SEISMIC DESIGN AND ANALYSIS OF REINFORCED SLOPES

AND HIGHWAY EMBANKMENTS

Evangelos KOUKOS1, Antonia BAGGOU2

ABSTRACT

Highway embankments are often constructed using soil strengthening techniques by means of horizontally

placed strips of geosynthetic reinforcement over successive layers of compacted fill material. HA 68/94 Advice

Note gives guidance on design methods for determining the required reinforcement of highway slopes, based on

a limit equilibrium approach applied on a two-part wedge mechanism. This type of mechanism has been found to

be particularly suited to the analysis of reinforced soil, as it can take any form provided that the inter-wedge

boundary is vertical and that the base of the lower wedge intersects the toe of the slope.

The currently proposed by HA68/94 approach provides a simple method for obtaining safe and economical

design solutions, yet limited to the evaluation of the reinforced slope design requirements with respect to static

loading only. Nonetheless, seismic loading conditions also constitute special conditions to which the reinforced

slopes may be subjected and would require consideration, in addition to the static loading, when performing

internal stability evaluation. Extending the two-part wedge mechanism design approach to cover an earthquake

loading scenario is of particular interest in this study.

This paper presents the results of an extensive parametric study, performed on a reinforced embankment of

varied geometric characteristics and soil properties subjected to a range of seismic accelerations. The

recommended design charts and tables, provide a very useful tool for practising engineers in their preliminary

design check as they allow simple hand calculations to be carried out for a quick estimate of the reinforcement’s

total force and layout geometry required to support an embankment slope under seismic loading.

Keywords: Reinforced slopes; Embankments; Internal stability; Two-part wedge mechanism; Geogrids

1. INTRODUCTION

A limit equilibrium approach is adopted by HA 68/94 for evaluating the internal stability of an

embankment based on a two-part wedge mechanism. This type of mechanism has been found to be

particularly suitable to reinforced soil as it can accommodate any reinforcement layout geometry and

is preferred as it provides a simple method for obtaining safe and economical design solutions.

1.1 Definition of Two-Part Wedge Mechanism

A typical geometry of the two-part wedge mechanism is shown in Figure 1. The mechanism assumes a

bilineal failure surface to simulate potential slip surfaces within reinforced soil mass and may take any

form provided that the inter-wedge boundary is vertical, and that the base of the lower wedge

intersects the toe of the slope. Three cases of two-part wedge mechanisms can be considered

depending on whether the inter-wedge boundary lies to the left or right of the crest and also whether

the upper wedge outcrops above the slope or on the slope face.

1Ministry of Infrastructure and Transport, Directory of Road Infrastructure, Civil Engineer MSc, Athens, Greece,

ekoukos75@gmail.com 2Ministry of Infrastructure and Transport, Directory of Road Infrastructure, Civil Engineer MSc, Athens, Greece,

abaggou@yahoo.gr

mailto:ekoukos75@gmail.com

2

u

2T

U2

N'2

K2

2R'

T1

1U

N'1

1K

R'1

W2

Q2N'

U

T

12

12

12

12R'

K12

12K

R'121

1

12

12

12

T

U

N'

W

Q

Wedge 1

Wedge 2

Wedge 2

Wedge 1

Competent bearing material

q

c', φ'

γ, r

X

Y

H

Figure 1. Geometry of two-part wedge mechanism and forces acting on wedges

The internal and external forces acting on the two wedges are also shown in Figure 1. It is assumed

that driving forces, such as soil’s self weight and any surcharge loads, are in limit equilibrium with

resisting forces such as soil shear strength and reinforcement force. The general formula derived when

resolving forces parallel and perpendicular to the lower surface of each wedge is too complex and

cannot be solved without any assumption. When the conservative assumption of a zero inter-wedge

friction angle is made, the formula is considerably simplified. In this case, the total quantity of

horizontal reinforcement force required for equilibrium, Ttot, is given by the following expression:

𝑇𝑡𝑜𝑡 = 𝑇1 + 𝑇2 =

=[(W1+Q1) (tan θ1−tan φ′1)+(U1 tan φ′1−Κ1)/ cos θ1]

(1+tan θ1 tan φ′1)+

[(W2+Q2)(tan θ2−λs tan φ′2)+λs (U2 tan φ′2−Κ2)/ cos θ2]

(1+λs tan θ2 tan φ′2) (1)

where T1,2 is the sum of reinforcement forces acting on each wedge, W1,2 is the weight of each wedge,

Q1,2 is the total surcharge force on each wedge, U1,2 is the total surcharge force on each wedge, K1,2 is

the cohesion force acting on base of each wedge, λs is the base sliding factor, θ1,2 is the base angle of

each wedge and φ’1,2 is the effective angle of friction acting on the base of each wedge.

