1 Introduction

Columns are commonly used to improve soft soils by increasing the bearing capacity, reducing total and differential settlements, and enhancing stability. Recently, columns have been increasingly used to support embankments over soft soils, especially when the requirements for settlement control and/or construction time are strict. Different types of columns have been used in practice, such as stone columns, deep mixed (DM) columns, jet grouted columns, vibro-concrete columns (VCCs), cement-fly ash-gravel (CFG) columns, T-shaped DM columns and rammed aggregate piers [6, 25,26,27,28,29, 32, 35, 39, 45]. Furthermore, composite columns have been increasingly used in practice, since they take advantages of positive effects of the individual components (e.g., concrete piles for strength and stiffness, and DM columns for increasing soil strength).

As a composite column, stiffened deep mixed (SDM) column is formed by inserting a precast concrete core pile into the center of a DM column immediately after the construction of the DM column (see Fig. 1) [40]. Concrete core pile is installed to increase the strength and stiffness of the column and reduce the ground settlement, meanwhile, DM column is used to increase the skin friction of the core pile shaft and reinforce the surrounding soil. Moreover, the bearing capacity provided by SDM columns was similar to the cast-in-place piles with the same diameter and length, while the SDM column can save nearly 30% cost [30].

Fig. 1
figure 1

SDM column-supported embankment over soft soil

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In the past decade, the behavior of SDM column-reinforced soft soil was investigated mostly by means of experimental and numerical studies [31, 33, 40]. Tanchaisawat et al. [31] experimentally found that the shear strength of the interface between concrete core pile and DM column was increased linearly with the unconfined compressive strength of soil–cement. Vootttipruex et al. [33] indicated that the settlement on SDM column was 40% less than that of DM columns due to its higher stiffness. Ye et al. [40] found that the critical height of soil arching in the SDM column-supported embankment had a good consistency with the clear spacing between the concrete core piles based on the three-dimensional finite element analysis.

The system of SDM column-reinforced soft soil is composed of three materials (i.e., core pile, DM column, and soft soil) and two interfaces (i.e., interface between core pile and DM column and interface between DM column and surrounding soil). Due to the complicity of this system, there have been only a few publications dealing with the theoretical analysis of SDM column-reinforced soft soil [20, 30, 37]. Wu et al. [37] proposed a method to calculate the bearing capacity of SDM column by assuming that the failure in the surrounding soil triggered the failure of SDM column. Jamsawang et al. [20] and Raongjant and Jing [30] indicated that the lateral bearing capacity of SDM column was 11–15 times and 3–4 times greater than that of DM column under static and cyclic loading, respectively.

At present, there is no applicable design method for calculating the settlement of SDM column-reinforced soft soil. Many studies have been conducted focusing on the settlement calculation of other type of column-reinforced soft clay. Han [17] established a method for calculating the settlement of stone column-reinforced foundation. Zhang et al. [41] developed an analytical solution for the settlement of a composite foundation reinforced with stone columns considering that the modulus of column had reduction effects on column settlements and bulging. Abdelkrim and Buhan [1] presented an elastoplastic homogenization method applied to a soil reinforced by regularly distributed columns. Castro and Sagaseta [7] presented an approximate solution to calculate the settlement of rigid footing resting on a stone column-reinforced soft soil by converting the column group to an axisymmetric problem with an equivalent single column with the same cross-sectional area.

Horikoshi and Randolph [18] proposed a simplified design method to calculate the settlement of piled rafts. Han and Wayne [16] demonstrated that the method proposed by Horikoshi and Randolph [18] for piled rafts can also be used to calculate the settlement of DM column-reinforced soft soil. Yao et al. [38] conducted a 1-g physical model test for soft ground with deep mixed columns considering the effect of column length, area replacement ratio and surcharge load on foundation settlement. Chai et al. [8] proposed a method to calculate the settlement of DM column-reinforced foundations underlain by a soft soil, in which the penetration of the columns was considered by treating the lower portion of the reinforced zone as an “unreinforced” layer. Therefore, it is necessary to develop a calculation method for the settlement of SDM column-reinforced soft soil.

In this paper, a theoretical analysis was conducted to derive a solution of the settlement of stiffened deep mixed column-supported embankment over soft clay. The unit cell concept is adopted to establish the calculation model and the subsoil is assumed to settle under one-dimensional condition. The settlement of subsoil consists of three components, i.e., the compression of the soil within length of stiffened core piles, the compression of the soil from core pile base to SDM column base and the compression of the soil below SDM column base. The upward and downward penetrations of concrete core pile were considered in the analysis. To validate the developed analytical method, three-dimensional (3-D) finite element models with different configurations of SDM columns were created and the comparison between the analytical predictions and the numerical results were discussed. A parametric study based on the derived solution was performed to investigate the influence factors of DM column modulus, length of core pile and DM column, diameter of DM column and core pile, and interaction coefficient on the settlement characteristic of SDM column-reinforced soil, and some recommendations are summarized for its application. Finally, the design charts were drawn for ease of application in design. Two case histories of SDM column-supported embankment over soft soil were introduced and the design method was applied to predict the ground final settlements.

2 Derivation of settlement of the foundation

2.1 Calculation model

The unit cell concept is adopted to establish the calculation model. The unit cell analysis is valid only when it is applied at the embankment centerline where the lateral displacements are zero, but it cannot successfully predict the settlement on other places under embankment where the lateral displacements took place. However, in design of an embankment over soft soil, engineers pay more attention on the settlement of ground surface at the centerline of embankment, since it is the maximum value along the embankment width.

Figure 2 shows the unit cell of SDM column-reinforced soil in the analysis, in which the unit cell is divided into three regions in the cross section: Region I (the region from ground surface to core pile base), Region II (the region from core pile base to DM column base), and Region III (the region of soft soil below DM column base). In Fig. 2, l1, l2 and l3 are the original thicknesses of Region I, Region II, and Region III, respectively; \(l_{1}^{{\prime }}\), \(l_{2}^{{\prime }}\) and \(l_{3}^{{\prime }}\) are the thicknesses of Region I, Region II, and Region III, respectively, when the unit cell is subjected to a surface pressure P; D and d are the diameters of DM column and core pile; and De is the equivalent diameter of influence zone, in which De = 1.05B for columns in a triangular pattern; De = 1.13B for columns in a square pattern; and B is the column spacing.

Fig. 2
figure 2

Unit cell of SDM column-reinforced soil: a before settlement, b after settlement

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The main assumptions made in the analysis are summarized as: (a) the concrete core pile, the DM column and the subsoil behave as isotropic linear-elastic materials; (b) the radial deformation is ignored and only vertical strain is considered; (c) the surrounding soil and the DM column in Region I and Region II are treated as a composite soil with equivalent constrained modulus.

Based on the division of the regions in the unit cell of the analytical model, the total settlement of SDM column-reinforced subsoil can be expressed as:

$$S_{\text{total}} = S_{\text{I}} + S_{\text{II}} + S_{\text{III}} ,$$
(1)

in which Stotal is the total settlement; SI is the compression of Region I; SII is the compression of Region II; and SIII is the compression of Region III. Referring to Fig. 2, \(S_{\text{I}} = l_{1} - l_{1}^{{\prime }}\), \(S_{\text{II}} = l_{2} - l_{2}^{{\prime }}\), and \(S_{\text{III}} = l_{3} - l_{3}^{{\prime }}\). The following sections discuss the derivations of settlement calculation for each Region.

2.2 Compression of soil in Region I

In the system of SDM column-supported embankment, gravel cushion is commonly placed on the top of pile head to adjust stress transfer between pile and soil. Due to the great difference in stiffness between pile and soil, more vertical stress is concentrated onto the core pile when it is subjected to the embankment load. As a result, the core pile would penetrate upward into the cushion layer and downward into the DM column under the embankment load. This phenomenon is called stress concentration effect and has been observed in the previous studies on pile-reinforced soil with a gravel cushion on the top surface [44]. On the other hand, the differential settlement between the pile and the soil results in the shear stress developing in the fill, which would increase the stress carried by the pile and reduce the stress carried by the soil. This stress transfer phenomenon is referred to as soil arching effect. To simplify the analysis, this study did not consider the soil arching effect (i.e., the shear stress developed in the gravel fill is neglected) but considered the stress concentration effect. In the unit cell model, the modulus of gravel cushion is considered but the embankment fill is equivalent to a uniform pressure applied on the cushion surface. This model is conservative as the stress imposed on the soil is overestimated. The derivation process is introduced in this section.

Considering the core pile penetration effect (see Fig. 2), one can obtain:

$$S_{\text{I}} = l_{1} - l_{1}^{{\prime }} = \delta_{\text{up}} + \delta_{\text{down}} + \delta_{\text{core}} ,$$
(2)

in which \(\delta_{\text{up}}\) is the upward penetration of the core pile; \(\delta_{\text{down}}\) is the downward penetration of the core pile; and \(\delta_{\text{core}}\) is the compression of the core pile. Since the surrounding soil in the shallow portion settles more than the core pile under the embankment load, negative frictional force occurs on the core pile shaft. Considering the existence of neutral point due the negative friction on the shaft of core pile, Eq. (2) is rewritten as:

$$S_{\text{I}} = S_{\text{su}} + S_{\text{sd}} = \left( {\delta_{\text{up}} + \delta_{1} } \right){ + }\left( {\delta_{\text{down}} + \delta_{2} } \right),$$
(3a)
$$S_{\text{su}} = \delta_{\text{up}} + \delta_{1} ,$$
(3b)
$$S_{\text{sd}} = \delta_{\text{down}} + \delta_{2} ,$$
(3c)

in which \(S_{\text{su}}\) and \(\delta_{1}\) are the compressions of the surrounding soil and the core pile above the neutral point, respectively; and \(S_{\text{sd}}\) and \(\delta_{2}\) are the compressions of the surrounding soil and the core pile below the neutral point, respectively.

Figure 3a shows the distribution of the skin frictional stresses along the core pile under different embankment height based on the field measurements by Wang et al. [34]. The skin frictional stress of core pile was calculated by the equation, \(\tau_{z} = {{\Delta F} \mathord{\left/ {\vphantom {{\Delta F} {\left( {u_{\text{p}}\Delta z} \right)}}} \right. \kern-0pt} {\left( {u_{\text{p}}\Delta z} \right)}}\), in which \(\Delta F\) is the change in axial force at two adjacent depths (kN); \(\Delta z\) is the distance between the two adjacent depths (m); and, \(u_{\text{p}}\) is the perimeter of core pile. The absolute values of the skin frictional stress (i.e., negative and positive skin friction) along the core pile increased with the increase in embankment height while the depth of neutral point was almost kept constant during the whole loading procedure. It should be realized that the location of neutral point dynamically changes with time and fill load. In this field test, the fill height was only 3.5 m. The small embankment loading results in a relative stable location of neutral point in the whole filling process. This phenomenon has also been found by Kong [24]. For a better presentation, the skin frictional stress along the core pile shaft was normalized to a ratio of the skin frictional stress at a given depth to the maximum positive skin friction. It is worth noticing from Fig. 3b that the skin frictional stress along the core pile had a consistent normalization with depth and the distribution of the skin frictional stress along the core pile could be approximated as a linear distribution. Therefore, the skin frictional stress along the core pile is assumed to be a linear distribution (see Fig. 4) in this analysis.

Fig. 3
figure 3

Consideration of skin friction distribution along concrete core pile

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Fig. 4
figure 4

Simplification of skin friction distribution along concrete core pile

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The basic Coulomb friction model was used to calculate the negative skin friction between the core pile and the DM column. The interface friction angle is considered, but the cohesion of DM column/core pile interface is ignored. The interfacial cohesion would decrease under certain conditions under the long-term condition. For example, when the interface is fully soaked, the interfacial cohesion would decrease [10, 14]. When a relative large displacement between core pile and DM column develops, the interfacial cohesion would also decrease due to the full slip [5, 21]. From a practical point of view, it is conservative to ignore the cohesion of DM column/core pile interface for a long-term performance. The maximum negative skin friction is expressed as follows [4]:

$$\tau_{0} = K\sigma_{\text{s}} \tan \varphi_{\text{i}}$$
(4)

in which \(\tau_{0}\) is the negative friction at the core pile head; \(\varphi_{\text{i}}\) is the interface friction angle between the core pile and the DM column; \(\sigma_{\text{s}}\) is the vertical stress on the top of surrounding soil; K is the in situ lateral earth pressure coefficient. The in situ lateral earth pressure coefficient is complicated and is dependent on soil type, pile material, and method of pile installation [36]. For a convenient use in design, K value is recommended in a range of lateral earth pressure coefficient at rest (i.e., K = 1 − sinφ, where φ is the effective friction angle of deep mixed soil) to passive lateral earth pressure coefficient [i.e., K = tan2(45° + φ/2)]. If there is no test result available to determine φ, 30° is recommended to use in design [2, 13, 43]. Therefore, K value is in a range of 0.5–3 when φ = 30°. For a displacement core pile, as driving process causes the soil to be displaced radially, a relatively large K value is recommended, and for a non-displacement and/or partial displacement core pile, as the soil is removed, a relatively small K value is recommended. However, a selection of specific value still relies on engineers’ experience and judgment. Three case histories were selected to verify the applicability of the proposed method to determine K value. Wu et al. [37] conducted a field investigation in which the core pile was driven directly into a DM column. Based on the field results, the back-calculated K was 1.9 by taking φ = 30° and φi = 18°), which approaches the passive lateral earth pressure coefficient [i.e., K = tan2(45° + φ/2) = 3]. Ding et al. [11] reported a field test in which the core pile was installed by a partially bored method. Based on the field results, the back-calculated K was 0.6, close to the lateral earth pressure coefficient at rest (i.e., K = 1 − sinφ = 0.5). Voottipruex et al. [33] presented a field pullout test of core pile which was driven directly into a DM column, and the back-calculated K was 1.3 which is also within the proposed range.

To demonstrate the feasibility of Eq. (4) for estimating the maximum negative friction on the core pile shaft, a comparison between the field test results by Wang et al. [34] and the calculated negative friction by Eq. (4) was conducted. The measured skin frictional stresses of core pile were obtained based on the measured vertical stress of the pile head and the measured point close to the pile head using the equation of \(\tau_{z} = {{\Delta F} \mathord{\left/ {\vphantom {{\Delta F} {\left( {u_{\text{p}}\Delta z} \right)}}} \right. \kern-0pt} {\left( {u_{\text{p}}\Delta z} \right)}}\). The meanings of notations have been explained previously. In the analysis, the interface friction angle of the DM column and the core pile was taken as 18° [31]. Considering the core pile was directly pushed into the ground, resulting in increasing the lateral earth pressure in the field, the K value was larger than the coefficient of earth pressure at rest. The K value was assumed to be passive lateral earth pressure coefficient in the analysis. It can be seen from Fig. 5 that the calculated skin friction agreed well with the measured results under the different fill load.

Fig. 5
figure 5

Comparison of skin friction between measured and calculated results

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According to the triangular pattern assumption of the skin friction stress along the core pile, the concrete core skin friction is written as:

$$\tau (z) = - \tau_{0} (1 - {z \mathord{\left/ {\vphantom {z {l_{0} }}} \right. \kern-0pt} {l_{0} }}),$$
(5)

in which \(\tau (z)\) is the skin friction stress at a depth of z (i.e., a negative value means the negative friction, vice versa); \(l_{0}\) is the depth of equal settlement plane, and z is the depth from ground surface.

Based on the force equilibrium in vertical direction, both the SDM column and the equivalent surrounding soil carry the applied loads, and thus,

$$P = m\alpha \sigma_{\text{p}} + (1 - m\alpha )\sigma_{\text{s}} .$$
(6a)

Setting the stress concentration ratio \(n = {{\sigma_{\text{p}} } \mathord{\left/ {\vphantom {{\sigma_{\text{p}} } {\sigma_{\text{s}} }}} \right. \kern-0pt} {\sigma_{\text{s}} }}\), one can obtain

$$\sigma_{\text{p}} = \frac{nP}{(n - 1)\alpha m + 1},$$
(6b)
$$\sigma_{\text{s}} = \frac{P}{(n - 1)\alpha m + 1},$$
(6c)

in which P is the applied pressure on the whole area of the unit cell; \(m\) is the replacement ratio of SDM column, i.e., \(m = {{D^{2} } \mathord{\left/ {\vphantom {{D^{2} } {D_{\text{e}}^{2} }}} \right. \kern-0pt} {D_{\text{e}}^{2} }}\); \(\alpha\) is the core pile area ratio in cross section of SDM column, i.e., \(\alpha = {{d^{2} } \mathord{\left/ {\vphantom {{d^{2} } {D^{2} }}} \right. \kern-0pt} {D^{2} }}\); \(\sigma_{\text{p}}\) and \(\sigma_{\text{s}}\) are the average vertical stress on the top of core pile and the equivalent surrounding soil, respectively.

The equivalent constrained modulus of surrounding soil and DM column in Region I (\(E_{\text{I}}^{\text{eq}}\)) is determined based on the area-weighted average of DM column socket and surrounding soil:

$$E_{\text{I}}^{\text{eq}} = m(1 - \alpha )E_{\text{DM}} + (1 - m)E_{\text{sI}} ,$$
(7)

in which \(E_{\text{I}}^{\text{eq}}\) is the equivalent constrained modulus of surrounding soil and DM column; \(E_{\text{DM}}\) is the constrained modulus of DM column; \(E_{{{\text{s}}{\rm I}}}\) is the constrained modulus of subsoil in Region I.

Figure 6 shows a slice element of the unit cell with a thickness of dz at a depth z. Based on the equilibrium of force in the slice element in a vertical direction, one can obtain:

$$\sigma_{\text{sz}} \left( {A_{\text{e}} - A_{\text{p}} } \right) - \left( {\sigma_{\text{sz}} + {\text{d}}\sigma_{\text{sz}} } \right)\left( {A_{\text{e}} - A_{\text{p}} } \right) - \tau_{z} u_{\text{p}} {\text{d}}z = 0,$$
(8)

in which \(\sigma_{\text{sz}}\) is the average vertical stress on surrounding soil and DM column at depth z; \(A_{\text{e}}\) and \(A_{\text{p}}\) are the areas of influence zone and core pile, respectively; and \(u_{\text{p}}\) is the perimeter of core pile.

Fig. 6
figure 6

Force analysis on calculation unit

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Setting \(\lambda = {{u_{\text{p}} } \mathord{\left/ {\vphantom {{u_{\text{p}} } {\left( {A_{\text{e}} - A_{\text{p}} } \right)}}} \right. \kern-0pt} {\left( {A_{\text{e}} - A_{\text{p}} } \right)}}\), Eq. (8) is rearranged as:

$${{{\text{d}}\sigma_{\text{sz}} } \mathord{\left/ {\vphantom {{{\text{d}}\sigma_{\text{sz}} } {{\text{d}}z}}} \right. \kern-0pt} {{\text{d}}z}} + \lambda \tau_{z} = 0.$$
(9)

Using the boundary condition, \(\sigma_{\text{sz}} = \sigma_{\text{s}}\), at z = 0, Eq. (9) is solved as:

$$\sigma_{\text{sz}} = {{(\lambda \tau_{0} z^{2} )} \mathord{\left/ {\vphantom {{(\lambda \tau_{0} z^{2} )} {(2l_{0} )}}} \right. \kern-0pt} {(2l_{0} )}} - \lambda \tau_{0} z + \sigma_{\text{s}} .$$
(10)

Thus, the compression of subsoil in Region I can be derived as:

$$S_{\text{su}} = \int_{0}^{{l_{0} }} {\left( {{{\sigma_{\text{sz}} } \mathord{\left/ {\vphantom {{\sigma_{\text{sz}} } {E_{\text{I}}^{\text{eq}} }}} \right. \kern-0pt} {E_{\text{I}}^{\text{eq}} }}} \right){\text{d}}z} = \frac{1}{{E_{\text{I}}^{\text{eq}} }}\left( {\sigma_{\text{s}} l_{0} - \frac{{\lambda \tau_{0} }}{3}l_{0}^{2} } \right),$$
(11a)
$$S_{\text{sd}} = \int_{{l_{0} }}^{{l_{1} }} {\left( {{{\sigma_{\text{sz}} } \mathord{\left/ {\vphantom {{\sigma_{\text{sz}} } {E_{\text{I}}^{\text{eq}} }}} \right. \kern-0pt} {E_{\text{I}}^{\text{eq}} }}} \right){\text{d}}z} = \frac{1}{{E_{\text{I}}^{\text{eq}} }}\left( {\frac{{\lambda \tau_{0} }}{{6l_{0} }}l_{1}^{3} - \frac{{\lambda \tau_{0} }}{2}l_{1}^{2} + \sigma_{\text{s}} l_{1} - \sigma_{\text{s}} l_{0} + \frac{{\lambda \tau_{0} }}{3}l_{0}^{2} } \right).$$
(11b)

All the above notations have been explained previously.

At depth z, based on the vertical force equilibrium on core pile shaft, the vertical stress in core pile can be expressed as:

$$\sigma_{\text{pz}} = \sigma_{\text{p}} + A_{\text{p}}^{ - 1} \int_{0}^{z} {\tau_{z} u_{\text{p}} {\text{d}}z} = \sigma_{\text{p}} + \gamma (\tau_{0} z - {{\tau_{0} z^{2} } \mathord{\left/ {\vphantom {{\tau_{0} z^{2} } {(2l_{0} )}}} \right. \kern-0pt} {(2l_{0} )}}),$$
(12)

in which \(\gamma = {{u_{\text{p}} } \mathord{\left/ {\vphantom {{u_{\text{p}} } {A_{\text{p}} }}} \right. \kern-0pt} {A_{\text{p}} }}\). Similarly, the compression of core pile in Region I can be derived as:

$$\delta_{1} = \int_{0}^{{l_{0} }} {\left( {{{\sigma_{\text{pz}} } \mathord{\left/ {\vphantom {{\sigma_{\text{pz}} } {E_{\text{p}} }}} \right. \kern-0pt} {E_{\text{p}} }}} \right){\text{d}}z} = \frac{1}{{E_{\text{P}} }}(\sigma_{\text{P}} l_{0} + {{\gamma \tau_{0} l_{0}^{2} } \mathord{\left/ {\vphantom {{\gamma \tau_{0} l_{0}^{2} } 3}} \right. \kern-0pt} 3}),$$
(13a)
$$\delta_{2} = \int_{{l_{0} }}^{{l_{1} }} {\left( {{{\sigma_{\text{pz}} } \mathord{\left/ {\vphantom {{\sigma_{\text{pz}} } {E_{\text{p}} }}} \right. \kern-0pt} {E_{\text{p}} }}} \right){\text{d}}z} = \frac{1}{{E_{\text{P}} }}\left( { - \frac{{\gamma \tau_{0} }}{{6l_{0} }}l_{1}^{3} + \frac{{\gamma \tau_{0} }}{2}l_{1}^{2} + \sigma_{\text{P}} l_{1} - \sigma_{\text{P}} l_{0} - \frac{{\gamma \tau_{0} }}{3}l_{0}^{2} } \right).$$
(13b)

The upward penetration of core pile can be determined as:

$$\delta_{\text{up}} = p_{\text{c}} (\sigma_{\text{p}} - \sigma_{\text{s}} ),$$
(14a)

where \(p_{\text{c}}\) is the upward penetration when an unit force is exerted on the top of core pile, and it can be obtained as:

$$p_{\text{c}} = {{l_{\text{GC}} } \mathord{\left/ {\vphantom {{l_{\text{GC}} } {E_{\text{GC}} }}} \right. \kern-0pt} {E_{\text{GC}} }},$$
(14b)

in which \(l_{\text{GC}}\) is the thickness of gravel cushion, and \(E_{\text{GC}}\) is the constrained modulus of cushion.

The downward penetration of core pile can be determined as:

$$\delta_{\text{down}} = p_{\text{s}} (\sigma_{{{\text{p}}l_{1} }} - \sigma_{{{\text{s}}l_{1} }} ),$$
(15a)

in which \(\sigma_{{{\text{p}}l_{1} }}\) is the vertical stress at the bottom of core pile; \(\sigma_{{{\text{s}}l_{1} }}\) is the vertical stress at the bottom of subsoil; and \(p_{\text{s}}\) is the downward penetration when an unit force exerted on the bottom of core pile which can be calculated based on the method proposed by Chen [9],

$$p_{\text{s}} = {{(1 - \mu_{0}^{2})\omega \sqrt {A_{\text{p}}}} \mathord{\left/{\vphantom {{(1 - \mu_{0}^{2})\omega \sqrt {A_{\text{p}}}} {E_{0}}}} \right. \kern-0pt} {E_{0}}}$$
(15b)
$$E_{0} = (1 - {{2\mu_{0}^{2} } \mathord{\left/ {\vphantom {{2\mu_{0}^{2} } {(1 - \mu_{0} )}}} \right. \kern-0pt} {(1 - \mu_{0} )}})E_{\text{II}}^{\text{eq}}$$
(15c)

in which \(\mu_{0}\) is the Poisson’s ratio of DM column; \(\omega\) is a parameter decided by the shape of loading plane; and \(E_{\text{II}}^{\text{eq}}\) is the equivalent constrained modulus of soil in Region II.

Thus, by combining Eqs. (11a), (13a), (14a), and (3b), one can obtain:

$$({{(\sigma_{\text{s}} l_{0} - {{\lambda \tau_{0} l_{0}^{2} } \mathord{\left/ {\vphantom {{\lambda \tau_{0} l_{0}^{2} } 3}} \right. \kern-0pt} 3})} \mathord{\left/ {\vphantom {{(\sigma_{\text{s}} l_{0} - {{\lambda \tau_{0} l_{0}^{2} } \mathord{\left/ {\vphantom {{\lambda \tau_{0} l_{0}^{2} } 3}} \right. \kern-0pt} 3})} {E_{\text{I}}^{\text{eq}} }}} \right. \kern-0pt} {E_{\text{I}}^{\text{eq}} }} = p_{\text{c}} (\sigma_{\text{p}} - \sigma_{\text{s}} ) + {{(\sigma_{\text{p}} l_{0} + {{\gamma \tau_{0} l_{0}^{2} } \mathord{\left/ {\vphantom {{\gamma \tau_{0} l_{0}^{2} } 3}} \right. \kern-0pt} 3})} \mathord{\left/ {\vphantom {{(\sigma_{\text{p}} l_{0} + {{\gamma \tau_{0} l_{0}^{2} } \mathord{\left/ {\vphantom {{\gamma \tau_{0} l_{0}^{2} } 3}} \right. \kern-0pt} 3})} {E_{\text{p}} }}} \right. \kern-0pt} {E_{\text{p}} }}$$
(16)

Similarly, combining Eq. (11b), Eq. (13b), Eq. (15a), and Eq. (3c) results in

$$\begin{aligned} & \frac{1}{{E_{\text{I}}^{\text{eq}} }}\left( {\frac{{\lambda \tau_{0} }}{{6l_{0} }}l_{1}^{3} - \frac{{\lambda \tau_{0} }}{2}l_{1}^{2} + \sigma_{\text{s}} l_{1} - \sigma_{\text{s}} l_{0} + \frac{{\lambda \tau_{0} }}{3}l_{0}^{2} } \right) \\ & \quad = \frac{1}{{E_{\text{p}} }}\left( { - \frac{{\gamma \tau_{0} }}{{6l_{0} }}l_{1}^{3} + \frac{{\gamma \tau_{0} }}{2}l_{1}^{2} + \sigma_{\text{p}} l_{1} - \sigma_{\text{p}} l_{0} - \frac{{\gamma \tau_{0} }}{3}l_{0}^{2} } \right) \\ & \quad \quad + p_{\text{s}} \left[ {\sigma_{\text{p}} - \sigma_{\text{s}} + (\gamma + \lambda )\tau_{0} l_{1} - \frac{{(\gamma + \lambda )\tau_{0} l_{1}^{2} }}{{2l_{0} }}} \right]. \\ \end{aligned}$$
(17)

Substituting Eq. (4) into Eqs. (16) and (17), the stress concentration ratio can be expressed as:

$$n = \frac{{\sigma_{\text{p}} }}{{\sigma_{\text{s}} }} = \frac{{\theta_{1} }}{{\theta_{2} }} = \frac{{\theta_{3} }}{{\theta_{4} }},$$
(18)

in which \(\theta_{1} = \frac{{l_{0} }}{{E_{\text{I}}^{\text{eq}} }} + p_{\text{c}} - \frac{{\lambda K\tan \varphi_{\text{i}} }}{{3E_{\text{I}}^{\text{eq}} }}l_{0}^{2} - \frac{{\gamma K\tan \varphi_{\text{i}} }}{{3E_{\text{p}} }}l_{0}^{2}\); \(\theta_{2} = p_{\text{c}} + \frac{{l_{0} }}{{E_{\text{p}} }}\); \(\theta_{4} = p_{\text{s}} + \frac{{(l_{1} - l_{0} )}}{{E_{\text{p}} }}\)

$$\begin{aligned} \theta_{3} & = \frac{1}{{E_{\text{I}}^{\text{eq}} }}\left( {\frac{{\lambda Kl_{1}^{3} \tan \varphi_{\text{i}} }}{{6l_{0} }} - \frac{{\lambda Kl_{1}^{2} \tan \varphi_{\text{i}} }}{2} + l_{1} - l_{0} + \frac{{\lambda Kl_{0}^{2} \tan \varphi_{\text{i}} }}{3}} \right) \\ & \quad - \frac{1}{{E_{\text{p}} }}\left( { - \frac{{\gamma Kl_{1}^{3} \tan \varphi_{\text{i}} }}{{6l_{0} }} + \frac{{\gamma Kl_{1}^{2} \tan \varphi_{\text{i}} }}{2} - \frac{{\gamma Kl_{0}^{2} \tan \varphi_{\text{i}} }}{3}} \right) \\ & \quad \quad - p_{\text{s}} \left( { - 1 + (\gamma + \lambda )Kl_{1} \tan \varphi_{\text{i}} - \frac{{(\gamma + \lambda )Kl_{1}^{2} \tan \varphi_{\text{i}} }}{{2l_{0} }}} \right). \\ \end{aligned}$$

In Eq. (18), only \(l_{0}\) is the unknown variation. By solving Eq. (18), \(l_{0}\) and n can be obtained. Then \(\sigma_{\text{p}}\) and \(\sigma_{\text{s}}\) can also be solved by Eqs. (6b) and (6c). One can obtain \(S_{\text{su}}\), \(S_{\text{sd}}\), \(\delta_{1}\), \(\delta_{2}\) from Eqs. (11a), (11b) and Eqs. (13a), and (13b), respectively. The settlement in Region I can be calculated as:

$$S_{\rm I} = S_{\text{su}} + S_{\text{sd}} = {{({{\lambda \tau_{0} l_{1}^{3} } \mathord{\left/ {\vphantom {{\lambda \tau_{0} l_{1}^{3} } {(6l_{0} ) - {{\lambda \tau_{0} l_{1}^{2} } \mathord{\left/ {\vphantom {{\lambda \tau_{0} l_{1}^{2} } 2}} \right. \kern-0pt} 2} + \sigma_{\text{s}} l_{1} }}} \right. \kern-0pt} {(6l_{0} ) - {{\lambda \tau_{0} l_{1}^{2} } \mathord{\left/ {\vphantom {{\lambda \tau_{0} l_{1}^{2} } 2}} \right. \kern-0pt} 2} + \sigma_{\text{s}} l_{1} }})} \mathord{\left/ {\vphantom {{({{\lambda \tau_{0} l_{1}^{3} } \mathord{\left/ {\vphantom {{\lambda \tau_{0} l_{1}^{3} } {(6l_{0} ) - {{\lambda \tau_{0} l_{1}^{2} } \mathord{\left/ {\vphantom {{\lambda \tau_{0} l_{1}^{2} } 2}} \right. \kern-0pt} 2} + \sigma_{\text{s}} l_{1} }}} \right. \kern-0pt} {(6l_{0} ) - {{\lambda \tau_{0} l_{1}^{2} } \mathord{\left/ {\vphantom {{\lambda \tau_{0} l_{1}^{2} } 2}} \right. \kern-0pt} 2} + \sigma_{\text{s}} l_{1} }})} {E_{\rm I}^{\text{eq}} }}} \right. \kern-0pt} {E_{\rm I}^{\text{eq}} }}.$$
(19)

2.3 Compression of soil in Region II

The unconfined compressive strength of DM column is typically within a range of 0.4–1.5 MPa for the soft clays in practice [12]. The elastic modulus of DM columns could be estimated based on the typical relationship of E = 100qu [12, 19]. Thus, the elastic modulus of DM columns may be within a range from 40 to 150 MPa. Due to the relative low modulus difference between the DM columns and the surrounding soil as compared with the difference between rigid columns and the surrounding soil, it is assumed that the DM column and the surrounding soil settles under an equal strain condition in the analysis of the compression of the soil in Region II. Zhang et al. [42] also made an equal strain assumption when deriving the consolidation solution of a composite foundation with short DM columns and long PVDs and a good agreement was obtained as compared with the field data. The equivalent modulus of the soil in Region II is determined based on the area-weighted average of the constrained modulus of the DM column and the surrounding soft soil:

$$E_{\text{II}}^{\text{eq}} = mE_{\text{DM}} + (1 - m)E_{{{\text{s}}{\rm I}{\rm I}}} ,$$
(20)

in which \(E_{\text{II}}^{\text{eq}}\) is the equivalent constrained modulus of the improvement area in Region II; and \(E_{{{\text{s}}{\rm I}{\rm I}}}\) is the constrained modulus of soil.

The stress solution for a three-layer system proposed by Jones [22] was used to calculate the vertical stresses on the top and bottom of Region II. The solution to the compression of the soil in Region II can be given as:

$$S_{\text{II}} = {{\left[ {\left( {\sigma_{1} + \sigma_{2} } \right)l_{2} } \right]} \mathord{\left/ {\vphantom {{\left[ {\left( {\sigma_{1} + \sigma_{2} } \right)l_{2} } \right]} {\left( {2E_{\text{II}}^{\text{eq}} } \right)}}} \right. \kern-0pt} {\left( {2E_{\text{II}}^{\text{eq}} } \right)}},$$
(21)

in which \(\sigma_{1}\) is the vertical stress on the bottom of Region I; and \(\sigma_{2}\) is the vertical stress on the bottom of Region II.

2.4 Compression of soil in Region III

The compression of soil in Region III is computed using Mindlin’s solution:

$$S_{\text{III}} = {{\eta \sigma_{2} l_{3} } \mathord{\left/ {\vphantom {{\eta \sigma_{2} l_{3} } {E_{\text{III}} }}} \right. \kern-0pt} {E_{\text{III}} }},$$
(22)

in which \(\eta\) is the coefficient of average superimposed stress; and \(E_{\text{III}}\) is the constrained modulus of soil in Region III.

Based on the above derivations, the solutions to the compressions of the three regions are developed. Substituting the compressions of the three regions into Eq. (1), the total settlement of the soil reinforced by SDM column can be obtained.

3 Model validation and parametric study

3.1 Model validation

To validate the derived solution of the settlement of SDM column-reinforced soil, a three-dimensional (3-D) finite element analysis incorporated in the software ABAQUS was adopted. The results calculated by the proposed analytical method were compared with the results by the 3-D numerical analysis. Three typical conditions were considered in the numerical models: (a) the SDM column with equal length of DM column and core pile were seated on a firm soil (Model 1); (b) the SDM column with DM column longer than core pile were seated on a firm soil (Model 2); (c) there was a weak underlying stratum below the base of SDM column with long DM column and short core pile (Model 3).

Figure 7 shows the configuration of Model 3. The subsoil consisted of a 15-m-thick soft soil underlain by a 5-m-thick firm soil layer. The SDM column had DM column with a diameter of 500 mm and a length of 10 m and concrete core pile with a diameter of 250 mm and a length of 6 m. The columns were installed in a square pattern at a spacing of 1.8 m. A 0.3-m-thick gravel cushion and 4.0-m-thick embankment fill were placed on the top of SDM column-reinforced soft soil. The other two numerical models (i.e., Model 1 and Model 2) had a same geological condition but different SDM column configuration. Figure 8 shows the mesh of Model 1. The 3-D finite element mesh consisted of 5714 C3D8 (i.e., an eight-node brick, trilinear displacement) solid elements for the meshes of core pile and 13039 C3D8R solid elements (i.e., an eight-node brick, trilinear displacement, reduced integration) for the meshes of cushion, DM column, and subsoils. Table 1 shows the geometrical information of the three numerical models. In some cases, the pile is modeled using structural element (for example, beam element or pile element) for simplicity and improving computation efficiency. The beam element is unable to obtain the stress distribution throughout the cross section. In this study, as it needs to obtain the stresses distributed on the cross section of pile to calculate the stress concentration ratio, the solid element instead of the structural element is used to simulate the pile. Furthermore, as the deformation of pile itself is very small under the embankment, the pile is modeled as elastic material with Young’s modulus and Poisson’s ratio. The sand cushion, the DM column, and the subsoil were modeled as linearly elastic–perfectly plastic materials with Mohr–Coulomb failure criteria [15, 40]. In the elastic part, the elastic modulus of DM column was determined based on the typical relationship of E = 100qu, and the Poisson’s ratio was estimated based on the empirical values. As the unconfined compressive strength of 1 MPa is very common in practice, E = 100 MPa was used in the numerical analysis. In the plastic part, the friction angle was taken as 30° and the cohesion was estimated as half of the unconfined compressive strength (500 kPa) in this study. Various studies investigated the friction angle of soil–cement [2, 23, 43], and it is typically in a range of 20°–35°. The parameters of concrete core pile can be obtained based on the criteria of concrete strength grades. The concretes with strength grades of C20–C35 (unconfined compression strengths from 20 to 35 MPa) are usually used to produce the core pile. Their corresponding elastic moduli are from 2.55 to 3.15 GPa, and the Poisson’s ratio is typically from 0.1 to 0.22. In the numerical analysis, elastic modulus of 2.55 GPa and Poisson’s ratio of 0.15 were used. The interface contact was created between the concrete core pile and DM column with interfacial friction angle of 18° [31] and the shear stiffness coefficient \(K_{\text{ss}}\) = 80 MPa/m [31, 40]. Table 2 shows the material properties used in the numerical models. The four side boundaries of models were fixed in the lateral displacement but allowed to move freely in the vertical direction. The bottom boundary was fixed in all three directions (i.e., x, y, and z directions). After completion of the initial geostatic stress balance, the sand cushion and embankment fill were placed on the base of SDM column-reinforced soil sequentially.

Fig. 7
figure 7

Unit cell model for floating SDM pile supported embankment (not to scale; unit: m): a cross section, b plan view

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Fig. 8
figure 8

3-D finite element mesh (unit: m)

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Table 1 Geometrical information of numerical models
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Table 2 Main physico-mechanical parameters of materials in numerical analysis
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Figure 9 shows the settlements predicted by the numerical analysis and the developed solution. It can be seen that in the three models, the compressions in Region II and III by the numerical analysis and the developed solution agreed well with each other, while the compression in Region I by the developed solution was greater than that by the numerical analysis. In general, the proposed method properly revealed the deformation behavior of SDM column-supported embankment over soft soil.

Fig. 9
figure 9

Comparison of predictions by 3-D analysis and developed solution

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3.2 Parametric study and discussion

To investigate the settlement characteristics of SDM column-reinforced soft soil, a parametric study was conducted in this section. In the baseline case, the soft soil consisted of a 15-m-thick soft soil. SDM columns were installed in a square pattern with a spacing of 1.8 m. The DM column had a diameter of 500 mm and a length of 10.0 m and the core concrete pile had a diameter of 250 mm and a length of 6.0 m. The interface friction angle was 18°. The gravel cushion had a thickness of 0.3 m and the embankment had a thickness of 4.0 m. The moduli of the soft soil, the DM column, the core pile and the gravel cushion were 5, 100, and 28,000, 30 MPa, respectively. In the following analysis, the influence factors of DM column modulus, length of core pile and DM column, diameter of DM column and core pile, and the interaction coefficient of the interface between DM column and core pile on the settlement characteristic of SDM column-reinforced soil were explored. In each case, only one influence factor was changed compared with the baseline case and a settlement ratio which is defined as a ratio of the settlement of the case with a changed parameter to that of the baseline case was introduced to investigate the role the influence factor played in the system. Based on the proposed method, the settlement of the baseline case was S0 = 131.5 mm.

Figure 10 shows the change of settlement ratio with different influence factors. Figure 10a–c shows that the settlement of SDM-reinforced soft soil can be reduced by increasing the modulus, length, and diameter of DM column. Increasing the length of DM column had a significant influence on mitigating the settlement. Figure 10d, e shows that the settlement of SDM-reinforced soft soil was reduced by increasing the length of core pile, and the change in the diameter of core pile had less influence on the settlement than the change in the length of core pile. Voottipruex et al. [31] also indicated that the section area of concrete core pile had a slight effect on the settlement of SDCM column based on the numerical parametric analysis. However, when the length of core pile was greater than 9 m and the diameter of core pile changed from 0.15 to 0.25 m, which corresponded to the ratio of DM column length to core pile length of 0.9, and the area ratio of core pile in the cross section of SDM from 10 to 25%, minor influences on mitigating settlement were obtained. Voottipruex et al. [33] found that the effective length ratio ranges from 0.57 to 0.85 in terms of the bearing capacity of SDM column. From a practical point of view, increasing the length of core pile is the preferred option on mitigating the settlement of soft soil when adopting the SDM column, and the optimal length and diameter of core pile should be considered in design.

Fig. 10
figure 10

Influence of different parameters on settlement: a modulus of DM column, b length of DM column, c diameter of SDM column, d length of core pile, e diameter of core pile, f interface friction angle

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Figure 10f shows that the settlement of ground was reduced slightly with the increase of interface friction angle \(\varphi_{\text{i}}\). It is because that the increase in the settlement due to the decrease in interface friction was small as compared with the total settlement. To clearly detect the effect of the interface, a ratio of the settlement in Region I between the case with a changed parameter to that of the baseline case was calculated alone, which is also included in Fig. 10f. It can be seen that the settlement ratio of Region I changes from 1.06 to 0.87 with an increase in interface friction angle from 6° to 30°. Therefore, the influence of interface friction angle is not as remarkable as the influence factors shown in the preceding Fig. 10a–e. This finding was similar to the numerical results by Voottipruex et al. [33].

4 Design method

Based on the above derivations, the settlements in Region II and Region III can be directly obtained based on the developed solutions, while the calculation of the settlement in Region I was complex. For a convenient application in design, this section developed a design procedure for the calculation of the settlement in Region I based on the derived solution. The proposed design method is applicable to the cases when the core pile length is smaller or equal to the length of DM column.

The settlement of Region I can be solved by Eq. (19). A 0.3-m-thick gravel cushion is typically used in practice, and thus the thickness of gravel cushion is kept constant as 0.3 m in the design method. Substituting Eqs. (4) and (6c) into Eq. (19), setting \(a = {{\lambda k\tan \varphi_{\text{i}} } \mathord{\left/ {\vphantom {{\lambda k\tan \varphi_{\text{i}} } 3}} \right. \kern-0pt} 3}\), \(b = {{\gamma k\tan \varphi_{\text{i}} } \mathord{\left/ {\vphantom {{\gamma k\tan \varphi_{\text{i}} } 3}} \right. \kern-0pt} 3}\), and ignoring the terms with \(E_{\text{p}}\) as they were small enough due to the large magnitude of \(E_{\text{p}}\) as the denominator, Eq. (19) is converted as follows:

$$S_{\rm I} = P\psi = P\left( {\frac{{{{al_{1}^{3} } \mathord{\left/ {\vphantom {{al_{1}^{3} } {(2l_{0} ) - 1.5al_{1}^{2} + l_{1} }}} \right. \kern-0pt} {(2l_{0} ) - 1.5al_{1}^{2} + l_{1} }}}}{{{{\alpha m(l_{0} - al_{0}^{2} )} \mathord{\left/ {\vphantom {{\alpha m(l_{0} - al_{0}^{2} )} {p_{\text{c}} }}} \right. \kern-0pt} {p_{\text{c}} }} + E_{\rm I}^{\text{eq}} }}} \right),$$
(23a)
$$\delta l_{0}^{3} + \zeta l_{0}^{2} + \varepsilon l_{0} + \xi = 0,$$
(23b)

in which \(\delta = a(2 + 2{{p_{\text{s}} } \mathord{\left/ {\vphantom {{p_{\text{s}} } {p_{\text{c}} }}} \right. \kern-0pt} {p_{\text{c}} }})\); \(\zeta = {{ - \delta } \mathord{\left/ {\vphantom {{ - \delta } a}} \right. \kern-0pt} a}\); \(\varepsilon = 2l_{1} - 3al_{1}^{2} - 6p_{\text{s}} (a + b)E_{\rm I}^{\text{eq}}\); \(\xi = al_{1}^{3} { + }3p_{\text{s}} (a + b)l_{1} E_{\rm I}^{\text{eq}}\). Setting \(c = {{l_{0} } \mathord{\left/ {\vphantom {{l_{0} } {l_{1} }}} \right. \kern-0pt} {l_{1} }}\) and thus Eqs. (23a) and (23b) is calculated as:

$$S_{\rm I} { = }P\psi = P\left( {\frac{{{{al_{1}^{2} } \mathord{\left/ {\vphantom {{al_{1}^{2} } {(2c) - 1.5al_{1}^{2} + l_{1} }}} \right. \kern-0pt} {(2c) - 1.5al_{1}^{2} + l_{1} }}}}{{{{\alpha m(cl_{1} - ac^{2} l_{1}^{2} )} \mathord{\left/ {\vphantom {{\alpha m(cl_{1} - ac^{2} l_{1}^{2} )} {p_{\text{c}} }}} \right. \kern-0pt} {p_{\text{c}} }} + E_{\rm I}^{\text{eq}} }}} \right),$$
(24a)
$$\delta l_{1}^{3} c^{3} + \zeta l_{1}^{2} c^{2} + \varepsilon l_{1} c + \xi = 0.$$
(24b)

A design procedure is recommended herein to determine \(\psi\), which is also summarized in the flow chart as shown in Fig. 11. The design charts corresponding to the flow chart are presented in Fig. 12. For the charts illustrated with a linear distribution, it is reasonable to extend the lines if the variable is not in the domain presented in the chart. The design procedure is referred as follows.

Fig. 11
figure 11

Proceeding of design curves of Region I

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Fig. 12
figure 12figure 12

Settlement design curves of Region I

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  1. 1.

    Choose the trial parameters of SDM column-reinforced soil, including d, D, B, l1, and l2. Then calculated the replacement ratio of SDM column, m, the core ratio,\(\alpha\). Based on the strength of DM column, estimate the coefficient of earth pressure and the interface friction angle of DM column and core pile, \(\varphi_{\text{i}}\).

  2. 2.

    Use the parameters of \(\varphi_{\text{i}}\), d and \(\alpha m\) to calculate \(b\) and \(a\).

  3. 3.

    Calculate \(E_{\rm I}^{\text{eq}}\), \(E_{{{\rm I}{\rm I}}}^{\text{eq}}\) and \(p_{\text{c}}\) from Eqs. (7), (20) and (14b), respectively. For multiple soil layers, \(E_{{{\text{s}}{\rm I}}} = {{(E_{{{\text{s}}1}} l_{11} + E_{{{\text{s}}2}} l_{12} + \cdots + E_{{{\text{s}}n}} l_{1n} )} \mathord{\left/ {\vphantom {{(E_{{{\text{s}}1}} l_{11} + E_{{{\text{s}}2}} l_{12} + \cdots + E_{{{\text{s}}n}} l_{1n} )} {l_{1} }}} \right. \kern-0pt} {l_{1} }}\), in which \(E_{\text{si}}\) and \(l_{{1{\text{i}}}}\) is the constrained modulus and thickness of each layer in Region I. Determine \(p_{\text{s}}\) and \({{p_{\text{s}} } \mathord{\left/ {\vphantom {{p_{\text{s}} } {p_{\text{c}} }}} \right. \kern-0pt} {p_{\text{c}} }}\) from the design chart (Fig. 12a).

  4. 4.

    Use Fig. 12b, c to obtain \(u_{1}\), \(u_{2}\), \(u_{3}\), and \(u_{4}\). Calculate \(u = {{({{(u_{1} - u_{2} )} \mathord{\left/ {\vphantom {{(u_{1} - u_{2} )} {u_{3} }}} \right. \kern-0pt} {u_{3} }} - u_{4} )} \mathord{\left/ {\vphantom {{({{(u_{1} - u_{2} )} \mathord{\left/ {\vphantom {{(u_{1} - u_{2} )} {u_{3} }}} \right. \kern-0pt} {u_{3} }} - u_{4} )} {l_{1}^{2} }}} \right. \kern-0pt} {l_{1}^{2} }}\).

  5. 5.

    Use Fig. 12d–f to determine \(v_{1}\) to \(v_{6}\). Calculate \(v = {{{{((v_{1} + v_{2} )} \mathord{\left/ {\vphantom {{((v_{1} + v_{2} )} {(2 + 2{{p_{\text{s}} } \mathord{\left/ {\vphantom {{p_{\text{s}} } {p_{\text{c}} }}} \right. \kern-0pt} {p_{\text{c}} }})}}} \right. \kern-0pt} {(2 + 2{{p_{\text{s}} } \mathord{\left/ {\vphantom {{p_{\text{s}} } {p_{\text{c}} }}} \right. \kern-0pt} {p_{\text{c}} }})}} + {{(v_{3} - v_{4} )} \mathord{\left/ {\vphantom {{(v_{3} - v_{4} )} {v_{6} }}} \right. \kern-0pt} {v_{6} }} - v_{5} )} \mathord{\left/ {\vphantom {{{{((v_{1} + v_{2} )} \mathord{\left/ {\vphantom {{((v_{1} + v_{2} )} {(2 + 2{{p_{\text{s}} } \mathord{\left/ {\vphantom {{p_{\text{s}} } {p_{\text{c}} }}} \right. \kern-0pt} {p_{\text{c}} }})}}} \right. \kern-0pt} {(2 + 2{{p_{\text{s}} } \mathord{\left/ {\vphantom {{p_{\text{s}} } {p_{\text{c}} }}} \right. \kern-0pt} {p_{\text{c}} }})}} + {{(v_{3} - v_{4} )} \mathord{\left/ {\vphantom {{(v_{3} - v_{4} )} {v_{6} }}} \right. \kern-0pt} {v_{6} }} - v_{5} )} {l_{1} }}} \right. \kern-0pt} {l_{1} }}^{3}\)

  6. 6.

    Based on step 5, use Table 7 to obtain the solution \(x\), and \(c = x + {1 \mathord{\left/ {\vphantom {1 {(3al_{1} )}}} \right. \kern-0pt} {(3al_{1} )}}\).

  7. 7.

    Use Fig. 12g, h to determine s1 and s2, and then the settlement coefficient of region I can be calculated as \(\psi = {{s_{1} } \mathord{\left/ {\vphantom {{s_{1} } {s_{2} }}} \right. \kern-0pt} {s_{2} }}\),

  8. 8.

    The settlement of Region I can be obtained by \(S_{\rm I} = P\psi\).

5 Application of the proposed method

5.1 Test embankment in AIT campus, Thailand [33]

Vootttipruex et al. [33] conducted a large scale field test of SDM column-supported embankment at the northern part of the Asian Institute of Technology (AIT) Campus, Thailand. Figure 13 shows the configuration of the test embankment. Vootttipruex et al. [33] introduced the geotechnical conditions and the test procedures on this site in detail. Therefore, this section only provides a brief summary. The test embankment consisting of a 1-m-thick sand cushion and 5-m-thick embankment fill was constructed with end slopes of 1:1 and side slopes of 1:1.5. The test embankment had base dimensions of 21 m × 21 m and top dimensions of 9 m × 6 m. The SDM columns with a diameter of 0.6 m and a length of 7 m were installed in a square pattern at a spacing of 2.0 m. The concrete core piles had a square cross section of 0.22 m × 0.22 m and a length of 6 m. Table 3 shows the main properties of the soil layers and the SDM columns used in the field test, which were provided by Vootttipruex et al. [33].

Fig. 13
figure 13

Test embankment (unit: m, after Vootttipruex et al. [33])

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Table 3 Main properties of materials in the test
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The side length of core pile d = 0.22 m, so \(\alpha m\) = 0.0121. Considering the core pile was directly pushed into the soil, a relative large K was selected herein (i.e., it is assumed to be the coefficient of passive lateral earth pressure in the design). The friction angles of DM column was taken as 30° [2, 13, 23, 43] and the interfacial friction angel between the DM column and the core pile was taken as 18° [31], and it can be calculated that \(b\) = 5.8, and \(a\) = 0.055.

The equivalent modulus of Region I is 5.0 MPa. The equivalent modulus of Region II is 5.5 MPa. The Poisson’s ratio \(\mu_{0}\) is 0.33, \(p_{\text{c}} = {{l_{\text{SC}} } \mathord{\left/ {\vphantom {{l_{\text{SC}} } {E_{\text{c}} }}} \right. \kern-0pt} {E_{\text{c}} }}\) = 0.0333, using d = 0.22 m, \(E_{{{\rm I}{\rm I}}}^{\text{eq}}\) = 5.5 MPa, and it can be obtained from Fig. 12a that \(p_{\text{s}}\) = 0.046, \({{p_{\text{s}} } \mathord{\left/ {\vphantom {{p_{\text{s}} } {p_{\text{c}} }}} \right. \kern-0pt} {p_{\text{c}} }}\) = 1.38.

The length of Region I (l1) is 6 m, using \(a\), \(\alpha m\), \(E_{{{\rm I}{\rm I}}}^{\text{eq}}\),\(p_{\text{s}}\), it can be obtained from Fig. 12b, c that \(u_{1}\) = 1, \(u_{2}\) = 1, \(u_{3}\) = 0.043, \(u_{4}\) = 110.2. So \(u = {{({{(u_{1} - u_{2} )} \mathord{\left/ {\vphantom {{(u_{1} - u_{2} )} {u_{3} }}} \right. \kern-0pt} {u_{3} }} - u_{4} )} \mathord{\left/ {\vphantom {{({{(u_{1} - u_{2} )} \mathord{\left/ {\vphantom {{(u_{1} - u_{2} )} {u_{3} }}} \right. \kern-0pt} {u_{3} }} - u_{4} )} {l_{1}^{2} }}} \right. \kern-0pt} {l_{1}^{2} }}\) = -−3.06. Similarly, it can be obtained from Fig. 12d–f that \(v_{1}\) = 216, \(v_{2}\) = 400, \(v_{3}\) = 6.1, \(v_{4}\) = 6.6, \(v_{5}\) = 445.2, \(v_{6}\) = 0.043. So \(v = {{({{(v_{1} + v_{2} )} \mathord{\left/ {\vphantom {{(v_{1} + v_{2} )} {(2 + 2{{p_{\text{s}} } \mathord{\left/ {\vphantom {{p_{\text{s}} } {p_{\text{c}} }}} \right. \kern-0pt} {p_{\text{c}} }})}}} \right. \kern-0pt} {(2 + 2{{p_{\text{s}} } \mathord{\left/ {\vphantom {{p_{\text{s}} } {p_{\text{c}} }}} \right. \kern-0pt} {p_{\text{c}} }})}} + {{(v_{3} - v_{4} )} \mathord{\left/ {\vphantom {{(v_{3} - v_{4} )} {v_{6} }}} \right. \kern-0pt} {v_{6} }} - v_{5} )} \mathord{\left/ {\vphantom {{({{(v_{1} + v_{2} )} \mathord{\left/ {\vphantom {{(v_{1} + v_{2} )} {(2 + 2{{p_{\text{s}} } \mathord{\left/ {\vphantom {{p_{\text{s}} } {p_{\text{c}} }}} \right. \kern-0pt} {p_{\text{c}} }})}}} \right. \kern-0pt} {(2 + 2{{p_{\text{s}} } \mathord{\left/ {\vphantom {{p_{\text{s}} } {p_{\text{c}} }}} \right. \kern-0pt} {p_{\text{c}} }})}} + {{(v_{3} - v_{4} )} \mathord{\left/ {\vphantom {{(v_{3} - v_{4} )} {v_{6} }}} \right. \kern-0pt} {v_{6} }} - v_{5} )} {l_{1}^{3} }}} \right. \kern-0pt} {l_{1}^{3} }}\) = −1.52. It can be obtained from Table 4 that \(x\) = −0.56, \(c = x + {1 \mathord{\left/ {\vphantom {1 {(3al_{1} )}}} \right. \kern-0pt} {(3al_{1} )}}\) = 0.45.

Table 4 Settlements by the proposed method and by the field monitor data (mm)
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Using c, a, l1, it can be obtained from Fig. 12g, h that \(s_{1}\) = 4.7, \(s_{2}\) = 8.8. So \(\psi = {{s_{1} } \mathord{\left/ {\vphantom {{s_{1} } {s_{2} }}} \right. \kern-0pt} {s_{2} }}\) = 0.53, the maximum loading is 80 kPa, so the settlement of Region I is \(S_{\rm I} { = }P\psi\) = 80 × 0.53 = 42.4 mm.

According to Jones’s solution, the key parameters can be summarized as follows: ① \(k_{ 1} = {{E_{1} } \mathord{\left/ {\vphantom {{E_{1} } {E_{2} }}} \right. \kern-0pt} {E_{2} }}\) = 5/5.5 = 0.91, ② \(k_{ 2} = {{E_{2} } \mathord{\left/ {\vphantom {{E_{2} } {E_{3} }}} \right. \kern-0pt} {E_{3} }}\) = 5.5/9 = 0.61, ③ \(H = {{h_{1} } \mathord{\left/ {\vphantom {{h_{1} } {h_{2} }}} \right. \kern-0pt} {h_{2} }}\) = 6/3 = 2, ④ \(a_{ 1} = {r \mathord{\left/ {\vphantom {r {h_{2} }}} \right. \kern-0pt} {h_{2} }}\) = 11.85/3 = 3.95.

Using linear interpolation, the vertical stress of Region II can be obtained as \(\sigma_{1}\) = 0.91 × 80, \(\sigma_{2}\) = 0.89 × 80, and the vertical stress on the bottom of medium stiff clay is 0.85 × 80, so the settlement of Region II is \(S_{\text{II}} = {{\left[ {\left( {\sigma_{1} + \sigma_{2} } \right)l_{2} } \right]} \mathord{\left/ {\vphantom {{\left[ {\left( {\sigma_{1} + \sigma_{2} } \right)l_{2} } \right]} {\left( {2E_{\text{II}}^{\text{eq}} } \right)}}} \right. \kern-0pt} {\left( {2E_{\text{II}}^{\text{eq}} } \right)}}\) = 13.1 mm. The settlement of medium stiff clay is 25.3 mm.

The coefficient of average superimposed stress \(\eta\) in third layer is 0.7, so the settlement of Region III is \(S_{\text{III}} = {{\eta \sigma_{2} l_{3} } \mathord{\left/ {\vphantom {{\eta \sigma_{2} l_{3} } {E_{\text{III}} }}} \right. \kern-0pt} {E_{\text{III}} }}\) +25.3 = 108.4 mm.

Figure 14 shows the variation of the ground settlement with time. According to the Asaoka method [3], the final settlement can be predicted based on the field data (see Fig. 11b), \(s_{\infty }\) = 161.3 mm. Table 4 presents the comparison of the settlement calculated using the proposed method with that using Asaoka method and they agreed well with each other.

Fig. 14
figure 14

Final settlement predicted by Asaoka method

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5.2 Field test in Nanjing, China [34]

Wang et al. [34] introduced two sections of field test of SDM column-supported embankment over soft soil at Nanjing Surrounding Expressway. Figure 15 shows the cross section of the two field tests. The SDM columns with a diameter of 0.5 mm were installed in a triangle pattern to a depth of 19.5 m. The column spacing was 1.5 m and 1.7 m in Site A and Site B, respectively. The width of embankment was 34.5 m with a side slope of 1:1.5. Table 5 shows the material properties in each test site.

Table 5 Main parameters of subsoils
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Table 6 shows the final settlements predicted by the proposed method and the Asaoka method [3]. The good agreements were obtained. Based on the application in the two field tests, it can be concluded that the proposed method is feasible for calculation of the SDM column-supported embankment over soft soil.

Table 6 Comparison of proposed method with field data (mm)
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6 Conclusion

Assuming the soil to be settled as one-dimensional deformation, this paper proposed an analytical method for calculating the settlement of SDM column-supported embankment over soft clay. The proposed method was verified by a comparison with the 3-D numerical model and field data. A parametric study was conducted to investigate the settlement characteristics of SDM column-supported embankment over soft soil. Based on the analysis and discussion, the following conclusions can be drawn.

  1. (a)

    Considering the upward and downward penetration effects of concrete core pile and the geometry characteristics in the cross section, the unit cell model was established, and the solution of the settlement of SDM column-reinforced soil was developed. The results by the developed solution agreed well with the results by the 3-D finite element models with three typical configurations.

  2. (b)

    The settlement of SDM-reinforced soft soil is reduced by increasing the modulus, diameter and length of DM column, the length of core pile. The cross section area of core pile and the interface friction angle had a minor influence on the settlement. From a practical point of view, increasing the length of core pile is the preferred option on mitigating the settlement of soft soil when adopting the SDM column, and the optimal length and diameter of core pile should be considered in design.

  3. (c)

    The design procedure as well as design charts were developed for the calculation of settlement of SDM column-reinforced soft sol. The design method was applied to two case histories of SDM column-supported embankments and a good agreement was found between the predicted settlements and the field measurements.