The Segalman-Reese Method for RMS von Mises Stress in Random Vibration

Structures subjected to random vibration environments are routinely evaluated for stress using the von Mises criterion, which condenses the full multiaxial stress state into a single scalar suited for comparison against a yield or fatigue allowable. But von Mises stress is a nonlinear function of the stress components, and this creates a trap in frequency-domain analysis. The RMS von Mises stress is not obtained by computing the RMS of each stress component and then substituting those RMS values into the von Mises formula.

Segalman, Fulcher, Reese & Field of Sandia National Laboratories resolved this problem with an elegant and computationally efficient method, published in 2000 and now implemented in several commercial finite element codes. This post walks through the method and shows why the naive component-RMS shortcut can err by a large margin.

The Von Mises Stress as a Quadratic Form

Collect the six independent Cauchy stress components at a point into a vector:

The square of the von Mises stress may then be written compactly as a quadratic form:

where $\mathbf{A}$ is a constant, symmetric matrix:

The quadratic character is the crux of the matter. Expectation is a linear operator, so the mean square of a linear response quantity follows directly from the response PSD. The mean square of a quadratic quantity involves the covariances among all of the stress components, including their cross terms and relative phase.

The Segalman-Reese Formulation

For a linear structure analyzed by modal superposition, the stress vector at a given location is expanded in terms of the modal stress vectors $\boldsymbol{\Psi}_i$ and the modal coordinates $q_i(t)$:

Substituting into the quadratic form and taking the expected value:

The matrix $T_{ij} = \boldsymbol{\Psi}_i^T \mathbf{A} \boldsymbol{\Psi}_j$ depends only on the modal stresses, computed once per location from the finite element model. The modal covariance matrix $\Gamma_{ij} = E[q_i q_j]$ is obtained by integrating the cross-spectral density of the modal coordinates over frequency:

where the modal cross-spectra follow from the frequency response functions $H_i(f)$ of each mode acting on the input force or base-acceleration spectral density. Note that $\Gamma_{ij}$ is computed once for the entire structure; only the small $T_{ij}$ matrix changes from point to point. The RMS von Mises stress is then simply:

The result is statistically exact for the mean square, requires no time-domain simulation, and is a cheap post-processing step on top of a standard random vibration modal analysis. The off-diagonal $\Gamma_{ij}$ terms carry the modal cross-correlation, which becomes significant when modes are closely spaced relative to their damping bandwidths.

Why the Naive Shortcut Fails

Consider a simple biaxial illustration: two in-plane normal stress components $\sigma_x(t)$ and $\sigma_y(t)$, each zero-mean Gaussian with the same RMS value $\sigma_0$, all shear components zero. The plane-stress von Mises relation is $\sigma_{vm}^2 = \sigma_x^2 + \sigma_y^2 – \sigma_x \sigma_y$. Taking expectations:

where $\rho$ is the correlation coefficient between the two components. The naive approach substitutes RMS values directly into the von Mises formula and always returns $\sigma_0$, blind to the correlation.

The Probability Distribution Question

A related subtlety: even when the applied random excitation is Gaussian, the von Mises stress is not Gaussian, nor is it Rayleigh-distributed. Its square is a quadratic form in Gaussian variables, which follows a generalized chi-square distribution. Segalman and his coauthors addressed this in a companion paper, providing a method for estimating the full probability distribution of von Mises stress so that peak values and exceedance probabilities can be assessed rationally.

This matters in practice because the familiar 3-sigma design habit is inherited from Gaussian thinking. Multiplying the RMS von Mises stress by three does not correspond to the same exceedance probability that 3-sigma implies for a Gaussian variable. For fatigue work, the distribution of von Mises stress cycles likewise differs from the narrowband Rayleigh assumption underlying many spectral fatigue methods, and appropriate care is warranted when feeding RMS von Mises values into damage calculations.

Practical Notes

The Segalman-Reese method is implemented in ANSYS Mechanical as the default combination for equivalent stress in PSD (random vibration) analyses, and equivalent formulations appear in other commercial codes. A few points for practitioners:

First, verify which combination method your solver uses for equivalent stress in random vibration results; older approaches based on component-RMS substitution still circulate. Second, retain enough modes: the modal stress matrix $T_{ij}$ converges more slowly than displacement quantities, and residual vector or mode acceleration corrections can be important for stress accuracy. Third, remember that the method delivers the RMS of von Mises stress; the choice of peak factor for a design margin is a separate decision that should acknowledge the non-Gaussian distribution discussed above.

Is the Method Frequency-Domain Only?

Not strictly. The combination formula itself is domain-agnostic: it requires only the modal covariance matrix $\Gamma_{ij} = E[q_i q_j]$, and nothing in the quadratic form cares how that matrix was obtained. In the original paper $\Gamma_{ij}$ comes from integrating the modal cross-spectral densities over frequency, because the target application is stationary random vibration specified by a PSD. But $\Gamma_{ij}$ could equally well be estimated from time-domain modal coordinate histories, whether simulated or measured, and the combination step would be identical.

