ESTIMATION OF HYPERBOLIC STRESS-STRAIN PARAMETERS FOR GYPSEOUS SOILS

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1 ESTIMATION OF HYPERBOLIC STRESS-STRAIN PARAMETERS FOR GYPSEOUS SOILS Dr. Ahmed H. Abdul-kareem Lecurer Dep. of Civil Engineering Universiy of Anbar Ahmed Helal Assisan Lecurer Dep. of Civil Engineering Universiy of Anbar ABSTRACT: The hyperbolic model is a simple sress-srain relaionship based on he concep of incremenally nonlinear elasic behavior. The hyperbolic sress-srain relaionship was developed for use in finie elemen analysis of sresses and movemens in earh masses. To esimae hyperbolic parameer values required for nonlinear finie elemen analysis, daa used from he riaxial compression ess for he gypseous soils exposed o he effec of drying and weing cycles carried ou by (Mohammed, 1993). From hese daa, he parameers (C, φ, K, n, R f ), which are required by Duncan-Chang model, 197 can obained for analyses of dams, excavaions and various ypes of soil-srucure ineracion problems. In addiion, i can be found ha he primary loading modulus, K, he exponen number, n, and he failure raio, R f, have random values during reweing cycles for CU and UU riaxial compression ess. الخلاصة: ان النم وذج ذو المقط ع الزاي د ه و عب ارة ع ن علاق ة ب سيطة ب ين الاجه اد والانفع ال تعتم د عل ى مفه وم الت صرف التزاي دي اللاخط ي الم رن. ه ذه العلاق ة وظف ت آ ي ت ستخدم ف ي طريق ة تحلي ل العناص ر المح ددة للاجهادات وحرآات الكتل الترابي ة. لتخم ين ق يم متغي رات المقط ع الزاي د المطلوب ة للتحلي ل اللاخط ي للعناص ر المح ددة اخ ذت معلوم ات م ن فحوص ات الان ضغاط ثلاث ي المح اور للت رب الجب سية المعرض ة لت ا ثير دورات التجفي ف والترطي ب والمنج زة م ن قب ل (1993.(Mohammed, وم ن ه ذه المعلوم ات ت م الح صول عل ى المتغي رات ) f (C, φ, K, n, R المطلوب ة لنم وذج model, 197 Duncan-Chang لتحلي ل ال سدود والحفريات والعديد من أنواع مشاآل التداخل بين التربة والمنشا. Creaed by Neevia Personal Converer 1 rial version hp://

2 بالا ضافة الى ذلك وجد ان معامل التحميل الابتداي ي (K) العدد التكاملي( n ) ونسبة الفشل ) f R) له ا ق يم عشواي ية خلال دورات اعادة الترطيب ضمن فحوص ات الان ضغاط ثلاث ي المح اور الغي ر مب زول بنوعي ة غي ر المنضم (UU) والمنضم.(CU) Inroducion: The hyperbolic sress-srain relaionship was developed for use in finie elemen analyses of sresses and movemens in earh masses. In he en years since is developmen, he model has been used in analyses of a large number of dams, braced and open excavaions, and a variey of ypes of soil-srucure ineracion problems. In is original form, as described by Duncan and Chang (197)(1), he hyperbolic model employed angen values of Young's modulus (E ), which varied wih he magniudes of he sresses, and consan values of Poisson's raio. The Young's modulus relaionships remain he same as described by Duncan and Chang (197)(1). The principal advanage of he hyperbolic model is is generaliy. I can be used o represen he sress-srain behavior of soil ranging from clays and sils hrough sands, gravels and rockfills. I can be used for parly sauraed or fully sauraed soils, and for eiher drained or undrained loading condiions in compaced earh maerial or naurally- occurring soils. Experience wih reaing hese various ypes of problems, and he accumulaed background of sress-srain parameer values for a wide variey of soils, provide a useful base for furher applicaions. To esimae hyperbolic sress-srain parameers required for finie elemen analysis, he daa colleced ino riaxial compression ess of gypseous soils carried ou by (Mohammed, 1993)(2). Sress-Srain Relaionships: The hyperbolic sress-srain relaionships are developed for incremenal analyses of soil deformaions where nonlinear behavior is modeled by a series of linear incremens. The relaionship beween sress and srain is assumed o be governed by he generalized Hook's Law of elasic deformaions. For plane srain condiions his relaionship may be expressed as follows (3): Creaed by Neevia Personal Converer 2 rial version hp://

