predicting bending failure of cdm columns under embankment
TRANSCRIPT
Computers and Geotechnics 91 (2017) 169–178
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Computers and Geotechnics
journal homepage: www.elsevier .com/locate /compgeo
Research Paper
Predicting bending failure of CDM columns under embankment loading
http://dx.doi.org/10.1016/j.compgeo.2017.07.0150266-352X/� 2017 Elsevier Ltd. All rights reserved.
⇑ Corresponding author.E-mail addresses: [email protected] (J.-c. Chai), [email protected]
(S. Shrestha), [email protected] (T. Hino).
Jin-chun Chai a,⇑, Sailesh Shrestha b,⇑, Takenori Hino b, Takemasa Uchikoshi c
aGraduate School of Science and Technology, Saga University, Saga, Japanb Institute of Lowland and Marine Research, Saga University, Saga, Japanc JIP Techno Science Corporation, Tokyo, Japan
a r t i c l e i n f o a b s t r a c t
Article history:Received 29 January 2017Received in revised form 15 July 2017Accepted 21 July 2017Available online 27 July 2017
Keywords:CDM columnsBending momentSoft soilsFinite element analysis
The mechanism of bending failure and the magnitude of the bending moment in a column formed bycement deep mixing (CDM) under an embankment load were investigated by a series of three-dimensional (3D) finite element analyses. Based on the numerical results, a design method to considerthe bending failure of the CDM column has been established. The usefulness of the proposed methodswas verified using the results of two centrifuge model tests of the embankments on the CDM columnimproved soft model ground model reported in the literature, and the columns had tensile cracks orshowed complete failure.
� 2017 Elsevier Ltd. All rights reserved.
1. Introduction
Cement deep mixing (CDM) is a process that normally formssoil-cement columns in the ground and is an effective and eco-nomic method to improve soft ground [1–3]. Both the dry mixing(cement powder) and wet mixing (cement slurry) can be used toform the soil-cement columns. The wet mixing under a lower pres-sure is called slurry double mixing (SDM) [4], and under a higherpressure, it is called jet grouting [5–7].
Current design methodologies for embankments on CDM col-umns consider only improving the subsoil bearing capacity of theground and the safety factor against slip circle failure (Fig. 1). How-ever, there are reported results of centrifuge model tests indicatingthat the bending failure of the columns is an important failuremechanism [8–12]. Fig. 2 shows an example of bending failure ofCDM columns from the centrifuge model test by Kitazume andMaruyama [10]. The bending failure of the CDM columns underan embankment load has also been reported by other researchersusing numerical analyses [13–17].
The only design method available to consider the bendingfailure of CDM columns under an embankment loading is that pro-posed by Kitazume and Maruyama [10]. The method considers thesituation where CDM columns are installed only under the toe anda portion of the slope range of an embankment, and assumes thatthe earth pressure from the centre of an embankment to the block
with the CDM columns is in an active state, and the earth pressurefrom the surrounding soil outside the toe of the embankment tothe block with the CDM columns is in a passive state. For a realfield case, neither active nor passive earth pressure can bemobilized before the bending failure of a column. Goh et al. [18]proposed a method to estimate the maximum bending momentof a single pile/column located under the toe of the embankment,which cannot be directly applied for group pile/column cases.Therefore, there is no current rational and practical design methodto consider the bending failure of CDM columns under an embank-ment load.
In this paper, a series of 3D finite element analyses were firstconducted to investigate the factors affecting the bending failureof the end bearing CDM columns under an embankment load usinga verified numerical procedure. Then, based on the results of thenumerical investigation, a design method has been proposed topredict the maximum bending moment in the CDM columns underthe toe of the embankment. Finally, the usefulness of the designmethod was verified with the results of two centrifuge model testsreported in the literature.
2. Numerical investigation
2.1. Numerical modelling
The numerical investigation was conducted with assumedplane strain type three-dimensional (3D), full-scale embankments.The deformation boundary conditions adopted were plane strain,
Fig. 1. Slip circle failure of an embankment on the CDM column improved deposit.
Fig. 2. Centrifuge test results of the bending failure of the model soil-cementcolumns [10].
