Seismic

5/24/2018 Seismic

1/13

Engineering Structures 27 (2005) 616628

www.elsevier.com/locate/engstruct

Seismic structural stability of concrete gravity dams considering transientuplift pressures in cracks

Farrokh Javanmardi, Pierre Lger, Ren Tinawi

Department of Civil Engineering, cole Polytechnique de Montral, University of Montreal, P.O. Box 6079, Station CV, Montral, Qubec, Canada H3C 3A7

Received 23 August 2004; received in revised form 29 November 2004; accepted 3 December 2004

Abstract

A theoretical model is developed for transient water pressure variations along a tensile seismic concrete crack with known crack wallmotion history. Experimental tests are performed to validate the proposed model. Experimental and numerical results show that water canpenetrate into new seismic cracks making them partially saturated over a length L sat. The magnitude ofL satand the total water uplift forceacting on a crack wall are decreased by crack opening and increased by crack closing. The model is then implemented in a nonlinear discretecrack finite element program for seismic analysis of concrete dams. A 90 m high gravity dam subjected to two different ground accelerationsis analysed. The magnitude of uplift force in the opening mode of the crack is small such that the downstream sliding safety factor (SSF)during crack opening is similar to the SSF assuming zero uplift force in the crack. Although the transient uplift force during crack closingreduces the upstream SSF, compared to a similar value assuming zero uplift force in the crack, its magnitude still remains larger than theminimum downstream SSF corresponding to the crack opening mode. 2005 Elsevier Ltd. All rights reserved.

Keywords:Dams; Concrete; Cracking; Seismic stability; Water uplift pressure; Seismic water crack interaction model

1. Introduction

A linearly varying static uplift pressure along thedamfoundation interface (in the case of no drainagesystem) is defined in dam safety guidelines [13]. Howeverthere is not a unified definition for the uplift pressureduring earthquakes when seismic cracks develop along thedamfoundation interface or in the dam body. Dam safetyguidelines are developed with the objective of ensuringthat while concrete dams could suffer cracking and damage

during the maximum design earthquake (MDE) they mustmaintain a stable condition to retain the reservoir. TheMDE oscillatory motions could initiate and propagate newcracks in mass concrete or activate opening and closingcycles along existing lift joints and cracks. Due to dynamic

Corresponding author. Tel.: +1 514 340 4711x3712; fax: +1 514 3405881.

E-mail addresses:[email protected] (F. Javanmardi),[email protected] (P. Lger), [email protected] (R. Tinawi).

0141-0296/$ - see front matter 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2004.12.005

pressure variations at the crack mouth in contact with thereservoir, and due to the rapid crack wall motions duringan earthquake, the water pressure inside a propagating oran existing crack becomes a transient variable modifiedfrom its initial hydrostatic condition. It was shown bylaboratory experimental tests [4,5] that water pressuremay decrease during opening and increase during crackclosing (Fig. 1). An accurate evaluation of the seismic upliftpressure along cracks is one of the most important aspectsin stability evaluation of concrete gravity dams during

earthquakes. Yet, due to the lack of historical, experimentaland numerical evidences, different hypotheses for evaluatingseismic uplift pressure along a crack have been retained,in dam safety guidelines, ranging from full crack mouthreservoir hydrostatic pressure [1] to zero pressure [2].

The cracking response of concrete dams using a discretecrack or a smeared crack finite element approach has beenextensively investigated by research in the last few decades.Computer programs have been developed for nonlinearanalysis of concrete dams subjected to earthquake loads.
http://www.elsevier.com/locate/engstructhttp://www.elsevier.com/locate/engstruct

5/24/2018 Seismic

2/13

F. Javanmardi et al. / Engineering Structures 27 (2005) 616628 617

Fig. 1. Static and transient uplift pressures.

Although it is anticipated that water pressure may changeduring an earthquake, due to the complexity of the problemtransient water pressure in concrete cracks remains a majorsource of uncertainty in time history analysis of crackedconcrete gravity dams subjected to dynamic loading.

