Applying the hyperbolic method and Ca/Cc concept for settlement prediction of complex organic-rich soil formations

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Applying the hyperbolic method and C

_a

/C

_c

concept for settlement prediction of complex organic-rich soil formations

Mosleh A. Al-Shamrani*

Civil Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia Received 11 November 2003; accepted 14 July 2004

Available online 11 September 2004

Abstract

The hyperbolic method has proved applicable for settlement prediction in complex soil formations. However, when an appreciable portion of the settlement is due to secondary compression, settlement prediction based on the slope of the initial linear portion of the hyperbolic plot requires a correction factor that would be different for different amounts of secondary compression. Although inverse of the slope of the final linear portion of the hyperbolic curve can provide reasonable estimates of total settlements, including secondary compression, establishing the curve slope requires data beyond 90% consolidation, and this renders the hyperbolic method less useful for practical applications. This paper examined the feasibility of predicting total settlements of heterogeneous soils, which also exhibit prominent secondary compression behaviour, by utilizing the hyperbolic method for estimating ultimate primary consolidation settlement and the time for its completion, and theCa/Cc concept of compressibility for predicting secondary compression. The applicability of the proposed procedure was examined using the results of laboratory tests and an instrumented test embankment constructed on a typical sabkha formation. The combined use of the hyperbolic method and theCa/C_cconcept has been found to provide reasonable estimates of total settlements where the hyperbolic curve was initially concave downward. Contrarily, for hyperbolic plots that do not show convex curvature toward the origin, the hyperbolic method predicted a significantly erroneous time for the completion of primary consolidation, hence leading to unreasonable estimates of secondary compression. Thus, where settlement versus time fully obeys the rectangular hyperbolic relationship, total settlement, inclusive of secondary compression, can simply be estimated as the reciprocal of the slope of the fitted straight line.

Keywords:Hyperbolic method;Ca/Ccconcept; Settlement; Heterogeneous soils; Primary consolidation; Secondary compression; Sabkha soils;

Test embankment

1. Introduction

Commonly ultimate primary consolidation settlement is obtained using Terzaghi’s conventional one- dimensional model. However, particularly for hetero-

doi:10.1016/j.enggeo.2004.07.004

* Fax: +966 1 4677008.

E-mail address:[email protected].

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geneous soils the discrepancy between predicted values and actual in situ settlements is evident. This discrepancy primarily arises from the fact that the compressibility parameters are obtained in the laboratory from samples of relatively small size that are more homogeneous compared to heterogeneous field sediments in which various soil types may be interlayered at random and may occur without exhibiting any real stratification.

The two-constant hyperbolic form of mathematical relation has been found to offer a practical tool for describing a multitude of physical phenomena in geotechnical engineering including: nonlinear stress–

strain behaviour (Konder, 1963; Duncan and Chang, 1970; Sridharan and Rao, 1972; Boscardin et al., 1990; Stark et al., 1994), and heave of expansive soils (Dakshanamurthy, 1978; Sivapullaiah et al., 1996).

Tan (1971),Kodandaramaswamy and Narasimah Rao (1980), and Narasimah Rao and Somayajulu (1981) has applied the hyperbolic fitting method for estimation of settlement, where the relationship between settlement d and time t is assumed to follow a hyperbolic curve given by the equation

d¼ t

BþAt: ð1Þ

Rearranging terms, Eq. (1) can be rewritten as t

d ¼BþAt: ð2Þ

Eq. (2) is the equation of a straight line, whereA and B are the slope and the intercept of the straight line, respectively. If settlementd versus timet obeys the rectangular hyperbolic relationship, the transformed hyperbolic plot of t/d against t should be a straight line and the ultimate total settlement is obtained from the asymptotic line to the hyperbola.

Taking the limits of Eq. (2) as t approaches infinity, the total settlement is given by 1/A, which is the reciprocal of the slope of the straight line.

Sridharan and Sreepada Rao (1981)andSridharan et al. (1987)applied Eq. (2) to obtain the coefficient of consolidation. The relationship shown in Fig. 1a between the average degree of consolidation, Uav, and the time factor,Tv, as obtained from Terzaghi’s theory for constant/linear pore water pressure distribution and double drainage, was plotted in the form of Tv/UavversusTvas shown inFig. 1b. The hyperbolic

curve is initially concave downward, but in the range 0.286VTvV0.848, which corresponds to 60%b Uavb90%, the relationship of Tv/Uav against Tv is approximately linear (Sridharan et al., 1987). Tan (1993) suggested that, with no appreciable loss of accuracy, the linear segment of the theoretical hyperbolic curve can be taken to start at Uav= 50% and described by

T_v Uav

¼bþaT_v; ð3Þ

whereb anda are, respectively, the intercept and the slope of the linear portion of the theoretical hyperbolic plot shown inFig. 1b. The average slope of the initial linear segment,ai, is equal to 0.824F0.04 (Sridharan and Sreepada Rao, 1981). Beyond Uav of 90%, the hyperbolic plot diverts slightly upward over a narrow range, leading to another linear relationship at a slope of 1, except for soils containing organic materials for which the line is diverted inwards due to the effects of secondary compression (Tan, 1994).

