Lecture Note 6. Moment-Curvature (M-φ) Relation - I

Transcription

1 7 Letre Note 6 oment-crvatre (-φ) Relation - I -φharateriti oniering IS: 456: The atal moment-rvatre relationhip o R.C. primati etion i obtaine rom tre-train iagram o onrete an teel. Starting rom the bai eqation, expreion or the axial ore an moment arring apait o the etion are allate in nonimenional orm. The eqation o the tre train iagram o the paraboli part or onrete i σ ε ε (1) σ ε0 ε0 bjet to the ollowing limiting onition: ( Fig. 1 an ) 1. at ε 0, σ 0,. at ε ε 0, σ σ, σ 3. at ε ε 0, 0. ε The permiible ompreive tre in onrete are oniere a: σ Uner an loaing onition, the etion nergoe train an oneqent tree. A linear train itribtion over the epth o ro-etion i ame or tre itribtion in onrete an teel(fig.1 an ), giving rie to two poible ae: 1. 0 ε 4 ε 0 an. ε 0 ε 4 ε ε ε σ For ε < or 0.00 ε

2 8 Fe50 Fig. 1 Stre train iagram or teel (Fe 50) Fig. Stre train iagram or onrete b IS oe

3 ε E Fig 3 Stre train iagram o teel (Fe415) For Fe415 ε σ (Pa) For Fe500 ε σ (Pa)

4 30 Cae I Cae - II X X t D X D X t D X D D (i) Cro etion D (i) Cro etion ε 3 ε 4 ε 0 ε 3 ε 4 ε 1 ε 0 ε4 ε0 ε 1 ε ε0 ε4 ε (ii) Strain iagram (ii) Strain iagram 3 4 σ 3 4 k ' D kd kd (iii) Stre iagram (iii) Stre iagram T (iv) Fore iagram C 1 C C 1 C 3 C T (iv) Fore iagram Fig. 3 Stre train itribtion or ae I an ae II

5 31 Letre Note 7 oment-crvatre (-φ) Relation - II Cae I ( 0 ε 4 ε 0 ) Aoring to Fig. 3, 4 x k ε ε + ε 4 Where k i the ratio o the netral axi epth to the eetive epth. ε 1 an ε 4 are the extreme iber train in ro etion. For the given etion, tenile ore or the tenile reinorement, T P t bd P t bde ε Where E ε 0.87 Pt Perentage o tenile teel b With o the etion D Overall epth o the etion E ol o elatiit o teel Stre in tenile teel ε train in tenile teel Charateriti trength o teel Compreive ore e to paraboli part o the tre-train iagram o onrete : 3 ε 4 50 σ kbd 10 ε 4 3 C1 Where σ Ultimate 8 a be trength o onrete in ompreion. Compreive ore e to preene o ompreion teel i, C P bd 3 P bd E ε 3 Where E ε P Perentage o ompreive teel Stre in ompreive teel 3 ε 3 Strain in ompreive teel Uing non-imenional ore parameter one ma get,

6 3 υ N N 0 C + C T bd 1 σ k P P ε 4 ' 1 3 t ε ' σ σ.() Where 3 E ε E ε 0.87 N Axial thrt N 0 Ultimate axial thrt ( σ bd) Similarl, nonimenional moment parameter, µ 0 bd σ k (1 X t X) P3 ε' 4 ε' 4 { (1 X t ) ε' 1 ε' 4} ε' (3) σ Where ε ' ε 4 4 x 10 3 ε ' 1 ε 1 x 10 3 Bening moment 0 Ultimate bening moment (σ bd ) Cae II ( ε 0 ε 4 ε ) ε0 ε0 Aoring to Fig. 3, k' k ε + ε ε 4 4 For the given etion ompreive ore e to paraboli part o the tre-train iagram o onrete i: C 1 k σ bd 3 Compreive ore e to ompreion teel, i preent, i: C P bd 3 Compreive ore e to traight part o the tre-train iagram: C 3 σ (k-k )bd Tenile ore e to tenile teel: T P t bd From the above vale o C 1, C, C 3 an T, one ma get an, ν N N 0 1 P k k' + 3 σ 3 Pt σ. (4) ' µ ( ) ( )( k + k'(1 Xt) k + K K' (1 X ) + K' K 1 3 (1 X t X) pc σ 3 ) t +. (5) To ati the eqation (), (3), (4) an (5) ollowing onition are to be lille. 1. E ε

