What are we working on?

What would pressure and heat do to this tube?

Predict stresses in a closed-end pressure tube. Separately compare heating a sliding tube with heating one whose axial ends cannot move. Before inspecting any FEA contour, establish the expected loads, signs and scale.

Review the physics before the answer.

All values are synthetic assumptions. Constant isotropic elasticity; uniform temperature; transverse surfaces traction-free in the thermal cases. Outside gauge pressure stays zero.

Apply is explicit. Draft values do not alter the working record until applied. Any applied edit removes the previous assumption review and invalidates mismatched retained FE evidence.

Derive, then substitute.

Compatibility and equilibrium determine the pressure field. Pressure tractions determine A and B. Closed-end thrust sets axial stress. Thermal boundary conditions decide whether expansion or compression results.

\epsilon_r=u_r',\quad\epsilon_\theta=\frac{u_r}{r},\quad\epsilon_z=\mathrm{const},\qquad \frac{d\sigma_r}{dr}+\frac{\sigma_r-\sigma_\theta}{r}=0

Long isotropic elastic tube, away from end effects. Tension is positive.

A=\frac{p_i a^2-p_o b^2}{b^2-a^2},\quad B=\frac{a^2b^2(p_i-p_o)}{b^2-a^2},\qquad\sigma_r=A-\frac{B}{r^2},\quad\sigma_\theta=A+\frac{B}{r^2}

Pressure-only Lamé field; radial tractions are minus gauge pressure.

\sigma_z=A,\qquad u_r(r)=\frac{(1-2\nu)Ar+(1+\nu)B/r}{E}

Closed ends with pressure thrust, not open ends or imposed plane strain.

\sigma_z=E\left(\frac{du}{dx}-\alpha\Delta T\right),\quad u_{\mathrm{free}}(L)=\alpha\Delta T L,\quad\sigma_{\mathrm{fixed}}=-E\alpha\Delta T

Uniform temperature; transverse surfaces traction-free. Cases are not combined with pressure.

Challenge the calculated field.

Test both pressure tractions, end-force equilibrium, radial differential equilibrium and the thermal limiting cases. These algebraic checks can expose implementation errors. They do not validate real material or model-form assumptions.

Identity-check tolerance: 10⁻¹⁰, for floating-point consistency only—not a universal engineering acceptance limit. The published Python verification also contains nine executable tests.

Compare with a different calculation—not with a confident answer.

The retained Python calculation assembles and solves a radial FE stiffness matrix plus a constant axial-strain degree of freedom. It is an educational implementation, not ANSYS or CalculiX.

\mathbf K=\sum_e\int_e\mathbf B^T\mathbf D\mathbf B\,2\pi rL\,dr,\qquad\mathbf K\mathbf q=\mathbf f

Elements	Stress L₂ error	Outer displacement error
4	1.3332%	0.012724%
8	0.6668%	0.003182%
16	0.3334%	0.000796%
32	0.1667%	0.000199%
64	0.0834%	0.000050%

Separate free/fixed thermal FE cases agree to floating-point precision. They are not combined-load exchanger qualification.

The retained numerical values only apply to the exact reference inputs. Changed inputs require a new FE calculation; this page does not silently reuse the old PASS.

A bounded conclusion. An explicit next question.

Equation consistency

PASS

FE comparison for active inputs

Retained reference only

Assumption review

NOT REVIEWED

Physical validation

NOT_ESTABLISHED

Hardware safety

NOT_ESTABLISHED

Next engineering action: establish the real alloy, temperature-dependent properties, allowables, actual tube-sheet restraint, load combinations and relevant failure modes. Then select and qualify the conventional model and compare with independent physical evidence. A larger model or finer mesh cannot supply missing requirements.