CE 641 Lecture 3 Strength Characteristics of Reinforced Earth

Department of Civil EngineeringDepartment of Civil Engineering

IIT IIT GuwahatiGuwahati

Strength Characteristics: Reinforced Earth

� Behavior of reinforced earth and

determination of strength characteristics

� Static Triaxial Compression test

� Soil samples are reinforced with discs or rings

or fibers of metallic or non-metallic materials

8/16/20138/16/20138/16/20138/16/2013CE 641 Lecture 3: Strength Characteristics of CE 641 Lecture 3: Strength Characteristics of CE 641 Lecture 3: Strength Characteristics of CE 641 Lecture 3: Strength Characteristics of

Reinforced EarthReinforced EarthReinforced EarthReinforced Earth2222

� Loaded to failure under different confining

pressures

� Furnish

� Stress-strain characteristics of reinforced earth

� Strength envelope for reinforced soil

Basic Concept

� Strength of reinforced soil [Schlosser

and Long (1978)]

� Two-dimensional element of cohesionless soil

(φ – Angle of internal friction)

� Reinforcement layers at vertical spacing S



A B

Sz

σ1

E

� Reinforcement layers at vertical spacing Sz

� Subjected to a confining pressure of σ3

� Vertical stress of magnitude σ1 is applied

and increased to failure

� Linear wedge (FE of length b) failure

assumed to take place at an angle θ with the

vertical

CD

σ3 σ3

σ1

θ

F

b

Basic Concept


and Long (1978)]

� Free-body diagram of the upper section of

the failed soil mass during equilibrium

� Considering only the wedge GEF



A B

σ1

σ3

σ3

EG

στ

Tb

� Considering only the wedge GEF

� σ3 on AG and BE balances each other

� Equilibrium of the wedge

� GF: Confining stress σ3

� GE: Vertical stress σ1

� BC: Normal stress σn and shear stress τ

� Total reinforcement force T

� Summation of the forces in various reinforcing

layers intersected by FE

σn

τ

F

σ1

σ3

EG

σn

τ

T

F

b

Basic Concept


and Long (1978)]

� Equilibrium equations

� ΣV = 0

ΣH = 0



σ1

σ3

EG

σn

τ

T

F

τ cos θ

σn sin θ

1. sin . cos . sinnb b bσ θ τ θ σ θ= +

b

. cos . sin . cosb T b bσ θ τ θ σ θ+ + =� ΣH = 0

� At failure, Mohr’s circle cuts the failure

envelope

� Combine the equations

σ1

σ3

EG

σnτ

T

F

τ sin θ

σn cos θ

b

3. cos . sin . cosnb T b bσ θ τ θ σ θ+ + =

tannτ σ φ=

( ) ( )1 3

tan tan

tan sin

T

b

θ φ θ φσ σ

θ θ

+ += +

Reinforcement Failures

� Two types of failure

� Rupture of reinforcement

� Governed by the tensile capacity of reinforcement

per unit length (RT)

� Maximum tensile force generated at the verge of

rupture of reinforcement



σ1

σ3

EG

Tbrupture of reinforcement

� No. of reinforcement

in the reinforced soil mass

� Slippage of reinforcement

� Governed by the total friction force generated in

various layers of reinforcement at the verge of

failure

. cosT

z

R bT

S

θ=

σn

τ

T

F

b

*1

cos2 . . r

z

bT f b

S

θσ=

Width of reinforcement (Strip reinforcements)

Coefficient of interface friction

Friction force developed on one side

of the reinforcement

Reinforcement Failure: Rupture

� Rupture failure of reinforcement

� Total tensile capacity of the

reinforcements (all reinforcements

intersected by FE) gets exceeded

� T = Total tensile capacity as f(RT)



σ1

σ3

EG

T

( )tanR θ φ+

θ

� Identification of the rupture plane

� Inclination θ at which σ1 is maximum

� On substitution

σn

τ

F

( ) ( )1 3

tan tan

tan sin

T

b

θ φ θ φσ σ

θ θ

+ += +

( )1 3

tan

tan

T

z

R

S

θ φσ σ

θ

+ = +

1 0 45 / 2σ

θ φθ

∂= ⇒ = −

∂

( ) ( )21 3 tan 45 / 2T zR Sσ σ φ= + +

. cosT

z

R bT

S

θ=

Reinforcement Failure: Rupture


� Relationship between vertical and

horizontal stress

N � Flow value of the soil mass



σ1

σ3

EG

Tθ

( )1 3 T zR S Nφσ σ= + 2tan 452

Nφφ

= +

� Nφ� Flow value of the soil mass

� For an unreinforced c-φ soil, relation

between the principal stresses

� Analogically

� Pseudo cohesion generated in a

cohesionless soil due to reinforcement

σn

τ

F

1 3 2N c Nφ φσ σ= +

2

T

z

Rc N

Sφ=


� Shear strength envelopes of

unreinforced and reinforced

soils are same (tan φ)

� Additional strength is

imparted by the reinforcement

as pseudo-cohesion (c)

Reinforcement Failure: Sliding/Slippage

� Sliding failure of reinforcement

� Total friction capacity of the

reinforcements (all reinforcements

intersected by FE) gets exceeded

� T = Total tensile capacity as f(f*)



σ1

σ3

EG

T

( )tan tanθ φ θ+

θ

� Identification of the rupture plane

� Inclination θ at which σ1 is maximum

σn

τ

F

( ) ( )1 3

tan tan

tan sin

T

b

θ φ θ φσ σ

θ θ

+ += +

( )

( )1 3 *

tan tan.

tan2 .1 .

tan

r

z

b f

S

θ φ θσ σ

θ φ

θ

+=

+−

1 0 45 / 2σ

θ φθ

∂= ⇒ = −

∂*

1cos

2 . . rz

bT f b

S

θσ=

Reinforcement Failure: Sliding/Slippage

� Sliding failure of reinforcement

� On substitution



σ1

σ3

EG

Tθ

3 '1 3*2 .

1 .r

z

NN

b fN

S

φφ

φ

σσ σ= =

−

� No content of pseudo-cohesion

� Improvement in the internal friction

angle of reinforced soil

� Angle of internal friction of the

reinforced soil mass

σn

τ

F

zS

'RN Nφ φ φ φ> ⇒ >

( )' 2tan 45 / 2RNφ φ= +

Mohr’s Circle Representation



Slippage FailureRupture Failure

Precedence of Failure

� Which failure will prevail???

� Slippage failure prevails

� Total friction force generated due to interface friction < Total tensile capacity

of the reinforcements



*1

. coscos2 . . T

rR bb

f bS S

θθσ <

3 '1 3*2 .

1 .r

NN

b fN

φφ

σσ σ= =

−

� Any special implication of the above relationship???

� With regard to reinforced retaining walls !

� In a natural state, confining pressure (σ3) is a function of depth

12 . . rz z

f bS S

σ <

*

3 *

2 .. 1

2 . .

rT

z

r

b fR N

S

b f N

φ

φ

σ

−

<

2 .1 .r

z

b fN

Sφ−

Precedence of Failure



Slippage Failure

� Increase in soil strength due to reinforcement

� Increased friction angle φR or pseudo-cohesion c

� Contributory factors

� Shear strength of sand

� Tensile strength, frictional strength and distribution of reinforcement

Rupture Failure



Thank You for Patient Hearing

CE 641 Lecture 3 Strength Characteristics of Reinforced Earth

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