Author Contributions
Conceptualization, Z.C. and N.S.; methodology, Z.C. and N.S.; data curation, J.Z. and Z.L.; software, C.W. and Z.L.; validation, Z.C., N.S. and C.Z.; formal analysis, N.H.; investigation, C.W.; resources, Z.C.; writing—original draft preparation, C.W.; writing—review and editing, N.H.; visualization, J.Z.; supervision, Z.C.; project administration, Z.C.; funding acquisition, Z.C. All authors have read and agreed to the published version of the manuscript.
Figure 1.
Schematic diagrams and practical photos of spherical inflatable membrane structures: (a) A schematic diagram of structural shape and composition; (b) Spherical inflatable membrane structure with cross cables; (c) Spherical inflatable membrane structure with radial cables.
Figure 1.
Schematic diagrams and practical photos of spherical inflatable membrane structures: (a) A schematic diagram of structural shape and composition; (b) Spherical inflatable membrane structure with cross cables; (c) Spherical inflatable membrane structure with radial cables.
Figure 2.
A plot of spans and rise–span ratios for practical spherical inflatable membrane structures displayed in
Table 1.
Figure 2.
A plot of spans and rise–span ratios for practical spherical inflatable membrane structures displayed in
Table 1.
Figure 3.
Distribution and numbering of pressure measuring taps.
Figure 3.
Distribution and numbering of pressure measuring taps.
Figure 4.
Photos of experiment models: (a) H/L = 0.25; (b) H/L = 0.33; (c) H/L = 0.50.
Figure 4.
Photos of experiment models: (a) H/L = 0.25; (b) H/L = 0.33; (c) H/L = 0.50.
Figure 5.
Boundary layer simulation results: (a) Wind velocity and turbulence intensity profiles; (b) Wind velocity power spectrum.
Figure 5.
Boundary layer simulation results: (a) Wind velocity and turbulence intensity profiles; (b) Wind velocity power spectrum.
Figure 6.
Comparison of the mean pressure coefficients along centerline of the model (
H/
L = 0.50) [
8,
9,
29].
Figure 6.
Comparison of the mean pressure coefficients along centerline of the model (
H/
L = 0.50) [
8,
9,
29].
Figure 7.
A flow chart of nonlinear dynamic time–history analysis.
Figure 7.
A flow chart of nonlinear dynamic time–history analysis.
Figure 8.
Contours of membrane stress with different mesh levels.
Figure 8.
Contours of membrane stress with different mesh levels.
Figure 9.
Modal shapes of spherical inflatable membrane structures: (a) 1st and 2nd horizontal modes (without cable); (b) 3rd vertical mode (without cable); (c) 1st and 2nd horizontal modes (cross cable); (d) 3rd vertical mode (cross cable); (e) 1st and 2nd horizontal modes (radial cable); (f) 3rd mode vertical (radial cable).
Figure 9.
Modal shapes of spherical inflatable membrane structures: (a) 1st and 2nd horizontal modes (without cable); (b) 3rd vertical mode (without cable); (c) 1st and 2nd horizontal modes (cross cable); (d) 3rd vertical mode (cross cable); (e) 1st and 2nd horizontal modes (radial cable); (f) 3rd mode vertical (radial cable).
Figure 10.
The influence of different parameters on the first 40 natural frequencies: (a) Span; (b) Rise–span; (c) Internal pressure; (d) Cable configuration.
Figure 10.
The influence of different parameters on the first 40 natural frequencies: (a) Span; (b) Rise–span; (c) Internal pressure; (d) Cable configuration.
Figure 11.
Contours of the maximum displacement component: (a) u component; (b) v component; (c) w component.
Figure 11.
Contours of the maximum displacement component: (a) u component; (b) v component; (c) w component.
Figure 12.
Mean deformation profiles in different sections (amplification coefficient a =5): (a) XZ-plane; (b) YZ-plane.
Figure 12.
Mean deformation profiles in different sections (amplification coefficient a =5): (a) XZ-plane; (b) YZ-plane.
Figure 13.
Contours of the maximum displacement under different wind velocities (L = 60 m, H/L = 0.5, p = 300 Pa).
Figure 13.
