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4rf aE

i—y

4R^g E

Si tdy

NB: F, f, and g are negative since towards the z-axis negative.

NB: F, f, and g are negative since towards the z-axis negative.

From the general equation (1.105) and the data in Table 1.9, we can derive the distribution t (y) when the bar is with null or finite thickness at its end y = i. When the bar is of finite thickness we refer to the truncated bar class. Its length is given as the same length i as that of an untruncated bar.

Denoting t0 the thickness at y = 0, and ti the thickness at y = i, we briefly list hereafter the cantilever geometries resulting from integration. 1. Equal constraint flexure of a rod and concentrate force F: We obtain t2 dy — 32G = 0. The solution

' = '0 {> — [> — ©ir — — = —H (1.106a)

is a truncated cubical parabola whose vertex is outside the rod, at yv = i/(1 — t\/t0). If the rod is with a null thickness at its end (ti = 0) the result, first derived by Clebsch [34], is a cubical parabola, term that he introduced because of the even symmetry.

2. Equal constraint flexure of a beam and concentrate force F: We obtain tjy - aF =

is a truncated parabola (Fig. 1.45). If tf = 0, then t2 = tg (1 - y/l) is a parabola. 3. Parabolic flexure of a rod and concentrate force F: We obtain t3 jy - = 0-The solution t = t0 1 -

0 0

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