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What are the partial differential equations for shear and bending moment diagrams? 
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#1
Nov2012, 06:24 AM

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I'm trying to write a very simple program to generate shear and bending moment diagrams for beams.
The thing is that I don't know what kind of loading that the beam may see so I want to be able to write it as generically as possible. I'm trying to find a web resource (not a book, unless it's an ebook or PDF) for the differential equations or for hints on how I can derive it from first principles. (I'm guessing that it has something to do with Newton's Second Law of Motion?) If anybody can help point me in the right direction, that would be greatly appreciated. Just googling "partial differential equations solid mechanics" brings up a lot of links ABOUT PDEs in solid mechanics but not the equations themselves (whereas I'm actually looking for the PDEs). Thanks. 


#2
Nov2012, 08:19 AM

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PF Gold
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Shear force and bending moment in prismatic beams are generally governed by ordinary differential equations, rather than PDEs. Try googling EulerBernoulli beam, or bending of beams.
In a nutshell: if y = deflection, theta = slope, M = moment, and V = shear, and the beam runs along the xaxis, then dy/dx = theta d(theta)/dx = M/EI, where E is Young's modulus, I = 2nd moment of area of the beam crosssection. dM/dx = V dV/dx = load on the beam. 


#3
Nov2012, 08:20 AM

P: 24

Thanks.



#4
Nov2012, 09:23 AM

P: 5,462

What are the partial differential equations for shear and bending moment diagrams?
There are no simple equations that hold for the length of a beam.
You either have to 1) Solve a set of ordinary differential equations, set up for different portions of the beam. 2) Use the impulse function, otherwise known as macaulay's method or macaulay's brackets. The second programs well in a computer and does provide a single equation to work on. Google will find plenty on this. 


#5
Nov2012, 09:28 AM

P: 24

The thing is that I don't know what kind of loading conditions (on the beam) will be thrown at me, so the debate now is whether I want to go with the singularity function or effectively end up (accidently) writing a FEA code that might be discretized in both time and space. (and then trying to figure out how I want to do the spatial discretization) (it's for a class that I am taking AT work...) 


#6
Nov2012, 09:58 AM

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Macaulay was designed to make things easy. It has been discussed many times at PF for example.



#7
Nov2012, 10:23 AM

P: 24

what happens say...for the distributed loading if it wasn't a constant value but rather some function of x? Or that you can pretty much have anything under the sun in terms of the shape and size of beam, where the loading might occur, supports, multiple supports. It's a freeforall smorgasbord where anything goes (in nearly any combination). And the code should be flexible enough that no matter what you throw at it  it should be able to handle it. (And that it'll actually come up with the singularity function formulation by itself and solve it by itself based on the inputs that you're feeding it (forces, supports, and positions of both).) So, if you to write it for the generic case (no case structure); how would you go about doing that? (I didn't think you could unless you basically start coding up "element types" which dictate the formulation for the different singularity functions, but please correct me if I'm wrong in my way of thinking about this problem.) 


#8
Nov2012, 10:34 AM

P: 5,462

The example I showed used specific figures. Create a variable for each one and use symbolic calculation. Standard cases might be point load, constantDL, triangularDL, parabolic DL and so on. You have to do this to tell the program about the geometry in any case. I also think you are being too ambitious if you think you can create a simple but all singing all dancing beam program. You would effectively be coding up KleinLogel, Pilkey or Roark. You will probably find some code segments in Numerical Recipes Press, Flannery, Teukolsky and Vetterling Cambridge University Press 


#9
Nov2012, 10:42 AM

P: 24

Like I said, I think that if I want my program to do what I am hoping for it to do  it'll end up being an FEA code (almost entirely by accident). "Accident" being that I didn't really set out to write a FEA code, but because I don't know what kind of loadings and supports that I might get, it ends up being one in order for it to be the most general/generic case. *sigh*...thanks for the help though guys. 


#10
Nov2012, 10:42 AM

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If you don't know what kind of loads the beam will have, then your code must include sections which handle each type of load. For most beam loadings, the following are generally encountered:
1. concentrated point load at a given location. 2. concentrated moment at a given location. 3. constant distributed load over a given portion of the beam. 4. distributed load with a linearly varying load distribution (triangular distribution). Other types of loading (e.g., parabolic distribution, etc.) can be derived, but they are somewhat less common. 


#11
Nov2012, 10:45 AM

P: 24

(That's the reason why I need it to be able to handle a wide variety of loading conditions beacuse of the final application of the program.) And by building in all of these flexibilities into the code, then yes, it would run slower than if it was just the straight singularity functions, but I can do it for any ladderframe based vehicle basically. (SUVs, crossovers, pickup trucks, anyduty trucks (light, medium, heavy), and some military vehicles too.) 


#12
Nov2012, 10:46 AM

P: 5,462

Stream King's list still will still be far from comprehensive since the beam may be loaded eccencentrically, or at some other angle to transversally or in torsion.
Further you must consider fixity conditions at the supports. FE will not help directly with any of this unless you are prepared to vary your approximating functions. Edit : Torsion is highly significant with ladder frames. 


#13
Nov2012, 10:48 AM

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#14
Nov2012, 10:52 AM

P: 5,462

You may be able to achieve something useful with a parametric drawing program such as DesignView, Imagination Engineer (came with MathCad once) or other.



#15
Nov2012, 11:50 AM

P: 24

Thanks.



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