A study on the tree dimensional effect of seepage force on the stability of cofferdam

An example of problem associated with boiling type of failure inside cofferdam for Daiichi-Shinkawa Bridge was introduced and the cause of the problem was adequately determined and discu

A STUDY ON THE THREE-DIMENSIONAL EFFECT OF SEEPAGE FORCE ON THE

STABILITY OF COFFERDAM

NOVEMBER 2010 MASTER OF ENGINEERING

DANG CHI LIET M095610 TOYOHASHI UNIVERSITY OF TECHNOLOGY

Cofferdams are temporary structures built to create dry conditions during construction in riverbeds and in the bottom of lakes With such structures, the inflow of groundwater as well as free water can be presented and these structures are designed to resist the lateral earth pressures and water pressures For excavation depths of up to 12

m the braced-sheet piled cofferdam is generally more economical than the cellular and caisson type cofferdams They are thus frequently employed during the construction of piers and abutments of bridges with medium span The collapse of cofferdams could occur due to the upward seepage force at the base even though they are designed to adequately resist the lateral thrust of the soil and water Such a failure is generally regard as boiling type or piping type of failure Sometimes a small leakage of water can initiate such a failure, which eventually leads to the sudden disintegration of the whole base Even with noticeable deformation, a cofferdam can generally be considered to perform successfully if dry condition is prevailed during the construction works and the inflow of water is smaller than the amount pumped out at the site A boiling type of failure only can be detriment to the performance of the cofferdam but also can affect the construction schedule and create changes in the design of the permanent structures During the boiling phenomenon, the upward seepage force is large enough to carry the sand and silt size particles with the discharge It is also a progressive type of failure wherein sudden flooding can occur inside the cofferdam with the continuous discharge of soil grains Thus a piping connection is made under the tip of the sheetpiles between the inside and the outside of the cofferdam Once boiling has occurred at the base floor, the original consistency, strength, and the stiffness of the natural ground is lost and this results in an inadequate bearing capacity at the base level Additionally, the influence of boiling can cause irreparable and severe damages to the ongoing construction works in the nearby structures, as well as the traffic and other human activities in the neighboring area Thus it is always important to ensure that boiling type

of failures do not occur in cofferdams

This study aims to investigate the properly mathematical formulation of soil materials and conditions for analysis of the transient response behavior of excavation ground base subjected to upward seepage force, excess pore water pressure increasing

by excavation process in cofferdam or auxiliary structures of foundation constructions

governing equations from the theoretical calculation of flow problem and those formulations based on finite element method for steady-state flow problem analysis is compiled

The purpose in this work is also to consider the stability analysis methods of cofferdam proposed previously A detailed study was carried out to clarify the cause of boiling type of failure in braced sheetpiled cofferdam as used for the construction of bridge piers The finite element method of analysis conducted here for better understanding the seepage boiling failure phenomenon includes the seepage analysis in 2-D plane condition as well as 3-D condition An example of problem associated with boiling type of failure inside cofferdam for Daiichi-Shinkawa Bridge was introduced and the cause of the problem was adequately determined and discussed by review from the analytical results of the problem by self-resetting program based on finite element method Additionally, the author conducted a series of calculations for influence factors

on boiling type failure inside cofferdam and found out general trend for the effect of each influence factors from the plots and described their relationships Furthermore, the author adequately proposed a simplified estimation method for design and check out the stability of cofferdam from the analytical results of the program code

Chapter 1 INTRODUCTION

1.1 General Introduction 1

1.2 Brief Literature Review of Previous Studies on Boiling Type Failure of Cofferdams 2

1.3 Composition of the Present Thesis 5

Chapter 2 THEORY OF SEEPAGE COMPUTATIONS 2.1 Darcy’s Law 6

2.2 Steady-State Flow Equations 7

2.3 Boundary Condition for Flow Problem 10

Chapter 3 FUNDAMENTALS FOR GROUND STABILITY ANALYSIS OF COFFERDAM 3.1 Introduction 11

3.2 Finite Element Method for Flow Problem 12

3.2.1 Governing Equations for Flow Problem 12

3.2.2 Formulation of Governing Equation in Finite Element 12

3.2.3 Weak Form of Boundary Problem for Flow Behavior 15

3.2.4 Interpretation of Weak Form of Governing Equation for Flow Problem.16 Chapter 4 A STUDY ON THE THREE-DIMENSIONAL EFFECT OF SEEPAGE FORCE ON THE STABILITY OF COFFERDAM 4.1 Design Method Used in Japan and Authorized by the Japanese Road Association 20

4.2 The Definition of Safety Factors Employed in This Study 21

4.3 Investigation of the Boiling Type of Failure inside the Cofferdam 23

4.4 Condition of the Damaged Ground 25

4.5 Investigation of the Factor of Safety 27

4.6 Parametric Investigation and Study of the Influence Factors on the boiling Type of Failure inside Cofferdam 31

4.6.3 Effect of Excavation Area and Anisotropic Permeability Layer 34

4.6.4 Effect of Weight of Footing Construction inside Cofferdam 35

4.6.5 Summary 37

4.7 A Simplified Estimation Method for the Factor of Safety against Boiling Type of Failure 38

4.7.1 Influence Factor of Shape of Cofferdam 45

4.7.2 Influence Factor of Sheetpile Penetration Depth of Cofferdam 53

4.7.3 The effect of Depth of Excavation 60

4.7.4 The Effect of Size of Excavation 69

Chapter 5 CONCLUSIONS AND RECOMMENDATIONS 5.1 Conclusions 77

5.2 Recommendations 77

ACKNOWLEDGEMENT 79

REFERENCES 80

Fig 1.1.1 Side view of new Daiichi-Shinkawa Bridge 2

Fig 2.2.1 Components of discharge velocity at six faces of an element of soil 8

Fig 4.1.1 Illustrated flow net and distribution of hydraulic potential in 2-D condition

