International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

System reliability of suspension bridges considering static divergence and flutter Øiseth, Ole; Rönnquist, Anders; Naess, Arvid Jul 31, 2015

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
53032-Paper_282_Ronnquist.pdf [ 11.12MB ]
Metadata
JSON: 53032-1.0076150.json
JSON-LD: 53032-1.0076150-ld.json
RDF/XML (Pretty): 53032-1.0076150-rdf.xml
RDF/JSON: 53032-1.0076150-rdf.json
Turtle: 53032-1.0076150-turtle.txt
N-Triples: 53032-1.0076150-rdf-ntriples.txt
Original Record: 53032-1.0076150-source.json
Full Text
53032-1.0076150-fulltext.txt
Citation
53032-1.0076150.ris

Full Text

12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015System Reliability of Suspension Bridges Considering StaticDivergence and FlutterOle ØisethAssociate Professor, Department of Structural Engineering, Norwegian University ofScience and Technology, Trondheim, NorwayAnders RönnquistAssociate Professor, Department of Structural Engineering, Norwegian University ofScience and Technology, Trondheim, NorwayArvid NaessProfessor Department of Mathematical Sciences, Norwegian University of Science andTechnology, Trondheim, NorwayABSTRACT: The Hardanger Bridge has recently been constructed in Norway. The main span is 1310 m,and since the bridge has only two traffic lanes and one lane for bicycles and pedestrians, it is very slenderand susceptive to wind induced vibrations. The reliability of the Hardanger Bridge considering staticdivergence and flutter is studied in this paper. An extensive finite element model of the structure is usedto obtain the still-air dynamic properties of the bridge while results from wind tunnel tests are used torepresent the aerodynamic properties in terms of aerodynamic derivatives and static load coefficients. Themean wind velocity at the site is represented by a Gumbel distribution and the scatter in the experimentalresults of the aerodynamic derivatives and static load coefficients are modelled as correlated Gaussianvariables. It is shown that by exploiting the regularity of the probability of failure, the system reliabilitycan be predicted using a reasonable amount of simulations.There exists numerous studies of bridge deck flut-ter, but less work have been put into assessing theprobability of flutter failure, or the flutter reliabil-ity of the structure. A study of the flutter reliabilityof the Great Belt Bridge is reported by Ostenfeld-Rosenthal et al. (1992). The reliability of the bridgehas been assessed using wind tunnel test results ofthe critical velocity as basis. The basic randomvariables includes the 10 minutes mean wind ve-locity, variability in the observed critical velocityin the wind tunnel tests, possible influence fromturbulence on the critical velocity and structuraldamping. A similar study for the Yangpu Bridgein Shanghai is presented by Ge et al. (2000). Thestudy is based on a closed form solution of thecritical flutter speed. Four basic random variableshave been introduced to take into account uncer-tainties in the critical velocity, scaling of the sectionmodel, the basic wind speed and the gust speed.More recent studies show that several still-air vi-bration modes may participate in the flutter motion.If multi-mode effects are likely it is difficult to usethe critical velocity measured in the wind tunnel us-ing a section model or closed form solutions of theflutter velocity as basis since only one horizontal,vertical and torsional mode can be modelled prop-erly. The reliability of the Vincent Thomas Sus-pension bridge taking into account multi-mode ef-fects have been assessed in a study presented byPourzeynali and Datta (2002). Variation in the massand stiffness properties of the bridge is modelledby multiplying the entire mass and stiffness matrixby normally distributed random variables such thatthe standard deviation and the mean value of thecritical velocity can be obtained. The random vari-able that represent the variation of the critical ve-112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015locity due to uncertainties in the mass and