Structural Analysis of Historical Constructions - Modena, Lourenço & Roca (eds) © 2005 Taylor & Francis Group, London, ISBN 04 1536 379 9

Experimental testing in support of a mechanics-based procedure for the seismic risk evaluation of unreinforced masonry buildings

L.F. Restrepo-V élez European School for Advanced Studies in Reduc/ion of Seismic Risk (ROSE), University of Pavia, Italy

G. Magenes Departmenl of Slruclural Mechanics, University of Pavia and ROSE School, Ilaly

ABSTRACT: A procedure based on simplified mechanics concepts and considering the main sources of uncertainty affecting risk assessment was developed, the objective being the loss estimation at urban or regional scale, going from the computation of the seismic risk for a class of buildings, to a general estimation for the building stock within a geographical region. The procedure shares the advantages and alleviates, at least partially, the disadvantages identified in existing procedures: it is based on displacement/drift demand; it includes the uncertainty in demand and capacity, and allows the computation of the probability of leveIs of damage for classes ofbuildings. A possible extension ofthe procedure to include complex out-of-plane failure mechanisms is discussed, for which the results of a series of 1:5 scale tests on stone masonry are presented and the preliminary findings are discussed.

INTRODUCTION

Unreinforced masonry has been used in the past as the main construction material for alI types of buildings, some ofwhich still exist nowadays. A large percentage of the stock of old buildings has a relevant impor­tance as historical buildings, and their conservation and maintenance are at the centre of many researches and government programs.

Seismic load is amongst the main hazards affect­ing historical masonry buildings, which in conjunction with the inherent high vulnerability of this type of structures, leads to the existence of considerable leveIs of seismic risk. In order to make possible the esti­mation of the leveI of seismic risk at regional scale, several analytical tools have been developed around the world, which, using different assumptions and approximations, render di fferent degrees of accuracy. Besides, some of those tools require a large amount of resources, in terms of money, time and computa­tional effort, in order to be properly implemented and effectively used (Restrepo-V élez 2003).

A new procedure is presented, based on simpli­fied mechanics concepts and considering the main sources of uncertainty affecting the risk assessment. The objective is the loss estimation at regional or urban scale, going from the basic computation of the probability offailure for a specific class ofbuildings, to a more general estimation for the building stock

within a geographical region. The proposed proce­dure shares the advantages and alleviates, at least partially, the disadvantages identified in many exist­ing procedures (Restrepo-V élez & Magenes 2004b) because it is based on displacement/drift demand, it includes the uncertainty in the demand as well as in the capacity, and it allows the computation of the probability of damage/failure for classes ofbuildings. Additionally, the new procedure includes the compu­tation of the probability of failure for both in-plane and out-of-plane response.

For in-plane mechanisms the demand is represented by the displacement response spectrum obtained from regional probabilistic seismic hazard studies, which is defined by the median spectrum and by the corre­sponding scatter. Regarding the limit states, appropri­ate median values corresponding to each given drift limit state are used, depending on the type and qual­ity of masonry. For the out-of-plane mechanisms, the procedure is restricted so far to simple one-way bend­ing mechanisms. Analogous procedures for other wall configuration and failure modes are currently under development, for which a series of static 1:5 scale tests on dry stone masonry have been carried out, with the aims of understanding the kinematics of these mechanisms, and developing appropriate ana­lytical expressions for the evaluation of their seismic response. Preliminary results of the test program are presented and the usefulness of these results in the

1079

development of the required analytical expressions to evaluate the collapse multipliers is discussed.

In the last part of the paper the main findings and conclusions are discussed.

2 THE MeBaSe PROCEDURE

The mechanics-based procedure for the seismic risk assessment of unreinforced masonry buildings (MeBaSe) has been developed at the University of Pavia (Restrepo-Vélez 2003 , Restrepo-Vélez & Magenes 2004a, b), sharing some desirable features already included in existing procedures and propos­ing new elements to overcome identified deficiencies: it is based on displacementldrift demand; it includes the uncertainty in demand and capacity, and allows the computation of the probability of leveis of damage for classes ofbuildings.

MeBaSe stems from a displacement-based approach for vulnerability evaluation of classes ofbuildings pro­posed by Calvi (1997 , 1999), but additional aspects have been developed in order to consider most of the identifiable sources of uncertainty, recognized to come from the demand, the structural capacity and the dynamic response of the structure.

A fundamental feature of MeBaSe is the direct computation of the seismic risk, avoiding the usual two-stages procedure of estimating first the vulnera­bility of the structures and then convoluting it with the seismic hazard in order to get the corresponding seismic risk.

The probability PJ of attaining or exceeding a spec­ified limit state is given by Equation I, where F D

represents the cumulative distribution function (CDF) of the demand being less or equal to the displace­ment demand b.Ls for a given period h s, and h.lr and f r are the probability density functions (PDF) for the displacement response conditional to some specified effective period and for the effective period, respectively.

PI = f f[1 - FD(b. Ls/Tl.s )}fL\ts!TLJTLs dMT (I) 1' 6

Tn the MeBaSe procedure F D is computed assuming a lognormal distribution and using the median values of the displacement response spectrum obtained with a probabilistic seismic hazard analysis, in which it is assumed that the aleatory component of the uncer­tainty related to the seismic action has been already included. As a consequence, the standard deviation used to compute F D represents just the epistemic com­ponent of the uncertainty existing in the definition of the spectrum. For out-of-plane mechanisms, the seis­mic demand is computed with the equation proposed in Eurocode 8 for non-structural elements (CEN 2001).

b.Ls and TLS are random variables for which the corresponding PDF functions are rarely, or almost never available. Different procedures can be applied to estimate in an approximate way the CDF functions, from which the PDF can be numerically evaluated. In MeBaSe these functions are computed by means of the fi rst order reliabi li ty method (FORM), which is a simpler and computationally faster alternative to Monte Carlo methods, and renders an adequate levei ofaccuracy, at least within the scope ofthis application (Restrepo-Vélez & Magenes 2004b).

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 Equation 2:

(2)

By assigning parametric values to a in a suitable range, for example from 0.01 sec to 4.0 sec, and run­ning FORM analysis for each discrete value of a, the result will be the probability of TLs being less or equal than a , i.e. the CDF of TLS. The required G limit state functions of b. and T are computed according to the type of mechanism.

2.1 Limit statefunctionsfor in-plane mechanisms

For the case ofin-plane fa ilure mechanisms, three limit state are considered in MeBaSe , namely LSI- LS2 for which just slight structural and non-structural damages occur, LS3 for which moderate structural damage and extensive non-structural damage occurs, and LS4 for which the collapse ofthe building is considered.

Equations 3 and 4 are used to compute the G limit state functions for the case of in-plane failure mecha­nisms, where the vector of random variables is formed by the total height ofthe building class h" the height ofthe openings at the failing storey hsp , the drift limit at yield 8y , the drift limit at the specified limit state 8Ls, the resistant area of the walls in the direction of minimum strength An" the ratio of areas of walls in both directions Ym, the referential shear strength of the masonry ' km, and the correction facto r </>c. This correction factor allows to express the tri-dimensional response of a building by means of a simplified bi­dimensional model, based solely on the shear strength of the walls (Restrepo-Vélez & Magenes 2004a, b), and it is computed with Equation (5), where Lw is the length of the piers and Lr is the total length of the perimeter walls.

