Dissipation testing is interpreted to yield a coefficient of consolidation, by comparison .... mation analysis, several Python files were programmed to establish the ...
Mahmoodzadeh, H. et al. (2014). Ge´otechnique 64, No. 8, 657–666 [http://dx.doi.org/10.1680/geot.14.P.011]
Numerical simulation of piezocone dissipation test in clays H . M A H M O O D Z A D E H , M . F. R A N D O L P H a n d D. WA N G
The piezocone is one of the most widely used in-situ tools with which to characterise the soil profile, distinguishing different types of sediment, and to provide quantitative data on the soil strength and consolidation characteristics. For fine-grained soils where penetration occurs under nominally undrained conditions, consolidation characteristics are obtained from dissipation tests conducted with the piezocone brought to rest. Dissipation testing is interpreted to yield a coefficient of consolidation, by comparison of the excess pore pressure decay with theoretical solutions. It is generally acknowledged that the coefficient of consolidation derived from in-situ dissipation tests will be greater, by a factor of perhaps 2 to 10, than values derived from oedometer tests on samples recovered from the corresponding depth. Apart from potential differences in horizontal and vertical permeability, differences arise from the influence of the rigidity index on the initial excess pore pressure field and the more complex stress paths undergone by soil elements surrounding the piezocone by comparison with an oedometer test. In order to advance the science underlying interpretation of dissipation testing, a careful parametric study has been undertaken using finite-element analysis, with the soil modelled as modified Cam Clay. The aim has been to derive an ‘operative’ coefficient of consolidation, linked to the consolidation and swelling parameters of the soil model, thus allowing quantitative treatment of the difference in coefficients of consolidation derived from laboratory oedometer or in-situ dissipation testing. The effect of partial consolidation during the penetration phase is also considered. KEYWORDS: clays; consolidation; finite-element modelling; penetrometers; silts; site investigation
INTRODUCTION The cone penetrometer is the most widely used onshore and offshore site investigation tool. A cone penetrometer equipped with a pore pressure transducer (i.e. a piezocone) is able to provide data on the consolidation characteristics of the soil, by measuring the generation and dissipation of excess pore pressure. Dissipation tests are usually conducted by halting the penetrometer at the desired depth and monitoring dissipation of the excess pore pressure generated during the penetration phase. Interpretation of the dissipation test is complicated by a number of factors related to the elasto-plastic response of soils. In particular, although it is widely accepted that coefficients of consolidation deduced from dissipation tests are significantly greater than those obtained from conventional laboratory oedometer tests, no framework has been developed linking field values of consolidation coefficient to the elasto-plastic compressional response of soil. Both the initial distribution of the excess pore pressure and subsequent dissipation are affected by factors such as the pre-yield (quasi ‘elastic’) and plastic stiffnesses of the soil, the degree of overconsolidation in situ, and the extent of any consolidation during penetration. Owing to the large deformations within the soil induced by cone penetration, a number of simplified decoupled approaches have been proposed to tackle the penetration and dissipation phases separately: (a) for nominally undrained penetration, the initial distribution of excess pore pressure may be predicted by analytical methods, such as cylindrical or spherical cavity expansion (Torstensson, 1977; Randolph & Wroth, 1979; Teh & Houlsby, 1991) or the more sophisticated strain path method
(Baligh & Laevadoux, 1980; Teh & Houlsby, 1991). (b) Dissipation of excess pore pressure is predicted using finiteelement (FE) or finite-difference approaches based on Biot’s theory or Terzaghi–Rendulic consolidation theory (Levadoux & Baligh, 1986; Teh & Houlsby, 1991). The shortcoming of the Terzaghi–Rendulic theory is that the total stresses during the dissipation phase are assumed to be constant. Recently, coupled effective stress FE approaches have been developed to reproduce the overall penetration and dissipation procedure (Chai et al., 2012; Yi et al., 2012). The theoretical advantage of the effective stress approaches over the decoupled approaches is that both the initial distribution and evolution of pore pressures during the dissipation phase are obtained by means of a constitutive model that can capture the effects of stress history, material state and elastoplastic deformation of soil. The reliability of the effective stress approach depends largely on the soil model. Some models employed in previous studies, such as the elasticperfectly plastic model with Drucker–Prager yield criterion in Yi et al. (2012), do not capture differences in soil compressibility during (plastic) compression and swelling. This paper presents results of coupled pore fluid diffusion and effective stress FE analyses, incorporating the modified Cam Clay (MCC) constitutive model, simulating cone penetration and dissipation testing in fine-grained soil. The purpose is to estimate an operative coefficient of consolidation from a piezocone dissipation test, and to relate the value to what would be obtained from a conventional oedometer or Rowe cell consolidation test. Traditional small strain FE methods cannot simulate extended cone penetration because of entanglement of soil elements around the cone. Therefore, a modified small strain (MSS) method proposed by Mahutka et al. (2006) and a large deformation finite-element (LDFE) approach based on frequent mesh regeneration were employed to overcome mesh distortion. Although the main focus of the paper is on the dissipation phase, the normalised penetration resistance profile is presented for the purpose of validation.
Manuscript received 20 January 2014; revised manuscript accepted 10 July 2014. Published online ahead of print 8 September 2014. Discussion on this paper closes on 1 January 2015, for further details see p. ii. Centre for Offshore Foundation Systems, the University of Western Australia, Crawley, Australia.