3 σ x σ y τ xy = (1 ν ) ν ε E x ν + (1 ν ) ε ( 1 ν y )(1 ν ).(1) (1 2ν )/ 2 γ xy where: σ x, σ y and τ xy = are he incremens of sress during a sep of analysis. ε x, ε y and γ xy = are he corresponding incremens of srain. E = is he angen value of Young's modulus. υ = is he angen value of Poisson's raio. The value of boh E and υ in each elemen change during each incremen of loading in accordance wih he calculaed sresses in ha elemen, in order o accoun for hree imporan characerisics of he sress-srain behavior of soil, namely non lineariy, sress-dependency, and inelasiciy. The procedures used o accoun for hese characerisics are described in he following paragraphs. Nonlinear Sress-Srain Curves Represened By Hyperbolas: Kondner (1963)(4) showed ha he sress-srain curves for many of soils, boh clay and sand, can be approximaed reasonably accurae by hyperbolas like he one shown in figure (1-A). This hyperbola can be represened by an equaion of he form: ( σ 1 ε σ ) = 3 1 ε + E ( σ σ ) i 1 3 ul (2) These hyperbolas have wo characerisics which make heir use convenien: 1. The parameers which appear in he hyperbolic equaion have physical significance. E i is he iniial angen modulus or iniial slope of he sresssrain curve, and (σ 1 - σ 3 ) ul is he asympoic value of sress difference which is relaed closely o he srengh of he soil. The value of (σ 1 - σ 3 ) ul is always greaer han he compressive srengh of he soils. 2. The values of E i and (σ 1 - σ 3 ) ul for a given sress-srain curve can be deermined easily. If he hyperbolic equaion is ransformed as shown in figure (1-B), i represens a linear relaionship beween [ε/ (σ 1 - σ 3 )] and ε. Creaed by Neevia Personal Converer 3 rial version hp://

4 Thus, o deermine he bes-fi hyperbola for he sress-srain curve, values of [ε/ (σ 1 - σ 3 )] calculaed from he es daa are ploed agains ε. The bes-fi sraigh line on his ransformed plo corresponds o he bes-fi hyperbola on he sresssrain plo. This research, daa from he riaxial compression ess of gypseous soils exposed o reweing cycles presened by researcher (Mohammed, 1993)(2) is used o re-plo he sress-srain relaions o calculae he required nonlineariy parameers. Figures (2) o (7) show sress-srain relaionships for riaxial compression ess of gypseous soils exposed o reweing cycles carried ou by he researcher (Mohammed, 1993)(2). The sress-srain relaions obained in figures (2) o (7) have been processed as shown in figures (8) o (13). For each sress-srain curve, he values of [ε/ (σ 1 - σ 3 )] are calculaed for each single curve, and hen, [ε/ (σ 1 - σ 3 )] are ploed agains ε. For each of hese ploed curves he values of (a) and (b) are obained, where (a) is he value of he (y-axis) inercep and (b) is he slope of he curve as explained in figure (1-B). as:- The value of he iniial modulus of elasiciy (E i ) is obained as: E 1 i a =...(3) and he value of asympoic ulimae deviaor sress (σ 1 - σ 3 ) ul is obained ( 1 σ ) 1 3 ul = b σ...(4) The failure raio (R f ) is evaluaed as: R f ( σ σ ) 1 3 f ( σ σ ) 1 3 ul =....(5) The variaion of (σ 1 - σ 3 ) f wih σ 3 is represened by he familiar More- Coulomb srengh relaionship, which can be expressed as follows: Mohr-Coulomb failure crierion, hen: 2ccosφ+ 2σ sinφ ( σ σ ) = f 1 sinφ.. (6) In which c and φ are he cohesion inercep and he fricion angle. Creaed by Neevia Personal Converer 4 rial version hp://

5 Sress-Dependen Sress-Srain Behavior: For all soils, excep fully sauraed soils esed under unconsolidaedundrained condiions, an increase in confining pressure will resul in a seeper sress-srain curve and a higher srengh and he values of E i and (σ 1 -σ 3 ) ul herefore increase wih increasing confining pressure. This sress-dependency is aken ino accoun by using empirical equaions o represen he variaions of E i and (σ 1 -σ 3 ) ul wih confining pressure (1),( 3). The variaion of E i wih σ 3 is represened by an equaion suggesed o be Janbu (1963)(5) of he form:- E i n σ = K P 3 a P a.(7) The variaion of E i wih σ 3 corresponding o his equaion is shown in figure (14). The parameer K in equaion (7) is he modulus number, and n is he modulus exponen. Boh facors are dimensionless numbers. P a is amospheric pressure, inroduced ino he equaion o make conversion from one sysem of unis o anoher more convenien sysem. The values of K and n are he same for any sysem of unis, and he unis of E i are he same as he unis of P a. Daa of figures (8) o (13) are used o plo he relaions beween he values of E i and σ 3 for each series of ess as shown in figures (15) o (2). The value of K is obained by aking he value of iniial modulus corresponding o one uni of confining pressure, while n is evaluaed o be he slope of he ( E i - σ 3 ) relaion, which is a sraigh line on a log-log scale. Relaionship Beween E and The Sresses: The insananeous slope of he sress-srain curve is he angen modulus, E. By differeniaing equaion (2) wih respec o ε and subsiuing he expressions of equaions (5), (6), (7) ino he resuling expression for E, he following equaion can be derived: E R (1 sinφ)( σ σ ) f 1 σ = 1 KP 3 2ccosφ+ 2σ sinφ a P 3 a n..(8) Creaed by Neevia Personal Converer 5 rial version hp://

