DIRECT DYNAMIC FORCE MEASUREMENT ON SLABS IN SPILLWAY

STILLING BASINS

By Alberto Bellini and Virgilio Fiorotto2

ABSTRACT: This paper presents a new experimental procedure aimed at defining the global instantaneousuplift force, acting on slabs at the bottom of stilling basins. The global uplift force on slabs of differentdimensions is measured for Froude numbers of the incident flow ranging from 5 to 12. The measurementswere made in the zone of hydraulic jump, where high turbulence yields the maximum uplift force. Thefluctuating force per unit area is shown to depend on the slab shape and on the statis~ic~1 structure of theinstantaneous spatial distribution of turbulent pressures at the bottom of the hydrauhc Jump. ThIS paperprovides direct experimental evaluation of a criterion for the .design of the protection slab.s in ~pillway stillingbasins. It is shown that rectangular slabs with the longer sIde placed along the flow dIrectIOn (where thetransverse direction is maintained to a technically minimum length) are most suitable for the design of thelining of spillway stilling basins.

scale of the pressure fluctuations; Y 1 = depth of the incidentflow; 'Y, 'Yc = specific weight of water and concrete; andC;, C7, = Ii. P';",.I( v2/2g) = positive and negative pressurecoefficients defined by pressure differences Ii. P,;",x above themean value (Toso and Bowers 1988). Notice that for anchoredstructures the failure strength of anchors must be transformedinto equivalent weight from which to compute the equivalentthickness s = s' + nAaj('Yc - 'Y)I/" with s' the thicknessof the slab, n the number of anchors, A and a" the area andthe admissible tension of each anchor, respectively.

This criterion was defined by theoretical and experimentalstudies of the influence of the instantaneous pressure distri­bution above and below the slabs on the instantaneous upliftforce. The results of the analysis can be summarized as follows(Fiorotto and Rinaldo 1992a): (1) The reduction of the pres­sures propagating below the slab due to friction can be ne­glected for a safe design for even long-term operation ofspillway basins; (2) the persistence of fluctuating pressures ata given point (time microscale) is greater than the propagatIontime of pressures between the joints; and (3) hydraulic res­onance does not occur for slab dimensions compatible withpractical applications. From this consideration we can. arguethat for practical purposes underpressure propagates ITIstan­taneously without damping effects. Fig. 1 shows the concep­tual scheme used in order to define the stability criterion,(1), which is obtained assuming the instantaneous balancebetween forces acting over and under the slab:

INTRODUCTION

One of the most important steps in the design of spillwaystilling basins is the definition of the maximum instantaneousuplift force produced by turbulent pressure fluctuations actingon the slabs (Fiorotto and Rinaldo 1992a). Cases of damageon chutes and spillway basins [e.g., Malpaso (Sanchez Bri­biesca and Viscaino 1973), Tarbela, and Karnafuli dams(Bowers and Toso 1988, 1990)] have clearly highlighted therelevance of this problem. It is known in fact that fluctuatingpressures at the bottom of the hydraulic jump are the primarycause of the failures (Bowers and Tsai 1969; Sanchez Bn­biesca and Viscaino 1973; Rinaldo 1985, 1986; Bowers andToso 1988, 1990; Fiorotto and Rinaldo 1988, 1992a, b; Fior­otto 1990). The process of generating uplift force is, however,quite complex. Instantaneous pressure differentials either oc­cur between the chute drain openings and upper surface ofthe chute slab or cause damage of joint seals, thereby prop­agating underpressures (Fig. 1). Although the pressures aredamped in the propagation through the drain system or alongthe soil-structure contact surface, the net difference betweenthese pressures and those acting on the surface of the slabsmay exceed the weight of chute slabs (Rinaldo 1985, 1986;Fiorotto and Rinaldo 1988; Toso and Bowers 1988; Oi Santoet al. 1991).

A design criterion based on the uplift induced by the severepressure fluctuations associated with energy dissipation in theregion of hydraulic jump was proposed by Rinaldo (1985) andlater refined by Fiorotto and Rinaldo (1992a). The criterionrelies on the following expression, which relates the equiva­lent thickness of the linings to geometric and hydrodynamicparameters:

Fm"x(t) = [F,,(t) - F,,(t)] = F;"ax + 'Ysl,l,

v'= D(q + C,~ h 2g f),. + 'Y sf),. (2)

(1)S ( I, I, I,) (C + C _ ) _'Y_-,- > D -, -, - "+,,v-/2g Y 1 A, A, 'Y, - 'Y

where S = equivalent thickness of the linings; v2/2g = in­flowing velocity head; n = uplift coefficient; Ln L, = longi­tudinal and transverse length of the protection, or span be­tween the joints; An Av = longitudinal and transverse integral

'Dept. of Civ. and Envir. Engrg .. Univ. di Trento. via Mesiano diPovo 77 1-3~050, Trento, Italy.