7 33. E ε 0.87 Ptting ierent vale o ν in the eqation () an (4) an aigning ome peii vale o ε 4 (ε ), orreponing vale are allate. The train o onrete being known or ame, the rotation apait o the trtre i allate. Correponing to thi rotation apait, moment arring apait o the trtre i allate rom the eqation (3) an (5). For peii vale o ompreion an tenile teel, the vale o ε (ε 1 +ε 4 ) an µ ( / 0 ) are allate. Governing eqation or Φharateriti oniering ACI Coe: Reerene: P. Deai an S. Krihnan, Eqation or the tre train rve o onrete Jornal o Amerian Conrete Intitte, Vol. 61, pp , The eqation o the tre-train rve given b ACI Coe i ε σ Eε ε 0... (6) ε ε ε0 ε 0 Where, E σ ε 0 initial tangent mol, ame to be twie the eant mol at maximm tre σ, ε 0 train at maximm tre. Stre(N/mm ) Strain( 10 3 ) Fig 4 Stre train iagram or onrete (ACI Coe) The vale o tree an train are taken a per ACI oe.

8 34 Strain itribtion over the epth o etion i oniere a linear. Aoring to Fig. 4, ε4 k ε + ε Where 4 k Ratio o the natral axi epth to the eetive epth. (ε an ε 4 are the extreme iber train in ro etion.) For the given etion, tenile ore or the tenile reinorement, T P t b P t be ε Where E ε 0.87 Pt Perentage o tenile teel b With o the etion D Overall epth o the etion E ol o elatiit o teel Stre in tenile teel ε Strain in tenile teel Charateriti trength o teel Compreive ore e to paraboli part o the tre-train iagram o onrete : C 1 σ kb ε 0 ε ln 1+ ε 0 Where σ Ultimate 8 a be trength o onrete in ompreion. Compreive ore e to the preene o ompreion teel, i C P b 3 P b E ε 3 Where E ε P Perentage o ompreive teel Stre in ompreive teel 3 ε 3 Strain in ompreive teel Uing non-imenional ore parameter, N ν No C1+ C T σ b ε 0 ε 4 P 3 Pt k ln 1 + ε + 4 ε.(7) 0 σ σ Where 3 E ε E ε 0.87 N Axial thrt N 0 Ultimate axial thrt ( σ bd) Similarl, nonimenional moment parameter,

9 35 µ 0 bd Where, σ ( ) 1 X k ε k ε ε ( 1 X ) t X PE 4 ln 1 4 0tan t + ε ε + ε4 ε0 ε4 ε0 σ (8) Bening moment 0 Ultimate bening moment (σ bd ) To ati the eqation (7) an (8) ollowing onition are to be lille. 1. E ε E ε 0.87

10 36 Letre Note 8 Behavior o RC ember: Flexre Eqivalent Compreion Blo Total ompreion, C Area o (ABFE-EFD) Th, C x b 0.36 x b xb xb 3 Taking moment abot EF x (0.57 x) 0.36 xbx ( kx ) xb b 1 k To in epth o NA TC t At xb tat x 0.36 b For balane etion, x 0.87 ta t 0.36 b (i)

11 37 Limiting vale o X / ε ; Where Ε 1.15E Again, x ε ε + ε ε 5 10 pa Th, x an be obtaine a given below: Tpe o teel Fe50 Fe415 Fe500 ε x x The ompreive ore e to onrete will be:c 0.36 x 0.36 b x Where, F 0.36 F b Tpe o teel Fe50 Fe415 Fe500 F oment apait 0.87 A ( L. A.) t 0.87 At ( 0.416x ) x 0.87 At( ) 0.87 At 0.87 At[ ][From eqn (i)] 0.36 b 0.87 At[ At ] b

12 38 p 0.87 pb [1 ] p p b in term o onrete ( 0. x ) 0.36 x b 416 x x b Th, Kb where, x K x Tpe o teel Fe50 Fe415 Fe500 K inimm epth or given K b K b For Fe50, K 0.15, For Fe415, K 0.14, b b Expreion or Steel Area or Balane Singl Reinore Setion From the eqilibrim onition: T C 0.87 A t bx At 0.36 x / k b 0.87

