Natural frequencies of arch dam reservoir system—by a mapping finite element method

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 10, 719-734 (1982)

NATURAL FREQUENCIES OF ARCH DAM RESERVOIR SYSTEMS-BY A MAPPING FINITE ELEMENT METHOD R. NATH* Faculty

of' Engineering, Queen Mtrry College, Uniuersiry of London, London, England SUMMARY

A simple mapping finite element method is used to calculate the coupled natural frequencies and mode shapes of realistic arch dam reservoir systems in which the dam is circular cylindrical with non-uniform cross-section. This method, in which both the dam and the reservoir domains are mapped into geometrically simpler shapes using cylindrical-polar transformations, is found to give accurate results, achieved simply and economically. Results of analysis show that hydrodynamic interaction has a substantial effect on the coupled natural frequencies and mode shapes; also that the effect of water compressibility in the type of dams considered can be ignored without significant loss of accuracy. A simple method is also presented for predicting the water compressibility effect before

undertaking detailed response analysis.

1. INTRODUCTION

The response of an arch dam impounding a reservoir to earthquake excitation is essentially a n elastohydrodynamic phenomenon in which the dam, the reservoir and the soil interact with each other during vibration. The overall response of the dam is thus determined by the interaction or coupling that occurs between these three aspects comprising the total 'system'. F r o m physical considerations it is clear that the extent of interaction in a given system will depend upon the geometric a n d material properties of the three constituent aspects comprising that system. Obviously, therefore, the analysis of such systems for response prediction is difficult owing mainly to the awkward configurations which real arch d a m reservoir systems usually have, and it is compounded by the energy dissipating mechanisms that are usually present in the system. The complexity of the system considerably simplifies when the solid aspects (i.e. dam a n d soil) are assumed to be infinitely stiff, because the system then degenerates t o the uncoupled hydrodynamic aspect only. A number of investigators' - 4 have considered this uncoupled problem, the solution of which leads t o the frequency response of uncoupled hydrodynamic pressures acting on the darn. Clearly, a t a given exciting frequency the resultants of these pressures can be applied to the d a m (now admitting its elasticity) to calculate its deformation a t that frequency. This approach, however, is too simplistic and a t best represents the first approximation to the real situation, because it does not adequately account for the interaction between the dam and the reservoir. Considering the inherent complexity of practical arch d a m reservoir systems, it is clear that the finite element method would suggest itself as possibly the only solution strategy for such problems. For relatively thick dams, arch-gravity type for example, thick shell elements may be used to represent the dam. In the case of relatively thin dams, on the other hand, finite elements based on thin shell theories (e.g. Love,5 Fliigge6) may be preferred. As for representing the hydrodynamic aspect, we have the following conventional alternatives: (a) standard finite elements derived from the scalar wave equation governing hydrodynamic pressure in the reservoir, or (b) boundary solution processes. In the case of large aspect ratio? systems, and practical systems often have large or very large aspect ratios, (a) leads to a large number of equations making

* Senior Lecturer in Civil Engineering. t Delined :IS [he r;ilio: length of reservoir,'heigli~ of dam. 0098-8847/82/050719-16$01.60 @ 1982 by John Wiley & Sons, Ltd.

Receioed 23 July 1981 Rrcised 11 Junuary 1982

720

B. NATH

solution computationally expensive; (b), on the other hand, provides an elegant means whereby the threedimensional geometry of the reservoir is reduced to a two-dimensional one for the purpose of mathematical manipulation; its disadvantage, however, is that since collocation methods are usually applied to find element properties, fully coupled unsymmetric matrices are generated and these cannot be processed by the readily available codes written for the sparse symmetric matrices which arise in the standard FEM. Awkward configurations of the system may also complicate matters, although this can be substantially remedied by using a process' in which standard finite elements are coupled to boundary elements. When the central angle of a circular cylindrical arch dam is 90 degrees, valley walls radial and the reservoir bottom flat, the mathematics of the uncoupled hydrodynamic aspect simplifies considerably; it then becomes relatively easier to find a closed-form solution for the uncoupled hydrodynamic pressures generated in the r e ~ e r v o i r .Ignoring ~ soil interaction Porter and Chopra' use such a solution to analyse the coupled behaviour of a dam of this geometry by modelling the dam with curved thick shell elements. Hall and Choprag consider the coupled response of a realistic arch dam using finite elements to represent both the dam and the reservoir. In this paper attention is focussed onto the determination of the coupled natural frequencies and mode shapes of realistic circular cylindrical arch dams, ignoring soil interaction and using novel mapping finite elements both for the dam and the reservoir. Our objective here is twofold: firstly, to find the low coupled natural frequencies of the system relative to a range of geometric parameters, and, secondly, to quantify the conditions under which water compressibility effects cease to be significant, so that the much simpler 'added mass' solution" may be implemented for response analysis. Clearly, in order to obtain a complete solution to the total problem of vibrating arch dam reservoir systems, tht soil aspect and its interaction should also be included in analyses. Work along this line is currently in progress and findings will be reported in due course.

2. THEORY 2.1. The structural aspect We will assume that the dam is relatively thin so that thin shell theories'. ' may be used to predict its behaviour. Consider the portion of a thin circular cylindrical shell, Figure l(a), in which the central angle is

i shell,

/

H -

2'

3

1

L

(a1 (bl Figure 1. (a) Portion of the middle surface ofa circular cylindrical shell. (b) Mapping of(a)in the z-Y plane: displacement component along the normal to the plane

IV

is

NATURAL FREQUENCIES OF ARCH DAM RESERVOIR SYSTEMS

72 1

28 and R denotes the radius of the middle surface. u, c and w denote the components of displacement along the axial, tangential and radial directions respectively. Assuming the shell to be linearly elastic, isotropic and homogeneous, we can show from Love's theory5 that its constitutive relationship is

(4 = CDI {&I in which [D] denotes the elasticity m a t r i ~ , ~

1.i

Ma K

= { N , N , N,,

Mx

= {ex e, ex,

k, k x J T

J T

and {El

kx

du

ex = -

ax

av

1 ?u

ex= = -+-.ax R act

kx

=

a2 w

p

N,, N , and N,, in equation ( 2 )denote the in-plane stress resultants per unit length; M,, M , and M,, denote the stress moments, also per unit length.5 Clearly, all the quantities in the above equations are expressed in a system of cylindrical-polar coordinates, (r,r , x), in which the radial direction is invariant (Y = R ) owing to the thin shell presumption. Furthermore, the problem domain is defined by the limits on the co-ordinates ct and x, so that the shell naturally maps into a rectangle in the a-x plane as shown in Figure l(b); obviously, the middle surface of the shell coincides with the a-x plane and shell thickness maps along the normal to the plane of paper. It follows therefore that the shell of Figure l(a) can be analysed by discretizing its mapping, Figure l(b), into finite elements, provided that the stiffness properties of these elements are derived from the strain energy integral

u = -2IJ., ( &iI T cDIjeJRdctdx

(5)

in which the integration refers to the entire element area. Obviously, a similar scheme will be implemented for deriving the consistent mass matrix for the discretized mapped shell.

2.2 The hydrodynumic aspect Figure 2 shows the geometry of the reservoir-dam system under consideration, in which the dam is circular cylindrical and the valley walls rigid and radial. We will assume'the water in the reservoir to be inviscid but linearly compressible. Then, with respect to the Cartesian framework (x,y , z), we can show that steady-state hydrodynamic pressure, p , generated in the reservoir in excess of static pressures will be governed by the

122 scalar wave equation’.

B. NATH lo

{Vx}’{ V, p } + Ap = 0 in R,

(6)

I. = (o/c)’

(9)

in which o and c respectively denote circular frequency of oscillation and acoustic velocity in water; R, denotes the reservoir domain in the physical (x,y, z ) space.

rigid valley wall.S, (a1

Figure 2. (a) Plan of reservoirdam system considered (symmetric half only; no variation in dam thickness along 2). (b) Elevation of(a) at a-a. (c) Elevation of dam as viewed from b

In order to achieve computational economy coupled with large or very large aspect-ratio capability relative to the standard FEM, we will now map R, into R, in the logarithmically condensed* image or uspace by implementing the following orthogonal transformation (see Figure l(a)):

x=x

( 1Oa)

y = exp (r)cos c(

(lob)

z = exp (r) sin cx

(10 4

It is then a simple matter to demonstrate” that equation (6) transforms in the u-space as 2 2 p s 2 p dZp --++-,+k-+kRp dr2 SCC &x2

=0

in R,

in which

k = exp(2r) * The theory of logarithmic condensation in the cylindrical-polar and spherical-polar u-spaces has been described in detail in References 11 and 12 respectively.

NATURAL FREQUENCIES OF ARCH DAM RESERVOIR SYSTEMS

123

We will approximate the curved valley wall S 3 , Figure 2, as a radial 'staircase'. Then, it can be verified that the transformation of R, using equations (10) leads to the mapped reservoir domain R, shown in Figure 3. Clearly, therefore, the solution of equation (6) in the physical domain R, with the prescribed boundary conditions is equivalent to solving equation (11) in the mapped domain SZ, subject to the appropriately

/+----+-

Figure 3, Physical coupling between the mapped dam and the mapped reservoir

transformed boundary conditions. It may be noted in this context that boundary conditions will undergo a transformation in the u-space only when they are prescribed in the physical space in terms of the gradient(s) of the field variable.'

2.3. Coupling between the structural and hydrodynamic aspects Figure 3 shows, typically, the physical coupling between the mapped circular cylindrical arch dam and the mapped reservoir which it impounds (see Figure 2). This figure also shows a typical finite element discretization which was used for solution. It can be verified that when p and y are varied, the discretized mapped dam of Figure 3 generates a wide range of valley shapes; Figure 4 shows the valley profiles at r = r l , generated in this way, considered in this work. The equation of coupled free motion of the dam can be written as", l 4 [Kd-oZ Mdl i6) +LTllIP) = iO} (13) in which [Kd] and [Md] denote the stiffness and consistent mass matrices of the mapped dam, respectively; vector (6) lists the nodal displacements of the dam. The weighting matrix, [T1], transforms nodal hydrodynamic pressures, ( p } , into hydrodynamic forces acting at the dam nodes. As for the hydrodynamic phase, we note that coupling occurs through the boundary condition". l4

to be satisfied at all nodes defining the reservoir-dam interface, S , ; p denotes mass density of water. Then, the coupled free equation of motion for the hydrodynamic phase can be written as'" l 4 [K,-02

M,1 { P ) + w 2 Pl[T21{6)

=

lo)

(15)

724

B. NATH

or [Z-02

K ; ' M , ] ( p } + 0 2 r , [ K r - ' T,] {S} = (O}*

in which [ K , ] and [ M , ] denote the 'stifhess' and consistent mass matrices of the mapped reservoir, respectively; [Z] denotes the unit matrix. [T2],which is a weighting matrix of areas relating to the interfacial nodes, derives from the FE processing of the gradient of equation (14).

-

y=0.5,1.5.2.5

Figure 4. Valley shapes at r

=

r I as viewed from b (see Figure 2(a)): solid lines represent 'staircase' shapes generated by the discretized dam of Figure 3; dotted lines represent 'average' profiles

Equations (13) and (16) may now be combined to give the coupled frequency equation of the system as

The following boundary conditions were also implemented (Figure 2): p = 0 at x = H and on S ,

(18)

and

ap =0 an

at x

=0

and on S , and S ,

(19)

in which n denotes the direction of the outward normal to surfaces in the u-space. Equation (18) does not recognize the existence of gravity waves at the free surface; this condition is reasonable, however, since the effect of these waves becomes significant only at very low frequencie~.'~ Furthermore, the condition of equation (18) is consistent' with the neglect of the convective acceleration terms in the governing equation for the hydrodynamic phase (this neglect is implicit in equation (6)).

*

Preferred since in this version spurious reservoir frequencies can be suppressed more easily.

725

NATURAL FREQUENCIES OF ARCH DAM RESERVOIR SYSTEMS

3. PROBLEM SOLUTION The mapped domain of the dam, Figure 3, was discretized into four-noded rectangular plate bending elements including in-plane displacements. Standard polynomials' were used to approximate the variations of the three mutually independent displacements u, v and w. The following nodal degrees of freedom were considered: u,

v, w, - - . 1( u + ; ) R

and aw

ax

Eight-noded rectangular parallelepiped elements were used to represent the mapped reservoir as shown in Figure 3; matrices [ K , ] and [Mr] were found from the standard shape functions of this element and equation (11). 4. ACCURACY OF SOLUTION 4.1. Test example 1 Porter and Chopra' consider a system in which the dam is circular cylindrical, impounding a constant depth reservoir bounded by rigid, vertical and radial valley walls. Other system details are as follows: Young's modulus of dam = 5,000,000psi ( = 34.5 GN/m2) unit weight of dam = 150 pcf ( = 2400 kg/m3) Poisson's ratio of dam = 0.17 central angle (28)of dam = 90" velocity of sound in water = 4720fps ( = 1440m/s) unit weight of water = 62.5 pcf ( = 1000kg/m3) t , = 0.035H t 2 = 0.2H Treating the dam as a thick shell, curved thick shell elements are used by the authors to find its stiffness and mass matrices. When p = 45", a closed-form solution for the hydrodynamic aspect in this case can be found relatively e a ~ i l y Porter ;~ and Chopra use such a solution to represent the reservoir effects. Table I shows some natural frequencies (symmetric modes only) of the dam vibrating without water, computed by the proposed method, compared with the corresponding results by Porter and Chopra. Clearly, agreement is reasonably good considering that in the present analysis only 12 effective nodes were used to Table I. Uncoupled natural frequencies of the dam of Reference 8. p flat; symmetric modes only Source

i

0.5 0.5 Per cent difference 1.5

f3

f4

.r5

1.856 1.830 1.4

2.4 10 2.369 1.7

3.080 3.258 - 5.7

3.276 3.349 - 2.2

3.445 3.601 -4.5

A

0.593 0.649 - 9.4

0.754 0.878 - 16.4

1.096 1.089 0.6

1.227 1.354 - 10.3

1.378 1.589 - 15.3

0.4 18 0443 - 6.0

0.470 0.526 - 11.9

0.630 0.750 - 19.0

0.939 0.859 8.5

0.949 1,041 - 9.7

=

B

Per cent difference = 2.5 2.5 Per cent difference =

A

B

A: Reference 8; B: proposed method. u p = -(Edg/wd):.

H

fi

reservoir bottom

A B

1.5

.L

fi

= 45',

726

B. NATH

represent the mapped symmetric half of the dam; this compares with 36 used in Reference 8. Furthermore, discrepancies probably also reflect to some extent the conceptual differences between a thick shell and a thin shell analysis. Figure 5 shows the reduction in the natural frequencies (symmetric modes only) of the dam caused by hydrodynamic interaction when the reservoir is full. For y = 1.5 and 2.5 present results are seen to be in good agreement with those of Porter and Chopra in the lower modes; discrepancies in the higher modes may be attributed to the relatively small number of nodes used to represent the dam in the present analysis.

1;;:;; 1 : 1';

incompressible

20 4 0

-. .-

[

;

;

i

0

7l.-

1

3

2

3

13

i0 - 4 0 1 : ; F

"

,

*

+

4

5

1

2

3

4

5

~

~

~

1

2

3

4

s

4

5

y = 0.5

L

c

;

compressible

20

.-0

+

L

V 3

P O L

1

2

3

4

s

40

t 20

mode i -

0 1

2

3

4

s

1

2

3

y=2.5 ~

~

4 y:rl/H; 5 ~ ;

+proposed method,

ref.[81.

Figure 5. Reduction of the dam's natural frequencies (symmetric modes) caused by hydrodynamic interaction with the reservoir full. r , / H = 100 in the proposed method and r , / H = infinity in Reference 8. Note: in this case the reservoir bottom isPat so that valley wall S , is uertical

