The degree of slenderness in a column is expressed in terms of "slenderness ratio," defined below:

Transcription

1 Chapter 4 Desin of Colmns By rat Saatiol Introdtion The majority of reinfored onrete olmns in pratie are sbjeted to very little seondary stresses assoiated with olmn deformations. These olmns are desined as short olmns sin the olmn interation diarams presented in Chapter 3. Rarely, when the olmn heiht is loner than typial story heiht and/or the olmn setion is small relative to olmn heiht, seondary stresses beome sinifiant, espeially if end restraints are small and/or the olmns are not braed aainst side sway. These olmns are desined as "slender olmns." Fi. 3.1 eloqently illstrates the seondary moments enerated in a slender olmn by P-Δ effets. olmns resist lower axial loads than short olmns havin the same ross-setion. Therefore, the slenderness effet mst be onsidered in desin, over and above the setional apaity onsiderations inorporated in the interation diarams. The sinifiane of slenderness effet is expressed throh slenderness ratio. 4. ness Ratio The deree of slenderness in a olmn is expressed in terms of "slenderness ratio," defined below: ness Ratio: kl / r where, l is nspported olmn lenth; k is effetive lenth fator refletin the end restraint and lateral brain onditions of a olmn; and r is the radis of yration, refletin the size and shape of a olmn ross-setion Unspported Lenth, l The nspported lenth l of a olmn is measred as the lear distane between the nderside of the beam, slab, or olmn apital above, and the top of the beam or slab below. The nspported lenth of a olmn may be different in two orthoonal diretions dependin on the spportin elements in 1 Professor and University Researh Chair, Dept. of Civil Enineerin, University of Ottawa, Ottawa, CANADA 1

2 respetive diretions. Fire 4.1 provides examples of different spport onditions and orrespondin nspported lenths ( l ). Eah oordinate and sbsript x and y in the fire indiates the plane of the frame in whih the stability of olmn is investiated. l l l x l l y Fi. 4.1 Unspported olmn lenth, l 4.. Effetive Lenth Fator, k The effetive lenth fator k reflets the end restraint (spport) and lateral brain onditions of a olmn relative to a pin-ended and laterally braed "referene olmn." The referene olmn, shown in Fi. 4.(a), follows a half sine wave when it bkles, and is assined a k fator of 1.0. Therefore, the effetive lenth kl for this olmn is eqal to the nspported olmn lenth l. A olmn with flly

3 restrained end onditions develops the defleted shape illstrated in Fi. 4.(b). The portion of the olmn between the points of ontraflexre follows a half sine wave, the same defleted shape as that of the referene olmn. This sement is eqal to 50% of the nspported olmn lenth l. Therefore, the effetive lenth fator k for this ase is eqal to 0.5. Effetive lenth fators for olmns with idealized spports an be determined from Fi. 4.. It may be of interest to note that k varies between 0.5 and 1.0 for laterally braed olmns, and 1.0 and for nbraed olmns. A disssion of lateral brain is provided in Se. 4.3 to establish whether a iven olmn an be onsidered to be as part of a sway or a non-sway frame. kl = l l kl = 0. 5l kl l l l kl 0l =. kl = l Fi. 4. Effetive Lenth Fator k for Colmns 3

4 ost olmns have end restraints that are neither perfetly hined nor flly fixed. The deree of end restraint depends on the stiffness of adjoinin beams relative to that of the olmns. Jakson and oreland alinment harts, iven in Colmns 4.1 and 4. an be sed to determine the effetive lenth fator k for different vales of relative stiffnesses at olmn ends. The stiffness ratios ψ A and ψ B sed in Colmns 4.1 and 4. shold reflet onrete rakin, and the effets of sstained loadin. Beams and slabs are flexre dominant members and may rak sinifiantly more than olmns whih are ompression members. The reded stiffness vales reommended by ACI are iven in Colmns 4.3, and shold be sed in determinin k. Alternatively, Colmns 4.4 may be sed to establish onservative vales of k for braed olmns Radis of Gyration, r The radis of yration introdes the effets of ross-setional size and shape to slenderness. For the same ross-setional area, a setion with hiher moment of inertia prodes a more stable olmn with a lower slenderness ratio. The radis of yration r is defined below. r = I A (4-1) It is permissible to se the approximations of r = 0.3h for sqare and retanlar setions, and r = 0.5h for irlar setions, where h is the overall setional dimension in the diretion stability is bein onsidered. This is shown in Fi Fi. 4.3 Radis of yration for irlar, sqare and retanlar setions 4.3 Lateral Brain and Desination of Frames as Non-Sway A frame is onsidered to be "non-sway" if it is sffiiently braed by lateral brain elements like strtral walls. Otherwise, it may be desinated as a "sway" frame. Frames that provide lateral resistane only by olmns are onsidered to be sway frames. Strtral walls that appear in the form of elevator shafts, stairwells, partial bildin enlosres or simply sed as interior stiffenin elements provide sbstantial drift ontrol and lateral brain. In most ases, even a few strtral walls may be sffiient to brae a mlti-storey mlti-bay bildin. The desiner an sally determine whether the frame is non-sway or sway by inspetin the floor plan. Frames with lateral brain elements, where the total lateral stiffness of the brain elements provides at least six times the smmation of the stiffnesses of all the olmns, may be lassified as non-sway. ACI permits olmns to be desined as part of a non-sway frame if the inrease in olmn end moments de to seond-order Conrete Desin Handbook, Cement Assoiation of Canada, third edition, 60 Qeen Street, Ottawa, ON., Canada, K1P 5Y7,

5 effets does not exeed 5% of the first-order end moments (Se ). Alternatively, Setion of ACI defines a stability index "Q" (iven in Eq. 4.), where, Q 0.05 indiates a non-sway olmn. P Δo Q = (4.) V l s Where, P is total fatored axial load atin on all the olmns in a story, V s is total fatored story shear, Δ o is lateral story drift (defletion of the top of the story relative to the bottom of that story) de to V s. The story drift Δ o shold be ompted sin the modified EI vales iven in Colmns 4.3 with β d defined as the ratio of the maximm fatored sstained shear within a story to the maximm fatored shear in that story. If Q exeeds approximately 0., the strtre may have to be stiffened laterally to provide overall strtral stability. 4.4 Desin of Colmns Desin of a slender olmn shold be based on a seond-order analysis whih inorporates member rvatre and lateral drift effets, as well as material non-linearity and sstained load effets. An alternative approah is speified in ACI for olmns with slenderness ratios not exeedin 100. This approah is ommonly referred to as the "oment anifiation ethod," and is based on manifyin the end moments to aont for seondary stresses. The appliation of this proedre is otlined in the followin setions Colmns in Non-Sway Frames ness effets may be neleted for olmns in non-sway frames if the followin ineqality is satisfied: kl 34 1( 1 / ) (4-3) r Where 34 1 / ) 40 (4-4) ( 1 1 / is the ratio of smaller to larer end moments. This ratio is neative vale when the olmn is bent in doble rvatre and positive when it is bent in sinle rvatre. Fi. 4.4 illstrates olmns in doble and sinle rvatres. Colmns in non-sway frames are more stable when they bend in doble rvatre, with smaller seondary effets, as ompared to bendin in sinle rvatre. This is refleted in Eq. (4-3) throh the sin of 1 / ratio. For neative vales of this ratio the limit of slenderness in Eq. (4-3) inreases, allowin a wider rane of olmns to be treated as short olmns. olmns in non-sway frames are desined for fatored axial fore P and amplified moment. The amplified moment is obtained by manifyin the larer of the two end moments to aont for member rvatre and resltin seondary moments between the spports, while the spports are braed aainst sidesway. If ompted for the rvatre effet between the ends is smaller than the larer end moment, the desin is arried ot for. = δ (4-5) ns 5

6 The ritial olmn load, P (Eler bklin load) is; Cm δ ns = 1.0 (4-6) P P P π EI = (4-7) ( kl ) Fi. 4.4 Colmns in Sinle and Doble Crvatre EI in Eq. (4-7) is ompted either with de onsiderations iven to the presene of reinforement in the setion, as speified in Eq. (4-8), or approximately sin Eq. (4-9). EI 0.EI + EsI se = (4-8) 1 + β d where β d is the ratio of the maximm fatored axial dead load to the total fatored axial load. The moment of inertia of reinforement abot the ross-setional entroid (I se ) an be ompted sin Colmns E I EI = 1 + β (4-9) d Note that Eq. (4-9) an be simplified frther by assmin β d = 0.6, in whih ase the eqation beomes; EI = 0.5E I. 6

