ANALYSIS
AND
DESIGN
DR. MOHAMED I. AMER
Professor of Soil Mechanics and Foundations FACULTY OF ENGINEERING CAIRO UNIVERSITY
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
Table of Contents 1.
CHAPTER (1): LATERAL EARTH PRESSURE THEORIES
1.1 General ......................................... ................................................................ .............................................. .............................................. .............................. .......4 1.2 Types Of Earth Pressure........................................... .................................................................. .............................................. .......................5 1.3 Earth Pressure Theories ........................ ............................................... ............................................... .......................................... ..................5
1.3.1 Rankine's Theory ........................ ............................................... .............................................. .............................................. .......................5 1.3.2 Coulomb’s Theory ............................................. .................................................................... ............................................. ......................5 1.4 Assumptions And Analysis Of Rankine's Theory Theory ...................................... ................................................ ..........6
1.4.1 Assumptions.............................................................. ..................................................................................... ...................................... ...............6 1.4.2 Analysis.............................................................. ..................................................................................... ............................................. ......................6 1.5 Active & Passive Shear Planes ......................... ................................................. ................................................ ............................ ....10 1.6 Effect of wall yielding ................................................ ....................................................................... ............................................ .....................11 1.7 Comparison Among The Values of K ’s ’s ........................................... ............................................................... ....................12
1.7.1 Relation Between Wall Movement & K ’s ’s ........................................... .................................................... ......... 12 1.7.2 Earth Pressure In Cohesionless Soil.............................................. .............................................................. ................12 1.7.3 Earth Pressure Distribution ....................................................... ........................................................................... ....................13 1.8 Inclined Cohesionless Ground Surface in Rankine’s Analysis.......................... ..........................14 1.9 Distribution of Earth pressure ............................................. ..................................................................... ................................ ........15
1.9.1 Lateral Effect Of Ground Water On Earth Pressure ..................................... .....................................15 1.9.2 Effect of Surcharge ............................................ ................................................................... ........................................... ....................16 1.9.3 Effect Effec t of Multilayer Soil System............................................. System................................................................... ......................16 1.9.4 Effect Effec t of Inclination of Wall Back ............................................... ................................................................ .................17 1.10 Lateral Earth Pressure Against Rough Walls Coulomb’s Analysis ................17 1.11 Earth Pressure Distribution ................................................. ........................................................................ ............................... ........18
1.11.1 Effect of Ground Water on Earth Pressure ................................................. .................................................18 1.11.2 Effect of Surcharge ........................................... ................................................................. .......................................... ....................19 1.11.3 Effect of Inclined Back & Inclined Backfill Bac kfill .................................... ............................................... ...........19 1.12 Graphical Solution “Wedge Method” ....................................................... ................................................................ .........20
1.12.1 Step by Step Procedure ........................................................ ............................................................................... .......................21 1.13 Solved Problems............................................. ..................................................................... ............................................... ............................... ........22 1.14 Problems ............................................. .................................................................... .............................................. ........................................... ....................29
2.
Chapter (2): Analysis and Design of Retaining Walls
2.1 General .............................................. ..................................................................... .............................................. ............................................... ........................33 2.2 Types of Retaining walls ............................................................ .................................................................................... ............................ ....33
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
Table of Contents 1.
CHAPTER (1): LATERAL EARTH PRESSURE THEORIES
1.1 General ......................................... ................................................................ .............................................. .............................................. .............................. .......4 1.2 Types Of Earth Pressure........................................... .................................................................. .............................................. .......................5 1.3 Earth Pressure Theories ........................ ............................................... ............................................... .......................................... ..................5
1.3.1 Rankine's Theory ........................ ............................................... .............................................. .............................................. .......................5 1.3.2 Coulomb’s Theory ............................................. .................................................................... ............................................. ......................5 1.4 Assumptions And Analysis Of Rankine's Theory Theory ...................................... ................................................ ..........6
1.4.1 Assumptions.............................................................. ..................................................................................... ...................................... ...............6 1.4.2 Analysis.............................................................. ..................................................................................... ............................................. ......................6 1.5 Active & Passive Shear Planes ......................... ................................................. ................................................ ............................ ....10 1.6 Effect of wall yielding ................................................ ....................................................................... ............................................ .....................11 1.7 Comparison Among The Values of K ’s ’s ........................................... ............................................................... ....................12
1.7.1 Relation Between Wall Movement & K ’s ’s ........................................... .................................................... ......... 12 1.7.2 Earth Pressure In Cohesionless Soil.............................................. .............................................................. ................12 1.7.3 Earth Pressure Distribution ....................................................... ........................................................................... ....................13 1.8 Inclined Cohesionless Ground Surface in Rankine’s Analysis.......................... ..........................14 1.9 Distribution of Earth pressure ............................................. ..................................................................... ................................ ........15
1.9.1 Lateral Effect Of Ground Water On Earth Pressure ..................................... .....................................15 1.9.2 Effect of Surcharge ............................................ ................................................................... ........................................... ....................16 1.9.3 Effect Effec t of Multilayer Soil System............................................. System................................................................... ......................16 1.9.4 Effect Effec t of Inclination of Wall Back ............................................... ................................................................ .................17 1.10 Lateral Earth Pressure Against Rough Walls Coulomb’s Analysis ................17 1.11 Earth Pressure Distribution ................................................. ........................................................................ ............................... ........18
1.11.1 Effect of Ground Water on Earth Pressure ................................................. .................................................18 1.11.2 Effect of Surcharge ........................................... ................................................................. .......................................... ....................19 1.11.3 Effect of Inclined Back & Inclined Backfill Bac kfill .................................... ............................................... ...........19 1.12 Graphical Solution “Wedge Method” ....................................................... ................................................................ .........20
1.12.1 Step by Step Procedure ........................................................ ............................................................................... .......................21 1.13 Solved Problems............................................. ..................................................................... ............................................... ............................... ........22 1.14 Problems ............................................. .................................................................... .............................................. ........................................... ....................29
2.
Chapter (2): Analysis and Design of Retaining Walls
2.1 General .............................................. ..................................................................... .............................................. ............................................... ........................33 2.2 Types of Retaining walls ............................................................ .................................................................................... ............................ ....33
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
2.3 Design Procedure of Retaining Walls .............................................. .................................................................. ....................34 2.4 Proportioning of Retaining walls ......................... ................................................. ............................................... ......................... 34
2.4.1 Gravity Retaining walls ........................................... .................................................................. ..................................... ..............34 2.4.2 Cantilever Type Retaining wall .............................................. .................................................................... ......................36 2.5 Overall Stability of Retaining walls ............................................. .................................................................... .......................... ...37
2.5.1 Stability of Gravity Wall ......................... ................................................ .............................................. ............................... ........37 2.5.1.1 Check of stresses at critical cri tical sections s ections of gravity walls ........................... ................................ .....39 2.5.2 Stability of Cantilever Wall .................................................................... .......................................................................... ......40 2.5.3 Design of Critical Criti cal Sections Sec tions of Cantilever Walls Wall s .................................... ........................................... .......41 2.6
Solved Problems............................................. ..................................................................... ............................................... ............................... ........43
2.7 Mechanically Stabilized Earth Walls (MSE) .................................................. ....................................................... .....55
2.6.1 General ............................................ ................................................................... .............................................. ....................................... ................55 2.8 Problems ............................................. .................................................................... .............................................. ............................................. ......................62
3.
Chapter (3): Design of Sheetpile Walls
3.1 General .............................................. ..................................................................... .............................................. ............................................... ........................66 3.2 Advantages of Using Sheetpiles ............................................ ................................................................... ................................. ..........67 3.3 Types of Sheetpile Materials.............................................. ..................................................................... .................................... .............67 3.4 Statical Systems of Sheetpiles ............................................ ................................................................... .................................... .............68 3.5 Design of Cantilever Sheetpile Wall.......................................................... Wall...................................................................... ............69
3.5.1 Cantilever Sheetpile In Sand .............................................. ..................................................................... ........................... ....70 3.5.2 Cantilever sheetpile in clay cla y ............................................. .................................................................... ............................. ......73 3.6 Design of Anchored Sheetpile Wall ............................................ .................................................................... .......................... ..75
3.6.1 Condition of Free Earth Support ............................................. ................................................................... ......................76 3.6.2 Condition of Fixed Earth Support ............................................ ................................................................ ....................79 3.7 Anchors............................................ ................................................................... .............................................. ................................................ .........................82
3.7.1 Design of Anchor Plates ........................................... .................................................................. ................................... ............83 3.7.2 Length of Anchor Rod ............................................................. ................................................................................. ....................84 3.7.3 Check Of Stability Stabilit y of Anchoring at Lower Failure Plane:............................ ............................85 3.8 BRACED CUTS .............................................. ...................................................................... ............................................... ............................... ........86
3.8.1 Lateral Earth pressure ............................................ .................................................................. ....................................... .................86 3.8.3 Stability of Braced Cuts ................................................. ........................................................................ ............................... ........91 3.9 Solved Problems .............................................. ...................................................................... ............................................... ............................... ........96 3.10 Problems ............................................. .................................................................... .............................................. ......................................... ..................101
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
CHAPTER (1) LATERAL EARTH PRESSURE THEORIES
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
CHAPTER (1) LATERAL EARTH PRESSURE 1.1 GENERAL It is one of the most important applications of using shear strength parameters in the analysis of soil-structure interaction. Figure (1.1) presents examples of that applications and they are: 1. Basement walls. 2. Retaining walls. 3. Sheetpile walls. 4. Tunnels. 5. Sewage & water pipelines. 6. Pile foundations.
(2)
(1)
(5)
(4)
(3)
(6)
Fig. (1.1): Applications of retaining walls-soil-foundations interaction
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
1.2 TYPES OF EARTH PRESSURE - At-rest earth pressure When the earth retaining structure has no chance to move. ex. pile foundations - Active earth pressure When the earth retaining structure has the tendency to move away from backfill. ex. retaining walls - Passive earth pressure When the earth retaining structure has the tendency to move towards the backfill. ex. bridge abutments
1.3 EARTH PRESSURE THEORIES 1.3.1 Rankine's Theory
It deals with equilibrium of a soil element due to lateral movement of soil, see Fig. (1.2).
1
3
3
1
Fig. (1.2): Stresses on a soil element due to lateral movement 1.3.2 Coulomb’s Theory
It deals with the equilibrium of a wedge of soil which is about to fail, see Fig. (1.3).
W
E R
Fig. (1.3): Equilibrium of a soil wedge due to lateral movement
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
Both theories are dealing with the state of stresses in the soil element or soil mass just at the limit of equilibrium. Therefore, it called limiting equilibrium (or plastic equilibrium) analysis.
1.4 ASSUMPTIONS AND ANALYSIS OF R ANKINE'S THEORY 1.4.1 Assumptions 1. Equilibrium & analysis are for soil element. 2. The back of the retaining wall is smooth (i.e. no friction between the wall & backfill). 3. The back of the retaining wall is vertical. 1.4.2 Analysis
Assume two-dimensional element of soil is stressed to failure under principal stresses
σ1 &
σ3. Mohr-Coulomb failure criterion presents the state of stresses at the failure plane as shown in Fig. (1.4).
(a)
(b)
Fig. (1.4): Failure mechanism of two dimensional element a) Principal stresses on a soil element, b) Mohr-Columb failure criterion
The shear strength of the soil element is given by: s = c + σ tan φ ……………………………………….……
(1)
While the normal & shear stresses on the plane of failure are ( σ, τ) and given by:
σ = and, τ =
σ1 + σ3 σ1 − σ3 cos 2α ………………………………… + 2
σ1 − σ3 2
2
sin 2α …………………………..……………
(2) (3)
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
at failure τ is equal to s so, eqn (1) = eqn (3)
∴
σ1 − σ3 2
sin 2α = c +
{ σ +2 σ 1
3
+
σ1 − σ3 2
}
cos 2 α tan φ ……………………………….… (4)
solving using the trigonometric relations sin 2 = 2 sin cos 2 & cos 2 = 2 cos - 1
resulting in :
σ1 = σ 3 +
α
c + σ 3 tan φ sin α cos α − cos 2
α tan φ
……………………….… (5)
is the angle of inclination of the plane of failure. It happen when the stresses on the
plane of failure is in its limits, i.e. when the value of
σ1 cannot be increased any further.
In other words when : d dα
(sin α cos α – cos2α tan φ) = 0
cos2α – sin2α + 2 tan φ sin α cos α = 0 from which
α = 45 + ˚
φ
……………………….
(6)
2
Since the stresses are symmetrical with respect to the major and minor principal axes, we get two sets of failure planes making angles of (±
α)
with the major principal plane, see
Fig. (1.5).
Fig. (1.5): Failure planes of two
dimensional element
Assume an element of soil at a depth “h” from ground surface. The element is subjected to a vertical stress (σy) and a horizontal stress (σx). If the wall a-a is stand still in its position, there is no shear stress developed on the vertical and horizontal planes on the element. So, those planes are
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
principal planes and (σx) & (σy) are principal stresses. These stresses can be presented by Mohr’s Circle No. (1) and this case called AT-R EST CONDITION. If the frictionless wall a-a is allowed to move away from the soil mass, the value of the deformation continues,
σx soon
σx decreases. If
reaches a minimum value at which plastic equilibrium is
attained in the soil. Any additional movement of the wall away from the soil mass will result in failure of the soil. This failure is called ACTIVE FAILURE. a
y
h
x
a
(a)
Circle No. (3) Passive Failure
Circle No. (1)
Circle No. (2)
At-rest cond.
Active Failure
α = 45°+
φ 2
α = 45° -
C
X
y
X
φ 2
X
Max
Min
(b) Fig. (1.6): Mechanism of coefficients of earth pressure a) principal stresses on a soil element at depth h, b) Mohr-Columb failure criterion
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
Since failure is obtained by reducing
σx, it must be, then, the minor principal stress ( σ3), while σy is
the major principal strews (σ1).
The relation between σ1 and σ3 at failure is given by equation no.(7) as :
σ 1
Substituting
Or,
c + σ 3 tan φ
= σ 3 +
α = 45 +
sin α cos α − cos 2 α tan φ
φ 2
into eqn (7), get:
σ3 =σ1 tan2(45° -
φ
σX =σy tan2(45° -
φ
2
2
) – 2c tan (45 ° ) – 2c tan (45 ° -
1 − sin φ
In other words
1 + sin φ
σh
=
………………….....……(7)
− 2c
φ 2
φ 2
)
) ………………….…. (8)
1 − sin φ 1 + sin φ
σv
……………….…. (9)
Where: 1 − sin φ 1 + sin φ
= tan2(45° -
φ 2
) = constant = K
Where constant K is called the coefficient of lateral earth pressure In the case that is represented by circle No (2) in Fig (1.6) it is called the coefficient of active earth pressure, K a i.e.
and
K a =
σh
1 − sin φ 1 + sin φ
=
σv K a – 2c
K a
…………….…..…. (10)
If the wall is moved against the soil, so that, the soil is compressed laterally, the horizontal
pressure (σx) is increased until reaches its maximum value (σ1) just
before
failure and still it is a case of plastic equilibrium. Any further movement will result in a soil failure. This case is represented by circle No. (3) in Fig (1.7) and the failure is called PASSIVE FAILURE.
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
Equation (8) becomes:
σX =σy tan2(45° + tan2(45° +
φ 2
φ 2
) – 2c tan (45 ° +
1 + sin φ
)=
1 − sin φ
φ 2
) ………………….… (11)
= K p
and is called the coefficient of passive earth pressure =
1 + sin φ 1 − sin φ
And eq. (11) becomes
σh
=
σv K p + 2c
K p
………………….. (12)
Defination of at–rest condition is represented by circle No. (1) in Fig (1.6). The earth pressure, in this case, is called
AT – REST EARTH PRESSURE.
