Twisted lattice under shear, bending & torsion

[Raggedman]

I'm new to this forum and I hope I can contribute as well as ask the questions. I'm a design engineer but I'm a bit rusty when it comes to some structural calcs.

I've currently got a bit of a head scratcher. A lattice of parallel I-beams with known dimensions of 'n' number. They are joined at both ends by a theoretical member that effectively results in 'fixed' ends (though the end elements can be assumed to rotate to carry torsional forces and bending moments).

The shear at the centre line is known, as is the transverse bending moment and longitudinal torsion. What I need to do is break down the forces onto the individual beams and check for failure.

I can analysis a single beam without any problem, what I cannot do for sure is divide the forces onto each beam. I suspect that:
Shear force can be divided, assumed, to apply equally across all members.
Bending moment the same
Torsional moment will generate both a local torsion and shear depending on the distance from the centroid of rotation? From Massprop in AutoCAD I can produce a polar moment for the cross-section latice but as this isn’t a single member I’m thinking Shear=My/J won’t give correct answers?

Can anyone perhaps give me some insight in this?

I think this is just the kind of thing that a FEA program would analyse very quickly, but I’d rather understand this by hand.

 

[dhengr]

Raggedman:
You have to be more specific about how you apply each of the various loads; and what ties the ‘n’ beams together at their top and bottom flanges; and how they are really restrained at their supported ends. Plus dimensions on the whole sketch, beam sizes, t & b tie plate thickness, load magnitudes, etc. so that we can get a sense of the proportions and various loads and stress magnitudes.

‘Qty’ usually doesn’t mean a vert. load in the ‘y’ direction in my world, but it might in yours. The way you show it and Mbz, they would be applied to that one beam unless you have some very serious connectivity btwn. the beams at their t&b flanges and the beams are fairly closely spaced. The same can be said of Mt, given what little we know, the beams would roll up like a window shade.

A sketch always helps show your problem, you know the old sketch worth 1000 words thing, so you get points for that. But, you’re still missing many details in explaining your problem. How is Mbz applied, or is it caused by Qty, as in a fixed-fixed beam with a concentrated load at its center. Is the load Qty applied uniformly to all the beams, is the load itself stiff enough to do this or can it flex to match the beam deflections? There are just too many questions unexplained in your OP for a meaningful answer.

 

[Raggedman]

dhengr, thank you for your time.

The loads are produced from equations in design guidelines for the conditions experienced by the structure and are given as loads applied across the entire structure. However once the forces are given there isn't much guidance as how to apply them to the elements.

The shear loading is the only force I can see which isn't applied as a direct result of the action of the ending fixing elements I am fairly sure can be taken to we spread equally across all the cross-beams. This isn't strictly accurate but will likely be sufficient for this analysis.

The end fixing points are two large structural elements which the cross-beams are fully intergrated into. For the sake of achievable calculations it could be assumed that the joining elements will never distort themselves as they are so much more massive than the cross beams.

Mt is applied by the end-structures rotating on the x-axis, whilst Mb is applied by the end-structures bowing in along the z-axis.

For rough dimensions, the entire unit might 10,000mm long, the beams 2,000mm wide and spaced 1000mm apart. The beams may not have uniform dimensions but all will be close to 300mm web to 250mm flange. All thicknesses would be 6mm.

If I could break this down to each beam member then solving the rest would be easy enough.

 

[dhengr]

Raggedman:

“a bit rusty when it comes to some structural calcs.” may be a kind understatement on your part. You don’t have to be embarrassed about that, but please be honest about your structural engineering experience level, so we know how/where to start helping you. I’m not going to do your design problem for you, but you’re not explaining it well enough so I can really even start to help.

Do you know the terms obfuscation or hocus-pocus? You are doing a good job so far with your verbiage, not withstanding your first sketch. Your verbiage doesn’t really seem to match you sketch or make it clearer to understand. You certainly need to understand and be able to explain the loads and how they are applied to the individual beams, and what those mysterious beam supports are which impart the loads or take the beam reactions.

If your individual beams A, B, C, etc. are ‘2000mm wide and spaced 1000mm apart,’ that doesn’t match your sketch; then you say the bm. flg. is about 250mm wide x 6mm thick, and the beams are about 300mm deep, with a 6mm web. Is any of this correct, on my part? Put the dimensions right on your sketch and do a cross-section showing the end fixity condition with some detail. Where do the loads which you say are distributed along the whole 10,000mm length, of ‘n’ beams come from? And, is the ‘large structural element’ strong enough and stiff enough to accumulate and apply the loads to the individual beams? How long are the beams? What are the magnitudes of the loads, in lbs./ft., ft.-lbs./ft., etc.?

Are the beams actually spacers/spreader bms. holding two major structural elements apart/together, thus acting as axially loaded members, also. I can certainly imagine some sort of end support mechanisms either imparting fixed end moments Mbz, maybe different at each beam end, to the beams; or maybe inducing/causing that Mbz as part of its beam support function; and these moments might cause Qty (you called it shear, which it is). These mechanisms would have to rotate to impart the end moments to the beams, or remain fixed against rotation to induce an end moment as a beam reaction, maybe caused by some loads on the length of the beams. And, “ending fixing elements” and “The end fixing points are two large structural elements which the cross-beams are fully intergrated into.” just doesn’t explain how the moments are imparted to the beams or what the beam reactions do to the ‘fixing elements,’ or if this ‘large structural element’ is continuous btwn. the beams. Can those ‘end fixing elements’ rotate about both the Z & X axes to induce Mbz (a bending moment) and Mtx (a torsional loading) loadings on the individual beams, or again, is there some continuity in this ‘large structural element’ btwn. beams? These relative strengths and stiffnesses will be part of the engineering analysis. You can see it, we can’t, and your word picture must be crystal clear if you expect us to understand and help.

 

[Raggedman]

Thanks for the reply - I think my statement is fairly accurate and I've got and okay level of knowledge, though things like that are fairly relative.

I've an engineering masters and at least half of that was structural work, however it has been quite a few years since I've done anything more than hand-calcs for support lugs or beam theory on frames etc. I've got a good idea of the fundamentals of structural analysis but it has been a good while since I've used a lot of it. I'm not sure if that helps show my level of experience but if you were to suggest something that I didn't understand, then it would just be a matter sitting, breaking it down and understanding it. From first principles I can normally get to most things given time.

You've lost me a bit with the 'hocus-pocus' - though I'm not trying to confuse or hide the issue, rather the opposite: Instead I have a fairly complex real-life situation that I need to breakdown into an simplified form. The quality (or lack thereof) to the sketch was purely as a lack of time on my part and a desire to move forward with this. I believe that I have the understanding on how to apply the bending and shear forces to the beams, it's just the torsional force that has me stumped - calculating the actions on a single beam wouldn't be a problem for me.

I've attached an updated sketch and will give some additional background. There are obviously some assumptions that need to be made to prevent this developing into a very complicated analysis.

The beams join two floating pontoons, each pontoon is internally well braced and stiff - and could to my mind be consider to be rigid. The beams span the gap between the pontoons. The load is applied but the buoyancy of each pontoon acting up against the overall weight acting down.

Lloyds Register Rules gives three equations (based on craft dimensions, environment etc.) that provide the values for the overall forces acting on the spanned section;
Transverse bending moment
Vertical shear
Longitudinal torsion

The first two I think to be easily divided across beams (as no bias is given to the loading for location). Only the last one is trickier to define. The manner in which it is give however suggests to me that it is expected to be a fairly simple to distribute the torsional loading across the beams - if the span was solid it would be. The end terminations of the beams are harder to define in a real world application they are intregrated into the pontoon body, for this analysis I would expect it to be adequate to say they are fixed and immobile to the pontoon as forces should never cause them to become a plastic hinge.

I hope I've been able to clarify things somewhat.

 

[rb1957]

what is the axis of the torsion load ? i suspect it is "bow up/stern down" ie the two pontoons are racking against each other. if so the torsion would be applying couples to the beams (highest at the bow and stern, opposite at the bow and stern ... clear as mud ?) so each cross beam has a couple applied (same but opposite shears for either pontoon), which implies balancing fixed end moments, yes?

 

[dhengr]

Raggedman:

Actually, this is not a simple design problem, but we are trying to be conservative in simplifying it enough so we can tackle it by long hand methods, at least as a first step. But, the devil will be in the connection details. My comments on “obfuscation or hocus-pocus” come from such things as your saying that the beams are ‘2000mm wide and spaced 1000mm apart,’ when it now appears that what you meant was, ‘they span about 2000mm btwn. two pontoons, and are spaced at about 1000mm o/c along the pontoon’s length.’ You might get a easier cross beam end detail if the cross beams sat atop the pontoons and spanned from outside to outside. And, it should go without saying that if torsion controls you might be better off with closed sections for the cross beams.

