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#### Synopsis and Validation of Proposed Versatile Analytical Models for Advancing the Internal Stability Design of GMSE/GRS Geostructures

Synopsis and Validation of Proposed Versatile Analytical Models for Advancing the Internal Stability Design of GMSE/GRS Geostructures

John Ngaya Mukabi

R&D/Design Dept.

Kensetsu Kaihatsu Consulting Engineers Ltd.

Nairobi, Kenya

Abstract The findings from current, on-going and past performance evaluation based on measurements of well instrumented GMSE (geosynthetics mechanically stabilized earth) and GRS (geosynthetic reinforced soil) geostructures provide definitively indisputable evidence that the conventional design methodologies for internal stability are excessively conservative, particularly with regard to the prediction of the prevalent reinforcement/tensile loads under in-service working conditions. This culminates in the exorbitant and unjustifiable use of public funds that would have otherwise been saved and invested in other development projects. As part of an absolutely necessary mitigation measure, this paper, within the R&D (research and development) framework of developing a pragmatic performance based value engineering design approach, presents an introduction of a set of proposed sophisticated analytical models that expound and explicate the influence factors proposed in the quasi-empirically developed K-Stiffness working stress method that takes into account, the structural contribution of global and local stiffness, wall facing rigidity, batter angle and cohesion (quality of reinforced backfill geomaterial properties), among other minor factors. The proposed analytical models are mainly validated and calibrated on the basis of measurements and performance data derived from a wide range of well instrumented GMSE-GRS geostructures as well as comparative analysis with reference to results obtained through the application of other analytical and/or numerical models. The versatility of the proposed analytical models is also discussed in this paper and fastidiously demonstrated in the other related papers to be published, which are cited herein. It is envisaged that the proposed TACH-MD analytical models are expedient for design and advancing R&D for GMSE-GRS retaining walls and bridge abutments.

Keywords GMSE-GRS, analytical models, internal stability, influence factors, K-Stiffness, TACH-MD, performance based, VE, value engineering, AASHTO, reinforced backfill geomaterial.

INTRODUCTION

Core of conservative approach and versatility of proposed analytical models

The conservative approach mainly emanates from the fact that the true mechanisms and interactive behavior between soil and geosynthetics reinforcement in GMSE-GRS geostructures including retaining walls and bridge abutments, has yet to be fully elucidated. As a consequence, practically all design guidance documents including those that are popularly adopted such as the AASHTO LRFD Bridge Design Specifications, 2012, AASHTO 2002, 2007;

FHWA 2001; BS8006 1995; BS8006:2010, CFEM 2006;

Geoguide 6 2002, NCMA 2009; PWRC 2000, among others, provide guidelines and specifications that are, by any standards, exceedingly conservative. On the other hand, the K-stiffness Method, which was empirically developed and calibrated based on post-construction structural performance of in-service GMSE-GRS Walls and first proposed by [1], quantitatively captures the influence of soil properties, reinforcement properties and structural wall facings on the magnitude of reinforcement loads under operational conditions, which leads to reasonable values for load and resistance factors.

This Study, within the R&D framework of performance based value engineering (PB-VE) design approach, takes advantage of the prodigious advances made in computer science, modelling and the research and methods of accurate measurements of in-service and full-scale experimental GMSE-GRS retaining walls and bridge abutments to propose TACH-MD universal analytical models that have been predominantly developed on the basis of these advances. The proposed analytical models provide considerable solutions to the aforementioned challenges and are versatile in application with appreciably high degree confidence levels. Case examples of the pragmatic applications of the proposed models for parametric investigations, characterization of various types of geomaterials, generation of performance- based value engineering designs, as well as development of special/particular specifications and construction QCA procedures, are also introduced in related publications ([2], [3], [4], [5], [6], [7] and [8]). It is also derived and concluded that appropriate application of the proposed analytical models can be useful in the enhanced prediction and precision of the maximum reinforcement/tensile loads, distribution characteristics, degree and well defined limitations of contribution factors, explicate characterization of the correlations between the influence factors and the principal

design parameters including the GMSE-GRS wall height, , geosynthetics base design length, and the vertical reinforcement spacing, , development of appropriate VE

designs for GMSE-GRS geostructures, seismic analysis,

designation of appropriate serviceability state criteria and meticulous prediction of structural performance, among other diverse geotechnical engineering applications.

The intricate aspects and meticulous characterization of these influence factors are presented in a number of other related papers, soon to be published ([2] ~ [8]).

Limitations and incongruity of the conventional methods of GMSE-GRS design and internal stability analysis Comprehensive literature review carried out on the State of

the Art and State of the Practice international publications indicates that the design guidelines currently adopted by practically all transportation and infrastructure agencies clearly demonstrate the existence of certain limitations, which certainly require further R&D.

Some of these limitations, which have been outlined and/or discussed in detail in some of the State of the Art and State of the Practice international publications include: i) hitherto designs have been undertaken by manufacturers limiting them to proprietary philosophies, characteristics, approach and methodologies culminating in delayed development of generic design procedures; ii) due to the fact that soil-geosynthetics interactive mechanisms have not been well elucidated, design methods adopt extremely conservative procedures of determining the design parameters; iii) design methods consider simultaneous failure of both the reinforcing elements and reinforced backfill soils, which, from a scientific, soil mechanics and geotechnical engineering perspective, is highly unlikely; iv) design methods do NOT consider the effective structural and serviceability contribution of the wall facing; v) design methods do NOT consider the effective structural and serviceability contribution of incorporating geosynthetics reinforcement within the reinforced fill soil of GMSE-GRS geostructures; vi) compaction induced stresses are never considered in the design and analysis; vii) development of apparent cohesion and increased confining stress as a result of frictional, lateral and bearing restraint are never considered in the design and analysis; viii) no definitive models and/or appropriate procedures for determining optimal base design length and vertical reinforcement spacing exist; ix) lack of appropriate methods to determine required ultimate tensile and junctionstrengths of the reinforcing geosynthetics and/or elements, overtensioning and structural performance of GMSE-GRS RWs; x) practically all designs adopt a Rankine and/or Coulomb failure envelopes which do NOT consider the effective structural and serviceability contribution of incorporating geosynthetics reinforcement within the backfill soil of GMSE-GRS RWs; xi) no definitive models and/or appropriate procedures are incorporated for determining the required design reinforcement length for the base; xii) no definitive models and/or appropriate procedures are incorporated for determining the effects of wall facing inclination; and xiii) no scientific procedure and/or definitive models are incorporated for determining the required ultimate tensile and junction strengths of the reinforcing geosynthetics and/or elements, among other limitations.

Shortcomings of the limit equillibrium based methods

A number of disadvantages of limit equilibrium-based methods for internal stability design of geosynthetic reinforced soil walls which contribute to their poor prediction accuracy have been identified by various researchers and practicing engineers. For example, [9] identified the following shortcomings of limit equilibrium based methods: i) equilibrium is satisfied only for sliding mass modes of failure;

deformation is not considered; iii) in simplified methods, failure is allowed only on predefined surfaces; and iv)

kinematics are not considered so that some failure mechanisms may not be possible.

As elucidated by [10] it is more appropriate to understand that this general approach results in simple models that do not satisfy a consistent mechanics framework but nevertheless result in conservative (safe) designs. Furthermore, the complex interactions that develop between a structural facing (a common feature of permanent walls) and the soil and reinforcement cannot be captured using simple wedge or slip surface models based only on force equilibrium. Reference [10] further state that; the persistence of limit equilibrium- based models for the internal stability design of geosynthetic reinforced design in current design codes is largely the result of lack of an alternative analytical approach. Nevertheless, the earliest attempts in North America to improve the prediction accuracy of geosynthetic reinforcement loads under operational conditions recognized that reinforcement loads were a function of displacement and hence the tensile stiffness of the reinforcement is a fundamental property for design [11]. An obvious contradictory paradox shortcoming of limit equilibrium methods that consider only the strength of the reinforcing elements is that predicted loads under operational conditions will be the same for steel and relatively extensible polymeric materials provided they have the same strength and number of layers in the wall.

Development of the K-Stiffness Method

Reference [12] investigated current North American methods for the prediction of reinforcement loads and concluded that the AASHTO Simplified Method [13] gave results similar to those of the other methods, yet had the advantage of being simpler to use and more broadly applicable. Consequently, [12] adopted the AASHTO Simplified Method as the baseline of comparison for predicted reinforcement loads using the new working stress method, the development of which they had undertaken based on long-term structural performance evaluation and case study analyses of well instrumented GMSE-GRS geostructures. Figure 1a clearly demonstrates the magnitude of deviation of the AASHTO Simplified Method from the actual measured reinforcement loads in geosynthetic walls built in the field.

Advancing the internal stability design approach within the Performanced-Based Value Engineering (PB-VE) design framework

Performance-Based Design (PBD) fundamentally entails that, deformation in ground, pavement materials and geo- structures along with the reciprocal structural deformation and stress states, be comprehensively analyzed by adopting appropriate sophisticated analytical methods, in order to achieve the predesignated structural and serviceability levels/requirements. The basic prerequisite of the PBD is that the acceptable level of the damage criteria be specified in engineering terms such as displacements, elastic limit stress state and ductility/ strain limit based on the function; as well dynamic loading and/ or seismic response of the structure.

One of the main objectives of developing the proposed analytical models is to contribute to the advancement of the design and analysis of the internal stability of GMSE-GRS

geostructures to transcend the conventional simplistic methods.

The TACH-MD models proposed in this paper have been developed for both Geosynthetics Mechanically Stabilized Earth (GMSE) and Geosynthetics Reinforced Soil (GRS) geostructures including retaining walls and bridge abutments.

OVERVIEW OF THE K-STIFFNESS METHOD WITHIN THE CONTEXT OF THE PROPOSED TACH-MD

MODELS

Advantages of the K-Stiffness Method

GMSE-GRS walls are usually designed for working stress conditions simulating the post-construction state of in-service operations. [10] undertook comprehensive evaluation and concluded that one of the most effective working stress based methods is the empirically developed K-Stiffness Method. This method explicitly includes the influence of reinforcement stiffness and the structural facing amongst other contributions. Statistical analysis of the bias of measured to predicted load was used to demonstrate the improved accuracy of this new load design approach as demonstrated in Figure 1b.

Some of the inimitable aspects of this method include: i) an empirically based method developed for internal stability analysis from extensive case study analyses covering a wide range of GMSE-GRS wall types, facing rigidity, batter angles (facing inclinations), backfill soils and geosynthetic reinforcement types; ii) calibrated against more than fifty fully and partially well instrumented MSE-GRS walls constructed and constantly monitored over varying periods since 1986 to- date; iii) although other methods can be used to evaluate the potential for reinforcement rupture and pullout for the strength and extreme event limit states, only the K-Stiffness Method can be used to directly evaluate the potential for soil backfill failure and to design the wall internally for the service limit state; iv) discerns failure limit state of soil and reinforcement separately. Conventional methods including the AASHTO simplified method are based on active earth pressure theory or Coulomb wedge analysis and hence the soil and critical reinforcement layers are assumed to be simultaneously at incipient failure; v) explicitly includes the quantitative influence of reinforcement stiffness and structural facing contribution; vi) predicts reinforcement loads that are within a range of 10% of the post-construction operational actual measurements resulting in approximately 1/3 of the loads predicted using the models for the AASHTO simplified method. This culminates in more than 35% savings in the required reinforcement quantities.

Based on the foregoing attributes, the further refinement and advancement of the K-Stiffness method to satisfy the principal requirements of PB-VE designs certainly desirable.

Fig. 1. Measured reinforcement load versus calculated load using the: a) AASHTO/FHWA Simplified Method; and b) K-stiffness Method (Bathurst et al., 2010).

Applicabilibity of the K-Stiffness Method and contribution of proposed TACH-MD analytical models

The K-Stiffness Method is empirically based with parameters that have been determined by calibration against a large database of carefully constructed, instrumented and monitored GMSE-GRS retaining wall structures. An important implication of this approach is that the method can only be used for structures with properties and boundary conditions that fall within the envelope of case study properties that were used to perform the calibration. For

example, the wall heights in the database vary from = 3 = 12.6 . Hence, using the K-stiffness Method to

design walls of greater height should be carried out with

caution. Consequently, the models proposed in this and the related papers {[2] ~ [8]} enable probing of extrapolated application to greater heights (refer to Figures 3 and 4 as well as Figures 20 ~ 35 and the corresponding models defined in Equations 15 and 16 as well as Equations 33 ~ 50 presented in this paper). Also, the K-stiffness Method is applicable to walls built on typical competent foundations where the performance of the structures is not influenced by excessive settlements or failure of the foundation or wall toe. The TACH-MD models and methodology proposed in [7], [8] and [14] can be adopted to substantially mitigate this limitation.

On the other hand, most of the wall sections used to calibrate the K-Stiffness Method were constructed with a vertical face; with a limited number constructed with facing

batter angles () from: = 3Â° = 27 (refer to Tables III

and VI). As can be inferred, the analytical model proposed in

Equation 36 and used in generating the characteristic curves graphically presented in Figures 25 and 26 can be effectively used in determining the appropriate limiting boundaries of the batter angle as a function of the shear strength of the reinforced backfill geomaterial. Furthermore, most of the walls were constructed with a hard structural facing. The TACH-MD analytical models proposed in [8], [15], and [16] can be adopted in modelling and determining the parameters required to achieve an optimal internal stability design mainly in consideration of the backfill quality and properties of the reinforcing elements for varying facing rigidity. A total of 58 data points were collected from 13 wall sections built with cohesionless soils and 79 data points from sections built with cohesive-frictional soils. The K-Stiffness Method in its most current form accounts for the positive contribution of soil cohesion to reduce geosynthetic reinforcement loads [17]. Indeed, it is extremely important to investigate the use of cohesive soils since purely frictional (granular) soils are hardly available and ignoring the cohesive component of available cohesive-frictional soils will lead to uneconomical structures. Equations 37 and 38 and the results depicted in Figures 27 and 28 demonstrate that the proposed analytical models can be useful to the design engineer in determining the appropriate value of the cohesion influence factor to adopt depending on the quality of the backfill soil.

Finally, it is perceived that the K-Stiffness Method was developed to compute reinforcement loads used for the internal stability design of reinforced soil retaining walls. At present the method applies only to internal rupture (over- stressing) and pullout failure modes (or limit states) and does not extend to the design of other structural components of the GMSE-GRS wall system. The method has also yet to cater for seismic loads. It is envisaged that the models proposed in this Study for seismic analysis, which are briefly introduced in this paper under Section VIII and discussed in detail in [6] can be applicable in alleviation. Refer to the models defined in Equations 30 ~ 32/51 ~ 53 as well as Figures 18 ~ 19/36 ~ 37.

Consequently, the K-Stiffness Method, as described by [1], is limited to the design of internal stability based on the following conditions: i) design internal stability for GMSE walls up to 12m in height; ii) GMSE Walls designed using this method should be constructed on sound foundations that are not supporting other structures and whose anticipated settlement is not greater than 150mm; iii) the method should be used for the design of walls with a maximum facing batter

angle of: = 20Â°; iv) at present the method applies only to

internal rupture (over-stressing) and pullout failure modes (or

limit states); v) other failure modes related to facing column stability, external stability and possible failure mechanisms that pass partially through the reinforced soil mass are beyond the scope of the method. For these failure modes current limit equilibrium-based models together with conventional factors of safety are available; and vi) the influence of additional loads due to earthquake has yet to be addressed within the K-

stiffness Method framework. The necessity to review the foregoing limitations, as proposed in this Study, is therefore well appreciated.

Range of GMSE-GRS wall types and material properties studied within the K-Stiffness context

Key properties and parameters for each of the case histories referenced, including facing type, reinforcement geometry, reinforcement type, soil properties, and construction history, are discussed in detail by [18]. Wall reinforcement products included geotextiles and geogrids, different polymers (polypropylene (PP), high-density polyethylene (HDPE), and polyester (PET)), strip and continuous reinforcements, a range of tensile strengths from 12 to 200 kN/m, and reinforcement stiffness values from 90 to 7400 kN/m. Reinforcement vertical spacing varied from

0.3 to 1.6 m. Wall facing batter angles varied from 0Â° (vertical) to 27Â°, although most of the walls had facing batter angles of 5Â° or less. Wall heights varied from 4.0 to 12.6 m, with surcharge heights of up to 5.3 m of soil. Facing types included geosynthetic wrapped-face, welded wire, precast concrete panels, and modular concrete blocks (segmental retaining wall units). Estimated peak plane strain soil friction angles varied from 42 to 57Â°. Many of the conditions that are likely to be encountered in the field are included in the database of case histories described previously.

On the other hand, plane strain conditions are known to typically exist in reinforced soil walls. Peak plane strain friction angles for granular soils are larger than values from triaxial compression or direct shear testing and hence are less conservative for design. Furthermore, recent work indicates that the peak plane strain soil friction angle in calculations gives a better estimate of reinforcement loads, at least for geosynthetic walls ([19], [20], [21] and [22]). Peak friction angles reported in the source references from triaxial

compression tests (, in degrees) were corrected therefore

to peak plane strain friction angles using the equation first

proposed by [23]:

= 1.5 17 (1)

Based on interpretation of data presented by [24] and [25]

for dense sands, values of were calculated from peak direct shear friction angles, , reported in the source

references using the following relationship:

= 1(1.2) (2)

Accuracy of the K-Stiffness Method

The same database of measured reinforcement loads used to investigate the accuracy of the AASHTO Simplified Method was used by [17] to quantify the accuracy of the K-stiffness Method. The results are depicted in Figure 1a for the AASHTO Simplified Method and Figure 1b for the K-stiffness Method. Data for walls with cohesive-frictional and frictional soil backfills are included in these figures. The load bias statistics are almost the same for both data sets. The graphical plots in Figure 1b, for example, show that the data are distributed much closer to the 1:1 reference line for the K stiffness Method than the corresponding data using the current AASHTO Simplified Method, which also exhibit a wide range of scatter (Figure 1b). The data show that the K-Stiffness Method does well in accurately capturing the measured load data for a range of wall heights and reasonable estimates of toe stiffness.

Universal model defining the maximum reinforcement load

The following key factors are known to influence the

magnitude of the maximum reinforcement load, : ( i) height of the wall, and ny surcharge loads, ; (ii) global,

and local stiffness, of the soil reinforcement; (iii)

resistance to lateral movement caused by the stiffness of the

facing, and restraint at the wall toe, .; (iv) face batter, . (v) shear strength, and stressstrain,

behaviour which characterizes stiffness defined in terms of

elastic modulus, 0 of the soil; (vi) unit weight of the soil,

; and (vii) vertical spacing of the reinforcement, . These

factors are introduced analytically in the generalized universal

model delineated in Equation 3, which defines the maximum

load per running unit length of wall in a reinforcement layer :

= (3) where is the tributary area (equivalent to the vertical

spacing of the reinforcement in the vicinity of each layer when

analyses are carried out per unit length of wall); is the lateral earth pressure acting over the tributary area; is

the load distribution factor that modifies the reinforcement

load based on layer location; and is the influence factor that

is the product of factors that account for the effects of local and

global reinforcement stiffness, facing stiffness, face batter and

soil cohesion. On the other hand, the lateral pressure, is calculated as the average value over the height of the wall

according to the conventional earth pressure theories, hence:

= 1 ( + ) (4)

2

where is the lateral earth pressure coefficient, is the unit weight of the reinforced backfill, is the height of the wall, and is the equivalent height of uniform surcharge pressure (. . , = /). The coefficient of lateral earth pressure is calculated for a vertical GMSE Wall, i.e., batter angle equals to zero; ( = 0) using the Jaky equation ([26]):

function of depth, accounting for the reinforcement properties, load redistribution among layers, and foundation conditions. It is expressed here as a function of normalized depth below the

top of the wall ( + )/( + ), including the effect of the soil surcharge , and varies over the range 0 1. The modifier is an empirically determined parameter

that captures the effect the major wall components have on

reinforcement load development. These parameters are used to improve the correlation between predicted and measured reinforcement loads at working stress conditions based on examination of a large number of case studies and wide range

of database. For brevity, the influence factor in Equation 6

is used to represent the product of five factors expressed in

Equation 7 as follows:

= Ã— Ã— Ã— Ã— (7) Substituting for the modifier in Equation 6 with the

expanded factors from Equation 7 leads to an expanded

universal model defined in Equation 8.

= ( + ) Ã— [ Ã— Ã— Ã—

1

2

Ã— ] (8) The terms , , , are influence

factors that account for the effects of global and local

reinforcement stiffness, facing stiffness, face batter and soil cohesion. This model captures all qualitative effects due to reinforcement stiffness, soil strength, facing stiffness and reinforcement arrangement expected by reinforced soil wall design engineers. The reinforcement load will increase as soil

unit weight, and reinforcement spacing, increases.

Further details of the development of the original K-stiffness

Method can be found in publications by [1] and [27]. The implementation of the original K-stiffness Method for cohesionless backfill soils can be found in [28] and [29]. Details of each influence factor and its computational approach are described next.

= 0 = = 1 ( )

(5a)

Influence of global stiffness factor

where ()

is the peak plane strain friction angle of the

Parameter is a global stiffness factor that accounts for

reinforced backfill soil. Note that, the use of = 0 in this

proposed method does not imply that at-rest conditions exist

within the reinforced backfill. 0 is simply used as a familiar

the influence of the stiffness and passively, the spacing of the

reinforcement layers over the entire wall height and is calculated as follows:

index parameter to characterize soil behaviour. For battered

GMSE walls analyzed in this paper, a TACH-MD modified

= [ ]

(9)

version of the simplified form of the Coulomb model, recommended by AASHTO, is adopted. This TACH-MD modified version, defined in Equation 5b, is compatible with the K-Stiffness Method.

Here, is the global reinforcement stiffness whereas

and are constant coefficients equal to 0.25 each. The non- dimensionality of the expression for global stiffness factor

is preserved by dividing the global reinforcement stiffness by

2[()

+]

= 101 (atmospheric pressure). The global

=

( )

2(1+ )

(5b)

reinforcement stiffness accounts for the relative

stiffness of the walls and is computed as follows:

Substituting for in Equation 3 yields:

1

= [ ]

(10)

= 2 ( + ) (6)

Equation 6 contains an expression for reinforcement loads

that is similar to the conventional expression used in current limit equilibrium methods of analysis but represents the average load applied to the reinforcement layers rather than a load that increases linearly as a function of the vertical overburden stress. The empirical reinforcement load

distribution parameter is used to distribute the load as a

Here, = 2% is the tensile stiffness at the end of wall

construction of an individual reinforcement layer expressed in

units of force per unit length of wall: also note that: =

(, ) Ã— . (see Figure 2).

The practical result of the formulation for global stiffness

factor (Equation 10) is that as reinforcement stiffness increases and all other factors remain the same, reinforcement load

( in Equation 4-8) goes up. Parameter is a local

stiffness factor that accounts for the relative stiffness of the

2

= [(

) ( ) (

)] ( ) (14)

reinforcement layer with respect to the average stiffness of all

1

reinforcement layers and is expressed as:

The unit weight of the concrete, , is assumed to be

24kN/m3. The wall and foundation friction angles, ,

= [ ]

(11)

and , respectively, are assumed to be 2/3 of the backfill

where is a constant coefficient and is the local

reinforcement stiffness for reinforcement layer i calculated as:

angle of friction, . Based on these assumptions, it is clear from Equation 1 that the coefficient is dependent on only

= [ ]

(12)

the variable. Essentially, this equation indicates that as the

wall thickness increases (i.e., increases), its capacity to

carry active earth pressure increases.

In order to investigate the influence of wall facing stiffness on the structural performance of GMSE and GRS RW systems and verify and further authenticate the concept of wall facing contribution, [34] developed several models within the TACH- MD framework. In particular, two models of interest that are pertinent to the K-Stiffness Method, designed to quantitatively probe the structural wall facing contribution are: i) wall facing capacity global stiffness model; and ii) universal wall facing stiffness – maximum tensile load delection model. The models are defined in Equations 15 and 16, whilst the characteristic curves depicting these correlations are graphically plotted in Figures 3 and 4 for descriptions i) and ii), respectively.

[0.75452+0.32590.625]Fig. 2. Low-strain secant creep stiffness at t = 3443 h from laboratory

=

[1+() (15)

]

constant-load (creep) tests (Allen & Bathurst, 2013).

Influence of facing batter

Parameter in the K-stiffness equation accounts for the influence of the facing batter and is computed as:

() .

= 2.23[1.31Ã—103(1+)()] () (16) where, is the lateral deflection, is the wall facing contribution factor, is height of GMSE-GRS wall and ()

is the required reinforcement stiffness for modular concrete

= [

]

(13)

block (MCB).

where, is the horizontal component of active earth pressure coefficient accounting for wall face batter, is the

horizontal component of active earth pressure coefficient

(assuming the wall is vertical), and is a constant coefficient. The form of the equation shows that as the wall

face batter angle 0 (i.e. wall facing batter approaches the

700

600

Lateral Deflection, h (mm)

500

Influence of Wall Facing Stiffness on Lateral Deflection

for Varying GMSEWall Heights

GMSE: Geosynthetics Mechanically Stabilized Earth

H: Height of Retaining Wall (m) FSF: Facing Stiffness Factor

vertical) the facing batter factor 1. The value of the coefficient term is taken as 0.5.

400

= .

. Ã—

.

.

300 ,( ) =

. + . .

+

Influence of wall facing stiffness

It is conventionally considered that: i) facings are incorporated to prevent a spill out of the backfill material; ii) earth pressure at the facing should be as low as possible; and,

facings should be flexible enough to accommodate deformation of supporting ground. However, on the contrary, [30] and [31] reported that: i) facing is a very essential

200

100

0

0 5 10 15 20 25 30

GMSEWall Height, H (m)

structural component that actively confines the backfill and develops the necessary large tensile forces within the reinforcement; ii) earth pressure at facing should be high enough to provide sufficient confining pressure to the backfill; and, iii) the facing should be flexible enough to accommodate deformation of supporting ground during construction but should become rigid before service. Furthermore, [32] and [33] demonstrated that full-height or cast-in-place concrete facings increase the stability of geosynthetic reinforced retaining walls

and introduced a coefficient that defines this correlation

as expressed in the following Equation 14.

FSF=0.05 FSF=0.1 FSF=0.2 FSF=0.3 FSF=0.4 FSF=0.5 FSF=0.7 FSF=1

Fig. 3. Influence of wall facing stiffness on lateral displacement for varying wall heights (Mukabi et al., 2015).

The results in Figures 3 and 4 clearly indicate that: i) lateral deflection (straining deformation) decreases as wall facing stiffness increases implying that wall facing stiffness decreases deformation; and ii) wall facing stiffness increases the global stiffness of the retaining system (required tensile stiffness is reduced).

1000

Influence of Wall Facing Stiffness on Required Geosynthetics Stiffness for Varying Specification Lateral Deflection Magnitudes

GMSE: Geosynthetics Mechanically

Wall Fcaing Stiffness Factor, fs: Classic K-Stiff. Model

3.5

Influence of GMSE Wall Geometry (Column Thickness and Height) on the Wall Facing Stiffness Influence Factor of MCBs Derived from Classic K-Stiffness Model

MCB: Modular Concrete Blocks

Required Geosynthetics Stiffness, Jreq. (kN/m)

900

800

Stabilized Earth

H: Height of Retaining Wall (m)

FSF: Facing Stiffness Factor

3.0

= ; =

.

.

t = b = tCB: Thicknessof MCB FS: Facing Stiffness

UBL: Upper Boundary Limit

700

600

500

400

,( ) =

. + .

+

.

2.5

2.0

1.5

UBL for Wall Facing Stiffness Influence Factor

300

200

100

0

= . . Ã—

.

.

1.0

0.5

10 20 30 40 50 60 70 80

Lateral Deflection, h (mm)

0.0

0 0.2 0.4 0.6 0.8 1

Wall Facing Column Thickness, tCB (m)

FSF=0.05 FSF=0.1 FSF=0.2 FSF=0.3 FSF=0.4 FSF=0.5 FSF=0.7 FSF=1

H=4 H=7 H=9 H=12 H=15 H=20 H=25 H=30

Fig. 4. Influence of wall facing stiffness on required tensile stiffness and lateral displacement (Mukabi et al., 2015).

On the other hand, the K-Stiffness influence factor for facing

stiffness (rigidity), is computed from the model defined in Equation 17 as:

Fig. 5. Influence of GMSE-GRS Wall geometry (column thickness and height) on the wall facing stiffness influence factor derived from the classic K-Stiffness Model.

Influence of GMSE Wall Geometry (Column Thickness and Height) on the Wall Facing Stiffness Influence Factor of MCBs Derived from TACH-MD Model

Wall Fcaing Stiffness Factor, fs: TACH-MD Model

3.5

. . ;

= . .

MCB: Modular Concrete Blocks

= ( )

(17)

= .

.

.

t = tCB: Thicknessof MCB

In the latest version of the K-stiffness Method ([35]), the

value of facing column stiffness parameter is calculated as:

1.53

3.0

2.5

FS: Facing Stiffness

UBL: Upper Boundary Limit

=

3(. )

(18)

2.0

UBL for Wall Facing

Here, = thickness of the facing column, = height of the facing column (wall), and = elastic modulus of the

equivalent elastic beam representing the wall face. The two

expressions used to compute the facing stiffness factor show

that as the wall becomes higher () and less stiff (3), its rigidity becomes less and hence more load is carried by the

reinforcement layers (i.e. is larger). A numerical

investigation by [36] also predicted that reinforcement loads

will increase in a propped panel wall as the stiffness of the

1.5

1.0

0.5

0.0

Stiffness Influence Factor

0 0.2 0.4 0.6 0.8 1

Wall Facing Column Thickness, tCB (m)

facing decreases. This effect has been quantitatively H=4 H=7 H=9 H=12 H=15 H=20 H=25 H=30

demonstrated using measurements from a pair of full-scale

reinforced soil walls tests reported by [37]. Based on back-

analyses performed by [17] the coefficient terms and were determined to be 0.69 and 0.11, respectively.

A comparison of characteristic curves depicting the influence of GMSE-GRS wall geometry (column thickness and height) on the wall facing stiffness influence factor derived

Fig. 6. Influence of GMSE-GRS Wall geometry (column thickness and height) on the wall facing stiffness influence factor derived from the classic K-Stiffness Model.

