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CONTINUOUS PRECISE DISPLACEMENT MONITORINGUSING GPS FOR ASSESSING THE STABILITY OF SLOPES

Norikazu ShimizuProf. Dr. Eng.Norikazu Shimizu, Rock Mechanics,Yamaguchi University, Faculty ofEngineering, Civil and

Summary:Monitoring is important to assessing the stability of structures and to confirming the validity of the design

during the construction and operation of the structures. The ideal monitoring system for projects in Rock andGeotechnical Engineering should be able to continuously and automatically monitor the behavior of anextensive area in real time and with high accuracy. In addition, the costs should be low and the handling shouldbe easy.

Displacement monitoring using GPS (Global Positioning System) satisfies the above requirements. Theauthor and his associates developed a GPS displacement monitoring system. The system provides three-dimensional displacements automatically and continuously at all monitoring points with high accuracy, andusers can see the results on the web through the Internet in real time.

In this paper, the system and the procedure of displacement monitoring using GPS are outlined, andpractical applications for assessing the stability of slopes are demonstrated.

Keywords: Displacement monitoring, GPS, three-dimensional displacements, slopes, assessing the stability,Back analysis


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1. INTRODUCTION

Monitoring is important to assessing the stability of structures and to confirming the validity of the designduring the construction and operation of the structures. Monitoring is also useful for predicting the risks, formanaging the safe operation, and for reducing the project costs. The ideal monitoring system for projects in Rockand Geotechnical Engineering should be able to continuously and automatically monitor the behavior of anextensive area in real time and with high accuracy. In addition, the costs should be low and the handling shouldbe easy.

Displacement monitoring using GPS (Global Positioning System) satisfies the above requirements. GPSbegan to be used for displacement monitoring in the mid-1980s in the fields of Civil and Mining Engineering aswell as other related fields (e.g., Chrzanowski and Wells, 1988; Burkholder, 1988 and 1989). Since then,practical applications have been performed by many researchers (e.g., Hudnut and Behr, 1998; Gili et al., 2000;Malet et al., 2002; Kim et al., 2003; Tasci, 2008) and guidelines have been published for displacementmonitoring (e.g., US Army Corps of Engineering, 2002; Vermeer, 2002; Bond, 2004). ISRM has also approvedthe suggested method for monitoring rock displacements using GPS (Shimizu et al., 2014). Thistechnologyhasbecome a common engineering tool for monitoring displacements in rock and geotechnicalengineering projects.

The author and his associates have developed a displacement monitoring system using GPS (Iwasaki etal., 2003; Masunari et al., 2003; Shimizu and Matsuda, 2002, 2003). The system provides three-dimensionaldisplacements automatically and continuously at all monitoring points, and users can see the results on the webthrough the Internet in real time.It has been applied to monitor landslides, cut slopes, quarries, dams, tunnelentrances, viaducts, retaining walls, railway tracks, etc (Hirabayashi et al., 2008; Furuyama et al., 2014;Nakashima et al., 2011; Satoh et al., 2014; Shimizu et al., 2006; Shitano et al., 2014; Tosa et al., 2014).

One of the most important issues in the practical use of GPS is how to improve the measurementaccuracy. The author has also proposed methods for removing errors and estimating the real values ofmeasurements. Those methods have succeeded in providing measurement results on the mm level that are a fewtimes higher in accuracy than the results of the standard GPS. Another important issue concerning monitoringtechnologies is how to evaluate the measurement results for assessing the stability of structures.

In this paper, the system and the procedure of displacement monitoring using GPS are outlined, andpractical applications for assessing the stability of slopes are demonstrated.

2. GPS DISPLACEMENT MONITORINGSYSTEM

2.1. Outline of system

The author and his associates developed a GPS displacement monitoring system using an L1 signal, asillustrated in Figure 1 (Iwasaki et al., 2003; Masunari et al., 2003; Shimizu and Matsuda, 2002, 2003). Sensors,composed of an antenna and a terminal box, are set on measurement points and a reference point.They areconnected to a control box into which a computer, a data memory, and a network device are installed.The dataemitted from the satellites are received by the sensors and then transferred to the control box through cables. Theserver computer, which is located in an office away from the measurement area, automatically controls the entiresystem to acquire and then analyze the data from the control box. Then, three-dimensional displacements arecontinuously obtained every hour at all the monitoring points. The monitoring results are provided to users onthe web through the Internet in real time. The users only need to access the home page to see the monitoringresults.


