Probabilistic Seismic Assessment of RC Bridges: Part

ISTRUC-00050; No of Pages 10 Structures xxx (2015) xxx–xxx

Contents lists available at ScienceDirect

Structures journal homepage: http://www.elsevier.com/locate/structures

Probabilistic Seismic Assessment of RC Bridges: Part II — Nonlinear Demand Prediction Ricardo Monteiro a,b,⁎, Raimundo Delgado b, Rui Pinho c a b c

Institute for Advanced Study (IUSS) of Pavia, Italy University of Porto, Department of Civil Engineering, Porto, Portugal University of Pavia, Department of Civil Engineering and Architecture, Pavia, Italy

a r t i c l e

i n f o

Article history: Received 18 March 2015 Received in revised form 10 July 2015 Accepted 1 August 2015 Available online xxxx Keywords: Seismic assessment RC bridges Demand prediction Nonlinear static procedures Failure probability

a b s t r a c t One of the most determinant components of any seismic safety assessment procedure is the estimate of the nonlinear seismic demand, particularly if such procedure is probabilistic rather than deterministic. Indeed, the prediction of the seismic demand of RC structures is dictated by a number of variables, related to the seismic action or to the geometrical and material properties, with different relevancy and uncertainty levels. From a macro-viewpoint one can distinguish the different approaches for nonlinear demand prediction in two major groups: nonlinear static and dynamic. Whereas the latter is widely recognised as the most accurate approach, the former, particularly by means of nonlinear static procedures (NSPs), represents a valid alternative tool for performance-based seismic assessment of structures, quite accepted among the scientific community worldwide. Significant effort has therefore been made, over the past few years, to improve those simplified methods and several different proposals in terms of application methodology have come up. Simultaneously, validation studies have thoroughly been carried out, even though focusing mainly on regular, building structures, rather than typically irregular bridges, and have focused the validation through the direct comparison of response prediction. The capability of such procedures within a probabilistic framework that yields the collapse probability of randomly sampled RC viaducts is herein investigated. A commonly employed nonlinear static procedure (N2) is further validated against nonlinear dynamic analysis using two distinct probabilistic procedures, described and tested in a companion paper, which account differently for the uncertainty of the concerned variables (geometry, material properties, earthquake records, intensity level). A case study of seven bridge configurations, with different (ir)regularity levels, is considered together with a relatively large set of real earthquake records and random simulation is carried out using the Latin Hypercube Sampling scheme. The conclusions have further probed the suitability of nonlinear static analysis in estimating seismic demand by leading, under probabilistic conditions, to slightly conservative probabilities of collapse of RC bridges with respect to nonlinear dynamic analysis. © 2015 The Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.

1. Introduction Estimating the seismic demand in a structure subjected to earthquake loading is one of the key issues within any structural safety assessment endeavour. Even if naturally depending on the structural characteristics and on the seismic input itself, the main differences tend to be associated to the nature of the employed approach, static or dynamic. Each of those foresees multiple possibilities, in terms of concept, technique and procedure. It is commonly recognised that nonlinear dynamic analysis (NDA), which reproduces directly the real phenomenon, will produce more accurate results [14,40]. Nevertheless, several drawbacks may be pointed out when considering dynamic analysis, particularly related to the employment of ground motion records, to the structural modelling (including damping issues, post-elastic behaviour or corresponding energy dissipation during loading and unloading periods) among others. In addition, attention must also be ⁎ Corresponding author at: IUSS Pavia, Palazzo del Broletto, Piazza della Vittoria 15, 27100 Pavia, Italy.

paid to computational onus, which is well known to be significant in dynamic analysis. The commonly employed alternative to NDA is nonlinear static analysis, based on pushover analysis, which typically has the goal of somewhat enveloping all, or at least a considerable part, the possible dynamic analysis results at each intensity level [27,1]. This kind of outcome is accomplished through the application of an increasing lateral force or displacement vector to the structure. Even if inherently limited with respect to its dynamic counterpart, nonlinear static analysis constitutes nonetheless a very valuable tool within the framework of performance-based seismic assessment of structures [20,25] since it readily provides important insight into the main characteristics of the structural response when subjected to horizontal excitation. Consequently, in the past couple of decades considerable effort has been made in the development of credible, yet relatively simple, pushover-based methodologies, leading to the introduction of the so-called nonlinear static procedures (NSPs) in a number of guideline documents, such as ATC-40 [2,4] and FEMA-273 [3], or design codes, such as Eurocode 8 [8,9]. The main drawback of the pushover-based approaches is indeed the level of simplification, given that one assumes that the structural

http://dx.doi.org/10.1016/j.istruc.2015.08.001 2352-0124/© 2015 The Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Monteiro R, et al, Probabilistic Seismic Assessment of RC Bridges: Part II — Nonlinear Demand Prediction, Structures (2015), http://dx.doi.org/10.1016/j.istruc.2015.08.001