Influence of soil cohesion

The effect of soil cohesion is captured by the cohesion

(influence) factor computed as:

from the classic K-Stiffness and the TACH-MD models can be

inferred from Figures 5 and 6, respectively. It can be observed

= 1

(19)

that the characteristic curves from these graphs superimpose perfectly. It is further derived from the same figures that: i)

is highly dependent on the MCB facing thickness, and

wall height; and ii) as can be anticipated, the contribution of

the wall facing stiffness exhibits drastic reduction for thinner MCBs.

where the cohesion coefficient = 6.5. Examination of this equation with = 6.5 reveals that the practical limit 0 1 requires 0.153 . It is possible that a

combination of a short wall height and high cohesive soil

strength could lead to = 0. In practical terms this means that no reinforcement is required for internal stability. The

TACH-MD models proposed in this paper do cater for this anomaly.

Characterization of load distribution

The load distribution factor, that features in the principal universal Equations 3, 6 and 8 modifies the

reinforcement load, to effectively distribute and locate

the appropriate critical failure surface. This parameter is

computed as:

=

(20)

where, is the maximum reinforcement load in a particular layer , whilst is the maximum reinforcement

load for the entire system.

Locating appropriate critical failure surface

Comparison of the K-Stiffness and AASHTO simplified methods

The load distribution factor, plotted against

normalized height of wall in Figure 7 is essentially trapezoidal

in shape as originally proposed by [1] or can be assumed to be bi-linear as plotted Figure 8.

Fig. 7. Maximum load predictions (Tmax) using AASHTO simplified method and K-stiffness method compared with measured load values at EOC (H =

10.7 m) deduced from measured strains. Notes: for class A predictions, ps = 41Â°, tx = 38Â°, and = 20.4 kN/m3; for class B and C1 predictions, ps = 54Â°, tx =

47Â°, and = 22.0 kN/m3 (Allen & Bathurst, 2013).

Influence of toe resistance Maximum Reinforcement Load (MRL) distribution

Figures 8a and 8b demonstrate that the distribution of the maximum reinforcement load is significantly influenced by the magnitude of the toe resistance.

It can be extrapolated from these figures that: i) the magnitude of the toe resistance not only impacts on the maximum reinforcement load distribution but also contributes immensely on the shape of the characteristic curves; ii) the characteristic curves are practically trapezoidal but tend towards linear as the toe resistance increases quantitatively (also refer to Figure 9); and iii) the AASHTO Simplified Method exceedingly overestimates the MRL.

0 1 2 3 4 5 6 7 8 9 10

Reinforcement load, Tmax (kN/m)

Reinforcement load,Tmax (kN/m)

Fig. 8. Influence of (constant) toe stiffness on maximum reinforcement loads and comparison with predictions using: a) LHS – AASHTO Simplified Method and K-stiffness Method for wall with polyester (PET) reinforcement. (H = 6 m, Sv = 0.6 m, = 8 degrees). (after Huang et al. 2010); and b) RHS – K-stiffness Method for modular block wall with PET reinforcement. (Sv =

0.6 m, = 8 degrees) (Bathurst et al., 2010).

Effects of influence factors on maximum reinforcement load distribution

The findings from the R&D undertaken within the framework of the TACH-MD indicate that the influence factors have a significant effect on the distribution of the maximum reinforcement load magnitude and characteristics. An example is presented in Figure 9, which is a depiction of

the impact of the wall facing stiffness on the distribution of the MRL.

The characteristic curves plotted in Figure 9 were generated from model Equation 21. Also refer to Figures 13 ~ 16, presented in the subsequent Section III, which make a comparison of the measured and TACH-MD modelled results.

(0.62830.2792)

M. Internal soil failure limit state

An integral and unique feature of the K-stiffness Method is the introduction of an internal soil failure limit state. The calibration of the method has been based on the requirement that a contiguous failure mechanism must not develop through the reinforced soil zone. This has been achieved by limiting the maximum strain in the reinforcement to 3% based on load-

= 0.8580.234

= {(0.62830.2792)1 [

0.8580.234

(21)

]} (22)

time-strain performance of the reinforcing layers in geosynthetic reinforced soil walls. This is an important difference from the AASHTO Simplified Method and variants that assume that the soil and reinforcement reach failure

Modelled Characteristics of Maximum Reinforcement Loads Distribution Under the

Influence of GMSE/GRS Wall Height and Wall Facing Stiffness Factor

Normalized Maximum Reinforcement Loads, Tmax/Tmxmx

0.0 0.2 0.4 0.6 0.8 1.0 1.2

simultaneously. The latter is a rare, if not impossible scenario for extensible geosynthetic reinforcement materials. Hence, designing to prevent soil failure is rational and safe, and at the

0.0 = . . . . + .

Normalized Depth of GMSE/GRS Retaining Wall, DRW = (zi+S)/(H+S)

0.1

Ã—

.

same time ensures good performance as defined by the criteria identified earlier. In other words, designing to prevent failure

0.2

GMSE: Geosynthetics Mechanically

Stabilized Earth

0.3 GRS: Geosynthetics Reinforced Soil

0.4

. . + .

0.5

. .

Default WF Stiffness Curve

( = . )

of the soil in the reinforced soil zone deters the possiblility to reach a failure limit state for the reinforcement (rupture or over-stressing).

Pre-failure and failure mechanisms including the characteristics described in the foregoing paragraph are well elucidated based on the TACH-MD GMSE-GRS models developed for the K-Stiffness Method reported in this and the

=

. .

. .

related papers in [2], [3], [4], [5], [6], [7] and [8]. The related

0.6

0.7

0.8

0.9

1.0

papers, which further illuminate the versatility of the proposed models, focus on the following topics: i) correlation and characterization of principal influence factors [2] introduced in Section IV of this paper; ii) prediction and characterization of the maximum reinforcement load [3] introduced in Section V of this paper; iii) influence of quality of reinforced backfill geomaterials on the internal stability of GMSE-GRS retaining

walls [4] introduced in Section VI of this paper; iv) influence

fs=0.3 fs=0.4 fs=0.5 fs=0.588 fs=0.7 fs=0.8 fs=0.9 fs=0.98

Fig. 9. Maximum reinforcement load distribution inluence factor,

plotted as a function of wall facing stiffness and GMSE wall height for

vertical retaining walls wih cohesionless (granular) backfill ( = 0).

The following observations can be made from Figure 9: i)

the characteristic curves are essentially trapezoidal tending towards linear as the magitude of the wall facing influence factor increases, i.e., the wall facing stiffness is less or non- influencial; and ii) the MRL distribution and maximum values

of the vary with depth.

L. Calibration of the K-Stiffness classic models

The K-stiffness Method is an empirical-based working stress design method. The influence factors and coefficients which appear in the equations introduced above were determined by back-fitting to measured loads using conventional optimization schemes (see [33].) However, only reinforcement loads from walls that were judged to exhibit good performance were considered for the database used for calibration. Good performance was defined as: i) reinforcement strains are small (typically less than 3%); ii) creep strains and strain rates decrease with time (i.e. only primary creep occurs); iii) the wall backfill soil does not exhibit signs of failure (cracking, slumping, etc.); iv) for frictional soils, post-construction deformations, which are typically greatest at the wall top, are less than 30 mm within the first 10,000 h; and v) for cohesive-frictional soils, post- construction deformations are not greater than 300 mm or 3% of the height of the wall, whichever is less.

of load carrying capacity and global stiffness of composite GMSE-GRS mass oninternal stability [5] introduced in Section VII of this paper; v) seismic analysis in cogitation of the K-Stiffness influence factors [6] introduced in Section VIII of this paper; vi) application of TACH-MD GMSE-GRS analytical models in design and stability analysis [7] introduced in Section IX of this paper; and vii) establishment of appropriate serviceability criteria and prediction of structural performance [8] introduced in Section X of this paper.

VALIDATION OF THE TACH-MD MODELS DEVELOPED & PROPOSED WITHIN THE K-STIFFNESS

FRAMEWORK

Lucidity of developing the TACH-MD GMSE-GRS models for the K-Stiffness Method

The lucidity of developing the TACH-MD K-Stiffness models is predominantly based on: i) the limitations elucidated in Section IA. of this paper; ii) the fact that the K-Stiffness Method is empirically developed hence the necessity to determine mechanistic models that correlate and diversely explicate the K-Stiffness influence factors accordingly; iii) necessity of defining and correlating the K-Stiffness influence factors to the wall geometry parameters in order to determine the magnitude/degree of influence; iv) necessity to define and correlate the K-Stiffness influence factors in terms of the interactive parameters that characterize the interaction

between the backfill geomaterial and the geosynthetics reinforcing elements within the reinforced zone; and v) importance of defining and correlating the K-Stiffness influence factors to the wall carrying capacity and stiffness parameters in order to better comprehend the characteristics and importance of taking the influence factors into account in design and stability analyses.

As a fundamental example of the lucidity of developing the TACH-MD GMSE-GRS K-Stiffness models, the TACH- MD model equivalents to Equations 1 and 2, which broaden the limits/range of application to make an approximate

estimate of the plane strain soil friction angle, based on triaxial, or direct shear, results, are proposed in

Equations 23 and 24, respectively.

TABLE II. CASE HISTORIES FOR VERTICAL FACE GEOSYNTHETICS-REINFORCED SOIL WALLS WITH COHESSIVE (C) SOILS (UPDATED DATA FROM MIYATA AND BATHURST, 2007A)

= 0.25631.3878

Wall case history and year built

Wall height

Surcharge heightj

Backfill

Facing type

Geosynthetic

Performanc ei

Original reference

(m)

(m)

Soil unit weight

Fines content

Peak friction anglea (o)

Cohesion c (kPa)

Type

Designation

Index tensile strengthb

Tult

J2%c (kN/m)

Measured

Tmxmxd

(kN/m3)

(%)

tx or ds

ps

ps

(kN/m)

(kN/m)

(, 0.075

mm)

(c Â¼ 0)

(c . 0)

PWRI Wall

4

0.5

17

25

29

31

29

4.9

Gabion basket with soil-cemente

Extruded drawn uniaxial

EGG-8

71.4

400

8.2

PWRI et al. (1988)

(GW27) 1987

HDPE

PWRI Wall

Gabion basket with crushed

stone

10.6

x

(GW28) 1987

PWRI Wall

Wrapped- face with sand bags

11.2

x

(GW29) 1987

PWRI Wall

8

0

18.5

25

31

34

31

10

Hollow masonry concrete block with granular

EGG-4

59.8

380

1.98

Ochiai and Fukuda (1996)

(GW30) 1995

infill

Kagoshima

4

0.4

15.2

2040

39

46

44

4.9

Geotextile wrapped- face

EGG-8

71.4

400

0.92

Yamanouchi et al. (1986a)

Wall (GW31)

1984

Yamaguchi Wall (GW32)

sections L &

6

0.4

17.4

6

36

44

40

12.8

Incremental concrete panel

EGG-8

71.4

400

L: 0.48 R:

0.76

Yamanouchi et al. (1986b)

R 1985

LTRC

5.4

0

18.8

91

24

29

28

1.4

Hollow masonry concrete block with granular

Extruded drawn uniaxial

EGG-9

35.0g

100h

2

Farrag et al. (2004)

(GW33i) 1998

infill

HDPE

LTRC

EGG-10

70.0f

250h

4.53

(GW33ii) 1998

LTRC

EGG-11

114

500

5.75

(GW33iii)

1998

(GW40) 1992

4

0

16.2

8

38

42

40

2

Incremental concrete panel

Extruded drawn biaxial

EGG-15

32

170

0.83

PWRC (1995)

PP

(GW42) 1992

4

0

15.5

8

38

42

40

2

Plywood panel

Extruded drawn uniaxial

EGG-4

59.8

400

2.72

HDPE

34Â° (23)

= 1.61970.9019

(24)

Integral data adopted for development and calibration of

TACH-MD GMSE-GRS models

The integral data adopted for the development and calibration of the TACH-MD GMSE-GRS K-Stiffness models is mainly based on case histories as summarized and published by [36] for vertical/battered face geosynthetic-reinforced soil walls with cohesionless (granular) backfill and updated by [38] and [35] for vertical/battered face geosynthetic-reinforced soil

walls with cohesive ( ) soils as presented in Tables I, II

and III.

Incremental concrete panel

Wall case history and year built

Wall height

Surcharge height

Backfill

Facing type

Geosynthetic

Performancee

Original reference

(m)

(m)

Soil unit weight

Peak friction anglea (8)

T ype

Designation

Index tensile

J2 %c (kN/m)

Measured T mxmxd (kN/m)

strengthb

T ult

(kN/m)

(kN/m3 )

tx or ds

ps

T anque Verde Wall

4.7

0

19.6

53

53

Full-height propped concrete panel

Extruded drawn uniaxial HDPE

EGG-1

73

340

1.12

Bright et al. (1994)

(GW5) 1984

Algonquin Wall

6.1

0

20.4

40

43

Incremental concrete panel

EGG-1

67.8

500

3.8

Christopher (1993)

(GW8) 1988

Algonquin Wall

5.9

0

20.4

40

43

Wrapped- face

Nonwoven PET

geotextile

NWGT -1

19.3

175

3.5

x

(GW10)

1988

RMC Wall (GW14)

3

0

18

46

55

Full-height aluminium panel

Extruded drawn biaxial PP

EGG-2

12

9095

0.38

Bathurst and Benjamin (1990)

1989

RMC Wall (GW15)

3

0

18

46

55

Incremental aluminium panel

8793

0.53

Bathurst et al. (1993a)

1989

Fredericton, New

6.1

0

20.4

42

45

Full-height propped concrete panel

Extruded drawn uniaxial HDPE

EGG-1

73

500

2.5

Knight and Valsangkar (1993)

Brunswick Wall

(GW18)

1990

PWRI T est Wall

4

0

16.7

31

31

Incremental plywood panel

Extruded drawn biaxial PP

EGG-3

18.2

130

6.01

x

PWRI (1988)

(GW21)

1988

PWRI T est Wall

6

0

16.7

43

48

EPS block facing

Extruded drawn uniaxial HDPE

EGG-4

59.8

380

4.67

T ajiri et al. (1996)

(GW22)

1996

PWRI T est Wall

6

0

16.7

43

48

EGG-4

4.14

(GW23)

1996

PWRI T est Wall

6

0

16.7

43

48

Full-height propped concrete panel

EGG-4

3.12

(GW24)

1996

PWRI T est Wall

6

0

16.7

43

48

Solid masonry block

EGG-4

1.9

(GW25)

1996

WSDOT SR- 18

10.7

0

22

54

Hollow masonry block with granular infill

EGG-5

62.5

350

5.61

Allen and Bathurst (2006)

Wall

(GW26C)f

2005

EGG-6

71.1

415

EGG-7

117

660

WSDOT SR- 18

5.5

0

22

54

EGG-5

62.5

246

1.23

(unpublished data)

Wall (GW26D)

TABLE I. CASE HISTORIES FOR VERTICAL FACE GEOSYNTHETICS-REINFORCED SOIL WALLS WITH COHESIONLESS (GRANULAR) BACKFILL (DATA FROM ALLEN ET AL., 2002A, 2003 AND MIYATA AND BATHURST, 2007A)

aSee Appendix A for method to calculate peak plane strain friction angles. bFrom in-isolation constant- rate-of-strain test carried out at 10% strain/minute. cStiffness based on elapsed time to construct wall or 1000 h if this information not available. dMaximum reinforcement load in wall calculated using Equation

4. eSoilcement properties estimated from data reported by Yu et al. (1997). The equivalent elastic modulus of the soil-cement mixture in the wall was reduced to reflect the shorter curing time in the field. fFrom manufacturers literature. gEstimated from tensile test carried out at 2.5% strain/minute reported by Farrag et al. (2004) and increased to value expected for a test carried out at 10% strain/min based on data presented by Miyata and Bathurst (2007b) for EGG-4. hNo creep data available value estimated from EGG-10 and EGG-11 stiffness values and correlation between ultimate strength and secant stiffness value for uniaxial HDPE geogrids reported by Walters et al. (2002). i[Â¼ good performance, 3 Â¼ poor performance. Criteria for good performance of walls with c soils given in Section 4. jBroken back surcharge profile maximum equivalent uniform surcharge calculated using the total weight of the maximum surcharge over the reinforced soil zone divided by the width of the reinforced soil zone.

