Short Beam Bending: Roark's Formulas For Stress and Strain

[Saladsamurai]

Is anyone familiar with the Roark's text that can clarify some terms for me? I am trying to clarify what Roark is referring to in the above mentioned text in the section on short beams (in the 5th edition 7.10 and in the 7th edition 8.10). Throughout the text, he uses the terms "span" and "depth" and then in the next section he uses the terms "width" and "breadth" to add to my confusion. Nowhere can I find a schematic or definition that clarifies what dimensions these correspond to.

Referring to the attached figure, I have a beam whose dimension are:

x = 0.187
y = .440
z = .430

and I am trying to figure out which case/equation I should be using: Beams of Relatively Great Depth (section 7.10 5th) or Beams of Relatively Great Width (section 7.11 5th).

I think that I might want to be using equation in section 5.10 5th ed:

$$\sigma = \frac{W}{t}[1+0.26(\frac{e}{r})^{0.7}]\dots$$

but I certainly don't want to guess.

Any input is appreciated. Thanks

 

[Mech_Engineer]

I'm thinking your problem is more fundamental- it comes down to the fact that your block cannot be approximated as a "beam" due to the small span/height ratio. In Roark's 7th edition, Pg. 125 (first page of Ch. 8):

The formulas in this section are based on the following assumptions: ... (6) The beam is long in proportion to its depth, the span/depth ratio being 8 or more for metal beams of compact cross-section, 15 or more for beams with relatively thin webs, and 24 or more for rectangular timber beams. (7) The beam is not disproportiantely wide (see section 8.11 for a discussion on the effect of beam width).

I've attached a quick FEA analysis which shows your geometry, with the "beam" and an intermediate load on the top of it. You can see in the deformation result the "beam's" deformation along it's span (x-axis) is not uniform through it's "height" (thickness, y axis). This analysis is definitely not a good candidate for a beam deformation formula.

 

[Saladsamurai]

Hi Mech_Engineer

Thanks for doing that. I am in agreement that the beam is not really a beam in the simple sense. I was under the impression though that Roark addresses this problem in Section 8.10, 7th ed., page 167, last paragraph, 1st sentence:

For extremely short, deep beams ...

I just need to straighten out what he means by "short, deep." My 'beam' seems to be quite short in the x direction compared to the y AND z direction. I am assuming that by span/depth he means x/z (since he uses a gear tooth as an example on page 168, 2nd paragraph, 1st sentence). For this he uses the equation for σ in the OP (found on page 168).

I believe this is how I can resolve the 'not a beam' problem. Would you agree?

Thanks again for your input.

Edit: out of curiosity, what loading did you use? My actual loading is 1023 lbf located on the top surface at (x/2, z/2)

 

[Mech_Engineer]

I was under the impression though that Roark addresses this problem in Section 8.10, 7th ed., page 167, last paragraph, 1st sentence:

Ok I see it's a formula for analysis of beams with small length/height ratios like gear teeth. Seems valid for your purposes, I haven't used that specific equation set before.

I just need to straighten out what he means by "short, deep."

I'm thinking he is using "short" as a description of span (e.g. a "long" beam would be one with a large span, a "short" beam is one with a small span). I see where you find trouble in the book's terminology...

As for "deep," based on the nature of the analysis I think it actually means the beam's Y dimension or dimension which defines stiffness in primary bending direction; in your coordinate system the "Y" axis. Note that in the table on pg. 168 the text has the span denoted as distance "l" and it looks like "depth" as "d."

I am assuming that by span/depth he means x/z (since he uses a gear tooth as an example on page 168, 2nd paragraph, 1st sentence).

Actually I think span/depth is referring to the X/Y ratio in your coordinates.

Edit: out of curiosity, what loading did you use? My actual loading is 1023 lbf located on the top surface at (x/2, z/2)

:shy: I thought you might ask about that... I used 0.1 lbf just to get some numbers and put the load at about the 2/3 point on the block. Here's with 1023 lbf and a centered line load on top:

EDIT: By the way I don't think these results are all that useful by themselves; but if you have a more complex geometry representation you might try running it through an open-source FEA package to see if you can get an analysis which includes stress analysis of the beam's base.

 

[Saladsamurai]

Hi Mech_Engineer. Thanks again for your time. Interesting stuff. Perhaps since you seem to be experienced with FEA you could answer an auxiliary question that has just arisen as a result of our discussion. It has been a while since my FEA course and I cannot seem to figure this one out.

I know I am oversimplifying FEA, but as I understand it, if we wanted to apply it to the beam in question in 1 dimension, we would break the beam into N elements along x. We would then write the stiffness matrix out for each element and then assemble the N matrices into 1 global stiffness matrix.

All of the FEA problems I have solved have used simple Hooke's law style equations for each element. Now this type approach would not be appropriate for this kind of short beam would it? Meaning, breaking the short beam into many, many elements does not alone remedy the fact that it is a 'short' beam right?

Thanks again.

 

[Vadar2012]

I read the first post and started doing some simulations in strand7 for you while eating lunch at work. Then I read the rest of the posts...

Breaking the beam into many small elements does solve the problem, well not completely, but it does give much better results. I'm sure you know that the formulas used to approximate the maximum stress at each section along a beam are derived integrating along the height to give a nice average. For a beam with a large height, there's a lot more room for error than a small beam. The error is just going to compound over the height.

When you break the beam up, your calculating the values at each node instead of finding the average over the cross section.

 

[Mech_Engineer]

It depends on the complexity of beam element, some 2-D beam elements derived in class are usually just a linear algebra implementation of combination load beam theory; as such they are subject to any simplification limitations (such as when you want to ignore shear effects).

However, more advanced beam elements like for example ANSYS's 'Beam4' element (a 3-D structural beam element), there is a summary of it on Pg. 505 of ANSYS's APDL Theory Reference, can automatically take into account shear effects for shorter length/height ratios. Beam4 needs to be able to do this in the case where you are splitting a complex geometry into many small elements.

If you'e running FEA on what I would consider more of a block than a beam, it's best (IMO) to run a structural analysis using solid elements. For the purpose of simplification you might run a 2-D analysis (with specified element thickness) rather than 3-D (I did 3-D because it was fastest to set it up that way ironically).

 

[Studiot]

Machinery is not my bag so I'm glad real experts posted here first.
I like particularly the observation about gear teeth.
I also suggest looking at corbel design or mooring bollard design.

With transverse loading to any stubby object the greatest stress is suffered at the root of the projection, and this is reflected in stresses within the foundation.
You have not identified the connection of your projection to the foundation or whether it is monolithic, welded or whatever. This is vital.
Either way the analysis should consider the stresses within the foundation.
This is often the failure zone.
I also agree with ME that the analysis needs to be three dimensional as your loading
point-force introduces eccentricities.

Needless to say I do not have either the 5th or 7th ed of Roark - mine is the 6th and the analysis of the formula you originally presented appears on p203 -204.
Entry 21 of table 37, p740 is also relevant and offers a different formula derived from photoelastic studies.

The latter formula also appears in Pilkey (Formulas for Stress Strain and structural Matrices) in table 6-1, entry 3 page 288.

Original studies were presented by Jacobson : Proc Inst Mech Engrs 1955, as a series of charts.

Other analyses appear in Seely and Smith (Advanced Mechanics of Material) p396

And in Benham and Warnock (Mechanics of solids and Structures) p 407.