A performance-based adaptive methodology for the ... - Springer Link

Bull Earthquake Eng (2011) 9:1463–1498 DOI 10.1007/s10518-011-9260-8 ORIGINAL RESEARCH PAPER

A performance-based adaptive methodology for the seismic evaluation of multi-span simply supported deck bridges Donatello Cardone · Giuseppe Perrone · Salvatore Sofia

Received: 18 December 2009 / Accepted: 22 March 2011 / Published online: 9 April 2011 © Springer Science+Business Media B.V. 2011

Abstract A performance-based adaptive methodology for the seismic assessment of highway bridges is proposed. The proposed methodology is based on an Inverse (I), Adaptive (A) application of the Capacity Spectrum Method (CSM), with the capacity curve of the bridge derived through a Displacement-based Adaptive Pushover (DAP) analysis. For this reason, the acronym IACSM is used to identify the proposed methodology. A number of Performance Levels (PLs), for which the seismic vulnerability and seismic risk of the bridge shall be evaluated, are identified. Each PL is associated to a number of Damage States (DSs) of the critical members of the bridge (piers, abutments, joints and bearing devices). The IACSM provides the earthquake intensity level (PGA) corresponding to the attainment of the selected DSs, using over-damped elastic response spectra as demand curves. The seismic vulnerability of the bridge is described by means of fragility curves, derived based on the PGA values associated to each DS. The seismic risk of the bridge is evaluated as convolution integral of the product between the fragility curves and the seismic hazard curve of the bridge site. In this paper, the key aspects and basic assumptions of the proposed methodology are presented first. The IACSM is then applied to nine existing simply supported deck bridges, characterized by different types of piers and bearing devices. Finally, the IACSM predictions are compared with the results of nonlinear response time-history analysis, carried out using a set of seven ground motions scaled to the expected PGA values. Keywords Bridges · Seismic evaluation · Damage states · Adaptive pushover analysis · Capacity spectrum method · Nonlinear response time-history analysis

1 Introduction Recent earthquakes have repeatedly demonstrated the seismic vulnerability of existing bridges, due to their design based on gravity loads only or inadequate levels of lateral forces

D. Cardone (B) · G. Perrone · S. Sofia DiSGG—University of Basilicata, Potenza, Italy e-mail: [email protected]

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(Priestley et al. 1996). The seismic zonation of many European countries, moreover, has been revised recently, prescribing more severe Peak Ground Accelerations (PGAs) in several regions. Reliable methods for the assessment of the vulnerability and seismic risk of existing bridges, to be used within a maintenance and management framework of a road network, are then needed. It is well known that the most accurate method of analysis for the evaluation of the seismic response of structures is the Nonlinear response Time-History Analysis (NTHA). This technique, however, requires the preliminary selection of an appropriate set of ground motions and considerable computational efforts with the consequent difficulty of producing ready-to-use results. Nonlinear Static Procedures (NSPs) based on pushover analysis represent nowadays a reliable alternative to NTHA and are widely recognized as a practical, yet accurate, engineering tool for the assessment of existing structures. The general purpose of NSPs is the evaluation of the so-called Performance Point (PP), defined, in seismic assessment terms, as the damage level of the structure due to a seismic event of predefined seismic intensity. From a graphical point of view, the PP can be derived by intersecting the capacity spectrum of the structure, derived from pushover analysis, with the seismic demand of the expected ground motions, represented by suitable response spectra, in the so-called ADRS (Acceleration Displacement Response Spectra) format. The most widely used NSPs for the seismic assessment of structures are: the Capacity Spectrum Method (CSM) (ATC 1996), the Displacement Coefficient Method (DCM) (ATC 2005) and the N2 method (Fajfar 1999; CEN 2005). All these methods are based on the conversion of the pushover curve of the multi-degree-of-freedom (MDOF) nonlinear model of the structure into an equivalent single-degree-of-freedom (SDOF) idealized (e.g. bilinear) capacity curve, based on the sole first mode of the structure. It should be noted here that the idea of representing a structure (such a multi-storey building or a multi-span bridge) by an equivalent SDOF oscillator, whose characteristics are defined from a nonlinear static analysis of the corresponding MDOF structure, was first explored by Saiidi and Sozen (1981), who extended previous works by Biggs (1964) and Clough and Penzien (1975) on equivalent SDOF systems. The currently used NSPs basically differ in the bilinearization technique adopted and in the description of the seismic demand by either high-damped elastic or isoductile inelastic response spectra. NSPs have been originally developed and applied to buildings. Only recently the applicability of NSPs to bridges has been comprehensively examined (Paraskeva et al. 2006; Casarotti et al. 2005; Aydinoglu and Önem 2007). Traditional pushover analysis techniques, based on monotonically increasing invariant distributions of lateral load patterns are suitable for structures whose seismic response is dominated by the first mode of vibration. In bridges, the effects of higher modes are generally not negligible and the modes of vibration of the structure can change significantly during strong earthquakes, due to damage of structural members (piers, bearing devices, abutments, etc.). Recent studies on the importance of higher mode effects and the accuracy of NSPs based on single-mode pushover analysis (such as the N2 method adopted in the European seismic code) have been carried out by Isaković et al. (2003, 2008), Isaković and Fischinger (2006), on a number of continuous single column bent viaducts. The results of these studies clearly point out that the higher modes significantly influence the response of relatively long viaducts, regardless the seismic intensity and the structural characteristics of the bridge (stiffness ratio between deck and bent, eccentricity between centre of mass and stiffness, ratio between torsional and translational stiffness, type of constraints at the abutments), thus making the use of NSPs based on single-mode pushover analysis inadequate.