1.2 General Concepts of the Design Method

When designing a reinforced soil slope, both the total reinforcement force required Ttot and the overall

dimensions of the zone containing reinforcement, denoted by LT and LB in Figure 2, must be specified.

More specifically, Ttot governs the design tensile strength Td of each reinforcement layer as well as the

minimum number N of reinforcement layers to be used (N=Ttot/Td). Furthermore the determination of

the reinforced zone limits, at the top and at the bottom of the embankment, establishes the required

length of reinforcement at any depth of the slope.

All the above should be considered as a preliminary estimate of the minimum reinforcement

requirements. For practical reasons, it may be desirable to rationalize the final reinforcement lengths

and spacings, which could lead to a greater total quantity of reinforcement being used.

When selecting the design value of strength Td for a polymeric reinforcement, consideration should

also be given to the long term reduction of strength due to factors such as creep, installation damage,

weathering, chemical effects etc. In case of geogrids, the global reduction factor of strength may vary

from 2.5 to 10 according to the BBA certificates or manufacturer’s literature. However, for

preliminary design and routine calculations a safety factor of about 7.0 may be used as a conservative

value.

θ1

θ2

β

θ1

3

L

L

L

T mechanism

Reinforced

Zone

Competent bearing material

T

B

e

T mechanism

max

ob

1

Figure 2. General concepts of design method

1.2.1 Determination of Ttot

For each one of the three possible cases considered, a spreadsheet may be developed for calculating

Ttot using Εquation 1 and the appropriate algebraic expressions for the main variables as proposed by

HA 68/94 and shown in Figure 3. By a computer search the critical two-part wedge mechanism among

cases 1, 2 and 3 which requires the greatest horizontal reinforcement force Ttot (called the “Tmax

mechanism”) can be identified. Usually case 1 is the most common situation for identifying the “Tmax

mechanism”.

1.2.2 Determination of LT

The “Tmax mechanism” also governs the length of reinforcement LΤ at the top of the slope. More

specifically, as shown in Figure 2, the length of LT is set such that the uppermost reinforcement layer

has just sufficient length Le1, beyond the failure surface of the “Tmax mechanism”, to mobilize its pull-

out resistance. The pull-out length Le1 required is given by:

𝐿𝑒1 =𝑇𝑑

𝜆𝑝(𝜎′

𝑣 𝑡𝑎𝑛 𝜑′)

(2)

where Td is the design value selected for the reinforcement strength, λp is a non-dimensional pull-out

factor and σ’v represents the normal effective stress acting on the first layer of reinforcement beyond

the failure surface. According to HA 68/94 if LT is less than [LΒ-Χc], where Xc=H/tanβ, then LT should

be set equal to [LΒ-Χc], so that the rear boundary of the reinforcement zone becomes vertical.

1.2.3 Determination of LB

Assuming that a competent bearing material exists beneath the reinforcement zone, the key

mechanism for determining the required length of the reinforcement zone LB at the base of the

embankment is forward sliding on the basal layer of reinforcement. This is called the “Tob

mechanism”.

The “Tob mechanism” is essentially a two-part wedge mechanism with a flat base of the lower wedge

requiring precisely zero reinforcement force for stability. The “Tob mechanism” is simple to define and

locate using Εquation 1 and a trial and error approach by means of a computer search. The length LB is

set equal to the base width X of the “Tob mechanism”. In most cases the critical value of θ1 for

identifying the “Tob mechanism” may be assumed to be (45ο +φ’/2).

