The basic functions of highway drainage structures are (1) as hydraulic facil­ities to safely convey floods under highways during all but the most severe flooding conditions and (2) as portions of the highway to move highway traffic freely over stream channels. There are three general types of drainage struc­tures: bridges, box culverts, and pipe culverts. Conventionally, bridges refer to structures measuring more than 20 ft along the roadway centerline (AASHTO, 1979). Box culverts are usually built of concrete with rectangular openings. Pipe culverts can be in various geometric forms, such as circular, arch, etc., and can be made of several different materials, such as steel, cast iron, concrete, or plastic.

The design of highway drainage structures involves both hydraulic design and structural design. The discharge associated with the critical flood that starts to cause hazards to life, property, and stream stability is termed as the hydraulic design discharge. The process to select the design discharge and to perform the necessary hydraulic computations for a proposed highway struc­ture is called hydraulic design. In practice, the design discharge is one-to-one related to the design frequency through frequency analysis. Therefore, the de­sign event also can be characterized by the design frequency. In this example, the design frequency refers to an annual exceedance probability or its reciprocal, the design return period.

The example problem under consideration is to design a circular culvert under a two-lane highway. The culvert is 100 ft long. The equivalent average daily traffic is 3000 vehicles per day. The discount rate used is 7.125 percent, and the useful service life of the culvert structure is estimated to be 35 years. Detailed descriptions of this example are given by Tung and Bao (1990).

In this example, only the inherent hydrologic and parameter uncertainties are considered. The primary objectives are (1) to search for the optimal de­sign parameters associated with the minimum total annual expected cost for the culvert and (2) to investigate the sensitivity of the optimal design parame­ters with respect to (a) the hydrologic parameter uncertainty, (b) the length of streamflow records, (c) the distribution model of flood flow, and (d) the maximum flood-damage cost. More specifically, the optimal design parameters considered in this example are the optimal design return period T and the associated least total annual expected cost (LTAEC).

The estimated sample mean and sample standard deviation for the flood flow are 47.9 and 71.9 ft3/s, respectively. The skewness coefficient of streamflow for the original scale and log-transformed scale are assumed to be 0.5 and 0.2, respectively.

In the sensitivity analysis, the optimal total annual expected cost was calcu­lated for various record lengths n of 10, 20, 40, 60, and 100 years; for maximum
flood damage cost Dmax of $928, $1500, $3500, and $4500; and for flood flow dis­tribution models of normal, lognormal, Pearson type 3, and log-Pearson type 3 probability distributions.

The cost function representing the annual installation (first) cost of the cul­vert is derived on the basis of data from Corry et al. (1980) using regression analysis:

AFC = 1.0215 – 2.62 x 10-7qc2 (8.38)

where AFC is the annual first cost ($), and qc is the design discharge (ft3/s). The R2 of this regression equation is 0.976.

The damage function D(q), approximating the original discrete form in Corry et al. (1980) by a continuous function, can be expressed as

q ^ qmax

qc < q < qmax (8.39)

q < qc

where Dmax is the maximum flood damage cost, qmax is the flood magnitude corresponding to Dmax, and qc is the design discharge. It is understood that, in general, qmax will be increased as a result of raising the design discharge qc. The rate ofincrease in qmax will slow down as qc increases. The damage function used, for illustration, is shown in Fig. 8.13, in which qmax is determined from

qmax = 210 + qc – q094 (8.40)

Because of the complexity of the functional form of the objective function, it is difficult to solve Eqs. (8.31) and (8.33) analytically. Therefore, optimum search techniques are useful to solve the problem. However, gradient search techniques are inappropriate for use in this case because the gradient of the objective function is not easily computable. Among the search techniques that do not require knowledge of gradient of the objective function, Fibonacci search is an efficient technique to be used for this single-decision variable-optimization problem (Sivazlian and Stanfel, 1974).

Fibonacci search applies the sequential search strategy that successively re­duces the feasible decision variable interval to 1/FN its original size with just N function evaluations. The final decision-variable interval can be made as close to the optimal solution as the desired accuracy. FN is called the N th Fibonacci number in the Fibonacci sequence FN, for i = 0, 1, 2, 3,…, whose value is given by the recurrence relation

F0 = F1 = 1

Fi+1 = Ft + Fi-1 i > 1

The computational procedures for determination of the optimal return period corresponding to the optimal capacity in the risk-based design of a pipe culvert considering the hydrologic inherent and parameter uncertainties is illustrated in Fig. 8.14.

The optimal design frequency T * and the associated LTAEC under different record lengths n and streamflow probability distributions with maximum flood damage cost (Dmax = $928) are listed in Table 8.3. The values in the columns for n = 10-100 are calculated by considering hydrologic parameter uncertainty, whereas the values in the column with n =<x were calculated without consid­ering hydrologic parameter uncertainty.

Comparing the two design methods, the value of the LTAEC without con­sidering parameter uncertainty is always smaller than the value considering parameter uncertainty regardless of the probability distributions or values of Dmax. This observation shows that neglect of the hydrologic parameter uncer­tainty could lead to an underestimation of the total expected cost.

The value of LTAEC decreases as the record length increases. This is expected because the effect of hydrologic parameter uncertainty involved in estimating the second cost diminishes as the record length for streamflow gets longer. The difference in LTAEC values calculated by the two methods, for Dmax = $928 and n > 20, is only about 3 percent for any of the four probability distributions considered. However, the higher the value of Dmax, the more dominant the sec­ond cost becomes in the objective function evaluation. Therefore, the difference in LTAEC values by the two methods at the same record length will be larger as Dmax is increased.

Examining the T * values in Table 8.3, the difference in T * between the two methods is less than 20 percent in most cases. Also, for fixed distribution and sample size, the optimal T * increases as Dmax increases (see Table 8.4).

This confirms the original intuition. However, there does not exist the same consistent tendency in T * as with the LTAEC shown earlier. Therefore, when T * is considered as a criterion in the comparison of the two design methods, it is difficult to conclude which method tends to be more conservative.

Figure 8.15 shows of the total annual expected cost function, annual first-cost function, and annual second-cost function versus the design return period T with record length varying from 10 to 100 years at -Dmax = $4500 for the log­normal probability distribution. Similar behavior was observed for three other

types of distributions. From Fig. 8.15 it is clear that the annual second cost (ASC) and the total annual expected cost (TAEC) decrease as the record length increases. Therefore, the LTAEC will be smaller when the record length gets longer. However, the corresponding T *, as discussed earlier, may not necessarily

Record length (in years)

become smaller as the record length increases. The inconsistent behavior be­tween T * and LTAEC in comparing the two design methods is mainly attributed to the nonlinear and nonmonotonic relationship between T * and LTAEC.

It can be seen from Fig. 8.15 that the total annual expected cost (TAEC) curves are very flat in a range of design frequencies from 5 to 20 years for this example. Therefore, from a practical point of view, a pipe culvert could be overdesigned about 5 to 10 years above the optimal design frequency to give more confidence in the safety protection of the structure with only a small fraction of extra annual capital investment.

Besides the economic factors that can be quantified in monetary terms in the design of hydrosystems, there are other intangible factors that are noncommen­surable and cannot be quantified. Some of the intangible factors might work against the principle of economic efficiency. Examples of intangible factors that are considered in the design and planning of hydrosystems may be potential loss of human lives, stability of water course, impacts on local society and environ­ment, health hazards after floods, litigation potential, maintenance frequency of the systems, and others. The conventional optimal risk-based design yields the most economically efficient system, which may not be acceptable or feasible when other intangible factors are considered.

