Evaluation of the accuracy of the Multiple Support Response Spectrum (MSRS) method

Transcription

1 Evaluaton of the accuracy of the Multple Support Response Spectrum (MSRS) method K. Konakl Techncal Unversty of Denmark A. Der Kureghan Unversty of Calforna, Berkeley SUMMARY: The MSRS rule s a response spectrum method for analyss of multply supported structures subjected to spatally varyng ground motons. Ths paper evaluates the accuracy of the MSRS rule by comparng MSRS estmates of mean peak responses wth correspondng exact mean values obtaned by tme-hstory analyss wth ensembles of smulated support motons. The smulated support motons are realzatons of an array of nonstatonary processes wth a specfed coherency functon, generated wth a smulaton approach developed elsewhere by the authors. These sets of motons are characterzed by consstent varablty at all support ponts, and thus, are approprate as nput for statstcal analyss. The structural systems consdered are four brdge models selected to have vastly dfferent structural characterstcs. The responses examned are per drfts, whch are quanttes partcularly mportant n performance based desgn of brdges. Results ndcate that the MSRS method s a relable analyss tool. Keywords: brdges, earthquake response, ground moton spatal varablty, MSRS rule, response spectrum 1. INTRODUCTION Response spectrum methods are wdely used n engneerng practce for lnear sesmc analyss of buldngs and brdges. An mportant advantage of the response spectrum approach over Response Hstory Analyss (RHA) s that t provdes a statstcal characterzaton of the response, not controlled by a partcular selecton of ground motons. Furthermore, ts smple and fast mplementaton s appealng from a desgn vewpont and allows extended parametrc analyss. For multply supported structures subjected to spatally varyng ground motons, Der Kureghan and Neuenhofer (1992) developed the Multple Support Response Spectrum (MSRS) method based on prncples of random vbraton theory. The present authors have recently generalzed and extended ths method to allow consderaton of all response quanttes and to account for the pseudo-statc contrbutons of truncated modes (Konakl and Der Kureghan, 211a). The MSRS rule evaluates the mean peak response n terms of the response spectra and mean peak ground dsplacements at the support ponts of the structure, and the coherency functon that characterzes the spatal varablty of the support motons. The MSRS method has been used by a growng number of researchers (e.g. Loh and Ku, 1995; Kahan et al., 1996; Yu and Zhou, 28) and has been adopted by sesmc codes (Eurocode 8, 1998). As wth other response spectrum methods, the MSRS rule nvolves fundamental assumptons and approxmatons rooted n the theory of statonary random vbratons. These nclude the assumptons that the ground moton s a broadband process wth a strong moton duraton several tmes longer than the fundamental perod of the structure and that the spatal varablty of the ground moton random feld s descrbed by a smooth coherency functon. The error encountered by use of MSRS s unknown and can be on ether conservatve or unconservatve sde. In ths paper, errors n the MSRS estmates of the mean peak per drfts of four brdge models are evaluated by comparsons wth the respectve exact quanttes obtaned from lnear RHA wth statstcally consstent nputs. The support exctatons n the RHA approach are synthetc arrays of spatally varyng motons smulated wth the method developed by Konakl and Der Kureghan (212). Comparsons between results from consstent MSRS and RHA analyss are performed for ground moton random felds characterzed by

2 dfferent frequency contents and coherency functons. 2. STRUCTURAL RESPONSE TO SPATIALLY VARYING SUPPORT MOTIONS Consder a lumped mass lnear structural model wth N unconstraned degrees of freedom (DOF) and m support DOF. Let u k (t) (k=1,..., m) denote the prescrbed support exctatons. Assumng classcal dampng, let s k (t) denote the normalzed response of mode (=1,..., N) to the kth support moton, obtaned as the soluton to s k 2 k ( t) 2ζ ω s ( t) ω s ( t) u ( t) (2.1) k k where ω and ζ respectvely denote the correspondng modal frequency and dampng rato of the fxed base structure. Neglectng dampng forces assocated wth the support DOF, a generc response quantty of nterest, z(t), can be expressed n