1.
SEISMIC SLOPE STABILTY AND ANALYSIS
OF THE UPPER SAN FERNANDO DAM
James Dismuke
The University of California at Davis
ECI 284, Winter Quarter, 2002
Professor Jeremic
2.
ABSTRACT
Static evaluation procedures of slope stability are well defined, and suitably accurate and
precise. Simple seismic evaluation of slope stability can be performed using a
pseudostatic analysis, but the results are misleading. In 1965, Newmark made the
analogy that a slope is like a block sliding on an inclined plane. Newmark was able to
reasonably compute the displacement due to shaking by double integrating portions of
actual acceleration time histories that exceeded the yield acceleration. In 1978, Makdisi
and Seed improved upon this procedure by accounting for the dynamic response of the
embankment. During the San Fernando Earthquake of 1971, the Upper San Fernando
Dam, despite a large pseudostatic factor of safety, failed. In 1973, Seed performed a
dynamic analysis of the embankment and computed displacements that closely agreed
with the observed deformations. Wu, in 2001, performed a nonlinear effective stress
finite element analysis that predicted displacements almost exactly as those measured.
3.
1.0 INTRODUCTION
Evaluating the stability of slopes is one of the most important activities of a geotechnical
engineer (Kramer, 1995). A stable slope under static conditions is thought of as having a
resistance to sliding greater than the driving forces that exist due to the slope geometry.
However, seismic loading of a slope can induce greater driving forces that will
potentially make a once stable slope unstable. A slope that fails under seismic loading
usually does so rapidly, and due to the nature of earthquakes, unexpectedly with
potentially great losses. An estimated 56% of the total cost of the damage caused in the
1964 Alaska earthquake was due to earthquake induced slope failures (Youd, 1978).
Seismic slope stability is usually performed using a pseudostatic analysis, but results
from a pseudostatic analysis can be somewhat misleading. Better approximations of
seismic slope stability can be calculated using finite element analyses that compute the
displacement due to seismic slope failure. The Upper San Fernando Dam failure is an
excellent case history that demonstrates the use of finite element analysis to predict
deformation.
2.0 STATIC AND PSEUDOSTATIC SLOPE STABILITY ANALYSES
The procedures for static slope stability are well established, and have been concisely
documented by Duncan (1987). The stability of a slope is represented by a factor of
safety, which is the ratio of the shear strength along a critical failure surface and the shear
stress induced on that failure surface by the slope. Thus, a factor of safety greater than 1
means that more than enough available shear strength exists to resist a slope failure.
Duncan recommends that evaluation of a slope focus on defining geometry, shear
strengths, unit weights, and pore water pressures. The various methods available for
slope stability analysis yield similar results (within ± 6%), and selection of the method is
less important than choosing accurate parameters.
Evaluation of seismic slope stability revolves principally around four key issues: 1)
identification of the critical mechanisim of failure, 2) the geometry of the slope, 3) the
4.
seismicity and dynamic response of the site, and 4) the resistance of the slope to the
critical failure (Seed, 2000). Seismic slope stability is evaluated using a “pseudostatic”
analysis where the failure mass is assumed to be horizontally accelerated by the seismic
coefficient, k (in units of gravity), appropriately chosen for the expected seismicity of the
site. Terzaghi (1950) originally recommended using k = 0.1, 0.2, and 0.5 for severe,
violent destructive, and catastrophic earthquakes, respectively. In California, k is usually
within the range of 0.15 to 0.3.
Selection of k is the most important, and most difficult, factor to determine in a
pseudostatic analysis. An earthquake may be capable of producing a certain maximum
acceleration, but this acceleration may act for less than a portion of a second. The factor
k in a pseudostatic analysis is almost always less than the anticipated maximum
acceleration, thus it is very possible that a slope with a factor of safety greater than 1 for a
chosen k, will fail because the slope was analyzed with a horizontal acceleration less than
the acceleration experienced in the field. Many researchers have recommended different
factors of safety for various k to attempt to remedy this situation, as shown in Figure 1.
Figure 1, Pseudo-Static Coefficient, k vs Recommended Pseudo-Static Factor of Safety
Source: geohazards.cr.usgs.gov/pubs/ofr/98-113/figure2.html
5.
A pseudostatic analysis is simple to do, and very straightforward. However, the difficulty
of interpreting k and the factor of safety warrant the use of other methods.
3.0 CALCULATING DISPLACEMENTS
3.1 Newmark’s Sliding Block Analysis
Newmark (1965) first attempted to analyze seismic slope stability by quantifying the
amount of displacement due to a failure. Newmark made the analogy that the sliding
mass on the failure surface is similar to a block resting on an inclined plane, as in Figure
2.
Figure 2, Newmark’s Sliding Block Analogy
Source: Kramer (1995)
He found that increasing the amount of shaking decreased the factor of safety, as
expected, and that for a given frictional resistance, there was one particular k that
produced a factor of safety of 1. The coefficient at which the factor of safety is 1 is the
yield coefficient, ky. When a block was subjected to acceleration greater than the yield
acceleration, the block moved. The permanent displacement of the block is found by
double integrating the acceleration time history where the acceleration exceeded the yield
acceleration, as shown in Figure 3. Newmark made the assumption that the block only
moves down the slope, i.e. the block will not move up the slope if the yield acceleration
is exceed in the negative direction.
6.
Figure 3, Double Integrating Acceleration to Calculate Displacement
Source: http://www.consrv.ca.gov/dmg/pubs/sp/117/chap_5.htm
Further studies using the sliding block analogy show that the displacement is sensitive to
the yield acceleration, and small differences in this can cause large differences in the
predicted displacement.
3.2 Makdisi and Seed’s Simplified Procedure (1978)
Makdisi and Seed (1978) used a sliding block analysis to compute permanent
deformation of earth dams and embankments by making assumptions about the dynamic
response of the soil. Newmark’s sliding block analogy assumes that the deformation will
be rigid and perfectly plastic, as shown in the stress strain curve in Figure 4.
Figure 4, Perfectly Plastic Stress Strain Behavior
Source: Kramer (1995)
7.
A slope, however, is compliant and will deform during shaking. Thus, it is possible for
adjacent portions of the sliding mass to be out of phase; different areas of the slope may
be accelerating in different directions (See Figure 5).
Figure 5, Frequency Effect of Seismic Shaking – Compliance
Source: Kramer (1995)
In Makdisi and Seed’s procedure, the yield acceleration is computed for the failure
surface using the dynamic yield strength, which is approximately 80% of the static
strength. The dynamic response of the soil is accounted for by using an acceleration
ratio, ayield/amax that varies with depth below the crest of the embankment, The results of
several tests were normalized with the base peak acceleration and fundamental period of
the embankment to develop the charts shown in Figure 6.
Figure 6, Slope Displacement Normalized for Peak Acceleration and Fundamental Period
Source: Kramer (1995)
8.
4.0 CASE HISTORY: BACK CALCULATION OF THE EARTHQUAKE
RESPONSE OF THE UPPER SAN FERNANDO DAM
4.1 Pseudostatic Analysis
The Upper and Lower San Fernando Dams, located in Southern California, were
constructed of hydraulic fill on native alluvial soils. During the M6.6 San Fernando
earthquake of 1971, the upper dam settled as much as 3 feet and moved laterally almost 6
feet, despite a pseudostatic factor of safety much greater than 1. Seed (1973) was able to
accurately calculate the response of the dam using a dynamic analysis, and Wu (2001)
back calculated deformations even more accurately using a nonlinear effective stress
finite element analysis.
Figure 7, Upper San Fernando Dam Area Map
Source: www.topozone.com, USGS San Fernando Quadrangle
Pseudostatic analyses indicate that both upper and lower dams were safe using the
selected seismic coefficient k = 0.15, as shown in Figure 8. Figure 8 also indicates that
the yield coefficient would have to be as large as 0.43 to 0.55, which is very large in
comparison to the values used in design practice today. This exemplifies the difficulty in
interpreting the results of a pseudostatic analysis. Seed (1973) performed an extensive
investigation of the upper dam failure, and was able to fairly accurately calculate a
similar response.
9.
Figure 8, Factors of Safety for the Upper and Lower San Fernando Dams
Source: Seed et al (1973)
4.2 Seed et al (1973) Dynamic Analysis
Seed’s procedure is summarized as follows:
1) Evaluate the static stresses in the embankment.
2) Determine the earthquake motion for the rock below the embankment.
3) Evaluate the response of the embankment to the base motion.
4) Determine the dynamic stresses in the embankment.
5) Use lab testing to evaluate the dynamic response of the embankment soils.
6) Evaluate the deformation of the embankment.
Seed classified the soils in the dam into 5 groups, as shown in Figures 9 and 10. There is
no recorded motion at the upper dam, so the nearby Pacoima Dam time history was
scaled to 0.6g for use as the rock motion. The calculated induced shear strains of 12-16%
correspond to 4½ to 6 feet of lateral movement of the crest, which is very close to the
observed movement of the dam during the San Fernando Earthquake.
10.
Figure 9, Input Motions and Results of Seed (1973) Analysis
Source: Seed et al (1973)
Figure 10, Soil Parameters of the Upper San Fernando Dam
Source: Seed (1973)
4.3 Wu (2001) Nonlinear Effective Stress Finite Element Analysis
Wu (2001) performed a nonlinear effective stress finite element analysis to compute the
deformations of the upper dam. Wu used a hyperbolic shear stress-shear strain model for
the soil, and Modified Martin Finn Seed model for pore pressures. The final shape of the
deformed embankment was determined using post-liquefaction and strain softened
strengths of the embankment materials.
4.3.1 Hyperbolic Shear Stress-Shear Strain Model
Wu’s hyperbolic model of shear stress-shear strain is given by:
11.
γ
τ
γ
τ
ult
xy
G
G
max
max
1+
= Where
f
ult
R
Gmax
=τ
γ is the shear strain, and Rf is a modulus reduction factor.
4.3.2 Martin Finn Seed Pore Pressure Model
Under earthquake loadings, an embankment may experience undrained conditions and
the subsequent build up of pore pressures. Martin et al (1975) related the change in pore
pressure to plastic volumetric strain by p
vrEu ε∆=∆ . Er is the unloading-reloading
modulus for the soil at a given effective stress, and can be estimated using (N1)60
Standard Penetration Test blow counts. The volumetric strain increment, p
vε∆ , is a
function of the total volumetric strain, p
vε∆ and the current shear strain, γ.
4.3.3 Strain Softening and Post-Liquefaction Strength
Effective stress is lowered by the increase of excess pore pressures. This is modeled by
updating the shear modulus and ultimate shear strength with the current effective stress:
'
'
max
vo
v
update GG
σ
σ
= and '
'
vo
v
ultupdate
σ
σ
ττ =
The post-liquefaction residual strength is assigned to soils when liquefaction occurs. Wu
considered liquefaction to occur when the pore pressure ratio, '
vo
u
PPR
σ
= , was greater
than 0.95.
4.3.4 Results of the Analysis
Wu used a modified version of the Pacoima time-history for the rock motion and the
same 5 soil types as did Seed in 1973 (See Figure 11).
12.
Figure 11, Finite Element Mesh Showing Cross-Section Profile
Source: Wu (2001)
Wu’s analysis indicates areas of liquefaction similar to those calculated by Seed, as
shown in Figure 12.
Figure 12, Comparison of Liquefied Zones Computed by Wu (top) and Seed (bottom)
Source: Wu (2001), and Seed (1973)
Lastly, Wu’s computed deformations are almost identical to the measured displacements,
as shown in Figure 13.
13.
Figure 13, Comparison of Observed and Calculated Displacements
Source: Wu (2001)
5.0 CONCLUSION
Static evaluation procedures of slope stability are well defined, and suitably accurate and
precise. Simple seismic evaluation of slope stability can be performed using a
pseudostatic analysis, but the results are misleading. In 1965, Newmark made the
analogy that a slope is like a block sliding on an inclined plane. Newmark was able to
reasonably compute the displacement due to shaking by double integrating portions of
actual acceleration time histories that exceeded the yield acceleration. In 1978, Makdisi
and Seed improved upon this procedure by accounting for the dynamic response of the
embankment. During the San Fernando Earthquake of 1971, the Upper San Fernando
Dam, despite a large pseudostatic factor of safety, failed. In 1973, Seed performed a
dynamic analysis of the embankment and computed displacements that closely agreed
with the observed deformations. Wu, in 2001, performed a nonlinear effective stress
finite element analysis that predicted displacements almost exactly as those measured.
14.
REFERENCES
Duncan, Buchignani, De Wit, (1987), An Engineering Manual For Slope Stability
Studies, Virginia Polytechnic Institute and State University, Blacksburg, Virginia
Terzaghi, K, (1950), Mechanics of Landslides, Engineering Geology (Berkey) Volume,
Geological Society of America
Seed, R.B., (2000), Class Notes, Evaluation and Mitigation of Seismic Hazards Seminar,
UC Berkeley Extension, Berkeley, California
Kramer, S.L., (1995), Geotechnical Earthquake Engineering, Prentice Hall, Upper Saddle
River, New Jersey
Makdisi, Seed, (1978), “Simplified procedure for estimation dan and embankment
earthquake-induced deformations”, Journal of the Geotechnical Engineering Division,
ASCE, Vol. 104, No, GT7
Newmark, N., (1965), “Effects of Earthquakes on Dams and Embankments”,
Geotechnique, Vol. 15, No. 2
Youd, T.L., (1978), “Major Cause of Earthquake Failure is Ground Failure”, Civil
Engineering, ASCE, Vol. 48, No. 4
Wu, G., “Earthquake-Induced Deformation Analysis of the Upper San Fernando Dam
Under the 1971 San Fernando Earthquake”, Canadian Geotechnical Journal, Vol. 38,
2001
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