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Simulation method for calculating stress and deformation of highway embankment on frozen soil area
Cold Regions Science and Technology 42 (2005) 89 – 95 www.elsevier.com/locate/coldregions
Simulation method for calculating stress and deformation of highway embankment on frozen soil area Tie-hang Wanga,*, Chang-shun Hub, Ning Lic a
College of Civil Engineering, Xi’an University of Architecture and technology, Xi’an 710055, PR China b College of Highway Engineering, Chang’an University, Xi’an 710064, People’s Republic of China c Research Institute of Geotechnical Engineering, Xi’an University of Technology, Xi’an 710048, People’s Republic of China Received 17 January 2004; received in revised form 28 June 2004; accepted 28 June 2004
Abstract As the seasons go on, the variation of water content and temperature distribution of highway embankment is not regular in permafrost zone. It results in heterogeneous changes of the property of embankment soil. This leads to heterogeneous deformation of the embankment. If the deformation goes beyond the allowable value, the highway would be threatened by collapse. All of these phenomena should be taken into account with emphasis in the design of highway passing through the permafrost area. But at present no favorable measure has been derived. This paper bases on the elasto-plasticity theory and creep theory, by simulation of the variation of water content and temperature distribution and soil property, establishes a twodimensional model to calculate the stress and deformation of highway embankment in permafrost area. In this model the creep deformation, the self-deformation in volume and the instantaneous deformation are considered. Especially the creep and instantaneous deformation behaviors under the changing process of soil property with time are studied. D 2005 Elsevier B.V. All rights reserved. Keywords: Highway embankment; Soil; Permafrost; Creep deformation; Instantaneous deformation
1. Introduction Qinghai–Tibet Highway has been playing a vital role in transportation and economics of Tibet. The Qinghai–Tibet railway under construction will become another important access to Tibet. The both
* Corresponding author. E-mail address: [email protected] (T. Wang). 0165-232X/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2004.06.008
lines go across the permafrost zone of Qinghai–Tibet plateau. Unfortunately, the design of the ways confronts many difficulties, e.g. deformation and stress in way embankment could not be well harnessed. Owing to the lack of suitable calculation method, the design calculation on the basis of deformation and strength index becomes impossible. Owing to the stress and deformation could not be effectively solved in design, Qinghai–Tibet Highway damage has been frequently occurred along the
90
T. Wang et al. / Cold Regions Science and Technology 42 (2005) 89–95
permafrost section (760 km). Settlement, fluctuation, torsion, bending, transversal and longitudinal cracks of the highway embankment are the major defects (Dou et al., 2000), which seriously demolish the way embankment, and consequently diminish the benefit of the highway. It is significant to make further study on the stress and deformation of embankment on permafrost area, in order to fulfill the design method for reducing the damage of embankment so as to ensure the efficient traffic of both highway and railway. To date, though thermal elasto-plasticity theory has already been applied to the calculation, however, since certain creep properties and the changing process of soil property with time for permafrost soil (Wiebe et al., 1998; Simonsen and Isacsson, 1999; Wu and Ma, 1994) are not taken into consideration, therefore, it is not viable to the permafrost projects. Literatures about creep deformation have been launched by The Third International Conference of Soil Mechanics and Foundation Engineering (1953). Some of the literatures on creep issues about soil, snow and ice pointed out that the breakthrough on creep study would have significant impact on future research of soil mechanics. Afterward, various literatures on this issue began to appear, basically, on soil with certain structure and physical property. About the frozen soil, Vyalov and Abdel (1992), Rein (1985), Zhu and Carbee (1983), Sheng and Wu (1995), etc., conducted much study on the issue. The damage of way embankment on permafrost zone are mainly attributed to the changing process of the soil moisture content and thermal regime that result in the changes of soil properties. In discussing the stress and deformation of way embankment, it needs to take account the changing of moisture, thermal and soil property. By considering the changing process, this paper, based on the elasto-plasticity theory and creep theory, intends to establish a numerical model of stress and deformation for way embankment on permafrost area.
2. Deformation of embankment soil on permafrost area Under loading, the deformation of way embankment soil on permafrost area includes two types: instantaneous deformation and creep deformation.
The soil self-volume deformation resulting from frost heave and thaw collapse also ought to be considered. Hence the formula for determining the deformation of embankment soil on permafrost area consists of three terms. eðt Þ ¼ ee ðt Þ þ ec ðt Þ þ eo ðt Þ
ð1Þ
Where e(t) is the total strain in time t; e e(t) is instantaneous strain; e c(t) is creep strain; e o(t) is soil self-volume strain. e e(t) and e c(t) are caused by stress, e o(t) is caused by temperature changing. During the changing process of moisture and temperature, especially in freezing and melting process, the soil properties and stress change with time, which yields the complexity of deformation calculation. For simplicity, the duration of moisture and thermal changes is subdivided into a series of time interval Dt i (i=1,2,: : : , n). When Dt i is less enough, moisture content and temperature in Dt i could be assumed unchanged. Likewise the soil properties in Dt i could also be supposed unchanged. In this case, soil parameters and stress in each time interval have fix values. Thus, there has to be stress stage-leap phenomenon between time intervals, that means the progressively changing stress is simplified as stage-leap changing stress. As a result, the stress increment Dr i in each time interval can be defined. In accordance with the stress increment, deformation increment in each time interval is first calculated, then, accounted for a total deformation. In any time interval Dt i , from time t i1 to t i , by the Eq. (1), deformation increment De i is Dei ¼ Deei þ Deci þ Deoi
ð2Þ
2.1. Instantaneous deformation Instantaneous deformation consists of two parts: elastic deformation and plastic deformation. Taking plastic element as elastic one by adjusting elastic parameters to get approximate plastic deformation, the instantaneous deformation may be calculated by nonlinear elastic model. This paper, by simulating the changing property of soil on permafrost area, derives the instantaneous deformation.
