Shear in Beams 2. Reinforced Concrete Design. Shear Design Summary. Shear design summary More detail shear design. Shear span Deep beam WSD SDM

Transcription

1 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 LL n / φ Conrete: 0.29 f b d Steel: - Spaing: A v f d / Min. Stirrp: max A v / b Chk. light hear: 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

2 ก ก LL fll pan DL fll pan w L 2 (ก) ก ก Max. end LL half pan DL fll pan wl L 8 ( ) ก ก Max. 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)/ ton w L L/8 4.25(10)/ ton 32.25/ ton /φ Diagram : 5.31/ ton Shear2_11

3 ame olmn width 0.40 m / φ at d (0.84/5)( ) ton Shear trength of onrete 0.53 f b d (30)(64)/1, ton t Critial etion t 84 m Reqired /φ t t 6.25 t Spport Midpan I the ro etion large enogh? f b d (30)(64)/1, > ton OK n,max f b d (30)(64) /1, > ton d/ 2 60 m max Shear2_12 Minimm tirrp : (ACI ) 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 ) Rearranging give max A vfy 1.27(2,400) 32 m 0.2 f b (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 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) ton φ 15 Shear2_13

4 37.94 t 84 m Critial etion t 29.1 t /φ t t 6.25 t Spport 500 m Midpan x x m from pport Change to 29 m, ompte /φ A vfyd 1.27(2.4)(64) ton φ x m from pport Shear2_14 15 x 140 m 29 x 239 m RB9@0.11 RB9@0.15 RB9@ m 11@11 m 1 m Spport 7@15 m 8@29 m 500 m Midpan 0.11 m : @ m > 140 m 0.15 m : 142+7@ m > 239 m OK OK 0.29 m : @ m Shear2_15

5 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 / / / / Shear2_17

6 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 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

7 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

8 Definition of Deep Beam ACI 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

9 Shear Strength of Conrete, M d f + 176ρ b d 1.6 f b d d M M d where 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 b h minimm A v b v Shear2_25

10 5.6 ก ก กก ก 60 ก 3.6 ก 35. ก d 90. f 280 กก./. 2 f y 4,000 กก./ m 60 t 60 t 1.20 m h 100 m d 90 m 4DB36 5 m 5 m 35@ m 40 m 3.6 m 40 m 35 m (a) ก L n /h 360/ < 4.0 ก Shear2_26 (b) ก กก a a 0.5(1.20) 0.60 < [d 0.90 ] ก 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 (0.67) 1.83 < 2.5 OK d v d f ρ ω M 4(10.18) ρ w 35(90) Shear2_27

11 v ( ) [ ] 21.5 กก./. 2 Upper limit: v 1.6 f kg/m ก ก v b w d 21.5(35)(90)/1, (d) ก ก 2 Reqired n 102 φ ton 2.7 f b d (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 f d b 35. f y 4,000 กก./ y A v 5 A (90) v vh + h min A v b v max v d/ min A vh b h max h d/5 18. DB12 ก h 18. min A vh (35)(18) A vh 2(1.13) > OK Shear2_29

12 A vh ก A v A v [ ] ก ก DB12: A v 2(1.13) ก 2.26/ < [d/5 18.] min A v (35)(18) < [A v ] OK OK DB12 ก ก ก 12. DB12@ m DB12@0.12 4DB36 30@ m 40 m 3.6 m 40 m 35 m Shear2_ 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 8DB 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, ,000 y 34.5 m max x 0.75 x b 0.75(34.5) 25.9 m

13 x A m 2 max A 51.9 m 2 Real A m 2 ε ε (0.003) > ε f ' f 25.9 ε y y C max C 0.85bβ 1 (max x) 0.85(0.24)(35)(0.85)(25.9) A f y 12.56(4.0) 50.2 max T max C max T max A 51.9 m m f > 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) (4.0) (4.0) x ε > ε y ก ก 24.2 C 0.85f bβ 1 x 0.85(0.24)(35)(0.85)(24.2) C A f y (4.0) 50.2 T A f y 49.28(4.0) a 1 d 57 (0.85) (24.2) 46.7 m 2 2 M n 146.9(46.7)/ (57-6)/

14 1 M w M 8 2 (8) φ n 0.90(94.2) 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/( ) 2.7 / ก w L 1.5(2.7) 4.0 / () ก ก 42.4 t LC of pport max. hear envelope t 6.8 t Midpan 8.0 m SHD with DL+LL on entire pan 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 ( ) 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 / 1, ton d 72 m Reqired φ 13.9 t φ 0.5φ 6.8 t C L of pport Midpan

15 Reqired φ -φ ton Min φ 0.85(3.5)(35)(57)/1, ton Max φ (for d / 2) / 1, ton Sine 5.9 ton < Reqired φ < 28.9 ton, max d/2 USE DB10 tirrp: φ Av fyd m φ Critial etion USE 13 m from z 0 to 57 m from fae of pport φ φ Av fyd ton 13 From z 57 m, et φ n 22.1 φ z 57 + (400 72) ก ก ก (m) φ (ton) z (m) (d/2) 51.2 (NG) 22.1 (Max) (Min) 0 to @13m 4@15m 2@20m 8@25m 30 m 1 m

16 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 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 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 M 4h d Mm M N 8 m v / f ( A ) v 0.93 f 1+ N 35 g ( + N Ag) f v ( + N Ag) f v ( N + ), กก./. 2 ( N - ), กก./. 2

17 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 b w where d / 5 45 m min A vh 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 f for 2 d d 2.1 f max v 2.7 f n