A probabilistic framework for Performance-Based Wind Engineering

EACWE 5 Florence, Italy 19th – 23rd July 2009 Flying Sphere image © Museo Ideale L. Da Vinci

A probabilistic framework for Performance-Based Wind Engineering F. Petrini, M. Ciampoli, G. Augusti Sapienza Università di Roma, Italy – [email protected]– Sapienza Università di Roma, – [email protected]– Sapienza Università di Roma – [email protected]

Keywords: Aeolian risk, Performance-Based Design, Performance-Based Wind Engineering, Suspension Bridge.

ABSTRACT It is widely recognized that the most rational way of tackling the risks of engineered facilities and infrastructures subject to natural and man-made phenomena, and reducing them, both in designing new facilities and in rehabilitating or retrofitting the existing ones, is Performance-Based Design, usually indicated by the acronym PBD (but a better term would be “Performance-Based Engineering”). The basic concepts of PBD have been applied for almost 30 years in the nuclear power plant industry; later, PBD has been formalized and developed primarily in seismic engineering but has been extended to other engineering field, like Blast Engineering and Fire Engineering. Wind engineering appears of great potential interest for further developments of PBD. The first steps in this direction go back to an Italian research project, in which the expression "Performance-Based Wind Engineering" (PBWE) was coined. In this paper, the approach proposed by the Pacific Earthquake Engineering Research Center for Performance-Based Earthquake Engineering (PBEE) is extended to the case of PBWE. The general framework of the approach is illustrated and applied to an example case: the assessment of the collapse and out-of-service risks of a long span suspension bridge.

Contact person: Francesco Petrini, Sapienza Università di Roma, via Eudossiana 18, Rome, Italy, phone: +390644585265 FAX: +39064884852. E-mail [email protected]

1. INTRODUCTION A modern approach to wind engineering should consider performances as key objectives of structural design. Hence optimal design procedures should be developed in the framework of Performance-Based Design, or, better, Performance-Based Engineering: the relevant structural performance requirements should be satisfied with a sufficiently high probability throughout the whole life-cycle. In other structural engineering fields, like seismic engineering, has been recognized that the Performance-Based approach it is a fundamental tool for the risk reduction and management (Pinto et al. 2004). Under this consideration, advanced probabilistic procedures has been developed and improved for the Performance-Based Earthquake Engineering (PBEE). The similarities between the seismic and the aeolian risk, suggest that analogues procedures can be developed for wind engineering in order to conduct the design by the Performance-Based approach. Many studies have been carried out on the performance assessment of structures subject to wind action. Among the others: Unanwa et al. (2000) developed a procedure to assess the wind damage band for the hurricane damage prediction, focusing on low and medium rise buildings; Khanduri and Morrow (2003) examined the vulnerability of buildings to windstorms; Ellingwood et al. (2004) developed a procedure for the fragility assessment of light frame wood constructions subject to hurricanes; Garciano et al. (2005) developed a procedure for the assessment of critical performances of wind turbines subject to typhoons; Pastò and de Grenet (2005) dealt with the problem of wind-induced risks in case of bridges; Pagnini (2005) focused the attention on the reliability of structures with uncertainty parameters under wind action; Norton (2007) proposed a systematic approach to assess the structural performances under hurricanes; Zhang et al. (2008) determined the probability density functions of the basic random parameters of the wind field, and proposed a procedure for the reliability assessment of tall buildings. From the present authors, after a preliminary introduction (Paulotto et al., 2004) in which the term Perform-ance-Based Wind Engineering (PBWE) was coined, studies on PBWE have been performed within the research project PERBACCO (Bartoli et al. eds., 2006), with particular reference to tall buildings. In the PERBACCO report volume, Augusti and Ciampoli (2006) introduced the use of the PEER equation (Porter 2003) in PBWE procedures. Later Sibilio and Ciampoli (2007) studied the performance of a footbridge subject to turbulent wind by using advanced Monte Carlo techniques; Petrini et al. (2008a) proposed a risk assessment procedure for the study of the fatigue damage to the hangers of a long span suspension bridge under wind action, and in another study (Petrini et al., 2008b) considered the serviceability and ultimate performances of the same bridge. In other recent contributes (Petrini et al. 2009), the importance of the various uncertainties related to the under-wind response of structure has been investigated. The goal of this paper is to give further contribution toward the development of probabilistic procedures for the application of Performance-Based Design concepts to wind engineering. The resulting PBWE procedure is applied to an example case: the assessment of the failure probability due to flutter instability and of the out-of-service risk of a long-span suspension bridge. 2. SOURCE OF UNCERTAINTY IN WIND ENGINEERING. The problem of PBWE has to be faced in probabilistic terms, due to the stochastic nature of both resistance and loading parameters. In characterizing the wind field and the corresponding actions, different sources of uncertainty are to be taken into account (Figure 1). The environment defines the wind field in the location of the structure as if it were absent (free-field wind); i.e. in the environment the basic site-dependent parameters of the wind field (mean value of the velocity in each direction, turbulence intensity, dominant direction of strong winds, ...) are not influenced by the presence of the structure, while they can be affected by the interaction with other environmental agents; a typical example is the interaction between wind and waves in offshore

