Radial load vs. deflection in pressurized hose?

I would buy or build one and measure it. I would NEVER try to calculate it, and I wouldn't trust any such calcs I were to see. Too many variables; by the time they are all accounted for, the purported rationale may be too complicated to be understandable.

Deformation of bellows

I would buy or build one and measure it. I would NEVER try to calculate it, and I wouldn't trust any such calcs I were to see. Too many variables; by the time they are all accounted for, the purported rationale may be too complicated to be understandable.

To be honest I am quite surprised by the comment TORNADO made for several reasons:

- If one looks at the documents presented by the company BOA a full manual for computing bellows in details is available

- If the manual is not available in the Catalogue pages from almost all manufacturers 3 stiffnesses for steel or copper alloys are indicated

- If those "measured" values are available then with help of a simple model deflections can be estimated ( I dare not say "computed") with a quite good accuracy. Anyway good enough for the requirements of an engineer considering the applications and the OP demand .

- Only one sample will not be enough to characterize the transverse stiffness due to all dispersions involved.

- A model for such computations has NOT to be complex and I presume the following one can be understood without difficulty. An estimation is always possible as long as the uncertainty level is accepted.

If "Tornado" is , basically, not wrong when he writes that there are many parameters to be considered he is wrong in not accepting a "simplified model" for the computation of a deflection model which is "good enough" for what the goal is.

Some of the parameters are:

-manufacturing tolerances

-wall thickness modified by the manufacturing process

-temperature

-material and dispersion of its own mechanical properties as the Young modulus

-number of layers in the wall

-a. s. o. !

Many of those parameters have a limited impact on the result and thus can be in a first approximation neglected. Engineering is a precise profession but accepts the "relative" , no engineer woks in the "absolute". For instance temperature has an effect via the Young modulus but its effect is limited, only about 300ppm/°C (0.03 %/°C).

This means that it is possible for a limited accuracy to develop simplified models for deflection's estimation.

Bellows configuration is a series of elastic ring membranes connected with zones of higher stiffness so that a lumped parameter model C an be build up.

The different deformations are in following pictures:

Mentioned stiffness values are indicated in the Data sheets for one wave:

-

- In axial direction C ax [N/mm]

Stretched Compressed

- In radial direction C rad [N/mm]

Shifted

- In angular deformation C a [Nm/°]

Bend

Indicated stiffness values are results of measurements done by manufacturer and are a representative value but have a dispersion which is seldom if ever indicated.

Depending on the uncertainty accepted level either C an those values be used with a simple model or a complex and expensive measurement is required. I say complex and expensive since

measurements have unfortunately also limitations : ALL dimensions and mechanical properties are NOT constant so that even a measurement will be subject to a dispersion and this C an be reduced ONLY by an important number of samples measured with great C are and has thus an impact on the cost due to time investment and required fixtures.

Assuming that the wished value is more an indication which will be accepted with a dispersion the values in the tables can be used or a simulation with a FEA model.

The last has also draw backs, in a FEA model the wall thickness is assumed constant but it is not and the wall thickness is present in the stiffness computation at the 3^rd power so that even small variations lead to not unimportant results changes.

However in both approaches the overall cost is less than for a true experimental determination.

Pictures show the 3D model of a bellows and the result of a FEA simulation for the radial displacement case.

The bellows has following dimensions:

Di=53.2 ± 0.3; De=75±0.5; P=4.1; s(wall thickness)=0.25 all in [mm] and only 1 layer

According to the catalogue for steel 316 the following stiffnesses can be used:

C ax=136 N/mm/wave C a=1.234 Nm/°/wave C rad=52966 N/mm/wave

Tolerances are less 1% of nominal value so that their influences will be small. In the deformation of a membrane the diameters are present power 2 so that maximal probable influence could be 0.9%.

The FEA analysis gave a quite good result for C rad (difference 10%) and good linearity as well in axial as in radial loading but gave a wrong value for C ax, much too big. This shows that a FEA model could be for such a situation not always the right choice.

If we want to have the stiffness for the situation from OP description it is easier to compute making the remark that for half the bellows we have only a radial force applied. The bellows being open at both ends (it is a hose) there is no axial force as indicated in following picture which shows the most complex case.

We have to consider the load F and the simple lumped model as follows:

"a series of hinges with C a as stiffness and at a distance in between of P, the wave length."

as following sketch shows:

Let us consider the wave "j" with the bellows under load F.

F will induce a bending moment proportional to the distance between centre at right end and centre of wave "j".

