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Structural rehabilitation—A semi-Markovian decision approach
Structural Safety, 8 (1990) 255-262
255
Elsevier
STRUCTURAL REHABILITATION--A SEMI-MARKOVIAN DECISIO N A P P R O A C H * Elisa GuagenU and Chlara Molina Department of Structural Engineering, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy (Accepted December 1989)
Key words: existing structures; earthquake; damage; cost; decision analysis.
ABSTRACT A semi-Markovian approach is proposed for programming the strengthening of existing unsafe buildings in seismic zones. It is aimed at optimizing a functional which represents the "'total rewards'" of a strengthening strategy, i.e. inclusive of all future rewards (cost and benefits) associated to damage or rehabilitation. The reward associated to structural rehabilitation is defined based on the physical damage process in the building. The influence of damage-stochastic-process hypotheses on future rewards computation is evaluated. A simple example shows that even if social impact and loss of human lives are disregarded, an early strengthening intervention (before and not after a next earthquake) is "convenient" on the basis of the reward model introduced.
1. INTRODUCTION One aspect of seismic risk prevention consists in the strengthening of existing unsafe buildings. The choice of the strengthening strategy depends on many factors--t~chnical, economical, social, etc.--and also on the stochastic model assumed to interpret the seismic history and the life of the building. This latter dependence, in particular, is discussed herein, within the limits of a general framework proposed in order to approach the problem. Even though a mechanical theory is not available up to now to evaluate neither the seismic history at a given site nor the damage history in an old building (typically masonry), both processes can be modeled according to observed behaviour, physical guesses and empirical laws. * Paper presented at the Euromech 250 Colloquium on Nonlinear Structural Systems under Random Conditions, Como, Italy, June 19-23, 1989. 0167-4730/90/$03.50 © 1990 - Elsevier Science Publishers B.V.
256 In particular the building damage can be described as a function of the earthquake size [1], while the earthquake sequence can be described as a semi-Markovian process [2]. In evaluating the effectiveness of a strengthening strategy, the problem arises of comparing costs and benefits, defined in one word as rewards, occurring at different time instants. A classical problem consists in comparing the cost of a reinforcing intervention, today, with the expected cost of seismic damage in the future; a discount factor is introduced in order to construct the "present" expected value, as is usually done in economic evaluations. Many years ago [3] cost-benefit analysis has shown that a minimum-cost strategy exists, i.e. it is "convenient" to increase the seismic safety coefficient (thus the initial cost) in order to decrease the present expected value of total cost (initial cost plus future damage cost). Because different stochastic models for representing seismic occurrences predict different probabilities of a seismic event at a given time, the model influences the judgement about the effectiveness of a strategy. A study has been done [4,5] if a renewal process holds: the present expected cost of a strategy changes with the failure time distribution, even if the return period of the earthquake sequence is given, as well as the coefficient of variation too. The strategy considered there was simply repairing after an earthquake. Prevention strategies have also been analyzed by the authors, but under stationary conditions [6]. At present research is dealing with general prevention strategies for a non-Markovian process of earthquake occurrence [7,8]. In the present paper a semi-Markovian process is assumed to represent both the seismic history and the building life; it allows to consider a rather general family of prevention strategies under non-stationary conditions. In order to compare different strategy choices a reward model is introduced that allows: - to take into account together with the building damage process due to earthquake occurrences, the continuous deterioration process due to aging; to introduce strengthening interventions either actually made in a precise time instant or continuously carried out in time. A simple numerical example analyzes the dynamic aspect of an optimum-reward strategy. Such an optimum may simply consist in a reinforcing intervention carried out in advance. While classical analysis leads to the result that delayed strengthening works are convenient, the present example, because of memory in the stochastic process, emphasizes that the benefit of avoided losses through an early intervention can be very high. Therefore early strengthening works can be convenient. This is especially true when an hypothetical gap in seismic history results in a high probability (in the semi-Markovian model) of an impending earthquake -
2. SEMI-MARKOVIAN APPROACH
In order to define the life of a building in a seismic zone, the local earthquake process is needed together with the building vulnerability and rehabilitation criterion. The local earthquake process is assumed, as mentioned above, to be semi-Markovian [2]. The vulnerability criterion is defined by physical properties such as type and quality of resisting system, conventional safety factor, foundations, plan, elevation, etc. [9,10]. The damage process turns out to be semi-Markovian [11]. Let us now see how a semi-Markovian framework can include the rehabilitation planning. A semi-Markovian process (s-MP) is a one-step memory process describing the evolution of a system that changes its state at each transition. In the problem considered here:
257 the system is the sample building; the state of the system is defined by the damage level which the system belongs to; a state transition is a change of level. The behaviour of an s-MP can be interpreted as follows. Whenever the system enters state i, its first action is to choose the next state j according to the transition probabilities Pij- Then the system establishes how much time is spent in state i from the holding time densities hij(x). Therefore an s-MP is defined by the following quantities: (a) initial conditions, i.e. initial state i and the time t o already elapsed in i since t = 0; (b) transition probability matrix -
