# Traditionally, Concrete Gravity Dams Have Been Designed and Analyzed by a Simple Procedure

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ANALYTICAL AND NUMERICAL STUDIES OF DAM RESERVOIR INTERACTION IN CONCRETE GRAVITY DAMS

Paulo Marcelo VIEIRA RIBEIRO

Doctorate Student, Universidade de Brasília – UnB

Selênio Feio DA SILVA

Lineu José PEDROSO

Professor, Universidade de Brasília – UnB

Brazil

1.INTRODUCTION AND HYSTORICAL NOTES

The large amount of water stored in a dam makes its rupture to be catastrophic. Project, design and execution are crucial for this kind of structure. Exceptional loadings must be taken in account during design phase, because of its highly destructive potential and social importance. According to [1] concerns about seismic safety of concrete dams have been growing in recent years, partly because the population at risk in downstream locations of major dams continues to expand, and also because it is evident that seismic design concepts, used when most existingdams were built,have been proved to be inadequate.

Dams are particularly different from other types of structures. Unlike typical uncoupled systems, analysis of the dynamic response of this type of problem usually arises into a complete system interaction, made by dam-reservoir-foundation effects. According to [2] a complete analysis of the earthquake response of concrete gravity dams presents a formidable problem, with major difficulties arising from the interaction between the dam and the water stored in the reservoir. Fluid domain pressure field depends on the structural displacement, which in turn depends on the forces exerted by the fluid.

The problem of dam-reservoir interaction has been widely studied in the last decades. The first attempt to solve this problem was made by [3]. He considered a dam accelerated at its base as perfect rigid body with a continuum infinite length reservoir. Because of the perfect rigidity of the dam, its acceleration along the structure was considered constant and equal to the foundation’s acceleration. For these conditions, Westergaard found out a solution for the pressure filed, which acts on the dam’s` face when it is under a seismic excitation. This introduced the well known added mass concept, which consisted in the representation of the inertia effects produced by the reservoir. This is the simplest form of treating the dam-reservoir interaction problem (Fig. 1).

Fig. 1

Representation of the Pseudo-Static design procedure

Dams were initially designed with this rigid body assumption. Until four or five decades ago, the only consideration given to earthquakes in design of concrete dams was to apply a static horizontal force specified as a fraction of the weight of the structure, to represent the seismic design loading. In later stages of its use, this equivalent static loading design procedure often included an additional mass, provided by the Westergaard’s solution to represent the inertial resistance of the water [1].

This traditional design procedure (also known as the pseudo-static or seismic coefficient method) was applied worldwide in dam design. According to [4], “in spite of its limitations, this procedure was an excellent piece of work for its time, since it provided a basis for computing earthquake forces, considering them in the design of dams”. However, the well studied example of Koyna Dam (Fig. 2), located in India and subjected to the 1967 earthquake, demonstrated that the assumptions made in the traditional design procedure may underestimate earthquake forces, and lead to structural damages during this event. This earthquake, with accelerations around 0.5g, caused significant structural damage to the dam, including horizontal cracks on the upstream and downstream faces of a number of nonoverflow monoliths [5].

Fig. 2

Koyna Dam – India (CEE, 1990)

Major improvements over the rigid dam approach occurred when the dynamic effects of the interaction were considered. The first improvement was to convert the equivalent static uniform force to a fundamental mode shape related action and the inclusion of fluid compressibility effects [1]. Later, dynamic amplification of ground motion was applied considering earthquake response spectrums [5]. With the advances in response methods, foundation interaction effects, reservoir bottom absorption and contributions of the higher vibration modes were included in the analysis [6]. This procedure is well known as the pseudo-dynamic approach, were resulting dynamic forces are applied to an equivalent static model for stress evaluation (Fig. 3).