7 Coeffiient C m is eqal to 1.0 for members with transverse loads between the spports. For the more ommon ase of olmns withot transverse loads between the spports; C m 1 = (4-10) Where, 1 / is positive if the olmn is bent in sinle rvatre. When the maximm fatored end moment is smaller than the minimm permissible desin moment,min, speified in Eq. (4-11), the manifiation applies to,min. P ( h ) (4-11),min + where h is the ross-setional dimension in inhes in the diretion of the eentriity of load. For olmns for whih,min is hiher than, the vales of C m, in Eq. (4-10) shold either be taken 1.0 or determined based on the ompted ratio of end moments ( 1 / ). One the amplified moment is obtained, the desiner an se the appropriate interation diarams iven in Chapter 3 to determine the reqired perentae of lonitdinal reinforement Colmns in Sway Frames Colmns in sway frames are desined for the fatored axial load P and the ombination of fatored ravity load moments and manified sway moments. This is speified below, and illstrated in Fi ns + δ s 1s = (4-1) ns + δ s s = (4-13) where, 1ns and ns are end moments de to fatored ravity loads; and 1s and s are sway moments normally ased by fatored lateral loads. All of these moments an be obtained from a firstorder elasti frame analysis. anified sway moments δ s 1s and δ s s are obtained either from a seond order frame analysis, with member flexral riidity as speified in Colmnss 4.3, or by manifyin the end moments by sway manifiation fator δ s. The sway manifiation fator is allated either as iven in Eq. (4-14) or Eq. (4-15). s δ s s = s (4-14) P P 1 Q s δ s s = s (4-15) 7

8 However, if δ s ompted by Eq. (4-15) exeeds 1.5, δ s s shall be allated either throh seond order analysis or sin Eq. (4-14). Fi. 4.5 Desin moments in sway frames In a sway frame, all the olmns of a iven story partiipate in the sway mehanism, and play roles in the stability of individal olmns. Therefore, Eq. (4-14) inldes P and P whih ive the smmations of fatored axial loads and ritial loads for all the olmns in the story, respetively. The ritial olmn load P an be ompted sin Eqs. (4-7) throh (4-9) with the effetive lenth fator k ompted for nbraed olmns (for sway frames) and β d as the ratio of the maximm fatored 8

9 sstained shear within the story to the maximm total fatored shear in the story. Eq. (4-14) provides an averae δ s for all the olmns in a story. Therefore, it yields aeptable reslts if all the olmns in a story ndero the same story drift. When sinifiant torsion is antiipated nder lateral loadin, a seond order analysis is reommended for findin the amplified sway moment, δ s s. The manifiation of moments throh Eq. (4-15) is appliable only if the sway manifiation fator δ s does not exeed 1.5. If it does, then either the seond-order analysis or Eq. (4-14) shold be employed (Se ). The sidesway manifiation disssed above is intended to amplify the end moments assoiated with lateral drift. Althoh the amplified end moment is ommonly the ritial moment for most sway olmns, olmns with hih slenderness ratios may experiene hiher amplifiation of moments between the ends (rather than at the ends) bease of the rvatre of the olmn alon the olmn heiht. This is assmed to or when the ineqality iven in Eq. (4-16) is satisfied. l r > 35 P f ' A (4-16) The manifiation of moment de to the rvatre of olmn between the ends is similar to that for braed olmns in non-sway frames. Therefore, if Eq. (4-16) is satisfied for a olmn, then the olmn shold be desined for fatored axial fore P and manified desin moment ( ) ompted sin Eqs. (4-5) and (4-6), with 1 and ompted from Eqs. (4-1) and (4-13). Sometimes olmns of a sway frame may bkle nder ravity loads alone, withot the effets of lateral loadin. In this ase one of the ravity load ombinations may overn the stability of olmns. The redtion of EI nder sstained ravity loads may be another fator ontribtin to the stability of sway olmns nder ravity loads. Therefore, ACI reqires an additional hek to safeard aainst olmn bklin in sway frames nder ravity loads alone (Se ). Aordinly, the strenth and stability of strtre is reonsidered dependin on the method of amplifiation sed for sway moments. If a seond order analysis was ondted to find δ s s, two additional analyses are neessary sin the reded stiffness vales iven in Colmns 4.3 with β d taken as the ratio of the fatored sstained axial dead load to total fatored axial load. First, a seond-order analysis is ondted nder ombined fatored ravity loads and lateral loads eqal to 0.5% of the ravity loads. Seond, a first-order analysis is ondted nder the same loadin ondition. The ratio of lateral drift obtained by the seond-order analysis to that obtained by the first-order analysis is reqired to be limited to.5. If the sway moment was amplified by omptin the sway manifiation fator iven in Eq. 4.14, as opposed to ondtin seond order analysis or sin Eq. (4-15), then δ s ompted by sin the ravity loads ( P and P orrespondin to the fatored dead and live loads) is reqired to be positive and less than or eqal to.5 to ensre the stability of the olmn. If the sway moment was amplified sin Eq. (4-15), then the vale of Q ompted sin P for fatored dead and live loads shold not exeed

