In the first-order methods, the performance function W (X), defined on the basis of the loading and resistance functions g(XL) and h(XR), are expanded in a Taylor series at a reference point. The second – and higher-order terms in the series expansion are truncated, resulting in an approximation involving only the first two statistical moments of the variables. This simplification greatly en­hances the practicality of the first-order methods because, in many problems, it is rather difficult, if not impossible, to find the PDF of the variables, whereas it is relatively simple to estimate the first two statistical moments. The pro­cedure is based on the first-order variance estimation (FOVE) method, which is summarized below. For a detailed description of the method in uncertainty analysis, readers are referred to Tung and Yen (2005, Sec. 5.1).

The first-order variance estimation (FOVE) method, also called the vari­ance propagation method (Berthouex, 1975), estimates uncertainty features of a model output based on the statistical properties of the model’s stochas­tic basic variables. The basic idea of the method is to approximate a model involving stochastic basic variables by a Taylor series expansion. Consider that a hydraulic or hydrologic performance function W (X) is related to K stochastic basic variables because W(X) = W(X1,X2,…,XK), in which X = (X1, X 2,…, XK )t, a K-dimensional column vector of variables in which all Xs are subject to uncertainty, the superscript t represents the transpose of a matrix or vector. The Taylor series expansion of the performance function W (X) with respect to a selected point of stochastic basic variables X = xo in the parameter space can be expressed as

in which wo = W(xo), and є represents the higher-order terms. The partial derivative terms are called sensitivity coefficients, each representing the rate of change in the performance function value W with respect to the unit change of the corresponding variable at xo.

Dropping the higher-order terms represented by є, Eq. (4.19) is a second – order approximation of the model W(X). Further truncating the second-order terms from it leads to the first-order approximation of W as

or in a matrix form as

W (X) = wo + sO (X – x o)

where sO = VxW(xo) is the column vector of sensitivity coefficients with each element representing dW/дXk evaluated at X = xO. The mean and variance of W by the first-order approximation can be expressed, respectively, as

In matrix forms, Eqs. (4.22) and (4.23) can be expressed as

in which fix and Cx are the vectors of the means and covariance matrix of the stochastic basic variables X, respectively.

Commonly, the first-order variance estimation method consists of taking the expansion point xo = (Ax at which the mean and variance of W reduce to

{Aw & g(Mx) — w (4.26)

and aw & s*CxS (4.27)

in which s = Vx W(p, x) is a K-dimensional vector of sensitivity coefficients eval­uated at xo = fAx. When all stochastic basic variables are independent, the variance of model output W could be approximated as

K

aw ^2 skal =s * Dxs (4.28)

k=1

in which Dx = diag(a12, af,…, aK)isaK x K diagonal matrix of variances of the involved stochastic basic variables. From Eq. (4.28), the ratio sak/Var( W) indi­cates the proportion of the overall uncertainty in the model output contributed by the uncertainty associated with the stochastic basic variable Xk.

The MFOSM method for reliability analysis first applies the FOVE method to estimate the statistical moments of the performance function W (X). This is done by applying the expectation and variance operators to the first-order Taylor series approximation of the performance function W (X) expanded at the mean values of the stochastic basic variables. Once the mean and stan­dard deviations of W (X) are estimated, the reliability is computed according to Eqs. (4.9) or (4.10), with the reliability index ^MFOSM computed as

Amfosm = , (4.29)

Vs1 Cx s

where fxx and Cx are the vectors of means and covariance matrix of stochas­tic basic variables X, respectively, and s = VxW(px) is the column vector of sensitivity coefficients with each element representing d W/дXk evaluated at X = l^x.