Contours of the maximum displacement under different wind velocities (L = 60 m, H/L = 0.5, p = 300 Pa).
Figure 14.
Contours of the maximum membrane stress under different wind velocities (L = 60 m, H/L = 0.50, p = 300 Pa).
Figure 14.
Contours of the maximum membrane stress under different wind velocities (L = 60 m, H/L = 0.50, p = 300 Pa).
Figure 15.
The maximum wind-induced responses with different spans (H/L = 0.50, p = 300 Pa, Uh = 15 m/s): (a) Displacement; (b) Stress.
Figure 15.
The maximum wind-induced responses with different spans (H/L = 0.50, p = 300 Pa, Uh = 15 m/s): (a) Displacement; (b) Stress.
Figure 16.
The maximum wind-induced responses with different rise–span ratios (L = 60 m, p = 300 Pa, Uh = 15 m/s): (a) Displacement; (b) Stress.
Figure 16.
The maximum wind-induced responses with different rise–span ratios (L = 60 m, p = 300 Pa, Uh = 15 m/s): (a) Displacement; (b) Stress.
Figure 17.
The maximum wind-induced responses with different internal pressures (L = 60 m, H/L = 0.50, Uh = 15 m/s and 20 m/s): (a) Displacement; (b) Stress.
Figure 17.
The maximum wind-induced responses with different internal pressures (L = 60 m, H/L = 0.50, Uh = 15 m/s and 20 m/s): (a) Displacement; (b) Stress.
Figure 18.
Contours of maximum displacement of structures with different cable configurations (L = 60 m, H/L = 0.50, p = 300 Pa): (a) Without cables; (b) Cross cables; (c) Radial cables.
Figure 18.
Contours of maximum displacement of structures with different cable configurations (L = 60 m, H/L = 0.50, p = 300 Pa): (a) Without cables; (b) Cross cables; (c) Radial cables.
Figure 19.
Contours of maximum membrane stress of structures with different cable configurations (L = 60 m, H/L = 0.50, p = 300 Pa): (a) Without cables; (b) Cross cables; (c) Radial cables.
Figure 19.
Contours of maximum membrane stress of structures with different cable configurations (L = 60 m, H/L = 0.50, p = 300 Pa): (a) Without cables; (b) Cross cables; (c) Radial cables.
Figure 20.
The influence of cables under different wind velocities: (a) Displacement; (b) Stress.
Figure 20.
The influence of cables under different wind velocities: (a) Displacement; (b) Stress.
Figure 21.
Contours of gust response factors: (a) Displacement-based; (b) Stress-based.
Figure 21.
Contours of gust response factors: (a) Displacement-based; (b) Stress-based.
Figure 22.
Comparisons of the gust response factors obtained by the proposed empirical formula and nonlinear time–history analysis: (a) Displacement; (b) Stress.
Figure 22.
Comparisons of the gust response factors obtained by the proposed empirical formula and nonlinear time–history analysis: (a) Displacement; (b) Stress.
Figure 23.
Comparisons between the equivalent response results and nonlinear dynamic analysis results: (a) Displacement; (b) Stress.
Figure 23.
Comparisons between the equivalent response results and nonlinear dynamic analysis results: (a) Displacement; (b) Stress.
Figure 24.
Comparisons between the equivalent response results and nonlinear dynamic analysis results under different cable configurations: (a) Displacement; (b) Stress.
Figure 24.
Comparisons between the equivalent response results and nonlinear dynamic analysis results under different cable configurations: (a) Displacement; (b) Stress.
Table 1.
Engineering information of spherical inflatable membrane structures.
Table 1.
Engineering information of spherical inflatable membrane structures.
