Question 6.2: Three beams have same length, allowable stress and bending m...
Three beams have same length, allowable stress and bending moment (Figure 6.11). Find the ratios of weights of circular and rectangular beam with respect to square beam.
As for the three cases, allowable stress, bending moment and section modulus are same, that is Z_1=Z_2=Z_3 .
For square cross-section,
Z_1=\frac{b d^2}{6}=\frac{m \times m^2}{6}=\frac{m^3}{6}
For rectangular cross-section,
Z_2=\frac{b d^2}{6}=\frac{b(2 b)^2}{6}=\frac{2 b^3}{3}
For circular cross-section,
Z_3=\frac{\pi d^3}{32}
Equating Z_1 \text { and } Z_2 , we get
\frac{m^3}{6}=\frac{2 b^3}{3}
or m³ = 4b³
Substituting, we get b = 0.63m. Similarly, equating Z_1=Z_3 , we get
\frac{m^3}{6}=\frac{\pi d^3}{32}
or d = 1.19 m
Now,
\begin{aligned}\frac{\text { Weight of square beam }}{\text { Weight of rectangular beam }} & =\frac{\text { Cross-section of square beam }}{\text { Cross-section of rectangular beam }} \\ & =\frac{m^2}{2 b^2}=\frac{m^2}{2(0.63 m)^2} \end{aligned}
which gives ratio of weights of square beam to rectangular beam as 1/0.79. Similarly,
\begin{aligned} \frac{\text { Weight of square beam }}{\text { Weight of circular beam }} & =\frac{\text { Cross-section of square beam }}{\text { Cross-section of circular beam }} \\ & =\frac{m^2}{(\pi / 4) d^2}=\frac{m^2}{(\pi / 4) \times(1.19 m)^2} \end{aligned}
which gives ratio of weights of square beam to circular beam as 1/1.1.