Yield Line Analysis of a Moment End-Plate 4

[rhollman]

I'm designing a moment end-plate for a WF beam framing into a WF column. The geometry of the top flange connection doesn't match any of those in AISC Design Guide 4 or 16, for which you would simply use the formulas provided in the Tables. I can't find any information on the derivation of the formulas in those tables, which I understand were arrived at through YLA. I understand how YLA works in a concrete slab or plate when using a unit displacement for external work, but does anyone have information on how to perform YLA with a unit rotation? How were the formulas derived in Table 3-2 AISC Design Guide 16 for example?

 

[canwesteng]

I don't think you need to consider some unit rotation here for your yield line analysis - just make sure that under your factored bolt tension all possible/reasonable yield lines have enough capacity that the plate doesn't yield. Your sketch looks like you're more concerned about stiffness - believe DG 16 recommends designing to at least 50% of the moment capacity of the member to consider the connection fixed.

 

[JoshPlumSE]

One of the paper's that the design guide is based on has been very helpful in the past for this sort of thing. I'm not sure that it has your exact configuration. But, my guess is that it will be helpful.

Emmet Sumner's PhD thesis, "Unified Design of Extended End- Plate Moment Connections Subject to Cyclic Loading" June 17, 2003

If you don't have the time / energy to go through this whole process, you might take just 50% of the capacity of an 8 bolt stiffened connection. That should be conservative and it would be a whole lot easier on you.

 

[rhollman]

Design Guide 16 gives a very brief overview of how YLA is used to derive the equations in the tables, and it states that the external work is calculated using a unit virtual rotation (eq. 2.6). I want the connection to be a moment connection so I guess I'm concerned about the stiffness, but I'm more interested in designing the end plate thickness for strength (and to make it thick enough to avoid prying action).

Sumner's thesis doesn't really seem to reveal anything more than what's already in the Design Guide...no derivations - just that YLA is used by equating internal and external work for an assumed yield line configuration.

Yea JoshPlum, that's basically what I ended up doing for time's sake. But I really want to learn how to use YLA in moment end-plates so that I can handle any other atypical situation I might come across. I think I could apply it to other situations if I could just find a detailed derivation of the formulas in the Design Guide.

 

[T_Bat]

In the past I've used the paper by Dowswell (2010) that is referenced in the prying action section of the 14th edition steel manual (attached). I agree that JoshPlum's method would work for quick and dirty but the Dowswell approach has saved me on some pretty gnarly connections in the past.

 

[JoshPlumSE]

I really want to learn how to use YLA in moment end-plates so that I can handle any other atypical situation I might come across. I think I could apply it to other situations if I could just find a detailed derivation of the formulas in the Design Guide.

I had the same idea once.... And, I thought the PhD thesis was helpful. But, maybe it was another paper by Sumner that went into more detail about the derivations. Sorry if I gave you the wrong citation. I'm at a new company and I don't have all the old references and PDF files I used to have.

You might check the references cited in the thesis and the design guide. I'm pretty sure it had Sumner's name on it and also Tom Murray's. It was definitely Virginia tech research. Used it to derive the yield line equations for some of the simpler cases, just to make sure I understood the theory properly. Also, it had more bolt arrangements than were given in the design guides. So, that also helped for some of those special cases.

 

[rhollman]

JoshPlum: Every paper I found said something along the lines of "Equating the internal work and the external work results in the following expression for the strength of the end-plate..." I'm trying to learn how the internal work is calculated, not a the formula resulting from the calculation,which is all I can find in any of the sources. I didn't find an actual calculation in Sumner, Murray, or Srouji's papers - just formulas that result from the calculation.

T_Bat: I can perform a YLA for tension, like in the Dowswell paper. But it's my understanding that a moment end-plate YLA is performed differently, mainly by assuming a unit virtual rotation rather than a unit virtual displacement. Thanks for the reference though, that might come in handy some day.

 

[Agent666]

rhollman, I believe that they are essentially the same thing, as it cancels out of the math at the end of the day when you equate the internal work to external work. After all the capacity is the same irrespective if you use a unit displacement or unit rotation. I had some university notes that explained it from first principles, I'll try find them. I used to use it all the time, but stopped and over time have forgotten the techniques involved.

 

[JoshPlumSE]

I've got my calculation at home I'll scan it and post it when I get a chance. It's the simplest case because I wasn't 100% sure what I was doing and the simple case made it a lot easier to match the DG16 formulas.

