DISPLACEMENT-BASED DESIGN OF CONCRETE TILT-UP FRAMES ACCOUNTING FOR FLEXIBLE DIAPHRAGMS

13 th Wold Confeence on Eathquake Engineeing Vancouve, B.C., Canada August 1-6, 24 Pape No. 154 DISPLACEMENT-BASED DESIGN OF CONCRETE TILT-UP FRAMES ACCOUNTING FOR FLEXIBLE DIAPHRAGMS Pey ADEBAR 1, Zhao GUAN 2, and Kenneth ELWOOD 3 SUMMARY This pape pesents a simplified pocedue to estimate the amplification of inelastic difts in concete tiltup fames due to steel deck oof diaphagms designed to emain elastic duing the design eathquake. The pocedue assumes that oof diaphagm displacements elative to the gound ae independent of all stength, hile oof diaphagm displacements elative to the alls ae popotional to all stength. Results obtained using this simplified pocedue ae in good ageement ith esults obtained fom nonlinea dynamic analysis. A ational basis to decide hen concete tilt-up fames must meet seismic design equiements fo cast-in-place fames is also pesented. INTRODUCTION The technique of casting concete panels flat and then lifting (tilting) them upight to fom alls oiginated in Califonia about 5 yeas ago as a method to constuct solid einfoced concete alls fo industial buildings. Today, tilt-up concete alls ae commonly used thoughout the US and Canada to constuct aehouses, shopping centes, office buildings, schools and many othe types of buildings. The alls that ae used in moden tilt-up buildings often have vey lage openings fo indos and doos, esulting in all panels that ae actually multi-stoy fames. Solid tilt-up all panels o all panels ith small openings ae inheently stiff and stong. The seismic design of these elements involves connecting enough all panels togethe to pevent excessive ocking, and poviding adequate connection beteen the all panels and the oof diaphagm. Expeience fom ecent Califonia eathquakes has shon that the eak link is often the out-of-plane connection beteen the alls and the oof diaphagm, paticulaly timbe oofs (Hambuge and McComick [5]). It is common fo moden tilt-up buildings to have lage openings along an entie side of the building, o on moe than one side. This occus, fo example, hen thee ae offices along the font of aehouses, and ith stoe fonts in shopping centes. The concete tilt-up fames aound these openings must esist inplane seismic foces in the same ay as cast-in-place einfoced concete fames; hoeve, thee ae to significant diffeences beteen typical tilt-up fames and typical cast-in-place fames. 1 Pofesso, Dept. Civ. Eng., Univ. of Bitish Columbia, Vancouve, Canada. adeba@civil.ubc.ca 2 Gad. Res. Asst., Dept. Civ. Eng., Univ. of Bitish Columbia, Vancouve, Canada. zhaoguan@telus.net 3 Asst. Pof., Dept. Civ. Eng., Univ. of Bitish Columbia, Vancouve, Canada. elood@civil.ubc.ca

The fist diffeence is that the seismic design/detailing pocedues used fo tilt-up fames is much less stingent than pocedues used fo cast-in-place fames. Table 1 compaes the Canadian code equiements fo modeately ductile cast-in-place fames and common pactice in esten Canada fo tilt-up fames (Weile [9]). As a esult of these diffeences, concete tilt-up fames ill have much less inelastic dift capacity than cast-in-place fames. Table 1: Compaison of Canadian code equiements fo modeately ductile cast-in-place fames and common pactice in Canada fo tilt-up fames. Issue Canadian concete code equiements fo modeately ductile fames Common pactice in Canada fo tilt-up fames Inelastic mechanism Plastic hinging limited to beams; columns must be about 2% stonge than beams. Plastic hinging not esticted; nomally in columns. Columns Anti-buckling ties (hoops) spaced at 6d b at column ends. No max. tie spacing; min. spacing equal to panel thick. egadless of d b Beams Hoops spaced at 8d b each end ove length equal to tice beam depth. Hoops spaced at d/2 (plastic hinging not expected). Beam Column Joints Joint shea stess limited to.85 f c (MPa); hoizontal joint einf. designed fo shea. No joint shea stess limit; column ties povided. The second significant diffeence beteen tilt-up and cast-in-place fames is that cast-in-place building systems include diaphagms that ae essentially igid, hile tilt-up building systems often have flexible diaphagms. In esten Canada, steel deck diaphagms ae nomally used, and these diaphagms ae constucted in such a ay that the diaphagms ill likely emain elastic duing the design eathquake. As a esult of the diaphagm emaining elastic hen the concete alls yield, the tilt-up fames ill be subjected to much lage dift demands than a linea analysis ould suggest. In buildings ith vey lage diaphagms, the inelastic displacements may be as much as 1 times the displacement detemined by a linea analysis, and this incease must be accounted fo. The cuent pape pesents a ational basis to decide hen concete tilt-up fames must meet the seismic design equiements fo cast-in-place fames athe than common pactice fo tilt-up alls, and pesents a simplified method to estimate the inceased inelastic dift demands on concete tilt-up fames due to flexible steel deck diaphagms. INELASTIC DRIFT CAPACITY OF TILT-UP FRAMES Tests on beam-column assemblies of typical concete tilt-up fames have ecently been conducted by De et al. [2]. Five tests ee completed on thee types of specimens ith vaying column tie spacing (1, 2, 3 mm). In all tests, the inelastic mechanism involved a plastic hinge in the column. A evie of the oiginal expeimental data indicates the capacities summaized in Table 2. Table 2: Measued capacities of tilt-up columns by De et al., [2]. Column tie spacing (mm) 1 2 3 Total Disp. Capacity (mm) 13 14 75 75 93 Inelastic Disp. Capacity (mm) 19 119 54 57 75 Inelastic Rotation Capacity (ad).4.44.2.21.28 detemined using an assumed hinge length of h/2 = 2 mm.

