Question

A spur gear of average radius 2.5 in and bevel gear of average radius 5.0 in are mounted on a shaft of diameter 0.5 in. The shaft is supported at two points using bearings 2. Draw the free body diagram Draw the shear force and bending moment diagram. Calculate the maximum stress in the shaft. At which point the maximum stress occurs. Figure not to scale, all dimensions in inches. E 30e6 psi a. b. c. 500 lb 100 lb 200 lb Bevel Gear Spur Gear 5.012.010.0

Answer 1

(a) In the given problem, the bearings will act as the simple supports to the beam and the equivalent free body diagram is drawn as follows: 200 lb 500 lb ch 100 lb Ru (100x5)=500 lb.in Ry2 25 in 12 in 10 in (b) Calculate the bearing reactions as follows: Note: Horizontal reactions i.e. R and Ry, has nothing to do with shear force diagram and bending moment diagram, therefore, only vertical reactions are calculated here. Consider equilibrium of forces in vertical direction: EF, =0 Ry1 + Ry2 - 200- 500 = 0 R, + R2 = 700 ...... (1) Take moment about B: M,=0 (2005) lb.in -(500x12) lb.in + 500 lb-in+R, X22) = 0 Ry2 = 204.54 lb [from equation (1)] Ryt + R2 = 700 R2 + 204.54= 700 Rri = 495.45 lb

Draw the shear force diagram and bending moment diagram as follows: 200 lb 500 lb 500 lb.in AC 495.45 lb 5 in 12 in 204.54 lb 10 in -200+495.45 = 295.45 lb 295.45 lb SFD -200 lb 295.45-500 = 204.55 lb 2045.4+500 = 2545.4 lb.in 204.54x10= 2045.4 lb.in BMD -200x5 = -1000 lb.in

(c) Refer to the SFD and BMD drawn above. There will be both shear stress and bending stress generated in the shaft. Calculate the maximum shear stress in the shaft as follows: 4 average - 4Prex 3 3A 4 Pret 3xD Here, P is the maximum shear stress in the shaft and A is the area of cross-section. From SFD, P =295.45 lb (In section BC) = 4x 295.45x4 max -= 2006.28 psi = 2.00628 ksi 3x a X0.5 Calculate the maximum bending stress in the shaft as follows: ( max Z ID Here, Z is the section modulus of the shaft. From BMD, M.mx = 2545.4 lb.in. (At point C) - 2545.4x32 = 207417.85 psi=207.417 ksi * AX0.5 Note: Maximum shear stress is negligible in comparison to maximum bending stress and, therefore, can be neglected. Hence the maximum stress in the shaft is 207.417 ksi at point C.