 

[rhollman]

I think I just figured it out...
1. small angle approximation says tan(theta)~(theta), which means (theta) = (delta)i/hi, where hi = distance from center of rotation to point where (delta)i is measured (bolt locations). Therefore (delta)i = (theta)*hi
2. Unit virtual rotation implies (theta) = 1, which means (delta)i = hi
3. Equate internal and external work using "hi" instead of a unit displacement = 1 at each bolt.

I did this and matched the formula given Table 3-2 Design Guide 16.

 

[JoshPlumSE]

Yes, I believe that is the key assumption. I have a page from one of those documents that implies this in the calcs I put together. But, I didn't write down where I copied the image from.

 

[BAretired]

The internal work is the sum of m.φ.L for every yield line where m is the plastic moment of the plate per unit length, φ is the angle the plate bends and L is the length of yield line.

The yield line method is a very useful technique for analyzing steel plates but is is an upper bound solution which means that, if you do not choose the correct yield line pattern, your solution will be unsafe.

In your diagram, you are showing too many yield lines, so you would get an unsafe result.

If you simplify your diagram to include only one bolt located at the center of gravity of your two bolts, you end up with three yield lines, two negative and one positive. I suggest you start with that assumption and massage it from there.

BA

 

[BAretired]

In Table 3-2 (see attached), assume the plate deforms one unit. The slope of each section between yield lines is then 1/p[sub]f[/sub], 1/s and 2/g respectively. If the plate has a plastic moment of m which is equivalent to Fy*t[sup]2[/sup]/4 then we can write an expression for Internal Work, I.W.

I.W. = 2*m.b[sub]p[/sub](1/pf+1/s) +4*m*2/g(pf+s)

Adding the term h[sub]1[/sub], we get an expression for moment in the beam based on plate yielding which is in agreement with the expression given in Table 3-2.



BA

 

[BAretired]

I understand how YLA works in a concrete slab or plate when using a unit displacement for external work, but does anyone have information on how to perform YLA with a unit rotation? How were the formulas derived in Table 3-2 AISC Design Guide 16 for example?

Unit rotation has nothing to do with the problem. The moment between beam and column is simply 2*P*h[sub]1[/sub] where P is the tension in one bolt at failure and h[sub]1[/sub] is the lever arm. If the load P is governed by bolt strength, then Yield Line Theory does not apply; however, if P is governed by yielding of the plate, then Internal Work and External Work are calculated in the usual way, just as you stated above using unit displacement.

The AISC Design Guide has produced formulas without much explanation of their derivation. Perhaps they thought it was obvious, but I believe it would have been more helpful to the reader if they had explained it a little more thoroughly.

BA

 

[rhollman]

BAretired: According to AISC, unit rotation has everything to do with the problem (see attached excerpt). And I think my yield lines are realistic. They match a similar layout shown in the Design Guide for an eight-bolt stiffened end plate (see attachment). Assuming the yield lines intersect the centroid of the bolt group doesn't make any sense because the centroid of the bolt group is in the same location as the stiffener. Plus every configuration in the Design Guide shows the yield lines propagating out from the bolts.

If anyone could provide a derivation for "Y" in Table 3-3 Design Guide 16, that would be very clarifying.

 

[rhollman]

(see attachment)

 

[T_Bat]

When I use the Dowswell paper for moment connections I calculate the resulting forces on the bolts based on CASE II from AISC assuming the neutral axis at the COG of the bolt group. Then I calculate prying action on the tension bolts using the Dowswell paper. This definitely isn't as rigorous a solution as developing my own yield lines but I think it's a useful approach for weird stiffened connections.

 

[BAretired]

BAretired: According to AISC, unit rotation has everything to do with the problem (see attached excerpt). And I think my yield lines are realistic. They match a similar layout shown in the Design Guide for an eight-bolt stiffened end plate (see attachment). Assuming the yield lines intersect the centroid of the bolt group doesn't make any sense because the centroid of the bolt group is in the same location as the stiffener. Plus every configuration in the Design Guide shows the yield lines propagating out from the bolts.

If anyone could provide a derivation for "Y" in Table 3-3 Design Guide 16, that would be very clarifying.

Okay, I looked only at Table 3-2, not Table 3-3. I will have to have another look at it. I notice in your latest post that several of the terms have been scratched out in the expression for "Y". Is that official or just someone doodling?

I do not have time to study this any more right now. If anyone else wants to get into the fray, be my guest.

BA

 

[rhollman]

Yea BAretired, that's just doodling. Thank you for your input.

 

[rhollman]

T_Bat: The bolt strength isn't what I'm asking about - I can handle that part. My question is about the end-plate strength.