Experiments of creep rate for over-consolidated clay under plain strain condition and a simple correlation

ABSTRACT

Soft foundations of high embankments exhibit significant time-dependent effect on deformation in decades after construction, and there is a lack of simple method for predicting the creep rate of over-consolidated soft clay under plain strain condition. In this paper, laboratory creep tests under plain strain consolidation with different principal stress ratio were shown. The variation of stress in zero strain direction (σ_y) under different degree of consolidation was analysed. Linear relationship between (σ_y) and the applied principal stress σ_x (or σ_z) was found. And then, the creep rate of volumetric strain (C_α) and vertical strain (β) were calculated and linked to volumetric stress defined ratio of over-consolidation (OCR_p) and stress ratio (K). Finally, the new parameter K∙OCR_p was found to have a good correlation with the creep rate (C_α or β).

KEYWORDS:

• Over-consolidation
• plain strain
• creep
• stress ratio

1. Introduction

Since a booming development of infrastructure constructions from 1980s in China, some unsatisfied performances in many geotechnical projects were exposed after the 30 ~ 40 years’ service period, e.g. large settlement of soft foundations for expressways and buildings without extra applied loading, especially in coastal areas (Chen et al., 2014; Li et al., 2012). One of the main reasons is the marine clayey foundations exhibited significant time-dependent effect on deformation under a fixed overburden load, termed secondary consolidation or creep. Secondary consolidation is an active topic in the past three decades (Nash et al., 1992; Li et al., 2009; Zhu, Zhang, et al., 2016a; Z.Z. Li et al., 2017; Xu et al., 2017; Yin & Feng, 2017; Gu et al., 2019; Lei et al., 2020; Sahib & Robinson, 2020; Zhou et al., 2020) and studied by many researchers considering shear modulus (Mesri et al., 1990), swelling behaviour (Feng et al., 2017; Vergote et al., 2020), freeze–thaw process (J. Li et al., 2020), undrained behaviour (Han et al., 2021), reconstituted clay (Wu et al., 2019) and nonlinear creep (Zhu, Yin, et al., 2016b; Feng et al., 2017).

Generally, classic existing models of secondary consolidation can be divided into three categories: three-dimensional (3D) model, theoretical model and empirical model. The 3D models, for example, proposed by Adachi and Oka (1982) can capture both deformation and strength characteristics of soil and also be able to consider very complex boundary conditions. However, because of mathematical difficulties the solutions of 3D models need finite element methods (FEM) and might not be applicable to some engineers. The theoretical model was originally developed by Taylor and Merchant (1940) and further studied by Gibson and Lo (1961) and other experts (Kurz et al., 2016; Yao & Fang, 2020). However, the material parameters used in the theoretical model could not be easily obtained from standard soil tests which hinders its application in engineering. The empirical model based on the simple hypothesis: secondary consolidation separately happens after the end of primary consolidation (EOP), was initially proposed by Buisman (1936). In this model, only the coefficient of secondary consolidation is used, denoted by C_α which is defined as the slope of the compression line in ε-log(t) (or e-log(t)) curve after EOP. Empirical model is widely used by engineers in designing, and during and after construction to evaluate the creep deformation as its simplicity.

As for empirical model, there are two shortcomings being ignored for years. First, the clayey deposits are needed to be preloaded by surcharge, vacuum load or combine surcharge and vacuum load before the construction of the upper pavement in order to reduce the post-construction settlement of soft foundation. This induces an over-consolidated (OC) state of clay after removing the applied load. The soft foundation may have significant deformation even ever experienced a considerable maximum past effective stress (Hu et al., 2016; Shi et al., 2018; Silvestri, 1980; Vergote et al., 2020; Yuan et al., 2015), i.e. with a large value of over-consolidate ratio (OCR). The second issue is that the soft foundation under high embankment is of plain strain condition typically. Whereas, the current empirical models cannot directly consider the influence of both OCR and plains strain condition in their models yet. This study investigated the creep rate for OC clay under plain strain condition through laboratory tests, and a new simple empirical correlation between the creep rate and the degree of over-consolidation and principal stress ratio (K) was proposed.

