Critical Height of Slopes in Homogeneous Soil: the Variational Solution

Transcription

1 Critical Height of Slope in Homogeneou Soil: the Variational Solution Chen, Rong State Key Laboratory of Coatal and Offhore Engineering & Intitute of Geotechnical Engineering, Dalian Univerity of Technology, Dalian, Liaoning Province, P. R. China, Luan, Mao-tian State Key Laboratory of Coatal and Offhore Engineering & Intitute of Geotechnical Engineering, Dalian Univerity of Technology, Dalian, Liaoning Province, P. R. China He, Zhi-hong State Key Laboratory of Coatal and Offhore Engineering & Intitute of Geotechnical Engineering, Dalian Univerity of Technology, Dalian, Liaoning Province, P. R. China ABSTRACT Thi paper utilize the variational limiting equilibrium (LE) procedure, formulated by Baker and Garber (1977, 1978), to deal with the critical height of homogeneou lope. Reconidering a vertical cut in coheive frictionle oil, the critical height i identical with the one obtained by upper bound analyi, thu providing an illutrative example to upport the equivalence between the variational LE procedure and upper bound theorem of platicity. Comparion alo how that reult determined by preent analyi and other claical method are lightly different from each other. Taking into account tenion crack and it effect on the maximum height of lope, it i found that in ome limiting cae, the depth of tenion crack may extend to a coniderable value up to 5 percent of the total height. Therefore, it deerve careful attention in engineering practice. KEYWORDS: lope tability; critical height; tenion crack; variational method; limit equilibrium INTRODUCTION Within the framework of limit equilibrium approach, Baker and Garber (1977, 1978), Baker (1981) preented a variational approach to analyze the tability of lope. The main advantage of thi approach i that it i free from any a prior kinematical or tatic aumption, which i the cae

2 Vol. 14 [009], Bund. M for many exiting method in the literature. For intance, Taylor method (1937) made arbitrary aumption on critical lip urface and normal tre ditribution along it; Morgentern and Price (1965) preumed the relation between the horizontal force and the vertical hear force on the ide of the lice. It can be hown that thee aumption are not only unneceary but alo rationally unjutified. In fact, by introducing the notion of afety functional and minimizing thi functional with repect to both potential lip urface and normal tre ditribution, lope tability analyi i tatically determined, provided the overall force and moment equilibrium equation are atified and everal boundary condition are conidered. The aforementioned method, however, hare a common feature: the concept of afety factor, uually defined a the ratio of the total hear trength available on the lip urface to the total hear trength required for equilibrium, i employed to ae the tability of lope. It i noted that the factor of afety i of poor phyical ignificance ince it in eence characterize a fictitiou material with mobilized trength. On the contrary, it appear more traightforward and meaningful to take the lope critical height h cr a a meaure of it afety. The preent analyi i aimed at the critical height of lope in homogeneou oil, following the variational limit equilibrium method formulated by Baker and Garber (1977, 1978). The mathematical formulation and determination of the problem i firt briefly etablihed, and then numerical reult and dicuion are preented. VARIATIONAL ANALYSIS Mathematical formulation of the baic problem Fig. 1 how a typical lope inclined at i with the horizontal direction. Soil profile i identified by the unit weight γ, coheion c, friction angle ϕ and the tenile trength T. h i the height of lope. ζ denote depth of tenion crack. It hould be noted that, a hown by Baker (003), conideration of tenion crack allow it only occur at endpoint and internal crack are excluded in a ingle tet body. No pore water preure and external load are introduced; the collape of lope i only due to gravity. Limit equilibrium method employ the following element: Satifaction of failure criteria τ=f (σ) along the lip urface. Here, Coulomb failure condition i adopted. Where τ τ( x) repectively; ψ = tanφ τ = c + σψ (1) = and σ = σ( x) are the ditribution of hear and normal tre along ( ) y x, Satifaction of the overall force and moment equilibrium equation for the liding ma y x. bounded by lope urface y( x ) and lip urface ( ) H = ( τcoα σin α)d= 0 (.a)

3 Vol. 14 [009], Bund. M 3 ( ) ( τinα σ co α)d x γ d 0 x1 (.b) V = + y y x = ( ) x ( ) (.c) x1 M = ( τ in α + σ co α ) x τ co α σ in α y d γ y y xdx = 0 where i the arc length along y( x ), α i the lope angle of ( ) y x. Eq. (.a) and (.b) are the horizontal and vertical force equilibrium equation, repectively, and Eq. (.c) i the moment equilibrium equation. Introducing Coulomb failure criteria into the three equilibrium equation and combining the geometrical relation coα = d x / d ; inα = y d x/ d, after implification, one arrive at x x1 x x1 ( ) H = c+ σ ψ y dx= 0 (3.a) x x1 ( ) γ ( ) V = σ ψ y cy y y dx= 0 (3.b) ( ) ( ) ( ) M = cxy + σx ψ y + 1 cy σ y ψ y γ y y x dx = 0 (3.c) Figure 1: Baic convention and definition It doe not matter that the height of the lope h i not explicitly included in the above equation. Still the height h, for which the lope can attain while in a tate of limit equilibrium, depend on the two kinematical and tre function y( x ) and σ ( x), repectively. Thu, h can be viewed a a functional of two function. The critical height h cr i the maximum value of h : h = max h{ y( x); σ( x)} = h[ y ( x), σ ( x)] (4) cr cr cr ( y, σ )

4 Vol. 14 [009], Bund. M 4 where ycr ( x ) and cr ( x) ditribution along it. σ repreent, repectively, the critical lip urface and normal tre Conequently, the baic problem can be tated a follow: To find a pair of function, y( x ) and σ ( x), which realize the maximum value h cr of the functional h, imultaneouly ubject to the three equilibrium equation, Eq. 3. Following the ame procedure a ued by Baker and Garber (1977, 1978), Baker (1981), the baic problem can be tranformed to a tandard ioperimetric problem of the calculu of variation, G y( x); σ ( x). which, in turn, i equivalent to the problem of extremizing an auxiliary functional [ ] x x G = g d x = V + λ H + λ M = { σ( ψ y + 1) + cy γ( y y ) + λ[ c + σ( ψ y )] x x1 + λ [ cxy + σ x( ψ y + 1) cy σ y( ψ y ) γ ( y y) x]}dx (5.a) ubject to the two integral contraint Eq. (3.a) and (3.c) and with the additional condition max G = 0 (5.b) ( y, σ ) Here, vertical force equilibrium equation i elected a the integral object, horizontal force and moment equilibrium equation a the integral contraint. Thi election i arbitrary, though. The parameter λ 1 and λ are Lagrange undetermined multiplier. Variational determination of the problem To obtain the critical value of h, Euler equation, neceary condition for the exitence of an extreme, ha to be atified for the function g, namely g d g = 0 σ dx σ g d g = 0 y dx y (6.a) (6.b) A hown by Baker and Garber (1977, 1978), Baker (1981) the firt Euler equation give the family of the potential lip urface, while the econd one lead to the form of the normal tre acting along the critical lip urface. Deduction will not be detailed here ince the proce i imilar with Baker. Only relevant reult are preented below. ψβ Ae, ψ 0 ρ = I1, ψ = 0 (7)

