# Theory of shells , Membrane theory

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1. Oct 26, 2014

### picovish

I am working on a design of a spherical dome.

I tried to understand membrane theory with no success.Can anyone please help me with an clear explanation to derivation of membrane theory of shells.

2. Oct 26, 2014

### OldEngr63

That is a pretty fuzzy question, but here are a few thoughts.

A typical membrane is a thin, flexible element like a rubber sheet. Thus a filled rubber balloon is a membrane shell.

Try drawing a Mohr's Circle for an element on the surface of a balloon. What you should find is that there is equal normal stress on all four sides of your stress block, no matter how it is oriented.

Does this get you started on membrane theory?

3. Oct 27, 2014

### picovish

yes,thank you.

yup , i agree. That is a fuzzy question.

I do understand the assumptions involved in membrane theory for designing a thin concrete shell structure. But i do not follow the derivation of it. I am doing my masters in structural engineering and I have never had classes in deferential geometry before.

( The following pages are not subject to copyrights.
1. Theory of Plates and Shells by Stephen Timoshenko
2. Published: 1940 )
why does the normal forces ,shear forces (pg 429 )bending and twisting moment have the term (1 - z/r) (pg 430)
Can you please explain pg 431.

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4. Oct 27, 2014

### picovish

There are a lot of video lectures for beam bending theory and classical plate bending theory, but there are almost none for theory of shells

5. Oct 27, 2014

### OldEngr63

Before tackling Timoshenko, let me suggest that you look at Den Hartog's Advanced Strength of Materials, McGraw-Hill, 1952, Ch III, "Membrane Stresses in Shells." On p. 78, Den Hartog addresses a spherical tank design problem, so this might be particularly relevant.

Specifically to your question from Timoshenko, "why does the normal forces ,shear forces (pg 429 )bending and twisting moment have the term (1 - z/r) (pg 430)," the answer is that Timoshenko is expressing the variation the stress through the thickness of the shell, much like assuming a linear variation of strain and stress through the depth of a beam in ordinary beam theory.

6. Nov 1, 2014

### NumericalFEA

Generally speaking, an optimal design of a shell would ensure that stresses caused by the bending moments are small in comparison with the stresses caused by membrane forces. That would ensure that the shell would have good load bearing capacity and at the same time its thickness can be small compared with the shell overall span (as a result, the shell weight can be reduced). There is a special class of shells called shallow: the shell curvature is small, so it is "almost a plate" (so to say), but at the same time the distribution of internal stresses is such that the compression forces play the major role in the stress distribution under loading. You may want to have a look at the following webpage: http://members.ozemail.com.au/~comecau/quad_shell_shallow_shell.htm
(at the bottom of the page there is a reference to a book devoted, in particular, to that subject).

7. Nov 2, 2014

Thankyou

8. Nov 2, 2014

### picovish

I am going through the book "THIN SHELL CONCRETE STRUCTURES", Billington. It is quite good.

9. Nov 2, 2014

### NumericalFEA

You are welcome. One of the reasons why I gave reference to that book is because it contains, in particular, working source codes that you may use and/or adapt for your problem (the theoretical background is also profound and presented accordingly).

10. Nov 12, 2014

### picovish

Membrane theory derivation is not hard to follow at all. I think I did get a little rusty.

I will be soon designing an RC dome. Since it is gonna be hemispherical ,there will not be any additional forces due to edge effects. But still I would like to know more about edge effects.

I am reading edge effects on shell structures. Can you suggest any literature for reading.

11. Nov 12, 2014

### picovish

Thank you. The derivation was not tough to follow.

But I do have a question. Why does normal stress on all four sides of stress block has to be equal.

I think hoop stress and meridian stress will have different values.

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12. Nov 20, 2014

### picovish

I am now doing an analysis on spherical domes with fixed edges. I am quite confused on applying the formula for forces and moments in a spherical shell due to edge forces. please help.

13. Nov 20, 2014

### picovish

Can somebody please tell me what units these values in the formula are?

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