~Dept. of Civ. Engrg .. Univ. di Trieste, piazzale Europa I 1-34100.Trieste. Italy.

Note. Di~cussion open until March I, 1996. To extend the closingdate one month, a written request must be filed with the ASCE Managerof Journals. The manuscript for this paper was submitted for review andpossible publication on April 19. 1994. This paper is part of the Journalof Hydraulic Engineering, Vol. 121. No. 10, October, 1995. ©ASCE,ISSN 0733-9429/95/0010-0686-0093/$2.00 + $.25 per page. Paper No.~222.

686/ JOURNAL OF HYDRAULIC ENGINEERING / OCTOBER 1995

where F:nax = maximum value of the fluctuating part of ~he

total uplift force, and the other symbols assume the mea01~g

indicated in Fig, 1. To assure slab stability the maximum uphftforce must be less than the slab weight: Flllax < 'Y ,51,1, so that(1) holds. It is evident, from Fig. L that n is a function ofthe instantaneous pressure distribution over the slab that fluc­tuates in the range between Po (x , y, t )min and Po (x, y, t )",ax'

The spatial distribution of the fluctuating pressure dependson the statistical characteristics of the pressure field at thebottom inside the hydraulic jump region (Fiorotto and Rin­aldo 1992b).

The aim of the present paper is to provide direct experi­mental evaluation of the uplift coefficient n. The task is ac­complished through a new experimental setup that allowsdirect measurement of the force acting on the lining slabs andsimultaneously the pressure at the bottom in the hydraulicjump region. The new experimental setup is obtained by iso-

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a

L.E.

P,,(x,y, t)

I.,D ( t) _ R (t) F(t) =F'(t) +-y81.,1l/r" x,y, 0 x,y, /

POmas - ~'<~K' { Po(x, y, t)ma., =Po + -yet ;;POm;n - '.' t ~ Po(x, y, t)

-""'" I ~ Po(x, y, t)m'. ~ 1'0 - l C; ;;

~::::::::::::=:-,....,., x P" (x, y, t) =Po(O, y, t) +18 =Po (lx , y, t) + 1 8Ix

FIG. 1. Conceptual Scheme of Instantaneous Uplift Force Generated by Unbalanced Pressure Distributions Acting on Faces of LiningSlabs

t~flOW pIpe

=~ .. tvo

(a)~to

Inl.t tankr-----

r-r" .A'.i_III ;if} /'-..., ~fon:. Got.~transducer

dota acquls:s,...r:~:::.r

~I~0 1 2 3m,

'%

(b) .:;;~:L '-- ----"'~""~"'~ot_'.__'

I----.."oblo I..,gh~

FIG. 2. Experimental Installation: (a) Test Facilities; (b) Experimental Details

lating a portion of the bottom of the flume suitably connectedto a strain gauge and accurately measuring the pressure fluc­tuations by two pressure transducers. The measurements weremade for several slabs, with different lengths Ix and Iv andFroude numbers of incident flow in the range from 5 to 12.

EXPERIMENTAL METHODS

Experiments were performed in a flume, 4-m long, 0.3-mwide and 0.5-m high, operated at the Hydraulics Laboratoryof the University of Trento, Fig. 2(a). A sliding gate wasoperated to control the hydraulic characteristics of the inci­dent flow, while at the end of the channel a gate was usedto localize the position of the jump over the test area that isshown in Fig. 2(a). In this area a steel frame 200-mm long,I50-mm wide and 5-mm deep with a centered rectangularhollow was inserted in the flume floor. Inside the hollow wasput a movable slab of aluminum 3-mm thick with the upperface at the same level [up to O(lO~:;) m in accuracy] of theflume floor. To properly locate slabs of different dimensions,different frames (with the same overall dimensions but dif­ferent hollows) was inserted in the flume floor. The dimen­sions of the hollows were such as to leave a gap of about 2mm along the boundaries. Through the gap the fluctuatingpressures at the bottom of the hydraulic jump propagates in

the 2-mm-thin water layer under the slab [Figure 2(b)]. Pre­vious experimental results showed that inside the water layerthe fluctuating pressures acting along the boundary of theslab are conveyed with very small friction effects (Fiorottoand Rinaldo 1992a).

The alignment of the upper face of the movable slab withthe bottom of the flume is a crucial problem because even asmall inaccuracy alters the pressure field around the slab,which induces errors in the measured force. For this reason,the movable slab was fixed to the force transducer (Trans­america Instrument model TF-02) with a special device thatallows the accurate adjustment of its position. The adjustmentwas performed acting on four micrometric adjusting screws[Fig. 2(b)].

