Varied cross section at discrete points - hand calculation method 2

[skeletron]

I'm wondering if there is a way to modify a Moment Distribution Method to account for a change in cross section. For example, a slab clear spans between 8' wide thickened bands over the support. I set up a moment distribution tool to handle multiple spans but I'm missing out on the accuracy of the thickened section between clear spans. Is there a way to hand calc or simplify the analysis through spreadsheet application? I think the obvious answer is "jump to the software," but as a curious mind and a novice concrete designer, I'd like to keep my *ahem* dignity before having to rely on the machine (but eventually I will turn to it...). I thought I could potentially modify my moment distribution spreadsheet to have stiffness factors at the support conditions that replicate the thickened band. I'm not sure if applying this is worth it or even accurate.

 

[dhengr]

Skeletron:
Take a look at Newark’s Methods, his numerical integration methods. They handle beams and columns with changing cross sections, and the like, quite easily once you learn the approach to the methods. The textbook I used to learn from was “Numerical Analysis of Beam and Column Structures,” by William G. Godden, pub. by Prentice-Hall. I’ve used this approach over the years on some fairly complex structures with good success and results which proved to be very good. I’ve seen the approach presented elsewhere also.

 

[Agent666]

https://www.youtube.com/channel/UCGlAo3l4aRZRyuxQBWOn3JQ/videos

somewhere in the post tensioning lecture series I seem to remember this chap went through working out the distribution factors accounting for for this type of effect. Don't ask me which number it was though (sorry), somewhere in the later lectures when he started going through some real world examples of full frames is as far as I can narrow it down by memory.

 

[Celt83]

Look into the Hardy Column Method/Analogy, there is also a book on moment distribution by Gere that covers non-prismatic beams.

I have a post on the hardy column stuff which had some great info from other folks here. will add a link when I find it.

Open Source Structural Applications:

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

 

[KootK]

I guarantee that there is a way to do what you wish to with the moment distribution method. As a result of Celt's thread, I think, I picked up a copy of the book Moment Distribution by James Gere. It's astounding the range of complexity that is treated via moment distribution in that book and with the moment distribution method. I recommend acquiring a copy if you mean to get deep into the weeds with moment distribution.

 

[KootK]

This is the Hardy Cross Column Analogy: Link.

It's less elegant but could one not also simply introduce analysis nodes at the changes in cross section? The efficacy of this would depend on how your spreadsheet is set up I suppose.

 

[rb1957]

are these "thickening spans" fully effective ?

Is the slab very wide, supported on several supports, with thickening between the supports ? And you want to account for this thickening in the stiffness (and deflection) of the slab ? How significant is the thickening ? double the thickness ? 1/2 ?? What proportion of the slab span ?

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

 

[Celt83]

Here is a link to the thread: Link
Also attached to this post should be a spreadsheet I created to do the distribution factors

Quick notes:
- You need to also do the Fixed end moments for the non-prismatic beams as well as the distribution factors, Hardy Column can be used to get both.
- if this is a slab/column system you need to add sections of infinite stiffness at the slab/column overlaps and introduce a torsional member factor to the joints

KootK:
not sure introducing nodes would be the way to go if doing moment distribution, certainly a decent option if doing stiffness matrix.





Open Source Structural Applications:

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

 

[KootK]

not sure introducing nodes would be the way to go if doing moment distribution, certainly a decent option if doing stiffness matrix.

I must know the nature of your objection. Theoretical? Logistical?

@skeletron: I phone photo'd the relevent sections of the Gere book for you: Link

 

[Celt83]

Logistical you need to do the extra step of getting the deflection at the cross section transitions and do iterative distribution passes to arrive at the correct final end moments, The Gere book presents a method for doing it that way I believe, similar to process on an elastic support.

Open Source Structural Applications:

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

 

[KootK]

Thanks Celt83. I'd not considered it explicitly before but, for basic MDM, I guess that it is assumed that all of the joints are fixed against translation. I haven't done a real MDM since college and tend to always think of things in terms of the stiffness matrix method.

 

[skeletron]

Wow. Thanks for the responses. I wasn't sure if this post would generate any interest. Clearly I was wrong! I'll be looking at the Hardy Cross Column analogy and the Gere text to see if I can yield anything from them. I remember coming across a "simple beam analysis" spreadsheet as a young EIT, and thought it was crazy and the most advanced thing ever. Once I started digging into the code, I realized that it was just Newmark's method with some Roark formulas. Breaking that barrier has really allowed me to dig into numerical methods to approximate and solve otherwise complex situations. Fast forward to yesterday's query, and I think I have another barrier to break through. Awesome!

@rb1957: The situation I simplified in the description is this:

4 suspended slabs with clear spans running E-W between slab bands. Slab bands are 2x thick as suspended slabs and run about half of the clear span length. Slab bands run N-S, supported on columns.
Here was my design process for those interested:
1) Find an upper bound for the clear span. Maximum positive moment = wL²/8. Negative moments estimated to be wL²/12
2) Use the Approximate Moment Coefficients applied to the 4-clear span continuous beam. These didn't necessarily apply because the spans vary in some locations by more than 20%.
3) Run a Moment Distribution spreadsheet to capture the continuous span behaviour.

I knew that #1 was an upper bound that wouldn't quite capture the negative moment in the interior spans.
I didn't have high regard for the results for #2 because the varying span lengths.
#3 captured the essence of the continuous behaviour and let me run skip loading cases to really get a picture of the negative moment. However, what I was leaving out was that my "pins" were actually 8' wide sections that were 2x as thick as the slab. So I (apparently) wasn't capturing the critical negative moment at the support as accurately as I should. A colleague ran it in an analysis program, increasing the stiffness 4' on either side of the pin.

 

[r13]

I knew that #1 was an upper bound that wouldn't quite capture the negative moment in the interior spans.

You are exactly right. I usually started sizing member by setting M = wL²/10, for concern of interior span, and occasionally for excessive moment due to stress reversal in lateral load analysis.

 

[BridgeSmith]

The simplest approach (my opinion) would be to envelope the analysis by running the MDM twice using the smaller section for the whole span, and larger section for the whole span separately. Taking the maximum reactions from either one should be conservative. You can get a sense of how conservative by comparing reactions at the same points between the two. If they're not much different, you could use the larger reactions from each model at each of the critical locations, and call it a day. If you decide to do a more rigorous analysis, they would provide you with upper and lower bounds to check that analysis.

Rod Smith, P.E., The artist formerly known as HotRod10

 

[Celt83]

I wasn't sure if this post would generate any interest.

yeah you hit a topic that has become a bit of an obsession for me in recent months/years. I'm part way thru the general derivation for a 3 section simply supported beam with variable loading, the piecewise formulas become a huge pain to keep track of once you hit the slope/rotation as they branch to cover cases when the variable load is in or out of the end zones.

Slope-Deflection and Direct Integration are other methods you could use to come up with the distribution factors but the Hardy Column Analogy is far more elegant.

Not sure if this is kosher but if you'd like to collaborate a bit on this feel free to open an "issue" on my GitHub to get a dialog started.

Open Source Structural Applications:

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

 

[skeletron]

Thanks @Celt83. I'd love to collaborate in the near future. I think this is the year I dip into Python.