線形座屈（固有値座屈）解析
Theory and Physics
Overview
Professor, are "linear buckling analysis" and "eigenvalue buckling analysis" the same thing?
They are the same. In the context of FEM, "linear buckling analysis," "eigenvalue buckling analysis," and "linearized pre-buckling analysis" all refer to the same method. It is a technique that creates a geometric stiffness matrix from the stress state of the reference configuration, solves an eigenvalue problem, and obtains the buckling load factor.
It's the $([K] + \lambda [K_\sigma])\{\phi\} = \{0\}$ we learned in the Euler buckling section, right? Does that mean applying it to general structures?
Exactly. Euler buckling was limited to the specific structure of a "column," but eigenvalue buckling analysis is a general method applicable to plates, shells, frames, and even their combined structures.
Mathematical Structure of Eigenvalue Buckling Analysis
Could you explain the general formula in a bit more detail?
It's a two-step procedure.
Step 1: Perform a static analysis under the reference load $\{F_{ref}\}$ to obtain the stress distribution $\{\sigma_{ref}\}$:
Step 2: Construct the geometric stiffness matrix $[K_\sigma]$ from the obtained stress and solve the eigenvalue problem:
$\lambda_i$ is the buckling load factor, and $\{\phi_i\}$ is the mode shape. If $\lambda_1 = 3.5$, it buckles at 3.5 times the reference load.
Correct. The important point here is that the static analysis in Step 1 is a linear static analysis. That means it does not consider large deformation effects. Under the assumption that stress increases proportionally with load, it predicts "when instability occurs."
That's why it's called "linear" buckling analysis.
Precisely. This "assumption of linearity" is the essential limitation of eigenvalue buckling analysis and simultaneously the source of its computational speed.
Why It Becomes an Eigenvalue Problem
First of all, why can buckling be formulated as an eigenvalue problem?
That's a fundamental question. Buckling is a phenomenon where "the deformation suddenly transitions from a state that was constant despite increasing load to a different deformation pattern." Mathematically, it is viewed as a bifurcation of the equilibrium path.
When scaling the load as $\lambda \{F_{ref}\}$, the overall stiffness becomes $[K_0] + \lambda [K_\sigma]$. Here, $[K_\sigma]$ is the geometric stiffness for the reference stress, so it scales proportionally with $\lambda$. The point where this overall stiffness becomes singular (det = 0) is the bifurcation point, and that $\lambda$ is the buckling load factor.
Finding the $\lambda$ where the determinant of $[K_0] + \lambda [K_\sigma]$ becomes zero... that takes the form of an eigenvalue problem!
Yes. It has exactly the same mathematical structure as vibration analysis $([K] - \omega^2 [M])\{\phi\} = \{0\}$. In vibration, the mass matrix $[M]$ is in that position; in buckling, the geometric stiffness matrix $[K_\sigma]$ is there.
Physical Meaning of the Geometric Stiffness Matrix
Could you explain the physical meaning of $[K_\sigma]$ a bit more?
$[K_\sigma]$ represents "how much work the current stress state does against a small displacement perturbation." If you deflect a beam element under compressive stress slightly sideways, the compressive force does work in the direction that increases the deflection. This acts as a negative stiffness.
To be specific, for a beam element under axial force $N$, when a small lateral deflection $\delta v$ occurs, the axial force does additional work of $N \cdot (\delta v')^2 / 2$. This "additional work due to stress" is what is matrix-formulated as $[K_\sigma]$.
Under compression ($N < 0$) stiffness decreases, under tension ($N > 0$) stiffness increases. That's why tensile members don't buckle, right?
That intuition is correct. However, there is a caveat. For example, even tensile members like prestressed cables can experience lateral buckling (a phenomenon essentially close to the vibration mode of a tensile member). Basically, you can understand it as "the part dominated by compressive stress is the starting point of buckling."
Premises and Limitations of Linear Buckling Analysis
Under what conditions does linear buckling analysis give the "correct answer"?
It is highly reliable when the following premises are met:
1. Pre-buckling deformation is small — The shape remains virtually unchanged.
2. Material is within the elastic range — No yielding at the buckling point.
3. Loads are proportional — All loads increase or decrease at the same ratio.
4. Initial imperfections are small — Deviations from the ideal shape are negligible.
What if these are not met?
The result of eigenvalue buckling analysis becomes an upper bound (unconservative estimate). That is, the actual collapse load is lower than the eigenvalue buckling load. How much lower specifically depends on the structure type:
| Structure Type | Reliability of Eigenvalue Buckling | Actual Collapse Load / Eigenvalue Buckling Load |
|---|---|---|
| Column (global buckling) | High | 0.85 ~ 1.0 |
| Flat Plate (in-plane compression) | Relatively High | 0.7 ~ 0.95 |
| Cylindrical Shell (axial compression) | Very Low | 0.2 ~ 0.5 |
| Stiffened Panel | Medium | 0.5 ~ 0.9 |
Cylindrical shells can drop to as low as one-fifth...?
That's why for buckling design of cylindrical shells, eigenvalue analysis alone is insufficient, and nonlinear analysis incorporating initial imperfections is essential. On the other hand, global buckling of columns can be estimated quite well with eigenvalue analysis. In practice, it is extremely important to understand the "reliability of eigenvalue analysis" according to the structure type.
Summary
To summarize the theory of eigenvalue buckling analysis...
It's important to recognize that eigenvalue buckling analysis is a "screening tool," not the "final answer."
Exactly. It is an optimal method for quickly finding "where is dangerous" in the early stages of design. However, don't forget that whether you can give a design OK based on it alone depends on the structure's imperfection sensitivity.
Origins of Eigenvalue Buckling Analysis and Hertz Contact
Eigenvalue buckling analysis (linear buckling analysis) was already formulated as a matrix eigenvalue problem in the 1950s-60s, before the FEM era. Shortly after Turner and Clough (1960) applied FEM to structural analysis, Martin (1965) introduced the geometric stiffness matrix, leading to the birth of linear buckling FEM. NASTRAN (1968) was the first to commercially implement this.
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, based on the assumption that "forces are applied slowly, so 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 stretch a spring, you feel a "force trying to return it," right? That is Hooke's law $F=kx$, the essence of the stiffness term. Now a question—an iron rod and a rubber band, which stretches more under the same force? Obviously the rubber band. This "resistance to stretching" is the 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 $f_s$ (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), the force with which the tire pushes 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 typical mistake 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 away. That's because the vibrational energy is converted into heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they deliberately absorb 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: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity or creep requires constitutive law extensions.
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
Internal Operation of Eigenvalue Solvers
What algorithm specifically solves $([K_0] + \lambda [K_\sigma])\{\phi\} = \{0\}$ and how does it work?
Eigenvalue problems in structural analysis are typically generalized eigenvalue problems. For buckling, when transformed to standard form:
$[K_0]$ is positive definite symmetric (if properly constrained), $[K_\sigma]$ is symmetric but indefinite. There are several efficient algorithms that utilize this structure.
Lanczos Method
I've heard the Lanczos method is standard in practice, why is that?
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