Modelling of the stochastic dynamic behaviour of the Bergsøysund Bridge: an application of the power spectral density method

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1 Proceeding of the 9th International Conference on Structural Dnamic, EURODYN 4 Porto, Portugal, June - Jul 4 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (ed.) ISSN: -9; ISBN: Modelling of the tochatic dnamic behaviour of the Bergøund Bridge: an application of the powepectral denit method Knut Andrea Kvåle, Ole Øieth, Ragnar Sigbjörnon, Department of Structural Eng., Facult of Engineering Science, NTNU, Rich. Birkelandvei A, 749 Trondheim, Norwa Facult of Civil Engineering, Univerit of Iceland, Hjardarhagi -6, 7 Rekjavik, Iceland knut.a.kvale@ntnu.no, ole.oieth@ntnu.no, ragnar.igbjornon@ntnu.no ABSTRACT: A tud on the hdrodnamic modelling of pontoon bridge i preented, and eemplified with the Bergøund Bridge. The tructural modelling i baed on the finite element method uing the FE-oftware ABAQUS; the fluid-tructure interaction, a well a the wave action, are modelled appling linearied potential theor repreented b the DNV HdroD WADAM oftware. The main emphai i put on the tochatic repone anali within the framework of the powepectral denit method. The accurac of the preented modelling i outlined. Convergence of the repone obtained b modal technique i dealt with. B mean of tate-pace repreentation and an iteration algorithm, the quadratic eigenvalue problem i olved. Contribution from the fluid-tructure interaction to the overall modal damping i preented and dicued, and i found to be ignificant. Critical damping ratio cloe to %, correponding to mode with natural frequenc around rad/, are found. The diplacement repone i conidered near-converged when approimatel mode are included. KEY WORDS: Floating bridge; The Bergøund Bridge; Modal dnamic; Stochatic dnamic behaviour; Modal anali. INTRODUCTION The Norwegian Public Road Adminitration (NPRA) i currentl working on plan for rebuilding Highwa E9 along the Norwegian wet-coat. Thi route tretche km between the citie Kritianand and Trondheim, and incorporate multiple croing of deep fjord, which toda are operated b eight ferr connection []. Floating bridge of the pontoon tpe are propoed a feaible option fouch croing. In connection with the NPRA project, the objective i to carr out verification of the accurac of the modelling method ued to ae the overall dnamic behaviour of floating tructure epoed to environmental action, epeciall due to locall wind generated ea wave. The Bergøund Bridge, hown in Figure and Figure, i a 9 m long floating bridge of the pontoon tpe, croing the trait between Apøa and Bergøa located on the north-wet coat of Norwa. Thi bridge make a ver intereting cae tud due to the abence of mooring. The bridge conit of tru work upported b 7 dicretel ditributed light-weight concrete pontoon. Furthermore, it i verticall, horiontall and aiall upported at the end. Figure. Overview of the Bergøund Bridge. Baed on NPRA drawing.. THEORETICAL MODELLING Equation of motion Within the framework of a Finite Element Method (FEM) formulation, the equation of motion for a floating tructure can be written a: (t ) + C u (t ) + K u(t ) = Ph (t ) Mu () where t i the time, Ph i the total hdrodnamic action, including the fluid-tructure interaction a well a the wave action, M, C and K are the tructural ma, damping and tiffne matrice, repectivel, and u (t ) i the diplacement vector. Hence, the floating element contribute with force from the interaction between the water and the tructure. Thee force are dependent on diplacement, velocitie and acceleration of the pontoon, which give rie to hdrodnamic ma, damping and retoring force. In the frequenc domain, thi reult in the following total tem matrice: Figure. The Bergøund Bridge. M (ω ) = M + M h (ω ) () C (ω ) = C + C h (ω ) () 9