Fig. 3

Representation of the Pseudo-Dynamic design procedure

According to [7] the seismic analysis of the gravity dam system interaction can be divided into five different levels, which depend on the degree of refinement required for the analysis. The preliminary level (level 1) consists on an initial investigation of the seismic aspects of the region, and defines if a further analysis is needed. The Pseudo-Static procedure (level 2) provides the dam-reservoir interaction effects for a rigid body movement towards an incompressible fluid reservoir. The Pseudo-Dynamic procedure (level 3) includes the fundamental mode shape response and the fluid compressibility effects in the previous level. More advanced variations are provided with the higher modes contributions as well as foundation interaction effects. Evaluations up to this point can be made with simplified analytical procedures, that provide hydrodynamic pressures and inertia forces that are statically applied to the model. The two final levels are respectively the time or frequency history linear (level 4) and non linear responses (level 5). These analysis often require the use of computers.

This work will provide a further understanding of the hydrodynamic pressure formulation involved in analysis levels 2 and 3. The mathematical assumptions as well as the analytical equations are described. The reservoir governing equation of motion and associated boundary conditions of the dam-reservoir system are established using an acoustic fluid model. Solutions for incompressible and compressible fluids, including rigid and flexible boundaries for the fluid and the fluid-structure interface are presented. These solutions are compared with results obtained from numerical analysis using the finite element method.

2.ANALYTICAL SOLUTIONSFOR THE 2D ACOUSTIC INTERACTION

According to [8] the fluid-structure interaction can be classified into three distinct categories: the acoustic fluid, the incompressible Navier-Stokes fluid, and the compressible Navier-Stokes fluid. This author indicates that the acoustic fluid model represents the simplest model of interaction, because of the imposed considerations, such as: inviscid fluid and particles undergoing only relatively small displacements. With these considerations it is assumed that the fluid transmits only pressure waves. Some applications examples of this theory include: pressure waves in a piping system, fluid sloshing in a tank and sound waves traveling trough fluid-solid media.

The following fundamental assumptions [9,2] are made for the solution of this problem:

1. The fluid is homogeneous, inviscid and linearly incompressible;
2. The flow filed is irrotational;
3. No sources, sinks or cavities are anywhere in the flow field;
4. The displacements and their spatial derivatives are small;
5. Effects of waves at the free surface of water are negligible;
6. The motion of the dam-reservoir system is two dimensional (the same for any vertical plane perpendicular to the axis of the dam);
7. The upstream face of the dam is vertical;
8. The reservoir extends to infinity in the upstream direction;
9. The depth of the reservoir is constant;
10. The deformations of the dam are approximated by a generalized coordinate mode shape conformal function;
11. Effects of the vertical component of ground motion are not considered.

The previous assumptions lead to a hydrodynamic pressure field, in excess of the hydrostatic pressure, governed by:

/ [1]

where:

/ [2]

corresponds to the two dimensional Laplace operator in cartesian coordinates and:

/ [3]

corresponds to the velocity of sound in the fluid,where is the bulk modulus of elasticity of the fluid and its mass density. Eq.[1] is the two dimensional wave equation governing the hydrodynamic pressure distribution in a linearly compressible fluid. For an incompressible fluid (where and therefore, ) Eq.[1] is simplified to the following expression:

/ [4]

which corresponds to the two dimensional Laplace equation governing the hydrodynamic pressure in an incompressible fluid.

Assuming that solutions of time harmonic pressure waves of frequency are given by Eq.[5]:

/ [5]

and substituting this last expression in Eq.[1] provides:

/ [6]

which corresponds to the well known Helmholtz differential equation, also known as reduced wave equation. Solutions for Eq. [6] are presented in the next items. The analytical formulations are based on the research developed by [10].

2.1.SOLUTION FOR A RIGID BOUNDARY IN A COMPRESSIBLE FLUID

The equation that governs this problem is given by Eq.[1]. For a rigid body it is assumed that the dam vibrates harmonically with the horizontal acceleration given by Eq.[7]. Fig. 4 illustrates the scheme and the boundary conditions.