Example 4.6 Manning’s formula for determining flow capacity of a storm sewer is

Q = 0.463n—1 D267 S 05

in which Q is flow rate (in ft3/s), n is the Manning roughness coefficient, D is the sewer diameter (in ft), and S is pipe slope (in ft/ft). Because roughness coefficient n, sewer diameter D, and sewer slope S in Manning’s formula are subject to uncertainty owing to manufacturing imprecision and construction error, the sewer flow capacity would be subject to uncertainty. Consider a section of circular sewer pipe with the following features:

Compute the reliability that the sewer capacity could convey a discharge of 35 ft3/s. Assume that stochastic model parameters n, D, and S are uncorrelated.

Solution The performance function for the problem is W = Q — 35 = 0.463n-1 D267 S0 5 — 35. The first-order Taylor series expansion of the performance function about no = xn = 0.015, Do = xd = 3.0, and So = xs = 0.005, according to Eq. (4.20), is

W ^ 0.463(0.015)—1(3)2’67(0.005)°’5 + (9Q/дn)(n — 0.015) + (9Q/дD)(D — 3.0)

+ (9 Q/д S )(S — 0.0005) — 35

= 41.01 — 2733.99(n — 0.015) + 36.50( D — 3.0) + 4100.99( S — 0.005) — 35 Based on Eq. (4.26), the approximated mean of the sewer flow capacity is

Xw ^ 41.01 — 35 = 6.01 ft3/s

Owing to independency of n, D, and S, according to Eq. (4.28), the approximated variance of the performance function W is

oQ ъ (2733.99)[6]Var(n) + (36.50)2Var(D) + (4100.99)2Var(S)

Since

Var(n) = (QnM-n)2 = (0.05 x 0.015)2 = (7.5 x 10—4)2 Var(D) = (QDjxD)2 = (0.05 x 3.0)2 = (1.5 x 10-1)2 Var(S) = (QS/xS)2 = (0.05 x 0.005)2 = 0.000252 = 6.25 x 10—8 the variance of the performance function W can be computed as

oQ ъ (2733.99)2(7.5 x 10—4)2 + (36.50)2(1.5 x 10-1)2 + (4100.99)2(2.5 x 10—4)2 = 2.052 + 5.472 + 1.032 = 35.23(ft[7]/s)2

Hence the standard deviation of the sewer flow capacity is V35.23 = 5.94 ft3/s.

The MFOSM reliability index is ^mfosm = 6.01/5.94 = 1.01. Assuming a normal distribution for Q, the reliability that the sewer capacity can accommodate a discharge of 35 ft3/s is

Ps = P [Q > 35] = $(^mfosm) = Ф(1.01) = 0.844 The corresponding failure probability is pf = Ф( —1.01) = 0.156.

Yen and Ang (1971), Ang (1973), and Cheng et al. (1986b) indicated that provided that ps < 0.99, reliability is not greatly influenced by the choice of distribution for W, and the assumption of a normal distribution is satisfactory. However, for reliability higher than this value (for example, ps = 0.999), the shape of the tail of a distribution becomes very critical. In such cases, accurate assessment of the distribution of W (X) should be used to evaluate the reliability or failure probability. The MFOsM method has been used widely in various hydrosystems infrastructural designs and analyses such as storm sewers (Tang and Yen, 1972; Tang et al., 1975; Yen and Tang, 1976; Yen et al., 1976), culverts (Yen et al., 1980; Tung and Mays, 1980), levees (Tung and Mays, 1981; Lee and Mays, 1986), floodplains (McBean et al., 1984), and open-channel hydraulics (Huang, 1986).

Example 4.7 Referring to Example 4.6, using the same values of the mean and stan­dard deviation for sewer flow capacity, the following table lists the reliabilities and failure probabilities determined by different distributional assumptions for the sewer flow capacity Q to accommodate the inflow discharge of 35 ft3/s.

As can be seen, using different distributional assumptions might result in signifi­cant differences in the estimation of failure probability. This results mainly from the fact that the MFOSM method solely uses the first two moments without taking into account the distributional properties of the random variables.

Assuming that stochastic parameters in the sewer capacity formula (that is, n, D, and S) are uncorrelated lognormal random variables, the sewer capacity also is a lognormal random variable. The following table lists the values of the exact reliabil­ity index and failure probability and those obtained from the MFOSM by Eq. (4.29) and (4.8). The table indicates that approximation by the MFOSM becomes less and less accurate as the computation approaches the tail portion of the distribution.

Application of the MFOSM method is simple and straightforward. However, it possesses certain weaknesses in addition to the difficulties with accurate es­timation of extreme failure probabilities mentioned earlier. These weaknesses include [8] 2 3

of the original performance function. In case the performance function is highly nonlinear, linear approximation of such a nonlinear function will not be accurate. Consequently, the estimations of the mean and variance of a nonlinear performance function will be less accurate. The accuracy asso­ciated with the estimated mean and variance deteriorates rapidly as the degree of nonlinearity of the performance function increases. For a linear performance function, the FOVE method would produce the exact values for the mean and variance.

4. Sensitivity of the computed failure probability to the formulation of the per­formance function W. Ideally, the computed reliability or failure probabil­ity for a system should not depend on the definition of the performance function. However, this is not the case for the MFOSM method. This phe­nomenon of lack of invariance to the type of performance function is shown in Figs. 4.4 and 4.5. The main reason for this inconsistency is because the MFOSM method would result in different first-order approximations for dif­ferent forms of the performance function. Consequently, different values of mean and variance will be obtained, resulting in different estimations of re­liability and failure probability for the same problem. This behavior of the MFOSM could create an unnecessary puzzle for engineers with regard to which performance function should be used to obtain an accurate estimation of reliability. This is not an easy question to answer, in general, except for a very few simple cases. Another observation that can be made from Figs. 4.4 and 4.5 is that the discrepancies among failure probabilities computed by the MFOSM method using different performance functions become more pronounced as the uncertainties of the stochastic basic variables get larger.

5. Limited ability to use available probabilistic information. The reliability index в gives only weak information on the probability of failure, and thus the appropriate system probability distribution must be assumed. Further, the MFOSM method provides no logical way to include available information on basic variable probability distributions.

From these arguments, the general rule of thumb is not to rely on the result

of the MFOSM method if any of the following conditions exist: (1) high accuracy

requirement for the estimated reliability or failure probability, (2) high non­linearity of the performance function, and (3) many skewed random variables involved in the performance function. However, Cornell (1969) made a strong defense for the MFOSM method from a practical standpoint as follows:

An approach based on means and variances may be all that is justified when one appreciates (1) that data and physical arguments are often insufficient to establish the full probability law of a variable; (2) that most engineering analyses include an important component of real, but difficult to measure, professional uncertainty; and (3) that the final output, namely, the decision or design parameters, is often not sensitive to moments higher than the mean and variance.

To reduce the effect of nonlinearity, one way is to include the second-order terms in the Taylor series expansion. This would increase the burden of analysis by having to compute the second-order partial derivatives. Another alternative within the realm of first-order simplicity is given in Sec. 4.5. Section 4.6 briefly describes the basis of the second-order reliability analysis techniques.

From Eqs. (4.1) and (4.4) one realizes that the computation of reliability requires knowledge of the probability distributions of the load and resistance or of the performance function W. In terms of the joint PDF of the load and resistance, Eq. (4.1) can be expressed as

in which f R L(r, t) is the joint PDF of random load L and resistance R, r and t are dummy arguments for the resistance and load, respectively, and (r1, r2) and (t1, t2) are the lower and upper bounds for the resistance and load, respectively. The failure probability can be computed as

This computation of reliability is commonly referred to as load-resistance interference.

