Stability analysis on Tingzikou gravity dam along deep-seated weak planes during earthquake

Abstract

The stability of a gravity dam against sliding along deep-seated weak planes is a universal and important problem encountered in the construction of dams. There is no recommended method for stability analysis of the dam on deep-seated weak planes under earthquake condition in Chinese design codes. Taking Tingzikou dam as an example, the research in this paper is focused on searching a proper way to evaluate the seismic safety of the dam against sliding along deep-seated weak planes and the probable failure modes of dam on deep-seated weak planes during earthquake. It is concluded that there are two probable failure modes of the dam along the main weak geological planes in the foundation. In the first mode, the concrete tooth under the dam will be cut and then the dam together with part foundation will slide along the muddy layer; in the second mode, the dam together with part foundation will slide along the path consist of the weak rock layer under the tooth and the muddy layer downstream the tooth. While there is no geological structure planes to form the second slip surface, the intersection of the main and the second slip surface is 40 to 80 m downstream from dam toe, and the angle between the second slip surface and the horizontal plane probably be 25 to 45 degrees.

Keywords

gravity dam / deep-seated weak planes / stability against sliding / earthquake

Introduction

The stability of a gravity dam against sliding along deep-seated weak planes is a universal and important problem encountered in the construction of dams. Biswas (1971) studied 300 failure dams. About 35% of them failed due to foundation problems [1]. Zhu and other researchers (1979) studied more than 100 dams in china. 92 dams of them were influenced by deep-seated sliding problems [2].

There are various methods to evaluate the stability of a structure against sliding. In the Gravity Dam Design Code in China, the multiple wedge analysis method is recommended to calculate the sliding safety factor along deep-seated planes. This method has been used in many projects. Much experience has been accumulated in the past decades. However, the stress and displacement distribution on sliding planes cannot be achieved by multiple wedge analysis. There are two commonly used methods at present for obtaining stress and displacement distribution in the foundation: response spectrum method and time history analysis method. After considering dam-foundation interaction, accurate stress and displacement distribution in foundation can be achieved by these two methods. In this paper the time history analysis method is employed in the calculation. And the visco-spring boundary is used to prevent wave reflection at the edge of discrete model [3,4].

The Tingzikou project is based on midstream of Jialing River in China. This project is built with the aims of flood control, irrigation, water supplying, power generation and shipping. The designed total storage capacity of the reservoir is 3.468 billion cubic meters with the normal storage level of 458 m. The general layout of the project is shown on Fig. 1. The terrane is titled to downstream of about 1 to 5 degrees. There is no fault but some cracks and several weak rock layers distributing in the foundation. Some muddy layers which have an important impact on the stability of the dam exist in rock foundation. This paper focuses on how to find the most likely failure mode of the overflow dam section during earthquake. The safety factors of different failure modes during earthquake are also estimated.

Safety factor along deep-seated planes during earthquake

If the failure surface is single plane, the safety factor is defined by the ratio of shear resistance and applied shear force along the plane as Eq. (1). It can be used in multiple wedge analysis or finite element analysis.

(1)K={\displaystyle \frac{fN+cA}{T}}.

If the failure surface is combined by two or more planes, there is no uniform method to calculate the safety factor along deep-seated planes. Eq. (2) is recommended by Pan when failure surface is combined by two planes under static load [2]. The safety factor is defined by the ratio of resisting moment and driving moment along failure surface. O₁ and O₂ are respectively point application of force N₁ and N₂ shown on Fig. 2. The intersection of OO₁ and OO₂ is regarded as the torque center. However, the stress distribution on AB and BC changes with time during the earthquake, so as to the location of O₁ and O₂. The safety factor at any certain moment during the earthquake can be obtained by this method, but the comparison between different moments and different failure modes is unreasonable. Compared with the complex calculation methods, simplified and uniform method that can meet comparison of different conditions may be more meaningful at present for the dynamic analysis of structure. In this paper, the scalar sums of normal stress and tangential stress along the failure surface are used to define the safety factor shown in Eq. (3).

(2)K={\displaystyle \frac{(f_{\mathrm{1}}{N}_{\mathrm{1}}+c_{\mathrm{1}}{L}_{\mathrm{1}})r_{\mathrm{1}}+(f_{\mathrm{2}}{N}_{\mathrm{2}}+c_{\mathrm{2}}{L}_{\mathrm{2}})r_{\mathrm{2}}}{T_{\mathrm{1}}{r}_{\mathrm{1}}+T_{\mathrm{2}}{r}_{\mathrm{2}}}},

(3)K={\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^n{\sigma }_i{l}_i{f}_i+{\displaystyle \sum _{i=\mathrm{1}}^n{l}_i{c}_i}}}{{\displaystyle \sum _{i=\mathrm{1}}^n{\tau }_i{l}_i}}},

where, K = safety factor against sliding, i = number of element, l = length or area of element, f = coefficient of internal friction of the material, c = cohesion of the material, σ = normal stress of element, τ = tangential stress of element. The safety factor at any certain moment can be obtained conveniently by this equation, and the comparison of sliding stability along deep-seated planes between different moments or different failure modes can be achieved.

During the earthquake, the value of safety factor defined by Eq. (3) changes with time. However, a certain value is necessary to describe the stability of a structure in the traditional method. In the history of safety factor, the minimum safety factor is the most proper value for evaluating the stability of structure under earthquake condition. So the minimum safety factor is chosen as the seismic safety factor of structure in this paper.

Nonlinear simulation of muddy layer

The simulation of nonlinear character of muddy layers is an important aspect in the numerical analysis. To get accurate stress distribution on muddy layers, the nonlinear characters of them should be simulated accurately. In this paper a kind of element fit for dynamic analysis is used to simulate the nonlinearity of muddy layers. The method to determine the stiffness of this element is the same with it of ordinary element. The constitutive matrix of this element is defined separately in normal and tangential direction. In normal direction, the compaction or expansion of layer is judged by the negative or positive of normal stress. The modulus under compaction condition is defined by Eq. (4) [5]. In tangential direction, the modulus is defined by Eq. (5) [6].

(4)D_{nn}=K_{ni}{\left(\mathrm{1}-{\displaystyle \frac{{\sigma }_{\mathrm{n}}}{V_{\mathrm{m}}{K}_{ni}+{\sigma }_{\mathrm{n}}}}\right)}^{-\mathrm{2}}·t,

(5)D_{ns}=K_{si}{\gamma }_{\mathrm{w}}{\left({\displaystyle \frac{{\sigma }_{\mathrm{n}}}{P_{\mathrm{a}}}}\right)}^n{\left(\mathrm{1}-{\displaystyle \frac{\tau ·R_{\mathrm{f}}}{{\tau }_{\mathrm{p}}}}\right)}^{\mathrm{2}}·t,

where, D_nn = modulus in normal direction under compaction, σ_n = normal stress, K_ni = initial stiffness in normal direction, V_m = maximum compressed thickness, t = thickness of element, D_ns = modulus in tangential direction, K_si = initial stiffness in tangential direction, γ_w = weight of water, P_a = atmospheric pressure, τ = tangential stress, R_f = stress ratio at failure, τ_p = crisis tangential stress.

This element can be used to simulate the shearing and compacting action of nonlinear material accurately. The thickness of element is determined by the actual thickness of the muddy layers. In this paper, the actual thickness of muddy layer is almost equal to or a little bit less than 10 cm. The thickness of muddy layer element in model is approximately taken as 10 cm.

Finite element model and parameters

Finite element model

An over-flow dam section with a crest thickness of 49.05 m and a base thickness of 95.12 m is simulated. The distance between the boundary and the dam is about 210 m in upstream and downstream direction, and about 150 m in vertical direction. The muddy layers are simulated with nonlinear elements, the other materials are simulated with isotropic elements with eight nodes or six nodes. The finite element model is shown on Fig. 3. A total of 49118 elements are used: 4246 elements to model the dam, and 44872 elements including 2264 nonlinear elements to model the foundation. The model contains 61643 nodal points.

Parameters of material

The parameters of materials used in analysis are given in Table 1 according to experiments. The parameters of muddy layer in the table are changed into parameters for nonlinear element used in numerical calculation. The modulus used in dynamic analysis is amplified by 1.3. The damping ratio is 0.05.

Loading and boundary condition

There exists primary stress consists of tectonic stress and geostatic stress in ground before structure is built. For the complexity and difficulty in simulation of tectonic stress, only geostatic stress is simulated in this paper. Other static loads including weight of dam and rock, pressure of normal reservoir water level and the corresponding tail water level, silt pressure and uplift pressure are considered.

The artificial seismic waves in stream direction and vertical direction are used in numerical calculation. The peak acceleration in stream direction is 0.107 g. The peak acceleration in vertical direction is two thirds of it in stream direction. For the mass of the foundation rock is included in the model, the response of acceleration on top of the foundation should be the same with original artificial seismic wave for the free field. By inversing analysis, the peak acceleration at the bottom of the model is 0.09 g. The seismic waves in stream direction and vertical direction are shown on Fig. 4.

The freedom in normal direction of both sides along dam axis is fixed. The visco-spring boundary is used on the upstream, downstream and bottom sides of the model. The inversion of artificial wave is changed into nodal forces and then input on nodes on the visco-spring boundary.

Probable paths of sliding

The foundation of Tingzikou dam is consists of several stratums with different thickness and modulus. The main types of rock is weak weathering sand stone. The main weak layers are made up of silt stone and clay stone that distributing in sand stone. Besides the weak layers, there are also some muddy layers in the foundation. The main weak planes distributing in the foundation are shown on Fig. 5. The range of these planes is listed in Table 2.

It is specified by Chinese design code for gravity dams that when there exist low angle dip weak planes in the foundation, the stability analysis of the structure along deep-seated planes should be performed in different sliding modes. According to the distribution of weak planes, the failure surface of Tingzikou dam may probably be consist of two parts: the main slip surface parallel to the weak layers and the second slip surface like DE shown on Fig. 9.

There are two probable failure modes of the dam along the main weak geological planes in the foundation. The result of model experiment by Yangtze River Scientific Research Institute is shown on Fig. 6 [7]. It implies that the concrete tooth under the dam will be cut and then the dam together with part foundation will slide along the muddy layer. The yield condition of plastic analysis is shown on Fig. 7. The muddy layer and area under tooth would yield during earthquake. It implies that the dam together with part foundation will slide along the path consist of the weak rock layer under the tooth and the muddy layer downstream the tooth.

For evaluating the local stability of the foundation, five weak planes are selected and the safety factor against local failure along these planes during earthquake is calculated. Eq. (3) in one element is used in the calculation of safety factor. The direction of tangential force is parallel with the weak plane. The minimum safety factor during earthquake of five weak planes is displayed on Fig. 8.

The minimum safety factor against local failure under tooth generally declines along the direction to downstream. Starting at the downstream side of the tooth, the value declines until the location is a certain distance away from the dam toe. And then the value begins to increase. Because of the poor strength of mud, the value on the muddy layer is clearly smaller than other geological structure planes. It implies that the muddy layer is the dominated geological structure plane. The change law of safety factor on the muddy layer is the same with other planes, but there is a smooth period with a length of about 60 m after dam toe.

The second slip surface

There is no geological structure plane to form the second slip surface. Different conditions should be calculated for the determination of the most dangerous second slip surface. In the trial calculating, Two factors of the second slip surface are verified: the dip angle β and the slip out point D shown on Fig. 9.

