EARTH PRESSURES AGAINST RIGID RETAINING …cibtech.org/sp.ed/jls/2015/03/285-JLS-S3-290-16-2.pdf · problems (including retaining walls) focus more attention on the properties and

Indian Journal of Fundamental and Applied Life Sciences ISSN: 2231– 6345 (Online) An Open Access, Online International Journal Available at www.cibtech.org/sp.ed/jls/2015/03/jls.htm 2015 Vol. 5 (S3), pp. 2456-2468/Morshedi et al.

Research Article

© Copyright 2014| Centre for Info Bio Technology (CIBTech) 2456

EARTH PRESSURES AGAINST RIGID RETAINING WALLS IN

THE MODE OF ROTATION ABOUT BASE BY APPLYING

ZERO-EXTENSION LINE THEORY

Morshedi S.M.1, Ghahramani A.2, Anvar S.A.2 and Jahanandish M.2 1Department of Geotechnical Engineering, Shiraz Univ., Shiraz, Iran

2Department of Civil Engineering, Shiraz Univ., Shiraz, Iran

*Author for Correspondence

ABSTRACT

This paper illustrates the use of the zero-extension line theory for determining distribution of active and

passive pressures from sand backfill behind a rigid retaining wall rotating about its base. These data

demonstrate a comparatively linear distribution in the active case. However, because of higher remaining

pressure on the lower wall (due to the lack of formation of the active state), the point of application of the

resultant force stands lower than H/3 (where H is the height of the wall). In the passive case, the pressure

distribution is approximately parabola-shaped. It also lacks a passive state formation on the lower wall,

which causes the location of the resultant force to increase in height. A comparison is made among the

obtained results, solutions resulting from Coulomb theory, and recent analytical and numerical research.

These comparisons show that the magnitude of the active force is nearly 20 percent greater than the

Coulomb solutions in the passive case. The resultant force is about 75 percent smaller than the Coulomb

solution, but agrees well with results obtained from Lancellotta and Shiau. Applying the Coulomb theory

is not reliable in either case.

Keywords: Zero-Extension Lines, Earth Pressure, Retaining Wall, Rotational Mode, Finite Difference

INTRODUCTION

Knowledge of the magnitude and distribution of earth pressures against retaining walls is important in the

design of many civil engineering structures. Common methods for stability analysis in geotechnical

problems (including retaining walls) focus more attention on the properties and behavior of soil in the

ultimate state. A number of these methods assume a failure surface and compute the necessary force for

moving the wedge above the assumed surface. In these methods, it is supposed that the soil fails in

accordance with the Mohr-Coulomb criteria. In the Coulomb limit equilibrium, the failure line in the

backfill consists of a straight line. However, Rankine assumed that the failure lines are formed throughout

the soil mass. Boussinesq examined the problem with the same assumption and consideration of the wall

roughness.

Various laboratory studies have assessed the classical theories of the lateral earth pressure as applied to

the design of retaining walls (Caltabiano et al., 1999; Fang and Ishibashi, 1986; Matsuo et al., 1978;

Sherif et al., 1982). These investigations demonstrated that the distribution pattern of lateral pressure

depended on the mode of wall movement. Therefore, the magnitude and point of application of the lateral

force for various wall movements (rotation about top, rotation about base and translation) are not equal,

and these amounts can be different from the Coulomb solutions (Fang and Ishibashi, 1986).