10 4.5 Colmn Desin Examples SLENDER COLUN EXAPLE 1 - Desin of an interior olmn braed aainst sidesway. Consider a 10-story offie bildin, laterally braed aainst sidesway by an elevator shaft (Q is ompted to be mh less than 0.05). The bildin has an atrim openin at the seond floor level with a two-story hih olmn in the openin to be desined. Desin the olmn for the nfatored desin fores iven below, obtained from a first-order analysis. The framin beams are 16 in wide and 0 in deep with 3 ft (anter-to-entre) spans. The beam depth inldes a slab thikness of 6 in. The story heiht is 14 ft (olmn heiht is 8 ft). It is assmed that the brain elements provide fll resistane to lateral fores and the olmns only resist the ravity loads. Start the desin with an initial olmn size of 0 in sqare. f = 6,000 psi for all beams and olmns; f y = 60,000 psi. Unfatored Loads Dead Load Live Load Axial load: 50 k 410 k Top moment: k-in -60 k-in Bottom moment: -848 k-in -540 k-in Note: oments are positive if onterlokwise at olmn ends. The olmn is bent in doble rvatre. l l l l l Colmn Example 1 10

11 Proedre Determine fatored desin fores: Note: 1 is the lower and is the hiher end moment. Callation i) U = 1.4D P = 1.4 P D = 1.4 (50) = 78 k = 1.4 D = 1.4 (1018) = 145 k-in 1 = 1.4 D1 = 1.4 (848) = 1187 k-in ACI Setion 9. Desin Aid ii) U = 1. D L P = 1. P D P L = 1. (50) (410) = 180 k = 1. D L = 1. (1018) (60) = 14 k-in 1 = 1. D L1 = 1. (848) (540) = 188 k-in Callate slenderness ratio kl / r i) Find nspported olmn lenth ii) Find the radis of yration iii) Find effetive lenth fator "k." This reqires the allation of stiffness ratios at the ends. First find beam and olmn stiffnesses. Note: Load Combination (ii) overns the desin. l = 8 0/1 = 6.3 ft r = 0.3 h = 0.3 (0) = 6 in (I ) beam = (I ) T-beam = 19,57 in 4 (I ) olmn = bh 3 /1=(0)(0) 3 /1 = 13,333 in 4 Craked (reded) EI vales: (EI) beam = (1,545)(19,57) = 30x10 6 k-in (EI) ol = (3,091)(13,333) = 41x10 6 k.in Fire 4.1 Cols. 4.3 (EI/ l ) beam = (30x10 6 ) / (3x1) = 109x10 3 k-in for both left and riht beams (EI/ l ) ol = (41x10 6 / (8x1) = 1x10 3 k-in for the atrim olmn to be desined. (EI/ l ) ol = (41x10 6 ) / (14x1) = 44x10 3 k-in for olmns above and below Ψ = (ΣEI/ l ) ol / (ΣEI/ l ) beam Ψ = [(EI/ l ) ol, above + (EI/ l ) ol, below ] / [(EI/ l ) beam, left + (EI/ l ) beam, riht ] Read k from Colmns 4.1 Ψ A = (44 + 1)x10 3 /( )x10 3 Ψ A = 1.7 = Ψ B for Ψ B = Ψ A = 1.7; selet k = 0.83 from Colmns 4.1 (Note that Colmns 4.4 ives a Cols. 4.1 Col

12 Compte the slenderness ratio Chek if slenderness an be neleted sin Eq.(4-3): Apply the limit of Eq. (4-4) onservative vale of k = 0.90) l = = 6.3 ft k l / r = 0.83 (6.3x1) / 6 = 45 kl 34 1( 1 / ) r ( / ) 40 Note 1 / = - 188/14 = (Bendin in doble rvatre) or, for Load Combination I; 1 / = /145 = [34-1 (-0.85)] = 44 > 40 se 40 kl / r = 45 > 40 (limitin ratio for neletin slenderness) Compte moment manifiation fator (δ ns ) from Eq. (4-6): Therefore, onsider slenderness. Cm δ ns = 1.0 P P i) Compte ritial load P from Eq (4-7) Use Eq. (4-8) to ompte EI. Assme.5% olmn reinforement, eqally distribted alon the perimeter of the sqare setion with γ = 0.75 where γ is the ratio of the distane between the entres of the otermost bars to the olmn dimension perpendilar to the axis of bendin. P π EI = ( kl ) E = 4415 ksi E s = 9,000 ksi (I ) olmn = 13,333 in 4 I se = 0.18 ρ t b h 3 γ (from Col. 4.5) Cols. 4.3 Cols. 4.5 I se = 0.18(0.05)(0)(0) 3 (0.75) = 405 in 4 β d = 1.D / (1.D + 1.6L) = 1. x 50 / (1. x x 410) = 64 / 180 = EI = (0.E I + E s I se ) / (1 + B d ) Alternatively, ompte EI from Eq,(4-9) Eq. (4-9) may frther be simplified by assmin a vale of β d = EI EI = = 0.5EI 1+ β d EI = [(0. x 4415 x 13,333) + (9,000 x 405)] / ( ) = 16 x 10 6 k-in EI = (0.4 x 4415 x 13,333) / ( ) EI = 16 x 10 6 k-in EI = 0.5 E I = 0.5 x 4415 x 13,333 EI = 15 x 10 6 k-in R