TABLE 4.1 Reliability Formulas for Selected Distributions
Distribution of W
Coefficient of variation Reliability ps — P (W > 0)
Probability density function fw(w)	Mean
Ow
Normal	Ф
°wl^w
2
МГп w Oln w
vaw-1
Lognormal	Ф
1 1 + в wo
g-fti-w-wo)
Exponential
1 – IG[a, в(w – $)]| Г(а)*
Gamma
a + в$
Bu (a, в)t B(a, в)
а a^—– (b — a) a + в	ав	(b – a)
Beta
a + в + 1 (a + в)^и
Triangular	for a w m
/ 1/2 /1 ab + am + bm V2 6^w J
a + m + b 3
_ (b – w)2 (b – a)(b – m) for m < w < b b – w b-a
for m w b
1 b – a V3 b + a
a + b 2
1
for a < w < b
Uniform
ba
*IG( ) — incomplete gamma function. t Bu(■) — incomplete beta function. SOURCE: After Yen et al. (1986).

Example 4.2 Consider the following joint PDF for the load and resistance: f R, L(r, t) = (r + t + r t)e-(r +t+rt) for r > 0,t > 0

Compute the reliability ps.

Solution According to Eq. (4.11), the reliability can be computed as

/* TO

/ [-(1 + t)e-(r +t+r t) ]0 dr

■J0

When the load and resistance are statistically independent, Eq. (4.11) can be reduced to

Ps = Г2 FL(r) fR(r) dr = Er [Fl(R)] (4.13a)

Jr 1

or Ps ^"2[1 – Fr(t)] fL(t) dt = 1 – El[Fr(L)] (4.13b)

Jt1

in which Fl() and Fr () are the marginal CDFs of random load L and resistance R, respectively, Er [Fl(R)] is the expected value of the CDF of random load over the possible range of the resistance, and El[Fr(L)] is the expected value of the CDF of random resistance over the possible range of the load. Similarly, the failure probability, when the load and resistance are independent, can be expressed as

Pf = 1 – Ps = Er[1 – Fl(R)] = El[Fr(L)] (4.14)

A schematic diagram illustrating load-resistance interference in the reliability computation, when the load and resistance are independent random variables, is shown in Fig. 4.2.

Example 4.3 Consider that the load and resistance are uncorrelated random vari­ables, each of which has the following PDF:

Load (exponential distribution):

f L(t) = 2e-2t for t > 0

Resistance (Erlang distribution):

f r(r) = 4re-2r for r > 0

Compute the reliability Ps.

Solution Since the load and resistance are uncorrelated random variables, the relia­bility ps can be computed according to Eq. (4.13a) as

/•TO

/ (4re-2r )(1 – e-2r) dr

Jq

1 + r) e-4r – (1 + 2r )e-2r

= 0.75

In the case that the PDF of the performance function W is known or derived, the reliability can be computed according to Eq. (4.4) as

п to

Ps = / fw(w) dw (4.15)

0

in which f w(w) is the PDF of the performance function.

Example 4.4 Define the performance function W = R – L, in which R and L are independent random variables with their PDFs given in Example 4.2. Determine the reliability ps using Eq. (4.15).

Solution To use Eq. (4.15) for the reliability computation, it is necessary to first obtain the PDF of the performance function W. Derivation of the PDF of W can be made based on the derived distribution method described in Tung and Yen (2005, Sec. 3.1) as follows: Define W = R – L and U = L from which the original random variables R and L can be expressed in terms of new random variables W and U as L = U and R = W + U. By the transformation of variables, the joint PDF of W and U can be expressed as

fw, u(w, u) = f R, L(r, t)| J |

in which the Jacobian matrix J is

The absolute value of the determinant of the Jacobian matrix | J | is equal to one. Hence the joint PDF of W and U is

fw, u(w, u) = f r(r) fL(t)| J | = f r(w + u) f l(u)(1) = 8(w + u)e 2(w+2u)

for – to < w < ж and u > 0. Because the marginal PDF associated with the perfor­mance function W is needed, it can be obtained from the preceding joint PDF as

From the derived PDF for W, the reliability can be computed as

In the conventional reliability analysis of hydraulic engineering design, un­certainty from the hydraulic aspect often is ignored. Treating the resistance or capacity of the hydraulic structure as a constant reduces Eq. (4.11) to

Ps = l fbU) di (4.16)

0

in which ro is the resistance of the hydraulic structure, a deterministic quan­tity. If the PDF of the hydrologic load is the annual event, such as the annual maximum flood, the resulting annual reliability can be used to calculate the corresponding return period.

