# Pressure exerted by a liquid on its container

For an individual project I'm working on, I've been trying to work out what the tensile strength would need to be for a spherical container to hold a given volume and density of liquid. The specific values are given below, but a general formula would be useful too.

Mass of liquid: 2 660 000kg
Density of liquid: 1000kg/m^3
Hence, volume of liquid: 2 660m^3
Hence, diameter of container: 17.2m
And surface area: 929.41m^2
Temperature of liquid: 36.5°C
Force exerted on container per m^2: ???
The container is exactly large enough to contain the liquid, and no larger.

I'm not sure what equations to use here. If any further information is needed to work out the answer, please let me know what it might be.

Bystander
Homework Helper
Gold Member
Material, for starters?

Of the container? It's with the intent of working out what kind of material it could be that I'm asking the question. I'm trying to work out what tensile strength would be required for this container, and to do that I need to work out the forces being exerted on it.

SteamKing
Staff Emeritus
Homework Helper
For an individual project I'm working on, I've been trying to work out what the tensile strength would need to be for a spherical container to hold a given volume and density of liquid. The specific values are given below, but a general formula would be useful too.

Mass of liquid: 2 660 000kg
Density of liquid: 1000kg/m^3
Hence, volume of liquid: 2 660m^3
Hence, diameter of container: 17.2m
And surface area: 929.41m^2
Temperature of liquid: 36.5°C
Force exerted on container per m^2: ???
The container is exactly large enough to contain the liquid, and no larger.

I'm not sure what equations to use here. If any further information is needed to work out the answer, please let me know what it might be.
This is not as simple as it seems.

A variety of liquids and gases can be stored in spherical pressure vessels, and their construction is governed by various design codes for pressure vessels and the usual formulas from strength of materials. In addition to the design of the skin of the tank itself, consideration must also be given to how the tank is to be supported on the ground or some other foundation, what type of environmental loads it must withstand, etc.

Obviously, the bottom of the tank is going to be subjected to the hydrostatic pressure of the contents of the tank, while the top receives very little loading from such pressure. Should you design for the minimum weight of the tank, a tank with a shell of constant thickness, what?

The article at the link describes the design process for a spherical tank designed to contain propane (under pressure):

http://www.pveng.com/FEA/FEASamples/Sphere/Sphere.php

Chestermiller
Mentor
If the liquid exactly fills the container, then I guess you are saying that the gauge pressure at the very top is zero, correct? In other words, the liquid is not put into the container under pressure?

That's right. I'm assuming no support, with the sphere lying on flat surface (capable of rolling) but no external factors except gravity itself. The greatest pressure on any square meter is the only one I need (so the pressure at the bottom, I suppose).

I am assuming no pressure except that of the liquid, yes.

Chestermiller
Mentor
That's right. I'm assuming no support, with the sphere lying on flat surface (capable of rolling) but no external factors except gravity itself. The greatest pressure on any square meter is the only one I need (so the pressure at the bottom, I suppose).

I am assuming no pressure except that of the liquid, yes.
That's not the only load on the sphere. For the sphere to be in equilibrium, there needs to be an upward loading on it. The pressure loading is easy, because it is perpendicular to the surface of the sphere (but it varies with depth). The external force might be more concentrated, and the stresses may be higher. What are your thoughts on how to analyze this problem? You can specify the loading pretty easily, but the stress analysis might be complicated.

You may also need to consider what happens (thermal expansion-wise) if the temperature changes.

That's not the only load on the sphere. For the sphere to be in equilibrium, there needs to be an upward loading on it. The pressure loading is easy, because it is perpendicular to the surface of the sphere (but it varies with depth). The external force might be more concentrated, and the stresses may be higher. What are your thoughts on how to analyze this problem? You can specify the loading pretty easily, but the stress analysis might be complicated.

You may also need to consider what happens (thermal expansion-wise) if the temperature changes.

Oh, yes. Allow normal force from the flat surface the sphere is resting on equal to the force of gravity pushing it down.
I was thinking I'd try to work out the maximum amount of horizontal (perpendicular to the flat surface) force there'd be at any one point on the surface of the container, but I'm not sure how to calculate the pressure on a fixed point or how to adjust the calculation to account for the shape of the container

I'm not going to bother with temperature expansion for the time being.

Chestermiller
Mentor
Oh, yes. Allow normal force from the flat surface the sphere is resting on equal to the force of gravity pushing it down.
I was thinking I'd try to work out the maximum amount of horizontal (perpendicular to the flat surface) force there'd be at any one point on the surface of the container, but I'm not sure how to calculate the pressure on a fixed point or how to adjust the calculation to account for the shape of the container
Well, if the pressure within the sphere were uniform and there were no contact force from the outside, there would be not problem. You could easily get the stresses in the shell. But, since the pressure is varying with depth, the stresses in the shell (longitudinal and latitudinal) will vary with latitude. On the outside of the shell, you won't have point loading, because there is going to be a circular contact patch of finite radius. This is a pretty complicated stress analysis. I haven't been able to think of any simplifying approximations yet. You might be able to solve this with a shell model.

Chet

Yes, that's what I've been struggling with. If you find a shell model that could give a decent approximation, let me know.

Chestermiller
Mentor
Yes, that's what I've been struggling with. If you find a shell model that could give a decent approximation, let me know.
You might want to try a packaged finite element code.

I don't know what that is. Can you give me a link or formula?

Thank you. I'm still working on it, but that link looks helpful.

Nidum