# Study on Concrete Workability Based on Comparison between the Minimum Paste Demand and the Closest Packing Density

Authors: Gaolong Zhang; Zhaoyu Yan; Fuqiang He; Ruipan Wang; Changping Chen
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

The mixture design method (MDM) has been used to design the concrete workability with the three-graded stone. The results indicated that the proportion of the stone corresponding to the minimum paste demand (Pmin) can meet that corresponding to the approximate closest packing density (ACPDmax). When the workability reaches its maximum value, the paste thickness (Tpaste) and mortar thickness (Tmortar) do not reach their respective extreme values. Therefore, the combined effect of the Tpaste and Tmortar needs to be considered when evaluating concrete workability. Compared with concrete designed by the maximum packing density, the concrete designed by the Pmin + ACPDmax has a lower cost; however, stone surface area (Sstone) and the average particle size of the stone (DA-stone) are increased by 8.6% and reduced by 5.7%, respectively. Therefore, the contradictory weakening effect of an increase in the Sstone and a decrease in the DA-stone on the interface transition zone in both cases requires further investigation.


---

## Body

## 1. Introduction

Since Powers [1] recognized that aggregate gradation has a great effect on the properties of concrete, the method of achieving the maximum packing density by changing aggregate gradation has become the most important design concept during the preparation of modern concrete. Powers et al. [1, 2] considered that in the case of the paste amount remaining the same, the larger the packing density is, the smaller the paste demand will be, the more excessive paste there will be, and the thicker the formed paste coating on the aggregate will be. Therefore, they thought that an increase in the packing density could increase the concrete workability while maintaining the same paste amount and water-to-binder ratio [3]. Additionally, the volume stability of concrete can also be increased with an increase in the packing density. Therefore, an aggregate system needs to achieve its maximum packing density. This may explain why many researchers use the closest packing aggregate system to design concrete, especially high-strength, high-performance concrete, and self-compacting concrete [4].Many models have been proposed to predict the packing density of aggregate, including the discrete model [2–5], the binary mixture model [1, 5, 6], the ternary mixture model [2, 7], the multicomponent mixture model [8, 9], the continuous model [2, 5, 6, 10], and so forth. In addition, some researchers even used the 3D computer simulation model [11–13], the digital image processing model [14, 15], and the discrete element model [16–19] to predict the packing density. Considering the interaction between aggregate particles, Dewar et al. [20] developed a model for predicting the packing density of multisize aggregate. De Larrard et al. [21] considered the interaction among aggregate particles and the compaction effect of the aggregate simultaneously and proposed a compressible packing model to predict the packing density. Kwan et al. [5] considered the effect of the shape factor and the convexity ratio of the aggregate and established a model for predicting the packing density of single-size aggregate.The above models are of great value for predicting and optimizing the packing density. However, the factors affecting the packing density including particle size range, gradation, particle shape, surface roughness, and various effects during the mixing and compaction (e.g., the wall effect and the loosing effect) [5, 13, 22]. Therefore, it is very difficult to predict the packing density by using a certain model to comprehensively evaluate the effect of all these factors. Even if these factors can be involved in a model, they are difficult to be quantified accurately, which may cause the prediction results to deviate greatly from the actual situation. In order to achieve the maximum packing density, it is necessary to further increase the content of finer particles. In this situation, the specific surface area of aggregate will increase, resulting in a paste thickness reduction, and thus concrete workability will decrease [23]. Therefore, it is necessary to consider whether a concrete will meet its minimum paste demand in the case of the maximum packing density of an aggregate system [24]. This means that when evaluating concrete workability, combined effect of the aggregate packing and paste amount on the workability should be considered. The volume of the paste filled among aggregate particles has been calculated by the void fraction of aggregate [1, 2, 25]. Some researchers attempted to calculate the volume of the filling paste by assuming the paste thickness and the shape of the aggregate [25]. However, the fluctuation in the paste thickness may make the calculated results vary over a relatively large range [25]. Ji et al. [26, 27] experimentally determined the packing density and the specific surface area of the aggregate and the filling paste amount to achieve the minimum paste demand [26, 27]. However, their method for testing the specific surface area is complicated [26], and they did not consider the closest packing density in their study. Wang et al. [28] presented an optimized method for designing ultrahigh performance concrete (UHPC) with high wet packing density, in which the multiple effects of solid and liquid phases on UHPC packing mode are considered, the designed UHPC contributes high packing density, leading to optimized pore structure and extraordinary compressive strength. This method may be a potential one for concrete workability design based on the consideration of the combined effects of solid particle packing and liquid.The use of two- or three-graded stone is a common selection in many concrete engineering. The closest packing density of two-graded stone can be determined by testing the mixed packing density of the two-graded aggregate with different proportions [26, 27]. Although Shi et al. [25] suggested that three-graded stone can be optimized by the simplex centroid design method (SCDM), there is no literature on how to get the closest packing density of a three-graded aggregate system. In fact, some researchers have used the SCDM for predicting the properties of a ternary system of cement-based materials [29–33]. Jiao et al. [29] studied the rheological properties of a coarse aggregate-sand-paste system by the SCDM. Douglas et al. [29–31] accurately predicted the strength of mortar and concrete in a ternary cementitious system using the SCDM. In addition, the SCDM can accurately predict the optimum composition of a ternary cementitious system of ladle slag-lime-quartz powder under autoclaved curing conditions [32] and the optimum composition of any ternary cementitious system when considering the expansion effect of the alkali aggregate reaction [33]. As a statistical method, the SCDM is one of the types of mixture design method (MDM). Another type of the MDM, extreme vertex mixture design method (EVMDM) in the MDM, is suitable for situations with constraints in which the extreme value differences in each component proportion are not completely the same, such as the coarse aggregate-sand-paste system in concrete [34].As reviewed above, how to experimentally realize the maximum packing density and minimum paste demand still has no good solution. Therefore, the difference between the concrete prepared based on the two situations cannot yet be effectively compared. This study fills a gap in current research by experimentally realizing the maximum packing density and the minimum paste demand based on EVMDM and SCDM, which use a limited number of trials and have no parameters assumptions and any models. A comparison between the maximum packing density and the minimum paste demand systems is presented. Additionally, unlike other studies in which only the effect of the paste coating on the workability is considered, the combined effect of thicknesses of the paste coating and mortar coating on the concrete workability is discussed in this study. Finally, a concrete mix design method based on the minimum paste demand and the approximate closest packing density is suggested.


## 2. Raw Materials and Experimental Method

### 2.1. Raw Materials

P.O. 42.5 cement, which conforms to the Chinese Standard GB175-2007 [35], grade II fly ash, which conforms to the Chinese Standard GB/T 1596–2017 [36], and grade S95 slag powder, which conforms to the Chinese Standard GB/T 18046–2008 [37], were used in this study. Polycarboxylate superplasticizer, which conforms to the Chinese Standard JB/T 223–2007 [38], was used in this experiment. The chemical compositions of the cement, fly ash, and slag powder are shown in Table 1.Table 1
Chemical compositions of the cement, fly ash, and slag power (by mass, %).
Material
SiO2
Al2O3
Fe2O3
CaO
MgO
SO3
K2O
f-CaO
Loss
Cement
25.54
8.13
5.59
52.27
2.3
2.16
0.64
1.22
1.75
Fly ash
70.59
15.16
2.10
2.01
0.65
0.43
4.17
2.24
5.92
Slag power
30.62
15.04
0.47
42.29
7.04
2.28
0.52
0.94
0.36The coarse aggregate used in the experiments is three-graded limestone crushed stone with single particle sizes of 5–10 mm (the small size stone), 10–20 mm (the middle size stone), and 16–31.5 mm (the large size stone). The particle size distributions were tested three times according to the Chinese National Standard GB14685-2011 [39], and the average values of the test results are taken as the final values. The particle size distributions of the three-graded stone are shown in Table 2 and the grading curves are shown in Figure 1.Table 2
Particle size distributions of the three-graded stone.
Sieve size (mm)
Large size stone
Middle size stone
Small size stone
Average value (g)
Coefficient of variation (%)
Cumulative percentage retained (%)
Average value (g)
Coefficient of variation (%)
Cumulative percentage retained (%)
Average value (g)
Coefficient of variation (%)
Cumulative percentage retained (%)
31.5
269
0.17
4.2
0
0
0
0
0
0
26.5
2030
0.19
35.5
0
0
0
0
0
0
19
3956
0.08
96.6
547
0.07
10.8
0
0
0
16
129
0.05
98.6
2725
0.03
64.6
0
0
0
9.5
56
0.11
99.4
1759
0.18
99.4
166
0.05
6.8
4.75
5
0.10
99.5
6
0.14
99.5
2010
0.06
89.4
2.36
1
1.18
99.5
1
0.29
99.5
240
0.21
99.3
Remaining
32
0.35
100.0
25
0.14
100.0
18
0.10
100.0
Total
6478
0.01
—
5063
0.04
—
2434
0.08
—Figure 1
Particle size distribution curves of the aggregate.The fine aggregate used here was river sand with the fineness modulus of 2.6. The compressed packing density, loose packing density, and the apparent density values of the sand are 1662 kg/m3, 1531 kg/m3 and 2622 kg/m3, respectively, tested according to the Chinese National Standard GB14684-2011 [40]. The particle size distribution of the sand was tested three times according to the Chinese National Standard GB14684-2011 [40], and the average value was taken as the final value. The particle size distribution of the sand is shown in Table 3, and the grading curve of the sand is shown in Figure 1.Table 3
Particle size distribution of the sand.
Sieve size (mm)
Average value (g)
Coefficient of variation (%)
Percentage retained (%)
Cumulative percentage retained (%)
4.75
1.0
0.073
0.2
0.2
2.36
40.0
0.034
8.0
8.2
1.18
67.0
0.042
13.4
21.6
0.60
148.5
0.013
29.7
51.3
0.30
149.0
0.019
29.8
81.1
0.15
75.0
0.037
15.0
96.1
Residue
19.5
0.091
3.9
100.0
Total
500.0
0.001
100.0
—


