= Lamé stress distribution in a pressurized cylindrical wall
{c}
{title2=$\tau_r=P-Q/r^2,\quad\tau_\phi=P+Q/r^2$}
For inner and outer radii $a,b$, with internal and external <pressures> $p_1,p_0$, the boundary tractions are $\tau_r(a)=-p_1$ and $\tau_r(b)=-p_0$. The <radial equilibrium of a thick elastic cylinder> gives $P=(p_1a^2-p_0b^2)/(b^2-a^2)$ and $Q=(p_1-p_0)a^2b^2/(b^2-a^2)$. The radial compressive pressure is $p(r)=-P+Q/r^2$. Since hoop stress is monotone in $r$ when $Q\neq0$, its maximum absolute value is the larger of $|P+Q/a^2|$ and $|P+Q/b^2|$. With negligible external pressure, its maximum is the tensile inner-wall value $p_1(a^2+b^2)/(b^2-a^2)$. The stress distribution itself is independent of elastic moduli in this linear homogeneous axisymmetric model; its associated displacement depends on them.
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