[10][11][12][13][14][15], Rosenblueth [16], Nath [17 ...€¦ · Westergaard's and Chopra's formulae, we propose the following relationship for the calculation of the hydrodynamic

Hydrodynamic pressures on dams during

earthquakes - approximate formulation

G. De Martino, M. Giugni, G. de Marinis

Dipartimento di Ingegneria Idraulica ed Ambientale 'G. Ippolito ',

Universita degli Studi di Napoli Federico //, via Claudio 21, 80125

Napoli, Italy

Abstract

Whenever carrying out safety inspections on dams under construction in officially declaredseismic areas or in any zones of this ilk, one must always take into account not only the normalstatic effects, but also those triggered by the earthquake. This means one must consider both theinertial effect of the structural mass as well as that of the water in the artificial reservoir.

In this paper, it is demonstrated that the hydrodynamic pressure values for rigid damsand incompressible liquid are lower than those which can be extrapolated from the modelsbased on the assumption of compressible liquid and both rigid and elastic dams (reservoir-daminteraction) and this observation becomes increasingly relevant the greater the height of thedam above 100 m.

Therefore, we propose some simple but technically reliable formulae for theassessment of hydrodynamic pressure both using the assumption of rigid dam and compressibleliquid as well as that of dynamic reservoir-dam interaction.

1. Hydrodynamic Pressures on a dam induced by earthquake

motions

An evaluation study [1][2][3][4] of dynamic water pressures against the wallsof a hydraulic container during an earthquake motion has already been carriedout by us (barrages for artificial storage capacity, both subsoil and suspendedwater reservoirs, head tanks, water clarification tanks, etc.).

For barrages (figure 1), a critical analysis has been carried out on thevarious methods used to determine the hydrodynamic pressures which can ariseon the dam face during earthquakes. These methods are based on frequentlydiffering trends of thought and relate to both dams with vertical upstream faceas well as to dams with sloping upstream face (Westergaard [5], von Karman,Bakhmeteff, Zangar [6], Housner [7], Kulmaci [8][9], Chopra

Transactions on the Built Environment vol 20, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509

622 Earthquake Resistant Engineering Structures

[10][11][12][13][14][15], Rosenblueth [16], Nath [17], Chwang, Bustamante,Taylor and others).

Figure I - Scheme of reference

Particular attention has been lent to the various simplified hypothesesintroduced by the writers as a basis for their studies (two-dimensional motion,non-viscous liquid, compressible or incompressible liquid, small amplitudemotion of the liquid particles, infinite or finite tank length, rigid dam, reservoir-dam interaction). To this end, a sensitivity test was performed on theparameters that the above inertia! effects depend on, both in the case of rigiddam (with liquid taken to be both incompressible and compressible) as well asin that of dynamic interaction of coupled reservoir-dam system (elasticstructure) [3].

Therefore, we feel it is appropriate to summarise the results of theabove study, given that they are used later on in this paper for the simplifiedformulae put forward by ourselves.

1.1 General findings

a) The hydrodynamic pressure distribution on the upstream face, assumedvertical, of a dam excited by seismic perpendicular horizontal acceleration onthe face itself, is approximately of a parabolic type and the peak coincides withthe lowest point in the reservoir bed upstream from the barrage. The origin ofthe resulting thrust, the extent of which is proportional to the square of thewater depth and obviously to the seismic coefficient (ratio between earthquakeacceleration and gravitational acceleration) can be roughly calculated to be at adistance from the bed equal to 2/5 of the water depth;

b) earthquake acceleration horizontal and perpendicular to the upstreamface brings about peaks in the inertial action of the water filling the reservoir.Some methods (Chopra [10] and Nath [17]), given a hypothesis of verticalacceleration and of a conservative system, lead us to assess hydrodynamiceffect on an upstream face which can be greater than those due to horizontalmovements of the same frequency. But various dissipating phenomena (forexample, the compressibility of the bedrock) generally bring about - as notedby Rosenblueth [16] and by Chopra [15] - a significant reduction in the


Earthquake Resistant Engineering Structures 623

hydrodynamic effects. Therefore, for practical purposes, in the case of verticalmotions, the assumption that the hydrodynamic thrusts are equal to thosededucible for acceleration horizontal and perpendicular to the upstream facecan be held to be reliable and in many cases can also lead to greater safetylevels;

c) the slope of the upstream face still causes a parabolic pressuredistribution, but it becomes less so compared to a vertical face, as the angle awidens (figure 1). For a#0°, however, according to certain methods (forexample, the Housner method [7] and the Zangar method [6]), the peakpressure would not be had any more for y=yo but at relative depths rangingroughly between 0.90 and 0.70 as a increases between 5° and 60°;

d) a sloping of the reservoir bed causes, from a technical viewpoint, areduction in the hydrodynamic thrust and, moreover, a shifting towards highervalues of peak frequencies of the system [18]. The assumption of a horizontalbed, therefore, would appear to ensure better safety standards.

2. The Rigid Models

2.1 Dams with vertical upstream face

In the case of a rigid structure firmly locked to the foundation (so the horizontalearthquake acceleration, taken to be perpendicular to the dam axis, is assumedto be evenly spread out over the whole barrage) and of incompressible liquid(as, for example in Zangar' s methods [6] - which current Italian Regulationsrefer to [19] - and Housner' s methods [7]), the hydrodynamic pressures ofseismic origin are not connected to the exciting frequency and therefore, to theperiod of the earthquake T, but only to the water depth y^, to the relative depthy/Vo, to the sloping angle a, and obviously, to the seismic coefficient C.

Consequently, the ratio between the inertial thrust So of the water on theupstream face and the hydrostatic thrust can be considered independent fromthe water depth and only a function of C and a.

In the case of a rigid dam and compressible liquid (for example, in themethods of Westergaard and Chopra, which only make reference to barrageswith a vertical upstream face), the hydrodynamic pressure is dependent not onlyon )'(„ y/>'o and C, but also on the ratio between the natural period of thereservoir T, and the earthquake period T and therefore on the coefficient 11:

T

The period of the reservoir T,, equal to 4y</Cp, Cp being the soundvelocity in water, can be computed with good accuracy (expressing T, inseconds and y« in metres) with the relationship [2] [13] [20]:



T - (2)

Consequently, the hydrodynamic pressure value is no longer found to bea linear function of the water depth. Therefore, the ratio So/Sj depends not onlyon the seismic coefficient, but also on the above-mentioned ratio Q. If we takeinto account the liquid compressibility, the extent of the hydrodynamic thrust isfound to increase with Q, in the range 1>Q>0 and therefore, given the sameearthquake period, to increase with the depth of the reservoir and, obviously,given the same y<,, with the reduction of the earthquake period.

To this end, having computed numerous calculations carried out withWestergaard's and Chopra's formulae, we propose the following relationshipfor the calculation of the hydrodynamic thrust on the upstream vertical face of abarrage:

c(3)

So, as can be gathered from figure 2, where we can see as a function ofQ the K ratio between the So thrust and the S, hydrostatic thrust, divided by theseismic coefficient C, this relationship allows us to assess So with deviations ofonly a few percentage units compared to the values deduced fromWestergaard's method.

5

K/C4

-Westergaard |

Formula (3) |

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 Q 1

Figure 2 - K/C ratio curve versus Q (rigid dam and compressible liquid)

Especially, from (3), we can see that S« increases with Q in a hyperbolicratio, taking, for Q>0.25, values which increase progressively as Q growscompared to the values gathered from methods which assume the liquid to beincompressible.



If we can assume the pressure distribution to be significantly parabolic,it is straightforward enough to deduce from (3) the extent of the hydrodynamicpressures at a general point of the face at depth y:

as well as the thrust at depth y:

. L 0")

y being the water specific weight.

2.2 Dams with sloping upstream face

In order to take into account the slope of the upstream face of the barrage, andtherefore of the conseguent reduction in the values of the hydrodynamicpressure and of the resulting thrust, many writers suggest introducing into theexpressions deduced for the vertical face, a derating coefficient, to bedetermined as a function of the angle a formed with the vertical axis.

To this intent and purpose, Housner [7], in the case of rigid dam andincompressible liquid, for 35°>oc>0°, suggests the following relationship:

P = — / sina | (4)co.ra 2

whilst for values of a greater than 35° the expression of (3 turns out to beextremely complex and has not been listed here for brevity's sake.

Zienkiewicz and Nath [21] suggest, in the same cases, with reference toZangar's experiments, the approximate relationship:

7C— a

-V2

From Kulmaci's method [8][9] (rigid dam and compressible liquid), wecan deduce the following expressions:

8 = / — for45°>o>0° (6)153.6



5 - COSOL for (6')

In the following table we have written out, as a function of the angle a,the values of the derating coefficient deducible from the relationships (4), (5),(6) and (6') as well as those recommended by the Italian Regulations.

oQ %Housner (g) || Nath (F)

0,005,00lorn15,00moo25.0030.0035.00moo45,0050.0055.0060.00

1.0000.9320.8760.8320.7970.7720.7560.7500.5530.5030.4610.4230.383

1.0000.9440.8890.8330.7780.7220.6670.6110.5560.5000.4440.3890.333

Kulmaci (8)| Ital.Reg.

1.0000.9670.9350.9020.8700.8370.8050.7720.7400.7070.6430.5740.500

1.0000.9450.905

0.811

0.608

0.405

£

1.0000.9500.9000.8500.8000.7500.7000.6500.6000.5500.5000.4500.400

From the data in the table above, we can easily see that when the angleof the upstream face is not greater than approximately 15°, the deviationsbetween the values of the derating factor provided by the methods quoted hereas well as those contained in the Italian Regulations are quite small. On theother hand, for a >15°, Kulmaci provides derating coefficient values which arestill greater than those which may be inferred from the formulae (4) and (5).

To this end, on the basis of the values given by the derating coefficientas a function of a and written out in the table, we propose the following simpleexpression for the assessment of the above coefficient which is designated by e:

(7)

whose values as a function of a can be also found in the table. If compared, wecan see that the deviations between (3, F, 8, the Italian Regulations and 8 arequite minimal, apart from those proposed by Kulmaci when a is greater than15°.

3. The Elastic Models

For gravity barrages with a vertical upstream face, excited by seismicacceleration horizontal and perpendicular to the dam axis, some mathematicalpatterns have been proposed which take into account the elasticity of thestructure, imagined to be like a cantilever beam of variable cross-section. Inparticular, if we make the assumption of interaction of reservoir-dam system



and of compressible liquid, (as in Chopra's methods [10][ 1 1 ][ 12][ 13][ 14] andNath's method [17]), as pointed out in the paper previously quoted [2|, thehydrodynamic pressures depend not only on C, y^, y/y^ and but also on the^r ratio between the period of the actual vibration of the structure, 7\. and thatof the reservoir, T,:

V r = r (8)

In (8), with reference to a horizontal vibratory motion, the period T\, inseconds, can be assessed with fair accuracy with the relationship [2]:

7 = — - (9)^ ^

(Hs being the height of the dam in metres), which can be considered to beaccurate when the ratio between the base and the height of the gravity barrageassumes values ranging from 0.7 and 0.8, as takes place generally in realbuilding processes.

As the two periods are respectively proportional to the height of thedam and to the water depth, the ratio Q^ is therefore an index of the filling ratioof the reservoir.

Given that, for practical purposes we need to work with the mostextreme conditions, that is to say, with maximum storage capacity in thereservoir, we will have to make reference to the value Q 0.68, for theassessment of hydrodynamic action, assuming in all accuracy that the height ofthe dam coincides with the depth of the reservoir (<T2r=340/500=0.68).

Having carried out numerous calculations, using the quite complexformulae provided by Chopra, we propose the following relationship for a rapidassessment of the S,,, significantly accurate in the range 0.80>O>0:

,, C

The above, as can be gathered from figure 3, in which the ratio K/C iswritten out as a function of Q, allows us to assess So with deviations of only afew percentage units compared to the values deducible from Chopra (and to beexact, negative for Q<0.14 and positive for values of Q>0.15).

Especially from (10), we can see that So increases with Q in ahyperbolic ratio taking on, for Q>0.25, values which become progressivelygreater than those attributed by Westergaard's method as the value of Q goesup.



For values of D included in the range 0.80 -0.90, we propose thefollowing expression:

K/C Chopra

Formula (10)

Formula (11)

-4-0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 ^

Figure 3 - K/C ratio curve versus Q (reservoir-dam interaction)

C

This allows for the assessment of the So with quite small deviationscompared to Chopra's values, and in particular positive for values of £2<0.88(figure 3), whilst for Q>0.90 - that is to say for barrages higher than 300 m -the extent of the hydrodynamic thrust provided in (11), tends towards lowervalues than those attributed by Chopra's formulae.

In figure 4 we have drawn up an abstract of the results of our analyticalstudy. In particular, in the above mentioned graph we have written out, withreference to a gravity dam with a vertical upstreame face, the values of the ratioK/C as a function of Q (in the range 1>£2>0), both with reference to the case ofrigid dam and compressible liquid (relationship (3)) and with reference tostructure-reservoir interaction (relationship (10) and (11)), as well as the K/Cvalue (which is, obviously, independent from Q) drawn from the ItalianRegulations in force (K/C=1.074) in the case of rigid dam and incompressibleliquid.

An examination of this graph will show that the effects of the watercompressibility and of the reservoir-dam interaction start to become tangible atvalues of Q>0.25. Therefore, hydrodynamic thrusts much greater could occurfor values of Q, greater than those mentioned above, than those assessablewithout taking into consideration fluid compressibility and structural elasticity.For example, for £2=0.75, the K/C ratio takes on the value of roughly 1.62according to (3), 2.46 according to (10), and 1.074 in the case of rigid dam and



K/C •4

Zangar

Formula (3)

J Formula (10)

... Formula (11)

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

Figure 4 - K/C ratio curve versus Q deduced from the Italian Regulations andfrom the approximate formulae proposed

incompressible liquid.For values of Q>1, the extent of the thrust tends to diminish, whilst for

£2=1 (earthquake period which coincides with that of the reservoir) thehydrodynamic actions tend to reach infinitely higher values (resonance).However, bearing in mind that for practical purposes, we can take as areference T values not lower than one second, excluding especially high dams(roughly 300 350 m high), the phenomenon of resonance should not occur[22].

From (10) and (11), if we still take the pressure distribution to besignificantly parabolic, we can deduce the following formulae:

nP = —4

^ =0.5Cy\-I

for 0<Q<0.80, and

4

for 0.80<S1<0.90.These allow us to assess the hydrodynamic pressure and thrust at the

general depth y.



We likewise believe that, for technical purposes and with fair accuracyone can still refer to the relationship (7) in order to take into account theinfluence of the slope of the structure's upstream face on the extent of thehydrodynamic pressure.

4. Approximate formulae for the assessment of hydrodynamic

pressure

From what has been said above, it is obvious that the extent of thehydrodynamic pressure caused by an earthquake motion, characterized byhorizontal acceleration perpendicular to the dam axis, on the upstream face of abarrage, assessed in the cases of rigid structure and incompressible liquid, canbe taken to be:

- lower than that deducible in the case of rigid dam and compressibleliquid, when the Q ratio takes on values greater than 0.25- 0.30;

- much lower than that deducible considering the reservoir-daminteraction.

Therefore, we believe that for the assessment above, reference shouldbe made to Westergaard's method when a barrage is assumed to be rigid and toChopra's when one cannot do anything else but assume that the structure isflexible.

In particular, we believe that for arch dams and earth darns, given theirspecific characteristics (barrage constraints in the first case where structuralelasticity is generally quite limited; material composition in the second casewhere any interaction effects should be absorbed locally), reservoir interactioncan be assumed to be negligible. On the other hand for gravity barrages, it isnot a good idea to overlook the elasticity of the structure, and therefore adynamic analysis of the reservoir-dam interaction should always be carried out.

To this intent, we propose the approximate expressions (3), (3') and(3") drawn up by ourselves in the case of rigid dam and compressible liquid,and (10), (101 and (10") for U<0.80 as well as (11), (IT) and (11") for0.90>£2>0.80, in the case of reservoir-dam interaction. These relationshipswere obtained by assigning values ranging between 0.2 and 2 seconds to theearthquake period T and values ranging between 10 and 350 m to the height ofthe barrage [2].

In the above relationships the So is a function of Q=Tj/T and therefore,of the predominant earthquake period, which the ground acceleration peakvalues correspond to. The earthquake period generally has values rangingbetween 0.5-H .5 seconds, especially as a function of the ground features.

For design purposes, many writers (Westergaard [5], Zangar [6], Bratu[22], Chopra [ 10]) believe that, for the assessment of hydrodynamic actions ondams during earthquakes, one can make accurate reference to a period ofroughly 1 second.



Therefore, with reference to the assumption of rigid dam with avertical or sloping face and compressible liquid, (3), (3') and (3") can bewritten out as:

S CK - __iL - 7_7 (j, (J40-yj""

/> = . Cy

As can be gathered from figure 5, in which the ratio K/C has beenwritten out as a function of the water depth to the maximum reservoir storagecapacity (and therefore, with good accuracy, of the height of the barrage), thedeviations contained in the values deducible from Westergaard's method andthose provided by (12) are quite small, and moreover contributes to highersafety standards as already has been made clear in figure 2.

5

K/C4 Zangar

Westergaard

Formula (12)

0 50 100 150 200 250 y j 350

Figure 5 - K/C ratio cun'e versus y,, (rigid dam and compressible liquid)

In figure 5, we have also written out the constant value of the ratio K/Cdeducible from the current Italian Regulations (based, as already mentioned, onthe hypothesis of rigid dam and incompressible liquid), equal to 1.074, whichfor dams of a height greater than 100 m comes to less (and increasingly so as y«goes up) compared to those provided in Westergaard\s method, and therefore,by the approximate formula proposed. Bearing in mind that for dams (evenearth ones) the height goes over 300 m, the hydrodynamic thrust can reachvalues equal to even 2^3 times those deducible from the hypothesis of rigidstructure and incompressible liquid.



For gravity dams higher than 100 m, given the sort of constructionthey are, it is not a good idea to overlook the elasticity of the structure and itwould be wise to refer to Chopra's methods, and therefore to the approximateformulae proposed which can be written, with reference to the assumption of

T=l s:

,(340 -y^)

accurate for dams of a maximum height of 275 m and:

(14")

for values of y« ranging from 275 to 300 m.In figure 6 we have written out the ratio K/C as a function of the water

depth deduced from formulae (13) and (14), as well as the constant value(K/C= 1.074) deducible from Zangar's method (which the Italian Regulation inforce are based on). From this figure, apart from noting the low deviationsbetween the values of the thrust So computed on the basis of approximateformulae and those assessable using Chopra's methods, starting with damshigher than 100 m, the considerable difference between the values provided by(13) and (14) and those deduced from the hypothesis of rigid dam andincompressible liquid, should become evident to us. For example, for a 300 mhigh gravity dam, the So obtains a value equal to roughly 5 times that calculatedwith the Zangar's method.



Zangar

Chopra

Formula (13)

Formula (14)

50 100 150 200 250 y (m)

Figure 6 - K/C ratio curve versus y<, (reservoir-clam interaction)

5. Conclusions

Even though, in view of the importance of dams, model tests, especially onvery tall structures, are always advisable, especially with a view to lookingmore deeply into the resonance factor, for an assessment of the hydrodynamicpressures which can be generated on the upstream face of a dam in a seismicarea, we propose these simplified but technically accurate formulae, both in thecase of rigid dam with compressible liquid as well as in the case of reservoir-dam interaction.

They allow for a simple determination of the hydrodynamic thrustswhich can be much greater than those deducible from the assumption of rigiddam and incompressible liquid, which for example the Italian Regulations inforce base themselves on.

References

[1] De Martino, G. & Giugni, M. Azioni idrodinamiche indotte da moti sismicisu strutture idrauliche, Giornale del Genio Civile, 1983, fascicolo l°-2°-3°,Gennaio-Febbraio-Marzo.

[2] De Martino, G. & Giugni, M. Effetti idrodinamici sulle dighe disbarramento durante terremoti, Giornale del Genio Civile, 1983, fascicolo4°-5°-6°, Aprile-Maggio-Giugno.

[3] De Martino, G. & Giugni, M. Azioni delle onde sismiche sui contenitoriidraulici interrati, G/orWf &W Gfmo Cav/c, 1989, fascicolo 10°-11°-12°.Ottobre-Novembre-Dicembre .

[4] De Martino, G. & Giugni, M. Azioni idrodinamiche su una diga in zonasismica. Proposta integrativa alia normativa vigente, Giornale del Genio



Civile, 1992, fascicolo 10°-1 r-12°, Ottobre-Novembre-Dicembre.[5] Westergaard, H.M. Water pressures on darns during earthquakes,

Transactions American Society of Civil Engineering, 1933, November.[6] Zangar, C.N. & Haefeli, R.J. Electric analog indicates effect of horizontal

earthquake shock on dams", Civil Engineering, 1952, April.[7] Housner, G.W. Dynamic pressures on accelerated fluid containers, Bulletin

of the Seismological Society of America, 1957.[8] Kulmaci, P.P. E'h\drodynamique des constructions hydrotechniques, Ed.

de L'Academie de Sciences d'U.R.S.S., Moscue, 1963.[9] Kulmaci, P.P. Methode pratique pour la determination de Faction de 1'eau

sur les constructions hydrotechniques massives soumises aux oscillations,Revue de I'lnstitut National pour la Recherche Scientifique 'V. E. Vedeev',1964, n. 74, Moscue.

[10]Chopra, A.K. Hydrodynamic pressures on dams during earthquakes,Journal of the Engineering Mechanics Division, A.S.C.E., 1967, December.

[1 l]Chopra, A.K. Reservoir-dam interaction during earthquakes, Bulletin of theSeisrnological Society of America, 1967, August.

[12]Chopra, A.K. Earthquake behavior of reservoir-dam systems, Journal ofthe Engineering Mechanics Division, A.S.C.E., 1968, December.

[13]Chopra, A.K. Earthquake response of concrete gravity dams. Journal of theEngineering Mechanics Division, A.S.C.E., 1970, August.

[14]Chakrabarti, P. & Chopra, A.K. Earthquake analysis of gravity dams,Earthquake Engineering and Structural Dynamic, 1973, vol. 2.

[15]Hall, J.F. & Chopra, A.K. Two-Dimensional Dynamic Analysis ofConcrete Gravity and Embankment Dams Including HydrodynamicEffects, Earthquake Engineering and Structural Dynamic, 1982, vol. 10.

[16]Rosenblueth, E.: Discussion on Hydrodynamic pressures on dams duringearthquakes by A. K. Chopra, Journal of the Engineering MechanicsDfvAs/^z, AS.C.E., 1968, August.

[17]Nath, B. Hydrodynamic pressures on high dams due to vertical earthquakemotions. Proceedings Institute of Civil Engineers, London, 1969, March.

[18]Ausilio, E., Conte, E. & Dente, G. Una procedura numerica per il calcolodelle pressioni idrodinamiche sulle dighe a gravita, XXV Convegno diIdraulica e Costruzioni hlrauliche, Torino, 1996.

[19]Decreto Ministerial 24/03/1982 Norme tecniche per la progettazione e lacostruzione delle dighe di sbarramento, Gazzetta Ufficiale del laRepubblica Italiana del 04/08/1982.

[20]Okamoto, S. Introduction to Earthquake Engineering, University of TokyoPress, 1973.

[21]Zienkiewicz, O.C. & Nath, B. Earthquake Hydrodynamic Pressures onArch Dams - an Electric Analogue Solution, Proceedings Institute of CivilEngineers, 1963, London, vol. 25, June.

[22]Bratu, C L'action hydrodynamique sur les barrages construits en regionsseismiques, X Convegno di Idraulica e Costruzioni Idrauliche, Cagliari,Febbraio 1967.


Documents

[10][11][12][13][14][15], Rosenblueth [16], Nath [17 ...€¦ · Westergaard's and Chopra's formulae, we propose the following relationship for the calculation of the hydrodynamic