21

Fig 4.2.1 Schematic diagram for the determination of factor of safety against boiling type of failure 22

Fig 4.2.2 Analytical condition for the determination of factor of safety against boiling type of failure 23

Fig 4.3.1 Cofferdam for Pier No 1 of Daiichi-Shinkawa Bridge 24

Fig 4.3.2 the installation of H-pile in cofferdam 24

Fig 4.3.3 Cofferdam just after the boiling type of failure occurred was filled with muddy water flow into the cofferdam 25

Fig 4.4.1 Soil profile of the damaged ground inside the cofferdam 26

Fig 4.4.2 the elevation of the damaged ground 27

Fig 4.4.3 Damaged cofferdam from boiling 27

Fig 4.5.1 Distribution of equi-potential line from 3-D FEM analysis of cofferdam for Pier No 1 of Daiichi-Shinkawa Bridge 29

Fig 4.6.1 Fundamental dimensions of the cofferdam employed in the parametric study

.32

Fig 4.6.1.1 Influence of analytical condition on the seepage inside and around cofferdam 33

Fig 4.6.2.1 Effects of the depth of permeable layer 34

Fig 4.6.3.1 Effects of excavation area and anisotropic permeability on the seepage inside and around cofferdam 35

Fig 4.6.4.1 Effects of footing on the seepage 36

Fig 4.6.4.2 Influence of some factors on the factor of safety against boiling type of failure in the cofferdam 36

Fig 4.7.1 Parameters employed in the program for the dimensions and soil properties of cofferdam 41

Fig 4.7.2 Safety factor in cofferdam as a function of the normalized horizontal space 44

Fig 4.7.1.1 Factors of safety of F sa and F sb as a function of aspect ratio b a/bb in

fundamental case 47

Fig 4.7.1.2 Factor of safety F sa as a function of aspect ratio b a/bb 47

Fig 4.7.1.3 Factor of safety F sb as a function of aspect ratio b a/bb 48

Fig 4.7.1.4 Seepage pressure ratio h a/H as a function of aspect ratio ba/bb 49

Fig 4.7.1.5 Maximum upward hydraulic gradient i max as a function of aspect ratio b a /b b

.49

Fig 4.7.1.6 the ratio (F sa)rectangle /(F sa)square as a function of aspect ratio b a /b b 50

Fig 4.7.1.7 the ratio (F sb)rectangle /(F sb)square as a function of aspect ratio b a /b b 51

Fig 4.7.1.8 the ratio of (i max)rectangle to (i max)square as a function of aspect ratio b a /b b 51

Fig 4.7.1.9 (h a /d i)rectangle/( h a /d i)square ratio as a function of aspect ratio b a /b b 52

Fig 4.7.2.1 Safety factor F sa as a function of the penetration depth ratio d i /b b 55

Fig 4.7.2.2 Safety factor F sb as a function of the penetration depth ratio d i /b b 56

Fig 4.7.2.3 Seepage force ratio h a/H as a function of the penetration depth ratio di/bb 56

Fig 4.7.2.4 Maximum upward hydraulic gradient i max as a function of the penetration depth ratio d i/bb 57

Fig 4.7.2.5 Factor of safety F sa normalized by its value as d i = b b/6 as a function of the penetration depth ratio d i/bb 58

Fig 4.7.2.6 Factor of safety F sb normalized by its value as d i = b b/6 as a function of the penetration depth ratio d i/bb 59

Fig 4.7.2.7 Maximum upward hydraulic gradient i max normalized by its value at d i = bb /6 as a function of the penetration depth ratio d i/bb 59

Fig 4.7.2.8 Seepage pressure ratio ha/di normalized by its value at d i = b b/6 as a function of the penetration depth ratio d i /b b 60

Fig 4.7.3.1 Safety factor F sa at the penetration depth ratio d i = b b/6 as a function of excavation depth ratio (d o -d i )/b b 62

Fig 4.7.3.2 Safety factor F sb at the penetration depth ratio d i = b b/6 as a function of excavation depth ratio (d o -d i )/b b 63

Fig 4.7.3.3 Maximum hydraulic gradient i max at the penetration depth ratio d i = b b/6 as a function of excavation depth ratio (d o -d i )/b b 64

Fig 4.7.3.5 the ratio (i max )*(b b/H) as a function of as a function of excavation depth

ratio (d o-di )/b b 65

Fig 4.7.3.6 the ratio (h a/di )*(b b/H) as a function of as a function of excavation depth ratio (d o-di )/b b 66

Fig 4.7.4.1 Safety factor F sa as a function of size ratio (b a/bb)i/(ba/bb)1 70

Fig 4.7.4.2 Safety factor F sb as a function of size ratio (b a /b b ) i /(b a /b b ) 1 71

Fig 4.7.4.3 Seepage pressure ratio as a function of size ratio (b a /b b ) i /(b a /b b ) 1 72

Fig 4.7.4.4 Maximum hydraulic gradient as a function of size ratio (b a /b b ) i /(b a /b b ) 1 72

Fig 4.7.4.5 the ratio as a function of size ratio (b 1 ( / ) ( / ) ( ) /( ) a b i a b sa b b sa b b F F a /b b ) i /(b a /b b ) 1 73

Fig 4.7.4.6 the ratio as a function of size ratio (b 1 ( / ) ( / ) ( ) /( ) a b i a b sb b b sb b b F F a/bb)i/(ba/bb)1 73

Fig 4.7.4.7 the ratio as a function of size ratio (b 1 max ( / ) max ( / ) ( ) /( ) a b i a b b b b b i i a/bb)i/(ba/bb)1 74

