NUMERICAL MODELING AND EXPERIMENTAL VALIDATION OF SEISMIC UPLIFT PRESSURE VARIATIONS IN CRACKED CONCRETE DAMS

Transcription

1 13 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August 1-6, 24 Paper No NUMERICAL MODELING AND EXPERIMENTAL VALIDATION OF SEISMIC UPLIFT PRESSURE VARIATIONS IN CRACKED CONCRETE DAMS Farrokh JAVANMARDI 1, Pierre LÉGER 2 and René TINAWI 3 SUMMARY A theoretical odel is developed for transient water pressure variations along tensile seisic concrete crack with known crack wall otion history. Experiental tests are perfored to validate the proposed odel. The odel is then ipleented in a nonlinear discrete crack finite eleent progra for seisic analysis of concrete das. Experiental and nuerical results show that water can penetrate in new seisic cracks aking the partially saturated over length L sat. The L sat and the agnitude of the total water uplift force acting on a crack wall are decreased by crack opening and increased by crack closing. A paraetric study of a 9 gravity da subjected to sinusoidal base accelerations with different frequencies (1Hz-1Hz and intensities (.2g-.35g shows that higher frequencies and saller acceleration aplitudes reduce the developed uplift force in a crack. Transient sliding safety factors (SSF are coputed as da stability indicators. INTRODUCTION Da safety guidelines are developed with the objective that concrete das could suffer cracking and daage during the axiu design earthquake (MDE but ust aintain a stable condition to retain the reservoir. The MDE oscillatory otions could initiate and propagate new cracks in ass concrete or activate opening and closing cycles along exiting lift joints and cracks. Due to dynaic pressure variations at the crack outh in contact with reservoir, and due to the rapid crack wall otions during an earthquake, the water pressure inside a propagating or an existing crack becoes a transient variable odified fro its initial value (zero or hydrostatic condition. An accurate evaluation of seisic uplift pressure along cracks, is one of the ost iportant aspects in stability evaluation of concrete gravity das during earthquakes. Yet, due to the lack of historical, experiental and nuerical evidences, different hypotheses to evaluate seisic uplift pressure along a crack have been retained in da safety guidelines ranging fro 1 Post-doctoral Fellow, Dept. of Civil, Mining and Geological Engineering, École Polytechnique de Montréal, Univ. of Montréal, Canada, Eail: farrokh@struc.polytl.ca 2 Professor, Dept. of Civil, Mining and Geological Engineering, École Polytechnique de Montréal, Univ. of Montréal, Canada, Eail: leger@struc.polytl.ca 3 Director of Research Office, Université du Québec à Montréal, Canada, Eail: tinawi.rene@uqa.ca

2 the full crack outh reservoir hydrostatic pressure (ICOLD 1986 [1] to zero pressure (USBR 1987 [2]. Further investigations are thus required to accept, reject, or adapt seisic uplift pressure propositions found in da safety guidelines depending on nuerical test results and related odels based on sound theoretical concepts validated by experients (Javanardi [3]. To this end, this paper presents a new dynaic water-crack interaction odel ipleented in a nonlinear finite eleent progra with gap-friction eleents to represent cracks and joints. The odel coputes the transient crack pressure spatial distribution and agnitude as a function of (a the crack outh opening displaceent, CMOD(t, and (b the crack outh pressure tie histories, P cr (t, assuing: (i 1D flow along the crack, (ii continuity condition with an incopressible fluid of a constant viscosity, (iii pressure-flow relations governed by the crack hydraulic conductivity accounting for the crack roughness, lainar or turbulent flow conditions according to Reynold s nuber, and cavitation, (iv ipervious crack walls, and (v residual crack aperture during cyclic otions (zero or larger. A paraetric study is perfored on a 9 high gravity da subjected to haronic ground acceleration to investigate water pressure variations in a crack located at the da-foundation interface as well as sliding safety factors for variations in (i the frequency of applied ground acceleration (1Hz to 1 Hz, (ii the axiu values of crack outh opening displaceent (.5 to 2, and (iii the residual crack aperture (.2 to 1. It is shown that during the crack opening ode, the seisic uplift pressure is significantly reduced along a crack such that downstrea sliding safety factors coputed, without considering uplift pressures in tensile cracks during the MDE are appropriate. REVIEW OF PREVIOUS WORK Da safety guidelines forulate the following hypotheses concerning transient seisic uplift pressures (Fig.1: Full uplift pressure in cracks initiated during the earthquake: according to ICOLD 1986 [1] assuing that pore pressure equal to the reservoir head is instantly attained in cracks is probably safe and adequate. Unchanged uplift pressure in existing cracks and joints during the earthquake: according to USACE 1995 [4] and FERC 22 [5] uplift pressure in cracks and joints should be assued to be unaffected by earthquake. Zero uplift pressure during earthquake: according to USBR 1987 [2] when a crack develops during an earthquake event, uplift pressure within the crack is assued to be zero. This assuption is based on studies that show the opening of a crack during an earthquake relieves internal water pressure, and the rapidly cycling nature of opening and closing the crack does not allow reservoir water, and the associated pressure, to penetrate. Zero or unchanged uplift pressure according to seisicity conditions: according to CDSA 1997 [6], in areas of low seisicity, the uplift pressure prior to the seisic event is norally assued to be aintained during the earthquake even if cracking occurs. In areas of high seisicity, uplift pressure on the cracked surface is assued to be zero during the earthquake when the seisic forces are tending to open the crack. USBR 1987 [2] hypothesis is independent of the crack wall otion frequency, while CDSA 1997 [6] introduces the notions of low and high seisicity that are not clearly defined and are explored in

3 ters of agnitude and frequency of applied ground accelerations and crack outh opening displaceent in this paper. H=86 9 F hd F hs 3 W F Ix (a Da safety guidelines : Seisic uplift pressure in cracks (b Initiation and propagation of new seisic cracks 7 γh=86 kpa γh=86 kpa L cr cracked uncracked U=111 kn U U 1 Initial uplift pressure distribution prior to earthquake 2 Uplift pressure is reduced to zero during the earthquake (USBR 1987, CDSA 1997 high seisic zones. 3 Uplift pressure reains unchanged during the earthquake (USACE 1995, FERC 22, CDSA1997 low seisic zone. γh=86 kpa U 4 Full uplift pressure is attained instantly in seisic cracks (ICOLD Fig. 1 Seisic uplift pressure in cracks: da safety guidelines. A rigorous deterination of uplift pressure in cracks of concrete das during earthquakes requires the forulation, experiental validation, and application of an appropriate water-crack interaction odel. However, as pointed out by differences in current da safety guidelines, there is not any universally recognised and validated odel to evaluate seisic water pressure variations along oving walls of cracks in a concrete da. This is due to the coplex nature of this fluid-structure interaction proble with oving boundaries. An analytical solution for water pressure variations in an existing crack due to crack wall otions was presented by Tinawi and Guizani [7]. They assued a water filled existing rectangular crack with constant length, sall aplitude crack wall otions (relative to their initial aperture, and one directional lainar water flow to siplify the solution. Based on the proposed forulation, equivalent added asses and daping were derived to be used in finite eleent coputer progra. The case study of a 55 high gravity das subjected to typical earthquake loading indicated that the axiu pressure in a 2 crack along the da base can be as high as 2 ties the hydrostatic pressure and the iniu pressure can be as low as cavitation pressure. Following a siplified nuerical odeling ethodology, Hall [8] accounted for the transient nature of seisic uplift pressures in seared crack analysis of arch das considering that hydrodynaic pressure variations due to da-reservoir interaction at the upstrea face are also acting at the ouths of cracks (joints. It was further assued that this pressure varies linearly along seared crack bands across da sections. No interaction between crack wall otions and related water pressure was considered. It was found that this dynaic internal water pressure effect was able to alter the coputed cracking pattern, crack opening and sliding responses. These results indicated the need for further developent of seisic water crack-interaction odel. Oachi et al. [9] perfored a series of

