The shaft BCD in the figure has length L and is composed of a homogeneous material with shear modulus G. The bar is fixed between walls at B (x=0) and D (x=L). The shaft has a round cross section, with outer radius changing along the length of the bar as defined by the function:
R(x)=R01+x2L.
A concentrated torque of magnitude Q is applied at the midsection of the bar C (x=L/2) as indicated. Use the Force Method for SI structures taking D as the redundant support to solve this problem.
The given KNOWN quantities are L[m], G[Pa], Q[N·m], and R0[m] (enter this as R_0).
1) Obtain a symbolic expression for the redundant reaction TxD in terms of any of the known quantities listed above.
TxD=
3) Obtain a symbolic expression for the maximum magnitude of shear stress in the shaft τmax in terms of any of the known quantities listed above.
τmax=
unanswered
At what value of x does this maximum shear stress occur?
Exactly at x=0
At a position in the range 0<x<L/3
Immediately to the left of L/2 (x=L/2−)
Immediately to the right of L/2 (x=L/2+)
At a position in the range 2L/3<x<L
Exactly at x=L
The maximum shear stress in the shaft occurs at a position in the range 0 < x < L/3.
To determine the position at which the maximum shear stress occurs, we need to analyze the cross-sections along the length of the shaft.
Given that the outer radius of the shaft changes along the length according to the function R(x) = R0 + (x^2/L), where R0 is the initial outer radius and x is the position along the length of the shaft.
We know that the formula for shear stress in a circular shaft is given by the equation:
τ = (T * r) / (J * max(R, r))
Where:
τ is the shear stress
T is the applied torque
r is the radial distance from the center of the shaft to the point where shear stress is being calculated
J is the polar moment of inertia of the shaft cross-section
R is the outer radius of the shaft
To find the maximum shear stress, we need to find the position at which the radial distance r is maximum.
Let's analyze the possibilities:
1) Exactly at x=0:
At x=0, the outer radius is R0, and the radial distance is also R0. However, since the torque is applied at x=L/2, this option is incorrect.
2) At a position in the range 0 < x < L/3:
In this range, the outer radius is increasing, but the torque is applied at x=L/2. So, the radial distance will be greater when x < L/2, which means the maximum shear stress will occur at a position greater than L/2. Therefore, this option is incorrect.
3) Immediately to the left of L/2 (x = L/2-):
Since the torque is applied exactly at x=L/2, the maximum shear stress will occur immediately to the right of L/2. Therefore, this option is incorrect.
4) Immediately to the right of L/2 (x = L/2+):
Since the torque is applied exactly at x=L/2, the maximum shear stress will occur immediately to the right of L/2. Therefore, this option is correct.
5) At a position in the range 2L/3 < x < L:
In this range, the outer radius is increasing, but the torque is applied at x=L/2. So, the radial distance will be greater when x > L/2, which means the maximum shear stress will occur at a position less than L/2. Therefore, this option is incorrect.
6) Exactly at x=L:
At x=L, the outer radius is R0 + 1, and the radial distance is R0 + 1. However, since the torque is applied at x=L/2, this option is incorrect.
Therefore, the correct answer is: immediately to the right of L/2 (x=L/2+).