T. Wang et al. / Cold Regions Science and Technology 42 (2005) 89–95
As for soil on permafrost area, elastic modulus E and Poisson’s ratio l are related with soil property, and are changing with the time. Within each time interval, moisture content and temperature is constant, and soil property is to be supposed unchanged. The parameters of soil in each time interval can be obtained. For the time intervals among which the moisture content and temperature are both stable, the soil sample whose temperature and moisture are of the average value should be taken for the test. The tested parameters can be regarded as those in the time intervals. The way embankment is a structure belonging to linear type. Its issues may be taken as plane strain issues. In this case, if the stress increment {Dr i } in any time interval Dt i and elastic parameters E i ,l i are known, the deformation increment aroused by Dr i in Dt i can be calculated.
1 Deiei ¼ ½Qi fDri g Ei
Deiei ¼ Deieix
Deieiy
Dcieixy gT
fDri g ¼ Drix
Driy
Dsixy
"
1 li ½Qi ¼ ð1 þ lÞi li 0
ð3Þ
T
li 0 1 li 0 0 2
#
Where E i is elastic modulus; l i is Poisson’s ratio. {Deiei } is instantaneous deformation increment matrix in Dt i ; De ixie is normal instantaneous deformation increment in x-axis of coordinate, and De iyie in y-axis; ie De ixy is shear deformation increment; {Dr i } is stress increment matrix in Dt i ; Dr ix is normal stress increment in x-axis of coordinate, and Dr iy in axis; Ds ixy is shear stress increment. In the loading process, if soil property changes, for the instantaneous deformation increment in time Dt i , apart from instantaneous deformation increment aroused by stress increment Dr i in Dt i obtained in Eq. (3), another instantaneous deformation increment aroused by the exerted stress increment prior to the time interval Dt i shall be taken into consideration. If
91
stress increment Dr l is exerted in any time interval Dt l , prior to time interval Dt i , instantaneous deformation increment resulting from Dr l may still occur in Dt i . If soil turns soft, soil elastic modulus in time interval Dt l+1 is less than that in Dt l . In this case, relationship between instantaneous deformation increment and stress increment is shown in Fig. 1. Instantaneous deformation is considered as elastic one. The relationship of stress and deformation turns to be a straight line, the straight lines 1 and 2 are associated with soil in Dt l and Dt l+1. After Dr l is exerted, deformation Delel occurs at time interval Dt l , and trajectory of deformation is shown as straight-line OA in Fig. 1. In time interval Dt l+1, soil turns soft, without previous deformation process, deformation le increment De l+1 resulting from Dr l will appear, and trajectory of deformation will be shown as straightline OB in Fig. 1. Because of the effect of previous deformation in Dt l , trajectory of deformation is no longer straight line OB, instead, it reaches B by passing A, exerting instantaneous deformation increlV e ment De l+1 in Dt l+1, trajectory of deformation is indicated as broken line OAB in Fig. 1. As in field engineering, soil properties changes progressively in process, instead of leap change, the actual trajectory of deformation is indicated as curve OB in Fig. 1. But it can be obtained from the Fig. 1 that the calculating result can not be changed if the curve is replaced by broken line OAB, which makes calculation simplified. From broken line OAB, derives instantaneous defor-
Fig. 1. Relationship between instantaneous deformation increment and stress increment.
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T. Wang et al. / Cold Regions Science and Technology 42 (2005) 89–95
lVe mation increment De l+1 in time interval Dt l+1 resulting from Dr l : lVe le le Delþ1 ¼ Delþ1 Del
¼
1 ½Qlþ1 Drl g ½Ql Drl g Elþ1 El 1
ð4Þ
If soil constantly turns soft, then, in any time interval Dt k after Dt l+1, instantaneous deformation increment resulting from Dr l is obtained.
1 ½Qk Drl DelVke g ¼ Delek Delek1 ¼ Ek 1 ½Qk1 Drl Ek1
ð5Þ
Let k=i, instantaneous deformation increment in time interval Dt i resulting from Dr l is:
1 1 ½Qil fDrl g DelVke ¼ ½Qi Drl Ei Ei1
ð6Þ
If the soil continues freezing after time interval Dt l , soil turns harder, then ice binding effect increases. The existing deformation is maintained in soil body as a type of membrane. In any time interval Dt i after Dt l , instantaneous deformation increment resulting from Dr l is zero. lVe Dei ¼ 0 ð7Þ In the case where soil elastic modulus first increases then decreases or conversely as Fig. 2, from Dt l till Dt i , in any time interval after Dt l , because the elastic modulus is bigger than that in Dt l , instantaneous deformation increment in this time interval resulting from Dr l is zero. From Dt k till Dt l , in any
Fig. 2. Relationship between modulus and time.
