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AISC Design Guide-9: How to interpolate rotation functions for other concent. mom. positons 1

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jochav52802

jochav52802

Structural
Nov 28, 2018
81
Good Morning,

Can anyone explain how I use the AISC Design Guide-9 diagrams in Appendix-C to determine the rotation and derivative functions for concentrated torsions applied at locations other than 0.1, 0.3 and 0.5 from the member end? The Design Guide doesn't appear to explain this anywhere and I have moments at 0.38 and 0.05 from my member end for instance.

I'd appreciate any help with this!

Best regards,

jochav52802
 
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Are you looking at Case 3 or Case 6, the general form rotation equations are at appendix section C.4 you need to differentiate them with respect to z, d/dz, 3 times to get the other needed equations.

units for my equations below:
T - Kip*in
G - Kip/in^2
J - in^4
l - in

Here are the equations I have for Case 3:
Python: 
        Theta:
        if 0 <= z_in <= (alpha*l):
            theta = ((T*l) / (G*J))*(((1.0-alpha)*(z/l))+(((a/l)*((sinh((alpha*l)/a)/tanh(l/a)) - cosh((alpha*l)/a)))*sinh(z/a)))
        else:
            theta = ((T*l) / (G*J))*(((l-z)*(alpha/l))+((a/l)*(((sinh((alpha*l)/a) / tanh(l/a))*sinh(z/a)) -(sinh((alpha*l)/a)*m.cosh(z/a)))))
            
        Theta':
        if 0 <= z_in <= (alpha*l):
            theta_prime = ((-1.0*T)/(G*J))*(alpha - 1.0 + (cosh(z/a)*((cosh((l*alpha)/a) - (sinh((l*alpha)/a)/tanh(l/a))))))
        else:
            theta_prime = -1.0*((T*(alpha - m.cosh(z/a)*sinh((l*alpha)/a)/tanh(l/a) + sinh(z/a)*sinh((l*alpha)/a))/(G*J)))
       
        Theta'':
        if 0 <= z_in <= (alpha*l):
            theta_doubleprime = -((T*sinh(z/a)*(cosh((l*alpha)/a) - sinh((l*alpha)/a)/tanh(l/a)))/(a*G*J))
        else:
            theta_doubleprime = (T*(-1.0*m.cosh(z/a) + sinh(z/a)/tanh(l/a))*sinh((l*alpha)/a))/(a*G*J)
       
        Theta''':
        if 0 <= z_in <= (alpha*l):
            theta_tripleprime = -((T*m.cosh(z/a)*(m.cosh((l*alpha)/a) - sinh((l*alpha)/a)/tanh(l/a)))/(G*J*a**2))
        else:
            theta_tripleprime = (T*(m.cosh(z/a)/tanh(l/a) - sinh(z/a))*sinh((l*alpha)/a))/(G*J*a**2)

Case 6: (double check this one my memory is failing me if I fully tested this one)
Python: 
H = (((1.0-cosh((alpha*l)/a)) / tanh(l/a)) + ((cosh((alpha*l)/a)-1.0) / sinh(l/a)) + sinh((alpha*l)/a) - ((alpha*l)/a)) / (((cosh(l/a)+(cosh((alpha*l)/a)*cosh(l/a))-cosh((alpha*l)/a)-1.0)/sinh(l/a))+((l/a)*(alpha-1.0))-sinh((alpha*l)/a))
	
theta:
if 0 <= z_in <= (alpha*l):
	thet = ((T*a) / ((H+1.0)*G*J))*((((H*((1.0/sinh(l/a))+sinh((alpha*l)/a)-(cosh((alpha*l)/a)/tanh(l/a))))+(sinh((alpha*l)/a) - (cosh((alpha*l)/a)/tanh(l/a)) + (1.0/tanh(l/a))))* (cosh(z/a)-1.0)) - sinh(z/a) + (z/a))
else:
	thet = (T*a / ((1+(1/H))*G*J))*((cosh((alpha*l)/a) - 1.0/(H*sinh(l/a)) +(cosh((alpha*l)/a)-cosh(l/a)+((l/a)*sinh(l/a))) / sinh(l/a)) +cosh(z/a)*((1.0-cosh((alpha*l)/a))/(H*tanh(l/a)) +(1.0-(cosh((alpha*l)/a)*cosh(l/a)))/sinh(l/a)) +sinh(z/a)*(((cosh((alpha*l)/a)-1.0)/H) + cosh((alpha*l)/a)) -(z/a))