And the coefficient of lateral earth pressure, in this
case, is denoted by (K 0) and is called At–Rest earth pressure coefficient. Since the At Rest condition is not a failure condition, K 0 cannot be calculated from plastic theory. K 0 is determined experimentally in the laboratory with the aid of Poisson’s ratio. A widely spread
formula of K 0 is: K 0 = 1- sin φ ……………………………. (13)
1.5 ACTIVE & PASSIVE SHEAR PLANES
Fig. (1.7): Active and passive shear planes
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
1.6 EFFECT OF WALL YIELDING
Fig. (1.7): Rotation of Frictionless Wall about the Bottom a) Active failure b) Passive failure
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
1.7 COMPARISON AMONG THE VALUES OF K ’s 1.7.1 Relation Between Wall Movement & K ’s K
9 8 7 6
K p
5 4 3 2
K 0
1 K a
Away from wall
-20
-15
-10
-5
0
5
10
15
Towards the wall
20
Fig. (1.8): Relation between Wall Movement and K ’s
1.7.2 Earth Pressure In Cohesionless Soil
Referring to eqn. (14):
σh
σv K a – 2c
=
Ka …………………….. (14)
For cohesionless soil c = 0
∴ σh
=
σv K a
for active pressure
σh
=
σv K p
for passive pressure
At any depth inside the soil mass
h
v
σv = γ h h
Therefore:
σh = γ h K a
in case of active
σh = γ h K p
in case of passive
In other words:
σe = γ h K a
= active earth pressure
σ p = γ h K p
= passive earth pressure
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
1.7.3 Earth Pressure Distribution
-2c tan (45°-
φ 2
)
Rankine’s active state
2c tan (45°-
φ 2
)
Rankine’s passive state
Fig. (1.9): Distribution of earth pressure
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
1.8 INCLINED COHESIONLESS GROUND SURFACE IN R ANKINE ’S ANALYSIS
Fig. (1.10): Earth pressure direction in Rankine’s analysis
With inclined ground the Rankine’s method considers
static
equilibrium
of
an
element
at
a depth y. The soil weight acts vertically, and the lateral earth pressure is conjugate to the weight as in Fig (1.10). Thus, the lateral earth pressure acts parallel to the ground surface. Note that the Rankine’s method assumes a frictionless wall, therefore, the stresses on the vertical face of the element are principal stresses. Rankine made an analytical solution of this case to obtain. Active pressure: K a = cos β
cos β −
cos 2 β − cos 2
φ cos β + cos 2 β − cos 2 φ cos β +
cos 2
Passive pressure : K p = cos β
β − cos 2 φ cos β − cos 2 β − cos 2 φ
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
1.9 DISTRIBUTION OF EARTH PRESSURE ea = γh K a
γ, φ,
Ea =
(K a)
= γh 2 K a 2
Ea
h dh
ea Figure (1.11): Distribution of earth pressure
1.9.1 Lateral Effect Of Ground Water On Earth Pressure
h1
γ, φ,
E1
c=0
e1
=
E2
h2
Ew e2
ew
ew
e2
Fig. (1.12): Effect of Ground Water on Earth Pressure
- Pressures e1 e2
- Forces = γh1 K a = (Σ γh) K a = ( γh1 + γ\h2) K a
E1 =
1 2
e1
e1 h1
+ e2
= γh1 K a + γ\h2 K a
Ew
= e1 + γ\h2 K a
Et = E1 + E2 + Ew
ew =
γwh2
e − e1
h2 = e1h2 + 2 2 2 1 1 = e h = γ h2 2 w 2 2 w 2
E2 =
h2
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
1.9.2 Effect of Surcharge
q = (t/m2)
e1
= q K a
e2
= (γh) K a
e3
= (q+γh) K a
1 , E1
h
E1 = e1 h
E2
E2 = 2 e1
Et
e2 e3
1 2
e2h
= E1 + E2
Figure (1.13): Effect of surcharge on distribution of earth pressure
1.9.3 Effect of Multilayer Soil System
e1
=
h
e2
= γ1 h1
e3
= e2 + γ2 h2
e4
= ( γ1 h1 γ2 h2
e5
= e4 γ3 h3
γ1, φ1
h1 1
e1
e2
γ2, φ2
h2 2
e4
e3
h3 3
e5
Fig. (1.14): Effect of multi layer soil system on earth pressure
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
1.9.4 Effect of Inclination of Wall Back
e E
= γhK a 1 = eh 2 =
h
E
R
W
W R
+
E
e Fig. (1.15): Effect of inclination of wall back on earth pressure
1.10 LATERAL EARTH PRESSURE AGAINST R OUGH WALLS COULOMB’S ANALYSIS Coulomb (1776) developed a method of determining lateral pressure that includes the effect of friction between the soil and the wall, see Fig. (1.16). A plane failure surface is assumed, and the lateral force required to maintain equilibrium is calculated. The procedure is repeated for several trial failure surfaces, and the one producing the critical force is selected. This general analysis procedure is a type of limiting equilibrium method. The method readily accomo a geometrically irregular backfill and sloping wall. i
Fig. (1.16): Lateral earth pressure against rough walls Coulomb’s analysis
δ
β
δ = angle of wall friction
sin 2 (β + φ)
K a =
K p =
⎡ sin(φ + δ)sin(φ − i) ⎤ sin 2 βsin(β − δ) ⎢1 + ⎥ sin(β − δ)sin(β + i) ⎥⎦ ⎢⎣ sin 2 (β + φ)
⎡ sin(φ + δ)sin(φ + i) ⎤ sin 2 βsin(β + δ) ⎢1 − ⎥ sin(β − δ)sin(β + i) ⎥⎦ ⎢⎣
2
2
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
1.11 EARTH PRESSURE DISTRIBUTION
δ
= angle of wall friction
φ
= e
=
γ h K a
E
=
eh
K a =
from Coulomb’s formula
h
l l a w h g u o r
δ
e
Fig. (1.17): Distribution of earth pressure, rough wall
1.11.1 Effect of Ground Water on Earth Pressure
e1
= γ h1 K a
e2
= e1 + γsup h2 K a
ew = γw h2
h1
δ
1
e1
h2
2
δ e2
ew
Fig. (1.18): Effect of Ground Water on Distribution of Earth Pressure, Rough Wall
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
1.11.2 Effect of Surcharge
e1 1
e1
= q K a
e2
= (q + γ h) K a
E1 = e1 h E2 = Et
q (t/m2)
(e2 – e1) h
E1
l l a w h g u o r
h
= E1 + E2
E2
δ
2
e2
Fig. (1.19): Distribution of earth pressure including surcharge load, rough wall
1.11.3 Effect of Inclined Back & Inclined Backfill i
e = γ h K a 1 E= eh 2
h
E
δ
δ e
Fig. (1.20): Effect of inclined back on distribution of earth pressure, rough wall
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
1.12 GRAPHICAL SOLUTION “WEDGE METHOD” Coulomb deals with the equilibrium of the wedge of soil as shown: Active Case – cohesionless soil
Ea Ea Datum B i
B
Ea
R
W
δ
φ
E
Free body diagram
Force polygon
Fig. (1.21): Graphical solution using wedge method
(Coulomb’s analysis)
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
1.12.1 Step by Step Procedure 1. Assume a trial plane failure surface (A-B) 2. Draw a free-body diagram of the assumed failure mass. 3. Draw a force polygon of the forces acting on the free-body diagram:
•
Weight
W
magnitude & direction are known.
•
Resultant
R
Direction is known
•
Active force
Ea
Direction is known
4. The magnitude of Ea required to close the force polygene is determined and represents
the lateral force that would be required to prevent failure along the assumed failure plane. 5. Steps 1 to 4 are repeated for other failure surfaces, the surface that yields the maximum
lateral force in the critical failure surface, and this force is the active lateral force.
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
1.13 SOLVED PROBLEMS Problem (1): For the retaining wall shown in Fig. (1). determine the magnitude, direction,
and point of application of the resultant force acting on it.
Fig. (1)
Solution
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
Problem (2): Draw the lateral earth pressure distribution on the wall shown in Fig. (2) and
determine the magnitude, direction, and point of application of the resultant force acting on it.
Fig. (2) Solution
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
Problem (3): Determine the total active earth pressure forces acting on the gravity type
retaining wall shown in Fig. (3).
Fig. (3)
Solution
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
Problem (4): For the same wall of problem (3) determine the intensity of uniformly
distributed surcharge placed on G.S. and extending from the wall back, that increases
the
earth
pressure
by
corresponding to case of problem (3).
Solution
only
65
percent
of
its
intial
value
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
1.14 PROBLEMS 1)
a- Using neat sketches and giving examples from practice, explain the lateral earth pressure in at rest, active and passive states. b- For the retaining walls shown in figure (1), draw the active lateral earth pressure diagrams. i- Neglecting the wall roughness. ii- Considering the wall roughness assuming Soil conditions:
φ = 2/3 φ
φ = 33° γ (above G.W.T) = 1.6 t/m 3 γ sat = 1.9 t/m3
Fig. (1)
2)
a- State the assumptions of Rankine’s theory of earth pressure then derive Rankine’s formula for earth pressure in case of cohesionless soil mass. b- calculate and plot the lateral earth pressure distribution on the wall shown in fig. (2) for the following backfill conditions.
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.1 Lateral Earth Pressure Theories
2
layer
c (kg/cm )
I
0.2
20
0.0
1.85
II
0.0
30
0.0
1.82
III
0.0
40
0.0
1.90
(cw,
γ (t/m
3
)
Assume the soil above G.W.T in layer II to be fully saturated.
Fig. (2)
3) For the smooth reataining wall shown in Fig. (3), determaine the earth pressure force acting on
the wall. If during a rain storm the tension cracks are considered to be filed with water, compute the percentage increase in the horizontal load acting on the wall due to the development of hydrostatic pressure.
Fig. (3)
4) Determine the resultant active earth pressure force acting on the wall shown in Fig. (4).
Fig. (4)
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
CHAPTER (2) ANLYSIS AND DESIGN OF R ETAINING WALLS
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
CHAPTER (2) ANALYSIS & DESIGN OF R ETAINING WALLS 2.1 GENERAL Retaining walls are structures that built for the purpose of supporting a vertical or nearly vertical backfill which, in turn, may support vertical loads. They may, also, be used to retain water or other materials.
2.2 TYPES OF R ETAINING WALLS There are many types of retaining walls. The most common types are seen in (Fig. 2.1): I) Gravity type
II) Semi – gravity type
III) Cantilever type
IV) Counterforted walls
(i) Gravity R.W.
(iv) Semi-Gravity R.W.
(iii) R.C Cantiliver R.W.
Fig. (2.1): Types of retaining walls
(ii) Counterforted R.W.
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
2.3 DESIGN PROCEDURE OF R ETAINING WALLS Design of retaining walls, generally, follows the following steps: 1. Analyzing the subsoil condition, 2. Establishment of surcharge loads, railways, highways … etc, 3. Select the type and tentative proportion of the retaining wall, 4. Compute earth pressure and other lateral pressures. Subsequently, calculate lateral forces, 5. Compute vertical forces, 6. Check the stability of the wall i) Sliding stability, ii) Overturning stability, iii) Check of bearing capacity and stresses on soil, 7. Design of structural elements, 8. Select drainage system in the wall or backfill, 9. Predict settlement and movement of wall, and 10. Eventually, draw concrete and reinforcement detailing.
2.4 PROPORTIONING OF R ETAINING WALLS 2.4.1 Gravity Retaining walls
Gravity walls are generally trapezoidal or with broken back to reduce the quantity of concrete. The base and other dimensions are to be chosen such that the resultant falls within the middle one third of the base (see Fig. 2.2). Because of the massive proportions of the wall, the resulting stresses in concrete are small, and, consequently, low strength concrete may generally be used for the wall construction. Critical sections will occur along the stem and/or through the toe.
50 cm h/12 2cm
100 cm
Fig (2.2): Empirical dimensions of gravity walls
Stem
h D/2 – D Heel Toe
D = h/8 – h/6
B = 0.50 - 0.80 h
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
The active earth pressure is to be calculated using either Coulomb or Rankine’s method. If the heel projection is small (10 – 15 cm), Coulomb method of analysis may be used for evaluating lateral pressure directly on the back of the wall. Rankine’s Solution may also be used on a vertical section through the heel. Figure 2.3 (a, b) presents a brief explanation for Coulomb's and Rankine’s analysis.
a) Coulomb’s Analysis
b) Rankine’s Analysis i
Eav
i
Ea
Eav Ea
i Eah
Ws W
W
e
e
X
X B
B
V= W + Eav
V= W + Ws + Eav
Eav
=
Ea sin (90- β +δ)
Eav
=
Ea sin i
Eah
=
Ea cos (90- β + δ)
Eah
=
Ea cos i
Fig (2.3): Pressures and forces on gravity walls
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
2.4.2 Cantilever Type Retaining wall 20-30 cm
h
D
a D = h/12 – h/10 B/3
B = 0.40 - 0.70 h
Fig (2.4): Proportioning of cantilever wall
The tentative dimensions shown above in Figure (2.4) are considered adequate in the absence of other data. These dimensions should be checked for structural and stability requirements. The top of the stem should not be less than (20 – 30 cm), so that proper pouring of concrete is possible. The base of the stem should be thick enough to resist excessive outward deformation of the stem and to satisfy the shear requirements without the use of shear reinforcement. The base slab dimensions are to be chosen in such a way to allow the resultant of the loads to fall within its middle third. Generally, the base with (B = a + b) is taken equal to (0.4 -0.7h). There are many combinations of “a” and “b” that provide successful solution. The location of the resultant force, however, is not sensitive to the toe distance “a” and, therefore, it is preferable to have a minimum value of “a” that satisfies the condition of allowable soil pressure at the toe and sliding stability of the walls as well.
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
2.5 OVERALL STABILITY OF R ETAINING WALLS 2.5.1 Stability of Gravity Wall
Referring to Fig. (2.3), the following steps are to be followed in order to assess the stability of Gravity type retaining wall. a) Compute and locate all forces and weights acting on the wall, than b) Check stability of the wall to satisfy the sliding condition:
F.S. =
F.S. =
……………… (1)
should be
≥ 1.5 if passive forces are neglected, and ≥ 2.0 if passive forces are considered
Sliding underneath the base should be considered in the resisting forces. In cohesionless soil: Fφ = ΣW’s tan δ
δ In cohesive soil:
=⅔φ
………………. (2) ………………. (3)
Fc = cw * B
………………. (4)
cw = ⅔ c
………………. (5)
where:
δ
: angle of wall friction
cw: adhesion between soil and wall c) Check the overturning stability of the wall by taking the moment of the
overturning and the stability forces about the toe of the wall. F.S. =
……………… (6)
Where Mr : resisting moment about point “o”. Mo: overturning moment about point “o”. F.S = should
≥ 1.5 in case of cohesionless soil. ≥ 2.0 in case of cohesive soil.
O
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
d) Check of Soil Pressure Under the wall base.
W3
Ea W5
W1
W2
Ep O
f 2 f 1
Fig (2.5): Loads that used in calculation of soil pressure under wall base
Referring to Fig. (2.5)
•
ΣW’s = N
•
ΣMo
•
eccentricity (e) =
•
f 1,2
=
=M
N
(1 ±
6e
)
B B ……………………………………………. (7) ƒ1 should not exceed qall ƒ2 should not be negative
If ƒ2 is negative, this means that the resultant lies outside the middle third. In this case the pressure on the soil is to be redistributed as following:
=
f on
soil
2 N 3e
&e ≥
B
6
≤
q all
……………………………………………… (8)
……………………………………………… (9)
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
2.5.1.1 Check of stresses at critical sections of gravity walls
a
a
b
b d
e
c
d
c
e
Fig (2.6): Critical sections to be checked due to lateral pressures
Selected sections are chosen as shown in fig (2.6). Normal and shear forces are computed at each section. Then, compressive, tensile and shear stresses are calculated at the critical section and compared with the allowable values for plain concrete. If are not safe, bigger dimensions of the wall are to be used.
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
2.5.2 Stability of Cantilever Wall
Same principles & procedures are followed in the stability analysis of cantilever retaining wall. It follows the following steps: a) Calculate weights, forces, and pressures acting on the retaining wall as presented
in Fig. (2.7). i
Eav
Ws
Ea
Eah W2
E p
W1 f 1
f 2
Fig (2.7): Loads & pressures on cantilever wall b) Sliding, Overturning and Stresses on soil are to be calculated and checked as
previously shown for gravity walls.
Note: keys may be made to develop passive pressure to increase sliding stability.
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
2.5.3 Design of Critical Sections of Cantilever Walls
Following the stability analysis of the retaining wall, the bearing pressure distribution is also, plotted. Then, the sections shown in fig (2.8) are designed as reinforced concrete sections. 1. Calculate the straining actions acting on each section as a result of the acting stresses & forces
as shown in Fig. 2.8. 2. Design each section for concrete dimensions and required reinforcement as shown in Fig. 2.9. 3. Draw a neat sketch for detailing the concrete & reinforcement as shown in Fig. 2.10.
E (3)
(2) (1)
(1)
(3)
(2)
qheel qtoe
Fig. (2.8): Sections to be designed and detailed
stresses from soil and concrete
weight of (3) concrete
2
W (1)
(2)
qheel
(1) Sec (1) – (1): Stem
Sec (2) – (2): Heel
(3) Sec (3) – (3): Toe
Fig. (2.9): Critical reinforced concrete sections at stem, heel, and toe to be designed
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
5 φ 12/
Secondary Rft ≥5φ12/
Main Rft. 5 φ 12/
5 φ 12/
5
12/
Fig. (2.10): Concrete dimensions and reinforcement details of cantliver wall
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
2.6 SOLVED PROBLEMS Problem (1): Check the overall and structural stability of the retaining wall shown in Fig. (1).
Fig. (1)
Solution
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
Problem (2): a- Discuss the realationship between the eccentricity of the resultant force
acting on the base of a retaining wall and the stress distribution under the reataining wall.
Solution
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
Problem (2): b- For the gravity retaining wall shown in Fig. (2):
i) Determine
the
minimum
distance
(x)
which
satisfies
the following conditions: - F.S. (overturning) > 2.0. - F.S. (sliding) > 1.5 ii) For the value of (x) determined in (i), find the normal stresses underneath the retaining wall.
Fig. (2)
Solution
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
Problem (3): Make a complete design for the cantilever retaining wall shown in figure (3).
The minimum factor of safety against sliding is required to be greater than or equal to 1.8.
Fig. (3)
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
Solution
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
2.7 MECHANICALLY STABILIZED EARTH WALLS (MSE) 2.6.1 General
Modern reinforced soil technology was developed in France by H. Vidal type 1960. This system is called reinforced earth and is shown in Fig. (2.11). Steel strips are used to reduce the earth pressure against the wall face. The design and construction of Vidal type reinforced earth walls are now well established, and many
thousands have been successfully built throughout the
world in the last decade. Other similar reinforcing systems have, also, been developed using steel bar meshes, grids, and gabions. The use of geotextiles as r einforcing elements started in the early 1970,s because of their resistance of possible corrosion over metalic reinforcement. Systems using sheets of geosynthetics rather than steel strips are shown in Fig. (2.12).
Fig. (2-11): Components of a reinforced earth wall
Fig. (2.12): Typical Geogrids Used as Soil Reinforcements
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
Applications
1. Tunnels and bridges
2. Abutment
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
3. Tunnels
4. Abutment
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
5. Miscellaneous
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
6. Reataining wall
7. Building
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
8. Construction of MSE walls
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
9. MSE Components
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
2.8 PROBLEMS 1)
a- Mention four practical uses of retaining walls. b- For the gravity type retaining wall shown in Fig. (1), make the necessary checks for the stability of the wall and for the internal stresses that develop along section 1-1. Suggest practical solution(s) if the stability of the wall is not completely fulfilled.
Fig. (1)
2)
a- Differntiate between gravity-type retaining walls and cantilever retaining walls with respect to material, stability, and dimensions. b- A gravity-type concrete retaining wall with a 6.0 m high vertical back is shown in Fig. (2) Data :
Layer (I) :
Unit weight
= 1.6 t/m3,
φ = 30° ,
c=0
Layer (1I):
Unit weight
= 1.9 t/m3,
φ = 10° ,
c = 0.32 kg/cm 2
Layer (III):
Dry unit weight = 1.6 t/m 3,
Saturated unit weight = 1.85 t/m3,
φ
E = 275 kg/cm 2
= 33° ,
qall = 2 kg/cm2
c = 0,
µ = 0.4
required : i) Check the stability of the wall.
ii) Determine the angle of tilt of the wall neglecting the passive resistance.
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
Fig. (2)
3)
a- Mention, using clear sketches, four practical solutions to avoid unsatisfactory stability against sliding in case of cantilever retaining walls. b- Make a complete design for the cantilever retaining wall shown in Fig. (3) which retains a dry sand fill. Data :
φ = 33° ,
γd= 1.6 t/m3,
Coefficient of base friction = 0.5
qall = 1.50 kg/cm 2 F.S. against sliding = 1.8
Fig. (3)
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.2 Analysis and Design of Retaining Walls
4) Make a complete design for the cantilever retaining wall shown in Fig. (4).
Fig. (4)
5)
a- Discuss the effect of the following on the stability of cantilever retaining walls; i) Existence of toe in front of the wall. ii) Increasing the heel length behind the wall.
b- For the retaining wall designed in Problem 3(b), if on the top of the ground runs a double railway line that can be substituted by a uniform surcharge of 5 t/m 2 extending over the heel zone only as shown in Fig. (5). Discuss, without calculations, its effect on the stability and design of the retaining wall.
Fig. (5)
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
CHAPTER (3) SHEETPILE WALLS & STRUTTED EXCAVATION (DESIGN METHODS)
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
CHAPTER (3) SHEETPILE WALLS & STRUTTED EXCAVATION (DESIGN METHODS) 3.1 GENERAL Sheetpile walls consist of a series of sheetpiles driven side by side into the ground forming a continuous vertical wall to retain an earth bank (see Fig. 3.1). They are often used for waterfront structures that may range from small pleasure-boat launching facilities to large dock facilities. Also, it is used to support the sides of deep excavations such as basements, tunnels, and utilities.…etc.
Anchor rod
sheetpile
Anchor wall (plate)
sheetpile
dredge level driven depth
Cantilever sheetpile
driven depth
Anchored sheetpile
Strut wale
From this analysis
sheetpile sheetpile
Fill dry side
Braced cuts (excavation) Strutted sheetpiles Fig. (3.1): Uses of sheetpile walls
wet side
Cofferdam
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.2 ADVANTAGES OF USING SHEETPILES 1. In contrast to the construction of other types of retaining wall, the building of sheetpile
walls does not usually require dewatering of the site. 2. They can be used as temporary structures. 3. They may be reusable. 4. Steel sheetpiles have light weights, resist high driving stresses and their length can be
increased.
3.3 TYPES OF SHEETPILE MATERIALS The most commonly used sheetpiles are 1. Wooden (timber) sheetpiles. 2. Precost concrete sheetpiles, and 3. Steel sheetpiles.
These types are shown in Fig. (3.2)
Fig (3.2): Various types of sheetpiles
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.4 STATICAL SYSTEMS OF SHEETPILES From the point of view of the statical system, sheetpiles can be classified into two types as following: 1- Cantilever sheetpiles. 2- Anchored sheetpiles. 2-1 free earth support. 2-2 fixed earth support.
The following sections illustrate the principles of the design of each of the previous types of the sheetpiles.
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.5 DESIGN OF CANTILEVER SHEETPILE WALL Cantilever sheetpile walls are recommended for walls of moderate height-about 10 ms or less measured above the dredge line. In these walls the sheetpiles act as a wide cantilever beam above the dredge line. The design and stability of the wall depend on the adequate embedded penetration depth into the soil below the dredge line in order to resist the lateral earth pressure developed by the backfill. Fig (3.3) shows the behavior and the statical system of this type of the wall.