According to the physics of the coupled problem''. l4 water compressibility has little effect on coupled response when the dam is flexible compared with the reservoir. However, as the dam becomes stiffer, compressibility effects become important and they lead to a greater reduction of the dam's coupled natural frequencies than when incompressibility is assumed. This aspect is vindicated by the proposed method for y = 0.5, which represents a stiff dam owing to increased arch action, as may be seen from Figure 5; in Reference 8, on the other hand, the compressible frequencies are almost identical to the corresponding incompressibility values for y = 0.5, as may also be seen from this figure. Good agreement was also achieved in the anti-symmetric modes. 4.2. Test example 2 The system considered has the same properties as that in Section 4.1, except that now the valley wall S , is curved as in Figure 2(c). Figure 6 shows the frequency response of the dam vibrating without water, computed by the proposed method using the mapped dam model of Figure 3, compared with the corresponding results

NATURAL FREQUENCIES OF ARCH DAM RESERVOIR SYSTEMS

7 27

obtained by the standard FEM using transformed plate bending elements.16 fl = 30", y = 0.5 and B = 60", y = 2.5 respectively represent the stiffest and the most flexible dams considered (also see Figure 4),and, as may be seen from Figure 6, agreement is good in both cases. Indeed, good agreement was achieved in all cases shown in Figure 4.

V = 0.5

c

O

. -

N

v)

c

l o b

B 2.0

1.0

1

0

b

;L 9 O 0

0

1.0

w -

2.0

w;'

- transformed o

plate bending element proposed method.

Figure 6. Undamped frequency response of dam, Figures 2 and 3, to an upstream-downstream ground acceleration of amplitude 1 ft/s2; reservoir empty ( 1 ft/s' = 0.305 m/s2)

5. COUPLED NATURAL FREQUENCIES AND MODE SHAPES O F THE SYSTEM The system considered has the properties given in Section 4.1, which, according to Porter and Chopra,8 may be regarded as being 'average' for arch dams. Valley wall S , is considered to be rigid, radial and curved as in Figure 2(c). In all cases the ratio, r 2 / H , which defines the remoteness of the imposed terminator" S , from the dam, was taken as 100 which is realistic, and the FE model of Figure 3 was used for solution. Also, in all cases the reservoir was assumed to be full since maximum interaction occurs in this condition. Three realistic radius to height ratios, namely y = 0.5, 1.5 and 2.5, and three values of the central angle, namely = 30,45 and 60" were considered; these values, together with the mapped FE model of Figure 3 (which was used in all cases), generate all the valley shapes at r = r l shown in Figure 4 (as viewed from b in Figure 2(a)).

728

B. NATH

For analytical purposes the excitation to which an arch dam reservoir system may be subject can be resolved into the upstreamclownstream, vertical and cross-stream components. Of these the first two, which excite the symmetric modes, generate greater response than the third.' For this reason the natural frequencies and mode shapes of the symmetric modes only were considered. Detailed frequency response analysis of the system has been considered elsewhere.". The uncoupled natural frequencies of the dam and the frequency reduction caused by reservoir interaction are given in Table I1 and Figure 7, respectively. From these data the following major observations may be made. (i) Hydrodynamic coupling or interaction leads to reduced natural frequencies of the dam, at all modes, compared with the corresponding in uucuo frequencies. Table 11. Uncoupled natural frequencies of the dam of Figures 2 and 3 found by the proposed method. Symmetric modes only

30"

0.5 1.5 2.5 0.5 1.5 2.5 0.5 1.5 2.5

45" 60"

1

2

3

2.441 0.829 0.559 1.940 0.726 0.491 1.733 0.682 0.474

4

4.185 1.887 0.965 3.138 1.109 0.656 2.722 0.915 0.618

5

1

6.114 2.077 1.350 3.998 1.499 0.858 3.181 1.344 0.794

2

3

6.577 2.215 1.528 4.412 1.733 1.104 3.570 1.354 1.018

4

7.609 2.610 2.100 5.103 2.300 1.248 4.052 1.788 1.039

5

1

2

3

4

5

1

2

3

4

5

r

v3L

0 0 C

.-0 c

u

u

3

V

E

1

2

3

1

2

3

4

5

1

4

5

1

2

3

4

5

3

4

5

0m 0I1 Q

L MODEi=

compressible;

2

+

incompressible

y = 0.5,1.5.2.5

Figure 7. Reduction of the dam's natural frequencies (symmetric modes) caused by hydrodynamic interaction. found by the proposed method. Reservoir full and r 2 / H= 100

