Section 1

Section 2

Section 3

Section 5

Section 6

Section 7

Section 8

Widht of the Beam 1) It is desirable that the width of the beam should be less than or equal to the width of its supporting structure like column width, or width of the wall etc. 2) Again, width to overall depth ratio is normally kept between 0.5 and 0.67. Selection of depths of beam d and D 1) The initial effective depth of the beam, however, is assumed to satisfy the deflection requirement depending on the span and type of the reinforcement. 2) The total depth D can be determined by adding 40 to 80 mm to the effective depth. 3) IS 456 stipulates the basic ratios of span to effective depth of beams for span up to 10 m as (Clause 23.2.1) Cantilever 7 Simply supported 20 Continuous 26 4) For spans above 10 m, the above values may be multiplied with 10/span in metres, except for cantilevers where the deflection calculations should be made. Further, these ratios are to be multiplied with the modification factor depending on reinforcement percentage and type. Figures 4 and 5 of IS 456 give the different values of modification factors.

Selection of the amount of steel reinforcement Ast 1) Minimum Reinforcement: The minimum reinforcement As is provided for creep, shrinkage, thermal and other environmental requirements irrespective of the strength requirement. The minimum reinforcement As to be provided in a beam depends on the fy of steel and it follows the relation: (cl. 26.5.1.1a of IS 456) 2) Maximum Reinforcement: The maximum tension reinforcement should not exceed 0.04 bD (cl. 26.5.1.1b of IS 456), where D is the total depth. 3) General Rule Besides satisfying the minimum and maximum reinforcement, the amount of reinforcement of the singly reinforced beam should normally be 75 to 80% of pt, lim. This will ensure that strain in steel will be more than (0.85fy/Es+0.002) as the design stress in steel will be 0.87 fy. Moreover, in many cases, the depth required for deflection becomes more than the limiting depth required to resist Mu, lim. Thus, it is almost obligatory to provide more depth. Providing more depth also helps in the amount of the steel which is less than that required for Mu, lim. This helps to ensure ductile failure. Such beams are designated as under-reinforced beams.

Grade of Concrete: Table 5 of IS 456 recommends M 20 as the minimum grade under mild environmental exposure (duralbility requirement) and other grades of concrete under different environmental exposures also. Grade of Stell Normally, Fe 250, 415 and 500 are in used in reinforced concrete work.Mild steel (Fe 250) is more ductile and is preferred for structures in earthquake zones or where there are possibilities of vibration, impact, blast etc. Effective Span Clause 22.2(a) of IS 456 recommends that the effective span is the lower of (i) clear span plus effective depth and (ii) centre to centre distance between two supports. Unit weight of reinforced concrete: With the unit weight of reinforced concrete as 25 kN/m3 (cl. 19.2.1 of IS 456):

fck fy

Grade of Concrete Grade of Steel Span of the Beam (center to center) 1 - Effective Depth (Section2.3) (based on deflection criteria) 1- Effective Span Leff Case 1: Clear span + Effective depth Case 2: Center to center distance Min of (case1 and case2) xumax/d ptlim (= Ast/bd*100) 1 - Ast Tensile Steel (75% to 80% of ptlim)

= = = =

20 415 8 400

(M20) (Fe415) m mm

d

Leff

= = = = = =

8100 8000 8000 0.48 0.96 0.72

mm mm mm

xumax/d ptlim Ast

Effective Depth based on modification factor - which is based on grade of steel and area of steel fs = 0.58 fy (Ast req/Ast prov) = 240.7 N/mm2 Modification factor: = From Fig. 4 of IS 456, the required modification factor is found to be for fs = 240.7 N/mm2 and Ast = 0.72 d 2 - Effective depth based on Mod factor Say = = 1.1

364 365

mm mm

Leff

2 - Effective Span Leff Case 1: Clear span + Effective depth Case 2: Center to center distance Min of (case1 and case2) Clear cover 1 - Total Depth of Beam Width of Beam Assume b/D ratio (between 0.5 to 0.67) Factored Bending Moment ((LL+DL)*1.5) 3 - Effective depth based on Mu = Mulim Revised d 2 - Total Depth of Beam

= = = = =

8065 8000 8000 50 415

mm mm mm mm mm

Cc D

b

= = =

250 mm 0.6 ok 115.08 kNm

Mu

d D

= = =

408.7 mm 410 mm 460 mm

Ast Pst d Ast Pst Mulim

Ast Based on d Percentage of steel 4 - Effective depth based on Ast and Assump Ast Based on d Percentage of steel Mulim

= = = = = =

966.52 mm2 0.94 % 450 mm 837.75 mm2 0.74 % 139.49 kNm

(Leff/20)

d area of steel (assume Ast req = Ast prov)

Xu/dComes from

Xumax

that value of xu which will satisf

Lever Arm:

Limiting value

(ignore 1.015)

Astlim

Mu

Xumaxthat value of xu which will satisfy assumptions (ii) and (vi) of sec. 3.4.2 and designate that by xu, max

Limiting value of xu

Astlim

fck fy b d D Ast

= = = = = =

20 415 300 550 600 1256 mm2

xumax/d xumax ptlim Mulim

(dep - fy) (dep - fy and fck)

= = = = = = = =

0.48 263.51 mm 0.96 % 250.06 kNm 0.38 209.94 mm 0.76 % 209.4 kNm

xu/d xu Ast/bd*100 Mu

fck fy b d D Ast Span xumax/d xumax ptlim Astlim Mulim (dep - fy) (dep - fy and fck)

= = = = = = = = = = = = = = = = = = = = =

20 415 1000 110 127 201 1.5 0.48 52.73 0.96 1051.61 33.38 0.09 10.08 0.18 7.7 27.3 18.2 18.2 1.7 169.2

Mpa Mpa mm mm mm mm2 m

127

mm % mm2 kNm

xu/d xu Ast/bd*100 Mu Wu (factored Load = Mu * 8/span2) W (Wu/1.5) W/sqm = W x b

mm % kNm kN/m kN/m KN/m2 KN/ft2 kg/ft2

Load LL (2 KN/m2) DL (25 KN/m3) Total (Wapp) Wapp/sqm = W x b

= = = = = =

2 3.175 5.175 5.4 0.5 50.6

KN/m KN/m KN/m KN/m2 KN/ft2 kg/ft2

200 kg/cm2 4150 kg/cm2 102 2.812148

Conversion - KN/m3 25 KN/m3 2500 kg/m3 70.8 kg/ft3 156.1 pf/ft3 For given thk = 29.50 kg/ft2 5 in

0.1 (1N to 0.1 kg weight = 0.1 x g = 0.1 * 10) 3.28084 (1m to 3.28084 ft) (7 lt = 0.25 cft) (1pound = 0.4536 kg)

Conversion - KN/m2 2 KN/m2 200 kg/m2 19 kg/ft2

(1m2 = 10.76 ft2) (1m3 = 35.31 ft3)

1 in = 25.4mm b D Asc 2 tor 16 Ast 2 tor 16 fck fy Cc Clear cover

= = = = = = =

228.6 457.2 402.1239 402.1239 20 415 25

in in mm2 mm2

mm

d xumax

= =

424.2 203.2374