3

Transportation Research Record: Journal of the Transportation Research Board, No. 2363, Transportation Research Board of the National Academies, Washington, D.C., 2013, pp. 3–11.DOI: 10.3141/2363-01

C. B. Farnsworth, School of Technology, Brigham Young University, 230 Snell Building, Provo, UT 84602. S. F. Bartlett and E. C. Lawton, College of Civil and Environmental Engineering, University of Utah, Suite 2000, 110 Central Campus Drive, Salt Lake City, UT 84112. Corresponding author: C. B. Farnsworth, cfarnsworth@byu.edu.

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 2 years (1). These lengthy EOP settlement durations could not be accommo-dated 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 3 to 6 months, which depends on the drain spacing. The Asaoka method for predict-ing 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 by 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 projec-tion date neared, an updated projection showed that additional settle-ment time was required. Geotechnical designers suspected that this phenomenon resulted from multiple layers consolidating at differ-ent rates, with some of the deeper, thicker layers consolidating more slowly. This “delayed” consolidation and its associated increase in construction time seriously affected the project schedule. The design-ers concluded that the Asaoka method was not valid for the sub-surface conditions found along parts of the I-15 alignment because of 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 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 com-plete geotechnical characterization of the subsurface soils, including identification of the subsurface stratification, the thickness of criti-cal layers, and compressibility and drainage properties of these lay-ers. The quality and quantity of the subsurface investigation greatly affects the settlement projections, and for time-critical projects, it is imperative that sufficient subsurface evaluations be performed to reduce uncertainties. For foundation soils that are to be treated with PV drains, this estimation also includes having an adequate knowl-edge of the horizontal drainage properties of the various soil layers. Current methods for obtaining this information include backcalcu-lation from field performance data, the cone penetration test (CPT) with piezometer probe (CPTU) pore pressure dissipation or other in situ permeability tests, and laboratory Rowe cell testing (5). Of

Estimation of Time Rate of Settlement for Multilayered Clays Undergoing Radial Drainage

Clifton B. Farnsworth, Steven F. Bartlett, and Evert C. Lawton

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 consoli-dation settlement is occurring in a multilayered system at varying rates. Semiempirical 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 reason-able characterization of the foundation soils’ horizontal drainage proper-ties and include these estimates in the time rate of settlement projections. Field monitoring of subsurface instrumentation is extremely valuable in providing additional information about the consolidation behavior of dif-ferent 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 projections of time rate of settlement obtained from semiempirical and finite difference methods to predict more accurately the time rate of consolidation behavior of multilayered foundation soils.

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 used to reduce construction time, thus minimizing disruption to the public and generally decreasing the cost of a project. In soft, low-permeability soils, prefabricated vertical (PV) drains are typi-cally used to decrease settlement duration. These drains allow dis-sipation of excess pore pressures 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, an accurate projection of the settlement duration is vital for project planning and construction. For example, during reconstruc-tion of I-15 through Salt Lake City, Utah (between 1998 and 2002),

4 Transportation Research Record 2363

these methods, the CPTU pore pressure dissipation 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 methods should consider the potential for various layers consolidating at different rates primarily because of differences in horizontal per-meability. 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 ver-ify initial settlement projections and design parameters and to release areas for subsequent construction. In essence, this is an application of the observational method (6), in which decisions or revisions of fast-paced construction schedules are made in response to the most recent field observations. Semiempirical 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 backcalculate the coefficient of consolidation. It is applicable to a homogeneous clay layer undergoing primary con-solidation settlement owing to the application of a constant load. This method follows the theory of consolidation introduced by Mikasa (7) that the one-dimensional (1-D) consolidation of clay is a function of compressive strain as opposed to the dissipation of excess pore water pressure used in the Terzaghi theory (8). Asaoka used the Mikasa theory because, by its 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 incor-rect 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, as 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 taken into account for an accurate estimate of the time rate of settlement.

In implementing the Asaoka method, settlement monitoring is generally performed at the surface by 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 and thus provides one 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 expressed as

S Si i= β + β − (1)0 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 numeri-cal 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

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

Settlement at time N-1 (m)

Set

tlem

ent

at t

ime

N (

m)

FIGURE 1 Numerically modeled settlement data plotted with Asaoka method to demonstrate nonlinearity, especially in early portion of data set.

Farnsworth, Bartlett, and Lawton 5

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 equa-tion can be used for multilayered systems. This general settlement prediction model is expressed as

S Si L i L∑= β + β − (2)0

where L represents the number of different layers. However, no guidance exists for determining the partial slopes, βL. Instead, the focus is on the single-layer application to interpreting and forecast-ing 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, and the foundation system was thus essentially treated as a single homogeneous clay layer. Unfortu-nately, 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 such data collection is necessary for using the Asaoka method. For these cases, data interpolation is a useful tool for creat-ing continuous curves and generating equal time intervals. The I-15 design team used this approach, fitting the field data with a theo-retical 1-D consolidation curve that was based on Terzaghi’s theory (8) and then interpolating the data to equal time increments before 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:

S U Sc t v c t= ( )( ) =∞ (3)

where Sc is the settlement at time t and Uv is the degree of consolida-tion with vertical drainage. 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 dem-onstrated 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 1-D consolidation, this assumption is essentially the equivalent of using strains that are constant throughout the compressible layer (10). However, strains are more closely related to the log of effective stress. This relationship means that the strains are greatest in the upper por-tions of the soil layer and decrease deeper in 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 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) (11):

U

T

Tv

v

v

=π

+π

100

4

14

(4)

0.5

2.8 0.179p

where Tv is the dimensionless time factor for two-way vertical drain-age and 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 set-tlement and Tv must be adjusted until a best fit of the field data is obtained.

To calculate the average degree of consolidation for radial drain-age (e.g., where PV drains have been used), the equation given by Barron may be used (12):

U erT F nr= − ( )−1 (5)8

where

F n nn

nn

n( )( )

( )( ) ( )=−

−

−

ln

1

3 1

4(6)

2

2

2

2p

and

Ur = degree of consolidation with radial drainage, Tr = radial drainage time factor, and n = the drain spacing ratio defined by

ndd

e

w

= (7)

where de is the equivalent diameter of influence and dw is the diam-eter of the drain. Tr is similar to the dimensionless vertical drainage time factor, Tv, but Tr is a function of the coefficient of horizon-tal 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 con-solidation, t. Values of Tr are related to these parameters by ch/de

2. As before, to match the field settlement data with the analytical curves, 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 consoli-dation theory, the accuracy of the results is therefore limited by the simplifying assumptions on which conventional theory is based, as noted earlier. However, this research has demonstrated that Asaoka projections relying solely on surface settlement data can provide rea-sonable 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 extremely different rates. For such cases, additional monitoring data and analytical approaches are required, as discussed later in the paper.

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 prefer-able 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 the MR s29-6-1 magnet extensometer is used as an example to show how to calculate the rate of consolidation for individual layers.

Before fill placement, a magnet extensometer was installed in the foundation soils. Nine spider magnets were strategically placed within

6 Transportation Research Record 2363

the subsurface, one at the base of the drill hole, another just beneath the ground surface, and seven others between targeting major soil boundaries (Figure 2). A spider magnet is designed to remain at the relative soil location at which it is installed. An extensometer probe subsequently measures the decreasing relative distance between adja-cent magnets, thus revealing the compression of the individual soil layers (13). Magnet elevations for this extensometer were predeter-mined by 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 set-tlement 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) (Figure 2).

The settlement at this location was caused by the placement of an embankment 12 m high (including surcharge) that was constructed over PV drain treated foundation soil. The settlement curves in Figure 3a 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 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 Mag-net 1) did not show any measurable settlement; thus, the elevation of these magnets did not change because of the placement of the embankment, as shown in Figure 3a. In contrast, the top two mag-nets (Magnet 7 and surface magnet) represent the total settlement of the soil profile because all measured foundation settlement occurred beneath the elevations of these two magnets. These data indicate that about 0.8 m of foundation settlement occurred over a 9-month period.

As indicated earlier, the Asaoka method is based on the assump-tion that the loading remains continuous throughout the settlement record. In this case, the Asaoka method can be applied either to the data starting from the beginning of embankment placement through September or to the data starting in September through the end of the record. However, the data for the second stage are 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 settle-ment plots in Figure 3b were obtained by differencing the total settle-ment measurements for each of the four major layers contributing to the foundation settlement.

Subsequently, the data in Figure 3b were used to make Asaoka projections for the individual layers shown in Figure 3a. Equation 5 was used to interpolate the data to equal time increments before the projections were completed. The percentages of EOP settlement on Day 279 (the final day within the record) were 99.6%, 98.2%, 84.2%, and 93.7%, for the ULBC, IB, LLBC, and PA layers, respectively. An Asaoka projection was also performed for the entire soil profile by using only surface settlement data and resulted in the founda-tion soils achieving 95.4% consolidation on Day 279. However, this percentage is misleading because it fails to account for the varying consolidation rates 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 overproject the actual level of con-solidation. In this case, the result would be additional settlement within the LLBC layer.

To maximize the effectiveness of the surcharge, the I-15 design team selected 98% EOP consolidation as the target value for start-ing surcharge removal and subsequent pavement construction (2). This value was calculated on the basis of Asaoka projections that used surface monitoring from a single settlement plate placed at the base of the fill because the use of magnet extensometers was rela-tively 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 assump-tions had been used. This research indicates that magnet extensom-eters should be deployed for heterogeneous, multilayered systems, especially for cases in which high-quality geotechnical data are not available to quantify adequately the coefficient of consolidation for the various layers. This is especially true for time-critical embank-ments at which 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 backcalculate the effective horizontal coefficient of consolidation, ch(e), for each consol-idating layer. The term “effective” is used because the backcalculated values represent the average horizontal coefficient of consolidation

FIGURE 2 Predominant subsurface layers shown with corresponding magnets (horizontal lines labeled “base” through “surface”) for MR s29-6-1 magnet extensometer.

PA

LLBC

IB

ULBC

Base

1

2

3

4

5

67Surface

Farnsworth, Bartlett, and Lawton 7

FIGURE 3 Graphs of (a) data for settlement versus time for MR s29-6-1 magnet extensometer and (b) total settlement record for each subsurface target interval plotted against logarithm of time.

(a)

(b)

8 Transportation Research Record 2363

for the selected interval between two respective magnets influenced by the disturbed zone that typically develops around the PV drains. The backcalculation is performed from magnet extensometer data for each soil layer by using the technique described by Bergado (14). This technique uses results from the Asaoka projection method and therefore maintains the same limitations as the Asaoka method, as previously identified. The simplified equation for this backcalculation is shown in Equation 8:

cF F

CCq

h en s

w

( )=

+

+

( ) (8)

12

where

Fn = drain-spacing correction, Fs = installation smear effects, C2 = drain-well resistance, qw = discharge capacity of drain, and C1 = radial distance and rate of drainage.

Backcalculated values of ch(e) using the settlement record from the MR s29-6-1 magnet extensometer were 38, 25, 10, and 16 mm2/min for the ULBC, IB, LLBC, and PA layers, respectively. The average value of ch(e) for the entire subsurface was 18 mm2/min. Those ch(e) values were useful because they represented estimates of the actual drainage properties for the PV drain treated soils, including installa-tion disturbance effects. For this example, the horizontal coefficient of consolidation for the LLBC layer was only one-fourth that for the ULBC layer.

Projections using finite difference Method

Terzaghi’s 1-D consolidation equation is a second-order partial differ-ential equation that can be solved in a variety of ways. Perrone devel-oped 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. The current authors believe that the finite difference method (FDM) offers the sim-plest 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 assump-tions are that (a) the coefficient of consolidation is constant, (b) the stress–strain behavior of the soil skeleton is linear and elastic, and (c) the vertical strain distribution in the soil profile must relate to the average degree of consolidation and the dimensionless time factor (T) in a specific way.

In relation to the first assumption, the coefficient of consolidation greatly decreases as the vertical effective stress reaches the preconsoli-dation 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.

As for 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 encountered with 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 com-pression. Thus, results from representative laboratory consolidation tests can be used to describe the nonlinear relationship between void ratio and effective vertical stress for recompression and virgin com-pression. However, when this is done, the recompression and virgin compression indices (i.e., cr and cc) should be corrected by using a method such as that developed by Schmertmann to obtain the slopes for field-corrected cr and cc values for each sublayer (19). In addition, 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 implemen-tation of the FDM because the strain is calculated between nodal points within the mesh. Therefore, no a priori assumption is needed in relation to the strain distribution that develops within the mesh.

For this research, the FDM was developed in spreadsheet format by 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 that uses two-way vertical drainage is shown in Equation 9:

ut

zu u u ut t t t t t( )( )

=∆∆

+ − ++∆ 2 (9)0, 2 1, 2, 0, 0,p

where

u = excess pore water pressure, Δt = factor equal to coefficient of vertical consolidation (cv)

multiplied by change in time (Δt), and Δz = change in depth.

In Equation 9, 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 by 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 1-D consolidation that considers only radial drainage is as follows:

put

ru u

u urr

u ut t

t tt t

t t( )

( )=

∆∆

+ +−

∆

−

++∆ 2

2(10)0, 2

3, 4,4, 3,

0, 0,

where

Δt = factor equal to coefficient of horizontal consolidation (ch) multiplied by change in time (Δt),

r = radius of drainage influence for PV drain, and Δr = change in radius.

Farnsworth, Bartlett, and Lawton 9

In Equation 10, Node 0 represents the selected node, Node 3 repre-sents 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 occurs. However, for relatively thick clay layers (3 to 5 m or greater) and with typical PV drain spacing (1.5-m triangular spacing), drain-age will occur predominantly in the radial direction; thus, vertical drainage was neglected for the analyses and results are 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). In addition, 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 by using methods that account for the geometry of the embankment or applied loading and layering of the foundation soils. For this research, a two-dimensional vertical stress distribution was developed by means of the Boussinesq solution for the calculation of vertical stress beneath the center of an embankment (21). It was further assumed, on the basis of the distribution of vertical stress, that the initial excess pore water pressure was equal to the change in vertical stress.

The use of the FDM to calculate the time rate of settlement of foun-dation 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 either 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 condi-tion at the completion of the ramp-loading sequence. In addition, if staged embankment construction is used, the staged loading can be modeled as a series of instantaneous loads that are placed at certain time intervals. 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. Backcalculated 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 distur-bance effects resulting from PV drain installation. If backcalculated ch(e) values are not available, then an assumption must be made about 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 tech-nique is the Rowe cell, which can be used to perform a 1-D labora-tory 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 dissipa-tion of the excess pore water pressure, the change in vertical effective stress, and the subsequent vertical strain and settlement as a function of time attributable to the placement of the embankment. The effec-

tive 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 follows:

e ccvt t

v t

∆ = ′σ′σ

( )

+∆log (11)

where

cc = compression index, σ′vt + Δt = effective vertical stress at some change in time, and σ′v(t) = effective vertical stress at initial given time.

For recompression, the same equation can be used, except that cr is substituted for cc. The vertical strain for each sublayer (εvi) is calculated from Equation 12:

eevi

o( )ε =

∆+1

(12)

where eo is the initial void ratio for recompression or the void ratio at the preconsolidation stress for virgin compression. The vertical settlement for each sublayer (Svi) is shown in Equation 13:

S Hvi vi i= ε (13)p

where Hi is the thickness of individual sublayers. 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 dur-ing construction to interpret field performance data by adjusting or cal-ibrating 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 backcalculated by using the settlement data shown in Figure 3b. 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 by using horizontal drainage (5).

The backcalculated ch(e) values were 26, 13, 5, and 9 mm2/min for the ULBC, IB, LLBC, and PA layers, respectively. These values were lower than those obtained by using the Asaoka backcalculation method, varying between approximately 30% and 50% smaller. In comparison, values of ch(e) for typical Lake Bonneville clay depos-its obtained in the laboratory by using a Rowe cell with horizontal drainage capabilities vary between about 4 and 90 mm2/min (5). The laboratory values provided a much wider range because samples from many depths were tested, while the backcalculated effective values represented an average across each layer. Furthermore, the FDM results showed 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 showed that the bottom three layers did not have nearly the dissipation of excess pore pressure as that originally obtained by using the Asaoka back projection method, especially the LLBC layer. The FDM results were approximately 7% to 25% smaller. These results further demonstrated

10 Transportation Research Record 2363

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, the authors concluded that the FDM, when properly calibrated, can be used to estimate reasonably the total settlement curve for a multilayered system consolidating at different rates. However, the use of magnet extensometer data provided a better understanding of the individual layers contribut-ing to the composite settlement profile and demonstrated 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 governs the critical path of that construc-tion. An accurate projection of the end of primary settlement is often much more critical than an accurate estimate of the magnitude of the total settlement. To provide accurate estimates of time rates of settle-ment, it is of foremost importance that an appropriate geotechnical investigation and subsurface characterization be performed and that subsequent design and construction techniques appropriately used.

The Asaoka projection method can be a valuable tool for estimating EOP consolidation settlement, but its accuracy is limited by its simpli-fying 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 in which 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 possible esti-mation of the level of consolidation for each subsurface layer. Such data can be used with backcalculation methods to provide, for each subsurface layer, the effective coefficient of consolidation, which considers disturbance effects resulting from PV drain installation.

The FDM is an underutilized numerical tool with the ability to pro-vide accurate estimates of the time rate of settlement of foundation soils with radial drainage. This accuracy results from the FDM’s dis-sipation of excess pore pressures and its relation to vertical strain being more correctly taken into account in the calculations. However, imple-mentation of the FDM requires a comprehensive characterization of the foundation soils to provide reliable projections. This research dem-onstrated that the FDM, coupled with magnet extensometer measure-ments, provides an accurate fit of the observed settlement behavior. This research suggests that the FDM be more universally applied.

The methods described in this paper should be used to provide more accurate estimates of time rates of settlement for layered clay systems with radial drainage, with the best method being the FDM coupled with magnet extensometer data. Maintaining a harmoni-ous balance between the geotechnical evaluations and the use of observational data is an important part of geotechnical engineer-ing. When used together appropriately, they provide the ability to achieve accurate and reliable estimates of the behavior of time rates of settlement for soft multilayered foundation soils.

FIGURE 4 Settlement profiles generated by finite difference technique (solid lines) with results from actual individual layers from MR s29-6-1 magnet extensometer as well as total settlement curve and numerically modeled results for entire profile.

Farnsworth, Bartlett, and Lawton 11

AcknoWledgMent

The authors acknowledge and thank the Research Division of the Utah Department of Transportation for its financial contributions to and technical support of this research project.

references

1. Farnsworth, C. B., S. F. Bartlett, D. Negussey, and A. W. Stuedlein. Rapid Construction and Settlement Behavior of Embankment Systems on Soft Foundation Soils. Journal of Geotechnical and Geoenvironmental Engineering, Vol. 134, No. 3, 2008, pp. 289–301.

