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2 span beam with hinge question

2 span beam with hinge question

2 span beam with hinge question

(OP)
I have two questions regarding a 2 span beam with a hinge.

The first question is if I have a 2 span beam as show in the attachment, how do I determine the bending moment on the beam if the length of both spans are L=10', the distance from the center roller is X=3' and w=10plf?

The second question is if I have the same beam as described above, how do I determine what X is in order to minimize the moment on the beam?

RE: 2 span beam with hinge question

Use the Method of Superposition... Is this a class assignment ? censored

RE: 2 span beam with hinge question

First off, this sounds somewhat like a homework problem, which is prohibited on this site.

RE: 2 span beam with hinge question

Hint: remember, the moment at the hinge always has to be zero.

RE: 2 span beam with hinge question

Additional hint: that turns the segment on the right into a simply supported beam and the segment on the left into a simply supported beam with a cantilever. And the moment at the end of the cantilever also has to be zero, which we already know it is since it's at the hinge.

You can probably take it from there...

RE: 2 span beam with hinge question

for me, the easy way to solve is using the unit load method ...
1) remove the central support, and solve statically; determine the deflection at the mid-support location.
2) without the applied loads, apply a unit load (opposite to the deflection from 1) and solve statically; determine the deflection at the mid-support.
3) then the mid-support reaction is deflection 1)/deflection 2), and find the rest of the reactions (and internal moements) statically.

Quando Omni Flunkus Moritati

RE: 2 span beam with hinge question

LOL,
This looks like a homework problem but I want to know the answer anyway.

RE: 2 span beam with hinge question

I just go to my steel book, use the equations there for the two cantilever load situations, and use superposition as previously mentioned. I do not try to reinvent the wheel whenever possible now that I'm licensed. However...

This has got to be a homework problem, and the OP is a sophomore, right?

Mike McCann
MMC Engineering

RE: 2 span beam with hinge question

Not quite sure what the question is... for a uniform load, your maximum moment will occur at mid support and will be 0.125ql^2. The point of zero shear will be at 0.25 span. What is the material? Steel, Concrete? Do you have alternate loading? This will have an impact on your moments, with no load and full load, your max moment will be at mid span, but of the same magnitude.

If you can use class 1 section, steel, and plastic design the moment will be 0.0858 ql^2 and there will be no impact on your max moment with alternate loadings, but your splice moment will change and your splice point should be at 0.172 of your span (I generally use approx 1/6 span).

Dik

RE: 2 span beam with hinge question

This has to be homework because the OP has only logged in twice, but out of curiosity, has anyone come up with an Excel solution to optimizing cantilever glulam beam systems?

RE: 2 span beam with hinge question

(OP)
This is not a homework problem. I am trying to study for the PE and saw a question similar to this but the question in the NCEES manual was just gave the influence line. I changed the problem around a bit so it was not the same but I want to learn how to perform an analysis.

RE: 2 span beam with hinge question

OK, what I was wondering was why you were asking the first question because it is simple statics.

The second question, I can't answer because it has been so long since I was in University. If I had a need for that concept on a project, I would just set up a simple Excel worksheet and iterate x manually until I got a fairly exact answer. But I don't see how this particular thing would ever be a question on a PE exam.

RE: 2 span beam with hinge question

(OP)
I could figure it out using STAAD but where is the fun in that!smile

RE: 2 span beam with hinge question

Thanks Mike... but, you scanned it upside down! That's what I like about plastic design...

Dik

RE: 2 span beam with hinge question

It has to be simple quadratic equation to find "x" to minimize the moment in the cantilever beam, but I don't have any idea - I still would like to know the answer.

RE: 2 span beam with hinge question

AELLC... are you talking about the Gerber approach to continuous glulam... I have one (originally for steel) that I wrote years ago... I'll see if I can dig it up. Should be easy to modify...

Dik

RE: 2 span beam with hinge question

I would think if you have the equations for the moment at each location, if you take its derivative and set it to zero, then solve for x, you can determine what x would minimize the moment at each location. Seems cumbersome though, there has to be a simpler way to do it.

RE: 2 span beam with hinge question

I spent a couple minutes trying to solve for X and determined it would take me longer than I cared to give it. lol. In practice I would do the iterative approach as AELLC mentioned, but that no help on the PE exam.

