3.6 Flexural, Design Example of Negative Moment Region

Transcription

1 3.0 CONCRETE STRUCTURES 3.1 Material Propertie 3.2 Fatigue Limit State 3.3 Strength Limit State 3.4 Flexure 3.5 Flexure Deign Example 3.6 Flexural, Deign Example o Negative Moment Region 3.7 Shear 3.8 Horizontal Interae Shear 3.9 Deign Example. AASHTO-PCI Bulb-Tee BT-72, Single Span With Compoite Dek, Deigned Aording to AASHTO LRFD Speiiation

2 3.0 CONCRETE STRUCTURES 3.1 Material Propertie. (LRFD Art. 5.4) Lo o pretre an be haraterized a that due to intantaneou lo and time-dependent lo. Loe due to anhorage et, rition and elati hortening are intantaneou. Loe due to reep, hrinkage and relaxation are time-dependent. Immediate loe beore traner Δ pt = Δ pes + Δ psr + Δ pcr + Δ pr2 Where: Δ pes = lo due to elati hortening (ki) Δ psr = lo due to hrinkage (ki) Δ pcr = lo due to reep o onrete (ki) Δ pr2 = lo due to relaxation o teel ater traner (ki) Notie that an additional lo our during the time between jaking o the trand and traner. Thi omponent i the lo due to the relaxation o teel at traner, Δ pr1 Three method o etimating time-dependent loe are provided in the LRFD Speiiation: (1) the approximate lump um etimate, (2) a reined etimate, and (3) the bakground neeary to perorm a rigorou time-tep analyi Creep (LRFD Art ) The Creep Coeiient or whih: t, t 1.9k k k k t (LRFD Eq ) i h td i V / S 1. 0 k (LRFD Eq ) k h H (LRFD Eq ) 5 k 1 i (LRFD Eq ) t ktd 1004 i 12 t i 20 (LRFD Eq ) k i ator or volume-to-urae ratio; k h i the humidity ator; k i the ator o onrete trength; k td i the development ator 3-1

3 3.1.2 Shrinkage (LRFD Art ) The train due to hrinkage: 3 k k k k (LRFD Eq ) h h td in whih: k h H (LRFD Eq ) Fig 3-1 Annual Average Ambient Relative Humidity in Perent (LRFD Fig ) Modulu o Elatiity & Rupture o Conrete (LRFD Art &6) Modulu o Elatiity E E 2.0 1w ' ,000K (or E = 2, ) (LRFD Eq ) For normal weight with w=0.145 k and up to 10 ki ompreive trength 1.5 E 33,000K w (or E = 1, ) (LRFD Eq. C ) 1 Modulu o Rupture For normal weight onrete: o Exept a peiied below o When ued to alulate the raking moment o a member in Artile

4 For light weight onrete: o For and-lightweight onrete o For all-lightweight onrete Pretreing Steel (LRFD Art ) Material Grade or Type Dia. IN Tenile Strength, pu (KSI) Yield Strength, py (KSI) Strand 250 KSI 270 KSI 1/4 to 0.6 3/8 to % o pu Bar Type 1, Plain Type 2, Deormed 3/4 to 1-3/8 5/8 to 1-3/ % o pu 80% o pu Table 3-1 Propertie o Pretreing Strand and Bar (LRFD Table ) 3.2 Fatigue Limit State (LRFD Art ) γ(δ) (ΔF) TH (LRFD Eq ) Reinoring Bar without a ro weld in the high-tre region: ( F) min (LRFD Eq ) TH y with a ro weld in the high-tre region: ( ) (LRFD Eq ) F TH min Pretreing Tendon 18ki or R>30 10ki or R<12 A linear interpretation may be ued or radii between 12 and 30 t. 3-3

5 3.3 Strength Limit State (LRFD Art ) Conventional Contrution: Reitane Fator o For tenion-ontrolled reinored onrete etion a peiied in Art o For tenion-ontrolled pretreed onrete etion a peiied in Art o For hear and torion: normal weight onrete lightweight onrete o For ompreion-ontrolled etion with piral or tie, a peiied in Artile , exept a peiied in Artile and b or Seimi Zone 2, 3 and 4 at the extreme event limit tate o For bearing on onrete o For ompreion in trut-and-tie model o For ompreion in anhorage zone: normal weight onrete lightweight onrete o For tenion in teel in anhorage zone o For reitane during pile driving o o For a pretreed member: l ) tl l 0.25 t (LRFD Eq ) ( For nonpretreed member: l ) tl l 0.15 t (LRFD Eq ) ( ε t = net tenile train in the extreme tenion teel ε l = ompreion-ontrolled train limit in the extreme tenion teel ε tl = tenion-ontrolled train limit in the extreme tenion teel 3-4

