Peak Factor and Random Fatigue

Abstract

This second chapter on random vibration is devoted to the analysis of the statistics of the random response in order to assess the reliability of the structure; two important failure mechanisms are examined: the failure by threshold crossing and fatigue. The first part of this chapter is devoted to the peak factor, that is the statistics of the extreme value of the random process during a given observation period. Approximate formulae are derived. The relationship between the peak factor and the so-called Response Spectrum traditionally used in earthquake engineering is elucidated. The second part of this chapter is devoted to random fatigue according to the linear damage theory (Miner). Classical results of uniaxial Gaussian stresses with zero mean are reviewed first; the case of biaxial loading is also addressed, by the definition of a special uniaxial equivalent von Mises stress; the finite element formulation of the method is outlined. A set of problems is proposed at the end of the chapter.

Le premier pas vers la

philosophie, c’est l’incrédulité.

Diderot, dernières paroles... (1784)

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

9.1 Introduction

In the previous chapter, we have analyzed how the statistics of the structural response can be evaluated from the statistics of the random excitation. If the excitation process is Gaussian with zero mean (and the structure behaves linearly), the statistical information about the random response is entirely contained in the PSD functions, from which one can calculate the RMS value and the central frequency of the response, but this is not enough to assess the reliability of the structure. In this chapter, we consider the two most important failure mechanisms: threshold crossing and fatigue.

9.1.1 Threshold Crossings

In some situations, the design must guarantee that the stress level or the displacements remain below a given limit (e.g. vibration amplitude of a rotor remaining lower than the existing gap with the stator, or the stress level remaining below the yield stress or a given regulatory limit for a design earthquake in a nuclear plant). In these circumstances, the designer is interested in the probability distribution of the largest value $x_{max}$ of the response over the duration of the excitation. This largest value is related to the RMS value $\sigma _x$ by

$$\begin{aligned} x_{max}= \eta _e \ \sigma _x \end{aligned}$$

(9.1)

where $\eta _e$ is the peak factor (it is dimensionless). $\eta _e$ is an increasing function of the duration of the phenomenon (it also depends on the probability of exceedance that is accepted). This chapter considers only the simplest case of a stationary Gaussian process of zero mean.

9.1.2 Fatigue

In many cases, the structural failure occurs after a fairly large number of cycles, while the nominal stress never exceeds the yield stress of the material. This situation is called high-cycle fatigue; the structure remains linear and, if the excitation is Gaussian, the response is also Gaussian. This chapter considers high-cycle fatigue of uniaxial as well as multi-axial stress fields; however, the discussion is limited to stress fields with zero mean. It has been observed that a constant compression stress does not affect the endurance limit, while a constant tension stress reduces the endurance limit. The models describing the effect of a constant stress are not examined in this chapter.

9.2 Peak Factor

This section analyzes the probability distribution of the largest extremum over some duration $T$. We begin with the distribution of the maxima.

9.2.1 Maxima

Fig. 9.1

Sample of duration $T$, of a stationary Gaussian process $x(t)$ of zero mean and RMS value $\sigma$. Probability density functions of the process $x(t)$, of the maxima $b$ (the curve shown is a Rayleigh distribution which corresponds to a narrow band process). Probability density function of the largest maximum $x_{max}(T)$ over the duration $T$. The peak factor is defined by $x_{max}(T)= \eta _e(T) \ \sigma$.

Consider the Gaussian stationary random process of zero mean $x(t)$ (Fig. 9.1); the distribution of the maxima $b$ is fairly complicated and depends on the bandwidth of the process. It can be shown that the probability density function $q(\eta )$ of the reduced maxima $\eta =b/\sigma$ is given by (Cartwright & Longuet-Higgins 1956)

$$\begin{aligned} q(\eta )=\frac{1}{(2\pi )^{1/2}}\left[ \varepsilon e^{-\eta ^2/2\varepsilon ^2}+(1-\varepsilon ^2)^{1/2}\eta e^{-\eta ^2/2}\int _{-\infty }^{\eta (1-\varepsilon ^2)^{1/2}/\varepsilon }e^{-x^2/2} dx\right] \end{aligned}$$

(9.2)

where $\varepsilon$ is a parameter which characterizes the bandwidth of the process and can be expressed in terms of the spectral moments according to

$$\begin{aligned} 0 \le \varepsilon ^{2} = 1 - \frac{{m_{2}^{2} }}{{m_{0} m_{4} }} = 1 - \left( {\frac{{\nu _{0} }}{{\nu _{1} }}} \right) ^{2} \le 1 \end{aligned}$$

(9.3)

According to the discussion of Sect. 8.5, the average number of maxima per unit of time, $\nu _1$ tends to be close to the central frequency $\nu _0$ for a narrow band process, and $\varepsilon ^2$ is close to 0. On the contrary, $\nu _1\gg \nu _0$ for a wide band process and $\varepsilon ^2$ is close to 1. For a wide band process, the maxima can assume negative as well as positive values (as illustrated in Fig. 8.7b) and, when $\varepsilon \rightarrow 1$, $q(\eta )$ is Gaussian; for $\varepsilon \rightarrow 0$, it becomes identical to the Rayleigh distribution

$$\begin{aligned} q(\eta )=\eta \exp (-\frac{\eta ^2}{2}) \qquad (\eta \ge 0) \end{aligned}$$

(9.4)

represented in Fig. 8.9.

9.2.2 First-Crossing Problem

Let $x(t)$ be a stationary Gaussian process of zero mean and standard deviation $\sigma$. The reliability $W(T,\eta )$ is the probability that $x(t)$ remains in the safe domain $|x(t)|<\eta \sigma$ during the observation period:

$$\begin{aligned} W(T,\eta )=\mathrm{Prob }\{|x(t)|<\eta \sigma \,;\ 0\le t < T \} \end{aligned}$$

(9.5)

$W(T,\eta )$ represents the fraction of samples which have not left the safe domain after the duration $T$.

The probability density function of the first-crossing time is

$$\begin{aligned} p_1(T)=- \frac{\partial W(T,\eta )}{\partial T} \end{aligned}$$

(9.6)

because $p_1(T) dT$ represents the probability that the first-crossing occurs in the time interval $[T,\,T+dT[$.

Alternatively, $W(T,\eta )$ is the probability that the largest extremum during the observation period $T$ remains below $b=\eta \sigma$, that is (by definition) the probability distribution function of the largest extremum (during the observation period $T$).¹ It follows that the probability density function of the peak factor is

$$\begin{aligned} p_e(T,\eta )= \frac{\partial W(T,\eta )}{\partial \eta } \end{aligned}$$

(9.7)

$p_e(T,\eta )d\eta$ represents the probability that the largest extremum during the observation period $T$ belongs to the interval $[\eta ,\eta +d\eta [$.

The literature about the first-crossing problem is considerable and can only be touched in this text. A model for the reliability can be built by considering the extreme point process $y(n)$ constructed as indicated in Fig. 9.2; if the process is narrow band, the time interval between two consecutive points is $\varDelta \simeq 1/2\nu _0$ ($\nu _0=\omega _0/2\pi$ is the central frequency).

Fig. 9.2

(a) Sample of a narrow band process $x(t)$. (b) Maxima and minima; the time interval between two maxima is $\simeq 1/\nu _0$. (c) Absolute extrema of the process, $y(n)=|x(t)|_{max}$ the time interval between two consecutive points is $\varDelta \simeq 1/2\nu _0$.

If $h(n)$ stands for the probability that the $n^{\text{ th }}$ extremum is the first one beyond the threshold $\eta \sigma$ (under the condition that all previous ones are below the threshold),

$$\begin{aligned} h(n)=P[y(n)>\eta \sigma \ | \bigcap _{i=1}^{n-1} y(i)< \eta \sigma ] \end{aligned}$$

(9.8)

The reliability can be written

$$\begin{aligned} W(T,\eta )=\prod _{n=1}^N[1-h(n)] \end{aligned}$$

(9.9)

where $N$ is the total number of extreme points over the duration $T$.² From this expression, various models can be built by making assumptions on the way the extrema occur.

The simplest of all possible assumptions is that of independent extrema; in this case, the condition disappears from Eq. (9.8). If, in addition to this, the process is assumed narrow band, the maxima are distributed according to the Rayleigh distribution (9.4) and $N=2\nu _0T$ is the number of half-cycles over the duration $T$. It follows that

$$\begin{aligned} h(n)=P[y(n)>\eta \sigma ]=\int _\eta ^\infty xe^{-x^2/2} dx= e^{-\eta ^2/2} \end{aligned}$$

(9.10)

and

$$\begin{aligned} W(T,\eta )=\prod _{n=1}^N[1-h(n)]=(1-e^{-\eta ^2/2})^N \end{aligned}$$

(9.11)

For large values of $N$, using $\lim \nolimits _{n\rightarrow \infty }(1+\frac{x}{n})^n=e^x$, this result may be written alternatively

$$\begin{aligned} W(T,\eta )=\exp [-Ne^{-\eta ^2/2}]=\exp [-2\nu _0Te^{-\eta ^2/2}] \end{aligned}$$

(9.12)

This is the simplest model for the reliability. The assumption of independent extrema may be criticized, particularly for narrow band signals, because the extrema (which belong to the envelope process) tend to vary smoothly rather than being independent, as illustrated in Fig. 9.2. However, this model is conservative and tends to overestimate the peak factor. The probability density function of the peak factor is obtained by partial derivation with respect to $\eta$. There is a single parameter which is the number of half-cycles, $N=2\nu _0T$.

9.2.3 Peak Factor

Figure 9.3 shows $W(N,\eta )$ for several values of $N$; the reliability is by definition the probability distribution function of the peak factor and $p_e(N,\eta )$ is given by Eq. (9.7); the maximum value of $p_e(N,\eta )$ tends to increase with $N$ and it occurs for larger values of $\eta$ (typically between 3 and 5); the dispersion of $p_e$ about its maximum decreases with $N$. The average and standard deviation of the peak factor can be approximated by (Davenport 1964)

$$\begin{aligned} E[\eta _e]\simeq (2 \ln N)^{1/2}+\frac{\gamma }{(2 \ln N)^{1/2}} \end{aligned}$$

(9.13)

$$\begin{aligned} \sigma [\eta _e]\simeq \frac{\pi }{\sqrt{6}}\frac{1}{(2 \ln N)^{1/2}} \end{aligned}$$

(9.14)

where $\gamma =0.5772$ (Euler’s constant).

9.3 Response Spectrum

In earthquake engineering, there is a tradition of using the concept of response spectrum. For a given accelerogram$\ddot{x}_0$ (Fig. 9.4), the displacement spectrum, $S_d(\omega _n,\xi )$, is defined as the maximum of the absolute value of the relative displacement $y$ of a single d.o.f. oscillator of natural frequency $\omega _n$ and damping $\xi$ in response to the acceleration $\ddot{x}_0$. The relative displacement $y$ is solution of Eq. (8.24); $S_d(\omega _n,\xi )$ is regarded as a function of $\omega _n$, with $\xi$ as a parameter (for every different value of $\omega _n$ and $\xi$, the relative response $y(t)$ is different, and so is $|y|_{max}=S_d(\omega _n,\xi )$). The most frequent representation is a log-log diagram of the pseudo-velocity spectrum, defined by

$$\begin{aligned} S_v(\omega _n,\xi )=\omega _n S_d(\omega _n,\xi ) \end{aligned}$$

(9.15)

The pseudo-acceleration spectrum is defined by

$$\begin{aligned} S_a(\omega _n,\xi )= \omega _n S_v(\omega _n,\xi )=\omega _n^2 S_d(\omega _n,\xi ) \end{aligned}$$

(9.16)

$S_v$ is different from the maximum velocity of the response, but usually $S_a$ is close to the maximum absolute acceleration of the oscillator. In the log-log diagram of the pseudo-velocity spectrum, constant values of the relative displacement ($S_d$) and of the acceleration ($S_a$) appear as straight lines.

Fig. 9.3

(a) Reliability $W(N,\eta )=\exp [-Ne^{-\eta ^2/2}]$ for various values of the number of half-cycles $N$ (the reliability is by definition the probability distribution function of the peak factor). (b) Corresponding probability density function of the peak factor ($p_e=\partial W/\partial \eta$) based on independent extrema. The maximum of $p_e$ occurs for typical values of $\eta _e$ between 3 and 5.

Fig. 9.4

(a) Artificial earthquake accelerogram. (b) Response $y$ of the single d.o.f. oscillator for specific values $(\omega _n,\xi )$ of the natural frequency and the damping ratio; $S_d(\omega _n,\xi )=|y|_{max}$, the maximum of the absolute value. (c) Pseudo-velocity spectrum $S_v(\omega _n,\xi )=\omega _n S_d(\omega _n,\xi )$, for two values of the damping ratio (the two curves have the same high frequency asymptote).

For a given site, the expected characteristics of the ground acceleration (maximum acceleration, duration, frequency content) depend on the seismicity of the site and on the local geological conditions. Standard shapes of response spectra have been defined (Fig. 9.5), corresponding to various local soil conditions (bedrock, alluvion); they are normalized with respect to the local seismicity, expressed by the maximum acceleration for the site (often called “ZPA”: zero-period acceleration, corresponding to the high frequency asymptote of the response spectrum). The maximum acceleration is generally expressed as a fraction of the acceleration of gravity, $g$; $a_{max}$ can be 0.15$g$ in areas of moderate seismicity and 0.3$g$ and above in area of high seismic activity; the duration is typically between 10 and 30 sec.³

Fig. 9.5

Horizontal design response spectra of the US NRC Regulatory Guide 1.60, normalized to a maximum ground acceleration of ZPA=1$g$.

The regulatory design response spectra are supposed to define an envelope for all possible ground motions at one site; they depend on the return period considered. In practice, for the design of nuclear power plants, two sets of response spectra are defined for a site: the Operating Basis Earthquake is the earthquake that the plant is likely to experience once during its lifetime, and after which it is supposed to restart after only minor repairs. The Safe Shut-down Earthquake is the maximum possible earthquake for the site, for which the plant must be safely shut down and maintained in a cool state, without any significant release of reactivity; no restart is expected.

9.3.1 Maximum Structural Response

The historical reason for developing response spectra is that, if the structural response is dominated by a single mode, the maximum response can be directly evaluated from the response spectrum. Indeed, referring to Eq. (7.21), the modal response (of mode $i$) is governed by

$$\begin{aligned} \mu _{i}\ddot{z_{i}}+2\xi _i\mu _{i}\omega _{i}\dot{z_{i}}+\mu _{i}\omega _{i}^{2}z_{i} =\varGamma _i\ddot{x}_{0} \end{aligned}$$

(9.17)

and the structural response

$$\begin{aligned} \mathbf y =\varvec{\phi }_{i} z_i \end{aligned}$$

(9.18)

From the definition of the response spectrum, it follows that the maximum structural response of mode $i$ is given by

$$\begin{aligned} \mathbf y _i^{max}=\varvec{\phi }_{i}\, \frac{\varGamma _i}{\mu _{i}}\, S_d(\omega _i,\xi _i) \end{aligned}$$

(9.19)

where $S_d(\omega _i,\xi _i)$ is the value of the response spectrum for the natural frequency $\omega _i$ and the damping $\xi _i$ of the dominant mode.

If several modes contribute to the structural response, the maximum response of each mode can still be evaluated in the same way. For well separated modes, we have seen in Sect. 8.9 that the modal responses may be treated as independent and the global response may be evaluated by the SRSS rule:

$$\begin{aligned} y^{{max}} = \left\{ {\sum \limits _{i} {[y_{i}^{{max}} ]^{2} } } \right\} ^{{1/2}} \end{aligned}$$

(9.20)

This combination rule may lead to substantial errors for closely spaced modes. Alternative combination rules have been proposed, but this topic is outside the scope of this textbook.

An elegant alternative to the use of response spectra would be to replace the specification of the ground motion by a finite duration stationary random process of specified power spectral density; the ground acceleration would be defined by a PSD $\varPhi _0(\omega )$ and a window function. However, the response spectra being defined by the regulatory authorities, the definition of a PSD consistent with a given response spectrum remains a subject of discussion. The relationship between the PSD and the response spectra is outlined below.

