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Robin91
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Homework Statement
a) Determine what the thickness should be in a closed tube versus an open tube to have the same twist angle
b) Determine what the thickness should be in a closed tube versus an open tube to have the same max shear stress
G=20GPa
T=50Nm
tr=1mm (for the open tube)
Also see the attachment
Homework Equations
a)
theta=(TL)/(GJ)
For closed:
J=pi/2(Ro^4-Ri^4)
Ro=Outer radius
Ri=inner radius
For open:
J=st^3/3 => J=1/3(D+tr)=1/3*pi*(D+1)
where:
s= circumference of circle (2pi*rm), rm is the radius up to the middle of the bar (between inner and outer, so 1/2D+1/2t)
t=thickness
b)
For closed:
tau=T*Ro/J
For open:
tau=T*t/J
The Attempt at a Solution
I calculated the J for the open and closed tube. However, I get a fourth degree equation, because I tried to substitute Ro=tl+D/2 into the equation to calculate J. After expansion I had terms containing tl^4, tl^3, tl^2 and tl, which resulted in a really long derivation of tl, however, I don't think that is necessary for this assignment. For the shear it was even worse, the resulting equation for tl didn't even fit on my paper.
Do I miss something here?
Thanks in advance,
Robin
P.s. Sorry for the equations written in this way, Latex didn't seem to work, it gave errors (while I was sure I typed it correctly)
Attachments
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