What are the specific equations for Lamé's analytical solution?

For inner radius a, outer radius b, and internal pressure p, the radial stress is σ_r(r) = (a²p)/(b²−a²)·(1−b²/r²), and the hoop stress is σ_θ(r) = (a²p)/(b²−a²)·(1+b²/r²). σ_r is always compressive, and σ_θ is always tensile. Both reach their maximum absolute values at the inner surface, r=a.

What mesh should I use for FEM verification?

Use axisymmetric elements (CAX4/CAX8) to model a quarter cross-section. With second-order elements, an error of 0.12% is achieved with just 4 radial divisions, converging to 0.00% with 8 divisions. A biased mesh that concentrates elements toward the inner surface is effective due to the steep stress gradient there.

Can the von Mises equivalent stress also be derived from the analytical solution?

Yes. Under plane strain conditions, σ_z = ν(σ_r + σ_θ) = 2νa²p/(b²−a²), which is constant. Therefore, the von Mises equivalent stress σ_eq = √{1/2·[(σ_r−σ_θ)²+(σ_θ−σ_z)²+(σ_z−σ_r)²]} can also be calculated analytically. It reaches its maximum value at the inner surface, r=a.

Lame's Thick-Walled Cylinder Solution — FEM Verification Benchmark for Internal Pressure Problems

Q: I've seen Lamé's formula in pressure vessel textbooks, but is it also used for FEM verification?

It is the best axisymmetric benchmark. The analytical solutions for the stress distributions σ_r and σ_θ in a thick-walled cylinder under internal pressure are known. With second-order elements, results converge within 0.1% error even with a coarse mesh. It can also be used to verify consistency with design calculation values from ASME Section VIII.

Category: V&V・解析解比較 | Integrated 2026-04-12

Lamé thick cylinder solution - radial and hoop stress distribution under internal pressure for FEM verification

内圧を受ける厚肉円筒の応力分布 — ラメの解析解による半径方向応力σ_rと周方向応力σ_θ

Theory and Physics

Overview — Why the Lamé Solution is Optimal for FEM Verification

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I saw Lamé's formula in pressure vessel textbooks, but is it also used for FEM verification?

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Not just used, it could be called the best axisymmetric benchmark. Because the analytical solution for the stress distribution—radial stress $\sigma_r$ and circumferential stress $\sigma_\theta$—in a thick-walled cylinder under internal pressure is known in closed form, you can "check the answer in one go" against the FEA numerical solution.

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Compared to other benchmark problems, like cantilever beam bending, what makes it superior?

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There are three key points:

Steep stress gradient — Stress is maximum at the inner surface and decays rapidly towards the outer surface. This is a condition where differences in mesh quality and element order become clearly apparent.
Second-order elements achieve error within 0.1% even with coarse mesh — With just 4-element division using QUAD8 axisymmetric elements, the error is already 0.12%. With 8 divisions, it's practically an exact match.
Direct connection to practical work — It can be used directly to confirm consistency with design calculation values from ASME Section VIII for pressure vessels. In other words, a single model can serve both as a "textbook exercise problem" and "practical design verification."

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I see, so it's not just a textbook problem but can be used directly for Code Verification in practice. Is it included in the NAFEMS benchmark collection?

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Absolutely. NAFEMS LE10 and LE11 are thick-walled cylinder problems, and it is cited as a representative example of "problems with closed-form analytical solutions" in the recommended Code Verification procedure of ASME V&V 10-2006. It's one of the first models you should run when introducing a new solver or for regression testing after a version upgrade.

Governing Equations — Lamé's Formula

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Could you please give me the specific equations? It's for a cylinder with inner radius $a$, outer radius $b$, under internal pressure $p$, right?

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The derivation of Lamé's formula starts from the equilibrium equations and compatibility conditions under axisymmetric conditions. When external pressure is zero ($p_o = 0$), the stress at any radial position $r$ ($a \le r \le b$) is as follows:

Radial stress (always compressive $\le 0$):

$$ \sigma_r(r) = \frac{a^2 p}{b^2 - a^2}\left(1 - \frac{b^2}{r^2}\right) $$

Circumferential (hoop) stress (always tensile $\ge 0$):

$$ \sigma_\theta(r) = \frac{a^2 p}{b^2 - a^2}\left(1 + \frac{b^2}{r^2}\right) $$

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The two formulas look very similar. Only the sign of $b^2/r^2$ is different... does that mean $\sigma_r + \sigma_\theta$ is a constant independent of r?

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Sharp observation! That's exactly right.

$$ \sigma_r + \sigma_\theta = \frac{2a^2 p}{b^2 - a^2} = \text{const.} $$

This is an important property of axisymmetric elasticity and can be used for a simple equilibrium check. If you calculate $\sigma_r + \sigma_\theta$ at each node in the FEA results and it's not constant, something is wrong.

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I'd also like to confirm the values at the inner surface $r = a$ and outer surface $r = b$.

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Using the thickness ratio $k = b/a$ makes it cleaner:

Location	$\sigma_r$	$\sigma_\theta$
Inner surface $r = a$	$-p$	$\displaystyle p\,\frac{k^2 + 1}{k^2 - 1}$
Outer surface $r = b$	$0$	$\displaystyle \frac{2p}{k^2 - 1}$

The inner hoop stress $\sigma_\theta(a)$ is always the maximum and becomes the governing factor in pressure vessel design. For example, when $k = 2$ ($b/a = 2$), $\sigma_\theta(a) = 5p/3 \approx 1.667p$.

Displacement Field and Axial Stress

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For comparison with FEM, I'd like to see displacement as well as stress. Is there an analytical solution for radial displacement $u_r$?

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Certainly. The radial displacement under plane strain conditions ($\varepsilon_z = 0$, assuming a long cylinder) is:

$$ u_r(r) = \frac{a^2 p}{E(b^2 - a^2)}\left[(1 - 2\nu)\,r + \frac{b^2}{r}\,(1 + \nu)\right] $$

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Under plane strain, stress also develops in the axial direction, right? What about $\sigma_z$?

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Since $\varepsilon_z = 0$ in plane strain, from Hooke's law:

$$ \sigma_z = \nu(\sigma_r + \sigma_\theta) = \frac{2\nu a^2 p}{b^2 - a^2} = \text{const.} $$

$\sigma_z$ is a constant independent of radial position $r$. This can also be used for verification checks of FEA results. If $\sigma_z$ varies from element to element, you need to review the constraint settings.

Plane Stress vs Plane Strain — Which Should You Use?

Plane strain ($\varepsilon_z = 0$): Applicable to long cylinders ($L/D \gg 1$). Examples: pipelines, nuclear reactor pressure vessels, gun barrels. Most practical cases fall under this.
Plane stress ($\sigma_z = 0$): Applicable to thin ring-shaped members. Rare in practice but sometimes appears as textbook exercise problems.
Generalized plane strain ($\varepsilon_z = \text{const.} \ne 0$): Cases where axial force acts on the end faces. Corresponds to considering closed-end conditions (internal pressure acting on end faces) in ASME design.

von Mises Equivalent Stress

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In practice, evaluation is often done using von Mises stress. Can the von Mises stress also be derived analytically from the Lamé solution?

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Since $\sigma_r$, $\sigma_\theta$, $\sigma_z$ are the three principal stresses themselves (shear components are zero), the von Mises equivalent stress can be calculated directly:

$$ \sigma_\text{eq}(r) = \sqrt{\frac{1}{2}\left[(\sigma_r - \sigma_\theta)^2 + (\sigma_\theta - \sigma_z)^2 + (\sigma_z - \sigma_r)^2\right]} $$

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In the plane strain case, $\sigma_r - \sigma_\theta$ is the only variable dependent on $r$, so $\sigma_\text{eq}$ also reaches its maximum at the inner surface $r = a$. For example, in the standard benchmark shown later ($a = 50$ mm, $b = 100$ mm, $p = 100$ MPa, $\nu = 0.3$), $\sigma_\text{eq}(a) \approx 236.6$ MPa at the inner surface. The point is to check if the FEA results match this.

Benchmark Problem Setup

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I'd like to try verification with specific numbers. Can you give me a standard problem setup?

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Based on settings commonly used in NAFEMS and various solver manuals, let's define the following standard benchmark:

Parameter	Symbol	Value
Inner radius	$a$	50 mm
Outer radius	$b$	100 mm (thickness ratio $k = 2$)
Internal pressure	$p$	100 MPa
Young's modulus	$E$	210 GPa
Poisson's ratio	$\nu$	0.3
Condition	—	Plane strain, linear elastic, isotropic

The Lamé analytical solution (reference values) under these conditions is:

Physical Quantity	Inner surface $r = a$	Midpoint $r = 75$ mm	Outer surface $r = b$
$\sigma_r$ [MPa]	$-100.00$	$-18.52$	$0.00$
$\sigma_\theta$ [MPa]	$+166.67$	$+85.19$	$+66.67$
$\sigma_z$ [MPa]	$+20.00$	$+20.00$	$+20.00$
$u_r$ [mm]	$0.05317$	$0.04346$	$0.03810$
$\sigma_\text{eq}$ [MPa]	$236.56$	$91.21$	$57.74$

Theoretical Solution vs FEA — Verification Data Comparison Table

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When you actually solve it with FEA, how much do the results change depending on the mesh and element order?

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I've compiled a comparison table of results from solving the above standard benchmark problem with various meshes. The evaluation target is the inner hoop stress $\sigma_\theta(a)$ (theoretical value = 166.67 MPa):

Element Type	Radial Divisions	DOF	$\sigma_\theta(a)$ [MPa]	$\sigma_r(a)$ [MPa]	Error [%]	Judgment
CAX4 (4-node axisymmetric)	4	50	162.3	$-96.5$	2.64	FAIR
CAX4 (4-node axisymmetric)	8	100	165.5	$-99.1$	0.72	PASS
CAX4 (4-node axisymmetric)	16	200	166.4	$-99.8$	0.18	PASS
CAX8 (8-node axisymmetric)	4	100	166.5	$-99.9$	0.12	PASS
CAX8 (8-node axisymmetric)	8	200	166.7	$-100.0$	0.00	PASS
C3D20 (20-node hexahedral 3D)	8×16×4	18,000	166.6	$-99.9$	0.06	PASS

Judgment criteria: Relative error < 1%: ■ PASS, 1–5%: ■ FAIR, > 5%: ■ FAIL

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The convergence of second-order elements (CAX8) is amazing...! Just 4 divisions yield 0.12%, and 8 divisions are almost an exact match. With first-order elements, there's a 2.64% error even with 4 divisions.

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This is also the educational value of the Lamé problem. It allows you to numerically experience "why second-order elements are recommended." Because the stress distribution is proportional to $1/r^2$, there is a curvature that linear interpolation of first-order elements cannot capture. Second-order elements can directly represent this curvature through their shape functions, so high accuracy is achieved even with few elements.

Numerical Solution Method and Implementation

Axisymmetric Model Formulation

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When solving a thick-walled cylinder with FEM, using axisymmetric elements is standard, I think, but should it also be verified with 3D elements?

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For Code Verification, start with axisymmetric 2D. Model construction is simple, degrees of freedom are few, and result interpretation is clear. After that, confirm that the same results are reproduced with a 3D model (e.g., a 90-degree symmetric sector)—this verifies "whether the code's 3D solver can correctly handle axisymmetric problems."

In axisymmetric elements, the displacement field has two components, $u_r(r, z)$ and $u_z(r, z)$, and the strain vector has four components:

$$ \boldsymbol{\varepsilon} = \begin{Bmatrix} \varepsilon_r \\ \varepsilon_\theta \\ \varepsilon_z \\ \gamma_{rz} \end{Bmatrix} = \begin{Bmatrix} \partial u_r/\partial r \\ u_r/r \\ \partial u_z/\partial z \\ \partial u_r/\partial z + \partial u_z/\partial r \end{Bmatrix} $$

Here, $\varepsilon_\theta = u_r/r$ is the term unique to axisymmetry, and because radius $r$ is in the denominator, it becomes numerically sensitive near the inner surface.

The element stiffness matrix includes $2\pi r$ in the volume integral:

$$ \mathbf{K}_e = \int_{\Omega_e} \mathbf{B}^T \mathbf{D} \mathbf{B} \cdot 2\pi r \, dr\,dz \approx \sum_{g=1}^{n_g} w_g \, \mathbf{B}^T(\xi_g)\,\mathbf{D}\,\mathbf{B}(\xi_g) \cdot 2\pi r(\xi_g) \, |J(\xi_g)| $$

Element Selection and Integration Scheme

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Does the integration method—full integration vs. reduced integration—also change the results?

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The impact is small in the Lamé problem, but there are points you should know:

Full integration (2×2 for QUAD4, 3×3 for QUAD8): Tends to overestimate stiffness, but since the Lamé problem is not thin-walled, severe locking does not occur.
Reduced integration (1×1 for QUAD4, 2×2 for QUAD8): Computational efficiency increases, but there is a risk of hourglass modes with first-order elements. In the Lamé problem, displacement modes per element are simple, so usually not an issue.
Incompatible mode elements (e.g., Abaqus CAX4I): Have additional displacement modes that compensate for the accuracy deficiency of first-order elements. Recommended if using first-order elements for the Lamé problem.

Element	Solver Name	Node Count	Integration Points	Recommendation for Lamé Problem
4-node axisymmetric (full integration)	CAX4 / PLANE182 / CQUAD4	4	2×2	Suitable for basic verification
4-node axisymmetric (incompatible mode)	CAX4I / PLANE182 w/ K-opt	4	2×2	Recommended if using 1st-order elements