Maximizing Bending Moment in Simply Supported Beams: An Experimental Approach

[SteamKing]

All you have to do is compare the expressions for deflection for a point load versus the distributed load.

 

[oxon88]

ok Thank you. so my figures for deflection look acceptable?

 

[SteamKing]

Yes.

 

[oxon88]

Q3. What I value for the beam would be required to halve the maximum
deflection of the beam?

A. In order to halve the maximum deflection of the beam, the value of I would need to be doubled to 300x10^-6m⁴, this is because the deflection δ is inversely proportional to I

 

[oxon88]

does the above answer look acceptable?

 

[SteamKing]

Yes.

 

[oxon88]

ok thanks.

Are you able to provide some guidance on the last question?

4. The calculation for the maximum bending moment is to be verified
experimentally using a strain gauge bonded to the outer surface of the
beam, at the point where the maximum bending moment occurs.
Derive an equation which could be used to calculate the bending
moment from the measured strain value. State the meaning of all
symbols used in your equation.

 

[SteamKing]

Can't help with this part. Don't know anything about strain gauges and have never worked with them.

 

[oxon88]

Im not sure the strain gauge part is all that relevant. Just looking to Derive an equation which could be used to calculate the bending moment from the measured strain value.


the strain is the ratio of the deflection to the radius of curvature


e = δ / R = Mδ / EI


e=strain
δ=Deflection
E=young's modulus
I=the second moment of area of section about neutral axis
M=the bending moment at the section concerned
R=radius of curvature of bent beam

 

[Big Jock]

Which formulas did you use for the various deflections? I am having a real nightmare trying to work this out

 

[SteamKing]

Which formulas did you use for the various deflections? I am having a real nightmare trying to work this out

Re-read the earlier posts in this thread. There is a reference to the deflection formulas.

 

[Big Jock]

Still a little lost with this. The question asks to Plot a graph of the deflection along the length of the beam
(calculate the deflection at 1 m intervals). So unsure which formulas to use for the 5kN udl the 20 and 30 kN point loads. So if that deflection or maximum deflection and why would you need the slope for this isn't that something completely different from the asked question?

 

[SteamKing]

It's very simple. The problem asks that you calculate and plot the values of the beam deflection at 1 m intervals along the length of the beam. Sure, the beam will have a maximum deflection associated with a particular loading, but the beam deflection is not zero at other points on the beam. The formulas in the beam table allow one to select a distance from one end of the beam at which deflection is to be calculated. The beam table includes formulas for the deflection due to a UDL and point loads. Remember, the total deflection due to several different loads is the sum of the deflection due to each load by itself. This part of the question is basically plug and chug and plot the results.

 

[Big Jock]

I know this but unsure which formulas to use for the 5kN udl and the 20 and 30kN point loads for the distances of x=0-10 this is where my problem lies. I think the udl formula should be w/EI(Lx^2/12-x^4/24-L^3/24) and the point load formula w/EI(x^3/12-L^2x/16) is this correct? I so I then have all the various deflection point for all the info then sum them together?...

 

[SteamKing]

I don't recognize your formulas for the deflections. Did you read the earlier posts in this thread? If you go to Post #26, the reference there gives deflection formulas for cantilever beams on p. 1 and simply supported beams on p. 2.

 

[Big Jock]

Yeah seen them so reference to the attachment beam 7 formula 2 and beam 8 formula 2 would that be correct and give me all the info I need for my graph? The other thing there are two separate formulas for beam 7 formula 2 which one do you use?

 

[SteamKing]

Yeah seen them so reference to the attachment beam 7 formula 2 and beam 8 formula 2 would that be correct and give me all the info I need for my graph? The other thing there are two separate formulas for beam 7 formula 2 which one do you use?

It shouldn't be this hard. For the point load, the range of x is given for which each formula is applicable (e.g., formula #1; 0 < x < a). You use the formula which is appropriate for the value of x at which you are calculating deflection. Since you are calculating deflection along the entire length of the beam, you will have used both formulas by the time you complete your calculations.

 

[Big Jock]

Think I have it. Been working on the udl first, for the first metre of the udl y=6.8125x10^-5(micromillimters) would this be the correct measurement for the y-axis or should it be converted to mm?

 

[SteamKing]

Think I have it. Been working on the udl first, for the first metre of the udl y=6.8125x10^-5(micromillimters) would this be the correct measurement for the y-axis or should it be converted to mm?

Be careful with your units. I checked you calculation and I come up with a deflection of 6.8125*10-3 m, which is equal to 6.8 mm. If you are not sure, post your work here where I can check it in detail.

 

[Big Jock]

5x1/24x200x10^9x150x10^-6(10^3-2x10x1^2+1^3) was what I did. Which I have re done instead of 5 the udl should have been 5000 which gave the figure you quoted. Many, many thanks SteamKing for you patience over this. Glad I asked though as I have learned how to do this now correctly with your help

 

[SteamKing]

You have to be careful in reading the beam diagram. The UDL is 5 kN/m, not 5 N/m.

 

[Big Jock]

oxon88 and SteamKing are the values in the table correct in post 30. I am using these as reference while I' am working them out??...

 

[SteamKing]

They seem to be OK.

 

[Big Jock]

That's what I have found so far. Having a little trouble getting the 30kN load deflection figures and wanted to double check that these were the values I was looking for...

 

[SteamKing]

If you can't resolve your problem, post a sample calculation, like you did for the UDL.

 

[Big Jock]

30000x7x2/6x200x10^9x150x10^-6(10^2x2^2-7^20) Think my problem is what the values of a and b are I have been alternating between 3 and 7 and plugging these into the formulas mentioned but not having much joy whichever way I do it...

 

[SteamKing]

30000x7x2/6x200x10^9x150x10^-6(10^2x2^2-7^20) Think my problem is what the values of a and b are I have been alternating between 3 and 7 and plugging these into the formulas mentioned but not having much joy whichever way I do it...

Your calculation isn't coming out correct because you are not using the right equation.

It looks like you are trying to determine deflection for x = 2 m

Use Eq. 7, first part from the attachment, since 0 < 2 < 7:

y = [(P*b*x)/(E*I*L)]*(L^2 - x^2 - b^2)

Part of the confusion is the table uses 'l' for L, and it can be overlooked easily. You forgot to divide by L and in the parenthesis part, you multiplied L^2 by x^2 when you should have subtracted.

 

[Big Jock]

Thanks very much your spot on and all my numbers all look like the others listed and so des my graph. Many thanks again for your patience.

 

[Big Jock]

SteamKing which formula would you use to find the I value to half the maximum deflection in the beam? Which I guess is the sum which at 5m is 49.20mm...

 

[SteamKing]

Look at the formulas you used to calculate deflection. Where is the I used in all of the formulas? What relative value of I would cause the deflection of the beam to be half of its original value?