A PROBABILISTIC APPROACH FOR SEISMIC ASSESSMENT OF R.C. STRUCTURES: APPLICATION TO HIGHWAY BRIDGES

Transcription

1 13 th World Conferene on Earthquake Engineering Vanouver, B.C., Canada August 1-6, 004 Paper No. 601 A PROBABILISTIC APPROACH FOR SEISMIC ASSESSMENT OF R.C. STRUCTURES: APPLICATION TO HIGHWAY BRIDGES Gianmaro de FELICE 1, Renato GIANNINI and Alessandro RASULO 3 SUMMARY The Effetive Fragility Analysis (EFA) methodology is applied to the assessment of seismi reliability of three reinfored onrete bridges along Italian highway network. The seleted bridges are different for strutural sheme (ontinuous/simply supported dek) and pier type (single stem/frame piers). The reliability evaluation onsiders different soures of unertainty: in the seismi input, through the use of different aelerograms for the dynami analysis, in the strutural behaviour, through the use of a refined non-linear finite element model, and in the ultimate state modelling. The EFA proedure implemented in this study redues the number of simulations required for the reliability analysis using a surfae response tehnique only for random variables affeting the strutural dynami behaviour, whilst state-limit randomness is treated expliitly during simulations. The final result of the analysis is the bridge fragility urve as funtion of spetral aeleration and, through its onvolution with loal hazard, the risk. Those results are briefly disussed with respet to the following aspets: omparison of bridge vulnerability in relation of different pier and dek strutural models, influene of material randomness on final reliability, relative importane of seismi input respet to mehanial and epistemi unertainty. INTRODUCTION In this paper, the EFA reliability proedure (Effetive Fragility Analysis) developed in [1] is applied to the assessment of seismi fragility of some bridges along Italian highway network. The method, modified and presented in detail in [], is based on the different treatment of unertainty oming from the random variables governing the dynami response of the struture, hereafter said external, and from those governing the limit states for whih the fragility urve is built, hereafter said internal. In order to study the dependene from the external variables, olleted in the vetor y, the proedure uses the response surfae method [3], [4]. This method ahieves a simple funtional relationship, usually of the polynomial type, between the external variables and a measure of seismi response through the run of a limited number of non-linear dynamial analysis. 1 Assoiate Professor, Dipartimento di Strutture, Università degli Studi Roma Tre, Roma, Italia Professor, Dipartimento di Strutture, Università degli Studi Roma Tre, Roma, Italia 3 Researh Fellow, Dipartimento di Meania, Strutture, Ambiente e Territorio, Università degli Studi di Cassino, Cassino (FR), Italia

2 On the other hand, the dependene on the internal variables, olleted in the vetor x, do not requires the exeution of extra dynami analysis on the struture sine the probability to undergo the limit state is expliitly obtained using standard reliability methods under the hypothesis of independene of ollapse mehanisms. This approah signifiantly redues the number of numerial analysis required by a diret use of standard reliability tehniques, like in MonteCarlo simulations. ANALYSIS OF EXTERNAL RANDOMNESS The aim of the analysis is to build the seismi fragility urve of the single bridge, or, when onsidering also the hazard at the site, to ahieve the seismi risk. The fragility urve is obtained from the definition of the response surfae of the reliability index, β, as a funtion of external random variables y representing both strutural unertainty and seismi input. Assuming a quadrati form in y, the response surfae an be written as: ( ) = θ + θi yi + θij yi y j + δ ε β y 0 + (i,j=1,,k) (1) i i j where are present k+ random variables: k are required to represent the strutural properties of the system, whilst the extra two variables are required to represent the input unertainty (deriving from the fat that experiments are arried out in a not-homogeneous way due to different seletion of the aelerogram in non-linear dynami analysis) and the model error, onsidering all the other soures of randomness. The definition of the unknown parameters, θ, is ahieved through a simulation plan on y. The outomes of the simulations an be expressed in a ompat format as: β = Y θ + Bδ + ε () where β is the vetor of numerial tests, Y and B are matries of known values, while ε and δ represent the vetors of random variables due respetively to model error and input unertainty. In partiular Y is defined in aordane with the seleted simulation plan on strutural external variables y as: v( y 1 ) Y = M v( y n ) where n is the number of experiments and y) = [ 1 y L y y y y L y ] v is an explanatory ( 1 m 1 1 m funtion of the polynomial terms present in (1), whilst B distributes the soure of input unertainty (in our ase the seleted aelerograms) to the general plan of experiments. The error terms ε and δ are modelled assuming a spherial multivariate normal distribution: ( 0, σ I ), δ = N( 0, σ ) ε = N (3) ε ne δ I na In this ase the unknowns needed in order to define the response surfae are θ, σ ε, σ δ. The problem has been solved maximizing the likelihood funtional through an iterative proedure [5]: 1 ( β, C Y) = C exp[ ( β Yθ) C ( β Yθ) ] L (4) YY YY YY T where C YY = BB σ δ Iσ ε is the orrelation matrix. Numerial experiments have been planned assuming a Central Composite Design (CCD) sheme with a stellar two-level part (where the variables are hanged one at one time for a total of k+1 experiments,