Fig 4.7.4.8 the ratio as a function of size ratio (b 1 a ( / ) a ( / ( / ) /( / ) a b i a b b b b b h H h H ) a /b b ) i /(b a /b b ) 1 75

Table 1.2.1 was prepared for the summary of literature review 4

Table 4.5.1 Calculation parameters for boiling type of failure in some cofferdams for bridge construction in Japan 30

Table 4.5.2 Calculation results for boiling type of failure in some cofferdams for bridge construction in Japan 30

Table 4.6.1 Analytical conditions for case study of the influence factors on boiling type of failure in the cofferdams for bridge construction 31

Table 4.7.1 Parameters using in the calculation of the effect of the dimensions of horizontal space 43

Table 4.7.1.1 Calculation results for the effect of shape of cofferdam 44

Table 4.7.2.1 Calculation results for the effect of penetration depth of sheetpile 54

Table 4.7.3.1 Calculation results for the effect of excavation depth 61

Table 4.7.4.1 Values of Shape factors, Penetration depth factors, and Reference value of seepage force and maximum hydraulic gradient 68

Table 4.7.4.1 Calculation results for the effect of excavation size 69

γ′ - submerged unit weight of soil [L MT ]

γw - unit weight of water [L-2MT-2]

A - base area of the soil prism [L2]

aa, ab - dimensions of footing [L]

ba, bb - excavation width [L]

d - penetration depth of sheetpile [L]

d´ - excavation depth [L]

Fs - factor of safety against boiling

F sa - factor of safety against boiling derived from the balance of seepage force and gravity force on the prism of soil mass

F sb - factor of safety against boiling derived from the comparison of maximum upward hydraulic gradient of groundwater, imax with its critical value ic

G s - specific gravity of soil grains

H – hydraulic potential head [L]

h a - average groundwater potential head which correspond to the pressure applied to the

bottom of the soil prism [L]

i c - critical hydraulic gradient

imax - maximum upward hydraulic gradient at the surface

kh, kv - horizontal and vertical coefficient of permeability [LT-1]

l – permeable layer thickness [L]

n - porosity

SPT - Standard Penetration Test

U - upward seepage force acting on the soil prism [L-1MT-2]

V - volume of the soil prism [L3]

ν - seepage velocity [L/T]

W′ - submerged weight of the prism [L-1MT-2]

Chapter 1 INTRODUCTION

1.1 General Introduction

Cofferdams are temporary structures built to create dry conditions during construction in riverbeds and in the bottom of lakes With such structures, the inflow of groundwater as well as free water can be presented and these structures are designed to resist the lateral earth pressures and water pressures For excavation depths of up to 12

m the braced-sheet piled cofferdam is generally more economical than the cellular and caisson type cofferdams stated by Teng [25] They are thus frequently employed during the construction of piers and abutments of bridges with medium span McNamee [15] indicated that the collapse of cofferdams could occur due to the upward seepage force at the base even though they are designed to adequately resist the lateral thrust of the soil and water Such a failure is generally regard as boiling type or piping type of failure Sometimes a small leakage of water can initiate such a failure, which eventually leads to the sudden disintegration of the whole base Even with noticeable deformation, a cofferdam can generally be considered to perform successfully if dry condition is prevailed during the construction works and the inflow of water is smaller than the amount pumped out at the site A boiling type of failure only can be detriment to the performance of the cofferdam but also can affect the construction schedule and create changes in the design of the permanent structures

During the boiling phenomenon, the upward seepage force is large enough to carry the sand and silt size particles with the discharge It is also a progressive type of failure wherein sudden flooding can occur inside the cofferdam with the continuous discharge

of soil grains Thus a piping connection is made under the tip of the sheetpiles between the inside and the outside of the cofferdam Once boiling has occurred at the base floor, the original consistency, strength, and the stiffness of the natural ground is lost and this results in an inadequate bearing capacity at the base level Additionally, the influence of boiling can cause irreparable and severe damages to the ongoing construction works in the nearby structures, as well as the traffic and other human activities in the neighboring area Thus it is always important to ensure that boiling type of failures do not occur in cofferdams

As stated earlier, a boiling type of failure occurred in the cofferdam of Pier No.1 at the downstream side of the new Daiichi-Shinkawa Bridge; see Fig 1.1 It was also stated that the cofferdam was designed with a minimum safety factor of 1.5 against

boiling type of failure in accordance with the ‘Directions for Road Earthworks’ of the

Japanese Road Association (JRA) [21] It should be noted that almost all cofferdams in

Japan are designed in accordance with the ‘Directions for Road Earthworks’ of JRA

However, in this particular site boiling failure has occurred and this has severely damaged the braced sheetpiles and the excavated ground base Thus, a detailed study is carried out in this thesis to clarify the cause of boiling type of failure in braced sheetpiled cofferdam as used for the construction of bridge piers The finite element method of analysis conducted here includes the seepage analysis in 2-D plane condition

as well as 3-D condition

Fig 1.1.1 Side view of new Daiichi-Shinkawa Bridge (after Miura 1999)

1.2 Brief Literature Review of Previous Studies on Boiling Type Failure of Cofferdams

The influence of the seepage force on the safe design of excavations with cofferdams is first addressed by Terzaghi [23] From the limited model tests conducted

on sheetpiled cofferdam, Terzaghi observed that the boiling type of failure in the heaved zone was confined to a soil prism adjacent to the wall Further, for the case of a row of sheetpiles in a uniform soil, the upheaved action is assumed to extend to a depth expect

to the pile penetration d and the width of the zone is taken as d/2 At the commencement

of failure the effective vertical stress within the zone is approximately zero on any

horizontal section Also, the lateral effective stress on the sides of the prism of soil influenced by the upheaval is also approximately zero