6 Segmental Contrution (LRFD Table ) Flexure Normal Weight Conrete Fully Bonded Tendon Unbonded or Partially Bonded Tendon Sand-Lightweight Conrete Fully Bonded Tendon Unbonded or Partially Bonded Tendon v Shear Table 3-2 Reitane Fator or Joint in Segmental Contrution. 3.4 Flexure (PCI 8.2) Stage o Loading Figure 3-2 (PCI Fig ) Loading Stage o a Preat Pretreed Conrete Bridge Beam 3-5

7 3.4.2 Allowable Steel Stre by LRFD (LRFD Table ) Condition Stre-Relieved Strand and Plain High-Strength Bar Tendon Type Low Relaxation Strand Deormed High-Strength Bar Pretenioning Immediately prior to traner ( pbt ) 0.70 pu 0.75 pu -- At ervie limit tate ater all loe ( pe ) 0.80 py 0.80 py 0.80 py Pot-tenioning Prior to eating-hort-term pbt may be allowed At anhorage and oupler immediately ater anhor et Elewhere along length o member away rom anhorage and oupler immediately ater anhor et 0.90 py 0.90 py 0.90 py 0.70 pu 0.70 pu 0.70 pu 0.70 pu 0.74 pu 0.70 pu At ervie limit tate ater loe ( pe ) 0.80 py 0.80 py 0.80 py Allowable Conrete Stre by LRFD (Other than egmentally ontruted bridge) Stre limit or onrete (LRFD Art ) For Temporary Stree Beore Loe Fully Pretreed Component 1. Compreion or pretenioned or pot-tenioned member, 0.60 i 2. Tenion: (LRFD Table ) a) In preompreed tenile zone without bonded reinorement, N/A b) In area other than the preompreed tenile zone and without bounded reinorement, i 0. 2 ki ) in area with bounded reinorement (reinoring bar or pretreing teel) uiient to reit the tenile ore in the onrete omputed auming an unraked etion, where reinorement i proportioned uing a tre o 0.5 y, not to exeed 30 ki, 0.24 i ki d) or handling tree in pretreed pile, i ki Stre limit or onrete at Servie Limit State or ully pretreed omponent (LRFD Art ): 1. Compreive tre limit at ervie limit tate (SLS-I) ater loe (LRFD Table ): 3-6

8 a) In other than egmentally ontruted bridge due to the um o eetive pretre and permanent load, 0.45 b) In egmentally ontruted bridge due to the um o eetive pretre and permanent load, 0.45 ) due to the um o eetive pretre, permanent and tranient load a well a hipping and handling, 0.60 w 2. Tenile tre limit at ervie limit tate (SLS-III) ater loe (LRFD Table ): a) or omponent with bonded pretreing tendon or reinorement that are ubjeted to not wore than moderate orroion ondition, 0.19, ki b) or omponent with bonded pretreing tendon or reinorement that are ubjeted to erve orroive ondition, , ki ) or omponent with unbonded pretreing, no tenion i allowed Deign Proedure Generally, the tenile tree at midpan due to ull dead and live load plu eetive pretre (ater loe) ontrol the deign. 1. Compute the tenile tre due to beam el-weight plu any other non-ompoite load uh a the dek, dek orm, haunhe, diaphragm, et., i any, applied to the beam etion only. 2. Compute the tenile tre due to uperimpoed dead load plu live load (Standard Speiiation) or 0.8 live load (LRFD Speiiation) applied to the ompoite etion. 3. The net tre, b, due to load in Step 1 plu 2, minu the allowable tenile tre i the tre that need to be oet by pretreing: Pe A Pee S b where P e i the eetive pretre, e i trand eentriity at midpan, and A and S b are beam area and bottom iber modulu. Solve or P e. The etimated number o tand P e / (area o one trand) ( pe ), where pe i the eetive pretre ater all loe whih may be approximated a 160 ki or Grade 270 trand. 4. Perorm a detailed alulation o pretre loe and repeat Step 3 i neeary. 5. Chek tree at the end (traner length) and midpan at releae and at ervie. Chek tree at the harp point at releae. Under typial load ondition, tree at harp point 3-7

9 do not govern at ervie load and are thereore not heked. Determine the amount o harping and/or debonding required to ontrol tree at the end o the beam. Thi may be done by omputing a required e or the eleted P e when draping i ued, or by omputing the required P e or a given e when debonding i ued. 6. Chek trength. 7. I neeary, revie number o tand and repeat Step 4 and Strand Conideration Harped Strand When onrete tree exeed allowable limit, trand harping beome an attrative option to redue pretre eentriity. The deigner hould be amiliar with the pratie and limitation o loal produer when onidering whether or not the alulated ore and harp angle an be tolerated. The ollowing are ome option to onider i the holddown ore exeed that whih the abriator an aommodate: 1. Split the trand into two group with eparate hold-down. 2. Change lope o harp by moving harp point loer to enterline o the beam, or by lowering harp elevation at beam end, or both. Alo, onider uplit ore and harp angle. 3. Dereae the number o harped trand. 4. Ue debonding intead o harping or ombine debonding with harping to redue harping requirement Debonded Strand Polyvinyl hloride polymer (PVC) An alternative to trand harping i to redue the total pretre ore by debonding ome trand at the end o member. Ater pretre i releaed to the onrete member, the debonded length o the trand ha zero tre. Strand debonding may be more eonomial or ome preat produer than harping. However, deigner hould take into aount the eet o the redution o preompreion, (P/A), a well a the lo o the vertial omponent o pretre whih ontribute to hear reitane near the member end. In addition, the alulated trand development length at the end o a debonded 3-8