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Different pushover analysis techniques, based on multimodal and/or adaptive approaches, are then needed, to account for the higher modes effects and the redistribution of inertia forces caused by structural damage and the associated changes in the vibration properties of the structure (Kappos and Paraskeva 2008). In the modal pushover analysis (MPA) proposed by Chopra and Goel (2002), a number of simultaneous pushover analyses are carried out separately for each significant mode. The contributions from individual modes to the seismic response of the structure are combined using appropriate combination rules (SRSS or CQC). Recently, Paraskeva and Kappos (2010) proposed some changes to MPA, based on the use of the so-called modal inelastic deformed shape of the structure, in lieu of the elastic modal shape, to adapt the use of MPA to the seismic assessment of bridges. Aydinoglu (2003), Aydinoglu and Önem (2007) proposed the so-called Incremental Response Spectrum Analysis (IRSA), wherein, each time a significant change of stiffness occurs in the structure, an elastic modal spectrum analysis is performed, taking into account the changes in the dynamic properties of the structure. First adaptive approaches have been presented by Reinhorn (1997) and Elnashai (2001). More recently, an innovative Displacement-based Adaptive Pushover (DAP) technique has been proposed by Antoniou and Pinho (2004a,b), in which a set of lateral displacements (rather than forces) is monotonically applied to the structure. The displacement pattern is updated at each step of the analysis, based on the current dynamic characteristics of the structure. Pinho et al. (2007) demonstrated that the use of the DAP technique can lead to the attainment of significantly improved predictions, compared to conventional (i.e. force-based) adaptive pushover procedures, which match very closely results from nonlinear response time-history analysis, even and especially for irregular bridges. An adaptive re-interpretation of the CSM has been also recently proposed by Casarotti and Pinho (2007) for bridge applications. The proposed method, called ACSM (Adaptive Capacity Spectrum Method), features two types of “adaptiveness”. First, the pushover algorithm. Indeed, the DAP analysis is adopted to better estimate the inelastic bridge behaviour, independently of structural regularity. The second element of adaptiveness of the ACSM resides in the way the pushover curve is converted in the equivalent SDOF idealized capacity curve. Indeed, all the equivalent SDOF quantities vary based on the current deformed shape of the bridge, in accordance with the principles of the Direct Displacement-based Design (DDBD) method (Priestley et al. 2007). In the traditional CSM, on the contrary, the conversion is carried out based on the sole elastic modal properties of the structure. In this paper, a performance-based adaptive methodology for the evaluation of the seismic vulnerability and seismic risk of highway bridges is proposed. In the current version, the proposed methodology is specialised to multi-span bridges, with either simple (isostatic) or continuous (hyperstatic) deck supported on bearings, although it could be extended to other bridge types simply by revising some modelling assumptions, as discussed below. In the first part of the paper, the background and implementation of the proposed methodology are presented. In the second part of the paper, the proposed methodology is applied to a set of nine simply supported deck bridges of the Italian A16 Napoli–Canosa highway, characterised by different types of piers and bearing devices. The effectiveness of the proposed methodology in reproducing the global behaviour and local phenomena of existing bridges is assessed by comparing the expected bridge response with the results of accurate nonlinear response time-history analyses.

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2 Description of the proposed methodology 2.1 Key aspects of the proposed methodology The proposed methodology is based on an Inverse (I) application of the ACSM. For this reason, the acronym IACSM is used to identify the proposed methodology. Contrary to ACSM, however, the IACSM is not iterative and does not require the bilinearization of the capacity curve of the equivalent SDOF model of the bridge. In the direct application of the ACSM, indeed, the main objective is to find the performance point (i.e. the damage state) of the structure under a predefined seismic intensity (i.e. a given PGA) using an over-damped demand spectrum derived by an iterative procedure on an idealized bi-linear capacity curve (ATC 1996). The main objective of the IACSM, instead, is to evaluate the seismic intensity (i.e. the PGA) of the expected ground motions, corresponding to pre-determined damage states of the structure, identified by given performance points on the capacity curve of the bridge. The inelastic deformed shape of the bridge corresponding to each damage state, therefore, is already known at the beginning of the analysis. As a consequence, the equivalent damping ratio of the bridge can be directly evaluated by properly combining (see Sect. 2.3.) the damping contributions of the single bridge components. In the proposed methodology, two Displacement Adaptive Pushover (DAP) analyses are performed to derive the capacity curves of the bridge in the longitudinal and transverse direction, respectively. The DAP technique has been preferred to the other conventional (i.e. force-based) adaptive pushover techniques, to better estimate the inelastic deformed shape of the bridge and the distribution of the shear forces between the structural elements. Compared to multiple-run pushover techniques (such as the MPA and IRSA methods), the single-run DAP technique results decidedly more suitable for an inverse application of the CSM. Moreover, it does not require the definition of any reference point. Basically, the DAP algorithm consists of four steps: (i) definition of a nominal displacement vector (Ui,0 , i = 1, …, n), (ii) computation of a load increment factor (λk ), (iii) calculation of a normalized modal scaling vector (Di,k , i = 1,…, n) and (iv) update of the displacement vector (Ui,k , i = 1, …, n), where Ui and Di represent the absolute displacements of the center of mass of the ith span of the bridge and n stands for the total number of spans. Whilst the first stage is carried out only once, at the start of the analysis, its three remaining counterparts are repeated at every (kth) step of the nonlinear static analysis procedure. A uniform deformed shape of the bridge has been assumed as nominal displacement vector. The normalized modal scaling vector Di,k , which defines the shape (not the magnitude) of the displacement increment vector, is computed at the start of every load increment as: Di,k =

