Velocity Method for Creep of Clayey Soils Researcher, Central Soil & Materials Research Station, Ministry of Water Resources, New Delhi, India. Email: tewatiask_2000@yahoo.co.in Dwarka Nath Bhargav Former Professor and Chief Engineer of UP irrigation, Water Resources Development Training Centre, IIT Roorkee, India. Email: rohaninthehouse@yahoo.com Gopal Chauhan Professor and former Director, Water Resources Development Training Centre, IIT, Roorkee, India. Email: gopalfwt@iitr.ernet.in Asuri Sridharan Hon. Professor, Department of Civil Engineering, and former Deputy Director of Indian Institute of Science, Bangalore, India. Email: asuri@civil.iisc.ernet.in |

ABSTRACT

Physics defines *creep* as the slow deformation of a body under constant stress where creep may generally exhibit itself as a straight line on δ-log(*t*) plot where δ is the experimental deformation and *t* is the experimental time, both measured from the instant when the constant stress was applied. But the problems in geotechnical engineering may be more practical, complex and down to earth. The creep of clayey soils poses such special problem where this instant is not known. In primary consolidation stress is being gradually but non-linearly transmitted from pore water to soil skeleton and hence for post-consolidation creep in clays, the real time of onset is obscure. A method is derived for the graphical integration of straight line semi-log plot and using the same, time-of-origin effects have been examined in relation to the creep settlement of clays, where it is shown that anomalous variations from a normal creep slope may arise as a result of incorrect assumptions, or lack of knowledge, of the true time of initiation of creep. A new method, named as velocity method, is proposed for evaluating the coefficient of secondary consolidation, α, when the time and settlement at the instant of load increment are not known. The proposed method is faster and better than the conventional δ-log(t) method. More precise definitions are suggested to distinguish creep from the secondary consolidation.

KEYWORDS: coefficients, consolidation, secondary compression, laboratory test, slope, settlement, load, drainage, velocity, creep, graphical integration

Introduction

When the excess pore pressure due to consolidation has been dissipated, the change in voids ratio continues, but generally at a reduced rate. This phenomenon is known as secondary consolidation or creep during which some of the highly viscous water between the points of contact is forced out from between the grains. During creep there is plastic re-adjustment of the soil particles and of the adsorbed water due to the continued stress, and the progressive fracture of some of the particles. In many soil deposits creep forms a substantial part of the settlement. Since it is not governed by the dissipation of excess hydrostatic pressure, Terzaghi’s theory of consolidation cannot be applied to find the rate of secondary consolidation or creep.

Secondary consolidation essentially starts coming into play in the range of primary consolidation itself. This is also the reason why the experimental time consolidation curve of Terzaghian soils is in agreement with Terzaghi’s theory of consolidation only up to U=50-60%, where U is degree of consolidation (Tewatia 1998). A pressure voids ratio diagram based on final voids ratio in a consolidation test includes also a part of creep settlement that occurs under various pressure increments. For highly organic soils, highly micaceous soils, and some loosely deposited clays secondary consolidation may constitute a major part of the total settlement.

It is often assumed that creep proceeds linearly with the logarithm of time. The creep usually appears as a straight line, or a series of straight lines, with different slopes in the plot of voids ratio as a function of logarithm of time, and may be represented by the following equation:

Where De is change in void ratio, a is coefficient representing rate of secondary consolidation; *t*_{2} is total elapsed time since the load was applied to the soil and time *t*_{1} is the period required for the hydrodynamic consolidation (i.e. primary consolidation to be nearly complete, and may be taken as the time corresponding to 90% consolidation; it is considered as the time for the onset of creep).

Time-of-origin effects have been examined in relation to the creep settlement of rock-fill dams
by Parkin (1990), where it is shown that anomalous variations from a normal creep slope arise as a
result of incorrect assumptions, or lack of knowledge, of the true time of initiation of creep.
Creep is the slow deformation of a body under constant stress, but in primary consolidation stress
is being gradually but non-linearly transmitted from pore water to soil skeleton and hence for
post-consolidation creep in clays, the real time of onset is obscure (Parkin 1990). There is,
therefore, need to evaluate the coefficient of creep, a, without using the time of initiation of creep. A method for the same is suggested here which not only evaluates a without knowing the time and settlement at the instant of load increment but can evaluate this time also. The paper makes use of the mathematical expression derived for the graphical integration of straight line semi-log plot
to show that a straight line on δ-log(t) plot has to be a straight line on δ-log(*v*) plot, where *v* (= dδ/dt) is the experimental velocity of settlement. The slopes of both of the plots
have to be same in magnitude. Results have been compared with the conventional method using a wide
variety of clays ranging from kaolinite (liquid limit = 39%) to bentonite (liquid limit = 495%).

