COMBINING SPATIAL COMPONENTS IN SEISMIC DESIGN

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1 Transactons, SMRT- COMBINING SPATIAL COMPONENTS IN SEISMIC DESIGN Mchae O Leary, PhD, PE and Kevn Huberty, PE, SE Nucear Power Technooges Dvson, Sargent & Lundy, Chcago, IL 6060 ABSTRACT Accordng to Reguatory Gude.9 [], the square root of the sum of the squares (SRSS) method and the rue are acceptabe means of combnng spata components n sesmc desgn. The two methods are compared n order to carfy ther proper mpementaton and to dentfy the advantages of each approach wth respect to the fnte eement-based desgn process common throughout the nucear ndustry. The excess conservatsm and the manpuaton of the sgns of mutpe response parameters requred for the adequate mpementaton of the SRSS method mae the rue a better approach for desgn. INTRODUCTION The two-step method s a common ndustry approach to ncorporate the effects of so-structure nteracton (SSI) anayss nto the desgn of new concrete nucear structures. The frst step s to perform the SSI anayss and the second step s to mae an equvaent statc sesmc fnte eement mode based on the SSI anayss resuts. Typcay the noda zero perod acceeratons (ZPA) from the SSI anayss resuts are mutped by ther trbutary mass to determne equvaent statc sesmc forces assocated wth a horzonta or vertca nput moton. These forces are apped to a three-dmensona equvaent statc sesmc fnte eement mode and combned usng the SRSS method or the percent combnaton rue n accordance wth Reguatory Gude.9 []. The two-step method offers a fexbe and practca approach to new desgn. If a coarser mesh s requred n the SSI anayss mode due to computatona mtatons, for exampe, the two-step method can accommodate a more refned mesh n the desgn mode. Snce a of the equvaent statc forces assocated wth the noda acceeratons act n phase, t s generay accepted that the two-step approach s conservatve []. The fowchart n Fgure summarzes ths concrete desgn process. Two acceptabe methods for combnng spata components n sesmc desgn (SRSS and rue) n Reguatory Gude.9 [] are compared n order to carfy ther proper mpementaton and to dentfy the advantages of each approach wthn the context of the desgn process common throughout the nucear ndustry. COMBINATION OF THREE-DIMENSIONAL EARTHQUAKE EFFECTS AND MULITPLE RESPONSE PARAMETERS In accordance wth Reguatory Gude.9 [], the SRSS combnaton of three-dmensona earthquae effects s R= RI, () I = where R s any response of nterest and R I s the combned response for the I th component of sesmc nput moton (e.g., x, y, or z component). ASCE 4-98 [4] adds that the SRSS method shoud ncude postve and negatve permutatons of the response.

2 SSI anayss Extract maxmum noda ZPAs and mutpy by the trbutary mass n an equvaent statc sesmc FE mode Post-process oad combnatons Determne requred renforcement that satsfes statc and dynamc demand for a structura members Fgure : Sesmc anayss and desgn process for concrete nucear structures. The rue s defned n Reguatory Gude.9 [] as ( ) R = R + R + R () where R, R, and R are the maxmum responses of the structure caused by each of the three earthquae components such that R R R. In ASCE 4-98 [4], a possbe twenty-four permutatons of the rue are made expct by the foowng expresson: [ ] or [ ] or [ ] R =± R ± R ± R ± R ± R ± R ± R ± R ± R. () The prncpe advantage of equaton () over () s that the drectona nature of the oad s made expct n the expresson. Moreover, the response R from equaton () w aways be equa to or greater than the response R from equaton (). Therefore, when the rue s nvoed throughout the remander of ths paper, the twenty-four responses contaned n equaton () are mped. As ndcated n On the Correct Appcaton of the Rue [], f the methodoogy prescrbed n ASCE 4-98 [4] for determnng the maxmum sesmc desgn forces s foowed, the rue s neary aways conservatve reatve to the SRSS method. Secton..7.. of ASCE 4-98 [4] states that when there s more than one response parameter, such as coumn axa force and moment, to be used n the desgn cacuaton, the combned vaue sha be cacuated. In other words, whether the rue or SRSS method s used to determne the co-drectona response from three earthquae components, the maxmum axa force sha be combned wth the maxmum moment for desgn. By contrast, secton C4.. of ASCE 4- (draft verson) [5] recommends n cases nvovng mutpe nteractng desgn parameters the maxmum vaue of each desgn parameter [s consdered] together wth the vaues of the other parameters that correspond to the same drectona combnaton. Indeed, the purpose of a sesmc mode s to conservatvey represent maxmum oadng condtons. A bref revew of tme hstory anayss resuts reveas that the maxmum forces from mutpe parameters rarey, f ever, occur at the same tme step. Snce the maxmum axa force amost never occurs at the same tme as the