9.3.2 Relation Between $S_v(\omega ,\xi )$ and $\varPhi _0(\omega )$

If the acceleration time-history $\ddot{x}_{0}$ is known, the response spectrum $S_d(\omega ,\xi )$ is entirely determined; the reverse is not true. However, for a lightly damped oscillator, the response is strongly influenced by the energy contained in the excitation in the vicinity of the natural frequency of the oscillator. If one assumes that the oscillator response is a stationary random process of duration $T$, an approximate relationship between $S_v(\omega ,\xi )$ and $\varPhi _0(\omega )$ may be obtained by the following steps:

• 1. According the the white noise approximation, the RMS response of the oscillator of frequency $\omega _n$ is related to the PSD of the excitation by Eq. (8.30)

  $$\begin{aligned} \sigma ^2=\frac{\pi \varPhi _0(\omega _n)}{2\xi \omega _n^3} \end{aligned}$$

  (9.21)

• 2. The maximum response is related to the RMS response by the peak factor $\eta _e(N)$ which depends on the number of half-cycles. For a narrow band process, the central frequency may be approximated by the natural frequency of the oscillator, $\nu _0\simeq \omega _n/2\pi$. Thus, $N=2\nu _0T \simeq \omega _nT/\pi$ and the average value of the peak factor, $E[\eta _e]$ may be evaluated from (9.13).

• 3. Finally, by definition, the response spectrum and the peak factor are related by

  $$\begin{aligned} S_d(\omega _n,\xi )=\sigma E[\eta _e] \end{aligned}$$

  (9.22)

  This leads to

  $$\begin{aligned} \varPhi _{0} (\omega _{n} ) = \frac{{2\xi \omega _{n}^{3} }}{\pi }\left[ {\frac{{S_{d} (\omega _{n} ,\xi )}}{{E[\eta _{e} ]}}} \right] ^{2} \end{aligned}$$

  (9.23)

  This simple model neglects all the transient effects present in the oscillator response, and is based on several simplifying assumptions; however, the estimator may be improved iteratively to fit the average response spectrum of a set of artificially generated accelerograms.⁴

9.4 Random Fatigue

9.4.1 S–N Curve

The characterization of the uniaxial fatigue behavior of materials is done by subjecting a sample to an alternating sine loading of constant amplitude $S$ and counting the number of cycles $N$ until the start of the first crack. The test is repeated for increasing values of the stress and the results are presented in the form of a curve $S(N)$ which is known as the Wöhler curve of the material, or simply the “$S\!-\!N$” curve. Some materials have an endurance limit, $S_e$, which is the limit stress under which the material can sustain an infinite number of cycles without initiation of a crack; some other materials do not have an endurance limit. For a large class of materials, the $S\!-\!N$ curve may be approximated by⁵

$$\begin{aligned} NS^{\beta }=c \end{aligned}$$

(9.24)

where $\beta$ and $c$ are material constants ($5<\beta <20$). The present discussion is focused on alternating stress fields and does not discuss the situation where the alternating stress is superimposed on a constant stress. The random stress is assumed to be a zero mean stationary Gaussian process.

9.4.2 Linear Damage Theory

Fatigue life for complex load histories can be treated by a cumulative damage analysis; the linear damage theory (also known as Palmgren-Miner criterion) assumes that $n_i$ cycles at a stress level $S_i$ produce a damage $D_i=n_i/N_i$, where $N_i=cS_i^{-\beta }$ is the number of cycles of the S–N curve under the constant amplitude $S_i$. According to the linear damage theory, the total damage associated with various levels of stress may be added linearly, producing a total damage

$$\begin{aligned} D=\sum _i \frac{n_i}{N_i} \end{aligned}$$

(9.25)

and the crack initiates when the total damage reaches $D=1$. This criterion does not take into account the order of application of the various stress levels; it is known to be inaccurate, but it has the enormous advantages of being simple and of relying on constant amplitude tests for which experimental data are widely available. Although inadequate as an absolute indicator of fatigue resistance, the linear damage $D$ provides a valuable relative information and can be used for design purposes: a design leading to a smaller value of $D$ is probably a better design. If the stress histories are available, the counting of the stress cycles can be done according to the rainflow method. In the present discussion which is concerned with random stress fields, the distribution of the stress cycles is derived from the PSD of the random stress.

9.4.3 Uniaxial Loading

Let $x(t)$ be a uniaxial Gaussian stress of zero mean and PSD $\varPhi (\omega )$. The material has a S–N curve of the form (9.24) and behaves according to the linear damage theory (9.25).

In the classical theory of random fatigue, the counting of the cycles is related to the occurrence of maxima. To account for the fact that fatigue damage is related to tension rather than compression stresses, it is assumed that every maximum with a positive amplitude $b$ contributes for one cycle to the damage, and that a maximum with a negative amplitude does not contribute to the damage. Thus, the damage associated with one cycle of amplitude $b$ is

$$\begin{aligned} D=\frac{1}{N}=\frac{1}{cb^{-\beta }}=b^{\beta }c^{-1}\qquad (b>0) \end{aligned}$$

(9.26)

where $N$ is the number of cycles of the S–N curve corresponding to the stress level $b$. If the average number of maxima per unit of time is $\nu _1$ [given by Eq. (8.42)], the average number of maxima per unit of time in $[b,b+db[$ is $\nu _1q(b)db$, where $q(b)$ is the probability density function of the maxima, given by Eq. (9.2). This leads to a rate of damage $b^{\beta }c^{-1}\nu _1q(b)db$. The expected damage per unit of time is obtained by integrating the damage rate over all possible values of the stress amplitude $b$,

$$\begin{aligned} E[D]=\nu _1c^{-1}\int _0^{\infty }b^{\beta }q(b)\,db \end{aligned}$$

(9.27)

Introducing the reduced stress, $\eta =b/\sigma _x$,

$$\begin{aligned} E[D]=\nu _1c^{-1}\sigma _x^\beta \int _0^{\infty }\eta ^{\beta }q(\eta )\,d\eta \end{aligned}$$

(9.28)

where $\sigma _x=m_0^{1/2}$ is the standard deviation (RMS value) of the stress. For a narrow band process, $q(\eta )$ is the Rayleigh distribution (9.4) and the rate of maxima $\nu _1$ may be approximated by the central frequency $\nu _0$, leading to

$$\begin{aligned} E[D]=c^{-1}\nu _0\sigma _x^\beta \int _0^{\infty }\eta ^{\beta +1}e^{-\eta ^2/2}d\eta \end{aligned}$$

or

$$\begin{aligned} E[D]=c^{-1}\nu _0\sigma _x^\beta \,2^{\beta /2}\varGamma (\beta /2+1) \end{aligned}$$

(9.29)

where $\varGamma (.)$ is the Gamma function.⁶ Upon substituting $\sigma _x=m_0^{1/2}$ and $\nu _0=\frac{1}{2\pi }(m_2/m_0)^{1/2}$,

$$\begin{aligned} E[D]=c^{-1}\frac{2^{\beta /2}}{2\pi }\varGamma (\beta /2+1)m_0^{(\beta -1)/2}m_2^{1/2} \end{aligned}$$

(9.30)

This result was first derived by (Miles 1954). It can also be used as an approximation for a wide band process, but it tends to be conservative. Alternative prediction models have been proposed, based on other spectral moments, but they are outside this introductory text.

Note that Eq. (9.29) shows that the damage is proportional to $\sigma _x^\beta$, where $\beta$ is the power of the S–N curve. In fatigue testing, it is often necessary to increase the level of excitation to reduce the duration of the tests; if the PSD $\varPhi _0(\omega )$ of the excitation is multiplied by a scalar $a$ while keeping the frequency content unchanged, the moment $m_0$ will also be multiplied by $a$ and the RMS response $\sigma _x$ by $a^{1/2}$. Thus, the damage per unit of time will be multiplied by $a^{\beta /2}$ and the duration of the test leading to a given damage will be reduced accordingly (Problem 9.2).

9.4.4 Biaxial Loading

Biaxial stress states are very important, because cracks often initiate at the surface, where the stress is biaxial. The von Mises criterion has been found to correlate fairly well with a lot of experimental data for biaxial stress states. The equivalent von Mises stress is defined by a quadratic equation,

$$\begin{aligned} s_c^2=s_x^2+s_y^2-s_xs_y+3s_{xy}^2 \end{aligned}$$

(9.31)

or

$$\begin{aligned} s_c^2=\mathbf s ^TQ\mathbf s \end{aligned}$$

(9.32)

where $\mathbf s =(s_x,s_y,s_{xy})^T$ is the stress vector and $Q$ is defined by

$$\begin{aligned} Q=\left[ \begin{array}{cc@{\quad }c} 1 &{} -1/2 &{} 0 \\ -1/2 &{} 1 &{} 0 \\ 0 &{} 0 &{} 3 \end{array} \right] \end{aligned}$$