3 where k is the number of random variables) and a omplete fatorial two-level part, with k ombinations [3]. Aiming not to reprodue the same seletion of strutural external variables for all the aelerograms onsidered, a bloking tehnique has been used, subdividing fatorial experiments in as many bloks as the number of onsidered aelerograms. In the appliation arried out, the seismi response of the bridges has been onsidered affeted by four external variables: three are strutural: the total mass, M, the steel yielding tension, f y, the onrete strength, f, while the fourth is a measure of the seismi input sine represent the spetral aeleration at the first natural period of the struture, S a, to whih the examined aelerograms have been saled. The expliit form of response surfae in this study is thus: β = θ + θ ln( f + θ ln( M ) + θ ln( f y 1 ) + θ ln( f + θ ln( S 8 a ) ln( M ) + θ ln( f ) + θ ln( M ) + θ ln( S ) + θ ln( f ) ln( f 13 y 9 y 3 ) ln( S a y ) + θ ) + θ a ln( f ln( M ) ln( S ) + θ ln( f ) 5 ) ln( M ) + θ a ) + δ + ε + θ ln( f 11 6 ln( f y ) ) ln( S + a ) + (5) where the logarithms of external variables are onsidered. The seismi input is represented by the eight aelerograms reported in Table 1 and Figure 1, randomly seleted from PEER database ( with the onstraint of onsidering intermediate soil (B type aording to EC8 lassifiation) with a magnitude in the range and a fous-tosite distane in the range 0-40 km. The seleted time histories have been subdivided in two groups of four reords eah, aording to the bloking sheme. Table 1. Aelerograms used in dynami analysis Id Earthquake Time Station Reg. M R [km] PGA [g] I Group 1 Chalfant Valley 1986/07/1 14:4 LakeCrowley - Shehorn R Cape Mendoino 199/04/5 18:06 Shelter Cove Airport Koaeli, Turkey 1999/08/17 Goynuk Loma Prieta 1989/10/18 00:05 Gilroy Array # II Group 5 Northridge 1994/01/17 1:31 LA - Chalon Rd Northridge 1994/01/17 1:31 LA - N Faring Rd San Fernando 1971/0/09 14:00 Castai - Old Ridge Route Friuli, Italy 1976/05/06 14:00 Tolmezzo Td [s] The experiment plan requires therefore repeating the 8 numerial analyses for eah group with a total of 56 runs. LIMIT STATES AND INTERNAL RANDOMNESS As antiipated, in this study the limit state formulation has been analysed through a funtion of the internal variables, x, where are present both the strutural demand and apaity. Aording to the reliability proedure developed, within the numerial run the demand is a deterministi funtion in time D(t) while the apaity an be represented as