2. Test programme

2.1. Apparatus and sample preparation

A test equipment for creep under plain strain condition called ‘plain strain creep consolidometer’ (Li et al., 2015, 2016, 2014) was developed as illustrated in Figure 1. The consolidometer can apply vertical (Z direction) load σ_z step by step using dead weight and lateral (X direction) load σ_x through pressurised water contained in a rectangular rubber-chamber to the soil samples. In Y direction, two rigid steel boards were used so that the strain in this direction can be set to zero during test. The stress in Y direction σ_y can be measured by a stress sensor. The settlement in Z direction and volume of water drained out can be measured and recorded by computer through a data logger.

Figure 1. Illustration of plain strain creep consolidometer

The soil sample adopted in this test has a height of 100 mm in Z direction, width of 100 mm in Y direction and length of 50 mm in X direction. The undisturbed soil samples were cut by a rectangular sampler and then placed into a model box and saturated for more than 24 h before testing. Then, strip pore papers were applied on the sample surface to provide peripheral drainage, and the soil sample was set up into a rubber which has the same dimensions. Finally, the sample was put into the consolidometer and the test was started under two-way drainage conditions. Figure 2 shows the photos of the test process.

Figure 2. Photos of the test process

2.2. Soil used and cases tested

Undisturbed marine clay called Yuedong Clay obtained from Shanfen area in Guangdong province which is located at the southern part of China was used in the tests. Yuedong clay is one of the three typical kinds of clay including Shanghai clay and Tianjin Binhai clay in China. The undisturbed soils were taken from the depth of approximately 6–9 m and sampled using a tube with 140 mm in diameter and 200 mm in height. The two ends of the sampling tube were closed with caps and then moved to the laboratory very soon after sampling. The soil samples were stored in the water to maintain its saturation. The basic soil properties of Yuedong clay are listed in Table 1. The cases tested were divided into three groups consisting of seven tests totally. There is only one test conducted in the first group with normally consolidated (NC) sample as comparative test. The rest six tests were separated into two groups in which over-consolidated samples were used, and the principal stress ratio in these two groups was controlled as K = σ_x/σ_z = 0.5 and 0.33, respectively. The cases and procedures of the tests are shown in Table 2.

Table 1. Basic soil properties of Yuedong clay

Property	Value
Initial water content, w0	56 ~ 60%
Compression index, Cc	0.5 ~ 0.7
Recompression index, Cs	0.05 ~ 0.08
Liquid limit, wL	62.5
Plastic limit, wP	37.2
Coefficient of consolidation, cv100-200(10^−3 cm^2/s)	1.05

Table 2. Detailed steps for reloading

Group	Test	Loading procedure (σX, σZ)
		Initial loading 24 h	Maximum preloading	Unloading 24 h	Consolidation, 168 h
					Loading(or Reloading) step
					1	2	3	4
1 (K = 0.5)	-	(25,25)	-	-	(50,100)	-(100,200)^^	-(150,300)	-(200,400)
2 (K = 0.5)	a	(25,25)	(200,400)	(25,25)	(50,100)	-(100,200)	-(150,300)	-(200,400)
	b		(150,300)		(50,100)	-(100,200)^#	-(150,300)
	c		(100,200)		(50,100)	-(100,200)
3 (K = 0.33)	a		(100,300)		(40,120)	-(60,180)	-(80,240)	-(100,300)
	b		(80,240)		(40,120)	-(60,180)	-(80,240)
	c		(60,180)		(40,120)*	-(60,180)

Note: designation of secondary consolidation step, e.g. ^{^} is T1-2, ^# is T2b-2 and * is T3c-1.

2.3. Test procedure

After preparation of soil sample, the tests were conducted by following the four procedures:

(1) Initial loading: Before the main test, the soil samples were applied an initial load of (σ_x, σ_z) = (25, 25) kPa to discharge the extra water and air between the sample and the rubber membrane, and to prepare a uniform initial condition for all tested samples.

(2) Preloading: This procedure only existed in groups 2 and 3. For secondary consolidation tests under OC state in groups 2 and 3, the soil samples were first preloaded to a certain maximum stress by multi-stages under specified stress ratio. The preloading time is 24 h for every loading stage which is enough for completion of primary consolidation. The maximum preloading stress are (200, 400), (150, 300) and (100, 200) kPa corresponding to 2a, 2b and 2 c, and (100,300), (80, 240), (60, 180) kPa corresponding to 3a, 3b, 3 c.

(3) Unloading: In groups 2 and 3, the soil samples were unloaded to (25, 25) kPa again for 24 h.