Di,k max Di,k

(1)

In order for such modal scaling vector to reflect the actual stiffness state of the structure, as computed at the end of the previous load increment, an eigenvalue analysis is first carried out to determine the modal shapes (ij ) and modal participations factors (j ) of the structure. Modal results are combined using the SRSS combination rule: m 2 m 2 Di,k = j,k ij,k Sd (Tj,k ) Dij,k = (2) j=1

j=1

where Sd (Tj,k ) represents the displacement response spectrum ordinate corresponding to the period of vibration of the jth mode and m stands for the total number of modes.

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The displacement vector at a given step k of the DAP analysis is updated according to an incremental response control algorithm: Ui,k = Ui,k−1 + λk Di,k Ui,0

(3)

until a predefined analysis target or numerical failure is reached. The DAP algorithm has been implemented in SeismoStruct (SeismoSoft 2006), which can be freely downloaded from the Internet. Full details on this computer package can be found in its accompanying manual (SeismoSoft 2006). One of the key aspects of the proposed methodology is the definition of a number of Performance Levels (PLs), for which the vulnerability and seismic risk of the bridge shall be evaluated. Each PL is associated to a number of Damage States (DSs) of the critical members of the bridge, identified by a series of points on the DAP curve of the bridge. The IACSM provides the earthquake intensity levels (PGA) corresponding to the attainment of the selected DSs, using over-damped elastic response spectra as demand curves. The seismic vulnerability of the bridge is then described by means of fragility curves, derived based on the PGA values previously obtained for each DS. Finally, a seismic risk index is evaluated as convolution integral of the product between the fragility curves and the seismic hazard curve of the bridge site. As far as the bridge modelling is concerned, the so-called Structural Components Modelling (SCM) approach (Priestley et al. 1996) has been followed, in which a number of idealized structural subsystems are connected to resemble the general geometry of the bridge. The phenomenological response of each structural subsystem is provided in the form of member end force-deformation relationships. 2.2 Bridge modelling assumptions According to the SCM approach, the bridge structure can be divided in a number of independent rigid diaphragms, modelling the bridge decks, mutually connected by means of a series of nonlinear springs, modelling bearing devices, piers and abutments (see Fig. 1). The translational and rotational mass of each deck (equal to μ · L and μ · L3 /12, respectively, where L is the total length of the deck and μ the seismic mass per unit of length) are lumped in the centre of mass of each deck. If the total mass of the pier is more than 1/5 of the mass of the part of the deck carried by that pier, a tributary mass of the pier (equal to the sum of the mass of the pier cap and one third of the mass of the pier shaft) can be taken into account. Table 1 summarises the basic modelling assumptions adopted for each bridge component, to describe their monotonic and cyclic behaviour, within nonlinear static and nonlinear dynamic analysis, respectively. As can be seen in Table 1, decks and foundations have been considered infinitely rigid and resistant. The hypothesis of high seismic resistance of decks and foundations is related to their low seismic vulnerability, compared to piers, abutments and bearing devices. The hypothesis of high elastic stiffness, on the other hand, is substantiated by current seismic codes [e.g. see EC8-part 2 (CEN 2004)], which gives the possibility to assume a rigid deck model when the deformation of the deck in a horizontal plane is negligible compared to the displacements of the pier tops. This is always valid in the longitudinal direction of approximately straight bridges. In the transverse direction, the deck can be assumed as rigid, according to EC8, if L/B < 4 or in general if Dd /da < 0.2, where L and B are the length and the width of the deck, respectively, Dd and da are the maximum difference and the average of the displacements in the transverse direction of all the pier tops under the transverse seismic action. It is worth to note that the aforesaid conditions for the applicability of the rigid deck model are always

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(a) Pushing action

Pushing action Deck mass

Bearing

Pier mass

Bearing

Pier mass Pier

ag

Joint

Deck mass

Pulling action

Pier

L

(b)