### 2.2. Mixing Design of the Coarse Aggregate System

In order to obtain better contour plots, the pivot point enhanced simplex centroid design method (ppe-SCDM) was used to design the three-graded stone system (L-M-S system), and 10 groups of different proportions of the coarse aggregate can be obtained, as shown in Table4.Table 4
Proportions and properties of the designed coarse aggregate system.
No.
Stone proportions, %, by mass
Apparent density (kg/m3)
Loose packing density (kg/m3)
Compressed packing density (kg/m3)
Large size stone (16–31.5 mm)
Middle size stone (10–20 mm)
Small size stone (5–10 mm)
1
50.0
0.0
50.0
2698.3
1576.8
1657.9
2
16.7
16.7
66.7
2746.1
1536.3
1704.4
3
0.0
50.0
50.0
2794.0
1588.3
1679.8
4
0.0
100.0
0.0
2792.0
1503.4
1663.9
5
100.0
0.0
0.0
2650.0
1398.5
1532.1
6
33.3
33.3
33.3
2723.7
1557.7
1731.2
7
0.0
0.0
100.0
2764.3
1430.0
1576.7
8
66.7
16.7
16.7
2709.6
1483.1
1610.9
9
16.7
67.7
16.7
2761.9
1579.2
1696.2
10
50.0
50.0
0.0
2705.9
1458.5
1582.8


### 2.3. Concrete Mix Proportion

To determine the proportions of the three-graded stone in the L-M-S system based on workability (slump and slump flow), 10 groups of coarse aggregate with different proportions were designed by the ppe-SCDM. The binder dosage in the concrete was fixed at 420 kg/m3 of which 60% was cement, 30% was fly ash, and 10% was slag powder, according to the durability design requirements for mass concrete, which are not within the scope of this study. The water-binder ratio (W/B) was calculated according to the National Standard of China JGJ55-2011 [41], the W/B and sand-to-aggregate ratio (S/A) were fixed at 0.36 and 0.39, respectively. The dosage of the polycarboxylate superplasticizer (type I, solids content 20%) was 0.8% of the binder, by mass. The proportions of the concrete mixtures are shown in Table 5.Table 5
Proportions of the three-graded stone and concrete mix proportions in the L-M-S system.
No.
1
2
3
4
5
6
7
8
9
10
Proportions of the three-graded stone, %, by mass
Large size stone
40.0
46.7
20.0
20.0
33.3
40.0
26.7
26.7
60.0
20.0
Middle size stone
40.0
26.7
60.0
40.0
33.3
20.0
26.7
46.7
20.0
20.0
Small size stone
20.0
26.7
20.0
40.0
33.3
40.0
46.7
26.7
20.0
60.0
Mix proportion, kg/m3, by mass
Cement
252.0
252.0
252.0
252.0
252.0
252.0
252.0
252.0
252.0
252.0
Fly ash
126.0
126.0
126.0
126.0
126.0
126.0
126.0
126.0
126.0
126.0
Slag power
42.0
42.0
42.0
42.0
42.0
42.0
42.0
42.0
42.0
42.0
Sand
713.2
713.2
713.2
713.2
713.2
713.2
713.2
713.2
713.2
713.2
Large size stone
446.2
520.6
223.1
223.1
371.9
446.2
297.5
297.5
669.3
223.1
Middle size stone
446.2
297.5
669.3
446.2
371.9
223.1
297.5
520.6
223.1
223.1
Small size stone
223.1
297.5
223.1
446.2
371.9
446.2
520.6
297.5
223.1
669.3
Water
151.2
151.2
151.2
151.2
151.2
151.2
151.2
151.2
151.2
151.2For the purpose of determining the binder dosage and S/A used in concrete, the EVMDM was adopted to design the paste-sand-stone ternary system (P-S-S system). The binder dosage was in the range of 380–420 kg/m3, the S/A varied from 0.35 to 0.45, and thus, 13 groups of concrete mixtures were obtained, as presented in Table 6. The W/B and the proportion of the three-graded stone were both fixed at 0.36, and the proportion of the large size stone: middle size stone: small size stone was equal to 0.32 : 0.34 : 0.34 (corresponding to the minimum paste demand, see Section 3.1.2 for the details). The proportions of the cement, fly ash, and slag powder in the binder were kept the same as those used in the L-M-S system. The dosage of polycarboxylate superplasticizer (type II, solids content 15%) was 1.6% of the binder, by mass.Table 6
Mixture design of the P-S-S system.
No.
Binder
Sand
Stone
S/A (%)
%, by mass
kg/m3
%, by mass
kg/m3
%, by mass
kg/m3
1
22.1
390.4
32.9
788.8
45.0
1080.3
0.42
2
23.8
420.0
26.9
645.9
49.3
1182.9
0.35
3
21.5
380.0
34.8
835.7
43.6
1047.6
0.44
4
23.3
410.4
32.3
774.0
44.5
1068.0
0.42
5
22.9
403.6
28.9
693.9
48.2
1157.4
0.37
6
23.0
406.3
26.9
645.9
50.1
1201.5
0.35
7
23.8
420.0
33.6
806.2
42.6
1022.7
0.44
8
22.7
400.8
30.9
741.9
46.4
1113.0
0.40
9
21.5
380.0
28.4
681.8
50.1
1201.5
0.36
10
22.6
398.3
34.8
835.7
42.6
1022.7
0.45
11
23.3
410.4
28.9
693.9
47.8
1148.1
0.38
12
22.6
399.5
32.9
788.8
44.5
1068.0
0.42
13
22.1
390.4
29.7
711.8
48.2
1157.4
0.38


### 2.4. Experimental Method

According to the China National Standard GB14685-2011 [39], the loose packing density, compressed packing density, and apparent density of the above 10 groups of three-graded stone were tested. The tested results are shown in Table 4. The concrete slump and slump flow tests were carried out in accordance with the National Standard of China GBT50080-2016 [42].


## 2.1. Raw Materials

P.O. 42.5 cement, which conforms to the Chinese Standard GB175-2007 [35], grade II fly ash, which conforms to the Chinese Standard GB/T 1596–2017 [36], and grade S95 slag powder, which conforms to the Chinese Standard GB/T 18046–2008 [37], were used in this study. Polycarboxylate superplasticizer, which conforms to the Chinese Standard JB/T 223–2007 [38], was used in this experiment. The chemical compositions of the cement, fly ash, and slag powder are shown in Table 1.Table 1
Chemical compositions of the cement, fly ash, and slag power (by mass, %).
Material
SiO2
Al2O3
Fe2O3
CaO
MgO
SO3
K2O
f-CaO
Loss
Cement
25.54
8.13
5.59
52.27
2.3
2.16
0.64
1.22
1.75
Fly ash
70.59
15.16
2.10
2.01
0.65
0.43
4.17
2.24
5.92
Slag power
30.62
15.04
0.47
42.29
7.04
2.28
0.52
0.94
0.36The coarse aggregate used in the experiments is three-graded limestone crushed stone with single particle sizes of 5–10 mm (the small size stone), 10–20 mm (the middle size stone), and 16–31.5 mm (the large size stone). The particle size distributions were tested three times according to the Chinese National Standard GB14685-2011 [39], and the average values of the test results are taken as the final values. The particle size distributions of the three-graded stone are shown in Table 2 and the grading curves are shown in Figure 1.Table 2
Particle size distributions of the three-graded stone.
Sieve size (mm)
Large size stone
Middle size stone
Small size stone
Average value (g)
Coefficient of variation (%)
Cumulative percentage retained (%)
Average value (g)
Coefficient of variation (%)
Cumulative percentage retained (%)
Average value (g)
Coefficient of variation (%)
Cumulative percentage retained (%)
31.5
269
0.17
4.2
0
0
0
0
0
0
26.5
2030
0.19
35.5
0
0
0
0
0
0
19
3956
0.08
96.6
547
0.07
10.8
0
0
0
16
129
0.05
98.6
2725
0.03
64.6
0
0
0
9.5
56
0.11
99.4
1759
0.18
99.4
166
0.05
6.8
4.75
5
0.10
99.5
6
0.14
99.5
2010
0.06
89.4
2.36
1
1.18
99.5
1
0.29
99.5
240
0.21
99.3
Remaining
32
0.35
100.0
25
0.14
100.0
18
0.10
100.0
Total
6478
0.01
—
5063
0.04
—
2434
0.08
—Figure 1
Particle size distribution curves of the aggregate.The fine aggregate used here was river sand with the fineness modulus of 2.6. The compressed packing density, loose packing density, and the apparent density values of the sand are 1662 kg/m3, 1531 kg/m3 and 2622 kg/m3, respectively, tested according to the Chinese National Standard GB14684-2011 [40]. The particle size distribution of the sand was tested three times according to the Chinese National Standard GB14684-2011 [40], and the average value was taken as the final value. The particle size distribution of the sand is shown in Table 3, and the grading curve of the sand is shown in Figure 1.Table 3
Particle size distribution of the sand.
Sieve size (mm)
Average value (g)
Coefficient of variation (%)
Percentage retained (%)
Cumulative percentage retained (%)
4.75
1.0
0.073
0.2
0.2
2.36
40.0
0.034
8.0
8.2
1.18
67.0
0.042
13.4
21.6
0.60
148.5
0.013
29.7
51.3
0.30
149.0
0.019
29.8
81.1
0.15
75.0
0.037
15.0
96.1
Residue
19.5
0.091
3.9
100.0
Total
500.0
0.001
100.0
—