To express the reliability in terms of stochastic variables in load and resis­tance functions, Eq. (4.11) can be written as

in which f (xL, xR) is the joint PDF of model stochastic basic variables X. For independent stochastic basic variables X, Eq. (4.17) can be written as

K

П fk (Xk) d xr (4.18)

k=m+1

in which fk ( ) is the marginal PDF of the stochastic basic variable Xk.

The method of direct integration requires the PDFs of the load and resistance or the performance function to be known or derived. This is seldom the case in practice, especially for the joint PDF, because of the complexity of hydrologic and hydraulic models used in design. Explicit solution of direct integration can be obtained for only a few PDFs, as given in Table 4.1 for the reliability ps. For most other PDFs, numerical integration may be necessary. Computation­ally, the direct integration method is analytically tractable for only very few special combinations of probability distributions and functional relationships. For example, the distribution of the safety margin W expressed by Eq. (4.5) has a normal distribution if both load and resistance functions are linear and all stochastic variables are normally distributed. In terms of the safety factor expressed as Eqs. (4.6) and (4.7), the distribution of W (X) is lognormal if both load and resistance functions have multiplicative forms involving lognormal stochastic variables. Most of the time, numerical integrations are performed for reliability determination. When using numerical integration (including Monte Carlo simulation described in Chap. 6), difficulty may be encountered when one deals with a multivariate problem. Appendix 4A summarizes a few one­dimensional numerical integration schemes.

Example 4.5 Referring to Example 4.1, the stochastic basic variables n, D, and S in Manning’s formula to compute the sewer capacity are independent lognormal random variables with the following statistical properties:

Compute the reliability that the sewer can convey the inflow discharge of 35 ft3/s.

Solution In this example, the resistance function is R(n, D, S) = 0.463 n-1 D2 67S0 5, and the load is L = 35 ft3/s. Since all three stochastic parameters are lognormal random variables, the performance function appropriate for use is

W(n, D, S) = ln(R) – ln(L)

= [ln(0.463) – ln(n) + 2.67 ln(D) + 0.5ln(S)] – ln(35)

= — ln(n) + 2.67 ln(D) + 0.5 ln(S) – 4.3319

The reliability ps = P [W(n, D, S) > 0] then can be computed as follows:

Since n, D, and S are independent lognormal random variables, ln(n), ln(D), and ln( S) are independent normal random variables. Note that the performance function W(n, D, S) is a linear function of normal random variables. Then, by the reproductive property of normal random variables as described in Sec. 2.6.1, W(n, D, S) also is a normal random variable with the mean

Pw = – Mln(n) + 2.67Pln(D) + °.5Mln(S) – 4.3319

and the variance

Var( W) = Var[ln(n)] + 2.672Var[ln( D)] + 0.52Var[ln( S)]

From Eq. (2.67), the means and variances of log-transformed variables can be obtained as

Var[ln(n)] = ln(1 + 0.052) = 0.0025 Pln(n) = ln(pn) – 0.5 Var[ln(n)] = -4.201

Var[ln(D)] = ln(1 + 0.022) = 0.0004 Pln(D) = ln(pD) – 0.5 Var[ln(D)] = 1.0984

Var[ln(S)] = ln(1 + 0.052) = 0.0025 Pln(S) = ln(pg) – 0.5 Var[ln(S)] = —5.2996

Then the mean and variance of the performance function W (n, D, S) can be computed as

pw = 0.1517 Var(W) = 0.005977

The reliability can be obtained as

ps = P (W > 0) = Ф f—) = ф( 10,1517 ^ = Ф(1.958) = 0.975 awJ VV0.005977 J

In reliability analysis, Eq. (4.3) alternatively can be written in terms of a per­formance function W (X) = W (XL, XR) as

ps = P [W(Xl, Xr) > 0] = P [W(X) > 0] (4.4)

in which X is the vector of basic stochastic variables in the load and resistance functions. In reliability analysis, the system state is divided into the safe (sat­isfactory) set defined by W (X) > 0 and the failure (unsatisfactory) set defined by W (X) < 0 (Fig. 4.1). The boundary that separates the safe set and failure set is a surface, called the failure surface, defined by the function W(X) = 0, called the limit-state function. Since the performance function W(X ) defines the condition of the system, it is sometimes called system-state function.

The performance function W(X) can be expressed differently as

Wi(X) = R – L = h(XR) – g(XL) (4.5)

W2(X) = (R/L) – 1 = [h(Xr)/g(Xl)] – 1 (4.6)

W3(X) = ln(R/L) = ln[h(Xr)] – ln[g(Xl)] (4.7)

Referring to Sec. 1.6, Eq. (4.5) is identical to the notion of a safety margin, whereas Eqs. (4.6) and (4.7) are based on safety factor representations.

Example 4.1 Consider the design of a storm sewer system. The sewer flow-carrying capacity Qc (ft3/s) is determined by Manning’s formula:

Qc = 0463xcD8/3 S1/2 n

where n is Manning’s roughness coefficient, Xc is the model correction factor to account for the model uncertainty, D is the actual pipe diameter (ft), and S is the pipe slope (ft/ft). The inflow Ql (ft3/s) to the sewer is the surface runoff whose peak discharge can be estimated by the rational formula

Ql =кLCiA

in which Xl is the correction factor for model uncertainty, C is the runoff coefficient, i is the rainfall intensity (in/h), and A is the runoff contributing area (acres). In the reliability analysis, the sewer flow-carrying capacity Qc is the resistance, and the peak discharge of the surface runoff Ql is the load. The performance functions can be expressed as one of the following three forms:

W1 = Qc – Ql = 0463XcD8/3S1/2 – XLCiA n

W2 = Qc – 1 = 0463XcD8/3S 1/2x-1C-1i-1 A-1 – 1 2 Ql n c L

W3 = ln ^= ln(0.463) – ln(n) + ln(Xc) + 3 ln(D) + 1 ln(S) – ln(XL)

– ln(C) – ln(i) – ln( A)

Also in the reliability analysis, a frequently used reliability indicator в is called the reliability index. The reliability index was first introduced by Cornell (1969) and later formalized by Ang and Cornell (1974). It is defined as the ratio of the mean to the standard deviation of the performance function W (X), which is the inverse of the coefficient of variation of the performance function W (X),

в ___ gw

aw

in which gw and aw are the mean and standard deviation of the performance function, respectively. From Eq. (4.8), assuming an appropriate probability
density function for the random performance function W (X), the reliability then can be computed as

Ps = 1 – Fw(0) = 1 – Fw(-в) (4.9)

in which Fw( ) is the cumulative distribution function of the performance func­tion W, and W’ is the standardized performance function defined as W’ = (W – pw)/aw. The expressions of reliability ps for some distributions of W(X) are given in Table 4.1. For distributions not listed, expressions can be found in Sec. 2.6. For practically all probability distributions used in the relia­bility analysis, the value of the reliability ps is a strictly increasing function of the reliability index в. In practice, the normal distribution is used commonly for W(X), in which case the reliability can be computed simply as

Ps = 1 – Ф(-в) = ф(в) (4.10)

where Ф( ) is the standard normal CDF the table for which is given in Table 2.2. Without using the normal probability table, the value of Ф( ) can be computed by various algebraic formulas described in Sec. 2.6.1.