CONCRETE THERMAL STRAIN, SHRINKAGE AND CRACKING ANALYSIS FOR THE PANAMA CANAL THIRD SET OF LOCKS PROJECT

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1 CONCRETE THERMAL STRAIN, SHRINKAGE AND CRACKING ANALYSIS FOR THE PANAMA CANAL THIRD SET OF LOCKS PROJECT Vik Iso-Ahola, P.E. 1 Bashar Sudah, P.E. 2 Vincent Zipparro, P.E. 3 ABSTRACT The Panama Canal Authority (ACP) has undertaken the Panama Canal Expansion Program to increase the Canal s capacity in order to meet the continuous growth in the number of transits and vessel size. The expansion of the Canal involves the construction of two new lock facilities, one on the Atlantic side and another on the Pacific side each with three chambers; the excavation of a new Pacific access channel to the new locks, and widening and deepening of the existing navigational channels and entrances; and increasing the elevation of Gatun Lake s maximum operating level. Two-dimensional and three-dimensional incremental finite element thermal analyses were performed using ANSYS software to estimate the temperature distribution within the new lock walls, lock heads, crossunders, central connections, and chamber conduits which consist of reinforced mass concrete structures. The estimated temperatures from the finite element model were used to estimate the thermal strains and potential for cracking using procedures outlined in ACI 207. The overall evaluation was used to determine optimal concrete placement temperatures, contraction joint spacing, and to comply with the Employer s Requirements regarding concrete temperature gradient limitations. Potential for cracking due to drying shrinkage was also evaluated and crack depths were estimated based on the anticipated moisture distribution within the concrete structures. This paper presents the thermal strain, drying shrinkage strain, and cracking potential analyses that have been performed for the new lock walls, lock head structures, and related concrete structures for the Panama Canal Third Set of Locks Project. The results of these analyses were used as key inputs to concrete mixes and their placement temperatures which are designed to withstand for 100 years the deleterious effects of seawater and load cycling of hydrostatic pressures during filling & emptying of lock chambers. INTRODUCTION Completion of the new Pacific and Atlantic Lock Complexes for the Panama Canal Expansion Project (illustrated in Figure 1) includes construction of several massive concrete sections that consist of lock walls, lock heads, central connections, and 1 Principal Engineer, MWH Americas Inc., Walnut Creek, California, vik.iso-ahola@mwhglobal.com 2 Structural Engineer, MWH Americas Inc., Walnut Creek, California, bashar.sudah@mwhglobal.com 3 Design Engineer, Panama Canal Third Set of Locks Project, MWH Americas Inc., Chicago, Illinois, vincent.j.zipparro@mwhglobal.com Concrete Thermal Strain 345

2 crossunders. These structures are being constructed using two different concrete mix types, a Structural Marine Concrete (SMC) mix and an Interior Mass Concrete (IMC) mix. A typical concrete section consists of IMC encapsulated by SMC facing. The SMC facing is typically 60 cm thick while the IMC varies in thickness based on the geometry of the structures. A typical lock wall monolith (Figures 2 and 3 in the following section) is approximately 18 meters wide, 30 meters high and 29 meters long. Each lock wall monolith contains two 6.5 meter high culverts; the main and secondary culverts are 8.3 and 7 meters wide, respectively. The culvert walls vary in thickness from 1.5 meters (center wall) to 4 meters. The wall stem thickness ranges from 12 meters at the bottom to 2 meters at the top. The designed lift heights range from 2 meters (culvert) to 3.75 meters (wall stem), and are constructed with IMC and SMC facing. Another feature of the lock structures include crossunders that provide utility and personnel access underneath the lock chambers and are constructed of SMC (Figure 4). Figure 1. Artistic Rendering of the Panama Canal Third Set of Locks Project The lock head structures (Figure 5) that house the lock chamber rolling gates are approximately 38.4 meters high, 67 meters wide, and 20 meters in section length, with wall thicknesses varying from roughly 6.6 to 14 meters. Similar to the lock wall structures, the lock head structures are constructed with IMC and SMC facing. The lock head structures are designed with thick concrete sections that provide housing for the rolling gates when they are in the open position, and protected dry bays that allow for maintenance and access to the gates, which are approximately 33 meters high by 58 meters long and either 8 or 10 meters wide. The culverts within the lock wall sections are part of the filling and emptying system that routes water from the lock chambers to either the Water Savings Basins (WSB) adjacent 346 Innovative Dam and Levee Design and Construction