10 trand i required to be doubled by the Standard Speiiation. Debonded trand have been hown by reent tudie, Ruell and Burn (1993, 1994-A and 1994-B), to perorm well and their ue i enouraged whenever poible. The Standard Speiiation do not ontain peii requirement regarding the maximum number and ditribution o debonded trand. However, Artile o the LRFD Speiiation provide the ollowing rule i debonded pretreing trand are ued: 1. The number o partially debonded trand hould not exeed 25% o the total number o trand. 2. The number o debonded trand in any horizontal row hall not exeed 40% o the trand in that row. 3. Debonded trand hould be ymmetrially ditributed about the enterline o the member. 4. Exterior trand in eah horizontal row hould be ully bonded. However, thee rule appear to be too onervative aording to urrent pratie in everal tate and the reent tudie by Ruell and Burn (1993, 1994-A and 1994-B), and other Minimum Strand Cover and Spaing The Standard Speiiation require a minimum onrete over over trand o 1.50 in. The LRFD Speiiation are unlear regarding onrete over over pretreing tand in preat onrete beam. For preat oit orm panel (tay-in-plae dek panel), the minimum over i 0.80 in. and or member ubjet to exterior expoure, the minimum i 2.0 in. regardle o whether the member i preat or at-in-plae. It i reommended here to ue the 1.50 in. minimum over peiied in the Standard Speiiation or bridge beam. The Federal Highway Adminitration ha approved ue o ½ in. diameter trand at a paing o 1.75 in., and 0.6 in. diameter trand at 2.00 in. on enter. A a reult, box beam, or example, may have two layer o ½ in. diameter trand in the bottom lange uing one o the alternative pattern. I the vertial trand paing i deired to be 2 in., the bottom lange thikne may have to be inreaed to atiy the minimum over requirement Nominal Flexural Reitane Required Parameter The average tre in bonded pretreing teel, 3-9

11 p pu 1 k (LRFD Eq ) d p Auming retangular etion behavior, the neutral axi depth: A p pu 1 1 A A pu b kap d p (LRFD Eq ) where = ditane between the neutral axi and the ompreive ae A p = area o pretreing teel pu = peiied tenile trength o pretreing teel A = area o mild teel tenion reinorement = tre in the mild teel tenion reinorement at nominal lexural reitane A = area o ompreion reinorement = tre in the mild teel ompreion reinorement at nominal lexural reitane b = width o ompreion o lange k = ator related to type o trand: py = (LRFD Eq ) pu = 0.28 or low relaxation trand py = yield trength o pretreing teel d p = ditane rom extreme ompreion iber to the entroid o the pretreing trand α 1 = tre blok ator peiied in LRFD Art (=.85 i ompreive trength not exeeding 10.0 ki and redued at a rate o 0.02 or eah 1.0 ki in exe o 10.0 ki; 0.75) The depth o the ompreion blok, a = β 1. I a > h (depth o the ompreion lange), langed etion behavior mut be ued with alulated by: A p pu A 1 A b 1 w ka p p pu b bw h 1 (LRFD Eq ) d where b w = width o web Flanged Setion (LRFD Eq ) 3-10

12 a a a M n Ap p d p A d A d a 2 b b h w h 2 Where p = average tre in pretreing teel a = depth o the equivalent tre blok = (β 1 ) = area o non pretreed tenion reinorement A d = ditane rom extreme ompreion iber to the entroid o nonpretreed tenile reinorement A = area o ompreion reinorement d = ditane rom extreme ompreion iber to the entroid o nonpretreed ompreion reinorement Fatored lexural reitane: M (LRFD Eq ) r M n Where = reitane ator a peiied in Art Maximum and Minimum Reinorement Limit Maximum Limit PROVISION DELETED IN Minimum Limit Unle otherwie peiied, at any etion o a lexural omponent, the amount o pretreed and nonpretreed reinorement hould be adequate to develope a atored lexural reitane, M r, at leat equal to the leer o (1) 1.33 time the atored moment required by the appliable trength load ombination and (2) M r. The LRFD Speiiation give a imilar proedure or omputing the raking moment, M r. M r r pe S M dn Sn S (LRFD Eq ) r where S, S n = ompoite and nonompoite etion modulu, pe = ompreive tre in onrete due to eetive pretre ore at extreme iber o etion; r = modulu o rupture = LRFD Art between 0.24 and 0.37 γ 1 = lexural raking variability ator (1.2 or preat egmental truture and 1.6 or all other onrete truture), γ 2 = pretre variability ator (1.1 or bonded tendon and 3-11

