retaining wall calculation note

CDR Contract No. 4354: HYDRO AGRICULTURAL DEVELOPMENT OF SOUTH LEBANON IRRIGATION AND WATER SUPPLY SCHEME – CONVEYOR 800

CALCULATION NOTE: RANDOM STONE RETAINING WALLS AT SM2 EMBANKMENT RESERVOIR

Contents

Calculation Note for Random Stone Retaining Walls at South Marjaayoun 2 Embankment

Reservoir.......................................................................................................................................................................... 4

1. Introduction....................................................................................................................................................... 4

2. Computation Parameters.............................................................................................................................4

2.1 Wall Properties........................................................................................................................................4

2.2 Loads............................................................................................................................................................ 5

2.3 Soil Parameters....................................................................................................................................... 5

3. Random Stone Retaining Wall Design Method..................................................................................6

3.1 Calculation of Horizontal Thrust.....................................................................................................7

3.1.1 Horizontal Active Soil Thrust.......................................................................................................8

3.1.2 Horizontal Active Surcharge Thrust..........................................................................................9

3.2 Overturning Analysis............................................................................................................................9

3.3 Sliding Analysis.....................................................................................................................................10

3.4 Eccentricity of Resultant Forces Analysis.................................................................................11

3.5 Allowable Bearing Pressure Analysis.........................................................................................11

4. Random Stone Retaining Wall Calculations......................................................................................13

4.1 Constants................................................................................................................................................. 13

4.2 Geometry of the Wall......................................................................................................................... 13

4.3 Stability Check.......................................................................................................................................14

4.3.1 Combination 1.................................................................................................................................. 15

4.3.1.1 Soil Parameters........................................................................................................................... 15

4.3.1.2 Horizontal Thrust.......................................................................................................................15

4.3.1.3 Overturning Analysis................................................................................................................16

4.3.1.4 Sliding Analysis............................................................................................................................18

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4.3.1.5 Eccentricity of Resultant Forces Analysis........................................................................19

4.3.1.6 Allowable Bearing Pressure Analysis................................................................................19

4.3.2 Combination 2.................................................................................................................................. 21

4.3.2.1 Soil Parameters........................................................................................................................... 21






4.3.3 Seismic Calculation.........................................................................................................................27

4.3.3.1 Soil Parameters........................................................................................................................... 27






5. Appendixes...................................................................................................................................................... 33

5.1 Appendix I – Plan and Sections.....................................................................................................33

5.2 Appendix II – Partial Factors Applied on Actions, Soil Parameters and Resistance

34

5.3 Appendix III – Laboratory Test Results.....................................................................................36

5.3.1 Angle of Friction of Soil................................................................................................................37

5.3.2 Unit Weight of Soil..........................................................................................................................40

5.3.3 Specific Gravity of Rock................................................................................................................45

5.3.4 Field Classification and Description of Rock and Soil.....................................................52

5.4 Appendix IV - Allowable Bearing Pressure Calculation......................................................69

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5.5 Appendix V – Internal Stability Check........................................................................................73

5.6 Appendix VI - Stability Analysis....................................................................................................77

5.7 Appendix VII – Formulas for Seismic Calculations...............................................................81

5.7.1 Horizontal Active Soil Thrust.....................................................................................................81

5.7.2 Horizontal Active Surcharge Thrust.......................................................................................82

5.7.3 Dynamic Active Earth Pressure Coefficient for Soil.........................................................82

5.7.4 Overturning moment.....................................................................................................................83

5.7.5 Resisting Force................................................................................................................................. 84

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Calculation Note for Random Stone

Retaining Walls at South Marjaayoun 2

Embankment Reservoir

1. Introduction

This chapter introduces the stability design required for the random stone walls at South

Marjaayoun 2 embankment reservoir.

The design of random stone walls depends mainly on its massive weight to resist failure

from overturning and sliding forces, ensuring that this weight is uniformly distributed to the

soil beneath the wall to avoid its collapse.

The wall design was done referring to the British Standard BS EN1997-1 (Eurocode 7:

Geotechnical Design – Part I: General Rules), BS EN1998-5 (Eurocode 8: Design of Structures

for Earthquake Resistance – Part V: Foundations, Retaining Structures and Geotechnical

Aspects), the technical specification volume 2.2 section 6 and the final detailed design report

for Conveyor 800 Project, Section 5.9.

Eurocode adopt a common design philosophy based on the use of separate partial

factors rather than global factors of safety; this is a substantial departure from much

traditional geotechnical design practice as embodied in BS Codes such as BS 8004.

2. Computation Parameters

The design of earth retaining walls requires the consideration of the interaction between the

ground and the structure. To perform the design of the earth retaining walls a set of

equilibrium calculations is to be done. Wall properties, loads and soil characteristics are the

main parameters to be taken into consideration in designing the retaining walls.

2.1 Wall Properties

Random stone walls used for retaining the fill will be stone masonry walls comprised of rock

materials packed with mortar joints. According to Technical Specification Volume 2.2 Section

6, each stone shall be of minimum mass 500 kg and minimum dimension 600 mm.

The density of the designed wall was measured to be 2.4 Ton/m3 according to laboratory

test results (Appendix III).

The following figure represents the geometrical parameter of the retaining wall:

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Figure 01 - Geometrical Parameters of Wall

2.2 Loads

The pressures acting on the retaining wall are:

1. Uniform surcharge pressure Q acting on the entire ground surface behind the wall. Q

is assumed to be 0.5 Ton/m2.