Two pressure taps were inserted on the bottom of the chan­nel in the central part of the test area. The pressure taps,inserted on both sides of the slab, have a diameter of 2 mmand were connected to the transducer by a rigid tube of 4­mm internal diameter and O.4-m length. The pressure trans­ducers used in this study were Transamerica Instruments BHL­4260 adjusted in the range of 0-250 mbar of relative differ­ence pressure. In the linear working range (25-225 mbar)the transient time to the Heaviside function with amplitudeof 50 mbar (range of variation of the maximum and minimumvalue of the fluctuating pressure respect to the mean value)

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is lower than the microscale time of the pulsating pressuremeasured by Abdul Khader and Elango (1974) and Fiorottoand Rinaldo (l992b).

A computer was linked to the transducer via a 16 channelanalog-digital board made by Data Translation DT 280l-A.Amplifier and conditioner were used to adjust the signal out­put of the transducer to resolution and acceptable range ofanalogical to digital board. The sampling was accomplishedby the I.L.S. program made by Signal Technology, imple­mented on a PC, and the data were stored in the hard diskto perform further elaborations. For each configuration setup,seven experiments of 0.5 h each, for a total of 3.5 h wereperformed. The duration of the experiments was fixed afterhaving carefully evaluated the rate of convergence of n. Wereturn to this argument in the following section.

The maximum discharge of the flume was 50 Lis. Specialcare has been devoted to the evaluation of the discharge andof the water depth at the jump toe. The discharge was mea­sured using a Venturi meter located in the inflow pipe andindirectly measuring the water level in the inlet tank. Theerror in measuring the discharge was estimated to be lowerthan 1.3%.

The upstream mean water depth used to compute the meanfeatures (e.g., v2/2g, F, R) was measured using stilling wellslocated along the section at the toe of the jump; the fluctua­tion of the toe around this section made the definition of itsinitial section a difficult task. Furthermore, because of thesmall gate openings, the cross-wave patterns downstream fromthe gate made the measurement of the average flow depth adifficult undertaking. Although small in height, the cross­waves can not be suppressed completely, particularly thosegenerated at the two sides of the channel. The estimated errorin Froude number of the incident flow including the effectsof discharge, depth, and channel geometry amounts to themaximum of 5%; while the error in the kinetic head wasestimated lower than 7%. Velocity measurements with a Pitottube were performed to check the uniform distribution of thevelocity transverse to the flux direction.

For the experiments 20 different rectangular and squareslabs with 50 'S I, :S 200 mm and 25 'S L" 'S 150 mm wereused.

The slabs were put in the hollow and by operating the gateat the end of the flume the hydraulic jump was adjusted tothe position that induced the maximum force on the slab.The optimal position was obtained by several preliminaryexperiments carried out for each slab and Froude number.In doing so, the distance between the toe of the jump andthe center of the slab, Xc was in the range of 1.5-2.5Y2 whereY~ is the conjugate depth in the jump. The values used inseveral experiments are shown in Table 1. The longitudinalextension of this zone of maximum turbulence was estimatedas 30% of the jump length (Vasiliev and Bukreyev 1967)about eight times the incident flow depth (Toso and Bowers1988).

The rectangular slabs were put with the greater dimensionplaced along the flow direction. Theoretical analysis had shown,in fact, that this is the most suitable condition for reducing

the overall size of the protection works (Fiorotto and Rinaldo1992a).

In the experiments, the signals from the force transducerand from the two pressure gauges were sampled. Fig. 3 showsthe spectra of the fluctuating pressure and of the forces actingon 150 x 200 mm2 (maximum area) and on 25 x 50 mm2

(minimum area) slabs, From the figures one can observe thatcompared to pressures the forces have a broader spectra lead­ing to a sampling rate of 50-100 Hz and 100-150 Hz, forpressures and forces, respectively, Amplitude analysis of forcepower spectrum suggests that the fluctuation energy is mainlyconcentrated at the lower end of the spectrum, with a smallenergy spreading toward frequencies larger than the char­acteristic pressure frequencies [Figs. 3(b and c)]. This follows

FIG. 3. Spectra of Fluctuating Pressure and Force for Froude ==10: (a) Spectrum of Pulsating Pressure; (b) Spectrum of FluctuatingForce for 200 x 150 mm2 Slab; (c) Spectrum of Fluctuating Forcefor 50 x 25 mm2 Slab

y,(cm) l'(m/s) Froude Reynolds(1 ) (2) (3) (4)

1.96 4.3 10 85,0001.92 5.2 12 100,0002.95 4.0 7.5 120,0003.62 3.4 5.7 123,0002.25 4.1 '1',.7 93,000

TABLE 2 Average Parameters in Tested Experimental ConditionTABLE 1 Average Distance x and Measured Values of Ayc

Ay(cm) Ay(cm)Froude XC /Y2 minimum maximum

(1 ) (2) (3) (4)

10 1.8 9.6 10.512 2.1 9.8 10.77.5 2.0 9.9 11.15.7 1.8 9.3 10.38.7 1.9 9.5 10.6

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from a weak nonlinearity in the physical process that controlsthe propagation of underpressures below the slab. The ex­periments were carried out for Froude numbers of the up­stream flow ranging from 5.7 to 12 and Reynolds numberclose to 100,000, while prototypes show Reynolds numbersranging from 107-108 leading to Reynold's scales of 100: 1 or1,000:1 (Table 2).