(3)

r 2 1 ( K ),v, G =~-AKT 1+ 2 -a (4)

6" 4 2 Ao m I ' m A (I + )r g J[ 'fie 111 Ym km

1080

Parapet wall Simply·supported Non· Loadbearing parapet wall

Simply·supported Loadbearing \ValI

Figure I. One-way bending mechanisms included in lhe MeBaSe procedure.

ifJc = 5.53 ( LfrT ) + 0.46 (5)

In Equations 3 and 4, K\ and K2 are used to com­pute the fraction of elastic and inelastic components of the displacements, according to the interstorey heights and the distribution of masses (R estrepo-Vélez 2003) and K\ and K2 are computed with Equations 6 and 7, respectively, where Wj is the weight at the storey i, W{ is the total weight ofthe building and h j is the height of the storey i from the ground levei :

K = I I wf.~

(j=m ~h .W . L.. ) )

(6)

j=1

f.w, K - j=m , -l.5 (7)

It is worth pointing out the dependence of t:>.LS on h s, as expressed in Equation 4, which implies that the CDF for t:>.LS is a condit ional di stribution .

2.2 Limit statefimctions for out-of-plane mechanisms

The out-of-plane mechanisms included in MeBaSe are restricted so far to the one-way bending mechanisms shown in Figure I, whose dynamic response has been described by Doherty et ai (2002) and Griffith et ai (2003) by means of a tri-linear model as the one shown in Figure 2, which is defined by the force required to initiate the incipient rocking motion F;, or "Rigid threshold", the ultimate static displacement at the point of instability t:>. 1I, and the displacements t:>.\ and t:>. 2 that are estimated from the material properties and the state of degradation ofthe cracked section at the pivot points.

F

Figure 2. Tri-Iinear simplified model for oUI-of-plane one-way bending mechani sms.

The corresponding limit state functions to compute the CDF of TLs and t:>. LS are given by Equations 8 and 9, where the vector of random variables is formed by P2 wh ich is the ratio between t:>. 2 and t:>. ,,, the height of the wall h, <fi that is a facto r between 0.8 and 1.0 used to express the displacement at a given limit state t:>. LS as a fraction of t:>. {" the thickness of the wall t and the ratio of the overburden load and the weight of the wall II!. Ln the case of out-of-plane walls the only limit states considered so far in the formulation are non-collapsed or collapsed. Griffith et ai (2003) have shown that the secant stiffness at displacement t:>. 2 can be used as a valid para meter, when the main objective is to determine whether a wall will collapse or not.

G ( 1( ' P,ifJÔ)1 Jy, Tu = ifJ(l-p,Xl+'I')gt -a

(8)

Góu

=Tfs ifJ(l-p,Xl+ 'I')gt -a , 1(' p , h

(9)

Here again the dependency of displacement on period, as shown in Equation 9, is underlined.

2.3 Seismic risk estimation aI regional scale

Having stated that the objective of the method is the risk assessment of classes of buildings at urban or regional scale, it is fundamental to define what is a building class within this context and to devise a way to apply the method to a building stock.

The definition ofthe building classes depends very much on the scope ofthe study and on the amount and quality of the available information . A good feature of MeBaSe is the flexibility it allows in the definition of the classes of buildings. For example, if a region is

1081

subdivided in n I zones, and the building stock within the particular zonei is subdivided in n 2 building classes CBj(ml,#s), e.g. according to the masomy type ml, and number of stories #s, by applying Equation I the prob­ability Pij ofreaching or exceeding the specified limit state ofinterest ("failure") for each CBj(mt,#s) in each zone i is computed. Then the probability of failure in the zone i for class CBj(ml,#s) is computed with Equation 10, where P mt,#s is the joint probability that a building within the building stock has a material mt and stories #s, which is inferred from the sample data gathered during a field survey. A specific new survey form was specifically designed for this methodology. Finally, the total probability of failure in the region for the class ofbuildings CBj (mt, #s) is computed with Equation l i .

3 COMPLEX OUT-OF-PLANE FAILURE MECHANISMS

(lO)

(11 )

As explained in Section 2, just simple unidirectional bending mechanisms for out-of-plane motion are included in the current version of MeBaSe. Nonethe­less, the occurrence of several different types of out-of-plane mechanisms has been widely reported.

Amongst the different out-of-plane failure modes proposed in the literature, particular attention is being placed to the definition of the corresponding ana­Iytical expressions to compute hs and 6 LS for the mechanisms shown in Figure 3.

Mechanisms A to G were studied by D' Ayala & Speranza (2003); mechanisms H are relevant to double-Ieafwalls. Equations 8 and 9 are applicable to mechanismsA, E, F, and H. For the remaining mech­anisms it is proposed to use also a tri-linearmodel, like the one shown in Figure 2, for which the corresponding equations are currently under development. The static rigid threshold force Fo and the ultimate displace­ment 6 u are computed from rigid-body limit analysis principies. The static force Fo can be expressed in terms of a multiplier of the effective mass of the wall (Fo =ÀMeg).

For what concerns 6 2, Doherty (2000) and Doherty et ai (2002) proposed ratios P2 obtained empirically from a series of dynamic tests, as function of the state of degradation of the mortar joints. More generally, the ratios P2 should depend on the type of masomy, the general state of degradation, possible preexisting damages and constructive defects, even for the case of dry masonry where by definition no mortar is present.

A

:'~'~~I ~~. !> !Ui) \1 ~ ~~)

1 ----" : c D " E,;,

;~~F ~~1ff:l , Hêee Figure 3. Out-of-plane failure mechanisms (afier D'Ayala & Speranza 2003).

Table I. P2 values for each out-of-plane failure mechanism and for each condition of the masonry.

Degradation Type of Good

Mechanism masonry* condition Moderate Severe

A, E,F, H A,B,E,F 0.50 0.60 0.70 C,D,G,H, 0.28** 0.40** 0.50** I,L,M

B1 , B2, C A,B,E,F 0.55 0.65 0.75 C,D,G,H, 0.35 0.45 0.55 I,L,M

D ,G A,B,E,F 0.40 0.50 0.60 C,D,G,H, 0.20 0.30 0.40 I,L,M

* Types of masonry according to lhe definitions given in GNDT (1993). ** Doherty (2000).

Besides, for each failure mechanism there should be a different ratio, also due to the geometry of the walls, and the shape and number ofboundaries in which fric­tion forces are generated. In Table 1 the proposed P2 values are presented.

1082

o co ú:l

Compute collapse mulliplier for H / ,

HZ or H3, compare with pre-selected mechanism and

select lhe lowest

Absence

Compute collapse Illultiplicr for C, HJ , H2orH3,

compare with pre -sc lccted mechanism and selecl lhe lowesl

Yes

The wa ll is a

·Presence o ~ _____ --,~~e~s~ chains. slee l ~ies or"»~N~o ____ _

conc:rele nng

egular distribution af

restr.tining de,vices on façade ?

beams?

Yes Good . ~

ed;::":I~~l~; ;~~jd0>--N=O---, restraint?

Good conncction wilh )>--'N-"o'"--_____ _

bOlh edge wall s?

"Staggering r~ltj~~ No slh low (slh < 0.8). ,.....,=-----,

?

Windows

Yes Is a comer or an" isolatcd bui lding?

No

CompUle coll apse muhiplier for C.

compart,: wi lh pre­se lccled mcchani sm and selec! lhe lowcsl

510p

Compute coll apsc muhiplicr for H I,

H2 ar H 3. compare wi lh prc-seleclcd mechanism and sc lccl lhe lowesl

I Restrain ing deviees are evenly di stributed in lhe longitudinal direction of the wall. for a given height

2 Correspond lO lhe Icnglh and lhe hcighl of lhe wall being considercd

J The likely mechanisms is a combinat ion of Fand 8 2. lo be conservilli ve lhe lower collapse Illultiplicr is selccted

4 The likely mechanisms is a combination of A and 82. to bc conservalive lhe lower eollapse multip li er is seleeled

Figure 4. Decision flowchart for out-of-plane failure mechan isms.

