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Random vibration of simple flexible arch dam reservoir systems from earthquakes
Random vibration of simple flexible arch dam reservoir systems from earthquakes C.Y. Yang, M. Debessay and W. G. Li Department of Civil Engineering, Universi(yof Delaware, Newark, DE 19716, USA
Nonstationary random vibration and hydrodynamic force solutions on arch dams from reservoir fluid during earthquakes are obtained and presented in normalized form for a group of general arch dam-reservoir-foundation systems. The solutions include nonstationary power spectral density and nonstationary mean square functions of the random earthquake generated vibration and hydrodynamic force on dams. The random earthquake ground acceleration is modelled by a nonstationary (or evolutionary) process due to PriestleyKanai-Tajimi and the random solution is based on the method by Shinozuka. The arch dam-reservoir geometry is restricted to a circular section of 90* central angle with two banks extending to infinity. The non-random but fundamental hydrodynamic solution for a rigid dam by Kotsubo is used as the basis of the random solution for the flexible arch dam. A key assumption of first symmetric mode shape leads to considerable simplicity in the formulation and the computational efforts. This simplified solution is a good approximation when certain limitations on the input random excitation is carefully observed.
INTRODUCTION 9One important problem in the seismic safety analysis of dams is the evaluation of the hydrodynamic force on dams from water in the reservoir during strong earthquakes. In the case of concrete arch dams, solutions of the problem have been presented by several well known investigators including Kotsubo t, Clough, Raphael and Mojtahedi 2, Craw ford3, Perumalswami and KaP, Porter and Chopra 5.6and Fok and Chopra 7. All these noted research work treat the earthquake input excitation as a definite or the so-called deterministic function of time. However, in a rational seismic safety analysis of any major structure, the uncertainty of future earthquakes must be taken into account and therefore dynamic structural response solutions to a definite earthquake ground motion excitation are not sufficient. To extend the deterministic dynamic structural analysis to the non-deterministic case, one can model the earthquake ground motion as a stochastic or random process, apply random vibration theory, find the random structural responses and finally estimate the probability of various modes of failure, under the condition that a given random earthquake motion has occurred. This classical random vibration approach of structural safety analysis is an improvement of the non-random approach because the uncertainties in amplitudes, duration and frequency contents of earthquake ground motion are taken into account in a rational way. This basic concept was explained clearly in a short technical note by Scanlan more than ten years ago 8. The analysis of random hydrodynamic force on concrete gravity dams subjected to nonstationary random earthquake excitation was first presented by Yang and Charito9 and and
Paper accepted October 1990. Discussion closes August 1991. 18
followed by Mills/Bria and Yang 1~ and Yang and Kuserk 11. For arch dam-reservoir systems, the only published work known to the authors are the random hydrodynamic force solution on rigid arch dams by Debessay and Yang 12 and the random displacement solution by Hollinger. In this paper, the author extend their previous work 12 of random hydrodynamic force solutLon on a rigid arch dam to the case of a flexible arch dam. In addition, the simplified nonstationary random process model of Priestley 13used in the previous work is improved by adding the Kanai 14 - Tajimi 15 filtered spectral density function for the random earthquake acceleration. New solutions of nonstationary power spectral density and mean square functions are obtained by a solution method of Shinozuka 16 as presented in detail in the recent book by Yang 17. The arch dam-reservoir geometry is shown in Fig.1. The horizontal section of the arch dam consists of concentric circular segments bounded by two banks extended to infinity with a central angle of 90*. The reservoir is assumed to have a full depth of compressible but nonviscous water. Under the horizontal ground acceleration in the downstream-upstream direction, the flexible elastic arch dam is assumed to deform in its first symmetric normal vibration mode. Vertical sections of the arch dam are assumed to be trapezoidal.
HYDRODYNAMIC FORMULATION The simplified problem of arch dam-reservoir-earthquake is shown in Fig. 1. The dam is assumed to be elastic with a 90* circular are between two rigidbanks. These banks and the full reservoir behind the dam-bank system are assumed to be infinite in extent in the positive x-direction. Water in the semi-infinite resrvoir of the depth H is assumed to be non-viscous but compressible. The foundation of the dam-reservoir system is assumed to be rigid. The earthquake ground acceleration fig (t) is assumed to be in the downstream-upstream orx
Probabilistic Engineering Mechanics, 1991, Vol. 6, No. 1
Random vibration of sbnple flexible arch dam reservoir systems from earthquakes: C. Y Yang et al Op (R, O,z) Or
--
-,,
o.
0 p (r,r00~x/4,z) = p ~ - ' 2
/ /////I// $r162
.%
(4)
(5)
t- 1'
0 p (r, O, z) = 0
a.a
** . . . . . . . . . . .
(6)
rO0 The system is assumed undisturbed until the earthquake ground acceleration as(t) occurs. The ground acceleration is modelled by the simple, but fundamental, nonstationary process, which is the zero start Gaussian stationary random process with a Kanai-Tajimi spectral density Ss(to). This completes the hydrodynamic formulation of the problem.
. - 4 ~ It
%%.
9D A M F L U I D
INTERACTION
In Fig. 1 let the relative radial displacement ap (0, z, t) of the elastic arch dam be approximated by its first symmetrical mode shape ~ (0, z) under the empty reservoir so that
Piaa
Fig. 1. Arch dam -reservior system ap (0, z, t) = Y (t) ~b(0, z) direction only and the dynamic system response is small such that water surface wave effect is negligible and the entire system response is in the linear range. Under these assumptions, the governing equation of fluid motion is the well known .wave equation in terms of either the velocity potential which is the standard choice in hydrodynamics or the hydrodynamic pressure p. The pressure p is used here for convenience together with the cylindrical coordinates r, 0, and z; the time t and the sound speed e as follows.
02p • 02p 0 2p 10Zp o,2 + 7-if-;+ ~ + o---Z = ~ o---7
(1)
Boundary conditions are: (2) the hydrodynamic pressure p is zero at the free surface, (3) the vertical acceleration which relates to vertical gradient of the pressure p is zero at the reservoir bottom, (4) the radial acceleration at the dam face which relates to the radial gradient of the pressurep is equal to the sum of the radial component of the upstream-downstream earthquake ground acceraltion/is (t) and the acceleration of the dam relative to its base ~, (5) the acceleration normal to the bank at the bank, which relates to the tangential (0-direction) gradient of the pressure p, is equal to the compenent of the earthquake ground acixleration normal to the bank at the bank, (6) by symmetry, the tangential acceleration which relates the tangential (0-direction) gradient of the pressure p along the centre line of the arch dam, along the x-axis with 0 = 0, is zero and finally, only outgoing pressure waves which propagate in the +x direction are considered on the basis of Sommerfcld's radiation condition, i.e., no in-coming or reflected waves. These bofindary conditions are given below. p (r, O, H) =" 0
(2)
Op(r, 0, 0) = 0
(3)
0z
(7)
where Y(t) is the generalized coordinate. Then the total radial acceleration of the dam/i (0, z, t,) can be expressed as the superposition of the radial component of horizontal ground acceleration as(t) cos0 and the relative radial acceleration ~, (0, z, t) by
il (o, z, t) = ~ ( 0 cos o + ff (o, z, t)
(s)
To formulate the equation of motion for the vibrating arch dam, the principle of virtual displacement, is applied as follows Is. Consider a small element of the dam specified by Rd0dz with mass per unit area m(0, z). The element is subjected to an inertia force of m ( O, z) Rd 0 dz [0 s (t) cos 0 + (0, z, t)] and the hydrodynamic forcep (0, z, t)Rd 0 dz, in addition to the hydrostatic and gravity force that are excluded from the present study. For this dynamic equilibrium element of the dam, a virtual displacement 6q,, which corresponds to a virtual generalized coordinate 6Y (t) by equation (7)is specified. Then the application of the principle, that the total virtual work done by all the forces in the arch clam during the specified virtual displacement is zero, gives the following equation
- f f p ( O , z , t ) R d 0 dz6 Y(p (0, z)
- f fro (0, z) n
d 0 dz [ lie (t) cos 0
+ i;'(t) ,~ (0,01 ~ r e (O,z) - KY (t) b Y (t) - C]; (t) 5 Y (t) = 0
(9)
The first term in equation (9) represents the virtual work done by all hydrodynamic forces; the second by all inertia forces; the third by internal stresses and the last by internal damping forces. The symbols K and C represent generalized stiffness and damping coefficient of the arch dam respectively. Can-
Probabilistic Engineerin 9 Mechanics, 1991, Vol. 6, No. 1
19
Random vibration of shnple flexible arch dam reservoir systems from earthquakes: C. Y Yang et al ceiling the common term 6Y (t) and using M for the generalized mass and P(t) for the generalized force in equation (9) defined by
M = f f m (0, z) dp2(0, z) R d 0 dz
The steady-state pressure response, equation (16) consequently can be obtained as the superpostion of responses to the respective excitations, the ground acceleration equation (15) and the motion of the dam relative to the ground of equation (17). Hence from equation (16),
(I0)
fi (0, z, to) exp (i to t) =Pa (0, z, to) exp (i to t)
and P (t) = - iig ( o f f
+Pb (0, Z, W) exp (i to t)
m (0, z) cos 0 r (0, z) n d 0 dz
+ fir (0, z, tO) ~ (to) exp (i to t) -ff
(11)
p (0, z, t) d~(0, z) R d 0 dz
or
f (0, z, to) =~d (0, z, to) +~b (0, z, to)
simplifies the equation to
+ Pr (0, z, tO) ~ (to) M}'(t) + CY(t) + KY(t) = P(t)
which is the standard equation for a simple oscillator. In terms of the natural circular frequency toa and damping ratio ~e, equation (12) becomes
}'(l) + 2~dtodY(l ) + to~Y(t).- P(t) ,
(13)
M
where (.Dd =
K~/-~-'M,
~ l -~ C/Ccr,
Ccr
= 2V/-~
(14)
Equation (13) is the coupled equation of motion for the dynamic interaction of the arch dam-reservior fluid system. The term interaction refers to the fact that the generalized force P (t), as given by equation (11) depends on the yet unknown hydrodynamic pressure p (0, z, t), which in turn depends on the unknown generalized coordinate Y (t).
in whichffa(0, z, to) a n d ~ (0, z, to) are the complex frequency response of hydrodynamic pressures acting on the rigid dam due to the rigid body excitation exp(i tot) of the dam and banks respectively. The last complex frequency response function in equation (18), if,(0, z, to), is due to the specified dam acceleration ~(0, z) exp (i to t) relative to the dam base. As response solutions to the same harmonic function of time exp (i to t), they are known in Refs 4 and 5 and are given as equations (A.2), (A.3) and (A.4) in the Appendix. Equation (18) is the key to the solution of the dynamic interaction problem, because with it, the additional un__knownfi (0, z, to) is expressed in terms of the only unknown Y (to), the complex frequency response of the generalized acceleration 17 (t) in equation (13). Note that Y (t) -- Y (to) exp (i to t)
I';(t) = i to Y(to) exp (i to t) = Y(to) exp (i to t) FREQUENCY SYSTEM
RESPONSE
OF DAM-FLUID
(21) SO
-
Y (to) = i OY(to) (to) = _ to2 ~ (to)
(16)
20
(22)
(23)
To carry out more details, substitutingp (0, z, to, t) of equation (16) together with equation (18) into P (0 of equation (11) and then in equation (13) gives
w h e r e f (0, z, to) is the complex frequency response function. From equations (7) and (8), it is clear that the total radial motion of the fluid is generated by the superpostion of two parts; the ground motion//s(t) cos 0 and the dam deformation ~(0, z, t) or l?(t) ~ (0, z). Under the steady-state condition iis(t) is given by equation (15) and }'(t)d~(O,z) = ~(to)exp(itot)d~(O,z)
(20)
l?(t) = (ico)2y(to) exp (i to t) = ~(to) exp (i to t)
(15)
and the steady-state pressure solution be p (0, z, to, t) = ~ (0, z, to) exp (i to t)
(19)
~
In the simple formulation of equation (13) together with the generalized mass M of equation (10) and generalized force P (t) of equation (11), it is assumed that the first symmetrical mode shape ~b(0, z), the natural frequency tod and damping ratio ~a are all available 3.5.6.20.To solve for the dam-fluid interaction response to a random ground acceleration input, the complex frequency response method is used. Let the steady-state ground acceleration ~Jg(t) = e~p (i to t)
(18)
(12)
(17)
Probabilislic Engineering Mechanics, 1991, 1Iol. 6, No. 1
MY(t) + 2~::,atoaMl~'(t)+ toa2MY (t) -- ~g (t) L - BI (co) exp (i to t)
+ B2 (co) Y (co) 0 2 exp (i co t)
(24)
RatMom vibration of slmple flexible arch dam reservoir systenm from earthquakes: C. Y. Yang et al where
L = f fro (0, z) d~(0, z) cos 0 R d 0 dz
(03) = f f
(25)
A (t, 03)*A (t, to') E [dOg (,co)*dog (to') ] = 0 for co * 03'
(0, 03)
(o, w)] r (O,z)R dO
whereA (t, 03) is a non-random modulation function of time t and frequency to and d0s(to ) is defined as a zero mean orthogonal random process such that the mathematical expectation
(26)
01)
when 03 = co', then
IA Ct, 03) 12E I B2 (03) = f f Pr (0, z, 03) d~(0, z) R d 0 0z
Using equations (19) through (23) in equation (24) gives the standard formulation of a simple oscillator
( M + Re [ B2 (03) ] ) ~"(t) +
(2~:103dm- 03Im [W2 (03)])l;(t)
+ to2aMY (t) -- - [L + B1 (03) ] exp (i 03 t)
(28)
where Re and I,,, represent the real and imaginar3/part respectively of the complex function B2(to). In the standard form of equation (28), the dynamic interaction between the arch dam and the fluid in the reservoir is explicitly revealed. The effects of the fluid in the reservoir on the deflection of the dam consists of three parts. The first is the added mass for excitation given by the term Bi (co) defined by equation (26). The second is the added mass for inertia given by Re[B2 (03)] defined by equation (27), and the third is the added damping given by - 03lm[B2 (co)], defined also by equation (27). All three parts are frequency dependent. The final solution for the complex_frequency response function for the generalized acceleration f'~ (03) is easily obtained by substituting Y (t), I; (t) and # of equation (19), (20) and (21) respectively into equation (28).
(03) =
C03)12 = Sg Ct,03)d03
(32)
(27)
L + B1(03) M [ (0)/03) 2 + 2i~d(03d/to ) _ 1 ] - B 2 (03)
(fi9)
In equations (31) and (32), the symbol "represents complex conjugate of the function and Ss(t, 03) is defined as the nonstationary power spectral density function representing the mean square distribution of iig(t) in frequency. When the modulation function A (t, co) in equation (30) takes on the special form of a constant with unit magnitude, then the integral representation, equation (30), degenerates into a stationary random process. From equation (32), whenA (t, 03) = 1, the nonstationary power spectral density function also degenerates into a stationary density function, Ss(03), independent of time t. Consequently, from equation (32),
03) Finally, by comparing equations (32) and (33), the following relation between the nonstationary spectral density function Ss(t,03) and the associated stationary spectral density function Sg(03) is obtained
Sg (t, 03) = [A (t, co) 12 Sg (03)
(34)
equations (30) and (34) constitute the key,ingredients of the nonstationary random process model for the ground acceleration ii,(t). When such a random process excitation is applied to the arch dam- reservoir system, the response hydrodymanic force f(t) is also a nonstationary random process. Thus, in a similar way, the response force f(t) has an integral representation, a modulation function B(t, o3) and a nonstationary power spectral density function St--(t, 03) as follows.
f (t)= f_~ ~oB (t,to) e i~t dF (60)
(35)
SF (t,
(36)
The other two relevant complex frequency response function for the generalized displacement Y (t) and velocity l';(t) can be obtained from equations (23) and (22) respectively. Finally, the complex frequency response of the hydrodynamic pressure can be obtained by equation (18) and (29).
RANDOM EXCITATION AND RESPONSE A nonstationary random process model based on Priestley's work ~3 as explained in Yang's book ~7, p.241 to 245, is used here for the ground acceleration /is(t). A brief outline of Priestley's model is given in the following. Let the ground acceleration tTs(t),-assumed to be a nonstationary random process with zero mean, be defined by the integral representation.
ag (0 = f = A (t, 03) e lot dog (03)
(30)
03) = I B (t, co) 12 SF (to)
The random response problem is now focused on the solution of the nonstationary power spectral density function of the hydrodynamic force, S~(t, co), in terms of the excitation spectral density function of the ground acceleration Ss (t, 03) and the system properties. The nonstationary response problem was solved by Shinozuka 16 and is briefly outlined in the following. It is well known that the excitation-response solution for a linear system follows the Duhamel's superposition integral in the time domain Is. For the dam-reservoir-earthquake problem here, the response hydrodynamic force f(t) can be obtained by summing up the effects from small increments of excitations hF (t- x )iig (x)d~ as follows.
Probabilislic Engineerin 9 Mechanics, 1991, Vol. 6, No. 1
21
Random vibration of shnple flexible arch dam reservoir systems from earthquakes: C. Y Yang et al
f (0 =f'ohF (t - x) ~s(~) dr,
(37a)
where h~(t) is the impulse response function for the hydrodynamic force on the arch dam, f (t). When the earthquake acceleration excitation iis(t) is a nonstationary random process defined by equations (30) and (34), the hydrodynamic force f(t) is also a nonstationary process defined by equations (35) and (36). Substituting iis(t) of equation (30) into equation (37a) and interchanging the order of integration gives f (t)-f- ~
( t - x , to)e-i~'hF(nC)d-c]d(lg(to)
where the impulse response function h~t) can be obtained by the inver_se Fourier transform of the complex frequency response F(to). (43)
B
F (03) exp ( i to t) dto --00
Finally, the stationary spectral density function Ss(to) of the ground acceleration in equation (42) is assumed to be the well known Kanai-Tajimi spectrum Is given by 1 + 4 ~ } (to/oJl) 2
(37b) Recall that in the case of stationary excitation and response, the Fourier transforms dF(to) and dog (to), and the spectral densities S~to) and Sx(to) are linked by the complex frequency response function F(to) as follows.
(to) = 7 (to) dog (to)
(38)
s, (to) = IF(to) ! 2sg (co)
(39)
,?
Substituting dOs(to) of equation (38) into equation (37b) and comparing the resulting equation (37b) with equation (35) gives
Sg (to) = [1 - (to/toj-) 2 ] 2 + 4 ~ i (to/toy) 2 So (44) in which ~j, tof and So arc the damping ratio, the central frequency and the spectral strength of the random ground acceleration respectively. The random hydrodynamic force response solution is now complete with equations (18) and (29), after integration over l__hereservoir depth, for the force complex frequency response F (to) in equation (45); equation (43) for the impulse response hr(t); equation (41) and (42)for the nonstationary power spectral density S~t, 03); and the following equation (46) for the mean square response.
(to) =
B (t, 03) = F (to)=1for4 (t - -q to) e -io~ hF ('c) d~
fo
H
C0, z, to) az
(45)
(40)
SubstitulingB(t,w) of equalion (40) and S~to) of equation
(41)
SF (/, 03) = [ f o t A (l - % to) e -i cox h v (T) ~ [ 2 Sg (to) Note that when A (t, co) = 1 and t ---, oo the intergral in equation..(41) reduces to the complex frequency response function F(to) and equation (41) reduces to the well known transfer relation in the statiq_nary case. Note also"that the complex frequency response F(to) and the impulse response h~(t) form a Fourier transform pair. For the present study, a simplification is made on the modulation functionA (t, to) in-equation (41) by assuming it to be a unit step function in time t and independent of frequency co. Such a simplified modulation function leads to the so-called zero-start stationary random process or pseudononstationary process, which is the simplest case in all the time variations. It also leads to conservative response solutions. With this simplification, the modulation functionA (tx, to) in equation (41) can be removed since in the range of integratiori for "I, (t- 1:) is always positive and consequently the function is always equal to unity. Thus equation (41) is reduced to (42) SF (t, to) [ f t h F (1;) exp ( - i to "~) dr [ 2 Sg (03)
"o
(46)
/i, o o
(39) into equation (36) gives the following input-output spectral realtion due to Shinozuka 16.
SF(trto ) dto
O~ ( t ) = J -
-
O0
The random hydrodynamic force analysis method outlined above applies also to random displacement of the dam, simply by replacing the force terms with symbols f and F by the displacement terms y and Yin equation (35) through(43 ) and equation (46).
NORMALIZED STOCHASTIC RESPONSES To better understanding the dynamic behaviour of the damfluid system and to facilitate numercial computations, all functions and variables are normalized and listed with definitions as follows. A. System Properties: tot = nc/2H = fundamental frequency of the reservoir fluid c = sound speed in fluid R = radius of the circular horizontal section of the arch clam H = height of dam and depth of fluid in the full reservoir COd= first symmetrical frequency of the arch dam with empty reservoir ~t = damping ratio of the arch dam with empty reservoir
22 Probabilislic Engineering Mechanics, 1991, Vol. 6, No. 1
Random vibration of sbnple flexible arch dam reservoirsystemsfrom earthquakes: C. Z Yang et al (~ = central frequency of the foundation in the Kanai-Tajimi spectrum for the random earthquake excitation ~j, = damping ratio of the foundation in the KanaiTajimi spectrum for the random earthquake excitation
(47) B. Normalized Variables: x = tot = normalized time [2 = to/to,. = normalized frequency with reservoir Qd = to/tot = normalized frequency with dam (48) C. Interaction Parameters: [2,- = to,-/tod= fluid-dam interaction parameter [2f = to/to = fiuid-foundation excitation parameter
mml
( - 1) '= +" ~n (2m - 1) 2 (1 - 16n 2)
~
+ iD,, (~.mR)]cos4nO sinctmU (54) oo
~,.([2)_ g.~,.(to)
B1 (f2) -
B1 (co) M
32~/'2wR2" ~ -
~
m=l n=O
Mg~2
( - 1 ) m I~n(-1) n
(2~--1") (1- 16n 2)
oo
-3___22R
(49) The normalized function of the dynamic system are obtained by using the above listed variables and parameters in equations (26) and (27), after using the pressure solution of equation (A.2), (A.3) and (A.4) in the Appendix. These are the added mass terms B1 (to) and R, [B2 (to)] and the added damping coefficient -toI,,[B2(to)] given by
n-0
E. x,.,, m-1 n*O
9 [ Cn (L~R) + iDn (X~R)] cos 4n 0 sin am H
where F, is the hydrostatic force wtF/2. Note that X,,,R is defined by Equation (A.8) as a function of the normalized frequency variable f~, the geometry ratio R/H and the dummy index m. Equations (54) and (55) give th._econtribution to the complex frequency response function F(f~) from the rigidbody motion of the dam only, F..,t([2); and from the motion of the dam relative to the ground, F,(f2), respectively. Again the effect of the bank motion is left out for simplicity. Hence, from equations (45) and (18), (Q) = Fd (n) + ~ ([2) Fr (n)
I,,,,,-[C,,(k,,,R)
B2 ([2) =
+
iD,,CkmR)]
3 2 w R 2 H oo t?l g 7~
[C,,(~..,R)
Note that the last term in equation (56) includes the complex frequency response function of the generalized acceleration Y (f~) for equation (29), representing the effect of structuralfluid interaction as explained previously in the section under title of dam-fluid interaction. The normalized impulse response function of the hydrodynamic force, based on equations (43) and (53) through (56), is
(I,,.)
rn - I n - O
iD.(XmR)]
+
(56)
(50)
B2 (~) M
-
(55)
(5!)
Note that from equation (A.8) in the Appendix, XmR is a function of the normalized frequency ~ and the contribution from the motion of the banks is left out in equation (50) for simplicity. The normalized, radial, generalized acceleration from equation (29) and displacement from equation (23) become
([2) = ~(co)
hr (1;) = ~
hF (t)
(57) 1 ~ ~ ([2) e i n ~ d [2 2~ . 1 _ oo The normalized Kanai-Tajimi power spectral denisty for the ground acceleration excitation, based on equation (44), is
ss ([2) = s~ (to)/so (L/M)
+ B1 ( ~ )
(52)
([2 [) r)-2 + 2ira ([2 [2r) -1 -- 1 -B2 (s
[1-
1 + (2~s n h i ) ~ (~/.)212 + (2~.f~2~f)2
(58) ~(n)
=
%2 ~ (~)
=
_ [2-z~ (n)
03)
Based on equations (45), (A.1), (A.2), (A.4) and (29), the normalized complex frequency response functions for the hydrodynamic forces on the arch dam are obtained and given as,
The normalized nonstationary spectral density function for the hydrodynamic force, based on equations (42), (57) and (58), is
s v Cx, Q) =
~ sv (t, to) F~So
Probabilislic Engineerin 9 Mechanics, 1991, Vol. 6, No. 1
23
Random vibration of sbltple flexible arch dam reservoir systems from earthquakes: C. Y. Yang et al Table 1. System properties tI = 91.44m = 300ft
Group R/H t.o6 Natural frequency of dam (radlsec)
2
1
3
2
1
3
0.5
1.5
2.5
0.5
1.5
2.5
76.9
24.7
17.3
46.2
14.8
10.4
wp Natural frequency of reservoir (rad,'sec)
~p - o~plo~5 o.)t
24.7
0.32
Resonant Freq. W: Fluid Dam Foundation
1.00
1.42
0.32
0.98
1.00
1.42
1.00
1.00
3.13
1.00
0.70
1.01
1.01
1.01
g2 o} (t) 2~F2sSoor (6o) 1 f
2~
~ SF(~,s
SYSTEM PROPERTIES EXCITATION
14.8
1.00
(59) Finally, the normalized mean square response of the hydrodynamic force, based on equations (46) and (59), is
AND INPUT
For the arch dam-reservoir fluid and random ground acceleration excitation system, an important part of the system properties are given in Table 1. In the table, the height H of the arch dam is selected to be 91~44m or 300ft and 152.40m or 500ft respectively. The radius to height ratio R/H includes 3 values, 0.5 for the large curvature arch, 1.5 for the typical arch and 2.5 for the fiat arch respectively. Corresponding to these geometric forms are the natural frequencies Od of the arch dam with empty reservoir; 76.90 rad/scc for the large curvature (or stif0 arch dam with R/H = 0.5 and 17.32 rad/sec for more flexible arch dam With R/H = 2.5, both for H = 91.44m = 300ft. For high dam of H = 152.40m = 500ft, the natural frequencies COdof the arch dam with empty reservoir are 46.14 rad/see for the stiff clam of R/H = 0.5 and 10.4 rad/sec for the soft dam ofR/H = 2.5. The natural frequencies Orfor the fluid in the full reservoir are 24.72 rad/sec for the low dam of 91.44m or 300ft and 14.83 rad/sec for the high dam of 152.40m or 500ft. The next 24
14.8
= O..~a'(.of
~t = 25 rad/sec. Natural frequency of foundation
0 } (~c) =
11= 152.40m = 500ft
1.00
9
1.00
1.00
3.13
1.00
0.70
1.70
1.70
1.70
row in Table 1 shows the ratios ~,. between the reservoir fluid frequency to,. and the arch dam frequency Od. The ratios are 0.32 for stiff dam 1.00 for typical dam and 1.42 for soft dam respectively. Note that these ratios are the same for low and high dams, because both or and Od are inversely proportional to the first power of the height H. The next row after f~,. in Table 1 shows the ratio f~l between the natural frequency of the reservoir f2 r and that of the foundation o f . For the typical foundation natural frequency 19 o f = 25 rad/see the ratio f~ f is 0.98 for low dam and 0.59 for high dam. Other system properties considered in the numerical analysis in addition to those given in Table 1 are the ratio bl/H between the top width bl and the height H of the arch clam (vertical section), the ratio b2/H between the bottom width b2 and the height H, the damping ratio ~d of the dam, the Young's modulus of the concrete E, the unit weight of concrete Wdand the damping ratio of the foundation ~f. Their typical values are selected and given in the following.
