Stability Analysis of Front Slopes at the Tunnel Entrance Based on Principle of Minimum Potential Energy
Xiangcan Wang1*, Kai Xing2, Hao Yang2, Qifang Zhan2 Binran Zhang2
1Post-Graduate, School of Civil Engineering, National Engineering Laboratory for High Speed Railyway Construction, Central South University, Changsha 410075, China, (Corresponding Author, e-mail: [email protected]) 2Post-Graduate, School of Civil Engineering, Central South University, Changsha 410075, China,
ABSTRACT
On the base of principle of minimum potential energy, the paper considers tunnel excavation’s and the pore water pressure’s effect and establishes the stability’s analysis model for homogeneous tunnel portal’s front slope whose sliding surface can be any shape. First, we regard sliding mass as a rigidity body, establish mass’s potential energy function by assuming any shape of sliding surface, and then obtain sliding mass’s displacement, which makes mass’s potential energy minimum according to principle of minimum potential energy. Next, we obtain the normal stress’s distribution and value through analyzing force and displacement’s relationship, then get resistance force and sliding force， and finally know the tunnel portal slope’s stability factor according to Mohr-Coulomb Criterion and safety factor’s definition. In the end, for one engineering project example’s application, the calculated result shows that safety factor of tunnel portal’s slope will be reduced in case of tunnel excavation and pore water pressure’s influence, and approximately agrees with widespread applied Bishop and Janbu’s result, which shows that our method is correct and rational in tunnel portal slope’s stability analysis application. KEYWORDS: principle of minimum potential energy; pore water pressure; tunnel portal front slope; stability; tunnel excavation; safety factor
INTRODUCTION
Tunnel portal section is always located at slopes which are eroded gravely by surface water and also develop much weathering fracture. Because of poor geological condition, portal’s excavation and insufficient embedded depth of portal section, the landslides easily occur and slope’s stability is hardly guaranteed. The scholars have still used the research method of slope’s stability in engineering geology to explore the tunnel portal slope’s stability. In the long term, ultimate equilibrium theory
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Vol. 21 [2016], Bund. 14 4512 and finite element simulation have always been applied to analyze the portal slope’s stability in tunnel design and construction. In ultimate equilibrium theory’s application, Liu et al.(2001) used circular destructed surface combined of ultimate equilibrium method to analyze the portal slope’s stability, and this kind of method can predicate the slope’s dangerous sliding surface, which can supply the basis for determining tunnel portal front slopes’ destruction and design properly. Kockar and Akgun (2003) used stereographic method and ultimate equilibrium method to calculate the portal slope’s safety factor. In finite-element numerical simulation’s application, Dong and Zhang(2007) calculated the slope’s nonlinear finite element model by means of strength reduction method, then got slope’s safety factor, sliding surface and destruction process by controlling the static analysis’s result’s non-convergence when the slope entered into a unstable condition. Jiang et al.(2015) applied finite element strength reduction method to dam of Jingping Hydropower Station, and assessed the slope’s stability in construction. Existing exploration mainly analyze the portal slope’s stability by means of finite element simulation and ultimate equilibrium theory. However, there is still few research for this problem from energy’s view. At same time, at home and abroad, some scholars’ researches of principle of minimum potential energy have showed that this principle is feasible in slope stability analysis. On the basis of previous work, considering the impact of tunnel excavation and pore water pressure, we use principle of minimum potential energy and propose one tunnel portal slope’s stability analysis model for homogeneous portal front slope with any sliding face’s shape.
THE PRINCIPLE OF MINIMUM POTENTIAL ENERGY
The principle of minimum potential energy means that: for all geometrical possible displacements, only the actual displacement makes total potential energy minimum. The method solves elastic mechanic problems by regarding displacement function as basic unknown. The difference between actual displacement and else possible displacements is that whether the static equilibrium condition could be satisfied.
Model analysis method
Model parameters and assumptions
The Fig.1 shows the homogeneous soil-slope where there could be any sliding surface at tunnel portal. The soil mass’s unit density is γ. The cohesion is c. The internal friction angle is φ.We choose portal slope’s possible sliding surface ABC, and assume that it’s curve equation is y = f (x) , AD’s is y = g(x) , DC’s is y = h(x) , A’s coordinate is (x1, y1 ), B’s is (x2 , y2 ), C’s is (x3 , y3 ) , D’s is
(x4 , y4 ) and E’s is (x5 , y5 ). In the condition where external resultant force R work, there will be d a small displacement d for the sliding mass. At this time, we regard whole sliding mass ABCD as a rigidity body, but consider that between sliding mass and sliding beds, there exists elastic compress deformation and potential energy stored which is for the reason of external resultant force R ’s work and its displacement. So we make the following assumptions for the model: 1: the sliding mass is rigidity body; the contact on sliding surface is elastic, and it’s elastic deformation can be simulated with series of springs whose stiffness is k . Vol. 21 [2016], Bund. 14 4513
2: for any differential plane dl on sliding surface, its normal stiffness k is proportional to dl ’s area; d 3: there exists a small displacement d for sliding mass ABCD under the action of external resultant force R , furthermore this displacement makes the whole sliding mass ABCD’s potential energy smallest.
Figure 1: Any Sliding Surface at Tunnel Portal Where Slope Is Homogeneous Soil-slope
Sliding mass’s total potential energy calculation
We suppose that when sliding mass begins to move, there is no deformation for itself which means that sliding mass is viewed as rigidity body, and elastic deformation will happen on sliding surface. Meanwhile, the elastic deformation on sliding surface is similar with linear spring whose stiffness is proportional to surface area. The sliding mass is effected by many kinds of external force including gravity, seismic force, external load and water pressure, and the sliding mass under all force’s work and sliding surface where elastic deformation occurs form an elastic conservative system. Given that sliding mass’s small displacement just occurs under all force’s effect, when the sliding system reaches the balance, the displacement right now will make whole system’s potential energy minimum. The result from sliding displacement is that there will be elastic strain energy on sliding surface. The potential energy dV stored in the differential arc length dl is:
' 2 2 ' 1 2 [f (x)] d f (x) dV = m⋅d + 2 − 2d d dl 1 ' 2 ' 2 1 2 ' 2 2 1+ [f (x)] 1+ [f (x)] 1+ [f (x)]
2 2 2 1 d [f ' (x)] + d − 2d d f ' (x) = ⋅m⋅ 1 2 1 2 ⋅dx (1) 2 2 1+ [f ' (x)] ' In Eq. (1), is the first derivative of curve is one constant related with material. and f (x) f (x) , m d1
d2 are respectively mass’s displacements along X-axis and Y-axis.
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The sliding system’s total potential energy V is the difference between stored strain potential energy in sliding surface’s elastic deformation and resultant external force’s work as follows:
2 ' 2 2 ' d x d 3 1 d1 [f (x)] + d2 − 2d1d2 f (x) V = dV − R ⋅d = ⋅m⋅ ⋅dx − R1 ⋅d1 − R2 ⋅d2 ∫ ∫x 2 2 2 1+ [f ' (x)] (2) Pore water pressure’s work
The slope’s rock-soil mass possesses one certain moisture content. The underground water’s impact on slope mass mainly includes effect of pore water and seepage. When pore water pressure’s effect on slope’s stability is analyzed, we generally consider that pore water pressure acts on rock-soil mass as external load, and the action’s location is on rock-soil mass’s frame and sliding surface. Its physical meaning is explicit enough. The pore water pressure’s power is: W = − un v dS u ∫ i i (3) The pore water pressure’s total work is: d = = = ⋅ d ⋅ W ∫Wu ∫∫ univids ∫ u ni didl t t s
x3 ' d1 f (x) d2 ' 2 = ∫u − 1+ [f (x)] dx ' 2 ' 2 x2 1+ [f (x)] 1+ [f (x)]
x3 = ⋅ ' − ⋅ ∫u (d1 f (x) d2 ) dx x2 (4) The pore water pressure u can be represented as follows: u = γ ⋅γ ⋅ z u (5)
In Eq. (5), γ u is pore water pressure coefficient; γ is soil mass’s volume density; z is the vertical distance between earth surface and any point below the earth. z(x) = g(x)− f (x)x x x 2 4 z(x) = h(x)− f (x)x x x 4 3 (6)
Substituting Eqs. (5) and (6) into Eq. (4),we obtain that pore water pressure’s work is:
x4 = g ⋅g ⋅ ' − ⋅ − W ∫ u [d1 f (x) d2 ] [g(x) f (x)]dx x2
x3 + g ⋅g ⋅ ' − ⋅ − ∫ u [d1 f (x) d2 ] [h(x) f (x)]dx x4 (7) From Eqs. (2) and (7),we can obtain that system’s total potential energy is: Vol. 21 [2016], Bund. 14 4515
2 ' 2 2 ' x 3 1 d1 [f (x)] + d2 − 2d1d2 f (x) ' ' V = ⋅m⋅ ⋅dx −R1d1 − R2d2 ∫x 2 2 2 1+ f ' (x) [ ] x4 x3 − g ⋅g ' − ⋅ − − g ⋅g ' − ⋅ − ∫ u [d1 f (x) d 2 ] [g(x) f (x)]dx ∫ u [d1 f (x) d 2 ] [h(x) f (x)]dx x2 x4 (8) d The displacement d ’ s solution
Eq. (8) is stable system’s potential energy when system reaches the ultimate balance. d Considering principle of minimum potential energy, in all possible displacement d , if system’s total d potential energy is minimum, the displacement d right now should meet the following equations: ∂V = 0 ∂d 1 ∂V = 0 ∂d 2 (9)
which means that:
2 x3 ' ' x4 x3 d1[f (x)] − d2 f (x) ' ' ' m dx − g u ⋅g ⋅ f (x)[g(x)− f (x)]dx − g u ⋅g ⋅ f (x)[h(x)− f (x)]dx − R1 = 0 ∫ ' 2 ∫ ∫ x2 1+ [f (x)] x2 x4 x3 ' x4 x3 d2 − d1 f (x) ' m dx + g u ⋅g [g(x)− f (x)]dx + g u ⋅g [h(x)− f (x)]dx − R2 = 0 ∫ ' 2 ∫ ∫ x2 1+ [f (x)] x2 x4
(10) Wh en we know any curve equation’s expression( f (x) , g(x) , h(x) ), we can solve the above equation set and get d1 , d2 that meet equations. We know that d1 and d2 are two function expressions whose denominatorare both m as follows: const1 = d1 m const2 d2 = m (11)
In Eq. (11), d and d are function values which don’t include m . So we can suppose that 1 2 d the angle between its direction and horizontal direction is α , and then get d ’s value. The resistance force’s derivation Vol. 21 [2016], Bund. 14 4516
According to the Mohr-Coulomb failure criterion, we assume that the angle between sliding surface’s tangent line and horizontal direction is θ (θ = arctan f ' (x)) and then obtain that：
1 f ' (x) cos(α −θ ) = cosα cosθ + sinα sinθ = cosα + sinα 2 2 1+ [f ' (x)] 1+ [f ' (x)] 1 f ' (x) sin(α −θ ) = sinα cosθ − cosα sinθ = sinα − cosα ' 2 ' 2 1+ [f (x)] 1+ [f (x)] (12)
The ultimate resistance force supplied by every small arcs on sliding surface is that:
dT1 = cos(α −θ )⋅{c ⋅dl + m⋅dl[d1 ⋅n1(x)+ d2 ⋅n2 (x)]⋅ tαnϕ} f ' (x) const2 (13) = cos(α −θ )c + const1− tαnϕdl ' 2 ' 2 1+[ f (x)] 1+[ f (x)]
The normal force supplied by every small arcs on sliding surface is that: f ' (x) const2 dN = const1− dl 2 2 1+ [f ' (x)] 1+ [f ' (x)] (14) = ⋅ α −θ = ' − ⋅ α −θ dT2 dN sin( ) [ f (x)const1 const2] sin( )dx (15)
So the total resistance force is:
Tresist = T1 +T2
x3 x3 2 = ∫ cos(a −θ )⋅c 1+ [f ' (x)] dx + ∫ cos(a −θ )⋅ tanϕ ⋅[f ' (x)const1− const2]dx x2 x2
x3 + ∫[f ' (x)const1− const2]⋅sin(a −θ )dx x2 (16)
Substituting Eq. (12) into Eq. (16), we obtain that: Vol. 21 [2016], Bund. 14 4517
x3 = ⋅ a + ' a Tresist c ∫ [cos f (x)sin ]dx x2
x3 1 f ' (x) + tanϕ cosa + sina [f ' (x)const1− const2]dx （17） ∫ ' 2 ' 2 x2 1+ [f (x)] 1+ [f (x)]
x3 1 f ' (x) + [f ' (x)const1− const2] sina + cosa dx ∫ ' 2 ' 2 x2 1+ [f (x)] 1+ [f (x)]
In Eq. (17), since there is no constant m related to material any more, we can absolutely complete the integral and get resistance force. We regard sliding mass as rigidity body before, so if the normal force along sliding surface is negative in calculation, we think the soil mass has been pulled apart. Sliding force’s derivation As shown in Fig.1, the sliding mass is divided into three parts: S2 , S3, S4 . ED’s equation is y = g(x) . BC’s equation is y = f (x) . DC’s equation is y = h(x) .
dFinduce = g ⋅dS ⋅sinα x2 S2 = [g(x)− y ]dx ∫ 2 x5 x4 = ( )− ( ) S3 ∫[g x f x ]dx x2 x3 S4 = ∫[h(x)− f (x)]dx x4 (18)
The sliding mass’s self-weight is: = ⋅γ = + + ⋅γ W S (S2 S3 S4) (19)
So the total sliding force is:
x2 x4 x3 F = W ⋅sinα = [γ(x)− y ]dx + [γ(x)− f (x)]dx + [h(x)− f (x)]dx⋅γ ⋅sinα induce ∫ 2 ∫ ∫ x5 x2 x4 (20)
Therefore, we can obtain that if f (x) , g(x) , h(x) are known, we can also complete this integral and then get the sliding force along the displacement ‘s direction. Safety factor’s solution Vol. 21 [2016], Bund. 14 4518
In this issue, the definitions of safety factor are a lot. Considering that the sliding ’s direction can’t be ensured for the reason that sliding surface’s equation could be any curve, we define the safety factor as the ratio of sliding resistance force and sliding force which are along the sliding d direction( displacement d ’s direction ).So that: T F = resist s F induce (21)
ENGINEERING PROJECT ANALYSIS
In order to verify the above method, we present one engineering example to analyze the tunnel portal’s front slope’s stability, and then prepare our method with traditional method to test its rationality. Engineering information
The tunnel is one highway separated tunnel with middle length, whose left tunnel is 678m(max buried depth:124m), and right tunnel is 662m(max buried depth:126m). The entrance portal is cutting face portal, while exit portal is bench portal. The entrance and exit’s engineering landform are both slopes, whose natural slope angle is approximately 300 ~ 500 . Fig. 2 is the geomorphologic map of the tunnel front slope.
Figure 2: Geomorphologic Map of The Tunnel Front Slope Vol. 21 [2016], Bund. 14 4519
Tunnel slope stability analysis
According to prospecting and drilling data, the slope’s possible destructed pattern is strong-weathered rock’s internal circular sliding. The slope’s volume density is γ=18. Soil’s cohesion is c=20. Soil’s internal friction angle is φ=37. The pore water pressure coefficient γ u =0.1. Fig. 3 is model’s calculation figure of the tunnel front slope.
