estimation of the time-rate of settlement …docs.trb.org/prp/13-1567.pdf · farnsworth, c.b.,...

Farnsworth, C.B., Bartlett, S.F., and Lawton, E.C. 1

ESTIMATION OF THE TIME-RATE OF SETTLEMENT FOR MULTI-LAYERED

CLAYS UNDERGOING RADIAL DRAINAGE

Authors:

Clifton B. Farnsworth, Ph.D., P.E. (Corresponding Author)

Assistant Professor

Construction Management

Brigham Young University

230 SNLB

Provo, Utah 84602

Ph: (801) 422-6494

F: (801) 422-0653

email: [email protected]

Steven F. Bartlett, Ph.D., P.E.

Associate Professor

Civil & Environmental Engineering

University of Utah

110 Central Campus Drive, Suite 2000

Salt Lake City, Utah 84112

Ph: (801) 587-7726

F: (801) 585-5477


Evert C. Lawton, Ph.D., P.E.

Professor

Civil & Environmental Engineering

University of Utah

110 Central Campus Drive, Suite 2000

Salt Lake City, Utah 84112

Ph: (801) 585-3947

F: (801) 585-5477


Paper Length:

6,197 Words

5 Figures

0 Tables

TRB 2013 Annual Meeting Paper revised from original submittal.


ABSTRACT

This paper demonstrates how the finite difference technique can be used to estimate the

time-rate of settlement for soft, compressible clayey soils treated with prefabricated vertical

drains, at sites where primary consolidation settlement is occurring in a multilayered system at

varying rates. Semi-empirical methods based on surface settlement monitoring have typically

been used to estimate the progression of primary consolidation settlement. However,

interpretation of such methods can be problematic for multilayered soil profiles. For such sites,

it is crucial to obtain a reasonable characterization of the foundation soils’ horizontal drainage

properties and include these estimates in the time rate of settlement projections. Field

monitoring of subsurface instrumentation is extremely valuable in providing additional

information regarding the consolidation behavior of different layers. When subsurface field

measurements are coupled with the proposed numerical method, far more reliable projections are

obtained. This paper focuses on how to integrate field and laboratory data with time-rate of

settlement projections obtained from semi-empirical and finite difference methods to more

accurately predict the time-rate of consolidation behavior of multilayered foundation soils.



INTRODUCTION

Construction of large embankments, or other heavy structures, atop soft, thick

compressible foundation soils requires considerable time to complete end of primary (EOP)

consolidation settlement. In urban environments rapid construction techniques are often utilized

to lessen construction time, thus minimizing disruption to the public and generally decreasing the

cost of the project. In soft, low permeability soils, prefabricated vertical (PV) drains are

typically used to decrease settlement duration. These drains allow dissipation of excess pore

pressures to occur primarily in the horizontal direction by shortening the drainage path and

therefore markedly decreasing the time to reach EOP consolidation.

Even if vertical drains are used, the time required to complete EOP consolidation

settlement can still be considerable, making this a critical path activity of many soft ground

construction projects. Thus, having an accurate projection of the settlement duration is vital for

project planning and construction. For example, during the reconstruction of I-15 through Salt

Lake City, Utah (1998-2002),without ground treatment the low permeability thick clayey soils

found in the underlying Lake Bonneville sediments were expected to produce lengthy primary

settlement durations greater than two years (1). These lengthy EOP settlement durations could

not be accommodated in the planned construction schedule without the use of ground treatment.

The installation of PV drains and extensive field monitoring of settlement progression

allowed for the successful completion of the project within the allotted time (2). The use of PV

drains decreased the time associated with primary settlement to about three to six months,

depending on the spacing. The Asaoka method for predicting settlement (3) was used as the

primary tool for forecasting the EOP consolidation date and correspondingly allowing surcharge

fill to be removed and paving operations to commence (4). Settlement projections were made

solely from the surface using settlement plates extending through the fill; thus, the projections

were based on the composite settlement of the foundation soils.

