Transient pressures in hydrotechnical tunnels .pdf

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EARTHQUAKE ENGINEERING A N D STRUCTURAL DYNAM ICS, VOL. 16, 523-539 (1988)

TRANSIENT PRESSURES IN HYDROTECHNICAL TUNNELS

DURING EARTHQUAKES

SLOBODAN B. KOJIC*

University of Southern California, Los Angeles, C A , U.S.A. , and Energoproject Co. , Belgrade, Yugoslavia

A N D

MIHAILO

D.

TRIFUNAC?

Department of Civil Engineering, University of Southern California, Los Anyeles , CA, U . S . A .

SUMMARY

Transient pressures generated by earthquake shaking in hydrotechnical tunnels are evaluated by the discrete Fourier

transform technique. The effects

of

the horizontal ground motion accelerating the closed dow nstream tunnel gate, as well

as the upstream dam face, and the influence of the vertical motion of the reservoir floor are considered in this analysis.A n

example

of

a typical bottom outlet

is

analysed by subjecting

it to

several computed accelerogram s.

It

is shown that

high

hydrodynamic pressures can be developed, several times larger

than

the hydrostatic pressure.

I N T R O D U C T I O N

Hydrotechnical tunnels, penstock and bottom outlets are common elements in many dam projects. Their

functions are to provide efficient and economical means of releasing the water from the reservoir according to

the desired dow nstream use for irrigation or

for

power generation. The conduits which lead the water to the

turbines are usually designed to withstand high hydraulic transient pressures arising in various turbine

operations. Surge tan ks are frequently used to protect the upstream par t of the conduit. For such systems there

is little need for analysis of hydrodynamic pressures due to earthquakes, althou gh som e pressure increase may

be expected. However, hydrotechnical tunnels and bottom outlets for irrigation purposes usually are not

designed for waterhammer effects like the turbine penstocks. In mo st cases such tunnels d o no t have surge

tanks or other openings, which would da m p the transient pressures caused by an earthquake . Usually, they

may have valves or gates located at the upstream intake, at an intermediate point an d at the downstream end.

Owing to various downstream demands, it may happen that the intermediate or the end gates are closed fo r

long periods of time, leaving the upstream condu it part u nder a full reservoir pressure. Under such conditions

in seismically active regions, an ear thq uak e may cause the water pressure to increase or decrease w ith respect to

the hydrostatic pressure or steady pressure conditions. Un derstanding of the transient hyd rodynam ic pressure

caused by earthquakes is of interest for the proper design approach

to

these structures. Failure of the

hydrotechnical tun nel during a n earth qua ke can initiate erosion of su rrou nd ing material and consequently,

depending where a break occurs, it may cause increased uplift un de r the dam , dam abu tm ent failure, stilling

basin

or

spillway damage, hydroelectric power plant break

or

a crash

of

any o ther vital component

of

the dam

system. Any of these events can be a starting point for a dam collapse.

Zienkiewicz' was am on g the first to point out the resonant effects in the bottom outlets due to harmonic

horizontal motion

of

the downstream gate. On a specific project O bradoviC2 carried out an e arth qu ake

*

Research Fellow, University of Southern California, and Staff Member, Energoproject Co.

Professor of Civil Engineering.

0098-8847/88/040523-17$08.50

988

by John Wiley

&

Sons, Ltd.

Received

13

Ma y

1987

Revised 6 October

1987


2/17

524

S.

B. KOJIC AN D

M.

. TRIFUNAC

response analysis of water in the bottom outlet. His model has included only horizontal downstream gate

motion in the generation of hydrody namic pressures due to earth quak es. The m ethod used in his analysis is the

method of characteristics. The results showed that high hyd rodynam ic pressures can be developed during an

earthquake and that these depend on earthquake amplitude and frequency content.

The present analysis has the following objectives.

(a) To illustrate the additional effects

of

upstream bound ary conditions, i.e. the influence of hydrody nam ic

pressures developed during earthquake response of dam and reservoir

floor

on generation of transient

pressures along the bottom outlet.

For

simplicity, the da m will be con sidered a s rigid, althou gh, in so m e cases,

its flexibility may not be igno red. Inclusion of the d am flexibility is possible via finite elemen t discretization of

the da m, for example, but it will not be studied here. The reservoir and the water in the condu it are assumed to

be compressible. The downstream boundary condition, horizontally moving gate, has been included also.

(b) To demonstrate the possibility of using the method of discrete Fourier transform in a hydraulic

transient problem via the fast Fourier transform algorithm.

(c) To examine the capabilities of the mathematical model an d of the p roposed numerical technique on a

realistic bottom outlet mo del.

DES CRIP TION OF T H E B O T T O M O U T L E T

The bo ttom outlets carry water from the reservoir to the river

or

to the irrigation channels downstream.

A

simplified schematic of this structure, with the dam and the stilling basin, is shown in F igure 1 . The bottom

, - I N T A K E GATE

CLOSED

D O W N S T R E A M GATE

x 2 , f

BOTTOM

OUTLET

BOTTOM

; HYP OCENTER

( a ) S E C T I O N

A - A

/

/

t EP I CENTER

( b ) L A Y O U T

Figure 1 .