time interval within Dt k and Dt k after Dt k , the same to above, instantaneous deformation increment in this time interval resulting from Dr k is zero, but in any time interval after Dt k , because the elastic modulus is minor than that in Dt k , instantaneous deformation increment in this time interval resulting from Dr k should be calculated by formula (6). After obtain instantaneous deformation increment in Dt i resulting from stress increment exerted in Dt l , similarly, obtain that in Dt i resulting from stress increment exerted in any time interval prior to Dt i , when added up, obtain instantaneous deformation increment in Dt i caused by stress increment exerted in all the time interval prior to Dt i : i1 e X DelVie Dei ¼ ð8Þ l¼1
By summation, the total instantaneous deformation increment in time interval Dt i is: e ie e ð9Þ Dei ¼ Dei þ Dei 2.2. Creep deformation In creep deformation developing process, soil freezing and thawing and water transfer in soil (Shoop and Bigl, 1997) all can result in changes of soil properties, so creep deformation calculation is very complicated. To simplified the calculation in this paper, creep deformation increment in Dt i caused by stress increment exerted in Dt l prior to Dt i is taken as modified creep deformation increment of soil at time Dt i caused by stress increment exerted in Dt l , modified coefficient relate to soil property change from Dt l to Dt i . Creep deformation calculation in each time interval has to take consideration of stress increment in this time interval and that of before. Based on above simplification, creep deformation increment in Dt i caused by stress increment exerted in Dt l is: ð10Þ Decil ¼ ecl ðtl ; ti Þ ecl ðtl ; ti1 Þ 1 Where ecl (t l , t i ) is creep deformation increment from time t l to time t i for the soil at Dt i , caused by stress increment exerted in Dt l ; t i , t i1 is time coordinate point around Dt i ; 1 is modified coefficient, if soil property unchanging, 1=1; if soil turns hard, 1b1; if soil turns soft, 1N1.
T. Wang et al. / Cold Regions Science and Technology 42 (2005) 89–95
According to Eq. (10) and Genga (G e H n a) equation (Du, 1987), obtain creep deformation increment in Dt i caused by stress increment exerted in Dt l as following, which is expressed in incremental form.
1 1 1 Decilx ¼ 1 Drlx Drlm 2 Gli Glði1Þ 1 1 þ1 ð11Þ Drlm kli klði1Þ
1 1 1 Decily ¼ 1 Drly Drlm 2 Gli Glði1Þ 1 1 þ1 ð12Þ Drlm kli klði1Þ 1 1 Dcclxy ¼ 1 ð13Þ Dslxy Gli Gliði1Þ Where Dr lx , Dr ly, Dr lxy : stress increment exerted in Dt l . Dl m : average stress; G li , k li , G l(i1), k l(i1) as: Gli ¼
1 m2 Aðtli Þcm þ Bðtli Þrlm sl li ¼ 1m2 cli cli
Glði1Þ ¼
kli ¼
sl clði1Þ
¼
1 m2
A tlði1Þ cm l ði1Þ þ B tl ði1Þ rlm 2 c1m l ði1Þ
rlm Dðtli Þ ¼ 1e elid m elid m
klðilÞ ¼
D tlði1Þ Drlm ¼ 1e elði1Þm elðilÞm
Where s l is shear stress strength in t l . r lm is average stress in t l . c li , e lid m are shear strain yield criterion and volumetric strain from time t l to time t i . t li is course of time from tl to ti. m 1, m 2, e are coefficients, obtained by tri-axial test. A(t), B(t), D(t) are time functions, on the basis of tested data, they can be expressed as Aðt Þ ¼
A0 1 þ y1 t a1
Bðt Þ ¼
B0 1 þ y2 t a2
D0 Dðt Þ ¼ 1 þ y 3 t a3
Where A 0, B 0, D 0, a i , d i (i=1,2,3) are tested coefficients. For the frozen soil, the available experiment has revealed the creep equation as: cli ¼ AðT ÞsB1 ðti tl ÞC
ð14Þ
Where A(T) is creep coefficient relating to temperature; B,C are coefficients relating to time and stress. After c lk is defined, G li , G l(i1) for frozen soil can be got. s1B l Gli ¼ AðT Þðti tl ÞC Glði1Þ ¼
s1B l AðT Þðti1 tl ÞC
Similarly, creep deformation increment in Dt i caused by stress increment exerted in any time interval prior to Dt i can be obtained. Creep deformation increment in Dt i caused by stress increment exerted in Dt i is: 1 1 ðDrix Drim Þ þ Drim ð15Þ Deicix ¼ 2Gi ki Deiciy ¼
1 1
Driy Drim þ Drim 2Gi ki
Dccixy ¼
93
1 Dsixy Gi
ð16Þ ð17Þ
According to Boltzman principle of superposition (Zhu, 1999), by summation get the creep total deformation increment in time interval Dt i as: i1 X 1 1 1 Decix ¼ 1 ðDrlx Drlm Þ 2 Gli Glði1Þ l¼1
1 1 þ Drlm þ Deicix ð18Þ kli klði1Þ Deciy
i1 X
1 1 1 Drly Drlm ¼ 1 2 Gli Glði1Þ l¼1
1 1 þ ð19Þ Drlm þ Deiciy kli klði1Þ
Dccixy ¼
i1 X 1 1 1 Ds1xy þ Dcicixy G G li l ð i1 Þ l¼1
ð20Þ
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T. Wang et al. / Cold Regions Science and Technology 42 (2005) 89–95
2.3. Self-volume deformation
relationship between element strain increment and nodal displacement increment is:
For way embankment soil on permafrost area, soil self-volume deformation is mainly related with frost heave and thaw collapse. Coefficient of frost heave is defined as the ratio of frozen expansion volume to that of unfrozen. Coefficient of thaw-subsidence is as the ratio of shrinking volume resulting from thaw collapse to the unthawed volume. Coefficient of frost heave and coefficient of thaw-subsidence both are of the selfvolume deformation coefficients, symbolized by g, which is mainly related with temperature change, and can be obtained by test (Hayhoe and Balchin, 1990; Xu et al., 1995). Soil self-volume deformation increment De ixo in Dt i caused by temperature change is: 8 9 8 o 9 < 13 gDfs = < Deix = Deo ð21Þ ¼ ð1 þ lÞ 13 gDfs : ; : Deoiy ; 0 ixy
fDei g ¼ ½ BfDyi g
ð26Þ
T fDyi g ¼ Dui ; DVi ; Duj ; DVj ; Dum ; DVm Where {Dd i }: nodal displacement increment array; Du is nodal displacement increment in x-axis of coordinate; DV is nodal displacement increment in yaxis; i,j,m are node numbers. Put Eq. (26) into Eq. (25), obtain: ZZ ½ BT ½ D Deci dxdy f DF g ¼ ½k fyDg
ZZ ZZ
½ BT ½ D De0i dxdy ½ BT ½ D Deei dxdy
ð27Þ
Where Df s is phase-change degree of soil water in Dt i .