As more intangible factors are considered in risk-based design, it becomes a multiobjective or multicriteria decision-making (MCDM) problem in which economic efficiency is one of many factors to be considered simultaneously. Use of a multiple-criteria approach enhances more realistic decision making, and the design frequency so determined will be more acceptable in practice and de­fensible during litigation or negotiation with others. Tung et al. (1993) adopted the MCDM framework to incorporate intangible factors in risk-based design of highway drainage structures through which a more defensible extended risk – based design frequency can be determined from integrated consideration of tangible and intangible factors.

In a risk-based design, in addition to quantitative measure of failure proba­bility and risk cost, consideration of intangible factors and societally acceptable risk issues should be included if possible. In the United States, the societally acceptable frequency of flood damage was formally set to once on average in 100 years (the so-called 100-year flood) in the Flood Disaster and Protection Act of 1973; however, the 100-year flood had been used in engineering design for many years before 1973. In this act, the U. S. Congress specified the 100-year flood as the limit of the flood plain for insurance purposes, and this has become widely accepted as the standard of hazard (Linsley and Franzini, 1979, p. 634). This acceptable hazard frequency was to be applied uniformly throughout the United States, without regard to the vulnerability of the surrounding land. The selection was not based on a benefit-cost analysis or an evaluation of probable loss of life. Linsley (1986) indicated that the logic for this fixed level of flood hazard (implicit vulnerability) was that everyone should have the same level of protection. Linsley further pointed out that many hydrologists readily accept the implicit vulnerability assumption because a relatively uncommon flood is used for the hazard level, and thus

The probability that anyone will ever point a finger and say “you were wrong” is equally remote. If the flood is exceeded, it is obvious that the new flood is larger than the 10-year or 100-year flood, as the case may be. If the estimate is not exceeded, there is no reason to think about it.

Mitigation of natural hazards requires a more rigorous consideration of the risk resulting from the hazard and society’s willingness to accept that risk.

In other cases of disaster, societally acceptable hazard levels also have been selected without formal evaluation of benefits and costs. For example, in the United States, dam-failure hazards are mitigated by designing dams where failure may result in the loss of life to pass the probable maximum flood. Also, in The Netherlands, coastal-protection works normally are designed by ap­plication of a semideterministic worst-case approach wherein the maximum storm-surge level (10,000-year storm surge) is assumed to coincide with the minimum interior water level.

In the design of the Eastern Schedlt Storm-Surge Barrier, the Delta Commit­tee in The Netherlands applied a simple risk-cost (in terms of lives) evaluation to set the design safety level. The Delta Committee set the total design load on the storm-surge barrier at the load with an exceedance probability 2.5 x 10-4 per year (i. e., the 4000-year water level) determined by integration of the joint prob­ability distribution among storm-surge levels, basin levels, and the wave-energy spectrum. A single-failure criterion then was developed for the functioning of all major components of the storm-surge barrier (concrete piers, steel gates, foundation, sill, etc.) under the selected design load. The failure criterion was tentatively established at 10-7 per year on the basis of the following reasoning. Fatality statistics for The Netherlands indicate that the average probability of death resulting from an accident is 10-4 per year. Previous experience has shown that the failure of a sea-defence system may result in 103 casualties. Thus a normal safety level can be guaranteed only if the probability of failure of the system is less than or equal to 10-7 per year. Comparison of the worst-case approach with the probabilistic-load approach resulted in a 40 percent reduc­tion in the design load when the actual, societally acceptable protection failure hazard was considered (Vrijling, 1993). This illustrates that when a compre­hensive risk assessment is performed, societally acceptable safety can be main­tained (and in some cases improved) while at the same time effectively using scarce financial resources. Some work on societally acceptable risk and intangi­ble factors can be found elsewhere (Jonkman et al., 2003; Vrijling et al., 1995).

8.4 Applications of Risk-Based Hydrosystem Design

In this section, two examples are described to illustrate the applications of risk-based design of hydrosystems. One is pipe culverts for highway drain­age, and the other is flood-damage-reduction projects implemented by the

U. S. Army Corps of Engineers. The first example involves optimal risk-based design considering only hydrologic inherent uncertainty, whereas the second example considers uncertainties from hydraulic and economic aspects.