3 to the lock structures (when they are used), or from Gatun Lake and chamber to chamber and to the Ocean when Lake to Ocean operations are used. Efficient routing of water requires a complex culvert geometry that includes curved conduits and connections which result in thick concrete sections (Figures 6 & 7) constructed with IMC and SMC facing. Mix designs for the IMC and SMC utilize onsite materials, local cement and pozzolan, and imported silica fume to produce mixes that meet ACP temperature and durability requirements, which stipulate a minimum 100-year life for the structures, including, but not limited to, protection of the reinforcing steel for resistance against corrosion from chloride (sea water) attack. THERMAL CRACKING EVALUATION In order to mitigate concrete cracking potential and meet ACP requirements for durability, a thermal cracking analysis was performed in order to select the optimal combination of concrete mixes and placement temperatures. Initially, a finite element thermal evaluation was performed to consider various temperatures and placement scenarios. Thereafter, both mass and surface gradient analyses, including estimated strain computations, were executed to perform the cracking evaluation. By combining the results from the finite element model with simplified strain computations, estimates of cracking potential for various combinations of mixes and placement temperatures were provided as changing geometry (e.g. over-excavation), mix designs, and cooling constraints were encountered during construction. The thermal studies were performed in general accordance with ETL (USACE, 1997). Thermal Finite Element Analysis Finite element thermal analysis was performed to estimate time-dependent temperature distributions and peak temperatures at specific points in both the Pacific and Atlantic lock complexes to verify ACP requirements for concrete temperature differentials and thereafter as input into thermal strain computations. Two-dimensional and three-dimensional finite element models for the incremental thermal analyses were created to represent the typical geometry of the Pacific and Atlantic lock walls, lock heads, crossunders, central connections, and chamber conduits. Using the computer program ANSYS Version 12.1, these models were developed to simulate phased construction of the concrete lifts, estimating the maximum temperature rise at critical locations in the structures. Representative finite element models for each structure analyzed are presented in Figures 1 to 6 below. Concrete Thermal Strain 347

4 Figure 2. Pacific Lock Wall Model Figure 3. Atlantic Lock Wall Model Figure 4. Cross-under Model Figure 5. Lock Head Model Figure 6. Central Connection Model Figure 7. Chamber Conduit Model Material properties used in the finite element thermal models were selected from laboratory test results and typical values published for mass concrete mixes with pozzolan and basalt aggregates, and are summarized in Table 1 below. 348 Innovative Dam and Levee Design and Construction

5 Table 1. Summary of Material Properties Interior Structural Properties Units Mass Marine Concrete Concrete (IMC) (SMC) Specific Heat (C h ) kj / kg C Thermal Conductivity (K) kj / m h C Density ( ) kg / m Diffusivity (h 2 ) m 2 / h x Adiabatic Temperature Rise C Ultimate Compressive Strength (F c ) MPa Ultimate Modulus of Elasticity (E c ) GPa Coefficient of Thermal Expansion (CTE) mil/ C Adiabatic temperature rise curves were developed in the laboratory for typical SMC and IMC mixes used in the lock structures. These curves were used to develop the concrete heat generation functions used to simulate heat rise within the finite element model (Figure 8). 60 Adiabatic Temperature Rise Curves 50 Temperature Rise ( C) Structural Marine Concrete Interior Mass Concrete Age (days) Figure 8. Adiabatic Temperature Rise Curves for Concrete The average daily temperatures at the Pacific and Atlantic sites, including the effects of the diurnal cycle, were applied as ambient temperatures at the air-exposed boundaries of Concrete Thermal Strain 349

6 the FEM models for every 4-hour time step. The average temperatures and normalized diurnal cycles for both sites are plotted in Figures 8 to 11 below. Temperature ( C) JAN FEB MAR APR Balboa Station ( ) Daily Maximum, Minimum and Average Air Temperatures MAY JUN JUL AUG SEP OCT NOV DEC Average Max Average Average Min Figure 9. Average Ambient Temperature es (Balboa Station Pacific Lock Site) Figure 10. Diurnal Temperature Cycle (Balboa Station Pacific Lock Site) Temperature ( C) Gatun Station ( ) Daily Maximum, Minimum and Average Air Temperatures JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Average Max Average Min Average Figure 11. Average Ambient Temperatures ( Gatun Station Atlantic Lock Site) Figure 12. Diurnal Temperature Cycle (Gatun Station Atlantic Lock Site) In addition, a convection boundary condition was applied at the concrete air-exposed The convection boundaries, simulating heat transfer based on averagee wind conditions. coefficient (film coefficient, h) for the thermal analyses was calculated as described in ETL (USACE, 1994). The resulting film coefficientss were calculated to be: Pacific 46.4 kj/h..m 2 C (concrete exposed to air, no formwork) 22.9 kj/h..m 2 C (using plywood formwork). Atlantic 45.0 kj/h..m 2 C (concrete exposed to air, no formwork) 22.6 kj/h..m 2 C (using plywood formwork). 350 Innovative Dam and Levee Design and Construction

7 Lift configuration n and lift heights used in the models varied from structure to structure, but were generally placed in 3 meter lifts. The modell inputs conservatively assumed thatt each subsequent lift was placed on the previous lift at 7 day intervals and that formwork was removed from each lift on the 7th day after placement. A typical heat distribution snapshot of peak temperatures generated in the lock wall after sequenced placement is presented in Figure 13 below. Figure 13. Temperature e Distribution Within Lock Wall Section (at t=100 days) Masss Gradient Strain Evaluation Once the estimated temperature distributions within the structures were determined in the finite element models, mass gradient strain evaluations were performed to check for cracking potential, both in the longitudinal and transverse directions of the analyzed cross sections. Strains were computed in accordance with ACI 207.2R, where peak temperatures and temperature differentials were used to evaluate the potential for thermally induced cracking. The equations used to estimatee thermally induced strain are presented below. Tensile stress = f t = K R K f c E c (Eq. 5-2 in section 5.2 of ACI 207.2R) Strain therm mal = K R K f (CTE) T Where K R K f c E c R = degree of structural geometry restraint expressed as a ratio f = degree of foundation restraint expressed as a ratio c = contraction if there were no restraint c = sustained modulus of elasticity of the concrete at the time when occurred and for the duration involved TT = difference between concrete peak temperature and final stable temperature CTE = Coefficient of Thermal Expansion c Concrete Thermal Strain 351

