Purpose of reinforcement P/2 P/2 P/2 P/2

Transcription

1 Department o Civil Engineering Purpose o reinorement Consider a simpl supported beam: P/2 P/ P/2 P/

2 Purpose o Reinorement Steel reinorement is primaril use beause o the nature o onrete tension apait 1. Flexural (tension) reinorement: provides neessar tension apait aross lexural raks 2. Shear reinorement (stirrups): arries shear aross inlined raks. 3. Compression reinorement: Inreases ompression apait o onrete i needed Support or stirrups Inreases setion dutilit Derease deletion Load (P) E (Failure) D (ield point) P C (Servie load) Δ B (raking point) A (beore raking) Deletion (Δ) 2

3 Point A (Beore raking): ε -No raks -Small strains -linear stress distribution ε<ε r < r Point B (Craking point): ε -Craks initiate -Tensile ore arried b reinorement -linear stress distribution ε r r Point C (Servie load point): ε -Craks propagate -Less onrete is eetive -Close to linear stress distribution ε>ε r s Point D (Yield point): -Reinorement reah ield ε -Less onrete is eetive -Close to linear stress distribution ε s = ε s = Point E (Failure point): -Crushing o onrete ε -Steel wa pass ield point -nonlinear stress distribution ε s > ε s = Failure point deines the ultimate apait o the reinored onrete setion ultimate limit state 3

4 Linear Stress Distribution Craking Moment M σ = σ M I M r t r = I M r where I g t M = I g g r 3 bh = ; 12 = h / 2; r t r = 0.7 = bh 2 ' ' ; h At the verge o raking: b t Usuall, when servie load is applied, M > M r and raks develop throughout the beam s length. ε r ε r 4

5 Basi Assumption in Flexure Theor Plane setions beore bending remain plane ater bending, i.e., linear strain distribution. Strains in reinorement and onrete are equal at the same level, i.e., peret bond. Neglet tensile strength in onrete Conrete ails at strain ε u = Flexural Capait (Strength) M n = Nominal moment apait, or nominal lexural strength. Nominal = alulated or theoretial φm n = Design strength, the reliable strength we an ount on or the real beam. φ = Capait redution ator, whih aounts or: Poor workmanship Conrete ompressive strength less than design value Inauraies in theories o analsis and design 5

6 Analsis o R/C Beams Consider a simpl supported beam reinored or positive moment, at ultimate ondition b b h d ε u h NA d d- C jd A s T s ε su >ε Strain Stress/Fore Deinitions b = beam width h = overall depth d = eetive depth, distane rom extreme ompression iber to entroid o tension steel A s = Area o tension reinorement ε u = Ultimate strain in onrete = ε su = Ultimate strain in steel = loation o neutral axis, distane rom extreme ompression iber to C = Conrete ompressive ore T s = Steel tension ore jd = Moment lever arm 6

7 Equivalent retangular stress blok (Ch. 4, art. 4.3) κ 3 C κ 2 a=κ 1 κ 2 C = κ 1 κ 3 b T s Atual T s Equivalent A = Conrete ompression area Conrete ompressive ore rom atual and equivalent must be equal Loation o ompression ore must also be equal 7

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9 ACI Retangular Stress Blok Also alled Whitne stress blok 0.85 a=β 1 C = 0.85 A β / 7( 30) 0.85 T s (MPa) Flexural Analsis 9

10 Ultimate Limit State (Flexural Capait) b ε u = d a=β 1 C = 0.85 A ε su T s Equilibrium: assume tension steel ields at ultimate, i.e., ε su >ε, stress in steel = C = T s 0.85 A = A s Retangular setion: A = ab; so ind a As a = 0.85 ; = a / ' b Compatibilit: Chek strain in tension steel ε su ε u d = ε su = ( )( ε u ) > ε? d Nominal and Design Moment Capait M n φm n = A s ( d ) = 0.85 ' = φa ( d ) = φ0.85 s β 1 A ( d ) ' A ( d ) = entroid o ompression ore = a/2 or retangular setions 10

11 Capait Redution Fator (φ) Wh Capait redution ator? Undersized members Bars plaed out o position Strength o onrete less than speiied ACI (9.2) ε su φ = > ε su > φ = ε su ε su = φ = 0.81 ε su < Setion is rejeted b the ACI ode Analsis Example 11