Geotechnical and Foundation Engineering (CON4342)

Slope Stability

Learning Outcome
 acquire the knowledge of stability analysis of soil slope;

Key Contents
• factor of safety
• Infinite slopes
• Finite slopes
• method of slices
• application of computer software in slope stability analysis

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Sections
1. Introduction
2. Factor of safety of slope
3. Stability of infinite slope
4. Stability of slope using mass procedure
5. Stability of slope using method of slice
6. Application of computer software in slope analysis

Reference: Das, B. M. (2006). Principles of Geotechnical Engineering. 6th Edition. Cengage Learning.

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1. Introduction

 Gravitational and seepage forces tend to cause instability in natural slopes;

excavation; embankments; earth dams.

Refer to figure 1
 rotational slips: circular arc or a non-circular curve.
 circular slips are associated with homogeneous soil conditions and
non-circular slips are associated with non-homogeneous conditions.
 Translational and compound slips occur where the form of the failure surface
is influenced by the presence of an adjacent stratum of significantly different
strength.
 Translational slips tend to occur where the adjacent stratum is at a
relatively shallow depth below the surface of the slope. In this case the
failure surface tends to be plane and roughly parallel to the slope.

Figure 1

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 Compound slips usually occur where the adjacent stratum is at greater

depth, the failure surface consisting of curved and plane sections.
 In practice, limiting equilibrium methods are used in the analysis of slope
stability. The problem is normally considered in two dimensions and plane
strain condition is assumed..

Figure 2 – A shallow slope failure in Hong Kong

Note the circular deep seated failure surface

http://dutcgeo.ct.tudelft.nl/allersma/hgball.htm

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2. Factor of Safety for Slopes

In limit equilibrium analysis, the factor of safety of slope (F) is given by the
following relationship:

(shearing resistance of the soil) / (Mobilized shear force) (1)

In terms of shear strength, F is usually defined as

F= f/m (2)

where  r = average shear strength along the potential failure surface

 m = average shear stress mobilized along the potential failure surface

The factors of safety for Hong Kong slopes recommended by the Geotechnical
Engineering Office are presented in Tables 1 and 2.

Note the following key words in the tables:

- “Ten years return period rainfall”
- “Risk to life”
- “Economic risk”
- “New slopes” vs “Existing slopes”

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Factor Safety for New Slopes

Table 1 (GEO 1984)

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Factor Safety for Existing Slopes

Table 2 (GEO 1984)

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The Stability of Hong Kong Slopes

The potential cause of slope failure are:

 Topography. The slopes in Hong Kong island are mostly steep and usually greater
than 30 degrees. Cut slopes up to 75 degrees also occur. Instability can be
triggered by removing the restraint at the toe of a slope.

 Geology. Both the granite and volcanic rocks have been heavily decomposed to
clayey silty sands (residual soils) up to 30 metres thick in places, and the
vegetation is sparse.

 Shear Strength. The values of c' and ' for the residual soil vary greatly with the
degree of saturation, Sr of the soil. The value of c' can drop to zero at low confining
stresses (shallow depths) in saturated soil.

 Rainfall and Groundwater. Summer typhoons result in periods of intense rainfall

which creates a saturated band in the soil. This infiltrates quickly downwards,
increasing pore-water pressures and creating instability. The groundwater regime
is often the only natural parameter that can be economically changed to increase
the stability of slopes.

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3. Stability of Infinite Slopes

Figure 3 shows an infinitely long slope of angle .

By considering the equilibrium of forces acting on the soil element abcd and
assuming there is no porewater pressure, it can be shown that the factor of safety
is:
cL
 HL cos  tan 
Ta cos 
F ; F
Tr HL sin 
c tan 
F  (3)
H cos  tan 
2
tan 

Analysis of infinite slope (without seepage)

Figure 3
c 1
For F =1; the critical Height Hcr at failure is H cr 
 cos  (tan   tan )
2

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For sands, c = 0, and the F = (tan )/(tan )

The value of F is independent of the height H and the dry density of the sand.
 It also indicates that the slope is stable as long as  < 0. When  = , F=1.
The value of  in this case is called the limiting angle.

 If the soil possesses cohesion and friction, i.e., a “c - ” soil, the depth of
the plane along which critical equilibrium occurs may be determined by
substituting F = 1 and H = Hcr into Eq. 3. Thus

c 1
H cr  (4)
 cos 2  (tan   tan )

Infinite Slopes With Seepage

Figure 4 shows an infinitely long slope in saturated soil which is subject to
steady state of downhill seepage, with the groundwater level at the surface.
Similarly:

c  ' tan
F  (5)
 sat H cos  tan   sat tan 
2

where ’ = submerged density of the soil

sat = saturated density of the soil

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Analysis of infinite slope (with seepage)

Figure 4

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Learning Activity 1 (exercise)

Learning Outcome : To determine the factor of safety of infinite slopes
Complete the following exercise by the students:
For the infinite slope shown in the Figure below, determine:
(a) The factor of safety against sliding along the soil-rock interface given H = 2.5
m
(b) The height, H, that will give a factor of safety of 2 against sliding along the
soil-rock interface.

c tan 
(i) F 
H cos 2  tan  tan 

Try this yourself

F = 1.18

c tan 
(ii) F 
H cos 2  tan  tan 
Try this yourself

H = 1.08m

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4. Stability of slope using mass procedure

(Finite Slopes with Circular Failure Surface)

4.1 General Modes of Slope Failure

 In general, slope failure occurs in one of the following modes (Figure
5)

 Failure above the toe of the slope is called a slope failure (Figure 5a).
The failure circle is referred to as a toe circle if it passes through the
toe of the slope and as a slope circle if it passes above the toe of the
slope.

 Failure at way above the toe is called shallow failure (figure 5b).

 Failure at some distance below the toe of the slope is called a base
failure (Figure 5c). The failure circle in the case of base failure is
called a midpoint circle.

Figure 5

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Procedure of Stability Analysis

 Mass Procedure,
 Method of Slices

 Mass Procedure In this case the mass of soil above the surface of sliding is
taken as a unit. This procedure is useful when the soil that forms the slope is
assumed to homogeneous.

• Method of Slices. In this procedure, the soil above the surface of sliding is
divided into a number of vertical parallel slices. The stability of each of the
slices is calculated separately. It can consider :
 non-homogeneity
 porewater pressure
 variation of the normal stress

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Location of the Most Critical Circle

 The slip circle with the lowest factor of safety for a given slope will allow the
engineer to decide whether the slope is safe or not.
 Trial and error. Various slip circles are considered, as shown in Figure 5.6a
The factor of safety is determined for each circle and the centre plotted with
the factor of safety marked along side it.
 The location of the centre of the most critical circle is at the centre of the
plot.

Figure 6

 More generally, some empirical rules have been formed regarding the
locations of the most critical circles both for cohesive soils and for soils
which have some frictional strength. These rules are:

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4.2 Mass Procedure of Stability Analysis (Circular Failure Surface)

(a) Slopes in Homogeneous Clay Soil with  =0 (Undrained Condition)

Clay soils slopes are analyzed for stability by a limit equilibrium method
under undrained condition. . The method is called total stress analysis.

The shear strength parameters used include results from the shear vane
test, the unconfined compression test, or the unconsolidated undrained
triaxial compression test on the clay soil. These parameters are cu with u
= 0.

Notes
- Porewater pressures in the slope are not taken into account since the
undrained cohesion, cu, is assumed to represent the shear strength of the
clay in the undrained condition.

This total stress analysis covers

 fully saturated clay under undrained conditions,
 Only moment equilibrium is considered
 Figure 7 shows a trial failure surface (centre O, radius r and arc length La).
Potential instability is due to the total weight of the soil mass (W) per unit
length above the failure surface.

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Figure 7

For equilibrium the shear strength which must be mobilized along the failure surface is
expressed as

m=  f /F = c u /F
Equating moments about O:
W d = (c u /F)L a r
and therefore
F = (c u L a r)/(W d) (6)

 The moments of any additional forces must be taken into account.

 In the event of a tension crack developed at top of the slope as shown in
Figure 8, the arc length La is shortened and a hydrostatic force will act
normal to the crack if it fills with water.

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Learning Activity 2 (example)

Learning Outcome : To understand the theory of analysis of finite slopes using mass
procedures.
A 45o slope is excavated to a depth of 8m in a deep layer of saturated clay of unit wight 19
kN/m3. The relevant shear strength parameters are cu = 65 kN/m2 and u =0 . Determine
the factor of safety for the trial failure surface specified below by considering equilibrium of
moments (area of ABCD = 70m2).

Equating Moment about O

W d = (cu/F)Lar
And therefore
F = (cuLar)/W d
W d = (19x70)x4.5 = 5985

CuLr = 65 x (12.1 x 89.5ox/180 ) x 12.1 = 14858.1

F = CuLr / W d
= 14858.1 / 5985 = 2.48

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Learning Activity 3 (example)

Learning Outcome : To understand the theory of analysis of finite slopes using mass
procedures.
The failure surface of the following slope lies within 2 layers of strata, unit weight of both
layers is 17.6 kN/m3.
The relevant shear strength parameters for the upper and lower layers are cu = 21 and 33.75
kN/m2 respectively with u =0 . Determine the factor of safety for the trial failure surface
specified below by considering equilibrium of moments (area of ABCD = 87m2).

Equating Moment about O

W d = (cu/F)Lar
And therefore
F = (cuLar)/W d

W d =87 x 17.6 x 2.75 = 4210.8

CuL r = [ 33.75 x (11.75 x 71.5ox/180 ) + 21.5 x (11.75 x

37ox/180 )]x11.75
= 7726.8

F = CuLr /W d = 7726.8/4210.8 = 1.83

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5. Stability of slope using method of slice

Method of Slices
 The above method of analysis, i.e., the mass procedure, assumes the soil is
homogeneous! However, most slopes are not made of homogeneous soils.
The mass procedure also does not take into account the effect of porewater
pressure and non-circular failure surfaces. The classical solution to these
problems is the method of slices first devised by W. Fellenius in Sweden
between 1914 and 1922.

Swedish (Fellenius) method of slices

Figure 8

Static forces on one slice

Figure 9

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 The potential failure surface AB is assumed to be a circular arc, centre 0,

radius r. This is found to be the common shape of failure surface.
 For any slice the angle of inclination of the failure surface to the horizontal is
i The failure surface is considered to be a plane over the span of the slice, l.
 The third dimension, the distance along the slope, is unity.
 Usually four to six slices are considered which are not necessarily of equal
width.
 The static forces on one slice are shown in Figure 9. In Fellenius' method or
the ordinary method of slices, all the interslice forces (the E and X forces) are
ignored and the slices are assumed to offer no support to each other to resist
slipping.

c' l  (W cos   ul ) tan '

F (7)
W sin 

where c’ and ' are respectively the cohesion and angle of internal. friction of soil with
respect to effective stress along the slip plane of the slice;
W is bh, the weight of the slice;
 is the average angle of the slope of the slip plane in the slice;
l is the span of slice on the slip plane; and
u is the average porewater pressure in the slice on the slip plane.

Summation of all the slices will give the factor of safety for the slope:

F  c' l  (W cos   ul ) tan ' (8)

W sin 

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Learning Activity 5 (example)

Learning Outcome: To understand the theory of analysis of finite slopes the method of slice.

[The pore pressure distribution may be represented by piezometric head (hw) or

pore pressure ratio ru.]
Shown below are the details of an existing slope. The soil properties are
=1820kg/m3; c’ = 7 kN/m2; ’ = 20o and no tension cracks have formed.
Check the stability of the bank along the slip surface shown (1.6; 0.93) using
Fellenius’s Method
 a) when there is no pore water pressure
 b) When the height of pore water is as shown
 c) When ru = 0.35

Slice hw
(m)
1 0.6
2 1.8
3 3
4 3.7
5 2

Slice No. α ( deg ) h(m) b ( m ) γ ( kN/m3 ) hw

1 -21 1.00 1.80 17.84
2 -4 2.60 1.80 17.84
3 14 3.70 1.80 17.84
4 33 3.70 1.80 17.84
5 58 2.00 1.80 17.84

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a) when there is no pore water pressure

F  c' l  (W cos   ul ) tan '

W sin 
3
Slice No. α ( deg ) h(m) b(m) γ ( kN/m ) ru or hw c' ( kN/m2 ) φ' ( deg ) u ( kN/m2 )
1 -21 1.00 1.80 17.84 0 7 20 0
2 -4 2.60 1.80 17.84 0 7 20 0
3 14 3.70 1.80 17.84 0 7 20 0
4 33 3.70 1.80 17.84 0 7 20 0
5 58 2.00 1.80 17.84 0 7 20 0

Slice No. (W cosα - ub secα ) tan φ ' W sin α Cb secα

1 10.9115 -11.5079 13.4964
2 30.3143 -5.8241 12.6308
3 41.9603 28.7438 12.9857
4 36.2682 64.7110 15.0238
5 12.3872 54.4650 23.7772
summation 131.8416 130.5878 77.9139

