Procedia Engineering 31 (2012) 502 – 508

1877-7058 © 2011 Published by Elsevier Ltd.doi:10.1016/j.proeng.2012.01.1059

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Procedia

Engineering Procedia Engineering 00 (2011) 000–000

www.elsevier.com/locate/procedia

International Conference on Advances in Computational Modeling and Simulation

Three-Dimensional Nonlinear Strain-Stress Analysis of Gravity Dam Base

Ze Lia, Zhilin Lianga*, Long Wangb aEngineering Mechanics Department of Kunming University of Science and Technology, Kunming,650500,China

bWater Resource and Hydropower Planning and Design General Institute，Beijing 100011,China

Abstract

This paper carries out the 3-D nonlinear numerical simulation of the dam body and base with the finite difference method in the consideration of the real condition of the concrete gravity dam of a hydroelectric station in Yunnan province. First of all, an integrated three-dimensional calculus model is set up, and the geological structure system is simulated. And then, the strain-stress relationship of the bedrock is simulated by the elastic-plastic constructive model and Mohr-Coulomb yield criterion. At last, the strain-stress distribution and the development law of plastic zone are analyzed under normal impounded level as well as overload level, and the weak point of the bedrock was thus exposed. © 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Kunming University of Science and Technology Keywords: Finite difference method, Gravity dam, Nonlinear Analysis, Strain-stress analysis.

1. Introduction

As an important type of dam in the dam construction of water conservancy and electric power, gravity dam is large in cubic content and has a relatively high demand for the base. With the development of dam construction, the geological condition of the dam base gets more and more complicated and the stability of the dam and base attracts more attention. According to statistics, of all the recorded cases of the devastation of gravity dam, 40% resulted from the destabilization of the dam base [1]. Rigid body limit equilibrium is now still the major method to analyze the stability of dam base. Besides, as the computer

* Corresponding author. Tel.: +86-0871-15887229621; fax: +86-0871-3398769. E-mail address: 441779143@QQ.com.

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technology develops, the methods of finite element and finite difference have been developed and improved [2].

A hydroelectric station is located on the middle reaches of Jinsha River. The dam is 129 m high. The water retaining structure is concrete gravity dam. The geological condition of the dam district is quite complicated. The base rock suffers various special geological conditions, there is squeezing rock foundation with developmental dislocation and mud, anti-dumping joint dislocation and extruded to form the deep foundation with deep sliding surface, steep dip in corrosion crack growth cracks and joints in the sand-like surface of lenses and other special geological defects, All these factors have adverse effects on the stability and deformation of the concrete dam body. All these factors have adverse effects on the stability and deformation of the concrete dam body. This paper carries out the 3-D nonlinear numerical simulation of the dam body and base with the Finite difference method. Then verify the rationality of the design, this paper carries out the 3-D nonlinear numerical simulation of the hydroelectric station to find the weakness of the bedrock, thus providing instruction for the design and construction.

2. The Constitutive Model and Yield Criterion of Rock Mass

The elastic-plasticity constitutive relationship of the dam base material is assumed for the model:

{ } [ ]{ }epD (1)

where { } and { } is the respectively to stress array and strain array, [ ]epD is the elastic-plasticity

array. A great many indoor and outdoor experimental results indicates that as to the weak interlayer of the

dam base rock, evident plastic yield shows itself under destruction without evident peak value. As a result, this paper adopts the ideal elastic-plasticity constitutive model to simulate the stress-strain relationship. But as to the relatively complete rock of the dam base, this paper adopts the widely-used Mohr-Coulomb yielding criterion [3-4]. This criterion shows the yielding condition of the rock in a quite real way, taking internal friction angle, cohesive strength into consideration, and thus approaching the real constructive nature of the rock [5-6].It is shown in Figure 1 and Figure 2. Where is the friction angle, c is the cohesive strength.

1 2 3 1 3 1 31 1( , , ) ( ) ( )sin cos 02 2

F c (2)

C

(n n

2

2

Fig.1 Mohr-Coulomb yielding criterion

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Fig.2 The shape of Mohr-Coulomb yielding criterion in the flat face and stress-space

3. The Three-Dimensional Stress-Strain Analysis of the Gravity Dam Base

3.1. The Calculation Model.

This paper selected a hydroelectric gravity dam for the study. The cross section of the dam body is shown in Figure 3 (a). The geological structure of the dam base is shown in Figure 3 (b). The structure is quite complicated: J11 represents the belt of fault zone，J2s represents the conglomerate zone, and F represents the fault zone.

