A Mechancs-Based Approach for Determnng Deflectons of Stacked Mult-Storey Wood-Based Shear Walls FPINNOVATIONS
Acknowledgements Ths publcaton was developed by FPInnovatons and the Canadan Wood Councl based on desgn and constructon practce and relevant research. Ths publcaton would not have been possble wthout the fnancal support of Forestry Innovaton Investment of the Provnce of Brtsh Columba. Authors: Grant Newfeld, B.Sc., M.Eng., P.Eng., StructEng., Read Jones Chrstoffersen Consultng Engneers Chun N, Ph.D., P.Eng., FPInnovatons Jasmne Wang, Ph.D., P.Eng., Canadan Wood Councl Revewers: Thomas Leung, Thomas Leung Engneerng Dsclamer The nformaton contaned n ths publcaton represents the latest research and techncal nformaton made avalable from many sources. It s the responsblty of all persons undertakng the desgn and constructon of the buldngs to fully comply wth the requrements of the Natonal Buldng Code of Canada and CSA standards. The authors, contrbutors, funders and publshers assume no lablty for any drect or ndrect damage, nury, loss or expense that may be ncurred or suffered as a result of the use of or relance on the contents of ths publcaton. The vews expressed heren do not necessarly represent those of ndvdual contrbutors, FPInnovatons, or the Canadan Wood Councl. Copyrght No porton of ths publcaton may be reproduced or transmtted n any form, or by any means mechancal, electronc, photocopyng, recordng or otherwse wthout the pror wrtten permsson of FPInnovatons and the Canadan Wood Councl.
INTRODUCTION The 009 edton of CSA Standard O86, Engneerng Desgn n Wood (CSA 009), provdes an equaton for determnng the deflecton of shear walls. It s mportant to note that ths equaton only works for a sngle-storey shear wall wth load appled at the top of the wall. Whle the equaton captures the shear and flexural deformatons of the shear wall, t does not account for moment at the top of the wall and the cumulatve effect due to rotaton at the bottom of the wall, whch would be expected n a mult-storey structure. In ths fact sheet, a mechancs-based method for calculatng deflecton of a mult-storey wood-based shear wall s presented. Deflectons of Sngle-Storey Wood-Based Shear Walls In CSA 086-09, the deflecton of a sngle-storey shear wall can be determned as follows: vh vh = + + 0. 005Hen + EAL B v H L d a [] In ths equaton, the frst term s derved from the flexural deflecton of a vertcal cantlever beam wth pont load appled at the top of the beam (Fgure ): VH vlh vh f = = = [] EI AL EAL E where V = H = E = pont load at the top of shear wall heght of the shear wall elastc modulus of boundary member I = moment of nerta A = cross-sectonal area of the boundary member L = length of shear wall
V H V M = V H Fgure. Shear and moment dagram of a sngle-storey shear wall. Deflectons of Stacked Mult-Storey Wood-Based Shear Walls For a stacked mult-storey shear wall, the effect of moment at the top of the wall and the cumulatve effect due to rotaton at the bottom of the wall need to be taken nto consderaton. The deflecton comprses fve parts: the deflecton due to bendng, panel shear, nal slp, wall anchorage system elongaton, and rotaton at the bottom of the shear wall. The nter-storey deflecton of the -th storey,, for a stacked mult-storey shear wall can be derved as follows: = b, + s, + n, + a, + r, [] where b, = nter-storey deflecton due to bendng of the -th storey s, = nter-storey deflecton due to panel shear of the -th storey n, = nter-storey deflecton due to nal slp of the -th storey a, = nter-storey deflecton due to vertcal elongaton of the wall anchorage system of the -th storey r, = nter-storey deflecton due to rotaton at the bottom of the shear wall of the -th storey. Deflecton due to bendng, b, Fgure shows the shear force and moment dstrbuton n a mult-storey shear wall. Unlke snglestorey shear walls where moment s zero at the top of the wall, there are also moments at the top of the walls n the mult-storey shear wall transferred from storeys above. Usng the engneerng mechancs, the flexural deflecton for a shear wall at the -th storey can be expressed as follows: V H M H b, = + [4] ( EI ) ( EI ) 4
where V = shear force of -th storey n = F = M = overturnng moment at top of -th storey n = V H = + H = heght of the shear wall of -th storey (EI) = bendng stffness of the shear wall of -th storey n = total number of storeys of the buldng Fgure. Shear and moment dagram of a stacked mult-storey shear wall. 5
