Seismic protection of constructed facilities: optimal use of resources

EISEVIER

Structural Safety 16 (1994) 91-109

Seismic protection of constructed facilities: optimal use of resources * G. Augusti

a,*, A. Borri b, M. Ciampoli

a

a Dipartimento

di Ingegneria Strutturale e Geotecnica, Uniuersitcidi Roma “La Sapienza”, via Eudossiana IS, 00184 Roma, Italy ’ Dipartimento di Ingegneria Civile, Uniuersitcidi Firenze, I/is di Santa Marta 3, 50139 Firenze, Italy

Abstract

The problem is tackled of the allocation of the resources assigned for upgrading an ensemble of buildings or other constructed facilities, whose seismic reliability is not considered sufficient. The resources are allocated in order to minimize the predicted value of either the economic losses, the number of endangered persons or the risk of severance of a network. The optimization procedure, although formulated in general terms, is illustrated with reference to simple examples, in which realistic values are attributed to the relevant parameters. These examples refer to masonry buildings, whose damages can be summed with each other to obtain the total damage of the ensemble, and to a highway network, whose critical elements are reinforced concrete bridges. The presence of several discrete quantities and discontinuous relationships requires the use of specific algorithms, and in particular of dynamic programming. Keywords: Seismic reliability;

1. Approaches

Upgrading;

to optimization

Optimization;

in structural

Resource allocation; Dynamic programming

design

Structural optimization (or optimal design) is by no means a new idea: indeed, in the last decades, it has been one of the most productive fields of research within structural engineering (for a recent critical review, see, e.g., [l]). In the classical approach, the material constitutive laws, the structural model and the relevant boundary conditions are fixed, and the design that optimizes an objective function is sought under some constraints: typical examples are the structures of minimum weight able to support given loads, the structures of given configuration and total volume with largest first eigenfrequency, etc. For the solution of these problems, in which all quantities are treated as * Discussion is open until June 1995 (please submit your discussion paper to the Editor, Ross B. Corotis). * Corresponding author. 0167-4730/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved 0167-4730(94)00029-P

SSDI

92

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Safety 16 (1994) 91-109

deterministic, sophisticated analytical and numerical techniques of constrained optimization have been elaborated and implemented. However, this approach tends to overlook a number of essential aspects of the physical-engineering problem. In particular, the uncertainty and randomness present in most of the relevant parameters are neglected and, as in all deterministic approaches, it is assumed that the optimal design provides absolute safety of the structures. On the contrary, the design of a structural system should be viewed as a decisional process involving a succession of choices under uncertainty [2,3]. In this context, the concept of expected utility of a structural design was developed, which takes into account the initial cost of the construction, the expected benefits in case of success, the running and maintenance costs (including the costs of inspections), and the expected losses in case of failure. Thus, identifying the optimal design with the design that maximizes the expected utility, a new approach to optimization was formulated, that takes into consideration the random uncertainties of the structural problem and introduces a non-zero probability of failure. Several refinements of the maximum utility design have also been suggested. For instance, it is possible to consider several types of failure (or limit conditions), and to take account of their different consequences (costs) in the objective function (e.g., [4]). Note however that in a complete design process several types of benefits and losses must be considered, which can be of very different and incommensurable natures: some are monetary, like the money earned from renting a building, or the direct and indirect economic losses of a construction failure; others, often called intangibles (like loss of life and limb, the artistic and historical value of an existing building, the social benefits deriving from public amenities, etc.) cannot be measured in monetary terms, and therefore cannot be combined with the previous ones, not even by attributing weights to each of them. Therefore, for the general optimal design problem a single objective function cannot be defined. Rather, it is necessary to set up two or more different optimization problems and let open the choice of the trade-offs between economic and intangible aspects: the concept of Pareto optima appear potentially useful in this respect [5,6]. Recently, it has been noted that the constraints of an optimal design problem can include the amount of the available resources. This is in particular relevant for the design of preventive interventions on existing buildings and other constructed facilities, aimed at reducing their risk (i.e. probability of damage) with respect to possible future situations [7,8]. In most of such cases, in fact, the amount of resources is limited, and a rational planning of interventions does require some form of optimization in order to maximize their effectiveness. This paper summarizes a series of research projects (some presented for the first time to an international readership) on the methodologies aimed at allocating in an optimal way the resources available for interventions on existing structures, with specific reference to seismic risk: the presented procedures, however, could be easily generalized to different hazard scenarios or to the optimal design of new structures. 2. Interventions on existing structures for seismic risk reduction Let us recall first that, by definition, the risk, or probability of damage, of an existing facility is the convolution of its vulnerability with the hazard. In turn, the hazard (which in the specific