Results from laboratory tests and reported field data have indicated that t/d versus t plots do not generally fit to a straight line (Sridharan and Sreepada Rao, 1981; Sridharan et al., 1987; Tan, 1993, 1994).

Consequently, ultimate settlement cannot always be determined from the hyperbolic-fitting method conventional inverse slope approach. Therefore, for the inverse slope method to work as a means of predicting ultimate settlement, and considering the characteristics of Terzaghi’s Tv/Uav versus Tv theoretical hyperbolic curve shown in Fig. 1b, Tan (1994) proposed that reasonable prediction of ultimate settlement can be obtained as the inverse of the slope of the initial linear portion of the hyperbolic plot fitted to actual settlement data,Ai, multiplied by the slope of the initial linear portion of the theoretical hyperbolic curve,ai.

However, the value of ai of 0.824 is based on Terzaghi’s theory, which ignores compressibility with time. Because settlement in some fine-grained soils, especially those containing an appreciable amount of organic matter, is due not only to primary consolidation but also to secondary compression, the ai/Ai

prediction would certainly underestimate the actual settlement. Therefore, especially in case of a meas- urable secondary compression, reasonable estimation of settlement based onAi would require a correction

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factor that would be unknown for different amounts of secondary compression.

Tan (1994) pointed out that because the slope of the final linear segment beyond 90% Uav, Af, is influenced by secondary compression, the inverse of Af would be an estimate of ultimate settlement including the effects of secondary compression.

However, even if the inverse of Af is considered to provide good estimates of total settlements, the use of Af has a number of restrictive limitations. Firstly, evaluation of Af requires data beyond 90% consolidation, and this renders the hyperbolic method less beneficial for practical field application (Tan, 1994).

Actually, the salient feature of the hyperbolic method is in estimating settlement only from settlement records sufficient to establish the linearity of the initial portion of the hyperbolic plot. Secondly, the hyperbolic method underestimates settlement when it is used to extrapolate future settlement for cases where the secondary compression index increases with time (Choi and Cepeda-Diaz, 1981). Thirdly, irrespective of the shape of the secondary portion of settlement versus logarithm of time curve, the total settlement predicted as 1/Afhas been found to be dependent on the time span over which the hyperbolic relationship

is fitted to the settlement data (Choi and Cepeda-Diaz, 1981). Hence, ift/dversustdoes not strictly follow a hyperbola, the predicted settlements would be an arbitrary quantity that is function of the number of data points utilized in the curve fitting.

It is evident therefore that for the case of soils which exhibit prominent secondary compression behaviour it would be both more rational and practical to apply the hyperbolic method only in estimation of ultimate primary consolidation settlement, and to utilize another approach for predicting secondary compression component. Based on the observation that the magnitude and behaviour of coefficient of secondary compression, Ca, with time is directly related to the magnitude and behaviour of compression index,Cc, with consolidation pressure,Mesri and Godlewski (1977) proposed the Ca/Cc concept of compressibility for predicting secondary compression.

According to this concept, knowing the void ratio versus logarithm of effective stress relationship at the end of primary consolidation together with the ratio Ca/Cc, which is considered constant for a given soil, it is possible to estimate secondary compression at all values of effective stress and throughout the secondary compression stage.

Fig. 1. Terzaghi’s theoretical one-dimensional consolidation curve: (a) average degree of consolidation-time factor curve; (b) transformed hyperbolic plot.

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It is the purpose of this paper to examine the applicability of combining the hyperbolic method and the Ca/Cc concept in the prediction of total settlements of heterogeneous soils with appreciable secondary compression. The hyperbolic method is utilized for the prediction of ultimate primary consolidation settlement and the time for its completion, while secondary compression is determined based on the Ca/Ccconcept of compressibility.A series of conventional and long-duration incremental loading one- dimensional consolidation tests was performed on undisturbed samples recovered from a typical compressible sabkha profile. The extreme heterogeneity of this type of soils gives rise to uncertainties in the analysis of their behaviour. Besides, estimation of sabkha settlements is further complicated by the occurrence of potential secondary compression associated with the relatively high organic content of the compressible layer.

The hyperbolic method was applied to the experimental data, and the predicted settlements were compared with settlement values obtained from the oedometer tests. Furthermore, the relationship

between the coefficient of secondary compression, Ca, and the compression index, Cc, was established, from which the values forCawere obtained and then used for predicting secondary compression. The applicability of the proposed approach was further ascertained using the results of an instrumented test embankment constructed on a typical sabkha soil profile.

2. Application to experimental data

2.1. Tested materials

The samples for this study were collected from different depths and locations in a compressible layer of a typical sabkha formation from the Jazan region on the Southwestern coast of Saudi Arabia. Sabkha formations are salt bearing arid climate deposits developed by the erosion of coastal deposits both by wind and extreme storm tides followed by a period of sedimentation (Akili, 1981). They are encountered in arid regions throughout the world, including Australia

Fig. 2. Profile characteristics in sabkha formations.