If we look at the tables given by manufacturers it is interesting to notice that C ax and C a are a lot smaller than C rad. The ratio is such that we can from the start neglect C rad' s effect.

The force acts only in radial direction so that there will not be any axial load.

The flexion is due to the bending moments which can be computed as

M_j = F*e_jax

e _jax= a cos (φo) + Σ_i=1^j-1P*cos(φ_i)

φ_i=Σ_nⁱ Δφ and Δφ_i=M_i*C a

We see that the functions are not explicit so that the simplest way is to use a spread sheet approach with a "seed" and successive corrections till the result is stable i.e. the differences between 2 successive computations are under the uncertainty limit we set from the beginning.

This approach has the advantage to be simple and to use data resulting from manufacturer 's measurements. My computations indicated a fast convergence after about 5 repetitions.

What we have to define from beginning is which is the accepted uncertainty the rest is a simple succession of computations without any complexity.

The following pictures show the displacements and angular values for a 10 waves bellows and for a 5 waves part with a radial force at free end.

Computations were done with the C a value indicated in the data sheet.

Picture shows the evolution of free end displacement as function of applied load and computed stiffness in N/mm. It is important to notice that although the material is still in the elastic range stiffness is NOT linear when stroke (and load) increase due to the angle effect and the bending moment values.

This picture shows how –for a 5 waves bellows- radial displacement and total angle evolutes as function of the distance to fixed point (at left end). Load was considered 10 N.

This means that a 10 waves bellows with a distance between ends ≈ 16 mm will introduce a force of 10 N in the structure.

I do NOT pretend that the way is a high precise solution but who needs a high precision ?

In engineering many times an order of magnitude is sufficient.

The approach is fast and requires low investments in comparison with a measurement which of course will give apparently more precise values but at which cost?

Nickname

Catspur,

I don't see what you are doing. If the end of the hose is free and you have water at 950 psi, ask any fire fighter what happens to the free end of a hose.

Back in my younger days as a fireman, we would never allow a hose under pressure not be restrained. The force of the water exiting the free end would cause a whipping motion.

I hope this isn't your case.

Best regards

I don't quite understand what you are looking for, either.

Is the free end of the hose pugged (static problem)? Then, is the so-called radial force actually a lateral load applied to the free end of the hose? in which case (for plugged condition) you are looking for the distance the hose will move for a given applied lateral force?

If the free end is open or has a nozzle attached, the number you are looking for cannot be dertermined with much confidence (too many variables, as mentioned by Toronado), and there will be some serious (undertiminable) whipping action as mentioned by akouvolo. Plus, your 950 psi probably does not exist at the (open) free end.

In answer to the questions about restraint. The "free" end of the pressurized hose, arranged horizontally, is connected to a vertical tank containing a water & vapor mix suspended through a load cell for active weight measurement. Hence the need for a compliant hose under pressure. The opposite "fixed" end of the hose is attached to a drain valve, well anchored. I seek to estimate the weight measurement error due to stiffness of the hose.

Catspur

I think the load cell deflects by only a small amount, and thus the force exerted by the hose would would be nearly constant. This would affect the total weight measurement of the tank + contents, but it would have very little effect on differential weight of the fluid at various levels in the tank. If that is what you really need, then there might not be a significant problem.

OK. Then, the Drain Tank is closed, so the problem is static (ignoring the velocity of the in-draining liquid). To assure a static problem take measurements periodically (with the drain valve closed intermittently) or maintain a slow drain rate.

The inside surface of the stretched side of the bent pipe is larger than that of the compressed side. According to one source I came across many many years back, using Archimedes' principal to integrate these pressures over their respective surfaces produces the net force you are looking for. However, there is also an (inclined) internal axial force (fluid pressure) with a horizontal component and a vertical component. I don't know how effectively this is taken care of in my colleague's Archimedes solution, but the vertical component is opposite the vertical component of the integrated inside surface pressure force.

I know the above is rather vague and unclear, but maybe it will trigger something in your or a colleague's head.

Personally, I suspect the friction forces are going to be more significant than the straightening effects of the pressurized flexible pipe .

Thank you all for the replies. It is interesting to note that even flex hose manufacturers are unresponsive on this question. As the product name implies, flexibility (variations based on internal pressure and length) appears to me as a fundamental parameter that a manufacturer should at least be able to estimate for a given application. Maybe this one is too esoteric to bother with.

I will adopt the Guru's advice and make a measurement. This will require making a hose purchase based on a best guess of length. Hope I get close.

Sincerely,

Catspur