Tensile reliability analysis for gravity dam foundation surface based on FEM and response surface method

Tong-chun LI, Dan-dan LI*, Zhi-qiang WANG

College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, P. R. China

Tensile reliability analysis for gravity dam foundation surface based on FEM and response surface method

Tong-chun LI, Dan-dan LI*, Zhi-qiang WANG

College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, P. R. China

In this study, the limit state equation for tensile reliability analysis of the foundation surface of a gravity dam was established. The possible crack length was set as the action effect and allowable crack length was set as the resistance in the limit state. The nonlinear FEM was used to obtain the crack length of the foundation surface of the gravity dam, and the linear response surface method based on the orthogonal test design method was used to calculate the reliability, providing a reasonable and simple method for calculating the reliability of the serviceability limit state. The Longtan RCC gravity dam was chosen as an example. An orthogonal test, including eleven factors and two levels, was conducted, and the tensile reliability was calculated. The analysis shows that this method is reasonable.

tensile reliability; foundation surface of gravity dam; nonlinear FEM; response surface method; Longtan RCC gravity dam

1 Introduction

Reliability calibration of gravity dams mainly focuses on reliability analysis of stability against sliding of the foundation, local weakness, tensile resistances of the dam heal, and compressive strength of the dam toe. In recent years, many studies on this issue have been conducted in China and elsewhere. Up to now, research on the reliability of gravity dams has mainly focused on the analysis of reliability against sliding, and there has been little research on tensile reliability (Hao et al. 2009). The main reason is that the material mechanics method, which does not allow for the occurrence of stress, is the basic analysis method according to theDesign Specifications for Concrete Gravity Dams(SETC 2000). The control standards of tensile stress corresponding to the material mechanics method are experiential, and the real stress characteristics and possibility of the destruction of the dam and foundation cannot be reflected accurately. If these standards are satisfied, uncontrollable destruction of the dam cannot occur, but the possible destruction mode and range cannot be described either. In order to reflect the real stress distribution of the dam, the results of finite element method (FEM) analysis are combined with the current criteria of the gravity dam, and control standards of upstream vertical stress of the gravity dam are stipulated as follows: (1) Taking the upliftpressure of the upstream face of the dam foundation into account, the distribution width of the tensile stress should be less than 0.07 times the width of the dam base, or the distance between the dam heel and centerline of the curtain. (2) Taking the uplift pressure of the upstream face of the dam into account, the distribution width of the tensile stress should be less than 0.07 times the distance between the upstream face of the calculation section and the centerline of drainage holes (pipes). These standards cannot be used directly for dam cracking examination, because a dam subjected to tension is not necessarily destroyed; even if the distribution width of tensile stress is large, the tensile stress still might not exceed the tensile strength.

In fact, cracking in the upstream foundation surface is allowable, and the dam may fall into the serviceability limit states of normal operation if the crack width meets the control standards described above. The method of tensile reliability analysis of the dam foundation surface in the serviceability limit state is introduced in this paper. The possible crack length was set as the action effect and the allowable crack length was set as the resistance in this limit state. The nonlinear FEM was applied to obtain the crack length of the foundation surface of the gravity dam (Frangopol and Imai 2000). The linear response surface method (Deng et al. 2005) was applied and the orthogonal test design method (Ren et al. 2005) was used to reduce the workload of finite element analysis. The Longtan RCC gravity dam was considered as an example, and the results of the analysis are provided.

2 Reliability analysis based on response surface method

At present, the methods for reliability analysis of the engineering structure include the first-order second-moment method, Monte-Carlo method, response surface method, and stochastic finite element method. The response surface method, which has been developed in recent years, is a comprehensive statistical test technique and is effective for large and complicated engineering structures (Wong 1985). It is used to handle the effects of different variables on a system or structure, or as a conversion between the input and output of the system or structure. It is described withnvariables,x1,x2,…,xn, as follows:Z=g(x1,x2,…,xn), whereZis the structure response. This functional relationship is implicit. In general, large amounts of simulations are necessary to obtain this function. The response surface method seeks to obtain the relationshipZ=g(x1,x2,…,xn) by means of regression and fitting based on limited testing, and then to replace the real curve surface in the reliability analysis.