Mathematical Derivation

Let the straight line creep portion in settlement versus log of time plot (say d-ln(*t*)
plot) be represented by the following equation

Where *a* and *b* are constants. Differentiating with respect to *t*, we obtain

Taking the natural logarithm of both sides

ln t = -ln v + ln a
| (4) |

Substituting ln(*t*) in equation (2), we get

Following inferences can be drawn from equations (2), (3) and (5).

- Comparing (2) with (5): If d
*versus*ln(*t*) plot is straight line then d-ln(*v*) plot is also a straight line having the same magnitude (but opposite in sign) of slope and*vice versa*. The difference between the intercepts of these two on d axis is =*a*ln(*a*). Thus one can be drawn if other is known. In fact equation (2) is graphical integration of equation (5) i.e. of*Y*versus d*Y*/d*X*curve to*Y*versus*X*curve in case of straight line semi-log plot, where*Y*is d,*X*is*t*and*v*is d*Y*/d*X*. - Elapsed time,
*t*, from the instant of load increment can be determined from equation (3). - However,
*a*(and hence a) can be determined from d-ln(*v*) plot without knowing the time of load increment.

Let e_{f }= final void ratio, h_{f }= final height and Dh = change in height of the specimen; then using the established relationships

Where *LC* is the least count of the dial gauge (= 0.002 mm or 0.0025 mm) and Dd is the change in settlement in terms of dial gauge reading. Therefore,

Where *a* is the slope of d-ln(*t*) plot from eq.(2), a is the coefficient of creep or secondary consolidation from eq.(1) and *c* is a constant as all other terms *e _{f}*,

The slope of d-log_{10}(t) plot (say *m*) is 2.3026 times of the slope of d-ln(*t*) plot, i.e.

m = 2.3026 a | (8) |

However, difference of intercepts on d axis between d-log_{10}(t) and d-log_{10}(*v*) plots remains same (= *a* ln(*a*)). Using eq.(8) it can be shown that

a ln(a) = m log_{10}(m/2.3026)
| (9) |

Eq.(9) is useful when log_{10} scale is used instead of natural log or ln
scale in the semi-log plot of d versus *t* and d versus *v*.

Consolidation Tests

Experimental data used has been obtained from numerous standard consolidometer tests performed at Central Soil and Materials Research Station, Delhi and Indian Institute of Science, Bangalore, on a variety of Indian soils viz. Narmada Soil (liquid limit 45% and plastic limit 24%), Kaolinite Soil (liquid limit 47% and plastic limit 30%), Dudhi Soil (liquid limit 54% and plastic limit 24%), Sawan-Bhadon Soil (liquid limit 61% and plastic limit 29%), Black Cotton Soil (liquid limit 69% and plastic limit 33%), Bentonite + Sand mixture (liquid limit 100% and plastic limit 30%) and Bentonite (liquid limit 495% and plastic limit 49%). Consolidation tests were carried out as per standard procedures on remolded saturated specimens. For this purpose soils passing through 2-mm sieve were used. The soil was packed into the ring taking density equal to 98% of the standard Proctor's maximum dry density. The consolidometer used was of fixed ring type with a diameter of 60 mm and a height of 20 mm. A load increment ratio of one was adopted and each load was maintained for sufficient time even upto 4 weeks until the compression virtually ceases. Vertical compression was measured using dial gauges with least count of 0.002 mm in case of Sawan Bhadon Dam Soil and 0.0025 mm in case of other soils. For more accuracy, when the rate of settlement is low, instead of noting settlement at certain intervals of time, time in seconds was noted at certain settlements when dial gauge needle coincides with the exact mark on the dial.

