NON-LINEAR ANALYSIS OF LIGHT-FRAMED WOOD BUILDINGS

Transcription

1 79 NON-LINEAR ANALYSIS OF LIGHT-FRAMED WOOD BUILDINGS Chukwuma G EKWUEME Ad Gary C HART SUMMARY The paper presets a oliear aalysis procedure that is effective for buildigs that utilize sheathed stud walls as the primary elemets i the lateral load resistig system. The load versus deformatio relatioships of the walls determie the earthquake respose of wood buildigs, ad these relatioships are govered primarily by the load versus slip relatioships of the sheathig-toframe coectors. A aalytical procedure that calculates the respose of timber walls is preseted. The techique calculates wall behavior i the oliear rage by equatig the work doe by the exteral lateral loads to the iteral eergy developed durig deformatio of the coectors. The procedure ca be applied to walls with ay spacig of studs, ad ay cofiguratio of iterior ad exterior ailig. Aalytical results are show to compare favorably with experimetal data ad ca provide iformatio o sheathed walls of various dimesios ad cofiguratios whe testig is ot feasible or ecoomical. The paper the presets a methodology for applyig the oliear properties of idividual walls to behavior of a etire wood-framed buildig. The approach, which is similar to that proposed for cocrete buildigs, icorporates the oliear ad dyamic properties of the various buildig compoets ad provides results for differet levels of earthquake loadig, as required by typical performace-based desig criteria. INTRODUCTION Majority of the buildigs i North America ad other regios aroud the world are costructed with timber, yet the procedures that structural egieers use to desig wood buildigs are primitive whe compared with the techiques used to desig cocrete, steel, or masory buildigs. The tremedous amout of damage to wood buildigs durig the Northridge earthquake revealed that the curret methods used to desig ad aalyze lightframed wood structures i regios of high seismic risk are iadequate. I additio, there is a tred i the structural egieerig professio towards performace-based desig, which requires aalyses that are more sophisticated as well as the determiatio of structural respose i the oliear rage. It is thus evidet that there eeds to be a improvemet i the techiques used to aalyze wood buildigs i seismic regios. This paper presets a procedure for the oliear aalysis of wood buildigs that utilize sheathed stud walls as primary lateral load resistig elemets. The relatioship betwee the slip of the sheathig-to-frame coectors ad wall displacemet is developed ad a computer program is created to perform the computatios required to obtai seismic respose of a wall from the load versus slip relatioships of the coectors. Oce the properties of the walls are kow, the performace of a buildig ca be determied usig a variety of aalysis methods. This paper presets a example of buildig respose calculated usig a static oliear aalysis, which is sometimes called a pushover aalysis. With this methodology, the properties of the walls are assembled to create a aalytical model that is subjected to progressively icreasig displacemets. The model is cotiually modified to accout for the softeig of the various compoets ad the resultig redistributio of lateral loads. (Applied Techical Coucil, 996, FEMA, 997) The aalysis provides a evelope to the oliear cyclicrespose of the buildig to earthquakes ad provides a tool for evaluatig buildig performace at various levels of displacemet. Hart Cosultat Group, Sata Moica, Califoria,USA. chukwuma@hartcosultatgroup.com Departmet of Civil ad Evirometal Egieerig, Uiversity of Califoria, Los Ageles, USA

2 NONLINEAR RESPONSE OF LIGHT-FRAMED SHEATHED WALLS Shear walls i light-frame timber costructio typically cosist of a wood frame made up of studs, a top plate, ad a sole plate; ad applied sheathig paels of plywood, gypsum wallboard, orieted strad board, or other such material. The sheathig paels are attached to the frame with coectors such as ails, staples or screws. Holdows are usually placed at the eds of the walls to resist overturig forces, ad achor bolts resist the shear force at the base. Figure shows the layout of a typical wood shear wall. Sheathig Paels Coectors Top Plate Studs Frame Sole Plate Holdow Achor Bolts Figure Compoets of a Typical Shear Wall i Light-Frame Buildig Costructio The lateral displacemet of a sheathed stud wall is due to cotributios from (i) the deformatio of the sheathig material; (ii) the distortio, or slip, of the sheathig-to-frame coectors; (iii) the deformatio ad slip of the holdows at the eds of the wall due to overturig momets; ad (iv) the slidig of the sole plate relative to the foudatio. The cotributio of the last two items to total displacemet is typically small for well-costructed walls with relatively stiff hold-dows. Thus, the first two parts act as sprigs i series so that the total displacemet at the top of the wall, T, is give by: T = S + C () where ad S is the displacemet due to the shear deformatio of the sheathig ad C is the displacemet due to coector slip. The shear deformatio of the sheathig is equal to: S = FH GtL () where H ad L are the height ad width of the sheathig pael, respectively; t is its thickess; ad G is the shear modulus of the sheathig material. Whe a sheathed stud wall is subjected to lateral displacemets, the wood frame deforms ito the shape of a parallelogram sice it possesses little lateral resistace. However, the sheathig paels have a large i-plae stiffess ad attempt to retai their rectagular shape. The coectors that attach the sheathig to the frame are thus subjected to distortio, or slip, due to the differece i deformatio betwee the flexible frame ad the stiff sheathig paels. Figure shows a sigle sheathig pael ad the pheomeo that occurs durig lateral displacemet. Sice each pael i a wall acts idepedetly, the respose of a wall with may paels is the sum of the respose of each idividual pael (icludig the deformatio of its coectors). Laboratory tests have show that the coectors at the corers of the frame slip approximately alog the diagoals of the sheathig pael [Tuomi ad McCutcheo, 978]. Thus, the relatioship betwee the corer coector slip, c, ad the displacemet at the top of the wall, C, is: c = C siα (3) 79

3 where α is the agle betwee the pael diagoal ad the vertical edge as show i Figure. The slip at ay locatio o a pael ca be obtaied from the slip of the corer coectors. The, usig Equatio (3), the slip for ay coector ca be calculated i terms of the displacemet at the top of the wall as follows: = f ( c ) = Q C (4) α H L (a) Udeformed Wall (b) Deformed Wall Figure Deformatio of a Sheathed Stud Wall For coectors alog the top ad bottom edges of the pael Q is give by: Q i = siα si α + cos α x / (5) ad for coectors alog the left ad right sides of the pael Q is give by: Q si j = α si α + cos α y / (6) The variables x ad y are the umber of coector spaces i the horizotal ad vertical directios, respectively. The parameter i is a horizotal idex used to idetify the ith coectio i the horizotal directio coutig from the corer ails; ad j is the vertical idex used to idetify the jth coectio i the vertical directio coutig from the corer ails. The slip of coectors located i the iterior of a pael ca be iterpolated from the coectors alog the edges. Therefore, for a vertical lie of iterior coectors: Q si j = α r si + cos y α r α x y / (7) where r x ad r y are the legth ad height reductio factors that are a ratio of the iterior coector layout to the exterior coector layout. To determie the respose of the wall i the oliear rage, the exteral work doe by the lateral force is equated to the iteral eergy created by the slip of the ails. The exteral work, W e, doe by the exteral lateral force, F, is the area uder the force versus displacemet curve. The iteral eergy or work, W i, is obtaied by 3 79

4 addig the iteral eergy developed by each coector. This is computed by summig the area uder each coector's load versus slip curve due to its slip at the give displacemet. It the follows that the exteral lateral force for a give top displacemet due to slip of the coectors is [Ekwueme ad Hart, 998]: F = d d C m = P d (8) Thus, if the load versus slip relatioship of the coectors is kow, the exteral lateral force at a give wall displacemet ca be calculated, sice the distortio of each coector, ca be computed from Equatios (3) through (7). Equatio (8) idicates that the wall displacemet resultig from the distortio of the coectors depeds o the load versus slip relatioship of the coectors. Sice this relatioship is typically oliear, a iterative solutio approach is required to determie what percetage of the total wall displacemet is caused by the slip of the coectors ad what percetage is due to the deformatio of the sheathig. A computer program, ISTAR-WD [Ekwueme ad Hart, 998], was developed to perform the above computatios to determie the respose of sheathed stud walls to lateral loads. For a specified value of total displacemet, T, the computer program calculates the displacemet due to coector slip, C, ad the exteral lateral force, F, is obtaied by umerically itegratig to satisfy Equatio (8). Usig the computed value of F, the displacemet due to shear deformatio of the pael is calculated from Equatio (). If the resultig total displacemet differs from the target displacemet by more tha a acceptable tolerace (0.% for ISTAR-WD) a ew estimate of C is computed ad the calculatios are repeated util covergece is achieved. VERIFICATION WITH EXPERIMENTAL DATA Results from computer aalyses usig ISTAR-WD were verified by comparisos with data from full-scale tests o light-framed shear walls. The comparisos allowed for a verificatio of the accuracy of the program's computatioal routies, ad provided bechmark values for the iput parameters, such as the variables that defie coector load-slip relatioships. Testig programs coducted by the Egieered Wood Associatio [Rose, 998] ad the City of Los Ageles [CoLA, 999] at the Uiversity of Califoria, Irvie provide abudat test data for verificatio. Tests were performed o 8-ft by 8-ft walls with various sheathig materials ad coector cofiguratios. Figure 3 ad 4 show that aalyses with ISTAR-WD are i excellet agreemet with experimetal data, ad provide accurate evelopes of the cyclic respose of walls sheathed with plywood ad gypsum wall board (GWB). I the experimets, sole plate slidig ad hold-dow slip ad deformatio cotributed about 3-4% of the total lateral displacemet. Therefore, igorig their effect i the aalysis was ot sigificat. A expoetial loadslip relatioship, recommeded by a umber of researchers [Schmidt ad Moody, 989; White ad Dola, 995] was used. The five-parameter mathematical represetatio is show i figure 5. Table provides the parameters used i the load-slip relatioship for the various sheathig materials ad ail sizes. 4 79

5 000 Experimet (COLA, 998) Aalytical (ISTAR-WD) Experimet (APA,998) Aalytical (ISTAR-WD) 6000 Force (lbs) Displacemet (iches) Figure 3 (a) Displacemet (iches) Force versus Displacemet Curves for 3/8" Plywood Walls with (a) 8d ails spaced at 6" ad " (b) 8d ails spaced at 3" ad ". (b) 7500 Experimet (CoLA, 999) 7500 Aalytical (ISTAR-WD) Aalytical (ISTAR-WD) Experimet (CoLA, 999) Force (lbs) 0 Force (lbs) /8" plywood with 6, (COLA Test #C) Displacemet (iches) (a) Displacemet (iches) Figure 4 Force versus Displacemet Curves for (a) /" GWB with -5/8" Drywall 7" ad (b) 5/8" GWB with -7/8" Drywall 4" (b) 5 79

6 Load, P K P K P = ( P + K ) y exp K P y ; Py K3 P = P K ( ) ; > 3 Slip, Figure 5 Expoetial Load versus Slip Relatioship for Sheathig-to-Frame Coectors Table Load versus- Slip Parameters for Sheathig-to-Frame Coectors Coector ad Sheathig Material P y (lbs) P (lbs) K (lbs/i) K (lbs/i) K 3 (lbs/i) 8d commo ails o 3/8 plywood /8" drywall ails o / GWB /8" drywall ails o 5/8 GWB ANALYSIS OF A WOOD-FRAMED APARTMENT BUILDING Figure 6 shows the floor plas of typical wood-framed buildig with eight towhouse apartmets. The properties of the walls are also show. A static oliear aalysis was performed usig the appropriate wall properties. The aalytical model was subjected to progressively icreasig displacemets usig a triagular load distributio. At each level of displacemet, the model was modified to accout for the softeig of the various compoets ad the resultig redistributio of lateral loads. The floor ad roof diaphragm were assumed to be stiff compared to the walls. Thus, forces were distributed to the walls i proportio to their stiffess at the give displacemet. The assumptio of a stiff diaphragm is cofirmed by tests o wood buildigs (Phillips, et. al., 993). I additio, observatios after earthquakes show that damage occurs much more frequetly to walls tha to the diaphragms of regularly shaped wood buildigs. This implies that while the walls are displaced ito the oliear rage, the diaphragms remai essetially elastic ad stiff whe compared to the softeed walls. Like most apartmet ad codomiium-type structures, the buildig is symmetrical i both major directios. Torsioal effects were therefore ot cosidered i the aalyses. For most detached residetial houses, ad other wood structures such as those characterized with tuck-uder parkig, the effects of torsio ca be sigificat. 6 79

