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Question 3 [27 Marks] A cantilever beam with a length of 15 meters and a circular cross section with a diameter (d) of 250 mm, respectively, carries a transverse end point load (P₁ = 25 kN) and a compressive load (P₂ = 1500 MN) on its free end that is applied through its centroid. (i) Starting from the general equation for bending, derive the equation for the maximum longitudinal direct stresses due to transverse concentrated load and calculate its maximum tensile and compressive values. [12] (ii) Develop an equation for the direct longitudinal stress due to the compressive end-load acting on the beam and calculate its numerical value. [2] (iii) By plotting these stresses on a diagram for the distribution of stress through the depth of the beam, determine the maximum direct stresses induced in the beam. [6] (iv) Use the plotted diagram to determine the location of the neutral axis with reference to the lower and upper surfaces of the beam cross-section. [7]