terms of the support motons, (t), and the normalzed modal responses, s k (t), as follows (Der Kureghan and Neuenhofer, 1992; Konakl and Der Kureghan, 211a): m m N z( t) a u ( t) b s ( t) (2.2) k k k1 k1 1 k k In the precedng equaton a k represents the response quantty of nterest when the kth support DOF s statcally dsplaced by a unt amount wth all other support DOF remanng fxed and b k represents the contrbuton of the th mode to the response z(t) arsng from the exctaton at the kth support DOF when s k (t) s equal to unty. Coeffcents a k and b k depend only on the structural propertes and can be computed by use of any conventonal statc analyss program (Konakl and Der Kureghan, 211a). The frst sngle-sum term n Eqn. 2.2 s the pseudo-statc component of the response,.e. the statc response of the system at each tme nstant when nerta and dampng forces are gnored; ths term s zero n the case of unform support motons. The second double-sum term s the dynamc component of the response,.e. the response of the structure to the dynamc nerta forces nduced by the support motons. Usng Eqn. 2.2 and the prncples of statonary random vbraton theory, Der Kureghan and Neuenhofer (1992) have shown that, for the case of translatonal support motons, the mean of the peak of the generc response quantty z(t) can be approxmately obtaned n the form u k m m m m N z( t) akalρu u uk,maxul,max akbljρu s uk,maxdl ω j,ζ j k l k lj E max k1 l1 m m N N b k k1 l1 1 j1 b lj ρ s s k lj D k k1 l1 j1 ω,ζ D ω,ζ l j j 1/ 2 (2.3) The precedng equaton represents the MSRS combnaton rule. The frst, double-sum term nsde the square brackets s the pseudo-statc component of the response, the thrd, quadruple-sum term s the dynamc component, and the second, trple-sum term s a cross term of the pseudo-statc and dynamc components. The mean of the peak response s gven n terms of the structural propertes, reflected n the coeffcents a k and b k, the mean peak ground dsplacements, u k, max, the ordnates of the mean D ω, ζ, for each support moton and each modal frequency and dsplacement response spectrum, k

3 dampng rato, and three sets of cross correlaton coeffcents: between the kth and lth support dsplacements, ρ u k s lj ρ u u k l, descrbng the correlaton, descrbng the correlaton between the kth support dsplacement and the response of mode j to the lth support moton, and ρ s k s lj, descrbng the correlaton between the responses of modes and j to the kth and lth support motons, respectvely. The coeffcents ρ are functons of the auto- and cross-power spectral denstes (PSDs) of the u k u l support motons, whereas the coeffcents ρ u k s lj and ρ s k s lj addtonally depend on the modal frequences and dampng ratos. The cross-psds of the support motons are gven n terms of the correspondng auto-psds and the coherency functon that models the spatal varablty of the ground moton random feld n the frequency doman. The auto-psd of each support moton s obtaned n terms of the specfed response spectrum (Der Kureghan and Neuenhofer, 1992). Thus, the set of response spectra for all support DOF (ncludng the lmts at nfnte perod, whch equal the respectve peak ground dsplacements) and the set of coherency functons for all pars of support motons represent a complete specfcaton of the nput ground motons requred n the MSRS analyss. 3. BRIDGE MODELS The structural systems consdered n ths study are dealzed models of four real brdges wth vastly dfferent structural characterstcs. The brdges have been desgned by the Calforna Department of Transportaton (Caltrans) and the correspondng brdge models have been developed accordng to Caltrans specfcatons (Caltrans SDC, 24). A bref presentaton of the models s provded n the followng; a detaled descrpton s gven n Konakl and Der Kureghan (211b). The Penstock Brdge, shown n Fgure 3.1 (upper left graph), s a four-span brdge wth one per per bent and a prestressed concrete box grder. The deck has a vertcal grade, varyng from.3% to 2.1%, and a constant horzontal curvature of radus R = 458m. The columns are consdered rgdly connected to the deck at the top and fxed n all drectons at the bottom. The ends of the brdge are supported on seat abutments. Followng Caltrans specfcatons, the horzontal response of the abutments s modeled through translatonal sprngs, whereas vertcal translatons are fully constraned. The