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Deflection Equations 2

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EngDM

EngDM

Structural
Aug 10, 2021
724
Hey all,

Anyone know of a resource to find a ton of different deflection equations? Everything I can find for a UDL not across the whole beam (and not located starting at a support) only gives the moment equation and no deflection.

Or if you know of a good resource where they go over the derivation of the deflection equations. For weird cases I typically use clearcalcs or RISA 2D for quick results, but it would be nice to have a spreadsheet that uses the superimpose method.
 
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The good old slope deflection equations will get you there. But it can be rather cumbersome to make into a spreadsheet.
The issue with just writing an equation for the deflection of say a simple beam with partial uniform load, is that the the solution is piecewise and constrained. Meaning that the point of maximum deflection, depends heavily on the loading parameters.

The best way to locate the maximum deflection is to use calculus on the deflected shape equation which is derived from slope-deflection differentials equations.
With mathcad or something similar it can be pretty straight forward to set up the triple integral. And solve for maximum displacement.

With excel or something numerical it is quite difficult.
 
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Check out Table 8.1 in Roark's Formulas for some pre-calculated formulas.

In spreadsheet form, using Newmark's "Numerical Procedure for Computing Deflections Moments and Buckling Loads" as a general guide for the method. It was a bit tricky to implement, but works for simple span.
 
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Another option would be to use the direct stiffness method and fixed end moments and such. This involves assuming a shape function for the beam usually cubic. You can get a closed form equation for deflected shape. Not too bad to put this into a spreadsheet.
 
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Ditto to Roark's and use of superposition formulas and symmetry.
 
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driftLimiter said:
With excel or something numerical it is quite difficult.
Click to expand...
not entirely accurate, excel without some VBA it can be bit of a pain but if you derive everything on a piecewise basis and combine into numerically continuous regions you can solve for the Shear and Slope roots to get the max/min moments and deflections.

Generalized python derivations for Point, Moment, UDL, and linearly varying (Trapezoidal) Loading for Euler-Bernoulli pinned-roller beams can be found here: Link
These were derived via direct integration with the assumption that positive loads act in the negative Y and the beams span in the positive X direction.
 
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driftLimiter said:
The best way to locate the maximum deflection is to use calculus on the deflected shape equation which is derived from slope-deflection differentials equations.
With mathcad or something similar it can be pretty straight forward to set up the triple integral. And solve for maximum displacement.
Click to expand...

I was hoping to be able to solve the integral algebraically and use that formula in the spreadsheet.
 
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Celt83 said:
Generalized python derivations for Point, Moment, UDL, and linearly varying (Trapezoidal) Loading for Euler-Bernoulli pinned-roller beams can be found here: Link
These were derived via direct integration with the assumption that positive loads act in the negative Y and the beams span in the positive X direction.
Click to expand...

Is this your toolbox? I see some gifs in there showing functionality, and I've opened it as a .py file but I don't get a GUI as shown in the gifs. Maybe I'm missing something.
 
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yeah, I would (I have) solve the beam in excel. For simply supported beams, with distributed load over any part of the span. Not too hard to extend for beams with 3 supports. Solve the moment and the displacement. With a little thought you may be able to show that max deflection is within the DL limits, and now you don't need to solve the complete beam ... you know the reactions, then you know the beam conditions at the limits of the DL, and with some smarts you can figure it out "easily".

Of course if accuracy is not prized then replace the DL with a point load and have done with it !

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.
 
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You may be looking at my older python library, have not updated it to the newer python version which generally breaks all the GUI stuff.

I originally did the derivations in VBA which can be found here: Link
 
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Ah I see, I haven't done much of anything with VBA.
 
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dauwerda said:
Table 3-23 of the AISC Steel Construction Manual has a decent amount of options.
Click to expand...

Yea these are the tables included in the HSC from CISC, but the one section I need (#4. in that table) doesn't provide a deflection equation haha.
 
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There is also the Beam Analysis tool by Alex Tomanovich here:Link
 
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EngDM said:
I was hoping to be able to solve the integral algebraically and use that formula in the spreadsheet.
Click to expand...

I think you could do this, but it would be a system of equations because the initial conditions that you need for the triple integration (some of them) are dependent on the loading.
 
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Thanks for posting the link Celt83.

Latest download for the continuous beam spreadsheet is:

Regarding finding the maximum deflection, I really think people are overcomplicating things.

Just create reasonably closely spaced output points, then the maximum deflection will be close enough to the actual maximum for all practical purposes.

Or if you really need to be more precise, find the two points with the greatest deflection and subdivide the output over that region.


Doug Jenkins
Interactive Design Services
 
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IDS said:
Regarding finding the maximum deflection, I really think people are overcomplicating things.
Click to expand...

Perhaps this is true. Using pre-defined stations (output points) should give a workable result in most cases. A fair point.

 
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IDS said:
Just create reasonably closely spaced output points, then the maximum deflection will be close enough to the actual maximum for all practical purposes.
Click to expand...

Would you not need an equation W.R.T. x for this though?
 
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EngDM said:
Would you not need an equation W.R.T. x for this though?
Click to expand...

I was assuming using a spreadsheet that allowed the number or position of output points to be specified, but for a simply supported span it's not that hard to set up the calculation from scratch:

1. Calculate the shear diagram.
2. Integrate for bending moment.
3. Assuming zero slope at end 1, integrate twice for slopes and deflections.
4. Find the slope at end 1 so that the deflection at end 2 is zero.
5. Adjust the slopes and deflections along the beam.

Doug Jenkins
Interactive Design Services
 
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