2 Proceeding of the 9th International Conference on Structural Dnamic, EURODYN 4 K = K + K (4) Here, ω i introduced a the frequenc variable. When auming harmonic ecitation, the following equation of motion for the coupled tem are obtained: h M( ω) u + C( ω) u + Ku= P( t) (5) where P () t repreent the wave ecitation force vector acting on the pontoon. Thi can be re-written in the frequenc domain a follow: S η (ω) [m 4 rad ] ω M( ω) u + iωc( ω) u+ Ku= P( ω) (6). Powepectral denit method A tated b Langen and Sigbjörnon [], the diplacement and force can be epreed within the theor of tochatic procee uing generalied harmonic decompoition: iω u() t = e t dz u ( ω) (7) iω p() t = e t dz p ( ω) () where Z u ( ω) and Z p ( ω) are the pectral procee correponding to the repone vector and the wave ecitation force vector, repectivel, and i. Equation (5) can, hence, be written a follow: ( i C K ) iωt iωt ω M( ω) + ω ( ω) + e dz u ( ω) = e dz p ( ω) H ( ω) (9) Here, the frequenc repone function H (ω) ha been introduced. Langen and Sigbjörnon [] epre the cropectral denitie of the diplacement u () t and the wave action p () t a: S ( ω) = E d ( ω) d H ( ω) u Zu Zu () S ( ω) = E d ( ω) d H ( ω) p Z p Zp () where the Hermittian operator [ i ] H ha been introduced a the comple conjugate and matri tranpoe and E[] i i the epectation operator. Combining thi with Equation (9) give: S u( ω) = H( ω) Sp ( ω) H( ω) () Furthermore, the coherence function and the correlation coefficient correponding to the component and of the diplacement vector proce u () t, with tandard deviance σ and σ a well a covarianceσ, are repectivel defined a: S ( ω) γ = S ( ω) S ( ω) () σ ρ = (4) σσ H Figure. One-dimenional ITTC wave pectral denit ued with the problem at hand, with ignificant wave height H =.4m.. Wave modelling The ea urface elevation i a calar quantit given a function of the location in pace and time t, and can be epreed mathematicall b the following Riemann-Stieltje integral [], [4]: where = { κ κ} i( ωt) η(, t) = κ e dzη ( κ, ω) (5) κ i the wave number vector, ω i the frequenc and Z η i the pectral proce of the ea urface. Fotationar and homogeneou random field, the pectral proce i related to the wave pectral denit a: H E dz η ( κ, ω) dz ( κ, ω) = dg (, ) d d r η η rη κ ω = S ηrη κ ω (6) where the indice r and correpond to two point in time and pace, Gηη denote the pectral ditribution, and S ηη the correponding pectral denit. In polar coordinate the wave number vector can be epreed a: { co in } κ = κ θ θ (6) where θ refer to the wave direction. Within the framework of Air wave theor, the wave number and wave frequenc are related through the diperion relation, which read out: ω = gκtanh( κh) (6) Here, g i the acceleration of gravit and h i the water depth. For deep water wave, thi epreion reduce to: ω ω = gκ κ = (6) g Hence, appling Air wave theor, the cro-pectral denit can be epreed a a function of wave frequenc and wave direction, Sηη = S (, ) ηη ωθ. The two-dimenional autopectral denit i obtained from the cro-pectral denit b merging the point r and. In a ingle point we get for a homogeneou tochatic wave field the following equation: 9

3 Proceeding of the 9th International Conference on Structural Dnamic, EURODYN 4 η, θ η, θ. Thi implie that we can epre the auto-pectral denit a a function of frequenc,, and direction, θ, i.e. η, θ ; commonl referred to a the directional wave pectral denit, and traditionall written a follow: S η ( ωθ, ) = S ( ω) D( ωθ, ) (7) η where S η ( ω ) i the o-called one-dimenional wave pectral denit and D( ωθ, ) i the directional ditribution. The cropectral denit of the water elevation can then be epreed a follow for deep water wave auming the directional function independent: S ηη ( ω) = S ( ω) D( θ) η π π ω ω ep i ( Δ coθ +Δinθ) dθ g () Here, Δ and Δ repreent the ditance between the point r and, and the mean wave direction i taken to be ero. The o-called ITTC wave pectral denit [5] wa applied a one-dimenional wave pectral denit in the current cae tud. Thi i a pecial cae of the Pieron-Mokowit tpe of wave pectral denit: where A and B are given b: A B S η ( ω) = ep 4 ω ω (9). A= α g, B= () H Here, g i the gravitational contant, H the ignificant wave height and α =. a contant called the Philip contant. Significant wave height H =.4m wa found uitable for the problem at hand. The reulting one-dimenional wave pectral denit i hown in Figure. The directional ditribution i commonl characteried b a bell haped function, with maimum value on the mean wave direction. For the problem at hand, the o-called co- ditribution wa applied [6]: Γ ( + ) θ θ D( θ ) = co π (.5) Γ + () with mean wave direction θ = and = 7. Baed on linear potential theor, the cro-pectral denit matri correponding to the wave ecitation can be written: H SQQ (, ) (, ) (, ) = Qr ωθ S ηη ωθ Q ωθ dθ () θ where Q r ( ωθ, ) refer to the directional wave ecitation tranfer function for pontoon no. r and Q ( ωθ, ) refer to the directional wave ecitation tranfer function for pontoon no.. The directional wave ecitation tranfer function are obtained appling potential theor a implemented in the computer package WADAM [7]. Thi reult in the following cro-pectral denit matri of wave ecitation: S QQ SQQ ( ω) S QQ ( ω) S ( ) Q7Q ω S ( ω) S ( ω) S ( ω) ( ω) ( ω) ( ω) SQQ S 7 QQ S 7 QQ 7 7 QQ QQ QQ 7 = ().4 Eigenvalue problem When damping i not neglected, the eigenvalue and eigenvector become comple. The eigenvalue problem then read out: [ λ M( ω) + λc( ω) + K] u= () Becaue the ma and damping matrice in the above equation are non-linear function of frequenc, the eigenvalue problem i in general non-linear, and need to be olved in an iterative manner (ee Appendi). The eigenvalue for an undercriticall damped and frequenc independent SDOF problem are found to be: λ = ξ ω ± ξ ω (4) r r r i r r Thi reult in the following relation: ωr = λr R( e λr ) ξr = λ r (5) where ω r and ξ r are the undamped natural frequenc and the critical damping ratio of mode r, repectivel. Equation (5) can be re-written: (6) u+ M Cu+ M Ku = M P To olve the eigenvalue problem, a tate-pace variable i introduced: u = u (7) Appling thi epreion, Equation (6) i rewritten in the tate pace a: u I u + = u M K M C u M p () On condened form, thi read out: ż+ A = Q (9) When olved for, thi reult in: N r = q t r eλ () r= where q r and λ r are the eigenvector and eigenvalue correponding to olution r in Equation (). It i aumed 9

4 Proceeding of the 9th International Conference on Structural Dnamic, EURODYN 4 that q r+ N i the comple conjugate of q r and that λ r + N i the comple conjugate of λ. Thi can be written compactl a: r u = Ψg () where Ψ i the modal tranformation matri containing the comple mode hape (both conjugate), and g i the generalied coordinate. Due to the frequenc dependence in the eigenvalue problem, it repreent a non-linear problem. Thi i olved b iteration in the current application. The iterative procedure ued to deal with thi i given in the appendi. COMPUTATIONAL MODELLING. Repone calculation An Abaqu/CAE tru model of the Bergøund Bridge b Hermtad [] wa ued. Thi model i hown in Figure 4. The tructural modal tem propertie, uch a mode hape and natural frequencie, were etracted from thi model. Contribution from the floating element to the overall tem tiffne, damping and ma, were calculated uing DNV HdroD WADAM. Thee contribution were thereafter tranformed to modal pace uing the modal hape retrieved from the finite element (FE) model. The etablihed tem propertie formed the bai for repone calculation, performed appling a Matlab [9] computer code peciall developed for thi purpoe. The outline of the proce i alo hown in Figure 5. For all reult preented in thi paper, the dr part of the tructure wa aigned damping correponding to Raleigh damping []: withα = β = 5. ωn ξ n = α β ω + () Figure 4. FE model ued for calculation of tructural propertie in Abaqu/CAE. n Figure 5. Connecting the domain. 4 RESULTS AND DISCUSSION 4. Repone pectral denitie The repone from the etimated wave ecitation pectral denitie introduced in Section. wa calculated uing the powepectral denit method. Figure 6 and Figure 7 how the reulting powepectral denitie for the repone in - direction and -direction, repectivel, of pontoon, 4 and 5, with correponding tatitic in Table and Table. The repone, indicated b it - and -component in Figure 6 and Figure 7, i in general found to have quite low correlation. The repone i enitive to the cret length of the wave, and the low correlation i upected to origin from the fact that the ea tate i baed on a rathehort-creted ea tate. Furthermore, it i een that the coherence i rather low, which i characteritic when the dominating wave period i a hort a in thi cae. Thi i important to account for in fatigue computation. 4. Natural frequencie, critical damping ratio and mode hape B emploing the algorithm introduced in Section.4, the etablihed tem matrice were ued to calculate the natural frequencie, damping coefficient and mode hape of the tem. The natural frequencie preented are the damped one, i.e. ωd = ξ ωn. The reulting mode hape were ued to ort the mode according to the kind of movement the repreented; horiontall tranveral (H), verticall tranveral (V) or a combination of the two (HV). Torional component were not conidered in thi urve. The natural frequencie and critical damping ratio for elected mode are preented in Table, whilt the correponding mode hape are preented in Figure - Figure. The critical damping ratio of mode around rad/ appear to reach value cloe to %. Compared to dteel tructure thi i ver high. In the contet of tem with ignificant hdrodnamic contribution, however, the damping value are not abnormall high. The relativel large damping i alo upported b the bluntne of the peak located around rad/ in the repone pectral denitie. 4 94

5 Proceeding of the 9th International Conference on Structural Dnamic, EURODYN 4 S S γ 4 S γ 5 γ S S S Figure 6. Calculated heave repone pectral denitie for the three midmot pontoon, No., No. 4. and No. 5. The cropectral denitie are repreented b imaginar (red) and real (blue) part. S γ 4.5 γ S 4 S 44 γ S 5 S 54 S Figure 7. Calculated -directional repone pectral denitie for the three midmot pontoon, No., No. 4. and No. 5. The cro-pectral denitie are repreented b imaginar (red) and real (blue) part. 5 95

6 Proceeding of the 9th International Conference on Structural Dnamic, EURODYN 4 Additionall, the critical damping ratio and natural frequencie baed on all eigenvalue ranging between and 7 rad/ are preented in Figure. Here, the damping i compared to the damping of a imilatem where the frequenc dependent damping contribution C (ω) i ecluded, to make it poible to tud the impact thi ha on the total damping. From the figure, it i evident that the hdrodnamic frequenc dependent damping contribute in a great etent to the overall damping of the tructure. Becaue the added hdrodnamic damping i non-ero onl for a quite narrowbanded frequenc region around - rad/, it i reaonable that the total damping converge toward the Raleigh damping for higher frequencie. Figure how a ditinctive preading of the critical damping ratio. Thi i upected to come from the originall phical nature of the hdrodnamic damping. Becaue the added damping i different for the variou DOF, it make ene that different amount of damping i contributed to the different mode. The total added damping for a ingle mode will be determined b the modal decompoition ued on the original damping, i.e. onl the added damping correponding to DOF active in the current mode will be contributing. Thi i in harp contrat to the Raleigh damping, which i added in a global manner to the entire tructure. Mode, with mode hape hown in Figure 9 and natural frequenc and critical damping ratio tated in Table, doe not how in Figure 6 or Figure 7. Thi i due to the limited frequenc range where S η ( ω ) and therefore alo S QQ ( ω) are non-ero. The one-dimenional wave pectrum ued, hown in Figure, reveal that the wave loading i in fact cloe to ero at the range of thi mode (.6 rad/). Table. Covariance and correlation coefficient for the - component of the repone, correponding to the pectral denitie hown in Figure 6. Component Covariance [ mm ] Correlation coefficient, 49.4 % 4,. 