Figure 3: Model’s Calculation Figure of The Tunnel Front Slope
In order to consider tunnel’s excavation’s effect and pore water pressure ‘s effect to slope stability, we calculate three different safety factors for excavation and pore water’s different states as follows: 1: Before the excavation; 2: Without consideration of pore water pressure after excavation; 3: With consideration of pore water pressure after excavation. And use Bishop and Janbu methods to calculate corresponding safety factors by rock-soil slope stability software. The result’s comparison is as follows:
Table 1: Safety Factor of The Tunnel Front Slope Condition Analysis method Safety factor Bishop 1.640 Janbu 1.663 Before excavation principle of minimum 1.587 potential energy Bishop 1.438 Without pore water Janbu 1.471 After excavation pressure principle of minimum 1.391 potential energy Vol. 21 [2016], Bund. 14 4520
Bishop 1.170 With pore water Janbu 1.152 pressure principle of minimum 1.125 potential energy
In table 1, we know that: before tunnel’s excavation, the safety factor is approximately 1.6 (far greater than 1), which means slope is stable; after excavation, without consideration of pore water pressure, the factor reduces obviously to 1.4 (still greater than 1), which means slope is stable so far. The result shows that tunnel’s excavation could affect slope’s stability greatly. Furthermore, after tunnel’s excavation, when considering pore water pressure, the factors are smaller which shows that pore water pressure is adverse to slope’s stability. For now, the safety factor reduces to approximately 1.15 (extremely similar with 1), which shows slope is in unstable critical state. The result agrees with sliding signs on the spot. The reason why tunnel’s excavation effects tunnel portal’s slope’s stability so much is that excavation can broke surrounding rock’s balance state, redistribute surrounding soil mass’s stress, and change sliding mass’s force. Pore water pressure’s effects to slope stability mainly conclude such two aspects: effect to rock-soil mass’ strength and to slope’s stress field. Pore water will reduce soil’s angle of internal friction and cohesion, and then resistance force, which is harmful to slope’s stability. Meanwhile, since pore water pressure work on slope’s latent sliding surface, for our model, it can change slope’s whole system potential, and finally effect the slope’s stability.
CONCLUSION
Principle of minimum potential energy is based on the fact that stable balance system’s potential energy is minimum, and that’s why it can be used to solve slope stability safety factor problem. This method’s calculation process is simple, and we don’t need to divide the mass into items, consider the force between items. Furthermore, we don’t need iteration, which means the result can be explicit. Principle of minimum potential energy has been applied successfully. On the base of general slope stability analysis of this principle，for homogeneous tunnel portal slope whose sliding surface can be any shape，we author consider the tunnel excavation’s effect and pore water pressure’s effect , and then establish one analysis method of tunnel portal slope’s stability. Through model’s building and engineering project’s verification, we can obtain the following conclusions: 1: the calculated safety factor by the method we propose is close to one by Bishop and Janbu methods, within the margin of error. So the rationality and accuracy of our method for tunnel portal slope’s stability can be verified. 2: the safety factor of portal slope deduces rapidly after tunnel portal’s excavation, which can affect the slope’s stability. So in practical engineering projects, before tunnel’s excavation, the portal slope must be reinforced to guarantee the construction. 3: the safety factor of portal slope deduces after we consider pore water pressure. The surface water and underground water are adverse to portal slope’s stability, and that’s why we need control water effectively in construction to guarantee the slope’s stability. 4: we apply our method to one engineering project, analyze its tunnel portal slope and get reasonable result which can prove our method has certain prospect in engineering’s application. Vol. 21 [2016], Bund. 14 4521
ACKNOWLEDGMENTS
The authors would like to acknowledge the project (50908234) supported by the National Natural Science Foundation of China, and the project (2011CB710604) supported by the National Basic Research Program of China, which collectively funded this project.
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© 2016 ejge Editor’s note. This paper may be referred to, in other articles, as: Xiangcan Wang, Kai Xing, Hao Yang, Qifang Zhang, Binran Zhang: “ Stability Analysis of Front Slopes at the Tunnel Entrance Based on Principle of Minimum Potential Energy” Electronic Journal of Geotechnical Engineering, 2016 (21.14), pp 4511-4522. Available at ejge.com.