Terzaghi called this upheaval phenomenon as “piping” in the textbook, while the author of this study preferred the term boiling or boiling type of failure for the reasons

as presented in the subsequent section

A factor of safety F s can thus be formulated based on the submerged weight of

the prism (W’=d2γ’/2) and the upward seepage force (U=dγw h a/2) acting on the prism as

Terzaghi intuitively assumed that the excess pressure head h a causing the seepage

is less than half the value of the water head difference i.e h a < h 1/2 The above

assumption is based on a two- dimensional analysis with flow net This design method proposed by Terzaghi [23] has been widely used in the design codes on cofferdams Subsequently, Marsland [16] conducted an extensive series of model tests both in dense and in loose homogeneous sand in an open excavation filled with water The conclusions of Marsland were that in loose sand, failure occurred when the seepage pressure at the pile tip is sufficient to lift the column of submerged sand near the wall of the cofferdam However, in the case of the dense sand the failure occurred when the hydraulic gradient at the excavated face reached a critical value This critical gradient for the failure in the dense sand is given by

where G s is the specific gravity of the sand grains; n is the porosity

Bazant [1] established a failure criterion with reference to the shear strength of the soil, by plotting the ratio of the excess hydrostatic pressure at the pile tip to the embedment depth of the pile, against, the internal friction angle of the soil

McNamee [15] also presented a general survey of the work carried out at the Building Research Station in England concerning the two-dimensional seepage flow around sheetpiled excavations in open water Two main types of failure were identified: local failure and general upheaval The local failure is most likely to begin at a point on the surface adjacent to the sheetpile as it is located within the shortest seepage path; the local failure was termed as failure by ‘piping’, and the more widespread type of failure

as ‘heaving’ McNamee [15] also presented charts to determine the exit gradients and

the factor of safety against boiling Richart and Schmertmann [19], Davidenkoff and Franke [6], Griffiths [9], King [13] and some other researchers produced several charts

to permit the designers, a rapid estimate of the earth pressure, water pressure and the safety factor of sheetpiled cofferdams with different shape and arrangement of sheetpiles

Milligan and Lo [17], Bauer [2], Bauer et al ([3], [4]) and Kaiser and Hewitt [11] presented case histories and results of laboratory tests on failures of cofferdams due to boiling These authors need both empirical and numerical methods such as the finite difference method and the finite element method The importance of the use of realistic factor of safety related to the shape of the cofferdams and the geological conditions were also emphasized by these authors

Table 1.2.1 was prepared for the summary of literature review

Analytical Method Analytical condition Design method Authors (year)

Field observation Model

study

FEM Others 2-d

plane

2-d axis symmetry 3-d Chart Formula

(1) Flow net (2) Relaxation method (3) Limit analysis (4) Method of conformal mapping

1.3 Composition of the Present Thesis

Present thesis consists of 5 chapters The composition of thesis is as follows: Chapter 2: In this chapter the theory of calculation of flow problem at steady- state in natural ground condition is introduced And boundary condition for steady-state seepage analysis problem is also proposed

Chapter 3: This chapter aims to investigate the properly mathematical formulation of soil materials and conditions for analysis of the transient response behavior of excavation ground base subjected to upward seepage force, excess pore water pressure increasing by excavation process in cofferdam or auxiliary structures of foundation constructions in sand layer below ground water table Fundamentals of finite element method and finite element method for flow problem are also introduced in this charpter The program code for calculation the stability of excavation base in cofferdam which combines governing equations from the theoretical calculation of flow problem and those formulations based finite element method for steady-state flow problem analysis is compiled

Chapter 4: The purpose in this chapter is to consider the stability analysis methods of cofferdam proposed in 4.1 and 4.2 sections An example of problem associated with boiling type of failure in side cofferdam for Daiichi-Shinkawa Bridge is introduced in 4.3 and the cause of the problem is adequately determined and discussed

by review from the analytical results of the problem by self-resetting program based on finite element method in 4.4 and 4.5 Additionally, the author conducts a series of calculations for influence factors on boiling type failure inside cofferdam in 4.6, and finds out trends for the effect of each influence factors from the plots and their relationships in 4.7 Furthermore, in section 4.7, the author adequately proposes a simplified estimation method for design and check out the stability of cofferdam from the analytical results of the program code Only steady state seepage analysis is concerned and a series of parametric study can be carried out to accomplish the subjects listed above

In chapter 5 the conclusion and recommendations drawn out from this study are presented.

Chapter 2 THEORY OF SEEPAGE COMPUTATIONS

2.1 Darcy’s Law

Darcy (1856) made observations of the rate of flow of water through saturated

granular soils and obtained the empirical relationship Darcy’s law states that there is a

linear relationship between hydraulic gradient and discharge velocity for any given soil

(representing a case of steady laminar flow at low Reynolds number)

where ν is discharge velocity or seepage velocity;

k is defined as the coefficient of impermeability of medium or hydraulic

gradient;

i is the hydraulic gradient

It is important to be noted that Darcy’s law or seepage velocity is an artificial

velocity since it equals to the quantity of water that percolates per unit of time across a

unit area of a section The average seepage velocity νs at which the quantity of water

flows can therefore be calculated as the discharge velocity divided by the volume

porosity of n of a plane section

ν

νs =

The hydraulic gradient results from a difference in the potential across an

element of the medium If Δh represents the total potential head loss of fluid through a

distance Δs then, in the limit below

The minus convention is denoting a negative gradient The term of h, hydraulic

head, is equal to the sum of kinetic energy, the piezometric pressure, and the position

where ρ is the bulk density of liquid and g is the acceleration of gravity Note

that the term h is sometimes called the groundwater head or piezometric head It may

easily be verified that the first term, which represents the kinetic energy, is in effect

usually negligible in seepage studies (Cadergren, 1977) Hence, Equation 2.1.4 becomes

