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Chapter 10

Q. 10.6

A thin-walled hollow tube AB of conical geometry with constant wall thickness t and average diameters d_a and d_b as shown in Figure 10.23 is subjected to a pure torsion, T. Calculate the strain energy due to torsion of the tube. Assume material is Hookean.

10.23

Step-by-Step

Verified Solution

For the thin-walled tube, let us calculate polar moment of inertia as follows:

J=\left\lgroup \frac{\pi}{32} \right\rgroup\left[(d+t)^4-(d-t)^4\right]

=\frac{\pi}{32} d^4\left[\left\lgroup 1+\frac{t}{d} \right\rgroup^4-\left\lgroup 1-\frac{t}{d} \right\rgroup^4\right]

=\frac{\pi}{32} d^4\left\{\left[1+4\left\lgroup \frac{t}{d} \right\rgroup +\cdots\right]-\left[1-4\left\lgroup \frac{t}{d} \right\rgroup +\cdots\right]\right\}

Neglecting higher powers of (t/d ) as t<<d , we get

J = \frac{\pi}{32} d^4\left\lgroup 8 \frac{t}{d} \right\rgroup

J=\frac{\pi}{4} d^3 t                 (1)

Let us place our coordinate x as shown in Figure 10.23. If d be the mean diameter at a distance x from end A, then

d=d_{ a }+\left\lgroup \frac{d_{ b }-d_{ a }}{L} \right\rgroup x              (2)

If we consider a differential portion of the hollow tube, we can consider its strain energy dU from Eq. (10.20) as

U_{\text {torsion }}=\frac{T^2 L}{2 G J}                (10.20)

d U=\frac{T^2 d x}{2 G J(x)}

The total strain energy of the tube is

U=\int_0^L \frac{T^2 d x}{2 G J(x)}=\left\lgroup \frac{T^2}{2 G} \right\rgroup \int_0^L \frac{4 d x}{\pi d ^3 t}

=\left(\frac{2 T^2}{\pi G t}\right) \int_0^L \frac{ d x}{\left[d_{ a }+\left\lgroup \frac{d_{ b }-d_{ a }}{L} \right\rgroup x\right]^3}

Therefore,          U=\left\lgroup \frac{2 T^2}{\pi G t} \right\rgroup \frac{1}{2}\left\lgroup \frac{L}{d_{ b }-d_{ a }} \right\rgroup\left[-\frac{1}{\left\{d_{ a }+\left\lgroup \frac{d_{ b }-d_{ a }}{L} \right\rgroup x\right\}^2}\right]_0^L

=\frac{T^2}{\pi G t} \cdot \frac{L}{\left(d_{ b }-d_{ a }\right)}\left[-\frac{1}{d_{ b }^2}+\frac{1}{d_{ a }^2}\right]=\frac{T^2}{\pi G t} \frac{L}{\left(d_{ b }-d_{ a }\right)} \frac{d_{ b }^2-d_{ a }^2}{d_{ a }^2 d_{ b }^2}

or            U=\frac{T^2 L}{\pi G t} \frac{d_{ b }+d_{ a }}{d_{ a }^2 d_{ b }^2}