Special case in rectangular plate bending
Special case in rectangular plate bending
(OP)
Greetings
I'm dealing with a case where a rectangular plate is subjected to a bending moment applied mid span of the plate along the shorter edge. I tried looking for a formula for this case but couldn't find any so hope you can help
I'm dealing with a case where a rectangular plate is subjected to a bending moment applied mid span of the plate along the shorter edge. I tried looking for a formula for this case but couldn't find any so hope you can help






RE: Special case in rectangular plate bending
Maine EIT, Civil/Structural.
RE: Special case in rectangular plate bending
RE: Special case in rectangular plate bending
Are you looking for the moment strength of the plate? That's going to be the yield strength x the plastic section modulus. Equation F11-1 in section F11 of the AISC-360 specification.
Maine EIT, Civil/Structural.
RE: Special case in rectangular plate bending
If the plate is just supported at the short ends, it is as TME says, but if supported all round, it is more complex, I don't know if Roarke covers it.
Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin
RE: Special case in rectangular plate bending
RE: Special case in rectangular plate bending
RE: Special case in rectangular plate bending
Quando Omni Flunkus Moritati
RE: Special case in rectangular plate bending
RE: Special case in rectangular plate bending
Also I cannot visualize how you could apply such a loading in a real situation. The problem is that the distributed moment cannot produce its effect near the boundary, as rotation and moment around the short side direction are zero there.
Note BTW that also most solutions for rectangular plates have no closed form equations: these are generally infinite series solutions, where the series expansion is replaced by a calculated coefficient like β. What you get on xcalcs (for the limited number of boundary and loading conditions provided) is an approximation by a polynomial expansion: it is also an equation, complex but usable.
prex
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RE: Special case in rectangular plate bending
this loading case can happen in a real situation as in a winch pulling some load while fixed to deck structure.
the effect of the moment on the edge will be a twisting moment. but all i need is equation of the moment produced in the plate.
thanks for the info about infinte series equations, those are numerical methods aren't they?
RE: Special case in rectangular plate bending
I agree, I haven't come across one.
You haven't told us how the moment is to be applied, at the moment it appears by magic. Is it limited in width or is it full width? it it a weldment causing it or a force couple or pressure couple?
FEA is probably the best way to go.
Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin
RE: Special case in rectangular plate bending
fea is the best way but I didn't study that. and thanks for clearing what afaik means.
RE: Special case in rectangular plate bending
This is one where you can make a model from cardboard or plastic foam board and see for yourself.
Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin
RE: Special case in rectangular plate bending
Maine EIT, Civil/Structural.
RE: Special case in rectangular plate bending
as i said earlier fea is the best for such situation but i cant use it now.
RE: Special case in rectangular plate bending
In short, you cannot apply the moment uniformly across the plate because it will not resist uniformly.
Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin
RE: Special case in rectangular plate bending
unless, of course the winch spans the plate is is uniformly attached to the plate ... big winch or small plate ?
Quando Omni Flunkus Moritati
RE: Special case in rectangular plate bending
Peddington: u can apply the moment uniformly but reactions will be nonuniform. it is not as complicated as u make it sound, there is such a case but for a circular plate which is why i cant use it.
rb: to be exact the winch is fixed over an h beam wrapped all around its sides which is in turn welded to the deck plate
RE: Special case in rectangular plate bending
In the circular plate case the load does not go all the way to the clamped edge and so there is warping of the plate.
Unsubscibing.
Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin
RE: Special case in rectangular plate bending
RE: Special case in rectangular plate bending
You should be able to consider half the plate, a/2 * b, clamped on three sides and free on the fourth side with a uniform moment of M/2 applied on the free edge. Because there is no deflection on the fourth edge, it could be taken as a simple support instead of a free edge.
That doesn't solve the problem, but it provides another model to be investigated.