fnte element model of the brdge conssts of 3 elements per per and 6, 8, 8 and 4 elements n spans 1, 2, 3 and 4, respectvely. Vertcal rgd frame elements are used to connect the tops of the pers wth the deck. Condensng out the rotatonal DOF and accountng for the constrants mposed by the rgd elements, the structure has 13 translatonal unconstraned DOF and 15 translatonal support DOF. The fundamental perod of the brdge model s T = 2.38s. The South Ingram Slough Brdge, shown n Fgure 3.1 (lower left graph), s a two-span brdge wth two pers per bent and a prestressed-concrete box grder. The deck has a vertcal grade, varyng from.52% to.85%, and a constant horzontal curvature of radus R = m. The columns are consdered rgdly connected to the deck at the top and fxed n all translatonal and rotatonal drectons at the bottom. The two ends of the brdge are supported on seat abutments. The fnte element model of the brdge conssts of 3 elements per per and 6 elements n each span. Vertcal rgd frame elements are used to connect the tops of the pers wth the deck. Condensng out the rotatonal DOF and accountng for the constrants mposed by the rgd elements, the structure has 55 translatonal unconstraned DOF and 12 translatonal support DOF. The fundamental perod of the brdge model s T = 1.24s. The Bg Rock Wash Brdge, shown n Fgure 3.1 (upper rght graph), s a three-span brdge wth three pers per bent and a prestressed concrete box grder. The longtudnal axs of the brdge, X, s a straght lne. The deck s characterzed by a constant profle grade of.5%. The pers are assumed to be rgdly connected to the deck at the top, whereas the bottom supports are fxed n all translatonal drectons and free n all rotatonal drectons. The two ends of the brdge are supported on seat abutments. The fnte element model of the brdge conssts of 3 elements per per and 4 elements per span. Vertcal rgd frame elements are used for the connecton of the upper column elements wth the grder

4 elements. Condensng out the rotatonal DOF and accountng for the constrants mposed by the rgd elements, the structure s modeled wth 89 translatonal unconstraned DOF and 24 translatonal support DOF. The fundamental perod of the structure s T =.61s. The Auburn Ravne Brdge, shown n Fgure 3.1 (lower rght graph), s a sx-span brdge wth two pers per bent and a prestressed-concrete box grder. The deck has a vertcal grade of.3% and a horzontal curvature of radus R = 1616m. The pers are consdered rgdly connected to the deck at the top, whereas the bottom supports are fxed n all translatonal drectons and free n all rotatonal drectons. The two ends of the brdge are supported on seat abutments. The fnte element model of the brdge conssts of 3 elements per per and 4 elements per span. The top of each per s connected wth the deck through two rgd frame elements: one vertcal and one n the drecton of the lne connectng the tops of the pers n the bent. Condensng out the rotatonal DOF and accountng for the constrants mposed by the rgd elements, the structure has 163 translatonal unconstraned DOF and 36 translatonal support DOF. The fundamental perod of the brdge model s T =.59s. In the RHA analyss, Raylegh dampng s assumed, wth the parameters adjusted so that the dampng ratos of the lower modes are close to 5%. Penstock Brdge Bg Rock Wash Brdge South Ingram Slough Brdge Auburn Ravne Brdge Fgure 3.1. Brdge models 4. GROUND MOTION INPUT The ground moton nputs n the RHA and MSRS analyses are consstent,.e., the response spectra and the coherency functon used to evaluate the terms n the MSRS rule represent average propertes of the ensembles of support exctatons used to perform RHA. Closely spaced records of ground motons are rare; furthermore, dstances between recordng statons dffer from the dstances between support ponts of specfc structural models to be analyzed. Thus, RHA for multply supported structures has to rely on synthetc ground motons. In ths study, the RHA nput conssts of synthetc arrays of motons generated usng the smulaton method developed n Konakl and Der Kureghan (212). Spatal varablty of ground moton arrays smulated wth ths method ncorporates the effects of () ncoherence,.e. the loss of coherency of sesmc waves wth

