Calculation Model for the Mixing Amount of Internal Curing Materials in High-strength Concrete based on Modified MULTIMOORA

2.2.1 OWA operator theory

There are subjective uncertainties and randomness in experts’ determining the evaluation index weights. The ordered weighted averaging (OWA) operator is a subjective assignment method that weakens the adverse effect of experts’ outliers through the combinatorial number formula or the normal distribution function [11]. Here are the steps of the OWA method:

1. Select Z experts to score index a_i, and the result is recorded as (e₁e₂ · · · e_z) of the index a_i given by Z experts, and arrange them in descending order to obtain a new array represented by vectors bi, bi = [b_l+1]_1×z = (b₁b₂ · · · b_z), l = 0, 1, 2, · · · z − 1.

2. Use the combinatorial number formula (2) to eliminate the effect of outliers and obtain the weighted vector w_i = [w_l+1]_1×z of the vector b_i

   (2)wl+1=Cz−1l∑k=0z−1cz−1k=Cz−1l2l−1,l=0,1,2,⋯z−1

3. Weight the evaluation indexes through the weighted vector w_i to obtain the absolute weight w_i of the index a_i

   (3)w¯i=∑l=1zwl+1⋅bl+1wl+10,1,l0,n−1

4. Calculate the subjective weight wi1of each index in the index set A

   (4)wi1=w¯i∑i=1mw¯ii=1,2,⋯m

2.2.2 Entropy weight theory

The entropy weight method is to determine the objective weight of an index [12]. The main steps are as follows:

1. Construct a judgment matrix and normalize the data using the range method

2. Determine the entropy value H_i of the i-th index

   (5)Hi=−1lnn∑j=1nfijlnfiji=1,2⋯m;j=1,2⋯n

3. Where: m represents the evaluation index, n represents the test scheme, fij=1+xij∑i=1m1+xij,xijrepresents

   the normalized result of the i-th index under the *j-th test scheme.

   Determine the weight: assume wi2as the entropy weight of the i-th evaluation index.

   (6)wi2=1−Hin−∑j=1mHi,i=1,2⋯m

2.2.3 Weight optimization model

The above weight determination methods respectively reflect the subjective and objective effects of the evaluation. To reduce such effects, the OWA operator method and the entropy weight method are coupled to establish a weight optimization model, namely:

Where, wi1andwi2respectively represent the weight of the i-th index calculated using the OWA operator method and the entropy weight method, and w_i represents the comprehensive weight of the i-th index.

4.1.1 Raw materials and mix proportion

See Table 1 for the mix proportion of internally cured concrete, in which the cement was P.O42.5 ordinary Portland cement; the fly ash was at Grade I with specific surface area of 455 m²/kg; the silica fume’s specific surface area was 20000 m²/kg; the fine aggregate was the Yaohe river sand with fineness modulus of 2.8, grading zone II; the coarse aggregate was the limestone gravel (5~20mm); QS80-20 polycarboxylic acid water reducing agent produced by Shanghai Softening Chemical was used, with water-reducing rate of 30% when its content accounted for 0.1% of the cement content; the super absorbent polymer had 60-100 meshes (particle size 0.01-0.25 mm), and the water absorption in the test was set to 25 times its mass. The test used 5-20mm gravel shale ceramsite produced by Yichang Baozhu Ceramsite Company, with apparent density of 1,570 kg/m³ and saturated water absorption of 12%.

The maximum water requirement for internal curing calculated according to the mix proportion was 32kg/m³. The initial mixing amount of internal curing materials [16] is shown in Table 1. Considering the mixing method of the internal curing water [17], it was designed that the total water-binder ratio of HSC 12.5- and HSC 25- groups remained unchanged. Partial coarse aggregates were replaced by LWA with equal quality, mixed with additional SAP. The test scheme was designed as shown in Table 1.