stiffnessproperties is further modified by multiplying it withrandom variables that take uncertainties in still-airdamping, finite element modelling, flutter deriva-tives and possible influence of turbulence on thecritical velocity. Theodorsen’s function was usedto model the flutter derivatives of the bridge deck.The flutter reliability of the Jiang Yin Bridge hasbeen studied by Cheng et al. (2005). Ten basic ran-dom variables have been used to model uncertain-ties in the structural properties of the bridge, twoto model the basic wind velocity and gust speed,one to model the uncertainty in the modal dampingratio and one related to uncertainties in the flutterderivatives. A Monte Carlo analysis of total damp-ing and flutter speed of the Messina strait bridge hasbeen presented byArgentini et al. (2014). The pro-posed methodology is used to study the sensitivityof the critical velocity with respect uncertainties instructural and aerodynamic modelling using simi-lar basic random variables as Cheng et al. (2005).Engineering judgement and experience have beenthe main source to define the statistical propertiesof the random variables in the studies mentionedso far. Another approach has been used in a studyof torsional flutter probability presented by Seo andCaracoglia (2011) where the scatter in the experi-mental results of the flutter derivatives have beenused to define the properties of the basic randomvariables.A suspension bridge represents a complicatedsystem from a reliability analysis point of view.There are a lot of limit state functions and de-pending on how the failure of the bridge is de-fined, the reliability needs to be assessed consid-ering it as combinations of series and parallel sys-tems. It is only requirements to the total proba-bility of failure that are relevant in design, whichimplies that the reliability needs to be assessed us-ing a methodology that can handle many marginalfunctions. A enhanced Monte Carlo method thatfulfils this requirement has been proposed by Naesset al. (2009). The robustness of the methodologyhas been proven for large systems by Naess et al.(2012). In this paper the enhanced Monte Carlo ap-proach is used to estimate the probability of failureof the Hardanger Bridge considering coupled flut-ter and static divergence. To model uncertaintiesinvolved in predicting the critical flutter velocity,scatter in the experimental results of the full set of18 aerodynamic derivatives have been used to de-fine 54 correlated normally distributed random vari-ables representing uncertainties in the experimentalresults. Twenty basic random variables have fur-ther been introduced to model the uncertainty in thestill-air damping ratios for the twenty still-air vibra-tion modes used as basis in the flutter assessment.The derivative of the overturning moment coeffi-cient is relevant for static divergence. The coeffi-cient has been modelled as a normally distributedvariable and its mean value and variance are ob-tained by simple regression to experimental results.It is shown that the enhanced Monte Carlo approachis robust for the case considered in this paper andthat the probability of failure fulfils the requirementPf < 10−71. SYSTEM RELIABILITYTo be able to assess the reliability of any structureit is necessary to define a set of safety marginsMi = Gi (X1, · · · ,XN) (1)Here Gi symbolizes the limit state function that de-fines the safety margin Mi as function of a vec-tor X = [X1,X2...XN ]T that contains n basic randomvariables. A high number of safety margins needsto be considered to predict the system reliabilityof a suspension bridge. Since we will only con-sider static divergence and flutter as possible failuremodes in this work, the problem can be modelled asa series system, which implies that the probabilityof failure can be obtained byp f = P[m⋃i=1(Mi < 0)](2)The probability of failure can be calculated ap-plying different methods. FORM and SORM arewell known methods Madsen et al. (2006) Melch-ers (1999), but they are most convenient when thereexists a rather simple closed form solutions of thelimit state