Statically Indeterminate Torsion Members

[aznkid310]

Homework Statement

A solid circular bar ABCD with fixed supports at ends A and D is acted upon by two equal and oppositely directed torques T_0. The torques are applied at points B and C, each of which is located at a distance x from one end of the bar. (The distance x may vary from zero to L/2).

a) For what distance x will the angle of twist at points B and C be a maximum

b) What is the corresponding angle of twist W_max?

Sorry I can't upload a picture, but basically there is a torque at B which is a distance x from end A of the bar, and another torque at C which is also a distance x to the other end D.

Homework Equations

Since it is fixed, then the sum of angles should be zero, correct?

So i would have to sum up the angles as follows: W_ab + W_bc + W_cd = W_ad ?

Also, the sum of torques should be zero.

Do i need to take an integral? If so, how should i approach this?

For each angle, I can make an imaginery cut and use the angle of twist formula, but this is where I am having trouble. Could someone walk me through this?

The Attempt at a Solution

Sum of torques: T_a - T_0 + T_0 - T_d = 0

T_a = T_d
W_ab = [(T_a)x]/GI
W_bc = [(T_a - T_0)]/GI
W_cd = [(T_a - T_0 + T_0)]/GI

where I = (pi/32)(d^4)

I don't believe this is correct

 

[mathmate]

> Since it is fixed, then the sum of angles should be zero, correct?
Correct, you could deduce that from symmetry (or anti-symmetry)

> Also, the sum of torques should be zero.
Correct, you could deduce that from
> ...two equal and oppositely directed torques T_0...If you look further using symmetry, what can you deduce for the angle of rotation at mid-length? Does that simplify your problem?

What can you deduce for the answer to the question:
> a) For what distance x will the angle of twist at points B and C be a maximum

 

[piercas]

. I would appreciate any help with this problem.

I would approach this problem by first understanding the concept of statically indeterminate torsion members. This means that the system cannot be fully analyzed using equilibrium equations alone and additional equations are needed to solve the problem.

To solve this problem, we can use the compatibility equation, which states that the sum of the angles of twist at all points in the bar must be equal to the total angle of twist. This means that W_ab + W_bc + W_cd = W_ad.

Next, we can use the equilibrium equation for torques, which states that the sum of all torques acting on the system must be equal to zero. This means that T_a - T_0 + T_0 - T_d = 0.

Using these two equations, we can solve for the unknown variables, x and W_max. We can also use the angle of twist formula, W = (TL)/GI, where T is the applied torque, L is the length of the bar, G is the shear modulus, and I is the polar moment of inertia.

To answer the questions posed in the problem, we need to find the value of x that will result in the maximum angle of twist, W_max. This can be done by taking the derivative of W with respect to x and setting it equal to zero. Solving for x will give us the value of x that will result in W_max.

In summary, to solve this problem we need to use the compatibility equation, the equilibrium equation for torques, and the angle of twist formula. It may also be helpful to draw a free body diagram and label all known and unknown variables.