## 2.2. Mixing Design of the Coarse Aggregate System

In order to obtain better contour plots, the pivot point enhanced simplex centroid design method (ppe-SCDM) was used to design the three-graded stone system (L-M-S system), and 10 groups of different proportions of the coarse aggregate can be obtained, as shown in Table4.Table 4
Proportions and properties of the designed coarse aggregate system.
No.
Stone proportions, %, by mass
Apparent density (kg/m3)
Loose packing density (kg/m3)
Compressed packing density (kg/m3)
Large size stone (16–31.5 mm)
Middle size stone (10–20 mm)
Small size stone (5–10 mm)
1
50.0
0.0
50.0
2698.3
1576.8
1657.9
2
16.7
16.7
66.7
2746.1
1536.3
1704.4
3
0.0
50.0
50.0
2794.0
1588.3
1679.8
4
0.0
100.0
0.0
2792.0
1503.4
1663.9
5
100.0
0.0
0.0
2650.0
1398.5
1532.1
6
33.3
33.3
33.3
2723.7
1557.7
1731.2
7
0.0
0.0
100.0
2764.3
1430.0
1576.7
8
66.7
16.7
16.7
2709.6
1483.1
1610.9
9
16.7
67.7
16.7
2761.9
1579.2
1696.2
10
50.0
50.0
0.0
2705.9
1458.5
1582.8


## 2.3. Concrete Mix Proportion

To determine the proportions of the three-graded stone in the L-M-S system based on workability (slump and slump flow), 10 groups of coarse aggregate with different proportions were designed by the ppe-SCDM. The binder dosage in the concrete was fixed at 420 kg/m3 of which 60% was cement, 30% was fly ash, and 10% was slag powder, according to the durability design requirements for mass concrete, which are not within the scope of this study. The water-binder ratio (W/B) was calculated according to the National Standard of China JGJ55-2011 [41], the W/B and sand-to-aggregate ratio (S/A) were fixed at 0.36 and 0.39, respectively. The dosage of the polycarboxylate superplasticizer (type I, solids content 20%) was 0.8% of the binder, by mass. The proportions of the concrete mixtures are shown in Table 5.Table 5
Proportions of the three-graded stone and concrete mix proportions in the L-M-S system.
No.
1
2
3
4
5
6
7
8
9
10
Proportions of the three-graded stone, %, by mass
Large size stone
40.0
46.7
20.0
20.0
33.3
40.0
26.7
26.7
60.0
20.0
Middle size stone
40.0
26.7
60.0
40.0
33.3
20.0
26.7
46.7
20.0
20.0
Small size stone
20.0
26.7
20.0
40.0
33.3
40.0
46.7
26.7
20.0
60.0
Mix proportion, kg/m3, by mass
Cement
252.0
252.0
252.0
252.0
252.0
252.0
252.0
252.0
252.0
252.0
Fly ash
126.0
126.0
126.0
126.0
126.0
126.0
126.0
126.0
126.0
126.0
Slag power
42.0
42.0
42.0
42.0
42.0
42.0
42.0
42.0
42.0
42.0
Sand
713.2
713.2
713.2
713.2
713.2
713.2
713.2
713.2
713.2
713.2
Large size stone
446.2
520.6
223.1
223.1
371.9
446.2
297.5
297.5
669.3
223.1
Middle size stone
446.2
297.5
669.3
446.2
371.9
223.1
297.5
520.6
223.1
223.1
Small size stone
223.1
297.5
223.1
446.2
371.9
446.2
520.6
297.5
223.1
669.3
Water
151.2
151.2
151.2
151.2
151.2
151.2
151.2
151.2
151.2
151.2For the purpose of determining the binder dosage and S/A used in concrete, the EVMDM was adopted to design the paste-sand-stone ternary system (P-S-S system). The binder dosage was in the range of 380–420 kg/m3, the S/A varied from 0.35 to 0.45, and thus, 13 groups of concrete mixtures were obtained, as presented in Table 6. The W/B and the proportion of the three-graded stone were both fixed at 0.36, and the proportion of the large size stone: middle size stone: small size stone was equal to 0.32 : 0.34 : 0.34 (corresponding to the minimum paste demand, see Section 3.1.2 for the details). The proportions of the cement, fly ash, and slag powder in the binder were kept the same as those used in the L-M-S system. The dosage of polycarboxylate superplasticizer (type II, solids content 15%) was 1.6% of the binder, by mass.Table 6
Mixture design of the P-S-S system.
No.
Binder
Sand
Stone
S/A (%)
%, by mass
kg/m3
%, by mass
kg/m3
%, by mass
kg/m3
1
22.1
390.4
32.9
788.8
45.0
1080.3
0.42
2
23.8
420.0
26.9
645.9
49.3
1182.9
0.35
3
21.5
380.0
34.8
835.7
43.6
1047.6
0.44
4
23.3
410.4
32.3
774.0
44.5
1068.0
0.42
5
22.9
403.6
28.9
693.9
48.2
1157.4
0.37
6
23.0
406.3
26.9
645.9
50.1
1201.5
0.35
7
23.8
420.0
33.6
806.2
42.6
1022.7
0.44
8
22.7
400.8
30.9
741.9
46.4
1113.0
0.40
9
21.5
380.0
28.4
681.8
50.1
1201.5
0.36
10
22.6
398.3
34.8
835.7
42.6
1022.7
0.45
11
23.3
410.4
28.9
693.9
47.8
1148.1
0.38
12
22.6
399.5
32.9
788.8
44.5
1068.0
0.42
13
22.1
390.4
29.7
711.8
48.2
1157.4
0.38


## 2.4. Experimental Method

According to the China National Standard GB14685-2011 [39], the loose packing density, compressed packing density, and apparent density of the above 10 groups of three-graded stone were tested. The tested results are shown in Table 4. The concrete slump and slump flow tests were carried out in accordance with the National Standard of China GBT50080-2016 [42].


## 3. Results and Discussion

### 3.1. Determination of the Mixing Ratio of the Three-Graded Stone

#### 3.1.1. Mixing Ratio of the Three-Graded Stone Based on the Maximum Packing Density

The packing density contour plots of the L-M-S system are shown in Figure2. It can be seen in Figure 2 that the compressed and loose packing density values of the L-M-S system reached 1736 kg/m3 and 1599 kg/m3, respectively. The proportions of the three-graded stone corresponding to the maximum compressed packing density (CPDmax), the approximate maximum compressed packing density (ACPDmax), the maximum loose packing density (LPDmax), and the approximate maximum loose packing density (ALPDmax) of the L-M-S system are given in Table 7. It can be seen in Table 7 that proportion regions of the three-graded stone of ALPDmax and ACPDmax do not have any overlapping area, and the proportion regions of the three-graded stone of LPDmax are not within that of ACPDmax. This means that the compressed packing density can reach its maximum only if these three types of stones are incorporated in a considerable proportion. That is, the smaller-size stones are gradually filled into the voids between the larger-size stones, so as to achieve the closest packing state. When the stones were loosely packed, large size stones cannot be incorporated to achieve the maximum loose packing density. This is because the inclusion of large size stones will form larger voids, and these larger voids cannot be filled very well without the compression.Figure 2
Contour plots of compressed packing density (a) and loose packing density (b) of the L-M-S system (kg/m3).
(a)
(b)Table 7
Stone proportions at different packing density (by mass).
Stone size (mm)
ACPD
max
ALPD
max
CPD
max
LPD
max
16–31.5
0.15–0.38
0–0.19
0.28
0
10–20
0.18–0.45
0.42–0.76
0.31
0.62
5–10
0.30–0.58
0.18–0.53
0.41
0.38
Note: the subscript “max” denotes the maximum value of the corresponding indicator;ACPDmax and ALPDmax are defined as the numerical sets with the deviation rate in the range of 1.0% from the maximum compressed packing density and the maximum loose packing density of the L-M-S system, respectively.


