Beam on top of beam, not connected, and of different length. 7

[jgrady]

http://www.eng-tips.com/viewthread.cfm?qid=110626

Hi all,

I was reading this thread, and one of the key points for the load to be shared was that beams must be spanning the same length. My question is, what if the top beam is only over the first 1/2 of the entire span. Does the load sharing still apply for the loads placed over that portion of the span?

Regards

 

[IDS]

Can it be that if the natural curvature of the upper beam is at a greater radius (flatter shape) than the lower beam that the forces on the upper beam cause it to conform to the lower beam's shape. The forces in the upper beam are modified by the lower beam, the upper beam no longer feels just forces from above. ???

I think that's a good summary of what happens.

It's also instructive to look at what happens from first contact, as the upper beam EI is reduced. Using the example of the upper beam half the length of the lower beam, symmetrically placed about the centre, with a full length UDL.

The upper beam first touches the lower when its EI reduces to 1/2.4 of the lower beam. At this stage there is no load transfer at midspan, and the upper beam moment is half that of the lower beam. Its EI is less than half, so its curvature is greater, and there is only a point contact.

The point contact continues until the EI factor reduces to 1/4. At this stage 20% of the UDL is transferred to the lower beam, the lower beam moment has increased, and the upper beam moment reduced such that the ratio is 4:1, so the curvatures are equal.

Any further reduction of the upper beam EI would further increase the curvature of the upper beam, but it is constrained by the lower beam, so the point contact spreads to a contact over some finite length.

From this point the contact length increases as the relative EI of the upper beam is reduced, as has been described before, but so long as the EI is greater than zero there must always be some non-contact zone at the ends of the beam.

Finally it should be said that we are talking ideal beams here, and the gap size as the contact zone extends is tiny. In any real beams the deviation of the beam surface from a smooth curve would result in multiple point contacts, rather than a single contact zone with two well defined gap lengths at the end. Also the "point contacts" would also be "small area contacts", because a point contact would result in infinite stress.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[jfmann]

For first contact, ratio of 1/2.4 is same as the 41-2/3-percent I reported, so at least we agree on starting point!

However there is no obvious reason to then claim that point contact (only) continues for all I-upper values down to 25-percent of I-lower........or that distributed contact then suddenly occurs
when I-upper = 25-percent of I-lower.

For I-upper values less than 41-2/3 percent of I-lower, it is much more logical to conclude that distributed contact occurs and becomes longer with lower values of I-upper.

Which then leads back to my example of I-upper = 40-percent of I-lower........for which moment in upper beam, M-upper = about 47-percent of M-lower (per my calculations)

Key to choosing the 40-percent value is that length of contact is short so that there is not much (none, I think) risk of shape of pressure distribution drastically changing results. However, if the claim of beams sharing moment in proportion to M/EI values when they have contact over some (any) finite length is valid in this case, then it should be valid for the 40-percent configuration.

If the 47-percent moment I calculate is essentially correct, then either the M/EI relation is not valid for this case (and I agree that would take some explaining!) or the curvatures are not the same (also problematic, but perhaps more likely)......or there is some other explanation that is not obvious now (most likely calculation error).

Considering that I calculate the exact relative I-upper value (41-2/3 percent) as Doug is reporting for initial contact provides some confidence level that my routine is on track......however, I acknowledge the possibility of calculation error.

My routine does not calculate length of contact exactly......however, for the I-upper-40 configuration, only short (or very short) length of contact is expected. For relatively "wide" range of contact lengths about midspan (up to 4-percent of span length for I-upper-40), changes in moment for each beam are very small. Most important is reaction force at midspan, which I calculate as 1.78-percent of total load (wL/2)........and then of course reduce the end reaction by half that amount.

John F Mann, PE

http://www.structural101.com

 

[jfmann]

Just some musings on fundamentals........though have not quite fitted all pieces into this puzzle.

