# Research on Structural Optimization of Prefabricated Components Based on Improved Immune Genetic Algorithm

Authors: Tingting Ma
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

On the premise of satisfying the design conditions and practical engineering needs, it is of practical significance to optimize the design of assembly components. Therefore, in this paper, based on the improved immune genetic algorithm, the structural optimization of prefabricated components is deeply studied, where it is converted into binary gene code, and the structural design requirements are introduced into the algorithm through structural layout vaccine and concrete strength vaccine. In addition, the section size of components is designed according to economic indicators, which has proved that the research is effective for the structural optimization method of prefabricated components.


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## Body

## 1. Introduction

Since the reform and opening up, China’s urbanization process has obviously accelerated, and a large number of rural people have flooded into cities with high economic development level. The limited urban land area and the expanding urban population have become one of the main social contradictions in today's developed cities in China, and more and more civil buildings adopt high-rise structures to ease the tension between population and living space [1–3]. At the same time, the traditional construction industry’s production is increasingly exposed to problems such as high energy consumption, heavy pollution, and long construction period. In addition, people’s awareness of environmental protection and the improvement of living quality have gradually made the production methods of site casting eliminated. Therefore, new architectural design and construction modes are needed to meet the social needs, and building industrialization and engineering structure optimization have become the main technical means to solve problems [4–6].Industrialization is a building production mode that integrates design, production, transportation, and construction and explores the deep integration of industrialization and informatization to achieve sustainable development [7, 8]. Its production mode is mainly characterized by standardized design, factory production, assembly construction, integrated decoration, and information management which is obviously improved in energy consumption, construction efficiency, environmental impact, and labor demand.In the process of building industrialization, prefabricated components are widely used. With the continuous expansion of engineering quantity and the increasing number of complex structural systems, how to meet the design conditions and the actual needs of the project in engineering practice and how to optimize its structure design are a problem worth studying. Therefore, in this paper, based on the goal of optimizing prefabricated components, an improved immune genetic algorithm is adopted, where the objective function is constructed, and the reliability, sensitivity, and dynamic balance of parameters of prefabricated structures are studied. Moreover, by designing the size of prefabricated components, the performance of components can be guaranteed, and the cost-economy index of components can be controlled, so that the optimal design of prefabricated components can be realized, and the optimal design basis can be provided for further application of prefabricated components in engineering practice. By constructing the objective function, the multiobjective is changed into a single objective, and a brand-new model to keep the dynamic balance between the objective function values is put forward, so as to control the performance of prefabricated components.


## 2. Objective of Structural Optimization in Prefabricated Components

Optimization of prefabricated concrete components mainly covers the following aspects: structural safety, service durability, component construction cost, production capacity of prefabricated component factory, site construction factors, component transportation, and site hoisting capacity [9–12]. From the perspective of function analysis, in order to ensure the stability of the performance of fabricated components, it is required that the sensitivity of components to design parameters should be as small as possible [13].


### 2.1. Safety of Prefabricated Components

As prefabricated components are mainly completed in prefabricated components factory, the water-cement ratio of prefabricated components, concrete grade, type of reinforced bar, and curing time will affect the quality of prefabricated components, and the quality of prefabricated components will affect the safety of the whole structure. The prefabrication quality shall meet the production requirements of relevant prefabricated components, so as to improve the safety of prefabricated components.


### 2.2. Economy of Prefabricated Components

The cost of prefabricated components is mainly composed of raw materials, manufacturing process, mold cost, and labor cost, among which the material cost of prefabricated components will greatly affect the selling price of prefabricated components. Therefore, considering the cost of building materials, the original design can be optimized to reduce unnecessary material loss, thus realizing the economy of building components.


### 2.3. Commonality of Prefabricated Components

On the premise of considering the safety of building components, the universality of prefabricated components should be met to the greatest extent; that is, the components are similar in size, which meets the requirements of common building modules, and the combination of components can form a variety of combination schemes, so as to realize the use of the least kinds of prefabricated components. After the types of prefabricated components are reduced, the manufacturing time and economic cost of prefabricated component templates can be reduced.


### 2.4. Convenient Transportation of Components

The dimension and length of prefabricated components shall be no more than 5 m, and the height shall be no more than 3 m, so as to meet the requirements of transportation and storage sites of prefabricated components; in addition, the dead weight of prefabricated components should not be greater than 6 tons, preferably within 4 tons, so as to meet the requirements of hoisting and temporary fixing of prefabricated components.


## 2.1. Safety of Prefabricated Components

As prefabricated components are mainly completed in prefabricated components factory, the water-cement ratio of prefabricated components, concrete grade, type of reinforced bar, and curing time will affect the quality of prefabricated components, and the quality of prefabricated components will affect the safety of the whole structure. The prefabrication quality shall meet the production requirements of relevant prefabricated components, so as to improve the safety of prefabricated components.


## 2.2. Economy of Prefabricated Components

The cost of prefabricated components is mainly composed of raw materials, manufacturing process, mold cost, and labor cost, among which the material cost of prefabricated components will greatly affect the selling price of prefabricated components. Therefore, considering the cost of building materials, the original design can be optimized to reduce unnecessary material loss, thus realizing the economy of building components.


## 2.3. Commonality of Prefabricated Components

On the premise of considering the safety of building components, the universality of prefabricated components should be met to the greatest extent; that is, the components are similar in size, which meets the requirements of common building modules, and the combination of components can form a variety of combination schemes, so as to realize the use of the least kinds of prefabricated components. After the types of prefabricated components are reduced, the manufacturing time and economic cost of prefabricated component templates can be reduced.