N A T U R A L FREQUENCIES OF ARCH DAM RESERVOIR SYSTEMS

729

(ii) The effect of water compressibility on the coupled frequencies is determined by the stiffness of the dam as defined by b and y. For a relatively stiff dam, /I = 30" and y = 0.5 for example, the compressibility effect is seen to be significantly more than when the dam is flexible, /? = 60" and y = 2.5 for example (this aspect will be discussed again in the next section). (iii) The compressible solution (Figure 7) indicates that the coupled fundamental frequency of the dam is reduced by 2&30 per cent, depending on dam stiffness, due to reservoir interaction. Uncoupled mode shapes were found to depend upon the cantilever stiffness of the dam compared with its arch stiffness. For low values of p arch stiffness is greater than cantilever stiffness, and, consequently, response was found to be primarily in the cantilever mode as we might have anticipated. With increasing values of /?, on the other hand, cantilever stiffness increases relative to arch stiffness, and when it exceeds arch stiffness,.the dam responds primarily in the arch mode. Figures 8 and 9, which respectively relate to the stiffest and the

I

r e s e r v o i r empty

I

1

reservoir full, compressible w a t e r

Figure 8 Some symmetric mode shdpes found by the proposed method,

' fi = 3 0 , 7 = 0.5

most flexible of the range of dams considered, confirm this observation. (In Figures 8 and 9 mode shapes have been drawn normal to the developed upstream face of the dam). Coupled mode shapes were found to be substantially different from the corresponding uncoupled mode shapes for relatively stiff dams, i.e. dams with low values of /?and y; the case of Figure 8 is typical. However, as the dam became increasingly more flexible, this difference was found to diminish as may be seen from the typical case of Figure 9. As we might have expected, no significant differences were observed between the compressible and incompressible mode shapes in the case of flexible dams; however, significant differences were observed in the case of stiff dams.

730

B. NATH

I r e s e r v o i r empty

i

reservoir f u l l , compressible water

i

Figure 9. Some symmetric mode shapes found by the proposed method, /I

=

60 , j. = 2.5

6. PREDICTION O F WATER COMPRESSIBILITY EFFECT For a given system it is usually difficult to predict, a priori, the extent to which the water compressibility effect may be important. The implementation of a compressible solution for detailed response analysis is generally more complex and computationally expensive than when incompressibility can be assumed without significant loss of accuracy, because, for example, in a compressible solution hydrodynamic radiation damping effects should be faithfully represented. In the incompressible ‘added mass’ type solution, on the other hand, hydrodynamic radiation damping does not arise, and, consequently, the treatment of the hydrodynamic aspect becomes considerably simpler by comparison, requiring only the determination of the [ K , ] matrix for the dam-reservoir interface. Indeed, in the case of complex or very complex system configurations simple electric analogue may be preferred to the FEM for finding the [ K , ] matrix for the reservoir-dam interface. Clearly, therefore, it would be useful to quantify the conditions under which a given system may be classified, accurately and easily, either as ‘compressible’ or as ‘incompressible’ before undertaking a detailed analysis. The ratio, o f / w ; , may be regarded as the primary parameter relating to the water compressibility effect.” The plots of Figure 10 obtained from a detailed parametric study show the relationship between this ratio and compressibility effect in the fundamental mode. These plots may be used to determine a priori whether a compressible or an incompressible solution should be implemented for the analysis of cylindrical dams considered.

NATURAL FREQUENCIES OF ARCH DAM RESERVOIR SYSTEMS

73 1

As a typical application, suppose that for the 'average' dam considered in Sections 4 and 5 we take: H = 1OOft ( = 303 m), y = 0.5 and p = 30". Then, from Table 11, cof = 303.47 rad/s; also, from Figure 11, o;= 2ncJ3.36H = 88.26radJs. Therefore, oflo; = 3.44, for which the relative difference between the compressible and incompressible solutions is 3.7 per cent, as given by the plot of /? = 30" in Figure 10. Thus, if

.

"3'

LO

5'0

F 0

3.0

2.0

1.0

;./'r. Figure 10. Computed relationship between dam stiffness and water compressibility effect; reservoir full

r 0"

l

l 15'

l

l

r 3 0'

l

l

l L5'

r

r

l 60'

B Figure 11. Some uncoupled natural frequencies of the reservoir

the level of significance is taken as, say, 5 per cent, then clearly the incompressible solution would be adequate in this case. Figures 7 and 10 show that for the range of circular cylindrical dams considered, and presumably for other dam geometries which can be approximated to these, water compressibility effect may be ignored without significant loss of accuracy (response studies"* also confirm this).

732

B. NATH

7. OBSERVATIONS 7.1. The hydrodynamic aspect Two different schemes, which are conceptually similar, can be implemented to map the reservoir domain of Figure 2: (i) logarithmic mapping," represented by equations (lo), which leads to the governing equation of equation (1 l), and (ii) mapping in the standard cylindrical-polar space (see Figure l(a)): x=x

(204

y

= rcosa

(20b)

z

= rsinu

(204

which leads to the governing equation (transformation of equation (6))

In both cases the reservoir domain, R, of Figure 2, maps into R, shown in Figure 3, its length being In(r2/rl) when (i) is implemented, and (r2- rl) in the case of (ii).Clearly, physical space is logarithmically condensed in (i), consequently permitting an accurate solution to be obtained with a relatively small number of nodes even when the physical aspect ratio, r2/H, is large or very large.". l 2 This facility is not offered by (ii). An interesting feature of mapping in this way, as can be seen from equations (6), (9) and (21), is that: with increasing radial distance from the source (dam), acoustic wave velocity in the mapped space continuously decreases according to cexp(r) in the case of (i), and according to c/r when (ii) is implemented. Although in both cases reducing wave velocity is apparently contrary to the concept of constant wave velocity in a linear, isotropic and homogeneous physical space, this is in reality a mathematical consequence of transformation which in no way invalidates problem solution in the mapped domain, using the FEM or otherwise. 7.2. The structural aspect For all the dam geometries considered the proposed shell mapping method was found to give accurate results. Currently a wider application of this method to shells generally is being investigated, and excellent results have been obtained in the case of spherical shells. The mathematics of the process suggests that the proposed mapping method can be applied to any shell whose constitutive relationships refer to its natural coordinates (the spherical-polar system, for example, is natural to spherical shells). In the conventional FE solution of thin shells16 the shell is represented as an assemblage of flat plate bending/stretching elements whose properties are transformed from the local to a common global framework. An inconsistency arises in this process when co-planar elements meet, and a fictitious rotational stiffness has to be introduced to remedy this. No such difficulty arises in the proposed method, in which, incidentally, the nodal degree of freedom is 5; this compares with 6 in the conventional FEM.16

8. CONCLUSIONS The coupled natural frequencies and mode shapes of realistic arch dam reservoir systems, in which the dam is circular cylindrical, has variable cross-section and is located in a V-shaped valley, have been obtained using a simple mapping finite element method. In this method both the dam and the reservoir domains are mapped, using cylindrical-polar transformations, into considerably simpler shapes compared with their respective physical configurations. Results obtained for a range of configurations show mainly that: (i) hydrodynamic interaction has a substantial effect on the coupled natural frequencies and mode shapes of the dam, and (ii) compressibility of reservoir water can be ignored without serious loss of accuracy in the case of circular cylindrical dams considered, and also presumably in the case of other dam geometries which can be approximated to these shapes.

NATURAL FREQUENCIES OF ARCH DAM RESERVOIR SYSTEMS

733

A simple method is also given for ascertaining, a priori, the effect of water compressibility in a given system before undertaking a detailed analysis. This, it is hoped, will be useful to designers and analysts in that, if little compressibility effect is indicated by this method, for example, then the much simpler added mass solution may be subsequently implemented for detailed response analysis. APPENDIX I velocity of sound in water Young’s modulus of dam circular frequency factor in the ith mode gravity acceleration height of dam mode number stiffness matrix of the mapped dam stiffness matrix of the mapped reservoir consistent mass matrix of the mapped dam consistent mass matrix of the mapped reservoir amplitude of hydrodynamic pressure cylindrical-polar co-ordinates radii of S , and S , respectively radius of the middle surface of a circular cylindrical arch dam circular cylindrical surface defining the upstream face of the dam (dam-reservoir interface) circular cylindrical surface defining the imposed far terminator thickness of dam components of dam displacement unit weight of dam rectangular Cartesian co-ordinates semi central angle of dam = r,/H = (w/c)2

mass density of water mapping of Qx physical problem domain circular frequency of oscillation coupled circular natural frequency of the dam in the ith mode due to reservoir interaction uncoupled circular natural frequency of the dam in the ith mode uncoupled circular natural frequency of the reservoir in the ith mode REFERENCES 1. S. Kotsubo, ‘Dynamic water pressure on dams during earthquakes’, Proc. 2nd world conf: earthquake eng., Tokyo, 2,799-814 (1960). 2. 0.C. Zienkiewicz and B. Nath, ‘Earthquake hydrodynamic pressures on arch dams-an electric analogue solution’, Proc. inst. cioif

eng. (London), 25, 165-176 (1963). 3. B. Nath, ‘Hydrodynamic pressures on arch dams during earthquakes’, Proc. 4th world con$ Earthquake eng., Santiago, Chile, 2, B-4, 97-105 (1969). 4. P. R. Perumalswami and L. Kar, ‘Earthquake hydrodynamic forces on arch dams’, J . eng. mech. dio. ASCE, 99, 965-977 (1973). 5. A. H. E. Love, A Treatise on the Mathematical Theory of Elasticity’ 3rd edn., Cambridge University Press, 1920. 6. W. Fliigge, Stresses in Shells, 4th printing, Springer-Verlag, New York, 1967. 7. 0.C. Zienkiewicz, D. W. Kelly and P. Bettess, ‘The coupling of the finite element method and boundary solution procedures’, Int j . numer. methods eng., 11, 355-375 (1977). 8. C. S. Porter and A. K. Chopra, ‘Dynamic response of simple arch dams including hydrodynamic interaction’, Report N o . UCB/EERC-80/17, University of California, 1980 (see also Earthquake eng. struct. dyn., 9, 573-597 (1981)). 9. J. F. Hall and A. K. Chopra, ‘Dynamic response of embankment, concrete gravity and arch dams including hydrodynamic interaction’, Report N o . UCB/EERC-80/39, University of California, 1980. 10. B. Nath, ‘Dynamics of structure-fluid systems’, in Aduances in Hydroscience, Vol. 9, Academic Press, New York, 1973, pp. 85-1 19.

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1 1 . B. Nath, ‘Hydrodynamic pressure on arch dams-by a mapping finite element method’, Earthquake eng. struct. dyn., 9, 117-131 (1981). 12. B. Nath, ‘A novel spherical polar finite element for the solution of the steady-state scalar wave equation in three-dimensions’, Earthquake eng. struct. dyn., 9, 33-51 (1981). 13. S. A. Schelkunoff, Applied MathematicsJor Engineers and Scientists, Van Nostrand, New York, 1948. 14. B. Nath, ‘Coupled hydrodynamic response of a gravity dam’, Proc. inst. civil eng. (London),48, 245-257 (1971). 15. C.-Y. Liaw and A. K. Chopra, ‘Dynamics of towers surrounded by water’, Earthquake eng. struct. dyn., 3, 33-39 (1974). 16. 0.C. Zienkiewicz, The Finite Element Method in Engineering Science, 2nd edn., McGraw-Hill, London, 1971. 17. B. Nath and S. G. Potamitis, ‘Coupled response of arch dams including hydrodynamic and foundation interaction’, Proc. inst. civil eng. (London) (to be published). 18. B. Nath and S. G. Potamitis, ‘Coupled response of arch dams including hydrodynamic and soil interaction-by mapping finite elements’, Proc. 7th Euro. can$ Earthquake eng., Athens, 1982 (to be published). 19. 0. C. Zienkiewicz and B. Nath, ‘Analogue -~ procedure for determination of virtual mass’, J . hydraul. diu. ASCE, 90, No. HY5, 69-8 1 (1964). 20. A. K. Chopra, ‘Earthquake behaviour of reservoir4am systems’, J . eng. mech. diu. ASCE, 94, 1475-1500 (1968).