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(Akpokodje, 1985), and the United States (Kinsman, 1969; Lund et al., 1990), but are predominantly found in vast areas of the coasts of Middle Eastern and North African countries (Akili, 1981; Khan and Hasnain, 1981; Stipho, 1985; Dhowian et al., 1987).

Borehole data from various investigations (Dho- wian et al., 1987; Erol, 1989; Al-Shamrani and Dhowian, 1997) have revealed that sabkha sediments of Jazan region possess highly variable characteristics both laterally and vertically. Various soil types, primarily composed of clays, silts, fine sands, and organic matter are interlayered at random. Variations have been noted in layering, grain size, cementation, and organic content. Lateral variation appears to depend on proximity to the shoreline. The subsoil profiles in the coastal zones consist of loose fine sand, whereas the subsoil profiles of inland zones are characterised by very soft clays and silts with appreciable organic material. Vertical variation arises from the development sequence of the sabkha sediments, the prevailing depositional environment, and subsequent diagenesis (Akili, 1981).

Fig. 2 shows the idealised soil profile of Jazan sabkha. Three zones characterise the soil profile: (1) sabkha crust; (2) compressible sabkha complex; (3)

sabkha base. The sabkha crust is relatively thin, dry, silty sand with an average thickness of about 1.5 m, existing above the water table, which is 1.0–2.0 m of the ground surface. The sabkha base is a firm stratum consisting mainly of medium dense to dense sand of relatively low compressibility. The middle zone, referred to as the compressible sabkha complex, is a soft/loose material composed of soils varying from nonplastic fine sands to highly plastic organic clays. A grouping of these materials into three sublayers within the sabkha complex is attempted in Fig. 2. However, layering usually is not distinctively conspicuous since a variety of materials, interlayered at random, occur without clear stratification. The interlayering occurs at both the macroscale where the sublayer thickness are in the order of meters, and at the microscale where seams or lenses are a few millimeters thick penetrate the dominant stratum to form an extremely heterogeneous soil profile with respect to composition and stratification.

The grain size and plasticity characteristics of the sabkha complex, shown, respectively, in Fig. 3a and b, reflect the presence of a variety of soil types in the stratum. The organic matter contents of typical silty- clay and sandy soil profiles are shown in Fig. 4a. In

Fig. 3. Typical parameters for sabkha soils: (a) grain size distribution; (b) consistency limits.

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general, the organic matter in silty-clay profiles varies from 3% to 8% within the upper 10 m of the sabkha sediments, with occasional higher and lower values, as shown in the histogram given inFig. 4b.

2.2. Oedometer tests

The testing program included 36 incremental- loading consolidation tests on 70-mm diameter, 19- mm-thick undisturbed specimens. The specimens were loaded at a load increment ratio (LIR) of unity.

In the first set of 24 tests, each sample was left under load for a time sufficient for completion of primary consolidation as estimated by Casagrande’s logarithm of time fitting method, and secondary compression for at least one logarithmic cycle of time. If the inflection point between primary and secondary compression

stages, which is taken to indicate the end of primary consolidation, was not well defined, the test was terminated after a 1 week.

The second set of tests comprised six incremental loading long-term one-dimensional consolidation tests all performed under an initial load increment of 28 kPa, with the final load increment given by the expression 282ⁿ¹kPa, wheren is the test number.

The final load increment for the 4th test was, for instance, equal to 224 kPa. A load increment ratio (LIR) of unity was used and each load increment was sustained on the sample for a period of 1 week, except the last increment which was maintained for about 10 to 14 weeks. A third set of six tests was performed following the same testing procedure of the second set, except that the tests were carried out on samples recovered after field preloading of a sabkha profile.

Fig. 4. (a) Variation of organic matter with depth in sand and clayey sabkhas; (b) organic content of sabkha compressible layer.

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Typical relationships obtained between settlement and logarithm of time are shown inFig 5a. It is obvious that secondary compression represents a significant part of sabkha deformation. This is expected in view of the appreciable amount of organic content in the tested sabkha soil as shown inFig. 4. The results of most tests followed a standarddSTshape, and the inflection point between the primary and secondary stages was well defined. The settlement versus logarithm of time relationship of a few tests did not follow the classical S-shaped curve, and instead of showing a concave- down shape in the initial part exhibited a concave-up shape. Because of the high initial permeability, the primary consolidation was very rapid and the 24-h

increment duration encompassed two to three loga- rithms of time cycles of secondary compression. In most of the consolidation tests conducted, the rate of secondary compression was either constant or gradu- ally decreased with logarithm of time. Only in two tests, the result of one of which is shown inFig. 5a, did the rate of secondary compression increase with logarithm of time.

2.3. Applicability of hyperbolic method to experimental data

The results of the 36 tests conducted were plotted in transformed hyperbolic plots oft/dversust, examples

Fig. 5. (a) Typical consolidation curves for sabkha soils; (b) transformed hyperbolic representation of settlement versus time plots.

Fig. 6. Measured versus 0.824/Aihyperbolic method predictions: (a) total settlement; (b) ultimate primary consolidation settlement.

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of which are shown inFig. 5b for the experimental data ofFig. 5a. In order for the initial portion of the curves to be seen, only data points sufficient to establish the initial linear portion of the hyperbolic plots are shown.

A variety of shapes for the hyperbolic plot was observed with the hyperbolic plots differing primarily in the extent and degree of concavity of the initial portion of the curve. The variation among the results is expected, and essentially reflects the natural variation of the sabkha soils over the area of the site. However, irrespective of the various observed shapes of the settlement versus log time curves, the linearity was discerned for a substantial time interval for all the tested samples and for all the stress increments, suggesting the applicability of the hyperbolic method as a representation of the primary consolidation behaviour of the tested sabkha soils.

The slope of the initial linear portion of the hyperbolic plot, Ai, was graphically determined for all the hyperbolic curves of the tests conducted.Fig.

6a compares the measured total settlements,dt, from the laboratory tests, with the values estimated from the inverse of the measured slope of the initial linear segment, Ai, multiplied by the theoretical factor, a_i= 0.824 (that isd_t= 0.824/Ai). It is noted that a high proportion of the results lies in a fairly close band despite the distinct dissimilarities in the composition of the tested soils. However, it is apparent that the hyperbolic method substantially underestimated most of the measured total settlements. This is a clear indication of the deviation of the actual settlement–

time response of the tested soils from Terzaghi’s theoretical curve. The discrepancy between the measured and estimated total settlements is essentially due to the contribution of secondary compression.

Indeed, if longer periods of secondary compression were allowed in the tests, the discrepancy would certainly have been higher than that shown inFig. 6a.

The time for the end of primary consolidation,tp, was identified using the Casagrande logarithm of time fitting procedure. Accordingly, the corresponding ultimate primary consolidation settlement, dp, was determined from the settlement versus logarithm of time curve associated with each load increment and plotted against 0.824/AiinFig. 6b. It is noted that the data are closer to the equity line as compared to the case of total settlement shown inFig. 6a. This suggests that the inverse of the slope of the initial linear portion of the

hyperbolic plot fitted to actual settlement data, Ai, multiplied by the slope of the initial linear portion of the theoretical hyperbolic plot,ai, provides a good estimate of ultimate primary consolidation settlement. Further- more, the discrepancy between measured total settlements and values predicted as 0.824/Aiwould decrease as the contribution of primary settlement to total settlement increases. Although secondary compression slightly affects the value of the slope of the initial linear segment of the hyperbolic curve, Ai, and thus the predicted ultimate primary consolidation settlement, its main effect is on the slope of the final linear segment of the hyperbolic curve, Af, which decreases with increasing rate of secondary compression. This is demonstrated in Fig. 7, where it is seen that the ratio Af/Aiis inversely proportional to the ratio of secondary settlement,ds, to total settlement,dt.

The slope of the final linear segment of the hyperbolic plot, Af, was graphically determined for all the hyperbolic curves of the tests conducted, and settlement estimates obtained as the inverse of Af are compared with the observed settlement values inFig.

8. The agreement between the measured and estimated total settlements is excellent. The data are clustered along the equity line, unaffected by soil type, loading duration, and intensity of loading, all of which are known to affect the compressibility parameters. How- ever, despite the fact that the inverse of Afis seen to provide a good estimate of the total settlement, the use

Fig. 7. Relationship between the ratio of secondary compression to total settlement and the proportion of final and initial slopes of the hyperbolic curve.

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of Af in settlement prediction has a number of disadvantages that have already been stated at the outset of this paper.

2.4. Secondary compression based on Ca/Ccconcept The secondary compression index,Ca, is by far the most useful and commonly used parameter for

describing secondary compression (Mesri and God- lewski, 1977). The importance of this parameter stems from the fact that for some soils the parameter indicates a nearly constant value for a given load increment (Lo, 1961). Hence, the secondary compression, ds, can simply be determined from (Buisman, 1936):

ds¼ CaHp

1þe_plog tf

t_p

ð4Þ whereHpis the thickness of the soil layer at timetp,tp

is the time corresponding to the completion of primary consolidation, tf is the time at which the secondary compression settlement is to be computed, andepis the void ratio at the assumed end of primary consolidation and commencement of secondary compression.

According to theCa/Ccconcept of compressibility, the end of primary consolidation void ratio versus logarithm of effective stress relationship and the ratio Ca/Cccompletely defines the secondary compression behaviour (Mesri and Godlewski, 1977). An end-of- primary (EOP) consolidation curve is obtained by plotting effective stress, rvV, and the corresponding void ratio,ep, for each load increment on aneversus logrvV plot.Fig. 9a shows such a plot for one of the oedometer tests conducted in the present study. Shown

Fig. 8. Measured total settlements compared to 1/Af hyperbolic method predictions.

Fig. 9. (a) Typical end of primary void ratio versus logarithm of consolidation stress for sabkha soils; (b) compression index versus coefficient of secondary compression for sabkha soils.

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also areelogrvVcurves for void ratios corresponding to 10tpand 100tp. It is noted that at any given stress the slopes of the tangents on theelogrvV curve, which represent the values for compression index,Cc, do not change with elapsed time of loading. For the same test, the rate of secondary compression was also found to be constant with logarithm of time. This trend which has been observed almost for all the specimens tested, substantiates the suggestion made by Mesri and Godlewski (1977)that changes inCawith time reflect changes inCcwith elapsed time of loading.

For any given consolidation stress, the value ofCa

was obtained from the linear portion of the elog t curve immediately after the completion of primary consolidation and transition to secondary compression. The corresponding value ofCcfor the same load increment was obtained from the slope (i.e., tangent) of the end-of-primary (EOP)elogrvVcurve.Further pairs of correspondingCaandCcwere determined at a certain void ratio and time of elapsed loading. A plot of the corresponding values of Ca and Cc for the tested sabkha soils is shown inFig. 9b. Initially, the regression line does not pass through the origin but instead has a minimal positive intercept on the vertical axis.Mesri et al. (1990)presented experimental data for a variety of sands where the linear regression lines fitted to Ca versus Cc plots have nonzero positive intercepts. On the other hand,Fox et al. (1992)found that for peat the fitting lines do not pass through the origin but have a negative intercept on the Ca axis.

This might imply that, for highly organic plastic clays, with significant secondary compression, the intercepts on the Ca axis are negative, while for granular materials the intercepts are positive. However, in general, for the majority of inorganic soft clays the regression lines pass through the origin.

Since the value of the intercept of the regression line of the data of Fig. 9b is infinitesimally small (310⁴), it was justifiable to neglect it and take the relationship betweenCaandCcto pass through origin.

From the results ofFig. 9b, the slope of the best-fit line through the origin gives a value of 0.037 for theCa/Cc

ratio. This value is within the range from 0.03 to 0.05 reported for soft clays (Mesri and Castro, 1987) and higher than the range from 0.015 to 0.03 found for granular soils (Mesri et al., 1990). The tested sabkha soils consist primarily of silt-size grains, and the clay and sand percentages vary from 15% to 30% and 10%

to 25%, respectively. Evidently, the composition of the tested sabkha soils is reflected in the value ofCa/Cc

ratio being above the range reported for granular soils and within the limits suggested for soft clays.

2.5. End of primary consolidation using the hyperbolic method

According to Eq. (4), the magnitude and hence significance of secondary compression depends, besidesCa, ontpandtf. The timetfsimply represents the design life for which the secondary settlement is computed. The time for the end of primary consolidation, tp, is best identified by the measurement of pore water pressure. However, in the absence of pore water pressure measurements,tpis routinely obtained from curve fitting methods, of which Casagrande’s logarithm of time fitting method is considered the most widely used procedure. According to this method, settlement is plotted versus logarithm of time and the intersection of the tangent at the inflection point and the tangent to the final linear portion of the curve designates the completion of primary consolidation and beginning of secondary compression.

However, the establishment of the intersection point requires the settlement records to exceed the primary consolidation stage and advance into the secondary compression stage for an elapsed time sufficient to establish the tangent of the linear portion representing secondary compression. This actually abrogates the merit of the hyperbolic method of estimating primary consolidation settlement from settlement records only sufficient to establish the linearity of the initial portion of the hyperbolic plot. Furthermore, the log-time method is used, based on the assumption that a linear relation is observed between settlement and logarithm of time beyond the inflection point. However, field and laboratory results have indicated that for highly organic soils and some inorganic silts, the settlement versus logarithm of time curve does not have a linear segment during secondary compression at all, but instead exhibits continuous curvature (Lo, 1961; Mesri and Godlewski, 1977; Fox et al., 1992). In this case, tp

cannot be identified in the arbitrary manner of Casagrande’s procedure (Wahls, 1962; Mesri and Godlewski, 1977, Sridharan et al., 1987).

Experimental results provided by Sridharan et al.

(1987), for different types of soils and different

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loading conditions, indicated that even if the secondary portion of the settlement versus logarithm of time curve is nonlinear, when the curve is transformed into a time–settlement versus time hyperbolic plot, the initial linear portion of the hyperbolic curve can be identified and its slope,Ai, is given fromFig. 10as

Ai¼ t_p d_p B

t_p : ð5Þ

Rearranging the terms in Eq. (5) gives tp¼ Bd_p

1A_id_p: ð6Þ

However, as proposed byTan (1993)dpis equal to the inverse ofAimultiplied by the theoretical slopeai,

or dp¼ai

A_i: ð7Þ

Substituting Eq. (7) into Eq. (6)yields t_p ¼ a_iB

1a_i ð ÞAi

: ð8Þ

From Eq. (8), the magnitude of tp can be determined knowing only the slope and intercept of the initial straight line part, and hence the consolidation test need not be taken beyond the point at which the initial linear segment of the hyperbolic curve is established.

