Factor of Safety of a Consolidated Slope

8/6/2019 Factor of Safety of a Consolidated Slope

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Factor of safety of a consolidating slope withvertical drains

under the guidance of Project by

Dr. A. Krishnamoorthy Madan V R

Professor 090918017MIT Mtech 2nd year


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Contents

Introduction

Assumptions

Methodology involved

Factors affecting on vertical drains

Scope of present work

Analysis of slope for E=5000kN/m2, 25000kN/m2 and

50000kN/m2

Conclusion

Reference


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Introduction

Prefabricated vertical drains (PVDs) are often used to speed upconsolidation and to increase shear strength under embankments

on soft soil.

Installing a drain material vertically into the ground can shorten

the drainage path of soft clay deposit significantly, and combinedwith preloading, it can improve the stiffness and strength of the

ground substantially in a short period.

The installation of PVDs by means of a mandrel causes significant

disturbance in the soil surrounding the drain. The shape of themandrel is also rectangular and therefore, the shape of the actual

disturbed zone is close to a rectangle or an ellipse.


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Assumptions

Each single drain is assumed to work independently,

Soil has a constant permeability and the consolidation takes place

in a uniform soil column with linear compressibility characteristics

in the absence of lateral movement.

The stressstrain behaviour of natural soft soils is highly nonlinear

and very complex due to different fundamental features of soil,

such as anisotropy and creep.


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Methdology involved

An axisymmetric unit cell with the total radius, Rand its equivalent plane strain unit cell with half width, B. The effective diameter of adrain influence was taken to be De = 1.13S for a square configuration,

where S is the drain spacing. The equivalent drain radius (rw) and unit

cell radius (R) were calculated as 0.034 m and 0.565 m, respectively.

Using the permeability matching procedure proposed by Hird et al.(1992) the coefficient of permeability is matched while keeping thedrain spacing constant.


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Equations used in the analysis

Displacement u and excess pore pressurep within the finite elementcan be related to nodal displacement vector {un} and the nodal pore

pressure vector {pn} as

u = [Ns]{Un}

p = [Nf]{pn}

Ns is the shape function defining the displacement of the soil element

whileNfis the shape function defining the pore pressure distribution.

consolidation The elemental equation of consolidation proposed by Zeinkiewicz

(1977) can be expressed in matrix form as

Ks is the soil stiffness matrix andHis the flow matrix,L is the coupling

matrix which is formed from the equation


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Computation of stress

The stresses at any point in the element can be calculated by usingthe equation,

{} = [C] { } Or {} = [C] [B] {q} {} = is the vector of stresses.

{}T= { x, y, xy}

x, yare the normal stresses in x and y direction and xy is the

shear stress in the xy plane,{q} is the vector of nodal displacements,[C] is the stress strain matrix.


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Equation for Factor of Safety The trial slip surface is divided into n number of segments each of

length L. The overall factor of safety for a particular slip surface isobtained using the equation

F.S = Where f is the mobilized shear stress and is the shear strength of the

material.Liis the length of the ith

segment on a particular slip surface, thevalues of and fmay be expressed as

f= C + ni tan

= 0.5 (yi

xi

) sin 21

+ xyi

cos21

ni = 0.5(yi + xi)+0.5(yi xi)cos 21 - xyi sin21

where Cand are the effective cohesion and effective angle of internal

friction of the soil. ni is the normal stress acting on segment i. xi, yi and

xyi are the effective stress on the ith

segment.


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Co-efficient of permeability for variousspacing provided

Spacing (K pl /Kax)Coefficient of permeability,

K(m/day)

1m 0.2077 1.798x10-4

2m 0.0497 4.311x10

3m 0.02156 1.867x10

4m 0.0119 1.0305x10

5m 7.534x10-3 6.524x10

Without drain 8.66x10-5 8.66x105


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Factors affecting vertical drains

smear effect

drain spacing and equivalent drain diameter

effect of sand mat and

drainage boundary condition.


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Smear effect

The drain is installed by using a mandrel, which is pushed into theground. Then the mandrel is withdrawn, leaving the drain in subsoil. This process creates a completely disturbed zone around the drain, called thesmear zone.

Two parameters are needed to characterize the smear effect, namely, the

diameter of the smear zone (ds) and the hydraulic conductivity ratio (kh/ks), i.e., the value in the undisturbed zone (kh) over that in the smear zone

(ks).

Diameter of the smear zone, ds

, can be estimated as

ds = (2 to 3)dm

Where, dm = equivalent diameter of the cross-sectional area of the mandrel.

Whereas Kh/Ks = 10(constant)


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Effect of Sand mat Part or all of the water collected by the drain will flow to the ground

surface first, and then drain out by the outlet system, the sand mat. Since the

hydraulic conductivity of sand is considerably higher than that of clay,usually, in analysis, it is assumed that there is no hydraulic resistance in thesand mat. If a thick layer (more than 0.5 m) of clean sand (percentage finesless than 5%) is used.

Equivalent drain diameter And a new equation based on the finite-element analysis result has been

suggested as (Rixner et al. 1986)

Where w and t= width and thickness of a vertical drain, respectively.


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FINITE ELEMENT METHOD

The past few decades have witnessed significant advances in analyses of

the slopes using finite element method. The main advantage of FiniteElement Method is that

It can be used to calculate pore pressure, stresses and displacements inembankment slope.

To study the conditions during the construction, and also followingconstruction, as consolidation or swelling occur and excess pore pressuredissipate.


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Scope of present work

The factor of safety of a dry slope on soft saturated consolidatingsoil is obtained at various time intervals using finite element method ofanalysis. A computer program and the method proposed byKrishnamoorthy and Mishra (1998) for the analysis of the slope on drysoil is modified by Krishnamoorthy (2010) to obtain the factor of safety

of the slope on consolidating soil. The method of obtaining the factor ofsafety consists of two steps.

