# Inspection and Reconstruction of Metal-Roof Deformation under Wind Pressure Based on Bend Sensors

^{*}

## Abstract

**:**

## 1. Introduction

^{2}.

## 2. Fast Reconstruction Model for Deflection Distribution of Metal Roof

#### 2.1. A Fast Reconstruction Model for the Deflection Distribution of Metal Roof

#### 2.1.1. Governing Equations for Deflection Deformation of Thin Plate

- I
- Free edges (y = 0, y = l): the bending moment and shear force are zero, and formulated as:$$D(\frac{{\partial}^{2}w}{\partial {x}^{2}}+\mu \frac{{\partial}^{2}w}{\partial {y}^{2}})=0,D(\frac{{\partial}^{3}w}{\partial {x}^{3}}+(2-\mu )\frac{{\partial}^{3}w}{\partial x\partial {y}^{2}})=0.$$
- II
- Supported Edges (x = 0, x = b): the deflection and bending moment are zero:$$D(\frac{{\partial}^{2}w}{\partial {x}^{2}}+\mu \frac{{\partial}^{2}w}{\partial {y}^{2}})=0.$$

- (a)
- A thin rectangular plate with four sides simply supported, and subjected to uniform load q.
- (b)
- A rectangular plate with two sides x = 0 and x = b seen as simply supported sides, and the other sides y = 0 and y = l seen as generalized simply supported edges.

#### 2.1.2. Deflection Calculation of Part I

#### 2.1.3. Deflection Calculation of Part II

- (1)
- Moment of the two generalized simply supported edges is zero, i.e., $M=0{|}_{y=0,l}$

- (2)
- Deflection of the two generalized simply supported sides is assumed, i.e., $w={\alpha}_{0}\mathrm{sin}(\beta x){|}_{y=0,l}$, and ${h}_{n}(y)$ can be given:$${h}_{n}(y)={\alpha}_{0}\left\{\frac{\mathrm{cosh}{\lambda}_{n}l-1}{\mathrm{sinh}{\lambda}_{n}l}[(\frac{{\lambda}_{n}l}{\mathrm{sinh}{\lambda}_{n}l}-\frac{1}{1-\mu})\mathrm{sinh}{\lambda}_{n}y+{\lambda}_{n}y\mathrm{cosh}{\lambda}_{n}y]+\frac{1}{1-\mu}\mathrm{cosh}{\lambda}_{n}y-{\lambda}_{n}y\mathrm{sinh}{\lambda}_{n}y\right\}$$

#### 2.2. Analysis of Model Error

## 3. Simulations and Experiments

#### 3.1. Simulation and Discussions

#### 3.2. Experimental Platform

#### 3.3. Experimental Results and Discussion

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

## Appendix B

## References

- Huang, W.; Dong, W.M. Introduction for metal roofing development and application classification. Archit. Technol.
**2015**, 12, 1103–1107. [Google Scholar] - Luo, Y.F.; Zheng, X.J.; Guo, X.N.; Li, D.Z.; Hu, Z. Loading behavior analysis of an aluminum ally roof structure system under wind loads. J. Chongqing Univ.
**2013**, 36, 94–100. [Google Scholar] - Baskaran, A.; Molleti, S.; Ko, S.; Shoemaker, L. Wind uplift performance of composite metal roof assemblies. J. Archit. Eng.
**2012**, 18, 2–15. [Google Scholar] [CrossRef] - Long, W.Z. Discussion on improving metal roof anti wind technology The roof of Beijing Capital International Airport T3 do not be lifted by the wind fourth times again. China Constr. Metal Struct.
**2013**, 13, 62–68. [Google Scholar] - Ying, X.J.; Zhong, J.H. Analysis on damage cause of the metal roof for the stadium of qiongtai normal college & its reinforcement. China Build. Waterproofing
**2015**, 11, 14–17. [Google Scholar] - Li, Y.H.; Wang, Y.; Chase, J.G.; Mattila, J.; Myung, H.; Sawodny, O. Survey and introduction to the focused section on mechatronics for sustainable and resilient civil infrastructure. IEEE/ASME Trans. Mechatron.
**2013**, 18, 1637–1646. [Google Scholar] [CrossRef] - La, H.M.; Lim, R.S.; Basily, B.B.; Gucunski, N.; Yi, J.G.; Maher, A.; Romero, F.A.; Parvardeh, H. Mechatronic systems design for an autonomous robotic system for high-efficiency bridge deck inspection and evaluation. IEEE/ASME Trans. Mechatron.
**2013**, 18, 1655–1664. [Google Scholar] [CrossRef] - Graham, R.E.; Kamer, T.; Bob, W. Development of a multi-purpose wireless network for the structural health monitoring of a suspension bridge. In Proceedings of the IET Conference on Wireless Sensor Systems (WSS 2012), London, UK, 18–19 June 2012. [Google Scholar]
- Park, J.W.; Sim, S.H.; Jung, H.J. Displacement estimation using multimetric data fusion. IEEE/ASME Trans. Mechatron.
**2013**, 18, 1675–1682. [Google Scholar] [CrossRef] - Lin, C.J.; Yau, H.T.; Lee, C.Y.; Tung, K.H. System identification and semiactive control of a squeeze-mode magnetorheological damper. IEEE/ASME Trans. Mechatron.
**2013**, 18, 1691–1701. [Google Scholar] [CrossRef] - Fan, H.C.; Zhang, Z.; Zhu, X.J. Application of FBG in flexible surface test. Mech. Electr. Eng. Mag.
**2009**, 26, 40–43. [Google Scholar] - Chan, T.H.T.; Ashebo, D.B.; Tam, H.Y.; Yu, Y.; Chan, T.F.; Lee, P.C.; Gracia, E.P. Vertical displacement measurements for bridges using optical fiber sensors and CCD cameras. Struct. Health Monit.
**2009**, 8, 243–249. [Google Scholar] [CrossRef] - Jeong, U.; Cho, K.J. A novel low-cost, large curvature bend sensor based on a Bowden-cable. Sensors
**2016**, 16, 961. [Google Scholar] [CrossRef] [PubMed] - Marco, L.; Denis, P.; Alessio, V.; Candido, F.P.; Alessandro, C. Development of a Flexible Lead-Free Piezoelectric Transducer for Health Monitoring in the Space Environment. Micromachines
**2015**, 6, 1729–1744. [Google Scholar] - Chiolerio, A.; Roppolo, I.; Sangermano, M. Radical diffusion engineering: Tailored nanocomposite materials for piezoresistive inkjet printed strain measurement. RSC Adv.
**2013**, 3, 3446–3452. [Google Scholar] [CrossRef] - Gentner, R.; Classen, J. Development and evaluation of a low-cost sensor glove for assessment of human finger movements in neurophysiological settings. J. Neurosci. Methods
**2009**, 178, 138–147. [Google Scholar] [CrossRef] [PubMed] - Tognetti, A.; Lorussi, F.; Carbonaro, N.; de Rossi, D. Wearable Goniometer and Accelerometer Sensory Fusion for Knee Joint Angle Measurement in Daily Life. Sensors
**2015**, 15, 28435–28455. [Google Scholar] [CrossRef] [PubMed] - Stoppa, M.; Chiolerio, A. Wearable electronics and smart textiles: A critical review. Sensors
**2014**, 14, 11957–11992. [Google Scholar] [CrossRef] [PubMed] - Nakai, H.; Miki, T.; Ohgaki, K. Analytical method for critical strength of thin-walled steel frames. Mem. Fac. Eng. Osaka City Univ.
**1985**, 26, 233–250. [Google Scholar] - Zhong, Y.; Chen, J.Y.; Wang, S.Y. Analytical solution for rectangular thin cantilever plate. Chin. J. Comput. Mech.
**2006**, 23, 368–372. [Google Scholar] - Batista, M. New analytical solution for bending problem of uniformly loaded rectangular plate supported on corner points. IES J. Part A Civ. Struct. Eng.
**2010**, 3, 75–84. [Google Scholar] [CrossRef] - Nishawala, V.V. A Study of Large Deflection of Beams and Plates. Master’s Thesis, Rutgers State University, New Brunswick, NJ, USA, 2011. [Google Scholar]
- Van, G.; Robert, A. Analytical method for the construction of solutions to the Föppl-von Kármán equations governing deflections of a thin flat plate. Int. J. Non-Linear Mech.
**2012**, 47, 1–6. [Google Scholar] - Segovia, E.; Ferrer, B.; Ramis, J.; Martinez, J.; Arenas, J.P. Vibration Modal Analysis of a Thin Folded Elastic Structure using a Levy-type Solution. Int. J. Acoust. Vib.
**2012**, 17, 191–199. [Google Scholar] [CrossRef] - Song, Y.H.; Yang, L.M.; Wang, Q.S.; Li, Y.X. Deformation characteristic study of metal roof panel under wind uplift loading. Build. Struct.
**2015**, 45, 87–91. [Google Scholar] - Wang, X.B.; Chen, J.J.; Liang, Z.T. Randomicity analysis of the piezothermoelasticity intelligent thin plate by appling the numerical method. J. Xidian Univ.
**2008**, 35, 592–599. [Google Scholar] - Zhang, Z.R.; Zhang, X.W.; Lü, W.G. Numerical method based on compatible manifold element for thin plate bending. Chin. J. Mech. Eng.
**2010**, 23, 100–109. [Google Scholar] [CrossRef] - Turevsky, I.; Sankara, H.G.; Suresh, K. An efficient numerical method for computing the topological sensitivity of arbitrary shaped features in plate bending. Int. J. Numer. Methods Eng.
**2009**, 79, 1683–1702. [Google Scholar] [CrossRef] - He, X.M.; Zou, W.M. Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal.
**2009**, 70, 1407–1414. [Google Scholar] [CrossRef] - Banerjee, A.; Bhattacherya, B.; Mallika, K. Large deflection of cantilever beams with geometric non-linearity: Analytical and numerical approaches. Int. J. Non-Linear Mech.
**2008**, 5, 366–376. [Google Scholar] [CrossRef] - Simmonds, J.G. Exact Levy-type solutions for plate bending exist for transversely isotropic but not for general monoclinic materials. J. Elast.
**2004**, 75, 49–56. [Google Scholar] [CrossRef] - GB50896-2013 M.O. Technical Code for Application of Profiled Metal Sheets; Ministry of Housing and Urban-Rural Construction of the People’s Republic of China, and General Administration of Quality Supervision, Inspection and Quarantine of the People’s Republic of China: Beijing, China, 2014.
- GB50009 M.O. Load Code for the Design of Building Structures; Ministry of Housing and Urban-Rural Construction of the People’s Republic of China, and General Administration of Quality Supervision, Inspection and Quarantine of the People’s Republic of China: Beijing, China, 2012.
- Flex Sensor Manufacturers Spectra Symbol. Available online: http://www.spectrasymbol.com/flex-sensor (accessed on 12 April 2016).

**Figure 1.**The application of metal roof in buildings: (

**a**) Olympic Sports Center of Guiyang; (

**b**) Beijing-Capital International Airport; (

**c**) The production plant of Hubei Wanmeng CNC combination machine Limited company; (

**d**) Guangzhou International Convention and Exhibition Center.

**Figure 2.**Wind-uplift destruction of metal roofs; (

**a**) Beijing International Airport T3; (

**b**) The metal roof panels of Beijing International Airport T3 were split by wind.

**Figure 5.**Comparison of two simulation methods; (

**a**) Simulation results in COMSOL; (

**b**) Simulation results in MATLAB.

Material | Length l (mm) | Width b (mm) | Thickness d (mm) | Poisson Ratio | Young Modulus (Pa) E (N/m^{2}) |
---|---|---|---|---|---|

AA3004 Al-Mg-Mn alloy | 1700 | 400 | 1 | 0.3 | 7 × 10^{10} |

Roof Material | Lock Edge | Free Edge | Thickness | Poisson Ratio | Young’s Modulus |
---|---|---|---|---|---|

l (mm) | b (mm) | d (mm) | μ | E (N/m^{2}) | |

AA3004 Al-Mg-Mn alloy | 150 | 400 | 1 | 0.3 | 7 × 10^{10} |

Measured Points | Deflection (mm) | Max Relative Error (%) | |||||||
---|---|---|---|---|---|---|---|---|---|

M | 0 | 9.6 | 22.3 | 31.5 | 40.7 | 53.5 | 59.6 | ||

A | TV | 0 | 10.4 | 24.1 | 34.1 | 42.7 | 57.9 | 64.5 | 7.1% |

MV | 0 | 10.2 | 22.5 | 33.0 | 42.0 | 54.6 | 62.0 | ||

B | TV | 0 | 9.6 | 22.3 | 31.5 | 40.7 | 53.5 | 59.6 | 3.6% |

MV | 0 | 9.3 | 21.5 | 32.7 | 42.1 | 53.8 | 57.5 | ||

C | TV | 0 | 3.9 | 9.2 | 13.0 | 20.1 | 22.0 | 24.6 | 5.3% |

TV | 0 | 3.8 | 8.9 | 12.5 | 19.1 | 21.5 | 23.9 |

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**MDPI and ACS Style**

Yang, L.; Cui, L.; Li, Y.; An, C.
Inspection and Reconstruction of Metal-Roof Deformation under Wind Pressure Based on Bend Sensors. *Sensors* **2017**, *17*, 1054.
https://doi.org/10.3390/s17051054

**AMA Style**

Yang L, Cui L, Li Y, An C.
Inspection and Reconstruction of Metal-Roof Deformation under Wind Pressure Based on Bend Sensors. *Sensors*. 2017; 17(5):1054.
https://doi.org/10.3390/s17051054

**Chicago/Turabian Style**

Yang, Liman, Langfu Cui, Yunhua Li, and Chao An.
2017. "Inspection and Reconstruction of Metal-Roof Deformation under Wind Pressure Based on Bend Sensors" *Sensors* 17, no. 5: 1054.
https://doi.org/10.3390/s17051054