Performance-Based Wind Engineering: Towards a general procedure

Structural Safety 33 (2011) 367–378

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Performance-Based Wind Engineering: Towards a general procedure M. Ciampoli ⇑, F. Petrini, G. Augusti Department of Structural and Geotechnical Engineering, Sapienza Università di Roma, Italy

a r t i c l e

i n f o

Article history: Received 8 September 2009 Received in revised form 30 June 2011 Accepted 5 July 2011 Available online 17 August 2011 Keywords: Eolian risk Probabilistic risk assessment Performance-Based Design Performance-Based Wind Engineering Suspension bridge Out-of-service risk Flutter instability

a b s t r a c t It is widely recognized that the most rational way of assessing and reducing the risks of engineered facilities and infrastructures subject to natural and man-made phenomena, both in the design of new facilities and in the rehabilitation or retrofitting of existing ones, is Performance-Based Design, usually indicated by the acronym PBD (but a better term would be ‘‘Performance-Based Engineering’’). The first formal applications of PBD were devoted to seismic engineering and design; later it has been extended to other engineering fields, like Blast Engineering and Fire Engineering. Wind Engineering has appeared of great potential interest for further developments of PBD. The expression ‘‘Performance-Based Wind Engineering’’ (PBWE) was introduced for the first time in 2004 by an Italian research project. In this paper, the approach proposed by the Pacific Earthquake Engineering Research Center (PEER) for Performance-Based Earthquake Engineering 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. A discussion of the open problems and the relevance of various sources of uncertainty conclude the paper. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction A modern approach to structural design should not be limited to optimize safety vs. costs, but consider as key objectives the whole range of ‘‘performances’’ in a probabilistic context and throughout the whole life-cycle of the structure: this has led to the recent significant development of ‘‘Performance-Based Design’’ (PBD) procedures (for recent reviews, see e.g. [1] and [2]). The first formal applications of PBD were devoted to seismic engineering and design [3]. More recent, but rapidly expanding, are extensions of PBD to other problems, and in particular to Wind Engineering (the definition ‘‘Performance Based Wind Engineering’’, PBWE, appeared for the first time in 2004 [4]). The approach is not new. During the last decades, the problem of the assessment of structural safety under wind action has been studied extensively: advanced probabilistic approaches have been largely adopted in Wind Engineering due the aleatory nature of winds (e.g. [5–7]). Great efforts have been devoted to the probabilistic characterization of both wind actions (e.g. [8–12]) and response of structures with uncertain characteristics under wind (e.g. [13–16]). Also the performance assessment of structures subject to wind actions has been the object of many studies in the past decade. Among the others: Unanwa et al. [17] developed a procedure to ⇑ Corresponding author. Tel.: +39 06 44 585 300;, fax: +39 06 48 84 852. E-mail address: [email protected] (M. Ciampoli). 0167-4730/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.strusafe.2011.07.001

predict the wind damage to low and medium rise buildings subject to hurricanes; Rosowski and Ellingwood [18], Ellingwood et al. [19] and van de Lindt and Dao [20] focused on wood structure performances under synoptic winds or hurricanes; Khanduri and Morrow [21] studied the vulnerability of buildings to windstorms; Garciano et al. [22] developed a procedure for the assessment of the critical performances for wind turbines subject to typhoons; Bashor et al. [23], Bashor and Kareem [24], Tamura [25] and Kim et al. [26] focused on comfort of tall building occupants. Notwithstanding this large effort in probabilistic assessment of structural performances, there is not yet consensus on a general Performance-Based approach for the assessment of risk under wind action (‘‘Eolian risk’’). The goal of this paper is a self-contained presentation of the probabilistic procedure for the application of Performance-Based Design concepts to Wind Engineering that the authors have outlined in previous paper [1,27–30]. The procedure is then applied to an example case: the assessment of both the failure probability due to flutter instability and the out-of-service risk of a bridge design derived from a preliminary design of the suspension bridge over the Messina Strait (Italy) [31]. A discussion of the open problems concludes the paper. 2. Sources of uncertainty in Wind Engineering As previously stated PBWE must be tackled in probabilistic terms, due to the stochastic nature of both resistance and loading parameters. First, in characterizing the wind field and the corre-

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sponding actions, different sources of uncertainty are to be taken into account: they must be considered differently according to their relation to the structure (Fig. 1). By definition, in the environment the wind field is considered as if the structure were absent (free-field wind). 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, etc.) 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 the interaction between the relevant properties of both the structure and the wind field, as well as the presence of nearby structures, cannot be disregarded. Hence aerodynamic and aeroelastic phenomena are essential in determining the relevant

features of the wind effects. 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. As concern the characterization of the wind field in the environment, the uncertainty is due to: (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 incompleteness of data and information (the epistemic uncertainty); (iii) the modeling of wind actions and their effects on structural response (the model uncertainty, often included in the epistemic uncertainty). In the exchange zone, the interaction parameters are strongly dependent on the basic parameters that characterize the wind field

EXCHANGE ZONE

ENVIRONMENT

STRUCTURAL SYSTEM

Aerodynamic and aeroelastic phenomena Structural systems

Wind site basic parameters Wind action Site-specific Wind

Structural system as modified by service loads

Environmental effects (like waves)

Non environmental actions Types of uncertainties

1. 2. 3.

