A long circular shaft of length L with variable cross-section is fixed onto rigid walls as shown in Figure 1.1. The polar moment of inertia of the L/3-long segment (J1) with a larger cross section is three times larger than that of a 2L/3-long segment (J2) on the right. Assume the shear modulu is constant and equal to G for the entire shaft and that strains are small and linear elastic. L/3 a b L/3 L/3 Figure 1.1. Shaft with variable cross-section and external torque T. (a) If the shaft is subjected to an external torque T at point b, find the angle of twist of point b with respect to point a. (b) If the shaft is subjected to an external torque T at point a, as shown in Figure 1.2, find the angle of twist of point a with respect to point b. J2 L/3- L/3 L/3

Values are m= 360mm n= 375mm q= 24mm r= 18mm P= 0.6kN

Analyse the statically determinate bar illustrated below by expressing the loading as a single function using Macaulay brackets and the Dirac delta, integrating to find th axial force and integrating again to find the displacements, applying the boundary conditions appropriately. Find the axial force in the bar at point A and the displaceme at point B. The cross section of the bar is constant with EA = 18000 kN. a = 4 m, b = 2 m, c = 2 m and d = 4 m. w1 = 12 kN/m, w2 = 17 kN/m,, P1 = 12 kN and P2 = 19kN. a W1 L/2 W2 Multiple Choice Answers Multiple Choice Answer: Axial force at point A (kN, tension positive): a. 3.31 b. 31.97 c. 37.33 d. 31 Multiple Choice Answer: Displacement at point B (mm, positive to right): a. 0.0061 b. 0.0395 c. 0.0193 d. 0.0261 Axial force at point A (kN, tension positive): Displacement at point B (mm, positive to right): L/2 P1 P2 (type in your multiple choice answer, e.g. a, b, c or d) (type in your multiple choice answer, e.g. a, b, c or d)