5 Vol. 14 [009], Bund. M 5 σ Be γ A c + e (co β + 3ψ in β), ψ 0 ψ ψβ ψβ = 1+ 9ψ cβ + γρ β + I ψ = co, 0 (8) where A, B, I 1, and I are integration contant. Note that Eq. 7 and 8 are formulation of the family of the potential lip urface and it aociated normal tre ditribution in polar coordinate ytem, repectively. The following co-ordinate tranformation ha been utilized: 1 x= + ρin β = x c + ρin β (9.a) λ λ = ρ β = ρco β (9.b) 1 y co y c λ where ( ρβ, ) i the polar co-ordinate ytem with the centre ( x c, y c ) a hown in Fig.1. The following comment are relevant with repect to Eq. (7) and (8): (a) Focuing on the tructure of the function g, it i eay to ee that g i irrelevant to σ, the firt derivative of σ with repect to x, and i linear in y. Hence, according to Petrov (1968), g i degenerated. It i thi degeneration that, on the one hand, make the two Euler differential equation uncoupled, i.e. Eq (7), (8) can be deduced eparately from Eq (6.a), (6.b); on the other hand, make the olution under intene controvery on the exitence of extreme, a preented by De Joelin De Jong (1980, 1981) and Catillo and Luceño (198). (b) Eq. (7), (8) correpond to the cae λ 0, in which a rotational mode of failure occur. The cae λ = 0, aociated with the tranlational mode of failure a hown in Baker and Garber (1978), i not conidered in the preent paper. However, it i not difficult to obtain an analytical olution of the critical height for the limiting caeϕ = 0, i = 90 under λ = 0. The value of h cr under uch condition i 4 c / γ and the lip urface correpond to a traight line paing through the toe with a lope angle 45, which i conitent with Chen (1975) upper bound analyi concerning the tranlational mechanim of a vertical cut. (c) Whether the hape of lip urface i a log-piral type or a circular type ued to be the cynoure of dipute in the literature, Eq. (7) how that, for coheive frictional oil lope, lip urface i the arc of a log-piral, and it reduce to a circular arc for the cae of frictionle oil. Notwithtanding thi paper i not intended to be involved in thi dipute ince different method may lead to different critical lip urface, not to mention that compoite lip urface are more common in engineering practice. Still one hould keep in mind that for uch a lip urface to be valid it i the required final objective, e.g. the minimum factor of afety, the critical height etc., for the different method hould be compared, not the one calculated for arbitrarily lip urface. Now that the form of potential lip urface and ditribution of normal tre acting along it are obtained, the force and moment equilibrium equation can be integrated explicitly. Final form of the three equation i lited below.

6 Vol. 14 [009], Bund. M 6 The complete expreion i too lengthy and i not preented herein. H = V = M = 0 (10) Up to thi point, to obtain a particular olution one till need to conider the following relation. 1) Geometrical boundary condition yx ( = x) = yx ( = x) = 0 (11) 1 1 yx ( = x) = hcr ζ (1) Note that the two endpoint 1 and are not fixed, i.e. they are variable along the lope urface eparately. However, their ordinate are known. ) Tranverality condition Since the poition of the two endpoint 1 and are not known beforehand, it i neceary to apply the tranverality condition, whoe general form i given below: g [ g + ( Θ y ) ] x= x i = 0 y (13) where x i may be either x 1 or x. Θ i the certain curve along which the two endpoint lide. Subtituting relevant expreion into Eq. (13), and utilizing coordinate tranformation Eq. (9), after implification, it yield [ σ( β)(in β + ψ co β) + cco β γ( y y)in β] β i = 0 (14) Applying thi equation to the endpoint 1, and realizing that at thi point there i no tenion crack, i.e. y y = 0, one obtain c σ1 = tan β + ψ 1 (15) Again, applying Eq. (14) to the endpoint, and conidering the exitence of tenion crack, i.e. y y= ζ, a hown by Baker (1981), one can ue thi equation in two way-either to calculate σ where ζ i known or aumed: γς tan β c σ = tan β + ψ (16) or, to olve for the depth of tenion crack ζ where σ = T

7 Vol. 14 [009], Bund. M 7 ζ tan β + ψ c γ tan β γ tan β = T + (17) 3) Stre boundary condition Introducing the tre boundary condition σ = σ( β = β) into Eq. (8) and olving for B or I γ A c B= σ e (co β + 3ψ in β ) + e 1+ 9ψ ψ ψβ ψβ, ψ 0 (18.a) c I = cβ γρco β, ψ = 0 (18.b) tan β In concluion, there are 8 unknow (h cr, ζ, A, B, β 1, β, x 1, x ) and 8 applicable equation (3 equilibrium equation, geometrical boundary condition, 1 tre boundary condition and coordinate tranformation), hence, the baic problem i tatically determined. NUMERICAL RESULTS AND DISCUSSIONS It i impractical to obtain the analytical olution of thee unknow due to the appearance of everal nonlinear imultaneou equation. However, a trial-and-err proce can be utilized to get the numerical olution. Expreing all the quantitie in their non-dimenional form, for intance, length-type parameter take the expreion of l = γ L/ c and tre-type parameter of = S / c, and introducing the concept of tability factor N, non-dimenional parameter defined a N = γ hcr / c a firt ued by Taylor (1948), one realize typical reult howing the critical lip urface and normal tre ditribution for variou condition a illutrated in Fig.. It i of interet to pay attention to Fig. (c). A mentioned in the ection titled Determination of the problem, Joelin de Jong (1980) preented a imilar approach concerning the critical height of a vertical cut-off in coheive frictionle oil by the calculu of variation. On the ground of an out-of-boundary cup of the critical lip urface and of the mathematical violation in relevant with ufficient and neceary condition for the exitence of an extreme, he concluded that the variational approach did not produce an extreme. Fig. (c) clearly how that for a low frictional oil ma, ay ϕ = 0.1, the critical height of the vertical cut-off attain the value of 3.85 c / γ (in fact, it i eay to obtain a more intriguing olution for frictionle oil ma, i.e. the critical height i 3.83 c / γ, which i identical with Felleniu upper limit olution with a circular lip urface, while Joelin de Jong gave the value of c / γ ). It i noted that the two approache differ from each other mainly in two apect. i.e. Kötter equation and tre boundary condition in Joelin de Jong (1980) replace Coulomb failure criteria and tranverality condition a ued in preent paper, repectively. The following interpretation are relevant with repect to above obervation: (a) It eem difficult to jutify whether the obtained olution i an extreme or not in a rigorou mathematical background ince Joelin de Jong (1981) argued a variational fallacy produced by a degenerated intermediate functional. However, Catillo and Luceño (1983),