6 This equaion can be used o calculae he appropriae value of angen modulus for any sress condiions [σ 3 and (σ 1 - σ 3 )] if he values of he parameers K, n, c, φ, and R f are known. Resuls And Discussion of Hyperbolic Sress-Srain Parameers: This secion includes he resuls of hyperbolic sress-srain parameers exraced from he riaxial compression ess of differen gypseous soils exposed o he effec of weing and drying cycles and summarized in he ables (1) and (2). The variaion of he Duncan-Chang model parameers wih weing and drying cycles is presened in figures (21) o (23). From figures (21) o (23), i can be observed ha he primary loading modulus (K), he exponen number (n) and failure raio (R f ) have random values during reweing cycles for boh ess. Conclusions: The hyperbolic model is a simple sress-srain relaionship based on he concep of incremenally nonlinear elasic behavior. I is applicable o virually any ype of soil and o drained or undrained condiions. Experience in applying he hyperbolic model o analyses of dams, excavaions and various ypes of soilsrucure ineracion problems has shown ha i is useful for calculaion movemens in sable earh masses, and is no suiable for predicing insabiliy or collapse loads. Like any heory hypohesis of soil behavior, is successful applicaion requires he exercise of engineering judgmen. In addiion, i can be concluded ha he primary loading modulus, K, he exponen number, n, and he failure raio, R f, have random values during reweing cycles for boh ess. Creaed by Neevia Personal Converer 6 rial version hp://

7 Table (1): Hyperbolic model parameers for gypseous soil samples (he presen sudy). Disurbed Gypseous Soil Parameers (1) (15) (3) (45) (6) K n R f K ur c (kpa) φ (degree) γ kn/m Soil Locaion Al-Sherqa area Classified Soil Clayey sandy sil Tes Type UU-riaxial es Reference Mohammed, 1993 Table (2): Hyperbolic model parameers for gypseous soil samples (he presen sudy). Disurbed Gypseous Soil Parameers (1) (15) (3) (45) (6) K n R f K ur c (kpa) φ (degree) γ kn/m Soil Locaion Al-Sherqa area Classified Soil Clayey sandy sil Tes Type CU-riaxial es Reference Mohammed, 1993 Creaed by Neevia Personal Converer 7 rial version hp://

8 A- Real B-Transformed Figure (1): Hyperbolic represenaion of a sress-srain curve (afer Kondner, 1963) Deviaor sress (kpa) Gypseous Soil - ( 1 ) Confining Pressure= kpa Deviaor sress (kpa) Gypseous Soil - ( 15 ) Confining Pressure= kpa Figure (2): Sress-srain relaionship for CU- riaxial esing (Mohammed, 1993). Figure (3): Sress-srain relaionship for CU- riaxial esing (Mohammed, 1993). Creaed by Neevia Personal Converer 8 rial version hp://

9 Deviaor sress (kpa) Gypseous Soil - ( 3 ) Confining Pressure= kpa Figure (4): Sress-srain relaionship for CU- riaxial esing (Mohammed, 1993). Deviaor sress (kpa) Gypseous Soil - ( 3 ) Confining Pressure= kpa Figure (5): Sress-srain relaionship for UU- riaxial esing (Mohammed, 1993) Deviaor sress (kpa) Deviaor sress (kpa) Gypseous Soil - ( 45 ) Confining Pressure= kpa Figure (6): Sress-srain relaionship for UU- riaxial esing (Mohammed, 1993). 2 Gypseous Soil - ( 6 ) Confining Pressure= kpa Figure (7): Sress-srain relaionship for UU- riaxial esing (Mohammed, 1993). Creaed by Neevia Personal Converer 9 rial version hp://

10 .3.6 Srain / Deviaor sress ( x.1).2.1 Gypseous Soil - ( 1 ) Confining Pressure= kpa Srain / Deviaor sress ( x.1) Gypseous Soil - ( 15 ) Confining Pressure= kpa.1. Figure (8): Srain / Sress raio vs. srain for CU- riaxial esing (Mohammed, 1993).. Figure (9): Srain / Sress raio vs. srain for CU- riaxial esing (Mohammed, 1993)..3.6 Srain / Deviaor sress ( x.1).2.1 Gypseous Soil - ( 3 ) Confining Pressure= kpa Srain / Deviaor sress ( x.1) Gypseous Soil - ( 3 ) Confining Pressure= kpa.1. Figure (1): Srain / Sress raio vs. srain for CU- riaxial esing (Mohammed, 1993).. Figure (11): Srain / Sress raio vs. srain for UU- riaxial esing (Mohammed, 1993). Creaed by Neevia Personal Converer 1 rial version hp://