G. Augusti et al. /Structural

Safety 16 (1994) 91-109

93

case corresponds to the seismicity of the site) is the probability of occurrence of each potentially damaging event (earthquakes of different intensities); the vulnerability is the probability of each degree of damage which would be caused by each event. The seismic vulnerability of a structure is fully described by a set of fragility curves, each giving the probability of reaching a certain degree of damage versus the intensity of the earthquake. However, due to the lack of sufficient data and the difficulties of using directly the fragility curves, seismic vulnerability is often measured in an approximate way by a number (the vulnerability index [9]) or-even more simply-by attributing the structure to a vulnerability czuss [lO,ll]. In the following analyses, either the probabilistic model of the earthquake is assumed known, or, as often done in engineering practice, the calculations are performed with reference to earthquakes of specific intensities (design earthquakes). In any case, it is assumed that knowledge of the vulnerability allows prediction of the average damage caused by an earthquake of given intensity (which will be defined forecast damage); the expected damage can then be calculated by convoluting the forecast damages with the distribution of probabilities of occurrence of each earthquake. In Sections 3 and 4 of this paper, the optimization of the upgrading interventions on existing structures for seismic risk reduction will be referred to two alternative objective functions, namely direct economic costs and number of endangered persons, as typical instances of monetary and intangible aspects respectively: in other words, the procedure will be aimed at maximizing-for a given amount of employed resources-the predicted (forecast or expected) reduction of either the costs or the number of endangered persons. It is therefore necessary to introduce relations that give either quantity in function of the structural damage. The optimization processes are then pursued separately for each objective function: the results can only be compared. It is also necessary to define the possible interventions and quantify how much they cost and how much they reduce the vulnerability. To develop optimization algorithms applicable to ensembles of typical structures, only a few basic typical interventions must be defined. These needs have suggested first the development of the procedure with reference to cases in which previous knowledge allows the introduction of realistic, albeit approximate, relationships and parameters. In particular, Sections 3 and 4 refer to masonry buildings, for which a wealth of field investigations exists in Italy [9,11,12]. Whenever the existing knowledge is not satisfactory, e.g., in modelling the number of persons present in a building at the time of an earthquake and the number of endangered persons, simple assumptions have been introduced. In fact, the studies presented here were aimed more at formulating methods to tackle the problem, than at obtaining results of immediate practical use: in any case, the structure of the proposed procedures is such that more sophisticated relationships and more reliable data can be introduced, when available. Many non-linear and/or discontinuous relationships and definitions are included in the modelling of damage. Also the earthquake intensity is usually measured according to a discrete scale (Mercalli, MSK or analogous); hence its probability distribution appears as a mass function and not as a continuous density. Thus, it is evidently impossible to use differential maximization procedures, and specific algorithms must be proposed. If the vulnerability of the relevant buildings is defined by its inclusion in a vulnerability class (as illustrated in Section 3)

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Safety 16 (1994) 91-109

the resources are allocated to perform a specific intervention on as many as possible buildings of a class, and the optimal allocation can be found by means of a simple heuristic procedure [13]. If each building is labelled with its vulnerability index (as described in Section 4), an automatic computational algorithm must be used, which has been provided by dynamic programming, that is particularly suited for discrete optimization problems [3,7,8]. The extension of the procedure to cases in which the damages to the individual structures cannot be simply added to each other, but the vulnerability of their ensemble must be evaluated taking into account the system they form (as, e.g., it happens when the structure is an essential part of a lifeline network [14,15]) is discussed in Section 5. This section is developed with reference to a highway network, assuming reinforced concrete bridges as vulnerable elements; vulnerability, damage and interventions must be defined in a specific way. It is important to underline that in all developed examples, it is assumed that a preliminary investigation has shown an insufficient reliability, which led to the decision of assigning an appropriate amount of resources to improve the situation; then the available resources are actually employed, and the optimization regards their distribution among the structures and the possible types of interventions. No comparison is made with the possible alternative of saving the resources by not performing any intervention. Indeed, such a comparison appears meaningless: in fact, it would be possible when only economic aspects are considered, but even in that case indirect economic damages (like reduction of productivity of an affected industry or disruption of the economy of a whole area) cannot be realistically taken into account in the calculations. In any case, upgrading reduces intangible damages, and in particular damages to life and limb, but this gain cannot be compared with the intervention cost. 3. Allocation to vulnerability

classes

Assume (i) that the damage caused to a building by an earthquake can be described according to a discrete qualitative scale; and (ii) that all relevant buildings can be subdivided into a limited number of vulnerability clusses, each class X being associated with a damage probability matrix (DPM). By definition, each element Pi,(X) of the DPM corresponding to the vulnerability class X, is the probability that a building of that class undergoes a level j damage, if subjected to an earthquake of intensity i. For the sake of clarity, the procedure is illustrated with direct reference to an example pertaining to masonry buildings [13]: an eight-level scale of damage is adopted (ranging from no apparent damage to complete collapse), while the buildings are divided into four classes, denoted (in decreasing order of vulnerability) as A, B, C, D; the first three classes are analogous to the classes defined in the MSK Scale [lo], the fourth class (D) is an ideal class of earthquake-resistant structures, including buildings which belonged to A, B, or C and have been fully upgraded. Hence, P,(D) = 0 by assumption, while, for classes A, B, C, the DPM’s shown in Table 1 [ll] will be used. All significant modifications of the vulnerability of a building can be indicated by its initial and final class: e.g., AB, AC, . . . BD, . . . shall correspond to upgrading interventions, while are examples of degradation of the structure. In this treatment, however, only CB, BA,... positive (upgrading) variations are considered.

