Comparison of Radial Consolidation Behavior of Clay under Three Types of Cyclic Loading

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1 Civil Engineering Infrastrutures Journal, 51(1): 17 33, June 018 Print ISSN: 3-093; Online ISSN: DOI: /eij Comparison of Radial Consolidation Behavior of Clay under Three Types of Cyli Loading Amiri, A. 1*, Toufigh, M.M., Sadeghi Janat Abadi, S. 3 and Toufigh, V. 4 1 M.S., Department of Civil Engineering, Shahid Bahonar University, Kerman, Iran. Professor, Department of Civil Engineering, Shahid Bahonar University, Kerman, Iran. 3 M.S. Student, Department of Civil Engineering, Shahid Bahonar University, Kerman, Iran. 4 Assistant Professor, Department of Civil Engineering, Graduate University of Advaned Tehnology, Kerman, Iran. Reeived: 3 Apr. 017; Revised: 31 De. 017; Aepted: 01 Jan. 018 ABSTRACT: Vertial drains and stone olumns whih have been used in infrastruture onstrution for highways, ports, oastal regions, et., provide signifiant benefits for improving soil harateristis suh as reduing the drainage length and aelerating the onsolidation proess. So the investigation of the radial onsolidation is inevitable. Soils may be subjeted to yli loading suh as silos, tanks, et. This paper presents semi-analytial solutions for radial onsolidation and investigates the onsolidation behavior under three types of yli loading. Consolidation under yli loads was alulated using the superimposition rule. Barron (1948) and Olson (1977) have presented theories for alulating radial onsolidation under stati and ramp load respetively. In this study, by using a set of ontinuous stati loads or a series of infinite ramp loads, with alternatively positive and negative signs, we have extended these theories for retangular, triangular and trapezoidal yli loads. The obtained analyti results demonstrate that the average degree of onsolidation at the steady state depends on the integral of the load-time urve for eah yle and it inreases with inrease of the integral and the results indiate that hange in yle period of time does not effet on the time of getting steady state. Radial and vertial onsolidation under retangular yli loading have also ompared and the effet of the distane between vertial drains on the time of getting steady state have investigated. Keywords: Radial Consolidation, Retangular Cyli Loading, Superimposition Rule, Trapezoidal Cyli Loading, Triangular Cyli Loading, Vertial Drains. INTRODUCTION For onstrution of a struture on very thik saturated lay layers, the dissipation of exess pore water pressure requires a long period of time. In suh ases, the damage aused by differential settlements an be prevented by * Corresponding author amin.amiri1371@yahoo.om 17 allowing a portion of the settlement to take plae before the beginning of onstrution. Also, to aelerate the onsolidation proess, the useful tehnique of building sand drains an be used. To analyze the behavior of vertial drains, the unit ell theory was proposed by Barron

2 Amiri, A. et al. (1948) and Rihart (1957). Yoshikuni and Nakanodo (1974) theoretially treated the onsolidation proess by the vertial drain method, taking the well resistane into onsideration. Hansbo et al. (1981) proposed a theory that inorporates important parameters suh as vertial disharge apaity, remoulding effets during installation, and filter resistane. Many more studies have been published (Lu and Xie, 011; Walker, 011; Deng et al., 013; Covo- Torres et al., 015; Lu et al., 015). In pratial geotehnial engineering, soils beneath many strutures, suh as oil and water tanks, highway embankments, oean banks, et., undergo yli loading (Ying- Chun and Kang-He, 005). The disrepanies between the settlements obtained from theoretial preditions based on the theory of onsolidation under stati loads and the settlements obtained from field measurements have required onsideration of onsolidation under yli loadings (Wilson and Elgohary, 1974). Sine the pioneering work of Shiffman (1958), the hallenge of deriving an analytial solution for investigation onsolidation settlement of soils under time-dependent loading has aptured the attention of researhers. Shiffman studied onsolidation settlements due to the timedependent loading by dividing the linear loading into a series of step loads and used the rule of superimposing. Wilson and Elgohary (1974) proposed an analytial solution for the progress of onsolidation of elasti lays subjeted to retangular yli loading. Terzaghi s onventional theory extended by Olson (1977) to over the one dimensional onsolidation under ramp loading and many various solutions have been proposed for onsolidation of soils under yli loading based on different assumptions and onsiderations. Razouki and Shanz (011) presented a study of the one-dimensional onsolidation proess under haversine repeated loading with and without rest period. The analysis was arried out using a hybrid oupled, analytial and numerial impliit finite differene tehnique. Also Razouki et al (013) have presented an exat analytial solution of the nonhomogeneous partial differential equation governing the onventional one-dimensional onsolidation under haversine repeated loading. Lo et al (016) have presented a losed form solution for one-dimensional onsolidation in unsaturated soils under yli loading. The pore water and air pressures along with the total settlement was derived by employing a Fourier series representation in the spatial domain and a Laplae transformation in the time domain. Consolidation with vertial drains under time-dependent loads have also studied before. Lei et al. (016) have derived an analytial solution to predit onsolidation with vertial drains under impeded drainage boundary onditions and multi-ramp surharge loading. And many other theories have studied onsolidation behavior of lay under yli loading (Ouria et al., 013; Ni et al., 013; Abbaspour, 014; Speirs et al., 014). To make lear the possible engineering appliation of eah loading type, we an onsider a silo that is full in odd months of a year and it is empty in even months, in this situation the soil under the silo has been subjeted to retangular yli loading. The soil under a silo that is filled in one month onstantly and is getting emptied in other month, is subjeted to triangular yli loading. If there is a gap in time between the end of filling the silo and the beginning of emptying, the type of applied yli load is trapezoidal. This paper presents a semi-analytial solution for radial onsolidation under yli loading. This method has also been used by Toufigh and Ouria (009). In that researh, eah full yle of loading was replaed by a pair of stati loads with different signs. Based on Terzaghi s theory, the degree of 18

3 Civil Engineering Infrastrutures Journal, 51(1): 17 33, June 018 onsolidation was alulated for eah stati load and the results were superimposed. MATERIALS AND METHODS In this researh, a method based on a superimposition rule is employed to alulate the pore water pressure and the degree of onsolidation for radial onsolidation of lay under retangular, triangular and trapezoidal yli loading. It is assumed that the oeffiient of onsolidation vr is onstant during the onsolidation and this study is based on elasti onsolidation theory. Also, exess pore water pressure an dissipate only horizontally. Based on the previous setion, radial onsolidation of lay under onstant and ramp loading have been studied before and in the presented method, by using a set of ontinuous stati loads or a series of infinite ramp loads, with alternatively positive and negative signs, we an extend these theories for retangular, triangular and trapezoidal yli loading as shown in Figure 1. It is assumed that the full yle period for the three types of yli loading mentioned above, are equal (t). Other properties of these load ases are shown in Figure 1. Radial Consolidation under Retangular Cyli Loading Figure 1a shows a retangular yli loading system whih is adapted in figure 1b to a set of ontinuous stati loads. As an be seen, eah full yle of retangular yli load was replaed by a pair of stati loads with positive and negative signs. Finally, based on Barron s equation (Barron, 1948) and superimposition rule, the average degree of onsolidation at the end of n half yles of the over onsolidated lay layer, an be alulated by the following equation: U hn n n i ( 1) ( 1) U( Ti ) (1) i1 where T i = i T. For alulating Uhn, U(Ti) is required. Based on Barron s equation, U(Ti) an be expressed as: 8Ti UT ( i) 1exp Fn ( ) where n 3n 1 F( n) ln( n) n 1 4n re n r w () (3) (4) Radial Consolidation under Triangular Cyli Loading Figure 1 shows a triangular yli loading system whih is adapted in Figure 1d to a series of ramp loads. Eah full yle of triangular yli load was replaed by a pair of ramp loads with positive and negative signs. The degree of onsolidation an be defined based on Olson s equations. As in the previous setion, based on Olson s equations and superimposition rule, the average degree of onsolidation at the end of n half yles of the over onsolidated lay layer, an be alulated by Eq. (1). Based on Olson s equation in radial onsolidation, U(Ti) an be expressed as: For Ti < 0.5 T: U r Ti 1 exp AT. i A (5) T For Ti 0.5 T: U r where AT. 1 exp 1 exp AT. i AT. (6) 19

4 Amiri, A. et al. 0

5 Civil Engineering Infrastrutures Journal, 51(1): 17 33, June 018 Fig. 1. The three types of yli loading adapted by the superimposition rule Cvrti Ti re (7) Cvrt T r (8) e A (9) Fn ( ) Radial Consolidation under Trapezoidal Cyli Loading This ase is similar to the previous setion and we use Olson s equation and superimposition rule to alulate trapezoidal yli loading (Figures 1e and 1f). Average degree of onsolidation at the end of N yles of the over onsolidated lay layer, an be alulated by the following equation: U N N i 1 3( N i) 1 U(( N i 1)T ) U(( ) T) 3 (10) For this ase Olson s equation (Olson 1977) for radial onsolidation, U(Ti) has been modified into the following equations: T ForTi : 3 U r 3 3T i 1 exp AT. i A T T ForTi : 3 U r 3 AT. 1 exp 1 exp AT. AT. 3 i (11) (1) 1

