INTRODUCTION
Rubber dams are cylindrical membrane structures which are attached to a rigid
foundation along two of their generators and are inflated with water, air, or
a combination of water and air (Zhang et al., 2002).
They are basically simple and portable barriers made of rubberized material
and are used for various purposes such as irrigation, control flood, tidal defense,
water supply and recreational purposes (Watson et al.,
1999). This type of structure is considered as more economical compared
with the rigid type of control structures constructed from concrete, masonry
and steel (Tam, 1998). Some studies of their crosssectional
static profiles have been carried out in the past, both for cases when the dam
impounds water and when overflow occurs (Hsieh and Plaut,
1990; Wu and Plaut, 1996). After obtaining the equilibrium
configurations, Small vibrations have been observed on actual dams and on the
physical models about this configuration (Plaut and Leeuwrik,
1988; Mysore et al., 1998; Ergin
and Temared, 2002; Plaut and Cotton, 2005). To analyze
the static and dynamic behaviors of such structure, using the numerical methods
such as finite element method and handling the computer analysis procedures
based on these methods is enough suitable (Tou and Wong,
1987; Mackerle, 2000). ANSYS uses NewtonRaphsonâ€™s
method as a numerical technique for solving the nonlinear equilibrium equations.
The basic idea is to reduce the set of nonlinear equations into a set of linear
equations to solve equilibrium equations at small increments, the size of which
depends on nonlinearity of the problem (Bathe, 1996). In
present study, the finite element package ANSYS is used to perform dynamic analysis
of the linear vibration of these dams when impounding hydrostatic and parallel
flowing water and without external water.
From viewpoint of anchored system, rubber dams are categorized to singleanchor
rubber dams and doubleanchor rubber dams (Dakshina Moorthy
et al., 1995). Although most recentlybuilt rubber dams utilize the
singleanchor system which have fin at two attached ends of rubber sheets to
facilitate smooth overflow (Chanson, 1997), but for the
dams with high height due to more stability of doubleanchor rubber dams against
different loads, in this study in addition to studying the effect of various
parameters in dynamic nonlinear behaviors of a singleanchor dam in various
conditions the results which are obtained from analyzing a computer model of
doubleanchor dam are presented and compared with the research of Dakshina
Moorthy et al. (1995) and the reasons of the percentage changes of
the results in two compared research have been mentioned but because of the
little difference between results due to increasing accuracy of modeling of
the problem, it is concluded that ANSYS Software can be used to analyze the
dynamic behavior of the structures such as inflatable flexible rubber dams.
VIBRATIONS OF THE DOUBLE ANCHOR RUBBER DAM
In this study, to study the linear vibrations, the rubber dam is modeled as
an elastic shell inflated with air. The internal pressure is increased slowly
until it reaches the desired value. The natural frequencies of the model of
the doubleanchor rubber dam have been computed by using ANSYS software. The
assumed model of the rubber dam includes one rubber sheet attached to concrete
slab along two edges of sheet longitudinally. The dimensions of the structure
and the rubber properties of the rubber dam are selected as the modeled rubber
dam in the research of Dakshina Moorthy et al. (1995),
but the size and the type of meshing of the model is different. Figure
1 shows the model information of the inflatable rubber dam. So, in this
3D model, the rubber sheet is meshed to smaller elements by shell element, SHELL63,
4 nodes element of ANSYS is selected to apply the analysis. Each nodes of element
includes 6 degrees freedom that indicates 24 degrees freedom for each element.
The applied element is enlarged in Fig. 2.

Fig. 1: 
Dimensions and properties of rubber in the model of
the doubleanchor rubber dam 

Fig. 2: 
Element of shell63 in ANSYS 
RESULTS OF THE VIBRATION ANALYSIS OF THE DOUBLEANCHOR RUBBER DAM
By comparing the results of both models, it is concluded that the obtained
vibration frequencies of the structure with presence and absence of external
water are in close agreement with those obtained by Dakshina
Moorthy et al. (1995). The Maximum difference in the first five frequencies
is 9.4% without considering external water and 10.2% with the presence of the
external water. The difference in frequencies can be attributed to the size
of the considered mesh (960 elements in present study and 200 in previous Research)
and the type of the selected element in present case and 9node element in Dakshina
Moorthy et al. (1995). The frequencies were found to be reduced in
presence of external water as were found by Dakshina Moorthy. Figure
3 is clustered column that compare the values of the natural frequencies
of a doubleanchor rubber dam across five modes.

Fig. 3: 
Natural frequencies of a doubleanchor rubber dam when (a)
the dam is not impounding water (b) when the dam is impounding water on
one side in first five vibration modes via Dakshina Moorthy
et al. (1995) research and present study 

Fig. 4: 
The model information of a singleanchor rubber dam 
VIBRATIONS OF THE SINGLEANCHOR RUBBER DAM
The rubber dam with the singleanchor system is modeled as two sheets
of rubber, on lying on the top of the other and attaches together at the
ends, from one side is anchored to a rigid slab and from the other side
makes fins for preventing from severe impact of overflow to backside of
the dam. The model of the structure is meshed to 1500 elements of SHELL63
which is introduced earlier. Figure 4 shows the information
of the assumed model of the singleanchor rubber dam.
The singleanchor rubber dam is modeled to be analyzed. After running
the program, the results of the linear vibration of the membrane of the
structure is obtained to prevent from probable damages to the structural
system of the rubber dam due to these vibrations. To achieve this purpose,
the natural frequencies of the vibration and the mode shapes are established
in three various cases without external water, in presence of external
water and with parallel flowing of water.
RESULTS OF THE VIBRATION ANALYSIS OF THE SINGLEANCHOR RUBBER DAM
Without external water: For the vibration analysis, the rubber
dam that is not impounding water is clamped at the ends while the equilibrium
shape is obtained, then small threedimensional vibrations are considered
about the equilibrium configuration. The first four vibration frequencies
and mode shapes are computed. Figure 5 shows the variation
of the frequencies against internal pressure for the first four vibration
modes and Table 1 shows the corresponding frequencies
for P_{int} = 1 kPa and P_{int} = 30 kPa.

Fig. 5: 
Variation of vibration frequencies against internal
pressure from 0 to 30 kPa, without water 
Table 1: 
Natural frequencies (rad sec^{1}) of the rubber
dam in P_{int }= 1 kPa and P_{int }= 30 kPa for three
cases without water, with hydrostatic water and with water 

As shown in Fig. 5, the slopes of the curves of the
first four vibration modes decrease as the internal pressure increases.
Also the slope of the curve in mode 4 is more than the slope of mode 3.
So, this relation exists for mode 3 than mode 2 and mode 2 than mode 1.
The vibration frequencies increase with the internal pressure as one would
expect. The squares of the frequencies vary almost linearly with the internal
pressure. Figure 6 and 7 image the
first four vibration mode shapes for internal pressures of 1 and 30 kPa,
respectively. The first and second modes are symmetric and the third and
fourth modes are nonsymmetric longitudinally. Also the corresponding
profiles of the central cross section for these modes (solid curves) and
the cross sections at the distance of onequarter length from each end
(dashed curves) are shown in Fig. 6 and 7
for internal pressures of 1 and 30 kPa, respectively. For the modes that
are symmetric longitudinally, the two dashed curves are identical.

Fig. 6: 
First four vibration modes shapes and Cross sections
of modes at the center (solid curve) and at the quarterlengths from
the ends (dashed curve) in case of without external water, for P_{int}=
1 kPa 
With external water: For the case of the dam impounding water
on one side, the dam is anchored along the equilibrium cross sections
at its two ends and then water is applied on the anchored side with a
height less than the dry equilibrium height. The new equilibrium configuration
is obtained and small vibrations of the dam about this equilibrium shape
are analyzed. Figure 8 represents the vibration of the
first four frequencies with the external water head, for internal pressure
of 30 kPa.
In Fig. 8, the frequencies at first tend to decrease
and then increase slightly as the external water head increases. At the
first of the curves, the value of decrease in Mode 4 is more than the
other vibration modes. The frequencies for the higher modes (modes 3 and
4) show more variation compared to those for the lower modes (modes 1
and 2). The frequencies are high enough to justify the infinite frequency
limit assumption used to define the boundary condition on the free surface.

Fig. 7: 
First four vibration modes shapes and Cross sections
of modes at the center (solid curve) and at the quarterlengths from
the ends (dashed curve) in case of without external water, for P_{int}=
30 kPa 

Fig. 8: 
Variation of vibration frequencies with external water
(P_{int} = 30 kPa) 

Fig. 9: 
First four vibration modes shapes and cross sections
of modes at the center (solid curve) and at the quarterlengths from
the ends (dashed curve) in case of the presence of external water,
for P_{int}= 1 kPa 
Table 1 compares the vibration frequencies of the dam
with no external water to those for the dam in presence of external hydrostatic
pressure (with no added mass effect), for P_{int} =1 kPa and P_{int} = 30 kPa. The external head is 0.5 and 1.5 m, respectively. Also the frequencies
of the structure in the presence of water are lower than those in absence
of water by a maximum of 12.7% and the frequencies of the structure absence
of water are lower than those in presence of external hydrostatic pressure
by a maximum of 8.4% in Table 1. The first and second
modes are symmetric and the third and fourth modes are nonsymmetric longitudinally.
The corresponding first four modes are shown in Fig. 9 and 10. Also these figures show the cross sectional
behavior of the modes at the halflength (solid curves) and quarterlength
(dashed curves) from each end of the dam for internal pressures of 1 and
30 kPa, respectively.