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Control box

Server

Cables for power supplyand communication

GPSsensor

GPSsensor

GPSsensor

GPS sensor atreference point

Measurement area

(a) System (b) Sensor

Figure 1.GPS displacement monitoring system.

2.2. Error-correction methods

Standard deviations in the conventional GPS survey are 5mm in the horizontal direction and 10mm inheight when the baseline length is less than one kilometer.Precise monitoring for practical use, however, oftenrequires higher accuracy than this. The most important issue in the practical use of GPS is how to improve themeasurement accuracy.

Measurement results generally include random errors (noise) and bias errors. Random errors arise fromrandom fluctuations in the measurements.On the other hand, typical bias errors in GPS monitoring aretropospheric delays and other signal disturbances caused by obstructionsabove the antennas and multipatheffects. Since both random and bias errors affect the monitoring quality, it is recommended that appropriateerror-correction methods be applied to reduce such errors.

Generally, measurement value y is composed of exact value u, bias errors p and, and random error R,as follows (Shimizu et al., 1024; Shimizu and Nakashima, 2016):

RTpuy (1)

where p is an error due toa signal disturbance caused by obstructions above the antennas, is an error due totropospheric delays, and R is a random error due to receiver noise. Tropospheric delays occur when signals fromthe satellite reflect in the atmosphere due to meteorological conditions (Hoffman-Wellenhof et al., 1997; Misraand Enge, 2006).

Both bias errors p and can be reduced and/or corrected by the methods described in the next section.After removing the bias errors from the measurement value, an appropriate statistical model is applied toestimate the exact displacement.

2.2.1. Bias errors

Tropospheric delaysTropospheric delays occur as a result of refractive effects, due to air density, when signals propagate

through the troposphere extending from a height of about 0 to 11 km above the earth (see Figure 2(a)). The airdensity is a function of the pressure of dry gases and water vapor. This means that the measurement results aresubject to the influence of the meteorological conditions along the signal propagation path. When the differencein height between measurement points is more than a few tens of meters, it may not be possible to ignore the biasof the tropospheric delays in precise monitoring with an accuracy of millimeter level, even for short baselinelengths. An appropriate model, like the modified Hopfield model, the Saastamoinen model, etc. (Hoffman-Wellenhof et al., 2001; Misra and Enge, 2006), can be employed to correct such tropospheric delays (Shimizu etal., 2014). Tropospheric models are usually installed in the baseline analysis software. The meteorological data(atmospheric pressure, temperature, and humidity) should be measured as input data for it.


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Signal disturbances due to obstructions above antennasObstructions above an antenna (trees, slopes, etc.) may block the reception of antenna signals or may

cause disturbances to the signals(see Figure 2(b)); these overhead obstructions become error sources in GPSdisplacement monitoring. Antennas should be located in areas with a sufficiently open sky. When there areunavoidable obstructions above the antennas, it is a good practice to analyze the data without the signalstransmitted from the satellites behind the obstructions (Shimizu et al., 2014).

Multipath signals constitute the other error source (see Figure 2(c)). This is the phenomenon in whichsignals reach an antenna via two or more paths.Multipath signals are mainly caused by signal reflections fromobjects (buildings, walls, fences, etc.) and from the ground surface in the vicinity of the antennas. Naturally, theprimary defense against multipath signals is to position the antennas away from any reflective objects and to seta mask angle for cutting multipath signals.

Direct path

Indirect path

(a) Tropospheric delays (b) Signal disturbances due to overhead obstacles (c) Multipath effectsFigure 2. Error factors in GPS monitoring

2.2.2. Random errors

After removing the bias errors from the measurement value, equation (1) can be written as follows:RTp uyy )(~ (2)

In order to estimate the exact value u from the corrected measurement result with random errors, it isrecommended that an adequate statistical method be adopted.The author used “the trend model” (Kitagawa andGersch, 1984) for this purpose (Shimizu, 1999; Shimizu and Matsuda, 2002; Shimizu et al., 2014).

