Deflection 2

[beetlejuice1976]

Hi There,

I hope there is someone out there that can help me.

I am producing a timber beam / joist design spreadsheet and have very nearly completed it but I need a bit of help.

I can't seem to find a deflection formula for partial UDL's for a simply supported beam.

I need it to work out three areas, namely

'A' - deflection at x from RA before UDL
'B' - deflection at x from RA during UDL
'C' - deflection at x from RA after UDL.

All structural/civil formula books, etc seem to neglect this information.

I don't really want to use equivalent UDL as this does not give a true depiction of the scenario.

Thanking anyone in advance.

Regards
Steve

 

[Lion06]

I'm not sure I understand the relevance of before, during, and after UDL. Why would the deflection before the dead load be any different than the deflection after the dead load?

 

[40818]

Only if the the load put the beam beyond its elastic limit,(but you knew that already).
Steve, the information your lookinf ofr is readily available from any basic textbook or on the web. But you need to explain a bit more about the scenario.

 

[beetlejuice1976]

Thank you for a swift response.

http://www.awc.org/pdf/DA6-BeamFormulas.pdf

I have attached a link to a document I came across for formulas.

The deflection calculations I require are associated to Fig 2 from the pdf.

 

[Lion06]

I think you're going to have to write some kind of program in Excel (or whatever you're using).

 

[Lion06]

Using something like Castigliano's method.

 

[JStephen]

See Roark's Formulas For Stress and Strain. I think you'll have to superimpose two load cases to get this one. But it does include deflections as well. Or you can just integrate it out yourself.

 

[JAE]

Are you looking for deflections for this load condition (see attached)???

 

[nutte]

JAE, that sounds like the condition he's looking for.

Beetlejuice, you can get there by superimposing the loads. You're loaded from a to b. Use the deflection from Figure 3 in your attachment to get the deflection with load from zero to b. Then use the same figure to get the deflection with load from zero to a. Subtract the second from the first, and you have deflection with load from a to b.

 

[miecz]

Although superposition methods are close enough, they are approximate, as the point of maximum deflection is not the same for the different load types.

For an exact solution, you could use Moment-Area Theorems. First, write your moment as a function of x. That's given in NDS Figure 2. Then, integrate to find your end slope. By trial and error, find the point where your slope is zero; and intergrate to find the deflection at the zero slope point. For the solution (using Mathcad), see the attached file.

 

[nutte]

You couldn't use the delta-max equation for one, then superimpose the delta-max from the other, because they occur at separate points. However, for any given point, superposition would give you the exact deflection at that point. If this is for a spreadsheet, the deflection could be calculated at 100 points along the beam's span, with the maximum of those 100 deflections used. It might not be the "exact" maximum deflection, but it would be pretty close, and likely plenty precise enough for the application.

 

[frv]

miecz-

If you know the deflection as a function of x, then you can superimpose the deflections to get an exact answer*.

* Caveat: this only applies to an isotropic, linearly elastic material in the elastic loading range. NEITHER of which apply to wood.

 

[miecz]

nutte and frv

Your're both right. That will work. I hadn't looked at figure 3 and assumed it only gave maximum deflections.

 

[mudflaps]

Use Roak's book or AISC beam formulas with superimposition. That said, here's some wisdom from another old guy (not me).

1. For a simple span the maximum deflection ALWAYS occurs within the middle 1/3 span and never moves more than 1/6 span away from midspan, no matter what kind of downward loading you have.

2. Within the middle third the deflected shape is so flat the delta "Y" from mid span to 1/3 span isn't going to be very much.

3. Calculated deflections, especially with wood, should always be suspect, due to variations in assumed loading, material quality, etc.

4. Wood creeps like green concrete. How do you plan to account for that. It also takes a long term "set" in part of the deflected shape.

There are other issues too. Can you account for cantilever moments too? Moisture content?

Just seems like you are expecting a bit too much accuracy given the material being used. Steel and concrete give better results, except for the uncertainty of the loadings.

Happy programming.

Old CA SE

 

[beetlejuice1976]

Thank you all for your help, I have ordered roark's book as I feel that this resource will stand me in good stead.


Thanks again.

 

[rb1957]

i'm surprised you couldn't calculate these results from 1st principles ... it'd "only" take a page or two of basic math ... but then i am an "older" "crankier" "hand-calc'ing" "engineer" ...

 

[MechEng2005]

A star for rb1957 for being "older, crankier, hand-calc'ing engineer."

 

[beetlejuice1976]

Have calculation deflection at x for a Partial UDL from first principles.

I own many text books for analysis and design but every book omits these formulas but gives deflection calculations for every other load case.
Does anyone have any thoughts why ?

In reply to mudflaps, I do appreciate that deflection results for timber should be "taken with a pinch of salt" but while programming excel I still needed the calculated deflected shape.