Fig. 10: 
First four vibration modes shapes and cross sections
of modes at the center (solid curve) and at the quarterlengths from
the ends (dashed curve) in case of the presence of external water,
for P_{int} = 30 kPa 
So the horizontal line at the left of the dam indicates
the water height. For the first two modes, the two dashed curves are identical.
The cross sectional behavior of the structure in the presence of external
water, at halflength and quarterlength from the ends, is similar to
that of the dam in absence of water for both the internal pressures.
With parallel flow: The case of the rubber dam impounding water flowing
parallel to the dam is considered. This situation may occur for the dam installed
along a river. In this case, the inflatable rubber dam is restrained along the
equilibrium cross sections at its ends and then parallel flowing water is applied
on the anchored side with the same height as that in the case of hydrostatic
water.

Fig. 11: 
First four vibration modes shapes and cross sections
of modes at the center (solid curve) and at the quarterlengths from
the ends (dashed curve) in case of the presence of parallel flow of
1 m sec^{1}, for P_{int} = 30 kPa 
Small vibrations of the structure about this equilibrium shape are analyzed.
The flow introduces hydrodynamic pressure and thus the boundary conditions on
the structure change due to the fluidstructure interaction. As the earlier
cases, the vibration analysis was performed for internal pressures of 1 and
30 kPa. Flow velocities of 1 and 5 m sec^{1} were considered.
As seen in Table 2 and 3, the natural
frequencies of rubber dam when the dam is impounding water flowing parallel
to the dam is lower than that in the presence of water without parallel
flow for internal pressure of 30 KPa by maximum of 8% when velocity flow
equals to 1 m sec^{1} and 5.1% when velocity flow equals to 5
m sec^{1}.

Fig. 12: 
First four vibration modes shapes and Cross sections
of modes at the center (solid curve) and at the quarterlengths from
the ends (dashed curve) in case of the presence of parallel flow of
5 m sec^{1}, for P_{int} = 30 kPa 
Table 2: 
Natural frequencies (rad sec^{1}) of the rubber
dam with parallel flow, for P_{int}= 1 kPa 

Figure 11 and 12 show the effect
of the parallel flow on the vibration behavior of the rubber dam. The
direction of the flow is from nearer end to the farther end along the
length of the dam in these figures. Also in this case the first and second
modes are symmetric and the third and fourth modes are nonsymmetric longitudinally
as other cases.
Table 3: 
Natural frequencies (rad sec^{1}) of the rubber
dam with parallel flow, for P_{int} = 30 kPa 

The cross sectional behavior of the three model of the
rubber dam in the presence of external parallel flow, at halflength and
quarterlength from the ends is similar to that of the dam in the absence
of water.
CONCLUSIONS
In the present study, the dynamic behaviors of linear vibrations of the singleanchor
and the doubleanchor air rubber dams have been studied by applying engineering
ANSYS software. The doubleanchor rubber dams have been modeled and analyzed
by ANSYS and then the results obtained from this dynamic analysis are compared
with the results given by Dakshina Moorthy et al.
(1995). By this comparison it is found that the outputs of ANSYS model are
in agreement with previous researches which have used other softwares but the
results of this research is more accurate due to type of meshing and type and
size of selected elements. So ANSYS is too capable to perform the dynamic analysis
of vibrations of the rubber dam behaviors. The natural frequencies and the corresponding
mode shapes of the singleanchor dam for three cases of the dam without external
water, with external water and with parallel flow have been computed and then
compared. The vibration analysis of the structure makes use of equilibrium configurations
which obtained after performing the static analysis. The effects of external
water head and internal pressure are studied. When external hydrostatic pressure
(with no added mass effect) is applied to the body of dam, the frequency of
vibration of singleanchor rubber dam is more than the other cases and in the
case of the presence of water impounded by rubber dam, the frequency is less
than the others. Also by increasing internal pressure while the ultimate pressure
(30 kPa) is achieved, the value of natural frequencies increase too much in
mode 1 and this progressive increase in mode 3 and 4 is more than the other
modes. Meanwhile by increasing the velocity of parallel water, the variation
frequency slightly decreases. So in case of stationary water backside of the
dam this amount of decrease is more than the other ones. In first four modes
the amount and the form of deformations of crosssection is different together,
but in different crosssection, these variations of the mode shapes in mode
3 and 4 is observed more than mode 1 and 2. The presence of the external hydrostatic
water tends to reduce the natural vibration frequencies of the rubber dam. The
variations of frequency depend on the internal pressure and the external water
head. The relation between the frequency and internal pressure for the case
of without external pressure is nonlinear and straight and the charts of function
of frequencyinternal pressure for the first four modes are ascendant with reducer
slope. Also in the case of the presence of external water the charts of function
of frequencyexternal water for internal pressure of 30 kPa are descent with
reducer slopes and gradually tend to be steady. The effect of external water
on the frequency essentially is observed in the low pressure. Finally the case
of the rubber dam impounding water flowing parallel to the dam has been considered.
The hydrodynamic pressure due to the external parallel flow results in added
inertia (i.e., added mass) which reduces the vibration frequencies of the structure.
This reduction depends on the internal pressure, the external water head and
flow velocity. Overall, it can be concluded that a computer finite element model
can be applied to study the dynamic vibration of behavior of a hydraulic structure
such as rubber dam which is resulted due to different loads applied to the dam.