Hailstone-induced dynamic responses of pretensioned umbrella membrane structure

Abstract

The membrane structure is a flexible structure, which is easy to vibrate or even relax under dynamic load. Engineering accident analysis shows that the relaxation of membrane structure is more likely to lead to structural failure. In this article, the impact load problem is combined with the flexible structure to analyze the impact of hailstone impact load on the dynamic response of membrane structure. First, the umbrella membrane stretching device was designed and manufactured, and the hailstone impact test was carried out on the umbrella membrane structure with polyvinyl chloride membrane material. Dynamic response data, tension relaxation of side cables and vibration deformation of umbrella membrane structures impacted by hailstones with different sizes and different characteristic points were obtained. In the numerical analysis, the form-finding analysis of umbrella membrane structure is carried out by finite element method, and the transient impact analysis is conducted in LS-DYNA. Finally, the reliability of the research results is verified by comparing the numerical and experimental results. The general laws and conclusions are drawn and the disaster-causing mechanism of membrane structure impacted by hailstone is revealed. On the whole, although the probability of hailstone destroying the membrane material directly is very small, it will relax the membrane structure and affect the safety of membrane structure. The conclusions of this article provide a theoretical basis for the design and maintenance of membrane structures.

Keywords

dynamic response hailstone impact load membrane structure numerical simulation

Introduction

The polyvinyl chloride (PVC) fabric membrane structure is a new building structure widely used in large-span spatial structures in recent decades. However, membrane structure is a flexible structure with light weight, large span, and small stiffness, which is very sensitive to external loads, such as wind, rain, and hailstone (Hincz and Gamboa-Marrufo, 2016). Thus, membrane structure is prone to large vibration under impact loading, which goes against to the structural stability and even lead to structural failures (Liu et al., 2019a; Zhang et al., 2019; Zheng et al., 2019). The analysis of many engineering accidents shows the relaxation of membrane surface caused by lateral load (e.g. impact load) will greatly reduce the critical wind speed of membrane structure, which leads to the failure of membrane structure under wind load less than the design. For example, the membrane roof of Denver International Airport terminal was torn in a snowstorm in March 2003 (Xu, 2005); In 1974, the US Pavilion of Spokane exposition was damaged after a large area of relaxation under the load; in 1995, the membrane of the US Georgia bent roof was torn after a storm; and in September 1999, the Japanese Kumamoto bent roof was damaged during the typhoon (Yan and Ai, 2005). However, according to current membrane structure design specifications and standards (ASCE/SEI 55-10, 2010; CECS 158-2015, 2015), in membrane structural load calculation, only the wind load is considered in the combination of load effect but the impact load like rain and hailstone are ignored. Although the wind load is immediate cause of membrane structure damage. The impact load will lead to the relaxation of membrane surface, which will greatly reduce the critical wind speed of membrane structure, and it is a nonnegligible profound factor. That has gradually attracted researchers’ attentions.

In recent years, a number of investigators began to study structural stability and dynamic response of membrane structure through theoretical analysis, experimental research and numerical simulation. Chen and Zheng (2003), Jin (2008) investigated the stability of nonlinear circular membrane under concentrated force by theoretical analysis. Kang (2017) investigated the dynamic response of circular membrane by deriving and solving the governing equations of viscously damped free and forced vibrations by a closed form exact method. Li et al. (2017) studied the pre-stressed orthotropic circular membrane under impact load through wave theory of impact and the principle of virtual displacement, respectively.

The above research mainly focused on the circular membranes. There are more researchers who have extended this problem to the planar rectangular orthotropic membrane structures. Shin et al. (2004) analyzed the in-plane free vibration of in-plane motion planar rectangular membrane structure. Based on the von Kármán s theory, Liu et al. (2010, 2011) solved the governing equations of large amplitude nonlinear free vibration of isotropic and orthotropic planar rectangular membrane structure with four edges simply supported by direct integration method and L-P perturbation method, respectively. Liu et al. (2018a) investigated the nonlinear damped vibration of pretensioned planar rectangular orthotropic membrane structure under impact load through analytical, experimental, and numerical methods. Li et al. (2017, 2018) studied stochastic vibration of planar rectangular orthotropic membrane structure under impact load theoretically and experimentally. Zheng et al. (2019) investigated the dynamic response of planar rectangular orthotropic membrane structure excited by the heavy rainfall by analytical and experimental methods.