#### 3.1.2. Mixing Ratio of the Three-Graded Stone Based on the Concrete Workability

The contour plots of the concrete slump and slump flow are plotted in Figure3. From Figure 3, the maximum slump (Smax), the approximate maximum slump (ASmax), the maximum slump flow (Dmax), and the approximate maximum slump flow (ADmax) of concrete prepared using the L-M-S system can be obtained, as shown in Table 8. It can be seen from Tables 7 and 8 that the three-graded stone proportions of ACPDmax of the L-M-S system overlap with that of the (ASmax + ADmax). The proportions of the three-graded stone corresponding to Smax and Dmax of the L-M-S system are almost the same, and the proportion ranges of the three-graded stone corresponding to ASmax and ADmax in the L-M-S system are also almost the same. The proportions of the three-graded stone corresponding to Smax and Dmax in the L-M-S system fall within the overlapping region of the three-graded stone proportions among ACPDmax, ASmax, and ADmax of the L-M-S system. However, the proportions of the three-graded stone in the case of CPDmax and LPDmax are not within this range. This means that the proportions of the three-graded stone corresponding to Smax, Dmax and CPDmax satisfy that of ACPDmax in the L-M-S system and the compressed packing density is more suitable for characterizing the packing state of the coarse aggregate in concrete than the loose packing density.Figure 3
Contour plots of the slump (a) and slump flow (b) of the L-M-S system (mm).
(a)
(b)Table 8
Proportions of the three-graded stone at different slump, slump flow, and packing density of the L-M-S system (by mass).
Stone size (mm)
AS
max
AD
max
ACPD
max + ASmax + ADmax
S
max
D
max
16–31.5
0.28–0.36
0.28–0.37
0.28–0.36
0.32
0.33
10–20
0.31–0.39
0.30–0.38
0.31–0.38
0.34
0.33
5–10
0.28–0.38
0.29–0.38
0.30–0.38
0.34
0.34
Note:Smax and Dmax are the maximum slump and maximum slump flow of the L-M-S system; ASmax and ADmax are the approximate maximum slump values within a range of 10 mm from Smax and the approximate maximum slump flow values within a range of 30 mm from Dmax in the L-M-S system, respectively. ACPDmax is a set of compressed packing density values within a range of >99% CPDmax in the L-M-S system.It can be read in Figure2 that the compressed packing density in the case of Smax is 1730.6 kg/m3 in the L-M-S system, which is very close to that of CPDmax (1736 kg/m3). The surface area corresponding to Smax is 245.7 m2, and the surface area corresponding to CPDmax is 266.9 m2. This means that paste volume of the three-graded stone system corresponding to Smax is very close to the minimum paste filling volume, but the surface area corresponding to Smax is smaller than that corresponding to CPDmax in the L-M-S system. Therefore, the paste demand for the same paste thickness is correspondingly smaller. It can be seen from Figure 3 that the slump and slump flow corresponding to CPDmax are 173 mm and 321 mm, respectively, which are relatively smaller than Smax (196 mm) and Dmax (390 mm). This shows that with the same amount of paste, the aggregate system according to CPDmax needs more filling and coating paste than that corresponding to Smax, and thus there is a less excessive paste. When obtaining a certain targeted slump, the three-graded stone system corresponding to Smax has the minimum paste demand (PDmin) aggregate system in the L-M-S system.The effect of the particle sizes and surface areas of the three-graded stone respectively corresponding toPDmin and CPDmax on the interface transition zone (ITZ) is a key consideration. The ITZ of concrete has a great effect on the strength and durability of concrete [43–48]. When mixing fresh concrete, a water film is formed around the large aggregate particles, resulting in a much higher W/B around the large particles than that of the remaining portion. When other conditions are constant, the larger the aggregate size, the greater the local W/B in the ITZ, and the lower the strength and durability of the concrete [43, 49]. The surface area (Sstone) and the average particle size (DA-stone) of the three-graded stone corresponding to PDmin and CPDmax in the L-M-S system can be calculated according to equation (A.9) and equation (A.14) (see Appendix A), which are 245.7 m2, 15.33 mm, and 266.9 m2, 14.45 mm, respectively. Compared with PDmin, Sstone corresponding to the CPDmax is increased by 8.6%, and the DA-stone is reduced by 5.7%. The increase in the DA-stone leads to an increase in the weakening degree to the ITZ. However, the decrease in Sstone leads to a decrease in the ITZ quantity and thus reduces the weakening degree to ITZ. The weakening degree to ITZ depends on the combined effects of Sstone and DA-stone. Therefore, the effect on ITZ in the case of CPDmax and Pmin on the properties of concrete needs further study.It is clear from the above discussion that the proportion of the three-graded stone in the cases ofPDmin also satisfies ACPDmax; therefore, the proportion of the three-graded stone corresponding to PDmin was used to prepare concrete in this study.


### 3.2. Determination of the Binder Dosage and S/A Based on EVMDM

#### 3.2.1. The Contour Plots of Concrete Slump and Slump Flow of the P-S-S System

The contour plots of the slump and slump flow of the P-S-S system are shown in Figure4. It can be seen in Figure 4 that the varying trends of concrete slump and slump flow are mostly the same. With an increase in the paste volume, the slump and slump flow gradually increase, and with a decrease in the S/A, the slump and slump flow first increase and then decrease.Figure 4
Contour plots of the slump (a) and the slump flow (b) of the P-S-S system (mm).
(a)
(b)Based on the concrete encapsulation model [50], the paste thickness (Tpaste) coated on the surface of the aggregate, and the mortar thickness (Tmortar) coated on the surface of the coarse aggregate can be calculated according to equations (B.12) and (B.13) (see Appendix B), respectively, whose results are shown in Figure 5. It can be seen in Figure 5 that Tpaste increases gradually with an increase in the binder dosage and the decrease in S/A and varies between 30 and 59 μm, which is consistent with the range of Tpaste obtained by other researchers [51]. This is because the total surface area of aggregate can be significantly reduced by reducing S/A, and Tpaste increases with a decrease in S/A when the paste volume is constant. When the binder dosage is at the maximum value (420 kg/m3) and S/A is at the minimum value (0.35), Tpaste reaches its maximum value (approximately 59 μm). Tmortar increases with an increase in the binder dosage and S/A. When the binder dosage is at the maximum value (420 kg/m3) and S/A is 0.43 (near its maximum value, 0.45), Tmortar reaches its maximum value (approximately 1.7 mm).Figure 5
Contour plots ofTpaste (a) and Tmortar (b) of the P-S-S system.
(a)
(b)Associated with Figures4 and 5, it can be found that when the slump and slump flow reach their maximum values, Tpaste and Tmortar vary in the range of 50–55 μm and 1.3–1.4 mm, respectively, which is in the middle region of Tpaste and Tmortar, respectively. This indicates that Tpaste and Tmortar are not at their maximum or minimum values in the case of PDmin; however, Tpaste is closer to its maximum value (59 μm), and Tmortar is closer to its minimum value (1.0 mm). Jiao et al. [52] regarded that when S/A is fixed, the slump and slump flow are mainly affected by Tpaste. However, their study cannot reflect the combined effects of the paste volume and S/A on the concrete workability by changing the paste volume and S/A simultaneously. The results obtained from Figures 4 and 5 demonstrates that when considering the effect of Tpaste on the workability of concrete, the effect of Tmortar must also be considered at the same time, and EVMDM can determine the most reasonable range of Tpaste and Tmortar in a convenient way.


#### 3.2.2. Determination of the Binder Dosage and the S/A Based onPDmin of the P-S-S System

The paste-sand-stone proportions corresponding toSmax, ASmax, Dmax, and ADmax of the P-S-S system can be read in Figure 4, as shown in Table 9. It can be seen in Table 9 that the proportions of paste-sand-stone corresponding to Smax and Dmax are almost the same; the proportions of paste-sand-stone corresponding to ASmax and ADmax are very similar. When the binder dosage varies in the range of 411–420 kg/m3 and S/A varies from 0.37 to 0.39, the concrete workability reaches its approximate maximum value. When the binder dosage is 420 kg/m3 and S/A is 0.38, the slump and slump flow synchronously reach their maximum values, which are 239 mm and 560 mm, respectively.Table 9
Binder dosage and the S/A in the case ofSmax and ASmax of the P-S-S system (by mass).
Items
AS
max
S
max
AD
max
D
max
AS
max + ADmax
Binder (kg/m3)
S/A
Paste
0.230–0.238
0.238
0.233–0.238
0.238
0.233–0.238
411–420
0.37–0.39
Sand
0.279–0.301
0.289
0.280–0.293
0.287
0.280–0.293
Stone
0.461–0.483
0.473
0.469–0.481
0.475
0.469–0.481
Note:Smax and Dmax are the maximum slump and maximum slump flow values of the P-S-S system, respectively; ASmax and ADmax are the approximate maximum slump values within a range of 10 mm from the Smax and the approximate maximum slump flow values within a range of 30 mm from the Dmax in the P-S-S system, respectively.


