Torsion differential equation

[SebasSC]

hey everyone,

i have a channel beam under a linearly varying torsional moment

my question is:

given the torsion differential equation written as:
phi''[x] - [E*Jw/(G*Jt)]phi''''[x] = -[mx/(G*Jt)]

how must it be written for a load like the one in the attached figure?

in other words, what "mx" should i use? or should i divide the beam in half? i've tried several things but the solutions came out "suspicious" every time.

 

[structSU10]

you can use a singularity function to describe it. I think if m(x) = To + ((Tm-To)/(L/2))*x - 2*((Tm-To)/(L/2))*(x-(L/2))*<x-(L/2)>^0 that might do it.

 

[KootK]

I'd study half the beam considering it to be torsionally pinned at x=0 and torsionally fixed at x=L/2. Symmetry is your friend here.

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.

 

[SebasSC]

"I'd study half the beam considering it to be torsionally pinned at x=0 and torsionally fixed at x=L/2."

Why?

the case i'm studying is torsionally fixed at both ends (phi = 0) and in two separate scenarios it can be warping fixed (phi' = 0) and warping free (phi'' = 0)

i've tried to divide it in half considering phi(0)=phi'(0)=0 and phi'(L/2)=0 but the results were not good

 

[KootK]

Why = for the half span model, moment is a continuous function that can be handled relatively easily. If you deal with the beam in its entirety, you'll need a moment function that deals with the discontinuity at L/2. That's mathematically a bit trickier. You'll basically end up having to deal with two functions that get combined at some point. I believe that structSU10's solution is in this vein.

Be careful with torsional fixity. It requires rotation restraint and warping restraint. Regardless of your boundary condition at the real supports, torsional fixity can be assumed at L/2 in a half span model.

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.

 

[SebasSC]

that's what i don't understand. Warping fixity is pretty obvious, but why rotation fixity can be assumed at midlength in a half span model? i've seen that kind of solution somewhere but i didn't understood why that can be done.

 

[KootK]

You've kinda got me wondering about that now too. I'll noodle on it.

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.

 

[jayrod12]

Isn't that because the rotation on each side of the point at L/2 is the same?

But that's just something that quickly popped into my head, I'm no solid mechanics expert.

 

[WARose]

Instead of pouring that kind of time into it......why not make the torque=Tm along the full length and check it that way (there are spreadsheets/programs that can do that in seconds).

That is, unless this is a existing member that you are trying to make work/pass under a new load.

 

[rb1957]

this is almost anti-symmetric ... zero twist at one end, some CW twist in the middle, zero twist at the other end.

it'd react the same way if if were a constant moment (ie zero-some-zero twist) ... the amount of twist is likely to be higher for the tapering torque.

another day in paradise, or is paradise one day closer ?

 

[rb1957]

another way to look at the problem is a set of discrete torques ... as a way to sanity check the distributed loading results.

another day in paradise, or is paradise one day closer ?

 

[SebasSC]

so, this is what i've been trying:

I divided the beam in half and at L/2 applied warping fixity (but free rotation) so now i just have 3 boundary conditions

for the left part:
phi'[x] - [E*Jw/(G*Jt)]phi'''[x] = Tm (2 (1 - b) x/L + b)/(G*Jt), phi[0] = 0, phi'[0] = 0, phi'[L/2] = 0

and for the right part:
phi'[x] - [E*Jw/(G*Jt)]phi'''[x] = -Tm (2 (1 - b) (L - x)/L + b)/(G*Jt), phi[L] = 0, phi'[L] = 0, phi'[L/2] = 0

where b is T0/Tm.

The results now seem plausible but i don't know for sure if they are correct

what do you guys think about this?

 

[structSU10]

Did you try a singularity function? it creates a continuous function out of it. here is a link to its explanation if you didn't get what I wrote:

http://en.wikipedia.org/wiki/Singularity_function

it is a very easy way to accommodate difficult functions. I'm not sure how it shakes out for torsional functions, but for beam functions it works quite well.