4.1.2 Test methods and results

The compression test was performed in accordance with GB/T50081-2019 Standard for Test Methods of Concrete Physical and Mechanical Properties, and the antifreezing test and self-shrinkage test were performed in accordance with GBT 50082-2009 Standard for Test Methods of Longterm Performance and Durability of Ordinary Concrete. One-side freezing-thawing method was used to test the quality loss and relative dynamic elastic modulus of test blocks after 28 days of curing and 25 freezing-thawing cycles in the antifreezing test. In the self-shrinkage test, the non-contact method was adopted. Because the self-shrinkage was intense during the high-strength coagulation and at an early age, the measurement was carried out in the first 3 days when the shrinkage was basically stable. The final shrinkage value was determined by fitting. The test results are shown in Table 2.

Based on the analysis of the results in the table, it is found that: (1) the compressive strength is positively correlated with the relative dynamic elastic modulus and inversely correlated with the quality loss rate; the self-shrinkage is greatly affected by the internal curing; (2) compared with the baseline group, due to the diversion of additional water, the water-binder ratio increases and the strength of concrete in the test group (additional water diversion) decreases; however, because the pore structure can be improved by the secondary hydration through internal curing water release, its freezing resistance is enhanced; (3) compared with SAP, LWA is porous, low in strength and inadequate in water release [18]. so it has poor strength, shrinkage and freezing resistance; and (4) compared with the baseline group, the total water-binder ratio of HSC 12.5-and HSC 25- groups remains unchanged, and because the internal curing material can continuously maintain the cement during hydration, the performance of the two groups was better.

4.2.1 Establishment of evaluation indexes

In this paper, the performance of internally cured concrete was evaluated and analyzed based on the Multimoora theory with four indexes measured in the test as evaluation indexes.

4.2.2 Determination of the weight of indexes

1. Determination of index weight by the OWA operator method

   Six experts (university professors who had been engaged in concrete research for more than 10 years) were invited to score each index. The scoring range was limited to 0~5, with one decimal place reserved. The higher the evaluation score is, the greater the effect of the index on the overall performance of internally cured high-strength concrete is, see Table 3. Take the compressive strength index as an example, it was arranged subject to the descending order: b₁ = (4.5, 4.2, 4.0, 4.0, 3.8, 3.5), n=6. According to formula (2), the weighted vector was calculated as: w₁ = (0.03125, 0.15625, 0.3125, 0.3125, 0.15625, 0.03125). According to formula (3), the absolute weight w₁ = 4.0 was obtained.

   Similarly, the absolute weight of other indexes was: Quality loss w₂ = 0.628; relative dynamic elastic modulus w₃ = 1.469; self-shrinkage w₄ = 1.063. According to formula (4), the subjective weighted vector of each index was obtained:W₁ = (0.559, 0.088, 0.205, 0.148)

2. Determination of index weight by the entropy weight method

   First, the judgment matrix A was constructed according to the test results in Table 2. By normalizing the data of different indexes using the range method and eliminating the dimensional difference of indexes, the new judgment matrix X was constructed. The data of X judgment matrix are shown in Table 4.

   By solving the formulas (5)~(6), the index weight W₂ was obtained: W₂ = (0.274, 0.227, 0.364, 0.135)

3. By coupling W₁ and W₂ according to formula (7), the comprehensive weight of index W was obtained:

4.2.3 Sequencing of each test group’s scheme through Mulitimoora

When evaluating the overall performance of each test group through Mulitimoora, the test data should be normalized to eliminate the dimensional difference between indexes first. In this paper, the test data were normalized by the vector normalization method to obtain the matrix B. The data of the matrix B are shown in Table 5.

The test groups in the matrix B were sorted by the modified Mulitimoora ratio system method, reference point method and perfect multiplication method through formulas (8), (9) and (10), respectively. The rating values and ranking results of each test group were determined, and the comprehensive performance evaluation results of internally cured concrete are shown in Table 6.

The dominant theory [19] was adopted to finally rank the overall performance of internally cured high-strength concrete. In this paper, the ranking results were accumulated for sequencing and final ranking. The ranking results are shown in Table 7.