# Classification of the Structural Behavior of Tall Buildings with a Diagrid Structure: A Machine Learning-Based Approach

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Workflow

- Form generation through a parametric design tool;
- Structural analysis allowing for building weight, dead load and seismic load;
- Classification learning, to allow discriminating the building factors having an impact on the structural response under the considered loading conditions.

#### Architectural and Structural Modelling

^{2}, while a 2.50 kN/m

^{2}live load is considered; all floors receive 100% of the dead load and 20% of the live load. Since our objective is to ascertain the feasibility of the ML approach to link architectural and structural choices, we make here the conventional assumption to allow for a weak seismic event, which implies that a linear elastic response of the structure is taken into account. The response is therefore purposely assumed to be not significantly affected by modes of vibrations higher than the fundamental one in each principal direction. The total amount of the equivalent lateral force is computed according to Eurocode 8 and then applied at the center of mass of each story proportionally to the floor height and mass [34]. Seismic and other loads are combined with their deterministic value, i.e., in the load combination a factor of 1 is simply assumed. More precisely, the seismic force is computed by following this procedure: (i) the input acceleration is determined according to the design spectrum; (ii) the base shear force is computed; (iii) the said shear force is distributed among all stories. The design spectrum ${S}_{d}$ is set by the standard and it can be computed on the basis of: the design ground acceleration ${a}_{g}$; the soil factor S; the behavior factor q; the fundamental vibration period ${T}_{1}$ along the direction of interest; the periods ${T}_{B}$, ${T}_{C}$ and ${T}_{D}$, related to the start of the acceleration-constant branch, the start of the velocity-constant branch and the start of the displacement-constant branch in the design spectrum, respectively. As indicated in [34], the expression to be used in case of a low seismic event characterized by $T>{T}_{D}$, is:

## 3. Machine Learning

## 4. Results

#### 4.1. Feasibility of the ML Classification Approach

#### 4.2. Comparison among Different Classifiers

#### 4.2.1. K-Nearest Neighbours

#### 4.2.2. Support Vector Machine

#### 4.2.3. Decision Tree

#### 4.2.4. Ensemble Method

#### 4.2.5. Naive Bayes

#### 4.2.6. Discriminant Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CDF | Cumulative Distribution Function |

DA | Discriminant Analysis |

DT | Decision Tree |

EM | Ensemble Method |

kNN | k-Nearest Neighbor |

MCE | Minimum Classification Error |

ML | Machine Learning |

NB | Naive Bayes |

PCA | Principal Component Analysis |

SVM | Support Vector Machine |

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**Figure 1.**Generation of the 144 architectural models for the tall buildings considered in this work. (

**a**): perspective. (

**b**): plan view to get details of top and bottom floor plans, and direction of the applied seismic load.

**Figure 2.**Tall building geometry with floor slabs and diagrid structure. In the inset: detail of the inner columns and of the diagrid geometry on the outer surface.

**Figure 3.**Colour-coded diagram depicting the structural response, in terms of drift, for the considered 144 tall building models.

**Figure 4.**Expected design weight vs drift plot, with insets reporting the geometries of the forms featuring the best, average and worst performance.

**Figure 5.**5-class representation of the structural performance based on the drift and diagrid degree features.

**Figure 7.**Projections on different feature planes of the 5-class representation of the structural performance.

**Figure 8.**Exemplary results of the Bayesian optimizer for the SVM classifier, reporting the error for each single iteration.

**Figure 9.**kNN classifier: effect of randomization on the training and testing accuracies. The highlighted trials are respectively characterized by to the maximum (run 16), minimum (run 21), and average accuracy (run 47).

**Figure 10.**kNN classifier (run 21): confusion matrices relevant to the (

**a**) training and (

**b**) testing datasets.

**Figure 11.**Comparison between the performances of the kNN and DT algorithms, in terms of the CDFs of the accuracy during (

**a**) training and (

**b**) testing with the randomized datasets.

**Figure 12.**Patterns in the accuracies for 1000 randomized training and testing datasets, as obtained via the DT algorithm backed by a Bayesian optimizer.

