Machine Learning and Optimality in Multi Storey Reinforced Concrete Frames
Abstract
:1. Introduction
1.1. Initial Considerations
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
3.2. Other Constraints Considered for the Reinforced Concrete Elements
4. Objective Function
5. Discussion
5.1. Optimization Scenarios
- Clear column height: 3 m.
- RC forming cost: €75 per m2.
- Concrete cost (concrete grade C 25/30): €60 per m3.
- 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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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 |
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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
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 StyleBekas, 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