theta':
if 0 <= z_in <= (alpha*l):
	theta_prime = (-1.0*T*(-1.0 + cosh(z/a) + (-1.0 + (1.0 + H)*cosh((l*alpha)/a))*(sinh(z/a)/tanh(l/a)) - 1.0*H*(sinh(z/a)/sinh(l/a)) - 1.0*sinh(z/a)*sinh((l*alpha)/a) - 1.0*H*sinh(z/a)*sinh((l*alpha)/a)))/(G*(1.0 + H)*J)
else:
	theta_prime = 0

theta'':
if 0 <= z_in <= (alpha*l):
	theta_doubleprime = -((T*((cosh(z/a)*(-1.0 + (1.0 + H)*(cosh((l*alpha)/a))/tanh(l/a)))/a - (H*(cosh(z/a)/sinh(l/a)))/a + sinh(z/a)/a - (cosh(z/a)*sinh((l*alpha)/a))/a - (H*cosh(z/a)*sinh((l*alpha)/a))/a))/(G*(1.0 + H)*J))
else:
	theta_doubleprime = 0
   
theta''':
if 0 <= z_in <= (alpha*l):
	theta_tripleprime = -((T*(cosh(z/a)/a**2 + ((-1.0 + (1.0 + H)*(cosh((l*alpha)/a))/tanh(l/a))*sinh(z/a))/a**2 - (H*(sinh(z/a)/sinh(l/a)))/a**2 - (sinh(z/a)*sinh((l*alpha)/a))/a**2 - (H*sinh(z/a)*sinh((l*alpha)/a))/a**2))/(G*(1.0 + H)*J))
else:
	theta_tripleprime = 0

Open Source Structural Applications:
 
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Much appreciated Celt83!

Best regards,

jochav52802
 
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Jochav52802:
Years ago, Bethlehem Steel had a great publication, “Torsion Analysis of Rolled Steel Sections,” on the subject of Torsional Analysis, it was the bible at the time. Although I don’t have the later editions of this material or the AISC Design Guides, I believe that AISC basically took this publication over at some point in time, and I suspect it is part of their current Design Guide on the subject (Design Guide 9?). The format and equations that StructSU10 attached are essentially the same as those in my 1963 ed. It also had a good bibliography, for that time, for some deeper reading on the original thinking and development of this material; all the great old names and titles which no one, today, even seems to know ever existed. I can remember doing some of this same development years ago, and as for a better understanding of the basic subject, I don’t think Google will never replace touching and reading some of these authors and pages.
 
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Thanks a lot everyone,

I have to brush up on my differentiation skills to verify Celt83's equations. To make things simpler, I just clustered portions of a the equations that were composed of multiple constants as a single constant, which made differentiation much easier to see. All I needed to know was the derivative of a constant in front of z and the derivative of the hyperbolic trig functions as a function of z.

Man, now I know why we should just avoid torsion from the outset; what a pain this is!

Best regards,

jochav52802
 
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Hello All,

One more question, the Case-3 chart with Alpha set to 0.5 indicates that at midspan, theta' should be zero. When I take the first derivative of theta and plug in the z-value corresponding to the midspan of the beam, I don't get a theta' value equal to zero. This would seem to suggest that my theta' equation is wrong, but my theta'' and theta''' equations are correct as they match the stresses from example 5.1 of Design Guide-9.

I'd appreciate any advice here.

Thank you,

jochav52802
 
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Hello Celt83,

Would you mind clarifying what the "m" represents in your theta''' equation for case-3?

Additionally, would you explain why you multiplied your derivative formulas by -1? I'd imagine that it was for some sort of convenience, but am not seeing it.

Thank you,

jochav52802
 
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Take a look at the resource StructSU10 posted it has the derivatives in reduced form.

That m. Your are seeing was left over from my python code you can ignore it. Been awhile since I did those derivatives but I think one of the hyperbolics results in sign inverse when you do the derivative.

Open Source Structural Applications:
 
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Thank you Celt83,

I found an error in my excel formula for the Theta', so now everything's working. Thanks for the clarification on the "m" and just for you information, the derivative for cosh or sinh don't produce a negative sign, so I think you must have used the negative for convenience.


 
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