8 Prior to computing mass gradient strains, age based compressive strength curves based on laboratory data (Figure 14) were determined, which were then correlated to time dependent tensile capacity, creep, and modulus of elasticity functions. The correlations were based on either published relationships or curve fit plots from correlated laboratory data. Laboratory tested modulus of elasticity vs. compressive strength is presented in Figure Estimated Compressive Strength Compressive Strength, f' c (MPa) Interior Mass Concrete (183+77) Structural Marine Concrete ( ) Age (days) Figure 14. Estimated Compressive Strength of Concrete 50 Estimated Modulus of Elasticity Young's Modulus, E c (GPa) % of Ultimate Load 75% of Ultimate Load 100% of Ultimate Load Compressive Strength, f' c (MPa) Figure 15. Estimated Young s Modulus vs. Compressive Strength of Concrete 352 Innovative Dam and Levee Design and Construction

9 Using these time dependent properties, the sustained modulus (Schrader, 1985) of the concrete was computed for the approximate time period that elapsed from peak temperature to the stable mean annual temperature for select nodes in the FEM model. The sustained modulus was used to account for the change (increase) in modulus of elasticity for the evaluated time periods, but also incorporates the effects of stress relaxation due to creep, generally resulting in a net reduction in the elastic modulus. Thereafter, strain capacities for each concrete mix were computed using the sustained modulus (Table 2). Table 2. Tensile Strain Capacity of Interior Mass and Structural Marine Concrete Concrete Age Range (days) Initial Final E initial (GPa) Interior Mass Concrete E final (GPa) E sustained (GPa) Strain Capacity (10-6 ) E initial (GPa) Structural Marine Concrete E final (GPa) E sustained (GPa) Strain Capacity (10-6 ) From the ANSYS thermal model, temperature time histories were extracted to determine the maximum temperatures generated in the concrete during construction at critical locations. Figure 16 shows temperature time histories used to evaluate the Pacific Lock Wall. Concrete Thermal Strain 353

10 Figure 16. Temperature Time Histories for Pacific Lock Wall with Marine Concrete in the Right Culvert Wall The temperature time histories were used to estimatee the maximum temperature differential to the average annual ambient temperature at each location. Thereafter, the duration from the peak temperature after placement to the point when the selected node normalized to the mean annual temperature was determined. The temperaturee differentials weree used to calculate the strains in the concrete at each location using restraint factors and equations from ACI 207.2R. The allowable strains for the selected age of concrete (from temperature peak to mean annual) was then determinedd for the concrete mixes. The strain limit of each age range was then compared to the calculated strain to check for thermal cracking potential at each node. While maintaining a constantt coefficient of thermal expansion, the ACI R modification factors, K R and K f were input as the only variable parameters. At each node, geometric properties of each element were used to determine these modification factors. K f factors were calculated using ACI 207.2R, Equation 5-1, while K R factors were interpolated from tables developed by ACI and refined by Schrader. An example of these strain calculations is shown in Table 3 for the longitudinal direction of the Pacific Lock Wall. 354 Innovative Dam and Levee Design and Construction

11 Representative Nodes Location Table 3. Mass Gradient Cracking Analysis (Pacific Lock Walls) Rel Elev (m) Modification Factors (Long. Direction) Temperature Differential Age Dependent Strain Capacity K R K f Max T T Age Strain Strain ( C) ( C) Range Limit (10-6 ) Strain Demand (Longitudinal Direction) Percent Strain Cracking (days) (10-6 ) Base of Culverts % No Left Culvert Wall % No Right Culvert Wall % No Left Culvert Wall % No Right Culvert Wall % No Top of Culverts % No Lower Part of Stem % No Middle of Stem % No Top of Stem % No Surface Gradient Strain Evaluation In addition to the mass gradient thermal strain evaluation, a surface gradient strain evaluation was performed. The surface gradient evaluation considered the potential for development of surface cracks during the critical period in the days immediately after placement when the surface of the concrete cools and contracts more rapidly than the warmer interior mass concrete. Surface gradient strains were evaluated based on the difference between actual temperatures throughout a given cross section and the concrete placement temperature. The critical point in surface gradient strain evaluations required determining where stress in the concrete is zero, or where it switched from tension (at the surface) to compression (beneath the surface). By plotting balanced temperature differences through a given cross section (Figure 17), the depth at which this transition occurred was determined. This depth was subsequently used to calculate the strain modification factor, K R. for input into strain computations as defined in ACI 207.2R. For the surface gradient evaluation, age ranges during the curing process were used to determine the time dependent material properties for input into the calculation of strain capacity. A similar process to the mass gradient evaluation was then used to calculate the strain demand in the concrete and checked against the computed strain capacity. Concrete Thermal Strain 355