sites. The exchange zone is the region around the structure where the structure and the wind field are strongly correlated, and the effects of interaction between the relevant properties of both the structure and the wind field, as well as the presence of nearby structures, cannot be disregarded and aerodynamic and aeroelastic phenomena determine the relevant features of the wind effects. Moreover, also non environmental actions can influence the structural response by modifying the aerodynamic and aeroelastic characteristics of the structure; a relevant example is represented by the transit of trains on a railway bridge, as it determines a change of the dynamic characteristics of a flexible superstructure. The wind effects thus depend on the coupled system composed by the structure and the influencing actions. ENVIRONMENT

EXCHANGE ZONE

Aerodynamic and aeroelastic phenomena

Structural system

Wind site basic parameters Structural system as modified by service loads

Wind action Locationspecific wind Environmental effects (e.g. waves)

Service loads and other non environmental actions Types of uncertainties • Inherent • Epistemic • Model

Basic parameters

α

• Inherent • Epistemic • Model

Derived parameters

• Inherent • Epistemic • Model

β

Independent parameters

γ

Figure 1. Sources of uncertainty in wind engineering.

As concern the characterization of the wind field in the environment, the uncertainty is due to (Morgan and Henrion 1998, Der Kiureghian and Ditlevsen 2009): (i) the intrinsic variability of the basic parameters (the inherent or aleatory uncertainty), arising from the unpredictable nature of magnitude and direction of the wind velocity and turbulence intensity; (ii) the errors associated to the experimental measures and the incomplete data and information (the epistemic uncertainty); (iii) the modelling of wind actions and their effects on structural response (the model uncertainty). In the exchange zone, the interaction parameters are strongly dependent on the basic parameters that characterize the wind field; examples of derived parameters are the aerodynamic polar lines, the aeroelastic derivatives, the Strohual number. In order to derive the probability density functions of these parameters, the uncertainty of the basic parameters must be taken into account, while the uncertainty of the relevant parameters in the exchange zone has no influence on the basic parameters. Other parameters are independent on the basic parameters, examples are the mechanical and material properties of the structure. In what follows, the uncertain basic parameters in the environment are grouped in the vector α; the uncertain parameters of interest in the exchange zone are grouped in the two vectors of derived β and independent γ parameters (Figure 1). The vectors α and γ can be assumed not correlated with each other and not affected by the uncertainty of the parameters β, i.e. P(α β) = P(α γ ) = P(α )

(1)

P( γ β) = P( γ α ) = P( γ )

(2)

where P(·|·) is a conditional probability. In PBWE, conditional probabilities, conditional density functions and the total probability theorem are essential. Given three events A,B and C, the definition of conditional probability implies: P( A, B, C ) = P( A B, C ) ⋅ P( B C ) ⋅ P(C ) P( A, B C ) =