45ο+φ’/2

4

where:

Figure 3. Algebraic Definitions of two-part wedge geometry for the three cases considered (after HA 68/94)

Case 1 W1= ½ γ [(a+b)2 cotθ1-a2 cotβ]

W2= ½ γ b X

U1= ½ ru γ [d e+(d+b) f]

U2= ½ ru γ b m

K1= c’1 (e+f)

K2= c’2 m

Q1= q k

Q2= Not applicable

Case 2 W1= ½ γ (H-Y)2 cotθ1

W2= ½ γ [2XH-X2 tanθ2 –Η2 cotβ]

U1= ½ ru γ g (H-Y)

U2= ½ ru γ [(H-Y)(X+t)secθ2+j m)]

K1= c’1 g

K2= c’2 m

Q1= q s

Q2= q t

Case 3 W1= ½ γ b u

W2= ½ γ b X

U1= ½ ru γ b u secθ1

U2= ½ ru γ b m

K1= c’1 [u secθ1]

K2= c’2 m

Q1= Not applicable

Q2= Not applicable

a= (H-X tanβ)

b= (Η-Υ-a) = X-tanβ -Υ

d= k tanθ1

e= k secθ1

f= [(a+b)/sinθ1]-e = (g-e)

g= s secθ1 = (a+b)/sinθ1

j= t tanθ2

k= [(a+b) cotθ1]-[a cotβ] = (s+t)

m= √(X+Y)

s= (a+b) cotθ1

t= (k-s)

u= b/(tanθ1-tanβ)

X

Y

H

k

d

b

a

e

f

m

Case 1

1

2

q

X

Y

H

s

g

m

t

Case 2

1

2

q

j

X

Y

H

u

b

a

m

Case 3

1

2

q

θ1

θ1

θ1

θ2

θ2

θ2

β

β

β

5

2. SEISMIC ANALYSIS OF EMBANKMENTS

The stability of reinforced slopes should be checked for both static and earthquake loading. Seismic

loading acting on embankment slopes is usually incorporated into the calculations by design engineers

using the pseudostatic approach. Such approach was used in this paper to extend the two-part wedge

mechanism design method proposed by HA 68/94 for the earthquake case as well.

2.1 Seismic Stability of Slopes and Embankments

According to Greek Regulation EAK-2000 stability of embankments up to 15-m in height, subjected

to an earthquake load, shall be checked using an additional horizontal seismic acceleration αh acting on

their soil mass, which vary from αΒ at the base up to αK at the top of the embankment.

𝑎𝐵 = 0.50 𝑎 (3)

𝑎𝐾 = 𝑎𝐵 𝛽(𝛵) (4)

where α is the normalised seismic ground acceleration, equals to 0.16 for Seismic Zone I , 0.24 for

Seismic Zone II and 0.36 for Seismic Zone III and β(Τ) is the spectral magnification factor that

corresponds to the fundamental period of the structure. In the absence of a more accurate analysis, T =

2.5 (H/Vs) may be used, where, Vs is the average shear wave velocity in the embankment.

The horizontal component of the effective seismic acceleration acting at the embankment slope αh, is

considered to be the mean average of αΒ and αK. Thus,

𝛼ℎ =(𝛼𝛣+𝛼𝛫)

2= 0.25 𝛼 [1 + 𝛽(𝛵)] (5)

The vertical component of the effective seismic acceleration αv, is assumed to be half of the horizontal.

𝛼𝑣 = ±0.50 𝑎ℎ (6)

For embankments higher than 15-m, EAK-2000 suggests that a special geotechnical and seismic

design is required for estimating the design seismic acceleration acting on its mass. However, in the

absence of a detailed and complete seismological study, it is recommended that, the earthquake’s

horizontal components at the existing ground level may be calculated from the design response spectra

formula using suitable values for the factors involved.

2.2 Two-Part Wedge Mechanism Design Method for Seismic Loading

In order to expand the two-part wedge mechanism design method for the earthquake case, in addition

to the existing static forces, pseudostatic seismic forces Fh1 and Fh2 were considered acting at the

centroid of each one of the two wedges and were incorporated into the general formula of Ttot. The

magnitude of each seismic force can be calculated by multiplying the horizontal seismic coefficient αh

with the total weight of each sliding mass. The vertical component of the seismic acceleration was

neglected and was not considered in the calculations performed.