2. At rest earth pressure exerted on the back of the wall.

3. Bearing pressure of the soil supporting the retaining wall.

2.3 Soil Parameters

According to laboratory test results (Appendix III), the soil parameters used for the design of

the retaining wall are:

1. Soil density: ɣs = 1.8 Ton/m3.

2. Friction angle of soil: = 38Φ o.

3. Allowable bearing pressure: qa = 35 Ton/m2 (Calculated in Appendix IV).

4. Water density: ɣw = 0 Ton/m3, since the wall is provided with weep

holes (4in PVC Pipes) embedded in each concrete layer and distributed at 2 meter

apart in vertical and horizontal directions, then no water is collected behind the wall.

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3. Random Stone Retaining Wall Design Method

The failure of the retaining wall is a result of the instability of the earth mass, such as

overturning or a rotational failure, a translational failure and bearing failure. Where the

distributing moments on the wall exceed the restoring moments in the rotational failure and

the distributing forces exceed the restoring forces in translational failure.

According to BS EN1997-1, two separate calculations shall, in principle, be performed:

1. Combination 1: A1 “+” M1 “+” R1


Where set A includes the partial factors applied on actions or the effects of actions (Appendix

II, Table A.3), set M includes the partial factors applied for soil parameters (Appendix II,

Table A.4) and set R includes the partial resistance factors (Appendix II, Tables A.13 and

A.14).

Moreover, since the designed wall is located in an area affected by earthquakes, another

calculation must be performed to ensure the functionality of the wall during and after an

earthquake, without suffering significant structural damage. This calculation was done

according to BS EN1998-5.

The two calculations according to BS EN1997-1 are used for static conditions, whereas the

calculation according to BS EN1998-5 is used for seismic conditions.

The forces acting on the wall are mainly due to the soil retained behind the wall along with

the surcharge above it producing a horizontal thrust.

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3.1 Calculation of Horizontal Thrust

The total horizontal thrust exerted on the retaining wall is the summation of horizontal

active soil thrust and horizontal active surcharge thrust as shown in the following figure:

Figure 02 - Horizontal Thrust

Pha=Phs+Phq

Where:

Pha: Horizontal thrust

Phs: Horizontal active soil thrust

Phq: Horizontal active surcharge thrust

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3.1.1 Horizontal Active Soil Thrust

The retained soil will create earth pressure on the retaining wall that is known as the active

soil pressure resulting in a horizontal active soil thrust calculated by the following

equations:

Phs=γGKo γ dH

2

2(For Static Conditi ons)

Where

Phs: Horizontal active soil thrust [Ton/ml]

Ko: At rest earth pressure coefficient for soil Ko=1−sin∅ '

where tan ( ’) = tan ( ) / Φ Φ ɣ ’Φ , is the angle of friction of soil and Φ ɣ ’Φ is the

partial factor for angle of shear resistance (Appendix II, Table A.4)

ɣd: Design soil density = ɣɣ x ɣs [Ton/m3]

ɣɣ: Partial factor for weight density (Appendix II, Table A.4)

ɣs: Soil density [Ton/m3]

ɣG: Partial factor on permanent actions (Appendix II, Table A.3)

H: Height of Wall [m]

Phs=(1+K v )K γ sH 2

2(For SeismicConditions )(Appendix VII )

Where

K: Dynamic active earth pressure coefficient for soil

K=cos2(∅−θ)

cosθ cos(θ+δ)[1+√ sin(∅+δ)sin (∅−θ)cos (θ+δ) ]

2 where tanθ=

K h1−K v

δ : Angle of friction between the soil and the wall = 2/3 Φ

Kh: Horizontal seismic coefficient

Kv: Vertical seismic coefficient = 2/3 Kh

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3.1.2 Horizontal Active Surcharge Thrust

The uniform surcharge pressure acting on the entire ground surface behind the wall exerts a

horizontal active surcharge pressure calculated by the following equations:

Phq=γQK oQH (For Static Condition s)

Where

Phq: Horizontal active surcharge thrust [Ton/ml]

Q: Surcharge Pressure [Ton/m2]

ɣQ: Partial factor on variable actions (Appendix II, Table A.3)

Phq=(1+K v )KQH (For SeismicConditions)(Appendix VII )

3.2 Overturning Analysis

Overturning of the retaining wall is calculated by considering moments about the toe of the

wall (Point O, Figure 02), and shall satisfy the following:

M r>M o

Where:

Mr: Resisting moment per linear meter of wall length [Ton.m/ml]

Mo: Overturning moment per linear meter of wall length [Ton.m/ml]

The resisting moment (Mr) is calculated according to the following equations:

M r=γG×W T×xCG (For Static Conditions)

Where:

WT: Weight of the retaining wall per linear meter of wall length [Ton/ml]

W T=γGW× A

Where ɣGW is the density of the retaining wall [Ton/m3] and A is the cross

sectional area of the wall [m2]

xCG: Lateral distance from WT to the toe of the wall [m]

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M r=W T×xCG (For SeismicConditions)

The overturning moment (Mo) is calculated according to the following equations:

M o=M os+M oq(For Static Conditions)

Where:

Mos: Soil thrust overturning moment per linear meter of wall length [Ton.m/ml]

M os=Phs×(H3 −h) Where h is the inclination height [m]

Moq: Surcharge thrust overturning moment per linear meter of wall length

M oq=Phq×(H2 −h) [Ton.m/ml]

M o=M os+M oq+W T (K v xCG+Kh yCG) (For SeismicConditions )(Appendix VII )

Where:

YCG: Vertical distance from WT to the toe of the wall [m]

The moment due to passive pressure is neglected. The reason is because with a rotation type

failure mode, the wall may not move enough laterally to induce passive earth pressures.

3.3 Sliding Analysis

Sliding of the retaining wall occurs when the driving force (Pha) is greater than the resisting

force (Phr) according to the following equation:

Phr>Pha

Phr=μ ' γGW T (For Static Conditions)

Where:

’:μ Design friction coefficient ¿ tan∅ '

Phr=¿

Where:

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:μ Characteristic friction coefficient ¿ tan∅

The horizontal passive soil thrust is a favorable action since it increases the stability of the

wall against sliding. However it was neglected since there might be excavation behind the

wall.

In some situations, there may be adhesion between the bottom of the retaining wall and the

bearing soil. This adhesion is often neglected because the wall is designed for active

pressures, which is typically developed when there is wall translation. Translation of the

wall will break the adhesive forces between the bottom of the wall and the bearing soil and

therefore adhesion is often neglected.

3.4 Eccentricity of Resultant Forces Analysis

The eccentricity (e) of the resultant forces acting on the base of the wall from center of the

base must fall in the middle one third of the wall base as shown in the following equation:

−B6≤e≤

B6

Where:

B: Width at the base of the wall [m]

e: Eccentricity of resultant forces [m]

e=B2

−x where x is the difference in the opposing moments divided by WT

x=M r−M o

W T[m]

3.5 Allowable Bearing Pressure Analysis

Failure of the soil under the base of the retaining wall occurs when the bearing pressure ( )σ

under the base of the wall is greater than the design allowable bearing pressure of soil (q’a),

or when there is tension at the base.

Hence, the following equations shall be satisfied:

σ max<qa'

σ min>0

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Where:

qa'=qaγR

(For Static Conditions)

Where:

qa: Allowable bearing pressure of soil [Ton/m2]

γR: Partial bearing resistance factor (Appendix II, Table A.13)

qa'=qa(For SeismicConditions)

σ=γGW T

B (1± 6eB )(For Static Conditions)

σ=W T

B (1± 6eB ) (For SeismicConditions )

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4. Random Stone Retaining Wall Calculations

The random stone retaining wall of South Marjaayoun 2 embankment reservoir can be

divided into 5 sections according to retained fill height.

The sections named as A-A, B-B, C-C, D-D and E-E.

4.1 Constants

Density of the retaining wall: γGW = 2.40 Ton/m3

Surcharge Pressure: Q = 0.50 Ton/m2

Soil Density: γs = 1.80 Ton/m3

Angle of Friction of Soil: = 38ϕ ᵒ Allowable Bearing Pressure: qa = 35.00 Ton/m2

4.2 Geometry of the Wall

Referring to Figure 01 and Appendix I, the retaining wall geometry can be summarized in the

following table:

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Section A-A B-B C-C D-D E-E

Height of Wall (H) [m] 7.00 8.00 9.00 10.00 10.50

Height of the Top Section of the Wall (H') [m] 5.68 5.68 5.68 5.68 5.68

Inclination Height (h) [cm] 0.35 0.40 0.50 0.55 0.60

Width at the Middle of the Wall (W) [m] 3.50 3.50 3.50 3.50 3.50

Width at the Base of the Wall (B) [m] 3.48 4.00 5.00 5.50 6.00

Inclination Width (b) [m] 0.70 0.80 0.90 1.00 1.05

Thickness at the Top of the Wall (T) [m] 0.75 0.75 0.75 0.75 0.75

Thickness at the Base of the Wall (t) [m] 0.94 1.00 1.00 1.00 1.00

Lateral Distance from WT to the Toe of the

Wall (xCG) [m]2.43 2.82 3.56 3.92 4.28

Vertical Distance from WT to the Toe of the

Wall (yCG) [m]2.51 2.81 2.98 3.27 3.36

Cross Sectional Area of Wall (A) [m2] 16.10 20.34 26.05 31.53 35.13

4.3 Stability Check

According to BS EN1997 and BS EN1998, three separate calculations shall be performed for

checking the stability of the retaining wall:



3. Seismic Calculation

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4.3.1 Combination 1

4.3.1.1 Soil Parameters

The soil parameters integrated in all the calculations depend mainly on the combination

chosen. For combination 1, set M1, the partial factors for soil parameters are as follows

according to table A.4 (Appendix II):

Weight Density: γ γ=1

Angle of Shear Resistance: γ∅ '=1

And hence the design soil parameters are as follows:

Soil Density: γ d=γ γ×γ s=1 .80 Ton/m3

Angle of Friction of Soil: ∅ '=tan−1 tan∅ '=tan−1 tan∅γ∅ '

=¿38 .00° ¿

Friction Coefficient: μ '=tanϕ '=0 .78 At Rest Earth Pressure

Coefficient for Soil: Ko=1−sin∅'=0.38

4.3.1.2 Horizontal Thrust

The horizontal thrust exerted on the retaining wall is a combination of horizontal active soil

thrust and horizontal active surcharge thrust:

Pha=Phs+Phq

Where:




Where:

Phs=γGKo γ dH

2

2

Knowing that:

ɣG = 1.35 (A lateral earth load is an unfavorable action as it

causes the wall to have a greater tendency to overturn, according to

table A.3, Appendix II)

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Ko = 0.38

ɣd = 1.80 Ton/m3

H = Height of Wall [m]

Phq=γQK oQH

Knowing that:

ɣQ = 1.5 (Surcharge is an unfavorable action as it increases the

tendency of the wall to overturn, according to table A.3, Appendix II)

Q = 0.5 Ton/m2

Substituting the corresponding values of each section in the above equations, the following

table provides the horizontal thrust:


Horizontal Active Soil Thrust

(Phs) [Ton/ml]22.88 29.89 37.82 46.70 51.48

Horizontal Active Surcharge

Thrust (Phq) [Ton/ml]2.02 2.31 2.59 2.88 3.03

Horizontal Thrust (Pha)

[Ton/ml]24.90 32.19 40.42 49.58 54.51

4.3.1.3 Overturning Analysis

Taking into consideration the moments about the toe of the retaining wall (Point O, Figure

02), the overturning moments shall satisfy the following:

M r>M o

Where:

M r=γG×W T×xCG

Knowing that:

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ɣG = 1 (The weight of the wall is a favorable action as

increasing the weight of the wall increases its stability, according to


W T = γGW× A [Ton/ml]

ɣGW = 2.40 Ton/m3

A = Cross sectional area of the wall [m2]

xCG = Lateral distance from WT to the toe of the wall [m]

M o=M os+M oq

Knowing that:

M os = Phs×( H3 −h) [Ton.m/ml]

M oq = Phq×(H2 −h) [Ton.m/ml]

h = Inclination height [m]

The following table represents the values of the resisting and overturning moments

corresponding to the different sections available:


Weight of Wall per Linear Meter

of Wall Length (WT) [Ton/ml]38.64 48.82 62.52 75.67 84.32

Soil Thrust Overturning Moment

per Linear Meter of Wall Length

(Mos) [Ton.m/ml]

45.45 67.80 94.64 130.11 149.46

Surcharge Thrust Overturning

Moment per Linear Meter of

Wall Length (Moq) [Ton.m/ml]

6.36 8.31 10.38 12.84 14.08

Overturning Moment per Linear

Meter of Wall Length (Mo)

[Ton.m/ml]

51.81 76.11 105.02 142.95 163.54

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Resisting Moment per Linear

Meter of Wall Length (Mr)

[Ton.m/ml]

93.88 137.47 222.84 296.54 360.63

As summarized in the above table, the resisting moment is much greater than the

overturning moment for all the sections and thus the geometry of the wall is sufficient for

resisting overturning using combination 1 calculation.

4.3.1.4 Sliding Analysis

The Sliding of the retaining wall occurs when the driving force (Pha) is greater than the

resisting force (Phr) according to the following equation:

Phr>Pha

Where:

Phr=μ ' γGW T

Knowing that:




’μ = 0.78

Substituting the corresponding values in the above equation, the following table is obtained:



[Ton/ml]24.90 32.19 40.42 49.58 54.51

Resisting Sliding Force (Phr)

[Ton/ml]30.19 38.14 48.85 59.12 65.88

The resisting force is greater than the driving force for all the above sections and thus the

geometry of the wall is sufficient for resisting sliding using combination 1 calculation.

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4.3.1.5 Eccentricity of Resultant Forces Analysis

The eccentricity of the resultant force can be calculated as follows:

e=B2

−x [m]

Where:

B = Width at the Base of the Wall [m]

x=M r−M o

W T

The obtained eccentricity results in the following table should satisfy the following

condition:−B6≤e≤

B6


() [m] 1.09 1.26 1.88 2.03 2.34

Eccentricity (e) [m] 0.65 0.74 0.62 0.72 0.66

(B/6) [m] 0.58 0.67 0.83 0.92 1.00

The eccentricity for all the sections almost satisfies the above condition, thus the line of

action of forces falls within the middle third of the base as calculated for combination 1.

4.3.1.6 Allowable Bearing Pressure Analysis

The bearing pressure should satisfy the following conditions:

σ max<qa'

σ min>0

Where:

qa'=qaγR

Knowing that:

ɣR = 1 (According to table A.13, Appendix II)

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qa = 35 Ton/m2

σ=γGW T

B (1± 6eB )Knowing that:

ɣG = 1.35 (The weight of the wall is an unfavorable action as

increasing the weight of the wall increases the bearing pressure,

according to table A.3, Appendix II)

The table below summarizes the obtained results:


Minimum Bearing Pressure

(σmin) [Ton/m2.ml]-1.86 -1.89 4.41 3.98 6.40

Maximum Bearing Pressure

(σmax) [Ton/m2.ml]31.81 34.84 29.35 33.16 31.54

The bearing pressure almost satisfies the above conditions for all the sections and thus the

allowable bearing pressure of the soil beneath the wall is enough for resisting bearing

pressures using combination 1 calculation.

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4.3.2 Combination 2


The soil parameters integrated in all the calculations depend mainly on the combination

chosen. For combination 2, set M2, the partial factors for soil parameters are as follows

according to table A.4 (Appendix II):

Weight Density: γ γ=1

Angle of Shear Resistance: γ∅ '=1.25

And hence the design soil parameters are as follows:

Soil Density: γ d=γ γ×γ s=1 .80 Ton/m3

Angle of Friction of Soil: ∅ '=tan−1 tan∅ '=tan−1 tan∅γ∅ '

=32 .01 °

Friction Coefficient: μ '=tanϕ '=0 .63 At Rest Earth Pressure

Coefficient for Soil: Ko=1−sin∅'=0.47




Pha=Phs+Phq

Where:




Where:

Phs=γGKo γ dH

2

2

Knowing that:

ɣG = 1 (According to table A.3, Appendix II)