**Table 1.**Properties of the considered site in Tehran, Iran. Data adapted from the design spectrum in [34].

S | 1.15 |

${T}_{B}$ (s) | 0.2 |

${T}_{C}$ (s) | 0.6 |

${T}_{D}$ (s) | 2.0 |

${a}_{g}$ (g) | 0.35 |

q | 2 |

$\lambda $ | 1 |

$\beta $ | 0.2 |

**Table 2.**Statistical properties computed over the 1000 randomized runs via the optimized DT algorithm.

Accuracy Parameter | Training Set | Testing Set |
---|---|---|

Average | 0.9894 | 0.9517 |

Standard deviation | 0.0126 | 0.0435 |

Median | 0.9907 | 0.9722 |

Variance | 1.592 × 10${}^{-4}$ | 0.0019 |

**Table 3.**Features of the kNN models, and relevant accuracies achieved. Excl. feat. = excluded features (⋆ = top/bottom geometry, diagrid total length, maximum normal force).

Model | k | Distance Metric | Distance Weight | Optimizer | Iterations | Excl. Feat. | PCA | Training Accuracy (%) | Testing Acc. (%) | Training Time (s) |
---|---|---|---|---|---|---|---|---|---|---|

K1 | 1 | city block | inverse | grid search | grid div = 10 | - | no | 91.7 | 94.5 | 121.1 |

K2 | 2 | city block | squared inverse | Bayesian | 30 | - | no | 91.7 | 94.4 | 38.1 |

K3 | 2 | city block | inverse | random search | 30 | - | no | 91.7 | 94.4 | 15.9 |

K4 | 3 | city block | squared inverse | Bayesian | 50 | ⋆ | no | 96.3 | 97.2 | 46.3 |

K5 | 9 | city block | squared inverse | Bayesian | 50 | - | no | 91.7 | 97.2 | 64.5 |

K6 | 15 | Euclidean | squared inverse | Bayesian | 50 | ⋆ | yes | 88.0 | 97.2 | 54.3 |

K7 | 54 | Mahalanobis | squared inverse | Bayesian | 50 | - | yes | 88.0 | 94.4 | 55.9 |

**Table 4.**Features of the SVM models, and relevant accuracies achieved. Excl. feat. = excluded features (⋆ = top/bottom geometry, diagrid total length, maximum normal force).

Model | Kernel Function | Kernel Scale | Multi-Class Method | Optimizer | Iterations | Excl. Feat. | PCA | Training Accuracy (%) | Testing Acc. (%) | Training Time (s) |
---|---|---|---|---|---|---|---|---|---|---|

S1 | Gaussian | 15.96 | one-vs.-one | Bayesian | 40 | - | no | 95.4 | 94.4 | 134.6 |

S2 | linear | 10 | one-vs.-one | grid search | grid div = 10 | - | no | 95.4 | 94.4 | 2030 |

S3 | linear | 2.14 | one-vs.-one | random search | 40 | - | no | 95.4 | 94.4 | 223.5 |

S4 | quadratic | 133.56 | one-vs.-one | Bayesian | 40 | ⋆ | no | 95.4 | 91.7 | 139.8 |

S5 | Gaussian | 0.12 | one-vs.-all | Bayesian | 40 | - | yes | 86.1 | 91.7 | 137.8 |

S6 | linear | 99.79 | one-vs.-one | Bayesian | 40 | ⋆ | yes | 91.7 | 91.7 | 152.9 |

**Table 5.**Features of the DT models, and relevant accuracies achieved. Excl. feat. = excluded features.

Model | Max Split | Split Criterion | Surrogate Decision Split | Optimizer | Iterations | Excl. Feat. | PCA | Training Accuracy (%) | Testing Acc. (%) | Training Time (s) |
---|---|---|---|---|---|---|---|---|---|---|

T1 | 5 | Twoing | - | Bayesian | 40 | - | no | 93.5 | 100.0 | 29.7 |

T2 | 4 | Gini | max 10 surr. | random search | 40 | - | no | 93.5 | 100.0 | 17.4 |

T3 | 7 | Gini | all | random search | 40 | - | no | 93.5 | 100.0 | 18.8 |

T4 | 42 | Gini | all | random search | 40 | - | yes | 86.1 | 77.8 | 30.8 |

T5 | 5 | Gini | off | random search | 40 | top/bottom geom. | no | 93.5 | 100.0 | 45.0 |

**Table 6.**Features of the EM models, and relevant accuracies achieved. Excl. feat. = excluded features (⋆ = top/bottom geometry, diagrid total length, maximum normal force); NoL = number of learners; LR = learning rate.