Additionally to the defini tion of the mechanical model for the out-of-plane fa ilure modes, a tool is needed that helps to identify the most Iikely mech­anism to develop on the peripheral walls of a given building.

A decision flowchart, which is shown in Figure 4, is proposed for the identification of the out-of-plane failure mode that is Iikely to happen, accordi ng to the boundary conditions of the peripheral walls, the stag­gering ratio of masonry, the presence of ring beams, steel ties , openings, internai bearing walls, number of stories and aspect ratio of walls. The flowchart has been developed taking into consideration the analyti­cal results presented by D' Ayala & Speranza (2003), and from the results of a series of static tests, whose results are partially presented in the next section. The decision criteria inferred from the results obtained by D' Ayala & Speranza have been shaded, whi lst the remaining decision criteria have been derived from the results of the scale tests.

The decision flowchart has been devised to be applied to every peripheral wall of the building, for which the required data is also gathered with the survey form des igned along with the MeBaSe procedure.

4 DRY STONE MASONRY WALL TESTS

A series of static 1:5 scale tests on dry stone masonry walls have been performed at the ROSE School, Uni­versity of Pavia, with the main objective of verifying ex isting analytical expression for the computation of the collapse multiplier of the out-of-plane fail­ure mechanisms shown in Figure 3, and to develop analytical expressions for the computation of the collapse multiplier and the ultimate static displace­ment for each mechanism. A testing method already applied by Ceradini (1992) and Giuffré has been fo llowed.

No mortar was used in the models, which means that the shear strength along the joints is given purely by friction. The bricks were cut in marble, the selec­tion criteria being the accuracy on the cutting, the hardness and durability of the material, an appropri­ate friction coefficient. The dimensions of the bricks were h = 28 mm, t = 40 mrn and / = 80 mm. The test­ing device for the static tests was an incl ined plane machine built with steel profi les and with an alu­miniurn platform. The collapse multiplier was cal­culated from the inclination angle of the platform. Figure 5 shows a general view of the inclined plane machine.

In total , 42 conf igurations have been tested, vary­ing the length of the walls, the presence and posi tion of openings, the staggering ratio, the quality of the connection between walls, the existence of overbur­den loads in the out-of-plane and in the in-plane walls,

Figure 5. General view of the inclined plane machine for the static tests.

Figure 6. Test S3, modelling of out-of-p lane collapse mechanism G.

and the number of stories. The execution of some of the tests is illustrated in Figures 6 to 10.

Friction tests on bricks yielded coefficients in the range ofO.67 to 0.77, for vertical stresses in the range of 0.77 kPa to 18.4 kPa. Comparison of experimen­taI collapse multipliers with predictions of analytical formulae was made using a friction coefficient of 0.7. Table 2 shows some ofthe experimental collapse multipliers compared with those obtained analytically from the application of the equations proposed by D' Ayala & Speranza (2003) and with modified equa­tions developed during this research, as discussed in what follows.

In Figures 1i to 19 the general set up of each of the tests listed in Table 2 are illustrated, and the direction ofrotation ofthe inclined plane is also shown.

1084

Figure 7. Test S9, modelling of out-of-plane collapse mechanism B2.

Figure 8. Test S 13 , modelling of out-of-plane collapse

Table 2. Collapse multipliers for some of the out-of-plane mechanisms in Figure 3.

Collapse multiplier

Mechanism Modified (test) Experimental Analytical equation

G (SI) 0.254 0.222 0.235 G (S5) 0.349 0.187 0.333 G (S6) 0.208 0.279 0.183 A- B2 (S7) 0.291 0.438-0.673 0.294 D (Sll) 0.097 0.271 0.098 A- B2 (S20) 0.285 0.334-0.486 0.263 F (S26) 0.259 0.289 0.252 G (S34) 0.217 0.1 37 0.229 A (S35) 0.168 0.689 0.173

mechanism D. H=21 bricks

Figure 9. Test S32, modelling of out-of-plane collapse mechanism F.

