Irodov 1.311: The Cost of Curling a Flat Band – Amit & Deepali Sinha Foundation

Problem Irodov 1.311

A flat steel band of length $l = 2.0$ m, width $h = 6.0$ cm and thickness $\delta = 2.0$ mm is bent, while staying within its elastic limit, into a closed circular hoop. Find the work that must be performed to bend the band into this shape.

The finished hoop, radius R, center of curvature C. Every element ds=R dθ of the band (one is shaded) is bent to the same curvature, so it stores the same bending energy per unit length. — The finished hoop, radius $R$ , center of curvature $C$ . Every element $ds=R\,d\theta$ of the band (one is shaded) is bent to the same curvature, so it stores the same bending energy per unit length.

Given

$l = 2.0$ m
$h = 6.0$ cm
$\delta = 2.0$ mm

Find

A

, the work needed to bend the band into a hoop

Solution

Step 1The bending moment along the finished hoop

Bending the band to a circle of radius $R$ gives it constant curvature, $d^2y/dx^2 = 1/R$ , everywhere along its length. By the governing relation $N=EI\,d^2y/dx^2$ stated for these problems, the bending moment is therefore the same at every cross-section, and that constant value is

N = \frac{EI}{R}

(1)

Step 2Energy stored per unit length, and the total

A rod carrying the bending moment $N$ stores an elastic energy per unit length of its own arc that depends only on $N$ and on the stiffness $EI$ of its section.

\frac{dA}{ds} = \frac{N^2}{2EI}

Since the $N$ of (1) is the same everywhere, so is this energy density, and the total stored energy is the density times the band’s own length $l$ , the integral of $ds=R\,d\theta$ once around the ring:

A = \int_0^l \frac{N^2}{2EI}\,ds = \frac{N^2}{2EI}\,l.

Putting the constant moment of (1) into that expression, the stiffness $EI$ cancels once against the square in the numerator.

\begin{align*}A &= \frac{l}{2EI}\left(\frac{EI}{R}\right)^{2} \\ A &= \frac{EIl}{2R^{2}}\end{align*}

A = \frac{EIl}{2R^2}

(2)

Step 3The hoop’s radius, and the section’s moment of inertia

The band’s whole length becomes the hoop’s circumference, so

R = \frac{l}{2\pi}. \tag{3}

The band bends about the axis along its width $h$ , with the thickness $\delta$ in the bending plane, so its cross-section is the same rectangle-in-bending as the square rod of Problem 1.301, with $\delta$ in place of the side length:

I = \frac{h\delta^3}{12}. \tag{4}

Step 4Combining

Substituting the section’s $I$ from (4) and the hoop’s $R$ from (3) into (2), and carrying the cancellation of $l$ through to the end:

\begin{align*}A &= \frac{E}{2}\cdot\frac{h\delta^3}{12}\cdot l\cdot\left(\frac{2\pi}{l}\right)^2 \\ A &= \frac{E h\delta^3 l}{24}\cdot\frac{4\pi^2}{l^2} \\ A &= \frac{\pi^2 E h\delta^3}{6\,l}\end{align*}

Step 5Numbers

Steel: $E=2.0\times10^{11}$ Pa. With $h=0.060$ m, $\delta=2.0\times10^{-3}$ m, $l=2.0$ m:

A = \frac{\pi^2\times0.060\times(2.0\times10^{-3})^3\times2.0\times10^{11}}{6\times2.0} \approx 79\ \text{J}.

A = \frac{\pi^2 h\delta^3 E}{6\,l} \approx 0.08\ \text{kJ}

(5)

Check

$A\propto\delta^3$ : from $A=EIl/(2R^2)$ with $I=h\delta^3/12$ , the thickness enters only through $I$ , linearly in $\delta^3$ , so a band twice as thick needs eight times the work. Equivalently, the bending stress at fixed curvature is $\sigma=E\delta/(2R)$ , so the elastic energy density $\sigma^2/(2E)\propto\delta^2$ , and integrating that density over the band’s volume, itself proportional to $\delta$ , reproduces the same $\delta^3$ . A vanishingly thin band ( $\delta\to0$ ) costs vanishingly little work to curl, which matches the everyday experience of bending foil against bending a bar.

Remark

The total work depends only on the final curvature $1/R$ the band ends up with, not on how gradually or unevenly it got there, because the bending energy $N^2/(2EI)$ is a function of local curvature alone, a state quantity rather than a path quantity. That is precisely what let the energy integral behind (2) skip over the actual bending process and treat every element of the band as if it simply appeared already curved to its final, uniform radius.