The progressive nature of failure in soil and the necessity of knowing the load-deflection behavior at

loads other than the limit loads have caused researchers to focus on the strain field. Researchers’ studied

the directions where the linear strain is zero. These directions, called zero extension lines (ZEL), have

many applications in understanding soil deformation (James and Bransby, 1971; Roscoe, 1970). A simple

pattern of ZEL has been used for finding the strain field behind a model retaining wall, with good

agreement between the predictions and observations (James and Bransby, 1971). Success in the prediction

of a strain field by the ZEL led the researchers to implement this theory in obtaining the mobilized

strength at different points of the soil mass (Roscoe, 1970). This information was necessary for obtaining

the stress field. It was this idea that led to the development of the method of associated field, which works


Research Article


well in the prediction of load-deflection behavior (Atkinson and Potts, 1975; James et al., 1972; Serrano,

1972). However, this method has a few problems: 1) It requires the use of an iterative process of

computations to achieve convergence and compatibility between the and fields. 2) It requires

elaborate interpolation routines because the velocities and stresses were not computed at the same points.

Attempts have been made to find a way to calculate the stress field by the ZEL alone. Two alternatives

have been presented for this problem. In the first, stresses were calculated by considering the force

equilibrium of the soil elements between the ZEL (Habibagahi and Ghahramani, 1979). In the second,

equilibrium-yield equations written along the stress characteristics were transferred into the ZEL

directions. The second method has more applications and has been shown to lead to the same results as

the first one (Anvar and Ghahramani, 1997).

The present study deals with the determination of the magnitude and distribution of active and passive

pressures behind a rigid retaining wall rotating about its base. For this purpose, equilibrium-yield

equations along the ZEL were transferred into incremental form, then used in writing a computer program

in the MATLAB software environment for drawing ZEL mesh and calculating active and passive

pressures behind a wall. To assess the ZEL method, the obtained results from this method have been

compared with Coulomb-theory solutions.

Zero-Extension Line Theory

If the soil displacements in the x and z directions are represented by U and W respectively, strains are

represented by xzzx ,, , and compressive strain is considered positive, then:

x

W

z

U

z

W

x

Uxzzx ;; (1)

The Mohr circle of strain is shown in Figure1 . If is the angle of dilation of soil, then sin is

related to the volumetric strain, v , and the maximum shear strain, max , by

max

sin v

(2)

Figure 1: Mohr circle and principle directions of strains

It is assumed that the direction of 1 and 1 coincide. If the origin of lines is P on the Mohr strain

circle, then linear strain is zero along the PA and PB directions. These are the zero extension lines. It is


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clear that both of these lines make the angle 24 with the 1 direction. We can use the

Mohr strain circle and algebraic manipulations to determine the two strain characteristic directions. For

the positive direction,

tandx

dz (3)

For the minus direction,

tandx

dz (4)

Accordingly, the field of zero extension lines and strain characteristics coincide. Through the aid of the

matrix calculations, the following relation between U and W along both characteristic directions is

obtained:

. . 0dU dx dW dz (5)

From Equations (3), (4) and (5), there exists the possibility of calculating U and W at point C if these

parameters and are known at points A and B (Figure 2). Therefore, if the field of ZEL and the

displacements at the boundaries are known, the displacements at the interior points of the field can be

calculated.

Figure 2: Strain characteristics

Equilibrium Equations along Zel

Consider a soil element under stress state xzzx ,, , as shown in Figure 3. It is assumed that the soil is at

the limit state of equilibrium at the same time as is the angle that 1 direction makes with x axis.

Figure 3: Stress field and principle directions Figure 4: Mohr circle of stress


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The resulting Mohr circle is the same as shown in Figure 4, in which the origin of planes is P . Thus, there

are two directions for yield: PA and PB , both of which connect point P to the tangency points of yield

lines with the Mohr circle. These directions that make angle 24 with 1 axis are yield

lines, which coincide with stress characteristics.

The equilibrium equations of the soil element shown in Figure 3 when subjected to body forces X and

Z are as follows:

;x xz xz zX Zx z x z

(6)

Through the Mohr stress circle and algebraic manipulation, the following relationships between u and

along the stress characteristic lines are gained:

Along the plus characteristics tandxdz :

2 tan tan tan

tan

du u c d X dz dx Z dx dz

c cu c dx dz dx dz

z x z x

(7)

Along the minus characteristics tandxdz :

2 tan tan tan

tan

du u c d X dz dx Z dx dz

c cu c dx dz dx dz

z x z x

(8)

Assume that and

denote the plus and minus stress characteristics that make angle with 1

axis and that and

denote the plus and minus zero extension lines that make angle with 1 axis

(which coincide with 1 axis) (Figure 5).

