# Prediction of the Seismic Response of Multi-Storey Multi-Bay Masonry Infilled Frames Using Artificial Neural Networks and a Bilinear Approximation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Database of Infilled Frames, EDIF

- a—height to length ratio,
- b—ratio of moments of inertia of beam to column,
- g—ratio of column width to the thickness of masonry infill,
- r
_{c}—reinforcement ratio of column, - f
_{y}—yield strength of the reinforcing steel, - λ
_{h}—stiffness ratio (Equation (1)), - V—axial load on columns.

_{h}is a measure of the relative stiffness between the frame and masonry infill; a greater λ

_{h}corresponds to a more flexible frame. It can be determined using the following equation:

- h—height of frame between the beam axis,
- h
_{w}—height of masonry infill, - E
_{c}—modulus of elasticity of column, - E
_{i}—modulus of elasticity of masonry infill, - I—moment of inertia of the column,
- θ—whose tangent is equal to the relation between height and length of infill.

_{c}and BS

_{c}) and maximum point (IDR

_{m}and BS

_{m}) of the capacity curve. The first cracking point is characterized by a sudden decline in stiffness and the maximum point is associated with the maximum lateral capacity of the system. Post-ultimate behaviour could not be determined from the available data since that region was not observed in most of the tests. In Table 2 are given output data including both IDR

_{c}and BS

_{c}, and IDR

_{m}and BS

_{m}from neural network processing and they are expressed as dimensionless because of the simplicity in neural network modelling.

## 3. Neural Network Modelling

_{i}and number of outputs (in this study No = 1, neural networks processing was done always with only one output). As is visible in Figure 1, the most frequent values are three, five and eight neurons. Accordingly, further analysis was done with those three suggestions in order to obtain the best results with neural network processing.

## 4. Definition of Models for Numerical Nonlinear FEM Analysis and Neural Network Processing

_{1}was determined according to proposal of Stafford et al. [39]. It is based on the parameter λ

_{h}which presents a measure of the relative stiffness of the frame to infill. The reduced area A

_{ms2}of the compressed diagonal depends on the stiffness coefficient λ

_{h}, according to the recommendations of Decanini [40]. Corresponding deformations were determined according to the limit states: the start of reduction of the initial area A

_{ms1}corresponds to the deformation at the end of linear elastic behaviour (ε

_{m}/3) while the A

_{ms2}secondary area is reached at 70% of the maximum compressive stress and the associated strain corresponds to the 1.5 × ε

_{m}(Figure 5).

#### 4.1. Results of Analysis and Prediction of Neural Networks

- in the cracking area the infill wall retains 80% load capacity, while the RC frame assumes 20% load capacity (10% per column);
- in the area of the maximum strength, the infill takes on average 40%, while the frame or each of the columns assume 30% of the load capacity.

#### 4.2. Application of Neural Networks on Multi-bay Frames with Same Bay Length

- IDR
_{c,i}—inter-storey drift Ratio at cracking point of multi-bay infilled frame, - IDR
_{m,i}—inter-storey drift Ratio at maximum capacity point of multi-bay infilled frame, - BS
_{c,I}—base shear at first cracking point of multi-bay infilled frame, - BS
_{m,i}—base shear at maximum capacity point of multi-bay infilled frame, - IDRc
_{y}(NN)—inter-storey drift ratio at cracking point of one story one bay infilled frame obtained by NN - IDR
_{m}(NN)—inter-storey Drift Ratio at maximum point of one story one bay infilled frame obtained by NN - BS
_{c}(NN)—base shear at cracking point of one story one bay infilled frame obtained by NN - BS
_{m}(NN)—base shear at maximum point of one story one bay infilled frame obtained by NN - i = 2 ..., n—number of bays of multi-bay frame.

#### 4.3. Application of Neural Networks on Multi-Bay Frames with Different Bay Length

- IDR
_{cr,i}—inter-storey drift ratio at cracking point of multi-bay infilled frame with different bay length, - IDR
_{mr,i}—inter-storey drift ratio at maximum capacity point of multi-bay infilled frame with different bay length, - BS
_{cr,I}—base shear at first cracking point of multi-bay infilled frame with different bay length, - BS
_{mr,i}—base shear at maximum point of multi-bay infilled frame with different bay length, - IDR
_{cj}(NN)—inter-storey drift ratio at cracking point of one story one bay infilled frame obtained by NN - IDR
_{mj}(NN)—inter-storey drift ratio at maximum capacity point of one story one bay infilled frame obtained by NN - BS
_{cj}(NN)—base shear at cracking point of one story one bay infilled frame obtained by NN - BS
_{mj}(NN)—base shear at maximum point of one story one bay infilled frame obtained by NN - i = 2, ..., n—number of bays of multi-bay frame
- j = 1, ..., n—ordinal number of bay in multi-bay frame

## 5. Validation of Proposed Equations on Mult-Storey Multi-Bay Infilled Frames

^{2}. The Patras multi-storey muti-bay frame (Figure 13) consists of the three floors, of which only the ground floor and the first floor are filled with masonry infills (uneven distribution of masonry infill by height of the structure) with two bays. Since the bays have different bay length, for the application of approximation formulas it was necessary to use expressions (7–10). Although the second floor was a bare frame example, the same procedure was applied as in the IFS frame; floor masses are represented by a vertical load on the one-bay columns. The EC8 four-storey two-bay frame (Figure 14) was simplified by the use of approximate terms on a two one-storey one-bay frames of 4 and 6 m.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Results of neural network processing for 5 hidden neurons.(

**a**) for IDRc training set; (

**b**) for IDRc validation set; (

**c**) for IDRc testing set; (

**d**) for IDRm training set; (

**e**) for IDRm validation set; (

**f**) for IDRm testing set; (

**g**) for BSc training set; (

**h**) for BSc validation set; (

**i**) for BSc testing set; (

**j**) for BSm training set; (

**k**) for BSm validation set; (

**l**) for BSm testing set.

**Figure 4.**Stress-strain relation for three masonry infill types according to Kaushik [36].

**Figure 5.**Stress-strain relation for masonry axial compression and the definition of limit states corresponded to the variability of areas and related axial strain.

**Figure 6.**Capacity curves of RC infilled frames A (

**a**), B (

**b**) and C (

**c**) with and without masonry infill in comparison with NN results.

**Figure 11.**Multi-bay frame models and capacity curves for multi-bay frames with different bay length.

Author | Year | Laboratory | Scale | Load | No of Samples |
---|---|---|---|---|---|

Combescure [10] | 2000. | LNEC, Lisbon | 1:1.5 | C | 1 |

Colangelo [11] | 1999. | L’aquila, Italy | 1:2 | C | 11 |

Cavaleri [12] | 2004. | - | 1:2 | C | 1 |

Lafuente [13] | 1998. | U.C.V. Caracas, Venezuela | 1:2 | C | 10 |

Kakaletsis [14] | 2007. | - | 1:3 | C | 2 |

Dukuze [15] | 2000. | - | 1:3 | M | 23 |

Žarnić [16,17] | 1985. | Institute for Testing and Research in Materials and Structures (ZRMK), Ljubljana | 1:2 | C | 1 |

1992. | 1:3 | 3 | |||

Al-Charr [18] | 1998. | USACERL, Illinois | 1:2 | M | 2 |

Angel [19] | 1994. | University of Illinois, Champaign | 1:1 | C | 7 |

Mehrabi [20] | 1994. | University of Colorado, Boulder | 1:2 | C | 8 |

M | 3 | ||||

Crisafulli [21] | 1997. | - | 1:1.33 | M | 2 |

Fiorato [22] | 1970. | University of Illinois, Urbana | 1:8 | C | 3 |

Yorulmaz [23] | 1968. | University of Illinois, Urbana | 1:8 | M | 7 |

Benjamin [24] | 1958. | Stanford University, California | 1:2.94 | M | 5 |

1:1.33 | 2 | ||||

1:1 | 1 | ||||

1:4 | 5 | ||||

1:2.38 | 7 | ||||

Zovkić [25] | 2012. | Faculty of Civil Engineering, Osijek, Croatia | 1:2.5 | C | 9 |

Statistical Function | Input Data | Output Data | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

a | b | g | r_{c} | f_{y} (MPa) | λ_{h} | V (kN) | IDR_{c} | IDR_{c} | BS_{c}(kN) | BS_{c}(kN) | |

min | 0.33 | 0.60 | 1 | 0.01 | 203.37 | 1.78 | 0 | 0.01 | 0.03 | 55.9 | 76.95 |

max | 2.28 | 8.00 | 6.1 | 0.04 | 607 | 8.56 | 2343.75 | 0.55 | 2.91 | 2278.4 | 2563.2 |

average | 0.74 | 2.04 | 2.01 | 0.02 | 406.94 | 3.68 | 599.06 | 0.13 | 0.72 | 594.64 | 878.22 |

Parameters | ANN |
---|---|

Number of input layer units | 7 |

Number of hidden layers | 1 |

Number of hidden layer units | 3, 5, 8 |

Number of output layer units | 1 |

Learning rate | 0.01 |

Performance goal | 0 |

Maximum number of epochs | 10,000 |

No | Method and Reference | N_{h} | Number of N_{h} |
---|---|---|---|

1. | Hecht-Nielsen (1987) [30] | ≤2·N_{i} | 14 |

2. | Hush (1989) [30] | 3·N_{i} | 21 |

3. | Popovics (1990) [31] | (N_{i} + N_{o})/2 | 4 |

4. | Gallant (1993) [31] | 2·N_{i} | 11 |

5. | Wang (1994) [30] | 2·N_{i}/3 | 5 |

6. | Masters (1994) [30] | (N_{i} + N_{o})^{1/2} | 3 |

7. | Li (1995) [29] | ((1 + 8 N_{i})^{1/2} − 1)/2 | 3 |

8. | Tamura (1997) [29] | N_{i} + 1 | 8 |

9. | Lai (1997) [31] | N_{i} | 7 |

10. | Nagendra (1998) [31] | N_{i} + N_{o} | 8 |

11. | Zhang (2003) [29] | 2^{Ni}/n + 1 | 19 |

12. | Shibata (2009) [29] | (N_{i}·N_{o})^{1/2} | 3 |

13. | Sheela (2013) [29] | (4 N_{i}^{2} + 3)/(N_{i}^{2} − 8) | 5 |

ANN Model | ||||||
---|---|---|---|---|---|---|

Neural Network Label | Learning Algorithm | No. of Hidden Nodes | MAE ^{1} | RMSE ^{2} | MAPE ^{3} (%) | R ^{4} |

LM_3 | Levenberg-Marquardt | 3 | 13.709 | 0.926 | 13.157 | 0.829 |

LM_5 | 5 | 10.0265 | 0.481 | 11.633 | 0.919 | |

LM_8 | 8 | 12.189 | 0.558 | 17.844 | 0.886 |

^{1}Mean absolute error (MAE),

^{2}root mean squared error (RMSE),

^{3}mean absolute percentage error (MAPE) and

^{4}coefficient of correlation (R).

Masonry Infill Type | Compressive Strength | ||
---|---|---|---|

Masonry Unit f _{b} (MPa) | Mortar f _{m} (MPa) | Masonry Infill f _{k} (MPa)–Equation (2) | |

Weak—ytong block | 3 | 10 | 1.35 |

Medium—hollow clay block | 10 | 5 | 2.92 |

Strong—solid brick | 20 | 5 | 5.01 |

Masonry Infill | f_{k} (MPa) | E_{i} (MPa) | ε_{m} | ε_{u} | ε_{1} | ε_{2} | λ_{h} | f_{mθ}∗(MPa) | A_{ms1}(m ^{2}) | A_{ms2}(%A _{ms1}) |
---|---|---|---|---|---|---|---|---|---|---|

A | 2.92 | 1610 | 0.0030 | 0.0083 | 0.001 | 0.0045 | 2.47 | 0.273 | 0.494 | 76.37 |

B | 2.51 | 0.563 | 76.17 | |||||||

C | 2.48 | 0.637 | 76.28 |

_{k}—compressive strength of masonry; E

_{i}—modulus of elasticity of masonry; ε

_{m}—strain at maximum axial stress; ε

_{u}—ultimate strain; ε

_{1}—strut area reduction strain; ε

_{2}—residual strut area strain; λ

_{h}—relative panel-to-frame stiffness parameter; f

_{m}

_{θ}∗—compressive strength; A

_{ms1}—initial area of strut; A

_{ms2}—final area of equivalent diagonal strut.

Capacity Curve Data | A_IF | B_IF | C_IF | A_NN | B_NN | C_NN |
---|---|---|---|---|---|---|

IDR_{c} (%) | 0.04 | 0.054 | 0.067 | 0.038{5} | 0.053{2} | 0.066{1} |

IDR_{m} (%) | 0.49 | 0.50 | 0.51 | 0.46{8} | 0.5{2} | 0.54{6} |

BS_{c} (kN) | 357.92 | 447.04 | 545.96 | 363{2} | 462{3} | 565{3} |

BS_{m} (kN) | 663.83 | 719.69 | 815.47 | 658{1} | 742{3} | 818{1} |

^{1}—values in braces { } presents relative error from data obtained by neural networks in regard to results from numerical analysis.

Frame Type | Frame Combination | IDR_{c} (%) | IDR_{m} (%) | BS_{c} (kN) | BS_{m} (kN) |
---|---|---|---|---|---|

A | AA_IF | 0.053 | 0.47 | 656.83 | 1129.91 |

AAA_IF | 0.067 | 0.47 | 996.97 | 1617.87 | |

AA_BA (3)-(6) | 0.049{5} | 0.46{1} | 689.7{9} | 1118.6 {2} | |

AAA_BA (3)-(6) | 0.061{10} | 0.46{2} | 1016.4 {2} | 1579.2 {2} | |

B | BB_IF | 0.08 | 0.48 | 846.43 | 1253.28 |

BBB_IF | 0.093 | 0.47 | 1219.77 | 1827.99 | |

BB_BA (3)-(6) | 0.069{14} | 0.5{4} | 877.8{4} | 1261.4{1} | |

BBB_BA (3)-(6) | 0.085{9} | 0.5{6} | 1293.6{6} | 1780.8{3} | |

C | CC | 0.08 | 0.55 | 1005.25 | 1386.18 |

CCC | 0.107 | 0.55 | 1405.98 | 2046.39 | |

CC_BA (3)-(6) | 0.086{7} | 0.54{2} | 1045.25{4} | 1406.96{2} | |

CCC_BA (3)-(6) | 0.1056{1} | 0.54{2} | 1525.5{9} | 2045{1} |

^{1}—values in brackets { } represent relative error from data obtained by BA with respect to results from numerical analysis.

**Table 10.**Evaluation of accuracy of proposed bilinear approximation equations for different bay length for frames with two bays.

Frame Type | Frame Combination | IDR_{c} (%) | IDR_{m} (%) | BS_{c} (kN) | BS_{m} (kN) |
---|---|---|---|---|---|

A/B | AB_IF | 0.06 | 0.44 | 764.33 | 1198.25 |

BA_IF | 0.06 | 0.44 | 765.62 | 1198.61 | |

AB_BA (7)-(10) | 0.054{10} | 0.48{9} | 777.8{2} | 1173.4{2} | |

BA_BA (7)-(10) | 0.064{7} | 0.48{9} | 787.8{3} | 1199.8{0} | |

A/C | AC_IF | 0.067 | 0.48 | 829.41 | 1292.75 |

CA_IF | 0.08 | 0.48 | 898.88 | 1291.81 | |

AC_BA (7)-(10) | 0.058{13} | 0.5{4} | 870.5{5} | 1226.6{5} | |

CA_BA (7)-(10) | 0.077{3} | 0.5{4} | 890.8{1} | 1275.8{1} | |

B/C | BC_IF | 0.08 | 0.49 | 934.02 | 1359.93 |

CB_IF | 0.08 | 0.49 | 986.58 | 1357.18 | |

BC_BA (7)-(10) | 0.073{9} | 0.52{6} | 970.5{4} | 1314.6{3} | |

CB_BA (7)-(10) | 0.082{2} | 0.52{6} | 980.8{1} | 1337.4{1} |

^{1}—values in braces { } presents relative error from data obtained by BA in regard to results from numerical analysis.

**Table 11.**Evaluation of accuracy of proposed bilinear approximation equations for different bay length for frames with three bays.

