Question

g A steel water pipe has an inner diameter of 12 in. and a wall thickness of 0.25 in. Determine the longitudinal and hoop stresses developed in the wall of the pipe at point B for the two cases below. Also, for each case draw the state of stress on a volume element located on the wall at point B. (a) The valve A is opened and the flowing water has a pressure of 200 psi. (b) The valve A is closed and the water pressure is 250 psi.

Answer 1

a)
`\mathbf{\sigma _ 1 = 4800 psi}`

`\mathbf{ \sigma _2 = 0}`

b)
`\mathbf{\sigma _ 1 = 6000 psi}`

`\mathbf{ \sigma _2 = 3000 psi}`

Step-by-step explanation:

Given that:

diameter d = 12 in

thickness t = 0.25 in

the radius = d/2 = 12 / 2 = 6 in

r/t = 6/0.25 = 24

24 > 10

Using the thin wall cylinder formula;

The valve A is opened and the flowing water has a pressure P of 200 psi.

So;

`\sigma_(hoop) = \sigma _ 1 = (Pd)/(2t)`

`\sigma_(long) = \sigma _2 = 0`

`\sigma _ 1 = (Pd)/(2t) \\ \\ \sigma _ 1 = (200(12))/(2(0.25))`

`\mathbf{\sigma _ 1 = 4800 psi}`

b)The valve A is closed and the water pressure P is 250 psi.

where P = 250 psi

`\sigma_(hoop) = \sigma _ 1 = (Pd)/(2t)`

`\sigma_(long) = \sigma _2 = (Pd)/(4t)`

`\sigma _ 1 = (Pd)/(2t) \\ \\ \sigma _ 1 = (250*(12))/(2(0.25))`

`\mathbf{\sigma _ 1 = 6000 psi}`

`\sigma _2 = (Pd)/(4t) \\ \\ \sigma _2 = (250(12))/(4(0.25))`

`\mathbf{ \sigma _2 = 3000 psi}`

The free flow body diagram showing the state of stress on a volume element located on the wall at point B is attached in the diagram below