Capter 3 Sort Colum Desig By Noel. J. Everard 1 ad ose A. Issa 3.1 Itrodutio Te majority of reifored orete olums are subjeted to primary stresses aused by flexure, axial fore, ad sear. Seodary stresses assoiated wit deformatios are usually very small i most olums used i pratie. Tese olums are referred to as "sort olums." Sort olums are desiged usig te iteratio diagrams preseted i tis apter. Te apaity of a sort olum is te same as te apaity of its setio uder primary stresses, irrespetive of its legt. Log olums, olums wit small ross-setioal dimesios, ad olums wit little ed restraits may develop seodary stresses assoiated wit olum deformatios, espeially if tey are ot braed laterally. Tese olums are referred to as "sleder olums". Fig. 3-1 illustrates seodary momets geerated i a sleder olum by P-δ effet. Cosequetly, sleder olums resist lower axial loads ta sort olums avig te same ross-setio. Tis is illustrated i Fig. 3-1. Failure of a sleder olum is iitiated eiter by te material failure of a setio, or istability of te olum as a member, depedig o te level of slederess. Te latter is kow as olum buklig. Desig of sleder olums is disussed i Capter 4. Te lassifiatio of a olum as a sort olum or a sleder olum is made o te basis of its Slederess Ratio, defied below. Slederess Ratio: kl u r were, l u is usupported olum legt; k is effetive legt fator refletig ed restrait ad lateral braig oditios of a olum; ad r is te radius of gyratio refletig te size ad sape of a olum ross-setio. A detailed disussio of te parameters ivolved i establisig te slederess ratio is preseted i Capter 4. Colums wit slederess ratios less ta tose speified i Ses. 10.1. ad 10.13. for o-sway ad sway frames, respetively, are desiged as sort olums usig tis apter. 1 Professor Emeritus of Civil Egieerig, te Uiversity of Texas at Arligto, Arligto, Texas. Professor, Departmet of Civil ad aterials Egieerig, Uiversity of Illiois at Ciago, Illiois.
No-sway frames are frames tat are braed agaist sidesway by sear walls or oter stiffeig members. Tey are also referred to as braed frames. Sway frames are frames tat are free to traslate laterally so tat seodary bedig momets are idued due to P-δ effets. Tey are also referred to as ubraed frames. Te followig are te limitig slederess ratios for sort olum beavior: kl u No-sway frames: 34 1( 1 ) (3.1) r kl Sway frames: u (3.) r Were te term [ 34 1( 1 ) ] 40 ad te ratio 1 is positive if te member is bet i sigle urvature ad egative if bet i double urvature. 3. Colum Setioal Capaity Fig. 3-1 Failure odes i Sort ad Sleder Colums I sort olums te olum apaity is diretly obtaied from olum setioal apaity. Te teory tat as bee preseted i Setio 1. of Capter 1 for flexural setios, also applies to reifored orete olum setios. However, olum setios are subjeted to flexure i ombiatio wit axial fores (axial ompressio ad tesio). Terefore, te equilibrium of iteral fores ages, resultig i sigifiatly differet flexural apaities ad beavioral modes depedig o te level of aompayig axial load. Fig. 3- illustrates a typial olum setio subjeted to ombied bedig ad axial ompressio. As a be see, differet ombiatios of momet ad aompayig axial fore result i differet olum apaities ad orrespodig strai profiles, wile also affetig te failure modes, i.e., tesio or ompressio otrolled beavior. Te ombiatio of bedig momet ad axial fore tat result i a olum apaity is best preseted by olum iteratio diagrams. Iteratio diagrams are ostruted by omputig momet ad axial fore apaities, as sow below, for differet strai profiles. P = C + Cs 1 + Cs Ts (3-3) Cx + Cs 1x1 + Ts x3 = (3-4)
