Modeling of Stress- Strain Curves of Drained Triaxial Test on Sand

Transcription

1 Amerian Journal of Alie Sienes 3 (): 8-3, 6 ISSN Siene Publiations Moeling of Stress- Strain Cures of Draine Triaxial Test on San Awa Al-Karni an Abulhafiz Alshenawy Ciil Engineering Deartment, College of Engineering, King Sau Uniersity P.O. Box 8, Riyah 4, Saui Arabia Abstrat: This aer resents a hyerboli mathematial moel to reit the omlete stress-strain ure of raine triaxial tests on uniform ense san. The moel was forme in one equation with many arameters. The main arameters that are neee to run the moel are the onfining ressure, angle of frition an the relatie ensity. The other arameters, initial an final sloes of the stress strain ure, the referene stress an the ure-shae arameter are etermine as funtions of the onfining ressure, angle of frition an the relatie ensity using best fitting ure tehnique from the exerimental tests results. Draine triaxial tests were run on lean white uniform san to utilize an erify this moel. These tests were arrie out at four leels of onfining ressure of,, 3 an 4 kpa. This moel was use to reit the stress-strain ures for raine triaxial tests on quartz san at ifferent relatie ensity using the ata of Kouner []. The moel reitions were omare with the exerimental results an showe goo agreements of the reite results with the exerimental results at all leels of alie onfining ressures an relatie ensities. Key wors: Moeling, stress, strain, water, san, triaxial INTRODUCTION Mathematial moeling of stress-strain ures of soil behaior obtaine from triaxial tests gaine great interest uring the last three eaes ue to the nee for alibration of the reent onstitutie soil moels that use in the analysis an reition of the behaior of omlex soil strutures an soil/struture interation roblems using finite elements or finite ifferene metho. An oeriew of onstitutie moels for soils was gien by Lae []. The Hyerboli Mathematial moel is one of the simler moels that an simulate the nonlinear stress-strain ure of soil. Konner [3] roose a funtional form base on hyerboli stressstrain funtion whih eeloe later by Dunan an Chang [4] an extene by Kulhawy et al. [5], but this moel is limite to the harening art of the stressstrain relationshi. A ersatile moel resente by Rihar an Abbott [6] has been use to reresent the stress-strain setrum of ifferent tyes of onrete. This moel was use by Almusallam an Alsaye [7] to ature the omlete stress-strain ures (harening an softening arts) for normal, high strength an light weight onrete teste uner arious loaing onitions. The authors hae trie ifferent emirial moels to reit the omlete stress-strain ure an onlue that some moels nee to hae two ifferent formulas to generate the omlete stress-strain ure an other moels oul not take into aount the influenes of ifferent fators influening the stress-strain ures. Also, it was foun that some moels require omliate omutations to ealuate their arameters. Howeer, the moel resente by Almusallam an Alsaye [7] was foun to hae the ability to onsier the influene of ifferent arameters affeting the stressstrain ure harateristis with only single arameter, whih is the ultimate omressie strength of the onrete, neee to run the moel. The other arameters of the moel are etermine using the bestfitting ure tehnique as a funtion of the omressie strength of the onrete. The moel was resente by Almusallam an Alsaye [7] an foun to roie goo reitions of the exerimental results for harening an softening arts. Alshenawy [8] extene the aliation of the aboe moel to reit the omlete ifferential aity ressure-aity olume hange ure of an exansion of a thik-walle hollow yliner test for both oarse an fine Ottawa sans. Base on t the exerimental results, the onfining ressure was hosen to be the inut arameter to run the moel. The other arameters were exresse as a funtion of the onfining ressure an etermine using the best-fitting ure tehnique. The omutational results of the moel were then omare with the exerimental results an showe goo agreements for both oarse an fine sans at all leels of onfining ressures. The moel was foun to be able to reit the omlete ifferential aity ressure-aity olume hange ure at any alue of onfining ressure other than use in the tests. The main objetie of this aer is to utilize this moel to reit the stress-strain ure of onsoliate raine triaxial test on granular soil. The main Corresoning Author: Awa Al-Karni, Ciil Engineering Deartment, College of Engineering, King Sau Uniersity, P.O.Box 8, Riyah 4, Saui Arabia 8