F = (77.92 + 131.84)/ 130.59

= 1.63

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b) When the height (hw) of pore water is as shown

u = hw  w

F  c' l  (W cos   ul ) tan '

W sin 
3 2 2
Slice No. α ( deg ) h(m) b(m) γ ( kN/m ) hw c' ( kN/m ) φ' ( deg ) u ( kN/m )
1 -21 1.00 1.80 17.84 0.6 7 20 5.886
2 -4 2.60 1.80 17.84 1.8 7 20 17.658
3 14 3.70 1.80 17.84 3 7 20 29.43
4 33 3.70 1.80 17.84 3.7 7 20 36.297
5 58 2.00 1.80 17.84 2 7 20 19.62

Slice No. (W cosα - ub secα ) tan φ ' W sin α Cb secα

1 6.7810 -11.5079 13.4964
2 18.7175 -5.8241 12.6308
3 22.0891 28.7438 12.9857
4 7.9140 64.7110 15.0238
5 -11.8693 54.4650 23.7772
summation 43.6323 130.5878 77.9139

F = (77.92 + 43.6)/ 130.59

= 0.93

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c) When ru = 0.35
ru is the ratio of the pore water pressure to overburden pressure : ru = u / h. It is used
to calculate pore water in an embankment or fill.

u = ru  h

F  c' l  (W cos   ul ) tan '

W sin 
3 2 2
Slice No. α ( deg ) h(m) b(m) γ ( kN/m ) ru c' ( kN/m ) φ' ( deg ) u ( kN/m )
1 -21 1.00 1.80 17.84 0.35 7 20 6.244
2 -4 2.60 1.80 17.84 0.35 7 20 16.2344
3 14 3.70 1.80 17.84 0.35 7 20 23.1028
4 33 3.70 1.80 17.84 0.35 7 20 23.1028
5 58 2.00 1.80 17.84 0.35 7 20 12.488

Slice No. (W cosα - ub secα ) tan φ ' W sin α Cb secα

1 6.5297 -11.5079 13.4964
2 19.6524 -5.8241 12.6308
3 26.3613 28.7438 12.9857
4 18.2210 64.7110 15.0238
5 -3.0519 54.4650 23.7772
summation 67.7125 130.5878 77.9139

F = (77.92 + 67.71)/ 130.59

= 1.12

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Other Methods of Slices

Simplified Bishop,
Janbu,
Morganstera & Price
Hoek, Sarma
Table 3 lists the methods of slope stability analysis and recommendations
for use in Hong Kong by the Geotechnical Engineering Office. When
choosing a method of stability analysis for design, the probable mode of
failure of the slope must be considered. The method chosen should model
the failure mode.

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Learning Activity 6 (class discussion)

Learning Outcome
- To appreciate the mechanisms of slope failure
- To understand the meaning of factor of safety in soil slope analysis

 Identify the causes of failure / landslip for natural and man-made slopes in
Hong
 Define "Factor of Safety" for slopes in terms of shear strength.
 What are the factors that affect the value of “Factor of Safety”.
 Identify the potential causes for the slope stability problems in Hong Kong.
 Describe briefly, with the aid of diagrams, the different types of slope
movements according to their motions relative to the adjacent or underlying
soil stratum.
 Explain briefly, with the aid of sketches, on two methods which can increase
the factor of safety of a circular slip surface on a soil slope.

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Bishop's Simplified Method

 In this method, the interslice shear forces are ignored as they are equal and
opposite (i.e. X1 = X2), but the interslice normal forces are not (i.e., E1  E2)
(see Fig. 11). These assumptions are reasonable if conditions are uniform
and ru is constant.

1 [c' b  (W  ub) tan  ' ] sec 

F  (8a)
W sin  1
tan  tan  '
F

 Note : The computation using equation (8a) is commenced by assuming a

trial value F.

