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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 University230 SNLB
Provo, Utah 84602
Ph: (801) 422-6494
F: (801) 422-0653email: [email protected]
Steven F. Bartlett, Ph.D., P.E.
Associate ProfessorCivil & Environmental Engineering
University of Utah
110 Central Campus Drive, Suite 2000Salt Lake City, Utah 84112
Ph: (801) 587-7726
F: (801) 585-5477email: [email protected]
Evert C. Lawton, Ph.D., P.E.
Professor
Civil & Environmental EngineeringUniversity of Utah
110 Central Campus Drive, Suite 2000Salt Lake City, Utah 84112
Ph: (801) 585-3947
F: (801) 585-5477
email: [email protected]
Paper Length:
6,197 Words
5 Figures0 Tables
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ABSTRACTThis 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 verticaldrains, 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 additionalinformation 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 moreaccurately predict the time-rate of consolidation behavior of multilayered foundation soils.
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INTRODUCTIONConstruction 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 aretypically 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 consolidationsettlement 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 soilsfound 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 projectionswere 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 wasconcluded 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 differingconsolidation 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, thethickness 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 includeshaving 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
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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 methodsemployed 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 anevaluation of several potential EOP projection methods, their associated assumptions, andimplementation issues, progressing from simplified to more elaborate techniques.
ASAOKA PROJECTION METHODThe 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-empiricalmethods, 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 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 pressuredissipation 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 formany 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 betweenthe 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 thesurface 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 againstthe 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 endof primary settlement. The basic form of the resulting first order difference equation is:
S i = β 0 + β 1 S i-1 (1)
where S is the measured settlement at time i, and β 0 and β 1 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. Thetheoretical settlement data are plotted at equal time intervals starting from the initial settlement
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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:
S i = β 0 + ∑ β L S i- L (2)
where the subscript, L, represents the number of different layers. However, there is not any
guidance for determining the partial slopes, β L, 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 equationwas 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
Settlement at time N-1 (m)
S e t t l e m e n t a t t i m e N ( m
FIGURE 1 Numerically modeled settlement data plotted with the
Asaoka Method demonstrating nonlinearity, especially in the early
portion of the dataset.
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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 theirAsaoka projections.
The 1-D primary consolidation settlement at any time (t ) after a load has been applied is
estimated by the following equation: S c(t ) = U v S c(t =∞) (3)where S c is the settlement at time t , and U v is the degree of consolidation with vertical drainage.
It should be noted that this equation is fundamentally incorrect. Values of U v actually represent
the average degree of dissipation of excess pore water pressure within the compressible layer andnot 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 areconstant 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 ofexcess 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 verticaldrainage (e.g., where PV drains have not been used):
U v = 100 * [(4T v / π)0.5
] / [1 + (4T v / π)2.8
]0.179
(4)
where T v is the dimensionless time factor for two-way vertical drainage and is a function of thecoefficient 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 T v 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 PVdrains have been used), the equation given by Barron (12) may be used:
U r = 1 – e-8T r / F (n) (5)
where U r is the degree of consolidation with radial drainage, T r 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 = d e / d w (7)
and d e is the equivalent diameter of influence and d w is the diameter of the drain. T r is similar to
the dimensionless vertical drainage time factor, T v, but T r is a function of the coefficient of
horizontal consolidation, ch, the length of the horizontal drainage path, d e, (which is equal to twotimes the radius of an equivalent soil cylinder from which radial drainage occurs), and the time
of consolidation, t . Values of T r are related to these parameters by ch/d e2. As before, to match
the field settlement data with the analytical curve(s), trial values of EOP settlement and T r 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 thatAsaoka projections relying solely on surface settlement data can provide reasonable results for
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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 keylocations to measure the compression of individual clay layers within the soil profile. Thismonitoring 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 anexample 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 Bonnevilleclay (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 settlementcurves 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 thesemagnets 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 ofthese 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 beapplied 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 pavementconstruction. 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 settlementmeasurements for each of the four major layers contributing to the foundation settlement.
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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 day279 (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 soilsachieving 95.4% consolidation at day 279. However, this percentage is misleading because itfails 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
67Surface
FIGURE 2 Predominant subsurface layers shown with corresponding magnets
(horizontal lines labeled base through surface) for magnet extensometer MR s29-6-1.
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-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
10 100 1000
Days
S e t t l e m e n 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
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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 surfacemonitoring 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 appropriatemethods with consistent assumptions had been used. This research indicates that magnetextensometers 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 wherethe 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 magnetextensometer 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) = ( F n + F s) / [C 1 + (C 2 / qw)] (8)
where F n is a drain spacing correction, F s accounts for the installation smear effects, C 2 accounts
for drain well resistance, and qw is the discharge capacity of the drain. The C 1 term requires theslope of the projection line (i.e., β 1) 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 horizontalcoefficient of consolidation for the LLBC layer is only one-fourth that of the ULBC layer.
PROJECTIONS USING THE FINITE DIFFERENCE METHODTerzaghi’s 1-D consolidation equation is a 2
nd 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 differencemethod (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
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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 thevertical 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 tooverconsolidation 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, therecompression 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 thatthe 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 assumptionis 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 / (Δ z )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 verticalconsolidation, cv, multiplied by the change in time, Δt , and Δ z 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 ofvertical 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 theradial 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 horizontalconsolidation, 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 tocalculate 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
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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. Theincrease in effective vertical stress for each sublayer can be calculated using methods thataccount 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 wasfurther 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 iscommonly 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 loadcondition 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 actualloading 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 beselected 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 degreeof 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 measureof 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 timedue 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)
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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:
S vi = vi * i (13)The summation of settlement for all individual sublayers produces the total settlement at eachtime increment.
The FDM is particularly useful as an observational technique during construction tointerpret field performance data by adjusting or calibrating the model to match the subsurfacesettlement 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 backcalculation 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)
S e t t l e m e n t ( 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.
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Farnsworth, C.B., Bartlett, S.F., and Lawton, E.C. 14
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, and82.6% 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 areapproximately 7% - 25% smaller. This further demonstrates the importance of correctlymodeling 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, weconclude 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 anurban 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 foremostimportance 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 consolidationsettlement, 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 differentrates. 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 comprehensivecharacterization 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
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Farnsworth, C.B., Bartlett, S.F., and Lawton, E.C. 15
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 UtahDepartment of Transportation for their financial contributions and technical support to thisresearch project.
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