A 12 foot pole rigidly fixed to the ground is loaded with a 300 lb horizontal force at its top. The bottom half of the pole is hollow and the half is solid, as shown in the diagram on the left.

What is the maximum bending stress at any location in the pole?

The maximum bending stress is a direct function of the maximum bending moment and the member cross section, as given by,

     σ = My / I

Since there are two different cross sections, both will need to be checked.

First, the moment distribution from the bottom to the top of the pole needs to be determined so that the maximum can be identified. The pole can be considered as a simple cantilevered beam which makes finding the moment diagram simple. If a cut is made at an arbitrary location, x, then the moment at that location is

     M_aa = (300 lb) (12 - x ft)(12 in/ft)
            = 43.2 - 3.6x  kip-in

The maximum in the bottom section is 43.2 kip-in at the ground. The maximum for the top section is 21.6 kip-in at the mid-point joint.

The moment of inertia for both sections are,

     I_top = πd⁴/64 = π2⁴/32 = 0.7854 in⁴

     I_bottom = π/64 (3⁴ - 2.5⁴) = 2.059 in⁴

The stress in both sections are

     σ_top-max = My / I = 21.6(1)/0.7854
                 = 27.50 ksi

      σ_bottom-max = My / I = 43.2(1.5)/2.059
                       = 31.47 ksi

The maximum stress occurs in the bottom section.

      σ_max = 31.47 MPa