Model | Ensemble Method | Max Split | NoL/LR | Optimizer | Iterations | Excl. Feat. | PCA | Training Accuracy (%) | Testing Acc. (%) | Training Time (s) |
---|---|---|---|---|---|---|---|---|---|---|

E1 | RUSBoost | 38 | 71/0.94 | Bayesian | 40 | - | no | 98.1 | 100.0 | 123.3 |

E2 | RUSBoost | 23 | 324/0.46 | grid search | grid div = 10 | - | no | 97.2 | 97.2 | 128.3 |

E3 | AdaBoost | 3 | 20/0.10 | random search | 40 | - | no | 98.1 | 100.0 | 72.0 |

E4 | RUSBoost | 14 | 12/0.98 | Bayesian | 40 | top/bottom geom. | no | 96.3 | 97.2 | 80.8 |

E5 | RUSBoost | 25 | 65/0.85 | Bayesian | 40 | ⋆ | no | 98.1 | 100.0 | 105.9 |

E6 | RUSBoost | 106 | 124/0.07 | Bayesian | 40 | ⋆ | yes | 85.2 | 94.4 | 129.4 |

E7 | Bag | 66 | 61/1 | Bayesian | 40 | - | yes | 88.0 | 94.4 | 99.2 |

**Table 7.**Features of the NB models, and relevant accuracies achieved. Excl. feat. = excluded features (⋆ = top/bottom geometry, diagrid total length, maximum normal force).

Model | Distribution | Kernel Type | Support | Optimizer | Iterations | Excl. Feat. | PCA | Training Accuracy (%) | Testing Acc. (%) | Training Time (s) |
---|---|---|---|---|---|---|---|---|---|---|

N1 | kernel | Gaussian | unbounded | Bayesian | 30 | - | no | 90.7 | 91.7 | 85.5 |

N2 | kernel | Box | positive | Bayesian | 30 | - | no | 92.6 | 88.9 | 109.7 |

N3 | kernel | Box | positive | grid search | grid div = 10 | - | no | 92.6 | 88.9 | 13.7 |

N4 | Gaussian | Triangle | positive | Bayesian | 30 | - | yes | 82.4 | 83.3 | 36.6 |

N5 | kernel | Gaussian | unbounded | Bayesian | 30 | ⋆ | no | 91.7 | 88.9 | 66.3 |

**Table 8.**Features of the DA models, and relevant accuracies achieved. Excl. feat. = excluded features (⋆ = top/bottom geometry, diagrid total length, maximum normal force).

Model | Discriminant | Optimizer | Iterations | Excl. Feat. | PCA | Training Accuracy (%) | Testing Acc. (%) | Training Time (s) |
---|---|---|---|---|---|---|---|---|

D1 | linear | Bayesian | 40 | - | no | 93.5 | 86.1 | 47.8 |

D2 | diag. quadratic | Bayesian | 40 | - | yes | 85.2 | 83.3 | 46.8 |

D3 | linear | Bayesian | 40 | top/bottom geom. | no | 94.4 | 91.7 | 45.3 |

D4 | linear | Bayesian | 40 | ⋆ | no | 95.4 | 91.7 | 47.9 |

D5 | diag. quadratic | Bayesian | 40 | ⋆ | yes | 85.2 | 83.3 | 48.7 |

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**MDPI and ACS Style**

Kazemi, P.; Ghisi, A.; Mariani, S.
Classification of the Structural Behavior of Tall Buildings with a Diagrid Structure: A Machine Learning-Based Approach. *Algorithms* **2022**, *15*, 349.
https://doi.org/10.3390/a15100349

**AMA Style**

Kazemi P, Ghisi A, Mariani S.
Classification of the Structural Behavior of Tall Buildings with a Diagrid Structure: A Machine Learning-Based Approach. *Algorithms*. 2022; 15(10):349.
https://doi.org/10.3390/a15100349

**Chicago/Turabian Style**

Kazemi, Pooyan, Aldo Ghisi, and Stefano Mariani.
2022. "Classification of the Structural Behavior of Tall Buildings with a Diagrid Structure: A Machine Learning-Based Approach" *Algorithms* 15, no. 10: 349.
https://doi.org/10.3390/a15100349