11 .6.6 Srain / Deviaor sress ( x.1) Gypseous Soil - ( 45 ) Confining Pressure= kpa Srain / Deviaor sress ( x.1) Gypseous Soil - ( 6 ) Confining Pressure= kpa.1.1. Figure (12): Srain / Sress raio vs. srain for UU- riaxial esing (Mohammed, 1993).. Figure (13): Srain / Sress raio vs. srain for UU- riaxial esing (Mohammed, 1993). Figure (14): Variaion of iniial Tangen modulus wih confining Pressure (afer Duncan, 1981). Creaed by Neevia Personal Converer 11 rial version hp://

12 ( 1) ( 15 ) Iniial Modulus (kpa) 1 Iniial Modulus (kpa) Confining Pressure (kpa) Confining Pressure (kpa) Figure (15): Iniial modulus vs. confining pressure for CU-es (Mohammed, 1993). Figure (16): Iniial modulus vs. confining pressure for CU-es (Mohammed, 1993). ( 3 ) ( 3 ) Iniial Modulus (kpa) 1 Iniial Modulus (kpa) Confining Pressure (kpa) Figure (17): Iniial modulus vs. confining pressure for CU-es (Mohammed, 1993) Confining Pressure (kpa) Figure (18): Iniial modulus vs. confining pressure for UU-es (Mohammed, 1993). Creaed by Neevia Personal Converer 12 rial version hp://

13 ( 45 ) ( 6 ) Iniial Modulus (kpa) 1 Iniial Modulus (kpa) Confining Pressure (kpa) Figure (19): Iniial modulus vs. confining pressure for UU-es (Mohammed, 1993) Confining Pressure (kpa) Figure (2): Iniial modulus vs. confining pressure for UU-es (Mohammed, 1993) UU-Tes ( Mohammed, 1993) CU-Tes (Mohammed,1993) 2.5 UU-Tes ( Mohammed, 1993) CU-Tes (Mohammed,1993) 15 Modulus Number ( K) Exponen Number ( n) Weing and Drying s Weing and Drying s Figure (21): Variaion of primary loading modulus number wih reweing cycles. Figure (22): Variaion of exponen number wih reweing cycles. Creaed by Neevia Personal Converer 13 rial version hp://

14 UU-Tes ( Mohammed, 1993) CU-Tes (Mohammed,1993) 1. Failure Raio ( Rf) Weing and Drying s Figure (23): Variaion of failure raio wih reweing cycles. REFERENCES 1- Duncan J., and Chang, C., (197): Non-Linear analysis of Sress and Srain in Soil J. S. M. F. E. A. S. C. E., Vol.96 Sm5. 2- Mohammed, R.K. (1993), Effec of Weing and Drying on Engineering Characerisics of Gypseous Soil, M.Sc. Thesis, Building and Consrucion Dep., Collage of Engineering Universiy of Technology, Baghdad. 3- Wong, K.S. and Duncan J. (1974): Hyperbolic Sress-Srain Parameer for Non-Linear Finie Elemen Analysis of Sress and Movemens in Soil Masses,Repor No.TE74-3, Universiy of California, Berkely, July. 4- Kondner, R. (1963): Hyperbolic Sress-Sraim Response of Cohesive Soils, J. S. M. F. D. ASCE, Vol.89, SM1. 5- Janbu, N. (1963): Soil compressibiliy as Deermined by Oedomeer and Triaxial Tess, European Conference on Soil Mechanics and Foundaion Engineering, Wissbaden, Germany, Vol.1, PP Duncan, J. (1981): Hyperpolic Sress-Srain Relaion Ship, Proc. of Workshop on Limi Equilibrium, Plasiciy and Generalized Sress-Srain in Geoechnical Eng. ASCE. Creaed by Neevia Personal Converer 14 rial version hp://

15 Noaions: A Hyperbolic consan for sress-srain relaionship. B Hyperbolic consan for sress-srain relaionship. C Cohesion. UU Unconsolidaed undrained (riaxial es). CU Consolidaed undrained (riaxial es) E i Iniial angen modulus. E Tangenial modulus. K Modulus number. n Exponen deermining rae of variaion of E i wih σ 3. R f Failure raio. P a amospheric pressure. ε Srain. γ Uni weigh. σ 1, σ 3 Major and minor principal sresses φ Angle of shear resisance (inernal fricion σ 1 -σ 3 Deviaor sress (σ 1 -σ 3 ) f Deviaor sress a failure (σ 1 -σ 3 ) ul Asympoic value of deviaor sress Creaed by Neevia Personal Converer 15 rial version hp://