Draw every possible alternative that your wording might allow/suggest, and you might see why we could be unsure of what you really meant. Again, you are looking at it, but we can’t see it from here, so you must be very careful that your verbiage only allows one word picture to be drawn, out of many. There isn’t anything wrong with a free hand sketch, but proportions are important, so reasonable scale and dimensioning is important, and that almost always shows things better than our verbiage. Finally, you can’t afford to be lazy about your sketches, if you don’t have time, why should we? You’re the one asking for free help, so don’t expect us to waste our time trying to figure out what you mean; that just leads to a bunch of half-a$$ed guesses/answers which may actually head you off in the wrong direction. Enough, already, DH!

I think Rb1957 has it about right, but I’ll try a slightly different tack since we are talking nautical here. Assume the torsional axis is approx. at midship lengthwise, and across the ship; further, assume that starting in the upper left corner of your plan view and working clockwise, the corners of the ship are lettered C,D,E & F, thus the lower left corner will be F. I suspect Lloyd’s is considering a wave amplitude and spacing, then (you said) as a function of ship size, and I think primarily length, then width, then load on the ship as relates buoyancy (pontoon depth in the water), corners C&E may be high, while corners D&F are low on the waves. Now, given a pontoon length you can start to put a number on the slope of the pontoons and a relative vert. displacement btwn. corners D&E or F&C, and a relative end connection rotation at each beam; and the width of the ship or span length of the cross beams and this relative rotation and vert. displacement induces torsion in the cross beams and also induces a secondary end moment (and shears) in the cross beams. The cross beams will all see the same torsional loading if the pontoon is stiff enough. And, the center most cross beams will see smaller or no secondary moment. I haven’t run any numbers on this, so you should. It would have been helpful if you had attached a couple pages of Lloyd’s Register Rules, formulas and explanation (their rational). In any case I’ll bet you’ll find the above embedded in their rational and formulas.

Another consideration is the rigidity of the loads on the boat as relates to this wave action. If you assume a very flexible and compliant load you just have pounds/sq.ft. on the cross beams, and the beam bending moments and shears seem fairly straight forward. But, a very stiff load might give you some very high shears at one end of the outermost end cross beams. Finally, it would seem that you have partially fixed cross beams, not fixed ends. Can your connection detail really transmit that fixed end moment into the pontoon without damaging it, are there stiffeners at each cross beam to take this moment into the pontoon? What is that partially fixed end moment? That’s a function of the stiffness of the cross beams, their deflection under load and their slope at their end connections, and how much these moments tend to cause the pontoons to roll in the water under these loads.

So, all-in-all, not a simple problem..., but I think you can start to get your arms around it with some long hand calcs. and some good experience and engineering judgement. Then be careful of the details because fatigue and any notch sensitive details could be killers.

 

[Raggedman]

"what is the axis of the torsion load ?"
*As the equations are for the spanned section I would take it to be around the midships of the spans at half the height of the beams, the centroid of the overall section.


"i suspect it is "bow up/stern down" ie the two pontoons are racking against each other."
*Yes, worst case would be the sea coming in at a diagonal lifting one pontoon bow and the opposite pontoon's stern. Whilst the alternate bow and sterns were in the trough of the wave.

"if so the torsion would be applying couples to the beams (highest at the bow and stern, opposite at the bow and stern ... clear as mud ?)"
*Yes, I just wasn't sure how to divide the torsional moment between all the beams. I'd considered that it may be sufficient to consider the forward and aft beam as the only two beam involved - if the span has the strength with two it could be taken that the full assembly will be strong enough.

"so each cross beam has a couple applied (same but opposite shears for either pontoon), which implies balancing fixed end moments, yes?"
*That sounds sensible - I might need to have a look at this on paper.

Cheers - any further thoughts very welcome.

 

[rb1957]

i think the torsion would ab applied as a linearly varying distributed force (+ve on one bow, -ve at the stern) reacted by an opposite distribution on the other pontoon.

or simplier, as you note, a couple between the fwd and aft most beams ... i think you'll find that this isn't too conservative ... and a bit of conservatism is good, no?