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GEOSYNTHETICS ENGINEERING: IN THEORY AND PRACTICE
Prof. J. N. Mandal
Department of Civil Engineering, IIT Bombay, Powai , Mumbai 400076, India. Tel.022-25767328email: cejnm@civil.iitb.ac.in
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
Module-1LECTURE- 3
INTRODUCTION TO REINFORCED EARTH
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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)
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
Anisotropic Cohesion Concept (Vidal and Schlosser, 1969)
At higher stress level, failure occurs by rupture of the reinforcement.
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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)
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
)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
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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.
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
Slippage of reinforcement (Hausmann, 1976)
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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).
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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.
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
BASIC DESIGN OF REINFORCED SOIL WALL
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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.
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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.
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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)
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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.
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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,
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
Reinforced Earth (After Vidal, 1966)
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
Please let us hear from you
Any question?
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
Prof. J. N. Mandal
Department of civil engineering, IIT Bombay, Powai , Mumbai 400076, India. Tel.022-25767328email: cejnm@civil.iitb.ac.in
Prof. J. N. Mandal, Department of Civil Engineering, IIT Bombay
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