(4) Consolidation (loading or reloading): After unloading, the soil samples were reloaded step by step to conduct consolidation tests under OC state and every step lasted for 168 h (7 days). In group 1, the 168 hours’ consolidation started immediately after initial loading step. The detailed reloading steps can be found in Table 2.

3. Test results and analysis

3.1. Stress in zero stain (Y) direction

Under plain strain state, the stress in Y direction (σ_y) reacts as a result of applied stress (σ_x, σ_z). Figure 3 shows the results of σ_y from test T1 for NC soil and T2a for OC soil during consolidation stages. The loading (or reloading) processes in these two tests are exactly the same. First of all, σ_y decreases with time for both NC and OC soils. After a load of (σ_x, σ_z) was applied, the soil will immediately yield a high total stress due to increase of excess pore pressure (u) resulting in a high value of σ_y. With consolidation process, σ_y gradually decreases to a steady state finally with dissipation of u. In one-dimensional consolidation test under k₀ condition, the lateral stress σ_h for NC soil is smaller than that for OC soil (Cai et al., 2011; El-Sekelly et al., 2015). A similar conclusion can be drawn with respect to σ_y from Figure 3. Generally, the horizontal stress σ_y for OC soil is 30 ~ 50 kPa higher than that for NC soil under same external loading.

Figure 3. Comparison of stress in zero strain direction (σ_y) under normal and over-consolidation states

The results of σ_y as a function of time during consolidation for OC soils from groups 2 and 3 are shown in Figs. 4 and 5, respectively. Under same external loading, e.g. (100,200) kPa in T2a-2, T2b-2 and T2c-2, the sample which experienced larger maximum preloading has higher value of σ_y. The final value of σ_y at end of test (i.e, 10000^th min) is equal to 91, 79 and 64.5 kPa in T2a-2, T2b-2 and T2c-2, respectively. As well known in one-dimensional consolidation, k₀ is increasing with increases of over-consolidation ratio (OCR); thus, σ_h is bigger corresponding to larger OCR. As for σ_y under plain strain condition, apparently this conclusion is also true.

Figure 4. Variations of σ_y with time for OC soils in test 2a, b and c

Figure 5. Variations of σ_y with time for OC soils in test 3a, b and c

The relationship between final σ_y (i.e. σ_y at 10000^th min) and σ_z for K = 0.5 and K = 0.33 are shown in Figs. 6 and 7, respectively. First, it can be seen clearly that the σ_y varies with σ_z linearly. The slope of σ_y ~ σ_z curve is affected uniquely by the stress ratio (K). The slope of σ_y ~ σ_z curve for K = 0.5 is about 0.4 (Figure 6(a)) and the value is about 0.34 for K = 0.33 (Figure 7(a)). In addition, the results from T1, T2 and T3 in Figs. 6 and 7 indicate that the slope of σ_y ~ σ_z curve is independent of consolidated state (NC or OC) and degree of consolidation (i.e. the maximum experienced preload). However, the consolidated state has vital impacts on the magnitude of σ_y. Under same magnitude of external loading, OC samples (T2a, b, c) have higher values of σ_y than NC samples (T1). Moreover, comparing T2a, 2b and 2 c in Figure 6(a) (or T3a, 3b and 3 c in Figure 7(a)), it was found that if the sample experienced higher maximum preloading, it yields a larger value of σ_y. The σ_y as a function of σ_x was also shown in Figure 6(b) and Figure 7(b); the slope is 0.8 and 0.67 for K = 0.5 and 0.33, respectively. The value of σ_y is close to σ_x in this study. However, σ_y is the minor principal stress in these two tested cases.

Figure 6. Relationship between final σ_y and (a) σ_z, (b) σ_x for K = 0.5

Figure 7. Relationship between final σ_y and (a) σ_z, (b) σ_x for K = 0.33

3.2. Compression and creep coefficient

The volumetric strain (ε_v) was calculated by using the volume of water drained out. ε_v as a function of time during reloading stages for both T1 and T2 were plotted in Figure 8. Without surprise, the volumetric strain for NC clay is several times of that for OC clay. And, higher maximum preload in the history yields less volumetric strain under same external loading. Similar results were obtained from tests T3. It is noted here that the friction between the soil sample (membrane) and the rigid board in Y direction was not included in analysis. As mentioned above, for OC clay, the σ_y is larger than NC clay; thus, OC clay will have larger friction than NC clay. The friction effect on creep deformation was not identical for OC and NC clay, and the creep deformation was underestimated as a result.

Figure 8. Variations of volumetric strain with time for tests T1 and T2 (a, b, c)

The volumetric creep coefficient, designated as C_α, can be estimated by analysing the ε_v ~ log (t) curve. The slope of the straight line in the end stage of ε_v ~ log (t) curve was calculated as C_α. Figure 9 shows an example of the graphical procedure for determining C_α. The values of C_α were evaluated by this graphical method for all steps in groups 2 and 3 (Table 3), and the values were used in later section. By the same method, the vertical creep coefficient β was calculated using vertical strain and listed in Table 4. In Figure 10, the variations in β with OCR (σ_z,max/σ_z) for tests 2 and 3 together with one-dimensional (1D) creep tests from Li et al. (2012) were plotted. For both plain strain and 1D condition, the creep rate decreases with increase of OCR. However, under 1D condition, the vertical creep rate is generally higher than that under plain strain condition. In plain strain condition, there is an extra active stress, σ_x, which will compress the soil in X direction and yield lateral compression. As a result, the vertical creep rate under 1D condition is higher than that under plain strain condition with respect to a same magnitude of OCR.

Table 3. Calculated results of volumetric creep coefficient and volumetric stress defined over-consolidation ratio

Group	Test	Cα(10^−2), OCRP
		Steps
		1	2	3	4
1 (K = 0.5)	-	1.000, 0.730	1.000, 0.744	1.000, 0.814	1.000, 0.773
2 (K = 0.5)	a	0.192, 4.845	0.372, 2.142	0.603, 1.405	0.656, 0.985
	b	0.262, 3.460	0.467, 1.604	0.829, 1.029
	c	0.404. 2.127	0.692, 1.044
3 (K = 0.33)	a	0.513, 2.591	0.638, 1.723	0.913, 1.281	1.173, 1.008
	b	0.570, 2.042	0.812, 1.353	0.573, 0.991
	c	0.693, 1.512	0.706, 1.010

Table 4. Calculated results of vertical creep coefficient and volumetric stress defined over-consolidation ratio

Group	Test	β(10^−2), OCRP
		Steps
		1	2	3	4
2 (K = 0.5)	a	0.026, 4.845	0.064, 2.142	0.096, 1.405	0.150, 0.985
	b	0.035, 3.460	0.085, 1.604	0.022, 1.029
	c	0.047. 2.127	0.172, 1.044
3 (K = 0.33)	a	0.059, 2.591	0.159, 1.723	0.484, 1.281	0.683, 1.008
	b	0.093, 2.042	0.287, 1.353	0.573, 0.991
	c	0.184, 1.512	0.431, 1.010

Figure 9. Graphic procedure for determining C_α: plot the ε_v ~ log (t) curve firstly, project a straight line along the final linear part of the ε_v ~ log (t) curve secondly, and calculate the slope of the straight line finally

Figure 10. Vertical creep coefficient β for OC clay under plain strain and 1D condition

3.3. New method to predict creep coefficient

In this section, the over-consolidation ratio was innovatively defined using the mean volumetric stress (Li et al., 2014) under plane strain condition and designated as OCR_P. The novel parameter OCR_P can be calculated as

(1) $\mathrm{O}\mathrm{C}{\mathrm{R}}_{\mathrm{P}}=\frac{{\sigma }_{\mathrm{x},MAX}+{\sigma }_{\mathrm{y},MAX}+{\sigma }_{z,MAX}}{{\sigma }_{\mathrm{x}}+{\sigma }_{\mathrm{y}}+{\sigma }_z}$(1)

where the subscript ‘MAX’ refers to the maximum stress that the soil experienced before unloading. The representative value of σ_y used in calculation was obtained at the end of each step (i.e. final σ_y). The stress history of σ_x, σ_y and σ_z is able to be included into degree of consolidation through this definition.

The values of OCR_P for all steps in tests of groups 1, 2 and 3 were calculated by Eq. (1)(1) $\mathrm{O}\mathrm{C}{\mathrm{R}}_{\mathrm{P}}=\frac{{\sigma }_{\mathrm{x},MAX}+{\sigma }_{\mathrm{y},MAX}+{\sigma }_{z,MAX}}{{\sigma }_{\mathrm{x}}+{\sigma }_{\mathrm{y}}+{\sigma }_z}$(1) and listed in Table 3 together with estimated C_α obtained in the above section. The value of OCR_p is from 1 to 5 in this study which covers the common range of over-consolidation ratio. The values of C_α and β were linked to OCR_P, and the results are shown in Figure 11(a, b), respectively. In general, C_α and β are decreasing with increase of OCR_P significantly for both K = 0.5 and K = 0.33, respectively. The value of C_α decreases by 58.3% as OCR_P increases to 2.6 from 1 for K = 0.33. And, C_α decreases by 75% as OCR_P increases to 4.9 from 1 for K = 0.5. Similar results were observed in β ~ OCR_p curve.

