Plain & Reinforced Concrete-1 CE3601 Lecture # 13, 14 &15 13th to 20th April 2012 Flexural Analysis and Design of Beams (Ultimate Strength Design of Beams)

Plain & Reinforced Concrete-1 Load Carried by the Beam Beam Supporting One-way Slab (ly/lx > 2) Exterior Beam Interior Beam lx lx ly Width of slab supported by interior beam = lx Width of slab supported by exterior beam = lx/2 + Cantilever width

Plain & Reinforced Concrete-1 Load Carrie by the Beam Beam Supporting Two-way Slab (ly/lx≤ 2) Exterior Long Beam 45o Exterior Short Beam lx Interior Long Beam lx ly ly Interior Short Beam

Plain & Reinforced Concrete-1 Load Carrie by the Beam Beam Supporting Two-way Slab (ly/lx≤ 2) contd… Shorter Beams For simplification this triangular load on both the sides is to be replaced by equivalent UDL, which gives same Mmax as for the actual triangular load. lx/2 45o lx/2 Area of Square

Plain & Reinforced Concrete-1 Load Carried by the Beam Beam Supporting Two-way Slab (ly/lx≤ 2) contd… Equivalent Rectangular Area 45o 45o lx Factor of 4/3 convert this VDL into UDL. Equivalent width supported by interior short beam Equivalent width supported by exterior short beam Cantilever

Plain & Reinforced Concrete-1 Load Carried by the Beam Beam Supporting Two-way Slab (ly/lx≤ 2)contd… Exterior Long Beam lx/2 ly - lx lx/2 Supported Area lx lx ly

Plain & Reinforced Concrete-1 Load Carried by the Beam Beam Supporting Two-way Slab (ly/lx≤ 2)contd… Exterior Long Beam Factor F converts trapezoidal load into equivalent UDL for maximum B.M. at center of simply supported beam. where For Square panel R = 1 and F = 4/3

Plain & Reinforced Concrete-1 Load Carried by the Beam Beam Supporting Two-way Slab (ly/lx≤ 2)contd… Exterior Long Beam lx/2 ly - lx lx/2 Equivalent width Equivalent width ly + Cantilever (if present)

Plain & Reinforced Concrete-1 Load Carried by the Beam Beam Supporting Two-way Slab (ly/lx≤ 2) contd… Interior Long Beam lx/2 ly - lx lx/2 Equivalent width Equivalent width ly

Plain & Reinforced Concrete-1 Wall Load (if present) on Beam tw (mm) UDL on beam H (m) (kN/m)

Plain & Reinforced Concrete-1 Wall Load on the Lintel Equivalent UDL on lintel if height of slab above lintel is greater than 0.866L 0.866L 60o 60o kN/m L tw = wall thickness in “mm” L = Opening size in “m” If the height of slab above lintel is less than 0.866L Total Wall Load + Load from slab in case of load bearing wall UDL = (Equivalent width of slab supported) x (Slab load per unit area) = m x kN/m2 = kN/m

Plain & Reinforced Concrete-1 Slab Load per Unit Area Top Roof Slab Thickness = 125 mm Earth Filling = 100 mm Brick Tiles = 38 mm Dead Load Self wt. of R.C. slab Earth Filling Brick Tiles Total Dead Load, Wd = 554 kg/m2

Plain & Reinforced Concrete-1 Slab Load per Unit Area (contd…) Top Roof Live Load WL = 200 kg/m2 Total Factored Load, Wu

Plain & Reinforced Concrete-1 Slab Load per Unit Area (contd…) Intermediate Floor Slab Thickness = 150 mm Screed (brick ballast + 25% sand) = 75 mm P.C.C. = 40 mm Terrazzo Floor = 20 mm Dead Load Self wt. of R.C. slab Screed Terrazzo + P.C.C Total Dead Load, Wd = 633 kg/m2

Plain & Reinforced Concrete-1 Slab Load per Unit Area (contd…) Intermediate Floor Live Load Occupancy Live Load = 250 kg/m2 Moveable Partition Load = 150 kg/m2 WL = 250 + 150 = 400 kg/m2 Total Factored Load, Wu

Plain & Reinforced Concrete-1 Slab Load per Unit Area (contd…) Self Weight of Beam Service Self Wight of Beam = b x h x 1m x 2400 Kg/m Factored Self Wight of Beam kN/m Self weight of beam is required to be calculated in at the stage of analysis, when the beam sizes are not yet decided, so approximate self weight is computed using above formula.

Plain & Reinforced Concrete-1 Bar Bending Schedule Serial # Bar Designation Number of Bars Length of one Bar Dia of bar Weight of Steel Required Shape of Bar #10 #13 #15 #19 #25 1 M-1 2 S-1 3 H-1 Σ Total weight of steel = 1.05Σ , 5% increase, for wastage during cutting and bending

Plain & Reinforced Concrete-1 Bar Bending Schedule (contd…) Bent-up Bar h h 45o L Additional Length = 0.414 h Total Length = L + 0.414 h

Plain & Reinforced Concrete-1 Bar Bending Schedule (contd…) 90o-Standard Hooks (ACI) L db R = 4db for bar up-to #25 R = 12db R = 5db for bar #29 & #36 Total Length = L + 18db For R 4db

Plain & Reinforced Concrete-1 Bar Bending Schedule (contd…) 180o-Standard Hooks (ACI) L db Same as 90o hook 4db Total Length = L + 20db

Plain & Reinforced Concrete-1 Bar Bending Schedule (contd…) Example: Prepare bar bending schedule for the given beam. Clear cover = 40 mm 2-#10 #10 @ 180 c/c 1-# 15 2-#20 570 2-#20 +1-#15 4000 Longitudinal Section 228

Plain & Reinforced Concrete-1 Bar Bending Schedule (contd…) Example: Prepare bar bending schedule for the given beam. Clear cover = 40 mm 2-#10 (H-1) #10 @ 180 c/c 375 (S-1) 2-#20 (M-1) 1-# 15 (M-2) 228 Cross Section

Plain & Reinforced Concrete-1 Bar Bending Schedule (contd…) Example: M-1 = 4000 + 2 x 228 – 2 x 40 + 2 x (18 x 20) = 5096 M-1 M-2 h = 375 – 2 x 40 – 2 x 10 – 2 (15/2) = 260 h M-2 = 4000 + 2 x 228 – 2 x 40 + (0.414 x 260) x 2 = 5091 H-1 H-1 = 4000 + 2 x 228 – 2 x 40 = 4376

Plain & Reinforced Concrete-1 Bar Bending Schedule (contd…) Example: Prepare bar bending schedule for the given beam. Clear cover = 40 mm Shear stirrups a Number of Bars = 4000 / 180 +1 = 24 Round-up a = 375 – 2 x 40 – 10 = 285 mm b = 228– 2 x 40 – 10 =138 mm b Total length of S-1 = 2 (138 + 285 + 18 x 10) = 1206 mm

Plain & Reinforced Concrete-1 Bar Bending Schedule Serial # Bar Designation Number of Bars Length of one Bar (m) Dia of bar Weight of Steel Bars Shape of Bar #10 #15 #20 1 M-1 2 5.096 20 24 M-2 4.591 15 7.2 3 H-1 4.376 10 6.9 4 S-1 1.206 22.7 1.05 Σ 31.1 7.6 25.2 4376 4376 4376 138 285 Total Weight = 64 kg

Plain & Reinforced Concrete-1 Design of Singly Reinforced Beam by Strength Method (for flexure only) Data: Load, Span (SFD, BMD) fc’, fy, Es Architectural depth, if any Required: Dimensions, b & h Area of steel Detailing (bar bending schedule)

Plain & Reinforced Concrete-1 Design of Singly Reinforced Beam by Strength Method (contd…) Procedure: Select reasonable steel ratio between ρmin and ρmax. Then find b, h and As. Select reasonable values of b, h and then calculate ρ and As.

Plain & Reinforced Concrete-1 Design of Singly Reinforced Beam by Strength Method (contd…) Using Trial Dimensions Calculate loads acting on the beam. Calculate total factored loads and plot SFD and BMD. Determine Vumax and Mumax. Select suitable value of beam width ‘b’. Usually between L/20 to L/15. preferably a multiple of 75mm or 114 mm. Calculate dmin. hmin = dmin + 60 mm for single layer of steel hmin = dmin + 75mm for double layer of steel Round to upper 75 mm

Plain & Reinforced Concrete-1 Design of Singly Reinforced Beam by Strength Method (contd…) Decide the final depth. For strength For deflection Architectural depth Preferably “h” should be multiple of 75mm. Recalculate “d” for the new value of “h”

Plain & Reinforced Concrete-1 Design of Singly Reinforced Beam by Strength Method (contd…) Calculate “ρ” and “As”. Four methods a) b) Design Table Design curves Using trial Method c) d)

Plain & Reinforced Concrete-1 Design of Singly Reinforced Beam by Strength Method (contd…) Check As ≥ As min. As min = ρmin bd (ρmin = 1.4/fy to fc’ ≤ 30 MPa) Carry out detailing Prepare detailed sketches/drawings. Prepare bar bending schedule.

Plain & Reinforced Concrete-1 Design of Singly Reinforced Beam by Strength Method (contd…) Using Steel Ratio Step I and II are same as in previous method. Calculate ρmax and ρmin & select some suitable “ρ”. Calculate bd2 from the formula of moment Select such values of “b” and “d” that “bd2” value is satisfied. Calculate As. Remaining steps are same as of previous method.

Concluded