2 Am. J. Alie Si., 3 (): 8-3, 6 arameters inole are the onfining ressure, the angle of frition an the relatie ensity. The other arameters were ealuate as a funtion of these arameters using best-fitting ure tehnique. The results of the suggeste moel was omare with the exerimental results an foun to be in a goo agreement with the harening an softening arts of the ure. Proose moel of stress-strain ure: Basially, the suggeste moel onsists of one equation whih an be written for the triaxial test on soil to relate the hange in eiator ertial stress ( ) to the ertial strain ( ) in the following form: ( K K ) = + K / n n ( K K ) () + f o As shown in Fig., the arameters K, K are the initial an final sloes of the stress strain ure resetiely, the arameter f o is a referene stress an n is a ure-shae arameter gien as: ln m n = f K ln fo K K where m is a onstant an f an be gien as: f = where is the eak eiator ertial stress, () (3) is the orresoning ertial strain an an be written as: fo = K K (4) The aantage of this moel is its ability to reit the omlete stress-strain ure for both harening an softening arts. Exerimentation: A set of onsoliate raine triaxial tests was arrie out in this stuy to alibrate the roose moel. The tests were run on loal white uniform san an the grain size istribution ure is shown in Fig.. The san grains are almost roune with an aerage grain size iameter of about.5 mm. The major roerties of this san are resente in Table. Aoring to the Unifie Classifiation System, this soil is oorly grae san (SP). The exeriments were erforme using triaxial testing aaratus using onsoliate raine onition. The triaxial seimen was reare at a high relatie ensity of 95% using slit mol with 35.5 mm in iameter an 7 mm in height. The alue of the ore water ressure arameter (B) was not less than 97% for goo egree of saturation. In the exerimental rogram, the seimens were teste at four leels of onfining Deiator Stress, f o K K n Confining Pressure, Fig. : Illustration of the moel's arameters Fig. : Graation ure of the teste san... ressure of,, 3 an 4 kpa at a onstant rate of islaement of.3 m se. Ealuation of the moel arameters: The arameters that neee to be etermine in orer to exeute the moel in Eq. () are: K, K, f o,,, m an n. These arameters were obtaine base on the exerimental results as funtions of the main arameters whih inlue the onfining ressure, the angle of frition an the relatie ensity using the best-fitting ure tehnique. i. The arameter K: The initial sloe (K) of the exerimental stress-strain ures, whih reresent the initial tangent moulus of elastiity of the san, oul be exresse as a funtion of the onfining ressure ( ) by a linear equation (Fig. 3) as: 9

3 Am. J. Alie Si., 3 (): 8-3, 6 5 Fitting alues Dr Fitting alues -5 5 K (kpa) 5 K = R =.98 K (kpa) - -5 K = D r D r R = Effetie onfining ressure, (kpa) Fig. 3: Variation of K with onfining ressure at D r =95% -5 Fig. 6: Variation of K with relatie ensity at =7 kpa 6 K x 3 (kpa) Fitting alues K = 8348Dr Dr R =.976 f (kpa) Fitting alues fo = R = Fig. 4: Variation of K with relatie ensity at =7 kpa K (kpa) D r Effetie onfining ressure, (kpa) K = R =.9655 Fitting alues -35 or as a funtion of the relatie ensity aoring to the Fig. 5: Variation of K with onfining ressure at ata of Kouner [] as shown by Fig. 6, as D r =95% K = D r D r (8) Effetie onfining ressure, (kpa) Fig. 7: Variation of f o with onfining ressure at D r =95% K = (5) Figure 4 illustrates the ariation of K against the relatie ensity (D r ) using the ata of Kouner [] where K exresse as: K = 8348D 86755D (6) r Where the units of K an are in kpa. r ii. The arameter K : The final sloes (K ) of the exerimental ures an be exresse as a funtion of the onfining ressure by linear equations, as shown by Fig. 5. K an be exresse as: K = (8.5 5) (7)

4 Am. J. Alie Si., 3 (): 8-3, 6 4 Fitting alues f o (kpa) f o = D r R =.9767 Deiator stress, (ka) Tests Results m= m=3 m=5 m= D r Fig. 8: Variation of f o with relatie ensity at =7 kpa 5 Chart Title Vertial strain, Fig. : Comarison between the reite stressstrain ures at ifferent alues of m at D r =95% an =4 kpa Peak Vertial Strain, 5 5 = 5.58Dr Dr R =.9867 Tests Results Fitting Values Deiator stress (ka) (kpa) 4 3 Suggeste moel Fig. 9: Variation of eak ertial strain with relatie ensity at =7 kpa Where the units of K an D r are in kpa. iii. The arameter f o : Figure 7 shows a linear relationshi between the referene stress (f o ) an the onfining ressure whih oul be exresse as: fo = (9) or as a funtion of the relatie ensity aoring to the ata of Kouner [] as shown in Fig. 8 as f = 644.7D + 5 () o Where the units of f o an r are in kpa. i. The arameter : Aoring to the tests results of Kouner [], was foun to ary with the ariation of the relatie ensity as shown in Fig. 9 aoring to the following relationshi 5 5 axial strain(%) Fig. : Comarison between the reite an the exerimental results of the eiator stress at D r =95% 5.58Dr 46.94Dr 9.4 = + () This relationshi was use in the alibration of the moel arameter m using the results of tests that onute in this stuy at a relatie ensity of 95% an foun to gie goo results.. The arameter : Aoring to Mohr-Coulomb failure riteria, the relationshi between the eak eiator stress an the onfining ressure for san (at C=, where C is the soil ohesion) is gien by sinφ = sinφ ()

