Hyperbolic constitutive model for tropical residual soils

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Hyperbolic constitutive model for tropical residual soils

  1. 1. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 3, May - June (2013), © IAEME121HYPERBOLIC CONSTITUTIVE MODEL FOR TROPICAL RESIDUALSOILSNagendra Prasad.K1, Sulochana.N21Professor, Dept. of Civil Engineering, SV University, Tirupati, India, (corresponding author)2Sulochana.N , Research Scholar, Dept. of Civil Engineering, SV University College ofEngineering, Tirupati & Lecturer in Civil Engineering, Govt. Polytechnic for Women,Palamaner, Chittoor District, A.P.ABSTRACTThe stress-strain response of natural soils depends on soil state, stress history anddrainage conditions. Many constitutive models are available for describing the stress-strainrelationship for different soil types. It is desirable to have a comprehensive model, based onsound principles of continuum mechanics, capable of describing the soil behaviour under anytype of loading. The model parameters involved in such models most often, require elaborateexperimental procedures to evaluate them. There are many instances when a problem posedto an engineer may not necessarily require such a complex material model. For example, asimple undrained analysis may be sufficient for the immediate or end of construction (thiswill be always critical condition) of structures on clayey soils. Depending on specific fieldsituation, it may be possible to analyze the problem with much simpler model. Therefore,there is a need to develop a realistic and simple model whose parameters can be determinedeasily with simple procedures. The cardinal aim of the present paper is to develop a simpleconstitutive relationship using hyperbolic approach, based on analysis of test results on fivedifferent types of soils. Combination of stress ratio and mean principal stress is identified tocapture the strain softening behaviour of residual soils. The model developed is applied topredict the stress-strain response for other soils found in literature. The model predictions arequite comparable and model parameters are easily determinable.Keywords: Tropical residual soils, hyperbolic model, stress-strain-pore pressure response,yield stress, confining pressure.INTERNATIONAL JOURNAL OF CIVIL ENGINEERING ANDTECHNOLOGY (IJCIET)ISSN 0976 – 6308 (Print)ISSN 0976 – 6316(Online)Volume 4, Issue 3, May - June (2013), pp. 121-133© IAEME: www.iaeme.com/ijciet.aspJournal Impact Factor (2013): 5.3277 (Calculated by GISI)www.jifactor.comIJCIET© IAEME
  2. 2. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 3, May - June (2013), © IAEME1221. INTRODUCTIONSoils are very complicated engineering materials, whose constitutive responsedepends on many compositional and environmental factors. The availability of high-speedcomputers and powerful numerical techniques (such as the finite element method) makes itpossible to incorporate the non-linear behaviour of materials into the analysis of soil systemsand soil structure interaction problems. Some advanced soil models have been proposed forthe non-linear stress-strain behaviour of soils, including the hypo elastic models, the hyperelastic models and the plasticity models. However, these models require the determination ofmany parameters for the investigated soils.The explicit nature of stress-strain response of tropical residual soil mostly dependson fabric and nature of cement bonding in addition to the usual factors such as current state,stress history, stress path and drainage conditions. Despite the availability of quite a goodnumber of constitutive relations concerning the behaviour of clays, there are still a largenumber of problems which have not been satisfactorily tackled. Among these, strainsoftening behaviour of tropical residual soils during deformation process is of importance.Strain softening is an important phenomenon causing concern as regards the design problemsassociated with estimation of bearing capacity, stability and deformation.Most often tropical residual soils are treated as overconsolidated soils because theyalso exhibit similar features like strain softening, higher initial stiffness, etc. But a closerstudy of the test results of different tropical residual soils found in literature would reveal thatthe behaviour of these soils in undrained shear is very much different from that ofuncemented overconsolidated soils. Most important difference is that softening here isassociated with continued positive pore pressures whereas softening in overconsolidated soilsis associated with negative pore pressures. Probably, this is because there is an additionalcomponent of resistance from cementation bonds.Thus there is a need for development of a realistic and simple model comprising ofeasily determinable constitutive parameters which is capable of capturing the most importantaspects of the behaviour. For instance, the hyperbolic elastic models (Duncan and Chang,1970) are still widely used in the non linear finite element analysis of uncemented soils is onesuch example. The