Reliability Analysis of Free Jet Scour Below Dams

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1 Entropy2012, 1, ; oi: /e Article OPEN ACCESS entropy ISSN eliability Analysis of Free Jet Scour Below Dams Chuanqi Li *, Shuai Wang an Wei Wang School of Civil Engineering, Shanong University, Jinan 25001, China; s: (S.W.); (W.W.) * Author to whom corresponence shoul be aresse; lichuanqi@su.eu.cn; Tel.: ; Fax: eceive: 29 October 2012; in revise form: 3 December 2012 / Accepte: 11 December 2012 / Publishe: 1 December 2012 Abstract: Current formulas for calculating scour epth below of a free over fall are mostly eterministic in nature an o not aequately consier the uncertainties of various scouring parameters. A reliability-base assessment of scour, taking into account uncertainties of parameters an coefficients involve, shoul be performe. This paper stuies the reliability of a am founation uner the threat of scour. A moel for calculating the reliability of scour an estimating the probability of failure of the am founation subjecte to scour is presente. The Maximum Entropy Metho is applie to construct the probability ensity function (PDF) of the performance function subject to the moment constraints. Monte Carlo simulation (MCS) is applie for uncertainty analysis. An example is consiere, an there liability of its scour is compute, the influence of various ranom variables on the probability failure is analyze. Keywors: free over fall jet; scour; reliability analysis; maximum entropy metho; Monte Carlo simulation 1. Introuction Scour ownstream of ams is one of the main concerns in hyraulic engineering. The transfer of water to the ownstream river may scour the am founation an the ownstream riverbe. In the long term, this scour process may create structural safety problems. Hence, accurate preictions of time evolution an ultimate scour epth are require [1]. A number of equations an moels have been evelope in the past to preict the potential free jet scour epth below ams an several stuies have

2 Entropy 2012, been carrie out to estimate ownstream scour below ams [2 6].These scour preiction equations are eterministic in nature, an o not account for the uncertainties of the parameters involve in the scour epth equations. Toay's most popular evaluation methos are of empirical an semi empirical nature. They o not fully escribe the physical backgroun of the phenomenon. A physical base scour preiction moel, the Comprehensive Scour Moel, has been evelope base on a parametric escription of the main physical processes responsible for scour [7 8].The physically base approach gives a much more reliable scour estimation than simple empirical formulas. The inherent uncertainties in various moel parameters suggest that am founation safety woul be ensure in a probabilistic sense. eliability analysis basically provies a quantitative estimation of am safety against scouring. Several authors have aresse reliability analysis of scouring in recent years. Muzzammil has presente a methoology for the reliability assessment of scour ownstream of a ski-jump bucket using the first-orer reliability metho (FOM) [9]. Yanmaz has assesse the reliability of the scouring of founations ownstream of outlet facilities [10]; reliability computations are carrie out by simulating the safety margin using the Monte Carlo technique. Ghahfarokhi uses the probabilistic metho to estimate plunge pool erosion ownstream of a ski jump bucket [11]. Hyraulic variables involve in plunge pools, such as ischarge, flow epth an velocity, are stochastic in nature, an therefore may be represente by relevant probability istributions. The maximum entropy metho (MEM), which is base on Shannon's measure of uncertainty, has been use for estimating istribution functions [12 1]. MEM is regare as the most unbiase estimation for the PDF, which means the most probable PDF from all the PDF uner the moment's constraint since it is maximally noncommittal with regar to missing information. The objective of this stuy is to present a methoology for quantifying the probability of am failure ue to scour. In orer to evaluate the stability of am founation, the paper combine the maximum entropy principle with Monte Carlo metho. The probability ensity function (PDF) of performance function is calculate base on maximum-entropy metho. Monte Carlo simulation (MCS) is use to generate ranom samples of the variables accoring to the given probability istributions. 2. Scour Moel of Free Over Fall Jet Free over fall jet scour over the top of arch ams. With free over fall over the top, the jet woul fall very near the base of the am; a eep scour hole (plunge pool) in the riverbe may pose safety problems for the am founation. Hyraulic, geotechnical an morphologic factors control the epth of plunge pool. The mechanism of scour is an extremely complex process ue to the influence of various hyraulic, hyrological an geological factors. There are many formulae evelope over the years to preict the maximum scour epth ownstream of a hyraulic structure. No formulas are presently available to accurately etermine the maximum epth of the plunge pool. However, base on laboratory stuies an fiel information obtaine in several projects, a number of empirical formulas have been evelope for preicting the maximum epth of scour of a stable plunge pool. Among these are the ones propose by Veronese [15], Mason [16] an Chen [17]. Bollaert [7 8] presents a physically base moel for evaluation of rock scour ue to high-velocity jet impact. Figure 1 shows a free jet ischarging over a am. The Veronese equation yiels an estimate of scour measure from the tail water surface to the bottom of the plunge pool. This equation is given as:

3 Entropy 2012, t q H m 1.9 (1) where: m is the maximum scour epth, t is the tail water epth, q is the water ischarge per unit with, H is the elevation ifference between reservoir an tail water (total water hea). The Veronose equation was publishe about 60 years ago an has been recognize as a significant basis for scour epth investigations. Figure 1. Definition sketch of free over fall jet. Base on prototype ata from ams in China, Chen [17] propose the following formula: m kq H t (2) where k is the scouring coefficient, varying with capacity of rocky riverbe to resist scour, for soli rock k for meium rock k , an for soft or broken rock k [18]. q an Hare the same as in Equation (1).Because of its convenient an concise form, formula (2) is wiely use in China to estimate the maximum scour epth at ownstream of hyraulic structures or spillways. The istance for maximum scour epth from the am ( L ) can be written as a function of the ischarge per unit with q an the elevation ifference. Accoring to the Design Specifications for Concrete Arch Dams (SL ), the istance of the maximum scour point is given as follows [19]: L q (3) where: L is the istance of maximum scour point from the am toe, is the elevation ifference between the top of am an the riverbe.

4 Entropy 2012, Scouring eliability of Dam Founation 3.1. Performance Function To perform the reliability analysis of founation scour, the failure an safety state of a am founation shoul be ientifie via the performance function. A performance function can first be efine as: g X X S X () where X X1, X2,, Xn is the vector of the input parameters, represents the resistance to failure of the structure, S represents the action causing failure. Both the action an resistance are, generally, functions of other basic variables whose probability istributions or statistical properties may be reasonably estimate. g X is the performance function with g X 0 inicating satisfactory performance (stability), g X 0 inicating unsatisfactory performance (failure), while the limit state is gx 0. For the am founation safety, it is important to ensure that the scour will not progress upstream to the extent that the safety of the structure might be in anger. The am founation safety can be evaluate by using the ratio of the istance for maximum scour epth ( L ) to the maximum scour epth ( m ). DefiningC L m,c can be use as the criterion of the effect of local scour against the stability of am, an C shoul be usually more than 3 to to ensure the am founation safety [18 19].In this stuy, the empirical scour moel will be use. Depening on the criteria aopte to efine the failure of the founation, the performance function g(x) expresses in terms of margin safety can be efine as: g X LCm 2.3 q C( kq H t) (5) Hence the reliability of the am founation can be efine as: 0 m 0 Ps P g X P LC (6) In a eterministic approach, the values of L an m are constant forgiven esign conitions an the value of g X nees to be greater than zero. In fact, most of the variables affecting scouring are uncertainties. Therefore, for the worst combination of scouring variables, g X may attain a negative value. The failure probability, following integral: where f P f,can be expresse in terms of the performance function by the 0 P 1P P g X 0 f (7) f s enotes the probability ensity function (PDF) of the performance function,, an the integral is carrie out over the failure omain. Equation (7) is nonlinear; therefore, an approximate technique shoul be evelope to etermine failure probability.

5 Entropy 2012, Uncertainty Analysis Dam founations may be seriously scoure uner jet action. The egree of scour epens on the characteristics of the jet leaving the top of ams, the epth of the tail water, an the properties of the be material, which are uncertain in nature. The Equations (2) an (3) are eterministic; they on't account for uncertainties in the moel parameters, or the hyraulic variables. In fact, hyraulic variables involve in plunge pools, such as ischarge, flow epth, an velocity, are stochastic in nature, which may be represente by relevant probability istributions. In this stuy, Monte Carlo simulation (MCS), as a significant sampling technique, is use to quantify the uncertainty of esire uncertain ranom variables. Monte Carlo simulations a powerful analysis tool that involves a ranom number generation an simulates the behavior of a variable when the ata is insufficient to make ecisions. In Monte Carlo simulation, probability istributions are assume for the uncertain variables for the system being stuie. anom values of each of the uncertain variables are generate accoring to their respective probability istributions.. Maximum Entropy Distribution.1. Determination of Density Function The maximum entropy metho is base on the concept that the istribution that maximizes the information entropy is the statistically most likely to occur. Here the maximum entropy metho is use to approximate the PDF of the performance function. Shannon [20] efine entropy as a measure of uncertainty about a ranom variable. Base on the entropy principle propose by Shannon entropy istributions are efine to be those which maximize the information entropy measure: ( )ln[ ( )] H f f f where f is the probability ensity function of the performance function,, an is the integral omain. Jaynes [21] formulate the maximum entropy principle as a rational approach for choosing a consistent probability istribution, amongst all possible istribution, that contains a minimum of spurious information. The principle states that the most unbiase estimate of a probability istribution is that which maximizes the entropy subject to constraints supplie by the available information, e.g., moments of a ranom variable. The maximum entropy metho of estimating f is state as follows: ln max H f f (9) (8) Subject to f 1 f i i f i 2,3,, m (10)

