Plastic Analysis and Design of Steel Plate Shear Walls

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1 7 Platc Analy and Degn of Steel Plate Shear Wall Jeffrey Berman Department of Cvl, Structural & Envronmental Engneerng, Unverty at Buffalo Reearch Supervor: Mchel Bruneau, Profeor Summary A reved procedure for the degn of teel plate hear wall propoed. In th procedure, the thckne of the nfll plate found ung equaton that are derved from platc analy of the trp model, whch wdely accepted for the repreentaton of teel plate hear wall. Equaton are derved from a bac teel plate hear wall confguraton but are hown to be conervatve for more complex confguraton. Fundamental platc collape mechanm for everal wall confguraton are alo gven. Introducton Steel plate hear wall (SPSW) have ometme been ued a the lateral load retng ytem n buldng. Untl recently, the falure mode of SPSW wa condered to be the out-of-plane bucklng of the nfll plate. Th led engneer to degn heavly tffened plate that offered lttle economc advantage over renforced concrete hear wall. However, a Baler (96) demontrated for plate grder web, the pot-bucklng tenon feld acton of SPSW can provde ubtantal trength, tffne and ductlty. The dea of utlzng the pot-bucklng trength of SPSW wa frt developed analytcally by Thorburn et al., (983) and verfed expermentally by Tmler and Kulak (983). Stude performed to evaluate the trength, ductlty, and hyteretc behavor of uch SPSW degned wth untffened nfll plate have demontrated ther gnfcant energy dpaton capablte (Tmler and Kulak 983) and ubtantal economc advantage (Tmler 998). At the tme of th wrtng, there are no U.S. pecfcaton or code addreng the degn of SPSW. The 00 Canadan tandard CAN/CSA S6-00 (CSA 00) now ncorporate mandatory claue for the degn of SPSW. One of the model recommended to repreent SPSW the trp model developed by Thorburn et al., (983). Th model generally recognzed for provdng relable aement of ultmate trength. In th tudy, the trp model ued a a ba to nvetgate the feablty of platc analy a an alternatve for the degn of SPSW. Fundamental platc collape mechanm are decrbed for ngle tory and multtory SPSW wth ether mple or rgd beam-tocolumn connecton. Thrut Area : Semc Retroft of Acute Care Faclte 35

2 Sngle Story Frame wth Smple Beam-to-Column Connecton Conder the frame wth nclned trp and mple beam-to-column connecton hown n Fgure. When the hear force,, dplace the top beam by a value uffcently large to yeld all the trp, the external work done equal to (Fgure ). d d α Zone h n α Zone Pn-Ended Beam Zone 3 h d θ F t α L co α L d Pn-Ended Strp h n α Fgure. Strp model of ngle tory wall Fgure. Sngle tory collape mechanm If the beam and column are aumed to reman elatc, ther contrbuton to the nternal work may be neglected when compared to the nternal work done by the brace. Hence, the nternal work [n b A t F y n (α)], where n b the number of trp anchored to the top beam, A t the trp area, F y the trp yeld tre, and α the angle of nclnaton of the trp meaured from vertcal. Th reult can be obtaned by the product of the yeld force tme the yeld dplacement of the trp, but for mplcty t can alo be found ung the horzontal and vertcal component of thee value. Note that the horzontal component of the yeld force of the trp on the column cancel (the force on the left column do negatve nternal work and the force on the rght column do potve nternal work) and the vertcal component of all the yeld force do no nternal work becaue there no vertcal deflecton. Therefore, the only nternal work done that due to the horzontal component of the trp yeld force anchored to the top beam. Equatng the external and nternal work gve: ( α) = n b F t n () Ung the geometry hown n Fgure, n b = [L co (α)] / and the trp force F t agan F y t. Subttutng thee nto Eq. and knowng that (/) n (α) = co (α) n (α), the reultng bae hear relatonhp : = Fy t L n( α) () 36

3 Sngle Story Frame wth Rgd Beam-to-Column Connecton In ngle tory SPSW havng rgd beam-to-column connecton (a oppoed to mple connecton), platc hnge alo need to form n the boundary frame to produce a collape mechanm. The correpondng addtonal nternal work 4 M p θ, where θ = h S the tory dplacement over the tory heght and M p the maller of the platc moment capacty of the beam (M pb ) or column (M pc ). For mot ngle-tory frame that are wder than tall, f the beam have uffcent trength and tffne to anchor the tenon feld, platc hnge wll typcally form at the top and bottom of the column and not n the beam. The ultmate trength of a ngle-tory SPSW n a moment frame wth platc hnge n the column become: ( α) 4 M pc = Fy t L n + (3) h In a degn proce, falure to account for the addtonal trength provded by the beam or column reult n larger plate thcknee than neceary. In turn, larger thckne would tranlate nto lower ductlty demand n the wall and frame member. Therefore, neglect of addtonal trength provded by the beam or column could be condered a conervatve approach. Multtory Frame For multtory SPSW wth pn-ended beam, platc analy can alo be ued to predct the ultmate capacty. The purpoe here not to preent cloed-form oluton for all poble falure mechanm, but to dentfy ome key platc mechanm that hould be condered n etmatng the ultmate capacty of a SPSW. Thee could be ued to defne a derable falure mode n a capacty degn perpectve and/or to prevent an underable falure mode, a well a to complement tradtonal degn approache. In oft-tory platc mechanm (Fgure 3), the platc hnge that would form n the column at the mechanm level could be ncluded n the platc analy. Calculatng and equalng the nternal and external work, the followng general expreon could be ued for oft-tory n whch all flexural hnge develop n column: n j= j = F y t L n ( α) + M h pc (4) where j are the appled lateral force above the oft-tory, t the plate thckne at the oft-tory, M pc the platc moment capacty of the column at the oft-tory, h the heght of the oft-tory, and n the total number of tore. Note that only the appled lateral force above the oft-tory do external work and they all move the ame dtance (). The nternal work done only by the trp on the oft-tory telf and by column hnge formng at the top and bottom of the oft-tory. Ung the above equaton, the poblty of a oft-tory mechanm hould be checked at every tory n whch there a gnfcant change n plate thckne or column ze. Addtonally, the oft-tory mechanm ndependent of the beam connecton type (mple or rgd) becaue hnge mut form n the column, not n the beam. A econd (and more derable) poble collape mechanm nvolve unform yeldng of the plate over every tory (Fgure 4). For th mechanm, each appled lateral force move a dtance Thrut Area : Semc Retroft of Acute Care Faclte 37

4 = θ h, and doe an external work equal to θ h, where h the elevaton of the th tory. The nternal work done by the trp of each yeldng tory. It mportant to note that the trp force actng on the bottom of a tory beam do potve nternal work and the trp force actng on top of the ame beam do negatve nternal work. Therefore, the nternal work at any tory equal to the work done by trp yeld force along the bottom of the tory beam mnu the work done by trp yeld force on the top of the ame beam. Th ndcate that n order for every plate at every tory to contrbute to the nternal work, the plate thcknee would have to vary at each tory n drect proporton to the demand from the appled lateral force. Even wth th n mnd, th mechanm provde nght nto the capacty and falure mechanm of the wall. The general equaton for the ultmate trength of a multtory SPSW wth mple beam-to-column connecton and th platc mechanm (equatng the nternal and external work) : n = h n = = F y ( t t ) L h n ( α) + (5) where h the th tory elevaton, n the total number of tore, and t the thckne of the plate on the th tory. After examnng reult of everal dfferent puhover analye for uch multtory SPSW, t ha been oberved that the actual falure mechanm typcally omewhere between a oft-tory mechanm and unform yeldng of the plate on all tore. Fndng the actual falure mechanm dffcult f hand calculaton are performed. Therefore, a computerzed puhover analy hould be ued. However, the mechanm decrbed above wll provde a rough etmate of the ultmate capacty. They wll alo provde ome nght a to whether a oft tory lkely to develop (by comparng the ultmate capacty found from the oft tory mechanm wth that of the unform yeldng mechanm) Column Platc Hnge (If Applcable) θ - Soft Story Beam Platc Hnge (If Applcable) Strp Yeldng + Column Platc Hnge θ + Column Platc Hnge (If Applcable) Fgure 3. Soft-tory mechanm Fgure 4. Unform yeldng mechanm 38

5 Propoed Degn Procedure Ung the reult of the platc analye decrbed prevouly, the nfll plate of SPSW can be zed to contently acheve the dered ultmate trength. The procedure mple, even for a multtory SPSW, and neglectng the contrbuton of platc hnge n beam and column wll alway gve a conervatve degn n the cae of rgd beam-to-column connecton. The propoed procedure requre the degner to:. Calculate the degn bae hear, and dtrbute t along the heght of the buldng a decrbed by the applcable buldng code;. Ue the followng equaton to calculate the mnmum plate thcknee requred for each tory: t = F y L Ω (6) n ( α) where Ω the ytem overtrength factor decrbed n FEMA 369 (BSSC 00) and the degn tory hear found ung the equvalent lateral force method; 3. Develop the trp model for computer (elatc) analy ung the equaton from Tmler et al., (983) to calculate the angle of nclnaton of the trp; 4. Degn beam and column accordng to capacty degn prncple (to nure the utmot ductlty) or other ratonal method ung plate thcknee pecfed (n cae thee thcknee exceed the mnmum requred for practcal reaon); 5. Check tory drft agant allowable value from the applcable buldng code; Concluon Platc collape mechanm for ngle and multtory SPSW wth mple and rgd beam-to-column connecton have been nvetgated and mple equaton that capture the ultmate trength of SPSW have been developed. Ung the reult of thee platc analye, a new procedure for the zng of the nfll plate ha been propoed. The propoed procedure allow the engneer to control the ultmate falure mechanm of the SPSW, and drectly account for tructural overtrength. Acknowledgement Th reearch wa carred out under the upervon of Dr. Mchel Bruneau, and upported n full by the Multdcplnary Center for Earthquake Engneerng Reearch. Reference Baler K (96): Strength of plate grder n hear. ASCE Journal of the Structural Dvon, 87 (7), CSA (00): Lmt tate degn of teel tructure. Standard CAN/CSA S6-00, Canadan Standard Aocaton, Wllowdale, Ontaro, Canada. Thrut Area : Semc Retroft of Acute Care Faclte 39

6 BSSC (00): NEHRP Recommended provon for emc regulaton for new buldng and other tructure: Part--Commentary. Report FEMA 369, prepared by the Buldng Semc Safety Councl for the Federal Emergency Management Agency, Wahngton, DC. Thorburn LJ, Kulak GL, Montgomery CJ (983): Analy of teel plate hear wall. Structural Engneerng Report No. 07, Department of Cvl Engneerng, Unverty of Alberta, Edmonton, Alberta, Canada. Tmler PA, Kulak GL (983): Expermental tudy of teel plate hear wall. Structural Engneerng Report No. 4, Department of Cvl Engneerng, Unverty of Alberta, Edmonton, Alberta, Canada. Tmler PA (998): Degn procedure development, analytcal verfcaton and cot evaluaton of teel plate hear wall tructure. Earthquake Engneerng Reearch Faclty Techncal Report No. 98-0, Department of Cvl Engneerng, Unverty of Brth Columba, ancouver, Brth Columba, Canada. 40

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