3. Model for calculating stress and deformation of way embankment From above formulas, each ingredient of deformation increment in Eq. (2) is got. Rewrite Eq. (2) as: Deiei ¼ Dei Deci Deoi Deei
ð22Þ
put Eq. (22) into Eq. (3) gives:
fDri g ¼ ½ D fDei g Deci Deoi Deei
Where [k]: element rigidity matrix, as: ZZ ½k ¼ ½ BT ½ D½ Bdxdy The second, third and fourth items in right side of the formula (27) can be respectively regarded as load increments reflecting creep deformation, selfvolume deformation and instantaneous deformation.
ð23Þ
u Dpci
¼
u
¼
ZZ ZZ
½ BT ½ D Deci dxdy ¼ A½ BT ½ D Deci
Where [D]=E i [ Q]1. Stress and deformation of way embankment on permafrost area can be solved by finite element method. Based on virtual work principle (Zhu, 1999), obtain the formula of nodal force increment: ZZ ½ BT fDri gdxdy ð24Þ fDF g ¼
Where [B]: geometrical matrix.Put Eq. (23) into Eq. (24) gives: ZZ f DF ge ¼ ½ BT ½ DfDei gdxdy ZZ
ð25Þ ½ BT ½ D Deci þ Deoi þ Deei Þdxdy
Where {Dpci }u is element nodal load increment array reflecting creep deformation increment in Dt i ; {Dp0i }u is element nodal load increment array reflecting selfvolume deformation increment in Dt i ; {Dpei }u is element nodal load increment array reflecting instantaneous deformation increment in Dt i , caused by stress increment exerted in all the time interval prior to Dt i .
Dp0i
u DPie
¼
ZZ
½ BT ½ D De0i dxdy ¼ A½ BT ½ D De0i ½ BT ½ D Deei dxdy
¼ A½ BT ½ DT DeeVi
T. Wang et al. / Cold Regions Science and Technology 42 (2005) 89–95
95
Combining the element rigidity matrix and element nodal load increment array, the overall rigidity matrix and overall nodal load increment array can be taken. Then, deformation increment of way embankment on permafrost area in any time interval Dt i can be obtained by the following equation of finite element method:
caused self-volume deformation and the instantaneous deformation are considered. Especially the creep and instantaneous deformation behaviors in the condition of soil property changes with time are studied with emphasis.
½ K fDyi g ¼ fDppi g þ Dpci þ Dp0i þ Dpei
Acknowledgements ð28Þ
Where [K] is overall rigidity matrix, composed of element rigidity matrix; {Dd i } is nodal displacement increment array in Dt i ; {Dppi } is overall nodal load increment array caused by external force in Dt i ; {Dpci }, {Dp0i }, {Dpei } are overall nodal load increment array reflecting creep deformation, self-volume deformation and instantaneous deformation caused by prestage stress. Solving Eq. (28) gives deformation increment in any time interval Dt i , integrate deformation increment of all time intervals to get the total deformation.
4. Summary The way embankment damage on permafrost zone is chiefly caused by the changes of soil moisture, temperature distribution and soil property, which are resulted from the brutal climate. Changing process of water content and temperature distribution in way embankment on permafrost zone possesses many varieties, which results in non-uniform changes in soil property, and non-uniform deformation. If the non-uniform deformation is greater than allowable value, the way embankment will suffer a collapse state. The changing process in soil property and the non-uniform deformation should be significantly taken into account in way embankment design in permafrost zone. Basing on the elasto-plasticity theory and creep theory, by simulating the changing process of water content and temperature distribution and soil property, this paper establishes a twodimensional model to calculate the stress and deformation of way embankment on permafrost zone. In this model the creep deformation, the temperature-
The project is supported by the National Natural Science Foundation of China. No. 50308024.
