# Machine Learning and Optimality in Multi Storey Reinforced Concrete Frames

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Initial Considerations

_{concrete}, C

_{steel}reinforcement, and C

_{formwork}, respectively, are the costs of concrete, steel reinforcement, and formwork in RC elements [2].

#### 1.2. Programming Logic Followed for the Construction of the FEM Algorithm

Beam nodes second floor: [3:(numberofstoreys + 1):(lastnode − (numberofstoreys − 2))]

Beam nodes third floor: [4:(numberofstoreys + 1):(lastnode)]

## 2. Optimization Procedure and Variables

- Variable related to the form of the frames whose change influences the number of bays (eight possible choices leading to a total number of beam-column elements between five and 19).
- Variables related to the lengths of the beams. Each front beam length is considered to have a value between 3 and 7.5 m, with a step size of 0.5 m.
- Variables related to the cross sections of the beams of each storey that compose the structural frames. For all the frame scenarios, the following beam cross sections were considered: b = 350 mm h = 550 mm ρ = 1%, b = 350 mm h = 550 mm ρ = 2%, b = 350 mm h = 550 mm ρ = 3%, b = 350 mm h = 550 mm ρ = 4%, b = 350 mm h = 550 mm ρ = 5%, b = 350 mm h = 550 mm ρ = 6%, b = 350 mm h = 600 mm ρ = 1%, b = 350 mm h = 600 mm ρ = 2%, b = 350 mm h = 600 mm ρ = 3%, b = 350 mm h = 600 mm ρ = 4%, b = 350 mm h = 600 mm ρ = 5%, and b = 350 mm h = 600 mm ρ = 6% (where: b is the smaller dimension of the cross section, h is the larger dimension of the cross section, and ρ is the steel reinforcement ratio of the cross section).
- Variables related to the cross sections of the columns (each storey is examined separately) that compose the structural frames. For all the frame scenarios, the following column cross sections were considered: b = 350 mm h = 350 mm ρ = 1%, b = 350 mm h = 350 mm ρ = 2%, b = 350 mm h = 350 mm ρ = 3%, b = 350 mm h = 400 mm ρ = 1%, b = 350 mm h = 400 mm ρ = 2%, b = 350 mm h = 400 mm ρ = 3%, b = 400 mm h = 400 mm ρ = 1%, b = 400 mm h = 400 mm ρ = 2%, b = 400 mm h = 400 mm ρ = 3%, b = 400 mm h = 450 mm ρ = 1%, b = 400 mm h = 450 mm ρ = 2%, b = 400 mm h = 450 mm ρ = 3%, b = 450 mm h = 450 mm ρ = 1%, b = 450 mm h = 450 mm ρ = 2%, b = 450 mm h = 450 mm ρ = 3%, b = 450 mm h = 500 mm ρ = 1%, b = 500 mm h = 500 mm ρ = 1%, and b = 500 mm h = 550 mm ρ = 1%.

## 3. Reinforced Concrete Design Constraints

#### 3.1. Modeling the RC Interaction Diagrams as a Separate Constraint

_{rd,max}), point 2: (N

_{2}, M

_{2}), and point 3: (M

_{bal}, N

_{bal}) (Where: N

_{rd}is the ultimate axial resistance of the RC cross section, N

_{bal}is the axial resistance value at the point of balanced failure, M

_{bal}is the moment resistance value at the point of balanced failure, and M

_{rd}is the moment of resistance of the RC cross section) [15,16,17,18,19]. The points are connected to each other in a consecutive order and this results in the creation of three lines (Figure 2), which are modelled as constraints representing bounds that must not be exceeded by any combination of N

_{sd}and M

_{sd}(Where: M

_{sd}is the design moment of the cross section and N

_{sd}is the design axial load of the cross section).

_{sc}is the inner force generated by the reinforcement at the compression zone, f

_{s}is the steel reinforcement stress, f

_{y}is the characteristic yield strength of the reinforcement, and f

_{yd}is the design yield strength of the reinforcement (f

_{yd}= f

_{y}/1.15).

