軸対称解析
Theory and Physics
What is an Axisymmetric Problem?
Professor, what is "axisymmetric" analysis? Can you solve a 3D problem in 2D?
Yes. When the geometry, material, loads, and boundary conditions are all symmetric about an axis of rotation, a 3D problem can be reduced to an analysis of a 2D cross-section. It's the assumption that there is no variation in the $\theta$ direction in cylindrical coordinates $(r, \theta, z)$.
What kind of structures are axisymmetric?
Actually, there are a huge number:
- Pressure Vessels — Cylindrical shells, heads, nozzles (when uniform in the circumferential direction)
- Flanges & Bolts — Analysis of axial force in fastened assemblies
- Bearings — Contact of inner/outer rings and rolling elements
- Pistons & Cylinders — Pressure analysis in internal combustion engines
- Internal Pressure in Pipes — Lamé's problem for thick-walled cylinders
- Rotating Bodies — Centrifugal force in disks, flywheels, turbine rotors
It's essential for pressure vessel design then.
Axisymmetric analysis is also recognized in ASME standards for pressure vessels. Compared to modeling an entire pressure vessel in 3D, axisymmetric cross-section analysis achieves equal or better accuracy with less than 1/100th the DOF.
Governing Equations
What are the stress components in an axisymmetric problem?
In cylindrical coordinates, there are four stress components:
Plane stress and plane strain had 3 components, but axisymmetric has 4?
Correct. A characteristic of axisymmetry is the existence of hoop stress $\sigma_\theta$. When there is a radial displacement $u_r$, the circumferential strain is:
$u_r / r$! So strain is simply displacement divided by radius. Is this a geometric effect?
Perfect understanding. The circumference of a circle with radius $r$ is $2\pi r$, so if the radius increases by $u_r$, the circumference becomes $2\pi(r + u_r)$. The strain is $\Delta L / L = u_r / r$. This $1/r$ term is the core of axisymmetric analysis and is a feature not present in plane stress/strain.
Constitutive Law
Hooke's law for axisymmetry:
$$ \begin{Bmatrix} \sigma_r \\ \sigma_\theta \\ \sigma_z \\ \tau_{rz} \end{Bmatrix} = \frac{E}{(1+\nu)(1-2\nu)} \begin{bmatrix} 1-\nu & \nu & \nu & 0 \\ \nu & 1-\nu & \nu & 0 \\ \nu & \nu & 1-\nu & 0 \\ 0 & 0 & 0 & \frac{1-2\nu}{2} \end{bmatrix} \begin{Bmatrix} \varepsilon_r \\ \varepsilon_\theta \\ \varepsilon_z \\ \gamma_{rz} \end{Bmatrix} $$
It has the same form as the 3D Hooke's law. 4 components, but essentially 3D.
Yes. Axisymmetry is "2D in appearance, 3D in essence." The stress state is fully 3D, and $\sigma_\theta$ is never zero (except at $r = 0$).
Thick-Walled Cylinder (Lamé's Problem)
Is there a theoretical solution for a thick-walled cylinder under internal pressure?
Lamé's problem is the most fundamental theoretical solution for axisymmetry. For inner radius $a$, outer radius $b$, and internal pressure $p$:
$$ \sigma_r = \frac{p a^2}{b^2 - a^2} \left(1 - \frac{b^2}{r^2}\right) $$
$$ \sigma_\theta = \frac{p a^2}{b^2 - a^2} \left(1 + \frac{b^2}{r^2}\right) $$
The hoop stress is maximum at the inner surface ($r = a$) and decreases towards the outer surface. The inner surface is in tension, and the tension decreases as you approach the outer surface.
Maximum hoop stress at the inner surface:
$$ \sigma_{\theta,max} = p \frac{a^2 + b^2}{b^2 - a^2} $$
Hooke's law for axisymmetry:
It has the same form as the 3D Hooke's law. 4 components, but essentially 3D.
Yes. Axisymmetry is "2D in appearance, 3D in essence." The stress state is fully 3D, and $\sigma_\theta$ is never zero (except at $r = 0$).
Is there a theoretical solution for a thick-walled cylinder under internal pressure?
Lamé's problem is the most fundamental theoretical solution for axisymmetry. For inner radius $a$, outer radius $b$, and internal pressure $p$:
The hoop stress is maximum at the inner surface ($r = a$) and decreases towards the outer surface. The inner surface is in tension, and the tension decreases as you approach the outer surface.
Maximum hoop stress at the inner surface:
For thin walls ($b \approx a$), this approaches $\sigma_\theta \approx pD/(2t)$ (the thin-walled cylinder formula). This is the best benchmark problem for verifying FEM axisymmetric analysis.
Summary
Let me organize the theory of axisymmetric analysis.
Key points:
- If geometry, loads, and materials are rotationally symmetric, it can be solved in 2D — DOF reduced to less than 1/100th.
- Four stress components — $\sigma_r, \sigma_\theta, \sigma_z, \tau_{rz}$. Hoop stress $\sigma_\theta$ is unique.
- $\varepsilon_\theta = u_r / r$ — Geometric circumferential strain.
- Essentially a 3D stress state — 2D in appearance but 3D in essence.
- Verify with Lamé's problem — Internal pressure in a thick-walled cylinder is the best benchmark.
For pressure vessel design, there's no reason not to use axisymmetric analysis. Equivalent accuracy at 1/100th the cost of 3D.
Exactly. However, non-axisymmetric loads (nozzle loads, seismic loads) or non-axisymmetric geometry (holes, discontinuities) require 3D analysis. An efficient approach is to screen with axisymmetric analysis and perform detailed 3D analysis only on necessary parts.
Historical Origin of Axisymmetry
The theoretical foundation of axisymmetric analysis dates back to the 1850s when G.B. Airy introduced stress functions. In the 1960s, Clough & Rashid formulated ring elements, reducing flop counts to about 1/10th of 3D analysis. It remains an active technique in the design of pressure vessels and nuclear reactor containment vessels.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, which is the assumption that "forces are applied slowly enough that acceleration can be ignored". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. So, a question — an iron rod and a rubber band, which stretches more under the same force? Obviously the rubber. This "resistance to stretching" is Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness ≠ strong". Stiffness is "resistance to deformation", strength is "resistance to failure"; they are different concepts.
- External Force Term (Load Term): Body force $f_b$ (e.g., gravity) and surface force $f_s$ (e.g., pressure, contact force). Think of it this way — the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common pitfall here: getting the load direction wrong. Intending "tension" but it becomes "compression" — sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — intentionally absorbing vibrational energy to improve ride comfort. What if damping were zero? Buildings would continue shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the stress-strain relationship is linear.
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity or creep, constitutive law extensions are needed.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit inconsistency when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation). |
| Elastic Modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence. |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system. |
Numerical Methods and Implementation
Axisymmetric Analysis by FEM
What are the characteristics of FEM elements for axisymmetry?
Axisymmetric elements are 2D elements in the $(r, z)$ plane, but their internal formulation includes a weight of $2\pi r$. This corresponds to the volume obtained by rotating the element 360° around the axis.
Element Formulation
Integration of the axisymmetric element stiffness matrix:
The $2\pi r$ (circumference) replaces the $t$ (thickness) in plane problems.
At $r = 0$ (on the axis of rotation), $2\pi r = 0$. Is that okay?
That's a very good question. Integration at $r = 0$ has a singularity. In practice, the following measures are taken:
- Gauss integration points are not placed on $r = 0$ (integration points are inside the element).
- At nodes on $r = 0$, $\varepsilon_\theta = u_r / r$ becomes $0/0$, but by L'Hôpital's rule, $\varepsilon_\theta = \varepsilon_r$.
- Practically, it's not a problem, but using extremely fine mesh near $r = 0$ can sometimes cause numerical instability.
Element Names by Solver
| Element | Nastran | Abaqus | Ansys |
|---|---|---|---|
| 4-node quadrilateral | CQUADX4 / CTRIAX | CAX4, CAX4R, CAX4H | PLANE182 (KEYOPT3=1) |
| 8-node quadrilateral | CQUADX8 | CAX8, CAX8R | PLANE183 (KEYOPT3=1) |
| 3-node triangle | CTRIAX6 | CAX3 | PLANE182 (degenerated) |
Abaqus's CAX is short for "Continuum AXisymmetric", right? CPS is plane stress, CPE is plane strain, CAX is axisymmetric.
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