Direct Dynamic Force Measurement on Slabs in ...

DIRECT DYNAMIC FORCE MEASUREMENT ON SLABS IN SPILLWAY STILLING BASINS

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By Alberto Bellini and Virgilio Fiorotto2 ABSTRACT: This paper presents a new experimental procedure aimed at defining the global instantaneous uplift force, acting on slabs at the bottom of stilling basins. The global uplift force on slabs of different dimensions is measured for Froude numbers of the incident flow ranging from 5 to 12. The measurements were made in the zone of hydraulic jump, where high turbulence yields the maximum uplift force. The fluctuating force per unit area is shown to depend on the slab shape and on the statis~ic~1 structure of the instantaneous spatial distribution of turbulent pressures at the bottom of the hydrauhc Jump. ThIS paper provides direct experimental evaluation of a criterion for the .design of the protection slab.s in ~pillway stilling basins. It is shown that rectangular slabs with the longer sIde placed along the flow dIrectIOn (where the transverse direction is maintained to a technically minimum length) are most suitable for the design of the lining of spillway stilling basins.

INTRODUCTION

One of the most important steps in the design of spillway stilling basins is the definition of the maximum instantaneous uplift force produced by turbulent pressure fluctuations acting on the slabs (Fiorotto and Rinaldo 1992a). Cases of damage on chutes and spillway basins [e.g., Malpaso (Sanchez Bribiesca and Viscaino 1973), Tarbela, and Karnafuli dams (Bowers and Toso 1988, 1990)] have clearly highlighted the relevance of this problem. It is known in fact that fluctuating pressures at the bottom of the hydraulic jump are the primary cause of the failures (Bowers and Tsai 1969; Sanchez Bnbiesca and Viscaino 1973; Rinaldo 1985, 1986; Bowers and Toso 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 of the chute slab or cause damage of joint seals, thereby propagating underpressures (Fig. 1). Although the pressures are damped in the propagation through the drain system or along the soil-structure contact surface, the net difference between these pressures and those acting on the surface of the slabs may exceed the weight of chute slabs (Rinaldo 1985, 1986; Fiorotto and Rinaldo 1988; Toso and Bowers 1988; Oi Santo et al. 1991). A design criterion based on the uplift induced by the severe pressure fluctuations associated with energy dissipation in the region of hydraulic jump was proposed by Rinaldo (1985) and later refined by Fiorotto and Rinaldo (1992a). The criterion relies on the following expression, which relates the equivalent thickness of the linings to geometric and hydrodynamic parameters: I, + C _ ) _'Y_ - ,S - > D ( -, -I,, -I,) (C "+,, v-/2g

Y1

A,

A,

'Y, - 'Y

(1)

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 di Povo 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 closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on April 19. 1994. This paper is part of the Journal of Hydraulic Engineering, Vol. 121. No. 10, October, 1995. ©ASCE, ISSN 0733-9429/95/0010-0686-0093/$2.00 + $.25 per page. Paper No. ~222.

scale of the pressure fluctuations; Y 1 = depth of the incident flow; 'Y, 'Yc = specific weight of water and concrete; and C;, C7, = Ii. P';",.I( v2 /2g) = positive and negative pressure coefficients defined by pressure differences Ii. P,;",x above the mean value (Toso and Bowers 1988). Notice that for anchored structures the failure strength of anchors must be transformed into equivalent weight from which to compute the equivalent thickness s = s' + nAaj('Yc - 'Y)I/" with s' the thickness of the slab, n the number of anchors, A and a" the area and the admissible tension of each anchor, respectively. This criterion was defined by theoretical and experimental studies of the influence of the instantaneous pressure distribution above and below the slabs on the instantaneous uplift force. 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 of spillway basins; (2) the persistence of fluctuating pressures at a given point (time microscale) is greater than the propagatIon time of pressures between the joints; and (3) hydraulic resonance does not occur for slab dimensions compatible with practical applications. From this consideration we can. argue that 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 balance between forces acting over and under the slab: Fm"x(t)

= [F,,(t) - F,,(t)] = F;"ax +

= D( q

+

C,~

h

v' 2g f),.

+

'Y sf),.

'Ysl,l,

(2)

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 uphft force must be less than the slab weight: F lllax < 'Y ,51,1, so that (1) holds. It is evident, from Fig. L that n is a function of the 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 depends on the statistical characteristics of the pressure field at the bottom 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 allows direct measurement of the force acting on the lining slabs and simultaneously the pressure at the bottom in the hydraulic jump region. The new experimental setup is obtained by iso-

686/ JOURNAL OF HYDRAULIC ENGINEERING / OCTOBER 1995

J. Hydraul. Eng., 1995, 121(10): 686-693

L.E.

P,,(x,y, t)

a

I., D

(


r" x,y,

t) _ R ( t ) 0 x,y,

~'.,h, = 2.6

Os 0.15

0.15

o

0.10


0.05

0.0

a

0.00

1.00

2.00

FROUOE· a.7

o

I.IY,=~.2

o I.IY,=3.3 • I.IY,=1.7

5.00

IY /Yl

0.20

1./Y,=6.a " 1./Y,=5.9 "" 1./Y,.5.1

0.05

~.OO

3.00

o

0.10

1./Y.-5.5 " 1./y.-~.8 "" 1./y,=~.1 o 1./Y,-3.5 o 1./Y,=2.8 • 1./Y,-1.~

b

>.,/Y,· ~.5

0.00 0.00

1.00

2.00

3.00

~.OO

5.00

6.00

IY /Y1

0.25 FROUOE=10

Os

Ay/Y, .5.1

0.20

0.15

0.15

c

0.10

o 1./y,-a.g

0.05

" 1./Y,=7.8 "" 1./Y,-6.7 o Is /Y,=5.6 o Is/y,-~.4 . ls /Y,=2.2

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

0.05

0.00 L_---.~----.---':=:;::==__. 6.00 a.oo ~.OO 0.00 2.00

d

2.00

0.25

~.OO

6.00

6.~

5.1 2.6 a.oo

IY/Y1

IY /Y1

OS

o I,lY,· o I,lY,. • I,lY,·

FROUOE-12 0.20 0.15

o

0.05

e FIG. 5.

0.0 0.00

Is/y,·10.~

" Is/Y,= /1 Is/Y,o Is/Y,= o Is/Y,· .ls/Y,= 2.00

4.00

9,1 7.8 6.5 5.2 2.6

6.00

a.oo

IY /Y1

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 boundaries are strongly correlated. In such a way the variability of n depends on the slab length. In agreement with the previous theoretical analysis the maximum value of n is observed for I, = 3X-X' On the other hand, the curve marked with circles represents the case in which pressure fluctuations in transverse direction and between longitudinal boundaries are weakly correlated. In such a case neither t/X- x or I)X-" playa promHowever, in inent role in reaching the maximum value of agreement with the theoretical analysis, the maximum value of n is observed for I, = 5X-X' The previous considerations are also in agreement with numerical results obtained using experimental correlation functions of the pressure fluctuations at the bottom inside the hydraulic jump region (Fiorotto 1990; Fiorotto and Rinaldo 1992a, b). From Fig. 5 we can argue that the most suitable form of the slabs (for a given area) is rectangular, with width kept to the technical minimum, as can be deducted from the property

n.

that

n is much more sensitive to modifications of I,.,

with X-,

> X- x • Fig. 7 shows the experimental frequency distributions of the reduced variable Z = F'(t)/(I/-1.J,. for Froude number equal to 10 and slab dimension of 200 x 150 mm 2 (maximum area) and 50 x 25 mm 2 (minimum area). Skewness and kurtosis are also reported in Fig. 7. The force was measured for a total sampling time of 3.5 h. The frequency distribution of Z is wel1 represented by a Gaussian distribution fitted to the data with the maximum likelihood method. A more accurate test of normality can be done looking at the 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 for a Froude number of 10 and slabs areas of 200 x 150 mm 2 and 50 x 25 mm 2 . In all the experiments, Sand K were found in the range of - 0.65 < S < 0.3 and 3.1 < K < 4.5, depending JOURNAL OF HYDRAULIC ENGINEERING / OCTOBER 1995/691

J. Hydraul. Eng., 1995, 121(10): 686-693

0.25

Os 0.20 0.15


0.10 5.2 2.6

0.05

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

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

SKEWNESS KURTOSIS

The probability density functions of the force show negligible skewness for all the slabs and kurtosis close to the value characteristic for the Gaussian distributions. The deviation from the expected value of the skewness observed for the small slab is probably due to the sensitivity of the high moments to small measurement errors. This is the a posteriori confirmation 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 validity of the assumptions used by Fiorotto and Rinaldo (1992a) in order to obtain (4). For purposes of illustration we present in this section the design of a stilling basin. The inflow depth and velocity are assumed equal to 0.8 m and 16 mis, respectively. The resulting Froude number is 5.7. From Toso and Bowers (1988, Table 1), the maximum C;" C7, values are found to be approximately 0.9. The stilling basin is built with concrete slabs of2.0-m width and 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 model scale effects, the differences between n, and Om' and the usual safety coefficient for these works, we suggest the introduction of an overall safety coefficient equal to 1.5. The final value of is then 0.165. The equivalent thickness of the lining (where equivalent thickness means that if the structure is anchored to underlying rock, the failure strength of the anchors must be transformed into equivalent weight) is given by (1); assuming the ratio "VI be - "V) equal to 2/3 we obtain s ~ 2.5 m.