13 1,0 or unbonded tendon), γ 3 = ratio o peiied minimum yield trength to ultimate tenile trength o the reinorement (0.67 or A615, Grade 60 reinorement, 0.75 or A706, Grade 60 reinorement, 1.0 or pretreed onrete truture). Contrary to the Standard Speiiation, the LRFD Speiiation require that thi riterion be met at all etion. 3.5 Flexural Deign Example Deign Example Deign Requirement 1 Figure 3-3 Doe the midpan etion have adequate lexural trength to reit a atored load moment, M u = 4,900 kip-t? Standard Speiiation Aume the depth o the ompreion blok, a, to all within the top lange. Compute the average tre in the pretreing teel at ultimate load, * 9-17 with, = 270 ki, = 0.28, β 1 = 0.76, and: * u, uing STD Eq. * A bd There or, * 0.28 u ki. 3-12

14 * * A u 46 The ompreion blok depth, a 7.47 in b 0.85 Thi i larger than the lange thikne (5.50 in.). Thereore, the etion behave a a langed etion, with b = in. b = 2(5) = in. t = 5.50 in. A = 0.85 (b b )/ * u = in. 2 Thu A r = A * A = in. 2 * The orreponding teel index, A r u / b d index o Thi exeed the maximum teel. Thu, the etion mut be deigned a an overreinored etion. Uing [STD Eq. 9-23], 4, 301 t-kip. Note that when reinorement amount greater than the maximum limit are ued, their eetivene i igniiantly diminihed. Suh deign i rare a it i generally uneonomial. The deign apaity, 4,301 t-kip, i le than the required apaity o 4,900 t-kip. The apaity may be improved by inreaing the value. Inreaing Would redue the reinorement index and improve the lever arm ditane between the enter o the trand group and the enter o the ompreion blok. Ue value o = 8,500 pi. Thi igniiantly larger value than 5,800 pi wa hoen or the purpoe o omparion o the reult with LRFD Speiiation and train ompatibility olution given later. The * value o β 1 and u Beome 0.65 and 255 ki. The orreponding a = 5.18 in., whih i le than 5.5 in. Thereore, M n = 5,009 Ft-kip whih i aeptable. The teel index * * u 0.12 whih i muh lower than the limit It hould be noted that it i not unuual to have lexural trength rather than ervie tre ontrol the deign o adjaent box beam bridge. LRFD Speiiation Ue LRFD Eq. ( ) M n with A p = in. 2 pu = 270 ki β 1 = 0.76 = 5.8 ki (b b w ) = 38 in. h = 5.50 in. 3-13

15 b w = in. k = 2( ) = 0.28 and d p = in., the neutral axi depth = in. and /d p = Thi i greater than the maximum value o The etion i over-reinored and LRFD Eq. (C ), whih i idential to Standard Speiiation Eq. 9-23, mut be ued. The reulting M n would thereore be idential to that obtained earlier. I the value i inreaed to 8.5 ki, the neutral axi depth = in., and /d p = 0.41 whih i lightly le than the maximum value Thu, the etion i under-reinored and LRFD Eq. ( ) may be ued. Subtituting into thi equation with a = β 1 = 9.68 in., p = 270(1 0.28(14.89)/36.13) = 239 ki, M = 4,557 t-kip. Thi value i le than the apaity needed. Note that the value o a, p and dierent rom the orreponding Standard Speiiation reult. Deign Requirement 2 n M n are oniderably Aume that the trand development length = 7 t or bonded trand and 14 t or debonded trand. Determine the envelope o the lexural apaity along the pan length. Aume 12 o the 46 trand are debonded a hown in Figure 3-5. Note that even though 14 t i a very onervative etimate o development length, it ha little impat on the lexural trength o the member Comparative Reult Table 3-3 ompare reult o the train ompatibility approah or = 5.8 ki and 8.5 ki with the reult o lexural deign o the example o Setion uing both the Standard Speiiation and the LRFD Speiiation. Note that the omparion are or an untopped box beam and not the beam with dek lab in the preeeding etion. = 5.8 ki = 8.5 ki STD Spe. LRFD Spe. Strain Comp. STD Spe. LRFD Spe. Strain Comp. Neutral axi depth,, in Compreion blok depth, a, in. Steel tre at ultimate lexure, ki M n t-kip 4,301 4,301 4,505 5,009 4,557 5,106 95% 95% 100% 98% 89% 100% 3-14

