[S0739-3717(00)00504-3]

In their recent paper 1 Segalman and coauthors estimate the probability distribution for the von Mises stress resulting from Gaussian random loadings of zero mean. Although they do not explain clearly what use is intended for this “important result for reliability of structures,” we anticipate that it must be somehow related either to fatigue or to the estimation of the extreme value over some return period (peak factor). The problem with the von Mises stress when it is defined in the time domain as it is in this paper is that:

• • it is always positive and does not reduce itself to the applied alternating stress in the simple uniaxial case.

• • its frequency content is not consistent with the frequency content of the stress components and the natural frequencies of the structure.

These drawbacks which are related to the quadratic form of the von Mises stress can be removed by an alternative definition in the frequency domain 2,3: If σ is the stress vector, the von Mises stress $σc$ is defined in the time domain by
$σc2=σTQσ=Trace{Q[σσT]}$
(1)
where Q is a constant matrix. Taking the expected value, we get
$E[σc2]=Trace{QE[σσT]}$
(2)
We recognize $E[σσT]$ as the covariance matrix of the stress vector, related to the PSD matrix of the stress vector $Φσσω$ by
$E[σσT]=∫−∞∞Φσσωdω$
(3)
From Eq. (2) and (3), we have
$E[σc2]=∫−∞∞Φcωdω=TraceQ∫−∞∞Φσσωdω$
(4)
where $Φcω$ is the PSD of the equivalent von Mises stress. Equation (4) is exact and does not involve any assumption. Next, we can define a Gaussian random process of zero mean by
$Φcω=Trace{QΦσσω}$
(5)
We call it “equivalent von Mises stress.” It is obvious that Eq. (4) is satisfied. Unlike the definition in the time domain as in 1, the foregoing random process is alternating, it reduces itself to the component stress in the uniaxial case and it is consistent with the frequency content of the stress components. From the PSD defined by Eq. (5), classical uniaxial random fatigue life prediction methods can be applied 2,3 and the peak factor formulae can also be used 3. The foregoing formulation in the frequency domain has been found extremely fast and useful in predicting the critical areas of structures subjected to random loading. It is worth pointing out that the spectral formulation can be extended to the multiaxial rainflow method 4 and to the Crossland failure criterion 5.

By Dan Segalman, Garth Reese, Richard Field, Jr., and Clay Fulcher and published in the January 2000 issue of the JOURNAL OF VIBRATION AND ACOUSTICS, Vol. 122, No. 1, pp. 42–48.

e-mail: scmero@ulb.ac.be, http://www.ulb.ac.be/scmero

1.
Segalman
,
D.
,
Reese
,
G.
,
Field
,
R.
, Jr.
, and
Fulcher
,
C.
,
2000
, “
Estimating the Probability Distribution of von Mises Stress for Structures Undergoing Random Excitation
,”
ASME J. Vibr. Acoust.
,
122
, pp.
42
48
.
2.
Preumont
,
A.
, and
Pie´fort
,
V.
,
1994
, “
Predicting Random High Cycle Fatigue Life with Finite Elements
,”
ASME J. Vibr. Acoust.
,
16
, pp.
245
248
.
3.
Preumont, A., Random Vibration and Spectral Analysis, Kluwer Academics, Dordrecht, 1994.
4.
Pitoiset, X., Preumont, A., and Kernilis, A., 1998, “Tools for a Multiaxial Fatigue Analysis of Structures Submitted to Random Vibration,” in Proceedings European Conference on Spacecraft Structures, Materials and Mechanical Testing, Braunschweig, Germany, November, 1998.
5.
Pitoiset, X., and Preumont, A., 1999. “Spectral Methods for a Multiaxial Random Fatigue Analysis of Metallic Structures,” submitted to Int. J. Fatigue, December, 1999.