13 ii) Compte C m from Eq. (4-10): P = π EI / (k l ) P = π x 16 x 10 6 / ( 0.83 x 6.3 x 1) P = 301 k C m = / 0.4 C m = (-0.85) = 0.5 < 0.4 se iii) oment manifiation fator Compte amplified moment from Eq. (4-5) Chek aainst minimm desin moment as per Eq. (4-11). δ ns 0.4 = = (0.75 )( 301) = δ = 1.55 (14) = 343 k-in ns,min P ( h ) ,min = 180 ( x 0) =1536 k-in Selet reinforement ratio and desin the olmn setion: Use Colmn Interation Diarams R and R for eqal reinforement on all sides and interpolate for γ = 0.75 (assmed above) A) Compte B) Compte K n= P f n ' A n R n= ' f A h C) Read ρ for K n and R n vales from the interation diarams D) Compte reqired A st from A st = ρ A E) Find olmn reinforement = 343 k-in >,min = 1536 k-in Desin for = 343 k-in Note: γ = 0.75 allows for more than 1.5 in lear over reqired for interior olmns, not exposed to weather. P / φ 180 / 0.65 K n= = = 0.8 ' f A (6 )( 0 ) n / φ 343 / 0.65 R n= = = 0.11 ' f A h (6 )( 0 ) ( 0 ) For K n = 0.8 and R n = 0.11 Read ρ = for γ = 0.7 and ρ = 0.09 for γ = 0.8 Interpolatin; ρ = for γ = 0.75 (Note that the reqired steel ratio of 3% is slihtly hiher that the.5% assmed for omptin EI. No revision is neessary). Reqired A st = x 400 in. = 1.0 in. Try # 9 bars; 1.0 / 1.0 = 1.0 Use 1 # 9 Bars Flexre 9 Colmns R and R Colmns R and R

14 SLENDER COLUN EXAPLE - Desin of an exterior olmn in a sway frame. A typial floor plan and a setion throh a mlti-story offie bildin are shown below. Desin olmn 3-A at the rond level for ombined ravity and east-west wind loadin. The reslts of first-order frame analysis nder fatored load ombinations are iven in the soltion. f' = 6,000 psi; f y = 60,000 psi. Colmn Example 14

15 Proedre Consider the appliable load ombinations: The strtre is not braed aainst sidesway. Therefore, the olmn will be desined onsiderin the loads that ase sidesway. Note that sidesway in this strtre is ased by wind loadin. No sinifiant sidesway is antiipated de to ravity loads sine the strtre is symmetri. However, the possibility of sidesway instability nder ravity loads alone shall be investiated as per Se Callation a)load ombinations that inlde wind; Comb. I: U = 1.D + 1.6L r + 0.8W Comb. II: U = 1.D + 1.6W + 1.0L + 0.5L r Comb.III: U = 0.9D + 1.6W b) Load ombinations for ravity loads; Comb. IV:U = 1.4D Comb. V: U = 1.D + 1.6L + 0.5L r Comb. VI: U = 1.D+ 1.6 L r + 1.0L ACI Setion Desin Aid Usin the preliminary olmn setion iven in the fire, determine the effetive lenth fator k for eah olmn at the rond level. This reqires the omptation of beam and olmn stiffnesses. Note: All olmns have the same setion. I beam = 87,040 in 4 (for T-setion) I ol = (0)(0) 3 /1 = 13,333 in 4 Find reded EI vales from Col. 4.3 for 6.0 ksi onrete; (E I) beam = 1545 I beam = (1545)(87,040) = 134 x 10 6 k-in (E I) ol = 3091 I ol = (3091)(13,333) = 41 x 10 6 k-in (EI/ l ) beam = 134 x 10 6 / ( x 1) = 507,576 k-in (EI/ l ) ol, typial = 41 x 10 6 / (10 x1) = 341,667 k-in (EI/ l ) ol, atrim = 41 x 10 6 / (18 x1) = 189,815 k-in Cols. 4.3 Cols. 4.3 Fator k reflets olmn end restraint onditions and depends on relative stiffnesses of olmns to beams at top and bottom joints. Ψ = (3EI/ l ) ol / (3EI/ l ) beam Ψ = [(EI/ l ) ol, above + (EI/ l ) ol, below ] / [(EI/ l ) beam, left + (EI/ l ) beam, riht ] Read k from Colmns 4. and 4.1. i) For exterior olmns (olmns on lines A and D): Ψ A = (341, ,815) / 507,576 = 1.05 Ψ B = Ψ A = 1.05; from Cols. 4.: k = 1.35 (for nbraed frames) k = 0.78 for a braed olmn, from Cols This vale is ompted for frther manifiation of moments, if neessary for olmn 3-A as per Se Cols. 4. Cols