Since most of the energy is located at small frequencies,the process that creates the uplift force is mainly controlledby the dynamic of the larger eddies, which are scaled ac­cording to the Froude law (Vasiliev and Bukreyev 1967),whereas Reynolds number influences are limited to fine-scalelocalized effects. However, to take into account that Reynoldsscale effects could influence the experimental results, a suit­able safety coefficient was introduced (see the slab designexample in the following section).

When the flow steady-state condition was reached, sam­pling of the force acting on the slab as well as the pressureat the bottom was started. To stabilize the variances of forceand pressure a sampling time of 1,800 s (30 min) was chosen.Furthermore, to compute the uplift coefficient 0, (3), theinstantaneous maximum uplift force and pressure was re­corded. The measurements were repeated for a total samplingtime of 3.5 h; for every measure the slab was removed andinstalled again and the flow condition accurately checked.For each 0.5-h long experiment, the signals were recordedand the statistics as well as the maximum and minimum valueswere computed with reference to the complete set of exper­iments (seven experiments each one 0.5-h long for a total of3.5 h).

The distance of the toe of the jump from the gate is in therange of 15-30 times the sluice gate opening; so the boundarylayer can be considered no fully developed at the beginningof the jump (Leutheusser and Kartha 1972; Wilson and Turner1972). From experimental results and for a given Froude num­ber, the condition of undeveloped flow tends to slightly in­crease the value of the standard deviation of the pressurefluctuation and slightly decrease C;" C;, with respect to thedeveloped condition (Toso and Bowers 1988).

EXPERIMENTAL RESULTS AND ANALYSIS

To estimate the uplift coefficient 0, the following quantitiesare computed from the sampled data: (1) The maximum de­viation of the uplift force from its mean value; (2) variance,skewness, and kurtosis coefficients of the force; (3) the max­imum positive and negative pressure fluctuations with respectto the mean value; and (4) variance, skewness, kurtosis, andcorrelation coefficient of pressures measured at the two pres­sure gauges.

The limit slab-stability condition is obtained by equatingthe maximum uplift force to the submerged slab weight:F~,,,, = s(-yc - 'Y)l)v' Computing the slab thickness from theprevious expression' and replacing it into (1) we obtain thefollowing expression for the uplift coefficient:

(3)

~~ere, to ~oint out that the computation of the uplift coef­fiCIent reqUIres, the direct experimental measurements of F' ,C;, and C-;" 0 are replaced by Om'

Toso and Bowers (1988) and Farhoudi and Narayanan (1991)showed that the duration of experiments involving the mea­sure .of the .m~~imum instantaneous value of force and pres­sure IS prohibItIve. Increasing Cp-values with the test duration~ere observed and a 24-h test duration was used by the men­tioned authors to stabilize the results and obtain sufficientlyaccurate extreme Cp-values.

Theoretical analyses (Fiorotto 1990; Fiorotto and Rinaldo1992a), have shown that the uplift coefficient can be relatedto the standard deviation of fluctuating force acting on theslab a F and the standard deviation of the pulsating pressurea p through the following expression, where 0 is replaced byOs:

(4)

Eq. (4) can be inferred from (3), assuming forces and pres­sures normally distributed and noting that the extreme valuesin a normally distributed population are proportional to thestandard deviation.

Figs. 4(a and b) show n,l0shm versus the acquisition timefor two representative slabs. Os converges rapidly to the sta­bl.e value n'hm (here represented by the value of n, computedWIth reference to 20 h of acquisition). Differences smallerthan 2.5% are observed after 0.5 h and after 3-4 h the dif­ferences are less than 0.5%; however, differences less than5% are obtained after 5 min. The differences observed forsmall acquisition time are probably due to slight oscillationsin the jump position during the test. On the other hand Omconverges slowly and then the possibility to use n, insteadof Om in (1) should be carefully evaluated. To evaluate thedifference between Om and nn long-duration experimentswere carried out. The test durations were 20 h and the sam­pling frequency was 150 Hz. Om and n, were computed asfollows: (1) The sample was divided into groups of durationof 0.5,2, 5, 10 and 20 h; (2) for each sequence the value ofOm was computed, selecting the maximum uplift force andthe maximum and minimum pressure fluctuation; (3) withreference to the same groups, the values of the standard de­viation of force and pressures were evaluated, and subse­quently by (4) the value of n,.

