Beam Formulas for Multiple Point Loads. 1

[truckdesigner]

Can anyone out there please help me?

For ages now I've been looking for simple formulas for calculating Mmax for simple beams with more than 2 point loads, for example multiple beams on girders. Text books etc only ever seem to list formulas for 2 loads.

What if I have 5, 6, 7 etc?

I did find one set of notes on-line that gave me the formula for 4 equally spaced point loads (equal loads): 3*Pl*L/5, but would like to know for more than 4 loads.

Appreciate any assistance.

Paul.

 

[IDS]

Creating a matrix and inverting it seems to be a rather sophisticated technique for solving a relatively simple problem.

For a "hand" calculation, where we have access to an intelligent computing device inside our heads, yes. On the other hand if you want to program a dumb external computing device to do it, it actually gets quite complex if you have non-uniform distributed loads over part of the length of the beam. Splitting the beam into short sections, either loaded over the full length or just at the ends, then solving the resulting matrix equations, is actually a reasonably simple way to do it, with the added advantage that it becomes trivial to extend the system to deal with continuous beams, and relatively simple to extend to 2D frame analysis.

My continuous beam spreadsheet Here goes the opposite way. It treats the beam between end supports as a simply supported beam, then finds the internal reaction loads required to get zero (or specified) deflection at the supports, but the complexity of the code for both approaches is probably about the same, and the computation time is negligible either way.

Buggar - there is a Visual Basic forum here, also VBA and spreadsheet forums which cover VBA, which is just Basic built into MS Office.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[IFRs]

What's wrong with superposition? I think I learned it as a 2nd year engineering student.

 

[rapt]

BAretired,

A point load over a support has no effect on the moment or shear diagram! Leaving it out would have no effect on the calculations. The "simplification" to equivalent UDL's is inherently dangerous (but not a lot) as it requires "fiddles in some areas to get the correct answer, so you need to know if your answer is correct before you start.

RE matrix, why would you do it for a simply supported beam unless you had to do it a lot of times and your loads were not symmetric or you already had a computer program to do it (that's a good idea if you do a lot of them!)?

 

[BAretired]

Teguci,

With multiple variable loads at multiple variable locations, it seems to me that a generic formula involving summations and boolean functions could be written, but I believe it would be more cumbersome than simply calculating the maximum moment as the area under the shear diagram while the value of shear remains positive.

rapt,

A point load over each support has no effect on the moment or shear diagram but it does affect the end reactions which are consistent with that of a uniform load. I see nothing "inherently dangerous" about the simplification, particularly if the number of point loads is large.

I agree that a computer program using matrix inversion will save time when there are a lot of beams to design, but it doesn't answer the question posed by the OP. Moreover, those who do not understand statics should not be using such programs. That would be dangerous.

BA

 

[dhengr]

BA:
There is lots of ‘dangerous’ out there these days, and it seems to pass as real engineering. It just staggers my imagination these days that almost nobody wants to really understand the fundamentals of what they are doing. Just give them a formula that they can plug numbers into or the exact code paragraph which tells them what to do, without any understanding of the intent of that section of the code, and they are on to their next problem without having learned a damn thing about real engineering. Just plug and grind without any understanding of the basics. And, better yet, if you can offer a computer program which does that math and takes the code into account, then they are really happy, no need to even lift a pencil. I think this is really a sad state of affairs.

 

[KootK]

I'm not sure if OP is interested in point loads of varying magnitude at irregular locations. If so, I don't think it can be made any simpler than:

1) Construct a shear diagram to ascertain point of max moment.

2) Apply formula below to each point load for moment contribution at max moment location.

Of course, now you've arrived at a place where the effort involved would start to rival that required to just do it from fundamentals or run it through a coded algorithm.

If one requires max deflection, and a midspan estimate isn't sufficient, then you're definitely in software territory.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[rapt]

dhengr and BAretired,

Completely agree. Knowledge and understanding of statics is essential before using anything automated.

I continually emphasis with RAPT software that designers should not be using it until they can regularly complete the same designs by hand calculation and have a full understanding of the design code they are using and the detailing requirements for structures. No computer program that I know of can completely analyse, design and detail concrete members, especially more complex concrete member variations. The designer must be able to do it and understand it all so that they can apply the results of the automated calculations in their final design decisions and detailing of the structure.

