Question 7.13: A water main of 500 mm internal diameter and 20 mm thick is ...

Given :
Internal dia.,          D_i= 500  mm = 0.5  m
Thickness of pipe, t = 20 mm
∴ Outer dia.,     D_0 = D_i + 2 × t = 500 + 2 × 20 = 540  mm = 0.54  m
Weight density of cast iron = 72000 N/m³
Weight density of water = 10000 N/m³
Internal area of pipe       = \frac{\pi}{4}D_i^2=\frac{\pi}{4} \times 0.5^2 = 0.1960  m^2
This is also equal to the area of water section.

∴ Area of water section= 0.196 m²
Outer area of pipe = \frac{\pi}{4}D_0^2=\frac{\pi}{4}\times 0.54^2  m^2

∴ Area of pipe section = \frac{\pi}{4}D_0^2  –  \frac{\pi}{4}D_i^2 \\ \quad \quad \quad \quad \quad =\frac{\pi}{4}[D_0^2-D_i^2] = \frac{\pi}{4}[0.54^2-0.5^2]=0.0327  m^2

Moment of inertia of the pipe section about neutral axis,

I=\frac{\pi}{64}[D_0^4-D_i^4]=\frac{\pi}{64}[540^4-500^4]=1.105\times 10^9  mm^4

Let us now find the weight of pipe and weight of water for one metre length.
Weight of the pipe for one metre run

= Weight density of cast iron × Volume of pipe
= 72000 × [Area of pipe section × Length]
= 72000 × 0.0327 × 1                    (∵ Length = 1 m)
= 2354 N

Weight of the water for one metre run

= Weight density of water × Volume of water
= 10000 × (Area of water section × Length)
= 10000 × 0.196 × 1 = 1960 N

∴ Total weight on the pipe for one metre run

= 2354 + 1960 = 4314 N

Hence the above weight is the U.D.L. (uniformly distributed load) on the pipe. The maximum bending moment due to U.D.L. is w × L²/8, where w = Rate of U.D.L. = 4314 N per metre run.

∴ Maximum bending moment due to U.D.L.,

M=\frac{w\times L^2}{8}=\frac{4314\times 10^2}{8} \quad (∵  L=10  m) \\ = 53925  Nm = 53925 × 10^3  N mm

Now using          \frac{M}{I}=\frac{\sigma}{y}

∴                           \sigma=\frac{M}{I}\times y

The stress will be maximum, when y is maximum. But maximum value of

y=\frac{D_0}{2}=\frac{540}{2}=270  mm

∴           y_{\max}=270  mm

∴ Maximum stress, \sigma_{\max}=\frac{M}{I}\times y_{\max}=\frac{53925 \times 10^3}{1.105 \times 10^9}\times 270 \\  \quad \quad \quad \quad \quad \quad =\pmb{13.18  N/mm^2.}