BA
RE: Special case in rectangular plate bending
BA: why did u divide M by half when taking half the plate and why do you think it won't solve the problem,I believe timoshinko mentioned such method in his book "theory of plates and shells"
RE: Special case in rectangular plate bending
BA
RE: Special case in rectangular plate bending
Yield line design would be another attractive option. However, I'm hesitant to go that route unless I feel very confident that I've got the right yield line model. And for me, that usually means that it's a very simple scenario or I've seen the model in print some place before.
For designs where I want reasonable proportions but I don't want to invest the time required to do FEM, I've had some success treating steel plates as assemblages of orthogonal, discrete beam "strips". Essentially a grid analysis yielding a lower bound capacity. This requires some redistribution capacity in the system and, as a result, I tend to be quite conservative with my weld designs.
The greatest trick that bond stress ever pulled was convincing the world it didn't exist.
RE: Special case in rectangular plate bending
instead of a line moment, i'd suggest a line couple (up on one edge, down on the other). maybe model the deckplate from the winch CL (where not much should be happening) so your plate would have two sides "fixed" and two SS (maybe 4 sides fixed ?, but i think the winch CL is SS) and an applied line force somewhere across the panel.
you might find more plate loading solutions in Roark's reference ... google "Moody Rectangular Plates" (seriously!).
FE should be a reasonable solution. how torisionally rigid are the edge frames ?
Quando Omni Flunkus Moritati
RE: Special case in rectangular plate bending
It seems to me that you have been working on this winch supported on a barge deck problem for some time now, and have asked essentially the same question several different ways, hoping to find a grand formula that will answer all. Please reread your earlier threads and answer some of the questions some of us have already asked and you have ignored. Also, provide some meaningful sketches of the actual situation, loads, dimensions, etc., because a winch of any real size should probably not (would likely not) be loading a plate the way you are suggesting. You have not structured the foundation (the barge deck structure) for the winch properly if it is. The base on any meaningful winch will be quite stiff itself. And, it will impart a substantial shear reaction to the deck, along with the reactions to the line pull which is some distance above the deck, imparting a moment which I suppose is what you’ve been talking about. You have to take these reaction loads into some real structural members under the deck plate, I would think.
RE: Special case in rectangular plate bending
KOOTK: u mean to divide the plate into smaller strips and divide my load equally by those strips?
rb: ba's method is sound and mentioned by timoshinko; however through my discussion with the guys above it turns out my loading case is different. I'm upploading some drawings and sketches to clear things up. I now believe my loading is more of a varying line load along the longer edges. so i will need an equation for a varying load and then use ba's method.
dhengr: this is more like an exercise for me by my boss and it goes aside whenever we have some real job so that is why i'm still on it. for now i'm only concerned with bending reactions and i will supply some drawings and sketches hope they can clear things up.
RE: Special case in rectangular plate bending
RE: Special case in rectangular plate bending
RE: Special case in rectangular plate bending
RE: Special case in rectangular plate bending
The drawings are a little puzzling. Why are you adding four new beams instead of just two? Why are you cutting the 635 deeps beams to receive the 150 x 90 angles instead of adding the beam below the angles and if necessary fitting stiffener plates between the angles?
The left ends of the 635 beams seem to be supported on the side shell plate. The right side seems to be unsupported. In short, the framing is not clear and until it is, it is difficult to recommend a design approach.
BA
RE: Special case in rectangular plate bending
i'll upload a better detail of the beam
RE: Special case in rectangular plate bending
The four 635 deep beams are supported by an unyielding support on the left end and by a deflecting "large L angle" of unknown size and unknown span on the right. Is the large angle supported at each bulkhead? The lack of clarity has nothing to do with converting from autocad to PDF.
Plate analysis is used for a plate of uniform thickness. Your plate is reinforced with angle stiffeners in one direction and four hefty beams in the other. It will not behave as a plate. It will behave as a beamed structure.
BA
RE: Special case in rectangular plate bending
Quando Omni Flunkus Moritati
RE: Special case in rectangular plate bending
presumably the shear loadpath into the deck plate is plenty strong (it usually is).
is 40 tons the working load, or some ultimate design load ?
