Deflection of a dome from a point load

[Tad5]

Has anyone seen a formula for calculating the center deflection of a dome, if you know the dome thickness and material, from a point load towards the dome from the convex side at the center of the dome?

 

[msquared48]

Seems like this would be easy to model and solve in RISA 3D with a series of arch strips.

Mike McCann
MMC Engineering
Motto: KISS
Motivation: Don't ask

 

[IRstuff]

Isn't there something like that in Roark's?

TTFN

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[IDS]

There doesn't seem to be anything in Roarke.

An axi-symmetric analysis in a FEA program would give you the answer for a point load at the crown.

For a point load anywhere else you would need to do a full 3D analysis.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[IRstuff]

What about Table 30, Case 3B?

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[IDS]

What about Table 30, Case 3B?

In Roark?

Which Edition?

Mine (Edition 7, 2002) only goes up to Table 17.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[IRstuff]

6th edition, page 547 Table 30 Case 3b Shallow spherical shell, point load at pole,edge fixed and held.

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[IDS]

6th edition, page 547 Table 30 Case 3b Shallow spherical shell, point load at pole,edge fixed and held.

Found it.

Table 13.3 - 3b Page 611 in Edition 7 (International Edition)

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[Tad5]

Thank you all for your help. I do not have accesss to the Roark's book. Can you enlighten me with the formula? Incidentally I failed to mention that I also know the dimensions of the dome.

 

[jte]

Library?

If you're going to be doing stuff like this as a professiona, a copy of Roark's Formulas for Stress and strain is a good addition to your personal collection. Just remember that Roark is not the author (since the 5th ed) - you won't find the book by looking for him as an author.

jt

 

[IRstuff]

The compilation was done by Warren C. Young.

TTFN

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[IDS]

The formulas in Roark involve tabulated coefficients and are best taken directly from the book. It's available for about $50, and is really very good value.

Unless the dome is very shallow the best table to use would be Case 2 from Table 13 (Seventh Edition), rather than Case 3. The description is "Partial spherical shell, load P concentrated on small circular area of radius r0 at pole; any edge support".

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[IFRs]

Roarks is also available for MathCAD, I have found it to be useful, including this case.

 

[Tad5]

Well it looks like I gotta buy Warren's book. Merry Christmas to me. Hope I can find a used one cheap.

 

[IRstuff]

you can try:
AddALL
Bookfinder
BestBookBuys
AbeBooks

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[GregLocock]

If you are a memeber of an electronic library such as Knovel they may well have it.

Incidentally the deflection of almost any structure to a point load is infinite.

Cheers

Greg Locock


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[IDS]

Even if you get the book (and you should), it's an interesting and worthwhile exercise to analyse it in an FE program, and compare it with the values given by Roark. I suggest starting with an axisymmetric analysis with say 8 elements, and subdivide until you get no change in the deflection. Also compare with 4 node and 8 node elements, and compare axisymmetric with a full 3D analysis with plate/shell elements and/or brick elements.

Incidentally the deflection of almost any structure to a point load is infinite.

Surely undefined, rather than infinite?

In "practice" on Earth the deflection of the structure would be zero, and the deflection of the load would be about 6400 km

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[chicopee]

Membrane analysis as you would have on pressure vessels heads?

 

[rconner]

At least for historical purposes, might look into the design of the U.S. Capitol Dome mentioned at

http://www.aoc.gov/cc/capitol/dome.cfm

etc., and specifically how it was designed to support the many ton surcharge weight of the “Statue of Freedom” and base right at the top/middle.
While Walter/Bullfinch/Meigs et al may have had arguably primitive design knowledge (and built as you will see with cast iron arches), I think these arches have basically supported the dome and superstructure vs all/whatever loads applied since ~1866. [It appears according to

http://onlinelibrary.wiley.com/doi/10.1002/1099-1794(200006)9:3<183::AID-TAL139>3.0.CO;2-F/abstract

, they may well carry that load for good while longer (I’ve seen in the news some modern domes, with one would think their modern design tools and constructions, have not been so lucky!)]
Everyone have a very Merry Christmas.