ussd-2014_fea of dynamic behavior of large dams

USSD will insert footer text here 1

FINITE ELEMENT ANALYSIS OF DYNAMIC BEHAVIOR OF LARGE DAMS

W.P. Kikstra1 F. Sirumbal1

G. Schreppers1,2 M. Partovi1

ABSTRACT In the design of new dams and in the assessment of existing dams, the dynamic response under earthquake loadings is a critical factor. Increased processing power and improved graphical capabilities of modern computers make that Finite Element Analysis can be used nowadays efficiently to predict the response of 3-dimensional dams. In such analyses the non-linear behavior of structure and foundation may be considered, such as non-linear foundation-dam interaction, opening and closing of joints, damage propagation in the dam and non-linear compaction of foundation. For high-dams the compressibility of the fluid and energy absorption by the reservoir are important aspects to be considered. In this paper different solutions procedures, such as frequency response analysis, transient analysis and hybrid-frequency-time-domain analysis will be discussed. In particular, the procedures for hybrid-frequency-time-domain analysis will be described and illustrated.

INTRODUCTION Dynamic fluid-structure interaction problems consist of coupled systems which usually describe different physical phenomena connected one to the other through a common surface called fluid-structure interface. The coupling of the systems is produced through this interface where unknown interaction forces are produced. Each physical domain is described by its corresponding differential equations of motion, which in most of the cases are expressed in terms of different dependent variables. These equations are coupled through the boundary conditions imposed in the fluid-structure interface, making the motion of one domain dependent on the motion of the others. For the seismic analysis of dam-reservoir interaction problems, although the fluid displacement remains small, the interaction with the structure is considerable. The base acceleration load is applied in the rock foundation, and as a consequence, the soil seismic waves propagate and produce the structural vibration of the dam. The structural motion of the dam generates pressures and wave propagation in the reservoir. Consequently, the reservoir waves interact with the dam generating pressure loads on the dam’s surface. For this reason, none of the two equations of motion (structure and fluid) can be solved independently of the other.

1 TNO DIANA BV, Delftechpark 19a, 2628 XJ Delft, the Netherlands, [email protected]. 2 corresponding author.

mailto:[email protected]

2 Dams and Extreme Events

In this paper the schematized arch dam-reservoir as defined by the formulators of Theme A of the 12th ICOLD Benchmark Workshop on Numerical Analysis of Dams [1] was taken as subject for different types of analysis using the program DIANA [2].

FLUID-STRUCTURE INTERACTION The formulation used by DIANA to couple the Finite Element equations of motion for Fluid-Structure Interaction (FSI) problems is a mixed displacement – scalar potential approach, which defines the solid variables in terms of displacement degrees of freedom (DOF), and the fluid variables in terms of pressure DOF. This definition of the fluid domain using Acoustic Finite Elements is called the Eulerian pressure formulation. One of the advantages of this type of formulation is the simple description of the fluid domain using a single scalar pressure variable ( ). This reduces considerably the number of variables of the system since only one DOF per node is required to describe the motion of the fluid domain. Taking into account that for dam-reservoir interaction problems the fluid motion is not substantial but small, considerable simplifications can be made in its equation of motion formulation. Those simplifications are a consequence of the following hypotheses assumed in DIANA FSI models:

• Small displacement amplitudes • Small velocities (convective effects are omitted) • Inviscid (viscous effects are neglected) • Small compressibility (variation of density is small) • No body forces in the fluid

Based on these hypotheses, the scalar fluid wave equation of motion is defined by Eq. (1).The wave speed (c), defined in terms of the fluid density ( ) and bulk modulus ( ), is defined by Eq. (2). (1)

(2)

In the same way, the FSI condition expressed in Eq. (3) relates the fluid gradient pressure in the normal direction ( ) to the interface surface (ΓI) with the structure acceleration vector ( ). on ΓI (3) In addition to the fluid-structure interface (ΓI) condition defined in Eq. (3), Fig. 1 shows three types of boundary conditions which can be defined in DIANA FSI models. It is possible to specify two types of boundary conditions for the free surface of the reservoir (Γs). The first and simplest one is a consequence of neglecting the effect of the surface waves, prescribing a pressure equal to zero in the horizontal top free surface, as expressed in Eq. (4). This essential or Dirichlet type of boundary condition is the one used in the benchmark case study. Additionally, in DIANA it is possible to define a


second type of boundary condition takes into account the pressure caused by the free surface gravity waves. on Γs (4) Two types of infinite extent boundary condition (Γe) are considered. The first one defined by Eq. (5.a) prescribes the hydrodynamic pressure equal to zero at a distance equal to the reservoir length. The second one defined by Eq. (5.b) is a radiation boundary theoretically located at an infinite distance from the dam, which ensures that no incoming waves enter into the system (only outgoing waves).