Bottom

outlet, dam and reservoir with earthquake ground accelerations: u; and a:


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HYDROTECHNICAL TUNNELS DURING EARTHQUAKES 525

outlet intake is usually located at the b ottom of the reservoir close to the d am upstream face. The intak e sliding

gate stops water flow when the b ot tom outlet is serviced or rep aired. An inte rme diate valve helps in closing of

the intake gate. The downstream gate regulates the water discharge according to the downstream

requirements. Before entering into the river the water is passed through the stilling basin whose role is to

decrease the water velocity. The cross section of the conduit is usually circular and it is made of reinforced

concrete. The conduit may be lined by steel,

if

water velocities and pressures a re high.

Th e gates, concrete a nd steel lining are, in general, dimen sioned t o withstand the full reservoir pressure. O nly

the intermediate valve is checked fo r hydrod ynam ic effects to en able its closure in th e flowing water with high

velocity

.

If the dam is located in a seismically active region, it is of interest for the gene ral da m safety to de termine the

hydrodynamic, transient pressure along the bottom outlet during an earthquake. The case when the

downstream gate is closed and the bottom outlet is under the full reservoir pressure will be considered.

Without loss of generality the bot tom outlet axis is assumed to be perpendicular to the da m upstream face.

Under these conditions the whole system, the dam, the reservoir bottom and the downstream gate, is

exposed to the ground motion.

T H E M A T H E M A T IC A L M O D E L A N D T H E S O L U T I O N P R O C E D U R E

One-dimensional wave equation for viscous po w

water assumed to be linearly compressible and viscous, is described by the following wave e q ~ a t i o n : ~

The hydrodynamic pressure associated with small amplitude, irrotational, one-dimensional mo tion, and for

where p ( s , r is the hydrodynamic pressure, in excess of hydrostatic pressure, alo ng the b otto m outlet, as a

function of the space coo rdin ates and time r , c is com pression wave velocity in water, and R describes friction

losses.

Equation (1) is called the waterhammer equation and it is used in hydraulic engineering for analysis

of

unsteady, transient flow through closed

conduit^.^

The friction

R

is assumed to be the same as fo r the steady-

state

flow

in con duits, i.e. the D arcy-Weisbach formula is used for comp uting th e friction losses.

R will

be

presented in this transient problem, caused by an earthquak e, in a somew hat different form t o that which is

normally used in waterhammer analysis (mean discharge3 is assumed to be equal t o zero). Fo r laminar flow, R

can be shown to be

and for turbulent flow,

fc

R = - -

c 2D

(3)

where v = p/p , the kinematic viscosity, p is the absolute water viscosity, p is the water mass density,

D

is the

condu it diameter,f'is the friction factor dependent on the conduit roughness and Reynolds number i t can be

determined from the M oody diagram4), and v is the water particle velocity in the cond uit. Th us, the hyp erbolic

partial differential equation (1) is linear for laminar flow an d non-linear for turbu lent flow.

The w ater particle velocity u is assumed

to

be equal to the ground velocity induced by an earthquake. Th e

earthquake ground motion accelerates the downstream gate, the dam and reservoir bottom and they all

generate the pressure waves which propagate through the bottom outlet.

A representative Reynolds number, for the peaks of the smallest recorded e arth qu ake velocities (Trifunac

and Brady,' v

x

0.01 m/s), and for the small bottom outlet diameters (D =

0.5

m), is abo ut 4000.This suggests


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526

S.8 .

KOJIC

A N D

M. .

RIFUNAC

that the water motion can be turbulent. However, during e arth qua ke shaking the water velocity is expected t o

change sign frequently, while it oscillates abo ut zero. Consequently, it may b e assumed th at most of the time

the water motion will be laminar. Therefore, R defined by equation

(2),

and the description of the pressure

change given by the linear version of equation ( l ) ,

will

be used.

The non-linear waterhamm er equation, presented by Streeter and W ylie4 in a somewhat different form , can

be solved by the method of characteristics. A solution is given in the time domain with an approximate

integration of the friction term.

Some experiments and analyses' have shown that th e influence of water friction in the hydraulic transient

problems is not very significant for the amplitudes of the hy drodyna mic pressures. However, the small viscous

term in equation ( 1 ) is retained to improv e numerical stab ility and the efficiency of the metho d app lied fo r the

solution.

Meth od of solution: The Fourier transform technique

Equation 1 ) for the laminar flow can be solved by the method of Fourier transforms. This method uses

a

property of linear time-invariant systems tha t, for the steady-state ha rmo nic exc itation, th e response is also the

steady-state harmonic motion a t the same frequency.6 So, if the harm onic excitation is the real part of e'"', then

the response p l ( s , ) is the real part of p (s, w)eio'f,where

p (s, 0)

is the complex frequency response function

which describes the frequency dependence of the response amp litude an d phase. Once this function

(s,

0 )

has

been obtained

for

a range of frequencies, the response to an arbitrary gro un d motion

a ,(t)

can be obtained by

the Fourier synthesis of the responses to the individual harmonic components, i.e.

where

i is the imaginary unit, =

x , y

are the horizontal and vertical com ponents of ground motion, and A,(to) is the

Fourier transform of a ,(t),

T , is the duration of the ground motion. The total response p ( s ,

)

to simultaneous horizontal and vertical

components of ground motion is obtained as

( 6 )

Evaluation of the integrals given by equations (4 )a nd 5) s perform ed in discrete form, using the fast Fou rier

P ( S ?

t ) =

P , t )+ P Y b , )

transform (F FT ) algorithm.'