Figure 11. (a) C_α and (b) β as a function of OCR_P.

As an indicative result, engineers can control the creep deformation and rate by properly applying a larger magnitude of surcharge preloading in practice. It is also observed that the relationship between C_α and OCR_P is nonlinear, and the reduction rate of C_α is decreasing significantly with increase of OCR_P. Thus, excessive preloading beyond a certain point might not be an economic and reasonable choice to reinforce the post-construction performance of a soft foundation.

The stress ratio K has significant effect on creep rate under plain strain condition. It is clear that C_α is higher for K = 0.33 than that for K = 0.5. This result indicated that reduction of lateral confining pressure will cause an increase of total volumetric strain. Similar to C_α, β is larger for smaller value of K, i.e. 0.33. As mentioned above, smaller σ_x yields less lateral compression or more lateral extension, and thus a faster vertical creep rate.

In order to consider the effect of K, we plotted C_α as a function of K∙OCR_P for test 1 (NC clay), 2 and 3 (OC clay) in Figure 12. It was interestingly found that there is a clear correlation between C_α and K∙OCR_P which can be expressed by

(2) $C_{\alpha }=a(\mathrm{K}\cdot \mathrm{O}\mathrm{C}{\mathrm{R}}_{\mathrm{P}}{)}^b$(2)

Figure 12. (a) C_α and (b) β as a function of K∙OCR_P.

where a and b are constants, equal to 0.38 and −0.9, respectively, for Yuedong clay. By Eq. (2(2) $C_{\alpha }=a(\mathrm{K}\cdot \mathrm{O}\mathrm{C}{\mathrm{R}}_{\mathrm{P}}{)}^b$(2) ), the creep coefficient of volumetric strain can be approximated with known value of OCR_P and K. Note that Eq. (2)(2) $C_{\alpha }=a(\mathrm{K}\cdot \mathrm{O}\mathrm{C}{\mathrm{R}}_{\mathrm{P}}{)}^b$(2) is also validated for NC clay when OCR_p = 1.0.

The relationship between β and K∙OCR_P can be expressed as

(3) $\beta =m(\mathrm{K}\cdot \mathrm{O}\mathrm{C}{\mathrm{R}}_{\mathrm{P}}{)}^n$(3)

where m and n are constants, equal to 0.055 and −2.0, respectively. The variations of β are similar to C_α. Based on the analysed results, we can conclude that for the clay used in this study, OCR_P ≈ 2.0 is an economic value for preloading design under which the soil yields a relatively acceptable deformation after unloading.

4. Conclusions

In this study, laboratory creep tests were presented and the relationship between the creep rate and stress condition was investigated. Based on the tests results and theoretical study, the following conclusions can be drawn.

(1) The stress in zero strain direction σ_y for over-consolidated clay is higher than normal consolidated clay, and higher with larger degree of consolidation. The ratio of σ_y/σ_z is uniquely related with principal stress ratio (K), and the ratio is 0.4 and 0.34 for K = 0.5 and 0.33, respectively.

(2) The creep rate under one-dimensional k₀ condition is larger than those under plain strain condition. Stress ratio K has significant effect on creep rate, and smaller value of K will yield larger creep rate as a result of less lateral constraint.

(3) The creep rate of OC clay under plain strain condition can be predicted reasonably by the novel parameter K∙OCR_P, where OCR_P is the volumetric stress defined ratio of over-consolidation. OCR_P ≈ 2.0 was suggested for economic preloading design.

Disclosure statement

No potential conflict of interest was reported by the authors.

Data availability statement

The data that support the findings of this study are available from the corresponding author, Y. Zhou, upon reasonable request.

Additional information

Funding

Financial support for this investigation was provided by the National Natural Science Foundation of China (Grant No. 51908235) and the Innovation Funds Plan of Henan University of Technology (Grant No. 2020ZKCJ05).