μL

Bearing

Bearing Pulling action

Deck mass

Joint

Joint

μL

L

μL μ L2 /12

Joint

Joint

Bearing

Bearing

Bearing

Deck mass

ag

Pier mass

μL μ L2 /12

Joint

Bearing

Pier mass Pier

Pier

Fig. 1 Schematization of the bridge structure for seismic analysis in the a longitudinal and b transverse direction

F Joint Deck 1

D

Deck 2 Gap Pier

Bearing

Fig. 2 Modelling of joints

satisfied, with adequate margin, for the nine simply supported deck bridges examined in this study, which are representative of the Italian bridge inventory (ASPI 1992). Therefore, it can be concluded that the hypothesis of infinitely rigid deck is, in most of the cases, appropriate for simply supported deck bridges. In first approximation, also the hypothesis of infinitely rigid foundation is reasonable in many cases. Again, this is substantiated by current seismic codes [see EC8-part2 (CEN 2004)], which state that, generally, pier and abutments shall be assumed as fixed to the foundation soil. Soil-structure interaction effects should be used only when the displacement due to soil flexibility is greater than 30% of the total displacement at the center of mass of the deck. The hypothesis of infinitely rigid and resistant foundation may be removed in the next versions of the proposed methodology, by introducing appropriately defined soil springs. Possible effects due to the closure of the joints have been taken into account in the analyses. Indeed, gap closure results in pounding between adjacent decks and/or between deck and abutment, which can determine a significant redistribution of the seismic forces between piers and abutments. Joints have been modelled with compression-only translational (longitudinal direction) and rotational (transverse direction) link elements, with an initial gap assigned based on the clearance of the joint (see Fig. 2). In the application of the proposed procedure (see Sect. 3), a gap of 20–50 mm has been considered, based on the actual width of the expansion joints of the bridges examined. Seat-type abutments founded on piles have been considered, since they represent the abutment type most widely used in Italy. The seismic response of the abutments in the longitudinal

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Infinitely rigid and resistant Lumped translational and rotational mass

Compression-only translational (long. dir.) and rotational (transv. dir.) springs with gap

Transv. dir.: infinitely rigid and resistant

Deck

Joints

Abutments

Bearings

Piers

Infinitely rigid and resistant

Foundations

Post-failure pier-deck sliding considered

Flexural behaviour derived from moment-curvature analysis Shear strength and P- effects considered

Beam with plastic hinges at the end(s)

Long. dir.: effects of piles-ground and backfill-abutment interaction considered

Modelling assumptions

Component

Viscous, hysteretic or frictional cyclic behaviour, depending on bearing type and displacement amplitude

Takeda degrading-stiffness hysteretic model

Long. dir.: kinematic multilinear plastic model

Rigid behaviour after gap closure

Elastic-plastic with hardening backbone curve

Multilinear backbone curve

–

Cyclic behaviour (NTHA)

Diaphragm behaviour

Long. dir.: elastic-perfectly-plastic backbone curve

Monotonic behaviour (NSA)

Table 1 Basic modelling assumptions considered in the nonlinear static analysis (NSA) and nonlinear time-history analysis (NTHA) of bridges

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F

Deck Abutment

Back Fill

Back Wall

Bearing

Piles

Pushing action

D

Pulling action

Fig. 3 Mechanical behaviour of abutment under pushing and pulling action

direction has been captured with a couple of nonlinear springs, characterised by two different elastic-perfectly-plastic backbone curves (see Fig. 3), modelling the pushing and pulling action of the abutment, respectively. The longitudinal response of the abutment is based on the interaction between bearing devices, joint gap, abutment back wall, abutment piles and soil backfill material. Prior to gap closure, the deck force is transmitted through the bearing devices to the abutment wall and subsequently to piles and backfill, in a series system. After gap closure, the deck pushes directly on the abutment back wall, with the risk of mobilizing the passive backfill pressure. In this study, the horizontal stiffness and ultimate strength of the abutment in the pushing direction have been derived from a combination of design recommendations (Caltrans 2006) and experimental test results on seat-type abutments with piles (Maroney et al. 1993). They are expressed as a function of the abutment back wall dimensions (w: width, hw : height) as follows: hw Kpush = ki · w · (4) 1.7 hw (5) Rpush = hw · w · pi · 1.7 in which ki and pi are two coefficients assumed equal to 11.5 (KN/ mm)/m and 239 kPa, respectively, according to the Caltrans (2006) recommendations. The horizontal stiffness of the abutment-pile system in the pulling direction has been assumed equal to 7·np KN/mm, np being the number of piles, in accordance with (DesRoches et al. 2003). The horizontal strength of the abutment-pile system in the pulling direction has been taken equal to the ultimate resistance of the pile group. Piers have been modelled with nonlinear springs characterised by a bilinear backbone curve. In this study, the lateral force-displacement relationships of the piers have been derived based on preliminary elasto-plastic pushover analyses of the piers, schematized as elastic beams with plastic hinges at the end(s). In the pushover analysis, the top displacement of each pier is progressively increased, tracking the formation of plastic hinges up to their ultimate rotation capacity or the attainment of the pier shear strength. P − effects due to gravity loads are considered in the analysis. First, a moment-curvature analysis of the critical cross sections of the pier is performed, considering the axial force due to gravity loads and the effects of concrete confinement and steel strain-hardening. In this study, reference to the model by Mander et al. (1988) and Menegotto and Pinto (1973) has been made for confined/cover concrete and steel, respectively. The moment-curvature relationship thus obtained is properly bilinearized (see Fig. 4a) and then converted in the moment-rotation behaviour of

