Slope Deflection Method for Beam with Varying Cross Section 1

[odu0485]

Hi,

Attached is an example problem in which I am trying to use the slope deflection method to determine moments in the indeterminate beam shown. The beam consists of a W12x50 (EI) with a W14x53 (1.38 EI) used from point B to point D. The problem I am trying to figure out how to handle by hand is the fact that the cross section change occurs at a point other than at a support. I also built this model in STAAD software to see the results and it shows a negative moment at point C of 8.9 kip-ft. Can anyone that has good knowledge of slope deflection tell me what I may be missing or doing wrong? It may have to do with fixed-end moments, not sure.

Thanks for your time!

 

[rb1957]

as drawn, your problem doesn't have fixed end moments (but simple supports). fixed moments (as an input to slope/deflection) do look "messed" ... Me = 3.75? I have Ma = -3.75 and maybe Mb = -2.92 and Me = 6.67 (in this example they should sum to zero).

I haven't used slope deflection in 30+ years, but I'd solve using unit force. You can easily calculate the deflection of the beam with the middle support removed; the varying I is a "wrinkle" but that's all (you'll need to analyze the beam in three segments, each with constant I). Then calculating the deflections due to a unit load at the middle support is equally easy. And so the solution is uncovered, the reaction at the middle support.

I think you could also use the three moment equation, and there are no doubt many graphical methods.

If you add fixed ends, the problem becomes triply redundant.

another day in paradise, or is paradise one day closer ?

 

[WARose]

For such a situation (i.e. where the EI isn't constant), I would think the moment-area method would be more appropriate. But it's been a while since I've done such a analysis by hand.

If you have confidence in the STAAD model....why even do it by hand?

 

[JoshPlumSE]

I'm not a slope deflection guy. But, it's obvious that there has to be a negative moment at point C. Isn't it?

 

[dhengr]

Odu0485:
Actually, the best approach to solving these kinds of problems long hand, is to use Newmark’s Methods, a numerical integration technique, which will handle all of the issues you are facing, and has other applications too. It is fairly straight forward and easy to use once you learn the basics. Actually, it is a numerical take-off on some of the older methods, like Slope Deflection, but in a tabular form which makes for fairly easy calcs. and accounting for member section variations, load variations, and the like. I am not trying to discourage a good fundamental understanding of many of the older std. methods, they are important in having a good background and understanding of Structural Engineering methods which are the basis of our methods and fundamental knowledge needed for our work efforts.

 

[Celt83]

points B and D do not have 0 deflection therefore you need to include that component in your slope deflection equations as you have them written now they have been assumed to either act as a pinned support or be a location of 0 deflection.

EDIT: It also doesn't appear you've accounted for the beam self weights but that is maybe on purpose.

Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering

 

[JoshPlumSE]

I'll second dhengr's suggestion. The Newmark method (which I do with the assistance of Excel) is probably the best way I've seen to do a "long hand" solution to members of varying cross section.

If you're interested, you might check out the AISC design guide on tapered members which has a good example or two in the appendix demonstrating this method.

 

[rb1957]

yeah, but this example is just a step change ... easy to hand calc without learning a numerical integration technique. The beam AE is just three segments ... a bunch of boundary conditions but nothing a hour or so should solve.

another day in paradise, or is paradise one day closer ?

 

[WARose]

I ran it myself on STAAD and came out with the max. positive moment being at Support C and the value being 7.65 ft-k. And the max. negative moment being at the midpoint of that 15' segment and its value is -5.56 ft-k.

[red]EDIT[/red]: Interesting thing about it is: the selection of the W14x53 makes the moment at the ends of the W14 (i.e. points B & D) very near zero. Any different and that changes.

 

[dhengr]

Rb1957:
Let us know, in an hour, how you are doing, then again in 2 and 3 hours. I’m not suggesting you will learn Newmark’s Method in only a couple hours, and it takes some practice to become proficient, but it is really a quick and powerful calculating tool for some of these kinds of problems, and a number of other applications as well, once you learn how to use it. It’s been quite a while since I last used it, so I couldn’t just sit down and do a problem, for the fun of it, without some serious study and thought, but I’ve used it on some pretty complex problems with very good success. Look the method up, and read a little on it, it can do a lot of things which you just can’t do in closed form or by std. integration methods. Of course, when I learned how to use it we didn’t have easy access to large computers, powerful software or FEA software. But, when you see how those are sometimes (and can be) misused by some people, it scares the hell out of you. Their modeling is so messed up to start with, that they may get many digit answers, but they have little relationship to the real structure and its actions.

 

[rb1957]

that's what I was suggesting ... that it'd take some time to learn a new "complicated" analysis, instead of "labouriously" solving a bunch (6?) of simultaneous equations (which most HS math students should be able to do).

though it may be a good tool to have in your tool-box (though many of our older hand calcs seem to have passed their "best by" date ... unfortunately IMO)

another day in paradise, or is paradise one day closer ?

 

[271828]

If I had to use a manual method, I'd use the Force Method, aka the Flexibility Method aka the Method of Consistent Deformations. Release the vertical reaction at C. Compute the deflection at C using virtual work. That wouldn't be very difficult.

 

[IDS]

For this application a reasonably simple hand method using slope deflection is:

- Find end reactions for a single span supported at the two ends
- Find the slope and deflection at changes of section and supports assuming zero slope at the left hand end
- Rotate for zero deflection at the right hand end, and find deflection at the central support location.
- Find the deflection at the central support location for a unit upward force.
- Find the central support reaction for zero deflection.
- Find shears, moments and deflections at all points of interest by adding the central reaction force results to the single span results.

I have attached a spreadsheet that follows that process.

I get a moment of -9.06 kip.ft at the central support. I presume the difference from the STAAD result is because I used EI values of 1000 and 1380,rather than the actual section stiffness values, which presumably do not have a ratio of exactly 1.38. Also STAAD may have included shear deflections, which can be added to the spreadsheet solution if you have the shear areas.

If adding in GA or support stiffness values note that the code requires the units to be in a recognised format. If you have any problems look at the unit list on the Ext Unit List sheet, or ask here.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[IDS]

If I had to use a manual method, I'd use the Force Method, aka the Flexibility Method aka the Method of Consistent Deformations. Release the vertical reaction at C. Compute the deflection at C using virtual work. That wouldn't be very difficult.

Beat me by 14 seconds

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[odu0485]

Celt83:

Good catch. Once I include the chord rotation parameter in the slope deflection equations to account for deflection at points B & D, how do I solve for that parameter?

So far, I have 5 unknowns (rotations at all 5 points) and 5 moment equilibrium equations. But when I add in the 2 unknowns for chord rotation, how do I get 2 more equilibrium equations so I can solve?

 

[Celt83]

been a long time since I used slope deflection could you get two more equations for consistent deformations at B & D.
slope @ left of Joint B = slope @ right of Joint B
slope @ left of Joint D = Slope @ right of Joint D

Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering