Torsion - AISC Design Guide 9 Formulas for Graphs? 3

[DAVE9999]

I am working on a Mathcad file for calculating torsional stresses. Looking at AISC Design Guide 9, is there a way to calculate the values in the graphs in Appendix B instead of looking at the graph to come up with the values?

Thank you

 

[JoshPlumSE]

The Steel Construction Institute produces a similar guide titled Design of Members Subject to Combined Bending and Torsion. It has an appendix with all the solutions to the differential equations.

If you want to do it by hand, then I believe the solution will always have the form:

Phi = Phi_h + Phi_p

Phi = twist angle

Phi_h = homogeneous solution to differential equation
Phi_h = A*Sinh(Lambda*z)+B*Cosh(Lambda*z) + C

Phi_p = Particular solution
Phi_p = C1 + C2*z+C3*Z^2 + ...

You need to set the values of the constants base on the known conditions (i.e. when z = 0, z = L... sometimes even z = 0.5*L).

 

[271828]

See App. C.4, Page 110.

 

[271828]

BTW, Josh is correct also.

Beware that the non-homogeneous equation has t(z) on one side. This is the *internal* torque, not the applied torque as indicated in DG9 Eq. 3.2, etc.

It takes a while to solve for the particular solution. I solved one of these manualy a couple of weeks ago for a case outside DG9 and it took about 5 pages by the time I correctly solved for Phi_p. Took me about 3 tries before I made it through without a goofy mistake. Not easy for someone who doesn't solve ODEs like this on a routine basis. Selection of Phi_p is also not trivial because some problems require what they call the "modification rule" to the method of undetermined coeff.

 

[miecz]

Appendix C has the formulas for [Θ]. The note at the top of page 110 says to take derivatives to get [Θ]', [Θ]'' and [Θ]'''. See the attached sheet for Case 4.

 

[DAVE9999]

Thank you for the help! I will give it a shot and if it doesn't work I'll resort to looking up the values from the charts

 

[DAVE9999]

Still working on this, does anyone have the formulas for the derivatives of ? to get ?', ?'' and ?''' for Case 3 (pin-pin, concentrated torque) ?

 

[miecz]

I posted the solution yesterday, but my posting was deleted from the thread. Attached is a solution to case 3 for a concentrated torque. No one has checked or verified this solution, so be cautious.

As a side note, someone out there is deleting my postings. This has happened at least a half dozen times. It is very disconcerting to have a posting deleted without ever learning why. It would be helpful if, when someone red flags a post, that the reason got passed on.

 

[DAVE9999]

That is strange why your posts are deleted. I agree, if someone is deleting it, they should explain why, that is poor communication!

Thanks for posting the solution. I agree with the solution but when z=0, I get ?'=0 (on paper and in mathcad) and that doesn't match up with what you show on page 11: a flange stress due to St.venant torsion of 4.53ksi (so ?' must not be zero) as well as in the AISC Design guide example.

What am I doing wrong to come up ?'=0 when z=0?

 

[miecz]

Can you post your formula for [Θ]'? From the graphs for case 3, [Θ]' shouldn't be zero.

 

[DAVE9999]

i couldn't get the file upload to work, i have a pdf of my mathcad file and when i hit upload it wouldn't work

i was getting the same formula you had for ?'and I agree, ?' shouldn't be zero from the graphs

your formula for ?' looks like it would be zero when z=0, what do you get for ?' when z=0?

 

[HeeeyDave]

Heeeeeeey Dave-
I'd be glad to help you.
Post your formula

 

[miecz]

Here are some intermediate values...

To Upload a file
Click "or Uploadyour File..."
Click "Browse"
Highlight your pdf and click "Open"
Click "Upload File"
Click "Click Here..."

 

[DAVE9999]

I get the 'page cannot be found' window when i try uploading but i got it working now (thanks HeeeyDave), i was mixing up location of the torque and the location of the stresses

thank you (again) for your help, hopefully i will have this working now

 

[HeeeyDave]

HEEEeeeeyy Dave!!!
I think you are using ? = 0, and I do not think that is correct buddy !!!

 

[WillisV]

Sorry to resurrect an old thread, but the solutions to the derivatives for all cases were provided in an Engineering Journal article in 2002 that can be downloaded here:

http://www.aisc.org/assets/0/544/546/670/59ce12b7-6d14-4861-89bf-ae2b7d4a12bc.pdf