Prestress Modal Analysis
Theory and Physics
What is Prestressed Modal Analysis?
Professor, what is "prestressed modal analysis"?
It's natural frequency analysis in the presence of initial stress (prestress). When a structure is under tensile or compressive forces, its natural frequencies change. It's the same principle as a string's pitch rising when tightened.
Like a guitar string. Increasing the tension raises the vibration frequency.
Exactly. Conversely, compressive forces lower the vibration frequency. As compressive force approaches the critical buckling load, the frequency approaches zero. Frequency zero = buckling.
Governing Equation
Eigenvalue problem including prestress:
$[K_\sigma]$ is the geometric stiffness matrix (stress stiffness), same as in buckling analysis.
Buckling is $([K_0] + \lambda [K_\sigma])\{\phi\} = \{0\}$, and vibration is $([K_0] + [K_\sigma] - \omega^2 [M])\{\phi\} = \{0\}$. $[K_\sigma]$ is common!
Perfect observation. Buckling and vibration share the same geometric stiffness matrix. Tensile prestress makes $[K_\sigma] > 0$, increasing overall stiffness and raising frequency. Compressive prestress makes $[K_\sigma] < 0$, decreasing overall stiffness and lowering frequency.
Application Examples
| Structure | Type of Prestress | Effect on Frequency |
|---|---|---|
| String / Cable | Tension | Frequency increases with tension |
| Rotating Disk | Centrifugal Force (Tension) | Frequency increases with rotation speed |
| Turbine Blade | Centrifugal Force | Frequency changes with rotation |
| Compression Column | Axial Compression | Frequency decreases with compression |
| Prestressed Concrete Beam | Tension (PC Steel) | Frequency slightly increases with tension |
| Membrane Structure (Tent) | In-plane Tension | Frequency increases with tension |
So centrifugal prestress is important for rotating body vibrations.
In turbine blades and rotating disks, centrifugal force creates tensile prestress, raising the frequency. This is called spin softening/hardening. Since frequency changes with each rotation speed, it's necessary to evaluate natural frequencies at each speed.
FEM Procedure
1. Static Analysis (Preload) — Determine initial stress (compression, tension, centrifugal, etc.)
2. Geometric Stiffness Matrix Formation — Calculate $[K_\sigma]$ from the static analysis stress
3. Eigenvalue Analysis — Use $[K_0] + [K_\sigma]$ as stiffness to find natural frequencies
That's almost the same procedure as buckling analysis.
Summary
Let me organize prestressed modal analysis.
Key points:
- Initial stress changes natural frequency — Increases with tension, decreases with compression
- $[K_0] + [K_\sigma] - \omega^2 [M] = 0$ — Eigenvalue problem with added geometric stiffness
- Buckling and vibration share the same $[K_\sigma]$ — Frequency becomes zero at buckling point
- Centrifugal prestress is important for rotating bodies — Spin softening/hardening
- Two-step process: static analysis → eigenvalue analysis — Same procedure as buckling analysis
The relationship "frequency becomes zero at buckling point" is profound. Vibration and buckling are connected by a single theory.
It's one of the most beautiful relationships in structural mechanics. VCT (Vibration Correlation Technique) utilizes this relationship to non-destructively predict buckling load from changes in frequency.
High Pitch of a Taut String and Vibration Frequency of a Compressed Spring
Applying tension to a string instrument's string raises its natural frequency. Conversely, applying compressive load lowers the natural frequency, reaching zero at the buckling load (Pcr). This relationship is expressed as f²=f₀²(1-P/Pcr), allowing the compressive load P (prestress amount difficult to measure) to be inversely calculated from the measured natural frequency. This principle is actually used for tension management in bridge cables.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Haven't you experienced your body being thrown forward during sudden braking? 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, assuming "forces are applied slowly so acceleration is negligible". 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", right? That's 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. 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" — 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), the force of the tire pushing 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 rotate 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 vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — intentionally absorbing vibration 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 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, stress-strain relationship is linear
- Isotropic material (unless specified otherwise): Material properties are independent of direction (anisotropic materials require separate tensor definition)
- Quasi-static assumption (for static analysis): Ignores inertial/damping forces, considers only equilibrium 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 extension
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting mm, unify loads/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
Solver Settings
Please teach me the settings for prestressed modal analysis.
Nastran
```
SOL 103
CEND
SUBCASE 1
LOAD = 100 $ Preload
METHOD = 10 $ Eigenvalue Analysis
SPC = 1
```
Setting a load in SOL 103 automatically makes it a prestressed modal analysis.
Abaqus
```
*STEP
*STATIC
*CLOAD
...
*END STEP
*STEP
*FREQUENCY
20, ,
*END STEP
```
Place a Frequency step after the Static step. Stress from the previous step is automatically reflected in the geometric stiffness.
Ansys
```
/SOLU
ANTYPE, STATIC
PSTRES, ON ! Activate stress stiffness
SOLVE
FINISH
/SOLU
ANTYPE, MODAL
MODOPT, LANB, 20
SOLVE
```
In Ansys, PSTRES, ON is mandatory, right? Same as for buckling.
Forgetting PSTRES, ON means stress stiffness isn't calculated, resulting in "normal" natural frequencies with zero prestress effect. Same pitfall as buckling.
Types of Preload
| Preload | Setting Method | Example |
|---|---|---|
| Axial Force | Concentrated load or distributed load | Tension cable, compression column |
| Centrifugal Force | RFORCE (Nastran), *DLOAD CENTRIFUGAL (Abaqus) | Rotating body |
| Temperature | TEMP load | Frequency change due to thermal stress |
| Internal Pressure | PLOAD4 / *DLOAD P | Vibration of pressurized vessels |
There's also prestress from temperature?
When a constrained structure undergoes temperature change, thermal stress occurs, affecting frequency. In steel beams during fire, temperature rise generates axial compressive force, leading to frequency decrease → buckling.
Summary
Let me organize the numerical methods for prestressed modal analysis.
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