Plotting of Observations

The observations were plotted for the analysis. In case of Sawan Bhadon Dam Soil dial gauge readings are directly plotted which decrease with increasing time. In case of other soils settlement in terms of dial gauge units is plotted which increases with increasing time. The settlement is calculated by subtracting the initial dial gauge reading at the time of load increment from the one corresponding to any time. Time in minutes is plotted for all soils. Units of velocity are dial gauge units per second in case of Sawan Bhadon soil and dial gauge units per minute in case of other soils. In case of Sawan Bhadon soil log(*v*) is plotted on linear scale while in other soils *v* is plotted on log scale. However, it is clear from equations (1), (2), (6) etc. that the slope of δ-log(*t*) plot is not affected by the units of time. Similarly, the slope of δ-log(*v*) plot is not affected by the units of velocity, as the units of slope per log cycle in both the cases are the units of δ i.e. dial gauge units. But for intercept on δ axis, this axis has to be same in both the cases. This is *y*-axis where ln(1) or log_{10}(1) is zero. Therefore, if *t* is in seconds then *v* should be in dial gauge units per sec; and if t is in minutes then *v* should be in dial gauge units per minute. Also, *t* and *v* both should be on ln scale or both of them should be on log_{10} scale. Shown in the Figures is log_{10} scale but in Table 1, ‘a’ is on ‘ln’ scale. In case of Sawan Bhadon soil table shows the intercept on δ axis (467) when *v* is in dial gauge units per second as is shown in fig 4. However, if *v* is in dial gauge units per minute the figure remains the same. Only difference is that δ axis shifts towards left by an amount of log_{10} (60) i.e. at –1.78 and then intercept of the creep line on it is 398, which is clear from Figure 4. The intercept remains same whether it is ln or log_{10} plot. For example, using ln plot slope 16.9 from Table1 and shifting towards left by ln(60), the intercept is = 467 – 16.9 ln(60) = 398. Different types of plots and units are used to experimentally justify the paper from all angles and to make the idea thoroughly clear to the readers.

Results and Discussion

Figures 1 through 10 and Table 1 show the results of various soils, where *a* is the slope and *b* is the intercept on δ-axis of the straight line of δ-log(t) and δ-log(*v*) plots. The values of *a* and *b* have been obtained from computerised plots using the best curve fitting methods. It is clear from the velocity plot of Sawan Bhadon soil, Figure 4 (Tewatia 1998), that secondary consolidation or creep under gradually increasing effective stress is already in progress in the range of primary consolidation. Time of origin effects give least or almost no error in case of Kaolinite soil where hardly any time taken for primary consolidation is observed and creep starts almost at the time of load increment. Here ‘*a*’ is almost same for both the plots and difference in *b* is almost __a____ ln( a)__. In other soils error in ‘

**Figure 1.** Kaolinite Soil (LL = 47%, PL = 30%). Semi-log plot of d versus *t*.
The d is in dial gauge units (one unit = 0.0025mm) and *t* is in minutes.

**Figure 2.** Kaolinite Soil (LL = 47%, PL = 30%). Semi-log plot of d versus *v*. The d is in dial gauge units (one unit = 0.0025mm) and *v* is in dial gauge units/minute.

**Figure 3.** Sawan-Bhadon Dam Soil (LL = 61%, PL = 29%). Semi-log plot of d versus t. The d is dial gauge reading (one unit = 0.002mm) and t is in minutes.

**Figure 4.** Sawan-Bhadon Dam Soil (LL = 61%, PL = 29%). Linear plot of d versus log_{10}(*v*). The d is dial gauge reading (one unit = 0.002mm) and *v* is in dial gauge units/sec. (Tewatia 1998).

**Figure 5.** Black-Cotton Soil (LL = 69%, PL = 33%). Semi-log plot of d versus *t*. The d is in dial gauge units (one unit = 0.0025mm) and t is in minutes.

**Figure 6.** Black-Cotton Soil (LL = 69%, PL = 33%). Semi-log plot of d versus *v*. The d is in dial gauge units (one unit = 0.0025mm) and *v* is in dial gauge units/minute.

**Figure 7.** Bentonite + Sand Mixture (LL = 100%, PL = 30%). Semi-log plot of d versus *t*. The d is in dial gauge units (one unit = 0.0025mm) and *t* is in minutes.

**Figure 8.** Bentonite + Sand Mixture (LL = 100%, PL = 30%). Semi-log plot of d versus *v*. The d is in dial gauge units (one unit = 0.0025mm) and *v* is in dial gauge units/minute.

