Question 3.20: Compute the maximum stress in a stepped flat plate subjected...

Objective: Compute the maximum stress in the stepped flat plate in Figure 3–32.

Given: The layout from Figure 3–32. Force = F = 9800 N
Using the notation from Figure 3–31(a): widths H = 12.0 mm; h = 9.0 mm
Plate thickness: t = 6.0 mm
Fillet radius at the step: r = 1.50 mm

Analysis: The presence of the change in width at the step causes a stress concentration to occur. The geometry is of the type shown in Figure 3–31(a) and Appendix A18–2 that we will use to find the value of K_t for this problem. That value is used in Equation (3–33) to compute the maximum stress.

\sigma_{\max }=K_{t} \sigma_{\text {nom }} \quad \text { or } \quad \tau_{\max }=K_{t} \tau_{\text {nom }}                    (3.33)

Results: The values of the ratios \mathrm{H} / \mathrm{h} and r / h are required:
\begin{aligned}H / h &=12.0 \mathrm{~mm} / 9.0 \mathrm{~mm}=1.33 \\r / h &=1.5 \mathrm{~mm} / 9.0 \mathrm{~mm}=0.167\end{aligned}
The value of K_{t}=1.83 can be read from Appendix A18-2.
The nominal stress is computed for the small section having a cross section of 6.0 \mathrm{~mm} by 9.0 \mathrm{~mm}.
A_{\text {net }}=h \cdot t=(9.0 \mathrm{~mm})(6.0 \mathrm{~mm})=54.0 \mathrm{~mm}^{2}
The nominal stress is:
\sigma_{\text {nom }}=F / A_{\text {net }}=(9800 \mathrm{~N}) /\left(54.0 \mathrm{~mm}^{2}\right)=181.5 \mathrm{~N} / \mathrm{mm}^{2}=181.5 \mathrm{MPa}
The maximum stress in the area of the fillet at the step is:
\sigma_{\max }=K_{t} \sigma_{\text {nom }}=(1.83)(181.5 \mathrm{MPa})=332 \mathrm{MPa}

Comments: The maximum stress of 332 MPa occurs in the fillet area at both the top and bottom of the small section.
A short distance to the right of the fillet, the local stress reduces to the nominal value of 181.5 MPa. The stress in the larger section is obviously much smaller because of the larger cross-sectional area of the plate. Note that specifying a smaller fillet radius would cause a much larger maximum stress because the curve for K_t increases sharply as the ratio of r/t decreases. Modestly smaller maximum stress would be produced for larger fillet radii.