Reliability-Based Verification of Serviceability Limit States of Dry Deep Mixing Columns

Embankment foundations with dry deep mixing columns typically need to be designed with respect to the following three aspects of serviceability:

The framework presented accounts for the first two aspects, which are formulated as limit state functions $G_1$ and $G_2$. Differential settlements can occur on two scales. On the small scale, differential settlement can occur between the columns. In the authors’ opinion, a rational approach to deal with this issue simply is to repair the unevenness as it occurs during construction. On the larger scale, differential settlement can occur either across or along the embankment. Broms (1991) reported the results of a long-term field test in Skå-Edeby, Sweden, which indicated that the differential settlement across an embankment is minor, assuming that the shear strength is not attained along the perimeter of the column-reinforced soil volume under the embankment. However, differential settlement along the embankment is caused by inherent horizontal variability in the soil properties under the embankment. Evaluating this variability is a challenging engineering problem; therefore, it is left for future studies.

The occurrence of settlement of an embankment founded on dry deep mixing columns is affected by the consolidation behavior of the soft soil, the extent of the soil improvement, and the effect of the curing on the column stiffness. This section presents the modeling of these aspects.

Settlement of soft soil can be divided into three components: elastic settlement caused by elastic deformation of the soil occurring without any change in moisture content, primary consolidation settlement caused by pore water seeping out of the pressurized soil, and secondary consolidation settlement caused mainly by creep. Because the primary consolidation settlement is the most important component, this study mainly disregards the other two.

However, settlement of columns is a more complex process. The current practice is to describe the column compressibility with a modulus of elasticity; however, this is a considerable simplification for this three-phase material. Because the dry deep mixing column is not completely saturated, some compression will be due to static compaction (reduction of gas volume). The remaining compression can be described as elastic compression, primary consolidation, and secondary consolidation. Of these, the elastic component generally is negligible, and the primary consolidation is the most prominent. The effect of secondary consolidation in improved soil is a current research topic (e.g., Venda Oliveira et al. 2017).

This paper modeled the complex settlement behavior of soil reinforced by deep mixing columns with an updated version of the well-established Voigt model, which assumes an equal strain in the soil and the column as well as no lateral deformation; Broms (1991) and Han (2015) discussed these assumptions. The calculated settlement from the model is denoted primary settlement, $s$. The update concerns the use of a composite vertical coefficient of consolidation, derived in accordance with Wijerathna et al. (2017).

The Voigt model implies that occurring settlements are evaluated using area-weighted average values of the column modulus of elasticity, ${\overline{E}}_c$, and the constrained (oedometer) modulus of the soil, ${\overline{M}}_s$ (Federal Highway Administration 2013; Han 2015). The total primary settlement after a long time can be found by integrating over the thickness of the improved soft soil stratumwhere $\mathrm{\Delta }\sigma ={\gamma }_{\mathrm{emb}}{h}_{\mathrm{emb}}$ = load of embankment, where ${\gamma }_{\mathrm{emb}}$ = unit weight of embankment material, and $h_{\mathrm{emb}}$ = embankment height; and $a_z=a_c/a_{\mathrm{tot}}$ = area ratio of columns to total horizontal area at depth $z$ (subsequently, $a$ without subscript $z$ is used for convenience when not using a varied area ratio with depth). The model assumes end-bearing columns, but extension of the model to account for floating columns, as detailed by Kitazume and Terashi (2013), should be straightforward.

