RELIABILITY ANALYSIS ON HYDRAULIC DESIGN OF OPEN CHANNEL

By

Ke-Zhong HUANG, Associatie ProfessorDepartement of River Mechanics and River Engineering Wuhan Institute of Hydraulic and Electric Engineering

Wuhan, CHINA

ABSTRACT

In this paper the reliability on hydraulic design of a trapeziodal open channel designed to convey a specified discharge from sluice gates is assessed by three different methods, which are the first-order second-moment analysis, the inequality analysis, and the Monte Carlo method. The principles of the methods are applicable to the reliability analysis for any hydraulic system and for other technical fields.

INTRODUCTIONAnalyses of uncertainties and risk in hydraulics have been

developed during the past thirteen years. Although hydraulicians have made significant contributions in the reliability of storm sewer design, pipe design, and of the theory of safety facrtors (e.g., Yen and Ang, 1971 ; Tang and Yen, 1972; Yen and Tang, 1976; yen,1977,1979), reliability of the hydraulic design of open channels has not been studied. It should be mentioned that uncertainties are unavoidable in an open channel to convey a specified discharge released from sluice gates or weirs.

Engineers encounter many factor which factors whith possess some degree of uncertainties affecting the determination of the channel capacity. Uncertainties occur in the hydraulic factors in the discharge formulas of the open channels and sluice gates or weirs. These factors include the roughness coefficient of the open channel, the discharge coefficient of teh sluice gates or weirs, etc. In the traditional deterministic design approaches, such uncertainties are not accounted for. Consequently, the safety factor is evaluated in a deterministic manner despite its inherent probabilistic nature. In this paper three different methodes are presented to assess the probability of safety of the capacity of a trapezoidal open channel to convey a specified design discharge released from a set of sluice gates.

1

THE FIRST - ORDER SECOND - MOMENT ANALYSIS

In general, L represents the loading and R is the resistance. Here the former means the discharge from the sluice into the channel. The latter means the channel capacity. The probability of safety and the probability of failure (i.e., risk) are defined as

...........................................................

and...............................

in which is the safety factor. Since both and are random variables, therefore, if the probability density functions are known, the probability of safety can be evaluated. For example, if and are statisfically independent variates of normal distribution, . 1 yields

......................

in which and are the mean values of the resistance and loading, respectively; and are their standard deviations : and are the coeficiens of variation; and denotes the cumulative standard normal distribution evaluated at . Likewise, if and are statistically independent variates of lognormal distribution, we have

..........................................

The first – order simplification of .4 is

..............................................................

The formula for discharge through a set of identical sluice gates is

2

(1)

(2)

(3)

(4)

(5)

(6)

............................................

in which is the head above the crest at the upstream side of the gates; is the coefficient of discharge; is a submerge coefficient; is the

width of each gate; is the height of the sluice opening; and is gravitational acceleration. The parameters , , , , and are assumed to be indepandent random variables. By using the first-order approximation of Taylor’s series expansion (Benjamin and Cornell, 1970) for . 6, we obtain the mean of the loading, , as

............................................and

.....................

Traditionally, the hydraulic design of open channel deals with uniform flow using manning’s formula. For trapeziodal open channels, the discharge in SI units is

...................................

in which is the deepth; b is the bottom width; m is the slope (horizontal to vertical) of the channel banks; i is the longitudinal slope of the channel; and n is the Manning roughness coefficient. The variables h, b, m, i, and n are assumed to be random and independent. Using the first order approximation of Taylor’s series expansion for . 9 we have

...........................

and

3

(7)

(8)

(9)

(10)

..................................................................

In order to calculate the values of the mean and coefficient of variation of , λ, , , , , , , , and , triangular distribution is assumed for each of them in terms of their respective design value as the mode and range from to , which are determined through other information such as the experimental error, constructional error, and posibble variations due to siltation and erosion. The triangular probability density is

..............

The mean and coefficient of variation of the triangular distribution can be computed from the following equations :

.................................................................

.............................................

