We were discussing the basic concept of
thin cylindrical and spherical shells
,
stresses in thin cylindrical shells
and derivation of expression forcircumferential stress or Hoop stressandlongitudinal stressdeveloped in the wall of cylindrical shell in our previous posts.
Today we will derive here the expression for stress developed in the wall of thin spherical shell, with the help of this post.
Before going ahead, we will first remind here the fundamental of a thinspherical shell
Thin spherical shell is also termed as a pressure vessel and such vessels are usually used in various engineering applications such as for storing the fluid under pressure. Air receiver tank is one of the best examples of thin spherical shells.
A spherical shell will be considered as thin spherical shell, if the wall thickness of shell is very small as compared to the internal diameter of the shell.
Wall thickness of a thin spherical shell will be equal or less than the 1/20 of the internal diameter of shell.
Circumferential stress or Hoop stress
Stress acting along the circumference of thin spherical
shell
will be termed as circumferential stress or hoop stress.
Let us consider here following terms to derive the expression for
circumferential stress or hoop stress
developed in the wall of
thin spherical
shell.
P = Internal fluid pressure
d = Internal diameter of
thin spherical
shell
t = Thickness of the wall of
thin spherical
shell
σ =
Circumferential stress or hoop stress
developed in the wall of thin spherical shell
Thin spherical shell bursting will take place if force due to internal fluid pressure, acting on the wall of thin spherical shell, will be more than the resisting force due to
circumferential stress or hoop stress
developed in the wall of thin spherical shell.
In order to secure the expression for
circumferential stress or hoop stress
developed in the wall of thin spherical shell, we will have to consider the limiting case i.e. force due to internal fluid pressure should be equal to the resisting force due to
hoop
stress developed in the wall of thin spherical shell.
Force due to internal fluid pressure = Internal fluid pressure x Area on which fluid pressure will be acting
Force due to internal fluid pressure = P x (π/4) d2
Resisting force due to
hoop
stress = σ x π d t
As we have seen above, we can write following equation as mentioned here.
Force due to internal fluid pressure = Resisting force due to
longitudinal
stress
P x (π/4) d2= σ x π d t
σ = P x d / (4 t)
Do you have suggestions? Please write in comment box.
We will now discuss the basic concept of thick cylindrical and spherical shell, in the category of strength of material, in our next post.
Reference:
Strength of material, By R. K. Bansal
Image Courtesy: Google
也读
你将如何考虑墙很薄。什么dimensions should wall have to be considered as thin ??
Reply删除If it's internal diameter is greater than or equal to 20 times it's wall thickness, it is considered thin
删除thank you sir
Reply删除What's the longitudinal stress in this case?
Reply删除