Uncertainty Evaluation in Slope Failure using
Centrifuge Model Tests
Takayuki SHUKU a, Shin-ichi NISHIMURA a and Ikumasa YOSHIDA b a
Grad. Sch. of Environmental and Life Science, Okayama University, Japan b
Dept. of Urban and Civil Engineering, Tokyo City University, Japan
Abstract. Strong nonlinear behavior such as slope failure involves large uncertainties. Even if we conduct model tests on slope failure under the same test conditions, we cannot reproduce the same results although the linear behavior can be reproduced with relative ease. This kind of uncertainties is well known as “aleatory uncertainty” and they should be distinguished clearly from the epistemic uncertainty in reliability-based design. However, characteristics of aleatory uncertainty have not been studied very well so far. This study quantitatively investigates the aleatory uncertainties involved in failure behavior of slopes thorough centrifuge model tests. The centrifuge model tests were conducted under the same test conditions twenty times to quantitatively investigate the variability of slip lines.
Keywords. slope failure, uncertainty, centrifuge model tests
1. Introduction
In reliability-based designs, engineers have to quantitatively evaluate uncertainties involved in design process. Although many sources of uncertainty exist, they are generally categorized as “aleatory” or “epistemic” (e.g. Kiureghian and Ditlevsen, 2009). The word aleatory means throwing a dice in Latin, and aleatory uncertainty is considered as inherent variability of phenomena. Thus uncertainties are categorized as aleatory if we cannot estimate the possibility if reducing them. While the word epistemic means knowledge in Greek, and that uncertainty is considered as being arisen from lack of knowledge or data. Thus uncertainties are categorized as epistemic if we can estimate a possibility to reduce them by gathering more data or by improving models. Although it is convenient to categorize uncertainty as aleatory or epistemic, their differences have not been clarified either and have not been focused on in geotechnical reliability-based design.
A typical uncertainty categorized as aleatory includes chaotic behavior in which a very small change in initial conditions creates significantly different results. Large deformation and collapse behavior of geotechnical structures are considered as then chaotic behavior.
Geotechnical design based on factor of safety has been transferred to performance based design, and deformations of structures have been beginning to be specified as performance criteria like serviceability limit states. To achieve more efficient and economical geotechnical design, aleatory uncertainty has to be investigated and applied in geotechnical reliability-based design.
This study has investigated aleatory uncertainty involved in slope failure through a series of centrifuge model tests. Kaolin clay was used for the material of model slopes. We observed slip lines and estimate stochastic characteristics of them. Spatial variations of water contents in the slope were also studied to investigate variability of initial condition of the model slopes.
2. Centrifuge Model Tests
A series of centrifuge model tests was conducted to evaluate aleatory uncertainty, which is variability of slope failure in this study. All slope models used in this experiment had the same geometry and were built within a same box. The model slopes were subjected to increasing centrifuge acceleration until it reached 150 G © 2015 The authors and IOS Press.
This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License.
over two minutes. Details of the test setup are presented in this section.
2.1. Geotechnical Centrifuge Apparatus
The centrifuge tests were conducted with a beam centrifuge at Okayama University, Japan. This apparatus is designed to apply centrifuge acceleration up to 560 G at a nominal radius of 0.3 m. The slope models were placed on a swing-up bucket, so that the soil surface remains always perpendicular to the direction of the acceleration. A general view of the centrifuge is shown in Figure 1.
2.2. Slope Models
A strong box, which is manufactured from aluminum alloy, and the length, width, and height of its inside are 120, 40, and 100 mm respectively, was used to contain the model slopes. A transparent acrylic board was used on
one side of the box to observe the failure behavior of the slope during the testing. Silicone grease was applied to the inside of the box to minimize wall friction with the model slopes. A non-slip strip was placed on the bottom of the box to stop slippage between the box and the model slopes.
All slope models were constructed with a height of 50 mm, and the slope gradient was 63 degrees. The schematic illustration of the model slope and the box are shown in Figure 2. The kaolin clay, which is extensively used in geotechnical experiments for many purposes, was used for the slope materials.
2.3. Construction and Testing Procedure
To construct uniform slope models as much as possible, the model slopes were carefully prepared and constructed. The procedure of construction of the model slopes and the centrifuge testing are as follows:
(1) Slurry of kaolin clay with an initial water content of 60% was prepared and cast into the box until it was filled with the slurry. To reduce entrapped air, the box was vibrated by hand during the casting. Two container boxes filled with the slurry of kaolin were prepared.
(2) Those two boxes, which have the same weight, were mounted on the buckets in the centrifuge and were accelerated to 300 G over 5 minutes. The slurry in the boxes was consolidated by its self-weight for two hours under 300 G.
(3) The two container boxes were taken from the bucket of the centrifuge apparatus, and a model slope (Figure 2) was carefully made by cutting the consolidated kaolin clay in the box. Two slope models were constructed every test.
(4) One model slope and a box for counter weight, which have exactly the same weight as the box of the model slope, were mounted on the buckets, and the centrifugal acceleration was raised by following the acceleration process shown in Figure 3. After the acceleration reached 150 G, the model was held at a constant acceleration, which is 150 G, for two minutes to allow equalization of the load. At the end of the Figure 1. A view of the centrifuge.
100 120 (Unit : mm) 40 20 60 40 10
testing, the centrifuge was slow down gradually until the spinning is full stopped. This procedure is summarized as a flowchart in Figure 4.
Since the front of the container box is made of transparent acrylic boards, failure process and deformation of the model slopes can be observed with a digital video camera during the spinning. In addition, slip lines observed along the front side of containers were carefully traced onto a piece of sheet and then digitized. All slip lines observed in the slope models were visible and they are easy to trace. After that, small soil blocks were sampled at various locations to confirm the distribution of water content in the
model slopes to study variability of the initial conditions of the model slopes.
3. Test Results
3.1. Variability of Slip Lines
A typical deformation of the model and a slip line are shown in Figure 5. A dashed line indicates the initial shape of the model slope and the red line indicates the slip line. Although some small slip surfaces were observed, we focused on just one representative slip surface which assumed to be appeared firstly.
A failure process was as follows: several seconds after the acceleration reached 150 G, (1) the model slope started to deform, and a few tension cracks at the middle of the crest appeared, and then (2) the toe of the slope was displaced to the front. Although the slip surface was not able to be observed with digital video camera during the model test, it can be assumed that slip failure occurred at this time. After that, (3) the slope continued to deform and other tension cracks which close to slope shoulder was appeared.
The above centrifuge models test under the same experimental conditions were conducted twenty times in total, and almost the same failure process explained before was observed. All test results are summarized in Figure 7. All slip lines Figure 3. Acceleration process of the centrifuge test.
Figure 4. Flowchart of the testing procedure.
Figure 5. A typical deformation of the model slope.
Figure 6. All test results on slip lines.
0 50 100 150 200 0 50 100 150 200
Centrifugal acceleration (m/sec
2 )
Elapsed time (sec)
0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 1.0 2.0 3.0 4.0 5.0 : Before : After : Slip line H (cm) L (cm) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 1.0 2.0 3.0 4.0 5.0 H (cm) L (cm)
pass through the toe of the model slope.
We defined the parameters on slip lines as L1 and L2, which are shown in Figure 7, and studied the variability of those values. Figure 8 shows the histograms of L1 and L2. The value of L1 ranges from 3.2 to 4.8 cm, and the most probable value, mean and standard deviation are 4.1, 3.85, and 0.33 cm respectively. While the value of L2 ranges from 6.8 to 7.8 cm, and the most probable
value, mean and standard deviation are 7.2, 7.23, and 0.20 cm respectively. The standard deviation of L1 is larger than that of L2. These values can be considered as aleatory uncertainty involved in slope failure and need to be considered when we design natural slopes and earthen embankments. 3.2. Distribution of Water Content
Distribution of water contents along slope height for 20 tests is summarized in Figure 9. Except the surface of the model, the water content increases with height. This result implies that there was little migration of pore water in the model during the test. This is because if the pore water migrated, water migration from the top to the bottom would occur and the water contents would decrease with height. The mean and standard deviation of the water content at each depth are summarized in Table 1. In addition, a histogram of water content at H = 1.5 cm is shown in Figure 10. The standard deviation at each height is very small and shows almost the same value. This means we were able to construct relatively uniform slope models in the test.
Figure 9. Distribution of water content along slope height. Table 1. Mean and standard deviation of water content.
38.0 40.0 42.0 44.0 46.0 48.0 0.0 1.0 2.0 3.0 4.0 5.0
H
(cm)
Water content (%)
Figure 7. Definition of L1 and L2.
Figure 8. Variability of L1 and L2.
Height (cm) PP VV 4.50 43.2 0.64 3.50 43.8 0.73 2.50 43.1 0.73 1.50 43.0 0.79 0.50 42.4 0.72 3.2 3.6 4.0 4.4 0 2 4 6 8 P = 3.78 V = 0.27 㻌 㻌 Frequency L1 (cm) P = 7.23 V = 0.20 6.4 6.8 7.2 7.6 8.0 0 2 4 6 8 㻌 㻌 Frequency L2 (cm)
Figure 10. Distribution of water content along slope height. The coefficient of variation (C.V.) for L1, L2 and water content are summarized in Table 2. The C.V. of the water content is almost the same value each other and relatively small. The C.V. of location of slip lines, L1 and L2, show larger values compared to those values of water contents.
Relationships between slip lines (L1 and L2) and water contents are shown in Figure 11. The dashed lines indicate the regression equations. The values of L1 and L2 scatter with water contents, and it seems like that L1 and L2 are uncorrelated with the water content because the values of R2 are small values. Therefore, the variability on slip lines (L1 and L2) can be considered as aleatory uncertainty involved in slope failure.
4. Summary
This study has investigated the uncertainty involved in slope failure through centrifuge model tests. The model tests on uniform slope of kaolin clay were conducted twenty times, and the variability of slip lines was investigated. The variability of slip line locations L1 and L2 can be considered as aleatory uncertainty involved in slope failure. This kind of uncertainty should be considered in geotechnical design of natural slopes and earthen embankments when displacements are specified as the serviceability limit state.
Further studies should be done to understand the characteristics of aleatory uncertainty on a deeper level.
References
Kiureghian, A. D. and Ditlevsen, O. (2009). Aleatory or epistemic? Does it matter?, Structural Safety, 31(2009), 105-112. 41.0 42.0 43.0 44.0 45.0 0 2 4 6 8 H = 1.5cm P = 43.0 V = 0.789
Frequency
㻌 Water content (%)Table 2. Coefficient of variation
PP VV C.V. L1 3.78 0.27 0.071 L2 7.23 0.20 0.028 w 4.5cm 43.2 0.64 0.015 3.5cm 43.8 0.73 0.017 2.5cm 43.1 0.73 0.017 1.5cm 43.0 0.79 0.018 0.5cm 42.4 0.72 0.017 (a) L1, H = 4.5 cm (b) L2, H = 1.5 cm
Figure 11. Relationship between slip lines and water content 41.0 42.0 43.0 44.0 45.0 3.0 3.5 4.0 4.5 H = 4.5cm L1 (cm) Water content (%) L1 = 0.0379w + 2.144 (R2 = 0.01) 41.0 42.0 43.0 44.0 45.0 6.0 6.5 7.0 7.5 8.0 H = 1.5cm L2 (cm) Water content (%) L2 = -0.086w - 10.933 (R2 = 0.11)
