Question

A prestressed concrete beam 150 mm $\times$ 300 mm supports a live load 5 kN/m over a simple span of 8 m. It has a parabolic cable having an eccentricity of 75 mm at mid-span and zero at the ends. The prestressing force required to maintain the net resultant stress at the bottom fibre at mid-span as zero under the action of DL + LL + prestress is

• A. 239 kN
• B. 293 kN
• C. 302 kN
• D. 392 kN

Answer 1

The correct option is D 392 kN

$\begin{array}{cc}\text{Area,}A& =45\times {10}^{3}m{m}^2\\ & \text{Section modulus,}\\ Z& =\frac{B{d}^2}{6}=\frac{150\times {300}^2}{6}\\ & =2.25\times {10}^{6}m{m}^2\\ \text{Dead load}& =0.15\times 0.30\times 1\times 25\\ & =1.125KN/m\\ & \text{For bottom fibre stress to be zero}\\ \frac{P}{A}+\frac{Pe}{Z}& =\frac{M_{DL}+M_{LL}}{Z}\\ M_{DL}+M_{LL}& =\frac{(1.25+5)\times 8^2}{8}\\ & =49kN-m\\ e& =75mm\\ \therefore P& =\frac{49\times {10}^6}{2.25\times {10}^6[\frac{1}{45\times {10}^3}+\frac{75}{2.25\times {10}^6}]}\\ & =392kN\end{array}$