Research Outline

This research utilizes the NCTU model wall facility to investigate the earth pressures against a non-yielding wall. The at-rest earth pressure theory and experimental findings associated with vibratory compaction are summarized in Chapter 2. Details of the NCTU non-yielding model wall system and the vibratory compactor used for the experiments are discussed in Chapter 3. Test results regarding the characteristics of backfill and soil density control are introduced in Chapter 4.

To investigate the earth pressure and dynamic stress path induced by compaction, the backfill was prepared by air-pluviated method and compaction method. A vibratory compactor was employed to densify the cohesionless fill. Experimental results of the earth pressures and stress path due to vibratory compaction are reported in Chapter 5.

1.3 Organization of Dissertation

This paper is divided into the following parts:

1.Review of theories regarding the earth pressure at-rest and past investigations about soil compaction. (Chapter 2)

2.Description of National Chiao Tung University non-yielding retaining-wall facility. (Chapter 3)

3.Backfill characteristics and soil density control. (Chapter 4)

4.Experimental results of earth pressure and dynamic stress path due to vibratory compaction, and the Comparison of Theoretical and Experimental Stress Paths.

(Chapter 5)

5.Conclusions. (Chapter 6)

Chapter 2

Literature Review

To improve its engineering properties, contractors are generally required to compact the loose soils to increase their unit weights and reducing settlements.

Previous studies associated with the compaction-induced effects such as the change of soil density, the change of stresses in the soil mass and mechanism of soils under compaction are discussed in this chapter.

2.1 Earth Pressure At–Rest

2.1.1 Coefficient of Earth Pressure At–Rest

As shown in Fig. 2.1(a), a soil element A located at depth z is compressed by the overburden pressure

σ

_v =

γ z

. During the formation of the deposit, the element A is consolidated under the pressure

σ

_v. The vertical stress induces a lateral deformation against surrounding soils due to the Poisson’s ratio effect.

Over the geological period, the horizontal strain is kept to be zero and the surrounding soil would develop a lateral stress to counteract the lateral deformation. A stable stress state will develop that the principal stresses acts

σ

1 and

σ

₃ on the vertical and horizontal planes, as shown in Fig. 2.1(b).

The soil in a state of static equilibrium condition is commonly termed as the

K

o condition. Donath (1891) defined the ratio of the horizontal stress

σ

_h to vertical stress

σ

_v is as the coefficient of earth pressure at-rest, Ko, or

v h

K

o

σ

=

σ (2.1)

since

σ

_v = , then

γ z σ

_h =

K

_o

γ z

, where

γ

is the unit weight of soil.

For an isotropic soil element shown in Fig. 2.2, if the soil behaved as an ideal elastic material, based on the mechanics of materials, the lateral strain

ε

y

can be expressed as:

_y ^y ( _{x z})

E

E ν σ σ

ε

=

σ

− + (2.2)

or

_h ^h

(

_{h v}

) E

E

ν σ σ

ε

=

σ

− +

(2.3)

where E is the elastic modulus and

ν

is the Poisson’s ratio of the soil.

Base on the definition of the at-rest condition, the lateral strain would be zero (εh= 0) under the application of stress state and the . Then the Eq. 2.3 can be written as:

v o

h

K

σ

σ

=

_h

= 1 ( K

_{o v}

− K

_{o v}

−

_v

) = 0 E

σ ν σ νσ

ε (2.4)

ν ν

= − 1

K

o (2.5) It should be mentioned that Eq. 2.5 is applicable for the isotropic and elastic materials only. However, the behavior of soil element is more complex and far from these assumptions. It is evident that the relationship between Ko and elastic parameter,

ν

of Eq. 2.5 is not practical for predicting in-situ horizontal stress.

2.1.2 Jaky’s Formula

strength properties of a soil and Ko. The empirical relationship to estimate Ko of coarse-grained soil is discussed in the following section.

Mesri and Hayat (1993) reported that Jaky (1944) established a relationship between Ko and maximum effective angle of internal friction

φ

by analyzing a talus of granular soil freestanding at the angle of repose. Jaky (1944) supposed that the angle of repose is analogous to the angle of internal friction

φ

. This is reasonable for a sedimentary, normally consolidated material. Jaky (1944) reasoned that the sand cone OAD in Fig. 2.3 is in a state of equilibrium and its surface and inner points are motionless. The horizontal pressure acting on the vertical plane OC is the earth pressure at-rest. Slide planes exist in the inclined sand mass. However, as OC is a line of symmetry, shear stresses can not develop on it. Hence OC is a principal stress trajectory. Based on the equations of equilibrium, Jaky expressed the coefficient of earth pressure at-rest Ko with the angle of internal friction,

φ

:

φ φ φ

sin 1

3sin 1 2 ) sin 1

( +

− +

o =

K

(2.6)

In 1948, Jaky presented a modified simple expression given by Eq. 2.7.

K

_o =1−sin

φ

(2.7)

Mayne and Kulhawy (1982) reported that, the approximate theoretical relationship for Ko for normally consolidated soils supposed by Jaky appears valid for cohesionless soils. Using Jaky’s equation to estimate the in-situ lateral earth pressure is reliable for most engineering purposes.

2.2 Plane Strain Condition

In many soil mechanics problems, a type of state-of-stress that is often encountered is the plane strain condition. Referring to Fig. 2.4, for the strip footing the strain in the y direction at any point P in the soil mass is equal to zero (εy = 0). The normal stress σy at all sections in the xz plane (i.e., normal to the y axis) are the same,and the shear stresses (τyx = 0,τyz = 0) on these sections are zero. Under a plane-strain state of stress, the normal and shear stresses on the plane normal to the x axis are equal to σx and τxz. Similarly, the normal and shear stress on the plane normal to the z axis are σz and τzx. The relationship between the normal stresses can be expressed as

(

x z

y ν σ σ

)

σ

= +

(2.8) where ν is Poisson’s ratio.

2.3 Distribution of Contact Stress over Footings

In calculating the vertical stress σz acting between a footing and soil, it is generally assume that the foundation of a structure is flexible and the contact stress σx is uniform. The actual nature of the distribution of contact stress will depend on the stiffness of the foundation and the soil on which the nature of the foundation is resting.

In Fig.2.5 (a), when a flexible foundation resting on a cohesionless soil, the distribition of contact pressure will be uniform. However, the maximum settlement will in the center of the foundation.This occus decause the soil located at the edge of foundation lacks lateral confining pressure and hence

will settle uniformly. The maximum contact pressure will be on the center of foundation.

2.4 Effects of Soil Compaction on Earth Pressure

Compaction of a loose soil can produce a stiff, settlement-free and less permeable mass. It is usually accomplished by mechanical means that cause the density of soil to increase. At the same time the air voids were reduced. It had been realized that the compaction of backfill material has important effects on the earth pressure in the soil mass.

Several theories and analytical methods had been proposed to analyze the residual lateral earth pressures induced by soil compaction. Most of these theories introduced the idea that compaction represented a form of over consolidation, where stresses resulting from a temporary or transient loading condition were retained following the removal of this load.

2.4.1 Study of Broms

Considering placement and compaction of horizontal layers of backfill adjacent to a non-deflection vertical wall, Broms (1971) proposed an analytical procedure based on the concept of hysteretic loading and unloading behavior.

The stress path of hysteretic model that Broms’ analytical procedure based is shown in Fig. 2.6a. Consideration of a soil element existed at some depth of backfill, the initial horizontal stress state of the element can be illustrated as

σ

hi

= K

o

σ

vi which is shown at point A in Fig. 2.6. When the compactor was positioned immediately above the soil element, an increase of the vertical stress results in an increase in horizontal stress on the basis of the assumption of no

lateral yield. The stress state can be expressed as

σ

hm

= K

o

σ

vm (point B). As the compactor moves off the fill, a subsequent decrease in vertical effective stress (unloading) results in no lateral stress decrease until a limitation (Kr-line) is reached (point C). The assumption is made that the maximum value of the horizontal stresses induced by compaction sustained until the vertical stress is reduced below a critical value at point C as shown in Fig. 2.6. After that, further unloading results in a decrease in horizontal stress through the stress path as

σ

hf

= K

r

σ

vi (point D) until the original vertical stress is reached. Kr is the coefficient of lateral earth pressure (

^K

^o

^{≤ K}

^r

^{≤ K}

^p, where Kp = coefficient of passive earth pressure). Broms (1971) assumed that Kr equals to 1/Ko. Compared with the residual horizontal stress,

σ

hf and initial horizontal stress,

σ

hi at the same vertical effective stress. It is obviously that the

σ

hf is much higher than

σ

hi. The process of soil compaction would result in a higher residual horizontal stress exists.

For a deeper soil element, the vertical stress on the soil element increases under the roller load from A^’ to B^’, and upon unloading the full maximum horizontal load (

σ

hm) is retained. Therefore, a critical depth zc will exist, where the stress state after compaction will return exactly to point C^’. The critical depth zc can be expressed as follows:

r vm o

c

K

z K

γ

=

σ (2.10)

where

σ

vm

= γ z + Δσ

v,

γ z is the vertical stress due to the weight of soil, and Δσ

v is the temporary increase in vertical stress at depth z due to the compactor.

Using the method proposed by Broms to calculate the compaction-induced

earth pressure involves incremental analysis of the stresses resulting from the placement and compaction of each layer of backfill. Compaction at any point is modeled as the application of a transient increase in vertical effective stress (

Δσ

v) caused by the compaction vehicle as determined by simple Boussinesq elastic analysis, followed by subsequent removal of the transient vertical load.

The horizontal effective stresses due to the transient compaction loading, as well as those due to surcharge increases as a result of fill placement, are then determined by the model shown in Fig. 2.6.

Considering the effect of placing and removing a compactor at the surface of the fill, the distribution of lateral pressure due to compaction proposed by Broms (1971) is shown in Fig. 2.7 (a). Before compaction is applied to the fill, the soil element is under the condition of at-rest, and the horizontal pressure is equal to Ko

σ

v (curve 1). The application of the compactor leads to an increase in vertical stress which decreases with depth. The maximum horizontal pressure can be calculated with Ko

σ

vm, where

σ

vm equals to

σ

v

+ Δσ

v and

Δσ

v is the increase in vertical stress at any depth due to the compactor (curve 2). As the compactor is removed, the backfill below the critical depth retains the increased horizontal stress and the fill above the critical depth reduces its horizontal stress to Kr

σ

v (curve 3). Based on the above discussions, as the backfill is compacted at the surface, the profile of the pressure distribution is indicated by the shaded area in Fig. 2.7 (a).

In reality, compaction is carried out regularly on thin layers of fill up the back of the retaining wall. The residual lateral pressure distribution is then given by the locus of the point A as the surface of the fill moves upward. A simplified distribution is illustrated in Fig. 2.7 (b).

2.4.2 Study of Duncan and Seed

Duncan and Seed (1986) presented an analytical procedure for evaluation of peak and residual compaction-induced stresses either in the free field or adjacent to vertical, non-deflecting soil-structure interfaces. This procedure employs a hysteretic Ko

-loading model shown in Fig. 2.8. The model is adapted to incremental analytical methods for the evaluation of peak and residual earth pressures resulting from the placement and compaction of soil. When the surcharge is applied on the soil surface, it will increase the vertical stress and the horizontal stress. In Fig. 2.8, as the virgin loading is applied on the soil, both

σ

v and

σ

h increase along the Ko -line (Ko

= 1-sin φ

).

Nevertheless, when the surcharge is removed,

σ

v and

σ

h would decrease along the virgin unloading path. As virgin reloading was applied again, the increment of earth pressure is less than that induced by the first virgin loading.

The hysteretic model may be applied to the analysis of compaction as represented by a transient, moving surficial load of finite lateral extent by directly modeling loading due to increased overburden as an increase in vertical effective stress (Δ

σ ’

v). To model compaction loading in terms of the peak virgin, compaction-induced horizontal stress increase (Δ

σ ’

h,vc,p) is defined as the horizontal effective stress which would be induced by the most critical positioning of the compactor. The Δ

σ ’

h,vc,p could be evaluated by the simple elastic analysis if the soil had been previous uncompacted (if the soil had no

“lock-in” residual stresses due to previous compaction). While the hysteretic model is applied to the analysis of compaction loading cycle, the Δ