4 C = C(t, x) ε (6) where C(t, x) is a funtion of time and internal variables representing the best estimate of strutural apaity, and ε is the apaity model error. In partiular ε has been assumed a log-normal random variable, Fig 1. Response Spetra of the eight aelerograms. with unitary mean and oeffiient of variation (.o.v.) to be seleted on the basis of the model auray. The limit states onsidered to represent the bridge pier ollapse are flexure and shear. In the flexure limit state, deformation apaity has been measured by the hord rotation θ through the expression: L L s p C y ( u - y ) 1-3 θ = χ + χ χ Lp (7) Ls where χ u, χ y are respetively ultimate and yielding urvature of the ritial setion, L s is the distane from the point of ounter-flexure and L p is the length of the plasti hinge. In our ase the plasti hinge length has been estimated, aording with [6], with the following formula: Lp = 0.08Ls + 0. f yd (8) where the seond term aounts for the strain penetration effets due to anhored longitudinal rebars (d is their diameter, f y their yielding strength in MPa). Equation (7) is based on a well-established model of onentrated plastiity. The hoie to express the limit state in terms of hord rotation rather than in terms of setion urvature is due to the aim to obtain a more stable result, not affeted by possible loalization of plasti deformations indued by the integration algorithm used by the omputational model. The omparison of the seleted model with the result of experimental test on flexural ollapse show a signifiant satter induing to selet a high error term ε θ expressed by a.o.v. equal to 0.5. Equation (7) an also be rewritten in a more general form in order to inlude the eventual yielding of the setion at the other side of the element, and therefore reads: 3 C y ( u - y ) 1-4 = L Lp θ χ + χ χ Lp (9) L

5 but in this ase the demand term has to be expressed as: Dθ = θi + θ j /, where θ i, θ j are respetively the hord rotations of the setion under study and the one at the other end of the strutural element of length L. For what onerns the shear limit state, the apaity model is the one presented in [7] with the modifiations suggested in [8]. For this limit state the demand is represented by the shear arried by the pier. Aordingly the apaity is expressed by the following expression: C { VC ( t) + VS VN ( t) } ε V = (10) V + again given as a the produt of a best estimate of apaity times an error term due to model auray. Shear apaity here is the sum of three main ontributions: A V C = 0.8A gk( t) f ; V S = f yd ot(30 ) s sw ; V tan ( ) N = N α t (11) representing respetively the effet due to the onrete in tension, the transversal reinforement and the presene of an arh ation ativated by the axial load. In equation (11) the meaning of variables is the following: A g is the shear effetive area, f is the onrete strength, k(t)=k(µ(t)) is a oeffiient aounting for the derease of onrete ontribution with dutility µ, A sw is the transversal reinforement area, s the stirrup distane, f y is the steel yielding strength, D is the net length of onrete in tension measured in the diretion of shear stress, N is the axial load and α(t) is the angle between the ompression strut and the axis of the element. Curvature dutility demand an be omputed as a funtion of yielding urvature χ y, end rotations of the strutural element, θ i, θ j, of element length L, distane of the ritial setion from point of ounterflexure and plasti hinge length, L p : θi + θ / µ = 1+ L ; (1) χ L L j * y p * p * 3Lp where Lp = Lp 1. 4Ls Aording on a limited number of experimental/theoretial omparison arried out by the shear model developers [8], the error term, ε V,.o.v. has been assumed equal to 0.3. Table. Internal random variables and log-normal parameters No Random Varaiable µ.o.v. 1 ε θ Model error for flexural Limit State ε V Model error for shear Limit State ε χy Yielding urvature ε χu Ultimate urvature ε Lp Plasti hinge length Aside the ε θ and ε V model errors, the ultimate and yielding urvatures of ritial setions χ u, χ y and the plasti hinge length, L p, have been treated as internal random variables in order to represent unertainties on the estimates of their value, espeially the plasti hinge length that has to be onsidered as a onventional rather than a physial quantity. In all the above ases randomness has been treated assuming the parameter as a best estimate value times a log-normal flutuation with a given.o.v. as reported in Table.

