Lamé stress distribution in a pressurized cylindrical wall (${\tau }_r=P-Q/r^2,{\tau }_{\varphi }=P+Q/r^2$)

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 $P=(p_1{a}^2-p_0{b}^2)/(b^2-a^2)$ and $Q=(p_1-p_0)a^2{b}^2/(b^2-a^2)$. The radial compressive pressure is $p(r)=-P+Q/r^2$. Since hoop stress is monotone in $r$ when $Q\ne 0$, 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.