0.003 C s1.a. Tesio otrolled Balaed setio Trasitio zoe P Compressio otrolled T s C s C x x 3 x 1 b Cross-Setio t = y =0.005 t Strai Distributio Stress Distributio Fig. 3- Aalysis of a olum setio 3..1 Colum Iteratio Diagrams Te olum axial load - bedig momet iteratio diagrams iluded erei (Colums 3.1.1 troug Colums 3.4.4) oform fully to te provisios of ACI 318-05. Te equatios tat were used to geerate data for plottig te iteratio diagrams were origially developed for ACI Speial Publiatio SP-7 3. I additio, omplete derivatios of te equatios for square ad irular olums avig te steel arraged i a irle ave bee publised i ACI Corete Iteratioal 4. Te origial iteratio diagrams tat were otaied i SP-7 were subsequetly publised i Speial Publiatio SP-17A 5. Te related equatios were derived osiderig te reiforig steel to be represeted as follows: (a) For retagular ad square olums avig steel bars plaed o te ed faes oly, te reiforemet was assumed to osist of two equal ti strips parallel to te ompressio fae of te setio. (b) For retagular ad square olums avig steel bars equally distributed alog all four faes of te setio, te reiforemet was osidered to osist of a ti retagular or square tube. () For square ad irular setios avig steel bars arraged i a irle, te reiforemet was osidered to osist of a ti irular tube. Te iteratio diagrams were developed usig te retagular stress blok, speified i ACI 318-05 (Se. 10..7). I all ases, for reiforemet tat exists witi te ompressed portio of te dept perpediular to te ompressio fae of te orete (a = β), te ompressio stress i te steel was redued by 0.85 f to aout for te orete area tat is displaed by te reiforig bars witi te ompressio stress blok. Te iteratio diagrams were plotted i o-dimesioal form. Te vertial oordiate [ K = P ( f A ) ] represets te o-dimesioal form of te omial axial load apaity of te g 3 Everard ad Coe. Ultimate Stregt Desig of Reifored Corete Colums, ACI Speial Publiatio SP-7, 1964, pp. 15-18. 4 Everard, N.J., Axial Load-omet Iteratio for Cross-Setios Havig Logitudial Reiforemet Arraged i a Cirle, ACI Strutural Joural, Vol. 94, No. 6, November-Deember, 1997, pp. 695-699. 5 ACI Committee 340, Ultimate Stregt Desig Hadbook, Volume, Colums, ACI Speial Publiatio 17-A, Ameria Corete Istitute, Detroit, I, 1970, 6 pages.
setio. Te orizotal oordiate [ R = ( f ) ] represets te o-dimesioal omial bedig momet apaity of te setio. Te o-dimesioal forms were used so tat te iteratio diagrams ould be used equally well wit ay system of uits (i.e. SI or i-poud uits). Te stregt redutio fator (φ) was osidered to be 1.0 so tat te omial values otaied i te iteratio diagrams ould be used wit ay set of φ fators, sie ACI 318-05 otais differet φ fators i Capter 9, Capter 0 ad Appedix C. It is importat to poit out tat te φ fators tat are provided i Capter 9 of ACI 318-05 are based o te strai values i te tesio reiforemet fartest from te ompressio fae of a member, or at te etroid of te tesio reiforemet. Code Setio 9.3. referees Setios 10.3.3 ad 10.3.4 were te strai values for tesio otrol ad ompressio otrol are defied. It sould be ote tat te eetriity ratios ( e = P ), sometimes iluded as diagoal lies o iteratio diagrams, are ot iluded i te iteratio diagrams. Usig tat variable as a oordiate wit eiter K or R ould lead to iauraies beause at te lower eds of te diagrams te e lies overge rapidly. However, straigt lies for te tesio steel stress ratios f s f y ave bee plotted for assistae i desigig splies i te reiforemet. Furter, te ratio f s f y =1. 0 represets steel straiε y = f y Es, wi is te boudary poit for te φ fator for ompressio otrol, ad te begiig of te trasitio zoe for liear irease of te φ fator to tat for tesio otrol. I order to provide a meas of iterpolatio for te φ fator, oter strai lies were plotted. Te strai lie forε t = 0. 005, te begiig of te zoe for tesio otrol as bee plotted o all diagrams. For steel yield stregt 60.0 ksi, te itermediate strai lie for ε t = 0. 035 as bee plotted. For Steel yield stregt 75.0 ksi, te itermediate strai lie for ε t = 0. 038 as bee plotted. It sould be oted tat all strais refer to tose i te reiforig bar or bars fartest from te ompressio fae of te setio. Disussios ad tables related to te stregt redutio fators are otaied i two publiatios i Corete Iteratioal 6,7. I order to poit to desigs tat are proibited by ACI 318-05, Setio 10.3.5, strai lies for ε = 0.004 ave also bee plotted. Desigs tat fall witi te ofies of te lies for ε = 0. 004 ad t K less ta 0.10 are ot permitted by ACI 318-05. Tis iludes tesio axial loads, wit K egative. Tesio axial loads are ot iluded i te iteratio diagrams. However, te iteratio diagram lies for tesio axial loads are very early liear from K = 0. 0 to R = 0. 0 wit [ K = A f ( f A ) ]. Tis is disussed i te ext setio. st y g t 6 Everard, N. J., Desigig Wit ACI 318-0 Stregt Redutio Fators, Corete Iteratioal, August, 00, Vol. 4, No. 8, pp 91-93. 7 Everard, N. J., Strai Related Stregt Redutio Fators (φ) Aordig to ACI 318-0, Corete Iteratioal, August, 00, Vol. 34, No. 8, pp. 91-93.
Straigt lies for K max are also provided o ea iteratio diagram. Here, K max refers to te maximum permissible omial axial load o a olum tat is laterally reifored wit ties oformig to ACI 318-05 Setio 7.10.5. Defiig K 0 as te teoretial axial ompressio apaity of a member wit R = 0. 0, K max = 0. 80K0, or, osiderig ACI 318-05 Eq. (10-), witout te φ fator, Te, P, max = 0. 8 [ 0. 85 f ( Ast ) + f y Ast ] (3-5) K = P (3-6) max f A max g For olums wit spirals oformig wit ACI 318-05 Setio 7.10.4, values of iteratio diagrams are to be multiplied by 0.850.80 ratio. K max from te Te umber of logitudial reiforig bars tat may be otaied is ot limited to te umber sow i te illustratios o te iteratio diagrams. Tey oly illustrate te type of reiforemet patters. However, for irular ad square olums wit steel arraged i a irle, ad for retagular or square olums wit steel equally distributed alog all four faes, it is a good pratie to use at least 8 bars (ad preferably at least 1 bars). Altoug side steel was assumed to be 50 peret of te total steel for olums avig logitudial steel equally distributed alog all four faes, reasoably aurate ad oservative desigs result we te side steel osists of oly 30 peret of te total steel. Te maximum umber of bars tat may be used i ay olum ross setio is limited by te maximum allowable steel ratio of 0.08, ad te oditios of over ad spaig betwee bars. 3.. Flexure wit Tesio Axial Load ay studies oerig flexure wit tesio axial load sow tat te iteratio diagram for tesio axial load ad flexure is very early liear betwee R o ad te tesio axial load value K t, as is sow i Fig. 3-3. Here, R 0 is te value of R for K = 0. 0, ad K = A f ( f A ) t st y g Fig. 3.3 Flexure wit axial tesio