reasons for using hyperbolic models are ascribed to its simplicity and welldefined constants associated with the model. It is well known that hyperbolic model wasoriginally formulated to fit the undrained triaxial test results with only two constants to bedefined. It subsequently grew in strength and came to be applied to realistic boundary valueproblems involving both drained and undrained conditions with corresponding modificationsusing incremental approach. The inherent capability of the hyperbolic form to capture thesoftening behaviour of tropical residual soils has not been attempted in the past (NagendraPrasad et al. 1999).2. BACKGROUND INFORMATIONIn a tropical region, residual soil layers can be very thick, sometimes extending tohundreds of meters before reaching un-weathered rock. Unlike the more familiar transportedsediment soil, the engineering properties and behaviour of tropical residual soils may varywidely from place to place depending upon the rock of origin and the local climate duringtheir formation; and hence are more difficult to predict and model mathematically. Despitetheir abundance and significance, our knowledge and understanding of these soils is not as
  3. 3. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 3, May - June (2013), © IAEME123extensive as that of transported sediment soil (Huat et al. 2012). However, with respect toresidual soil, both its interaction mechanism and its failure behaviour in soil composites arenot well understood due to limited study (Mofiz et al. 2010).Tropical soils appear in large regions of the world and have been less studied thansoils from temperate climates, particularly with respect to critical state and limit stateconditions. Most geo-materials are structured in nature and this natural structure affects thebehavior of tropical soils. Structural features affecting soil behavior include soil cementationand soil fabric.Sarma et al. (2008) observed that the consolidation properties of soils indicate aninsight on the compressibility behaviour of soils with associated expulsion of water.However, determination of such properties involves considerable time, cost and rigoroustesting process. Further, natural state of partial saturation and soil-moisture is not simulatedin the standard consolidation procedures. The sampling technique is also not specific for theOedometer tests and sampling disturbance influences the results considerably. As such,modified methodologies of Odometer test for field simulation as well as simple correlationsof the consolidation parameters with fundamental properties are always preferred bypractising engineers.Karmakar et al. (2004) brought out that the soil undergoes both elastic and plasticdeformation when subjected to loading. The basic requirement for integrated analyses ofmovements and failure of a soil mass is a constitutive relationship capable of modellingstress-strain behaviour of soil up to and beyond failure. Development of such a relationshipgenerally involves separating the elastic and plastic behaviour. This is achieved using a well-defined curve known as the yield locus located in a shear stress-normal stress space. If thestress state of a soil plots inside the yield locus, it is considered to be elastic and undergoesrecoverable deformation. On the other hand, if a particular stress path puts the stress state ofthe soil on or outside the yield locus, plastic or irrecoverable deformation of soil occurs.Elasto-plastic constitutive models help to distinguish between the recoverable andirrecoverable deformations for understanding the stress strain behaviour of soil duringloading and unloading. In order to develop a simple framework, a mechanistic approach isneeded based on well planned experimental investigation.3. SCOPE OF THE PAPERParticularly soils in the Southern Indian Region are residual in nature (those derivedby in-situ weathering of rocks). In residual soils the particles and their arrangement wouldhave evolved progressively as a consequence of physical and chemical weathering. Althoughthe geological study of the formation and structure of in-situ residual soils is well advanced,the simple and rapid methods to analyze and assess the engineering properties of these soilshave not received the same level of attention. This is in contrast to the situation whilesedimentary soil deposits are encountered. Quite often cementation in rock would be leftbehind due to varied degrees of weathering.The objective of this paper is to develop a simple practical procedure for representingthe nonlinear, stress dependent, inelastic stress-strain behaviour of tropical residual soilduring undrained shear. Accordingly, the relationship described has been developed in such away that values of the required parameters may be derived from the results of the standardlaboratory triaxial tests. The formulations are proposed within the framework of a hyperbolicrelation of stress ratio (q/p) and also of effective mean principal stress (p) with strain. The
  4. 4. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 3, May - June (2013), © IAEME124formulation incorporates stress dependency, nonlinearity, strain softening aspects of thebehaviour quite effectively. It must be pointed out that this is an attempt to circumvent thedifficulty associated with the choice of complicated constitutive models while dealing withundrained field situations.4. EXPERIMENTAL INVESTIGATIONS4.1 Residual Soils TestedIn order to understand the mechanisms involved in shear and compression behaviourin relation to naturally sedimented soils, a detailed experimental program has beenundertaken on undisturbed soil samples extracted from regional soil deposits in Tirupati andits surroundings. These soil deposits are residual in nature which has been subjected to anumber of wetting and drying cycles. Owing to increase in construction activity concerningthese soils, there is a need to comprehensively understand the mechanisms involved in shearand compression behaviour. The investigation considers the laboratory testing onrepresentative field samples (both undisturbed and remolded) which have been extractedfrom the bottom of test pits of depths ranging from 1.8m to 3.5m. It may be seen that thesesoils represent wide spectrum of residual soils encountered in practice in the region. Theliquid limit values range from 27-92 and fine fraction ranging from 32-79. The basic soilproperties of the soils considered are shown in table 1.Table 1: Soil PropertiesS.No. DescriptionVinayakaNagarGayathriNagarReniguntaMuni ReddyNagarTiruchanurDepth of Sampling, m2.70 3.50 2.50 1.80 2.701 % Gravel 3.00 1.00 6.60 0.50 7.002 %Sand 46.00 43.00 63.40 57.50 48.003 % Silt + Clay 51.00 56.00 30.00 42.00 45.004 Liquid limit (%) 42.00 33.00 92.00 27.00 55.005 Plastic limit (%) 30.10 22.17 45.30 20.75 35.246 Plasticity index (%) 11.90 10.83 46.70 6.25 19.767 Void ratio (eo) 0.610 0.601 0.620 0.605 0.5508 Percent < 425µ 68 79 32 77 509Modified liquidlimit, (WL )M %28.56 26.00 29.44 21.00 27.5010 eo/eLM 1.080 0.800 0.785 0.750 0.85011 IS Classification CI CL SC SC SC12 Field density, kN/m319.58 19.85 19.62 19.18 19.5413Natural moisturecontent, %16.70 17.73 19.86 16.25 14.9214Yield stress σy inkPa80 76 67 64 80
  5. 5. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 3, May - June (2013), © IAEME1254.2 Analysis of Test DataThe test results are analysed using the effective mean normal and deviatoric stressparameters p and q as given by:32 31 σσ +=p (1)31 σσ −=q (2)Where σ1 = axial stresses on a cylindrical sample.σ3 = radial stresses on a cylindrical sample.The deviatoric strain is expressed by:)2(3231 εεε −=s (3)31 2εεε +=v (4)Where ε1 = axial strainsε3 = radial strains.For undrained tests, εv = 0 and hence εs = ε1. Axial strains were measured externally and thedeviatoric stresses were calculated from the readings of pressure controller and the currentsample area using conventional area correction.Figures 1 and 2 shows the stress-strain-pore pressure response of tropical residualsoils for two confining pressures (50kPa and 100kPa) of two soils. From the figures 1 and 2,it may be observed that the strain softening is associated with positive pore pressure. Theother soils are also following the same trend. It is well known that it is not possible to get aunique plot of q/po versus εs for tropical residual soils while it is possible in the case ofnormally consolidated clays. This may be attributed to the fact that the evolution ofcementation bond resistance and subsequent softening during deformation process is notproportional to the initial confining pressures, there by being more predominant for lowconfining pressures in comparison to the equivalent unbonded response.Effective stress paths of the two samples are presented in figures 3 and 4 for theconfining pressures tested. The other soils are also following the same trend. These stresspaths indicate that mean effective stress decreases during strain hardening and strainsoftening process. The specimens tested under different confining pressures tend to reach thecritical states corresponding to remoulded situation if cementation bonds were not present. Itturns out that critical state is approached only slowly at large strains. The results indicate thatit is the type of soil that determines the critical state parameters and not the initial state orcementation bonding. The results show that the value of stress ratio (η=q/p) upon reachingrespective peak values remains nearly constant for two confining pressures as indicated infigures 3 and 4. It may be further observed from figures 1 and 2 that the pore water pressurecontinues to be positive even in the softening region indicating that the behaviour is notsimilar to that of overconsolidated soils as is frequently reported. Strain softening associatedwith positive pore pressures is perhaps peculiar feature concerning the behaviour of tropicalresidual soils. This may be ascribed to the additional stress transfer on to the pore pressure asa consequence of debonding with progressing shearing. This stress transfer seems to occur insuch a way that the value of η remains fairly a constant with distortional strain.
  6. 6. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 3, May - June (2013), © IAEME126Fig.2 Stress-strain-pore pressure response ofGayathri nagar soilFig.4 Effective stress paths ofGayathri nagar soilFig.3 Effective stress paths ofVinayaka nagar soilFig.1 Stress-strain-pore pressure responseof Vinayaka nagar soil
  7. 7. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 3, May - June (2013), © IAEME127An examination of data obtained from experimental results of tropical residual soilsindicates that p versus εs (figures 5 and 6) and η versus εs (figures 7 and 8) relations arehyperbolic. Other soils are also following the same trend. These observations form the basisfor the formulations proposed in this paper. The two constant hyperbolic relations have beenutilized with advantage to analyze the consolidated undrained triaxial test results. Variationof stress ratio (q/p) and the mean principal stress with deviatoric strain in terms of hyperbolicrelation takes the form as)( 22 ssXba εεη+= (5)Fig.6 Mean principal stress-strainresponse of Gayathri Nagar soilFig.7 Stress ratio-strain responseof Vinayaka Nagar soilFig.8 Stress ratio-strain responseof Gayathri Nagar soilFig.5 Mean principal stress-strainresponse of Vinayaka Nagar soil
  8. 8. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 3, May - June (2013), © IAEME128)()(33 ssoXbappεε+=− (6)Where η = q/pεs = shear strainpo = Initial mean principal stressEquations 5 and 6 can be transformed into the linear form as presented below in order to beable to make them suitable for experimental verification.)( 22 ssXba εηε+= (7))()(33 sosXbappεε+=−(8)The experimental data of five soils is plotted in the form represented by equations 7and 8 and are shown in figures 9 and 10. A good straight line can be fitted to the experimentaldata between εs/η versus εs and εs/(po-p) versus εs for all the soils of selected confiningpressures. This is a good indication of the applicability of the form proposed to represent thestress-strain response of tropical residual soils.Elimination of εs in equations 5 and 6 yields:)1()(1)(2233bappbppaoo−=−−− η(9)which describes the undrained stress path of tropical residual soil.Fig.9 Transformed stress ratio-strain curvesFig.10 Transformed meanprincipal stress-strain curves
  9. 9. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 3, May - June (2013), © IAEME129For meaningful application of the relations proposed it is necessary to determine theexact nature of the parameters a2, b2, a3 and b3 in relation to the confining pressurenormalized with yield stress. Figures 11-14 shows the variation of these constants with theconfining pressure normalized with yield stress.Experimental results indicate that it is convenient to express the parameters a3 and b3in terms of initial confining pressures normalized with yield stress in the form of a powerfunction and a2 and b2 in the form of linear relationship with confining pressure normalizedwith yield stress on log scale (Equations 10 and 11).Fig.12 Variation of parameter b2 withconfining pressure normalized with yieldstressFig.13 Variation of parameter a3 withconfining pressure normalized with yield stressFig.14 Variation of parameter b3 withconfining pressure normalized with yield stressFig.11 Variation of parameter a2 withconfining pressure normalized with yieldstress
  10. 10. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 3, May - June (2013), © IAEME130488.0ln326.02 +=yopaσ(10a)481.0ln143.02 +=yopbσ(10b)28.13 109.0−=yopaσ(11a)87.13 011.0−=yopbσ(11b)The hyperbolic constants a2, b2, a3 and b3 as obtained by the equations mentionedabove are used to compute the stress-strain response. In computing the above parameters, inaddition to the experimental results, data from the literature related to tropical residual soils isalso used. The test data of reddish lateritic soil (Futai et al. 2004) which is sampled at 1mdepth and having yield strength of 100kPa. This soil is tested under different confiningpressures ranging from 25kPa to 400kPa. The experimental data is plotted in the transformedhyperbolic form with εs / η and εs / (po-p) on y-axis and deviatoric strain on x-axis and arepresented in figures 15 and 16. The computed and the observed plots of q versus εs of soilstested are presented in figures 17 and 18 respectively. The close agreement between thecomputed and experimental results seems to confirm the applicability of the hyperbolicmodel for the tropical residual soils in undrained shear. Other soils are also following thesame trend. However, these formulations may not be applicable for very low confiningpressures where the pore pressure response does not follow a hyperbolic variation with strain.Fig.15 Transformed stress ratio-straincurves of lateritic soilFig.16 Transformed mean principalstress-strain curves of lateritic soil
