Stress in a Thick Pipe

Pressurize a thick pipe and its wall stretches unevenly: most at the inside surface. The equation for that stretch has a special symmetry, so powers of the radius solve it. That is why thick pipes fail from the inside.

How it works

Euler equations

In one sentence

An Euler equation $ax^2y'' + bxy' + cy = 0$ has solutions $y = x^m$ with $am(m - 1) + bm + c = 0$ ; distinct real roots $m_1, m_2$ give $y = c_1x^{m_1} + c_2x^{m_2}$ for $x > 0$ .

General formula

{\htmlData{sym=1}{r}}^2{\htmlData{sym=0}{u}}'' + {\htmlData{sym=1}{r}}\,{\htmlData{sym=0}{u}}' - {\htmlData{sym=0}{u}} = 0 \;\Rightarrow\; {\htmlData{sym=0}{u}}({\htmlData{sym=1}{r}}) = {\htmlData{sym=2}{A}}\,{\htmlData{sym=1}{r}} + \frac{\htmlData{sym=3}{{B}}}{\htmlData{sym=1}{{r}}}

Every term scales the same way when $r$ doubles, so powers of $r$ keep their shape: plug one in and the equation shrinks to a quadratic for the power.

What the symbols mean

$u$
Radial displacement
How far each layer of the wall moves outward.
$r$
Radius
Distance from the pipe's center.
$A$
Stretching part
Displacement that grows with r.
$B$
Inner-pressure part
Displacement that grows toward the inner wall.

The wall's radial displacement $u(r)$ satisfies an Euler equation: each derivative comes with a matching power of $r$ .
Try $u = r^m$ : then $m(m - 1) + m - 1 = 0$ , so $m = \pm 1$ .
The general solution is $u = Ar + B/r$ ; the inner and outer pressures fix $A$ and $B$ .
The $B/r$ part is why stress concentrates at the inner wall: it grows as $r$ shrinks.

Learn the math

Euler EquationsDiff Eq · Lesson 38, Series Solutions