concrete thermal strain, shrinkage and … thermal strain, shrinkage and cracking analysis ......

Concrete Thermal Strain 345

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, [email protected] 2 Structural Engineer, MWH Americas Inc., Walnut Creek, California, [email protected] 3 Design Engineer, Panama Canal Third Set of Locks Project, MWH Americas Inc., Chicago, Illinois, [email protected]

346 Innovative Dam and Levee Design and Construction

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


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 1110-2-542 (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.


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.


Table 1. Summary of Material Properties

Properties Units

Interior Mass

Concrete (IMC)

Structural Marine

Concrete (SMC)

Specific Heat (Ch) kJ / kg·°C 0.83 0.83 Thermal Conductivity (K) kJ / m·h·°C 3.74 3.74 Density (ȡ) kg / m3 2508 2523 Diffusivity (h2) m2 / h x 10-3 1.79 1.78 Adiabatic Temperature Rise °C 26.8 52.3 Ultimate Compressive Strength (Fc) MPa 30.3 59.2 Ultimate Modulus of Elasticity (Ec) GPa 38.9 43.3 Coefficient of Thermal Expansion (CTE) mil/°C 8.0 8.0

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).

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

0

10

20

30

40

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(°C)

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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 15.

Figure 14. Estimated Compressive Strength of Concrete

Figure 15. Estimated Young’s Modulus vs. Compressive Strength of Concrete

0

10

20

30

40

50

60

70

80

1 10 100 1000

Com

pres

sive

Stre

ngth

, f' c

(MPa

)

Age (days)

Estimated Compressive Strength

Interior Mass Concrete (183+77)

Structural Marine Concrete (300+56+19)

0

5

10

15

20

25

30

35

40

45

50

0 5 10 15 20 25 30 35 40 45 50 55 60

Youn

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Mod

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, E c

(GPa

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Compressive Strength, f'c (MPa)

Estimated Modulus of Elasticity

Ec @ 25% of Ultimate Load




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

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 Age Range (days) Interior Mass Concrete Structural Marine Concrete

Initial Final Einitial (GPa)

Efinal (GPa)

Esustained (GPa)

Strain Capacity

(10-6)

Einitial (GPa)

Efinal (GPa)

Esustained (GPa)

Strain Capacity

(10-6) 0 1 0.0 9.9 4.7 57 0.0 24.7 11.0 92 1 3 9.9 18.3 12.9 50 24.7 37.0 28.2 77 3 7 18.3 25.3 19.3 54 37.0 42.4 35.8 95 7 14 25.3 32.9 25.0 67 42.4 43.6 38.6 119 14 28 32.9 38.9 30.1 82 43.6 43.3 38.7 140 28 90 38.9 41.7 32.0 98 43.3 42.8 37.0 171 90 180 41.7 42.3 33.6 100 42.8 42.8 36.9 174 180 365 42.3 42.5 33.1 104 42.8 42.8 36.1 180

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Table 3. Mass Gradient Cracking Analysis (Pacific Lock Walls)

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, KR. 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.

Base of Culverts 1.3 1.00 0.93 41.5 14.8 23 - 365 121 110.2 91% NoLeft Culvert Wall 4.8 1.00 0.48 59.1 32.4 6 - 180 196 124.3 63% NoRight Culvert Wall 4.8 1.00 0.55 60.1 33.4 7 - 180 192 147.5 77% NoLeft Culvert Wall 8.23 1.00 0.55 60.2 33.5 7 - 180 192 147.8 77% NoRight Culvert Wall 8.23 1.00 0.48 58.8 32.1 5 - 90 192 123.3 64% No

Top of Culverts 11.4 1.00 0.89 42.0 15.3 15 - 365 130 108.3 83% NoLower Part of Stem 18.4 0.41 0.35 45.4 18.7 29 - 365 116 21.8 19% No

Middle of Stem 25.9 0.11 0.35 46.1 19.4 19 - 365 125 6.0 5% NoTop of Stem 34.1 0.01 0.35 58.5 31.8 2 - 49 206 0.9 0% No

Strain Demand (Longitudinal Direction)

Modification Factors (Long. Direction)

KR KfStrain (10-6)

Percent Strain

CrackingLocationRel Elev

(m)Max T (°C)

ǻT (°C)

Age Range (days)

Strain Limit (10-6)

Representative Nodes Temperature Differential

Age Dependent Strain Capacity

356

The cPacif

InitialTime

(days)

0124714285690

180(1) Tem(2) Pos

Figure

calculated sufic Lock Wal

Table

)

Final Time

(days)

Einitial

(GPa)

1 0.002 17.724 23.227 28.61

14 31.5428 32.4956 31.9090 30.89180 30.57365 30.40

meprature differencesitive is tension and n

e 17. Surfac

urface gradiell are summa

4. Surface G

Efinal

(GPa)Creep

F(k) (

17.72 35.023.22 6.928.61 5.131.54 3.932.49 3.131.90 2.630.89 2.330.57 2.130.40 2.130.23 2.1

e from the balanced tnegative is compress

Innova

e Gradient T

ent strains acarized in the

Gradient Ana

Esus

MPa)

Compressive Strength

(MPa)

8.2 14.2619.7 20.8424.5 30.2528.3 39.7129.6 51.3429.5 59.2428.6 65.8228.1 67.5727.4 68.4526.8 69.33temperature (zero stsion

ative Dam a

Temperature

cross the firse table below

alysis (First Tensile

Strength(MPa)

H L/

1.01 0.4 41.60 0.39 42.48 0.48 33.41 0.61 34.59 0.75 25.42 0.78 26.12 0.93 16.31 1.1 16.40 1.33 16.50 1.45 1

tress temperature)

and Levee D

es Across Co

st lift of the lw.

Lift of Left

/H h/H Kr

45 1.00 0.9346 1.00 0.9438 1.00 0.9230 1.00 0.9024 1.00 0.8823 1.00 0.8819 1.00 0.8516 1.00 0.8314 1.00 0.7912 1.00 0.78

Design and C

oncrete Secti

left culvert w

Culvert Wa

!T(1)

(°C)

Incrementa!T

(°C)

2.52 2.529.77 7.2516.03 6.2617.71 1.6816.28 -1.439.34 -6.945.58 -3.763.48 -2.092.21 -1.270.83 -1.38

Constructio

ion

wall for the

all) l

% Capacity Crackin

15% No Cra77% No Cra95% No Cra79% No Cra52% No Cra18% No Cra4% No Cra0% No Cra1% No Cra1% No Cra

on

ng

ackackackackackackackackackack


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.

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.

70%

75%

80%

85%

90%

95%

100%

0 5 10 15 20 25 30

Rela

tive

Hum

idity

(%)

Depth from surface(cm)

Relative Humidity Vs. Depth in Concrete

36 Days 72 Days

180 Days 365 Days


Figure 19. Estimated Drying Shrinkage Strains with 14-Day Moist Cure Period

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.

0

100

200

300

400

500

600

700

800

0 50 100 150 200 250 300 350

Dryi

ng S

hrin

kage

Str

ain

(mill

iont

hs)

Sample Age (days)

Drying Shrinkage Strain (14-Day Moist Cure)

50% RH

60% RH

70% RH

80% RH

90% RH

0

100

200

300

400

500

600

0 50 100 150 200 250 300 350

Dryi

ng S

hrin

kage

Str

ain

(mill

iont

hs)

Sample Age (days)

Drying Shrinkage Strain (28-Day Moist Cure)

50% RH60% RH70% RH80% RH90% RH


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)

Depth from Surface

(cm)

Age Range (days)

Drying Duration

(days)

Relative Humidity

(%)

Strain (10-6)

Incremental Strain (10-6)

Esus

(GPa)

Incremental Stress (MPa)

Cumulative Stress (MPa)

Predicted Tensile

Strength (MPa)

% of Capacity

Crack / No Crack

14 - 28 0 - 14 79% 154 154 38.7 6.0 6.0 5.4 110% Crack28 - 56 14 - 42 78% 261 107 38.0 4.1 10.0 6.1 164% Crack56 - 90 42 - 76 77% 311 50 38.0 1.9 11.9 6.3 189% Crack90 - 180 76 - 166 77% 331 20 36.9 0.7 12.7 6.4 198% Crack

180 - 365 166 - 351 76% 365 34 36.1 1.2 13.9 6.5 214% Crack14 - 28 0 - 14 91% 66 66 38.7 2.6 2.6 5.4 47% No Crack28 - 56 14 - 42 90% 118 52 38.0 2.0 4.5 6.1 74% No Crack56 - 90 42 - 76 89% 149 31 38.0 1.2 5.7 6.3 91% No Crack90 - 180 76 - 166 87% 187 38 36.9 1.4 7.1 6.4 111% Crack

180 - 365 166 - 351 85% 228 41 36.1 1.5 8.6 6.5 132% Crack14 - 28 0 - 14 94% 44 44 38.7 1.7 1.7 5.4 31% No Crack28 - 56 14 - 42 93% 83 39 38.0 1.5 3.2 6.1 52% No Crack56 - 90 42 - 76 92% 108 25 38.0 0.9 4.1 6.3 66% No Crack90 - 180 76 - 166 90% 144 36 36.9 1.3 5.5 6.4 85% No Crack

180 - 365 166 - 351 88% 182 38 36.1 1.4 6.8 6.5 105% Crack14 - 28 0 - 14 96% 29 29 38.7 1.1 1.1 5.4 21% No Crack28 - 56 14 - 42 95% 59 30 38.0 1.1 2.3 6.1 37% No Crack56 - 90 42 - 76 94% 81 22 38.0 0.8 3.1 6.3 49% No Crack90 - 180 76 - 166 92% 115 34 36.9 1.3 4.4 6.4 68% No Crack

180 - 365 166 - 351 90% 152 37 36.1 1.3 5.7 6.5 88% No Crack14 - 28 0 - 14 98% 15 15 38.7 0.6 0.6 5.4 11% No Crack28 - 56 14 - 42 97% 36 21 38.0 0.8 1.4 6.1 23% No Crack56 - 90 42 - 76 96% 54 18 38.0 0.7 2.1 6.3 33% No Crack90 - 180 76 - 166 94% 86 32 36.9 1.2 3.2 6.4 51% No Crack

180 - 365 166 - 351 92% 122 36 36.1 1.3 4.5 6.5 70% No Crack14 - 28 0 - 14 98% 15 15 38.7 0.6 0.6 5.4 11% No Crack28 - 56 14 - 42 97% 36 21 38.0 0.8 1.4 6.1 23% No Crack56 - 90 42 - 76 97% 41 5 38.0 0.2 1.6 6.3 25% No Crack90 - 180 76 - 166 95% 72 31 36.9 1.1 2.7 6.4 42% No Crack

180 - 365 166 - 351 93% 106 65 36.1 2.3 5.1 6.5 78% No Crack

Drying Shrinkage Strain (14-Day Moist Cure, Tensile Strength = 11% of Compressive Strength)

0

1

2

5

3

4


Table 6. Drying Shrinkage Cracking Analysis (28-Day Moist Cure)

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 water-bearing 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, 2005-2009, Temperatura Horaria Promedio, Estacion Balboa FAA, Periodo 2005-2009, Departamento de Ambiente, Agua y Energia, Division de Agua, Seccion de Recursos Hidricos

Depth from Surface

(cm)

Age Range (days)

Drying Duration

(days)

Relative Humidity

(%)

Strain (10-6)

Incremental Strain (10-6)

Esus

(GPa)

Incremental Stress (MPa)

Cumulative Stress (MPa)

Predicted Tensile

Strength (MPa)

% of Capacity

Crack

28 - 56 0 - 28 79% 150 150 38.0 5.7 5.7 6.1 93% No Crack56 - 90 28 - 62 78% 186 36 38.0 1.4 7.1 6.3 112% Crack90 -180 62 - 152 77% 214 28 36.9 1.0 8.1 6.4 127% Crack

180 - 365 152 - 337 76% 237 23 36.1 0.8 8.9 6.5 137% Crack28 - 56 0 - 28 91% 64 64 38.0 2.4 2.4 6.1 40% No Crack56 - 90 28 - 62 90% 85 21 38.0 0.8 3.2 6.3 51% No Crack90 -180 62 - 152 87% 121 36 36.9 1.3 4.6 6.4 71% No Crack

180 - 365 152 - 337 85% 148 27 36.1 1.0 5.5 6.5 85% No Crack28 - 56 0 - 28 94% 43 43 38.0 1.6 1.6 6.1 27% No Crack56 - 90 28 - 62 93% 59 16 38.0 0.6 2.2 6.3 36% No Crack90 -180 62 - 152 91% 84 25 36.9 0.9 3.2 6.4 49% No Crack




180 - 365 152 - 337 93% 69 22 36.1 0.8 2.6 6.5 39% No Crack

4

5

Drying Shrinkage Strain (28-Day Moist Cure, Tensile Strength = 11% of Compressive Strength)

0

1

2

3


Autoridad del Canal de Panama, 2008, RFP-76161 - 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 1110-2-365, Engineering and Design Nonlinear, Incremental Structural Analysis of Massive Concrete Structures, 31 December 1994

U.S. Army Corps of Engineers (USACE), 1997, ETL 1110-2-542, “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, 1992.

URS Holdings, Inc., 2007, Table 4-42, Chapter 4, Category III Environmental Impact Study, Panama Canal Expansion Project, July 2007

Hydromechanical Analysis 363

HYDROMECHANICAL ANALYSIS FOR THE SAFETY ASSESSMENT OF A GRAVITY DAM

Maria Luísa Braga Farinha1

Eduardo M. Bretas2 José V. Lemos3

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, 1700-066 Lisboa, Portugal, [email protected]. 2 PhD, Graduate Student, Universidade do Minho, Departamento de Engenharia Civil, P-4800-058 Guimarães, Portugal, [email protected]. 3 Senior Research Engineer, Concrete Dams Department, LNEC – National Laboratory for Civil Engineering, Av. Brasil 101, 1700-066 Lisboa, Portugal, [email protected].


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 2006. The controlled first filling of the reservoir ended in April 2006.

a

d

b

g

c

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


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:

lpakq j

!"= 3 (1)

where kj = a joint permeability factor (also called joint permeability constant), whose theoretical value is 1/(12 ȝ) being ȝ 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 1.002 × 10-3 N.s/m2 and thus the joint permeability factor is 83.3 Pa-1s-1. The hydraulic aperture to be used in Equation 1 is given by: aaa !+= 0 (2) where a0 = aperture at nominal zero normal stress and a! = joint normal displacement taken as positive in opening. A maximum aperture, amax, is assumed, and a minimum value, ares, 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


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 cross-joints, 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 200.0 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.

200 m

80 m

33.8 m

Concrete: unit weight = 2400 kg/m3 Young´s modulus = 30 GPa Poisson’s ratio = 0.2

Foundation blocks: unit weight = 2650 kg/m3 Young´s modulus = 10 GPa Poisson’s ratio = 0.2

Foundation discontinuities: kn = 1 or 10 or 100 GPa/m ks = 0.5 kn ĳ = 30°

Figure 2. Discontinuum model of Pedrógão dam foundation and material properties.


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 (kn) of 10 GPa/m, a joint shear stiffness (ks) of 5 GPa/m, and a friction angle (ĳ) 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 (kn 5 times higher and 5 times lower than that assumed in the base run). Using the following equation,

skEE nRRM

111 += (3)

where ER is the modulus of deformation of the rock matrix, kn 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 kn = 10 GPa/m an equivalent deformability of 8.33 GPa and the stiffest foundation, with kn = 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 ıH/ıV = 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


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 (a0 and ares ), which correspond to an equivalent permeability of the rock mass of 5.0 × 10-7 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 water-inflow 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 amax and ares were assumed and a0 was that which, in each analysis, led to the recorded discharge (a0 = 0.1313 mm for kn = 50 GPa/m, a0 = 0.17 mm for kn = 10 GPa/m, and a0 = 0.4287 mm for kn = 2 GPa/m and ares = 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.

RESULTS ANALYSIS Fluid flow 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)).


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) constant joint aperture

b) hydromechanical interaction

c) hydromechanical interaction, no drainage system

max flow rate = 2.011E-05each line thick = 3.000E-06



a) b)

c)

Figure 5. Flow rate for full reservoir (flow rate is proportional to line thickness; flow rates below 3.0 × 10-6 (m3/s)/m (0.18 (L/min)/m) are not represented).


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 bi-linear 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.


0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0

Distance from the heel along the base of the dam (m)

Dom

ain

pres

sure

(x 1

0-1 M

Pa)

bi-linear distribution of uplift pressures

linear distribution of uplift pressures

constant joint aperture constant joint aperture, no drainage systemHM interaction (kn = 2 GPa/m) HM interaction, no drainage system (kn = 2 GPa/m)HM interaction (kn = 10 GPa/m) HM interaction, no drainage system (kn = 10 GPa/m)HM interaction (kn = 50 GPa/m) HM interaction, no drainage system (kn = 50 GPa/m)

Figure 6. Water pressure along the dam/foundation joint and comparison with both bi-

linear and linear distribution of water pressures. Figure 7 shows the comparison between water pressures along the dam/foundation interface calculated with both UDEC and DEC-DAM, for the case of joint normal stiffness (kn) of 10 GPa/m and of both operational and non-operational drainage systems. In the former case, there is an overall good match between the curves, except in the vicinity of the drain due to the small difference in the location assumed in the numerical representation.

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0

Distance from the heel along the base of the dam (m)

Dom

ain

pres

sure

(x 1

0-1 M

Pa)

DEC-DAM, no drainage system

UDEC

DEC-DAM

UDEC, no drainage system

Figure 7. Water pressure along the dam/foundation joint, calculated with both UDEC and

DEC-DAM.


STABILITY ANALYSIS Strength reduction method The UDEC model developed, with joint normal stiffness of 10 GPa/m, was used to assess the stability of the dam/foundation system for the four different possible sliding failure scenarios shown in Figure 8. Scenarios a) and d) concern only the dam/foundation joint. Sliding along this interface is the most probable failure scenario in dam foundation rock masses containing widely spaced discontinuities, none of which are unfavourably oriented. Pedrógão dam is embedded in the foundation, and therefore the resistance to sliding is high. Scenario d) neglects the resistance of the rock wedge at the toe of the dam, in order to take into account a possible excavation downstream, close to the toe of the dam. Scenario b) involves both the dam/foundation joint and the rock mass joint dipping 25° towards upstream, which was purposely included in the model for stability analysis. This hypothetical situation may simulate a combined mode of failure, where the failure path occurs both along the dam/foundation interface and through intact rock, in geology where the rock is horizontally or near horizontally bedded and the intact rock is weak (USACE 1994). In scenario c), sliding along the inclined rock mass joint is prevented, assuming that the behaviour of this joint is elastic.

a) dam/foundation interface b) dam/foundation interface and rock mass joint

downstream from the dam dipping 25° towards upstream

c) dam/foundation interface, preventing slip on the rock mass joint downstream from the dam dipping 25° towards upstream

d) dam/foundation interface, neglecting the resistance of the rock wedge at the toe of the dam

Figure 8. Analysed failure modes.


Analysis was carried out with the method of strength reduction, typically applied in foundation design. An initial friction angle of 35° was assigned to the rock mass discontinuities, dam foundation interface and dam lift joints, and zero cohesion and zero tensile strength were assigned to the dam/foundation joint, involved in the failure modes. The model was first run until equilibrium, then the fluid flow analysis was switched off and, from this step, water pressures were kept constant. For each failure scenario, the friction angle of the discontinuities highlighted in Figure 8 was gradually reduced until failure (the reduction coefficient was applied to tan ĳ). The failure indicator was the horizontal crest displacement. Analysis was carried out assuming that the reservoir was at the RWL or at the MWL, and that the drainage system was either operational or non-operational. Stability analysis results are shown in Figure 9 and in Table 1. In Figure 9, friction angles in the x-axis are shown in reverse order, for ease of analysis.

a) b)

0.0

10.0

20.0

30.0

40.0

5.010.015.020.025.030.035.040.0

Friction angle (degrees)

Hor

izon

tal d

ispla

cem

ent a

t cre

st

(mm

)

0.0

10.0

20.0

30.0

40.0

5.010.015.020.025.030.035.040.0


Hor

izon

tal d

ispla

cem

ent a

t cre

st

(mm

)

c) d)

0.0

2.0

4.0

6.0

8.0

10.0

5.010.015.020.025.030.035.040.0


Hor

izon

tal d

ispla

cem

ent a

t cre

st

(mm

)

2.0

4.0

6.0

8.0

10.0

20.025.030.035.040.0


Hor

izon

tal d

ispla

cem

ent a

t cre

st

(mm

)

RWL, with drainage RWL, no drainage system RWL, failure MWL, with drainage MWL, failure Figure 9. Variation in crest horizontal displacement due to reduction of the friction angle

on highlighted joints, for the failures modes shown in Figure 7. In the four analysed failure modes, the dam foundation system is unstable when the reservoir is at the MWL and the drainage system is non-operational, and therefore, these situations are not shown in Figure 9. For the same reservoir level, in both scenarios a) and c) the dam/foundation system remains stable when the drainage system is working properly, while in scenario b), as shown in Figure 9, failure occurs for a friction angle of around 27.5° (safety factor F = 1.4). In scenario d) the dam is unstable for friction angles lower than 34.5° when the reservoir is at the MWL (F = 1.01).


Table 1. Comparison of friction angles for which failure occurs calculated with the hydromechanical model and with the limit equilibrium method.

Hupstream.

(m) Hdownstream

(m) Drainage system

River bottom downstream

from the dam *

Friction angle Limit

equilibrium ** UDEC

failure last stable

84.8 60.0 not operative 1) 27.8° 34.2° 34.5° (RWL) 2) 11.1° - 22.6° 18.4° 19.3°

operative 1) 21.2° 21.3° 22.4° 2) 8.2° - 17.1° 14.0° 14.5°

91.8 67.8 not operative 1) 45.6° unstable (MWE) 2) 27.8° - 40.6° unstable

operative 1) 32.4° 34.5° 34.7° 2) 18.2° - 28.1° 26.6° 28.3°

* Downstream from the dam the river bottom is: 1) at the same level as the dam/foundation interface (51.0 m) – scenario d)

2) at its actual level (59.5 m) – scenario b) ** For failure scenario b), results are shown considering full passive force or only 1/3 of the passive force Comparison of the UDEC results with those obtained using the limit equilibrium method Table 1 shows the comparison between the UDEC results and those from the equilibrium method, for failure modes b) and d). In the analysis in which the stabilizing effect of the rock wedge downstream from the dam is taken into account, the study was done assuming either full development of passive pressure, which is improbable as it requires large structure displacements, or the development of one-third of the passive pressure, which is more realistic. Results show that the dam is stable when the reservoir is at the RWL, even when the drainage system is inoperative. When the reservoir is at the MWL, the safety factor is lower than 1 when: i) the drainage system is inoperative and the resistance from the rock wedge downstream from the dam is neglected (F = 0.69); and ii) the drainage system is inoperative and only one third of the passive force is considered in the analysis (F = 0.82). Failure mode d) is the only one which enables UDEC analysis to be verified, as the same results must be obtained for similar loads with both the UDEC and limit equilibrium analysis. Indeed, when the reservoir is at the RWL and the drainage system is operative almost the same friction angles were obtained (21.2° in the limit equilibrium analysis and between 21.3° and 22.39° in the UDEC analysis). A difference as low as around 2° is obtained in similar conditions, but with the reservoir at the MWL (32.4° in the limit equilibrium analysis and between 34.47° and 34.73° in the UDEC analysis). However, when the drainage system is inoperative, the friction angles obtained in the UDEC analysis (34.21° - 34.47°) are higher than that given by the limit equilibrium method (27.8°). This difference can be explained by the higher uplift pressures obtained in the UDEC analysis, when compared with those given by the linear distribution of water pressures between the reservoir and the tailwater, assumed in the limit equilibrium analysis. This difference in water pressures is shown in Figure 10. A limit equilibrium analysis carried out assuming a resultant of the uplift pressure 24 % higher than that


given by the linear distribution of water pressures would lead to the same friction angle at failure as the UDEC analysis (assuming that in the UDEC model failure occurs for a friction angle of 34.3°). In the analysis in which it is assumed that downstream from the dam the reservoir is at its actual level, the UDEC results are within the range of friction angles given by the limit equilibrium method, when only part or full passive force is considered, but are closer to those obtained for one third passive force.

NPA

33.8 m

9.0 m

NPA

33.8 m

9.0 m

34.7°34.5°32.4°operative(NME)

unstable45,6°not operative67.891.8

22.4°21.3°21.2°operative(NPA)

34.7°34.5°32.4°operative(NME)

unstable45,6°not operative67.891.8

22.4°21.3°21.2°operative(NPA)

0.09 MPa

0.338 MPa

linear distribution ofuplift pressures

hydromechanical model

RWL

Figure 10. Comparison between the UDEC results and those from the limit equilibrium method.

CONCLUSION

This paper presents a study on seepage in Pedrógão dam foundation using a discontinuum model, which was developed taking into account recorded data and information provided from tests carried out in situ. Analysis of seepage was done using both UDEC and DEC-DAM codes, which take into account the coupled hydromechanical behaviour of rock masses. Stability analyses were carried out for different failure scenarios and with different assumptions about uplift pressures and joint shear strength. Some of the analyzed scenarios are highly unfavourable hypothetical situations, as in this dam the resistance to sliding is high. Results allowed us to quantify the influence of water pressures on the stability of the dam. This result draws attention to the importance of using recorded water pressures for the sliding safety assessment of existing dams, as recommended by the European Club of ICOLD (2004). The uplift water pressure along the dam base is always of concern to the stability of concrete dams and is usually prescribed in design codes assuming a bi-linear uplift distribution to account for the relief drains. The study presented here shows that results depend mainly on the joint normal stiffness and on joint aperture. The comparison between the results obtained with the codes UDEC and DEC-DAM showed that there is a good match between water pressures calculated along the dam foundation joint, with both operational and non-operational drainage systems.


Discontinuum models are difficult to apply in most practical cases, because jointing patterns are very complex and there is usually a lack of data on hydraulic properties of the discontinuity sets. Among these parameters are the orientation and spacing of discontinuities, and the hydromechanical characterization data, namely joint normal stiffness, joint apertures and residual aperture, which is not readily available. However, such models which simulate the hydromechanical interaction are relevant in stability analysis, and the uncertainty in the different parameters, can be overcome by performing stability analysis assuming that each parameter may vary within a credible range. Flow in fractured rock masses is mainly three-dimensional. However, in dam foundations the flow is mainly in the upstream-downstream direction, and therefore 2D analysis may be considered adequate in most cases. For arch dams, 3D analysis is necessary, but coupled fracture flow modelling of an arch dam foundation would imply representing a network of joints from various sets, which would be computationally prohibitive. The alternative is to use 3D mechanical models, in which only the discontinuities involved in possible failure modes are represented, and the water pressures are obtained from 3D equivalent continuum models. In dam stability evaluation, the main advantage of using a 2D hydromechanical discontinuum code instead of the limit equilibrium method is that it allows the study of a wider range of failure modes. In addition, this type of code enables displacements to be calculated in seismic analysis, in contrast to what happens with the limit equilibrium approach. This type of analysis is particularly useful when the foundation contains more than one material or is made up of a combination of intact rock, jointed rock and sheared rock, as, in these cases, the overall strength of the foundation depends on the stress-strain characteristics and compatibility of the various materials. It is also relevant in those cases in which controls of maximum displacement, needed to ensure proper function and safety, may prevail over safety factor requirements. In 3D, discontinuum models are particularly adequate for scenarios of foundation failure, as limiting equilibrium procedures, like those proposed by Londe (1973), make basic assumptions about the forces acting on the independent volumes of rock that may become kinematically unstable, and are thus much simplified.

ACKNOWLEDGEMENTS Thanks are due to EDIA, Empresa de Desenvolvimento e Infra-Estruturas do Alqueva, SA for permission to publish data relative to Pedrógão dam.

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