Ko = 0.47

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ɣd = 1.80 Ton/m3


Phq=γQK oQH

Knowing that:

ɣQ = 1.3 (According to table A.3, Appendix II)

Q = 0.5 Ton/m2





(Phs) [Ton/ml]20.73 27.07 34.26 42.30 46.63


Thrust (Phq) [Ton/ml]2.14 2.44 2.75 3.05 3.21


[Ton/ml]22.86 29.51 37.01 45.35 49.84




M r>M o

Where:

M r=γG×W T×xCG

Knowing that:



ɣGW = 2.40 Ton/m3

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M o=M os+M oq

Knowing that:

M os = Phs×( H3 −h) [Ton.m/ml]

M oq = Phq×(H2 −h) [Ton.m/ml]









(Mos) [Ton.m/ml]

41.17 61.42 85.72 117.86 135.38




6.74 8.80 11.00 13.60 14.93

Overturning Moment per Linear


[Ton.m/ml]

47.91 70.22 96.73 131.46 150.30



[Ton.m/ml]

93.88 137.47 222.84 296.54 360.63



resisting overturning using combination 2 calculation.

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Phr>Pha

Where:

Phr=μ ' γGW T

Knowing that:


’μ = 0.63




[Ton/ml]22.86 29.51 37.01 45.35 49.84


[Ton/ml]24.15 30.51 39.08 47.29 52.70


geometry of the wall is sufficient for resisting sliding using combination 2 calculation.



e=B2

−x [m]

Where:


x=M r−M o

W T



B6

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() [m] 1.19 1.38 2.02 2.18 2.49

Eccentricity (e) [m] 0.55 0.62 0.48 0.57 0.51

(B/6) [m] 0.58 0.67 0.83 0.92 1.00

The eccentricity for all the sections satisfies the above condition, thus the line of action of

forces falls within the middle third of the base as calculated for combination 2.



σ max<qa'

σ min>0

Where:

qa'=qaγR

Knowing that:

ɣR = 1 (According to table A.13, Appendix II)

qa = 35 Ton/m2

σ=γGW T

B (1± 6eB )Knowing that:


The table below summarizes the obtained results:



(σmin) [Ton/m2.ml]0.55 0.81 5.26 5.23 6.95

Maximum Bearing Pressure 21.63 23.60 19.75 22.29 21.16

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(σmax) [Ton/m2.ml]

The bearing pressure satisfies the above conditions for all the sections and thus the


pressures using combination 2 calculation.

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4.3.3 Seismic Calculation


The soil parameters integrated in all seismic calculations are used by their characteristic

values as described in section 4.1 of this calculation note. Moreover, seismic coefficients will

be used in the below calculations. These coefficients are as follows according to the final

detailed design report for Conveyor 800 Project, Section 5.9:

Horizontal seismic coefficient: Kh=0.13

Vertical seismic coefficient: K v=23Kh=0.09

And hence the remaining soil parameters are as follows:

Characteristic Friction

Coefficient: μ=tanϕ=0.78

Angle of friction between

the soil and the wall: δ=23∅=25.33 °

Angle Used for Computing K: θ=tan−1( Kh1−K v )=8 .10 °

Dynamic Active Earth

Pressure Coefficient for Soil:

K=cos2(∅−θ)


2=0.30




Pha=Phs+Phq

Where:



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Where:

Phs=(1+K v )K γ sH2

2

Knowing that:

Kv = 0.09

K = 0.30

ɣs = 1.80 Ton/m3


Phq=(1+K v )KQH

Knowing that:

Q = 0.5 Ton/m2





(Phs) [Ton/ml]14.55 19.01 24.06 29.70 32.75


Thrust (Phq) [Ton/ml]1.16 1.32 1.49 1.65 1.73


[Ton/ml]15.71 20.33 25.54 31.35 34.48




M r>M o

Where:

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M r=W T×xCG

Knowing that:


ɣGW = 2.40 Ton/m3



M o=M os+M oq+W T (K v xCG+Kh yCG)

Knowing that:

M os = Phs×(H3 −h) [Ton.m/ml]

M oq = Phq×( H2 −h) [Ton.m/ml]


Kh = 0.13

yCG = Vertical distance from WT to the toe of the wall [m]








(Mos) [Ton.m/ml]

28.91 43.13 60.20 82.76 95.07




3.64 4.76 5.94 7.35 8.06

Overturning Moment per Linear 53.30 77.63 109.68 147.98 171.21

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[Ton.m/ml]



[Ton.m/ml]

93.88 137.47 222.84 296.54 360.63



resisting overturning under seismic conditions.




Phr>Pha

Where:

Phr=¿

Knowing that:

μ = 0.78




[Ton/ml]15.71 20.33 25.54 31.35 34.48


[Ton/ml]22.55 28.49 36.49 44.16 49.21


geometry of the wall is sufficient for resisting sliding under seismic conditions.



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e=B2

−x [m]

Where:


x=M r−M o

W T



B6


() [m] 1.05 1.23 1.81 1.96 2.25

Eccentricity (e) [m] 0.69 0.77 0.69 0.79 0.75

(B/6) [m] 0.58 0.67 0.83 0.92 1.00

The eccentricity for all the sections almost satisfies the above condition, thus the line of

action of forces falls within the middle third of the base as calculated under seismic

conditions.



σ max<qa'

σ min>0

Where:

qa'=qa

Knowing that:

qa = 35 Ton/m2

σ=W T

B (1± 6eB )The table below summarizes the obtained results:

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(σmin) [Ton/m2.ml]-2.11 -1.97 2.15 1.95 3.46

Maximum Bearing Pressure

(σmax) [Ton/m2.ml]24.30 26.38 22.86 25.56 24.64

The bearing pressure almost satisfies the above conditions for all the sections and thus the


pressures under seismic conditions.