As monitoring progressed, the design-build team noted problems with their Asaoka

projections. Typically as the original EOP projection date neared, an updated projection showed

that additional settlement time was required. Geotechnical designers suspected that this

phenomenon resulted from multiple layers consolidating at different rates, with some of the

deeper, thicker layers consolidating more slowly. This “delayed” consolidation and its

associated increase in construction time seriously impacted the project schedule. It was

concluded that the Asaoka method was not valid for the subsurface conditions found along parts

of the I-15 alignment due to the heterogeneity in drainage properties of the multilayered profile.

This paper summarizes subsequent research efforts to analyze the effects of differing

consolidation rates within a subsurface profile on the projection of EOP primary consolidation

settlement and to develop a more reliable projection method for such conditions (5).

A critical step in the estimation of settlement behavior is a complete geotechnical

characterization of the subsurface soils, including identifying the subsurface stratification, the

thickness of critical layers, and compressibility and drainage properties of these layers. The

quality and quantity of the subsurface investigation greatly impacts the settlement projections,

and for time-critical projects it is imperative that sufficient subsurface evaluations be performed

to reduce uncertainties. For foundation soils to be treated with PV drains this also includes

having an adequate knowledge of the horizontal drainage properties of the various soil layers.

Current methods for obtaining this information include back calculation from field performance

data, CPTU pore pressure dissipation or other in situ permeability tests, and laboratory Rowe



Cell testing (5). Of these methods, the CPTU pore pressure dissipation test appears to be the

most widely used technique for measuring the horizontal coefficient of consolidation.

The accuracy of any EOP settlement projection is a function of the predictive methods

employed and their simplifying assumptions. For some geologic environs with multilayer

deposits, the evaluation method(s) should consider the potential for various layers consolidating

at different rates, primarily due to differences in horizontal permeability. This paper contains an

evaluation of several potential EOP projection methods, their associated assumptions, and

implementation issues, progressing from simplified to more elaborate techniques.

ASAOKA PROJECTION METHOD

The progression of consolidation settlement is often monitored to verify initial settlement

projections and design parameters and to release areas for subsequent construction. In essence,

this is an application of the Observational Method (6), where decisions or revision of fast-paced

construction schedules are made according to the most recent field observations. Semi-empirical

methods, such as the Asaoka method, are attractive because they rely on observed settlement

data to make EOP projections that can be updated as more data become available.

The Asaoka method can be used to forecast the amount of EOP settlement and to back

calculate the coefficient of consolidation. It is applicable to a homogeneous clay layer

undergoing primary consolidation settlement owing to the application of a constant load. This

method follows the theory of consolidation introduced by Mikasa (7) that the 1-D consolidation

of clay is a function of the compressive strain, as opposed to the excess pore water pressure

dissipation used in Terzaghi theory (8). Asaoka used Mikasa theory because by being developed

from compressive strain, it was directly linked to the settlement. However, Asaoka considered

that a uniform strain develops throughout the clay profile, which is an incorrect assumption for

many situations. Duncan clearly demonstrated that this common assumption significantly

reduces the accuracy of the estimated time rate of settlement (9). When the strains decrease with

depth, which they typically do, the consolidation occurs more rapidly than when the strains are

uniform, when drainage occurs only at the top of the layer. Therefore, the relationship between

the degree of consolidation and the actual strain profile must be accounted for to accurately

estimate the time rate of settlement.

In implementing the Asaoka method, settlement monitoring is generally performed at the

surface using settlement plates that measure the composite vertical compression of the

foundation soils. The data analysis for this method involves selecting settlement data at

successive equal time steps. The settlement for the current time step (N) is then plotted against

the settlement for the previous time step (N–1). As settlement progresses the difference between

successive readings decreases and upon completion of settlement the values are equal.

Therefore, the best-fit line through these points intercepts a 1:1 sloped line, thus, providing the

ability to estimate both the projected magnitude of settlement and the time remaining to the end

of primary settlement. The basic form of the resulting first order difference equation is:

Si = β0 + β1 Si-1 (1)

where S is the measured settlement at time i, and β0 and β1 are the intercept and slope,

respectively, of the plotted line (3).

