# Fast Seismic Assessment of Built Urban Areas with the Accuracy of Mechanical Methods Using a Feedforward Neural Network

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Parametric Generation of Structures

_{x}, n

_{y}, n

_{z}= the number of spans in X, Y and Z, respectively, and s

_{x}, s

_{y}, s

_{z}= the number of steps in X, Y and Z, respectively. According to this formula, T = 9 × 9 × 3 × 24

^{9}× 24

^{9}× 10

^{3}and therefore, T > 10

^{18}.

^{18}possibilities, these are created using random values within the given ranges. In some studies, certain combinations of parameters have been filtered out to avoid computing structures that are not represented in a given database [41]. However, in this work, all possible combinations are computed. While this might be less effective computationally, it allows the model to be more general or inclusive, provided that the results are still satisfactory. Figure 2 shows a visualization of some of the generated structures.

#### 2.2. Push-Over Analysis

- -
- All supports are rigidly clamped to their foundation.
- -
- Slabs are computed as horizontal diaphragms, as in [45].
- -
- -
- -
- As recommended in [44], the nonlinear behavior of RC elements is simulated by defining plastic hinges within them, according to the ASCE-41-13. In a similar way to [44,48], PM2M3 plastic hinges are introduced in the columns, while the M3 type is used in the beams. They are introduced at the ends of the beams and the columns, as in [49] and as recommended in Eurocode 8 (EC-8) [50].
- -
- The capacity curves are not truncated by shear or flexural failure and, therefore, remain in the ductile domain at all times.
- -
- The contribution of the infill walls is not considered, as in [51].
- -
- Gravitational loads (G) are also obtained from the buildings’ data and the CTE. These are combined according to the seismic combinations and coefficients established in the Spanish seismic code (NCSE-02) [46], as shown in Equation (2).$$G=W+D+0.3Q$$
- -
- Push-over loads are applied to all nodes in the XZ/YZ plane proportional to the loaded slab weight and are determined by the following formula:$${F}_{L}={Z}_{L}\xb7{G}_{s}$$
_{L}is the total amount of horizontal force applied to each level, Z_{L}is the height in meters from the ground to the slab where the load is being applied, and G_{s}is the combination of the structural weight, dead loads and live load of the slab, as described above. Note that although EC-8 obliges calculating two load patterns (the other is a load pattern proportional to the first mode of vibration of the building), in [13], it was demonstrated that for prismatic low-rise structures, there are no substantial differences between the two. - -
- The control point for displacements is located at the highest, most central point of the structure.

#### 2.3. Curve Data Processing

#### 2.3.1. Curve Pre-Processing

#### 2.3.2. Curve Post-Processing

#### 2.4. Artificial Neural Network

#### 2.4.1. Loss Function and Error Measurements

_{i}) are each one of the expected output values and (y′

_{i}) are the corresponding predicted values. This value will be calculated for each training sample and back-propagated into the network to serve as criteria for weight optimization. After intense training, the MAE value will be as low as possible (and thus the prediction error will be minimum). Both errors are similar, but, on the one hand, RMSE penalizes large errors more severely, whereas MAE provides a linear penalization; on the other hand, RMSE is less intuitive to interpret and is sensitive to the number of samples used for training. In [53], this is thoroughly analyzed and it is suggested that error metrics based on absolutes rather than squares can perform better in regression problems. For all of these reasons, MAE was chosen in this study over RMSE. Squared mean error is also very commonly used in neural networks, but was discarded due to poor results in preliminary testing.

_{d}), allow the separate evaluation of (i) how well the curves fit one another in terms of shape and (ii) how well the network has predicted the end of the curve.

#### 2.4.2. Network Architecture

#### 2.4.3. Activation Function

#### 2.4.4. Optimizer

#### 2.4.5. Preliminary Adjustment of Stochastic Gradient Descent (SGD) Parameters

#### 2.4.6. Network Architecture Configuration

#### 2.4.7. Network Parameter Fine-Tuning

## 3. Results

_{d}) are shown in Figure 18, Figure 19, Figure 20 and Figure 21, respectively. The average fitted area error obtained for capacity curves is 2.35%, while the full area error is 2.65%. Last displacement error (L

_{d}) is 20.81% on average.

_{d}), the thresholds chosen to provide a visual illustration are L

_{d}< 5%, L

_{d}~21.32% (average of the distribution) and L

_{d}~60%, as shown in Figure 25, Figure 26 and Figure 27. More data on these specific samples can be consulted in Table A2 of the Appendix A.

## 4. Discussion of Results

#### Future Work

## 5. Conclusions

- -
- ANN provide an accurate approximation method for the nonlinear static push-over calculation of low-rise structures within a wide range of sizes and geometric configurations.
- -
- The accuracy of the method successfully addresses the shortcomings of current macro-seismic approaches, while remaining fast and efficient.
- -
- Stress-deformation curves in a plastic regime can be predicted with ANN in one go for entire buildings using only basic geometric parameters. For low-rise structures, this work achieves a curve area error below 2.7% and a resolution of up to 100 points.
- -
- The relative simplicity of the ANN architecture required to predict the capacity curves of low-rise buildings makes a strong case for the future research of high-rise structures using deeper networks and larger datasets.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Sample #147 | Sample #1547 | Sample #1355 | ||
---|---|---|---|---|

Error data | MAE | 0.0097 | 0.0048 | 0.00072 |

Full area error | 5.21% | 2.13% | 0.49% | |

Fitted area error | 5.48% | 2.27% | 0.80% | |

L_{d} error | 13.3% | 9.52% | 7.14% | |

Input parameters | Spans in X (m) | 6.4, 5.4 | 6.43, 5.39 | 6.7, 4.1, 4.8, 6.9, 4.3, 4.5, 6.6 |

Spans in Y (m) | 5.2, 4.3, 5.8, 3.1, 5.3, 3.2, 4.1 | 4, 4.5, 4.8, 3, 3.3, 3.2, 5.3, 4.9, 5.4 | 3.2, 4.8 | |

Spans in Z (m) | 0.7, 3.5, 3.3, 3.4 | 0.7, 3.4, 3.2, 3 | 0.6, 3.9, 3.1 | |

Wide load B. B.? | Yes | No | No | |

Load B. B. dim | 60 × 30 cm | 30 × 60 cm | 30 × 60 cm | |

Non-load B. B. dim | 30 × 30 cm | 30 × 30 cm | 30 × 30 cm | |

Supports dim | 30 × 30 cm | 30 × 30 cm | 30 × 30 cm | |

Slab thickness | 30 cm | 30 cm | 30 cm |

Sample #1926 | Sample #1971 | Sample #639 | ||
---|---|---|---|---|

Error data | MAE | 0.0179 | 0.0065 | 0.0074 |

Full area error | 12.44% | 3.62% | 2.75% | |

Fitted area error | 2.21% | 2.26% | 1.99% | |

L_{d} error | 56.2% | 22.7% | 3.70% | |

Input parameters | Spans in X (m) | 7.7, 4.5, 5.6, 4.9 | 6.1, 5.2 | 6.3, 4.5, 4.2, 5.9 |

Spans in Y (m) | 4.8, 5.6, 5.5 | 3.8, 4.8, 4.7, 5.3, 5.0, 5.2, 5.1 | 5.1, 4.6, 4.3, 5.3, 4.8, 4.7 | |

Spans in Z (m) | 0.6, 3.9, 3.1 | 0.7, 3.4, 3.1, 3.2 | 0.6, 3.65, 3.35, 3.4 | |

Wide load B. B.? | Yes | No | Yes | |

Load B. B. dim | 60 × 30 cm | 30 × 60 cm | 60 × 30 cm | |

Non-load B. B. dim | 30 × 30 cm | 30 × 30 cm | 30 × 30 cm | |

Supports dim | 30 × 30 cm | 30 × 30 cm | 30 × 30 cm | |

Slab thickness | 30 cm | 30 cm | 30 cm |

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**Figure 2.**Randomly generated structures. (

**a**) Sample #1355, (

**b**) sample #147, (

**c**) sample #1926 and (

**d**) sample #639. For each structure, a vector containing the 30 values that result from applying the above parameters is stored and will be used as the input of the neural network (after normalization).