P( A, B, C ) = P( A B, C ) ⋅ P( B C ) P(C )

(3) (4)

Equations (1)-(4) can be applied to the vectors of basic, derived and independent parameters, whose joint probability can be expressed as:

P(α, β, γ ) = P(β α, γ ) ⋅ P(α γ ) ⋅ P( γ ) = P(β α, γ ) ⋅ P(α ) ⋅ P( γ )

(5)

Let us consider a collection E = [E1, E2, …, EN]T of mutually exclusive and collectively exhaustive events. For any other event A, the total probability theorem states that: N

P( A) = ∑ P( A Ei ) ⋅ P( Ei ) i =1

(6)

If continuous random variables X and Y are considered, the total probability theorem yields the complementary cumulative distribution function (CCDF) of X: ∞

G( x) = ∫ G( x y ) ⋅ f( y ) d y

(7)

−∞

where the integration is performed over the entire range of Y, and the conditional CCDF G(x|y) is the probability that X will be larger than or equal to x when Y is equal to y, namely G( x y ) = P[ X ≥ x Y = y]

(8)

Considering a vector θ of uncertain independent parameters, equation (7) becomes ∞

G( x) = ∫ G( x θ) f (θ) dθ

(9)

−∞

If:

θ = [α β γ ]T

(10)

and X is a random parameter that describe the structural response under wind actions, equation (9) becomes ∞

G( x) = ∫ G( x α, β, γ ) ⋅ f(α, β, γ ) d α d β d γ = −∞

∞

= ∫ G( x α, β, γ ) ⋅ f(β α, γ ) ⋅ f(α ) ⋅ f( γ ) d α d β d γ −∞

(11)

3. A PROCEDURE FOR PERFORMANCE-BASED WIND ENGINEERING (PBWE) A central objective of a procedure for Performance-Based Wind Engineering is the assessment of the adequacy of the structure through the probabilistic description of a set of decision variables DVs. Each DV is a (quantitative) measure of a specific structural performance, that can be defined in terms of the main interest of the society or the stakeholder. Examples of DVs are the number of lives lost during tornados, the economic losses resulting from wind-storms, the exceeding of a (collapse or serviceability) limit state, the discomfort of the occupants, the length of the out-of-service time. The starting points of the procedure are the relationships, expressed in probabilistic terms, between the performances specific to the considered construction (no collapse, occupant safety, accessibility, full functionality, limited displacements or accelerations, etc.) and different “intensities” of the wind action. With reference to a specific performance, the structural risk may be conventionally measured by the probability of exceeding a relevant value of the corresponding DV; this probability can be expressed in terms of a mean annual frequency, and evaluated by taking into account the Aeolian hazard (i.e. the occurrence of wind actions of specified intensity and characteristics at the site), the calculated structural response and damages, and the correlation between the attained damage level and the relevant DV. The structural design can be optimized by applying a decisional strategy to the risk analyses, with the objective of minimizing the total risk or of maximizing an utility function. In general terms, a procedure of PBWE consists of intermediate steps aimed at - defining the Aeolian hazard at the site, in terms of wind intensity and/or parameters of the wind velocity field. This part generally (referring to PBEE) consists in two steps: o a certain observation period of time is chosen; o the hazard is characterized by the so called “hazard curve” which states the mean frequency of overcoming the various earthquake intensity during the observed period; - analyzing the structural response, mainly in the context of stochastic dynamics; - defining and evaluating indicators of the structural damage (intended as an unacceptable performance), considering performances related to safety and functionality or comfort; - defining the decisional variables that are appropriate to quantify the performances required for the structure, in terms of consequences of damage (restoration costs, costs due to loss or deterioration of service, personal damages, alterations of users comfort, etc.); - evaluating the structural risk by the probabilistic characterization of the decision variables; - optimizing design, that is minimizing risk, by appropriate techniques of decision analysis. In general, the assessment of the Aeolian hazard requires the use of the most efficient techniques for modelling wind actions on a slender structure, and the choice of the intensity parameter vector IM whose stochastic characteristics are sufficient to fully describe the Aeolian hazard at the site. IM is chosen by considering the characteristics of the wind turbulence, direction and arrival processes, the relevant features of the construction site, the relevant physical phenomena that influence the specific performance under consideration. Attention shall be focused on the minimum information that are needed to characterize satisfactorily the wind field, since today available data (essentially turbulence spectra and maps of expected wind speeds) are rather limited. The probabilistic modelling of the structural response requires the choice of the relevant engineering demand parameter vector EDP (interstorey drifts, accelerations and velocities of selected points, stresses and displacements, etc.). The damage evaluation requires the choice (and probabilistic characterization) of the damage parameter vector DM, that is able to quantify the structural damage due to wind actions in relation to the considered performances. The choices of EDP and DM are strongly dependent on the considered structural type and performances. Different parameters can be assumed as DM: they can be defined by a combination of relevant EDPs (like the Park&Ang index that quantifies the structural damage to a rc component), or by other parameters, representing, for example, the damage to the partition walls in a building structures as a function of the interstorey drift. However, in the latter case it is usually rather difficult