Learning Activity 7 (example)

Learning Outcome: to understand the theory of analysis of finite slopes using mass
procedures and the method of slice.
Analyse the stability of the slope below using Bishop’s simplified method. =
2100 kg/m3; c’ = 8 kN/m2; ’ = 32o and no tension cracks have formed.
(1.095)

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Slice No. α ( deg ) h(m) hw ( m ) b(m)

1 49.5 1.33 0.4 1.90
2 44.7 3.20 1.6 1.90
3 38.2 4.62 3.2 3.10
4 31.4 5.28 4.5 3.10
5 25 5.40 4.9 3.10
6 18.8 5.10 5 3.10
7 13.1 4.60 4.7 3.10
8 7.3 3.60 4 3.10
9 1.6 2.40 2.8 3.10
10 -4.1 0.84 1.2 3.10

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Bishop's Method (Berry & Reid page 219)

Slice No. α ( deg ) h(m) hw ( m ) b(m) γ ( kN/m3 ) c' ( kN/m2 ) φ' ( deg ) u ( kN/m2 ) A = [ c'b + (w - ub ) tan φ ' ] sec α
1 49.5 1.33 0.4 1.90 20.58 8 32 3.92 66.2758
2 44.7 3.20 1.6 1.90 20.58 8 32 15.68 105.1936
3 38.2 4.62 3.2 3.10 20.58 8 32 31.36 188.6233
4 31.4 5.28 4.5 3.10 20.58 8 32 44.1 175.5765
5 25 5.40 4.9 3.10 20.58 8 32 48.02 162.2561
6 18.8 5.10 5 3.10 20.58 8 32 49 140.7025
7 13.1 4.60 4.7 3.10 20.58 8 32 46.06 122.1368
8 7.3 3.60 4 3.10 20.58 8 32 39.2 93.1363
9 1.6 2.40 2.8 3.10 20.58 8 32 27.44 67.3494
10 -4.1 0.84 1.2 3.10 20.58 8 32 11.76 35.5978

Try Fs = 1
B= (tan φ ' tanα )/F A
Slice No. W sin α
1+B
1 0.73 38.2737 39.5454
2 0.62 65.0001 88.0132
3 0.49 126.4465 182.2739
4 0.38 127.0984 175.5039
5 0.29 125.6454 145.5959
6 0.21 116.0220 104.8555
7 0.15 106.6314 66.5155
8 0.08 86.2335 29.1833
9 0.02 66.1940 4.2752
10 -0.04 37.2670 -3.8316
Sum Σ = 894.8121 Sum Σ = 831.9303

Fc = 1.076

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Janbu's Routine Analysis for the Stability of Slopes with Non-Circular Failure Surface
 Whilst the Bishop's simplified method was originally developed for analysis
with assumed circular failure surfaces, the Janbu's Rigorous and Routine
methods are suitable for non-circular slip surfaces.
 Non-circular slip surface failures are common forms of landslip because the
shape of the failure surface is often controlled by geological planes of
weakness within the soil mass. For example, in Hong Kong slope failures in
extremely decomposed granite may follow relict joints in this residual soil.

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6. Application of computer software in slope analysis

e.g. Slope 2000, Slope/W etc
(Any GEO approved slope analysis software will serve the
purpose, the input interface may be different but the
underlying principles are similar)
The following example adopts Slope/W 2012 (student version) for illustration.

Step 1
Keyin
Analysis

Remember to set the PWP conditions

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Set
Scale

Set
Axes

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Step 2
KeyIn
Materials

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Step 3
Draw
Region

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Draw
Materials

Step 4
Draw
Pore water pressure

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Step 5
Draw
Slip surface
Grid

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Step 6
Draw
Slip surface
Radius

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Step 7
Solve Manager
successful

unsuccessful

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Disclaimer- the author has tried his best to indicate all references but there is no
guarantee that all materials cited can be included.

Further References
1. Das, B. M. Principles of Geotechnical Engineering. 6th Edition. Cengage Learning.
2. Craig, R. F. Soil Mechanics. 7th Ed, E & FN Spon.
3. Whitlow,R. Basic Soil Mechanics, 2nd Edition, Prentice Hall.
6. Barnes, G.E. Soil Mechanics Principles and Practices, MacMilan.
7. Berry, P.L. & Reid, D. An Introduction to Soil Mechanics, McGraw-Hill Book
Company
8. Budhu, M. Soil mechanics & Foundations, 2nd Ed., John Wiley & Sons
9. Smith G.N. Elements of Soil Mechanics, 6th Ed, BSP Professional Books.
10. Smith M.J. Soil Mechanics, Longman.
11. Sutton, B.H.C. Solving Problems in Soil Mechanics, 2nd Ed, Longman.
12. ELE International. Catalogue in Laboratory Testing.
13. Geotechnical Engineering Office, (1995), Geoguide 5 - Guide to Slope Maintenance,
CED, The Government of Hong Kong Special Administrative

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