geosynthetics engineering: in theory and practice 3.pdf · a = cross sectional area of cylindrical...

GEOSYNTHETICS ENGINEERING: IN THEORY AND PRACTICE

Prof. J. N. Mandal

Department of Civil Engineering, IIT Bombay, Powai , Mumbai 400076, India. Tel.022-25767328email: [email protected]

Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay

Module-1LECTURE- 3

INTRODUCTION TO REINFORCED EARTH


RECAP of previous lecture…..

Sustainability using geosynthetics

Major applications

Lessons learned from failure and success

Key benefits

Introduction to reinforced earth

Basic concepts and mechanisms of reinforced earth(partly covered)


Anisotropic Cohesion Concept (Vidal and Schlosser, 1969)

At higher stress level, failure occurs by rupture of the reinforcement.


For unreinforced soil, 1 = 3 kp

The failure envelope forreinforced soil is defined by,

1r = 1 + Δ1 = 3 kp + Δ1

They compared the above equation with Rankine-Bellequation for c- soils,1r = 3 tan2 (45°+/2) + 2 cr tan (45°+/2) = 3 kp + 2 cr kp

Equating the above two equations, cr = Δ1/ 2kp

LCPC cohesion theory (Schlosser and Long, 1973)

Where, cr = Reinforcement induced cohesion


σ1r = Major principal stress σ3 =Minor principal stress R = Resultant force F = Resultant tensile force within failure planeA = Cross sectional area of cylindrical sample

Coulomb Analysis (Schlosser and Long, 1973)


From the force triangle,

Assume vertical spacing between reinforcements “h” and tensile strength of a reinforcement = T,

Combining the above equations,

By differentiating, σ1r will be maximum when

)tan(AtanAF r13

Th

tanAF

)cot(tanhT

3r1

2

45


)2

45cot(2

45tanhT

3r1

245tan

245tan

hT

3r1

hTkk p3pr1

Again, 1r = 1 + Δ1 = kp 3 + Δ1

Therefore,hTk p1

hT

2k

k2hTk

k2c p

p

p

p

1r


NSW Cohesion Theory (Hausmann, 1976)

1ar3 k

rp3pr3p1 kk)(k

From Rankine-Bell equation for c-ϕ soil ,

pr3p1 kc2k

Comparing the above two equations,

2k

c prr


A = Cross sectional area of the strip

Additional Stress in the soil due to inclusion of reinforcement = σr

Considering failure by rupture of reinforcement (Hausmann,1976),

σ = Ultimate tensile strength or yield strength of reinforcement

Therefore, σr. B. H = σ. A

ppr

r kBH2A

2k

c

BHA

r


Failure conditions for variable σr (Hausmann, 1976)

At low stress level, failure in a reinforced earth tends tooccur by slippage. If friction along the reinforcement isproportional to vertical stress and cohesion remains zero,

F = friction factor characterizing reinforcement interaction withthe cohesionless soil. If F = Ka, ϕr = 90º, failure occurs byrupture rather than slippage.


Slippage of reinforcement (Hausmann, 1976)


Due to lateral expansion, shear stress (τ) acts between soiland the reinforcement. Magnitude of the shear stressdepends on the angle of wall friction, δ, between soil andreinforcement as well as on relative movement between soiland reinforcement.

The shear stress is zero at the mid point and at the end ofthe strip arriving maximum value of σ1 tan δ.

The uneven distribution of shear stress and imperfecttransfer should be taken into account by considering areinforcing efficiency (e).


Composite Mohr Envelope (Hausmann, 1976)

At higher stress level, failure occurs by rupture of the reinforcement.

At low stress level, failure in a reinforced earth tends to occur by slippage.


BASIC DESIGN OF REINFORCED SOIL WALL


The reinforced earth concept can be used for retaining walls.For basic understanding, a model reinforced soil wall isconstructed using paper and gravel/sand.

For internal stability, we have to determine the length ofreinforcement (L), overlap length (Lo) and spacing of thereinforcement (Sv). For external stability, we have to check for sliding,overturning and bearing capacity failures.


Failure line (BC) dividesthe reinforced fill intotwo zones:

(A) Active zone, and

(B) Restraining zone

Cross section of a reinforced soil wall In active zone, the shearing stresses from the soil to thereinforcement act towards the facing element. This meansthat reinforcement is pulled towards the facing element. In restraining zone, the shear stresses from the soil toreinforcement act away from the facing, thereby tend to holdthe reinforcement in soil.


From both experimental and field measurements, it has beenobserved that the tensile force in reinforcement is low or evenzero at the facing, and increases to a maximum value at adistance from the facing with (+dT/dl) and again decreasestowards zero at the free end of reinforcement with (-dT/dl).

Mobilization of tensile strength along the length of reinforcement (After Schlosser and Long 1974)


Coulomb Force Method:

Coulomb’s wedge failure in reinforced earth wall with force directions

(After Schlosser and Vidal 1969)

Force triangle on the active wedge


From the force triangle, )tan(WTr

Now, W = 1/2 γH2 tan (90° - θ)

Therefore,

)tan(cot H 21)tan() - (90° tanH

21T 22

r

0ddTr

Tr is maximum when θ = (45° + /2)

2a

22(max)r HK

21

245tanH

21T

This resultant tensile force is distributed in all the reinforcement layers.


Triangular distribution where tension increases with depthand becomes maximum at the bottom layer (i = n) (Schlosserand Vidal, 1969).

Tensile force distributions along depth

n = total number of strips

i = numbering of layersfrom top


vamaxn HSK1n

nTT

vS

Hn

The maximum tension per meter run of the wall develops atthe bottom layer and can be determined using principle ofequal angle triangles,

v

v

i

n

iSnS

TT

n

iTT ni

Now, 2a(max)rn321 HK

21TT..........TTT

Therefore,

2a

n HK21)n.........321(

nT


Rankine Force Method:It is assumed that the ith reinforcement layer resists earthpressure for the zone falls between Sv(i-1/2) and Sv(i+1/2). Theconcept of Rankine’s method is shown in the Figure below.

(After Schlosser and Vidal, 1969)Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay

2

1iSKP va21ia

2

1iSKP va21ia

Therefore, the ith layer is subjected to lateral active earthpressure as shown in Figure below.

Active earth pressure in the zone for ith layer reinforcement


Ti = Developed tension in reinforcement = Area of active earth pressure diagram

2vavvai SiKS

21i

21iSK

21T

vv Si

vvai SKT

Vertical stress at ith layer,

Therefore,


Reinforced Earth (After Vidal, 1966)


Please let us hear from you

Any question?


Prof. J. N. Mandal

Department of civil engineering, IIT Bombay, Powai , Mumbai 400076, India. Tel.022-25767328email: [email protected]


geosynthetics engineering: in theory and practice 3.pdf · a = cross sectional area of cylindrical...

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