#### 3.2.3. Determination of the Binder Dosage and the S/A Based on the Targeted Workability of the P-S-S System

A typical contour plot of the targeted slump (Star, assumed to be 180–200 mm) and targeted slump flow (Dtar, assumed to be 350–450 mm) is shown in Figure 6. It can be seen in Figure 6 that when Star and the Dtar are synchronously met (the red part in Figure 6), the proportion of the minimum paste amount is 0.215, the proportion of the sand and the stone drops within the regions of 0.284–0.304 and 0.481–0.501, respectively. Namely, the binder dosage is 380 kg/m3, and S/A is from 0.36 to 0.39. It can also be read from Figure 6 that when the proportions of the paste, sand, and stone are in the range of 0.226–0.238, 0.317–0.338, and 0.436–0.448, respectively, the concrete workability can also meet the targeted requirements. Correspondingly, the binder dosage varies between 399 and 420 kg/m3, and S/A varies within the range of 0.42–0.44.Figure 6
Contour plot of the concrete slump and slump flow (mm, slump = 180–200 mm, slump flow = 350–450 mm).The compressed packing density of the sand-stone system with different S/A values is shown in Figure7. It can be seen in Figure 7 that the sand-stone system has the maximum packing density when S/A is 0.41. Therefore, S/A corresponding to CPDmax is not the same as that corresponding to PDmin. Corresponding to PDmin, the binder dosage is 420 kg/m3 and S/A is 0.38, which are not the same as those of the maximum packing density of the sand-stone system. In this situation, Tpaste is approximately 49.5 μm and Tmortar is approximately 1.33 mm. When taking S/A (0.41) corresponding to the maximum packing density of sand-stone system and when the binder dosage is 420 kg/m3, Tpaste and Tmortar are approximately 44 μm and 1.49 mm, respectively, while the slump and slump flow are 210 mm and 405 mm, respectively, which are far from those corresponding to PDmin in the P-S-S system. This shows that the combined effect of an increase in Tmortar and a decrease in Tpaste significantly reduces the concrete workability. Therefore, it is not enough to study the concrete workability only based on Tpaste. The combined effect of Tpaste and Tmortar on the workability should be considered. Cementitious materials will cause early thermal expansion and shrinkage in the hydration process, which will also produce a drying shrinkage in a long-term drying environment [53]. When W/B is constant, reducing the content of cementitious materials can effectively reduce the effect of the paste on the volume stability of concrete [23]. Therefore, one can control Tpaste at a lower level by keeping Tmortar within a reasonable range.Figure 7
Relationship between S/A and the compressed packing density of the sand-stone system. Note: the proportion of the three-graded stone corresponding toCPDmax of the L-M-S system was used.S
stone of the three-graded stone corresponding to Star and Smax values of the P-S-S with different binder dosages and S/A and CPDmax values with S/A = 0.41 are shown in Table 10. It can be seen in Table 10 that Sstone value corresponding to CPDmax is larger than that corresponding to Star and Smax. In the case of Star, Sstone with smaller binder dosage is larger than that with large binder dosage. Therefore, it also has a larger ITZ. However, in the case of the maximum packing density of sand-stone system, DA-stone of the three-graded stone decreases by 5.7%, which makes it more difficult to form a water film compared with Smax situation and thus reduces the effect of the particle size on W/B in ITZ. When W/B is constant, reducing the content of cementitious materials can effectively reduce the effect of the paste on volume stability of concrete [23]. To achieve the targeted workability requirements of a 180–200 mm slump and a 350–450 mm slump flow, the binder dosage = 380 kg/m3 and S/A = 0.36–0.39 can be selected to design the concrete with the lowest cost. It is worth noting whether the P-S-S system based on the PDmin is consistent with the P-S-S system according to the best durability needs further research. It can be recommended that EVMDM is also suitable for durability design based on adjustments of the P-S-S system.Table 10
TheSstone based on PDmin of the P-S-S system.
Binder content (kg/m3)
S/A
S
stone (m2)
Comment
420
0.37
253.6
420
0.38
249.6
PD
min
420
0.39
245.5
420
0.41
257.9
CPD
max
411
0.37
255.3
411
0.38
251.2
PD
min
411
0.39
247.2
411
0.41
259.6
CPD
max
380
0.36
265.3
380
0.38
257.0
PD
min
380
0.39
252.8
380
0.41
265.6
CPD
max
399
0.42
237.1
399
0.43
233.0
PD
min
399
0.44
228.9
399
0.41
261.9
CPD
max


## 3.1. Determination of the Mixing Ratio of the Three-Graded Stone

### 3.1.1. Mixing Ratio of the Three-Graded Stone Based on the Maximum Packing Density

The packing density contour plots of the L-M-S system are shown in Figure2. It can be seen in Figure 2 that the compressed and loose packing density values of the L-M-S system reached 1736 kg/m3 and 1599 kg/m3, respectively. The proportions of the three-graded stone corresponding to the maximum compressed packing density (CPDmax), the approximate maximum compressed packing density (ACPDmax), the maximum loose packing density (LPDmax), and the approximate maximum loose packing density (ALPDmax) of the L-M-S system are given in Table 7. It can be seen in Table 7 that proportion regions of the three-graded stone of ALPDmax and ACPDmax do not have any overlapping area, and the proportion regions of the three-graded stone of LPDmax are not within that of ACPDmax. This means that the compressed packing density can reach its maximum only if these three types of stones are incorporated in a considerable proportion. That is, the smaller-size stones are gradually filled into the voids between the larger-size stones, so as to achieve the closest packing state. When the stones were loosely packed, large size stones cannot be incorporated to achieve the maximum loose packing density. This is because the inclusion of large size stones will form larger voids, and these larger voids cannot be filled very well without the compression.Figure 2
Contour plots of compressed packing density (a) and loose packing density (b) of the L-M-S system (kg/m3).
(a)
(b)Table 7
Stone proportions at different packing density (by mass).
Stone size (mm)
ACPD
max
ALPD
max
CPD
max
LPD
max
16–31.5
0.15–0.38
0–0.19
0.28
0
10–20
0.18–0.45
0.42–0.76
0.31
0.62
5–10
0.30–0.58
0.18–0.53
0.41
0.38
Note: the subscript “max” denotes the maximum value of the corresponding indicator;ACPDmax and ALPDmax are defined as the numerical sets with the deviation rate in the range of 1.0% from the maximum compressed packing density and the maximum loose packing density of the L-M-S system, respectively.


### 3.1.2. Mixing Ratio of the Three-Graded Stone Based on the Concrete Workability

The contour plots of the concrete slump and slump flow are plotted in Figure3. From Figure 3, the maximum slump (Smax), the approximate maximum slump (ASmax), the maximum slump flow (Dmax), and the approximate maximum slump flow (ADmax) of concrete prepared using the L-M-S system can be obtained, as shown in Table 8. It can be seen from Tables 7 and 8 that the three-graded stone proportions of ACPDmax of the L-M-S system overlap with that of the (ASmax + ADmax). The proportions of the three-graded stone corresponding to Smax and Dmax of the L-M-S system are almost the same, and the proportion ranges of the three-graded stone corresponding to ASmax and ADmax in the L-M-S system are also almost the same. The proportions of the three-graded stone corresponding to Smax and Dmax in the L-M-S system fall within the overlapping region of the three-graded stone proportions among ACPDmax, ASmax, and ADmax of the L-M-S system. However, the proportions of the three-graded stone in the case of CPDmax and LPDmax are not within this range. This means that the proportions of the three-graded stone corresponding to Smax, Dmax and CPDmax satisfy that of ACPDmax in the L-M-S system and the compressed packing density is more suitable for characterizing the packing state of the coarse aggregate in concrete than the loose packing density.Figure 3
Contour plots of the slump (a) and slump flow (b) of the L-M-S system (mm).
(a)
(b)Table 8
Proportions of the three-graded stone at different slump, slump flow, and packing density of the L-M-S system (by mass).
Stone size (mm)
AS
max
AD
max
ACPD
max + ASmax + ADmax
S
max
D
max
16–31.5
0.28–0.36
0.28–0.37
0.28–0.36
0.32
0.33
10–20
0.31–0.39
0.30–0.38
0.31–0.38
0.34
0.33
5–10
0.28–0.38
0.29–0.38
0.30–0.38
0.34
0.34
Note:Smax and Dmax are the maximum slump and maximum slump flow of the L-M-S system; ASmax and ADmax are the approximate maximum slump values within a range of 10 mm from Smax and the approximate maximum slump flow values within a range of 30 mm from Dmax in the L-M-S system, respectively. ACPDmax is a set of compressed packing density values within a range of >99% CPDmax in the L-M-S system.It can be read in Figure2 that the compressed packing density in the case of Smax is 1730.6 kg/m3 in the L-M-S system, which is very close to that of CPDmax (1736 kg/m3). The surface area corresponding to Smax is 245.7 m2, and the surface area corresponding to CPDmax is 266.9 m2. This means that paste volume of the three-graded stone system corresponding to Smax is very close to the minimum paste filling volume, but the surface area corresponding to Smax is smaller than that corresponding to CPDmax in the L-M-S system. Therefore, the paste demand for the same paste thickness is correspondingly smaller. It can be seen from Figure 3 that the slump and slump flow corresponding to CPDmax are 173 mm and 321 mm, respectively, which are relatively smaller than Smax (196 mm) and Dmax (390 mm). This shows that with the same amount of paste, the aggregate system according to CPDmax needs more filling and coating paste than that corresponding to Smax, and thus there is a less excessive paste. When obtaining a certain targeted slump, the three-graded stone system corresponding to Smax has the minimum paste demand (PDmin) aggregate system in the L-M-S system.The effect of the particle sizes and surface areas of the three-graded stone respectively corresponding toPDmin and CPDmax on the interface transition zone (ITZ) is a key consideration. The ITZ of concrete has a great effect on the strength and durability of concrete [43–48]. When mixing fresh concrete, a water film is formed around the large aggregate particles, resulting in a much higher W/B around the large particles than that of the remaining portion. When other conditions are constant, the larger the aggregate size, the greater the local W/B in the ITZ, and the lower the strength and durability of the concrete [43, 49]. The surface area (Sstone) and the average particle size (DA-stone) of the three-graded stone corresponding to PDmin and CPDmax in the L-M-S system can be calculated according to equation (A.9) and equation (A.14) (see Appendix A), which are 245.7 m2, 15.33 mm, and 266.9 m2, 14.45 mm, respectively. Compared with PDmin, Sstone corresponding to the CPDmax is increased by 8.6%, and the DA-stone is reduced by 5.7%. The increase in the DA-stone leads to an increase in the weakening degree to the ITZ. However, the decrease in Sstone leads to a decrease in the ITZ quantity and thus reduces the weakening degree to ITZ. The weakening degree to ITZ depends on the combined effects of Sstone and DA-stone. Therefore, the effect on ITZ in the case of CPDmax and Pmin on the properties of concrete needs further study.It is clear from the above discussion that the proportion of the three-graded stone in the cases ofPDmin also satisfies ACPDmax; therefore, the proportion of the three-graded stone corresponding to PDmin was used to prepare concrete in this study.


