Q.An R.C. beam 200 mm wide and 500 mm deep effective is reinforced with 3 nos. of 16 mm dia bars. Find the moment of resistance of the beam by using working stress method. Concrete grade M20 and mild steel reinforcement are used.
SOLUTION:
WIDTH b= 200mm
EFFECTIVE DEPTH d = 500 mm
AREA OF TENSION STEEL Ast = 3× π /4 ×16² = 603.2mm²
1. PERMISSIBLE STRESSES.
σcbc = 7 N/mm²
σst = 140 N/mm²
2. MODULAR RATIO
•m = 280/3σcbc
= 280/ 3×7
= 13.33
•Critical depth of neutral axis NC = 0.4d
3. DEPTH OF NEUTRAL AXIS
Equating the moments of areas of compression and tension zone about the neutral axis
bn²/2= m.Ast(d – n)
1/2×200 n² = 13.33 x 603.2 (500 – n)
100 n² = 4.02×10⁶ – 8040.7 n
n² +80.4 n – 40203= O
n= (- 80.4 + √80.4² +4 × 40203)/2
n = 164.3 mm
critical depth of neutral axis Nc= 0.4×500 = 200mm
since n < nc, the section’s is under Reinforced
4. MOMENT OF RESISTANCE
MR = σst . Ast (d-na/3)
= 140×603.2 (500-164.3/3)
=37.6×10⁶ N-mm = 37.6kNm
MOMENT OF RESISTANCE = 37.6 kNm
5. MAXIMUM LOAD:
let w KN/m be the uniformly distributed load the beam can carry maximum Bending moment = wI²/8
=w×5²/8
=3.125
Equating maximum bending moment to the moment of Resistance Of the section
3.125 w = 37.6
w = 37.6/3.125
= 12.032kN/m ( including self weight)