6 Amiri, A. et al. APPLICATION OF THE PRESENTED METHODS The proedure of using the presented methods for alulating radial onsolidation of lay is desribed in the following example. Radial onsolidation alulation of OC lay layer with vr = m /day, rw = 0.38 m, re = 1.5 m and t = 30 days is examined. All three types of yli loading applied to the lay layer are shown in figure1. In order to alulate the average degree of onsolidation, the time fator for a full yle and other required parameters are alulated: T t vr re (1.5) re 1.5 n 4 r 0.38 w n 3n 1 F ( n) ln( n) n 1 4n 4 3 (4 ) (4 ) ln(4) A.6869 Fn ( ) Average degree of onsolidation at the end of the first half yle: a) For retangular yli loading (by using Eq. (1)): Uh 1 U( T1) U(0.5 T ) U(0.5T) would be obtained from Eq. (): 4T Uh 1 U(0.5 T ) 1 exp Fn ( ) b) For triangular yli loading (by using Eq. (1)): Uh 1 U( T1) U(0.5 T ) U (0.5T) would be obtained from Eq. (6): U h1 1 A. T A. T exp 1 e xp AT. Average degree of onsolidation at the end of the first full yle: a) For retangular yli loading (by using Eq. (1)): U 1 Uh U( T ) U(0.5 T ) As presented previously, the value of U (0.5T) is alulated. U (T) an be alulated by using Eq. (): 8T UT ( ) 1 exp Fn ( ) So, U1 an be alulated: U b) For triangular yli loading (by using Eq. (1)): U 1 Uh U( T ) U(0.5 T ) The value of U(0.5T) as previous setion is alulated. U(T) an be alulated by using Eq. (6): UT ( ) 1 AT. AT. exp 1 ex p. AT So, U1 an be alulated: U ) For trapezoidal yli loading (by using Eq. (10)): U 1 U( T ) U( T / 3) From Eq. 1: UT ( ) 1 3 AT. exp 1 exp AT AT. 3 UT ( / 3) 1 3 A. T A. T exp 1 exp AT. 3 3 So, U1 an be alulated: U Above proedure an be repeated for the next yles for alulating the degree of onsolidation. The results of the alulations are shown in Table 1. 5

7 Civil Engineering Infrastrutures Journal, 51(1): 17 33, June 018 Table 1. The results of presented example N Time UN (day) Retangular Cyli Loading Triangular Cyli Loading Trapezoidal Cyli Loading RESULTS AND DISCUSSION Figure illustrates average degree of onsolidation for the above mentioned yli loadings. The soil properties and geometri harateristis are the same as the previous setion. Load harateristis are also the same as appliation exept the full yle period. From Figure it may be seen that for radial onsolidation under retangular and triangular yli loading (with equal integral of load-time urve for eah yle) with equal yle period, the average of degree of onsolidation are equal at the steady state. For radial onsolidation under trapezoidal yli loading and with yle period equal to above states, the average of degree of onsolidation is more than them at the steady state (beause the integral of load-time urve for trapezoidal yli loading for eah yle is more than retangular and triangular yli loading). 3

8 Amiri, A. et al. Fig.. The average degree of onsolidation: a) T = 10 days, b) T = 30 days The degree of onsolidation of soil under trapezoidal yli loading in radial onsolidation is less than the retangular yli loading from the beginning and beomes more than it after a period of time. The degree of onsolidation under triangular yli loading in radial onsolidation is less than the loading states mentioned above. The time of getting to the steady state in radial onsolidation of soil under triangular and trapezoidal yli loading are approximately equal and more than if under retangular yli loading. Figures 3-5 were plotted to investigate the influene of type of soil on radial onsolidation under yli loading. Geometri harateristis are the same mentioned before and soil properties are shown in Table. Soil type Typial value (m /s) 10-4 Table. Soil properties v kh/kv Adopted Value of v Referene Experimental Value Referene Vertial Consolidation (m /day) 10-4 Radial Consolidation (m /day) 10-4 Soft blue lay Wallae and Otto (1964) 1.5 Basett and Brodie (1961) Boston blue lay Organi silt 40±0-10 Ladd and Lusher (1965) Lowe, Zaheo, and Feldman (1964) Haley and Aldrih (1969) Tsien (1955)