References Dou, M., Hu, C., Zhang, Y., 2000. Analysis research on typical structure of pavement and embankment on permanent permafrost: study on Qinghai–Tibet highway status and failure mechanism. Chang’an University, Xi’an. Du, Y., 1987. Rheology Theory of Soil Mechanics. Science Press, Beijing. Hayhoe, H.N., Balchin, D., 1990. Field frost heave measurement and prediction during periods of seasonal frost. Canadian Geotechnical Journal 27 (3), 393 – 397. Rein Jr., R.G., 1985. Correspondence of creep data and constant strain-rate data for frozen silt. Cold Regions Science and Technology 11 (2), 187 – 194. Sheng, Y., Wu, Z., 1995. Creep rupture behaviour of frozen sand under two-stage loading. Journal of Glaciology and Geocryology 17 (4), 334 – 338. Shoop, S.A., Bigl, S.R., 1997. Moisture migration during freeze and thaw of unsaturated soils: modeling and large scale experiments. Cold Regions Science and Technology 25 (1), 33 – 45. Simonsen, E., Isacsson, U., 1999. Thaw weakening of pavement structures in cold regions. Cold Regions Science and Technology 29 (2), 135 – 151. Vyalov, S.S., Abdel, D.A., 1992. Methods of determination of rheological characteristics of clay soils, Osnovaniya. Fundamenty i Mehanika Gruntov 5, 2 – 5. Wiebe, B., Graham, J., Xiangmin, G., et al., 1998. Influence of pressure, saturation and temperature on the behaviour of unsaturated sand-bentonite. Canadian Geotechnical Journal 35 (2), 194 – 205. Wu, Z., Ma, W., 1994. Frozen Soil Strength and Creep. Lanzhou University Press, Lanzhou. Xu, X., Jiacheng, W., Lixin, Z., 1995. Frozen Soil Expansion and Salt Expansion Mechanism. Science Press, Beijing. Zhu, B., 1999. Bulk Concrete Temperature Stress and Temperature Control. China Electricity Press, Beijing. Zhu, Y., Carbee, D.L., 1983. Creep behaviour of frozen silt under constant uniaxial stress. 4th Int. Conf. on Permafrost.
Simulation method for calculating stress and deformation of highway embankment on frozen soil area Tie-hang Wanga,*, Chang-shun Hub, Ning Lic a
College of Civil Engineering, Xi’an University of Architecture and technology, Xi’an 710055, PR China b College of Highway Engineering, Chang’an University, Xi’an 710064, People’s Republic of China c Research Institute of Geotechnical Engineering, Xi’an University of Technology, Xi’an 710048, People’s Republic of China Received 17 January 2004; received in revised form 28 June 2004; accepted 28 June 2004
Abstract As the seasons go on, the variation of water content and temperature distribution of highway embankment is not regular in permafrost zone. It results in heterogeneous changes of the property of embankment soil. This leads to heterogeneous deformation of the embankment. If the deformation goes beyond the allowable value, the highway would be threatened by collapse. All of these phenomena should be taken into account with emphasis in the design of highway passing through the permafrost area. But at present no favorable measure has been derived. This paper bases on the elasto-plasticity theory and creep theory, by simulation of the variation of water content and temperature distribution and soil property, establishes a twodimensional model to calculate the stress and deformation of highway embankment in permafrost area. In this model the creep deformation, the self-deformation in volume and the instantaneous deformation are considered. Especially the creep and instantaneous deformation behaviors under the changing process of soil property with time are studied. D 2005 Elsevier B.V. All rights reserved. Keywords: Highway embankment; Soil; Permafrost; Creep deformation; Instantaneous deformation
1. Introduction Qinghai–Tibet Highway has been playing a vital role in transportation and economics of Tibet. The Qinghai–Tibet railway under construction will become another important access to Tibet. The both
* Corresponding author. E-mail address: [email protected] (T. Wang). 0165-232X/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2004.06.008
lines go across the permafrost zone of Qinghai–Tibet plateau. Unfortunately, the design of the ways confronts many difficulties, e.g. deformation and stress in way embankment could not be well harnessed. Owing to the lack of suitable calculation method, the design calculation on the basis of deformation and strength index becomes impossible. Owing to the stress and deformation could not be effectively solved in design, Qinghai–Tibet Highway damage has been frequently occurred along the
90
T. Wang et al. / Cold Regions Science and Technology 42 (2005) 89–95
permafrost section (760 km). Settlement, fluctuation, torsion, bending, transversal and longitudinal cracks of the highway embankment are the major defects (Dou et al., 2000), which seriously demolish the way embankment, and consequently diminish the benefit of the highway. It is significant to make further study on the stress and deformation of embankment on permafrost area, in order to fulfill the design method for reducing the damage of embankment so as to ensure the efficient traffic of both highway and railway. To date, though thermal elasto-plasticity theory has already been applied to the calculation, however, since certain creep properties and the changing process of soil property with time for permafrost soil (Wiebe et al., 1998; Simonsen and Isacsson, 1999; Wu and Ma, 1994) are not taken into consideration, therefore, it is not viable to the permafrost projects. Literatures about creep deformation have been launched by The Third International Conference of Soil Mechanics and Foundation Engineering (1953). Some of the literatures on creep issues about soil, snow and ice pointed out that the breakthrough on creep study would have significant impact on future research of soil mechanics. Afterward, various literatures on this issue began to appear, basically, on soil with certain structure and physical property. About the frozen soil, Vyalov and Abdel (1992), Rein (1985), Zhu and Carbee (1983), Sheng and Wu (1995), etc., conducted much study on the issue. The damage of way embankment on permafrost zone are mainly attributed to the changing process of the soil moisture content and thermal regime that result in the changes of soil properties. In discussing the stress and deformation of way embankment, it needs to take account the changing of moisture, thermal and soil property. By considering the changing process, this paper, based on the elasto-plasticity theory and creep theory, intends to establish a numerical model of stress and deformation for way embankment on permafrost area.
2. Deformation of embankment soil on permafrost area Under loading, the deformation of way embankment soil on permafrost area includes two types: instantaneous deformation and creep deformation.
The soil self-volume deformation resulting from frost heave and thaw collapse also ought to be considered. Hence the formula for determining the deformation of embankment soil on permafrost area consists of three terms. eðt Þ ¼ ee ðt Þ þ ec ðt Þ þ eo ðt Þ
ð1Þ
Where e(t) is the total strain in time t; e e(t) is instantaneous strain; e c(t) is creep strain; e o(t) is soil self-volume strain. e e(t) and e c(t) are caused by stress, e o(t) is caused by temperature changing. During the changing process of moisture and temperature, especially in freezing and melting process, the soil properties and stress change with time, which yields the complexity of deformation calculation. For simplicity, the duration of moisture and thermal changes is subdivided into a series of time interval Dt i (i=1,2,: : : , n). When Dt i is less enough, moisture content and temperature in Dt i could be assumed unchanged. Likewise the soil properties in Dt i could also be supposed unchanged. In this case, soil parameters and stress in each time interval have fix values. Thus, there has to be stress stage-leap phenomenon between time intervals, that means the progressively changing stress is simplified as stage-leap changing stress. As a result, the stress increment Dr i in each time interval can be defined. In accordance with the stress increment, deformation increment in each time interval is first calculated, then, accounted for a total deformation. In any time interval Dt i , from time t i1 to t i , by the Eq. (1), deformation increment De i is Dei ¼ Deei þ Deci þ Deoi
ð2Þ
2.1. Instantaneous deformation Instantaneous deformation consists of two parts: elastic deformation and plastic deformation. Taking plastic element as elastic one by adjusting elastic parameters to get approximate plastic deformation, the instantaneous deformation may be calculated by nonlinear elastic model. This paper, by simulating the changing property of soil on permafrost area, derives the instantaneous deformation.
T. Wang et al. / Cold Regions Science and Technology 42 (2005) 89–95
As for soil on permafrost area, elastic modulus E and Poisson’s ratio l are related with soil property, and are changing with the time. Within each time interval, moisture content and temperature is constant, and soil property is to be supposed unchanged. The parameters of soil in each time interval can be obtained. For the time intervals among which the moisture content and temperature are both stable, the soil sample whose temperature and moisture are of the average value should be taken for the test. The tested parameters can be regarded as those in the time intervals. The way embankment is a structure belonging to linear type. Its issues may be taken as plane strain issues. In this case, if the stress increment {Dr i } in any time interval Dt i and elastic parameters E i ,l i are known, the deformation increment aroused by Dr i in Dt i can be calculated.
1 Deiei ¼ ½Qi fDri g Ei
Deiei ¼ Deieix
Deieiy
Dcieixy gT
fDri g ¼ Drix
Driy
Dsixy
"
1 li ½Qi ¼ ð1 þ lÞi li 0
ð3Þ
T
li 0 1 li 0 0 2
#
Where E i is elastic modulus; l i is Poisson’s ratio. {Deiei } is instantaneous deformation increment matrix in Dt i ; De ixie is normal instantaneous deformation increment in x-axis of coordinate, and De iyie in y-axis; ie De ixy is shear deformation increment; {Dr i } is stress increment matrix in Dt i ; Dr ix is normal stress increment in x-axis of coordinate, and Dr iy in axis; Ds ixy is shear stress increment. In the loading process, if soil property changes, for the instantaneous deformation increment in time Dt i , apart from instantaneous deformation increment aroused by stress increment Dr i in Dt i obtained in Eq. (3), another instantaneous deformation increment aroused by the exerted stress increment prior to the time interval Dt i shall be taken into consideration. If
91
stress increment Dr l is exerted in any time interval Dt l , prior to time interval Dt i , instantaneous deformation increment resulting from Dr l may still occur in Dt i . If soil turns soft, soil elastic modulus in time interval Dt l+1 is less than that in Dt l . In this case, relationship between instantaneous deformation increment and stress increment is shown in Fig. 1. Instantaneous deformation is considered as elastic one. The relationship of stress and deformation turns to be a straight line, the straight lines 1 and 2 are associated with soil in Dt l and Dt l+1. After Dr l is exerted, deformation Delel occurs at time interval Dt l , and trajectory of deformation is shown as straight-line OA in Fig. 1. In time interval Dt l+1, soil turns soft, without previous deformation process, deformation le increment De l+1 resulting from Dr l will appear, and trajectory of deformation will be shown as straightline OB in Fig. 1. Because of the effect of previous deformation in Dt l , trajectory of deformation is no longer straight line OB, instead, it reaches B by passing A, exerting instantaneous deformation increlV e ment De l+1 in Dt l+1, trajectory of deformation is indicated as broken line OAB in Fig. 1. As in field engineering, soil properties changes progressively in process, instead of leap change, the actual trajectory of deformation is indicated as curve OB in Fig. 1. But it can be obtained from the Fig. 1 that the calculating result can not be changed if the curve is replaced by broken line OAB, which makes calculation simplified. From broken line OAB, derives instantaneous defor-
Fig. 1. Relationship between instantaneous deformation increment and stress increment.
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T. Wang et al. / Cold Regions Science and Technology 42 (2005) 89–95
lVe mation increment De l+1 in time interval Dt l+1 resulting from Dr l : lVe le le Delþ1 ¼ Delþ1 Del
¼
1 ½Qlþ1 Drl g ½Ql Drl g Elþ1 El 1
ð4Þ
If soil constantly turns soft, then, in any time interval Dt k after Dt l+1, instantaneous deformation increment resulting from Dr l is obtained.