Figure lO. Test S35, modelling of out-of-plane collapse mechanisms A- B2.

Figure li. Outline ofTest SI , mechanism G.

H = 21 bricJa

Figure 12. Outline oftest S5 , mechanism G.

1085

H =21 brlcks

Figure 13 . Outline oftest S6, mechanism G.

Figure 14. Outline oftest S7, mechan ism A- B2.

L = 12 bricks L =10bricks

Figure 15. Outline oftest SI I, mechanism D.

Figure 16. Outline of test S20, mechanism A- B2.

Positionof baringplld: 5mmfrom infernal cdllc

L=l brick

VertiCJÚ r Cllction

toborizonW motion

Figure 17. Outline of test S26, mechanism F.

PandORa!

move borízontsUy

H=21 brícks

Vertical reaetioa

Figure 18. Outline oftest S34, mechanism G.

1086

VertícsJ reaction

Figure 19. Outline oftest S35 , mechanism A.

The partial results obtained so far indicate that the theoretical results predict within a 10% error the value ofthe collapse multiplier for some mechanisms, whilst for other mechanisms the collapse multiplier measured during the tests is greatly overestimated (more than 200% of error), showing that a modifi­cation in the application of the analytical formulae is necessary.

For mechanisms type G (tests SI , S5, S6, S34) the theoretical results render maximum errors of 86.6% and 58.4% for tests S5 and S34, respectively; for the other two tests the error is lower than 25%, indicating anyhow a low levei of accuracy. A modified equa­tion was developed by following a different approach, using a partia I eificiency factor for the coefficient of friction, as a function of the aspect ratio of the wall and the magnitude of the vertical loads acting on the front and side walls of the mechanism.

During the tests it was observed that the friction along the "cylindrical hinge lines" does not fully develop, due to a component of the relative rotation ofthe bricks in contact, which causes uplifting ofthe bricks, changing a surface-contact to a single-point­contact. It was also observed that the longer the wall the more pronounced were the uplifting effects and hence the lower the friction developed. Moreover, the position and magnitude of the applied vertical loads transmitted by the floor joists influence also the effec­tiveness of friction , increasing the friction forces and moments when the load is placed on the side wall but decreasing the friction when the load is applied on the front wall. Equation 12 was developed to compute an empirical partial eificiency factor 0.peJ for mechanism G, as a function of the equivalent length of the front wall Leq and the length of the brick forming the wall I. Leq is obtained with Equation 13, where L is the length of the front wall , PJ is the applied load on the front wall, Ps is the applied load on the side walls, y is the specific weight of masonry, I and h are the length

0.9

0,8

0,7

0,6

'" d 0,5

0,4

0,3

0,2

• •

Qp,! = 1,0-0,062L.q/l

R' = 0,8074

• • •

0,1

O+----------------r--------------~ •

5,0 10,0

L,qll

15,0

Figure 20. Partial efficiency facto r rlpef for mechanism G.

and height of the brick, respectively, t is the thickness of the wall and the s is the overlapping length of the bricks.

L,q O re! = I.O-O.062-1- ~O

PI P, L =L+-s--s

' q yflh yfth

(12)

(13)

Figure 20 shows the empirical relationship esti­mated for 0.peJ from the results obtained during the tests.

With regards to tests S7 and S20, the failure mode is a combination of mechanisms A and B2, being the results for the experimental collapse multiplier much lower than the predictions computed with the equa­tions for either mechanisms. This could be explained considering how the equations rely on friction resis­tance as an additional source of strength. As can be seen in Figure 7, the collapse involves part ofthe edge walls, and the mechanism has a rotational component that causes uplifting of the bricks on the edge walls, reducing the effectiveness of the restraint offered by friction . If the corresponding equations are modi­fied by neglecting friction resistance along the edge walls , which means that the wall is considered as free standing (Mechanism A with no lateral restraints), the predictions for case S7 and S20 become closer to the experimental results within a 8.4% ofmaximum error.