function, which is not the case for ourproblem. Another more attractive alternative is to212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015use crude Monte Carlo simulations, but since weare dealing with a system with high reliability thiswill simply require to much computational effort.This can be overcome by applying an importancesampling technique, but this might be cumbersomefor a system with several marginal functions thatprovides contributions to the probability of failurefrom several places in the space defined by the ran-dom variables. Naess et al. (2009) have suggestedan enhanced Monte Carlo approach that overcomesthis difficulty. The key step is to shift the marginalfunctions towards the mean value of the safety mar-gin to get more hits in the failure domain.Mi (λ ) = Mi−µMi (1−λ ) (3)The next step is to study the behaviour of the prob-ability of failure as function of the λ introduced inEq. (3). It turns out that the behaviour of the func-tion Pf (λ ) is highly regular, which implies that theprobability of failure of the system, Pf (λ = 1) canbe obtained by extrapolation. Naess et al. (2009)have proposed to use the following approximationp f (λ )≈ qexp{−a(λ −b)c}, λ0 > λ > 1 (4)The parameters q,a,b,c are determined by a leastsquares fit to data obtained by Monte Carlo simula-tions. Assuming a sample of N realizations of thebasic random variables the probability of failure asfunction of λ can be estimated as:pˆ f (λ ) =N f (λ )N· (5)Here N is the number of realizations of the basicrandom variables, and N f (λ ) is the number of out-comes for which failure of the system occur. Anapproximation of the 95 percent confidence inter-val for the estimate of pˆ f (λ ) can be obtained by:C± (λ ) = pˆ f (λ )[1±1.961√p f (λ )N](6)The 95 percent confidence interval for the probabil-ity of failure can then be obtained by curve fittingthe expression presented in Eq. (4) to the data forthe confidence intervals and evaluate the expressionwhen λ = 12. BRIDGE AERODYNAMICSStatic divergence and flutter are the most relevantinstability phenomena for suspension bridges witha wedged shaped box girder. It is therefore neces-sary to obtain the marginal functions that describestheir safety margins.2.1. Coupled flutterThe stability of an aeroelastic second order sys-tem, where N still-air vibration modes are usedas generalised degrees-of-freedom, can be pre-dicted considering the following quadratic eigen-value problem:(S2nM˜0 +Sn(C˜0− C˜ae(K))+(K˜0− K˜ae(K)))Zn = 0(7)Here, M˜0 represents the generalized structural massmatrix, C˜0 denotes the generalized damping matrixand K˜0 represents the generalized structural stiff-ness matrix, where the subscript 0 indicates that thematrices contain properties obtained in still air. Thematrices C˜ae(K) and K˜ae(K) symbolises the aero-dynamic damping and stiffness matrix in general-ized coordinates, respectively and are related to theself-excited forces generated by the motion of thecross-section.The self-excited forces acting on a bridge decksection are commonly represented by the aerody-namic derivatives introduced in bridge engineer-ing by Scanlan and Tomko (1971). For a two-dimensional bridge deck section (see Figure 1), thiscan be expressed in matrix notation as follows:q(t) = Cae(K)u˙(t)+Kae(K)u(t) (8)Kae(K) =12ρV 2K20 0 0 00 P4∗ P6∗ BP3∗0 H6∗ H4∗ BH3∗0 BA6∗ BA4∗ B2A3∗(9)Cae(K) =12ρV KB0 0 0 00 P1∗ P5∗ BP2∗0 H5∗ H1∗ BH2∗0 BA5∗ BA1∗ B2A2∗(10)q=[qx qy qz qθ]T, u=[ux uy uz uθ]T(11)312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Here, V is the mean wind velocity; ρ is the airdensity; B is the width of the cross section; K =Bω/V is the reduced circular frequency of motion;the vector u contains the displacements along thegirder, where ux symbolizes the longitudinal dis-placement, uy the transverse horizontal displace-ment, uz the transverse vertical displacement anduθ The rotation of the girder. The displacementsare positive in the same direction as the forces dis-played in Figure 1. P∗n , H∗n , A∗n n ∈ [1,2, . . .6] arethe dimensionless aerodynamic derivatives, whichare characteristic cross-sectional properties