Dip angle of the second slip surface

Two conditions are chosen to ensure the dip angle of the second slip surface. In one condition, the dam together with part foundation slip out at the dam toe; in the other condition, the dam together with part foundation slip out at 60 m away from dam toe. The safety factor of ADE shown on Fig. 9 is calculated to ascertain the angle β. The comparison of safety factor of different angle β is shown on Fig. 10. If the structure slip out at dam toe, the safety factor is smaller when β is between 15 to 25 degrees. If the structure slip out at 60 m away from dam toe, the safety factor is smaller when β is between 25 to 45 degrees. The structure is more likely to slip out at 60 m away from dam toe.

Slip out point

The slip out point is signed as D. For different location of D, the dynamic safety factor of ADE is calculated when β is 25 degrees. Result is shown on Fig. 11. When D is about 40-80 m away from dam toe, the safety factor is generally the same and is smaller than other conditions.

Critical slip surface

According to the research in Sections 5 and 6, there are two possible failure modes. In the first mode, the concrete tooth under the dam will be cut and then the dam together with part foundation will slide along the muddy layer; in the second mode, the dam together with part foundation will slide along the path consist of the weak rock layer under the tooth and the muddy layer downstream the tooth. The most dangerous second slip surface is titled to upstream of 25 degrees with a slip out point 60 m away from dam toe. Two probable critical slip surfaces are displayed on Fig. 12. The safety factor of two probable critical slip surfaces during earthquake is shown on Fig. 13. The safety factor is smaller in the second failure mode. And the minimum safety factor of the structure in the second failure mode is 2.21.

Conclusions

1) In the foundation, the minimum safety factor against local failure on geological planes generally declines along the direction to downstream until to a certain distance away from dam toe. And then the value increases.

2) The safety factor of slip surface indicate that the structure is more likely to slip out at a certain distance away from dam toe, this distance may be related to the structure type and the location of geological planes.

3) The dip angle of the most dangerous second slip surface is relevant to the location slip out point.

4) The safety factor of slip surface is smaller when the second slip surface are titled to upstream of about 25-45 degrees on the most probable of slip out point in Tingzikou project.

5) There are two probable failure modes for Tingzikou project. In the first mode, the concrete tooth under the dam will be cut and then the dam together with part foundation will slide along the muddy layer; in the second mode, the dam together with part foundation will slide along the path consist of the weak rock layer under the tooth and the muddy layer downstream the tooth. The analysis shows that the second failure mode is more likely to happen.

References

[1] Zhou Z, Zhou F, Pan J X, Wang Z Y. Study on stability against deep sliding of gravity dam. Rock and Soil Mechanics, 2008, 29(6): 1719-1722 (in Chinese)

[2] Pan J Z. Deep-seated sliding stability of gravity dam. Sichuan Water Power, 1983, 1: 6-19 (in Chinese)

[3] Liu J B, Lu Y D. A direct method for analysis of dynamic soil-structure interaction. China Civil Engineering Journal, 1998, 31(3): 55-64 (in Chinese)

[4] Du X L, Zhao M, Wang J T. A stress artificial boundary in FEA for near-field wave problem. Chinese Journal of Theoretical and Applied Mechanics, 2006, 38(1): 49-56 (in Chinese)

[5] Bandis S C, Lumsden A C, Barton N R. Fundamentals of rock joint deformation. International Journal of Rock Mechanics and Mining Sciences, 1983, 20(6): 249-268

[6] Clough G W, Duncan J M. Finite element analyses of retaining wall behavior. Journal of Soil Mechanics and Foundation Engineering Division, ASCE, 1971, 97(12): 1657-1673

[7] Jiang X L, Sun S W, Zhu J B. Model test of stability against deep sliding in Tingzikou dam. Journal of Yangtze River Scientific Research Institute, 2010, 27(9): 65-69 (in Chinese)

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