**Figure 6.**Re-sampling of capacity curves of structures #11 (

**a**) and #34 (

**b**) with a resolution of 100 points and an interval of 0.01. Curve points to the left of the dashed line correspond to the original capacity curve after normalization. Curve points to the right are included in order to complete short curves with a stretch of constant shear value.

**Figure 10.**Varying hidden layer sizes: (

**a**) h

_{1}= 30, (

**b**) h

_{1}= 65, (

**c**) h

_{1}= 100, (

**d**) h

_{1}= 135.

**Figure 17.**Average, maximum and minimum validation loss, and average training loss/epochs for final scheme (30-65-65-100).

Parameter | Minimum Value | Maximum Value | Value Step |
---|---|---|---|

Number of spans in X | 1 | 9 | 1 |

Number of spans in Y | 1 | 9 | 1 |

Number of spans in Z | 1 | 3 | 1 |

Dimensions of all spans in X | 4 m | 8 m | 0.166 m |

Dimensions of all spans in Y | 3 m | 6 m | 0.125 m |

Dimensions of all spans in Z | 3 m ^{(a)} | 3.4 m | 0.040 m |

Height of suspended ground floor | 0.6 m | 0.8 m | 0.025 m |

Load-bearing beams: width | 30 cm | 60 cm ^{(b)} | - ^{(c)} |

Load-bearing beams: height | 30 cm | 60 cm | - ^{(c)} |

Non-load-bearing beams: width | 30 cm | 30 cm | - ^{(c)} |

Non-load-bearing beams: height | 25 cm | 30 cm | - ^{(c)} |

Supports: width in X | 30 cm | 30 cm | - ^{(c)} |

Supports: width in Y | 30 cm | 30 cm | - ^{(c)} |

Slab thickness | 25 cm | 30 cm | - ^{(c)} |

^{(a)}Ground floor height is always 3.4.

^{(b)}Wide load-bearing beams are randomly considered when the span length is shorter than 6 m.

^{(c)}Frames dimensions are first calculated according to their span, then the most restrictive section for each type (load-bearing, non-load-bearing, supports, slabs) is used for the whole structure.

Longitudinal Rebar | Transversal Rebar | Cover | |||
---|---|---|---|---|---|

Top | Bottom | Ø | Spacing | ||

Deep load-bearing beams 30 × 40 cm | 2 Ø12 mm | 4 Ø16 mm | 2 Ø6 mm | 20 cm | 3 cm |

Deep load-bearing beams 30 × 60 cm | 2 Ø12 mm | 5 Ø16 mm | 2 Ø6 mm | 15 cm | 3 cm |

Wide load-bearing beams 60 × 30 cm | 3 Ø12 mm | 5 Ø16 mm | 2 Ø6 mm | 15 cm | 3 cm |

Non-load-bearing beams 30 × 25 cm | 2 Ø12 mm | 2 Ø12 mm | 2 Ø6 mm | 20 cm | 3 cm |

Non-load-bearing beams 30 × 30 cm | 2 Ø12 mm | 2 Ø12 mm | 2 Ø6 mm | 20 cm | 3 cm |

Columns 30 × 30 cm | 2 Ø12 mm | 2 Ø12 mm | 2 Ø6 mm | 15 cm | 3 cm |

Network Architecture | |||
---|---|---|---|

Layers | X | h1 | Y |

Layer size | 30 | 65 | 100 |

Network Parameters | |||

Layers | X | h1 | Y |

Activation | - | tanh (Hyperbolic Tangent) | sigmoid |

Weight initialization | - | Random seed range (0, 0.1) | Random seed range (0, 0.1) |

Bias initialization | - | Random seed range (0, 0.1) | Random seed range (0, 0.1) |

Training Parameters | |||

Epochs (ep) | 200 | ||

Batch size | 12 | ||

Shuffle samples at each epoch | Yes |

Lr (Learning Rate) | Decay | M (Momentum) | |||
---|---|---|---|---|---|

MAE loss | MAE loss | MAE loss | |||

0.15 | 0.0159 | Lr/800 | 0.0158 | 0.80 | 0.0160 |

0.20 | 0.0157 | Lr/1000 | 0.0156 | 0.85 | 0.0156 |

0.25 | 0.0156 | Lr/1200 | 0.0155 | 0.90 | 0.0150 |

0.30 | 0.0156 | Lr/1400 | 0.0154 | 0.95 | 0.0181 |

0.35 | 0.0154 | Lr/1600 | 0.0150 | Nesterov | 0.0149 |

0.40 | 0.0157 | Lr/1800 | 0.0152 | ||

0.45 | 0.0159 | Lr/2000 | 0.0154 | ||

Fixed parameters: Decay = Lr/1000 M = 0.9 | Fixed parameters: Lr = 0.35 M = 0.9 | Fixed parameters: Lr = 0.35 Decay = Lr/1600 |

Network Parameters | |||
---|---|---|---|

Layers | X | h1 | Y |

Activation | - | tanh (Hyperbolic tangent) | sigmoid |

Weight initialization | - | Random seed range (0, 0.1) | Random seed range (0, 0.1) |

Bias initialization | - | Random seed range (0, 1) | Random seed range (0, 1) |

Training Parameters | |||

Lr (Learning rate) | 0.35 | ||

Decay | Lr/(8.ep) | ||

M (Momentum) | Nesterov | ||

Epochs (ep) | 800–1200 | ||

Batch size | 12 | ||

Shuffle at each epoch | Yes |

Network Architecture | ||||||
---|---|---|---|---|---|---|

Layer scheme and size | MAE loss | |||||

X | h1 | Y | Validation error | Training error | ||

30 | 30 | 100 | 0.0153 | |||

30 | 65 | 100 | 0.0135 | |||

30 | 100 | 100 | 0.0132 | |||

30 | 135 | 100 | 0.0131 | 0.0115 | ||

30 | 170 | 100 | 0.0132 | |||

30 | 205 | 100 | 0.0133 | |||

X | h1 | h2 | Y | Validation error | Training error | |

30 | 30 | 30 | 100 | 0.0133 | ||

30 | 30 | 65 | 100 | 0.0137 | ||

30 | 30 | 100 | 100 | 0.0137 | ||

30 | 55 | 80 | 100 | 0.0128 | ||

30 | 65 | 65 | 100 | 0.0126 | 0.0106 | |

30 | 65 | 100 | 100 | 0.0131 | ||

30 | 100 | 100 | 100 | 0.0134 | ||

X | h1 | h2 | h3 | Y | Validation error | Training error |

30 | 30 | 30 | 30 | 100 | 0.0137 | |

30 | 30 | 30 | 65 | 100 | 0.0133 | 0.0107 |

30 | 30 | 65 | 65 | 100 | 0.0138 | |

30 | 65 | 65 | 65 | 100 | 0.1407 |

Network Architecture | |||||
---|---|---|---|---|---|

Layer scheme and size | MAE loss | ||||

X | h1 | h2 | Y | Validation error | Training error |

30 | 65 | 65 | 65 | 0.0127 | |

30 | 65 | 65 | 82 | 0.0127 | 0.0105 |

30 | 65 | 65 | 100 | 0.0126 | 0.0106 |

30 | 65 | 65 | 135 | 0.0130 | |

30 | 65 | 100 | 135 | 0.0126 | 0.0107 |

Network Architecture | ||||
---|---|---|---|---|

Layers | X | h1 | h2 | Y |

Layer size | 30 | 65 | 65 | 100 |

Network Parameters | ||||

Layers | X | h1 | h2 | Y |

Activation | - | tanh | tanh | sigmoid |

Weight initialization | - | Random (0, 0.1) | Random (0, 0.1) | Random (0, 0.1) |

