Mathematical Problems in Engineering

Volume 2018, Article ID 8743505, 12 pages

https://doi.org/10.1155/2018/8743505

## Study on Viscoelastic Deformation Monitoring Index of an RCC Gravity Dam in an Alpine Region Using Orthogonal Test Design

^{1}College of Hydraulic & Environmental Engineering, China Three Gorges University, Yichang 443002, China^{2}State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China

Correspondence should be addressed to Zhiyong Wan; nc.ude.uhw@nawgnoyihz

Received 15 January 2018; Accepted 26 March 2018; Published 9 May 2018

Academic Editor: Xander Wang

Copyright © 2018 Yaoying Huang and Zhiyong Wan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The main objective of this study is to present a method of determining viscoelastic deformation monitoring index of a Roller-compacted concrete (RCC) gravity dam in an alpine region. By focusing on a modified deformation monitoring model considering frost heave and back analyzed mechanical parameters of the dam, the working state of viscoelasticity for the dam is illustrated followed by an investigation and designation of adverse load cases using orthogonal test method. Water pressure component is then calculated by finite element method, while temperature, time effect, and frost heave components are obtained through deformation statistical model considering frost heave. The viscoelastic deformation monitoring index is eventually determined by small probability and maximum entropy methods. The results show that (a) with the abnormal probability 1% the dam deformation monitoring index for small probability and maximum entropy methods is 23.703 mm and 22.981 mm, respectively; thus the maximum measured displacement of the dam is less than deformation monitoring index, which indicates that the dam is currently in a state of safety operation and (b) the obtained deformation monitoring index using orthogonal test method is more accurate due to the full consideration of more random factors; the method gained from this study will likely be of use to diagnose the working state for those RCC dams in alpine regions.

#### 1. Introduction

A Roller-compacted concrete (RCC) dam is constructed with the Roller-compacted placement method in thin layers of dry lean concrete, composed of mixed sand aggregate and cement [1]. In recent years, RCC dams have been commonly applied in dam domains because of its low cost, rapid construction, and a better control of heat generation of concrete. However, the safety of RCC dams during operation period is extremely crucial, especially in alpine regions. In order to ensure the safety operation of dams over the long-term service period, it is necessary to monitor the dam working performance by carrying out real-time analysis of monitoring data [2, 3]. As the deformation monitoring index is a crucial index to evaluate and monitor dam safety, it is of high engineering significance for the safety operation of dams. Generally, the deformation monitoring index is an alarm or extreme value of deformation under possible loads according to the* Technical Specification for Concrete Dams Safety Monitoring* [4].

The deformation monitoring index of dams may be primarily determined using two approaches. Initially, the deformation monitoring index is obtained through the mining of dam deformation information using mathematical model on the basis of the existing monitoring effect quantity. Prior studies on monitoring index determined by mathematical model have focused on the methods of confidence interval, small probability, and statistical model [5, 6] which have been commonly used in numerous projects. However, due to the lack of consideration about spaciousness and fuzziness as well as randomness for multipoint monitoring in these methodologies, in recent years, Lei et al. [7] proposed the early warning index of spatial deformation of high concrete dams based on deformation entropy; G. Yang and M. Yang [8] proposed the determination method of multistage warning indicators for the overall deformation of concrete dam regarding fuzziness and randomness; Qin et al. [9] combined comprehensive block displacement with multidimensional confidence region method to diagnose the safety of concrete double-curvature arch dam in Sichuan Province, China. Although it is convenient for engineers to determine deformation monitoring index by these mathematical models, the calculated displacement is the only alarm value or extreme value when the dam has experienced the extreme load case combinations in the long-term monitoring data.