12 Figure 17. Surface Gradient Temperature es Across Concrete Section The calculated surface gradient strains across the first lift of the left culvert wall for the Pacific Lock Wall are summarized in the table below. Table 4. Surface Gradient Analysis (First Lift of Left Culvert Wall) Initial Time (days) Final Time (days) E initial (GPa) E final (GPa) E sus Creep F(k) ( MPa) Compressive Strength (MPa) Tensile Strength (MPa) H L/ /H h/h K r T (1) ( C) Incremental T % Capacity Cracking ( C) (1) Temeprature differencee from the balanced temperature (zero stress temperature) (2) Positive is tension and negative is compression % No Crack 77% No Crack 95% No Crack 79% No Crack 52% No Crack 18% No Crack 4% No Crack 0% No Crack 1% No Crack 1% No Crack 356 Innovative Dam and Levee Design and Construction

13 DRYING SHRINKAGE CRACKING EVALUATION As exposed faces of the freshly placed cure and the moisture in the concrete normalizes with the humidity in the ambient air, shrinkage caused by this loss of moisture produces differential strains from the concrete surface to the interior mass, which can potentially produce shrinkage cracks. The surfaces open to air in the lock chambers (exposed up to 1 year prior to filling and operation) were subject to drying shrinkage, requiring a cracking potential evaluation to determine whether reinforcing steel would be exposed to chloride attack and loss in durability. Identifying cracking potential and providing mitigations was critical since exposed and untreated cracks in the lock chambers would be exacerbated by the continuous filling and emptying of the locks during canal operations. In order to determine the cracking potential of the designed concrete mixes, drying shrinkage strain computations required estimation of the relative humidity within the concrete blocks from the surface to the interior of the concrete, including the change in humidity within the concrete over time. The relative humidity at depths from the concrete surface to the interior is shown in Figure 18 for concrete ages ranging from 36 to 365 days for a typical concrete mix tested in the laboratory. 100% Relative Humidity Vs. Depth in Concrete 95% Relative Humidity (%) 90% 85% 80% 36 Days 72 Days 75% 180 Days 365 Days 70% Depth from surface(cm) Figure 18. Estimated Relative Humidity vs. Depth in Concrete The drying shrinkage strains at different relative humidity values were estimated from actual 28-day drying shrinkage lab data points provided for the SMC mix, which were derived from two separate moist cure periods of 7 and 28 days. These data points were used as a basis for developing typical drying shrinkage strain curves for 14 and 28 day moist cure periods, as shown in Figures 19 and 20, respectively. Concrete Thermal Strain 357

14 Drying Shrinkage Strain (14-Day Moist Cure) Drying Shrinkage Strain (millionths) % RH 60% RH 70% RH 80% RH 90% RH Sample Age (days) Figure 19. Estimated Drying Shrinkage Strains with 14-Day Moist Cure Period 600 Drying Shrinkage Strain (28-Day Moist Cure) 500 Drying Shrinkage Strain (millionths) % RH 60% RH 70% RH 80% RH 90% RH Sample Age (days) Figure 20. Estimated Drying Shrinkage Strains with 28-Day Moist Cure Period In addition, the strain evaluation assumed a concrete splitting tensile strength equal to 11%, and computed a sustained modulus using the modulus vs. compressive strength curve (Figure 15) in order to determine strain capacity and tensile strength. 358 Innovative Dam and Levee Design and Construction

15 The strain and tensile stress induced by the drying shrinkage was then calculated across the evaluated section at increasing increments of age and compared against the estimated strain capacity and tensile strength at the corresponding age. Strains were evaluated in 1 cm intervals from the concrete surface to depths where strain capacity exceeded drying shrinkage strain (thus no cracking). The drying shrinkage strain evaluation compared differences in cracking for a 14-day moist cure period (required curing period) versus a 28-day moist cure period. The comparative evaluation showed that, by extending the curing period by 14 days to a total of 28 days, shrinkage strains and predicted cracking depth was noticeably reduced. Results of the comparison are summarized in Tables 5 and 6. Table 5. Drying Shrinkage Cracking Analysis (14-Day Moist Cure) Drying Shrinkage Strain (14-Day Moist Cure, Tensile Strength = 11% of Compressive Strength) Depth from Surface (cm) Age Range (days) Drying Duration (days) Relative Humidity (%) Strain (10-6 ) Incremental Strain (10-6 ) E sus (GPa) Incremental Stress (MPa) Cumulative Stress (MPa) Predicted Tensile Strength (MPa) % of Capacity Crack / No Crack % % Crack % % Crack % % Crack % % Crack % % Crack % % No Crack % % No Crack % % No Crack % % Crack % % Crack % % No Crack % % No Crack % % No Crack % % No Crack % % Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack Concrete Thermal Strain 359

