materials Review Modeling Stone Columns Jorge Castro Department of Ground Engineering and Materials Science, University of Cantabria, Avda. de los Castros s/n, 39005 Santander, Spain; [email protected]; Tel.: +34-942-201813 Received: 4 May 2017; Accepted: 7 July 2017; Published: 11 July 2017 Abstract: This paper reviews the main modeling techniques for stone columns, both ordinary stone columns and geosynthetic-encased stone columns. The paper tries to encompass the more recent advances and recommendations in the topic. Regarding the geometrical model, the main options are the “unit cell”, longitudinal gravel trenches in plane strain conditions, cylindrical rings of gravel in axial symmetry conditions, equivalent homogeneous soil with improved properties and three-dimensional models, either a full three-dimensional model or just a three-dimensional row or slice of columns. Some guidelines for obtaining these simplified geometrical models are provided and the particular case of groups of columns under footings is also analyzed. For the latter case, there is a column critical length that is around twice the footing width for non-encased columns in a homogeneous soft soil. In the literature, the column critical length is sometimes given as a function of the column length, which leads to some disparities in its value. Here it is shown that the column critical length mainly depends on the footing dimensions. Some other features related with column modeling are also briefly presented, such as the influence of column installation. Finally, some guidance and recommendations are provided on parameter selection for the study of stone columns. Keywords: stone columns; encased stone columns; geosynthetic; numerical modeling; critical length; parameter selection 1. Introduction Ground improvement using stone columns, also known as granular piles or aggregate piers, is one of the most popular techniques to improve soft soils for the foundation of embankments or structures. These are vertical boreholes in the ground, filled upwards with gravel compacted by means of a vibrator. The idea of improving soft soils for foundation purposes using granular inclusions is relatively old. It is documented [1] that in 1839 in Bayonne (France), the French colonel Burbach used for the first time sand piles as deep foundations instead of the classical wood piles that rapidly degrade with fluctuations of the ground water level. However, it was not until the 50 s of the last century when stone columns started to be used. The ground improvement technique started as an extension of traditional vibro-compaction (deep compaction) to non-granular soils, whose low permeability and cohesion do not allow for a quick rearranging of soil particles in a denser configuration. Stone columns act mainly as inclusions with a higher stiffness, shear strength and permeability than the natural soil. Consequently, they improve the following aspects: • • • • • The bearing capacity The stability of embankments and natural slopes Final settlement Degree of consolidation Liquefaction potential Materials 2017, 10, 782; doi:10.3390/ma10070782 www.mdpi.com/journal/materials Materials 2017, 10, 782 2 of 23 The reduction of the liquefaction potential is beyond the scope of the present paper. Please refer to [2,3] for further information on that topic. The present paper focuses on the other four improvements, particularly, the settlement reduction. The modeling strategy for stone columns should be chosen depending on which of the previous improvements is to be analyzed. In extremely soft soils, stone columns are not suitable because their continuity, stability, geometric shape, etc. cannot be guaranteed. The undrained shear strength (cu ) of natural soft soil is generally used as the limiting parameter for stone column feasibility. A limiting value around 5–15 kPa [4] may be adopted. To improve the lateral confinement of stone columns in those extremely soft soils, encasing the columns with geotextiles or other geosynthetics, such as geogrids, has proven to be a successful technique in recent times (e.g., [5]). Rigid or semi-rigid inclusions (e.g., adding lime or cement) is another common alternative (e.g., [6,7]). Han [8] summarizes different ground column technologies. Ground improvement using stone columns, either ordinary or encased stone columns, requires a considerable amount of columns or, at least, a group of columns. This implies a complex modeling process of the real geometry. This paper provides some guidelines for obtaining these simplified geometrical models. Some other features related with column modeling are also briefly presented, such as the critical length of the column and the influence of column installation. Additionally, some guidance and recommendations are provided on parameter selection to study stone columns. Here, the word “modeling” is understood in a broad sense, covering geometrical, mechanical, geotechnical and installation features of stone columns. The increase in computer power and the availability of finite element codes makes numerical analyses very appealing in geotechnical design. They usually lead to more detailed studies but require a clear conception of the modeling techniques. Ground improvement techniques, such as stone columns, are more and more popular due to the increasing occupation of natural soft soils and environmental concerns [9,10]. Within these current trends, the review of the modeling techniques for stone columns seems interesting and useful. Besides, in some instances, there is some confusion about geometrical models and their application. For example, the results for an isolated column under concentrated load just on top of the column cannot be directly extrapolated for a large group of columns under distributed uniform load. 2. Geometrical Models To simplify the real geometry of the problem that usually involves a considerable amount of columns (e.g., Figure 1a) and to be able to deal with the problem, the following simplified geometrical models are usually adopted: • • • • • • • • “Unit cell” in axial symmetry (Figure 1b). Only one column and its corresponding surrounding soil are studied. It may be useful to study just a horizontal slice of the unit cell, rather than the whole length. Longitudinal gravel trenches (Figure 1c). The stone columns are transformed into longitudinal gravel trenches to study the problem in plane strain conditions. Cylindrical rings of gravel (Figure 1d). The columns are transformed into cylindrical rings of gravel to study the problem in axial symmetry. Homogenization or equivalent homogeneous soil (Figure 1e). The columns and the surrounding soil are transformed into a homogeneous soil with equivalent improved properties. Three-dimensional (3D) model of a row or slice of columns (Figure 1f). Geometrical models for small groups of stone columns. “Unit cell” with constant lateral pressure (triaxial conditions). Isolated column. Materials 2017, 10, 782 3 of 23 Materials 2017, 10, 782 3 of 23 (a) (b) (c) (d) (e) (f) Figure 1. Main geometrical models for stone column studies: (a) Full 3D model; (b) Unit cell; Figure 1. Main geometrical models for stone column studies: (a) Full 3D model; (b) Unit cell; (c) (c) Longitudinal gravel trenches; (d) Cylindrical gravel rings; (e) Equivalent homogenous soil; (f) 3D slice Longitudinal gravel trenches; (d) Cylindrical gravel rings; (e) Equivalent homogenous soil; (f) 3D slice of columns. of columns. The two latter cases, namely “unit cell” under triaxial conditions and isolated column, do not usually real problems, but they have been used for conditions laboratory tests [11–13]). The case The twoappear latterincases, namely “unit cell” under triaxial and(e.g., isolated column, do not of an isolated column is widely used for field tests (usually plate load tests) for the sake of simplicity usually appear in real problems, but they have been used for laboratory tests (e.g., [11–13]). The case (e.g., [14]). of an isolated column is widely used for field tests (usually plate load tests) for the sake of simplicity As a simple introductory comparison between the different geometrical models, Table 1 (e.g., [14]). summarizes the suitability of some of these models to study the improvements achieved with a stone As a simple introductory comparison between the different geometrical models, Table 1 column treatment for the foundation of an embankment. summarizes the suitability of some of these models to study the improvements achieved with a Tabletreatment 1. Suitability of the simplified geometrical to study different features of a stone column stone column for foundation of anmodels embankment. treatment for the foundation of an embankment. Table 1. Suitability of simplified geometrical models to study different features of a stone column Geometrical Model Final Settlement Consolidation Stability treatment for the foundation of an embankment. Unit cell *** *** Gravel trenches ** ** ** Geometrical Model Final Settlement Consolidation Stability Homogenization ** * * Unit cell 3D slice *** *** *** *** *** Gravel trenches ** ** ** *** Completely suitable, ** Moderately suitable, * Slightly suitable, - Not suitable. Homogenization ** * * 3D slice *** *** *** All the geometrical models listed above are valid for non-encased stone columns. However, for Completely suitable, Moderately suitable, * way Slightly suitable, -the Notcylindrical suitable. encasement encased stone*** columns, there is not**yet any satisfactory to convert (geosynthetic) that surrounds the column for the cases of longitudinal gravel trenches and cylindrical ringsthe of gravel. All geometrical models listed above are valid for non-encased stone columns. However, for encased stone columns, there is not yet any satisfactory way to convert the cylindrical encasement 3. Unit Cell (geosynthetic) that surrounds the column for the cases of longitudinal gravel trenches and cylindrical The unit cell model is the most widely used for theoretical analyses and it is reviewed in detail, rings of gravel. for example, in [15]. The basis for the simplified model is the usage of a great number of columns, 3. Unit Cell distributed in a wide area under a uniform load. This is the case, for example, in the central uniformly part of an embankment on soft ground improved with stone columns. In these situations, the The unit cell model is the most widely used for theoretical analyses and it is reviewed in detail, behavior of all the columns is the same and then, it is enough to study the behavior of just one column for example, in [15]. The basis for thesoil simplified is the usage of a great number of columns, with the corresponding surrounding (tributarymodel area). Due to symmetry conditions, at the external uniformly a wide under a uniform load. Thisand is the case, for example, in theare central lateraldistributed boundary ofinthe unit area cell, only vertical displacements only vertical water seepage part of an embankment on soft ground improved with stone columns. In these situations, the behavior allowed. Stone columns are generally uniformly distributed in the triangular or of square grids. Thus, with the the of all the columns is the same and then, it is enough to study behavior just one column tributary area of natural soil for each column is a hexagon or a square, respectively. To allow for axial corresponding surrounding soil (tributary area). Due to symmetry conditions, at the external lateral symmetry is transformed a circle (cylinder) of the same boundary of theconditions, unit cell, the onlytributary verticalarea displacements andinto only vertical water seepage are (crossallowed. sectional) area. Therefore, the diameter of the unit cell is equal to = 1.05 − 1.13 for triangular Stone columns are generally uniformly distributed in triangular or square grids. Thus, the and square grids, respectively, where is the centre-to-centre distance between columns (Figure 2). tributary area of natural soil for each column is a hexagon or a square, respectively. To allow for Figure 3 shows the unit cell model with axial symmetry that can be studied in two dimensions. axial symmetry conditions, the tributary area is transformed into a circle (cylinder) of the same (cross-sectional) area. Therefore, the diameter of the unit cell is equal to de = 1.05 − 1.13 s for triangular and square grids, respectively, where s is the centre-to-centre distance between columns (Figure 2). Figure 3 shows the unit cell model with axial symmetry that can be studied in two dimensions. Materials 2017, 10, 782 4 of 23 Materials 2017, 10, 782 4 of 23 Materials 2017, 10, 782 4 of 23 s s rc rc re re rc rc re re s s Figure 2. Simplification of the unit cell to axial symmetry conditions. Figure 2. Simplification of the unit cell to axial symmetry conditions. Figure 2. Simplification of the unit cell to axial symmetry conditions. Figure 3. Unit cell model for analytical solutions. Figure3.3.Unit Unit cell cell model solutions. Figure modelfor foranalytical analytical solutions. The unit cell model is used for most of the existing analytical solutions (e.g., [16,17]). In the analytical solutions, thereisisused usually further simplifying assumption for the geometry, which to The unit cell model for amost of the existing analytical solutions (e.g., [16,17]). In isthe The unit cell model is used for most of the existing analytical solutions (e.g., [16,17]). In the assume that the behavior horizontal slice is independent, i.e., the stresses arewhich neglected. analytical solutions, there of is each usually a further simplifying assumption forshear the geometry, is to analytical solutions, there isofusually a further simplifying assumption for the which Numerical [17,18] show that shear stresses are usually for geometry, distributed loads. is to assume that analyses the behavior each horizontal slice is independent, i.e.,negligible the shear stresses are neglected. assume that the behavior of each horizontal slice is independent, i.e., the shear stresses are neglected. Balaam andanalyses Booker [17,18] [19] is ashow notable to theare simplifying assumption neglectingloads. shear Numerical thatexception shear stresses usually negligible for of distributed Numerical analyses [17,18] show that shear stresses are usually negligible for distributed loads. Balaam stressesand because they study total exception length of the unitsimplifying cell as a whole. Nevertheless, the solution Balaam Booker [19] is a the notable to the assumption of neglecting shear and stresses Booker [19] is athey notable to theof simplifying assumption of neglecting stresses requiresbecause numerical integration, makes the solution complex for practical purposes. studyexception the which total length the unit cell as a whole. Nevertheless, theshear solution Most analytical solutions focus on reduction caused bypractical stone columns [16,17,19]), requires numerical integration, which makes thecell solution for purposes. because they study the total length ofthe thesettlement unit as a complex whole. Nevertheless, the(e.g., solution requires but stone columns also act as vertical drains and, complex therefore, they dissipate excess pore(e.g., pressures. The Most analytical solutions focus on the settlement reductionfor caused by stone columns [16,17,19]), numerical integration, which makes the solution practical purposes. consolidation process may be studied independently from the settlement analysis using solutions for but stone columns also act as vertical drains and, therefore, they dissipate excess pore pressures. The Most analytical solutions focus on the settlement reduction caused by stone columns vertical drainsprocess [20,21].may Hanbeand Ye [22] and Castro and Sagaseta [23] showed that the consolidation consolidation studied independently from the settlement analysis using solutions for (e.g., [16,17,19]), but stone columns also act as vertical drains and, therefore, they dissipate excess pore processdrains around stone Han columns is slightly because of the vertical stresses vertical [20,21]. and Ye [22] and different Castro and Sagaseta [23]distribution showed thatofthe consolidation pressures. The consolidation process may be studied independently from the settlement analysis using betweenaround soft soil and columns stone columns and different they proposed specific solutions to study the process stone is slightly because of the analytical distribution of vertical stresses solutions forsoft vertical drains [20,21]. and Yeproposed [22] andspecific Castro andLogar Sagaseta [23] to showed that the consolidation process around stoneHan columns. recently, Pulkoanalytical and [24] have developed between soil and stone columns and theyVery solutions study thea consolidation process around stone columns is slightly different because of the distribution of vertical fully coupled process solutionaround for thestone consolidation assuming the and soil Logar as a poroelastic and consolidation columns.process Very recently, Pulko [24] have medium developed a stresses soft soil and stone columns and they proposed specific solutions to study the between stone column as an elastoplastic material. The solution is the very accurate but requires numerical fully coupled solution for the consolidation process assuming soil as a analytical poroelastic medium and inversion of the Laplace transform. the consolidation process stonematerial. columns. Very recently, Pulko and Logar [24] have developed the stone column as anaround elastoplastic The solution is very accurate but requires numerical Extending analytical solutions for ordinary stone columns to encased stone columns is quite and inversion of the Laplace transform. a fully coupled solution for the consolidation process assuming the soil as a poroelastic medium straightforward (e.g., [25]). Equilibrium and compatibility conditions of the geosynthetic encasement are Extending analytical solutions for ordinary stone columns to encased stone columns is quite the stone column as an elastoplastic material. The solution is very accurate but requires numerical those of a thin-walled tube, or better said, those of a flexible membrane because the encasement does straightforward (e.g., [25]). Equilibrium and compatibility conditions of the geosynthetic encasement are inversion of the Laplace transform. not usually support compressive stresses. The of internal andmembrane external pressures areencasement here denoted as those of a thin-walled tube, or better said, those a flexible because the does Extending analytical solutions for ordinary stone columns to encased stone columns is quite not usually support compressive stresses. The internal and external pressures are here denoted as straightforward (e.g., [25]). Equilibrium and compatibility conditions of the geosynthetic encasement are those of a thin-walled tube, or better said, those of a flexible membrane because the encasement does not usually support compressive stresses. The internal and external pressures are here denoted Materials 2017, 10, 782 5 of 23 10, 782 5 of 23 as σrcMaterials and σ2017, (Figure 4). Thus, the equilibrium and compatibility conditions of the rs , respectively geosynthetic encasement are the following, respectively: and , respectively (Figure 4). Thus, the equilibrium and compatibility conditions of the geosynthetic encasement are the following, respectively: Tg σrc = = rc + σrs + Tg = Jg = (1) (1) sr rc (2) (2) where sr is the radial displacement of the column and the encasement, Jg is the hoop or circumferential where is the radial displacement the force column andgeosynthetic. the encasement, is the stiffness of the geosynthetic and Tg is the of hoop at the The units of Jhoop Tg are g andor stiffness of thethe geosynthetic is the hoop force the geosynthetic. The units of force circumferential per length (F/L) because thicknessand of the geosynthetic is at usually negligible. The equation and the are forcestress per length because thickness of the negligible. The that relates radial of the(F/L) column (σrcthe ) with that of thegeosynthetic soft soil (σrsis) usually and allows the confining equation that relates the radial stress of the column ( ) with that of the soft soil ( ) and allows the effect of the encasement to be included is obtained by combining Equations (1) and (2): confining effect of the encasement to be included is obtained by combining Equations (1) and (2): σrc == Jg sr + + σrs rc2 (3) (3) rs sr rc Tg rc Tg Figure 4. Equilibrium and compatibility conditions for the geosynthetic surrounding an encased Figure 4. Equilibrium and compatibility conditions for the geosynthetic surrounding an encased stone column. stone column. The unit cell model is also very useful for numerical analyses because numerical simulations allowunit more complex be studied the analytical solutions, suchnumerical as layered ground, The cell modelfeatures is alsotovery usefulthan for with numerical analyses because simulations advance constitutive models or cyclic loads (e.g., [26–29]). Nowadays, 3D numerical codes are fairly allow more complex features to be studied than with the analytical solutions, such as layered ground, easy to access and to use; so, there are some authors that study the unit cell as a full 3D problem, i.e.,fairly advance constitutive models or cyclic loads (e.g., [26–29]). Nowadays, 3D numerical codes are considering the hexagonal or square prism. Nevertheless, the differences between the full 3D prism and easy to access and to use; so, there are some authors that study the unit cell as a full 3D problem, i.e., the two-dimensional (2D) model (cylinder) in axial symmetry are negligible [30,31]. The simplicity of considering the hexagonal or square prism. Nevertheless, the differences between the full 3D prism the unit cell model has recently led to highly advanced numerical models, such as that presented by and the two-dimensional (2D) model (cylinder) in axial symmetry are negligible [30,31]. The simplicity Indraratna et al. [32], where the column is modelled using 2D discrete elements to represent the gravel of theparticles. unit cellThis model recentlyanalysis led to highly advanced models, as that presented type has of numerical can be regarded onlynumerical as explorative and forsuch research purposes. by Indraratna etmodel al. [32], where theofcolumn 2Ddepth discrete elements represent the The unit cell or the model a slice of is themodelled unit cell atusing a specific are also used fortosmall-scale gravellaboratory particles.studies This type numerical analysis can be regarded only as explorative and for research (e.g., of [33,34]). The unit cell also allows floating columns to be that do not reach a rigid purposes. The unit cell model or the model of a slice ofstudied, the uniti.e., cellcolumns at a specific depth are also used for substratum. In thisstudies case, the(e.g., settlement due to column punching into the underlying layer and the small-scale laboratory [33,34]). deformation the underlying layer should be considered [35]. The cell model only valid when The unit cellofalso allows floating columns to be studied, i.e.,unit columns thatisdo not reach a rigid the ratio between the width of the loaded area and the thickness of the soft soil layer ( / ) is high substratum. In this case, the settlement due to column punching into the underlying layer and the enough to ensure confined (oedometric) conditions. On the contrary, the applied load is distributed deformation of the underlying layer should be considered [35]. The unit cell model is only valid when with depth, in a roughly trapezoidal manner, its value decreasing with depth. Figure 5 presents some the ratio between the of width of the loaded area andfor theend-bearing thickness and of the soft soil layer[36–38]. (B/H)The is high specific proposals load distribution width depth floating columns enough to ensure confined (oedometric) conditions. On the contrary, the applied load is distributed unit cell model for these cases usually overestimates the settlement because the reduction of the applied with depth, a roughly trapezoidal manner, its value decreasing with depth. Figure 5 presents some verticalin stress with depth is not reproduced. specific proposals load width depth for end-bearing floating columnsand [36–38]. Finally, theofunit celldistribution is an appropriate simplified geometrical modeland to study the settlement its evolution with time at the center of an embankment, obviously it is because not valid the for studying theof the The unit cell model for these cases usually overestimatesbut the settlement reduction stability of the lateral slopes (Table 1). For similar reasons, it does not allow the lateral spreading and applied vertical stress with depth is not reproduced. the contribution of a geosynthetic reinforcement that acts as a “blanket” or “bridging layer” over the and Finally, the unit cell is an appropriate simplified geometrical model to study the settlement columns and soft soil foundation (geosynthetic reinforced and column supported embankments, its evolution with time at the center of an embankment, but obviously it is not valid for studying the stability of the lateral slopes (Table 1). For similar reasons, it does not allow the lateral spreading and the contribution of a geosynthetic reinforcement that acts as a “blanket” or “bridging layer” over Materials 2017, 10, 782 6 of 23 Materials 2017, 10, 782 6 of 23 the columns and soft soil foundation (geosynthetic reinforced and column supported embankments, GRCSE) GRCSE) to be studied [31]. To study the lateral spreading of of thetheembankment, unitcell cellmodel model may to be studied [31]. To study the lateral spreading embankment, the the unit Materials 2017, 10, 782 by substituting 6 of 23 [39]. be improved by substituting the horizontally-fixed external lateral elastic springs may be improved the horizontally-fixed external lateral boundary boundary byby elastic springs [39]. GRCSE) to be studied [31]. To study the lateral spreading of the embankment, the unit cell model may be improved by substituting the horizontally-fixed external lateral boundary by elastic springs [39]. (a) (b) Figure 5. Stress distribution beneath footings: (a) End-bearing columns; (b) Floating columns. Figure 5. Stress distribution beneath footings: (a) End-bearing columns; (b) Floating columns. (a) 4. Longitudinal Gravel Trenches (b) Figure 5. Stress distribution beneath footings: (a) End-bearing columns; (b) Floating columns. 4. Longitudinal Trenches ManyGravel geotechnical problems fulfill plane strain conditions. Thus, when stone columns are used those problems, for example, for the foundation of an embankment for a linear infrastructure, it is 4.inLongitudinal Gravel Trenches Many geotechnical problems fulfill plane strain conditions. Thus, when stone columns are used useful to study the problem under 2D plane strain conditions, transforming the columns into in thoselongitudinal problems, for example, for the foundation of an embankment for acolumns linear infrastructure, Many geotechnical problems fulfill strainofconditions. Thus, stone used gravel trenches (Figure 6). plane The width the trenches and when the spacing between are them are in those problems, for example, for the foundation of an embankment for a linear infrastructure, it is useful to study the problem under 2D plane strain conditions, transforming the columns part of the unknown parameters that should be properly estimated to be equivalent. Besides, itit isis into useful tonecessary study the under 2D plane conditions, transforming the columns into longitudinal gravel trenches (Figure 6). The widthstrain of these the trenches and the trenches. spacing between them are usually to problem alter the gravel parameters for equivalent gravel Furthermore, longitudinal gravel trenches (Figure 6). The width of the trenches and the spacing between them are to appropriately model the consolidation process, it is also estimated necessary toto modify the permeability of part of the unknown parameters that should be properly be equivalent. Besides, it is part of the unknown parameters that should be properly estimated to be equivalent. Besides, it is the natural soft soil. usually usually necessary to alter the the gravel parameters theseequivalent equivalent gravel trenches. Furthermore, necessary to alter gravel parameters for for these gravel trenches. Furthermore, to appropriately modelmodel the consolidation process, it itis isalso to modify modifythe the permeability to appropriately the consolidation process, alsonecessary necessary to permeability of of the natural the softnatural soil. soft soil. Figure 6. Longitudinal gravel trenches to model a stone column treatment in plane strain. 4.1. Settlement 6. Longitudinal gravel trenchesto tomodel model aastone treatment in plane strain. strain. Figure Figure 6. Longitudinal gravel trenches stonecolumn column treatment in plane The classical proposal for this simplified model is by Van Impe and De Beer [40], who developed an analytical 4.1. Settlementsolution to study the settlement for this case. Their proposal [40] is to keep the (drained) 4.1. Settlement properties of the soil and column and transform each row of columns in a longitudinal gravel trench The classical of the same area. proposal for this simplified model is by Van Impe and De Beer [40], who developed Theanclassical proposal thisthe simplified model by Van Impe and De [40], who developed analytical solution tofor study settlement forcolumn thisis case. Their proposal [40] is Beer to keep thefactor (drained) The most important parameter in a stone treatment is the area replacement ( ), properties of theto soil and and transform each row ofTheir columns in acolumns: longitudinal an analytical study the settlement fororthis case. [40] is togravel keep trench the (drained) whichsolution represents the areacolumn of soft soil replaced displaced by theproposal stone of the area.and column and transform each row of columns in a longitudinal gravel trench properties of same the soil The most important parameter in a stone column = =treatment is the area replacement factor ( (4)), of the same area. which represents the area of soft soil replaced or displaced by the stone columns: The most important parameter in a stone column treatment is the area replacement factor (ar ), The proposal by [40] seems reasonable because it allows the area replacement factor ( ) and the which represents the area of softtosoil replaced or=displaced by thedisadvantages: stone columns: (4) = two major soil and gravel parameters be kept. However, it has The proposal by [40] seems reasonable because Ac it allows rc2 the area replacement factor ( ) and the a = = r soil and gravel parameters to be kept. However,A it has two r2 major disadvantages: e (4) e The proposal by [40] seems reasonable because it allows the area replacement factor (ar ) and the soil and gravel parameters to be kept. However, it has two major disadvantages: Materials 2017, 10, 782 Materials 2017, 10, 782 • • 7 of 23 7 of 23 The resulting thickness of the gravel trenches is usually small, and consequently, the gravel The resulting thickness of the gravel trenches is usually small, and consequently, the gravel trenches are very slender. trenches are very slender. The confining conditions of the columns thesame sameasas lateral confinement of the gravel The confining conditions of the columnsare arenot not the thethe lateral confinement of the gravel trenches [30] (Figure 7). trenches [30] (Figure 7). Materials 2017, 10, 782 7 of 23 The resulting thickness of the gravel trenches is usually small, and consequently, the gravel trenches are very slender. The confining conditions of the columns are not the same as the lateral confinement of the gravel trenches [30] (Figure 7). Figure 7. Different confinement and seepage conditions for columns in axial symmetry and for Figure 7. Different confinement and seepage conditions for columns in axial symmetry and for longitudinal gravel trenches in plane strain. longitudinal gravel trenches in plane strain. Regarding the first bullet point, there is no restriction on increasing the thickness of the gravel trenches, while proportionally increasing the spacing between the longitudinal trenches to keep Regarding the first bullet point, there is no restriction on increasing the thickness of the gravel constant. A reasonable value for the thickness of the gravel trenches may be the diameter of the trenches, while proportionally increasing the spacing between the longitudinal trenches to keep columns. 7. Differentvalue confinement and thickness seepage conditions forgravel columns trenches in axial symmetry and the for diameter of ar constant.The AFigure reasonable for the of the may second bullet point causes that the matching of the settlement between the be real situation longitudinal gravel trenches in plane strain. and the plane strain model is not accurate enough, particularly when plastic strains appear in the the columns. and then, differences thematching column confinement are notable. Plasticthe in the Thecolumns second bullet point causes thatinthe of theonsettlement between real situation and Regarding thethe first bullet point, there is no restriction increasing the thickness ofstrains the gravel columns arewhile common and, if theincreasing column only deforms elastically, the designtrenches is overconservative. trenches, proportionally the spacing between the longitudinal to keep the plane strain model is not accurate enough, particularly when plastic strains appear in the columns Tanconstant. et al. [30] analytical proposals equivalent properties of the A presented reasonable two value for the thickness of to theobtain gravelthe trenches may be the diameter of gravel the and then, the differences in the column confinement are notable. Plastic An strains in the columns are trenches and the surrounding soil, but neither of them is totally satisfactory. analytical equation columns. common and, if the column only elastically, the design is overconservative. Tan may be The derived for bullet the equivalent elastic of the trenches [41], but the thatreal is only validet if al. [30] second point deforms causes thatmodulus the matching ofgravel the settlement between situation presented analytical proposals to obtain the equivalent of the gravel trenches and the plane strain model is not accurateFor enough, particularly when plastic strains ininthe the two column and the soil deform elastically. general cases,properties comparing the unit cellappear model axialand the columns andthe then, in the column are notable. Plasticstarting strains symmetry unitthecell of the is gravel trenches isconfinement very useful (Figure 8).equation Thus, from an surrounding soil,and but neither ofdifferences them totally satisfactory. An analytical mayin bethe derived for columnsproposal are common and, if the column only deforms elastically, the design is overconservative. analytical for the parameters of the gravel trenches, they may be further adjusted or tuned the equivalent elastic modulus of the gravel trenches [41], but that is only valid if the column and the Tan et the al. [30] presented twounit analytical to obtainTo thefitequivalent properties the gravel to match settlement of the cell inproposals axial symmetry. the settlement in theofplastic range, soil deformtrenches elastically. For general cases, comparing the unit cellsatisfactory. model in An axial symmetry and the unit and the surrounding soil, but neither of them totally analytical equation adjusting the friction angle of the gravel trenches seemsisthe most appropriate alternative [41]. Once cell of the gravel trenches is very useful (Figure 8). Thus, starting from an analytical proposal may be derived for the equivalent elastic modulus of the gravel trenches [41], but that is only valid the parameters have been calibrated using the unit cell case as an auxiliary problem, they mayif be for the the column and the soil deform elastically. generaladjusted cases, comparing theto unit cell model in axial parameters of the they mayHowever, beFor further oristuned match the settlement used for thegravel full 2Dtrenches, plane strain model. the calibration tailor-made or specific for each of the symmetry and the unit cell of the gravel trenches is very useful (Figure 8). Thus, starting from an Thus,symmetry. it varies, forTo example, the value theplastic appliedrange, load [42]. unit cellcase. in axial fit the with settlement inofthe adjusting the friction angle of the analytical proposal for the parameters of the gravel trenches, they may be further adjusted or tuned gravel trenches seems the mostofappropriate [41].ToOnce parameters have been to match the settlement the unit cell inalternative axial symmetry. fit thethe settlement in the plastic range,calibrated using the unit cell the case as anangle auxiliary problem, they may used for the alternative full 2D plane strain model. adjusting friction of the gravel trenches seems thebe most appropriate [41]. Once the parameters have been calibratedorusing the unit case as anThus, auxiliary problem,for they may be with the However, the calibration is tailor-made specific forcell each case. it varies, example, used for the full 2D plane strain model. However, the calibration is tailor-made or specific for each value of the applied load [42]. case. Thus, it varies, for example, with the value of the applied load [42]. Figure 8. Calibration of the parameters of longitudinal gravel trenches using the unit cell model as an auxiliary problem. 4.2. Consolidation TheFigure coefficient of horizontal permeability (or the coefficient of consolidation ) of the 8. Calibration of the parameters of longitudinal gravel trenches using the unit cell model as an Figure 8. Calibration of parameters of longitudinal trenches using the under unit cell model as an natural soil should bethe adjusted to properly reproduce gravel the consolidation process plane strain auxiliary problem. conditions because, similarly to the column confinement, the seepage problem is different in axial auxiliary problem. 4.2. Consolidation symmetry and plane strain conditions (Figure 7). Hird et al. [43] and Indraratna and Redana [44] The coefficient of horizontal permeability 4.2. Consolidation (or the coefficient of consolidation ) of the natural soil should be adjusted to properly reproduce the consolidation process under plane strain The coefficient of horizontal permeability consolidation cvhin) axial of the natural conditions because, similarly to the column kconfinement, the seepageofproblem is different h (or the coefficient and plane strain conditions (Figure Hird et al. [43]process and Indraratna Redana [44]conditions shouldsymmetry be adjusted to properly reproduce the 7). consolidation under and plane strain soil because, similarly to the column confinement, the seepage problem is different in axial symmetry and plane strain conditions (Figure 7). Hird et al. [43] and Indraratna and Redana [44] proposed analytical Materials 2017, 10, 782 8 of 23 expressions to modify k h , based on the comparison between analytical solutions for vertical drains for both cases (axial symmetry and plane strain). When using numerical analyses, the analytical values of k h [43,44] may be tuned using the unit cell as an auxiliary problem (Figure 8) to have a better matching of the results. As it is difficult to match the whole consolidation curve, it may be convenient to match the values of the degrees of consolidation that are relevant for the study, usually around 80–95%. 4.3. Stability The plane strain model is useful for studying the global stability, or, for example, the stability of the lateral slopes of an embankment (Table 1). In this case, the plane strain model should match the average shear strength along the slip line of the real situation. Here, it is important to distinguish between limit equilibrium analyses (e.g., method of slices) or full stress-strain analyses (e.g., finite differences or finite element method). Stress-strain analyses are able to capture the stress concentration on the columns. Thus, if the stress concentration on the columns is properly reproduced (depending on the rigidity of soil and gravel), the results are satisfactory just keeping constant the area replacement ratio ar and the resistance properties (shear strength) of soil and gravel (e.g., [45]). For limit equilibrium analyses, the stress concentration on the longitudinal gravel trenches should be artificially generated because the vertical stress is directly the weight above the studied point. Priebe [35] proposed altering the real profile of the embankment, so the modeled height of the embankment is the real one times the stress concentration either on the columns or on the soft soil. Another alternative is to modify the friction angle of the longitudinal gravel trenches to match the average shear strength along the slip line [15]. A very difficult task is to evaluate the stress concentration on the columns because it depends on many factors, such as the drainage conditions (undrained, partial drainage or fully drained), the depth and inclination of the slip line, the specific position of the columns beneath the slope and the applied load. Recent studies [45] show that in situations close to failure, there is no stress concentration on the columns, i.e., the vertical stress on soil and columns is very similar. Thus, assuming no stress concentration may be generally advisable and in any case, it is an assumption on the safe side. Finally, rigid inclusions, such as deep mixing columns, may fail not only by shearing as stone columns (shear failure along the slip line), but they may also fail by tilting or bending [46]. For widely spaced columns, soil may also extrude between columns [47]. When columns break by bending, it is necessary to consider their moment of inertia [48] or the measured bending failure for encased stone columns. 5. Homogenization The homogenization method consists in replacing the stone columns and the soft soil by an equivalent homogeneous soil with improved properties. This equivalent homogeneous soil occupies the zone treated with stone columns. This model simplifies enormously the geometry of the problem. For example, when the problem itself has a highly complex geometry and the zone treated with columns is just a part of the problem, this method is highly advisable (e.g., [49]). For the design stage, this method allows the area replacement ratio (ar ) to be varied without changing the geometry of the model, just the material parameters. 5.1. Equivalent Parameters The most straightforward proposal for obtaining the improved parameters of the equivalent homogeneous soil is just to average the soil and column parameters weighted by their corresponding areas though ar . Thus, for the elastic modulus, the weighted average is: Em = Es (1 − ar ) + Ec ar (5) Materials 2017, 10, 782 9 of 23 Materials 2017, 10, 782 9 of 23 However, this can only be regarded as a first approximation. A more detailed analysis shows that the equivalent homogenous soil[51–53]) shoulduse be anisotropic naturemedia to account for the column orientation [50]. Some authors (e.g., theories for by periodic to propose analytical Materials 2017, 10, 782 9 of 23 orientation [50]. from Some authors (e.g., for periodic media to propose analytical transformations the columns and[51–53]) the soft use soil theories (heterogeneous periodic composite material at the transformations from the columns and the soft soil (heterogeneous periodic composite material microscale) to a homogeneous equivalent soil (homogenous material at a macroscale). They a orientation [50]. Some authors (e.g., [51–53]) use theories for periodic media to propose analyticaluse at the microscale) to a homogeneous equivalent soil (homogenous material at a macroscale). They use a macroscopic strengthfrom criterion of the and anisotropic homogeneous equivalent soil to evaluate the transformations the columns the soft soil (heterogeneous periodic composite material at bearing the microscale) to a homogeneous equivalent (homogenous material atand a macroscale). Theythe use a for macroscopic criterion of the anisotropic homogeneous equivalent soil to evaluate bearing capacity. Thestrength practical application of thesesoiltechniques is complex they are only valid macroscopic strength criterion the anisotropic homogeneous equivalent soil evaluate bearing capacity. Thesmall practical application these techniques is complex and theythe areto only valid for sufficiently sufficiently values of the ofof scale factor, i.e., the ratio between micro andthe macroscales, capacity. The practical application of these techniques is characteristic complex and they are only validproblem for ratio small values of the scale factor, i.e., ratio between the micro and macroscales, forthe example, the for example, the ratio between the the column spacing and the size of whole sufficiently small values of the scale factor, i.e., the ratio between the micro and macroscales, between the column spacing and the characteristic size of the whole problem to be studied. to be studied. for example, the ratio between the column spacing and the characteristic size of the whole problem as for gravelgravel trenches, a simplea method using numerical In aasimilar similarmanner manner as the for longitudinal the longitudinal trenches, simple when method when using to be studied. analyses is tuning or the improved parameters of the equivalent soil usingsoil theusing unit cell numerical analyses is adjusting tuning or adjusting the improved parameters of the equivalent the In a similar manner as for the longitudinal gravel trenches, a simple method when using as ancell auxiliary problem (Figure 9). Ng and Tan [54] have calibrated the improved parameters unit as an auxiliary problem (Figure 9). Ng and Tan [54] have calibrated the improved parameters numerical analyses is tuning or adjusting the improved parameters of the equivalent soil using the for several cases and tabulated their results asas a reference. The accuracy of the matching may for several cases tabulated their results aand reference. The accuracy ofthe the matching may be high unit cell as anand auxiliary problem (Figure 9). Ng Tan [54] have calibrated improved parameters and, yet the specific of the their excess poreaspressure between columns may not bemay modeled using for several cases values and tabulated results a reference. The accuracy of the matching be high and, yet the specific values of the the pore pressure columns be modeled using homogenization technique, the excess average degree of between consolidation ormay thenot settlement rate may be the homogenization technique, average degree of consolidation or the settlement rate may the homogenization technique, the average degree of consolidation or the settlement rate maysoil be [54]. proper calibration of the the parameters of the theequivalent equivalent homogeneous soil [54]. correctly capture after a proper calibration of parameters of homogeneous correctly capture after a proper calibration of the parameters of the equivalent homogeneous soil [54]. cases, it may be enough enough to match just the values of the degree of consolidation consolidation that are For most cases, For most cases, it may be enough to match just the values of the degree of consolidation that are relevant for the study, usually around 80–95%. Figure 10 shows a simple example of application for relevant for the study, usually around 80–95%. Figure 10 shows a simple example of application for the extension of an an airport airport runway, runway, whose embankment had to be founded founded on ground ground improved improved with the extension of an airport runway, whose embankment had to be founded on ground improved with floating stone columns. floating stone columns. Figure 9. Calibration of the parameters of the equivalent homogeneous soil using the unit cell model Figure 9. 9. Calibration of of the the parameters parameters of of the the equivalent equivalent homogeneous homogeneous soil soil using using the the unit unit cell cell model model Figure as anCalibration auxiliary problem. as an auxiliary problem. as an auxiliary problem. Figure 10. Example of calibration of the parameters of the equivalent homogeneous soil using the unit cell model as an auxiliary problem. Figure 10. Example of calibration of the parameters of the equivalent homogeneous soil using the unit method reproduce physical phenomena at the local level, the such as FigureThe 10. homogenization Example of calibration of cannot the parameters of the equivalent homogeneous soil using unit cell model as an auxiliary problem. stress concentration on the column or radial drainage towards the column, but the overall response cell model as an auxiliary problem. The homogenization method cannot reproduce physical phenomena at the local level, such as stress concentration on the column or radial drainage towards the column, but the overall response Materials 2017, 10, 782 10 of 23 The homogenization method cannot reproduce physical phenomena at the local level, such as stress concentration on the column or radial drainage towards the column, but the overall response of the system on a larger scale may be correctly reproduced after calibrating the parameters of the equivalent homogeneous soil. Materials 2017, 10, 782 10 of 23 5.2. Specific Constitutive Model of the system on a larger scale may be correctly reproduced after calibrating the parameters of the The idea of developing an advanced, specific constitutive model to represent the behavior of equivalent homogeneous soil. the periodic material formed by the columns and the corresponding surrounding soil was initially formulated by Schweiger and Pande [55]. The constitutive model assumed that both soil and column 5.2. Specific Constitutive Model undergo theThe same strains, there slippage between soil and column and the internally uses idea of developing is anno advanced, specific constitutive model to represent themodel behavior of the independent and existing constitutive models for the soil and the column and ensures that there is periodic material formed by the columns and the corresponding surrounding soil was initially formulated by Schweiger and Pande [55]. The constitutive model assumed that both soil and column equilibrium of horizontal stresses at the soil-column interface [55]. The model was later improved, undergo same strains, there is no slippage between soil andthe column thebeen modelapplied internallytouses particularly its the numerical implementation [56]. Recently, ideaand has deep soil and existing constitutive modelsmodels for the soil and the column is mixing independent columns using advanced constitutive and with the aimand of ensures using itthat forthere complex 3D equilibrium of horizontal stresses at the soil-column interface [55]. The model was later improved, geometries [57]. However, the numerical stability of its formulation and its range of application are particularly its numerical implementation [56]. Recently, the idea has been applied to deep soil still limited. mixing columns using advanced constitutive models and with the aim of using it for complex 3D geometries [57]. However, the numerical stability of its formulation and its range of application are 6. Gravel stillRings limited. The idea of transforming the stone columns into gravel rings, tubes or cylindrical trenches 6. Gravel Rings follows the same concept as for the longitudinal gravel trenches (Section 4) but for problems with The idea of transforming the stone columns into gravel rings, tubes or cylindrical trenches axial symmetry rather than plane strain conditions. Thus, the model is used for problems with follows the same concept as for the longitudinal gravel trenches (Section 4) but for problems with axial symmetry, suchrather as circular embankments or Thus, circular storage tanks. The model the axial symmetry than plane strain conditions. the model is used for problems with implies axial transformation of each group of columns into an equivalent gravel ring with the same area to keep symmetry, such as circular embankments or circular storage tanks. The model implies the ar constant (Figure 11). In the caseofofcolumns Figureinto 11, an theequivalent 8 neighboring columns of same the central column are transformation of each group gravel ring with the area to keep constant (Figure 11). In the case of Figure 11, the 8(tneighboring columns of thearea. central column arethat tr is transformed into an equivalent ring with a thickness the same Assuming r ) that gives transformed into an small enough, its value is:equivalent ring with a thickness ( ) that gives the same area. Assuming that is small enough, its value is: tr = d2c /rr (6) = / (6) The distance of the gravel ring to the central point (rr ) is the weighted average of the distance of The distance of the gravel ring to the central point ( ) is the weighted average of the distance of the 8 columns to the center. √ the 8 columns to the center. rr = 1 + 2 s/2 (7) = 1 + √2 s/2 (7) For a group of onlyof4only columns, the weighted average proposalofof other For a group 4 columns, the weighted averageisisdirectly directly rr = = ss [58]. [58]. The The proposal authors other [59] is slightly different and implies that the gravel ring surrounds the same area as that authors [59] is slightly different and implies that the gravel ring surrounds the same area as of the that of the by the 8 columns: square formed bysquare the 8formed columns: √ rr ==2s/ (8) (8) 2s/√ π For practical purposes, differencesare arenegligible. negligible. For practical purposes, the the differences Figure 11. Transformation of stone columns in equivalent gravel rings. Figure 11. Transformation of stone columns in equivalent gravel rings. Materials 2017, 10, 782 11 of 23 Mitchel and Huber [58] seem to be the first authors to use this model. They used it in combination with theMaterials finite2017, element 10, 782 method to study a field case (the foundation of a water treatment 11 of 23 plant). 10, 782 of 23 gravel ContraryMaterials to the2017, model of the longitudinal gravel trenches, here the confining conditions of11the and Huber [58] seem to be the authorscolumns to use thisand model. usedto it in combination rings seem toMitchel be somehow similar to those of first the stone it isThey enough maintain the values Mitchel Hubermethod [58] seem be theafirst to use this model. usedtreatment it in combination with the finiteand element totostudy fieldauthors case (the foundation of They a water plant). of ar and of the drained properties of the soil and the columns to obtain satisfactory results [60,61]. with the to finite element method to studygravel a field case (the foundation of a conditions water treatment plant). Contrary the model of the longitudinal trenches, here the confining of the gravel As for the longitudinal gravel isofnot yet any satisfactory to convert thegravel cylindrical Contrary of trenches, thesimilar longitudinal gravel trenches, here theand confining conditions of the rings seemtotothe bemodel somehow tothere those the stone columns it way is enough to maintain the encasement (geosynthetic) that surrounds the columns for this case. rings seem to be of somehow similar to those stone and to it obtain is enough to maintain the values of and the drained properties ofof thethe soil and columns the columns satisfactory results values As of for and of the drained properties of the soilisand to obtainway satisfactory results [60,61]. the longitudinal gravel trenches, there notthe yetcolumns any satisfactory to convert the 7. Three-Dimensional of Columns [60,61]. Asencasement for theSlice longitudinal gravelthat trenches, therethe is columns not yet any cylindrical (geosynthetic) surrounds for satisfactory this case. way to convert the cylindrical encasement (geosynthetic) that surrounds the columns for this case. There are problems that fulfill plane strain conditions but for the stone column treatment, such 7. Three-Dimensional Slice of Columns as an embankment for a linear infrastructure. In these cases, analyzing a slice or a row of columns 7. Three-Dimensional Slice of Columns There are problems that fulfill plane strain conditions but for the stone column treatment, such is very useful when 3D that numerical analyses (Figure 12). Nowadays, the highofcomputer power areusing problems fulfill plane strain conditions for the astone such as an There embankment for a linear infrastructure. In these cases,but analyzing slicecolumn or a rowtreatment, columns is and the very availability of 3D numerical codes makes this simplified geometrical model very appealing as anuseful embankment for 3D a linear infrastructure. In these12). cases, analyzingthe a slice a row ofpower columns when using numerical analyses (Figure Nowadays, highor computer andis useful when 3D numerical analyses (Figure 12). Nowadays, theorhigh computer power and because the itvery isavailability relatively simple and it does not require any transformation determination of equivalent of using 3D numerical codes makes this simplified geometrical model very appealing the availability of 3D numerical codes makes this simplified geometrical model very appealing because it is relatively simple and it does not require any transformation or determination of a stone parameters [48,62]. Furthermore, it allows any of the features or improvements achieved with because it is relatively simple and it does not require any transformation or determination equivalent parameters [48,62]. Furthermore, it allows any of the features or improvements achieved column treatment to be studied, such as consolidation, deformations and stability (Table 1), of and it is equivalent [48,62].toFurthermore, it allows any of the deformations features or improvements achieved with a stone parameters column treatment be studied, such as consolidation, and stability (Table 1), valid for both encased and non-encased columns. withitaisstone treatment benon-encased studied, suchcolumns. as consolidation, deformations and stability (Table 1), and validcolumn for both encased to and For and theFor ofcase aforsquare gridgrid or mesh enough to model justofhalf of of a row of itcase is the valid both encased andor non-encased columns. of a square meshofofcolumns, columns, ititisisenough to model just half a row columnscolumns dueFor todue symmetry conditions (Figure 13). On the contrary, for a triangular grid of columns, the to case of a square grid or (Figure mesh of13). columns, is enough model justgrid halfofofcolumns, a row of symmetry conditions On the itcontrary, for to a triangular to study symmetry conditions (Figure 13). the contrary, for a triangular grid of columns, itcolumns is necessary at least two different rows ofOn columns (a(ahalf of of each is enough) (Figure 13). 13). it is necessary to due study at least two different rows of columns half each is enough) (Figure it is necessary to study at least two different rows of columns (a half of each is enough) (Figure 13). Figure 12. Finite elementmodel model of of of columns. Figure 12. Finite element ofaa3D 3Dslice slice columns. Figure 12. Finite element model of a 3D slice of columns. Figure 13. Numerical model of a slice of columns for triangular and square grids. Figure 13. Numerical model of a slice of columns for triangular and square grids. Figure 13. Numerical model of a slice of columns for triangular and square grids. 8. Isolated Column 8. Isolated Column For the sake of simplicity, some experimental tests, either in the field or in the laboratory, are 8. Isolated Column For the of simplicity, some experimental tests, eitherThe in the field or in the laboratory, are performed onsake an isolated or single stone column (e.g., [63,64]). load is usually applied only on performed on an isolated single column (e.g., [63,64]). load is usually only on the column surface (Figure or 14a) or onstone an area slightly higher than The the column. In theseapplied cases, the area For the sake of simplicity, some experimental tests, either in the field or in the laboratory, are the column surface or on an higher the thanarea the of column. In these cases, area replacement ratio ( (Figure ) may 14a) be defined asarea the slightly ratio between the column and the the loaded performed on an isolated or single stone column (e.g., [63,64]). The load is usually applied only on replacement ratio footprint ( ) may area be defined as the between the areaisofusually the column and the which loaded area or the footing [61]. Thus, forratio isolated columns, around 100%, the column surface (Figure 14a) or on an area slightly higher than the column. In these cases, the area the footing footprint area [61]. isolated columns,the loaded is usually isarea notor a realistic situation because it is Thus, more for efficient to increase areaaround ( << 100%, 100%)which [61]. replacement ratio (a ) may be defined as the ratio between the area of the column and the loaded is not a realistic << 100%) [61]. area r situation because it is more efficient to increase the loaded area ( or the footing footprint area [61]. Thus, for isolated columns, ar is usually around 100%, which is not a realistic situation because it is more efficient to increase the loaded area (ar << 100%) [61]. Loading Materials 2017, 10, 782 12 of 23 the surroundings of the columns is beneficial because it increases the horizontal stresses at the lateral Materials 2017, 10, 782 12 of 23 boundaries of the columns and increases their confinement. TheLoading resultsthe ofsurroundings research performed on isolated columns sometimes directly stresses extrapolated to of the columns is beneficial becauseare it increases the horizontal at the lateral boundaries ofconfusion the columnsand and non-accurate increases their confinement. other cases, leading to some predictions. Some authors (e.g., [15] (p. 28)) The results bulges of research on isolated columns sometimes directly extrapolated show that the column (or performed notably expands radially) atare their upper part, specifically at to an upper other cases, leading to some confusion and non-accurate predictions. Some authors (e.g., [15] (p. 28)) zone with a length of two or three times the column diameter (2–3 dc ). Hughes and Withers [63] show that the column bulges (or notably expands radially) at their upper part, specifically at an upper measured a length of 4 column diameters forcolumn the upper bulging for an column (ar = 100%) zone with a length of two or three times the diameter (2–3 zone ). Hughes andisolated Withers [63] measured throughalaboratory tests. That is valid forupper casesbulging with load the single column, i.e.,through dc = B. In the length of 4 column diameters for the zone only for anon isolated column ( = 100%) laboratory tests. is valid forfor cases withcases, load only the single column, i.e., = .the In the next zone next section, it will beThat shown that other it isonmore meaningful to define bulging section, it will be shown that for other cases, it is more meaningful to define the bulgingcases, zone using using the footing width (B) instead of the column diameter [61] because in those the column footing width ( ) instead of the column diameter [61] because in those cases, the column bulging bulgingthe may be deeper [3] (p. 114) [15] (p. 28) (Figure 14). Besides, the failure mechanism of the may be deeper [3] (p. 114) [15] (p. 28) (Figure 14). Besides, the failure mechanism of the columns may columns may not be bulging, for example, it may be shearing [65]. not be bulging, for example, it may be shearing [65]. (a) (b) (c) (d) Figure 14. Column deformation for different configurations: (a) Isolated column with = 100%; Figure 14. Column deformation different configurations: (a) Isolated ar = 100%; < for 100%; (c) Groups of columns beneath footing; (d)column Column with treatment (b) Isolated column with (b) Isolated column with a < 100%; (c) Groups of columns beneath footing; (d) Column treatment r beneath an embankment. beneath an embankment. Isolated columns are generally used to study their bearing capacity. The maximum vertical stress that a stone column may bear is usually given as: Isolated columns are generally used to study their bearing capacity. The maximum vertical stress (9) 20 that a stone column may bear is usually given as: where is the undrained shear strength of the surrounding soft soil. Equation (9) is derived assuming max that the column is in an active state with an active earth pressure coefficient of around = 1/3 and that σvc ≈ 20c (9) u the radial cavity expansion factor is around 6–8 (e.g., [63,66]). Equation (9) is strictly valid for isolated with = 100%, but when used for larger loaded areas, is conservative it neglects where cucolumns is the undrained shear strength of the surrounding softitsoil. Equation because (9) is derived assuming the increase in the radial stress due to vertical loading of the soil surrounding the column. Besides, that the column is in an active state with an active earth pressure coefficient of around k ac = 1/3 and Equation (9) assumes perfect undrained conditions for the soil surrounding the column and some that the radial cavity expansion factor is around 6–8 (e.g., [63,66]). Equation (9) is strictly valid for drainage could be expected near the granular column. For common cases, Equation (9) gives a vertical isolatedload columns withforareach = 100%, when usedtons for [15] larger loaded areas,field it istests conservative because it supported stone but column of 20–50 (p. 6). Sometimes, are performed neglectstothe increase invertical the radial duebytoone vertical of the soil surrounding theofcolumn. check the exact load stress supported isolatedloading column with = 100%. The approach the assumes bearing capacity a footing asconditions the sum of for the the contribution of each column may Besides,considering Equation (9) perfectofundrained soil surrounding the column and usually be overconservative. For footings, an improvement for field tests (plate load tests) on an some drainage could be expected near the granular column. For common cases, Equation (9) gives isolated column may be achieved by using a larger loaded area, so same as in the a vertical load supported for each stone column of 20–50 tons [15] (p. is6).theSometimes, field tests footing [3] (p. 165). are performed to check the exact vertical load supported by one isolated column with ar = 100%. The approach of of considering 9. Groups Columns the bearing capacity of a footing as the sum of the contribution of each column mayStone usually betreatments overconservative. Forused footings, improvement for as field tests (plate load column are traditionally beneathan large loaded areas, such embankments, tests) onbut anthey isolated column may be achieved by using a larger loaded area, so a is the same are also used beneath footings, when the applied loads are not high [67]. Forr these cases, theas in the footing homogenization [3] (p. 165). technique is valid [68]. Besides, the plane strain model using gravel trenches 9. Groups of Columns Stone column treatments are traditionally used beneath large loaded areas, such as embankments, but they are also used beneath footings, when the applied loads are not high [67]. For these cases, Materials 2017, 10, 782 13 of 23 the homogenization technique is valid [68]. Besides, the plane strain model using gravel trenches (Figure 1c)Materials is also2017, valid for strip footings. On the other hand, when the footing could be13converted into 10, 782 of 23 a circular footing (e.g., a square footing), the simplified geometrical model in axial symmetry that uses (Figure 1c) is also valid for strip footings. On the other hand, when the footing could be converted gravel rings is also applicable [60,61] (Figure 15). For the latter case (i.e., axial symmetry conditions), into a circular footing (e.g., a square footing), the simplified geometrical model in axial symmetry the authorthat hasuses recently proposed an additional model thatcase assumes that all the columns gravel rings is also applicable [60,61] simplified (Figure 15). For the latter (i.e., axial symmetry may be converted into central the an same area (Figure 15d), keeping ar constant conditions), theaauthor hascolumn recently with proposed additional simplified model that assumes that all [61]. The the columns maymodel be converted intoitamay central the same area (Figure solutions 15d), keeping main advantages of this are that becolumn used towith developed analytical [38] and that it constant [61]. The main advantages of this model are that it may be used to developed analytical is also applicable to encased columns. For encased columns, it is also necessary to transform the elastic solutions [38] and that it is also applicable to encased columns. For encased columns, it is also propertiesnecessary of the encasement, the factor Jgof/rthe kept constant [69]. this central c isencasement, ⁄ is of to transform so the that elastic properties so that the The factorvalidity kept column model for small groups of columns is based on the small influence on the footing constant [69]. The validity of this central column model for small groups of columns is based on thesettlement small influence footing settlement (usually less than 10%)position of the number of columns and their [61]. This (usually less than 10%)onofthethe number of columns and their beneath the footing position beneath the footing [61]. This simplified central column is only valid to study the footing simplified central column is only valid to study the footing settlement and should not be used for settlement and should not be used for other features, such as the consolidation process or the bending other features, such consolidation process or the bending moments in the footing. moments in as the the footing. (a) (b) (c) (d) Figure 15. Simplified geometrical model for groups of stone columns beneath a square footing: Figure 15.(a)Simplified for groups of stone columns beneath a square footing: 3D model; (b)geometrical Gravel rings; (c)model Homogenization; (d) One central column. (a) 3D model; (b) Gravel rings; (c) Homogenization; (d) One central column. 10. Critical Column Length The concept of critical length is used for stone columns as in piles (e.g., [65]). Thus, for columns 10. Critical Column Length longer than the critical length, the settlement reduction or the bearing capacity of the footing does The concept of change criticalorlength used for stone columns incritical pileslength (e.g., of [65]). Thus, for for columns not notably improve.isFor non-encased stone columns,asthe the columns settlement reduction is around 1.5 − 2 [61,70]. The critical length for encased columns usually longer than the critical length, the settlement reduction or the bearing capacity ofisthe footing does higher, around 2 − 3.5 for situations [69]. The main controlling parameter of the critical not notably change or improve. Forcommon non-encased stone columns, the critical length of the columns for column length is the extension of the load, i.e., , but the specific value of the critical column length settlementalso reduction is around 1.5 − 2 B [61,70]. The critical length for encased columns is usually depends on other parameters, such as and the soil and column properties [61,65,69,70]. higher, aroundIt 2is− 3.5 B for common situations [69]. Thegive main controlling worth noting that many authors (e.g., [71–74]) critical lengths as parameter a function ofof thethe critical column diameter, but the column length to diameter ratio has a minor influence by itself [61,69]. The column length is the extension of the load, i.e., B, but the specific value of the critical column length origin the misinterpretation factthe thatsoil the first were for[61,65,69,70]. the case of also depends onofother parameters, could such lie asinarthe and andproposals column[63,64] properties an isolated column with load only on the column surface, i.e., = . It is worthFor noting that many authors (e.g., [71–74]) give critical lengths as a function of the column settlement reduction, the critical column length is related to the pressure bulb that the diameter, but thegenerates column(Figure length16a), to diameter ratio bearing has a minor influence itself [61,69]. footing while for footing capacity, the criticalby length depends on The the origin of failure mechanism (Figure thethat critical settlement[63,64] reduction is longer, thatcase is usually the misinterpretation could lie in16b). theAs fact thelength firstfor proposals were for the of an isolated the considered Forcolumn large loaded areas (high of ), the critical column length is higher column with load onlyvalue. on the surface, i.e., dvalues c = B. than the soft layer thickness, and consequently, the concept of critical or optimal length does not apply. For settlement reduction, the critical column length is related to the pressure bulb that the footing The mechanisms in Figure 16 also explain the influence of additional columns outside the generates footing. (FigureColumns 16a), while forthefooting capacity, theline critical length depends outside footing bearing are crossed by the slip or failure mechanism; so, on theythe failure contribute to the bearing capacity. On the contrary, they hardly influence the settlement reduction if mechanism (Figure 16b). As the critical length for settlement reduction is longer, that is usually the considered value. For large loaded areas (high values of B), the critical column length is higher than the soft layer thickness, and consequently, the concept of critical or optimal length does not apply. The mechanisms in Figure 16 also explain the influence of additional columns outside the footing. Columns outside the footing are crossed by the slip line or failure mechanism; so, they contribute to the bearing capacity. On the contrary, they hardly influence the settlement reduction if they are outside the pressure bulb. The only minor beneficial effect for the settlement reduction is that they distribute Materials 2017, 10, 782 Materials 2017, 10, 782 14 of 23 14 of 23 they are outside the pressure bulb. The only minor beneficial effect for the settlement reduction is that theyover distribute the applied load over Therefore, a slightly wider area.the Therefore, when the bearing capacityin terms of the applied load a slightly wider area. when bearing capacity is defined is defined in the terms of a critical settlement, theoutside effect of the extrafooting columnsisoutside the footing[65]. is only a critical settlement, effect of extra columns only marginal marginal [65]. The presented analysis on critical column length is based on a homogeneous soft soil layer; The presented analysis on critical column length is based on a homogeneous soft soil layer; for for layered soils, this analysis directly applicable. layered soils, this analysisisisnot not directly applicable. 16. Conceptual justification of critical column length in a homogenous soil layer: (a) for Figure 16.Figure Conceptual justification of critical column length in a homogenous soil layer: (a) for settlement reduction; (b) for bearing capacity. settlement reduction; (b) for bearing capacity. 11. Column Installation Effects 11. Column Installation Effects Column installation alters the surrounding soil, especially when the columns are installed by vibrodisplacement. However, it is commonly accepted that in a stone column treatment, installation Column installation alters the surrounding soil, especially when the columns are installed by effects are less significance and the main improvement is caused by the inclusion of gravel. The search vibrodisplacement. However, commonly accepted that in installation a stone column treatment, for more accurate designs it hasisled to more detailed analyses of the effects [75–78]. Notableinstallation pore pressuresand are consistently in the field due to column [79]. effects areexcess less significance the main measured improvement is caused by theinstallation inclusion ofHowever, gravel. The search most cases, these pore quicklydetailed dissipate analyses [80]. Column especially by for more inaccurate designs has pressures led to more of installation, the installation effects [75–78]. vibrodisplacement, also leads to an increase in the horizontal stresses [75]. This is usually quantified Notable excess pore pressures are consistently measured in the field due to column installation [79]. through the earth pressure coefficient, [3,75,76]. The increase in effective mean stress leads, in turn, However,toinanmost cases, these pore pressures dissipate [80]. Column installation, especially by increase in the soft soil stiffness [3] (p. quickly 138). The excess pressures and the in remolding caused by column [75]. installation to an quantified vibrodisplacement, alsopore leads to an increase the horizontal stresses This islead usually instantaneous reduction of the undrained shear strength just afterin installation the pore through the earth pressure coefficient, K [3,75,76]. The increase effective[77,81]. meanAs stress leads, in turn, pressures dissipates and the effective mean stress increases, the value of the undrained shear strength to an increase in the soft soil stiffness [3] (p. 138). recovers and usually increases above its initial value [77,81]. Special care must be taken with soil The remolding excess pore pressures and Poker the vibration remolding caused column in sensitive soils [67,77]. is necessary forby proper column installation compaction, butlead to an excessive use of the of poker many repenetrations (“overworking”) may result in high[77,81]. excess pore instantaneous reduction thewith undrained shear strength just after installation As the pore pressures and disturbance of sensitive or overconsolidated soil layers, such as a dry, stiff upper crust pressures dissipates and the effective mean stress increases, the value of the undrained shear strength that is common is soft soils and is beneficial for load distribution [79]. recovers and The usually increases above its initial value [77,81]. Special care must be taken with soil remolding caused by column installation also alters the permeability of the soft soil in the remoldingvicinity in sensitive soils [67,77]. isisnecessary propersoft column compaction, but of the column. This zone,Poker whosevibration permeability lower than for the natural soil, is usually “smear” [82,83]. Therepenetrations properties of the smear zone, specially its sizeresult and permeability, excessive called use ofthethe pokerzone with many (“overworking”) may in high excess pore largely studied for verticalor drains [84–86]. Another related problemsuch is column [83]. pressures have andbeen disturbance of sensitive overconsolidated soil layers, as aclogging dry, stiff upper crust Gravel compaction during column installation and the high hydraulic gradients at the soil-column that is common is soft soils and is beneficial for load distribution [79]. interface inevitably produce a migration of clay particles into the pores of the granular column, The remolding caused by column installation alsoofalters the permeability soft soil in the notably reducing the permeability of an external annulus the column. This phenomenonofis the normally important in geosynthetic because the acts as asoft filter.soil, is usually vicinity ofless the column. This zone,encased whosecolumns permeability is geosynthetic lower thanusually the natural The introduction of installation effects in the modeling stone columns is a complex called the “smear” zone [82,83]. The properties ofnumerical the smear zone, of specially its size and permeability, process. Some attempts have been made using field measurements and back-fitting the results [75] have been largely studied for vertical drains [84–86]. Another related problem is column clogging [83]. Gravel compaction during column installation and the high hydraulic gradients at the soil-column interface inevitably produce a migration of clay particles into the pores of the granular column, notably reducing the permeability of an external annulus of the column. This phenomenon is normally less important in geosynthetic encased columns because the geosynthetic usually acts as a filter. The introduction of installation effects in the numerical modeling of stone columns is a complex process. Some attempts have been made using field measurements and back-fitting the results [75] or using previous numerical analyses that simulate column installation as a cylindrical cavity expansion [78]. From a practical point of view, column installation may be considered altering the at-rest earth pressure coefficient of the natural soft soil. Priebe’s theoretical solution [16] considers, Materials 2017, 10, 782 15 of 23 for example, a value of 1. Published values [59] vary between 0.4 and 2.5, with average values slightly above 1. Obviously, the earth pressure coefficient after column installation should depend on the initial value, the construction method and the type of soil. Finally, if the numerical model does not consider a stress-dependent constitutive model for the soft soil, its stiffness should be increased correspondingly [76]. 12. Properties of the Columns The modeling process of the soft soil is beyond the scope of this review and it is broadly analyzed elsewhere (e.g., [87,88]). Some authors [79,89] warn about the importance of secondary compression in some soils and the limited capacity of stone columns to reduce it. Gravel properties are also well-analyzed (e.g., [90–92]), but the properties of the gravel for stone columns are not so readily available nor are they generally measured for each project. For stone columns, the gravel should be clean, preferably crushed stone, hard, unweathered, free from organics or other deleterious materials and its degradation using the Los Angeles testing machine should be less than a 45% of loss [15] (p. 117). Its grain size should be between 12 and 75 mm. For bottom feed columns, the maximum size is usually limited to 50 mm to avoid obstructions in the feeding tube. The relative density of the gravel in the stone columns is not usually measured and it may vary along the length of the column, in a similar manner as the column diameter. A proper stone column construction should achieve relative densities of the gravel above 75% [15]. Herle et al. [93] measured values close to 100%. The friction angle of the columns (φc ) has a notable influence on the results of a stone column treatment. Its value decreases with the confining pressure (σc ). Using samples in a dense state, Herle et al. [93] measure values up to 60◦ for low confining pressures (σc = 50 kPa) and values around 50◦ for medium confining pressures (σc = 200 kPa) (Table 2). However, when choosing a value of the friction angle to model stone columns, it is necessary to consider the following points: • • • • Columns are usually under triaxial conditions and for granular soils, the friction angle measured by direct shear tests is slightly higher than that measured using triaxial tests [94]. Values in Table 2 are peak values, but plastic strains in the columns may be important; so, it is recommended to reduce peak friction values by 5–7% to obtain approximate residual values [3]. The validity of some theoretical methods and their success in predicting field measurements may be linked to the use of conservative values of the friction angle. Published numerical analyses consider values of 40–45◦ (Table 3). Table 2. Stress-dependent peak friction angles of dense gravel [93]. Type of Gravel φc,max (◦ ) σc,max (kPa) φc,min (◦ ) σc,min (kPa) Remarks Crushed lime stone River gravel River gravel, sub-round Rivel gravel, sub-round Rivel gravel, crushed Dolomite Dolomite Sandstone Basalt Basalt Basalt 63.1 58.8 57.1 59.2 60.4 64.0 54.0 60.1 64.2 71.8 70.0 50 50 50 50 50 15 15 27 27 8 8 53.8 51.9 50.9 53.2 55.2 43 40 45.6 45.6 45.6 51.1 200 200 200 200 200 500 500 695 695 240 120 DS, Vibro SC DS, Vibro SC DS, Vibro SC, d60 /d10 = 2.6 DS, Vibro SC, d60 /d10 = 2.1 DS, Vibro SC TX, γ = 17 kN/m3 , [90] TX, γ = 15 kN/m3 , [90] TX, [92] TX, [92] TX, d50 = 30 mm, [91] TX, d50 = 30 mm, [91] DS: Direct shear test; TX: Triaxial test; Vibro SC: dense gravel for vibrated stone columns [93]; d60 /d10 : uniformity coefficient. Materials 2017, 10, 782 16 of 23 Table 3. Parameters used to model stone columns in numerical analyses. Reference φc (◦ ) ψc (◦ ) Ec (MPa) ν (-) m (-) γd /γsat (kN/m3 ) [59] [68] [70] [74] [73] [30] [54] [49] [78] [95] [96] 41 35 40–35 45 48 40 40 35 42 46 45 3 0 26 0 5 12 10 15 29.2 67.5 50 100 2.5 30 30 25 35 22 70 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.2 0.15 0.2 0.59 0.3 0.25 0.3 18.6/21.6 19/19 16/15/15 -/15 20/23.5 16/19 19/19 The reference pressure for the stiffness is 100 kPa; m: Exponent of the power law used to reproduce the stress dependent stiffness. The Young’s modulus of the columns is usually between 25 and 100 MPa (Table 3) and it also varies with the confining pressure. A hyperbolic power law is sometimes used to reproduce the stress dependent stiffness of the gravel of the columns (e.g., [58,95,96]). The common value of the exponent of the power law is around 0.3 (Table 3). The unit weight of the gravel does not require further comment, except for the cases where the columns do not reach a rigid substratum and they are installed by vibrodisplacement (e.g., [49]). In those cases, the addition of gravel and its corresponding weight may cause additional deformation in the underlying layer that is not improved with stone columns. For example, [49] used non-realistic high values, namely γsat = 23.5 kN/m3 to account for this effect. Regarding encased stone columns, the material used to construct them is usually sand instead of gravel. The available information is more limited, but it seems obvious to use lower values of both the friction and dilatancy angles (Table 4). Table 4. Parameters used to model encased stone columns in numerical analyses. Reference φc (◦ ) ψc (◦ ) Ec (MPa) ν (-) m (-) γd /γsat (kN/m3 ) [73] [97] [98] [28] [5] [31] [48] [69] 48 40 38 40 32.5 35 38 40 4 10 5 0 0 10 5 9 40 15.5 80 30 40 30 0.3 0.3 0.3 0.2 0.3 0.33 - 16/-/23 -/18 -/20 19/20 -/22 -/20 m: Exponent of the power law used to reproduce the stress dependent stiffness. 13. Modeling the Geosynthetic Encasement As shown in Figure 4, the geosynthetic encasement may be modeled as a flexible membrane that does not support compressive stresses, has a negligible thickness and behaves as a linear elastic material with a modulus of Jg = 1000–5000 kN/m. Regarding its tensile strength, for modeling purposes, it may usually be enough to verify that it is far from being reached. The tensile strength is usually reached for circumferential strains of around 5–10%, which implies strength values of 100–300 kN/m [5]. It is common that geosynthetic may be anisotropic and then different properties should be input for each direction. Some other geosynthetic features, such as creep and damage during installation, are usually considered through reduction factors. Recent numerical analyses [99] model numerically the creep behavior of geosynthetics. Materials 2017, 10, 782 17 of 23 In those numerical analyses where the geosynthetic is modeled as a continuum element of small thickness, it is necessary to ensure that it does not support compressive stresses, and hence, bending moments. Its Young’s modulus should be Jg divided by its thickness. Little attention is usually paid to the Poisson’s ratio of the geosynthetic (νg ), but it may have important consequences on the results [100]. When the encasement is modeled as a membrane, it is common to study the two directions of the geosynthetic independently, without interaction between them (Equation (3)). That means νg = 0. On the other hand, when the geosynthetic is modeled as a continuum element, it is necessary to specify a value for νg . In some cases (e.g., [31]), a common value of 0.3 is input. That implies that, although the geosynthetic that surrounds the columns does not support compressive stresses in the vertical direction, it compresses vertically (vertical strains) and then, that vertical compression leads to a radial expansion of the geosynthetic (horizontal or circumferential strains) due to Poisson’s effect. Thus, the geosynthetic encasement decreases its lateral confinement to the column due to this radial expansion. Common geosynthetics for column encasement are woven geotextiles. For woven geotextiles, the two directions work nearly independently; so, it seems logical to use values of the Poisson’s ratio close to 0. Soderman and Giroud [101] propose values of νg = 0.1 for woven geotextiles and νg = 0.35 for non-woven geotextiles. 14. Conclusions This paper reviews the main modeling techniques for stone columns, both ordinary or non-encased stone columns, and geosynthetic-encased stone columns. The paper tries to encompass the more recent advances and recommendations in the topic. There exist several simplified geometrical models. The suitability of each of them depends on the type of process to be studied, e.g., bearing capacity or settlement, and the type of analyses, e.g., analytical or numerical in 2D or 3D. For numerical analyses of a problem that fulfils plane strain conditions but for the stone column treatment, such as an embankment for a linear infrastructure, the simplified geometrical model of a three-dimensional slice of columns is recommended because it does not require any transformation of the problem parameters. For more simplified models for the stone column treatment, such as gravel trenches or homogenization, calibrating or tuning the parameters using the unit cell as an auxiliary problem is recommended. The model of an isolated column with load just on top of it may be useful for field tests, but it is not a common situation in real cases because loading the soil that surrounds the column is beneficial. The behavior of an isolated column is different from that of a column within a large group under distributed load. Those differences should be considered when extrapolating results. For groups of columns, the critical column length depends mainly on the loading area. For non-encased columns, its value is around twice the footing width and for encased columns is slightly higher. Some guidance is provided to consider installation effects and to model the column and the geosynthetic encasement. Column installation usually increases the horizontal stresses and that is often accounted for using a high value of the earth pressure coefficient, typically around 1. On the other hand, the relevance of the Poisson’s ratio of the geosynthetic is taken into account and an appropriate value for a woven geotextile could be close to 0, i.e., the two directions work nearly independently. Finally, laboratory tests are not commonly performed to characterize stone column properties and some column parameters published in the literature are tabulated as a reference. Acknowledgments: The author thanks all the institutions that have allowed him to work on stone columns and all the people that have collaborated in the research on stone columns. The author thanks the Spanish Society for Soil Mechanics and Geotechnical Engineering (SEMSIG) for the invitation to give a lecture on stone columns in the 10th National Symposium on Geotechnical Engineering, held in A Coruña in October 2016. The present paper is based on that lecture and the corresponding conference paper [102] (In Spanish). The author acknowledges the funding received through the following projects: “Improvement of soft soils with stone columns for foundation of embankments. Analysis of the process and design criteria” for Materials 2017, 10, 782 18 of 23 the Spanish Ministry of Public Works (03-A634), “An integrated calculation procedure for stone columns, considering the influence of the method of installation” for the Spanish Ministry of Science and Innovation (BIA2009-13602) and “The critical distance in rock fracture” for the Spanish Ministry of Economy, Industry and Competitiveness (BIA2015-67479-R). The author is very thankful to the Group of Geotechnical Engineering of the University of Cantabria, particularly to C. Sagaseta, J. Cañizal, A. Da Costa, A. Cimentada and M. Miranda. The research on stone columns has been contributed by the international cooperation of H. Schweiger, B. McCabe, M. Karstunen and N. Sivasithamparam. The author acknowledges the participation in the following projects supported by the European Commission: the Marie Cure Research Training Network entitled “Advanced Modelling of Ground Improvement on Soft Soils (AMGISS)” (MRTN-CT-2004-512120) and the Industry-Academia Partnerships and Pathways project on “Modelling Installation Effects in Geotechnical Engineering (GEO-INSTALL)” (PIA-GA-2009-230638). The following companies have also contributed to the research: Keller Group, mainly through E. Carvajal, G. Vukotić and J. Wehr; Huesker Inc., mainly through I. Diego; and Geocisa, particularly for the field measurements carried out in Sueca, near Valencia (Spain). Conflicts of Interest: The author declares no conflict of interest. The founding sponsors had no role in the design of the review; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results. List of Symbols ar cu dc d10 , d60 k m pa pa (0) pa (z) r s sr sz sz0 t u A B E H Jg K K0 L Tg b g ν σ σc ϕ ψ Area replacement ratio: Ac /Ae Undrained shear strength Column diameter Diameters corresponding to 10% and 60% finer in the particle-size distribution, respectively Hydraulic conductivity/permeability Exponent of the power law used to reproduce the stress dependent stiffness Uniform applied vertical pressure Uniform applied vertical pressure at the surface Reduced value using a trapezoidal distribution of the uniform applied vertical pressure at a depth z Radius Centre-to-centre column spacing Radial displacement of the column Settlement Settlement without columns Thickness Pore water pressure Cross-sectional area Footing width Young’s modulus Soft soil layer thickness Encasement stiffness Coefficient of lateral earth pressure Coefficient of lateral earth pressure at rest (Column) length Tensile hoop force at the encasement Settlement reduction factor: β = sz /sz0 Unit weight Poisson’s ratio Stress Confining pressure Friction angle Dilatancy angle Materials 2017, 10, 782 19 of 23 Subscripts c, s, g, e d m max min r r, θ, z sat x, y, z ¯ (upper bar) column, soil, encasement or geosynthetic, loaded area or unit cell dry equivalent homogeneous soil maximum value minimum value equivalent gravel ring Cylindrical coordinates Saturated Cartesian coordinates average value along the radius Abbreviations DS G.W.T. 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