Hoop Stress in Thin Spherical Shells



Application of hoop or circumferential in thin spherical shells has prime significance in daily life.

Pressurized gas filled in shell
Image a spherical shell in which we have pumped a gas up to significant pressure. Due to this pressurized gas, the shell will undergo volumetric expansion and stresses will be developed. 

Whenever we have to account for the stresses develop in any shape, we use a hypothetical cutting plane, to divided our object into sections and then examine the nature of stress developed in analytical method. However, for the experimental stress analysis there are in-numerous methods and each method can be exploited for determining specific forces.

Expression:

Sphere being cut into halves
Consider a spherical shell with is cut into half using an imaginary cutting plane. Now the sphere is made into two halves and we analysis on one of the half.

The stress will have two components. One component along x-axis and one along y-axis. We will consider the stress along the x-axis only because the other component will give a zero resultant upon integration. So we will have to take the projection of area in the x-direction.



Light being shined on the object
Imagine some beams of light of approaching our reduced sphere from the x-direction and a screen is placed at the back of sphere at some distance. The shadow formed at the screen will be the shape of a circle. Therefore, the pressure will be acting upon this projected area inside the sphere.




Let,
t be the thickness
D be the diameter
p be the pressure
Area resisting the force
P be the force
σc be the circumferential or hoop developed
The force acting F=pressure*projected area
                                =p*(π/4)*D2   
The area resisting the force developed= π*D*t
Therefore, resisting force=sigma…* πDt
Under equilibrium, equating the force generated due to pressurized gases to the stress developed…
                                P*(π/4)D2=σc* (πDt)
                                σc=p*D/(4*t)
Finally, the above equation gives the hoop stress.







Location: Phagwara, Punjab 144401, India

Comments

  1. Unknown20 August 2016 at 12:10

    Any comments or suggestions are warmly welcomed. For any query feel free to drop at e-mail at er.agarwal03@gmail.com.

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