9 Civil Engineering Infrastrutures Journal, 51(1): 17 33, June 018 Fig. 3. The average degree of onsolidation under retangular yli loading: a) T = 5 days, b) T = 10 days 5

10 Amiri, A. et al. Fig. 4. The average degree of onsolidation under triangular yli loading: a) T = 5 days, b) T = 10 days Fig. 5. The average degree of onsolidation under trapezoidal yli loading: a) T = 10 days, b) T = 0 days 6

11 Civil Engineering Infrastrutures Journal, 51(1): 17 33, June 018 The results indiate that the time of reahing steady state dereases with inrease of onsolidation oeffiient and hange in yle period of time does not affet the time of getting steady state. Figures 6-8 illustrates the degree of onsolidation of the three types of soil in radial and vertial onsolidation under retangular yli loading to investigate the effet of using vertial prefabriated drains or stone olumns to aelerate the onsolidation of soft soils. The alulation of vertial onsolidation under retangular yli loading is based on the study of Toufigh and Ouria (009). The oeffiient of onsolidation of eah soil is shown in Table and the soil properties and geometri harateristis for eah ase are given in Table 3. As it an be seen from Table, for organi silt and soft blue lay, the ratio of radial oeffiient of onsolidation to vertial oeffiient of onsolidation is 1.5 and it an be seen from Figures 3 and 5 that when the ratio of drainage length of radial onsolidation to vertial beomes, the average degree of onsolidation for radial onsolidation is less than for vertial onsolidation. 7

12 Amiri, A. et al. Fig. 6. The average degree of onsolidation under retangular yli loading for soft blue lay: a) ase 1, b) ase, ) ase 3, d) ase 4 Table 3. Load and geometri harateristis Case Cyle Period Radial Consolidation Vertial Consolidation (day) rw (m) re (m) Hdr (m)

13 Civil Engineering Infrastrutures Journal, 51(1): 17 33, June 018 9

14 Amiri, A. et al. Fig. 7. The average degree of onsolidation under retangular yli loading for Boston blue lay: a) ase 1, b) ase, ) ase 3, d) ase 4 30

15 Civil Engineering Infrastrutures Journal, 51(1): 17 33, June 018 Fig. 8. The average degree of onsolidation under retangular yli loading for organi silt: a) ase 1, b) ase, ) ase 3, d) ase 4 For Boston blue lay, the ratio of radial oeffiient of onsolidation in to vertial oeffiient of onsolidation is 1.7 and when the ratio of drainage length of radial onsolidation in to vertial beomes, the average of degree of onsolidation for radial onsolidation is equal to vertial from beginning to steady state (ase 3). Therefore radial onsolidation and vertial onsolidation under retangular yli loading beome equal when: / 1 re H dr 1.5 k / k (13) h v where re : is radius of the equivalent irle (= de/), Hdr : is the length of vertial drainage, kh: is the horizontal oeffiient of permeability and kv: is the vertial oeffiient of permeability. 31