The expression for the total quantity of horizontal reinforcement force required Ttot given in HA 68/94

can be then modified, for the case of seismic loading, as follows:

𝑇𝑡𝑜𝑡 = 𝑇1 + 𝑇2 + 𝐹ℎ1 + 𝐹ℎ2 =

=[(W1+Q1) (tan θ1−tan φ′1)+(U1 tan φ′1−Κ1)/ cos θ1]

(1+tan θ1 tan φ′1)+

[(W2+Q2)(tan θ2−λs tan φ′2)+λs (U2 tan φ′2−Κ2)/ cos θ2]

(1+λs tan θ2 tan φ′2)+ 𝛼ℎ (𝑊1 + 𝑊2)(7)

where Fh1,2 is the horizontal seismic force acting at the centroid of each wedge.

6

3. PAPAMETRIC ANALYSIS

This paper presents the results of an extensive parametric study, with more than 240 different cases

analyzed, accounting for a reinforced embankment slope of varied geometric characteristics and soil

properties, subjected to a range of seismic accelerations. Since the scope of this study was to give

guidance on the design requirements for the strengthening of new earthworks, effort was made to

analyse typical cases of embankments that are most likely to be encountered in actual case studies.

3.1 Assumptions and limitations of the analysis

The following assumptions were considered for the parametric analysis performed:

External Geometry

A new highway embankment of 10-m, 20-m and 30-m in height, with a slope angle β ranging from 40o

to 70o, constructed over horizontal ground of a competent bearing material, was considered for the

analysis.

Fill Material

A fill material with a unit weight γ of 20 kN/m3 and typical shear strength properties was considered in

the parametric analysis. More specifically since effective friction angle φ’ usually lies in the range 30o-

35o for granular fills and in the range 20o–25o for low plasticity (i.e. PI

7

3.2 Modifications in the general formula of Ttot

Due to the assumptions made above, Equation 7 now simplifies to:

𝑇𝑡𝑜𝑡 = 𝑇1 + 𝑇2 + 𝐹ℎ1 + 𝐹ℎ2 =[𝑊1 (𝑡𝑎𝑛 𝜃1−𝑡𝑎𝑛 𝜑′1)]

(1+𝑡𝑎𝑛 𝜃1 𝑡𝑎𝑛 𝜑′1)+

[𝑊2 (𝑡𝑎𝑛 𝜃2−𝜆𝑠 𝑡𝑎𝑛 𝜑′2)]

(1+𝜆𝑠 𝑡𝑎𝑛 𝜃2 𝑡𝑎𝑛 𝜑′2)+ 𝛼ℎ (𝑊1 + 𝑊2) (8)

It is evident that the calculation of Ttot under seismic loading depends, apart from the design seismic

acceleration of the slope αh, on the strength of the fill φ’ as well as on the weight of the two wedges

and the geometry of the failure zone (θ1, θ2).

A spreadsheet was developed, based on the equation above, in order to identify Tmax and Tob

mechanisms and calculate Ttot and LB respectively. More specifically, by a trial and error process the

geometry of the failure zone in terms of θ1, θ2 and X was located and the greatest required horizontal

force as well as the maximum required length of reinforcement at the base of the embankment among

cases 1, 2 and 3 were specified for a given embankment geometry and material fill properties.

Spreadsheet validation was also performed by comparison with known results as those available by the

HA68/94 design tables for static loading conditions.

4. RESULTS

The results of the parametric study are presented in the form of design charts and tables (see

Appendix). For a given seismic acceleration αh input values of the embankment slope angle β and the

effective angle of the fill material φ’ would only be required to estimate the reinforcement’s total force

and layout geometry to satisfy internal stability requirements. Obviously linear interpolation is always

recommended for calculating intermediate values.

4.1 Results of Parametric Analysis

After careful evaluation of the results of the parametric analysis, it was verified that the total

reinforcement force required for stability Ttot is directly proportional to the square of the height H of

the embankment. In particular, it was found that the total required reinforcement force was four times

more for a embankment with a height of 20-m and nine times more for a embankment with a height of

30-m compared to the total reinforcement force Ttot required for a 10-m height embankment. Thus, for

presentation purposes, it was convenient to non-dimensionalise the calculated values of Ttot by using

the total force coefficient parameter Kmax where:

𝐾𝑚𝑎𝑥 = 𝑇𝑡𝑜𝑡 (0.50 𝛾 𝛨2)⁄ (9)

Similarly, it was found convenient to normalize the resulting values for the required lengths of the

reinforcement at the bottom and at the top of the embankment with respect to the height H, as it was

proven that the derived lengths LB and [LT-Le1] were directly proportional to the height of the

embankment for the cases analyzed.

