1

Stability of dam abutment including seismic loading

Morteza Sohrabi Gilani Civil Engineering Department Sharif University of Technology Azadi Ave.- Tehran- Iran mortezasohrabigilani@Yahoo.com

Rupert Feldbacher Institute of Hydraulic Engineering and Water Resources ManagementGraz University of Technology Stremayrgasse 10/II 8010 Graz rupert.feldbacher@tugraz.at

Gerald Zenz Institute of Hydraulic Engineering and Water Resources ManagementGraz University of Technology Stremayrgasse 10/II 8010 Graz gerald.zenz@tugraz.at

1. Introduction One of the most important aspects in the stability analysis of arch dams, which has been encountered for many years, is the stability of the abutment. This study is aimed to evaluate within the Tenth Benchmark Workshop on Numerical Analysis of Dams-Theme C, the abutment stability of Luzzone arch dam under static and seismic loadings. At first the three dimensional model of the dam has been transferred for being applicable in the finite element program of Abaqus 6.7. With the FEM the interface forces between concrete dam and wedge are calculated for the required loading cases. The stability analysis of the given wedge is evaluated by Londe method. 2. System Assumption 2.1. Luzzone dam The Luzzone dam is a double curved concrete arch dam which was initially built in the sixties. The dam was heightened within the ninetieths. The total height of the dam is 225 m. Figure 1 shows the Luzzone dam.

Fig. 1: Luzzone dam

2

2.2. Wedge definition For the benchmark in the left bank of the dam are two geological joints. With these joints a wedge is defined and has a potential to slide under arch dam and uplift loading. To verify about this situation a stability assessment is necessary. The volume of the wedge has been estimated as 361092.1 m . The wedge position is shown in figure 2.

Fig.2: Finite element model of dam and foundation and Geometry of the wedge 2.3. Material properties The material property of the mass concrete dam and foundation are defined as: Concrete of dam: Density ( ) = 2400 kg/m3 Poisson ratio ( ) = 0.167 Modulus of elasticity (E) = 27 GPa Rayleigh damping coefficients: =0.6 and =0.001 Foundation rock: Density ( ) = 2600 kg/m3 Poisson ratio ( ) = 0.2 Modulus of elasticity (E) = 25 GPa Rayleigh damping coefficients: =0.6 and =0.001 Water: Density ( ) = 1000 kg/m3

``

Fig 3: Material properties of dam and foundation

It should be mentioned that in calculating the interface forces between dam and wedge, only the stiffness of foundation is considered and density of it is taken as zero. In other words a massless foundation is considered.

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2.4. Loading The static and seismic load cases are considered to calculate the dam-foundation interface forces. In the self-weight condition the dam is considered monolithic and isotropic material behavior is used. Under the reservoir full condition the hydrostatic pressure is applied to the dams upstream surface according to the programs loading definition.

Fig 4: cross section of the dam

For seismic analysis, three stochastically independent acceleration time histories are used according to the data provided by the formulator. These accelerations are scaled according to the peak ground accelerations of these components are: Downstream-upstream (X- direction) = 0.16 g Vertically upwards (Y- direction) = 0.1063 g Cross valley direction (Z- direction) = 0.16 g In wedge stability assessment uplift pressure at the wedge interfaces is considered very conservative, and all planes are under full uplift pressure. 3. Calculation Procedure Figure 2 presents the finite element model of the dam. This model is created within Abaqus 6.7 and linear elements (C3D8) are used to define dam and foundation body. The dam-foundation interface is modeled as a joint with a high friction coefficient to reduce relative displacement between dam and foundation to a minimum. The resultant forces transmitted between dam and wedge are computed as the sum of pressure and shear stresses at the wedge dam interface. For the seismic analysis direct time history approach is used and hydrodynamic pressure is computed by Westergaards added mass method. According to westergaard, the hydrodynamic pressures that the water exerts on the dam during an earthquake are the same as if a certain body of water moves back and forth with the dam whiles the remainder of the reservoir is left inactive. The added mass per unit area of the

upstream wall is given in approximate form by the expression )(87 yhh www , where w is the density

of water. It should be mentioned that in calculating the interface forces between dam and wedge, only the stiffness of foundation is considered and density of it is taken as zero. In other words a massless foundation is considered.

1385 m

1510 m

1610 m

4

4. Stability of wedge The next step for the analysis is to evaluate is to evaluate the wedge stability. For this purpose Londe method for stability of rock slopes is used and some simplifying assumptions are made. The volume of the wedge is limited by intersections of three planes (Planes Plane1, Plane2 and Plane3 in the figure 2). This assumption is conservative as the natural surfaces are generally irregular [2]. The wedge is considered as a rigid body and the geometry of the wedge would not change during application of the forces throughout the investigation. Cohesion and tensile strength are neglected in the contact planes and therefore, it is supposed that the friction between surfaces is the only parameter that can resist sliding. It is supposed that the moments of the forces have negligible influences and can be ignored. The applied forces can be categorized as:

- Weight of the wedge (Ww) - The thrust force which is the resultant force at the dam wedge interface. This force is time

dependent and its magnitude and direction will change by time. So this force can be defined by Fx, Fy and Fz.

- The forces due to uplift pore pressure: U1, U2 and U3 which are applied to the planes 1, 2 and 3 respectively- these forces do not change during investigation.

- Seismic forces due to applied earthquake: the three components of seismic forces are considered as the max, may and maz which m is the mass of the wedge and ax, ay are az are acceleration time histories which were defined before.

- The reaction of planes: due to applied forces, three reaction forces will develop on the planes (N1, N2 and N3). As mentioned before these forces can only be compressive. Tensile forces, which mean that the plane is open, are not acceptable and will lead to a different sliding mode respectively exclude sliding in the decoupling plane due to tensile forces.

For wedge stability evaluation at first the three plane reaction forces are to be calculated by solving static equilibrium equations in three direction x, y and z, Figure 5-a. For this wedge geometry and applied forces, due to equilibrium condition and calculated plane reaction forces, eight cases are possible. Table 1 shows all possibilities.

- Case 1: All plane reaction forces are compressive: all planes are in contact and the wedge is perfectly stable.

- Case 2: The reaction force of plane 1 is tensile, but the other two reaction forces are compressive

(N1>0, N2

5

- Case 4: Sliding along the intersection of plane 1 and 3 (N3>0, N10 and N30, N3>0 and N20, N3>0 and N10, N2>0 and N3

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Case Nature of sliding Contact faces Open Faces Diagram

1 No sliding 1, 2, 3

2 intersection of Planes 2,3 2, 3 1

3 intersection of Planes 1,3 1, 3 2

4 intersection of Planes 1,2 2, 3 3

5 in plane 3 3 1, 2

6 in plane 2 2 1, 3

7 in plane 1 1 2, 3

8 in space 1, 2, 3

Table 1: all possible movement cases of the wedge

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Wedge Stability Safety Factor

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

Time(s)

SF

Fig. 6: Safety factor of the wedge stability during seismic loading 5. Displacement of the wedge To calculate the displacement of the wedge in the first step acceleration of the wedge is calculated in the x,

y and z direction. The magnitude of the acceleration is wedgetheofmass

ForcegStabilizinForceDriving and its direction

is being defined due to the movement case. (For one plane sliding the resultant acceleration is decomposed

due to the applied forces, for example in case 2, m

NFaand

mNFa yyyxxx

)tan()tan( 33 == , but for two planes sliding the direction is the intersection of corresponding planes which is constant) Displacement of the wedge is the double integration of this computed acceleration. The integration should continue till the velocity in the considered direction vanishes. Figure 7 shows the calculated displacement for the wedge in the x and y directions. The calculated displacement is reported in the table 3. 6. Conclusion Under the assumption of a rigid body wedge the analysis is carried out for dead weight, water loading and uplift. No variation of the earthquake acceleration along the valley is assumed. The uplift pressure at the wedge interface is very conservative, as it would never be the case that the entire planes are under full uplift. With the help of FEM the wedge is suggested to be analyzed as deforming body and with this the stability of the abutment. However, this assumption used normally, was out of scope of this benchmark.

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ax

-2.4

-2

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

12.9

8

13.0

0

13.0

2

13.0

4

13.0

6

time (s)

ax (m

/s2)

ay

-2.4

-2

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

12.9

8

13.0

0

13.0

2

13.0

4

13.0

6

13.0

8

13.1

0

time (s)

ay (m

/s2)

Vx

-0.01

0

0.01

0.02

0.03

0.04

12.9

8

13.0

0

13.0

2

13.0

4

13.0

6

time (s)

Vx (m

/s)

Vy

-0.01

0

0.01

0.02

0.03

0.04

12.9

8

13.0

0

13.0

2

13.0

4

13.0

6

13.0

8

13.1

0

time (s)

Vy (m

/s)

Ux

0

0.5

1

1.5

2

2.5

12.9

8

13.0

0

13.0

2

13.0

4

13.0

6

time (s)

Ux

(mm

)

Uy

0

0.5

1

1.5

2

2.5

12.9

8

13.0

0

13.0

2

13.0

4

13.0

6

13.0

8

13.1

0time (s)

Uy (

mm

)

Fig 7: acceleration, velocity and displacement of the wedge in x and y direction 7. Acknowledgement The support of this work by the research project Design of Hydraulic Structures by Pyry Energy Ltd is gratefully acknowledged. 8. References 1. Dr. Russell Michael Gunn, Computational aspects of analysis and design of dams, Tenth benchmark workshop on numerical analysis of dams. 2. P. Londe, Analysis of the stability of rock slopes, published in Quarterly Journal of Engineering Geology and Hydrogeology, vol. 6, issue 1, p. 93-124, 1973. 3. Prof. N. M. Newmark, effect of earthquakes on dams and embankments, published in Geotechnique, Milestones in Engineering, vol. 15, No. 2, p. 109-129, 1965. 4. Abaqus version 6.7-EF Documentation.

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Table 2: Dam thrust force, Contact plane forces, driving and stabilizing forces

Fx (M

N)

Fy (M

N)

Fz (M

N)

FR (M

N)

Alpha

BetaJh (M

N)

J1 (MN)

J2 (MN)

D(MN)

S(MN)

JhJ1

J2Fsw

4182.656955.55

9787.7112715.09

50.331.0

38457.940.00

3000.417576.19

26928.54No

YesNo

Fsw+hyd

20776.2518618.85

47437.3055032.82

59.5131.9

58094.6110705.24

0.0022488.11

48174.18No

No

Yes0

20776.2518618.85

47437.3055032.82

59.5131.9

58094.0610681.88

0.0022496.20

48157.43No

No

Yes0.01

20777.5818620.77

47439.5055035.87

59.5131.9

58095.3210684.14

0.0022497.30

48159.89No

No

Yes0.02

20777.4218622.47

47441.7055038.28

59.5131.9

58096.7810685.94

0.0022496.88

48162.18No

No

Yes.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.12.97

4175.1237.95

5814.407158.23

54.389.5

20469.260.00

0.009418.90

14332.73No

YesYes

12.983086.85

1736.032000.55

4067.5129.5

60.618315.04

0.000.00

12414.5812824.33

No

YesYes

12.992477.41

2694.97585.01

3707.119.1

42.615917.85

0.000.00

15936.0611145.80

No

YesYes

132358.44

2485.851018.86

3574.8816.6

43.516396.92

0.000.00

17697.2711481.24

No

YesYes

13.013150.52

1056.291366.05

3592.7222.3

71.519023.48

0.000.00

18587.2513320.39

No

YesYes

13.024858.73

726.154860.24

6910.6144.7

98.521815.56

0.000.00

19206.4515275.42

No

YesYes

13.036682.26

2663.268792.95

11360.5250.7

111.724983.98

0.000.00

16940.4717493.97

No

YesYes

13.048511.44

4980.6314115.90

17219.4655.1

120.329517.55

0.000.00

16190.5620668.41

No

YesYes

13.0510874.14

7630.2220813.70

24691.6457.5

125.135570.08

0.000.00

17440.4224906.44

No

YesYes

13.0614152.58

10742.6229353.90

34312.5558.8

127.243341.38

0.000.00

19660.0630347.96

No

YesYes

13.0717461.17

14013.0638231.60

44304.7859.6

128.752829.17

0.000.00

22397.9136991.38

No

YesYes

13.0820452.24

17602.6947234.90

54399.3160.3

130.760393.94

1965.010.00

23308.6543664.21

No

No

Yes13.09

23248.7821197.32

55629.3063909.71

60.5132.4

66254.518045.91

0.0023968.14

52025.71No

No

Yes13.1

25308.3024014.10

61647.8070835.29

60.5133.5

69672.1213678.51

0.0025137.37

58362.74No

No

Yes13.11

26704.2725861.84

65403.8075230.38

60.4134.1

72166.3517512.70

0.0024744.46

62793.95No

No

Yes13.12

27772.2126909.53

67836.3078084.46

60.3134.1

74540.9220171.95

0.0026162.33

66318.66No

No

Yes.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.31.94

29496.0027019.54

68269.0079124.75

59.6132.5

74669.5320777.64

0.0030442.51

66832.83No

No

Yes31.95

29382.0926892.51

67984.1078793.10

59.6132.5

74448.2720627.06

0.0030340.11

66572.46No

No

Yes31.96

29279.7226774.81

67722.4078488.95

59.6132.4

74245.4120487.84

0.0030248.38

66332.93No

No

Yes31.97

29190.2926667.86

67487.4078216.34

59.6132.4

74063.7320361.68

0.0030168.61

66117.38No

No

Yes31.98

29114.3226572.85

67281.2077977.69

59.6132.4

73904.7520249.94

0.0030101.21

65927.82No

No

Yes31.99

29051.8926490.00

67104.5077773.69

59.6132.4

73769.0720152.87

0.0030046.24

65764.85No

No

Yes32

29002.5026419.23

66957.1077603.96

59.6132.3

73656.5520070.34

0.0030003.21

65628.27No

No

Yes

StabilityandDrivingForce

LossofContactPlaneResultsCase

ResultantForces(Damthrust)

ResultantForces(Damthrust)

ContactPlaneForces

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Ux(mm) Uy(mm) Uz(mm) UR alpha beta

Fsw 0 0 0 0 0 0 3.55Fsw+hyd 0 0 0 0 0 0 2.14

0 0 0 0 0 0 0 2.140.01 0 0 0 0 0 0 2.140.02 0 0 0 0 0 0 2.14. . . . . . . .. . . . . . . .

12.97 0 0 0 0 0 0 1.5212.98 0 0 0 0 0 0 1.0312.99 0.01 0.01 0 0.02 0 28.49 0.7013 0.06 0.10 0 0.12 0 29.44 0.65

13.01 0.17 0.29 0 0.33 0 31.18 0.7213.02 0.36 0.55 0 0.65 0 33.05 0.8013.03 0.59 0.86 0 1.04 0 34.46 1.0313.04 0.81 1.16 0 1.42 0 34.85 1.2813.05 0.96 1.43 0 1.72 0 33.72 1.4313.06 0.96 1.65 0 1.91 0 30.12 1.5413.07 0.96 1.81 0 2.05 0 27.82 1.6513.08 0.96 1.93 0 2.16 0 26.33 1.8713.09 0.96 2.01 0 2.23 0 25.41 2.1713.1 0.96 2.01 0 2.23 0 25.41 2.3213.11 0.96 2.01 0 2.23 0 25.41 2.5413.12 0.96 2.01 0 2.23 0 25.41 2.53

. . . . . . . .

. . . . . . . .31.94 0.96 2.01 0 2.23 0 25.41 2.2031.95 0.96 2.01 0 2.23 0 25.41 2.1931.96 0.96 2.01 0 2.23 0 25.41 2.1931.97 0.96 2.01 0 2.23 0 25.41 2.1931.98 0.96 2.01 0 2.23 0 25.41 2.1931.99 0.96 2.01 0 2.23 0 25.41 2.1932 0.96 2.01 0 2.23 0 25.41 2.19

ResultsCaseWedgedisplacements

Factorofsafety

Table 3: displacement of the wedge in the x and y direction