95

G. Augusti et al. /Structural Safety 16 (1994) 91-109

Table 1 Damage probability matrices PJX)

of masonry buildings (from [ll])

CLASS X

A

MSK intensity i

6

7

8

6

7

8

6

7

8

0.15 0.19 0.25 0.19 0.12 0.07 0.03 0.00

0.07 0.12 0.16 0.20 0.21 0.17 0.05 0.02

0.01 0.03 0.05 0.06 0.07 0.12 0.32 0.34

0.33 0.25 0.25 0.10 0.05 0.02 0.00 0.00

0.20 0.26 0.26 0.13 0.08 0.05 0.02 0.00

0.04 0.11 0.20 0.16 0.14 0.13 0.12 0.10

0.64 0.24 0.08 0.03 0.01 0.00 0.00 0.00

0.52 0.24 0.15 0.05 0.03 0.01 0.00 0.00

0.06 0.24 0.20 0.17 0.11 0.10 0.09 0.03

Damage level j

1 2 3 4 5 6 7 8

C

B

Realistic costs have been estimated (as percentages of the construction cost), both for restoring a building of each class to its original condition after a level j damage, and for each type of upgrading intervention. For simplicity’s sake, all these percentages (and the construction cost per unit building volume) have been assumed to be constant irrespective of the actually involved volume. The restoration costs after an earthquake of any relevant intensity can be forecast-for each class of buildings-by multiplying the probabilities of Table 1 by the unit restoration costs estimated for each damage level, and summing up the columns: the results (again in percent of the construction cost) are shown in Table 2. A preventive upgrading intervention changes the class of the building, and therefore the forecast cost to be read in Table 2: Columns (2)-(4) of Table 3 show the unit gains 6ri due to each type of intervention, forecast for each given

Table 2 Forecast restoration

costs of masonry buildings (in percent of the construction

MSK intensity i CLASS X

cost)

6

7

8

C

56.4 41.0 30.3

D

0.0

73.3 48.8 34.7 3.3

98.3 75.8 64.0 8.3

A B

Table 3 Forecast (Sri) and expected (Sr,) unit economic gains; cost C, and efficiency G, of interventions

(2) 616

(1) Intervention

AB AC AD BC BD CD

(3)

(4)

fjr8

(5)

(6)

(7)

Sr7

Sr,

Cl

GC

26.1

24.5 38.6

22.5 34.3

14.8 24.2

23.3 33.3

0.64 0.73

56.4 10.7 41.0 30.3

70.0 14.1 45.0 31.4

90.0 11.8 67.5 55.7

51.2 9.3 36.2 27.0

56.6 28.3 43.3 26.6

0.90 0.33 0.84

15.4

1.01

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G. Augusti et al. /Structural Safety 16 (1994) 91-109

intensity i, i.e. the differences between the forecast restoration costs without and with the intervention indicated in Column (1). Multiplying these forecast gains by the probabilities of occurrence ri of the relevant earthquake during the design life of the building, and summing up, the expected unit gains Sr, can be calculated: the values shown in Column (5) of Table 3 have been calculated introducing the probabilities: rg = 0.5;

7rT= 0.2;

7rs = 0.1

which, assuming a loo-year lifetime, correspond approximately to the seismicity of many areas in Central Italy. Finally, Column (6) of the same Table 3 shows the estimated (deterministic) costs C, (once more, as percentages of the construction cost) of each intervention, and Column (7) the ratio G, between the values in Columns (5) and (6), i.e. the expected efficiency of each type of intervention. It can be noted that, with the used numerical values (realistic, even if derived from rough estimates) most values of G, are smaller than one, i.e. no economic advantage should be expected from preventive interventions. However, as already discussed, a number of considerations invalidate such a conclusion: it is therefore assumed that preventive interventions are actually performed and their optimal choice is sought. No formal optimization procedure is necessary for this choice. In fact, the larger or smaller efficiency of an intervention depends on the relative values of the ratio G,: such a comparison is easily achieved by drawing (as it has been done in Fig. la) straight lines with slopes equal to the values of G,. Remember that, once the buildings of the examined ensemble have been assigned to a class, the procedure does not distinguish between individual buildings but can only refer to fractions of the volume of each class. Then, the choice of the interventions to be performed in this case does not present any difficulty: in fact, Fig. l(a) shows immediately that the most convenient interventions are, in the order, CD, AD and BD (while in Fig. 2 a more complicated case will be found). Therefore, if the amount of available resources is comparatively small, they are used to bring into class D the largest possible volume of buildings belonging to class C; if more money is available than necessary to upgrade all buildings of class C, the extra resources can be employed for intervention AD; then, if also class A can be fully upgraded, further resources can be employed for intervention BD. It is thus possible to calculate the total gain SR, = C,6r, - V,, where 6r, and V, are the unit gain and the volume of each intervention, and the total expenditure H = C&C, * V,. Examples of plots of the total gain 6R, and of the volume V, of each intervention versus the amount of money H available for preventive upgrading are shown in Figs. l(b) (solid line) and l(c); these plots have been calculated introducing the unit construction cost: C, = 300 000 Lire/m3, and the following volumes of buildings of each vulnerability class: V’ = 441854 m3;