Figure 1. Fluid and solid domains, fluid-structure interface and boundary conditions.

on Γe (5a) on Γe (5b) Finally, two types of bottom boundary condition (Γb) are considered. The first one is setting the gradient of the pressure in the normal direction equal to zero, as expressed in Eq. (6.a). The second one defined by Eq. (6.b) is a radiation boundary which takes into account the energy absorption of the bottom materials in terms of the wave reflection coefficient ( ) defined as the ratio between the amplitudes of the incident pressure wave over the reflective pressure wave. on Γb (6a)

on Γb (6b)

DISCRETE FEM FORMULATION OF THE COUPLED SYSTEM

The discrete formulation of the equation of motion of the fluid domain is obtained by the application of the Galerkin method on the fluid wave equation defined in Eqs. (1) and (2), the FSI condition defined by Eq. (3), and the boundary conditions defined by Eqs. (4) to (6). As a consequence, the matrix equations given by Eqs. (7) to (11) are obtained in


terms of the nodal hydrodynamic pressure ( ) and structural displacement ( ) vectors, in addition to their correspondent shape function matrices ( ).

(7) (8)

(9)

(10)

(11) Similarly, the non-linear discrete equation of motion of the structure is defined in Eq. (12). The coupling between Eqs. (7) and (12) is described by the interaction matrix ( ) expressed in Eq. (11) as a surface integral over the fluid-structure interface area. (12)

SOLUTION OF THE COUPLED SYSTEM USING THE HFTD METHOD

First, the fluid equation of motion defined by Eq. (7) is solved in the frequency domain to obtain the hydrodynamic pressure amplitude vector ( ) in terms of the structural displacement amplitude vector ( ), as expressed in Eqs. (13) and (14). By doing this, a dynamic linear behavior is assumed for the fluid domain only.

(13) (14) On the other hand, the structure non-linear equation of motion defined in the time domain by Eq. (12) is re-formulated in Eq. (15) based on relative coordinates ( ). In this way, the earthquake ground acceleration vector ( ) is introduced as external loading. (15) The non-linear internal force of the structure is expressed in Eq. (16) as the difference between the linear internal force and an unknown pseudo force vector. The pseudo force concept is graphically shown in Fig. 2.


Figure 2. Concept of the pseudo force vector used in the HFTD method.

(16)

Eq. (16) is substituted into Eq. (15) to obtain the HFTD pseudo-linear equation of motion of the dam in the time domain given by Eq. (17). The pseudo force vector in the right-hand side of Eq. (17) is not known a priori but determined through an iterative procedure.

(17)

The HFTD method solves the pseudo-linear equation of motion in the frequency domain. Therefore Eq. (18) is obtained after transforming Eq. (17) to the frequency domain. The hydrodynamic pressure vector, which is already defined in the frequency domain by Eqs. (13) and (14), is replaced into Eq. (18) to obtain the final form of the HFTD pseudo-linear equation of motion, defined by Eqs. (19) to (21).

(18)

(19) (20) (21)

Eq. (19) shows that the effect that the reservoir has in the seismic response of the dam is given by the fluid contribution of additional mass ( ) and additional damping ( ), which are defined as the real and imaginary part of the product , respectively. is defined in the frequency domain, and for that reason and are identified as the frequency dependent properties of the dynamic dam-reservoir interaction systems. The HFTD method in DIANA is implemented in combination with the mode superposition method by introducing the change of variables defined by Eq. (22). Consequently, the pre-multiplication of Eq. (19) by the transposed matrix of modal vectors ( ) results in the equation of motion defined by Eqs. (23) to (29), in terms of the generalized modal coordinates vector ( ).

(22) (23) (24) (25)


(26) (27) (28) (29)

The equation of motion in the generalized modal coordinates constitutes a condensed system of equations in which the unknowns are reduced to a number of chosen modes of vibration. In the general case only the frequency independent matrices and are diagonal, which means that the system is not uncoupled but reduced.

The HFTD solution of the generalized pseudo-linear equation of motion in the frequency domain defined by Eq. (23) previously requires expressing in the frequency domain the ground acceleration signal and the pseudo force. This is done by means of the direct Discrete Fourier Transform (DFT) expressions shown in Eqs. (30) and (31) for the ground acceleration and pseudo force, respectively.