givenas functions of the reduced frequency of motion.Figure 1: Aerodynamic forces acting on a cross sec-tion of a bridge deck. The section shown is from theHardanger Bridge.The solution of Eq. 7 gives 2N eigenvalues, Sn,and corresponding eigenvectors Zn, where N is thenumber of degrees-of-freedom. The roots are ingeneral complex and the systems ability to dissipateenergy can be assessed by studying the real part ofits eigenvalues. If µn is negative, the solution showsexponential convergence corresponding to positivedamping of the aeroelastic system. On the otherhand, if µn is positive, the solution exhibits expo-nential divergence, which is sometimes interpretedas negative damping of the aeroelastic system. seeØiseth and Sigbjörnsson (2011) for further details2.2. Static divergenceAnother important aeroelastic instability phe-nomenon for suspension bridges is static diver-gence. This is very similar to lateral torsional buck-ling of beams and simply implies that the systemhas zero total stiffness. Since the instability phe-nomenon is static such that the reduced velocityis infinitive, it is not possible to use experimen-tally determined aerodynamic derivatives to assesthe critical velocity. Using quasi steady theory toquantify the self-excited forces and assuming that itis the first symmetric torsional vibration mode thatis most important, the following closed form solu-tion of the critical velocity can be obtained.VCR = Bωi√√√√√√√√2M˜0iρB4C′ML∫0φ2θ (x)dx(12)Here, M˜0i represent the modal mass for the torsionmode, φθ (x) the rotation of the cross section alongthe girder and C′M the derivative of the static over-turning moment coefficient with respect to the an-gle of the attack.3. CASE STUDY: THE HARDANGERBRIDGEThe Hardanger Bridge displayed in Figure 2 isthe longest bridge in Norway and among the topten longest suspension bridges in the world. It hasa total length of 1800 m including a main span of1310 m. To assess the reliability of a large structuresubjected to environmental actions is challengingsince there are a lot of sources of inaccuracies thatshould be taken into account. It is natural to dividethe basic random variables into three groups, (a) un-certainties in the still-air properties of the bridge,(b) uncertainties involved in the aerodynamic mod-elling and (c) uncertainties related to the mean windvelocity at the site.Figure 2: The Hardanger Bridge412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20153.1. Still-air dynamic propertiesA detailed finite element model of the Hardan-ger Bridge is displayed in Figure 3. The entirebridge is modelled using a total of 1358 beam el-ements. Since a lot of simulations needs to be car-ried out to estimate the probability of failure it isreasonable to use still-air vibration modes as gener-alized coordinates to obtain the equation of motionfor a reduced system see, Øiseth et al. (2010) formore details. It is the still-air vibration modes withlowest natural frequency that will most likely con-tribute to the flutter vibration mode and for a singlespan suspension bridge it is well known that it isthe first symmetric vertical and torsional vibrationmodes that play the fundamental role in the fluttermotion. It has, however been shown in a numberof case studies that several still-air vibration modesmight contribute to the flutter motion, which im-plies that great care needs be put into the selec-tion process. The first horizontal, symmetric ver-tical and torsional vibration modes are displayed inFigure 3.A suspension bridge is simple to model relativeto other civil engineering structures. The stiffnessof the components are well defined and most ofthe total mass is construction steel. Each of themain cables represents a huge parallel system sincethey consists of 10032 steel wires. Since the steelwires are very long it seems reasonable to assumea low correlation of the relevant variables betweenthe wires in the parallel system, which implies thatthe coefficients of variation of the area, weight andaxial stiffness of the main cables are close to zero.The same argument applies for the box