Understanding hoop stress and its significance for the effective pressures needed for smaller and larger girth penises

In physical modelling, “a cow can be approximated as a sphere with a radius of one metre”, is a long-standing joke or meme among mathematicians and physicists. Physical models don’t always need to be perfectly accurate, but can serve as intuition pumps and help us reason by analogy. 

Let’s think of the tunica albuginea of the penis as a cylinder. This is only a slight simplification. It has a wall in the middle separating two halves, but we’ll ignore this because it doesn’t really matter for our purposes here. We want a model of how the tunica albuginea is affected by a pressure differential between inside and outside - i.e. as a cylindrical pressure vessel, similar to a hot water boiler or a storage tank for propane. Because it actually is a pressure vessel. 

When you get an erection, your blood pressure inflates the corpora cavernosa inside the tunica albuginea, working against the air pressure on the outside. This air pressure is nothing to sneeze at. If you pull the air out of a tin can, the air pressure on the outside will crush it in a heartbeat. Your internal pressure (systolic) is somewhere around 5 inHg (130mmHg - higher for some, lower for some, but this is the ballpark number if you’re a healthy adult male). Note: this is how much higher your internal blood pressure is, compared to the surrounding air pressure. 

Now say you get erect and start pumping your penis with a vacuum pump. This lowers the external air pressure. The partial vacuum does not magically “pull on” your skin - it just makes the pressure differential between the inside and outside of your stiff tunica albuginea greater. Let’s say you pump to -10 inHg, and that you are still erect so that the inside of your penis experiences the full +5inHg blood pressure. Now, the pressure differential between inside and outside the tunica will be 15 inHg. (This is one important reason for why it’s good to be erect when you start pumping, and why you ideally want to keep your erection during pumping so your penis has the full +5 inHg internal pressure the whole time). 

So… how much force does this put on your tunica albuginea? This is different in different directions, actually. There is one form of stress “girthwise” and another form of stress lengthwise. The circumferential stress is called “hoop stress” and the longitudinal stress is sometimes called “axial stress” (because it’s along the axis of the cylinder), but I prefer to just call it longitudinal. 

Now, in material science, there are formulas for all sorts of things, and we have very good formulas indeed for modelling circumferential hoop stress and longitudinal stress in thin-walled pressure vessels. (The designation of a pressure vessel or cylinder as "thin-walled" hinges on the relationship between its radius (r) and wall thickness (t). Generally, a vessel is considered thin-walled if the ratio of its radius to its wall thickness is greater than or equal to 10:1. This criterion allows for simplifications in stress calculations because it assumes that the stress across the wall thickness is uniformly distributed. The tunica is approximately 0.8mm thick, with some individual variance, and the penis has a radius significantly larger than 8mm, so the criterion is met.)

So, what’s the formula for hoop stress and longitudinal stress? I’m glad you asked! 

Hoop stress " σθ" is calculated as 

σθ = (p * r) / t

where p is the pressure differential between inside and outside, r is the internal radius of the cylinder, and t is the wall thickness. 

Longitudinal stress is calculated as:

σL = (p * r) / (2t)

Now, I’m not going to toss in values here immediately - just notice an important detail: When the radius increases, the hoop stress increases. Hoop stress is directly proportional to the radius. If pressure inside the penis and in the vacuum chamber are kept the same,  and thickness of the tunica wall is the same, then a penis that is 40% larger in radius will experience 40% greater hoop stress. The same is true of the longitudinal stress. 

What does this mean for pressure? Well, let’s compare someone who is 4” circumference and someone who is 6” circumference. Circumference scales linearly with radius (2pi being a constant), so someone with a 6” circumference has a 50% larger radius than someone with a 4” girth. How much higher pressure must the 4” dude use to get the same hoop stress as the 6” dude? 

Let’s say both have an interior pressure of 5 inHg. The 6” girth dude uses a vacuum pressure of 5 inHg, making the total pressure differential 10 inHg. Let’s look only at the product (p * r). The radius is 50% higher. For the hoop stress to be equal, we get the following: 

(p * r)=(10*1.5r)

Let’s solve for p (the total pressure differential the smaller guy needs to use)

p= (10*1.5r)/r   (notice that when I express the radius of the larger guy as an increment of the smaller guy’s radius, it simplifies away and I don’t need to use an actual value)

p=15

Subtract the 5 inHg internal pressure, and the smaller guy needs to use a pressure of -10inHg in the vacuum pump. 

Ok, I made those numbers really easy to apply on purpose. Let’s use some less extreme values for girth: 

One guy is 5.5” girth and another is 4.5”. The larger guy uses -10inHg for a total pressure differential of 15 inHg. We express the larger guy’s radius as a multiple of the smaller guy’s: 5.5/4.5 = 1.222

The radius simplifies away and p=15*1.222=18.33 

Subtract the internal pressure, and you get 13.3 inHg. 

The smaller guy needs to use a vacuum pressure of -13 inHg to experience the same hoop stress as the larger guy who uses -10 inHg, all else equal. 

 

But “all else equal” is of course an assumption we need to motivate. Is it true that larger guys have tunicas of the same thickness as smaller guys? Well, that’s an empirical question and I don’t think we have that answer. We know the thickness of the tunica albuginea increases with age from 400–450 μm in young men to more than 900 μm in elderly men, and the thickness is in the denominator (divisor, some people say), so this means the older you are, the more pressure you will need to apply to get the same hoop stress, provided the collagen has the same properties (which might not hold true, actually - collagen synthesis changes with age). But as for thickness of the tunica, we don’t know if it varies with girth. It might, and it might not. 

 

But if it does not, then it follows that larger guys need less negative pressure to get the same circumferential hoop stress in a vacuum pump as a smaller guy. Correspondingly, smaller guys need to use more pressure than larger guys to get the same hoop stress. Life is unfair, is it not? It would appear it’s prima facie easier to gain girth if you are already girthy. 

 

But I would like to make one other point again: It’s the sum of the internal positive pressure, and the absolute value of the external negative pressure which matters for hoop stress. If you begin pumping soft, you are at a 5 inHg disadvantage from the get-go. You also probably shouldn’t get in the pump and forget about mental arousal and erection and let the pump do the work. It’s probably beneficial to try and maintain arousal and erection - perhaps by wanking the tube a little back and forth to some stimulating input. And why not try doing sit-ups while in the tube, while tensing your thighs? These increase the internal pressure by a lot, giving you a pulse of higher blood pressure. I’m not saying you should go ahead and do this - there might be pelvic floor issues I’m not considering here - but it’s an idea that needs to be explored I think. 

 

I hope this has been an elucidating piece of material science, physics and maths, and that you were able to follow the calculations and conclusions. Let me know if you think I’m missing something, or if you have a piece to add to the puzzle! 

 

Oh, a little ps: 

What I described here is the reason that an aortic aneurysm will invariably get worse and worse. Since the internal pressure remains constant, once the aorta starts getting wider and wider, the hoop stress just keeps increasing, making the rate of widening increase, etc…