/ [7]

Fig. 4

Dam-reservoir interaction scheme and boundary conditions

where indicates the maximum amplitude of the ground acceleration. Application of the four boundary conditions results in:

/ [8]

The boundary conditions listed in Eq.[8] can be readily applied into the Helmholtz differential equation given by Eq.[6]. The solution of this pressure field is given by [10]:

/ [9]
/ [10]

which is the spatial solution of the pressure field in a linear compressible fluid under a rigid harmonic boundary excitation. The solution of an incompressible fluid is achieved by setting the limit . Thus:

/ [11]

It should be noted that Eq.[9] is frequency dependent, while Eq.[11] it is not. Also, Eq.[11] is clearly a particular case of the compressible solution. This last solution approaches the incompressible equation with the limit .

Analysis of Eq. [9] indicates that real valued pressures are expected as long as the following condition is observed:

/ [12]

which results in:

/ [13]

therefore Eq.[9] produces real values as long as the frequency is smaller orthe vibration period is greater than the value correspondingto in the right hand side of Eq.[13]. This term represents the corresponding frequency and the period values of the transverse direction of the reservoir. Substituting in the previous expression results in:

/ [14]

If and consequently then, resulting in resonance. If the conditions given by Eq.[14] are not satisfied, then the solution is made of a real and an imaginary part.

The work developed by [9] provides another way of the dealing with Eq.[9]. This author divides the hydrodynamic pressure function in two expressions, with one being related to imaginary and real values and the other being exclusively related to real valued pressures. Thus:

/ [15]

where is the smallest integer value for which . It should be noted that when the first series vanishes and only real valued pressures are obtained. This corresponds to the case where is less than the first fundamental frequency of the reservoir. If is larger than this value, then will be larger than 1, and both series will be present.A convenient representation of the second series of Eq.[15] is given by the ratio . Fig. 5 illustrates the hydrodynamic pressure distribution over the fluid structure interface for different values of this parameter.

Fig. 5

Hydrodynamic pressure distribution over the fluid-structure interface

(real valued pressures)

Fig. 5 indicates that the non dimensional pressure suffers significant influence from the parameter. When then and the solution approaches the incompressible fluid behavior. However, higher pressure fields are expected with the increase in this parameter (when the vibration period approaches the reservoir period ), indicating that fluid compressibility produces important effects that cannot be neglected. Resonance is expected when this parameter approaches . Fig. 6 illustrates the hydrodynamic pressure behavior for values of . It should be noted the increasing pressure distribution for a minor variation of this parameter.

Fig. 6

Hydrodynamic pressure distribution over the fluid-structure interface

(real valued pressures)

The real valued pressures are conditioned to a vibration period greater than . In this case the hydrodynamic pressure distribution is in phase with the exciting boundary. For the solution provides both real and imaginary parts, indicating that the response has an in-phase and an out-of-phase component.

2.2.SOLUTION FOR A FLEXIBLE BOUNDARY IN A COMPRESSIBLE FLUID

The equation that governs this problem is given by Eq.[1]. It is assumed that the dam vibrates harmonically, with a prescribed mode shape normalized at , producing a resulting horizontal acceleration given by:

/ [16]

where the prescribed mode shape is an arbitrary function.

Fig. 7 illustrates the scheme and the boundary conditions needed for this solution. The boundary conditions , and are the same from the previous solution. However now depends on the prescribed mode shape function, which provides a varying acceleration, starting from zero at and reaching a maximum equal to at the dam’s crest ().

Fig. 7

Dam-reservoir interaction scheme and boundary conditions

Application of the four boundary conditions results in:

/ [17]

The boundary conditions listed in Eq.[17] can be readily applied into the Helmholtz differential equation given by Eq.[6]. The solution of this pressure field is given by [10]:

/ [18]
/ [19]

which is the spatial solution of the pressure field in a linear compressible fluid under a flexible boundary excitation. If is set to 1, then Eq.[19] is reduced to:

/ [20]

which can be readily substituted in Eq.[18], resulting in the rigid boundary solution. Thus it is concluded that Eq.[18] represents a general solution, being valid for both rigid and flexible boundaries, as well as compressible and incompressible behaviors. The incompressible solution is achieved with . This provides:

/ [21]