Figs. 4(c and d) show the ratio Om/n, as a function of theexperiment duration for the same slabs used in Figs. 4(a andb). O,)n, decreases uniformly with the experiment duration~eaching the value of about 1.1 after 20 h. Large spreadingIn the values of Om/Os is observed for small-duration testsbecause the sampling does not have enough duration to ac­curately capture extreme values of force and pressures. Forlong-duration tests the small difference between 0 . and 0can be explained as follows. Since the pressure fi~ld is n~'thomogeneous (Vasiliev and Bukreyev 1967), a, varies alongthe longitudinal direction. The variance of the pressure isevaluated in the central part of the slab where its value ismaximum [about 10% larger than the mean value over theslab (Abdul Khader and Elango 1974)]. The overestimationof a p together with the accurate evaluation of a r leads to areduction of 0 (n, < 0). On the other hand, the maximumpressure difference measured at two fixed points can be smallerthan that effectively observed on the slab leading to an over­estimation of 0 in (3), (Om ~ 0). For these reasons thecoeffi.cient Om is expected to be larger than n, even for long­duratIOn experiments (n,. :5 0 :5 Om)'

No~ably, the value of n, that corresponds to experimentduratIOns of 0.5 h is in good agreement with the value of 0obtained after 20 h. m

Finally, we conclude that the value of n, computed withreference to 0.5-h experiment durations is a good estimationof the uplift coefficient to be used in (1) in order to computethe ~Iab thickness that assures stability of the stilling basin.

FIgs. 5 and 6 show the results of experiments carried outwith diffe~ent sla~ ?imensions and Froude numbers. In Fig.5, the upltft coeffICIent n, computed by (4) is plotted versusIV/YI for constant values of I)YI and Froude numbers. Thetransverse integral scale of the pressure fluctuations, com­puted using pressures measured at two points and assuming

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1 h

• 2h 5h 10 h• : • ••• 20 h• I •• •• t •• •• ••• •I I

10 100 1000 10000

Experiment Duration (min.)

••

I•

30 min•

5 min

-15 min

30 min- I 1~ 2.h 5~ l~h 20h" . . •- • • •• • •• • •- •I •

I I I

1.5

1.4 -

1.3 -

a '" 1.2 -....E

1.1 -a

1.0

0.9 -

0.8

1.050

1.030

Ji 1.010'"a....<I> 0.990a

0.970

0.950

b10 100 1000 10000

Experiment Duration (min.)

30 min• 1 h 2h• •• 5h

I • • 10 h• •

I I I I 20 h•I • •..I • • .• ••

5min15 min

- •.30 min

•- lh 2h

I • 5h 10h 20h-I ! :-

- •I I I

1.050

1.030

.E 1.010aV>....a'" 0.990

0.970

0.950

a

1.5

1.4 -

1.3 -

a'" 1.2 -....

E1.1 -a

1.0

0.9 -

0.8 I I

C 0.1 1 10 100 d 0.1 1 10 100

Experiment Duration (h) Experiment Duration (h)

FIG. 4. For Froude = 10: (a) Us/U." versus Experiment Duration for Slab Dimension 50 x 25 mm2 ; (b) U.!!},. versus Experiment Durationfor Slab Dimension 200 x 150 mm2

; fc) Urn/U. versus Experiment Duration for Slab Dimension 50 x 25 mm2 i'Td) Urn/H. versus ExperimentDuration for Slab Dimension 200 x 150 mm2

an exponential covariance function (Fiorotto and Rinaldo1992b), is also shown in Fig. 5. Fig. 6 shows the uplift coef­ficient n, versus 1,1Y I for constant values of [,./Y I and forFroude number equal to 12. Theoretical analyses pointed outthat the overall behavior of n can be explained as follows:

1. For I,. < A, (Fig. 5), or I. < A, (Fig. 6), when the ratioI./A,. or I,IA, decreases the correlation between forcesacting under and over the slab (Fli and F" in Fig. I, onewith opposite direction respect to the other), increasesand the total uplift force per unit area decreases. At thelimit when I,./A I' or I,IA x -7 0 the forces are perfectlycorrelated (IF"I- IF"J -7 0, Fig. I) and n tends to zero.