Following a simplified numerical modelling methodol-ogy, Hall [6]accounted for the transient nature of the seis-mic uplift pressures in smeared crack analysis of arch damsby considering that hydrodynamic pressure variations due todamreservoir interaction at the upstream face are also act-ing at the mouth of cracks (joints). It was further assumed

that this pressure varies linearly along smeared crack bandsacross the dam sections. No interaction between crack wallmotions and related water pressure was considered. It wasfound that this dynamic internal water pressure effect wasable to alter the computed cracking pattern, crack openingand sliding responses. Ohmachi et al. [7] performed a se-ries of shaking table experiments to measure water pressureinside narrow cavities like cracks (30 cm long) inserted insmall submerged acrylic blocks (approx. 50 cm3), but nocrack wall relative motion (openingclosing) was permitted.These results were used by Zhang and Ohmachi [8] who pro-posed that the hydrodynamic pressure acting at nodei in ahorizontal crack filled with water can be expressed as

Pi (t)= [Pstat + Pwest(t)+ax(t)xci ] (1)

where Pstat is the hydrostatic uplift pressure at thecrack mouth, Pwest is the dynamic crack mouth pressureapproximated by Westergaards theory, is the water massdensity,ax is the total horizontal dam acceleration at nodei ,andxci is the distance between the crack mouth and node i .Seismic smeared crack analysis of the Konya dam using theproposed model, assuming that hydrodynamic pressure incracks develops instantaneously upon cracking, showed thathydrodynamic pressure inside cracks tends to increase thecrack length. A basic limitation of this approach is that the

effect of crack wall motions on the transient seismic waterpressure along the crack is neglected.

The hydrodynamic pressure in a crack growing in brittlematerials, under cyclic loading, is also studied by researchersin materials science. Yi et al. [9] developed a fluid pressurerelation assuming one-dimensional flow along the crack thatis applicable to a crack of arbitrary shape. The results werecombined with a fracture mechanics model to compute thecrack tip stress intensity. The test results have shown slowercyclic crack growth rate in wet brittle materials comparedto the cyclic crack growth rate in dry conditions. The model

was also modified to consider cavitation near the crack tipby considering pressure balance in this region. However, theproposed modified method was developed assuming a smallcrack length (Lcr = 20 mm)that is not applicable for thecase of cracks in concrete dams where crack lengths aremuch longer.

An analytical solution for water pressure variation inan existing crack due to crack wall motion was presentedby Tinawi and Guizani [10]. They assumed a water filledexisting rectangular crack with constant length, smallamplitude crack wall motions (relative to their initialaperture), and one-directional laminar water flow to simplify

the solution. On the basis of the proposed equation,equivalent added masses and damping were derived that canbe used in a finite element based computer program. Thecase study of a 55 m high gravity dam subjected to typicalearthquake loading shows that the maximum pressure canbe as high as 2 times the static pressure and the minimumpressure can be as low as the cavitation pressure.

Slowik and Saouma [5] measured the water pressure de-veloped along a concrete crack in a dynamic wedge splittingtest. They also developed a theoretical model to simulate thetest results [11]. The model developed is applicable to theinitial crack propagating phase during an earthquake but thesubsequent crack closing and opening was not considered.

5/24/2018 Seismic

3/13

618 F. Javanmardi et al. / Engineering Structures 27 (2005) 616628

Seismic watercrack interaction was investigated theoret-ically and experimentally [12]. The experimental results fornew crack and existing crack tests and the theoretical modeldeveloped for simulation of test results were presentedin [4].The variation of the water pressure along new developingcracks in concrete gravity dams is presented herein. The pro-

cedure for computing dynamic water pressure using crackwall motion time history is first presented briefly. The imple-mentation of the proposed method in a finite element basedcomputer program is then explained. A 90 m high concretegravity dam subjected to two typical North American earth-quakes is analysed considering transient water crack inter-actions. The results of these analyses are compared with theresults of similar analyses where there is no hydrodynamicpressure in the cracks.

2. Water flow and water pressure in cracks with moving

walls

Propagating cracks during earthquakes have oscillatoryopening and closing motions.Fig. 2shows a phenomeno-logical description of water flow and corresponding pressurein a propagating crack during its first two successive open-ing and closing cycles. Neglecting water flow across crackwalls (impermeable walls), water exhibits a 1D flow alongthe crack.Fig. 2(a) shows the water flow along the crackduring its first opening cycle. The existing water pressure atthe crack mouth pushes water into the crack and water flowsfrom the crack mouth toward the crack tip. Water fills a partof the void developed due to crack opening and saturates,partially, the propagating crack. The magnitude of the pres-

sure developed along the saturated length (Lsat)is not con-stant. At the crack mouth, it is equal to the reservoir pressurepstat, and it is decreasing from the crack mouth toward thecrack tip (pressure losses due to flow) to become equal to thevoid pressure at the end of the saturated region. The magni-tude of the void pressure is determined by the existence ofair in the developed void. In a case where there is no air inthe concrete, the void pressure decreases to the water vapourpressure (cavitation) and water vapour fills the void. If thereis air in the concrete pores, then depending on the ratio ofair to void volume, the void pressure will vary between thewater vapour pressure and zero.

The volume of void decreases as the crack closingmode begins. The water flow from crack mouth towardcrack tip continues during the closing mode as long as thecrack closing velocity is small. On increasing the crackclosing velocity, the existing water between crack wallsis pressurized (like water squeezed between two closingparallel plates). The water flow changes direction. It nowflows in two opposite directions from a stagnation pointalong the saturated part of the crack (Fig. 2(b)). The valueofL satstill increases due to water flow from the stagnationpoint toward the crack tip. The water pressure is maximumat the stagnation point and it decreases from this pointtoward the saturated region boundaries to become equal

to the pressure at these two points. The location of thestagnation point,L sat, and the pressure magnitude along thecrack change with time and are functions of the crack closingvelocity, crack aperture, crack roughness, crack length, andexisting crack mouth pressure.

An important aspect related to water flow in an oscillating

crack is that a complete hydraulic closure of a crack isalmost impossible. A small space remains between crackwalls even after they get in mechanical contact. Compressingcrack walls reduces the crack aperture, but due to crackwall asperities and the existence of free sedimentary material(from crack wall local crushing of concrete particles), itnever becomes zero. A residual hydraulic crack aperture,ures, is defined as a function of the crack wall surfacegeometric properties (u = umech + ures). Thus, aftermechanical contact of crack walls, a small amount of waterexists in the space between the walls.

In subsequent opening cycles, new voids are developingalong the crack. Water flow occurs along the crack from the

crack mouth toward the crack tip (Fig. 2(c)). Water flow canonly fill the voids close to the crack mouth; the voids in therest of the crack remain unfilled andL satdecreases. The onlydifference between the first opening and subsequent openingcycles is the existence of some water in the residual openingthat was already filled due to crack closing. The water flowand corresponding pressure during the second closing cycleis therefore similar to that of the first closing cycle exceptthat there is already some water in the wetted unsaturatedregion. Thus, Lsat may be longer at the end of the secondclosing cycle (Fig. 2(d)).

3. Theoretical modelling

Exact theoretical modelling of the water flow in crackswith moving walls, considering the possibility of cavitationand two-phase flow along the crack, is a very complexproblem. But with some assumptions, it is possible toformulate a simpler, yet representative problem. The mainassumptions retained herein are that:

A tapered crack with known crack length, Lcr, is usedto model the crack geometry and rigid body crack wallmotions,(u(x, t)= u(L, t)x/L).

Crack walls are impermeable. Water flow is one dimensional along the crack length. There is no significant water flow in the unsaturated part

of the crack. The water vapour pressure is taken equal to zero. The pressure gradient expressions for laminar and

turbulent flow,Q(x, t), in the saturated region of a roughcrack are [13,14]

dP(x, t)

dx

laminar

=6[1+8.8(k/2u(x, t))1.5]

Q(x, t)

u3

(x, t)

(2)

5/24/2018 Seismic

4/13


Fig. 2. Water flow and water pressure along a new crack with oscillatory crack wall motions.

dP(x, t)

dx

turbulent

=

16

log 1.9k/2u(x,t)

2 Q2(x, t)

u3(x, t)(3)

whereis the dynamic viscosity of water,u is the crackaperture, k/2u(x, t)is the relative crack roughness (taken

as 0.5 for rough concrete tensile cracks), and is the massdensity of water.

A computer program is developed for numericalintegration of Eqs. (2) and (3) along the crack saturatedlength, while the flow continuity and pressure compatibilityare satisfied [12]. Using this computer program, waterpressure variations along a crack with known crack wallmotion historyu(x, t), crack mouth pressure Pcrm(t)(equalto the hydrostatic value Pstat or to the hydrostatic plushydrodynamic values), and crack length (Lcr) can becomputed. This program is applicable for new cracks as wellas initially saturated existing cracks.

4. Experimental program

Experimental tests were performed to measure waterpressure along concrete cracks with moving walls. Theconcrete specimens and test set-up are shown in Fig. 3.

The concrete specimens (0.15 m 0.55 m 1.5 m)are fixed to a very stiff steel supporting structure andare attached to the shake table by a rigid link to applydisplacements. A small notch was created in the specimensduring pouring of concrete to induce a crack at a certainlevel. A small steel tank was added in front of the notchto approximately model the reservoir in a real dam. Anadditional water tank pressurized with an air compressor isused to provide the desired static water pressure in the notch.A thin waterproof membrane is glued to the concrete surfacearound the notch and the expected crack path. Four holes,that intercept the estimated crack trajectory, were used tomeasure the pressure from pressure transducers (PT) at the

5/24/2018 Seismic

5/13


(a) Specimen (dimensions in m). (b) Test set-up.

Fig. 3. Experimental study: (a) specimen geometry; (b) test set-up.

top of the specimens. Using this experimental set-up, fivenew specimens are tested. Water pressure variations alongthe propagating crack and subsequent closing and openingharmonic motions are measured. Letting the crackedspecimens be completely saturated, more than 220 tests havebeen done. Harmonic oscillatory displacement control testswith different frequencies,crack opening displacements, andcrack mouth pressures were performed. Existing crack andnew crack test results are used to validate the proposed

theoretical model [4,12].Fig. 4(a) shows the measured pressure as well as

computed pressure for a new crack test ( Pcrm = 100 kPa,f = 2 Hz). Initially in the absence of a crack, the pressurein all PT is zero. The pressure drops in PT2, PT3, and PT4during the crack initial propagation confirm the developmentof voids along a length, while the unchanged pressure inPT5 (zero) shows that this hole is not yet crossed by thecrack. The minimum pressure in this case is 60 kPa.The observed high frequency pressure oscillations in PT3are due to air bubble collapse confirming the occurrenceof cavitation. Replacing the measured PT3 response by an

equivalent smoothed pressure variation, and neglecting thedifferences for the region of negative pressure (cavitation)that comes from the zero-pressure assumption for cavitationin computed results, the computed pressures are similar tothe measured pressures.

The water pressure profiles along the crack for fourdifferent stages of crack motion, including cracking (t =2.69 s), first closing (t = 2.85 s), second opening(t = 3.05 s), and second closing (t = 3.35 s), areshown in Fig. 4(b). During the test, PT1 (Pcrm) exhibitssome variations due to opening and closing of the notchand the water tank. To eliminate this pressure variationeffect, the measured pressures are normalized such that

Pcrm(t) remains unchanged (100 kPa). The measuredpressure profiles along the crack are basically similar to thecorresponding phenomenological pressure profiles shown inFig. 2. More detailed evaluation of the computed resultsapplying the proposed model have shown that they aregenerally in good agreement with the new and existing cracktest results [4,12].

5. Finite element analysis of a dam considering dynamicuplift pressure

In a simplified approach, to compute the uplift pressurealong a seismic crack in a concrete dam, an uncoupledhydro-mechanicalanalysis may be utilized. The crack mouthopening displacement (CMOD) time history response couldbe determined by a discrete crack finite element analysisignoring transient water pressure variations and then usedas input data in the proposed crackwater interactionmodel. However, water pressure variations inside the crackmodify the crack wall motions and an uncoupled hydro-

mechanical analysis is not always realistic (at least inclosing mode, as will be shown later). It is thus requiredto adjust the computed CMOD response while consideringdynamic water pressure variations in the crack by an iterativeprocedure. A coupled watercrack interaction model wasthus implemented in INTRFACE, a finite element computerprogram for seismic analysis of a gravity dam usingdiscrete gap-friction elements to model cracks and joints[15].

The dynamic equilibrium equations of the dam subjectedto seismic excitation can be expressed as

[M]{u} + [C]{u} + {R} = [M]{r}{ug} + {fstat} (4)

5/24/2018 Seismic

6/13


Fig. 4. Computed and measured results for a typical new crack test ( Pcrm = 100 kPa, f = 2 Hz): (a) time variations of CMOD, computed and measured

pressures along the new developing crack; (b) measured pressure variations along the new developing crack.

where [M] is the mass matrix, [C]is the damping matrix,{R} is the restoring force vector, {fstat} is the vector ofpre-seismic applied force, {u}, {u}, {u} are displacement,velocity, and acceleration vectors, respectively, {ug}is thevector of ground accelerations, and {r} is the unit vectorspecifying the active dynamic degrees of freedom. Thedynamic water pressure in a crack, expressed as a forcevector {W pr} (Fig. 1), can be treated as an additionalrestoring force in Eq.(4) that can be written as

[M]{u} + [C]{u} + {R+W pr} =[M]{r}{ug}+ {fstat}. (5)

The NewmarkBeta integration method was adoptedherein to integrate Eq. (5) for the cracking analysis ofgravity dams using gap-friction elements. The restoringforce and water pressure force are functions of the gapelement relative displacements that are not known a prioriduring the integration such that an iterative procedure isrequired to solve Eq. (5). The modified NewtonRaphsonmethod is used for iterations in each mechanical time step.

The rapid increase in uplift pressure in the crack during

the closing mode is similar to a mechanical impact wherelarge contact forces develop at contact time. While themagnitude of the contact force in a mechanical impactis proportional to the contact deformations, the upliftforce developed during crack closing (hydro-mechanicalcontact) is mainly a function of crack closing velocityand displacement. The large contact force during impactleads to high frequency motions subsequently appearing inthe displacement (velocity) response. The convergence inmechanical impact analysis can be achieved by reducing thecomputational time step to reduce the changes in magnitudeof contact deformation and force in each time step. However,reducing the computational time step leads to larger iterative

crack closing velocity that increases uplift pressure inhydro-mechanical contact and decreases the convergencerate or even perturbs the water pressure inside the crack.Both phenomena, mechanical and hydro-mechanical impact,may occur during crack closing in a concrete dam andan appropriate computational time step should be used toachieve convergence.

Hydrodynamic damreservoir interaction increases ordecreases water pressure on the upstream face and thecrack mouth in contact with the reservoir. In this work, theWestergaard formulation is used to represent this additionalpressure:

Pwest(t) =7

8H

1

y

H

12

ug (6)

where is the mass density of water, H is the height ofthe reservoir, y is the crack mouth height from the bottomof the reservoir, and ug is the horizontal earthquake groundacceleration. The crack mouth pressure (Pcrm) for waterpressure computations along the crack is then computedfrom

Pcrm(t)= Pstat+ Pwest(t) (7)

wherePstatis the hydrostatic pressure at the crack mouth.

6. Damfoundationreservoir system analysed

The effect of dynamic uplift pressure on the seismicbehaviour of a typical 90 m high concrete gravity damresting on a rigid foundation, with the base width of70 m, is investigated. The dam geometry and the finiteelement mesh used to model the dam body are shownin Fig. 5. Gap-friction elements are located along thedamfoundation interface to model crack propagation along

5/24/2018 Seismic

7/13


Fig. 5. Typical 90 m high concrete gravity dam for applications.

the damfoundation interface. The tensile strength andcohesion are taken as zero and a friction angle of 45 isused. In the analysis, the concrete modulus of elasticity istaken as 27.7 GPa, the Poissons ratio as 0.20, and the massdensity as 2400 kg/m3. A stiffness proportional viscousdampingmodel, with a value of 5% in the fundamental modeof the dam, has been considered. The first three periodsof vibration of the damfoundationreservoir system are:T1 =0.270 s,T2 =0.093 s, andT3 =0.055 s. To investigatewater pressure variations in successive opening and closingof cracks during an earthquake, two different earthquakeinput motions are considered. The first ground motion usedis an Eastern Canadian record from the 1988 Saguenayearthquake, and the second record is a typical WesternNorth America record, from the 1957 Taft earthquake.These two records were scaled to obtain peak groundaccelerations of 0.25g. The two scaled records as well astheir response spectra are shown in Fig. 6. The Easternrecords have typically high frequency contents comparedwith the Western North America records (Fig. 6(c)).

6.1. Analysis procedure

The structural analyses of the dam subjected to theearthquake base accelerations, assuming zero uplift pressure

along the damfoundation interface (dry conditions), arefirst performed to compute the crack lengths. Coupledanalyses with hydrodynamic water pressure along the crack(wet conditions) are then performed. A time step equal to0.005 s is used in the analysis using the Saguenay recordwhile a larger time step (0.02 s) is used for the Taft record.In the coupled analysis, water pressure along the uncrackedpart is assumed to be equal to zero; therefore the structuralresponses of the dam from dry and wet analyses shouldbe the same until initiation of a crack occurs. A residualopening is also considered in the hydraulic computation toaccount for the possibility of water flow during mechanicalcontact of crack walls.

6.2. Earthquake response analysis1988 Saguenay earth-quake record

The results of dam analysis without uplift pressure (dryanalysis) show that cracking along the damfoundationinterface starts att =6.25 s and the computed crack lengthis 14 m. Hydrodynamiccoupled analysis of the cracked damwith the assumed ures = 0.5 mm leads to a crack lengthof 14 m. As soon as cracking starts at t = 6.25 s waterpenetrates inside the crack, saturates a small part of the crackclose to the crack mouth, and the water pressure develops

5/24/2018 Seismic

8/13


Fig. 6. Base motion records for the first 15 s: (a) Saguenay accelerogram; (b) Taft accelerogram; (c) response spectra.

Fig. 7. Computed water pressure variations along the crack during the 1988 Saguenay record (Lcr = 14 m).

along the saturated area of crack. A three-dimensional viewof the pressure variations along the crack developed duringthe earthquake is presented inFig. 7. The pressurized length

inFig. 7represents Lsat which is increasing with time dueto the successive opening and closing of the crack during theearthquake. The detailed pressure profiles along the crack for

5/24/2018 Seismic

9/13


Fig. 8. Computed pressure profiles along the crack for three differentopeningclosing cycles: (a) opening att =6.54 s and subsequent closing att =6.59 s; (b) opening att =8.10 s and subsequent closing att =8.23 s;(c) opening att= 9.92 s and subsequent closing at t= 10.02 s.

three different openingclosing cycles are shown in Fig. 8.Comparing the pressure profiles during the crack opening, itis concluded that the saturated length during crack openingis smaller than the saturated length during closing. The

pressure varies almost linearly from the existing pressure atthe crack mouth to zero pressure at the end of the saturatedarea. The magnitude of the closing pressure also changesin different closing cycles. In this special case, while themaximum developed pressure is 5000 kPa at t = 6.59 s, itreduces to 2000 and 1400 kPa att =8.23 s andt =10.02 s,respectively.

To investigate the effects of developed uplift pressureon dam response, the computed CMODs from dry and wetanalyses are shown in Fig. 9(a). The variations of upliftforce in the crack are computed, by integrating the existingpressure along the saturated length of crack (Fig. 9(b)).

Fig. 10(a) and (b) show similar graphs with more details forthe interval oft=6 s tot=8 s.Comparisons of CMOD variations from two analyses

in Figs. 9(a) and10(a) show that they are similar duringcrack opening while the closing response of the crack isaffected by the developed uplift pressure in wet conditions.Comparing CMOD variations with uplift force variations, itis clear that the uplift force (and water pressure) increasesduring crack closing and uplift reach a maximum local peakduring every closing cycle.

However the average magnitude of uplift force increaseswith time (Fig. 9(b)). The developed uplift force is stillsignificantly smaller than the uplift force assuming full uplift

pressure or linearly varying pressure along the whole cracklength.

From Fig. 10(a) and (b) three different phases can berecognized for the CMOD and developed uplift pressure(force) during the earthquake. These are: (i) crack wallopening phase, (ii) crack wall closing phase, and (iii)

crack walls in mechanical contact phase. The direction ofearthquake loads and developed pressure profiles changefrom one phase to the other phase as follows:

Crack opening phaseWhen dam inertia forces are oriented downstream, the

crack is opened due to dam deformations. The cracksaturated length (where water pressure exists) and theassociated uplift force decrease significantly. The pressurevaries almost linearly from the crack mouth to become zeroat the end of saturated length and the rest of the crackremains unpressurized. The developed uplift force in thiscase is small such that crack opening, and therefore the dam

structural response, is similar to that of a dam without upliftforce in the crack.

Crack closing phaseThis phase occurs due to dam deformation when dam

inertia forces are oriented upstream. The saturated lengthis increasing during the crack closing. Comparing CMODvariations with uplift force variations it is clear that upliftforce (and so water pressure) increases during crack closingand uplift have a maximum local peak during every closingcycle. The developed uplift force exerts external forces oncrack walls that reduce the crack closing velocity (slope of

CMOD curve) compared to that for a dam with no upliftpressure.

Mechanical contact phaseIf the developed uplift force is high enough it may

prevent the crack wall from having mechanical contact;otherwise, due to the mechanical contact, the crack closingvelocity decreases instantaneously. The uplift forces (oruplift pressures) fall from their peak values to smallermagnitudes and remain almost constant as long as the crackwalls are in contact.

6.3. Earthquake response analysisTaft earthquake record

For this earthquake record, cracking along the damfoundation interface starts at t = 3.2 s and the computedcrack length is 35 m. Assuming a 0.5 mm residual openinga hydro-mechanical coupled analysis was performed. Thethree-dimensional view of pressure variations along thecrack is shown in Fig. 11 and CMOD and uplift forcevariations are shown in Fig. 12. The magnitude of thedeveloped pressure and uplift force in this case are largerthan the magnitudes of similar quantities occurring duringthe Saguenay earthquake record. The main reason is thelarger crack openings in the Taft case where the maximum

5/24/2018 Seismic

10/13


Fig. 9. Analysis results for the Saguenay earthquake record from t =5 s tot =15 s: (a) CMODs for wet and dry cracks; (b) uplift force; (c) sliding safetyfactors for wet and dry cracks.

Fig. 10. Details of analysis results for the Saguenay earthquake record from t =6 s tot =8 s: (a) CMODs for wet and dry cracks; (b) uplift force; (c) slidingsafety factors for wet and dry cracks.

crack opening is 5 mm compared to the 0.8mm opening withthe Saguenay record. The dominant frequencies of crackopening and closing motion from both Saguenay and Taftcases are almost equal to the first natural frequency of thedam (f =3.7 Hz). Thelarger crack opening results in larger

crack closing velocity, that produces higher pressure with theTaft record.

The developed uplift force in the crack is high enough toprevent crack wall impact at some closing cycles as showninFig. 12(b).

5/24/2018 Seismic

11/13


Fig. 11. Computed water pressure variations along the crack during the Taft 1952 record (Lcr = 35 m).

Fig. 12. Analysis results for the Taft earthquake record from 0 to 16 s (a, c, e) and from 6 to 8 s (b, d, f): (a, b) CMODs for wet and dry cracks; (c, d) upliftforce; (e, f) sliding safety factors for wet and dry cracks.

7. Dam sliding stability

Sliding stability under seismic excitations is one of themost important concerns for gravity dams. It is important toinvestigate the effect of dynamic uplift pressure variationson the sliding stability of a gravity dam. In the absence ofcohesion, a transient value of the sliding safety factor (SSF)

could be defined as

SSF(t) =

V(t)

H(t)(8)

where is the friction coefficient,

V(t) is the summa-tion of vertical forces acting on the dam (including upliftpressure), and H(t)is the summation of horizontal forces

5/24/2018 Seismic

12/13


Fig. 13. Extreme uplift pressures along a cracked dam during an earthquake.

acting on the dam. SSF(t)becomes a useful indicator of slid-ing stability for comparing different seismic uplift pressuremodelling assumptions. Assuming that = 1.0, SSF(t)isthus computed for the dry and wet crack conditions for thetwo earthquake records. The equal friction factor assumptionin wet and dry conditions is only for comparison purposes,

while in an actual case they may be different. The results areshown in Figs.9(c),10(c),12(e), and12(f). The SSFs fromwet and dry analyses in Figs. 9(c) and10(c) are basicallysimilar due to the small magnitude of the uplift forces dur-ing the earthquake. InFig. 12(e) and (f) the SSFs from wetand dry analyses are also similar for the heel crack openingmode. This is the critical condition for sliding stability ashydrostatic, hydrodynamic, and concrete inertia forces areall pointing in the downstream direction. The uplift forcedeveloped during the crack opening mode considering thewatercrack coupling effect is small such that it does notchange the computed SSF as compared to the dry conditionone. The increase in uplift force during the crack closingmode slightly reduces the SSF (wet condition) compared tothe SSF with a dry condition. However, SSFs during crackclosing are still much larger than SSFs during the crackopening mode because during crack closing the concrete in-ertia (hydrodynamic) forces are pointing in the upstream di-rection while the hydrostatic force is pointing downstream.

8. General remarks on crack uplift pressures during

earthquakes

The uplift pressures in the cracks of concrete damsduring an earthquake are not constant and vary due to crack

wall relative motions. As soon as a seismic crack developsalong the damfoundation interface it is assumed that theexisting static uplift pressure is relieved along the openedsection. The extreme seismic uplift pressure profiles along acrack located at the damfoundation interface are shown inFig. 13. It is assumed that water pressures in the uncracked

part do not change. The definition of an equivalent staticuplift pressure to be used in simplified pseudo-static orpseudo-dynamic analysis of concrete dams would be useful.Uplift pressure develops along a part of a seismic crack;therefore the zero-uplift-pressure assumption [2] may notalways be accurate for computing the structural responseof cracked dams during earthquakes. The minimum SSFof a concrete dam occurs in downstream sliding during thecrack opening mode. A practical and simplified method forcomputing the saturation length and corresponding upliftpressure during the crack opening mode was presentedin [4], to be used in pseudo-static or pseudo-dynamicanalysis of concrete dams.

9. Summary and conclusions

A theoretical model is developed to compute upliftpressure variations along concrete cracks with moving walls.Experimental crack test data are used to verify the proposedmodel. The model is implemented in a finite elementcomputer program for dynamic analysis of gravity damsconsidering hydro-mechanical watercrack coupling. Theresults of analyses of a typical 90 m high concrete gravitydam subjected to two typical earthquake records show thatwater can penetrate into part of a seismically initiated crack

and saturate it partially. During the crack opening modethe saturated length is small, from a few centimetres toa few metres, depending on the opening velocity and themagnitude of the opening and crack mouth pressure. Waterpressure decreases along this length from the crack mouthpressure to the existing void pressure at the end of thesaturation length. The uplift forces developed are small andthe modelling assumption of zero uplift pressure in a seismiccrack in the tensile opening mode appears to be justified.The magnitude of the crack closing and its velocity have themost important effects on the magnitude of the seismic upliftforce in the closing mode. The saturation length and the

uplift force in the crack closing mode increase in successivecycles. The maximum pressure can be high locally and thesaturated length can be increased up to several metresstill smaller than the crack length in the system analysed.Because the pressure tends to develop in a region close to thecrack mouth, detrimental effects for the global dam stabilityare unlikely to occur (ex. wedge effect propagating a crackfilled with water as it closes).

The seismic uplift force during the heel crack openingmode is very small relative to the dam weight. On thebasis of the limited experimental and numerical evidencepresented in this paper, and pending further investigations, itappears that the critical SSF of the dam against downstream

5/24/2018 Seismic

13/13


sliding could thus be computed by considering zero upliftpressure in the crack region subjected to seismic tensileopening.

Acknowledgments

The authors would like to thank Mr. Martin Leclerc,Research Engineeringat cole Polytechnique, who providedvery valuable assistance during the laboratory experimentsand the post-processing of the related results. The writersalso gratefully acknowledge the financial support providedby the Natural Sciences and Engineering Research Councilof Canada (NSERC), Hydro-Qubec and Alcan as wellas the Quebec Funds for Research on New Technologies(FQRNT).

References

[1] ICOLD (International Commission on Large Dams). Earthquakeanalysis for dams. Bulletin 52, Paris; 1986.[2] USBR (US Bureau of Reclamation). Design of small dams. 3rd ed.

Colorado (USA): Denver; 1987.[3] USACE (US Army Corps of Engineers). Engineering and design,

gravity dams. EM 1110-2-2200, Washington, DC, USA; 1995.[4] Javanmardi F, Lger P, Tinawi R. Seismic water pressure in cracked

concrete gravity dams: experimental study and theoretical modeling.Journal of Structural Engineering ASCE 2005;131(1):13950.

[5] Slowik V, Saouma VE. Investigation on cracking of concrete withapplication to the seismic safety of dams. In: Proc. Euro-C 1994.p. 67988.

[6] Hall JF. Efficient non-linear seismic analysis of arch dams.Earthquake Engineering and Structural Dynamics 1998;27:142544.

[7] Ohmachi T, Zhang H, Yabuki N, Tsukada N. Experimental studyof hydrodynamic pressure inside narrow cavities. Dam EngineeringJSDE 1998;8:3540.

[8] Zhang H, Ohmachi T. 2 Dimensional analysis of seismic cracking inconcrete gravity dams. Dam Engineering JSDE 1998;8:93101.

[9] Yi KS, Dill SJ, Dauskardt RH. Sub critical crack growth in glassesunder cyclic loads: effect of hydrodynamic pressure in aqueousenvironments. Acta materialia 1997;45(7):267184.

[10] Tinawi R, Guizani L. Formulation of hydrodynamic pressures incracks due to earthquakes in concrete dams. Earthquake Engineeringand Structural Dynamics 1994;23:699715.

[11] Slowik V, Saouma VE. Water pressure in propagating concrete cracks.Journal of Structural Engineering ASCE 2000;126(2):23542.

[12] Javanmardi F. Transient uplift pressures in seismic cracks induced inconcrete dams: shake table experiments, theoretical modeling, andcase study. Ph.D. thesis. Montreal (Canada): Department of CivilEngineering, cole Polytechnique de Montral; 2003.

[13] Louis C. A study of groundwater flow in jointed rock and its influenceon the stability of rock masses. Rock mechanic progress report 10,Imperial College, London, England; 1969.

[14] Wittke W. Rock mechanics, Theory and applications with casestudies. Berlin (Germany): Springer-Verlag; 1990. p. 1919.

[15] Fronteddu L, Lger P, Tinawi R. Static and dynamic behaviour ofconcrete lift joints interfaces. Journal of Structural Engineering ASCE1998;124(12):141830.

Seismic

Documents

Transcript of Seismic