V, = 197 169 m3;

V, = 223543 m3.

(These values have been estimated for the historic centre of Priverno, a town in the Province of Latina, approximately 100 km south of Rome [13].)

G. Augusti et al. /Structural Safety 16 (1994) 91-109

0.8 -0.6 --

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

(4 120, 100..

6R

80.. 60-40..

H (IO’ Lire) 0

20

40

60

80

100

120

140

0

20

40

60

80

100

120

140

0.8

0.6

0.4

0.2

0.0

Fig. 1. Interventions distributed among building vulnerability classes, optimized with respect to direct economic costs: (a) Efficiency of interventions; (b) Expected gain (solid line) and comparison with gain expected from the solution optimized with respect to saved lives (dotted); (c) Interested volumes.

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G. Augusti et al. /Structural

Table 4 Assumed ratios nj between endangered

Safety 16 (1994) 91-109

and present people

Damage level j

1

2

3

4

5

6

7

8

‘i

0.00

0.00

0.00

0.04

0.12

0.24

0.40

0.80

In this way, the intervention diagram, optimized with respect to the direct economic losses, has been constructed in function of the available resources. The interventions can be optimized with respect to other objectives, for instance with respect to the decrease of the number of persons endangered by an earthquake. In want of reliable models for the number of persons present and endangered by an earthquake, the calculations for this optimization have been developed assuming that (i) 0.017 persons/m3 inhabit the buildings (this value corresponds to the average density given by Italian statistics), (ii) 60% of the inhabitants are present in the buildings at the time of the earthquake, and (iii) the ratios between the number of endangered and present persons are given, for each level of damage, by Table 4. Table 5 has been calculated in perfect analogy to Table 3. Namely, Columns (2)-(4) show, for each intervention, the corresponding number 6ni of saved people (i.e. the reduction of endangered people) per unit volume, forecast for each earthquake intensity, while Column (5) shows the expected unit number an, of saved people, assuming the already reported probabilities of occurrence. Finally, Column (7) shows the efficiency of each intervention in terms of saved people, i.e. the ratio G, between the expected unit gain of Column (5) and the intervention cost of Column (6): note that in the present case G, is a ratio between two incommensurable quantities, which can be used only for comparative purposes. The last two rows of Table 5 show the differences in gains and costs between different interventions on class A, and the corresponding ratios G,, which will be necessary to construct the optimal intervention diagram in the present case. The most convenient intervention is AB (Fig. 2(a)). However, if more resources are available than necessary to upgrade to class B the whole class A, it becomes next convenient not to intervene on more volumes, but to substitute intervention AC to A?: the intervention diagram

Table 5 Forecast (hi)

and expected (Sn,)

(1) Intervention

AB AC AD BC BD CD

Interv. Substn. AB + AC AC+AD

number of saved people; cost C, and efficiency G, of interventions (2) an, 0.045 0.060 0.063 0.015 0.018 0.003

(3) Sn, 0.100 0.133 0.143 0.033 0.043 0.010

(4)

(5)

(6)

Sn8

SnP

Cl

(7) G,

0.321 0.419 0.549 0.098 0.228 0.130

0.075 0.098 0.115 0.024 0.040 0.016

23.3 33.3 56.6 28.3 43.3 26.6

0.00320 0.00296 0.00203 0.00084 0.00093 0.00062

Ah, 0.023 0.017

AC, 10.0 23.3

G” 0.00230 0.00073

G. Augusti et al./Structural Safety 16 (1994) 91-l 09

99

Cl . Vl I

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

(4

0

20

40

60

80

100

120

140

1.0 CD

V, (10” m3) 0.8 --

CD AD

AC-AD

0.6 --

BD H (IO' Lire) 100

120

I 140

(4 Fig. 2. Interventions distributed among building vulnerability classes, optimized with respect to the decrease in the number of persons endangered by an earthquake (saued people): (a) Efficiency of interventions; (b) Expected gain (solid line) and comparison with gain expected from the solution optimized with respect to economic costs (dotted); (c) Interested volumes.

100

G. Augusti et al. /Structural

Safety 16 (1994) 91-109

is constructed as indicated in Figs. 2(b) and 2(c), taking into account that the efficiency of the substitution AB 4AC is G, = 0.0023 (Table 5, Col. 7). If the whole class A can be upgraded to class C, the next convenient intervention is BD (G, = 0.00093), then the substitution of AD to AC (G, = 0.00073), and finally CD (G, = 0.00062). The optimal intervention diagram is thus completed (Figs. 2(b) and 2(c)). Both objectives of the performed optimizations should be taken into account in the final choice of the solution, although an overall optimum cannot be defined. To this purpose, Figs. l(b) and 2(b) show also, in dotted lines, respectively (Fig. l(b)) the total economic gain of the solution optimized in terms of saved people, and (Fig. 2(b)) the people saved by the optimal economic solution. Although no general conclusions can be drawn, it can be noted that in this case the solution optimized with reference to saved people is close to optimal with respect to economic costs (Fig. l(b)), while the reverse is not true (Fig. 2(b)).

4. Allocation to individual buildings In this section, assume that the seismic vulnerability of each building is known and measured by a number I/ (the vulnerabilityindex). As illustrated in detail in [9] and elsewhere, the vulnerability index can be evaluated by means of a form listing a number of items that characterize the seismic response of the structure; the survey team classifies each item on a simple scale; the index is then obtained by summing up the values associated to the condition of each item. When, as in the examples illustrated in this section, the vulnerability of masonry buildings is concerned, the most important items of the form are the connection between the walls, the stiffness of the horizontal diaphragms and the lateral strength of the structural system. In the development of the examples, it has been assumed that the vulnerability index V, the earthquake intensity and the average degree of damage of a building are related by the deterministic curves shown in Fig. 3, which have been obtained by a statistical analysis of the damages caused by several recent Italian earthquakes [12]. The -degree of damage D is set 1.00 -

V.“”

0 Fig. 3. Expected

damage

;0 of masonry

lb0 buildings

150 vs. vulnerability

200

250

300

index, for several MSK intensities

(from [12]).

G. Augusti et al. /Structural

101

Safety 16 (1994) 91-109 1.07

3.0-

n

cd& 0% 2.0. 0.6..

0.2..

D 0.07 0.0

0.2

0.8

0.6

0.4

1.0

0.0

0.2

0.4

0.6

0.8

1.0

09

(a)

Fig. 4. Assumed relationships (masonry buildings): (a) ratio between cost of damages C, and construction cost C, vs. damage degree D; (b) ratio n between number of endangered and present people (0.6 X inhabitants) vs. damage degree D.

equal to unity in case of total damage (collapse). It is thus possible to calculate the forecast damage of each building for any given earthquake intensity: a more complete evaluation should take into account the uncertainties in the vulnerability-intensity-damage relationship. As in Section 3, direct economic costs and number of endangered persons have been taken as objective functions. The assumed relationships between the degree of damage D and respectively the monetary losses and the number of endangered persons n are shown in Fig. 4. In the numerical calculations, the cost of construction C, per unit volume has been assumed 200000 Lire/m3 for all relevant buildings, and the number of people present at the moment of the earthquake 0.6 x 0.017 persons/m3, as in Section 3. Three possible intervention types have been defined, namely: (i) in L (light intervention) the horizontal connections between orthogonal walls are secured; (ii) M (medium intervention) includes also the strengthening of the horizontal diaphragms; (iii) finally H (heavy intervention) includes also an increase in the overall strength against horizontal actions. By definition, each type of intervention brings the corresponding items of the vulnerability form to the highest level: the vulnerability index of the building decreases correspondingly. The unit costs assumed in the calculations are reported in Table 6. Define return gk(Cljrn of an intervention of cost C, performed on the mth building, the decrease in the expected damages when an earthquake of intensity i occurs (k = c will indicate economic returns, k = u return in terms of saved people). In other words, the return is equal to Table 6 Assumed construction Construction

CC 200 000

and intervention

costs per unit volume of masonry buildings (Lire/m3)

Intervention

CL

CM

20000

40 000

cH 80000

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G. Augusti et al./Structural

Safety 16 (1994) 91-109

the difference between the damages that would occur without any intervention and after having performed the intervention of cost C,. For discrete types of interventions (like the quoted three types L, M, H), the return functions are multiple step functions. With reference to the forecast return under an earthquake of given intensity i, the optimization problem can be formulated as follows: ??

maximize the total return (for either k = c or k = v>:

??

subject to:

where the index m = 1, 2,. . . , N indicates the building, and the index i = 1, 2, 3,4 the relevant interval of seismic intensity. The maximum can instead be sought of the expected total return: i

m

where 7ri, is the probability of occurrence of an earthquake

of intensity i at the site of building

m.