16 Compte the slenderness ratio ii) For interior olmns (olmns on lines B and C): Ψ A = (341, ,815)/(507, ,576) = 0.5 Ψ B = Ψ A = 0.5; from Cols. 4.; k = 1.15 (for nbraed frames) Cols. 4. Compte ritial load P from Eq. 4.7 and EI from either Eq. 4.8 or 4.9. Note, if Eq. 4.9 is sed for simpliity with β d = 0 (sine wind loadin is a short term load) 0.4EI EI = = 0.4EI 1+ β d i) For exterior olmns (olmns on lines A and D): E = 4415 ksi for f' = 6 ksi For sway olmns; EI = 0.4E I = 0.4(4415)(13,333) = 3.5 x 10 6 k-in l = (18) (1) - 3 = 184 in P = π EI/(k l ) = π (3.5x10 6 )/(1.35x184) = 3759 kips for a sway frame Cols. 4.3 For braed olmns, Eq. 4.9 an be simplified by sbstittin β d = 0.6. Then; EI = 0.5 E I For braed olmns; EI = 0.5E I = 0.5(4415)(13,333) = 14.7 x 10 6 k-in P for braed olmns may be needed if frther manifiation of moments is reqired as per Se P = π EI/(k l ) = π (14.7x10 6 )/(0.78x184) = 7044 kips for braed olmns. ii) For interior olmns (olmns on lines B and C): P = π EI / (k l ) = π (3.5 x 10 6 ) / (1.15 x 184) = 5180 kips for a sway frame. ΣP = 10 (3759) + 10 (5180) = 89,390 kips Compte manified sway moment δ s s Under Load Combination I. Condt first-order frame analysis sin Load Combination I, and the stiffness vales speified in Cols Note: Conterlokwise moment at olmn end is positive. i) Load Comb. I: U = 1.D + 1.6L r + 0.8W Load 1.D +1.6L r 0.8W P (kips) 45 ±1 Corner Colmn P (kips) 68 ±1 Ede Colmn P (kips) 1134 ±4 Interior Colmn ( ) top (k-in) -196 ±765 Colmn 3-A ( ) bot (k-in) Colmn 3-A -196 ±

17 Compte sway manifiation fator δ s from Eq This reqires the omptation of ΣP f in addition to ΣP Obtained in the previos step. Sway manifiation fator δ s : ΣP f = 4 (45 + 1) + 10 (68 + 1) + 6 ( ) = 15,516 kips δ s = 1 / [1 - ΣP f / [(0.75)ΣP ] = 1/[1 15,516 /(0.75 x 89,390)] = δ s 1s = 1.30 x 765 = 995 k-in δ s s = 1.30 x 1111 = 1444 k-in Compte desin moments 1 and 1 = 1ns +δ s 1s = = 91 k-in = ns +δ s s = = 738 k-in Fi. 4.5 Chek if frther manifiation of moments is reqired for Colmn 3-A de to the rvatre of olmns between the ends as per Se Compte manified sway moment δ s s Under Load Combination II. l > r 35 P /(f' A P = = 694 k l /r = 184 / (0.3 x 0) = / 694/(6x(0) ) = 65.1 > 30.7 Therefore, no frther manifiation is reqired. ii) Load Combination II: U=1.D+1.6W+1.0L+0.5L r ) Condt first-order frame analysis sin Load Combination I, and the stiffness vales speified in Cols Note: Conterlokwise moment at olmn end is positive. Load P (kips) Corner Colmn P (kips) Ede Colmn P (kips) Interior Colmn ( ) top (k-in) Colmn 3-A ( ) bot (k-in) Colmn 3-A 1.D +1.0L 1.6W + 0.5L r 493 ±4 845 ± ± ± ± Compte sway manifiation fator δ s from Eq This reqires the omptation of ΣP f in addition to ΣP Obtained in the previos step. Sway manifiation fator δ s : ΣP f = 4 ( ) + 10 ( ) + 6 ( ) = 19,560 kips δ s = 1 / [1 - ΣP f / (0.75ΣP )] = 1/[1 19,560 /(0.75 x 89,390)] = 1.41 δ s 1s = 1.41 x 1530 = 157 k-in δ s s = 1.41 x = 3111 k-in