Figure 5: ZEL and stress characteristics lines

For any function f of the variable x , z then takes its directional derivatives with respect to the plus and

minus stress characteristics and ZEL. Thus, the following equations are obtained:

cos sin.

cos sin

f f

x

f f

z

(9)


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cos sin.

cos sin

f f

x

f f

z

(10)

From Eqs. (9) And (10), yield-equilibrium equations along stress characteristics and extensive algebraic

and trigonometric manipulations, the following equations containing the differential of u and along and

are be obtained:

For plus direction,

2 tan tan . .

1tan . tan tan .

cos

1tan .

cos

du u c d d X dz dx

Z dx dz u c d d

cdc d

(11)

For minus direction,

2 tan tan .

1tan . tan tan .

cos

1tan .

cos

du u c d d X dz dx

Z dx dz u c d d

cdc d

(12)

where

1 sin .sin sin sin cos; ;

cos .cos cos .cos cos

(13)

These equations are the stress equilibrium equations along the ZEL. It is worth noting that Eqs. (11) And

(12) on ZEL reduce exactly to Eqs. (7) and (8) on stress characteristics if , which holds for the

associative flow of soils. This creates an analytical tool with the known values of the stress state u and ,

displacements U and W and coordinates x and z at the points A and B along minus and plus ZEL

directions. We can find these parameters at the point C (Figure 2).

Practical Examples and Discussions

Equations (3), (4), (5), (11) and (12) were transferred into incremental form, then used in writing a

computer program in the MATLAB software environment for drawing ZEL mesh and calculating of the

active and passive pressures behind a wall. For these purposes, we need a dilation angle and an initial

friction angle for the backfill material. A dense sand, when exposed to the shear, shows a peak and critical

shear strength. As shown by Cole (1967), the angle of dilation remains constant during a large portion of

a shear test; hence, the results presented in Figure 6 were considered as the input for the ZEL program.

This gives a 15 and an initial

25 at a void ratio of 0.534.

The passive and active cases simulate the loading and unloading conditions on an element of soil.

Because of this, the results of shear tests simulating the loading condition cannot be used to predict active

behavior (Ghahramani and Clemence, 1980). The results of several tests indicate that, although the peak

of sin for the active and passive cases is nearly equal, the shear strain for the passive peak is 3 to 8


Research Article


times larger than the shear strain for the active peak of sin . Therefore, it is recommended to use shear

strain results such as in Figure 6 for the passive case and to use the same curve but a reduced on the

order of 1/3 for the active case.

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4

Shear Strain ( )

sin

j

-8.00E-02

-6.00E-02

-4.00E-02

-2.00E-02

0.00E+00

0 0.1 0.2 0.3 0.4

Shear Strain ( )

Vo

lum

etric

Stra

in ( n

)

(a) (b)

Figure 6: a) sin ; b) curves for a dense sand (Cole, 1967)

A general calculation process is as follows: a rotation angle, , is applied to the wall. From the

displacements (U & W) at the points along the wall, we can obtain the displacements throughout the ZEL

mesh. After the shear strain is calculated at the points of mesh from Eq. (1), the angle of internal friction

developed at these points can be obtained from a simple shear test. Finally, from the developed friction

angle and Eqs. (11) and (12), the magnitudes of u and at each point of the mesh and therefore the

pressures behind the wall can be determined.

To assess and compare the results obtained from the ZEL program with the classical theory of earth

pressure, two examples have been examined for the active and passive cases. In these examples, the

height of the wall ism2 , the unit weight of the backfill material is

3

5.18 mKN, the intensity of surcharge is

2

5 mKN and the number of mesh divisions considered is12 .