As already stated, the non-linearity and discontinuity of the relevant relationships do not allow the use of differential maximization procedures. Noting that the objective function Fkli or Fk is the sum of as many quantities as the buildings, each in turn a function of the resources assigned to the mth building only, the optimization process is a multi-stage decisional process, which can be tackled with a comparatively small number of operations by dynamic programming [3,7,8]. Figure 5 shows the optimal allocation of 120 resource units (T.u.), with respect to F, and to F,,, among 30 buildings located in three different areas of Umbria, a region of Central Italy, where the loo-year probabilities of occurrence shown in Table 7 had been estimated: details on the volume and vulnerability of the buildings are given in [7,8]. When trying to pursue at the same time both the economic and the saved-people criteria, one could be content with a solution that is nearly optimal from the one of the viewpoints, but at the same time yields significant advantages from the other viewpoint. For instance, following an idea presented in an earlier paper [5], an attempt can be made starting from the design point that corresponds to the optimal solution from the economic viewpoint (0 in Fig. 6), and determining the paths along which the decrease in the economic return F, is smallest in comparison with the increase in saved lives: the procedure is stopped when the probability of failure attains the largest admissible value, which remains to be decided upon by the user. The same procedure can be followed assuming the solution corresponding to the largest number of saved lives as a starting point and determining the path which corresponds to the slowest decrease of F,. An alternative approach is to seek the so-called Pareto optima, corresponding to the points of the boundary line of the region of all possible solutions in the F,-F, plane: by definition, in

G. Augusti et al./Structural Safety 16 (1994) 91-109

103

8 6

Fig. 5. Allocation of 120 resource units that maximize the decrease in either the expected total economic loss (F,) or the expected number of endangered persons VU) in 30 masonry buildings.

Table 7 Assumed lOO-year probabilities

of earthquake

occurrence

i

1

MSK Intensity

7-7.5

2 7.5-8

3 8-8.5

4 > 8.5

Buildings

Site (town)

Tlt?I

=2m

r3rn

=4nI

m = l-10 m = 11-20 m = 21-30

Bastia Umbra Cittci di Castello Cascia

0.39

0.23

0.19

0.12

0.18

0.11

0.13 0.13 0.08

0.17 0.12 0.11

.. .

.I t

:

;‘.

:‘a; .

I *

.,I :. .

Fig. 6. Pareto optimal solutions: points belonging to the boundary line between the max F, and max F, points. All possible allocations among an ensemble of 10 masonry buildings have been calculated and plotted [8].

104

G. Augusti et al. /Structural

Safety 16 (1994) 91-109

fact, the boundary line corresponds to optimal combinations of pairs of F, and F, values. As an example, Fig. 6 shows such region and boundary calculated by an exaustive search on an ensemble of 10 buildings [S].

5. Allocation to the critical elements of a lifeline system How to obtain the most efficient reduction of seismic risk from the available resources is a significant problem also with regard to lifeline and highway systems. As a matter of fact, recent earthquakes have confirmed, on the one hand the vulnerability of these systems and on the other the large direct and indirect consequences of their failure on the economy, as well as on the lives and the well being, of whole regions, even beyond the area directly affected by the earthquake. A lifeline system can be in general modelled as a redundant network, comprising a number of vulnerable elements, that may themselves be complex redundant structural or mechanical systems. Therefore, differently from the case of buildings tackled in Sections 3 and 4, the study of the conditions of malfunction and of the appropriate measures for reducing the relevant risks requires a system approach: in fact, the consequences of failure of a vulnerable element depend significantly on the relevance of that element in the logical diagram of the network. This section describes briefly an elementary treatment of this problem, namely an optimal allocation procedure formulated with reference to a specific case: the planning of preventive upgrading interventions on the bridges included in a portion of a highway network [14,15]. To start from the simplest way of approach, the function of the network is identified with ensuring a connection between a source node S and a destination node D. Thus, failure is by definition the severance of this connection: it has been assumed that this can happen only because one or more bridges fails (i.e., that the bridges are the only vulnerable elements of the network), and the optimization has been aimed at reducing the probability of such failure, as presented in greater detail elsewhere [16]. To this aim it is necessary first to identify the interested network and its vulnerable elements, and to define the network topology (usually described by its minimal cut sets or its minimal path sets), which depends on the connections between the elements and on the assumed functionality condition. If the vulnerability of the critical elements is also evaluated, it is then possible to derive the reliability of the network as a whole. To elaborate a strategy for its improvement, it is also necessary to estimate the costs and the benefits of possible preventive measures in terms of their effects on the vulnerability of critical elements and of the whole system. The bridges considered in the examples are reinforced concrete girder bridges: the decks are simply supported on reinforced concrete piers. Their seismic vulnerability is defined by fragility curves, which had been previously evaluated by a Monte Carlo procedure which made use of Importance Sampling and Directional Simulation [17]. The fragility curve used in the numerical elaboration relates the peak ground acceleration (taken as measure of the earthquake intensity) and the probability of overcoming an appropriate threshold value of an indicator of the damage level in the critical sections of the piers (details on the structural data and the vulnerability analyses are reported in [18]). The bridges have been considered as originally designed for a