2

R. Monteiro et al. / Structures xxx (2015) xxx–xxx

behaviour obtained from horizontal pseudo-static loading is equivalent to its dynamic counterpart. Although such approaches have focused mainly on the application to seismic assessment of buildings rather than bridges, recent studies, listed in the subsequent section, have been made to explicitly verify the application of NSPs to individual and populations of bridge structures. However, these studies have been directed mainly at the direct validation of the nonlinear static procedures in predicting the seismic demand whereas the focus herein is the assessment of the performance of such procedures within simplified probabilistic safety assessment approaches. A companion paper has fully described and applied two probabilistic procedures for numerical seismic safety assessment of bridges based on different uncertainty models employing nonlinear dynamic analysis. On the other hand, the present paper makes use of such validated simplified approaches and addresses the performance of a commonly employed nonlinear static procedure within such probabilistic framework by direct comparison with nonlinear dynamic results. The study also includes the investigation of the use of different static analysis versions corresponding to different types of pushover load distributions. The parametric comparison of the static and dynamic approaches is then carried out. 2. Nonlinear static seismic demand Nonlinear static procedures are nowadays extensively spread within the earthquake engineering community mainly due to its simplicity and potentially expedite application when assessing a large number of structures [7,42]. Although these methods were initially thought having the application to buildings in mind, their extension to bridge structures has been further and further investigated and can become particularly useful in e.g. road network analyses [43]. A number of very recent endeavours have dealt with the application of NSPs to bridges (e.g. [12, 22,26,32,34]). The aforementioned methods are essentially based on the computation of the pushover curve, a representation of the nonlinear forcedeformation behaviour of the structure. With the use of appropriate transformation relationships the capacity curve of the single-degreeof-freedom (SDOF) system equivalent to the original multi-degree-offreedom (MDOF) structure may then be obtained. At this point the main differences between the proposed methods arise, such as the contemplation or not and choice for a reference node; the way of reducing the demand spectrum so as to account for the hysteretic energy dissipation or the consideration of higher modes of vibration, on the way to determine the displacement demand for a given ground motion. There are quite a few available procedures to carry out nonlinear static analysis. These range from truly simplified methods, based on coefficients, to more complex, including nonlinear and modal effects. A first group of proposed NSPs corresponds to the earliest attempts to propose a static method that would quickly lead to accurate results. Such pioneering procedures are the Capacity Spectrum Method (CSM), introduced by Freeman [23] and implemented in ATC-40 guidelines, and the N2 method [21] included in the recommended simplified procedures of the European Design Code [8]. These first proposals are appealing mainly for the simplicity of the method and usually consider either the first mode proportional or uniform load distribution for the pushover curve computation. Recently, an improved version of the CSM has been presented in FEMA-440 guidelines [4] and mainly consists of the update of the prescribed empirical relations to determine both the equivalent viscous damping and spectral reduction factor. Also the N2 method has witnessed improved versions better suited to irregular 3D buildings. Later, the Displacement Coefficient Method (DCM) [3] has been proposed within the NEHRP Guidelines for the Seismic Rehabilitation of Buildings [3]. Whereas ATC-40 referred only to concrete structures DCM is applicable to a wide range of structural systems and computes the elastic response, which is then tuned by a series of coefficients that account for the nonlinear behaviour. DCM assumes therefore a certain empirical character, which distinguishes it from the previous ones.

An additional group of procedures includes more recent approaches to nonlinear static analysis, such as the Modal Pushover Analysis (MPA) [14,15], the Adaptive Modal Combination Procedure (AMCP), proposed by Kalkan and Kunnath [24] and later adapted to bridges by Shakeri et al. [39] and the Adaptive Capacity Spectrum Method (ACSM), developed by [10,11]. All of them come with improvements, mainly the inclusion of higher modes contribution or, as in the ACSM, an alternative way of addressing the reference node issue. The application of methods which try to implement new features may generate controversy and it may be argued that the original concept of simplicity is being left behind towards a complexity level increasingly similar to nonlinear dynamic analysis. Such perspective is to the authors' opinion debatable as there may be so-called advanced NSPs, MPA or AMCP, which foresee several repeated steps whereas others, such as ACSM, bring no additional effort into the analysis since higher modes are included in the analysis per se when computing the pushover curve. The judgement of the importance of an increase in the computational onus will be left to the assessment of the corresponding performance augment. Recent studies [28,34] have addressed the validation of different NSPs in estimating the seismic response of single bridges or fragility functions of populations of bridges. The conclusions pointed out to N2, ACSM or modified MPA as the most accurate nonlinear static procedures to predict the seismic response of large numbers of randomly generated RC bridges. Moreover, the increased accurateness when using improved methods (ACSM, MPA) did not seem particularly advantageous when compared to the employment of N2, which is numerically less onerous. Similar findings were also observed for RC building frames in a study by Pinho et al. [35]. Based on such, the N2 method was selected herein to test the use of nonlinear static analysis in the probabilistic seismic safety assessment of bridges.

2.1. N2 method The N2 method was proposed by Fajfar and Fischinger [21] and features the use of conventional force-based pushover analysis of a MDOF model combined with the inelastic response spectrum analysis of its equivalent SDOF system. Similarly to CSM, the N2 method may also be formulated in the acceleration-displacement format thus providing a visual interpretation of the procedure. However the main difference with respect to the former is that N2 makes use of ductility-dependent inelastic spectra rather than over-damped elastic ones. Its implementation in the bridges chapter of Eurocode 8 foresees the computation of at least two pushover curves: one derived with a loading vector shape factor that is uniform along the deck and the other where the first mode proportional shape is used instead. An additional possibility considered in the past by the authors of this study was to employ the envelope of those two load shapes but no major improvement was found. The capacity curve is represented by plotting the base shear force as a function of the reference node displacement, which is recommended by the code as the centre of mass of the deformed deck. The performance point is determined through the computation of the target displacement of the equivalent SDOF system obtained through the intersection between the initial period of vibration of the latter and an inelastic spectrum derived as a function of the ductility achieved by the system. A non-compulsory iterative procedure to match the target displacement obtained for the equivalent SDOF system with that employed in the determination of the idealised elasto-perfectly plastic force-displacement relationship is also proposed and was indeed considered in this work. Two different loading possibilities have been considered in the application of the N2 method: (i) uniform load distribution and (ii) first mode load distribution. Due to the fact that the application of the method depends on the chosen reference node, each of the modalities was carried out considering the reference node as both the deck's centre of mass or maximum displacement.



3

Fig. 1. Failure probability: local uncertainty approach (left) and global uncertainty approach (right).

Further investigation of the nonlinear static N2 method within the simplified probabilistic approach has been carried out by adding a version based on adaptive pushover analysis instead of the originally prescribed conventional, following the promising results of recent studies. Adaptive pushover analysis has been originally proposed by Reinhorn [36] and Elnashai [19] and more recently improved by Antoniou and Pinho [1] in an innovative displacement-based variant of the method (DAP – displacement based adaptive pushover). In this version the pushover analysis algorithm is fully adaptive due to the impossibility of a fixed force pattern, characteristic of conventional pushover forcebased methods, to accomplish the collapse mode characteristic of a bridge. A study by Casarotti et al. [13] has showed that the employment of the DAP algorithm leads to better estimates of the inelastic deformed pattern, as well as of the distribution of base forces at a given inelasticity level, independently of structural regularity. 2.2. Failure probability In order to obtain the probability of collapse employing nonlinear static based seismic demand, two methodologies are used. These methodologies correspond to different uncertainty models and were described and investigated in detail in the companion paper, resulting in a general agreement between their output. Whereas the detail is to be found in such study a very brief summary is provided herein for the sake of completeness and discussion in terms of integration with nonlinear static analysis. 2.2.1. Local uncertainty model This approach was originally developed in the work by Costa [16] and is focused on the definition of the probability density function of the capacity and the seismic demand, which will then be convolved. In order to do so a relationship intensity measure-engineering demand parameter (IM-EDP) is used — the vulnerability function. In this particular study the considered IM is the peak ground acceleration whereas the EDP is the ductility in curvature at the bottom of the piers. When using nonlinear static analysis the IM is simply given by the spectral

ordinate (T = 0 s) of the selected record. For a predefined number of IM levels corresponding to different return periods and probabilities of occurrence, different real ground motion records or corresponding response spectra are scaled to match the corresponding peak ground acceleration (PGA). Nonlinear static analysis is then carried out using each of the scaled spectra yielding a sample of the seismic demand that accounts for record-to-record variability. For each IM level the mean EDP is computed and a polynomial function is adjusted to those points to define the aforementioned vulnerability function (curve 3 in Fig. 1 (left), which briefly illustrates the probabilistic framework). The full procedure corresponding to the local uncertainty approach, which is schematically depicted in Fig. 1, starts by characterising the seismic hazard with its IM probability density function (1). Analogously, for every IM level the seismic demand is obtained, in terms of the chosen EDP. The vulnerability function (3) is therefore defined and used to define the statistical distribution of the demand (4). At the same time and independently (hence ‘local uncertainty’) the distribution of the capacity (2) is defined in terms of the same EDP using a random simulation scheme — herein Latin Hypercube [5,29–31] was adopted. Finally, the convolution of capacity with the demand (5) will yield the intended failure probability. All the details can be found in the companion paper. 2.2.2. Global uncertainty model An alternative approach to the local uncertainty based failure probability computation considers the uncertainty of the different variables in a global manner i.e. simultaneously rather than independently, within a global procedure carried out fully at each iteration step. When using this approach the statistical distributions of the different variables are defined (material properties, IM, ground motion record) and the assessment procedure is carried out completely and but individually. Using again the random simulation technique each variable is randomly simulated; a nonlinear dynamic analysis is carried out on the bridge modelled with such parameters and the structural demand and capacity are determined (each simulation corresponds to one point in the curves 1 to 4 of the local uncertainty framework). Finally, the difference

Label 222 Label 2222222

REGULAR Label 232

SEMI REGULAR / IRREGULAR

Label 123

Label 3332111

Label 213

Label 2331312

Fig. 2. Case study bridge configurations.


4


1000 Mean Spectrum NEHRP Spectrum

900 800

Sd (mm)

700 600 500 400 300 200 100 0

0

1

2

3

4

5

6

T (s) Fig. 3. NERHP displacement spectrum for Los Angeles and individual and mean displacement spectra of the set of 10 ground motion records.

between capacity and demand is calculated and a statistical distribution is adjusted to the variable given by such difference (μR − μS). The failure probability will be given by the ordinate of the corresponding cumulative density function for the null abscissa, as illustrates according to Fig. 1 (right). 2.2.3. Pushover based failure probability For the sake of computational onus, the calculation of the failure probability using the demand predicted with nonlinear static analysis was carried out using the local uncertainty model only. By defining the new static-based vulnerability functions (see Section 4.1) the failure probability can be readily obtained by replacing the ones previously NDA r 2 =0.9962 2 max mod r =0.99461 max uni r 2 =0.99805 2 centr mod r =0.99461 2 centr uni r =0.99805 adapt r 2 =0.97838

Ductility demand

50

3. Case study The comparison of the use of different approaches for the seismic demand estimate within different safety assessment models has been carried out using the same set of bridge structures and real earthquake records employed and described in detail in the companion paper. A short summary is provided next, focusing on the aspects particularly relevant when using nonlinear static procedures. Seven bridge configurations have been considered, corresponding to two bridge lengths (viaducts with four and eight 50 m spans), with regular, irregular and semi-regular layout of the piers' height and continuous deck-abutment connections supported on piles with bilinear NDA r 2 =0.99733 2 max mod r =0.98512 max uni r 2 =0.99795 2 centr mod r =0.98936 2 centr uni r =0.99795 adapt r 2 =0.97487

90

123 - P2

80 70

Ductility demand

60

obtained with nonlinear dynamic analysis. On the other hand, for the sake of completeness, the results were in any case compared with the global uncertainty approach using nonlinear dynamic analysis, which had yielded conservative results with respect to the local uncertainty model. The employment of the chosen nonlinear static procedure can foresee different hazard representation modes i.e. the response spectra. According to its formulation the N2 method should be employed with a smoothed design code spectrum as it depends on the identification of the corner period to define the simplified inelastic response. A more realistic possibility for assessment purposes would be to use the real spectra of the records however such option would render the definition of the corner period less appropriate and could compromise the correct application of the NSP. Based on such considerations the smoothed design spectrum has been employed (further details can be found in Section 3). The adoption of a different NSP that would make use of real spectra can represent a future possibility to overcome such feature of this study.