Aleatory Epistemic Model

1. 2. 3.

Basic parameters ( IM )

Aleatory Epistemic Model

Derived parameters ( IP)

1. 2. 3.

Aleatory Epistemic Model

Independent parameters (SP )

2

Exchange zone

Stru

ctu

1

Environment

re

Vm+ v(t)

r

rive

Turbulent wind velocity profile

Vm

Mean wind velocity profile

Fig. 1. Sources of uncertainty in Wind Engineering.

M. Ciampoli et al. / Structural Safety 33 (2011) 367–378

and the structural behavior. Examples of derived parameters in case of suspension bridges are the aerodynamic coefficients (whose plots are sometimes referred to as ‘‘polar lines’’), the aerodynamic derivatives (or ‘‘aeroelastic derivatives’’), the Strouhal number (see [32] Chapters 4 and 6). 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. The parameters that characterize the structural behavior, like the mechanical and material properties of the structure, are independent of the basic parameters of the wind field, and often their inherent uncertainty can be considered negligible. In what follows, the uncertain basic parameters in the environment are grouped in the intensity measure vector IM; the uncertain parameters of interest in the exchange zone are grouped in the two vectors of derived interaction parameters IP and independent structural parameters SP (Fig. 1). The vectors IM and SP can be assumed not correlated with each other and not affected by the uncertainty of the parameters IP, i.e.

PðIMjIPÞ ¼ PðIMjSPÞ ¼ PðIMÞ

ð1Þ

PðSPjIPÞ ¼ PðSPjIMÞ ¼ PðSPÞ

ð2Þ

where P(|) is a conditional probability. In PBWE procedures, conditional probabilities, conditional density functions and the total probability theorem are essential. From elementary theory of probability, given Eqs. (1), (2), the joint probability of IM, IP and SP is:

PðIM; IP; SPÞ ¼ PðIPjIM; SPÞ  PðIMjSPÞ  PðSPÞ ¼ PðIPjIM; SPÞ  PðIMÞ  PðSPÞ

ð3Þ

and the complementary cumulative distribution function (CCDF) of a continuous random variable X dependent on the basic, derived and independent parameters is:

GðxÞ ¼ PðX > xÞ Z 1 GðxjIM; IP; SPÞ  f ðIM; IP; SPÞ dIM dIP dSP ¼ 1 Z 1 GðxjIM; IP; SPÞ  f ðIP jIM; SPÞ  f ðIMÞ ¼ 1

 f ðSPÞ dIM dIP dSP

ð4Þ

where f() is a probability density function (PDF) and f(|) a conditional probability density function (CPDF). 3. A procedure for Performance-Based Wind Engineering (PBWE) The procedure for PBWE presented in this paper has been obtained by extending the approach proposed for Performance-Based Seismic Design (PBSD) by the researchers of the Pacific Earthquake Engineering Research Center (PEER) (see e.g. [3]). The central objective of any procedure for Performance-Based Design 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 interest of the stakeholder or the society in general. In Wind Engineering, relevant examples of DVs are the number of lives lost during windstorms, the economic losses resulting from windstorms, the exceeding of a (collapse or serviceability) limit state, the discomfort of the occupants, the length of the out-of-service time due to a windstorm, etc. The starting points of the procedure are the relationships, expressed in probabilistic terms, between the performances specific

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to the considered construction (no collapse, occupant safety, accessibility, full functionality, limited displacements or accelerations, etc.) and different ‘‘intensities’’ or mean return periods of the wind action. With reference to a specific performance, usually the structural risk is conventionally measured by the probability of exceeding a relevant value of the corresponding DV. This probability of exceedance is expressed in terms of a mean annual frequency or a complementary cumulative distribution function (CCDF), that are evaluated by taking into account the Eolian hazard (i.e. the frequency of 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 should be optimized by applying a decisional strategy to the risk analyses, with the objective of minimizing the total risk or of maximizing a utility function. Thus, in general terms, a procedure of PBWE should consist of the following steps. First, the Eolian hazard at the site, in terms of wind intensity and/or parameters of the wind velocity field shall be defined. The assessment of the Eolian hazard requires the use of efficient techniques for modeling wind actions on a slender structure, and the choice of the intensity measure vector IM whose stochastic characteristics are able to describe sufficiently and efficiently the Eolian hazard at the site. Hence IM must be chosen by considering the characteristics of the wind turbulence, direction and arrival processes, the relevant features of the construction site, the structural properties and the interaction phenomena (wind turbulence, vortex shedding, aeroelastic phenomena, aerodynamic effects). However, the choice must be done by paying attention to minimizing the needed information, since today the available data (essentially turbulence spectra and maps of expected wind speeds) are still rather limited. Then, the interaction phenomena shall be modeled in probabilistic terms. This implies the choice and probabilistic characterization of the interaction parameter vector IP, that allow to take into account the relevant aspects of the interaction between the environment and the structure in the exchange zone. The following step is the analysis of the structural response, mainly in the context of stochastic dynamics. The probabilistic modeling of the structural response requires the choice of the relevant engineering demand parameter vector EDP (inter-storey drifts, accelerations and velocities of selected points, stresses and displacements, etc.). The evaluation of the structural damage (intended as unacceptable performance) requires the choice (and probabilistic characterization) of a 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 one or a combination of relevant EDPs (e.g. the inter-storey drift), or by other parameters, representing e.g. the damage to the partition or façade walls in a building as a function of the inter-storey drift. However, in the latter case it is usually rather difficult to establish sound correlations between the evaluated EDPs and the chosen DMs. The following step is the definition of the decision variables DVs that are appropriate to quantify the performances required for the structure, in terms of consequences of damage (personal damages, restoration costs, costs due to loss or deterioration of service, alterations of users comfort, etc.). The decision variables DVs must distinguish between low and high performances or performance levels: the former (low performances) imply possible consequences on structural and personal safety (e.g. partial or total collapse and permanent damages); the latter (high performances) affect only serviceability and comfort (e.g. small displacements, limited vibrations, and wind discomfort also in the area around the construction) [33,34]. For low performances, the significant DV