2. For I, » A, (Fig. 5) or I, » A, (Fig. 6), when theratio /)A, or I,IA x increases, the uplift coefficient de­creases. At the limit when (lA, -7 00 or I,IA x -7 00,

n -7 O. Pressures measured at distances much largerthan the directional integral scales are statistically in­dependent, so that we observe the occurrence of un­correlated spots of negative and positive fluctuatingpressures (respect to the mean value), with area of orderAxA, acting over the slab and along its boundaries. Theyare statistically independent in such a way that negativeand positive pressure spots are statistically balanced,leading to a reduction of the global uplift coefficientwhen the slab dimension increases (Feller 1968). In fact,as can be argued from the scheme in Fig. 1, when I. -7 00

F", Fli -7 0, and as consequence n -7 O.3. The maximum value of n is reached for slab dimensions

roughly in the range 2 < //A < 4 (UA x or /)A", de­pending on the slab shape). With reference to the scheme

690 I JOURNAL OF HYDRAULIC ENGINEERING I OCTOBER 1995

shown in Fig. 1, when I, > 2A x the pressures actingalong the boundaries x = 0 and x = I, are uncorrelatedwith pressures acting in the central part of the slab, Insuch a way positive pressure fluctuations at the slabboundaries may be contemporaneous with negativepressure fluctuations in the central part of the slab lead­ing to the maximum difference between forces aboveand below the slab. The dimension of the zone wheremaximum negative pressures at the slab center are un­correlated with maximum positive pressures at theboundaries is roughly AX" Hence, as the slab length in­creases, the number of uncorrelated spots increases,leading to the reduction of the uplift coefficient n (case2). Consequently the maximum value of n is expectedto occur in the range of 2 < I,IA x < 4.

To summarize, for (lA" and Froude number constants, nfirst increases with UA x ' reaching the maximum value, andthen decreases tending to zero for I, -7 x (Fig. 6). The sameconclusion can be obtained assuming UA, and Froude num­ber constants and allowing I,./A, variable (Fig. 5).

Our experimental data are in the range of 0 < I,./A, < 2(Fig. 5) and 0 < (lA, < 10 (Fig. 6) while the ratio betweenthe integral scales in longitudinal and transverse directions isin the range of 5-6.

Notably, Fig. 5 covers the range of variability of the slabdimensions compatible with practical applications, I" I,. s10 m.

As an example, we discuss in the following the case in whichF = 12, A, = 5.3y, and A, = YI (Fig. 6). The curve markedwith squares represents the case in which pressure fluctuations

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0.20 0.20

OSFROUOE • 5.7 >.,h, = 2.6 Os

0.15 0.15

0.10 o 1./Y.-5.5 0.10 o 1./Y,=6.a

" 1./y.-~.8 " 1./Y,=5.9

"" 1./y,=~.1 "" 1./Y,.5.1

0.05 o 1./Y,-3.5 0.05o I.IY,=~.2

o 1./Y,=2.8 o I.IY,=3.3

• 1./Y,-1.~ • I.IY,=1.7

0.0 0.00

a 0.00 1.00 2.00 3.00 ~.OO 5.00 b 0.00 1.00 2.00 3.00 ~.OO 5.00 6.00

IY/Yl IY/Y1

0.20 FROUOE· a.7 >.,/Y,· ~.5

6.00 a.oo

IY/Y1

o 1./Y,=10.2" I,lY,. 9.0"" I,/Y,= 7.7o I,lY,· 6.~

o I,lY,. 5.1• I,lY,· 2.6

~.OO2.00

FROUOE=10 Ay/Y, .5.1

0.05

0.15

d

0.25

Os0.20

6.00 a.oo

IY/Y1

o 1./y,-a.g" 1./Y,=7.8"" 1./Y,-6.7o Is/Y,=5.6o Is/y,-~.4

.ls/Y,=2.2

~.OO2.00

0.10

0.15

0.00 L_---.~----.---':=:;::==__.0.00

0.05

c

0.25FROUOE-12

OS0.20

0.15

o Is/y,·10.~

" Is/Y,= 9,1/1 Is/Y,- 7.8o Is/Y,= 6.5

0.05 o Is/Y,· 5.2.ls/Y,= 2.6

0.0e 0.00 2.00 4.00 6.00 a.oo

IY/Y1

FIG. 5. Uplift Coefficient fi. Computed by Eq. 4: (a) Froude = 5.7; (b) Froude = 7.5; (c) Froude = 8.7; (d) Froude = 10; (e) Froude = 12

in transverse direction and between longitudinal boundariesare strongly correlated. In such a way the variability of ndepends on the slab length. In agreement with the previoustheoretical analysis the maximum value of n is observed forI, = 3X-X' On the other hand, the curve marked with circlesrepresents the case in which pressure fluctuations in trans­verse direction and between longitudinal boundaries are weaklycorrelated. In such a case neither t/X- x or I)X-" playa prom­inent role in reaching the maximum value of n. However, inagreement with the theoretical analysis, the maximum valueof n is observed for I, = 5X-X' The previous considerationsare also in agreement with numerical results obtained usingexperimental correlation functions of the pressure fluctua­tions at the bottom inside the hydraulic jump region (Fiorotto1990; Fiorotto and Rinaldo 1992a, b).