16 Table 3-3 (PCI ) Flexural Capaity Predition by Variou Method The table learly how the advantage o uing the aurate train ompatibility approah. For = 5.8 ki, the approximate approah utilize an equation that i not even a untion o the teel provided. For = 8.5 ki, the Standard Speiiation give reult that are muh loer to the train ompatibility approah than the reult o the LRFD Speiiation. Part o the reaon i the etimation o the neutral axi depth whih i exeive, reulting in a low teel tre and a orrepondingly low M. Some deigner ompound the error reulting rom the approximate proedure by lumping all pretenioning teel in a etion into a ingle loation or the purpoe o etablihing the eetive depth. Thi i inorret. Only the reinorement near the tenion ae o the member hould be onidered in determining the teel tre uing Eq. [STD 9-17] and [LRFD ]. n 3.6 Flexural Deign Example o Negative Moment Region Strength Deign Where ontinuity at interior upport under live load and ompoite dead load i deired at interior upport, negative moment reinorement may be provided within the at-inplae dek lab. The negative moment etion i deigned a a reinored etion uing the ompreive trength o the beam onrete regardle o the trength o the at-inplae onrete. Ue the width o the bottom lange a the width o the onrete ompreive tre blok, b. Determine the required teel in the dek to reit the total atored negative moment, auming that the ompreion blok i uniorm: R n M u (PCI Eq ) 2 bd where R n M u d = trength deign ator = total atored negative moment = ditane rom extreme ompreion iber to entroid o the negative moment reinoring or preat beam bridge made ontinuou = trength redution ator = 0.9 Thi value i onitent with at-in-plae onrete ontrution, rather than = 1.0 or preat member. Thi i reaonable a the main reinorement i plaed in the ield. 3-15

17 Etimate the required area o teel uing the ollowing equation: 1 2mR n 1 1 (PCI Eq ) m y where y y m (PCI Eq ) = yield tre o non-pretreed onventional reinorement = ompreive onrete trength at 28 day or the beam The teel area, A bd. Alternatively, A may be determined uing one o everal approximate method. For example, A M u 0.9 d (PCI Eq ) y The above equation implie that the lever arm between the tenion and ompreion tre reultant i approximately 0.9d. The deign moment trength, M n, may be omputed by: a M n A y d (PCI STD Eq. 8-16) 2 where A y a = depth o ompreion blok = 0.85 b = area o nonpretreed tenion reinorement A (PCI STD Eq. 8-17) I the depth o the ompreion blok i larger than the thikne o the bottom lange, langed etion analyi imilar to that ued or the poitive moment etion will need to be done Standard Speiiation Reinorement Limit Maximum reinorement ,000 b (STD Eq. 8-18) y 87,000 y 3-16

18 max b (STD Art ) Minimum reinorement The total amount o nonpretreed reinorement hould be adequate to develop an ultimate moment at the ritial etion at leat 1.2 time the raking moment. The raking moment may be alulated a or a pretreed onrete etion exept pe = 0. M 1. 2 n M r LRFD Speiiation Reinorement Limit Maximum reinorement PROVISION DELETED IN 2005 Minimum reinorement [LRFD Art ] Requirement or minimum reinorement in the negative moment region are the ame a in the poitive moment region in Setion Servieability The dek lab i not pretreed and thereore i not ubjeted to the tenile tre limit peiied under ervie load ondition or pretreed onrete member. Ditribution o the lexural reinorement in the dek lab hould be heked in order to ontrol raking. The bet rak ontrol i obtained when the teel reinorement i well ditributed over the zone o maximum onrete tenion [STD ]. Several bar at moderate paing are more eetive in ontrolling raking than one or two larger bar o equivalent area. Crak width i ontrolled by: teel tre thikne o onrete over area o onrete urrounding eah individual reinoring bar urae ondition o the reinoring bar The Standard Speiiation ue the ollowing approah or rak ontrol. The tenile tre in the mild reinorement at ervie load, hould not exeed: 3-17

19 z / 3 y d A (STD Eq. 8-61) where d A Z = depth o onrete rom extreme tenion iber to enter o bar = area o onrete having the ame entroid a the tenile reinorement and bounded by the urae o the ro-etion and a traight line parallel to the neutral axi, divided by the number o bar = z = rak width parameter In ituation where the onrete urae i ubjet to evere expoure ondition, a maximum value o Z = 130 kip/in. i ued in deign. For moderate expoure ondition, a maximum value o Z = 170 kip/in. i ued. Several bar at moderate paing are more eetive in ontrolling raking than one or two larger bar o equivalent area. Thereore, the rak ontrol aording to LRFD Speiiation i a ollow: The paing o mild teel reinorement in the layer loet to the tenion ae hall atiy the ollowing 700 e 2d (AASHTO Eq ) d 1 0.7( h d ) γ e = expoure ator (1.0 or Cla 1 expoure ondition and 0.75 or Cla 2), d = thikne o onrete over, = alulated tenile tre in mild teel reinorement at the ervie limit tate not to exeed 0.60 y (ki), h = overall thikne Fatigue in Dek Reinorement The longitudinal dek reinorement in the negative moment zone over the pier mut be heked or atigue. Thi portion o the dek i likely to rak due to ervie load and the teel tre range may be igniiant. The tre range in reinorement i limited by: 3-18

20 r min 8 (STD Eq. 8-60) h where min r/h = tre range = algebrai minimum tre level, poitive i tenion, negative i ompreion = ratio o bae radiu to height o rolled-on tranvere deormation; i the atual value i not known, 0.3 may be ued. 3.7 Shear For tre alulation aording to the LRFD Speiiation, the peial atigue truk loading mut be introdued to the ontinuou truture. Figure 3-4 (Figure ) Type o Craking in Conrete Beam LRFD Speiiation There are two method o hear deign preented in the LRFD Speiiation. The mot general method i the trut-and-tie model. Thi model an be applied to any deign ituation, inluding member with irregular ro-etion or diontinuitie. It i alo ued to deign a member or all load eet, not jut hear. The method ued or typial hear deign i the etional deign model, or modiied ompreion ield theory developed by Collin, Mithell and other. Thi method i baed on the variable angle tru model in whih the inlination o the diagonal ompreion ield i allowed to vary. Thi dier rom the approah ued in the Standard Speiiation in whih thi angle i alway aumed to be 45. Thi i epeially igniiant or pretreed onrete member where the inlination i typially 20 to 40 degree due to the eet o the pretreing ore. (The Modiied Compreion Field Theory (MCFT) i a general model or the loaddeormation behavior o two-dimenional raked reinored onrete ubjeted to hear. 3-19

21 It model onrete onidering onrete tree in prinipal diretion ummed with reinoring tree aumed to be only axial.) Thi model alo dier rom the hear deign method ound in the Standard Speiiation beaue the onrete ontribution, V, i attributed to tenion being arried aro the ompreion diagonal. The ontribution ha been determined experimentally and ha been related to the train in the tenion ide o the member. In general, the higher the train in the tenion ide at ultimate, the wider the hear rak, and in turn the maller the onrete ontribution. It i igniiant to note that the onrete ontribution, V, i what et the etional deign model apart rom the trut-and-tie model. Both model are baed on the variable-angle tru analogy in whih a onrete member reit load by a tru ompoed o onrete ompreion trut and teel tenion tie. While thi model i an eetive tool in etimating the hear apaity o onrete member, it ha been ound to underetimate V when ompared to tet reult. Thereore, the etional deign method an be expeted to give higher apaitie than the trut-and-tie model. - LRFD Region required tranvere reinorement: V u > 0.5 (V + V p ) (LRFD Eq ) - LRFD Minimum tranvere reinorement: A v b v 0 (LRFD Eq ) y - LRFD Maximum paing o tranvere reinorement: I v u < 0.125, then max = 0.8d v 24 inh (LRFD Eq ) I v u 0.125, then max = 0.4d v 12 inh (LRFD Eq ) - LRFD Shear Stre on Conrete: Vu V p vu (LRFD Eq ) b d v v 3-20

22 Figure 3-5 The LRFD Speiiation, Artile introdue the etional deign model. Subetion 1 and 2 deribe the appliable geometry required to ue thi tehnique to deign web reinorement. The nominal reitane i taken the leer o: V n V V V, or, (LRFD Eq ) p V n b d V (LRFD Eq ) v v p where b v d v = eetive web width = eetive hear depth LRFD Eq. ( ) repreent an upper limit o V n to aure that the onrete in the web will not ruh prior to yield o the tranvere reinorement. The LRFD Speiiation deine the onrete ontribution a the nominal hear reitane provided by the tenile tree in the onrete. Thi reitane i omputed uing the ollowing equation: V b d (LRFD Eq ) v v β ator indiating ability o diagonally raked onrete to tranmit tenion and hear 3-21

23 I minimum amount o tranvere reinorement, 4.8 (LRFD Eq ) (1 750 ) I not, (LRFD Eq ) (1 750 ) (39 ) x (LRFD Eq ) The unit ued in the LRFD Speiiation are kip and inhe. The ator i equal to 1 1,000 whih overt the expreion rom pi to ki unit or the onrete ompreive trength. The ontribution o the web reinorement i given by the general equation: V Av ydv ot ot in (LRFD Eq ) where the angle, and, repreent the inlination o the diagonal ompreive tree meaured rom the horizontal beam axi and the angle o inlination o tranvere reinorement to longitudinal axi.. For ae o vertial web reinorement, the expreion or V impliie to: V Av ydv ot (LRFD Eq. C ) Tranvere hear reinorement hould be provided when: V u p 0.5 V V (LRFD Eq ) 3-22

