Reinfored Conrete Deign Shear in Beam 2 Shear deign mmary More detail hear deign Shear pan Deep beam Mongkol JIRAACHARADET S U R A N A R E E UNIERSITY OF TECHNOLOGY INSTITUTE OF ENGINEERING SCHOOL OF CIIL ENGINEERING Shear Deign Smmary WSD SDM Shear: DL + LL Shear: 1.4 DL + 1.7 LL n / φ Conrete: 0.29 f b d Steel: - Spaing: A v f d / Min. Stirrp: max A v / 0.0015 b Chk. light hear: 0.795 f bd max d/2 60 m Chk. heavy hear: 1.32 f bd max d/4 30 m Conrete: 0.53 f b d Steel: n - Spaing: A v f y d / Min. Stirrp: max A v f y / 3.5 b Chk. light hear: 1.1 f bd max d/2 60 m Chk. heavy hear: 2.1 f bd max d/4 30 m
ก ก LL fll pan DL fll pan w L 2 (ก) ก ก Max. hear @ end LL half pan DL fll pan wl L 8 ( ) ก ก Max. hear @ midpan ( ) Shear fore envelop w L 2 wl L 8 EXAMPLE 6-2 More Detailed Deign of ertial Stirrp SDM The imple beam pport a niformly ditribted ervie dead load of 2 t/m, inlding it own weight, and a niformly ditribted ervie live load of 2.5 t/m. Deign vertial tirrp for thi beam. The onrete trength i 250 k, the yield trength of the flexral reinforement i 4,000 k. DL 2 t/m LL 2.5 t/m d 64 m w 1.4(2) + 1.7(2.5) 7.05 t/m w L 1.7(2.5) 4.25 t/m L 10 m 30 m w L/2 7.05(10)/2 32.25 ton w L L/8 4.25(10)/8 5.31 ton 32.25/0.85 37.94 ton /φ Diagram : 5.31/0.85 6.25 ton Shear2_11
ame olmn width 0.40 m / φ at d 37.94 (0.84/5)(37.94 6.25) 32.62 ton Shear trength of onrete 0.53 f b d 0.53 250 (30)(64)/1,000 16.09 ton 37.94 t Critial etion 32.62 t 84 m Reqired /φ 16.09 t 0.5 8.05 t 6.25 t Spport Midpan I the ro etion large enogh? + 2.1 f b d 16.09+ 2.1 250 (30)(64)/1,000 79.84> 32.62 ton OK n,max + 1.1 f b d 16.09+ 1.1 250 (30)(64) /1,000 55.6> 32.62 ton d/ 2 60 m max Shear2_12 Minimm tirrp : (ACI 11.5.6.3) USE RB9 : A v 2(0.636) 1.27 m 2, f y 2400 k A v,min 0.2 f b f y (ACI Eq. 11-13) Rearranging give max A vfy 1.27(2,400) 32 m 0.2 f b 0.2 250 (30) bt not le than max A f 1.27(2,400) 3.5b 3.5(30) v y 29 m Ue max 29 m < [d/2 64/2 32 m] < 60 m Compte tirrp reired at d from pport A f d 1.27(2.4)(64) / φ 32.62 16.09 v y 11.8 m Ue RB9@0.11m. Change paing to 15 m where thi i aeptable, and then to the maximm paing of 29 m. Compte /φ where an be inreaed to 15 m. A vfyd 1.27(2.4)(64) + + 16.09 29.1 ton φ 15 Shear2_13
37.94 t 84 m Critial etion 32.62 t 29.1 t /φ 16.09 t 0.5 8.05 t 6.25 t Spport 500 m Midpan x 37.94 29.1 x 500 140 m from pport 37.94 6.25 Change to 29 m, ompte /φ A vfyd 1.27(2.4)(64) + + 16.09 22.82 ton φ 29 37.94 22.82 x 500 239 m from pport 37.94 6.25 Shear2_14 15 m @ x 140 m 29 m @ x 239 m RB9@0.11 RB9@0.15 RB9@0.29 20 m 11@11 m 1 m Spport 7@15 m 8@29 m 500 m Midpan RB9 @ 0.11 m : 20+1+11@11 142 m > 140 m RB9 @ 0.15 m : 142+7@15 247 m > 239 m OK OK RB9 @ 0.29 m : 247 + 8@29 479 m Shear2_15
Shear Span (a M / ) Ditane a over whih the hear i ontant a P P a Shear +P Diagram + - -P M a Moment Diagram + Shear2_16 Crak Pattern in Several Length of Beam Span Mark (m) a/d 1 0.90 1.0 2 1.15 1.5 3 1.45 2.0 4 1.70 2.5 5 1.95 3.0 6 2.35 4.0 7/1 3.10 5.0 8/1 3.60 6.0 10/1 4.70 8.0 9/1 5.80 7.0 Shear2_17