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(a)

1471

(b)

M

(c)

F

F

rK

Fy

Mu My

High shear resistance

Kp2

K

D

Flexural behaviour with P-Δ effects

Mr

Kp3

Buckling and lap-splice effects

Φ Φy

Kp1 Low shear resistance

D

K μ

Φu

Fig. 4 Pier modelling: a moment-curvature analysis of the critical section, b force-displacement behaviour taking into account possible shear failure, c Takeda degrading-stiffness-hysteretic model

the plastic hinge, whose length (Lp ) has been evaluated according to the formula by Priestley et al. (1996): Lp = 0.08L + 0.022fyh dbl ≥ 0.044fyh dbl

(6)

where L is the distance between the centre of the plastic hinge and the first counterflexure point in the elastic deformed shape of the pier, fyh and dbl are the yielding stress and the minimum diameter of the steel longitudinal reinforcement, respectively. In this phase, possible premature failure due to lap splice effects or buckling of longitudinal bars can be considered (see Fig. 4a) through simple semi-empirical relationships (Priestley et al. 1996). After that, the shear resistance of the pier (Vd ) is computed, based on the relationship proposed by Priestley et al. (2007): Vd = Vc + Vs + Vp

(7)

in which Vc is the shear resistance of concrete, Vs the contribution of the transverse reinforcement and Vp is the shear absorbed by the axial load. More details on the three shear components (Vc , Vs and Vp ) can be found in Priestley et al. (2007). Finally, the shear resistance, expressed as a function of the pier top displacement, is compared to the flexural behavior of the pier derived from elasto-plastic pushover analysis, in order to determine the actual flexural/shear behavior of the pier (see Fig. 4b). As far as the cyclic behaviour of the piers is concerned, reference to the Takeda degradingstiffness-hysteretic model has been made (see Fig. 4c). The Takeda degrading-stiffness-hysteretic model is very similar to the multi-linear Takeda model (Takeda et al. 1970) but has additional parameters to control the degrading hysteretic loop. It is particularly well suited for reinforced concrete members and it is based on the observation that unloading and reverse loading tend to be directed toward specific points, called pivot points, in the force-deformation plane. The Takeda degrading-stiffness-hysteretic model does take into account the strength degradation due to P-delta effects (not shown in Fig. 4c), while it neglects the strength degradation due to cyclic effects near collapse. The Takeda degrading-stiffness-hysteretic model is fully de scribed in (Dowell et al. 1998). The nonlinear behaviour of the bearing devices is defined according to the bearing type under consideration (see Fig. 5). If necessary, a post-failure frictional behaviour, corresponding to sliding between deck and pier cap, is considered. Based on a comprehensive survey of the Italian highway bridge stock (ASPI 1992), which comprises many bridges realised between 1960s and 1970s, five different types of bearing

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F

F

Fu Ffr

Ffr

D

(a)

(b)

F

F

Ffr K,ξ

D

(c)

D

D

(d)

Fig. 5 Force-displacement behaviour of different types of bearing devices: a steel hinges and dowel steel bars, b sliding bearings, c steel Pendulum and roller bearings and d neoprene pads

devices have been identified, i.e.: (i) steel hinges, (ii) dowel steel bars, (iii) sliding bearings, (iv) steel pendulum and roller bearings, (v) neoprene pads. Steel hinges and dowel steel bars are assumed to remain linear elastic up to failure (Fig. 5a), which is usually brittle, being due to the attainment of the shear strength (Fu ) of the device. The shear stiffness has been estimated based on the geometric details available. During the analysis, the maximum shear force has been monitored. When the shear strength was prematurely exceeded, a post-failure frictional behaviour, corresponding to sliding between deck and pier cap, has been considered (Fig. 5a). Reference to a Coulomb (rigid-perfectly-plastic) model has been made to describe the frictional behaviour of sliding bearings, steel pendulum and roller bearings (Fig. 5b, c). As known, the friction coefficient (μfr ) depends on many parameters and factors, such as: bearing type, sliding interfaces, state of lubrication and maintenance, bearing pressure, air temperature, sliding velocity, etc. (Dolce et al. 2005). As a consequence, a proper choice of the friction coefficient shall be made, based on the data available (design specifications, code requirements, manufacturer datasheets, common practice, etc.). In the application of the proposed procedure to the selected case studies (see Sect. 3), a friction coefficient of 10% has been assumed for sliding and roller bearings (Fig. 5b), while it has been neglected for steel pendulum bearings (Fig. 5c). Moreover, based on the geometric characteristics of the examined bearing devices, the maximum displacement capacity of sliding/roller bearings and steel pendulum bearings has been assumed equal to 150 and 100 mm, respectively. A linear visco-elastic behaviour has been considered for neoprene pads (Fig. 5d), whose shear stiffness (K) is evaluated based on the dimensions (cross section area and thickness) of the pads and shear modulus (G) of neoprene. In this study, a shear modulus of 1 MPa and a viscous damping ratio (ξ ) of 6% have been assumed as typical values of neoprene pads and then adopted in the examples of application presented in Sect. 3. The horizontal strength of the bearing system has been computed as the lowest between the shear resistance of neoprene