For continuous beam.......moment for each beam segment is of course the same at interior support. Slope is also the same at end of each beam segment at this support (that is how segments are "continuous"). Yet curvature for each segment (at support) is function of stiffness which is function of E, I and L for each segment.

Curvature is defined as change in slope over distance.

In 'my' book (vintage, though I suspect fundamentals remain yet relevant!).......change in slope over distance is equal to area under the M/EI diagram over that same distance.

"Beam on elastic foundation" requires differential equation to solve.

If my calculations are correct (and that is yet to be proven), my guess at this point is that the lack of physical connection between beams may be involved.

However........of interest (and perhaps confusion) is the classic example for "shear flow".......where one upper "plank" slides relative to lower "plank". Only problem there is that these planks are always shown to be in full contact along entire length.........and shown to be same length (at least before deflection of the assembly).

John F Mann, PE

http://www.structural101.com

 

[jfmann]

Wrapping this up, at least to a "point".........looks like I owe Doug a beer!

Using only point load reaction at midspan.......I calculate that "perfect" contact occurs only at midspan, down to I-upper-25 (that is I-upper = 25-percent of I-lower).......for which configuration the midspan reaction is (somewhat amazingly) exactly 20-percent of total load on upper beam (wL/2). Also, at midspan, M-lower (moment is lower beam) is very near to being exactly 25-percent of I-lower........which, interestingly, is not the case for greater values of I-upper.

At I-upper-24........small amount of load distribution (along lower beam, near midspan) occurs........and, yes indeed, M-upper = 24-percent M-lower

So there it is (well not entirely, but mostly!).......and I have learned something along the way.

John F Mann, PE

http://www.structural101.com

 

[jfmann]

But wait!.......there's more.......oh geez........as close as I can get to "pure" contact for I-upper-24 (I-upper = 24-percent I-lower).......I end up with the following;

Reaction at midspan = 21.01 percent of total load
Reaction at 0.49 span (that is, one-percent of upper-beam-span to left of midspan)......and at 0.51 span (mirror image) = 0.36 percent of total load (very small)

Reason that some load distribution must be included is that, when total reaction is attempted only at midspan, with otherwise "perfect" contact (with lower beam) at midspan......some "over-deflection" occurs adjacent to midspan, meaning that some contact (and load distribution) must occur with (to) lower beam adjacent to midspan.

However.........although M-upper is 24.01 percent of M-lower at 0.49 span..........M-upper is only 23.59 percent of M-lower at midspan.......and, this 23.59 percent value is very insensitive to changes in "trial" load distribution pattern around midspan (which occurs during "convergence" process of calculation).

Of course (and as indicated by my evolving calculation results throughout) this may be calculation "tolerance". However, it is consistent with previous results showing that, as I-upper is reduced further, M-upper as percentage of M-lower drifts further away from the I-upper to I-lower ratio........such that I am going to reserve complete judgment on the overall solution until I can check out more refined results for lower I-upper values more thoroughly.........AND double-check that end-rotation of upper beam is still "unobstructed" at ends.

John F Mann, PE

http://www.structural101.com

 

[IDS]

John - Thanks for the beer (or did you take it back before I had a chance to drink it? ).

I think something you may be missing is that when the EI of the top beam is low enough for a finite contact length at mid-span there is still a point load transfer at the ends of the contact zone.

In the case of the top beam EI being 24% of the lower the contact length is 0.0505103 m either side of mid-span. For an applied UDL of 20 kPa the contact pressure is 16.129032 kPa (20 x 1/1.24). This gives a contact force of 1.63 kN from the UDL over the full contact zone. The required force for equilibrium is 21.38 kN, so the missing force is supplied by a point force of 9.88 kN at each end of the contact zone. If the top beam EI was increased towards 25% of the lower beam the contact length reduces towards zero, and the point force at the ends increases towards 10.

