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Chapter 10: Problem 55

A thin-walled spherical pressure vessel having an inner radius \(r\) and thickness \(t\) is subjected to an internal pressure \(p .\) Show that the increase in the volume within the vessel is \(\Delta V=\left(2 p \pi r^{4} / E t\right)(1-\nu) .\) Use a small-strain analysis.

Short Answer

Expert verified
\(\Delta V = (2p 蟺 r^4 / Et)(1 - 谓)\)

Step by step solution

01

Understand the Problem

The problem explains itself as a situation where internal pressure \(p\) inflates a spherical pressure vessel of inner radius \(r\) and thickness \(t\). The objective is to determine the increase in volume using given parameters.
02

Use Formula for Strain

The first step is to establish the strain in the spherical vessel due to the applied pressure. This can be done by using the formula for radial strain (蔚), which is given by 蔚 = stress/Young's modulus, or \(蔚 = 蟽/E\). Here, the stress 蟽 is equal to the pressure \(p\). Now, knowing that the radial stress on thin shells is given by: \(蟽 = pr/t\), we place this in the equation thus getting: \(蔚 = pr/(Et)\).
03

Calculate the Change in Volume

We can determine the change in volume of the vessel using the formula \(\Delta V = V(蔚(1 - 谓))\). Substituting the formula for the volumetric volume of a sphere, \(V = 4/3 蟺r^3\), the formula can be rewritten as \(\Delta V = 4/3 蟺r^3(蔚(1 - 谓))\). By substituting the value of \(蔚\) that we obtained from step 2, we obtain: \(\Delta V = (4/3 蟺 r^3)*((pr / Et)*(1- 谓))\).
04

Simplify Final Expression

The final step consists of simplifying the above expression to the desired form. We do this by simplifying terms, and we obtain: \(\Delta V = (2p 蟺 r^4 / Et)(1 - 谓)\), which is the increase in the volume inside the vessel, as required.

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Most popular questions from this chapter

If the 2-in.-diameter shaft is made from brittle material having an ultimate strength of \(\sigma_{u l t}=50 \mathrm{ksi},\) for both tension and compression, determine if the shaft fails according to the maximum-normal- stress theory. Use a factor of safety of 1.5 against rupture.

The state of strain at a point on a wrench has components \(\epsilon_{\mathrm{k}}=120\left(10^{-6}\right), \epsilon_{\mathrm{y}}=-180\left(10^{-6}\right), \gamma_{\mathrm{BV}}=150\left(10^{-6}\right)\). Use the strain-transformation equations to determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In Each case specify the orientation of the clement and show how the strains deform the element within \(x-y\) plane.

The state of strain at the point has components of \(\epsilon_{x}=200\left(10^{-6}\right), \epsilon_{y}=-300\left(10^{-6}\right),\) and \(\gamma_{x y}=400\left(10^{-6}\right)\). Use the strain-transformation equations to determine the equivalent in-plane strains on an element oriented at an angle of \(30^{\circ}\) counterclockwise from the original position. Sketch the deformed element due to these strains within the \(x-y\) plane.

The internal loading at a critical section along the steel drive shaft of a ship are calculated to be a torque of \(2300 \mathrm{lb} \cdot \mathrm{ft},\) a bending moment of \(1500 \mathrm{lb} \cdot \mathrm{ft},\) and an axial thrust of 2500 lb. If the yield points for tension and shear \(\operatorname{are} \sigma_{Y}=100 \mathrm{ksi}\) and \(\tau_{Y}=50 \mathrm{ksi},\) respectively, determine the required diameter of the shaft using the maximum-shear-stress theory.

The thin-walled cylindrical pressure vessel of inner radius \(r\) and thickness \(t\) is subjected to an internal pressure \(p .\) If the material constants are \(E\) and \(\nu,\) determine the strains in the circumferential and longitudinal directions. Using these results, compute the increase in both the diameter and the length of a steel pressure vessel filled with air and having an internal gauge pressure of 15 MPa. The vessel is \(3 \mathrm{m}\) long, and has an inner radius of \(0.5 \mathrm{m}\) and a thickness of \(10 \mathrm{mm} . E_{\mathrm{st}}=200 \mathrm{GPa}, \nu_{\mathrm{st}}=0.3\).
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