to establish sound correlations between the evaluated EDPs and the chosen DMs. The decision variable DVs, that quantify the performances, can be distinguished between those corresponding to (a) possible consequences on structural and personal safety (low performance levels), and (b) effects only on serviceability and comfort, including the area around the structures (high performance levels) (Paulotto et al., 2004; Augusti and Ciampoli, 2006). For low performances, the significant DV can be identified with the cost necessary to restore the construction to the undamaged state; correspondingly, DM is the set of damages to be restored, and EDP can be identified with the most significant response parameters for the specific case (peak displacement or acceleration at the building top, overall action at the base, local pressure, etc.). Appropriate relationships between any DM and the relevant EDP allow to evaluate the damage states corresponding to given values of the response parameter EDP, and also the resulting losses if the relationships between DV and DM are considered. High performances are related to the users’ comfort/discomfort and, in case of buildings, to inconvenient alterations of the wind field in pedestrian areas around the construction. Appropriate discomfort criteria, that is, statements specifying the maximum acceptable frequencies of occurrence for various degrees of discomfort, can be defined. Analogously, if the structural risk is quantified by the exceedance probability of a limit state, the distinction between low and high performances can be interpreted considering: - the ultimate limit states (ULS) as related to low performances: examples are the attainment of the capacity of any significant part of the structure, the fatigue collapse of some elements, the instability of parts or of the whole structure, etc.; - the serviceability limit states (SLS) as related to high performances: examples are excessive deformations or vibrations compromising the use of the structure or its function in service. The optimization of design requires the development and implementation of a decisional strategy aimed at minimizing the total risk or at maximizing an appropriately defined utility. The essential DV can be identified with the economical losses due to windstorms, evaluated taking into account the whole lifetime of the construction. A formal procedure for PBWE can be obtained by extending the approach proposed by the Pacific Earthquake Engineering Research Centre for PBEE (Porter, 2003, Mohele et al 2005, Kunnath 2007). An overall review can be found in (Augusti and Ciampoli, 2008). With respect to the PEER approach, in the case of the PBWE, proper stochastic parameters IP which are able to describe the interaction effects between the wind and the structure have to be considered. The structural risk is evaluated as the mean annual frequency λ of exceeding a threshold level of DV:

λ( DV ) = ∫ ∫ ∫ ∫ ∫ P( DV DM ) ⋅ P( DM EDP) ⋅ ⋅ P( EDP IM,IP, γ ) ⋅ g(IP IM, γ ) ⋅ g(IM )

(12)

⋅ g( γ ) ⋅ d DM ⋅ d EDP ⋅ d IM ⋅ d IP ⋅ d γ where: DM is a damage measure; EDP is the corresponding engineering demand parameter, representing the structural response; the basic parameters α characterizing the Aeolian hazard (Sec. 2) are described by a vector IM of Intensity Measures; the derived parameters β (Sec. 2) are collected in the vector of the Interaction Parameters IP, that is the vector containing the set of aerodynamic and aeroelastic parameters that allow to take into account the relevant aspect of the interaction between the environment and the structure; γ is the vector of the parameters characterizing the structural systems and non environmental actions; finally g(IP|IM,γγ) is the conditional probability density function of IP. A reduced version of the equation (12) (see equation (14)), can be derived from the equation (11), were the EDP was represented from a generic response parameter X and the probability G of overcoming a threshold was considered instead of the mean annual frequency λ.

Through equation (12), the problem of risk assessment is disaggregated in the following elements: - site and structure-specific hazard analyses, that is, the assessment of the probability density functions g(IM), g(γ) and g(IP|IM,γ); - structural analysis, aimed at assessing the probability distribution function of the structural response P(EDP|IM,IP,γ) conditional on the parameters characterizing the wind field and the structural properties; - damage analysis, that gives the damage probability P(DM|EDP) conditional on EDP; - finally, loss analysis, that is the assessment of P(DV|DM). Under these assumptions, a flowchart similar to the PEER flowchart (Mitrani-Reiser 2007) can be defined for the proposed PBWE procedure (Fig. 2). Site hazard analysis

Wind-structure interaction analysis

Structural analysis

Damage analysis

Loss analysis

g(IM|O,D)

g(IP|IM,γ)

P(EDP|IM,IP)

P(DM|EDP)

P(DV|DM)

Facility info

Decisionmaking

O, D

g(IM)

g(IP)

P(EDP)

P(DM)

P(DV)

O location D design

IM: intensity measures

IP: interaction parameters

EDP: engineering demand parameter

DM: damage measure

DV: decision variable

Select O, D

Figure 2. Flowchart of the PBWE procedure.

With respect to the PEER approach, the step of probabilistic characterization of the interaction parameters IP has been introduced. This step requires the assessment of the probabilistic correlation between IP and IM and can be based on either wind tunnels tests (Diana et al 1995) or CFD analyses (Morghenthal, 2006). As previously stated, equation (12) can be simplified by assuming that the chosen EDP is a measure of the structural damage (that is, EDP ≡ DM), and expressing the performance by a Limit State (LS). The risk assessment is thus based on the evaluation of the mean annual frequency λ(LS) of exceedance, given by

λ( LS ) = ∫ ∫ ∫ ∫ P( LS EDP) ⋅ ⋅ P( EDP IM,IP, γ ) ⋅ g(IP IM, γ ) ⋅ g(IM )

(13)

⋅ g( γ ) ⋅ d EDP ⋅ d IM ⋅ d IP ⋅ d γ If the limit state is quantified in terms of EDP, the whole procedure simply requires the evaluation of the mean annual frequency of exceedance: ∞

λ( EDP) = ∫ P ( EDP IM,IP, γ ) ⋅ g (IP IM, γ ) ⋅ g (IM ) ⋅ g ( γ ) ⋅ dIM ⋅ dIP ⋅ dγ

(14)

−∞

There are several methods to compute the integrals (12)-(14). In the numerical example below (Sec. 5), a crude Monte Carlo simulation will be used.

4. MODEL OF THE WIND FIELD The three components of the wind velocity field Vx(j), Vy(j), Vz(j) in each point j (the variation with time is omitted for simplicity) of a structure can be expressed as the sum of a mean (time-invariant) value and a turbulent component u(j), v(j), w(j) with zero mean. Vm(j) can be determined from a database of values recorded at or near the site, and evaluated as the record average over a proper time

interval (e.g. 10 minutes); its probability distribution function is given by:

 1  V k  P(Vm ) = 1 − exp −  m    2  σ  

(15)

The variation of Vm with the height z over a horizontal terrain of homogeneous roughness can be described by the logarithmic law (Lungu and Rackwitz, 2001; Solari and Piccardo, 2001):