bl/H = 0.042, ~t = 0.05,
wd = 150pcf,
b2/H = 0.167 E = 7.20xl0Spsf = 34.5 N/m 2 ~i = 0.1, 0.5 and 0.9
= 2,403 kg/cm The natural frequencies Od of the arch dam with empty reservoir in Table 1 are taken from Ref. 6, checked closely for the case of R/H = 1.5 with those given by Ref. 20 and with those computed by the simple method of Ref. 3. Finally, the first symmetrical mode shape of the arch dam (0, z) for R/H = 1.5 is taken from Ref. 20 as
r (o, z) = r (o) r (z) where
Probabilistic Engineering Mechanics, 1991, Vol. 6, No. 1
r
(0) = b o + blO + b 2 0 2 + b 3 0 3 b 4 0 4 + b 5 0 5
Ramtom vibration of shnple flexible arch dam reservoir systems from earth@takes: C. Y. Yang et al
~"~1 = 0.005~3
30
A
20
2O
,...,
v
to
to
0 0
5
!o
200
400
0
600
0
I1
~'~f 9 0.59
,1o
2
4
fl
~'~f = 1.46
30
20
20
v to
~"~f = 0,296
30
v
to
I0
0
I0
0
-
0
4
-' 0
0
2
4
n
Fig. 3. htput Kanai-Tajlmi spectra Sg (f~) = Sg (to)/So VS if2 = to~to r for different f~f = codto$. Legend ~j = O.1 solM curve _ _ ~i = 0.5 dashed curve . . . . . ; ~i = 0.9, dashed curve . . . . . . .
|.$
given by equations (30), (34), (44) and (58). The Kanai-Tajimi spectral density function, based on equation (58) is shown in Fig. 3 for four values of the foundation frequency ratio Qy and three values of foundation damping ratio ~f. From Fig. 3, it is obvious that for very small damping (~y = 0.1), the spectral peak is narrow and pronounced. As the .damping ratio increases, the spectral distribution becomes broad banded and consequently the foundation resonant effect decreases. Also, Fig. 3 shows that the effect of varying the foundation frequency ratio f~f is to change the location o f the spectral peak. For very small f2f (0.0059), the peak is located near f2 = 180. As g2f increases to 1.48, the peak moves to f~ = 0.7. Since the range of siginificant frequency ratio lies between fl = 0 to about f2 = 5.0, (see the bottom rows in Table 1 with further explanations in the next section), the case of f2f = 0.0059 is effectively a white noise input according to equation (58).
.R/H= 0.5
R/).I = t5 ""~.,.
e/H 5.?.s
O.S-
0
-0.$ 0.0
0.2
m
0.4
m
0.6
I
O.B
0 Fig. 2. M o d e shapes r
INTERACTION
= a o + a l ( z / H ) + a 2 ( z / H ) 2 + a3 ( z / H ) 3
The constants are b0 = 0.99693, b3 = -14.47143, a0 = - 0.00007, a3 = - 0.12814
bx = 0.15592, b4 = 68.79296, ax = 0.05946,
b2 = 9.09407 bs = --49.11577 a~ = 1.06969
For arch clams with other R / H geometry ratios, the mocle shapes are given in Ref. 21. These mode shapes are shown in Fig.2. All natural frequencies and'mode shapes have been obtained by a finite element method in Ref. 6 and 20. Input random excitation is modelled by a nonstationary random process with a unit step time modulation function and a stationary Kanai-Tajimi spectral density function as
AND RESONANCE
The interaction among the arch dam, the reservoir fluid and the foundation is best described by a set of normalized frequency variables and parameters as follows. g2 = oJ/to~, f~, = ~/r Qf = torltof, f~ a= r162 Q ~ ~= (o/oJ,, ~ f = editor where co, co, cod and (,of are the frequency of the excitation, frequency of the reservoir, frequency of the dam and frequency of the foundation respectively. The ratios g2,., ~ f represent the interaction between the reservoir and the dam, and between the reservoir and the foundation respectively. Various interaction ratios of f~, and g2y are given in Table 1. The resonant conditions are most clearly indicated by the normalized variable f2 and products fK2r anti Qf. The uncoupled resonant conditions are first pointed out as follows.
Probabilistic Engineerin9 Mechanics, 1991, Vol. 6, No. I
25
Ramtom vibration of sbnple flexible arch dam reservoir systems from earthquakes: C. Y Yang ct al
Od
0
i
= .
Od
0
!
PV~ = 0.~. fir = 0.~
= 0.32
.
1-
;
o
I
i
,
0
s
I
o
'
,
i
Od
lo
0
2
1 v
,,, I
,
5
flu
~
4
:,
,:,
S
6
7
0
'
200
t
2
:5
4
'
,
'
'
C
$"
I
P . ~ = 2.$. Or = t ~
R,/H = 2.5: O r = 1.42
C
S
.;
100-
I>-
ILA.
) 0
"~'r
a
- -
--
0
~
'
s
[
f~
Fig. 4. Complex frequency response of hydrodynamic force F ( ~ ) = g-ff (co)/Fs and displacement t" (s = co=,~'(co) v s n co/corattd f~a = (o/coafor stiff dams of f2 r = co,/co,t= 0.32 and flexible dams off2 r = 1.42. ~ d O.05 for all dams.
When g2 = 1.0, resonance o f the reservoir fluid occurs When f ~ ,. = 1.0, resonance of the dam without fluid occurs When
dation col = 25 rad/sec. For the soft dam with R/H = 2.5 and H = 91.44m or 300 ft. f2 g2 r = 1.42 fl f2
QQj, = 1.0 resonance of the foundation occurs. It is important to note that, for the typical arch dam ofR/H = 1.5 and H = 91.44m or 300ft in Table 1, f2, -, f2j, ~ 1.0, so that the three resonant peaks otZa coupled system response are all located near f~ = 1.0 in the normalized frequency abscissa. The deviations from the uncoupled resonant frequencies represent system interaction effects. For the stiff dam with R/H = 0.5, and H = 91.44m or 300ft., g2, = 0.32 so that resonance of the dam occurs at g2 f~ r = 0.32g2 = 1.0 f~
=
3.11
Consequently, for the stiff dam, two seperate resonant peaks occur, one near g2 = 1.0 for the resonance of the reservoir and the foundation, the other near Q = 3.11 for the resonance o f the dam. Note that the typical natural frequency of the foun26
=
=
1.0
0.7
The resonant peak for the dam is located near fl = 0.7. The other two resonant peaks for the reservoir and foundation respectively are again near f~ = 1.0. Again cos = 25 tad/see is used here and in Table 1. Finally, it is instructive and also useful to note that, based on equation (44)when the natural frequency of the foundation co.f approaches to infinity, then S(co) --* So, and the KanaiTajimi spectrum becomes a white noise with a constant strength of So.
NUMERICAL RESULTS Based on the study of interaction and resonance o f various arch dam-reservoir-foundation systems in the previous section, three groups of problems as given in Table 1 are selected for numerical computations. The three groups are characterized by the dam geometry parameters R/H, with 0.5, 1.5 and 2.5 respectively. Furthermore, each of the three groups is
Probabilislic Engineering Mechanics, 1991, Vol. 6, No. 1
Random vibration of shnple flexible arch dam reservoir systems from earthquakes: C. Y Yang ct al divided into two parts, one for low dam of 91.44m or 300ft for which the foundation frequency parameter ~ f = 0.98, the other for high dam of g2 s = 0.59. The damping ratio for the dam ~ = 0.05 is used for all cases. A typical resonant frequency of cot = 25 rad/sec and a small damping ratio of ~t = 0.1 are selected for the input Kanai-Tajimi spectrum simulating random earthquake ground accelerations. The choice of to t = 25 rad/sec reflects a typical case according to the survey of Ref. 19. The unusually small damping of ~ t = 0.1 corresponds to a narrow banded spectral density with a pronounced peak so that the foundation resonant effect will be brought out dearly in the response solution. Figure 4 shows the comp.].ex frequency response function for the hydrodynamic force F(_~.) and the displacement of the dam at the center top crown Y (f~) respectively by the two figures at the top row. Two peaks appear clearly in both force and displacement response. The first peak located at f2 = to/oJr = 1.0 indicates the resonant effect of the reservoir. The second peak located at about s = 2.5 indicates the resonant effect of the dam with the added mass from water. A second horizontal scale is given at the top of each figure by the variable ~ a = to/o~a. The resonant frequency of the dam without water occurs when f~a = 1.0 corresponding to ~ = 3.13 in this case where g2~= to,,ttod = 0.32 (since g2 = color = (O)/(.Od) (tod/tor) = f2,df]r = 1.0/0.32). These two figures at the top row of Fig. 4 are for stiff arch dams with the geometric parameter R.H. = 0.5 (corresponding to f~, = 0.32). The effect of increasing dam flexibility is shown by the two figures at the second row of Fig. 4 for which R/H = 2.5 (corresponding to f2 = 1.42). Again two peaks show in both the force and the displacement
complex frequency response. The location of the peak corresponding to reservoir resonance remains unchanged (f~ = 1.0) whereas that corresponding to dam resonance moves to f~ *, 0.5 (f~ ,~*, 0.7). Figure 5 shows the power spectral density for the hydrodynamic force SF(x, ~) and displacement St(x, f~) respectively by the two figures at the top row. The input random excitation is characterized by the foundation damping ratio of ~ f = 0.1 and the resonant frequency of tot = 25 rad/see. so that the foundation frequency parameter f2t = to,/tot = 0.59 for high dams of 500ft (152.4m) with tot = 14.8 rad/sec. (See Table 1). In both the top two figures, three spectral peaks appear. The first and the third peak correspond to reservoir (f~ = 1.0) and dam (f~ = 2.5) respectively as in the case of complex frequency response solution of Fig. 4. The middle peak located at about f~ = 1.7 corresponds to the input spectral peak (since to/tot = 1.0, todto/ = 0.59 then S'-2= to/tot = (to/tot)/(to,/toI) = 1.0/0.59 = 1.7). Again these top two figures correspond to stiff arch dams ~vith R/H = 0.5 (f2r = 0.32). For more flexible dams with R/H = 2.5 (Wr = 1.42), the spectral density for hydrodynamic force and displacement respectively, are shown by the two figures at the second row of Fig. 5. A comparison of the top row figures with those of the second row shows that as the dam flexibility increase from R/H = 0.5 to 2.5, the resonant frequency for the dam with added mass decreases from f~ = 2.5 to f~ = 0.5. The corresponding spectral peak (at ~ = 0.5) becomes the dominant one relative to the reservoir resonant peak at f] = 1.0 and the input spectral peak at f2 = 1.7. Also shown in Fig. 5 is the nonstationarity of the special response at x = 50 and x = 100. It is clear that
Fig. 5. Spectral density of hydrodynamic force SF (% f2) = g2SF(t, ~)/ F 2 So aml displacement Sy (% g)) = to4 Sift, to) So V S f2 = to/torATx= oh t = 50 attd 100. Damping ratio of dam ~a = 0.05, of foundation ~ t
2S
20 R/~=0.S. ~ = 0 . 3 2
R / ~ = 0..5, 9 r = 0.32. C)f = 0.$9
20,
nf=0.59
15. Vl
vi-> . 10~ tn
10T=5o
$0
i
0
~
1"=50
1.=1oo
!
O'
,
2
0
4 n
SO
30000. T=50
T=50
7-100 v
t,-
= 2.~ n r = 1.42. t.~! = 0.,~9
0
F-
i
2 N
R/H= 2...% f~r =1.42 D-l=0.fq
>,.
(/1
20-
o -~
20000-
10000-
0 _~
0
!
2
3
.q
Probabilislic Engineerin9 Mechanics, 1991, Vol. 6, No. 1
27
Random vibration of shnple flexible arch dam reservoir systen~ from earthquakes: C. Y. Yang et al
$
6-
4.
4
z 03~1
[]t.'..O !~
I,d~
b"
/I/H = 0.$. nr ,,: 0 . ~
!"4=0.~_
2 I r~'-
n,-.o_.gJ
:~
o
R/'H z 0.5, fir = 0.32 0
~o
150
150
0
T
T
I')t = 0.S9
p.L=.o . ~
. A
500-
>..
2"
PI
= 2.5. n t z t 4 2
- z~.
T
t,2 "%'"o.._.~
T
Fig.
6. Mean square o f hydrodynamic force a~- (x) = g2 o ~ o ~, (x) = co 3 o {, ( 0 / 2 ~ So vs tbne T t = wr t for xd = 0.05 aJ~x~ = 0.1
as the dam flexibility increases the transient time to reach stationarityincreases. Figure 6 shows the mean square response for the hydrodynamic force o ~" (x) and displacement o ~ (x) respectively as a function o f time x. The two figures at the top row correspond to stiff arch dams of R/H = 0.5 (f2 , = 0.32)
4000
%.,
~'~I z 0.0059
attd
displacement
whereas those two at the second row correspond to more flexible dams of R]H = 2.5 ( ~ , = 1.42). A comparison of figures between these two rows shows that as the flexibility of the dam increases, the mean square hydrodynamic force decreases somewhat. The mean square deflection, on the other hand, increases greatly. The transient time to station-
9~"~f ,~ 0.296
9 I000
AIC
(t)/2~F2s to= So
4OOO
C Vl 1000
v)
!
0
o
m
e
2
L o
., I 2
N
~ f = 1.48
~"~fz O.$g
I000 4000
|eoee
0
.J ,Ir~ .... o
l, ! 2
Io
n
Fig. Z Spectral density of displacemeut S Y (t, IV) = oJ~ Sy (t, to)/So at x = tOrt = 50for different We] = wr/w~ attdx~. All for dams with R/H = 1.5 f2t = f~t / g2a = 1.0 and ~ = 0.05. Legend E_i= 0.1; solid curve xi = 0.5 dashed curve - - - ; xi = 0.9, dashed curve . . . .
28
.
Probabilislic l?,ngineering Mechanics, 1991, Voi. 6, No. 1
Random vibration of simple flexible arch dam reservoir systems from earthquakes: C. Y. Yang et al
~'~f "" 0,0059
200
#
I-V ell
~']f = 0.296
300
,~
I00
200
b"
b
100
t
,
50
I
I
100
150
,
0
200
,
0
I
~
50
#
200 N
b
i
100
T
300
I
l
l
150
200
T
~"~f = 0.59
~"~f = 1.48 4000
t-2000
b
I00
0
!
0
50
,
.
I
100
,
I
150
i
D
200
I
0
T
50
,
I.
100
,
I
150
,
,, 200
"T
Fig. 8. Mean square displacement o ~ ('c) = co 3, o ~ ( 0 / 2 ~ So for different f2y = co#cofand ~f. All for dams with R/H = LS, f~z = cor6Oa=l.0and~a= 0.05. arity increases with increasing fiexiblity of the dam as in the case of Fig. 5 fromx = 50 tox = 100. There are clear transient peaks settling down to stationary in the case of stiff arch dams in both mean square force and displacement in the top r~w figures but not in the case of more flexible dams. Also shown in Fig. 6 are the comparison between f2f = 0.59 for a high dam and Of = 0.98 for a low dam (based on a typical foundation resonant frequency cot = 25 rad/sec 0. In three but not all of the four figures the mean squre responses for high dams (f~f = 0.59, col = 25 rad/sec., solid lines) are smaller than that corresponding to low dams (Qf = 0.98, col = 25 rad/see., dashed lines). This result cannot be explained straight forwardly because in the normalization of the mean square responses, the dam height is involved indirectly through the resonant frequency of the reservoir co, (= ~xC/2H). The normalized solution given in Fig. 6 can be converted to corresponding dimensional mean squares for a better comparison. Figure 7 shows the power spectral displacement response S~(x, f2) for various input random excitations defined by the foundation resonant frequency Qf and damping ~f; all for high arch dams of 500ft (152.4 m), medium stiffness of R/H = 1.5 (f2r = 1.0), and at time t = 50. First, the effect of
increasing damping from ~.r = 0.1, 0.5 to 0.9 is to reduce the magnitude of the spectral response and flatten the spectral peak in particular. Secondly, the damping effect increases as the foundation resonance parameter increases from f~l = 0.0059, 0.296, 0.59 to 1.48. Thirdly, as f~f increases, the spectral displacement increases for a given damping ~f. The location of the spectral peak remains at about f~ = 0.7, corresponding to the resonant frequency of the dam with added mass. The case for f~f = 0.0059 approaches thatof a white noise input as a limit (see equation (58)). Figure 8 shows the mean square displacement response o ~('0 for various input random excitations defined by the resonant frequency of f~f and damping ~l; again all for high dams of 500ft (152.4m) and medium stiffness ofR/H = 1.5 (Qr = 1.0). Similarly to the previous Fig. 7, the damping effect is to reduce the mean square displacement and is more pronounced as the frequency ~ f increases from the limiting white noise case of 0.0059 to 1.48. Again, as f~l increases, the mean square displacement increases, because the peak of the input excitation moves closer to the system resonant frequencies.
Probabilistie Engineerin9 Mechanics, 1991, Vol. 6, No. 1
29
Random vibration of shnple flexible arch dam reservoir systems from earthquakes: C. Y Yang et al
60-
5O
9
IVHs " t.O. I ~
,~
fir
z tO, | =
9 I.S
O.OS
l~
..'p 5O.
'J~"
20.
.J~,J
i,.
Legend
Ill
9
co~,o,_._~.
I0-
j]
0 0
. ~-~-'-_.._~ I
2
9
IL 3
..-. 4
recognized as an important design measure, but the random vibration and random h y d r o d y n a m i c force solutions presented here for the flexible arch dam problem are new. These random solutions are needed for a probability based seismic reliability analysis and design of the arch darn-reservoir systems. The relatively simple solutions presented here, of course, are subjected to several important assumptions and restrictions including linear elastic clam structure, rigid foundation and side walls and the first symmetric mode of dam vibration. Most of these assumptions and restrictions can be partially or totally eliminated with added complications in analysis and computations for future work. Here in concluding this work, the assumption of the first symmetric mode shape is pointed out again as a precaution for practical engineering application. The present solutions have high accuracy only when the earthquake excitation spectrum lies primarily below and near the first natural frequency or above four times the first natural frequency of the dam.
ACKNOWLEDGEMENT Fig. 9. Comparison of complex frequency acceleration Y (f2) = ~" (to) between our calculated shzgle mode solution with finite element solution of Porter arm Chopra (page 83 R e f 6)
This research work is supported in part by the National Science Foundation Grant No. ECE 850-1580 with the University of Delaware. The authors are grateful for the support.
Figure 9 shows a comparison on the complex frequency acceleration response between the first symmetrical mode approximation and the multiple mode solution of Ref. 6. The comparison is generally satisfactory for frequency ratio f~d = to/rod < 1.0 and > 4.0. The largest discrepancy lies in the neighbourhood of 1.5 and 3.0. This observation suggests that the approximate solutions should give satisfactory answers qualitatively. However, in case the input excitation has dominant peaks in the neighborhood of one and half times or three times the first symmetrical frequency of the dam without water (to = 1.5 rod or 3.0 rod), then the accuracy of the approximate solutions deteriorates and a multiple mode solution should be used.
REFERENCES
SUMMARY AND CONCLUSIONS A simple arch dam-reservoir system is selected to study the stochastic vibration of the dam and the hydrodynamic force response on the dam to a nonstationary random excitation process, simulating earthquake ground acceleration. The dam is assumed to be elastically deformable but is restricted to the first symmetric mode shape for simplicity. The random earthquake acceleration is assummed to be in the horizontal upstream-downstream direction and is modelled by the simplest nofi-stationary random proces"s which is the zero-start stationary process with a Kanai-Tajimi spectral density function. The stochastic vibration solutions and the hydrodynamic force response solutions are presented by the nonstationary power spectral density functions and the mean square functions all in general nondimensional forms, in addition to the non-random solutions of complex frequency response functions. The range of problems studiedinclude high and low dams an/l geometrically stiff (with large curvature) and flexible (with small curvature) dams. For all these problems, the locations of important resonant peaks are identified in both the non-random complex frequency response functions and the random spectral density functions. The vibration of the clam and the hydrodynamic force response on dams during earthquake motions have long been 30
1 Kotsubo, A. External forces on arch dams during earthquakes, Faculty of EngineeringMemoirs, 1961, 20 (4), Kyushu University, Fukuoka, Japan 2 Clough, R. W., Raphael, J. M., and Mojtahedi, S. ADAP-A computer program for static and dynamic analysis of arch dams, Rept. No. EERC
73-14, Collegeof Engineering,Universityof California,Berkeley,California, June 1973 3 Crawford,C. C. Earthquakedesignloadingsfor thin archdam~,Proceedings, 3rd WorldConference on EarthquakeEngineering, 1965, III, IV172-185, New Zealand 4 Perumalswami,P. R. and Kar,L Earthquakebehaviourof archdam-reservoir systems,Proceedings5th WorldCo~[erenceon EarthquakeEngineering, June, 1973 5 Porter,C. S. and Chopra,A. K. Ilydrodynamieeffectsin dynamicresponse of simple arch dams, Earthg. Engr. and Structural Dynamics, 1982, 10 0), May-June,471-432 6 Porter, C. S. and Chopra, A. K. Dynamicresponseof simple arch dams including hydrodynamieinteraction, EERC Report No. UCBIEERC80117, Univ. of Californiaat Berkeley,July, 1980 7 Fok, K. L and Chopra,A. K. Earthquakeanalysisof arch damsincluding dam-water interaction, reservoirboundary absorption and foundation flexibility,EarthquakeEngineering and StructuralDynamlcs, 1986, 14, 155-184, 8 Seanlan,R. II. On earthquakeloadings for structural design,Earthquake Engineering and StructuralDesign, 1977, 5, 203-205 9 Yang, C. Y. and Charito, V. Randomhydrodynamicforce on dams from earthquakes, Journal of the Engineering Mechanics Division, ASCE, February 1981, 107 (EMI) 10 Mills-Bria and Yang, C. Y. Nonstaioaaryresponseof concrete gravity dam-reservoirto randomvertical earthquakeexcitation,Proceedingsof the 8th WorldConferenceon EarthquakeEngineering, July 1984, VIII 11 yang, C. "I(.and Kuserk,R. Transientrandominteractionof dam reservoir systems,ProbabillstleEngineeringMechanics, Mar. 1987, 2 (1), 1625 12Debussey,M. and Yang,C. Y.Nonstationaryhydrodynamicforcesoa arch dams from randomearthquakes,Procee&'ngsof the 3rdNational Earthquake Engineering Conference, Charleston,SC, August 1986 13 Priestley, M. B. EvolutionaJyspectra and nonstationaryprocess,Joun RoyaIStatistlcalSoc~, 1965, Series B, 27 (2), 204-237 14 Kanal, Semi-empiricalformula for the seismic characteristics of the ground, Univ. TokyoBull. EarthquakeRe~ lnsL 1957,35, 309-325 15 Tajimi,II. A statistical methodof determiningthe maximumresponseof a building structureduringan earthquake,Proc.2nd WorldConf.Earthq. Eng., Tokyoand Kyoto,July 1960 II, 781-798 16 Shinozuka, M. Random processes with evolutionary power, Your, Engr. Mech. Div., ASCE, August 1970, 96, (EM4), Proe. Paper 7444, 543-545 17 Yang, 12. Y. Random Vibratlon of Structures, John Wiley International
Probabilislic Engineerin9 Mechanics, 1991, Vol. 6, No. 1
Science Publishing Co., 1986
Ramtom vibration of sbnple flexible arch dam reservoir systems fivm earthquakes: C. Y Yang et al 18 Clough, R. W. and Penzien, J. Dynamics of Structure, McGraw Hill Pub. Co., 1975 19 Lai, P. Statistical charaaerization of stone ground motions using power spectral density function, Bulletin of the selsmologicalSoclety ofAmerlca, Feb., 1982, 72 (1), 259-274 20 Kar, L EarthquakeBehavlor of Arch Dam ReservoirSystems, Doctor's Thesis, South Dakota School of Mines and Technology, 1972 21 Wells, J., Bhobe, R. and Yang, C. Y. Transient response of dam reservoir systems to random earthquake excitation, Probabilisdc Engineerlng Mechanics, September 1989, 4, (3) 142-149
APPENDIX: HYDRODYNAMIC SOLUTIONS ~(O,z, to) = ~d(O,z,r
PRESSURE
+ ~ ( O , z , to)
+ ~(60)pr (0, Z~03)
(A.1)
where
~xR q l (z,n- 1)2 -- ((o/to,)z I x=R =~--ff
(A.8)
= dimensionless parameter of the frequency ratio to/to, and dam geometry R/H respectively.
f ~./4f Iron = H1 ,do d o H d~(0, z) cos (4n 0) cos (ct m z) d 0 dz (A.9) dp(0, z) = first symmetrical mode shape of arch dam. The remaining symbols, are given in terms of Bessel functions of the first kind of order n, J,, ('Z.mR),Bessel runelions of the second kind of order n, Yn(~,~R) and modified Bessel functions of the second kind of order n, K,,(s
c~ (XmR) =
~d (o, z, co) -
16 vr2 to R ~
"[ Cn()vaR )
g~2 +
~
[A.(K,,,R)J4.(XmR)+Bn(~.mR)Y4n(KmR) ]
(-1) men (-1).
(A.10)
R [A 2 (XmR) + Bn2 (~-mRI
m-12 ?o- (2n-~-l) 5-1-6n2)
D,, (Lm R) =
iD. (XmR)] c o s 4 n 0 cosctmz (A.2)
[B,,(k,,,R)J4,,(~mR)-An(~.mR)Y,t,,(LmR)]
(A.11)
~.,~R [A ~ (~.,~R) + B n2 (XmR)] where
An(~'mR) = J4n-l (~'mR) - J 4 n + l (~m R) (A.12)
Bn (~.~ R) = Y4n - 1 (~m R) - Y4. +1 (~'mR) (A.3)
+ iV,,, (XmR)] cos 4n 0 cos Ctmz}
oo
r (0, z, to) =
(A.13) The above equations for (7, (X,,,R) and D,, (X,,,R) apply for m < mr, where mr is the largest integer "m" satisfying the inequality (w/c) < a,,,. For m > rex,
oo
166o R
g4. (XmR) m-ln-O
cn ( ~ R) = Xm R [K4n_1 (Lm R) + K4n + 1 (~'mR) ]
9 [Cn(Z.mR) + iDn (kmR)] cos4n0 cOSamZ
(A.14)
(A.4) Various symbols in equations (A.2), (A.3) and (A.4) are defined as follows. t"
e n = ] 1, n = 0 t 2, n,, 0
tar -
(A.15)
For m < mk, (- 1)" E,, (X,,R) - (2mI) ~-mR { sin [X~R sin ( n / 4 - 0)]
+ sin [ k~ R sin (a/4 + 0) ] }
~c
2H
(A.16)
_
fundamental frequency of the water in the reservoir (2m -
am-
(A.5)
D. (~mR) = 0
1) ~r
(A.6) (A.7)
(- I)m { F m (k,,, R) = (Z,n -1) ~.mR
cos [
sin
O) ]
2//
Probabilislic Engineering Mechanics, 1991, Vol. 6, No. 1
31
Ranclon, vibration of sbnple flexible arch dam reservoir systems from earthquakes: C. Y. Yang ct al
+ cos
[ kmR sin (u/4
(A.17)
+0)]}
Umn(kmR) = (-1)"(-1)"{ (~, a) 7". (~m) c. (xm R)
(16n2+ 4 k 2 - 1) (16n2-4k 2_ 4k - 1) (16n2 _ 4k 2 + 4k- 1)
E~ (~.,R) =
+ ~-A, (XmR) Dn (KinR)
(A.lS) Vm, (XmR) = (-1)"(-1)~ ( (2m- 1) 7". (~,. R)
(A.20)
for m > mt
-
- (- 1)m [ e -~Rs~n(~4-~ (2.7: i72-~ R
+ e- ~ R sin (g./4 + 0)]
(A.21)
Dn (X,. R) F m (~.rnR) = 0
(A.22)
- ~ A. (XmR) G (XmR)
(A.19)
where
-(-1)~'
u,,,. G,, R) - ~ T - T) C~(~ R) G (~ R) (A.23)
k-0
32 Probabilislic Engineering Mechanics, 1991, Vol. 6, No. 1
v,,, (MR) = o
(A.24)
Nonstationary random vibration and hydrodynamic force solutions on arch dams from reservoir fluid during earthquakes are obtained and presented in normalized form for a group of general arch dam-reservoir-foundation systems. The solutions include nonstationary power spectral density and nonstationary mean square functions of the random earthquake generated vibration and hydrodynamic force on dams. The random earthquake ground acceleration is modelled by a nonstationary (or evolutionary) process due to PriestleyKanai-Tajimi and the random solution is based on the method by Shinozuka. The arch dam-reservoir geometry is restricted to a circular section of 90* central angle with two banks extending to infinity. The non-random but fundamental hydrodynamic solution for a rigid dam by Kotsubo is used as the basis of the random solution for the flexible arch dam. A key assumption of first symmetric mode shape leads to considerable simplicity in the formulation and the computational efforts. This simplified solution is a good approximation when certain limitations on the input random excitation is carefully observed.
INTRODUCTION 9One important problem in the seismic safety analysis of dams is the evaluation of the hydrodynamic force on dams from water in the reservoir during strong earthquakes. In the case of concrete arch dams, solutions of the problem have been presented by several well known investigators including Kotsubo t, Clough, Raphael and Mojtahedi 2, Craw ford3, Perumalswami and KaP, Porter and Chopra 5.6and Fok and Chopra 7. All these noted research work treat the earthquake input excitation as a definite or the so-called deterministic function of time. However, in a rational seismic safety analysis of any major structure, the uncertainty of future earthquakes must be taken into account and therefore dynamic structural response solutions to a definite earthquake ground motion excitation are not sufficient. To extend the deterministic dynamic structural analysis to the non-deterministic case, one can model the earthquake ground motion as a stochastic or random process, apply random vibration theory, find the random structural responses and finally estimate the probability of various modes of failure, under the condition that a given random earthquake motion has occurred. This classical random vibration approach of structural safety analysis is an improvement of the non-random approach because the uncertainties in amplitudes, duration and frequency contents of earthquake ground motion are taken into account in a rational way. This basic concept was explained clearly in a short technical note by Scanlan more than ten years ago 8. The analysis of random hydrodynamic force on concrete gravity dams subjected to nonstationary random earthquake excitation was first presented by Yang and Charito9 and and
Paper accepted October 1990. Discussion closes August 1991. 18
followed by Mills/Bria and Yang 1~ and Yang and Kuserk 11. For arch dam-reservoir systems, the only published work known to the authors are the random hydrodynamic force solution on rigid arch dams by Debessay and Yang 12 and the random displacement solution by Hollinger. In this paper, the author extend their previous work 12 of random hydrodynamic force solutLon on a rigid arch dam to the case of a flexible arch dam. In addition, the simplified nonstationary random process model of Priestley 13used in the previous work is improved by adding the Kanai 14 - Tajimi 15 filtered spectral density function for the random earthquake acceleration. New solutions of nonstationary power spectral density and mean square functions are obtained by a solution method of Shinozuka 16 as presented in detail in the recent book by Yang 17. The arch dam-reservoir geometry is shown in Fig.1. The horizontal section of the arch dam consists of concentric circular segments bounded by two banks extended to infinity with a central angle of 90*. The reservoir is assumed to have a full depth of compressible but nonviscous water. Under the horizontal ground acceleration in the downstream-upstream direction, the flexible elastic arch dam is assumed to deform in its first symmetric normal vibration mode. Vertical sections of the arch dam are assumed to be trapezoidal.
HYDRODYNAMIC FORMULATION The simplified problem of arch dam-reservoir-earthquake is shown in Fig. 1. The dam is assumed to be elastic with a 90* circular are between two rigidbanks. These banks and the full reservoir behind the dam-bank system are assumed to be infinite in extent in the positive x-direction. Water in the semi-infinite resrvoir of the depth H is assumed to be non-viscous but compressible. The foundation of the dam-reservoir system is assumed to be rigid. The earthquake ground acceleration fig (t) is assumed to be in the downstream-upstream orx
Probabilistic Engineering Mechanics, 1991, Vol. 6, No. 1
Random vibration of sbnple flexible arch dam reservoir systems from earthquakes: C. Y Yang et al Op (R, O,z) Or
--
-,,
o.
0 p (r,r00~x/4,z) = p ~ - ' 2
/ /////I// $r162
.%
(4)
(5)
t- 1'
0 p (r, O, z) = 0
a.a
** . . . . . . . . . . .
(6)
rO0 The system is assumed undisturbed until the earthquake ground acceleration as(t) occurs. The ground acceleration is modelled by the simple, but fundamental, nonstationary process, which is the zero start Gaussian stationary random process with a Kanai-Tajimi spectral density Ss(to). This completes the hydrodynamic formulation of the problem.
. - 4 ~ It
%%.
9D A M F L U I D
INTERACTION
In Fig. 1 let the relative radial displacement ap (0, z, t) of the elastic arch dam be approximated by its first symmetrical mode shape ~ (0, z) under the empty reservoir so that
Piaa
Fig. 1. Arch dam -reservior system ap (0, z, t) = Y (t) ~b(0, z) direction only and the dynamic system response is small such that water surface wave effect is negligible and the entire system response is in the linear range. Under these assumptions, the governing equation of fluid motion is the well known .wave equation in terms of either the velocity potential which is the standard choice in hydrodynamics or the hydrodynamic pressure p. The pressure p is used here for convenience together with the cylindrical coordinates r, 0, and z; the time t and the sound speed e as follows.
02p • 02p 0 2p 10Zp o,2 + 7-if-;+ ~ + o---Z = ~ o---7
(1)
Boundary conditions are: (2) the hydrodynamic pressure p is zero at the free surface, (3) the vertical acceleration which relates to vertical gradient of the pressure p is zero at the reservoir bottom, (4) the radial acceleration at the dam face which relates to the radial gradient of the pressurep is equal to the sum of the radial component of the upstream-downstream earthquake ground acceraltion/is (t) and the acceleration of the dam relative to its base ~, (5) the acceleration normal to the bank at the bank, which relates to the tangential (0-direction) gradient of the pressure p, is equal to the compenent of the earthquake ground acixleration normal to the bank at the bank, (6) by symmetry, the tangential acceleration which relates the tangential (0-direction) gradient of the pressure p along the centre line of the arch dam, along the x-axis with 0 = 0, is zero and finally, only outgoing pressure waves which propagate in the +x direction are considered on the basis of Sommerfcld's radiation condition, i.e., no in-coming or reflected waves. These bofindary conditions are given below. p (r, O, H) =" 0
(2)
Op(r, 0, 0) = 0
(3)
0z
(7)
where Y(t) is the generalized coordinate. Then the total radial acceleration of the dam/i (0, z, t,) can be expressed as the superposition of the radial component of horizontal ground acceleration as(t) cos0 and the relative radial acceleration ~, (0, z, t) by
il (o, z, t) = ~ ( 0 cos o + ff (o, z, t)
(s)
To formulate the equation of motion for the vibrating arch dam, the principle of virtual displacement, is applied as follows Is. Consider a small element of the dam specified by Rd0dz with mass per unit area m(0, z). The element is subjected to an inertia force of m ( O, z) Rd 0 dz [0 s (t) cos 0 + (0, z, t)] and the hydrodynamic forcep (0, z, t)Rd 0 dz, in addition to the hydrostatic and gravity force that are excluded from the present study. For this dynamic equilibrium element of the dam, a virtual displacement 6q,, which corresponds to a virtual generalized coordinate 6Y (t) by equation (7)is specified. Then the application of the principle, that the total virtual work done by all the forces in the arch clam during the specified virtual displacement is zero, gives the following equation
- f f p ( O , z , t ) R d 0 dz6 Y(p (0, z)
- f fro (0, z) n
d 0 dz [ lie (t) cos 0
+ i;'(t) ,~ (0,01 ~ r e (O,z) - KY (t) b Y (t) - C]; (t) 5 Y (t) = 0
(9)
The first term in equation (9) represents the virtual work done by all hydrodynamic forces; the second by all inertia forces; the third by internal stresses and the last by internal damping forces. The symbols K and C represent generalized stiffness and damping coefficient of the arch dam respectively. Can-
Probabilistic Engineerin 9 Mechanics, 1991, Vol. 6, No. 1
19
Random vibration of shnple flexible arch dam reservoir systems from earthquakes: C. Y Yang et al ceiling the common term 6Y (t) and using M for the generalized mass and P(t) for the generalized force in equation (9) defined by
M = f f m (0, z) dp2(0, z) R d 0 dz
The steady-state pressure response, equation (16) consequently can be obtained as the superpostion of responses to the respective excitations, the ground acceleration equation (15) and the motion of the dam relative to the ground of equation (17). Hence from equation (16),
(I0)
fi (0, z, to) exp (i to t) =Pa (0, z, to) exp (i to t)
and P (t) = - iig ( o f f
+Pb (0, Z, W) exp (i to t)
m (0, z) cos 0 r (0, z) n d 0 dz
+ fir (0, z, tO) ~ (to) exp (i to t) -ff
(11)
p (0, z, t) d~(0, z) R d 0 dz
or
f (0, z, to) =~d (0, z, to) +~b (0, z, to)
simplifies the equation to
+ Pr (0, z, tO) ~ (to) M}'(t) + CY(t) + KY(t) = P(t)
which is the standard equation for a simple oscillator. In terms of the natural circular frequency toa and damping ratio ~e, equation (12) becomes
}'(l) + 2~dtodY(l ) + to~Y(t).- P(t) ,
(13)
M
where (.Dd =
K~/-~-'M,
~ l -~ C/Ccr,
Ccr
= 2V/-~
(14)
Equation (13) is the coupled equation of motion for the dynamic interaction of the arch dam-reservior fluid system. The term interaction refers to the fact that the generalized force P (t), as given by equation (11) depends on the yet unknown hydrodynamic pressure p (0, z, t), which in turn depends on the unknown generalized coordinate Y (t).
in whichffa(0, z, to) a n d ~ (0, z, to) are the complex frequency response of hydrodynamic pressures acting on the rigid dam due to the rigid body excitation exp(i tot) of the dam and banks respectively. The last complex frequency response function in equation (18), if,(0, z, to), is due to the specified dam acceleration ~(0, z) exp (i to t) relative to the dam base. As response solutions to the same harmonic function of time exp (i to t), they are known in Refs 4 and 5 and are given as equations (A.2), (A.3) and (A.4) in the Appendix. Equation (18) is the key to the solution of the dynamic interaction problem, because with it, the additional un__knownfi (0, z, to) is expressed in terms of the only unknown Y (to), the complex frequency response of the generalized acceleration 17 (t) in equation (13). Note that Y (t) -- Y (to) exp (i to t)
I';(t) = i to Y(to) exp (i to t) = Y(to) exp (i to t) FREQUENCY SYSTEM
RESPONSE
OF DAM-FLUID
(21) SO
-
Y (to) = i OY(to) (to) = _ to2 ~ (to)
(16)
20
(22)
(23)
To carry out more details, substitutingp (0, z, to, t) of equation (16) together with equation (18) into P (0 of equation (11) and then in equation (13) gives
w h e r e f (0, z, to) is the complex frequency response function. From equations (7) and (8), it is clear that the total radial motion of the fluid is generated by the superpostion of two parts; the ground motion//s(t) cos 0 and the dam deformation ~(0, z, t) or l?(t) ~ (0, z). Under the steady-state condition iis(t) is given by equation (15) and }'(t)d~(O,z) = ~(to)exp(itot)d~(O,z)
(20)
l?(t) = (ico)2y(to) exp (i to t) = ~(to) exp (i to t)
(15)
and the steady-state pressure solution be p (0, z, to, t) = ~ (0, z, to) exp (i to t)
(19)
~
In the simple formulation of equation (13) together with the generalized mass M of equation (10) and generalized force P (t) of equation (11), it is assumed that the first symmetrical mode shape ~b(0, z), the natural frequency tod and damping ratio ~a are all available 3.5.6.20.To solve for the dam-fluid interaction response to a random ground acceleration input, the complex frequency response method is used. Let the steady-state ground acceleration ~Jg(t) = e~p (i to t)
(18)
(12)
(17)
Probabilislic Engineering Mechanics, 1991, 1Iol. 6, No. 1
MY(t) + 2~::,atoaMl~'(t)+ toa2MY (t) -- ~g (t) L - BI (co) exp (i to t)
+ B2 (co) Y (co) 0 2 exp (i co t)
(24)
RatMom vibration of slmple flexible arch dam reservoir systenm from earthquakes: C. Y. Yang et al where
L = f fro (0, z) d~(0, z) cos 0 R d 0 dz
(03) = f f
(25)
A (t, 03)*A (t, to') E [dOg (,co)*dog (to') ] = 0 for co * 03'
(0, 03)
(o, w)] r (O,z)R dO
whereA (t, 03) is a non-random modulation function of time t and frequency to and d0s(to ) is defined as a zero mean orthogonal random process such that the mathematical expectation
(26)
01)
when 03 = co', then
IA Ct, 03) 12E I B2 (03) = f f Pr (0, z, 03) d~(0, z) R d 0 0z
Using equations (19) through (23) in equation (24) gives the standard formulation of a simple oscillator
( M + Re [ B2 (03) ] ) ~"(t) +
(2~:103dm- 03Im [W2 (03)])l;(t)
+ to2aMY (t) -- - [L + B1 (03) ] exp (i 03 t)
(28)
where Re and I,,, represent the real and imaginar3/part respectively of the complex function B2(to). In the standard form of equation (28), the dynamic interaction between the arch dam and the fluid in the reservoir is explicitly revealed. The effects of the fluid in the reservoir on the deflection of the dam consists of three parts. The first is the added mass for excitation given by the term Bi (co) defined by equation (26). The second is the added mass for inertia given by Re[B2 (03)] defined by equation (27), and the third is the added damping given by - 03lm[B2 (co)], defined also by equation (27). All three parts are frequency dependent. The final solution for the complex_frequency response function for the generalized acceleration f'~ (03) is easily obtained by substituting Y (t), I; (t) and # of equation (19), (20) and (21) respectively into equation (28).
(03) =
C03)12 = Sg Ct,03)d03
(32)
(27)
L + B1(03) M [ (0)/03) 2 + 2i~d(03d/to ) _ 1 ] - B 2 (03)
(fi9)
In equations (31) and (32), the symbol "represents complex conjugate of the function and Ss(t, 03) is defined as the nonstationary power spectral density function representing the mean square distribution of iig(t) in frequency. When the modulation function A (t, co) in equation (30) takes on the special form of a constant with unit magnitude, then the integral representation, equation (30), degenerates into a stationary random process. From equation (32), whenA (t, 03) = 1, the nonstationary power spectral density function also degenerates into a stationary density function, Ss(03), independent of time t. Consequently, from equation (32),
03) Finally, by comparing equations (32) and (33), the following relation between the nonstationary spectral density function Ss(t,03) and the associated stationary spectral density function Sg(03) is obtained
Sg (t, 03) = [A (t, co) 12 Sg (03)
(34)
equations (30) and (34) constitute the key,ingredients of the nonstationary random process model for the ground acceleration ii,(t). When such a random process excitation is applied to the arch dam- reservoir system, the response hydrodymanic force f(t) is also a nonstationary random process. Thus, in a similar way, the response force f(t) has an integral representation, a modulation function B(t, o3) and a nonstationary power spectral density function St--(t, 03) as follows.
f (t)= f_~ ~oB (t,to) e i~t dF (60)
(35)
SF (t,
(36)
The other two relevant complex frequency response function for the generalized displacement Y (t) and velocity l';(t) can be obtained from equations (23) and (22) respectively. Finally, the complex frequency response of the hydrodynamic pressure can be obtained by equation (18) and (29).
RANDOM EXCITATION AND RESPONSE A nonstationary random process model based on Priestley's work ~3 as explained in Yang's book ~7, p.241 to 245, is used here for the ground acceleration /is(t). A brief outline of Priestley's model is given in the following. Let the ground acceleration tTs(t),-assumed to be a nonstationary random process with zero mean, be defined by the integral representation.
ag (0 = f = A (t, 03) e lot dog (03)
(30)
03) = I B (t, co) 12 SF (to)
The random response problem is now focused on the solution of the nonstationary power spectral density function of the hydrodynamic force, S~(t, co), in terms of the excitation spectral density function of the ground acceleration Ss (t, 03) and the system properties. The nonstationary response problem was solved by Shinozuka 16 and is briefly outlined in the following. It is well known that the excitation-response solution for a linear system follows the Duhamel's superposition integral in the time domain Is. For the dam-reservoir-earthquake problem here, the response hydrodynamic force f(t) can be obtained by summing up the effects from small increments of excitations hF (t- x )iig (x)d~ as follows.
Probabilislic Engineerin 9 Mechanics, 1991, Vol. 6, No. 1
21
Random vibration of shnple flexible arch dam reservoir systems from earthquakes: C. Y Yang et al
f (0 =f'ohF (t - x) ~s(~) dr,
(37a)
where h~(t) is the impulse response function for the hydrodynamic force on the arch dam, f (t). When the earthquake acceleration excitation iis(t) is a nonstationary random process defined by equations (30) and (34), the hydrodynamic force f(t) is also a nonstationary process defined by equations (35) and (36). Substituting iis(t) of equation (30) into equation (37a) and interchanging the order of integration gives f (t)-f- ~
( t - x , to)e-i~'hF(nC)d-c]d(lg(to)
where the impulse response function h~t) can be obtained by the inver_se Fourier transform of the complex frequency response F(to). (43)
B
F (03) exp ( i to t) dto --00
Finally, the stationary spectral density function Ss(to) of the ground acceleration in equation (42) is assumed to be the well known Kanai-Tajimi spectrum Is given by 1 + 4 ~ } (to/oJl) 2
(37b) Recall that in the case of stationary excitation and response, the Fourier transforms dF(to) and dog (to), and the spectral densities S~to) and Sx(to) are linked by the complex frequency response function F(to) as follows.
(to) = 7 (to) dog (to)
(38)
s, (to) = IF(to) ! 2sg (co)
(39)
,?
Substituting dOs(to) of equation (38) into equation (37b) and comparing the resulting equation (37b) with equation (35) gives
Sg (to) = [1 - (to/toj-) 2 ] 2 + 4 ~ i (to/toy) 2 So (44) in which ~j, tof and So arc the damping ratio, the central frequency and the spectral strength of the random ground acceleration respectively. The random hydrodynamic force response solution is now complete with equations (18) and (29), after integration over l__hereservoir depth, for the force complex frequency response F (to) in equation (45); equation (43) for the impulse response hr(t); equation (41) and (42)for the nonstationary power spectral density S~t, 03); and the following equation (46) for the mean square response.
(to) =
B (t, 03) = F (to)=1for4 (t - -q to) e -io~ hF ('c) d~
fo
H
C0, z, to) az
(45)
(40)
SubstitulingB(t,w) of equalion (40) and S~to) of equation
(41)
SF (/, 03) = [ f o t A (l - % to) e -i cox h v (T) ~ [ 2 Sg (to) Note that when A (t, co) = 1 and t ---, oo the intergral in equation..(41) reduces to the complex frequency response function F(to) and equation (41) reduces to the well known transfer relation in the statiq_nary case. Note also"that the complex frequency response F(to) and the impulse response h~(t) form a Fourier transform pair. For the present study, a simplification is made on the modulation functionA (t, to) in-equation (41) by assuming it to be a unit step function in time t and independent of frequency co. Such a simplified modulation function leads to the so-called zero-start stationary random process or pseudononstationary process, which is the simplest case in all the time variations. It also leads to conservative response solutions. With this simplification, the modulation functionA (tx, to) in equation (41) can be removed since in the range of integratiori for "I, (t- 1:) is always positive and consequently the function is always equal to unity. Thus equation (41) is reduced to (42) SF (t, to) [ f t h F (1;) exp ( - i to "~) dr [ 2 Sg (03)
"o
(46)
/i, o o
(39) into equation (36) gives the following input-output spectral realtion due to Shinozuka 16.
SF(trto ) dto
O~ ( t ) = J -
-
O0
The random hydrodynamic force analysis method outlined above applies also to random displacement of the dam, simply by replacing the force terms with symbols f and F by the displacement terms y and Yin equation (35) through(43 ) and equation (46).
NORMALIZED STOCHASTIC RESPONSES To better understanding the dynamic behaviour of the damfluid system and to facilitate numercial computations, all functions and variables are normalized and listed with definitions as follows. A. System Properties: tot = nc/2H = fundamental frequency of the reservoir fluid c = sound speed in fluid R = radius of the circular horizontal section of the arch clam H = height of dam and depth of fluid in the full reservoir COd= first symmetrical frequency of the arch dam with empty reservoir ~t = damping ratio of the arch dam with empty reservoir
22 Probabilislic Engineering Mechanics, 1991, Vol. 6, No. 1
Random vibration of sbnple flexible arch dam reservoirsystemsfrom earthquakes: C. Z Yang et al (~ = central frequency of the foundation in the Kanai-Tajimi spectrum for the random earthquake excitation ~j, = damping ratio of the foundation in the KanaiTajimi spectrum for the random earthquake excitation
(47) B. Normalized Variables: x = tot = normalized time [2 = to/to,. = normalized frequency with reservoir Qd = to/tot = normalized frequency with dam (48) C. Interaction Parameters: [2,- = to,-/tod= fluid-dam interaction parameter [2f = to/to = fiuid-foundation excitation parameter
mml
( - 1) '= +" ~n (2m - 1) 2 (1 - 16n 2)
~
+ iD,, (~.mR)]cos4nO sinctmU (54) oo
~,.([2)_ g.~,.(to)
B1 (f2) -
B1 (co) M
32~/'2wR2" ~ -
~
m=l n=O
Mg~2
( - 1 ) m I~n(-1) n
(2~--1") (1- 16n 2)
oo
-3___22R
(49) The normalized function of the dynamic system are obtained by using the above listed variables and parameters in equations (26) and (27), after using the pressure solution of equation (A.2), (A.3) and (A.4) in the Appendix. These are the added mass terms B1 (to) and R, [B2 (to)] and the added damping coefficient -toI,,[B2(to)] given by
n-0
E. x,.,, m-1 n*O
9 [ Cn (L~R) + iDn (X~R)] cos 4n 0 sin am H
where F, is the hydrostatic force wtF/2. Note that X,,,R is defined by Equation (A.8) as a function of the normalized frequency variable f~, the geometry ratio R/H and the dummy index m. Equations (54) and (55) give th._econtribution to the complex frequency response function F(f~) from the rigidbody motion of the dam only, F..,t([2); and from the motion of the dam relative to the ground, F,(f2), respectively. Again the effect of the bank motion is left out for simplicity. Hence, from equations (45) and (18), (Q) = Fd (n) + ~ ([2) Fr (n)
I,,,,,-[C,,(k,,,R)
B2 ([2) =
+
iD,,CkmR)]
3 2 w R 2 H oo t?l g 7~
[C,,(~..,R)
Note that the last term in equation (56) includes the complex frequency response function of the generalized acceleration Y (f~) for equation (29), representing the effect of structuralfluid interaction as explained previously in the section under title of dam-fluid interaction. The normalized impulse response function of the hydrodynamic force, based on equations (43) and (53) through (56), is
(I,,.)
rn - I n - O
iD.(XmR)]
+
(56)
(50)
B2 (~) M
-
(55)
(5!)
Note that from equation (A.8) in the Appendix, XmR is a function of the normalized frequency ~ and the contribution from the motion of the banks is left out in equation (50) for simplicity. The normalized, radial, generalized acceleration from equation (29) and displacement from equation (23) become
([2) = ~(co)
hr (1;) = ~
hF (t)
(57) 1 ~ ~ ([2) e i n ~ d [2 2~ . 1 _ oo The normalized Kanai-Tajimi power spectral denisty for the ground acceleration excitation, based on equation (44), is
ss ([2) = s~ (to)/so (L/M)
+ B1 ( ~ )
(52)
([2 [) r)-2 + 2ira ([2 [2r) -1 -- 1 -B2 (s
[1-
1 + (2~s n h i ) ~ (~/.)212 + (2~.f~2~f)2
(58) ~(n)
=
%2 ~ (~)
=
_ [2-z~ (n)
03)
Based on equations (45), (A.1), (A.2), (A.4) and (29), the normalized complex frequency response functions for the hydrodynamic forces on the arch dam are obtained and given as,
The normalized nonstationary spectral density function for the hydrodynamic force, based on equations (42), (57) and (58), is
s v Cx, Q) =
~ sv (t, to) F~So
Probabilislic Engineerin 9 Mechanics, 1991, Vol. 6, No. 1
23
Random vibration of sbltple flexible arch dam reservoir systems from earthquakes: C. Y. Yang et al Table 1. System properties tI = 91.44m = 300ft
Group R/H t.o6 Natural frequency of dam (radlsec)
2
1
3
2
1
3
0.5
1.5
2.5
0.5
1.5
2.5
76.9
24.7
17.3
46.2
14.8
10.4
wp Natural frequency of reservoir (rad,'sec)
~p - o~plo~5 o.)t
24.7
0.32
Resonant Freq. W: Fluid Dam Foundation
1.00
1.42
0.32
0.98
1.00
1.42
1.00
1.00
3.13
1.00
0.70
1.01
1.01
1.01
g2 o} (t) 2~F2sSoor (6o) 1 f
2~
~ SF(~,s
SYSTEM PROPERTIES EXCITATION
14.8
1.00
(59) Finally, the normalized mean square response of the hydrodynamic force, based on equations (46) and (59), is
AND INPUT
For the arch dam-reservoir fluid and random ground acceleration excitation system, an important part of the system properties are given in Table 1. In the table, the height H of the arch dam is selected to be 91~44m or 300ft and 152.40m or 500ft respectively. The radius to height ratio R/H includes 3 values, 0.5 for the large curvature arch, 1.5 for the typical arch and 2.5 for the fiat arch respectively. Corresponding to these geometric forms are the natural frequencies Od of the arch dam with empty reservoir; 76.90 rad/scc for the large curvature (or stif0 arch dam with R/H = 0.5 and 17.32 rad/sec for more flexible arch dam With R/H = 2.5, both for H = 91.44m = 300ft. For high dam of H = 152.40m = 500ft, the natural frequencies COdof the arch dam with empty reservoir are 46.14 rad/see for the stiff clam of R/H = 0.5 and 10.4 rad/sec for the soft dam ofR/H = 2.5. The natural frequencies Orfor the fluid in the full reservoir are 24.72 rad/sec for the low dam of 91.44m or 300ft and 14.83 rad/sec for the high dam of 152.40m or 500ft. The next 24
14.8
= O..~a'(.of
~t = 25 rad/sec. Natural frequency of foundation
0 } (~c) =
11= 152.40m = 500ft
1.00
9
1.00
1.00
3.13
1.00
0.70
1.70
1.70
1.70
row in Table 1 shows the ratios ~,. between the reservoir fluid frequency to,. and the arch dam frequency Od. The ratios are 0.32 for stiff dam 1.00 for typical dam and 1.42 for soft dam respectively. Note that these ratios are the same for low and high dams, because both or and Od are inversely proportional to the first power of the height H. The next row after f~,. in Table 1 shows the ratio f~l between the natural frequency of the reservoir f2 r and that of the foundation o f . For the typical foundation natural frequency 19 o f = 25 rad/see the ratio f~ f is 0.98 for low dam and 0.59 for high dam. Other system properties considered in the numerical analysis in addition to those given in Table 1 are the ratio bl/H between the top width bl and the height H of the arch clam (vertical section), the ratio b2/H between the bottom width b2 and the height H, the damping ratio ~d of the dam, the Young's modulus of the concrete E, the unit weight of concrete Wdand the damping ratio of the foundation ~f. Their typical values are selected and given in the following.
bl/H = 0.042, ~t = 0.05,
wd = 150pcf,
b2/H = 0.167 E = 7.20xl0Spsf = 34.5 N/m 2 ~i = 0.1, 0.5 and 0.9
= 2,403 kg/cm The natural frequencies Od of the arch dam with empty reservoir in Table 1 are taken from Ref. 6, checked closely for the case of R/H = 1.5 with those given by Ref. 20 and with those computed by the simple method of Ref. 3. Finally, the first symmetrical mode shape of the arch dam (0, z) for R/H = 1.5 is taken from Ref. 20 as
r (o, z) = r (o) r (z) where
Probabilistic Engineering Mechanics, 1991, Vol. 6, No. 1
r
(0) = b o + blO + b 2 0 2 + b 3 0 3 b 4 0 4 + b 5 0 5
Ramtom vibration of shnple flexible arch dam reservoir systems from earth@takes: C. Y. Yang et al
~"~1 = 0.005~3
30
A
20
2O
,...,
v
to
to
0 0
5
!o
200
400
0
600
0
I1
~'~f 9 0.59
,1o
2
4
fl
~'~f = 1.46
30
20
20
v to
~"~f = 0,296
30
v
to
I0
0
I0
0
-
0
4
-' 0
0
2
4
n
Fig. 3. htput Kanai-Tajlmi spectra Sg (f~) = Sg (to)/So VS if2 = to~to r for different f~f = codto$. Legend ~j = O.1 solM curve _ _ ~i = 0.5 dashed curve . . . . . ; ~i = 0.9, dashed curve . . . . . . .
|.$
given by equations (30), (34), (44) and (58). The Kanai-Tajimi spectral density function, based on equation (58) is shown in Fig. 3 for four values of the foundation frequency ratio Qy and three values of foundation damping ratio ~f. From Fig. 3, it is obvious that for very small damping (~y = 0.1), the spectral peak is narrow and pronounced. As the .damping ratio increases, the spectral distribution becomes broad banded and consequently the foundation resonant effect decreases. Also, Fig. 3 shows that the effect of varying the foundation frequency ratio f~f is to change the location o f the spectral peak. For very small f2f (0.0059), the peak is located near f2 = 180. As g2f increases to 1.48, the peak moves to f~ = 0.7. Since the range of siginificant frequency ratio lies between fl = 0 to about f2 = 5.0, (see the bottom rows in Table 1 with further explanations in the next section), the case of f2f = 0.0059 is effectively a white noise input according to equation (58).
.R/H= 0.5
R/).I = t5 ""~.,.
e/H 5.?.s
O.S-
0
-0.$ 0.0
0.2
m
0.4
m
0.6
I
O.B
0 Fig. 2. M o d e shapes r
INTERACTION
= a o + a l ( z / H ) + a 2 ( z / H ) 2 + a3 ( z / H ) 3
The constants are b0 = 0.99693, b3 = -14.47143, a0 = - 0.00007, a3 = - 0.12814
bx = 0.15592, b4 = 68.79296, ax = 0.05946,
b2 = 9.09407 bs = --49.11577 a~ = 1.06969
For arch clams with other R / H geometry ratios, the mocle shapes are given in Ref. 21. These mode shapes are shown in Fig.2. All natural frequencies and'mode shapes have been obtained by a finite element method in Ref. 6 and 20. Input random excitation is modelled by a nonstationary random process with a unit step time modulation function and a stationary Kanai-Tajimi spectral density function as
AND RESONANCE
The interaction among the arch dam, the reservoir fluid and the foundation is best described by a set of normalized frequency variables and parameters as follows. g2 = oJ/to~, f~, = ~/r Qf = torltof, f~ a= r162 Q ~ ~= (o/oJ,, ~ f = editor where co, co, cod and (,of are the frequency of the excitation, frequency of the reservoir, frequency of the dam and frequency of the foundation respectively. The ratios g2,., ~ f represent the interaction between the reservoir and the dam, and between the reservoir and the foundation respectively. Various interaction ratios of f~, and g2y are given in Table 1. The resonant conditions are most clearly indicated by the normalized variable f2 and products fK2r anti Qf. The uncoupled resonant conditions are first pointed out as follows.
Probabilistic Engineerin9 Mechanics, 1991, Vol. 6, No. I
25
Ramtom vibration of sbnple flexible arch dam reservoir systems from earthquakes: C. Y Yang ct al
Od
0
i
= .
Od
0
!
PV~ = 0.~. fir = 0.~
= 0.32
.
1-
;
o
I
i
,
0
s
I
o
'
,
i
Od
lo
0
2
1 v
,,, I
,
5
flu
~
4
:,
,:,
S
6
7
0
'
200
t
2
:5
4
'
,
'
'
C
$"
I
P . ~ = 2.$. Or = t ~
R,/H = 2.5: O r = 1.42
C
S
.;
100-
I>-
ILA.
) 0
"~'r
a
- -
--
0
~
'
s
[
f~
Fig. 4. Complex frequency response of hydrodynamic force F ( ~ ) = g-ff (co)/Fs and displacement t" (s = co=,~'(co) v s n co/corattd f~a = (o/coafor stiff dams of f2 r = co,/co,t= 0.32 and flexible dams off2 r = 1.42. ~ d O.05 for all dams.
When g2 = 1.0, resonance o f the reservoir fluid occurs When f ~ ,. = 1.0, resonance of the dam without fluid occurs When
dation col = 25 rad/sec. For the soft dam with R/H = 2.5 and H = 91.44m or 300 ft. f2 g2 r = 1.42 fl f2
QQj, = 1.0 resonance of the foundation occurs. It is important to note that, for the typical arch dam ofR/H = 1.5 and H = 91.44m or 300ft in Table 1, f2, -, f2j, ~ 1.0, so that the three resonant peaks otZa coupled system response are all located near f~ = 1.0 in the normalized frequency abscissa. The deviations from the uncoupled resonant frequencies represent system interaction effects. For the stiff dam with R/H = 0.5, and H = 91.44m or 300ft., g2, = 0.32 so that resonance of the dam occurs at g2 f~ r = 0.32g2 = 1.0 f~
=
3.11
Consequently, for the stiff dam, two seperate resonant peaks occur, one near g2 = 1.0 for the resonance of the reservoir and the foundation, the other near Q = 3.11 for the resonance o f the dam. Note that the typical natural frequency of the foun26
=
=
1.0
0.7
The resonant peak for the dam is located near fl = 0.7. The other two resonant peaks for the reservoir and foundation respectively are again near f~ = 1.0. Again cos = 25 tad/see is used here and in Table 1. Finally, it is instructive and also useful to note that, based on equation (44)when the natural frequency of the foundation co.f approaches to infinity, then S(co) --* So, and the KanaiTajimi spectrum becomes a white noise with a constant strength of So.
NUMERICAL RESULTS Based on the study of interaction and resonance o f various arch dam-reservoir-foundation systems in the previous section, three groups of problems as given in Table 1 are selected for numerical computations. The three groups are characterized by the dam geometry parameters R/H, with 0.5, 1.5 and 2.5 respectively. Furthermore, each of the three groups is
Probabilislic Engineering Mechanics, 1991, Vol. 6, No. 1
Random vibration of shnple flexible arch dam reservoir systems from earthquakes: C. Y Yang ct al divided into two parts, one for low dam of 91.44m or 300ft for which the foundation frequency parameter ~ f = 0.98, the other for high dam of g2 s = 0.59. The damping ratio for the dam ~ = 0.05 is used for all cases. A typical resonant frequency of cot = 25 rad/sec and a small damping ratio of ~t = 0.1 are selected for the input Kanai-Tajimi spectrum simulating random earthquake ground accelerations. The choice of to t = 25 rad/sec reflects a typical case according to the survey of Ref. 19. The unusually small damping of ~ t = 0.1 corresponds to a narrow banded spectral density with a pronounced peak so that the foundation resonant effect will be brought out dearly in the response solution. Figure 4 shows the comp.].ex frequency response function for the hydrodynamic force F(_~.) and the displacement of the dam at the center top crown Y (f~) respectively by the two figures at the top row. Two peaks appear clearly in both force and displacement response. The first peak located at f2 = to/oJr = 1.0 indicates the resonant effect of the reservoir. The second peak located at about s = 2.5 indicates the resonant effect of the dam with the added mass from water. A second horizontal scale is given at the top of each figure by the variable ~ a = to/o~a. The resonant frequency of the dam without water occurs when f~a = 1.0 corresponding to ~ = 3.13 in this case where g2~= to,,ttod = 0.32 (since g2 = color = (O)/(.Od) (tod/tor) = f2,df]r = 1.0/0.32). These two figures at the top row of Fig. 4 are for stiff arch dams with the geometric parameter R.H. = 0.5 (corresponding to f~, = 0.32). The effect of increasing dam flexibility is shown by the two figures at the second row of Fig. 4 for which R/H = 2.5 (corresponding to f2 = 1.42). Again two peaks show in both the force and the displacement
complex frequency response. The location of the peak corresponding to reservoir resonance remains unchanged (f~ = 1.0) whereas that corresponding to dam resonance moves to f~ *, 0.5 (f~ ,~*, 0.7). Figure 5 shows the power spectral density for the hydrodynamic force SF(x, ~) and displacement St(x, f~) respectively by the two figures at the top row. The input random excitation is characterized by the foundation damping ratio of ~ f = 0.1 and the resonant frequency of tot = 25 rad/see. so that the foundation frequency parameter f2t = to,/tot = 0.59 for high dams of 500ft (152.4m) with tot = 14.8 rad/sec. (See Table 1). In both the top two figures, three spectral peaks appear. The first and the third peak correspond to reservoir (f~ = 1.0) and dam (f~ = 2.5) respectively as in the case of complex frequency response solution of Fig. 4. The middle peak located at about f~ = 1.7 corresponds to the input spectral peak (since to/tot = 1.0, todto/ = 0.59 then S'-2= to/tot = (to/tot)/(to,/toI) = 1.0/0.59 = 1.7). Again these top two figures correspond to stiff arch dams ~vith R/H = 0.5 (f2r = 0.32). For more flexible dams with R/H = 2.5 (Wr = 1.42), the spectral density for hydrodynamic force and displacement respectively, are shown by the two figures at the second row of Fig. 5. A comparison of the top row figures with those of the second row shows that as the dam flexibility increase from R/H = 0.5 to 2.5, the resonant frequency for the dam with added mass decreases from f~ = 2.5 to f~ = 0.5. The corresponding spectral peak (at ~ = 0.5) becomes the dominant one relative to the reservoir resonant peak at f] = 1.0 and the input spectral peak at f2 = 1.7. Also shown in Fig. 5 is the nonstationarity of the special response at x = 50 and x = 100. It is clear that
Fig. 5. Spectral density of hydrodynamic force SF (% f2) = g2SF(t, ~)/ F 2 So aml displacement Sy (% g)) = to4 Sift, to) So V S f2 = to/torATx= oh t = 50 attd 100. Damping ratio of dam ~a = 0.05, of foundation ~ t
2S
20 R/~=0.S. ~ = 0 . 3 2
R / ~ = 0..5, 9 r = 0.32. C)f = 0.$9
20,
nf=0.59
15. Vl
vi-> . 10~ tn
10T=5o
$0
i
0
~
1"=50
1.=1oo
!
O'
,
2
0
4 n
SO
30000. T=50
T=50
7-100 v
t,-
= 2.~ n r = 1.42. t.~! = 0.,~9
0
F-
i
2 N
R/H= 2...% f~r =1.42 D-l=0.fq
>,.
(/1
20-
o -~
20000-
10000-
0 _~
0
!
2
3
.q
Probabilislic Engineerin9 Mechanics, 1991, Vol. 6, No. 1
27
Random vibration of shnple flexible arch dam reservoir systen~ from earthquakes: C. Y. Yang et al
$
6-
4.
4
z 03~1
[]t.'..O !~
I,d~
b"
/I/H = 0.$. nr ,,: 0 . ~
!"4=0.~_
2 I r~'-
n,-.o_.gJ
:~
o
R/'H z 0.5, fir = 0.32 0
~o
150
150
0
T
T
I')t = 0.S9
p.L=.o . ~
. A
500-
>..
2"
PI
= 2.5. n t z t 4 2
- z~.
T
t,2 "%'"o.._.~
T
Fig.
6. Mean square o f hydrodynamic force a~- (x) = g2 o ~ o ~, (x) = co 3 o {, ( 0 / 2 ~ So vs tbne T t = wr t for xd = 0.05 aJ~x~ = 0.1
as the dam flexibility increases the transient time to reach stationarityincreases. Figure 6 shows the mean square response for the hydrodynamic force o ~" (x) and displacement o ~ (x) respectively as a function o f time x. The two figures at the top row correspond to stiff arch dams of R/H = 0.5 (f2 , = 0.32)
4000
%.,
~'~I z 0.0059
attd
displacement
whereas those two at the second row correspond to more flexible dams of R]H = 2.5 ( ~ , = 1.42). A comparison of figures between these two rows shows that as the flexibility of the dam increases, the mean square hydrodynamic force decreases somewhat. The mean square deflection, on the other hand, increases greatly. The transient time to station-
9~"~f ,~ 0.296
9 I000
AIC
(t)/2~F2s to= So
4OOO
C Vl 1000
v)
!
0
o
m
e
2
L o
., I 2
N
~ f = 1.48
~"~fz O.$g
I000 4000
|eoee
0
.J ,Ir~ .... o
l, ! 2
Io
n
Fig. Z Spectral density of displacemeut S Y (t, IV) = oJ~ Sy (t, to)/So at x = tOrt = 50for different We] = wr/w~ attdx~. All for dams with R/H = 1.5 f2t = f~t / g2a = 1.0 and ~ = 0.05. Legend E_i= 0.1; solid curve xi = 0.5 dashed curve - - - ; xi = 0.9, dashed curve . . . .
28
.
Probabilislic l?,ngineering Mechanics, 1991, Voi. 6, No. 1
Random vibration of simple flexible arch dam reservoir systems from earthquakes: C. Y. Yang et al
~'~f "" 0,0059
200
#
I-V ell
~']f = 0.296
300
,~
I00
200
b"
b
100
t
,
50
I
I
100
150
,
0
200
,
0
I
~
50
#
200 N
b
i
100
T
300
I
l
l
150
200
T
~"~f = 0.59
~"~f = 1.48 4000
t-2000
b
I00
0
!
0
50
,
.
I
100
,
I
150
i
D
200
I
0
T
50
,
I.
100
,
I
150
,
,, 200
"T
Fig. 8. Mean square displacement o ~ ('c) = co 3, o ~ ( 0 / 2 ~ So for different f2y = co#cofand ~f. All for dams with R/H = LS, f~z = cor6Oa=l.0and~a= 0.05. arity increases with increasing fiexiblity of the dam as in the case of Fig. 5 fromx = 50 tox = 100. There are clear transient peaks settling down to stationary in the case of stiff arch dams in both mean square force and displacement in the top r~w figures but not in the case of more flexible dams. Also shown in Fig. 6 are the comparison between f2f = 0.59 for a high dam and Of = 0.98 for a low dam (based on a typical foundation resonant frequency cot = 25 rad/sec 0. In three but not all of the four figures the mean squre responses for high dams (f~f = 0.59, col = 25 rad/sec., solid lines) are smaller than that corresponding to low dams (Qf = 0.98, col = 25 rad/see., dashed lines). This result cannot be explained straight forwardly because in the normalization of the mean square responses, the dam height is involved indirectly through the resonant frequency of the reservoir co, (= ~xC/2H). The normalized solution given in Fig. 6 can be converted to corresponding dimensional mean squares for a better comparison. Figure 7 shows the power spectral displacement response S~(x, f2) for various input random excitations defined by the foundation resonant frequency Qf and damping ~f; all for high arch dams of 500ft (152.4 m), medium stiffness of R/H = 1.5 (f2r = 1.0), and at time t = 50. First, the effect of
increasing damping from ~.r = 0.1, 0.5 to 0.9 is to reduce the magnitude of the spectral response and flatten the spectral peak in particular. Secondly, the damping effect increases as the foundation resonance parameter increases from f~l = 0.0059, 0.296, 0.59 to 1.48. Thirdly, as f~f increases, the spectral displacement increases for a given damping ~f. The location of the spectral peak remains at about f~ = 0.7, corresponding to the resonant frequency of the dam with added mass. The case for f~f = 0.0059 approaches thatof a white noise input as a limit (see equation (58)). Figure 8 shows the mean square displacement response o ~('0 for various input random excitations defined by the resonant frequency of f~f and damping ~l; again all for high dams of 500ft (152.4m) and medium stiffness ofR/H = 1.5 (Qr = 1.0). Similarly to the previous Fig. 7, the damping effect is to reduce the mean square displacement and is more pronounced as the frequency ~ f increases from the limiting white noise case of 0.0059 to 1.48. Again, as f~l increases, the mean square displacement increases, because the peak of the input excitation moves closer to the system resonant frequencies.
Probabilistie Engineerin9 Mechanics, 1991, Vol. 6, No. 1
29
Random vibration of shnple flexible arch dam reservoir systems from earthquakes: C. Y Yang et al
60-
5O
9
IVHs " t.O. I ~
,~
fir
z tO, | =
9 I.S
O.OS
l~
..'p 5O.
'J~"
20.
.J~,J
i,.
Legend
Ill
9
co~,o,_._~.
I0-
j]
0 0
. ~-~-'-_.._~ I
2
9
IL 3
..-. 4
recognized as an important design measure, but the random vibration and random h y d r o d y n a m i c force solutions presented here for the flexible arch dam problem are new. These random solutions are needed for a probability based seismic reliability analysis and design of the arch darn-reservoir systems. The relatively simple solutions presented here, of course, are subjected to several important assumptions and restrictions including linear elastic clam structure, rigid foundation and side walls and the first symmetric mode of dam vibration. Most of these assumptions and restrictions can be partially or totally eliminated with added complications in analysis and computations for future work. Here in concluding this work, the assumption of the first symmetric mode shape is pointed out again as a precaution for practical engineering application. The present solutions have high accuracy only when the earthquake excitation spectrum lies primarily below and near the first natural frequency or above four times the first natural frequency of the dam.
ACKNOWLEDGEMENT Fig. 9. Comparison of complex frequency acceleration Y (f2) = ~" (to) between our calculated shzgle mode solution with finite element solution of Porter arm Chopra (page 83 R e f 6)
This research work is supported in part by the National Science Foundation Grant No. ECE 850-1580 with the University of Delaware. The authors are grateful for the support.
Figure 9 shows a comparison on the complex frequency acceleration response between the first symmetrical mode approximation and the multiple mode solution of Ref. 6. The comparison is generally satisfactory for frequency ratio f~d = to/rod < 1.0 and > 4.0. The largest discrepancy lies in the neighbourhood of 1.5 and 3.0. This observation suggests that the approximate solutions should give satisfactory answers qualitatively. However, in case the input excitation has dominant peaks in the neighborhood of one and half times or three times the first symmetrical frequency of the dam without water (to = 1.5 rod or 3.0 rod), then the accuracy of the approximate solutions deteriorates and a multiple mode solution should be used.
REFERENCES
SUMMARY AND CONCLUSIONS A simple arch dam-reservoir system is selected to study the stochastic vibration of the dam and the hydrodynamic force response on the dam to a nonstationary random excitation process, simulating earthquake ground acceleration. The dam is assumed to be elastically deformable but is restricted to the first symmetric mode shape for simplicity. The random earthquake acceleration is assummed to be in the horizontal upstream-downstream direction and is modelled by the simplest nofi-stationary random proces"s which is the zero-start stationary process with a Kanai-Tajimi spectral density function. The stochastic vibration solutions and the hydrodynamic force response solutions are presented by the nonstationary power spectral density functions and the mean square functions all in general nondimensional forms, in addition to the non-random solutions of complex frequency response functions. The range of problems studiedinclude high and low dams an/l geometrically stiff (with large curvature) and flexible (with small curvature) dams. For all these problems, the locations of important resonant peaks are identified in both the non-random complex frequency response functions and the random spectral density functions. The vibration of the clam and the hydrodynamic force response on dams during earthquake motions have long been 30
1 Kotsubo, A. External forces on arch dams during earthquakes, Faculty of EngineeringMemoirs, 1961, 20 (4), Kyushu University, Fukuoka, Japan 2 Clough, R. W., Raphael, J. M., and Mojtahedi, S. ADAP-A computer program for static and dynamic analysis of arch dams, Rept. No. EERC
73-14, Collegeof Engineering,Universityof California,Berkeley,California, June 1973 3 Crawford,C. C. Earthquakedesignloadingsfor thin archdam~,Proceedings, 3rd WorldConference on EarthquakeEngineering, 1965, III, IV172-185, New Zealand 4 Perumalswami,P. R. and Kar,L Earthquakebehaviourof archdam-reservoir systems,Proceedings5th WorldCo~[erenceon EarthquakeEngineering, June, 1973 5 Porter,C. S. and Chopra,A. K. Ilydrodynamieeffectsin dynamicresponse of simple arch dams, Earthg. Engr. and Structural Dynamics, 1982, 10 0), May-June,471-432 6 Porter, C. S. and Chopra, A. K. Dynamicresponseof simple arch dams including hydrodynamieinteraction, EERC Report No. UCBIEERC80117, Univ. of Californiaat Berkeley,July, 1980 7 Fok, K. L and Chopra,A. K. Earthquakeanalysisof arch damsincluding dam-water interaction, reservoirboundary absorption and foundation flexibility,EarthquakeEngineering and StructuralDynamlcs, 1986, 14, 155-184, 8 Seanlan,R. II. On earthquakeloadings for structural design,Earthquake Engineering and StructuralDesign, 1977, 5, 203-205 9 Yang, C. Y. and Charito, V. Randomhydrodynamicforce on dams from earthquakes, Journal of the Engineering Mechanics Division, ASCE, February 1981, 107 (EMI) 10 Mills-Bria and Yang, C. Y. Nonstaioaaryresponseof concrete gravity dam-reservoirto randomvertical earthquakeexcitation,Proceedingsof the 8th WorldConferenceon EarthquakeEngineering, July 1984, VIII 11 yang, C. "I(.and Kuserk,R. Transientrandominteractionof dam reservoir systems,ProbabillstleEngineeringMechanics, Mar. 1987, 2 (1), 1625 12Debussey,M. and Yang,C. Y.Nonstationaryhydrodynamicforcesoa arch dams from randomearthquakes,Procee&'ngsof the 3rdNational Earthquake Engineering Conference, Charleston,SC, August 1986 13 Priestley, M. B. EvolutionaJyspectra and nonstationaryprocess,Joun RoyaIStatistlcalSoc~, 1965, Series B, 27 (2), 204-237 14 Kanal, Semi-empiricalformula for the seismic characteristics of the ground, Univ. TokyoBull. EarthquakeRe~ lnsL 1957,35, 309-325 15 Tajimi,II. A statistical methodof determiningthe maximumresponseof a building structureduringan earthquake,Proc.2nd WorldConf.Earthq. Eng., Tokyoand Kyoto,July 1960 II, 781-798 16 Shinozuka, M. Random processes with evolutionary power, Your, Engr. Mech. Div., ASCE, August 1970, 96, (EM4), Proe. Paper 7444, 543-545 17 Yang, 12. Y. Random Vibratlon of Structures, John Wiley International
Probabilislic Engineerin9 Mechanics, 1991, Vol. 6, No. 1
Science Publishing Co., 1986
Ramtom vibration of sbnple flexible arch dam reservoir systems fivm earthquakes: C. Y Yang et al 18 Clough, R. W. and Penzien, J. Dynamics of Structure, McGraw Hill Pub. Co., 1975 19 Lai, P. Statistical charaaerization of stone ground motions using power spectral density function, Bulletin of the selsmologicalSoclety ofAmerlca, Feb., 1982, 72 (1), 259-274 20 Kar, L EarthquakeBehavlor of Arch Dam ReservoirSystems, Doctor's Thesis, South Dakota School of Mines and Technology, 1972 21 Wells, J., Bhobe, R. and Yang, C. Y. Transient response of dam reservoir systems to random earthquake excitation, Probabilisdc Engineerlng Mechanics, September 1989, 4, (3) 142-149
APPENDIX: HYDRODYNAMIC SOLUTIONS ~(O,z, to) = ~d(O,z,r
PRESSURE
+ ~ ( O , z , to)
+ ~(60)pr (0, Z~03)
(A.1)
where
~xR q l (z,n- 1)2 -- ((o/to,)z I x=R =~--ff
(A.8)
= dimensionless parameter of the frequency ratio to/to, and dam geometry R/H respectively.
f ~./4f Iron = H1 ,do d o H d~(0, z) cos (4n 0) cos (ct m z) d 0 dz (A.9) dp(0, z) = first symmetrical mode shape of arch dam. The remaining symbols, are given in terms of Bessel functions of the first kind of order n, J,, ('Z.mR),Bessel runelions of the second kind of order n, Yn(~,~R) and modified Bessel functions of the second kind of order n, K,,(s
c~ (XmR) =
~d (o, z, co) -
16 vr2 to R ~
"[ Cn()vaR )
g~2 +
~
[A.(K,,,R)J4.(XmR)+Bn(~.mR)Y4n(KmR) ]
(-1) men (-1).
(A.10)
R [A 2 (XmR) + Bn2 (~-mRI
m-12 ?o- (2n-~-l) 5-1-6n2)
D,, (Lm R) =
iD. (XmR)] c o s 4 n 0 cosctmz (A.2)
[B,,(k,,,R)J4,,(~mR)-An(~.mR)Y,t,,(LmR)]
(A.11)
~.,~R [A ~ (~.,~R) + B n2 (XmR)] where
An(~'mR) = J4n-l (~'mR) - J 4 n + l (~m R) (A.12)
Bn (~.~ R) = Y4n - 1 (~m R) - Y4. +1 (~'mR) (A.3)
+ iV,,, (XmR)] cos 4n 0 cos Ctmz}
oo
r (0, z, to) =
(A.13) The above equations for (7, (X,,,R) and D,, (X,,,R) apply for m < mr, where mr is the largest integer "m" satisfying the inequality (w/c) < a,,,. For m > rex,
oo
166o R
g4. (XmR) m-ln-O
cn ( ~ R) = Xm R [K4n_1 (Lm R) + K4n + 1 (~'mR) ]
9 [Cn(Z.mR) + iDn (kmR)] cos4n0 cOSamZ
(A.14)
(A.4) Various symbols in equations (A.2), (A.3) and (A.4) are defined as follows. t"
e n = ] 1, n = 0 t 2, n,, 0
tar -
(A.15)
For m < mk, (- 1)" E,, (X,,R) - (2mI) ~-mR { sin [X~R sin ( n / 4 - 0)]
+ sin [ k~ R sin (a/4 + 0) ] }
~c
2H
(A.16)
_
fundamental frequency of the water in the reservoir (2m -
am-
(A.5)
D. (~mR) = 0
1) ~r
(A.6) (A.7)
(- I)m { F m (k,,, R) = (Z,n -1) ~.mR
cos [
sin
O) ]
2//
Probabilislic Engineering Mechanics, 1991, Vol. 6, No. 1
31
Ranclon, vibration of sbnple flexible arch dam reservoir systems from earthquakes: C. Y. Yang ct al
+ cos
[ kmR sin (u/4
(A.17)
+0)]}
Umn(kmR) = (-1)"(-1)"{ (~, a) 7". (~m) c. (xm R)
(16n2+ 4 k 2 - 1) (16n2-4k 2_ 4k - 1) (16n2 _ 4k 2 + 4k- 1)
E~ (~.,R) =
+ ~-A, (XmR) Dn (KinR)
(A.lS) Vm, (XmR) = (-1)"(-1)~ ( (2m- 1) 7". (~,. R)
(A.20)
for m > mt
-
- (- 1)m [ e -~Rs~n(~4-~ (2.7: i72-~ R
+ e- ~ R sin (g./4 + 0)]
(A.21)
Dn (X,. R) F m (~.rnR) = 0
(A.22)
- ~ A. (XmR) G (XmR)
(A.19)
where
-(-1)~'
u,,,. G,, R) - ~ T - T) C~(~ R) G (~ R) (A.23)
k-0
32 Probabilislic Engineering Mechanics, 1991, Vol. 6, No. 1
v,,, (MR) = o
(A.24)