  11. 11. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 3, May - June (2013), © IAEME131Fig.17 Experimental and predicted stress-strain curves of Vinayaka Nagar soilFig.18 Experimental and predicted stress-strain curves of Gayathri Nagar soilFig.19 Experimental and predicted stress-strain curves of saprolitic soil
  12. 12. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 3, May - June (2013), © IAEME1324.3 Application to other experimental investigationsIt is desirable to consider the proposed mathematical form in relation to otherpublished literature in order to assess its general applicability. The test data of tropicalresidual saprolitic soil (Futai et al. 2004) which is sampled at 5m depth and having yieldstrength of 260kPa is examined in this connection. This soil is tested under differentconfining pressures ranging from 25kPa to 690kPa. The equations 10 and 11 are used topredict the stress-strain characteristics. The close agreement between predicted andexperimental values of saprolitic soil (Futai et al. 2004) is once again well demonstrated bycomparative plot shown in figure 19.5. CONCLUDING REMARKSBased on the analysis of test results of carefully planned experimental programme, thefollowing concluding remarks may be made.1. The strain softening behaviour associated with positive pore water pressures noticedin the residual soils can be captured using hyperbolic approach with appropriatemodifications.2. The combination of stress ratio (η=q/p) and mean principal stress (p) is used torepresent the non-linear stress dependent behaviour of residual soils.3. Four parameters are involved in the proposed hyperbolic model which can bedetermined from simple consolidated undrained triaxial tests and one dimensionalcompression tests.4. The model parameters are found to have functional relationship with the yield stressvalue (σy) in one dimensional oedometer compression.5. The model developed has been applied to other soil data and the applicability isevidenced from the model predictions being in close agreement with observedbehaviour.REFERENCES1. Bujang B.K. Huat, David G. Toll, Arun Prasad (2012) - Handbook of Tropical ResidualSoils Engineering- Published 24th May 2012 by CRC Press.2. Duncan J.M. and Chang C.Y. (1970) - Nonlinear analysis of stress and strain in soils.Journal of the Soil Mechanics and Foundations Division, ASCE, 1970, 96, No. SM5,1629-1653.3. Karmakar1, S., Sharma.J. and Kushwaha.R.L.(2004),” Critical state elasto-plasticconstitutive models for soil failure in tillage – A review”, Canadian biosystemsengineering, volume 46. 2004.4. M. M. Futai, M. S. S. Almeida, and W. A. Lacerda, (2004) Yield, Strength, andCritical State Behavior of a Tropical Saturated Soil, Journal Of Geotechnical AndGeoenvironmental Engineering © ASCE / November 2004, 1169 -11795. Mofiz M. and Mohammad Nurul Islam M. (2010)- Assess the Stress-Strain andInterfacial Frictional Behaviour of Nonwoven Geotextile Reinforced Residual Soils-GeoFlorida 2010: Advances in Analysis, Modeling & Design (GSP 199) © 2010ASCE.
  13. 13. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 3, May - June (2013), © IAEME1336. Nagendra Prasad, K., Srinivasa Murthy, B.R., Sitharam, T.G., and Vatsala, A. (1999)-Hyperbolic stress-strain and pore pressure response of sensitive clays – IndianGeotechnical Journal, 29 (3), 221-241.7. Sarma, M.D. & D. Sarma,D. (2008) “Prediction of Consolidation Properties ofPartially Saturated Clays” The 12thInternational Conference of InternationalAssociation for Computer Methods and Advances in Geomechanics (IACMAG) 1-6October, 2008 Goa, India8. Nagendra Prasad.K, Manohara Reddy.R, Chandra.B and Harsha Vardhan Reddy.M,“Compression Behaviour of Natural Soils”, International Journal of Civil Engineering& Technology (IJCIET), Volume 4, Issue 3, 2013, pp. 80 - 91, ISSN Print:0976 – 6308, ISSN Online: 0976 – 6316.9. Nagendra Prasad.K, Sivaramulu Naidu.D, Harsha Vardhan Reddy. M and Chandra.B,“Framework for Assessment of Shear Strength Parameters of Residual TropicalSoils”, International Journal of Civil Engineering & Technology (IJCIET), Volume 4,Issue 2, 2013, pp. 189 - 207, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316.

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