-
-
Pij = Pr[next state j / p r e s e n t state i] ; (c) the probability density function (p.d.f.) hgj(x) of the holding times "r~j; ¢ij is defined as the conditional inter-occurrence time between two transitions, if the present state is i and the next state is j. All quantities defining seismic risk can be calculated according to the theory of s-MP [12]. Because of the terms involving memory, computation is based on recursive integral equations; therefore a convenient analytical tool is the Laplace Transform (L.T.). As a consequence of the definition of an s-MP ((a)-(c) above), a dynamic programming model can be defined to represent the building life, containing, together with the damage process due to aging or to earthquake occurrence, the unknown rehabilitation process; thus two types of transition can be distinguished: damage transitions (due to earthquake occurrence or aging); rehabilitation transitions (due to prevention, repairing or rebuilding interventions). Let us assume that when i < j, state i corresponds to less deterioration than state j. Then the transition i - ~ j > i is a damage transition while the transition j - ~ i < j is a rehabilitation transition. The quantities Pij and hgj(x) with i > j define the unknown decisional process. The quantities p~j and h~j(x) with i < j define the damage process; they are drawn from the local earthquake process and from the building's vulnerability [11]. -
-
3. THE SEMI-MARKOVIAN REWARD MODEL A question arises concerning sequential decisions, i.e. which sequence of decisions should be assumed, bearing in mind that each decision affects all future programming. Of course, the aim is to make decisions which yield the highest possible reward, i.e. to realize a profit or at least avoid a loss. A reward model, which allows to evaluate the effectiveness of prevention strategies, can always be associated with the behaviour of an s-MP. In particular a reward model that conforms with the problems considered here can be defined as follows. It will be assumed that, when an s-MP occupies state i before the transition to state j, it earns a reward with a rate called the yield rate Yig; when a transition is actually made at a certain time x, the process earns a reward called bonus big. The parameters Yij and bij have to be derived from the building's vulnerability. Let the system be placed in state i at t = 0, and t o be the time elapsed since the last transition occurred. The expected present value of the reward the process will generate in a time interval of
258 length t is then given by the following integral equation: x
Ei[t, to, 7] = ~.,Pij(to)ftt dxhij(x, tO)fo do e-V°yij(o + to) J +~.,pij(tO)fotdXhij(X, to)(foXdoe-'°yij(O+to)+e-'Xbij(x+to)} J
fo
+ Y'~Pij(to) dxhij(x, to) e-'XEj(t-x, 7) J
(1)
where ~, is the discount factor introduced to compare non-contemporaneous rewards and
hig(X, to)=
h,(t+to) oo
Pij
f h.(x)dx
,
Pij(to) =
to
f h,(x) d x to
Ejp#f
to
h,j(x)dx
The three summations in eqn. (1) can be interpreted as follows. The first transition out of state i can occur either before or after the time t. The first summation represents the discounted yield rate contribution if the first transition out of state i occurs after t; the second summation represents the discounted yield rate and bonus contribution from the first transition out of state i if it occurs before t; the third summation represents the discounted reward contribution from future transitions after the first transition out of state i, if it occurs before t. The assumptions on which the reward model definition (1) is based, can be summarized as follows: damage transitions occur either because of aging or depending on the s-MP introduced to represent the seismic history and on the vulnerability properties of the sample building; rehabilitation transitions can be made instantaneously at any time, by means of the bonus model; rehabilitation transitions can be made continuously in time between two transitions, by means of the yield-rate model. When t is large, the expected reward (1) depends directly on the L.T. of the holding time densities. It assumes the following form: -
-
-
Ei(to,)')=Y'~Pij(to) J
/o dXl_Hi(to) h3(v)
d°e-V°Yij(°+to) +e
+ Y"P'J(t°) 1 - Hi(to) e-Vt°Eg(y) J
big(X+to)
} (2)
where the star denotes the L.T. and
f0'
Hi(t, to)= Ep,(to) h,(x, to) dx J In classical seismic risk analysis the following assumptions hold: - damage to the building is due only to earthquakes; - the earthquake sequence is represented by a stationary Poisson process; earthquakes of a certain size are considered to be randomly distributed in a background process of all earthquakes;
259 rehabilitation interventions are carded out immediately whenever a damage occurs; any rehabilitation intervention upgrades the sample building to the initial state of the building, before damage. In this case eqn. (2) reduces to -
-
E=b/
(3)
r
where T is the return period of earthquakes and b/T represents an annual mean reward associated to the strategy assumed. It is noted that, when a stationary risk evaluation is required, i.e. when the time t 0, at which the risk evaluation is carried out, is a sampling point chosen at random over a very long time period, then the s-MP introduced to represent the building life becomes a so called "in equilibrium" process [12], which has the same stationary properties as a Poisson process. In particular, the expected reward Ei(t o, ~,) can be calculated on the basis of a mean annual reward, as in eqn. (3). As a conclusion, the two expected present values (1) and (2), respectively generated in a limited or in an unlimited interval, define the objectives to be optimized in the dynamic programming model. The first must be considered when dealing with time-limited programming problems, typically when a fixed budget is disposable to obtain a final system state. The second must be considered when dealing with stationary planning strategies, that is, when an annual rehabilitation budget is available.