## 2.4. Convenient Transportation of Components

The dimension and length of prefabricated components shall be no more than 5 m, and the height shall be no more than 3 m, so as to meet the requirements of transportation and storage sites of prefabricated components; in addition, the dead weight of prefabricated components should not be greater than 6 tons, preferably within 4 tons, so as to meet the requirements of hoisting and temporary fixing of prefabricated components.


## 3. Optimization of Prefabricated Components Based on Improved Immune Genetic Algorithm

### 3.1. The Optimization Goal

This chapter introduces in detail the process of optimization of prefabricated concrete structure by using improved immune genetic algorithm. With the idea of combining global optimization with local optimization, the optimization of prefabricated components is regarded as the overall optimization of the structure, which is mainly used to control the overall index of the structure, while the optimization of section size of component can be regarded as local optimization of structure, which is mainly used to control the bearing capacity of the components [14–17].The optimization of structure is divided into two levels: the first level is the optimization of concrete grade and structure layout, and the second level is the optimization of section size of components. The first-level optimization is realized by transforming specific problems into binary gene codes and carrying out population evolution, through structural layout vaccine, concrete strength vaccine, etc. In the optimization process, internal force analysis and section size adjustment need to be repeated until the total cost of the structure converges. There are two reasons for optimizing the section size of members as a second-order variable:(1)
The influence of the section size of most components on the overall index of the structure is not as obvious as that of the structural layout.(2)
If all the cross-sectional dimensions are converted into binary codes as variables at the same level as the structural layout, the number of generations required for convergence will increase exponentially, and the efficiency of the algorithm will be low.Therefore, in this study, section optimization is carried out simultaneously in the process of structural optimization, and the interaction between the overall layout of the structure and the bearing capacity of members is considered, so that the optimization results are more reasonable.


### 3.2. Optimization Process of the Structural Model

#### 3.2.1. Construction of Fitness Function

Fitness function is the main criterion for genetic algorithm to detect the effect of individual optimization, and it is also the main index for individual selection. Therefore, the structural reliability function is constructed as the objective function, and the optimization effect is evaluated. The reliability of prefabricated structure is expressed as(1)R=∫⋯∫Ωfxdx,where fx is the joint probability density function of the basic random parameter vector x=x1,x1,…,xnT, whose parameters are loads or random quantities representing component characteristics, and Ω=x|gx>0gx is a functional function or a state function, and the two states of the structure are(2)gx⩽0,invalid,gx>0,safe.Considering that the reliability is insensitive to the change of design parameters in the design requirements of prefabricated structures, the random parameter vectorx and functional function gx are expressed as(3)x=xd+εrr,gx=gdx+εgrx.Among them,ε is a parameter with small value, and the part with subscript d represents the deterministic part of random parameter. The part with subscript r represents the random part of the random parameter, and its mean is 0.Reliability indexβ is a standardized random variable, and its probability distribution function can be approximately expanded into a standard normal distribution function according to Edgeworth series, namely,(4)Fy=Φy−φy1θg3!σg3H2y+14!ηgσg4−3H3y+⋯,where φy is the probability density function of the standard normal distribution function Φy and Hjy is Hermite polynomial. According to (4), the probability density function is generally asymmetric when the function whose probability distribution is unknown is expanded into the expression of the standard normal distribution. Therefore, the systematic reliability can be expressed as(5)Rβ=Pgx>0=1−F−β.Obviously, (5) is an approximate expression of (1). However, when (5) is used to calculate the reliability, when R > 1 occurs, the following empirical formula is adopted for correction:(6)R∗=Rβ−Rβ−Φβ1+Rβ−Φβββ.Edgeworth series can accurately approximate the true distribution of any random variable, and there is no need to restrict the distribution type and excitation type of random parameters in the derivation process, so its result is very close to the engineering practice. Usually, the first four terms of Edgeworth series have high accuracy, which can fully meet the needs of general engineering practice. Therefore, in this paper, the first four terms of (4) are used for reliability calculation and sensitivity analysis.The objective function of prefabricated components proposed in this paper is for the mean pointx¯, the basic random parameter vector x, so it is near the mean point x¯, and the sensitivity of reliability to the mean and variance of x is(7)∂R∂x¯T=∂Rβ∂β∂β∂μg∂μg∂x¯T,(8)∂R∂Varx=∂R∂β∂β∂σg+∂R∂σg∂σg∂Varx.The sensitivities∂R/∂x¯T and ∂R/∂Varx of reliability can be obtained by substituting the known conditions and the calculation results of reliability analysis into (7) and (8).When reliabilityR > 1, the sensitivity of reliability to β is(9)∂R∗∂β=∂R∂β+A×ββ−1C−11+Cββ+1+C1+Cβln1+Cβ+C21+Cββ+1.Among them,A=∂R/∂β−φβ;C=Rβ−Φβ.In order to ensure that the fabricated structure has good durability and does not suffer from performance degradation due to the change of external environment, it is required that the sensitivity of the structure to design parameters should be as small as possible. In order to meet the requirements of economic indicators such as structural lightweight, the following objective function is established:(1)
According to the previous idea of modeling, the minimum joint sensitivity of reliability is taken to the mean point of design variables as the first objective function:(10)f1x=m∑i=1n∑∂R∂μxi2,∈s.t.R−R0⩾0,qx=0,hx⩾0.Among them,x=μx1,μx2,…,μxnT;R0 is the reliability that should be satisfied; R is the reliability calculated by the above Edgeworth series or empirical modified formula; qx is the equality constraint matrix; hx is the inequality constraint matrix.(2)
For the economic index such as the smallest volume, establish the second objective function:(11)f2x=mV,s.t.qx=0,hx⩾0.Among them,V is the volume of the structure.Compared with the volume, the reliability sensitivity function value (generally 0.1) is small, so the following model is proposed:(12)f1′x=10Mf1x,M=lgf2x−lgf1x.Among them,lg is a common logarithm and [ ] is a Gaussian rounding symbol.No matter how the design variablex changes, the new function value f1′x is always in the same order of magnitude as the function value f2x, and the function value can be adjusted adaptively to achieve dynamic balance between the objectives. The implementation uses f2x instead of the original f1x for subsequent operations, which can not only achieve the dynamic balance between two objectives but also adapt to the case of more than three objective functions.According to the image set method, it can be transformed into the following single-objective problem:(13)mfx=∑k=1nwkfkx,s.t.R−R0⩾0,qx=0,hx⩾0,where wk⩾0 is the weighting factor of subobjective function fkx and depends on the order of magnitude and importance of each subobjective function value. wk can be determined by image set method in weighting group method, that is,(14)wk=fn+1−kxk∗−fn+1−kxn+1−k∗∑k=1nfkxn+1−k∗−fkxk∗;k=1,…,n,where n is the number of subobjective functions and xk∗ is the optimal solution vector of the k-th subtarget.


### 3.3. Improved Immune Genetic Algorithm

After the objective function of the optimized prefabricated component is constructed, the structural influence parameters are coded for population evolution. Prerequisites such as design requirements are introduced into the algorithm by vaccination, and the search mode of the algorithm is optimized to improve the overall adaptability. Finally, the section size of the fabricated structural members is optimized according to the economic indicators, and the structural optimization of prefabricated components based on improved immune genetic algorithm is completed [18–20].


#### 3.3.1. Problem Coding

Whether the wall is arranged in the layout of prefabricated structure is a BOOL discrete variable. The binary gene coding features are just in line with the characteristics of assembly structure layout optimization variables. Therefore, this paper adopts the traditional binary coding, which contains the following information:(1) The Location of the Wall. In this paper, a 1-digit binary code is used to represent the wall layout of a grid, as shown in Figure 1. The dotted line position in the figure indicates the axis network where shear walls can be arranged, so 12-digit code is needed to indicate the wall position of this floor. Full line means to decorate the frame beam, and green line means to decorate the shear wall.Figure 1
Schematic diagram of structural layer.(2) The Length of the Wall. In this paper, the 3-digit binary code is used to represent the length of the wall in the grid. 00 represents the minimum wall length Lmin, which is 8 times the wall thickness; 11 indicates the maximum acceptable wall length Lmax of the structure. Other codes determine the wall length by linear interpolation and multiple of 50. For example, when Lmin equals 1600 mm and Lmax and LMAX equal 3500 mm, the wall length represented by code 010 is L = LMIN + 2 (LMAX-LMIN)/7 = 2143 mm.(3) Strength Grade of Concrete. In this paper, 3-digit codes are used to represent the concrete strength grade of the first floor, and the codes 000 to 111 range from C30 to C65.


#### 3.3.2. Immune Operator

(1) Structural Layout Vaccine. The structural layout vaccine is used to eliminate the discontinuity of the wall from top to bottom in the structure; that is, it is considered that if the wall layout as shown in Figures 2(a) and 2(b) appears in the structure, it is invalid to continue to optimize the individual. At this time, the corresponding codes “1010” and “0011” in individuals are antigens, and there are many choices of corresponding vaccines. As shown in Figures 2(c) and 2(d), there are two optional vaccines, and their codes are “1100” and “1110.”Figure 2
Wall layout of antibodies and vaccines.
(a)(b)(c)(d)There are five codes {0000, 1000, 1100, 1110, and 1111} for vaccines that can meet the continuous structural layout from top to bottom. This code set can be regarded as a temporary vaccine bank. Structural vaccine detects 100% of the vertical layout codes of walls in all individuals. Once the structural layout is found to be discontinuous, a temporary vaccine bank will be generated, and a vaccine will be randomly selected for vaccination. Vaccination of structural vaccine is always carried out in the whole evolution process, ensuring that all individuals of each generation population are always effective.(2) Concrete Strength Grade Vaccine. The concrete strength grade vaccine is used to eliminate the situation that the concrete grade in the upper layer of the structure is higher than that in the lower layer, and it is also used to avoid invalid operation. In this paper, the concrete strength grade vaccine directly adopts the lower concrete strength grade code. For example, the concrete strength grade code of an individual appears as “001010.” For a two-story structure, the meaning of this code is that the strength grade of the bottom concrete is C35, and that of the second-floor concrete is C40. However, such concrete grade distribution does not conform to the structural design habits. Therefore, the code “010” representing the strength of two-story concrete can be regarded as antigen, the code “001” representing the strength of bottom-story concrete can be regarded as vaccine, and the strength code of inoculated concrete can be changed to “001001”; that is, the second-floor adopts C35 concrete grade.The layout vaccine and concrete strength grade vaccine can be checked and vaccinated after the code is formed, which is called “Class I vaccine.” The overall vaccine is called “Class II vaccine” and can only be checked and vaccinated after the individual optimization is completed. Because “Class I vaccine” is used to avoid invalid operation, while “Class II vaccine” only replaces the individual who does not meet the overall index of the structure after changing the concrete strength grade, the fitness of the population will generally not be obviously degraded after vaccination, and the immune genetic algorithm in this paper is not necessary for “immune selection.”The convergence condition is that the difference of total structural cost for three consecutive times is less than 1%. Grid search method is an organic part of the improved immune genetic algorithm in this paper. Only after the grid search method is completed can the fitness of individuals be calculated. Figure3 is the overall flowchart of the improved immune genetic algorithm in this paper.Figure 3
Illustration of improved immune genetic algorithm.The change of cross section size will inevitably lead to the redistribution of internal force of the structure, so it needs to be completed through repeated cyclic iterations. The specific process is shown in Figure4:(1)
Set the initial section of the structure.(2)
Make analysis of internal force of the structure.(3)
Optimize the section size of the component.(4)
Judge whether the convergence condition is met; if it satisfied, the section optimization is finished; otherwise, return to step (2).Figure 4
Schematic diagram of immune genetic algorithm flow.