170 J.-c. Chai et al. / Computers and Geotechnics 91 (2017) 169–178
but 3 rows of CDM columns were modelled explicitly, i.e., in themodel, CDM columns interact with the surrounding soil in a 3Dmanner. The adopted model is shown in Fig. 3. The modelled area
Fig. 3. Plain strain type 3D model of an embankmen
had an overall length of 80 m, a width that was 3 times the spacingbetween CDM columns and a vertical thickness of 35 m from theground surface. Even though real CDM columns are cylindrical, asquare cross-section was used to model the CDM columns to sim-plify the meshing process. The side length of the cross-section ofthe square column was determined under an equal bending stiff-ness condition (EI, E is Young’s modulus and I is the moment ofinertia of the cross-sectional area of a column). Referencing CDMcolumns constructed in Saga, Japan, a diameter of 1.2 m wasadopted as a basic case and the corresponding side length of asquare section was 1.05 m. The boundary conditions at the leftand the right (x direction) and the front and the back (y direction)boundaries vertical displacement was allowed, while the horizon-tal displacement was fixed. At the bottom boundary both the hor-izontal and vertical displacements were fixed. Both the groundsurface and the bottom boundaries (sand layer) were defined aspermeable and other boundaries were defined as impermeable.Ten-node tetrahedron elements with excess pore water pressuredegrees of freedom at all nodes were used to represent the founda-tion soil while similar elements without excess pore water pres-sure degrees of freedom were used to represent theembankment. For the adopted model, the total number of nodes(vertex plus side nodes) was approximately 182,000, and the totalnumber of elements was approximately 127,000. The Plaxis 3D(2013 version) program was used for conducting the simulation.The detailed geometry of the assumed basic case is shown inFig. 4. The embankment height was 6.0 m, with a side slope of1:1.8 (V:H), and the thickness of soft clay layer was 10.0 m. Thearea improvement ratio (a) of the CDM columns was 30%. For thecolumns arranged in a square pattern, a is defined as [19]:
a ¼ pD2
4S2ð1Þ
where D is the diameter of the column and S is the centre-to-centrespacing between the columns.
The soft clay was modelled using the Soft Soil Model (SSM) [20],and the embankment and the sand layer were modelled with thelinear elastic model obeying the Mohr-Coulomb failure criterion.The CDM columns were treated as a linear elastic material. The
t on the CDM column improved subsoil deposit.
Fig. 4. Cross-sectional and plan view of the assumed basic case.
J.-c. Chai et al. / Computers and Geotechnics 91 (2017) 169–178 171
stage construction procedure was used to simulate the embank-ment loading in 12 different phases, and each phase was loadedwith a fill thickness of 0.5 m. In the simulation, each phase had atime period of 5 days. The load was increased linearly with time,and the consolidation was simulated simultaneously. After theembankment was constructed, the consolidation was simulatedfor a period of an additional year. For all cases, the primary consol-idation was almost finished one year after the end of the embank-ment construction. The coupled consolidation analysis with theupdated nodal coordinate option was adopted.
(a)
6.5 m
9.5 m
Soft clayStiff clay
Stiff clay
Sand
Stiff clay
Sand
To 23 m
Inclino-meter
6 m
I1 I2
CL
Column
Fig. 5. (a) Cross-section of the embankment and (b) comparison of the
The suitability of the numerical modelling procedure and ofadopted soil models was verified by simulating a test embankmenton a CDM column to improve soft clayey subsoil in Saga, Japan [21]and some laboratory centrifuge model tests reported in the litera-ture [22]. Since this study is focused on the bending deformation ofthe columns, an example comparison of the measured and simu-lated profiles of the lateral displacements of a column under thetoe of the test embankment reported by Chai et al. [21] is givenin Fig. 5.
The adopted values of the model parameters for the basic caseare listed in Table 1. The values were determined in reference tothe soil properties at a test embankment site in Saga, Japan,reported by Chai et al. [21]. The clayey soil deposited at the siteis called Ariake clay, which is a marine clay with a natural watercontent that is normally more than 100%, The value of the slopeof compression line (k) in the e-ln(p0) plot (e is the void ratio andp0 is the effective compression stress) is approximately 0.4–1.3(compression index, Cc, of approximately 1.0–3.0) [23,24]. Thebasic case is considered to represent the weaker soft clay deposit,while the stiffer cases are simulated by increasing the effectivecohesion (c0) and reducing the value of k, as detailed in a later sec-tion. For the soft soil model, the value of the slope of theunloading-reloading line in the e-ln(p0) plot, j, was assumed tobe 1/10 of k. The value of Poison’s ratio (m) was assumed to be0.15 for all the soft soil layers and the CDM columns. The valuesof permeability in the horizontal direction (kh) were set as 1.5times the corresponding values of permeability in the verticaldirection (kv) [25]. The ground water level was assumed to be1.0 m below the ground surface. The values of the coefficient ofthe earth pressure at-rest, Ko of 0.6 for the soil layers from theground surface to a 2 m depth (OCR > 1.5) and 0.5 for the remain-ing soil layers were adopted. The values of kv and kh listed in Table 1are initial values, and they were allowed to vary with void ratio eduring the consolidation process, according to Taylor [26] equa-tion. The constant Ck in Taylor’s equation was set as, Ck = 0.5eo (eois the initial void ratio) [27]. The unconfined compression strength(qu) of the CDM column was assumed to be 1000 kN/m2, and theYoung’s modulus (E) was estimated as 100qu [28,29]. The
(b)
-20
-15
-10
-5
0 0 20 40 60 80 100
Measured (I1) Measured (I2) FEM
Lateral displacement (mm)
At elapsed time559 days
lateral displacement from the FEM 3D with measured data [21].
Table 1Model parameters for the basic full-scale embankment.
Depth (m) Soil strata E (kN/m2) m c0 (/0) (kN/m2) j kb M e0 ct (kN/m3) kv (10�4 m/day) kh (10�4 m/day)
0.0–1.0 Surface soil – 0.15 5 0.0435 0.435 1.64 1.61 16.0 6.0 9.01.0–2.0 Soft clay-1 – 0.15 5 0.0652 0.652 1.64 1.61 16.0 6.0 9.02.0–5.0 Soft clay-2 – 0.15 5 0.087 0.87 1.64 2.86 14.0 4.4 6.65.0–7.0 Soft clay-3 – 0.15 5 0.0652 0.652 1.64 2.45 14.5 5.6 8.47.0–8.0 Soft clay-4 – 0.15 5 0.0521 0.521 1.64 2.45 14.5 5.6 8.48.0–9.0 Soft clay-5 – 0.15 5 0.0434 0.434 1.64 2.12 15.0 5.6 8.49.0–10.0 Soft clay-6 – 0.15 5 0.0347 0.347 1.64 2.12 15.0 5.6 8.410.0–35.0 Sand 40,000 0.10 20 (35) – – – 0.76 19.0 2500 2500
Embankment 3000 0.45 20 (35) – – – 0.5 19.0 – –
Note: c = unit weight; m = Poisson’s ratio; c0 = cohesion; /’ = internal friction angle; E = Young’s modulus; kb = value of slope of the consolidation line in the e-ln(p’) plot (e isvoids ratio and p’ is effective mean stress) for the basic case; j = slope of the rebound line in the e-ln(p’) plot; and M = the slope of critical state line (CSL) in the p’–q plot (q isdeviator stress).
Fig. 6. Yield surface of the Soft Soil Model in the p0-q plane.
Fig. 7. Plan view of the DCM the arrangement of the CDM columns and cross-section of a 6 m height embankment.
172 J.-c. Chai et al. / Computers and Geotechnics 91 (2017) 169–178
permeability of the columns was assigned the same values as thecorresponding untreated soils.
From SSM, the undrained shear strength (su) of the soil can becalculated as:
su ¼ MMC
2p0MCðOCReÞK ð2Þ
p0MC ¼ p0
eM2 þ g2
e
M2 þM2MC
!K
ð3Þ
OCRe ¼r0vOCRð1þ2Knc
o Þ3 þ c0 cot/0
p0e
!ð4Þ
where ge is the initial stress ratio (qe/p0e), qe is the initial deviator
stress, p0e is the initial mean effective stress, M is the slope of the
critical state line (CSL) in the p0 – q plot, MMC is the slope of theMohr-Coulomb failure criteria in the p0 – q plane, p0
MC is the equiv-alent mean stress on the MMC line (Fig. 6), K = 1 � j/k, OCR is theover consolidation ratio, /0 is the effective internal friction angleof soil, and r0
v is the initial vertical effective stress. In SSM, M ismainly a function of the coefficient of the at-rest earth pressure atthe normal consolidated state, Knc
o . The interpretation of some ofthe parameters in Eqs. (2)–(4) are illustrated in Fig. 6.
2.2. Assumed other cases
The following factors/variables have been investigatednumerically.
(1) Geometrical variables
(a) Embankment height: 4, 6 and 8 m (the top width andslope angle were fixed).
(b) Thickness of soft clay layer (Hs): 6, 8, 10, 14 and 20 m.(c) Area improvement ratio (a) by the CDM column: 10%,
15%, 20%, 25% and 30%. As illustrated in Fig. 7, reducingthe value of a increased the spacing between thecolumns.
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
00 10 20 30 40
Soft Clay
Dep
th (m
)su (kN/m2)
c' = 0 kN/m2
c' = 5 kN/m2
c' = 10 kN/m2
c' = 15 kN/m2
sumin
Zone for calculating suavg
2 m
2 m
Fig. 8. Simulated value of su of the model ground.
Table 2Cases in
No.
1234567891011121314151617181920
J.-c. Chai et al. / Computers and Geotechnics 91 (2017) 169–178 173
(d) Diameter of CDM column: 0.6, 0.8, 1.0 and 1.2 m.
(2) Soil properties(a) Undrained shear strength (su) of the soft clay layer:
Variations in su was simulated by varying the effective cohesion(c0) in Eq. (4). The simulated cases are c0 = 0, 5, 10, and 15 kN/m2.Using the parameters of the basic case, for c0 = 0–15 kN/m2, theSSM simulated variations of su with respect to the depth aredepicted in Fig. 8.
(b) Compressibility of the soft clay:
The values of k were varied to investigate the effect of the com-pressibility of soft soil layer. Designated the values of k for the
vestigated.
Height of embankment(m)
Thickness of soft soil, Hs (m) Area improvement(%)
4 10 306 10 308 10 304 10 306 10 308 10 304 10 306 10 308 10 306 10 256 10 206 10 156 10 106 6 306 8 306 14 306 20 306 10 306 10 306 10 30
basic case as kb, the k/kb ratios considered were k/kb = 0.25, 0.5and 1.0.
All cases investigated are summarized in Table 2.
2.3. Numerical results
The results presented are focused on the maximum bendingmoment in the CDM column under the toe of the embankment,which is the critical location for the bending failure of the CDM col-umn. The bending moment in the column was calculated using thesimulated distribution of stresses in the column.
(1) Effect of undrained shear strength (su)
For a natural clayey deposit, the values of su vary with thedepth. It is desirable to define a representative value of su to pro-pose a design method that considers the tensile failure of theCDM columns. Although two possible options can be considered,(1) the average value and (2) the minimum value, as illustratedin Fig. 9, the two profiles may have the same average value (suavg)but different minimum values (sumin). It is considered that neithersuavg nor sumin can serve as a representative value. Therefore, aparameter (sua) that represents both the effects of the averageand the minimum value of su is introduced and designated the rep-resentative value of su as:
sua ¼ ðsuavg � suminÞ0:5 ð5ÞThen, to calculate suavg, the applicable range must be considered.
Considering that for many natural deposits, the thickness of thecrust layer is 1–2 m and the minimum value of su occurs belowthe crust layer, it is proposed to use the average values of a zone(range) from 2 m above to 2 m below the minimum value of su asillustrated in Fig. 8. For the distributions of su in Fig. 8, the esti-mated values of sua are summarized in Table 3, which were usedto analyse the results of the finite element analysis (FEA). Usingthe other conditions of the basic case, but varying the value of c0
for the soft clay from 0 to 15 kN/m2 (sua = 9.6–19.4 kN/m2), theeffect of sua on the maximum bending moment (Mmax) in theCDM columns under the toe of the assumed embankment is com-pared in Fig. 10. It can be seen that increasing sua significantlyreduced the values of Mmax. Increasing sua from 9.6 to 19.4 kN/m2
reduced Mmax from approximately 31.6 to approximately7.5 kN-m in the column.
ratio, a k/kb (kb from Table 1) Diameter of the column, D (m) c0 (kN/m2)
1.0 1.2 0, 5, 10, 151.0 1.2 0, 5, 10, 151.0 1.2 0, 5, 10, 150.5 1.2 0, 5, 100.5 1.2 0, 5, 10, 150.5 1.2 0, 5, 10, 150.25 1.2 0, 50.25 1.2 0, 5, 10, 150.25 1.2 0, 5, 10, 151.0 1.2 51.0 1.2 51.0 1.2 51.0 1.2 51.0 1.2 51.0 1.2 51.0 1.2 51.0 1.2 51.0 1.0 51.0 0.8 51.0 0.6 5
-10
-8
-6
-4
-2
00 10 20 30 40
sua = 9.6 kN/m2
sua = 12.7 kN/m2
sua = 16.0 kN/m2
sua = 19.4 kN/m2
Bending moment (kN-m)
Dep
th (m
)
H = 6 m, D = 1.2 m
b = 1.0/
Fig. 10. Effect of the ground sua on the bending moment in the column.
Table 3Summary of the representative values of the undrained shearstrength (sua).
c0 (kN/m2) sua (kN/m2)
0 9.65 12.710 16.015 19.4
-30
-20
-10
0 20 40 60 80
Profile 1
su (kN/m2)D
epth
(m)
Profile 2
Fig. 9. Illustration of the su distribution patterns.
0 0.1 0.2 0.3 0.40
10
20
30
40
50
60
y = 7.1x-0.82
Area improvement ratio (
Mm
ax (
kN-m
)
H = 6 m, sua = 12.7 kN/m2
b = 1.0
D = 1.2 m
Fig. 11. Mmax - a relationship.
-5
-4
-3
-2
-1
00 50 100 150 200 250 300 350 400
Dep
th (m
)
Stiffness index (I r )
sua = 9.6 kN/m2
b = 1.0b = 0.5b = 0.25
Fig. 12. Distributions of the stiffness index with depth.
174 J.-c. Chai et al. / Computers and Geotechnics 91 (2017) 169–178
(2) Effect of area improvement ratio (a)
Using the basic case conditions but varying a from 10% to 30%,the maximum bending moments (Mmax) in the column versus thevalue of a is depicted in Fig. 11. The relationship can be expressedby a power equation as:
Mmax ¼ 7:1 a�0:82ðMmax in kN�mÞ ð6Þ(3) Effect of stiffness index of soft subsoils (Ir)
Both the stiffness of the soft ground and the CDM column willinfluence the Mmax in the CDM column. Since the stiffness of theCDM columns is normally much higher than that of soft clays,the Mmax is more sensitive to the stiffness of soft clay layers. Thestiffness index (Ir) is used to represent the effect of the stiffnessof the soft soil. The expression for Ir is as follows:
Ir ¼ Gsu
ð7Þ
G ¼ E50
2ð1þ mÞ ð8Þ
where G is the shear modulus of the soil, E50 is the secant modulusof the soil corresponding to 50% of the shear strength, and m is Pois-son’s ratio of the soil.
The value of Ir of the assumed ground was determined from theresults of the numerical triaxial tests by Plaxis 3D. Severalresearchers have justified the effectiveness of the numerical simu-lation of triaxial tests to study the stress-strain behaviour of thesoft clay and the cement treated clay [30,31]. Values of E50 and suwere calculated from the simulated stress-strain curves. Fig. 12
Table 4Summary of the representative values of the stiffness index (Ira).
k/kb sua (kN/m2) Ira
1.0 9.6–19.4 650.5 9.6–19.4 1300.25 9.6–19.4 255
0 0.2 0.4 0.6 0.8 1 1.2 1.40
10
20
Diameter (D)
Mm
ax /
D 4
H = 6 m, b= 1.0
sua = 12.7 kN/m2,
Fig. 14. Effect of the column diameter on the ratio of Mmax/D4.
-15
-10
-5
0 10 20 30
H s = 6 m H s = 8 m H s = 10 m H s = 14 m H s = 20 m
Bending moment (kN-m)
Dep
th (m
)
H = 6 m, sua = 12.7 kN/m2,
b= 1.0
D = 1.2 m
J.-c. Chai et al. / Computers and Geotechnics 91 (2017) 169–178 175
shows the variations of Ir with depth for the assumed 10 m thicksoft clay layer. The sua value was 9.6 kN/m2 but with different com-pression indexes, i.e., k/kb = 0.25, 0.5 and 1.0. Similar to su, the rep-resentative value of the stiffness index (Ira) is defined as:
Ira ¼ ðIravg � IrminÞ0:5 ð9Þwhere the subscript ‘‘min” means minimum and ‘‘avg” means aver-age. The value of Iravg was calculated from a range of �2.0 m fromthe location of Irmin. For all the analysed cases, the values of the rep-resentative stiffness index are summarized in Table 4. Fig. 13 showsthe effect of Ir on the simulated bending moment in the columnunder the toe of the assumed embankment. The higher the Ir, thelower the Mmax was in the column.
(4) Effect of diameter (D) of the column
In the moment-area theory for a beam, the bending equation isexpressed as:
MI¼ E
Rð10Þ
where R is the radius of curvature, I is the moment of inertia of thecross-sectional area, E is the Young’s modulus, andM is the bendingmoment on the beam. Eq. (10) indicates that for two beams formedby a given material, if the ratio of M/I is the same, the curvature ofthe beams will be the same. Then, for a cylinder column, I / D4, andM/I /M/D4. The numerical results indicate that for a given value ofa, the diameter of the columns does not significantly influence thecurvature (R) of the deformed column under the toe of the embank-ment. Therefore,Mmax/D4 was used in the analysis. Fig. 14 shows the
-10
-8
-6
-4
-2
0 0 10 20 30Bending moment (kN-m)
Dep
th (m
)
H = 6 m, D = 1.2 msua = 12.7 kN/m2,
b = 1.0b = 0.5b = 0.25
Fig. 13. Effect of stiffness on the bending moment in the column.
-20
Fig. 15. Effect of the thickness of soft deposits (length of end bearing columns) onthe bending moment in the column.
relationship of Mmax/D4 versus D. It can be seen that the value ofMmax/D4 is nearly a constant for the cases investigated.
(5) Effect of thickness of soft deposit (Hs)
To investigate the effect of the thickness (Hs) of a soft subsoillayer, the Hs was varied from 6.0 m to 20.0 m. In the cases whereHs was less than 10.0 m, the corresponding soft clay layer(s) waschanged to sand layer(s). In the cases when Hs was more than10.0 m, clay layers were added by changing the correspondingsand layer(s) to clay layer(s). Considering the fact that for mostthick clayey deposits, the compressibility of the clay layers gradu-ally reduces with depth, the added clay layers used a value of k of0.282 and 0.217 for the depths of 10.0–14.0 m and 14.0–20.0 m,respectively, while the values of the other parameters were setto be the same as the values of the clay layer from 9.0 to 10.0 mdepth (Table 1).
Fig. 15 compares the simulated distributions of the bendingmoment in the CDM columns under the toe of a 6.0 m highembankment. It can be observed that the thickness of soft deposit
176 J.-c. Chai et al. / Computers and Geotechnics 91 (2017) 169–178
does not have a significant effect on the Mmax for the assumed con-ditions, i.e., the stiffness increased with depth.
3. Proposed design method
3.1. Main influencing factors
From the numerical results presented in the above, the factorsthat influence the maximum bending moment (Mmax) in CDM col-umns under the toe of an embankment are: (1) the embankmentload (pem), (2) undrained shear strength (su) and rigidity index(Ir) of the subsoil, (3) area improvement ratio (a) by the CDM col-umns, and (4) the diameter (D) of the CDM column. To propose adesign methodology, all these factors must be considered.
3.2. Key parameters
Two new dimensionless key parameters are introduced, theload/strength ratio (Pn) and the normalized maximum bendingmoment in the CDM column (Mn), with their definitions explainedin below.
(1) Load/strength ratio (Pn)
Since the effects of the maximum embankment load (pem) andundrained shear strength (su) of the soft subsoil are relevant fac-tors, a parameter, the load/strength ratio, is defined as:
Pn ¼ pem
suð11Þ
Generally, the larger the Pn, the larger the bending moment inthe CDM column will be.
(2) Normalized maximum bending moment (Mn)
From the numerical results, the effect of the main influencingfactors on the maximum bending moment (Mmax): (1) the load/strength ratio, Pn = pem/su, (2) the area improvement ratio (a) ofthe columns, and (3) the diameter (D) of the columns, have beenquantified. Then, assuming the effects of these parameters areindependent, a normalized maximum bending moment (Mn) isdefined as:
Mn ¼ Mmaxa0:82Do
PnD4pa
ð12Þ
where pa is the atmospheric pressure and Do is a constant of 1.0 m.The Do and pa are included to non-dimensionalize Mn.
0 2 4 6 8 10 12 14 16 180
0.001
0.002
0.003
0.004
0.005
0.006
(Mn
)
(Pn )
Ir = 65.0Ir = 130.0Ir = 255.0
Nor
mal
ized
max
imum
ben
ding
mom
ent
Ratio of load to undrained shear strength
Ir = 65
Ir = 130
Ir = 255
Fig. 16. Pn – Mn relationships.
3.3. Proposed design chart
With a few trials, it was found that the numerical results can besummarized in Pn �Mn relationships with Ir as an independentparameter, as shown in Fig. 16, which is the proposed design chart.The chart is applicable only for the following ranges of parameters:
(1) Load/strength ratio, Pn: 4–16;(2) Area improvement ratio, a: 10–30%;(3) Diameter of the column, D: 0.6–1.2 m;(4) Stiffness index, Ir: 65–255.
3.4. Design procedure
The main steps to use the proposed design chart are as follows:
(1) Step 1: Calculate pem. With the known height of an embank-ment (H) and the unit weight of the fill (ct), the maximumembankment load (pem) is calculated as
pem ¼ ct � H ð13Þ(2) Step 2: Calculate Pn. Estimate the representative value of the
undrained shear strength (sua) of a deposit with Eq. (5) usingthe su profile and computing Pn as pem/sua.
(3) Step 3: Estimate Ira using Eq. (9).(4) Step 4: ObtainMn. With known values of Pn and Ira,Mn can be
obtained from the design charts of Pn – Mn. Then, calculateMmax:
Mmax ¼ MnPnD4pa
Doa0:82 ð14Þ
(5) Step 5: Calculate the maximum tensile stress in the CDMcolumn, rtmax. Finally, evaluate the possibility of bendingfailure of the CDM column.
rtmax ¼ MmaxD2I
� r0a ð15Þ
where r0a is the effective axial stress in the column at the location
where the maximum bending moment occurred. To be practical it isproposed to use the initial vertical effective stress at the middle ofthe column under the toe of an embankment. If rtmax is greater thanthe tensile strength of the CDM column, the CDM column will failby bending. For example, if the CDM columns have a compressivestrength of 1000 kN/m2, then the tensile strength will be approxi-mately 100 kN/m2 (1/10 of the compressive strength) [29,32]. Ifrtmax > 100 kN/m2, bending failure occurs.
4. Verification of the proposed design method
In the literature, the results of two centrifuge model test withthe bending failure of the model soil-cement columns have beencollected. For these two cases, the columns were installed onlyunder the toe and the slope of the model embankments, which isdifferent from the assumed full improvement condition adoptedin the numerical investigation. The full improvement aims toincrease the stability of the embankment and control settlement.While partial improvement is mainly focused on increasing thefactor of safety of an embankment system. Shrestha et al. [22]reported results of a 3D finite element simulation of embankmentson a CDM column improved soft subsoil and noted that increasingthe size of the CDM column improved zone under the embank-ment, which can reduce the maximum bending moment in the col-umn. However, when the zone under the toe and the slope of anembankment was improved, further increasing the size of theimproved zone has only marginal effects on the maximum bending
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moment of the column under the toe of the embankment. There-fore, the results of these two model tests can be used to validatethe proposed design method.
4.1. Brief descriptions of each test
(1) Model test by Inagaki et al. (2002) (M-1)
The geometry of the centrifuge model test reported by Inagakiet al. [11] (Case 2 in their paper) and particular physical andmechanical properties of the model ground are shown in Fig. 17.The value of the representative sua was approximately 18 kN/m2,and the representative Ira was chosen as 65, the lowest value inthe design chart (Fig. 16). The unconfined compressive strength
Fig. 17. Cross–section of the embankment and soil profile for the model test byInagaki et al.
(a) (b)
Fig. 18. (a) Cross–section of the embankment and soil profile for the model test byKitazume and Maruyama [9,10] and (b) the measured su profile of the modelground.
Table 5Predicted results of the centrifuge model tests.
Cases pem (kN/m2) a (%) D (m) sua (kN/m2) Ira Pn
M-1 110.5 20 1 18 65 6.14M-2 50 28 1 4.4 65 11.4
of the soil-cement columns (qu) of 219–302 kN/m2 was reported.Then, the estimated tensile strength is approximately 21.9–30.2 kPa (approximately 1/10 of the value of qu). The centrifugegravity adopted was 50g.
(2) Model test by Kitazume and Maruyama (2006; 2007) (M-2)
One of the centrifuge model tests reported by Kitazume andMaruyama [9,10] (Case 8 in their paper) is analysed here. Thegeometry of the model test is shown in Fig. 18(a). The undrainedshear strength of the model ground was measured at two depths,as shown in Fig. 18(b). Here, the measured smaller value of thesu of 4.4 kN/m2 is chosen as the representative value. The modelground was very weak, and an Ira of 65 was assumed; however,the actual value of Ira might be lower than 65 (lower boundary ofthe proposed design chart). As a result, the predicted maximumtensile stress may be lower. It will be explained later that evenusing Ira = 65, the bending failure was predicted.
The reported unconfined compression strength (qu) of the soil-cement column was 409 kN/m2, and the reported tensile strength(rt) was 132 kN/m2. Because a small carbon rod with a diameterof 2 mm (dimension in centrifuge scale) was inserted inside the20 mm diameter model soil-cement column. However, it appearsthat the value of rt was evaluated by simply adding the bendingstrength of the soil-cement column and the carbon rod. For thiskind of composited column, the failure process will be a progres-sive manner, i.e., the two materials may not fail simultaneously.The tensile strength of the soil-cement column is 1/10 of its com-pression strength, which is approximately 40.9 kPa. The rt of thecomposited column can be in the range of 40.9–132 kPa. The cen-trifuge gravity adopted was also 50g.
4.2. Predicted rtmax and FS
Using the available parameters of the model tests, from the pro-posed design method, Mmax was calculated for the two cases usingEq. (14). Then, the value of rtmax was obtained using Eq. (15). Theconditions and the analysed results of pem, sua, Ira, Pn and rtmax forthe two model tests are summarized in Table 5.
For the two model tests, the predicted maximum tensile stres-ses (rtmax) in the soil-cement columns were nearly equal to or lar-ger than the values of the tensile strength (rt f) of the columns. Ithad been reported that for Case M-1, cracks on the columns wereobserved, and for Case M-2, the columns failed through bending.Therefore, the predictions agree with the observed results.
5. Conclusions
The bending failure mechanism and maximum bendingmoment in cement deep mixing (CDM) formed column under anembankment load was investigated by a series three-dimensional(3D) finite element analyses (FEA). Based on the results of theFEA, the following findings/conclusions can be obtained.
(1) The main factors influencing the value of the maximumbending moment in CDM columns are as follows: (a) magni-tude of an embankment load (pem), (b) undrained shear
Predicted maximumtensile stress rtmax (kN/m2)
Tensile strengthrtf (kN/m2)
FS
Predicted Observed
26 21.9–30.2 (26.1) 1.0 Cracks123 40.9–132.0 (81.5) Fail Fail
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strength (su) and rigidity index (Ir) of the soft deposit, (c) thearea improvement ratio by the CDM column (a), and (d) thediameter (D) of the column. The effect of each factor wasevaluated quantitatively.
(2) A design method to consider the bending failure of CDM col-umns under an embankment load is proposed. The methodconsiders all the main influencing factors on the maximumbending moment in the CDM column, which are summa-rized into three dimensionless parameters, namely, the loadto shear strength ratio, (Pn = pem/su), Ir, and the normalizedmaximum bending moment in the CDM column. Using thismethod, with known values of pem, a, D and su and Ir, the fac-tor of safety of the CDM column against bending (tensile)failure can be predicted.
(3) Using the results of two centrifuge model tests of embank-ments on cement mixing formed model columns improvedsoft subsoils reported in the literature, the usefulness ofthe proposed method was verified. For the two cases consid-ered, tensile cracks or failure of the model columns werereported, and the predicted results agree with the testresults.
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