Question 6.10: Review Example Analysis 9. Verify that in the special cases ......
Review Example Analysis 9. Verify that in the special cases when the inclination angle θ of the bending axis has the values zero degrees (0°) and 30 (30°), that the analysis of Example Analysis 9 is consistent with Example Analysis 4 (θ = 0°) and Example Analysis 8 (θ = 30°), respectively.
Consider again the distances a, b, and c from the centers of bolts A, B, and C to the bending axis as in Figure a. In Eq. (e) of Example Analysis 9, we see that a, b, and c are expressed as:
\alpha =R\sin \left[\left\lgroup\frac{\pi }{3}-\theta \right\rgroup \right], b =R\sin \left[\left\lgroup\frac{2\pi }{3}-\theta \right\rgroup \right], c=R\sin\left[\pi -\theta \right] (a)
where θ is the inclination angle for the bending axis as shown in Figure a. By inspection of Figure a we see that when θ = 0° (as in Example Analysis 4) a, b, and c are:
\alpha =\left\lgroup\frac{\sqrt{3} }{2} \right\rgroup R,b =\left\lgroup\frac{\sqrt{3} }{2} \right\rgroup R, and c =0 (b)
Similarly, when θ = 30° (as in Example Analysis 8),a, b, and c are seen to be:
\alpha =\left\lgroup\frac{1 }{2} \right\rgroup R, b=R, c =\left\lgroup\frac{1 }{2} \right\rgroup R (c)
If we assign θ to be 0° in Eq. (a), a, b, and c are:
\alpha =R\sin\left\lgroup\frac{\pi }{3} \right\rgroup= R\sin 60^{\circ }=\frac{\sqrt{3} }{2R} \\ b=R\sin\left\lgroup\frac{2\pi }{3} \right\rgroup= R\sin 120^{\circ }=\frac{\sqrt{3} }{2R}\\ c= R\sin \pi =R\sin 120^{\circ }=0 (d)
These results are identical to the values in Eq. (b). If we assign θ to be 30° (or π/6 radians) in Eq. (a), a, b, and c are:
These results are identical to the values in Eq. (c). Finally, by reflective symmetry we see in Figure a that the distances, say d, e, and f, from the bolt centers to the bending axis are equal to a, b, and c, respectively. Therefore, the foregoing analysis is similarly applicable for bolts D, E, and F.