ariation in Shear Strength with a/d for retanglar beam Shear-ompreion trength Flexral moment trength Failre moment a Deep beam Shear-tenion and hear-ompreion failre Inlined raking trength, Flexral failre Diagonal tenion failre 0 1 2 3 4 5 6 7 a/d Shear2_18 DEEP BEAM Brnwik Bilding. Note the deep onrete beam at the top of the grond olmn. Thee 168-ft beam, pported on for olmn and loaded by loely paed faia olmn above, are 2 floor deep. Shear tree and failre mehanim were tdied on a mall onrete model. (Chiago, Illinoi) Shear2_19
Shear2_20 Deep Beam Deep beam are trtral element loaded a beam in whih a ignifiant amont of the load i tranferred to the pport by a ompreion thrt joining the load and the reation. When hear pan a M / to depth ratio < 2 Mehanim: Ue both horizontal and vertial may prevent rak Compreive trt If nreinfored, large rak may open at lower midpan. Shear2_21
Definition of Deep Beam ACI 10.7.1 Deep beam are member loaded on one fae and pported on the oppoite fae o that ompreion trt an develop between the load and the pport, and have either: (a) lear pan, L n, eqal to or le than for time the overall member depth; or h L n / h 4 L n (b) region with onentrated load within twie the member depth from the fae of the pport. P x h x < 2 h Shear2_22 Deign Criteria for Shear in Deep Beam Bai Shear Strength: φ n where n + Loation for Compting Fatored Shear: (a) Simply Spported Beam (Critial etion loated at ditane z from fae of pport) - z 0.15L n d for niform loading - z 0.50a d for onentrated loading (b) Contino Beam Critial etion loated at fae of pport Limitation on Nominal Shear Strength n,max 2.7 f bd Shear2_23
Shear Strength of Conrete, M d 3.5 2.5 0.50 f + 176ρ b d 1.6 f b d d M M d where 1.0 3.5 2.5 2.5 If ome minor nightly raking i not tolerated, the deigner an e Simplified method: 0.53 f bd Shear Reinforement, A v 1+ L n / d A vh 11 L n / d fyd v 12 + h 12 A v ก (. 2 ), A vh ก (. 2 ) v ก ก (.), h ก (.) Shear2_24 Minimm Shear Reinforement d maximm v 30 m 5 and d maximm h 30 m 5 minimm A vh 0.0015 b h minimm A v 0.0025 b v Shear2_25
5.6 ก ก กก ก 60 ก 3.6 ก 35. ก d 90. f 280 กก./. 2 f y 4,000 กก./. 2 1.20 m 60 t 60 t 1.20 m h 100 m d 90 m 4DB36 5 m 5 m 35@10 3.5 m 40 m 3.6 m 40 m 35 m (a) ก L n /h 360/100 3.6 < 4.0 ก Shear2_26 (b) ก กก a 1.20. 0.50a 0.5(1.20) 0.60 < [d 0.90 ] ก 0.60. ก () ก ก ก ก กก 1.7 LL 1.7(60) 102 ก ก กก ก M 102(60) d 102(90) 0.67 M ก 3.5 2.5 3.5 2.5(0.67) 1.83 < 2.5 OK d v d 1.83 0.50 f + 176 ρ ω M 4(10.18) ρ w 35(90) 0.0129 Shear2_27
v ( ) 176 0.0129 1.83 0.50 280+ 0.67 1.83[8.37 + 3.39] 21.5 กก./. 2 Upper limit: v 1.6 f 1.6 280 26.77 kg/m ก ก v b w d 21.5(35)(90)/1,000 67.8 (d) ก ก 2 Reqired n 102 φ 0.85 120 ton 2.7 f b d 2.7 280(35)(90) /1,000 n,max 142 >120 OK ก n ก > (120 > 67.8) ก ก Shear2_28 (e) ก ก L n /d 4 : Av 1 + Ln / d Avh 11 Ln / d 12 + 12 f d b 35. f y 4,000 กก./. 2 2 120 67.8 52.2 y A v 5 A 7 52.2 12 12 4.0(90) v vh + h min A v 0.0025 b v max v d/5 18. 0.145 min A vh 0.0015 b h max h d/5 18. DB12 ก h 18. min A vh 0.0015(35)(18) 0.945 2 A vh 2(1.13) 2.26 2 > 0.945 2 OK Shear2_29
