Abstract
Seismic safety evaluation of earth works on liquefiable ground currently uses ∆umethod or seismic coefficient method to calculate the factor of safety in Japan.
Recently, coupled effective stress dynamic analysis is gradually used to calculate theresidual displacement induced by the earthquake including the possible liquefaction.
This paper proposes to use seepage analysis, fully coupled effective stress dynamic
analysis, and shear strength reduction technique to calculate the residual
displacement and the factor of safety of embankments. We analyzed the Lower andUpper San Fernando Dam using the proposed approaches. The major slide for the
Lower San Fernando Dam and the residual displacement for the Upper San
Fernando dam were re-produced well.
Keywords: embankment, liquefaction, slope stability, finite element method, limit
equilibrium method, seismic design.

1

Introduction

Seismic coefficient method is conventionally used to calculate the factor of safety ofembankment stability assuming a circle slip surface; for liquefiable embankments,
the ∆u-method is used to estimate the seismic stability of embankments in Japan. In
the ∆u-method, empirical analysis is used to estimate excess pore water pressure
induced by earthquake, and conventional stability analysis is used to calculate thefactor of safety. Because of potential limitations in the seismic coefficient method
and the called ∆u-method, dynamic analysis procedure has been proposed toevaluate behaviours of embankments under strong earthquakes [1, 2].
Dynamic analysis procedure is mainly used to evaluate the residual displacementof embankments under strong earthquakes, and the seismic stability of embankments
is mainly evaluated based on the calculated residual displacement. However, forliquefiable embankments, most sliding failures occurred after main shaking of1

earthquake [2]. The Lower San Fernando Dam is a famous example [3]. It isnecessary for the liquefiable embankments to calculate the stability after theearthquake considering the excess pore water pressure induced by the main shakingof earthquake.
This paper proposes to use finite element approaches to evaluate the seismic
safety of embankments under strong earthquake; in details, we use finite element
seepage analysis to calculate the pore water pressure before earthquake, use fully
coupled dynamic elastoplastic finite element approach to calculate the residual
displacement and excess pore water pressure induced by strong earthquake, and use
static elastoplastic finite element method with shear strength reduction technique toevaluate the safety of embankments after strong earthquake considering the porewater pressure calculated by finite element seepage analysis and excess pore waterpressure calculated by fully coupled dynamic elastoplastic finite element approach.
The factor of safety of embankments of the finite element approaches is comparedwith those of conventional ∆u-method or seismic coefficient method.

2

Analysis Procedure

2.1 Analysis Procedure
The analysis procedure involves a series of steps, and it is basically similar to thatadvised by Seed et al. [1, 2].
(1) Select the cross-section of the embankment to be analyzed, input motion, andwater level for seismic design.
(2) Discrete the selected cross-section using finite element mesh.
(3) Identify the parameters necessary for the analyses, included hydraulicproperties for seepage analysis, parameters of selected constitutive models
used in static and dynamic finite element analyses.
(4) Conduct finite element seepage analysis to evaluate the distribution of porewater pressure under the water level selected in step (1) before the earthquake.
(5) Calculate the stresses existing in the embankment before the earthquake
considering the pore water pressure evaluated in step (4). We use total stressanalysis to evaluate the initial stress in the embankment, considering the porewater pressure as equivalent nodal loads, and using an elastic-perfectly plastic
constitutive relationship defined by a yield surface of Mohr-Coulomb typeand a plastic potential of Drucker-Prager type.
(6) Conduct fully coupled effective stress dynamic finite element analysis toevaluate the residual displacement and excess pore water pressure induced by
the input motion selected in step (1). We use the Pastor-Zienkiewicz model
for all types of soils.
(7) Evaluate the seismic stability of the embankment using the sum of the porewater pressure before the earthquake and the excess pore water pressure
induced by the earthquake. We use the elasto-perfectly plastic finite element
method in conjunction with shear strength reduction technique and limit
equilibrium method to evaluate the seismic stability.

22.2 Analysis Methods
We use VisualFEA+VGFlow [9] a finite element seepage analysis software toevaluate the pore water pressure before the earthquake, UWLC [10] a fully coupledeffective stress dynamic finite element program to calculate the residual
displacement and excess pore water pressure in the embankment induced by theearthquake, and GeoFEAS [11] a static elastoplastic finite element program tocalculate the initial stress in the embankment and to estimate the stability of theembankment.
2.2.1 Fully Coupled Dynamic Finite Element Analysis
A basic fully coupled effective-stress procedure is formulated based on the balanceof momentum and conservation of masses. For a saturated soil subjected to cyclicloading, the dynamic equations of motion can be written following Biot. After the
spatial finite element discretization of the displacement and excess pore waterpressure, the governing equations for the solid-fluid mixture and fluid phase at any
time can be written as:&& + Ku − Qp = f u
Mu(1)

Q T u& + Hp + Sp& = f p(2)

where M is the global mass matrix, K is the global stiffness matrix, Q is thecoupling matrix, H is the permeability matrix, S is the compressibility matrix, u isthe displacement vector, p is the excess pore water pressure vector , f u is the force
vector for the solid-fluid mixture, and f p is the force vector for the fluid phase.
To complete the numerical solution, it is necessary to integrate the ordinary
differential Equations (1) and (2) in time by one of the many available schemes. The
unconditionally stable generalized Newmark method is used in UWLC [10]. In thismethod, the acceleration, velocity, and displacement at time station t n +1 are linked
with those at time station t n :&& n+1 = u&& n + ∆u
&& n
u

(3)

&& n ∆tu& n+1 = u& n + (1 + θ1 )∆u(4)

u n +1 = u n + ∆u& n ∆t +1
(1 + θ 2 )∆u&& n ∆t 2
2

(5)

The rate of the excess pore water pressure change and the excess pore water pressure
at time station t n +1 are given by:
p& n +1 = p& n + ∆p& n(6)3

p n +1 = p n + (1 + θ )∆p& n ∆t(7)&& n and ∆p& n are as yet unknown quantities. The parameters
In Equations (3)-(7), ∆uθ1 , θ 2 , and θ are usually chosen as θ1 ≥ θ 2 ≥ 1 2 and θ ≥ 1 2 for unconditionallystability of the scheme. Substituting Equations (3)-(7) into Equations (1)-(2), we canobtain:12M + 2 θ 2 ∆t K − θ∆tQ T



 ∆u
&&  ψ u   n  =  np+1 θ − θ∆tH + S  ∆p& n  ψ n +1 
θ1 − θ∆tQ(8)

The residual on the right hand of Equation (8) is given by:&& n+1 − ∫ B Tn +1σ ′n +1 dΩ + Q n +1p n +1
ψ un+1 = f nu+1 − M n +1u

(9)

θ p
(
f n+1 − Q Tn +1u& n +1 − H n +1p n +1 − S n +1p& n+1 )
θ1

(10)

Ω

ψ np+1 = −Equation (8) is nonlinear, and it is solved by initial stiffness method in UWLC [10].
Soil behaviour under earthquake loading is complex. It is essential that theconstitutive relationship used in the fully coupled effective-stress dynamic finite
element analysis should be able to capture important features of soil response undercyclic loading such as dilatancy, hysteresis loops, and permanent deformation.
UWLC uses the generalized plasticity model [6, 7] for sand.
2.2.2 Static Elastoplastic Analysis
We use GeoFEAS to calculate the initial stresses in the embankment before theearthquake, and to evaluate the slope stability of the embankment. Though
GeoFEAS is a static total-stress finite element code, it takes the pore water pressure
as an external force. Consequently, it can be used to estimate the influence of theexcess pore water pressure induced by the earthquake on the shear strength and thenon the stability of the embankment.
The initial stresses in the embankment before the earthquake were calculated
taking the self weight of soils and the pore water pressure as the external forces. The
pore water pressure was calculated using the steady seepage analysis. The method to
calculate the equivalent nodal forces of the pore water pressure is shown in section
2.3 of this paper. The soils were considered as an elastic-perfectly plastic material,
defined by a yield function of Mohr-Coulomb type, and a plastic potential of
Drucker-Prager type.
The stability of the embankment was also estimated using a static elastoplastic
finite element analysis in conjunction with the shear strength reduction technique. In

4the elastoplastic finite element method in conjunction with the shear strength
reduction technique, the safety factor emerges naturally from the analysis without
the user having to commit to any particular form of the slip surface in advance [12].
The excess pore water pressure induced by the earthquake was considered as anexternal force.

2.3 Pore Water Pressure
To calculate the equivalent nodal forces induced by the pore water pressure, first, we
calculate the body forces (buoyancy and seepage forces) at the Gaussian integration
points using the shape function and nodal pore water pressure. The body forces in xand y-directions are given by Equations (1) and (2) respectively:

γ w i x = −γ w

γ w i y + γ w = −γ w

n
∂N
∂p
∂h
= −∑ i p i=−∂x∂x
i =1 ∂x

(11)

n
∂N
∂p
∂h
+γ w = −= −∑ i p i∂y∂yi =1 ∂y(12)

where i x and i y are the hydraulic gradient in the x- and y-directions respectively, his the hydraulic head, p is the pore water pressure, γ w is the unit weight of water,
N i is the shape function, pi is the nodal pore water pressure, and n is the numberof the nodes of one element.
Second, the equivalent nodal forces of the pore water pressure are given by:

f xp = − ∫ N T

∂p
dΩ
∂x

(13)

f yp = − ∫ N T

∂p
dΩ
∂y

(14)

Ω

Ω

If Equations (11)-(14) are used to calculate the equivalent nodal forces of thepore water pressure at nodes, it is not necessary to consider the water pressure on theboundary as an external distributed load.
When limit equilibrium methods are used to evaluate the seismic stability
considering the pore water pressure including the pore water pressure before theearthquake and the excess pore water pressure induced by the earthquake, we haveto calculate the pore water pressure on the slip surface using the pore water pressure
at the nodes of the finite element mesh. Assuming an arbitrary point A located on
the slip surface, we can calculate the pore water pressure at the point A using thefollowing steps:(1) Pick up the element inside that the point A locates using the coordinates ofthe point A ( x A , y A ) .

5

(2) Calculate the local coordinates (ξ A , η A ) of the point A. We can directly
calculate the local coordinates for 4-node isoparametric element; however,iteration is necessary for 8-node isoparametric [5].
(3) Calculate the pore water pressure at the point A using the shape function andnodal pore water pressure using Equation (15):np A = ∑ N i (ξ A , η A ) p i

(15)

i =1

3

The Lower and Upper San Fernando Dams

The Lower and Upper San Fernando Dams were located northwest of Los Angles,
and they were within about three kilometres of each other. Both of the dams wereconstructed in the period 1915-1925. However, the stability of the Lower Dam had
been evaluated in 1966, and it had been considered to be adequately resistant toearthquake effects.
The Lower San Fernando Dam was essentially a hydraulic fill embankment
capped by a potpourri of wagon-dumped and rolled fills. The embankment was
about 43.7m high at its maximum section and was founded on 9.5m recent alluvium
deposits consisting of stiff clay with lenses of sand and gravel. A typical crosssection of the Lower Dam is shown in Figure 1.
The Upper San Fernando Dam was constructed using a semi-hydraulic filltechnique. The embankment was about 24.4m high and was constructed on 15.2 m
recent alluvial deposits consisting of stiff clay and clayey gravel. A typical crosssection of the Upper Dam is shown in Figure 2.
During the 1971 San Fernando Earthquake, the Lower San Fernando Damexperienced a major slide in the upstream shell, as shown in Figure 3. In comparison,
the Upper San Fernando Dam moved 1.5-2.1m downstream as the results of theshaking; the movement was significant, but limited, as shown in Figure 4.
A detailed description of the construction and history of the Lower and Upper
San Fernando Dams and their seismic behaviour during the 1971 San Fernando
Earthquake can be found in Reference [3].

Figure 1: Typical cross-section of the lower San Fernando Dam (after Reference [3]).

6Figure 2: Typical cross-section of the upper San Fernando Dam (after Reference [3]).

Figure 3: Cross-section of the lower San Fernando Dam through slide area andreconstructed cross section (after Reference [3]).

Figure 4: Measured movement at the upper San Fernando Dam (after Reference [4]).

74Model and Input Motion

Figure 5(a) shows the material zones of the typical cross-section of the Lower San
Fernando Dam. The finite element mesh of the typical cross-section are shown in
Figure 5(b).
Figure 6(a) shows the material zones of the typical cross-section of the Upper San
Fernando Dam. The finite element mesh of the typical cross-section are shown in
Figure 6(b).
The modified Pacoima Dam accelerogram with a peak acceleration of 0.6g
(Figure 7) as originally used by Seed et al. [1] was used as the input motion at thebase of the model.

(a) Material zones
AB(b) Finite element meshes

Figure 5: Material zones of the Lower San Fernando Dam and finite element meshes.

(a) Material zones
A

C
D

B

(b) Finite element meshes

Figure 6: Material zones of the Upper San Fernando Dam and finite element meshes.

8

ACC (g)

0.6
0.4
0.2
0
-0.2
-0.4
-0.6
0

5

10

15

20

Time (s)Figure 7: Input motion, modified Pacoima Dam accelerogram.

5

Constitutive Relationship and Parameters

The same number is used for the same material of the Lower and Upper San
Fernando Dams in Figure 5(a) and 6(a). The permeability of all kinds of materials
used for the steady seepage analysis before the earthquake is listed in Table 1. Table
1 also shows the parameters used to calculate the initial stresses in the embankments
before the earthquake and to evaluate the safety factor of the embankment slopes
after the excess pore water pressure has induced by the earthquake. Table 2 indicatesthe parameters of the generalized plasticity model used for the fully coupled
dynamic effective-stress analysis. The parameters except Mg and Mf borrowed from
Reference [7], and Mg of 1.375 borrowed from Reference [8].
Material
number
1
2
3
4
5
6
7
8
9

E
(MPa)
102
102
89
98
153
153
153
102
102

ν

c
(kPa)

0.286
0.286
0.286
0.286
0.286
0.286
0.286
0.286
0.286

0.0
0.0
0.0
0.0
0.0
20.0
20.0
0.0
0.0

φ
(˚)
34
34
34
34
34
37
37
34
34

γ
(kN/m3)
22.0
22.0
19.2
19.2
19.2
20.3
20.3
22.0
22.0

ky
kx
-5
(10 m/s) (10 m/s)
0.5
0.2
0.5
0.2
10.0
4.0
10.0
4.0
0.5
0.2
1.0
0.4
1.0
0.4
1.0
0.2
1.0
0.2
-5

Table 1: Material properties.

6

Numerical Results of the Lower San Fernando Dam

6.1 Seepage Analysis and Initial Stress Analysis
Figure 8 shows the contours of the pore water pressure in the embankment at thesteady state before the earthquake.

9

Material
K0
number
1
80
2
80
3
70
4
75
5
120
6
120
7
120
8
80
9
80

G0

Mg

Mf

αg, αf

β0

β1

H0

120
120
105
112
180
180
180
120
120

1.375
1.375
1.375
1.375
1.375
1.375
1.375
1.375
1.375

1.032
1.032
0.570
0.825
1.242
1.242
1.242
1.032
1.032

0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45

4.2
4.2
4.2
4.2
4.2
4.2
4.2
4.2
4.2

0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2

467.0
467.0
408.3
408.3
700.0
700.0
700.0
467.0
467.0

HU0
(MPa)
40.0
40.0
35.0
37.5
60.0
60.0
60.0
40.0
40.0

γ, γu

2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0

Table 2: Parameters of the generalized plasticity model.

Figure 8: Contours of pore water pressure at steady state for the Lower San
Fernando Dam.

6.2 Fully Coupled Dynamic Finite Element Analysis
Figure 9 shows the deformed mesh at 100s after the beginning of the earthquake.
The deformation in the upstream shell was so large that we can consider that a
failure has occurred. The lateral displacement and its velocity at the toe of the
upstream shell are shown in Figure 10. Even after the shaking of the earthquake, thelateral displacement was continually developed with a velocity of about 0.2m/s.
During the main shaking to 10s, the times of the maximum velocity of the lateraldisplacement at the toe of the upstream shell was corresponding to the times of the
maximum acceleration of the input motion. After the main shaking, the velocity wassignificantly reduced from 18s to 25s. Although the velocity changed small from 25s
to 100s, the velocity was reduced from 0.24m/s at the time of 35s to 0.14m/s at thetime of 55s; however, the velocity was increased to 0.21m/s during 2s at about 55s.
This increase in the velocity was considered to be induced by the re-distribution of
the excess pore water pressure in the upstream shell. Additionally, the time of theincrease in the velocity was consistent with the time of 60s when the failure
occurred, reported in Reference [1].

10Lateral Displacement (m)Figure 9: Deformed mesh of the Lower San Fernando Dam at 100s.

0
-5
-10
-15
-20
-25
0

20

40

60

80

100

80

100

Time (sec)Lateral velocity (m/s)(a) Lateral displacement.
1
0.5
0
-0.5
-1
0

20

40

60
Time (sec)(b) Velocity of lateral displacement.

Figure 10: Time histories of lateral displacement and its velocity at the upstream toeof the Lower San Fernando Dam.

The acceleration responses at representative elevations along the centreline of thecross-section: the crest of the embankment and the top and bottom of the core wall
are shown in Figure 11. Compared with the input motion in Figure 7, the highfrequency contents in the input motion were damped out because of the Rayleighdamping and hysteresis damping. The acceleration at the top of the core walldecreased significantly because the soils were loose and liquefied during theearthquake. The acceleration at the crest was amplified again because a dry rolled
fill exists at the top of the embankment. The stiffness of the dry rolled fill was notexperienced a significant decrease during the earthquake.
Figure 12 shows the distribution of the excess pore water pressure ratio: the ratioof the pore water pressure to the initial vertical effective stress, at 10s and 60s. The
excess pore water pressure ratio in the upstream and downstream shell below thephreatic surface was quite high up to about 0.7. This indicate that the hydraulic fillmaterials in these regions may have either liquefied or been weakened severely.
Additionally, the excess pore water pressure ratio in the upstream shell was still high11

ACC (m/s/s)

8
4
0
-4
-8
0

2

4

6

8

10

12

14

16

18

20

14

16

18

20

14

16

18

20

Time (sec)(a) At the crest of the embankment.

ACC (m/s/s)

4
2
0
-2
-4
0

2

4

6

8

10

12

Time (sec)(b) At the top of the core wall.

ACC (m/s/s)

4
2
0
-2
-4
0

2

4

6

8

10

12

Time (sec)(c) At the bottom of the core wall.

Figure 11: Time histories of acceleration at representative elevations.
even at 60s after the beginning of the earthquake. In the upstream and downstream
level ground, the excess pore water pressure was reached unity; however, nosignificant lateral displacement was calculated because of the absence of a highdriving shear stress.
Figures 13 and 14 show the time histories of the excess pore water pressure andexcess pore water pressure ratio at two typical points A and B located in the
upstream shell. The details of the location of the points A and B can be found Figure
7. Point A locates at the upstream shell and near the core wall; point B locates in the
middle of the upstream shell. Both of the two points locate at the bottom of the
upstream shell. The excess pore water pressure at point A was about two times ofthat at point B because the initial vertical effective stress at point A was significantly
larger than that at point B. Consequently, the excess pore water pressure ratio at
point A was significantly smaller than that at point B. The excess pore waterpressure began to dissipate after the main shaking of the earthquake, and the excess
pore water pressure of about 100 kPa dissipated from 20s to 100s. In comparison,12

(a) At the time of 10s.

(b) At the time of 60s.

Figure 12: Contours of excess pore water pressure in the Lower San Fernando Dam.
the excess pore water pressure did not decreased until 100s after the beginning of theearthquake. Because the upstream boundary under the water level was assumed to bedrained in the fully coupled dynamic effective-stress analysis, the water during thedissipation of the excess pore water pressure in the upstream shell was flowed to
upstream. Because the excess pore water pressure at point A was 150 kPa larger thanthat at point B, the dissipation of the excess pore water at point B was cancelled out
by the increase of the excess pore water pressure due to the higher excess pore waterpressure in the zone at the right of point B. Consequently, the excess pore waterpressure at point B did not decrease after the main shaking of the earthquake within

PWP (kPa)

400
300
200
100
0
0

20

40

60

80

100

80

100

Time (sec)PWP Ratio(a) Excess pore water pressure.
0.5
0.4
0.3
0.2
0.1
0
0

20

40

60
Time (sec)(b) Excess pore water pressure ratio.

Figure 13: Time histories of excess pore water pressure and excess pore waterpressure ratio at point A.
13

PWP (kPa)

180
120
60
0
0

20

40

60

80

100

80

100

Time (sec)PWP Ratio(a) Excess pore water pressure.
1.0
0.8
0.6
0.4
0.2
0.0
0

20

40

60
Time (sec)(b) Excess pore water pressure ratio.

Figure 14: Time histories of excess pore water pressure and excess pore waterpressure ratio at point B.
the calculation duration of 100s. Moreover, the excess pore water pressure at point B
increased about 2.5 kPa at the time of about 60s when the main slide occurred in the
upstream shell.

7

Numerical Results of the Upper San Fernando Dam

7.1 Seepage Analysis and Initial Stress Analysis
Figure 15 shows the contours of the pore water pressure in the embankment at thesteady state before the earthquake.

7.2 Fully Coupled Dynamic Finite Element Analysis
Figure 16 shows the deformed mesh at 100s for the Upper San Fernando Dam after
the beginning of the earthquake. The deformation in the upstream shell of the Upper
San Fernando Dam was significantly smaller than that of the Lower San Fernando
Dam. Figure 17 shows the time histories of the lateral displacement at thedownstream berm and at the downstream toe. The lateral displacement stopped withthe finish of the main shaking of the earthquake. The calculated lateraldisplacements were 2.5m at the downstream berm and 1.2m at the downstream toe;they were consistent with the observation of 2.2m at the downstream berm and 1.1m
at the downstream toe.

14

Figure 15: Contours of pore water pressure at steady state for the Upper San
Fernando Dam.

EPWP Ratio

Figure 16: Deformed mesh of the Upper San Fernando Dam at 100s.

0.8
0.6
0.4
0.2
0
-0.2
0

20

40

60

Time (sec)

EPWP Ratio(a) At the downstream berm.
0.8
0.6
0.4
0.2
0
-0.2
0

20

40

60

Time (sec)(b) At the downstream toe.

Figure 17: Time histories of lateral displacement at representative points of theUpper San Fernando Dam.

Figure 18 shows the distribution of the excess pore water pressure ratio in the Upper
San Fernando Dam at 10s and 60s. The excess pore water pressure ratio in the
upstream shell of the Upper San Fernando Dam is about 0.6, and the excess pore15

water pressure ratio kept still high 60s after the beginning of the earthquake. The
excess pore water pressure ratio in the Upper San Fernando Dam is about 0.1 lower
than that in the Lower San Fernando Dam. Because the Lower San Fernando Dam is
25m higher than the Upper San Fernando Dam, the shear stress before theearthquake in the Lower San Fernando Dam is larger than that in the Upper San
Fernando Dam. This is the one of the main reason that a large slide occurred in the
upstream of the Lower San Fernando Dam but only some residual displacement
occurred in the Upper San Fernando Dam.
Figure 19 shows the time histories of the excess pore water pressure ratio at
representative points A and B in the downstream shell and points C and D in the
upstream shell of the Upper San Fernando Dam.

(a) At the time of 10s.

(b) At the time of 60s.

EPWP Ratio

Figure 18: Contours of excess pore water pressure in the Upper San Fernando Dam.

0.8
0.6
0.4
0.2
0
-0.2
0

20

40

60

Time (sec)

EPWP Ratio(a) At point A in the downstream shell.
0.8
0.6
0.4
0.2
0
-0.2
0

20

40
Time (sec)(b) At point B in the downstream shell.
(Continued to the next page)

16

60

EPWP Ratio

0.8
0.6
0.4
0.2
0
-0.2
0

20

40

60

Time (sec)

EPWP Ratio(c) At point C in the upstream shell.
0.8
0.6
0.4
0.2
0
-0.2
0

20

40

60

Time (sec)(d) At point D in the upstream shell.

Figure 19: Time histories of excess pore water pressure ratio at representative points
in the Upper San Fernando Dam.

PWP (kPa)

150
100
50
0
0

20

40

60

Time (sec)(a) At point A in the downstream shell.

PWP (kPa)

150
100
50
0
-50
0

20

40

60

Time (sec)(b) At point B in the downstream shell.

Figure 20: Time histories of excess pore water pressure at points A and B in thedownstream shell of the Upper San Fernando Dam.

17

Figure 20 shows the time histories of the excess pore water pressure at points A and
B in the downstream shell of the Upper San Fernando Dam. The initial verticaleffective stresses at the two points are almost the same as one another, and theexcess pore water pressure at the two points was also almost identical to one another.
The excess pore water pressure at point A almost did not dissipate during 40s after
the main shaking of the earthquake. In comparison, the excess pore water pressure atpoint B dissipated slightly because point B is nearer the drained downstream
boundary. The significant re-distribution of the excess pore water pressure happenedin the upstream shell of the Lower San Fernando Dam did not occur in thedownstream shell of the Upper San Fernando Dam.

8

Stability Analyses

The seismic coefficient method and ∆u-method were used the safety factor of theLower and Upper San Fernando Dams. Here, the simplified Bishop’s method wasused to calculate the safety factor. In the seismic coefficient method, the horizontal
seismic coefficient was assumed to 0.15, and the consolidated undrained shear
strength: c=57.5kPa, φ =20º for materials 1-5 and 8 and 9, c=74.4kPa, φ =27º formaterials 6 and 7. The calculated factor of safety of the upstream shell of the Lower
San Fernando Dam was 1.156, which was consistent with 1.20 reported by Seed etal. [1]. Obviously, the seismic coefficient method can not explain the major slide
occurred in the upstream shell of the Lower San Fernando Dam. The factors ofsafety of the upstream and downstream shells of the Upper San Fernando Dam were
2.163 and 2.005 respectively. In the ∆u-method in this paper, the pore waterpressure in the steady state condition and the excess pore water pressure weresummed up and used to calculate the safety factor. The calculated factor of safety of
the upstream shell of the Lower San Fernando Dam was 1.356. The factors of safetyof the upstream and downstream shells of the Upper San Fernando Dam were 1.253
and 1.164 respectively.
The finite element method in conjunction with the shear reduction technique wasalso used to calculate the factor of safety. The consideration of the pore waterpressure was the same as that in the ∆u-method. The calculated factor of safety of
the upstream shell of the Lower San Fernando Dam was 0.88. This can explain why
the major slide occurred in the upstream shell of the Lower San Fernando Dam
during the 1971 San Fernando Earthquake. And the sliding surface can be estimatedusing the incremental shear strain at the last step of the incremental calculation of
the finite element analysis. Figure 21 shows that, for the Lower San Fernando Dam,

Figure 21: Incremental shear strain at the last step of the finite element method in
conjunction with shear strength reduction technique.
18the siding surface first occurred near the toe of the upstream. This was consistent
with the failure mechanism observed, as shown in Figure 3.

9Conclusions

Seismic safety evaluation procedure currently used in Japan first calculates thefactor of safety using ∆u-method, in which the excess pore water pressure is simply
evaluated based on the simple liquefaction evaluation, or seismic coefficientmethod assuming a circular slip surface. If the calculated factor of safety is smaller
than the planned one, dynamic analysis has to be conducted to calculate the residual
displacement under strong earthquake. However, because the slip surface is mostpossibly non-circular for liquefiable embankments, and because the seismic
characteristics are not considered for ∆u-method or seismic coefficient method,
the residual displacement cannot be calculated with a satisfactorily confidence. This
paper proposes to use finite element approaches to evaluate the seismic safety ofembankments under strong earthquake; in details, we use seepage analysis tocalculate the pore water pressure before earthquake, use fully coupled dynamic
elastoplastic finite element approach to calculate the deformation and excess porewater pressure induced by strong earthquake, and use static elastoplastic finite
element method with shear strength reduction technique to evaluate the safety ofembankments after strong earthquake considering the pore water pressure andexcess pore water pressure calculated by finite element seepage analysis and fully
coupled dynamic elastoplastic finite element approach. The factor of safety ofembankments of finite element approaches is compared with those of conventional∆u-method or seismic coefficient method.
The proposed approaches are used to re-analyze the famous Lower and Upper
San Fernando Dams. During the San Fernando earthquake of February 9, 1971
(magnitude 6.6), a major slide occurred in the upstream slope of the Lower San
Fernando Dam; slide movements also caused a downstream movement of about
1.5m in the embankment of the Upper San Fernando Dam; however, a major slide
did not occur. For the Lower San Fernando Dam, numerical results can re-produce
the major slide with a large residual displacement and a factor of safety after theearthquake is 0.88 when the excess pore water pressure due to the liquefaction wasconsidered. For the Upper San Fernando Dam, the calculated residual displacement
after the earthquake was consistent with the observations, and the factor of safety forthe downstream shell is larger than one even the excess pore water pressure wasconsidered.

References[1]

H.B. Seed, I.M. Idriss, K.L. Lee, F.I. Makdisi, “Dynamic analysis of the slidein the Lower San Fernando Dam during the earthquake of February 9, 1971”,
Journal of the Geotechnical Engineering Division, ASCE, 101(9), 889-911,
1975.

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