Boundary condit ions

The downstream gate is assumed to be vertical, flat and rigid. G rou nd motion is assumed to be transferred

directly to this gate. Under these conditions the hydrodynam ic pressure for s = 0,at th e gate, can be expressed

as

This equation follows from D'Alembert's principle applied to an infinitesimal fluid element at the interface

between water and the gate.

At the intake s

= L,

the hydrodynamic pressures in the bottom outlet and the reservoir are the same:

p(L,

)

=

PXR(i,

j ,

+

P'R

( 2 , )

(8)

where p t Y

X,

j ,

)

is the hydrodynamic pressure in the reservoir at the intake of bottom outlet , with

coordinates

(3,

j), ue to horizontal

or

vertical ground motion.


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HYDROTECHNICAL TUNNELS DU RIN G EARTHQUAKES 527

Cho pra and his co-workers*.' analysed generation of the hydrody nam ic pressures in the reservoir due to the

dam an d reservoir bottom motion durin g earthquak es. In this illustration of the botto m outlet pressures, the

dam will be assumed to be rigid and with vertical upstream face an d the solution given by Fenves an d Chop ra,'

and by Ro senblueth" will be used for approximate determination o fp i( , x , y, t).A more refined determination

of

p k ( x ,

y,

t)

would require the three-dimensional analysis of the dam and waves propagating through the

basement and reservoir.

The hydro dynam ic pressure in the reservoir,

p k ( x ,

y,

t ) ,

in excess

of

the hyd rostatic p ressure, is govern ed by

the two-dimensional wave eq uation which is valid for small, irrotationa l, inviscid m otion,

where

c

is the velocity of sound in water. The pressure gradient at the vertical, upstream da m face is

By approximate modelling of the interaction between the reservoir water and the reservoir bottom,

considering only the vertically prop agating pressure waves, the bo und ary co ndition a t that interface can then

be expressed as follows:

where

q

=

p / ( p F c F )

nd

c F

=

J ( EF /p F ) .

EF is he Young's m odulus ofelasticity and p Fis he mass density of the

reservoir bottom material.

At

the free surface the following condition is satisfied:

P R ( x , H,

)

= Patm

(12)

and without loss of generality

pat,,,=

0.

The governing e quation

(9)

and the boundary conditions, equations

lo),

(1 1) and

(12),

are linear and the

same solution technique, explained previously, can be applied. The hyd rodynam ic pressure

p i

x ,

y,

t )

within

the reservoir domain du e to harmonic ho rizontal ground motion a;

=

1 eitiJt,s

where

i

nd , are comp lex valued and frequency dependen t eigenvalues and eigenfunctions respectively of

the impounded reservoir water. They are given by the following equations:

A

-

wq

1, o q

n = l , 2 , 3

. . .

2i j~,H =

and

I,

is defined as an integral of the eigenfunction, over the reservoir d epth, i.e.

and

K,

is given by

(17)

= j 2 - w 2 / c 2 ,

n = 1 , 2, 3, . . .

The exponential function e - "nXdecreases ressure with increasing

x

because of the assum ption of an infinite

reservoir in the upstream direction.


6/17

528

S.

B. KOJIC

AND M . D. TRIFUNAC

For the vertical harmo nic motion of the reservoir bottom , a',(t)

=

1 eirfjr,he hydrodynamic pressure in the

reservoir is

(0

sin-

( H

- y )

Equations (13) and (18) without time functions eiCurepresent the complex frequency response functions

p R ( x ,y, to)

for hyd rodynam ic pressure in the reservoir domain . Modu li of these functions are given in Figure 2

for the range up to

25

Hz and for several values of the wave reflection coefficient9

c1

defined by

1 - q c

1 + q c

a=

\

I

I

2

3 4

w/wp

Figure 2. Influence of reservoir bottom absorption on the hydrodynamic pressure in frequency domain at the bottom outlet intake:

a)

horizontal motion of the rigid dam; (b) vertical m otion of th e reservoir floor


7/17

HYDROTECHNICAL TUN NELS D URING EARTHQUAKES

529

This coefficient represents the ratio of the am plitude of the reflected hydrody nam ic pressure wave to the

amp litude of the vertically propag ating pressure wave incident o n the reservoir floor. The p lotted values are

scaled by the hydrostatic pressure and the horizontal frequency axis is scaled by the fundamental reservoir

frequency.

Complex valued frequency response function for bottom outlet

The hydrodynamic pressure along the bottom outlet due to harmonic ground motion can be expressed as

p ( x , )= p ( s ,

co)eio)f,

= x , y (20)

The complex valued frequency response function p ( s ,

o )

s obtained by solving equa tion (l),with the bound ary

conditions expressed by equations

(7)

and (8).

