Plate section modulus 3

[strlengr7]

I have a 4x8 plate that will span 4 feet across two beams. Then a load will be applied to the mid span. When calculating the section modulus to determine Mn=FyZ, how would you determine what b should be used in Z=bd^2/4?

 

[rb1957]

my 2nd thoughts ... maybe the /4 is a plastic bending factor ?

my 2c ... i think even if the plate is just resting on the two supports, that given enough deflection (to invalidate plate bending) it would develop membrane stresses, and react these as a contact force against the supports.

one way to visualise what's happening to the plate is to divide it up into strips (nominally) 1" wide. the central strip has the load applied to it, and reacts this at the supports and shear into the adjacent strips, maybe a sine wave dist'n ? the adjacent strips see this load and react it at the supports, and shear into the next strip; and so on. "play" with the amplitude of the sine waves, each strip would react a portion of the load. the deflection of the adjacent strips either side of the load point should be similar to the deflection of the central strip ... the plate will try to make a reasonably symmetric -ve mound around the load.

i wonder what our irish friend, Tim O'Shenk, would make of this ??

or you could use FE.

or you could simply test it ...

 

[TLHS]

If I needed more capacity than is afforded by assuming a 45 degree load distribution, and the plate is well restrained I'd likely try to use yield line theory.

But yeah, if it were an initial design (ie not retrofitting to something existing) and the thickness coming out of the calcs wasn't crazy I'd likely just assume a force distribution in the plate.

I'd also be concerned about deflection if you're spanning this far with plate. In this sort of situation it may make sense to go with a thinner plate and stiffeners.

 

[MikeDB]

I would assume an effective plate width "b" equal to the width of the tire plus 12 times the plate thickness. One could use the plastic section modulus, but since this is a moving load with people involved, I would want my results to be on the conservative side. It is not uncommon to use a safety factor of 3 to 5 with construction loads.

 

[TLHS]

Hey Mike, I'm curious if the 12 times the thickness comes from somewhere, or is it just a conservative assumption you use?

 

[nutte]

TLHS, Blodgett uses 12 times the plate thickness in an example on page 6.6-6.

 

[chrislaope]

Take b=12 times plate thickness may be more easily to be justified. However it is apparently toooooooooooo much conservative.

I have just repeated another test, I happen to have a 30.625"x20"x16g(measured t=0.059")plate at hand. I simply supported this plate along the two 30.625" edges (support width = 30.625") with a span=18" (no support along the 18" edge). Now I put 25.8 lb weight on center point of the plate with the center point area = 1.25" Diameter area, then I got the measured deflection = 0.27". From theoretical calculation, with b=18", the deflection will be calculated as 0.35". This repeated test again proves that even 45 degree rule is still conservative enough.

 

[TLHS]

Yeah, for a plate with a point load where the span is some amount shorter than the supported length you're going to get two way action locally to resist a point load, so you're going to be stronger at ultimate failure than a straight up 45 degree distribution.

I suspect for a point load that the two possible governing yield line patterns are those shown in the attached. The first one will only happen if your plate is narrow enough in the non-spanning direction to allow the other yield line mechanism to intersect with the side. You can see that you gain capacity from where your hinges develop on the sides.

There are a couple of extra patterns to check (yield lines intersecting back to the corners would be one I'd need to check because your outside edge is free to rotate), but I don't see
them being lower load failures.

 

[BAretired]

The name of the thread is "Plate Section Modulus", not Equivalent Width for Deflection". If you consider an equivalent width of 4' and calculate the section modulus, the resulting stress will be lower than the maximum stress and higher than the minimum.

Stress in the immediate vicinity of the load will be much larger than that in more remote regions...in both directions.

BA

 

[paddingtongreen]

There is a question here as to whether a quick and dirty estimate will do or if sophisticated analysis is required. The greater the deflection the greater the slope to be climbed by the jack. If we are to get into sophisticated analysis, we need to know if the plate is just resting on the supports, or if it is bolted. If bolted we could go into membrane theory, provided the supports could resist the horizontal component. Another question for me, how near the right and left hand edges can the jack approach?

With the load at the center, the greatest deflection will be at the same place. The deflection will be a little less ahead of, behind and to the left and right of the center. It will be more reduced a little more forward, back, left and right of the center. A cross section through the center, in both directions will be a curve with the low point at the center. Cross sections further forward and back, left and right will be similarly curved, but the low point will be higher than the center section. This implies that there will be high lines running diagonally from near the center, similar to an upside down vaulted cathedral, to perhaps the corners but perhaps not that far out. This is why I said earlier that the corners try to lift up. Anyone with Roarke can check me on this. If the corners lift, the ridges go towards the last contact point with the support. The point of this is that the load from the jack and the whole weight of the plate is carried by only the center part that is in contact with the support.