16 Table 6. Drying Shrinkage Cracking Analysis (28-Day Moist Cure) Drying Shrinkage Strain (28-Day Moist Cure, Tensile Strength = 11% of Compressive Strength) Depth from Surface (cm) Age Range (days) Drying Duration (days) Relative Humidity (%) Strain (10-6 ) Incremental Strain (10-6 ) E sus (GPa) Incremental Stress (MPa) Cumulative Stress (MPa) Predicted Tensile Strength (MPa) % of Capacity % % No Crack % % Crack % % Crack % % Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack % % No Crack Crack SUMMARY AND CONCLUSIONS Lock wall and Lock head structures for the Panama Canal Third Set of Locks project were analyzed for thermal stresses imposed during early placements of the massive concrete sections, providing guidance on mix design, placement temperature, and configuration to produce stress levels that minimized cracking in the critical waterbearing structures. By combining the finite element thermal analysis with spreadsheet based strain limit calculations, efficient re-evaluations were performed as additional concrete mix material property data was produced during construction. This methodology allowed for quick judgments and changes to be made for concrete placement temperatures, lift heights, and other recommendations during the fast-paced design-build construction. Similarly, drying shrinkage cracking potential for air-exposed lock chamber surfaces was evaluated to determine cracking extent and provide recommendations for minimization the potential for cracking. The cracking potential evaluations ultimately provided optimization of mix designs and construction methodology to produce concrete durable enough to meet stringent criteria for the project s 100 year design life. REFERENCES American Concrete Institute (ACI) September 2007, ACI 207.2R-07, Report on Thermal and Volume Change Effects on Cracking of Mass Concrete Autoridad del Canal de Panama, , Temperatura Horaria Promedio, Estacion Balboa FAA, Periodo , Departamento de Ambiente, Agua y Energia, Division de Agua, Seccion de Recursos Hidricos 360 Innovative Dam and Levee Design and Construction

17 Autoridad del Canal de Panama, 2008, RFP Design and Construction of the Third Set of Locks, Appendix A, Climatological Data from Balboa FAA, Volume VI- Reference Documents, Part 7 - Hydrometeorological Report, September 2008 Schön, J.H., 1996, Physical Properties of Rocks: Fundamentals and Principles of Petrophysics, PermagonPress Schrader, Tatro, 1985, "Thermal Considerations for Roller-Compacted Concrete", ACI Journal, March-April 1985 U.S. Army Corps of Engineers (USACE), 1994, ETL , Engineering and Design Nonlinear, Incremental Structural Analysis of Massive Concrete Structures, 31 December 1994 U.S. Army Corps of Engineers (USACE), 1997, ETL , Thermal Studies of Mass Concrete Structures, 30 May 1997 USBR (U.S. Bureau of Reclamation) 1981, A Water Resources Technical Publication, Engineering Monograph No.34, Control of Cracking in Mass Concrete Structures, Revised Reprint 1981 USBR, 1992, Concrete Manual, Pt. 2, A Manual for the Control of Concrete Construction, US Department of the Interior, Bureau of Reclamation, URS Holdings, Inc., 2007, Table 4-42, Chapter 4, Category III Environmental Impact Study, Panama Canal Expansion Project, July 2007 Concrete Thermal Strain 361

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19 HYDROMECHANICAL ANALYSIS FOR THE SAFETY ASSESSMENT OF A GRAVITY DAM Maria Luísa Braga Farinha 1 Eduardo M. Bretas 2 José V. Lemos 3 ABSTRACT This paper presents a study on seepage in a gravity dam foundation carried out with a view to evaluating dam stability for the failure scenario of sliding along the dam/foundation interface. A discontinuous model of the dam foundation was developed, using the code UDEC, and a fully coupled mechanical-hydraulic analysis of the water flow through the rock mass discontinuities was carried out. The model was calibrated taking into account recorded data. Results of the coupled hydromechanical model were compared with those obtained assuming either that the joint hydraulic aperture remains constant or that the drainage system is clogged. Water pressures along the dam/foundation interface obtained with UDEC were compared with those obtained using the code DEC-DAM, specifically developed for dam analysis, which is also based on the Discrete Element Method but in which flow is modelled in a different way. Results confirm that traditional analysis methods, currently prescribed in various guidelines for dam design, may either underestimate or overestimate the value of uplift pressures. The method of strength reduction was used to estimate the stability of the dam/foundation system for different failure scenarios and the results were compared with those obtained using the simplified limit equilibrium approach. The relevance of using discontinuum models for the safety assessment of concrete dams is highlighted. INTRODUCTION Gravity dams resist the thrust of the reservoir water with their own weight. The flow through the foundation, in the upstream-downstream direction, gives rise to uplift forces, which, in turn, reduce the stabilizing effect of the structures weight. Due to the great influence that uplift forces have on the overall stability of gravity dams, the distribution of water pressures along the base of the dam should be correctly recorded, in operating dams, and as accurately predicted as possible, using numerical models, at the design stage or for dams in which additional foundation treatment is required. Stability analysis of gravity dams for scenarios of foundation failure is often based on simplified limit equilibrium procedures. Equivalent continuum models of the rock mass foundation can be employed to assess the safety of concrete dams, complemented with 1 Research Engineer, Concrete Dams Department, LNEC National Laboratory for Civil Engineering, Av. Brasil 101, Lisboa, Portugal, lbraga@lnec.pt. 2 PhD, Graduate Student, Universidade do Minho, Departamento de Engenharia Civil, P Guimarães, Portugal, eduardombretas@gmail.com. 3 Senior Research Engineer, Concrete Dams Department, LNEC National Laboratory for Civil Engineering, Av. Brasil 101, Lisboa, Portugal, vlemos@lnec.pt. Hydromechanical Analysis 363

20 interface elements to simulate the behaviour of joints, shear zones and faults along which sliding may occur. More advanced analysis, however, is carried out with discontinuum models which simulate the hydromechanical interaction, which is particularly important in this type of structure. These models take into account not only shear displacements and apertures of the foundation discontinuities, but also water pressures within the dam foundation. Discrete element techniques, which allow the discontinuous nature of the rock mass to be properly simulated, are particularly adequate to assess the safety of concrete dams. This study was carried out with data obtained from Pedrógão gravity dam (Figure 1), the first roller compacted concrete (RCC) dam built in Portugal, located on the River Guadiana. The dam is part of a multipurpose development designed for irrigation, energy production and water supply (Miranda and Maia 2004). It is a straight gravity dam with a maximum height of 43 m and a total length of 448 m, of which 125 m are of conventional concrete and 323 m of RCC. The dam has an uncontrolled spillway with a length of 301 m with the crest at an elevation of 84.8 m, which is the retention water level (RWL). The maximum water level (MWL) is 91.8 m. The foundation consists of granite with small to medium-sized grains and is of good quality with the exception of the areas located near two faults in the main river channel and on the right bank, where the geomechanical properties at depth are weak. The construction of the dam began in April 2004 and work was concluded in February The controlled first filling of the reservoir ended in April d g c b a Figure 1. Pedrógão dam. Downstream view from the right side of the uncontrolled spillway and average position of the main sets of rock joints in relation to the dam. In order to analyse seepage in some foundation areas and to interpret recorded discharges, a two-dimensional equivalent continuum model was developed, in 2006, in which the main seepage paths, identified with in situ tests, were represented (Farinha 2010; Farinha et al. 2007). This model allowed recorded discharges during normal operation to be accurately interpreted and thus it was used to calibrate the parameters of the discontinuous hydromechanical model of Pedrógão dam foundation presented in this paper. Analysis was carried out with the code UDEC (Itasca 2004), in which the medium is represented as an assemblage of discrete blocks and the discontinuities as boundary 364 Innovative Dam and Levee Design and Construction

21 conditions between blocks. Water pressures along the dam/foundation interface obtained with UDEC were compared with those obtained using the code DEC-DAM, which is being developed as part of a PhD thesis currently being written by the second author, for the safety assessment of gravity dams. This code is also based on the Discrete Element Method but the flow is modelled in a different way. Results of the coupled hydromechanical model were compared with those obtained with a simple hydraulic model, in which the joint hydraulic aperture remains constant. The method of strength reduction was used to estimate the stability of the dam/foundation system for different failure scenarios, and the results were compared with those obtained using the simplified limit equilibrium approach. HYDROMECHANICAL DISCONTINUUM MODEL Fluid flow analysis with both UDEC and DEC-DAM The code UDEC allows the interaction between the hydraulic and the mechanical behaviour to be studied in a fully-coupled way. Joint apertures and water pressures are updated at every timestep, as described in Lemos (1999) and in Lemos (2008). It is assumed that rock blocks are impervious and that flow takes place only through the set of interconnecting discontinuities. These are divided into a set of domains, separated by contact points. Each domain is assumed to be filled with fluid at uniform pressure and flow is governed by the pressure differential between adjacent domains. Total stresses are obtained inside the impervious blocks and effective normal stresses at the mechanical contacts. Flow is modelled by means of the parallel plate model, and the flow rate per model unit width is thus expressed by the cubic law. The flow rate through contacts is given by: q = k j a 3 p l (1) where k j = a joint permeability factor (also called joint permeability constant), whose theoretical value is 1/(12 ) being the dynamic viscosity of the fluid; a = contact hydraulic aperture; p = pressure differential between adjacent domains (corrected for the elevation difference); l = length assigned to the contact between the domains. The dynamic viscosity of water at 20 C is N.s/m 2 and thus the joint permeability factor is 83.3 Pa -1 s -1. The hydraulic aperture to be used in Equation 1 is given by: a = a 0 + a (2) where a 0 = aperture at nominal zero normal stress and a = joint normal displacement taken as positive in opening. A maximum aperture, a max, is assumed, and a minimum value, a res, below which mechanical closure does not affect the contact permeability. The code DEC-DAM allows both static and dynamic analysis by means of the Discrete Element Method, and has been used to investigate failure mechanisms of reinforced Hydromechanical Analysis 365

22 gravity dams (Bretas et al. 2010). In both of the above-mentioned codes, the medium is assumed to be deformable and the flow is dependent on the state of stress within the foundation. The main difference between both codes relies on the hydraulic-mechanical data model, mainly on the representation of block interaction. Regarding modelling of the hydraulic behaviour, DEC-DAM considers flow channels, where the flow rate is determined, and hydraulic nodes, where water pressures are calculated. The flow channels correspond to the mechanical face-to-face contacts, while the hydraulic nodes correspond to the sub-contacts where the mechanical interaction between blocks takes place. The main advantage of the approach used in DEC-DAM is that the mechanical actions of the water are obtained from the integration of a trapezoidal diagram of water pressures (rectangular diagrams are used in UDEC), allowing greater accuracy even when a coarse mesh is used. Both the above-mentioned codes allow the modelling of grout and drainage curtains, which is necessary in order to study seepage in concrete dam foundations. Model description The discontinuous model developed to analyse fluid flow through the rock mass discontinuities is shown in Figure 2. In a simplified way, only two of the five sets of discontinuities identified at the dam site were simulated: the first joint set is horizontal and continuous, with a spacing of 5.0 m, and the second set is formed by vertical crossjoints, with a spacing of 5.0 m normal to joint tracks and standard deviation from the mean of 2.0 m. The former attempts to simulate the sub-horizontal set of discontinuities g) and the latter the sub-vertical set b), both of which are shown in Figure 1. An additional rock mass joint was assumed downstream from the dam dipping 25 towards upstream, necessary to the stability analysis for failure scenarios of sliding along foundation discontinuities. The foundation model is m wide and 80.0 m deep. The dam has the crest of the uncontrolled spillway 33.8 m above ground level and the base is 44.4 m long in the upstream-downstream direction. In concrete, a set of horizontal continuous discontinuities located 2.0 m apart was assumed to simulate dam lift joints. The UDEC model has 611 deformable blocks divided into 2766 zones, and 3451 nodal points, and the DEC-DAM model has 611 deformable blocks m 80 m Concrete: unit weight = 2400 kg/m 3 Young s modulus = 30 GPa Poisson s ratio = 0.2 Foundation blocks: unit weight = 2650 kg/m 3 Young s modulus = 10 GPa Poisson s ratio = 0.2 Foundation discontinuities: k n = 1 or 10 or 100 GPa/m 200 m k s = 0.5 k n = 30 Figure 2. Discontinuum model of Pedrógão dam foundation and material properties. 366 Innovative Dam and Levee Design and Construction

23 Both dam concrete and rock mass blocks are assumed to follow elastic linear behaviour, with the properties shown in Figure 2. Discontinuities are assigned a Mohr-Coulomb constitutive model, complemented with a tensile strength criterion. In a base run, a joint normal stiffness (k n ) of 10 GPa/m, a joint shear stiffness (k s ) of 5 GPa/m, and a friction angle ( ) of 35 were assumed at the dam lift joints, at the foundation discontinuities and at the dam/foundation interface. Both at the dam lift joints and at the dam/foundation interface cohesion and tensile strength were assigned 2.0 MPa. In rock joints, cohesion and tensile strength were assumed to be zero. Figure 3. Block deformation (magnified 3000 times) due to dam weight, hydrostatic loading and flow. To take into account the uncertainty in joint normal stiffness, new analysis was carried out assuming rock masses with different deformability (k n 5 times higher and 5 times lower than that assumed in the base run). Using the following equation, 1 E RM 1 1 = + (3) E k s R n where E R is the modulus of deformation of the rock matrix, k n is the fracture normal stiffness, and s is fracture spacing, the rock mass in which the normal stiffness of discontinuities is assumed to be 2 GPa/m has an equivalent deformability of 5 GPa, that with k n = 10 GPa/m an equivalent deformability of 8.33 GPa and the stiffest foundation, with k n = 50 GPa/m, an equivalent deformability of 9.6 GPa. Sequence of analysis Analysis was carried out in two loading stages. Firstly, the mechanical effect of gravity loads with the reservoir empty was assessed. In the UDEC model, an in-situ state of stress with an effective stress ratio H / V = 0.5 was assumed in the rock mass. The water table was assumed to be at the same level as the rock mass surface upstream from the dam. Secondly, the hydrostatic loading corresponding to the full reservoir was applied to both the upstream face of the dam and reservoir bottom. Hydrostatic loading was also applied to the rock mass surface downstream from the dam. In this second loading stage, mechanical pressure was first applied, followed by hydromechanical analysis. In both Hydromechanical Analysis 367