The real underlying assumption is stationarity, not the domain. The formulation treats $\Gamma_{ij}$ as constant, so a genuinely transient event such as a shock would require a time-varying covariance $\Gamma_{ij}(t)$, at which point the method loses its elegance. And in a true time-domain analysis the method is largely unnecessary anyway: if the stress time histories are available, the von Mises stress can be evaluated directly at every time step and its RMS taken with no combination problem to solve. The Segalman-Reese machinery exists precisely because a frequency-domain analysis never produces those time histories — only spectral densities — and the quadratic nonlinearity blocks the direct route. In short, it is a stationary random vibration method expressed in the frequency domain because that is the only setting where the problem it solves actually arises.

From Stress Margins to Fatigue: The Preumont–Piefort EVMS PSD

Segalman-Reese delivers a single statistical moment — the mean square of von Mises stress — which makes it a stress-margin tool. Fatigue assessment by rainflow cycle counting demands more: the distribution of cycle ranges depends on the spectral content of the stress process, not merely its RMS. The method as published therefore cannot feed a rainflow count on its own. The bridge to fatigue is the equivalent von Mises stress (EVMS) PSD of Preumont and Piefort, which uses exactly the same quadratic-form machinery but declines to integrate everything down to one number. Instead, it defines a spectral density frequency by frequency:

where $\mathbf{S}_{ss}(f)$ is the 6×6 cross-spectral density matrix of the stress components at the point of interest. In the modal framework of this post, the EVMS PSD takes an especially convenient form built from the same ingredients already computed:

with $T_{ij} = \boldsymbol{\Psi}_i^T \mathbf{A} \boldsymbol{\Psi}_j$ as before. Only the real part of the modal cross-spectra survives because $\mathbf{A}$ and $T_{ij}$ are symmetric. By construction, integrating $S_{vm}(f)$ over frequency recovers the Segalman-Reese mean square exactly:

The two methods are thus siblings: Segalman-Reese is the zeroth spectral moment of the Preumont–Piefort PSD. But the EVMS carries the full spectral shape, so all of its spectral moments are available — and that is precisely what spectral fatigue methods consume. The EVMS PSD can be handed directly to Dirlik, Lalanne, Tovo-Benasciutti, or the Meta-Dirlik correction to estimate the rainflow cycle range distribution and Miner damage, exactly as one would for a uniaxial stress PSD. The multiaxial random stress state is collapsed to an equivalent uniaxial Gaussian process, and the familiar spectral fatigue pipeline takes over from there.

What the EVMS Is — and Is Not

Conceptually, the EVMS defines a hypothetical zero-mean Gaussian process whose PSD reproduces the second-moment spectral content of the von Mises quadratic form. This is a deliberate modeling choice with three consequences worth understanding.

The EVMS is a Gaussian fiction. The true von Mises process is non-negative and non-Gaussian, as discussed above. The EVMS replaces it with a signed, zero-mean Gaussian surrogate. Since Dirlik-type formulas are derived under a Gaussian assumption, the pairing is internally consistent — but it is an engineering model, not an exact statistical result, and validation studies show it is generally accurate to slightly conservative for broadband Gaussian environments.

It sidesteps the sign problem. Von Mises stress discards sign, which is fatal for rainflow counting performed literally on a von Mises time history: compression and tension excursions fold onto one another and the cycle ranges are corrupted. The EVMS never constructs the von Mises history at all; it works entirely in spectral densities, where the difficulty does not arise. (The time-domain analogue for proportional loading is the signed von Mises stress, with the sign borrowed from the dominant principal or hydrostatic stress.)

Nonproportional loading is its limit. When the principal stress axes rotate during the response — closely spaced modes with differently oriented modal stresses, or multiple uncorrelated excitation axes — any scalar-equivalent approach degrades, because a single equivalent process cannot represent damage that accumulates differently on differently oriented planes. In that regime, critical plane methods (Findley, Fatemi-Socie, and their spectral counterparts) are the more defensible tool, at the cost of a search over candidate plane orientations. A practical screening check is the degree of proportionality of the stress state: when one modal stress pattern dominates the response at the location of interest, the EVMS is on firm ground.

References

D.J. Segalman, C.W.G. Fulcher, G.M. Reese & R.V. Field Jr., An Efficient Method for Calculating R.M.S. von Mises Stress in a Random Vibration Environment, Journal of Sound and Vibration, Vol. 230, No. 2, 2000 (originally presented at IMAC XVI, 1998; Sandia Report SAND98-0260), https://www.osti.gov/servlets/purl/573295
D.J. Segalman, G.M. Reese, R.V. Field Jr. & C.W.G. Fulcher, Estimating the Probability Distribution of von Mises Stress for Structures Undergoing Random Excitation, ASME Journal of Vibration and Acoustics, Vol. 122, No. 1, 2000.
A. Preumont & V. Piefort, Predicting Random High-Cycle Fatigue Life with Finite Elements, ASME Journal of Vibration and Acoustics, Vol. 116, No. 2, 1994.
X. Pitoiset & A. Preumont, Spectral Methods for Multiaxial Random Fatigue Analysis of Metallic Structures, International Journal of Fatigue, Vol. 22, No. 7, 2000.
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