Explanation Given information: A thin-walled spherical pressure vessel having an inner radius r and thickness t . The spherical pressure vessel subjected to an internal pressure p . Calculation: The thin wall analysis is used to find the normal stress. For the plane stress σ 3 = 0 . Find the normal stress as shown below. σ 1 = σ 2 = p r 2 t Here, p is the gage pressure, r is the inner radius of the pressure vessel, and t is the thickness of the pressure vessel. Apply Hooke’s law as shown below. Find the normal strain ε 1 . ε 1 = 1 E ( σ 1 − ν ( σ 2 + σ 3 ) ) Here, E is the modulus of elasticity and ν is the Poisson’s ratio. Substitute p r 2 t for σ 1 , p r 2 t for σ 2 , and 0 for σ 3 . ε 1 = 1 E ( p r 2 t − ν ( p r 2 t + 0 ) ) = p r 2 t E ( 1 − ν ) Find the normal strain ε 2 . ε 2 = 1 E ( σ 2 − ν ( σ 1 + σ 3 ) ) Substitute p r 2 t for σ 1 , p r 2 t for σ 2 , and 0 for σ 3 . ε 2 = 1 E ( p r 2 t − ν ( p r 2 t + 0 ) ) = p r 2 t E ( 1 − ν ) Find the normal strain ε 3 . ε 3 = 1 E ( σ 3 − ν ( σ 1 + σ 2 ) ) Substitute p r 2 t for σ 1 , p r 2 t for σ 2 , and 0 for σ 3 . ε 3 = 1 E ( 0 − ν ( p r 2 t + p r 2 t ) ) = − ν p r t E Volume of the spherical pressure vessel. V = 4 3 π r 3 Change in volume of the spherical pressure vessel. V + Δ V = 4 3 π ( r + Δ r ) 3 = 4 3 π r 3 ( 1 + Δ r r ) 3 Here, Δ V < < V and Δ r < < r

9th Edition

ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

See similar textbooks

Concept explainers

Strain energy

The term strain indicates the variation in an object's length, area, and volume when the object is subjected to an external force. The external force may be of normal, shear, moment, torque, etc. The strain in an object depends on the types of external f…

Videos

Question

Chapter 10.6, Problem 10.55P

To determine

To show that: The increase in the volume within the vessel is ΔV=2pπr4Et(1−ν) using small-strain analysis.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.6, Problem 10.54P

Chapter 10.6, Problem 10.56P

Students have asked these similar questions

Correct detailed answer and complete fbd only. I will upvote. The compound shaft, composed of steel,aluminum, and bronze segments, carries the two torquesshown in the figure. If TC = 250 lb-ft, determine the maximumshear stress developed in each material (in ksi). The moduliof rigidity for steel, aluminum, and bronze are 12 x 106 psi, 4x 106 psi, and 6 x 106 psi, respectively

Correct answer and complete fbd only. I will upvote. A flanged bolt coupling consists of two concentric rows of bolts. The inner row has 6 nos. of 16mm diameterbolts spaced evenly in a circle of 250mm in diameter. The outer row of has 10 nos. of 25 mm diameter bolts spaced evenly in a circle of 500mm in diameter. If the allowable shear stress on one bolt is 60 MPa, determine the torque capacity of the coupling. The Poisson’s ratio of the inner row of bolts is 0.2 while that of the outer row is 0.25 and the bolts are steel, E =200 GPa.

Correct answer and complete fbd only. I will upvote. The shaft carries a total torque T0 that is uniformly distributedover its length L. Determine the angle of twist (degrees) of the shaft in termsif T0 = 1.2 kN-m, L = 2 m, G = 80 GPa, and diameter = 120 mm

Chapter 10 Solutions

Mechanics of Materials

Ch. 10.3 - Prove that the sum of the normal strains in...Ch. 10.3 - The state of strain at the point on the arm has...Ch. 10.3 - Prob. 10.3P Ch. 10.3 - Prob. 10.4P Ch. 10.3 - 10-5. The state of strain at the point on the gear...Ch. 10.3 - Use the strain transformation equations and...Ch. 10.3 - Use the strain transformation equations and...Ch. 10.3 - Prob. 10.8P Ch. 10.3 - Use the strain transformation equations to...Ch. 10.3 - Use the strain- transformation equations to...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

To show that: The increase in the volume within the vessel is Δ V = 2 p π r 4 E t ( 1 − ν ) using small-strain analysis.

Solution Summary: The author explains how a thin-walled spherical pressure vessel has an inner radius r and thickness t.

Concept explainers

Videos

To show that: The increase in the volume within the vessel is ΔV=2pπr4Et(1−ν) using small-strain analysis.

Want to see the full answer?

Chapter 10 Solutions