P Dredge
Pp D Insufficient penetration
Bigger penetration but still insufficient
D
Sufficient enetration
Pp
Penetration De th
(a) Earth Movement
P
P
P
P
Pp D
D Elastic Line
D
Exact Earth Pressure
P
P`
Simplified Earth Pressure
t
F D Idealized Earth Pressure
(b) Resisting lateral loads Fig (3.3): Behavior and statical system of cantilever sheetpile walls
to
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
Consider a sheetpile wall loaded with a horizontal concentrated load P as shown in Fig (3.3-b). In order to fulfill the condition of equilibrium, a certain movement must be permitted by the wall. The movement of the wall should be in a way such that a point of rotation D is developed. The pressure at the point D is zero. It is too much simplification if an idealized pressure distribution is considered in the calculations, i.e. the passive pressure P p is replaced by a concentrated load F. It is found that the ratio t / to = 1.2. However this ratio is 1.1 in case of earth and/or water pressure instead of the load P. The max bending moment was found to be almost identical in both cases of loading. Fig (3.4) shows the earth pressure and bending moment diagram in case of the sheetpile penetrating a sand deposite. 3.5.1 Cantilever Sheetpile In Sand
The following is the analysis of the pressures, forces and bending moment on the sheetpile wall. Analysis will be relating to Fig (4).
L1
e1
z L
L2
Sand , , c=0 sat , , c=0
Pa La e2
u t
x
o Pp e3
s D
xs
F
Fig (3.4): Earth pressure and bending moment diagram
e1 = γ L1K a e2 = (γ L1 + γ′L2) K a
γ′ = effective unit weight = γsat - γw
at point o: enet = e p – ea = 0
γ L1K a + γ′(L2 + u) K a = γ′u K p γ L1K a + γ′L2K a + γ′u K a - γ′u K p = 0 e2 - γ′u (K p - K a) = 0 u=
e
2
γ ′( K p − K a )
the slope of the line e2-o-e3 is 1: γ′(K p – K a )
∴ ∴
e3 = xγ′ (K p-K a) P p =
x2
2
γ′ (K p-K a)
Mmax.
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
in order to have the system in equilibrium; ΣMD=0 x Pa (La + u + x) = P p 3 P p (La + u + x) = x 3 =
γ′(K p – K a)
6 P p ( L a
γ ′( K p
+ u + x) − K a )
x3 6
solve for x then: to = u + x and
t = 1.25 to
The maximum bending moment happens at the point of zero shear (s). This point is found to be below the point of zero pressure (o) with distance x 3 . The point of zero shear has the forces Pa = P p , i.e. the net force = 0. i.e.
γ′
xs = Mmax = Ms = Pa (La + u + x s) -
γ′
x s2
2
x s2
2
(K p – K a) = Pa 2Pa
γ ′( K p − K a
(K p – K a)
)
xs
3
Allowable flexural stress of steel sheetpile = σall Section modulus required per unit length of the structure = Z Z – modulus
≡ section modulus =
M max
σall
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.5.1.1 Summary
- Step-By-Step Procedure (cantilever sheetpile in sand) 1.
Calculate K a, K p, K r = (K p – K a)
using Rankine’s approach
2.
Calculate e1, e2
3.
Determine the depth u
4.
Calculate Pa, La
5.
Calculate e3, P p → in terms of x
6.
Take moment about D → to determine x
7.
t = 1.25 (u + x)
8.
Determine point of zero shear → xs
9.
Calculate the maximum moment at point of zero shear Ms
10. Calculate the section modulus Z = Ms / σall 11. Choose the suitable section, using appendix “A”
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.5.2 Cantilever sheetpile in clay
L1
e1
z Sand ,
L
L2
e3 to
e2
La
Pp c,
D
F
e1 = γ L1 K a e2 = (γL1 + γ′L2) K a The net pressure at the dredge line: at any distance z > L1 + L2 active pressure = [ γ L1 + γ′ L2 +
γ′c (z – L1 – L2)] K a – 2c
in case φ = 0 → K a = 1 passive pressure = γ′c (z – L1 – L2) K p + 2c
K a
K p
in case φ = 0 → K p = 1
∴ the net pressure = passive press – active pressure e3 = 2c + γ′c (z – L1 – L2) + – γL1 – γ′L2 – γ′c (z – L1 – L2) + 2c = 4c – ( γL1 + γ′L2) in case of equilibrium MD = 0 Pa (La + to) = P p
to 2
= [4c – ( γL1 + γ′L2)]
t o2 2
solve for to and, t = 1.5 t o - From equilibrium in the horizontal direction ΣFH = 0 obtain the value of F. - Draw the shearing force diagram. - Calculate the max bending moment at the point of zero shear. M max - Section modulus Z = .
σall
3.5.2.1 Summary
c=0
Pa
Fig (3.5): Earth pressure diagram
t
, ,
Clay ′c , c,
=0
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
Step-By-Step Procedure (cantilever sheetpile in clay) 1.
Calculate K a
2.
Determine e1, e2
3.
Calculate Pa, La
4.
Calculate e3
5.
Calculate P p
6.
Take the moment about D → get to
7.
t = 1.5 t o
8.
Σ FH = o → get F
9.
Draw S.F.D. & B.M.D.
10. Calculate Zreq =
as a function of to
M max
σall
11. Choose the suitable section from Appendix “A”
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.6 DESIGN OF ANCHORED SHEETPILE WALL When the height of the backfill material behind a cantilever sheetpile wall exceeds about 10m, it becomes more economical to tie the sheetpile wall near the top to anchor plates, anchor walls, or anchor piles. This is referred to as anchored sheet piling or anchored bulkhead. Anchors minimize the lateral deflection and, in turn, the bending moment and the depth of penetration of the sheetpiles. There are two basic methods of analyzing the anchored sheetpile walls: (i) the free earth support method & (ii) the fixed earth support method
Anchor Tie Rod
Anchor Tie Rod
deflection
deflection Moment
dredge line
Moment dredge line
(a) free-earth support
(b) fixed-earth support
Fig (3.6): Basic methods of analyzing the anchored sheetpile walls
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.6.1 Condition of Free Earth Support
This condition takes place in case of: 1. the driven depth is relatively short. 2. the soil is relatively compressible, or 3. the sheetpile is relatively stiff.
i) In sand e1 = γ L1 K a L
e2
P p = γ′
L24 2
3
l2
e1
Pa
z
ez
La
e2 L3 to
(K p – K a)
L4
∑Mo′ = 0
2
L2
γ ′( K p − K a )
e3 = γ′ L4 (K p – K a)
P p(
o` z
e2 = (γL1 + γ′ L2) K a L3 =
l1
F
L1
Pp e3
Fig (3.7): Earth pressure distribution in
L4 + L3 + L2 + l2)
sand, (free earth support)
– Pa (L2 + l2 – La) = 0
γ ′L24 2 2
(
3
3
L4 + L3 + L2 – l2) – Pa (L2 + l2 – La) = 0 → f ( L 4 ) → L4
to = L3 + L4 Actual depth (t) = 1.25 theoretical depth (t o) ∑ Horizontal load = 0 F + P p – Pa = 0 → determine F (force/unit length) It was found that the maximum bending moment occurs at point z which located at a depth z from the ground surface such that L1 < z < (L1 + L2)
∴ point z is the point of zero shear ∴
1 2
e1 L1 + e1 (z – L1) +
γ′ 2
(z – L1)2 K a = F → determine z
maximum moment = M max = M z Section modulus =
M max
σall
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
ii) In Clay L1
e1, e2 → are as before
O`
e3 = net pressure
l2
e1
z
= passive pressure – active pressure = 2c + ( γ ′2
l1
F
L
L2
Pa
to) × 1
z
– [γ1L1 + γ 1′ L2 +
γ ′2
e3
to) – 2c]
= 4c – ( γ1L1 + γ1′ L2)
to
for static equilibrium
ez
La
e2 Clay c, 2
Pp
e3
Fig (3.8)
∑ M′o = 0 Pa (L2 + l2 – La) = P p (
to 2
+ L2 + l2)
Pa (L2 + l2 – La) = to × e3 (
to 2
+ L2 + l2)
Solve for to → P p = to (e3) ∑ Horizontal forces = 0 F = Pa – P p → get F
Max. moment occurs at a distance3907hsd z at which L 1 < z < (L1 + L2) from ground surface.
z is the joint of zero shear at which: F=
1 2
e1 L1 + e1 (z - L1) +
1 2
γ1′
(z – L1)2 K a → solve for z
M z is the max bending moment, and Section modulus =
Mz
σall
Sand , 1
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.6.1.1 Summary
- Step-by-step design procedure (Free-earth support) In sand
In clay
1. Calculate K a , K p
1. Calculate K a
2. Calculate e1 , e2
2. Calculate e1 , e2
3. Calculate L3
3. Calculate e3
4. Determine Pa , La
4.
