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RC03-01

3 -1. 矩形斷面之撓曲強度 Strength of Rectangular Section in Bending. - Load Transferred to Beam from Slabs - Reinforced Concrete Beam Behavior - Moment Strength of Tension-Reinforced Beams - Maximum & Minimum Reinforcements - Design of Tension-Reinforced Beams. RC03-01.

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RC03-01

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  1. 3-1 矩形斷面之撓曲強度 Strength of Rectangular Section in Bending - Load Transferred to Beam from Slabs - Reinforced Concrete Beam Behavior - Moment Strength of Tension-Reinforced Beams - Maximum & Minimum Reinforcements - Design of Tension-Reinforced Beams RC03-01

  2. Cast-in-place with span ratio m = S/L < 0.5 S = Span in short direction L = Span in long direction L Floor load = w kg/sq.m Tributary area = SL sq.m Load on beam = wS kg/m S S S Load Tranferred to Beam from One-Way Slab單向版 RC03-02

  3. Floor load = w kg/sq.m S Tributary area = 0.5SL sq.m Load on beam = 0.5wS kg/m L 預鑄混凝土版Precast Concrete Slab RC03-03

  4. Short span (BC): Floor load = w kg/sq.m Tributary area = S2/4 sq.m Load on beam = wS/4 wS/3 kg/m D C 45o 45o S 45o 45o A B C C B B L Long span (AB): Floor load = w kg/sq.m Tributary area = SL/2 - S2/4 = sq.m Load on beam kg/m 雙向版Two-way Slab Span ratio m = S/L RC03-04

  5. w L Elastic Bending (Plain Concrete) Working Stress Condition C T = As fs 荷重下之樑行為Behavior of Beam under Load RC03-05

  6. 脆性破壞模式Brittle failure mode ecu= 0.003 C Crushing T = As fs 韌性破壞模式Ductile failure mode ec < 0.003 C T = As fs RC03-06

  7. 1) Minimum Depth (for deflection control) oneway slab L/20 L/24 L/28 L/10 BEAM L/16 L/18.5 L/21 L/8 2) Temperature Steel (for slab) b SR24: As = 0.0025 bt SD30: As = 0.0020 bt SD40: As = 0.0018 bt fy > 4,000 ksc: As = 0.0018ด 4,000 bt t As fy 樑設計之規定Beam Design Requirements RC03-07

  8. 3) 最少鋼筋量 Minimum Steel (for beam) As min = minbwd As To ensure that steel not fail before first crack 4) 混凝土保護層 Concrete Covering 箍筋stirrup 縱向鋼筋 Durability and Fire protection > 4/3 max. aggregate size 5) 鋼筋間距 Bar Spacing RC03-08

  9. Modular ratio (n): 樑彎矩之工作應力設計法WSD of Beam for Moment Assumptions: 1) Section remains plane 2) Stress proportioned to Strain 3) Concrete not take tension 4) No concrete-steel slip RC03-09

  10. 有效深度Effective Depth (d) : Distance from compression face to centroid of steel d 開裂轉換斷面Cracked transformed section strain condition force equilibrium compression face C kd N.A. d jd b RC03-10

  11. Compression in concrete: C kd N.A. jd Tension in steel: Equilibrium SFx= 0 : Compression = Tension Reinforcement ratio: 1 RC03-11

  12. kd d 2 Analysis: know r find k 1 2 Design: know fc , fs find k 2 應變協和Strain compatibility: RC03-12

  13. Steel: SR24: fs = 0.5(2,400) = 1,200 ksc SD30: fs = 0.5(3,000) = 1,500 ksc SD40, SD50: fs = 1,700 ksc Example 3.1: = 150 ksc , fs = 1,500 ksc 允許應力Allowable Stresses Plain concrete: Reinforced concrete: RC03-13