Quando Omni Flunkus Moritati
RE: Special case in rectangular plate bending
Come-on, get with it and start thinkin a little bit. Don’t keep showing the same sketches or sections without providing any new or meaningful design info. You showed almost the same damn inadequate sketch of the winch in one of your earlier threads without a bit of meaningful info. on it either. We do understand what a winch is to the extent that your sketches show it. Get the spec. sheets of on the winch, and post them here so we can see it, since you don’t seem able to pick the important info. off of them. How high, above the deck, is the 40T cable force applied? You actually have several darn smart people here trying to help you, but you keep mis-directing the discussion by ignoring their questions. This is not likely a plate bending problem, however much you want to learn about plate theory. What does this winch have to do with the funny shaped cut-out just above it in your plan view? Is that the base for a crane of some sort? What is the 2680x2800mm doubler pl. intended to do? What is its thickness and how is it attached to the deck pl. be specific and complete? How is the winch attached to the barge deck, show this in detail, with dimensions, etc. This attachment had better relate to the new added beams, or you may have to make some adjustments. Show a large section through the added beams, dimensions, welding, etc. The lower added beam will be loaded upward by the winch base, and the upper added beam will be loaded downward by the winch base. Show a large section of the new ‘large L angle’ with all dimensions, welding, reaction points, etc. We are not nit-picking here, we are trying to get a sense of the proportions and details of the problem. We are trying to understand the framing system and how the winch loads will be put into the deck structure, and it better not involve a bunch of pl. bending or pl. action. If you want to put a name on your barge deck system, it is a gridwork of ribs and beams, or an orthotropic deck.
RE: Special case in rectangular plate bending
plate analysis is indeed used for stiffened plates and plated structures as u might see in many books and standard so wold u explain why its not applicable here.
RB: the winch is seated on h beams fixed around its four edges. the h beams are then bolted to the deck plate. consider everything above the deck plate as rigid. the 40 tons is ultimate design load. im uploading another sketch of loads causing the bending moment induced by the h beam into the deck structure.
it looks like u both advise neglecting the plate in the analysis. won't that affect the analysis in a big way? its like your saying the plate is taking none of the load.
I should thank you both for your patience with me so far
RE: Special case in rectangular plate bending
"the winch is seated on h beams fixed around its four edges. the h beams are then bolted to the deck plate." that maybe so, but your sketch shows the winch having side plates so there's a more direct loadpath into the sides (with the tapering bending field reactions (as opposed to the ends where the applied moment is reacted as a couple). if the base of the winch is fastened around into the H-angles then there's a story there. In the end it's your analysis; remember we could just be a bunch of lunes, drooling and typing in our bath-robes.
btw, what's your level of education, experience ? it seems to me that you're trying to do the "right" thing, or else you've gotten caught in an inspection and someone has said "show me". attaching a 40ton winch is no small matter. it could well be that rules of thumb and general experience can produce an acceptable design ... but they can always overlook things.
a small thing, the bolts attaching to the deck are carrying shear and tension loads.
Quando Omni Flunkus Moritati
RE: Special case in rectangular plate bending
"your sketch shows the winch having side plates so there's a more direct loadpath into the sides" i'm not sure i get what u mean by the side plates,are u talking about the shell side plate? there are no side plates around the winch other wise.
actually, I tried dealing with the problem as a system of beams but my boss,a naval architect, dismmised it as a wrong approach. his method is to get the bending moment for a plate portion between transverse frames and then divide it by the combined section modulous of the plate and longitudinal stiffeners. i have an experience of only 4 months and under grad education in mechanical engineering if ur asking.
dhenger: for some reason your comment didnt show up before, i'm sorry if my supplied details are lacking and if i'm frustrating u all but problem is now the actual vessel is gone and this is the only info i can get.