A vh ก A v A v 5 2.26 7 12 + 18 12 [ ] 12 0.145 0.145 0.0732 0.172 5 ก ก DB12: A v 2(1.13) 2.26 2 ก 2.26/0.172 13.1. < [d/5 18.] min A v 0.0025(35)(18) 1.58 2 < [A v 2.26. 2 ] OK OK DB12 ก ก ก 12. DB12@0.18 90 m DB12@0.12 4DB36 30@12 3.6 m 40 m 3.6 m 40 m 35 m Shear2_30 6.1 ก ก ก ก ก ก ก ก ก ก ก ก ก 1.5 f 240 กก./. 2 f y 4,000 กก./. 2 4DB20 8DB20 7.7 m lear 65 m 4DB20 8DB20 6 m 57 m 8.0 m / 35 m a) ก กก x b 6,120d 6,120 (57) 6,120+ f 6,120 + 4,000 y 34.5 m max x 0.75 x b 0.75(34.5) 25.9 m
x 25.9. A 12.56 m 2 max A 51.9 m 2 Real A 49.28 m 2 ε 0.003 25.9 6 ε (0.003) 0.0023 > ε 0.0020 f ' f 25.9 ε y y C max C 0.85bβ 1 (max x) 0.85(0.24)(35)(0.85)(25.9) 157.2 A f y 12.56(4.0) 50.2 max T max C 157.2 + 50.2 207.4 max T 207.4 max A 51.9 m 49.28 m f 4.0 2 2 > OK y (b) ก M n ก ก ก ก β 0.85 f b 1x + A f y A f y 0.85(0.24)(35)(0.85x) + 12.56(4.0) 49.28 (4.0) x 24.2.. 24.2 6.0 ε 0.003 0.0023 > ε y ก ก 24.2 C 0.85f bβ 1 x 0.85(0.24)(35)(0.85)(24.2) 146.9 C A f y 12.56 (4.0) 50.2 T A f y 49.28(4.0) 197.1 a 1 d 57 (0.85) (24.2) 46.7 m 2 2 M n 146.9(46.7)/100 + 50.2(57-6)/100 94.2 -
1 M w M 8 2 (8) φ n 0.90(94.2) 84.8 - w 10.6 / ก ก w L 1.5w D w 1.4w D + 1.7(1.5w D ) ก ก w D 10.6/(1.4+2.55) 2.7 / ก w L 1.5(2.7) 4.0 / () ก ก 42.4 t LC of pport max. hear envelope + 6.8 t 6.8 t Midpan 8.0 m SHD with DL+LL on entire pan - 42.4 t Max. hear at pport: wl 10.6(8) 42.4 ton 2 2 Max. hear at midpan when half LL on pan: wl 10.6(8) 6.8 ton 8 8 Critial etion from fae of pport d 57 m, pport width 30 m Therefore ompte at 57+30/2 72 m (42.4 6.8) 42.4 72 36.0 ton 4(100) ( f b d) Shear trength of onrete φ φ 0.53 w 42.4 t Fae of pport Critial etion 36.0 t 0.85 0.53 240 35 57 / 1,000 13.9 ton d 72 m Reqired φ 13.9 t φ 0.5φ 6.8 t C L of pport Midpan
Reqired φ -φ 36.0-13.9 22.1 ton Min φ 0.85(3.5)(35)(57)/1,000 5.9 ton Max φ (for d / 2) 0.85 1.1 240 35 57 / 1,000 28.9 ton Sine 5.9 ton < Reqired φ < 28.9 ton, max d/2 USE DB10 tirrp: φ Av fyd 0.85 2 0.78 4.0 57 13.7 m φ 22.1 @ Critial etion USE 13 m from z 0 to 57 m from fae of pport φ φ Av fyd 0.85 2 0.78 4.0 57 23.2 ton 13 From z 57 m, et φ n 22.1 φ z 57 + (400 72) 36.0 6.8 5.1 ก ก ก (m) φ (ton) z (m) 13.7 15 20 25 28.5 (d/2) 51.2 (NG) 22.1 (Max) 20.1 15.1 12.1 10.6 5.9 (Min) 0 to 57 79 135 169 186 238 6@13m 4@15m 2@20m 8@25m 30 m 1 m
Shear Strength of Member nder Combined Bending and Axial Load Axial Compreion N Simplified method: 0.53 (1 + ) f bw d 140A where N Fatored axial ompreive load A g Gro area of the onrete etion More detailed eqation: g d (0.5 f + 176 ρw ) bw d 0.93 f ' bw d M Replae M with M m, where h/2 h N M A 4h d Mm M N 8 a/2 C d - a/2 T 7d/8 d/8 a h a [ΣM A 0] T d M N 2 2 2 N (pper limit) 0.93 f 1+ bw d 35 A g N Axial Tenion 0.53(1 + ) 35 A f b d w g v ρwd 0.50 f + 176 M 4h d Mm M N 8 m v / f ( A ) v 0.93 f 1+ N 35 g ( + N Ag) f v 0.53 1 140 0.53 ( + N Ag) f v 0.53 1 35 56.3 35.2 ( N + ), กก./. 2 ( N - ), กก./. 2
Strength - Contino Beam Simplified method: More detailed proedre: 0.53 f b d w d 0.50 f + 176ρw bw d 0.93 f bw d M Strength - Contino Beam A f d v y Minimm Shear Reinforement: min A v 0.0015 b w where d / 5 45 m min A vh 0.0025 b w 2 where 2 d / 3 45 m Limitation on Nominal Shear Strength Nominal tre v n n / (φ b w d) max Ln vn 2.1 f for d < 2 Ln Ln max vn 0.18 10 + f for 2 d d 2.1 f max v 2.7 f n