My Air Mattress as a Pressure Vessel

[Tasker]

I wanted to make a free body diagram of my air mattress. The mattress is comprised of air cells. Each air cell is a 30 inches in length with a diameter of about 4 inches. Each cell is inflated to about 0.4 psi.

If I wanted to accurately model the whole bed (not just one individual cell), wouldn't I have to account for each cell as a thin-walled pressure vessel? That is, the whole bed doesn't have a reactive pressure of 0.4 necessarily right?

Say I wanted to find out the largest force the bed could handle. Wouldn't I be wrong to say that the limiting equation is P=F/A? That is, if my bed is 30 inches x 80 inches, and each air cell is filled to 0.4 psi, (and assuming a person is perfectly distributed), I would be wrong to say that my bed could support 960lbs?

 

[LightHero]

The answer is yes, you would be wrong to say that your bed could support 960lbs. The limiting equation for pressure vessels is P = S x E x (D/t), where P is the maximum pressure, S is the allowable stress of the material, E is the modulus of elasticity, D is the outside diameter, and t is the wall thickness. In addition, the area of the vessel (A) must take into account the number of cells in the mattress, as well as the shape of the mattress, as the pressure is distributed across the entire mattress. Therefore, to accurately model the whole bed, you would need to take into account the number of cells, the shape of the mattress, and the material properties of the air cells, in order to accurately calculate the maximum pressure the mattress can handle.