girder of thebridge since it is reasonable to assume a low corre-lation of the steel properties along the girder. Thisimplies that it will be possible to predict natural fre-quencies, mode shapes and generalized propertieswith high accuracy. It is therefore chosen to neglectuncertainties related to structural properties in thiswork. This choice is also partly supported by theresults presented by Cheng et al. (2005).The uncertainty involved in the prediction ofthe still-air damping is however much larger. Thestill-air damping ratios are therefore modelled aslog-normally distributed variables with mean value0.5% and coefficient of variation 0.4a b c Figure 3: FE model of the Hardanger bridge: (a) thefirst symmetric horizontal vibration mode, (b) the firstsymmetric vertical vibration mode, (c) the first symmet-ric torsional vibration mode3.2. Aerodynamic modellingExtensive wind tunnel testing was carried out byHansen et al. (2009) in the design of the HardangerBridge to ensure a superior performance in strongwinds. The aerodynamic derivatives of the bridgedeck is shown in Figures 4 and 5. As can be seenin the figures there are some scatter in the data,which represents an uncertainty in the modellingof the aerodynamic actions. The blue curves havebeen obtained by multivariate multiple regressionusing the MATLAB function mvregress.m MAT-LAB (2012). Each of the aerodynamic derivativesis curve fitted using a second order expression onthe formP∗1 (Vˆ ) = β0 +β1Vˆ +β2Vˆ 2 (13)The outputs from the mvregress.m function are avector of 54 β¯n values and the covariance matrix ofthe parameters. This implies that the coefficients inthe least squares fit β¯ can be modelled as a mul-tivariate normal distribution β = N(β¯ ,σ). Here β¯represent the mean value and σ the covariance ma-trix. Realizations of the basic variables can thenbe obtained by the MATLAB function mvnrnd.m.The light grey curves in Figure 4 and 5 represent100 realizations of different curve fits to the data. Itseems like the grey curves represents the scatter inthe experimental data in a fair manner.512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015The static overturning moment is strongly relatedto static divergence. The static overturning momentqθ (α) as function of angle of attack α is defined asqθ (α) = 1/2ρB2CM(α). Data for the overturningmoment coefficient as function of angle of attack isdisplayed in Eq. (6). As can be seen in Figure 12 itis the derivative of the coefficient that is relevant forstatic divergence. It is chosen to curve fit the datausing a straight line CM(α) = C¯M +C′M ·α such thatthe expected value and variance of the of the slopecoefficient C′M can be used to model the scatter inthe experimental data. Ten realizations of possiblefits to the data are displayed in Figure 6, and as canbe seen, the grey curves seems to fairly representthe scatter in the experimental data1 1.5 2−3−2−101P 1*2 3 4−1.5−1−0.500.5P 5*1 1.5 2−0.2−0.100.10.2P 2*1 1.5 2−1001020H5*2 3 4−15−10−50H1*1 1.5 2−0.500.511.5H2*1 1.5 2−4−202A5*V/(B⋅ω)2 3 4−4−3−2−10A1*V/(B⋅ω)1 1.5 2−0.6−0.4−0.200.2A2*V/(B⋅ω)Figure 4: Aerodynamic derivatives related to displace-ments (See Eq. ( 9 )). The dots represents e experimen-tal data while the grey lines represent realisations ofpossible fits to the data3.3. Mean wind velocityThe wind conditions in the Hardanger fjord wasmonitored for 4 years as a part of the planning of thebridge. The data obtained at the site was combinedwith long term data from nearby permanent weatherstations to predict the annual extreme value distri-bution of the 10 minutes mean wind velocity. Sincethe bridge is located in a narrow fjord, the strongwinds are nearly perpendicular to the bridge, which1 1.5 2−2−101P 4*2 3 4−2−101P 6*1 1.5 2−1−0.50P 3*1 1.5 2−10010H6*2 3 4−10−505H4*1 1.5 20510H3*1 1.5 2−2−101A6*V/(B⋅ω)2 3 4−4−202A4*V/(B⋅ω)1 1.5 2024A3*V/(B⋅ω)Figure 5: Aerodynamic derivatives related to velocities(See Eq. ( 10 )). The dots represents experimental datawhile the grey lines represent realisations of possiblefits to the data−0.1 −0.05 0 0.05 0.1−0.1−0.0500.050.1α (rad)C M(α)Figure 