CHAPTER 5
G
RANULAR
C
OLUMNS
Several types of granular columns are currently used to improve bearing soils for shallow foundations in one or more of the following ways: (a) increase ultimate bearing capacity, (b) reduce compressibility, (c) increase the rate of settlement in saturated soils, (d) reduce liquefaction potential, (e) increase lateral resistance, (f) increase uplift capacity, and (g) increase the stability of natural or fill slopes. The applicability of each type of granular column is summarized in Table 5.1.
TABLE 5.1. Comparison of Engineering Properties Improved by Various Granular Columns
Increase Reduce
Type of Ultimate Magnitude Increase Provide Increase Reduce Increase
Granular Bearing of Rate of Uplift Lateral Liquefaction Slope
Column Capacity Settlement Settlement Resistance Resistance Potential Stability Stone Column Significant Significant Significant None Moderate Significant Significant
Sand Column Significant Significant Significant None Some Significant Significant Geopier RAP Significant Significant Significant Significant Significant Significant Significant Impact RAP Significant Significant Significant None Moderate Significant Significant Gravel Drain Some Some Significant None Not Applicable Significant Not Applicable
Sand Drain Some Some Significant None Not Applicable Significant Not Applicable
Type of Improvement
Although the methods of installation and types of columnar materials can vary significantly among the different types of granular columns, all types have the following basic features:
1. A single vertical, cylindrical column or group of columns consisting of granular or chemically-stabilized granular material are created within the ground. Typically about 10 to 40% of the volume of the native soil is replaced or displaced by the granular columns within the reinforced zone.
2. The columns are usually stronger, stiffer, and more permeable than the pre-existing natural or fill soil into which they are installed (hereinafter called matrix or native soil).
3. When large areal coverage is provided, such as beneath a long embankment or a mat foundation, a triangular pattern is typically used (Fig. 5.1). Sometimes a square pattern is used.
4. For support of individual footings, a variety of patterns can be used depending on the shape and size of the footing, as well as the types and degrees of improvement needed in the soil. Some common patterns for individual footings are illustrated in Fig. 5.2.
s s s
TRIANGULAR
SQUARE
s s 0 .8 6 6 sFIGURE 5.1 Triangular and square patterns of columnar reinforcement
FIGURE 5.2 Some common patterns of granular columns to support individual footings (from Geopier Foundation Company 1998)
5.1 TYPES OF GRANULAR COLUMNS
The primary types of granular columns in common use today are stone columns, sand columns, Rammed Aggregate Piers, gravel drains, and sand drains. The important characteristics of each of these types are described in the following sections.
5.1.1 Stone Columns
Stone columns are installed using either vibratory, rotary, or ramming techniques. The conventional stone column method was developed in Germany in the 1950s as an extension of the vibro-compaction process (see Section 4.2). Conventional stone columns are installed using the same type of horizontally vibrating probe used in vibro-compaction. Either replacement (wet) or displacement (dry) techniques can be used. In the vibro-replacement method (Fig. 5.3a), a hole is created in the ground to the desired depth by water jetting from the vibratory probe (ASCE 1987). The uncased hole is flushed out and stone is added in increments through the annular space between the probe and the enlarged hole. The stone is installed in about 12 to 48 in. (0.3 to 1.2 m) thick lifts by a combination of vibration from the probe and lowering the probe into the stone. During this process, soft matrix soils may collapse into the hole. If so, the continuing water upflow carries the collapsed material to the ground surface, allowing the stone to expand farther outward until equilibrium is reached. The diameter of the column varies with depth, generally being larger at the top, bottom, and at softer soil layers. Vibro-replacement is best-suited for sites with soils having undrained shearing strengths in the range of about 300 to 1,000 psf (15 to 50 kPa) and a high groundwater table. In the 1970s and early 1980s, vibro-replacement was the only method used to construct stone columns in the U.S., although dry methods were used elsewhere (Barksdale and Bachus 1983, Goughnour 1997). However, since that time environmental constraints have complicated the disposal of the large amounts of silt and clay-laden effluents generated during the wet process. An additional problem is the ponding of water on the ground surface, which can disrupt work and slow production. In response to these constraints and problems, the use of dry techniques is now common in the U.S.
In the vibro-displacement technique, the vibrating probe displaces the soil laterally as it is advanced into the ground, usually with the aid of compressed air through the tip of the probe (ASCE 1987). In the top-feed method illustrated in Fig. 5.3b, the probe is removed from the hole after reaching the desired depth, backfill is dropped in the annular space between the probe and enlarged hole, and the probe is lowered again to displace the stone laterally and downward. This process is repeated in lifts to create the stone column. This top-feed dry method is best suited for sites with a deep groundwater table and firm soils with undrained shearing strengths from about 600 to 1,200 psf (30 to 60 kPa) and low sensitivity. Beginning about 1976, bottom-feed equipment and methods have developed to extend the use of vibro-displacement methods to soft and loose saturated soils (Jebe and Bartels 1983). In the bottom-feed methods, the probe remains in the hole while the stone is discharged through the probe. The columns created by the dry techniques are usually smaller in diameter than those created by the wet process because no matrix material is removed from the hole in the dry process. The vibro-stitcher described in Section 4.2 and depicted in Fig. 4.17, which uses a bottom-feed method, can also be used to install conventional stone columns to a lesser depth than the traditional vibroflot (also described in Section 4.2).
(a)
(b)
FIGURE 5.3 Construction of a stone column by top-feed (a) vibro-replacement (from Baumann and Bauer 1974), and (b) vibro-displacement (from Barksdale and Dobson 1983).
FIGURE 5.4 Perspective view of impeller used in construction of Rotary Stone Columns (from Goughnour 1997)
The rotary method of installing stone columns was developed in the mid 1990s as an alternate technique for use in soft cohesive soils and loose silty or clayey sands (Goughnour 1997). This method was developed to reduce the problems associated with contamination of the stone by intermixing with matrix soil that occur in vibratory installation methods. The heart of this system is an impeller that consists of two symmetrically located logarithmic spiral sections (Fig. 5.4). The impeller is driven by a centrally-placed drive shaft and fits closely beneath the bottom of a feed pipe. Stone is fed down the annular space between the feed pipe and the shaft. During rotation of the impeller, stone is thrown radially outward while additional stone falls from the feed pipe into the pockets behind the logarithmic spiral sections. The main components of the machine used to install rotary stone columns (Fig. 5.4a) are a carrier (crane) and mast (construction leads), a hopper and winch arrangement for stone delivery, a probe which includes a vibratory driver/extractor, a rotary hydraulic motor, an airlock and chute, and an impeller fitted at the bottom. The top portion of the probe is shown in Fig. 5.4b.
The normal procedures used to build a rotary stone column are as follows (Goughnour 1997): 1. The feed pipe and the impeller are positioned over the location where the stone column is to
be installed.
2. The feed pipe is lowered into the ground. During this process air pressure is applied to the interior of the empty feed pipe, the impeller is rotating, and the vibratory driver/extractor is turned on.
3. When the desired depth is reached, the penetration, vibration, and impeller rotation are all stopped.
FIGURE 5.5 Equipment used to construct rotary stone columns (from Goughnour 1997): (a) Overall view of rig, and (b) detail of top part of probe.
