Question

A reinforced concrete beam, size 200 mm wide and 300 mm deep overall is simply supported over a span of 3 m. It is subjected to two point loads P of equal magnitudes placed at middle third points. The two loads are gradually increased simultaneously. Beam is reinforced with 2 HYSD bars of 16 mm diameter placed at an effective cover of 40 mm on bottom face and nominal shear reinforcement. The characteristic compressive strength and the bending tensile strength of the concrete are 20.0 N/mm${}^2$ and 2.2 N/mm${}^2$ respectively.
The theoretical failure load of the beam for attainment of limit state of collapse in flexure is

• A. 23.7 kN
• B. 25.6 kN
• C. 28.7 kN
• D. 31.6 kN

Answer 1

The correct option is D 31.6 kN

Effective depth
d = 300 - 40
= 260 mm


Maximum depth of Neutral Axis
$x_{umax}=0.48$
d = 0.48$\times$ 260
= 124.8 mm
Actual depth of Neutral Axis ($X_u$)
C = T
$0.36f_{ck}.b.x_u=0.87f_y.A_{st}$
$0.36\times 20\times 200x_u=0.87\times 415\times (\pi \frac{{16}^2}{4}\times 2)$
$x_u=100.82mm$
$x_u<x_{umax}$ under - reinforced section

Moment of Resistance
$M=0.36f_{ck}.b.x_u\times (d-0.42x_u)$
$=0.36\times 20\times 200\times 100.82(260-0.42\times 100.82)$
$=31.599\times {10}^6$ N
= 31.6 kN-m

At the time of limit state of collapse, M.O.R will be equal to max moment generated in the beam.
Maximum moment generated
= 1000 P
$=31.599\times {10}^6$ N-mm
P = 31599.41 N = 31.6 kN