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Pushover analysis of the bridge

I. Define a number of Damage States (DSs) II. Convert the Pushover Curve in the Adaptive Capacity Spectrum

II.a Actual deformed shape of the bridge at each (k-th) step of analysis III.a Equivalent Damping ξPP III.b Reduction Factor η(ξPP)

III. Get the Demand Spectrum (for each DS) IV.a Effective Period T PP

IV. Evaluate PGA (for each DS)

IV.b Target Acceleration Sa,PP IV.c Spectral Acceleration Sa1(TPP)

V. Generate Fragility Curves (for each DS)

V.a Lognormal Cumulative Probability Function

VI. Evaluate Seismic Risk (for each DS) VI.a Seismic Hazard Curve

Fig. 6 Flowchart of the proposed methodology for the evaluation of the vulnerability and seismic risk of bridge structures

pads and the friction resistance between neoprene and concrete sliding surfaces. In the case studies examined in Sect. 3, the shear resistance of neoprene pads has been associated to the attainment of a shear strain of 150%. The friction coefficient between neoprene and concrete has been taken equal to 70%. 2.3 Evaluation of bridge vulnerability and seismic risk The first step of the proposed procedure (see Fig. 6) is to define a number of performance levels, for which the vulnerability and seismic risk of the bridge shall be evaluated. Each PL is associated to a number of Damage States (DSs) of the critical members of the bridge (piers, abutments and bearing devices), identified by a series of Performance Point (PP) on the DAP curve of the bridge. Herein, the PLs are divided in three groups (see Table 2), based on the consequences in terms of damage that the attainment of each PL can produce. Obviously, this division is only formal and it is made only for more clarity. Any point of the DAP curve of the bridge can be arbitrarily selected at the beginning of the analysis. The first group (PL1) includes post-earthquake Damage States (D1j in Table 2) in which only very limited structural damage has occurred and although some minor structural repairs may be appropriate, these generally do not require any traffic interruptions. Examples of DSs of the first group are: (i) pier yielding, (ii) attainment of the horizontal strength in neoprene pads (iii) closure of the joints, etc. The second group (PL2) includes post-earthquake Damage States (D2j in Table 2) in which significant damage to some structural elements has occurred but large margin against either partial or global collapse still remains. The overall risk of life-threatening injury as a result of structural damage is expected to be low. Whilst the damaged structure is not an imminent collapse risk, it would be prudent to implement shortly structural repairs. This may require traffic interruptions or the installation of temporary bracing systems. Examples of DSs of the second group are: (i) attainment of the ultimate displacement capacity of sliding/roller

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Table 2 Selected performance levels (PLs) and corresponding damage states (DSs) PL1 (slight damage state)

PL2 (moderate damage state)

PL3 (severe damage state)

Joint closure (DS11)

Pier 50% ultimate ductility (DS21)

Failure of fixed hinges (DS31)

Pier yielding (DS12)

Displacement capacity of sliding/roller bearings (DS22) Active resistance of abutment (DS23)

Pier ultimate ductility (DS32)

Maximum force of neoprene pads (DS13)

Pier shear failure (DS33) Passive resistance of abutment (DS34) Deck unseating (DS35)

bearings, which is generally associated to the occurrence of large residual displacements of the deck, (ii) significant ductility demands (e.g. 50% of the ultimate ductility) in piers, (iii) attainment of the active resistance of abutments (pulling action), etc. The third group (PL3) includes post-earthquake Damage States (D3j in Table 2) in which the structure continues to support gravity loads but retains no margin against collapse. Aftershock activity could induce the partial or total collapse of the bridge. Extensive damage to the structure has occurred, that implies significant degradation in the stiffness and strength of the lateral-force resisting system and large permanent lateral deformations of the structure. Significant risk of life-threatening injury exists. The structure may be technically repairable but costs could be very high and the closure of the bridge for long time is inevitable. Examples of DSs of the third group are: (i) collapse of steel hinges or dowel bar connections, (ii) pier collapse due to the attainment of its ultimate ductility or shear strength, (iii) attainment of the passive resistance of abutments (pushing action), (iv) incipient unseating of the deck, etc. The second step of the procedure is to convert the pushover curve of the nonlinear MDOF model of the bridge into an equivalent SDOF adaptive capacity spectrum. The latter is stepby-step derived by calculating the displacement (Sd,k ) and the acceleration (Sa,k ) of the equivalent SDOF system based on the actual deformed shape of the bridge at each (k-th) step of analysis, according to the fundamental equations of the DDBD method (Priestley et al. 2007) particularized to the bridge model considered:

2 + I · δ2 m · D j j j j,k j,k

Sd,k = (8) m · D j,k j j Sa,k =

Vb,k Me,k · g

(9)

where Vb,k is the base shear of the bridge, mj and Dj,k are the translational mass (μ · L) and the horizontal displacement of the centre of mass of the jth deck, respectively, Ij and δj,k are the rotational mass (μ · L 3 /12) and the rotation around the vertical axis of the jth deck, respectively, and Me,k is the effective mass of the entire bridge:

j mj · Dj,k (10) Me,k = Sd,k The third step of the procedure is to determine the seismic demand associated to each DS, represented by a reference over-damped elastic response spectrum, whose seismic intensity (PGAPP ) is still unknown at this step of the analysis. This step requires the evaluation of the

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equivalent viscous damping of the bridge associated to each DS. To this end, the following routine is followed: (i) find out the PP of the pushover curve corresponding to the selected DS, (ii) enter the pushover database to determine the corresponding deformed shape of the bridge and the actual displacement of each structural member, (iii) evaluate the equivalent damping of each structural member, (iv) combine the damping contributions of each structural member to get the equivalent viscous damping of the entire bridge. The equivalent damping of the bearing devices can be calculated through the well-known Jacobsen approach (Jacobsen 1930): ξb,j (PP) =

Evisc + Ehyst + Efr 2π · Fb,j · db,j

(11)

in which Evisc , Ehyst and Efr identify the energy dissipated by the device, through its viscous, hysteretic or frictional behaviour, in a cycle of amplitude db,j , being db,j the displacement of the device at the selected DS and Fb,j the corresponding force level. As far as piers are concerned, reference has been made to the following relationship: 1 (1 − r ) ξ p, j (PP) = 1− √ − rμ (12) π μ which relates the equivalent hysteretic damping of the pier to its displacement ductility (μ) and post-yield hardening ratio (r). The aforesaid relationship has been derived by Kowalsky et al. (1995), by applying the Jacobsen’s approach to the Takeda degrading-stiffness-hysteretic model. The equivalent damping of each pier-bearings system is then computed, by combining the damping values of pier and bearing devices in proportion to their individual displacements: ξj (PP) =

ξb,j · db,j + ξp,j · dp,j db,j + dp,j

(13)

Finally, the equivalent damping ratios of the pier-bearings systems are combined to provide the equivalent damping ratio of the bridge as a whole, for the selected PP. The approach followed is to weigh the damping values of the single pier-bearings systems in proportion to the corresponding force levels (Fj = Fb,j = Fp,j neglecting pier mass):

n

n j=1 ξj · Fj j=1 ξj · Fj ξPP = n = (14) Vb j=1 Fj Once the equivalent damping of the bridge has been determined, the corresponding demand spectrum can be derived from the reference 5%-damping normalized response spectrum, using a proper damping reduction factor (Cardone et al. 2008). As already suggested in previous studies on the seismic design of bridges (Kowalsky et al. 1995; Kowalsky 2002), in the proposed procedure the damping reduction factor adopted in an old version of the Eurocode 8 (CEN 1998):

7 η= (15) (2 + ξ P P ) has been employed, with the limitation η > 0.50, corresponding to a limit damping ratio of approximately 25%. The fourth step of the procedure is to determine the PGA values associated to each DS. From a graphical point of view, this can be done by a translation of the over-damped Normalised Response Spectrum (NRS in Fig. 7a) to intercept the Adaptive Capacity Spectrum

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(a) Sa (g) Sa1 (TPP,ξPP)

P(D≥DS|PGA)

High-damping NRS

1

ACS

Sa,PP PGAPP

(b)

5%-damping NRS

0.5

PP

PP

Demand spectrum Sd,PP

PGAPP

Sd

PGA

Fig. 7 a Evaluation of the PGA associated to a selected performance point (PP) and b derivation of the corresponding fragility curve

(ACS in Fig. 7a) in the Performance Point (PP). From an analytical point of view, the PGA associated to a given PP can be determined as the ratio between the acceleration of the ACS corresponding to that PP (Sa,PP in Fig. 7a) and the normalized spectral acceleration at the effective period of vibration (TPP ) and global equivalent damping (ξPP ) associated to the selected PP (Sa1 (TPP , ξPP ) in Fig. 7a): PGAPP =

Sa,PP Sa1 (TPP , ξPP )

(16)

being: TPP = 2π ·

M∗ = 2π · K∗

Sd,PP g · Sa,PP

(17)