The following diagrams summarize the results of the parametric analysis performed:

Figure 4. Variation of Kmax against embankment slope angle β for different seismic coefficients

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

40 45 50 55 60 65 70

Km

ax

β (deg)

αh=0.10φ=25φ=30φ=35φ=40

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

40 45 50 55 60 65 70

Km

ax

β (deg)

αh=0.20φ=25φ=30φ=35φ=40

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

40 45 50 55 60 65 70

Km

ax

β (deg)

αh=0.30

φ=25φ=30φ=35φ=40

8

Figure 5. Variation of LB/H ratio against embankment slope angle β for different seismic coefficients

Figure 6. Variation of [LB- Le1]/H ratio against embankment slope angle β for different seismic coefficients

The following conclusions can be drawn after evaluating the results above:

Influence of fill material

The total reinforcement force required for stability highly depends on the shear strength of the fill

material. For example as shown in Table 1, in the case of an embankment with slope angle of 70o, Kmax

can be reduced over 50% when using a fill material with an effective friction angle of 40o instead of

25o.

The required length of reinforcement at the base of the embankment is also significantly affected by

the shear strength of the fill material. More specifically for the case described above, LB/H is increased

about 2.5 times for seismic acceleration αh=0.10 and almost 7.0 times for seismic acceleration αh=0.30

when a fill material with an effective friction angle of 25o instead of 40o is used.

Table 1. Influence of fill material strength in case of a very steep slope (β=70ο).

αh Kmax LΒ/H

(φ’=25ο) (φ’=40ο) (φ’=25ο) (φ’=40ο)

0.10 0,408 0,177 1,06 0,42

0.20 0,516 0,243 1,82 0,53

0.30 0,657 0,324 4,84 0,72

Influence of slope angle

Embankments with very steep slopes require a larger quantities of reinforcement force for stability but

surprisingly reduced lengths of reinforcement at the base when compared to embankments with flatter

slopes, as shown in the example presented at Table 2. The latter conclusion, also observed in the case

of static loading, is attributed to the shallower failure planes that are developed below the slope face in

case of steep slopes.

0.00

0.50

1.00

1.50

2.00

2.50

40 45 50 55 60 65 70

L B/H

β (deg)

αh=0.10φ=25φ=30φ=35φ=40

0.00

0.50

1.00

1.50

2.00

2.50

40 45 50 55 60 65 70

L B/H

β (deg)

αh=0.20

φ=25φ=30φ=35φ=40 0.00

1.00

2.00

3.00

4.00

5.00

6.00

40 45 50 55 60 65 70

L B/H

β (deg)

αh=0.30

φ=25φ=30φ=35φ=40

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

40 45 50 55 60 65 70

[LT

-L e

1]/H

β (deg)

αh=0.10

φ=25

φ=30φ=35

φ=40

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

40 45 50 55 60 65 70

[LT

-L e

1]/H

β (deg)

αh=0.20

φ=25

φ=30

φ=35

φ=40

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

40 45 50 55 60 65 70

[LT

-L e

1]/H

β (deg)

αh=0.30

φ=25φ=30φ=35φ=40

9

Table 2. Influence of slope angle in case of fill effective friction angle φ’=30ο.

αh Kmax LΒ/H

(β=40ο) (β=70ο) (β=40ο) (β=70ο)

0.10 0.151 0.317 1.14 0.73

0.20 0.268 0.409 1.50 1.09

0.30 0.424 0.524 2.36 1.94

Influence of seismic acceleration

It is obvious after evaluating the results of Tables 1 and 2 that Kmax increases almost proportionally to

the seismic coefficient αh for all cases analyzed. On the other hand, LB/H increases substantially for

embankments with fill materials of reduced strength when they are subjected to strong seismic

accelerations.

It should be noted that even for the lowest considered seismic acceleration, the estimated

reinforcement force and lengths greatly exceed those derived in HA68/94 for the case of static loading.

This observations highlights the importance of the current work as in the majority of practical cases,

earthquake loading controls the design.