4. NUMERICAL EXAMPLES
Two simple rehabilitation strategy examples are considered in order to show that, when certain conditions are statified on the sample building's vulnerability and on the seismicity of the zone the building belongs to, then reinforcing interventions can be advantageous. These interventions either involve an upgrading to a more resistant category of building or are carried out in advance. First, the case is considered in which only two states are defined: P12, Pll 1 -P12, h12 and hll define the damage transitions. It is supposed that rehabilitation works are carded out immediately whenever damage occurs; this means that: =
yij=O,
b11=b12=b22=O,
p21 =p12,
h~'l(~,) = h~'2(~/) = 1
Under these hypotheses h~2(/0, T) Ea[t0, Y] =p12b21 1 _Pllh~1(r)_Pa2h~2(y )
=P12b21dP(to,Y)
(4)
Now two cases are considered where state 1 has different vulnerability properties: I a denotes a less vulnerable state, 1~ a more vulnerable. The question is as follows: when is rehabilitation to state I a more advantageous than rehabilitation to state 1B? In other words, when are more expensive rehabilitation works "convenient"? On the basis of eqn. (4) the answer is, when the following condition holds
p(~)l"(")'~(~)(to, y ) < /"(~)1"(~)~'(~)(to, ~/) 12 t"21 ~ " 1 2 u21 ~"
(5)
260
This result, obtained here as a particular case of the general reward model introduced earlier, coincides with an analogous result obtained by a direct method of computation in Ref. [13]. As a second example, a case is considered in which only one type of earthquake and three states of the building system are defined. State 1 consists of a-seismic constructions, i.e. when the earthquake occurs, the sample building of state 1 suffers damage but does not collapse. State 2 consists of non a-seismic constructions, i.e. when the earthquake occurs, the sample building of state 2 collapses in state 3. State 3 consists of constructions in a completely deteriorated state. The following rehabilitation strategy is defined: repairing the damaged buildings of state 1 and upgrading the collapsed buildings to state 1 immediately after the earthquake; reinforcing all the buildings of state 2 up to state 1 instantaneously in a time ~. These hypotheses imply that -
-
oo
Pll=I,
Pz1=f
f~(x) dx,
hl,=L(x),
h:i=8(x-'7),
bal, b2a, b31 constant,
P23 = 1 - P 2 1 ,
h23 =
f,(x)
fo2(X)
P31 = 1
U(x-~),
h31 = 8 ( x )
(6)
b23 = 0
where £ ( x ) is the p.d.f, of interoccurrence time between two earthquakes. As a consequence
E2(to=O, 3',r)=(b21+E1)e-V~f~ f,(x) dx+(b31+Ea) f02 (x) e-'Xdx where b l l f , * (3')
E1- 1-£*('f) The question is as follows: under what conditions does a moment ~ exist in which it is "convenient" to carry out reinforcing works? When is this moment anticipated with respect to
E2
P
Fig.
1.
Present
expected
value of E2(t 0 = 0, y, ( ) w h e n r is e x p o n e n t i a l l y - d i s t r i b u t e d . (a) b31 < (1 + TT)b21 + bll; (b) b31 > (1 + yT)b21 + bll.
