Force couple acting on rigid frame
Force couple acting on rigid frame
(OP)
How would I go about solving for the moment(s) at the base where this structure is fixed? From looking at examples of frames, a few assumptions are typically made to reduce the indeterminacy of the problem. For this case, can I assume the reactions acting opposite to the applied 1430 forces are also 1430? The problem I run into with this is that, if I assume that the horizontal member has a point of inflection at the midway point, how do I resolve the internal forces when "cutting it in half" and doing an force equilibrium analysis to find the moment at the base? If I already have assumed the applied force and the reaction are equal, then any internal force in the beam would have to be zero in a force balance, in which case your moment is just 1430*25.5. I don't think that is correct, the horizontal member connecting the two should 'stiffen' and lessen the bending moment logically. What am I assuming incorrectly? Wouldn't there also be a torsional component as well?






RE: Force couple acting on rigid frame
Unless you have a 3D frame package, I would just do 1430 lbs x 25.5 ft .
RE: Force couple acting on rigid frame
RE: Force couple acting on rigid frame
RE: Force couple acting on rigid frame
I would argue that XR's solution basically is the analytical solution. Unless your columns are monstrous HSS, the response of the frame should be very accurately represented by 1430 lbs x 25.5 ft. From a global frame torsion perspective, flexure in the the columns is analogous to warping torsion and torque in the columns is analogous to St.Venant. As usual, warping dominates for open sections.
As with all problems, you can always drill deeper analytically. There's no need here for practical design purposes however.
For the loads shown, I don't think that the horizontal member would stiffen the system much really. That said, it may have some other benefits from a stability perspective etc.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Force couple acting on rigid frame
You'd need to do a 3D stiffness matrix, having 6 DOFs at each of the two nodes. By symmetry you could do 6 DOFs.
I'm assuming you know how to get shear, flexural, and axial stiffnesses.
Torsional stiffness is GJ/L if torsionally pinned (warping is unrestrained).
If warping is restrained, torsional stiffness is:
et1 and et2 are numbers from 0 to 1, 0 being a 100% warping unrestrained end connection and 1 being 100% restrained.
lambda is dimensionless: Length * sqrt(GJ/ECw)
RE: Force couple acting on rigid frame
But, as KootK said, accounting for this is not going to change much unless the columns have huge torsional stiffnesses. The solution will show you that the bending moment is almost exactly 1430 lbs * 25'6" at the column bases.
RE: Force couple acting on rigid frame
Ay and By should be very small, and will probably induce small fixed end moments about the z-axis)
there will also be another fixed end moment, along the zed axis (driven by the applied x-load) ... moments at a and b will be equal and opposite.
The problem (in my mind) with applied torsion couples is that they induce equal and opposite reactions at other places in the structure that you can't see from a simple FBD ... the simple reaction to your applied load is Ax = Bx = 1430 lbs ... but then you ask yourself "what is happening in the vertical members ? they look (taken as individuals) like cantilever beams ...
another day in paradise, or is paradise one day closer ?
RE: Force couple acting on rigid frame
1. -All members W10x30
Column:
Moment at top = 11.96 k-in
Moment at base = 402 k-in
Torsion = 7.28 k-in
Shear = 1.35 kip
Beam:
End Moment = 7.28 k-in
Torsion = 11.96 k-in
Shear = 0.08 kip
2. - All members HSS6x6x3/8
Column:
Moment at top = 73.1 k-in
Moment at base = 143 k-in
Torsion = 69.5 k-in
Shear = 0.71 kip
Beam:
End Moment = 69.5 k-in
Torsion = 73.1 k-in
Shear = 0.72 kip
As expected, the torsional stiffness plays a large part for closed sections, and is negligible for open sections.
RE: Force couple acting on rigid frame
RE: Force couple acting on rigid frame
Just did some matrix operations with Excel.
Here's what I get Using HSS2.5x2.5x.25 and dimensions in inches...
Column:
Moment at top = 6.09 k-in
Moment at base = 11.6 k-in
Torsion = 5.87 k-in
Shear = 0.70 kip
Beam:
End Moment = 5.86 k-in
Torsion = 6.09 k-in
Shear = 0.73 kip
RE: Force couple acting on rigid frame
"W10x30" is an I-beam, yes? ... it would matter which axes was bending in the "beam", strong or weak axes.
the center of the beam (and the foot of the vertical) remain fixed in location (by symmetry), so the lateral deflection of the beam matches the lateral deflection of the vertical post. so you could solve it with a very large beam, and a wimpy little post ... but it doesn't sound like a very robust solution ...
another day in paradise, or is paradise one day closer ?
RE: Force couple acting on rigid frame
RE: Force couple acting on rigid frame
another day in paradise, or is paradise one day closer ?
RE: Force couple acting on rigid frame
with the lifting fork adjusted all the way to the end, it generates a force couple on the carriage (tubed structure we've discussed here), and I'm trying to discern the stress at which point it attaches to the vertical column.
RE: Force couple acting on rigid frame
1. The torsional stiffness of the beam allows the column to behave more like a cantilever with it's free end restrained against rotation.
If the beam had no torsional stiffness, the column would act as a normal cantilevered beam.
2. The torsional stiffness of the columns allows the beam to act more like a beam with end rotations fixed when it's ends are being translated in opposite transverse directions.
If the columns had no torsional stiffness the beam would be a straight line, but it has reverse curvature when there is torsional stiffness.
RE: Force couple acting on rigid frame
1) in isolation I see the beam working like a, ummm, beam ... bending under the applied load, cantilevered at the CL ... I won't call that torsion on the beam. but my question was why would one shape be more effective than the other in reacting shear ?
my question 2) was more about how do you orient the W section in the beam ? is the applied load bending about the strong axes or weak ?
we agree, if the beam was a "noodle", the the vertical posts would behave like cantilevers.
another day in paradise, or is paradise one day closer ?
RE: Force couple acting on rigid frame
RE: Force couple acting on rigid frame
It's not the beam's weak axis flexural stiffness that causes the shear to be dumped into the beam, it's the two points I explained above. Let me try to be more clear.
1. First imagine the beam has high torsional stiffness (i.e. the tip of the cantilevered column resists rotation).
As the cantilever column deflects, the tip rotates, inducing torsion on the beam. Because of the beams torsional stiffness, it attracts more load and the cantilevered column attracts less load as well as developing a reverse moment at it's tip.
2. Imagine the column has no torsional stiffness. As the beam ends translate transversely, the column freely twists and the beam stays in a straight line.
Now add torsional stiffness to the column, and it resists the twisting, which puts the beam into an S bend reverse curvature.
It's not that the beam is more effective in shear, it's that the torsional resistance of both the beam and the column stiffens up the joints.
RE: Force couple acting on rigid frame
RE: Force couple acting on rigid frame
the big difference (I see) between your runs is the shear carried by the beam (instead of being reacted at the ground) ... this is flexure of the beam. And so the interconnection with torsion ... a torsionally strong beam allows the post to develop more top end moment, less ground shear reaction. similarly a torsionally strong post allows the beam react more fixed end moment, to behave more like a cantilever.
I don't know your sections well so I asked why would one section be so much stiffer in bending, and recongising the I beam has a strong and a weak axes, then I asked which way round is it in your run.
another day in paradise, or is paradise one day closer ?
RE: Force couple acting on rigid frame
I changed the load to 1 kip and all the member lengths to 10".
Everything normal, the beam takes 32% of the shear.
Now, leaving everything else the same, if I change the following:
Column torsional stiffness = ∞: Beam takes 82% of shear.
Beam flexural stiffness = ∞: Beam takes 34% of shear.
Column torsional stiffness = 0, Beam takes 0 shear.
Beam flexural stiffness = 0, Beam takes 0 shear.
Column torsional stiffness = ∞ and beam flexural stiffness = ∞, Beam takes 100% of shear.
The flexural strength has some part in attracting the shear, but it is primarily the torsional stiffness.
RE: Force couple acting on rigid frame
a flexurally rigid up-right would react all the shear (zero beam shear),
a torsionally rigid beam would ... ?
a flexurally weak up-right would make the beam react all the shear, (may well collapse),
a torsionally weak beam would ... ?
another day in paradise, or is paradise one day closer ?