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The effects of rigidity index (Ir), Poisson ratio (), coefficient of earth pressure at rest (K0), overconsolidation ratio (OCR) and penetration rate on the dissipation response are studied. The results of the FE analyses are also compared with data from centrifuge tests performed on kaolin clay covering different penetration rates (Mahmoodzadeh & Randolph, 2014). An operative coefficient of consolidation for the interpretation of the dissipation test, differing from what would be obtained from an oedometer or Rowe cell test, is suggested. A normalised time based on the operative coefficient of consolidation is shown to unify dissipation curves following undrained penetration in normally consolidated clay. Guidance is provided for adjustments to the normalised dissipation graph to account for higher values of OCR and K0, and the potential errors noted in respect of partial consolidation during the penetration phase. FINITE-ELEMENT MODELLING The penetration of a cone and the following dissipation phase was investigated using MSS and LDFE analyses based on the commercial package Abaqus/standard. The soil in the axisymmetric problem was discretised with quadratic quadrilateral elements using reduced integration (CAX8RP in Abaqus). The penetrometer was idealised as an impermeable rigid body owing to its much greater stiffness than the soil. Interaction between the penetrometer and soil was modelled with smooth (zero interface friction) contact. A separate study showed that the cone–soil interface condition had only a very minor effect on the normalised pore pressure dissipation curves for times beyond 20% consolidation. The soil domain was extended at least 40 times the cone shaft diameter (D) in the horizontal and vertical directions, which was sufficient to avoid boundary effects during the dissipation stage. Roller constraints were specified on the base and sides of the soil domain. The excess pore pressure was allowed to dissipate at the soil surface, while the base and sides were taken as impermeable. The numerical simulations were conducted in terms of model piezocone tests in a centrifuge at an acceleration of 110g, rather than the typical 36 mm diameter piezocone used in the field, in order to make straightforward comparisons with the tests by Mahmoodzadeh & Randolph (2014). The soil sample comprised normally consolidated kaolin clay. The cone penetrometer had a shaft diameter of D ¼ 10 mm and a tip angle of 608. It incorporated a 1 .5 mm thick highdensity polypropylene filter element located 1 mm behind the cone shoulder (i.e. measuring u2). Cone penetration tests were performed at six different rates (0 .0045,1 mm/s) to explore undrained and partially drained responses of the soil. Dissipation tests, with the piezocone brought to rest, were performed at a model depth of 160 mm (depth of maximum area of the cone, i.e. the shoulder position). In the FE analyses, the penetrometers were pre-embedded to a depth of 10D and penetrated to a depth of 16D (160 mm equivalent to 17 .6 m prototype) prior to the dissipation phase. Based on a large number of trial analyses, the penetration distance of 6D was sufficient to achieve steady conditions in terms of both penetration resistance and the resulting excess pore pressure field. Note that although the simulated piezocone, with equivalent prototype diameter of 1 .1 m, is much larger than normal field piezocones, all results are interpreted in appropriate non-dimensional forms (Palmer, 2008) applicable to any size of cone or engineering parameters for the soil. MSS method Only limited penetration of the probe may be simulated in conventional small strain FE analysis due to the serious
mesh distortion in the zone ahead of the probe. To overcome this problem, an artificial rigid tube with a diameter of D/20 was attached to the cone tip (Fig. 1) with no vertical constraint. As for the cone, the interface between the tube and soil was taken as fully smooth and the tube surface as impermeable. This method was originally presented by Mahutka et al. (2006) and also adopted by Yi et al. (2012) for cone penetration. The slight modification in geometry provides essentially similar boundary conditions of zero horizontal displacement or pore fluid flow close to the axis of symmetry and avoids undesirable mesh distortion since the soil domain mesh can slide along the central tube and piezocone. LDFE procedure Although the MSS method has been used in research with simple soil models, its accuracy has not been tested rigorously. The main concern is whether the addition of the tube affects the dissipation response and excess pore pressure field, especially when an advanced soil model is incorporated. For validation purposes, LDFE analyses were also undertaken, following the so-called ‘remeshing and interpolation technique with small strain’ approach (Hu & Randolph, 1998; Wang et al., 2010). No artificial tube is assumed in the LDFE simulation; instead, the element entanglement around the cone is avoided by periodic mesh generation. The deformed soil geometry was remeshed after every cone displacement of 0 .1D; that is, a total number of 60 incremental steps for the cone penetration distance of 6D.
5 mm
30
°
0·25 mm
Fig. 1. Modified small strain FE model
NUMERICAL SIMULATION OF PIEZOCONE DISSIPATION TEST IN CLAYS Abaqus was employed to generate the mesh and conduct updated Lagrangian calculations at each incremental step. To facilitate automatic and continuous running of large deformation analysis, several Python files were programmed to establish the FE model and to extract key data from the result files. The detailed procedure of Abaqus-based LDFE analysis and mapping of field variables can be found in Wang et al. (2010, 2011, 2013). Material properties Although elastic-perfectly plastic models with Drucker– Prager yield criteria have been used in coupled FE analysis of piezocones (Yi et al., 2012), the MCC model is recognised to be more appropriate to quantify the consolidation behaviour of clay. The standard MCC model was incorporated into both the MSS and LDFE analyses. Note that although the standard MCC model has a circular shape in the deviatoric plane, Abaqus includes a feature that allows other shapes to be simulated. A separate study showed that reducing the ratio of the strengths in triaxial extension and compression had negligible effect on the normalised pore pressure dissipation curves. For the kaolin clay used in the centrifuge tests of Mahmoodzadeh & Randolph (2014), the MCC parameters are as listed in Table 1 (Stewart & Randolph, 1991). The coefficient of earth pressure at rest was expressed in terms of the friction angle, 9, and OCR as (Mayne & Kulhawy, 1982; Chang et al., 1999) K 0 ¼ (1 sin 9)OCRsin 9
(1)
Other parameters, such as the initial void ratio (e0) and profile of undrained shear strength for triaxial compression conditions, were obtained from standard relationships for MCC (Wroth, 1984; Wood, 1990; Chang et al., 1999). Both the cone penetration resistance and the dissipation response depend on the rigidity index, Ir ¼ G/su, where G is the elastic shear modulus. The latter quantity may be expressed as G¼
3(1 2) p9(1 þ e) 2(1 þ ) k
(2)
where e is the current void ratio and p9 is the mean effective stress. For convenience, relevant parameters deduced from the MCC model, and values at the relevant depth of 17 .6 m, are appended to Table 1.
Determination of soil permeability Dissipation of the excess pore pressure around the piezocone is governed by the consolidation characteristics of the Table 1. Characteristics of the kaolin clay Properties (after Stewart & Randolph, 1991) Angle of internal friction, 9 Void ratio at p9 ¼ 1 kPa on virgin consolidated line, eN Slope of normal consolidation line, º Slope of swelling line, k Plastic compression ratio, ¸ ¼ 1 – k/º Poisson ratio, Submerged density
Value 238 2 .252 0 .205 0 .044 0 .79 0 .3 630 kg/m3
Deduced MCC values at depth of 17 .6 m (dissipation depth) Vertical effective stress, v9 Undrained shear strength, su, at OCR ¼ 1 Rigidity index, G/su, at OCR ¼ 1
109 kPa 26 .4 kPa 75