## 3.1.1. Mixing Ratio of the Three-Graded Stone Based on the Maximum Packing Density

The packing density contour plots of the L-M-S system are shown in Figure2. It can be seen in Figure 2 that the compressed and loose packing density values of the L-M-S system reached 1736 kg/m3 and 1599 kg/m3, respectively. The proportions of the three-graded stone corresponding to the maximum compressed packing density (CPDmax), the approximate maximum compressed packing density (ACPDmax), the maximum loose packing density (LPDmax), and the approximate maximum loose packing density (ALPDmax) of the L-M-S system are given in Table 7. It can be seen in Table 7 that proportion regions of the three-graded stone of ALPDmax and ACPDmax do not have any overlapping area, and the proportion regions of the three-graded stone of LPDmax are not within that of ACPDmax. This means that the compressed packing density can reach its maximum only if these three types of stones are incorporated in a considerable proportion. That is, the smaller-size stones are gradually filled into the voids between the larger-size stones, so as to achieve the closest packing state. When the stones were loosely packed, large size stones cannot be incorporated to achieve the maximum loose packing density. This is because the inclusion of large size stones will form larger voids, and these larger voids cannot be filled very well without the compression.Figure 2
Contour plots of compressed packing density (a) and loose packing density (b) of the L-M-S system (kg/m3).
(a)
(b)Table 7
Stone proportions at different packing density (by mass).
Stone size (mm)
ACPD
max
ALPD
max
CPD
max
LPD
max
16–31.5
0.15–0.38
0–0.19
0.28
0
10–20
0.18–0.45
0.42–0.76
0.31
0.62
5–10
0.30–0.58
0.18–0.53
0.41
0.38
Note: the subscript “max” denotes the maximum value of the corresponding indicator;ACPDmax and ALPDmax are defined as the numerical sets with the deviation rate in the range of 1.0% from the maximum compressed packing density and the maximum loose packing density of the L-M-S system, respectively.


## 3.1.2. Mixing Ratio of the Three-Graded Stone Based on the Concrete Workability

The contour plots of the concrete slump and slump flow are plotted in Figure3. From Figure 3, the maximum slump (Smax), the approximate maximum slump (ASmax), the maximum slump flow (Dmax), and the approximate maximum slump flow (ADmax) of concrete prepared using the L-M-S system can be obtained, as shown in Table 8. It can be seen from Tables 7 and 8 that the three-graded stone proportions of ACPDmax of the L-M-S system overlap with that of the (ASmax + ADmax). The proportions of the three-graded stone corresponding to Smax and Dmax of the L-M-S system are almost the same, and the proportion ranges of the three-graded stone corresponding to ASmax and ADmax in the L-M-S system are also almost the same. The proportions of the three-graded stone corresponding to Smax and Dmax in the L-M-S system fall within the overlapping region of the three-graded stone proportions among ACPDmax, ASmax, and ADmax of the L-M-S system. However, the proportions of the three-graded stone in the case of CPDmax and LPDmax are not within this range. This means that the proportions of the three-graded stone corresponding to Smax, Dmax and CPDmax satisfy that of ACPDmax in the L-M-S system and the compressed packing density is more suitable for characterizing the packing state of the coarse aggregate in concrete than the loose packing density.Figure 3
Contour plots of the slump (a) and slump flow (b) of the L-M-S system (mm).
(a)
(b)Table 8
Proportions of the three-graded stone at different slump, slump flow, and packing density of the L-M-S system (by mass).
Stone size (mm)
AS
max
AD
max
ACPD
max + ASmax + ADmax
S
max
D
max
16–31.5
0.28–0.36
0.28–0.37
0.28–0.36
0.32
0.33
10–20
0.31–0.39
0.30–0.38
0.31–0.38
0.34
0.33
5–10
0.28–0.38
0.29–0.38
0.30–0.38
0.34
0.34
Note:Smax and Dmax are the maximum slump and maximum slump flow of the L-M-S system; ASmax and ADmax are the approximate maximum slump values within a range of 10 mm from Smax and the approximate maximum slump flow values within a range of 30 mm from Dmax in the L-M-S system, respectively. ACPDmax is a set of compressed packing density values within a range of >99% CPDmax in the L-M-S system.It can be read in Figure2 that the compressed packing density in the case of Smax is 1730.6 kg/m3 in the L-M-S system, which is very close to that of CPDmax (1736 kg/m3). The surface area corresponding to Smax is 245.7 m2, and the surface area corresponding to CPDmax is 266.9 m2. This means that paste volume of the three-graded stone system corresponding to Smax is very close to the minimum paste filling volume, but the surface area corresponding to Smax is smaller than that corresponding to CPDmax in the L-M-S system. Therefore, the paste demand for the same paste thickness is correspondingly smaller. It can be seen from Figure 3 that the slump and slump flow corresponding to CPDmax are 173 mm and 321 mm, respectively, which are relatively smaller than Smax (196 mm) and Dmax (390 mm). This shows that with the same amount of paste, the aggregate system according to CPDmax needs more filling and coating paste than that corresponding to Smax, and thus there is a less excessive paste. When obtaining a certain targeted slump, the three-graded stone system corresponding to Smax has the minimum paste demand (PDmin) aggregate system in the L-M-S system.The effect of the particle sizes and surface areas of the three-graded stone respectively corresponding toPDmin and CPDmax on the interface transition zone (ITZ) is a key consideration. The ITZ of concrete has a great effect on the strength and durability of concrete [43–48]. When mixing fresh concrete, a water film is formed around the large aggregate particles, resulting in a much higher W/B around the large particles than that of the remaining portion. When other conditions are constant, the larger the aggregate size, the greater the local W/B in the ITZ, and the lower the strength and durability of the concrete [43, 49]. The surface area (Sstone) and the average particle size (DA-stone) of the three-graded stone corresponding to PDmin and CPDmax in the L-M-S system can be calculated according to equation (A.9) and equation (A.14) (see Appendix A), which are 245.7 m2, 15.33 mm, and 266.9 m2, 14.45 mm, respectively. Compared with PDmin, Sstone corresponding to the CPDmax is increased by 8.6%, and the DA-stone is reduced by 5.7%. The increase in the DA-stone leads to an increase in the weakening degree to the ITZ. However, the decrease in Sstone leads to a decrease in the ITZ quantity and thus reduces the weakening degree to ITZ. The weakening degree to ITZ depends on the combined effects of Sstone and DA-stone. Therefore, the effect on ITZ in the case of CPDmax and Pmin on the properties of concrete needs further study.It is clear from the above discussion that the proportion of the three-graded stone in the cases ofPDmin also satisfies ACPDmax; therefore, the proportion of the three-graded stone corresponding to PDmin was used to prepare concrete in this study.