24 Where the reation ore in the diretion o the applied hear introdue ompreion into the end region o a member, the loation o the ritial etion or hear hall be take a d v rom the internal ae o the upport. To determine the nominal reitane, the deign engineer mut determine and rom the LRFD Speiiation, Artile For mildly reinored onrete etion, the value o and are and 45 repetively. Thee will produe reult imilar to the Standard Speiiation. However, or pretreed onrete, the engineer an take advantage o the preompreion and ue lower angle o, whih optimize the web reinorement Deign Proedure To deign the member or hear, the deigner irt determine the atored hear due to applied load at the etion under invetigation. The value or d v i generally taken rom midpan lexural apaity alulation, where d v = d a/2. The hear ontribution rom any harped trand, V p, i then omputed. In lieu o more involved proedure, hall be determined a: M u 0.5N u 0.5Vu Vp ot Ap po d v (LRFD Eq ) E A E A p p The peiiation indiate that the area o pretreing teel, A p, mut aount or lak o development near the end o pretreed beam. Any mild reinorement or trand in the ompreion zone o the member, whih i takena one-hal o the overall depth (h/2), hould be negleted when omputing A and A p or ue in thi alulation. Thi i very important when evaluating member with harped trand, ine near the end o typial beam, harped trand are near the top o the beam. Beaue o thi, it i reommended that the traight and harped trand be onidered eparately in the analyi. It i the phyial loation o eah trand that i important and not the entroid o the group. The variable, po, repreent the tre in the pretreing trand when the tre in the urrounding onrete i zero. For the uual level o pretreing, 0.7 pu may be ued. 3-23

25 Figure 3-6 The value o orreponding to v/ and ε x i ompared to the aumed value o. I the value math, V i alulated uing Eq. ( ) whih the value o rom the table. I they do not math, the value o taken rom the table i ued or another iteration. O the quantitie omputed thu ar, only ε x will hange with a new value or, o the eort required or additional iteration i minor. Ater V ha been omputed, V i alulated uing Eq. ( ). The quantity o hear reinorement i then alulated uing Eq. (C ) with the value o rom the table. Ater determining the amount o hear reinorement needed, the deigner hould hek the maximum paing allowed by the peiiation a given in Artile Alo, the amount o hear reinorement hould be heked to enure that it i equal to or larger than the minimum value required by the peiiation, whih i: A v b v 0 (LRFD Eq ) y In region o high hear tree, the longitudinal (lexural) reinorement mut alo be able to arry the additional tre due to hear, i.e., the horizontal omponent o the diagonal ompreion ield. Thereore, the amount and development o the longitudinal reinorement mut atiy Eq. ( ): M u Nu Vu A y Ap p 0.5 Vp 0.5V ot (LRFD Eq ) dv v Satiying thi equation i very important or pretreed onrete beam, epeially near non-ontinuou upport where a ubtantial portion o the pretreing trand are 3-24

26 harped and the traner length o the trand extend into the pan. Harped trand are not eetive in ontributing to thi longitudinal reinorement requirement ine they are above midheight o the member. The LRFD Speiiation require that thi riterion be heked at the ae o the bearing. At thi etion, whih uually lie within the traner length o the trand, the eetive pretreing ore in the trand i not ully developed. Thu, the term p hould be alulated a a portion o the eetive pretre ore baed on linear variation tarting rom zero at the end o the beam to ull eetive pretre at the traner length. The deigner hould not be onued by the term p, whih generally reer to the pretre ore at Strength Limit State, beaue the trand at thi etion do not have enough development length to provide uh level o pretre. I the trand are well anhored in a diaphragm at the end o the member, the tre in the trand, p, an be onidered to equal the tre in the trand at Strength Limit State. Thi approah o varying p to aount or lak o development i preerred over the method implied by the deinition o A p or the lak o development. 3.8 Horizontal Interae Shear Shear Frition (PCI 8.5) Theory Cat-in-plae onrete dek deigned to at ompoitely with preat onrete beam mut be able to reit the horizontal hearing ore at the interae between the two element. The bai trength equation or the deign o the interae between the dek and beam i: V (STD Eq. 9-31a) u V nh where V u V nh = atored hear ore ating on the interae = trength redution ator = nominal hear apaity o the interae Deign i arried out at variou loation along the pan, imilar to vertial hear deign. Theoretial alulation o the hearing ore ating on the interae at a given etion i not imple beaue the etion doe not behave a a linear elati material near ultimate apaity. I it did, the hear tre, horizontal or vertial, at any iber in a ro-etion would be alulated rom the amiliar equation: VQ v h (PCI Eq ) Ib Where V = vertial hear ore at the etion I = moment o inertia b = etion width at the iber being onidered 3-25