On the other side, the deformation extreme value can also be determined on the basis of the structural numerical techniques of the dam body and its foundation. Numerous scholars have attempted, in terms of finite element numerical techniques, to assemble the adverse loads of dams that may occur during operation period, which makes up for the shortage of the adverse load combination cases of monitoring data series observed by precise instruments. For instance, Wu et al. [10] proposed that the structural characteristics of RCC dams should be divided into three stages: elasticity, elastic-plasticity, and unstable failure. The deformation monitoring index of dams was determined through the introduction of the statistical model [11–14], and the method was successfully applied to Shapai RCC arch dam in Sichuan province, China. Considering the complexity characteristics of RCC structure [15], such as the complex anisotropy, Gu [16] defined the diagnostic index of three-stage safety deformation from the perspective of yield ratio mutation feature of layer surface and foundation plane, and then the theory was successfully applied to Longtan dam project in Guangxi province, China.

Prior studies on monitoring index have focused on adverse load combinations that offer a simple combination by the accumulation of adverse loads rather than forming a full consideration on the actual situations (e.g., loads, boundary conditions, and uncertain parameters). For instance, the Canadian Turkstra combination rule is customarily used in the engineering field, which holds that the maximum effect value of the load combination manifests when a variable load reaches the maximum value in the design benchmark period and the other variable loads are in the form of instantaneous value [10]. The form of load combination is pretty rough with the characteristics of subjective factor; in particular, the running state of dams presents the characteristics of fuzziness and randomness; thus the load combination cases selected are not always representative. Moreover, with the presentation of the complex behavior of dams in alpine regions, only few scientific publications have been published concerning deformation monitoring index of dams in alpine regions. Therefore, a typical combination of adverse case sample is selected using orthogonal test design method [17] so as to determine deformation monitoring index of dams in alpine regions.

In this study, a typical RCC gravity dam in an alpine region is taken as a case, and the typical water retaining dam block is selected as an analysis object. A combination sample of adverse cases is designed using orthogonal test method; the total effect quantity (i.e., deformation) sample is obtained through statistical model and finite element method, and then the viscoelastic deformation monitoring index for the dam in an alpine region is eventually determined based on the small probability and maximum entropy methods.

#### 2. A Method of Determining Viscoelastic Deformation Monitoring Index for RCC Gravity Dams

##### 2.1. Orthogonal Test Design

Orthogonal test design, as a highly efficient way capable of dealing with multifactor tests, is commonly adopted to arrange and analyze datasets by means of selecting a reasonable orthogonal table based on levels and factors [17]. In considerable combinations, it is convenient to conduct a test with the method employed to select a representative combination with the characteristics of “uniform dispersion” and “neatly comparable.” Simultaneously, it may make the test cases with approaches of different combinations analyzed comprehensively available, which is characterized by less times, high execution efficiency, and convenient operation.

##### 2.2. Effect Quantities of Adverse Load Combination

###### 2.2.1. Deformation Statistical Model Considering Frost Heave

In the case of an RCC dam in an alpine region, the dam crest displacement is affected by water pressure, temperature, time effect, and frost heave [6, 18, 19] when the dam crest has no thermal insulation measures. For this purpose, the modified deformation statistical model for the concrete dams considering frost heave in alpine regions is expressed as follows:where represents the displacement caused by dam body and its foundation deformation under the action of a water load; represents the displacement caused by the temperature change of dam body concrete and dam foundation rock; denotes the time effect (time-dependent) displacement which is employed to represent the creep of dam concrete and bedrock, as well as the plastic joint deformation; (i.e., ) denotes the frost heave deformation of dams in alpine regions, including a periodic term and a hysteresis term . The water pressure, temperature, and frost heave are assumed to be elastic components, while the plastic components caused by water pressure, temperature, and frost heave are all included in the time effect component. These components [6, 18] can be written as follows:where is the water pressure component regression coefficient; for a gravity dam and for an arch dam; is the water depth on the monitoring day and is the water depth of the beginning day; and and are temperature component regression coefficients. or corresponds to a year and half-year cycle, respectively, and , , and are time effect component regression coefficients. denotes the cumulative days from the beginning day to the monitoring day, denotes the days from the beginning day to the calculated day; denotes the number of days from the beginning day to the monitoring day divided by 100; denotes the number of days from the beginning day to the calculated day divided by 100.

represents the frost heave factor to distinguish the periodic function in the same model; represents the number of days from the beginning day to the monitoring day; represents the number of days from the beginning day of the analyzed monitoring series to the beginning day of the negative temperature in the same year; and are the frost heave component periodic regression coefficients; , and are the frost heave component hysteresis regression coefficients; and represents temperature hysteresis factor, denotes the effect of time on temperature hysteresis, and denotes the number of days at the average temperature before the hysteresis time is affected. Generally, as a unit of weeks, days, or months, the hysteresis days are determined by the measured temperature and hysteresis temperature of dams.