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5. Appendixes

5.1 Appendix I – Plan and Sections

The following figures show the plan layout and the different sections of the retaining wall:

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5.2 Appendix II – Partial Factors Applied on Actions, Soil Parameters

and Resistance

The following tables represent the partial factors on actions or the effects of actions (Table

A.3), the partial factors for soil parameters (Table A.4), the partial resistance factors for

retaining structures (Table A.13) and the partial resistance factors for slopes and overall

stability (Table A.14).

The tables are presented in Annex A of BS EN1997.

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5.3 Appendix III – Laboratory Test Results

The values of the angle of friction of soil and the unit weight of the retaining wall and of the

retained soil were computed according to tests done in our laboratory.

The test results are listed below:

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5.3.1 Angle of Friction of Soil

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5.3.2 Unit Weight of Soil

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5.3.3 Specific Gravity of Rock

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5.3.4 Field Classification and Description of Rock and Soil

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5.4 Appendix IV - Allowable Bearing Pressure Calculation

The allowable bearing pressure (qa) was calculated according to the following equation of BS

EN1997:

qa=c ' N c bc sc ic+q'N qbq sq iq+0.5 γ dB

'N γ bγ sγ iγ

Since the cohesion of soil (c’) is equal to 0, the equation of qa will be:

qa=q'Nqbq sqiq+0.5 γ dB

' N γbγ sγ iγ

Where:

ɣd: Design soil density = 1.80 Ton/m3

’:φ Design angle of friction of soil = 38˚

q’: Design effective overburden pressure at the level of wall base [Ton/m2]

q '=γ d×t where t is the thickness of buried soil at the wall base

[m]

Nq,ɣ: Bearing capacity factors

Nq=eπ tan∅ ' tan2(45+∅ '

2)

N γ=2(N q−1) tan∅'

bq,ɣ: Inclination of base factors

bq=bγ ¿(1−α tan∅ ')2 where is the inclination of the wallα [m]

sq,ɣ: Shape factors

sq=sγ=1 (For foundations where the length is much greater than the width)

iq,ɣ: Load inclination factors

iq=(1−FHFV

)2

iγ=(1−F HFV

)3

Where FH/FV is the ratio of horizontal forces to vertical forces

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FH=Fhs+Fhq [Ton/ml]

Fhs=K o γdH

2

2[Ton/ml]

Fhq=KoQH [Ton/ml]

Ko: At Rest Earth Pressure Coefficient for Soil = 0.38

Q: Surcharge Pressure = 0.5 Ton/m2

H: Height of Wall [m]

FV=W T=γGW× A [Ton/ml]

WT: Weight of the retaining wall per linear meterof wall length [Ton/ml]

ɣGW: Density of the retaining wall = 2.40 Ton/m3

A: Cross sectional area of the wall [m2]

B’: Effective foundation width [m]

B'=B−2e

B: Width at the base of the wall [m]

e: Eccentricity of resultant forces [m]

e=B2

−x where x=MFr−MFo

W T is the difference in the opposing

moments divided by WT [m]

MFr=FV× xCG [Ton.m/ml]

MFo=Fhs×(H3 −h)+Fhq×( H2 −h) [Ton.m/ml]

xCG: Lateral distance from WT to the toe of the wall [m]

h: Inclination height [m]

Substituting the above values, the following table summarizes the results of the allowable

bearing pressure (qa) for each different wall section:

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Design Effective Overburden Pressure

Design Effective Overburden Pressure (q') [Ton/m2] 1.69 1.8 1.8 1.8 1.8

Horizontal and Vertical Forces

Horizontal Active Soil Force (Fhs) [Ton/ml] 16.95 22.14 28.02 34.59 38.14

Horizontal Active Surcharge Force (Fhq) [Ton/ml] 1.35 1.54 1.73 1.92 2.02

Horizontal Force (FH) [Ton/ml] 18.29 23.68 29.75 36.51 40.15

Vertical Force (FV) [Ton/ml] 38.64 48.82 62.52 75.67 84.32

Factors

Bearing Capacity Factor (Nq) 48.93 48.93 48.93 48.93 48.93

Bearing Capacity Factor (Nɣ) 74.90 74.90 74.90 74.90 74.90

Inclination of Base Factor (bq) 0.85 0.85 0.85 0.85 0.85

Inclination of Base Factor (bɣ) 0.85 0.85 0.85 0.85 0.85

Shape Factor (sq) 1.00 1.00 1.00 1.00 1.00

Shape Factor (sɣ) 1.00 1.00 1.00 1.00 1.00

Load Inclination Factor (iq) 0.28 0.27 0.27 0.27 0.27

Load Inclination Factor (iɣ) 0.15 0.14 0.14 0.14 0.14

Effective Foundation Width

Overturning Moment per Linear Meter of Wall

Length (MFo) [Ton.m/ml]37.91 55.76 77.02 104.94 120.10

Resisting Moment per Linear Meter of Wall Length

(MFr) [Ton.m/ml]93.88 137.47 222.84 296.54 360.63

( ) [m]ẍ 1.45 1.67 2.33 2.53 2.85

Eccentricity (e) [m] 0.29 0.33 0.17 0.22 0.15

Effective Foundation Width (B') [m] 2.90 3.35 4.66 5.06 5.71

Allowable Bearing Pressure

Allowable Bearing Pressure (qa) [Ton/m2] 43.73 46.08 59.10 60.28 67.54

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As a result, a minimum value for the allowable bearing pressure was assumed to be 35

Ton/m2.

Moreover, as recommended by the Engineer, the wall is laid on soil replacement fill with an

average of depth 3.5 meters beneath heel of wall, and thus the allowable bearing pressure is

achieved.

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5.5 Appendix V – Internal Stability Check

Internal stability check should be studied to avoid the failure of the retaining wall and to

ensure that all the stones comprising the wall are acting together. Knowing that the

minimum thickness of the stones used in the construction of the retaining wall is 60cm, it is

enough to study the stability of the retaining wall against sliding across the joints, i.e. every

60cm. Moreover, since the stones in each row are of variable thicknesses, this will create

keys between rows that will produce additional sliding resistance.

Referring to this calculation note, combination 1 has shown to be the most critical regarding

stability against sliding, so the internal stability checks against sliding can be performed only

on this combination.

Considering (i) to be the height from the top of the wall, the resisting force against sliding at

height (i), (Phr(i)), must be greater than the sliding force at the same height, (Pha(i)), according

to the following equation:

Phr (i )>Pha(i)

Where:

Phr (i )=μ ' γGW T (i)

Knowing that:




W T (i) = γGW× A(i) [Ton/ml]

ɣGW = 2.40 Ton/m3

A(i) = Cross sectional area from the top of the wall to height i

from top of wall [m2]

’μ = 0.78

Pha (i)=Phs(i)+Phq (i)

Phs (i)=γGKo γ d i

2

2

Knowing that:

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ɣG = 1.35 (A lateral earth load is an unfavorable action as it

causes the wall to have a greater tendency to overturn, according to


Ko = 0.38

ɣd = 1.80 Ton/m3

Phq (i)=γQKoQi

Knowing that:

ɣQ = 1.5 (Surcharge is an unfavorable action as it increases the

tendency of the wall to overturn, according to table A.3, Appendix II)

Q = 0.5 Ton/m2

The sliding force is independent of the geometry of the section but yet depends on the height

at which the force is calculated. Consequently the sliding force results are the same for all

sections at each height.

Whereas the resisting force depends on the area of section from top of wall to height i, and

for simplicity the calculation will be applied only for the smallest area.

Substituting the corresponding values in the above equations, the following table is

obtained:

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Sections A-A,

B-B, C-C, D-D &

E-E

A(i)

[m2]

WT(i)

[Ton/ml]

Phr(i)

[Ton/ml]

Phs(i)

[Ton/ml]

Phq(i)

[Ton/ml]

Pha(i)

[Ton/ml]

i = 0.6 m 0.59 1.41 1.10 0.17 0.17 0.34

i = 1.2 m 1.31 3.15 2.46 0.67 0.35 1.02

i = 1.8 m 2.21 5.30 4.14 1.51 0.52 2.03

i = 2.4 m 3.27 7.86 6.14 2.69 0.69 3.38

i = 3.0 m 4.51 10.82 8.46 4.20 0.86 5.07

i = 3.6 m 5.92 14.20 11.09 6.05 1.04 7.09

i = 4.2 m 7.49 17.98 14.05 8.24 1.21 9.45

i = 4.8 m 9.24 22.17 17.32 10.76 1.38 12.14

i = 5.4 m 11.15 26.77 20.91 13.62 1.56 15.17

i = 6.0 m 13.22 31.73 24.79 16.81 1.73 18.54

i = 6.6 m 15.31 36.75 28.71 20.34 1.90 22.24

i = 7.2 m 17.71 42.51 33.21 24.21 2.08 26.28

i = 7.8 m 20.59 49.42 38.61 28.41 2.25 30.66

i = 8.4 m 23.59 56.62 44.24 32.95 2.42 35.37

i = 9.0 m 26.89 64.54 50.42 37.82 2.59 40.42

i = 9.6 m 30.49 73.18 57.17 43.04 2.77 45.80

The resisting force is greater than the driving force at all the heights, thus the internal

stability of the wall is sufficient for resisting sliding.

As mentioned in the method statement of the earth embankment reservoirs, the

construction of the wall must be followed by successive backfilling behind the rock layers,

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and thus precautions must be taken into consideration so that the compaction

procedure does not affect the internal stability of the wall. To ensure that, roller must

be kept at a distance “d” from the back of the wall, where d is calculated as per the

following equation (According to Reinforced Concrete Designers Handbook, 10th

Edition, By Reynolds and Steedman):

F s=KoN

d+0.5b

Knowing that:

F s = Sliding Force due to Roller [Ton/ml]

N = Weight of Roller [Ton]

Ko = Active Earth Pressure Coefficient for Soil

b = Width of Roller [m]

The roller to be used during compaction will be of maximum weight of 9 Ton, and a width of

2 m, and thus (N) will be taken as 9 Ton and (b) as 2 m.

According to the above table, the minimum difference between the resisting sliding force

(Phr) and the sliding force (Pha) is found to be 1.44 Ton/ml (Excluding the last layer which

will be compacted using plate compactor only), and thus (Fs) will be taken as 1.44 Ton/ml.

Substituting the values of Fs, Ka, N and b in the above equation we obtain d equal to 1.38m.

Thus by maintaining the compactor at a distance of 1.5m behind the back of the wall, its

internal stability will be ensured during construction phase, where the rest of the material

will be compacted with plate compactor.