5. The air pressure from the feed pipe is released, the airlock is opened, and stone is fed from the hopper through the airlock into the feed pipe.
6. The airlock is closed and air pressure is re-established within the feed pipe.
7. With the impeller rotating and vibration applied as necessary, the feed pipe is raised in response to hydraulic pressure on the impeller motor. The pressure must be maintained as closely as possible to the target pressure, which is site-specific and depends on field conditions and desired column diameter. The target pressure may be to be varied if field conditions change on site.
8. When the feed pipe is empty, the hydraulic pressure on the impeller motor drops and does not rebuild. At this point the pipe is no longer lifted, and the vibration and impeller are stopped.
9. Repeat steps 4 through 8 until the column is constructed to the desired elevation.
Stone columns have also been installed using a technique called dynamic replacement., which is a combination of stone columns and dynamic compaction, as shown in Fig. 5.6 (Gambin 1984, Liausu 1984). In this method, stone is placed in a layer on the ground surface and compacted using a heavy tamper in the manner described for dynamic compaction (Section 4.1). Soil improvement occurs not only at the location of the stone columns, but also between column locations owing to horizontal densification of the soil.
FIGURE 5.6 Construction of stone columns by dynamic replacement (modified from Liausu 1984).
5.1.2 Sand Columns
Sand columns – also known as sand compaction piles – have been used extensively in Japan and other parts of Asia to reinforce soft and loose soils. The methods of installation used to create sand columns can be classified into the following three primary types (Japanese Geotechnical Society 1998):
1. Vertical vibration and compaction with enlargement of the column diameter by repeatedly supplying sand and then pulling and re-penetrating the casing pipe (Fig. 5.7a).
2. Vertical vibration and compaction with enlargement of the column diameter by a hydraulic compaction device at the end of the casing pipe (Fig. 5.7b).
3. Vertical vibration and compaction with enlargement of the column diameter by horizontal vibration of the end of the casing pipe (Fig. 5.7c).
FIGURE 5.7 Common methods used to install sand columns (from Japanese Geotechnical Society 1998)
5.1.3 Rammed Aggregate Piers
Rammed Aggregate Piers® (RAPs) were developed as a proprietary product of Geopier Foundation Company and the technology was and is protected by U.S. and international patents (e.g. Fox and Lawton 1993). RAPs can be installed using either replacement (Geopier® ) or displacement (Impact® ) methods. The Geopier method was developed in the late 1980s and the Impact method in the early 2000s. Although the main patent on the Geopier method has expired, there are auxiliary patents on the Geopier method and patents on the Impact method still in effect. There are two primary types of RAPs – compressive and uplift. The major steps used to construct compressive Geopier RAPS are illustrated in Figs. 5.8 and 5.9 and summarized as follows:
1. A cylindrical cavity is formed in the soil by drilling (Fig. 5.8a). The cavity is typically in the range of 24 to 36 in. (0.61 to 0.91 m) in diameter and 7 to 30 ft (2 to 9 m) in length.
2. Aggregate is placed at the bottom of the hole and is compacted by repeated ramming using a specially designed tamper with a beveled head (Fig. 5.8b,c). High frequency, low amplitude energy for this process is supplied by a skid loader, a backhoe, or an excavator (Fig. 5.9). The "bulb" created from this process provides a firm foundation on which to construct the remainder of the pier and is especially important in soft and loose soils.
3. The main body of the pier is then constructed in a similar manner by placing loose aggregate in the hole and compacting it in 12 in. (0.30 m) or thinner lifts to the desired height (Fig. 5.8d). The height of a pier is typically two to eight times its diameter.
An uplift pier is constructed in a similar manner with the following additional steps. After the bottom bulb is constructed (step 2), an uplift assembly consisting of a horizontal steel plate and attached vertical threaded steel bars (Fig. 5.10) is set in hole and rests on the bottom bulb. A spacer is used to hold the threaded bars apart during construction of the main body of the pier in the same manner as described is step 3 above. In permanent applications, the uplift plate and threaded bars may be galvanized to reduce long-term corrosion of the steel. The uplift bars or dowels spliced to the uplift bars extend upward into a reinforced concrete footing and are bonded to the concrete, forming an integral foundation system with the footing. Uplift piers also provide substantial resistance to compressive and lateral forces and displacements.
Well-graded gravelly aggregate (normally highway base course material) is typically used as the pier aggregate above the groundwater table, with open-graded aggregate normally used below the groundwater table. Owing to the confinement provided by the adjacent matrix soil and the high energy used to compact the piers, high densities are achieved within the compacted pier aggregate (typically more than 100% of modified Proctor maximum dry density). In addition, the adjacent matrix soils are substantially prestressed and prestrained. Measurements in matrix soils adjacent to Geopier elements have shown that installation of a pier can increase the horizontal stresses as far away as 10 ft (3 m) or more, and that the horizontal stresses immediately adjacent to the piers can reach the limiting passive condition.
FIGURE 5.8 Steps in construction of a compressive Rammed Aggregate Pier® using the Geopier system (from Geopier Foundation Company 1998)
FIGURE 5.9 Installation of Uplift Geopiers with excavators supplying the energy (courtesy of Geopier Foundation Co., Mooresville, NC)
Linear elements can also be constructed by first excavating a trench using a backhoe or excavator. Aggregate is then compacted to fill the trench in thin lifts using the same procedures described above for columnar piers. Linear elements are used to support walls and long, thin rectangular foundations and typically have nominal widths of 18 to 30 in. (0.46 to 0.76 m).
The process for installing Impact Piers is summarized as follows (see Figs. 5.11 and 5.12): 1. A specially designed mandrel and tamper foot is driven into the ground using a strong static
force augmented with dynamic vertical impact energy to depths typically ranging from about 10 to 40 ft (3 to 12 m). A sacrificial cap prevents soil from entering the tamper foot and mandrel during driving. The process displaces soils laterally, densifying and reinforcing adjacent existing soils. No spoils are generated during the process.
2. After driving to the design depth, aggregate is placed through the mandrel and is then compacted by raising the mandrel about 3 ft (0.9 m) and then driving the beveled tamper foot back down 2 ft (0.6 m) using static down force and dynamic vertical ramming, forming a 1-ft (0.3-m) thick compacted lift.
FIGURE 5.11 Schematic illustration of the Impact Pier construction process (courtesy of Geopier Foundation Co., Mooresville, NC)
FIGURE 5.12 Installation of Impact Piers (courtesy of David Plehn, Geopier Northwest, Sandy, UT)
5.1.4 Sand and Gravel Drains
The main purpose of sand and gravel drains is to provide rapid drainage of saturated soils during precompression (Chapter 6), earthquakes, or other phenomena in which significant excess pore water pressures may develop. Sand and gravel drains are typically 8 to 20 in. (200 to 500 mm) in diameter and are spaced anywhere from 5 to 20 ft (1.5 to 6.0 m). Since drainage is the primary desired engineering characteristic, compaction of the sand or gravel is not a priority; rather, excessive compaction may reduce the permeability and hence reduce drainage capability of the drain. Therefore, only sufficient compaction necessary to ensure continuity of the drain is typically undertaken.
Many methods of installation are used to construct sand and gravel drains, which are generally classified into two groups - displacement and non-displacement. The equipment used to install these types of drains include closed-ended mandrel, screw-type auger, continuous flight hollow stem auger, internal jetting, rotary jet, and Dutch jet-bailer. Additional details on the installation can be found in Ladd (1986).
5.2 ENGINEERING CHARACTERISTICS, RESPONSE, AND BEHAVIOR 5.2.1 Stress Concentration
When compressive loads are applied to a bearing soil reinforced with granular or chemically stabilized columns, the vertical stress induced on top of the substantially stiffer columns is much greater than on the more compressible matrix soil at the same level. This concept is illustrated using a spring analogy in Fig. 5.13 for a centric vertical load applied to a rigid foundation supported by a Geopier element. In this analogy, each spring represents the force generated over the same amount of contact area. It is well-known from basic physical laws that the force induced in a linearly elastic spring is equal in magnitude to the spring constant times the amount of deflection of the spring and acts in the direction opposite to the direction of the spring deflection (F = -Kx). For the conditions shown in Fig. 5.13, the footing will settle the same amount at all points. Hence the force induced in the stiff spring representing a granular column will be substantially greater than the forces generated in the springs representing the matrix soil. The sum of these spring forces must equal the applied load P. If these forces are divided by the same contact area, it is concluded that the stresses induced on the columns are greater than those induced on the matrix soil. Stress concentration also occurs beneath flexible foundations but to a lesser degree. Concentration of stresses on the stiffer columns is a key factor in controlling foundation settlement, increasing lateral resistance of footings, and stabilizing slopes with columnar reinforcement.
The concepts of unit cell, area replacement ratio and stress concentration are illustrated in Fig. 5.14 for columnar reinforcement arranged in a large square array at a center-to-center spacing of s. The unit cell is comprised of a single column and the corresponding tributary matrix soil. The boundaries of the unit cells are shown with dashed lines in Fig. 5.12a. The area of a column is designated Ac , the area of the matrix soil within the unit cell Am , and the total area of the unit cell A . The area replacement ratio (Ra ) and the stress concentration ratio (Rs ) are given by the following equations:
FIGURE 5.13 Spring analogy illustrating force and stress concentration on granular columns (modified from Geopier Foundation Company 1998)
FIGURE 5.14 Illustration of (a) unit cell and area replacement ratio (horizontal section), and (b) stress concentration and stress concentration ratio (vertical section)
A A R c a (5.1) m c s q q R (5.2)
For an individual footing, Ac is the area of all the columns supporting the footing and A is the total area of the footing. If the footing is rigid, as is the case for most individual footings composed of reinforced concrete, Rs = ksc / ksm , where ks is vertical subgrade modulus (Lawton et al. 1994). Since ks varies as a function of the width of the loaded area (see Eqs. 3.51-3.54), the values of ksc and ksm used must correspond to the appropriate loaded areas of the columns and the matrix soil. ksc can be obtained by performing a plate load test on top of a test or production column, with the diameter of the plate the same as the nominal diameter of the column. ksm can be estimated by performing a similar plate bearing test on the matrix soil at the elevation of the top of the columns and using established scaling relationship to calculate ksm for the width and shape of the actual footing. ksm can also be estimated from the results of in-situ or laboratory tests on the matrix soil (see Bowles 1975, pp. 516-518).
The following equations for the stresses induced on the top of columns (qc ) and the matrix soil (qm ) as a function of the average applied stress (q0 ) can be obtained by summing forces in the vertical direction and satisfying static equilibrium (Aboshi et al. 1979):
(5.3) (5.4) where c and m are stress concentration factors for the column and matrix materials,
respectively.
Both qc and qm depend on Rs , which must be estimated. The following theoretical equations from Aboshi et al. (1979) can be used to estimate the stress concentration ratio at yield (Rsy ). For friction-only columnar and matrix materials ( , c = 0):
45 2
tan
45 2
tan
45 2
tan2 2 2 m o c o sy c o R (5.5)For friction-only columnar material (c , cc = 0) and a saturated cohesive matrix soil in the
unconsolidated-undrained condition (m = 0, cm = sum ):
c
um my
o sy s q R tan2 45 2 12 (5.6)
c s a s c q R R R q q 0 0 1 1
m s a m q R R q q 0 0 1 1 1In a design situation where settlement is to be estimated, these equations are of limited value because the actual state of stress should be well below the level at which yield or failure occurs.
Typical values of Rs vary depending on the type of granular columns, applied stress level, duration of the load, stiffness of the matrix soil, and flexibility of the foundation applying the load to the columns. Rs is greater for rigid foundations than for flexible foundations (all other factors being the same). The settlement patterns of a rigid footing and a flexible foundation subjected to a centric, vertical load are compared in Fig. 5.15 for the same magnitude of settlement at the locations of the columns. The settlement within the matrix soil is greater for the flexible foundation than for the rigid footing, and thus the matrix soil carries more load (and hence the granular columns carry less load) for the flexible foundation. Measured values of Rs
for stone columns and sand columns generally range from 2.5 to 5.0 for typical values of applied stress. In design, Rs is usually conservatively assumed to be about 2 or 3. In tests with rigid foundations supported by Geopier elements, measured stress concentration ratios varied from about 8 to 40 at typical applied stress levels (Lawton 1999). Rs is usually estimated to be about 10 to 15 for the design of Geopier-supported foundations. It appears that Rs increases with increasing applied load up to a critical value of applied stress where it reaches a maximum (Han and Ye 1991). At stresses higher than the critical value, Rs decreases with increasing load. Furthermore, Rs usually increases with time at constant applied stress, probably owing to greater secondary compression within the matrix soil than the granular columns.
Base of Rigid Foundation after Settlement Base of Flexible Foundation after Settlement Base of Rigid and Flexible Foundation before Settlement
FIGURE 5.15 Settlement patterns for rigid and flexible foundations supported by granular columns and subjected to a centric, vertical load
5.2.2 Settlement
The inclusion of columnar reinforcement reduces the magnitude of settlement and generally increases the rate of settlement compared to the unreinforced soil. Numerous methods varying in complexity from simple approximations to sophisticated numerical analyses have been used to estimate the magnitude of settlement for foundations bearing on columnar-reinforced soil. Some of these methods are used exclusively for certain types of columnar reinforcement. Only a general overview of some of the most commonly used methods will be described here. Discussions of additional methods can be found, for example, in ASCE (1987) and Barksdale and Bachus (1983).
One of the simplest methods to estimate the settlement of structures founded on column-reinforced bearing soil is the equilibrium method. Although this method was originally developed to estimate primary consolidation settlement of saturated clays reinforced with sand columns (Aboshi et al. 1979), it can be applied to any type of matrix soil and drainage conditions. However, the equilibrium method is valid only where the columnar reinforcement extends throughout the entire thickness of the compressible material or throughout the depth where most of the strain occurs (approximately two times the width of a circular or a square foundation and about four times the width of a strip foundation). The steps in the equilibrium method are summarized as follows:
1. Estimate the stress induced on the matrix soil at the bearing level (qm ) using Eq. 5.4.
2. Calculate the settlement of the structure as if there is no columnar reinforcement and the average stress at the bearing level is qm rather than q0 . Use the actual dimensions and shape of the foundation.
The following example is given to illustrate the method. Example Problem 5.1
Given: A proposed highway embankment will be40-m (131-ft) wide at its crest and 5-m (33-ft) tall (He ) with 2H:1V side slopes and an average unit weight (e ) of 20 kN/m3 (127 pcf). The
embankment will be constructed on the surface of a normally consolidated clay stratum that is 10-m (33-ft) thick with an average saturated unit weight ( ) of 7.0 kN/m3 (45 pcf), an existing average void ratio of 1.30, and a virgin compression index (Cc ) of 0.30. The clay stratum is underlain by hard and impervious bedrock. The properties of the clay stratum will be assumed constant throughout the clay layer to simply the calculations. Granular columns 1.0 m (3.3 ft) in diameter will be arranged in a square array at a spacing (s) of 1.5 m (4.9 ft) and will extend the entire height of the clay stratum. The stress concentration ratio (Rs ) is estimated to be 5.
2 1 2 1 10 m 5 m 40m NC CLAY BEDROCK EMBANKMENT
Required: Estimate the ultimate primary consolidation settlement (Sc ) of this embankment along its centerline both without and with the columnar reinforcement using the equilibrium method.
Solution:
Sc without columnar reinforcement
Sc for the clay layer without reinforcement will be calculated by subdividing the clay stratum into ten 1.0-m (3.3-ft) thick sublayers and calculating and summing Sc for each sublayer. In equation form:
i n i ci c S S 1where n is the total number of sublayers (ten in this case) and i identifies individual sublayers. The solution is shown in tabular form below. It should be noted that in real practice, a greater number of sublayers should be used, which can be easily accomplished using a spreadsheet program. Only ten sublayers are used here because of space limitations. For simplicity, buoyancy effects produced by submergence of the lower portion of the embankment as settlement occurs will be ignored (these effects are usually minor).
For a normally consolidated clay:
0 0 0 log 1 v vf i i ci ci e H C S where H0i = initial height of sublayer i
v0i = initial effective vertical stress at the midheight of sublayer i = zi vf = final effective vertical stress at the midheight of sublayer i = v0i + vi vi = total vertical stress induced by load from embankment
v will be calculated using the following equation (Osterberg 1957):
n m n n m m n m q v 1 1 0 tan tan 2 where m = a / z n = b / za = horizontal width of one slope = 10 m (33 ft)
b = half the width of the embankment crest = 20 m (66 ft)
q0 = applied stress at bottom of full-height portion of embankment = eHe = (20)(5) = 100 kPa (2.09 ksf)
Therefore, Sc = 0.91 m (3.0 ft) without reinforcement.
Sc with columnar reinforcement
Sc for the clay layer with reinforcement will be calculated in the same manner except that qm will be used in place of q0 . First, the area replacement ratio (Ra ) will be calculated using the unit cell concept because the columnar reinforcement is of large areal extent.
Area of the unit cell, Acell = s2 = (1.5)2 = 2.25 m2 (24.2 ft2)
Ac = 0.25(1.0)2 = 0.7854 m2 (8.45 ft2)
Ra = Ac / Acell = 0.7854 / 2.25 = 0.3491 Now calculate qm using Eq. 5.4:
0.4173
41.73kPa(872psf) 100 1 100 q Hoi zi ′v0i vi ′vfi Scii (m) (m) (kPa) m n (kPa) (kPa) (m)
1 1 0.5 3.5 20.00 40.00 100.00 103.50 0.1919 2 1 1.5 10.5 6.67 13.33 99.99 110.49 0.1333 3 1 2.5 17.5 4.00 8.00 99.95 117.45 0.1078 4 1 3.5 24.5 2.86 5.71 99.88 124.38 0.0920 5 1 4.5 31.5 2.22 4.44 99.74 131.24 0.0808 6 1 5.5 38.5 1.82 3.64 99.54 138.04 0.0723 7 1 6.5 45.5 1.54 3.08 99.26 144.76 0.0656 8 1 7.5 52.5 1.33 2.67 98.89 151.39 0.0600 9 1 8.5 59.5 1.18 2.35 98.44 157.94 0.0553 10 1 9.5 66.5 1.05 2.11 97.89 164.39 0.0513 Sci = 0.9103
The solution is set up in the same tabular form as before.
Summary of Answers:
Using columnar reinforcement would result in an estimated 58% reduction in stress induced in the matrix soil. The ultimate primary consolidation settlement is expected to be reduced from 0.91 m (3.0 ft) to 0.58 m (1.9 ft), a 36% reduction. Note that the reduction in Sc is less than the reduction in stress because Sc is proportional to the logarithm of induced vertical stress rather than linearly proportional to induced vertical stress. For very small values of induced stress and very high values of initial effective stress (deep compressible layers), the percent reduction in settlement approaches the average percent reduction in induced stress (Barksdale and Bachus 1983). Note also that the magnitude of settlement for thick embankments constructed on soft clays is significant even for substantial reinforcement. Generally the columnar reinforcement will also substantially increase the rate at which the settlement occurs. This increase in the rate of settlement is especially important in embankment and fill construction wherein most of the settlement must be allowed to occur before any structures (pavement systems, bridge abutments, buildings, etc.) are founded on or within the embankment or fill.
Priebe (1988) has developed design charts for estimating the reduction in settlement for long granular columns. Charts were provided for both one-dimensional and three-dimensional settlement (Figs. 5.15 to 5.17). The charts for one-dimensional settlement (Figs. 5.15 and 5.16) are appropriate when the width of the loaded area (B ) is very large in comparison to the height of the compressible layer, and the columns extend throughout the height of the compressible layer. Fig. 5.18 applies to isolated footings supported by granular columns.
Using Priebe’s Charts for Settlement from One-Dimensional Strains
One-dimensional strains occur when the width of the loaded area is infinitely wide. This condition is approximated when the width of the loaded area is very large in comparison to the height of the compressible layer. The results in Fig. 5.16 are based on the assumption that the
Hoi zi ′v0i vi ′vfi Sci
i (m) (m) (kPa) m n (kPa) (kPa) (m)
1 1 0.5 3.5 20.00 40.00 41.73 45.23 0.1450 2 1 1.5 10.5 6.67 13.33 41.73 52.23 0.0909 3 1 2.5 17.5 4.00 8.00 41.71 59.21 0.0690 4 1 3.5 24.5 2.86 5.71 41.68 66.18 0.0563 5 1 4.5 31.5 2.22 4.44 41.62 73.12 0.0477 6 1 5.5 38.5 1.82 3.64 41.54 80.04 0.0415 7 1 6.5 45.5 1.54 3.08 41.42 86.92 0.0367 8 1 7.5 52.5 1.33 2.67 41.27 93.77 0.0329 9 1 8.5 59.5 1.18 2.35 41.08 100.58 0.0297 10 1 9.5 66.5 1.05 2.11 40.85 107.35 0.0271 Sci = 0.5767
granular columns are incompressible, which is obviously not the case. Fig. 5.17 is used to correct for the compressibility of the granular columns. The procedure for estimating settlement for this case is as follows:
1. Calculate A/Ac = 1 / Ra
2. Determine the ratio of one-dimensional compression (constrained) moduli for the columnar material to the matrix soil (Dc / Dm ). This value can be estimated (it is approximately equal to the stress concentration ratio, Rs ), or one-dimensional compression tests can be conducted on specimens of the two materials.
3. From Fig. 5.17 with Dc /Dm and ′c , determine the additional area ratio [(A/Ac ) ] calculated in step 1.
4. Add the value of (A/Ac ) calculated in step 3 to the true area ratio (A/Ac ).
5. With the combined value of A/Ac calculated in step 4 and c , determine the settlement
improvement factor (n ) using Fig. 5.16. n is defined as the ratio of the settlement without granular columns to the settlement with granular columns.
6. Divide the value of settlement for no granular columns (calculated using any appropriate method) by the value of n found in step 5. This value is the estimated settlement with granular columns. In equation form:
FIGURE 5.17 Priebe’s chart for additional area ratio to account for compressibility of the granular columns (from Moseley and Priebe 1993)
FIGURE 5.18 Priebe’s chart to estimate settlement of an isolated footing supported by granular columns (modified from Moseley and Priebe 1993)
Using Priebe’s Charts for Settlement from Three-Dimensional Strains
Three-dimensional strains occur within the soil when the width of the loaded area is small in comparison to the height of the compressible layers. This commonly occurs when individual footings are supported by granular columns. The procedure for estimating settlement for this case is as follows:
1. Determine the number of granular columns in the group and the height-to-diameter ratio of the columns (Hc / dc ). With these values, find the settlement ratio S/S using Fig. 5.18. S/S
is defined as the settlement of the finite group (3-D strsains) divided by the settlement for an infinitely wide group (1-D strasins) under the same conditions.
2. Estimate S using the method described above for one-dimensional settlement.
3. Multiply S/S from step 1 by S from step 2 to obtain an estimate of the settlement for the
Example Problem 5.2
Given: A 20-ft thick layer of soft, normally consolidated, silty clay will be reinforced with 3.5-ft diameter granular columns spaced at 6.5 3.5-ft in a triangular pattern that extend the full height of the silty clay layer. The stress concentration ratio is estimated to be 5. A wide fill with a total thickness of 15 ft will be placed on the reinforced silty clay. The fill will consist of a 2.5-ft thick sand blanket (for drainage) overlain by granular embankment fill. Properties of the materials are provided in the figure below.
Required: (a) Estimate what the ultimate primary consolidation settlement of the fill would be if the silty clay layer were not reinforced with granular columns; (b) estimate the settlement of the fill for the reinforced silty clay layer using both Priebe’s method and the equilibrium method; and (c) compare answers from part b and comment.
Solution:
(a) Settlement of unreinforced silty clay layer
Sc for the silty clay layer without reinforcement will be calculated by dividing the layer into twenty 1.0-ft thick sublayers using the same method as in Example Problem 5.1. The vertical stress applied to the surface of the silty clay layer by the fill is calculated as follows:
12.5
120 2.5 108 1,770psf0
The unit weight of water is assumed to be 62.4 pcf, so the effective unit weight of the silty clay is: pcf 6 . 32 4 . 62 0 . 95 sat w (5.13 kN/m3)
Since the fill is wide, one-dimensional compression will be assumed, with the ultimate primary consolidation settlement for each sublayer calculated using the following equation:
i v i v i i ci ci q H e C t S 0 0 0 0 0 log 1 The solution is set up in tabular form.
Hoi zi ′v0i vi ′vfi Sci i (ft) (ft) (psf) (psf) (psf) (ft) 1 1.0 0.5 16.3 1,770.0 1,786.3 0.4759 2 1.0 1.5 48.9 1,770.0 1,818.9 0.3664 3 1.0 2.5 81.5 1,770.0 1,851.5 0.3165 4 1.0 3.5 114.1 1,770.0 1,884.1 0.2842 5 1.0 4.5 146.7 1,770.0 1,916.7 0.2604 6 1.0 5.5 179.3 1,770.0 1,949.3 0.2418 7 1.0 6.5 211.9 1,770.0 1,981.9 0.2266 8 1.0 7.5 244.5 1,770.0 2,014.5 0.2137 9 1.0 8.5 277.1 1,770.0 2,047.1 0.2027 10 1.0 9.5 309.7 1,770.0 2,079.7 0.1930 11 1.0 10.5 342.3 1,770.0 2,112.3 0.1844 12 1.0 11.5 374.9 1,770.0 2,144.9 0.1767 13 1.0 12.5 407.5 1,770.0 2,177.5 0.1698 14 1.0 13.5 440.1 1,770.0 2,210.1 0.1635 15 1.0 14.5 472.7 1,770.0 2,242.7 0.1578 16 1.0 15.5 505.3 1,770.0 2,275.3 0.1525 17 1.0 16.5 537.9 1,770.0 2,307.9 0.1476 18 1.0 17.5 570.5 1,770.0 2,340.5 0.1430 19 1.0 18.5 603.1 1,770.0 2,373.1 0.1388 20 1.0 19.5 635.7 1,770.0 2,405.7 0.1349 Sci = 4.3503
(b) Settlement of reinforced silty clay layer
Priebe’s Method
The shape of the unit cell for a triangular pattern of reinforcement (Fig. 5.1) is hexagonal with an area given by the following equation:
2
2 3
s Acell
The area of one column is: 2
4 c
c d
A
The area replacement ratio is calculated from Eq. 5.1 as: 262947 . 0 5 . 6 5 . 3 3 2 3 2 2 2 s d A A R c cell c a Calculate A/Ac as follows: 80304 . 3 5 . 3 5 . 6 3 2 3 2 1 2 2 c a c d s R A A 0 . 5 s m c R D D 40 c From Fig. 5.17: 1.16 c A A 96 . 4 16 . 1 80 . 3 c c A A A A From Fig. 5.16: n = 2.16
2.01ft (614mm) 2.16 4.3503 columns granular without columns granular with n S S c cEquilibrium Method Calculate qm using Eq. 5.4:
0.262947
5 1
1 0.487380 1 1 1 1 s a m R R
0.487380
862.662psf 770 , 1 0 c m q q (20.7 kPa)The solution is set up in tabular form with the ultimate primary consolidation settlement for each sublayer calculated using the following equation:
i v m i v i i ci ci q H e C t S 0 0 0 0 log 1 Hoi zi ′v0i vi ′vfi Sci i (ft) (ft) (psf) (psf) (psf) (ft) 1 1.0 0.5 16.3 862.7 879.0 0.4041 2 1.0 1.5 48.9 862.7 911.6 0.2964 3 1.0 2.5 81.5 862.7 944.2 0.2482 4 1.0 3.5 114.1 862.7 976.8 0.2176 5 1.0 4.5 146.7 862.7 1,009.4 0.1954 6 1.0 5.5 179.3 862.7 1,042.0 0.1783 7 1.0 6.5 211.9 862.7 1,074.6 0.1645 8 1.0 7.5 244.5 862.7 1,107.2 0.1531 9 1.0 8.5 277.1 862.7 1,139.8 0.1433 10 1.0 9.5 309.7 862.7 1,172.4 0.1349 11 1.0 10.5 342.3 862.7 1,205.0 0.1275 12 1.0 11.5 374.9 862.7 1,237.6 0.1210 13 1.0 12.5 407.5 862.7 1,270.2 0.1152 14 1.0 13.5 440.1 862.7 1,302.8 0.1100 15 1.0 14.5 472.7 862.7 1,335.4 0.1052 16 1.0 15.5 505.3 862.7 1,368.0 0.1009 17 1.0 16.5 537.9 862.7 1,400.6 0.0970 18 1.0 17.5 570.5 862.7 1,433.2 0.0933 19 1.0 18.5 603.1 862.7 1,465.8 0.0900 20 1.0 19.5 635.7 862.7 1,498.4 0.0869 Sci = 3.1830 Sc = 3.18 ft (969 mm) without reinforcement.(c) Comparison of answers from Priebe’s method and equilibrium method
Without reinforcement: Sc = 4.35 ft (1.33 m)
With reinforcement using Priebe’s method: Sc = 2.01 ft (644 mm) With reinforcement using Equilibrium method: Sc = 3.18 ft (969 mm)
The answers are significantly different using the two methods to estimate primary consolidation settlement for the reinforced soil. Priebe’s method predicts a 54% reduction in Sc whereas the equilibrium method predicts a 27% reduction.
Summary of Answers:
(a) Without reinforcement: Sc = 4.35 ft (1.33 m)
(b) With reinforcement using Priebe’s method: Sc = 2.01 ft (644 mm) With reinforcement using Equilibrium method: Sc = 3.18 ft (969 mm)
(c) The answers are significantly different using the two methods to estimate primary consolidation settlement for the reinforced soil. Priebe’s method predicts a 54% reduction in
Sc whereas the equilibrium method predicts a 27% reduction.
Example Problem 5.3
Given: A square footing bearing on the surface of the same reinforced silty clay layer from Example Problem 5.2 with the same q0 and the same Ra supported by four granular columns. Required: Calculate the ultimate primary consolidation settlement of this footing using Priebe’s method.
Solution:
The width (B) of a square footing that will have the same value of Ra when supported by four granular columns with dc = 3.5 ft (1.07 m) can be calculated as follows:
Area of the footing: Aftg = B2
Area of the four columns: 2 2
4 4 c c c d d A ft 10 . 12 262947 . 0 5 . 3 2 2 a c c ftg c a R d B B d A A R (3.69 m) 71 . 5 5 . 3 20 c c d H
From Fig. 5.18: 0.471
S S
squarefooting withgranular columns
0.471
3.1830
1.499ft
456mm
columns granular with width infinite columns granular with footing square c c c S S S S S
Final Answer: Sc = 1.50 ft = 18.0 in. (456 mm)
5.2.3 Ultimate Bearing Capacity
In most instances, the bearing soil for a foundation is supported by a group of reinforcing columns. The mechanisms by which failure occurs for a group of columns is quite complex and involves several types of interaction. Sometimes a foundation is supported by a single column, for which the mechanisms of failure are more straightforward. Thus, ultimate bearing capacity for single columns will be discussed first, which will then provide the basis to understand the more complex situation for groups of columns.
5.2.3.1 Single Columns
There are three possible mechanisms of failure for a single reinforcing column in a homogeneous soil mass, as illustrated in Fig. 5.19. The type of failure that would occur in any situation depends on characteristics of the columnar material and the matrix soil, and the area over which the load is applied (Fig. 5.20).
Bulging: Failure of a single granular column in weak matrix soil such as a soft clay will occur by bulging. Granular columns have little or no internal cohesion and therefore depend on lateral resistance provided by the matrix soil. As load is applied to the top of the column, the column will tend to push outward within the upper portion of the column. If the matrix soil is weak, bulging will occur within a height of about two to three times the diameter of the column if the matrix soil is relatively homogeneous within this zone. If the upper portion of the matrix soil consists of layers of relatively strong and weak soils, bulging may occur only in the weaker layers. If the load is applied over an area greater than the area of the column (Fig. 5.20a), the ultimate bearing capacity is increased in two ways: (a) Some of the load is carried by the matrix soil, lessening the load carried by the column; and (b) the stress carried by the matrix soil increases the confinement on the upper portion of the column.
General or Local Shear: It is unlikely that a single column will fail by general or local shear failure, but it may occur under the following conditions: (a) a very short column [Hc < (2 to 3)dc ] bearing on a rigid base, or (b) the columnar material is not significantly stronger than the matrix soil. Neither situation occurs very often in practice. These types of failure are similar to those that occur for shallow foundations bearing on homogeneous unreinforced soils.
columns. This mechanism of failure may also predominate in very short granular columns [Hc < (2 to 3)dc ] floating in the matrix soil (not bearing on a rigid base). Resistance to the applied load develops as skin resistance (shearing stresses) along the interface of the column and matrix soil, end bearing (normal stresses) at the bottom of the column, or a combination of both.
The best method to determine the ultimate bearing capacity of a single column is to load a prototype to failure. This is generally accomplished by conducting a plate load test (see Section 3.5.2 for details). However, this is an expensive and time-consuming process that typically requires generating about 100 to 500 kips (450 to 2200 kN) of reactive force. Thus, ultimate bearing capacity is usually estimated.
The following empirical formula can be used to estimate the ultimate bearing capacity of a single granular column within a matrix of soft clay (bulging failure):
qc-ult = Nscsum (5.7)
where Nsc = bearing capacity factor for a single column
sum = undrained shearing strength of the clay matrix
For vibro-replacement stone columns, Nsc has been found to range from about 18 to 25 (Barksdale and Bachus 1983, Mitchell 1981).
Many theories and analytical methods have been developed for the ultimate compressive capacity of a single granular column in saturated clay under undrained conditions. Summaries of many of these methods can be found in ASCE (1987), Barksdale and Bachus (1983), and Brauns (1978). Two theories will be described here for illustrative purposes.
FIGURE 5.19 Mechanisms of failure for single reinforcing columns: (a) Bulging, (b) general or local shear, and (c) punching
FIGURE 5.20 Types of loading for single reinforcing column: (a) Footing larger than column, (b) footing same size as column
Bell's (1915) method is the simplest and most conservative and is illustrated in Fig. 5.21. In this method, the weight of the matrix soil and the shearing stresses that develop along the column-matrix interface are ignored, as well as the distribution of stress within the columnar and matrix soils. Hence, the principal stresses in both the column and matrix soil adjacent to the interface are assumed to act in the vertical and horizontal directions. In the column adjacent to the interface, the major principal stress is in the vertical direction (qc-ult) owing to the high stress concentration on the column. The minor principal stress therefore acts in the radial direction and is designated r. For a granular column the cohesion intercept (c′c ) is zero, and the following
equation can be written based on the geometry of the Mohr's circle at failure in the column:
45 2
tan2 c r ult c q (5.8)The corresponding equation at failure in the clay adjacent to the interface is as follows:
um m r q 2s
(5.9)
Inserting Eq. 5.9 into Eq. 5.8 gives:
2
tan2
45 2
c um m ult c q s q (5.10)Normalized values of ultimate bearing capacity (Nsc = qc-ult /sum ) are plotted versus ′c for Bell's method assuming qm = 0 in Fig. 5.22. Note that Bell's method is very conservative compared to the other methods shown and compared to the typical range of values of Nsc
Hughes and Withers (1974) considered bulging failure of a single column to be similar to the expansion of a pressuremeter probe against the sides of a borehole. In a pressuremeter test, the radial resistance of the soil reaches a limiting value (rl) at which indefinite expansion occurs. If the soil is idealized as an elasto-plastic material, the limiting radial stress can be estimated from the following equation (Gibson and Anderson 1961):
m um m um ro rl s E s 1 2 ln 1 (5.11)where ln is the natural logarithm, ro is the initial radial stress in the matrix soil (prior to conducting the pressuremeter test), and sum , Em, and m are the undrained shearing strength, elastic modulus, and Poisson's ratio of the matrix soil, respectively. The following empirical equation for rl determined from the results of many quick-expansion pressuremeter tests can also be used: o um ro rl 4s u (5.12)
where ro is the initial effective radial stress and uo is the initial excess pore water pressure. Because the granular columnar material acts as a drain, uo can be reasonably taken as zero with little or no error introduced. Of course, rl can also be determined from pressuremeter tests conducted within the range of depths where bulging is expected occur (upper two to three diameters of the column).
Ignoring the shearing stresses that develop along the interface of the columnar and matrix soils and any stress carried by the matrix soil at the bearing level (qm), the ultimate bearing capacity can be calculated as follows:
45 2
tan2 c rl ult c q (5.13)Curves of qc-ult /sum versus c are shown in Fig. 5.22 using Eqs. 5.12 and 5.13 and two values of (ro + uo ): zero and 2.27sum .
Madhav and Vitkar's (1978) method can be used to estimate the ultimate bearing capacity for general shear failure within a long granular trench (rectangular prism) and an undrained clay matrix supporting a foundation. This method is described in Section 2.1. So far as the author knows, there is no method available for general shear failure for a single reinforcing column. Established methods for estimating the ultimate bearing capacity of piles can be extended to single reinforcing columns that fail by punching. These methods can be found in any standard foundation engineering book and will not be discussed here.
5.2.3.2 Groups of Columns
Hughes and Withers (1974) indicated that groups of granular columns within soft cohesive soils act independently when the center-to-center spacing of the columns is greater than about 2.5dc . This criterion also seems reasonable for other types of columns and matrix soils. For this case, the average ultimate stress that can be carried by the group (qult) is given by the following
expression:
a
m a ult c m m c ult c ult q R q R A A q A q q 1 (5.14)It should be noted that many of the methods developed for single columns ignore the confining pressure provided by qm that in a group of columns helps to resist bulging. This contribution may be significant in some instances and should not be arbitrarily ignored.
For closer spacings the columns act with the matrix soil as a composite material, and the group action of the reinforced zone must be considered. It is useful in some instances to consider this composite zone to have a single set of Mohr-Coulomb strength parameters calculated as follows:
c a c m a m
comp R R
tan1 tan 1 tan (5.15)
a
m a c comp c R c R c 1 (5.16)Note that the effects of stress concentration are included in Eq. 5.15 for comp . If stress concentration does not occur, such as when columnar reinforcement is used to stabilize natural slopes (to be discussed subsequently), values of c = m = 1 must be used in Eq. 5.15.
FIGURE 5.22 Comparison of predicted values by various methods of ultimate bearing capacity for a single granular column
within an undrained clay matrix (modified from Brauns 1978)
If the height of the columns is relatively short compared to the width of the foundation (Hc 1.5B ), qult for the group can be estimated using composite strength parameters and the method developed for overexcavation/replacement (Section 2.1). For taller columns (Hc > 1.5B), the failure mechanism suggested by Barksdale and Bachus (1983), as illustrated in Fig. 5.23, may be appropriate. In this method, the major principal plane at the bearing level is assumed to be horizontal, with the result that failure is initiated at one corner of the foundation and the failure surface extends underneath the foundation at an angle of = 45 + comp /2 relative to
horizontal to a point directly beneath the opposite corner of the foundation. This sliding wedge is resisted by passive lateral resistance within the adjacent matrix soil. This passive resistance is assumed to be the major principal stress in the matrix soil (1m) and the minor principal stress in the composite reinforcing zone (3c ), and qult is the major principal stress for the composite reinforced zone. If the weight of the sliding wedge is ignored, which is conservative, the corresponding equations are as follows:
Granular Columns 284
0.5 tan
tan2
45 /2
2 tan
45 /2
3 1m c q mB o m cm o m (5.17) tan2 2 tan 3c comp ult c q (5.18)
where q is the effective vertical surcharge pressure at the bearing level. The term
q0.5mBtan
corresponds to the average vertical normal stress acting along the vertical face of the sliding wedge and is valid for either a shallow or deep groundwater table. The value of m used in these cases should correspond to the appropriate groundwater and drainage conditions. That is, the saturated unit weight should be used for a shallow groundwater table and undrained (short-term) conditions; the effective (buoyant) unit weight should be used for a shallow groundwater table and drained (long-term) conditions; and the total (wet) unit weight should be used for a deep groundwater table. If the groundwater table is located within the height of the sliding wedge, the appropriate value of average vertical stress along the vertical face owing to the weight of the matrix soil below the bearing level should be used in place of 0.5m B tan and added to q. Total vertical stress should be used for undrained conditions and effective vertical stress for drained conditions, with the exception that q should be the effectivesurcharge pressure for undrained and drained conditions.
According to Wissmann (2007), there are four potential failure modes for bearing capacity of a group of columns that must be checked (Fig. 5.24): (a) Bulging failure of individual piers, (b) shearing below the tips of individual piers, (c) general shearing within the pier-reinforced soil, and (d) shearing within the unreinforced soil below the bottom of the pier-reinforced zone. The procedures recommended for each method are summarized here, with additional details available in the reference.
Bulging failure of individual columns was discussed previously in Section 5.2.3.1. qc-ult can be estimated using Eq. 5.13 with rl equal to the limiting passive pressure within the matrix soil
estimated from Rankine’s equation.
For shearing at the bottom of individual columns, the ultimate force that can be carried at the top of a column (Qc-ult) can be expressed by the following equation:
c tip ult c sc ult c Q Q W Q , (5.19)
Where Qsc is the shaft resistance along the vertical interface of the column and the matrix soil,
Qc-ult,tip is the ultimate force that can be carried at the tip (bottom) of the column, and W′c is the buoyant weight of the column itself.
FIGURE 5.24 Potential modes of failure for groups of Geopier elements supporting a foundation (courtesy of Geopier Foundation Co., Davidson, North Carolina)
Qsc and Qc-ult,tip can be calculated from the following equations:
sc sc
sc f A
Q (5.20)
where fsc is the shaft resistance along the vertical column-matrix soil interface and Asc is the area along the vertical interface that shearing resistance develops for the selected interval of depth.
tip c tip ult c tip ult c q A Q , , ,
where qc-ult,tip is the ultimate stress that can be carried at the tip of the column and Ac,tip is the area of the column at the tip. qc-ult,tip can be determined using any method typically used for piles. Shearing within the column-reinforced soil can be determined using composite strength parameters for the column-reinforced zone (Eqs. 5.15 and 5.16) and assuming general shear failure. However, some judgment must be exercised with respect to the value of stress concentration ratio, Rs, used to calculate c and m (Eqs. 5.3 and 5.4), because Rs will decrease
with depth within the column.
According to Wissmann (2007), shearing within the unreinforced soil below the bottom of the column-reinforced zone can be conservatively calculated using the following method. First assume that the vertical stress induced along the interface of the bottom of the reinforced zone and the underlying unreinforced soil can be estimated using the 2:1 method. Designating q0 as the average bearing pressure induced at the base of the footing and qbottom as the induced vertical stress at the bottom of the reinforced zone, qbottom can be calculated from q0 as follows:
B H
L H
BL q qbottom 0 (5.21)where B is the width of the footing, L is the length of the footing, and H is the height of the column-reinforced zone. The ultimate bearing pressure at the bottom of the column-reinforced zone (qult,bottom) is then calculated for a fictitious footing of width (B + H) and length (L + H) using any method applicable for general shear failure. The average ultimate bearing stress at the bottom of the actual footing, qult, can then be calculated from Eq. 5.21 as follows:
BL H L H B qqult ult,bottom (5.22)
5.2.4 Liquefaction Potential
Columnar reinforcement can mitigate the potential for liquefaction of saturated granular matrix soils in the following four ways (Baez 1995):
4. Reduce the shearing stress carried by the matrix soil
The effectiveness of each type of granular column in reducing liquefaction potential by the four possible ways described above is summarized in Table 5.2.
Table 5.2 Comparison of Effectiveness of Various Types of Granular Columns in Controlling Liquefaction Potential
Increase Stress
Type of Levels in
Granular Adjacent
Column Matrix Soil
Stone Column Moderate
Sand Column Some
Geopier RAP Significant
Impact RAP Moderate
Gravel Drain Slight
Sand Drain Slight
Some Slight Slight Significant Some Moderate Some Stresses in Matrix Soil Significanta Significanta Significanta Significanta Radial Drainage Moderate Increase Density of Adjacent Matrix Soil Significant Moderate Significanta Significant Moderate
Method of Controlling Liquefaction Potential Reduce Buildup of
Excess Pore Water Pressures via
Reduce Shearing
Significanta
aSignificant so long as the permeability of the columnar material is greater than 200 times the permeability of the
matrix soil, the columnar material acts as a filter to prevent intrusion of finer particles into the columns, and there is sufficient void space in the columns to handle the volume of inflowing water required to keep the excess pore water pressures to a sufficiently low level.
Granular columns must meet three requirements if they are to reduce significantly the excess pore water pressures generated in potentially liquefiable saturated granular soils during an earthquake (Seed and Booker 1976, Sonu et al. 1993): (1) To act as a free drain, the columnar material must have a permeability of at least 200 times that of the matrix soil; (2) the columns must effectively filter the matrix soil to prevent clogging by intrusion of finer particles into the columns; and (3) the columns must have sufficient void space to handle the volume of inflowing water required to keep the excess pore water pressures to an acceptably low level.
The increase in density of the matrix soil adjacent to stone columns and Impact RAPs can be significant. This also occurs in other types of granular columns to a lesser extent (see Table 5.2). Baez (1995) evaluated this effect for vibratory stone columns at 18 sites based on an evaluation of nearly 400 sets of pre-treatment and post-treatment data from standard penetration and cone penetration tests. Empirical correlations were developed for post-treatment normalized SPT blowcounts as a function of pre-treatment values for area replacement ratios from 5 to 20%. Results of the empirical study are presented graphically in Fig. 5.25 and are valid for uniform to medium silty sands with less than 15% fines and little or no clay content. When used in conjunction with Seed's method, this design chart can be used to estimate the area replacement ratio needed to achieve a desired factor of safety against liquefaction (see discussion of liquefaction potential in Section 3.4.1).
The increase in horizontal stresses within the matrix soil adjacent to Geopier elements can be significant. This also occurs in other types of granular columns to a lesser extent (see Table 5.2). There may also be some minor increases in the vertical stresses. This increase in stress level helps to reduce the potential for liquefaction by increasing the confining stresses, particularly for vertical shaking (see Fig. 3.55).
During horizontal ground shaking, lateral shearing stresses are generated in the soil. Within a columnar reinforced soil, a concentration of shearing stresses on the columns occurs similar to the concentration of normal stresses for vertical loads discussed previously. This results in a reduction in the shearing stresses within the potentially liquefiable matrix soil. The magnitude of this reduction depends on the area replacement ratio and the ratio of the shear modulus for the granular columns to that for the matrix soil (RG = Gc /Gm), according to the following equation:
1
1 1 G a m R R (5.23)where is the average shearing stress generated by the earthquake. Eq. 5.23 can be rearranged to solve for area replacement ratio:
1 1 1 m G a R R (5.24)
The factor of safety against liquefaction is generally defined as the available cyclic shearing strength divided by the cyclic shearing stress expected to be generated by the earthquake (Eq. 3.6). The pre-improvement factor of safety (FSpre ) can be introduced into Eq. 5.24 as m /, resulting in the following equation for the area replacement ratio required to product a
post-treatment factor of safety of one: 1 1 1 1 pre G a FS R R (5.25)
Eq. 5.25 is shown graphically in Fig. 5.26. 5.2.5 Slope Stability
Columnar reinforcement can be used to increase the stability of man-made and natural slopes. In situations where fill or other load is placed on top of the columns, such as occurs when stabilizing a soil deposit and then placing an embankment on top of the reinforced soil (for example, see Example Problem 5.1), the concentration of stress on the columns provides a very efficient and cost-effective reinforced zone. When stabilizing natural slopes or landslides, however, stress concentrations generally do not occur and the method is less efficient but still technically effective.
Most slope stability analyses conducted in engineering practice involve the use of a two-dimensional computer program. Although all slope stability problems are three-two-dimensional, two-dimensional analyses are usually conducted for three primary reasons:
0 2 4 6 8 10 12 14 16 18 20 Pre-Improvement Normalized SPT Blowcount, Npre (blows/ft) 0 5 10 15 20 25 30 35 40 Blowcount, N post (blows/ft) Pos t-Improvement No rmali zed SP T Ra = 5% Ra = 10% Ra = 15% Ra = 20%
FIGURE 5.25 Prediction of post-improvement normalized SPT blowcount as a function of pre-improvement normalized SPT blowcount for uniform fine to medium
silty sands with less than 15% fines and little or no clay content (from Baez 1995)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Pre-Improvement Factor of Safety Against Liquefaction, FSpre 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Factor of Safety
Against Liquefaction of One
Re qu ir ed R a for Post-Improvement RG = 2 RG = 3 RG = 4 RG = 5 RG = 7 RG = 10 RG = 15 RG = 20 RG = 30 RG = 40
FIGURE 5.26 Required area replacement ratio to achieve a post-treatment factor of safety of one against liquefaction considering only redistribution of shearing stress (from Baez 1995)
1. Many slopes are long in comparison to their height and width and thus are reasonably represented as two-dimensional problems.
2. Three-dimensional analyses require about an order of magnitude more time to perform than two-dimensional analyses.
3. Use of a two-dimensional analysis is nearly always conservative.
Two methods for performing slope stability analyses involving columnar reinforcement are the composite strength method and the profile method. Either method can be used with standard slope stability computer programs. In the composite strength method, which is valid only when stress concentration is not present, the reinforced zone is treated as a single composite material with one set of Mohr-Coulomb strength parameters (comp and ccomp) and a composite unit weight
(comp ). Some judgment is required to determine how far outside the edge of the outermost row
of columns the reinforced zone should extend. ccomp can be calculated from Eq. 5.16 and comp from Eq. 5.15 with c = m =1. comp can be calculated from the following
equation:
a
m a c comp R 1R (5.26)The values used for c and m should correspond to the location of the groundwater table and type
of strength analysis