However, high velocity usually occurs in some situations, for example, adjacent

to the end of cut-off walls and in the vicinity of wells Modifications can be made to the

permeability to allow for high velocities (Volker, 1975)

The components of the seepage velocities ν are as follows

h k x h k y h k z

ννν

Where kx, ky, and kz are the component permeability These equations only hold

when the x-, y-, and z-axes coincide with principal permeability axes (Bear, 1972)

To solve problems of seepage flow, Darcy’s flow equation alone is not

sufficient It only gives three relations between four unknown quantities, the three

components of the velocity vector and the potential head An additional equation may

be obtained by realizing that the flow has to satisfy the principle of the conservation of

mass This leads to an equation of continuity for steady-state flow

2.2 Steady-State Flow Equations

Consider the steady-state flow through a small elementary volume of porous

media (see Fig 2.2.1) By assuming that the fluid flow through porous media is

incompressible, the principle of conservation of mass stipulates the sum of mass flow

into the three faces is equal to the sum of mass flow out the three opposite faces Since

the derivation could be restricted to water having a constant density, the conservation

principle could be enforced by ensuring that the total volume of water flowing into or

out of the element of soil is equal

The discharge velocity vector at the center point (x = y = z = 0) of the soil

element in Fig 2.2.1 has components νx, νy, and νz in the x-, y-, and z-directions,

respectively At the center of the element, the rate of flow in x-direction is νxdydz,

where dydz is the area of the element perpendicular to the x-direction Consequently, on

the plane x = - 0.5dx, the velocity in x-direction is

2

x x

dx x

ν

ν −∂

The rate of flow of water, entering the element across this plane, is given by the velocity

times the cross-sectional area as shown below

(

2

x x

dx x

dx x

ν

ν +∂

The net rate of flow into or out of the soil element in x-direction is the algebraic sum of

the flow rates in or out

x dxdydz x

ν

∂

z dxdydz x

ν

∂

Fig 2.2.1 Components of discharge velocity at six faces of an element of soil

In the y- and z-directions, the net volume of water flowing per unit of time into or out of

the element soil is then

According to the conservation principle the change per unit of time of the volume of

water in the element must be equal to zero Hence, we obtain

This is the steady-state continuity equation

Differentiating Eq.2.1.6 with respect to x, y, and z respectively and combining with Eq

On the assumption that kx, ky, and kz are constants in the x-, y-, and z-directions,

respectively, Eq 2.2.7 becomes

x x

If a soil anisotropic, it may be treated as if it were isotropic by introducing the following

transformation of the x-, y-, and z- dimensions

k

Where ko is any reference permeability, such as ko = 1 m/s or ko= kz

For example, if ko = kz = kv and kx = ky = kh, then the transformation of dimensions are as

follows

2.3 Boundary Condition for Flow Problem

The difference types of boundary conditions, which can apply in steady-state

flow, ca be grouped as follows

• The potential head of water takes known values on the boundary such as

water table ground water table

• For impermeability boundary no water crosses the boundary Since the

flow of water is proportional to the potential gradient, this is satisfied by setting∂ ∂ =h/ n 0, where n is in the direction normal to boundary

• A similar condition applies when the magnitude of the flow crossing the

boundary is known This is represented by the normal gradient∂ ∂h/ n

taking a specified value ∂ ∂ =h/ n - (velocity normal to boundary divided

by permeability normal to the boundary)

Of course, these conditions must be held on both internal and external

boundaries

Chapter 3 FUNDAMENTALS FOR GROUND STABILITY

ANALYSIS OF COFFERDAM

3.1 Introduction

In the analysis of the behavior of the ground and soil structures, soil material

must be treated as porous medium of multi-phase: solid, liquid, and gas phases Due to

the difficulty of formulation of multi-phases material, there are some variations of the

formulations for soil material, where simplified sets of governing equations are

employed depending on the mechanical condition and feature of calculation of problems

to be solved For example, some steady-state problem of ground deformation may be

analyzed using sample mono-phase formulation with sufficient accuracy And transient

behavior of saturated soil can be analyzed also using simplified model even under static

or quasi-dynamic condition The importance of interaction between the solid phase and

fluid phase depends on physical properties of soil layer, permeability, and/or required

feature of calculation results

The target in this study is the transient response behavior of excavation base

inside and around cofferdam in isotropic homogeneous sand layers to effect of seepage

force The boiling type failure inside cofferdam is estimated corresponding to seepage

force by increasing upward hydraulic gradient, water head loss between inside and

outside of sheetpile type cofferdam while excavating The effect of plane shape of

analytical conditional or the concentrated hydraulic gradient at the corner of braced

sheetpile type cofferdam is clearly stated that is quite significant in the cofferdam

analysis problem The consideration of balance of gravity force on the prism of soil

mass adjacent to sheetpile and seepage force was also determined having the

relationship to the boiling type of ground failure inside cofferdam In this problem the

interaction between fluid phase and solid phase as well as fluid flow in porous medium

is quite important Excess pore water pressure at a point on ground surface or under

footing of cofferdam resulting boiling type of failure may be caused as a function of the

uplift seepage influent more by excavation level and sheetpile penetration depth The

passive earth pressure acting on sheetpile wall, mechanism, and interaction of the angle

of friction at the soil-structure associating with deformation and failure are out of scope

of this study

3.2 Finite Element Method for Flow Problem

3.2.1 Governing Equations for Flow Problem

General form of the governing equations for flow problem can be expressed as

, , , ,

(a f ij j)i+(b f i )i+c f i i+df + = e 0 (3.2.1.1)