6 THE VALLONE DEL DUCA VIADUCT The viadut is along the highway A16 Napoli-Canosa, in the Southern Italy, between Benevento and Avellino. The bridge has been reently retrofitted restraining the dek gaps and introduing seismi isolators at pier aps, but has been studied in the as built onditions. The bridge has 6 spans, eah 3 m long with a strutural sheme of a simple supported beam. The dek is omposed by three pre-stressed beams 1.9 m high, onneted by 5 traversal link and a slab 0 m thik and 9.54 m wide. The r.. piers have a single stem with a retangular solid setion m and a total longitudinal reinforement of 40 rebars φ8. Figure. Vallone Del Dua viadut. Mehanial properties of materials have been obtained from the original design doumentation and from experimental test arried out when the retrofitted was planned. Conrete strength, on the basis of ore samplings at pier base, has been evaluate on average about f = 45 MPa and.o.v. equal to The reinforing steel is lassified Aq50, a mild steel type following old Italian regulation (Cirolare Min. LLPP N. 147 del 1957) with an ultimate strength not less than 500 MPa, an yielding strength not less than 70 MPa and a ultimate strain not less than 16%. Statistial studies on Aq50 grade steel permit to evaluate average strength and.o.v. both at yielding: f y = 370 MPa.o.v. = 0.08, and at ultimate f u = 545 MPa.o.v. = In the following the reliability results for three piers are illustrated (see Fig. ), namely n 3, 16 m height, n 4, 9.5 m height and n 5, 5.5 m height.

7 Pier 3 h=16 m P f 0.4 Pier 4 h=9.5 m 0. Pier 5 h=5.5 m S a (g) Fig. 3 Fragility urves of three pier of Vallone del Dua via- The final result, in terms of fragility urve, is illustrated in Fig. 3. The differenes an be explained sine in pier 5, the squattest, failure is reahed essentially in shear with a strong orrelation of f on failure probability, whilst in pier 3, the more slender, failure is reahed essentially in flexure. The omparison of fragility urves shows that most vulnerable pier is the tallest. Convolving the fragility of that pier with the seismi hazard at site we obtain a probability of ollapse evaluated in 100 years equal to.%. Fig. 4 The Olmeta Viadut

8 THE OLMETA VIADUCT The Olmeta viadut (see Fig.4) is along A1 highway between Bologna and Florene, in the Central Italy. It has been built during early 60, without seismi provisions. The dek has five pre-stressed beams with a span 1.16 m long and a slab 9.60 m wide exluding sidewalks about 70 m wide. The piers have a framed struture with five olumns with a setion m eah and a spaing of.40 m among them, linked together by a ap beam with a setion m. The pier height varies to a maximum of 1 m. The dek is simply supported by piers. Pier reinforement onsists of 1 rebars φ16 longitudinally and 7 1.8E-1 1 β north3 S.Fernando North8 M endoino Friuli Koaelii Chalfant Gilroy 9.9E E E P f Φ( β) Φ( β δ ε) Φ( β+δ+ε) Φ( β δ) Φ( β+δ) Sa (g) Fig. 5 Olmeta viadut, pier n. : (left) Reliability index β for different aelerograms vs. the spetral aeleration; (right) Fragility urve and sensitivity of error model and variability of response. Sa (g) φ10 stirrups every 16 m transversally. In 1974 the viadut has been retrofitted: slab has been rebuilt and pier olumns have been jaketed by a 10 m thik onrete layer reinfored longitudinally by 6 rebars φ0 and transversally by φ10 stirrups every 40 m, but the analysis has been arried out in the as built status. In the following the results are reported only for pier n (olumn 9 m high). Figure 5 (left) sows the variability of the reliability index evaluated in the entral point of the experiment design for the different aelerograms used in the analysis; the plots in figure 5 (right) show the fragility urve of pier n and the influene of model errors and of the spread of response evaluated at ± one standard deviation of r.v. ε and δ. THE CADRAMAZZO VIADUCT The Cadramazzo viadut is along highway A3 between Udine and Tarvisio, in North of Italy, and has been built after the Friuli earthquake aording to some seismi regulation developed at that time. The viadut has a ontinuous dek for a total of 15 spans eah of them 40 m long. The dek is onstituted by a prestressed multiellular setion stiffened by transversal beams at supports on piers. Piers have a irular hollow setion with a 3.8 m external diameter and different heights ranging from 3 to 4.9 m. Tall piers have an initially solid setion for a length of.5 m that beomes hollow at higher levels with a onrete thikness different for eah pier, but in the range m. Seismi reliability of the pier has been onsidered only in the transversal diretion (orthogonal to bridge axis). In the numerial model material inelastiity has been onsidered only in the piers, while ap beams