It should be noted that Eq.[18] is frequency dependent, while Eq.[21] it is not. Also, Eq.[21] is clearly a particular case of the compressible solution. This last solution approaches the incompressible equation with the limit .The same conditions for real valued pressures of the rigid boundary solution remain valid. Thus Eq.[12], Eq.[13] and Eq.[14] are also applied in this case. The hydrodynamic pressure can also be divided in two parts [9]:

/ [22]

where is the smallest integer value for which . It should be noted that when the first series vanishes and only real valued pressures are obtained. A convenient representation of the second series of Eq.[22] is given by the ratio . Fig. 5 illustrates the hydrodynamic pressure distribution over the fluid structure interface for different values of this parameter. In these graphics the fundamental mode shape function provided by [11] is adopted:

/ [23]

Fig. 8

Hydrodynamic pressure distribution over the fluid-structure interface

(real valued pressures)

Fig. 8 indicates that the non dimensional pressure suffers significant influence from the parameter. When then and the solution approaches the incompressible fluid behavior. However, higher pressure fields are expected with the increase in this parameter (when the vibration period approaches the reservoir period ), indicating that fluid compressibility produces important effects that cannot be neglected. Resonance is expected when this parameter approaches . Fig. 9 illustrates the hydrodynamic pressure behavior for values of . It should be noted the increasing pressure distribution for a minor variation of this parameter.

Fig. 9

Hydrodynamic pressure distribution over the fluid-structure interface

(real valued pressures)

3.NUMERICAL ANALYSES AND ANALYTICAL VALIDATION

In this item results from analytical and numerical analyses using the finite element method are compared. A fully coupled solution is used in the numerical procedure. The major objective is to validate the proposed analytical formulations for both rigid and flexible boundaries. Effects of the fluid-structure interaction are verified in the frequency and hydrodynamic pressure distribution for corresponding system eingenvalues and eigenvectors.

3.1.RIGID BOUNDARY MODEL

A finite element model was constructed (based on Fig. 10 scheme) in order to evaluate the rigid body interaction behavior. This system consists on a rigid structure in contact with a reservoir and attached to equally distributed springs.

Fig. 10

Rigid body interaction scheme

Material properties used in this analysis are given by Table 1. Modal analysis results for the first five corresponding frequencies and mode shapes are illustrated on Table 2. It should be noted that the first frequency corresponds to a very small value when compared to the other frequency results. In fact it is a value that approaches the uncoupled structural frequency (0.1266 ).

Table 1

Material Properties

Parameter / Value
Structure / Elasticity modulus () / 2.10 x 1012
Poison coefficient () / 0.30
Density () / 7800
Fluid / Sonic velocity () / 1500
Density () / 1000
Spring / Individual stiffness () / 20,000

Table 2

Coupled System Frequencies and Mode Shapes (numerical results)

Mode / / Mode shape
1 / 0.1187 /
2 / 59.6337 /
3 / 62.0590 /
4 / 66.4635 /
5 / 72.5519 /

An initial analysis of the first mode indicates a 6.2% reduction of the uncoupled structural frequency, resulting in an added mass distribution. This frequency value can also be verified by means of the following equation:

/ [24]

where:

/ [25]

In other words this means that an additional mass induced by the vibrating fluid should be added to the structural mass. This additional mass is provided by integration of the hydrodynamic pressures over the fluid-structure interface. Thus:

/ [26]

A plot of Eq.[24] for a series of values of is illustrated on Fig.11. It can be noted that only a single solution exists for this problem. This value corresponds to the coupled system frequency solution for an added mass effect.

Fig. 11

Analytical solution for the coupled frequency (added mass effect)

The numerical solution for the first frequency is equal to 0.1187 and the analytical solution provides 0.1186 , resulting in almost identical values.If the analytical result is substituted in Eq. [9], then the corresponding pressure field can be evaluated. Fig. 12 illustrates this pressure distribution. It should be noted that this figure is also identical to the first mode of Table 2. However, the numerical analysis indicates more than one solution for the coupled problem. In fact this solution provides an infinite number of frequencies for this problem. A verification can be made by substituting the second frequency value from the numerical analysis in Eq.[9]. But it should be noted that real and imaginary parts are expected, since the limit frequency of Eq.[14] provides: