---
title: "Pipe wall stress for 3 s valve closure"
subject: "Hydraulics"
chapter: 4
key: "semester_4/hydraulics:valve-closure-3s-stress-in-pipe-wall"
source: https://www.bctengineer.com/q/semester_4/hydraulics/valve-closure-3s-stress-in-pipe-wall
site: BCT Engineering
---

# Pipe wall stress for 3 s valve closure

Hydraulics · Chapter 4: Unsteady flow in pipes

- Asked in: 2072 Magh, 6 marks
- Video: [Derivation On Sudden Closure Of Valve When Pipe Is Elastic - Pipe Network and Water Hammer Effect](https://www.youtube.com/watch?v=2iDmOeajflE) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/valve-closure-3s-stress-in-pipe-wall
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-4

## Question

Water flows through a 25 cm diameter 1500 m long pipe at rate of 75 lps. The static pressure of water in the pipe is 200 m at the downstream end of the pipe and the thickness of the pipe material is 6 mm. If a valve at the downstream end closed in 3 sec estimate the stress in the pipe wall. Take Bulk modulus of water = $2.2\times10^9$ N/m² and Young's modulus of elasticity of steel = $2.1\times10^{11}$ N/m².

## Short answer

Velocity is 1.528 m/s, wave celerity 1237.5 m/s; 2L/c = 2.42 s < 3 s, so closure is gradual. Pressure rise Δp = ρLV/tc = 0.764 MPa; total pressure 2.726 MPa gives hoop stress σ = pD/2t = 56.8 MPa.

## Answer

The valve closure time is compared with the critical time $2L/c$. If the closure is slower than $2L/c$, the pressure rise is found from the gradual-closure formula $\Delta p=\rho L V/t_c$. Hoop (circumferential) stress in the wall is then $\sigma=pD/2t$ using the total pressure.

#### Given data
$D=0.25$ m, $L=1500$ m, $Q=0.075$ m³/s, $t=6$ mm, $t_c=3$ s, static head = 200 m, $K=2.2\times10^9$ N/m², $E=2.1\times10^{11}$ N/m².

#### Step 1: Velocity
$$
A=\frac{\pi}{4}(0.25)^2=0.04909\ \text{m}^2,\qquad V=\frac{Q}{A}=\frac{0.075}{0.04909}=1.528\ \text{m/s}
$$

#### Step 2: Wave celerity in elastic pipe
$$
c=\sqrt{\frac{K/\rho}{1+\dfrac{KD}{Et}}}=\sqrt{\frac{2.2\times10^9/1000}{1+\dfrac{2.2\times10^9\times0.25}{2.1\times10^{11}\times0.006}}}=1237.5\ \text{m/s}
$$

#### Step 3: Type of closure
$$
\frac{2L}{c}=\frac{2\times1500}{1237.5}=2.42\ \text{s}<t_c=3\ \text{s}
$$
So the closure is gradual (slow), and the Joukowsky value $\rho cV$ does not apply.

#### Step 4: Pressure rise
$$
\Delta p=\frac{\rho L V}{t_c}=\frac{1000\times1500\times1.528}{3}=763944\ \text{N/m}^2=0.764\ \text{MPa}\ (=77.9\ \text{m of water})
$$

#### Step 5: Stress in pipe wall
Static pressure $p_s=\rho g h=1000\times9.81\times200=1.962$ MPa.

Total pressure $p=p_s+\Delta p=1.962+0.764=2.726$ MPa.
$$
\sigma=\frac{pD}{2t}=\frac{2725944\times0.25}{2\times0.006}=56.79\ \text{MPa}
$$
The stress due to the static head alone is 40.88 MPa; the water hammer adds 15.92 MPa.

**Answer: pressure rise = 0.764 MPa (77.9 m); maximum hoop stress in pipe wall = 56.8 MPa (about 57 N/mm²).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How do you decide whether closure is rapid or gradual?
   A. Compare pipe thickness with diameter
   B. Compare closure time t_c with 2L/c
   C. Compare velocity with 1 m/s
   D. Compare static head with 100 m
   Answer: B. If t_c is longer than 2L/c the closure is gradual and the Joukowsky value does not apply. (Key point: Closure type)

2. Here 2L/c = 2.42 s and t_c = 3 s. Which pressure-rise formula applies?
   A. rho L V / t_c
   B. rho c V
   C. K V / c
   D. rho g H
   Answer: A. Since t_c > 2L/c the closure is slow, so dp = rho L V / t_c = 0.764 MPa. (Key point: Result of check here)

3. Which pressure is used in the hoop stress formula?
   A. Static pressure plus water hammer rise
   B. Static pressure minus the rise
   C. Water hammer rise only
   D. Static pressure only
   Answer: A. The wall carries the total pressure: 1.962 + 0.764 = 2.726 MPa. (Key point: Total pressure for stress)

4. The hoop stress in a thin-walled pipe is given by
   A. pt/2D
   B. pD/2t
   C. pD/4t
   D. 2pD/t
   Answer: B. Hoop stress sigma = pD/(2t), giving about 56.8 MPa here. (Key point: Hoop stress)

## Related in Hydraulics

- [Water hammer in 1.2 m steel pipe, wall thickness](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Awater-hammer-steel-pipe-thickness-1200mm) — Hydraulics, chapter 4
- [Diameter to thickness ratio for stress limit](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Apipe-diameter-thickness-ratio-hoop-stress) — Hydraulics, chapter 4
- [Hammer pressure for 4 s and 60 s closure](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asteel-pipeline-61cm-hammer-4s-60s) — Hydraulics, chapter 4
- [Water hammer in 20 m steel pipeline with valve](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Awater-hammer-20m-pipeline-pressure-time-curves) — Hydraulics, chapter 4
- [Water hammer pressure and stress, 2300 m pipeline](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Awater-hammer-2300m-pressure-stress-1725m) — Hydraulics, chapter 4
- [Rapid closure time and peak hammer pressure](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Acast-iron-pipe-rapid-closure-time-and-length) — Hydraulics, chapter 4

## Related in other subjects

- [Water pipe thickness from pressure head](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Awater-pipe-thickness-pressure-head) — Strength of Materials, chapter 5
- [Water hammer in 200 mm pipe](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Awater-hammer-200mm-pipe) — Hydro Power, chapter 1
- [Pipe outlet minimum water level, 50 lps](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Apipe-outlet-min-water-level-50lps) — Irrigation and Drainage Engineering, chapter 7
- [Economic diameter factors; penstock wall thickness (3 m)](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aeconomic-diameter-and-wall-thickness-factors) — Hydropower Engineering, chapter 6