MATHEMATICAL MODEL TO PREDICT COMPRESSION INDEX OF UNIFORM LOOSE SAND IN COASTAL AREA OF DEGEMA, RIVERS STATE OF NIGERIA

http://www.iaeme.com/IJARET/index.asp 86 [email protected]

International Journal of Advanced Research in Engineering and Technology

(IJARET) Volume 6, Issue 12, Dec 2015, pp. 86-103, Article ID: IJARET_06_12_009

Available online at

http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=6&IType=12

ISSN Print: 0976-6480 and ISSN Online: 0976-6499

© IAEME Publication

MATHEMATICAL MODEL TO PREDICT

COMPRESSION INDEX OF UNIFORM

LOOSE SAND IN COASTAL AREA OF

DEGEMA, RIVERS STATE OF NIGERIA

Eluozo. S. N

Subaka Nigeria Limited Port Harcourt Rivers State of Nigeria

Director and Principal Consultant Civil and Environmental Engineering,

Research and Development

Ode T

Department of Civil Engineering, faculty of Engineering

Rivers State University of Science and Technology Port Harcourt

ABSTRACT

Predicting the compression index applying mathematical model for loose

dense sand has been thoroughly developed, this is to monitor the rate of

compression during settlement of loose dense sand, the model were generated

to monitor the compression index of uniform loose sand in coastal area of

Degema, the study express compression index at various depth within the

specified range, the generated model produced simulation values compared

with the measured values, both parameters developed faviourable fits, the

compression index expressed linear increase to the optimum level at different

depth, the study has also express the rate of homogeneity of the strata in

various formations, these developed model will definitely be applied to predict

the compression index for uniform loose sand under the influences of

settlement in caring any impose load .

Key words: Mathematical Model, Compression Index and Uniform Loose

Sand

Cite this Article: Eluozo. S. N and Ode T, Mathematical Model To Predict

Compression Index of Uniform Loose Sand In Coastal Area of Degema,

Rivers State of Nigeria. International Journal of Advanced Research in

Engineering and Technology, 6(12), 2015, pp. 86-103.

http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=6&IType=12

Mathematical Model To Predict Compression Index of Uniform Loose Sand In Coastal Area

of Degema, Rivers State of Nigeria


1. INTRODUCTION

Compression of soil structural setting basic aspect of soil deformation,, compression

of such behaviour normally forms the base of modelling thus the stress-strain

relationships of soils (e.g., Pestana and Whittle, 1995; Hong and Onitsuka, 1998;

Potts and Zdravkovic 1999; Baudet and Stallebrass, 2001; Liu et al, 2011 Martin et

al 2013). The perceptive of compression behaviour has consequently in normal

setting been imperative to geological and geotechnical engineering practice, the

research for such soil condition has been on course for many decades (e.g., Skempton,

1944; Bowles, 1989; Butterfield and Baligh, 1996; Desai, 2001; and Chai et al, 2004).

Stress is one of the common conditions which soils have been subjected; the

surroundings in which they are shaped thus the time that has lapsed on the

geotechnical time scale over numerous steps of their formation have been recognized

as prospective factors in their compressibility. It has been observed that soils age and

creep over time; hence, bonds build up at particle contacts in natural clay, which can

also be considered as “structured clay” (Leonards, 1972; Leroueil et al., 1979;

Michell, 1996; and Shibuya, 2000; etc.). The resistances of soil formation are known

to be responsible for the various conditions in the engineering behaviour of natural

soils; these are between the structured and the destructured (reconstituted) states

(Leroueil et al., 1979 and 1983; Hanzawa and Adachi, 1983; Leroueil and Vaughan,

1990; Mitchell, 1996; and Shibuya, 2000 Martin et al 2013). reconstituted clay is

intrinsic base on compression curve such as (devoid of soil structure) and is normally

applied as a frame of reference for the behaviour of naturally and artificially soils

formation (Burland, 1990; Nagaraj and Miura, 2001; Nagaraj et al., 1990 and 1998)

these are some conditions that couild be applied as a basis for modelling the

behaviour of soils in the structured state (e.g., Liu et al, 2000; Masín, 2007;

Hinchberger and Qu, 2009; Horpibulsuk et al, 2007, 2010; 2013; Suebsuk et al., 2010

and 2011). Moreover, reconstituted clay including its characteristic are applied as a

liner for landfill and a fill for reclaimed area, and the compressibility of the

reconstituted clay is one of the required design parameters. In his fortieth Rankine

lecture, Burland (1990) introduced the concept of void index and performed a

systematic study on the compression behaviour of clays via the void index. The

current research is carried out based on Burland’s original work and subsequent

research by 136 others (e.g., Amorosi and Rampello, 2007; Bobet et al, 2011; Hong et

al, 2012).