RE: 2 span beam with hinge question

dik:

Print it out and turn it around! pc

Mike McCann
MMC Engineering

RE: 2 span beam with hinge question

ameyer...

The splice location for a uniform load would be at 0.25L. The reaction would be 3/8 the total load with 1.25 being the centre reaction, and the max positive moment would occur at 3/8 of span, or 3/4 of span to PZS. The max +ve moment would be R^2/(2*q). Simple statics for this problem...

Dik

RE: 2 span beam with hinge question

@dik

I never heard of Gerber.

It was years ago (before we had computers in the office), we had all these glulam beam roof systems in warehouse type buildings, and of course someone demanded that we come up with the most economical (smallest weight of glulam overall) solution.

We got really burnt out from doing so many of those by calculator and pencil/paper.

RE: 2 span beam with hinge question

Actually, it was even more complicated because it was the double cantilever system with drop-in beams.

RE: 2 span beam with hinge question

One thing I tried, but have not verified if it is indeed the answer, is the following:

I assumed if you made the maximum positive moment in the backspan equal the maximum negative moment from the cantilever, you wouldb minimizing the moment.

From that, using beam tables, I set the moment equations at those two locations equal to one another. You get an equation that has a^4, so I solved with mathcad and got .168L

I ran it in FEA, and got .17355L, so I tried it another way - I noticed in FEA that the positive moment in the beams are equal even when I vary the right span, so I set the right positive moment equal to the cantilever moment, and got 0.1715L, which was still not correct.

So I am stuck...

RE: 2 span beam with hinge question

OK, I set it up with Excel, and got 0.1715L

RE: 2 span beam with hinge question

AELLC,

Years ago I read in Engineering News Record that the greatest structural analysis software in existence is...Excel. I can't disagree.

RE: 2 span beam with hinge question

(10'-1.175')*105 = 927 ... no? where'd 435 come from ??

we've got two 10' spans, yes? total load = 105*20 = 2100
LH+RH = 435+1231 = 1666 ??

if the RH beam is effectively SS, then the LH is statically solvable, with a tip load of w(10-x)/2 (and the UDL)
then LH reaction will be w(10+x)(10-(10+x)/2)/10 - w(10-x)/2*x/10
and the moment (in the LH beam) will be M(y) = LH*y - wy^2/2
maximum at y = LH/w = (10+x)(10-x)/20 - (10-x)x/20 = (10-x)/2

Quando Omni Flunkus Moritati

RE: 2 span beam with hinge question


Archie264,

Yes, and the irony is I refused to learn it at first because I thought it was simply accounting mathematics - oriented, and had no idea it had all the Boolean logic and table lookup (and even some database) functions we need. But back in those days we used Enercalc a lot.

RE: 2 span beam with hinge question

rb1957,

Sorry, I forgot to write divide by 2 for the half-span. =But the 435# applied at the cant end is correct.

RE: 2 span beam with hinge question

Pick a cantilever length; solve for Pos M's and Neg M; size the beam; and move on. You can live with 95% solution and spend a lot less time (and the clients money) than trying to determine the "optimal" solution. Besides the detailer will appreciate have 3'-0 or 3'-6 rather than 3'-4_29/32".

gjc

RE: 2 span beam with hinge question

mtu,
We were approaching this as an academic exercise. It took me less than as minute to iterate the x-value. Now that I have it set up, I can use it in the future with ease.

RE: 2 span beam with hinge question

AELLC,

Yeah, I prefer Excel to MathCAD. I find it easier to use and even think it might be more powerful. Others' mileage may vary...

RE: 2 span beam with hinge question

The OP is an interesting academic problem. And, anyone who wants to pretend to be an engineer or is going to take the PE exam should probably not need a computer to solve it. However, the exercise and thought process would do them some good in becoming thinking engineers. For the beam arrangement shown, I’ve been using x or a = .1716(length, l) and M = .0858wl^2, for the last 50 years. I had a 4' long slide rule, so I could read it pretty well to four places with the aid of a machinists micrometer, wink, wink. These are tabulated in different books for various beam arrangements. I understand that STAAD or various spreadsheets can really expedite our work, but it seems they are turning us into a bunch of non-thinking, non-reasoning computer operators, not better engineers with good experience and judgement.