Bias initialization | - | Random (0, 1) | Random (0, 1) | Random (0, 1) |

Training Parameters | ||||

Lr (Learning rate) | 0.35 | |||

Decay | Lr/(8.ep) | |||

M (Momentum) | Nesterov | |||

Epochs (ep) | 1200 | |||

Batch size | 12 | |||

Shuffle samples at each epoch | Yes |

Parameter Variation | MAE Loss (1200 Epochs) | |
---|---|---|

Validation error | Training error | |

Initial conditions (no variation) | 0.0126 | 0.0103 |

Adadelta* instead of SGD | 0.0136 | 0.0114 |

SGD without Nesterov (M = 0.9) | 0.0128 | 0.0107 |

Lr (Learning rate) = 0.25 | 0.0128 | 0.0110 |

Lr (Learning rate) = 0.45 | 0.0127 | 0.0106 |

Decay = Lr/(6·ep) | 0.0127 | 0.0108 |

Decay = Lr/(10·ep) | 0.0128 | 0.0106 |

Sigmoid activation in hidden layers | 0.0147 | 0.0130 |

Relu activation in hidden layers | 0.1406 | 0.1358 |

Batch size = 1 (minimum) | 0.0131 | 0.0102 |

Batch size = 6 | 0.0129 | 0.0102 |

Batch size = 24 | 0.0124 | 0.0103 |

Batch size = 36 | 0.0131 | 0.0114 |

Network Architecture | ||||
---|---|---|---|---|

Layers | X | h1 | h2 | Y |

Layer size | 30 | 65 | 65 | 100 |

Network Parameters | ||||

Layers | X | h1 | h2 | Y |

Activation | - | tanh | tanh | sigmoid |

Weight initialization | - | Random (0, 0.1) | Random (0, 0.1) | Random (0, 0.1) |

Bias initialization | - | Random (0, 1) | Random (0, 1) | Random (0, 1) |

Training Parameters | ||||

Lr (Learning rate) | 0.35 | |||

Decay | Lr/(8.ep) | |||

M (Momentum) | Nesterov | |||

Epochs | 1200 | |||

Batch size | 24 | |||

Shuffle samples at each epoch | Yes |

Epoch | Average Training Error | Average Validation Error | Overfitting Ratio |
---|---|---|---|

100 | 0.0144 | 0.0151 | 0.9536 |

200 | 0.0132 | 0.0139 | 0.9470 |

300 | 0.0124 | 0.0133 | 0.9366 |

400 | 0.0119 | 0.0128 | 0.9301 |

500 | 0.0116 | 0.0126 | 0.9228 |

600 | 0.0114 | 0.0125 | 0.9111 |

700 | 0.0112 | 0.0124 | 0.9048 |

800 | 0.0110 | 0.0123 | 0.8947 |

900 | 0.0109 | 0.0123 | 0.8878 |

1000 | 0.0108 | 0.0123 | 0.8790 |

1100 | 0.0107 | 0.0123 | 0.8707 |

1200 | 0.0106 | 0.0123 | 0.8631 |

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## Share and Cite

**MDPI and ACS Style**

de-Miguel-Rodríguez, J.; Morales-Esteban, A.; Requena-García-Cruz, M.-V.; Zapico-Blanco, B.; Segovia-Verjel, M.-L.; Romero-Sánchez, E.; Carvalho-Estêvão, J.M.
Fast Seismic Assessment of Built Urban Areas with the Accuracy of Mechanical Methods Using a Feedforward Neural Network. *Sustainability* **2022**, *14*, 5274.
https://doi.org/10.3390/su14095274

**AMA Style**

de-Miguel-Rodríguez J, Morales-Esteban A, Requena-García-Cruz M-V, Zapico-Blanco B, Segovia-Verjel M-L, Romero-Sánchez E, Carvalho-Estêvão JM.
Fast Seismic Assessment of Built Urban Areas with the Accuracy of Mechanical Methods Using a Feedforward Neural Network. *Sustainability*. 2022; 14(9):5274.
https://doi.org/10.3390/su14095274

**Chicago/Turabian Style**

de-Miguel-Rodríguez, Jaime, Antonio Morales-Esteban, María-Victoria Requena-García-Cruz, Beatriz Zapico-Blanco, María-Luisa Segovia-Verjel, Emilio Romero-Sánchez, and João Manuel Carvalho-Estêvão.
2022. "Fast Seismic Assessment of Built Urban Areas with the Accuracy of Mechanical Methods Using a Feedforward Neural Network" *Sustainability* 14, no. 9: 5274.
https://doi.org/10.3390/su14095274