2. Saye, S. R., C. C. Ladd, P. C. Gerhart, J. Pilz, and J. C. Volk. Embank-ment Construction in an Urban Environment: The Interstate 15 Experi-ence. Proc., Foundations and Ground Improvement, ASCE Specialty Conference, Blacksburg, Va., June 9–13, 2001, pp. 842–857.

3. Asaoka, A. Observational Procedure of Settlement Prediction. Soils and Foundations, Vol. 18, No. 4, 2008, pp. 87–101.

4. Bartlett, S., G. Monley, A. Soderborg, and A. Palmer. Instrumentation and Construction Performance Monitoring for I-15 Reconstruction Project in Salt Lake City, Utah. In Transportation Research Record: Journal of the Transportation Research Board, No. 1772, TRB, National Research Council, Washington, D.C., 2001, pp. 40–47.

5. Farnsworth, C. B., and S. F. Bartlett. Evaluation of Methods for Determin-ing Horizontal Drainage Properties of Soft Clayey Soils. Report UT-08.11. Utah Department of Transportation, Salt Lake City, 2008.

6. Peck, R. B. Advantages and Limitations of the Observational Method in Applied Soil Mechanics. Geotechnique, Vol. 19, No. 2, 1969, pp. 169–187.

7. Mikasa, M. The Consolidation of Soft Clay—A New Consolidation Theory and Its Application. Civil Engineering in Japan, Vol. 4, 1965, pp. 21–26.

8. Terzaghi, K. Erdbaumechanik auf Bodenphysikalischer Grundlage (in German). Franz Deuticke, Leipzig, Germany, 1925.

9. Duncan, J. M. Limitations of Conventional Analysis of Consolidation Settlement. Journal of Geotechnical Engineering, Vol. 119, No. 9, 1993, pp. 1333–1359.

10. Lawton, E. C. Practical Foundation Engineering Handbook. McGraw-Hill, New York, 2000.

11. Sivaram, B., and P. Swamee. A Computational Method for Consolida-tion Coefficient. Soils and Foundations Journal, Vol. 17, No. 2, 1977, pp. 48–52.

12. Barron, R. A. Consolidation of Fine-Grained Soils by Drain Wells. ASCE Transactions, Vol. 113, 1948, pp. 718–742.

13. Bartlett, S. F., and C. B. Farnsworth. Monitoring and Modeling of Inno-vative Foundation Treatment and Embankment Construction Used on the I-15 Reconstruction Project, Project Management Plan and Instrument Installation Report. UT-04.19. Utah Department of Transportation, Salt Lake City, 2004.

14. Bergado, D. T. Asian Experiences on the Use of Prefabricated Vertical Drains in Soft Bangkok Clay. Proc., 1st International Symposium on Lowland Technology, Saga, Japan, Nov. 1998, pp. 49–60.

15. Perrone, V. J. One Dimensional Computer Analysis of Simultaneous Consolidation and Creep of Clay. PhD dissertation. Virginia Polytechnic Institute and State University, Blacksburg, Va., 1998.

16. Harr, M. E. Foundations of Theoretical Soil Mechanics. McGraw-Hill, New York, 1966.

17. Das, B. M. Advanced Soil Mechanics. Hemisphere Publishing, New York, 1983.

18. Terzaghi, K., and O. K. Frolich. Theory of Settlement of Clay Layers. Leipzig, Germany, 1936.

19. Schmertmann, J. H. The Undisturbed Consolidation Behavior of Clay. ASCE Transactions, Vol. 120, 1955, pp. 1201–1233.

20. Scott, R. F. Principles of Soil Mechanics. Addison-Wesley, Reading, Mass, 1963.

21. Boussinesq, J. Application des Potentials a L’etude de L’equilibre et du Mouvement des Solides Elastiques (in French). Gauthier-Villars, Paris, 1863.

The Transportation Earthworks Committee peer-reviewed this paper.