Question 4.CA.7: An open-link chain is obtained by bending low-carbon steel r......
An open-link chain is obtained by bending low-carbon steel rods of 0.5-in. diameter into the shape shown (Fig. 4.43a). Knowing that the chain carries a load of 160 lb, determine (a) the largest tensile and compressive stresses in the straight portion of a link, (b) the distance between the centroidal and the neutral axis of a cross section.
a. Largest Tensile and Compressive Stresses. The internal forces in the cross section are equivalent to a centric force P and a bending couple M (Fig. 4.43b) of magnitudes
P = 160 lb
M = Pd = (160 lb)(0.65 in.) = 104 lb·in.
The corresponding stress distributions are shown in Fig. 4.43c and d. The distribution due to the centric force P is uniform and equal to \sigma_0=P / A . We have
\begin{aligned} & A=\pi c^2=\pi(0.25 \text { in. })^2=0.1963 in ^2 \\ & \sigma_0=\frac{P}{A}=\frac{160 lb }{0.1963 in ^2}=815 psi \end{aligned}
The distribution due to the bending couple M is linear with a maximum stress \sigma_m=M c / I . We write
\begin{aligned} I & =\frac{1}{4} \pi c^4=\frac{1}{4} \pi(0.25 in .)^4=3.068 \times 10^{-3} in ^4 \\ \sigma_m & =\frac{M c}{I}=\frac{(104 lb \cdot in .)(0.25 in .)}{3.068 \times 10^{-3} in ^4}=8475 psi \end{aligned}
Superposing the two distributions, we obtain the stress distribution corresponding to the given eccentric loading (Fig. 4.43e). The largest tensile and compressive stresses in the section are found to be, respectively,
\begin{aligned} & \sigma_t=\sigma_0+\sigma_m=815+8475=9290 psi \\ & \sigma_c=\sigma_0-\sigma_m=815-8475=-7660 psi \end{aligned}
b. Distance Between Centroidal and Neutral Axes. The distance y_0 from the centroidal to the neutral axis of the section is obtained by setting \sigma_x=0 in Eq. (4.50) and solving for y_0 :
\sigma_x=\frac{P}{A}-\frac{M y}{I} (4.50)
\begin{aligned} 0 & =\frac{P}{A}-\frac{M y_0}{I} \\ y_0 & =\left(\frac{P}{A}\right)\left(\frac{I}{M}\right)=(815 psi ) \frac{3.068 \times 10^{-3} in ^4}{104 lb \cdot in .} \\ & =0.0240 in. \end{aligned}