The method should be as simple as possible and flexible enough to reflect different real curve surfaces while selecting the expression form of the response surface (Kim and Na 1997; Bucher and Bourgund 1990). To bring the response surface function close to the real limit state function, the first-degree polynomial function is usually selected (Rajashekhar and Ellingwood 1993). For a situation withnrandom variables,

The coefficientai(i=0, 1,…,n) is determined as follows: Consideringμxi, the mean value ofxi, to be central,n+1 sampling points are selected from (μxi?f′σxi,μxi+f′σxi), whereσxiis the standard deviation ofxi, andf′ is the parameter for determination of marginal value, usually between 1 and 3. After the determination of the coefficientaiwith then+1 function valuesg(x) at sampling points, the response surface function is determined, and the reliability indexβand testing point, which is closer to the origin in the limit state surface, are obtained. Then, the testing point is considered the center, new sampling points are selected, and undetermined coefficients are determined by the same method. It is very important to conduct as few tests as possible but to achieve the goal of high precision at the same time.

In this study, the orthogonal test design method was used to arrange the response surface test. In the test, variables that might influence the reliability index, including the size of the structure and the geometrical character, are called factors. The specific condition of each factor is called a level. To determine the level of a factor, the values beyond the region of (μ?σ,μ+σ) are abandoned based on theσstatistics principle, whereμandσare the mean and standard deviation of the variable, respectively. According to the factors of the response equation as well as the level and number of the unknown factors, the scheme of the orthogonal test is selected. In the orthogonal test, each level of each factor and each combination of any two factors at different levels appear at the same frequency, so the test can comprehensively reflect the effect of each factor and level on the reliability index, and effectively reduce the number of tests.

Finally, the reliability indexβand design point, which is the closest one to the origin on the limit state surface, are calculated based on the reliability analysis with the generalized random space method (Zhao and Wang 1996). The calculation formula is

3 Nonlinear contact algorithm for crack length calculation

Pairs of contact points are assigned in the potential fracture area of the contact interface between the gravity dam and the bedrock, and the finite element hybrid method is adopted to deal with the frictional contact problem (Zhao et al. 2006), in which the initial tensile forces are allowable. The action force is decomposed into the external force and the contact forceon the contact interface, and the displacements of the contact interface and the contact force are considered mixed variables. With the displacement of the contact interface being the basic unknown variable, and the contact forces at nodes in the local coordinate system of the contact region being iterative variables, nonlinear iteration is carried out and restricted within the contact interface, and complicated nonlinear frictional contact is reflected by the changes of the contact force. Then, the iterative calculation of contact states and contact forces is performed. This makes the iterative calculation simpler, greatly improving the computational efficiency (Zheng and Das 2000; Refaat and Meguid 1998). In the process of calculation, it is unnecessary to specify coefficients of normal stiffness and tangential stiffness; the effect of human factors on the results and the embedding problem of other methods can thus be avoided.

At the beginning of calculation, various elements of the flexibility matrix are pre-calculated. In each step of calculation, the deformation modes of the contact interface, such as the opening, closing, and slip, are considered, the Mohr-Coulomb criterion is used to describe the friction effect of the contact interface, and the contact force at the node is obtained by an iterative solution based on the state assembly of the contact flexibility matrix. Finally, the contact force at the node is added to the total cumulative load array to obtain an integrated solution. Specific calculation methods and formulas can be found in the literature (Zhao and Wang 2006).

Another advantage of this method is that the uplift pressure can be considered to be acting directly on the dam and the surface of the bedrock. If the foundation surface does not crack, the uplift pressure can be reflected by the contact force between two contact interfaces; if the foundation surface cracks, the uplift pressure is equivalent to the external load on the bedrock and the dam, reflecting the splitting effect of water on the foundation surface.

4 Tensile reliability analysis of foundation surface of gravity dam

The analysis of tensile reliability of a gravity dam based on a combination of FEM and the response surface method is described above. The details of the processes are as follows:

(1) Based on the number of random variables and a reasonable scheme of orthogonal tests, the number of groups of FEM calculation is considered. The mean value of each variable is taken as the initial value of each random variable for iterative calculation, and their standard deviations are determined.

(2) Each group is calculated with the FEM, and the crack length of the foundation surface is calculated with the nonlinear contact algorithm.

(3) The limit state equation is obtained through regression and fitting, and the reliability index and values of random variables at testing points are calculated.

(4) These steps are repeated until the relative error of the reliability index between the two successive calculations is within a reasonable range. In this study, the relative error limit was set at 1%.

5 Case study

5.1 Validation of response surface method

We performed a case study to demonstrate the superiority of the response surface method. Fig. 1 is a calculation diagram of a plane frame structure, a building with three spans and twelve stories. The height of the building was 48 m, and the span lengths were 7.5 m, 3.5 m, and 7.5 m. The structure was composed of five kinds of rods, with different cross-section areas, denoted as 1 through 5. The elastic modulus of each rod was 2.0× 107kN/m2. The section areas of the rodsAi(i=1, 2,…,5)and the external loadPwere considered random variables. All the variables were normally distributed except the variableP, which was log-normally distributed. The mean values ofAi(i=1, 2,…,5) were 0.25 m2, 0.16 m2, 0.36 m2, 0.20 m2, and 0.15 m2, and the mean value ofPwas 30.0 kN. The standard deviations ofAi(i=1, 2,…,5)were 0.025 m2, 0.016 m2, 0.036 m2, 0.020 m2, and 0.015 m2, and the standard deviation ofPwas 7.50 kN.

Fig. 1 Calculation diagram

Based onCode for Design of Steel Structures(MC 2003), the maximum allowable horizontal displacement at pointBis=0.096 m, whereHis the height of the structure, which is 48 m in this case, so the limit state equation is as follows:

whereuBis the real horizontal displacement at pointB.

According to the literature (Tong and Zhao 1997), the reliability calculation requires 41 FEM analyses and three iterations. However, with the present method, the reliability is calculated with eight FEM analyses and three iterations. The reliability indexβand the design values of the variables,(i=1, 2,…,5) andP?, obtained with the two schemes are shown in Table 1. The present method is rational, and can effectively reduce the workload of numerical analysis.

Table 1 Reliability index and design values obtained with different schemes

5.2 Analysis of tensile reliability of Longtan Dam

The Longtan dam is located in Tian’e Country, in the Guangxi Autonomous Region. Itis the key project for power transmission from the west to the east of China. The dam is a grade I structure. It is a roller-compacted concrete gravity dam with a normal impounded level of 400 m above the mean sea level, a base level elevation of 210.0 m, and a dam height of 196.5 m. The dam was classified in three parts, I, II, and III, according to the different material characteristics, shown in Fig. 2.

A 2-D finite element model was established, as shown in Fig. 3. Thex-axis is the direction from upstream to downstream, and they-axis is the vertical direction. The grid graph contains 7 792 nodes and 7 562 elements. The different colors correspond to different materials.

Fig. 2 Material partition of dam (Unit: m)

Fig. 3 Grid graph of dam section

The different engineering loads on the dam are the following:

(1) Dead weight load: The density of concrete is 2.45 kg/m3.

(2) Hydrostatic pressure: The density of water is 1 kg/m3.

(3) Wind and wave load: The average elevation of the reservoir bottom in the water area is 280 m. For a basic load combination, the calculated wind speed is 24 m/s; for an accidental load combination, the calculated wind speed is 14 m/s. The length of the wind area is 2 km, and the prevailing wind is southwesterly.

(4) Sediment load: The centennial elevation of silt sediment is 287.6 m, the density of sediment is 1.22 kg/m3, and the internal friction angle of sediment is 24o.

(5) Uplift pressure: The calculation coefficients are determined according toDesign Specifications for Concrete Gravity Dams(SETC 2000).

Random properties of variables are listed in Table 2. The wind and wave load and sediment load have little effect on calculation results, so they are not treated as random variables. In Table 2,ρis the density of concrete;α1andα2are the upstream and downstream uplift reduction coefficients, respectively;Ei(i= I, II, and III) is the elastic modulus of different material partitions;Hsis the upstream water level;Hxis the downstream water level;cis the cohesive force;fis the friction factor; andftis the tensile strength. All the variables are normally distributed exceptc, which is log-normally distributed.

Table 2 Properties of variables

According toDesign Specifications for ConcreteGravity Dams(SETC 2000), the distribution width of tensile stress should be less than 0.07 times the width of the dam base or the distance between the dam heel and the centerline of the curtain. The distance is considered the resistance term and was 27 m in this study, so the limit state equation of tensile reliability was as follows:

wherexis the crack length.