5. Determine P p in terms of L 4
5. ∑ FH = 0 → get F
6. Take moment about o′ → get L4
6. Forces = 0 at z (P.O.Z. shear) → get z
7. ∑ FH = 0 → get F
7. Mmax = M z
8. Force = 0 at z → get z
8. Choose sec. having modulus =
9. Max. moment = M z
9. Design the anchorage
10. Choose the section according to M max section modulus =
σall
11. Design the anchorage
∑ M′o = 0 → get to
M max
σall
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.6.2 Condition of Fixed Earth Support
When using the fixed earth support method, one assumes that the toe of the sheetpile is restrained from rotation. This condition takes place if: 1. The piling is driven to a sufficient depth. 2. The soil under the dredge line is strong enough to assure the condition of fixation, or 3. The piling is relatively flexible.
A
L1
F
O`
L
L2
l2
e1
z
e z
z
l1
Pa La
e2
L3 L4
t
e3
I
L5
H H` B
BMD Fig (3.9): Earth pressure distribution in sand, (fixed earth support)
e1 = γ L1 K a e2 = e1 + γ′ L2 K a L3 =
e2
γ ′( K p − K a
)
According to φ → get
e3 =
e 2 (L 3
− L5 )
L3
L5 L1
+ L2
→ get L5
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
Divide the pressure diagram into two parts at point I. Internal force P 1 will appear. Neglecting the part of the diagram below point H, its resultant P 2 shall be applied at H.
For the upper part of the diagram A-I: Taking moment about
z
o′
P1 (L5 + L2 + l2) = Pa (L2 – La + l2) Solve for P1 → get F = P a – P1
L5 1
For the lower part of the diagram I-H: F1 = F2 =
1 2 1 2
L3
e3 (L3 – L5) e4 L4 =
1 2
γ′ L24 (K p – K a)
Taking moment about H
∴P1 (L3 – L5 + L4) + F1 (
2 3
(L3 – L5) + L4) = F2
L4
Fig (3.10): Analysis of sheetpile wall, (fixed earth upport)
3
solve for L4 t = 1.25 (L 3 + L4) maximum bending moment will occur in the upper part (A-I), z is the point of zero shear at a depth z from ground surface. At z : F = Max moment
1 2
e1 L1 + e1(z – L1) + = M z
Section modulus =
M max
σall
1 2
γ′ (z – L1)2 K a → Solve for z
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.6.2.1 Summary
-Step-by-step Procedure: Fixed Earth Support (Equivalent beam solution) 1. Estimate K a, K p. 2. Calculate e1 , e2. 3. Calculate L3. 4. From Fig. (3.11), according to φ → get L5. 5. Calculate e3. 6. Divide the pressure diagram into two parts at the point of inflection (I). 7. For the upper part
∑ M′o = 0 → get P1.
8. For the upper part ∑ FH = 0 → get F. 9. For the lower part ∑ MH = 0 → get L4. 10. T = 1.25 (L 3 + L4). 11. Determine the point of zero shear at the upper part. 12. Calculate Mmax , 13. Section modulus =
M max
σall
Fig (3.11)
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.7 ANCHORS The general types of anchor used in sheetpile walls presented in Fig. 3.12 and they are as follows: 1. Anchor plates and beams (dead man). 2. Tie backs. 3. Vertical anchor sheetpiles. 4. Anchor beams supported by batter piles (compression & tension).
wale
Anchor rod
Anchor plate
Tie rod or Cable (strand) grout
2. Tie back
1. Anchor plate
Tie rod
Tie rod
Anchor beam beam
sheetpile
compression pile
3. Sheetpile
4. Pile Fig (3.12): Types of anchors used in sheetpile wall
tension pile
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.7.1 Design of Anchor Plates Plates a) As a continuous wall
Force acting on the plate are:
d
i. Tension in the tie rod F / unit length.
F
ii. Active earth thrust.
h
iii. Passive earth thrust.
Assuming the depth d is known for equilibrium of the wall;
ep
The net thrust ( → ) = the force F ( ← ) at depth “d”.
en
Fig (3.13)
At depth “d”:
ea = γ d K a e p = γ d
K p
en = γ d (
n K p
n
n is a factor of safety = 1.5 - K a)
net thrust = γ d h ( F =
γ d h (
K p
n
K p
- K a)
n
- K a) F
h= γd(
K p
n
− K a
F = [force]/unit length
)
Maximum bending moment = M I-I K p h h MI-I = γd ( - K a) * * n 2 4 F*h = 8 where F is the force/unit length b) As isolated wall
The proper spacing between anchor rods = 2.5 → 4 ms The force in the rod = F (force/unit length) × spacing At depth d;
the net pressure = γd (
K p
1.5
- K a)
assuming square plate of dimensions h
Fig (3.14) h
×
ea
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
at equilibrium
γd (
K p
n
- K a) h2 = F F
h= γd (
K p
n
− K a
F = [force]
)
3.7.2 Length of Anchor Anchor Rod
C
A
E IV
I
H
II III
safe
F
partly safe
D
t
Fig (15) 45+( /2)
B Zone I: wedge ABC
is define by the plane surface of rupture behind the sheetpile. This wedge will move causing the active earth pressure acting on the wall. If the anchor wall is constructed in this Zone it will move with the sheetpile and consequantly gives no resistance. Zone II
is the wedge defined by the plane of the natural slope and the plane of rupture which is kept in place through the existance of the sheetpile wall. The resistance of the anchor plate in this zone will be considerably reduced. Zone III
if the anchor plate is constructed in this zone the passive rupture surface CF will interfere with the active rupture surface CB and resistance of the anchor plate will be reduced. Zone IV
is the safe zone in which the anchor plate possess full capacity.
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.7.3 Check Of Stability of Anchoring at Lower Failure Plane:
A
C
E
P
FEXIST D W PA
R A
W2
R
45+( /2)
Fig (16)
B
The soil wedge ABE with dead weight W 1 and rupture surface BE loads the sheet piling and supports itself on the anchoring section BECD with force R a. The anchoring section BECD is defined by the rupture surface BE and a failure plane extends from the lower edge of the anchor plate to the lower edge of the sheetpile. This section is being held in place by the reaction R. The anchoring is stable when the average force F acting on the anchoring section BECD in the direction of the anchor is greater than the actual or existing anchor force that results from the calculation of the sheetpile. F.S. =
F possible Fexist
≥ 1 .5
If this is not the case, the anchor should be lengthened. FORCE
MAGNITUDE
W1 + W2
known
known
Pa
known
known
P
known
known
R
unknown
known
F
unknown
known
From this analysis
From sheet pit analysis
F possible
DIRECTION
Fig (17)
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.8 BRACED CUTS Sometimes construction work requires ground excavation with vertical or near-vertical force; for example, basements of building or underground transportation facilities at shallow depth below the ground surface. The vertical forces of cuts need to be protected by temporary bracing systems to avoid failure that may be accompanied by considerable settlement or by bearing capacity failure of nearby foundations. Strut wale sheetpile
strut sheetpile
1
wale
1
Section Plan 1-1
Elevation
Fig (18)
3.8.1 Lateral Earth pressure
a) Retaining wall
b) Braced cut
Fig (19)
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
The nature of yielding of braced cuts is different than that of the retaining wall. Therefore, the expected earth pressure diagram developed on the wall of the braced cut is much different from that applied on a retaining wall.
3.8.1.1 One-Layer System: Peck (1969)
0.25H
0.25H
0.50H
H 0.75H
ea
0.25H
ea
ea = 0.65γHK a
ea
ea = γH-4c or = 0.3 γH
ea = 0.2γH → 0.4γH ≅ 0.3γH
which ever higher a) Moist or Dry Sand
b) Soft to Medium Clay
c) Stiff Clay
(γh/c)>4
(γh/c) ≤ 4
Fig (20)
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.8.1.2 Two-Layer System: Peck (1943)
If layers of both sand and clay are encountered, they can be replaced by one clayey layer has an equivalent value of cohesion (φ = 0 concept);
σeq =
1 2H
2
[ γs Hs K s tan φs + Hc n qu] Where:
H
= total height of the cut
γs
= unit weight of sand Hs
Sand
Hs = height of sand layer
s
H
K s = coefficient of lateral earth pressure of sand layer ( ≈ 1)
φs
Clay c
Hc
= angle of friction of sand
Hc = height of clay layer n = coefficient
of
progressive
failure (ranges from 0.5 → 1;
Fig (21)
average value 0.75) qu = unconfined compression strength of clay Also, an average unit weight γav is expressed as follows:
γav
=
1 H
[γs Hs + γc Hc]
where γc is the saturated unit weight of clay layer Once the equivalent values of cohesion and unit weight are determined, the pressure envelopes in clay can be used to design the cuts.
