Question

A vertical load P is applied at the center A of the upper section of a homogeneous
frustum of a circular cone of height h, minimum radius a, and maximum radius b.
Denoting by E the modulus of elasticity of the material and neglecting the effect
of its weight, determine the deflection of point A.
1) neglecting the effect of its weight only due to p
2) consider the effect of its weight (p+w)

Answer 1

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$\boxtimes$ 501?

find the deflection of paint "A"

Deflection of paint "A" is

Nothing but overall

deflection of the member (s)

For varying cross-section

Deflection.

$$V_A = s = \int_0^h \frac{P_n dn}{A_n E_n} = \int_0^h \frac{P_y dy}{A_y E_y}$$ $$\therefore A_y = \pi (c D^2) = \pi (c D^1 + D^1 D^2)^2$$

"D'D" is to be evaluated

"A_y" is the Area at

particular distance

y from reference

From the similar triangles.

$$\frac{b - a}{h} = \frac{D D^1}{h - y}$$ $$D D^1 = \frac{(b - a)}{h} (h - y)$$ $$\begin{array}{l} \therefore A_y = \pi \left[ a + \frac{(b - a)}{h} (h - y) \right]^2 \\ A_y = \pi \left(b + \frac{(a - b)}{h} y\right)^2 \\ v_y = \delta = \int_0^h \frac{\rho \, dy}{\pi E \left(b + \frac{(a - b)}{h} y\right)^2} \\ v_A = \frac{h}{(a - b)} \cdot \frac{p}{\pi E} \left[ \frac{-1}{\left[ b + \frac{(a - b)}{h} y \right]^0} \right] \\ = \frac{\rho h}{\pi E (a - b)} \left[ -\frac{1}{a} + \frac{1}{b} \right] \\ = \frac{\rho h}{\pi E (a - b)} \left[ \frac{-b + a}{a b} \right] \\ \end{array}$$ $$\boxed{V_A = \frac{\rho h}{\pi E a b}}$$

Deblession of point "A" $\boxed{V_A = \frac{\rho h}{\pi E a b}}$

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