In the case oftest S35 , a verticalload was added to the front wall simulating an unrestrained floor system (Figure 10) and causing a failure mode with mecha­nism type A. The added vertical load resulted in an enhancement ofthe friction effect. lndeecl, it was seen that the uplifting effect ofthe bricks on the lateral walls was not as pronounced as in tests S7 and S20, the bricks tended to slide in proximity of the vertical edge, and

1087

0,7

0.6

0,5

';;. 0,4

d 0,3

0,2

0.1

0,0 0,0

•

• •

rlp,/= 1,0-0. 185U" R2 = 0.77

1,0 2,0 3,0

Uh

4,0 5,0 6,0

Figure 21 . Partial efficiency factor Q pej for mechanism A.

the height of wall taking part to the mechanism was also larger. Iffriction resistance is fully considered, the computed collapse multipl ier strongly overestimates the experimental result. If a partial efficiency jactor for friction is introduced which takes into account the aspect ratio and the flexib ility of the wall, the analyt­ical result can better match the experiment within a 3% erro r.

A similar concept to the one used for mechanism G was implemented here in the development of the corresponding partial efficiency jactor for friction, being in this case just a function of the length of the wall L and the height ofthe failing portion h, according to Equation 14 and Figure 21.

L n p'/ = 1.0 - 0.185 h <: O (14)

The experimental collapse multiplier for mecha­nism D (test S 11) is much lower than the analytical prediction, possibly due to the fact that an effect of friction res istance along the crack line is included in the equation by D ' Ayala & Speranza. In fact, when friction is fully neglected, the analytical collapse multi­plier is 0.096 compared to the measured value ofO.097, suggesting that for this type of mechanisms the con­tribution offriction along the crack line is negligible.

Finally, for what concerns mechanism F (test S26), where the roof acts as a restraint, the original equation only slightly over-estimates the collapse multiplier. This is due to the hypothesis on the position of the roof load, which is assumed to be centred across the thickness of the wall, whilst considering the correct eccentric position ofthe model (5 mm from the inter­nai face) it is possible to match the experiment more

closely. Clearly, this is not an issue related to the the­oretical model but to the possibility of estimating the exact eccentricity of the roof vertical load, which can be determined with some accuracy in an experiment, but could be quite uncertain for a real building.

5 CONCLUSIONS

The general formulation and development of a pro­cedure for the seismic risk assessment of classes of unreinforced masonry buildings, based on concepts of structural mechanics, has been presented.

Among the most relevant features ofthe method are the use of displacement response spectrum to represent the seismic demand, the inclusion of the uncertain­ties coming from seismic demand, structural capacity and dynamic response, and the use ofmechanical cri­teria for the definition of the structural capacity. A crucial feature that is being currently developed is the inclusion ofthe most common out-of-plane failure mechanisms, for which just simple one way bending mechanisms are included so far in the formulation. For what concerns complex out-of-plane fa ilure mech­anisms, in this paper the proposal of extending the tri-linear model given by Doherty et ai (2003) and Griffith et ai (2003) for simple mechanisms, to be used also for complex mechanisms, is considered. AIso regarding complex out-of-plane motion, a deci­sion flowchart for the identification ofthe most likely fa ilure mechanism has been presented.

An essential element for the definition of the tri­linear model is the static threshold. To provide a reference for the evaluation ofthe static threshold as a function ofthe collapse multiplier of each out-of-plane failure mechanism considered, a series of static 1:5 scale tests were carried out, aiming to the validation of analytical expressions recently proposed by D ' Ayala & Speranza (2003). From the analysis ofpartial results it appears that, while for some mechanisms the analytical predictions give a good match with the experiments, for other mechanisms they strongly overestimate the static threshold. This may be possibly due to the way in which the friction res istance fo rces are considered, and a modification of the formulae was therefore introduced in order to match the results obtained dur­ing the tests. It must be pointed out, however, that, although maximum care was put in the application of the cri teria by D' Ayala and Speranza, the pre­liminary results discussed herein may be affected by the independent interpretation and judgment of the authors , and additional research is needed and is being carried out to better understand the phenomena, in order to confirm or modify the decision flowchart and the analytical expressions to compute the static thresholds.

1088

The proposed risk assessment procedure is cur­rently being applied to selected areas of the city of Benevento, Italy, to test the capabilities of the method in the estimation of the seismic risk at urban scale.

ACKNOWLEDGEMENTS

The present research was funded within the rNGV­GNDT 2000- 2003 Framework Program as a part of the Project "Traiano".

REFERENCES

Calvi, G.M. 1997. Un Metodo agli Spostamenti per la Valutazione della Vulnerabilità di Classi di Edifici. I.;lngegneria Sismica in Italia, Proc VIll Convegno Naziona/e A NIDlS, Taormina.

Calvi , G.M. 1999. A displacement-based approach for vul­nerability evaluation of classes of buildings. Journa/ o[ Ear/hquake Engineering 3(3): 411-438.

CEN. 200 I. Eurocode 8: Design ofStructures for Earthquake Resistance. Part I: General Rules, Seismic Actions and Rules for Buildings. Draft 4, Final Project Team Draft (Stage 34).

Ceradini , V 1992. Modellazione e Sperimentazione per lo Studio della Struttura Muraria Storica. Tesi di Dollora/o. Dipartimento di lngegneria Strutturale e Geotecnica, Università degli Studi di Roma " La Sapienza" (in ltalian).

D'Ayala, D. & Speranza, E. 2003. Definition of Collapse Mechanisms and Seismic Vulnerability ofMasonry Struc­tures. Ear/hq/lake Spec/ra 19(3): 479- 509.

Doherty, K.T. 2000. An Investigation of the Weak Links in the Seismic Load Path of Unreinforced Masonry Build­ings. PhD Thesis. School of Civil and Environmental Engineering, Adelaide University.

Doherty, K.T. , Griffith, M.C., Lam, N. & Wilson, J. 2002. Displacement-based Seismic Analysis for Out-of-plane Bending of Unreinforced Masonry Walls. Ear/hquake Engineering and S/ruc/ura/ Dynamics 31 : 833- 850.

GNDT. 1993. Rischio sismico di edifici pubblici-Parte I Aspetti metodologici. Consiglio Nazionale delle Richerche - Gruppo Nazionale per la Difesa dai Terre­moti , Roma.

Griffith, M.C. , Magenes, G., Melis, G. & Picchi , L. 2003 . Evaluation of out-of-plane stability ofunreinforced masonry walls subjected to se ismic excitation. Journa/ on Ear/hquake Engineering, Specia/lssue 1(7): 141- 169.

Restrepo-Vélez, L.F. 2003 . A simplified mechanics-based procedure for the seismic risk assessment ofunreinforced masonry buildings. Repor/, ROSE School - University of Pavia.

Restrepo-Vélez, L.F. & Magenes, G. 2004. A mechanics­based procedure for the seismic risk assessment of masonry buildings at urban scale. Proc Xl Convegno Naziona/e AN/DIS, Genova.

Restrepo-Vélez, L.F. & Magenes, G. 2004. Simplified Pro­cedure for the Seismic Risk Assessment of Unreinforced Masonry Buildings. Proc 13/h World Con[erence on Ear/hquake Engineering, Vancouver.

Soresi , C. 2003. Statistical analysis of the strength and deformability properties of some masonry typologies. Laurea Thesis in Civil Engineering, University of Pavia (i n ltalian).

1089