In order to simulate real dam and foundation of the stress-strain, the three-dimensional stress-strain calculation demands a relatively wide model range and considers the influence of the key geological body and structure. The calculating range in this paper is as follows: taking 1.5 of the height of the gravity dam both on the upper and lower reaches along the river. The depth of bedrock range is 124m high. The whole space of calculation is divided by eight-node hexahedral element discretization, and some local irregular space is divided by five-facet triangular tetrahedral element [6-7]. The model disperses into 22 332 nodes and 16 311 elements in total. The calculation makes use of the elastic-plasticity constitutive model to simulate the nonlinear nature of the concrete material and bedrock materials. The whole model of the three-dimensional grid is shown in Figure 4. For the sake of analysis, ten characteristic points are picked on the dam site, where the displacement and stress of the points is carried out.

Fig.3 (a) the cross section of the dam; (b) the geological structure of the dam

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Fig.4 The 3-D grid of the dam and rock

3.2. Material Parameter and Calculation Load.

The material parameter of the rock is shown in Table 1. The major material of the dam is C15, C20 concrete. The compression strength of C15 is 7.5MPa, tensile strength is 0.9 MPa, Young's modulus is 22GPa, Poisson’s ratio is 0.167 and gravity is 24.0 kN·m-3 . The compression strength of C20 is 12.5MPa, tensile strength 1.30 MPa, Young's modulus is 28GPa, Poisson’s ratio 0.167 and gravity 24.0 kN·m-3.

Table 1 the mechanics parameters of the rock mass of the dam base

Rock mass Gravity/

[kN·m-3] Young's modulus/ Pa Poisson ratio μ

Shear strength Compression strength/ MPa f′ C′/MPa

J2s2 22.0 4.0 0.30 0.230 0.6 5.0

J2s3 26.0 8.0 0.22 1.400 1.12 70.0

J2s4 25.0 5.0 0.24 0.620 0.69 40.0

crush zone 24.0 3.0 0.29 0.035 0.3 12.0

fracture 24.0 1.0 0.29 0.000 0.27 \

The load combination of the calculation is normal work condition, the load combination is include the weight of the dam body, the water pressure of the normal impounded level on both upper and lower dam faces, initial gravity of the rock mass, sediment stress and uplifting pressure. The water level of the upper reaches is 1134.00, and that of the lower reaches are 1036.78, the height of sediment is 1082.00. The sediment buoyant unit weight is 9.5kN/m3, and the sediment internal friction angle is 24°.

4. Calculation Results and Analysis

The stress-strain distribution of the dam body and base has been calculated. The displacement and stress are showed in Table 2 in details. Under normal work condition, the dam body displacement agrees with the general law. The whole deformation trend of the dam body is: the dam axis tilts towards the lower reaches. The largest displacement of the dam body along the river is 32.2mm, and takes place at the dam top. The largest vertical displacement is -21.60mm, takes place at the dam top. The horizontal

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displacement of the dam body is quite limited, the greatest compressive stress at the dam is -6.1MPa; the greatest principal tensile stress is 0.7 MPa, and takes place at the superficial layer of the heel of dam. Due to the small size of these regions, the whole stress distribution of the dam is not greatly influenced.

At the same normal impounded level condition, the contour of the displacement of the rock mass is shown in Figure 5, and the contour of stress of the rock mass is shown in Figure 6 and Figure 7. The largest displacement along the river is 18.07mm; the largest vertical displacement is -19.2mm. Because of the low counter cutting strength and large width (9.0mm) of the erosion conglomerate zone (J2s2-4), the left and right rocks disturbed each other under gravity along the J2s2-4 with the largest displacement 3.00mm. Under the pressure of the water along the river, the hanging layer of fault and heading layer of fault disturb each other with the largest displacement 4.0mm.

Table 2 the stress and displacement of the characteristic points

Characteristic

points

Displacement/mm Stress/MPa

Along the river Vertical direction x y xy 1 3

1 22.975 -14.996 -1.076 -3.842 -1.256 -0.590 -4.127

2 26.514 -18.709 -1.125 -1.767 -0.155 -0.483 -1.803

3 29.142 -20.162 -0.496 -0.733 0.015 -0.205 -0.734

4 31.154 -20.854 -0.174 -0.487 0.005 -0.110 -0.487

5 32.235 -21.052 0.003 -0.034 0.001 0.003 -0.034

6 32.240 -21.605 0.002 -0.034 0.000 0.002 -0.034

7 31.115 -21.580 -0.136 -0.370 0.206 -0.016 -0.491

8 28.649 -21.992 -0.432 -0.734 0.546 -0.016 -1.149

9 24.119 -19.683 -1.234 -2.213 1.655 0.002 -3.449

10 22.233 -17.342 -6.965 -4.787 5.089 -0.672 -6.10

Fig.5 (a) displacement along the river of the dam rock; (b) Vertical displacement of the dam rock (unit: m)

Fig. 6 (a) the normal stress along the river of the dam base; (b) the vertical normal stress of the dam base (unit: MPa)

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Fig. 7 The shear stress

xy contour of the dam base (unit: MPa)

In order to know about the developing trend of the plastic zone of the rock mass under overloaded condition, this paper conducts the overload calculations under different water levels. This paper calculates the distribution of rock mass plastic zone under 1.0, 1.5, 2.5, and 3.5 normal impounded levels; under different overload impounded level, the dam base rock deformed plastically in different degrees and zones. As is shown in Figure 8 and Figure 9, under the 1.0 normal impounded level, the plastic zones are distributed in fracture, the J11 crush zone in the rock above fracture and the erosion zone J2s2-2、J2s2-3、J2s2-4. Thanks to the fact that these zones don’t form the connected zone, the safety of the dam is not greatly influenced. Under 1.5 normal impounded level, the plastic zone is slightly expanded towards the deep layer compared with the 1.0 condition. Under 2.5 normal impounded level, the erosion J2s2-5 strip zone deforms greatly, and the strip zone deformed plastically before deformed further. Under 3.5 normal impounded level, apart from the great plastic deformation of J11 crush zone and the J2s2-2 、J2s2-3 、J2s2-

4 、J2s2-5, the rock of the fracture formed a connected plastic zone between the upper and lower reaches, and the plastic deformation takes place across the dam, leading to the loss of bearing capacity.

Fig.8 (a) the whole plastic zone of dam rock mass under 1.0 water head; (b) the whole plastic zone of dam rock mass under 1.5 water head

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Fig.9 (a) the whole plastic zone of dam rock mass under 2.5 water head; (b) the whole plastic zone of dam rock mass under 3.0 water head

5. Conclusions

The bearing capacity of the gravity dam rock mass is a widely concerned issue. This paper takes the real condition of a hydroelectric station, makes use of the finite difference method to make a three-dimensional numerical simulation, and draws the following conclusions:

(1) Under normal impounded level, the stress of the dam body and rock mass and the displacement agrees with the general law: the largest compression stress and tensile stress distributed on the heel of dam and dam site, and the largest displacement takes place at the dam top. The rock mass zones attracting major attention is the erosion conglomerate strip zone J2s2-4 located directly under the dam base and the fracture F. The shear strength of these zones is quite low and the rock body long them disturb each other.

(2) The weakest part of the rock mass is the crush zone and the erosion strip zone with low shear strength, and thus suffering the plastic yielding damage before other parts.

(3) From the overload damage degree we can see that as the water pressure increases, the plastic yielding occurs in deep base of the erosion strip zone and develops fast. As the water pressure further increases, the shear failure zone appears in the concrete around the dam site and the plastic failure zone of the heel of dam expanded. The yielding zone of the face of the constructive base is enlarged from the heel of dam and dam site to the inner part of the dam till the plastic deformation connecting the upper and lower reaches appears and the bearing capacity of the whole structure is lost.

Acknowledgements

It is a project supported by National Natural Science Foundation of China (grant number: 50839003, 51009074, 11072092) and Yunnan Province Application Basic Research Fund (grant number: 2008ZC028M).

References

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