Frame Type | Frame Combination | IDR_{c} (%) | IDR_{m} (%) | BS_{c} (kN) | BS_{m} (kN) |
---|---|---|---|---|---|

A/B/C | ABC_IF | 0.08 | 0.45 | 1197.65 | 1848.31 |

ABC_BA (7)-(10) | 0.073{8} | 0.5{11} | 1286.3{7} | 1746{6} | |

BCA_IF | 0.085 | 0.47 | 1275.44 | 1857.63 | |

BCA_BA (7)-(10) | 0.084{1} | 0.5{6} | 1296.3{2} | 1772.4{5} | |

CAB_IF | 0.09 | 0.48 | 1285.47 | 1840.39 | |

CAB_BA (7)-(10) | 0.093{4} | 0.5{4} | 1306.6{2} | 1795.2{3} |

^{1}—values in braces { } presents relative error from data obtained by BA in regard to results from numerical analysis.

Buildings | a | b | g | r_{c} | f_{y} | λ_{h} | N | |
---|---|---|---|---|---|---|---|---|

IFS | 0.87 | 1 | 2.66 | 1.76 | 240 | 2.65 | 21.2 | |

Patras | 4m | 0.875 | 0.76 | 3.57 | 2.00 | 555.0 | 2.22 | 194.9 |

6m | 0.583 | 0.76 | 3.57 | 2.00 | 555.0 | 2.12 | 146.7 | |

EC8 | 4m | 0.875 | 0.76 | 3.57 | 2.35 | 553.5 | 2.47 | 233.3 |

6m | 0.583 | 0.76 | 3.57 | 2.35 | 553.5 | 2.38 | 309.5 |

Buildings | V_{1} (kN) | V_{2} (kN) | V_{3} (kN) |
---|---|---|---|

IFS | 21.2 | 21.2 | - |

Patras | 194.85 | 292.49 | 146.66 |

EC8 | 233.26 | 464.02 | 309.52 |

**Table 14.**Evaluation of accuracy of proposed bilinear approximation equations for multi-storey multi-bay frames.

Buildings | Approximation | IDR_{c} (%) | IDR_{m} (%) | BS_{c} (kN) | BS_{m} (kN) |
---|---|---|---|---|---|

IFS | IFS_NN | 0.08 | 0.65 | 7.8 | 10.3 |

IFS_IF | 0.104 | 0.65 | 14.82 | 17.51 | |

IFS_BA (7)-(10) | 0.104{0} | 0.67{3} | 13.7{8} | 17.13{2} | |

Patras | Patras 6m_NN | 0.09 | 0.7 | 290 | 420 |

Patras 4m_NN | 0.05 | 0.54 | 250 | 380 | |

Patras_IF | 0.11 | 0.69 | 446.24 | 711.25 | |

Patras_BA (7)-(10) | 0.105{5} | 0.62{9} | 515{14} | 686{4} | |

EC8 | EC8 4m_NN | 0.05 | 0.54 | 250 | 380 |

EC8 6m_NN | 0.07 | 0.73 | 290 | 450 | |

EC8_IF | 0.071 | 0.635 | 511 | 695 | |

EC8_BA (7)-(10) | 0.08{11} | 0.615{3} | 451.15{13} | 701.65{1} |

^{1}—values in braces { } presents relative error from data obtained by BA in regard to results from numerical analysis.

© 2019 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**

Kalman Šipoš, T.; Strukar, K.
Prediction of the Seismic Response of Multi-Storey Multi-Bay Masonry Infilled Frames Using Artificial Neural Networks and a Bilinear Approximation. *Buildings* **2019**, *9*, 121.
https://doi.org/10.3390/buildings9050121

**AMA Style**

Kalman Šipoš T, Strukar K.
Prediction of the Seismic Response of Multi-Storey Multi-Bay Masonry Infilled Frames Using Artificial Neural Networks and a Bilinear Approximation. *Buildings*. 2019; 9(5):121.
https://doi.org/10.3390/buildings9050121

**Chicago/Turabian Style**

Kalman Šipoš, Tanja, and Kristina Strukar.
2019. "Prediction of the Seismic Response of Multi-Storey Multi-Bay Masonry Infilled Frames Using Artificial Neural Networks and a Bilinear Approximation" *Buildings* 9, no. 5: 121.
https://doi.org/10.3390/buildings9050121