°

0.06 3.40

CONCLUSIONS 5.00

0.00 -0.20

-0.10

0.10

Z

0.20

a 12.00

p*

SKEWNESS KURTOSIS

-0.633 3.47

°

ACKNOWLEDGMENTS The writers are indebted to Andrea Rinaldo for continuous advice and support during the progress of the work. The writers wish to thank also the anonymous reviewers for their helpful and very useful comments and suggestions.

10.00 8.00

APPENDIX I.

6.00 4.00 2.00 0.00 -0.20

The following conclusion can be drawn from the present study: (1) The more convenient shape of stilling basin slabs is rectangular with the larger dimension along the flow direction and the transverse dimension maintained to the technical minimum; and (2) the equivalent thickness should be designed 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.

-0.10

0.00

0.10

Z

0.20

b FIG. 7. Probability Density Function of Force Fluctuation for Froude = 10: (a) Slab Dimensions = 200 x 150 mm2 ; (b) Slab Dimensions = 50 x 25 mm 2

on the slab dimensions and position with respect to the zone where the maximum difference in pressure occurs being the statistical properties of force and pressure influenced by the distance from the jump toe XC'

REFERENCES

Abdul Khader, M. H., and Elango, K. (1974). "Turbulent pressure field beneath a hydraulic jump." 1. Hydr. Res., 12(4), 469-489. Bowers, C. E., and Toso, J. (1988). "Karnafuli project, model studies of 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 spillway stilling basins." 1. Hydr. Div., ASCE, 95(6), 2071-2079. Oi Santo. A., Petrillo, A.. and Piccinnini, A. F. (1990). "Studio delle sollecitazioni idrodinamiche sulle piastre costituenti il fonda dei bacini di 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 XXII Convegno di Idraulica e Costruzioni Idrauliche. Vol. \, 283-294. Cosenza. Italy (in Italian). Fiorotto. V.. and Rinaldo. A. (1988). "Sui dimensionamento delle pro-

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J. Hydraul. Eng., 1995, 121(10): 686-693


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 linings design in spillway stilling basins." 1. Hydr. Engr., ASCE. 118(4),578596. Fiorotto. V., and Rinaldo. A. (1992b). "Turbulent pressure fluctuations under hydraulic jumps." J. Hydr. Res.. 30(4). 499-520. Leutheusser, H. J., and Kartha. V. C. (1972). "Effect of inflow condition on the hydraulic jump." J. Hydr. Div., ASCE, 98(8), 1367-1386. Rinaldo, A. (1985). "Un criterio per il dimensionamento delle protezioni di 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 stilling basins." Excerpta, 1. 81-89, Padova, Italy. Sanchez Bribiesca. J. S.. and Viscaino. A. C. (1973). "Turbulent effects on the lining of stilling basin." ICOLD llth Congr. Madrid. Spain, Q. 41. Vol. 2. Toso. J .. and Bowers. E. C. (1988). "Extreme pressure in hydraulic jump stilling basin." 1. Hydr. Engrg.. ASCE. 114(8), 829-843. Vasiliev. O. F.. and Bukreyev. V. I. (1967). "Statistical characteristics of pressures fluctuations in the region of hydraulic jump." Proc., 12th COllgr. Int. Assoc. of Hydr. Res., Fort Collins, Colo., Vol. 2.1-8. Wilson. E. H .. and Turner. A. A. (1972). "Boundary layer effects on hydraulic jump location." J. Hydr. Div., ASCE, 98(7), 1127-1142.

APPENDIX II.

K Ix

I,. n Po Po S s s'

NOTATION

The following symbols are used in this paper: A C"

F F'

area of anchor; dimensionless pressure coefficient;

0",

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,,; .f!slim

uplift coefficient estimated using IT F and IT,,; and uplift coefficient evaluated after 20-h experiment duration.

JOURNAL OF HYDRAULIC ENGINEERING / OCTOBER 1995/693

J. Hydraul. Eng., 1995, 121(10): 686-693