40

30

20

60

213 - P2

50 40 30 20

10

10 0

0

100

200

300

400

500

600

700

800

900

0

1000

0

100

2

400

500

600

700

800

900

1000

900

1000

45

NDA r 2 =0.99403 2 max mod r =0.99707 max uni r 2 =0.99848 2 centr mod r =0.99707 2 centr uni r =0.99848 adapt r 2 =0.99424

40 35

222 - P2


40 35

Ductility demand

45

Ductility demand

300

Peak ground acceleration (cm/s )

50

30 25 20 15

30

232 - P2

25 20 15 10

10

5

5 0

200

2


0

100

200

300

400

500

600

700

800

2


900

1000

0

0

100

200

300

400

500

600

700

800

2


Fig. 4. Nonlinear static vulnerability functions: 3-pier configurations.



behaviour. The bridges are depicted in Fig. 2 where the label numbers 1, 2 and 3 stand for pier heights of 7, 14 and 21 m, respectively. The employed ground motion set is defined by an ensemble of ten records selected from a suite of historical earthquakes [37] scaled to match the 10% exceeding probability in 50 years (475 years return period) NEHRP uniform hazard spectrum for Los Angeles, as depicted in Fig. 3, which corresponds in the current endeavour to the IM level 1.0. The ground motions were obtained from California earthquakes with a magnitude range of 6–7.3 recorded on firm ground at distances of 13– 30 km; their significant duration [6] ranges from 5 to 25 s, whilst the PGA (for intensity level 1.0) varies from 0.23 to 0.99 g. The mean spectrum has therefore been used as required by the N2 method together with the assumption that the record-to-record variability is implicitly considered by the matching to the average of the real records. With respect the adopted structural modelling, plastic hinge models were used in order to reduce the amount of time required by the significant number of nonlinear static and dynamic analysis. The nonlinear dynamic seismic demand on the bridge models was evaluated with assumed equivalent viscous damping of 2%, using the software PNL [16–18,41], which was developed by the local research group at the University of Porto. On the other hand, the software package SeismoStruct [38], whose accuracy in predicting the seismic response of bridge

5

structures has been demonstrated through comparisons with experimental results derived from pseudo-dynamic tests carried out on large-scale models [10], was employed in the running of the conventional force-based and displacement-based adaptive pushover analyses [1,33], which are not enabled by the PNL software. Detailed information related to the modelling of the specific structural elements (piers, deck, connections and abutments) can be found in the companion paper. 4. Safety assessment — comparative study Two different procedures for the failure probability computation have thus been considered (numerical safety assessment using a local uncertainty model or a global simulation uncertainty model with Latin Hypercube Sampling) together with nonlinear static or dynamic analysis for the estimate of the seismic demand. The suitability of the nonlinear static predictions has been investigated by direct comparison with the failure probability obtained with nonlinear dynamic analysis. 4.1. Failure probability with local uncertainty model When using nonlinear static analysis as an alternative to the dynamic counterpart, pushover analysis is used for the prediction of the structural demand and the definition of the vulnerability function. Within the

55

Ductility demand

40 35

2222222 - P4 70

Ductility demand

45


80


50

30 25 20 15

60

2331312 - P4

50 40 30 20

10

10 5 0

0 0

100

200

300

400

500

600

700

800

2

900

1000

0

100


200

300

400

500

600

700

800

Peak ground acceleration (cm/s 2)

900

1000

35


30

Ductility demand

25

3332111 - P5

20

15

10

5

0

0

100

200

300

400

500

600

700

800

900

1000

2

Peak ground acceleration (cm/s ) Fig. 5. Nonlinear static vulnerability functions: 7-pier configurations.


6


selected nonlinear static procedure, for each IM level, herein expressed in PGA, the structure is pushed until a level corresponding to the considered PGA. The performance point is found by approximate inelastic spectral reduction and the chosen EDP (ductility in curvature) is measured. At this stage, there is the need to associate a specific peak ground acceleration to a specific pushing level. Such task may be accomplished by means of the response spectrum that is used within the nonlinear static procedure. For each IM level the response spectrum will therefore be scaled so that its zero ordinate (PGA) matches such level. The validity of such approach is logical but arguable and shall be taken into account when analysing differences in vulnerability function predictions. Recalling the considerations in Section 2, more than one nonlinear static procedure has proved itself as acceptably able to predict the response of bridges subjected to earthquake loading. Consequently, one of the traditional methods, implemented in the European code [9], the N2 method, has been chosen to obtain pushover-based vulnerability curves. The pushover analysis may be conventional or adaptive and both possibilities have been used. According to the recommendations on the application of the N2 method, two loading distributions – uniform and 1st mode proportional (mod and uni) – have also been tested. Additionally, two reference nodes have been considered: centre of mass of the deck and maximum modal displacement (centr and max), leading to a total

of four versions using conventional pushover and one corresponding to the adaptive pushover, which skips the need for a reference node or load pattern. The comparison between the ‘static’ vulnerability functions, coming from the different versions, is represented together with the one corresponding to dynamic analysis for each of the tested configurations in Figs. 4 and 5. Due to the large number of results only the curves corresponding to the governing pier (the one with highest ductility demand) are presented. The observation of the plots referring to the shorter configurations denotes that the pushover-based vulnerability functions are generally conservative, overpredicting the dynamic analysis response regardless the location of the governing pier or the ductility level. For some cases there is slight underprediction of the nonlinear dynamic response for low IM levels (up to approximately 0.3 g) which is very likely due to the nonlinear regression analysis fitting algorithm. Although not represented for each configuration, the agreement between different curves inferior for the pier with higher ductility demand, especially for the configurations 123, 213 and 232, where the differences among piers are higher. Larger dispersion among different pushover versions can be found for irregular configurations whereas for semi-regular and regular configurations there is higher agreement between different versions of the method as well as overprediction. Indeed, the irregular