5 Am. J. Alie Si., 3 (): 8-3, 6 Deiator stress, kpa Test results from Koerner (97) Suggeste moel results Vertial strain (a) D r =.3 Deiator stress, KPa Test results from Koerner (97) Suggeste moel results Vertial strain (b) D r =38.5 Deiator stress, kpa Test results from Koerner (97) Suggeste moel results....3 Vertial strain () D r =59.3 Deiatori stress, kpa Test results from Koerner (97) 3 Suggeste moel results Vertial Strain () D r =.745 Deiator stress, kpa Test results from Koerner (97) Proose moel results Vertial strain (e) Dr=.86 Fig. : Comarison between the reite an the exerimental results of the eiator stress at onfining ressure of 7 kpa Where φ is the angle of frition of the san. i. The arameter m: The arameter m will be ealuate after the alibration of the moel. ii. The arameter n: The arameters n an be alulate from Eq. (). Calibration of the roose moel: The alibration of the moel eens on the etermination of the arameter m sine the other arameters are ealuate iretly from the tests results. This was arrie out by testing the moel for ifferent alues of m as shown by Fig. using the tests results at onfining ressure of 4 kpa, the best alue was foun to be when m=7.

6 It was foun that the best alue that is aliable to all the onfining stress leels with marginal eiation from the test results when m=3 as shown by Fig.. Hene a alue of m of m=3 was use to erify the moel. Verifiation of the roose moel: To erify the roose moel, it was use to reit the stress-strain ures for quartz san teste using onsoliate raine triaxial tests that were ublishe by Kouner []. The reition of the stress-strain ures at ifferent relatie ensities ia the roose moel is shown Fig.. This figure shows that the reite ures by the suggeste moel are in a goo agreement with the exerimental ures in both harening an softening arts for all leels of the relatie ensity. In the ase of ense san or high onfining ressure where the stress-strain ure has a well efine eak alue an at high strain alue of about 5%, the reite eiator stress alues beome lower than those of the exerimental results as the strain inreases where the exerimental alues stay almost onstant as the strain inreases. Howeer, in most of the ratie roblems, we may not nee to go beyon a ertial strain of 5%. This onlues that this simle moel is effiient in reiting the stress-strain ure of the sany soil at any alue of onfining ressure an relatie ensity using the onsoliate raine triaxial test. CONCLUSION In this aer, a simle hyerboli mathematial moel is roose to generate the omlete eiator ertial stress s. ertial strain ure of the onsoliate raine triaxial test on san. The moel has the aantage of onsiering the influene of ifferent fators affeting the stress-strain ure harateristis inluing the onfining ressure, angle of frition an relatie ensity. The moel was alibrate an erifie using two sets of ata of onsoliate raine triaxial tests at ifferent leels of onfining ressures an relatie ensity. The first set of ata onsists of four tests on ense sany samles that were teste at relatie ensity of 95% an at four leels of onfining ressure of,, 3 an 4 kpa. The other set onsists of fie tests that were ublishe by Kouner [] at onfining ressure of 7 kpa an ifferent relatie ensities of.3, 38.5, 59.3, 74.5 an 8.6. The three main arameters that were use to run the moel inlue the onfining ressure, the angle of frition an the relatie ensity. The other arameters of the moel were etermine using the best fitting ure tehnique as funtion of these main arameters. The moel reition ures were omare with the exerimental ones an foun to roie goo agreements at all the leels of onfining ressure an relatie ensity for the harening an softening arts of the stress-strain ure. Am. J. Alie Si., 3 (): 8-3, 6 3 List of Symbols D r = relatie ensity E = oi ratio f = the referene stress f = onstant Gs = seifi graity K = the initial sloe of the stress strain ure K = the final sloe of the stress strain ure n = a ure-shae arameter m = onstant φ =angle of frition γ =ry unit weight = eiator ertial stress = the eak eiator ertial stress = the ertial strain =the eak ertial strain = onstant REFERENCES. Kouner, R.M., 97. Effet of artile harateristis on soil strength. J. Soil Mehanis an Founation Diision, ASCE, 96 (SM4): Lae, P.V., 5. Oeriew of onstitutie moels for soils. Geotehnial Seial Publiation No. 8: Soil Constitutie Moel Ealuation, Seletion an Calibration. ASCE, : Konner, R.L., 963. Hyerboli stress-strain resonse: Cohesie soils. J. Soil Mehanis an Founation Diision, ASCE, 89(SM): Dunan, J.M. an C.Y. Chang, 97. Nonlinear analysis of stress an strain in soils. J. Soil Mehanis an Founation Diision, ASCE, 96 (SM5): Kulhawy, F.H. an J.M. Dunan, 97. Stresses an moements of Oroille am. J. Soil Mehanis an Founation Diision, ASCE, 98 (SM7): Rihar, R.M. an B.J. Abbott, 975. Versatile elasti-lasti stress-strain formula. J. Engg. Meh., ASCE, : Almusallam, T.H. an S.H. Alsaye, 995. Stressstrain relationshi of normal, high-strength an lightweight onrete. Mag. Conrete Res., 47: Alshenawy, A.,. Mathematial moeling of aity exansion test on ry san. Emirates J. Engg. Res., Unite Arab Emirates Uniersity, 7: 7-4.