Calculating Angle of Twist and Maximum Shear Stress in a Torsional Load

[aznkid310]

Homework Statement

Unfortunately, I don't have a picture to upload, so I'll describe it the best that I can.

A prismatic bar AB of length L and solid circular cross section (diameter d) is loaded by a distributed torque of constant intensity t per unit distance. Determine the angle of twist W between the ends of the bar.

Homework Equations

The Attempt at a Solution

d(Torque) = tdx --> Torque T = integral (from 0 to L) [tdx] = tL

W = int(0 to L) [T(x)dx/GI(x)] , where G = shear modulus, I = polar moment of inertia

Is my T(x) equal to T = tL ?

How do I find I? I realize that I = int over the area [x^2 dA], where x is the distance from the center to dA

Can i just use the formula I = [(pi)r^4]/4 for a solid bar?

 

[Pyrrhus]

if the bar is prismatic it means its cross section is constant along its length.

For this case a differential twist angle will be

$$d \phi = \frac{T(x)dx}{GI_{polar}}$$

Is my T(x) equal to T = tL ?

No, T = f(x). It's not constant for this case.

How do I find I? I realize that I = int over the area [x^2 dA], where x is the distance from the center to dA

Yes, but that will just make your work longer.

Can i just use the formula I = [(pi)r^4]/4 for a solid bar?

That formula is wrong. That's the area moment of inertia of the section. Look up the polar moment of inertia.

 

[aznkid310]

Sorry i typed it in wrong. It should be I = (pi/2)(r^4)

Could you get me started on finding T(x)?

 

[Pyrrhus]

Could you get me started on finding T(x)?

What do you understand by "... a distributed torque of constant intensity t per unit distance..." ?

 

[aznkid310]

that the torque varies linearly with distance?

 

[Pyrrhus]

that the torque varies linearly with distance?

so T(x) = ?

 

[aznkid310]

T(x) = tx?


W = (t/GI)int(0 to L) [ xdx]

= (tL^2)/2GI

= [16tL^2]/[(pi)Gd^4]

 

[diyana]

hi...how do i get the angle of twist (in radian) if the question give the revolutions number?

for more understanding, here is the question :

the ship at the surface,A has just started to drill for oil on the ocean floor at a depth of 1500m. knowing that the top of the 200-mm-diameter steel drill pipe (G=77.2GPa) rotates two complete revolutions before the drill bit at the bottom,B starts to operate, determine the maximum shear stress caused in the pipe by torsion.

i know that angle of twisT,@ = TL/JG where J= (pi/2)(c^4) ...c=outermost radius.

and when i got the T (torsion), i can use the equation ; shear stress = Tc/J.

am i correct here? i just got confuse how do i get the T..