10 0

10

max mod max uni centr mod centr uni adapt NDA

123

10 -5

213 10 -10

10

Failure probability 10

-15

-1

Failure probability

10

0

10 -20

10

-25

10

-30

-2

10 -35

10

-3

P1

P2

10

-40

10

-45

10 -50

P3

Pier

10

-2


10 -1

10 -3

10

P3

232

Failure probability

Failure probability

10

P2

10 0


222

-1

P1

Pier

0

10


-4

P1

P2

P3

10

-2

10

-3

10 -4

P1

Pier

P2

P3

Pier

Fig. 6. Failure probability, for short configurations, using different vulnerability functions.


R. Monteiro et al. / Structures xxx (2015) xxx–xxx 10 0

10 0

10

10

10

10

10

-5

-10

-15


2222222

Failure probability

Failure probability

10

7

-20

P2

P3

P4

P5

P6

10

-10

10

-15

10

-20

10

-25

10

-30

10

-35

10

-40

10

-45

10 -50

-25

P1

-5

P7

max mod max uni centr mod centr uni adapt NDA P1

2331312

P2

P3

P4

P5

P6

P7

Pier

Pier 10

10

0

-5

Failure probability

10 -10

10

-15

10

-20


10 -25

10

3332111

-30

P1

P2

P3

P4

P5

P6

P7

Pier Fig. 7. Failure probability, for long configurations, using different vulnerability functions.

configuration 213 is the only for which underprediction is verified unless adaptive pushover is used, which calls for attention when dealing with highly irregular configurations using nonlinear static analysis. Further to the differences between the several pushover approaches the adaptive analysis is generally the most conservative, always overestimating the dynamic analysis predictions, particularly for the piers with higher ductility demand. Regarding the different conventional pushover versions, the modal loading shape appears to yield more overpredicting results, closer to the adaptive pushover ones. Although

a modal load pattern will require a previous eigenvalue analysis, the increase in the accuracy of the results may compensate particularly for irregular configurations whereas an easier uniform shape can suffice for regular structures. For what concerns longer deck and regular configurations, herein represented by bridge 2222222, the results are similar to what has been observed for the shorter ones, which could be expected, as the “disruption” introduced by higher deck length is somewhat balanced by the regularity of the piers. The results are however steadier from

Table 1 Failure probability for 3-pier configurations.

123

213

222

232

P1 P2 P3 P1 P2 P3 P1 P2 P3 P1 P2 P3

max mod

max uni

centr mod

centr uni

adapt

NDA

5.11E−02 1.42E−02 1.80E−02 4.02E−46 2.21E−02 2.56E−04 1.94E−02 3.89E−02 1.94E−02 2.85E−02 3.48E−02 2.86E−02

7.05E−02 3.85E−03 2.17E−03 3.46E−24 7.48E−03 4.19E−19 1.23E−02 2.08E−02 1.23E−02 1.93E−02 1.47E−02 1.93E−02

5.11E−02 1.42E−02 1.80E−02 1.19E−45 3.90E−02 7.78E−04 1.94E−02 3.89E−02 1.94E−02 2.85E−02 3.48E−02 2.86E−02

7.05E−02 3.85E−03 2.17E−03 3.46E−24 7.48E−03 4.19E−19 1.23E−02 2.08E−02 1.23E−02 1.93E−02 1.47E−02 1.93E−02

7.88E−02 6.45E−03 5.26E−03 9.25E−03 2.30E−01 1.77E−03 1.82E−02 3.05E−02 1.85E−02 2.91E−02 2.64E−02 2.93E−02

2.75E−02 3.96E−03 3.82E−03 6.17E−05 1.34E−02 4.09E−04 6.15E−04 2.55E−03 6.16E−04 7.50E−04 5.41E−03 7.51E−04


8


Table 2 Failure probability for 7-pier configurations.

P1

P2

P3

P4

P5

P6

P7

2222222 max mod

3.90E–22

2.13E–03

1.76E–02

3.00E–02

1.77E–02

2.19E –03

8.67E –22

max uni

2.85E–07

8.16E–04

2.54E–03

2.93E–03

1.83E–03

3.74E –04

7.38E –10

centr mod

3.90E–22

2.13E–03

1.76E–02

3.00E–02

1.77E–02

2.19E –03

8.67E –22

centr uni

2.85E–07

8.16E–04

2.54E–03

2.93E–03

1.83E–03

3.74E –04

7.38E –10

adapt

3.57E–03

1.65E–02

4.60E–02

6.62E–02

4.61E–02

1.66E –02

3.60E –03

NDA

4.68E–07

2.97E–04

1.41E–03

2.43E–03

1.42E–03

2.98E –04

4.70E –07

max mod

7.87E–04

1.69E–03

4.36E–04

1.44E–02

2.10E–25

1.32E –27

1.30E –46

max uni

2.10E–04

8.66E–05

3.31E–08

5.44E–03

1.05E–24

2.55E –25

4.55E –44

centr mod

8.37E–04

2.90E–03

1.58E–03

4.79E–02

7.35E–25

1.82E –27

1.04E –46

centr uni

1.38E–05

2.79E–05

1.12E–07

7.89E–03

1.13E–24

2.98E –25

5.30E –44

adapt

1.48E–02

1.64E–02

8.88E–03

1.56E–01

3.41E–05

3.30E –02

2.36E –05

NDA

1.16E–07

1.01E–03

2.44E–03

1.25E–02

1.98E–04

5.58E –03

3.54E –10

max mod

7.32E–03

2.41E–02

1.68E–02

1.75E–03

8.17E–05

8.48E –27

2.21E –27

max uni

2.39E–03

3.88E–03

1.46E–03

3.00E–05

2.71E–09

2.95E –27

1.42E –25

centr mod

2.36E–02

8.04E–02

5.76E–02

9.32E–03

6.71E–03

6.02E –27

3.70E –27

centr uni

6.22E–03

1.22E–02

6.47E–03

1.07E–03

8.94E–04

5.56E –27

1.42E –25

adapt

2.37E–02

3.29E–02

1.53E–02

2.32E–03

3.12E–03

3.69E –27

3.40E –08

NDA

2.93E–03

1.38E–02

1.27E–02

3.76E–04

2.26E–04

4.96E –05

4.24E –07

2331312

3332111

The failure probability of the different bridge piers obtained with the employment of static and dynamic vulnerability functions are compared in Figs. 6 and 7 and Tables 1 and 2, in which the rows corresponding to the critical piers are highlighted in grey. Due to the nature of the probabilistic assessment procedure (Fig. 1) the differences that have been found between such vulnerability curves may assume higher or lower

10 0


10

-1

10

-2

10

-3

Failure probability

pier to pier with the previous main observations still standing, regardless of the demand level. Adaptive pushover analysis is again overestimating, sometimes heavily, whereas the different versions of conventional pushover do not differ considerably with a tendency for the uniform load shape together with the centre of mass of the deck as reference node to be closer to the dynamic results. With respect to irregular configurations, although the adaptive pushover analysis keeps overestimating results for the majority of the cases, for some piers (short and close to the abutments) it is actually the most accurate (e.g. P5, P6 and P7 in configuration 2331312). The utility of the adaptive pushover is thus associated again to the irregular configurations. For most of the remaining piers the 1st mode proportional load shape version, using the deck centre of mass, is the most accurate. It is additionally noteworthy that for the critical pier, corresponding to the higher ductility demand, the nonlinear dynamic analysis predictions are bounded by the two aforementioned pushover variants, something that has been verified for all the configurations. It is thus possible, at least for the pier with highest demand, to enclose the dynamic vulnerability functions by the predictions of two most promising pushover techniques. The use of pushover analysis in the estimation of vulnerability functions proves itself practical and useful at least for an expedite estimate of the ductility demand level that should be expected for a certain configuration. However the use of conventional pushover with the consideration of the maximum modal displacement as reference node does not represent a proper alternative. Adaptive pushover may be used in irregular and longer configurations for short, abutment-near piers, so as to complement or even replace inaccurate vulnerability functions obtained with conventional pushover analysis.

123

213

222

232

2222222

2331312

3332111

Bridge configuration Fig. 8. Critical pier failure probability, per bridge configuration.



9

Table 3 Failure probability for the critical pier.


123

213

222

232

2222222

2331312

3332111

P1

P2

P2

P2

P4

P4

P2

5.11E−02 7.05E−02 5.11E−02 7.05E−02 7.88E−02 2.75E−02

2.21E−02 7.48E−03 3.90E−02 7.48E−03 2.30E−01 1.34E−02

3.89E−02 2.08E−02 3.89E−02 2.08E−02 3.05E−02 2.55E−03

3.48E−02 1.47E−02 3.48E−02 1.47E−02 2.64E−02 5.41E−03

3.00E−02 2.93E−03 3.00E−02 2.93E−03 6.62E−02 2.43E−03

1.44E−02 5.44E−03 4.79E−02 7.89E−03 1.56E−01 1.25E−02

2.41E−02 3.88E−03 8.04E−02 1.22E−02 3.29E−02 1.38E−02

importance in the computation of the failure probability depending on the shape of the seismic action probability density function. The first and general observation that can be drawn from Fig. 6 is the confirmation that nonlinear static analysis is conservative with respect to nonlinear dynamic predictions, particularly for regular, shorter configurations. Adaptive and conventional pushover analyses coupled with 1st mode proportional load shape tend to yield higher failure probabilities. Larger differences are observed, as expected, for the piers with highest ductility demand. For the two irregular configurations, in some of the piers, the adaptive pushover denotes higher performance. Numerically, based on Table 1, one can see how the different pushoverbased versions overestimate the nonlinear dynamic analysis of around one order of magnitude, particularly for the regular configurations. Furthermore, from a numerical viewpoint the fluctuation of results around the same order of magnitude are not highly significant. In terms of bridge safety assessment, such differences are given by the convolution of very low probability densities of capacity and demand. With respect to longer configurations, results tend to be more uniform and the main differences take again place for piers with higher demand. Even though nonlinear dynamic analysis is not as overestimated as for short configurations the adaptive pushover estimates still yield the highest collapse probabilities confirming the trend for pushoverbased assessment to play a conservative role. This is certainly reassuring with respect to the use of preliminary, simplified procedures but also opens the way to possible use of reduction factors. Nevertheless the adaptive pushover is confirmed as the only able to reproduce the dynamic counterpart for some of the non-critical piers of the two irregular configurations. Fig. 8 and Table 3 summarise the results, presenting the failure probability of the critical pier of each bridge all together. Given that the relevance of the reference node in the final results has been found to be minor such comparison for the entire set of bridges has only been based on different load patterns. It is confirmed by looking at the comparison for the entire set of bridges that regular configurations feature lower collapse probabilities, even though associated to larger variability in terms of different pushover approaches. Regarding the type of pushover to use, if a simple static

Failure probability

10

analysis is intended, the modal load pattern does not introduce significant advantage when compared to the uniform one, which is also easier to apply. Adaptive pushover is generally the most overpredicting, getting closer to the nonlinear dynamic counterpart for the highly irregular configurations. No particularly important differences are found concerning the length of the configuration: short bridges are slightly less safe than the long ones together with a little less dispersion within different versions. 4.2. Comparison of approaches Fig. 9 illustrates the comparison of the static failure probabilities with the dynamic ones, obtained with local and global uncertainty models (the latter are reproduced from the companion paper). For the sake of simplicity, only static failure probabilities based on conventional pushover, with uniform load pattern and central reference node, and adaptive pushover versions are presented. The general observation is that pushover-based estimation of seismic demand leads, in most of the cases, to higher failure probability especially when using adaptive analysis, which is frequently associated to a failure probability of one order of magnitude above. When compared to the global uncertainty dynamic failure probabilities, which were observed to play on the safe side, the static based approaches still play an overestimating role. Interestingly, the irregular configurations are the only for which the static and dynamic approaches cross or get closer. These observations reinforce the supporting role played by nonlinear static analysis in the use of probabilistic methods when compared to the dynamic counterpart, serving as providers of expedite conservative estimates. 5. Concluding remarks This study, which complements a companion paper, has addressed the use of nonlinear static (pushover) analysis within a previously validated nonlinear static procedure (N2 method) to obtain seismic demand estimates for simplified probabilistic assessment of reinforced concrete bridges. Static analysis based failure probabilities have

0

10

-1

10

-2

10 -3

10

-4

10

-5

NDA local uncertainy NDA global uncertainy Pushover (conventional) Pushover (adaptive) 3332111

232

2222222

222

2331312

213

123

Bridge configuration Fig. 9. Failure probability, according to bridge configuration, for static and dynamic analysis based versions.


10


therefore been calculated for a case study made up of seven bridge configurations and compared to dynamic analysis based approaches, using two uncertainty models. The observation of the results has indicated a tendency for the static analysis approaches to be conservative, at times highly, even when applied with conventional uniform loading shape. The use of nonlinear static analysis for the seismic demand estimate has therefore been confirmed as a valid approach and can become particularly useful to obtain quick conservative estimates of the structural vulnerability of large numbers of structures. The adaptive pushover is more suitable for the prediction of the behaviour of irregular configurations whereas conventional pushover, preferably with modal load pattern, is enough to get rather acceptable estimates for regular configurations. Within methodologies such as the probabilistic ones, in which the seismic demand needs to be estimated through a large number of nonlinear analyses, this feature constitutes an important asset, at least for a preliminary analysis. This represents an important outcome if one keeps in mind the need for sound methodologies that are less complex and time consuming than nonlinear dynamic analysis, to be incorporated in seismic design and/or assessment of structures by practitioners. Indeed, the use of pushover analysis is many times the preferred tool of the practitioners that face the need to characterise the nonlinear behaviour of structures, as it is conceptually easy to grasp and simple to apply. To this end, the proposed study has provided guidance to some of the limits and advantages of applying such tools to RC bridges with a view also to the employment of more complex probabilistic methodologies. Acknowledgements Financial support to the first author was provided by the Portuguese Foundation for Science and Technology (FCT — Fundação para a Ciência e a Tecnologia) Grant number SFRH/BPD/78467/2011. References [1] Antoniou S, Pinho R. Development and verification of a displacement-based adaptive pushover procedure. J Earthq Eng 2004;8(5):643–61. [2] Applied Technology Council, ATC. Seismic evaluation and retrofit of concrete buildings, ATC-40 report. California, USA: ATC; 1996. [3] Applied Technology Council, ATC. NEHRP guidelines for the seismic rehabilitation of buildings, rep. no. FEMA-273, Washington, DC, USA. 1997. [4] Applied Technology Council, ATC. Improvement of nonlinear static seismic analysis procedures, rep. no. FEMA-440, Washington, DC, USA. 2005. [5] Ayyub BM, Lai KL. Structural reliability assessment using Latin hypercube sampling. Proceedings of the ICOSSAR'89, August 8–11, San Francisco, USA; 1989. [6] Bommer JJ, Martínez-Pereira A. The effective duration of earthquake strong motion. J Earthq Eng 1999;3(2):127–72. [7] Bonstrom H, Corotis RB. Building portfolio seismic loss assessment using the firstorder reliability method. Struct Saf 2015;52:113–20. [8] CEN. Eurocode 8: design of structures for earthquake resistance — part 1: general rules, seismic actions and rules for buildings. EN 1998–2, Comité Européen de Normalisation, Brussels, Belgium; 2005. [9] CEN. Eurocode 8: design of structures for earthquake resistance — part 2: bridges. EN 1998–2, Comité Européen de Normalisation, Brussels, Belgium; 2005. [10] Casarotti C, Pinho R. Seismic response of continuous span bridges through fibrebased finite element analysis. J Earthq Eng Eng Vib 2006;5(1):119–31. [11] Casarotti C, Pinho R. An adaptive capacity spectrum method for assessment of bridges subjected to earthquake action. Bull Earthq Eng 2007;5(3):377–90. [12] Casarotti C, Monteiro R, Pinho R. Verification of spectral reduction factors for seismic assessment of bridges. Bull N Z Soc Earthq Eng 2009;42(2):111. [13] Casarotti C, Pinho R, Calvi GM. Adaptive pushover based methods for seismic assessment and design of bridge structures. ROSE research report no. 2005/06. Pavia, Italy: IUSS Press; 2005. [14] Chopra AK, Goel RK. A modal pushover analysis procedure for estimating seismic demands for buildings. Earthq Eng Struct Dyn 2002;31:561–82.