Vm ( z ) =

 z  u* ⋅ ln   k  z0 

Where u* is the friction or shear velocity, given by: √(0.006)·Vm(z = 10 m), k is the von Karman’s constant, set equal to 0,4 and z0 is the roughness length. The turbulent components of the wind velocity are modelled as Gaussian ergodic independent processes (Solari and Piccardo, 2001); they are completely characterized by the power spectral density matrix [S]. The diagonal terms Sii(n, z) (i = u, v, w) of [S] are given by the normalised half-side von Karman’s power spectral density:

n S ii (n, z )

σ

2 i

=

4 ni ( z )

[1 + 70,8 n (z )] 2 i

5/6

(17)

Where n is the current frequency (in Hz), z is the height (in m), σ i2 is the variance of the velocity fluctuations, given by σ i2 = [6-1.1 arctan (log (z 0 ) + 1.75)] ⋅ u*2 ⋅ α i

αu=1.00, αv=0.75 and αw=0.25, u* is the friction or shear velocity (in m/s), ni(z) is a non-dimensional height-dependent frequency given by ni ( z ) =

n Li ( z ) Vm ( z )

In the example case, the integral scale Li(z) of the turbulent components have been derived according to the procedure given in (ESDU, 2001).

5. APPLICATION OF PBWE TO THE DESIGN OF A LONG SPAN SUSPENSION BRIDGE In order to check the validity of the proposed procedure for PBWE, a case example is considered: a proposed long-span suspension bridge, that has been and still is the subject of many studies in Italy (Bontempi et al. 2008a, 2008b, Bontempi 2006). The main span of the bridge is 3300 m long; including the side spans, the total length is 3666 m. The cross section of the deck is 60 m wide serving both roadways and railway traffic. Each roadway is composed by three lanes, 3.75 m wide (two driving lanes and one emergency lane) and the railway has two tracks. The towers are 383 m high and the bridge suspension system relies on two pairs of steel cables, each with a diameter of 1.24 m and a total length, between the anchor blocks, of approximately 5000 m. As indicated in Sec. 3, the structural performances are classified as high and low performances. In what follows, the serviceability of the bridge under wind actions is investigated by considering the behaviour of the deck in service, while, with regard to structural safety, the flutter stability is considered. Adopting the Limit State format, hence the simplified form of PBWE [equation (14)], the first step of the procedure requires the choice of appropriate EDPs (i.e. DMs), that are able to quantify the considered performances.

For high performances, relevant EDPs are deck translational and rotational accelerations, deck rotational velocity, deck twist that could generate a misalignment of the rails, etc. In the example case, three EDPs have been considered: the vertical acceleration (Av), the rotational velocity (Vrot) and the component Al of the acceleration in the longitudinal direction. They are referred to the center of mass of the cross section. Two high performance levels have been identified (SLS-1 and SLS-2): full serviceability (both train and vehicle transit is allowed) and partial serviceability (only train transit is possible). The two levels are identified by different thresholds. With reference to the low performance criterion (avoiding the flutter instability), the damping δ of a relevant displacement time-history (for example, the vertical displacement of the mid-span section) is assumed as EDP; δ must be positive. For each performance, failure is attained when the peak value (with reference to the whole bridge deck) of the relevant EDP overcomes the corresponding threshold (that defines the “failure criterion”) during the considered windstorm. The threshold values are summarized in Table 1.

Table 1. Considered performances and failure criteria. Limit State Performance EDP Threshold max {Vrot} = 0.04 rad/s SLS-1 Full serviceability max {Al} = 2.5 m/s2 Vrot max {Av} = 0.9 m/s2 SLS Al Partial serviceability max {Vrot} = 0.043 rad/s Av SLS-2 (only railway traffic max {Al} = 2.5 m/s2 is allowed) max {Av} = 1.1 m/s2 Total Preservation (structural + of structural min {δ} = 0 ULS ULS-FL aerodynamic) integrity damping δ The analyses have been carried out in time domain on a complete FE model of the bridge, second order effects are taken into account. The incident turbulent wind velocity time-histories have been generated as components of a multivariate, multidimensional Gaussian stationary stochastic process; the Weighted Amplitude Wave Superposition method (W.A.W.S.) and a Proper Orthogonal Decomposition (P.O.D.) of the PSD matrix (Carassale and Solari, 2006) have been adopted. Starting from the wind velocity time histories, the wind actions have been evaluated by aeroelastic theory (Salvatori and Borri, 2007). With regard to low performance (ULS-FL), the wind flow has been modelled as non turbulent. The probabilistic characterizations of the considered EDPs have been obtained by Monte Carlo simulation. The application of the PBWE procedure would require, for each considered performance, the evaluation of the mean annual frequency of exceeding the threshold value of each response parameter [equation (14)]. In the numerical example, the IP parameters have been considered deterministic, and IM is reduced to one parameter, that is, the mean wind velocity evaluated at 10 meters height (V10), assumed as a random variable, with the Weibull annual CDF given by equation (15). The parameters σ and k have been set equal to 6.02 m/s and 2.02 according to a database available for the site of the bridge; the roughness length z0 has been set equal to 0.05 m.

5.1

High performances

The annual probability densities and distribution functions for two (Vrot and Av) of the three considered EDPs, evaluated by Monte Carlo simulation (500 runs), are reported in Figure 5, for what concerns the third EDP (Al), it is always lower than the given threshold; hence the corresponding serviceability criterion is automatically satisfied in the considered example. This would suggest that the criterion does not appear significant for the design optimization.

In Figures 4 (a), (b), the mean annual frequencies λ(EDP), evaluated by equation (15) for each EDP, are reported. The values of the mean annual frequencies corresponding to the threshold values reported in Table 1 give the probability of unsatisfaction of the corresponding performance (Table 2). In the same Table 2, the values of the probability of exceeding any performance criterion for the same limit state are reported, for both SLS-1 and SLS-2. 70

0.12

Vrot [m/s]

SLS-1

0.10

SLS-2

100%

0.08 0.06 0.04 0.02

Vm [m/s] 0.00

0%

(a)

0.101

Other

0.093

0.084

0.076

0.067

0.059

35

0.051

30

0.042

25

0.034

20

0.025

15

0.017

10

0.008

5

0.000

0

0

(b) 210

4.00

Av[m/s2]

3.50

SLS-1

SLS-2

100%

3.00 2.50 2.00 1.50 1.00 0.50

Vm [m/s]

0%

(c)

Other

0.428

0.392

0.357

0.321

0.285

0

35

0.250

30

0.214

25

0.178

20

0.143

15

0.107

10

0.071

5

0.036

0

0.000

0.00

(d)

1.0

1.0

0.8

0.8

λ(EDP)

λ(EDP)

Figure 3. Results of Monte Carlo simulation: (a) – (c) maximum values of Vrot, and Av; (b) – (d) corresponding histograms and cumulative distribution functions.

0.6

0.6

0.4

0.4

0.2

0.2

0 0

0.02

0.04

0.06

0.08

0.10

EDP = Vrot - DM = max (vrot) [rad/s]

0.12

0 0

1

2

EDP = Av - DM= max (av) [m/s2]

(a)

Figure 4. Mean annual frequencies (a) λ( Vrot), (b) λ(Av).

(b)

3

In the analyses illustrated so far, only the randomness of the mean value of the wind velocity has been considered; however, in Sec. 2, it has been recognized that also the uncertainty of the roughness length z0 should be taken into account. As a preliminary check of its relevance, Monte Carlo simulations have been repeated, assuming three different values of the roughness length: z0 = 0.05 m; z0 = 0.10 m; z0 = 0.20 m. In Figure 5, the three resulting functions λ(Vrot) are reported: it appears evident that it is mandatory to take into account the uncertainty of the roughness length in assessing the risk of the bridge. However, the characterization of the PDF of z0 is a very complex problem; research is still needed on this subject.

Limit state

SLS

SLS-1

SLS-2

Table 2. “Failure” probabilities. Probability of Failure unsatisfaction of each criterion performance criterion Vrot ≥ 0.04 rad/s 0.08178 2 Al ≥ 2.5 m/s ≅0 Av ≥ 0.9 m/s2 0.09112 Vrot ≥ 0.043 rad/s 0.07009 Al ≥ 2.5 m/s2 ≅0 Av ≥ 1.1 m/s2 0.05607

Probability of unsatisfaction of any performance criterion 0.11215

0.08411

1.0 z0=0.2 z0=0.1 z0=0.05

λ(EDP)

0.8 0.6 0.4 0.2 0 0

0.04

0.08

0.10

0.12

0.14

0.16

EDP = Vrot ; DM= max (vrot) [rad/s]

Figure 5. Fragility curves λ(Vrot) for three values of the roughness length.

5.2

Low performance

In evaluating the flutter condition, the polar lines in Figure 6(a) have been considered. By means of a time-domain analysis based on the Quasi-Steady (QS) theory for the evaluation of the aeroelastic forces, the critical flutter velocity Vcrit equal to 70 m/s has been obtained. The probability of loss of the structural integrity as a consequence of flutter instability is equal to the probability that the mean wind velocity exceeds Vcrit = 70 m/s. Note that the design guidelines for this bridge indicate 57 m/s as the design wind velocity; therefore the corresponding performance criterion would be satisfied in a deterministic framework, since Vcrit > 57 m/s. However, in deriving Vcrit, the uncertainty of IP (in this case, the polar lines) has been disregarded. The relevance of the uncertainty of the aerodynamic polar lines has been investigated by a parametric analysis, adopting a linear combination of the two sets of polar lines A and B (Figure 6):

cL (α ) = cL_B (α ) ⋅ PL + cL_A (α ) ⋅ (1 − PL ) cM (α ) = cM_B (α ) ⋅ PM + cM_A (α ) ⋅ (1 − PM )

where: ci_j (i = L, M and j = A, B) is the aerodynamic coefficient i, corresponding to the polar type j; α is the generic angle of attack; PL and PM are combination parameters, which vary between 0 and 1. Resulting flutter velocities are shown in Figure 7; the points marked by a cross have been evaluated by the FE analysis, the others have been derived by linear interpolation. The response surface in Figure 7 shows that the uncertainty of the interaction parameters cannot be disregarded; in particular, the influence of the uncertainty of the moment coefficients cM is more relevant than the uncertainty of the lift coefficients cL. 0.3

0.3

0.2

0.2

0.1

0.1

0 -10

-8

-6

-4

Drag

-2 0 -0.1

2

4

6

8 [deg]

10 -10

-8

-6

-4

0 -2 0 -0.1

-0.2

-0.2

-0.3

-0.3

-0.4

-0.4

2

4

6

8 10 [deg]

-0.5

-0.5 Lift

Drag

Moment

(a) Figure 6. Polar lines: type A (a), type B (b).

Lift

Moment

(b)

Vcrit 400 300 200 100 01

1 0.5

0.5 PL

PM 0 = computed by the

FE model

0 = obtained by minimum

squares method

Figure 7. Critical flutter velocities: response surface obtained by varying the polar lines.

6. CONCLUSIONS In this paper the assessment of structures subject to wind actions has been considered in the framework of Performance-Based Wind Engineering (PBWE).

The approach proposed by the Pacific Earthquake Engineering Research Center for Performance-Based Earthquake Engineering has been extended into PBWE: the structural risk is identified with the mean annual frequency of exceeding a threshold level of a Decision Variable. The general aspects of the proposed approach have been outlined and exemplified with reference to a specific case: the assessment of the out-of-service and collapse risks of a long span suspension bridge. The main results of the numerical calculations have been illustrated and discussed. It is evident that PBWE is feasible, but to make it more reliable it is essential to improve the probabilistic description of the parameters of the wind field at the site and the phenomena that represent the interaction between the wind actions and the structure. This will require much further research work.

7. ACKNOWLEDGEMENTS The research presented in this paper have been developed in the Wi-POD Project (2008-2010) and other research projects in wind engineering, partially financed by the Italian Ministry for Education, University and Research (MIUR). Fruitful discussions with Prof. Franco Bontempi are also acknowledged.

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