5.9 % 4, % 5, % 5, % 5,5 79. % Table. Covariance and correlation coefficient for the - component of the repone, correponding to the pectral denitie hown in Figure 7. Component Covariance [ mm ] Correlation coefficient, 79.7 % 4, % 4,4 64 % 5, % 5, % 5,5 7. % Table. Natural frequencie and damping ratio foelected mode. H refer to horiontal mode hape, V to vertical mode hape and H/V to mied modal deformation. Mode no. Damped natural frequenc [rad/] Critical damping ratio [%] Tpe.66.7 H..7 H/V..7 V H/V H/V V H/V H H.5. H Figure how the critical damping ratio which are retrieved from olution of the eigenvalue problem for all the frequenc value ranging between and 4 rad/. To be able to etract eigenvalue correponding to the ame eigenvector for all the frequenc value, an algorithm earching foimilar eigenvector for different frequencie wa emploed, i.e. b finding the eigenvector for ω k and ω k + that reulted in the i j j larget calar product, qk q k+. Eigenvector j forω k + ( q k+ ) i coinciding mot with eigenvector i for ω k ( q k ) wa rearranged o that it alo for ω k + wa the i th eigenvector. Thi wa performed for all frequencie along the dicretied frequenc ai. The eigenvalue were thereafter arranged correpondingl. ω =. rad/ 4 6 ω 6 =.497 rad/ 4 6 Figure. Pure verticall tranveral mode hape. ω =.665 rad/ 4 6 ω 9 =.55 rad/ 4 6 ω =.4457 rad/ 4 6 ω =.47 rad/ 4 6 Figure 9. Pure horiontall tranveral mode hape. 6 96

7 Proceeding of the 9th International Conference on Structural Dnamic, EURODYN 4 ω =. rad/ 4 6 ω 4 =.5 rad/ 4 6 ω 5 =.69 rad/ 4 6 ω 7 =.99 rad/ 4 6 ω =. rad/ 4 6 ω 4 =.5 rad/ 4 6 ω 5 =.69 rad/ 4 6 ω 7 =.99 rad/ 4 6 Figure. Tranveral mode hape, with both horiontal and vertical component. Normalied pectral denit, n mode Normalied pectral denit, n mode n = 5 n = n = n = n = Normalied pectral denit, 5 mode Figure. Modal convergence of the real part of the cropectral denit between degree of freedom and. The pectral denitie are normalied with repect to the maimum real value of the reference pectral denit with 5 mode. ξ n..5 Including C(ω) Vertical mode Horiontal mode Vert. and hor. mode Raleigh damping Ecluding C(ω) ω n [rad/] Figure. Critical damping ratio from the comple eigenvalue olution. The hdrodnamic ma and tiffne contribution are included in both cae, and onl the hdrodnamic damping contribution differ. Re(λ)/ λ Mode (.66 rad/) Mode (. rad/) Mode (. rad/) Mode 9 (.5 rad/) Figure. Normalied real value of eigenvalue a function of frequenc for mode,, and 9. The olution of the eigenvalue problem i indicated with a circle for each mode. Normalied pectral denit, n mode Normalied pectral denit, n mode n = 5 n = n = n = n = Normalied pectral denit, 5 mode Figure 4. Modal convergence of the real part of the cropectral denit between degree of freedom and. The pectral denitie are normalied with repect to the maimum real value of the reference pectral denit with 5 mode. 7 97