8 Vol. 14 [009], Bund. M 8 Baker, Luceño and Catillo (1983), Baker and Frydman (1983), Lehchinky, Baker and Silver (1985) demontrated that the variational limit equilibrium method wa, in eence, equivalent with the upper bound theorem of platicity. Thi explain the identity between preent olution and Felleniu, and alo, etablihe an uncontroverial alternative to verify that the application of calculu of variation in limit equilibrium procedure i feaible. (b) A Baker and Frydman (1983) implicated that Kötter' equation along the potential lip urface, in fact, wa atified by the econd Euler equation (Eq. 6(b) in preent paper) and coordinate tranformation. However, Kötter' equation in Joelin de Jong (1980) might be quetionable ince he treated x and y coordinate a both function of only a ingle parameter α, which led to the vanihing of the polar radiu ρ. Furthermore, variational limit equilibrium analyi only employ force and moment equilibrium equation globally, not at each point. Stre admiibility condition at the cret reulted in an unreaonable lip urface (i.e. multi-value y( x )) in De Jong (1980). Figure : Typical lip urface and normal tre ditribution

9 Vol. 14 [009], Bund. M 9 Figure 3: Stability factor for un-cracked lope Figure 4: Stability factor for oil with zero tenile trength Fig. 3 depict the tability factor N a a function of ϕ and i (friction angle and lope inclination repectively) for the cae of no tenion crack. Comparing the preent reult with upper bound analyi given by Chen (1975), one realize that the value of N vary lightly between different method. It i expected that variational approach obtain exactly the ame theoretical value of N with an upper bound analyi, however, numerical olution determine that thi i not the cae. Conidering the formulation of tenion crack, it i worth noting that the ignificance of tenion crack i till not emphaized. Some invetigator have come to a concluion that the influence of tenion crack on lope i negligible. Fig. 4 preent N a a function of ϕ and i for the cae of a oil with zero tenile trength. One hould note that it i applicable to deignate zero tenile trength for material epecially like oil. The dahed line in thi figure repreent the ratio of the depth of tenion crack to the critical height under uch condition. It i obviou that the larget depth of tenion crack, extending to a coniderable value up to 5% of the lope height, occur for a vertical cut, i.e. i =90. A friction angle ϕ increae, the ratio ζ / N decreae a expected, while the maximum value of the tenion crack increae, a hown in Fig. 5. The dahed line in Fig. 5 give the theoretical formula determining the depth of tenile crack due to a cut in coheive oil, a ued by Chowdhury (1978). z c c π 1 = tan + ϕ γ 4 (19)