This model is composed of system equation (3) and observation equation (4), as follows:

nnk vu (3)

nnn wuy ~ (4)where un represents the estimates for the exact values of the displacements and is the corrected measurementresults represented by equation (2). The measurement interval is t and subscript n denotes progressing time t (t= nt). is the operator for the finite difference (un= un-un-1) and k means the rank “k” difference.Equation (3)is a kind of probability finite difference equation for rank k. vn and wn are white noises with an average value of0, a standard deviation of , and an observation error with a standard deviation of .

The trend model can yield good estimates for exact displacements. Through many experiments andpractical applications, it has been proven that the system can detect displacements of 1-2 mm and displacementvelocities of 0.1 mm/day (Shimizu, 1999; Shimizu and Matsuda, 2002; Shimizu et al., 2014).

2.3.Verification: field experiments

In order to verify the monitoring system using GPS and the trend model, an experiment was conductedaccording to the following procedure:

Move the antennas carefully by hand and give the prescribed displacements daily (see Figure 3).Monitor the displacements every hour with the monitoring system and apply the trend model.The results are shown in Figure 4 and Table 1. The squares are the given (exact) displacements, and the

solid line in Figure 4 shows the smoothing results obtained by applying the trend model to the raw measurementdata. Comparing the monitoring results with the exact values, both are found to be in good agreement in Figure4. Table 1 shows the detailed results of the comparison.


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The discrepancy between the monitoring and the exact values falls within one millimeter. This means thatthe system with the trend model can detect displacements in millimeter order.

Antenna

Displacement

Second day

Disp

lace

men

t

First day

Day

(a) Antenna (b) Given displacements

Figure 3. Experiment for investigating measurement accuracy of system

ExactMeasured

Figure 4. Experiment results: exact displacements and displacements smoothed by trend model

Table 1. Comparison between exact and measured displacements

Exact value (mm) GPS with trend model (mm) |GPS-Exact | (mm)12.5 11.8 0.79.6 8.7 0.97.4 8.3 0.95.6 4.8 0.84.3 5.1 0.83.3 3.0 0.32.5 2.6 0.12.0 1.9 0.1

3. UNSTABLE STEEP SLOPE

The GPS displacement monitoring system (see Figure 1) is applied to monitor the displacements of anunstable steep slope along a road. Since local slope failures have occurred several times over the last 20 years,displacement monitoring has been conducted by borehole inclinometers and surface extensometers. Some of theinstruments, however, have occasionally not worked well due to large deformations, and it has been difficult toperform the monitoring continuously.In order to overcome such trouble, the GPS monitoring system has beenapplied for continuous monitoring (Furuyama et al., 2014).

3.1. Monitoring site

Figure 5(a) presents photographs of the slope and the monitoring area.The slope is composed of mainlytuff and sandstone, and its bedrock is granite. The surface is covered with a colluvial deposit.The left side of theslope, as seen in the photograph in Figure 5(a), has been gradually failing over the last 20 years, and a concreterock-shed tunnel has been constructed to cover the road and to protect it.

Two antennas were set at the top of the slope to monitor displacements, and another antenna was set at afixed point in a stable area as a reference point, denoted by K-1, beneath the slope. The monitoring points, G-1and G-2, were set on the left and right sides of the slope, respectively (see Figure 5(b)).


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The distance between monitoring point G-1 and reference point K-1 was 221 m, while that betweenmonitoring point G-2 and the reference point was 258 m. The differences in height between the two points andthe reference point were 103 m and 112 m, respectively.

G-１

G-2

sensor

G-１

3

sensor

G-2

(a) Slope and monitoring area (b) GPS sensors at G-1and G-2

Figure 5.Monitoring site; slope beside road.

3.2. Monitoring results

Three-dimensional displacements were continuously measured every hour using the GPS displacementmonitoring system. The monitoring results at monitoring point G-1 are shown with an hourly amount of rainfallin Figure 6. Small displacements of less than 2-3 mm/month were generated at monitoring point G-1 during thelow rainfall period from March to early June. Whenever heavy rain fell from July to October, the displacementgradually increased. Eventually, it reached 355 mm in the north direction, 234 mm in the west direction, and 137mm in settlement by the end of October. However, displacements at G-1 gradually converged and became stableafter this rainfall period.

On the other hand, no remarkable displacements were measured at G-2 in this period. This means that theright-hand side of the slope is seen to be more stable than the left-hand side.

Figure 7(a) shows the displacement vectors in the plan view of the slope, while Figures 7(b) and 7(c)show the vertical sections including G-1 and G-2, respectively. Both directions of vectors for G-1 and G-2almost coincided with the steepest direction of the slope in the plan view. The direction of the displacement at G-1 was toward the front of the slope in the vertical section until the middle of August. After heavy rainfall at theend of August and early September, the direction changed to be parallel to the slip plane of the slope, and it wastoward the front of the slope again after the last large displacement of October 23. On the other hand, thedirection of the displacement at G-2 was parallel to the slip plane at the top of the slope in the vertical section.

11/03/13 19/04/13 28/05/13 06/07/13 14/08/13 22/09/13 31/10/13 09/12/13 17/01/140102030405060

-1000

100200300400500 23/103/9

31/8

6/7

19/6

σ=1.6(mm)

Latit

ude(

mm

)

Rain

fall(

mm

/hou

r)

(a) Displacement in direction of latitude


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11/03/13 19/04/13 28/05/13 06/07/13 14/08/13 22/09/13 31/10/13 09/12/13 17/01/140102030405060

σ=2.0(mm)

-500-400-300-200-100

0100

Long

itude

(mm

)

Rain

fall(

mm

/hou

r)

(b) Displacement in direction of longitude

11/03/13 19/04/13 28/05/13 06/07/13 14/08/13 22/09/13 31/10/13 09/12/13 17/01/140102030405060σ=4.5(mm)

-500-400-300-200-100

0100

Hei

ght(m

m)

Rain

fall(

mm

/hou

r)

(c) Settlement

Figure 6. Displacements at measurement point G-1.

0 200Displacement [mm]

(a) Plan view


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Welded tuff

Tuffaceoussandstone

Slip plane

Colluvial deposite

11/33/9

23/1017/1

0m 50m 100m

200mm

Granite

Slip plane

Granite

Colluvial deposite

11/36/9

17/111/9

0m 50m 100m

200mm

(b) Section A-A’ (c) Section B-B’

Figure 7. Transitions of displacement vectors.

3.3. Evaluation of results

Regulations (criteria) for traffic safety along this road have been given in Table 2. When the displacementvelocity (mm/day) is beyond 10 mm/day and the total amount of continuous rainfall exceeds 100 mm, one laneof the roadway is temporarily closed. And when the displacement velocity (mm/day) is over 20 mm/day and thetotal amount of continuous rainfall exceeds 200 mm, both lanes of the road are temporarily closed.

Figure 8 shows the velocity of the displacement (mm/day) obtained from the monitoring displacement.During this period, the road was closed a few times. The GPS monitoring results are seen to have providedeffective information for managing traffic safety, by comparing them with the above criteria.

Table 2. Criteria for assessing stability

Criteria Safety measureLevel 1 Displacement > 10mm/day One lane of the road is closedLevel 2 Displacement > 20mm/day Both lanes of the road are closed

23/08/13 02/09/13 12/09/13 22/09/13 02/10/13 12/10/13 22/10/13 01/11/13-10

0

10

20

30

40

50

-100

0

100

200

300

400

500

Total displacement (m

m)

Displacement velocity (mm/day)

Displacement (G1)

Displacement velocity (G1)

Criteria Level 1: 10mm/d

Criteria Level 2: 20mm/d

Figure 8. Displacement and velocity (mm/h) at G-1.

4. LARGE SLOPE OF LIMESTONE QUARRY

4.1. Monitoring site

The area of the quarry is approximately 2,000 m in length by 600 m in width (see Figure 9). The quarrycontains 4 billion tons of limestone reserves. The height of the slope at present is above 150 m, but the finalheight will be 300 m. The height of each bench is 15 m and the average angle of the slope is 60 degrees. Onebench is excavated every year and a half.

The antennas (sensors) are set on three measurement points, No.26, No.29, and No.31, located around thecrest of the slope (see Figure 10). Another sensor, No.28, is set at the bench mark point. All sensors receive the


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data transmitted from the GPS satellites and transfer them through a cable to a personal computer in thecontrol office. The cable also provides power to the receivers. The computer has automatic control over theentire system and analyzes the data in order to obtain displacements at all the measurement points (Hirabayashiet al., 2008).

No.26 (Measurement point) No.29 (Measurement point)

No.31 (Measurement point) No.28 (Reference point)

Figure 9. Limestone quarry

No.31 No.26 No.29 No.28PC monitor in thecontroll officePC monitor incontrol office

Figure 10. Locations of GPS sensors

4.2. Monitoring results

Three-dimensional displacements were continuously measured every hour. Figure 11 shows a portion ofthe continuous monitoring results at measurement point No.26 for three and half years. The red lines show theresults of the data smoothed by the trend model. A small displacement toward the east and upward was detectedduring this period.


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latit

ude

[mm

]

k=1

τ=0.04mm

σ=3.9mm

k=1

τ=0.01mm

σ=4.3mm

k=1

τ=0.02mm

σ=5.1mm

2/6/2003 6/5/2005

22/11/2005 31/12/2007

8/11/20051/5/20052/9/2003 3/12/2003 4/3/2003 4/6/2004 4/9/2004 5/12/2004 6/8/2005

22/2/2006 25/5/2006 25/8/2006 25/11/2006 25/2/2007 28/5/2007 28/8/2007

(a) Displacement in direction of latitude

long

itude

[mm

]

k=1

τ=0.04mm

σ=3.5mm

k=1

τ=0.03mm

σ=4.4mm

k=1

τ=0.05mm

σ=4.8mm

2/6/2003 6/5/2005

22/11/2005 31/12/2007

8/11/20051/5/20052/9/2003 3/12/2003 4/3/2003 4/6/2004 4/9/2004 5/12/2004 6/8/2005

22/2/2006 25/5/2006 25/8/2006 25/11/2006 25/2/2007 28/5/2007 28/8/2007

(b) Displacement in direction of longitude

heig

ht [m

m]

k=1

τ=0.04mm

σ=4.8mm

k=1

τ=0.10mm

σ=8.4mm

k=1

τ=0.03mm

σ=10.5mm

2/6/2003 6/5/2005

22/11/2005 31/12/2007

8/11/20051/5/20052/9/2003 3/12/2003 4/3/2003 4/6/2004 4/9/2004 5/12/2004 6/8/2005

22/2/2006 25/5/2006 25/8/2006 25/11/2006 25/2/2007 28/5/2007 28/8/2007

(c) Vertical displacement

Figure 11. Measurement results by GPS displacement monitoring system (No.26, June 2003- Dec. 2007)

4.3. Comparisonofmeasured and analytical displacements

In order to interpret the measurement results for assessing the stability of the slope, a three-dimensionalelastic analysis is conducted. The measured displacements are compared with the analytical results. The modelfor the analysis is shown in Figure 12. The area of the model is 4001080678 m. FLAC3D (Itasca, 2005) isused for the analysis. The input parameters are shown in Table 3. The excavation process for the analysis isdivided into five steps, as illustrated in Table 4 and Figure 13.

Figure 14showsa comparison of the measured and the analytical displacements in plan view and sectionA-C, respectively. Elastic displacements tend to be directed upward and backward to the slope. The direction ofthe measured displacement at No.31 almost coincides with the direction obtained from the elastic analysis.Thismeans that the slope around No.31 shows elastic behavior and is then stable.

On the other hand, the displacement measured at No. 26 shows inelastic behavior. The direction of themeasured displacement differs from the direction obtained from the elastic analysis. The real displacement goesdownward although the elastic one goes upward. The measured displacement at No.29 repeats this elastic andinelastic behavior.


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Since the total amount of those displacements is small, the slope is stable at the present time. It should becarefully monitored, however, to see whether such behavior will cause the instability of the slope.

No.29

face ground plane

No.31 No.26

Plan view

Figure 12. Model for three-dimensional elastic analysis

Table 3. Input parameters for elastic analysis Table 4. Excavation steps and periods

Modulus of elasticity [GPa] 10Poisson’s ratio 0.25Unit weight [KN/m3] 26.5

STEP1 6/2003-5/2004STEP2 6/2004-5/2005STEP3 6/2005-11/2005STEP4 12/2005-9/2006STEP5 10/2006-11/2007

No.31 No.26 No.29

Figure 13. Excavation steps in analysis


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2mm

No.29

No.26 No.31

No.31

2mm

No.26

2mm

No.29

2mm

Elastic analysis

Measurement

A sectionB section

C section

Figure 14. Comparison of measured and analytical displacements

4.4. Assessment of stability

4.4.1. Outline of method

The purpose of the method applied to the measured displacements here is to detect a potential failuremode, namely, plane sliding, wedge sliding and toppling, and a potential failure area (Matsuda and Shimizu,2004). The method uses geometrical data on the slope and its rock discontinuities (the strike, the dip, etc.).

Figure 15 shows the distribution of the poles of the discontinuities located in a rock slope. The slope faceis represented as a great circle in this figure. The poles, which are plotted within the crescent-shaped regionformed by the friction circle and the daylight envelope, indicate kinematically feasible sliding planes. Thefriction angle of the discontinuities is assumed to be 30 degrees.

When a displacement is measured on the slope, the pole corresponding to the direction of the measureddisplacement can be plotted on a stereonet. The feasible sliding planes can then be detected around the polerepresenting the direction of the displacement.

On the other hand, if the direction of the measured displacement coincides with the direction of theintersection line of a rock wedge, wedge failure may be occurring. Plotting the direction of the displacementsobtained by continuous monitoring on the stereonet leads to the discovery of which rock wedge has the potentialto slide, as shown in Figure 16.


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Daylight envelopeMeasured displacement vector

Friction angle 30°

Friction circle

Slope face

Area including kinematically possiblesliding planes of which the directionscoincide with the directions of themeasured displacements

(a) Plane failure (b) Measured displacement on a stereonetFigure. 15 Graphic representation of discontinuities together with displacement vector on stereonet (plane

failure)Direction of slide wedge

Great circle of discontinuous plane 1

Great circle of discontinuous plane 2

Direction of measured displacement

Trace of the direction of displacementby continuous monitoring

(a) Wedge failure (b) Measured displacement on a stereonetFigure 16. Identification of sliding wedge from direction of measured displacement (wedge failure)

4.4.2. Application of method to measured displacements

Since the measured displacement at No. 26 differs from the elastic one, as shown in Figure 14, themethod outlined in 4.4.1 is applied to investigate whether this indicates the plane failure mode or not. The polesof the incrementally measured displacements and the total measured displacements are plotted in Figures 17(a)and 17(b), respectively. The number of each pole represents the displacement measured during (1) June 2003-May 2004, (2) June 2004- May 2005, (3) June 2005-Nov. 2005, (4) Nov. 2005-Sept. 2006, and (5) Oct. 2006-Nov. 2007.

The poles of the measurement displacements are not concentrated in an unstable region (the crescent-shaped region), and since the amount of displacement is sufficiently small, it is found that no plane slide occursand that the slope around No. 26 is stable. Similar results are obtained at No.29.


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2mm 2mm

2mm 2mm

①

② ③

④

⑤

①

②

③④

⑤

①②③

④

⑤

①

②③

④⑤

①

③④

②⑤

①③④

②

⑤

B sectionB section

(a) Poles of incremental displacements (b) Poles of total displacements

Figure 17. Poles of displacement vectors on stereonet for detecting plane failure mode from measurements(No.26)

5. BACK ANALYSIS FOR ASSESSING STABILITY OF CUT SLOPE

5.1. Monitoring site and results

The slope in this study is composed of highly weathered sandstone and slate. The height and width of theslope are about 120m and 200m, respectively (see Figure 18). Figure 19 shows the vectors of the measureddisplacements at the measurement points set on the slope. From Figure 19, the occurrence of a large movementof the slope has been detected around measurement points 7, 8, and 9.


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Figure 18. Monitoring slope

1-10: Measurement pointson the slope

Scale

Displacement

Scale

(a) Plan view (b)Vertical section (a-a’)

Figure 19. Displacement vectors on slope measured by GPS

5.2. Back-analysis method

5.2.1.Modeling

The stress-strain relationship containing anisotropic parameter m is adopted here. It can simulate thedeformational behavior of the sliding of jointed rock masses (Sakurai et al., 1986; Sakurai 1987).

Assuming thatrock masses are equivalent continuous bodies, the incremental stress and strain relationshipcan be expressed in the x'-y' local coordinate system (Figure 20).

}{][}{ D (5)

where

22

2

21000101

21][

m

ED (6)

Hence, it is transformed into the x - y global coordinates as follows:

}{][}{ D (7)

whereTTDTD ][][][][ (8)

[T] is a transformation matrix expressed as


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22

22

22

sincoscossincossincossin2cossincossin2sincos

T (9)

where, is the angle between the x’ and x axes.

Figure 20. Coordinate system

If )1(2/1 m , then equation (6) represents an isotropic elastic material. An isotropic parameters mand can be back-calculated from the field measurement results.

5.2.2. Procedure of back analysis for assessing slope stabilityThe mechanical constants, including the anisotropic parameters and the initial state of the stresses, are

back-calculated so as to minimize the following value:

.min2

1

M

i

mi

ci uu (10)

where uim and ui

c represent the measured and the computed displacements, respectively. M is the numberof measurement points.

The stability of slopes is usually assessed by means of the factor of safety; thefactor of safety is defined asthe ratio of the strength against the stress occurring in the ground. In order to evaluate the factor of safety, wemust first know the strength of the rock masses.Therefore, a back analysis should be used to determine thestrength parameters of the cohesion and the internal friction angle of the rock masses. The procedure of the backanalysis for calculating the strength parameters and the factor of safety is as follows (Sakurai and Shimizu, 1992;1994):

(1) Determine Young's modulus E and anisotropic parameter m by a back analysis of the measureddisplacements (see equation(10)).

(2) Calculate shear modulus G as follows:

G=mE (11)

(3) Critical shear strain is defined as the ratio of shear strength c against the shear modulus; determine itthrough laboratory tests (Figure 21) (Sakurai, 1993).Shear strength parameter c will thus be obtained as

0 Gc (12)

(4) Assuming internal friction angle , determine cohesion c as (Figure 22)

cc

cos

sin1 (13)

(5) Then, calculate the factor of safety by a common limit equilibrium method using c and determined bymeans of the above procedure.


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GraniteTuffDepositsSandstoneSiltstoneClaystone, MudstoneSandClay, Silt

Shear modulus G (MPa)

Criti

cal S

hear

stra

ing 0

(%)

Figure 21. Critical shear strain versus shear modulus

G Shear modulus

Criti

cal s

hear

stra

in

(a) Relationship between shear modulus and critical strain (b) Shear strength

Figure 22. Schematic diagram for critical shear strain and shear strength


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5.3. Backanalysis and assessmentof slope stability

In order to obtain the mechanical constants of a rock mass, the back analysis outlined in the last section isapplied to the measured displacements taken with GPS. From the measurement results shown in Figure 19, theback analysis can be conducted as a two-dimensional problem in vertical section a-a' of the slope. Figure23

shows the finite element mesh. The back-analyzed mechanical constants are shown in Table 5. Thedisplacements calculated with the back-analyzed mechanical constants are compared with the measured values.From Figure 24 the calculated and the measured displacements agree well. The distributions of displacement andthe maximum shear strain of the slope are estimated on the basis of the back analysis results shown in Figures 25and 26, respectively.

A potential sliding surface may exist in this slope, as shown in Figure 26. The factor of safety for theblock of rock mass at the boundary of the sliding surface and the ground surface is calculated by the limitequilibrium method using the back-analyzed mechanical constants given in Table 5. In this case, the value of thefactor of safety is 1.1.

For increasing the safety of the slope, therefore, the part of the rock denoted by the dotted zone in Figure27 was excavated from the slope, and the slope angle was changed from 1:1 to 1:1.5. After changing the slopeangle, the GPS displacement measurements were continued at measurement points 2, 12 and 15 in order tomonitor the slope stability. The displacements converged and the slope construction was completed safely.

Zone 1

Zone 2

Zone 3

Zone 4

Zone 5

Zone 6

Scale

Figure 23. Finite element mesh

DisplacementMeasurementBack analysis

10cm Scale

Figure 24. Comparison between calculated and measured displacements

DisplacementScale

Figure 25. Displacements calculated using back-analyzed materialconstants


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Scale

Figure 26. Maximum shear strain distribution

Excavated area for safety of the slopebased on results of back analysis

Original shape

Potential sliding surface

Final shape2, 12, 15 : Measurement points

Scale

Figure 27. Potential sliding surface estimated by back analysis

Table 5. Results of back analysis

Young’s modulus E 94 MPaUnit of weight(assumed) 19.6 KN/m3

Poisson’s ratio (assumed) 0.3Anisotropic parameter m

m1 (Zone 1) 0.385 (isotropic)m2 (Zone 2) 0.360 (2=45deg.)m3 (Zone 3) 0.100 (3=35deg.)m4 (Zone 4) 0.050 (4=35deg.)m5 (Zone 5) 0.012 (5=20deg.)m6 (Zone 6) 0.005 (6=20deg.)

Shear modulus (sliding zone)G4 (Zone 4) 4.7 MPaG5 (Zone 5) 1.1 MPa

Critical shear strain(Zone 4) 1.0 %(Zone 5) 1.6 %

Internal friction angle(assumed) 25 deg.Cohesion

c4 (Zone 4) 30 KPac5 (Zone 5) 11 KPa

Factor of safety Fs 1.1


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6. CONCLUDING REMARKS

In this paper, the system of displacement monitoring using GPS was outlined together with a procedure toimprove accuracy. The monitoring procedure is summarized as the ISRM Suggested Method for MonitoringRock Displacements using GPS, and this technology will be further expanded as a standard tool in Rock andGeotechnical Engineering.

Three practical applications of the system have been demonstrated, and the stability of slopes wereassessed based on the monitored displacements, as follows:

1. The first case study of a steep slope along a traffic road: three-dimensional displacementscould be continuously measured with high accuracy. The monitoring system was availableeven under conditions of heavy rain, and it provided useful data for assessing the stability ofthe slope. The discussion for interpreting the monitoring results was extended to therelationship between the intensity of the rainfall and the increase in displacements. The GPSmonitoring results provided effective information for managing traffic safety, comparing thecriteria for road closures.2. The second case study of a limestone quarry: three-dimensional displacements weremeasured continuously for more than 5 years. The systemwas reliable for the continuous long-term monitoring of large slopes. It was found that the measured displacements exhibited elasticbehavior at one measurement point and inelastic behavior at other measurement points after acomparison with a three-dimensional elastic analysis. The direction of the inelasticmeasurement displacement was not concentrated in an unstable region on the stereonetrepresenting the rock discontinuities existing in the slope, and there were no signs ofinstability.3. The third case study of a large cut slope: a back-analysis of displacements measured by GPSwas applied to estimate the strength parameters of rock masses of the slope. In order to assessthe stability of the slope, the factor of safety was calculated using the back-analyzed strengthparameters. The original design was changed to increase the safety based on the back-analysisresults.

7. ACKNOWLEDGEMENTS

This research has been partially supported by JSPS KAKENHI (Grant-in-Aid for Scientific Research,Japan Society for the Promotion of Science) Grant Number 25350506. The author wishes to express hisappreciation and thanks to his colleagues and students for the contributions they made to the field work and theanalysis of the data. The author also thanks Ms. H. Griswold for proofreading this paper.

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