4 shaking table experients to easure water pressure inside narrow cavities like cracks (3 c long inserted in sall suberged acrylic blocks (approx. 5 c 3, but no crack wall relative otions (openingclosing was peritted. These results were used by Zhang and Ohachi [1] who proposed that the hydrodynaic pressure acting at node i in a horizontal crack filled with water can be expressed as: P i (t = [P stat +P cr (t + ρ a x (t x ci ] ( 1 where P stat is hydrostatic uplift pressure at crack outh, P cr is the dynaic crack outh pressure approxiated by Westergaard s theory, ρ is the water density, a x is the total horizontal da acceleration at node i, and x ci is the distance between the crack outh and node i. Seisic seared crack analysis of Konya da using the proposed odel, assuing that hydrodynaic pressure in cracks develops instantaneously upon cracking, showed that hydrodynaic pressure inside cracks tends to increase the crack length. A basic liitation of this approach is that the effects of crack wall opening and closing otions on the reduction and increase of seisic water pressure were neglected. Slowik and Saoua [11] easured the developed water pressure along a concrete crack in a dynaic wedge splitting test. They also developed a theoretical odel to siulate the test results. The developed odel is applicable for the initial crack propagating phase during an earthquake but the subsequent crack closing and opening is not considered. Moreover, this odel was not applied in the context of seisic crack propagation and stability evaluation of concrete das. Seisic water-crack interaction was also investigated theoretically and experientally by Javanardi et al. as discussed below [3,12]. WATER FLOW AND WATER PRESSURE IN PROPAGATING CRACKS Propagating cracks during earthquakes have oscillatory opening and closing otions. Figure 2 shows a phenoenological description of water flow and corresponding pressure in a propagating crack during its first two successive opening and closing cycles. Neglecting water flow across crack walls (ipereable walls, water exhibits a 1D flow along the crack. Figure 2.a shows the water flow along the crack during its first opening cycle. The existing water pressure at the crack outh pushes water into the crack and water flow fro crack outh toward crack tip. Water fills a part of the void developed due to crack opening and saturates, partially, the propagating crack. The agnitude of the developed pressure along the saturated length (L sat is not constant. At the crack outh, it is equal to the reservoir pressure p stat, and it is decreasing fro crack outh toward crack tip (pressure losses due to flow to becoe equal to the void pressure at the end of the saturated region. The agnitude of the void pressure is deterined by the existence of air in the developed void. In a case where there is no air in the concrete, the void pressure decreases to water vapour pressure (cavitation and water vapour fills the void. If there is soe air, fro existing air in concrete pores (holes, then depending on the ratio of air to void volue, the void pressure will change between the water vapour pressure and zero. The volue of void decreases as the crack closing ode begins. The water flow fro crack outh toward crack tip continues during the closing ode as long as the crack closing velocity is sall. By increasing the crack closing velocity, the existing water between crack walls is pressurized (like water squeezed between two closing parallel plates. The water flow changes direction. It now flows in two opposite directions for a stagnation point along the saturated part of the crack (Fig. 2.b. The L sat still increases due to water flow fro the stagnation point toward the crack tip. The water pressure is axiu at the stagnation point and it decrease fro this point toward the saturated region boundaries to becoe equal to the pressure at these two points. The location of the stagnation point, L sat and the pressure agnitude along the crack are changing with the tie and are functions of the crack closing velocity, crack aperture, crack roughness, crack length, and existing crack outh pressure.

5 CMOD(t 1 Water front velocity=v w (t 1 Crack front velocity=v cr (t 1 (a Initial cracking t=t 1 Water flow due to existing reservoir pressure fills the developed void due to crack propagation P stat P v L sat (t 1 x L cr (t 1 Fracture process zone (FPZ CMOD(t 2 water vapour pressure Stagnation point v w (t 2 x c Void filled with air and/ or water vapour (b Crack closing t=t 2 Saturation length (L sat increases in closing ode due to water flow that fills the existing void P stat L sat (t 2 x L cr (t 2 FPZ P v x c (c Further crack extension in opening t=t 3 CMOD(t 3 v w (t 3 v cr (t 3 Saturation length ay decrease copared to closing ode due to fast crack opening P stat L sat (t 3 L cr (t 3 x FPZ P v x c CMOD(t 4 Stagnation point v w (t 4 (d Crack closing t=t 4 Saturation length increases again in closing ode due to water flow that fills the void P stat L sat (t 4 x L cr (t 4 FPZ P v x c Fig. 2 Water flow and water pressure along a new crack with oscillatory crack wall otions. An iportant aspect related to water flow in an oscillating crack is that a coplete hydraulic closure of a crack is alost ipossible. A sall space reains between crack walls even after they get in echanical contact. Copressing crack walls reduces crack aperture, but due to crack wall asperities and the existence of free sedientary aterial (fro crack wall local crushing of concrete particles, it never becoes zero. A residual hydraulic crack aperture,, is defined as a function of the crack wall surface geoetric properties (u = u ech + ; Fig.5. Thus, after echanical contact of crack walls, a sall aount of water exists in the space between the walls.

6 In subsequent opening cycles, new voids are developing along the crack. Water flow occurs along the crack fro crack outh to crack tip (Fig. 2.c. Water flow can only fills the voids close to the crack outh, the voids in the rest of the crack reain unfilled and L sat decreases. The only difference between the first opening and subsequent opening cycles is the existence of soe water in the residual opening that was already filled due to crack closing. The water flow and corresponding pressure during the second closing cycle is siilar to that of the first closing cycle except that there is already soe water in the unsaturated region. Therefore, L sat ay be longer at the end of the second closing cycle. THEORETICAL MODELING Exact theoretical odeling of the water flow in cracks with oving walls, considering the possibility of cavitation and two phase flow along the crack, is a very coplex proble. But with soe assuptions, it is possible to forulate a sipler, yet, representative proble. The ain assuptions retained herein are that: A tapered crack with known crack length, L cr, is used to odel the crack geoetry and rigid body crack wall otions, (u(x,t=u(l,t*x/l, Crack walls are ipereable, Water flow is one diensional along the crack length, There is no significant water flow in the unsaturated part of the crack, Water vapour pressure is taken equal to zero, The pressure gradient expressions for lainar and turbulent flow, Q(x,t, in the saturated region of a rough crack are (Louis [13]: dp(x, t dx la in ar = 6µ 1.5 [ ( k / 2u(x, t ] Q(x, t 3 u (x, t ( 2 dp(x, t dx ρ 2 Q (x, t = ( turbulent 1.9 u (x, t 16 log k / 2u(x, t where µ is the water dynaic viscosity, u is the crack aperture, k/2u(x,t is the relative crack roughness (taken as.5 for rough concrete tensile cracks, and ρ is the water ass density. A subroutine called DUP_CRACK is developed for nuerical integration of Eqs (2,3 along the crack saturated length, while the flow continuity and pressure copatibility are satisfied. Using this subroutine, water pressure variations along a crack with known crack wall otion history u(x,t, crack outh pressure P cr (t (equal to the hydrostatic value P stat or to the hydrostatic plus hydrodynaic values, and crack length (L cr can be coputed. This subroutine is applicable for new cracks as well as initially saturated existing cracks. EXPERIMENTAL PROGRAM Experiental tests were perfored to easure water pressure along concrete cracks with oving walls. The concrete speciens and test set up are shown in Fig. 3. The concrete speciens (.15 x.55 x 1.5 are fixed to a very stiff steel supporting structure and are attached to the shake table by a rigid link to apply displaceents. A sall notch was created in the speciens during pouring of concrete to induce a crack at a certain level. A sall steel tank was added in front of the notch to approxiately odel the

7 reservoir in a real da. An additional water tank pressurised with an air copressor is used to provide the desired static pressure in the notch. A thin waterproof ebrane is glued to the concrete surface around the notch and the expected crack path. Four holes, that intercept the estiated crack trajectory, were used to easure the pressure fro pressure transducers (PT at the top of the speciens. Using this experiental set up, five new speciens are tested. Water pressure variations along the propagating crack and subsequent closing and opening haronic otions are easured. Letting the cracked speciens be copletely saturated, ore than 22 tests have been done. Haronic oscillatory displaceent control tests with different frequencies, crack opening displaceents and crack outh pressures were perfored. Existing crack and new crack test results are used to validate the proposed theoretical odel..55 LVDT Pressure transducer (PT Load cell Air copressor inlet Water tank PT5 PT4 PT3 PT2 PT1 hole Φ= pipe Φ=2.5.2 crack path 1.5 Shake table Steel frae Sall water tank Reaction frae (a Specien (diensions in (b Test set up Fig. 3 Typical specien geoetry and experiental test set up. Figure 4.a shows the easured pressure as well as coputed pressure for a new crack test (P cr =1kPa, f=2hz. Initially in the absence of a crack, the pressure in all PT is zero. The pressure drops in PT2, PT3, and PT4 during the crack initial propagation confirs the developent of voids along a length, while the unchanged pressure in PT5 (zero shows that this hole is not yet crossed by the crack. The iniu noralised pressure in this case is -6 kpa. The observed high frequency pressure oscillations in PT3 are due to air bubble collapse confiring the occurrence of cavitation. Replacing the easured PT3 response by an equivalent soothed pressure variation, and neglecting the differences for the region of negative pressure (cavitation that coes fro the zero pressure assuption for cavitation in coputed results, the coputed pressures are siilar to the easured pressures. The water pressure profiles along the crack for four different stages of crack otion, including cracking (t=2.69 sec, first closing (t=2.85 sec, second opening (t=3.3 sec, and second closing (t=3.35 sec are shown in Fig 4.b. During the test, PT1 (P cr exhibits soe variations due to opening and closing of the notch and the water tank. To eliinate this pressure variation effect, the easured pressures are noralized such that P cr (t reains unchanged (1kPa. The easured pressure profiles along the crack

8 are basically siilar to the corresponding phenoenological pressure profiles shown in Fig 2. More detailed evaluation of coputed results applying the proposed odel have shown that they are generally in good agreeent with the new and existing crack test results (Javanardi [3,12]. (a CMOD ( Pressure (kpa Pressure (kpa Equivalent curve Oscillation due to collapse of air bubbles Tie (sec Test results PT2 PT3 PT4 PT5 Coputed results PT2 PT3 PT4 P (kpa P (kpa P (kpa P (kpa (b PT2 PT3 PT4 t=2.85 sec t=3.5 sec t=3.35 sec PT5 t=2.69 sec Fig. 4 Coputed and easured results for a typical new crack test (P cr = 1 kpa, f=2 Hz. FINITE ELEMENT ANALYSIS OF DAM CONSIDERING DYNAMIC UPLIFT PRESSURE In a siplified approach, to copute the uplift pressure along a seisic crack in a concrete da, an uncoupled hydro-echanical analysis ay be utilized. The CMOD tie history response could be deterined by a discrete crack finite eleent analysis ignoring transient water pressure variations and then used as input data in the proposed crack-water interaction odel. However, water pressure variations inside the crack odify crack wall otions and an uncoupled hydro-echanical analysis is not realistic (at least in closing ode, as it will be shown later. It is thus required to adjust the coputed CMOD response while considering dynaic water pressure variations in the crack by an iterative procedure. A coupled water-crack interaction odel was thus ipleented in INTRFACE, a finite eleent coputer progra for seisic analysis of gravity da using discrete gap-friction eleents to odel cracks and joints (Fronteddu [14]. The dynaic equilibriu equations of the da subjected to seisic excitation are expressed as: [ M ]{} u& + [ C]{} u& + { R} = [ M]{}{ r & } + { } & u g f stat ( 4 where [M] is the ass atrix, [C] is the daping atrix, {R} is the restoring force vector, {f stat } is the vector of pre-seisic applied force, { u }, {} u&, {} & u& are displaceent, velocity and acceleration vectors, respectively, {& u& g } is the vector of ground acceleration, and {r} is the unit vector specifying the active