5. CONCLUSIONS

A preliminary estimate of the reinforcement’s total force and layout geometry required to support a

new embankment slope of certain characteristics under seismic loading is proposed in this paper. The

design charts and tables suggested, in coordination with the design tables given by HA 68/94 for the

case of static loading, can be a very useful tool when performing a preliminary design check to obtain

safe and economical solutions which provide internal stability of the slope. The research also

highlighted that for a given seismic acceleration both the slope geometry (height and inclination) as

well as the strength of the fill play an important role in the evaluation of the reinforced slope design

requirements, affecting significantly the cost of the construction.

6. ACKNOWLEDGMENTS

The authors would like to express their gratitude to the Ministry of Infrastructure and Transport of

Greece for giving them the opportunity to participate and present their study at the 16th European

Conference on Earthquake Engineering.

7. REFERENCES

British Standarts Institution (1994). BS8006 Code of Practice for Strengthened/Reinforced Soils and Other Fills.

EAK 2000 (2000). Greek Code for Seismic Resistance Structures

FHWA-NHI-00-043 (2001). Mechanically Stabilized Earth Walls and Reinforced Soil Slopes Design and

Construction Guidelines

HA 68/94 (1994). Design methods for the reinforcement of highway slopes by reinforced soil and soil nailing

techniques. Design Manual for Roads and Bridges, Vol. 4, Section 1, Part 4, HMSO

Konstantinidis G., (2010). Guidelines for the Design of Reinforced Embankments of the Egnatia Odos A.E.,

Proceedings of the 6th Hellenic Conference on Geotechnical and Geo-environmental Engineering, 29 September

- 01 October, Volos, Greece.

Mouratidis A, (2007). Road Construction: The construction of road works, University Studio Press, Thessaloniki

10

APPENDIX

The results of the parametric analysis are summarized in the Tables below. For each case the geometry

of “Tmax mechanism” (giving X/H, Y/H, θ1) and the corresponding total force coefficient parameter

Kmax are provided as well as the minimum lengths of the reinforcement zone LB and [LT-Le1] required

for internal stability. (Note: The values of [LT-Le1] were determined geometrically given the geometry

of “Tmax mechanism”).

u

Le1

BL

LT

c'=0

φ'

γ

r =0

H

Y

X

Table I. Two-part wedge solutions for seismic loading with αh=0.10

(ru=0, λs=0.8, c’=0, i=0, θ2≥0)

β φ’ Kmax X/H Y/H θ1 LB/H (LT-Le1)/H

40o

25o 0.259 0.765 0 46.0o 1.47 0.54

30o 0.151 0.617 0 46.0o 1.14 0.39

35o 0.071 0.449 0 45.1o 0.85 0.26

45o

25o 0.293 0.650 0 47.0o 1.37 0.58

30o 0.187 0.543 0 48.0o 1.05 0.45

35o 0.105 0.416 0 47.8o 0.82 0.32

40o 0.047 0.285 0 47.2o 0.58 0.21

50o

25o 0.322 0.553 0 48.0o 1.29 0.61

30o 0.219 0.465 0 49.2o 0.97 0.49

35o 0.137 0.373 0 49.9o 0.77 0.38

40o 0.075 0.277 0 50.3o 0.59 0.27

55o

25o 0.347 0.468 0 48.9o 1.22 0.64

30o 0.247 0.395 0 50.3o 0.90 0.52

35o 0.166 0.325 0 51.5o 0.71 0.42

40o 0.103 0.253 0 52.4o 0.56 0.32

60o

25o 0.369 0.387 0 49.4o 1.16 0.66

30o 0.272 0.332 0 51.3o 0.84 0.55

35o 0.193 0.279 0 53.0o 0.65 0.45

40o 0.128 0.221 0 54.1o 0.52 0.37

65o

25o 0.389 0.316 0 50.0o 1.11 0.68

30o 0.296 0.273 0 52.3o 0.78 0.57

35o 0.217 0.230 0 54.1o 0.59 0.48

40o 0.153 0.187 0 55.6o 0.47 0.40

70o

25o 0.408 0.247 0 50.4o 1.06 0.70

30o 0.317 0.215 0 52.9o 0.73 0.60

35o 0.241 0.183 0 55.0o 0.54 0.51

40o 0.177 0.151 0 57.0o 0.42 0.43

θ1

θ2 β

11

Table II. Two-part wedge solutions for seismic loading with αh=0.20

(ru=0, λs=0.8, c’=0, i=0, θ2≥0)