(9.33)

If one considers this quadratic relationship in the strict sense, one would normally have to work in the time domain rather than in the frequency domain,⁷ and the Gaussian property would also be lost. Alternatively, in what follows, we consider it in the mean square sense to define a new Gaussian scalar random process. Taking advantage of the relationship

$$\begin{aligned} \mathbf s ^TQ\mathbf s =\mathrm{Trace }\{Q[\mathbf ss ^T]\} \end{aligned}$$

(9.34)

the mean square value of the von Mises stress reads

$$\begin{aligned} \mathrm{E }[s_c^2]=\mathrm{E }[\mathbf s ^TQ\mathbf s ]=\mathrm{Trace }\{Q\mathrm{E }[\mathbf ss ^T]\} \end{aligned}$$

(9.35)

where the expectation $\mathrm{E }[.]$ applies only to the random quantities. $\mathrm{E }[\mathbf ss ^T]$ is the covariance matrix of the stress vector, related to the PSD matrix of the stress vector, $\varPhi _{ss}(\omega )$ by

$$\begin{aligned} \mathrm{E }[\mathbf ss ^T]=\int _{-\infty }^{\infty }\varPhi _{ss}(\omega )d\omega \end{aligned}$$

(9.36)

Thus, if one defines a equivalent random stress $s_c(t)$ as a zero mean Gaussian process with PSD

$$\begin{aligned} \varPhi _c(\omega )= \mathrm{Trace }\{Q\,\varPhi _{ss}(\omega )\} \end{aligned}$$

(9.37)

this process has indeed the same mean square value as the von Mises stress,

$$\begin{aligned} \mathrm{E }[s_c^2]=\int _{-\infty }^{\infty }\varPhi _c(\omega )d\omega \end{aligned}$$

(9.38)

Since a zero mean Gaussian process is entirely defined by its PSD, Eq. (9.37) may be regarded as the definition of an equivalent uniaxial alternating von Mises stress. Although this process varies noticeably from the time-histories defined by Eq. (9.31), it is meaningful from the physical point of view, because:

• (i) In the uniaxial case where only a single stress component $s_x\ne 0$, $\varPhi _c(\omega )=\varPhi _x(\omega )$, so that $s_c=s_x$.

• (ii) The process $s_c$ defined by (9.37) does have components at all the natural frequencies of the structure, unlike the time-histories obtained by (9.31).

9.4.5 Finite Element Formulation

The stress vector at one point (or within one finite element) is related to the modal amplitudes by

$$\begin{aligned} \mathbf s =S \, \mathbf z \end{aligned}$$

(9.39)

where $S$ is the $(3\times m)$ modal stress matrix at this point. The corresponding $(3\times 3)$ PSD matrix of the stress vector is

$$\begin{aligned} \varPhi _{ss}(\omega )=S \,\varPhi _{z}(\omega )\, S^T \end{aligned}$$

(9.40)

where $\varPhi _{z}(\omega )$ is the $(m \times m)$ PSD matrix of the modal responses, computed according to Sect. 8.8. $\varPhi _c(\omega )$ can be calculated by (9.37) and the procedure for uniaxial stress discussed above applies.

9.5 Problems

P.9.1 Show that, according to the simplified model relating the response spectrum and the power spectral density developed in Sect. 9.3.2, the amplitudes of the response spectrum for two different values of the damping ratio are related by

$$\begin{aligned} \frac{S_d(\omega _n,\xi _1)}{S_d(\omega _n,\xi _2)}\approx \left( \frac{\xi _2}{\xi _1}\right) ^{\frac{1}{2}} \end{aligned}$$

P.9.2 Accelerated fatigue test: a specimen must be subjected to a fatigue endurance test of duration $T$ with a stationary random excitation of prescribed PSD $\varPhi (\omega )$. In order to reduce the duration of the test, it is considered to scale up the excitation. Determine the scaling factor of the PSD, in order to produce the same damage in the reduced time $T/\alpha$.