The point forces at the ends of the contact zone allow all the requirements of equilibrium and compatibility to be met precisely.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[jfmann]

Doug......sounds like you are assuming uniform contact pressure, and then adding point loads (somewhat arbitrarily) to make up difference.

Although I have been "tripped up" on first-pass at logic with this analysis........it appears that distributed load should not be uniform, but should taper off away from peak pressure at midspan.

I submit that supposed "point loads" for I-upper between 24 or 25ish and 41-2/3 percent of I-lower are in fact very short distributed loads. Length of distribution appears to be increasing exponentially.

The only reason that "point load" appears to provide solution, is that length of distribution is so small (for higher values of I-upper in this range) that we do not "see" distribution with precision of accounting we use for values.

Any greatly different distribution does not appear consistent with both beams being in continuous contact, which is apparently driving proportionate load sharing in contact region.

Have to look back to find total length of beam you are using........for I-upper-24, what is contact length you calculate in terms of percentage of total span?

John F Mann, PE

http://www.structural101.com

 

[IDS]

John - yes that is what I am doing, but no, it isn't arbitrary, it is the only way to satisfy both equilibrium and compatibility.

Over the length of the contact zone compatibility requires that the applied load over any length is distributed to the two beams in the ratio of their EI values. If the applied load is uniform, the load transfer between the two beams must also be uniform. This leaves a force to be transferred to maintain equilibrium. It can only be transferred where the beams are in contact, but it cannot be transferred anywhere within the contact zone, so it must be applied at the ends of the contact zone, where the curvatures of the two beams, moving away from the CL, start to differ.

This is only exactly true for our ideal beams. In real beams there will be some local deformation so that "point" loads are distributed to bring the contact stress down below the yield stress. There will also be some transverse strain which this analysis ignores. Curvature of the two beams will be exactly the same at their contact surface, but the curvature of their neutral axes will be slightly different. But this non-linear behaviour is equally true of a point load at mid-span with no contact length.

For my last quoted example I used a 10 m lower beam and 5 m upper, so the total contact length was 2.0204% of the upper beam length. I used a lower beam EI of 10,000 kN.m2, but this does not affect the contact length.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[jfmann]

Doug......I must disagree with your assumed distribution of reaction force..........so then, by the same logic, if the load on upper beam were highly irregular.....say, with completely non-uniform shape......would you say that shape of pressure within contact area must somehow be the same irregular distribution?......of course not.

I have checked reaction force for I-upper values between upper limit of I-upper = 41-2/3 percent of I-lower (first contact)........and I-upper of 25-percent of I-upper, which for some inexplicable reason (for now) is a clear break-point between; (1) Point-only contact (and reaction force) and, (2) Distributed-pressure contact for lower values of I-upper. I have verified this by moving edge of potential distributed contact zone closer and closer and closer to midspan to try and find another point of contact (adjacent to midspan) .....and, amazingly, it does not happen for any values as close to midspan as you care to get. Also, the midspan reaction (applied by lower beam) for I-upper-25 is an exact 20-percent of total load, which also is surprising.......though I suspect that for fixed dimensions of this configuration the "stars" just align perfectly for that relative value of I-upper.

For I-upper less than 25-percent of I-lower.......some length of contact and distributed contact-pressure occurs. It is reasonable that character of this contact pressure occurs in a way that demonstrates a transition consistent with behavior for point-load-only contact ......that is, contact pressure should have large peak pressure at midspan that reduces down to zero at edge of contact area. In fact, that is what I find for I-upper-24.5 and I-upper-23, with results listed below. Of interest is that rate of pressure-reduction (from peak to zero) is not straight-line.......in fact reduction is very rapid, such that pressure most likely conforms to an exponential function (which is not surprising at all).

For I-upper = 24.5 percent of I-lower
Length of contact (centered at midspan) = 2.08 percent of L-upper (which is L-lower /2)
Total reaction force = 20.683 percent of total load (wL/2)
Peak node force at midspan = 99.40 percent of total reaction force
M-upper at edge of contact = 24.72 percent of M-lower
M-upper at midspan = 24.29 percent of M-lower

For I-upper = 23 percent of I-lower
Length of contact (centered at midspan) = 5.50 percent of L-upper (which is L-lower /2)
Total reaction force = 22.77 percent of total load (wL/2)
Peak node force at midspan = 94.30 percent of total reaction force
M-upper at edge of contact = 23.26 percent of M-lower
M-upper at midspan = 22.21 percent of M-lower









John F Mann, PE

http://www.structural101.com

 

[IDS]

..........so then, by the same logic, if the load on upper beam were highly irregular.....say, with completely non-uniform shape......would you say that shape of pressure within contact area must somehow be the same irregular distribution?......of course not.

Why of course not? For an ideal beam (plane sections remain plane, etc) that is exactly what I would expect. For a real beam that is a close approximation to what I would expect.

At the contact points both beams have the same slope and curvature, but a different rate of change of curvature. There must be a point load at this location so that the rate of change of curvature after contact is equal for the two beams. Working the other way, the beams while they are in contact must have equal slope, curvature, and rate of change of curvature. For them to separate their rates of change of curvature must change at a point, so there must be a point load transfer.

For your second example my spreadsheet gives:
Top moment/Bottom moment at both locations: Exactly 0.23 (to machine precision)
Gap at both locations: 0.000% of total deflection (1.09E-14 m at both locations to be exact).
Contact length = 4.083% of upper beam length
Contact pressure = (1/0.23)/(1/0.23 + 1)
Point loads at end of contact zone = WL/2 x 0.19495

Entering your loads into my spreadsheet I get almost the same moments as you (differences in the last S.F.) and a gap of -0.28% and-0.27% at the ends of the contact zone and mid-span (i.e. the top beam is calculated to be below the lower beam).

Similar findings for the first example, but the difference is less because the contact length is less.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[jfmann]

Doug.........I continue to disagree about shape of pressure within contact zone, and doubt there is any sudden concentration of load at edge of contact zone....I have to believe pressure develops similar to "beam on elastic foundation" which has smooth transition.

You seemed to be claiming that shape of load on upper beam should define shape of pressure within contact zone........and I was suggesting that does not make sense if you consider distributed load with highly irregular shape (like up-down roller coaster shape) or even just series of highly unequal point loads. It seems most logical for pressure to develop smoothly as for any spring-type support.

If your model (like mine) requires that assumed shape of pressure distribution be input, then solution may not be completely accurate (of course we are talking minute details here, though the principles at work are of interest).

Do not see how rate of change of curvature for each beam would be equal at edge of contact zone since such rate must be different outside edge of contact zone for beams with different curvatures to merge with the same curvatures. Only way they could have same rate of change for curvature......at edge of contact zone...... would be if they already had same rate of change outside of contact zone, and such rate remained the same within contact zone.

After further refinement of my model.......for I-upper (23-percent)......I get contact length of 5.2-percent of upper beam length (only slightly reduced from previous results). Total reaction force remains the same at 22.77-percent of total load on upper beam. I am using discrete points along beam (close spacing near midspan) with peak reaction force at midspan, tapering away. Solution requires highly concentrated reaction force at midspan to ensure smooth contact along contact zone. There are no gaps. However, total reaction force is remarkably constant within variations of node point locations.

May take this up again during some blizzard, but have to take a break for now........has been fun!.......oh and if you are near Belmar NJ sometime, will be happy to buy you that brew!



John F Mann, PE

http://www.structural101.com

 

[IDS]

May take this up again during some blizzard, but have to take a break for now........has been fun!.......oh and if you are near Belmar NJ sometime, will be happy to buy you that brew!

Likewise if you are ever passing through Sydney.

Not a bad discussion for such a "simple" question!

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/