J. Cent. South Univ. (2012) 19: 2047-2053 

DOI: 10.1007/s11771-012-1243-7

Attenuation-type and failure-type curve models on accumulated pore water pressure in saturated normal consolidated clay

ZHAO Chun-yan(赵春彦)

School of Civil Engineering, Central South University, Changsha 410075, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2012

Abstract: Based on dynamic triaxial test results of saturated soft clay, similarities of variations between accumulated pore water pressure and accumulated deformation were analyzed. The Parr’s equation on accumulated deformation was modified to create an attenuation-type curve model on accumulated pore water pressure in saturated normal consolidation clay. In this model, dynamic strength was introduced and a new parameter called equivalent dynamic stress level was added. Besides, based on comparative analysis on variations between failure-type and attenuation-type curves, a failure-type curve model was created on accumulated pore water pressure in saturated normal consolidation clay. Two models can take cycle number, coupling of static and dynamic deviator stress, and consolidation way into consideration. The models are verified by test results. The correlation coefficients are more than 0.98 for optimization of test results based on the two models, and there is good agreement between the optimized and test curves, which shows that the two models are suitable to predict variations of accumulated pore water pressure under different loading cases and consolidation ways. In order to improve prediction accuracy, it is suggested that loading cases and consolidation ways should be consistent with in-situ conditions when dynamic triaxial tests are used to determine the constants in the models.

Key words: saturated normal consolidation clay; equivalent dynamic stress level; accumulated pore water pressure model; attenuation-type curve; failure-type curve

1 Introduction

In railway and urban transit engineering, structures such as tunnels, pile foundations and subgrades bear not only static loads induced by deadweight of structures but also cyclic traffic loads. Traditional calculation methods for post-construction settlement usually only considering static loads and traffic loads were often treated as equivalent static loads so that mechanism of settlement induced by traffic loads was not considered. Dynamic stress induced by traffic loads changed around a certain standard stress, which was defined as static deviator stress induced by traffic loads. Then, the half amplitude of dynamic stress was defined as cyclic dynamic stress [1]. When a train passes by, the static deviator stress remains unchanged while the cyclic dynamic stress varies and makes the total stress possess cyclic property.

Previous studies [2-4] showed that the settlement of structures in high-speed railways was very large and some even exceeded limit value of safe running. Besides high-speed railways, settlement induced by traffic loads was also obvious in ordinary railways and highways with soft clay ground [5-7]. Thus, the long-term cyclic action of traffic loads deserves much attention when the post-construction settlement is calculated. It was shown by a lot of theoretical studies and test results [8-11] that the traffic loads would lead to excessive pore water pressure in saturated clay. Therefore, the reasonable models on accumulated pore water pressure in saturated clay are very important to calculate the post-construction settlement.

The pore water pressure in soils under cyclic dynamic loads has been widely studied. YASUHARA et al [12] held that variation of pore water pressure was not influenced by loading mode. Actually, the variations of pore water pressure under monotonic and cyclic dynamic loads couldn’t be expressed by the same equation [13]. RAMSAMOOJ and ALWASH [14] thought that the pore water pressure was only related to vertical stress and soil parameters; The model proposed by AZZOUZ et al [15] could only calculate the pore water pressure at the failure of soil while could not reflect variations of pore water pressure at different stages. Some other pore water pressure models were presented based on dynamic triaxial tests on saturated clays [16-18]. It was known from many test results that the accumulated pore water pressure was influenced by many factors, such as cycle number, dynamic stress level, static deviator stress level, and consolidation way. In this work, based on dynamic triaxial test results of saturated soft clay, attenuation-type and failure-type curve models were studied on accumulated pore water pressure in saturated normal consolidated clay.

2 Accumulated pore water pressure models

Reference [19] showed that critical dynamic stress existed under dynamic loads on soils. When the dynamic stress was less than the critical value, variation curves of accumulated pore water pressure showed an attenuation- type trend [3]; when the dynamic stress was larger than the critical value, a failure-type trend was shown [1].

