Moment/Shear/Deflection/Rotation Diagrams

[hardyworld]

Greetings:

I am trying to put together a program that will work in many many instances. I've worked it out so that for a given "element" (a given section of a continuous beam) I know the Moment, Shear, deflection, and rotation of both end points AND all loads applied onto the element.

How do I calculate the Moment, Shear, Deflection, and rotation for any point within my element? I seem to have all the information I could possibly need to produce these 4 diagrams. I cannot seem to find the answer I am looking for.


If you are somewhat confused as to what I am calling an element: it's basically a section of a beam. Say I have a 30' long simply supported beam with a uniform load throughout. My element dilemma is that: I know the M,V,?, and ? at both the 1/3 and 2/3 points of the beam, but cannot find these values within the element (between the 1/3 and 2/3 points).

 

[EdR]

Probably the easiest method to do what you want is to use the method of integration. Check any Mechanics of Materials text book.....

Ed.R.

 

[rb1957]

"method of sections" ... look into any structures text for how to analyze a beam ... it's easy, like EdR says, it's all about integration ... shear force integrates to moments, momments integrate to slopes, slopes to deflection.

i've done this (not the achievement of my life) in excel to help solve redundant beams ... yeah, i know i can download it from several places, but it was slow and a minor challenge!

 

[asixth]

Everything is an intergration of something else. To calculate the shear of the element. Take the shear at node A and add the applied load to calculate the shear at intermediate points of the beam.

Eg. Take an element of length [l] loaded by [w]

For Shear:

Vintermediate=Va+w*x

where Va is shear at node A

For Moment

Mintermediate=Ma+Va*x+w*x/2

For Rotation

Sigmaintermediate=sigmaa+Ma/E/I*x+Va/E/I*x^2/2+w/E/I*x^3/6

And likewise for displacement however I am not going to write it out.

How are you writing your program and what elements are you programming and what codes do you intend to include. I have only put together a Matlab routine, I can upload my code if you need any help. Let us know how it goes.

Don't forget axial forces if you are programming 3d beam elements.

 

[rb1957]

the only catch is calculating the constants of integration (to fully define the function (slope or deflection) from the relevant boundary conditions

 

[atee001]

Hi all,

As most suggest using a free body diagram and an integration scheme to fet forces/displacements within the element.

However, once we use an integration scheme we are doing essentially conventional mechanics.

What am trying to question is,whether is there any procedure that one could get exact solutions within the leemnt using the interpolation/shape functions for the corresponding element (beam element here)?

I know the solution using interpolation/shape functions would not be exact-is there any scheme to obtain an exact solution using Finite element procedures than the free body diagram?

 

[rb1957]

if you're saying you have the FE output (at the nodes obviously) and want to interpolate between the nodes, i think shape functions will give you exactly what the FE calculated (though it is an approximation of the real world).

if you're saying you know the free body diagram of the "element" (in the real world), then applying the equations of equilibrium (integrating shears to moments, etc) will give you the exact solution. equations of equilibrium will also work with the free body from the FE (though of course this is an approximation of the real world).

 

[Stazz]

I made a spreadsheet that does exactly this. Just treat each element as simply supported and apply the end Shears, Moments, Theta's and Delta's (from the FE analysis) to the element. Then integrate using all the loads in between. Since the member is simply supported then it shouldn't be too redundant and the constants of integration should work out to be the end shears, moments, etc. that you came up with from the FE analysis.

Of course integration is only exact if you truely integrate it which is difficult to program so you'll probably have to use Newtonian integration (Take it piece by piece). The spreadsheet I made could go up to 1000 pieces and that was more than suffice.

I would also recommend for a truely exact solution to use the equations given in the AISC manual and super impose each case (Point load, distributed load, point moment). It all depends on how your loads are defined for this to be feasable, plus the AISC manual doesn't provide equations for theta and point moments so further research is required for that.

 

[hardyworld]

Thanks Stazz. I came to the realization a while ago that I'll end up doing just as you describe: including using Newtonian integration. As long as my error is no more .01", I'm sure everything will work out great.

 

[rb1957]

you don't Have to (use Newtonian integration) ... you can divide your beam into a number a sections to catch the step changes in shear (and possibly moment) then indefinate integration then boundary conditions to determine the integration constants

 

[Dinosaur]

All you need to do is apply the shape functions used to derive your beam element to get the basic shape produced by the end deflections and rotations. Then superimpose the deflected shape of a fixed-fixed beam subject to the loading you have. This is basically what Stazz is doing by applying the rotations and deflections and loading to a SS beam. However, in his method, the ends will be allowed to rotate out of the shape you are required to keep. If you don't understand what this means, use his method to analyze the three parts of the beam you describe and then tell me why the rotations at the ends do not mate. Good Luck!

 

[Stazz]

Dinosaur: Did you mean "why the rotations at the ends do not mate? or do not match?

no wonder why you're extinct

 

[Dinosaur]

Stazz,

I suffer from "Fat finger syndrome" and often what I mean to type is not what makes it onto the page. However, I don't recall which word I wanted to use.

But the point remains the same, in the last step you have to superimpose the interior loading onto a fixed-fixed beam element to maintain compatability between elements.

 

[Stazz]

Oh I see, I assumed that these end moments would force the beam into the required shape by themselves, but what your saying is that I have to also apply the end rotations and keep them fixed?

I think these 2 methods are exactly the same. I'll have to do an exeriment.

 

[hardyworld]

Each element's end moments and deflections must be applied to keep continuity with adjacent elements. I think rotations would be automatically met in this case.