Pressure in two Different Sized Plastic Bottles

[Michael Strong]

I have two different sized plastic bottles of the same material and with the same wall thickness and I raise the pressure inside the bottles at the same rate then which bottle will rupture first? The bigger bottle or the smaller one?

I would think that both bottles would rupture at about the same pressure because the bottle walls are the same and they are made out of the same material.
Is there any theory that states otherwise? Or what could be some factors that would cause one bottle to rupture before the other one?

 

[Bystander]

See "hoop stress."

 

[billslugg]

The hoop stress equation relates the tension in a curved member to the radius of curvature and internal pressure.
Pressure = Tension/radius
Suppose you have a plastic wall that can withstand an ultimate tension in a one inch wide strip of say 100 pounds. If the large bottle had a diameter of 6 inches then the burst pressure would be equal to 100 pounds per inch/ 3 inches or 33 psi.
A bottle of diameter 3 inches would have a burst pressure of 100 pli/1.5 in or 67 psi.

So, for a given wall thickness, the larger the bottle, the less pressure it can hold.

Imagine the problem of putting a one mile diameter clear glass dome in space with an internal pressure equal to Earth's sea level.
In that case the tension in a one inch wide strip of the glass dome would be equal to 14.7 psi times 31680 inches or 466,000 pounds. Really strong window glass has an ultimate yield strength of about 20,000 psi. A one inch wide strip would have to be at least 23 inches thick. With the customary 2.5 safety factor, the space colony would require a window that was about 5 feet thick.