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 σallow throughout its length.

The radius y as a function of x.
Answer to Problem 11.1RP
The radius y as a function of x is
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:
Calculate the moment as shown below:
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:
Here, c is the radius of section.
Substitute y for c in Equation (1).
Find the value of section modulus S as shown in below:
Here, I is the moment of inertia and c is the centroid of section.
Substitute
Calculate the allowable bending stress
Here, M is the moment.
Substitute
Hence, the radius y as a function of x is
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