Steel Design Report

“

NEW WORDS Reinforced Earth A composite material having facing panels, soil reinforcement strips and select fill as basic components.

Highway Embankment A raised earthen structure used for increasing the level of pavement from the ground due to a number of factors.

Study and Design of Reinforced Earth Retaining Wall with steel Reinforcement for Highway Embankment TABLE OF CONTENTS Why highway embankment is needed to be strengthened?_____________________________________________3 What is done to strengthen the embankment?_________3 Which method is the best?_______________________________4 What is a Reinforced Earth?______________________________5 Introduction to reinforced soil structure________________6 The Reinforced Earth Retaining Wall_____________________6 Mechanism Of Reinforced Soil____________________________7 Shear Box Analogy Concept______________________________8 What can be used as a Reinforcement?________________10 1._________________________________________________Geosynthetics ________________________________________________________________10 2._____________________________________________Reinforcing strips ________________________________________________________________10

Stability of Retaining Walls______________________________11 Metallic Strip Reinforcement____________________________12 APPENDIX A (Review of Shear Strength)_______________22 APPENDIX B (Rankine’s Theory of Lateral earth Pressure)_________________________________________________26

Table Of Figures Figure a External stabilization...........................................6 Figure b Internally stabilized reinforced soil.......................6 Figure c (a) Element of unreinforced soil, (b) Element of reinforced soil............................................................7 Figure d Slopes showing failure surface.............................8 Figure e shear box test on unreinforced and reinforced soil...............................10

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Figure Figure Figure Figure Figure

f External stability checks (after Transportation Research Board, 1995)...11 g Steel reinforced earth retaining wall...................................................12 h Analysis of a Reinforced Earth retaining Wall.......................................14 i Understanding the equation of Lateral pressure due to Surcharge.........15 j Calculation for Overturning..................................................................20

PART 1

Introduction

Why highway embankment is needed to be strengthened? A highway embankment is a structure (usually made up of earth in Pakistan) which has a sufficient height to separate the pavement built over it from the ground level. The reasons why it is necessary to make it strong are given below: •

It has to bear the forces of water during the flood.

•

It has to bear huge loading of traffic with large number of repetitions.

•

To avoid the Land sliding.

•

Over-burden

•

Dead and Live Load Surcharge

•

Earth Pressure

•

Hydrostatic Pressure

•

Seismic Loads

•

Construction Loads etc…..

What is done to strengthen the embankment? There are a large number of solutions to the problem of strengthening an earthen embankment. These include: •

Giving slopes on both sides of embankment when embankment height is sufficiently high.

• •

Ability of Plant Roots to Strengthen Soil Influence of Short Polymeric Fibers on Crack Development in Clays

•

Recycled Plastic Soil Nails Provide Slope Stabilization Project

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• • • • •

In-situ densification of soils. Ground improvement and modification. Reinforced soil. Grouting. Grading and other soil improvement methods

Sheet piling Sheet piling may be composed of steel, timber or concrete piles, with each pile being linked to the next to form a continuous wall. Sheet pile walls are sufficiently watertight for most practical purposes. Grouting covers different injection techniques of special liquid or slurry materials called grouts into the ground for the purpose of improving the soil or rock. Muckshift can be described as the excavation and exchange of unsuitable soil regions by more qualified ones. It is a kind of large scale land clearance. Sandbagging Sandbags can be used for preventing a leach ate discharge downstream of the embankment site. Geosynthetical structures Rock facing / rock riprap slope surface consists of rock or cobble fills, no special slope surface treatment is necessary. Downstream slopes with outer sand and gravel should be protected against erosion especially during flood.

Which method is the best? Most of these methods work in different problems in different situations and therefore none can be said as best. Another reason for this is that other than reinforcing the earth and making a composite material called reinforced earth, all the above mentioned methods require having slopes of embankment for an economical design. There can be a situation where one wants to have an embankment with angle of repose 90 degrees. This can be achieved by employing the reinforced earth concept for the embankment.