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5.6 Appendix VI - Stability Analysis

Stability analysis was performed using the software TALREN 4 in compliance with the

European code.

The factor of safety obtained is 1.80 which is higher than the required factor of 1.5 for

overall stability.

The results of the analysis are shown below:

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5.7 Appendix VII – Formulas for Seismic Calculations

According to BS EN1998-5 (Eurocode 8: Design of Structures for Earthquake Resistance –

Part V: Foundations, Retaining Structures and Geotechnical Aspects), chapter 7, paragraph

7.3.2.2 (Pages below), the seismic action is represented by a set of horizontal and vertical

static forces equal to the product of the gravity forces and a seismic coefficient, where these

forces are considered as acting upward or downward so as to produce the most unfavorable

effect.

The gravity forces acting on the retaining wall are the surcharge (Q) and the weight of the

wall (WT), as a result, the horizontal force representing the seismic action is Kh WT and the

vertical forces are Kv Q and Kv WT.

Knowing that the design of the wall was performed according to combination 1 and 2 and

then checked for seismic calculations, then increasing any of the unfavorable actions and

coefficients used in seismic calculations will not affect the design of the wall.

5.7.1 Horizontal Active Soil Thrust


Part V: Foundations, Retaining Structures and Geotechnical Aspects), the total design force

acting on the retaining structure (Ed) is given by:

Ed=12γ ¿ (1∓K v )K H

2(Defined below in Equation E.1, Annex E)

Where:

H: Wall height [m]

γ ¿: Soil unit weight [Ton/m3] (Defined below in

Equation E.5, Annex E)

Kv: Vertical seismic coefficient

K: Earth pressure coefficient (static + dynamic) (Defined below in section 5.6.3)

Since the horizontal active soil thrust is considered as unfavorable action, then despite the

direction of the vectors of seismic acceleration coefficients, using the formula

Phs=(1+K v )K γ sH2

2

(Defined above in section 3.1.1), will ensure a safe design as the value of the unfavorable

action is being increased.

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5.7.2 Horizontal Active Surcharge Thrust

The vertical force representing the seismic action (Kv Q) is considered acting downward so

as to increase the unfavorable horizontal active surcharge thrust, thus the total surcharge

load is (1+Kv)Q and the horizontal active surcharge thrust defined in section 3.1.2 was

calculated according to the following formula:

Phq=(1+K v )KQH

5.7.3 Dynamic Active Earth Pressure Coefficient for Soil


Part V: Foundations, Retaining Structures and Geotechnical Aspects), Annex E, Equation E.2

(Pages below), the dynamic active earth pressure coefficient for soil may be computed from

the Mononobe and Okabe formula as follows:

K=sin2(ѱ+∅ d

' −θ)

cosθ sin2ѱ sin (ѱ−θ−δ d)[1+√ sin(∅ d' +δ d)sin(∅ d

' −β−θ)sin(ѱ−θ−δ d)sin (ѱ+β) ]

2

Where:

∅ d' : Design value of the angle of shearing resistance of soil

∅ d' =tan−1( tan∅ 'γ∅ ' )

∅ ' : Angle of Shearing Resistance

γ∅ ': Partial Factor for the Angle of Shearing Resistance

δ d: Design Value of the Friction Angle between the soil and the wall

δ d=tan−1( tan δγ∅ ' )

δ : Friction Angle between the Ground and the Retaining Wall

θ: Angle defined as tanθ=Kh1± K v

(Since increasing θ increases the value of K, then

for more safety we used the value of the maximum θ resulting from a

maximum tan θ, and thus we defined tanθ=K h1−K v

)

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Kh: Horizontal seismic coefficient

Kv: Vertical seismic coefficient

Ѱ : Inclination Angle of the Back of the Wall from the Horizontal Line

β: Inclination Angle of the Backfill Surface from the Horizontal Line

Substituting the following values:

∅ '=∅γ∅ '=1}∅ d

' =∅∧δ d=δ

Ѱ=90 ° (In our case Ѱ is slightly greater than 90 , but since increasing ⁰ Ѱ

decreases the value of K, then using a value of Ѱ= 90 will be more safe)⁰

β=0 °

The above formula may be written as:

K=sin2(90 °+∅−θ)

cosθ sin290 ° sin(90°−θ−δ)[1+√ sin (∅+δ)sin (∅−0 °−θ)sin(90°−θ−δ)sin (90 °+0 °) ]

2

Knowing that sin (90 °−x )=cos x, then the latter formula is simplified to:

K=cos2(∅−θ)


2

5.7.4 Overturning moment

The overturning moment is the sum of the moments due to the horizontal active soil thrust,

horizontal active surcharge thrust and horizontal and vertical seismic forces resulting from

the weight of the wall.

Since the overturning moment is unfavorable action, then the moments due to the seismic

forces are taken positive, thus the vertical force (Kv WT) is considered acting downward,

whereas the horizontal force (Kh WT) is considered acting opposing the toe of the wall.

As a result, the overturning moment (Mo) defined in section 3.2 above is:

M o=M os+M oq+W T (K v xCG+Kh yCG)

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5.7.5 Resisting Force

The force resisting sliding is equal to the product of the gravity force due to the weight of the

wall and a friction coefficient for soil. Under seismic state, the weight of the wall produces

additional vertical force (Kv WT) and horizontal force (Kh WT). In this case, the vertical

seismic force was considered acting upward and the horizontal force acting toward the toe

of the wall since the resisting force is a favorable action.

As a result, the resisting force (Phr) defined in section 3.3 above is:

Phr=¿

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retaining wall calculation note

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