## 3.1. The Optimization Goal

This chapter introduces in detail the process of optimization of prefabricated concrete structure by using improved immune genetic algorithm. With the idea of combining global optimization with local optimization, the optimization of prefabricated components is regarded as the overall optimization of the structure, which is mainly used to control the overall index of the structure, while the optimization of section size of component can be regarded as local optimization of structure, which is mainly used to control the bearing capacity of the components [14–17].The optimization of structure is divided into two levels: the first level is the optimization of concrete grade and structure layout, and the second level is the optimization of section size of components. The first-level optimization is realized by transforming specific problems into binary gene codes and carrying out population evolution, through structural layout vaccine, concrete strength vaccine, etc. In the optimization process, internal force analysis and section size adjustment need to be repeated until the total cost of the structure converges. There are two reasons for optimizing the section size of members as a second-order variable:(1)
The influence of the section size of most components on the overall index of the structure is not as obvious as that of the structural layout.(2)
If all the cross-sectional dimensions are converted into binary codes as variables at the same level as the structural layout, the number of generations required for convergence will increase exponentially, and the efficiency of the algorithm will be low.Therefore, in this study, section optimization is carried out simultaneously in the process of structural optimization, and the interaction between the overall layout of the structure and the bearing capacity of members is considered, so that the optimization results are more reasonable.


## 3.2. Optimization Process of the Structural Model

### 3.2.1. Construction of Fitness Function

Fitness function is the main criterion for genetic algorithm to detect the effect of individual optimization, and it is also the main index for individual selection. Therefore, the structural reliability function is constructed as the objective function, and the optimization effect is evaluated. The reliability of prefabricated structure is expressed as(1)R=∫⋯∫Ωfxdx,where fx is the joint probability density function of the basic random parameter vector x=x1,x1,…,xnT, whose parameters are loads or random quantities representing component characteristics, and Ω=x|gx>0gx is a functional function or a state function, and the two states of the structure are(2)gx⩽0,invalid,gx>0,safe.Considering that the reliability is insensitive to the change of design parameters in the design requirements of prefabricated structures, the random parameter vectorx and functional function gx are expressed as(3)x=xd+εrr,gx=gdx+εgrx.Among them,ε is a parameter with small value, and the part with subscript d represents the deterministic part of random parameter. The part with subscript r represents the random part of the random parameter, and its mean is 0.Reliability indexβ is a standardized random variable, and its probability distribution function can be approximately expanded into a standard normal distribution function according to Edgeworth series, namely,(4)Fy=Φy−φy1θg3!σg3H2y+14!ηgσg4−3H3y+⋯,where φy is the probability density function of the standard normal distribution function Φy and Hjy is Hermite polynomial. According to (4), the probability density function is generally asymmetric when the function whose probability distribution is unknown is expanded into the expression of the standard normal distribution. Therefore, the systematic reliability can be expressed as(5)Rβ=Pgx>0=1−F−β.Obviously, (5) is an approximate expression of (1). However, when (5) is used to calculate the reliability, when R > 1 occurs, the following empirical formula is adopted for correction:(6)R∗=Rβ−Rβ−Φβ1+Rβ−Φβββ.Edgeworth series can accurately approximate the true distribution of any random variable, and there is no need to restrict the distribution type and excitation type of random parameters in the derivation process, so its result is very close to the engineering practice. Usually, the first four terms of Edgeworth series have high accuracy, which can fully meet the needs of general engineering practice. Therefore, in this paper, the first four terms of (4) are used for reliability calculation and sensitivity analysis.The objective function of prefabricated components proposed in this paper is for the mean pointx¯, the basic random parameter vector x, so it is near the mean point x¯, and the sensitivity of reliability to the mean and variance of x is(7)∂R∂x¯T=∂Rβ∂β∂β∂μg∂μg∂x¯T,(8)∂R∂Varx=∂R∂β∂β∂σg+∂R∂σg∂σg∂Varx.The sensitivities∂R/∂x¯T and ∂R/∂Varx of reliability can be obtained by substituting the known conditions and the calculation results of reliability analysis into (7) and (8).When reliabilityR > 1, the sensitivity of reliability to β is(9)∂R∗∂β=∂R∂β+A×ββ−1C−11+Cββ+1+C1+Cβln1+Cβ+C21+Cββ+1.Among them,A=∂R/∂β−φβ;C=Rβ−Φβ.In order to ensure that the fabricated structure has good durability and does not suffer from performance degradation due to the change of external environment, it is required that the sensitivity of the structure to design parameters should be as small as possible. In order to meet the requirements of economic indicators such as structural lightweight, the following objective function is established:(1)
According to the previous idea of modeling, the minimum joint sensitivity of reliability is taken to the mean point of design variables as the first objective function:(10)f1x=m∑i=1n∑∂R∂μxi2,∈s.t.R−R0⩾0,qx=0,hx⩾0.Among them,x=μx1,μx2,…,μxnT;R0 is the reliability that should be satisfied; R is the reliability calculated by the above Edgeworth series or empirical modified formula; qx is the equality constraint matrix; hx is the inequality constraint matrix.(2)
For the economic index such as the smallest volume, establish the second objective function:(11)f2x=mV,s.t.qx=0,hx⩾0.Among them,V is the volume of the structure.Compared with the volume, the reliability sensitivity function value (generally 0.1) is small, so the following model is proposed:(12)f1′x=10Mf1x,M=lgf2x−lgf1x.Among them,lg is a common logarithm and [ ] is a Gaussian rounding symbol.No matter how the design variablex changes, the new function value f1′x is always in the same order of magnitude as the function value f2x, and the function value can be adjusted adaptively to achieve dynamic balance between the objectives. The implementation uses f2x instead of the original f1x for subsequent operations, which can not only achieve the dynamic balance between two objectives but also adapt to the case of more than three objective functions.According to the image set method, it can be transformed into the following single-objective problem:(13)mfx=∑k=1nwkfkx,s.t.R−R0⩾0,qx=0,hx⩾0,where wk⩾0 is the weighting factor of subobjective function fkx and depends on the order of magnitude and importance of each subobjective function value. wk can be determined by image set method in weighting group method, that is,(14)wk=fn+1−kxk∗−fn+1−kxn+1−k∗∑k=1nfkxn+1−k∗−fkxk∗;k=1,…,n,where n is the number of subobjective functions and xk∗ is the optimal solution vector of the k-th subtarget.