To account for the effect of the curing time on the occurring settlements, and to facilitate modeling of residual settlements that occur after completion of the embankment, the rate of consolidation under the embankment needs to be established. For this purpose, Wijerathna et al. (2017) derived an expression of a composite vertical coefficients of consolidation, $c_{v,\mathrm{comp}}$, based on Terzaghi’s one-dimensional consolidation equation and an equal strain assumption. The expression considers a potential difference in hydraulic conductivity and stiffness between the columns and the natural soil, which has been noted for dry deep mixing columns (e.g., Terashi and Tanaka 1983; Baker 2000; Åhnberg 2003), such thatwhere $A=a_s/a_c$ = ratio of area of surrounding soil to area of columns; $R_k=k_c/k_s$ = ratio of hydraulic conductivity of columns to hydraulic conductivity of natural soil; and $c_{v,c}$ and $c_{v,s}$ = coefficients of vertical consolidation of columns and natural soil, respectivelywhere $g$ = gravitational acceleration. According to Wijerathna et al. (2017), their proposed expression for $c_{v,\mathrm{comp}}$ agreed well with the physical model tests performed by Horpibulsuk et al. (2012). The derived $c_{v,\mathrm{comp}}$ can be applied straightforwardly in the expression for the adjusted time factor $T_v$ that is used to calculate the average degree of vertical consolidation under the embankment:where $h_{\mathrm{dr}}$ = longest drainage path; and $t$ = time.

The curing of the columns affects the occurring settlement rate because the column stiffness increases with time. In the absence of studies of this phenomenon for dry deep mixing columns, herein it was assumed that the stiffness increases proportionally to the unconfined compressive strength of the columns. [For concrete, stiffness has been shown to improve faster than the strength (De Schutter and Taerwe 1996), but this fact was not considered for reasons of simplicity.] According to measurements reported by Åhnberg (2006), the unconfined compressive strength of the columns improves according to the following empirical relationship:where $q_{u,28}$ = measured unconfined compressive strength of columns 28 days after installation; and $t$ is measured in days. With the assumed proportionality, the average modulus of elasticity of the columns can be described bywhere ${\overline{E}}_{c,28}$ = measured average modulus of elasticity of columns 28 days after installation. Similar empirical relationships for $q_{u,c}(t)$ were described by Horpibulsuk et al. (2003).

Because both the column stiffness increase and the consolidation of the soil stratum are time-dependent processes, the effect of the curing time on the developed settlements is accounted for by dividing the consolidation time into $n_t$ intermediate time steps [$t_{i–1}$, $t_i$] with the residual settlement starting at the end of construction, $t_{\mathrm{EC}}$, and ending at the end of the service life, $t_{\mathrm{ESL}}$. Reformulating Eq. (1), this gives a stepwise calculation of the residual settlementswhere [${\overline{E}}_c(t_i)+{\overline{E}}_c(t_{i-1})]/2$ = average column modulus of elasticity during time step $[t_{i-1},t_i]$ and $U(t_i)-U(t_{i-1})$ = change in average degree of consolidation during this time step.

With this analytical model of the occurring residual settlements under the embankment, a serviceability limit state for the maximum allowable residual settlement can be described bywhere $s_{r,\mathrm{allow}}$ = predefined allowable settlement value; and $S_r$ = probability distribution of predicted residual settlement, obtained by evaluating Eq. (7) with the relevant random variables.

Because yielding of the dry deep mixing columns may cause excessive deformation, an additional serviceability limit state can be established to avoid this issue. This effectively creates a boundary within which the settlement model of Eq. (1) is applicable. First, the compressive strength of the columns can be described with the Mohr–Coulomb failure criterion as (Alén et al. 2005; Larsson 2006)where ${\phi }_c^{\prime }$ = effective friction angle of columns; $c_c^{\prime }$ = effective cohesion of columns; coefficient of passive earth pressure $K_p=(1+\sin {\phi }_c^{\prime })/(1-\sin {\phi }_c^{\prime })$; and ${\sigma }_{h,c}^{\prime }$ = horizontal effective stress acting on column. Assuming that the installation of the columns does not affect the stress situation, ${\sigma }_{h,c}^{\prime }$ can be expressed as follows, using Rankine’s theory of lateral pressure:where ${\sigma }_{h,c}^{\prime \{0\}}$ = horizontal effective stress on column prior to loading, which is assumed to be equal to corresponding vertical effective stress ${\sigma }_{v,s}^{\prime \{0\}}$ if the virgin soil is expected to behave almost like a heavy liquid; $K_0$ = coefficient of earth pressure at rest; and $\mathrm{\Delta }{\sigma }_{v,s}$ = increase in vertical effective stress in soil due to loading. According to Alén et al. (2005), a bilinear stress–strain model can be used instead of a unilinear model to capture better the long-term stiffness reduction for high loads and thereby avoid underestimating the deformation; its application within the presented framework is straightforward.

As long as the axial stress in the columns, ${\sigma }_{v,c}$, does not exceed $f_c^{\prime }$, the respective increases in vertical effective stress in the columns and the soil ($\mathrm{\Delta }{\sigma }_{v,c}$ and $\mathrm{\Delta }{\sigma }_{v,s}$) will depend on the applied load $\mathrm{\Delta }{\sigma }_v$, the ratio ${\overline{E}}_{c,28}/{\overline{M}}_s$, and $a_z$ (Broms 1991; Baker 2000)The equations are derived from the Voigt model’s equal-strain assumption, distributing the vertical load with respect to area ratio, i.e., $\mathrm{\Delta }{\sigma }_v=\mathrm{\Delta }{\sigma }_{v,c}{a}_{\mathit{z}}+\mathrm{\Delta }{\sigma }_{v,s}(1-a_{\mathit{z}})$.

Combining Eqs. (9)–(11), a maximum allowable increase in column stress before yielding occurs can be described aswhere ${\sigma }_{v,c}^{\prime \{0\}}$ = vertical effective stress prior to loading. A serviceability limit state function then can be defined asAttainment of this limit state implies that the columns are close to yielding, which may cause excessive settlement that is not captured by the other limit state, $G_1$. Because the favorable effect of ${\sigma }_{h,c}^{\prime }$ is smallest close to the surface, as typically is the column shear strength as well, it is proposed that this limit state be evaluated with respect to the geotechnical conditions just below the groundwater level.

Because the two serviceability limit states concern the development of excessive settlement under the embankment, it is proposed that they should be evaluated as a correlated series system, such that the system probability of failure iswhich can be evaluated using Monte Carlo simulation. Generating $N$ samples of each random variable in $\overline{\mathbf{X}}$ and collecting them in a matrix $\overline{\mathbf{x}}$, $p_F$ then is approximated aswhere $I_F$ is the indicator function of limit state attainment, i.e., $I_F(\overline{\mathbf{x}})=1$ if either of the limit states $G_j(\overline{\mathbf{X}})$ is violated, and $I_F(\overline{\mathbf{x}})=0$ otherwise. This system formulation facilitates a design that avoids unsatisfactory structural behavior (excessive settlement) with a predefined acceptable target failure probability, $p_{\mathrm{FT}}$. In essence, by assigning a value to $p_{\mathrm{FT}}$, a required area ratio ($a$) of the columns can be computed.

To overcome the challenge of predicting the column properties in advance, Larsson and Bergman (2015) proposed an observational design approach, in which early property estimations are verified by quality control of the actual column properties during or after construction. A column penetration test can be used to investigate the properties of dry deep mixing columns (Axelsson and Larsson 2003; Liyanapathirana and Kelly 2011; Federal Highway Administration 2013). The column penetration test measures the penetration force, $F_c$, as a cylindrical penetrometer with two horizontal vanes is pushed through the column; $F_c$ is converted into tip resistance, $q_c$, by considering the penetrating cross-sectional area of the penetrometer. These measurements can serve as indirect observations on the column strength and deformation properties through established empirical transformations.

The quality control provides new information, $Q$, according to which the design can be adjusted, typically by increasing or decreasing $a$ or adjusting the binder content. This allows the designing engineer to determine an economical design that accommodates the actual ground conditions at the site. The procedure is in line with the observational method, first described by Peck (1969).

Mathematically, the effect of the new information, $Q$, on the calculated $p_F$ can be described with a limit state function of its own, herein denoted $h$ (Rackwitz and Schrupp 1985). Considering epistemic uncertainties and transformation errors in the control method, the possible outcome of the planned quality control, $q_c^{\{\mathrm{obs}\}}$, is viewed as a random variable, such that $q_c$ measurements become uncertain observations on $c_{c,28}^{\prime }$ and $E_c$ (Larsson 2006; Bergman et al. 2013):where $T_{c_c^{\prime }}=n_{c_c^{\prime }}{\epsilon }_{\mathrm{tr}}^{\{c_c^{\prime }\}}$ = empirical transformation of $q_c$ values to $c_c^{\prime }$ with associated transformation error ${\epsilon }_{\mathrm{tr}}^{\{c_u\}}$; and $T_{E_c}=n_{E_c}{\epsilon }_{\mathrm{tr}}^{\{E_c\}}$ = transformation of $q_c$ values to observations on $E_c$ with associated transformation error ${\epsilon }_{\mathrm{tr}}^{\{E_c\}}$. The factors $n_{c_c^{\prime }}$ and $n_{E_c}$ have been found to be rather uncertain. A reasonable value for $n_{c_c^{\prime }}$ based on the available research literature is $n_{c_c^{\prime }}=13/0.3=43.3$, where 13 transforms empirically the measured tip resistance to undrained shear strength, based on reviews and numerical analyses by Liyanapathirana and Kelly (2011), and 0.3 is recommended in Swedish practice (Larsson 2006) to account for expected drained conditions at yielding (because the column penetration test measures undrained shear strength). The value of $n_{E_c}$ depends on the column strength interval and the soil type (Larsson 2006); the Federal Highway Administration (2013) recommends $n_{E_c}=150$ as a reasonable estimate for dry deep mixing. The transformation errors were not studied explicitly in this paper, but were judged conservatively to be on the order of 20% coefficient of variation. With further research on these issues, it should be possible to reduce the transformation errors.

To accept the installed columns, a predefined quality threshold for the required tip resistance needs to be exceeded; this acceptance event can be described by the limit state functionwhere ${\tau }_{\mathrm{req}}^{\{q_c\}}$ = required threshold value of tip resistance. Therefore, the probability of unsatisfactory performance of the quality-controlled system can be writtenBased on this formulation, the required threshold level that is needed to satisfy a desired $p_{\mathrm{FT}}$ can be computed by solving the following equation for the only unknown, ${\tau }_{\mathrm{req}}^{\{q_c\}}$:The theoretical background of the establishment of such reliability-based alarm thresholds was presented by Spross and Gasch (2019). Conceptually, the solving of Eq. (24) implies a truncation of the probability distribution of $q_c^{\{\mathrm{obs}\}}$ at the threshold ${\tau }_{\mathrm{req}}^{\{q_c\}}$ in such a way that the $p_{\mathrm{FT}}$ is satisfied, given that the $q_c$ measurements exceed the threshold. For a computed ${\tau }_{\mathrm{req}}^{\{q_c\}}$, the probability of satisfying this threshold, i.e., the probability of verifying that the initial design acceptable, then can straightforwardly be calculated as

The design concept implies that, in principle, a substantial range of area ratios ($a$) is possible, but the quality control will determine whether the executed design is acceptable. A bold initial design (small $a$) will give a smaller initial cost but a higher probability of failing the quality control [Eq. (25)], the occurrence of which would require installation of additional columns at some cost. [In principle, the optimal initial $a$ can be seen as a decision-theoretical problem, the solution of which is beyond the scope of this paper; Spross and Johansson (2017) gave details of this issue.] The practical implementation of the design concept to dry deep mixing columns is illustrated in the following design example.

To illustrate the design procedure, a practical example based on real case data for the soil characterization is presented. The same site was used to illustrate Spross and Larsson’s (forthcoming) design procedure for surcharges on vertical drains. A 23-m-wide and $2.5\text{-}\mathrm{m}\text{-}\mathrm{high}$ road embankment was to be constructed on 8.5 m of very soft clay in the south of Stockholm County, Sweden. Fig. 1 shows a critical cross section for which the dry deep mixing column design was prepared with end-bearing columns. The example considered only the design task of selecting a single area ratio $a$ (i.e., no variation of area ratio with depth was considered), and the establishment of threshold values ${\tau }_{\mathrm{req}}^{\{q_c\}}$ for the quality control, whereas other design parameters such as column radius, amount of mixing, binder recipes, and binder content were assumed to be fixed. Settlement of the dry crust and the till layer was disregarded. The result was compared with an executed quality control of dry deep mixing columns installed at the construction site.

Soil samples were available for the critical section, from which probability distributions of the unit weight of the clay ${\gamma }_s$ and the soil modulus ${\overline{M}}_s$ were evaluated with Eqs. (14)–(18). Their distributions represented the uncertainty of the respective mean value. Because the columns were end-bearing, $\mathrm{\Delta }{\sigma }_{v,s}$ was assumed to be constant with depth because the columns were substantially stiffer than the surrounding clay; the potential trend with depth therefore also was disregarded for the soil parameters. The other relevant geotechnical parameters were assessed as detailed in Table 1. Two-way drainage was assumed for the consolidation of the soft soil.

The road embankment was assumed to be completed $t_{\mathrm{EC}}=90$ days after the column installation, at which time the residual settlement started to develop [Eq. (7)]. Primary consolidation was assumed to be finished at $t_{\mathrm{ESL}}=\mathrm{1,000}$ days.

In this example, the $p_{\mathrm{FT}}$ was set to 5% for the system, which corresponds to suggestions by Akbas and Kulhawy (2009). The $s_{r,\mathrm{allow}}$ was set to 5 cm. For each random geotechnical parameter in Table 1, 50,000 samples were generated with crude Monte Carlo simulation. Fig. 2 shows the calculated failure probabilities for the respective limit states $G_1$ and $G_2$, as well as the system probability of failure $p_F$ [Eq. (19)], for a range of area ratios.

The value of $a$ that corresponded to the assigned $p_{\mathrm{FT}}$ was close to 0.37, indicating that the initial design should have an area ratio less than this value. This is an effect of the conditional criterion for observing a tip resistance above the established threshold, i.e., $q_c^{\{\mathrm{obs}\}}\ge {\tau }_{\mathrm{req}}^{\{q_c\}}$ in Eq. (24), which allows the designer to use a bolder design than the $a=0.37$ associated with the $p_{\mathrm{FT}}$. In this case, the bearing capacity ($G_2$) is clearly the governing failure mode, because this limit state contributes the most to the calculated $p_F$. This indicates that any monitoring in the quality control should be directed to gaining information about the parameters affecting this limit state. Consequently, an alarm threshold should be established for $c_{c,28}^{\prime }$ (since there is little to gain from monitoring of ${\overline{E}}_{c,28}$).

The quality control was planned to be performed with the column penetration test, to obtain observations of $c_c^{\prime }$ [Eq. (21)]. In this example, the transformation error of this control method, ${\epsilon }_{\mathrm{tr}}^{\{c_u\}}$, was judged to be normally distributed with unit mean and 20% coefficient of variation.

To establish the threshold for the quality control, Eq. (24) was solved for ${\tau }_{\mathrm{req}}^{\{q_c\}}$. To find ${\tau }_{\mathrm{req}}^{\{q_c\}}$, the generated sample distribution of $q_c^{\{\mathrm{obs}\}}$ is truncated until the calculated $p_F$ equals $p_{\mathrm{FT}}$. In this paper, a tolerance of 10% in the error $e=|p_F–p_{\mathrm{FT}}|/p_{\mathrm{FT}}$ was accepted in the derivation of ${\tau }_{\mathrm{req}}^{\{q_c\}}$. Because the design parameter of interest is the area ratio $a$, corresponding thresholds ${\tau }_{\mathrm{req}}^{\{q_c\}}$ can be established for a range of potential area ratios.

The designing engineer then can choose an area ratio that corresponds to the client’s risk appetite: Fig. 3(a) shows the interdependence of ${\tau }_{\mathrm{req}}^{\{q_c\}}$ and $a$, illustrating how a bolder design (smaller $a$) requires higher quality of the installed columns. The probabilities of violating the threshold for the considered area ratios are shown in Fig. 3(b). The optimal combination of ${\tau }_{\mathrm{req}}^{\{q_c\}}$ and $a$ depends on the cost of the prepared action plan that is put into operation if the quality control indicates substandard column quality; its complete evaluation is not part of this paper. From a risk management perspective, the final design decision of $a$ should be made by the risk owner, i.e., the person who is financially responsible for the quality of the embankment foundation works [this can be the contractor or client, depending on contractual arrangement (Spross et al. 2018)]. In this example, a probability of success of 90%, i.e., $P$ (alarm) $=10\%$, was judged to be acceptable, which gives $a=0.35$ [Fig. 3(b)] and a corresponding ${\tau }_{\mathrm{req}}^{\{q_c\}}=1.2\mathrm{MPa}$ [Fig. 3(a)].

The quality of six columns was investigated close to the analyzed critical embankment section using a column penetration test in predrilled, centered holes 28 days after installation. Fig. 4 shows an example of the measured $F_c$ for one of the columns. Accounting for the penetrating area of the penetrometer ($A_{\mathrm{pen}}={\mathrm{6,750}\mathrm{mm}}^2$), the observed tip resistances for the six columns, measured 1 m into the columns from the groundwater level, were $q_c^{\{\mathrm{obs}\}}=F_c/A_{\mathrm{pen}}=2.44$, 1.63, 2.74, 3.85, 4.14, and 5.04 MPa, giving a mean value of ${\overline{q}}_c^{\{\mathrm{obs}\}}=3.31\mathrm{MPa}$, which was within the 1st and 99th percentiles of the $q_c$ used in the design phase [Fig. 3(a)]. Because the observed mean value exceeded the established ${\tau }_{\mathrm{req}}^{\{q_c\}}=1.2\mathrm{MPa}$, the quality of the columns was found to be satisfactory, ensuring that the probability of violating the analyzed system of serviceability limit states was less than $p_{\mathrm{FT}}$.

In the illustrative example, column yielding turned out to be the governing failure mode from a system perspective. However, running the same simulation for other column lengths showed that for longer columns, the residual settlement may be the governing failure mode. From the simulations in Fig. 5, it is clear that if the columns had been about 20 m or longer, the residual settlement would have had a nonnegligible effect on the $p_F$ for these geotechnical conditions. This shows that both failure modes are relevant to analyze, although yielding in the column top can be expected to be the governing failure mode for shorter columns.

It also is clear that a considerable volume of soil needs to be improved for short columns as well, to avoid column yielding and the subsequent expected excessive embankment settlement that occurs when the linear-elastic settlement model [Eq. (1)] no longer is valid. An obvious solution is to use higher $a$ at shallow depth, alternating long and short columns. This can be accounted for straightforwardly in the model, because Eq. (7) integrates over the depth $z$ and allows different area ratios at different depths ($a_z$) and the limit state for column yielding ($G_2$) is evaluated at a shallow depth. There also are other options for reducing the effects of column yielding: columns combined with mass stabilization of the top layers, reinforcing columns with a stiffer core (e.g., Zhang et al. 2020), and installing T-shaped columns (e.g., Phutthananon et al. 2020).

Although all these options are viable options to reduce the issue of column yielding, they are not easy to model analytically, but rather require numerical models. Numerical models also are required to capture well the effect of secondary consolidation settlement and spatial variability. An engineer designing an embankment foundation with dry deep mixing therefore faces a considerable challenge if they want both to capture uncertainties by applying a reliability-based design method and to model structural behavior with high detail by using a numerical model. Considering the large number of simulations that is needed in a probabilistic evaluation, highly detailed models normally are not feasible in practice if reliability-based design methods are to be applied. However, some rather complex phenomena and effects can be taken into account in an analytical model, although with some degree of conservatism to err on the safe side: for example, if a sequential loading scheme is applied in a staged construction, the authors suggest assuming full loading as early as Day 28 (the day of the control measurements), because the embankment construction is expected to start after the measurements have been conducted. Moreover, the increased settlement caused by secondary consolidation possibly might be accounted for by reserving a margin for this by setting the allowable residual settlement, $s_{r,\mathrm{allow}}$, closer to 0 in the evaluation of the limit state [Eq. (8)]. On the other hand, curing of the dry deep mixing columns will with time improve the $c_c^{\prime }$. The observed column quality at Day 28 therefore can serve as a conservative assumption.