_{s}, and f

_{sc}are known, the axial and moment resistance of the cross section can easily be computed by the following generalized formulae [9,11,16,17,18,19]:

_{1}, k

_{2}are the characteristic ratios of the stress block, F

_{c}is the inner force generated by the concrete section, F

_{sc}is the inner force generated by the reinforcement at the compression zone, F

_{st}is the inner force generated by the reinforcement at the tension zone, A

_{s}

_{1}is the cross sectional area of the steel reinforcement at the compression zone, and A

_{s}

_{2}is the cross sectional area of the steel reinforcement at the tension zone [16,17,18,19]: Moreover:

_{bal}and M

_{bal}[11,16,17,18,19]:

_{sd}and M

_{sd}:

#### 3.2. Other Constraints Considered for the Reinforced Concrete Elements

_{max}= 8%.

## 4. Objective Function

_{i}represents all the constraints that concern the structural checks of the beam and column elements according to Eurocode 2 [16]. These checks are inside the two aforementioned generic data structures. The value of the factors p

_{i}is conditional. Whenever a constraint is satisfied, the value of the factor p

_{i}that relates to a particular constraint is equal to zero, whereas, whenever a constraint is violated, the factor p

_{i}that relates to a particular constraint has a very high value, exceeding the highest possible cost of the frame. Through this methodology, they function as conditional penalties and this leads to the evolutionary exclusion of undesired solutions.

## 5. Discussion

#### 5.1. Optimization Scenarios

- Clear column height: 3 m.
- RC forming cost: €75 per m
^{2}. - Concrete cost (concrete grade C 25/30): €60 per m
^{3}. - RC reinforcement cost per kg (rebar steel grade S500): €4708.2.
- RC cover: 35 mm.

#### 5.2. Optimization Results and Conclusions

#### 5.3. Machine Learning Applied on the Optima

- Optimal column area prediction network: network train ratio = 50%, network validation ratio = 25%, network test ratio = 25%, number of neurons = 900, number of hidden layers = 2, and transfer function = tan-sigmoid.
- Optimal number of bays prediction network: network train ratio = 50%, network validation ratio = 25%, network test ratio = 25%, number of neurons = 600, number of hidden layers = 2, and transfer function = log-sigmoid.

## 6. Further Discussion on the Results

## Author Contributions

## Conflicts of Interest

## Appendix A

Scenario | Number of Storeys | Load (kN/m) | Frame Length | Column 1 1st Storey | Beam 1 1st Storey | Column 2 1st Storey | Beam 2 1st Storey | Column 3 1st Storey | Beam 3 1st Storey | Column 4 1st Storey | Beam 4 1st Storey | Column 5 1st Storey | Beam 5 1st Storey | Column 6 1st Storey | Beam 1 6st Storey | Column 7 1st Storey | Number of Bays | Beam Length 1 | Beam Length 2 | Beam Length 3 | Beam Length 4 | Beam Length 5 | Beam Length 6 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 15 | 15 | 0.129 | 1.000 | 0.129 | 1.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 2 | 8.077 | 6.923 | 0.000 | 0.000 | 0.000 | 0.000 |

2 | 2 | 35 | 15 | 0.129 | 1.000 | 0.129 | 1.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 2 | 7.500 | 7.500 | 0.000 | 0.000 | 0.000 | 0.000 |

3 | 2 | 55 | 15 | 0.129 | 1.000 | 0.129 | 1.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 2 | 6.923 | 8.077 | 0.000 | 0.000 | 0.000 | 0.000 |

4 | 2 | 75 | 15 | 0.129 | 1.000 | 0.129 | 1.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 2 | 7.826 | 7.174 | 0.000 | 0.000 | 0.000 | 0.000 |

5 | 2 | 15 | 25 | 0.129 | 1.000 | 0.129 | 1.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 2 | 12.500 | 12.500 | 0.000 | 0.000 | 0.000 | 0.000 |

6 | 2 | 35 | 25 | 0.129 | 1.000 | 0.129 | 1.000 | 0.129 | 1.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 3 | 8.721 | 7.558 | 8.721 | 0.000 | 0.000 | 0.000 |

7 | 2 | 55 | 25 | 0.129 | 1.000 | 0.169 | 1.000 | 0.169 | 1.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 3 | 8.523 | 7.950 | 8.520 | 0.000 | 0.000 | 0.000 |

8 | 2 | 75 | 25 | 0.129 | 1.000 | 0.226 | 1.000 | 0.190 | 1.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 3 | 8.784 | 7.433 | 8.784 | 0.000 | 0.000 | 0.000 |

9 | 2 | 15 | 35 | 0.129 | 1.000 | 0.129 | 1.000 | 0.129 | 1.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 3 | 11.667 | 10.000 | 13.333 | 0.000 | 0.000 | 0.000 |

10 | 2 | 35 | 35 | 0.129 | 2.000 | 0.129 | 2.000 | 0.129 | 2.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 3 | 12.000 | 11.000 | 12.000 | 0.000 | 0.000 | 0.000 |

11 | 2 | 55 | 35 | 0.129 | 1.000 | 0.169 | 1.000 | 0.190 | 1.000 | 0.148 | 1.000 | 0.130 | 0.000 | 0.000 | 0.000 | 0.000 | 4 | 8.750 | 9.375 | 9.375 | 7.500 | 0.000 | 0.000 |

12 | 2 | 75 | 35 | 0.129 | 2.000 | 0.237 | 2.000 | 0.237 | 2.000 | 0.226 | 2.000 | 0.148 | 0.000 | 0.000 | 0.000 | 0.000 | 4 | 7.955 | 9.545 | 9.545 | 7.955 | 0.000 | 0.000 |

13 | 3 | 15 | 15 | 0.129 | 1.000 | 0.129 | 1.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 2 | 7.500 | 7.500 | 0.000 | 0.000 | 0.000 | 0.000 |

14 | 3 | 35 | 15 | 0.129 | 1.000 | 0.129 | 1.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 2 | 8.077 | 6.923 | 0.000 | 0.000 | 0.000 | 0.000 |

15 | 3 | 55 | 15 | 0.129 | 1.000 | 0.226 | 1.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 2 | 7.800 | 7.200 | 0.000 | 0.000 | 0.000 | 0.000 |

16 | 3 | 75 | 15 | 0.169 | 1.000 | 0.290 | 1.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 2 | 8.125 | 6.875 | 0.000 | 0.000 | 0.000 | 0.000 |

17 | 3 | 15 | 25 | 0.129 | 2.000 | 0.129 | 2.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 2 | 12.964 | 12.038 | 0.000 | 0.000 | 0.000 | 0.000 |

18 | 3 | 35 | 25 | 0.129 | 1.000 | 0.148 | 1.000 | 0.148 | 1.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 3 | 8.553 | 7.895 | 8.553 | 0.000 | 0.000 | 0.000 |

19 | 3 | 55 | 25 | 0.136 | 1.000 | 0.237 | 1.000 | 0.226 | 1.000 | 0.148 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 3 | 8.523 | 7.950 | 8.523 | 0.000 | 0.000 | 0.000 |

20 | 3 | 75 | 25 | 0.148 | 2.000 | 0.290 | 2.000 | 0.263 | 2.000 | 0.237 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 3 | 8.333 | 7.639 | 9.028 | 0.000 | 0.000 | 0.000 |

21 | 3 | 15 | 35 | 0.129 | 1.000 | 0.129 | 1.000 | 0.129 | 1.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 3 | 11.667 | 11.667 | 11.667 | 0.000 | 0.000 | 0.000 |

22 | 3 | 35 | 35 | 0.129 | 1.000 | 0.190 | 1.000 | 0.190 | 1.000 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 3 | 12.209 | 10.581 | 12.209 | 0.000 | 0.000 | 0.000 |

23 | 3 | 55 | 35 | 0.169 | 1.000 | 0.190 | 1.000 | 0.226 | 1.000 | 0.226 | 1.000 | 0.190 | 1.000 | 0.130 | 0.000 | 0.000 | 5 | 7.609 | 6.594 | 7.609 | 6.594 | 6.594 | 0.000 |

24 | 3 | 75 | 35 | 0.148 | 2.000 | 0.226 | 2.000 | 0.226 | 2.000 | 0.226 | 2.000 | 0.237 | 2.000 | 0.237 | 2.000 | 0.226 | 6 | 5.904 | 4.639 | 5.904 | 6.326 | 6.326 | 5.904 |

Scenario | Column 1 2nd Storey | Column 2 2nd Storey | Column 3 2nd Storey | Column 4 2nd Storey | Column 5 2nd Storey | Column 6 2nd Storey | Column 7 2nd Storey | Column 1 3rd Storey | Column 2 3rd Storey | Column 3 3rd Storey | Column 4 3rd Storey | Column 5 3rd Storey | Column 6 3rd Storey | Column 7 3rd Storey | Cost (€) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 4564.126 |

2 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 4671.232 |

3 | 0.129 | 0.148 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 4797.345 |

4 | 0.129 | 0.190 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 5039.027 |

5 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 6665.498 |

6 | 0.129 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 7192.480 |

7 | 0.129 | 0.129 | 0.129 | 0.136 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 7435.705 |

8 | 0.129 | 0.148 | 0.129 | 0.169 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 7869.759 |

9 | 0.129 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 9021.519 |

10 | 0.129 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 9589.716 |

11 | 0.129 | 0.129 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 10,244.155 |

12 | 0.129 | 0.148 | 0.169 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 11,055.325 |

13 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 6847.017 |

14 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 7004.584 |

15 | 0.129 | 0.148 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 7284.292 |

16 | 0.129 | 0.190 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 7758.18 |

17 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.000 | 9993.10 |

18 | 0.129 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.129 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 10,806.77 |

19 | 0.129 | 0.190 | 0.148 | 0.129 | 0.000 | 0.000 | 0.000 | 0.129 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 11,389.19 |

20 | 0.148 | 0.237 | 0.226 | 0.169 | 0.000 | 0.000 | 0.000 | 0.129 | 0.148 | 0.148 | 0.129 | 0.000 | 0.000 | 0.000 | 12,307.80 |

21 | 0.129 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 0.129 | 0.129 | 0.129 | 0.129 | 0.000 | 0.000 | 0.000 | 13,544.88 |

22 | 0.148 | 0.169 | 0.129 | 0.148 | 0.000 | 0.000 | 0.000 | 0.148 | 0.129 | 0.129 | 0.148 | 0.000 | 0.000 | 0.000 | 14,623.87 |

23 | 0.129 | 0.129 | 0.190 | 0.129 | 0.129 | 0.148 | 0.000 | 0.129 | 0.148 | 0.130 | 0.130 | 0.129 | 0.148 | 0.000 | 16,036.79 |

24 | 0.129 | 0.129 | 0.136 | 0.169 | 0.187 | 0.187 | 0.148 | 0.129 | 0.129 | 0.136 | 0.129 | 0.148 | 0.129 | 0.129 | 17,457.05 |

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**Figure 1.**Generalized depiction of the node indices’ numbering logic of a frame with multiple bays and three storeys.

Number of Storeys | Loading | Loading | Loading | Loading | Length of Frame |
---|---|---|---|---|---|

2 | 15 kN/m | 35 kN/m | 55 kN/m | 75 kN/m | 15 m |

3 | 15 kN/m | 35 kN/m | 55 kN/m | 75 kN/m |

Number of Storeys | Loading | Loading | Loading | Loading | Length of Frame |
---|---|---|---|---|---|

2 | 15 kN/m | 35 kN/m | 55 kN/m | 75 kN/m | 25 m |

3 | 15 kN/m | 35 kN/m | 55 kN/m | 75 kN/m |

Number of Storeys | Loading | Loading | Loading | Loading | Length of Frame |
---|---|---|---|---|---|

2 | 15 kN/m | 35 kN/m | 55 kN/m | 75 kN/m | 35 m |

3 | 15 kN/m | 35 kN/m | 55 kN/m | 75 kN/m |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bekas, G.K.; Stavroulakis, G.E.
Machine Learning and Optimality in Multi Storey Reinforced Concrete Frames. *Infrastructures* **2017**, *2*, 6.
https://doi.org/10.3390/infrastructures2020006

**AMA Style**

Bekas GK, Stavroulakis GE.
Machine Learning and Optimality in Multi Storey Reinforced Concrete Frames. *Infrastructures*. 2017; 2(2):6.
https://doi.org/10.3390/infrastructures2020006

**Chicago/Turabian Style**

Bekas, Georgios K., and Georgios E. Stavroulakis.
2017. "Machine Learning and Optimality in Multi Storey Reinforced Concrete Frames" *Infrastructures* 2, no. 2: 6.
https://doi.org/10.3390/infrastructures2020006