13 39 At Now, aming p 100 ; p / k b 100 Th, / x x p 100k Th the perentage o balane reinorement p B will be a hown in the table below: Steel x/ p B (or 0) p Fe Fe Fe Where p B Balane perentage o teel Expreion or x / or a given b, & x x b x x x b ( 1.) 6. b 1 Th, x an be on ot rom,, b & Dobl Reinore Beam Thi tpe o beam i neear i 1. Depth i retrite or arhitetral point o view. oment i high 3. oving loa 4. Dtlit i reqire 5. Seimi reitant 6. At pport o ontino beam. / The train in teel an be allate rom the relation o: ε (1 ) x

14 40 Two ae ma arie regaring the tre in reinorement 1. Strain o ompreion teel i reahe at iel train.. Strain o onrete i reahe to iel train o & train o ompreion teel i below o iel train. Anali & Deign o Dobl Reinore Beam 1. Strain ompatibilit metho ing bai eqation.. Steel Beam Theor. 3. Ue o eign ai (SP-16) Strain Compatibilit etho Step1: Chooe the vale o x. Ame the onrete ail at a ompreive train o Then raw the train itribtion o the etion. Step: Callate train & orreponing tre in the tenile teel. T t A t Step3: Callate train & orreponing tre in the ompreive teel. C A Step4: Callate the ompreive ore in the onrete bloc xb Step5: Callate total ompreive ore a C C + C Step6: Che i T C, then ame netral axi epth x i OK. Otherwie hooe another itable x o that T C Step7: C / ( k x) + C ( )

15 41 Letre Note 9 Behavior o RC ember: Flexre Steel Beam Theor 1 aximm moment o the onrete beam an arr. oment apait o the teel Beam / ( ) The total moment will be: 1 + where A ( Singl reinore beam reahe ltimate train in t Steel Beam Theor Step or anali Step1: Callate b (Fe50) or onrete ailre a a ingl reinore beam. Th, b (Fe415) b (Fe500) Step: Determine the balane p p t 1.97 or Fe50

16 4 A Step3: Determine t 1 p b t 100, Then C or Fe415 or Fe500 / Step4: Fin aitional moment, A( ) A Step5: Fin total moment rom ompreion ailre + 1 Step6: Fin b tenion ailre; A At t At At1 Step7: Fin the total moment b tenion ailre, / ( 0. )( ) 1 + At 87 t Step8: leer vale o & t For given vale o, b, an grae o onrete an teel, in ot Step-1. Fin l k b or onrete ailre. A & At Step-. Callate A t1 oniering balane reinore etion. Step-3. Fin Step-4. Callate At / 0.87 A A + A t t1 t t / ( ) 0 A ( ) Step-5. Fin SP-16. Otherwieε whih epen on '. Fin the orreponing tree rom table F o x /. The vale o then an be on rom the tre-train iagram o teel.

17 43 Step-6. Fin Th A t & A A A t ( 0.87 ) an be on ot Deign Ai (SP-16) Table are eaier than hart Chart 19 &0 give A t vale or (- / ) an Fe50 onl. For other grae o teel, it i to be moiie b table G on page 13 o SP16 Table (45-56) give vale or iret eign o obl reinore beam. p A 100 b p & p t At pt 100 where, A t At1 + At b p p + p t t1 t are obtaine rom the ollowing expreion. We know, pt / (lim) + b(0.87 )( ) 100 (lim) / pt Or, + (0.87 )(1 ) b b 100 Now, an p p t p p + p t1(lim) t t 0.87 aximm & minimm Tenion teel in Beam: aximm teel p p b Here inreae with p in a paraboli relation. For 0 & Fe415, 3.6 p 0.75p

18 44 x From T Cwe an have: 0.36 b t A t x t p For Fe415 & 0: x 0.58 p Dtilit an be meare b rvatre. ε θ x ϕ ε 1 Strain in ompreion ibre Th, R Depth o NA x 1 θ ε φ R x x 0.58 p p Th, Dtilit Hene, thogh 1 p inreae with the inreae o p bt tilit (i.e., rvatre) ereae. Hene, IS oe (Clae 6.5.1) ha pt the pper limit o tenion & ompreion teel a 4%. inimm Steel inimm teel i reqire to take are tilit & hrinkage o onrete. A per lae 6.5.1

