Statics: When the reactions depend on the displacements

[Juanda]

@T1m0 you pretty much nailed the solution and even found it in Roark's which follows the same procedure you described in this thread. You gave me some hope back on tackling a similar problem. Let me introduce it and tell me if you'd be interested in giving it a whack.

For a while, I have been interested in knowing how pressurized tanks can be modeled with this kind of concept where the reactions depend on the deformation.
There are some similarities to the beam we've been discussing where the deformation causes the apparition of a torque that wasn't initially there. In pressurized tanks, the pressure makes it expand which increases the area where the pressure acts (assume pressure to be constant). So using equilibrium equations in the undeformed state is only an approximation. The actual deformation and stress will be greater just as what happened with the beam in this post.

I tried to tackle this problem already in this post but I got lost before understanding the matter. Do you know how to find a solution for such a situation?


EDIT.
Now that I think of it with the new perspective gained from this post I think this is how I'd do it.
Assuming there is a $k$ which relates the linear behavior between the pressure inside the tank and the radius, a constant pressure $P$, an area of the interior of the spherical tank $A$, the initial radius of the tank $r_i$ when there is no pressure and the actual radius $r$ when the pressure is applied:
$$PA=k(r-r_i)\rightarrow P(4\pi r^2)=k(r-r_i)$$
I don't know how to derive that $k$ now or if it's even linear so that might change but the overall procedure feels OK and in line with what we've been doing with the beam. That is, making equilibrium in the deformed state.
What do you think?

 

[Juanda]

EDIT.
Now that I think of it with the new perspective gained from this post I think this is how I'd do it.
Assuming there is a $k$ which relates the linear behavior between the pressure inside the tank and the radius, a constant pressure $P$, an area of the interior of the spherical tank $A$, the initial radius of the tank $r_i$ when there is no pressure and the actual radius $r$ when the pressure is applied:
$$PA=k(r-r_i)\rightarrow P(4\pi r^2)=k(r-r_i)$$
I don't know how to derive that $k$ now or if it's even linear so that might change but the overall procedure feels OK and in line with what we've been doing with the beam. That is, making equilibrium in the deformed state.
What do you think?

For greater clarity of what I'm trying to say, the "usual method" would be to make an equilibrium on the undeformed state. That would originate the following:
$$PA_i=k(r-r_i)\rightarrow P(4\pi r_i^2)=k(r-r_i)\rightarrow r=\frac{P(4\pi r_i^2)+kr_i}{k}$$
Using the equilibrium equation in the deformed state should yield greater deformation (and therefore, greater stress too).
$$PA=k(r-r_i)\rightarrow P(4\pi r^2)=k(r-r_i) \rightarrow 4P\pi r^2-kr+kr_i=0\rightarrow$$
$$\rightarrow r=\frac{-(-k) \pm \sqrt{(-k)^2-4(4P\pi)(kr_i)}}{2(4P\pi)}$$
I'm concerned by the fact that the result could be a complex number for certain input numbers which should be valid.
From the two solutions due to the square root, it seems the negative one is the correct one.
For as long as the second equation results in a real number, it's greater than the solution with the equilibrium in the undeformed state.
For smaller pressures, the results from the two equations are very close. As pressure increases, the results start to differ until the solution becomes a complex number which I don't understand how to interpret.
You can use this Desmos link to play around with the values if you want.

 

[T1m0]

Juanda, There is a Section in Roark's book on Thin Walled pressure vessels that gives a fairly simple formula for the radial expansion of a spherical tank as a function of internal pressure. The effect of this expansion on the pressure will strongly depend on whether the fluid inside the tank is a liquid or a gas. You can look at a number of websites regarding the topic of fluid compressibility; e.g.:

https://phys.libretexts.org/Bookshe...tatic_Fluids/27.5:_Compressibility_of_a_Fluid

 

[Juanda]

Juanda, There is a Section in Roark's book on Thin Walled pressure vessels that gives a fairly simple formula for the radial expansion of a spherical tank as a function of internal pressure.

Chapter 13. Shells of Revolution; Pressure Vessels; Pipes
It contains information about the spherical case which I believe is the simplest because of the present symmetries. However, the explanations are so concise that I can't make much out of them. It's a common problem I have with this book.
Still, one of the formulas shown there is related to what I'm trying to understand. To be precise:
$$\Delta R_2=\frac{qR_2^2(1-\nu)}{2Et}$$
The results will vary depending on whether

• we consider $R_2$ as the initial dimension which is known (the usual procedure) resulting in a linear result.

OR

• we consider $R_2$ the final dimension (what we have been discussing in the thread) resulting in a quadratic equation.

This is similar to what's shown in post #32 so the same questions remain.

 

[T1m0]

Juanda, you can find a fairly thorough explanation of the pressure vessel formulas in

https://pkel015.connect.amazon.auck...lasticity_Applications_03_Presure_Vessels.pdf

I am not aware of any analytical solutions to the pressure vessel problems for the large displacement case. This may be because most materials used in pressure vessels will experience yielding before the displacements get large. I suppose a really soft material might deform significantly before yielding. However, these materials tend to have more complicated stress-strain laws than Hooke's law.

In general, there are few analytical solutions to large displacement problems. The usual (and easiest) way to attack these problems is to use structural finite element analysis software (e.g., ANSYS, ABAQUS, etc). Although the full versions of these programs are very expensive, there are student versions that are free or very inexpensive.

 

[Juanda]

Juanda, you can find a fairly thorough explanation of the pressure vessel formulas in

https://pkel015.connect.amazon.auck...lasticity_Applications_03_Presure_Vessels.pdf

I am not aware of any analytical solutions to the pressure vessel problems for the large displacement case. This may be because most materials used in pressure vessels will experience yielding before the displacements get large. I suppose a really soft material might deform significantly before yielding. However, these materials tend to have more complicated stress-strain laws than Hooke's law.

In general, there are few analytical solutions to large displacement problems. The usual (and easiest) way to attack these problems is to use structural finite element analysis software (e.g., ANSYS, ABAQUS, etc). Although the full versions of these programs are very expensive, there are student versions that are free or very inexpensive.

That was a much lighter read. I appreciate it. It is similar to what's shown in Mechanics of Materials by Barry J. Goodno and James M. Gere.

I wouldn't know how to apply what we discussed in this thread to it though. You mention how those solutions are only for small displacements. However, the equations we used for the beam are also related to small displacements. It's just that by studying equilibrium in the deformed state it's possible to find a more accurate solution which can be critical in some cases when that increment in accuracy reveals that the stress is greater than initially calculated.