27 Q = irt moment o the area above (or below) the iber being onidered However, at ultimate ondition, the material i no longer elati and the onrete may be raked at the etion being onidered. Further, the ompoite ro-etion onit o two dierent type o onrete with dierent propertie. Thereore, appliation o the above equation to deign at ultimate, without modiiation, would yield quetionable reult. Loov and Patnaik (1994) determined that the above equation may yield adequate reult i both the raked etion moment o inertia and area moment o a tranormed ompoite etion are ued. The etion would be tranormed uing the lab-to-beam modular ratio ued in lexural deign by the allowable tre method. However, thi approah i till too ompliated. It onue the alulation at two limit tate: ervie and ultimate. Kamel (1996) ued equilibrium o ore to how that: h b v v V / jd (PCI Eq ) where V d jd b v = atored vertial hear at the etion in quetion = eetive depth o the member = ditane between the tenion and ompreion reultant tree in the etion. Thi i the ame ditane a d v ued in the LRFD Speiiation. = etion width at the interae between the preat and the at-in-plae onrete. It i important to undertand that b v i not the web width. Another important iue i whih load hould be ued to alulate V u at a etion. Neither the Standard Speiiation nor the LRFD Speiiation give guidane in thi regard. While mot deigner would ue all load to ompute V u, a trong ae an be made or exluding the el-weight o the preat onrete member, and the weight o the dek ine they are preent prior to ompoite ation taking eet. Some deigner and agenie, uh a the Illinoi Department o Tranportation, ue only the ompoite load, whih inlude the uperimpoed dead load (barrier, wearing urae, et.) and the live load. Fortunately, the amount o reinorement required, even with onideration o all load, i reaonable in pratial appliation. To determine the hear apaity o the interae, the LRFD Speiiation ue a orm o the well-etablihed hear rition theory, while the Standard Speiiation ue an empirial approah baed on everal invetigation, or example, Birkeland and Birkeland (1966), Mat (1968), Kriz and Rath (1965) and Hobek, et al (1969). It i not poible to diretly ompare the reult o the two peiiation beaue the method ued in the Standard Speiiation i tated in term o vertial hear while in the LRFD Speiiation i tated in term o horizontal (interae) hear. 3-26

28 3.8.2 LRFD Speiiation LRFD Speiiation give no guidane or omputing horizontal hear due to atored load. The ollowing ormula may be ued a diued in Setion with the ubtitution d v or jd: V u vuh or dvbv V u Vh (PCI Eq ) dv where v uh = horizontal atored hear ore per unit area o interae V u = atored vertial hear ore at peiied etion due to uperimpoed load d v = the ditane between reultant o tenile and ompreive ore = b v = interae width V h = atored horizontal hear ore per unit length o interae Required trength nominal trength, or: v uh Av Vn Equivalent to: V h V n (PCI Eq ) where V n where μ A v A v P y = nominal hear reitane o the interae urae A A P (LRFD Eq ) = v v y = oheion ator = 0.10 or thi ae = rition ator = 1.0 or thi ae = interae area o onrete engaged in hear traner = area o hear reinorement roing the hear plane within area = permanent net ompreive ore normal to the hear plane (may be onervatively negleted) = yield trength o hear reinorement Typially, the top urae o beam i intentionally roughened to an amplitude o 1/4 in. Thereore, or normal weight onrete at againt hardened, roughened, normal weight onrete, the above relationhip may be redued to the ollowing ormula: uh v y v v 0.1 A / A (PCI Eq ) A b / where the minimum v 0.05 v y Nominal hear reitane o the interae plane i the leer o: 3-27

29 V ni K 1 A, and, (LRFD Eq ) v Vni K2 A v (LRFD Eq ) While the LRFD Speiiation require that minimum reinorement be provided regardle o the tre level at the interae, deigner may hooe to limit thi reinorement to ae where v i greater than 0.10 ki. Thi would be onitent with the Standard Speiiation, the / uh ACI Code and other reerene. It would eem to be impratial and an unneeary expene to provide onnetor in a number o ommon appliation, uh a preat tay-in-plae panel i the interae tre i lower than 0.10 ki. K 1 = ration o onrete trength available to reit interae hear (LRFD Art ) K 2 = limiting interae hear reitane (LRFD Art ) B5.2-1, 2, or 3 Table B5.2-1 Figure 3-7 (LRFD Figure CB5.2-5) Flow Chart or Shear Deign o Setion Containing at Leat Minimum Tranvere Reinorement 3-28

30 Table B5.2-1 Value o θ and β or Setion with Tranvere Reinorement Table B5.2-2 Value o θ and β or Setion with Le than Minimum Tranvere Reinorement Biaxial Flexure 1 P rxy I the atored axial load i not le than 0.10φ Ag: AASHTO Eq. ( ) P P P rx ry 0 In whih: ' P 0.85 ( A A A ) A A ( E ) AASHTO Eq. ( ) 0 g t p y p pe p u I the atored axial load i le than 0.10φ A g : M M ux rx M M uy ry 1.0 AASHTO Eq. ( ) 3-29