RE: Special case in rectangular plate bending
""your sketch shows the winch having side plates so there's a more direct loadpath into the sides" i'm not sure i get what u mean by the side plates,are u talking about the shell side plate? there are no side plates around the winch other wise." ... like i say, your earlier sketch, the hand drawn one (4:47 on the 24th), shows the winch axle supported by two plates on the sides, so this is where the load will prefer to go (stiffer loadpath). now granted the winch will have a base plate and this will distribute some the load along the ends (the couple reaction to the applied moment) but is it "butch" enough to be considered infinitely rigid ?
we taking about reacting the load as a system of beams. If your boss wants you to do something with plate moments then he's taking a different tack to us. he should help you ...
1) solve this problem his way, and
2) show you why his way is preferred (is it simpler than beams ?). i suspect he's thinking there are simple conservative assumptions that make the job easy to solve with conservative results.
Quando Omni Flunkus Moritati
RE: Special case in rectangular plate bending
What transverse frames? You have a rigid plate element on one side and a "large angle" on the other side which spans from bow to stern. If you had a rigid plate element on both sides, the problem would be a little easier, but we have no idea how the large angle is going to deflect under the applied forces.
Where is the large angle supported? Where is the ribbed deck supported? At present, it appears they both span the entire length of the ship. How can you determine the behavior of members without knowing the span?
BA
RE: Special case in rectangular plate bending
as for my boss, i spoke to him today he said that treating the system as beams makes the problem simpler and is not the professional way to do it as the plate is contributing to holding the load. he also said that the plate needs to be taken into consideration if the h beam under the winch is laid on a plate portion that is not supported by a beam since plates won't transfer loads.
is it anyway practical to lay the beams supporting a winch directly over a deck structure without som mounting plate?
ba:both the large angle and ribbed deck span the entire length of the ship. do u think the problem with this large l angle is that it will cause the side fixed to the shell plate to be loaded more than the beams on the side of the large l angle?
RE: Special case in rectangular plate bending
the problem becomes very difficult very quickly since you have a continuous plate supported at many places, loaded at a bunch of other ones. i'd suggest FEA is the only reasonable way to analyze this. that said, there are always simplfying assumptions ... you might say that each H-frame load is reacted by a circle of plate, simplify to SS, and you can quickly get a plate bending moment.
Quando Omni Flunkus Moritati
RE: Special case in rectangular plate bending
No! That is not my concern. I am still trying to understand your framing plan which is not at all clear. Are the ribs and large angle supported vertically by FR-4, FR-6 etc.? The thing marked 'Bulkhead'...is that FR-5?
If the ribbed deck and large angle are spanning the entire length of the ship, a dimension which you have not provided, I would be concerned that deflections would be excessive. If, on the other hand the ribs and angle are supported by bulkheads at closer centers, that is another matter.
BA
RE: Special case in rectangular plate bending
No one is planning to do your design and analysis for you. But, we can’t even really give you very good direction if you can’t give some more meaningful info. You would be surprised how much an experienced engineer can glean from properly proportioned and dimensions sketches and other significant basic info. What thickness are the doubler pl. and deck pl? It isn’t that the pls. don’t come into play, its just that they probably shouldn’t come into play the way you were going about it earlier. How are all of these various pieces welded together? That will determine how they come into play in your design and analysis.
I’m not a navel architect or engineer, but I’ll bet your boss and the rest of us are talking approx. the same engineering language, some terminology may need to be ironed out. There might be ship hull forces and stresses which must be combined with the new, more concentrated, stresses from this new winch loading. You do have to make some educated/experience based judgements here, which aren’t really covered in any text book or by any magic formula. Please, please, tell us how high the cable (40T tension force) is above the deck so we can get a feel for the magnitude of the moment. Then tell us the size of the winch base and the bolt spacing and locations on the deck. Reread my last post for some of its actual engineering meaning, and answer the questions. These answers are needed to start to formulate an approach to solving this problem. How thick are the doubler pl. and deck pl., and how are the longitudinal angle ribs and the new large L beams welded to the deck, weld sizes, etc. You need this info. to start to make a judgement about how these elements work with the deck pl. as built-up structural members. Where are the winch data sheets, does anyone have any design drawing on the barge, what about your ship yard? You have to locate the winch hold down bolts with respect to the ribs and large L beam to start making sense out of this problem.
