Creep Buckling
Theory and Physics
What is Creep Buckling?
Professor, what's the difference between "creep buckling" and regular buckling?
Regular buckling occurs instantaneously — buckling deformation begins the moment the load exceeds the critical value. On the other hand, creep buckling progresses slowly over time. Even if the load is lower than the elastic buckling load, creep deformation accumulates over a long period and eventually leads to buckling.
It buckles even below the elastic buckling load!? That's scary.
Creep is a phenomenon where materials deform over time in high-temperature environments. Strain continues to increase even under constant stress. The accumulation of this creep strain gradually changes the shape of the structure and destabilizes it, which is creep buckling.
Fundamentals of Creep
Please teach me the basics of the creep phenomenon.
Creep strain under constant stress $\sigma$ and constant temperature $T$ progresses in three stages:
1. Primary Creep (Transient Creep) — Strain rate decreases over time
2. Secondary Creep (Steady-State Creep) — Strain rate is constant. The longest stage
3. Tertiary Creep (Accelerating Creep) — Strain rate increases, leading to final rupture
The steady-state creep strain rate is often expressed by Norton's (power-law) rule:
Here, $A, n$ are material constants, $Q$ is activation energy, $R$ is the gas constant, and $T$ is absolute temperature.
With $\sigma^n$ and $n$ around 3~8 for steel, if stress doubles, creep speed becomes 8~256 times faster! The sensitivity to stress is incredibly high.
Exactly. That's why stress redistribution is important in creep buckling. The initial elastic stress distribution becomes uniformized over time due to creep relaxation. This process changes the structural behavior.
Mechanism of Creep Buckling
How does creep buckling occur?
There are two mechanisms.
1. Bifurcation-Type Creep Buckling — Similar bifurcation to elastic buckling occurs, but with a time delay. Under compressive stress, bending deformation gradually increases due to creep, and buckling occurs abruptly at a certain point.
2. Pseudo-Buckling (creep buckling by deflection amplification) — Initial imperfection-induced bending deformation is amplified over time by creep. There is no clear bifurcation point; the moment deformation exceeds an allowable value is defined as "buckling".
So in pseudo-buckling, "deformation becoming too large" is the definition of buckling.
Yes. The "critical time" for creep buckling is often defined by how many times the displacement has increased from its initial value. For example, the time when "displacement becomes 5 times the initial value" is taken as the critical time.
Concept of Critical Time
What exactly is "critical time"?
It is the "time to buckling" corresponding to the load level $P/P_{cr}$ (ratio to elastic buckling load).
In Hoff's classical result (1958), the creep buckling time for a column with initial imperfection:
The closer the load is to the elastic buckling load, the shorter $t_{cr}$ is; the lower the load, the longer $t_{cr}$ is.
So even at $P/P_{cr} = 0.5$, buckling is possible if enough time passes?
Theoretically, yes. However, when $P/P_{cr}$ is low, $t_{cr}$ may exceed the structure's lifespan (decades). In that case, creep buckling is not a practical concern.
Fields Where Creep Buckling is a Problem
In what kind of structures is creep buckling a problem?
| Structure | Temperature Range | Typical Load |
|---|---|---|
| Thermal Power Plant Boiler Tubes | 500–600°C | Internal Pressure + Self-Weight |
| Nuclear Reactor Vessels | 300–600°C | Internal Pressure + Thermal Stress |
| Jet Engine Casings | 600–1000°C | Internal Pressure + Centrifugal Force |
| High-Temperature Chemical Plants | 400–900°C | Internal Pressure + Self-Weight |
| Concrete Columns (Long-Term) | Ambient Temperature | Sustained Compressive Force |
Does concrete also creep at ambient temperature?
Concrete creeps even at ambient temperature (drying creep). In columns and walls subjected to large sustained loads over the long term, additional eccentricity due to creep reduces buckling resistance. Design codes (e.g., Eurocode 2) consider the effect of long-term loads using creep coefficients.
Summary
Let me organize the theory of creep buckling.
Key Points:
- Creep buckling is time-dependent buckling — Buckling can occur even below the elastic buckling load given sufficient time.
- Norton's Law $\dot{\varepsilon}_{cr} = A\sigma^n$ — Creep rate proportional to stress raised to the power $n$.
- Critical Time — Time to buckling corresponding to load level.
- Important for high-temperature environment structures — Boilers, reactors, turbines, chemical plants.
- Creep buckling is also a problem for concrete under long-term loads.
Adding the dimension of time makes buckling problems much more complex.
Yes. Elastic buckling is a binary question of "whether the load exceeds the critical value," but creep buckling is a continuous problem of "when buckling occurs." It must be judged in relation to the design life.
Creep Buckling and the Challenger Lesson
Creep buckling is a phenomenon where deformation progresses over time under constant stress at high temperatures, eventually leading to buckling. In the investigation of the 1986 Space Shuttle Challenger accident, the direct cause was the creep deformation of the rubber O-rings due to low temperatures on the launch pad. However, subsequent research revealed that the aluminum shell of the solid rocket booster also had a tight margin against its creep buckling design limit.
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 assumes "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 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, question — an iron rod and a rubber band, which stretches more under the same pulling 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 $f_b$ (gravity, etc.) and surface force $f_s$ (pressure, contact force, etc.). Think of it this way — the weight of a truck on a bridge is a "force acting on the entire contents" (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 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 vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — deliberately absorbing vibration energy to improve ride comfort. What if damping were zero? Buildings would keep shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is important.
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 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 and 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 load/elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Note unit system inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note 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
Numerical Methods for Creep Buckling
How do you solve creep buckling with FEM? It's not a problem that can be solved in one shot like eigenvalue buckling, right?
Exactly. Creep buckling is a problem requiring time integration. Instead of an instantaneous determination like eigenvalue buckling, it tracks the evolution of deformation over time.
Basic Solution Methods
Procedure:
1. Initial State — Calculate the elastic response at the moment load is applied.
2. Time Integration — Calculate the increment of creep strain at each time step and update stress.
3. Equilibrium Iteration — Satisfy equilibrium using the Newton-Raphson method at each step.
4. Buckling Detection — Detect sudden increase in displacement or loss of tangent stiffness.
How do you perform time integration for creep strain?
The Implicit Euler Method is the most stable. Creep strain increment over time step $\Delta t$:
$\sigma_{n+1}$ is unknown (stress at the next step), so iteration is required. Newton-Raphson iteration must be performed at each time step.
Is the size of the time step important?
Very important. Especially as buckling approaches, deformation speed increases rapidly, so adaptive time integration that automatically reduces the time step is desirable. Abaqus's *VISCO step does this automatically.
Abaqus
In Abaqus, use the *VISCO step for structural analysis including creep:
```
*MATERIAL, NAME=Steel_creep
*ELASTIC
200000., 0.3
*CREEP, LAW=NORTON
1.0e-20, 5.0, 0.0
*Related Topics
なった
詳しく
報告