Deflection thin wall arch 2

[nickjk]

I am currently designing a project that uses a thin walled cylinder and was asked to come up with the following info.
I started to solve the problem but feel I may be in left field because of the thickness of the cylinder.

I have a thin wall cylinder (.874" I.D. & .020" wall). The I.D. of the cylinder contacts 20 equally spaced .0625 diameter pins. The pins are in a fixed position that are assumed not to move. I will provide a uniform pressure to the O.D. of the cylinder to deflect the arch between the pins .005"

See attachment for illustration.

I need to determine the pressure required to deflect the cylinder or arch .005" between the pins and determine the stress when deflected.

I started to solve this problem by analyzing the cylinder between pins as shown in section Z-Z (see of attached drawing.)
Because of symmetry I was able to divide the section in half and use Castigliano's Method which I plugged in the deflection and solved for the force. I came up with a calculated resultant force of 2,881 lbs. If I use deflections formulas for straight beam with both ends fixed I come up with similar results. I was going to continue by calculating the maximum bending stresses both compression and tension and transverse shear stress.

I was told originally I may be able to solve this by using hoop stress calculations by finding the difference in length and solving. In the segment between pins I come up with .0005" difference before and after compressed. This method gave me a calculated resultant force of 1,044 lbs.

Any info on the proper method to solve this problem or examples would greatly be appreciated.

I have reviewed Roark's and all the engineering books I could find, but do not know enough about the exceptions to rules.

Thanks in advance.

Nickjk

 

[ishvaaag]

Pilkey's (16.1c) equation gives how much the neutral axis is off the center of your section, for your case the 1.9 per thousandth of the depth or 0.38% of half the depth. Hence, a linear assumption of the distribution of the stresses in bending is not going to be much off-the-mark, and neither are going to be all those the concomitant effects of shear in the section, and others (everything is inter-related in beam-arch theory). Since as well the ratio span to depth of one substituting fixed straight beam is near 7, neither can we say your purportedly substitutory fixed beam is still a deep beam, also causing nonlinear distribution of the stresses along the depth. All this brings the problem to get an approximate answer quite likely under 5% to 10% error using a model either from a fixed arch or a flat fixed beam (or plate). May later try to see the differences, mainly on the standing compressive force on the flatly curved beam.

Also, a 2D or 3D FEM model can give further assurance.

 

[ishvaaag]

As well, what above ... mainly in the small deflections realm. At .005" wanted deflection on a .02" deep cylindrical plate the deflections are not properly small deflections, if not overly large ones. Hence, use of a program that accounts for material nonlinearity, geometrical nonlinearity, specifically at large deflections range is warranted for the case. On some assumptions one might still get some approximate results with some program for elastic plus p-delta calculation, better if of those retaking the deformed shape and rebuilding the matrix. If not, one can take out the axial stiffness eaten by the arch compression (that not hoop, as will see in the following paragraph) by dismissing a central core at yield stress equal to the compression, and using the remaining unyielded section for the calculations (arch, beam or whatever).

Also, upon compression and since seating upon a polygonal array of unyielding supports, hoop stresses are simply not possible, because the reaction supports take out the problem of such solution. Arches or plate arches fixed at the supports are correct, from symmetry. And then everything of what above.

 

[prex]

What you have is a circular arch with opening angle equal to [π]/10 and fixed ends. As it is in fact a tube (relatively long in the axial direction) you should substitute EI (per unit length) with Et³(1-[ν]²)/12 , though this won't make a very big difference (~10% on deflections).
In the first site below, under Arches -> Circular -> Fixed-fixed -> Unif.rad.load , you find a sheet for calculating stresses and deflections for such an arch.
However the Roark also has it, it is, in my 5th Ed., case 5h of Table 18; it doesn't provide general stress and deformation equations though.

prex

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[nickjk]

ishvaaag and prex

I would like to thank ishvaaag and prex for your help and support. Yesterday was spent trying to prove how a certain theory would apply to this application. prex I hope to dive into your suggestions this weekend.

This forum is such a powerful tool with brilliant minds on hand. Just wanted to let you know how much your support is appreciated.

Thank you again.

Nickjk