Chapter 11, Problem 1RP

The cantilevered beam has a circular cross section. If it supports a force P at its end, determine its radius y as a function of x so that it is subjected to a constant maximum bending stress σ_allow throughout its length.

To determine

The radius y as a function of x.

Answer to Problem 1RP

The radius y as a function of x is $\underset{\_}{{\left[\frac{4P}{\pi {\sigma }_{\text{allow}}}x\right]}^{\frac{1}{3}}}$

Explanation of Solution

Given information:

The force is P.

Calculation:

Sketch the free body diagram of cantilever beam as shown in Figure 1:

Let, M is the moment acting cantilever beam and V is the shear force.

Consider the length is x.

Refer to Figure 1:

Calculate the shear force as follows:

$V=wx$

Calculate the moment as shown below:

$\begin{array}{c}M=wx\times \frac{x}{2}\\ =\frac{w{x}^2}{2}\end{array}$

Sketch the calculated values as shown in Figure 2.

Write the section properties as follows:

Calculate the moment of inertia (I) as shown in below:

$I=\frac{\pi }{4}{c}^4$ (1)

Here, c is the radius of section.

Substitute y for c in Equation (1).

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

Find the value of section modulus S as shown in below:

$S=\frac{I}{c}$ (2)

Here, I is the moment of inertia and c is the centroid of section.

Substitute $\frac{\pi }{4}{y}^4$ for I and y for c in Equation (2).

$\begin{array}{c}S=\frac{\frac{\pi }{4}{y}^4}{y}\\ =\frac{\pi }{4}{y}^3\end{array}$

Calculate the allowable bending stress $\left({\sigma }_{\text{allow}}\right)$ as a function of x can be obtained by using flexural formula.

$\sigma =\frac{M}{S}$ (3)

Here, M is the moment.

Substitute $Px$ for M and $\frac{\pi }{4}{y}^3$ for S in Equation (3).

$\begin{array}{c}{\sigma }_{\text{allow}}=\frac{Px}{\frac{\pi }{4}{y}^3}\\ \frac{\pi }{4}{y}^3\left({\sigma }_{\text{allow}}\right)=Px\\ y={\left[\frac{4P}{\pi {\sigma }_{\text{allow}}}x\right]}^{\frac{1}{3}}\end{array}$

Hence, the radius y as a function of x is $\underset{\_}{{\left[\frac{4P}{\pi {\sigma }_{\text{allow}}}x\right]}^{\frac{1}{3}}}$.