6: Overturning moment coefficientimplies that directional effects can be disregarded.Some variation of the mean wind velocity along thebridge is expected, but this is not included in thiswork3.4. Monte Carlo SimulationsRealizations of the critical flutter velocity,VCR,F(x), have been obtained using the fist 20still-air vibration modes as generalized coordinates.A total of 74 basic random variables have beenincluded in the analysis, whereas 54 are corre-lated normally distributed variables representingthe coefficients in the curve fits to the aerodynamicderivatives. The remaining 20 are log-normally dis-tributed variables representing the still-air dampingrations. A full quadratic response surface was fit-ted to 10.000 simulations where the covariance ma-trix has been increased by a factor of 2. The co-variance matrix was scaled to make sure that bothlow and high critical velocities are accurately repre-612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015sented by the response surface. The critical veloc-ities predicted using the response surface is com-pared to results from Eq. (7) in Figure 7. As can beseen from the figure, the quadratic response surfaceis able to capture the critical velocity accurately.The response surface was used to generate 10 mil-lion samples of the critical flutter velocity VCR,F(x).The results seems to follow a log-normal probabil-ity density function with mean value 79.6 m/s andcoefficient of variation 0.034.Realizations of the critical velocity with respectto static divergence VCR,SD(x) can be obtained di-rectly from the closed form presented in Eq. 12.The derivative of the overturning moment coeffi-cient, C′M is the only basic random variable consid-ered in the modelling and its mean value and vari-ance are obtained by linear regression. Ten millionsamples where also generated for the critical veloc-ity with respect to static divergence.As shown in Eq. (2), the system reliability is de-fined by the union of the two marginal functionsM1 = V −VCR,F(x) and M1 = V −VCR,SD(x), rep-resenting flutter and static divergence respectively.The probability of failure Pf (λ ) as function of λis presented in Figure 8. The data for the proba-bility of failure and the confidence intervals havebeen estimated applying Eq. (5) and Eq. (6) re-spectively. The function presented in Eq. (4) havebeen fitted to the data to obtain the probabilityof failure Pf = 2.4 · 10−8 and its confidence inter-val C± = [1.0 · 10−8...4.4 · 10−8]. As expected theHardanger Bridge has a high reliability, which iswell within the requirement Pf < 10−7.4. CONCLUDING REMARKSThe probability of failure of the HardangerBirdge considering flutter and static divergence aspossible failure modes has been analysed in this pa-per. A total of 74 basic random variables have beenused to model the uncertainties involved in predict-ing the critical flutter velocity, whereas 54 are re-lated to the scatter in the experimental results ofthe flutter derivatives while the remaining 20 arerelated to the still-air damping ratios. The slopeof the overturning moment coefficient was used tomodel the uncertainties in predicting the criticalvelocity for static divergence. Since the Hardan-60 70 80 90 100 1106065707580859095100105110Eigenvalue calculationsResponse surfaceR2=0.99Figure 7: Comparison of critical flutter velocities pre-dicted using the response surface and results obtainedsolving the eigenvalue problem0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110−810−610−410−2λP f(λ)  Pf(λ)Pf(λ)C+C−C−C+pf(λ=1)Figure 8: The system reliability Pf (λ ) as function of λger Bridge represents a very reliable system, a en-hanced Monte Carlo approach has been success-fully applied to estimate the probability of failureof the system. It is concluded that the methodologyis robust, and that several marginal functions can behandled in a simple manner.Several uncertainties have been neglected in thispaper. The structural properties have been con-sidered deterministic, which is a fair approxima-tion since there is a low uncertainty in the parame-ters relevant when predicting