3 maxmum bendng moment, the revsed ASCE 4- (draft verson) [5] approach resuts n more reasonabe renforcement whe mantanng the conservatsm requred n sesmc desgn. Under the ASCE 4- (draft verson) [5] approach, however, the SRSS method s amost aways more conservatve than the rue, as can be seen n the foowng exampe. COLUMN EXAMPLE Consder a smpe 0 ft coumn wth a ft by ft cross secton. The sesmc responses due to the x, y, and z drecton exctatons are each equa to 0 ps apped at the top of the coumn: R x = R y = R z = 0 ps. As can be seen n Tabe, the rue yeds the maxmum snge response parameter hghghted n red, but these parameters nteract wth ower correspondng response parameters. Subsequenty, M x and M y ponts from rue may fa wthn the baxa nteracton surface where the SRRS ponts do not due to the hgher correspondng axa oad as shown schematcay n Fgure. Furthermore, f there s a torsona component to the sesmc response, the maxmum torson does not tend occur n the same oad combnaton as the maxmum shear force under the rue. Agan, whe the maxmum torson due to the SRSS method may be ower than the rue, the correspondng shear force s typcay hgher and subsequenty more crtca. P M x x M y Pane of P = P y P M x Pane of P = 0 Pane of P = 9. Pane of P = 4 M y Pane of P = 9. SRSS P M x M y M x M y Fgure : Schematc baxa nteracton curve. Pane of P =

4 Tabe : Mutpe response parameters (absoute vaues) for a 0 ft coumn where R x = R y = R z = 0 ps. Combnaton P M x (p-ft) M y (p-ft) V x V y SRSS SHEAR WALL FINITE ELEMENT MODEL The test mode shown n Fgures 4 s three stores hgh wth three bays n both drectons. There s a twenty foot span between each of the three bays. The three foors are spaced at 8 feet and the sabs are 0 nches thc. The three openngs shown n Fgure 4 are ony found on the second foor. A was, ncudng the two nteror shear was are two feet thc. For ths study, the fxed base mode s frst subected to the three orthogona tme hstores shown n Fgure 5. The maxmum absoute acceeratons from a three tme hstory anayses are extracted and combned by the SRSS method. These three orthogona acceeratons are mutped by ther trbutary masses to determne an x, y, and z drecton equvaent statc force. These three forces are then combned by means of the SRSS method and the rue. Athough the fxed-base acceeratons are not dentca to SSI anayss, the process of determnng equvaent statc sesmc forces s the same. y z x Fgure : Fnte eement test mode.

5 Fgure 4: Mode foor pan and eevaton. Acceeraton (g) Tme (sec) a.) X drecton exctaton. Acceeraton (g) Tme (sec) b.) Y drecton exctaton.

6 Acceeraton (g) Tme (sec) c.) Z drecton exctaton. Fgure 5: Tme hstores used n the fxed base tme hstory anayses. RESULTS The maxmum anaytca resuts for vertca axa forces n the nteror wa dentfed n Fgure are shown n Fgure 6. In order to compare the accuracy of the oad path, the maxmum tme hstory anayss resuts are shown aong wth the rue and the SRSS method resuts. Snce the three tme hstory cases are anayzed separatey, ony the resuts from the domnant drectona oad are shown. The maxmum anaytca resuts for vertca axa forces n the exteror wa dentfed n Fgure are shown n Fgure 7. Tabes and provde the sum of the eement forces aong the bottom of the nteror and exteror shear was hghghted n Fgure, respectvey for the oad combnatons where the n-pane shear forces are greatest n magntude. The n-pane moment s cacuated by summng the moment of the vertca eement forces of the bottom row of eements about the geometrc center of the wa: n w M n pane = F x = (4) where F represents the force assocated wth the bottom face of the n th eement, w s the entre wa ength, and x s the force ocaton reatve to the center ne of the wa. Athough the sgn of the SRSS axa forces are postve across the entre ength of the shear wa, t s assumed that a the forces on one sde of the geometrc center of the wa are postve and a the forces on the other sde are negatve n determnng the n-pane moment. None of the resuts ncude the effect of the dead oad on the structure. Ony the anaytca resuts of the equvaent statc nerta oads are shown.

7 SRSS Y drecton tme hstory Fgure 6: Maxmum vertca axa eement forces (ps/ft) n the nteror wa for the rue, the SRSS method, and the y drecton tme hstory anayss at tme step seconds SRSS X drecton tme hstory Fgure 7: Maxmum vertca axa eement forces (ps/ft) n the exteror wa for the rue, the SRSS method, and the x drecton tme hstory anayss at tme step.45 seconds.