9 dynaic degrees-of-freedo. The dynaic water pressure in a crack, expressed as a force vector{wpr}, can be treated as an additional restoring force in Eq. (4 that can be written as: [ M ]{} u& + [ C]{} u& + { R + Wpr} = [ M]{}{ r & } + { } & u g f stat ( 5 The α integration ethod was adopted herein to integrate Eq. (5 for the cracking analysis of gravity das using gap-friction eleents. The restoring force and water pressure force are functions of the gap eleents relative displaceents that are not known a priori during the integration such that an iterative procedure is required to solve Eq. (5. The odified Newton-Raphson ethod is used for iterations in each echanical tie step. Due to the rapid increase in uplift pressure in crack during the closing ode, it was found that the iterative solution procedure exhibit soeties a convergence proble (especially for initially saturated cracks. This proble is siilar to a echanical ipact where huge contact forces develop at contact tie leading to high frequency otions subsequently appearing in the displaceent (velocity response and perturbing water pressures. This proble needs further investigation to develop a robust nuerical solution procedure. However, nuerical investigations have indicated that water-crack interaction could be properly odeled for potential discontinuities (cracks that are initially in an unsaturated condition being initiated and subsequently propagated by the earthquake. Hydrodynaic da-reservoir interaction increases or decreases water pressure on the upstrea face and the crack outh in contact with the reservoir. Herein, the Westergaard approach is used to represent this additional pressure: P west y (t = ρ H 1 & u g ( 6 8 H where ρ is the ass density of water, H is the height of reservoir, y is the crack outh height fro the botto of the reservoir, and u& & g is the horizontal earthquake ground acceleration. The crack outh pressure (P cr, for water pressure coputations along the crack is then coputed fro: P cr (t = Pstat + P west (t ( 7 TRANSIENT SEISMIC WATER PRESSURE IN CRACKED DAM To investigate water pressure variations in successive opening and closing of cracks during earthquake, a typical 9 gravity da section subjected to very low frequency sinusoidal base acceleration (f =1Hz with a peak ground acceleration PGA=.2g is first analysed. Low frequency otions proote water penetration in the crack. The da geoetry and the assued concrete properties are shown in Fig. 5. Gap eleents are located along the da-foundation interface to odel crack propagation. The structural response of the da is first coputed assuing zero initial uplift pressure (dry condition, the coputed crack length is 14. A coupled analysis with dynaic water pressure along the crack is then perfored (wet condition leading also to a crack length of 14. A.5 residual opening is assued. Figures 6.a,c,e show CMOD for these analyses and uplift forces (pressures fro the coupled hydro-echanical (wet condition. Figures 6.b,d,f show these response quantities in ore details for the last two sec.

10 5 Concrete properties: Mass density ρ c =24 kg/ 3 Elastic odulus E=2769 MPa Poisson's ratio ν=.2 Gap-friction eleent Da (C,φ f t = Da-reservoir free vibration periods T 1 =.27 sec T 2 =.93 sec T 3 =.55 sec 9 H=86 F hd 2 =.543ρH & u g W=7451 kn Crack haronic otion u ax u(t ures x c u(t = u res L cr 7 P west = ρhu& g 8 F hs =36277 kn P stat = ρgh oving crack wall at t fixed crack wall Q(x,t u + 2 ax sin 2πft x = 86kPa P cr = P west +P stat L sat (t crack && u g = a sin 2πft Haonic base acceleration 7 Lcr U=U(t zero uplift assuption along uncracked part Rigid base Fig. 5 Typical 9 da for applications. Coparisons of CMODs show that the opening responses are siilar for dry and wet conditions but that the closing responses are different. The uplift force during the opening ode is so sall that it does not affect the crack opening response but the uplift force increases as crack closing starts. The uplift force in closing ode is considerably larger than the uplift force in opening ode. This uplift force build-up slows down the crack closing velocity (slope of CMOD curve and changes the crack closing response. However, the uplift force is not large enough to prevent crack walls fro echanical ipact. As soon as two crack walls hit each other, the crack closing velocity is instantaneously reduced causing a significant decrease in water pressure along the crack. Water pressure variations becoe alost linear along the crack saturated length as along as crack walls reains in contact. Pressure variations along the crack (Fig. 6e show the water penetration (pressure build-up along the crack in successive crack opening-closing cycles. Figure 7, representing the 3D pressure variations along the crack, illustrates this phenoenon ore clearly. The saturated length (L sat increases fro. to 4.6. Water pressure variations fro 2 sec to 4 sec and fro 8 sec to 1 sec are also shown in ore details in Figs 7.b,c. Although the agnitudes of uplift pressure and L sat in closing ode are different, pressure variations during opening are siilar. This is an iportant aspect that was used to derive a siplified ethod for seisic uplift pressure evaluation in opening crack ode (Javanardi [3].

11 CMOD ( Tie (sec (a wet dry CMOD ( closing echanical ipact opening Tie (sec (b Uplift force (kn 8 4 Uplift force (kn 8 4 Pressure (kpa Tie (sec (c xc=. x c =.5 x c =1. x c =2. xc=3. Pressure (kpa Tie (sec (d Tie (sec (e Tie (sec (f SSF wet dry SSF Tie (sec (g Tie (sec Fig. 6 Coupled analysis of a 9 da subjected to a sinusoidal base acceleration. (h ( & (t =.2g sin 2πft ; f=1 Hz u g

12 (a (b (c Fig. 7 Three diensional presentation of pressure variations along the crack of a 9 da subjected to sinusoidal base acceleration ( & (t =.2g sin 2πft ; f=1 Hz. u g

