# Ovalisation of a ring

1. Oct 3, 2005

### lalbatros

Hello,

I am looking for a book or a web site giving the differential equations for the ovalisation of a ring.
In principle these are six differential equations:

one for the compression force along the ring
one for the shear force
one for the bending moment
one for the radial displacementof the ring
one for the derivative of the radial displacement
one for the longitudinal displacement

these are similar to the equations of a beam, the difference is that it is a ring.
I tried to write these equations by myself, but I have some doubts about signs and how bending moment and displacements relate to each other.

I would also be interrested in the more general theory for shell deformations: where is there a good text on the web?
With such a reference I hope to be able to work out such a particular problem in a more systematical way.

Appendix: the equations I would like to check
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$$\frac{dT}{d\Theta} = -F-R\ p_\bot$$

$$\frac{dF}{d\Theta} = T-R\ p_\|$$

$$\frac{dM}{d\Theta} = R\ T$$

$$\frac{dp}{d\Theta} = -w-\frac{MR²}{EI_z}$$

$$\frac{dw}{d\Theta} = p$$

$$\frac{dv}{d\Theta} = w$$

with the following meaning:

T: shear stress
F: compression stress
M: bending moment
$$I_z$$: moment of inertia of the material cross-section for the ring bending
$$\ p_\bot$$: radial force acting of the ring (depends on the angle)
$$\ p_\|$$: longitudinal force acting on the ring (depends on the angle)