Question

3- The shaft shown in the figure rotates with constant angular velocity and is transmitting a torque from gear B to gear C through the forces Exo and Ez as shown. The shaft is subject to combined bending and torsion due to the forces shown. The weights of the shaft and pulleys may be neglected and the supports can exert only concentrated force reactions. The radius of the gear at B is 50 mm and that at C is 75 mm. The force Ex= 1 kilo Newton (kN). You are required to: i. Determine the position of the most heavily stressed point in that section. Neglect shear stresses due to transverse shear forces compared to shear stresses due to twisting moment. ii. Determine the stress components at the most heavily stressed point in that section, and hence determine the maximum shear stress at this point in terms of the shaft diameter "d”. iii. If the shaft is made of steel Tal = 80 MPa, determine a suitable shaft diameter to avoid failure based maximum shear and maximum distortion energy theories. (45%) 350 mm D 400 mm 200 mm B 75 mm 50 mm

Answer 1

Given rB = 50mm, rc = 75mm, Ey = I KN As the Gear B and Gear c wtates at same angular speed, Torque Torquec Ey x ro Fzx rc Fzx75 B 1x50 F₂ = 0.667 KN Torque Transmitted by shaft, T = Fyx dp = 1x50 - 50 kN-mm = 50x102 N-mm x-y plane | Ray Roy В c A ka 2004 - Fy Roy: 400 ile 350 EMA=0: 200 fy= 950 Rpy 200x1 = 0.21 kN 950 0.79 KN 158 kN-mm Ray = Fy - Roy 73,5 KNam + Mez Ray x 200 = 0.79x200 - 158 K N-mm Ma Royx350 = 0.21x350 = 73.5 kN-mm X-Z plane F =0.667 KN EMA=O: 450 Roz = 600 fz Roz= 600x 0.667 = 0.421 KN 950 RAZ= Fz - RO2= 0.667-0.421 0.246 KN A B. D +200-14 400 с >k 350 RAL 'Roz Mej RD4 * 350 - 0.421 x 350 = 147.35 KN-mm MBy – Raz X200 - 0.246x200 = 49.2 kN-mm + Resultant Bending moment Mo = √ MBy²+M B 2² √ 49.2² + 158² 165.48 KN-mm Critical section is B MC: Muyt + M c2² = √147, 35² + 73.5² 164.66 KN-mm

Point (ان | T = 800000 Td 3 Bis Critical: 16T 16 x 50 x103 nd3 32MB 32 x 165.48 x103 td nd3 Tc3 06: 5295360 Tid3 ii) Max Shear stress Theory: Tmax = √ (b)+22 tid V( 5295360)2 + 800000 80- 2765901.18 3 X Tid d=22,243 mm Shaft cliometer Distortion Energy Theory: Tum = 10b²+ 372 5295366 + 3x8000002 ad3 Cum 5473649.34 ad 3 Tmax : 0.577 Ovin 80 - 0.577 x 5473649.37 ad3 da 23,249 mm