16 Amiri, A. et al. CONCLUSIONS The following onlusions may be drawn from this study: 1. The average of degree of onsolidation in steady state depends on the integral of the load-time urve for eah yle and it inreases with inrease of the integral.. The degree of onsolidation of soil under trapezoidal yli loading is less than for the retangular yli load from the beginning and beomes more than it after a period of time. Degree of onsolidation is less than the loading states mentioned above for triangular yli loading. 3. The time of reahing steady state, is lower for retangular yli loading than the other types of yli loading that have been investigated in this researh. 4. Change in yle period of time does not effet on the time of getting steady state. 5. Radial onsolidation and vertial onsolidation under retangular yli loading beome equal when the ratio of drainage length of radial onsolidation to vertial is about times the ratio of radial oeffiient of onsolidation to vertial. REFERENCES Abbaspour, M. (014). An investigation of onsolidation proess under triangular yli loading by numerial and experimental analysis, Eletroni Journal of Geotehnial Engineering, 19, Barron, R. (1948). Consolidation of fine-grained soils by drain wells, Transportation, ASCE, 113, Paper No Basett, D. and Brodie, A. (1961). A study of Matabithual varved lay, Ontario Hydro Researh News, 13, 1-6. Covo-Torres, Á., Eljaiek-Urzola, M. and Vivas-Reyes, R. (015). Radial onsolidation for equal strain with resistane in the vertial drain, Soil and Tillage Researh, 145, Das, B.M. (013). Advaned soil mehanis, CRC Press. Deng, Y.-B., Xie, K.-H. and Lu, M.-M. (013). Consolidation by vertial drains when the disharge apaity varies with depth and time, Computers and Geotehnis, 48, 1-8. Haley, X. and Aldrih, X. (1969). Engineering properties of foundation soils at Long Creek-Fore river areas and Blak Cove, Report to Maine State Highway Company. Hansbo, S., Jamiolkowski, M. and Kok, L. (1981). Consolidation by vertial drains, Geotehnique, 31, Ladd, C. and Lusher, U. (1965). Engineering properties of the soils underlying the MIT ampus, Researh Report, R Lei, G., Fu, C. and Ng, C.W. (016). Vertial-drain onsolidation using stone olumns: An analytial solution with an impeded drainage boundary under multi-ramp loading, Geotextiles and Geomembranes, 44, Lo, W.-C., Sposito, G., Lee, J.-W. and Chu, H. (016). One-dimensional onsolidation in unsaturated soils under yli loading, Advanes in Water Resoures, 91, Lowe III, J., Zaheo, P.F. and Feldman, H.S. (1964). Consolidation testing with bak pressure, Journal of the Soil Mehanis and Foundations Division, ASCE, 90, Lu, M., Wang, S., Sloan, S.W., Sheng, D. and Xie, K. (015). Nonlinear onsolidation of vertial drains with oupled radial vertial flow onsidering well resistane, Geotextiles and Geomembranes, 43(), Lu, M., Xie, K. and Wang, S. (011). Consolidation of vertial drain with depth-varying stress indued by multi-stage loading, Computers and Geotehnis, 38, Ni, J., Indraratna, B., Geng, X.-Y., Carter, J.P., and Rujikiatkamjorn, C. (013). Radial onsolidation of soft soil under yli loads, Computers and Geotehnis, 50, 1-5. Olson, R.E. (1977). Consolidation under timedependent loading, Journal of the Geotehnial Engineering Division, 103, Ouria, A., Desai, C.S. and Toufigh, V. (013). Disturbed state onept based solution for onsolidation of plasti lays under yli loading, International Journal of Geomehanis, 15(1), Razouki, S.S., Bonnier, P., Datheva, M. and Shanz, T. (013). Analytial solution for 1D onsolidation under haversine yli loading, International Journal for Numerial and Analytial Methods in Geomehanis, 37(14), Razouki, S.S. and Shanz, T. (011). Onedimensional onsolidation under haversine repeated loading with rest period, Ata Geotehnia, 6(1),

17 Civil Engineering Infrastrutures Journal, 51(1): 17 33, June 018 Rezaei, Z. (016). Consolidation of elasti and inelasti lays under triangular yli loading, M.S. Thesis, Univrrsity of Shahid Bahonar, Kerman, Iran (In Persian). Rihart, F.E. (1957). A review of the theories for sand drains, Florida Engineering and Industrial Experiment Station, College of Engineering, University of Florida. Shiffman, R. (1958). Consolidation of soil under time-dependant loading and varying permeability, Highway Researh Board Proeedings, 37, Speirs, A.D., Beaulé, P.E., Ferguson, S.J. and Frei, H. (014). Stress distribution and onsolidation in artilage onstituents is influened by yli loading and osteoarthriti degeneration, Journal of Biomehanis, 47(10), Toufigh, M.M. and Ouria, A. (009). Consolidation of inelasti lays under retangular yli loading, Soil Dynamis and Earthquake Engineering, 9, Tsien, S. (1955). Stability of marsh deposits, Highway Researh Board Bulletin, 15, Walker, R.T. (011). Vertial drain onsolidation analysis in one, two and three dimensions, Computers and Geotehnis, 38, Wallae, G. and Otto, W. (1964). Differential settlement at selfridge air fore base, Journal of the Soil Mehanis and Foundations Division, 90, Wilson, N. and Elgohary, M. (1974). Consolidation of soils under yli loading, Canadian Geotehnial Journal, 11, Ying-Chun, Z. and Kang-He, X. (005). Study on one-dimensional onsolidation of soil under yli loading and with varied ompressibility, Journal of Zhejiang University Siene, 6, Yoshikuni, H. and Nakanodo, H. (1974). Consolidation of soils by vertial drain wells with finite permeability, Soils and Foundations, 14,