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PART 2

Reinforced Earth What is a Reinforced Earth? A Reinforced Soil System (RSS) is a composite material which has the following basic components; •

Facing Panel (Commonly made of concrete, steel plate, wire mesh, block etc…)

•

Soil Reinforcement Strips (Galvanized steel, geotextiles, etc)

•

Select Fill (Cohesionless soil meeting specific defined requirements)

The frictional forces created when combining the select fill with the flexible metallic or non-metallic reinforcing strips result in a robust structural material, commonly known as Reinforced Earth. The strips are attached to a front facing panel, which may be manufactured from concrete or steel. The facing material selected is generally dependent on it having sufficient durability to accommodate the design life of the structure, and also meet the aesthetic needs of the project. The Reinforced Earth monolithic mass acts cohesively and supports it’s own weight and any applied loads which may include all the forces as described in part 1 of this report. The forces induced in the steel strips can be precisely calculated and depend on ;•

Strip geometry

•

Strip frictional characteristics

•

Vertical soil pressure on the strip

•

Strength and stiffness characteristics on the strip

Importantly, the durability of the structure relies heavily on the ability of the soil reinforcement strip to maintain a level of tensile strength in the operational environment for the duration of the structure’s design life. The strip made up of steel, if used, is therefore designed to include a sacrificial 5 | Page

steel thickness, which predicts the amount of strip corrosion throughout the design life of the structure. This is achieved by controlling the environment in which the strip will be operating. The select fill, whilst having certain physical requirements that ensure it is activated in forming part of the structural mass, is also required to have electrochemical characteristics that also ensures that corrosion of the strip is not excessive or beyond the allowance made in the strip design. Furthermore, the strip is coated with zinc galvanizing for further protection. The final length and frequency of the soil reinforcement strips is a function of the combinations of geometric and physical properties of the structure and the applied design loads. Whilst the facing to the Reinforced Earth wall technically does not take on a structural role in support of the loads, it obviously forms an important part in the wall in preventing the erosion of backfill, supporting the soil reinforcement and weathering the local environment. Typically, for roads projects, concrete is the only economical material that can achieve the necessary 100 year design life without the need for any continuous maintenance or repair. The facing also forms the most visual aspect of the structure and is often required to have some aesthetic appeal, particularly in urban areas. Concrete can lend itself readily to the provision of architectural and aesthetic requirements. The facing panels can however, often be a complex component to manufacture as each facing panel may have very individual characteristics with respect to its geometry, finish or cast-in inclusions.

Introduction to reinforced soil structure Reinforced soil structures are fundamentally different from conventional earth retaining systems which are externally stabilized in thatFigure a External stabilizationthey utilize a different mechanism for support and are internally stabilized. An externally stabilized system uses an external structural wall against which stabilizing forces are mobilized, for example, gravity retaining walls and excavations supported with strutting (fig A) An internally stabilized system involves reinforcements installed within & extending beyond the potential failure mass (tied back, reinforced soil walls, and soil-nailed excavations). With this system, the interactions between the reinforcements and soil (to mobilize the tensile capacity of closely spaced 6 | Page

reinforcing elements) eliminate the need for a structural wall or a support (figure B)

The Reinforced Earth Retaining Wall According to Reinforced Earth company,

Figure b Internally stabilized reinforced soil

®

Reinforced Earth retaining walls are gravity structures consisting of alternating layers of granular backfill and reinforcing strips with a modular precast concrete facing. They are used extensively in transportation and other civil engineering applications. Because of its high load-carrying capacity, Reinforced Earth is ideal for very high or heavy-loaded retaining walls. The inherent flexibility of the composite material makes it possible to build on compressible foundation soils or unstable slopes. These performance advantages combined with low materials volume and a rapid, predictable and easy construction process make Reinforced Earth an extremely costeffective solution over conventional retaining structures.

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Mechanism Of Reinforced Soil The mechanism of reinforced soil can be explained in simple terms considering an element of cohesionless soil shown in figure C(a) . If a vertical stress is applied on the soil it deforms both laterally and vertically and reaches a new equilibrium. If reinforcement in the form of plane sheet is introduced in the sample before the application of vertical stress on the sample,

Figure c (a) Element of unreinforced soil, (b) Element of reinforced soil

deformations are restrained due to the interaction b/w the soil and the reinforcement to some extent as in figure C (b). introduction of reinforcement generates inward lateral stress, which resists the shear stresses that are generated when a vertical stress (sigma 1) is applied. If there is no significant deformation, the lateral stress is equal to Ko times the vertical stress and this condition prevails at higher vertical stresses also.

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The shear stress at the interface of soil and reinforcement generates strains in the reinforcement and tensile force is mobilized in the reinforcement. If the reinforcement force exceeds the tensile capacity of sheet reinforcement, rupture failure occurs. Secondly, it is likely that a slip occurs b/w soil & reinforcement if deformation are high or interface is smooth. These two conditions viz tesile failure & pullout failure need to be examined to ensure stability of reinforced soil structures.

Shear Box Analogy Concept These shear box test simulates the mechanism and behavior of both unreinforced and reinforced soils. Figure on the left shows slope of unreinforced soil and on the right shows slope of reinforced soil.

Figure d Slopes showing failure surface

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What can be used as a Reinforcement? 1. Geosynthetics Figure 2 shear box test on unreinforced and reinforced soilASTM

has defined a geosynthetic as a planar product manufactured from a polymeric material used with soil, rock, earth, or other geotechnical-related material as an integral part of a civil engineering project, structure, or system. Geotextiles: A geotextile is a permeable geosynthetic made of textile materials Geotextile in fabric form are being used as a basal reinforcement of embankment and fills on the ground Geogrids: Geogrids are primarily used for reinforcement; they are formed by a regular network of tensile elements with apertures of sufficient size to interlock with surrounding fill material. These are made up of High Density Polyethylene (HDPE) and interconnected longitudinal and transverse Member. These are made from sheets of polymer by punching holes and stretching the sheets in one or two direction. Geocomposits: Geotextiles and related products such as nets and grids can be combined with geomembranes and other synthetics to take advantage of the best attributes of each component. These products are called geocomposites. 10 | P a g e

2. Reinforcing strips Reinforced members are composed of thin wide steel or aluminum strips called ties. The flexibility of reinforcing strips and their tensile strengths are essential elements. The reinforcing strips are made of mild galvanized steel, stainless steel or aluminum alloy. Bolts and nuts for fixing the ties are made of the same material as that of the reinforcing strips. The durability of the strips depends on the chemical and electro-chemical behavior of these metals when in contact with soil particles

Works Cited Introduction to Soil Reinforcement and Geosynthetics [Book] / auth. Babu Visakumar. - to be added : to be added, to be added. - Vol. to be added.

PART 3

Stability of Retaining

Walls A retaining wall may fail in any of the following ways (see figures): •

It may overturn about its toe.

•

It may slide along its base.

•

It may fail due to the loss of bearing capacity of the soil supporting the base.

•

It may undergo deep-seated shear failure.

•

It may go through excessive settlement.

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Figure f External stability checks (after Transportation Research Board, 1995)

When a weak soil layer is located at a shallow depth—that is, within a depth of 1.5 times the width of the base slab of the retaining wall—the possibility of excessive settlement should be considered. In some cases, the use of lightweight backfill material behind the retaining wall may solve the problem.

Retaining Walls

PART 4

with Metallic Strip Reinforcement

As discussed before, the reinforced earth retaining wall can deploy steel and geotextiles as reinforcements. In this part we will be looking at the design of reinforced earth wall using steel as reinforcement. A basic model of steel reinforced earth retaining wall is shown in figure (f).

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Figure 0 Steel reinforced earth retaining wall

It can be seen that it consists of: •

Granular soil as backfill

•

Thin, wide steel reinforcing strips placed at regular intervals and

•

A cover or skin on the front face of the wall

Calculation of active horizontal and vertical earth pressures Figure (g) shows a retaining wall with granular backfill having a unit weight of γ1 and a friction angle of ф1. Below the base of the retaining wall, the in-situ soil has been excavated and recompacted, with granular soil used as backfill. Below the backfill the in-situ soil has the unit weight of γ2, friction angle of ф2 and cohesion of c2’. A surcharge having intensity of q per unit area lies atop the retaining wall, which has reinforcement ties at depths z= 0, Sv, 2Sv, . . . , NSv. The height of the wall is NSv = H. According to Rankine active pressure theory (see Appendix B for details),

Where = Rankine active pressure at any depth z For dry granular soilswith no surcharge at the top, = 0, = and

Ka = tan

2

(45 – ф1’/2). Thus,

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When surcharge is added at the top, as shown then =

(1) +

Due to soil only

(2)

Due to Surchar ge

According to Laba and Kennedy (1986),

(For z≤2b’) And

(For z›2b’) Also when a surcharge is added at the top the lateral pressure at any depth is = + (2) Due to soil only

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Due to Surchar ge

Figure h Analysis of a Reinforced Earth retaining Wall

According to Laba and Kennedy (1986)

= M[2q/π(β-

↑ (In radians) Where

M = [1.4 – (0.4 /0.14H)] ≥ 1 Tie Force The tie force per unit length of wall developed at any depth z (figure f) is T = active earth pressure at depth z x area of the wall to be supported by the tie

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Figure i Understanding the equation of Lateral pressure due to Surcharge

T=) Factor of Safety against Tie Failure The reinforcement ties at each level, and thus the walls, could fail by either, (a) Tie breaking or (b) Tie pullout

Yield or breaking strength of each tie / maximum

force in any tie

/) Where = width of each tie = thickness of each tie = yield or breaking strength of the tie material A factor of safety of about 2.5 – 3 is generally recommended for ties at all levels. Reinforcing strips at any depth z will fail by pullout if the frictional resistance developed along the surfaces of the ties is less than the force to which the ties are being subjected. The effective length of the ties along 16 | P a g e

which frictional resistance is developed may be conservatively taken as the length that extends beyond the limits of the Rankine active failure zone, which is the zone ABC in figure(). Line BC makes an angle 45 + ф1’/2 with the horizontal. Now the maximum friction force that can be realized for a tie at depth z is

Where = effective length = effective vertical pressure at a depth z

фu = soil -

tie friction angle

thus the factor of safety against tie pullout at any depth z is

FS(P) = FR / T ) Total length of Tie The total length of ties at any depth is

L = lr + le l

Where r = length within the rankine failure zone

le = effective length } + )} The Design Procedure with summary of steps Following is a step-by-step procedure for the design of reinforced-earth retaining wall General

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Determine the height of the wall, H, and the properties of the granular .)backfill material, such as the unit weight () and the angle of friction (ф1 Obtain the soil-tie friction angle, (фu) FS(P) |

and the required value of FS

Exhibit 1

Given Data And general Requirements Height of the Embankment 8 cohesion 52 Unit Weight of Granular Backfill (γ 1 ) Angle of Friction of Backfill (φ 1 ')

Soil-Tie Friction Angle (φ u') Factor of Safety

16.6 30 20

against Tie-Breaking FS(B) Factor of Safety

3

FS(P) Horizontal Tie Spacing (SH) vertical Tie Spacing (SV)

3

against Tie Pull-out

α Angle β

Angle

to Soil Only σ'a(1) Total Lateral

σ'a Tie Force T

Pressure

Thickness of ties

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degrees degrees n/a n/a metres