13 DAM SLIDING STABILITY Sliding stability under seisic excitations is one of the ost iportant concerns for gravity das. It is iportant to investigate the effect of dynaic uplift pressure variations on the sliding stability of gravity da. In the absence of cohesion, a transient evaluation of the sliding safety factor (SSF could be defined as: µ. V(t SSF (t = ( 8 H(t where µ is friction factor, V (t is suation of vertical forces acting on the da (including uplift pressure, and H (t is suation of horizontal forces acting on the da. SSF(t becoes a useful indicator of sliding stability to copare different seisic uplift pressure odeling assuptions. Assuing that µ=1., SSF (t is thus coputed for the dry and wet crack conditions as explained before. The results are shown in Figs 6.g,h. SSF fro wet and dry analyses in Fig 6.h are basically siilar for the heel crack opening ode. This is the critical condition for sliding stability as hydrostatic, hydrodynaic and concrete inertia forces are all pointing in the downstrea direction. The developed uplift force during the crack opening ode considering the water-crack coupling effect is so sall that it does not change the coputed SSF as copared to the dry condition. The increase in uplift force during the crack closing ode slightly reduces the SSF (wet condition copared to SSF with a dry condition. However, SSF during crack closing are still uch larger than SSF during the crack opening ode because during crack closing the concrete inertia (hydrodynaic forces are pointing in the upstrea direction while the hydrostatic force is pointing downstrea. PARAMETRIC STUDY According to the proposed odel, the ajor paraeters that affect the agnitude of seisic pressure in a crack of a gravity da subjected to sinusoidal base acceleration are: (i the axiu crack opening CMOD ax, (ii the residual crack opening, (iii the crack oscillating frequency f. A series of analyses have been done to investigate the effects of these paraeters on the seisic uplift pressure at the base of the 9 da of Fig. 5. Haronic base accelerations at 1 Hz and 1 Hz with PGAs adjusted to produce crack wall otions with CMODs equal to.5 and 2 are considered (Table 1. The analyses are perfored over the necessary duration to reach a steady state uplift force; the 1 Hz excitation case is analyzed for 1 sec, while the 1 Hz case is analyzed for 4 sec. For hydraulic coputation, three different residual openings ( ;.2,.5, and 1. are considered. The results including (i the axiu closing pressure, (ii the axiu uplift force in the crack, (iii the axiu L sat in closing ode, (iv the iniu L sat in opening ode, (v the inial value of SSF in, and (vi the crack length are suarized in Table 1. Coparisons of the results lead to the following conclusions: Effect of axiu crack opening, CMOD ax : By increasing CMOD ax the closing pressure and axiu L sat in closing ode are increasing (for a constant frequency, an increase in CMOD ax increases the crack closing velocity producing larger water pressure. The agnitude of the axiu uplift force in closing ode is also increasing. The agnitude of crack opening in a gravity da subjected to earthquake loading depends on the earthquake intensity. Therefore larger dynaic uplift forces are expected in cracks of concrete das during the earthquake as its intensity increases. However, even for very low frequency otions and large CMOD with long tie period for water

14 penetration, the results indicate that heel tensile crack opening (coinciding with critical downstrea sliding condition reduces the uplift forces to a negligible value. Effect of frequency of crack otion, f: Due to increase in crack otion frequency fro 1Hz to 1 Hz, the axiu closing pressure increases locally. But due to a reduction in crack saturation length the axiu uplift force during high frequency closing ode decreases. Therefore the developed uplift forces in concrete gravity das subjected to earthquake ground otions with high frequency content are saller copare to uplift forces in concrete das subjected to ground otions with lower frequency content. Effect of residual crack opening, : By increasing, the agnitude of seisic uplift pressures decreases but the agnitudes of the uplift forces reain alost unchanged due to the increase in crack saturation length. Therefore, the developed uplift force along the crack is not very sensitive to the assued residual crack opening. Table 1 Paraetric analysis results of 9 da subjected to sinusoidal base acceleration. f=1 Hz, PGA=.2 g, CMOD ax =.5 CMOV ax =3.1 /sec f=1 Hz, PGA=.32 g, CMOD ax =2. CMOV ax =12.6 /sec f=1 Hz, PGA=.35g, CMOD ax =.5 CMOV ax =31.4 /sec Max closing press. (kpa Max closing uplift (kn Max saturation length ( Min saturation length ( Min opening uplift (kn Max crack length ( % Cracked base SSF in (downstrea <1. <1. < Dry % Cracked base crack SSF in 1.2 < According to the results of all analyses, the saturation length in crack opening ode is sall copared to the length of crack. The developed uplift force in crack tensile opening ode is so sall that the iniu SSF of the da in a wet condition is not affected by the developed uplift force. SUMMARY AND CONCLUSION A theoretical odel is developed to copute uplift pressure variations along concrete cracks with oving walls. Experiental crack test data are used to verify the proposed odel. The odel is ipleented in a finite eleent coputer progra for dynaic analysis of gravity das considering hydro-echanical water-crack coupling. The results of analyses of a typical 9 da subjected to low (1Hz and high frequency (1Hz sinusoidal base accelerations show that water can penetrate into part of a seisically initiated crack and saturates it partially. During the crack opening ode the saturated length is sall (fro few centietres to few eters depending on the opening velocity, agnitude of the opening and crack outh pressure. Water pressure decreases along this length fro crack outh pressure to the existing void pressure at the end of the saturation length. The developed uplift forces are sall and the odeling assuption of zero uplift pressure in a seisic crack in tensile opening ode appears to be justified.

15 The axiu crack opening and frequency of crack oscillatory otions have the ost iportant effects on the agnitude of seisic uplift force in closing ode, while the assued residual crack opening does not change it significantly. The saturation length and the uplift force in crack closing ode are increasing in successive cycles to reach a steady state. The axiu pressure can be very high locally and the saturated length can be increased up to several eters, still saller than the crack length. Because the pressure tends to develop in a region close to the crack outh, detriental effects for the global da stability are unlikely to occur (ex. wedge effect propagating a crack filled with water as it closed. It is shown than the seisic uplift force during the heel crack opening ode is very sall relative to the da weight. Based on the liited experiental and nuerical evidences presented in this paper, and pending further investigations, it appears that the critical SSF of the da against downstrea sliding could thus be coputed by considering zero uplift pressure in the crack region subjected to tensile opening. REFERENCES 1. ICOLD (International Coission on Large Das. Earthquake analysis for das. Bulletin 52, Paris, USBR (US Bureau of Reclaation. Design of sall das. 3 rd Edition, Denver, Colorado, USA, Javanardi F. Transient uplift pressures in seisic cracks induced in concrete das: shake table experients, theoretical odeling, and case study. Ph.D. Thesis, Departent of Civil Engineering, École Polytechnique de Montréal. Montreal, Canada, USACE (US Ary Corps of Engineers. Engineering and design, gravity das. EM , Washington, D.C. USA, FERC (Federal Energy Regulatory Coission. Engineering guidelines for the evaluation of hydropower projects. Washington, D.C., USA, CDSA (Canadian Da Safety Association. Da safety guidelines and coentaries. Edonton, Alberta, Canada, Tinawi R, Guizani L. Forulation of hydrodynaic pressures in cracks due to earthquakes in concrete das. Earthquake Engineering and Structural Dynaics 1994; 23: Hall J. F. Efficient non-linear seisic analysis of arch das. Earthquake Engineering and Structural Dynaics 1998; 27: Ohachi T, Zhang H, Yabuki N, Tsukada N. Experiental study of hydrodynaic pressure inside narrow cavities. Da Eng. JSDE 1998; 8: Zhang H, Ohachi T. 2 Diensional analysis of seisic cracking in concrete gravity das. Da Eng. JSDE 1998; 8: Slowik V, Saoua VE. Water pressure in propagating concrete cracks. Journal of Structural Engineering ASCE 2; 126(2: Javanardi F, Léger P, Tinawi R. Seisic water pressure in cracked concrete gravity das: experiental study and theoretical odeling. Subitted to Journal of Structural Engineering ASCE, Louis C. A study of groundwater flow in jointed rock and its influence on the stability of rock asses. Rock Mechanic Progress Report 1, 1969, Iperial College, London, England. 14. Fronteddu L, Léger P, Tinawi R. Static and dynaic behaviour of concrete lift joints interfaces. Journal of Structural Engineering ASCE 1998; 124(12: