## Uplift force and momenta on a slab subjected to hydraulic jump

## Fuerza de levantamiento y momentos en una losa sometida a salto hidráulico

## DOI:

https://doi.org/10.15446/dyna.v83n199.52252## Palabras clave:

Uplift pressure, hydrodynamics uplift, slabs, physical model, stilling basins, hydraulic jump. (en)Presiones de levantamiento, levantamiento hidrodinámico, losas, modelo físico, tanques de amortiguación, salto hidráulico. (es)

## Descargas

**DOI:** https://doi.org/10.15446/dyna.v83n199.52252

**Uplift
force and momenta on a slab subjected to hydraulic jump**

**Fuerza de levantamiento y momentos en una losa
sometida a salto hidráulico **

**Mauricio González-Betancourt **

*Investigador del convenio
COLCIENCIAS- Servicio Nacional de Aprendizaje SENA y miembro del Grupo de
investigación del Posgrado en Aprovechamiento de Recursos Hidráulicos,
Universidad Nacional de Colombia, Colombia. magonzalezb@sena.edu.co;
mao275@yahoo.com*

**Received: August 03 ^{th}, 2015.
Received in revised form: February 14^{th}, 2016. Accepted: July 18^{th},
2016.**

**This work is licensed under a** Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

**Abstract**

As part of the search for safe and
economic design criterion to line slabs of stilling basins, the present study
is one of the first to calculate the center of pressure, uplift forces, and
momenta from a spatiotemporal analysis of the pressures measured above and
below instrumented slabs in a physical model. Controlled release of the
waterstops, and variation in the dimensions of expansion joints and in the gap
between foundation and the lining slab were carried out in order to consider
their effects on the magnitudes of uplift forces and momenta. An offset of the
center of pressure from the slab's center of gravity was identified. The objective
of this work was to consider the failure mechanism induced by momentum in the
slab's design. Design criterion to make the lining slab's thickness to length
between 6 and 12 times the incident flow depth, is proposed, and this is
compared to other design criteria.

*Keywords*: Uplift pressure; hydrodynamics uplift; slabs, physical model;
stilling basins; hydraulic jump.

**Resumen
**Buscando un
criterio de diseño seguro y económico para las losas de los tanques de
amortiguación, se calcularon fuerzas de levantamiento, centros de presión y
momentos a partir de un análisis espacio-temporal de las presiones medidas
encima y debajo de losas instrumentadas en un modelo físico. La liberación
controlada de los sellos y la variación en las dimensiones de las juntas y de
la separación entre la cimentación y la losa, fueron realizadas para considerar
sus efectos sobre la fuerza de levantamiento y los momentos. Se identificó
desplazamiento del centro de presión respecto al centro de gravedad de la losa.
Para considerar el mecanismo de falla inducido por el momento en el diseño de
losas, objetivo del trabajo, se propone un criterio de diseño para el espesor
equivalente de las losas con longitudes entre 6 y 12 veces la profundidad del
flujo incidente y se compara con otros criterios.

*Palabras clave*: Presiones de levantamiento; levantamiento
hidrodinámico; losas; modelo físico; tanques de amortiguación; salto
hidráulico.

**1. Introduction**

Outlet works conduit typically requires dissipation of excess kinetic energy to prevent downstream channel erosion as this flow often discharges at a high velocity. An energy dissipator, such as a stilling basin, is used to retard the fast moving water by creating a hydraulic jump [1]. The design of an energy dissipating structure need criteria to avoid cavitation, abrasion, internal erosion, hydrodynamic uplift, etc.

Historical reporting of the failure of slabs in flumes and stilling basins, with an equivalent thickness ranging from 0.3 m to 4 m, has show the hydrodynamic uplift to be a structural design problem [2-5]. Equivalent thickness is the real slab thickness, and anchors are defense against uplift.

Since the early 60s, design criteria for lining slabs to calculate the uplift force and equivalent thickness have been proposed. The most recognized design criteria were based on stochastic analysis of the pressure and force fluctuations at the floor of the hydraulic jump in a physical model [6-10]. A summary of the criteria can be found in references [11-12]. They determine the uplift force taking into account the length (L) and width (W) of the slab. Researchers agree with the criteria that the length of the slab in direction of flow has an inverse relation to the uplift force [6,7,9,10,12]. However, there is no complete agreement as to the slab's width influence on the uplift force. Bellin and Fiorotto [10] suggest building the slab with the minimum width technically possible, and other criteria indicate otherwise [8,9,12].

There are differences between design
criteria due to simplifications in the physical and conceptual models that were
supported. For the same condition, the slab's equivalent thickness that was
calculated using existing criteria shows large differences. Thus, it is
difficult to choose criteria that guarantee the stability of the slab with the
lowest cost and, generally, the designer chooses the more conservative
criteria. According to Khatsuria [11], the vast variation between criteria
points to the fact that this science is still in an evolutionary stage. While Bollaert
[13] expressed that "*despite major advances in measurement
technology and data acquisition, a safe and
economic design method for any kind of concrete lined stilling basin is still
missing today. Especially the dynamic or even transient character of pressure
pulsations as a function of their two-dimensional spatial distribution above
and underneath the lining is not fully assessed and implemented in existing
design methods. *[13]".

In addition, the physical models that supported the design criteria were not simulated in their true scale joints, waterstops, slab thickness (s), and gap between soil foundation and the concrete slab (d) since the pressure drop through the joints to the foundation was depreciated. In the prototype, gap d and width joint (e) can change by sources of natural movement, internal erosion, etc. Furthermore, the position and number of sensors reported by the references [6-10] were not sufficient to be able to accurately estimate both the uplift force and its center of pressure.

To date, the evidence suggests that hydrodynamic uplift is also influenced by the interaction of fluid with joint and detachment of the waterstops [13-19]. However, these factors have not been considered in the design criteria of the slab subjected to a hydraulic jump with horizontal aprons.

Previous work was observed in which joints and waterstops act as pressure filter fluctuations generated in the flume [18]. Also, the joints generate a time delay between the entry of the pressure wave at the joint and its arrival below the slab. It leads to the pressure differential between top and bottom of the slab and, therefore, the uplift force emerges. The interactions between the joints and the main stream alter the amplitude of pressure wave below the slab. Joints and waterstops promote generating pressure gradients below the slab and some instants, the pressure gradients will have positive or negative linear correlation. With only one open transversal joint, the pressure below the slab was uniform. With two or more open joints, pressure gradients below the slab were generated.

No uniformity in pressure over a slab
leads to the consideration of the failure mechanism induced by momentum of
force. If a slab turns a little due to momentum, offset between slabs occurs
and the stagnation point increases the uplift pressure and drag force. Failure
or loss of intimate contact with the
soil is typically the result of a slab overturning its downstream contact point
[1]. The forces of interest in a momentum balance are slab weight (F_{weight}),
strength of the anchors, the resultant forces F^{-} and F^{+} of field pressure acting on the surface above and below the slab, and drag
force (F_{R}). The net uplift force (F_{net}) is a vector sum
of the forces F^{+} and F^{-}, which could have a center of
pressure in a different coordinate to the center of gravity of the slab that
changes over time (Fig. 1).

Therefore, in this study the author determined
the forces F^{+}, F^{-} and F_{net} from a spatial
integration of pressure fields and balance forces. The author established the
pressure fields from measured pressures with multiple sensors above and below a
slab that was subject to hydraulic jump. The center of pressure to be able to
calculate the momenta of force F^{+} and F^{-} was also
identified, and the momentum Ma in terms of the downstream contact point of the
slab, and the momentum Mc, in terms of the upstream contact point of the slab (Fig.
1) was calculated. The author also contemplated the variation of magnitude of
the force F^{+} and its center of pressure by the joints, the damage of
the waterstops, and the variation of gap d and e.

Multiple tests including physical and hydrodynamic variations were evaluated. They were analyzed and helped improve the understanding of the uplift hydrodynamic of the lining slabs. The author present the design criterion that has an equivalent thickness of the slabs with lengths between 6 and 12 times the depth of the incident flow. This paper is based on the author's PhD thesis [18].

**2. Materials and methods **

Pressures were measured above and below slabs in a physical model that was made in the Hydraulic Laboratory at the "University of Valle" in Cali, Colombia (Fig. 2). The physical model contains slabs fixed under the horizontal flume floor at different distances from the load tank. The flume was 0.5 m high, 8m long, and 0.35 m wide. The slab and its details such as expansion joints, slab thickness, gaps d, and e, were simulated with several acrylic boxes with dimensions from largest ((L+)*(W+)*(s+d), internal dimensions) to smallest (L*W*s, external dimensions; Table 1; Figs. 1, 2).

The length (L) of
individual slabs ranged from 6 to 12 times the incident flow depth (y_{1}),
and the slab's width was approximately half its length. The gap d was
possible by interposing aluminum sheet rings 1 mm in diameter and thickness
that were required to achieve the desired separation.

The flume floor was
drilled and slotted to provide continuity to pressure taps and joints (Fig. 3).
The model slabs were fixed to the basin to prevent motion. The coupling between
elements in the system was monitored to avoid stagnation points by the offset
between edges, as was recommended and studied in references [19-20]. The offset
was estimated to be 10^{-5}m. This could be the case for the prototype
due to imperfections in the finishing of the slabs and their rearrangement by
natural movement.

The
flow entering the hydraulic jump was full and partially developed with a
Reynolds number in the boundary layer (Re_{x}) between 300,000 and
17,380,000. The first slab (S1) was located at a distance at which the Re_{x} was between 300,000 and 660,000 (transition), the second slab 2 (S2) was in the
area of Re_{x} between 4,150,000 and 9,130,000, and the third slab (S3)
was the farthest from the load tank with Re_{x} between 7,900,000 and
17,380,000. The experimental design varied the state of development of the
boundary layer since the magnitude of pressure fluctuation depended on whether
the flow is fully developed or undeveloped [11]. However, the flow regime was
always turbulent with Reynolds numbers between 90,000 and 200,000. Incident
flow velocities (V_{1}) ranged from 1.65 and 5.76 m/s.

The pressure was measured with 32 Motorola sensors (MPXV 4006GC7U, range 0-6 kpa and accuracy ±5 %) and with circular pressure taps of 2 mm in diameter above and below the slab with two distributions (Fig. 4). The first sensors distribution (D1) selected 16 pressure taps above and below the slab that had an equal distribution (Fig. 4a, sensor symbol "•" and "■"). The second sensors distribution (D2) selected 8 pressure taps that were located above the slab in the central line (Fig. 4a; sensor symbol "•"). When these sensors failed, the pressure taps next to the longitudinal joint were used (Fig. 4a; sensor symbol "■"). Furthermore, 24 pressure taps below the slab and the bottom of the joints were implemented to achieve a higher resolution of the pressure field (Fig. 4b; sensor symbol "♦").

Every five working hours, in accordance with the methodology proposed in the reference [21], the pressure measurement system was dynamically calibrated, and dynamic uncertainty was, on average, 8.82%. The signs that were acquired with a data acquisition system (DAQ National Instruments, NI SCXI: 1000, 1102B, 1600, 1300) were sent to a laptop. The sampling frequency (fs) was 200 Hz and was limited by the data acquisition system available. In addition, it held the sampling theorem (avoiding aliasing) and improved the resolution in the time of the digitized signal (5 ms). In signal processing, a median digital filter was used to remove frequency components that were not part of the phenomenon since it closely recovers the original signal while removing noise [22]. According to the analysis of the frequency signal and the dynamic characteristics of the pressure measurement system, the cutoff frequency of the digital filter was 10.5 Hz (window median filter equal to nineteen). Thus, the typical overshoot of the pressure measurement system in response to a sudden change of pressure (pressure fluctuation) was minimized. "Overshoot is the amount of output measured beyond the final steady output value in response to a step change in the measurand" [23].

In each test, the slab type (S1A, S1A*,
S1B, S2, and S3: Table 1), the open
joint(s), and the Fr_{1}, were selected. Hydrodynamic variations
included a minimum seven different Fr_{1} between 2.3 and 10 for each
physical variation.

Physical variations include the controlled release of the waterstop(s) in: a) one of the four joints, front transverse joint (FTJ), rear transverse joint (RTJ), or longitudinal joint (LJ); b) two joints simultaneously, longitudinal joints (LJ), or transverse joints (TJ); c) all joints (AJ).

The flow depth was measured on a) FTJ, b) LJ in the middle of the slab, c) RTJ. At each point, the minimum and maximum depth detected in 30 seconds were measured with depth gages that had a 300 mm range and an accuracy of ± 0.2 mm. The discharge was regulated between 8.15 and 14.1 Gallon/min, as was the vertical gate of the load tank (1.8m high, 1m long and 0.35 m wide) between 2.5 and 5.5 cm. The discharge was measured from an Omega flow meter (FMG-901). The flow rate was measured with a Prandtl tube at a point located 0.05 m upstream of the front transverse joint, and the measurement error was 4%.

The test was run over 15 minutes and the data acquisition was performed during in the last five minutes. Data acquisition time was mainly associated with extensive data and the number of tests explored (420). One test had 60,000 discrete samples in which the pressure fields were analyzed.

The hydraulic jump with rectangular weirs of heights ranging from 5 to 20 cm at the end of the flume was induced. Because, in general, the highest pressure fluctuations are reported in the first third of the length of the free hydraulic jump [8,24-30], the slabs were located under 30% of its length. Some tests in the slab "S1" had a submerged hydraulic jump.

**3. Data processing and results**

This researcher evaluated each sampling
instant from the 58 test samples that had an S1A configuration, the forces F^{+},
F^{-}, F_{net}, their center of pressure over the slab
surfaces, and the momenta Ma and Mc. Using data processing, the pressure fields
from the pressure measured above and below the slab in each sampling instant
were adjusted. According to the theory of Riemann integration, to obtain the
total force vectors F^{+} and F^{- }(eq. 1 and eq. 2), the area
of slab was discretized and the sum of the partial pressures was computed.

Where, P^{-} and P^{+} are the pressures above and below the slab's surfaces. (L/186 x B/115) are area
elements in which the slab area (A_{L}) was
discretized.

The x-coordinate of center of pressure (intersection of the resultant force and the surface's line of action) was obtained with eq. 3.

Where x_{i} is x coordinate of
the center of the differential element of area.

The forces F^{+ }and F^{-}'s
centers of pressure were different to their centers of gravity, which affect
momenta that induce rotation of the slab. The offset percentage of the center
of pressure from the center of gravity in x-coordinate was calculated using eq.
4 (Fig. 5).

Where x_{c} is x-coordinate of
the slab's center of gravity (or centroid).

The author calculated the momenta
balances with respect to the slab's extreme contact points. The clockwise
momentum was positive. The F_{net} and net momenta (Ma_{net }and
Mc_{net}) determine the possibility of the slab's uplift or rotation
(Fig. 1). Then, the maximum value of F_{net}, Ma_{net, }Mc_{net} in each test was selected and expressed in
terms of net instability dimensionless
coefficients (F_{NM}*, Ma_{NM}*, |Mc_{NM}*|;
eq. 5 -7).

Where g is the specific weight of water and g is the gravity.

The coefficients F_{NM}*
were lower than the coefficient Ma_{NM}* and |Mc_{NM}*| (Fig.
6), which show that the mechanism of initial failure is the slab's rotation.
The dotted boundary was called the enveloping curve of instability net coefficients (C_{net}) and
it will be used to predict the uplift hydrodynamic in the next subsection. Two
values of |Mc_{NM}*| that fall outside the area
of the curve (Fig. 6). These values manifest the
combination of a great uplift force with a great center of pressure offset from
the center of gravity below the slab (Fig. 6).

To expand upon the above,
the author analyzed the impact on uplift force of the variations of Re_{x},
the gap d, the gap e, and the controlled release of the waterstops. It was considered that these physical variables only affect pressure fields below the slab, i.e. the force F^{+}. Thus, in each sampling instant from the 420 tests we
calculated the force F^{+} (eq. 2) and then expressed it in the
form of a dimensionless coefficient, according to eq. 8.

Where was calculated as the average of three minimum depths measured on FTJ, LJ, RTJ with a baseline below the slab.

To avoid the mistake of
basing the analyses on a spurious value, ten maximum values identified in each
test were plotted (Figs. 7-10). For same Froude number, coefficient F^{*} varied as a result of pressure fluctuations of the hydraulic jump and the
geometric variations made in each test.

The incidence of the
state of development of flow over force F^{+} was not clear. The force F^{+} is proportional to the Froude
number and, generally, it was lower than twice the average hydrostatic force
below the slab (; Fig. 7).

The effect on the force F^{+} by detachment of the waterstops, the change of the gaps e, and d can
be deduced from Figs. 8-10. These showed the maximum F^{+} found in the
different tests with S1 configuration and open transversal joints (Fig. 8),
open longitudinal joints (Fig. 9), and all open joints (Fig. 10). Furthermore,
a slab configuration with gap e of 0.5 mm and 2 mm; and gap d of 0.2
mm, 0.5 mm, and 1 mm was also considered

As a
result of these tests, it is possible to say that the narrowest joint
(e=0.5 mm) induced a greater uplift force F^{+ }(Fig. 8-10). Tests
with open traversal joints induced major uplift forces
under the slabs, and these were followed by tests with all open joints.

The incidence of gap d on uplift force was not clear. For the condition of open transverse joints an inverse relationship between the uplift force and the gap d was observed (Fig. 8).

However, occasionally the author identified a proportional relationship between F+ and gap d in tests with longitudinal joints or all open joints (Figs. 8, 9). The greatest uplift forces obtained from the combination of gap e of 0.5 mm and gap d of 0.5 mm, and the results showed in references [15,16], led to us leaving the hypothesis open (narrower gap d leads to a hydraulic jack).

The study of net uplift force and its point of application showed that it is necessary to consider the equivalent thickness of lining slabs in design criteria as well as the failure mechanism induced by momentum. Also, detachment of the waterstops, the size and the orientation joints that have an effect on the uplift force, and consequently influence its momentum should also be considered.

**4. Estimation of hydrodynamic uplift**

The author proposes experimental tests and design criterion to estimate the equivalent thickness of a slab with lengths between 6 and 12 times the incident flow depth. The design of concrete slabs for hydrodynamic loading focuses on the determination of the maximum possible destabilizing force and momentum. In this criterion, uplift force and momenta are considered by the dimensionless design coefficient , calculated by eq. 9.

Where, C_{net} is the instability net coefficients, which can be obtained from Fig. 6's
envelope curve. "" is an dimensionless experimental coefficient that takes into account the increase of the force F^{+} and its momentum from the
detachment of the waterstops, and the size and the orientation joints. "" is a dimensionless experimental coefficient that takes into account extreme instabilities.

To
obtain the coefficients and , the force F^{+} found in
each sampling instant from 420 tests was used to calculate the momenta Ma^{+} and Mc^{+}. The largest momenta Ma^{+} and Mc^{+} in each
test were selected and expressed in form of dimensionless coefficients (eq. 10 and eq. 11).

The largest absolute value of the coefficients Ma* and Mc* in each test were identified and called coefficient M (eq. 12).

M coefficients
from 58 tests with an S1A configuration were added to the subscript r (M_{r}).
A curves adjustment over the maximum value of M_{r} and M vs. Fr_{1} was plotted (Fig. 11). The M curves that were proposed
as lines ensured that the slope is always positive as the uplift force F^{+} is proportional to Fr_{1}. Furthermore, since the positive and negative maximum pressure coefficient increases over time [8,27], the
line fits the data conservatively to
compensate for the short-time data acquisition (5 min).

The
amplification factor that depends on Fr_{1} is calculated
according to equation 13 (Fig. 11).

M_{max }were associated with the extreme
instabilities and the low probability of occurrence (Fig. 11). To determine the coefficient the same
methodology used to determine was implemented, and the M curve was compared with a curve fit to M_{max }(eq. 14, Fig. 11).

The design coefficient in the Fr_{1} function is presented in Fig. 12. The buoyancy is considered by the design
coefficient if the real thickness used in the prototype is less than the
calculated thickness scaling model to prototype (Table 1). If the above are fulfilled, the thickness of
the slab can be calculated with equation 15.

Where is the specific weight of concrete. In a scenario in which the slab could be immersed in a water-sediment mixture (for example, during the flushing of sediments from reservoirs), it is necessary to replace the specific weight of water by mixture (). The above equation intends to compensate for the increase in the buoyancy force on the slab by the increased density of the mixture.

The slab thickness is most often selected empirically as achieving stability to resist uplift force alone with this parameter requires a heavy slab, and this is not always possible. In these cases, the methodology of equivalent thickness through the anchor should be used [31]. Slab thickness and the anchor could be calculated with eq. 16

The left side of eq. 16 is considered the hydrodynamic uplift. "" is the drainage coefficient, and it can be used as a means to theoretically reduce up to 50% of the pressure uplift (= 0.5; [32-33]). The right side of eq. 16 represents the forces that counteract the uplift force, including the weight of the concrete slab and the force that provides the anchor. Where, "" is the tensile stress between steel-slab, is the number of steel bars, and is the area of the steel bar. For safety purposes, double the area of the anchor steel is assumed to design slabs in stilling basins [31].

Underdrains, anchors, cutoffs, and slab
thickness are all provided to stabilize the slabs [32]. A slab that is about
600 mm thick is the minimum recommended [33], and this shall be determined by
analyzing hydrostatic uplift and an elastic foundation analysis [32]. Uplift
momenta and misalignment between slabs can be prevented by adding steel in the
partial contraction deboned joint located the transversal joint (the joint with
slip dowels). Thus, the deboned joint allows the expansion or contraction of
the slab while the steel counteracts the shear loads void misalignment between
slabs [32,34-35]. Also, safety reinforcement counteracts the momentum of force
F_{R}.

**5. Analysis**

To ensure that the physical model represents the prototype, there must be geometrical, kinematic, and dynamic similarity [36-38]. Therefore, it is necessary to consider the effects of scale before using any lining slabs design criteria.

In hydraulic jump, when the large eddies are well reproduced in a model, the representation of turbulence is nearly achieved since the eddies are the energy carriers [11]. The large eddies in the turbulent flow are proportional to the principal dimension of the flow field, and they ensure correct simulation using a geometrically similar model [11]. The author recommends using a geometric scale more generous than 1:50 [36].

Viscous effects can lead to scale
effects, especially in models based on Froude, for which the Reynolds number is
always less than the prototype [11]. In tests with an Fr_{1} between
2.3 and 7.15, viscous effects and scale effects in terms of void fraction,
bubble count rate, and bubble chord time distributions were overcome. Their
influence was at least minimized since in the model the flow depth was greater
than 30 mm, and there was a turbulent flow with Reynolds numbers greater than
100,000 [39,36-41]. When the physical model does not take into account the
concentration of air in the evaluation of the pressures, it may require an
additional safety factor. Pinheiro [25] found that increased air concentration
decreases the mean pressure value and the standard deviation of force.

According to Lopardo *et al.* [29], the pressure data collected in a physical model with a
free vertical gate system and horizontal flume can be extrapolated to sloping
channels in hydraulic jump stilling basins. That is as long as the error is on
the side of safety. Similarity law of fluctuating pressure spectrum in the
strongly rolling area agrees with gravity law [41]. Thus, data from basic
research provides pressure values that are useful for stilling basin predesign,
despite the limits in scale similarity.

Viscous
and inertial forces dominate physical processes involving flow through small
cracks or joints along a channel boundary [19]. The dimensions used to simulate
the expansion joints (0.5 mm - 2 mm) lead us to suggest the use of a geometric
scale more generous than 1:30. To establish similarity between the roughness
implemented in the general contour of the model (coated in acrylic: n_{Manning} = 0.009) and prototype (coated concrete: n_{Manning} = 0.014), the
geometric scale should be 1:14. Thus, the Manning roughness scale factor limits
the geometric scale [36]. When this parameter has no similarity, the model is
rougher than its prototype, and scale effects are generated.

Inside the joints and the d gap, as
with the cavities, constrictions, and junctions, the propagation speed of
pressure wave (c) becomes a function of the variation of fluid density
(compressibility), and area (distensibility) increases in pressure [42]. For a
given Fr_{1}, the effects of the Reynolds number on the two-phase flow
properties are particularly notable in the developed shear layer [38]. Thus,
aeration in a prototype will be higher than in the model, and a similar or
smaller "c" can be expected in the prototype. Pressure waves experience
diffraction, interference, reflection, and refraction that depend of solid material in the boundary. These
can alter the propagation speed, amplitude and the transmitted energy [14,18].
In the prototype, the boundary may be concrete and soil. The boundary was
acrylic in the model, in which "c" may decrease and energy losses in the
reflection of the pressure wave increase. Nevertheless, in a prototype designed
with the criterion and a scale more generous than 1:14, the impact of the last
variables are less in the resonance phenomena and persistence time of the net
uplift pressure due to the length of the slabs [27], the hydraulic jump's low
frequency [27,31,37], and the role of the joints as frequency filters [17,18].
The pressure amplification under the slab for the effects of fluid, joints
interaction, and pressure waves at frequencies below 10.5 Hz were considered as
part of this criterion.

In
these cases that Fr_{1} is greater than 7.16, and/or the stilling basin
demands a greater geometric scale than 1:15 to be able to establish similarity.
The criterion proposed can be used for the predesign of lining slabs. "For a joint length of 18 m and *c* of 100 m/s, one obtains a resonance frequency < 3 Hz [13]". If the resonance
is demonstrated, the transient approach needs a quantification of pressure
amplification inside the joints, which can be the use of an appropriate
pressure amplification coefficient [13]. To verify resonance inside the joints
and the gap d, a model scale 1:1, pressure sensors with high natural
frequency and with a damping close to 0.6 to prevent overshoot error are
required. To check
the stability of the slabs in the physical model of the prototype, a scale more
generous than 1:14 is recommended. Given the opportunity to try alternatives,
analyze different solutions, view operating
conditions in extreme situations, and eventually, reduce the risk [43], physical modeling allows
details to be refined so to as find a safer and more economical project.

To compare the developed criterion with
other traditional criteria to estimate the slab equivalent thickness, some
scale effects are ignored and two illustrative examples are shown. The author
consider the hydraulic jump stilling basin with the following details. The
first example: y_{1}=2.43m, y_{2}=31.45m (major conjugate of
the hydraulic jump). Size of panel monolith: W=11.5 m (W=4.4y_{1}) and
L= 25 m (L= 9.76y_{1}). V_{1} = 46.1 m/s and Fr_{1} =
9.44. Second illustrative example: y_{1}=0.45m, y_{2}=3.54 m,
W= 4.5 m (W=4.4y_{1}), L=2 m (L=10y1), V_{1} = 16.5 m/s and Fr_{1} = 5.5. In the computation, the submerged specific weight is 1.6 ton/m^{3},
and a safety factor and an operating drainage system are not considered.

Hydrostatic uplift is evaluated using
three conditions: spillway design flood,
stilling basin empty, and sudden drawdown following design flood [11,33].
Usually, the first condition gives the maximum uplift force in the slab close
to the hydraulic jump [11]. The latter is due to the pressure of tail water
level (Tw), which is transmitted by a saturated foundation that has greater
fluid pressure on the slab. The criteria belonging to Hajdin [7] and Toso *et al.* [8] was used, which are based on
the measurement of fluctuating pressures, and the criteria of Farhoudi *et al.* [9] and Bellin *et al.* [10], which are based on direct
measurement of fluctuating force. In all these studies, except Bellin *et al.* [10], propagation of fluctuating
pressures below the apron was not considered. For the criteria to be applied,
the influence of the length and width of the slab are taken into account by
using the coefficient of spatial correlation (, the force coefficient (), or the uplift coefficient (. This paper also considers the dimensionless pressure coefficient
(Cp), which is based on the maximum positive (Cp^{+}) and negative
pressure (Cp^{-}) deviation from the mean pressure, and the coefficient
based on root-mean-square pressure fluctuations (Cp') that is reported in the
references [7-10, 27, 30].

In each example, two computations to
evaluate Bellin *et al.* [10] and
Toso's [8] criteria with two Cp were used as it is important acknowledge the
importance of a correct selection pressure coefficient when estimating the equivalent thickness. The slab equivalent thickness computation using various
relationships is presented in Table 2.

The first
computation of the two illustrative Cp examples was based on Cp^{+} and
Cp^{- }experimental data. In this case, the equivalent thickness estimated by criteria from Bellin *et al.* [10], Hajdin [7], and Farhoudi *et al.* [9] were similar. The equivalent thickness that was estimated by the
criterion proposed in this paper (last row, table 2) is greater than those
mentioned above while Toso *et al.*'s
criterion [8] was most conservative.

In the second computation in the two
illustrative examples, Cp was based on the suggestions from each group of
researchers. Bellin *et al.* [10]
suggest assuming the pressure coefficients Cp+, Cp- =1 in case of a lack of
experimental data while Toso *et al.* [10] suggest Cp 0.9 for the incident Froude number between 7.7 and 10. In this
case, the criteria were very conservative.

The Toso [8] criteria may be safely assumed, but it is conservative for a large slab that presents the compensation of the pressure pulses on its upper face. In the hydraulic jump, the pressure pulses on big slabs are uncorrelated since the slab length is larger than the integral scale of the pressure fluctuations. The integral scale is thereby defined as the distance on which, on average, two pressure pulses become fully uncorrelated. In other words, it defines the maximum possible area in which a pulse may reasonably act [13,17].

Thus, extreme pulses recorded by a single sensor in the large slab are not a representative sample of the pressure fields above and below the slab to accurately calculate the uplift force [13,18]. Above the large slab, a positive or negative pulse is a local effect.

In the first illustrative example, for the first computation and the discarded Toso criterion, hydrostatic uplift was the most critical condition for the slab's stability. This clearly shows that the release of waterstops at two different points in the floor stilling basin leads to critical situations that deserve to be paid more attention from engineers.

In this study, a slab was located in the highest pressure fluctuations zone of the hydraulic jump as the
study of Farhoudi *et al.* [9]
and Bellin *et al.* [10] indicated. Differential
heads resulting from the sloping water surface of the jump can cause a
circulating flow under the slab if leakage is allowed to enter the joint at the
downstream end of the basin and to flow out of the joint at the upstream end
[32]. Therefore, it is necessary in future research to
mount several slabs along the hydraulic jump and to vary the number of waterstops detached
on two different slabs: one at the area of maximum pressure
fluctuation in the toe jump and the other at the area of maximum depth.

The criterion proposed contributed to the
search for a safe and economic design method for a concrete lined stilling
basin because it considers uplift force and momenta. It was computed from a
spatiotemporal analysis of the pressure fields that was measured above and
below the instrumented slabs in a physical model. Furthermore, the criterion
was supported by a large amount of experimental information, for which details
that had previously been poorly studied such as joints, waterstops, and gaps
between foundations and the concrete slab were considered. The design
coefficient that is supplied by a curve in an Fr_{1}function
facilitates the designer's application of the criterion.

**6. Conclusions **

The waterstops, the size, and the orientation joints have an effect on the uplift pressures, and consequently, influence the magnitude of uplift forces and momenta. The narrower joints and open traversal joints generated major uplift forces below the slabs. An offset of the center of pressure from the center of gravity in the flow direction increased the momenta Ma and Mc by up to 30%. Thus, it was necessary to consider the failure mechanism induced by momentum in the design criteria.

Considering the maximum force and momenta in each test, a design coefficient was found that defines the equivalent thickness depending on the incident Froude number between 3 and 10. It considers the effects of the offset on the center of pressure from the slab's center of gravity. This is generated by the influence of waterstops, joints, and hydraulic jump macroturbulence with full and partially developed inflow. According to changes in the hydrodynamic conditions and the physical model's characteristics, the study involves slabs with lengths between 6 and 12 times the incident flow depth.

The
author discuss the scale effect that is inherent to the physical model and
conclude that the experimental results are useful for: a) Designing the lining
slabs in hydraulics structures that are similar to the model using scale more
generous than 1:14, and an Fr_{1} between 2.3 and 7.15; b) Predesigning
the lining slabs in stilling basins that require a geometric scale greater than
1:15 in order to establish similarity and/or an Fr_{1} between 7.16 and
10.

**Acknowledgments**

This research project was partially funded by COLCIENCIAS. The author would like to give a special thanks to Efraín del Risco and Lilian Posada for their valuable suggestions and insights in this field. The experiments necessary for the development of this research were undertaken in the Hydraulic Laboratory at the University of Valle in Cali, Colombia. The author sincerely appreciate the support.