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soil, specifically the permeability and compressibility. Permeability is a function of the void ratio, which suggests that the permeability at a specified position is not constant, especially when the soil undergoes significant compression or swelling. In this study, the permeability was deduced from Rowe cell tests conducted over a range of effective stresses, using the relationship k ¼ ªw cv =E0
(3)
where ªw is unit weight of porous water, E0 is the onedimensional (1D) modulus and cv is the coefficient of consolidation deduced from the Rowe cell test (1D conditions with compression and pore water flow in the vertical direction). The compressibility of the soil depends on the strain path. Since the soil in the Rowe cell moves along a path parallel to the isotropic virgin consolidation line (VCL) in e–lnp9 space, the compressibility is related to º as 1 º ¼ E0 (1 þ e) v9
(4)
where v9 represents the vertical effective stress level. The Rowe cell testing data measured by Richardson (2007) for kaolin clay were fitted with pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (5) cv ¼ 1 þ 0.14 v9 =31.54 with units of cv and v9 being mm2/s and kPa, respectively. The profile is very close to previous correlations (e.g. House et al., 2001; Randolph & Hope, 2004). For the soil under a K0 state, the relationship between void ratio and vertical effective stress is 1 þ 2K 0 (6) e ¼ (eN C) º ln v9 3 where C is the distance between the VCL and normal consolidation line (NCL), C ¼ 0 .048 based on the parameters listed in Table 1. Substituting equations (4)–(6) into equation (3), the permeability is written as a function of the void ratio k¼
ªw º(1 þ 2K 0 ) 94.61(1 þ e) exp [(eN C e)=º] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0.42 exp [(eN C e)=º] 3 1þ 1 þ 2K 0
(7)
Note the units of permeability in the equation are mm/s. VERIFICATION Comparison of penetration resistance with previous methods To verify the MSS and LDFE approaches, the computed normalised penetration resistances were compared with those obtained from analytical and numerical methods within the framework of total stress analysis (i.e. in terms of a cone factor, Nkt, the ratio of net cone resistance, qnet, to undrained shear strength, su). The OCR was taken as 1 and K0 ¼ 0 .61 by equation (1). As listed in Table 1, Poisson ratio was constant in the soil; therefore the rigidity index varied with depth. In the numerical analyses the piezocone was penetrated at a velocity of v ¼ 1 mm/s to simulate penetration under undrained conditions. The empirical criterion for undrained conditions is vD/cv . 30 (Finnie & Randolph, 1994; Randolph & Hope, 2004) with cv being the coefficient of consolidation obtained from an oedometer or Rowe cell test at the given effective stress level. This criterion is
MAHMOODZADEH, RANDOLPH AND WANG
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satisfied over the relevant depth (and hence geostatic vertical effective stress and resulting cv) range. In Fig. 2, the cone factors, Nkt, from previous studies are presented for the rigidity index corresponding to the final depth of penetration of 16D (Ir ¼ 75). There is excellent agreement between the resistance–displacement curves of the LDFE and MSS methods, with a cone factor of 8 .78 approached at a displacement of 5D. The computed cone factor is close to 8 .82 as predicted by Lu et al. (2004) and 8 .57 from Abu-Farsakh et al. (2003), but 10 .2% higher than the fitting equation by Walker & Yu (2006). Comparison of dissipation graph with centrifuge tests Figure 3 compares the numerical dissipation graphs and the centrifuge test data with a penetration rate of 1 mm/s (Mahmoodzadeh & Randolph, 2014). The penetration rate was sufficiently fast to ensure undrained conditions. The soil properties used in both the MSS and LDFE analyses are listed in Table 1. The numerical analyses show a small rise in excess pore pressure initially, with a maximum at a time of about 5 s. By comparison, the centrifuge data show a
much more subtle initial rise, although the actual maximum also occurs after a few seconds’ delay. The form of curve, with an initial rise in pore pressure, is common in the field, particularly as the OCR increases (Burns & Mayne, 1998; Chai et al., 2012). In Fig. 3(b), dissipation times have been normalised as T ¼ cvt/D2, based on a vertical coefficient of consolidation of cv ¼ 0 .128 mm2/s estimated from the Rowe cell data for the vertical effective stress (109 kPa) at the final penetration depth of the cone and the cone diameter, D. In this figure, and similar figures from the parametric study presented later, the excess pore pressure, ˜u, has been normalised as ˜u/˜umax, where ˜umax is an equivalent (backward extrapolated) initial or maximum excess pore pressure using the root time approach described by Sully et al. (1999). The excess pore pressure is first plotted against the square root of time since the end of penetration, similar to Taylor’s method for interpreting t50 values from a 1D consolidation test. The dissipation graph, following some initial redistribution, is usually a straight line and the initial excess pore pressure, ˜umax, is defined by back-extrapolation of the line to t ¼ 0 (Mahmoodzadeh & Randolph, 2014). Overall, good agreement was achieved between both numerical methods and the centrifuge test data.
10 9
Cone factor, Nkt
8 7 6 5
MSS LDFE Lu et al. (2004) Abu-Farsakh et al. (2003) Walker & Yu (2006)
4 3 2 1 0
0
4 2 3 Normalised penetration distance, w/D
1
5
6
Fig. 2. Normalised penetration resistance for smooth cone
Excess pore pressure: kPa
200 180 160
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140
LDFE
120
Centrifuge
100 80 60 40 20
Normalised excess pore pressure: kPa
0 0·1
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10
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1 (1 þ ) 1 ¼ E0 3(1 ) K iso
30 20 10 0 0·1
ESTIMATION OF COEFFICIENT OF CONSOLIDATION IN NORMALLY CONSOLIDATED CLAY More parallel calculations were performed using the MSS and LDFE approaches separately, with variations of rigidity index, OCR and K0. Similar to the results in Fig. 2 and Fig. 3, both approaches predicted very similar results in respect of penetration resistance profiles and dissipation graphs, highlighting the reliability of the MSS approach. To save computational effort, only the MSS approach was used for the remaining simulations discussed here. The soil properties were maintained as in Table 1 unless otherwise stated. The factors affecting the dissipation response, and consequently the coefficient of consolidation, include: the rigidity index, initial stress state and plastic deformation behaviour. These factors may have combined impacts, which lead to difficulty in separating each effect. A normalised dissipation time capturing all possible factors is introduced for normally consolidated clay (i.e. OCR ¼ 1), in which an operative coefficient of consolidation rather than cv from the Rowe cell data is considered. The operative coefficient of consolidation is estimated by fitting the dissipation test result to the proposed normalised dissipation graph. In addition, the permeability can then be estimated from the operative coefficient of consolidation using an appropriate value for the soil compressibility. The radial distribution of excess pore pressure created by piezocone penetration, and in particular the subsequent dissipation process, may be estimated with surprising accuracy by assuming simple radial displacement of the soil and flow of pore water (Randolph, 2003). Therefore, the operative coefficient of consolidation obtained from the dissipation test is essentially that associated with 1D radial consolidation. If the soil was regarded as an elastic material, the soil (1D) compressibility would be written as (Davis & Poulos, 1968)
1
10 100 Normalised time, T ⫽ cvt/D2 (b)
1000
10 000
Fig. 3. Dissipation graph from the numerical modelling and the centrifuge test
(8)
where Kiso is the elastic bulk modulus for isotropic compression. For the MCC model (for elastic conditions, inside the yield envelope), Kiso may be related to the swelling parameter, k, and the mean effective stress, so that the operative coefficient of consolidation is
NUMERICAL SIMULATION OF PIEZOCONE DISSIPATION TEST IN CLAYS kE0 3(1 ) (1 þ e0 )p9 k ¼ 1þ k ªw ªw
(9)
A subscript, h, is used to distinguish this from the value obtained from a conventional oedometer test. In certain soils it may be necessary to account for anisotropic permeability (with kh used for ch, and kv for the interpretation of oedometer tests), although only isotropic permeability is considered here. In reality the dissipation process is accompanied by a combination of plastic deformation near the piezocone, and unloading further away (Fahey & Lee Goh, 1995). Following a large amount of trial computations, the operative coefficient of consolidation that leads to a unified normalised dissipation graph is modified as 3(1 ) (1 þ e0 )p9 k 1 þ kÆ º(1Æ) ªw
(10)
where Æ is a weighting factor defining the contribution of elastic and plastic behaviour, with a default value of 0 .5 (see later justification). The normalised dissipation time is related to the operative coefficient of consolidation as proposed by Teh & Houlsby (1991), although here defined in terms of the diameter, D, rather than the radius of the piezocone, so ch t (11) T ¼ 2 pffiffiffiffi D Ir
The effect of the rigidity index The initial distribution of the excess pore pressure around the cone is one of the key factors controlling the dissipation time. The size of the zone in which the excess pore pressure is generated during penetration, and thus the scale of the drainage path, increases with the rigidity index (Teh & Houlsby, 1991). With other factors unchanged, a higher rigidity index causes longer dissipation time, as indicated in equation (11). In the MCC model, the rigidity index of soil may be changed by adjusting Poisson ratio. In the following MSS analyses, OCR was taken as 1 and Poisson ratio was changed from 0 .2 to 0 .45, covering a range of rigidity index from 122 to 17 (see equation (2)) for the dissipation depth of 16D with D ¼ 10 mm and acceleration of 110g, so equivalent depth of 17 .6 m. The value of the rigidity index is in terms of the geostatic stresses prior to insertion of the cone at the dissipation depth of 16D. Figure 4 compares the distribution of excess pore pressure at the beginning of the dissipation phase for Ir ¼ 17 and 122. The outer contour represents an excess pore pressure of 10% of the maximum excess pore pressure generated, which is considered as representative of the size of the pore pressure Δu/Δumax
0·9 0·9 0·1 0·2 0·3 0·4 0·5 0·4 0·8 0·6 0·5 0·6 0·7 0·7 0·8
Ir ⫽ 17
0·3
0·2
0·1
Ir ⫽ 122
Fig. 4. Distribution of excess pore pressures at the beginning of dissipation
Effect of º and k The operative coefficient of consolidation in equation (10) is a function of both k and º. In contrast, the vertical coefficient of consolidation from a Rowe cell test, cv , depends on º only, as indicated in equation (4). To evaluate the effect of º and k on the dissipation curve, either k or º was kept constant and the other parameter changed, with OCR ¼ 1. Although the permeability deduced from an oedometer test varies with soil compressibility and hence º, for this parametric study a constant permeability equal to the value estimated at the dissiaption depth, k ¼ 1 .13 3 107 m/s, was assumed within the whole soil region to avoid combined effects of permeability and º. Figure 6(a) shows the dissipation responses at the u2 position when k ¼ 0 .044 and º changes from 0 .088 to 0 .352 (k/º ranging from 0 .5 to 0 .125). Variation of º also changes the rigidity index through changing the initial void ratio (equation (2)), which is taken into account when normalising the dissipation time by equation (11). To achieve a normalised dissipation graph, the weighting factor in equation (10) was selected as Æ ¼ 0 .5. The resulting normalised dissipation graph is presented in Fig. 6(b). The contributions of k and º 200
Excess pore pressure: kPa
ch ¼
zone. Fig. 5(a) shows the dissipation responses at the u2 position for values of rigidity index from 17 to 122. The increase in Poisson ratio results in a reduction of the size of the excess pore pressure zone generated, which tends to reduce the dissipation time. On the other hand, the ratio E0 / Kiso from equation (8) decreases with increasing Poisson ratio, causing a reduction of the coefficient of consolidation for a given mean effective stress (equation (9)). Solutions based on 1D radial dissipation of the excess pore pressure (Randolph & Wroth, 1979) seem to agree well with the results of the dissipation tests. If the dissipation time is normalised based on equation (11), with the radial operative coefficient of consolidation determined through equation (10), the normalised dissipation graphs for different rigidity indices converge to a unified curve, as shown in Fig. 5.
180 160 140 120 100 80 60 40 20 0 0·1
Normalised excess pore pressure: kPa
ch ¼
661
Ir ⫽ 17 Ir ⫽ 35 Ir ⫽ 75 Ir ⫽ 122 1
10 Time, t (s) (a)
100
1000
0·01 0·1 Normalised time, T* ⫽ cht/(D2冪Ir) (b)
1
100 90 80 70 60 50 40 30 20 10 0 0·001
Ir ⫽ 17 Ir ⫽ 35 Ir ⫽ 75 Ir ⫽ 122
Fig. 5. Dissipation response at different rigidity indices
MAHMOODZADEH, RANDOLPH AND WANG 180
160
160
140 120 100 80 60 40 20 0 0·1
Normalised excess pore pressure: kPa
Excess pore pressure: kPa
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k ⫽ 0·044 k/λ ⫽ 0·125 k/λ ⫽ 0·215 k/λ ⫽ 0·5 1
10 Time, t: s (a)
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k ⫽ 0·044 k/λ ⫽ 0·125 k/λ ⫽ 0·215 k/λ ⫽ 0·5 0·01 0·1 Normalised time, T* ⫽ cht/(D2冪Ir) (b)
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140 120 100 80
λ ⫽ 0·205
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Excess pore pressure: kPa
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100 90 80 70 60 50 40 30 20 10 0 0·001
λ ⫽ 0·205
k/λ ⫽ 0·215 k/λ ⫽ 0·33 k/λ ⫽ 0·5 0·01 0·1 Normalised time, T* ⫽ cht/(D2冪Ir) (b)
1
Fig. 6. Variation of dissipation response with º
Fig. 7. Variation of dissipation response with k
on the dissipation process, and thus the appropriate value of Æ, can be interpreted from the consolidation path in e–lnp9 space; this will be discussed later in the context of different OCRs. Similar analyses were performed with º ¼ 0 .205 and k varing from 0 .044 to 0 .1025 (k/º ranging from 0 .21 to 0 .5). The change of k will also cause variation of the rigidity index. Based on a weighting factor Æ ¼ 0 .5 in equation (10), the dimensional dissipation graphs in Fig. 7(a) are normalised to a unique curve in Fig. 7(b). The value of ch deduced from a dissipation response is obtained by varying the value to match the normalised graph in Fig. 6(b) or Fig. 7(b). The soil permeability where the dissipation test is performed is then available through equation (10). In an oedometer test, the relationship between the void ratio and the mean effective stress is parallel to the VCL with the slope of º. Therefore, the operative coefficient of consolidation is related to cv from an oedeometer test as Æ 3(1 ) º cv (12) ch ¼ (1 þ ) k
The rigidity index is affected by K0, ranging between 63 and 92 as K0 is increased from 0 .5 to 1. In turn, the value of Ir is incorporated in the normalised dissipation time – see equation (11). An increase in K0 leads to higher initial pore pressures at the end of penetration, as shown in Fig. 8(a). The normalised dissipation responses for K0 ¼ 0 .61 to 1 match each other in Fig. 8(b), whereas the response for K0 ¼ 0 .5 plots slightly lower than the others. Since most normally consolidated clays have K0 . 0 .5, the value of K0 (for OCR ¼ 1) has minimal influence on the normalised dissipation response.
For the kaolin clay with characteristics shown in Table 1 and OCR ¼ 1, the operative coefficient of consolidation is about 3 .5 times the coefficient of consolidation from a Rowe cell test. Effect of the initial stress anisotropy The initial stress anisotropy affects the cone penetration resistance and initial distribution of excess pore pressure around the cone. The coefficient of earth pressure is a function of the friction angle of the soil and OCR as shown in equation (1). However, in order to separate the effect of different factors, K0 was varied artificially in the range 0 .5–1 in this section, maintaining OCR ¼ 1. All other material properties, including the friction angle, remained as in Table 1 and the soil permeability was fixed at k ¼ 1 .13 3 107 m/s.
OTHER PARAMETERS AFFECTING THE DISSIPATION GRAPH Effect of overconsolidation ratio Theoretically, the excess pore pressure generated around a cone comprises two components: one caused by the octahedral normal stress that results from outward displacement of the soil and the other associated with the shear deformation of the soil adjacent to the cone body. The octahedral component of pore pressure increases (Torstensson, 1977; Wroth & Houlsby, 1985) and the shear component at the u2 position decreases with increasing OCR (Mayne & Bachus, 1988). In order to quantify the effects of OCR, a series of analyses were conducted for OCR between 1 and 5. To generate a constant OCR throughout the depth, the soil was assumed weightless and a pressure loading equal to the vertical effective stress (109 kPa) at the dissipation depth for normally consolidated clay was applied on the soil surface. The OCR was varied by changing the size of the initial yield surface, and the horizontal effective stress was adjusted to give K0 consistent with equation (1). The resulting rigidity index reduced from 75 for OCR ¼ 1 to 29 for OCR ¼ 5. The permeability was kept as 1 .13 3 107 m/s. The combined effects of OCR and rigidity index caused a significant increase of the initial excess pore pressure as
Normalised excess pore pressure: kPa
Excess pore pressure: kPa
NUMERICAL SIMULATION OF PIEZOCONE DISSIPATION TEST IN CLAYS 220 200 180 160 140 120 100 80 60 40 20 0 0·1
K0 ⫽ 0·5 K0 ⫽ 0·61 K0 ⫽ 0·8 K0 ⫽ 1 1
10 Time, t: s (a)
100
1000
0·01 0·1 Normalised time, T* ⫽ cht/(D2冪Ir) (b)
1
100 90 80 70 60 50 40 30 20 10 0 0·001
K0 ⫽ 0·5 K0 ⫽ 0·61 K0 ⫽ 0·8 K0 ⫽ 1
Fig. 8. Effect of stress anisotropy on dissipation response
220 200 180 160 140 120 100 80 60 40 20 0 0·1
OCR ⫽ 3 OCR ⫽ 2 OCR ⫽ 1
OCR ⫽ 5
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10 Time, t: s (a)
100
significantly during the combined penetration and dissipation stages at high OCR than for normally consolidated clay. Fig. 10 shows the paths in e–p9 space at different OCRs for a point located at ,0 .25D horizontal distance from the (final) u2 position. For a given vertical effective stress, increasing OCR from 1 to 5 leads to higher mean effective stress (by a factor close to 5 during the dissipation phase) and lower void ratio (with specific volume decreasing by about 15 %). Therefore, based on the change of p9 and e dissipation should occur around four times as fast for OCR ¼ 5 compared with OCR ¼ 1, which is consistent with Fig. 9(a). In Fig. 10, during dissipation the path in e–p9 space lies closely parallel to the unloading–reloading line (slope of k) initially, gradually steepening towards the VCL (slope of º) near the end of dissipation, particularly for low values of OCR. Essentially, the weighting factor Æ in equation (10) defines the contributions of k (volumetric strains within the yield envelope) and º (volumetric strains expanding the yield envelope). If all stresses were to fall inside the yield surface throughout the dissipation, Æ would be unity; on the other hand, Æ should reduce to zero if the yield surface is expanding, as indicated by the consolidation paths in Fig. 10. For relevant normally and lightly overconsolidated clays, the evolution of the stress state whithin the region of generated excess pore pressure will encompass both responses, suggesting that Æ should lie between zero and one. For the normally consolidated clay highlighted in Fig. 6 and Fig. 7, Æ ¼ 0 .5 seems to unify the dissipation graphs. This value of Æ also works well for higher OCR, as shown below, provided some adjustment is made to the mean effective stress, p9, in equation (10). Figure 9(b) showed that higher OCR leads to faster normalised dissipation response, suggesting that the coefficient of consolidation has been underestimated. This is consistent with the average mean effective stress during the dissipation phase being much greater than the in-situ value for high OCR clays (see Fig. 10). A simple approach is to replace the in-situ mean effective stress by an average between the CSL and VCL lines or, to sufficient accuracy, the value on the NCL at the initial void ratio. This simplification avoids any correction for normally consolidated conditions, essentially replacing p9 in equation (10) by p9(OCR)¸. Fig. 11 shows the normalised dissipation graphs after correcting the operative coefficient of consolidation in this way. All dissipation responses now join a single curve for normalised times greater than about 0 .01. Correction of the operative mean effective stress, and hence the coefficient of consolidation, during the dissipation phase is particularly important if the hydraulic conductivity of the soil is to be estimated from the dissipation response.
1000
100
1·5
90 70 60 50 40
OCR ⫽ 1, 2, 3 and 5
30 20
OCR ⫽ 2
1·3 1·2
Start of dissipation End of dissipation
OCR ⫽ 3 OCR ⫽ 5
1·1 1·0
10 0 0·0001
OCR ⫽ 1
1·4
80
Void ratio, e
Normalised excess pore pressure: kPa
Excess pore pressure: kPa
OCR increased from 1 to 3. However, with further increase in OCR to 5, the rise in octahedral excess pore pressure was essentially compensated by reduction of the shear induced excess pore pressure at the u2 position (Fig. 9(a)). The normalised dissipation time in Fig. 9(b) was calculated for ch based on the initial void ratio and mean effective stress, just as for the case of normally consolidated clay. However, the mean effective stress changes much more
663
Start of penetration CSL
0·001 0·01 0·1 Normalised time, T* ⫽ cht/(D2冪Ir) (b)
Fig. 9. Effect of OCR at u2 position
1
0·9 10
100 Mean effective stress, p⬘: kPa
NCL
VCL
1000
Fig. 10. Paths in e–p9 space during penetration and dissipation at various OCRs
MAHMOODZADEH, RANDOLPH AND WANG 100 90 80 70 60
OCR ⫽ 1, 2, 3 and 5
50 40 30 20 10
0 0·0001
0·001 0·01 0·1 Normalised time, T* ⫽ cht/(D2冪Ir)
1
Fig. 11. Normalised dissipation graphs after effective stress correction
The effect of partial drainage during penetration The soil conditions during a standard cone penetration test are usually undrained in clays, allowing dissipation tests to be interpreted as discussed. However, the coefficients of consolidation for silts or sandy silts will typically exceed 1000 m2/year (3 .2 3 105 m2/s), resulting in partial drainage during penetration. The drainage condition during cone penetration depends on the normalised velocity V¼
vD cv
(13)
100 90 80
v ⫽ 0·15 mm/s Centrifuge MSS
70 60 50 40 30 20 10 0 0·001
0·01 0·1 Normalised time, T* ⫽ cht/(D2冪Ir)
1
100 90
v ⫽ 0·015 mm/s
80
Centrifuge MSS
70 60 50 40 30 20 10 0 0·001
0·01 0·1 Normalised time, T* ⫽ cht/(D2冪Ir)
CONCLUSIONS The link between coefficients of consolidation measured in field piezocone dissipation tests and in laboratory oedometer tests is complicated by the elasto-plastic response of soil. The aim in this study has been to quantify that link, within a relatively simple constitutive model for the soil, exploring how different parameters affect the link. The study leads to a quantitative framework, allowing insight into relevant values of consolidation coefficient for wider application, such as moving boundary problems (including penetrometer testing) where the degree of consolidation is affected by the rate of movement. Coupled FE analyses were performed to study the dissipation test following piezocone penetration. The soil was modelled using modified Cam Clay (MCC) with the cone surface considered to be smooth. Large deformation FE and modified small strain (MSS) methods were used to simulate the penetration and dissipation phases. The results of this study were verified by comparison with previous studies and centrifuge tests. It was shown that the simpler MSS approach could represent the penetration of the cone and the following dissipation test with good accuracy.
Normalised excess pore pressure: kPa
Normalised excess pore pressure: kPa
Normalised excess pore pressure: kPa
Normalised velocities of 0 .1 and 30 were suggested to define the limits of drained and undrained regions, based on the penetration responses of shallow circular foundations (Finnie & Randolph, 1994), with the limits broadly confirmed for piezocone tests (Randolph & Hope, 2004). The effects of partial consolidation can be explored by either increasing the coefficient of consolidation or reducing the rate of penetration. Mahmoodzadeh & Randolph (2014) conducted a series of centrifuge tests in which the drainage conditions were varied from fully undrained to drained by varying the penetration rate from 1 to 0 .0045 mm/s.
The tests were simulated numerically by varying the penetration rate in the MSS FE analyses. Fig. 12 compares the normalised dissipation graphs from the MSS analyses and centrifuge tests at four penetration rates. Good agreement was achieved, suggesting that the MSS approach has the potential to capture the penetration and dissipation processes under partial drained penetration. The numerical dissipation responses at different penetration rates are presented in Fig. 13. From Fig. 13(b), the normalised dissipation time with v ¼ 0 .0045 mm/s is about three times larger than that after undrained penetration. If no account is taken of partial consolidation for soils where vD/cv is less than about unity (corresponding here to a velocity of v , 0 .1 mm/s) the coefficient of consolidation and permeability will tend to be underestimated.
Normalised excess pore pressure: kPa
Normalised excess pore pressure: kPa
664
1
100 90
v ⫽ 0·045 mm/s
80
Centrifuge MSS
70 60 50 40 30 20 10 0 0·001
0·01 0·1 Normalised time, T* ⫽ cht/(D2冪Ir)
100
1
v ⫽ 0·0045 mm/s
90
Centrifuge
80
MSS
70 60 50 40 30 20 10 0 0·001
0·01 0·1 Normalised time, T* ⫽ cht/(D2冪Ir)
Fig. 12. Comparison between normalised dissipation graphs from centrifuge tests and MSS
1
NUMERICAL SIMULATION OF PIEZOCONE DISSIPATION TEST IN CLAYS Excess pore pressure: kPa
180 v ⫽ 1 mm/s
160 140
v ⫽ 0·15 mm/s
120 100
v ⫽ 0·045 mm/s
80 60 40
v ⫽ 0·015 mm/s
20
v ⫽ 0·0045 mm/s
0
1
10
100
1000
Normalised excess pore pressure: kPa
Time, t: s (a)
665
ACKNOWLEDGEMENTS This work forms part of the activities of the Centre for Offshore Foundation Systems (COFS) at the University of Western Australia, currently supported as a node of the Australian Research Council Centre of Excellence for Geotechnical Science and Engineering, and by the Lloyd’s Register Foundation. This project has received additional support from the Australia–China Natural Gas Technological Partnership Fund. The first author is grateful for support from an International Postgraduate Research Scholarship and University Postgraduate Award from the University of Western Australia and Benthic Geotech PhD Scholarship from Benthic Geotech Pty Ltd.
100
NOTATION
90
C
80 70 60 50 40
v ⫽ 0·0045, 0·015, 0·045, 0·15 and 1 mm/s
30 20 10 0 0·001
0·01 0·1 Normalised time, T* ⫽ cht/(D2冪Ir) (b)
1
Fig. 13. Penetration rate effects from model piezocone tests in normally consolidated clay
The source of the difference between the consolidation coefficient cv measured in an oedometer or Rowe cell test and the value deduced from a piezocone dissipation test was explored through a detailed parametric study of the input MCC parameters. An ‘operative’ coefficient of consolidation was introduced based on the factors affecting the compressibility of the soil. The effect of the rigidity index on the dissipation response was studied by varying Poisson ratio in the model. The results confirmed the time normalisation proposed by Teh & Houlsby (1991), incorporating the square root of the rigidity index. The relative effects of ‘elastic’ volumetric strains within the MCC yield envelope, and ‘plastic’ strains during expansion of the yield envelope, were explored by varying k and º. This resulted in a recommendation to adopt a consolidation coefficient based on the geometric mean of the elastic and plastic 1D compressibility. The normalised dissipation response was found to shift to the left with increasing OCR, suggesting that the operative consolidation coefficient should be increased. Most of this change is attributable to an increase of the mean effective stress during penetration (and following dissipation) at high OCR, because of the negative shear-induced component of excess pore pressure. A practical solution to consider this effect was presented, basing the operational consolidation coefficient on the mean effective stress on the normal consolidation line, rather than the in-situ value. Finally, it was shown that the coupled FE analysis accurately captured the effects of partial consolidation during penetration on the subsequent dissipation response. Similarly to the results from centrifuge tests (Mahmoodzadeh & Randolph, 2014), the numerical modelling also showed that not considering the partial consolidation effect for soils with normalised velocity, vD/cv, less than unity will result in underestimation of the coefficient of consolidation and permeability.
ch cv D E0 e eN G Ir K Kiso K0 M Nkt p9 q su T t V v Æ ªw ˜u ˜umax k ¸ º v9 9
distance between virgin consolidation line and normal consolidation line operative coefficient of consolidation coefficient of consolidation cone diameter one-dimensional modulus void ratio void ratio at p9 ¼ 1 kPa on virgin consolidated line elastic shear modulus rigidity index soil permeability elastic bulk modulus for isotropic compression coefficient of earth pressure at rest slope of the critical state line in the q–p9 plane capacity factor of cone mean effective stress deviatoric stress undrained shear strength normalised dissipation time consolidation time normalised velocity penetration velocity of cone weighting factor unit weight of water excess pore pressure equivalent (backward extrapolated) initial or maximum excess pore pressure slope of swelling line ¸ plastic compression ratio slope of normal consolidation line Poisson ratio vertical effective stress internal friction angle
REFERENCES Abu-Farsakh, M., Tumay, M. & Voyiadjis, G. (2003). Numerical parametric study of piezocone penetration test in clays. Int. J. Geomech. 3, No. 2, 170–181. Baligh, M. M. & Laevadoux, J.-N. (1980). Pore pressure dissipation after cone penetration, Research Report R80-11. Cambridge, MA, USA: Department of Civil Engineering, MIT. Burns, S. E. & Mayne, P. W. (1998). Monotonic and dilatory porepressure decay during piezocone tests in clay. Can. Geotech. J 35, No. 6, 1063–1073. Chai, J., Sheng, D., Carter, J. P. & Zhu, H. (2012). Coefficient of consolidation from non-standard piezocone dissipation curves. Comput. Geotech. 41, 13–22. Chang, M.-F., Teh, C. I. & Cao, L. (1999). Critical state strength parameters of saturated clays from the modified Cam clay model. Can. Geotech J. 36, No. 5, 876–890. Davis, E. H. & Poulos, H. G. (1968). The use of elastic theory for settlement prediction under three-dimensional conditions. Ge´otechnique 18, No. 1, 67–91, http://dx.doi.org/10.1680/geot.1968. 18.1.67. Fahey, M. & Lee Goh, A. (1995). A comparison of pressuremeter and piezocone methods of determining the coefficient of consolidation. In The pressuremeter and its new avenues (ed. G. Ballivy), pp. 153–160. Rotterdam, the Netherlands: Balkema.
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Finnie, M. S. & Randolph, M. F. (1994). Punch-through and liquefaction induced failure of shallow foundations on calcareous sediments. Proceedings of the international conference on behaviour of offshore structures, BOSS ’94, Boston, USA, pp. 217–230. Oxford, UK: Pergamon Press. House, A. R., Oliveira, J. R. M. S. & Randolph, M. F. (2001). Evaluating the coefficient of consolidation using penetration tests. International Journal of Physical Modelling in Geotechnics 1, No. 3, 17–25. Hu, Y. & Randolph, M. F. (1998). A practical numerical approach for large deformation problems in soil. Int. J. Numer. Analyt. Methods Geomech. 22, No. 5, 327–350. Levadoux, J.-N. & Baligh, M. M. (1986). Consolidation after undrained piezocone penetration. I: prediction. J. Geotech. Engng ASCE 112, No. 7, 707–726. Lu, Q., Randolph, M. F., Hu, Y. & Bugarski, I. C. (2004). A numerical study of cone penetration in clay. Ge´otechnique 54, No. 4, 257–267, http://dx.doi.org/10.1680/geot.2004.54.4.257. Mahmoodzadeh, H. & Randolph, M. F. (2014). Penetrometer testing – effect of partial consolidation on subsequent dissipation response. J. Geotech. Geoenviron. Engng ASCE 140, No. 6, 04014022, http://dx.doi.org/10.1061/(ASCE)GT.1943–5606.0001114. Mahutka, K.-P., Konig, F. & Grabe, J. (2006). Numerical modeling of pile jacking, driving and vibratory driving. Proceedings of the international conference on numerical modelling of construction processes in geotechnical engineering for urban environment, pp. 235–246. London, UK: Taylor and Francis. Mayne, P. W. & Bachus, R. C. (1988). Profiling OCR in clays by piezocone. Proceedings of the 1st international symposium on penetration testing, Orlando, USA (ed. J. De Ruiter), pp. 20–24. Rotterdam, the Netherlands: Balkema. Mayne, P. W. & Kulhawy, F. H. (1982). K0–OCR relationships in soil. J. Geotech. Engng ASCE 108, No. 6, 851–872. Palmer, A. C. (2008). Dimensional analysis and intelligent experimentation. Singapore: World Scientific. Randolph, M. F. (2003). Science and empiricism in pile foundation design. Ge´otechnique 53, No. 10, 847–875, http://dx.doi.org/ 10.1680/geot.2003.53.10.847. Randolph, M. F. & Hope, S. (2004). Effect of cone velocity on cone resistance and excess pore pressures. Proceedings of international symposium on engineering practice and performance of soft deposits, Osaka, Japan, pp. 147–152. Randolph, M. F. & Wroth, C. P. (1979). An analytical solution for
the consolidation around a driven pile. Int. J. Numer. Analyt. Methods Geomech. 3, No. 3, 217–229. Richardson, M. (2007). Rowe cell test on kaolin clay, COFS internal report. Crawley, Australia: the University of Western Australia. Stewart, D. P. & Randolph, M. F. (1991). A new site investigation tool for the centrifuge. In Centrifuge 91: proceedings of the international conference on centrifuge modelling, Boulder, USA (eds H. Ko and F. G. McLean), pp. 531–538. Rotterdam, the Netherlands: Balkema. Sully, J. P., Robertson, P. K., Campanella, R. G. & Woeller, D. J. (1999). An approach to evaluation of field CPTU dissipation data in overconsolidated fine-grained soils. Can. Geotech. J. 36, No. 2, 369–381. Teh, C. I. & Houlsby, G. T. (1991). An analytical study of the cone penetration test in clay. Ge´otechnique 41, No. 1, 17–34, http:// dx.doi.org/10.1680/geot.1991.41.1.17. Torstensson, B. A. (1977). The pore pressure probe, Paper No. 34, pp. 34 .1–34 .15. Oslo, Norway: Nordiske Geotekniske Mote. Walker, J. & Yu, H. S. (2006). Adaptive finite element analysis of cone penetration in clay. Acta Geotechnica 1, No. 1, 43–57. Wang, D., Hu, Y. & Randolph, M. F. (2011). Keying of rectangular plate anchors in normally consolidated clays. J. Geotech. Geoenviron. Engng ASCE 137, No. 12, 1244–1253. Wang, D., Randolph, M. F. & White, D. J. (2013). A dynamic large deformation finite element method based on mesh regeneration. Comput. Geotech. 54, 192–201. Wang, D., White, D. J. & Randolph, M. F. (2010). Large deformation finite element analysis of pipe penetration and large-amplitude lateral displacement. Can. Geotech. J. 47, No. 8, 842–856. Wood, D. M. (1990). Soil behaviour and critical state soil mechanics. Cambridge, UK: Cambridge University Press. Wroth, C. P. (1984). The interpretation of in situ soil tests. Ge´otechnique 34, No. 4, 449–489, http://dx.doi.org/10.1680/ geot.1984.34.4.449. Wroth, C. P. & Houlsby, G. T. (1985). Soil mechanics – Property characterization and Analysis procedures. Proceedings of 11th international conference on soil mechanics and foundation engineering, vol. 1, pp. 1–54. Rotterdam, the Netherlands: Balkema. Yi, J. T., Goh, S. H., Lee, F. H. & Randolph, M. F. (2012). A numerical study of cone penetration in fine-grained soils allowing for consolidation effects. Ge´otechnique 62, No. 8, 707–719, http://dx.doi.org/10.1680/geot.8.P.155.
(2014). Ge´otechnique 64, No. 8, 680 [http://dx.doi.org/10.1680/geot.2014.64.8.680]
Corrigendum Mahmoodzadeh, H., Randolph, M. F. & Wang, D. (2014). Numerical simulation of piezocone dissipation test in clays. Ge´otechnique 64, No. 8, 657–666, http://dx.doi.org/10.1680/geot.14.P.011. On page 659, equation (7) is incorrect; it should read as follows. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ª º(1 þ 2K ) 0.42exp[(eN C e)=º] 0 w 1þ (7) k ¼ 1.06 3 105 (1 þ e)exp[(eN C e)=º] 1 þ 2K 0 The correct value was used in the numerical analyses and the authors believe there was no error in calculating the permeability. On page 661, nine lines from the bottom of the right-hand column; on page 662, three lines from the top of the right-hand column; and on page 663, in the second line of the left-hand column, ‘k ¼ 1 .13 3 107 m/s’ should read ‘k ¼ 1 .02 3 109 m/s in prototype’. On page 661, nine lines from the bottom of the right-hand column, ‘dissiaption’ should read ‘dissipation’. The vertical axis labels for Figs 3b, 5b, 6b, 7b, 8b, 9b, 11, 12 and 13b should not include ‘kPa’ as a unit of measurement since these axes represent normalised values.
680
[http://dx.doi.org/10.1680/geot.2014.64.10.848]
Corrigendum where is Poisson’s ratio; e0 is the initial void ratio; p9 is the mean effective stress; k is the (isotropic) permeability; ªw is the unit weight of water; and k and º are the gradients for swelling and consolidation in the modified Cam Clay model. A weighting factor, Æ, was introduced defining the contribution of elastic and plastic behaviour, and the paper went on to argue that a value of Æ ¼ 0 .5 led to a unified dissipation curve as input parameters such as , k and º were varied. However, once the correct values of e0 are allowed for, it turns out that a value of Æ ¼ 0 .75 proves superior in order to unify the dissipation responses. The affected figures are provided here; for completeness Figure 3(b) is also included since, although it was not affected by the error, the published version of the paper had an incorrect axis for the normalised time, T ¼ cvt/D2, where t is the time and D is the diameter of the cone. The normalised p time ffiffiffiffi in the remaining figures is given by T ¼ ch t=D2 I r , where Ir is the rigidity index. The value of Æ ¼ 0 .75 implies that the dissipation response is dominated by elastic response of the soil with only minor effects of plasticity. Nevertheless, the influence of plasticity increases T 50 to about 0 .08, compared with the value of 0 .061 derived by Teh and Houlsby (1991).
Mahmoodzadeh, H., Randolph, M. F. & Wang, D. (2014). Numerical simulation of piezocone dissipation test in clays. Ge´otechnique 64, No. 8, 657–666, 680, http://dx.doi.org/ 10.1680/geot.14.P.011. The authors are embarrassed to admit to a systematic error that affected the normalised plots of pore pressure dissipation and one of the primary conclusions from the above paper. The error was in the calculation of the value of the consolidation coefficient relevant for the various numerical analyses, where the authors had failed to take account of changes in initial void ratio, and hence also rigidity index, that resulted from varying different input parameters. The authors would like to take this opportunity to provide corrected versions of the affected figures, and also revise the conclusion regarding the relationship between the coefficients of consolidation that result from (a) laboratory oedometric compression (cv) and (b) piezocone dissipation testing (ch). The paper presented a relationship for ch, within the idealisations of modified Cam Clay and assuming isotropic permeability, of Æ 3(1 ) (1 þ e0 )p9 k 3(1 ) º ¼ cv (14) ch ¼ 1 þ kÆ º(1Æ) ªw (1 þ ) k
180
200
160
MSS
140
LDFE
120
Centrifuge
Excess pore pressure: kPa
Excess pore pressure: kPa
200
100 80 60 40 20 1
10
100 Time, t: s (a)
1000
160 140 120 100 80 60 40 20
10000
0 0·1
Ir ⫽ 17 Ir ⫽ 34 Ir ⫽ 73 Ir ⫽ 119 1
10 Time, t: s (a)
100 90 80
MSS
70
LDFE
60
Centrifuge
Normalised excess pore pressure
Normalised excess pore pressure
0 0·1
180
50 40 30 20 10 0 0·001
0·01 0·1 Normalised time, T ⫽ cvt/D2 (b)
1
10
100
100 90 80 70 60 50 40 30 20 10 0 0·001
Ir ⫽ 17 Ir ⫽ 34 Ir ⫽ 73 Ir ⫽ 119 0·01
0·1
Normalised time, T*⫽ cht/(D2冪Ir) (b)
Fig. 3. Dissipation graph from the numerical modelling and the centrifuge test
Fig. 5. Dissipation response at different rigidity indices
848
1000
1
200
Excess pore pressure: kPa
160 140 120 100 80 60 40 20 0 0·1
Normalised excess pore pressure
849
220
κ ⫽ 0·044 κ/λ ⫽ 0·125 κ/λ ⫽ 0·215 κ/λ ⫽ 0·5 1
10 Time, t: s (a)
100
90 80 70 60 50 30 20 10
κ ⫽ 0·044 κ/λ ⫽ 0·125 κ/λ ⫽ 0·215 κ/λ ⫽ 0·5
0 0·001
160 140 120 100 80 60 40 0 0·1
1000
100
40
180
20
Normalised excess pore pressure
Excess pore pressure: kPa
CORRIGENDUM 180
0·01
0·1
70 60 50 40 30 20 10
Excess pore pressure: kPa
Excess pore pressure: kPa
120 100 80 λ ⫽ 0·205 κ/λ ⫽ 0·215 κ/λ ⫽ 0·33 κ/λ ⫽ 0·5 1
10 Time, t: s (a)
100
Normalised excess pore pressure
Normalised excess pore pressure
80 70 60 50
20 10 0 0·001
λ ⫽ 0·205
κ/λ ⫽ 0·215 κ/λ ⫽ 0·33 κ/λ ⫽ 0·5 0·01
180 160
OCR ⫽ 3 OCR ⫽ 2 OCR⫽ 1
140 120 100 80
OCR ⫽ 5
60 40 0 0·1
1000
90
30
1
20
100
40
0·01 0·1 Normalised time, T * ⫽ cht/(D2冪Ir) (b)
200
140
0 0·1
K0 ⫽ 0·5 K0 ⫽ 0·61 K0 ⫽ 0·8 K0 ⫽ 1
220
160
20
1000
Fig. 8. Effect of stress anisotropy on dissipation response
180
40
100
80
Normalised time, T * ⫽ cht/(D 冪Ir) (b)
60
10 Time, t: s (a)
90
2
Fig. 6. Variation of dissipation response with º
1
100
0 0·001
1
K0 ⫽ 0·5 K0 ⫽ 0·61 K0 ⫽ 0·8 K0 ⫽ 1
0·1 2
Normalised time, T * ⫽ cht/(D 冪Ir) (b)
Fig. 7. Variation of dissipation response with k
1
1
10 Time, t: s (a)
100
1000
100 90 80 70 60 50 40
OCR ⫽ 1, 2, 3 and 5
30 20 10 0 0·0001
0·001 0·01 0·1 Normalised time, T * ⫽ cht/(D2冪Ir) (b)
Fig. 9. Effect of overconsolidation ratio (OCR) at u2 position
1
CORRIGENDUM
Normalised excess pore pressure
850 100 90 80 70 60 OCR ⫽ 1, 2, 3 and 5
50 40 30 20 10
0 0·0001
0·001 0·01 0·1 Normalised time, T * ⫽ cht/(D2冪Ir)
1
Fig. 11. Normalised dissipation graphs after effective stress correction 100
90
Normalised excess pore pressure
Normalised excess pore pressure
100 ν ⫽ 0·15 mm/s
80
Centrifuge MSS
70 60 50 40 30 20 10 0 0·001
0·01 0·1 Normalised time, T * ⫽ cht/(D2冪Ir) (a)
ν ⫽ 0·045 mm/s
80
Centrifuge MSS
70 60 50 40 30 20 10 0 0·001
1
0·01 0·1 Normalised time, T * ⫽ cht/(D2冪Ir) (b)
1
100
Normalised excess pore pressure
100
Normalised excess pore pressure
90
ν ⫽ 0·015 mm/s
90 80
Centrifuge
70
MSS
60 50 40 30 20 10 0 0·001
0·01 0·1 Normalised time, T * ⫽ cht/(D2冪Ir) (c)
90
ν ⫽ 0·0045 mm/s
80
Centrifuge
70
MSS
60 50 40 30 20 10 0 0·001
1
0·01
0·1
1
Normalised time, T * ⫽ cht/(D2冪Ir) (d)
Fig. 12. Comparison between normalised dissipation graphs from centrifuge tests and MSS
Normalised excess pore pressure
Excess pore pressure: kPa
180 ν ⫽ 1 mm/s
160 140
ν ⫽ 0·15 mm/s
120 100
ν ⫽ 0·045 mm/s
80 60 40
ν ⫽ 0·015 mm/s
20
ν ⫽ 0·0045 mm/s
0
1
10
100 Time, t: s (a)
1000
100 90 80 70 60 50 40
ν ⫽ 0·0045, 0·015, 0·045, 0·15 and 1 mm/s
30 20 10 0 0·001
0·01 0·1 Normalised time, T * ⫽ cht/(D2冪Ir) (b)
Fig. 13. Penetration rate effects from model piezocone tests in normally consolidated clay
1
