Tools · Structural dynamics
Pipe span natural frequency
First four bending modes of a straight pipe span between supports, the quick check behind span-length decisions in piping vibration work.
Defaults: 6" Sch 40 carbon steel, water-filled. For gas service set contents density to the gas density (or 0).
| Mode | λL | Frequency (Hz) | CPM |
|---|
The formulas
fi = ((λiL)² / 2πL²) · √(EI / m)
I = π(Do⁴ − Di⁴)/64 m = pipe + contents + insulation mass per metre
λL comes from each support condition's frequency equation: nπ for pinned–pinned, cosh·cos = 1 for fixed–fixed and free–free, cosh·cos = −1 for a cantilever, tan = tanh for fixed–pinned.
Real pipe supports are neither perfectly pinned nor perfectly fixed, so the true first mode usually sits between the pinned–pinned and fixed–fixed results. That band is a useful sanity check against FEA or a field measurement. Heavy valves or flanges in the span, bends, and support flexibility all pull the frequency down; for those, use FEA.
Method & sources
What it calculates
- Euler–Bernoulli beam theory: fi = ((λiL)²/2πL²)·√(EI/m).
- λL from each support condition's characteristic equation, solved by root-finding to 6 significant figures (π·n; cosh·cos = ±1; tan = tanh).
- Mass per metre = pipe steel + contents + user-entered extra (insulation, cladding); contents add mass but no stiffness.
Assumptions & limits
- No shear deformation or rotary inertia: accurate for slender spans (L/D above about 10). Short, stubby spans read high.
- Ideal supports. Real supports are flexible, so measured frequencies usually sit below the fixed–fixed value; Wachel et al. give field correction factors.
- No internal pressure, axial load, bends or concentrated masses (valves, flanges).
Verified: 6" Sch 40, water-filled, 3 m pinned–pinned gives 39.01 Hz (hand calculation: 39.01 Hz). Every mode shape was checked numerically against its boundary conditions.
Sources
- Blevins, R. D. (1979, reprinted 2001). Formulas for natural frequency and mode shape. Krieger
- Rao, S. S. Mechanical vibrations. Pearson
- Wachel, J. C., Morton, S. J. and Atkins, K. E. (1990). Piping vibration analysis. Proceedings of the 19th Turbomachinery Symposium, Texas A&M University, 119-134 ↗
- ASME B36.10M. Welded and seamless wrought steel pipe ↗