24 stages, vertical displacements at the base of the model and horizontal displacements perpendicular to the lateral model boundaries were prevented. Regarding hydraulic boundary conditions, joint contacts along the bottom and sides of the model were assumed to have zero permeability. The drainage system was simulated assigning a hydraulic head along the drains equal to one third of the sum of the hydraulic head upstream and downstream from the dam. On the rock mass surface, the head was 33.8 m upstream from the dam, and 5.0 m downstream. Figure 3 shows a detail of dam and foundation deformation due to the simultaneous effect of dam weight, hydrostatic loading and flow. Hydraulic parameters The model hydraulic parameters (a 0 and a res ), which correspond to an equivalent permeability of the rock mass of m/s, were adjusted from a two-dimensional equivalent continuum model previously developed, which had been calibrated taking into account recorded discharges (Farinha et al. 2007). It was assumed that the grout curtain was 10 times less pervious than the surrounding rock mass. The in situ borehole waterinflow tests performed (test procedures described in detail in Farinha et al. (2011)), led to the conclusion that the main seepage paths crossed the drains at between 3.0 and 8.0 m down from the dam/foundation interface. In order to simulate this area where the majority of the flow is concentrated, it was assumed that the horizontal discontinuity located 5.0 m below the dam/foundation interface was 8 times more pervious than the other rock mass discontinuities, in the area underneath the dam and crossing the grout curtain. In every run, with different joint stiffnesses, the same a max and a res were assumed and a 0 was that which, in each analysis, led to the recorded discharge (a 0 = mm for k n = 50 GPa/m, a 0 = 0.17 mm for k n = 10 GPa/m, and a 0 = mm for k n = 2 GPa/m and a res = 0.05 mm). In this way, the same situation is simulated with different models, which enables comparison of water pressures and apertures along the base of the dam or along other rock mass discontinuities. Fluid flow analysis RESULTS ANALYSIS Results of fluid flow analysis carried out with the UDEC model, with the reservoir at the RWL, both with constant joint hydraulic aperture and taking into account the hydromechanical interaction are shown in Figures 4 and 5. Figure 4 shows the percentage of hydraulic head contours within the dam foundation (percentage of hydraulic head is the ratio of the water head measured at a given level, expressed in metres of height of water, to the height of water in the reservoir above that level). In Figure 5, the line thickness is proportional to the flow rate in the fracture.when the coupling between stress and flow is taken into account, the loss of hydraulic head is concentrated at the grout curtain s area, below the heel of the dam, and the maximum water pressure is around 10 % higher (Figure 4 a) and b)). Without drainage, the hydraulic head decreases gradually below the base of the dam (Figure 4 c)). 368 Innovative Dam and Levee Design and Construction

25 a) b) c) a) constant joint aperture b) hydromechanical interaction c) hydromechanical interaction, without drainage system Figure 4. Percentage of hydraulic head contours for full reservoir. a) b) max flow rate = 2.011E-05 each line thick = 3.000E-06 max flow rate = 2.089E-05 each line thick = 3.000E-06 c) a) constant joint aperture b) hydromechanical interaction c) hydromechanical interaction, no drainage system max flow rate = 4.966E-06 each line thick = 3.000E-06 Figure 5. Flow rate for full reservoir (flow rate is proportional to line thickness; flow rates below (m 3 /s)/m (0.18 (L/min)/m) are not represented). Hydromechanical Analysis 369

26 Figure 5 shows that the majority of the flow is concentrated in the first two vertical joints upstream from the heel of the dam, and that this water flows towards the drain, or towards downstream in the foundation with no drainage system, along the joint of higher permeability that crosses the grout curtain, which simulates the main seepage paths. When the hydromechanical interaction is taken into account, flow rates are higher at lower levels and a higher quantity of water flows into the model through the second vertical joint upstream from the heel of the dam, rather than through the first as is the case in the run where joint aperture remains constant. This depends on the increase in water pressure in a given vertical joint, which causes the closure of adjacent vertical joints. The maximum flow rate is slightly higher when the interaction is taken into account (it varies from around 1.21 to 1.25 (L/min)/m). The quantity of water that flows through the model in the analysis with no drainage system and constant joint aperture is 0.57 (L/min)/m. This increases by around 248 %, to 1.40 (L/min)/m, in the case of the most deformable foundation, and decreases by around 26 %, to 0.42 (L/min)/m, in the case of the stiffest foundation. Water pressures along the dam/foundation joint The variation of water pressures along the dam/foundation joint is shown in Figure 6, along with a comparison of water pressures along the dam/foundation joint with both bilinear and linear uplift distribution, usually used in stability analysis of dams with and without drainage systems, respectively. Results obtained with the foundations of different deformability are presented. In the hydraulic analysis in which the HM effect is not taken into account, variations in uplift pressures along both the interface and the foundation discontinuities are the same regardless of the foundation deformability, because the joint hydraulic aperture remains constant. Figure 6 shows that variations in water pressures are highly dependent on the pressure on the drainage line. Upstream from this line, water pressures are higher for more deformable foundations. Downstream from the drainage line, on the contrary, water pressures are higher for stiffer rock masses. Along the dam/foundation joint, if all the drains are clogged, the highest water pressures are obtained with the stiffest foundation, and the lowest with the most deformable rock mass. In the case of drained foundations, the water pressure curves are close to the bi-linear distribution. In this case, computed water pressures between the heel of the dam and the drainage line are lower than those given by the bi-linear distribution, whereas between the drainage line and the toe of the dam they are higher, except for the most deformable foundation. In the case of the stiffest foundations with no drainage system, calculated uplift pressures are lower than those obtained with the linear distribution, to a distance of around 8.0 m from the heel of the dam, and downstream from this point they are considerably higher. At the dam/foundation joint end close to the toe of the dam, UDEC water pressures are higher than those assumed with the linear distribution of pressures, due to the presence of the rock wedge downstream from the dam. For the most deformable foundation, the linear distribution of uplift pressures greatly overestimates pressures along the base of the dam, with the exception of an area with a length of around 6.0 m, close to the toe of the dam. 370 Innovative Dam and Levee Design and Construction