Desig values for flexure wit tesio axial load a be obtaied usig te equatios: K [ 1. 0 R R ] = K (3-7) t 0 R [. K ] = 10 (3-8) Ro Kt Also, te tesio side iteratio diagram a be plotted as a straigt lie usig R0 ad K t, as is sow i Fig. 3.3. 3.3 Colums Subjeted to Biaxial Bedig ost olums are subjeted to sigifiat bedig i oe diretio, wile subjeted to relatively small bedig momets i te ortogoal diretio. Tese olums are desiged by usig te iteratio diagrams disussed i te preedig setio for uiaxial bedig ad if required eked for te adequay of apaity i te ortogoal diretio. However, some olums, as i te ase of orer olums, are subjeted to equally sigifiat bedig momets i two ortogoal diretios. Tese olums may ave to be desiged for biaxial bedig. A irular olum subjeted to momets about two axes may be desiged as a uiaxial olum ated upo by te resultat momet; = + φ = + (3-9) u ux uy x y For te desig of retagular olums subjeted to momets about two axes, tis adbook provides desig aids for two metods: 1) Te Reiproal Load (1P i ) etod suggested by Bresler 8, ad ) Te Load Cotour etod developed by Parme, Nieves, ad Gouwes 9. Te Reiproal Load etod is more oveiet for makig a aalysis of a trial setio. Te Load Cotour etod is more suitable for seletig a olum ross setio. Bot of tese metods use te oept of a failure surfae to reflet te iteratio of tree variables, te omial axial load P ad te omial biaxial bedig momets x ad y, wi i ombiatio will ause failure strai at te extreme ompressio fiber. I oter words, te failure surfae reflets te stregt of sort ompressio members subjet to biaxial bedig ad ompressio. Te bedig axes, eetriities ad biaxial momets are illustrated i Fig. 3.4. 8 Bresler, Boris. Desig Criteria for Reifored Colums uder Axial Load ad Biaxial Bedig, ACI Joural Proeedigs, V. 57, No.11, Nov. 1960, pp. 481-490. 9 Parme, A.L. Nieves, J.. ad Gouwes, A. Capaity of Reifored Retagular Colums Subjeted to Biaxial Bedig. ACI Joural Proeedigs, V. 63, No. 9, Sept. 1966, pp.911-93.
y e x x P e y x y x = P e y y = P e x Fig. 3.4 Notatios used for olum setios subjeted to biaxial bedig A failure surfae S 1 may be represeted by variables P, e x, ad e y, as i Fig. 3.5, or it may be represeted by surfae S represeted by variables P, x, ad y as sow i Fig. 3.6. Note tat S 1 is a sigle urvature surfae avig o disotiuity at te balae poit, wereas S as su a disotiuity. (We biaxial bedig exists togeter wit a omial axial fore smaller ta te lesser of P b or 0.1 f A g, it is suffiietly aurate ad oservative to igore te axial fore ad desig te setio for bedig oly.) Fig. 3.5 Failure surfae S 1 Fig. 3.6 Failure surfae S 3.3.1 Reiproal Load etod I te reiproal load metod, te surfae S 1 is iverted by plottig 1P as te vertial axis, givig te surfae S 3, sow i Fig. 3.7. As Fig. 3.8 sows, a true poit (1P 1, e xa, e yb ) o tis reiproal failure surfae may be approximated by a poit (1P i, e xa, e yb ) o a plae S 3 passig troug Poits A, B, ad C. Ea poit o te true surfae is approximated by a differet plae; tat is, te etire failure surfae is defied by a ifiite umber of plaes.
Poit A represets te omial axial load stregt P y we te load as a eetriity of e xa wit e y = 0. Poit B represets te omial axial load stregt P x we te load as a eetriity of e yb wit e x = 0. Poit C is based o te axial apaity P o wit zero eetriity. Te equatio of te plae passig troug te tree poits is; 1 P i 1 1 1 = + (3-10) P P P x y o Were: P i : approximatio of omial axial load stregt at eetriities e x ad e y P x : omial axial load stregt for eetriity e y alog te y-axis oly (x-axis is axis of bedig) P y : omial axial load stregt for eetriity e x alog te x-axis oly (y-axis is axis of bedig) P o : omial axial load stregt for zero eetriity Fig. 3.7 Failure surfae S3,, wi is reiproal of surfae S1 Fig. 3.8 Grapial represetatio of Reiproal Load etod For desig purposes, we φ is ostat, te 1P i equatio give i Eq. 3.9 may be used. Te variable K = P (f A g ) a be used diretly i te reiproal equatio, as follows: 1 K i 1 1 1 = + (3-11) K K K x y o Were, te values of K refer to te orrespodig values of P as defied above. Oe a prelimiary ross setio wit a estimated steel ratio ρ g as bee seleted, te atual values of R x ad R y are alulated usig te atual bedig momets about te ross setio X ad Y axes, respetively. Te orrespodig values of K x ad K y are obtaied from te iteratio diagrams preseted i tis Capter as te itersetio of appropriate R value ad te assumed steel ratio urve for ρ g. Te, te
value of te teoretial ompressio axial load apaity K o is obtaied at te itersetio of te steel ratio urve ad te vertial axis for zero R. 3.3. Load Cotour etod Te load otour metod uses te failure surfae S (Fig. 3.6) ad works wit a load otour defied by a plae at a ostat value of P, as illustrated i Fig. 3.9. Te load otour defiig te relatiosip betwee x ad y for a ostat P may be expressed odimesioally as follows: α + x y = ox oy α 1 (3-1) For desig, if ea term is multiplied by φ, te equatio will be uaged. Tus ux, uy, ox, ad oy, wi sould orrespod to φ x, φ y, φ ox, ad φ oy, respetively, may be used istead of te origial expressios. Tis is doe i te remaider of tis setio. To simplify te equatio (for appliatio), a poit o te odimesioal diagram Fig. 3.10 is defied su tat te biaxial momet apaities x ad y at tis poit are i te same ratio as te uiaxial momet apaities ox ad oy ; tus x = ox (3-1) y oy or; = β ad = β (3-13) x ox y oy Fig. 3.10 Load otour for ostat P o failure surfae
I pysial sese, te ratio β is te ostat portio of te uiaxial momet apaities wi may be permitted to at simultaeously o te olum setio. Te atual value of β depeds o te ratio P P og as well as properties of te material ad ross setio. However, te usual rage is betwee 0.55 ad 0.70. A average value of = 0.65 is suggested for desig. Te atual values of β are available from Colums 3.5. Te load otour equatio give above (Eq. 3-10) may be writte i terms of β, as sow below: log 0.5logβ + oy log 0.5logβ x y = ox 1 (3-14) A plot of te Eq. 3-1 appears as Colums 3.6. Tis desig aid is used for aalysis. Eterig wit x ox ad te value of β from Colums 3.5, oe a fid permissible y oy. Te relatiosip usig β may be better visualized by examiig Fig. 3.10. Te true relatiosip betwee Poits A, B, ad C is a urve; owever, it may be approximated by straigt lies for desig purposes. Te load otour equatios as straigt lie approximatio are: i) For ii) For y oy oy 1 β = + x y x ox ox β oy (3-13) ox (3-14) y oy ox 1 β = x + y x ox oy β For retagular setios wit reiforemet equally distributed o all four faes, te above equatios a be approximated by; For b 1 β oy = y + x (3-15) β y oy or y b x ox x were b ad are dimesios of te retagular olum setio parallel to x ad y axes, respetively. Usig te straigt lie approximatio equatios, te desig problem a be attaked by overtig te omial momets ito equivalet uiaxial momet apaities ox or oy. Tis is aomplised by; (a) assumig a value for b (b) estimatig te value of β as 0.65 () alulatig te approximate equivalet uiaxial bedig momet usig te appropriate oe of te above two equatios (d) oosig te trial setio ad reiforemet usig te metods for uiaxial bedig ad axial load. Te setio ose sould te be verified usig eiter te load otour or te reiproal load metod.
3.4 Colums Examples COLUNS EXAPLE 1 - Required area of steel for a retagular tied olum wit bars o four faes (slederess ratio foud to be below ritial value) For a retagular tied olum wit bars equally distributed alog four faes, fid area of steel. Give: Loadig P u = 560 kip ad u = 390 kip-i. Assume φ = 0.70 or, Nomial axial load P = 5600.70 = 800 kip Nomial momet = 3900.70 = 5600 kip-i. aterials Compressive stregt of orete f = 4 ksi Yield stregt of reiforemet f y = 60 ksi Nomial maximum size of aggregate is 1 i. Desig oditios Sort olum braed agaist sidesway. Proedure Determie olum setio size. Determie reiforemet ratio ρ g usig kow values of variables o appropriate iteratio diagram(s) ad ompute required ross setio area A st of logitudial reiforemet. A) Compute B) Compute P K = f R = f A g Calulatio Give: = 0 i. b = 16 i. P = 800 kip = 5600 kip-i. = 0 i. b = 16 i. A g = b x = 0 x 16 = 30 i. K = R = 800 4 ( )( 30) 5600 ( 4)( 30)( 0) = 0.65 = 0. - 5 0-5 C) Estimate γ γ = 0.75 0 D) Determie te appropriate For a retagular tied olum wit bars iteratio diagram(s) alog four faes, f = 4 ksi, f y = 60 ksi, ad a estimated γ of 0.75, use R4-60.7 ad R4-60.8. For k = 0.65 ad R = 0. E) Read ρ g for k ad R values from Read ρ g = 0.041 for γ = 0.7 ad appropriate iteratio diagrams ρ g = 0.039 for γ = 0.8 Iterpolatig; ρ g = 0.040 for γ = 0.75 F) Compute required A st from A st =ρ g A g Required A st = 0.040 30 i. = 1.8 i ACI 318-05 Setio 10. 10.3 Desig Aid Colums 3.. (R4-60.7) ad 3..3 (R4-60.8)
COLUNS EXAPLE - For a speified reiforemet ratio, seletio of a olum setio size for a retagular tied olum wit bars o ed faes oly For miimum logitudial reiforemet (ρ g = 0.01) ad olum setio dimesio = 16 i., selet te olum dimesio b for a retagular tied olum wit bars o ed faes oly. Give: Loadig P u = 660 kips ad u = 790 kip-i. Assume φ = 0.70 or, Nomial axial load P = 6600.70= 943 kips Nomial momet = 4000.70= 3986 kip-i. aterials Compressive stregt of orete f = 4 ksi Yield stregt of reiforemet f y = 60 ksi Nomial maximum size of aggregate is 1 i. Desig oditios Slederess effets may be egleted beause kl u is kow to be below ritial value Proedure Determie trial olum dimesio b orrespodig to kow values of variables o appropriate iteratio diagram(s). A) Assume a series of trial olum sizes b, i ies; ad ompute A g =b, i. 4 384 B) Compute 943 P K = ( 4)( 384) f = 0.61 3986 C) Compute ( 4)( 384)( 16) R = f A = 0.16 g Calulatio P = 943 kips, = 3986 kip-i. = 16 i. f = 4 ksi, f y = 60 ksi ρ g = 0.01 6 416 943 4 ( )( 416) = 0.57 3986 ( 4)( 416)( 16) = 0.14 8 448 943 4 ( )( 448) = 0.53 3986 ( 4)( 448)( 16) = 0.14 D) Estimate - 5 γ 0.7 0.7 0.7 D) Determie te appropriate iteratio For a retagular tied olum wit bars diagram(s) alog four faes, f = 4 ksi, f y = 60 ksi, ad a estimated γ of 0.70, use Iteratio Diagram L4-60.7 E) Read ρ g for k ad R values For γ = 0.018 0.014 0.011 0.7, selet dimesio orrespodig to ρ g earest desired value of ρ g = 0.01 Terefore, try a 16 x 8-i. olum ACI 318-05 Setio 10. 10.3 Desig Aid Colums 3.8. (L4-60.7)
COLUNS EXAPLE 3 - Seletio of reiforemet for a square spiral olum (slederess ratio is below ritial value) For te square spiral olum setio sow, selet reiforemet.. Give: Loadig P u = 660 kips ad u = 640 kip-i. Assume φ = 0.70 or, Nomial axial load P = 6600.70= 943 kips Nomial momet = 6400.70= 3771 kip-i. aterials Compressive stregt of orete f = 4 ksi Yield stregt of reiforemet f y = 60 ksi Nomial maximum size of aggregate is 1 i. Desig oditios Colum setio size = b = 18 i Slederess effets may be egleted beause kl u is kow to be below ritial value Proedure Determie reiforemet ratio ρ g usig kow values of variables o appropriate iteratio diagram(s) ad ompute required ross setio area A st of logitudial reiforemet. A) Compute P K = f B) Compute R = f A g Calulatio P = 943 kips = 3771 kip-i. = 18 i. b = 18 i. A g =b = 18 18=34 i. K = R = 943 4 ( )( 34) 3771 = 0.73 ( 4)( 30)( 18) = 0.16 C) Estimate - 5 18-5 γ γ = 0.7 18 D) Determie te appropriate For a square spiral olum, f = 4 ksi, iteratio diagram(s) f y = 60 ksi, ad a estimated γ of 0.7, use Iteratio Diagram S4-60.7 ad S4-60.8 E) Read ρ g for k ad R values. For k = 0.73 ad R = 0.16 ad, γ = 0.70: ρ g = 0.035 γ = 0.80: ρ g = 0.031 for γ = 0.7: ρ g = 0.034 A st = 0.034 30 i. = 1.8 i ACI 318-05 Setio 10. 10.3 Desig Aid Colums 3.0. (S4-60.7) ad 3.0.3 (S4-60.8)
COLUNS EXAPLE 4 - Desig of square olum setio subjet to biaxial bedig usig resultat momet Selet olum setio size ad reiforemet for a square olum wit ρ g 0.04 ad bars equally distributed alog four faes, subjet to biaxial bedig. Give: Loadig P u = 193 kip, ux = 1917 kip-i., ad uy = 769 kip-i. Assume φ = 0.65 or, Nomial axial load P = 1930.65= 97 kips Nomial momet about x-axis x = 19170.65= 949 kip-i. Nomial momet about y-axis y = 7690.65= 1183 kip-i. aterials Compressive stregt of orete f = 5 ksi Yield stregt of reiforemet f y = 60 ksi Nomial maximum size of aggregate is 1 i. Proedure Assume load otour urve at ostat P is a ellipse, ad determie resultat momet x from y r = x + A) Assume a series of trial olum sizes, i ies. Calulatio For a square olum: =b ( 949) + ( 1183) = 3177 r = kip-i. 14 16 18 B) Compute A g =, i. 196 56 34 C) Compute D) Compute K R f P = ()( 196) 97 5 = 0.30 3177 = ()( 5 196)( 14) f = 0.3 97 5 ()( 56) = 0.3 3177 ()( 5 56)( 16) = 0.16 97 5 ()( 34) = 0.18 3177 ()( 5 34)( 18) = 0.11 E) Estimate - 5 γ 0.64 0.69 0.7 F) Determie te appropriate iteratio diagram(s) E) Read ρ g for R ad k values, For γ = 0.60, For γ = 0.70, ad For γ = 0.80 Iterpolatig for γ i step E For a retagular tied olum wit f = 5 ksi, f y = 60 ksi. Use Iteratio Diagrams R5-60.6, R5-60.7, ad R5-60.8. 0.064 0.030 0.01 0.048 0.06 0.011 0.058 0.06 0.01 Terefore, try = 15 i. ACI 318-05 Setio Desig Aid Colums 3.3.1 (R5-60.6), 3.3. (R5-60.7), ad 3.3.3 (R5-60.8)
