Equations for Ring Ovalisation and Shell Deformations: A Comprehensive Guide

[lalbatros]

Hello,

I am looking for a book or a website 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.

Thank for your help !


Appendix: the equations I would like to check
==================================

$$\frac{dT}{d\Theta} = -F-R\ p_\bot$$

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

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

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

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

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

with the following meaning:

T: shear stress
F: compression stress
M: bending moment
w: radial displacement
v: longitudinal displacement
R: radius of the ring
E: elastic modulus
$$I_z$$: 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)

 

[Achy47]

Hello there,

Thank you for your interest in the ovalisation of a ring and the corresponding differential equations. I am happy to provide some helpful resources for you to further your understanding of this topic.

Firstly, for the specific equations you have listed, I would recommend checking out the book "Theory of Elasticity" by Timoshenko and Goodier. It covers the equations for the deformation of rings, as well as a more general theory for shell deformations.

Another useful resource is the website of the International Association for Shell and Spatial Structures (IASS). They have a section on their website dedicated to publications, where you can find articles and books related to shell structures and their deformations. This may be a good place to find more specific information on the ovalisation of rings.

Additionally, I would suggest looking into finite element analysis software, which can help you solve complex deformation problems for various structures, including rings. Some popular software options include ANSYS, Abaqus, and COMSOL.

I hope these resources will be helpful to you in your research. Best of luck with your work!

 

[ravenprp]

Hello,

Thank you for reaching out and sharing your interest in finding a comprehensive guide for the equations of ring ovalisation and shell deformations. I completely understand your need for a reliable resource to help you better understand and solve these equations.

One book that I would highly recommend is "Theory of Elasticity" by Timoshenko and Goodier. It covers the mathematical theory of elasticity and includes detailed derivations and explanations of equations for various deformations, including those for rings and shells.

In addition, there are several online resources that can also be helpful in understanding and solving these equations. Some good options include "Engineering Mechanics of Deformable Solids" by William F. Hosford and "Mechanics of Structures" by David W. A. Rees. Both of these resources provide comprehensive coverage of the equations for ring ovalisation and shell deformations, along with examples and exercises to help you practice and apply the concepts.

I hope these recommendations are helpful and that you are able to find the resources you need to solve your particular problem in a systematic way. Best of luck in your studies!