Discussion
A bracket is attached to a vertical column by means of two identical rivets U and V separated by a distance of \(2a\) = 100 mm, as shown in the figure. The permissible shear stress of the rivet material is 50 MPa. If a load \(P\) = 10 kN is applied at an eccentricity \(e = 3\sqrt{7}\,a\), the minimum cross-sectional area of each of the rivets to avoid failure is ___________ mm2.
Answer: A
Each rivet needs at least 800 mm2.
The rivets are one above the other, a distance \(a = 50\) mm from their centroid. The load gives each rivet a direct shear \(P/2 = 5\) kN downwards, and the moment \(Pe\) gives each a secondary shear at right angles to the line of rivets, that is horizontal:
\[ \begin{aligned} F_{s} &= \frac{Pe\,a}{2a^{2}} = \frac{P e}{2a} = \frac{10 \times 3\sqrt{7}}{2} \\ &= 15\sqrt{7} \approx 39.7\ \text{kN} \end{aligned} \]
The two components are perpendicular, so each rivet carries
\[ \begin{aligned} F &= \sqrt{5^{2} + (15\sqrt{7})^{2}} = \sqrt{25 + 1575} = 40\ \text{kN} \\ A &= \frac{40000}{50} = 800\ \text{mm}^{2} \end{aligned} \]
200 mm2 ignores the moment, \(100\sqrt{7}\) and 25 mm2 do not follow from combining the two shears. Adding the components directly would apply only if the rivets were side by side.
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