Additionally, in (1), with respect to a definite step function, is defined byin which °C is the reference temperature of the concrete, denotes the concrete temperature, and the function “differential” is a Dirac function due to a jump when ; thus, is referred to as the unit step function.

###### 2.2.2. Effect Quantities

The basic loads consist of water pressure, uplift pressure, sediment pressure, and other loads regarding the permanent cases during the operation period of dams. Some analysis is conducted by selecting and combining these loads which have a strong influence on structure deformation. Due to the complexity of load combination under adverse cases, in this study, the water pressure and uplift pressure are recognized as the investigated loads, and then the orthogonal test method, according to the range of loads and levels as well as factors, is introduced to assemble loads under adverse cases. Combined with (1), together with finite element numerical techniques, it is feasible to obtain total effect quantity samples.

To be specific, effect quantities consist of water pressure component, temperature component, time effect component, and frost heave component in this study. Water pressure component is calculated using the finite element method, and the transversely isotropic mechanical parameters in the finite element model are obtained from inversion; while the temperature component is calculated through statistical model expression form, as shown in (3), time effect and frost heave components are also calculated by statistical model expressions, as shown in (4) and (5)-(6), respectively.

##### 2.3. Deformation Monitoring Index of Dams

*(1) Small Probability Method*. Effect quantities under each adverse load combination are obtained through the 3D finite element method and deformation statistical model of an RCC dam; thus the sample space is developed; moreover, the Kolmogorov-Smirnov (K-S) distribution test method is utilized to diagnose the probability density function on the basis of the eigenvalues (i.e., mean and standard deviation) of samples. Here is considered as the deformation monitoring index value; the running state of a dam will be abnormal provided that (measured data) is greater than (monitoring index); the abnormal probability of dam deformation can be given by

As the abnormality of a dam is regarded as a small probability event, the abnormal probability is conventionally determined by the project grade or rank; thus the deformation monitoring index is written as , where , are mean and standard deviation of samples, respectively.

*(2) Maximum Entropy Method*. The maximum entropy method is able to be adopted when the distribution function for the actual monitoring effect quantities can not fully conform to the classical distribution function. There is no need to assume the distribution function type of samples in advance; the probability density function with higher precision can be obtained directly based on the numerical eigenvalues of samples [20]. The calculation steps are as follows.

*Step 1. *According to the principle of maximum entropy, when the probability distribution reaches the minimum deviation under the given constraint condition of sample information, the entropy will reach the maximum. The objective function and constraints are written asAdditionally, where denotes the *-*th sample value; represents sample size; is the domain of integration and generally it can be approximated as ; and are sample mean value and standard deviation, respectively; ; is -th origin moment; is the order of origin moment.

*Step 2. *According to a set of sample information , the origin moment of displacement sample is calculated, and the Lagrange multiplier method is employed to solve the maximum value of the entropy . The maximum entropy probability density function is, therefore, expressed aswhere and represent Lagrange multiplier coefficients.

Hence, according to (10a) and (10b) and (12), the following equation can be obtained.

The Lagrange multiplier coefficients of the aforementioned formula are estimated by optimization algorithm.

*Step 3. * is assumed as the deformation monitoring index; the abnormal or dangerous possibility for dams can be described by

The probability of abnormal behavior for dams is rather low, the abnormal probability is commonly regarded as 1% or 5% considering structural importance (rank or grade), and then the deformation monitoring index value can be obtained based on the property of the inverse cumulative distribution function.

The flow diagram of implementation for viscoelastic deformation monitoring index of dams is shown as Figure 1.