8 Tabe : Maxmum eement forces summed up aong the bottom of the nteror shear wa Load combnaton Equvaent statc (-)40-00-(-)40 Equvaent statc SRSS Tme hstory at t = sec ( ) Tme hstory at t = sec (SRSS) Out-ofpane shear In-pane shear Axa force Out-of-pane moment (p-ft) In-pane moment (p-ft) Tabe : Maxmum eement forces summed up aong the bottom of the exteror shear wa Load combnaton Equvaent statc (-)00-40-(-)40 Equvaent statc SRSS Tme hstory at t =.45 sec ( ) Tme hstory at t =.45 sec (SRSS) Out-ofpane shear In-pane shear Axa force Out-of-pane moment (p-ft) In-pane moment (p-ft) DISCUSSION The axa eement force pots of the , SRSS, and tme hstory oad cases shown n Fgures 6 and 7 are smar n terms of magntude and oad dstrbuton. The maxmum mutpe response parameters reported n Tabes and echo the nteracton shown n Tabe for the coumn exampe. Whe the maxmum snge response occurs n one of the twenty-four permutatons of the rue, t nteracts wth ower correspondng response parameters than the SRSS oad. Moreover, the crtca response parameters n the drecton of the earthquae oad, ncudng the n-pane shear, and n-pane moment forces, are smar for both the rue and the SRSS method. The correspondng SRSS axa forces n both the nteror and exteror was are we over twce the magntude of the rue axa forces, however. As a resut, the SRSS method s more an to usng a combnaton rue for desgnng a shear wa for axa force and moment. Asde from the excess conservatsm of the SRSS method, the eement force pots shown n Fgures 6 and 7 ndcate a potenta probem n ts mpementaton. As expaned above, the ony way to obtan the n-pane moment from the eement resuts s to manuay reverse the sgn of the axa force on ether sde of the geometrc center of the shear wa. Wth more compcated structures typca n nucear concrete desgn n whch mutpe was and sabs frame nto shear was wth openngs, ths approach

9 cannot be easy mpemented, however. The actua drectona oad path s requred to determne the npane moment. Fnay, a resuts combned by the SRSS method are postve or negatve. But as secton cut force Tabes and revea, mutpe response parameters from drectona oads do not necessary have the same sgn. Therefore, uness the anayst manpuates the sgn of response parameters as part of a postprocessng routne, the SRSS method potentay msses oad combnatons n whch the sesmc response parameter nteracts constructvey wth dead or ve oad response parameters, for exampe. Fgure 8 ustrates the sgn conventon for she eements. Tabes 4 and 5 st the permutatons of the sesmc response for she eements and frame eements that are mssed by the postve and negatve mpementaton of equaton (). Athough the number of requred permutatons for frame eements n Tabe 5 s greater than the number of she permutatons, the permutatons requred to account for n-pane moment when the SRSS method s used are not ncuded Tabe 5. (-)N (-)N (-)N Tenson Compresson Tenson Compresson (compresson) (-)N (compresson) (-)M Compresson Tenson (+)M (-)M Compresson (+)M Tenson (+)N (+)N (tenson) (+)N (+)N (tenson) (+)M (-)M Fgure 8: She eement sgn conventon. Tabe 4: Requred she eement permutatons for SRSS. Permutaton N M N M M F V V +SRSS SRSS Not captured Not captured

10 Tabe 5: Requred frame eement permutatons for SRSS. Permutaton P M x M y +SRSS SRSS Not captured Not captured Not captured Not captured Not captured Not captured CONCLUSION Ths study has confrmed the conservatsm of the rue reatve to the SRSS method for a snge response parameter as we the conservatsm of the SRSS method when mutpe response parameters are consdered. Both methods of combnng spata components of earthquaes n determnng equvaent statc oads are acceptabe accordng to Reguatory Gude.9 [] and both methods adequatey enveop the crtca forces from tme hstory anayss. In ths respect, the rue s preferred for desgn snce the fna renforcement w be ess congested than a desgn based on the SRSS method. Both methods aso requre permutatons of sgns to adequatey capture the most crtca oad combnatons. The advantage of the rue s that the permutatons of the oads are requred before anayss, whch s a reatvey smpe tas. The SRSS method, on the other hand, requres manpuaton of the sgn of ndvdua response parameters outsde of the fnte eement anayss. And for a more compcated structure usng the SRSS method, there may be no cear way of evauatng n-pane moment. In concuson, the drectona nature of the oads and the reatve smpcty of ts mpementaton mae t preferabe to the SRSS method. ACKNOWLEDGMENTS The authors gratefuy acnowedge Andrew Bomqust and Wam Godfrey of Sargent & Lundy for ther assstance n preparng the fnte eement modes and certan fgures. REFERENCES [] U.S. Nucear Reguatory Commsson. (0). Reguatory Gude.9, Combnng Moda Responses and Spata Components n Sesmc Response Anayss, Revson. [] Watns, D., Gürbüz, O., and Ma, T. (006). Two-step method of sesmc anayss, Proc., Frst European Conference on Earthquae Engneerng and Sesmoogy, Geneva, Swtzerand. [] Ne, J., Morante, R., Mranda, M., and Braverman, J. (00). On the correct appcaton of the rue for combnng responses due to three drectons of earthquae oadng, Proc., ASME 00 Pressure Vesses and Ppng Dvson, Beevue, WA. [4] Amercan Socety of Cv Engneers. (000). ASCE 4-98: Sesmc Anayss of Safety-Reated Nucear Structures, Revson. [5] Amercan Socety of Cv Engneers. (0). ASCE 4-: Sesmc Anayss of Safety-Reated Nucear Structures, Draft.