Help - Need Equation for Round Cantilever Beam

Hi. I found this link to beam boy in another post while browsing the site... Hope it works.

Have fun!
Stress calculator for beams free download

Cool, I love all these calculators & stuff!

Hi Oomsarel,

I wanted to have a look at the "Beamboy" but it was impossible to download it. Do you have a download already? I would be interested to see what is proposed as solutions.

Could it be possible to get it from you?

Thanks

Young's Modulus is the ration of stress over strain = E = δ/ε

Therefore ε = δ/E

δ Bending = (My)/I

M = Bending moment max for cantilever = F x Lmax (length of beam)

y = distance from neutral axis outer edge is where the highest stress occurs.

I = Second Moment of Area for Circle = (πd⁴)/64

I think your statement: ε = (F * L * 6) / (Y * W * H^2), is a bad way of writing or understanding the principles, I found this a little miss-leading to begin. If you were to write it in terms of stress and strain beforehand, it would be much easier for you to understand and derive the equations, when you know what the components are.

Please note this is only for small strains anything larger becomes more complicated.

Thanks, Andy

What I am trying to do is substitute an equation in an Excel spreadsheet. The output of that equation is expressed in strain and I just need to convert the formula I cited at the original post from a rectangular beam to a round beam.

It is a case of expanding the basic formulas I gave above to give:

ε = δ/E where = My/I

For max strain y (distance from neutral axis) is equal to 0.5d

Therefore ε = (32FL)/Eπd³ At any other diameters ε = (64FLy)/Eπd⁴

Strain is a ratio and should be unitless, by substituting the units of the variables and eliminating any that divide through you should end up with nothing as a formula check.

(32FL)/Eπd^{3 =} Nm/((N/m²)m³) = Nm³/Nm³ = 0

Thanks, Andy

Strain= (F*L)/(Y* (π/32)*d^3)

same meanings as you used d= rod diameter in the section where strain is required.

L = distance of applied force to considered section.

The equation you use is ONLY for a cantilever valid!

For other configurations you should use:

Strain= Mb/(Y*(π/32)*d^3) where Mb= bending moment in considered section.

Thanks, guys.

One question; what is "n" in the equation?

The system I am trying to model is a round cantilever rod that is fixed to a solid surface (ground). There is a strain gauge mounted on the rod .375" up from the point where the rod intersects the solid surface. I need to know what diameter to select for the 316 stainless steel rod for maximum gain, but still keep a safety factor for the material.

L would be the distance between the center of the strain gauge and the applied force. Is that correct? Does the distance between the applied force and the ground point matter in that calculation. I am thinking it does not.

n is pi, it is just the symbols in the editor are terrible and π looks like n.

As strain is a ratio E = δ/ε, you use max stress to work out your diameter δ = (My)/I

δ = yield strength, M = FL for cantilever, y = 0.5 d for max stress, I = second moment of area for circle (pi.d⁴⁾/64

Rearrange to give d³ = (32FL)/(δπ). Apply the SF to your load F beforehand

Remember to keep continuity of units throughout.

Thanks, Andy

It's pi. Copied in from the palette via the toolbar. Between ο and ρ (bottom left corner). But it doesn't look too realistic so I tend to type pi.

L would be the distance between the center of the strain gauge and the applied force. Is that correct? Yes

Does the distance between the applied force and the ground point matter in that calculation. I am thinking it does not. That's correct. I assume L is >> your 0.375", then the strain will be only a little higher at ground level.

I need to know what diameter to select for the 316 stainless steel rod for maximum gain, but still keep a safety factor for the material. It depends on the length and applied force. Not sure what you're trying to do, but I suggest strain not above say 1/2000 (0.0005) so with Y ~ 200000 MPa, stress = 100 MPa, reasonable working stress for stainless.

For high gain do work with a half-bridge that means put 2 strain gauges at 180° and use them in a push-pull connection. Depending on the kind of load you have and on maximal probable load you can go till 2.5E-3 for the strain value. If the load is variable you should consider that fatigue limit in case on high number of cycles (> 50.000). Up to 20.000 cycles you can accept 0.9 of the minimal yield limit and work with correspondent strain. The way your cantilever is fixed in the "ground" could influence the strain distribution in the measuring section. Local distribution can be very different from the one computed with above formula which is valid ONLY far from the force introduction. It could be perhaps of interest to place your strain gauge (or gauges) not very near to the anchoring zone.

The surface area of the round bar is =PI*radius* radius = surface area, just like your W*H, I think that is the answer unless a round bar has higher rigidity than a rectangle or a square ?

YT Lim

I am sorry but I do not understand is your comment related to the remarks I made?

If so I still do not understand what you mean. Please be more explicit.

Bending Stress is related to with the second moment of area about its axis:

Rectangle Second Moment of Area = (b.d³)/12

Circle Second Moment of Area = (pi.d⁴)/64

And bending stress is (My)/I

y being the distance from the neutral axis. The original equation (F * L * 6) / (Y * W * H^2) is only relevant for the strain at the top or face of the rectangular section, where y is half of H so the equation becomes:

(F*L*0.5*H*12)/(Y*W*H^3), H cancels with H^3 to give H^2 and 12*0.5 = 6

Giving the original equation (F * L * 6) / (Y * W * H^2)

Hence for the round bar cantilever when y = 0.5D, I = (pi.d⁴)/64

The eqaution becomes (32FL)/(pi.d⁴) or for excel (32*F*L)/(pi*d^4)

I am sorry to say that you have ^4 in stead of ^3 for the d in your last formula. You can recognize it making a dimension analysis [F*L]=F*L [pi*d^4]= L^4 and F*L/L^4=F/L^3 or stress is measured in F/L^2 for instance in N/mm^2 or psi or ksi.

If the person who asked for help has to use "STRAIN gages" then he is interested in STRAIN since this is what the gage is sensitive to. If he only wants to determine the dimension of the section then it has to be done on the basis of the stress.

In fact stress or strain this is the same formula the difference is only the division by the Young modulus in order to convert stress in strain.

That is correct thanks. That is just why workings are so important.

Thanks, Andy

I agree with Andy. I don't think the author is interested in Strain, but rather stress. Follow Andy's line of reasoning and you'll have a much better understanding of Stress & Strain.

I have entered all the formulas from AISC's Allowable Strees Design handbook's
Beam Diagrams and formulas on a spreadsheet, referenced to a bare set of inputs. This has become a free beam calculator that I use with first principles, and can solve most bending problems with it at a moments notice. The beam diagrams and formulas are old as the hills and can be forund in most any book that has a discussion of strength of materials. Accompany that with a good set of steel tables and you're set to go.