The values of tp obtained from Eq. (8) and from Casagrande’s method are compared in Fig. 11a. It is noted that the hyperbolic method gave lower tp

values. This agrees with the observation made by Sridharan et al. (1995) that the graphical methods that depend on the later stages of consolidation for curve fitting (like Casagrande’s method), where secondary compression effect dominates, give lower values of the coefficient of consolidation (i.e., higher values of tp).

2.6. Hyperbolic method and Ca/Cc concept for estimation of total settlements

With tp determined from Eq. (8) and Ca found from the ratioCa/Ccexpressed inFig. 9b, secondary settlements,ds, provided by Eq. (4)are compared with the measured values in Fig. 11b. Although Fig. 11a

Fig. 10. Determination oftpbased on hyperbolic relationship.

Fig. 11. (a)tpfrom Casagrande and hyperbolic fitting methods; (b) secondary compression based ontpfrom Casagrande and hyperbolic fitting methods.

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indicates rather significant discrepancy in the values of tp obtained from the hyperbolic and Casagrande fitting methods, it seen from Fig. 11b that the secondary compressions obtained based on tp from the two methods are quite comparable. For all the tests conducted, the total settlements,d_t, was computed as the sum of ultimate primary settlements,dp, given by Eq. (7)and secondary settlements,ds, provided by Eq.

(4). Fig. 12 shows that the agreement between the measured and computed total settlement values is remarkably good.

3. Application to in situ settlements

The applicability of the proposed approach of combined use of the hyperbolic method and theCa/ Cc concept for settlement prediction was further validated by considering the results of an instrumented test embankment constructed on a typical sabkha formation. The preload was a two-stage embankment. The first stage was a 1.0-m-high fill and covered an area of 40 by 40 m. A second stage 2.0 m high was built to attain a central 3.0-m fill height over an area of 15 by 15 m. Prior to placing the fill, four boreholes were drilled at the location of the instrumented test embankment. The borehole logs indicated that the compressible layer is about 10 m thick and is covered with a sabkha crust layer of 1.0-m thickness. Details of the test embankment, the

location of the field instrumentation, and characteristics of the sabkha complex at the specific location of the embankment are found elsewhere (Dhowian et al., 1987).

The loading history and the corresponding time–

settlement plot recorded under the centre of the embankment are illustrated in Fig. 13a and b, respectively. Because the initial part of the curve was influenced by breaks in the loading process, the zero time corresponding to an instantaneous loading was taken equal to half the construction period of 1 week. The excess pore pressure recorded by a piezometer installed under the centre of the embankment at a depth of 8.75 m below the ground surface is shown in Fig. 13c.

It can be observed from Fig. 13b that the total settlement was about 197 mm at the end of the observation period. The portions of observed settlement due to primary consolidation and secondary compression are 153 and 44 mm, respectively.

Separation of the observed total settlement into primary and secondary components was established based on the premise that the primary consolidation was concluded by the complete dissipation of mid- plane excess pore pressures, as measured by installed piezometers. According to the piezometer records given in Fig. 13c, the end-of-primary consolidation stage was reached approximately 18 days after construction of the embankment.

3.1. Ultimate primary consolidation settlement predictions

The hyperbolic plot of t/d versust corresponding to the settlement records ofFig. 13b is shown inFig.

13d. It is noted that after an upward concave initial portion the data are approximately linear, indicating that the field settlement versus time relationship follows a rectangular hyperbola. Because only one data point lies above the straight line, the conclusion that the initial part is concave upward is speculative.

Besides, the initial portion of the curve cannot be defined accurately due to the breaks in the construction of the embankment.

Despite the distinct dissimilarities in the composition of the sabkha soils, as indicated by the straight line correlation, the hyperbolic representation given in Fig. 13d is considered exceptionally

Fig. 12. Measured total settlements compared to predictions from combined use of the hyperbolic method and theCa/Ccconcept.

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good. The slope of the fitted line was found to be approximately the same, regardless of whether the entire data was fitted or only data at the initial stage was considered. Thus, Ai and Af were considered equal for all subsequent computations. From Fig.

13d, the slope of the fitted line,Ai, was found to be 0.0048/mm. Substituting this value into Eq. (7), along with ai= 0.824, gave an estimate of ultimate primary consolidation settlement of 172 mm.

Although the recorded value is overestimated by about 12.4%, this estimate is considered reasonably good in view of the complexity of the sabkha soils, including its extreme heterogeneity and appreciable secondary compression.

