# Influence of Friction Coefficient between Cable and Membrane on Wind-Induced Response of Air-Supported Membrane Structures with Oblique Cable Net

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Coefficients of Friction Test between Cable and Membrane Materials

^{2}(side length is 63 mm) which is pasted on the same side-length flat sheeting so that the total sheeting mass is (200 ± 2) g.

## 3. Model and Methods

#### 3.1. Model Establishment

^{3}, the elastic modulus in the warp and weft directions is E

_{x}= E

_{y}= 9.43 × 10

^{8}N/m

^{2}, and the Poisson’s ratio is ν

_{x}= ν

_{y}= 0.31; the cross-section size of the steel cable is 707 mm

^{2}, the density is 7850 kg/m

^{3}, the elastic modulus of the cable is E = 1.1 × 10

^{11}N/m

^{2}, and the Poisson’s ratio is ν = 0.30.

#### 3.2. Methods

- The contact surface is smooth and continuous.
- Friction on contact surfaces obeys Coulomb’s law of friction.
- The membrane material is an orthotropic elastic material, and the warp and weft directions of the material are always perpendicular before and after deformation.

**K**

_{i}is the global mass matrix of object i;

**P**

_{i}is the global external load vector of object i;

**R**

_{i}is the contact force vector of object i;

**U**

_{i}is the node displacement vector of object i.

_{n}is the normal contact stiffness; u

_{n}is the contact gap size; ε is the intrusion tolerance; λ

_{k}is the component of the Lagrange multiplier at the kth iteration step.

**M**

_{i}and

**C**

_{i}are the global mass matrix and the global damping matrix of object i, respectively; ${\ddot{\mathit{U}}}_{i}and{\dot{\mathit{U}}}_{i}$ are the node acceleration vector and the node velocity vector of object i, respectively.

**0**. Consequently, the calculation result of Equation (4) will converge to that of Equation (1). Therefore, the problem of poor stability of calculation and nonconvergence of results by using the static calculation method has been solved, and the correctness and reliability of its results have been verified [18].

## 4. Results and Discussion

#### 4.1. Initial Form Analysis

#### 4.2. Cable-Membrane Contact under Equivalent Static Wind Load Action

_{z}is the wind vibration coefficient at height z, according to the technical specification for membrane structures (CECS 158-2015) [29]. The recommended value of the wind vibration coefficient for air-supported membrane structures is 1.2–1.6, and 1.5 is assumed as the wind vibration coefficient in this paper; μ

_{s}is the volume factor of the structure; μ

_{z}is the wind pressure height factor; μ

_{s}μ

_{z}is the average wind pressure coefficient; the average wind pressure coefficient in this paper is obtained from the wind tunnel test at 180° wind angle [18], the results of which are shown in Figure 7; w

_{0}is the reference wind pressure.

#### 4.2.1. Membrane Surface

#### 4.2.2. Cable Net

#### 4.2.3. Separated Region, Slip Region, and Contacted Region between Cables and Membranes

## 5. Conclusions

- The test method adopted in this paper can effectively measure the friction coefficient between the cable and the membrane. The static friction coefficient between the cable and the membrane observed in this study ranges from 0.27 to 0.73, while the dynamic friction coefficient ranges from 0.24 to 0.71. Static friction coefficients are greater than dynamic friction coefficients.
- The contact relationship between the membrane and the cable net is established by utilizing contact pairs and adopting the ANSYS software; the calculation results indicate that this contact model can accurately simulate the sliding and separation between the membrane and the cable net of air-supported membrane structures with an oblique cable net.
- The conventional binding model is incapable of accurately simulating the actual connection state between the cable net and the membrane. For an air-supported membrane structure with an oblique cable net, when employing the binding model, the axial force of the restrained cable elements is overestimated by approximately 26%. Furthermore, the displacement and von Mises stress on the membrane’s windward region are underestimated by approximately 26.9% and 19.6%, respectively, when the binding model is considered.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Coefficient of friction test between cable and membrane materials: (

**a**) Membrane materials; (

**b**) Test instrument.

**Figure 4.**Cloud chart of membrane surface displacement (unit: m): (

**a**) μ = 0.2; (

**b**) μ = 0.44; (

**c**) μ = 0.68; and (

**d**) μ = ∞*.

**Figure 5.**Stress cloud chart of membrane surface of initial form (unit: Pa): (

**a**) μ = 0.2; (

**b**) μ = 0.44; (

**c**) μ = 0.68; and (

**d**) μ = ∞*.

**Figure 6.**Cloud chart of cable net axial force (unit: N): (

**a**) μ = 0.2; (

**b**) μ = 0.44; (

**c**) μ = 0.68; and (

**d**) μ = μ = ∞*.

**Figure 8.**Cloud chart of x-component membrane surface displacement (unit: m): (

**a**) μ = 0.2; (

**b**) μ = 0.44; (

**c**) μ = 0.68; and (

**d**) μ = ∞*.

**Figure 9.**Cloud chart of y-component membrane surface displacement (unit: m): (

**a**) μ = 0.2; (

**b**) μ = 0.44; (

**c**) μ = 0.68; and (

**d**) μ = ∞*.

**Figure 10.**Cloud chart of z-component membrane surface displacement (unit: m): (

**a**) μ = 0.2; (

**b**) μ = 0.44; (

**c**) μ = 0.68; and (

**d**) μ = ∞*.

**Figure 11.**Stress cloud chart of membrane surface under equivalent static wind load action (unit: Pa): (

**a**) μ = 0.2; (

**b**) μ = 0.44; (

**c**) μ = 0.68; and (

**d**) μ = ∞*.

**Figure 12.**Cloud chart of cable net axial force (unit: N): (

**a**) μ = 0.2; (

**b**) μ = 0.44; (

**c**) μ = 0.68; and (

**d**) μ = ∞*.

**Figure 14.**The separated region, slip region, and contact region of cable and membrane under different friction coefficients: (

**a**) μ = 0.2; (

**b**) μ = 0.44; and (

**c**) μ = 0.68.

Membrane Materials | Static Friction Coefficient | Dynamic Friction Coefficient |
---|---|---|

America Seaman BRITE membrane material Tedlar^{®} PVF | 0.27 | 0.24 |

Duraskin PVDF6915 | 0.36 | 0.28 |

Kebao PVDF | 0.48 | 0.36 |

Huifeng 250PVDF | 0.60 | 0.44 |

Huifeng 450PVDF | 0.53 | 0.44 |

Xingyida PVDF | 0.53 | 0.44 |

Hongtai PVDF | 0.73 | 0.65 |

Tianjinlong PVDF | 0.73 | 0.71 |

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**MDPI and ACS Style**

Lai, G.; He, Y.; Zhao, Y.; Zhang, L.
Influence of Friction Coefficient between Cable and Membrane on Wind-Induced Response of Air-Supported Membrane Structures with Oblique Cable Net. *Buildings* **2023**, *13*, 649.
https://doi.org/10.3390/buildings13030649

**AMA Style**

Lai G, He Y, Zhao Y, Zhang L.
Influence of Friction Coefficient between Cable and Membrane on Wind-Induced Response of Air-Supported Membrane Structures with Oblique Cable Net. *Buildings*. 2023; 13(3):649.
https://doi.org/10.3390/buildings13030649

**Chicago/Turabian Style**

Lai, Guangxin, Yanli He, Yanguo Zhao, and Limei Zhang.
2023. "Influence of Friction Coefficient between Cable and Membrane on Wind-Induced Response of Air-Supported Membrane Structures with Oblique Cable Net" *Buildings* 13, no. 3: 649.
https://doi.org/10.3390/buildings13030649