Active Case: The active earth pressure distribution behind a wall rotating about its base is generally

believed to be hydrostatic, and experimental evidence has proven this for most of the wall depth (Sherif et

al., 1984). However, the stress condition near the bottom of the wall demonstrates a more complicated

situation.

Figure 7a shows typical change of active pressure distribution as a function of wall rotation angle. As is

shown in Figure 7b, the active earth pressure decreases rapidly soon after the wall starts to rotate up to 05.1 . It then increases at a much slower rate as the wall rotation angle increases. When the upper

parts of the wall enter the active state at a rotation angle of about 3.0 in Figure 7a, the remaining higher

pressures at the lower parts of the wall lead the point of application of the resultant force to a lower

position. In these stages, the top of the wall translation is about 0.005 H. This implies that the bottom

point of the wall will never be able to completely enter the active stage because it requires a great

translation, which opposes the adequate serviceability of the retaining wall. This problem was also

pointed out by Sheriff et al., (1984). The obtained magnitude of the total active thrust from the ZEL

analysis for this wall is kN4.10 , which differs from the Coulomb solution of

KN45.8 by about %20 .

Figure 7c shows the changing point of application of the resultant force as a function of wall rotation

angle. It is necessary to mention that the Coulomb theory does not directly determine the lateral pressure

distribution, but the triangular shape of the linear backfill can be shown. From this interpretation of the

Coulomb theory, the point of application of the resultant force is m74.0 from the base. Because of higher


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pressure at the lower part of the wall, the point of application falls to a lower location m72.0 from the

base of the wall. According to the Coulomb theory, the resultant force is inclined at an angle w from

perpendicular to the back face wall, while the results given from ZEL analysis show that this angle is a

little smaller than w .

If, in the active state, the shear strength is equal through the whole height of the wall (i.e., max ), then

the results obtained from the ZEL analysis will precisely match the Coulomb solution (see Figure 7a).

Such an occurrence, however, is practically impossible.

In addition, a continuous mass of dense sand will degenerate into quasi-rigid blocks separated by rupture

surfaces soon after the shear strain in the mass exceeds that required to cause the sand to reach its peak

stress ratio condition.

Figure 8a shows the initial and deformed ZEL net for the rotation angle 05.1 of a m2 wall. In the

active case, shown in Figure 8b, the ZEL nets for a smooth wall and a rough wall are the approximately

the same.

This implies that the roughness of the wall has very little effect on the active pressures. To demonstrate

this aspect, results are given in Table 1 for ZEL analysis at different magnitudes of w , along with

solutions from Coulomb theory and Lancellotta’s equation (Lancellotta, 2002).

Table 1: Active pressure Coefficient

w ak

Coulomb Solution Lancellotta ZEL Analysis

0 0.1918 0.1918 0.2287

1/3 0.1787 0.1740 0.2210

1/2 0.1772 0.1667 0.2235

2/3 0.1787 0.1601 0.2291

1 0.1861 0.1523 0.2433

Passive Case: Figures 9a and 9b show the changes of passive pressure distribution and passive resultant

force, respectively, as functions of wall rotation angle. It is evident from Figure 9b that the passive

pressure first increases due to shearing and then decreases, similar to the results of the simple shear test.

In this case, the formation of a fully passive state at the wall base is very difficult, if not impossible. At a

rotation angle of about 0.1 , the upper part of the wall reaches the passive state, while the mobilized

friction angle in the lower parts is much less than max .

The strain is high near the top of the wall but becomes much smaller at greater depth. As a consequence,

high stresses will be generated at the top of the wall and cause the resultant force to rise to higher

location.

At later stages, strain in the sand near the top of the wall is large and falls below its peak value. Thus

the stresses at the top of the wall decrease below their early peak value. No rupture surfaces are predicted

near the toe of the wall, even after 5 of wall rotation ( 1.0 ).

As seen in Figure 9a, the passive pressure distribution is not simply triangular. This change of earth

pressure distribution causes the resultant force to rise to a higher location. Figure 9c shows the change of

the point of application of resultant force as a function of wall rotation angle. If according to Figure 9b,

the pressure distribution for 4.2 is chosen as the passive case, and the necessary displacement for

the formation of passive state is about 0.04H, which equals cm0.8 for a

m2 wall.


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0 5 10 15 20 25-2.5

-2

-1.5

-1

-0.5

0

Active Pressure (Kpa)

He

igh

t o

f th

e W

all

(m

)

Coulomb Solution

ZEL Analysis with

ZEL Analysis for

ZEL Analysis for

ZEL Analysis for

ZEL Analysis for

ZEL Analysis for

max

(a)

0 0.5 1 1.5 2 2.5 3 3.50

5

10

15

20

25

To

tal

Ac

tiv

e T

hru

st

(KN

)

Rotation Angle (Deg.)

ZEL Analysis

Coulomb Solution

(b)

0 0.5 1 1.5 2 2.5 3 3.50.71

0.72

0.73

0.74

0.75

0.76

0.77

0.78

0.79

Po

int

of

Ap

pli

ca

tio

n f

rom

th

e B

as

e (

m)


ZEL Analysis

Coulomb Solution

(c)

Figure 7: a) Distribution of Active Pressure at Different Wall Rotation Angle; b) Active Resultant

Force versus Wall Rotation Angle; c) Point of Application of Resultant Force versus Wall Rotation

Angle


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-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-2.5

-2

-1.5

-1

-0.5

0

Distance from the Back of the Wall (m)

He

igh

t o

f t

he

Wa

ll (

m)

Mixed Zone

Rankine Zone

Goursat Zone

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-2.5

-2

-1.5

-1

-0.5

0


He

ig

ht o

f th

e W

all (m

)

(a) (b)

Figure 8: a) Initial and Deformed ZEL Net at Active Condition with 2 w; b) ZEL Net with 0w

The resultant force from this analysis is KN24.435 which, in comparison with the Coulomb theory equal

to KN24.770 , demonstrates the overestimation and lack of safety of the Coulomb solutions. The

Coulomb theory estimated greater passive force of about %75 , which was confirmed in previous

investigations (Fang and Ishibashi, 1986; Ghahramani and Clemence, 1980; Habibagahi and Ghahramani,

1979; Harr, 1966; Jahanandish, 1988; Jahanandish et al., 1989; James and Bransby, 1970; James and

Bransby, 1971; James et al., 1972; Lancellotta, 2002; Matsuo et al., 1978; Roscoe, 1970; Serrano, 1972;

Sherif et al., 1984; Sherif et al., 1982).

The point of application of the resultant force was calculated at m73.0 from the base, whereas the

Coulomb theory was calculated m74.0 from the base.

Figure 10a shows the initial and deformed ZEL net for the rotation angle 2.1 for the

m2 wall. In the

passive case, shown in Figure 10b, the ZEL net for a smooth and a rough wall are different. This implies

that the roughness of the wall has a strong effect on the passive pressures. This condition is for the case

2 w , but as shown in Figure 10c, the error of the Coulomb method increases with the w increase,

and when w becomes greater than 2 , the error amount of this method goes beyond %100 .

Furthermore, the results obtained from ZEL analysis for different magnitudes of w with solutions

resulted from numerical and analytical solutions of Lancellotta and Shiau (Shiau et al., 2008) are given in

Table 2.

Table 2: Passive pressure Coefficient

w

pk

Coulomb

Solution Lancellotta

Shiau ZEL Analysis

LB UB

0 5.2138 5.2138 5.2585 5.2896 5.2135

1/3 10.1502 8.5405 8.3928 9.7448 7.8761

1/2 15.9174 10.3508 11.2902 13.1728 9.6899

2/3 29.005 12.0727 15.5012 17.8164 11.8064

1 122.9841 13.8985 29.37 33.6216 14.5544


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0 100 200 300 400 500 600 700-2.5

-2

-1.5

-1

-0.5

0

Passive Pressure (Kpa)

He

igh

t o

f th

e W

all

(m

)

Coulomb Solution

ZEL Analysis with

ZEL Analysis for

ZEL Analysis for

ZEL analysis for

ZEL Analysis for

ZEL Analysis for

max

(a)

0 1 2 3 4 5 6 7 8100

200

300

400

500

600

700

800

To

tal

Pa

ss

ive

Th

rus

t (K

N)


ZEL Analysis

Coulomb Solution

(b)

0 1 2 3 4 5 6 7 80.68

0.7

0.72

0.74

0.76

0.78

0.8

0.82

Po

int

of

Ap

plic

atio

n f

rom

th

e B

ase

(m)


ZEL Analysis

Coulomb Solution

(c)

Figure 9: a) Distribution of Passive Pressure at Different Wall Rotation Angle; b) Passive Resultant

Force versus Wall Rotation Angle; c) Point of Application of Resultant Force versus Wall Rotation

Angle


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0 0.5 1 1.5 2 2.5 3 3.5-2.5

-2

-1.5

-1

-0.5

0

0.5


He

igh

t o

f t

he

Wa

ll (

m)

Rankine Zone

Goursat Zone

Mixed Zone

0 0.5 1 1.5 2 2.5 3-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0


He

ig

ht o

f th

e W

all (m

)

(a) (b)

0

200

400

600

800

1000

1200

1400

1600

1800

0 5 10 15 20 25 30

Friction Angle Between Wall and Soil (Deg.)

Pass

ive

Th

rust

(K

pa)

Coulomb Results Calculated From ZEL

(c)

Figure 10: a) Initial and Deformed ZEL Net at Passive Condition with 2 w; b) ZEL Net

with 0w ; c) Variation of Coulomb Solutions and ZEL Results versus Friction Angle between Wall

and Backfill

It is necessary to mention that the numerical analysis of Shiau is related to the translational mode, while

previous investigations demonstrated that the pressure values in this mode are greater than the rotational

one.

CONCLUSIONS

On the basis of the results obtained from the examined examples with the ZEL program, one can conclude

that:

In the mode of rotation about the base for a dense sand backfill, the formation of the active and passive

states occur at the displacement of about H04.0 and H005.0 from the top of the wall, respectively,

which agrees with the presented magnitudes in the geotechnical references.

Because the formation of the active state at the wall base is practically impossible, the presented values

of the active force by the Coulomb theory are less than the real ones, and the results obtained from ZEL

analysis are greater by about %20 .


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The existence of remaining higher pressures at the lower part of the wall causes the point of

application of the resultant force to fall to a lower location, and the pressure distribution is approximately

linear.

The resultant force direction makes a 3.17 angle with the vertical direction crossed to the back

surface of the wall, which is less than the soil-wall friction angle 35.212 .

In the passive case, because of the lower levels of displacement at the lower parts of the wall, the

applied pressures are much less than the anticipated pressures by the Coulomb theory. The results

obtained from the ZEL analysis demonstrate that the Coulomb solutions are about %75 greater than the

real quantities.

The passive pressure distribution has a parabola shape, and the point of application of the resultant

force is about 3H from the base of the wall.

When it is assumed that the max has been mobilized through the whole height of the wall, the results

obtained from the ZEL analysis are equal to the Coulomb solution in the active case. In the passive case,

the Coulomb solutions are greater by about %60 than the ZEL results.

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EARTH PRESSURES AGAINST RIGID RETAINING …cibtech.org/sp.ed/jls/2015/03/285-JLS-S3-290-16-2.pdf · problems (including retaining walls) focus more attention on the properties and