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can be identified with the cost necessary to restore the construction to the undamaged state (or rebuild it in case of collapse); correspondingly, DM is the set of damages to be restored, and EDP includes the most significant response parameter for the specific case (peak displacement or acceleration at the building top, overall action at the base, local pressure, etc.). The shortcoming of this definition is the impossibility of including in it the ‘‘intangible’’ losses, i.e. the losses that cannot be measured in ‘‘monetary’’ terms. 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 [35]. Using the ‘‘limit states’’ approach (i.e. if the structural risk is quantified by the probability of exceeding a limit state), ultimate limit states (ULS) are 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.), while serviceability limit states (SLS) are related to high performances (examples are excessive deformations or vibrations compromising the use of the structure or its function in service). The structural risk is then evaluated on the basis of the probabilistic characterization of the decision variables. Appropriate relationships between any DM and the relevant EDP allow evaluating the damage states corresponding to given values of the response parameter EDP, and also the resulting losses, taking into consideration the relationships between DM and DV. According to the usual definition of risk as the convolution of hazard, vulnerability and exposure [36], the relationships between DM and DV take into account the exposure that reflects the consequences of damage. In the PBWE procedure adopted in this paper (as in the PEER’s approach), the structural risk is defined as the probability of exceeding a threshold level of a relevant DV:

GðDVÞ ¼

Z Z Z Z Z

GðDV jDMÞ  f ðDM jEDP Þ

 f ðEDPjIM;IP; SPÞ  f ðIP jIM; SPÞ  f ðIMÞ  f ðSPÞ  dDM  dEDP  dIM  dIP  dSP

ð5Þ

where, as already stated: DM is a damage measure; EDP is the corresponding engineering demand parameter, representing the structural response; the basic parameters characterizing the Eolian hazard (Section 2) are described by a vector IM of intensity measures; the vector of the interaction parameters IP contains 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; SP is the vector of the parameters characterizing the structural systems and non environmental actions. By means of Eq. (5), the problem of risk assessment is disaggregated into the following elements: site and structure-specific hazard analyses, that is, the assessment of the probability density functions f(IM), f(SP) and f(IP|IM, SP); structural analysis, aimed at assessing the probability density function of the structural response f(EDP|IM, IP, SP) conditional on the parameters characterizing the wind field and the structural properties; damage analysis, that gives the damage probability density function f(DM|EDP) conditional on EDP; finally, loss analysis, that is the assessment of G(DV|DM). Under these assumptions, a flowchart similar to the PEER flowchart can be defined for the proposed PBWE procedure (Fig. 2). 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 or CFD techniques [37,38]. Moreover, the characterizations of the

environment (in terms of IM) and of the structural system (in terms of SP) are made separately, the former depending on the candidate location of the structure, and the latter depending on the candidate design configuration. Eq. (5) 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 probability of exceedance, given by

GðLSÞ ¼

Z Z Z Z

GðLSjEDPÞ  f ðEDPjIM;IP; SPÞ  f ðIP jIM; SPÞ

 f ðIMÞ  f ðSPÞ  dEDP  dIM  dIP  dSP

ð6Þ

If the limit state is quantified in terms of EDP, the whole procedure simply requires the evaluation of the probability of exceedance:

GðEDPÞ ¼

Z

1

GðEDP jIM;IP; SPÞ  f ðIPjIM; SP Þ  f ðIMÞ  f ðSPÞ

1

 dIM  dIP  dSP

ð7Þ

There are several methods for computing the integrals (5–7). In the numerical example below (Section 4), a crude Monte Carlo simulation will be used. The final step of the procedure is the optimization of design, that is the minimization of risk, by appropriate techniques of decision analysis. The optimization of design in PBWE would require the development and implementation of a decisional strategy aimed at minimizing the total risk or at maximizing an appropriately defined utility. In the high performance case, the relevant DVs could be identified with the economical losses due to windstorms, evaluated taking into account the whole lifetime of the construction; in the low performance case, casualties and losses of lives would be involved. The ethical and practical difficulties of taking both aspects into account add to the overwhelming analytical difficulties and to the lack of reliable data: as suggested in [39], it is realistic to leave the intangible and monetary aspects separate, and open the final decision. These aspects are currently being developed [40]. Moreover, it has to be noted however that the ‘‘complete’’ decisional process is seldom fully pursued in PBSD, and likewise in PBWE: actual procedures, like those discussed and presented in this paper, omit the last step, i.e. are limited to evaluating the risk, and at most to comparing the risk associated to different design alternatives. 4. Model of the wind field The modeling of the wind field assumed in this paper is summarized in this section: it is described in detail in several texts, e.g. [32,41]. A discussion of the influence of the uncertainties in the considered model and the effect of alternative models is reported in Section 6. Indicating by x, y, z respectively the horizontal axis orthogonal to the structure (in the example case, a suspension bridge), the horizontal axis parallel to the structure and the vertical axis, the three components of the wind velocity field Vx(j), Vy(j), Vz(j) in each point j(x, y, z) of the structure (the indication of the dependence on time is omitted for simplicity of notation) can be expressed as the sum of a mean value Vm(j), that is time-invariant and depends only on the height z above terrain of point j, and turbulent components u(j), v(j), w(j) with zero mean. It is assumed that Vm(j) = Vm(z) is parallel to the x axis and its value is 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 min) at a proper height (e.g. 10 m); the probability distribution function of Vm(z) is assumed as [42]:

"

 k # 1 V m ðzÞ PðV m ðzÞÞ ¼ 1  exp  2 r

ð8Þ

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Hazard analysis

f (IM|O) Environment info

f (IM )

O

Aerodynamic analysis

Structural analysis

Damage analysis

Loss analysis

f (IP|IM,SP)

f (EDP|IM,IP,SP)

f (DM|EDP)

f (DV|DM ) Decisionmaking

IM: intensity measures O: location D: design SP: structural system parameters

f (IP)

G(EDP)

G (DM )

G (DV )

IP: interaction parameters

EDP: engineering demand parameters

DM: damage measures

DV: decision variables

Select O, D

f (SP)

D Structural system info

f (SP|D) Structural characterization Fig. 2. Performance-Based Wind Engineering flowchart.

The variation of Vm(z) with the height z above a horizontal terrain of homogeneous roughness is well described by the logarithmic law [11,42]:

V m ðzÞ ¼

  u z  ln z0 Ka

ð9Þ

where Ka is the von Kármán’s constant, usually set equal to 0.41; z0 is defined as the ‘‘roughness length’’; u is the friction or shear velocity (in m/s), given by: [(F)1/2  V10], where F is a coefficient dependent on the roughness length z0. The turbulent components of the wind velocity are modeled as Gaussian ergodic independent processes [11]. By considering the wind acting on N points, and neglecting the component v, the turbulent components u and w are completely characterized by the power spectral density matrices [S]i (i = u, w). The diagonal terms (auto-spectra) Sij ij ðnÞ of [S]i (j = 1, 2,. . ., N) can be expressed in terms of normalized one-side power spectral density ([11,43]) as

nSuj uj ðnÞ

r2u

¼

nSwj wj ðnÞ

r2w

6:868  nu ½1 þ 10:302 n2u ðzj Þ5=3

¼

6:103 nw ½1 þ 9:1545 n2w ðzj Þ5=3

ð10Þ

ð11Þ

where n is the current frequency (in Hz); z is the height above terrain (in m), r2u and r2w are the variances of the velocity fluctuations, given by

r2u ¼ ½6  1:1 arctanðlogðz0 Þ þ 1:75Þ  u2

ð12Þ

rw ¼ 0:5 ru

ð13Þ

ni(z) is a non-dimensional height-dependent frequency given by

ni ðzÞ ¼

nLi ðzÞ V m ðzÞ

ð14Þ

In the example case, the integral scale Li(z) of the turbulent components for i = u, w have been derived according to the procedure given in [44].

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 bridge design derived from the 2005 preliminary design of the suspension bridge over the Messina Strait [31] has been considered as a case example (Fig. 3). To develop this example, a number of simplified assumptions are introduced into the general procedure described so far. For details of the computations, see [45]. The main span of the bridge is 3300 m; the total length, including the side spans, is 3666 m. The cross section of the deck (61 m wide) is composed of three box elements, the external ones for the roadways and the central one for the railway. Each roadway is composed of 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; the bridge suspension system is made by two steel cables, each with a diameter of 1.24 m and a total length, between the anchor blocks, of approximately 4700 m, and by 121 pairs of rope hangers. As indicated in Section 3, the structural performances are distinguished between high and low performances. In what follows, the serviceability of the bridge under wind actions (high performances) is investigated by considering the behavior of the deck in service, while, with regard to structural safety (low performances), the flutter stability is considered. A more detailed description of the required performances for this complex structure can be found in [46]. Adopting the Limit State format, hence the simplified form of PBWE [Eq. (7)], 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 could be the deck rotational velocity, the deck translational and rotational accelerations, the deck twist that could generate a misalignment of the rails, etc. In the example case, three EDPs have been considered: the rotational velocity (Vrot) and the components of the acceleration in the longitudinal Al and vertical Av directions, all referred to the centroid (center of mass) of the cross section. Two high performance levels (SLS-1 and SLS-2) have been identified, that correspond to full serviceability (SLS-1: both train and

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M. Ciampoli et al. / Structural Safety 33 (2011) 367–378

+383 m

+383 m +63 m

+52 m

+118 m

+77 m

+53 m

183 m

S

183 m

777 m

3300 m

627 m

C

Main suspension cables (twin)

variable

50.63 m

2.78 m

Hangers

12.00 m

2.69 m

7.18 m

Wind barriers Service lane

Roadway grider

3.67 m

Railway grider

14.74 m

8.48 m 53.08 m 61.00 m

Fig. 3. Characteristics of the long-span suspension bridge considered as an example case (2005 preliminary design).

vehicle transits are allowed) and partial serviceability (SLS-2: only train transit is allowed). The two levels are identified by different thresholds. In the low performance criterion (ULS-FL: avoidance of flutter instability), the motion must not diverge: the EDP is identified with the total damping d of a relevant displacement time-history, i.e. the vertical displacement of the mid-span section, that 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. Relevant threshold values are summarized in Table 1. The analyses have been carried out in time domain on a finite element model of the bridge; the total number of elements (beams, no compression cable elements and gaps) is 1614, and the number of nodes is 1140. A Newmark’s time integration scheme has been adopted, to take into account nonlinear and second order effects. The incident turbulent wind velocity time series have been generated as components of a multivariate, multidimensional Gaussian stationary stochastic process; the Weighted Amplitude Wave Superposition method (WAWS) and a Proper Orthogonal Decomposition (POD) of the PSD matrix [43] have been adopted. Starting from the wind velocity time series, the wind actions have been

evaluated by aeroelastic theories [47,48]. With regard to low performance (ULS-FL) and the determination of the flutter velocity, the wind flow has been modeled as non turbulent (laminar). The PBWE procedure requires, for each considered performance, the evaluation of the annual probability of exceeding the threshold value of each response parameter [Eq. (7)]. In the numerical example, the IP and SP parameters have been considered deterministic, and IM is reduced to one parameter, namely the mean wind velocity evaluated at the height of 10 m, that is assumed as a random variable, with the Weibull CDF given by Eq. (8). In order to describe the 10-min average wind velocity, the parameters r 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. This value, that corresponds to a terrain of second category according to Structural Eurocode EN 1991 [49], is higher than the currently adopted value for over-water roughness length: however, it has been specifically indicated in the guidelines for the design of the bridge (see Section 6.1 for an investigation of the effect of this value). Starting from the mean wind velocity evaluated at the height of 10 meters, the mean wind velocity Vm(zdeck) at the height of the deck zdeck (equal at midspan to 77 m) is evaluated by Eq. (9).

Table 1 Considered performances and failure criteria. Limit state

Performance

EDP

Threshold

SLS

SLS-1 SLS-2

Vrot Al Av

max {Vrot} = 0.040 rad/s max {Al} = 2.5 m/s2 max {Av} = 0.7 m/s2 max {Vrot} = 0.043 rad/s max {Al} = 2.5 m/s2 max {Av} = 1.0 m/s2

ULS

ULS-FL

Full serviceability Partial serviceability (only railway traffic is allowed) Preservation of structural integrity

Total (structural + aerodynamic) damping d

min {d} = 0

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Obviously, considering just one parameter as the only source of uncertainty in characterizing the wind field represents a drastic simplification: however, in the Authors’ opinion this choice does not affect the validity of the proposed procedure. The developed numerical calculations shall be considered as a check of its feasibility; the role of different sources of uncertainty will be considered if the real problem of the optimal design of the considered bridge should be faced with. Moreover, in parallel with the development of the procedure, a huge effort is currently devoted to collect data about the probabilistic characterization of the relevant parameters, together with their correlation. In the example case, only the aleatory uncertainty has been considered; however in the following, some remarks about the relevance of the epistemic and model uncertainties will be illustrated.

5.1. Results for high performances (serviceability) The annual probability densities and distribution functions of the three considered EDPs, evaluated by Monte Carlo simulation (500 runs), are reported in Fig. 4. From Fig. 4c it is evident that the longitudinal acceleration Al 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: therefore, it has been neglected in the successive elaborations. In Figs. 5 and 6, the fragility curves of the bridge deck, evaluated considering three different threshold values of the two relevant EDPs (the rotational velocity Vrot and the vertical acceleration Av) are shown. By definition, the fragility functions P(EDP|IM) define the probability of violation of a threshold: those shown in Figs 5 and 6 have been assessed by following a procedure analogous to the procedure described in [50]. For each EDP, a performance indicator Yi has been introduced, defined as

Yi ¼

EDPi EDPi

ð15Þ

where EDPi is the threshold value corresponding to the relevant performance criterion. Under these assumptions it is possible to write the fragility function as follows:

PðEDPi P EDPi jIMÞ ¼ PðY i P 1jIMÞ ¼ PðIM P IMðY i ¼ 1ÞÞ

ð16Þ

where IM(Yi = 1) is the value of IM for which Yi = 1. IM(Yi = 1) is assumed as a stochastic variable described by a Lognormal distribution whose median and fractional standard deviation have been computed by using the results of Monte Carlo simulation. In Fig. 7, the fragility curves evaluated for the threshold values corresponding to each full serviceability criterion (Table 1, SLS-1) are shown (thin lines): it can be noted that for small wind velocities, the Av performance criterion (denoted by +) prevails, while for large velocities the Vrot criterion (denoted by ) prevails. The serviceability of the bridge should be evaluated with respect to violation of either performance criterion: the corresponding fragility curve, derived directly from results of Monte Carlo simulation, is plotted as a bold line in the same Fig. 7. In Figs 8 (a), (b), (c), the complementary cumulative probability distribution functions G(EDP), evaluated by Eq. (7) for each EDP, are reported. From these functions, the exceedance probabilities corresponding to the threshold values reported in Table 1 (i.e. the probabilities of not satisfying the corresponding performances) are derived. These values are reported in Table 2, together with the values of the probability of exceeding any performance criterion for the same limit state, evaluated for both SLS-1 and SLS-2.

373

5.2. Results for low performance (avoidance flutter instability) In evaluating the flutter condition, the aerodynamic coefficients in Fig. 9a have been considered. By means of a time-domain analysis based on the Quasi-Steady (QS) theory for the evaluation of the aeroelastic forces [47,51], the time series of the vertical displacement at bridge midspan for several values of Vm(zdeck) have been obtained [Fig. 10a]. Damping has been evaluated as the parameter d of the envelope function:

qðtÞ ¼ q  q0  edt

ð17Þ

of the time series. d represents the total damping of the structural system, which is the sum of the structural component (assumed to be constant and equal to 0.5%) and the aerodynamic component. In Fig. 10(b), the total damping d is shown as a function of Vm(zdeck). The low performance criterion: d > 0 is satisfied for Vm(zdeck) < 70 m/s. Hence the critical flutter velocity is: Vcrit = 70 m/s. 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 70 m/s. Note that the design guidelines for this bridge [31] indicate 57 m/s as the limit flutter wind velocity; therefore the corresponding performance criterion would be satisfied in a deterministic framework, since Vcrit > 57 m/s. In Section 6, the role of the uncertainties of the model assumed for evaluating Vcrit and of the aerodynamic coefficients will be discussed. 6. Investigation on further sources of uncertainty 6.1. Uncertainty of the roughness length In the analyses illustrated in Section 5.1, only the randomness of the mean value of the wind velocity has been considered. In Section 2, it has been recognized that also the uncertainty of the roughness length z0 should be taken into account; however, very little research has been so far performed to characterize the variability of z0. 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 (the value assumed in Section 5.1); z0 = 0.10 m; z0 = 0.20 m. In Fig. 11, the three resulting functions G(Vrot) are reported: the large effect of variations in the roughness length on the risk of the bridge appears evident (the risk decreases when z0 is increased). 6.2. Uncertainty of the aerodynamic coefficients In Section 5.2, only the variation of the mean wind velocity Vm has been considered in deriving the critical (flutter) velocity Vcrit, and the uncertainty of the interaction parameters IP (in this case, the aerodynamic coefficients) has been disregarded. The relevance of the uncertainty of the aerodynamic coefficients has been investigated by a parametric analysis, adopting a linear combination of the two sets of aerodynamic coefficients A and B (Fig. 9), that were derived in experimental tests on models at different scales [29,52]:

cL ðaÞ ¼ cL B ðaÞ  PL þ cL A ðaÞ  ð1  PL Þ cM ðaÞ ¼ cM B ðaÞ  PM þ cM A ðaÞ  ð1  PM Þ

ð18Þ

where: ci_j (i = L, M and j = A, B) is the aerodynamic coefficient i, corresponding to the set j; a is the generic angle of attack; PL and PM are combination parameters, which vary between 0 and 1. The resulting flutter velocities are shown in Fig. 12; 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 Fig. 12 shows that the uncertainty of the interaction parameters should not be disregarded; in particular, the influence of the uncertainty of the

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M. Ciampoli et al. / Structural Safety 33 (2011) 367–378

(a)

70

(b)

0.12

Vrot [rad/s]

SLS-1 SLS-1

SLS-2 SLS- 2

100%

0.10 0.1 0.08 0.08 0.06 0.04 0.02

Vm (zdeck ) [m/s]

0.101 0.101 Other Other Other

0.084 0.084

0.093 0.093

0.076 0.076

35

0.059 0.059

30

0.067 0.067

25

0.051 0.051

20

0.042 0.042

15

0.034 0.034 0.034

10

0.017 0.017 0.017 0.025 0.025 0.025

5

0.000 0.000 0.000 0.008 0.008 0.008

0

0%

0

0 0.00

210

(c)

(d)

3.00

Al [m/s2]

100% 100%

2.50 SLS-1

SLS-2

2.00 1.50 1.00 0.50

Vm (zdeck ) [m/s]

(e)

0% 0.428 0.428

Other Other

0.392 0.392

0.357 0.357

0.321 0.321

0.285 0.285

0

35

0.250 0.250

30

0.214 0.214

25

0.178 0.178

20

0.134 0.143

15

0.107 0.107

10

0.071 0.071

5

0.036 0.036

0

0.000 0.000

0.00

70

4.00

Av [m/s2]

3.50

SLS-1

(f)

SLS-2

100% 100%

3.00 2.50 2.00 1.50 1.00 0.50

Vm (z deck ) [m/s]

0.00

Other Other Other

2.456 2.456

2.251 2.251

2.047 2.047

1.637 1.637

1.842 1.842

1.433 1.433

35

1.228 1.228

30

1.024 1.024

25

0.819 0.819

20

0.614 0.614

15

0.410 0.410

10

0.205 0.205

5

0.000 0.000

0

0

0% 0%

Fig. 4. High performances: results of Monte Carlo simulations (500 runs): (a), (c), (e): maximum values of Vrot, Al and Av; (b), (d), (f): corresponding histograms and cumulative distribution functions.

moment coefficient cM is more relevant than the uncertainty of the lift coefficient cL. 6.3. Uncertainty of the wind-action model Also the uncertainties in modeling the aeroelastic actions play a major role: this question has been investigated with some detail in [28] and [47]. Lacking a quantitative treatment, four simplified models of increasing complexity have been successively considered, namely: the non-aeroelastic (NO) theory (aeroelastic effects are disregarded, and the angle of incidence of the wind changes with time only due to wind turbulence); the steady (ST) theory (the angle of incidence changes with time due to both the incident wind turbulence and the torsional rotation – twist – of the deck);

the quasi-steady (QS) theory (the instantaneous aeroelastic forces acting on the structure are the same that act on the structure when it moves with constant translational and rotational velocities, with values equal to the actual instantaneous ones: the angle of incidence changes with time due to both the incident wind turbulence and the torsional rotation – twist – of the deck, but is computed by considering the relative velocity between the wind and the deck); the modified quasi-steady (QSM) theory (the aerodynamic lift and moment coefficients are considered variable with time and measured by wind tunnel tests [52]). The analyses have been performed for mean incident wind speeds equal respectively to 21, 45 and 57 m/s. It has been noted (see [47]) that, when the action model complexity increases (from NO to QSM), both the mean values and the variances of the deck response parameters (displace-

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M. Ciampoli et al. / Structural Safety 33 (2011) 367–378

1

(a) 1.0

P( Vrot > v rot*|V m(zdeck))

0.9

0.8

0.8

G(EDP)

0.7 0.6

0.6

0.4

0.5 0.4

0.2 0.3 0.2

0 0

0.1

0.02

V m(zdeck) [m/s] 0 0

5

10

15

20

25

30

0.04

0.06

0.08

0.10

0.12

EDP = Vrot - DM = max (vrot) [rad/s] 35

(b) 1.0 0.8

G(EDP)

Fig. 5. Fragility curves for EDP = Vrot and three different threshold values – D: mrot = 0.02 rad/s; s: mrot = 0.03 rad/s; h: mrot = 0.04 rad/s (full serviceability).

1

0.6

0.4

P( A v > a v*|V m(z deck ))

0 .9

0.2

0 .8 0 .7 0 .6

0

0.10

0 .5

0.20

0.30

0.40

0.50

EDP = Al - DM= max (Al ) [m/s 2 ]

0 .4

(c) 1.0

0 .3 0 .2

Vm(z deck ) [m/s]

0 0

5

10

15

20

25

30

35

G(EDP)

0.8

0 .1

0.6

0.4

Fig. 6. Fragility curves for EDP = Av and three different threshold values – D: am = 0.45 m/s2; s: am = 0.63 m/s2; h: am = 0.90 m/s2 (full serviceability).

0.2

0 1

0

1

* +

0.9

2

3

EDP = A v - DM= max (a v) [m/s2 ] Fig. 8. Exceedance probabilities (a) G(Vrot), (b) G(Al), (c) G(Av).

0.8 0.7

+*

0.6 0.5

Table 2 ‘‘Failure’’ probabilities.

0.4 0.3

Limit state

Failure criterion

Probability of not satisfying each performance criterion

Probability of not satisfying either performance criterion

Vrot P 0.04 rad/s Al P 2.5 m/s2 Av P 0.7 m/s2 Vrot P 0.043 rad/s Al P 2.5 m/s2 Av P 1.0 m/s2

0.0720 ffi0 0.150 0.0620 ffi0 0.0920

0.1541

0.2 0.1

Vm(zdeck ) [m/s]

SLS

SLS-1

0 0

5

10

15

20

25

30

35

Fig. 7. Comparison between the fragility curves corresponding to each full serviceability criterion (+: Av; : Vrot) and (in bold line) to the violation of either performance criterion.

SLS-2

0.0980

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M. Ciampoli et al. / Structural Safety 33 (2011) 367–378

0.3

(a)

1.0

z0=0.2 z0=0.1 z0=0.05

0.2 0.8

0.1

-8

-6

-4

-2

0

2

4

6

-0.1

8

10

[deg]

-0.2

G(EDP )

0 -10

0.6

0.4

-0.3 0.2

-0.4 -0.5

Drag

Lift

Moment

0

0

0.02

0.3

(b)

0.06

0.04

0.10

0.08

0.12

EDP = Vrot - DM = max ( vrot) [rad/s] Fig. 11. Exceedance probability functions G(Vrot) for three values of the roughness length.

0.2 0.1 0

-10

-8

-6

-4

-2

0

2

4

6

-0.1

8

10

[deg]

Vcrit

-0.2

400 -0.3

300

-0.4 -0.5

Drag

Lift

Moment

200

Fig. 9. Polar lines (plots of aerodynamic coefficients): (a) basic; (b) alternative.

100

(a)

0 1

0.525

q [m] 0

1

0.520

0.5

0.5

PPMM

q=q+q0· e-δt 0.515

0

0.510

= computed by the FE model

PL P L

0

= obtained by minimum squares method

Fig. 12. Critical flutter velocities: response surface obtained by varying the polar lines.

0.505

t t[sec] [sec] 0.500 600 600

(b)

700

800

900 900

1000 1000

ments, rotations, velocities) decrease. The same trend appears also in the distributions along the whole length of the bridge of the maximum transversal and vertical displacements.

1.5 Damp (%)

7. Concluding remarks

1.0 0.5 0.0 0

10

20

30

40

50

60

70

80

-0.5 Wind Velocity [m/s]

-1.0 -1.5

Total

Structural

Aerodynamic

Fig. 10. (a) Typical time series of the vertical displacement at bridge midspan for a given value of Vm; (b) damping as functions of Vm.

This paper essentially aims at setting up a roadmap towards a general procedure for the assessment of the performances of structures subject to wind actions. The approach proposed by PEER for Performance-Based Seismic Engineering has been extended to Performance-Based Wind Engineering (PBWE): the structural risk is identified with the mean annual probability 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. In the final Section 6, a few qualitative results on the effects of some

M. Ciampoli et al. / Structural Safety 33 (2011) 367–378

sources of uncertainty have been presented: their great importance is evident. It can be concluded 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 collecting and elaborating many more experimental data, and much further research work also to improve the framework proposed in this paper, including optimization. Acknowledgements The researches presented in this paper have been developed within 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). This paper is an extended and updated version of the papers presented at ICOSSAR’09 [27], COMPDYN’09 [28] and IMECE’09 [30], and is a synthesis of the results illustrated in [45]. References [1] Augusti G, Ciampoli M. Performance-based design in risk assessment and reduction. Probabilist Eng Mech 2008;23(4):496–508. [2] Augusti G, Ciampoli M. Performance-based design as a strategy for risk reduction: application to seismic risk assessment of composite steel-concrete frames. In: Korytowski A, Malanowski K, Mitkowski W, Szymkat M, editors. System modeling and optimization – 23rd IFIP TC7 Conference. Springer Verlag; 2009. p. 3–22. [3] Porter KA. An overview of PEER’s performance-based engineering methodology. In: Armen Der Kiureghian A, Madanat S, Pestana JM editors. Applications of statistics and probability in civil engineering – proceedings of the ninth international conference on applications of statistics and probability in civil engineering ICASP9; San Francisco, CA, USA. Rotterdam: Millpress; 2003. p. 973–80. [4] Paulotto C, Ciampoli M, Augusti G. Some proposals for a first step towards a performance based wind engineering. In: Proceedings of the IFEDinternational forum in engineering decision making; First Forum, December, 5–9, Stoos, Switzerland; 2004. . [5] Kareem A. Wind effects on structures: a probabilistic viewpoint. Probabilist Eng Mech 1987;4(2):166–200. [6] Schuëller GI, editor. A state-of-the-art report on computational stochastic mechanics. Probabilist Eng Mech 1997;12(4):197–321. [7] Kareem A. Numerical simulation of wind effects: a probabilistic perspective. J Wind Eng Ind Aerodyn 2008;96:1472–97. [8] Davenport AG. On the assessment of the reliability of wind loading on low buildings. J Wind Eng Ind Aerodyn 1983;11(1–3):21–37. [9] Kasperski M, Niemann HJ. A general method of estimating unfavorable load distributions for linear and non-linear structural behavior. J Wind Eng Ind Aerodyn 1992;43(1–3). [10] Holmes JD. Wind loading of structures, application of probabilistic methods. Prog Struct Eng Mater 1998;1(2):193–9. [11] Solari G, Piccardo G. Probabilistic 3-D turbulence modeling for gust buffeting of structures. Probabilist Eng Mech 2001;16:73–86. [12] Minciarelli F, Gioffrè M, Grigoru M, Simiu E. Estimates of extreme wind effects and wind load factors: influence of knowledge uncertainties. Probabilist Eng Mech 2001;16:331–40. [13] Kareem A. Aerodynamic response of structures with parametric uncertainties. Struct Safety 1988;5:205–25. [14] Solari G. Wind-excited response of structures with uncertain parameters. Probabilist Eng Mech 1997;12(2):75–87. [15] Pagnini LC, Solari G. Serviceability criteria for wind-acceleration and damping uncertainties. J Wind Eng Ind Aerodyn 1998;74–76:1067–78. [16] Zhang L, Jie L, Peng Y. Dynamic response and reliability analysis of tall buildings subject to wind loading. J Wind Eng Ind Aerodyn 2008;96:25–40. [17] Unanwa CO, McDonald JR, Mehta KC, Smith DA. The development of wind damage bands for buildings. J Wind Eng Ind Aerodyn 2000;84:119–49. [18] Rosowski DV, Ellingwood BR. Performance-based engineering of wood frame housing: fragility analysis methodology. J Struct Eng 2002;128(1):32–8. [19] Ellingwood BR, Rosowsky DV, Li Y, Kim JH. Fragility assessment of light-frame wood construction subjected to wind and earthquake hazards. J Struct Eng 2004;130(12):1921–30. [20] van der Lindt JW, Dao TN. Performance-based wind engineering for woodframe buildings. J Struct Eng 2009;135(2):169–77. [21] Khanduri AC, Morrow GC. Vulnerability of buildings to windstorms and insurance loss estimation. J Wind Eng Ind Aerodyn 2003;91:455–67. [22] Garciano LE, Maruyama O, Koike T. Performance-based design of wind turbines for typhoons. In: Augusti G, Schuëller GI, Ciampoli M, editors. Safety and reliability of engineering systems and structures – proceedings of the 9th international conference on structural safety and reliability ICOSSAR05; Rome, Italy. Rotterdam: Millpress; 2005 (Paper in CD-ROM).

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