Unfortunately the opposite seems to be becoming the norm, where if you do not know how to do it, software will do it for you.

 

[medeek]

Using basic statics the principle of superposition it should be possible to come up with a generic equation for any loading however the equations will probably become quite large as the number of loads gets larger. I've actually had to do this sort of thing for my truss calculator when I added in point loads a while back. With point loads the shear and moment diagrams are not continuous functions so for each segment you have to define a unique function (equations) for shear, moments and deflections.

A confused student is a good student.
Nathaniel P. Wilkerson, PE

http://www.medeek.com

 

[BAretired]

If I remember correctly, a stepped function can be represented by a Fourier Series. This makes it possible to differentiate or integrate it directly. It has been a few years since I looked at that, so I am a little rusty at the moment.

BA

 

[IDS]

If I remember correctly, a stepped function can be represented by a Fourier Series. This makes it possible to differentiate or integrate it directly. It has been a few years since I looked at that, so I am a little rusty at the moment.

I suppose you could, but evaluating the step functions directly (which is what Macaulay's method does) seems much simpler.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BAretired]

IDS,

You are probably right, Doug. I was just thinking of the remark made by KootK:

If one requires max deflection, and a midspan estimate isn't sufficient, then you're definitely in software territory.

If the deflection can be described as a continuous function by means of a Fourier Series, perhaps the maximum deflection can be determined without resorting to software. But in any case, it is easy enough to find the maximum deflection in the standard way by simply testing values each side of the peak.

BA

 

[Teguci]

BARetired,
Yes, since we are talking about a simply supported beam, then the reactions are easily calculated and the beam diagrams can be procedurally generated by hand. The math gymnastics I mentioned, aren't appropriate until you have an indeterminate beam. Of course, now that I've developed the tool, I have no problem using it for the simple problems too.

 

[rb1957]

I'm being unabashedly lazy (in not reading all the posts) ...
have we considered, as a simplification of the problem, replacing many point loads with a uniform load ?

another day in paradise, or is paradise one day closer ?

 

[SlideRuleEra]

What the OP is asking about is such a special case (equal point loads, equally spaced) that it is easy to bound the solutions to the problem.

In the hand drawn example I provided a total of 500 lb. is distributed over a 12 ft. long beam, the maximum bound is 1500 ft-lb (one 500 lb. point load at the center, PL/4).

The minimum bound is 750 ft-lb for an infinite number of equal point loads, equally spaced that total 500 lb. (actually a UDL).

Any number of equally spaced, equal point loads that total 500 lb. will fall between 750 ft-lb and 1500 ft-lb. The more points, the closer the approximation of a UDL, therefore the closer to 750 ft-lb. Semi-graphical integral calculus.

http://www.SlideRuleEra.net

http://www.VacuumTubeEra.net

 

[IFRs]

Is it correct to use MathCAD, create an array that steps along the member calculating the bending moment or shear for each load and then get the MAX of M1 + M2 + M3 etc?

 

[KootK]

Is it correct to use MathCAD, create an array that steps along the member calculating the bending moment or shear for each load and then get the MAX of M1 + M2 + M3 etc?

Sure. This is precisely what I do for the general case. It's somewhat approximate but, then, what isn't?

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[rb1957]

why do you think superposition is approximate ?

if you have multiple loads on a beam, then you can add together the effects of each individual load.

this is (really) a simple problem.

another day in paradise, or is paradise one day closer ?

 

[KootK]

why do you think superposition is approximate ?

I don't think that superposition is approximate. I think that, if you're stepping through values at discrete points to find the maximums, it's quite likely that you'll end up with something close to, but not exactly, the maximum. Such is the case with all things FEM-ish.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[IFRs]

In my MathCAD worksheet, I divide the beam into one inch increments. Yes it yields an approximate solution in the sense that it is not "exact" but in all reality it is darn good enough.

 

[TLHS]

Even mathematically exact solutions are approximations, given the assumptions used in defining an engineering problem.