18 Compte desin moments 1 and Chek if frther manifiation of moments is reqired for Colmn 3-A de to the rvatre of olmns between the ends as per Se Compte manified sway moment δ s s Under Load Combination II. Condt first-order frame analysis sin Load Combination I, and the stiffness vales speified in Cols Note: Conterlokwise moment at olmn end is positive. Compte sway manifiation fator δ s from Eq This reqires the omptation of ΣP f in addition to ΣP Obtained in the previos step. Compte desin moments 1 and Chek if frther manifiation of moments is reqired for Colmn 3-A de to the rvatre of olmns between the ends as per Se = 1ns +δ s 1s = = 3913 k-in = ns +δ s s = = 4867 k-in l > r 35 P /(f' A ) P = = 869 k l /r = 184 / (0.3 x 0) = / 869/(6x(0) ) = 58. > 30.7 Therefore, no frther manifiation is reqired. iii) Load Comb. III: U = 0.9D + 1.6W Load 0.9D 1.6W P (kips) 58 ±4 Corner Colmn P (kips) 45 ±4 Ede Colmn P (kips) 790 ±8 Interior Colmn ( ) top (k-in) -97 ±1530 Colmn 3-A ( ) bot (k-in) Colmn 3-A -97 ± Sway manifiation fator δ s : ΣP f = 4 (58 + 4) + 10 (45 + 4) + 6 ( ) = 10,676 kips δ s = 1 / [1 - ΣP f / (0.75ΣP )] = 1/[1 10,676 /(0.75 x 89,390)] = 1.19 δ s 1s = 1.19 x 1530 = 181 k-in δ s s = 1.19 x = 644 k-in 1 = 1ns +δ s 1s = = 793 k-in = ns +δ s s = = 3616 k-in l > r 35 P /(f' A P = = 476 k l /r = 184 / (0.3 x 0) = / 476/(6x(0) ) = 78.6 > 30.7 Therefore, no frther manifiation is reqired. ) Fi Fi

19 Chek the stability of olmn nder ravity loads only (Load ombinations IV, V and VI) as per Se Consider fatored axial loads and bendin moments obtained from a firstorder frame analysis, ondted sin the flexral riidities iven in Cols. 4.3 Note: Conterlokwise moment at olmn end is positive. iv) Load Comb. IV: U = 1.4D Load 1.4D P (kips) 40 Corner Colmn P (kips) 703 Ede Colmn P (kips) 19 Interior Colmn ( ) top kip-in -151 Colmn 3-A ( ) bot kip-in -151 Colmn 3-A Cols. 4.3 Compte the sway manifiation fator δ s Critial load, from earlier allation. Sway manifiation fator δ s : ΣP f = 4 (40) + 10 (703) + 6 (19) = 16,01 kips ΣP = 10 (3759) + 10 (5180) = 89,390 kips δ s = 1 / [1 - ΣP f / (0.75ΣP )] = 1 / [1 16,01 / (0.75 x 89,390)] = Chek the stability of olmn nder Load ombination V as per Se Consider fatored axial loads and bendin moments obtained from a firstorder frame analysis, ondted sin the flexral riidities iven in Cols. 4.3 Note: Conterlokwise moment at olmn end is positive. δ s = 1.31 <.5 O.K. v) Load Comb. V:U =1.D + 1.6L + 0.5L r Load P (kips) Corner Colmn P (kips) Ede Colmn P (kips) Interior Colmn ( ) top kip-in Colmn 3-A ( ) bot kip-in Colmn 3-A 1.D + 1.6L + 0.5L r Cols. 4.3 Compte the sway manifiation fator δ s Critial load, from earlier allation. Sway manifiation fator δ s : ΣP f = 4 (568) + 10 (976) + 6 (1687) =,154 kips ΣP = 10 (3759) + 10 (5180) = 89,390 kips δ s = 1 / [1 - ΣP f / (0.75ΣP )] = 1 / [1,154 / (0.75 x 89,390)] = 1.49 δ s = 1.49 <.5 O.K