**Figure 9.** Bentonite (LL = 495%, PL = 49%). Semi-log plot of d versus *t*. The d is in dial gauge units (one unit = 0.0025mm) and *t* is in minutes.

**Figure 10.** Bentonite (LL = 495%, PL = 49%). Semi-log plot of d versus *v*. The d is in dial gauge units (one unit = 0.0025mm) and *v* is in dial gauge units/minute (Tewatia 1998).

Distinction between Creep and Secondary Consolidation

This paper and particularly Figure 4 (where for the first time secondary consolidation has been quantitatively isolated) make it most immediately obvious that any experimental consolidation settlement under constant loading may be assumed to be the summation of 4 types

- Initial Compression
- Primary, Hydrodynamic or Terzaghian Compression
- Creep or creep-like slow deformation under gradually increasing effective stress
- Creep under constant effective stress

Hence, creep may be defined as the compression of soil skeleton under constant effective stress. Secondary compression may be defined as any extra compression other than initial and primary compression. Secondary consolidation is the summation of type 3 and type 4.

Conclusions

A semi-log plot of settlement versus velocity has been proposed for evaluating the coefficient of creep or secondary consolidation, without knowing its time of initiation. A few characteristics of this consolidation have been shown in this paper. A method is proposed to obtain δ–log(*t*) plot from the δ-log(*v*) plot. The velocity method is faster and better than the δ-log(*t*) method. Results have been compared with the conventional methods using a wide variety of clays ranging from kaolinite (liquid limit = 39%) to bentonite (liquid limit = 495%). It is shown that anomalous variations from a normal creep slope may arise as a result of incorrect assumptions, or lack of knowledge, of the true time of initiation of creep settlement. Secondary compression may be defined as any extra compression other than initial and primary compression. It is the summation of the creep or creep like settlements that occur under constant and gradually increasing effective stress. The mathematical derivation has been used here for clayey soils but it applies to all such straight line semi-log plots everywhere.

Acknowledgement

First author is grateful to Prof. (Emeritus) Bharat Singh, Ex. Vice Chancellor of Roorkee University, India, for his valuable advice in this work.

References

Parkin, A. K., 1990, “Creep of Rockfill,” Chapter 9, Advances in rockfill structures (E. Maranha das Neves, Eds). NATO Advanced Study Institute, Series E, No. 200, Lisbon, Portugal, pp 221-237.

Tewatia, S. K., 1998, "Evaluation of True *C*n and Instantaneous *C*n, and Isolation of Secondary Consolidation," American Society for Testing and Materials, Geotechnical Testing Journal, GTJODJ, Vol. 21, No. 2, pp. 102-108.

**Table 1.** The slope ‘*a*’ and intercept ‘*b*’ values of
d-ln(*t*) and d-ln(*v*) plots of the five soils.

Shown in table, for each soil, is % error in ‘*a*’ which is equal to % error in a.

Kaolinite Soil LL = 47% PL = 30% |
% error in slope,a = -1.2 |
d-ln(t) plot |
a | 5.77 |

b | 127.4 | |||

d-ln(v) plot |
a | -5.70 | ||

b |
137.7 | |||

| ||||

Sawan-Bhadon Dam Soil LL = 61%, PL = 29% |
% error in slope,a = -7.5 |
d-ln(t) plot |
a | -18.1 |

b | 458.2 | |||

d-ln(v) plot |
a | 16.9 | ||

b | 467.0 | |||

| ||||

Black-Cotton Soil LL = 69% PL = 33% |
% error in slope,a = 6.9 |
d-ln(t) plot |
a | 22.8 |

b | 240.2 | |||

d-ln(v) plot |
a | -24.5 | ||

b | 310.2 | |||

| ||||

Bentonite +Sand Mixture LL = 100% PL = 30% |
% error in slope,a = -18.4 |
d-ln(t) plot |
a | 13.38 |

b | 356.0 | |||

d-ln(v) plot |
a | -11.3 | ||

b | 404.1 | |||

| ||||

Bentonite LL = 495% PL = 49% |
% error in slope,a = 9.7 |
d-ln(t) plot |
a | 235.4 |

b | -811.9 | |||

d-ln(v) plot |
a | -260.6 | ||

b | 433.3 | |||

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