Mathematical Problems in Engineering

Volume 2019, Article ID 9575897, 14 pages

https://doi.org/10.1155/2019/9575897

## ECC Design Based on Uniform Design Test Method and Alternating Conditional Expectation

^{1}Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, 650550 Kunming, China^{2}Country Garden Real Estate Limited Company of Yunnan Province, 650000 Kunming, China

Correspondence should be addressed to Zude Ding; moc.361@tdsvdzd

Received 2 July 2019; Revised 23 August 2019; Accepted 3 September 2019; Published 1 October 2019

Academic Editor: Alessandro Formisano

Copyright © 2019 Xiaoqin Li et al. 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

Engineered cementitious composites (ECC) have higher ultimate tensile strains than normal concrete. The mechanical properties of ECC strongly depend on raw materials and the mix proportions. The uniform design test method and alternating conditional expectation, which is a nonparametric regression analysis method, were used to design the ECC mix proportion. According to the regression analysis, the optimized W/B, S/B, and F/B ranges could be obtained as 0.35–0.42, 0.25–0.3, and 0.02, respectively. The tested proportions for validation were randomly adopted within the range of W/B, S/B, and F/B. The uniaxial compression, tension, and four-point bending tests were conducted to verify the material behaviour of the designed ECC. Results showed that all the specimens had large ultimate tensile strains and high fracture energy capacities, and strain hardening was also observed. The fibers were found to be uniformly distributed in the specimens by using a scanning electron microscope. This paper may provide theoretical and practical guidance for the ECC and other cement-based material mix proportion design.

#### 1. Introduction

Concrete is widely used in civil engineering. However, concrete is brittle, and its toughness decreases with its increasing strength, which may lead to cracking in structures [1–3]. Engineered cementitious composites (ECC) were developed due to the advantages in toughness and energy absorption capacities. The ultimate tensile strain of ECC is 1–8%, whereas that of normal concrete is only about 0.01%. Moreover, the tensile ductility of ECC is nearly 300 times that of normal concrete [4]. Polyvinyl alcohol (PVA) fiber, which is normally used in ECC [5], is an acid/alkali-resistant material with high elastic modulus, high tensile strength, and low industrial cost. In the current paper, ECC is referred to as PVA-ECC only, and the fiber volume ratio of ECC is 2% or less [6].

The mechanical properties of cement-based materials depend on their raw materials. Based on the chosen local materials, the ECC material properties are mainly controlled by mix proportions. The mix proportion design methods, which have already been used in the ECC design, include mixing proportion table method [7], absolute volume method [8], idealized comprehensive test design method, and orthogonal experimental design method [9]. For the mixing proportion table method [7], existing mix proportions of ECC from literatures and their performances, such as the specimen tensile strains, need to be carefully reviewed and listed firstly. Then, one or several mix proportions with better specimen performances can be selected accordingly, and validation tests must be conducted based on the chosen mix proportions. However, the mixing proportion table method is not scientific and rigorous enough as the raw materials are different from the literatures. The absolute volume method [8] is based on the normal concrete mixture design standard [10]. The per cubic meter of concrete mixture is equal to the sum of the absolute volume of each material and the small volume of air. According to the code JGJ 55-2011 [10], the process of absolute volume method is as follows: (1) calculate the water-binder ratio based on the strength of concrete; (2) according to the water-binder ratio and the kinds of mineral admixture (fly ash, silica fume, etc) added, the dosage of mineral admixture can be obtained according to the Table 3.0.5-1 listed in the code; (3) according to the expected slump and the aggregate size, the unit water dosage can be obtained according to Table 5.2.1-2; (4) the dosage of sand based on the sand ratio can be calculated according to Equation 5.5.2 in the code; (5) 1% volume air need to be considered without air-entraining agent; (6) the initial mix proportions can be achieved, and the final mix proportions need to be done after testing and adjusting. Normal concrete mix proportions can be properly designed with this method, but large errors may exist when using this method to design ECC mix proportions due to the differences of the raw materials between normal concrete and ECC. With the idealized comprehensive test design method, a number of tests need to be conducted with different level combinations of every test factors. However, this method for mix proportion design is difficult to be achieved when the numbers of the test factors and levels are over 3. The orthogonal experimental design method [9] is much more widely used in statistical tests and design because of its effective performance in terms of dealing with multifactor experiments and screened optimum levels [11, 12]. However, researchers should have a clear view of the relationships between the objective function and the related factors when performing the test data regression analysis. This relationship must be implicit before the regression analysis [13].

The four design methods listed above all have disadvantages for the ECC design. In the current paper, the uniform design test method and the alternating conditional expectations were used for the ECC design based on the local materials in Kunming, Yunnan province, China. The uniform design test method performs more effectively. For instance, if *s* is the number of the chosen design factors and *q* is the number of factor levels, the number of tests required is when the idealized comprehensive test design method is used for the ECC design. The number of tests needed is , if the orthogonal experimental design method is used, whereas the number of tests required is only *q* when the uniform design method is chosen. Meanwhile, the alternating conditional expectation analysis used in this paper does not require a predefined explicit relationship between the objective function and the related factors. The analysis only depends on the results of the uniform design tests.