Compared with planar membrane structures, the space membrane structures (i.e. saddle membrane structures, umbrella membrane structures and pneumatic membrane structure; Pagitz and Pellegrino, 2007) are more extensively in application and deserve to be studied. Liu et al. (2017) and Xu et al. (2011) investigated the nonlinear wind-induced aerodynamic stability of orthotropic saddle membrane structure. Rizzo et al. (2011) and Wu et al. (2015) studied the aero-elastic instability mechanism of hyperbolic paraboloid membrane structures by wind tunnel experiment. Zhu and Yang (2010) investigated the saddle membrane structure response under the actions of wind-structure coupling and wind-rain coupling by means of finite element method. Liu et al. (2019b) studied the large deflection nonlinear damped vibration of orthotropic saddle membrane structures excited by hailstone impact load by analytical and numerical methods. Hincz and Gamboa-Marrufo (2016) and Yu et al. (2005) investigated the pressure coefficients of tensile umbrella membrane structures subjected to wind loads by wind tunnel test.

With the development of finite element method, the numerical simulation software LS-DYNA is widely used in the research of impact load (Seica et al., 2019). Chordiya and Goel (2019) simulated free fall drop weight impact on aluminum cenosphere syntactic foam using LS-DYNA. Isotropic symmetric glass fiber–reinforced polymer (GFRP) laminate under impact load was investigated in LS-DYNA (Rawat et al., 2017). At present, the researches on hail impact load mainly focus on rigid materials (Anghileri et al., 2005; Tang et al., 2017) investigated the damages of composite aluminum alloy plate under high-velocity hailstone impact. By applying smooth particle hydrodynamics (SPH) method in LS-DYNA, Lavoie et al. (2011) studied the hailstone impact on components of aircraft. Barauskas and Abraitiene (2007) studied bullet impact composite woven fabrics structure in LS-DYNA. By comparing with test, the sliding friction coefficients of bullet and woven fabrics structure was obtained. Liu et al. (2020) investigated tensile saddle fabric membrane structure under hailstone impact load. The results indicated that the influence of hail load on membrane structure cannot be ignored.

In summary, most of those researchers are more focused on membrane material mechanics tests, planar membrane structures, or wind load. Few studies have been done on impact loads of space membrane structures. Thus, the further researches on impact load of membrane structures are necessary to fill the gaps in membrane structures load analysis. The hailstone load impacting on rigid material is an important issue, which has been studied deeply in aircraft or high-speed train. However, there are few reports about hailstone impacting on flexible structures (e.g. membrane structure) at natural landing speed. Therefore, the umbrella membrane structure, which is widely used in engineering, is taken as the research object to explore the influence of hailstone impact on the vibration and stability of membrane structures. In this article, the umbrella membrane structure, which is widely used in engineering, is taken as the research object to explore the influence of hailstone impact on the vibration and stability of membrane structures.

Experimental study

Experimental specimen

The PVC fabric membrane which has been widely implemented in engineering is selected in the experiment. The planar size of the umbrella membrane structure is 1.5 m × 1.5 m and the height is 0.5 m, see Figure 1. The parameters of membrane material are shown in Table 1. In membrane stability investigation, load case can be simplified by considering the symmetry aspects of membrane (Chen et al., 2017; Eriksson and Nordmark, 2016). In the experiment, the whole membrane structure is tensioned. The umbrella membrane structure is symmetrical when stress distribution is good. Therefore, according to the principle of symmetry, a quarter of the membrane surface was chosen to be loaded. The dynamic response data (such as displacement) of the membrane node is of fundamental significance for the stability evaluation of the whole membrane surface (Chen et al., 2019). Thus, the dynamic response of the feature points (as shown in Figure 1) on membrane surface is monitored by laser displacement sensors (as shown in Figure 2). The 6×19 IWRC wire rope is adopted to the tension cables, density 7850 kg/m³, Young’s modulus 1.5 × 10⁵ MPa, and diameter 12 mm.

Figure 1.

The umbrella membrane structure: (a) specimen and (b) schematic diagram.

Table 1.

Membrane parameters of umbrella membrane structure specimen.

Membrane parameters	Data
Thickness	0.80 mm
Area density	950 g/m²
Elastic modulus (warp/weft)	1720/1490 MPa

Figure 2.

The umbrella membrane structure specimen on the tension device.

Experimental device

Membrane structure is formed by tensioning steel wire ropes, which provide the structure with desired stiffness (Chen et al., 2020). As shown in Figures 2 to 4, the umbrella membrane structure is stretched by the screw rods connected steel wire rope and fixed on the tension device. The pretension force of cables can be adjusted by screw rods and monitored by the tensile sensors. Four tensile sensors are placed at four corners cables of the umbrella membrane structure. Tensile sensors 1 and 2 are placed at the corners cables which are close to feature point E and F, respectively. Tensile sensors 3 and 4 are installed in the positions of 1 and 2 symmetry, respectively.

Figure 3.

Schematic diagram (a) and physical picture of tension device (b).

Figure 4.

Umbrella membrane structure tensile corner in detail.

In experiment, the hailstones were regarded as spheres and generated by silica gel molds (Figure 5(a)) in various diameters. The red dye was added to silica gel molds for highlighting the hailstones and fishing lines were pre-embedded in silica gel molds of every hailstone (Figure 5(b)). In order to make the impact experiment more approximative to natural conditions and reduce the damage of hailstones in acceleration process, the hailstones were accelerated by free falling. In order to aim hailstones to membrane, hailstones were located and hanged with gauze screen (Figure 6). When experiment started, all hailstones were released by cutting fishing line at the same time.

Figure 5.

Molds and hailstones: (a) hailstones with various diameters and (b) silica gel molds.

Figure 6.

Hailstones gauze screen.

Load case

In impact experiment, the membrane pretension force is set to 3 MPa and the pretension force of side cables is set to 7 kN. According to National Standard of hailstone (GB/T 27957-2011, 2011) and considering extreme load cases, the hailstone diameter was set to 1.7, 2.5, 3.0, 4.5, and 6.0 cm. And, the hailstones were set to hit the midpoint of the quarter membrane (i.e. feature point D). The parameter of hailstones with different diameter is shown in Table 2. According to the hailstone diameter, the load cases are represented by symbols as D17, D25, D30, D45, and D60.

Table 2.

Parameter of hailstones with different diameters.

Diameter (cm)	1.7	2.5	3.0	4.5	6.0
Density (g/cm³)	0.90
Mass (g)	2.31	7.36	12.72	42.94	101.78
Velocity (m/s)	15.32	17.93	19.30	25.63	34.51

Experimental results

Experimental results of displacement time histories

Exporting the experiment data of displacement sensor, we obtain the displacement time histories of feature points of umbrella membrane under hailstone impact. Taking feature point D as example, the displacement time histories of different hailstones are shown in Figure 7.

Figure 7.

The displacement time histories of feature point D of different hailstones.

From Figure 7, we can see that the large deflection vibration occurs caused by hailstone impact. And, the displacement rapidly increases to its maximum when the membrane impacted by hailstone. Soon afterward, with the increasing of time, the displacement decreases gradually until it reaches zero. Namely, the impact process is in a short time, but the vibration resulting from the impact is sustaining a long time.

Experimental results of maximum displacement

The maximum displacement results of each point are listed in Table 3 and plotted in Figures 8 and 9.

Table 3.

Maximum displacement results (mm) of each point.

Feature points	Load case
	D17	D25	D30	D45	D60
A	0.92	2.16	3.35	7.22	10.63
B	1.28	2.55	3.91	8.37	12.47
C	1.61	2.92	4.72	9.76	14.02
D	1.99	3.54	5.45	11.03	16.81
E	1.40	3.30	4.16	10.00	14.78
F	1.59	3.15	4.25	10.08	15.22

Figure 8.

The maximum displacement results of different points under impact of different hailstones.

Figure 9.

The maximum displacement results of different hailstones and feature points.

Maximum displacement results of different hailstone diameters

The dynamic response of membrane under different diameter hailstones impact is analyzed by applying feature point D as control variable. The maximum displacement results of feature point D under impact of different diameter hailstones are shown in Figure 8. From Figure 8, we can see that the maximum displacement of membrane obviously grows with increasing hailstone diameter and linear growth occurs when diameter is 2.5 to 6.0 cm.

Maximum displacement results of different feature points

Applying hailstone diameter as control variable, the dynamic response of different feature points on membrane are analyzed. The maximum displacement results of different feature points (the hailstone diameter is 6.0 cm) are shown in Figure 9.

From Figure 9, we can see that the displacement is positively correlated with the distance. The displacement of feature point A (i.e. the farthest feature point from the impact point) is minimum. And, the maximum displacement occurs at feature point D (i.e. the impact point). The feature points E and F are symmetry points, so the results are approximate.

Tension relaxation of side cables

The side cables tension is monitored by the tensile sensors throughout the experiment process. The tension relaxation percentage was calculated by dividing the tension difference before and after the experiment by the tension value before the experiment. The side cables’ tensile force comparison before and after the impact is shown in Table 4. The membrane exhibits a strong nonlinear characteristic after being stressed, especially stress relaxation after Tension (Lei and Wu, 2013; Xu et al., 2016). In this experiment, the tensile force monitored by the tensile sensors reduced in a varying degree. However, the stress relaxations of sensors 1 and 2 which are close to the loaded area are significantly larger. Therefore, we can conclude that the hailstone impact did increase the stress relaxation, which is adverse to the safety and stability of membrane structure (Zhang et al., 2014).

Table 4.

The comparison of tensile forces of side cables before and after the impact.

Tensile force (kN)	Sensor 1	Sensor 2	Sensor 3	Sensor 4
Before experiment	7.00	7.00	7.00	7.00
After experiment	6.08	6.12	6.39	6.33
Tension relaxation ((Before − After)100%/Before)	13.14%	12.57%	8.71%	9.57%

Numerical simulation

The general purpose transient dynamic finite element software ANSYS/LS-DYNA was extensively used to transient dynamics studies, especially the impact problem (Her and Liang, 2004; Shokrieh and Javadpour, 2008). In this article, the dynamic process of hailstone impacting umbrella membrane structure was simulated by applying the implicit–explicit sequential solution method of the finite element analysis software ANSYS/LS-DYNA. The numerical simulation follows the following assumptions:

It is assumed that the side cable and membrane can only sustain tension rather than pressure, which is consistent with the umbrella membrane structure in experimental study.

The side cable and membrane are assumed as linear elastic body in working condition.

It is assumed that the membrane is articulated to the column and side cables without relative displacement.

Element and material parameter

According to the implicit–explicit sequential solution method (Zheng et al., 2013), In the implicit analysis, the element SHELL181 is applied to simulate membrane, the element LINK10 is applied to simulate cables, the element SOLID185 is applied to simulate hailstone. In the explicit analysis, the element SHELL163 is applied to simulate membrane, the element LINK167 is applied to simulate cables, the element SOLID164 is applied to simulate hailstone (Gil and Bonet, 2006). Except Poisson’s ratios, which are quoted from Liu et al. (2018b) and Liu et al. (2020), the material parameters adopted in numerical simulation are according to the experiment. The element and material parameters are shown in Table 5.

Table 5.

Element and material parameters.

Element	Thickness, h	Density, ρ	Young’s modulus, E_x	E_y	Poisson’s ratio, v_xy
SHELL163	0.8 mm	950 kg/m³	1720 MPa	1490 MPa	0.34
LINK167	–	7850 kg/m³	1.5 × 10⁵ MPa	–	–
SOLID164	–	900 kg/m³	53 MPa	53 MPa	0.30

Modeling and mesh generation

According to the experiment, we build a 1.5-m-long plane square with a 0.3-m-diameter circular hole in the middle and divide the plane into four blocks along two diagonal lines. Due to the symmetry of the model, only four quarters of the hailstone load on the membrane surface is considered. Subsequently, a sphere is established at 0.5 m above the central point of one quarter of the plane. The three-node surface element is adopted to generate the mesh of the plane (i.e. membrane surface) and the hexahedral elements is adopted to generate the mesh of sphere (i.e. hailstone). The two-node element is adopted to generate the mesh of the side and diagonal lines (i.e. cables).

Form-finding analysis

The umbrella membrane structure is represented by a regular surface with negative Gaussian curvature, and its boundary is a set of regular curves, whose curvature depends on the cable elements (Viglialoro and González Murcia, 2017). The form-finding procedure is an important issue of tensile membrane structures, which decide the structural form and force distribution (Asadi et al., 2018). The prestress of umbrella membrane structure is applied by cooling method and support displacement method. The temperature load is applied to the shell elements to simulate the tension force, and initial strain is applied to the link elements to simulate the tension force of the side cables. The circle is lifted 0.5 m in the Z-direction, and fixed constraints are applied to four corners. The form-finding analysis of membrane surface is solved by means of equilibrium iteration. Stress stiffness effects and large deflection effects are activated. The convergence criteria label set to force, value 0.01 (Wang, 2007). The form-finding result of umbrella membrane structure is shown in Figures 10 to 12. As shown in Figures 11 and 12, due to the hole in middle of the umbrella-shaped membrane structure, unavoidable stress concentration appears in the central area of the membrane surface. Except that, the stress distribution of the main part of membrane surface ranges from 3.23 to 3.09 MPa, with a relative difference of 4.33%. And, the stress distribution of the side cables ranges from 6921.17 to 7032.85 N, with a relative difference of 1.58%. The stress distribution of the main part of membrane surface and the side cables meet the requirement of the maximum stress difference within 10% (Wang, 2007). Therefore, the form-finding analysis results are satisfactory with the experiment.

Figure 10.

The geometric model of umbrella membrane structure and hailstone.

Figure 11.

Stress distribution of membrane (the unit is Pa).

Figure 12.

Stress distribution of side cables (the unit is Pa).

Loading and contact definitions

In explicit dynamic analysis, the impact load was defined by setting the contact of membrane and hailstone. And, the contact between membrane and hailstone is defined as automatic surface-to-surface contact. Hailstone was set to be master segment, and membrane was set to be slave segment. In *CONTROL_CONTACT, scale factor for sliding interface penalties is set 0.1, which is default. In order to avoid penetration and ensure convergence, full check of initial penetration is performed. The initial velocity is applied to hailstone instead of the free falling process, and the hailstone is defined as a rigid body to simplify the calculation. In explicit dynamic analysis, initial time step size is determined by LS-DYNA, and the scale factor for computed time step is 0.9.

Numerical solution and results analysis

Numerical solution

The numerical solution results are read by post-processing program LS-PREPOST. The dynamic response process of membrane surface impacted by single hailstone is shown in Figure 13. The position of feature points in numerical simulation corresponds to the ones in experiment. The time histories of the vertical displacement of feature point D impacted by single hailstone are shown in Figure 14.

Figure 13.

Dynamic response process of membrane impacted by 6 cm diameter hailstone: (a) initial time when hailstone contacts membrane surface, (b) when the displacement of the membrane is maximum, and (c) when hailstone is bounced up.

Figure 14.

Time histories of the vertical displacement of feature point D impacted by different hailstones.

The process of hailstone impacting on membrane surface is essentially energy conversion (Zheng et al., 2013). Because the membrane structure cannot bear bending moment, it can only resist load through deformation (Liu et al., 2019a). According to Figures 13 and 14, we can draw some conclusions:

In the initial time when hailstone contacts membrane surface, stress concentration occurs in the impact point and the surrounding areas. And, the hailstone begins to decelerate and the vertical displacement of membrane begins to increase.

In a very short time, the membrane vertical displacement increases to maximum and the hailstone velocity decreases to zero. As the displacement increases, stress wave on membrane surface spreads out and the stress reaches the maximum.

In the process of the impact, the kinetic energy of hailstone is converted into the deformation and vibration of membrane. The hailstone is bounced up because the reaction force and the contact process between hailstone and membrane is ended. At the same time, the stress wave diffuses to all around, but rebounds at the boundary cables. Meanwhile, the membrane begins to vibrate freely, and the vibration dampens.

Results comparison of simulation and experiment

The dynamic response results of numerical simulation are read by post-processing program LS-PREPOST. Dynamic response results of numerical simulation are analyzed by controlling variable method. First, by applying feature point D as control variable, the maximum displacement results in feature point D of simulation and experiment were compared (Figure 15). Then, by applying hailstone diameter as control variable, the comparison of simulation results and experiment results of different feature points on membrane under impact of 3.0 cm diameter hailstone were shown in Figure 16.

Figure 15.

Dynamic response results of simulation and experiment of feature point D (different hailstone diameters).

Figure 16.

Dynamic response results of simulation and experiment of different feature points (hailstone diameter 3.0 cm).

A brief analysis of Figures 15 and 16 leads to the following conclusions:

In Figure 15, the relative errors between simulation and experiment of feature points from A to F are 9%, 8%, 14%, 16%, 17%, and 16%, respectively, and the average error is 13.3%. In Figure 16, the relative errors between simulation and experiment of hailstones diameter from 1.7 to 6.0 cm are 15%, 18%, 16%, 15%, and 8%, respectively, and the average error is 14.4%. The membrane material in experiment is a polyester fabric, which is coated with polymer coating. The influence of coating is not considered in the numerical analysis, which may account for the error. The dynamic response laws that reflected by simulation results are basically consistent with the experimental ones, and the relative error is in the limit of tolerance.

The maximum displacement of membrane increases with increasing hailstone diameter. That is because the mass and velocity of hailstone are increasing with the increase in diameter, that means the kinetic energy carried by hailstone is enlarging with that.

The displacement of feature point D is the maximum and the displacement of feature points decreases as the distance from feature point D increases. When hailstone impact on feature point D, the hailstone kinetic energy is converted into the membrane vibration through the contact area. Then, the vibration energy diffuses outward with the centering on point D. Because of the energy loss in energy propagation of membrane vibration, the displacement of feature points decreases as the distance from membrane boundary decreases, which reflects the nonlinear vibration of membrane.

Conclusion

The dynamic response of pretensioned umbrella fabric membrane structure subjected to hailstone impact is studied by means of numerical simulation and experimental study. The following key conclusions can be drawn from the analysis of experimental and numerical simulation results.

There is no visible damage on the membrane surface after the experiment, which indicates that the hailstone impact will not lead to the directly structure failure of the membrane structure when the diameter of hailstone is less than 6 cm.

Through data analysis, it can be seen that the large deflection vibration occurs on the membrane under the action of hailstone. This will lead to plastic strain of membrane material and relaxation of membrane structure, and the maximum relaxation rate is 13.14%, which will aggravate the fatigue of membrane material and reduce the service life of membrane structure.

The stiffness of membrane structure is provided by the pretension. In order to ensure the structural safety and stability, the pretension detection and re-tension are necessary for membrane structure after loaded by hailstone.

The simulation results are basically consistent with the experimental ones, and the relative error is in the limit of tolerance, which increases the reliability of experimental and numerical results. It also shows that the numerical analysis scheme of this article can be used to study similar problems in the future to save the experimental cost.

The results of this article provide a theoretical basis for the dynamic design and maintenance of space membrane structures.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (project number 51608060), Guangdong Basic and Applied Basic Research Foundation (project number 2019A1515011063), and Research projects supported by Chengdu University of Technology (project number YJ2019-JX001 and YJ2017-JD002).

ORCID iD

Fan Wang

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