TABLE III. CASE HISTORIES FOR WALLS WITH A FACING BATTER

Wall case history and year built

Wall height

Surcharge heightb

Backfill

Facing

Geosynthetic

Performanc ee

Original reference

(m)

(m)

Soil unit weight

Fines content

Friction anglea (o)

Cohesion c (kPa)

Type

Batter angle (o)

Type

Designation

Index tensile strength Tult

J2%c (kN/m)

Measured

Tma x

(kN/m3)

(%)

tx or ds

ps

ps

(kN/m)

(kN/m)d

(c Â¼ 0)

(c . 0)

Oslo, Norway,

4.8

J: 1.75

17

3

41

46

46

0

Welded wire

27

Extruded drawn uniaxial

EGG-12

53.1

350

J: 1.47 N:

2.21

Fannin and Hermann (1990)

Section J and

N: 1.3

HDPE

N (GW7)

1987

Algonquin,

6.1

1.2

20.4

0

40

43

43

0

Hollow modular masonry block with granular

3

Woven PET

WGG-1

39.2

200

1.26

Bathurst et al. (1993b)

USA (GW9)

infill

geogrid

1988

Rainer

12.6

2.7

21.1

0

45

54

54

0

Wrapped face

3

Woven PP geotextile

WGT-1

31

120

5.2

Allen et al. (1992)

Avenue, USA

(GW16) 1989

WGT-2

62

190

WGT-3

92/p>

340

Woven PET

WGT-4

186

1000

geotextile

Vicenza, Italy

4

2.4

21.1

10

47

57

55

8

Welded wire

5

Extruded drawn uniaxial

EGG-13

56.3

300

0.87

Carruba et al. (1999)

(GW20hp)

HDPE

1998

Vicenza, Italy

4

2.5

21.1

10

47

57

55

8

Welded wire

5

Extruded drawn PP

EGG-14

20.9

90100

0.55

(GW20pp)

1998

PWRI Wall

4.5

0.5

18.6

25

24

27

24

4.9

Gabion basket with sand bags

6

Extruded drawn uniaxial

EGG-4

59.8f

380

1.6

x

Nakane et al. (1990);

Onodera et al. (2004)

(GW35) 1990

HDPE

PWRI Wall

6.3

0

16

0

41

46

45

2

Sand bags (not wrapped)

11

EGG-4

59.8f

380

1.56

Miki et al. (1992)

(GW36)

1991g

Kagoshima

6

0.9

14.7

6

45

52

51

2.5

Wrapped- face with sand bags

17

EGG-8

71.4f

400

1

Yamanouchi et al. (1986a)

Wall (GW37)

1984

Nagano Wall

7.5

0.5

16.1

75

25

27

25

12

Wrapped- face with sand bags

11

EGG-8

71.4f

400

2

Itoh et al.

(GW38) 1987

-1988

Saitama Wall

8.7

0.9

19.7

6

44

53

51

17

Wrapped- face with sand bags

27

EGG-4

59.8f

380

0.68

Abe et al.

(GW39) 1989

-1989

(GW41) 1992

4

0

16

8

38

42

40

2

Sand bags

11.3

EGG-4

59.8f

400

1.68

PWRC (1995)

( > 0) UPDATED FROM ALLEN ET AL. (2002A, 2003)

a tx, ds are reported peak friction angle from triaxial compression tests and direct shear tests, respectively. ps is peak plane strain friction angle. bFrom in-isolation constant-rate-of-strain test carried out at 10% strain/minute. cStiffness based on elapsed time to construct wall or 1000 h if this information not available. dMaximum reinforcement load in wall calculated using Equation 4; e[Â¼ good performance, 3

Â¼ poor performance. Performance criteria for walls with granular backfill as defined by Allen and Bathurst (2002b). fIn previous related paper by Miyata and Bathurst (2007b) this wall was referred to as GW26.

aSee Appendix A for method to calculate peak plane strain friction angles. bBroken back surcharge profile maximum equivalent uniform surcharge calculated using the total weight of the maximum surcharge over the reinforced soil zone divided by the width of the reinforced soil zone. cStiffness based on elapsed time to construct wall or 1000 h if this information was not available. dEnd-of-construction values corresponding to S < 1 m. e[Â¼ good performance, 3 Â¼ poor performance of walls with c soils based on criteria given in Section 4. fHDPE geogrid tensile strength adjusted to values estimated from 10% strain/minute tests;. gDate of construction not reported by Miki et al. (1992) assumed as 1991.

Comparison of measured vs. TACH-MD modelled data

Comparison of measured data vs. results that were modelled based on TACH-MD GMSE-GRS models developed for the K-Stiffness Method is the most fundamental approach adopted in the validation and calibration of the proposed models. In so doing, four principal parameters including: i) the secant stiffness at the end of construction

(EOC) or @ 2% strain, @=2% ; ii) maximum reinforcement (tensile) load, ; iii) global maximum reinforcement load, ; and iv) the maximum load distribution influence factor, are adopted. Under this topic, comparison of small strain secant stiffness, @0.1% characteristics with progressive straining, 0.1% ; and

seismic analysis using plots of characteristic curves generated

from TACH-MD GMSE-GRS models is also made employing measured data and modelled results.

Note that the TACH-MD GMSE-GRS models for the K- Stiffness Method were specifically developed for the modular concrete blocks (MCBs), which are the most commonly used for wall facings in Kenya.

Tables IV ~ VI present a summarized tabulation of measured and modelled results derived from the case history data shown in Tables I ~ III. O the other hand, graphical comparisons are depicted in Figures 10 ~ 28. In general, it can be derived that, the characteristic curves generated from the TACH-MD GMSE-GRS models in cogitation of similar conditions exhibit an appreciably good agreement with the measured data and are compatible (harmonious) with the classic K-Stiffness models.

Tabulated summary of measured vs. modelled results

The measured and TACH-MD modelled results are summarized in Tables IV ~ VI.

TABLE IV. COMPARISON OF MODEL COMPUTED RESULTS AND MEASURED DATA FROM TABLE I

TABLE V. COMPARISON OF MODEL COMPUTED RESULTS AND MEASURED DATA FROM TABLE II

Case histories for vertical face geosynthetic-reinforced soil walls with cohesive backfill soil

Wall case history and year built

Wall height

Surcharge heightj

Backfill

Facing type

Geosynthetic

Equation 7

Equation 8

Original reference

Measured

Computed

Measured

Computed

(m)

(m)

Soil unit weight

Fines content

Peak friction anglea (o)

Cohesion c (kPa)

Type

Designation

Index tensile strengthb

Tult

J2%c (kN/m)

Tmax (kN/m)

(kN/m3)

(%)

tx or ds

ps

ps

(kN/m)

(H,Dtmax)

(H,)

(, 0.075

mm)

(c Â¼ 0)

(c . 0)

PWRI Wall

4

0.5

17

25

29

31

29

4.9

Gabion basket with soil-cemente

Extruded drawn uniaxial

EGG-8

71.4

400

431

8.2

1.11

1.83

PWRI et al. (1988)

(GW27) 1987

HDPE

PWRI Wall

Gabion basket with crushed

stone

10.6

(GW28) 1987

PWRI Wall

Wrapped- face with sand bags

11.2

(GW29) 1987

PWRI Wall

8

0

18.5

25

31

34

31

10

Hollow masonry concrete block with granular infill

EGG-4

59.8

380

373

1.98

2.07

3.64

Ochiai and Fukuda (1996)

(GW30) 1995

MQC

Kagoshima

4

0.4

15.2

2040

39

46

44

4.9

Geotextile wrapped- face

EGG-8

71.4

400

431

0.92

1.11

1.13

Yamanouchi et al. (1986a)

Wall (GW31)

1984

Yamaguchi Wall (GW32)

sections L &

6

0.4

17.4

6

36

44

40

12.8

Incremental concrete panel

EGG-8

71.4

400

431

L: 0.48 R:

0.76

1.60

1.93

Yamanouchi et al. (1986b)

R 1985

LTRC

5.4

0

18.8

91

24

29

28

1.4

Hollow masonry concrete block with granular infill

Extruded drawn uniaxial HDPE

EGG-9

35

100

241

2

1.45

2.81

Farrag et al. (2004)

(GW33i) 1998

MQC

LTRC

EGG-10

70

250

424

4.53

(GW33ii) 1998

LTRC

EGG-11

114 500

631

5.75

(GW33iii)

1998

(GW40) 1992

4

0

16.2

8

38

42

40

2

Incremental concrete panel

Extruded drawn biaxial

EGG-15

32

170

224

0.83

1.11

1.29

PWRC (1995)

PP

(GW42) 1992

4

0

15.5

8

38

42

40

2

Plywood panel

Extruded drawn uniaxial

EGG-4

59.8

400

373

2.72

1.11

1.29

HDPE

td rowspan=”3″>

Extruded drawn PP

Case Histories for Walls with a Facing Batter (Ã¸ > 0)

Wall case history and year built

Wall height

Surcharge heightb

Backfill

Facing

Geosynthetic

Equation 7

Equation 8

Original reference

Measured

Computed

Measured

Computed

(m)

(m)

Soil unit weight

Fines content

Friction anglea (o)

Cohesion c (kPa)

Type

Batter angle (o)

Type

Designation

Index tensile strength Tult

J2%c (kN/m)

Tmax (kN/m)

(kN/m3)

(%)

tx or ds

ps

ps

(kN/m)

(H,Dtmax)

(H,)

(c Â¼ 0)

(c . 0)

Oslo, Norway,

4.8

J: 1.75

17

3

41

46

46

0

Welded wire

27

Extruded drawn uniaxial

EGG-12

53.1

350

339

J: 1.47 N:

2.21

1.31

1.40

Fannin and Hermann (1990)

Section J and

N: 1.3

HDPE

N (GW7)

1987

Algonquin,

6.1

1.2

20.4

0

40

43

43

0

Hollow modular masonry block with granular infill

3

Woven PET geogrid

WGG-1

39.2

200

264

1.26

1.62

2.03

Bathurst et al. (1993b)

USA (GW9)

MQC

1988

Rainer

12.6

2.7

21.1

0

45

54

54

0

Wrapped face

3

Woven PP geotextile

WGT-1

31

120

218

5.2

3.12

2.08

Allen et al. (1992)

Avenue, USA

(GW16) 1989

WGT-2

62

190

384

WGT-3

92

340

530

Woven PET

WGT-4

186

1000

940

geotextile

Vicenza, Italy

4

2.4

21.1

10

47

57

55

8

Welded wire

5

Extruded drawn uniaxial

EGG-13

56.3

300

355

0.87

1.11

0.79

Carruba et al. (1999)

(GW20hp)

HDPE

1998

Vicenza, Italy

4

2.5

21.1

10

47

57

55

8

Welded wire

5

EGG-14

20.9

90100

158

0.55

1.11

0.79

(GW20pp)

1998

PWRI Wall

4.5

0.5

18.6

25

24

27

24

4.9

Gabion basket with sand bags

6

Extruded drawn uniaxial

EGG-4

59.8

380

373

1.6

1.24

2.41

Nakane et al. (1990);

Onodera et al. (2004)

(GW35) 1990

HDPE

PWRI Wall

6.3

0

16

0

41

46

45

2

Sand bags (not wrapped)

11

EGG-4

59.8

380

373

1.56

1.67

1.89

Miki et al. (1992)

(GW36)

1991g

Kagoshima

6

0.9

14.7

6

45

52

51

2.5

Wrapped- face with sand bags

17

EGG-8

71.4

400

431

1

1.60

1.47

Yamanouchi et al. (1986a)

Wall (GW37)

1984

Nagano Wall

7.5

0.5

16.1

75

25

27

25

12

Wrapped- face with sand bags

11

EGG-8

71.4

400

431

2

1.95

4.37

Itoh et al.

(GW38) 1987

-1988

Saitama Wall

8.7

0.9

19.7

6

44

53

51

17

Wrapped- face with sand bags

27

EGG-4

59.8

380

373

0.68

2.23

1.96

Abe et al.

(GW39) 1989

-1989

(GW41) 1992

4

0

16

8

38

42

40

2

Sand bags

11.3

EGG-4

59.8

400

373

1.68

1.11

1.29

PWRC (1995)

TABLE VI. COMPARISON OF MODEL COMPUTED RESULTS AND MEASURED DATA FROM TABLE III

Case histories for vertical face geosynthetic-reinforced soil walls with cohesionless (granular) backfill

Wall case history and year built

Wall height

Surcharge height

Backfill

Facing type

Geosynthetic

Equation 7

Equation 8

Original reference

Measured

Computed

Measured

Computed

(H,Dtmax)

(H,)

(m)

(m)

Soil unit weight

Peak friction anglea (8)

Type

Designation

Index tensile

J@ 2 %c (kN/m)

Tmxmx (kN/m)

Tmax (kN/m)

strengthb

Tult

(kN/m)

(kN/m3)

tx or ds

ps

Measured

Computed

Measured

Computed

Measured*

Computed

Tanque Verde Wall

4.7

0

19.6

53

53

Full-height propped concrete panel

Extruded drawn uniaxial HDPE

EGG-1

73

340

439

1.12

1.58

0.90

1.29

1.09

Bright et al. (1994)

(GW5) 1984

Algonquin Wall

6.1

0

20.4

40

43

Incremental concrete panel

EGG-1

67.8

500

413

3.8

2.08

3.04

1.62

2.24

Christopher (1993)

(GW8) 1988

Algonquin Wall

5.9

0

20.4

40

43

Wrapped- face

Nonwoven PET

geotextile

NWGT-1

19.3

175

148

3.5

2.02

2.80

1.57

2.16

(GW10) 1988

RMC Wall (GW14)

3

0

18

46

55

Full-height aluminium panel

Extruded drawn biaxial PP

EGG-2

12

9095

101

0.38

1.03

0.30

0.86

0.79

Bathurst and Benjamin (1990)

1989

RMC Wall (GW15)

3

0

18

46

55

Incremental aluminium panel

8793

0.53

1.05

0.42

0.86

0.79

Bathurst et al. (1993a)

1989

Fredericton, New

6.1

0

20.4

42

45

Full-height propped concrete panel

Extruded drawn uniaxial HDPE

EGG-1

73

500

439

2.5

2.05

2.00

1.62

2.10

Knight and Valsangkar (1993)

Brunswick Wall

(GW18) 1990

PWRI Test Wall

4

0

16.7

31

31

Incremental plywood panel

Extruded drawn biaxial PP

EGG-3

18.2

130

142

6.01

1.48

4.81

1.11

1.83

PWRI (1988)

(GW21) 1988

PWRI Test Wall

6

0

16.7

43

48

EPS block facing

Extruded drawn uniaxial HDPE

EGG-4

59.8

380

373

4.67

2.07

3.74

1.60

1.99

Tajiri et al. (1996)

(GW22) 1996

PWRI Test Wall

6

0

16.7

43

48

Incremental concrete panel

EGG-4

4.14

2.06

3.31

1.60

(GW23) 1996

PWRI Test Wall

6

0

16.7

43

48

Full-height propped concrete panel

EGG-4

3.12

2.04

2.50

1.60

(GW24) 1996

PWRI Test Wall

6

0

16.7

43

48

Solid masonry block

EGG-4

23

1.9

1.99

1.52

1.60

1.99

Model Compatible Facing

(GW25) 1996

WSDOT SR- 18

10.7

0

22

54

Hollow masonry block with granular infill

EGG-5

62.5

350

387

5.61

3.44

4.49

2.69

2.07

Allen and Bathurst (2006)

Wall (GW26C)f

2005

EGG-6

71.1

415

429

EGG-7

117

660

644

WSDOT SR- 18

5.5

0

22

54

EGG-5

62.5

246

387

1.23

1.82

0.98

1.48

1.26

(unpublished data)

Wall (GW26D)

Measured vs. modelled results for EOC secant modulus

Figure 10 makes a comparison between measured data and results derived on the basis of TACH-MD GMSE-GRS models developed for the secant stiffness determined at the end of

construction, as defined in Equation 25.

0.8145

= 13.323.

(25)

End of Construction (EOC) Secant Stiffness, JEOC (kN/m)

1100

1000

Comparison Between Measured and Modelled EOC Secant Stiffness

EOC: End of Construction (J@2% Strain)

Vert: Facing @ Vertical (=0)

conformity with the universal Equation 8, the GMSE-GRS wall height, surcharge, wall facing type, backfill geomaterial quality, vertical reinforcement spacing and facing batter angle

900

800

700

600

500

400

300

200

100

.

= . .

Influence of Wall Facing Type

have a significant influence on maximum reinforcement load.

4) Measured vs. modelled results for Tmxmx

Figure 12 provides a comparison between measured data and TACH-MD modelled results based on Equation 27 for the

global maximum reinforcement load, . Similar observations can be made to those irradiated for te in

the preceding sub-Section 3.

0

0 20 40 60 80 100 120 140 160 180 200

=

(0.000142+0.0013+0.9538) (27)

2.34760.851

Ultimate (Index) Tensile Strength, Tult.(kN/m)

Comparison of Measured and Model Computed Global Maximum

Measured Vert. Cohesionless

Measured Vert. Cohesive

Modelled Vert. Cohesive

Measured Facing Battered

Modelled Vert. Cohesionless

Modelled Facing Battered

Reinforcement Load, Tmxmx

Modelled Global Maximum Reinforcement Load, Tmxmx (kN/m)

6

Fig. 10. Comparison between measured and TACH-MD modelled Secant

Stiffness @ EOC (End of Construction), .

It can be noted, from the results provided in Figure 10, that:

i) the measured and modelled results show a close agreement; and ii) the wall facing type has an appreciable influence on secant stiffness measured at the end of construction.

Measured vs. modelled results for Tmax

The comparison between measured data and TACH-MD results modelled from Equation 26 for the maximum

reinforcement load, , is presented in Figure 11.

Note that . = 0.8 in all model equations that contain the

5

4

Perfect Superimposition for Solid Modular Concrete Block

3

2

.

in this paper.

1 Deviation Influenced Mainly by Wall Facing Type and Multiple Other Factors

term

= {[1.8729()

0

(0.002220.0022+0.7313)

0.7808]

} Ã— [

.

] (26)

0 1 2 3 4 5 6

Measured Global Maximum Reinforcement Load, Tmxmx (kN/m)

Comparison of Measured and Model Computed Maximum

Reinforcement Load, Tmax

6

Modelled Maximum Reinforcement Load, Tmax (kN/m)

Deviation Predominantly VF: Vertical Facing

1:1-Perfect Superimposition Measured vs. Modelled

Fig. 12. Comparison between measured and TACH-MD modelled global

Influenced by Low

Angles of Internal Friction, for Cohesive

5 Backfill (Material Quality)

4

3

2

G: Granular C: Cohesive

BF: Battered Facing

maximum reinforcement (tensile) load, .

Measured vs. modelled results for Dtmax

The comparison between measured data and TACH-MD results modelled from Equation 28 for the distribution of

maximum reinforcement load, is graphically presented in

Figures 13 ~ 16 principally as a function of the wall facing

stiffness influence factor for varying reinforced backfill geomaterial type and batter angles.

(0.62830.2792)

= 0.8580.234

(28)

1

Deviation Influenced Mainly by Wall Facing Type and Multiple Other Factors

0

0 1 2 3 4 5 6

1:1-Perfect Superimposition

VFG Eqn. 7

VFG Eqn. 8

VFC Eqn. 7

VFC Eqn. 8

BF Eqn. 7

BF Eqn. 8

Measured Maximum Reinforcement Load, Tmax (kN/m)

It can be noted, that: i) when compared under similar

structural conditions, the measured and modelled results exhibit an appreciably good agreement; ii) the wall facing stiffness has significant influence on the magnitude and characteristics of the maximum reinforcement load distribution; iii) the nature/quality of backfill geomaterial and batter angle also do have appreciable influence on the MRL distribution.

Fig. 11. Comparison between measured and TACH-MD modelled maximum

reinforcement (tensile) load, .

It can be noted, that: i) for higher reinforced backfill shear

strength (angle of internal friction/shearing resistance) under similar structural conditions, the measured and modelled results show an appreciably good agreement; and ii) in

Comparison of Modelled and Measured Results of Maximum Reinforcement Loads Distribution Under the Influence of GMSE/GRS Wall Height and Wall Facing Stiffness Factor Vertical Retaining Walls with Cohesionless (Granular) Backfill (c = 0): Generic Curves

Normalized Maximum Reinforcement Loads, Tmax/Tmxmx = Dtmax

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0

Comparison of Modelled and Measured Results of Maximum Reinforcement Loads Distribution Under the Influence of GMSE/GRS Wall Height and Wall Facing Stiffness Factor Vertical Retaining Walls with Cohesive Backfill (c > 0): Generic Curves

Normalized Maximum Reinforcement Loads, Tmax/Tmxmx = Dtmax

0.0 0.2 0.4 0.6 0.8 1.0 1.2

= . . . . + .

Ã—

.

0.0

= . .

. . + . Ã—

Normalized Depth of GMSE/GRS Retaining Wall, DRW = (zi+S)/(H+S)

0.1

No Wall Facing Stiffness Effect

0.2 ( )

GMSE: Geosynthetics

0.3 Mechanically Stabilized Earth

GRS: Geosynthetics

Reinforced Soil

0.4

Default WF Stiffness Curve

( = . )

Normalized Depth of GMSE/GRS Retaining Wall, DRW = (zi+S)/(H+S)

0.1

0.2

0.3

0.4

No Wall Facing Stiffness Effect

( )

.

Default WF Stiffness Curve

( = . )

0.5 =

0.6

. . . . + .

. . . .

0.5

0.6

. . + .

. .

=

. . . .

0.7

0.8

0.9

1.0

Meas.CL-Vert. fs=0.3 fs=0.4 fs=0.5 fs=0.588 fs=0.7 fs=0.8 fs=0.9 fs=0.98

0.7

0.8

0.9

1.0

GMSE: Geosynthetics Mechanically Stabilized Earth

GRS: Geosynthetics Reinforced Soil

Meas. CH-Vert. fs=0.3 fs=0.4 fs=0.5 fs=0.588

Fig. 13. Comparison between measured and TACH-MD modelled results for

maximum reinforcement (tensile) load distribution inluence factor,

plotted as a function of wall facing stiffness and GMSE wall height for

vertical retaining walls wih cohesionless (granular) backfill ( = 0).

Comparison of Modelled and Measured Results of Maximum Reinforcement Loads

Distribution Under the Influence of GMSE/GRS Wall Height and Wall Facing Stiffness Factor

Battered Retaining Walls with Cohesionless (Granular) Backfill (c = 0): Generic Curves

Normalized Maximum Reinforcement Loads, Tmax/Tmxmx = Dtmax

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0

fs=0.7 fs=0.8 fs=0.9 fs=0.98

Fig. 15. Comparison between measured and TACH-MD modelled results for

maximum reinforcement (tensile) load distribution inluence factor,

plotted as a function of wall facing stiffness and GMSE wall height for

vertical retaining walls wih cohesive backfill ( > 0).

Comparison of Modelled and Measured Results of Maximum Reinforcement Loads

Distribution Under the Influence of GMSE/GRS Wall Height and Wall Facing Stiffness Factor Battered Retaining Walls with Cohesive Backfill (c > 0): Generic Curves

Normalized Maximum Reinforcement Loads, Tmax/Tmxmx = Dtmax

= .. .. .. .. + ..

Ã—

..

0.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2

. . + . Ã—

Normalized Depth of GMSE/GRS Retaining Wall, DRW = (zi+S)/(H+S)

0.1

0.2

0.3 No Wall Facing Stiffness Effect

( )

0.4

0.5

. . + .

Default WF Stiffness Curve

( = .. )

Normalized Depth of GMSE/GRS Retaining Wall, DRW = (zi+S)/(H+S)

0.1

0.2

0.3

0.4

No Wall Facing Stiffness Effect

( )

= . .

.

Default WF Stiffness Curve

( = . )

=

. .

. .

0.5

. .

. . + .

0.6 . .

GMSE: Geosynthetics Mechanically

0.7 Stabilized Earth

=

0.6

. . . .

GRS: Geosythetics Reinforced Soil

0.8

0.9

1.0

Meas. CL-Batt. fs=0.3 fs=0.4 fs=0.5 fs=0.588 fs=0.7 fs=0.8 fs=0.9 fs=0.98 fs=0.3 fs=0.4 fs=0.5

Fig. 14. Comparison between measured and TACH-MD modelled results for

0.7

0.8

0.9

1.0

GMSE: Geosynthetics Mechanically Stabilized Earth

GRS: Geosynthetics Reinforced Soil

Meas. CH-Batt. fs=0.3 fs=0.4 fs=0.5 fs=0.588 fs=0.7 fs=0.8 fs=0.9 fs=0.98

maximum reinforcement (tensile) load distribution inluence factor,

plotted as a function of wall facing stiffness and GMSE wall height for

battered retaining walls wih cohesionless (granular) backfill ( = 0).

Fig. 16. Comparison between measured and TACH-MD modelled results for

maximum reinforcement (tensile) load distribution inluence factor,

plotted as a function of wall facing stiffness and GMSE wall height for

Figure 16, which compares measured and TACH-MD

modelled results for cohesive backfill with battered facing shows that most data points plot along the maximum

distribution value of = 1.

battered retaining walls wih cohesive backfill ( > 0).

Measured vs. modelled results for secant modulus

decay characteristics with progressive straining

The comparison between measured and TACH-MD modelled results for the decay characteristics for reinforcement secant stiffness with progressive straining are graphically plotted in Figure 17 for three different initial stiffness values. The modelled curves are generated from Equation 29.

@0.1% = 63.38(0.0009940)() + 571.29(0) 3326.3 (29)

The results in Figure 27 show a perfect superimposition for

the lowest initial stiffness value of 0 = 600 and a very good agreement for the other two curves; 0 800 and 0 1000 . The fact that the reinforcement secant stiffness decays rapidly within the initial zone of small strains 0.6% tending to a residual state as the straining progresses, in

all cases, can also be derived from this figure.

Comparison Between Measured and TACH-MD Modelled: Low Strain Secant Stiffness – Strain Correlations

0.7

Seismic Force Coefficient, K

0.6

0.5

0.4

0.3

0.2

Influence of GMSE-GRS Wall Geometry and Reinforced Backfill Geomaterial

Shear Strength on the Seismic Force Coefficient

Secant Stiffness, J(,t) (kN/m)

1400

1200

1000

@ . % = . . + . .

Jo: Initial Low Strain Secant Stiffness Meas. : Measured Data

=

.

I&

TC

G: Ism

:

e

TACH

ik&

-M

ulle

D M

r (1

odel

998)

T

ACH

-MD

Mo

del

Equa

tion

=

.

.

.

+ .

0.1

0

Mod.: TACH-MD Modelled Curves

800

0 0.05 0.1 0.15 0.2 0.25

Facing Wall Column Thickness to GMSE-GRS Wall Height Ratio, t/H

600 =20 I&G

=20 TC =25 I&G

=25 TC =30 I&G

=30 TC =35 I&G

400

=35 TC =40 I&G

=40 TC

200

0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Strain, (%)

Fig. 18. Influence of quality of reinforced backfill geomaterial defined in terms of shear strength (angle of internal friction) and GMSE-GRS wall geometry on the seismic force coefficient: TACH-MD modelled results, Koseki et al., 2006 (also refer to Figure 36).

Coefficient of Active Pressure vs. Angle of Internal Friction Under the Influence of Seismic Acceleration

0.9

FI = 50

TACH-MD Model Equation

Meas. Jo1 Meas. Jo2 Meas. Jo3

0.8

ru = 0.25

= . . + . + . ) + . . + . + .

Mod. Jo1 Mod. Jo2 Mod. Jo3

Fig. 17. Comparison between measured and TACH-MD modelled low strain secant stiffness strain correlations.

Comparison of modelled results for seismic analysis

Development of TACH-MD models that can be adopted for seismic analysis for the K-Stiffness Method has also been undertaken. Some of the models that have been developed and proposed are introduced in this paper in the generalized Equations 30 ~ 32 and in the particular Equations 51 ~ 53. Equation 30 is adopted in investigating the influence of reinforced backfill shear strength and the GMSE-GRS wall geometry on the seismic force coefficient, K. The results

Coefficient of Seismic Earth Pressure, Ka

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

a: Seismic Acceleration FI: Facing Inclination

ru: Constant Static Pore Pressure Ratio Cet.: Cascone et al. (1995)

TC: TACH-MD Model

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Angle of Internal Friction for Reinforced Backfill Geomaterial, ()

a=0g Cet. a=0.1gCet. a=0.2gCet. 0.3gCet. a=0g TC a=0.1g TC a=0.2g TC 0.3g TC

reported by [39] and [40] are compared, in Figure 18, to those generated from the TACH-MD model Equation 30. The harmonic agreement is seen to be exemplary.

Fig. 19. Influence of quality of reinforced backfill geomaterial defined in terms of shear strength (angle of internal friction) and seismic acceleration on the coefficient of seismic earth pressure: TACH-MD modelled results, Koseki et al., 2006 (also refer to Figure 37).

2.2765

= 0.000259

( ) 0.49( ) + 2.1392

(30)

Comparison of classic K-Stiffness and TACH-MD models

The other approach that was adopte in validating the TACH-MD GMSE-GRS models developed for the K-Stiffness

The influence of seismic acceleration and backfill properties on coefficient of seismic pressure can be computed from the TACH-MD model Equations 31 and 32. The results, showing a perfect agreement, are compared in Figure 19 to those reported by [41] and [40].

= 0.3818(2.2693)( ) + 1.5686(2.3868) (31)

= (7.83333 0.42 + 0.7017 +

Method is based on the comparative analysis of the TACH- MD and classic K-Stiffness principal models, the results of which are presented in Figures 20 ~ 29.

