Hi everybody! I really need help with this problem as soon as possible...
SI SHAFT WITH VARIABLE SECTION
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) = R0/SQUARE R {[1 + (x/2L)]}
A concentrated torque of magnitude Q is applied at the midsection of the bar C (x = /L2) 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[Nám], and R0[m] (enter this as R_0).
#1:
Obtain a symbolic expression for the redundant reaction Tx(D) in terms of any of the known quantities listed above.
Tx(D) = ?
#2
Obtain a symbolic expression for the maximum magnitude of shear stress in the shaft τ max in terms of any of the known quantities listed above.
T max = ?
To solve the problem, we need to apply the Force Method for SI structures. The Force Method involves the following steps:
1. Determine the redundants: In this problem, the redundant support is at point D. The redundant reaction at point D is denoted as Tx(D).
2. Obtain the governing differential equation: In this case, we need to derive the differential equation by equating the internal moment to the applied torque at the midsection of the bar C. This differential equation relates Tx(D) to the applied torque Q.
3. Solve the differential equation: Once we have the differential equation, we can solve it to obtain the symbolic expression for Tx(D).
Now, let's proceed with solving the problem step by step.
#1: Obtaining a symbolic expression for the redundant reaction Tx(D):
Step 1: First, we need to derive the differential equation governing the system. The internal moment M(x) at any section x within the bar can be calculated using the formula:
M(x) = G * A(x) * φ(x)
where A(x) is the area of the cross-section at x, and φ(x) is the derivative of the displacement function.
For a round cross section, the area A(x) is given by:
A(x) = π * R(x)^2
where R(x) is the radius of the cross-section at x.
Step 2: Now, we can calculate the derivative of the displacement function φ(x) with respect to x. Since we have a concentrated torque at the midsection of the bar C (x = L/2), we can use the displacement function φ(x) = 0 for x ≠ L/2, and φ(x) ≠ 0 for x = L/2.
Step 3: The applied torque Q at point C creates a Bending Moment at the same point C. In this case, the Bending Moment M(x) is given by M(x) = Q.
Step 4: Equate the internal moment equation and the bending moment equation to obtain the governing differential equation:
G * A(x) * φ(x) = Q
Step 5: Substitute the expressions for A(x) and φ(x) into the differential equation:
G * π * R(x)^2 * φ(x) = Q
Step 6: Substitute the expression for R(x) = R0/SQUARE R [1 + (x/2L)] into the differential equation:
G * π * (R0/SQUARE R [1 + (x/2L)])^2 * φ(x) = Q
Step 7: Solve for φ(x):
φ(x) = Q / [G * π * (R0/SQUARE R [1 + (x/2L)])^2]
Step 8: The redundant reaction Tx(D) can be obtained by integrating φ(x) from x = 0 to x = L:
Tx(D) = ∫[0,L] φ(x) dx
Now you need to perform the integration from 0 to L and substitute the values of Q, G, R0, SQUARE R, and L in the equation to obtain the symbolic expression for Tx(D).
#2: Obtaining a symbolic expression for the maximum magnitude of shear stress in the shaft τmax:
Once we have the expression for Tx(D), we can use it to calculate the maximum magnitude of shear stress in the shaft. The shear stress τ(x) at any section x within the bar can be calculated using the formula:
τ(x) = (Tx(D) / A(x))
where A(x) is the area of the cross-section at x.
The maximum magnitude of shear stress τmax occurs at the section with the minimum area, which corresponds to the point D (x = L).
Substitute the expression for Tx(D) and the expression for A(x) into the equation τ(x), and evaluate τ(x) at x = L to obtain the symbolic expression for τmax.
Remember to substitute the values of Tx(D), G, R0, SQUARE R, and L into the equation to obtain the symbolic expression for τmax.
I hope this helps you get started with solving the problem! Let me know if you need any further assistance.