AL In Equaton 4, the moment of nerta, I =, apples to a shear wall where a dscrete hold-down anchor system (Fgure ) s used and wood end studs at both ends of the wall are reled on to resst the overturnng moment. Fgure. Dscrete hold-downs n a shear wall. Where a contnuous rod n leu of dscrete hold-down s used n a shear wall, the tenson and compresson forces due to overturnng moment wll be ressted by the contnuous steel rod and wood end studs (see Fgure 4), respectvely. As a result, the transformed bendng stffness of the shear wall, EI tr, should be used n Equaton 4. It can be obtaned as follows: E = E c [5] E E t n = [6] c At, tr = At n [7] y tr = Ac A t, tr Lc + A c [8] I tr ( L y ) t, tr ytr + Ac c tr = A [9] Where I tr s the transformed moment of nerta of the shearwall and the tenson chord area s transformed by n = E t / E. c 6
Ls Tenson Chord Et, At Compresson Chord Ec, Ac ytr Lc Fgure 4. Shear wall secton.. Deflecton due to panel shear, s, The deflecton due to panel shear can be obtaned as follows: V H s, = [0] L Bv, where L = length of the shear wall of -th storey B v, = shear-through-thckness rgdty of the sheathng of -th storey. Deflecton due to nal slp, n, The deflecton due to nal slp can be obtaned as follows:, = 0. 005H e, [] n n where e n,- = nal deformaton of -th storey 4. Deflecton due to vertcal elongaton of the wall anchorage system, a, The deflecton due to vertcal elongaton of the wall anchorage system can be obtaned as follows: H d a, = a, [] L where d a, = total vertcal elongaton of the wall anchorage system of -th storey 7
5. Deflecton due to rotaton at the bottom of the wall, r, For a shear wall at -th storey of a mult-storey, both the cumulatve rotaton due to bendng, = the cumulatve rotaton due to wall anchorage system elongaton, = α, from the storeys below θ, and contrbute to the rotaton of the shear wall at -th storey, whch n turn affects the nter-storey deflecton of the wall at -th storey. Fgure 5 shows the effect of the cumulatve rotaton due to bendng and the effect of the cumulatve rotaton due to wall anchorage system elongaton. Assumng rotatons between the adacent storeys are contnuous, the nter-storey deflecton due to the rotaton at the bottom of -th storey can be wrtten as: r, = H = M H V H d + = + a, θ H α H + H [] = = ( EI) ( EI) = L + + + + Level Level Level (-) Level (-) a) cumulatve rotaton due to bendng b) cumulatve rotaton due to wall anchorage system elongaton Fgure 5. Deflecton due to rotaton at the bottom of the wall. In platform wood-frame constructon, the stacked shear walls are separated by daphragm at each storey. Ths dscontnuty has a drect mpact on the rotaton of a stacked mult shear wall at the bottom of -th storey. If the daphragm s flexble n the out-of-plane drecton, the daphragm wll lkely rotate the same amount as the shear wall below. In ths case, t can be assumed that the shear wall at the bottom of -th storey wll have the same rotaton as that at the top of (-)-th storey, and the amount of rotaton from the storey below can be determned by Equaton. On the other hand, f the daphragm s rgd n the out-of-plane drecton, t s unlkely that the daphragm wll rotate the same amount as the shear wall below. In ths case, the rotaton of shear wall due to storeys below can be gnored. In 8
desgn, t s the udgment of desgn engneers to decde how much the rotaton due to lower storeys should be taken nto consderaton. By takng nto consderaton the rotaton of the storeys below, ths method should provde a conservatve approach for determnng deflectons and nter-storey drfts. On the other hand, ths approach may underestmate the shear wall stffness and overestmate the buldng perod, and therefore potentally underestmate the base shear the buldng may experence durng an earthquake f one s to use a perod determned by a mechancs-based method other than the Code-based perod. However, ths may be addressed by the perod cut-off mposed by the Code for base shear calculaton. A comparatve analyss conducted by the Canadan Wood Councl looked at dfferent methods for deflecton calculaton, ncludng the mechancs-based approach whch takes nto account the rotaton from the storeys below and the approach whch excludes the rotaton. Ths study showed that the perods calculated usng both methods were sgnfcantly hgher than Code-based perod for the desgn example, and suggested that usng the Code perod s a reasonable assumpton for the ntal desgn. However, t wll be mportant that desgners confrm that ths s a reasonable assumpton by calculatng the perod. It s recommended n ths study that the mechancs-based approach (takng nto account the rotatonal effects) be used for determnng drfts, stffnesses, and buldng perods for wood-based shear walls. Ths should lead to a reasonable dstrbuton of forces for a sesmc force resstng system of wood-based shear walls and should provde a conservatve approach for determnng deflectons and drfts. References Canadan Standards Assocaton (CSA). 009. CAN/CSA O86-09 Engneerng desgn n wood. Rexdale, Ont. 9
A Mechancs-Based Approach for Determnng Deflectons of Stacked Mult-Storey Wood-Based Shear Walls 0