Doubly reinforced beams...PRC-I
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Doubly reinforced beams have both tension and compression reinforcement, allowing for a shallower beam depth than a singly reinforced beam. There are two cases for the behavior of doubly reinforced beams at ultimate loading: 1) Case I occurs when both tension and compression steel yield. The neutral axis depth can be calculated and the moment capacities from compression steel, concrete, and tension steel determined. 2) Case II occurs when only the tension steel yields, and the compression steel does not yield. The strain in the compression steel must be calculated. The document discusses the behavior of doubly reinforced beams under ultimate loading conditions for both cases when compression steel does and does not yield. It provides equations to calculate forces, strains, and moment
Doubly reinforced beams...PRC-I
- 1. Plain and Reinforced Concrete- 1 Doubly Reinforced Beams ByEngr.RafiaFirdous
- 2. Plain & Reinforced Concrete-1 Doubly Reinforced Beams “Beams having both tension and compression reinforcement to allow the depth of beam to be lesser than minimum depth for singly reinforced beam” • By using lesser depth the lever arm reduces and to develop the same force more area of steel is required, so solution is costly. • Ductility will be increased by providing compression steel. • Hanger bars can also be used as compression steel reducing the cost up to certain cost. • For high rise buildings the extra cost of the shallow deep beams is offset by saving due to less story height.
- 3. Plain & Reinforced Concrete-1 Doubly Reinforced Beams (contd…) • Compression steel may reduce creep and shrinkage of concrete and thus reducing long term deflection. • Use of doubly reinforced section has been reduced due to the Ultimate Strength Design Method, which fully utilizes concrete compressive strength. Doubly Reinforced Beam
- 4. Behavior Doubly Reinforced Beams Tension steel always yields in D.R.B. There are two possible cases: 1. Case-I Compression steel is yielding at ultimate condition. 2. Case-II Compression steel is NOT yielding at ultimate condition.
- 5. Behavior Doubly Reinforced Beams Cc T = Asfs N.A. εcu= 0.003 Strain Diagram Internal Force Diagram εs h c d b 0.85fc a Whitney’s Stress Diagram (d-d’) fs d εs’ fs’ Cs d – a/2 T = Asfs Cs=As’fs’ Cc=0.85fc’ba fs=Esεs fs’=Esεs’
- 6. Behavior Doubly Reinforced Beams (contd…) Case-I Both Tension & Compression steel are yielding at ultimate condition fs = fy and fs’=fy Location of N.A. Consider equilibrium of forces in longitudinal direction sc CCT yscys f'Aba'f85.0fA b'f85.0 f'AA a c yss 1β a c and
- 7. Case-I Both Tension & Compression steel are yielding at ultimate condition (contd…) c d'c 0.003 'εs εcu= 0.003 Strain Diagram εs c εs’ d’ B D E C A Δ ABC & ADE c d'c 0.003'εs 1 1 s β β c d'c 0.003'ε a d'βa 0.003'ε 1 s If εs’ ≥ εy compression steel is yielding. If εs’ < εy compression steel is NOT yielding. (1)
- 8. Case-I Both Tension & Compression steel are yielding at ultimate condition (contd…) Cc T = Asfy N.A. Internal Force Diagram (d-d’) Cs d – a/2 T = total tensile force in the steel 21 TTT T1 is balanced by Cs T2 is balanced by Cc s1 CT c2 CT
- 9. Case-I Both Tension & Compression steel are yielding at ultimate condition (contd…) Cc T = Asfy N.A. Internal Force Diagram (d-d’) Cs d – a/2 Moment Capacity by Compression Steel 'dd'f'A'ddCM yssn1 'ddT1 Moment Capacity by Concrete 2 a dT 2 a dCM 2cn2 2 a dTTM 1n 2 2 a d'f'AfAM ysysn 2
- 10. Case-I Both Tension & Compression steel are yielding at ultimate condition (contd…) Total Moment Capacity 21 nnn MMM 2 a d'f'AfA'dd'f'AM ysysysn
- 11. Case-II Compression steel is not yielding at ultimate condition fs = fy and fs’< fy 'εE'f ss b'f85.0 'f'AfA a c ssys 1β a c and a d'βa 600'f 1 s Location of N.A.
- 12. Concluded