σ ’

h,vc,p should be transformed to an equivalent peak vertical load increment (Δ

σ ’

v,e,p) calculated as

o p vc h p

e

v

K

' , ' ,

, ,

σ

= ^Δ

σ

Δ

(2.11)

calculated lateral stress increase rather than directly calculated peak vertical stress increase multiplied by Ko, Ka or some other coefficient. Seed and Duncan (1983) concluded that either in the free field, or at or near vertical, nondeflecting soil/structure interfaces, Δ

σ ’

h,vc,p resulting from surficial compaction loading can be calculated directly by simple elastic analysis. The parameter of Poisson’s ratio, ν for surficial compaction loading may be chosen according to the empirically derived relationship

( 0 . 5 ) 2

1

o

o ν

ν

ν

= + −

(2.12)

where

o o

o

K

K

= +

ν

1

K

o

= 1-sin φ

Seed and Duncan (1983) also brought up a simple hand calculation procedure which results in good agreement with the incremental procedure described above. In Fig. 2.9, it is apparent the simple hand solution has a good agreement with the incremental procedure.

2.4.3 Study of Peck and Mesri

Based on the elastic analysis, Peck and Mesri (1987) presented a calculation method to evaluate the compaction-induced earth pressure. The lateral pressure profile can be determined by four conditions on

σ

_h, as illustrated in Fig. 2.10 and summarized in the following.

1. Lateral pressure resulting from the overburden of the compacted backfill,

σ

_h =(1−sin

φ

)

γ z

(2.13)

2. Lateral pressure limited by passive failure condition,

σ_h

= tan

²

( 45 +

φ

/ 2 )

γ

z

(2.14)

3. Lateral pressure resulting from backfill overburden plus the residual horizontal

stresses,

σ_h

= −

φ γ

z + ( 5

^φ

− 1 ) Δ

σ_h

4

) 1 sin 1

(

^1.2sin

(2.15)

where Δ

σ

h is the lateral earth pressure increase resulted from the surface compaction loading of the last backfill lift and can be determined based on the elastic solution.

4. Lateral pressure profile defined by a line which envelops the residual lateral pressures resulting from the compaction of individual backfill lifts. This line can be computed by Eq. 2.16.

σ φ

( 5 5

φ

)

γ

4

sin

1 − −

_1.2sin

Δ =

Δ z

h

(2.16)

Fig. 2.10 indicates that near the surface of backfill, from point a to b, the lateral

pressure on the wall is subject to the passive failure condition. From b to c, the overburden and compaction-induced lateral pressure profile is determined by Eq. 2.15.

From c the lateral pressure increases with depth according to Eq. 2.16 until point d is reached. Below d, the overburden pressure exceeds the peak increase in stress by compaction. In the lower part of the backfill, the lateral pressure is directly related to the effective overburden pressure.

2.4.4 Study of Chen and Fang

Chen and Fang (2008) reported some experiments in nonyielding retaining wall at National Chiao Tung University to investigate influence of earth pressure due to vibratory compaction. Air-dry Ottawa sand was used as backfill material. Vertical and horizontal stresses in the soil mass were measured in loose sand and compacted sand.

Based on his test results, Chen and Fang (2008) proposed the following conclusions:

(1) after compaction, the lateral stress measured near the top of backfill is almost identical to the passive earth pressure estimated with Rankine theory. The compaction-influenced zone rises with rising compaction surface. Below the compaction-influenced zone, the horizontal stresses converge to the earth pressure at-rest, as indicated in Fig. 2.11 and Fig.2.12; (2) when total (static + dynamic) loading due to the vibratory compacting equipment exceeds the bearing capacity of foundation soils, the mechanism of vibratory compaction on soil can be described with the bearing capacity failure of foundation soils; (3) the vibratory compaction on top of the backfill transmits elastic waves through soil elements continuously. For soils below the compaction-influenced zone, soil particles are vibrated. The passive state of stress among particles is disturbed. The horizontal stresses among soil particles readjust under the application of a uniform overburden pressure and constrained lateral deformation, and eventually converge to the at-rest state of stress.

