterizio di buona qualità, pieno o semipieno (% foratura ≤ 45%). M : Muratura in laterizio con foratura > 45%. N : Pareti in calcestruzzo non armato O : Pareti in calcestruzzo armato e ben collegato; nel caso non sia possibile rilevare o avere notizie sulla presenza di armatura, assegnare la tipologia N.

Strutture orizzontali • • • • • • • • • •

A : Solai con orditura principale e secondaria in legno. B : Come sopra con catene o tiranti. C : Solai con putrelle metalliche e voltine o tavelloni. D : Come sopra con catene o tiranti. E : Solai laterocementizi o solette in calcestruzzo con cordolo ammorsato o inserito nella muratura. F : Volte (a botte, a padiglione, ecc.) senza catene. G : Come sopra con catene. H : Combinazione di volte spingenti (tipo F) con solai piani. I : Combinazione di volte non spingenti (tipo G) con solai piani. L : Altra tipologia strutturale per le strutture orizontali.

Coperture • • •

M : Coperture in legno spingenti per i soli carichi verticali, generalmente costituite da due falde inclinate senza trave di colmo o con trave di colmo deformabile (rapporto luce/altezza l/h > 20) e senza catene o tiranti (Fig. 7a, 7b). N : Coperture in legno con travi inclinate ad una falda oppure a due falde e con muro di spina (Fig. 7c, 7d) oppure con trave di colmo poco deformabile (rapporto l/h < 20) (Fig 7e). O : Coperture in legno a spinta eliminata (Fig. 7f) o con travi orizzontali (Fig. 7g). A2.6

Annex 2

• • • • • •

P : Coperture con solai laterocementizi o solette in calcestruzzo. Q : Coperture con orditura principale in putrelle metalliche, spingente (analoga al tipo M per il legno). R : Coperture con orditura principale in putrelle metalliche, non spingente (analoga ai tipi N ed O per il legno). S : Combinazione dei tipi suddetti con almeno un tipo spingente (M, Q). T : Combinazione dei tipi non spingenti suddetti (N, O, P, R). U : Altra tipologia strutturale per le coperture. h

l = luce trave l/h > 20 Fig. 7a

Fig. 7b

Fig. 7c

Fig. 7d

h

l = luce trave l/h < 20 Fig. 7e

Fig. 7f

Fig. 7g

Cordoli di piano (caselle 149 a 156): Si riporta se il edificio presenta cordoli perimetrali in calcestruzzo ed il numero di piani per i quali si ripete.

8.

SEZIONE 7: CARATTERISTICHE GENERALI DELL’EDIFICIO

Regolarità pianta (caselle 159 a 166): L’assegnazione di un edificio alle varie classi aviene sulla base della più sfavorevole delle condizioni poste dai parametri B1 e B2 nel modo seguente: B1 = a / l *100

B2 = b / l * 100

a

a

a

b

b

l

l

l a

a

b

b

l

l

A2.7

Annex 2

Orditura solai (caselle 169 a 176): Si riporta la direzione prevalente dell’orditura del solai per ogni piano. Corpi aggiunti (caselle 179 a 186): Si intende riportare la esistenza di corpi precari aggiunti successivamente, magari di altezza inferiore e male ammorsati all’edificio originale. Restringimenti (caselle 189 a 196): Si riportano eventuali restrizioni nell’altezza del edificio in ogni piano. Sopraelevazione precarie (caselle 199 a 206): Si riporta se esistono elementi aggiunti recentemente (ultimi 50 anni), non ben integrati nel sistema strutturale. Solai sfalsati (caselle 209 a 216): Si intende riportare quando il solaio di uno degli edifici adiacenti è nel terzo centrale dell’interpiano. Demolizione elementi portanti (caselle 219 a 226): Si riportano eventuali interruzioni a piano terra dei maschi murari in facciata o eliminazione di setti portanti all’interno (normalmente a beneficio di esercizi commerciali). Aggetti ed elementi svettanti (caselle 229 a 236): Si riportano elementi pericolanti come balconi in pietra degradati, piccoli corpi abusivi, cornicioni particolarmente aggettanti, camini in cattivo stato di manutenzione, ogni altro elemento ritenuto pericoloso. Pianta schematica tipica: Si deve disegnare la pianta tipica dell’edificio schematicamente, identificando con un indice le pareti. Non è necessario disegnare l’edificio in scala. Indicare il piano asunto come piano tipico. Conviene trascurare gli elementi accessori dell’edificio, quali portici, garage, ecc., che non si sviluppano per l’intera altezza del corpo principale. Pareti perimetrali: Per ogni parete perimetrale indivuata nella pianta schematica si deve riportare: • • • • • • • • • • •

Indice parete (colonna 239): Si riporta l’indice della parete secondo la pianta schematica. Spessore t e rapporto Lm/Lo(colonne 240 a 255): Si riporta lo spessore de la parete t ed il rapporto tra la lunghezza totale dei maschi Lm ed la lunghezza totale della parete Lo in ogni livello. Direzione (colonna 256): Si riporta X o Y della parete tenendo conto se la parete è parallela oppure perpendicolare alla facciata. Lunghezza Lo (colonne 257 a 259): Si riporta la lunghezza totale della parete. Piani edificio contiguo (colonna 260): Si riporta il numero minimo di piani degli edifici contigui. # piani a pareti uguali (colonna 261): Si riporta il numero di piani per cui la pianta è uguale. Distribuzione regolare di catene (colonna 262): Si riporta se le catene sono regolarmente distribuite nella lunghezza della parete. Numero di pareti interne collegate (colonna 263): Si riporta quanti pareti interne sono collegate alla parete perimetrale. Buon collegamento con pareti laterali (colonna 264): Si riporta se c’è un buon collegamento tra la parete perimetrale e le parete laterale in modo tal che sia possibile ipotizzare un comportamento unitario tra le pareti collegate. Numero di aperture allineate verticalmente con distanza L1 < L2 (colonna 265): Si riporta il numero di colonne di aperture allineate verticalmente, per cui la distanza L1 < L2 secondo il seguente schema. Distanza Lvar (colonne 266 a 268): Si riporta la distanza Lvar secondo il seguente schema:

A2.8

Annex 2 Lvar

Lvar

Lvar

L1

Numero di aperture allineate verticalmente con distanza L1 < L2 = 0

L2


L1

L2


Superficie media coperta (caselle 488 a 507): Si definisce superficie media coperta l’area coperta al lordo delle pareti dai seguenti elementi strutturali: solai, coperture, volte, porticati, scale esterne. Altezza media interpiano (caselle 510 a 521): La misura deve essere comprensiva di metà spessore dei due solai nel caso della altezza media interpiano. Altezza aperture (caselle 524 a 535): Si riporta la altezza media delle aperture nel piano (finestre, porte, ecc.). Area totale degli elementi resistenti (caselle 538 a 565): L’area degli elementi resistenti è misurata tra gli interassi dei muri ortogonali di intersezione. L’area degli elementi inclinati di un angolo α rispetto alla direzione considerata va moltiplicata per Cos2α.

9.

SEZIONE 8: MECCANISMI DI RIBALTAMENTO ASSOCIATI A DANNI ESISTENTI

Meccanismo (caselle 569 a 576): Si reporta il meccanismo di ribaltamento associato al danno esistente, tenendo conto dei dieci meccanismi in Fig. 9. Lunghezza (caselle 577 a 588): Si riporta la lunghezza L del meccanismo associato tenendo conto della Fig. 9. Altezza (caselle 589 a 600): Si riporta la altezza h del meccanismo associato tenendo conto della Fig. 9. Spessore (caselle 601 a 608): Si riporta lo spessore t del meccanismo associato tenendo conto la Fig. 9.

A2.9

Annex 2

t

t

h

h

1

2

L

L

t 2/3h’

h

h

L t

3

4

L

t t

h h

5

L

6

L

t

t

L h L

h

8

7

A2.10

Annex 2

t t L h

h L

9

10 Fig. 9

10.

SEZIONE 9: SEZIONE DI CONTROLLO

Scheda controllata (caselle 609 a 614): Si riporta la data in cui la scheda è stata controllata e verificata la non mancanza de nessuna informazione fondamentale. Si riporta anche il nome e cognome della persona che l’ha controllata. Scheda nella database (caselle 615 a 620): Si riporta la data in cui la scheda è stata inserita nella database ed anche il nome e cognome della persona che l’ha inserita.

A2.11

Annex 2

A2.12

Annex 2

A2.13

Annex 2

A2.14

A Simplifies Mechanics-based Procedure for the Seismic Risk Assessment of Unreinforced Masonry Buildings

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