Start with what you know, the winch and winch force. What’s the moment caused by the 40T cable force, (40T)(dist. above deck), are there any variations to this? How is the moment taken out and into the deck, how many bolts, spacing, lever arm btwn. bolt rows, front and back? Can the deck pl. take these bolt loads upward and the line of compression downward, or do I need to add some pl. or structure under the deck pl. to help distribute these bolt loads? Can I situate the winch so the loads go directly into the new large L beams? I would guess that’s how and why the shipyard located these new beams, to relate to the winch base. Then what bending loads do these bolt forces induce in the rib/deck pl. or large L beam/deck pl. built-up beams?
RE: Special case in rectangular plate bending
i just thought of this, can't the transverse beams transfer the h-beam loads to the sorrounding longitudinal beams and thencontinue the problem like the loads are distributed among the longitudinal beams alone? it is a simplifying assumption offcourse i know that.
RE: Special case in rectangular plate bending
they are indeed supported by a bulkheads at closer centers
RE: Special case in rectangular plate bending
Quando Omni Flunkus Moritati
RE: Special case in rectangular plate bending
Assume the force is H and is located a distance h above the deck. Then the moment is H*h. This will be delivered by the two beams shown in green to the ribbed deck. It will result in a couple, i.e. two forces, one up, one down to the green beams. The magnitude of each force will be H*h/d where d is the distance between the two green beams. If I am not mistaken, d = 914 + 216 + 1075 = 2205 mm. By the way, you should dimension to the nearest millimeter, not fractions of millimeters.
Those forces are applied to the continuous beam supported by the bulkheads shown in red. The continuous beam is the ribbed slab. The number of ribs effective in resisting the load is perhaps six (judgment call).
The span is approximately 1830 mm (distance between red lines). In the worst case, the load is approximately at midspan, so simple span M = PL/4 = H.h.L/4 where L = 1830 mm. In the other case, simple span M = Pab/L. From this, knowing the properties of the ribbed deck, you can work out the bending stresses.
You can refine your analysis with a 2D frame program but I would be inclined to accept the above approximation as sufficiently accurate for your purposes.
BA
RE: Special case in rectangular plate bending
BA
RE: Special case in rectangular plate bending
Oops...that should be M = PL/4 = H.h.L/4d.
BA
RE: Special case in rectangular plate bending
this there a direct connection between the beams under the deck and the H-frame above it ?
Quando Omni Flunkus Moritati
RE: Special case in rectangular plate bending
here u are talking about the shear force reight?
RE: Special case in rectangular plate bending
note, this is taking the overturning moment out at the ends, and neglecting (conservatively) the contribution of the tapering forces along the sides. which is what i'd do for a quick answer, but in this case i see the loadpath to the sides as being more direct than the ends. but it's a matter of judgement.
Quando Omni Flunkus Moritati
RE: Special case in rectangular plate bending
However you cut it, BA has the right idea. You (or the OP’er.) can FEA the hell out of it if you wish, and it’ll still lead essentially to what BA has shown, because that’s how it will act, once we have some meaningful details/facts on the matter. The only thing I would add the BA’s comments is that the OP’er. had better pay some attention to the continuity of the longitudinal angle ribs/deck action over bulkhead ‘FR4 ‘, since both winch reaction forces, fore and aft, will cause increased tension on the bottom leg (horiz. leg) of the angle ribs under the front of the winch. I have some trouble imagining that the boss would have nixed BA’s presentation, if it had actually been offered in the first place. In fact, looking at his post 26AUG,1:01, 3rd para.; what he says the boss told him to do, is essentially BA’s presentation. And, more of the same in his post 26AUG,17:33, 2nd para. Again, this is an orthotropic deck plate or gridwork of beams problem, and only secondarily a plate problem.