1 ½Qk Drl DelVke g ¼ Delek Delek1 ¼ Ek 1 ½Qk1 Drl Ek1
ð5Þ
Let k=i, instantaneous deformation increment in time interval Dt i resulting from Dr l is:
1 1 ½Qil fDrl g DelVke ¼ ½Qi Drl Ei Ei1
ð6Þ
If the soil continues freezing after time interval Dt l , soil turns harder, then ice binding effect increases. The existing deformation is maintained in soil body as a type of membrane. In any time interval Dt i after Dt l , instantaneous deformation increment resulting from Dr l is zero. lVe Dei ¼ 0 ð7Þ In the case where soil elastic modulus first increases then decreases or conversely as Fig. 2, from Dt l till Dt i , in any time interval after Dt l , because the elastic modulus is bigger than that in Dt l , instantaneous deformation increment in this time interval resulting from Dr l is zero. From Dt k till Dt l , in any
Fig. 2. Relationship between modulus and time.
time interval within Dt k and Dt k after Dt k , the same to above, instantaneous deformation increment in this time interval resulting from Dr k is zero, but in any time interval after Dt k , because the elastic modulus is minor than that in Dt k , instantaneous deformation increment in this time interval resulting from Dr k should be calculated by formula (6). After obtain instantaneous deformation increment in Dt i resulting from stress increment exerted in Dt l , similarly, obtain that in Dt i resulting from stress increment exerted in any time interval prior to Dt i , when added up, obtain instantaneous deformation increment in Dt i caused by stress increment exerted in all the time interval prior to Dt i : i1 e X DelVie Dei ¼ ð8Þ l¼1
By summation, the total instantaneous deformation increment in time interval Dt i is: e ie e ð9Þ Dei ¼ Dei þ Dei 2.2. Creep deformation In creep deformation developing process, soil freezing and thawing and water transfer in soil (Shoop and Bigl, 1997) all can result in changes of soil properties, so creep deformation calculation is very complicated. To simplified the calculation in this paper, creep deformation increment in Dt i caused by stress increment exerted in Dt l prior to Dt i is taken as modified creep deformation increment of soil at time Dt i caused by stress increment exerted in Dt l , modified coefficient relate to soil property change from Dt l to Dt i . Creep deformation calculation in each time interval has to take consideration of stress increment in this time interval and that of before. Based on above simplification, creep deformation increment in Dt i caused by stress increment exerted in Dt l is: ð10Þ Decil ¼ ecl ðtl ; ti Þ ecl ðtl ; ti1 Þ 1 Where ecl (t l , t i ) is creep deformation increment from time t l to time t i for the soil at Dt i , caused by stress increment exerted in Dt l ; t i , t i1 is time coordinate point around Dt i ; 1 is modified coefficient, if soil property unchanging, 1=1; if soil turns hard, 1b1; if soil turns soft, 1N1.
T. Wang et al. / Cold Regions Science and Technology 42 (2005) 89–95
According to Eq. (10) and Genga (G e H n a) equation (Du, 1987), obtain creep deformation increment in Dt i caused by stress increment exerted in Dt l as following, which is expressed in incremental form.
1 1 1 Decilx ¼ 1 Drlx Drlm 2 Gli Glði1Þ 1 1 þ1 ð11Þ Drlm kli klði1Þ
1 1 1 Decily ¼ 1 Drly Drlm 2 Gli Glði1Þ 1 1 þ1 ð12Þ Drlm kli klði1Þ 1 1 Dcclxy ¼ 1 ð13Þ Dslxy Gli Gliði1Þ Where Dr lx , Dr ly, Dr lxy : stress increment exerted in Dt l . Dl m : average stress; G li , k li , G l(i1), k l(i1) as: Gli ¼
1 m2 Aðtli Þcm þ Bðtli Þrlm sl li ¼ 1m2 cli cli
Glði1Þ ¼
kli ¼
sl clði1Þ
¼
1 m2
A tlði1Þ cm l ði1Þ þ B tl ði1Þ rlm 2 c1m l ði1Þ
rlm Dðtli Þ ¼ 1e elid m elid m
klðilÞ ¼
D tlði1Þ Drlm ¼ 1e elði1Þm elðilÞm
Where s l is shear stress strength in t l . r lm is average stress in t l . c li , e lid m are shear strain yield criterion and volumetric strain from time t l to time t i . t li is course of time from tl to ti. m 1, m 2, e are coefficients, obtained by tri-axial test. A(t), B(t), D(t) are time functions, on the basis of tested data, they can be expressed as Aðt Þ ¼
A0 1 þ y1 t a1
Bðt Þ ¼
B0 1 þ y2 t a2
D0 Dðt Þ ¼ 1 þ y 3 t a3
Where A 0, B 0, D 0, a i , d i (i=1,2,3) are tested coefficients. For the frozen soil, the available experiment has revealed the creep equation as: cli ¼ AðT ÞsB1 ðti tl ÞC
ð14Þ
Where A(T) is creep coefficient relating to temperature; B,C are coefficients relating to time and stress. After c lk is defined, G li , G l(i1) for frozen soil can be got. s1B l Gli ¼ AðT Þðti tl ÞC Glði1Þ ¼
s1B l AðT Þðti1 tl ÞC
Similarly, creep deformation increment in Dt i caused by stress increment exerted in any time interval prior to Dt i can be obtained. Creep deformation increment in Dt i caused by stress increment exerted in Dt i is: 1 1 ðDrix Drim Þ þ Drim ð15Þ Deicix ¼ 2Gi ki Deiciy ¼
1 1
Driy Drim þ Drim 2Gi ki
Dccixy ¼
93
1 Dsixy Gi
ð16Þ ð17Þ
According to Boltzman principle of superposition (Zhu, 1999), by summation get the creep total deformation increment in time interval Dt i as: i1 X 1 1 1 Decix ¼ 1 ðDrlx Drlm Þ 2 Gli Glði1Þ l¼1
1 1 þ Drlm þ Deicix ð18Þ kli klði1Þ Deciy
i1 X
1 1 1 Drly Drlm ¼ 1 2 Gli Glði1Þ l¼1
1 1 þ ð19Þ Drlm þ Deiciy kli klði1Þ
Dccixy ¼
i1 X 1 1 1 Ds1xy þ Dcicixy G G li l ð i1 Þ l¼1
ð20Þ
