Ovalisation or a ring: diferential equations

[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)

 

[AndyW]

Hello there,

Thank you for your interest in the ovalisation of a ring and for sharing your equations with us. I can recommend a few resources that may help you in your research.

Firstly, for a more general understanding of shell deformations, I would suggest looking into the book "Theory of Elasticity" by Timoshenko and Goodier. This book covers the theory of elasticity in detail and includes a chapter on shell deformations.

For a more specific focus on the ovalisation of a ring, I would recommend the book "Elasticity: Theory, Applications, and Numerics" by Martin H. Sadd. This book has a chapter specifically on the deformation of rings and includes the equations you have mentioned in your post.

If you prefer online resources, I would suggest looking into the "Theoretical and Applied Mechanics" section of the University of Oxford's Department of Engineering Science website. They have a section on shell theory which may be helpful for your research.

I hope these resources will be helpful to you in your research. Good luck!

 

[sir-pinski]

Hello,

Thank you for your question. I am not an expert in this specific topic, but I can provide some general information about differential equations and their application to ovalisation or ring deformations.

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. In the context of ovalisation or ring deformations, these equations can be used to model the behavior of the ring under different forces and stresses. The six equations you have listed are a good starting point, but it is important to note that they may not be the only equations needed to fully describe the behavior of the ring. Depending on the specific characteristics and conditions of the ring, additional equations may be necessary.

As for finding a book or website that specifically focuses on the differential equations for ovalisation or ring deformations, I would recommend searching for textbooks or research papers on structural mechanics or elasticity. These topics often cover the use of differential equations in modeling the behavior of various structures, including rings and shells. Additionally, you may find helpful resources on websites or forums dedicated to engineering and mechanics.

I would also suggest consulting with a professor or expert in this field to ensure the accuracy of your equations. They may also be able to provide additional resources or guidance for your specific problem.

I hope this helps and good luck with your research!