huangyhg

double cantilevered beam equations
i am looking for some help in designing a double cantilevered beam. the member i am inquiring about is the beam on a "t" shaped structure and an "offset t" structure. the structure is designed as a cantilevered column with a beam on top. the beam on the "t" shaped structure has spans equal in distances on each side of the column. the beam on the "offset t" has one span significantly longer on one side of the column than the other. in both cases the connection of the beam to the column is designed to carry the moment at the location and the moment is then added to the moment in the cantilevered column.
with all of that said i have always designed the beam as two independent cantilevered beams with uniformly distributed loads utilizing m = w*l^2/2 and deflection = (w*l^4)/(8*e*i).
i am beginning to believe that utilizing those equations yields a conservative solution. while i am ok with a conservative solution i am one who wants to ensure i am designing as tight as possible to be as economical as possible.
as such i am starting to believe that the moment and the deflection equations could be lowered due the beam being a single member the moments should transfer through the member and start counter acting/reducing each other. similarly the deflection on one side should start counteracting/reducing the deflection on the other. i believe that as justification you can use many items as an argument. the first is using common sense and look at a see-saw or other type structures and you can see the counter acting affects. the second is a simple span beam with an overhang past one support and other multi span members where loads from other spans create reductions in adjacent spans.
i am sure you can put it in some finite element software and get an answer, but i feel you still should know what is governing the solution and why you get the answer you do. i also understand the fact that i need to take into consideration unbalanced loading, but that is not my main concern.
what i am looking for is a moment and a deflection equation for this condition. anyone have these equations and a source? i believe the equations would look something like the equations found in case 6 of table 3-23 of the 13th edition of the aisc steel construction manual; however adjusted for the actual beam support condition.
also i am open to thoughts on my position of the moments and the deflections being reduced. anyone have a legitimate reason why i should not consider them as counter acting?

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i think that for a t-condition, the beam deflection could exceed that of a fixed cantilever. while i agree that one cantilever in theory "counteracts," as you put it, the other - you need to consider any unbalanced loads. in the worse case there is full dead, live and possibly even wind on one side, and little to no load or maybe even additive wind on the other cantilever. the support will rotate due to an unbalacnced condition, increasing the deflection. the net unbalanced moment is taken into the column. in a perfectly balanced case, there would, in theory, be no moment into the top of column.
the way you have calculated the moments in the beams is correct on each side. it just depends on the length of the cantilevers. and the deflection calculation should be correct for balanced loading with equal cantilevers, as that condition is the same as a fixed cantilever, and no moment is transferred to the column. for unequal cantilevers, the deflection is more complicated, as it involves sidesway and depends on the relative stiffness of the cantilevered beams and column.
the effect of the unbalanced load and moment at the top of the cantilevered column can have a significant impact on the design of the column and the anchor rod anchorage.
dik
i would say that if you consider them as counteracting, then the typical fixed end cant will give you the right answer.
once you consider the unbalanced loading, then it will give you unconservative results. the reason is that you need to account for the lack of rigidity of the column. if you load one side and not the other, you're cranking an unbalanced moment into the column, which will cause rotation of the column (adding to your cantilever vertical deflection) and lateral translation of the column (which may or may not be an issue for your condition). additionally, this lateral translation will add to your second order effects for the column.
i would also like to clarify that the statement in my first paragraph is based on an ideal condition, where the column is erected perfectly plumb and has no initial crookedness, and the beams are exactly equal in length with no initial sweep, and the load is applied exactly through the web of both beams. once you start deviating from that then all of your initial assumptions go out the window. this could be critical for this condition as just a small out-of-plumb dimension could have a big impact on the column.
must keep unbalanced load conditions in mind when design the beam-column connection, the column, and footing.
if the structure is made of concrete, the cantilever would deflect more than the calculated amount due to cracks.
if you are holding pipe then empty vs fill conditions may unbalance the balanced case. more information is needed regarding the load. also, consider seisimc and wind (may have been mentioned above)
the statics are not difficult. each cantilever has a moment of mb = wl^2/2. the column has a maximum moment of mc = mb1(total) - mb2(dead).
rotation at the top of column, rc = mc*h/(e*ic).
deflection of either beam = w*l^4/(8*e*ib) + rc*l.
to find maximum deflection of either beam, you must consider unbalanced loading.
ba
i guess let me clarify my question some. i understand i have to keep in mind unbalanced loading and the rotation of the column. i am not looking to know the overall deflection at the end of the beam due to the framing system.
what i am looking for is the deflection just of the beam due to the loading of the beam. meaning i have items attached to the beam that only allow a certain deflection. as such i need to ensure the bema in not deflecting more than that amount, due to the vertical loading.
i know that due to the unbalanced loading will create the maximum deflection. however my unbalanced loading still results in loading that has one side fully loaded and the other with around 50% of the other side's loads. as such i still feel that if you were to look at a beam that has no rotation at its fixed connection that the beam would have some counter acting affects due to be loaded on each side (just as in a multi span beam). basically this is a multi-span beam just with a single fixed connection at mid span. i can then later add in the affects of the rotation of the connection point and any column deflection as well, (but again i am interested only in the moment and the deflection of the beam before all of that occurs).

the entire structure is statically determinate, so i guess i am not seeing the problem you are having. if both beams are of equal length and have equal load, the column moment will be zero. both beams will have a fixed end moment of wl/2 where w = wl and the deflection at any point of either beam will be the same as a beam with fixed end.
if one beam has more load or greater length or both, the column will rotate in the direction of that beam. the fixed end of the other beam will rotate upward the same amount, so its deflection will be reduced by rc*x where rc is the rotation of the column and x is the distance along the cantilever.
ba
ba:
i think you are helping his hw. he needs to review "structural analysis on static determinate structures". also, he didn't point out what is the material the t is made of, or he wanted to design a double cantilever such as boom on tower crane, or else.

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