The columns had 2M (2 mm dia.) vetical einfocing bas at the compession face that buckled at failue. Thus the tie spacing coesponds to 5d b, 1d b and 15d b. The taditional tie spacing to pevent buckling is beteen 6d b = 12 mm and 8d b = 16 mm, thus the 1 mm spacing is clealy ithin this limit, and the 2 and 3 mm spacing clealy exceed the limit. This explains hy the otational capacities of the specimens ith 2 and 3 mm tie spacing ae consideably less. The expeimental esults can be summaized vey simply as the inelastic otational capacity is.4 adians hen the tie spacing is ithin the anti-buckling limit, and the inelastic otational capacity is.2 adians hen the tie spacing exceeds the anti-buckling limit (i.e., do not confom to the equiements fo cast-inplace fames). No axial compession as applied in the tests because of limitations in the test set-up; hoeve, typical columns in tilt-up fames ae not subjected to high levels of axial compession. The addition of significant axial compession ould educe otational capacities. It is inteesting to compae the esults fom the cuent tests ith the acceptance citeia given in FEMA 356 fo einfoced concete columns. The inelastic otation limit fo collapse pevention pefomance level vaies beteen.2 and.1 adians fo columns ith confoming ties, and beteen.1 and.5 adians fo columns ith nonconfoming ties. The limits ithin FEMA 356 ae intended to be consevative fo columns ith significant axial compession. The inelastic dift capacity of the tilt-up fame (measued at the oof level) can be detemined fom the inelastic otation capacity of individual panel components (columns and beams) using a push-ove analysis of the paticula all panel. Often, the inelastic mechanism is hinging in the columns at the top of the loe level (the base is assumed to be pinned), in hich case the inelastic dift capacity of the entie fame is simply the inelastic otation capacity of the column (.2 o.4 ad) times the effective height of the column (distance fom the base to the cente of the hinge). 24 Canadian Concete Code The 24 Canadian Concete Code (Clause 21.7.1.2) equies that tilt-up all panels ith openings be designed to the equiements fo Modeately Ductile Fames (ductility foce eduction facto R d = 2.5) hen the maximum inelastic otational demand on any pat of the panel exceeds.2 adians, and in no case shall the inelastic otational demand exceed.4 adians. As these ae the fist povisions to be incopoated in the code fo tilt-up fames, the inelastic otational limits ee puposely made less estictive. It is expected that these limits ill be educed somehat in futue editions of the code. STEEL DECK DIAPHRAGMS The oof diaphagms that ee assumed in the cuent study coespond to typical esten Canadian pactice; Weile [9]. The depth of the steel deck as 1½ in. (38 mm), and the joists ee spaced at 6 ft (1.83 m). 22 gauge (.76 mm), 2 gauge (.91 mm) o 18 gauge (1.22 mm) ee used depending on the size of the oof. To diffeent gauges ee used fo each oof size, ith 2% of the oof aea nea the ends of the diaphagm (high shea zone) having the thicke gauge. The chod angles along the alls also depended on the oof size, and ee eithe 3 x 3 x 1/4 in., 4 x 4 x 3/8 in. o 5 x 5 x ½ in. The side laps in the deck units ee connected by button punching, and the deck as ac-spot elded to the fame. The spacing of the button punching and elding vaied ith the deck gauge. The mass on all oofs as assumed to consist of 2 psf (.94 kpa) dead load, and 1 psf (.47 kpa) live load (25% sno load). In Canada, the in-plane stiffness of steel deck diaphagms is usually detemined accoding to CSSBI [1] hich is based on the pocedues in the 1982 Seismic Design fo Buildings (Ti-Sevices Technical Manual). The method assumes that a steel deck diaphagm is analogous to a plate gide having its eb in a hoizontal plane to esist the lateal foces applied to the building. The chod angles povided at the

edge of the deck ae the flanges of the gide. The deflection of the steel deck diaphagm is assumed to be made up of a flexual component f, and a eb (shea) component s. The flexual component is detemined using conventional beam deflection fomulae ith the flexual igidity EI calculated fom the coss-sectional aeas of the chod angles. The eb deflection depends on the shea defomation of the steel deck, the flexibility of the deck-fame connections (ac-spot elds), and the amount of slip of the side lap connections beteen deck units. Table 3 summaizes the diaphagm deflections due to a lateal load equal to the dead load plus 25% sno load on the oof fo some example oof sizes. In all cases shon in Table 3, the deck as assumed to be 2/22 gauge, and the chod angle had a coss-sectional aea of 2.9 in. 2 Table 3: Flexual and shea deflections in example oof diaphagms. Roof Size f Building s (W x L) (ft) Peiod 1 (s) (in.) (in.) 1 x 1.23.85.42 1 x 2 3.6 3.41.85 3 x 3 6.7 7.67 1.16 2 x 3 9.11 7.67 1.26 1 x 3 18.22 7.67 1.52 1 Calculated using Eq. [9]. A seies of tests ee conducted by Temblay et al. [7] to investigate the ductility of steel deck diaphagms. They found that hen side lap connections beteen deck units consist of button punches and the deck is ac-spot elded to the fame, the diaphagm fails in a vey bittle manne. The esidual stength educes to zeo ith minimal enegy dissipation. They ecommend that such diaphagms be designed to emain elastic. The Canadian steel design code CSA S16.1-21 equies a capacity design check to veify that the shea stength of steel deck diaphagms is geate than the actual lateal capacity of the vetical bacing system to ensue that inelastic action is avoided in the steel deck diaphagm. DYNAMIC ANALYSIS OF TILT-UP BUILDINGS A study as undetaken to investigate the behavio of some typical esten Canadian tilt-up buildings in ode to bette undestand these systems, and to develop a simplified pocedue to estimate the inelastic demands on tilt-up fames. The buildings ee assumed to be single-stoey ith steel deck diaphagms of vaying sizes. Thoughout this pape, oof length denotes span length of diaphagm o length of out-ofplane alls; and oof idth denotes depth of diaphagm (plate gide) o length of in-plane alls. The simplified models used fo each component of the tilt-up buildings ae descibed belo. In-plane Walls With Openings A typical all panel ith openings is shon in Fig. 1 (Weile [9]). Sectional analysis of the beams and columns of this fame indicated that an effective flexual igidity E c I e equal to.25 E c I g is appopiate to account fo the eduction in stiffness due to flexual cacking. A pushove analysis of the fame indicated that the membe shea foces ae small enough to ignoe the effect of shea cacking (Guan [4]). A linea analysis of the fame, accounting fo the joint size, indicates that the in-plane foce at the level of the oof to cause a unit in-plane deflection at the fame is 8.46 kn/mm. Thus a single sping ith a stiffness of 8.46 kn/mm can be used to epesent the elastic esponse of the to-stoy fame in a simple model. To othe simila size all panels ith diffeent opening sizes ee also included in the cuent study and the coesponding stiffness of these is 1.27 and 13.47 kn/mm. The total mass of the all panel shon in Fig. 1 is 16 tonnes, and the masses of the othe to panels ee 1 and 26 tonnes. In the single

75 mm 15 19 sping model of the tilt-up panel, the distibuted mass of the all is epesented by a lumped mass at the end of the sping equal to 5% of panel mass. P 2 kn/m 15 A 3 A B 9 72 9 2-2M 1M @ 4 2-15M 15 mm 2-2M 2-15M 2-15M 2-15M 2-2M 2-15M 2-15M 2-2M 9 mm 1M TIES @ 4mm A - A B - B Fig. 1: Example concete tilt-up fame used in the cuent study. Note that 1M column ties ae spaced at 19 mm in plastic hinge egion (fom Weile [9]). Out-of-Plane Walls In a detailed model of tilt-up buildings, the out-of-plane alls could be modeled as vetical beams pinconnected to the floo slab and the oof diaphagm. Such a model ould be necessay, fo example, to study the out-of-plane connection foce beteen the alls and oof diaphagm. As the natual fequency of the oof diaphagm spanning beteen in-plane alls is much loe than the natual fequency of the outof-plane bending of the all panels, the influence of the out-of-plane alls can be accounted fo simply by adding half the mass of the out-of-plane all to the oof diaphagm. In the cuent study, a mass of 89 kg/m as added fo each out-of-plane all. Steel Deck Diaphagm The level of complexity needed to model a steel deck diaphagm depends on the pupose of the analysis. Temblay et al. [7] used a lage numbe of tuss elements ith a stiffness degading hysteesis esponse based on thei shea element test esults in ode to simulate the nonlinea shea behavio of steel deck diaphagms. As the focus of the cuent study as to detemine the influence of flexible oof diaphagms on the inelastic all demand, a simple model could be used. The oof diaphagm as modeled using the Ti-Sevices plate gide analogy. This appoach as also used by Temblay [8] to investigate the influence of flexible diaphagms on vetical bacing systems in one-stoy steel buildings. A model ith to spings epesenting the tilt-up alls on eithe end of the diaphagm, and a beam (plate gide) epesenting the diaphagm is shon in Fig. 2(a).

k M/L k M/2 E I, GA, L M/2 (a) 2kW M 2kWk / ( 2k W+k ) M M + M (b) (c) Fig. 2: Simple models fo dynamic analysis of tilt-up buildings: (a) plate gide model of diaphagm ith all spings, (b) to degee of feedom model ith fist mode of diaphagm epesented as a sping, (c) equivalent single degee of feedom model. Figue 3 shos the modal displacements fo the 25 NBCC spectum (Vancouve Site Class E) of a tiltup building ith a oof that is 2 ft ide by 3 ft long. Fo this symmetic stuctue, only the symmetic (odd numbeed) modes ae pesent, and the fist mode clealy dominates the esponse of the system. Linea and nonlinea time histoy analyses indicate that at the point of maximum displacement of the alls, the fist mode clealy dominates. Thus cetain aspects of the inelastic demands on the alls can be studied ith an even futhe simplified model of the diaphagm. The fist mode esponse of the diaphagm can be epesented by a single sping as shon in Fig. 2(b). If the oof deflections ae due to flexue, the mass paticipation in the fist mode ill be 81% of the total unifomly distibuted mass. On the othe hand, if shea defomations ae significant, a lage potion of the unifomly distibuted mass ill paticipate in the fist mode. Thus 1% of the oof mass M as used in the simple to-sping model..35.3 Modal Displacement (m).25.2.15.1.5 3d Mode 1st Mode 5th Mode -.5 Fig. 3: Example modal displacements in a tilt-up building ith flexible diaphagm. Results fom the plate gide model of the diaphagm indicate that the lage inelastic demands on the tiltup alls occu hen the oof span is lage and the flexual defomations of the oof ae dominate. Thus it is easonable to develop a simplified model of the diaphagm by fist conveting the total fist mode defomations of the diaphagm into an equivalent flexual mode. The stiffness of the sping is detemined

so that the peiod of the sping, ith a lumped mass equal to 1% of the oof mass, is equal to the fist mode peiod of a simply suppoted beam ith unifomly distibuted mass, and ith an equivalent flexual stiffness that esults in the coect total (flexue plus shea) mid-span displacement: 4 π EI k = 3 L f f + s [1] hee EI is the flexual igidity of the oof calculated fom the chod angles, L is the span of the oof, and the typical atios of flexual displacement to total displacement of the oof f / ( f + s ) is shon in Fig. 4 fo diffeent oof spans and oof span-to-idth atios. f / ( f + S ) 1..8.6.4.2 L=1ft L=2ft L=3ft L=5ft L=4ft. 1 2 3 4 Roof Span-to-Width Ratio Fig. 4: Flexual potion of oof diaphagm deflections fo diffeent size steel deck diaphagms. Nonlinea Dynamic Analysis The plate gide plus all sping model shon in Fig. 2(a) and the to-sping model shon in Fig. 2(b) ee used to conduct nonlinea dynamic analysis. In the fist phase of the study (Guan [4]), five eathquake ecods ee used. To of the ecods, LA24 (1989 Loma Pieta) and SE32 (1985 Valpaaiso Seattle aea), ee fom Phase 2 of the FEMA/SAC Steel Poject, and ee scaled to a 2% in 5-yea pobability of occuence. The emaining thee eathquakes VAN1, VAN2 and VAN3 ee scaled to the 25 NBCC design spectum fo Vancouve Site Class C (vey dense soil). In the second phase, fou additional eathquake ecods, modified to fit the 25 National Building Code of Canada design spectum fo Vancouve Site Class E (soft soil), ee used. This includes to Loma Pieta ecods (LPEW and LPNS) and to San Fenando aea ecods (SFCT316 and SFCT317). The acceleation spectum and displacement spectum fo one of these ecods is shon in Fig. 5. Compute pogam CANNY-99 (Li [6]) as used fo all nonlinea analyses. 5% Rayleigh damping as used fo the fist to modes. The hysteesis model that simulated the pinched nonlinea behavio of the steel deck diaphagm as used in the to-sping model. A compaison of the theoetical hysteesis cuves and the test esults of Essa et al. [3] shoed good ageement (Guan [4]). The nonlinea foce-displacement elationships of the concete tilt-up fames ee simulated using a hysteesis model that assumes elasticpefectly plastic behavio ith the eloading slope educed depending on the maximum displacement in the pevious cycles.

Acceleation (cm/s/s) 1 8 6 4 2 2 4 6 (a) Fundamental Peiod (s) Displacement (cm) 14 12 1 8 6 4 2 2 4 6 (b) Fundamental Peiod (s) Fig. 5: Acceleation and displacement specta fo eathquake ecod SFCT316 fit to 25 NBCC Vancouve Site Class E design spectum (shon as bold line). ANALYSIS RESULTS: EQUAL ROOF DISPLACEMENT Figue 6 shos a typical esult obtained ith the plate gide model of the oof diaphagm. The deflected shape of the linea stuctue at the instance of maximum demand is shon. That is, the maximum oof deflection/foce and maximum all deflection/foce occued simultaneously. The deflected shape of the same stuctue hen the stength of the alls as limited to one-half the maximum elastic foce demand is also shon in Fig. 6. It can be seen that the aveage deflection of the distibuted mass on the oof diaphagm is appoximately equal in the to cases..3 Elastic Wall Displacement (m).2.1 Inelastic Wall Fig. 6: Typical maximum displacements in a tilt-up building ith elastic alls (linea analysis) and inelastic alls (nonlinea analysis). To futhe investigate this phenomenon, non-linea analyses ee done using the to-sping model shon in Fig. 2(b) ith the sping stiffnesses and the masses based on actual building popeties. In these analyses, the atio of oof mass to all mass (M /M ) vaied fom 1 to 2, and the atio of oof-sping stiffness to all-sping stiffness (k /2k ) vaied fom 1 to 4. Fo each building, a linea time-step analysis and nonlinea time-step analysis as done. In the latte, the stength of the alls as set at half the maximum foce duing the coesponding linea analysis. That is, the nonlinea analysis as done fo buildings ith alls designed using a foce eduction facto of 2.. The maximum displacements of the oof masses duing the nonlinea analyses and the maximum displacements of the oof masses duing the linea analyses ae compaed in Fig. 7(a). In most cases, the oof displacements hen the alls ae inelastic ee ithin 75 to 1% of the oof displacements detemined fom a linea analysis. That is, the oof displacements ee geneally smalle hen the alls ee inelastic.

Roof Displacement (m) (In-elastic Wall) 1.6 1.2.8.4 1..75 Wall Displacement (m) (Inelastic).3.25.2.15.1.5 1..4.8 1.2 1.6 (a) Roof Displacement (m) (Elastic Wall)..5.1.15 (b) Wall Displacement (m) (Elastic) Fig. 7: Compaison of maximum displacements detemined fom nonlinea analysis and linea analysis: (a) oof displacements, (b) all displacements. The maximum total displacements of the inelastic alls ae compaed ith the all displacements detemined fom a linea analysis in Fig. 7(b). In this case, the displacements of the inelastic alls ae alays lage than the linea analysis indicates. Depending on the flexibility of the diaphagm, the maximum displacements of the inelastic alls may be much lage than the elastic analysis suggests. While in eality the incease is a fixed amount elated to the displacements of the diaphagm, if the incease is expessed as a atio, the numbe ould be vey lage fo cases that the linea analysis indicates small all displacements. This is consistent ith the vey lage ductility atios obseved by Temblay [8] in the analysis of vetical bacing in single-stoey steel stuctues ith flexible oof diaphagms. SIMPLIFIED PROCEDURE TO ESTIMATE INELASTIC WALL DRIFT The displacement of the alls can be calculated fom the folloing expession: ( ) = [2] hee is the deflection of the oof elative to the top of the all. The objective is to estimate the maximum hen the alls ae inelastic. As discussed above, the maximum oof displacement detemined fom a linea analysis of the stuctue, denoted as δ, can be used as a safe (lage) estimate of at the instance of maximum inelastic all displacement. Since the oof is assumed to emain linea, the displacement of the oof elative to the all can be detemined if the foce in the oof is knon. ( ) is the deflection of the oof elative to the gound, and ( ) The oof foce at the instance of maximum inelastic all displacement ill be less than the maximum oof foce hen the alls ae linea-elastic. Yielding of the alls ill isolate the oof fom the gound excitation and pevent significant foce incease. A simple appoach is to assume that the eduction in oof foce due to yielding of the alls is popotional to the eduction in all foce. This can be expessed as: V V i y = α [3] Ve Ve hee V i is the foce in the oof at the instance of maximum inelastic displacement of the alls, V e is the maximum foce in the oof based on a linea analysis, V y is the yield stength of the alls, V e is the

maximum foce in the alls based on a linea analysis, and α is a constant. Thus the actual displacement of the oof elative to the all at the instance of maximum all displacement is given by: V Vy α [4] V V i ( ) = ( δ δ ) = ( δ δ ) hee δ and δ ae the oof displacement and all displacement detemined fom a linea analysis. Results fom numeous nonlinea analysis using the to-sping model summaized in Fig. 8 fo diffeent all types indicate that α vaies fom about.5 to 1.5, and that the aveage value of alpha is about.8. A value of α = 1. as used fo simplicity and to compensate fo the diffeence beteen and δ. Thus the maximum (total) displacement of the inelastic alls can be estimated as: e e Vy = δ ( δ δ ) [5] V e 2 1.5 α 1.5 1 2 3 4 T (s) Fig. 8: α factos in Eq. [3] detemined fom nonlinea analysis. The total displacement of the inelastic alls is made up of an elastic potion, equal to the yield displacement of the all, and an inelastic potion: = + [6] y i The yield displacement of the all is given by: Vy y = δ [7] V e Substituting Eqs. [6] and [7] into Eq. [5] and eaanging, gives the folloing simplified expession fo the maximum inelastic all displacement:

i Vy = δ 1 [8] Ve The displacement of the oof diaphagm δ in Eq. [8] can be detemined fom a numbe of diffeent linea analyses, such as the equivalent single degee of feedom model shon in Fig. 2(c), o can be detemined diectly fom the spectal displacement S d if the fundamental peiod of the building is knon. FEMA 356 suggests the folloing expession fo the fundamental peiod of a one-stoy building ith a single span flexible diaphagm: T =.1δ +. 78δ [9] hee δ and δ ae the in-plane all displacement and mid-span oof diaphagm displacement in inches, due to a lateal load equal to the eight of the diaphagm. A compaison of the peiods detemined fom Eq. [9] and the plate gide model shon in Fig. 2(a) indicates vey good ageement. Table 3 pesents some esults obtained using Eq. [9] fo some example buildings ith diffeent oof sizes and all panels as shon in Fig. 1. In pactice, the atio of all stength to maximum elastic demand (V y /V e ) is detemined fom the foce eduction facto R used to design the alls and any ove-stength of the alls, nomally expessed as the atio of nominal all stength to factoed load γ. The minimum value of γ is the invese of the esistance facto, and any factoed esistance geate than the factoed load ill incease this atio futhe. Thus the maximum inelastic all displacement fo a tilt-up building ith a flexible diaphagm can be expessed diectly as: i = S d γ 1 [1] R ( T ) hee T is the fundamental lateal peiod of the building accounting fo the flexible diaphagm, R is the foce eduction facto used to design the alls, and γ is the nominal ove-stength atio. The concept of estimating the inelastic displacement as the total displacement times ( 1 / R) is ell knon. The subtle but vey impotant diffeence in Eq. [1] is that the inelastic displacement is in the alls only, hile the total displacement S d (T) is of the all and flexible diaphagm. Accoding to Eq. [1], as the stength of the alls ae educed ( γ / R is educed), the elastic displacements of both the alls and oof decease, and the inelastic displacement of the alls must incease to make up the entie diffeence. This effect can be descibed as an amplification of the inelastic displacement in the alls due to the elastic diaphagm, as the all stength is educed. Unsymmetical Stuctues Tilt-up buildings often have lage openings on one side (e.g., font) of the building and fee smalle openings along the opposite side (e.g., back) of building. Yielding of the all panels ill be limited to one side of the diaphagm in these unsymmetical stuctues. The influence of alls yielding at only one end of the diaphagm as investigated using the beam (plate gide) model shon in Fig. 2(a). When only one all yields, the inetial foce in the oof shifts aay fom the cente of the oof toads the yielding all. As a esult, the inetial foce in the oof educes fo a given all stength and the displacements of the oof coespondingly educe. This effect can be accounted fo in a simple ational ay by educing the α facto in Eq. [4]. γ