2. GOVERNING EQUATION

02

2

dx

dck

dx

dcV

dx

cdVt o

(1)

Nomenclature

V = Velocity of fluid

k = Permeability

Vo = Void Ratio

Cc = Compression index

Z = Depth

002

2

dx

dcKV

dx

cdVt (2)

Eluozo. S. N and Ode T


Let

0n

n

n xaC

1

11

n

n

n xnaC

2

211 1n

n

n xannC

011

1

0

2

2

n

n

n

n

n

n xnaVxannVt (3)

Replace n in the 1st term by n+2 and in the 2

nd term by n+1, so that we have;

01120

1

0

2

n

n

no

n

n

n xanVxannVt (4)

i.e. 102 112 nn anKVannVt

(5)

12

1 102

nnVt

anKVa n

n

(6)

2

102

nVt

akVa n

n

(7)

for

Vt

aKVan

2,0 10

2

(8)

(9)

Subject equation (16) to the following boundary condition

HoCandoC 10

xVt

kV

aaxC

0

10

010 aaoC

i.e. 010 aa (10)

x

Vt

KV

aVt

VxC

0

101

!2

Ha

Vt

KVoC

1

01

!2

xVt

kV

aaxC

0

10




KV

HVta

0

1

(11)

Substitute (10) into equation (11)

01 aa

KV

HVta

0

0 (12)

Hence, the particular solution of equation (16) is of the form:

xVt

kV

KV

HVt

KV

HVtxC

0

00

10

0

xVt

KV

KV

HVtxC (13)

3. MATERIALS AND METHOD

Standard laboratory experiment where performed to monitor compression index of

loose dense sand at different formation, the soil deposition of the strata were collected

in sequences base on the structural deposition at different locations, this samples

collected at different location generated variations at different depth producing

deposition of stiff clay compression at different strata, the experimental result are

applied to be compared with the theoretical values to determined the validation of the

model.

4. RESULT AND DISCUSSION

Results and discussion are presented in tables including graphical representation of

compression index of loose dense sand

Table 1 Predictive Values of loose sand compression index at Different Depth

Depth [M] Predictive of loose sand Cc

0.2 0.002

0.4 0.004

0.6 0.0066

0.8 0.0088

1 0.011

1.2 0.0132

1.4 0.0154

1.6 0.0176

1.8 0.0198

2 0.022

2.2 0.0242

2.4 0.0264

2.6 0.0286

2.8 0.0308

3 0.033

3.2 0.0352

3.4 0.0376




3.6 0.0396

3.8 0.0418

4 0.044

4.2 0.0462

4.4 0.0484

4.6 0.0506

4.8 0.0528

5 0.055

Table 2 Predicted and Measured of compression index for loose sand at Different Depth