8 Proceeding of the 9th International Conference on Structural Dnamic, EURODYN 4 hdrodnamic damping, the modal damping ratio var quite much. B mean of tem identification appling repone meaurement, modal parameter uch a critical damping ratio, natural frequencie and modal hape need to be determined. The identified tem will then be ued to calibrate the preented computational hdrodnamic model, and thereb quantif the uncertaintie of the method applied. ACKNOWLEDGMENTS We would like to thank Sindre Hermtad for hi work on the Abaqu model ued. Thi ha upported the following work ignificantl. Thank are alo given to Profeor Bernt Leira, and Abdillah Suuthi a well a Sindre Hermtad for their work on the linear potential theor calculation performed in WADAM. Thi reearch wa carried out with financial upport from the Norwegian Public Road Adminitration. The author greatl acknowledge thi upport. 4. Figure 5. Global coordinate tem and elected global DOF. Modal convergence Scatter plot of cro powepectral denitie between ome choen DOF for different number of mode are hown in Figure and Figure 4. Here, the pectral denitie are normalied with repect to the maimum real value of the reference pectral denit, i.e. reulting from 5 included mode. The DOF ued are annotated in Figure 5. From thee figure it i oberved that the olution appear nearconverged when mode are included. It hould be noted that the number of mode required will depend on the quantit under conideration, e.g. the acceleration repone will require more mode than the diplacement repone. The ame applie for internal force. 5 CONCLUDING REMARKS Reult of numerical calculation of the repone of the Bergøund Bridge were preented, with emphai on repone pectral denitie reulting from the powepectral denit method a well a modal parameter reulting from the comple and non-linear eigenvalue problem. Quite low correlation value of the repone were found, which i tpical for repone ecited b fairl hort-creted wave. The diplacement repone olution appear nearconverged when approimatel mode are included. The current tud doe not include conideration regarding the modal convergence of (i) acceleration repone, which i an important apect regarding traffic afet; or (ii) internal force, which i crucial in a general tructural deign contet. Due to the fact that both thee quantitie are differential function of the diplacement, the error would tend to increae, which implie that more mode would be necear foufficient accurac. For mode around rad/, critical damping ratio of approimatel % were found from the olution of the quadratic eigenvalue problem. The bluntne of the peak in the repone pectral denitie qualitativel upport the damping level. Due to the phical nature of the added REFERENCES [] O. Elleveet, Project Overview Coatal Highwa Route E9. Norwegian Public Road Adminitration,. [] I. Langen and R. Sigbjørnon, Dnamik anale av kontrukjoner: Dnamic anali of tructure. Tapir, 979. [] R. Sigbjörnon, Stochatic theor of wave loading procee, Eng. Struct., vol., no., pp. 5 64, Jan [4] I. Langen and R. Sigbjörnon, On tochatic dnamic of floating bridge, Eng. Struct., vol., no. 4, pp. 9 6, Oct. 9. [5] J. Norwa, The Specialit Committee on Wave, in Final Report and Recommendation to the rd ITTC, pp [6] A. Nae and T. Moan, Stochatic dnamic of marine tructure. Cambridge Univerit Pre,. [7] WADAM, Verion.. DNV, P.O. Bo, N- Hovik, Norwa,. [] S. Hermtad, Dnamic Anali of the Bergøund Bridge in the Time Domain, Norwegian Univerit of Science and Technolog,. [9] MATLAB, verion..64 (Ra). The MathWork Inc., Natick, Maachuett,. [] J. L. Humar, Dnamic of tructure. CRC Pre,. APPENDIX The iteration algorithm ued to etablih the eigenvalue of the problem i hown below. Do for all n from to N:. Setω = ω, a the th iteration.. Solve the eigenvalue problem in Equation (), for the choenω.. Sort all the eigenvalue and the eigenvector correpondingl. th 4. Set ω equal the abolute value of the n reulting eigenvalue. 5. If the number of iteration ha eceeded a pecified value of maimum iteration, or the previoul found ω deviate with le than a pecified tolerance from the newω : et λ n equal the n th reulting eigenvalue, and q n equal the n th reulting eigenvector - and break iteration loop. If neither i true, do tep - 5 over. 9