There are several acoustic conditions that directly result in mechanical problems. The
reciprocating compressor problem is well known, and analysis techniques are long
established. Steamhammer and waterhammer problems and solutions are also readily
recognizable and solvable. More recently however, the prevention of dynamic problems
due to acoustic resonances has also received attention. The following case history
describes a typical example, where analysis beforehand would have tipped the designer
off to a condition that eventually led to failure. FE/Pipe finite element analysis was used
to verify that CAESAR II elbow loading conditions would produce the type of cracking
observed. A CAESAR II dynamic model of the piping system showed natural
frequencies at 5.06, 5.55 and 7.2 Hz. The fundamental acoustic natural frequency was
shown by BOS Fluids to be at 5.0 Hz. Leaving a gas relief system with mechanical and
acoustic natural frequencies that were identical resulted in the ultimate failure of this
Fluid conditions known to cause resonance-type problems in piping systems are outlined
1) Two phase flow. (Any volumetric percentage of liquid entrained in a gas
2) Misty Flow. (A subset of two-phase flow.)
3) Gas flows in a boiling or flashing (furnace) environment.
4) Relief Valve Firing
5) Valve Cycling
6) Tube Rupture
7) Slugging (Typically not associated with acoustic resonance however.)
8) Pressure Regulated Flows (A subset of Valve Cycling)
9) Furnace Burner Gas Flows
10) Choked Flow
Unexpected mechanical vibration problems can occur due to one or more of the
1) The piping system has low mechanical natural frequencies. (In the 5 Hz and
lower range, although sometimes “low” can be in the 20 Hz and lower range.
Low depends on the level of excitation present.) Low natural frequencies
commonly occur in hot piping systems that are excessively spring supported,
or in other piping systems that are inadequately supported.
2) Any of the above fluid events act on the system flow and produce a white
noise that will respond at the system acoustic natural frequency.
3) The acoustic natural frequency (determined by BOS Fluids) corresponds to a
mechanical natural frequency (calculated by CAESAR II).
Copyright 1999 by Paulin Research Group Page 2 of 2
Gas flows can be more troublesome than liquid flows because gas flows tend to have
lower acoustic natural frequencies. “Misty” flows usually have an even lower natural
frequency than a pure gas flow. Unexpected problems arise when low acoustic
resonances amplify the effect of the unbalanced pressure and low mechanical resonances
amplify the effect of the unbalanced displacement.
The following system experienced just such a problem.
When the relief valve failed the system vibrated, moving approximately 1.5” peak-topeak
in the vertical direction at the long riser section. Failure occurred at the first 45-
Several pictures of the cracked geometry are shown below.
Copyright 1999 by Paulin Research Group Page 3 of 3
There were parallel cracks on each side of the bend shifted closer to the inside (or
intrados) of the elbow. A finite element plot of the high stresses in a 45-degree elbow
loaded via an out-of-plane moment is shown below.
The finite element runs were made to verify that external loads would produce the type of
stress pattern needed to cause the cracking observed. The plots of the elbow shown
above do just that. The plot on the right shows the high stress zones exactly where the
parallel cracking in the actual 45-degree elbow occurred. Stress Intensification Factors
from the FE/Pipe finite element calculation for the 45-degree elbows matched the B31.3
Code values as shown below:
Outplane 1.948 1.60
Inplane 2.338 1.912
Plots of several of the system natural frequencies as calculated by CAESAR II are shown
in the figures below:
Copyright 1999 by Paulin Research Group Page 4 of 4
The CAESAR II model of the piping system was imported into BOS Fluids and the
following acoustic natural frequencies were generated from BOS Fluids.
The strongest acoustic mode exists at almost exactly 5 Hz. This system is clearly prone
to suffer acoustically excited mechanical resonance when the relief valve opens sending a
pressure wave back through the system. Pressure waves of this type require only 5 backand-
forth trips in the piping system to fully excite the mechanical natural frequency.
With a wavespeed of 600 ft./sec. the mechanical resonant multiplication of the response
is seen quickly.
Additional supports were added to remove the mechanical dynamic modes as best as
possible. A harmonic simulation using CAESAR was employed to simulate the required
dynamic excitation to cause the observed displacement. Markl’s equation to failure, and
an FE/Pipe validation of the piping stress intensification factors, were used to show that
thru-wall cracking at the observed displacements could occur in less than 10 minutes.
Copyright 1999 by Paulin Research Group Page 5 of 5
Markl’s equation to failure, which can be used with CAESAR II or B31.3 expansion
stresses, or FE/Pipe and ASME Section VIII Division 2, Pl+Pb+Q+F stresses is:
Sfail = 490,000 N-0.2 psi.