Substituting equation

20)

into equation

(1)

yields the one-dimensional Helmholtz equation

p( s , o) +

toR

p ( s , to) =

0

(5

The solution of equation

21)

is

p ( s ,

to)

=

C,

to)ews

C:

(to)

e -

ws ,

=

x ,

y

where w is given by

The coefficients C,

to)

and C:(to) are determined from the boundary conditions. Satisfying the boundary

condition at the dow nstream gate,expressed by equation

(7),

for harmonic ground motion

a,[)

= 1

e, leads

to

P

C i ( 0 )=

-

+c:(m),

=

x

y

W

Applying the upstream boundary condition given by equation

(8)

yields

where

piR& j ,

o) is evaluated by equ ation (1 3)

or (18).

Thus, for horizontal ground motion, the complex valued

frequency response function for the hydrodynamic pressure along the bottom outlet is given by

w ( l

+ e Z w L )

w

[ w

ewLp,(i,

, Q)- ]

and d ue to vertical ground motion, equation 22) combined with equations 24) and (25), for = y, yields

EXAMPLE

To illustrate this computational procedure, the bottom outlet, shown in Figure 1 (similar to the one

constructed on the Ch ira-Piura Project in Peru), has been chosen as an example. Th e botto m outlet has a

diameter

D

=

4.0

m an d length

L

= 300 m. At the intake the dep th of reservoir is

H =

100m. Th e compression

wave velocity of the water in the bottom outlet as well as in the reservoir is assumed to be the same,

c =

1300 m/s.

The complex valued frequency response functions

p ( s ,

(0 ) are computed for the range up to 25 Hz, at five


8/17

530 S.

B.

KOJIC A N D M. D . TRIFUNAC

sections, each located at a qu arter length of the b ottom outlet, and for three different cases of the unit harm onic

excitations. In the first case, shown in Figure 3(a), h e do wn stream g ate accelerates horizontally while the dam

and the reservoir floor do not move. The second case, show n in Figure 3(b), represents the horizontal mo tion of

the rigid dam w ithout m oving the downstream gate and the reservoir floor. Vertical mo tion of the reservoir

bottom corresponds to the third case of the unit h armon ic excitation, and is shown in F igure 3(c), with no

motion of the dow nstream gate and the da m. The plots in Figure 3 represent the m oduli of

p ( s , ( 0 )

scaled by the

hydrostatic p ressure

phs.

The frequency axis is scaled by the fund amen tal frequency of the b ottom outlet

Common features to all frequency response functions are the frequencies where peaks occur. These

represent characteristic frequencies of the water in the bottom outlet. For undamped motion they are

nC

o , B O = -n

n =

1, 3, 5,

2L

The characteristic frequencies

For the chosen parameters,

L

of the reservoir water with a non-absorbing boundary

( q = 0)

are

(29)

TLC

10

=

2H n

n = 1,

3,

5 , . . .

=

300

m

and

H

=

100 m, the ratio

W . R / W ~

=

3.

The frequency response functions differ

in

the width a t resonant peaks. 'For the case of the downstream gate

moving with unit acceleration, Figure 3(a), the broad est peak is encountered at the fun dame ntal frequency,

wFo

=

nc/2L. Unit horizontal rigid dam mo tion has generated the broadest peak at the second characteristic

frequency. This can be expected considering the value of the ratio o R/o.BOand the shape of the reservoir

frequency response function, sho wn in F igure 2(a), for the wave reflection coefficient

u

= 0.8

used in this

example. Similar effects are observed for the vertical motio n of the reservoir floor, Figure 3(c), where bro ad

peaks repeat at each reservoir characteristic frequency [see also Figure 2(b)]. The pressures in the frequency

domain also differ in phase.

Groun d m otions of the two recorded earthquakes, San Ferna nd o of February 9,1971 at Pacoima D am, and

Kern C ounty of July 21, 1952 at T aft Lincoln Sch ool Tunnel, have been used, with som e modifications, as

excitation functions for the outlet-dam-reservoir system. The modifications are mad e to represent the effects

of the plane P,SV and

SH

waves propagating throug h the half space.'

' *

he wavesare assumed to com e from

the earthquake hypocentre

in

the vertical plane Xl-X2, show n in Figu re l(b ), and with inci,dent angle

O , ,

which is seen in Figure l(a ). In the ho rizontal plane X l-X 3 the epicentral azimuth is defined by angle p.

Fo r the purpose of this example, the vertical recorded earthq uak e com pon ent

is

supposed to be associated

only with

P,

the largest horizontal w ith

SH

and another recorded horizontal acceleration with

SV

waves.

All

incident components are scaled arbitrarily in such a way that the largest component has the maximum

amp litude equal to 0.15

g.

Fou rier transfo rms of those accelerograms,

a,

are multiplied by the corresponding

transfer functions fiP.SV ,SHfo rccelerations du e to P, SV and SHwaves given by equ ations (A17) o (A19)

n

the

Appendix. The simulated accelerograms are then o btained alo ng coordin ate axes X1, X 2 and X 3 by inverse

Fourier transform of the resulting functions 6

.

Those are comp uted a t points

0

and

I,

shown in Figure

1,

for

two incident angles: do

=

30" and 8 ,

=

60 . A t the point

I

these accelerograms have accounted for the delay

(relative to point 0)d ep en din g on the incident angle O o the botto m outlet length L an d the chosen velocities of

longitudinal and transverse waves:

c L=

2500 m/s and

c T

= 1450 m/s, respectively. Th e resulting horizo ntal

(along the bottom outlet at p

= 45 ,

Figure l), and vertical earthquake ground motions at the point are

shown in Figure 4. This set of the simulated accelerograms is used to investigate possible effects of the phase

delay of the earthquake motions on the water response

in

the bottom outlet.

The hydrod ynam ic pressures in the time do ma in (for the first 30

s),

at five sections along the bottom outlet,

are evaluated by using the convolution integral given by equa tion (4).To preserve the stability of the num erical

procedu re som e investigation

of

the com plex frequency response functions may be necessary. Instability m ay

arise from the sh arp peaks at the characteristic frequencies of these functions and from the limited num ber o f

discrete points for their description. Stability can be achieved by increasing the num ber of po ints required by

the FFT algorithm, or by analytic integration in the vicinity of the high peaks of the frequency response


9/17

. 0

s =

L / 4

5 L / 2

_ _ _

s

.3L/4

5

w / w;o

.I

5

15

10

w /

w y

15

2 0

kd

2 0

2 0

Figure 3. Spatial variationsof the hydrodynamic pressure in the frequency domain a long the b ottom outlet: (a) unit horizontal motio n of

the downstream gate; (b) unit horizontal motion

of

the rigid dam; (c) unit vertical motion

of

the reservoir

floor


10/17

532

S.

B.KOJIC AND

M.

. TRIFUNAC

( a )

HORIZONTAL COMPONENT ( b ) VERTICAL COMPONENT

PACOIMA DAM, @=30

r

ACOIMA DAM, 4.30.

3r

- 3 1

L

N 3 - PACOIMA DAM, 8. =60 PACOIMA DAM, &60

u

\

E o

g

- 3 L

4: 3 r T A F T , 8,=30 T A F T ,

8.=30

a

w

J

w o

V

V

- 3 L

3r

L

T A F T , = S O 0 -

T A F T , e e = 6 0 a

t

-31 I I

I

I

I

I

I

L

0 5

10

I S 20

2 5

0

5

10

15 20

25

T I M E ,

s o c

T I M E , a e c

Figure

4.

Simulated accelerograms at the downstream gate

PACOIMA DAM, 8.=30 T A F T . 8o=3Oo

v;

W

LL - 2

13

v

v

-4

W

LL

L

PACOIMA DAM,

80=60

T A F T , e.=600

- 1 - 1

10 2 0 30 0 10 20 30

T I M E ,

s e c T I M E , s a c

Figure 5. Hydrodynamic pressures at the downstream gate generated by its motion


11/17


functions. Stability can be achieved also by increasing R in equation (l), .e. increasing th e w ater viscosity v

slightly above its real value. By doing this, sharp peaks are rounded and only 4096 points are used for

description of time or frequency functions an d without any significant loss in accuracy of the end results. Even

with the increased water viscosity, v, the hydrodynamic pressures, after the end of the excitation, may not

decrease much with time. Therefore, the special overlap-add procedu re to avoid end effects has been

applied.

The comp utation has been performed separately for each of the three b oundary conditions: motion of the

downstream gate, the dam motion and the reservoir floor motion. The simulated accelerograms shown in

Figure 4(a ) are used for the do wnstream gate excitation. The d am is subjected to two sets of the simulated

accelerograms for in-phase [shown in Figure 4(a)] and for phase shifted accelerations (see the Appendix).

Similarly, the reservoir bo ttom is excited by the com puted vertical com pon ents of accelerograms, for in-phase

[shown in Figure 4(b)] and for the phase shifted motions.

Figure

5

shows the variation of the pressure due to the downstream gate motions and associated

predominantly with the fundamental frequency of the system, although the presence of the second

characteristic frequency is noticeable. Fo r these mo derate accelerations (ab ou t 0.2

g),

the hydrodynamic

pressures have reached the value of the hydrostatic pressure (about

1

MN/m2).

The hydro dynam ic pressures caused by the horizon tal in-phase acceleration of the dam are show n in Figure

6(a). It is seen that the contrib utions t o the response com e mainly from the second characteristic frequency.

( b

PACOIMA DAM, &,=SOo

PACOIMA DAM

- 2

w t

4

t

A

T A F T ,

e.=600

2 -

0

- 2 -

- 4

0 10

20 30

0

10 20 30

T I M E , s e c T IME s a c

Figure

6.

Hydrodynamic pressures at the dow nstream gate generated by: (a) the horizontal motion of the dam ; (b) the vertical mo tion

of

the reservoir

floor


12/17

534

s.

B. KOJIC

A N D M. . T R I F U N A C

This is not surprising, because the complex frequency response function [Figure 3(b)] has the b roadest peak at

this frequency. The p ressures obtained from the accelerograms w ith shifted phase, not shown here, are smaller

than for the in-phase motion, shown in Figure 6(a). The highest amp litudes, ab ou t

1.8

MN /m2, are obtained

for the 'Pacoima Dam' excitation at incident angle

Bo =

60".

Substantially larger pressure amplitudes [shown in Figure 6(b)] result from the vertical motion of the

reservoir

floor.

This is expected

if

one no tices that the complex frequency response function, Figure 3(c), has

several broad peaks located at each of the reservoir characteristic frequencies. For the 'Pacoima Dam'

accelerogram s the pressures oscillate with the second chara cteristic frequency, while for the 'Taft' acc elerogram

both first and second m odes of vibration are present. The highest amplitudes, reaching

3 M N/ m 2

(three times

the hydrostatic pressure) are computed for the case

of

the 'Pacoima Dam' accelerogram with incident angle of

0,

= 30 .

The differences between the pressures com puted for in-phase and with phase shifted accelerograms

(not show n) are sm aller than in the above case.

Total hydrody namic pressures, show n in Figu re

7,

are computed by summ ing up the pressure time histories

obtained due to the gate motion, the dam motion and the reservoir floor mo tion. It is seen that in som e cases

the amplitudes are larger for the phase shifted than for the in-phase mo tion at po ints 0 and I. For earthquake

wavesarriving with

Bo

= 60", the presence of the fun dam ental frequency of the bottom outlet is noticeable. The

( 0 ) IN - PH ASE G R O U N D

M O T I O N

4 1 PACOIMA DAM, 8 o = 3 O o

N

E - 4 L

PACOIMA DAM, 80.60'

W

a

- 4 1

4 r

I

T A F T , 8 0 =3 0

4 t

( b ) GROUND MOTION WITH

SH IF T ED PH ASE

I

PACOIMA DAM,

&=3O0

I-

r

PACOIMA DAM,

&=60

L

T A F T , 8.=30

I-

T A F T , 8.=60

t

2

0

- 2

I I

I

I

I

L

0 10 20 30

T I M E ,

s e c

I I 1

I I

I

L

0 10

20 30

T I M E , s e c

Figure

7 .

Total hydrodynamic pressures at the downstream gate due to the gate, dam and reservoir floor motions


13/17


highest amplitudes, about 3.3 M N/m 2, are obtained for the 'Pacoima Dam' (0, = 30") accelerogram. All

pressure histories have shown no significant amp litudes decaying with time, even after th e earthq uak e m otions

completely stop. This is because of the sm all energy dissipation of the pressure waves in the bo ttom outlet.

Envelopes of the absolute maximum pressures computed from the pressure time histories, at five sections

along the bottom outlet, are shown in Figures 8(a) o (e) for the gate, dam and reservoir

floor

motions, and due

to fo ur earthqu ake excitation cases, discussed above (two with and two without the phase shift). It is noted that

the dam m otion and the reservoir floor mo tion increase the hydrody namic pressures substantially. The phase

shift of the accelerograms, caused by th e prop agati on effects from

0

o

I,

in som e cases decreases the pressures

by ab ou t 25 per cent [Figures 8 (b)an d (c)], and in other cases increases the pressures by

- 10

per cent [Figures

8( d) an d (e)]. At the section located at 4

L

from the downstream gate, the pressure am plitudes are decreased due

to a wave node located near this section.

Envelopes of the total, absolute maximum pressures are shown in Figure 9(a) for in-phase motion s and in

21

4

I I

I

I

I

I

s = o

L / 4 L / 2 3L/4 s = L

(GATE) (RESERVOIR)

Figure

8.

Envelopes of the absolute maximum hydrodynamic pressures along the bottom outlet: (a)horizontal mo tion of the downstream

gate; (b)horizontal in-phase motion of the dam; (c) horizontal mo tion o f the dam with shifted phase; (d) vertical in-phase mot ion of the

reservoir floor:

(e)

vertical motion of the reservoir floor with shifted phase


14/17

5

36

S.9. KOJIc

AND

M.

. TRIFUNAC

6

( b )

4 1

O L

I I

s = o

L / 4

L / 2

3L/4 s = L

( G A T E )

(RESERVOIR)

Figure 9. Envelopes of the total, absolute maximum hydrodynamic pressures along the b ottom outlet: (a) in-phase motion of the gate,

dam and reservoir floor; (b ) phase shifted motion of the gate, dam and reservoir

floor

Figure 9(b) or the phase shifted excitations. The simulated Pacoima Dam accelerogram

(0, = 30")

governs at

all sections. In this example for all earthquakes considered and at all sections, except at the bottom outlet

entrance, the hydrodynamic pressures range from 1.5 to 3.3 times the hydrostatic pressure of 0.98 MN/m2.

This indicates that, for even moderate earthquakes with peak acceleration - 0.2 g , large hydrodynamic

pressures

will

be developed within the first several seconds of excitation. This may cause non-linear response of

the water (not considered in this analysis) and consequently the walls of the bottom outlet may be exposed to

cavitation effects.

CONCLUSI ONS

This study examined the phenomenon of the transient pressures in hydrotechnical tunnels generated by

earthquakes. The method of the discrete Fourier transform has been applied and is shown to be useful in

solving hydraulic transient problems of this type.

An example of the bottom outlet has been investigated. To illustrate the phase delay of the earthquake

ground motions between the entrance and the end of the bottom outlet, the recorded accelerograms have been

reconstructed through wave propagation effects of P, SV and SH waves in the half space. From the examples

analysed, the differences in the hydrodynamic pressures, due to the phase delay

of

the excitation functions, are

10

to

25

per cent relative to the results computed for the in-phase motions. However, the site specific effects

of

the wave passage along the long bottom outlet will depend on the details of the local site geology, the ratio of

the outlet length L and the wave length of incident waves, and thus may be quite different in each particular

case (see for example Moeen-Vaziri and Trifunac *. 15).

The horizontal motion of the dam and the vertical motion of the reservoir floor have generated the

hydrodynamic pressure along the bottom outlet with substantially higher amplitudes than the motion of the

downstream gate itself. This

is

observed especially for the case of the vertical motion of the reservoir floor. In

other words, the hydrotechnical tunnel with closed downstream gate and under the reservoir hydrostatic

pressure represents an amplifier of the reservoir hydrodynamic pressures caused by earthquakes. Therefore,

the upstream boundary conditions cannot be omitted from the analysis of these structures.


15/17

HYDROTECHNICAL TUN NEL S DUR ING EARTHQUAKES 537

Along the length of the bo ttom outlet, hydrodynam ic pressures exceeded the hydro static pressure several

times. This indicates that, even for mod erate earthqua kes, cavitation may oc cur. In this case the bot tom outlet

lining should be designed to withstand larger positive pressures than those obtained in the linear response

analysis, because of the dy nam ic closure of cavitated regions,I6 and should be adequa tely stiffened to prevent

its buckling u nder the negative pressure.

The simple mathematical model presented in this paper can be further improved and extended by

introducing other b oun dary co nditions, like the surge tank located along the botto m outlet. This may decrease

the transient pressures and help decay the pressure oscillations after an earthquake.

ACKNOWLEDGEMENTS

S. B. Kojic is grateful to the University of S outhe rn California, Los Angeles, U.S.A. and Energo project Co.,

Hidroingineering D ivision, Belgrade, Yugoslavia, for their con tinuou s sup port during this work.

A P P E N D I X

In Figure 10 the coordin ate system at p oint 0 (from Figure 1) is repeated w ith additional designa tions for the

wave am plitudes and angles of incident an d reflected waves.

Accelerations due to plane P,

SV

and SH waves in the elastic half space can be computed from the

expressions given for their displacemen t amplitudes.'

* I 2

For point

I

(DX1, DX2

= 0,

DX3), located a t the free

surface, the ex pressions for accelerations

C

y V . S H i nhe

i =

1,2 and 3 directions are given in the following. In all

subsequent quantities representing the second derivative with respect to time, and designated by U in this

Appendix,

o

as been set to one an d omitted.

A ,

was also set equal t o o ne in all calculations, which here

address only the relative amplitudes at a given frequency, but at different locations.

Accelerations due to

P

waves

For

0

d

8, d 90"

accelerations in the X1 an d X 2 directions are

U p =

{ (A,

sin

0, A ,

sin

8,

A ,

cos

0,)

exp

(-

DX 1

K O

in

e, }

exp

(iot)

U,P=

{ (A, cos Oo

-

A , cos 0 , +A , sin 8,)exp ( - iDX IKo s in 0,)} exp ( iot )

( A l )

(A2)

where

8, =

O0 in0,

=

K - ' s h o o ,

K =

K , / K o , K O

=

o/c , and

K,

= o / c T

(1)

is the frequency of harmonic

waves while

cL

nd

c T

are the v elocities of long itudinal and transversal waves, respectively. Th e amp litude

t x 2

*'HYPO c E

N

T ER

Figure 10. Coordinate system with incident and reflected plane P, SV and SH waves


16/17

538 S.

B.

KOJIC A N D

M .

D. TRIFUNAC

ratios

of

the reflected waves an d the incident wave are

A ,

A ,

sin

28,

sin

20 ,

- cos2 20,

sin

20,

sin

20, + i z

cos2

20,

- -

and

AZ-

A ,

2 K sin

20,

cos

20,

sin 20, sin

20,

+

K 2

cos2

28;

-

Accelerations due to

S V

waves

When 0, < O , , [where

BCr

= sin-

( l / ~ ) ] ,

he accelerations are

USv

=

{ (-

A , cos

0,

A , sin

8, A ,

cos

0,)

exp

(

- DX I Ko s in

0,))

exp (iwt)

(A51

(A61

where 8, = Go, sin$, = KsinCt,, K

= c L / c T ,

K O=

K , ,

a n d

w,

c L nd cT are as previously defined. The

amplitude ratios

of

the reflected an d incident waves are

and

USv = { ( A , sin 8, - A , cos

8,

A, sin 0,) exp

( -

iDXlK, s in O , ) } exp ( io t )

- sin

48,

,

A ,

sin 20, sin

28 ,

+

i

cos

20,

A , -

sin

20,

sin

20 ,

-

cos2

20,

A , sin

28,

sin

20 , + i 2

cos2

28,

-

-

For 0, = Q,,,

0 ,

=

4 2 , and the amplitude ratios become

-=

- 1

,

A0

For

the case 0,

>

O,, accelerations are

U sv

=

{ -

A , cos

8,

exp

(-

D X l

K O

in

0,) +S

sin

0 ,

exp

[

- (DX1

K O

in 0, +

a ) ]

-

A , cos

0,

exp

[-

( DX1

K O

in

0,

+

2 a ) ] }

xp (iwt)

and

Uqv

= { A , sin Oo exp(- DX1

K O

in

8,) -

S c o s 8, exp [ ( DX1

K O

in 0,

+ a ) ]

- A , sin

8,

exp [- (DX

1

K O in

0,

+

2 a ) ] }

ex p (icot)

(A121

where cos0, = i(Ksin61,- l)/, si n0 ,

=

t i s indo ,

- A , sin 48,

[K

C O S ~8,

+4 ( ~ in Cto - 1)sin

20,

sin200]12

S =

2 ( ~

in2

eo

- 1)sin 28, sin

8

a

= t a n -

K c o s ~

8,

and

Accelerations due t o

S H

waves

This wave causes motion in the X 3 direction only, so tha t

U s H

=

{ 2 A o

exp

[

- (DX1KO in

do)]}

exp

( i w t )


17/17

HYDROTECHNICAL TUNNELS DURING EARTHQUAKES

539

In this case O 2

= e0,

A 2 = A o , A , = 0 and K O

=

o/cT

Equations (A l), (A2), (AS), (A6), (A ll ), (A12) and (A16) without exp (iwt) represent the transfer functions,

U

p,svorSHfor the har mo nic , plan e waves in the

i

=

1 , 2and 3 directions a t the free surface. These have been used

for the modification of the Pacoima Dam and Taft recorded accelerograms. This has been performed by

multiplication of the above transfer functions with the corresponding c om pon ents of recorded accelerograms

transferred in the frequency domain,

a a&,*

and

a

in the following way:

By the inverse Fou rier transfo rm of

d , 6

nd d , the simulated accelerograms are ob tained a t the specified

point on the free surface in the X1, X 2 and X 3 directions. Applying this procedure it was possible to sim ulate

the delay of ground motion between the intake and the end of the bottom outlet.

REFERENCES

1 .

0.C. Zienkiewicz, Hydrodynamic pressures due to earthquakes,

Wate r power constr. ,

15, Sept.

(1963).

2.

D. Obradovic, Analysis of transient phenomenon in the bottom outlet of the Haditha Dam (in Serbocroatian), Report by

3. M. H. Chaudhry, Applied Hydraulic Transients, Van Nostrand Reinhold, New York, 1979.

4.

V.

L.

Streeter and E. B. Wylie,

Fluid Mechanics,

8th edn, McGraw-Hill, New York,

1985.

5.

M.

D.

Trifunac and A.

G.

Brady, Correlations

of

peak acceleration, velocity and displacement with earthquake magnitude, distance

6.

S. H.

randall and W. D. Mark, Random Vibration in Mechanical Systems, Academic Press, New York, 1973.

7. J.

F. Hall, An FF T algorithm for structural analysis,

Earthquake eng. struct. dyn. 10, 797-811 (1982).

8.

A.

K.

Chopra, Hydrodynamic pressures on dams during earthquakes,

J . eng. mech. diu. ASCE, 93,

No.

EM6, 205-223 (1967).

9.

G. Fenves and A. K. Chopra, Effects of reservoir bottom absorption and dam-water-foundation rock interaction on frequency

10. E. Rosenblueth, Presion hydrodinamica en presas debida a

la

acceleracion vertical con refraccion en a1fonds,

2nd natl. cong. sers. eng.

1

.

J .

D. Achenbach, Wav e Propagation in Elastic Solids, North-Holland, Amsterdam, 1980.

12.

M.

D.

Trifunac,

A

note on rotational components of earthquake motions on ground surface for incident body waves,

Soil dyn.

13.

E. 0.Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, N.J.,

1974.

14.

N. Moeen-Vaziri and M. D. Trifunac, Scattering and diffraction of plane SH waves by two-dimensional inhomogeneities,Soil dyn .

15. N. Moeen-Vaziri and M. D. Trifunac, Scattering and diffraction of plane P and SV waves by two-dimensional inhomogeneities,Soil

16. 0. C.

Zienkiewicz,

D. K.

Paul and E. Hinton, Cavitation in fluid-structure response (with particular reference to dams under

Energoproject CO., Belgrade, Yugoslavia, Mar.

1982.

and site conditions,

Earthquake eng. struct . dyn. 4, 45 14 71 (1976) .

response functions for concrete gravity dams,

Earthquake eng. struct. dyn.

13,

13-31 (1985).

Veracruz, Mexico

(1968).

earthquake eng.

1 , 11-19 (1982).

earthquake eng. (in press).

dyn. earthquake eng. (in press).

earthquake loading), Earthquake eng. struct. dyn. 1 1 ,

46 34 81 (1983) .

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