There are several other things to consider in this beam arrangement problem. You should consider the potential of unbalanced loading. For all the fine tuning of the moment and splice location, you may not be able to find a W17.83x60.326 which has the exact Sx you need; and then you will not want the spice to fall at the exact location where a joist or truss frames onto the beam, since you don’t want a truss seat right on the splice. You don’t want a bunch of different length beams in the same area, so you’ll pick a few common splice locations. And finally, you have to pay attention to beam bracing at the canti. tip and over the column, at the -M.

RE: 2 span beam with hinge question

@dhhengr,

The first question seemed to be a basic EIT test question, and the second one, I doubt would be on the PE exam.

In defense of Excel vs old-fashioned thinking and slide rules, since I used slide rules in University until my senior year, I agree...however there's a lot of thing that Excel can be set up to check that would be too time-consuming to do even by slide rule or design chart.

I probably forgot the rules of thumb of which you are familiar, because I haven't done a cantilever system since 1980. But I still do cantilever beams on Excel, because in my type of work they have horribly complex loadings.

The skip loading is easy in Excel because it is a piece of cake to apply and remove span-specific loads.

What you advocate is typical of engineering practice before Excel was popular, but I guarantee you that it ensured that I was working on weekends instead of with my family.

On the other hand, I agree 100% that any engineer that uses STADD or Excel exclusively without never doing the old-fashioned methods will become merely an input technician. The trick is to find a good balance.

I used to like tables and charts but in my line of work, the loadings are just too complex to ever allow those to be accurate.

RE: 2 span beam with hinge question

If the cantilever length 'x' is correct, the remaining distance is not (1-x)L. It is L-x. So the diagram is wrong to start with.

BA

RE: 2 span beam with hinge question

BA,
hmmm, just noticed that - I didn't see that before. What I did was write down the problem in my own fashion, and correctly wrote down L-x.

RE: 2 span beam with hinge question

The moment in the right span is w(L-x)2/8. By symmetry, the positive moment in the left span is the same.

The load on the cantilever is equivalent to a concentrated load of wL/2 at the tip, so Mcant = wLx/2.

Moment will be minimized when the negative moment at Support 2 is equal to the positive moment in the other two spans,
i.e. when wLx/2 = w(L-x)2/8

Solving this quadratic equation, we find that x = 0.1516L which was found by dhengr above.

BA

RE: 2 span beam with hinge question

1st part, I solve for reactions by statics then find the area under the shear diagram for the cantilevered beam.
2nd part, iteration is good but complicated (for me) by compression face bracing field location limitations.

RE: 2 span beam with hinge question

Quote:

On the other hand, I agree 100% that any engineer that uses STADD or Excel exclusively without never doing the old-fashioned methods will become merely an input technician. The trick is to find a good balance.

It doesn't depend on the tools, it depends on how you use them. You can use Excel to plug numbers into someone else's spreadsheet, or you can use it to work from first principles, or anything in between. You can use STAAD to do a complete design for you, or you can use it to understand how the structure behaves under different conditions. The same applies to "hand calcs"; you can plug numbers into a formula found in a book, or you can work to improve your understanding of how the structure behaves.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

RE: 2 span beam with hinge question

Mike... Now you tell me... after I rotated my monitor 180 deg...

RE: 2 span beam with hinge question

BA, I thought dhengr got 0.1716L

I got 0.1715L by plugging into Excel, and structSU10 got 0.168, 0.1715, and 0.1735 depending on what computer method he used.

Was your 0.1516L a typo?

RE: 2 span beam with hinge question

Quote (The same applies to "hand calcs"; you can plug numbers into a formula found in a book, or you can work to improve your understanding of how the structure behaves)


The thing I like so much about Excel is you can goal-seek or adjust the inputs so easily to see what changes in how the structure behaves

RE: 2 span beam with hinge question

Thank you for pointing that out, AELLC. Yes indeed, my value of 0.l516L should have read 0.1716L.

BA

RE: 2 span beam with hinge question

Or to put it another way,
x = L(3 +/-√8)
= 5.828L or 0.17157L with only the latter value within bounds.

BA

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