From Fig. 5 we can argue that the most suitable form ofthe slabs (for a given area) is rectangular, with width kept tothe technical minimum, as can be deducted from the property

that n is much more sensitive to modifications of I,., with X-,> X- x •

Fig. 7 shows the experimental frequency distributions ofthe reduced variable Z = F'(t)/(I/-1.J,. for Froude numberequal to 10 and slab dimension of 200 x 150 mm2 (maximumarea) and 50 x 25 mm2 (minimum area). Skewness and kur­tosis are also reported in Fig. 7. The force was measured fora total sampling time of 3.5 h. The frequency distribution ofZ is wel1 represented by a Gaussian distribution fitted to thedata with the maximum likelihood method.

A more accurate test of normality can be done looking atthe skewness S and the kurtosis K. For the Gaussian distri­bution, these two parameters assume values of 0 and 3, re­spectively.

Fig. 7 shows the skewness, and the kurtosis, computed fora Froude number of 10 and slabs areas of 200 x 150 mm2

and 50 x 25 mm2. In all the experiments, Sand K were found

in the range of - 0.65 < S < 0.3 and 3.1 < K < 4.5, depending

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0.00 L-----.---,------r---,-----,.---,0.00 2.00 4.00 6.00 B.OO 10.00 12.00

Abdul Khader, M. H., and Elango, K. (1974). "Turbulent pressure fieldbeneath a hydraulic jump." 1. Hydr. Res., 12(4), 469-489.

Bowers, C. E., and Toso, J. (1988). "Karnafuli project, model studiesof spillway damage." J. Hydr. Engrg., ASCE, 114(5), 469-483.

Bowers, C. E., and Toso, J. (1990). "Closure to 'Karnafuli project.model studies of spillway damage.'" J. Hydr. Engrg., ASCE. 116(6).854.

Bowers, C. E., and Tsai, F. Y. (1969). "Fluctuating pressures in spillwaystilling basins." 1. Hydr. Div., ASCE, 95(6), 2071-2079.

Oi Santo. A., Petrillo, A.. and Piccinnini, A. F. (1990). "Studio dellesollecitazioni idrodinamiche sulle piastre costituenti il fonda dei bacinidi dissipazione a risalto." Alli XXIII Convegno di Idraulica e Costru­zioni Idrauliche, Firenze, Italy. Vol. 4. E127-E138 (in Italian).

Farhoudi, J., and Narayanan, R. (1991). "Force on slab beneath hy­draulic jump." J. Hydr. Engrg., ASCE. 117(1),64-81.

Feller. W. (1968). An introduction to probability theory and its appli­cations. John Wiley and Sons, New York, N.Y.

Fiorotto, V. (1990). "Un approccio bidimensionale allo studio della sta­bilita' delle protezioni di fondo in bacini di dissipazione." Alli XXIIConvegno di Idraulica e Costruzioni Idrauliche. Vol. \, 283-294. Co­senza. Italy (in Italian).

Fiorotto. V.. and Rinaldo. A. (1988). "Sui dimensionamento delle pro-

ACKNOWLEDGMENTS

APPENDIX I. REFERENCES

The probability density functions of the force show negli­gible skewness for all the slabs and kurtosis close to the valuecharacteristic for the Gaussian distributions. The deviationfrom the expected value of the skewness observed for thesmall slab is probably due to the sensitivity of the high mo­ments to small measurement errors. This is the a posterioriconfirmation of the good agreement between n, and Om'Hence the force is with good approximation normally dis­tributed leading to the a posteriori confirmation of the validityof the assumptions used by Fiorotto and Rinaldo (1992a) inorder to obtain (4).

For purposes of illustration we present in this section thedesign of a stilling basin. The inflow depth and velocity areassumed equal to 0.8 m and 16 mis, respectively. The resultingFroude number is 5.7. From Toso and Bowers (1988, Table1), the maximum C;" C7, values are found to be approxi­mately 0.9.

The stilling basin is built with concrete slabs of2.0-m widthand 4.5-m length, so IvlYI = 2.5 and UYI = 5.6. From Fig.5(a) we obtain 0 = 0.11. To take into account the modelscale effects, the differences between n, and Om' and theusual safety coefficient for these works, we suggest the in­troduction of an overall safety coefficient equal to 1.5. Thefinal value of ° is then 0.165.

The equivalent thickness of the lining (where equivalentthickness means that if the structure is anchored to underlyingrock, the failure strength of the anchors must be transformedinto equivalent weight) is given by (1); assuming the ratio "VIbe - "V) equal to 2/3 we obtain s ~ 2.5 m.