A scheme of an orthogonal test (Ren et al. 2005) that includes eleven factors and two levels was adopted. Twelve nonlinear FEM analyses were conducted and the response equation was obtained by regression.

Using this method, the reliability index and design values were calculated. The detailed iterative process is shown in Table 3. The reliability indexβ= 2.56. The design values and sensitive coefficients of variables are shown in Table 4. The variableHshas the most significant influence on the reliability.

Table 3 Iterative process of reliability

Table 4 Design values of variables

6 Conclusions

In this study, the analysis of tensile reliability of a gravity dam was performed with the nonlinear FEM based onDesign Specifications for Concrete Gravity Dams(SETC 2000). In this method, the limit state equation of tensile reliability is established with the possible crack length being considered the action effect and the allowable crack length the resistance. Because this limit state equation is implicit, the linear surface response method was applied and the orthogonal test design method was used. The number of FEM analyses was greatly reduced. The Longtan RCC gravity dam was considered as an example, and the analysisshows that this method is reasonable, but further research is needed on the determination of the allowable crack length.

Bucher, C. G., and Bourgund, U. 1990. A fast and efficient response surface approach for structural reliability problems.Structural Safety, 7(1), 57-66. [doi:10.1016/0167-4730(90)90012-E]

Deng, J., Gu, D. S., Li, X. B., and Yue, Z. Q. 2005. Structural reliability analysis for implicit performance functions using artificial neural network.Structural Safety, 27(1), 25-48. [doi:10.1016/j.strusafe.2004. 03.004]

Frangopol, D. M., and Imai, K. 2000. Geometrically nonlinear finite element reliability analysis of structural systems.Computer and Structures, 77(6), 693-709. [doi:10.1016/S0045-7949(00)00010-9]

Hao, Z. Q., Wu, L., and Guo, J. 2009. Reliability computation of stability for gravity dams based on FEM.China Rural Water and Hydropower, 5, 127-129. (in Chinese)

Kim, S. H., and Na, S. W. 1997. Response surface method using vector projected sampling points.Structural Safety, 19(1), 3-19. [doi:10.1016/S0167-4730(96)00037-9]

Ministry of Construction, P. R. China (MC). 2003.Code for Design of Steel Structures (GB50017-2003). Beijing: China Planning Press. (in Chinese)

Rajashekhar, M. R., and Ellingwood, B. R. 1993. A new look at the response surface approach for reliability analysis.Structural Safety, 12(3), 205-220. [doi:10.1016/0167-4730(93)90003-J]

Refaat, M. H., and Meguid, S. A. 1998. A new strategy for the solution of frictional contact problems.International Journal for Numerical Methods in Engineering, 43(6), 1053-1068. [doi:10.1002/(SICI) 1097-0207(19981130) 43:6<1053::AID-NME460>3.0.CO;2-L]

Ren, X. J., Shi, D. Y., and Wang, J. H. 2005. Positive cross method of responsive section experiment design.Shanxi Architecture, 31(2), 17-18. (in Chinese)

State Economic and Trade Commission, P. R. China (SETC). 2000.Design Specifications for Concrete Gravity Dams (DL5108-1999). Beijing: China Electric Power Press. (in Chinese)

Tong, X. L., and Zhao, G. F. 1997. The response surface method in conjunction with geometric method in structural reliability analysis.Civil Engineering Journal, 30(4), 51-57. (in Chinese)

Wong, F. S. 1985. Slope reliability and response surface method.Journal of Geotechnical Engineering, 111(1), 32-53. [doi:10.1061/(ASCE)0733-9410(1985)111:1(32)]

Zhao, G. F., and Wang, H. D. 1996. A practical analysis method for reliability of structures in generalized random space.China Civil Engineering Journal, 29(4), 47-51. (in Chinese)

Zhao, L. H., Li, T. C., and Niu, Z. W. 2006. Mixed finite element method for contact problems with friction and initial gaps.Chinese Journal of Geotechnical Engineering, 28(11), 2016-2018. (in Chinese)

Zheng, Y., and Das, P. K. 2000. Improved response surface method and its application to stiffened plate reliability analysis.Engineering Structures, 22(5), 544-551. [doi:10.1016/S0141-0296(98)00136-9]

*Corresponding author (e-mail:ldd0517@hhu.edu.cn)

Received Jan. 13, 2010; accepted Apr. 21, 2010