The validity of the hyperbolic method predictions of ultimate primary settlement was further investigated by considering predictions obtained from Terzaghi’s conventional one-dimensional model. The characteristics of the soils which compose the sublayers within the sabkha complex at the specific location of the embankment were obtained from the results of standard oedometer tests performed by Dhowian et al. (1987) on undisturbed samples of sabkha soils, and are shown in Fig. 14. The water table is at a depth of 1 to 2

m below the natural ground surface. The relationship between compression index, Cc, and dry density, cd, is shown in Fig. 15. The relatively high correlation coefficient r² of 0.92 indicates that the compression index, Cc, correlated well with the reciprocal of dry density, cd, for a wide range of Cc

values.

Based on the profile of cdwith depth, Cc values for sublayers were found and the initial void ratios were determined based on cd, considering an average specific gravity of 2.78 for the entire compressible sabkha layer. The compressible layer under the test embankment was divided into 10 sublayers. Referring to Fig. 14, the effective over- burden pressure was estimated at the middle of each sublayer, and the induced vertical stress increase at the middle of each sublayer due to embankment loading was computed using an elastic, semi-infinite medium.

The ultimate primary consolidation settlement under the centre of the embankment obtained from Terzaghi’s one-dimensional model was 233 mm.

This value overestimated the observed ultimate primary consolidation settlement by about 52%, which is obviously higher than the 12.4% hyperbolic

Fig. 13. Instrumented test embankment: (a) loading history; (b) measured settlements; (c) excess pore pressure; (d) hyperbolic plot for measured settlements.

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method overprediction. The significant overprediction of settlement based on conventional compressibility analysis is primarily due to the use of the laboratory-determined parameters. Because of their relatively small size, experimental samples were more homogeneous compared with field sediments with highly variable profiles.

3.2. Secondary settlement predictions

Knowing the compression index,Cc, the secondary compression index,Ca, may be obtained from theCa/ Ccratio expressed inFig. 9b. From the intercept of the hyperbolic plot with the ordinate ofFig. 13d, B was found to be 0.040 min/mm. Substituting this value into Eq. (8), along withai= 0.824 andAi= 0.0048/mm, gives a time for the end of primary settlement of 39 days. This value significantly overestimated the 18

Fig. 14. Soil profile under the test embankment.

Fig. 15. Compression index versus reciprocal of dry density for sabkha soils.

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days found based on the piezometer data shown in Fig. 13c.

An average smooth curve was fitted to the field settlement data as shown in Fig. 16. Using Casa- grande’s procedure, tp was found to be 26 days.

Although the actual 18-day period required for full dissipation of pore pressure was grossly overestimated by both methods, it is obvious that the log-time fitting method furnished a far better estimate fortpthan the hyperbolic method. However, it should be noted that the hyperbolic method found tpby considering only the first four readings while the log-time fitting method utilized the entire data points.

Sincetf/tphas the same value for all sublayers, it is possible to consider the profile as a single layer.

However, the soil profile consists of distinct soil deposits, each with a distinct value for Cc, which result in a significant variation in Ca. Therefore, the secondary compression of each sublayer was computed separately, and the total secondary compression was determined by the sum of secondary compressions of the individual sublayers.

From Eq. (4), withtfof 65 days,tpof 39 days and substituting for ep the in situ void ratio e0 for individual sublayers, the predicted secondary compression for the entire compressible sabkha layer was found to be 15 mm. This value significantly underestimated the measured value by about 66%. This is primarily due to the rather unrealistic time to the end of primary consolidation obtained from Eq. (8).

However, by using the actual tp of 18 days, the predicted secondary compression was found to be 34.5 mm, underpredicting the measured value by less than 22%. Therefore, considering the highly heterogeneous and complex sabkha soil profiles, it seems possible to suggest that the use of Ca/Ccconcept of compressibility could furnish adequate prediction of secondary compression of sabkha soils.

In fact, examination of the results ofFig. 11a has shown that the best agreements between tp values determined from Casagrande’s method and from Eq.

(8) were for the cases when the initial portions of the hyperbolic curves show convex curvature toward the origin, indicating a definite nonlinearity in the early portion of settlement versus time relationship. Contra- rily, the discrepancy between the values oftpobtained from the two fitting methods was large for test results whose hyperbolic plots were approximately fitted by a straight line over the entire data range. It is evident therefore that where the transformed hyperbolic plot deviates from the concave-down shape of the transformed theoretical curve shown in Fig. 1b and is approximately fitted by a straight line, Eq. (8) would give unreasonable value for tp.

Taking the inverse of the slope of the hyperbolic plot, Af, (which for Fig. 13d is equal to Ai) gave a total settlement of 208 mm. This prediction is considered exceptionally good as it overestimated the measured value by less than 6%. This agrees with the results of the experimental data shown in Fig. 8, in which the reciprocal of Afprovided good estimates of the measured total settlements. The relatively small difference between the measured total settlements and the 1/Af predictions may, in part, be attributed to the fact that the soil layer under the test embankment may still had a tendency of undergoing further settlements. Hence, the settlement value taken at the end of the observation period may be somewhere between the observed ultimate primary consolidation settlement and the potential total settlement.