7 ½ Plywood with 8d 6 ad 5 8 GW B with 7/8 drywall 4 ½ GWB with -5/8 drywall 4 N First Floor Pla Figure 6 Secod Floor Pla Typical Wood Towhouse Apartmet Buildig Figure 7 shows the results of the oliear static aalysis i the North-South directio. As has bee observed after earthquakes, the aalysis shows that the damage first occurs to the iterior gypsum wallboard i the first ad secod floors. Majority of the damage was icurred i the first floor, ad the imum base shear is attaied at the stregth limit state of the plywood i the first floor After attaiig the imum base shear, the stregth of the buildig degrades rapidly, because of the large quatity of gypsum wallboard that provides lateral resistace. The aalysis also showed that the buildig exhibits a soft-story pheomeo at large displacemets, with a more flexible first floor ad a relatively udamaged upper story. Plywood yields (st Floor) Degradig GWB (st Floor) Plywood begis to degrade (st Floor) / GWB begis to softe (st Floor) Figure 7 Lateral Respose of Example Buildig i North-South Directio 7 79

8 CONCLUSIONS The seismic respose of a sheathed stud walls ca be calculated from the load versus slip relatioships of the coectors that attach the sheathig to the frame. Aalyses with the computer program ISTAR-WD compare excelletly with the results of full-scale testig of light -framed sheathed walls. Special cosideratio is eeded for poorly costructed walls i which displacemets due to holdow slip ad deformatio are a sigificat amout of total wall displacemet. The properties obtaied from aalyses of idividual walls ca be assembled to create a aalytical model for determiig the oliear respose of a light framed wood buildig. A oliear static aalysis or pushover aalysis provides a tool for evaluatig the damage i a wood buildig at various levels of displacemet. Buildigs with sigificat torsioal respose may require additioal aalyses. I additio, sice wood buildigs may have a soft first story at large displacemets, the ivestigatio of buildig performace may be improved by modifyig the distributio of the lateral load to the floors of the buildig as the aalysis progresses. ACKNLOWLEDGEMENTS The authors are grateful to the APA - Egieered Wood Associatio ad the Uiversity of Califoria, Irvie for providig the data from the full-scale tests of light-framed sheathed walls. REFERENCES Applied Techical Coucil, "ATC-40, Seismic Evaluatio ad Retrofit of Cocrete Buildigs", Seismic Safety Commissio, State of Califoria, Report No. SSC 96-0, November, 996. Federal Emergecy Maagemet Agecy (FEMA), "NERHP Guidelies for the Seismic Rehabilitatio of Buildigs (FEMA Publicatio 73)", Buildig Seismic Rehabilitatio Project, October, 997. Ekwueme, C. G. ad Hart, G. C., "ISTAR-WD: A computer program for the Noliear Aalysis of Sheathed Stud Walls, Versio.0 - User's Maual", Hart Cosultat Group, March, 998. Phillips T. L., Itai, R. Y., ad McLea, D. I., "Lateral Load Sharig by Diaphragms i Wood-Framed Buildigs", Joural of the Structural Egieerig, Vol. 9, No. 5, May, 993, pp Rose, J. D., "Prelimiary Testig of Wood Structural Pael Shear Walls Uder Cyclic (Reversed) Loadig", APA- The Egieered Wood Associatio, Techical Services Divisio, Report 58, March, 998. Schmidt, R. J. ad Moody, R. C., "Modelig Laterally Loaded Light-Frame Buildigs", Joural of Structural Egieerig, Vol. 5, No., Jauary 989, pp Tuomi, R. L. ad McCutcheo, W. J., "Rackig Stregth of Light Frame Nailed Walls", Joural of the Structural Divisio, Proceedigs of the America Society of Civil Egieers, Vol. 04, No. ST7, July, 978, pp White, M. W. ad Dola, J. D., "Noliear Shear Wall Aalysis", Joural of Structural Egieerig, Vol., No., November,