# Research and Model Prediction on the Performance of Recycled Brick Powder Foam Concrete

Authors: Hongyang Xie; Jianjun Dong; Yong Deng; Yiwen Dai
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

In order to achieve resource conservation, protect the environment and realize the sustainable development of the construction industry, the low energy resource utilization of construction waste was explored. In this paper, the effect of air bubble swarm admixture, recycled brick powder admixture, water to material ratio, and HPMC content on the physical and mechanical properties of recycled brick powder foam concrete was investigated by conducting a 4-factor, 5-level orthogonal test with recycled brick powder as fine aggregate, and the effect of each factor on the physical and mechanical properties of recycled brick powder foam concrete was derived, and the optimum ratio of recycled brick powder foam concrete was determined by analysing the specific strength. Five machine learning models, namely, back propagation neural network improved by particle swarm optimization (PSO-BP), support vector machine (SVM), multiple linear regression (MLR), random forest (RF), and back propagation neural network (BP), were used to predict the compressive strength of recycled brick powder foam concrete, and the PSO-BP model was found to have obvious advantages in terms of prediction accuracy and model stability. The experimental results and prediction models can provide experimental and theoretical references for the research and application of recycled brick powder foam concrete.


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## Body

## 1. Introduction

With the development of urbanization, the amount of construction and demolition waste is increasing rapidly. In China, about 1.8 billion tons of construction waste are generated every year [1]. In the past, construction waste was often dumped or landfilled, which not only wasted land and resources but also polluted the environment. Xiao et al. [2] reported that, about 80% of construction waste was reusable construction waste, such as waste concrete, bricks, and glass. Therefore, it has become an effective way to use construction waste as recycled aggregate for cement-based materials. Recycled concrete aggregates are equipped with similar properties to natural aggregates, which have been widely used in concrete production. Previous studies [3–5] have reported that recycled brick powder (RBP) has similar characteristics as supplementary cementitious materials, such as high content of SiO2 and Al2O3. However, waste clay brick aggregates are of high crushing value and water absorption, which limits their large-scale application in concrete. Therefore, the effective utilization of clay bricks recovered from construction waste has become a top priority.Foam concrete takes the cement-based cementitious materials, admixtures, and so on, as the main cementing materials. The admixtures and water, or some granular lightweight aggregates, are added to make slurry. It is a lightweight concrete material that contains a large number of tiny, independent, and evenly-distributed bubbles formed by casting and curing at the construction site or in the factory [6]. In recent years, foam concrete has been widely used in the field of civil engineering as a building material with excellent performance. Foam concrete consumes a large amount of cementitious material [7], with more expensive costs and lower strength than structural concrete. Reducing the amount of cementitious materials while maintaining its light weight and increasing strength is an important direction for studying the mix ratio of foam concrete [8]. Studies have shown that adding fillers and improving the pore structure can effectively increase the strength of foamed concrete [9–13]. Domestic and foreign scholars have explored the possibility of replacing cementitious materials with various materials [14–18]. However, few reports have emerged on using recycled brick powder as a filler for foam concrete.In this paper, recycled brick powder was prepared based on the waste clay bricks produced by demolishing houses in rural areas. Based on the test of ordinary foam concrete, recycled brick powder and hydroxypropyl methyl cellulose ether (HPMC) foam stabilizer was added to explore the influence of bubble group content, recycled brick powder content, water-to-material ratio, and HPMC content on the compressive strength of recycled brick powder foam concrete. The pore size distribution and microscopic morphology were observed, the parameter sensitivity of various factors were analyzed through neural network and machine learning theory, and the compressive strength of recycled brick powder foam concrete cubes was predicted.


## 2. Experiment Overview

### 2.1. Raw materials

#### 2.1.1. Cement

The cement should also have high strength and not compromise the stability of the foam, based on compliance with the Chinese standard “General Silicate Cement” (GB175-2007) [19]. Conch brand P.O.42.5 ordinary Portland cement was used in this experiment. The chemical composition and basic properties of the cement are shown in Table 1.Table 1
Chemical composition of cement.
ComponentSiO2Fe2O3Al2O3CaOMgOSO3Content (%)224.25.2631.52.2


#### 2.1.2. Recycled Brick Powder

Crush the waste clay bricks to Table2 particles of about 4.75 mm with a hammer crusher and use a ball mill to grind them for one hour to obtain regenerated recycled brick powder particles of 10 to 80 μm. Figure 1 shows the waste clay bricks, and Figure 2 shows the recycled brick powder.Table 2
Basic properties of cement.
ItemIndexResultsJudgementInitial set time (min)≥60176QualifiedFinal set time (min)≤600310Qualified3d flexural strength (MPa)≥3.54.2Qualified28d flexural strength (MPa)≥6.57.1Qualified3d compressive strength (MPa)≥1722.3Qualified28d compressive strength (MPa)≥42.548.2QualifiedFigure 1
Waste clay bricks.Figure 2
Recycled brick powder.


#### 2.1.3. Foaming Agent

The HT composite foaming agent produced by Henan Huatai Building Materials Co., Ltd., was used, while its physical properties are shown in Table3.Table 3
Physical performance indicators of HT composite foaming agent.
HTAppearancesDensity (g/cm3)Solid content (%)pH valueDilution ratioSettlement distance for 1 h (mm)Bleeding rate for 1 h (%)Dark brown1.0523.89.215∼30≤50≤70The foaming agent performance test was carried out in accordance with the Chinese standard “Technical Regulations for Air-bubble Mixed Lightweight Soil Filling Engineering,” and the results showed that the HT compound foaming agent diluted 30 times had the lowest water secretion and the smallest settling distance. The preparation process is shown in Figure3.Figure 3
Foam preparation process.


#### 2.1.4. Foam Stabilizer

HPMC produced by Fuqiang Fine Chemical Co., Ltd., was adopted, and its physical performance indicators are shown in Table4. Figure 4 shows a photograph of the hydroxypropyl methyl cellulose ether (HPMC).Table 4
Physical performance indicators of HPMC.
Indicator resultsAppearancesViscosity (Pa s)PH valueWater content (%)White powder1506.52.1Figure 4
HPMC.


### 2.2. Calculation of mix Ratio

The commonly-used methods of concrete mix design comprised the volume method and mass method. Due to the light mass of bubbles, the volume method was adopted in this research after a comprehensive consideration [20]. According to formula (5.2.1-1) in the Chinese standard “Technical Specification for Foamed Mixture Lightweight Soil Filling Engineering,” the mix ratio calculation is conducted, and the expression is as follows:(1)mcρc+mwρw+mfρf+mfρf=1.In the formula,mc: cement dosage per cubic meter of foam concrete (kg); ρc: cement density (kg/m3); mw: water consumption per cubic meter of foamed concrete; ρw density of water (kg/m3), taking 1000 kg/m3; mf.: amount of air foams per cubic meter of foam concrete (kg); ρf.: density of air foams (kg/m3); mm: mixture amount of foam concrete per cubic meter (kg); ρm: density of the admixture (kg/m3).


### 2.3. Mixture Ratio Data of Ordinary Foam Concrete

To explore the performance difference between recycled brick powder foam concrete and ordinary foam concrete, ordinary foam concrete test blocks were made as a control group. The ordinary foam concrete adopted the orthogonal test of two factors and five levels, and the change range of the bubble swarm content was 3% to 7%, while the change range of the water-cement ratio was 0.45 to 0.65. The data is shown in Table5.Table 5
Mixture ratio data of ordinary foam concrete.
Group numberBubble volumes (%)Water cement ratioCement (g)Bubble groups (g)Water (g)130.45728.5521.86327.85230.50702.9521.09351.48330.55679.0820.37373.49430.60656.7819.70394.07530.65635.9019.08413.34640.45635.9025.44286.16740.50616.3024.65308.15840.55597.8823.92328.83940.60580.5223.22348.311040.65564.1522.57366.701150.45564.1528.21253.871250.50548.7327.44274.371350.55534.0226.70293.711450.60520.1326.01312.081550.65506.9525.35329.521660.45506.9530.42228.131760.50494.4229.67247.211860.55482.4928.95265.371960.60471.1228.27282.672060.65460.2827.62298.182170.45460.2832.22207.132270.50449.9331.50224.972370.55440.0330.80242.022470.60430.5630.14258.342570.65421.4829.50273.96


### 2.4. Test Pieces Preparation and Test Methods

Ordinary foam concrete specimens were made according to the following process: (1) Accurately weigh the quality of cement; (2) Add water of accurate quality into the mixer, and then quickly stir for 150 seconds; (3) Add the foaming agent aqueous solution to the cement foaming integrated machine, pressurize the air to foam, prepare to measure the mass of the bubble group, and quickly add it to the cement slurry of construction waste, and stir for 120 seconds. When the mixing is uniform and no bubbles have emerged on the surface of the slurry, the fluidity test is conducted immediately. After meeting the requirements of 160 to 200 mm, quickly pour it into a 100 mm × 100 mm × 100 mm mold, and gently flip the mold to prevent irregular test blocks: (4) After standing for 36 hours, scrape the surface of the mold and remove it with an air gun. To fulfill the strength requirements, it needs to be stored in a standard curing room with a temperature of 20 ± 2°C and a humidity of 95% or more, as well as normal curing for 28 days.The process of making recycled brick powder foam concrete specimens (1) was as follows: Accurately weigh the quality of cement, recycled brick powder and HPMC, and drily mix with a mixer for 150 seconds. The process from (2) to (4) was the same as that of the control group. The preparation process for the recycled brick powder foam concrete specimens in this paper is shown in Figure5.Figure 5
Test piece preparation process for recycled brick powder foam concrete.The dry density of the control and experimental group specimens were measured according to the Chinese standard “Foam Concrete” [21]; the determination of the compressive strength of the control group and the experimental group could refer to the Chinese standard “Standard for Test Methods of Mechanical Properties of Ordinary Concrete” and “Test methods of Autoclaved aerated Concrete” [22].


### 2.5. Orthogonal Mix Ratio Design

Recycled brick powder foam concrete adopted an orthogonal test of 4 factors and 5 levels, that was, the orthogonal test table was L25 (45). The four factors of bubble group content, recycled brick powder content, water-to-material ratio, and HPMC content were used, respectively, for experimental research. The specific level factor data is shown in Table 6.Table 6
Orthogonal test factor level of experimental group.
LevelFactorBubble volumes (A) (%)Recycled brick powder volume (B) (%)Water to material ratio (C) (%)HPMC content (D)13100.450.0324200.50.0435300.550.0546400.60.0657500.650.07


## 2.1. Raw materials

### 2.1.1. Cement

The cement should also have high strength and not compromise the stability of the foam, based on compliance with the Chinese standard “General Silicate Cement” (GB175-2007) [19]. Conch brand P.O.42.5 ordinary Portland cement was used in this experiment. The chemical composition and basic properties of the cement are shown in Table 1.Table 1
Chemical composition of cement.
ComponentSiO2Fe2O3Al2O3CaOMgOSO3Content (%)224.25.2631.52.2


### 2.1.2. Recycled Brick Powder

Crush the waste clay bricks to Table2 particles of about 4.75 mm with a hammer crusher and use a ball mill to grind them for one hour to obtain regenerated recycled brick powder particles of 10 to 80 μm. Figure 1 shows the waste clay bricks, and Figure 2 shows the recycled brick powder.Table 2
Basic properties of cement.
ItemIndexResultsJudgementInitial set time (min)≥60176QualifiedFinal set time (min)≤600310Qualified3d flexural strength (MPa)≥3.54.2Qualified28d flexural strength (MPa)≥6.57.1Qualified3d compressive strength (MPa)≥1722.3Qualified28d compressive strength (MPa)≥42.548.2QualifiedFigure 1
Waste clay bricks.Figure 2
Recycled brick powder.


### 2.1.3. Foaming Agent

The HT composite foaming agent produced by Henan Huatai Building Materials Co., Ltd., was used, while its physical properties are shown in Table3.Table 3
Physical performance indicators of HT composite foaming agent.
HTAppearancesDensity (g/cm3)Solid content (%)pH valueDilution ratioSettlement distance for 1 h (mm)Bleeding rate for 1 h (%)Dark brown1.0523.89.215∼30≤50≤70The foaming agent performance test was carried out in accordance with the Chinese standard “Technical Regulations for Air-bubble Mixed Lightweight Soil Filling Engineering,” and the results showed that the HT compound foaming agent diluted 30 times had the lowest water secretion and the smallest settling distance. The preparation process is shown in Figure3.Figure 3
Foam preparation process.


### 2.1.4. Foam Stabilizer

HPMC produced by Fuqiang Fine Chemical Co., Ltd., was adopted, and its physical performance indicators are shown in Table4. Figure 4 shows a photograph of the hydroxypropyl methyl cellulose ether (HPMC).Table 4
Physical performance indicators of HPMC.
Indicator resultsAppearancesViscosity (Pa s)PH valueWater content (%)White powder1506.52.1Figure 4
HPMC.


## 2.1.1. Cement

The cement should also have high strength and not compromise the stability of the foam, based on compliance with the Chinese standard “General Silicate Cement” (GB175-2007) [19]. Conch brand P.O.42.5 ordinary Portland cement was used in this experiment. The chemical composition and basic properties of the cement are shown in Table 1.Table 1
Chemical composition of cement.
ComponentSiO2Fe2O3Al2O3CaOMgOSO3Content (%)224.25.2631.52.2


## 2.1.2. Recycled Brick Powder

Crush the waste clay bricks to Table2 particles of about 4.75 mm with a hammer crusher and use a ball mill to grind them for one hour to obtain regenerated recycled brick powder particles of 10 to 80 μm. Figure 1 shows the waste clay bricks, and Figure 2 shows the recycled brick powder.Table 2
Basic properties of cement.
ItemIndexResultsJudgementInitial set time (min)≥60176QualifiedFinal set time (min)≤600310Qualified3d flexural strength (MPa)≥3.54.2Qualified28d flexural strength (MPa)≥6.57.1Qualified3d compressive strength (MPa)≥1722.3Qualified28d compressive strength (MPa)≥42.548.2QualifiedFigure 1
Waste clay bricks.Figure 2
Recycled brick powder.


## 2.1.3. Foaming Agent

The HT composite foaming agent produced by Henan Huatai Building Materials Co., Ltd., was used, while its physical properties are shown in Table3.Table 3
Physical performance indicators of HT composite foaming agent.
HTAppearancesDensity (g/cm3)Solid content (%)pH valueDilution ratioSettlement distance for 1 h (mm)Bleeding rate for 1 h (%)Dark brown1.0523.89.215∼30≤50≤70The foaming agent performance test was carried out in accordance with the Chinese standard “Technical Regulations for Air-bubble Mixed Lightweight Soil Filling Engineering,” and the results showed that the HT compound foaming agent diluted 30 times had the lowest water secretion and the smallest settling distance. The preparation process is shown in Figure3.Figure 3
Foam preparation process.


## 2.1.4. Foam Stabilizer

HPMC produced by Fuqiang Fine Chemical Co., Ltd., was adopted, and its physical performance indicators are shown in Table4. Figure 4 shows a photograph of the hydroxypropyl methyl cellulose ether (HPMC).Table 4
Physical performance indicators of HPMC.
Indicator resultsAppearancesViscosity (Pa s)PH valueWater content (%)White powder1506.52.1Figure 4
HPMC.


## 2.2. Calculation of mix Ratio

The commonly-used methods of concrete mix design comprised the volume method and mass method. Due to the light mass of bubbles, the volume method was adopted in this research after a comprehensive consideration [20]. According to formula (5.2.1-1) in the Chinese standard “Technical Specification for Foamed Mixture Lightweight Soil Filling Engineering,” the mix ratio calculation is conducted, and the expression is as follows:(1)mcρc+mwρw+mfρf+mfρf=1.In the formula,mc: cement dosage per cubic meter of foam concrete (kg); ρc: cement density (kg/m3); mw: water consumption per cubic meter of foamed concrete; ρw density of water (kg/m3), taking 1000 kg/m3; mf.: amount of air foams per cubic meter of foam concrete (kg); ρf.: density of air foams (kg/m3); mm: mixture amount of foam concrete per cubic meter (kg); ρm: density of the admixture (kg/m3).


## 2.3. Mixture Ratio Data of Ordinary Foam Concrete

To explore the performance difference between recycled brick powder foam concrete and ordinary foam concrete, ordinary foam concrete test blocks were made as a control group. The ordinary foam concrete adopted the orthogonal test of two factors and five levels, and the change range of the bubble swarm content was 3% to 7%, while the change range of the water-cement ratio was 0.45 to 0.65. The data is shown in Table5.Table 5
Mixture ratio data of ordinary foam concrete.
Group numberBubble volumes (%)Water cement ratioCement (g)Bubble groups (g)Water (g)130.45728.5521.86327.85230.50702.9521.09351.48330.55679.0820.37373.49430.60656.7819.70394.07530.65635.9019.08413.34640.45635.9025.44286.16740.50616.3024.65308.15840.55597.8823.92328.83940.60580.5223.22348.311040.65564.1522.57366.701150.45564.1528.21253.871250.50548.7327.44274.371350.55534.0226.70293.711450.60520.1326.01312.081550.65506.9525.35329.521660.45506.9530.42228.131760.50494.4229.67247.211860.55482.4928.95265.371960.60471.1228.27282.672060.65460.2827.62298.182170.45460.2832.22207.132270.50449.9331.50224.972370.55440.0330.80242.022470.60430.5630.14258.342570.65421.4829.50273.96


## 2.4. Test Pieces Preparation and Test Methods

Ordinary foam concrete specimens were made according to the following process: (1) Accurately weigh the quality of cement; (2) Add water of accurate quality into the mixer, and then quickly stir for 150 seconds; (3) Add the foaming agent aqueous solution to the cement foaming integrated machine, pressurize the air to foam, prepare to measure the mass of the bubble group, and quickly add it to the cement slurry of construction waste, and stir for 120 seconds. When the mixing is uniform and no bubbles have emerged on the surface of the slurry, the fluidity test is conducted immediately. After meeting the requirements of 160 to 200 mm, quickly pour it into a 100 mm × 100 mm × 100 mm mold, and gently flip the mold to prevent irregular test blocks: (4) After standing for 36 hours, scrape the surface of the mold and remove it with an air gun. To fulfill the strength requirements, it needs to be stored in a standard curing room with a temperature of 20 ± 2°C and a humidity of 95% or more, as well as normal curing for 28 days.The process of making recycled brick powder foam concrete specimens (1) was as follows: Accurately weigh the quality of cement, recycled brick powder and HPMC, and drily mix with a mixer for 150 seconds. The process from (2) to (4) was the same as that of the control group. The preparation process for the recycled brick powder foam concrete specimens in this paper is shown in Figure5.Figure 5
Test piece preparation process for recycled brick powder foam concrete.The dry density of the control and experimental group specimens were measured according to the Chinese standard “Foam Concrete” [21]; the determination of the compressive strength of the control group and the experimental group could refer to the Chinese standard “Standard for Test Methods of Mechanical Properties of Ordinary Concrete” and “Test methods of Autoclaved aerated Concrete” [22].


## 2.5. Orthogonal Mix Ratio Design

Recycled brick powder foam concrete adopted an orthogonal test of 4 factors and 5 levels, that was, the orthogonal test table was L25 (45). The four factors of bubble group content, recycled brick powder content, water-to-material ratio, and HPMC content were used, respectively, for experimental research. The specific level factor data is shown in Table 6.Table 6
Orthogonal test factor level of experimental group.
LevelFactorBubble volumes (A) (%)Recycled brick powder volume (B) (%)Water to material ratio (C) (%)HPMC content (D)13100.450.0324200.50.0435300.550.0546400.60.0657500.650.07