Comparison of global stiffness influence factor models Figure 20 depicts the characteristic curves generated from back-calculation using the classic K-Stiffness model Equation 8, whilst Figures 21 and 22 show the graphical plots based on the TACH-MD models for the global influence factor

generated from Equations 33 and 34, respectively.

0.393))( ) + 29.4673

1.0552

+ 3.2938 +

1.6045 (32)

= 0.28590.1411[0.0922(0.033)] Ã— [

.

] (33)

[0.0601(0.0232)]= 0.3143(0.0136)

Ã— [ . ]

(34)

Correlation between Global Stiffness Influence Factor and Maximum Reinforcement Load as Function of GMSE-GRS Wall Height: Derived from Classic K-Stiffness Model

0.70

It can be observed that the data from both models is in appreciably close proximity with all the data points plotting

=

+

Ã—

Ã— Ã— Ã—

within 10% difference (deviation). The slight difference is

Global Stiffness Influence Factor, g

0.60

0.50

0.40

0.30

0.20

Default Value:

=0.358

mainly attributed to the fact that default values were used for

the influence factors in the classic K-Stiffness model, whilst the data generated from the TACH-MD model adopted reciprocal variables. This fact can be ascertained by comparing the data point in yellow that is within the closest proximity to the default value whereby the deviation from the line of perfect superimposition is only 1.13%. Note that this data point was generated using default values for both models.

0.10

0.00

GMSE-GRS Heights

Out of Scope of K-Stiffness Calibration

0 1 2 3 4 5 6 7 8

Maxium Reinforcement Load, Tmax (kN/m)

0.40

Comparison between TACH-MD and Classic K-Stiffness Model for Global Stiffness Influence Factor as Function of GMSE-GRS Wall Height

Classic K-Stiffness Model

H=4 H=7 H=9 H=12 H=15 H=20 H=25 H=30

=

+

Ã— Ã— Ã— Ã—

Fig. 20. Characteristic curves generated from the classic K-Stiffness model back-calculation for the global stiffness influence factor.

Correlation between Global Stiffness Influence Factor and Maximum Reinforcement Load as Function of GMSE-GRS Wall Height: Derived from TACH-MD Model Eqn. 1

0.39

TACH-MD Model: Global Stiffness Influence Factor, g

0.38

Note

All Data Points Within 10%

0.52

0.50

Global Stiffness Influence Factor, g

0.48

0.46

= . . . .

Ã— .

0.37

0.36

Default Value Line:

=0.358

Default Value:

=0.358

Difference

0.44

0.42 0.35

0.40

0.38

0.36

Default Value:

=0.358

0.34

0.34

0.32

0 1 2 3 4 5 6 7 8

Maxium Reinforcement Load, Tmax (kN/m)

0.33

0.32

TACH-MD Model

= . . . .

Ã— .

H=4 H=7 H=9 H=12 H=15 H=20 H=25 H=30

Fig. 21. Derivation of th global stiffness influence factor based on TACH-

MD model equation 1 (Equation 33), which correlates with angle of internal friction of reinforced backfill geomaterialand GMSE wall height.

Correlation between Global Stiffness Influence Factor and Maximum Reinforcement Load as Function of GMSE-GRS Wall Height: Derived from TACH-MD Model Eqn. 2

0.52

0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40

Classic K-Stiffness Model: Global Stiffness Influence Factor, g

1:1 Line of Perfect Superimposition H=4 H=7 H=9 H=12

Fig. 23. Comparison between K-Stiffness and TACH-MD models for the global stiffness influence factor.

Comparison of wall facing stiffness influence factor models

0.50 = . . .

. + . . .

Ã— .

Comparison of the wall facing stiffness influence factor

Global Stiffness Influence Factor, g

0.48

0.46

0.44

0.42

0.40

0.38

0.36

. Ã—

Default Value:

=0.358

generated from the two models is made in Figure 24. The wall facing stiffness factor is plotted as a function of the wall facing column thickness in both cases. The characteristic curves for the K-Stiffness models are generated from Equations 17 and 18, whilst the ones for the TACH-MD model are generated from Equations 35. Note that the wall facing considered is the modular concrete block (CB) commonly used in Kenya.

0.34

0.32

0 1 2 3 4 5 6 7 8

Maxium Reinforcement Load, Tmax (kN/m)

Or,

= 0.13980.66

0.33

(35a)

H=4 H=7 H=9 H=12 H=15 H=20 H=25 H=30

Fig. 22. Comparison of results computed using TACH-MD model equation 2 (Equation 34).

The direct comparison of the results generated from the classic K-Stiffness and TACH-MD models for the global stiffness influence factor is made in Figure 23. Note that the scale has been blown up for clearer perception.

= 0.18220.44

0.33

(35b)

Comparison of Classic K-Stiffness and TACH-MD Models for Wall Facing Stiffness Influence Factor of MCBs

3.0

KC: Classic K-Stiffness Model

influence increases with increase in the magnitude of the batter angle.

Classic K-Stiff. Model: = ; =

Wall Fcaing Stiffness Factor, fs

2.5

.

.

MCB: Modular Concrete Blocks t = b = tCB: Thickness of MCB UBL: Upper Boundary Limit

1.1

Comparison of Classic and TACH-MD Models for Facing Batter Angle Factor as a Function of the Angle of Internal Friction of Reinforced Backfill and Wall Facing Batter Angle

. .

. .

1.0

TACH-MD Model: = . Ã— . + . + . .

2.0 TACH Model: = .

Facing Batter Angle Influence Factor, fb

; = . .

1.5 UBL for Wall Facing Stiffness

Influence Factor

0.9

0.8

1.0

0.5

Default Value Line:

0.7

0.6

0.5

Default Line:

=

Classic Model: =

Default Characteristic

Curves: = Â°

= .

0.0

0 0.2 0.4 0.6 0.8 1

0.4

AIF: Angle of Internal Friction

rbf: Reinforced Backfill

: Facing Batter Angle

=

+

+

Default Line:

= Â°

Wall Facing Column Thickness, tCB (m)

0.3

= Â°; =1 0

=

H=4 KC

H=4 TACH H=7 KC

H=7 TACH

25 30 35 40 45 50 55 60

H=9 KC H=9 TACH H=12 KC H=12 TACH

Angle of Internal Friction of Reinforced Backfill, rbf ()

Fig. 24. Comparison between K-Stiffness and TACH-MD models for the

=0: Class.

=0:TACH

=4: Class.

=4:TACH

=8: Class.

wall facing stiffness influence factor.

=8:TACH

=12: Class.

=12:TACH

=20: Class.

=20:TACH

The following observations can be made from Figure 24: i) the curves generated from both models show a perfect agreement (superimposition); ii) the contribution of the wall facing stiffness degrades with the increase in wall facing

Fig. 25. Comparison between K-Stiffness and TACH-MD odels for the wall facing batter angle influence factor as a function of the angle of internal friction of reinforced backfill and magnitude of wall facing batter angle.

Comparison of Facing Batter Angle Factor as a Function of the Angle of Internal Friction of Reinforced Backfill and Wall Facing Batter Angle Generated from Classic and TACH-MD Models

column thickness tending towards a residual state; iii) as

1.0

. + . + . .

expected, the degradation is critical (acute) for thinner walls

within the range of; < 0.06 for = 4 to <

Facing Batter Angle Influence Factor Generated From TACH-MD Model, [fb ]TACH

0.9

TACH-MD Model: = . Ã—

0.19 for = 12 implying that the rate of decay is also

dependent upon the height of the GMSE-GRS wall; iv) the

0.8

Classic Model: =

=

Default Characteristic

+ Curve: = Â°

Upper Boundary Limit (UBL) clearly demarcates both the

+

limit of the wall facing stiffness contribution and the end of critical degradation; and v) the default value line plotted at;

= 0.588, inidcates that this value is attained when the thickness is: = 0.25 for = 4 ; = 0.40 for

= 7 , = 0.54 for = 9 and = 0.73 for

= 12.

Comparison of wall facing batter angle influence factor

models

0.7

0.6

0.5

0.4

0.3

=

Significant Deviation

for = Â°

Characteristic

Curve

Zone of Perfect Superimposition:

. 1.0

AIF: Angle of Internal Friction rbf: Reinforced Backfill

: Facing Batter Angle

= Â°; =1 0

Figures 25 and 26 show the comparison of the results

generated from the classic K-Stiffness and the TACH-MD models for the wall facing batter angle influence factor. Whilst Figure 25 depicts this comparison for the batter angle factor plotted as a function of the angle of internal friction of the reinforced backfill geomaterials and batter angle, Figure 26 makes a direct comparison of the two sets of data against the line of perfect superimposition. The K-Stiffness curves were generated from model Equation 13, whilst the ones for the TACH-MD model were generated from Equation 36.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Facing Batter Angle Influence Factor Generated From Classic Model, [fb ]Classic

=0 =4 =8 =12 =20

Fig. 26. Comparison between K-Stiffness and TACH-MD models for the wall facing batter angle influence factor as a function of the angle of internal friction of reinforced backfill and magnitude of wall facing batter angle.

Comparison of cohesion influence factor models

The influence factor that measures the contribution of cohesion to the magnitude of the maximum reinforcement load and other parameters is graphically plotted in Figures 27 and

2

= 1.0967 Ã— 1050.9673 + 0.000275 +

1.0001(0.011) (36)

The results from these figures indicate that: i) perfect

superimposition is achieved for both models as the batter angle

28 for curves generated from both models. The data was generated adopting Equation 19 for the classic K-Stiffness model and Equations 37 and 38 for the TACH-MD model.

( + )

becomes lower implying that the K-Stiffness model was

calibrated for lower batter angles with an average of; =

= 1 6.5 Ã—

= 1 6.5 Ã—

11.5Â°, most of the walls of which were lower than this value

(+)

)

5 ( +

) 100 (37)

(refer to the data in Tables III and VI); ii) the = 20Â°curves

exhibit signicant deviation due to the reason given in i); note

14.673(0.0091

) Ã— ( + )

that the TACH-MD models were developed on the basis of a wider range of data; and iii) the magnitude and characteristics of the batter factor are significantly impacted by the shear strength of the reinforced backfill geomaterial whereby this

= 1 0.19681(0.0091

(38)

where, = 2.58150.6437

; = ( .

) Ã— ; :

Limited examples correlating the global, local and wall facing stiffness are subsequently presented.

Reinforcement Induced Cohesion (RIC).

In this case, the reinforcement induced cohesion (RIC) was

considered to be zero ( = 0) since this factor is not incorporated in the classic K-Stiffness model.

The cohesion influence factor is plotted as a function of both the cohesion intercept and the quality of reinforced backfill geomaterials delineated in terms of shear strength (angle of internal friction/shearing resistance). The results in Figures 27 and 28 depict that: i) the curves exhibit perfect superimposition for both models; ii) the influence of cohesion reduces as the cohesion intercept increases; and iii) as the shear strength increases, the contribution of the cohesion influence factor is reduced.

Comparison of Classic K-Stiffness and TACH-MD Models for the Cohesion Influence Factor

0.94

Correlating global and local stiffness influence factors

The correlation between the global and local stiffness influence factors as a function of the wall height is depicted in Figure 29; the curves are generated from Equation 39.

.

= {170.85(0.023)2 + 119.48(0.011) 20.1424} Ã— [ ] (39)

The following derivations can be made from Figure 29: i)

local stiffness increases with increasing global stiffness; ii) the rate of increase is higher for the local than the global stiffness factor; and iii) whilst the default value line for the local

stiffness influence factor intersects all curves including =

H = 7m (Height of GMSE-GRS Wall)

TACH-MD GMSE-GRS Models for K-Stiffness Method

1. = . Ã—

. .

4 ~ = 30, the global one is limited to = 12, which is

0.89

2. = . . Ã— +

the limiting calibration height of the K-Stiffness Method.

Cohesion Influence Factor, c

0.84 = . .

; = . Ã—

; = .

0.79

Correlation Between Local and Global Stiffness Influence Factors for Varying GMSE/GRS Wall Heights

+ . . . Ã—

Classic K-Stiffness Model

=

0.74

0.90

0.85

Intersection of Default Values

@ Default Height;

=

= . .

.

0.69

0.64

0.59

: Reinforcement Induced Cohesion

: Reinforcement Density

.: Ultimate Tensile Strength

: Vertical Reinforcement Spacing

: Consolidated Reduction Factor

4 6 8 10 12 14 16 18 20

Cohesion Intercept Value, c (kPa)

Local Stiffness Influence Factor, local

0.80

0.75

0.70

=0.82

Default Line

=34KC

=34TACH =40KC

=40TACH =48KC

=48TACH =60KC

=60TACH

0.65

Fig. 27. Comparison between K-Stiffness and TACH-MD models for the cohesion influence factor as a function of the cohesion intercept value and angle of internal friction of reinforced backfill.

Comparison of Classic K-Stiffness and TACH-MD Models for the Cohesion Influence Factor

0.60

=0.588

Default Line

0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48

Global Stiffness Influence Factor, g,

H=4 H=7 H=9 H=12 H=15 H=20 H=25 H=30

0.94

0.89

Classic K-Stiffness Model

=

KC: Classic K-Stiffness Model

: Reinforcement Induced Cohesion

: Reinforcement Density

.: Ultimate Tensile Strength

Fig. 29. Influence of GMSE-GRS wall column height and shear strength of reinforced backfill geomaterials on the cohesion influence factor.

: Vertical Reinforcement Spacing

CohesionInfluence Factor, c

: Consolidated Reduction Factor

0.84

Correlating global and wall facing stiffness influence

factors

0.79

0.74

H = 7m (Height of GMSE-GRS Wall)

TACH-MD GMSE-GRS Models for K-Stiffness Method

The reciprocal correlations of the global and wall facing stiffness influence factors are evaluated using Equations 40 and 41.

1. = . Ã—

. .

[0.0601(0.0232)]0.69 = 0.3143(0.0136)

Ã— [ . ]

2. = . . Ã— +

= . . ; = . Ã—

0.64

; = .

0.59

0.59 0.64 0.69 0.74 0.79 0.84 0.89 0.94

(40)

=

{[0.0601(0.0232)]1 [ ] Ã—

Cohesion Intercept Value, c (kPa)

=34 KC-TACH =40 KC-TACH =48 KC-TACH =60 KC-TACH 1:1 Line of Perfect Superimposition

Fig. 28. Superimposition omparison between K-Stiffness and TACH-MD models for the cohesion influence factor as a function of the angle of internal friction of reinforced backfill.

CORRELATION AND CHARACTERIZATION OF PRINCIPAL INFLUENCE FACTORS

Details under this topic are discussed in [2], which not only correlates all the principal influence factors but also provides particularization of their characteristics and quantification of their contribution to the internal stability of GMSE-GRS walls.

(0.3143(0.0136))

.

[ ]} (41)PREDICTION AND CHARACTERIZATION OF

THE MAXIMUM REIFORCEMENT LOAD

Preamble

Various TACH-MD GMSE-GRS universal models are proposed for defining and predicting the maximum

reinforcement load. The models correlate the to GMSE wall height, and maximum reinforcement load distribution

factor, to various other principal design parameters and

influence factors including: i) angle of internal friction of the

reinforced backfill, ; ii) elastic modulus, 0 ; iii) unit

weight, ; iv) coefficient of lateral pressure, ; v) ultimate

It can be inferred from Figure 31 that: i) maximum reinforcement load distribution characteristics are highly dependent on the shear strength of the reinforced backfill (RBF) geomaterial; ii) RBF geomaterials with friction angles

34Â° can only be characterized by the

factor

tensile (index) strength, . ; vi) ultimate load carrying

of

capacity, . ; vii) secant stiffness @ end of construction,

for GMSE-GRS walls that are below 3m in height ( 3)

= 48Â°, the

factor

; viii) global stiffness influence factor, (Equation 42);

whereas, for the default value of

ix) local stiffness influence factor, ; x) wall facing stiffness influence factor, ; xi) wall facing batter angle influence factor, ; and xii) cohesion influence factor, .