β φ’ Kmax X/H Y/H θ1 LB/H (LT-Le1)/H

40o

25o 0.408 0.890 0 42.1o 2.23 0.81

30o 0.268 0.754 0 44.0o 1.50 0.60

35o 0.157 0.604 0 44.5o 1.14 0.43

45o

25o 0.433 0.755 0 42.7o 2.13 0.84

30o 0.300 0.643 0 45.0o 1.40 0.64

35o 0.193 0.527 0 46.3o 1.05 0.48

40o 0.109 0.403 0 46.6o 0.81 0.35

50o

25o 0.454 0.634 0 43.1o 2.05 0.86

30o 0.327 0.548 0 46.0o 1.32 0.67

35o 0.223 0.455 0 47.6o 0.97 0.53

40o 0.141 0.361 0 48.7o 0.76 0.40

55o

25o 0.472 0.530 0 43.5o 1.98 0.88

30o 0.351 0.457 0 46.5o 1.25 0.70

35o 0.251 0.388 0 48.6o 0.90 0.57

40o 0.169 0.314 0 50.1o 0.70 0.45

60o

25o 0.488 0.439 0 43.9o 1.92 0.89

30o 0.372 0.381 0 47.1o 1.19 0.73

35o 0.275 0.325 0 49.6o 0.84 0.59

40o 0.195 0.267 0 51.4o 0.64 0.48

65o

25o 0.502 0.355 0 44.1o 1.86 0.91

30o 0.391 0.310 0 47.6o 1.14 0.75

35o 0.298 0.266 0 50.3o 0.78 0.62

40o 0.220 0.221 0 52.5o 0.59 0.52

70o

25o 0.516 0.278 0 44.4o 1.82 0.92

30o 0.409 0.244 0 48.1o 1.09 0.77

35o 0.319 0.210 0 51.0o 0.74 0.65

40o 0.243 0.179 0 54.0o 0.53 0.54

Table III. Two-part wedge solutions for seismic loading with αh=0.30

(ru=0, λs=0.8, c’=0, i=0, θ2≥0)

β φ’ Kmax X/H Y/H θ1 LB/H (LT-Le1)/H

40o

25o 0.605 1.036 0 35.2o 5.25 1.26

30o 0.424 0.891 0 40.0o 2.36 0.89

35o 0.278 0.740 0 42.2o 1.52 0.65

45o

25o 0.617 0.871 0 35.4o 5.16 1.27

30o 0.447 0.750 0 40.5o 2.26 0.92

35o 0.309 0.634 0 43.3o 1.41 0.69

40o 0.198 0.513 0 44.9o 1.04 0.52

50o

25o 0.627 0.730 0 35.5o 5.08 1.28

30o 0.466 0.635 0 40.8o 2.18 0.95

35o 0.335 0.537 0 44.0o 1.34 0.73

40o 0.229 0.439 0 46.0o 0.96 0.56

55o

25o 0.636 0.611 0 35.6o 5.01 1.30

30o 0.483 0.529 0 41.1o 2.11 0.97

35o 0.358 0.454 0 44.7o 1.27 0.76

40o 0.256 0.373 0 47.0o 0.89 0.60

60o

25o 0.644 0.505 0 35.7o 4.95 1.31

30o 0.498 0.437 0 41.3o 2.05 0.99

35o 0.379 0.376 0 45.3o 1.21 0.78

40o 0.280 0.315 0 48.0o 0.83 0.63

65o

25o 0.651 0.408 0 35.7o 4.89 1.32

30o 0.512 0.356 0 41.7o 1.99 1.00

35o 0.398 0.307 0 45.8o 1.15 0.80

40o 0.303 0.259 0 48.8o 0.77 0.66

70o

25o 0.657 0.319 0 35.8o 4.84 1.32

30o 0.524 0.278 0 41.9o 1.94 1.02

35o 0.415 0.241 0 46.1o 1.11 0.83

40o 0.324 0.206 0 49.5o 0.72 0.69