**References**

**[1]** Federal Emergency Management
Agency., Outlet works energy dissipators best practices for design, construction,
problem identification and evaluation, inspection, maintenance, renovation, and
repair. Technical manual, USA, FEMA P-679, pp. 1-66, 2010.

**[2]** Hepler, T.E. and Johnson, P.L.,
Analysis of spillway failure by uplift pressure. ASCE National Conference,
Colorado Springs, Colorado, august 8-12, pp 857-862, 1988.

**[3]** VSL International. Soil and
rock anchors, Examples from practice. Berne, Switzerland. [Online]. pp 19-20,
1992 [Consulted: 15th of august of 2009]. Available at:
www.vsl.net/Portals/0/vsl_techreports/PT_Ground_Anchors.pdf

**[4]** Bowers, C.E. and Tsai, F.Y.,
Fluctuating pressures in spillway stilling basins. J. Hydraulic Division, ASCE,
95(6), pp. 2071-2079, 1969.

**[5]** Bribiescas, S. and Capella, V.,
Turbulence effects on the linning of stilling basins. Proceedings ICOLD.
Congrés des Grands Barrages. Madrid. Q.41. R.83. pp. 1575-1592, 1973.

**[6]** Bribiescas,
S. and Fuentes, O., Tanques amortiguadores. Technical
Report, Project 7045. UNAM, México D. F. pp. 1-50, 1978.

**[7]** Hajdin, G., Contribution to the
evaluation of fluctuation pressure on fluid currents limit areas- based on the
pressures recorded at several points of the area. VIII Conference of Yugoslav
Hydraulics Association, Portoroz. 1982.

**[8]** Toso, J. and Bowers, C.,
Extreme pressures in hydraulic-jump stilling basins, J. of Hydraulic
Engineering, 114(8), pp. 829-843, 1988. DOI: 10.1061/(ASCE)0733-9429(1988)114:8(829)

**[9]** Farhoudi, J. and Narayanan, R.,
Force on slab beneath hydraulic jump. J. of Hydraulic Engineering, 117(1), pp.
64-82, 1991. DOI: 10.1061/(ASCE)0733-9429(1991)117:1(64)

**[10]** Bellin, A. and Fiorotto, V.,
Direct dynamic force measurements on slab in spillway stilling basin. J. of
Hydraulic Engineering. 121(10), pp. 686-693, 1995. DOI: 10.1061/(ASCE)0733-9429(1995)121:10(686)

**[11]** Khatsuria R.M., Hydraulics of
spillways and energy dissipators. New York: Marcel Dekker, 2005. pp 411-424.

**[12]** Pinheiro, A., Quintela, A. and
Ramos, C., Hydrodynamic forces in hydraulic jump stilling Basins. Proceedings
of the symposium on fundamentals and advancements in hydraulic measurements and
experimentation. ASCE. New York. USA. pp. 321-330, 1994.

**[13]** Bollaert, E.F.R., Dynamic
uplift of concrete linings: Theory and case studies, USSD Annual Meeting, April
24-26 2009, Nashville, United States. pp. 1-16, 2009.

**[14]** Bollaert, E.F.R. and Schleiss,
A.J., Transient water pressures in joints and formation of rock scour due to
high-velocity jet impact. Communication 13, Laboratoire de Constructions
Hydrauliques, Ecole Polytechnique Fédérale de Lausanne. 2002.

**[15]** del Risco,
E., Hurtado, E. y González, M., Estudio experimental de las presiones de
levantamiento bajo una losa con juntas transversales al flujo. Tecnología y
Ciencias del Agua. I(1), pp. 31-42, 2010.

**[16]** Hurtado,
E., del Risco, E. y González, M., Presiones medidas en la base de una losa de
fondo con juntas paralelas al flujo en un canal. Revista de la Facultad de
Ingeniería de la Universidad de Antioquia, 47, pp. 39-52, 2009.

**[17]** Melo, J., Pinheiro, A. y Ramos,
C., Forces on plunge pool slabs: Influence of joints location and width. J. of
Hydraulic Engineering, 132(1), pp. 49-60. 2006. DOI: 10.1061/(ASCE)0733-9429(2006)132:1(49)

**[18]** González,
M., Levantamiento de una losa de piso sometida a un flujo turbulento. PhD Tesis,
Facultad de Minas, Universidad Nacional de Colombia, Medellín, Colombia. 2013.

**[19]** Frizell, W.K., Uplift and crack
flow resulting from high velocity discharges over. Report DSO-07-07 U.S.
Department of the interior, Bureau of Reclamation. Denver, Colorado, 2007, pp. 2-5.

**[20]** del Risco,
E., Investigación experimental de la falla de losas de revestimiento de tanques
amortiguadores. MSc. Tesis, Facultad de Ingeniería. UNAM, México, 1983.

**[21]** González,
M. y Giraldo, S., Caracterización dinámica de sensores de presión utilizando el
principio de la botella de mariotte. Rev. Fac. Ing.
Univ. Antioquia, 71, pp. 146-156, 2014.

**[22]** Proakis,
J.G. and Manolakis D.G., Tratamiento Digital de Señales. Prentice Hall. España,
2000.

**[23]** The International Society of
Automation (ISA 37.16.01). A guide for the dynamic calibration of pressure
transducers. pp. 13-38, 2002.

**[24]** Vasiliev, O.F. and Bukreyev, V.I.,
Statistical characteristics of pressure fluctuations in the region of hydraulic
jump. Proceedings 12th IAHR Congress, Fort Collins, USA, pp. 1-8, 1967.

**[25]** Pinheiro,
A., Accoes hidrodinámicas em bacias de dissipacao de energia por ressalto. PhD.
dissertation, Universidad Técnica de Lisboa, Lisboa, Portugal, 353 P., 1995.

**[26]** Khader, M. and Elango, K.,
Turbulent pressure field beneath a hydraulic jump. J. of Hydraulic Research,
12(4), pp. 469-489, 1974. DOI: 10.1080/00221687409499725

**[27]** Fiorotto, V. and Rinaldo, A.,
Fluctuating uplift and linnings design in spillway stilling basins. J.
Hydraulic Enginnering ASCE. 118(4), pp. 578-596. 1992a. DOI: 10.1061/(ASCE)0733-9429(1992)118:4(578)

**[28]** Fiorotto, V. and Rinaldo, A.,
Turbulent pressure fluctuations under hydraulic jumps. J. of Hydraulic
Research, 130, pp. 499-520, 1992b. DOI: 10.1080/00221689209498897

**[29]** Lopardo, R., de Lio, J. and
Lopardo, M., Physical modelling and design estimation of instantaneous
pressures in stilling basins. Proceedings of the XXVIII IAHR Congress. Graz.
132 P., 1999.

**[30]** Mees, A.,
Estudo dos esforços hidrodinâmicos em bacias de dissipação por ressalto
hidráulico com baixo número de Froude. MSc. Thesis. Universidade Federal do Rio
Grande do Sul, Porto Alegre, Brasil. 2008.

**[31]** Fiorotto, V. and Salandin, P.,
Design of anchored slabs in spillway stilling basins. ASCE. J. of Hydraulic
Engineering. 126(7), pp. 502-512. 2000. DOI: 10.1061/(ASCE)0733-9429(2000)126:7(502)

**[32]** Blair, H.K. and Rhone, T.J.,
Design of small dams. Bureau of Reclamation, 3a ed., Washington, DC, cap. 9,
Spillways structural design details. 1987, pp. 429-434.

**[33]** Bureau of Indian Standards.,
Structural design of energy dissipators for spillways criteria. Doc. WRD
09(489). Preliminary Indian Standard. July 2007. Available at: www.bis.org.in/sf/wrd/WRD09(489).pdf

**[34]** Bussell, M. and Cather, R.,
Design and construction of joints in concrete structures, report 146.
Construction industry research and information association. London, UK,
pp.12-64, 1995.

**[35]** British Standard 8007., Code of
practice for design of concrete structures for retaining aqueous liquids. UDC
624.953, pp. 11-19. 1987.

**[36]** Chanson, H., The hydraulics of
open channel flow. Edward Arnold, London, UK. 1999.

**[37]** Murzyn, F., and Chanson, H.,
Free surface, bubbly flow and turbulence measurements in hydraulic jumps.
Report CH63/07, Div. of Civil Engineering, University of Queensland, Brisbane,
Australia, August, July. 2007. 116 P.

**[38]** Chanson, H., Turbulent
air-water flows in hydraulic structures: dynamic similarity and scale effects. Environ.
Fluid. Mechanics 9(2), pp. 125-142, 2009. DOI: 10.1007/s10652-008-9078-3

**[39]** Lopardo, R., Discussion:
Prototype measurements of pressure fluctuations in The Dalles Dam stilling
basin. Journal of Hydraulic Research, 48(6), pp. 822-823. 2010. DOI: 10.1080/00221686.2010.536432

**[40]** Novak P., Nalluri C. and
Narayanan R., Hydraulic Structures. Fourth Edition Taylor & Francis. New
York pp: 246-265. 2007

**[41]** JiJian, L. JiMin, W. and JinDe,
G., Similarity law of fluctuating pressure spectrum beneath hydraulic jump.
Chinese Science Bulletin. 53(14). pp. 2230-2238, 2008. DOI:
10.1007/s11434-008-0300-y

**[42]** Lightthill, J., Waves in
fluids. Cambridge University Press. pp. 89-202, 1979.

**[43]** Lopardo,
R., Algunos aportes de los modelos físicos en la optimización hidráulica de
grandes presas Argentinas. V Congreso argentino de presas y aprovechamientos
hidroeléctricos. San Miguel de Tucumán (Tucumán). pp. 1 -19, 2008.

**M. Gonzalez-Betancourt, **is BSc. in Agricultural Engineering jointly from
the Universidad Nacional de Colombia (Palmira campus), and Universidad del
Valle, Cali, Colombia, in 2008. He received his PhD in
Engineering with an emphasis on Hydraulic Resources in 2014 from the Universidad
Nacional de Colombia, Medellin, Colombia. Between 2008 and 2010, he worked at
the Hydraulic Laboratory at the Universidad del Valle. Since 2009 he has been
associated with the Posgrado en Aprovechamiento de Recursos Hidráulicos
research group [Postgraduate program on the use of Hydraulic Resources] at the
Universidad Nacional de Colombia. From 2014 to date, he has been a researcher
in Colombia subject to the terms of the joint agreement between the Organización
de Estados Iberoamericanos [Organization of Iberoamerican States], the Servicio
Nacional de Aprendizaje [National Apprenticeship Service] and COLCIENCIAS
[Administrative Department of Science, Technology and Innovation]. ORCID: 0000-0001-5485-8043

## Referencias

Federal Emergency Management Agency., Outlet works energy dissipators best practices for design, construction, problem identification and evaluation, inspection, maintenance, renovation, and repair. Technical manual, USA, FEMA P-679, pp. 1-66, 2010.

Hepler, T.E. and Johnson, P.L., Analysis of spillway failure by uplift pressure. ASCE National Conference, Colorado Springs, Colorado, august 8–12, pp 857-862, 1988.

VSL International. Soil and rock anchors, Examples from practice. Berne, Switzerland. [Online]. pp 19-20, 1992 [Consulted: 15th of august of 2009]. Available at: www.vsl.net/Portals/0/vsl_techreports/PT_Ground_Anchors.pdf

Bowers, C.E. and Tsai, F.Y., Fluctuating pressures in spillway stilling basins. J. Hydraulic Division, ASCE, 95(6), pp. 2071-2079, 1969.

Bribiescas, S. and Capella, V., Turbulence effects on the linning of stilling basins. Proceedings ICOLD. Congrés des Grands Barrages. Madrid. Q.41. R.83. pp. 1575-1592, 1973.

Bribiescas, S. and Fuentes, O., Tanques amortiguadores. Technical Report, Project 7045. UNAM, México D. F. pp. 1-50, 1978.

Hajdin, G., Contribution to the evaluation of fluctuation pressure on fluid currents limit areas- based on the pressures recorded at several points of the area. VIII Conference of Yugoslav Hydraulics Association, Portoroz. 1982.

Toso, J. and Bowers, C., Extreme pressures in hydraulic-jump stilling basins, J. of Hydraulic Engineering, 114(8), pp. 829-843, 1988. DOI: 10.1061/(ASCE)0733-9429(1988)114:8(829)

Farhoudi, J. and Narayanan, R., Force on slab beneath hydraulic jump. J. of Hydraulic Engineering, 117(1), pp. 64-82, 1991. DOI: 10.1061/(ASCE)0733-9429(1991)117:1(64)

Bellin, A. and Fiorotto, V., Direct dynamic force measurements on slab in spillway stilling basin. J. of Hydraulic Engineering. 121(10), pp. 686-693, 1995. DOI: 10.1061/(ASCE)0733-9429(1995)121:10(686)

Khatsuria R.M., Hydraulics of spillways and energy dissipators. New York: Marcel Dekker, 2005. pp 411-424.