Engineering Project | Span L (m) | Rise H (m) | Rise/Span H/L | Cable Configuration |
---|
Nalati Horse Dance Performance Hall | 80 | 30 | 0.38 | Radial cable |
SOCT Children’s Paradise | 54 | 15 | 0.28 | Cross cable |
Jinxiu Water Sports Carnival | 100 | 25 | 0.25 | Radial cable |
Yanjing Shenmuyuan Water Park | 110 | 35 | 0.32 | Cross cable |
Jiaozhou Sports Center | 108 | 33 | 0.31 | Radial cable |
Junmei Gymnasium | 90 | 23 | 0.25 | Cross cable |
Zibo International Convention and Exhibition Center | 98 | 35 | 0.36 | Cross cable |
Xiangshawan Desert Art Museum | 100 | 30 | 0.30 | Radial cable |
Large granary in Liaoning | 40 | 20 | 0.50 | Without cable |
Zhongwei Starry Sky Theater | 60 | 25 | 0.42 | Radial cable |
Inflated Airform for Pabco Gypsum near Las Vegas | 65 | 23.5 | 0.36 | Without cable |
The Double Membrane cover in Exeter Maine | 44 | 11 | 0.25 | Without cable |
Table 2.
Detailed parameters of the experiment models.
Table 2.
Detailed parameters of the experiment models.
No. | Span L (m) | Rise H (m) | Rise/Span H/L | Number of Taps |
---|
1 | 0.6 | 0.30 | 0.50 | 379 |
2 | 0.6 | 0.20 | 0.33 | 217 |
3 | 0.6 | 0.15 | 0.25 | 217 |
Table 3.
Material properties of the finite element model.
Table 3.
Material properties of the finite element model.
Membrane | | Cable | |
---|
Thickness | 1 × 10−3 m | Sectional area | 2 × 10−4 m2 |
Elastic modulus | 600 MPa | Elastic modulus | 1.5 × 105 MPa |
Poisson ratio | 0.32 | Poisson ratio | 0.3 |
Density | 1.38 kg/m2 | Density | 7850 kg/m2 |
Table 4.
The parameters of all cases.
Table 4.
The parameters of all cases.
Parameters | Range |
---|
Span L (m) | 60, 80, 100 |
Rise–span H/L | 0.50, 0.33, 0.25 |
Internal pressure p (Pa) | 300, 350, 400, 450 |
Cable configuration | Without cables, cross cables, radial cables |
Wind velocity Uh (m/s) | 10–20 (Interval of 1) |
Table 5.
Gust response factors for spherical inflatable membrane structures without cables.
Table 5.
Gust response factors for spherical inflatable membrane structures without cables.
Response | Internal Pressure | H/L = 0.25 | H/L = 0.33 | H/L = 0.50 |
---|
L = 60 m | L = 80 m | L = 100 m | L = 60 m | L = 80 m | L = 100 m | L = 60 m | L = 80 m | L = 100 m |
---|
Displacement | 300 Pa | 1.26 | 1.27 | 1.28 | 1.30 | 1.35 | 1.35 | 1.41 | 1.50 | 1.52 |
350 Pa | 1.24 | 1.25 | 1.26 | 1.25 | 1.32 | 1.32 | 1.40 | 1.50 | 1.50 |
400 Pa | 1.22 | 1.23 | 1.23 | 1.23 | 1.30 | 1.30 | 1.40 | 1.48 | 1.50 |
450 Pa | 1.20 | 1.21 | 1.21 | 1.21 | 1.28 | 1.28 | 1.40 | 1.48 | 1.50 |
Stress | 300 Pa | 1.19 | 1.20 | 1.20 | 1.21 | 1.22 | 1.23 | 1.24 | 1.24 | 1.24 |
350 Pa | 1.18 | 1.18 | 1.19 | 1.20 | 1.20 | 1.20 | 1.23 | 1.23 | 1.23 |
400 Pa | 1.16 | 1.15 | 1.17 | 1.18 | 1.19 | 1.20 | 1.23 | 1.23 | 1.23 |
450 Pa | 1.15 | 1.12 | 1.16 | 1.17 | 1.18 | 1.18 | 1.23 | 1.22 | 1.23 |
Table 6.
Recommendations of gust response factors and nonlinear adjustment factors.
Table 6.
Recommendations of gust response factors and nonlinear adjustment factors.
Response | Cable Configuration | Gust Response Factor | Nonlinear Adjustment Factor |
---|
Displacement | Without cable | | 1.03 |
Cross cable | | 1.04 |
Radial cable | | 1.07 |
Stress | Without cable | | 1.03 |
Cross cable | | 1.02 |
Radial cable | | 1.00 |