6 Entropy 2012, i where is the mean value of the performance function,, enotes the i-th central moment of ; m is the number of the given moment constrains. Max means that when the entropy reaches the f enotes the PDF of g X to be maximum, we obtain the best probability ensity function. etermine by the maximum entropy. The optimal solution of Equation (10) is the maximum entropy estimate of f. In many stuies, it was shown that the first four moments are sufficient to escribe a wie range of istribution types [22 23]. We use Lagrange's metho to solve for the PDF. The solution of Equation (9), subject to Equation (10), can be obtaine by using the metho of Lagrange multipliers. This yiels: i f( ) exp0 i (11) i1 where i, i 0,1, 2,3,, are Lagrange multipliers, which can be expresse in terms of constraints given by Equation(10).To compute a MEM ensity, we have to etermine the Lagrange multipliers,,, by the non-linear of equations Solving the Lagrangian Multipliers To fin the maximum entropy solution, we have to solve for the Lagrangian multipliers. Five Lagrangian multipliers i, i 0,1, 2,3,, can be obtaine by solving the following equations: r 0 ln exp i i1 i i exp i i1 i exp i i1 r i exp i i1 i exp i i1 r 2,3, (12) (13) (1) For a more convenient numerical solution, Equation (12) an (13) are change as follows: i exp i i exp i i1 i1 1 1 (15) r i exp i i1 r 1 r 2,3, r i exp i i1 Where the r are the resiuals that are reuce to near zero by a numerical technique. A solution can be obtaine by using nonlinear programming to obtain the minimum of the sum of the squares of the resiuals: (16)

7 Entropy 2012, min 2 2 r r1 (17) Convergence is achieve when 2 (12) is use to obtain Calculation of Failure Probability, or r, where is the specifie acceptable error. Equation Base on the probability ensity function f, the failure probability of the am founation can be calculate as: 0 f 0 i i1 P P( 0) exp[ ( ) i ] (18) where f enotes the probability ensity function (PDF) of the performance function,, an the integral is carrie out over the failure omain. 5. Example The application example correspons to the Ertan Dam. A reliability evaluation of the am founation subjecte to scouring was carrie out. The Ertan Damis an arch am on the Yalong iver, a tributary of the Yangtze iver in Sichuan Province, southwest China. The Ertan Dam is a 20 m high an 77.7 long ouble-curvature arch am. The am withhols a normal reservoir of m 3. The am has four ifferent ways to ischarge water ownstream. The am has the two large spillway tunnels (13 m 13.5 m), seven surface spillways, six mile outlets an four bottom outlets. The seven surface spillways can ischarge 6,260 m 3 /s. In this example, we stuy the scour of the surface spillways of Ertan am. The formula for ischarge per unit with is[2]: HW H W q m ( ) 2gH H H 2 3/2 W (19) where: is contraction coefficient, H is esign water hea, m is ischarge coefficient corresponing to H is water hea over the weir crest. W H, m , Three ranom variables are consiere: water level (H), scour coefficient (k) an the ownstream water epth ( t ). t can be calculate by the ownstream water level an the riverbe level. The istributions of three ranom variables are assume to obey the normal istribution base on the analysis an stuy of existing research results [25 27]. The means an stanar eviations of the ranom variables are liste in Table1.

8 Entropy 2012, Table 1. Parameters of ranom variables. Variables Distribution Mean value Stanar eviation Scour coefficient, k Normal water hea, H (m) Normal unit ischarge, q(m 3 /m.s) Normal Monte Carlo simulation is use to generate ranom samples of the variables accoring to the probability istributions. The values of the variables are ranomly generate accoring to their probability ensity functions. The performance function may be written as: q C kq H t ( ) (20) In this equation, 0 inicates failure. Consiering this single performance function, an estimate is require for the probability P 0. Taking C as 3, the PDF of the performance function can be expresse as Equation (21) accoring to the MEM: 3 2 f( ) exp( ) (21) The probability of failure is: 0 P f P( 0) f ( ) 0.51% (22) To investigate the effect of k, H an q on reliability, ifferent values of k, H an q was taken into account. The variations of the failure probability with respect to the scouring coefficient, ischarge per unit with, an water hea are shown in Figures 2. Figure 2. elationship between failure probability an scouring coefficient.

9 Entropy 2012, Figure 3. elationship between failure probability an ischarge per unit with. Figure. elationship between failure probability an water hea. 6. Conclusions In this stuy, a probabilistic moel in orer to assess the risk of am founation failure ue to scour action is presente. Uncertainty in the hyraulic variables involve in plunge pools, such as ischarge, flow epth, is explicitly consiere. Monte Carlo simulation (MCS) is applie to generate a large number of ranom sample values accoring to their respective probabilistic istributions. A limit state function has been formulate for the am founation safety uner jet action. The probability ensity function (PDF) of the performance function is calculate base on maximum-entropy metho (MEM).As an illustration of the evelope methoology, a jet scour case example is analyze. The example emonstrates how the reliability metho can be use to evaluate the risk of am founations subjecte to scour action. Acknowlegments This research was supporte in part by the Special Fun for Public Welfare Inustry (No ).

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