9 P f 6.E-0 5.E-0 4.E-0 3.E-0.E-0 1.E-0 0.E Sa [g] Fig. 6 Cadramazzo viadut: fragility of the bridge (thik line) and probability of failure at the entral point of experiment design (thin line) and dek have been modelled through elasti models. Supporting devies at the interfae between pier and dek have been modelled using neoprene elasti properties. Material properties have been assumed aording to values reported on original design doumentation as follows: onrete has an average strength f = 34.11Mpa and a.o.v. equal to 0.7, whilst steel has an average yielding strength f y = 430 Mpa and a.o.v. equal to Figure 7 shows fragility urve of the whole viadut (thik line) ompared to the one alulated on the basis of median value of random variables (thin line). The differenes between the two urves measures the relevane due to strutural random variables. CONCLUSIONS The risk assessment proedure developed for onrete strutures has been applied to three highway Italian viaduts, assumed to represent typial ases for diffusion of strutural sheme. Two are simply supported by single stem or frame piers, the third is a ontinuous bridge. The bridges have been modelled with a refined finite element analysis program [9] and different ollapse mehanism have been onsidered. The results obtained proof not only the effetiveness, but also the flexibility of the reliability tehnique implemented. Indeed omplex strutures have been analysed using realisti models for ations and apaity, inluding the assoiated unertainties that, as shown in Fig. 6, have a relevant effet on final value of fragility. Finally it has been possible to derive the sensitivity of failure risk upon single random variables, in order to plan a rational retrofit strategy. REFERENCES 1. Giannini R., Pinto P.E., de Felie G. Seismi reliability assessment of RC strutures. Submitted to Ase Jorunal of Strutural Engineering, de Felie G., Giannini R. A probabilisti approah for seismi assessment of r.. strutures: Theory. 13 th WCEE, Vanouver, Canada. Paper no. 600, Box G.E.P., Draper N.R. Empirial Model-Building and Response Surfaes, John Wiley & Sons, Franhin P., Lupoi A., Pinto P.E. Seismi fragility of reinfored onrete strutures using a response surfae approah. Journal of Earthquake Engineering, 003, 7: Searle S.R., Casella G., MCulloh C.E. Variane Components, Prentie-Hall, 199.

10 5. Priestley M.J.N., Seible F., Calvi G.M. Seismi Design and Retrofit of Bridges, John Wiley & Sons, Priestley M.J.N., Verma R., Xiao Y. Seismi shear strength of reinfored onrete olumns. ASCE Journal of Strutural Engineering, 1994, 10 (8): Kowalsky M.J., Priestley M.J.N. Improved analytial model for shear strength of irular reinfored onrete olumns in seismi regions. ACI Strutural Journal, 000, 97( 3): Petrangeli M., Pinto P.E., Ciampi V. Fibre Element for yli bending and shear of RC strutures. I: Theory, ASCE Journal of Strutural Engineering, 1999, 15(9):