The PGA values thus obtained represent an estimate of the median threshold value of the peak ground acceleration related to the selected PP. They can be used to derive a number of fragility curves (see Fig. 8b), which provide the probability of exceedance of the selected DS, as a function of the PGA of the expected ground motions. In the proposed procedure, the fragility curves are expressed by a lognormal cumulative probability function: 1 PGA P(D ≥ DS |PGA ) = (18) ln βc PGAPP in which P(•) is the probability of the Damage (D) being equal to or exceeding the selected DS, for a given seismic intensity (PGA), is the standard lognormal cumulative probability function, PGAPP the median threshold value of the seismic intensity associated to the selected PP and βc the lognormal standard deviation which takes into account the uncertainties related to input ground motion, bridge response, material characteristics, damage state definition, etc. According to previous studies (Dutta and Mander 1998; Basöz and Mander 1999; Kappos and Paraskeva 2008; Paraskeva and Kappos 2010), a value of βc equal to 0.6 can be assumed for existing RC bridges. Lower values of βc can be adopted for new bridges or, generally speaking, when the knowledge of the structural characteristics is accurate. The final step of the proposed procedure is the evaluation of the seismic risk associated to the selected DS, with the use of hazard maps, which provide the PGA values at the bridge site having a given probability of exceedance (e.g. 10%) in a given interval of time (e.g. 50 years). The seismic risk is obtained as convolution integral of the product between the

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(a)

(b)

Sa (g)

Vulnerability (V) 1 PP2

PGA3

PP1

PP3

PP2 PP3

0.5

PGA2 PGA1

PP1

PGA1 PGA2 PGA3

(c)

(d)

Hazard (P)

PGA

Ranking R>R*

PxV

10%

PGA PGA10%/50y

R 0.5 (see Fig. 19).

4 Conclusions A new methodology for the seismic evaluation of multi-span simply supported bridges has been proposed. The proposed methodology is based on an Inverse Adaptive application of the Capacity Spectrum Method. For this reason the acronym IACSM is used to identify the proposed methodology. The IACSM provides the PGA values associated to predefined Performance Points of the structure corresponding to different Damage States of the critical members of the bridge (piers, abutments, bearing devices, joints). Based on these PGA

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FIUMARELLA BRIDGE – A16 NAPOLI-CANOSA (km 95+0.93) D1

D2

J1

D3

J2

D4

J3

P2

d p (mm)

0 -25

P3

4000

4000

-50

8000

Fp (kN)

Fp (kN)

Fp (kN)

8000

SF

P3

P2

P1

P1

SH A2

A1

0

25

-4000

50

4000 d p (mm)

0 -50

-25

0

25

-4000

NTHA

8000

50

-50

-25

0

-4000

NTHA

IACSM

25

50 NTHA IACSM

IACSM

-8000

d p (mm)

0

-8000

-8000

Fig. 18 Seismic response of the Fiumarella bridge in the transverse direction at 0.60g PGA (DS32). Comparison between IACSM predictions and NTHA results (accelerogram n. 3) in terms of cyclic behaviour of piers CASTELLO BRIDGE – A16 NAPOLI-CANOSA (km 582+931)

A1

D1 SB

A2

D2

J1 NP

NP P1

NP/SB

SB

SW

200 PGA=0.65g

100 50

Ddeck (mm)

150

IACSM NTHA η=η(ξ) NTHA η=η(ξ)≥ 0.55

PGA=0.49g

0

D1

D2

Fig. 19 Seismic response of the Castello bridge in the transverse direction (DS22). Effects of the damping reduction factor on the accuracy of the results

values, a number of fragility curves are derived to describe the seismic vulnerability of the bridge from a probabilistic point of view. The seismic risk is then computed as convolution integral of the product between the seismic vulnerability of the bridge (expressed by the fragility curves associated to selected Damage States) and the seismic hazard of the bridge site (expressed by a suitable seismic hazard curve). The IACSM has been applied to a set of nine multi-span simply supported deck bridges of the Italian A16 Napoli–Canosa highway, differing in pier types, pier layout and pier-deck connections. The predictions of the IACSM have been compared with the results of Nonlinear response Time-History Analyses (NTHA), carried out using a set of seven accelerograms,

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compatible with the EC8-soilB response spectrum scaled to the PGA values provided by IACSM for selected Damage States. The comparison has been made in terms of maximum deformed shapes of the deck, joint displacements and top pier displacements. The comparison between IACSM predictions and NTHA results confirms the good accuracy of the proposed methodology in predicting the PGA values associated to slight-to-severe Damage States. In all the examples of application considered, indeed, the IACSM correctly identified the critical element of the bridge, where first a given damage state was reached. The displacement profile of the bridge predicted by the IACSM (including joint displacements, top pier displacements, bearing device displacements and deck rotations) is in good agreement with the maximum deformed shape of the bridge found with NTHA. Although the proposed methodology appears very promising, there are a number of aspects that require further investigation. In particular, the number of bridges considered was too small to examine the effectiveness of the method to address directional effects and soil-structure interaction effects. Curved bridges and skewed bridges, moreover, have not yet been studied. A related area of research will involve an investigation as to when the assumption of a rigid superstructure is valid, as such an assumption greatly simplifies the process for evaluating the inelastic displacement pattern of the bridge. Future research should focus also on the influence of different modeling approaches, more refined than that assumed in the current version of the method. Acknowledgments This work has been carried out within the S.A.G.G.I. research project, funded by the Italian Ministry for the University and the Research (MUR) and led by Autostrade per l’Italia S.p.A.

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