2.1 Attenuation-type curve model

Dynamic triaxial tests for Shanghai silt clay (with soil parameters listed in Table 1) under different static deviator stress levels and cyclic dynamic stress levels were done by HUANG et al [3, 20]. The samples were consolidated under confining pressure for 48 h and then static deviator stress was applied under undrained condition. After that, cyclic dynamic stress was applied. Test results are shown in Fig. 1 and Fig. 2. Here, η_s=; η_d=; q_s, q_d and  are static deviator stress, dynamic deviator stress and effective confining pressure, respectively.

The above tests of saturated soft clay were under isotropic consolidation. However, practically, clay was often consolidated under anisotropic state. Thus, cyclic dynamic tests under K₀ consolidation were done in this work. Besides, as for shallow saturated soft clay (with soil parameters listed in Table 2), low confining pressure was applied. The soil sample was undisturbed with 39 mm in diameter and 80 mm in height. Anisotropic consolidation of K₀=0.71 was first done and then cyclic dynamic stress was applied. The variations of accumulated strain and accumulated pore water pressure are shown in Figs. 3-6. Here, p′₀ is vertical effective consolidation pressure and A is the amplitude of dynamic stress.

Table 1 Soil parameters

Fig. 1 Variations of accumulated strain (=200 kPa)

Fig. 2 Variations of accumulated pore water pressure (=200 kPa)

Table 2 Soil parameters

Fig. 3 Variations of accumulated strain (=30 kPa, =42 kPa)

From micromechanics point of view, mechanism of accumulated deformation and that of accumulated pore water pressure generated in saturated clay under cyclic dynamic loads were different. However, by comparing the three pairs in Figs. 1-6, it was seen that the accumulated strain and accumulated pore water pressure were influenced by the same factors and also showed similar variations. Therefore, it was thought that the accumulated pore water pressure model could be created based on the accumulated strain model.

Fig. 4 Variations of accumulated pore water pressure (=30 kPa, =42 kPa)

Fig. 5 Variations of accumulated strain (=41 kPa,    =58 kPa, A=15 kPa)

Fig. 6 Variations of accumulated pore water pressure    (=41 kPa, =58 kPa, A=15 kPa)

PARR [21] did dynamic triaxial tests on London clay, and the following equation on the accumulated strain was presented as

                       (1)

where is the strain rate at the N-th cycle;  is the strain rate at the first cycle; C and ζ are test constants; N is the cycle number of dynamic loads.

Based on the comparative analysis of above tests, a similar equation on the accumulated pore water pressure was presented referring to Eq. (1) as

                       (2)

where  is the pore water pressure rate at the N-th cycle;  is the pore water pressure rate at the first cycle; a and b are test constants.

Equation (2) can be transformed into

                                (3)

ZHOU [18] pointed out that the influence of frequency on the accumulated pore water pressure was very little when it was larger than 0.1 Hz. The frequency of traffic loads was usually larger than 0.1 Hz and belonged to long-term loads and its influence can be ignored. Thus, Eq. (3) can be simplified as

u_N=au₁N ^b                                     (4)

where u_N is the accumulated pore water pressure at the N-th cycle; u₁ is the pore water pressure at the first cycle.

Considering the strength attenuation of soil under cyclic dynamic loads, q_dult was introduced and dynamic stress level D_d was defined as

                                  (5)

where q_dult is dynamic strength of soil under undrained consolidation. Reference [19] showed that dynamic strength was commonly 0.6-0.7 times the static strength. Here, the dynamic strength was taken as

q_dult=0.6q_ult                                     (6)

where q_dult is the consolidated undrained strength of soil.

From Fig. 2, it was known that the accumulated pore water pressure increased with static deviator stress. For the two cases of η_s=0.5, η_d=0.2 and η_s=0.25, η_d=0.25 under the same cycle number, the former generated larger accumulated pore water pressure than the latter despite that the dynamic load was less. This was due to the larger static deviator stress in the former case. The same result was verified by WANG and CHEN [1]. Therefore, an equivalent coefficient k₂ was introduced and the static deviator stress level was taken as a part of the dynamic stress level. An equivalent dynamic stress level D^* was defined as

                 (7)

where D_s is static deviator stress level defined by the following equation:

D_s=q_s/q_ult                                   (8)

Based on the analysis of the present work and the test results, an equation of the pore water pressure at the first cycle was created as

                 (9)

where k is a correction coefficient and reflects the influence degree of the equivalent dynamic stress level on the pore water pressure at the first cycle; k₃ is a coefficient representing the influence of the equivalent dynamic stress level on the shape of variation curve of pore water pressure at the first cycle.

When Eq. (9) was substituted into Eq. (4) and b was replaced by k₄, u_N would be calculated as

       (10)

Supposing k₁=ka, Eq. (10) was transformed into

       (11)

where k₄ is a coefficient representing the influence of the cycle number on the shape of variation curve of the accumulated pore water pressure; k₁, k₂, k₃ and k₄ can be obtained by dynamic triaxial tests.

2.2 Failure-type curve model

The dynamic triaxial tests were done by WANG and CHEN [1] on the Xiaoshan soft clay (with soil parameters listed in Table 3) in Hangzhou city of China.

Table 3 Soil parameters

The soil samples were consolidated at the confining pressure and then axial static deviator stress was applied without being drained. Finally, the sinusoidal dynamic loads were applied. Test results are shown in Figs. 7  and 8.

By the comparison between attenuation-type curves in Figs. 2, 4, 6 and failure-type curves in Figs. 7, 8, it could be known that: 1) Compared with the attenuation-type curves, the failure-type curves had more obvious inflexions; 2) For the attenuation-type curves, the accumulated pore water pressure went stable with cycle number; while for the failure-type curves, the accumulated pore water pressure followed a trend of infinite growth when the cycle number reached a certain value. Based on the two points, Eq. (11) was modified as

(12)

where K, K*, , K₂ and K₃ are test constants reflecting the shape of test curve; u_T is the pore water pressure at the inflexion of accumulated pore water pressure curve.

Fig. 7 Variations of accumulated pore water pressure (=100 kPa, η_s=0)

Fig. 8 Variations of accumulated pore water pressure (=100 kPa, η_s=0.2)

Supposing  and , then the

failure-type curve model on the accumulated pore water pressure can be expressed as

     (13)

where K₁, K₂, K₃ and K₄ can be obtained by the dynamic triaxial tests.

3 Calculation of consolidated undrained strength

In order to determine the dynamic strength in Eqs. (11) and (13), the consolidated undrained strength q_ult must be figured out first. Study showed that the consolidated undrained strength varied due to different consolidation ways [22].

3.1 Calculation of q_ult under isotropic consolidation

According to triaxial tests, the following equations can be derived from Fig. 9 based on the Mohr-Column rules [22] as

               (14)

                               (15)

where τ_f is shear strength under undrained consolidation; p_f is the vertical pressure at failure.

Fig. 9 Stress circle and strength line under undrained consolidation

When Eq. (15) was substituted into Eq. (14), τ_f could be obtained. Then, the following equation could be derived as

         (16)

3.2 Calculation of q_ult under anisotropic consolidation

Supposing that the vertical effective consolidation pressure on soil sample was  and the static lateral pressure coefficient was K₀, the effective consolidation confining pressure  would be . The sample was consolidated with  and . The increments of two principal stresses at failure were noted as Δσ₁ and Δσ₃. Then, the effective principal stresses at failure based on Ref. [22] were as follows:

                  (17)

                   (18)

where A_f is the pore water pressure coefficient at  failure.

The following equation can be derived at failure based on the Mohr-Column rules as

                 (19)

where c′ is the effective cohesion; φ′ is the effective internal friction angle.

Since the following equation can be derived from Eq. (19) as

                    (20)

Base on Eqs. (17) and (18), the following equation can be derived:

                (21)

When Eq. (18) is substituted to Eq. (20), the following equation can be obtained:

  (22)

When Eq. (21) is substituted into Eq. (22), q_ult can be finally figured out by

       (23)

4 Verification of models and application

According to the characteristics of the test curves, the attenuation-type curve model was chosen to optimize the data in Figs. 2, 4 and 6, and the failure-type curve model was chosen to optimize the data in Figs. 7 and 8.

The objective function was expressed based on the least square method as follows:

                             (24)

where J is objective function; u_i and  mean the i-th optimized value and measured value of accumulated pore water pressure, respectively. u_i is a function of undetermined constants k₁, k₂, k₃, k₄ or K₁, K₂, K₃, K₄. So, J is also a function of the above undetermined constants. Optimization process of undetermined constants would make the objective function J minimum.

The advanced fitting tool of non-linear curve fit in the OriginPro software was applied. Equations (11) and (13) were embedded into the OriginPro by being programmed in Origin C language. In the OriginPro, initial values of undetermined constants were first supposed, then iterations were done on the function until J arrived at minimum. Then, the optimized values of the undermined constants could be determined.

Equations (16) and (23) were chosen to calculate q_ult  based on consolidation ways. It should be noted that anisotropic consolidation also generated static deviator stress same with soil strength formed, which should be considered in the total static deviator stress. Here, c′, φ′, K₀ and A_f were taken as 10 kPa, 12?, 0.71 and 1.5, respectively, based on test results. q_dult could be obtained from Eq. (6) when q_ult was solved.

The correlation coefficients in Figs. 2, 4, 6, 7 and 8 are 0.993 9, 0.993 3, 0.993 0, 0.980 3, 0.984 5 and the comparisons of optimized curves and test curves are shown in Figs. 10-14. It can be seen that the two models in Eqs. (11) and Eq. (13) both show good prediction on the variations of accumulated pore water pressure under different loading cases and consolidation ways.

The correlation coefficients are more than 0.98 for optimization of test results based on the two models, and there is good agreement between optimized and test curves, which shows that the two models are suitable to predict variations of accumulated pore water pressure under different loading cases and consolidation ways.

Fig. 10 Variations of optimized and measured accumulated pore water pressure (＝200 kPa)

Fig. 11 Variations of optimized and measured accumulated pore water pressure (=30 kPa, =42 kPa)

Fig. 12 Variations of optimized and measured accumulated pore water pressure (=41 kPa, =58 kPa,  A=15 kPa)

Fig. 13 Variations of optimized and measured accumulated pore water pressure (=100 kPa, η_s=0)

Fig. 14 Variations of optimized and measured accumulated pore water pressure (=100 kPa, η_s =0.2)

In practical engineering, dynamic loads started when the consolidation was still not completed; while the static deviator stress induced by traffic loads and the dynamic loads were applied at the same time [1]. Therefore, the consolidation ways and loading conditions should be consistent with the practice. Besides, the above mentioned accumulated pore water pressure models were mainly aimed at saturated normal consolidated clay while not applicable for over-consolidated and under-consolidated soils. For the practical application, the attenuation-type or failure-type curve model should be chosen based on the shape of the test curves to do optimization. When the constants were determined by optimization, the models would be adopted to make long-term prediction on the accumulated pore water pressure.

5 Conclusions

1) The attenuation-type and the failure-type curve models are presented for accumulated pore water pressure in saturated normal consolidated clay. Cycle number, coupling of static and dynamic deviator stress, and consolidation way are taken into consideration in the models.

2) The correlation coefficients are more than 0.98 for optimization on test results based on the two models, and there is good agreement between optimized and measured curves, which shows that the two models are suitable to predict variations of accumulated pore water pressure under different loading cases and consolidation ways.

3) For the application of the two models, the consolidation ways and loading cases should be consistent with in-situ conditions when dynamic triaxial tests are used to determine the constants in the models. The attenuation-type or failure-type curve model should be chosen based on the shape of the test curves to do optimization. When the constants are determined by optimization, the models will be able to be adopted to make long-term prediction on the accumulated pore water pressure.

4) The two models presented can be applied to the calculation and prediction on long-term settlement in high-speed railway, urban transit and highway.

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(Edited by YANG Bing)

Foundation item: Project(2009AA11Z101) supported by National High Technology Research and Development Program of China; Project supported by Postdoctoral Science Foundation of Central South University, China; Project(2012QNZT045) supported by Fundamental Research Funds for Central Universities of China; Project(2011CB710601) supported by the National Basic Research Program of China

Received date: 2011-04-11; Accepted date: 2011-07-29

Corresponding author: ZHAO Chun-yan, PhD; Tel: +86-15116437321; E-mail: zcyzbb@163.com