Chen’s test results were compared with the design recommendations proposed by NAVFAC DM-7.2 (1982), Duncan and Seed (1986), Peck and Mesri (1987), and Duncan et al. (1991) as shown in Fig. 2.13. Parameter values used in the stress calculation including the unit weight

γ

, relative Dr, internal friction angle

φ

, wall friction angle

δ

, and cyclic compaction stress

σ

cyc are shown in Fig. 2.13. The horizontal pressure distribution suggested by the Navy Design Manual DM-7.2 was based on the analytical method proposed by Ingold (1979). The pressure distribution calculated with the method proposed by Duncan et al. (1991) was obtained from the

design chart for vibratory plates with a cyclic compaction stress q = 34.9 kN/m² (5 psi).

In

Fig. 2.13, Chen’s test data are in good agreement with the proposed design

methods. The horizontal stresses in the uppermost compacted lift are equal to or slightly less than the passive Rankine pressure. However, at a lower depth, the Chen’s test data are apparently lower than the calculated horizontal stresses. It is important that the application of Chen’s test findings are limited to estimating the horizontal stresses acting on a nonyielding wall induced by a small size vibratory hand tamper.

Chapter 3

Experimental Apparatus

To investigate the effects of vibratory compaction on the vertical stress σv

and horizontal stress σh in a cohesionless soil mass, the instrumented non-yielding model retaining wall facility at National Chiao Tung University (NCTU) was used. This chapter introduced the NCTU non-yielding retaining wall facility and the vibratory compactor. Chen and Fang (2008) described the facility consist of three components: (1) model retaining wall; (2) soil bin; and (3) data acquisition system. The details of the foregoing apparatuses are described in the following sections.

3.1 Model Retaining Wall

The model wall shown in Fig. 3.1 is 1,500 mm-wide, 1,600 mm-high, and 45 mm-thick. To achieve an at-rest condition, the wall material should be nearly rigid.

It is hoped that the deformation of the model wall could be neglected with the application of earth pressure. As indicated in Fig. 3.1, twenty-four 20 mm-thick steel columns were welded to the four sidewalls to reduce any lateral deformation during loading. In addition, twelve C-shaped steel beams were also welded horizontally around the box to further increase the stiffness of the box.

3.2 Soil Bin

To simulate a plane strain condition for model test, the soil bin is designed to minimize the lateral deflection of sidewalls. In Fig. 3.1, the soil bin was fabricated of steel plates with inside dimensions of 1,500 mm ×1,500 mm ×1,600 mm.

Assuming a 1,500 mm-thick cohesionless backfill with a unit weight γ = 17.1 kN/m³, and an internal friction angle φ = 41^o was pluviated into the soil bin. A 45 mm-thick solid steel plate with a Young’s modulus of 210 GPa was chosen as the wall material. The estimated deflection of the model wall would be only 1.22 × 10^-3 mm. Therefore, it can be concluded that the lateral movement of the model wall is negligible and an at-rest condition can be achieved.

The end-wall and sidewalls of the soil bin were made of 35 mm-thick steel plates.

Outside the steel walls, vertical steel columns and horizontal steel beams were welded to increase the stiffness of the end-wall and sidewalls. If the soil bin was filled with dense sand, the estimated maximum deflection of the sidewall would be 1.86 × 10^-3 mm. From a practical point of view, the deflection of the four walls around the soil bin can be neglected.

To investigate the distribution of horizontal earth pressure σh, soil pressure transducers (SPT) were attached to the model wall as illustrated in Fig. 3.2. Fifteen strain-gage-type transducers (Kyowa PGM-02KG, capacity = 19.62 kN/m²) were arranged within the central zone of the wall. The soil pressure transducer with the adapter is shown in Fig. 3.3. The diameter of the SPT sensing area is 12 mm. To investigate the development of vertical stress σv in the backfill, another series of soil pressure transducers (Kyowa BE-2KCM17, capacity = 98.1 kN/m²) were arranged behind the model wall. The transducers were used to measure the

To investigate the distribution of horizontal earth pressure σh, soil pressure transducers (SPT) were attached to the model wall as illustrated in Fig. 3.2. Fifteen strain-gage-type transducers (Kyowa PGM-02KG, capacity = 19.62 kN/m²) were arranged within the central zone of the wall. The soil pressure transducer with the adapter is shown in Fig. 3.3. The diameter of the SPT sensing area is 12 mm. To investigate the development of vertical stress σv in the backfill, another series of soil pressure transducers (Kyowa BE-2KCM17, capacity = 98.1 kN/m²) were arranged behind the model wall. The transducers were used to measure the

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