## 3.2. Determination of the Binder Dosage and S/A Based on EVMDM

### 3.2.1. The Contour Plots of Concrete Slump and Slump Flow of the P-S-S System

The contour plots of the slump and slump flow of the P-S-S system are shown in Figure4. It can be seen in Figure 4 that the varying trends of concrete slump and slump flow are mostly the same. With an increase in the paste volume, the slump and slump flow gradually increase, and with a decrease in the S/A, the slump and slump flow first increase and then decrease.Figure 4
Contour plots of the slump (a) and the slump flow (b) of the P-S-S system (mm).
(a)
(b)Based on the concrete encapsulation model [50], the paste thickness (Tpaste) coated on the surface of the aggregate, and the mortar thickness (Tmortar) coated on the surface of the coarse aggregate can be calculated according to equations (B.12) and (B.13) (see Appendix B), respectively, whose results are shown in Figure 5. It can be seen in Figure 5 that Tpaste increases gradually with an increase in the binder dosage and the decrease in S/A and varies between 30 and 59 μm, which is consistent with the range of Tpaste obtained by other researchers [51]. This is because the total surface area of aggregate can be significantly reduced by reducing S/A, and Tpaste increases with a decrease in S/A when the paste volume is constant. When the binder dosage is at the maximum value (420 kg/m3) and S/A is at the minimum value (0.35), Tpaste reaches its maximum value (approximately 59 μm). Tmortar increases with an increase in the binder dosage and S/A. When the binder dosage is at the maximum value (420 kg/m3) and S/A is 0.43 (near its maximum value, 0.45), Tmortar reaches its maximum value (approximately 1.7 mm).Figure 5
Contour plots ofTpaste (a) and Tmortar (b) of the P-S-S system.
(a)
(b)Associated with Figures4 and 5, it can be found that when the slump and slump flow reach their maximum values, Tpaste and Tmortar vary in the range of 50–55 μm and 1.3–1.4 mm, respectively, which is in the middle region of Tpaste and Tmortar, respectively. This indicates that Tpaste and Tmortar are not at their maximum or minimum values in the case of PDmin; however, Tpaste is closer to its maximum value (59 μm), and Tmortar is closer to its minimum value (1.0 mm). Jiao et al. [52] regarded that when S/A is fixed, the slump and slump flow are mainly affected by Tpaste. However, their study cannot reflect the combined effects of the paste volume and S/A on the concrete workability by changing the paste volume and S/A simultaneously. The results obtained from Figures 4 and 5 demonstrates that when considering the effect of Tpaste on the workability of concrete, the effect of Tmortar must also be considered at the same time, and EVMDM can determine the most reasonable range of Tpaste and Tmortar in a convenient way.


### 3.2.2. Determination of the Binder Dosage and the S/A Based onPDmin of the P-S-S System

The paste-sand-stone proportions corresponding toSmax, ASmax, Dmax, and ADmax of the P-S-S system can be read in Figure 4, as shown in Table 9. It can be seen in Table 9 that the proportions of paste-sand-stone corresponding to Smax and Dmax are almost the same; the proportions of paste-sand-stone corresponding to ASmax and ADmax are very similar. When the binder dosage varies in the range of 411–420 kg/m3 and S/A varies from 0.37 to 0.39, the concrete workability reaches its approximate maximum value. When the binder dosage is 420 kg/m3 and S/A is 0.38, the slump and slump flow synchronously reach their maximum values, which are 239 mm and 560 mm, respectively.Table 9
Binder dosage and the S/A in the case ofSmax and ASmax of the P-S-S system (by mass).
Items
AS
max
S
max
AD
max
D
max
AS
max + ADmax
Binder (kg/m3)
S/A
Paste
0.230–0.238
0.238
0.233–0.238
0.238
0.233–0.238
411–420
0.37–0.39
Sand
0.279–0.301
0.289
0.280–0.293
0.287
0.280–0.293
Stone
0.461–0.483
0.473
0.469–0.481
0.475
0.469–0.481
Note:Smax and Dmax are the maximum slump and maximum slump flow values of the P-S-S system, respectively; ASmax and ADmax are the approximate maximum slump values within a range of 10 mm from the Smax and the approximate maximum slump flow values within a range of 30 mm from the Dmax in the P-S-S system, respectively.


### 3.2.3. Determination of the Binder Dosage and the S/A Based on the Targeted Workability of the P-S-S System

A typical contour plot of the targeted slump (Star, assumed to be 180–200 mm) and targeted slump flow (Dtar, assumed to be 350–450 mm) is shown in Figure 6. It can be seen in Figure 6 that when Star and the Dtar are synchronously met (the red part in Figure 6), the proportion of the minimum paste amount is 0.215, the proportion of the sand and the stone drops within the regions of 0.284–0.304 and 0.481–0.501, respectively. Namely, the binder dosage is 380 kg/m3, and S/A is from 0.36 to 0.39. It can also be read from Figure 6 that when the proportions of the paste, sand, and stone are in the range of 0.226–0.238, 0.317–0.338, and 0.436–0.448, respectively, the concrete workability can also meet the targeted requirements. Correspondingly, the binder dosage varies between 399 and 420 kg/m3, and S/A varies within the range of 0.42–0.44.Figure 6
Contour plot of the concrete slump and slump flow (mm, slump = 180–200 mm, slump flow = 350–450 mm).The compressed packing density of the sand-stone system with different S/A values is shown in Figure7. It can be seen in Figure 7 that the sand-stone system has the maximum packing density when S/A is 0.41. Therefore, S/A corresponding to CPDmax is not the same as that corresponding to PDmin. Corresponding to PDmin, the binder dosage is 420 kg/m3 and S/A is 0.38, which are not the same as those of the maximum packing density of the sand-stone system. In this situation, Tpaste is approximately 49.5 μm and Tmortar is approximately 1.33 mm. When taking S/A (0.41) corresponding to the maximum packing density of sand-stone system and when the binder dosage is 420 kg/m3, Tpaste and Tmortar are approximately 44 μm and 1.49 mm, respectively, while the slump and slump flow are 210 mm and 405 mm, respectively, which are far from those corresponding to PDmin in the P-S-S system. This shows that the combined effect of an increase in Tmortar and a decrease in Tpaste significantly reduces the concrete workability. Therefore, it is not enough to study the concrete workability only based on Tpaste. The combined effect of Tpaste and Tmortar on the workability should be considered. Cementitious materials will cause early thermal expansion and shrinkage in the hydration process, which will also produce a drying shrinkage in a long-term drying environment [53]. When W/B is constant, reducing the content of cementitious materials can effectively reduce the effect of the paste on the volume stability of concrete [23]. Therefore, one can control Tpaste at a lower level by keeping Tmortar within a reasonable range.Figure 7
Relationship between S/A and the compressed packing density of the sand-stone system. Note: the proportion of the three-graded stone corresponding toCPDmax of the L-M-S system was used.S
stone of the three-graded stone corresponding to Star and Smax values of the P-S-S with different binder dosages and S/A and CPDmax values with S/A = 0.41 are shown in Table 10. It can be seen in Table 10 that Sstone value corresponding to CPDmax is larger than that corresponding to Star and Smax. In the case of Star, Sstone with smaller binder dosage is larger than that with large binder dosage. Therefore, it also has a larger ITZ. However, in the case of the maximum packing density of sand-stone system, DA-stone of the three-graded stone decreases by 5.7%, which makes it more difficult to form a water film compared with Smax situation and thus reduces the effect of the particle size on W/B in ITZ. When W/B is constant, reducing the content of cementitious materials can effectively reduce the effect of the paste on volume stability of concrete [23]. To achieve the targeted workability requirements of a 180–200 mm slump and a 350–450 mm slump flow, the binder dosage = 380 kg/m3 and S/A = 0.36–0.39 can be selected to design the concrete with the lowest cost. It is worth noting whether the P-S-S system based on the PDmin is consistent with the P-S-S system according to the best durability needs further research. It can be recommended that EVMDM is also suitable for durability design based on adjustments of the P-S-S system.Table 10
TheSstone based on PDmin of the P-S-S system.
Binder content (kg/m3)
S/A
S
stone (m2)
Comment
420
0.37
253.6
420
0.38
249.6
PD
min
420
0.39
245.5
420
0.41
257.9
CPD
max
411
0.37
255.3
411
0.38
251.2
PD
min
411
0.39
247.2
411
0.41
259.6
CPD
max
380
0.36
265.3
380
0.38
257.0
PD
min
380
0.39
252.8
380
0.41
265.6
CPD
max
399
0.42
237.1
399
0.43
233.0
PD
min
399
0.44
228.9
399
0.41
261.9
CPD
max


## 3.2.1. The Contour Plots of Concrete Slump and Slump Flow of the P-S-S System

The contour plots of the slump and slump flow of the P-S-S system are shown in Figure4. It can be seen in Figure 4 that the varying trends of concrete slump and slump flow are mostly the same. With an increase in the paste volume, the slump and slump flow gradually increase, and with a decrease in the S/A, the slump and slump flow first increase and then decrease.Figure 4
Contour plots of the slump (a) and the slump flow (b) of the P-S-S system (mm).
(a)
(b)Based on the concrete encapsulation model [50], the paste thickness (Tpaste) coated on the surface of the aggregate, and the mortar thickness (Tmortar) coated on the surface of the coarse aggregate can be calculated according to equations (B.12) and (B.13) (see Appendix B), respectively, whose results are shown in Figure 5. It can be seen in Figure 5 that Tpaste increases gradually with an increase in the binder dosage and the decrease in S/A and varies between 30 and 59 μm, which is consistent with the range of Tpaste obtained by other researchers [51]. This is because the total surface area of aggregate can be significantly reduced by reducing S/A, and Tpaste increases with a decrease in S/A when the paste volume is constant. When the binder dosage is at the maximum value (420 kg/m3) and S/A is at the minimum value (0.35), Tpaste reaches its maximum value (approximately 59 μm). Tmortar increases with an increase in the binder dosage and S/A. When the binder dosage is at the maximum value (420 kg/m3) and S/A is 0.43 (near its maximum value, 0.45), Tmortar reaches its maximum value (approximately 1.7 mm).Figure 5
Contour plots ofTpaste (a) and Tmortar (b) of the P-S-S system.
(a)
(b)Associated with Figures4 and 5, it can be found that when the slump and slump flow reach their maximum values, Tpaste and Tmortar vary in the range of 50–55 μm and 1.3–1.4 mm, respectively, which is in the middle region of Tpaste and Tmortar, respectively. This indicates that Tpaste and Tmortar are not at their maximum or minimum values in the case of PDmin; however, Tpaste is closer to its maximum value (59 μm), and Tmortar is closer to its minimum value (1.0 mm). Jiao et al. [52] regarded that when S/A is fixed, the slump and slump flow are mainly affected by Tpaste. However, their study cannot reflect the combined effects of the paste volume and S/A on the concrete workability by changing the paste volume and S/A simultaneously. The results obtained from Figures 4 and 5 demonstrates that when considering the effect of Tpaste on the workability of concrete, the effect of Tmortar must also be considered at the same time, and EVMDM can determine the most reasonable range of Tpaste and Tmortar in a convenient way.