#### 2. Uniform Design Test Method and Alternating Conditional Expectation

The uniform design test method was designed by Fang and Wang [14]. This test design method is based on the uniformity of the test points and is an application of the Monte Carlo method. The uniform tables and application tables [15] are used to design and organize the test data. The uniform table is a normalized table, and each table is a matrix with *n* rows and *m* columns. The expression of the uniform table is , where *U* is the uniform table; *n* is the number of tests required, which is normally 3–5 times that of the number for the chosen design factors; and *q* is the number of levels of the factors. Normally, *q* is equal to *n*, and *m* is the number of design factors. According to the number of the chosen design factors (*s*, *s* ≤ *m*), the corresponding application tables [15] are used to determine the chosen uniform table . Deviations are generally used to measure the uniformity of the points in the test domain in selecting the best level combination of uniformity [14]. The most widely used deviation uniformity justification is the centered L_{2}-discrepancy (CD_{2}) [14]. Equation (1) is used to map the *n* × *s* elements *u*_{ij} in the chosen uniform table to (0, 1), thus forming the relevant mapping table.where *i* = 1, 2, … , *s* and *j* = 1, 2, … , *n*. The deviation CD_{2} of the chosen uniform table is given as follows:

For the uniform tests with the *s* factors, the first step in the design process is to determine the variation range of each factor in the test, which is referred as (*i* = 1, 2, … , *s*), where *X*_{i min} and *X*_{i max} are the minimum and maximum values of the *i*th factor, respectively. The test design table can be obtained using the following equation:where *i* = 1, 2, … , *s*, which are the factor sequence numbers; *j* = 1, 2, … , *n* are the level sequence numbers; and *X*_{ij} is the *j*th level value of the *i*th factor. The uniform table and the column number in the application table are selected according to the number of factors and levels to generate the combinative table , which is a matrix with *n* rows and *s* columns. Thereafter, the test result table can be obtained, which is a matrix with *n* rows and 1 column according to the corresponding test results. Finally, the chosen regression analysis, such as alternating conditional expectations, is applied to find relationships between and .

Based on and , the alternating conditional expectation regression analysis [16] which is a nonparametric regression analysis can be applied. Compared with the parametric regression analysis, the derived function does not depend on the predefined function type but only depends on the test data and their sample size with the nonparametric regression analysis [17, 18]. The function could be in various types, which provides more general conclusions rather than parametric regression analysis [19, 20]. Nonparametric regression analysis should be used when the relationship between the test factors and results is not clear. The alternating conditional expectations aim to fit the data in an additive, nonlinear model in the form of equation (4):where *A* is the dependent variable, *x*_{i} is the regression variable, *θ*, *φ*_{1}, … , *φ*_{s} are the smooth and nonlinear transformations of the data, and *δ* is a random residual. Nonlinear transformations are mostly used in alternating conditional expectations to maximize the correlation between and based on the data feature of the combinative table and test result table . Using the transformation of the best-fitting additive model can converge the regression function with the optimal solution. Meanwhile, the stability and symmetry of the random error *ɛ*_{i} can be satisfied. The target function can be determined by using the following equation:where is the inverse function of *θ*. S-Plus [21] can provide the alternating conditional expectation regression process. Only a specific command is required to invoke the calculation to obtain the results, including the transformation data set with the maximal correlation for the combinative table and the test result table . Accordingly, the relationship between and can be determined via interpolation algorithm with the following equation:where is a nonlinear transformation correlation between the combinative table and the test target *A*; *x*_{i} is the test vector of each factor; *X*_{ij} is the maximum correlation transform data of the combinative table ; and *X*_{i} is the test factor. is given by the following equation:

The test target is given using the following equation:where *y*_{i} is the test result vector of the target and *A*_{t} is the transformed data of the test result . Based on the above calculations, a developed interpolation algorithm program in MATLAB can be implemented to complete the construction of the target function relationship.

#### 3. ECC Design Based on the Uniform Design Test Method and the Alternating Conditional Expectation

##### 3.1. Raw Materials

Local materials from Kunming, Yunnan province, China were used as the raw materials for the ECC design, including the P.O 42.5 cement, class F fly ash, and quartz sand. The material properties of the cement and fly ash are listed in Tables 1 and 2. The quartz sand with a grain size between 0.2 mm and 0.4 mm was used as fine aggregates. The PVA fiber produced in Japan was used, and its material properties are listed in Table 3. The water-reducing admixture is polycarboxylate superplasticizer, and its water-reducing rate is 20%. The water-reducing admixture is not set as the design factor. Therefore, the dosage of the water-reducing admixture is based on the sump test of ECC, which is case dependent. The final water-reducing admixture to cement mass ratios are between 0-–0.022.