  14. Moment arm distance : j d kd/3 M jd T = As fs Steel: Concrete: 抵抗彎矩Resisting Moment RC03-14

  15. R (kgf/cm2) fc (kgf/cm2) n fs=1,200 (kgf/cm2) fs=1,500 (kgf/cm2) fs=1,700 (kgf/cm2) 45 50 55 60 65 12 12 11 11 10 6.260 7.407 8.188 9.386 10.082 5.430 6.463 7.147 8.233 8.835 4.988 5.955 6.587 7.608 8.161 設計步驟Design Step:已知known M, fc, fs, n 1) Compute parameters RC03-15

  16. fs=1,200 (kgf/cm2) fs=1,500 (kgf/cm2) fs=1,700 (kgf/cm2) fc (kgf/cm2) n k j k j k j 0.897 0.889 0.888 0.882 0.883 45 50 55 60 65 12 12 11 11 10 0.310 0.333 0.335 0.355 0.351 0.265 0.286 0.287 0.306 0.302 0.912 0.905 0.904 0.898 0.899 0.241 0.261 0.262 0.280 0.277 0.920 0.913 0.913 0.907 0.908 設計參數Design Parameter k and j 1) For greater fs , k becomes smaller --> smaller compression area 2) j ป 0.9 --> moment arm j d ป 0.9d can be used in approximation design. RC03-16

  17. From 2) Determine size of section bd2 Such that resisting moment of concrete Mc= R b d 2ณ Required M b = 10 cm, 20 cm, 30 cm, 40 cm, . . . Usuallybป d / 2 : d = 20 cm, 30 cm, 40 cm, 50 cm, . . . 3) Determine steel area 4) Select steel bars and Detailing RC03-17

  18. Number of Bars Bar Dia. 1 2 3 4 5 6 RB6 RB9 DB10 DB12 DB16 DB20 DB25 0.283 0.636 0.785 1.13 2.01 3.14 4.91 0.565 1.27 1.57 2.26 4.02 6.28 9.82 0.848 1.91 2.36 3.53 6.03 9.42 14.73 1.13 2.54 3.14 4.52 8.04 12.57 19.63 1.41 3.18 3.93 5.65 10.05 15.71 24.54 1.70 3.82 4.71 6.79 12.06 18.85 29.45 RC03-18

  19. 樑深度之建議值Recommended depth for beam Simple supported One-end continuous Both-ends continuous Cantilever Member One-way slab L/20 L/24 L/28 L/10 L/16 L/18.5 L/21 L/8 Beam L = span length For steel with fy not equal 4,000 kg/cm2 multiply with 0.4 + fy/7,000 RC03-19

  20. w = 4 t/m 5.0 m Example: Working Stress Design of Beam Concrete: fc = 65 kgf/cm2 Steel: fs = 1,700 kgf/cm2 From table: n = 10, R = 8.161 kgf/cm2 Required moment strength M = (4) (5)2 / 8 = 12.5 t-m Recommended depth for simple supported beam: d = L/16 = 500/16 = 31.25 cm USE section 30 x 50 cm with steel bar DB20 d = 50 - 4(covering) - 2.0/2(bar) = 45 cm RC03-20

  21. Steel area: Select steel bar 4DB20 (As = 12.57 cm2) Moment strength of concrete: Mc = R b d2 = 8.161 (30) (45)2 = 495,781 kg-cm = 4.96 t-m < 12.5 t-m NG TRY section 40 x 80 cm d = 75 cm Mc = R b d2 = 8.161 (40) (75)2 = 1,836,225 kg-cm = 18.36 t-m > 12.5 t-m OK RC03-21

  22. Alternative Solution: From Mc = R b d2 = required moment M For example M = 12.5 t-m, R = 8.161 ksc, b = 40 cm USE section 40 x 80 cm d = 75 cm RC03-22

  23. Revised Design due to Support width Column width 30 cm 30 cm 30 cm 4.7 m clear span 5.0 m span Revised Design due to Self Weight From selected section 40 x 80 cm Beam weight wbm = 0.4 ด 0.8 ด 2.4(t/m3) = 0.768 t/m Required moment M = (4 + 0.768) (5)2 / 8 = 14.90 < 18.36 t-m OK Required moment: M = (4.768) (4.7)2 / 8 = 13.17 t-m RC03-23

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