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D

Fig. 7. Diagram of the example highway network [the critical elements (bridges) are indicated by squares].

horizontal ground acceleration of 0.10 g (in accord with the current Italian Regulations), or upgraded in one of three ways, which follow two different techniques, namely: jacketing of the piers with shotcrete cover and addition of longitudinal reinforcement to improve the pier flexural capacity and shear strength (the reinforcement is increased by 50% in intervention I; by 100% in intervention II); elimination of expansion joints between the decks and introduction of isolation/dissipation devices on piers to replace the existing supports (intervention III). Consider the network whose diagram is represented in Fig. 7: it contains ten bridge. Six different paths connect S and D: namely (6-7-8-9), (6-7-8-lo), (l-4-5-9), (l-4-5-10), (l-2-3-5-9), (l-2-3-5-10). An algorithm, again based on dynamic programming, has been formulated for choosing, given the total available resources C,,,, the interventions that maximize the reduction of the probability Pfail of loss of connection between source S and destination D under an earthquake of given peak ground acceleration a. The results of the optimization are shown in Table 8 and Fig. 8. Three values of a have been considered, namely 0.15 g, 0.25 g and 0.35 g; the optimal solution has been sought for a given amount C,,, of the available resources, measured by conventionally putting equal to 100 resource units (Y.u.) the construction cost of the most expensive bridge. The whole range of values of C,,, has been investigated from nil up to the value that would allow the most complete intervention (III) on all bridges, which was found equal to 92 r.u. Inspection of results such as those presented in Fig. 8 and Table 8 allows many interesting considerations and evaluations, which can help to understand and improve the design and policy choices. For instance, the convenience of an optimal versus a rule-of-thumb allocation of resources can be easily put in evidence: in the specific example, to quote only one possible comparison, if a = 0.25 g and intervention II is performed on all bridges, 62 r.u. are employed

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107

G. Augusti et al./Structural Safety 16 (1994) 91-109

l.OOE+OO- -_--...--... -,..

Pfd l.OOE-02 --

.---.-x............i,,,,_,,,,,. ..: ‘---!L

--‘.._ ._. ,_... l.OOE-04 --

:_..-., I___

a=0.35g “?... !., :...

a = 0.25g

,,._,

/Li_,__

1 i...;y -l.._..(__ _ :..- ___.., i..‘......?

l.OOE-06 --

a = O.ISg /_.__.. _ _.-.._._ .._..-.._._ _._.. _.._. . . -.-..-..-..-.-.-.-..-..-. -.-.-.

l.OOE-08 -I 0

10

20

30

40

50

60

70

80

90

I 100

C ava

Fig. 8. Probability Pfail of network failure vs. available resources (optimized interventions).

and Pfail is reduced from 0.59 to 0.21: but if the same 62 T.U.are distributed in the optimal way, Pfail becomes as low as 0.1110P3! It can be also noted that, when the resources are optimally allocated, the reduction of Pfail with C,,, is very slow beyond 68 T.u.: therefore, a sensible general policy of good exploitation of resources would allocate no more than 68 r.u. to the upgrading of bridges in the considered network. It is of some interest to distinguish the preferential paths automatically chosen by the if it is larger. optimization procedure: in the example, (6-7-8) if C,,, is rather small, (l-4-5) In conclusion, it is fair to say that, notwithstanding the many simplifying assumptions that have been assumed, the procedure could already be applied to real problems of resource allocation, when the reliability of the connection is the main concern. Further research is in progress for introducing alternative objective functions of the optimization process, and for taking account of several degrees of damage (of the elements and/or the network), related to the reduction of load capacity of bridges and of traffic capacity of the road network. The optimization of the interventions might then aim at, say, the largest expected traffic capacity between source and destination after the earthquake has occurred, or the minimum repair time necessary to restore the S-D connection when it is severed as a consequence of the earthquake.

6. Concluding

remarks

In this paper, several cases of reduction of the seismic risk of existing constructed facilities have been tackled. It has been shown that it is convenient to allocate the available resources in an optimal way, which depends on the objective pursued and on the building system: this is all the more true if the available resources are limited. Conversely, the minimum amount of resources necessary to reach a given objective can be obtained. The feasibility of procedures aimed at optimizing such allocation has been proved for objective functions of either economic

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or intangible type, in both cases of buildings, when the effects of the damages to the individual constructions can be summed, and of structures that are the critical elements of a network. The results so far obtained in this research are very promising: indeed, it appears that in several instances the ideas and procedures presented in this paper could easily be applied to specific cases, and give some help to the decision makers, who often face dramatic alternatives in the allocation of public and private resources. Future research should aim, on the one side, at more realistic and precise quantitative definitions of the upgrading interventions, of their costs and effects, and of all other relationships and data to be inserted in the numerical procedures. On the other side, further conceptual developments appear still important: in particular to select other significant objective functions and to elaborate algorithms applicable to more complex networks. It would also be of particular interest to investigate further how several objectives, possibly incommensurable with each other, can be taken into consideration at the same time.