261 E2
Fig. 2. Present expected value of E2(t 0 = 0, "y, ~) when $ is Gamma2-distributed. 2+~,T 2 (a) b31 < 2 b2a + ~,T + 4 bn ; (b) b31 >
2+ ~,T 2 2 b21 + y--f-~ bll;
(c) b3a>
3yT+4 6 4 b21+ 2 ( , / T + 4 ) bn ;
(d) baa >
(2+ v T ) 2 37T+4 6 4 b21 + bn > - -4 b21 + 2(~,T + 4) bll"
the return period T of earthquakes? The answer depends on f~. As an example, when "r is exponentially distributed, a reinforcing work is advantageous only at ;r = 0 and only if b31 > (1 +
"rT)b2x+ bll
where T denotes, as usually, the earthquake return period (Fig. 1). When r is Gamma2-distributed, i.e. when
f~(x)=p(px)e -px,
p=2/T
the expected reward E2(t 0 = 0, 3', rr) assumes the values shown in Fig. 2 for the four conditions on the parameters b~j there specified. An instant ~=
4 b n T + b21"yT2(T'y + 4) 2{ ('tT + 4)[2b31- (2 + ),T)b21 ] - 4bn )
anticipated with respect to the return period T, exists in which it is advantageous to carry out the reinforcing intervention if the following condition is satisfied b31 >
3vT + 4 6 4 bE1 + 2 ( v T + 4) bll
which coincides with condition (c) in Fig. 2. Note that condition (a) in Figs. 1 and 2 leads to the situation which characterizes the classical analysis.
262
5. CONCLUSIONS Consider a sample building in seismic region. A s e m i - M a r k o v i a n process is used to idealize local e a r t h q u a k e hazard. It is coupled with a building vulnerability criterion to construct a structural d a m a g e model, which turns o u t to be semi-Markovian. The vulnerability criterion is based on the m e c h a n i c a l a n d typological properties of the building; it leads to the definition of the p a r a m e t e r s b o t h in the d a m a g e process a n d in the rehabilitation one. A reward m o d e l is c o n s t r u c t e d to evaluate the effectiveness of risk mitigation strategies. The ultimate result is to identify the o p t i m a l s t r e n g t h e n i n g time. Simple examples show that optimal i n s t a n t is significantly better anticipated in this n e w f r a m e w o r k c o m p a r e d to classical theory.
REFERENCES 1 E. Guagenti and V. Petrini, I1 caso delle vecchie costruzioni: verso una legge danni-intensit/l, Conv. Naz. L'Ingegneria Sismica in Italia, Milan, 1989. 2 E. Guagenti, C. Molina and G. Mulas, Seismic risk analysis with predictable models, Earthquake Eng. Struct. Dyn., 16 (1988) 343-359. 3 E. Guagenti and V. Petrini, Comparative analysis of the seismic risk in sites of different seismicity, Earthquake Eng. Struct. Dyn., 5 (1977) 53-65. 4 L. Esteva, Bases para la formulacion de designo sismico, Rep. n. 182, Nat. Univ. of Mexico, Mexico, 1968. 5 E. Guagenti and C. Molina, Valutazione probabilistica di danni con rischio variabile, Rend. Ist. Lomb. Accad. Sci. Lettere, 113 (1979) 202-209. 6 E. Guagenti, A. Finzi and C. Molina, Un criterio di ottimizzazione dinamica in un problema di prevenzione del rischio sismico, Cong. Naz. Mecc. Teor. e Appl., Palermo, 1980, pp. 202-213. 7 E. Rosenblueth, Optimum reliabilities and optimum design, Structural Safety, 3 (1986) 69-83. 8 E. Rosenblueth and J.M. Jara, Constant versus time dependent seismic design coefficients, to be published. 9 D. Benedetti and V. Petrini, On seismic vulnerability of masonry buildings: proposal of an evaluation procedure, Industria Costruzioni, 18 (1984) 66-74. 10 P. Angeletti, A. Bellina, E. Guagenti, A. Moretti and V. Petrini, Comparison between vulnerability assessment and damage index--some results, Proc. 9th World Conf. on Earthquake Eng., Tokyo, 1988. 11 E. Guagenti, V. Petrini and A. Drei, Rischio sismico locale in ipotesi non poissoniana: confronto fra siti, AIMETA 88, Bad, 1988, pp. 663-666. 12 R.A. Howard, Dynamic Probabilistic Systems, Wiley, New York, 1971. 13 G. Grandori, Cost-benefit analysis in earthquake engineering, 6th Eur. Conf. Eng., Dubrovnik, 1978.
255
Elsevier
STRUCTURAL REHABILITATION--A SEMI-MARKOVIAN DECISIO N A P P R O A C H * Elisa GuagenU and Chlara Molina Department of Structural Engineering, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy (Accepted December 1989)
Key words: existing structures; earthquake; damage; cost; decision analysis.
ABSTRACT A semi-Markovian approach is proposed for programming the strengthening of existing unsafe buildings in seismic zones. It is aimed at optimizing a functional which represents the "'total rewards'" of a strengthening strategy, i.e. inclusive of all future rewards (cost and benefits) associated to damage or rehabilitation. The reward associated to structural rehabilitation is defined based on the physical damage process in the building. The influence of damage-stochastic-process hypotheses on future rewards computation is evaluated. A simple example shows that even if social impact and loss of human lives are disregarded, an early strengthening intervention (before and not after a next earthquake) is "convenient" on the basis of the reward model introduced.
1. INTRODUCTION One aspect of seismic risk prevention consists in the strengthening of existing unsafe buildings. The choice of the strengthening strategy depends on many factors--t~chnical, economical, social, etc.--and also on the stochastic model assumed to interpret the seismic history and the life of the building. This latter dependence, in particular, is discussed herein, within the limits of a general framework proposed in order to approach the problem. Even though a mechanical theory is not available up to now to evaluate neither the seismic history at a given site nor the damage history in an old building (typically masonry), both processes can be modeled according to observed behaviour, physical guesses and empirical laws. * Paper presented at the Euromech 250 Colloquium on Nonlinear Structural Systems under Random Conditions, Como, Italy, June 19-23, 1989. 0167-4730/90/$03.50 © 1990 - Elsevier Science Publishers B.V.
256 In particular the building damage can be described as a function of the earthquake size [1], while the earthquake sequence can be described as a semi-Markovian process [2]. In evaluating the effectiveness of a strengthening strategy, the problem arises of comparing costs and benefits, defined in one word as rewards, occurring at different time instants. A classical problem consists in comparing the cost of a reinforcing intervention, today, with the expected cost of seismic damage in the future; a discount factor is introduced in order to construct the "present" expected value, as is usually done in economic evaluations. Many years ago [3] cost-benefit analysis has shown that a minimum-cost strategy exists, i.e. it is "convenient" to increase the seismic safety coefficient (thus the initial cost) in order to decrease the present expected value of total cost (initial cost plus future damage cost). Because different stochastic models for representing seismic occurrences predict different probabilities of a seismic event at a given time, the model influences the judgement about the effectiveness of a strategy. A study has been done [4,5] if a renewal process holds: the present expected cost of a strategy changes with the failure time distribution, even if the return period of the earthquake sequence is given, as well as the coefficient of variation too. The strategy considered there was simply repairing after an earthquake. Prevention strategies have also been analyzed by the authors, but under stationary conditions [6]. At present research is dealing with general prevention strategies for a non-Markovian process of earthquake occurrence [7,8]. In the present paper a semi-Markovian process is assumed to represent both the seismic history and the building life; it allows to consider a rather general family of prevention strategies under non-stationary conditions. In order to compare different strategy choices a reward model is introduced that allows: - to take into account together with the building damage process due to earthquake occurrences, the continuous deterioration process due to aging; to introduce strengthening interventions either actually made in a precise time instant or continuously carried out in time. A simple numerical example analyzes the dynamic aspect of an optimum-reward strategy. Such an optimum may simply consist in a reinforcing intervention carried out in advance. While classical analysis leads to the result that delayed strengthening works are convenient, the present example, because of memory in the stochastic process, emphasizes that the benefit of avoided losses through an early intervention can be very high. Therefore early strengthening works can be convenient. This is especially true when an hypothetical gap in seismic history results in a high probability (in the semi-Markovian model) of an impending earthquake -
2. SEMI-MARKOVIAN APPROACH
In order to define the life of a building in a seismic zone, the local earthquake process is needed together with the building vulnerability and rehabilitation criterion. The local earthquake process is assumed, as mentioned above, to be semi-Markovian [2]. The vulnerability criterion is defined by physical properties such as type and quality of resisting system, conventional safety factor, foundations, plan, elevation, etc. [9,10]. The damage process turns out to be semi-Markovian [11]. Let us now see how a semi-Markovian framework can include the rehabilitation planning. A semi-Markovian process (s-MP) is a one-step memory process describing the evolution of a system that changes its state at each transition. In the problem considered here:
257 the system is the sample building; the state of the system is defined by the damage level which the system belongs to; a state transition is a change of level. The behaviour of an s-MP can be interpreted as follows. Whenever the system enters state i, its first action is to choose the next state j according to the transition probabilities Pij- Then the system establishes how much time is spent in state i from the holding time densities hij(x). Therefore an s-MP is defined by the following quantities: (a) initial conditions, i.e. initial state i and the time t o already elapsed in i since t = 0; (b) transition probability matrix -
-
-
Pij = Pr[next state j / p r e s e n t state i] ; (c) the probability density function (p.d.f.) hgj(x) of the holding times "r~j; ¢ij is defined as the conditional inter-occurrence time between two transitions, if the present state is i and the next state is j. All quantities defining seismic risk can be calculated according to the theory of s-MP [12]. Because of the terms involving memory, computation is based on recursive integral equations; therefore a convenient analytical tool is the Laplace Transform (L.T.). As a consequence of the definition of an s-MP ((a)-(c) above), a dynamic programming model can be defined to represent the building life, containing, together with the damage process due to aging or to earthquake occurrence, the unknown rehabilitation process; thus two types of transition can be distinguished: damage transitions (due to earthquake occurrence or aging); rehabilitation transitions (due to prevention, repairing or rebuilding interventions). Let us assume that when i < j, state i corresponds to less deterioration than state j. Then the transition i - ~ j > i is a damage transition while the transition j - ~ i < j is a rehabilitation transition. The quantities Pij and hgj(x) with i > j define the unknown decisional process. The quantities p~j and h~j(x) with i < j define the damage process; they are drawn from the local earthquake process and from the building's vulnerability [11]. -
-
3. THE SEMI-MARKOVIAN REWARD MODEL A question arises concerning sequential decisions, i.e. which sequence of decisions should be assumed, bearing in mind that each decision affects all future programming. Of course, the aim is to make decisions which yield the highest possible reward, i.e. to realize a profit or at least avoid a loss. A reward model, which allows to evaluate the effectiveness of prevention strategies, can always be associated with the behaviour of an s-MP. In particular a reward model that conforms with the problems considered here can be defined as follows. It will be assumed that, when an s-MP occupies state i before the transition to state j, it earns a reward with a rate called the yield rate Yig; when a transition is actually made at a certain time x, the process earns a reward called bonus big. The parameters Yij and bij have to be derived from the building's vulnerability. Let the system be placed in state i at t = 0, and t o be the time elapsed since the last transition occurred. The expected present value of the reward the process will generate in a time interval of
258 length t is then given by the following integral equation: x
Ei[t, to, 7] = ~.,Pij(to)ftt dxhij(x, tO)fo do e-V°yij(o + to) J +~.,pij(tO)fotdXhij(X, to)(foXdoe-'°yij(O+to)+e-'Xbij(x+to)} J
fo
+ Y'~Pij(to) dxhij(x, to) e-'XEj(t-x, 7) J
(1)
where ~, is the discount factor introduced to compare non-contemporaneous rewards and
hig(X, to)=
h,(t+to) oo
Pij
f h.(x)dx
,
Pij(to) =
to
f h,(x) d x to
Ejp#f
to
h,j(x)dx
The three summations in eqn. (1) can be interpreted as follows. The first transition out of state i can occur either before or after the time t. The first summation represents the discounted yield rate contribution if the first transition out of state i occurs after t; the second summation represents the discounted yield rate and bonus contribution from the first transition out of state i if it occurs before t; the third summation represents the discounted reward contribution from future transitions after the first transition out of state i, if it occurs before t. The assumptions on which the reward model definition (1) is based, can be summarized as follows: damage transitions occur either because of aging or depending on the s-MP introduced to represent the seismic history and on the vulnerability properties of the sample building; rehabilitation transitions can be made instantaneously at any time, by means of the bonus model; rehabilitation transitions can be made continuously in time between two transitions, by means of the yield-rate model. When t is large, the expected reward (1) depends directly on the L.T. of the holding time densities. It assumes the following form: -
-
-
Ei(to,)')=Y'~Pij(to) J
/o dXl_Hi(to) h3(v)
d°e-V°Yij(°+to) +e
+ Y"P'J(t°) 1 - Hi(to) e-Vt°Eg(y) J
big(X+to)
} (2)
where the star denotes the L.T. and
f0'
Hi(t, to)= Ep,(to) h,(x, to) dx J In classical seismic risk analysis the following assumptions hold: - damage to the building is due only to earthquakes; - the earthquake sequence is represented by a stationary Poisson process; earthquakes of a certain size are considered to be randomly distributed in a background process of all earthquakes;
259 rehabilitation interventions are carded out immediately whenever a damage occurs; any rehabilitation intervention upgrades the sample building to the initial state of the building, before damage. In this case eqn. (2) reduces to -
-
E=b/
(3)
r
where T is the return period of earthquakes and b/T represents an annual mean reward associated to the strategy assumed. It is noted that, when a stationary risk evaluation is required, i.e. when the time t 0, at which the risk evaluation is carried out, is a sampling point chosen at random over a very long time period, then the s-MP introduced to represent the building life becomes a so called "in equilibrium" process [12], which has the same stationary properties as a Poisson process. In particular, the expected reward Ei(t o, ~,) can be calculated on the basis of a mean annual reward, as in eqn. (3). As a conclusion, the two expected present values (1) and (2), respectively generated in a limited or in an unlimited interval, define the objectives to be optimized in the dynamic programming model. The first must be considered when dealing with time-limited programming problems, typically when a fixed budget is disposable to obtain a final system state. The second must be considered when dealing with stationary planning strategies, that is, when an annual rehabilitation budget is available.
4. NUMERICAL EXAMPLES
Two simple rehabilitation strategy examples are considered in order to show that, when certain conditions are statified on the sample building's vulnerability and on the seismicity of the zone the building belongs to, then reinforcing interventions can be advantageous. These interventions either involve an upgrading to a more resistant category of building or are carried out in advance. First, the case is considered in which only two states are defined: P12, Pll 1 -P12, h12 and hll define the damage transitions. It is supposed that rehabilitation works are carded out immediately whenever damage occurs; this means that: =
yij=O,
b11=b12=b22=O,
p21 =p12,
h~'l(~,) = h~'2(~/) = 1
Under these hypotheses h~2(/0, T) Ea[t0, Y] =p12b21 1 _Pllh~1(r)_Pa2h~2(y )
=P12b21dP(to,Y)
(4)
Now two cases are considered where state 1 has different vulnerability properties: I a denotes a less vulnerable state, 1~ a more vulnerable. The question is as follows: when is rehabilitation to state I a more advantageous than rehabilitation to state 1B? In other words, when are more expensive rehabilitation works "convenient"? On the basis of eqn. (4) the answer is, when the following condition holds
p(~)l"(")'~(~)(to, y ) < /"(~)1"(~)~'(~)(to, ~/) 12 t"21 ~ " 1 2 u21 ~"
(5)
260
This result, obtained here as a particular case of the general reward model introduced earlier, coincides with an analogous result obtained by a direct method of computation in Ref. [13]. As a second example, a case is considered in which only one type of earthquake and three states of the building system are defined. State 1 consists of a-seismic constructions, i.e. when the earthquake occurs, the sample building of state 1 suffers damage but does not collapse. State 2 consists of non a-seismic constructions, i.e. when the earthquake occurs, the sample building of state 2 collapses in state 3. State 3 consists of constructions in a completely deteriorated state. The following rehabilitation strategy is defined: repairing the damaged buildings of state 1 and upgrading the collapsed buildings to state 1 immediately after the earthquake; reinforcing all the buildings of state 2 up to state 1 instantaneously in a time ~. These hypotheses imply that -
-
oo
Pll=I,
Pz1=f
f~(x) dx,
hl,=L(x),
h:i=8(x-'7),
bal, b2a, b31 constant,
P23 = 1 - P 2 1 ,
h23 =
f,(x)
fo2(X)
P31 = 1
U(x-~),
h31 = 8 ( x )
(6)
b23 = 0
where £ ( x ) is the p.d.f, of interoccurrence time between two earthquakes. As a consequence
E2(to=O, 3',r)=(b21+E1)e-V~f~ f,(x) dx+(b31+Ea) f02 (x) e-'Xdx where b l l f , * (3')
E1- 1-£*('f) The question is as follows: under what conditions does a moment ~ exist in which it is "convenient" to carry out reinforcing works? When is this moment anticipated with respect to
E2
P
Fig.
1.
Present
expected
value of E2(t 0 = 0, y, ( ) w h e n r is e x p o n e n t i a l l y - d i s t r i b u t e d . (a) b31 < (1 + TT)b21 + bll; (b) b31 > (1 + yT)b21 + bll.
261 E2
Fig. 2. Present expected value of E2(t 0 = 0, "y, ~) when $ is Gamma2-distributed. 2+~,T 2 (a) b31 < 2 b2a + ~,T + 4 bn ; (b) b31 >
2+ ~,T 2 2 b21 + y--f-~ bll;
(c) b3a>
3yT+4 6 4 b21+ 2 ( , / T + 4 ) bn ;
(d) baa >
(2+ v T ) 2 37T+4 6 4 b21 + bn > - -4 b21 + 2(~,T + 4) bll"
the return period T of earthquakes? The answer depends on f~. As an example, when "r is exponentially distributed, a reinforcing work is advantageous only at ;r = 0 and only if b31 > (1 +
"rT)b2x+ bll
where T denotes, as usually, the earthquake return period (Fig. 1). When r is Gamma2-distributed, i.e. when
f~(x)=p(px)e -px,
p=2/T
the expected reward E2(t 0 = 0, 3', rr) assumes the values shown in Fig. 2 for the four conditions on the parameters b~j there specified. An instant ~=
4 b n T + b21"yT2(T'y + 4) 2{ ('tT + 4)[2b31- (2 + ),T)b21 ] - 4bn )
anticipated with respect to the return period T, exists in which it is advantageous to carry out the reinforcing intervention if the following condition is satisfied b31 >
3vT + 4 6 4 bE1 + 2 ( v T + 4) bll
which coincides with condition (c) in Fig. 2. Note that condition (a) in Figs. 1 and 2 leads to the situation which characterizes the classical analysis.
262
5. CONCLUSIONS Consider a sample building in seismic region. A s e m i - M a r k o v i a n process is used to idealize local e a r t h q u a k e hazard. It is coupled with a building vulnerability criterion to construct a structural d a m a g e model, which turns o u t to be semi-Markovian. The vulnerability criterion is based on the m e c h a n i c a l a n d typological properties of the building; it leads to the definition of the p a r a m e t e r s b o t h in the d a m a g e process a n d in the rehabilitation one. A reward m o d e l is c o n s t r u c t e d to evaluate the effectiveness of risk mitigation strategies. The ultimate result is to identify the o p t i m a l s t r e n g t h e n i n g time. Simple examples show that optimal i n s t a n t is significantly better anticipated in this n e w f r a m e w o r k c o m p a r e d to classical theory.
REFERENCES 1 E. Guagenti and V. Petrini, I1 caso delle vecchie costruzioni: verso una legge danni-intensit/l, Conv. Naz. L'Ingegneria Sismica in Italia, Milan, 1989. 2 E. Guagenti, C. Molina and G. Mulas, Seismic risk analysis with predictable models, Earthquake Eng. Struct. Dyn., 16 (1988) 343-359. 3 E. Guagenti and V. Petrini, Comparative analysis of the seismic risk in sites of different seismicity, Earthquake Eng. Struct. Dyn., 5 (1977) 53-65. 4 L. Esteva, Bases para la formulacion de designo sismico, Rep. n. 182, Nat. Univ. of Mexico, Mexico, 1968. 5 E. Guagenti and C. Molina, Valutazione probabilistica di danni con rischio variabile, Rend. Ist. Lomb. Accad. Sci. Lettere, 113 (1979) 202-209. 6 E. Guagenti, A. Finzi and C. Molina, Un criterio di ottimizzazione dinamica in un problema di prevenzione del rischio sismico, Cong. Naz. Mecc. Teor. e Appl., Palermo, 1980, pp. 202-213. 7 E. Rosenblueth, Optimum reliabilities and optimum design, Structural Safety, 3 (1986) 69-83. 8 E. Rosenblueth and J.M. Jara, Constant versus time dependent seismic design coefficients, to be published. 9 D. Benedetti and V. Petrini, On seismic vulnerability of masonry buildings: proposal of an evaluation procedure, Industria Costruzioni, 18 (1984) 66-74. 10 P. Angeletti, A. Bellina, E. Guagenti, A. Moretti and V. Petrini, Comparison between vulnerability assessment and damage index--some results, Proc. 9th World Conf. on Earthquake Eng., Tokyo, 1988. 11 E. Guagenti, V. Petrini and A. Drei, Rischio sismico locale in ipotesi non poissoniana: confronto fra siti, AIMETA 88, Bad, 1988, pp. 663-666. 12 R.A. Howard, Dynamic Probabilistic Systems, Wiley, New York, 1971. 13 G. Grandori, Cost-benefit analysis in earthquake engineering, 6th Eur. Conf. Eng., Dubrovnik, 1978.