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T. Wang et al. / Cold Regions Science and Technology 42 (2005) 89–95
2.3. Self-volume deformation
relationship between element strain increment and nodal displacement increment is:
For way embankment soil on permafrost area, soil self-volume deformation is mainly related with frost heave and thaw collapse. Coefficient of frost heave is defined as the ratio of frozen expansion volume to that of unfrozen. Coefficient of thaw-subsidence is as the ratio of shrinking volume resulting from thaw collapse to the unthawed volume. Coefficient of frost heave and coefficient of thaw-subsidence both are of the selfvolume deformation coefficients, symbolized by g, which is mainly related with temperature change, and can be obtained by test (Hayhoe and Balchin, 1990; Xu et al., 1995). Soil self-volume deformation increment De ixo in Dt i caused by temperature change is: 8 9 8 o 9 < 13 gDfs = < Deix = Deo ð21Þ ¼ ð1 þ lÞ 13 gDfs : ; : Deoiy ; 0 ixy
fDei g ¼ ½ BfDyi g
ð26Þ
T fDyi g ¼ Dui ; DVi ; Duj ; DVj ; Dum ; DVm Where {Dd i }: nodal displacement increment array; Du is nodal displacement increment in x-axis of coordinate; DV is nodal displacement increment in yaxis; i,j,m are node numbers. Put Eq. (26) into Eq. (25), obtain: ZZ ½ BT ½ D Deci dxdy f DF g ¼ ½k fyDg
ZZ ZZ
½ BT ½ D De0i dxdy ½ BT ½ D Deei dxdy
ð27Þ
Where Df s is phase-change degree of soil water in Dt i .
3. Model for calculating stress and deformation of way embankment From above formulas, each ingredient of deformation increment in Eq. (2) is got. Rewrite Eq. (2) as: Deiei ¼ Dei Deci Deoi Deei
ð22Þ
put Eq. (22) into Eq. (3) gives:
fDri g ¼ ½ D fDei g Deci Deoi Deei
Where [k]: element rigidity matrix, as: ZZ ½k ¼ ½ BT ½ D½ Bdxdy The second, third and fourth items in right side of the formula (27) can be respectively regarded as load increments reflecting creep deformation, selfvolume deformation and instantaneous deformation.
ð23Þ
u Dpci
¼
u
¼
ZZ ZZ
½ BT ½ D Deci dxdy ¼ A½ BT ½ D Deci
Where [D]=E i [ Q]1. Stress and deformation of way embankment on permafrost area can be solved by finite element method. Based on virtual work principle (Zhu, 1999), obtain the formula of nodal force increment: ZZ ½ BT fDri gdxdy ð24Þ fDF g ¼
Where [B]: geometrical matrix.Put Eq. (23) into Eq. (24) gives: ZZ f DF ge ¼ ½ BT ½ DfDei gdxdy ZZ
ð25Þ ½ BT ½ D Deci þ Deoi þ Deei Þdxdy
Where {Dpci }u is element nodal load increment array reflecting creep deformation increment in Dt i ; {Dp0i }u is element nodal load increment array reflecting selfvolume deformation increment in Dt i ; {Dpei }u is element nodal load increment array reflecting instantaneous deformation increment in Dt i , caused by stress increment exerted in all the time interval prior to Dt i .
Dp0i
u DPie
¼
ZZ
½ BT ½ D De0i dxdy ¼ A½ BT ½ D De0i ½ BT ½ D Deei dxdy
¼ A½ BT ½ DT DeeVi
T. Wang et al. / Cold Regions Science and Technology 42 (2005) 89–95
95
Combining the element rigidity matrix and element nodal load increment array, the overall rigidity matrix and overall nodal load increment array can be taken. Then, deformation increment of way embankment on permafrost area in any time interval Dt i can be obtained by the following equation of finite element method:
caused self-volume deformation and the instantaneous deformation are considered. Especially the creep and instantaneous deformation behaviors in the condition of soil property changes with time are studied with emphasis.
½ K fDyi g ¼ fDppi g þ Dpci þ Dp0i þ Dpei
Acknowledgements ð28Þ
Where [K] is overall rigidity matrix, composed of element rigidity matrix; {Dd i } is nodal displacement increment array in Dt i ; {Dppi } is overall nodal load increment array caused by external force in Dt i ; {Dpci }, {Dp0i }, {Dpei } are overall nodal load increment array reflecting creep deformation, self-volume deformation and instantaneous deformation caused by prestage stress. Solving Eq. (28) gives deformation increment in any time interval Dt i , integrate deformation increment of all time intervals to get the total deformation.
4. Summary The way embankment damage on permafrost zone is chiefly caused by the changes of soil moisture, temperature distribution and soil property, which are resulted from the brutal climate. Changing process of water content and temperature distribution in way embankment on permafrost zone possesses many varieties, which results in non-uniform changes in soil property, and non-uniform deformation. If the non-uniform deformation is greater than allowable value, the way embankment will suffer a collapse state. The changing process in soil property and the non-uniform deformation should be significantly taken into account in way embankment design in permafrost zone. Basing on the elasto-plasticity theory and creep theory, by simulating the changing process of water content and temperature distribution and soil property, this paper establishes a twodimensional model to calculate the stress and deformation of way embankment on permafrost zone. In this model the creep deformation, the temperature-
The project is supported by the National Natural Science Foundation of China. No. 50308024.
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