10 Vol. 14 [009], Bund. M 10 Figure 5: Effect of friction angle on tenion crack where z c i the depth of tenion crack. Baker (1981) regarded thi equation a the formula yielding the maximum poible height of a vertical cut ince the crack itelf might be conidered a a vertical cut. Owning to thi perpective, it eem reaonable that ζ obtained by the preent approach fall below the one by Eq. (19). Fig. 6 preent effect of tenile trength on tability factor N and tenion crack ς for the cae of ϕ = 5 and ϕ = 0. From thi figure it i een that to a coniderable accuracy both N and ζ are linear on T with cloe lope for variou i. Thi linear behavior make it poible to imulate the relationhip between N, ζ, and T in linear fitting. The identification of fitting will not be elaborated here. Eq. (17) clearly illutrate thi behavior between ζ and T. And for ome pecific value of ϕ, numerical analyi how that variation of β for different i i quite flatter, reulting in the light difference of lope of the linear relationhip. The analyi done o far lead to imilar reult with Baker (1981) aociated with the determination of the factor of afety. It i noted, however, that the two different problem, in effect, are nothing but the ame in the framework of limit equilibrium analyi. i.e. given ome parameter defining the problem, it i required to eek a minimal or maximal value X m of ome objective parameter X. Such parameter X, determined by ome pecific problem, could be the afety factor of lope, the critical height of a vertical cut, limiting earth preure (active or paive) of retaining tructure, ultimate bearing capacity of hallow foundation etc.

11 Vol. 14 [009], Bund. M 11 Figure 6: Effect of tenile trength on tability factor and tenion crack CONCLUSIONS In thi paper, variational limit equilibrium procedure i applied to determine the critical height of lope in homogeneou oil. Analyi how that the critical lip urface, in general, i log-piral. However, it reduce to an arc of circle in the limiting cae ϕ = 0. Alo typical reult jutifie that thi procedure i almot identical with upper bound analyi of platicity. Introducing the concept of tenile trength, it i poible to etimate the depth of tenion crack and it effect on lope. The exitence of tenion crack deerve careful attention ince calculation how that it may extend to a maximum depth up to 5% of the height of lope. Fig. 3 and Fig. 4 are the main end product of preent analyi. They provide a convenient and traightforward way to determine the maximum height of unreinforced lope for civil engineer and hence are ueful chart to engineering practice.

12 Vol. 14 [009], Bund. M 1 ACKNOWLEDGEMENT Thi paper i upported by National Natural Science Foundation of China under Grant No Thi upport i gratefully acknowledged. Alo the author i indebted to Prof. Rafael Baker who afford valuable material and kind intruction during the preparation of thi paper. REFERENCES [1] Baker, R. (1981) Tenile trength, tenion crack, and tability of lope, Soil and Foundation, Vol.1, No., pp [] Baker, R. (003) Sufficient condition for exitence of phyically ignificant olution in limiting equilibrium lope tability analyi, International Journal of Solid and Structure, Vol.40, pp [3] Baker, R. and Frydman, S. (1983) Upper bound limit analyi of oil with non-linear failure criterion. Soil and Foundation, Vol.3, No.4, pp [4] Baker, R. and Garber, M. (1977) Variational approach to lope tability, Proceeding of the 9the International Conference on Soil Mechanic and Foundation Engineering, Tokyo, Japan, Vol.(), pp. 9-1 [5] Baker, R. and Garber, M. (1978) Theoretical analyi of the tability of lope, Géotechnique, Vol.8, No.4, pp [6] Catillo, E. and Luceño, A. (198) A critical analyi of ome variational method in lope tability analyi, International Journal for Numerical and Analytical method in Geomechanic, Vol.6, pp [7] Catillo, E. and Luceño, A. (1983) Variational method and upper bound theorem, Journal of Engineering Mechanic, ASCE, Vol.109, No.5, pp [8] Chen, W. F. (1975) Limit Analyi and Soil Platicity, Amterdam: Elevier [9] Chowdhury, R. N. (1978) Slope Analyi, Amterdam: Elevier [10] De Joelin De Jong, G. (1980) Application of the calculu of variation to the vertical cut off in coheive frictionle oil, Géotechnique, Vol.30, No.1, pp [11] De Joelin De Jong, G. (1981) A variational fallacy. Géotechnique, Vol. 31, pp [1] Lehchinky, D., Baker, R. and Silver, M. L. (1985) Three dimenional analyi of lope tability, International Journal for Numerical and Analytical Method in Geomechanic, Vol. 9, pp [13] Morgentern, N. R. and Price, V. E. (1965) The analyi of the tability of general lip urface, Géotechnique, Vol.15, pp [14] Taylor, D. W. (1948) Fundamental of Soil Mechanic, New York: Wiley 009 ejge