## 3.2.1. Construction of Fitness Function

Fitness function is the main criterion for genetic algorithm to detect the effect of individual optimization, and it is also the main index for individual selection. Therefore, the structural reliability function is constructed as the objective function, and the optimization effect is evaluated. The reliability of prefabricated structure is expressed as(1)R=∫⋯∫Ωfxdx,where fx is the joint probability density function of the basic random parameter vector x=x1,x1,…,xnT, whose parameters are loads or random quantities representing component characteristics, and Ω=x|gx>0gx is a functional function or a state function, and the two states of the structure are(2)gx⩽0,invalid,gx>0,safe.Considering that the reliability is insensitive to the change of design parameters in the design requirements of prefabricated structures, the random parameter vectorx and functional function gx are expressed as(3)x=xd+εrr,gx=gdx+εgrx.Among them,ε is a parameter with small value, and the part with subscript d represents the deterministic part of random parameter. The part with subscript r represents the random part of the random parameter, and its mean is 0.Reliability indexβ is a standardized random variable, and its probability distribution function can be approximately expanded into a standard normal distribution function according to Edgeworth series, namely,(4)Fy=Φy−φy1θg3!σg3H2y+14!ηgσg4−3H3y+⋯,where φy is the probability density function of the standard normal distribution function Φy and Hjy is Hermite polynomial. According to (4), the probability density function is generally asymmetric when the function whose probability distribution is unknown is expanded into the expression of the standard normal distribution. Therefore, the systematic reliability can be expressed as(5)Rβ=Pgx>0=1−F−β.Obviously, (5) is an approximate expression of (1). However, when (5) is used to calculate the reliability, when R > 1 occurs, the following empirical formula is adopted for correction:(6)R∗=Rβ−Rβ−Φβ1+Rβ−Φβββ.Edgeworth series can accurately approximate the true distribution of any random variable, and there is no need to restrict the distribution type and excitation type of random parameters in the derivation process, so its result is very close to the engineering practice. Usually, the first four terms of Edgeworth series have high accuracy, which can fully meet the needs of general engineering practice. Therefore, in this paper, the first four terms of (4) are used for reliability calculation and sensitivity analysis.The objective function of prefabricated components proposed in this paper is for the mean pointx¯, the basic random parameter vector x, so it is near the mean point x¯, and the sensitivity of reliability to the mean and variance of x is(7)∂R∂x¯T=∂Rβ∂β∂β∂μg∂μg∂x¯T,(8)∂R∂Varx=∂R∂β∂β∂σg+∂R∂σg∂σg∂Varx.The sensitivities∂R/∂x¯T and ∂R/∂Varx of reliability can be obtained by substituting the known conditions and the calculation results of reliability analysis into (7) and (8).When reliabilityR > 1, the sensitivity of reliability to β is(9)∂R∗∂β=∂R∂β+A×ββ−1C−11+Cββ+1+C1+Cβln1+Cβ+C21+Cββ+1.Among them,A=∂R/∂β−φβ;C=Rβ−Φβ.In order to ensure that the fabricated structure has good durability and does not suffer from performance degradation due to the change of external environment, it is required that the sensitivity of the structure to design parameters should be as small as possible. In order to meet the requirements of economic indicators such as structural lightweight, the following objective function is established:(1)
According to the previous idea of modeling, the minimum joint sensitivity of reliability is taken to the mean point of design variables as the first objective function:(10)f1x=m∑i=1n∑∂R∂μxi2,∈s.t.R−R0⩾0,qx=0,hx⩾0.Among them,x=μx1,μx2,…,μxnT;R0 is the reliability that should be satisfied; R is the reliability calculated by the above Edgeworth series or empirical modified formula; qx is the equality constraint matrix; hx is the inequality constraint matrix.(2)
For the economic index such as the smallest volume, establish the second objective function:(11)f2x=mV,s.t.qx=0,hx⩾0.Among them,V is the volume of the structure.Compared with the volume, the reliability sensitivity function value (generally 0.1) is small, so the following model is proposed:(12)f1′x=10Mf1x,M=lgf2x−lgf1x.Among them,lg is a common logarithm and [ ] is a Gaussian rounding symbol.No matter how the design variablex changes, the new function value f1′x is always in the same order of magnitude as the function value f2x, and the function value can be adjusted adaptively to achieve dynamic balance between the objectives. The implementation uses f2x instead of the original f1x for subsequent operations, which can not only achieve the dynamic balance between two objectives but also adapt to the case of more than three objective functions.According to the image set method, it can be transformed into the following single-objective problem:(13)mfx=∑k=1nwkfkx,s.t.R−R0⩾0,qx=0,hx⩾0,where wk⩾0 is the weighting factor of subobjective function fkx and depends on the order of magnitude and importance of each subobjective function value. wk can be determined by image set method in weighting group method, that is,(14)wk=fn+1−kxk∗−fn+1−kxn+1−k∗∑k=1nfkxn+1−k∗−fkxk∗;k=1,…,n,where n is the number of subobjective functions and xk∗ is the optimal solution vector of the k-th subtarget.


## 3.3. Improved Immune Genetic Algorithm

After the objective function of the optimized prefabricated component is constructed, the structural influence parameters are coded for population evolution. Prerequisites such as design requirements are introduced into the algorithm by vaccination, and the search mode of the algorithm is optimized to improve the overall adaptability. Finally, the section size of the fabricated structural members is optimized according to the economic indicators, and the structural optimization of prefabricated components based on improved immune genetic algorithm is completed [18–20].


### 3.3.1. Problem Coding

Whether the wall is arranged in the layout of prefabricated structure is a BOOL discrete variable. The binary gene coding features are just in line with the characteristics of assembly structure layout optimization variables. Therefore, this paper adopts the traditional binary coding, which contains the following information:(1) The Location of the Wall. In this paper, a 1-digit binary code is used to represent the wall layout of a grid, as shown in Figure 1. The dotted line position in the figure indicates the axis network where shear walls can be arranged, so 12-digit code is needed to indicate the wall position of this floor. Full line means to decorate the frame beam, and green line means to decorate the shear wall.Figure 1
Schematic diagram of structural layer.(2) The Length of the Wall. In this paper, the 3-digit binary code is used to represent the length of the wall in the grid. 00 represents the minimum wall length Lmin, which is 8 times the wall thickness; 11 indicates the maximum acceptable wall length Lmax of the structure. Other codes determine the wall length by linear interpolation and multiple of 50. For example, when Lmin equals 1600 mm and Lmax and LMAX equal 3500 mm, the wall length represented by code 010 is L = LMIN + 2 (LMAX-LMIN)/7 = 2143 mm.(3) Strength Grade of Concrete. In this paper, 3-digit codes are used to represent the concrete strength grade of the first floor, and the codes 000 to 111 range from C30 to C65.


### 3.3.2. Immune Operator

(1) Structural Layout Vaccine. The structural layout vaccine is used to eliminate the discontinuity of the wall from top to bottom in the structure; that is, it is considered that if the wall layout as shown in Figures 2(a) and 2(b) appears in the structure, it is invalid to continue to optimize the individual. At this time, the corresponding codes “1010” and “0011” in individuals are antigens, and there are many choices of corresponding vaccines. As shown in Figures 2(c) and 2(d), there are two optional vaccines, and their codes are “1100” and “1110.”Figure 2
Wall layout of antibodies and vaccines.
(a)(b)(c)(d)There are five codes {0000, 1000, 1100, 1110, and 1111} for vaccines that can meet the continuous structural layout from top to bottom. This code set can be regarded as a temporary vaccine bank. Structural vaccine detects 100% of the vertical layout codes of walls in all individuals. Once the structural layout is found to be discontinuous, a temporary vaccine bank will be generated, and a vaccine will be randomly selected for vaccination. Vaccination of structural vaccine is always carried out in the whole evolution process, ensuring that all individuals of each generation population are always effective.(2) Concrete Strength Grade Vaccine. The concrete strength grade vaccine is used to eliminate the situation that the concrete grade in the upper layer of the structure is higher than that in the lower layer, and it is also used to avoid invalid operation. In this paper, the concrete strength grade vaccine directly adopts the lower concrete strength grade code. For example, the concrete strength grade code of an individual appears as “001010.” For a two-story structure, the meaning of this code is that the strength grade of the bottom concrete is C35, and that of the second-floor concrete is C40. However, such concrete grade distribution does not conform to the structural design habits. Therefore, the code “010” representing the strength of two-story concrete can be regarded as antigen, the code “001” representing the strength of bottom-story concrete can be regarded as vaccine, and the strength code of inoculated concrete can be changed to “001001”; that is, the second-floor adopts C35 concrete grade.The layout vaccine and concrete strength grade vaccine can be checked and vaccinated after the code is formed, which is called “Class I vaccine.” The overall vaccine is called “Class II vaccine” and can only be checked and vaccinated after the individual optimization is completed. Because “Class I vaccine” is used to avoid invalid operation, while “Class II vaccine” only replaces the individual who does not meet the overall index of the structure after changing the concrete strength grade, the fitness of the population will generally not be obviously degraded after vaccination, and the immune genetic algorithm in this paper is not necessary for “immune selection.”The convergence condition is that the difference of total structural cost for three consecutive times is less than 1%. Grid search method is an organic part of the improved immune genetic algorithm in this paper. Only after the grid search method is completed can the fitness of individuals be calculated. Figure3 is the overall flowchart of the improved immune genetic algorithm in this paper.Figure 3
Illustration of improved immune genetic algorithm.The change of cross section size will inevitably lead to the redistribution of internal force of the structure, so it needs to be completed through repeated cyclic iterations. The specific process is shown in Figure4:(1)
Set the initial section of the structure.(2)
Make analysis of internal force of the structure.(3)
Optimize the section size of the component.(4)
Judge whether the convergence condition is met; if it satisfied, the section optimization is finished; otherwise, return to step (2).Figure 4
Schematic diagram of immune genetic algorithm flow.


## 3.3.1. Problem Coding

Whether the wall is arranged in the layout of prefabricated structure is a BOOL discrete variable. The binary gene coding features are just in line with the characteristics of assembly structure layout optimization variables. Therefore, this paper adopts the traditional binary coding, which contains the following information:(1) The Location of the Wall. In this paper, a 1-digit binary code is used to represent the wall layout of a grid, as shown in Figure 1. The dotted line position in the figure indicates the axis network where shear walls can be arranged, so 12-digit code is needed to indicate the wall position of this floor. Full line means to decorate the frame beam, and green line means to decorate the shear wall.Figure 1
Schematic diagram of structural layer.(2) The Length of the Wall. In this paper, the 3-digit binary code is used to represent the length of the wall in the grid. 00 represents the minimum wall length Lmin, which is 8 times the wall thickness; 11 indicates the maximum acceptable wall length Lmax of the structure. Other codes determine the wall length by linear interpolation and multiple of 50. For example, when Lmin equals 1600 mm and Lmax and LMAX equal 3500 mm, the wall length represented by code 010 is L = LMIN + 2 (LMAX-LMIN)/7 = 2143 mm.(3) Strength Grade of Concrete. In this paper, 3-digit codes are used to represent the concrete strength grade of the first floor, and the codes 000 to 111 range from C30 to C65.


## 3.3.2. Immune Operator

(1) Structural Layout Vaccine. The structural layout vaccine is used to eliminate the discontinuity of the wall from top to bottom in the structure; that is, it is considered that if the wall layout as shown in Figures 2(a) and 2(b) appears in the structure, it is invalid to continue to optimize the individual. At this time, the corresponding codes “1010” and “0011” in individuals are antigens, and there are many choices of corresponding vaccines. As shown in Figures 2(c) and 2(d), there are two optional vaccines, and their codes are “1100” and “1110.”Figure 2
Wall layout of antibodies and vaccines.
(a)(b)(c)(d)There are five codes {0000, 1000, 1100, 1110, and 1111} for vaccines that can meet the continuous structural layout from top to bottom. This code set can be regarded as a temporary vaccine bank. Structural vaccine detects 100% of the vertical layout codes of walls in all individuals. Once the structural layout is found to be discontinuous, a temporary vaccine bank will be generated, and a vaccine will be randomly selected for vaccination. Vaccination of structural vaccine is always carried out in the whole evolution process, ensuring that all individuals of each generation population are always effective.(2) Concrete Strength Grade Vaccine. The concrete strength grade vaccine is used to eliminate the situation that the concrete grade in the upper layer of the structure is higher than that in the lower layer, and it is also used to avoid invalid operation. In this paper, the concrete strength grade vaccine directly adopts the lower concrete strength grade code. For example, the concrete strength grade code of an individual appears as “001010.” For a two-story structure, the meaning of this code is that the strength grade of the bottom concrete is C35, and that of the second-floor concrete is C40. However, such concrete grade distribution does not conform to the structural design habits. Therefore, the code “010” representing the strength of two-story concrete can be regarded as antigen, the code “001” representing the strength of bottom-story concrete can be regarded as vaccine, and the strength code of inoculated concrete can be changed to “001001”; that is, the second-floor adopts C35 concrete grade.The layout vaccine and concrete strength grade vaccine can be checked and vaccinated after the code is formed, which is called “Class I vaccine.” The overall vaccine is called “Class II vaccine” and can only be checked and vaccinated after the individual optimization is completed. Because “Class I vaccine” is used to avoid invalid operation, while “Class II vaccine” only replaces the individual who does not meet the overall index of the structure after changing the concrete strength grade, the fitness of the population will generally not be obviously degraded after vaccination, and the immune genetic algorithm in this paper is not necessary for “immune selection.”The convergence condition is that the difference of total structural cost for three consecutive times is less than 1%. Grid search method is an organic part of the improved immune genetic algorithm in this paper. Only after the grid search method is completed can the fitness of individuals be calculated. Figure3 is the overall flowchart of the improved immune genetic algorithm in this paper.Figure 3
Illustration of improved immune genetic algorithm.The change of cross section size will inevitably lead to the redistribution of internal force of the structure, so it needs to be completed through repeated cyclic iterations. The specific process is shown in Figure4:(1)
Set the initial section of the structure.(2)
Make analysis of internal force of the structure.(3)
Optimize the section size of the component.(4)
Judge whether the convergence condition is met; if it satisfied, the section optimization is finished; otherwise, return to step (2).Figure 4
Schematic diagram of immune genetic algorithm flow.


## 4. Case Analysis

Reinforced concrete is made of two kinds of materials to resist external loads. On the premise of meeting the requirements of the minimum reinforcement ratio and the maximum height of concrete compression zone, the amount of concrete and steel bars in the section is a trade-off.Take a simply supported beam with a width of 250 mm as an example. The beam span is 3000 mm, and the bending moment in the middle of the beam is 240 Kn/m. C30 concrete and HPB400 steel bar are used as materials, and only tensile steel bar is used. By the Code for Design of Concrete Structures (6.2.10–1),(15)M=αlfcbxh0−x2.The obtained height of concrete compression zone is(16)x=h0−h02−2Mα1fcb.According to formula (6.2.10-2) of Code for Design of Concrete Structures, it is obtained that(17)As=α1fcbxfy.Five cases of beam height from 450 mm to 650 mm were investigated according to step size of 50 mm. The cost of this simply supported beam is calculated according to the unit price of concrete 500 yuan/cubic meter and the unit price of steel 5000 yuan/ton. The reinforcement is shown in Table1.Table 1
The reinforcement of simply supported single-reinforced beam.
Beam depth h (mm)450500550600650Effective height of sectionh0 (mm)380430480560610Height of concrete compression zonex (mm)267205170137122The theory of accessories As (mm2)26502036168813561215The actual reinforcement4Ø254Ø208Ø183Ø253Ø183Ø203Ø184Ø20Actual reinforcement area As’ (mm2)27262036170213661256Assuming 2Ø14 for the upper frame and Ø14@200 for the stirrup, taking 2Ø12 for the waist rebar when the section height is less than 500 mm, and 4Ø12 for the waist rebar when the section height is not less than 500 mm, without considering the template and anchorage length, the cost of this beam is shown in Figure 5, which shows that when the section height is 450 mm, the reinforcement rate of the beam reaches 2.47%. Because the unit price of steel bar is much higher than that of concrete, the cost of the beam is much higher than that of other sections.Figure 5
Cost of simply supported beam.In the unit price of the two groups of materials, the lowest cost is the section height of 500 mm, with the beam reinforcement rate 1.63%. Meanwhile, when the beam section exceeds 500 mm, the cost of the beam rises again due to the increase in the number of waist tendons. Therefore, when the internal force of a component is determined, the cost of the component is completely determined by the section size, and there is always an intermediate value, which can minimize the cost of the component.The optimization of section size is completed by grid search method. The optimization variables of beam section include section height and section width, and the optimization variables of column section include section side lengths in different directions. To meet the requirements of modular system, the beam-column section size is a discrete variable with a step size of 50 mm, and the upper and lower limits of the variable are determined in advance. The beam sections that can be selected are shown in Table2 (where Y represents that it is optional, while N represents that it is not optional). According to the optional section combination, a beam section size solution can be formed, the trial calculation is carried out in turn, and the information of the section size corresponding to the lowest fraud is selected as the optimal solution.Table 2
Range of beam sections.
Width/high4004505005506006507007508008509009501000250YYYYYNNNNNNNN300NYYYYYYYNNNNN350NNNYYYYYYYNNN400NNNNYYYYYYYYYOnce the reinforcement ratio of the section reaches the minimum reinforcement ratio, exit the inner loop (i.e., the section height is no longer increased), and return to the outer loop (i.e., increase the beam section width by 50 mm) until all feasible solutions are calculated. So there is no need to continue to enlarge the section at this time. If these solutions are converted into point sets in plane space, a grid-like graph as shown in Figure6 will be formed. Taking the beam as an example, the section size optimization needs a two-layer nesting cycle to complete. First, fix the beam section width, and then increase the beam section height for trial calculation.Figure 6
Grid search route diagram.The optimization of section size of members starts from the top layer and goes down layer by layer. To satisfy the constraints, it is converted into the following form:(18)bx,i,m=bx,i+1,opt,bx,i,m=bx,i+1,opt,ti,m=ti+1,opt,where bx,i,min and by,imin are the optimal initial length of the i-th layer column, bx,i+1,opt,by,i+1,opt are the i + 1-th layer corresponding to the optimal location column length, ti,min is the optimal initial thickness of the wall at the i + 1-th layer, and ti+1,ppt is the optimal thickness of the wall at the corresponding position of the wall at the i + 1-th layer.


## 5. Conclusion

With the development of building industrialization technology, prefabricated buildings will become the main part of China’s future construction industry. Reasonable design of prefabricated components to ensure that they have sufficient mechanical properties and improve the economy is a hot topic of current research. In this paper, based on the improved immune genetic algorithm, the structural optimization of prefabricated components is deeply studied, where the structural optimization of prefabricated components is converted into binary gene code, and the structural design requirements are introduced into the algorithm through structural layout vaccine and concrete strength vaccine. According to the economic index, the prefabricated reinforced concrete simple-supported beam is optimized. The results show that when the internal force of the component is determined, the cost of the structural component is determined by the section size and there exists an intermediate value to reduce the cost of components to the lowest level, which proves the feasibility of optimizing the design of prefabricated structures through section size and provides a preliminary scheme for future designers of prefabricated structures.



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*Source: 4648112-2022-06-09.xml*