Depth [M] Predictive of loose sand Cc Measured Values of loose sand Cc

0.2 0.002 0.00211

0.4 0.004 0.00431

0.6 0.0066 0.00651

0.8 0.0088 0.00871

1 0.011 0.0109

1.2 0.0132 0.0131

1.4 0.0154 0.0153

1.6 0.0176 0.0175

1.8 0.0198 0.0197

2 0.022 0.0219

2.2 0.0242 0.0241

2.4 0.0264 0.0263

2.6 0.0286 0.0285

2.8 0.0308 0.0307

3 0.033 0.0329

3.2 0.0352 0.03511

3.4 0.0376 0.0373

3.6 0.0396 0.03951

3.8 0.0418 0.0417

4 0.044 0.0439

4.2 0.0462 0.04611

4.4 0.0484 0.04831

4.6 0.0506 0.0505

4.8 0.0528 0.05271

5 0.055 0.0549



0.2 0.00289

0.4 0.0056

0.6 0.0084

0.8 0.0112

1 0.014

1.2 0.0168

1.4 0.0196

1.6 0.0224

1.8 0.0252

2 0.026

2.2 0.03

2.4 0.0336





2.6 0.0364

2.8 0.0392

3 0.042

3.2 0.048

3.4 0.0476

3.6 0.0504

3.8 0.0532

4 0.056



0.2 0.00289 0.0028

0.4 0.0056 0.0056

0.6 0.0084 0.0084

0.8 0.0112 0.0112

1 0.014 0.014

1.2 0.0168 0.0167

1.4 0.0196 0.0188

1.6 0.0224 0.0221

1.8 0.0252 0.0262

2 0.026 0.024

2.2 0.03 0.034

2.4 0.0336 0.035

2.6 0.0364 0.038

2.8 0.0392 0.041

3 0.042 0.045

3.2 0.048 0.049

3.4 0.0476 0.051

3.6 0.0504 0.052

3.8 0.0532 0.054

4 0.056 0.058



0.2 0.013

0.4 0.024

0.6 0.036

0.8 0.052

1 0.066



0.2 0.013 0.0127

0.4 0.024 0.0237

0.6 0.036 0.0361

0.8 0.052 0.04988

1 0.066 0.07





0.2 0.0038

0.4 0.0076

0.6 0.011

0.8 0.015

1 0.019

1.2 0.0228

1.4 0.0266

1.6 0.0304

1.8 0.0342

2 0.038

2.2 0.0418

2.4 0.046

2.6 0.0495

2.8 0.052

3 0.057

Figure 8 Predicted and Measured of compression index for loose sand at Different Depth


0.2 0.0038 0.00378

0.4 0.0076 0.00758

0.6 0.011 0.0114

0.8 0.015 0.0152

1 0.019 0.0189

1.2 0.0228 0.02278

1.4 0.0266 0.0266

1.6 0.0304 0.0304

1.8 0.0342 0.03418

2 0.038 0.03798

2.2 0.0418 0.04178

2.4 0.046 0.04558

2.6 0.0495 0.04938

2.8 0.052 0.05318

3 0.057 0.05698



0.2 0.0088

0.4 0.017

0.6 0.0264

0.8 0.0352

1 0.044

1.2 0.052






0.2 0.0088 0.0059

0.4 0.017 0.0172

0.6 0.0264 0.02579

0.8 0.0352 0.03439

1 0.044 0.04299

1.2 0.052 0.05159



0.2 0.0024

0.4 0.004

0.6 0.006

0.8 0.008

1 0.01

1.2 0.012

1.4 0.014

1.6 0.016

1.8 0.018

2 0.02

2.2 0.022

2.4 0.024

2.6 0.026

2.8 0.028

3 0.03

3.2 0.032

3.4 0.034

3.6 0.036

3.8 0.038

4 0.04

4.2 0.042

4.4 0.044

4.6 0.046

4.8 0.048

5 0.05



0.2 0.0024 0.00206

0.4 0.004 0.00406

0.6 0.006 0.00606

0.8 0.008 0.00803

1 0.01 0.01

1.2 0.012 0.012

1.4 0.014 0.016

1.6 0.016 0.018

1.8 0.018 0.024

2 0.02 0.026

2.2 0.022 0.03

2.4 0.024 0.032



2.6 0.026 0.034

2.8 0.028 0.036

3 0.03 0.038

3.2 0.032 0.039

3.4 0.034 0.041

3.6 0.036 0.044

3.8 0.038 0.046

4 0.04 0.047

4.2 0.042 0.048

4.4 0.044 0.049

4.6 0.046 0.05

4.8 0.048 0.051

5 0.05 0.053



0.2 0.014

0.4 0.0208

0.6 0.0312

0.8 0.0416

1 0.052



0.2 0.014 0.01308

0.4 0.0208 0.02112

0.6 0.0312 0.03012

0.8 0.0416 0.04008

1 0.052 0.0576

Figure 1 Predictive Values of loose sand compression index at Different Depth

y = 0.011x - 9E-05 R² = 1

0

0.01

0.02

0.03

0.04

0.05

0.06

0 2 4 6

pre

dic

tive

val

ue

s fo

r lo

ose

de

nse

sa

nd

Depth [M ]

