Question 3.18: Compute the maximum stress in a round bar subjected to an ax...

Objective     Compute the maximum stress in the stepped bar shown in Figure 3-26.
Given            The layout from Figure 3-26. Force = F = 9800 N.
The shaft has two diameters joined by a fillet with a radius of 1.5 mm.
Larger diameter = D = 12 mm; smaller diameter = d = 10 mm.
Analysis         The presence of the change in diameter at the step causes a stress concentration to occur. The general situation is a round bar subjected to an axial tensile load. We will u.se the top graph of Figure A15-1 to determine the stress concentration factor. That value is used in Equation (3-27) to determine the maximum stress.

\sigma _{max}=K_t\sigma _{nom} or \tau _{max} =K_t\tau _{nom}                   (3-27)

Results          Figure A15-1 indicates that the nominal stress is computed for the smaller of the two diameters of the bar. The stress concentration factor depends on the ratio of the two diameters and the ratio of the fillet radius to the smaller diameter.

D/d = 12 \ mm/10 \ mm = 1.20
r/d = 1.5 \ mm/10 \ mm = 0.15

From these values, we can find that K_t = 1.60. The stress is

σ_{nom}= F/A = (9800N)/[π(10 \ mm)^2/4] = 124.8 MPa
σ_{max}= K_tσ_{nom} = (1.60)(124.8 \ MPa) = 199.6 MPa

Comments        The maximum tensile stress of 199.6 MPa occurs in the fillet near the smaller diameter. This value is 1.60 times higher than the nominal stress that occurs in the 10-mm-diameter shaft. To the left of the shoulder, the stress reduces dramatically as the effect of the stress concentration diminishes and becau.se the area is larger.