The point all of this is to give my reason for my conservative approximation. It is less generous than it sounds. In this case the width of the load plus 1'-0" (1/4 of the span).

Personally, I would be more concerned with the slope and whether the edge on the support would lift than I would the stress. Function must be catered for before we think of stress.

Michael.
Timing has a lot to do with the outcome of a rain dance.

 

[chrislaope]

Good point, BAretired.

I actually also considered about this stress concentration issue, that is why I am not solely rely only on deflection measurement test. This 45 degree result actually comes from my Finite element analysis model. The reason we have done so many analysi is we actually had taken several industry projects before, which involves a lot of point load on center of a steel plate problem. If take effective b=load width+12*t, or b=load width + 1/4 span, then these projects become a mission impossible. This forced us to dig into more detailed analysis to find the more truthful, more realistic solution. One of our example FEA result showes that once the point load width is equal or larger than 1/9 span, then the maximum concentrated stress in the vicinity of the point load is already smaller than the stress calculated from effective b = 45 degree rule (21.4 ksi vs. 24.0 ksi). Of course if the point load width shrinks toward zero, then the concentrated load will be surely excessive, however in this situation, the punching shear will control the design.

As I stated before, I am not against effective b=load width+12*t, or b=load width + 1/4 span. I think it is more easier to get justified however not practical in many circumtances. On the other hand, our 45degree approximation, it is more complicated and troublesome to get justified (our FEA analysis model is our legal support), however it does effectively solved a lot of practical engineering problems in real world.

 

[BAretired]

The yield line patterns provided by TLHS suggest to me that the most critical condition is the first diagram, i.e. a yield line occurs at midspan and runs the full length of the plate, in this case eight feet.

If the load is displaced by unit distance, the External Work is P*1 or P. If the span is 'a' and the width is 'b' and the yield moment of the plate is 'm' foot pounds per foot, then Internal Work = (m*b*2/a)2 = 4mb/a. In our case, b/a = 2 so I.W. = 8m.

Equating E.W. to I.W. we get m = P/8.

In the second diagram, there are four yield lines, two diagonal lines and two vertical lines. The vertical lines are negative bending. If x represents the distance between negative yield lines, then:

E.W. = P
I.W. = m(4a*2/x+2x*2/a) = 4m(2a/x + x/a)

Setting the last expression equal to zero, we find x = [√]2*a and I.W. = P/11.3 which is less critical than the previous case, i.e. the plate requires a smaller yield moment.

If we consider a third case where the plate is not held down along the long edges but permitted to lift off the supports, then x = a and m = P/8, which is the same as we obtained in the first case.

BA

 

[BAretired]

The second last paragraph should read:

Setting the derivative of the last expression equal to zero, we find x = √2*a and I.W. = P/11.3 which is less critical than the previous case, i.e. the plate requires a smaller yield moment.

BA

 

[BAretired]

The original question was:

When calculating the section modulus to determine Mn=FyZ, how would you determine what b should be used in Z=bd^2/4?

By yield line analysis, it appears that the value of b to use in this case is 8', but if 4b/a > 11.3 (i.e. aspect ratio b/a > 2.825 the second yield line pattern shown by TLHS would govern.

BA

 

[TLHS]

Thank you for working through that, BA

From a comfort and confidence standpoint with plate structures, yield line theory is very much the best tool I've encountered. It's a good balance between simple and practical. While I may make more conservative assumptions, it gives you something to fall back on that lets you work out your actual failure state. It lets you use a reasonable hand method to check finite element, and it also gives you a good basis that you can use to justify working around small local stress peaks in a finite element solution.

It's the only major analysis type I regularly use that wasn't even touched on in my classes back in university. I think it's ridiculous that, while plastic analysis of frames was done, nobody ever took the extra few minutes necessary to explain that it can be extended to plate structures.

 

[chrislaope]

Good job, BA, one more star for you, and one star for TLHS for suggesting yield line pattern.

Thanks.

 

[BAretired]

TLHS,
I agree that yield line theory is a useful method of tackling certain problems, particularly when checking steel plates but it must be used with caution.

Yield line theory always gives an upper bound to collapse load and there is no assurance of adequate strength in diagonal tension or deflection control.

The Hillerborg strip method gives a lower bound to collapse load and is deemed by many to be preferable to the yield line method for the design of concrete slabs.

BA