Predictive of loose sand Cc

Linear (Predictive of loose sand Cc)




Figure 2 Predicted and Measured of compression index for loose sand at Different

Depth


0

0.01

0.02

0.03

0.04

0.05

0.06

0 2 4 6

pre

dic

tive

an

d m

easu

red

val

ues

fo

r lo

ose

den

se

san

d o

n c

om

pre

ssio

n in

dex

Depth [ m]


Measured Values of loose sand Cc

y = 0.0141x - 0.0002 R² = 0.9973

0

0.01

0.02

0.03

0.04

0.05

0.06

0 1 2 3 4 5

pre

dic

tive

val

ues

fo

r lo

ose

den

se s

and

Depth [m]






Depth


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 1 2 3 4 5

pre

dic

tive

an

d m

easu

red

val

ue

s fo

r lo

ose

de

nse

san

d

on

co

mp

ress

ion

ind

ex

Depth [m]



y = 0.0179x2 + 0.0456x + 0.003 R² = 0.9989

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.2 0.4 0.6 0.8 1 1.2

pre

dic

tive

val

ues

fo

r lo

ose

den

se s

and

Depth [m]


Poly. (Predictive of loose sand Cc)





Depth


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 0.5 1 1.5

pre

dic

tive

an

d m

easu

red

val

ues

fo

r lo

ose

den

se

san

d o

n c

om

pre

ssio

n in

dex

Depth [m]



y = 0.019x - 2E-05 R² = 0.9996

0

0.01

0.02

0.03

0.04

0.05

0.06

0 1 2 3 4

pre

dic

tive

val

ues

of

loo

se d

ense

san

d

Depth[m]





Figure: 8 Predicted and Measured of compression index for loose sand at Different

Depth


0

0.01

0.02

0.03

0.04

0.05

0.06

0 1 2 3 4

pre

dic

tive

an

d m

easu

red

val

ues

fo

r lo

ose

den

se

san

d o

n c

om

pre

ssio

n in

dex

Depth [m]



y = 0.0437x - 1E-05 R² = 0.9995

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.5 1 1.5

pre

dic

tive

val

ues

fo

r lo

ose

den

se s

and

Depth [m]







Depth


0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.5 1 1.5

pre

dic

tive

an

d m

easu

red

val

ues

fo

r lo

ose

d

en

se s

and

on

co

mp

ress

ion

ind

ex

Depth [m]



y = 0.01x + 6E-05 R² = 1

0

0.01

0.02

0.03

0.04

0.05

0.06

0 1 2 3 4 5 6

pre

dic

tive

val

ues

fo

r lo

ose

den

se s

and

Depth [m]






Depth


0

0.01

0.02

0.03

0.04

0.05

0.06

0 2 4 6

pre

dic

tive

an

d m

easu

red

val

ues

fo

r lo

ose

d

en

se s

and

on

cm

pre

ssio

n in

dex

Depth [m]



y = 0.0129x2 + 0.033x + 0.0065 R² = 0.9984

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8 1 1.2

pre

dic

tive

val

ues

fo

r lo

ose

den

se s

and

Depth [m]


Poly. (Predictive of loose sand Cc)





Depth

The figure presented express the generated express the deposition of loose dense

sand in various depositions, figure one and two shows the uniformity of the deposition

structured in homogeneous condition, linear increase were observed in the

compression of the strata at different formation to the optimum depth porosity were

more experienced within the [0.2-1.4] these implies that the permeability in terms of

fluid flow experienced low deposition more, but the compression index of the

formation observed linear deposition to five metres, comparative analysis between

predictive and measured developed best fits as stated in figure two observing the same

linear increase between both parameters. Figure three and four observed different

depositions on compression index of the soil in different depth to optimum level, low

permeability were observed between [0.2-0.6]. while porosity express higher

deposition, thus pressure the deposition of compression in loose dense sand,

fluctuation in compression were observed on the exponential deposition of

compression index for loose dense sand to the optimum level. Comparative process

between predictive and measured values express similar fluctuation thus developed

best fits. While five and six were ex press slight different deposition compare to

previous figures, the prediction of the compression were monitored between the

specified ranged thus from [0.2-1.0M],gradual increase were experienced from the

lowest to the optimum level, the measured compared with predictive maintained

faviourable fits, figure seven and eight experienced slight higher porosity but

maintained linear homogeneity in compression to the optimum depth of prediction to

three metres, comparing these predictive values with experimental data best fits were

observed in figure eight, while figure nine and ten obtained exponential state to the

optimum level of loose dense soil at [1.0M], the formation observed homogeneous

setting as it was structured in the formation, compression of the loose dense sand were

pressured by these characteristics, the predictive were compared with the measured

values, both parameter express the best fits. Figure eleven and twelve express linear

increase of compression index, but the measured developed fluctuation from [2-6M],

these can be attributed to the depositional variation of some formation characteristic at

various depth thus predicted within the specified standard of loose dense sand.. Figure

thirteen and fourteen developed its compression between the specified ranged for

loose dense sand but at more shallow depth, these implies that the developed model

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0 0.5 1

pre

dic

tive

an

d m

easu

red

val

ues

fo

r lo

ose

den

se s

and

on

co

mp

ress

ion

in

dex

Depth [m]





can be applied at any depth for the determination of compression index, linear

increase were experienced between the predicted depth thus within the specified for

such type of formation, comparing predictive and measured both parameters express

faviourable fits validating the developed model for the study.

5. CONCLUSION

The loose dense sand compression in the developed model has been thoroughly

expressed, the compression index of the soil were to monitored at various depth to

predict within the specified range, several experts has been applying the experimental

approach to monitor, thus empirical solution has been applied to predict compression

index of the soil, but these conceptual framework using analytical method has not

been applied by other experts, this concepts developed the generated model that

produces predictive values for loose dense sand specified within the range for loose

dense sand , several prediction at different depth producing the specified compression

within the range for loose dense sand , the predictive values were compared with

measured valued , both parameters generated faviourable fit validating the developed

model for loose dense sand.

REFERENCES

[1] Amorosi, A and Rampello, S 2007, ‘An experimental investigation into the

mechanical behaviour of a structured stiff clay’, Géotechnique, vol.57,

pp.153-166.

[2] Baudet, BA and Stallebrass, SE 2001, ‘Modelling the destructuring of soft

natural clays’ Computer Methods and Advances in Geomechanics, vol.1,

pp.297-302.

[3] Bobet, A, Hwang, J, Johnston, CT and Santagata, M 2011, ‘One-dimensional

consolidation behavior of cement-treated organic soil’, Canadian

Geotechnical Journal, Vol.48 (7), pp.1100-1115.

[4] Bowles, JE 1989, Physical and Geotechnical Properties of Soils, McGraw-

Hill, New York

[5] Burland JB 1990, ‘On the compressibility and shear strength of natural soils’,

Géotechnique, vol.40, pp.329-378

[6] Chai, JC, Miura, N, and Zhu, HH 2004, ‘Compression and consolidation

characteristics of structured natural clays’, Canadian Geotechnical Journal,

vol.41, no.6, pp.1250-1258.

[7] Desai, CS and Toth, J 1996, ‘Disturbed state constitutive modelling based on

stress-strain and 505 non-destructive behaviour’, Int. J. of Solids and

Structures, vol.33, pp.1619-1650

[8] Hong, Z and Onitsuka, K 1998, ‘A method of correcting yield stress and

compression index 519 of Ariake clays for sample disturbance’, Soils and

Foundations, vol.38 (2), pp.211-222.

[9] Hanzawa, H and Adachi, K 1983, ‘Overconsolidation of alluvial clays’, Soils

and 513 Foundations, vol.23 (4), pp.106-118.

[10] Horpibulsuk, S, Liu MD, Liyanapathirana, S and Suebsuk, J 2010,

‘Behaviour of cemented clay simulated via the theoretical framework of the

SCC model’, Computers and Geotechnics, vol.37 (1), pp.1-9.

[11] Horpibulsuk, S, Yangsukaseam, N, Chinkulkijniwat, A 525, and Du, YJ

2011, ‘Compressibility and permeability of Bangkok clay compared with

kaolinite and bentonite’, Applied Clay Science, Vol.52, pp.150-159.




[12] Horpibulsuk, S, Rachan, R, Suddeepong, A, Liu, MD and Du, YJ 2013,

‘Compressibility of lightweight cemented clays’, Engineering Geology,

Vol.159, pp.59-66.

[13] Lancellotta, R and Pepe, C 1990, “Mechanical behaviour of upper Pisa clay”,

Internal report, Technical University of Torino

[14] Liu MD and Carter JP 1999, ‘Virgin compression of structured soils’,

Géotechnique, vol.49 (1), pp.43-57.

[15] Liu, MD, Carter, JP, Desai, CS and Xu, KJ 2000, ‘Analysis of the

compression of structured soils using the disturbed state concept’, Int. J. for

Numerical and Analytical Methods in Geomechanics, vol.24, pp.723-735.

[16] Leonard, GA 1972, Discussion of “Shallow foundation”, Proceedings of

ASCE Spec. Conf. on Perf. of Earth and Earth supported Struct., ASCE, New

York, vol.3, pp.169-173.

[17] Leroueil, S and Vaughan, PR 1990, ‘The general and congruent effects of

structure in natural soils and week rock’, Geotechique, 40, pp.467-488.

[18] Leroueil, S, Tavenas, F, Brucy, F, La Rochelle, P, and Roy, M 1979,

‘Behavior of destructured natural clays’. Journal of Geotechnical

Engineering, ASCE, vol.105 (6), pp.759-778.

[19] Leroueil, S, Tavenas, F, Samson, L, and Morin, P 1983, ‘Preconsolidation

pressure of Champlain clay. Part II: Laboratory determination’, Canadian

Geotechnical Journal, vol.20 (4), pp.803-816.

[20] Masín, D 2007, ‘A hypoplastic constitutive model for clays with meta-stable

structure’ Canadian geotechnical journal, vol.44 (3), pp.363-375.

[21] Mitchell, JK 1996, Fundamentals of soil behavior, John Willey&Sons Inc.,

New York, 437p

[22] Pestana, JM and Whittle, AJ 1995, ‘Compression model for cohesionless

soils’, Géotechnique, vol.45, pp.621-631.

[23] Potts, DM and Zdravkovic, L 1999, Finite Element Analysis in Geotechnical

Engineering: Theory, Thomas Telford, London

[24] Nagaraj, TS, Srinivasa Murthy, BR, Vatsala, A and Joshi, RC 1990,

‘Analysis of compressibility of sensitive clays’, Journal of Geotechnical

Engineering, ASCE, vol.116 (GT1), pp.105-118.

[25] Nagaraj, TS, Pandian, NS, and Narasimha Raju, PSR 1998, ‘Compressibility

behavior of soft cemented soils’, Geotechnique, vol.48(2), pp.281-287

[26] Shibuya, S 2000, ‘Assessing structure of aged natural sedimentary clays’,

Soils and Foundations, vol.40 (3), pp.1-16.

[27] Skempton, AW 1944, ‘Notes on compressibility of clays’, Qaurterly Journal

of the Geological Society, London, vol. 100(2), pp.119-135.

[28] Suebsuk, J, Horpibulsuk, S, and Liu, MD 2010, ‘Modified Structured Cam

Clay: A constitutive model for destructured, naturally structured and

artificially structured clays’, Computers and Geotechnics, vol.37 (7-8),

pp.956-968.

[29] Suebsuk, J, Horpibulsuk, S and Liu, MD 2011, 'A critical state model for

overconsolidated structured clays’, Computers and Geotechnics, vol.38,

pp.648-658

[30] Martin D. liu and Ziling Zhuang Suksun H 2013 Estimation of the

compression behaviour of Reconstituted clays Geology, 167 (December), 84-

94.

[31] S.O. Nkakini and Ndor .M.Vurasi, Ergonomic Evaluation of Lawn Mower

Operation for Comfort in Rivers State, Nigeria. International Journal of

Advanced Research in Engineering and Technology, 6(7), 2015, pp. 43-51