# Error Analysis Method of Geometrically Incomplete Similarity of End-Plate Connection Based on Linear Regression

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Derivation of Completely Similar Macro Conditions for Semi-Rigid Connections

_{1}, x

_{2}, x

_{3,}and x

_{4}, the equation representing M (moment) in the form of basic parameters is expressed as followed:

_{1}= 1, thus,

#### 2.2. Numerical Simulation Method

_{f}: the upper flange of the beam and 75 mm away from the end-plate; C

_{f}: the flange of the column and 75 mm away from the top of the beam; C

_{p}: the central position of the upper partition of the column; C

_{w}: the web of the column and 75 mm away from the surface of the upper partition; N

_{c}: the central position of the core area of the node; and E

_{p}: the upper and middle parts of the end-plate 27.5 mm away from the top of the beam.

## 3. Results

#### 3.1. Comparison of Semi-Rigid Node Indicators Under Completely Similar Conditions

#### 3.2. Effect of End-Plate Thickness on Similar Errors

## 4. Error Analysis Method

_{pi}is the predicted value of the node, y

_{i}is the known value of the node, and n is the number of node models.

## 5. Discussion

_{f}: $Y=-0.3727+0.0428x$. The correlation coefficient: 0.9349.

_{f}: $Y=-1.0157+0.1265x$. The correlation coefficient: 0.9823.

_{p}: $Y=0.3083-0.0335x$. The correlation coefficient: 0.8817.

_{w}: $Y=-0.9547+0.1104x$. The correlation coefficient: 0.9699.

_{c}: $Y=-0.0667+0.0061x$. The correlation coefficient: 0.6579.

_{p}: $Y=0.3089-0.0695x$. The correlation coefficient: 0.6445.

_{f}: the upper flange of the beam and 75 mm away from the end-plate; C

_{f}: the flange of the column and 75 mm away from the top of the beam; C

_{p}: the central position of the upper partition of the column; C

_{w}: the web of the column and 75 mm away from the surface of the upper partition; N

_{c}: the central position of the core area of the node; and E

_{p}: the upper and middle parts of the end-plate 27.5 mm away from the top of the beam.

## 6. Conclusions

- (1)
- Through the derivation, the completely similar conditions of the semi-rigid beam–column connection structure were obtained, and the constraints of the similar factors were clarified. This paper analyzed the limitations of the classic similarity theory and provided a theoretical basis for further research on similarity errors.
- (2)
- The numerical model proves that under the premise of meeting the completely similar conditions, the prototype and the model’s stress distribution law and development process are completely consistent.
- (3)
- This paper took the thickness of the end-plate as a similar factor, established six incomplete scale models, selected the stress development history of six points as the research object, analyzed, in detail, the influence of the end-plate thickness on the model’s stress development process and distribution when it does not meet the exact similar conditions, and summarized the stress development law of typical points of the end-plate connection.
- (4)
- This paper calculated the similarity errors due to the incomplete end-plate thicknesses. The analysis shows that the similarity error and the similarity factor are roughly monotonic and linear. A regression analysis was performed, and a regression formula for similar error and end-plate thickness was obtained. The correlation coefficient of the corresponding formula was calculated, and the results show that except for the stress at the end-plate points, the univariate linear regression equations at other points are more significant. Similar error analysis methods were provided which can be applied to other types of models.
- (5)
- The analysis results in this paper show that even if the stress distribution of the model only approximately meets the linear distribution law, the linear regression method can still reduce the incomplete similarity error to a large extent, which provides a reference for future study of the similarity error distribution law.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Physical Quantity 1 | Physical Quantity 2 |
---|---|

L: geometric dimension. | A: sectional area. |

g: acceleration of gravity. | γ: Poisson ratio. |

F: static load. | ρ: density. |

M: moment | σ: stress. |

E: elastic modulus. | ε: strain. |

G: shear modulus. | s: displacement. |

K: initial stiffness of semi-rigid joint. | θ: angle. |

I: moment of inertia | T: time. |

Basic Dimension | L | g | F | M | E | G | K | I |
---|---|---|---|---|---|---|---|---|

F | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |

L | 1 | 1 | 0 | 1 | −2 | −2 | 1 | 2 |

T | 0 | −2 | 0 | 0 | 0 | 0 | 0 | 1 |

Basic Dimension | A | γ | ρ | σ | ε | s | θ | T |

F | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |

L | 2 | 0 | −4 | −2 | 0 | 1 | 0 | 0 |

T | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 1 |

Factor | Similarity Ratio | Factor | Similarity Ratio |
---|---|---|---|

S_{L} | 1:2 | S_{A} | 1:4 |

S_{F} | 1:4 | S_{y} | 1:1 |

S_{M} | 1:8 | S_{σ} | 1:1 |

S_{E} | 1:1 | S_{ε} | 1:1 |

S_{G} | 1:1 | S_{s} | 1:2 |

S_{K} | 1:8 | S_{θ} | 1:1 |

Prototype | Component Modulus (mm) | Model | Component Modulus of Completely Similar Conditions (mm) |
---|---|---|---|

B_{p} | HN300 × 200 × 8 × 12 | B_{m} | 150 × 100 × 4 × 6 |

C_{p} | HN300 × 300 × 10 × 15 | C_{m} | 150 × 150 × 5 × 7.5 |

E_{p} | 250 × 500 × 16 | E_{m} | 125 × 250 × 8 |

BL_{p} | M20 | BL_{m} | M10 |

_{p}: beam of prototype; C

_{p}: column of prototype; E

_{p}: end plate of prototype; BL

_{p}: bolt of prototype; B

_{m}: beam of model; C

_{m}: column of model; E

_{m}: end-plate of model; BL

_{m}: bolt of model. HN: Narrow flange H-beam. M: Bolt specifications.

Index | Bolt | Beam and Column |
---|---|---|

Yield strength | 940 | 373.2 |

Ultimate strength | 1130 | 540.7 |

Yield strain | 0.05 | 0.08375 |

Ultimate strain | 0.1 | 0.3 |

Position | Prototype (mm) | Model (mm) |
---|---|---|

Beam | 30 | 15 |

Column | 30 | 15 |

End-plate | 8 | 4 |

Bolt | 2.8 | 1.4 |

Name | Status | 4 (mm) | 6 (mm) | 8 (mm) | 10 (mm) | 12 (mm) | 14 (mm) | 16 (mm) |
---|---|---|---|---|---|---|---|---|

B_{f} | before | −26.38% | −10.55% | 0.00% | 8.75% | 17.40% | 25.89% | 23.35% |

after | 6.23% | −1.04% | −3.03% | −3.22% | −3.31% | −3.24% | 7.86% | |

C_{f} | before | −44.40% | −28.07% | 0.00% | 17.22% | 41.65% | 87.37% | 100.94% |

after | −6.57% | 2.40% | −0.37% | 7.71% | 8.58% | −11.84% | −0.11% | |

C_{p} | before | 24.24% | 11.18% | 0.00% | −8.78% | −14.46% | −14.00% | −16.62% |

after | −6.81% | −0.45% | 4.03% | 6.11% | 5.09% | −2.07% | −6.15% | |

C_{w} | before | −60.19% | −32.18% | 0.00% | 22.44% | 44.27% | 62.22% | 68.29% |

after | 8.88% | 2.95% | −7.15% | −7.51% | −7.26% | −3.13% | 12.88% | |

N_{c} | before | −7.64% | −0.90% | 0.00% | 0.42% | 0.85% | 1.52% | 1.90% |

after | 3.41% | −2.11% | −1.79% | −0.99% | −0.20% | 0.35% | 1.19% | |

E_{p} | before | 15.17% | −15.96% | 0.00% | −73.79% | −73.85% | −64.23% | −57.81% |

after | −12.08% | 5.15% | −24.71% | 35.18% | 21.34% | −2.18% | −22.50% |

_{f}: the upper flange of the beam and 75 mm away from the end-plate; C

_{f}: the flange of the column and 75 mm away from the top of the beam; C

_{p}: the central position of the upper partition of the column; C

_{w}: the web of the column and 75 mm away from the surface of the upper partition; N

_{c}: the central position of the core area of the node; and E

_{p}: the upper and middle parts of the end-plate 27.5 mm away from the top of the beam.

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## Share and Cite

**MDPI and ACS Style**

Zhao, D.; Wang, Z.; Pan, J.; Wang, P.
Error Analysis Method of Geometrically Incomplete Similarity of End-Plate Connection Based on Linear Regression. *Appl. Sci.* **2020**, *10*, 4812.
https://doi.org/10.3390/app10144812

**AMA Style**

Zhao D, Wang Z, Pan J, Wang P.
Error Analysis Method of Geometrically Incomplete Similarity of End-Plate Connection Based on Linear Regression. *Applied Sciences*. 2020; 10(14):4812.
https://doi.org/10.3390/app10144812

**Chicago/Turabian Style**

Zhao, Dongzhuo, Zhan Wang, Jianrong Pan, and Peng Wang.
2020. "Error Analysis Method of Geometrically Incomplete Similarity of End-Plate Connection Based on Linear Regression" *Applied Sciences* 10, no. 14: 4812.
https://doi.org/10.3390/app10144812