And flow rate (gradient of function with coefficient a and b) is defined as:

In the case of heat flow problem, scalar function f is temperature, tensor coefficient a ij

consists of heat conductivity coefficients, vector coefficient c i is for heat convection or

advection, coefficient d is for heat radiation, coefficient e is related to heat radiation,

and h is heat capacity For the harmonic analysis of transient state of heat flow, where

the function is given as a function of exponent of time fe i tω , Eq.(3.2.1.1) is modified as

where h is capacity, ω is angular frequency, and the scalar function f and all

the coefficients may be no longer real, but complex The modified governing equation

and the definition of flow rate are equivalent to those of steady state Then, in the

following, only Eq.(3.2.1.1) is concerned

3.2.2 Formulation of Governing Equation in Finite Element

The governing equation for the continuum element, plate element and rod element

where the left side is for heat capacity, the first term on right side is for heat

conductivity, the second tern is for head advection and the third is for heat generation

In rectangular coordinate system, cylindrical coordinate system, and spherical

coordinate system, the coefficient tensors in Eq.(3.2.1.1) are given as follows:

[rectangular coordinates]

where the scalar S is thickness of the plate element The plate element is available

only under three-dimensional condition

The vectors Δ ξi and Δ  ( 1, 2,3)ηi i= corresponding to increment local normalized

coordinate axis ( ,ξ η) are given by

i i

η ξ

η ξ

where the angle θ between y -axis and η -axis is given by

where the scalar A is sectional area of the rod element The plate element is

available under three-dimensional and two-dimensional conditions

The vectors Δ ξi (i=1, 2, 3, or 1, 2) corresponding to the increment in local

normalized coordinate axis (ξ) are given by

2( ) [ ] [ ] 1

N

k e i

k

N x

The gradient of function in the coordinate system can be calculated as

3.2.3 Weak Form of Boundary Problem for Flow Behavior

On two types of boundary, corresponding boundary conditions are given as

follows:

(forced boundary condition; on Γ ) f f = f (3.2.3.1)

(natural boundary condition; on Γ ) n

,

where vector n is outer unit normal on boundary i

In the framework of finite element method, the weak form of the governing

equation is dealt with The weak form for the governing equation is given as

integrations inside analytical region and over boundaries

Newly introduced scalar functions v and v are arbitrarily selectable functions

The forced boundary condition is presumably satisfied by the choice of the function f ;

the second integration of the weak form vanishes The first and second term of the first

integration can be modified based on the Green’s formulae

, , ,

Because the forced boundary condition is satisfied on the boundaryΓ , the second f

integration can be eliminated by the choice of the function so that v=0 on the

boundary Γ And, further, the condition, f

which can be employed without loss of generality as both functions are arbitrary,

simplifies the Eq.(3.2.3.3 ) into

This is the final expression of the weak form of the governing equation

3.2.4 Interpretation of Weak Form of Governing Equation for Flow Problem

The integrations in the weak form of governing equation derived in the previous

section can be interpreted as the summation of the integrations conducted in each of

finite elements The values of function at nodes and shape function are defined with

sequential number of nodes in finite element For the scalar functions f andv , the

values at nodes in nth element are indicated as

where d is analytical dimension, o is the order of element, N ne n( ) is total number

of nodes in nth finite element and equals to ( o+1)d The number in box blanket is

sequential node number in finite element, which is calculated as

1 1

d

j j

Then the interpolation of functions are conducted as follows

The integration is conducted as follows:

(First term on left side)

The integration in the summation is calculated numerically It should be noted

that this integration forms N ne×N ne matrix designated as ⎡⎣A a e n( )⎤⎦ Finally Eq.(3.2.4.10)

is modified as

{ }( ) ( ) { }( ) , ,

1 1 1 ( ) ( ) ( ) 1

e

e n

ne n ne n e

e n e

1 1 1 ( ) ( ) ( ) 1

e

e n

ne n ne n e

e n e

e n

d o n k

( ) ( ) ( )( ) ( ) ( )( ) [ ] [ ] [ ] [ ]

1 1 1 ( ) ( ) ( ) 1

e

e n

ne n ne n e

e n e

( ) ( )( ) ( ) [ ] [ ]

1 1 ( ) ( ) 1

e

e n

ne n e

e n e

N

e n e n n

N N

( ) 1( )( ) ( ) [ ] [ ]

1 1 ( ) ( ) 1

b

nb n b

n n b

N

n N N

where N is the number of natural boundaries on finite elements, b N nb n( ) in the

number of nodes in a natural boundary and equals to (o+1)d−1, and Γnb n( ) denotes the

region of nth natural boundary The vector is calculated as

Since this matrix form equation must be satisfied for any types of vectors{ }v , then the

solution of the following matrix equation (simultaneous equations) gives the answer

Chapter 4 A STUDY ON THE THREE-DIMENSIONAL EFFECT OF

SEEPAGE FORCE ON THE STABILITY OF COFFERDAM

In this chapter, a method for estimating the three-dimensional seepage force in cofferdam is described Then, a serious of parametric calculations by using finite element method are conducted to estimate the effect of the influenced parameters on boiling type of failure by self-resetting program and the results are presented in the form

of tables and charts Next, from the tables and charts we prepared from the calculations results, a simplified method for the calculation of the factor of safety against the boiling type of failure inside cofferdam is derived and applied in some case of actual cofferdams for bridges construction projects in Japan The results from the simplified method are also compared to the previously calculated results of the safety of factor by

using 3-D finite element method and 2-D plane method proposed by the ‘Directions for

Road and Earthworks’ of Japanese Road Association (JRA) [21]

In the literature there seems to be some confusion in describing the failure phenomena associated with the seepage force In this study the technical terms are used

in accordance with the ‘Directions for Road and Earthworks’ by JRA and the

‘Thesaurus for Geotechnical Engineering’ published by Japanese Geotechnical Society (JGS) [22] Here, ‘piping’ is referred to as a local progressive failure of cohesive ground due to the concentrated hydraulic gradient resulting from the seepage As a contrast to this description of ‘piping’, the term ‘boiling’ is used for the general failure of a less cohesive ground (consisting of sand and/or silty sand) when the upward seepage force exceeds the gravity force The term ‘heaving’ is used for whenever there is an upheaval

of clayey soil associated with or without upward seepage force

4.1 Design Method Used in Japan and Authorized by the Japanese Road Association

In Japan, all designs of braced sheetpiled cofferdams are based on the ‘Directions

for Road Earthworks’ specified by JRA In these methods the two-dimensional seepage

analyses as described by Terzaghi and Peck [24] is employed for the estimation of the upward seepage force The average value of the potential head is used without any effort to find the precise distribution of the potential heads This approach seems to be

conservative for both cases (with and without excavation) as shown in Fig 4.1.1a and 1b As long as infinite horizontal boundaries are assumed in 2-dimensional plane condition, the excess potential head at the tip of the sheetpile is reasonably equal to, or, less than, a half of the head difference between the water level inside and outside of the cofferdam In this method, the factor of safety as calculated by equation (4.2.2) using

h a =H/2, is required to be more than 1.5 So, if we use an approximate value of the

submerged unit weight of soil as equal to the unit weight of water (γ’ = γw), then the sheetpile penetration depth required should be more than three quarter of the total head

(a) without excavation (b) with excavation

Fig 4.1.1 Illustrated flow net and distribution of hydraulic potential in 2-D condition; (a) without excavation; (b) with excavation

4.2 The Definition of Safety Factors Employed in This Study

In this study, the finite element analysis is carried out to find the distribution of the potential head inside and around the cofferdam The steady state distribution of the potential head is of concern in this analyses and the following governing equation is solved in two-dimensional plane condition as well as 3-D conditions obtained from the finite element analysis

where k x , k y and k z are Darcy’s permeability coefficients in x-, y- and z-directions,

respectively The program code is an ordinary one, but was developed by the authors themselves, where the steady state seepage flow is calculated numerically using quadrilateral elements with four Gaussian points for numerical integration The

applicability of this type of numerical analysis in determining the groundwater potential distribution around a cofferdam was verified with the field observation data by Furukawa [7]

The factor of safety against boiling inside cofferdam was calculated with the precise distribution of groundwater potential inside the cofferdam as obtained by the finite element analysis Two kinds of definitions for the safety factors were employed, and their applicability was also discussed The first type of definition is derived from the considerations of the seepage force and the gravity force on the prism of soil mass as adopted by Terzaghi and Peck [24] The schematic diagram for the determination of the

factor of safety is shown in Fig 4.2.1 The factor of safety F sa can be defined as shown

below

Fsa = W′ / F = γ′ V / ha A γw (4.2.2) where γ′ is the submerged unit weight of soil, γw is the unit weight of water, V is the

volume of the soil prism, A is the base area of the soil prism, and h a is the average groundwater potential head which correspond to the pressure applied to the bottom of

the soil prism The width of soil prism was assumed to be equal to (d-d′)/2, as shown in

Fig 4.2.1; in 2-D plane condition and 3-D condition; the plan view of the cofferdams can be strip, or rectangle

Fig 4.2.1 Schematic diagram for the determination of factor of safety against boiling type of failure

The second definition of the safety factor against boiling type of failure was derived from the comparison of the maximum upward hydraulic gradient of the

groundwater, i max with its critical value i c as follows

F sb = i c /i max (4.2.3)

The critical gradient i c was given by Marsland [16]

i c = (G s -1)(1-n) = γ′/γw (4.2.4) The location at which the maximum upward hydraulic gradient appeared at the surface

is also shown in Fig 4.2.2

4.3 Investigation of the Boiling Type of Failure Inside the Cofferdam

As an example of the boiling type failure, the accident, which occurred in the cofferdam in the Daiichi-Shinkawa Bridge is introduced; the failure was reported with the field exploration by Imafuku [10] The cause of the failure was investigated with the finite element analysis of the actual cofferdams built for the construction of the bridge

Fig 4.2.2 Analytical condition for the determination of factor of safety against boiling type of failure

The boiling type of failure occurred in the temporary cofferdam built for casting Pier No 1 on the downstream side Daiichi-Shinkawa Bridge The plan and side views are shown in Fig 4.3.1; the riverbed consisted of homogeneous clean sand up to an elevation of -15.06 m After the installation of the sheetpiles and the pumping of the inside water, the ground inside the cofferdam was excavated to about 4.7m in depth, and the steel piles of the old bridge were cut and removed in dry condition Then, six steel H-piles were driven with vibro-hammer (60 kW power) to provide a temporary platform above the cofferdam (See Fig 4.3.2) The installation of the H-piles was also carried out under dry condition with continuous pumping; just after the completion of the piling, a small amount of leakage of water mixed with sand and mud was noticed In the judgment of the responsible people concerned, the leakage was considered to be insignificant in amount and was not progressive, therefore, no treatment was provided and the inside of the cofferdam was kept dry over night with pumping At 11.00 A.M,

on the following day, 18 hours after the placing of the H-piles, the groundwater blew up and within 5 minutes the cofferdam was filled with muddy water (see Fig 4.3.3) At the time of the boiling phenomenon, below the sheetpiles the inside and outside of the

Fig 4.3.1 Cofferdam for Pier No 1 of

Daiichi-Shinkawa Bridge (a) plan view; (b) side view

Fig 4.3.2 The installation of H-pile in cofferdam

cofferdam should have been confining and in stability eroded by a progressive piping failure

Fig 4.3.3 Cofferdam just after the boiling type of failure occurred was filled with

muddy water flow into the cofferdam

4.4 Condition of the Damaged Ground

The profile of the damaged ground with SPT value is shown in Fig 4.4.1 The

riverbed consisted of a uniform clean sand layer of D50 = 0.12-0.16mm and U c = 2, which was underlain by laminated silt layers Artesian water head was not detected beneath the damaged cofferdam; the artesian water head rarely causes a hydraulic failure of cofferdam as reported by Keriba [11] Due to the disturbance from the failure, the foundation ground became loose and its bearing capacity was reduced as indicated

by the SPT values From the ground surface elevation after boiling as shown in Fig 4.4.2, the disturbance of the riverbed and the excavated ground was found to be most severe on upstream side and in the central part of the riverbed The most damaged part

of the cofferdam was also on the upstream side and in the central part of the riverbed, where steel sheetpiles sank about 5 cm and the foundation was loosened, so that some sheetpiles could be swayed by hand As shown in Fig 4.4.3, after the boiling phenomenon took place, although there were no structural abnormalities, such as plastic deformation and disconnection between sheetpiles, there was notable amount of leakage through some of the joints between the steel sheetpiles The damaged cofferdam in Pier

No 1 was later recovered as follows; first, the sheetpiles were penetrated to an

additional depth of 0.2 m and then the deep mixing method was employed to stabilize the soil layer in the range close to the level of the sheetpile tips (Fig 4.4.1) After this treatment the construction was restarted, and completed, without any further changes to the original design of the superstructures The design of the opposite side of the cofferdam for this bridge, that is the cofferdam for Pier No.2, was revised to avoid the reoccurrence of the boiling type of failure; that is, the penetration depth of the sheetpiles were extended to reach the silt layers underling the sand layer

Sand Silt Gravel

Silty sand Organic silt

Fig 4.4.1 Soil profile of the damaged ground inside the cofferdam

3 2

(b) (a)

Value of depth of penetration of pile due

to self weight of vibro-hammer and H-pile :

Location of H-pile :

1.5 2 0

2.0 1.5

2.8

4

Fig 4.4.2 The elevation of the damaged ground; (a) the depth of penetration of pile with self weight of vibro-hammer and H-pile, (b) the variation of elevation of riverbed outside cofferdam

Fig 4.4.3 Damaged cofferdam from boiling; during pumping noticeable leakage was

observed through some joints between sheet piles

4.5 Investigation of the Factor of Safety

For the design of the cofferdam for Pier No 1 on the new downstream side of the Daiichi-Shinkawa Bridge, it was assumed that the submerged unit weight γ′ = (γw = 9.81 kN/m3), the total head difference, H = 6.66 m, and the penetration depth of the sheetpile,

d = 5.34 m According to the ‘Directions for Road and Earthworks’, the calculated

factor of safety was equal to 1.60 The analytical results of the groundwater potential with the finite element method in three-dimensional condition are shown in Fig 4.5.1 Inside the cofferdam, the equipotential lines are somewhat crowded and the average

potential head associated with uplift seepage force, h a is rather larger than a half of the

total head difference, H/2 This larger uplift head h a resulted in the reduction of the safety factor from the designed value The factors of safety based on the 3-D finite

element analysis were F sa = 1.20 and F sb = 1.25

y

z

h=0.9H

h=0.8H h=0.7H h=0.6H h=0.5H

h=0.9H

h=0.8H h=0.5H h=0.3H

0.000m 5.200m

20.800m

x

y

h=0.9H h=0.8H h=0.6H

Fig 4.5.1 Distribution of equi-potential line from 3-D FEM analysis of cofferdam for Pier No 1 of Daiichi-Shinkawa Bridge

Also the analysis was repeated for some additional cofferdams listed in table 4.5.1 and the results are presented in Table 4.5.2 As shown in the Table 4.5.2, a safety factor

of 1.5 as required by the ‘Directions for Road and Earthworks’ was not conservative

enough to take account of the 3-D effect of seepage force There were two more examples of the boiling type of failure in cofferdams (Bridge A and Bridge B); in these two cases with boiling, and, other four cases without boiling listed at the bottom of

Table 4.5.2, the safety factor F obtained from the 3-D analysis was less than 1.5 The

calculated values were 1.024 for Bridge A and 1.033 for Bridge B, while the required

value by the ‘Directions for Roads and Earthworks’ was 1.5 For the case study of

Daiichi-Shinkawa Bridge, the value of the submerged unit weight of γ’ = γw = 9.81 kN/m3 was not a conservative selection, when the site exploration was not conducted in the riverbed for the determination of soil properties For instance, a submerged unit weight γ’ of 84% of γw , which is required even for the critical condition of F sa = 1.0 is not an unrealistic value

Table 4.5.1 Calculation parameters for boiling type of failure in some cofferdams for

bridge construction in Japan

Size of cofferdam

Size of footing

Water head difference

Penetration depth

Excavation level

Permeable layer

Name of

Bridge

Number of national road

(place) [ comment ]

'γ(kN/m3) FSa FSb

3* Bridge B [ see Suzuki (1990) ] 8.82 1.033 1.052 1.543

4* Bridge C [ incomplete penetration ] (9.80) 1.011 1.103 1.200

[ partly incomplete penetration under construction ]

(9.80) 1.020 1.093 1.237

5’ 〃 [ result of the analysis in

the design ] (9.80) 1.623 1.727 2.151