Fo one example stuctue, the esults fom fou diffeent eathquake ecods fit to the same spectum indicated that hen only one all yields, the maximum displacement of the inelastic all vaied fom 7 to 18% (aveage of 14%) of the maximum all displacement hen both alls yielded simultaneously. EXAMPLE Figue 9 summaizes pedictions fom the simplified pocedue fo one-stoy tilt-up buildings ith the all panel fames shon in Fig. 1, and 3 ft long by 2 ft ide steel deck diaphagms ee compaed ith pedictions fom nonlinea dynamic analysis using the beam (plate gide) model shon in Fig. 2(a). Eight buildings that ee identical except fo all stength ee investigated using the 25 NBCC design spectum fo Vancouve Site Class E. To eathquake ecods fit to that spectum ee used to make the nonlinea analysis pedictions..3 Roof Displacement (m).2.1 SFCT31 LPEW.5 1 1.5 2 2.5 (a) Ratio of Elastic Demand to Wall Stength Inelastic Wall Displacement (m).3.2 Eq. [1].1.5 1 1.5 2 2.5 (b) Ratio of Elastic Demand to Wall Stength Inelastic Rotation of Wall (ad).1.8.6.4.2.5 1 1.5 2 2.5 (c) Ratio of Elastic Demand to Wall Stength Fig. 9: Compaison of esults obtained fom poposed simplified method and nonlinea dynamic analysis fo to ecods fit to the same spectum.