RE: Special case in rectangular plate bending
I think this is what the OP's boss meant by plate analysis rather than looking at individual beams. But I don't really know as I am not clairvoyant. I am assuming that the 635 mm deep beams are infinitely rigid over their short length so that deflection of those beams can safely be ignored.
If I had known that the bulkheads were acting as bearing walls, I would have suggested the above analysis much earlier. I thought the span of the ribbed slab and the large angle was the overall length of the ship. As it is, the spans are much shorter, so the above seems like a sensible and simple solution.
BA
RE: Special case in rectangular plate bending
No! There is an applied moment of M which resolves into two equal and opposite forces of M/d.
Perhaps you are getting confused with my earlier suggestion of applying M/2 to each half of the plate which we were discussing. That was correct and I will leave you to sort that out in your head.
I am talking about the applied force H which your earlier sketch suggested was 40 Tonnes. It is also the value of the shear between the winch base and the deck but inside the deck, it is not a shear force. It may be a combination of shear, tension and compression...whatever is required to get the force into the structure of the ship.
BA
RE: Special case in rectangular plate bending
RE: Special case in rectangular plate bending
RE: Special case in rectangular plate bending
Quando Omni Flunkus Moritati
RE: Special case in rectangular plate bending
RE: Special case in rectangular plate bending
Quando Omni Flunkus Moritati
RE: Special case in rectangular plate bending
An unsupported rectangular plate with a moment applied mid span is not in equilibrium. Your first sketch in your Aug. 8 post showing a plate of length a and width b with a moment in the middle is not in equilibrium as pointed out by paddingtongreen the same day. If you and your boss don't know that, you both need to review elementary statics.
If that plate is simply supported along the short ends b, the reaction at each end is M/a and the moment at any point x from the one end is M.x/a - M(x>a/2) where (x>a/2) is a boolean function equal to 0 when x<a/2 and 1 when x>a/2. By inspection, the moment varies linearly from 0 at each end to M/2 at the middle. The moment per unit width varies from 0 to M/2b.
As rb indicated, it seems an odd way to solve it. I would go further and say it is not the way to solve it. Your boss should think it through once again.
BA
RE: Special case in rectangular plate bending
RE: Special case in rectangular plate bending
I have no issue with treating it as a composite section where the section properties can be calculated using standard methods, but in order to solve the problem, boundary conditions are required. This means knowing how the plate is supported. I thought this had been resolved when you advised that it was supported at bulkheads acting more or less like bearing walls. If that is the case, the solution I suggested earlier is a reasonably good approximation of the behavior of the deck in resisting the action of the winch.
It's not clear to me what the objection is.
BA
RE: Special case in rectangular plate bending
yesterday as i had some spare time i showed my boss the calculations using the method
you taught me here. he said that it is acceptable by him but he doubts its acceptable by the classification classes and that its not a good enough method. he is still insisting on his method and i honestly doubt that it's correct.
RE: Special case in rectangular plate bending
if your boss is insisting on a particular analysis path, maybe it time to say "yes sir". the key thing with any analysis is to draw FBDs of where the load is going; if you can do that then your analysis isn't too "wrong". the next thing to consider is strain energy, and whether the analysis is using the lowest total strain energy to react the loads (lower is typically "righter").
it'd be interesting so see a sketch of how your boss thinks the loads are reacted.
another day in paradise, or is paradise one day closer ?
RE: Special case in rectangular plate bending
BA
RE: Special case in rectangular plate bending
BA
RE: Special case in rectangular plate bending
another day in paradise, or is paradise one day closer ?
RE: Special case in rectangular plate bending
BA
RE: Special case in rectangular plate bending
RE: Special case in rectangular plate bending
[cyclops]
Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin