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Using a genetic algorithm owing to high nonlinearity of constraints, this paper first works on the optimal design of two-span continuous singly reinforced concrete beams. Given conditions are the span, dead and live loads, compressive strength of concrete and yield strength of steel; design variables are the width and effective depth of the continuous beam and steel ratios for positive and negative moments. The constraints are built based on the ACI Building Code by considering the strength requirements of shear and the maximum positive and negative moments, the development length of flexural reinforcement, and the serviceability requirement of deflection. The objective function is to minimize the total cost of steel and concrete. The optimal data found from the genetic algorithm are divided into three groups: the training set, the checking set and the testing set for the use of the adaptive neuro-fuzzy inference system (ANFIS). The input vector of ANFIS consists of the yield strength of steel, compressive strength of concrete, dead load, span, width and effective depth of the beam; its outputs are the minimum total cost and optimal steel ratios for positive and negative moments. To make ANFIS more efficient, the technique of Subtractive Clustering is applied to group the data to help streamline the fuzzy rules. Numerical results show that the performance of ANFIS is excellent, with correlation coefficients between the three targets and outputs of the testing data being greater than 0.99.

Genetic algorithms are search algorithms based on the mechanics of natural selection and natural genetics. They have been developed and were formally introduced in 1970s by Professor John Holland at the University of Michigan, who in 1975 published the ground-breaking book “Adaptation in Natural and Artificial System” [

The artificial neural network was originated by McCulloch and Pitts in 1943 [

Fuzzy sets were introduced by Zadeh [

Neuro-fuzzy systems are fuzzy systems which use ANNs theory in order to determine their membership functions and fuzzy rules by processing data samples. A specific approach in neuro-fuzzy development is the adaptive neuro-fuzzy inference system (ANFIS) first proposed by Jang [

Distinct from other authors’ works, this paper tries to combine the techniques of the genetic algorithm and ANFIS to optimally design reinforced two-span continuous reinforced concrete beams with rectangular cross- section. Based on the provisions of the ACI Building Code Requirements for Structural Concrete and Commentary [

Genetic algorithms were inspired by the evolution theory of “survival of the fittest”, proposed by Charles Darwin in 1860s. They simulate the survival of the fittest among individuals over consecutive generation and can solve both constrained and unconstrained optimization problems according to the “natural selection”. Genetic algorithms are less susceptible to getting stuck at local optima than traditional gradient search methods. This paper uses the Global Optimization Toolbox based on MATLAB [

Supposed that

Minimize

Such that

where

Most of the constraints built in this paper are nonlinear. The Global Optimization Toolbox based on MATLAB uses the augmented Lagrangian genetic algorithm [

The adaptive neuro-fuzzy inference system (ANFIS) consists of two components: fuzzy inference systems and neural networks. Using a given input/output data set, ANFIS constructs a fuzzy inference system whose membership function parameters are adjusted by a hybrid learning algorithm to approximate the precise value of the model parameters [

where A and B are the linguistic values defined by fuzzy sets in the antecedent, while

where

If the fuzzy inference is zeroth-order, then

Data clustering specifying each data point belonging to a cluster to some degree by a membership grade can identify natural groupings of data from a large data set to produce a concise representation of a system’s behavior. Based on the cluster information, a Sugeno-type fuzzy inference system that best models the data behavior can be generated. The data clustering technique adopted in this paper is “Subtractive Clustering” [_{ }(default 0.5), to specify the range of influence of a cluster center. The more neighboring data points a data point can enclose, the higher potential it has as a cluster center; 2) squash factor _{ }(default 0.5), to set the potential above which another data point will be accepted as a cluster center; 4) reject ratio _{ }(default 0.15), to set the potential below which a data point will be rejected as a cluster center.

The two-span continuous reinforced concrete beams with a rectangular section are subjected to a uniformly distributed load

Suppose that

1) Region I: If

2) Region II: If

needs to be provided, where

3) Region III: If

has to be provided to carry the difference and the spacing s must not be larger than

4) Region IV: If

The above statements can be summarized in

Because the reaction, in the direction of applied shear, introduces compression into the end regions of a member, the critical section can be assumed at a distance of

For simplicity, this paper assumes that the strain in the tension reinforcement is equal to 0.005; therefore, the section is tension-controlled, that is, the strength reduction factor for moment is fixed at 0.9, not a function of strain in the tension reinforcement any more. Accordingly, the constraint for both positive and negative moment takes the form

where

itive moment

When the strain in the tension reinforcement is equal to 0.005, the area of the reinforcement is of the form

where

where

and

According to the ACI Code, at least one-third of the total tension reinforcement provided for negative bending moment at the support should extend beyond the inflections point not less than the effective depth

where

The ACI Code indicates that wherever excessive deflection may adversely affect the service-ability of the structure at service loads, deflections under service load conditions must be computed. Creep and shrinkage will magnify the magnitude of deflection with time. Consequently, design engineers have to evaluate immediate as well as long-term deflection in order to ensure their values satisfy the maximum permissible criteria for the particular structure and its particular use. The additional deflection under sustained loading and long-term shrinkage in accordance with ACI procedure can be calculated by multiplying the immediate deflection by a factor

where

where

The given conditions for the optimal design of two-span continuous singly reinforced rectangular concrete beams with a rectangular cross-section are the span length L, uniformly distributed dead _{ }for the positive moment and the steel ratio _{ }for the negative moment. The concrete cover for the reinforcement is 4 cm and No. 3 vertical stirrups are used. The objective function is to find the minimum cost in New Taiwan Dollars of concrete and steel used in the two-span continuous beam. In Taiwan, the unit price of concrete is 1800 NT$/m^{3 }and the unit price of steel is 19.5 NT$/kgf. The optimal results found by the genetic algorithm consist of the minimum cost of the two-span continuous beam, the width ^{2} (40 ksi), 3500 kgf/cm^{2} (50 ksi) and 4200 kgf/cm^{2} (60 ksi) as well as three kinds of compressive strength ^{2} (3000 psi), 280 kgf/cm^{2} (4000 psi) and 350 kgf/cm^{2} (5000 psi). Three kinds of span length are chosen: 6 m, 8 m and 10 m; four kinds of uniformly distributed dead load

To run the genetic algorithm of the MATLAB software, some parameters need to be selected. Here are the values used in this paper: after a number of trials, the population size is set to be 20, crossover rate 0.8, and elite number 2. Furthermore, all the individuals are encoded as real numbers; “Rank” is used as the scaling function that scales the fitness values based on the rank of each individual; “Roulette” is the selection function to choose parents for the next generation; “Two-Point Crossover” is used as the crossover method to form a new child for the next generation; the “Adaptive Feasible Function” is chosen as the mutation function to make small random changes in the individuals and ensure that linear constraints and bounds are satisfied. The genetic algorithm is executed 30 times for each case, from which the best is selected. For the use of ANFIS, the total 108 cases of data are divided into 3 groups randomly by a computer algorithm: 64 cases of training data (60%), 22 cases of checking data (20%) and 22 cases of testing data (20%).

When using ANFIS with MATLAB, there are some restrictions: 1) only first- or zeroth-order Sugeno-type systems are supported; 2) there is only one single output; 3) each rule is of unit weight. The inputs of the adaptive neuro-fuzzy inference system consist of six elements:

Parameters Influence Range | m | b | r |
---|---|---|---|

0.1 | 0.7774 | 0.0011 | 0.8256 |

0.2 | 0.9005 | 0.0008 | 0.8211 |

0.3 | 0.9337 | 0.0006 | 0.8121 |

0.4 | 0.8988 | 0.0008 | 0.8474 |

0.5 | 0.8318 | 0.0012 | 0.8882 |

0.6 | 0.6519 | 0.0018 | 0.6029 |

0.7 | 1.0316 | −0.0003 | 0.9380 |

0.8 | 0.8869 | 0.0004 | 0.8208 |

0.9 | 0.9529 | 0.0003 | 0.9570 |

1.0 | 0.9106 | 0.0006 | 0.9745 |

1.1 | 1.0241 | −0.0001 | 0.9972 |

1.2 | 0.9946 | 0.0000 | 0.9958 |

1.3 | 0.9953 | 0.0000 | 0.9944 |

1.4 | 1.0087 | −0.0001 | 0.9983 |

1.5 | 1.0397 | −0.0002 | 0.9882 |

Parameters Influence Range | m | b | r |
---|---|---|---|

0.1 | 0.9119 | 0.0014 | 0.8175 |

0.2 | 0.9389 | 0.0003 | 0.6480 |

0.3 | 0.9165 | 0.0011 | 0.8494 |

0.4 | 0.9068 | 0.0011 | 0.7426 |

0.5 | 0.5709 | 0.0054 | 0.5486 |

0.6 | 0.7702 | 0.0026 | 0.8936 |

0.7 | 1.1192 | −0.0031 | 0.7251 |

0.8 | 0.7186 | 0.0028 | 0.7531 |

0.9 | 0.9394 | 0.0007 | 0.9334 |

1.0 | 0.9221 | 0.0009 | 0.9799 |

1.1 | 1.0350 | −0.0005 | 0.9888 |

1.2 | 0.9325 | 0.0008 | 0.9939 |

1.3 | 1.0109 | 0.0000 | 0.9970 |

1.4 | 1.0038 | 0.0000 | 0.9984 |

1.5 | 0.9484 | 0.0005 | 0.9751 |

_{ }and the minimum cost (10^{3} NT$) are shown in Figures 4-6, respectively. The correlation coefficients between the network outputs and targets are 0.9983, 0.9984 and 0.9996 for the steel ratios

Parameters Influence Range | m | b | r |
---|---|---|---|

0.1 | 0.9855 | 0.0505 | 0.9944 |

0.2 | 0.9736 | 0.1458 | 0.9925 |

0.3 | 0.9582 | 0.2669 | 0.9858 |

0.4 | 0.9832 | 0.0934 | 0.9912 |

0.5 | 0.9478 | 0.4552 | 0.9916 |

0.6 | 0.9431 | 0.3981 | 0.9919 |

0.7 | 1.0030 | 0.1634 | 0.9951 |

0.8 | 1.0744 | −0.6299 | 0.9962 |

0.9 | 0.9643 | 0.3517 | 0.9982 |

1.0 | 1.0350 | −0.1380 | 0.9959 |

1.1 | 1.0067 | −0.0733 | 0.9998 |

1.2 | 0.9931 | 0.0621 | 0.9994 |

1.3 | 1.0014 | 0.0182 | 0.9996 |

1.4 | 1.0012 | −0.0241 | 0.9996 |

1.5 | 0.9943 | 0.0509 | 0.9989 |

intercept b approximately equals 0.

Based on Figures 4-6 and Tables 1-3, the performance of ANFIS is satisfactory and considered to be excellent.

Outputs Influence Range | ρ_{1} | ρ_{2} | Cost |
---|---|---|---|

0.1 | 64 | 64 | 64 |

0.2 | 64 | 64 | 64 |

0.3 | 64 | 64 | 64 |

0.4 | 64 | 64 | 64 |

0.5 | 64 | 64 | 61 |

0.6 | 47 | 47 | 46 |

0.7 | 34 | 34 | 33 |

0.8 | 25 | 26 | 23 |

0.9 | 16 | 16 | 16 |

1.0 | 11 | 11 | 12 |

1.1 | 7 | 7 | 8 |

1.2 | 7 | 7 | 7 |

1.3 | 4 | 4 | 5 |

1.4 | 3 | 3 | 3 |

1.5 | 2 | 2 | 3 |

Inputs | Targets | Outputs | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

^{2}) | ^{2}) | w_{d} (ton/m) | L (m) | b (m) | d (m) | ρ_{1} | ρ_{2} | Cost (10^{3} NT$) | ρ_{1} | ρ_{2} | Cost (10^{3} NT$) |

2.8 | 0.28 | 2.7 | 6 | 0.2018 | 0.6309 | 0.0080 | 0.0149 | 6.904 | 0.0079 | 0.0147 | 6.870 |

3.5 | 0.21 | 2.3 | 6 | 0.2000 | 0.6617 | 0.0055 | 0.0102 | 6.051 | 0.0055 | 0.0103 | 6.094 |

4.2 | 0.35 | 2.1 | 8 | 0.2006 | 0.7178 | 0.0065 | 0.0121 | 9.115 | 0.0065 | 0.0118 | 9.279 |

This paper first uses the genetic algorithm to work on the optimal design of two-span continuous reinforced concrete beams with a rectangular section. The adaptive neuro-fuzzy inference system (ANFIS) is then built based on the data of the given conditions and optimal results of the genetic algorithm. The inputs of this model are the yield strength of steel, compressive strength of concrete, dead load (live load is fixed) and span length, width and effective depth of the beam; targets are the minimum cost, the steel ratios for the positive and negative moments. The inputs of ANFIS are different from the given conditions of the genetic algorithm, which makes ANFI more useful and flexible in the design of beams. This paper proves that ANFIS has excellent performance with correlation coefficients between outputs and targets of the steel ratios for positive and negative moments and the minimum cost of the testing data being 0.9983, 0.9984 and 0.9996, respectively. In addition, the influence ranges of a cluster center from 0.1 to 1.5 for “Subtractive Clustering” to estimate the number of clusters and the cluster centers are explored, among which the value of 1.4 can lead to the best results as a whole, as far as the performance of ANFIS is concerned. In the future, once the input data are provided, ANFIS could quickly yield the minimum cost, steel ratios for the positive and negative moments as well as the spacing of vertical stirrups in each region with high precision, which automatically accomplish the design of the continuous reinforced concrete beams. The ANFIS model for the design of beams is easily implemented and timesaving, because it does not need to build the tedious and complex constraints.