The simplified method pedictions ae shon as solid lines in Fig. 9, hile the esults fom the nonlinea dynamic analysis using the plate gide model shon in Fig. 2(a) ae shon as discete points. In the simplified method, the displacement of the oof is assumed to be independent of the all stength and is equal to S d (T), hile the nonlinea analysis indicates that the oof displacement is influenced somehat by the stength of the alls; see Fig. 9(a). In geneal, the simplified method pedicts an uppe-bound to the oof displacements. The pedictions of the inelastic all displacements fom Eq. [1] ae compaed ith the nonlinea dynamic analysis esults in Fig. 9(b). The to agee emakably ell. The inelastic displacements of the alls ee used to detemine the coesponding inelastic otation in the columns, and these ae shon in Fig. 9(c). Fo the fame shon in Fig. 1, the inelastic otation (inelastic dift) is equal to the inelastic displacement divided by 2.55 m, hich is the height fom the column base to the cente of the plastic hinge assumed to be.9/2 =.45 m don fom the undeside of the loe beams. CONCLUDING REMARKS The simplified method pesented in this pape can be used to quickly pedict the inelastic all displacement in a vaiety of buildings ith diffeent size steel deck diaphagms. Figue 1 shos the esults fo thee additional buildings ith 1 ft ide oof diaphagms that have diffeent spans. With the slende oof diaphagm, the inelastic displacements, and hence inelastic otations of the tilt-up alls, incease vey quickly if the alls ae pemitted to yield. Also shon in the figue (as dashed lines) ae the esults if the amplification due to the flexible diaphagm is ignoed. That is, if the total displacement of the all is assumed to be independent of the stength of the alls and equal to the value pedicted by a linea analysis. The incease in inelastic otation shon in Fig. 1(b) as dashed lines fo that case is due only to the eduction in elastic all displacements ith educing stength. Fo each of the oof sizes shon in Fig. 1, and many othe oof sizes, the elastic demand to all stength atio ( R γ ) at hich the inelastic otation eaches the.2 ad and.4 ad limits given in the 24 Canadian code ae summaized in Fig. 11. That is, fo values of R / γ and oof span that plot to the left of the line fo the paticula oof idth (W), the inelastic otations ill be less than.2 adian in Fig. 11(a) and less than.4 adians in Fig. 11(b). The esults in Fig. 11 indicate that in geneal, thee ae thee anges: (1) shot oof diaphagm spans hee the tilt-up fames do not need to be designed fo significant inelastic otations. In this case, common pactice fo tilt-up constuction ould suffice. (2) Long oof diaphagm spans hee the tilt-up fames must be designed to emain elastic, and (3) intemediate oof diaphagm spans hee an adequate combination of stength and ductility must be povided depending on the oof diaphagm dimensions. REFERENCES 1. CSSBI. Design of Steel Deck Diaphagms, Canadian Sheet Steel Building Institute. 1991 2. De, M., Sexsmith, R., and Weile, G., Effect of Hinge Zone Tie Spacing on Ductility of Concete Tilt-up Fame Panels, ACI Stuctual Jounal, V. 98, No. 6, Nov.-Dec. 21. 3. Essa, H.S., Temblay, R., Roges, C.A. Inelastic Seismic Behaviou of Steel Deck Roof Diaphagms Unde Quasi-static Cyclic Loading, Reseach Repot No. EPM/CGS-21-11, Dept. of Civil, Geo., and Mining Eng., Ecole Polytechnique, Monteal, 21. 4. Guan, Z., Displacement-based Design of Concete Tilt-up Walls, M.A.Sc. thesis, Dept. of Civil Eng., Univesity of Bitish Columbia, Mach 24, 161 pp. 5. Hambuge, R.O., and McComick, D.L. Implications of the Januay 17, 1994, Nothidge Eathquake on Tilt-up and Masony Buildings ith Wood Roofs. Stuctual Enginees Association of Nothen Califonia (SEAONC), May 1994, Semina Papes. San Fancisco, CA, pp. 243-255.

6. Li, Kangjing, CANNY 99 Manual, Singapoe, 1996, 24 pp. 7. Temblay, R., Roges, C.A., Essa, H.S., and Matin, E., Dissipating Seismic Input Enegy in Loise Steel Buildings Though Inelastic Defomations in the Metal Roof Deck Diaphagm, Poceedings of CSCE 4 th Stuctual Specialty Confeence, Monteal, June 22. 8. Temblay, R. and Stieme, S.F., Seismic Behavio of Single-Stoey Steel Stuctues ith a Flexible Diaphagm, Can J. of Civ. Eng., Vol. 23, 1996, pp. 49-62. 9. Weile, G., Chapte 13 Tilt-up Wall Panels, Concete Design Handbook, Cement Association of Canada, Ottaa, 1995. 1. Weile, G., Pesonal communication, 23..2 Wall Displacement (m).1 1 x 3 1 x 2 1 x 1 1 x 3 - No amplification..5 1. 1.5 2. 2.5 (a) Ratio of Elastic Demand to Wall Stength Inelastic Wall Rotation (Rad).4 1 x 3 1 x 2.2 1 x 1 1 x 3 - No amplification.5 1. 1.5 2. 2.5 (b) Ratio of Elastic Demand to Wall Stength Fig. 1: Example pedictions fom poposed simplified pocedue fo diffeent oof idths () and spans in ft.

Elastic Demand to Wall Stength 2 1.5 1 Inelastic Rotation less than.2 W=1 W=2 W=3 W=4 W=5 1 2 3 4 5 (a) Roof Span (ft) Elastic Demand to Wall Stength 2 1.5 1 W=1 W=2 W=3 W=4 W=5 Inelastic Rotation less than.4 1 2 3 4 5 (b) Roof Span (ft) Fig. 11: Example esults fom poposed pocedue - elationship beteen elastic demand to all stength atio and oof span fo diffeent oof idths (W) in ft fo: (a) inelastic otation of.2 ad, and (b) inelastic otation of.4 ad.