The secondary compression obtained from the hyperbolic method is simply the difference between the predicted total and ultimate primary settlements, and thus is equal to 36 mm. This prediction by the hyperbolic method is remarkably similar to that determined based on theCa/Ccconcept when tpwas taken from piezometer measurements.

Fig. 16. Log time curve fitted to measured field settlements under the centre of test embankment.

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4. Suggested procedure for settlement prediction

The first essential step is to plot the time–

settlement versus time relationship, and the range of the data should be sufficient to establish the initial portion of the hyperbolic plot. If the hyperbolic curve has a concave-down initial portion, the total settlement, d_t, is determined from a combined use of the hyperbolic method and the Ca/Ccconcept, and Eqs.

(4), (7), and (8) yield dt¼a_i

Ai

þ CaHp

1þep

log tfð1a_iÞAi

aiB

:

ð9Þ The values of Ai and B are determined from the hyperbolic plot, and the value ofCais obtained after establishing the relationship between Ca and Cc as, for instance, shown in Fig. 9b for the sabkha soils considered in this study.

For the case where the hyperbolic plot entirely follows a straight line, the end of primary consolidation settlement obtained using the hyperbolic method would be unreliable, and hence the total settlement, inclusive of secondary compression, is obtained from

d_t¼ 1 Ai

: ð10Þ

If the components of the total settlement are required, then the ultimate primary consolidation settlement is obtained from Eq. (7), and the secondary compression is simply expressed as

d_s¼1a_i Ai

: ð11Þ

It has been noted at the outset of this paper that using the inverse slope method involves a number of limitations, the least of which is that establishing the slope of the final portion of the hyperbolic plot requires data beyond 90% primary consolidation.

However, where the hyperbolic curve entirely follows a straight line, Af is essentially equal to Ai, and as expressed in Eq. (10) settlement computations may be based on Ai. Thus, the limitations associated with the use of Afbecome, to great extent, irrelevant.

An advantage of using Eq. (10) is that the total settlement is directly obtained without the need for it

to be partitioned into primary and secondary components. This is important considering the widely held view (Bjerrum, 1967; Crawford, 1986) that both primary and secondary consolidations begin simulta- neously with the application of a pressure increment, and hence it is not possible to know how much of the measured field settlement is due to primary consolidation and how much to secondary compression.

Indeed, in a nonhomogeneous soil formation, such as the one under consideration, it is to be expected that thin layer or layers with high permeability may have experienced significant secondary compression, while others may be still at the early stage of primary consolidation.

5. Summary and conclusions

In this paper, the feasibility of using the hyperbolic method and Ca/Cc concept in combination for settlement predictions of complex soil formations, displaying appreciable secondary compression, has been investigated. The hyperbolic method is used to provide an estimate of the ultimate primary consolidation settlement and the time for its completion, and secondary compression is calculated using theCa/Cc

concept of compressibility.

The applicability of the proposed procedure has been verified using experimental data from series of conventional and long-duration one-dimensional consolidation tests, conducted on undisturbed samples, as well as the results of an instrumented test embankment constructed on a typical compressible sabkha profile. The hyperbolic method estimate of the in situ ultimate primary consolidation settlement was also compared with the prediction obtained from Terza- ghi’s conventional one-dimensional model.

Provided that the hyperbolic plot has an initial portion that is concave downward, the proposed approach has been found to give reasonable estimates of total settlements. Ultimate primary consolidation settlement was reasonably well predicted from the inverse of the slope of the initial linear segment of the hyperbolic plot fitted to actual settlement data, Ai, multiplied by the slope of the initial linear portion of the theoretical hyperbolic curve, ai. With the secondary compression index, Ca, obtained using the Ca/Cc concept and the time

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corresponding to the end of primary consolidation determined from the hyperbolic method, a satisfac- tory agreement was found between measured and predicted secondary compression values.

However, for hyperbolic plots that entirely follow a straight line, the hyperbolic method was not reliably able to partition settlement into primary and secondary components. The time for the end of primary consolidation was considerably overpredicted, and hence theCa/Ccconcept of compressibility unreason- ably underestimated secondary compression. There- fore, where settlement versus time entirely obeys rectangular hyperbola, total settlement, inclusive of secondary compression, is satisfactorily obtained from the hyperbolic method as the reciprocal of the slope of the hyperbolic fitted straight line. If the components of total settlement are required, the ultimate primary consolidation settlement and secondary compression are obtained from Eqs. (7) and (11), respectively.

Analysis of the experimental data and settlement under the test embankment indicated that total settlements provided by Eq. (10) were in remarkably good agreement with the measured values. Further- more, because in this case the hyperbolic curve entirely is fitted to a straight line, Af is essentially equal toAi, and hence the limitations associated with the use ofAfbecome irrelevant.

Finally, the proposed procedure has shown to be applicable for evaluating field settlement of heterogeneous soils with appreciable secondary compression, where representative properties are difficult or even impossible to determine. However, it is worth noting that the hyperbolic method is observational in nature and cannot substitute the predictive methods needed at the design stage.

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