The following conclusion can be drawn from the presentstudy: (1) The more convenient shape of stilling basin slabsis rectangular with the larger dimension along the flow di­rection and the transverse dimension maintained to the tech­nical minimum; and (2) the equivalent thickness should bedesigned using (1) with the coefficient °obtained from Fig.5. A safety coefficient greater or equal to 1.5 is recommended.In case of a lack of experimental data, the pressure coeffi­cients C;, and C7, may be safely assumed as C;" c-;, = I.

The writers are indebted to Andrea Rinaldo for continuous adviceand support during the progress of the work. The writers wish to thankalso the anonymous reviewers for their helpful and very useful commentsand suggestions.

CONCLUSIONS

5.22.6

SKEWNESS 0.06KURTOSIS 3.40

0.05

0.10

5.00

0.00-0.20 -0.10 0.10 0.20

Za

12.00 SKEWNESS -0.633

p*KURTOSIS 3.47

10.00

8.00

6.00

4.00

2.00

0.00-0.20 -0.10 0.00 0.10 Z 0.20

b

0.25

Os0.20

0.15

FIG. 7. Probability Density Function of Force Fluctuation for Froude= 10: (a) Slab Dimensions = 200 x 150 mm2 ; (b) Slab Dimensions= 50 x 25 mm2

FIG. 6. Uplift Coefficient O. as Function of '.. for Froude Equalto 12 and '. Constant

on the slab dimensions and position with respect to the zonewhere the maximum difference in pressure occurs being thestatistical properties of force and pressure influenced by thedistance from the jump toe XC'

692/ JOURNAL OF HYDRAULIC ENGINEERING / OCTOBER 1995

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tezioni di fondo in bacini di dissipazione: nuovi risultati teorici e sper­imentali." Giornale del Genio Civile, Rome, Italy, (7-8-9), 179-201(in Italian).

Fiorotto. V., and Rinaldo, A. (1992a). "Fluctuating uplift and liningsdesign in spillway stilling basins." 1. Hydr. Engr., ASCE. 118(4),578­596.

Fiorotto. V., and Rinaldo. A. (1992b). "Turbulent pressure fluctuationsunder hydraulic jumps." J. Hydr. Res.. 30(4). 499-520.

Leutheusser, H. J., and Kartha. V. C. (1972). "Effect of inflow conditionon the hydraulic jump." J. Hydr. Div., ASCE, 98(8), 1367-1386.

Rinaldo, A. (1985). "Un criterio per il dimensionamento delle protezionidi fondo in bacini di smorzamento." Giornale del Genio Civile, Rome,Italy. (4-5-6), 165-186 (in Italian).

Rinaldo. A. (1986). "The structural design of the lining of spillway stillingbasins." Excerpta, 1. 81-89, Padova, Italy.

Sanchez Bribiesca. J. S.. and Viscaino. A. C. (1973). "Turbulent effectson the lining of stilling basin." ICOLD llth Congr. Madrid. Spain,Q. 41. Vol. 2.

Toso. J .. and Bowers. E. C. (1988). "Extreme pressure in hydraulicjump stilling basin." 1. Hydr. Engrg.. ASCE. 114(8), 829-843.

Vasiliev. O. F.. and Bukreyev. V. I. (1967). "Statistical characteristicsof pressures fluctuations in the region of hydraulic jump." Proc., 12thCOllgr. Int. Assoc. of Hydr. Res., Fort Collins, Colo., Vol. 2.1-8.

Wilson. E. H .. and Turner. A. A. (1972). "Boundary layer effects onhydraulic jump location." J. Hydr. Div., ASCE, 98(7), 1127-1142.

APPENDIX II. NOTATION

The following symbols are used in this paper:

A area of anchor;C" dimensionless pressure coefficient;

FF'

KIxI,.n

PoPoSs

s'

0",

.f!slim

total uplift force;fluctuating part of total uplift force;maximum value of fluctuating part of uplift force;kurtosis of normalized force fluctuation Z;length of slab in x-direction;length of slab in y-direction;number of anchors for each slab;pressure fluctuations above slab;pressure fluctuations under slab;skewness of normalized force fluctuation Z;equivalent thickness of slab;thickness of slab for anchored structures;flow velocity upstream of jump;flow depth upstream of jump;conjugate depth in jump;distance between toe of jump and center of slab;normalized force fluctuation;specific weight of water;specific weight of concrete;integral scale of pressure fluctuations in x-direction;integral scale of pressure fluctuations in y-direction;admissible tension of anchor;standard deviation of fluctuating force;standard deviation of pulsating pressures;uplift coefficient;uplift coefficient estimated using values of F' and C;"C,,;uplift coefficient estimated using IT F and IT,,; anduplift coefficient evaluated after 20-h experiment du­ration.

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