Pinheiro, A., Quintela, A. and Ramos, C., Hydrodynamic forces in hydraulic jump stilling Basins. Proceedings of the symposium on fundamentals and advancements in hydraulic measurements and experimentation. ASCE. New York. USA. pp. 321-330, 1994.

Bollaert, E.F.R., Dynamic uplift of concrete linings: Theory and case studies, USSD Annual Meeting, April 24-26 2009, Nashville, United States. pp. 1-16, 2009.

Bollaert, E.F.R. and Schleiss, A.J., Transient water pressures in joints and formation of rock scour due to high-velocity jet impact. Communication 13, Laboratoire de Constructions Hydrauliques, Ecole Polytechnique Fédérale de Lausanne. 2002.

del Risco, E., Hurtado, E. y González, M., Estudio experimental de las presiones de levantamiento bajo una losa con juntas transversales al flujo. Tecnología y Ciencias del Agua. I(1), pp. 31-42, 2010.

Hurtado, E., del Risco, E. y González, M., Presiones medidas en la base de una losa de fondo con juntas paralelas al flujo en un canal. Revista de la Facultad de Ingeniería de la Universidad de Antioquia, 47, pp. 39-52, 2009.

Melo, J., Pinheiro, A. y Ramos, C., Forces on plunge pool slabs: Influence of joints location and width. J. of Hydraulic Engineering, 132(1), pp. 49-60. 2006. DOI: 10.1061/(ASCE)0733-9429(2006)132:1(49)

González, M., Levantamiento de una losa de piso sometida a un flujo turbulento. PhD Tesis, Facultad de Minas, Universidad Nacional de Colombia, Medellín, Colombia. 2013.

Frizell, W.K., Uplift and crack flow resulting from high velocity discharges over. Report DSO-07-07 U.S. Department of the interior, Bureau of Reclamation. Denver, Colorado, 2007, pp. 2-5.

del Risco, E., Investigación experimental de la falla de losas de revestimiento de tanques amortiguadores. MSc. Tesis, Facultad de Ingeniería. UNAM, México, 1983.

González, M. y Giraldo, S., Caracterización dinámica de sensores de presión utilizando el principio de la botella de mariotte. Rev. Fac. Ing. Univ. Antioquia, 71, pp. 146-156, 2014.

Proakis, J.G. and Manolakis D.G., Tratamiento Digital de Señales. Prentice Hall. España, 2000.

The International Society of Automation (ISA 37.16.01). A guide for the dynamic calibration of pressure transducers. pp. 13-38, 2002.

Vasiliev, O.F. and Bukreyev, V.I., Statistical characteristics of pressure fluctuations in the region of hydraulic jump. Proceedings 12th IAHR Congress, Fort Collins, USA, pp. 1-8, 1967.

Pinheiro, A., Accoes hidrodinámicas em bacias de dissipacao de energia por ressalto. PhD. dissertation, Universidad Técnica de Lisboa, Lisboa, Portugal, 353 P., 1995.

Khader, M. and Elango, K., Turbulent pressure field beneath a hydraulic jump. J. of Hydraulic Research, 12(4), pp. 469-489, 1974. DOI: 10.1080/00221687409499725

Fiorotto, V. and Rinaldo, A., Fluctuating uplift and linnings design in spillway stilling basins. J. Hydraulic Enginnering ASCE. 118(4), pp. 578-596. 1992a. DOI: 10.1061/(ASCE)0733-9429(1992)118:4(578)

Fiorotto, V. and Rinaldo, A., Turbulent pressure fluctuations under hydraulic jumps. J. of Hydraulic Research, 130, pp. 499-520, 1992b. DOI: 10.1080/00221689209498897

Lopardo, R., de Lio, J. and Lopardo, M., Physical modelling and design estimation of instantaneous pressures in stilling basins. Proceedings of the XXVIII IAHR Congress. Graz. 132 P., 1999.

Mees, A., Estudo dos esforços hidrodinâmicos em bacias de dissipação por ressalto hidráulico com baixo número de Froude. MSc. Thesis. Universidade Federal do Rio Grande do Sul, Porto Alegre, Brasil. 2008.

Fiorotto, V. and Salandin, P., Design of anchored slabs in spillway stilling basins. ASCE. J. of Hydraulic Engineering. 126(7), pp. 502-512. 2000. DOI: 10.1061/(ASCE)0733-9429(2000)126:7(502)

Blair, H.K. and Rhone, T.J., Design of small dams. Bureau of Reclamation, 3a ed., Washington, DC, cap. 9, Spillways structural design details. 1987, pp. 429-434.

Bureau of Indian Standards., Structural design of energy dissipators for spillways criteria. Doc. WRD 09(489). Preliminary Indian Standard. July 2007. Available at: www.bis.org.in/sf/wrd/WRD09(489).pdf

Bussell, M. and Cather, R., Design and construction of joints in concrete structures, report 146. Construction industry research and information association. London, UK, pp.12-64, 1995.

British Standard 8007., Code of practice for design of concrete structures for retaining aqueous liquids. UDC 624.953, pp. 11-19. 1987.

Chanson, H., The hydraulics of open channel flow. Edward Arnold, London, UK. 1999.

Murzyn, F., and Chanson, H., Free surface, bubbly flow and turbulence measurements in hydraulic jumps. Report CH63/07, Div. of Civil Engineering, University of Queensland, Brisbane, Australia, August, July. 2007. 116 P.

Chanson, H., Turbulent air-water flows in hydraulic structures: dynamic similarity and scale effects. Environ. Fluid. Mechanics 9(2), pp. 125-142, 2009. DOI: 10.1007/s10652-008-9078-3

Lopardo, R., Discussion: Prototype measurements of pressure fluctuations in The Dalles Dam stilling basin. Journal of Hydraulic Research, 48(6), pp. 822-823. 2010. DOI: 10.1080/00221686.2010.536432

Novak P., Nalluri C. and Narayanan R., Hydraulic Structures. Fourth Edition Taylor & Francis. New York pp: 246-265. 2007

JiJian, L. JiMin, W. and JinDe, G., Similarity law of fluctuating pressure spectrum beneath hydraulic jump. Chinese Science Bulletin. 53(14). pp. 2230-2238, 2008. DOI: 10.1007/s11434-008-0300-y

Lightthill, J., Waves in fluids. Cambridge University Press. pp. 89-202, 1979.

Lopardo, R., Algunos aportes de los modelos físicos en la optimización hidráulica de grandes presas Argentinas. V Congreso argentino de presas y aprovechamientos hidroeléctricos. San Miguel de Tucumán (Tucumán). pp. 1 -19, 2008.

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