Timoshenko - Theory of Plates & Shells 1

[pperlich]

2nd Edition, Chapter 1, Article 1 the following statement is made and is key to the solution: "The lateral strain in the y direction must be zero in order to maintain continuity in the plate during bending..."

I don't grasp that concept. It seems to me like there could be a small amount of strain in the y direction, and rather than a loss of continuity, the cross section of the plate would end up as a trapezoid.

What am I missing?

Thanks in advance!

 

[GregLocock]

What happens at the corners?

Cheers

Greg Locock


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[dhengr]

Pperlich:
That statement is a simplification, an idealized condition, but pretty close to the truth vs. the real world or exact condition, and for a long or infinitely long plate; so that a strip of unit width can be used and analyzed as representing a typical strip anywhere in the plate length, removed from too near the ends of the plate or reaction locations. This simplification is akin to the simplification Timoshenko makes when talking about ‘Plane Stress’ and ‘Plane Strain,’ in chap. 2 of his “Theory of Elasticity” text. This was pretty much a basic assumption, a first step, in simplifying many of our daily engineering analysis problems, until we really do start thinking three dimensionally about a volume of material under load.

 

[pperlich]

dhengr,

Thanks for the reply. I guess that make sense. Just for fun I decided to run the math using the assumption that the stress in the y direction was 0. No surprise here, but I ended up with EI(d2w/dx2) = -M. So I guess the width of the plate makes the zero y-strain a more appropriate simplification than a zero y-stress simplification.

Greg, I don't know what you mean by your question exactly.

 

[GregLocock]

You can't have a lateral strain at an unrestrained corner, can you?

Cheers

Greg Locock


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[pperlich]

Greg: No, at the corner the lateral strain would be zero but that doesn't mean that away from the corners that the lateral strain is necessarily zero. Since the simplification of zero lateral strain is made, then the lateral stress must be non-zero. But at the unrestrained corner you can't have a lateral stress either, right?

 

[JStephen]

He's cutting a unit strip from a wide piece of plate. If the Y-strain varies through the thickness, that corresponds to the plate curling in the crosswise direction. In which case, you don't have cylindrical bending, you have dishing of the plate.

If you have a beam, you assume zero stress in the Y-direction and whatever strain occurs, there it is- the comopression flange can get wider and the tension flange can get narrower when bent. With a wide plate or wide bar, the compression and tension faces are tied together and can't move sideways relative to each other, so you assume zero strain rather than zero stress.

When I was in college in an ME lab, we did a strain-gauge test of a beam, which was a flat bar about an inch wide and 1/8" thick. Our strains were off somewhat from what they should have been, and it was only years later, I figured out why- from this effect. I believe Roark mentions it in connection with flat bars bent the easy way.

 

[pperlich]

Thanks JStephen! I didn't think about the curling aspect. That makes it make much more sense now! I remember doing a similar lab, but I don't recall what the lateral strain did. I still have all my notes. I'm half tempted to go look it up, lol.

 

[JStephen]

If I remember right, that throws in a factor of (1-nu^2) or about 10% difference on the deflection.
And now that I think about it, maybe we were measuring deflection of the beam, rather than strain with strain gauges- it's been a while.

 

[bridgebuster]

A little off topic but this thread calls attention to a typo in the book.

http://www.eng-tips.com/viewthread.cfm?qid=413308

 

[pperlich]

Thanks for the heads-up! I must be a serious nerd, because I love reading this book. Granted I'm only in Chapter 1 still, but its a good read. My old boss used to joke about how the ASME BPVC was "a good read". But this book actually is!

 

[btrueblood]

Yup. Ol' Tim O'Shenko had a pretty good way with the words, like most Irishmen.

(joke stolen shamelessly from Greg)

 

[JStephen]

Speaking of which- if you actually enjoy reading his stuff- get "History of the Strength of Materials" (was available cheap from Dover Books at one time, I assume it still is.)

 

[btrueblood]

I think his Theory of Plates and Shells was available on the (gutenberg project)? as a free pdf. Wherever. Greg pointed to the site awhile back, and I searched on the author and found the pdf file. It's in my download folder...which reminds me to transfer it to my personal thumb drive. Hmm, 55 Mb...need the big toe drive...

A quick google found this link.

http://155.207.34.6/files/Timoshenko.pdf

Some light beach reading.

 

[pperlich]

Call me old fashioned, but I like an actual book in my hand. I hate trying to read books on my computer/tablet/phone etc. That said I also have this book saved in pdf to my dropbox.

 

[btrueblood]

I cannot disagree with that pperlich. My wife had three of Timoshenko's texts, which mysteriously disappeared in our last move using a commercial hauler.

 

[Shu Jiang PEng SE]

The thickness of the plate thickness is assumed not change after the deformation. In other word, at any given cross section of the plate, the relative displacement of any layers of the plate along y direction is same. Therefore the strain of the plater in y direction is zero. In reality, there should have small plate thickness change and strain along y direction.

 

[MJCronin]

A groundbreaker in the Mechanical and Structural engineering field .....He wrote seminal works in the areas of engineering mechanics, elasticity and strength of materials,

Russian born Stephen P. Timoshenko is widely considered to be the father of Engineering Mechanics

https://en.wikipedia.org/wiki/Stephen_Timoshenko

MJCronin
Sr. Process Engineer

 

[pperlich]

@ jiang46602,
I appreciate your response, but in the text the plate thickness is in the z-direction. The unloaded plate is in the xy-plane.

 

[Shu Jiang PEng SE]

Sorry for the misleading. I read the document again and thought a possible explanation.

One implicit assumption of the analysis model of taking one strip of the plate as independent beam is the plate dimension along y direction is far larger than those in both x and z directions. The plane strain theory is applicable in this case and therefore the strain along y direction is zero.