(30)

(31)

Once Eq. (23) is solved and the generalized displacement in the frequency domain [ ] is obtained, the inverse DFT expression shown in Eq. (32) is used to determine the correspondent solution in the time domain [ ]. This operation is required to evaluate the linear ( ) and non-linear ( ) internal forces for each time step, and then calculate the new value of the pseudo force vector ( ) by means of Eq. (16). The process is repeated until the pseudo force of two consecutive iterations satisfy the convergence criteria for all the time steps considered in the analysis.

(32)

In conclusion, the HFTD method solves the coupled equations of motion in the frequency domain, and then, by means of the Fourier analysis, it carries out an iterative procedure in the time domain to include the structure non linear behavior.

HFTD METHOD IMPLEMENTATION ISSUES

There are two important implementation issues of the HFTD method: Extended Fourier Period and Time Segmentation Approach. The purpose of the HFTD method is to determine the actual transient response of the system subject to an arbitrary ground acceleration loading, and not the steady-state response of the system subject to a periodic loading. Therefore, to obtain an accurate approximation of the transient response, the period used in the Fourier transform should


be extended sufficiently long that the effects of the steady-state response and the loading periodicity become negligible, as described by Veletsos and Ventura [3]. The Fourier period ( ) should be defined as the summation of the ground acceleration duration ( ) plus an additional extended period called quiet zone ( ). The quiet zone is a band of zeros added at the end of the ground acceleration signal, which length depends on the free vibration properties of the system (fundamental period of vibration and viscous damping ratio of the structure). Eqs. (30) to (32) indicate that the DFT is applied to the earthquake ground acceleration (direct), the pseudo force (direct) and the response of the system (inverse), respectively. All these transformations should always be performed over the same Fourier period ( ), which is calculated using Eqs. (33) and (34), in terms of the fundamental vibration period ( ), the viscous damping ratio of the system ( ) and the time span of interest ( ) as described by Chavez and Fenves [4]. The time span of interest is the time for which the calculation of the system response is desired.

(33) (34)

Once the extended Fourier period ( ) is calculated, the number of time points ( ) required by the DFT in Eqs. (30) to (32) is determined with Eq. (35), where is the time step size of the ground acceleration signal discretization.

(35)

Eq. (34) shows that for long oscillation periods (low frequency systems) and low damping ratio, the quiet zone ( ) is longer. Veletsos and Ventura [3] explained that in the extreme case of a damping ratio equal to zero tends to infinity, which means that the HFTD method is not applicable for undamped systems. As mentioned before, the quiet zone is necessary to satisfy the initial conditions and consequently obtain a better approximation of the transient response instead of the steady state response. On the contrary, Darbre and Wolf [5] stated that “the use of Fourier transformation period which is too short provides inaccurate results, no matter that they be stable and apparently reasonable”. In conclusion, the insertion of a quiet zone should be understood as a numerical representation of what theoretically should be an infinity period in the analytic Fourier integral. The second HFTD implementation issue is the time segmentation approach. The HFTD procedure converges in a time progressive manner, and Darbre and Wolf [5] stated that “convergence at any time is reached only after the solution has converged at all previous times”. If the procedure described above is executed directly along the whole time span of interest, convergence and stability problems are likely to occur.


For this reason, the HFTD method is performed following a time segmentation approach in which the displacements and pseudo forces are evaluated for a limited number of consecutive time steps contained in time segments. Once convergence is achieved in all the time steps within a segment, the iterative procedure is repeated for the next time segment, until the total time span of interest is covered. In summary, the time span of interest has to be divided into time segments containing one or several time steps. Fig. 3 shows the kth time segment and the time steps contained inside. The initial points of time segments “k” ( ) and “k+1” ( ), the last converged time step ( ) of the previous iteration, the time step size ( ), and the time segment size ( ) can also be identified from Fig. 3. The initial time step of segment “k” is calculated with Eq. (36).

Figure 3. Graphic representation of the kth time segment. (36) For the first iteration of the first time segment only, the pseudo force vector is set equal to zero and transformed to the frequency domain using the Fourier transformation period (Tp). Once the pseudo linear system has been solved in the frequency domain, the generalized response is transformed back to the time domain. For the subsequent iterations, the segmentation approach requires that the pseudo forces and displacements in the time domain are calculated and updated only for the not converged time steps belonging to the current time segment ( ).The iterative process is repeated until the convergence criteria are fulfilled for all the time steps inside the current segment of analysis ( ). When all the time steps inside the current segment have converged ( ), the next segment is investigated. In this way, the iterative procedure is continued until the last segment of the time span of interest is covered.

