Question 11.4: A brass bar AB projecting from the side of a large machine i...
A brass bar AB projecting from the side of a large machine is loaded at end B by a force P = 7 kN acting with an eccentricity e =11mm (Fig. 11-26). The bar has a rectangular cross section with height h = 30mm and width b =15mm.
What is the longest permissible length L_{max} of the bar if the deflection at the end is limited to 3 mm? (For the brass, use E =110GPa.)
Use a four-step problem-solving approach.
1. Conceptualize:
Critical load: This bar is a slender column that is fixed at end A and free at end B. Therefore, the critical load (see Fig. 11-20b) is
P_{ cr }=\frac{\pi^{2} E I}{4 L^{2}} (a)
The moment of inertia for the axis about which bending occurs is
I=\frac{h b^{3}}{12}=\frac{(30 mm )(15 mm )^{3}}{12}=8.44 \times 10^{3} mm ^{4}Therefore, the expression for the critical load becomes
P_{ cr }=\frac{\pi^{2}(110 GPa )\left(8.44 \times 10^{3} mm ^{4}\right)}{4 L^{2}}=\frac{2.29 kN \cdot m ^{2}}{L^{2}} (b)
in which P_{cr} has units of kN and L has units of meters.
2. Categorize:
Deflection: The deflection at the end of the bar is given by Eq. (11-62), which applies to a fixed-free column as well as a pinned-end column:
\delta=e\left[\sec \left(\frac{\pi}{2} \sqrt{\frac{P}{P_{ cr }}}\right)-1\right] (c)
In this equation, P_{cr} is given by Eq. (a).
3. Analyze:
Length: To find the maximum permissible length of the bar, substitute for δ its limiting value of 3 mm. Also, substitute e = 11mm and P = 7 kN, and substitute for P_{cr} from Eq. (b). Thus,
3 mm =(11 mm )\left[\sec \left(\frac{\pi}{2} \sqrt{\frac{7 kN }{2.29 / L^{2}}}\right)-1\right]The only unknown in this equation is the length L (meters). To solve for L, perform the various arithmetic operations in the equation and then rearrange the terms. The result is
0.2727 = sec (2.746L) – 1
Use radians and solve this equation to get L = 0.243m. Thus, the maximum permissible length of the bar is
L_{\max }=0.243 m4. Finalize: If a longer bar is used, the deflection will exceed the allowable value of 3 mm.