flutter or static diver-gence. Variations in the mean wind velocity alongthe bridge deck has not been taken into accountand neither has possible directional effects. It isbelived that these effects are minor since the ap-plied Gumbel distribution is obtained for data mea-712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015sured perpendicular to the bridge and that it is themean wind velocity close to the mid-span that pro-vides the dominating contribution to the modal self-excited forces.A larger uncertainty is related to the fact that onlyone dataset of the aerodynamic derivatives havebeen considered. It is well known that the exper-imental results of the flutter derivatives may varydepending on which method or which aerodynamiclaboratory that has been used to obtain them. Theauthors will look into this in the near future.The probability of failure of the HardangerBridge was estimated to Pf = 2.4 · 10−8 which iswell within the requirement of Pf < 10−75. REFERENCESArgentini, T., Pagani, A., Rocchi, D., and Zasso, A.(2014). “Monte carlo analysis of total damping andflutter speed of a long span bridge: Effects of struc-tural and aerodynamic uncertainties.” Journal of WindEngineering and Industrial Aerodynamics, 128, 90–104 cited By (since 1996)0.Cheng, J., Cai, C., Xiao, R.-C., and Chen, S. (2005).“Flutter reliability analysis of suspension bridges.”Journal of Wind Engineering and Industrial Aerody-namics, 93(10), 757–775 cited By (since 1996)30.Ge, Y., Xiang, H., and Tanaka, H. (2000). “Applica-tion of a reliability analysis model to bridge flutterunder extreme winds.” Journal of Wind Engineeringand Industrial Aerodynamics, 86(2-3), 155–167 citedBy (since 1996)30.Hansen, S. O., Lollesgaard, M., Rex, S., Jakobsen, J. B.,and Hansen, E. H. (2009). “The hardanger bridge:Static and dynamic wind tunnel tests with a sectionmodel, revision 2.” Report, Svend Ole Hansen ApS.Madsen, H. O., Krenk, S., and Lind, N. C. (2006). Meth-ods of structural safety. Dover Publications, Mineola,N.Y.MATLAB (2012). version R2012a. The MathWorksInc., Natick, Massachusetts.Melchers, R. E. (1999). Structural reliability: analysisand prediction. Wiley, Chichester 2nd ed.Naess, A., Leira, B., and Batsevych, O. (2009). “Sys-tem reliability analysis by enhanced monte carlo sim-ulation.” Structural Safety, 31(5), 349–355 cited By(since 1996)32.Naess, A., Leira, B., and Batsevych, O. (2012). “Re-liability analysis of large structural systems.” Proba-bilistic Engineering Mechanics, 28, 164–168 cited By(since 1996)4.Øiseth, O., Rönnquist, A., and Sigbjörnsson, R. (2010).“Simplified prediction of wind-induced responseand stability limit of slender long-span suspensionbridges, based on modified quasi-steady theory: Acase study.” Journal of Wind Engineering and Indus-trial Aerodynamics, 98(12), 730–741 cited By (since1996)9.Øiseth, O. and Sigbjörnsson, R. (2011). “An alterna-tive analytical approach to prediction of flutter sta-bility limits of cable supported bridges.” Journal ofSound and Vibration, 330(12), 2784–2800 cited By(since 1996)4.Ostenfeld-Rosenthal, P., Madsen, H., and Larsen, A.(1992). “Probabilistic flutter criteria for long spanbridges.” Journal of Wind Engineering and IndustrialAerodynamics, 42(1-3), 1265–1276 cited By (since1996)22.Pourzeynali, S. and Datta, T. (2002). “Reliability anal-ysis of suspension bridges against flutter.” Journal ofSound and Vibration, 254(1), 143–162 cited By (since1996)26.Scanlan, R. H. and Tomko, J. J. (1971). “Airfoil andbridge deck flutter derivatives.” Journal of the Engi-neering Mechanics Division, 97, 1717–1737.Seo, D.-W. and Caracoglia, L. (2011). “Estimation oftorsional-flutter probability in flexible bridges consid-ering randomness in flutter derivatives.” EngineeringStructures, 33(8), 2284–2296 cited By (since 1996)5.8

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.53032.1-0076150/manifest

Comment

Related Items

Admin Tools

To re-ingest this item use button below, on average re-ingesting will take 5 minutes per item.

Reingest

To clear this item from the cache, please use the button below;

Clear Item cache