SOLUTION IN THE TIME DOMAIN

The sources of frequency dependence are fluid compressibility and radiation boundary conditions (infinite extent and bottom absorption). Therefore, frequency independent systems are obtained when the fluid of the reservoir is assumed to be incompressible ( ) and also the infinite extent and bottom boundary conditions of the reservoir are defined by Eqs. (5.a) and (6.a), respectively. For this particular case, the fluid domain


matrices defined by Eqs. (8) and (9) are equal to zero ( ). As a consequence, the fluid matrix defined by Eq. (14) only depends on , as expressed by Eq. (37) (It is not frequency dependent anymore).

(37)

The fluid added mass and damping matrices are obtained by replacing Eq. (37) into Eqs. (20) and (21), respectively. The fluid added matrices expressed in Eqs. (38) and (39) are replaced into Eq. (19) to obtain the pseudo linear equation of motion in the frequency domain, defined by Eq. (40).

(38) (39) (40)

From Eq. (39) it can be noticed that there is no additional out of phase response introduced in the system by the fluid. In fact, the dam-reservoir interaction pseudo-linear equation of motion expressed in Eq. (40) only has a fluid added mass term which is not frequency dependent. Therefore Eq. (40) can be transformed into Eq. (41) and be directly solved in the time domain applying any standard time integration scheme (f.e.: Newmark method)

(41)

THREE-DIMENSIONAL NUMERICAL CASE STUDY

The foundation-dam-reservoir interaction model generated in FX+ for DIANA is shown in Fig. 4. The mesh of the three components and their geometric characteristics are shown in Fig. 5.

Figure 4. Foundation-dam-reservoir interaction 3D model


Figure 5. Components and geometrical characteristics of the foundation-dam-reservoir

interaction system

The dam and the foundation are modeled with hexahedral and wedge solid finite element based on quadratic interpolation. On the other hand, the reservoir is modeled with three-dimensional flow elements based also on quadratic interpolation. The characteristics of the finite element mesh for each of the three components of the model are shown in Table 1. In addition, the material properties assigned to the dam, reservoir and foundation are listed in Tables 2, 3 and 4, respectively. Table 1: Characteristics of the Finite Element Model Mesh

Component Type DIANA element name Number of Elements Number of Nodes

Dam Solid 3D CHX60

712 3601 CTP45

Foundation Solid 3D CHX60

4896 23339 CTP45

Reservoir Flow 3D CTP15H

2670 11950 CHX20H

Table 2: Material parameters for the concrete dam

Parameter DIANA variable name Value / Type Units Modulus of elasticity YOUNG 2.7 x 1010 N / m2 Poison modulus POISON 1.67 x 10-1 - Density DENSIT 2.4 x 103 kg / m3

Rayleigh damping RAYLEI 5.71199 x 10-1 - 1.447 x 10-3 -

Table 3: Material parameters for the reservoir fluid

Parameter DIANA variable name Value / Type Units

Conductivity CONDUC 1 -

Sonic speed CSOUND 1.483 x 103 m / s

Density DENSIT 1.0 x 103 kg / m3


Wave reflection coefficient for infinite extent boundary

ALPHAB 0 -

Wave reflection coefficient for bottom absorption boundary

ALPHAB 5.0 x 10-1 -

Table 4: Material parameters for the foundation soil

Parameter DIANA variable name Value / Type Units

Modulus of elasticity YOUNG 2.5 x 1010 N / m2

Poison modulus POISON 2.0 x 10-1 -

Density DENSIT 0 kg / m3

The foundation-dam-reservoir interaction system is subject to a base acceleration with duration of 20 seconds, simultaneously applied in the three translational directions of the supports located in the foundation edges. The ground acceleration signals in the X, Y and Z directions are shown in Figs. 6 to 8, respectively. A factor of 0.1 times the gravity acceleration constant is applied to the three signals.

Figure 6. Ground acceleration excitation load signal in the X-direction

Figure 7. Ground acceleration excitation load signal in the Y-direction

Figure 8. Ground acceleration excitation load signal in the Z-direction


The transient responses of the two analysis cases shown in Fig. 9 are determined and studied.