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.8.1.3 multi-layer system are encountered in the cut, the average undrained
1
Clay
H2
cohesion can be expressed as:
cav =
Clay
H1
In a similar manner, when a number of clay layers
2
H
1
[c1 H1 + c2 H2 + … + cn Hn] H and the average unit weight can be determined from: 1 γav = [γ1 H1 + γ 2 H2 + … + γ n Hn] H then, the previous procedure will be applied.
Clay
Hn
n
Fig (22) 3.8.2 Design of various components of braced cuts (A) Struts
Struts are actually horizontal columns subject to bending. The load carrying capacity of columns l will depend on the, slenderness ratio . The slenderness ratio can be reduced by providing vertical r and horizontal supports at intermediate points. In case of braced cuts in clayey soils the depth of the first strut below ground surface should be less than the depth of tensil cracks Zc. 2c
Zc =
γ
or
Zc =
K a
(a)
(c)
(b)
Fig (23)
2c
γ
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
The force in struts are calculated s follow: 1. Draw the pressure envelope for the cut with the proposed positions of the struts are
indicated. 2. Divide the vertical cut into two simple cantilevers and a simple beam 3. Solve each individual beam and cantilever to determine the reaction A, B 1, C1, C2,
D1, and D. 4. Strut loads are:
PA = A(s) PB = (B1 + B2) (s) PC = (C1 + C2) (s) PD = D (s) Where s is the horizontal spacing between struts 5.
Choose the proper section from the steel manual.
(B) Sheetpiles 1. For each section (beam and cantilever) shown in fig (23.C) determine the maximum bending
moment. 2. Determine the maximum value of the maximum bending moment (M max) obtained in
step 1. 3. Obtain section modulus
Z=
M max
σall
4. Choose the proper section. (C) Wales 1. Wales can be treated as continuous horizontal members if they are spliced properly. 2. Conservatively they may be treated as simple beams pinned at the struts.
at level A
Mmax =
at level B
Mmax =
at level C
Mmax =
at level D
Mmax =
A(s)2 8 (B1 + B2 )s 2 8 (C1 + C2 )s 2 8 D(s) 2
3. Determine the section modulus for each wale M max Z=
σall
4. Choose proper section from steel manual.
8
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.8.3 Stability of Braced Cuts 3.8.3.1 Heave of the bottom of the cut in clay
Braced cuts in clay may become unstable as a result of the heaving of the bottom of the excavation. L
i) Terzaghi’s analysis: (1943)
Load/unit length of the cut causing the bottom heave = Q Q = B1 Hγ - cH where, B1 = 0.7 B c = cohesion The load Q can be treated as a load per unit length on a continuous foundation at the level bd (and a f) having a width of B1 = 0.7B.
Fig (24)
According to Terzghi’s bearing capacity theory, the net ultimate load-carrying capacity per unit length of this foundation can be given by: Qu = cNc B1 in case of
∴ F.S. =
, Nc = bearing capacity factor
φ = 0 → Nc = 5.7
Qu = 5.7 c B 1 Qu Q
=
5.7cB1 H(B1 γ − c)
=
5.7c c H(γ ) 0.7B
= F.S = 1.25 → 1.5
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
ii) Bjerrum and Eide analysis: (1956)
They proposed the following formula: FS =
cN c
γH
The bearing capacity factor Nc varies with the ratio of H/B and L/B (where L= Length of the cut). Nc can be estimated from Fig. (25) (at H/B & L/B)
c
Fig (25)
Variation of Nc with L/B and H/B Based on Bjerrum and Eid’s Rquation
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.8.3.2 Stability of the bottom of the cut in sand
Sheetpiles are sometimes driven for excavations that need dewatering. In such cases, the factor of safely against piping should be checked. “Piping is another term for failure by heave” Failure of the bottom (piping) may occur when high D
hydraulic gradient is set up as a result of the flow of water into the excavation. The factor of safety should be checked by drawing
h
flow nets and determining the maximum exit gradient [i
max (exit)]
that will occur at points A and B
Fig (26) i max (exit) =
h/Nd
A
B
L1 L2
a L3
where,
Flow of water
a ≡ length of the flow element at point A (or B) Fig (27)
Impervious Layer
Nd ≡ number of head drops in Fig (27) = 9 drops Fig (26)
The factor of safety against piping i cr ≥ 1.5 F.S. = i max(exit ) where, icr = critical hydraulic gradient = i cr =
−1 e +1
Gs
Gs = specific gravity of the solid particles of the soil e = void ratio of the soil Generally, critical hydraulic gradient for most soils ranges between 0.9 and 1.1 and usually taken as 1.0.
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
D
h
9
1
a 8 7
5
6
2 3 4
Impervious Layer Fig (27)
i) Harr (1962) method of analysis Referring to fig (26) For L3 = ∞ , knowing L 1, L2 The maximum exit hydraulic gradient can be determined from figures (28), (29) as follows: 1. Determine the modulus, m, by obtaining 2L 2/B (or B/2L2) and 2L1/B. 2. With the known modulus and 2L 1/B, determine L2*i max (exit)/h. 3. Having L2 & h, i
max (exit) can
be calculated.
4. The factor of safely against piping =
i cr
i
max (exit)
=
1 i
max (exit)
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
Fig (28): Variation of modulus (from Groundwater and Seepage, by M. E. Harr. Copyright © 1962 by McGraw-Hill. Used with the permission of the McGraw-Hill Book company)
t s i x e . x a m i * 2 L
Fig (29): Variation of maximum exit gradient with modulus (from Groundwater and Seepage, by M. E. Harr, Copyright © 1962 by McGraw-Hill. Used with the permission of the McGraw-Hill Book company)
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.9 SOLVED PROBLEMS
Problem (1): Fig. (1) shows a profile of the original ground surface abc where an anchored
sheetpile wall will be constructed. After driving the sheetpiles and before fixing the anchorage, the wall will be backfilled by dredged sand (
φ
= 32°,
γ
= 1.6 t/m3). i- Find the maximum height of sand fill behind the sheetpile wall that satisfies
the statical requirements for stability of the wall without any temporary supports and before fixing the anchorage. ii- For this height of sand, determin the maximum bending moment in the
sheetpile wall. iii- Make a complete deign of the anchored sid support system.
Fig. (1)
Solution
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
Problem (2): For the anchored sheetpile wall shown in Fig. (2), no embedment is possible
into the hard rock base. In order to provide stability for this wall, a compacted fill is to be placed in front of the wall. Two soils are available for such fill with the following properties.
3
3
Soil
(t/m )
Cost of 1 m of soil including transportation and compaction to the required density (L.E.)
A
33°
1.7
4
B
36°
1.8
5
Find which soil is more economical to be used in this project
Fig. (2)
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
Solution
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
Earth Reataining Structures Prof. Mohamed I. Amer
Ch.3 Design of Sheetpile Walls
3.10 PROBLEMS 1)
a- Sketch four cases in which sheet pile walls are used. b- Make a complete design of the cantilever sheet pile wall shown in Fig. (1).
Fig. (1)
2) a- What are the different materials used in sheet pile walls? What are the advantages of
steel sheet piles over other materials? b- Redesign the sheet pile wall shown in Fig. (1) if the groundwater table rises behind the wall to 3.0 m below the ground surface and is maintained at the excavation level in front of the wall. 3) a- Sketch the elastic lines and bending moment diagrams for the maximum height of
backfill that can be safely retained behind it. b- For the sheep pile wall shown in Fig. (2), find the maximum height of backfill that can be safely retained behind it.
Fig. (2)