Chapter 13, Problem 1RP

The post has a circular cross section of radius c. Determine the maximum radius e at which the load P can be applied so that no part of the post experiences a tensile stress. Neglect the weight of the post.

To determine

Find the maximum radius e at which the load P can be applied.

Answer to Problem 1RP

The maximum radius e at which the load P can be applied is $\underset{\_}{\frac{c}{4}}$.

Explanation of Solution

Given information:

The post has a circular cross section of radius c.

Neglect the weight of the post.

Calculation:

Sketch the Free Body Diagram of the applied load P as shown in Figure 1.

Apply the normal stress

${\sigma }_A=0$ (1)

Apply the stress formula as shown below.

${\sigma }_A=\frac{P}{A}+\frac{My}{I}$ (2)

Here, P is the applied load, A is the area of the cross section, M is the moment at the point, y is the centroid, and I is the moment of inertia.

Equate the Equation (1) and (2).

$\frac{P}{A}+\frac{My}{I}=0$ (3)

Determine the area of cross section as shown below.

$A=\pi r^2$

Here, r is the radius of the cross section.

Substitute c for r .

$A=\pi c^2$

Determine the moment of inertia as shown in below:

$I=\frac{\pi }{4}{r}^4$

Substitute c for r .

$I=\frac{\pi }{4}{c}^4$

Sketch the free body diagram of the moment at the point as shown in Figure 2.

Refer to Figure 2.

Find the moment using the calculation.

$M=Pe$

Substitute $-P$ for P, $\pi c^2$ for A, $\frac{\pi }{4}{c}^4$ for I, and Pe for M in Equation (3).

$\begin{array}{l}\frac{-P}{\pi c^2}+\frac{\left(Pe\right)c}{\frac{\pi }{4}{c}^4}=0\\ \frac{P}{\pi }\left(-\frac{1}{c^2}+\frac{4e}{c^3}\right)=0\\ -\frac{1}{c^2}+\frac{4e}{c^3}=0\\ \frac{4e}{c^3}=\frac{1}{c^2}\end{array}$

$e=\frac{c}{4}$

Therefore, the maximum radius e at which the load P can be applied is $\underset{\_}{\frac{c}{4}}$.