## 3. Experimental Results and Analysis

### 3.1. Orthogonal Experiment Results

The orthogonal test results of recycled brick powder foam concrete are shown in Table7.Table 7
Orthogonal test results of recycled brick powder foam concrete.
Group numberDry density (kg/m3)28 d compressive strength (MPa)Specific strength (N·m·kg−1)Group numberDry density (kg/m3)28 d compressive strength (MPaSpecific strength (N·m·kg−1)11171.106.075185.3014710.850.841186.8921210.255.804792.6115838.752.372830.703988.005.765833.3516806.802.112621.4141072.754.554241.3017952.502.822955.6451039.453.092975.8118857.602.472882.646997.155.175183.1719647.651.742686.567854.254.234947.3820889.401.251404.438680.853.114570.7621769.502.152794.679937.552.262410.2222975.852.582645.59101071.203.042838.8323851.151.511777.3611988.205.315369.9224675.201.962901.14121008.452.762736.9225786.600.961226.0413721.451.462020.72The orthogonal test results of ordinary foam concrete are shown in Table8.Table 8
Orthogonal test results of ordinary foam concrete.
Group numberDry density (kg/m3)28 d compressive strength (MPa)Specific strength (N·m·kg−1)Group numberDry density (kg/m3)28 d compressive strength (MPa)Specific strength (N·m·kg−1)1743.552.2903079.8214591.61.1821997.972722.382.2293085.6315590.31.0141717.773707.61.9982823.6316644.11.7342692.134714.92.0122814.3817628.71.5482462.225639.41.6682608.7018584.10.9491624.726536.00.8801641.7919599.71.2122021.017735.22.1092868.6120550.30.9191670.008797.12.4083020.9521608.41.2292020.059648.60.8531315.1422500.10.8751749.6510567.40.8301462.8123672.71.4622173.3311713.81.6572321.3824592.81.0731810.0512638.21.4712304.9225567.40.9511676.0713615.91.6312648.16According to the orthogonal test results in Tables7 and 8, the compressive strength is analyzed through the extremely poor method [23]. The larger the range value is, the greater the influence of this factor on the compressive strength would be. The results are shown in Tables 9 and 10.Table 9
Analysis of extreme differences in compressive strength of recycled brick powder foam concrete.
Bubble volumesRecycled brick powder volumeWater to material ratioHPMC contentK15.064.163.002.77K23.563.643.322.88K32.552.863.703.11K42.082.272.703.16K51.832.142.363.17Extreme difference value R3.222.021.350.40Rank1234Table 10
Analysis of extreme differences in compressive strength of ordinary foam concrete.
Bubble volumesWater cement ratioK12.041.56K21.421.65K31.391.69K41.271.27K51.121.08Extreme difference value R0.920.61Rank12The influence order of various factors on the compressive strength of recycled brick powder foam concrete can be obtained from Table9 as follows: bubble volume > recycled brick powder volume > the ratio of water to material > HPMC content. Table 10 shows that the amount of bubble exerts a greater effect on the compressive strength of ordinary foam concrete.The data in Tables9 and 10 is visualized, as shown in Figure 6.Figure 6
Changes in the influence of various factors on compressive strength.
(a)(b)(c)(d)Figure6(a) shows that the compressive strength of recycled brick powder foam concrete and ordinary foam concrete both show a decreasing trend with the increase of the amount of air bubble, which drop by 63.6% and 45.1%, respectively. The increase in the bubble volume poses a significant effect on reducing the compressive strength of recycled brick powder foam concrete. The main reasons are as follows: (1) the increase in the bubble volume is accompanied by the decrease in the amount of cementitious material, which are the main source of the strength of recycled brick powder foam concrete; (2) the increase in the bubble volume will increase the pore volume and produce numerous harmful pores, which will make the pore distribution in the specimen uneven and reduce the strength.Figure6(b) shows that the compressive strength of recycled brick powder foam concrete is decreased by 48.6% with the increase in the amount of recycled brick powder. The main reasons are as follows: (1) the increase in the amount of recycled brick powder is accompanied by the decrease in the amount of cement, which leads to the reduction of C-S-H hydration products and the decrease in strength; (2) the reduction of cement content has led to the deformation of a dense skeleton in part of the recycled brick powder; (3) the main component of recycled brick powder is glassy SiO2, which is an inert material. Although it contains a small amount of Al2O3, most of the substances do not participate in the early hydration reaction of cement-based materials, while the increase in the substitution rate will reduce the strength; (4) the mixture of recycled brick powder poses a dilution effect on the cement and increased the effective water-cement ratio.Figure6(c) shows that the compressive strength of recycled brick powder foam concrete and ordinary foam concrete are increased at first and then decreased with the increase in water-to-material ratio. The reasons are as follows: (1) when the water-to-material ratio is within the range of 0.45 to 0.55, the compressive strength of recycled brick powder foam concrete is increased with the increase of the water-to-material ratio. As recycled brick powder is of a certain pozzolanic activity, the water demand is more than that of virgin materials. As the amount of water is increased, the pozzolanic activity of recycled brick powder helps in generating more C-S-H gel, which further improves the microstructure of recycled brick powder foam concrete, thereby increasing its strength [24]. (2) When the water-to-material ratio is between 0.60 and 0.65, the compressive strength of recycled brick powder foam concrete is decreased as the water-to-material ratio is increased, which is mainly due to the following: (1) the amount of water is excessive. Due to the different densities, the denser components in the recycled brick powder are left underneath, and the bubbles and remaining cementitious materials are on the top, leading to delamination; (2) excessive water thickens the bubble group liquid film, which will accelerate the bleeding under the action of gravity, leading the bubble group to bursting prematurely or forming numerous harmful holes, thereby reducing the strength.Figure6(d) shows that the compressive strength of recycled brick powder foam concrete is increased with the increase of HPMC content, while the increasing rate is gradually slowed. The reasons are as follows: with the increase in the HPMC content, the performance of the bubble group is improved, the shape of the pores, the distance between the holes, and the pore diameter are all improved. Moreover, the bubble group is uniformly distributed in the slurry. Therefore, the strength is improved. However, as the strength of recycled brick powder foam concrete is mainly derived from cementitious materials, the upward trend tends to be flat.


### 3.2. Determination of the Optimal Mix Ratio

The specific strength is the ratio of the strength of a material to its dry density, and it is an important indicator to measure the light weight and high strength of the material [21]. The greater the specific strength is, the better the light-weight and high-strength performance of the material would be [25]. Table 7 shows that the specific strength of the test blocks under the mix ratios in the 3rd group is the largest, as the level of recycled brick powder foam concrete that fulfilled the requirements of light weight and high strength is A1B3C3D3. That is, the optimal mixing ratio is 3% of the bubble group content, 30% of the recycled brick powder content, 0.55 of the water-to-material ratio, and 0.05% of the HPMC content. It can be concluded from Table 8 that the specific strength of the test block is the largest under the mix ratios in the 2nd group, that is, the optimal mix ratio of ordinary foam concrete is 3% of the bubble group content and 0.50 water-cement ratio.For the optimal mix ratio of recycled brick powder foam concrete, the water-to-cement ratio is 0.55, while for the optimal mix ratio of ordinary foam concrete, the water-cement ratio is 0.50. That is, adding recycled brick powder requires more water to promote the complete hydration of the cementitious material. The dry density of the 3rd group of specimens in Table7 is 988 kg/m3, while the 28 d compressive strength is 5.76 MPa; The dry density of the 2nd group of specimens in Table 8 is 722.38 kg/m2, and the 28 d compressive strength is 2.229 MPa. The compressive strength of recycled brick powder foam concrete with 28 d compressive Strength is 2.58 times that of ordinary foam concrete, while the compressive strength of dry density recycled brick powder foam concrete is 1.37 times that of ordinary foam concrete. Although the addition of recycled brick powder will increase the dry density of foam concrete, the increase in compressive strength is even greater.


### 3.3. Microscopic Morphology Analysis

The optimal mix ratio test piece can be observed by the SU8100 scanning electron microscope, and the test results are shown in Figure7.Figure 7
Photograph of 28 d hydration of the specimen when scanned by electron microscope at 500x.The pore diameter of the samples was measured by the NanoMeasure software. Figure3 shows that the average pore diameter of the ordinary Figure 8 foam concrete is 109.73 μm, and the average pore diameter of the recycled brick powder foam concrete is about 72.17 μm, a reduction of 34.2%. The pore wall thickness in Figure 7(b) is obviously greater than that in Figure 7(a). Obvious micro-filling effects can be found. Therefore, the added recycled brick powder refines the pores, increases the compactness of the concrete matrix, and improves the compressive strength of the recycled brick powder foam concrete. When the multiplying factor is 50x, the pore diameter is large with obvious connecting holes in Figure 8(a). In Figure 8(b), the pore diameter is smaller, the pore structure is more uniform, the wall between the pores is dense, the communicating pores are less, and the closed pores are more. As a foam stabilizer, HPMC reduces the occurrence of harmful pores such as connecting holes in the test piece, and makes the pore distribution more uniform.Figure 8
Photograph of 28 d hydration of the specimen when scanned by electron microscope at 50x.


### 3.4. Correlation Analysis of Various Factors and Recycled Brick Powder Foam Concrete

To investigate the correlation between the compressive strength of recycled brick powder foam concrete and the bubble volumes, the amount of recycled brick powder, the water-to-material ratio, and the HPMC content, the Pearson correlation coefficient was used for analysis. The Pearson correlation coefficient [26] (also known as covariance correlation or Bravais–Pearson correlation) describes the degree of correlation between two fixed-ratio, linearly correlated variables (measurement value series) that are not less than affected by their units (also known as tightness). The Pearson correlation coefficient is calculated based on the covariance and product of mean difference of two variables, and its calculation formula is as follows:(2)r=covx,ySx•Sy=∑inxi−x¯•yi−y¯n•Sx•Sy.In the formula:covx,y is the covariance of variables x and y; Sx and Sy are the variances of variables x and y; n is the number of variables; x¯ and y¯ are the mean values of variables x and y, respectively. The value of the correlation coefficient is generally between −1 and 1. The greater the absolute value of the correlation coefficient is, the stronger the correlation of 2 variables would be. If the absolute value of the correlation coefficient is close to 1, it indicates that a perfect linear correlation exists between the two variables; if it is close to 0, no linear correlation exists between the two variables [27].In the Figure9, X1, X2, X3, and X4 represent the volume of bubble volumes, the volume of recycled brick powder, water-to-material ratio, and HPMC content, respectively. Y1 represents the compressive strength of recycled brick powder foam concrete. The order of correlation is as follows: bubble volume > recycled brick powder volume > the ratio of water to material > HPMC content, which is consistent with the results from the orthogonal test analysis. The amount of bubble volume poses a significant effect on the compressive strength of recycled brick powder foam concrete.Figure 9
Heat map of the correlation between each factor and the compressive strength of the cube.


### 3.5. PSO-BP (Back Propagation Neural Network Improved by Particle Swarm Optimization) Prediction

For the recycled brick powder foam concrete, due to the increase in its components, the hydration reaction mechanism is not completely clear. The influencing factors are complex and interactive, which manifests as a specific high-dimensional nonlinear law [28]. The BP neural network is an effective method to solve high-dimensional nonlinear problems. Among them, the PSO-BP [29] neural network defines the initial weights and thresholds of the BP neural network as the particles in the particle swarm optimization algorithm. All particles form a group, and the optimal solution is found through cooperation and information-sharing between individuals in the group. The defects of multiple local extreme points in the BP neural network and difficulty in establishing mathematical models can be avoided. The PSO-BP model consists of the following steps [29].Step 1.
Enter the mix ratio and compressive strength data of recycled brick powder foam concrete and randomly assign 70% of the data as the training set and 30% of the data as the test set.Step 2.
Initialize the parameters of the BP neural network.Step 3.
The initial weights and thresholds are extracted from the BP neural network, encoded as a set of particles in the PSO (group sizeN). Initialize the number of iterations t, maximum number of iterations tmax, inertia weight W, and acceleration factor c1 and c2.Step 4.
Assume the position of each particleXi and the velocity of each particle Vi in space.Step 5.
Evaluate the fitness value of all particles and obtain the best personal solutionpbest and the best group solution gbest.Step 6.
Usepbest and gbest to correct the velocity of each particle Vi(t + 1).Step 7.
Correct the position of each particleXi(t + 1).Step 8.
Repeat Step4 to Step 6 until t ≥ tmax.Step 9.
The optimized positiongbest can be obtained.Step 10.
The optimized BP neural network model is trained with the training set, and the accuracy of the model is verified with the test set.
The establishment process of the PSO-BP neural network in this paper is shown in Figure10.
Taking cement quality, bubble group quality, recycled brick powder quality, water and HPMC quality as input independent variables, and the compressive strength of recycled brick powder foam concrete as output dependent variables, a 5-1-11-1 PSO-BP neural network was established. The prediction of the cubic compressive strength of recycled brick powder foam concrete and the strength of the specimen are compared as shown in Figure11.
Figure11 shows that the predicted compressive strength of the PSO-BP model is close to the actual value, that is, the prediction result of the PSO-BP model is more accurate, with small errors and high fitting accuracy.Figure 10
Establishment process of PSO-BP neural network.Figure 11
PSO-BP model fitting effect diagram.


### 3.6. Model Performance Comparison

The prediction performance of the PSO-BP model was compared with various models, including the support vector machine (SVM), multiple linear regression (MLR), random forest (RF), and artificial neural network (BP) models. The fitting effect is shown in Figure12.Figure 12
Scattered distribution diagram of predicted and actual values of each model.
(a)(b)(c)(d)(e)Figure12 shows that most of the scattered data of the prediction model of PSO-BP are concentrated in and around the 100% regression line. While SVM, MLR, RF and BP scattered point data are very discrete, the data in Figure 11 is fitted according to the form of y=a0x+a1.. Compared with the four models in the control group, the a0 value of the PSO-BP model is closer to 1 and a1 values are closer to 0. The prediction accuracy of the PSO-BP model is higher and the error is smaller.


### 3.7. Statistical Indicators of the Model

In this paper, five statistics were selected to evaluate the prediction accuracy of the PSO-BP model and the control model, as the coefficient of determination (R2), the mean square error (RMSE), the mean error (MAE), the calculation formula of the average absolute deviation percentage error (RMAE), and benefit coefficient (E) [30–32]. The calculation formula is shown in Table 11.Table 11
Calculation formula of statistical indicators.
MeasureCalculationCoefficient of determination (R2)R2=1−1/N∑i=1NYi−Y¯2/1/N∑i=1NYi−Y¯2Root mean square error (RMSE)RMSE=1/N∑i=1NYi−Y¯2.Mean absolute error (MAE)MAE=1/N∑i=1NabsYi−yi.Root mean absolute error (RMAE)RMAE=100•MAE/y¯Coefficient of efficiency (E)E%=1−∑i=1NYi−Y¯2/∑i=1NYi−Y¯2.yi was the actual target value, y¯ was the average value of yi; Yi was the predicted value, Y¯ was the average of Yi; and N was the number of model data.According to the evaluation index formula describes in Table11, the calculation results are shown in Table 12.Table 12
R2, RMSE, MAE, RMAE, and E (%) results of different models.
ModelR2RMSEMAERMAEE (%)PSO-BP0.9570.4660.35211.67390.867SVM0.7820.6340.48215.97977.391MLR0.8530.5990.46415.39985.296RF0.6110.8270.68922.85871.945BP0.8880.4720.37212.32587.094The coefficient of determinationR2 is widely used in regression problems to estimate the correlation between the target value and the predicted value; RMSE and MAE are two standards used to measure the average error between the target and the predicted output; RMAE represents the average percentage size of the total absolute deviation error between the target value and the predicted output; the efficiency coefficient E (%) represents the prediction accuracy of the model. The value of R2 is close to 1, the values of RMSE, MAE, and RMAE are close to 0, and the value of E is more than 90%, which mean that the accuracy of the model is higher [33]. Table 12 has shown that the PSO-BP prediction model is of higher R2 and E (%) values than other models, as well as lower RMSE, MAE, and RMAE values. That is, the PSO-BP model can accurately predict the compressive strength of recycled brick powder foam concrete.As can be seen from Figure13, the relative error values of the PSO-BP model are mostly concentrated around the value of 0, and its performance is relatively stable, while the relative error values of the other models fluctuate widely. Combined with Figure 13 and Table 12, it can be seen that the PSO-BP model can make more accurate predictions and its performance is significantly better than the rest of the models.Figure 13
Relative error plots for each model.


## 3.1. Orthogonal Experiment Results

The orthogonal test results of recycled brick powder foam concrete are shown in Table7.Table 7
Orthogonal test results of recycled brick powder foam concrete.
Group numberDry density (kg/m3)28 d compressive strength (MPa)Specific strength (N·m·kg−1)Group numberDry density (kg/m3)28 d compressive strength (MPaSpecific strength (N·m·kg−1)11171.106.075185.3014710.850.841186.8921210.255.804792.6115838.752.372830.703988.005.765833.3516806.802.112621.4141072.754.554241.3017952.502.822955.6451039.453.092975.8118857.602.472882.646997.155.175183.1719647.651.742686.567854.254.234947.3820889.401.251404.438680.853.114570.7621769.502.152794.679937.552.262410.2222975.852.582645.59101071.203.042838.8323851.151.511777.3611988.205.315369.9224675.201.962901.14121008.452.762736.9225786.600.961226.0413721.451.462020.72The orthogonal test results of ordinary foam concrete are shown in Table8.Table 8
Orthogonal test results of ordinary foam concrete.
Group numberDry density (kg/m3)28 d compressive strength (MPa)Specific strength (N·m·kg−1)Group numberDry density (kg/m3)28 d compressive strength (MPa)Specific strength (N·m·kg−1)1743.552.2903079.8214591.61.1821997.972722.382.2293085.6315590.31.0141717.773707.61.9982823.6316644.11.7342692.134714.92.0122814.3817628.71.5482462.225639.41.6682608.7018584.10.9491624.726536.00.8801641.7919599.71.2122021.017735.22.1092868.6120550.30.9191670.008797.12.4083020.9521608.41.2292020.059648.60.8531315.1422500.10.8751749.6510567.40.8301462.8123672.71.4622173.3311713.81.6572321.3824592.81.0731810.0512638.21.4712304.9225567.40.9511676.0713615.91.6312648.16According to the orthogonal test results in Tables7 and 8, the compressive strength is analyzed through the extremely poor method [23]. The larger the range value is, the greater the influence of this factor on the compressive strength would be. The results are shown in Tables 9 and 10.Table 9
Analysis of extreme differences in compressive strength of recycled brick powder foam concrete.
Bubble volumesRecycled brick powder volumeWater to material ratioHPMC contentK15.064.163.002.77K23.563.643.322.88K32.552.863.703.11K42.082.272.703.16K51.832.142.363.17Extreme difference value R3.222.021.350.40Rank1234Table 10
Analysis of extreme differences in compressive strength of ordinary foam concrete.
Bubble volumesWater cement ratioK12.041.56K21.421.65K31.391.69K41.271.27K51.121.08Extreme difference value R0.920.61Rank12The influence order of various factors on the compressive strength of recycled brick powder foam concrete can be obtained from Table9 as follows: bubble volume > recycled brick powder volume > the ratio of water to material > HPMC content. Table 10 shows that the amount of bubble exerts a greater effect on the compressive strength of ordinary foam concrete.The data in Tables9 and 10 is visualized, as shown in Figure 6.Figure 6
Changes in the influence of various factors on compressive strength.
(a)(b)(c)(d)Figure6(a) shows that the compressive strength of recycled brick powder foam concrete and ordinary foam concrete both show a decreasing trend with the increase of the amount of air bubble, which drop by 63.6% and 45.1%, respectively. The increase in the bubble volume poses a significant effect on reducing the compressive strength of recycled brick powder foam concrete. The main reasons are as follows: (1) the increase in the bubble volume is accompanied by the decrease in the amount of cementitious material, which are the main source of the strength of recycled brick powder foam concrete; (2) the increase in the bubble volume will increase the pore volume and produce numerous harmful pores, which will make the pore distribution in the specimen uneven and reduce the strength.Figure6(b) shows that the compressive strength of recycled brick powder foam concrete is decreased by 48.6% with the increase in the amount of recycled brick powder. The main reasons are as follows: (1) the increase in the amount of recycled brick powder is accompanied by the decrease in the amount of cement, which leads to the reduction of C-S-H hydration products and the decrease in strength; (2) the reduction of cement content has led to the deformation of a dense skeleton in part of the recycled brick powder; (3) the main component of recycled brick powder is glassy SiO2, which is an inert material. Although it contains a small amount of Al2O3, most of the substances do not participate in the early hydration reaction of cement-based materials, while the increase in the substitution rate will reduce the strength; (4) the mixture of recycled brick powder poses a dilution effect on the cement and increased the effective water-cement ratio.Figure6(c) shows that the compressive strength of recycled brick powder foam concrete and ordinary foam concrete are increased at first and then decreased with the increase in water-to-material ratio. The reasons are as follows: (1) when the water-to-material ratio is within the range of 0.45 to 0.55, the compressive strength of recycled brick powder foam concrete is increased with the increase of the water-to-material ratio. As recycled brick powder is of a certain pozzolanic activity, the water demand is more than that of virgin materials. As the amount of water is increased, the pozzolanic activity of recycled brick powder helps in generating more C-S-H gel, which further improves the microstructure of recycled brick powder foam concrete, thereby increasing its strength [24]. (2) When the water-to-material ratio is between 0.60 and 0.65, the compressive strength of recycled brick powder foam concrete is decreased as the water-to-material ratio is increased, which is mainly due to the following: (1) the amount of water is excessive. Due to the different densities, the denser components in the recycled brick powder are left underneath, and the bubbles and remaining cementitious materials are on the top, leading to delamination; (2) excessive water thickens the bubble group liquid film, which will accelerate the bleeding under the action of gravity, leading the bubble group to bursting prematurely or forming numerous harmful holes, thereby reducing the strength.Figure6(d) shows that the compressive strength of recycled brick powder foam concrete is increased with the increase of HPMC content, while the increasing rate is gradually slowed. The reasons are as follows: with the increase in the HPMC content, the performance of the bubble group is improved, the shape of the pores, the distance between the holes, and the pore diameter are all improved. Moreover, the bubble group is uniformly distributed in the slurry. Therefore, the strength is improved. However, as the strength of recycled brick powder foam concrete is mainly derived from cementitious materials, the upward trend tends to be flat.


## 3.2. Determination of the Optimal Mix Ratio

The specific strength is the ratio of the strength of a material to its dry density, and it is an important indicator to measure the light weight and high strength of the material [21]. The greater the specific strength is, the better the light-weight and high-strength performance of the material would be [25]. Table 7 shows that the specific strength of the test blocks under the mix ratios in the 3rd group is the largest, as the level of recycled brick powder foam concrete that fulfilled the requirements of light weight and high strength is A1B3C3D3. That is, the optimal mixing ratio is 3% of the bubble group content, 30% of the recycled brick powder content, 0.55 of the water-to-material ratio, and 0.05% of the HPMC content. It can be concluded from Table 8 that the specific strength of the test block is the largest under the mix ratios in the 2nd group, that is, the optimal mix ratio of ordinary foam concrete is 3% of the bubble group content and 0.50 water-cement ratio.For the optimal mix ratio of recycled brick powder foam concrete, the water-to-cement ratio is 0.55, while for the optimal mix ratio of ordinary foam concrete, the water-cement ratio is 0.50. That is, adding recycled brick powder requires more water to promote the complete hydration of the cementitious material. The dry density of the 3rd group of specimens in Table7 is 988 kg/m3, while the 28 d compressive strength is 5.76 MPa; The dry density of the 2nd group of specimens in Table 8 is 722.38 kg/m2, and the 28 d compressive strength is 2.229 MPa. The compressive strength of recycled brick powder foam concrete with 28 d compressive Strength is 2.58 times that of ordinary foam concrete, while the compressive strength of dry density recycled brick powder foam concrete is 1.37 times that of ordinary foam concrete. Although the addition of recycled brick powder will increase the dry density of foam concrete, the increase in compressive strength is even greater.


## 3.3. Microscopic Morphology Analysis

The optimal mix ratio test piece can be observed by the SU8100 scanning electron microscope, and the test results are shown in Figure7.Figure 7
Photograph of 28 d hydration of the specimen when scanned by electron microscope at 500x.The pore diameter of the samples was measured by the NanoMeasure software. Figure3 shows that the average pore diameter of the ordinary Figure 8 foam concrete is 109.73 μm, and the average pore diameter of the recycled brick powder foam concrete is about 72.17 μm, a reduction of 34.2%. The pore wall thickness in Figure 7(b) is obviously greater than that in Figure 7(a). Obvious micro-filling effects can be found. Therefore, the added recycled brick powder refines the pores, increases the compactness of the concrete matrix, and improves the compressive strength of the recycled brick powder foam concrete. When the multiplying factor is 50x, the pore diameter is large with obvious connecting holes in Figure 8(a). In Figure 8(b), the pore diameter is smaller, the pore structure is more uniform, the wall between the pores is dense, the communicating pores are less, and the closed pores are more. As a foam stabilizer, HPMC reduces the occurrence of harmful pores such as connecting holes in the test piece, and makes the pore distribution more uniform.Figure 8
Photograph of 28 d hydration of the specimen when scanned by electron microscope at 50x.


## 3.4. Correlation Analysis of Various Factors and Recycled Brick Powder Foam Concrete

To investigate the correlation between the compressive strength of recycled brick powder foam concrete and the bubble volumes, the amount of recycled brick powder, the water-to-material ratio, and the HPMC content, the Pearson correlation coefficient was used for analysis. The Pearson correlation coefficient [26] (also known as covariance correlation or Bravais–Pearson correlation) describes the degree of correlation between two fixed-ratio, linearly correlated variables (measurement value series) that are not less than affected by their units (also known as tightness). The Pearson correlation coefficient is calculated based on the covariance and product of mean difference of two variables, and its calculation formula is as follows:(2)r=covx,ySx•Sy=∑inxi−x¯•yi−y¯n•Sx•Sy.In the formula:covx,y is the covariance of variables x and y; Sx and Sy are the variances of variables x and y; n is the number of variables; x¯ and y¯ are the mean values of variables x and y, respectively. The value of the correlation coefficient is generally between −1 and 1. The greater the absolute value of the correlation coefficient is, the stronger the correlation of 2 variables would be. If the absolute value of the correlation coefficient is close to 1, it indicates that a perfect linear correlation exists between the two variables; if it is close to 0, no linear correlation exists between the two variables [27].In the Figure9, X1, X2, X3, and X4 represent the volume of bubble volumes, the volume of recycled brick powder, water-to-material ratio, and HPMC content, respectively. Y1 represents the compressive strength of recycled brick powder foam concrete. The order of correlation is as follows: bubble volume > recycled brick powder volume > the ratio of water to material > HPMC content, which is consistent with the results from the orthogonal test analysis. The amount of bubble volume poses a significant effect on the compressive strength of recycled brick powder foam concrete.Figure 9
Heat map of the correlation between each factor and the compressive strength of the cube.


## 3.5. PSO-BP (Back Propagation Neural Network Improved by Particle Swarm Optimization) Prediction

For the recycled brick powder foam concrete, due to the increase in its components, the hydration reaction mechanism is not completely clear. The influencing factors are complex and interactive, which manifests as a specific high-dimensional nonlinear law [28]. The BP neural network is an effective method to solve high-dimensional nonlinear problems. Among them, the PSO-BP [29] neural network defines the initial weights and thresholds of the BP neural network as the particles in the particle swarm optimization algorithm. All particles form a group, and the optimal solution is found through cooperation and information-sharing between individuals in the group. The defects of multiple local extreme points in the BP neural network and difficulty in establishing mathematical models can be avoided. The PSO-BP model consists of the following steps [29].Step 1.
Enter the mix ratio and compressive strength data of recycled brick powder foam concrete and randomly assign 70% of the data as the training set and 30% of the data as the test set.Step 2.
Initialize the parameters of the BP neural network.Step 3.
The initial weights and thresholds are extracted from the BP neural network, encoded as a set of particles in the PSO (group sizeN). Initialize the number of iterations t, maximum number of iterations tmax, inertia weight W, and acceleration factor c1 and c2.Step 4.
Assume the position of each particleXi and the velocity of each particle Vi in space.Step 5.
Evaluate the fitness value of all particles and obtain the best personal solutionpbest and the best group solution gbest.Step 6.
Usepbest and gbest to correct the velocity of each particle Vi(t + 1).Step 7.
Correct the position of each particleXi(t + 1).Step 8.
Repeat Step4 to Step 6 until t ≥ tmax.Step 9.
The optimized positiongbest can be obtained.Step 10.
The optimized BP neural network model is trained with the training set, and the accuracy of the model is verified with the test set.
The establishment process of the PSO-BP neural network in this paper is shown in Figure10.
Taking cement quality, bubble group quality, recycled brick powder quality, water and HPMC quality as input independent variables, and the compressive strength of recycled brick powder foam concrete as output dependent variables, a 5-1-11-1 PSO-BP neural network was established. The prediction of the cubic compressive strength of recycled brick powder foam concrete and the strength of the specimen are compared as shown in Figure11.
Figure11 shows that the predicted compressive strength of the PSO-BP model is close to the actual value, that is, the prediction result of the PSO-BP model is more accurate, with small errors and high fitting accuracy.Figure 10
Establishment process of PSO-BP neural network.Figure 11
PSO-BP model fitting effect diagram.


## 3.6. Model Performance Comparison

The prediction performance of the PSO-BP model was compared with various models, including the support vector machine (SVM), multiple linear regression (MLR), random forest (RF), and artificial neural network (BP) models. The fitting effect is shown in Figure12.Figure 12
Scattered distribution diagram of predicted and actual values of each model.
(a)(b)(c)(d)(e)Figure12 shows that most of the scattered data of the prediction model of PSO-BP are concentrated in and around the 100% regression line. While SVM, MLR, RF and BP scattered point data are very discrete, the data in Figure 11 is fitted according to the form of y=a0x+a1.. Compared with the four models in the control group, the a0 value of the PSO-BP model is closer to 1 and a1 values are closer to 0. The prediction accuracy of the PSO-BP model is higher and the error is smaller.


## 3.7. Statistical Indicators of the Model

In this paper, five statistics were selected to evaluate the prediction accuracy of the PSO-BP model and the control model, as the coefficient of determination (R2), the mean square error (RMSE), the mean error (MAE), the calculation formula of the average absolute deviation percentage error (RMAE), and benefit coefficient (E) [30–32]. The calculation formula is shown in Table 11.Table 11
Calculation formula of statistical indicators.
MeasureCalculationCoefficient of determination (R2)R2=1−1/N∑i=1NYi−Y¯2/1/N∑i=1NYi−Y¯2Root mean square error (RMSE)RMSE=1/N∑i=1NYi−Y¯2.Mean absolute error (MAE)MAE=1/N∑i=1NabsYi−yi.Root mean absolute error (RMAE)RMAE=100•MAE/y¯Coefficient of efficiency (E)E%=1−∑i=1NYi−Y¯2/∑i=1NYi−Y¯2.yi was the actual target value, y¯ was the average value of yi; Yi was the predicted value, Y¯ was the average of Yi; and N was the number of model data.According to the evaluation index formula describes in Table11, the calculation results are shown in Table 12.Table 12
R2, RMSE, MAE, RMAE, and E (%) results of different models.
ModelR2RMSEMAERMAEE (%)PSO-BP0.9570.4660.35211.67390.867SVM0.7820.6340.48215.97977.391MLR0.8530.5990.46415.39985.296RF0.6110.8270.68922.85871.945BP0.8880.4720.37212.32587.094The coefficient of determinationR2 is widely used in regression problems to estimate the correlation between the target value and the predicted value; RMSE and MAE are two standards used to measure the average error between the target and the predicted output; RMAE represents the average percentage size of the total absolute deviation error between the target value and the predicted output; the efficiency coefficient E (%) represents the prediction accuracy of the model. The value of R2 is close to 1, the values of RMSE, MAE, and RMAE are close to 0, and the value of E is more than 90%, which mean that the accuracy of the model is higher [33]. Table 12 has shown that the PSO-BP prediction model is of higher R2 and E (%) values than other models, as well as lower RMSE, MAE, and RMAE values. That is, the PSO-BP model can accurately predict the compressive strength of recycled brick powder foam concrete.As can be seen from Figure13, the relative error values of the PSO-BP model are mostly concentrated around the value of 0, and its performance is relatively stable, while the relative error values of the other models fluctuate widely. Combined with Figure 13 and Table 12, it can be seen that the PSO-BP model can make more accurate predictions and its performance is significantly better than the rest of the models.Figure 13
Relative error plots for each model.


## 4. Conclusion

In this paper, experimental research on the mechanical properties of recycled brick powder foam concrete was carried out, and a neural network model was established to accurately predict the compressive strength based on the compressive strength data of recycled brick powder foam concrete, and the main conclusions reached are as follows.(1)
The range analysis was conducted on the orthogonal test data of recycled brick powder foam concrete, and the results have shown that the influence of various factors on the compressive strength of recycled brick powder foam concrete in descending order is the bubble volumes, the amount of recycled brick powder, water-to-material ratio, and HPMC content.(2)
Comparing the specific strength of each group of recycled brick powder foamed concrete finds that the optimal mixing ratio of recycled brick powder foam concrete fulfilling the requirements of light weight and high strength is 3% bubble group content, 30% recycled brick powder content, 0.55 water-to-material ratio, and 0.05% HPMC content.(3)
With the increase in the bubble volumes, the compressive strength of recycled brick powder foam concrete is decreased, while the reduction effect is significant; with the increase in the amount of construction waste, the compressive strength of recycled brick powder foam concrete exhibits a slow decline; with the increase in the water-to-material ratio, the overall compressive strength of recycled brick powder foam concrete is increased and decreased; with the increase in HPMC content, the compressive strength of recycled brick powder foam concrete is gradually increased, but the increased amount is not as huge.(4)
Two improvements of the compressive strength of recycled brick powder foam concrete is analyzed from a microscopic point of view. One is that the added recycled brick powder refines the pores and increases the compactness of the concrete matrix. Moreover, the foam stabilizer HPMC reduces the damage of the bubble group during the preparation of the test piece and reduces the generation of harmful holes, such as connecting holes.(5)
The PSO-BP model predicts the compressive strength of recycled brick powder foam concrete with high prediction accuracy and data reliability. It provides an effective method for exploring the changing law of the strength of recycled brick powder foam concrete under multiple factors.While the work in this paper is beneficial to the progress of subsequent foam concrete research, the following issues remain to be further explored due to the time and conditions of the tests and some human uncontrollable factors.(1)
Although the pore structure of recycled brick powder foam concrete has been observed by electron microscopy scanning in this paper, only a few specimens have been studied and future experiments related to the pore structure need to be designed in order to derive the specific influence of pore structure characteristics on the performance of recycled brick powder foam concrete.(2)
Due to time and equipment constraints, a large database was not constructed for neural network model training. With the continuous research on recycled brick powder foam concrete, a large amount of experimental data should be used to train the neural network model in the future, thus supporting the application of the constructed neural network model to engineering practice.



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*Source: 2908616-2022-03-30.xml*