Acknowledgement

The research reported in this paper has been supported of University and Research (MURST).

by grants from the Italian Ministry

References [l] M.Z. Cohn and AS. Dinovitzer, Application of structural optimization, J. Struct. Engrg., ASCE, 120 (1994) 617-645. [2] G. Augusti, A. Baratta and F. Casciati, Probabilistic Methods in Structural Engineering, Chapman and Hall, New York, 1984, Ch. 6. [3] S.S. Rao, Reliability-Based Design, McGraw-Hill, New York, 1992. [4] G. Augusti and F. Casciati, Multiple modes of structural failure: probabilities and expected utilities, ZCOSSAR ‘77 (Proc. 2nd Znt. Conf on Structural Safety and Reliability), Munich, Werner-Verlag, Dusseldorf, 1977, pp. 39-61. [5] G. Augusti, A. Borri and F. Casciati, Structural design under random uncertainties: economical return and intangible quantities, ZCOSSAR ‘81 (Proc. 3rd Znt. Conf on Structural Safety and Reliability), Trondheim, Elsevier, Amsterdam, 1981, pp. 483-494. [6] G. Augusti, Decisions and optimization, General Report for Session B7, CERRA-ZCASP6 (Proc. 6th Znt. Conf on Applications of Statistics and Probability in Civil Engineering), Mexico City, Mexico (1991), Vol. 3, pp. 283-293. [7] G. Augusti, A. Borri and E. Speranzini, Seismic vulnerability data and optimal allocation of resources for risk reduction, ZCOSSAR ‘89 (Proc. 5th Znt. Conf on Structural Safety and Reliability), San Francisco, 1989, pp. 645-652. [8] G. Augusti, A. Borri and E. Speranzini, Optimum allocation of available resources to improve the reliability of building systems, Reliability and Optimization of Structural Systems ‘90, Proc. 3rd ZFZP WG 7.5 Conf, Berkeley, California, USA, 1990, Springer, Berlin, 1991, pp. 23-32. [9] G. Augusti, D. Benedetti and A. Corsanego, Investigations on seismic risk and seismic vulnerability in Italy, ZCOSSAR ‘8.5 (Proc. 4th Znt. Conf on Structural Safety and Reliability), Kobe, Japan, 1985, Vol. 2, pp. 267-276. [lo] S.V. Medvedev and W. Sponheuer, Scale of seismic intensity 0964).

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[ll] F. Braga, M. Dolce and D. Liberatore, Southern Italy November 23, 1980 earthquake: a statistical study on damaged buildings and an ensuing review of the MSK-76 scale, 7th European Conf on Earthquake Engineering, Athens, 1982. [12] D. Benedetti and G.M. Benzoni, Seismic vulnerability index versus damage for unreinforced masonry buildings, Proc. Znt. Conf on Reconstruction, Restoration and Urban Planning in Seismic Prone Areas, Skopje, 1985, pp. 333-347. [13] G. Augusti and A. Mantuano, Sulla determinazione della strategia ottimale per la prevenzione del rischio sismico nei centri abitati: un nuovo approccio, L’Zngegneria Sismica in Ztalia 1991, Proc. 5th Italian Nat. Conf Earthquake Engrg., Palermo, 1991, pp. 117-128. [14] G. Augusti, A. Borri and M. Ciampoli, Optimal allocation of resources in repair and maintenance of bridge structures, Probabilistic Mechanics and Structural and Geotechnical Reliability, Proc. 6th ASCE Specialty Conf , Denver, Colo., 1992, pp. l-4. [15] G. Augusti, A. Borri and M. Ciampoli, Ottimizzazione degli interventi di adeguamento sismico delle reti autostradali, Giornate A.Z.C.A.P.‘93 “Le opere in cu. e c.a.p. nelle infrastrutture per la mobilita e il trasporto”, Pisa, 1993, pp. 27-36. [16] G. Augusti, A. Borri and M. Ciampoli, Optimal allocation of resources in the reduction of the seismic risk of highway networks, Engrg. Structures (Special Issue on Structural Safety and Reliability in Earthquake Engineering), in print. [17] M. Ciampoli, R. Giannini, C. Nuti and P.E. Pinto, Seismic reliability of non-linear structures with stochastic parameters by directional simulation, ZCOSSAR ‘89 (Proc. 5th Znt. Conf on Structural Safety and Reliability), San Francisco, 1989, pp. 1121-1126. [18] M. Ciampoli and G. Augusti, Seismic reliability assessment of retrofitted bridges, Structural Dynamics, Eurodyn ‘93 (Proc. 2nd European Conf on Structural Dynamics), Trondheim, Norway, 1993, Vol. 1, pp. 193-200.