[15] Chopra AK, Goel RK, Chintanpakdee C. Evaluation of a modified MPA procedure assuming higher modes as elastic to estimate seismic demands. Earthquake Spectra 2004;20(3):757–78. [16] Costa AG. Análise sísmica de estruturas irregulares. Ph.D. Thesis Porto, Portugal: University of Porto; 1989(in Portuguese). [17] Costa AG, Costa AC. Modelo Histerético das Forças-Deslocamentos Adequado à Análise Sísmica de Estruturas. Relatório Técnico, Núcleo de Dinâmica Aplicada. Lisbon: National Laboratory of Civil Engineering; 1987 (in Portuguese). [18] Delgado P, Costa A, Pinho R, Delgado R. Different strategies for seismic assessment of bridges — comparative studies. Proceedings of the 13th World Conference on Earthquake Engineering — 13WCEE, Vancouver, Canada; 2004. [19] Elnashai AS. Advanced inelastic static (pushover) analysis for earthquake applications. Struct Eng Mech 2001;12(1):51–70. [20] Fajfar P. A nonlinear analysis method for performance-based seismic design. Earthquake Spectra 2000;16(3):573–92. [21] Fajfar P, Fischinger M. N2 — a method for non-linear seismic analysis of regular buildings. Proceedings of the Ninth World Conference in Earthquake Engineering, Tokyo-Kyoto, Japan, 5. ; 1988. p. 111–6. [22] Isakovic T, Fischinger M. Higher modes in simplified inelastic seismic analysis of single column bent viaducts. Earthq Eng Struct Dyn 2006;35(10):95–114. [23] Freeman SA. Development and use of capacity spectrum method. Proceedings of the Sixth U.S. National Conference on Earthquake Engineering, Seattle, Oakland, USA; 1998. [24] Kalkan E, Kunnath SK. Adaptive modal combination procedure for nonlinear static analysis of building structures. J Struct Eng 2006;132(11):1721–31. [25] Kappos AJ, Panagopoulos G. Performance-based seismic design of 3D R/C buildings using inelastic static and dynamic analysis procedures. ISET J Earthq Technol 2004; 41(1):141–58. [26] Kohrangi M, Bento R, Lopes M. Seismic performance of irregular bridges—comparison of different nonlinear static procedures. Struct Infrastruct Eng 2014:1–19 (ahead-of-print). [27] Krawinkler H, Seneviratna GDPK. Pros and cons of a pushover analysis of seismic performance evaluation. Eng Struct 1998;20(4):452–64. [28] Monteiro R, Zhang X, Pinho R. Different approaches to derive analytical fragility functions of bridges. Proceedings of the Second European Conference on Earthquake Engineering and Seismology, Istanbul, Turkey; 2014. [29] Monteiro R. Using Latin hypercube sampling for probabilistic seismic assessment of RC bridges. Comput Struct 2015 submitted for publication. [30] Novák D, Teplý B, Keršner Z. The role of Latin hypercube sampling method in reliability engineering. Proceedings of the ICOSSAR 97, Kyoto, Japan; 1997. [31] Olsson A, Sandberg G, Dahlblom O. On Latin hypercube sampling for structural reliability analysis. Struct Saf 2003;25(1):47–68. [32] Paraskeva TS, Kappos AJ, Sextos AG. Extension of modal pushover analysis to seismic assessment of bridges. Earthq Eng Struct Dyn 2006;35(10):1269–93. [33] Pinho R, Casarotti C, Antoniou S. A comparison of single-run pushover analysis techniques for seismic assessment of bridges. Earthq Eng Struct Dyn 2007;36(10): 1347–62. [34] Pinho R, Monteiro R, Casarotti C, Delgado R. Assessment of continuous span bridges through nonlinear static procedures. Earthquake Spectra 2009;25(1):143–59. [35] Pinho R, Marques M, Monteiro R, Casarotti C, Delgado R. Evaluation of nonlinear static procedures in the assessment of building frames. Earthquake Spectra 2013; 29(4):1459–76. [36] Reinhorn A. Inelastic analysis techniques in seismic evaluations. Proceedings of the International Workshop in Seismic Design Methodologies for the Next Generation of Codes, Bled, Slovenia; 1997. p. 277–87. [37] SAC. Develop suites of time histories. Project task 5.4.1 draft report. Sacramento, CA: SAC Joint Venture; 1997. [38] Seismosoft. SeismoStruct — a computer program for static and dynamic nonlinear analysis of framed structures. available online from http://www.seismosoft.com; 2015. [39] Shakeri K, Tarbali K, Mohebbi M. Modified adaptive modal combination procedure for nonlinear static analysis of bridges. J Earthq Eng 2013;17(6):918–35. [40] Vamvatsikos D, Cornell CA. Incremental dynamic analysis. Earthq Eng Struct Dyn 2002;31(3):491–514. [41] Varum HSA. Modelo Numérico para a Análise Sísmica de Pórticos Planos de Betão Armado. M.Sc. Thesis Porto, Portugal: University of Porto; 1996(in Portuguese). [42] Zelaschi C, Monteiro R, Pinho R. Improved fragility functions for RC bridge populations. In: Papadrakakis M, Papadopoulos V, Plevris V, editors. 5th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2015), Crete Island, Greece, 25–27 May; 2015. [43] Zelaschi C, De Angelis G, Giardi F, Forcellini D, Monteiro R. Performance based earthquake engineering approach applied to bridges in a road network. In: Papadrakakis M, Papadopoulos V, Plevris V, editors. 5th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2015), Crete Island, Greece, 25–27 May; 2015.
