YIELD DISPLACEMENT ESTIMATES FOR DISPLACEMENT-BASED SEISMIC DESIGN OF DUCTILE REINFORCED CONCRETE STRUCTURAL WALL BUILDINGS

Transcription

1 3 th World Conerene on Earthqake Engineering Vanover, B.C., Canada Agst -6, 2004 Paper No. 035 YIELD DISPLACEMENT ESTIMATES FOR DISPLACEMENT-BASED SEISMIC DESIGN OF DUCTILE REINFORCED CONCRETE STRUCTURAL WALL BUILDINGS Tjen N. TJHIN, Mark A. ASCHHEIM, 2 and John W. WALLACE 3 SUMMARY This paper presents the reslts o an analtial std on ield rvatre or estimating eetive ield displaements or the design o dtile reinored onrete antilever strtral all bildings. Improved estimates o eetive ield rvatre ere obtained or retanglar and barbell ross setions based on standard moment rvatre analses. Variables onsidered inlde bondar and eb reinorement ratios, onrete ompressive and reinoring steel strengths, and axial load ratio. Formlas or estimating eetive ield rvatres and displaements are also presented, and an example is provided to illstrate the appliation o these estimates to the design o dtile reinored strtral all bildings. INTRODUCTION Ne displaement-based methods or seismi design [, 2, 3] rel on an estimate o the eetive ield displaement orresponding to the ndamental mode response o the strtre to be designed. The ield displaement is a relativel stable parameter that an be estimated based on kinemati relationships earl in the design proess, aonting or the strtral geometr, approximate distribtion o mass, material properties, and the nominal member dimensions o the strtral sstem []. In ontrast, onventional design approahes are based on the period o vibration, hih is prone to greater variation bease the stiness o the strtre is not knon initiall and tpiall ill var as the strengths o the strtral members are adjsted to obtain a strtre having the intended seismi perormane harateristis. The variation o the period ith hanges in strength tends to ase period-based seismi design approahes to reqire a larger nmber o iterations than are needed ith ield displaement approahes. Estimates o ield displaement sed or the design o dtile reinored onrete strtral all bildings are rrentl based on ield rvatre estimates that range beteen / l and / l or higher, here l is the length o the all in plan [4]. An analtial std as ndertaken in order to improve the preision o this estimate or retanglar setion and barbell setion alls having a large range o bondar and eb reinorement ratios, onrete ompressive strengths, and axial load ratios. This paper Gradate Researh Assistant, Universit o Illinois at Urbana-Champaign, Urbana, IL, USA 2 Assoiate Proessor, Santa Clara Universit, Santa Clara, CA, USA 3 Assoiate Proessor, Universit o Caliornia at Los Angeles, Los Angeles, CA, USA

2 presents the reslts o this analtial std on ield rvatre. Formlas or estimating eetive ield rvatres and displaements are presented together ith an example illstrating the appliation o these estimates to the design o dtile reinored onrete strtral all bildings. The disssion is limited to ll-height prismati antilever all sstems exhibiting dtile lexral response, or hih onl lexral deormations o the alls are onsidered. YIELD DISPLACEMENT OF DUCTILE REINFORCED CONCRETE WALLS Fig. (a) shos a tpial antilever all having height h and length l. Under lateral inertial ores (Fig. (b)), the roo displaement o this all at ield,, aonting or lexral deormations onl (Fig. ()) an be expressed b = κ h, () 2 here κ = ield displaement oeiient, and = eetive ield rvatre o the all ross setion at the base. The ield displaement oeiient, κ, depends on the rvatre distribtion along the height o the all (Fig. (e)), hih in trn depends on the lateral load distribtion and stiness distribtion over the all height. It also depends on seondar eets, sh as ondation rotation, shear distortion, and tension shit mehanism. The rvatre distribtion over the height o the all an be estimated sing simple beam theor as M ( x) / E I( x), here M (x) = moment at a setion loated x rom the base (Fig. (d)), and E I (x) = lexral rigidit at a setion loated x rom the base. n-th l i-th stor st V M h (a) Wall Elevation h e V (b) Lateral Fore Distribtion () Deleted Shape x Mx ( ) M (d) Moment Distribtion Fig. Response o a single all sstem at ield. (e) Idealized Crvatre Distribtion Vales o κ, ompted assming niorm E I(x) over the height o the all, niorm loor masses, niorm stor heights, and response in the ndamental mode, are given in Table. For irreglar all sstems, κ ma be determined b an elasti strtral analsis sing raked setion properties ith lateral ores applied in proportion to the ndamental mode shape amplitde and mass at eah loor. The eetive ield rvatre,, in Eq. () ideall orresponds to M / E I r, here M = eetive ield moment at the base o the all, and E I r = raked-setion lexral rigidit at the base o the all, rather than the instant hen the ield strain is reahed at the extreme tension reinorement. Varios estimates o the eetive ield rvatre,, have been proposed [e.g., 4, 5, 6], oten in the orm

3 κ =, (2) l here κ = eetive ield rvatre oeiient that depends primaril on the ross setional shape o the all, axial load level, and the amont, onigration, and ield strength o the longitdinal reinorement. Table Properties o Prismati Walls Responding in the Fndamental Mode n κ Γ α h e /h For retanglar ross setions, Wallae and Moehle [4] have reommended vales o κ in the range o to or Grade 60 (44 MPa) steel, or tpial levels o axial load and reinorement ratio. Vales or other ross setions have also been sggested b other researhers [e.g., 5, 6]. More generall, κ an be obtained rom a moment-rvatre analsis b linearl extrapolating the ield rvatre orresponding to the irst ield o longitdinal reinorement to the eetive ield moment, M, hih is the moment resistane orresponding to a predetermined rvatre, as illstrated in Fig. 2. The predetermined rvatre old orrespond to the development o the nominal lexral strength, as determined sing bilding odes sh as ACI-38 [7]. Moment, M EI r M M First Yield Point Predetermined Crvatre Crvatre, Fig. 2 General deinition o eetive ield rvatres. ESTIMATES OF EFFECTIVE YIELD CURVATURES Methodolog An analtial std as ondted to improve the preision o eetive ield rvatre estimates or retanglar and barbell all ross setions (Fig. 3(a)). Variables onsidered inlde axial load level, longitdinal bondar reinorement ratios, longitdinal eb reinorement ratio, speiied onrete

4 ompressive strength, and speiied steel reinorement ield strength. Common ranges o variables ere seleted or the std, as smmarized in Table 2. t ρt l 2d ρt l t l kd ε Paraboli Shape ρ t ρt l 2d l ρ t ρt l t l Centroid o Tension Steel d Gross-Setion Centroid (a) Idealized Cross Setions ε (b) Assmed Strain Proile at First Yield () Assmed Conrete Stress Distribtion at First Yield Fig. 3 Idealization and assmptions sed or determining eetive ield rvatres. Table 2 Smmar o the Variables Used ( ksi = MPa) Parameter Vale Cross Setion Retanglar, Barbell End (Bondar) Longitdinal Steel Ratio, ρ Retanglar: 0.25%, 0.5%, %, 2%, 3% Barbell: %, 2%, 3%, 4%, 5% Web Reinoring Steel Ratio, ρ" 0.25%, 0.3%, 0.4%, 0.5% Speiied Compressive Strength o Conrete, (ksi) 4, 5, 6 Speiied Yield Strength o Reinoring Steel, (ksi) 40, 60, 75 Axial Load Ratio, P/( A ) 0 to 0.2 Normalized Centroid o Bondar Reinorement, d/l 0.05, 0., 0.5 Normalized Flange Thikness, t /t (or Barbell onl) 2 Normalized Flange Depth, l /t (or Barbell onl),.5, 2 The eetive ield rvatre estimates ere determined sing standard moment-rvatre analses satising strain ompatibilit, material stress-strain relationships, and eqilibrim. The longitdinal bondar reinorement as assmed to be lmped at its entroid, and the longitdinal eb reinorement as assmed to be niorml distribted as a thin sheet (Fig. 3(a)). For barbell setions, the entroid o bondar reinorement oinides ith the entroid o the bondar region. The strain distribtion as assmed to be linear aross the setion (Fig. 3(b)). The stress-strain rve or onrete in ompression as assmed to be paraboli (Figs. 3() or 4(a)). The modls o elastiit o onrete in ompression, E, as 57,000 (in psi), as deined in ACI [7]. This modls is a seant modls representing the slope o line passing throgh the onrete stressstrain rve at 0.45, as illstrated in Fig. 4(a). The strain at the instant that the stress reahes, denoted ε, that orresponds to this deinition is , , and or eqal to 4, 5, and 6 ksi (28, 35, 42 MPa), respetivel. The onrete tensile strength as negleted. An elasto-plasti stress-

5 strain rve as sed or reinoring steel in both ompression and tension (Fig. 4(b)). The modls o elastiit o the steel, E, as taken as 29,000 ksi (200,000 MPa). s Conrete Stress, E ε = 2 ε - ε 2 ε Steel Stress, s E s 0.45 Note: No tension properties Note: Tension and ompression are deined properties are the same 0 ε Conrete Strain, ε E ε Steel Strain, ε s (a) Conrete (b) Reinoring Steel Fig. 4 Stress-strain relationships or onrete and reinoring steel sed in this std. The irst-ield rvatre,, as established based on the strain ondition shon in Fig. 3(b) as ε + ε =, () d here ε = / Es = steel ield strain, ε = onrete strain at the extreme ompression iber at the time longitdinal bondar reinorement strain reahes ε, and d = the distane rom the extreme onrete ompression iber to the entroid o the bondar reinorement. For higher axial load levels, the steel ield strain, ε, ma or at a ver high vale o ε. For these ases, the irst-ield rvatre is deined as rvatre orresponding to ε, i.e., the strain at the time the stress reahes. The eetive ield rvatre,, as obtained rom extrapolating the irst-ield rvatre to a point here the moment reahes ltimate strength, M, assming elasto-plasti response (Fig. 5), or M =, (2) M here M = moment resistane hen longitdinal bondar reinorement strain reahes ε. The ltimate lexral strength, M, is deined as the moment resistane orresponding to a onrete strain o at the extreme ompression iber.

6 Moment, M M EI r Idealized Observed M Eetive Yield Point First Yield Point Crvatre, Fig. 5 Deinition o eetive ield rvatres sed in this std. Reslts Charts ere generated to allo the eetive ield rvatre o retanglar and barbell all ross setions to be estimated diretl. Eah hart plots the ield rvatre oeiient as a ntion o the axial load level in a dimensionless orm or a nmber o longitdinal bondar reinorement ratios, a speiied longitdinal eb reinorement ratio, speiied onrete ompressive strength, and steel reinorement ield strength. Tpial harts are given in Fig. 6 or both retanglar and barbell ross setions. Given the ross setional shape, the axial load level, the longitdinal bondar reinorement ratio, ρ, and the longitdinal eb reinorement ratio, ρ ", the eetive ield rvatre oeiient an be interpolated beteen the rves. From the moment-rvatre analses, it is observed that eetive ield rvatre is nearl insensitive to onrete strength and eb reinorement ratio. It is also observed that both the retanglar and barbell ross setions exhibit similar reslts or the range o vales overed in the std. At earl stage o the design proess, the bondar and eb steel reinorement steel is sall not knon et. For design prposes, it is thereore pratial to deine eetive ield rvatre as a ntion o axial load level and steel reinorement grade; at a later stage the neessit o a onined bondar element an be addressed. The eetive ield rvatre oeiient or retanglar or barbell ross setions ma be estimated ith the olloing ormla ith errors tpiall less than 5 to 0%: P κ =.8ε (5) A It shold be noted that Eq. (5) is valid or ases here the axial load ratio, P /( A ), is not more than 0.2. In addition, Eq. (5) is appliable or barbell ross setions in hih l / t 2 and t / t 2. The same eqation proposed or both the retanglar and barbell ross setions provides lexibilit in satising the detailing reqirements or speial transverse bondar reinorement presribed in the perormane objetives or reqired b ode strain ompatibilit analsis provisions.

7 Yield Crvatre Coeiient, κ = l d = 0.05l, ρ = 0.25% = 4 ksi, = 40 ksi t tl 2d Yield Crvatre Coeiient, κ = l t = 2t, l = t, d = 0.05l, ρ = 0.25% = 4 ksi, = 40 ksi tl t l ρ = 0.25% ρ = 0.5% ρ = % ρ = 2% ρ = 3% Axial Load Level, P /( A ) t tl 2d l ρ = % ρ = 2% ρ = 3% ρ = 4% ρ = 5% Axial Load Level, P /( A ) t tl t l Yield Crvatre Coeiient, κ = l d = 0.05l, ρ = 0.25% = 5 ksi, = 60 ksi t tl 2d Yield Crvatre Coeiient, κ = l t = 2t, l = t, d = 0.05l, ρ = 0.25% = 5 ksi, = 60 ksi tl t l t ρ = 0.25% ρ = 0.5% ρ = % ρ = 2% ρ = 3% Axial Load Level, P /( A ) t tl 2d l ρ = % ρ = 2% ρ = 3% ρ = 4% ρ = 5% Axial Load Level, P /( A ) t tl l Yield Crvatre Coeiient, κ = l d = 0.05l, ρ = 0.25% = 6 ksi, = 75 ksi t tl 2d Yield Crvatre Coeiient, κ = l t = 2t, l = t, d = 0.05l, ρ = 0.25% = 6 ksi, = 75 ksi tl t l t ρ = 0.25% ρ = 0.5% ρ = % ρ = 2% ρ = 3% Axial Load Level, P /( A ) t tl 2d l ρ = % ρ = 2% ρ = 3% ρ = 4% ρ = 5% Axial Load Level, P /( A ) t tl l Fig. 6 Tpial harts o eetive ield rvatre oeiients ( ksi = MPa). APPLICATION TO DESIGN APPROACHES BASED ON YIELD DISPLACEMENT ESDOF Sstem or Approximating Dnami Response o Strtres Eqivalent Single-Degree-o-Freedom (ESDOF) representations have been sed in seismi evalation proedres, sh as the displaement oeiient and apait spetrm methods [8, 9], to estimate inelasti seismi demands o the strtre nder onsideration. In these proedres, the response o a mlti-degree-o-reedom (MDOF) model o the strtre is assmed to be predominantl in a single deleted shape throghot the response histor. ESDOF representations have also reentl been

8 emploed or determining the reqired strength and stiness to limit the maximm displaement response to a desired vale. The latter se ill be demonstrated b a design example presented in the next setion. Fig. 7 smmarizes an ESDOF ormlation adapted rom ATC-40 [8]. This ormlation ses the elasti irst mode shape onsidering raked-setion lexral stiness properties, reslting in a math beteen the ndamental period o the all sstem, T, assoiated ith the raked setion stiness, k, (Fig. 7(a)) and the initial period o the ESDOF model, T. Also, the height o the lateral ore resltant o the MDOF sstem, h e, is the same as the eetive height o the ESDOF sstem. m n n = m= m i Σ i = h he i k n m i i Base Shear, V Base Shear, V k Idealized V = C W V = W Observed m C k T = T = 2π W= n Σ i = m i g C g = C C α W = m g = µ µ = = Roo Displaement, Γ Displaement, (a) MDOF Sstem (b) ESDOF Sstem Fig. 7 Idealized load-deormation responses o a all sstem and the ESDOF sstem. As indiated in Fig. 7, the relationship beteen the roo displaement o a MDOF model o the all sstem at ield,, and the ield displaement o the orresponding ESDOF sstem,, is given as =, (6) Γ here Γ = the irst mode partiipation ator, allated ith the mode shape normalized sh that the mode shape amplitde at the roo, n, is eqal to one. The base shears o the ESDOF sstem, V, and the MDOF sstem, V, at ield are related b V V, (7) = α

9 here α is the irst mode eetive mass oeiient. Normalizing Eq. (7) b the orresponding eights o the sstems gives C C, (8) = α here C = shear oeiient o the MDOF sstem. V / W = the ield strength oeiient o the ESDOF sstem and C = V / W = the base For bildings onsisting o prismati alls having niorm stor heights and niorm loor masses and responding in lexre onl, Γ, α, and h e have the vales given in Table. For other ases, the vales o these terms an be determined sing standard ormlas. While not reqired or design, the period o vibration o the all sstem oten is o interest. Corresponding to raked-setion stiness o the all sstem, k, as deined in Fig. 7(a), this period ma be allated based on the properties o the ESDOF sstem as T = T = 2π. (9) C g The displaement response and peak drit o an ESDOF sstem,, sbjeted to grond motion exitation ma be estimated sing a dnami analsis sing a simpliied hstereti model. For slender reinored onrete strtral all bildings, an elasto-plasti model ith stiness degradation oten is siientl arate. For ases here onl smoothed ode spetra are available, the peak drit demand ma be estimated sing a sitable R-µ -T relationship developed or stiness-degrading osillators. Given the ESDOF drit estimate, the roo drit o the all sstem an be estimated as = Γ implies that the displaement dtilit o the ESDOF and MDOF sstems is eqal and is given as. This µ = =. (0) Yield Point Spetra or Estimating Seismi Demands and Reqired Lateral Strengths Yield Point Spetra (YPS) [0] plot rves o onstant displaement dtilit on the axes o ield strength oeiient and ield displaement, or SDOF osillators having a range o initial (elasti) periods o vibration, a speiied hstereti propert, and a speiied level o visos damping (Fig. 8). An one point on a YPS plot represents or qantities:, C, T, and µ. The irst three o these qantities are related aording to Eq. (9). Frthermore, knoledge o the ESDOF dtilit, µ, allos the peak displaements, and, to be estimated sing Eq. (0). YPS ma diretl be ompted or speii grond motion reords or estimated b appling previosl established R-µ -T relationships to elasti design response spetra. YPS ma be sed to estimate the peak displaement response o a SDOF sstem (or an ESDOF sstem) having a knon ield point. YPS ma also be sed or an inverse proess, i.e., to determine the ield point

10 or a ne design or to determine the level o strengthening and stiening reqired to rehabilitate an existing strtre to have aeptable seismi perormane. The inverse proess, or determining aeptable ield points or a desired perormane, is shematiall illstrated in Fig. 8. Yield Strength Coeiient, C µ = µ o T = 2π C o µ o = o o (given) o µ = µ = 4 µ = 2 µ = o (given) Yield Displaement, (mm) o o C o g Fig. 8 Example o YPS and its se or estimating ield strength oeiient (25.4 mm = in.). Design Example Bilding Desription The design o to barbell alls in an eight-stor strtral all bilding is presented to illstrate the se o the estimated ield displaement in design. To idential barbell setion alls are sed as the lateral load arring sstem or the N-S diretion, as indiated in Fig. 9. All loor-to-loor heights are 2 t (3.65 m). The onrete ompressive strength,, and the reinoring steel ield strength,, are 5 ksi (35 MPa) and 60 ksi (44 MPa), respetivel. Floor dead and live loads are 75 ps (8.4 kn/m 2 ) and 50 ps (2.4 kn/m 2 ), respetivel. The axial load at the base o the alls as estimated to be 0.5 A. Seismi eight, W, as assmed to ome rom the dead load onl, hih is 8(75 ps)(80 t)(60 t) = 5,20 kips (67,254 kn). 24 t (7.32 m) W 4 in. (356 mm) W 60 t (8.3 m) N 80 t (54.9 m) Fig. Tpial loor o the design example. Design Perormane Objetive The N-S diretion as designed to satis a lie saet perormane objetive, hih is somehat arbitraril assoiated ith a peak roo displaement limit eqal to 0.83% o the height o the bilding. The design earthqake assoiated ith this perormane objetive is represented b the smoothed elasti design spetrm shon in Fig. 0(a). This spetrm has the olloing parameters per IBC 2000 [] terminolog: T = 0.40 se, T o = 0. 2T = 0.08 se, S Ds = g, and S D = g. The simple R-µ -T s

11 relationships shon in Fig. 0(b) are sed to derive inelasti seismi demands rom the design spetrm. Onl translational response is onsidered in this design; the potential eets o aidental eentriit and rotational omponents o the grond motion on torsional response are negleted. Spetral Psedo Aeleration, S pa (g ) S DS = 0.937g 0.8 Response Modiiation Fator, R 0 8 µ = T o = 0.08 se T s = 0.40 se S DS = 0.377g T s = 0.40 se µ = 4 µ = 2 µ = Period, T (se) (a) Design Spetrm Period, T (se) (b) R-µ-T Relationships Fig. 0 Design spetrm and R-µ -T relationships sed in the design example. Design Callations The design approah desribed in the previos setions is sed to determine the seismi demand and reqired base shear or determining the reqired lexral strength at the base o eah all. For brevit, onl allations or reqired base shear and base moment o the alls are presented. The distribtion o lateral ores over the height o the bilding and to eah all at eah loor level ma ollo ode or other established proedres [e.g., 6, 2]. The reqired shear and moment apaities over the height o eah all shold be determined onsidering higher mode eets sing an o several established proedres [e.g., 3-6]. The eetive ield rvatre oeiient or the alls is estimated sing Eq. (5), as.8(60 ksi/29,000 ksi) (0.5) = B Eq. (2), the estimated ield rvatre is /(24 t) = rad/in. ( rad/mm). For an eight-stor bilding, the estimated ield displaement oeiient, κ, is aording to Table. Using Eq. (), the ield displaement,, is 0.295( rad/in.)(96 t) 2 = 6.0 in. (52 mm). Based on the design perormane objetive, the roo displaement limit,, is (0.83%)(96 t) = 9.6 in. (244 mm). Ths, the displaement dtilit limit, µ, is 9.6/6.0 =.6. From Table, the modal partiipation ator, Γ, mass oeiient, α, and eetive height, h e, or an eight-stor bilding are.445, 0.653, and 0.770, respetivel. The ESDOF displaement,, is (6.0 in.)/.445 = 4.5 in. (05 mm). The YPS or this design are shon in Fig., established b appling R-µ -T relationships to the design spetrm. The minimm ield strength oeiient, C, or µ =.6 and = 4.5 in. is obtained rom YPS as 0.3 (Fig. ). B Eq. (8), the reqired base shear oeiient, C, is 0.653(0.3) = Thereore, the base shear o the bilding is (5,20 kips) = 293 kips (5752 kn), the reqired base

12 shear o eah all is (293 kips)/2 = 646 kips (2876 kn), and the reqired moment at the base o eah all is 0.770(293 kips)(96 t)/2 = 47,790 k-t (64,790 kn-m). Yield Strength Coeiient, C C = 0.3 µ = 0. = 4.5 in. µ = µ = 2 µ = 4 µ = Yield Displaement, (in.) Fig. 2 YPS or the design example ( in. = 25.4 mm). CONCLUSION Improved estimates o eetive ield rvatre or retanglar and barbell ross setions ere presented or se in design o dtile reinored onrete antilever strtral alls. These estimates ere derived based on onsistent standard moment rvatre analses overing variation in bondar reinorement, eb reinorement, steel reinorement ield strength, onrete ompressive strength, axial load ratio, relative dimensions o the ross setion. A set o harts as developed to allo the eetive ield rvatre to be estimated diretl. Simple expression derived in terms o parameters sall knon at the earl design stage axial load and steel reinorement grade as also presented. A design example based on eqivalent single-degree-o reedom in onjntion ith Yield Point Spetra as provided to illstrate the appliation o these estimates to the design o dtile reinored strtral all bildings. REFERENCES. Ashheim, M. A., Blak, E. F. Yield point spetra or seismi design and rehabilitation, Earthqake Spetra, EERI, 2000; 6(2): Ashheim, M. A. The prima o the ield displaement in seismi design. Seond US-Japan Workshop on Perormane Based Design o Reinored Conrete Bildings, Sapporo, Japan, September 0-2, Pala, T. A displaement-osed seismi design o mixed bilding sstems. Earthqake Spetra, 2002; 8(4): Wallae, J. W., Moehle, J. P. Dtilit and detailing reqirements o bearing all bildings. Jornal o Strtral Engineering, 992; 8(6): Priestle, M. J. N., Koalsk, M. J. (998). Aspets o drit and dtilit apait o retanglar antilever strtral alls. Blletin o the Ne Zealand National Soiet or Earthqake Engineering, 998; 3(2): Pala, T. An estimation o displaement limits or dtile sstems. Earthqake Engineering and Strtral Dnamis, 2002; 3(3): ACI Committee 38. Bilding ode reqirements or strtral onrete (ACI 38-02) and ommentar (ACI 38R-02). Amerian Conrete Institte, Farmington Hills, MI, 2002.

13 8. ATC-40. Seismi evalation and retroit o onrete bildings. Applied Tehnolog Conil, Redood Cit, CA, FEMA 273. NEHRP gidelines or the seismi rehabilitation o bildings. Report No. FEMA 273, Federal Emergen Management Agen, Washington, DC, Ashheim, M. A., Blak, E. F. Yield point spetra or seismi design and rehabilitation. Earthqake Spetra, 2000; 6(2): International Conerene o Bilding Oiials (ICBO). International bilding ode. Whittier, CA, Tjhin, T. N., M. A. Ashheim, Wallae, J. W. Perormane-based seismi design o dtile reinored onrete all bildings. Report (nder revie), Universit o Illinois at Urbana- Champaign, Urbana, IL, Pala, T. The design o dtile reinored onrete strtral alls or earthqake resistane. Earthqake Spetra, EERI, 986; 2(4): Eberhard, M. O., Sozen, M. A. A behavior-based method to determine design shear in earthqakeresistant alls. Jornal o Strtral Engineering, ASCE, 993; 9(2): ACI Committee 368. Drat ommittee report: reommendations or proportion and design o reinored onrete sstems and elements. Amerian Conrete Institte, Detroit, MI, Ashheim, M., Tjhin, T., Comartin, C., Hambrger, R., Inel, M. The saled nonlinear dnami proedre. ASCE Strtres Congress, Nashville, TN, Ma 22-26, NOTATION A s = Area o longitdinal bondar (end) reinorement o a all. A = Cross setion area o a all. C = Base shear strength o a all or a all sstem, V, normalized b its seismi eight, W (sall termed base shear strength oeiient). C = Yield strength oeiient o the ESDOF model o a all sstem. d = Distane rom the extreme ompression iber to the entroid o longitdinal bondar reinorement. d = Distane rom the extreme onrete iber to the entroid o longitdinal bondar reinorement. E I(x) = Flexral rigidit at a all setion loated at a distane x rom the base. E I r = raked-setion lexral stiness at the base. E = Modls o elastiit o steel reinorement. g s = Conrete ompressive stress hen the extreme ompression iber reahes ε. = Speiied ompressive strength o onrete. = Speiied ield strength o longitdinal reinorement. = Aeleration o gravit. h e = Resltant o lateral ores measred rom the base o a all sstem (sall termed eetive h k height). = Total height o a all or a all sstem. = Lateral elasti raked-setion stiness o a all or a all sstem in elasti region hen sbjeted to a lateral ore distribtion proportional to the prodt o the ndamental mode shape amplitde and loor mass at eah loor.

14 k d = Netral axis or ompression zone depth o a all ross setion at ield. k = Lateral stiness o the ESDOF model o a all sstem. l = Depth o the bondar part o a bar-bell all ross setion. This dimension is parallel to the horizontal length, l. l = Horizontal length o a all. m i = Lmped mass o the i-th loor o an n-stor bilding. m = Seismi mass o the ESDOF model o a all sstem. M = Moment resistane orresponding to eetive ield rvatre o a all setion at the base,. M M = Moment resistane orresponding to irst-ield rvatre o a all setion at the base,. = Moment resistane orresponding to a onrete strain o at the extreme ompression iber. M (x) = Moment o a all at a setion loated at a distane x rom the all base. n = Nmber o stories. P = Axial load o a all. R = Ratio o the elasti strength demand to the ield strength oeiient (also knon as response modiiation oeiient in IBC 2000 []). S = Design spetral psedo aeleration at se period (parameter in IBC 2000 []). D S = Design spetral aeleration at short periods (parameter in IBC 2000 []). DS t = Thikness o the bondar o a bar-bell all ross setion. This dimension is parallel to the eb thikness, t. t = Web thikness o a all. T = First mode period o vibration o a all sstem in the design diretion based on rakedsetion stiness, k. T = Period o vibration o an ESDOF model o a all sstem. T s = harateristi period (parameter in IBC 2000 []). T o = 0.2T s (parameter in IBC 2000 []). V = Lateral base shear o a all or all sstem at ield hen sbjeted to a lateral ore distribtion proportional to the prodt o the ndamental mode shape amplitde and loor mass at eah loor (also termed base shear strength). V = Yield strength o the ESDOF model o a all sstem. W = Seismi eight o a all sstem. W = Seismi eight o the ESDOF model o a all sstem. x = Vertial distane o a all ross setion, measred rom the all base. α = First mode eetive mass oeiient. = Lateral roo displaement o a all or a all sstem at ield hen sbjeted to a lateral ore distribtion proportional to the prodt o the ndamental mode shape amplitde and loor mass at eah loor. = Lateral displaement o the ESDOF model o a all sstem at ield. = Maximm lateral roo displaement (also knon as peak drit) o a all sstem at ltimate. = Maximm lateral displaement o the ESDOF model o a all sstem.

15 ε = Conrete strain at the extreme ompression iber hen longitdinal bondar reinorement strain reahes ε. ε ε i κ = Yield strain o longitdinal reinorement. = Strain orresponding to peak stress,, in a niaxial stress-strain rve o onrete. = First mode shape amplitde at the i-th loor o an n-stor bilding. = First-ield rvatre o a all setion at the base. = Eetive ield rvatre o a all setion at the base. = Yield displaement oeiient o a all. κ = Yield rvatre oeiient o a all. Γ = First mode modal partiipation ator. µ = Displaement dtilit ator o a all or all sstem. ρ = Ratio o longitdinal bondar reinorement o a all = A s /( tl). " " ρ = Ratio o longitdinal eb reinorement o a all = A /( t l ). APPENDIX s Table Conversion Fators rom U.S. Cstomar to SI Units To Convert To Mltipl b inh (in.) millimeter (mm) 25.4 oot (t) meter (m) kilopond ore (kip or k) kiloneton (kn) kilopond ore per sqare inh (ksi) megapasal (MPa) pond per sqare oot (ps) megapasal (MPa) 47.88