The TACH-MD GMSE-GRS universal models for i) ~ xii)

are proposed in [3], which is the third part of this paper.

Influence of global stiffness and GMSE-GRS wall height on the maximum reinforcement load, Tmax

The application of the global stiffness factor and the

can be adopted for walls up to; = 12, which is the limiting

height of the K-Stiffness Method; and iii) the default value for

the factor can graphically be confirmed to be; =

0.7. Also refer to the subsequent Section VI. for further insight

on the influence of the shear strength of RBF geomaterials.

Influence of GMSE/GRS Wall Height and Angle of Internal Friction of Reinforced Backfill Geomaterial on Distribution of Maximum Reinforcement Load

Maximum Reinforcement Load Distribution Factor, Dtmax

0 1 2 3 4 5 6 7

GMSE-GRS wall height in the prediction of the maximum

reinforcement load can be made by adopting model Equation 42; the converse of which can be evaluated using Equation 43.

=

0 Default AIF Curve ,

(

= Â°) = . . .

5 Default Height (H = 7m)

Ã— . + .

. .

{[0.0922(0.033)]1(3.49770.1411)} Ã—

.

t = tCB: Thickness of Concrete Block (m)

GMSE: Geosynthetics Mechanically Stabilized Earth

GRS: Geosynthetics Reinforced Soil

AIF: Angle of Internal Friction

[Depth of GMSE/GRS Retaining Wall, H (m)

] (42) 10

: Upper Boundary Limit for Dtmax = 1.0

: Lower Boundary Limit for Dtmax = 0.58

[0.0922(0.033)]: Intersection Point for Default Values

= 0.28590.1411

Ã— [ . ] (43)

15

Figure 30 shows that: i) the upper and lower boundary

values of the maximum reinforcement load are primarily dictated by height of the GMSE-GRS wall with the resulting

20

increasing with the increase in height; and ii) for

12; 8

(walls calibrated for K-Stiffness).

Effect of Global Stiffness Influence Factor on Maximum Reinforcement Load for Varying GMSE Wall Height 25

30

UBLL: Upper Boundary

Limiting Line for

Max. Reinf. Load, Tmax

Maximum Reinforcement Load, Tmax (kN/m)

25

1

20 = 2 + Ã— Ã— Ã— Ã— Ã—

30

=30 =34 =38 =44 =48 =50 =55 =60

15

Default Wall Facing

Stiffness Value Factor, g = 0.358

10

5

0

LBLL: Lower Boundary Limiting Line for

Max. Reinf. Load, Tmax

Fig. 31. Influence of GMSE-GRS wall height and shear strength of

reinforced backfill geomaterial on the distribution of the maximum reinforcement load.

INFLUENCE OF QUALITY OF REINFORCED BACKFILL GEOMATERIALS ON THE INTERNAL STABILITY OF GMSE-GRS RETAINING WALLS

The quality of reinforced backfill geomaterial is evaluated mainly on the basis of the physical and mechanical

0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48

Global Stiffness Influence Factor, g

H=4 H=7 H=9 H=12 H=15 H=20 H=25 H=30

Fig. 30. Application of GMSE-GRS wall height and wall facing stiffness factor in the prediction of the maximum reinforcement/tensile load.

Variation of maximum load distribution factor with GMSE- GRS wall depth under the influence of the angle of internal friction (shearing resistance)

The results generated using Equation 44 are graphically plotted in Figure 31. The model is useful in predicting the value of the critical failure characteristics and the maximum reinforcement load distribution factor.

= 0.8580.234 Ã— [356.81 1,829 Ã—

properties including the angle of internal friction/shearing

resistance, , unit weight, , stiffness defined in terms of small strain elastic modulus, 0, cohesion intercept, and index parameters, mainly plasticity index, . Whilst limited

examples that validate the influence of quality of reinforced

backfill geomaterials are subsequently provided, the details regarding the same are discussed methodically in [4].

A number of researchers have shown that the type of backll had the most profound eect on various aspects of the behavior and performance of the GMSE-GRS geostructures ([14], [15], [16] and [42]). Also refer to Section VIII.

A. Influence of RBF geomaterial properties and GMSE-GRS

5 2

(0.62830.2792)

wall height on the global stiffness factor

(4Ã—10

0.0023+0.3643)]

(44)

One of the TACH-MD models that is applicable in

demonstrating the effect of the quality of the reinforced

backfill geomaterial and GMSE-GRS wall height on the global stiffness factor is defined in Equation 45.

= 0.3143(0.0136) Ã—

C. Influence of RBF geomaterial quality and GMSE-GRS wall height on the maximum reinforcement load

The influence of reinforced bakfill geomaterial quality on the maximum reinforcement load can be investigated through the use of the analytical model defined in Equation 48.

2 0.0023 +0.3643) [0.0601(0.0232)]

{356.81 1.829

(4Ã—105

}

Ã—

= {0.81031.3058[(1 Ã— 1042 0.0003 +

[ ] (45)0.0317) ]} Ã— [

] (48)

.

B. Influence of RBF geomaterial properties and GMSE-GRS

.

wall height on the equivalent wall facing stiffness factor

Equations 46 and 47 are adopted in characterizing the influence of RBF geomaterial properties on the equivalent wall facing stiffness influence factor as a function of GMSE-GRS wall height. The results generated using this equation are graphically plotted in Figure 32.

2.771

The results generated using Equation 48 are illustrated in

Figure 33. It can be seen that the shear strength of the backfill defined in terms of the angle of internal friction/shearing resistance has significant influence on the magnitude and characteristics of the maximum reinforcement load (MRL). It can further be derived that the resulting MRL reduces exponentially as the shear strength increases and that this characteristic is also considerably impacted by the height of

= {0.1592[455.25

Ã— the GMSE-GRS wall.

2 1

(6Ã—105 00035+1.052)]0.660.33}

(46)

30

Influence of Angle of Internal Friction of RBF Geomterial on Maximum Reinforcement Load for Varying GMSE Wall Heights

2 0.0023 +0.3643)

RBF/rbf: Reinforced Backfill

: Tmax Default Value

AIF: Angle of Internal Friction

1

= 2 + Ã— Ã— Ã— Ã— Ã—

= 356.81 1.829

(4Ã—105

Resulting Maximum Reinforcement Load, Tmax (kN/m)

(47) 25

: Adequate AIF Values to Achieve Tmax Default Value

= . . Ã— . + . Ã— .

20

Correlation Between Equivalent Wall Facing Influence Factor and Angle of Internal Friction of

Reinforced Backfill Geomaterial Under the Influence of GMSE-GRS Wall Height

5.0 15

.

Resulting Reinforcement Load Reducing with Increasing AIF and Reducing GMSE/GRS

Equivalent Wall Facing Stiffness Influence Factor, fs

4.5

4.0

= . . . Ã— Ã— + .

.

10

Height

Default Value for Angle of

Internal Friction =

. + .

. Ã—

3.5 = .

5

. .

= . .

3.0

2.5

2.0

Default Characteristic Curve:

= . ; = Â°

0

25 30 35 40 45 50 55 60

Angle of Internal Friction of Reinforced Backfill Geomaterial, rbf ()

H=4 H=7 H=9 H=12 H=15 H=20 H=25 H=30

1.5

1.0

0.5

0.0

Default Value Line

= .

20 25 30 35 40 45 50 55 60

Angle of Internal Friction of Reinforced Backfill Geomaterial, rbf ().

H=4 H=7 H=9 H=12 H=15 H=20 H=25 H=30

Fig. 33. Influence of quality of reinforced backfill geomaterial defined in terms of shear strength (angle of internal friction) and GMSE-GRS wall height on the maximum reinforcement load.

D. Influence of RBF geomaterial properties, GMSE-GRS wall column thickness and batter angle on the geosynthetics base design length

One of the useful models in investigating the influence of RBF on principal internal stability design parameters is

Fig. 32. Influence of quality of reinforced backfill geomaterial defined in

terms of shear strength (angle of internal friction) and GMSE-GRS wall height on the equivalent wall facing stiffness influence/contribution.

The characteristic curves depicted in Figure 32 show that: i)

defined in Equation 49, the results of which are presented in Figure 34.

= 1.229 Ã— {[0.0012 0.544]( ) + 0.0044 +

the equivalent wall facing stiffness increases with increasing magnitude of shearing resistance; ii) the contribution of the wall facing stiffness influence factor degrades as the GMSE- GRS height increases; and iii) it can be derived that the influence of the equivalent wall facing stiffness factor is

1.7112} Ã— (49)

effectively initiated when: = 34Â° = 4, =

36Â° = 7, = 40Â° = 12 , whilst = 50Â° = 30 , values which constitute the lower

boundary limits of the required angle of internal friction.

Determination of the Geosynthetics Base Design Length as a Function of the Angle of Internal Friction and the GMSE-GRS Wall Height

30

VIII. SEISMIC ANALYSIS IN COGITATION OF THE K- STIFFNESS INFLUENCE FACTORS AND PARAMETRIC

= . :

= . Ã— . . + . + . Ã—

Geosynthetics Base Design Length, LGD (m)

= Â°

25

20

15

10

5

0

Default Characteristic Curve:

= . ; =

Default Value:

= Â°

VARIABILITY

The inimitable TACH-MD analytical models that are applicable in carrying out rigorous seismic analysis in cogitation of the influence factors proposed in the K-Stiffness Method and a wide range of parametric variability are conscientiously examined in [6]. The models have also been developed to enable profound seismic analysis in rumination of reciprocal parametric effects (refer to Section III for validation).

Some of integral aspects ruminated include the seismic effects and/or interactive features regarding/involving: i) the

20 25 30 35 40 45 50 55 60

Angle of Internal Friction of Reinforced Backfill Geomaterial, rbf ()

H=4 H=7 H=9 H=12 H=15 H=20 H=25 H=30

Fig. 34. Influence of quality of reinforced backfill geomaterial defined in terms of shear strength (angle of internal friction) and GMSE-GRS wall height on the required geosynthetics base design length.

The results in Figure 34, for a vertical GMSE-GRS wall

[ = 90 ( = 0)] clearly indicate that the angle of internal friction has ramarkable effects on the geosynthetics basedesign length. The plot further shows that the required base design length is reduced as the shear strength of the RBF increases and the GMSE-GRS wall height reduces. Another interesting derivation is the fact that the popularly adopted

estimate; ( = 0.7) is only valid when; ( = 34Â°) as

demonstrated by the yellow marks with blue rings.

INFLUENCE OF ULTIMATE LOAD CARRYING CAPACITY AND GLOBAL STIFFNESS OF COMPOSITE GMSE-GRS MASS

ON INTERNAL STABILITY

The influence of ultimate load carrying capacity (ULCC) and global stiffness of the composite GMSE-GRS mass on the internal stability is punctiliously evaluated in [5]. An example of such application based on Equation 50 is provided in Figure

35. That ULCC has significant effect on the characteristics of the maximum reinforcement load can very well be appreciated.

= {[0.5741() 0.2969] (7 Ã— 1052

influence factors introduced in the K-Stiffness working stress Method that mainly take into account, the structural contribution of global and local stiffness, wall facing rigidity, batter angle and cohesion (reinforced backfill geomaterial properties), among other minor factors; ii) precise prediction and optimal quantification of the maximum reinforcement load and its distribution thereof; iii) reinforcement stiffness; iv) serviceability limit state; v) characteristics of yield and connection strengths of the reinforcing elements; vi) distribution attributes of the peak horizontal acceleration within the reinforced soil zone for varying magnitudes of reinforcement stiffness, wall geometry and quality of backfill geomaterials; vii) principal geosynthetics-soil interactive design parameters mainly the base design length and vertical reinforcement spacing; and viii) damping characteristics, among oter considerations [43]. A few examples of this initiative are provided in the subsequent sub-Sections A. ~ D.

Influence of GMSE-GRS wall geometry and reinforced backfill geomaterial properties on the force coefficient, K The TACH-MD analytical model that can be used for

reconnoitering the impact of GMSE-GRS wall geometry and quality of reinforced backfill geomaterials can be undertaken adopting Equation 30 (model is inset in Figure 36). The graphical curves depicted in Figure 36 were generated using this equation. It can be discerned that both these parameters have a linear influence on the seismic force coefficient as has been reported by various researchers ([39], [40], [44]). It can

0.0007 + 0.0342) .,

} Ã— [ .

] (50)

also be observed that as the shear strength of the reinforced

backfill, increases the wall facing thickness to height

[2.1257(0.0452)]ratio, tends to a limiting value within the 1st quadrant.

Influence of Ultimate Load Carrying Capacity on Maximum Reinforcement Load for Varying GMSE Wall Heights

30 = . . Ã— . + . .,

Ã— .

Influence of GMSE-GRS Wall Geometry and Reinforced Backfill Geomaterial

Shear Strength on the Seismic Force Coefficient

. .

.

1

= 2 + Ã— Ã— Ã— Ã— Ã—

Resulting Maximum Reinforcement Load, Tmax (kN/m)

25

0.7

Seismic Force Coefficient, K

0.6

= .

. + .

. .

20 UBLL: Upper Boundary Limiting Line for

Max. Reinf. Load, Tmax

15

10

., = .

Intermediary Boundary Limiting Line for Max. Reinf.

Load, Tmax

0.5

0.4

0.3

0.2

5

LBLL: Lower Boundary Limiting Line for

0 Max. Reinf. Load, Tmax

300 350 400 450 500 550

Ultimate Load Carrying Capacity (ULCC) Required to Sustain Tmax, qult.lc,Tm (kPa)

H=4 H=7 H=9 H=12 H=15 H=20 H=25 H=30

0.1

0.0

= .

0 0.05 0.1 0.15 0.2 0.25

Facing Wall Column Thickness to GMSE-GRS Wall Height Ratio, t/H

=20 =25 =30 =35 =40 =45 =50 =55

Fig. 35. Influence of ultimate load carrying capacity on the maximum reinforcement load for varying GMSE-GRS wall heights.

Fig. 36. Influence of quality of reinforced backfill geomaterial defined in terms of shear strength (angle of internal friction) and GMSE-GRS wall geometry on the seismic force coefficient: TACH-MD modelled results (refer to validation in Figure 18).

Influence of seismic acceleration and reinforced backfill geomaterial properties on seismic coefficient of active

0.6

TACH-MD Modelled Results: Influence of Wall Facing Stiffness Factor and GMSE-GRS Wall Geometry on the Seismic Force Coefficient

. . . .

. . .

pressure

The influence of seismic acceleration and quality of reinforced backfill can be explored by employing model Equations 31and 32. The results presented in Figure 37 demonstrate that both these parameters have significant effects on the magnitude and characteristics of the coefficient of seismic earth pressure whereby an increase in the quality of geomaterials (angle of internal friction/shearing resistance) culminates in a reduction of the resulting earth pressure whereas the converse applies to the seismic acceleration.