20 Chek the stability of olmn nder Load ombination VI as per Se Consider fatored axial loads and bendin moments obtained from a firstorder frame analysis, ondted sin the flexral riidities iven in Cols. 4.3 Note: Conterlokwise moment at olmn end is positive. vi) Load Comb. VI: U=1.D+1.6 L r + 1.0L Load 1.D+ 1.6 L r + 1.0L P (kips) 548 Corner Colmn P (kips) 900 Ede Colmn P (kips) 1514 Interior Colmn ( ) top kip-in Colmn 3-A ( ) bot kip-in Colmn 3-A Cols. 4.3 Compte the sway manifiation fator δ s Sway manifiation fator δ s : ΣP f = 4 (548) + 10 (900) + 6 (1514) = 0,76 kips ΣP = 10 (3759) + 10 (5180) = 89,390 kips Critial load, from earlier allation. δ s = 1 / [1 - ΣP f / (0.75ΣP )] = 1 / [1 0,76 / (0.75 x 89,390)] = 1.43 δ s = 1.43 <.5 O.K. Desin the Colmn for the overnin load ombination. Note: Conterlokwise moment at olmn end is positive. Selet the interation diarams iven in Colmns from Chapter 3 for eqal reinforement on all sides for γ = 0.80 (assmed) Compte; Compte; K n= R n= P f n ' A n ' f A h Smmary of Desin Loads: Load Combinations P (kn) ( ) (kn.m) I - U = 1.D + 1.6L r + 0.8W II - U =1.D+1.6W +1.0L+0.5L r III - U = 0.9D + 1.6W IV - U = 1.4D V - U =1.D + 1.6L + 0.5L r VI - U=1.D+1.6 L r + 1.0L For Load Combination II; P /φ 845/0.65 K n= = = 0.54 ' f A (6)(0) n /φ 4867/0.65 = = ' f A h (6)(0) (0) R n = 0.16 Colmns

21 Read ρ for K n and R n vales from the interation diarams For K n = 0.54 and R n = 0.16 Read ρ = Reqired A st = x 400 in. = 1.0 in. Try # 9 bars; 1.0 / 1.0 = 1.0 Try 1 # 9 Bars. Chek for Load Combination V; P /φ = ' f A 976/0.65 = (6)(0) K n = n /φ 03/0.65 = = ' f A h (6)(0) (0) 0.63 R n = For K n = 0.63 and R n = ρ = is sffiient Therefore, se 1 # 9 Bars Note: For frther details of ross-setional desin refer to Chapter 3. Colmns Colmns

22 4.6 Colmn Desin Aids Colmn Effetive Lenth Fator Jakson and oreland Alinment Chart for Colmns in Braed (Non-Sway) Frames 3 3 Gide to Desin Criteria for etal Compression embers, nd Edition, Colmn Researh Conil, Fritz Enineerin Laboratory, Lehih University, Bethlehem, PA, 1966

23 Colmns - 4. Effetive Lenth Fator Jakson and oreland Alinment Chart for Colmns in Unbraed (Sway) Frames 4 4 Gide to Desin Criteria for etal Compression embers, nd Edition, Colmn Researh Conil, Fritz Enineerin Laboratory, Lehih University, Bethlehem, PA,

24 Colmns Reommended Flexral Riidities (EI) for se in First-Order and Seond Order Analyses of Frames for Desin of Colmns f (ksi) E (ksi) I/I E I / I (ksi) Beams Colmns Walls (Unraked) Walls (Craked) Flat Plates Flat Slabs Notes: 1. The above vales will be divided by (1+β d ), when sstained lateral loads at or for stability heks made in aordane with Setion of ACI For non-sway frames, β d is ratio of maximm fatored axial sstained load to maximm fatored axial load assoiated with the same load ombination, β d = 1.D / (1.D + 1.6L).. For sway frames, exept as speified in Setion of ACI , β d is ratio of maximm fatored sstained shear within a story to the maximm fatored shear in that story. 3. The above vales are appliable to normal-density onrete with w between 90 and 155 lb/ft The moment of inertia of a T-beam shold be based on the effetive flane width, shown in Flexre 6. It is enerally sffiiently arate to take I of a T-beam as two times the I for the web. 5. Area of a member will not be reded for analysis. 4

25 Colmn Effetive Lenth Fator k for Colmns in Braed Frames 5

26 Colmns oment of Inertia of Reinforement abot Setional Centroid 5 5 This table is based on Table 1-1 of agreor, J.G., Third Edition, Prentie Hall, Enlewood Cliffs, New Jersey,