## 3.2.2. Determination of the Binder Dosage and the S/A Based onPDmin of the P-S-S System

The paste-sand-stone proportions corresponding toSmax, ASmax, Dmax, and ADmax of the P-S-S system can be read in Figure 4, as shown in Table 9. It can be seen in Table 9 that the proportions of paste-sand-stone corresponding to Smax and Dmax are almost the same; the proportions of paste-sand-stone corresponding to ASmax and ADmax are very similar. When the binder dosage varies in the range of 411–420 kg/m3 and S/A varies from 0.37 to 0.39, the concrete workability reaches its approximate maximum value. When the binder dosage is 420 kg/m3 and S/A is 0.38, the slump and slump flow synchronously reach their maximum values, which are 239 mm and 560 mm, respectively.Table 9
Binder dosage and the S/A in the case ofSmax and ASmax of the P-S-S system (by mass).
Items
AS
max
S
max
AD
max
D
max
AS
max + ADmax
Binder (kg/m3)
S/A
Paste
0.230–0.238
0.238
0.233–0.238
0.238
0.233–0.238
411–420
0.37–0.39
Sand
0.279–0.301
0.289
0.280–0.293
0.287
0.280–0.293
Stone
0.461–0.483
0.473
0.469–0.481
0.475
0.469–0.481
Note:Smax and Dmax are the maximum slump and maximum slump flow values of the P-S-S system, respectively; ASmax and ADmax are the approximate maximum slump values within a range of 10 mm from the Smax and the approximate maximum slump flow values within a range of 30 mm from the Dmax in the P-S-S system, respectively.


## 3.2.3. Determination of the Binder Dosage and the S/A Based on the Targeted Workability of the P-S-S System

A typical contour plot of the targeted slump (Star, assumed to be 180–200 mm) and targeted slump flow (Dtar, assumed to be 350–450 mm) is shown in Figure 6. It can be seen in Figure 6 that when Star and the Dtar are synchronously met (the red part in Figure 6), the proportion of the minimum paste amount is 0.215, the proportion of the sand and the stone drops within the regions of 0.284–0.304 and 0.481–0.501, respectively. Namely, the binder dosage is 380 kg/m3, and S/A is from 0.36 to 0.39. It can also be read from Figure 6 that when the proportions of the paste, sand, and stone are in the range of 0.226–0.238, 0.317–0.338, and 0.436–0.448, respectively, the concrete workability can also meet the targeted requirements. Correspondingly, the binder dosage varies between 399 and 420 kg/m3, and S/A varies within the range of 0.42–0.44.Figure 6
Contour plot of the concrete slump and slump flow (mm, slump = 180–200 mm, slump flow = 350–450 mm).The compressed packing density of the sand-stone system with different S/A values is shown in Figure7. It can be seen in Figure 7 that the sand-stone system has the maximum packing density when S/A is 0.41. Therefore, S/A corresponding to CPDmax is not the same as that corresponding to PDmin. Corresponding to PDmin, the binder dosage is 420 kg/m3 and S/A is 0.38, which are not the same as those of the maximum packing density of the sand-stone system. In this situation, Tpaste is approximately 49.5 μm and Tmortar is approximately 1.33 mm. When taking S/A (0.41) corresponding to the maximum packing density of sand-stone system and when the binder dosage is 420 kg/m3, Tpaste and Tmortar are approximately 44 μm and 1.49 mm, respectively, while the slump and slump flow are 210 mm and 405 mm, respectively, which are far from those corresponding to PDmin in the P-S-S system. This shows that the combined effect of an increase in Tmortar and a decrease in Tpaste significantly reduces the concrete workability. Therefore, it is not enough to study the concrete workability only based on Tpaste. The combined effect of Tpaste and Tmortar on the workability should be considered. Cementitious materials will cause early thermal expansion and shrinkage in the hydration process, which will also produce a drying shrinkage in a long-term drying environment [53]. When W/B is constant, reducing the content of cementitious materials can effectively reduce the effect of the paste on the volume stability of concrete [23]. Therefore, one can control Tpaste at a lower level by keeping Tmortar within a reasonable range.Figure 7
Relationship between S/A and the compressed packing density of the sand-stone system. Note: the proportion of the three-graded stone corresponding toCPDmax of the L-M-S system was used.S
stone of the three-graded stone corresponding to Star and Smax values of the P-S-S with different binder dosages and S/A and CPDmax values with S/A = 0.41 are shown in Table 10. It can be seen in Table 10 that Sstone value corresponding to CPDmax is larger than that corresponding to Star and Smax. In the case of Star, Sstone with smaller binder dosage is larger than that with large binder dosage. Therefore, it also has a larger ITZ. However, in the case of the maximum packing density of sand-stone system, DA-stone of the three-graded stone decreases by 5.7%, which makes it more difficult to form a water film compared with Smax situation and thus reduces the effect of the particle size on W/B in ITZ. When W/B is constant, reducing the content of cementitious materials can effectively reduce the effect of the paste on volume stability of concrete [23]. To achieve the targeted workability requirements of a 180–200 mm slump and a 350–450 mm slump flow, the binder dosage = 380 kg/m3 and S/A = 0.36–0.39 can be selected to design the concrete with the lowest cost. It is worth noting whether the P-S-S system based on the PDmin is consistent with the P-S-S system according to the best durability needs further research. It can be recommended that EVMDM is also suitable for durability design based on adjustments of the P-S-S system.Table 10
TheSstone based on PDmin of the P-S-S system.
Binder content (kg/m3)
S/A
S
stone (m2)
Comment
420
0.37
253.6
420
0.38
249.6
PD
min
420
0.39
245.5
420
0.41
257.9
CPD
max
411
0.37
255.3
411
0.38
251.2
PD
min
411
0.39
247.2
411
0.41
259.6
CPD
max
380
0.36
265.3
380
0.38
257.0
PD
min
380
0.39
252.8
380
0.41
265.6
CPD
max
399
0.42
237.1
399
0.43
233.0
PD
min
399
0.44
228.9
399
0.41
261.9
CPD
max


## 4. Conclusions

In this study, ppe-SCDM and EVMDM have been used to design the workability of concrete with the three-graded stone, which can experimentally realize the minimum paste demand and the maximum packing density of the three-graded stone in a convenient way.Sstone, Ssand, DA-stone, Tpaste, and Tmortar were calculated to discuss their effect on concrete workability. Some of the main conclusions obtained are as follows:(1)
The proportions of the three-graded stone in the case ofCPDmax and LPDmax are quite different. The compressed packing density is more suitable for characterizing the packing state of the three-graded stone in concrete than the loose packing density. The proportions of the three-graded stone in the case of Smax and Dmax of the L-M-S system fall within the overlapping region of the three-graded stone proportions among the ACPDmax, ASmax, and ADmax of the L-M-S system. However, the proportion of the three-graded stone corresponding to CPDmax is not within this overlapping region. This indicates that the proportion of the three-graded stone corresponding to Smax and Dmax can meet that corresponding to ACPDmax.(2)
When the binder dosage and S/A are in the range of 380–420 kg/m3 and 0.35–0.45, respectively, Tpaste and Tmortar vary between 30 and 59 μm and 1.2–1.7 mm, respectively. However, Tpaste and Tmortar vary in the range of 50–55 μm and 1.3–1.4 mm, respectively, when the slump and slump flow reach their maximum values. This means that when the workability reaches a maximum value, Tpaste and Tmortar do not reach their extreme values; however, Tpaste is closer to its maximum value, and Tmortar is closer to its minimum value. Therefore, it is very necessary to consider the combined effects of Tpaste and Tmortar on concrete workability, not just the effect of Tpaste.(3)
The MDM can determinePDmin and the optimal S/A under any target workability requirements. At the same time, the concrete designed by the MDM approximately reaches the ACPDmax value. Therefore, it can be expected that the concrete prepared using this method has a lower cost. However, compared with PDmin (meeting the ACPDmax), Sstone of the three-graded stone system corresponding to CPDmax is increased by 8.6%, and DA-stone is reduced by 5.7%. The increase of the particle size will cause an increase in the weakening degree of ITZ. However, the decrease in Sstone will reduce the weakening degree of ITZ. The weakening degree of ITZ depends on the combined effects of Sstone and DA-stone. Therefore, the contradictory weakening effect of an increase in Sstone and a decrease in DA-stone to ITZ in both cases requires further evaluation.



---

*Source: 8934684-2020-04-27.xml*