Question

A PSC beam, 200 mm wide and 300 mm deep, is used over an effective span of 6 m to support an imposed load of 4 kN/m. The density of concrete is $24kN/m^3$. At the quarter span section of the beam , the magnitude of eccentric prestressing force located 100 mm from the bottom of beam which would nullify the bottom fibre stress due to loading is ________kN.

• A. 183.6

Answer 1

The correct option is A 183.6
Stress at bottom , $f_b=\frac{P}{A}+\frac{Pe}{Z}-\frac{M_D}{Z}-\frac{M_L}{Z}$

Moment at $x=\frac{L}{4}$

$M=R_A\times \frac{L}{4}-w\times \frac{L}{4}\times \frac{L}{8}=\frac{wL}{2}\times \frac{L}{4}-w\times \frac{L}{4}\times \frac{L}{8}=\frac{3}{32}w{L}^2$

Self weight of beam , $W_d=yBD=24\times 0.2\times 0.3$

= 1.44 kN/m

$\therefore M_D=\frac{3}{32}\times \left(1.44\right)6^2=4.86$kNm

$M_L=\frac{3}{32}\times \left(4\right)6^2=13.5$ kNm

$\therefore f_b=\frac{P}{200\times 300}+\frac{P\times 50}{\frac{200\times {300}^2}{6}}-\frac{4.86\times {10}^6}{\frac{200\times {300}^2}{6}}-\frac{13.5\times {10}^6}{\frac{200\times {300}^2}{6}}=0$

$\Rightarrow P=183.6kN$