Thus, we can obtain the probability of safety from . 3 or 4 together with . 7, 8, 10, 11, 13, and 14.

THE INEQUALITY ANALYSIS

In the first-order second-moment analysis described above, the distributional types or and are assumed. Although the probability of failure is not sensitive to the type of distribution for a risk level of 10 -3 or larger (Ang, 1970; yen and Ang 1971), the acceptable probability of failure would be in the order of 10-4 or smaller for major hydraulic engineering designs in China. In orther to avoid the sensitivity to the types of distribution, we derived the greatest lower bound of the probability of safety by using the Chebyshev inequality.

The Chebyshev inequality can be written as

4

(11)

(12)

(13)

(14)

in which S is again the safety factor; and d is a real number for every . Let , and , then

and

It follows that

where

In order to obtain the greatest lower bound of the probability of safety, take , then

and

Hence, we obtain the greatest lower bound of the probability of safety on the basis of ,

....................................

where and can be assessed by using the first – order approximation of Taylor’s series expansion for :

.....................................................................................

................................................

The greates lower bound of probability of safety plays an important role in the reliability analysis on major hydraulic engineering design, when the data of and used directly for computing the mean and coefficient of variation can be collected.

5

(15)

(16)

(17)

MONTE CARLO METHOD

The safety probability evaluation by using the first-order second-moment method is sensitive to the type of distribution of , , and the first-order approximation of . 6 and 9. In order to avoid these sensitivities, the Monte Carlo method is applied to estimate the probability of safety.

We assume the distributions of , , , , , , , , , and to be tringular. The pseudo-random tringularly distributed numbers may be obtained through generating pseudo-random uniformly distributed numbers on a computer. A relation that connects random number x having a triangular distribution and number r having a uniform distribution (0,1) can be derived from the formula (Shreider, 1964)

......................................................................

. in which is teh triangular probability density ( . 12). Thus,

....................

A sample value of can be determined from . 6 and 9 in terms of the sample values of , , , , , , , , , and . Suppose that is the sample number of and is the number of . If is sufficiently large, on the basis of the law of large numbers, the probability of safety can be estimated as

.........................................................

It may be seen that this method prossess some merits in comparison with the first-order second-moment analysis as follows:

(a) The assumption of the types of distribution of and as well as the first-order approximation can be avoided.

(b) The theoritically, the types of distribution of and must be determined by deriving the distributions function from . 6 and 9 in terms of the types of distribution of and n, respectively, but

6

(18)

(19)

(20)

these integral operations are very difficult, if possible. Therefore, there is a contradiction of the distributional assumption in the first-order second-moment analysis. On the contrary, the Monte Carlo method may evade this fault in theory.

The deficiency of the Monte Carlo method is that it must assume the distributions for and n. Becase we do not have enough data to establish these distributions, the deficiency is unavoidable when the acceptable profitability of failure is in the other of 10-4 or smaller for major engineering designs.

EXAMPLE APPLICATION

A trapezoidal open channel is designed to convey a specified discharge released from a set of seven identical sluice gates. The data of the sluice gates and the trapezoidal open channel are given in Table 1.

TABLE 1. Parameter values of ExampleSluice Gates Open Channel

Factor Design Value Range Factor Design Value Rangew 4,50 m 4,46 m - 4,55 m 50.0 m 45.0 m – 55.0 m

4,20 m 4,16 m - 4,2 m 1.5 1.35 – 1.656,50 m 6,40 m - 6,60 m 20.00 x 10-4 (1.9 – 2.1) x 10-4

0,484 0,471 - 0,497 7.5 m 7.13 m – 7.88 m1 2.52 x 10-2 (2.32 – 2.75) x 10-2

7

Using the method of the first-order second-moment analysis, we

obtain . , , from

. 4.

A digital computer was used for the calculations of the Monte Carlo method. Taking , is obtained.

If , , , and are calcualted by using . 7, 8, 10, and 11, respectively, we obtain the greatest lower bound of probability of safety

from inequality 15. This result may be used as a reference value.

7