19 45 A b % or Fe % or Fe50 Neeit o minimm teel or hear Clae: inimm teel i neear to 1. Prevent brittle hear ailre, whih an or withot hear teel.. Gar againt an en ailre o a beam i onrete over brt an the bon to the tenion i lot. 3. Prevent ailre that an be ae b tenion e to hrinkage an thermal tree an internal raing in the beam. 4. Hol the reinorement in plae while poring the onrete. 5. At a the neear tie or the ompreion teel an make them eetive. inimm paing Av 0.4 bs 0.87 S v v ASv.175 b AS 90 v b or Fe415 AS 544 v b or Fe50 A v total area o tirrp S v Spaing bbreath o web at level o tenion aximm paing Sv 0.75 or vertial tirrp 0 or inline ( 45 ) 300 in an ae

20 46 Letre Note 10 Behavior o RC Beam: Shear Tpe o hear Failre Shear trength o RC Beam (Withot web Reinorement) Total reitane vz + va + v

21 47 vz va Shear in ompreion zone Aggregate interlo ore v Dowel ation rom longitinal bar 1. Tenile trength o onrete Aet inline raing loa. Longitinal reinorement (p) Retrain ra 3. Inreae in the epth o beam Ree hear tre at inline raing 4. Axial tenion Dereae inline raing loa 5. Axial ompreion Inreae inline raing loa Fntion & trength o web reinorement Fntion o web Reinorement Web reinorement i provie to enre that the ll lexral apait i evelope. At a lamp to keep hear ra rom wiening. Shear reite b v apart rom v, v & v. z a inreae a ra wien ntil ieling o tirrp & then tirrp provie v ontant reitant. Flexral Craing: Shear i reite b V z, V a, V & V Deigning to Reit Shear Shear Strength (ACI 318 Se 11.1) φv V [i.e., Capait eman] n

22 48 V atore hear ore at etion Vn Normal hear trength φ 0.85(Shear)-trength retion ator V V + V n V Normal hear reitane provie b onrete V Normal hear provie b the hear reinorement Shear Strength Provie b Conrete Bening onl Simple ormla: ore etaile: V b Eqn [11.3] 3.5 V V pw bw w 3.5 b Eqn [11.6] w b w V Note 1 Bening an Axial Compreion Simple ormla V N 1 + bw Eqn [11.4] 000A g 3.5 N bw A N i poitive or ompreion an ore etaile N g A g Eqn [11.8] are in pi. 4h m N Ue m in Eqn [11.6] with no limit bw 1 Eqn [11.8] 500Ag V + N i poitive or ompreion an N N A g are in pi.

23 49 Bening an Axial Tenion Simple ormla V 0 Deign hear reinorement or all hear V N A g N i negative or tenion an N A g bw are in pi. Lightweight Conrete: Shear Strength Provie b Shear Reinorement inimm Shear Reinorement: (11.5.5) 1 Reqire when V φ V Exept: (a)slab & Footing (b)conrete Joit Contrtion (eine 8.11) ( )Beam with h larger o 10".5 t 1/ b w

24 50 Tpial Shear Reinorement Stirrp - perpenilar to axi o ember (minimm labor - more material) ( in α + oα ) Av V ACI eqn α o 90 V A v Bent Bar (more labor - minimm material) ee reqire in ( in α + oα ) Av V ACI α 1. A 45 o V 41 v

25 51 Deign Proere or Shear 1. Callate V. Callate φv Eqn 11-3 or 11-5 (no axial ore) 3. Che 1 V φv I e, a web reinorement (go to 4) I no, one 1 4. I φv V φv Provie minimm hear reinorement b ( ) w Av A v min 50 or max or min Av 50bw Alo one max 4" (11.5.4) 5. I V φv, allate V (reqire) φ φ + φ V Vn V V V φv V φv V V φ Che, V 8 b (otherwie illegal) (11.5.4) w 6. Solve or reqire tirrp paing (trength) Ame # 3, #4, or #5 tirrp A v rom V 7. Che minimm teel reqirement (eqn 11-13) max A v 50b w 8. Che maximm paing reqirement (ACI ) I V 4 bw max 4" I V 4 bw max 1" Note: V 8 bw 4 (Illegal) 9. Ue mallet paing rom tep 6,7,8 Note: A pratial limit to minimm tirrp paing i 4 inhe.

26 5 Loation o aximm Shear or Beam Deign Non-pre-tree member: Setion loate le than a itane rom ae o pport ma be eigne or ame hear, V, a the ompte at a itane. Compreion an arrie loa iretl into pport. Compreion an arrie loa into pport Loation o aximm Shear or Beam Deign When: 1. The pport reation introe ompreion into the en region o the member.. No onentrate loa or within rom ae o pport. Loation o aximm Shear or Beam Deign