Determie reiforemet ratio ρ g usig kow values of variables o appropriate iteratio diagram(s) ad ompute required ross setio area A st of logitudial reiforemet. A) Compute P K = f B) Compute R = f A g A g = = (15) = 5 i. P = 97 kip r =3177 kip-i. K = R = 97 5 ()( 5) 3177 = 0.64 ()( 5 5)( 15) = 0.188 C) Estimate - 5 15-5 γ γ = 0. 67 15 D) Determie te appropriate For a retagular tied olum wit f = 5 iteratio diagram(s) ksi, f y = 60 ksi, ad γ = 0.67. Use E) Read ρ g for k ad R values from appropriate iteratio diagrams F) Compute required A st from A st =ρ g A g ad add about 15 peret for skew bedig Iteratio R5-60.6 ad R5-60.7. For k = 0.64, R = 0.188, ad γ = 0.60: ρ g = 0.043 γ = 0.70: ρ g = 0.034 for γ = 0.67: ρ g = 0.037 Required A st = 0.037 5 i. = 8.6 i Use A st 9.50 i. 10. 10.3 Colums 3.3.1 (R5-60.6) ad 3.3. (R5-60.7)
COLUNS EXAPLE 5 - Desig of irular spiral olum setio subjet to very small desig momet For a irular spiral olum, selet olum setio diameter ad oose reiforemet. Use relatively ig proportio of logitudial steel (i.e., ρ g = 0.04). Note tat k l u is kow to be below ritial value.. Give: Loadig P u = 940 kips ad u = 480 kip-i. Assume φ = 0.70 or, Nomial axial load P = 9400.70= 1343 kips Nomial momet = 4800.70=686 kip-i.. aterials Compressive stregt of orete f = 5 ksi Yield stregt of reiforemet f y = 60 ksi Nomial maximum size of aggregate is 1 i. Desig oditio Slederess effets may be egleted beause kl Ρ u is kow to be below ritial value Proedure Determie trial olum dimesio b orrespodig to kow values of variables o appropriate iteratio diagram(s). Calulatio P = 1343 kips, = 686 kip-i. f = 5 ksi f y = 60 ksi ρ g = 0.04 A) Assume a series of trial olum 1 sizes b, i ies; ad ompute A g =π(), i. 113 B) Compute R 686 = ()( 5 113)( 1) f = 0.101 16 01 686 ()( 5 01)( 16) = 0.043 0 314 686 ()( 5 314)( 0) = 0.01 C) Estimate - 5 γ 0.64 0.69 0.7 D) Determie te appropriate iteratio diagram(s) E) Read R ad ρ g values, after iterpolatio F) Compute G) Compute = P A g f k A g =, i. π For a irular olum wit f = 5 ksi, f y = 60 ksi. Use Iteratio Diagrams C5-60.6, C5-60.7, C5-60.7 ad C5-60.8. 0.90 1.14 1.3 1.5 0.90 1.14 1.4, i. 98 36 17 19.5 17.3 16.6 Terefore, try 17 i. diameter olum ACI 318-05 Setio Desig Aid Colums 3.15.1 (C5-60.6), 3.15. (C5-60.7), ad 3.15.3 (C5-60.8)
Determie reiforemet ratio ρ g usig kow values of variables o appropriate iteratio diagram(s) ad ompute required ross setio area A st of logitudial reiforemet. A) Compute P K = f B) Compute R = f A g = π K = R = 17 ()( 5 7) = 7 i. 1343 = 1.18 686 ()( 5 7)( 17) = 0.0356 C) Estimate - 5 17-5 γ γ = 0. 71 17 D) Determie te appropriate For a irular olum wit f = 5 ksi ad iteratio diagram(s) f y = 60 ksi. Use Iteratio C5-60.7. E) Read ρ g for k ad R values from For k = 1.18, R = 0.0356, ad appropriate iteratio diagrams γ = 0.71: ρ g = 0.040 F) Compute required A st from A st =ρ g A g Required A st = 0.040 7 i. = 9.08 i Colums Colums 3.15. (C5-60.7)