When successive data points are plotted they often do not follow the linear relationship

suggested by Asaoka, especially in the early part of the settlement history. Figure 1 shows an

example of this, using settlement data generated for a single layer with a numerical model. The

theoretical settlement data are plotted at equal time intervals starting from the initial settlement



reading. Even though the projection becomes more linear as the EOP consolidation settlement

nears, the nonlinearity in the early part of the projection unfortunately causes an increase in the

projected settlement data as more data are obtained and plotted.

Asaoka also suggested that a higher order autoregressive equation can be used for multi-

layered systems. This general settlement prediction model is expressed as:

Si = β0 + ∑ βL Si-L (2)

where the subscript, L, represents the number of different layers. However, there is not any

guidance for determining the partial slopes, βL, for Equation (2). Instead the focus is on the single

layer application to interpreting and forecasting field data (3). The higher order autoregressive

equation was not used by the I-15 designers for their predictions. Rather, the first order equation

was applied for forecasting, thus essentially treating the foundation system as a single

homogeneous clay layer. Unfortunately, for the multilayered system with differing consolidation

rates, the use of Equation (1) provided somewhat inaccurate results.

ASAOKA PROJECTIONS WITH DATA INTERPOLATION Field settlement data are often not gathered in equal time intervals, even though this is

necessary for utilizing the Asaoka method. For these cases, data interpolation is a useful tool for

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Settlement at time N-1 (m)

Se

ttle

me

nt

at

tim

e N

(m

)

FIGURE 1 Numerically modeled settlement data plotted with the

Asaoka Method demonstrating nonlinearity, especially in the early

portion of the dataset.



creating continuous curves and generating equal time intervals. The I-15 design team used this

approach, fitting the field data with a theoretical 1-D consolidation curve based on Terzaghi’s

theory (8), and then interpolating the data to equal time increments prior to performing their

Asaoka projections.

The 1-D primary consolidation settlement at any time (t) after a load has been applied is

estimated by the following equation:

Sc(t) = Uv Sc(t=∞) (3)

where Sc is the settlement at time t, and Uv is the degree of consolidation with vertical drainage.

It should be noted that this equation is fundamentally incorrect. Values of Uv actually represent

the average degree of dissipation of excess pore water pressure within the compressible layer and

not the average degree of primary consolidation settlement. Duncan demonstrated how the use

of the assumption shown in Equation (3) can provide unrealistic results, because conventional

theory assumes that the stress-strain behavior of the soil skeleton is linear and elastic (9). For

one-dimensional consolidation this is essentially the equivalent of assuming that the strains are

constant throughout the compressible layer (10). However, strains are more closely related to the

log of effective stress. This means that the strains are greatest in the upper portions of the soil

layer and decrease with depth within the layer. Duncan further showed that the dissipation of

excess pore water pressure needs to correspond to the actual strain profile to provide accurate

results (9).

Equation (4) from Sivaram and Swamee (11) is a best-fit approximation of Terzaghi’s 1-

D equation and can be used to calculate the average degree of consolidation for two-way vertical

drainage (e.g., where PV drains have not been used):

Uv = 100 * [(4Tv / π)0.5

] / [1 + (4Tv / π)2.8

]0.179

(4)

where Tv is the dimensionless time factor for two-way vertical drainage and is a function of the

coefficient of vertical consolidation, cv, the drainage path length, H, and the time of

consolidation, t. To match field settlement data with Equation (4), both the estimated EOP

settlement and Tv must be adjusted until a best fit of the field data is obtained.

To calculate the average degree of consolidation for radial drainage (e.g., where PV

drains have been used), the equation given by Barron (12) may be used:

Ur = 1 – e -8Tr / F(n)

(5)

where Ur is the degree of consolidation with radial drainage, Tr is the radial drainage time factor,

and F(n) is equal to:

F(n) = ln(n) * [n2 / (n

2 - 1)] – [(3n

2 - 1) / (4n

2)] (6)

and n is the drain spacing ratio defined by:

n = de / dw (7)

and de is the equivalent diameter of influence and dw is the diameter of the drain. Tr is similar to

the dimensionless vertical drainage time factor, Tv, but Tr is a function of the coefficient of

horizontal consolidation, ch, the length of the horizontal drainage path, de, (which is equal to two

times the radius of an equivalent soil cylinder from which radial drainage occurs), and the time

of consolidation, t. Values of Tr are related to these parameters by ch/de2. As before, to match

the field settlement data with the analytical curve(s), trial values of EOP settlement and Tr must

be made until a best fit is obtained.

Because Equations (4) and (5) are based on Terzaghi’s 1-D consolidation theory, the

accuracy of the results are therefore limited by the simplifying assumptions for which

conventional theory is based, as mentioned. However, this research has demonstrated that

Asaoka projections relying solely on surface settlement data can provide reasonable results for



sites with multiple layers consolidating at or near the same rate, but this is not true for sites

where multiple layers are consolidating at very different rates. For such cases, additional

monitoring data and analytical approach are required, as discussed subsequently.

ASAOKA PROJECTIONS WITH SUBSURFACE MEASUREMENTS During the I-15 Reconstruction Project, magnet extensometers were placed at key

locations to measure the compression of individual clay layers within the soil profile. This

monitoring strategy is preferable where clay layers are consolidating at markedly different rates

because this technique can provide more reliable estimates of the time rate of consolidation than

surface monitoring. Project data from magnet extensometer MR s29-6-1 will be used as an

example to show how to calculate the rate of consolidation for individual layers.

Prior to fill placement, a magnet extensometer was installed in the foundation soils. Nine

spider magnets were strategically placed within the subsurface, one at the base of the drill-hole

and another just beneath the ground surface, and seven others in between targeting major soil

boundaries (see Figure 2). Spider magnets are designed to remain at the relative soil location at

which they are installed. An extensometer probe subsequently measures the decreasing relative

distance between adjacent magnets, thus revealing the compression of the individual soil layers

(13). Magnet elevations for this extensometer were predetermined using a CPT profile. Figure 2

shows the CPT profile for this site and the corresponding magnet elevations in relation to the soil

boundaries. The four major layers contributing to the foundation settlement at this site include

the upper Lake Bonneville clay (ULBC), interbedded silts and sands (IB), lower Lake Bonneville

clay (LLBC), and deeper Pleistocene alluvium and clay (PA) (see Figure 2).

The settlement at this location was caused by the placement of a 12-m high embankment

(including surcharge) that was constructed over PV drain treated foundation soil. The settlement

curves in Figure 3(a) are calculated from the change in elevation with respect to time for each

magnet. Because the settlement measured at each magnet is cumulative, these plots show the

total amount of compression that occurred in all layers below each magnet position. Therefore,

magnets with the largest settlement are those positioned closest to the surface.

The embankment that caused this settlement was placed in two major stages, with the

second stage of construction beginning in September 1998. The bottom two magnets (base

magnet and magnet 1) did not show any measurable settlement; thus, the elevation of these

magnets did not change due to the placement of the embankment, as shown in Figure 3(a). In

contrast, the top two magnets (magnet 7 and surface magnet) represent the total settlement of the

soil profile because all of the measured foundation settlement occurred beneath the elevations of

these two magnets. These data indicate that there was about 0.8 m of foundation settlement that

occurred over a 9-month period.

As indicated earlier, the Asaoka method is based on the assumption that the loading

remains continuous throughout the settlement record. In this case, the Asaoka method can be

applied to the data starting from the beginning of embankment placement through September, or

it can be applied to the data starting in September through the end of the record. However, the

data for the second stage is generally more important because it represents the full loading

condition, and its settlement behavior controls the start date for subsequent pavement

construction. Because the settlement history is known in each major layer, the Asaoka method

can be used to estimate the total settlement, consolidation rates, and drainage properties for each

layer. The settlement plots in Figure 3(b) were obtained by differencing the total settlement

measurements for each of the four major layers contributing to the foundation settlement.



Subsequently, the data in Figure 3(b) was used to make Asaoka projections for the

individual layers shown in Figure 3(a). Equation (5) was used to interpolate the data to equal

time increments prior to completing the projections. The percentage of EOP settlement at day

279 (the final day within the record) was 99.6, 98.2, 84.2, and 93.7%, for the ULBC layer, IB

layer, LLBC layer, and PA layer, respectively. An Asaoka projection was also performed for the

entire soil profile using only surface settlement data and resulted in the foundation soils

achieving 95.4% consolidation at day 279. However, this percentage is misleading because it

fails to account for the varying consolidation rates actually occurring within each layer. The

individual layer results clearly identify that the different intervals are consolidating at different

rates. Surface settlement projections cannot account for these differences and therefore over

project the actual level of consolidation. In this case, the result would be additional settlement

within the LLBC layer.

PA

LLBC

IB

ULBC

Base

1

2

3

4

5

6 7 Surface

FIGURE 2 Predominant subsurface layers shown with corresponding magnets

(horizontal lines labeled base through surface) for magnet extensometer MR s29-6-1.



-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

10 100 1000

Days

Sett

lem

en

t (m

)

PA (1 to 3)

LLBC (3 to 4)

IB (4 to 5)

ULBC (5 to 6)

FIGURE 3 (a) Settlement versus time data for magnet extensometer MR s29-6-

1, (b) Total settlement record for each of the subsurface target intervals,

plotted against the logarithm of time.

PA

LLBC

IB

ULBC



To maximize the effectiveness of the surcharge, the I-15 design team selected 98% EOP

consolidation as the target value for starting surcharge removal and subsequent pavement

construction (2). This value was calculated based on Asaoka projections that used surface

monitoring from a single settlement plate placed at the base of the fill, since the use of magnet

extensometers was relatively limited for this project. However, the results clearly indicate that

better EOP estimates could have been made if layer-by-layer projections and more appropriate

methods with consistent assumptions had been used. This research indicates that magnet

extensometers should be deployed for heterogeneous, multilayered systems, especially for cases

where high quality geotechnical data are not available to adequately quantify the coefficient of

consolidation for the various layers. This is especially true for time critical embankments where

the subsurface soils are not fully characterized and estimates of drainage properties have not

been obtained or are poorly supported by the data at hand.

The magnet extensometer data can also be used to back-calculate the effective horizontal

coefficient of consolidation, ch(e), for each consolidating layer. The term “effective” is used

because the back calculated values represent the average horizontal coefficient of consolidation

for the selected interval between two respective magnets, influenced by the disturbed zone that

typically develops around the PV drains. The back calculation is performed from magnet

extensometer data for each individual soil layer using the technique described by Bergado (14).

This technique utilizes results from the Asaoka projection method, and therefore maintains the

same limitations as the Asaoka method, as previously identified. The simplified equation for this

back calculation is:

ch(e) = (Fn + Fs) / [C1 + (C2 / qw)] (8)

where Fn is a drain spacing correction, Fs accounts for the installation smear effects, C2 accounts

for drain well resistance, and qw is the discharge capacity of the drain. The C1 term requires the

slope of the projection line (i.e., β1) and the incremental time step (i.e., Δt) from the Asaoka

projection, along with the equivalent PV drain zone of influence (5).

Back calculated values of ch(e) using the settlement record from magnet extensometer MR

s29-6-1 are 38, 25, 10, and 16 mm2/min for the UBLC, IB, LLBC, and PA layers, respectively.

The average value of ch(e) for the entire subsurface is 18 mm2/min. The above ch(e) values are

useful because they represent estimates of the actual drainage properties for the PV drain treated

soils, including installation disturbance effects. For this particular example, the horizontal

coefficient of consolidation for the LLBC layer is only one-fourth that of the ULBC layer.

PROJECTIONS USING THE FINITE DIFFERENCE METHOD

Terzaghi’s 1-D consolidation equation is a 2nd

order partial differential equation that can

be solved in a variety of ways. Perrone developed a general finite-element 1-D consolidation

computer program for vertical drainage of multilayered systems (15). However, this program

does not address radial drainage in soils with PV drains. We believe that the finite difference

method (FDM) offers the simplest and most direct way of numerically modeling the

consolidation process for PV drain treated soil (16, 17). In addition, the properties required for

the FDM can be obtained from high-quality laboratory testing, or from in situ measurements that

are further verified and calibrated with magnet extensometer data for multilayered systems.

Regardless of the numerical approach, conventional theory (8, 18) has three important

assumptions that must be addressed to provide reliable estimates of the consolidation process (9).

These assumptions are that the coefficient of consolidation is constant, the stress-strain behavior



of the soil skeleton is linear and elastic, and how the vertical strain distribution in the soil profile

relates to the average degree of consolidation and the dimensionless time factor (T).

Regarding the first assumption, the coefficient of consolidation greatly decreases as the

vertical effective stress reaches the preconsolidation pressure and also varies as a function of

depth and with time in a given layer (9). These considerations are accommodated by the FDM

because its fundamental algorithm allows for material properties to change with respect to

overconsolidation ratio and effective vertical stress at each time step, for each sublayer or node.

Regarding the second assumption, soil behavior is actually nonlinear; thus, 1-D Terzaghi

consolidation theory (8) is not applicable for large-strain consolidation problems like those found

in highly compressible clays (9). Standard consolidation tests show that the change in void ratio

(or vertical strain) is proportional to the change in the logarithm of effective stress for

recompression and virgin compression. Thus, results from representative laboratory

consolidation tests can be used describe the nonlinear relationship between void ratio and

effective vertical stress for recompression and virgin compression. However, in doing so, the

recompression and virgin compression indices (i.e, cr and cc) should be corrected using a method

such as that developed by Schmertmann (19) to obtain the slopes for “field corrected” cr and cc

values for each sublayer. Also, a good definition of the overconsolidation ratio is needed so that

the appropriate compression index can be used to calculate the incremental settlement within

each sublayer.

Finally, the third assumption is not necessary in the implementation of the FDM because

the strain is calculated between nodal points within the mesh. Therefore, no a priori assumption

is needed regarding the strain distribution that develops within the mesh.

For this research, the FDM was developed in spreadsheet format using the equations for

1-D consolidation summarized by Das (17) for both the vertical and radial drainage cases (5).

The basic finite difference equation to express the dissipation of excess pore water pressure for

1-D vertical consolidation of a soil layer using two-way vertical drainage is:

u0,t+Δt = (Δt / (Δz)2) * (u1,t + u2,t - 2u0,t) + u0,t (9)

where u is the excess pore water pressure, Δt is a factor equal to the coefficient of vertical

consolidation, cv, multiplied by the change in time, Δt, and Δz is the change in depth. In this

equation, node 0 represents the selected node, node 1 represents the adjacent node directly

above, and node 2 represents the adjacent node directly below. With this equation, a linear set of

vertical nodes can be used to calculate the 1-D dissipation of excess pore pressures within the

subsurface profile considering only vertical drainage.

The FDM can also be used to estimate the dissipation of excess pore pressures for the

radial drainage case. The basic finite difference solution for one-dimensional consolidation

considering only radial drainage is:

u0,t+Δt = (Δt/(Δr)2)*{u3,t+u4,t+[(u4,t–u3,t)/2(r/Δr)]-2u0,t} + u0,t (10)

where u is the excess pore water pressure, Δt is a factor equal to the coefficient of horizontal

consolidation, ch, multiplied by the change in time, Δt, r is the radius of drainage influence for

the PV drain, and Δr is the change in radius. In this equation, node 0 represents the selected

node, node 3 represents the adjacent node directly to the left, and node 4 represents the adjacent

node directly to the right. From this equation, a linear set of horizontal nodes can be used to

calculate the 1-D radial dissipation of excess pore pressures within the subsurface profile.

For PV drain treated soil, both horizontal and vertical drainage occur. However, for

relatively thick clay layers (3 to 5 m, or greater) and with typical PV drain spacing (1.5 m



triangular spacing), drainage will occur predominately in the radial direction; thus, vertical

drainage was neglected for the analyses and results shown later.

The value of Δt / (Δr)2 in Equations (9) and (10) must remain less than 0.5 for

convergence of the solution. The best approximation of the solution occurs for Δt / (Δr)2 equal

to the ratio of 1/6 (20). Additionally, because consolidation is a highly nonlinear process, it is

important to subdivide relatively thick layers into sublayers approximately 0.3-m thick. The

increase in effective vertical stress for each sublayer can be calculated using methods that

account for the geometry of the embankment or applied loading and layering of the foundation

soils. For this research, a 2-D vertical stress distribution was developed using the Boussinesq

solution for the calculation of vertical stress beneath the center of an embankment (21). It was

further assumed that the initial excess pore water pressure was equal to the change in vertical

stress, based on the vertical stress distribution.

The use of the FDM to calculate the time-rate of settlement of foundation soils provides

the ability to replicate the actual fill placement process, which may take several weeks and is

commonly referred to as a ramp loading. The actual load can be adjusted at the appropriate time

steps within the finite difference model to represent the loading sequence or to use an average

loading condition for the duration of the ramp loading and then adjust the model to the final load

condition at the completion of the ramp loading sequence. Additionally, if staged embankment

construction is used, the staged loading can be modeled as a series of instantaneous loads that are

placed at certain intervals of time. Thus, the FDM has the inherent ability (because it is a time

stepping technique) to provide estimates of pore pressure dissipation for the anticipated or actual

loading scenario.

To implement the FDM for a PV drain treated soil, a horizontal 1-D finite mesh is created

for each sublayer and representative values of the horizontal coefficient of consolidation must be

selected for each sublayer. Back calculated values of ch(e) from magnet extensometer data are

especially useful, because they provide an average horizontal coefficient of consolidation for

specific layers, including any disturbance effects resulting from PV drain installation. If back

calculated ch(e) values are not available, then an assumption must be made regarding the degree

of disturbance and its effect upon the horizontal coefficient of consolidation. In most instances,

field performance data will not be available and the horizontal drainage properties of the soil

layers must be obtained in some other manner. The most common technique is with the CPTU

pore pressure dissipation test. However, other in situ permeability tests could also be performed.

An underutilized technique is utilization of the Rowe Cell, which can be used to perform a 1-D

laboratory consolidation test with radial drainage. Each of these techniques provides a measure

of the horizontal coefficient of consolidation.

Once all compressibility and drainage properties were defined for this research, the FDM

spreadsheet was used to calculate the dissipation of the excess pore water pressure, the change in

vertical effective stress, and the subsequent vertical strain and settlement as a function of time

due to the placement of the embankment. The effective vertical stress at each time step was

calculated as the average dissipated excess pore water pressure in the sublayer added to the

original in situ effective stress for hydrostatic conditions. The change in void ratio for virgin

compression during each time increment is calculated as:

e = cc log (vt+t / v(t)) (11)

For recompression, the same equation can be used, except that cr is substituted for cc. The

vertical strain for each sublayer is calculated from:

vi = e / (1 + eo) (12)



where eo is the initial void ratio for recompression or the void ratio at the preconsolidation stress

for virgin compression. The settlement for each sublayer is:

Svi = vi * i (13)

The summation of settlement for all individual sublayers produces the total settlement at each

time increment.

The FDM is particularly useful as an observational technique during construction to

interpret field performance data by adjusting or calibrating the model to match the subsurface

settlement measurements. Because magnet extensometer measurements were available for this

research, values of ch(e) for each layer were back calculated using the settlement data shown in

Figure 3(b). A trial and error method was used by varying ch(e) values for each layer until the

FDM model matched the observed settlement record, as shown in Figure 4. Other FDM model

parameters, including estimates of the initial effective vertical stress, OCR, and the field

corrected cr and cc for each layer, were obtained from laboratory Rowe Cell testing utilizing

horizontal drainage (5).

The back calculated ch(e) values were 26, 13, 5, and 9 mm2/min for the ULBC, IB, LLBC,

and PA layers, respectively. These values are lower than those obtained using the Asaoka back

calculation method, varying between approximately 30% - 50% smaller. Comparably, values of

ch(e) for typical Lake Bonneville clay deposits, obtained in the laboratory using a Rowe Cell with

horizontal drainage capabilities, vary between about 4 and 90 mm2/min (5). The laboratory

-0.80

-0.70

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0 50 100 150 200 250 300

Time (days)

Se

ttle

me

nt

(m)

layer 5-6

layer 4-5

layer 3-4

layer 1-3

Total Settlement

FIGURE 4 Finite difference generated settlement profiles (solid lines) with actual

individual layer results from magnet extensometer MR s29-6-1, as well as the

total settlement curve and numerically modeled results for the entire profile.



values provided a much wider range, because samples from many different depths were tested,

while the back calculated effective values represent an average across each layer. Furthermore,

the FDM results show the dissipation of excess pore pressure at day 279 as 99.3, 91.7, 63.4, and

82.6% for the ULBC, IB, LLBC, and PA layers, respectively. These results show that the bottom

three layers do not have nearly the dissipation of excess pore pressure as originally obtained

using the Asaoka back projection method, especially the LLBC layer. The FDM results are

approximately 7% - 25% smaller. This further demonstrates the importance of correctly

modeling pore pressure dissipation for time-rate of settlement calculations.

Figure 4 also shows the calculated composite settlement curve from the FDM calibrated

to the magnet extensometer data, demonstrating a good fit to the observed data. Thus, we

conclude that the FDM can be used to reasonably estimate the total settlement curve for a

multilayered system consolidating at different rates, when properly calibrated. However, the use

of magnet extensometer data provides a better understanding of the individual layers contributing

to the composite settlement profile and demonstrates the value of such data.

CONCLUSIONS

In many instances of highway embankment construction over soft soil sites within an

urban environment, the time required for primary consolidation settlement to occur governs the

critical path of the embankment construction. Having an accurate projection of the end of

primary settlement is often much more critical than having an accurate estimate of the magnitude

of the total settlement. To provide accurate time-rate of settlement estimates, it is of foremost

importance that an appropriate geotechnical investigation and subsurface characterization be

performed and that subsequent design and construction techniques are appropriately utilized.

The Asaoka projection method can be a valuable tool for estimating EOP consolidation

settlement, but its accuracy is limited by its simplifying assumptions. For foundations with fairly

uniform consolidation properties, the Asaoka method with curve-fitting techniques can be

effectively used for both vertical and radial drainage. However, this method loses accuracy for

cases when the foundation soils include multiple layers consolidating at substantially different

rates. For such cases, more rigorous methods, such as the FDM, are recommended.

The data obtained from magnet extensometers can greatly improve the accuracy of EOP

projections. The use of magnet extensometer data, in conjunction with the Asaoka method,

makes it possible to estimate the level of consolidation for each subsurface layer. Such data can

be utilized with back calculation methods to provide the effective coefficient of consolidation for

each subsurface layer, which considers disturbance effects resulting from PV drain installation.

The FDM is an underutilized numerical tool with the ability to provide accurate estimates

of the time-rate of settlement of foundation soils with radial drainage. This accuracy occurs

because the dissipation of excess pore pressures and its relation to vertical strain is more

correctly accounted for in the calculations. However, to implement the FDM a comprehensive

characterization of the foundation soils is required to provide reliable projections. This research

demonstrated that the FDM coupled with magnet extensometer measurements provides an

accurate fit of the observed settlement behavior. This research suggests that the FDM be more

universally applied.

The methods described within this paper should be utilized to provide more accurate

time-rate of settlement estimates for layered clay systems with radial drainage, with the best

method being the FDM coupled with magnet extensometer data. Maintaining a harmonious

balance between the geotechnical evaluations and the use of observational data is an important



part of geotechnical engineering. When used together appropriately, they provide the ability to

achieve accurate and reliable estimates of the time-rate of settlement behavior for soft

multilayered foundation soils.

ACKNOWLEDGEMENT

The authors wish to acknowledge and thank the Research Division of the Utah

Department of Transportation for their financial contributions and technical support to this

research project.

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