5 Sa, g Sa, g a g, g a g, g dstance as represented by random dfferences n the ampltudes and phases of the waves, () wave passage,.e. the determnstc tme delay n the arrval of sesmc waves at separate statons, and () varaton of sol condtons underneath the supports and the way t affects the ampltude and frequency content of the surface motons. For unform sol condtons, ths smulaton method only requres specfcaton of a seed accelerogram at a reference and a coherency functon that descrbes the spatal varablty of the ground moton random feld. Two approaches were developed n the aforementoned work: the condtonal smulaton method, whch preserves tme-hstory characterstcs of the specfed seed record and generates arrays of motons characterzed by ncreasng varablty wth dstance from the reference ; and the uncondtonal smulaton method, whch generates arrays of motons that preserve the overall temporal and spectral characterstcs of the specfed seed record and exhbt unform varablty at all s. Snce unform varablty s essental for consstent comparsons between the MSRS and RHA, the uncondtonal approach s employed n ths study. In ths study, arrays of spatally varyng ground motons are smulated for two seed accelerograms; the fault-normal components of the Hollster South & Pne (HSP) record from the 1989 Loma Preta earthquake, and the fault-normal component of the Pacoma Dam (PUL) record from the 1971 San Fernando earthquake. For each record, Fgures 4.1 and 4.2 show the acceleraton tme hstores and correspondng response spectra, respectvely. HSP record, FN component PUL record, FN component Fgure 4.1. Acceleraton tme hstores of seed records 1 HSP record, FN component 1 1 PUL record, FN component /2, Hz /2, Hz Fgure 4.2. Acceleraton response spectra of seed records The coherency model employed to descrbe the spatal varablty between the ground moton processes at two stes, k and l, s a functon of the frequency, ω, and s gven by γ kl 2 L adklω/ vs exp(ω dkl / vapp) (ω) exp (4.1) where a s an ncoherence parameter, d kl s the dstance between the stes k and l, vs s the average L shear wave velocty of the ground medum along the wave travel path, d kl s the projected algebrac horzontal dstance n the longtudnal drecton of propagaton of waves, and v app s the surface apparent wave velocty. In ths model, the modulus and phase shft of the coherency functon represent

6 the effects of ncoherence and wave passage, respectvely (Der Kureghan, 1996). For the ncoherence component, the model by Luco and Wong (1986) has been adopted. For all brdges, t s assumed that the waves propagate n the drecton from abutment 1 to the abutment at the other end of the brdge. The values of shear wave velocty and apparent wave velocty are taken to be vs 6m/s and v app 4m/s, respectvely. Analyses of recorded arrays have shown that the rate of decay of the ncoherence component descrbed by the ncoherence parameter can vary sgnfcantly between dfferent arrays (Harchandran and Vanmarcke, 1986; Abrahamson et al., 1991). Smaller values of the ncoherence parameter ndcate more coherent motons. To assess the effect of varatons n the ncoherence parameter, the values of a. 2 and a. 4 are consdered when the HSP record s used as seed. (Only a. 2 s consdered when the PUL record s used as seed.) Ensembles of 2 support moton arrays are smulated for each case of spatal varablty. It s of nterest to nvestgate how spatal varablty affects dfferences n the mean peak responses evaluated wth consstent tme-hstory and response spectrum analyses. For ths reason, the case of unform support exctatons s also examned. For each brdge model, the nput exctaton n ths case s the moton at a reference support from the ensemble of arrays smulated for a. 2. The reference supports are bent 3 for Penstock Brdge, bent 2 for South Ingram Slough Brdge, bent 2 for Bg Rock Wash Brdge and bent 4 for Auburn Ravne Brdge. The mean response spectra obtaned by averagng 5% damped spectra for all smulatons and all support ponts determne the respectve nput for the MSRS analyss. Averagng over all support motons s vald because under unform sol condtons the response spectra at all support ponts should be the same. Response spectra values for dampng ratos other than 5% are evaluated by adjustng the 5% damped spectral values accordng to Caltrans specfcatons (Caltrans SDC, 24). The coherency functon used to evaluate the MSRS correlaton coeffcents s the theoretcal model n Eqn Konakl and Der Kureghan (212) have shown that estmates of the coherency functon from ensembles of smulated arrays of motons are n good agreement wth the target theoretcal model. These coherency estmates are obtaned after smoothng and averagng over the ensemble of arrays, snce the non-smoothed coherency estmated from a sngle par of motons exhbts erratc behavor. In the case of unform exctatons, the MSRS rule reduces to the square-root of the quadruple-sum term representng the dynamc component of the response, whch has the same form as the well known CQC rule (Der Kureghan, 1981), but wth a more accurate approxmaton of the cross-modal correlaton coeffcents. 5. ASSESSMENT OF THE MSRS RULE BY COMPARISONS WITH RHA RESULTS Estmates of mean peak responses evaluated wth the MSRS formula gven by Eqn. 2.3 are compared wth the actual means of the temporal peaks obtaned from RHA usng the decomposton formula n Eqn For consstent comparsons, the same ntegraton method s used for the evaluaton of the th modal tme-hstory response, s k (t), as the th-mode spectral value, Dk ω, ζ. The RHA results have been valdated through comparsons wth tme-hstory analyss wth software OpenSees, whch performs ntegraton of the equatons of moton n matrx form wth Raylegh dampng. The responses examned are per drfts, whch are quanttes partcularly mportant from a desgn vewpont. On the bass of the equal dsplacement rule (Veletsos and Newmark, 196), for suffcently flexble structures, per drfts obtaned from lnear analyss can be used to approxmately evaluate nonlnear demands. Prelmnary analyss has ndcated that, for the per drfts of the specfc brdge models, consderng the frst 4 modes n the analyss s suffcent. For the four brdges, the results of the analyses are presented n Tables For each ground moton random feld, the tables lst mean peak responses evaluated wth the RHA and MSRS approaches, as well as the errors n the MSRS values f the RHA results are consdered exact. These errors are gven n parentheses next to

7 the MSRS estmates. For each brdge and ground moton random feld, mean values of the (algebrac) errors obtaned by averagng over all pers are also lsted. For a selected per of each brdge and for the ground moton felds characterzed by a. 2 (both seed records), Fgure 5.1 shows the tme hstores of the drft responses for the 2 smulated support moton arrays. In each graph, the 2 tme hstores are plotted and compared wth the MSRS estmates represented by the thcker horzontal lnes. These graphs provde an llustraton of the varablty of the peak responses over the ensemble of realzatons. Note that n ths fgure the MSRS estmates are practcally dentcal to the exact RHA mean peak values for the South Ingram Slough Brdge and the Auburn Ravne Brdge wth the PUL record as seed. Consderng the absolute values of the MSRS errors for ndvdual per drfts (lsted n Tables ), the maxmum error observed under unform support motons s 1.% (HSP seed, Auburn Ravne Brdge, bent 2: per 2), whereas under varable support motons ths s 12.3% (PUL seed, Penstock Brdge, bent 4). Consderng the absolute values of the average MSRS errors over all per drfts of each brdge (lsted n Tables ), the maxmum error observed under unform support motons s 2.4% (HSP seed, Auburn Ravne Brdge), whereas under varable support motons ths s 8.% (HSP seed, a. 4, Auburn Ravne Brdge). In most cases, the errors are negatve,.e. the response spectrum approach underestmates the tme-hstory response. Under unform support motons, the mean (standard devaton) of the absolute values of the errors over all pers and brdges s 2.4% (2.6%) for HSP as seed and 3.3% (2.5%) for PUL as seed. Under varable support motons, the mean (standard devaton) of the absolute values of the errors over all pers and brdges s 3.5% (4.3%) for HSP and a. 2, 6.% (4.3%) for HSP and a. 4, and 4.4% (3.6%) for PUL. The MSRS method s ntended for use n conjuncton wth smooth response spectra that represent broadband exctatons and a smooth coherency functon. In our analyss, jagged response spectra from relatvely narrowband exctatons were used. Furthermore, the smooth coherency functon used for evaluaton of the correlaton coeffcents n the MSRS analyss dffers from the actual coherency values for pars of smulated support motons, whch can exhbt large fluctuatons around the theoretcal model. Consderng these dfferences, the results of the MSRS analyss are found to be remarkably accurate. The errors tend to be larger under spatally varyng motons compared to the case of unform exctatons. Ths s because the case of varable support motons employs addtonal assumptons regardng the coherency model explaned above and nvolves addtonal approxmatons n representng the pseudo-statc component of the response and ts cross wth the dynamc component. 6. CONCLUSIONS The accuracy of the MSRS method n evaluatng mean peak responses of brdges subjected to spatally varyng ground motons was assessed through comparsons wth respectve results from consstent RHA. Consderng absolute values of the MSRS errors, the mean and standard devaton over all responses examned (per drfts of four brdge models) and non-unform ground moton felds consdered (three cases) were 4.6% and 3.7%, respectvely. The maxmum error observed was 12.3%, but n most cases, the errors were smaller than 1%. The good agreement between the two analyss approaches showed that the MSRS method s a relable tool for response spectrum analyss under dfferental support exctatons. REFERENCES Abrahamson, N. A., Schneder, J. F. and Stepp, J.C. (1991). Emprcal spatal coherency functons for applcaton to sol-structure nteracton analyses. Earthquake Spectra 7,1-28. Calforna Department of Transportaton (24). Caltrans Sesmc Desgn Crtera (SDC). Der Kureghan, A. (1981). A response spectrum method for random vbraton analyss of MDF systems. Earthquake Engneerng and Structural Dynamcs 9, Der Kureghan, A. (1996). A coherency model for spatally varyng ground motons. Earthquake Engneerng

8 and Structural Dynamcs 25, Der Kureghan, A. and Neuenhofer, A. (1992). Response spectrum method for multple-support sesmc exctaton. Earthquake Engneerng and Structural Dynamcs 21:7, Eurocode 8: Desgn of structures for earthquake Resstance-Part 2: Brdges (1998). Harchandran, R. S. and Vanmarcke, E. H. (1986). Stochastc varaton of earthquake ground moton n space and tme. Journal of Engneerng Mechancs 112, Kahan, M., Gbert, R.J. and Bard, P.Y. (1996). Influence of sesmc waves spatal varablty on brdges: a senstvty analyss. Earthquake Engneerng and Structural Dynamcs 25, Konakl, K. and Der Kureghan, A. (211a). Extended MSRS rule for sesmc analyss of brdges subjected to dfferental support motons. Earthquake Engneerng and Structural Dynamcs 4:12, Konakl, K. and Der Kureghan, A. (211b). Stochastc dynamc analyss of brdges subjected to spatally varyng ground motons. Report No. UCB/EERC-11/8, Earthquake Engneerng Research Center, Unversty of Calforna, Berkeley, Konakl, K. and Der Kureghan, A. (212). Smulaton of spatally varyng ground motons ncludng ncoherence, wave-passage and dfferental ste-response effects. Earthquake Engneerng and Structural Dynamcs 41:3, Loh, C.H. and Ku, B.D. (1995). An effcent analyss of structural response for multple-support sesmc exctatons. Engneerng Structures 17, Luco, J.E. and Wong, H.L. (1986). Response of a rgd foundaton to a spatally random ground moton. Earthquake Engneerng and Structural Dynamcs 14, Veletsos, A.S. amd Newmark, N.M. (196). Effect of nelastc behavor on the response of smple systems to earthquake motons. Proceedngs of the 2nd World Conference on Earthquake Engneerng, Japan, 2, Yu, R.F. and Zhou, X.Y. (28). Response spectrum analyss for non-classcally damped lnear system wth multple-support exctatons. Bulletn of Earthquake Engneerng 6, Table 5.1. Penstock brdge: Mean peak per drfts (n meters) from RHA and MSRS analyses and respectve MSRS errors seed: HSP unform motons varable motons, a. 2 varable motons, a. 4 RHA MSRS (% error) RHA MSRS (% error) RHA MSRS (% error) bent (.) ( 1.7) ( 1.5) bent ( 2.) ( 3.6) (1.) bent ( 4.6) ( 1.2) ( 7.5) average ( 2.2) ( 5.2) ( 2.7) seed: PUL unform motons varable motons, a. 2 RHA MSRS (% error) RHA MSRS (% error) bent (.4) ( 6.5) bent (.9) ( 2.9) bent ( 5.4) ( 12.3) average ( 2.2) ( 7.2) Table 5.2. South Ingram Slough Brdge: Mean peak per drfts (n meters) from RHA and MSRS analyses and respectve MSRS errors seed: HSP unform motons varable motons, a. 2 varable motons, a. 4 RHA MSRS (% error) RHA MSRS (% error) RHA MSRS (% error) bent 2: per (.) (.5) (.5) bent 2: per (.) (.5) (.5) average (.) (.5) (.5) seed: PUL unform motons varable motons, a. 2 RHA MSRS (% error) RHA MSRS (% error) bent 2: per (.) (.) bent 2: per (.) (.) average (.) (.)

9 Table 5.3. Bg Rock Wash Brdge: Mean peak per drfts (n meters) from RHA and MSRS analyses and respectve MSRS errors seed: HSP unform motons varable motons, a. 2 varable motons, a. 4 RHA MSRS (% error) RHA MSRS (% error) RHA MSRS (% error) bent 2: mddle ( 3.2).41.4 ( 2.4) ( 6.7) bent 2: sde ( 3.2).41.4 ( 2.4) ( 6.7) bent 3: mddle.7.71 (1.4) ( 1.7) ( 4.8) bent 3: sde.7.71 (1.4) ( 1.7) ( 4.8) average (.9) ( 2.1) ( 5.8) seed: PUL unform motons varable motons, a. 2 RHA MSRS (% error) RHA MSRS (% error) bent 2: mddle ( 2.6) (9.7) bent 2: sde ( 2.6) (9.7) bent 3: mddle (2.4) ( 5.) bent 3: sde (2.4) ( 5.) average (.1) (2.4) Table 5.4. Auburn Ravne Brdge: Mean peak per drfts (n meters) from RHA and MSRS analyses and respectve MSRS errors seed: HSP unform motons varable motons, a. 2 varable motons, a. 4 RHA MSRS (% error) RHA MSRS (% error) RHA MSRS (% error) bent 2: per ( 7.3) ( 3.) (.) bent 2: per ( 1.) (.) (.) bent 3: per ( 3.4).3.29 ( 3.3) ( 11.4) bent 3: per ( 3.5) (3.6).33.3 ( 9.1) bent 4: per ( 1.4) ( 2.9) ( 11.9) bent 4: per ( 1.4) ( 2.9) ( 12.2) bent 5: per (.) ( 4.3) ( 9.6) bent 5: per (.) ( 4.4) ( 9.8) bent 6: per (1.3) ( 8.2) ( 8.1) bent 6: per (1.3).6.55 ( 8.3) ( 8.2) average ( 2.4) ( 3.4) ( 8.) seed: PUL unform motons varable motons, a. 2 RHA MSRS (% error) RHA MSRS (% error) bent 2: per ( 3.8) (6.1) bent 2: per ( 3.8) (7.2) bent 3: per ( 7.5) ( 1.4) bent 3: per ( 6.7) (.) bent 4: per ( 5.4) (.) bent 4: per ( 5.5) ( 1.8) bent 5: per (.).7.72 (2.9) bent 5: per ( 1.5) (3.) bent 6: per (5.9) ( 6.) bent 6: per (5.9) ( 4.6) average ( 2.2) (.5)

10 seed: HSP, varable motons, a. 2 seed: PUL, varable motons, a. 2 Penstock Brdge, bent South Ingram Slough Brdge, bent 2: per Bg Rock Wash Brdge, bent 2: mddle per Auburn Ravne Brdge, bent 4: per Fgure 5.1. RHA and MSRS per drft responses