0.5

metres

0.075 10 3 0.1 0.543

metres metres metres

1.32

n/a radians

0.1357

radians

Lateral Pressure due to Surcharge σ'a(2) Lateral Pressure due

kN/m3

1

Width of Reinforcing

Strip (ω ) Load per unit area q Span of Load a Distance of Load from Wall b M

Units metres kN/m2

7.033141 130.4 137.4331 68.71657 0.011453

kN/m2 kN/m2 kN/m2 kN metres

(B)

and

Length within the rankine failure zone lr

0

le

4.625899

metres

4.625899 0.333

metres

Effective length of Ties Length of Ties Ka

L

metres

Internal Stability (see the attached Exhibit 2 and the result of internal stability analysis in succeeding pages)

Assume values for horizontal and vertical tie spacing. Also, assume the width of reinforcing strip, w, to be used. Calculate from given Eqs. Calculate the tie forces at various levels. For the known values of FS(B), calculate the thickness of ties, t, required to resist the tie breakout:

)/ The convention is to keep the magnitude of t the same at all levels, so in Eq. should equal (max). For the known values of Φu and FS(P), determine the length L of the ties at various levels. The magnitudes of S v , S H , t , , and L may be changed to obtain the most economical design. External Stability 1) Check for overturning, using Figure () as a guide. Taking the moment about B yields the overturning moment for the unit length of the wall:

Mo = Pa z’ Here,

Pa = active force =

Calculation shown for the above equation 19 | P a g e

8

soil only

⌠  5.478h ⋅ dh → 175.296= 175.296 ⌡ 0

⌠   ⌡

 1.4 − .96  ⋅ 1.66dH → 2.324H ⋅ + −11.382857142857142857 ⋅ ln ) H)   .14H  

2

⌠   1.4 − .04  ⋅  20 ⋅ ) 1.159085553− sin ) 1.159085553 ) ⋅ cos ) 2⋅ 0.099668652 ) ) dH → 1.9955799158760896291  π 3   .14H   ⌠ ⌡  1  1.4 − .04  ⋅  20 ⋅ ) 0.947871788− sin ) 0.947871788 ) ⋅ cos) 2⋅ 0.049958396 ) ) dH → 1.1424541308466395876 = 1.142      .14H   π  ⌡ 2

7

⌠   1.4 − .04  ⋅  20 ⋅ ) 0.460226744− sin ) 0.460226744 ) ⋅ cos) 2⋅ 0.016665124 ) ) dH → 0.1408975856346407028 = 0.141   π ⌠ 4  .14H    .04 20 ⌡  1.4 −  ⋅  ⋅ ) 0.768469142− sin ) 0.768469142 ) ⋅ cos ) 2⋅ 0.033320996 ) ) dH → 0.62900883954153347651 = 0.629 6     .14H   π  ⌡ 3

8 5

⌠⌠  20 .04.04  H → 0.36093894903972636611  ⋅ 20   1.4 ⋅  ⋅ ) 0.402613328 ⋅ ) 0.634315275 − sin ) 0.634315275 ) ⋅ cos 2⋅ 0.024994794 0.361 −− − sin ) 0.402613328 ) ⋅ cos ) 2⋅)0.014284743 ) ) )d)Hd→ 0.094927136841385213336 == 0.095  1.4     .14H π      .14H π    ⌡⌡ 7 4

⌠   ⌡

6

5

 1.4 − .04  ⋅  20 ⋅ ) 0.534998393− sin ) 0.534998393) ⋅ cos) 2⋅ 0.019997334) ) dH → 0.21938693099589765693 = 0.219    .14H   π  

………… /{[ 20 | P a g e

Exhibit 2

Overturning Active force

175.3

4.5

179.8

soil

surchar ge

total

Resisting Moment W1 x1

1726.4 6.5 11221. 6 11269. 6

W1x1 MR

kn m kn-m kn-m

Overturning Moment z' Mo

2.67 480.06 6

FOS

23.475 11

m

Figure j Calculation for Overturning

>

3

2) The check for sliding can be done by using

………… /{ Sliding Numerat 639.27 or 69 FOS

Where k

≈

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2/3

3.56 > 3

3) Check for ultimate bearing Capacity can be done by

The vertical stress at z = H is

So the factor of safety against bearing capacity failure is

Ultimate Bearing Capacity qu = 3805.2 Vertical stress at 8 m = 142.5 FOS = 26.7 > 5

Generally, minimum values of 3, 3, and = 3 to 5 are recommended.

APPENDIX A Review of Shear Strength Sources of stresses in the ground Geostatic stresses

τσxz xσ zx

γz z=

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Geostatic stresses are those that occur due to the weight of the soil above the point being evaluated. They are caused by gravity acting on soil or rock so the direct result is the vertical normal stress σz. This vertical normal stress indirectly produces horizontal normal stresses and shear stresses.

Exhibit 3: Three dimensional soil element (left), 2D soil element (middle), Induced horizontal stress in an unconfined soil element (right)

Horizontal stresses are either direct (e.g. braking forces from wheels of large truck) or induced (see figure (right)). Unlike the figure, real soils in the field are not unconfined and the adjacent elements of soil or rock also wish to expand but in opposite direction. These opposing forces may cancel each other (i.e. there may be no horizontal strain) or the horizontal strain may be much less than would occur in an unconfined sample. Either way, the result will be the formation of horizontal stresses in the ground. If ground surface is horizontal, the geostatic shear stresses on horizontal and vertical planes are all equal to zero.

τxz = τzx = τyz = τzy = τxy = τyx

(1)

Induced stresses These are the stresses due to external loads. Equation (1) will not be true for induced stresses.

Effective stress The compressive stress, σ, is carried partially by the solid particles & partially by the pore water. It is called the total stress because it is sum of the stresses carried by these two phases in the soil.

u = γ w zw Where γw = unit weight of water 23 | P a g e

zw = depth from the groundwater table to the point where pressure head is to be computed. The particle contact forces in an element of soil that is initially above the groundwater table will decrease if the groundwater rises above that element. This is because of the buoyancy force of water.

Vertical effective stress = σz = (σz) geostatic + (σz) induced - u

Horizontal effective stress = σx’ = σx – u σy’ = σy – u

Coefficient of lateral earth pressure The ratio of horizontal to vertical effective stresses is defined as the coefficient of lateral earth pressure, K.

Mohr’s circle It is used for computation of stresses acting on planes other than horizontal and vertical planes. It describes the 2D stresses at a point in a material. Each point on the circle represents the normal & shear stresses acting on one side of an element oriented at a certain angle. For example, in this mohr’s circle, points A & B represent (σz, τzx) and (σx,

τ

xz),

which are the stresses acting on an element aligned with the x and z axes, while points C & D represent the stresses on an element oriented as shown.

Principal stresses If the soil element is rotated to a certain angle, shear stresses will be zero on all four sides. The planes on each side of this element are represented by points E & F and are known as principal planes. Stresses acting along these planes are known as principle stresses. 24 | P a g e

σ

= major principal stress (greatest normal stress acting on any plane)

σ

= minor principal stress (smallest normal stress acting on any plane)

1

3

these two stresses act on right angle to each other. As told earlier, when the ground surface is level, the geostatic shear stresses on vertical & shear planes ar all zero. Therefore, these are the principal planes and the geostatic principal stresses act vertically and horizontally. If

σx

>

σz then σ3 = σz and σ1 = σx

If

σz

>

σx then σ3 = σx and σ1 = σz

Mohr’s circle for effective stress principle of effective stress applies only to normal stresses, not to shear stresses because the pore water can’t carry a static shear stress.

Shear failure in soils In soil, shear failure occurs when the shear stresses b/w the particles are such that they slide or roll pass each other. The shear stress primarily depends upon interactions b/w the particles, not on their internal strength. Shear strength = frictional strength + cohesive strength

Frictional strength The force that resists sliding is equal to the normal force multiplied by the coefficient of friction, µ. For soils, ф’ is used i.e. ф’ = tan-1 µ Frictional shear strength = effective stress acting on shear plane tan ф

The value of ф depends on both the frictional properties of the individual particles and interlocking b/w particles. It also depends upon mineralogy, shape gradation, void ratio & organic material in soil. Pore water pressure reduces the frictional shear strength.