1. the effective stresses at required points are obtained by finite elementmethod of analysis

2.3. the critical slip surface is located using Monte Carlo Technique

proposed by Greco(1996).


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This method and computer program is used to study the effectof vertical drains on the factor of safety of a consolidating slope. Thefactor of safety of the slope on a consolidating soil without verticaldrains is compared to the factor of safety of a slope with verticaldrains. In addition, the effect of spacing of vertical drains on factorsafety of slope is also studied; three types of soil having differentvalue of modulus of elasticity E i.e., E=5000kN/m, 25,000kN/m

and 50,000kN/m are considered for the analysis.

The factor of safety, pore pressure, effective vertical andhorizontal stresses at various time intervals are obtained and theeffect of vertical drains on these factors is critically discussed in thethesis.


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Embankment considered for the analysis


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The analysis is used to study the effect of vertical drains on factor of a slopeconstructed on soft consolidating soil. A slope considered for the study. Theembankment is 5.0m high and crest of the embankment is 20 m wide and theslopes have a gradient of 1:1.The vertical drains were installed as a square gridat a spacing of 1.0m spacing before embankment construction. Theembankment, which is made of granular fill, is modelled using the followingmaterial parameters:

Unit weight = 20kN/m

Modulus of elasticity for soil type 1 = 5000kN/m

Modulus of elasticity for soil type 2 = 25,000kN/m

Modulus of elasticity for soil type 3 = 50,000kN/m

Poissons ratio = 0.30

Density = 18kN/m

Effective cohesion = 20kN/m

Effective angle of internal friction = 30


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Finite element discretisation

1

1

5 . 0 m

2 5 . 0 m

1 0 . 0 m

y

A

B C

DE

F

P

1 4 . 0 m


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Trial slip circle ABCDE considered forthe analysis


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Factor of safety

Factor of safety of the slope at the end of construction with drain

is equal to 1.42 and it decreases to 1.30 without drain. Factor of safetyincreases with time due to dissipation of pore pressure. The factor ofsafety of the slope after the complete dissipation of pore pressure (atT=1000) is equal to 1.64.


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Effect of spacing

The factor of safety at the end of construction is equal to 1.42for s=1m, 1.39 for s=2m, 1.36 for s=3m, 1.34 for s=4m, 1.33 fors=5m and 1.30 for foundation soil without drain. I.e., as thespacing of the drain increases the factor of safety at the end ofconstruction decreases. The spacing of drains increases the time

required for consolidation also increases.


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Pore pressure

Pore pressure immediately after the end of construction is more at point Y

as compared to the point P. Pore pressure dissipate with time and becomes zeroafter the end of consolidation. At point P it can be observed that the porepressure is equal to 22.40kN/m2 (with drain) and 21.40kN/m2 (without drain)and pore pressure at point Y is equal to 81.5kN/m2 (with drain) and 79.5kN/m2

(without drain). Pore pressure at the end of consolidation is equal to zero.


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Horizontal stress

Horizontal stress at point P, at the end of construction is equalto 14.8kN/m2 (with drain) and 15.31kN/m2 (without drain),whereas at point Y, horizontal stress at the end of construction isequal to 3.33kN/m2 (with drain) and 9.48kN/m2 (without drain).


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Vertical stress

Vertical stress at point P, at the end of construction is equal to23.73kN/m2 (with drain) and 18.45kN/m2 (without drain), whereasat point Y, horizontal stress at the end of construction is equal to

67.85kN/m2

(with drain) and 54.68kN/m2

(without drain).


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Critical slip circle

critical slip surfaces obtained from the analysis at the end ofconstruction (T=0) and at the end of consolidation(T=1000days). The slip surface corresponding to drainedcondition is slightly different from the slip surface

corresponding to undrained condition.

S ttl t


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Settlement

It can be observed that the settlement of the embankmentwith drain is more than the settlement without drainimmediately after the construction. In case of foundation soilwithout drain the settlement at end of construction is equalto19mm and it settles upto73mm at the end of consolidation.


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Effect of youngs modulus on pore pressure

it can be observed from the figure that the time required forconsolidation is same for all the three values of E. This indicatesthat the modulus of elasticity of embankment has no effect on theconsolidation process.


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Effect of youngs modulus on horizontal stress

The horizontal stress decreases immediately after constructionfor both the cases of with and without drains. However E has nomuch effect on horizontal stress at the end of consolidation.


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Effect of youngs modulus on vertical stress

This shows that the vertical stress significantly increases with theincrease in value of E of embankment soil at the end of consolidation.


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Effect of youngs modulus on settlement

I.e. The settlement increases as the value of E increases for

both the cases of with and without drains.


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conclusion

Based on the analysis, it is concluded that

The proposed method can be used to obtain the factor safety, pore pressureand effective stresses for consolidating slope at various time intervals.

The method of locating critical slip surface is simple since it requires fewtrial slip surfaces.

Effective Horizontal stresses and vertical stresses will increase with timeduring end of consolidation.

The increase in the spacing has a minimal effect on the rate of consolidation.

The increase of settlement value at the end of construction is significant(about 17.6 and 19.4 cm) which implies a certain improvement for the

foundation soil properties by consolidation during construction period.


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Borges JL. Three dimensional analysis of embankments on soft soil incorporating vertical drains by finite elementmethod, computers and geotechnics 2004; 31:665-676.

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Greco VR. Efficient Monte Carlo technique for locating critical slip surface. ASCE Journal of GeotechicalEngineering 1996;122: 517-525.

Contd


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Factor of Safety of a Consolidated Slope