A very long thick-walled cylinder pipe is subjected to an internal pressure and a steady state temperature distribution with T_{i} and T_{o} temperatures at inner and outer surfaces respectively. Calculate the stress distribution in the cylinder.

Problem Type: Axisymmetric problem of thermo-structural coupling.

Geometry:

Given: Dimensions R1 = 1 cm, R2 = 2 cm;
Inner surface temperature T_{i} = 100 °C;
Outer surface temperature T_{o} = 0 °C;
Coefficient of thermal expansion α = 10^{ -6} 1/K;
Internal pressure P = 10^{6} N/m^{2};
Young's modulus E = 3·10^{11} N/m^{2};
Poisson's ratio ν = 0.3.

Problem: Calculate the stress distribution in the pipe.

Solution: Since none of physical quantities varies along z-axis, a thin slice of the cylinder can be modeled. The axial length of the model is arbitrarily chosen to be 0.2 cm. Axial displacement is set equal to zero at the side edges of the model to reflect the infinite length of the cylinder.

Results:

Temperature distribution in a cylinder pipe:

Stress distribution
in a cylinder pipe:

Radial and circumferential stress at r=1.2875 cm:

σ_{r} (N/m^{2})

σ_{q} (N/m^{2})

Theory

-3.9834·10^{6}

-5.9247·10^{6}

QuickField

-3.959·10^{6}

-5.924·10^{6}

See the Coupl2HT.pbm and Coupl2SA.pbm problems are for the corresponding heat transfer and structural analysis.

* Reference: S. P. Timoshenko and Goodier, Theory of Elasticity, McGraw-Hill Book Co., N.Y., 1961, pp. 448-449.