Cohesive strength Cohesion may be true or apparent. True cohesion is cementation and electrostatic forces. Apparent cohesion is negative pore water pressures, etc. 25 | P a g e

APPENDIX B

Rankin’s theory for Active Earth Pressures Active Earth Pressure Assumption  Frictionless wall

 Before the wall moves the stress condition is given by circle “a”  State of Plastic equilibrium represented by circle “b”. This is the “Rankine’s active state”  Rankine’s active earth pressure is given by

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∆L A'

A

σ

B'

' a

σo'

B

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z

 With geometrical manipulations we get:

σ 'a = σ 'o

1 −sin φ′ cosφ′ −2c' 1 +sin φ′ 1 +sin φ′

(

)

(

σ 'a = γz tan 2 45 − φ2′ −2c' tan 45 − φ2′  For cohesionless soil, c’=0

φ'

σ =σ tan ) 45 − ' a

' 0

2

2

)

Rankine’s Active Pressure Coefficient, Ka  The Rankine’s active pressure coefficient is given by: σ a' K a = ' = tan 2 45 − φ2′

σo

(

)

 The angle between the failure planes /slip planes and major principal plane (horizontal) is:

(

± 45 + φ2′

)

σ a' The variation of

28 | P a g e

)

With depth:

The slip planes:

Lateral Earth Pressure Distribution against Retaining Walls There are three different cases considered: Horizontal backfill Cohesionless soil Partially submerged cohesionless soil with surcharge Cohesive soil 29 | P a g e

Sloping backfill Cohesionless soil Cohesive soil Walls with Friction

Active Case

Horizontal backfill with Cohesionless soil

σ a = K aγz

Pa =

1 K aγH 2 2

Horizontal backfill with Cohesionless, partially submerged soil

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σ a' = K a [q + γH 1+γ ' ) z − H1 )] Horizontal backfill with cohesive soil

The

depth at which the active pressure becomes equal to zero (depth of tension crack) is

z0 =

2c ' γ Ka

For the undrained condition, φ = 0, then Ka becomes 1 c=cu. Therefore,

z0 =

(tan245° = 1) and

2cu γ

Tensile crack is taken into account when finding the total active force. i.e., consider only the pressure distribution below the crack

Active total pressure force will be 31 | P a g e

1 2c ' 2 2 ' Pa = K aγH − 2 K a c H + 2 γ Active total pressure force when φ = 0

2cu2 1 2 Pa = γH − 2cu H + 2 γ Sloping backfill cohesionless soil Earth pressure acts an angle of α to the horizontal

σ a' = K aγz Pa =

1 K aγH 2 2

This force acts H/3 from bottom and inclines α to the horizontal 2

K a = cosα ⋅

cosα − cos α − cos2 φ ′

cosα + cos2 α − cos2 φ ′

Sloping backfill cohesive soil

σ a' = γzK a = γzK a" cosα Depth to the tensile crack is given by

z0 =

K a" =

Ka cosα

2c' 1 + sin φ ' γ 1 − sin φ '

Friction walls Rough retaining walls with granular backfill. Angle of friction between the wall and the backfill is δ'

32 | P a g e

Case 1: Positive wall friction in the active case (+δ´)

Wall AB

A’B

causes a downward motion of soil relative to wall.

Causes downward shear on the wall (fig. b) Pa will be inclined δ’ to the normal drawn to the back face of the retaining wall Failure surface is BCD (advanced studies): BC curve & CD straight Rankine’s active state exists in the zone ACD

Case 2: Negative wall friction in the active case (-δ’) - Wall is forced to a downward motion relative to the backfill

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