0.5

Seismic Force Coefficient, K

0.4

0.3

0.2

0.1

0.0

= . .

t: Wall Facing Thickness H: GMSE-GRS Wall Height

: Seismic Horizontal Acceleration Coefficient

= .

= .

Default Line:

= .

. .

Default Curve:

=

= .

Default Line:

= .

+ .

1.6

Seismic Active Earth Pressure Coefficient, Ka

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Seismic Active Earth Pressure Coefficient vs. Angle of Internal Friction Under the Influence of Seismic Acceleration

= . . + . + . ) + . . + . + .

a: Seismic Acceleration FI: Facing Inclination

ru: Constant Static Pore Pressure Ratio

FI = 50

ru = 0.25

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Angle of Internal Friction for Reinforced Backfill Geomaterial, ()

a=0g a=0.1g a=0.2g a=0.3g a=0.4g a=0.5g

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Wall Facing Stiffness Influence Factor, fs

H=4; t/H=0.075 H=7; t/H=0.043 H=9; t/H=0.033 H=12; t/H=0.025

H=15; t/H=0.02 H=20; t/H=0.015 H=25; t/H=0.012 H=30; t/H=0.010

Fig. 38. Influence of wall facing stiffness factor and GMSE-GRS wall geometry on the seismic force coefficient: TACH-MD modelled results.

D. Influence of the wall facing stiffness influence factor on the seismic active earth pressure coefficient for varying magnitudes of seismic acceleration.

The influence of the wall facing stiffness factor on the seismic active earth pressure coefficient for varying magnitudes of seismic acceleration can be investigated through the application of the models delineated in Equations 52 and 53; for estimation and perfect fit solutions, respectively. The graph depicted in Figure 39 is plotted using Equation 53. This figure demonstrates that the wall facing stiffness

Fig. 37. Influence of quality of reinforced backfill geomaterial defined in

terms of shear strength (angle of internal friction) and seismic acceleration on the coefficient of seismic earth pressure: TACH-MD modelled results (refer to validation in Figure 19).

Influence of GMSE-GRS wall geometry and wall facing stiffness influence factor on the force coefficient, K

As one of the vast examples to be presented and discussed comprehensively in [6], which involve the influence factors proposed in the K-Stiffness Method, the models delineated in Equations 51 ~ 53 and the corresponding results plotted in

influence factor has considerable impact on the magnitude and characteristics of the resulting seismic active earth pressure coefficient notwithstanding the scale of the seismic acceleration. Nevertheless, it can be inferred that the degree of this influence is impacted by the magnitude of seismic acceleration.

=

0.023

0.3818(2.2693) [24.2980.1966(0.5462 )] +

Figures 38 and 39 are invoked accordingly.

0.023

0.000259 [24.2980.1966(0.5462

) 2.2765

]

=

( )

1.5686(2.3868) (52)

= (7.83333 0.42 + 0.7017 +

(0.54620.023)

)

0.023

0.393)) [24.2980.1966

] + 29.4673

0.49 [24.2980.1966(0.5462

] + 2.1392 (51)

1.0552 + 3.2938 + 1.6045 (53)

The characteristic curves depicted in Figure 38 represent

1.4

Influence of Wall Facing Factor and Seismic Acceleration on Seismic Active Earth Pressure Coefficient

. .

the degree of influence of the wall facing stiffness factor on the

Seismic Active Earth Pressure Coefficient, Ka

1.2

= . . + . + . ) . .

+ . . + . + .

magnitude of the seismic force coefficient for varying wall

facing column thickness to GMSE-GRS wall height ratio

1.0

. .

.

= . . .

+ . .

() . The following derivations can be made from this

figure: i) the influence of the wall facing stiffness factor is

definitively significant; ii) the force coefficient increases in

exponential proportion with the increase in ; iii) reduction in the ratio results in increased seismic force; iv) the

default seismic force coefficient value can be designated as:

= 0.167 for a GMSE-GRS height of = 7 and =

0.8

0.6

0.4

0.2

0.0

a: Seismic Acceleration

FI:Facing Inclination

ru: Constant Static Pore Pressure Ratio

=

= .

= Default Wall Height

Default Line:

= .

0.588; and v) the proposed model can be effectively adopted

for the intended purposes.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Wall Facing Stiffness Influence Factor, fs

a=0g a=0.05g a=0.1g a=0.15g a=0.2g a=0.3g a=0.4g a=0.5g

Fig. 39. Influence of wall facing stiffness factor and seismic acceleration on the seismic active earth coefficient: TACH-MD modelled results.

APPLICATION OF PROPOSED TACH-MD GMSE-GRS ANALYTICAL MODELS IN DESIGN AND STABILITY ANALYSES

Application of the proposed TACH-MD GMSE-GRS analytical models is to be discussed in detail in [7]. As can be derived, the analytical models expressed in Equations 54 and

55 correlating the maximum reinforcement load, to the geosynthetic base design length, and the vertical reinforcement spacing, , respectively; as a function of GMSE-GRS wall height, can be adopted in the prediction and characterization of . Note that the and are

most integral parameters in relation to the internal stability

design. The results generated from these models are depicted

in Figures 40 and 41 for the ~ and ~ , respectively.

= {[0.4846() 0.3888] [(0.00072

0.0284 + 2.5028) ]} Ã— [ ] (54)

ESTABLISHMENT OF APPROPRIATE SERVICEABILITY CRITERIA AND PREDICTION OF

STRUCTURAL PERFORMANCE

Reference [8] proposes and delineates analytical models that can be applied to establish the appropriate serviceability criteria depending on various prevalent conditions and variables. Analytical models that can be adopted in the prediction of the structural performance of GMSE-GRS geostructures are also proposed and discussed herein. Equations 56 ~ 58 and Figures 42 and 43 provide but a few of such applications ([45], [46]).

Universal model correlating maximum lateral deflection to wall facing stiffness factor, GMSE-GRS wall height and shear properties of reinforced backfill geomaterial Equation 56 defines the analytical universal model

correlating the maximum lateral deflection to the wall facing stiffness influence factor, GMSE-GRS wall height and the

.

shear properties of reinforced backfill geomaterial.

, =

= {[0.4846() 0.3888](0.00072

7 .1 5 31

(1.5151515 ( ))

2.1268

0.33

)

0.0284 + 2.028)[21.796[0.055]

]} Ã—

0.813 {0.65 [0.2106 (

.

[ ] (55)

0.0001] Ã— 0.0783} Ã— 2.0624 () (56)

Influence of GR Base Design Length/Height Ratio on Maximum Reinforcement Load for Varying GMSE Wall Heights

30

= + Ã— Ã— Ã— Ã— Ã—

Maximum Reinforcement Load, Tmax (kN/m)

25 = . . . . + .

.

Ã—

.

UBLL: Upper Boundary Limiting Line for

Max. Reinf. Load, Tmax

Influence of wall facing stiffness on structural performance and serviceability limit state

The results depicted in Figure 42 indicate that the maximum

= . .

20

.

= . Ã— . . + . + .

Default Value for Length

15 to Height Ratio; L/H = 0.47

lateral deflection is defined by both the wall facing stiffness,

and the height of the GMSE-GRS wall, .

LBLL: Lower Boundary

10 Limiting Line for

Max. Reinf. Load, Tmax

, =

[0.75452+0.3259+6.325]7.1531

() (57)

[1+{1.5151515[5

0.33

]}]

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Geosynthetics Reinforcement Base Design Length to GMSE Wall Height Ratio, LGD/H

H=4 H=7 H=9 H=12 H=15 H=20 H=25 H=30

Variation of Predicted Maximum Lateral Deflection with GMSE-GRS Wall Depth Under the Influence of Wall Facing Stiffness Factor

Predicted Maximum Lateral Deflection, max (mm)

0 100 200 300 400 500 600

Fig. 40. Correlation between maximum reinforcement load, and 0

geosynthetics-soil interaction principal design parameter (base design length,

, =

. + . + .

.

as a function of GMSE-GRS wall height.

Influence of Vertical Reinforcement Spacing on Maximum Reinforcement Load for Varying GMSE Wall Heights

5 Default Height (H = 7m)

+ .

.

30

UBLL: Upper Boundary Limiting Line for

Max. Reinf. Load, Tmax

25

Depth of GMSE/GRS Retaining Wall, H (m)

1

= 2 + Ã— Ã— Ã— Ã— Ã—

10

. .

= . . . Ã—

Maximum Reinforcement Load, Tmax (kN/m)

+ . . . . + . .

GMSE: Geosynthetics Mechanically Stabilized Earth

GRS: Geosynthetics Reinforced Soil

: Line of Maximum Deflection: , =

: Intersection Point for

Default Values

= . .

20

15

Default WF Stiffness Curve

15 ( = . )

Default Value for Vertical

10 Reinforcement Spacing; Sv = 0.74

5

LBLL: Lower Boundary 20

Limiting Line for

Max. Reinf. Load, Tmax

25

0

0.2 0.4 0.6 0.8 1.0 1.2 1.4

Vertical Reinforcement Spacing (VRS), Sv (m)

H=4 H=7 H=9 H=12 H=15 H=20 H=25 H=30

30

fs=0.3 fs=0.4 fs=0.5 fs=0.588 fs=0.7 fs=0.8 fs=0.9 fs=0.98

Fig. 41. Correlation between maximum reinforcement load, and geosynthetics-soil interaction principal design parameter (vertical

reinforcement spacing, ) as a function of GMSE-GRS wall height.

Fig. 42. Variation of predicted maximum lateral deflection, with depth of GMSE-GRS walls considering effects of the wall facing stiffness factor,

.

Nomograph for designating maximum deflection in relation to wall facing stiffness influence factor with GMSE-GRS wall structural thickness considerations Figure 43 is cogitated to be a useful nomograph for

specifying the maximum lateral deflection based on the magnitude of the wall facing stiffness influence factor as a function of the wall facing thickness. This figure clearly shows that the wall facing stiffness significantly impacts on the magnitude of lateral deflection and that the degree of this influence is highly dependent upon the thickness of the wall facing column.

factor for GMSE-GRS walls that are below 3m in

height ( 3) whereas, for the default value of

= 48Â°, the factor can be adopted for walls up to; = 12, which is the limiting height of the K-

Stiffness Method.

The equivalent wall facing stiffness increases with increasing magnitude of shearing resistance, whilst the contribution of the wall facing stiffness influence factor degrades as the GMSE-GRS height increases.

The influence of the equivalent wall facing stiffness

factor is effectively initiated when: =

= 0.14390.607

0.1871

(58)

34Â° = 4, = 36Â° = 7, =

Monograph for Predicting Manitude of Maximum Lateral Deflection as a Result of

Influence of Wall Facing Stiffness Factor and GMSE/GRS Wall Structural Thickness

1.4

= .

40Â° = 12 , whilst = 50Â° = 30 , values which constitute the lower boundary

limits of the required angle of internal friction.

Wall Facing Stiffnes Influence Factor, fs

1.2

1.0

0.8

0.6

0.4

0.2

0.0

, =

. .

. + . + .

+

Default Thickness Curve (t = 0.3m)

Line of Default WFSF:

= .

t = tCB: Thickness of Concrete Block (m) GMSE: Geosynthetics Mechanically

Stabilized Earth

GRS: Geosynthetics Reinforced Soil

: DefaultValues Intersection Point

The shear strength of the backfill has significant influence on the magnitude and characteristics of the maximum reinforcement load.

The angle of internal friction has remarkable effects on the geosynthetics base design length whereby the required base design length is reduced as the shear strength of the RBF increases and the GMSE-GRS wall height reduces. Another interesting derivation is the

fact that the popularly adopted estimate; ( = 0.7) is only valid when; ( = 34Â°) as

0 100 200 300 400 500 600 700

Maximum Lateral Deflection, max (mm)

t=0.2 t=0.25 t=0.3 t=0.35 t=0.4 t=0.45 t=0.5 t=0.75

Fig. 43. Nomograph for predicting the magnitude of maximum lateral deflection under the influence of wall facing stiffness factor and GMSE-GRS structural thickness.

CONCLUSIONS

Universal and versatile analytical models equipped with a variety of application modules have been proposed in this paper. Application of the proposed models has been practically manifested through graphical examples for the characterization of the influence factors and material properties as well generation of imperative design parameters. The design characteristic curves and parametric values generated based on the application of these models distinctly confirm the validity, lucidity and rationality of the proposed analytical models. In particular, the following conclusions can be derived.

The proposed TACH-MD GMSE-GRS analytical models developed for the K-Stiffness Method geo- mathematically confirm the validity of the influence factors and facilitate for their versatile application and parametric correlation.

The upper and lower boundary values of the maximum reinforcement load are primarily dictated by height of

the GMSE-GRS wall with the resulting

increasing with the increase in height; and that for

demonstrated in this paper.

That ultimate load carrying capacity has significant effect on the magnitude and characteristics of the maximum reinforcement load.

The influence of the wall facing stiffness factor on the force coefficient is definitively significant; implicit that increased facing stiffness exponentially reduces the force coefficient.

The wall facing stiffness influence factor has considerable impact on the magnitude and characteristics of the resulting seismic active earth pressure coefficient notwithstanding the scale of the seismic acceleration, the extent of which is dependent on the magnitude of seismic acceleration.

The influence factors have profound impact on the

geosynthetic base design length, and the vertical reinforcement spacing, parameters of which are

most integral in relation to the internal stability design.

The results depicted in Figure 42 indicate that the maximum lateral deflection is defined by both the wall

facing stiffness, and the height of the GMSE-GRS wall, .

Figure 43 is cogitated to be a useful nomograph for

specifying the maximum lateral deflection based on the magnitude of the wall facing stiffness influence factor as a function of the wall facing thickness. This figure

12; 8

in conformity with the results

clearly shows that the wall facing stiffness significantly

reported for the walls calibrated for the K-Stiffness Method.

3. The maximum reinforcement load distribution characteristics are highly dependent on the shear strength of the reinforced backfill (RBF) geomaterial; and that RBF geomaterials with friction angles of

34Â° can only be characterized by the

impacts on the magnitude of lateral deflection and that

the degree of this influence is highly dependent upon the thickness of the wall facing column.

It is pragmatically demonstrated that the models proposed in this paper have the propensity to enable probing of limits and the extrapolated application of the

K-Stiffness Method to transcend the existing limitations including seismic response analysis.

The proposed analytical models can be effectively used in determining the appropriate limiting boundaries of the batter angle as a function of the shear strength of the reinforced backfill geomaterial and in modelling and determining the parameters required to achieve an optimal internal stability design mainly in consideration of the influence factors and various other design parameters.

ACKNOWLEDGMENT

The author wishes to acknowledge, with utmost gratitude, Prof. Richard Bathurst and Prof. Fumio Tatsuoka for kindly sharing their research, which have provided exceptional insight in the comprehension and elucidation of the internal stability characteristics.

Sincere appreciation is also expressed to the Japan International Cooperation Agency (JICA), Japan Bank of International Cooperation (JBIC), Construction Project Consultants Inc., Kajima Corporation and Kajima Foundation for funding a substantial part of the study conducted in Africa. The author is also indebted to the Materials Testing & Research Department, Ministry of Transport & Infrastructure, Kenya, as well as the Research Teams of Kensetsu Kaihatsu Engineering Consultants Limited and the Kenya Geotechnical Society (KGS) for their relentless efforts in providing the due assistance that culminated in the successful compilation of this paper.

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