## Deflection Equations

## Deflection Equations

(OP)

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.

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.

## RE: Deflection Equations

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.

## RE: Deflection Equations

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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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.

## RE: Deflection Equations

I was hoping to be able to solve the integral algebraically and use that formula in the spreadsheet.

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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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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.

## RE: Deflection Equations

I originally did the derivations in VBA which can be found here: Link

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http://www-classes.usc.edu/engr/ce/457/moment_tabl...

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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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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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Latest download for the continuous beam spreadsheet is:

https://newtonexcelbach.com/2021/09/28/conbeamu-up...

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

http://newtonexcelbach.wordpress.com/

## RE: Deflection Equations

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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Would you not need an equation W.R.T. x for this though?

## RE: Deflection Equations

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

http://newtonexcelbach.wordpress.com/

## RE: Deflection Equations

In the majority of cases, an engineer should be able to come up with a pretty good approximation with a minimum of effort.

An equal and opposite moment applied to each end of a beam results in a constant moment over the span. Slope and deflection of the real beam are equal to the imaginary shear Vc and bending moment Mc of the conjugate beam. The M/EI curve is the load imagined to be carried by the conjugate beam.

Bending moment for the conjugate beam is Wc*L/8, which is the deflection of the real beam loaded with constant moment across the span.

Wc*L/8 = M/EI*L

^{2}/8 = M/EI*L^{2}/8 = WL/8EI*L^2/8 = wL^{4}/64EI.With a little practice, an engineer should be able to estimate deflections quite accurately. In those rare cases where it is necessary, one must sharpen the pencil a bit.

## RE: Deflection Equations

Design of Welded Structures

Design of Weldments

## RE: Deflection Equations

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EDIT: Of course one can apply the unit load theorem and compute deflection or rotation at discrete points by using the bending diagram, but that also requires a lot of algebra (or alternatively Excel) for any non-trivial structure.

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Here is a case of a partial uniform load on a beam. It would be good for a spread sheet I think, and you could use a separate line for each different kind of load. Two or three lines should be enough for most beams, but there is no limit. In the end, the spread sheet would sum the values at each station, showing a composite deflection diagram.

## RE: Deflection Equations

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^{2}b^{2}/3EIl, or 30*(10*20)^{2}/3EI*30= 13,333/EI, considerably more than my result 8160/EI but looks believable.The method needs more testing, but the OP should be optimistic about using a spreadsheet.

## RE: Deflection Equations

2) I would not blame you for doubting the accuracy of these equations. That said, I vetted the crap out of them and actually found bug in S-Frame in the process of doing so. I've also been using these equations for about 20 years in some MathCAD design sheets without incident.

3) Realistically, the hardest part of getting this right is typing out the equations properly.

4) The closed form solutions run wicked fast in MathCAD relative to iterative or discretized approaches which is helpful there.

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https://awc.org/wp-content/uploads/2021/12/AWC-DA6...

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Yes thank you, it's quite a good table, but in some cases, deflections are not displayed.

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I agree with BA, that this is typically not critical for design (so this may be wasted effort).

but I think we should learn things for ourselves (rather than to rely on what we read on the "interwebs").

I would (and have) solved the deflection of the beam mathematically in excel. Solve a couple of simple loadings and you can solve any load and determine the maximum deflection. And with a little more effort different end conditions.

And try things, like compare a full span UDL with a point load, compare a mid-span point load with a point load at "x".

"Hoffen wir mal, dass alles gut geht !"

General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

## RE: Deflection Equations

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I ran it in my ConBeamU spreadsheet (using the single span function) and got exactly the same results (to displayed precision).

Using the spreadsheet, if it is important to get the position of maximum deflection and/or bending moment the Excel solver does a good job of finding that. For the deflections I multiplied the deflection by 1E6 in the adjacent column, to save adjusting the solver tolerance, or I could have just set the deflection units to nanometres.

I also checked baretired's example and got significantly different results, but I will double-check the units on that one before posting the results.

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

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...gulp!

## RE: Deflection Equations

Deflections are found by fitting a cubic shape function to the deflections calculated at beam ends, so it is not surprising that there are small differences compared with our calculations, where we adjusted the beam end locations to find the "exact" maximum deflections.

Note that all these calculations are ignoring shear deflections, which would have a far bigger effect than the 6th significant figure.

BA - I'll post my results later. They were the same order of magnitude as yours :)

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Deflection Equations

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I set EI to 8160 and got identical results with the spreadsheet and FEA analysis:

With the distributed load: deflection at mid-span = 1.685, maximum moment = 166.7 kip ft.

With the point load at 10 ft: deflection at mid-span = 1.762, maximum moment = 200 kip ft.

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Deflection Equations

I agree with your maximum moments of 167'k and 200'k for the distributed and point load cases respectively, but the deflections can't be right, particularly if they are in feet.

For a maximum moment of 200'k, we could try W18x50 with I of 802in^4. EI=29,000k/in^2*802in^4 = 23.258e6 k-in^2 or 161,514 k-ft^2. Then delta = 13785/161,514 = 0.08535' = 1.02", very nearly L/360, which sounds a little more reasonable.

## RE: Deflection Equations

I set the EI to 8160 so the deflection would be 1 foot. Changing to 13,785 I get:

- At mid-span 0.997 ft

= At 13.923 ft (max. deflection point): 1.004 ft.

So that's pretty good :)

Using your realistic EI value I get a maximum deflection of 1.028 in.

By the way the units-aware version of my spreadsheet is very sensitive to the format of non-SI units, and it doesn't give a helpful message if it doesn't like the units, so I'm working on making it friendlier. I will post here when I have finished.

For now, units that work are:

EI: kipf.ft2

Load/m: kipf/ft

Moment: kip ft or kip in

(so EI needs a . between the units, but for moment it has to be a space)

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Deflection Equations

Interesting. 75% of all beams I size are deflection limited. This is residential and light commercial though. So much so, that I check deflection first on most beams.

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Uh, yeah, NO

Honestly, I can count on the one hand the number of times strength controlled an LVL or steel beam in residential or light commercial so I might as well check deflection first.Bearing stress is the next thing I check for wood construction.

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For download see:

https://newtonexcelbach.com/2023/01/20/using-conbe...

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Deflection Equations

The location of maximum deflection could have been solved by setting y' equal to zero, but it would require solving a cubic equation, so deflections were found at three locations. Maximum deflection is seen to be located between 13' and 15' from the left end.

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These are derived using a consistent sign convention of positive loading and reactions pointing in the positive y direction, positive moments are counter-clockwise, and positive internal shear points in the negative y direction. This results in beam rotations and deflections consistent with common practice. The derivations assume consistent units across the various inputs. Fixed end moment formulas are also included.