Figure 9. Foundation-dam-reservoir interaction analysis cases

Case I corresponds to the frequency independent system presented in the previous section, for which the fluid of the reservoir is assumed to be incompressible ( ) and the infinite extent and bottom boundary conditions of the reservoir are defined by Eqs. (5.a) and (6.a), respectively. This frequency independent analysis case is solved with both HFTD and Newmark methods, with the objective of assessing the accuracy of HFTD method. On the other hand, Case II corresponds to a frequency dependent system in which fluid compressibility and reservoir radiation boundary conditions are included. Reservoir bottom absorption (with ) and radiation boundary of infinite extent, defined by Eqs. (6.b) and (5.b), respectively, are included in the analysis. As it was previously explained, the transient analysis of this frequency dependent system cannot be solved by the standard Newmark method, only by the HFTD method.

Only the linear elastic results of both analysis cases are presented in the next section. Nevertheless, as explained in Sections 4 and 5, the non-linear behavior of the dam can also be obtained with the HFTD method for any of the two analysis cases presented in Fig. 9. For example, two interesting types of non-linear behavior which could be included in the analysis are the dam’s concrete cracking and base sliding. The analysis parameters used in HFTD and Newmark methods are presented in Tables 5 and 6, respectively.

Table 5: HFTD analysis parameters

Parameter Variable symbol Value / Type Units


Loading duration 20.00 s Time response of interest 22.00 s Fourier Period 31.66 s Time step size 0.01 s Number of Fourier time points 3166 - Number of time segments 11 - Time segment size 2 s Minimum loading excitation frequency included 0 Hz

Maximum loading excitation frequency included 20 Hz

Number of free vibration modes included 50 -

Table 6: Newmark analysis parameters Parameter Variable symbol Value / Type Units

Loading duration 20.00 s Time response of interest 22.00 s Time step size 0.01 s

Prior to the time domain analysis, the eigenvalues of the dam-foundation system without fluid reservoir (dry case) were determined. The first 10 mode-shapes and corresponding eigenvalues are presented in Figure 10.


Figure 10: Mode-shapes and eigenvalues dry case dam-foundation system

Case I was solved using both Newmark time integration and HFTD analysis. Figure 11 shows the amplitude of the displacement of the crest at the main section. Agreement of the displacement results obtained with both methods is very good.

Figure 11: Dam´s crest displacement amplitude [m] at main section for Case I with

HFTD and Newmark methods

The frequency dependent properties of Case II (Fig. 9) introduce more damping in the system which should lead to lower responses to the earthquake loading. Figure 12 shows that the crest displacement amplitude has considerably lower peaks compared to Case I. A 2D case study shows that reservoir fluid compressibility increases the response; that there is minimal influence of the radiation boundary if the length of the reservoir length is sufficient and that bottom absorption damps the response (Sirumbal, [6]).


Figure 13: Dam´s crest displacement amplitude [m] at main section for Cases I and II

with HFTD method

CONCLUSION

The given foundation-dam-reservoir system was analyzed for the full duration of the earthquake. With the HFTD method implemented in the standard version of DIANA for the case of frequency independent properties the same results could be reproduced as with implicit time stepping with Newmark’s method. With HFTD the effect of frequency dependent properties such as compressibility of fluid, reservoir bottom absorption and infinite extend reflection have been analyzed and quantified, resulting in an interesting method to be applied specially in the dynamic analysis of dam-reservoir interaction.

REFERENCES [1] Graz University of Technology, Institute of Hydraulic Engineering and Water Resources Management, 12th International Benchmark Workshop on Numerical Analysis of Dams – Theme A Fluid Structure interaction Arch Dam – Reservoir at seismic loading, 2-4 October 2013 in Graz, Austria. [2] TNO DIANA BV (2011). DIANA User’s Manual Rel. 9.4.4, Delft, The Netherlands. [3] Veletsos, A.S., and Ventura, C.E. (1985). Dynamic analysis of structures by the DFT method. J. Struct. Eng. ASCE, Vol.111, 2625-2642. [4] Chavez, J.W., and Fenves, G.L. (1993). Earthquake analysis and response of concrete gravity dams including base sliding. Report UBC/EERC-93/07, Earthquake Eng. Research Center, Univ. of California, Berkeley. [5] Darbre, G.R., and Wolf, J.P. (1988). Criterion of stability and implementation issues of hybrid frequency-time domain procedure for nonlinear dynamic analysis. Earthquake Engineering and Structural Dynamics, Vol.16, pp. 569-581. [6] Sirumbal, F. (2013) Numerical modeling of dam-reservoir interaction seismic response using the Hybrid Frequency-Time Domain (HFTD) method, Master thesis, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands.

ussd-2014_fea of dynamic behavior of large dams

Documents