Analysis of Mechanical Properties of Air-Ribbed Skeleton Membrane Structure

Abstract

The existing large inflatable membrane structure is complex in structure and takes a long time to be extended. In this paper, an air-ribbed skeleton membrane structure is proposed to meet the requirement of camp and emergency tents needing to be set up quickly and easily. Ten single-arch air ribs are arranged side by side to support the tarpaulin, and wind ropes are added to the end and side faces of the tarpaulin to remain stable. The finite element model of the air-ribbed skeleton membrane structure is established to analyze the stress and the displacement of the membrane structure under combined wind and snow loads. The maximum displacement (439.4 mm) and maximum stress (29.94 MPa) are both within the safe standard. The stress and the displacement of the membrane structure in the wind load case is affected by the angle of the wind. The value of maximum stress and displacement at the wind angle of 90° are both lower than those at 0° and 45° in the wind load case. It is advisable to align the site layout of the membrane structure at the wind angle of 90°. The effect of the angle of wind ropes on the stress and displacement of the membrane structure is also studied. The maximum stress and displacement in the case when the angle of wind rope is 30° is smaller than those in the case when the angle of wind ropes is 45°. It is recommended that the wind rope should be laid at 30° to reinforce the membrane structure.

1. Introduction

The air-ribbed skeleton membrane structure is made up of air ribs and a membrane. Some air ribs constitute frames that act as supports, and the membrane acts as a surface [1]. The new membrane structure features a simple design, modular expansion, ground assembly, rapid deployment, efficient transport capacity, and large-span spatial shaping [2]. At present, more attention has been paid to the air-ribbed skeleton membrane structure from various fields such as medicine, military, aviation, outdoor camping, and others. This subject is believed to have potential value [3].

A small membrane structure with air ribs can be rapidly extended in daily life [4,5]. When the span and area of the membrane structure are larger than 6 m and 50 m², a large amount of gas should be filled into the ribs to support the structure. With the increase in the size of the structure, it becomes more complex and more time is consumed to extend it. Therefore, a new inflatable membrane structure has to be designed [6].

Scholars have analyzed the strength and bearing capacity of air-ribbed membrane structures with different structural forms. Gong et al. [7] studied the collapse behavior of the inflatable single-arch structure by simulation and experiments; the stress at the junction is compared when the shape of the surface is different. Yang [8] proposed curved and arched membrane structures as the research objects. He explored the failure performance of the membrane structures while increasing the load gradually. Shan et al. [9] developed a membrane structure with upright folded air ribs and added a buffer gate structure based on the applicable scene. Wang et al. [10] established a model of an inflatable hangar, which is closely connected with upright arched air ribs. They analyzed the wind pressure distribution on the structure at different wind angles and evaluated the strength. Liu et al. [11] designed a structure whose main body was semi-elliptical, air-ribbed arches and the shape changed gradually. They analyzed the overall stress of the structure and modified the simulation model based on actual observation data. Huang et al. [12] established membrane structure models with arched and curved air ribs as supporting structures, respectively. Through load response analysis, they determined that the bearing capacity of arched air ribs was better than that of curved air ribs. He et al. [13] analyzed the structure of the arched air rib retractable membrane through the fluid–solid coupling analysis method. They obtained the wind vibration coefficient for each partition of the membrane structure. Li et al. [14] studied the influence of four factors (surface pressure, air rib pressure, the ratio of length and width, and the ratio of height and span) on the structural and mechanical properties. They found that the internal pressure had little impact on resisting snow load.

The continuous arch is used in the existing air rib membrane structure, which can be formed once it is inflated. However, it cannot be repaired immediately when the notch appears on the air ribs, leading to the failure of the membrane structure. Therefore, the anti-accident ability of the air rib membrane structure should be improved to ensure normal use. The optimization of the membrane structure is mainly based on the parameters of the main structure and the exploration of external factors that influence membrane structure failure. Consequently, establishing external structures similar to the wind rope is of great significance to improve the performance of the membrane.

ANSYS is widely used in the numerical simulations of membrane or air rib structures [15,16]. The advantages of ANSYS in solving conventional linear and coupled problems are simple modeling and high-speed calculation. The air-ribbed membrane structure is composed of flexible tarpaulin and air ribs, which deform seriously under load. Therefore, it is necessary to solve the nonlinear large deformation problem when calculating the air-ribbed membrane structure. The nonlinear analysis of ABAQUS covers material nonlinearity, geometric nonlinearity, and state nonlinearity. It has advantages in simulating large deformation problems and can better simulate the stress and displacement of membrane structures under load [17,18].

In order to realize the requirements of rapid deployment and convenient construction of the membrane structure, a new membrane structure with a single-arch air-ribbed frame is proposed in this paper. A model of the air plenum is established in ABAQUS to analyze its mechanical properties under wind and snow loads. We analyzed the stress and displacement of the membrane structure under the wind and snow load.

2. Single-Arch Membrane Structure Design Scheme

When designing the membrane structure, it is necessary to consider the expansion time and aeration volume of the structure and to maximize the effective use of space within the given gas capacity. Therefore, the connecting rod between the air ribs in the traditional air frame membrane structure has been eliminated and vertical, single-arch air ribs are selected as the supporting structure. The fixed deployment time of the air-ribbed skeleton membrane structure established in this manuscript is 45 min by 10 people, and the deployment time of an ordinary frame tent is 60 min by 20 people. The modified structure reduces the folding time of the tent and requires fewer people.

There are two types of arched air rib construction: vertical type and curved type. The vertical air frame membrane structure has a larger available space and is more suitable for complex terrain under the same conditions. Hence, the vertical type is chosen for the design. The membrane structure should be able to carry 30 people. According to the requirement of 2 square meters per person in an emergency house, the usable area of the membrane structure shall be 60 square meters. So, the expansion radius of the arched air rib is 4 m and the span is 8 m, shown in Figure 1. The distance between the two ends of the air rib is 18 m, and the distance between the two sides of the ground is 18 m. The section diameter of the air rib is 0.3 m, with a space of 2 m between air ribs. A total of 10 air ribs are arranged side by side to support the membrane structure.

The air-ribbed membrane structure, which consists of tarpaulin and air ribs, is depicted in Figure 2. The air ribs and the outer tarpaulin are fixed by Velcro binding. External loads such as snow and wind are applied on the surface of the tarpaulin membrane directly; then, these loads are transmitted from the membrane surface to the air rib, which acts as the primary load-bearing component of the entire structure. To reinforce the overall construction, wind ropes are added on both sides of the air ribs and tarpaulin. The wind ropes are made of polyester core rope. On the end face, the wind ropes are fixed in a manner resembling the letter “M”. The line between the connecting point on the end face of the tarpaulin and the original point is named line C. The angle between line C and the X axis is 45°. For the side face, ten wind ropes are attached on each side of the tarpaulin in parallel and the height of the connected point is 2829 mm (4000/√2).

3. Construction of Membrane Structure Model

The simulation model of the air-ribbed skeleton membrane structure is depicted in Figure 3. This model was generated using the FE analysis software ABAQUS 6.14(SIMULIA Company, Providence, RI, USA). The tarpaulin and air ribs are represented as membrane elements within the simulation. The material parameters can be found in

Table 1. The property parameters of membrane structure material.

Name	Material	Thickness (mm)	Density (g/cm3)	Elasticity Modulus (GPa)	Poisson’s Ratio	Breaking Strength (MPa)
tarpaulins	polyester filament Oxford cloth	0.27	1.000	0.940	0.35	178.33
air rib	high-strength fiber	2.60	1.069	0.470	0.32	119.47

.

There is a slight deformation between the outer surface of the air rib and the inner surface of the tarpaulin. The self-contact surface between the air rib and the tarpaulin is configured in the model to prevent mutual penetration. Because of the line contact between the air rib and tarpaulin, the nodes along the contact line are bonded together, meaning that the nodes have the same coordinates. The surface load is applied on the tarpaulin and transferred to the air rib through the connecting nodes between them. The wind rope is secured to the ground by ground nails, which is simplified such that the bottom of the wind rope is fixed entirely. The wind rope is configured as a truss element fastened onto the air rib by binding constraints, subjected solely to tension. Figure 4 displays the boundary conditions. The degree of freedom of the nodes at the end of the air rib and the tarpaulin’s edge is zero.

Air rib material and tarpaulin are ideal materials in terms of bending resistance. For simulation purposes, the tarpaulin and air rib are simulated by M3D4R units [19] and the wind rope is simulated by T3D2 units [20]. To enhance calculation accuracy and minimize the grid spacing between the tarpaulin and air ribs, the grid diagram of the model is shown in Figure 5 and the structure used has a total of 169,920 grid units.

4. Model Load Condition

When the stress corresponding to the wind and snow load is applied to the membrane structure, the membrane structure displaces under the force. The resulting displacements are analyzed to assess the structural safety. Additionally, the impact of the wind ropes used to fix the membrane on the mechanical properties of the structure is also examined in this study.

4.1. Wind Load

According to Figure 6, the wind load is applied on the structure at 0° and 90°. The basic wind pressure is 0.27 kN/m² (the speed of force-eight wind is 20.7 m/s). This determination is in accordance with the Code for Building Structure Loads (GB50009-2012) [21] and previous studies conducted by relevant scholars [22,23,24]. The standard wind pressure of a tarpaulin surface is determined as Formula (1):

$$w_k={\beta }_z{\mu }_s{\mu }_z{w}_0$$

(1)

where the variables are as follows:

w_k—standard value of wind load (kN/m²);

β_z—wind vibration coefficient at height z, 1.4;

μ_s—size coefficient of wind load (values are shown in

Table 2. The shape coefficient of wind load.

Location	Stoss Side	Lee Side	Side	Top
shape factor	+0.6	−0.5	−0.7	−0.8

);

μ_z—wind pressure height variation coefficient, taken as 1.0;

w₀—basic wind pressure (kN/m²).

4.2. Snow Load

The basic pressure of snow load is 0.5 kN/m², namely, the pressure of the return period of 50 years in North China is defined as average basic snow. According to the Code for Building Structure Loads (GB50009-2012), the snow load on the membrane structure canopy is calculated according to Formula (2):

$$s_k={\mu }_r{s}_0$$

(2)

where the variables are as follows:

s_k—standard value of snow load (kN/m²);

μ_r—distribution coefficient of roof snow cover;

s₀—basic snow pressure (kN/m²).

4.3. Load Condition Setting

The dead weight of the membrane structure is determined by gravity, with an acceleration of 9.8 m/s². The connecting line between any point on the roof of the tarpaulin and the original point is named line1. When the angle between line1 and the X axis is 45° to 135°, the area of the tarpaulin is named area-A. As depicted in Figure 7, the snow load is applied on area-A downwards, while the wind load is applied over the entire surface of the tarpaulin. In order to explore the impact of the wind rope on the membrane structure, three typical layout methods are considered when the wind angle is set at 0°, 45°, and 90°. The angle between the cable wind rope and the ground is set to be 45° or 30°. The load conditions of the eight cases that are set to be analyzed are shown in

Table 3. The load cases of air-ribbed membrane structure.

Name of Working Condition	Operating Condition Number	Combination of Loads
0° angle wind load	case 1	1.3 * $S_G$+ 1.5 * $W_K$
45° angle wind load	case 2	1.3 * $S_G$+ 1.5 * $W_K$
90° angle wind load	case 3	1.3 * $S_G$+ 1.5 * $W_K$
snow load	case 4	1.3 * $S_G$+ 1.5 * $W_K$
wind and snow load (90° angle wind load; the angle of wind rope is 45°)	case 5	1.3 * $S_G$+ 1.5 * $W_K$+ 1.5 * 0.7 * $S_K$
wind and snow load (90° angle wind load; the angle of wind rope is 30°)	case 6	1.3 * $S_G$+ 1.5 * $W_K$+ 1.5 * 0.7 * $S_K$
wind and snow load (0° angle wind load; the angle of wind rope is 45°)	case 7	1.3 * $S_G$+ 1.5 * $W_K$+ 1.5 * 0.7 * $S_K$
wind and snow load (0° angle wind load; the angle of wind rope is 30°)	case 8	1.3 * $S_G$+ 1.5 * $W_K$+ 1.5 * 0.7 * $S_K$

* $S_G$ is the self-weight of the membrane structure, $W_K$ is the standard value of wind load, and $S_K$ is the standard value of snow load.

.

The model is calculated by the display dynamics algorithm, which is especially suited for the structure’s oscillating contact and transient response. In this case, the stiffness of the structure changes significantly throughout the analysis, which can affect the nonlinear behavior and failure mode of the structure. Thus, it is more appropriate to calculate complex post-buckling processing using the display dynamics algorithm.

5. Analysis of Calculation Results

5.1. Analysis of Structural Mechanical Properties under Wind Load

5.1.1. Structural Stress Analysis

The air ribs are placed evenly and connected with others by the flexible tarpaulin. The air ribs have a higher stiffness compared to the tarpaulin due to their internal pressure. Flexible tarpaulins are prone to significant deformation under the influence of wind loads. The deformation of tarpaulin and air ribs is different. The stress distribution of tarpaulin under wind load is shown in Figure 8. The analysis results indicate that high stress occurs at the connection area between the air ribs and the tarpaulin. Specifically, the maximum stress appears at the connection between the air ribs and the end face of tarpaulin. The value of maximum stress is 80.40 MPa when the wind angle is 0°, 109.00 MPa at a wind angle of 45°, and 22.02 MPa at a wind angle of 90°. It should be noted that the maximum stress of the tarpaulin is higher at a wind angle of 45° compared with both 0° and 90° wind angles.

Figure 9 presents the stress distribution of the air ribs under wind load. The maximum stress is observed at the point of the deformation bending section. The angle between the point and the ground is 45°, and the distribution of stress is the same in all cases. The maximum stress values are as follows: 17.35 MPa at a wind angle of 0°, 19.61 MPa at a wind angle of 45°, and 17.97 MPa at a wind angle of 90°. The stress of other components is uniform, and the average stress of the air ribs in three cases is 11.34 MPa, 13.38 Mpa, and 11.99 MPa, respectively. Based on these results, it can be concluded that the air-ribbed skeleton membrane structure is safe under the action of wind load because the stress is less than the fracture strength.

The air ribs are made of annular film material, which is utilized to hold high-pressure air. The thickness of the film is much smaller than the average diameter of the air rib; so, it can be simplified to a thin-walled ring. The internal pressure of the air rib is 0.2 MPa (relative pressure). Many scholars have utilized the formula for thin-walled ring structures, as mentioned in references [25,26].

The axial and circumferential stresses of the air rib can be calculated as follows:

$${\sigma }_c=\frac{pbd}{2bt}=12.095 \mathrm{MPa}$$

(3)

$${\sigma }_a=\frac{p\pi d^2/4}{\pi \left(D^2-d^2\right)/4}=5.998 \mathrm{MPa}$$

(4)

where the variables are as follows:

σ_c—circumferential stress (Pa);

σ_a—axial stress (Pa);

p—relative pressure inside and outside the air rib (Pa);

b—length of single air rib (m);

D—outer diameter of air rib (m);

d—inner diameter of air rib (m);

t—wall thickness of air rib (m).

The primary stresses are calculated to be 13.72 MPa and 4.73 MPa, respectively, using Formulas (5) and (6):

$${\sigma }_1=\frac{{\sigma }_c+{\sigma }_a}{2}+\sqrt{\frac{{\left({\sigma }_c-{\sigma }_a\right)}^2}{2}}=13.72 \mathrm{MPa}$$

(5)

$${\sigma }_2=\frac{{\sigma }_c+{\sigma }_a}{2}+\sqrt{\frac{{\left({\sigma }_c-{\sigma }_a\right)}^2}{2}}=4.73 \mathrm{MPa}$$

(6)

The stress calculated by the simulation is in the same range as the theoretical calculations. In the model, the stress transferred from the tarpaulin is applied on the air ribs; thus, the simulation results typically match the maximum theoretical calculation value.

5.1.2. Analysis of Structural Displacement

According to Figure 10, the maximum displacement of the tarpaulin under wind load appears at the middle of the end face. The maximum displacement values at 0° wind direction angle and 90° wind direction angle are 406.1 mm and 317.9 mm, respectively. The displacement of the tarpaulin at the lateral and roof are less than 110 mm. At a wind direction angle of 45°, the maximum displacement is 439.4 mm. The maximum displacement of the roof is less than 110 mm, and the maximum displacement of the side reaches about 200 mm.

In the model, the edge of the air rib at the end and the tarpaulin on the end face are joined together. The top of parallelly arranged ribs is closely connected with the top of tarpaulins, and the stiffness of the ribs is greater than that of the flexible tarpaulins. The tarpaulin is supported by the ribs, which can minimize the displacement of the tarpaulin under wind load. As a result, the displacement of the tarpaulin is lowest on the top and end faces. However, there is a significant displacement in the center of the end face of the tarpaulin under wind load. This is the total displacement, which is the square root of the sum of squares of longitudinal displacement and lateral displacement. For the three cases (case 1, case 2, and case 3), the center of the end surface of the tarpaulin exhibits obvious displacement. The maximum displacement in case 2, specifically at the center of the end face of the tarpaulin, is larger than that in case 1 and case 3.

The displacements of an air rib in case 1, case 2, and case 3 are shown in Figure 11. In case 1, the maximum displacement of the air rib appears at the top of the windward end face, and the maximum displacement is 182.7 mm. The displacements of the other air ribs remain within 100 mm. The maximum displacement of the air rib appearing at the top of the windward end face in case 2 is 247.1 mm, and the displacement of other air ribs are within the range of 140 mm and larger than that of case 1. In case 3, the load applied on the top of the tarpaulin has the greatest influence on the displacement of the air ribs, and the maximum displacement of the air ribs occurs at the top of the central air ribs, with a value of 100.3 mm. The lateral displacement of the air ribs is less than 50 mm. Comparing the three cases, the maximum displacement of the air ribs in case 2 is larger than that in case 1 and case 3. In case 1, the load applied on the windward end face is resisted by the air rib on the end face alone. The direction of the load is along the positive direction of the Z axis such that the air rib on the end face is unstable and the displacement of the air rib on the end face reaches the maximum. In case 2, the wind force affects both the end face and the top of the structure due to the comprehensive effect of wind direction at 45°. In case 3, the wind load is transferred from the tarpaulin to the air ribs and all the air ribs resist the load. As a result, the influence of wind load on the rib on the end face is weak.

According to the guidelines of the Chinese Technical Specification for Membrane Structures (CECS158:2015) [27], the maximum displacement of an air-ribbed membrane structure under load and normal working air pressure should be less than 0.5 times the clear distance between the surface of the air-ribbed membrane and any internal or external objects in the undistorted state. In this case, the horizontal distance between the membranes is 1 m; thus, the maximum horizontal displacement is limited to 0.5 m. The maximum displacements in all cases are less than the designated deformation limit, both satisfying the design criteria.

5.2. Analysis of Mechanical Properties of Structures under Combined Loads

5.2.1. Structural Stress Analysis

In

Table 4. The stress of the tarpaulin and the air ribs in case 3, case 4 and case 5.

Working Condition	Stress of Tarpaulin (MPa)	Stress of Air Ribs (MPa)
case 3	22.02	17.97
case 4	29.94	18.76
case 5	23.28	20.66

, the stress of the tarpaulin and air ribs is presented for case 3, case 4, and case 5. It is observed that the stress of the tarpaulin and air ribs in case 3 is similar to that in case 5. The connection between the air ribs and the end surface of the tarpaulin produces the maximum stress of the tarpaulin. The maximum stress of the air ribs is at the bend at an angle of 45° to the ground. The snow load is applied on the top of the membrane structure and has a straining influence on the upward wind load. There is no impact on the load distribution on the end face of the membrane structure by snow load.

The maximum stress of the tarpaulin in case 4 appears at the end face of the tarpaulin and air ribs, which is shown in Figure 12. The value of maximum stress is 29.94 MPa. The stress distribution on other areas of the tarpaulin is relatively uniform, with an average stress of about 5 MPa. For the air ribs, the maximum stress is observed at the bending part with an angle of 45° to the ground. This maximum stress value is 18.76 MPa, and the average stress of the air ribs is 14.25 MPa.

The maximum stress of the tarpaulin in case 5 is 23.28 MPa, which is 6.66 Mpa less than the value in case 4. The maximum stress on the air ribs is 20.66 Mpa, which is 1.90 Mpa more than the value in case 4. The wind suction is applied on the top of the tarpaulin due to wind loads. The direction of wind suction is opposite to the snow load. The maximum stress of the tarpaulin in case 5 decreases because the snow load is resisted by the wind suction to a certain extent. In case 4 and case 5, the stress of the tarpaulin and air ribs is less than the material’s breaking strength. It demonstrates that the design satisfies the requirements when subjected to the combined effects of snow and wind loads.

5.2.2. Structural Displacement Analysis

The displacement results in case 3, case 4, and case 5 are displayed in

Table 5. The displacement of the tarpaulin and air ribs in case 3, case 4, and case 5.

Working Condition	Displacement of Tarpaulin (MPa)	Displacement of Air Ribs (MPa)
case 3	317.9	100.3
case 4	402.1	309.9
case 5	325.9	99.2

. The maximum displacement values are close in case 3 and case 5, and the maximum values appear at the end face. However, the distribution of displacements is different. The displacement of the roof in case 3 is larger than that in case 5. With the snow load, the direction of load on the roof changes from upwards to downwards, resulting in a reduction in roof displacement. Since the air ribs are the main support structure, the load will have a greater impact on the displacement of the air ribs and different loads will cause the position of the maximum displacement of the air ribs to be different. The maximum displacement of the air ribs generated in case 3 is at the top. To mitigate the displacement, the wind rope can be fixed on the first layer of the tarpaulin. The maximum displacement in case 5 occurs on the leeward side of the air ribs.

According to Figure 13, the tarpaulin exhibits a bagging effect in case 4. The maximum displacement appears at the middle-top of the tarpaulin, which is 407.1 mm. Additionally, there is a significant displacement region observed on the side of the tarpaulin of around 170 mm. The maximum displacement of the air ribs is 309.9 mm.

The tarpaulin is more sensitive to the change in vertical load. In case 5, where the snow load is applied on the top of the membrane structure, it exerts a restraining influence on the upward force caused by the wind. Thus, the maximum displacement in case 5 appears at the end face and is less than that in case 4. The variation range of the tarpaulin is also greater compared with the variation range of the air ribs.

The vertical height of the membrane structure is 4 m, the maximum height of the built-in items is 2 m, and the net distance between the objects in the membrane structure is 2 m. According to the design requirement, the maximum vertical deformation of the membrane structure is limited to 1 m. In case 4, the tarpaulin exhibits a maximum displacement of 407.1 mm, while the maximum displacement of the air ribs is 309.9 mm. However, it is important to note that the maximum displacements in case 3, case 4, and case 5 are all less than the designated deformation limit. This indicates that the design space within these cases falls within the specified criteria, ensuring the structure meets the required specifications.

5.3. Angle of Cable Wind Rope Effect on Mechanical and Structural Properties

Based on relevant literature and practical engineering experience, the wind rope angles commonly used for the air-ribbed skeleton membrane structure are 30° and 45° [28]. If the angle of the wind rope is less than 30°, the cable wind rope becomes excessively long, requiring a larger layout area and stricter requirements of the site environment. On the other hand, setting the cable wind rope at a 45° angle ensures that it is tangential to the surface of the tarpaulin. If the angle exceeds 45°, interference may occur between the wind rope and the tarpaulin. In this paper, the influence of two widely utilized wind rope angles (30° and 45°) on the mechanical characteristics of an air-ribbed membrane structure is compared and analyzed.

5.3.1. Structural Stress Analysis

As depicted in Figure 14, the stress on the tarpaulin is identical in case 5 and case 6. The high stress appears at the connection area between the tarpaulin and the air ribs. The stress of tarpaulin on the other area distributes uniformly, meaning the wind rope has less impact on the stress. The maximum stress appears at the connection area between the tarpaulin on the end face and the air ribs. The maximum stress in case 6 is larger than that in case 5 by 8.14 MPa. The stress of tarpaulin is shown in

Table 6. The maximum and average stress of the tarpaulin in case 4 and case 5.

Working Condition	Maximum Stress (MPa)	Average Stress of Stress Band (MPa)	Stress between End Face and Air Frame (MPa)
case 5	23.28	11.80	2.24
case 6	31.52	15.96	2.99
case 7	53.75	17.05	3.88
case 8	54.73	13.84	3.85

. The stress of each part of the membrane structure in case 5 is smaller than that in case 6, which indicates that the stress of the tarpaulin is smaller when the angle of the wind rope is set to be 45°. This is advantageous for ensuring the safety of the structure.

In both case 7 and case 8, the maximum stress on the tarpaulin is observed at the joint between the end surface of the tarpaulin and the air ribs. The maximum stress values are 53.75 MPa and 54.73 MPa, respectively. The average stress levels remain below 22 MPa and 18 Mpa, showing that the angle of the wind rope at 0° wind direction angle has no obvious effect on the structural stress and the structure is safe in both cases.

As shown in Figure 15, the stress distribution is the same in all four cases. The stress on the air ribs is uniform, with the maximum stress occurring in the bending region. The stress of the air ribs is shown in

Table 7. The maximum and average stress of the air ribs in case 5, case 6, case 7, and case 8.

Working Condition	Maximum Stress	Average Stress
case 5	20.66	12.29
case 6	18.99	11.85
case 7	21.60	13.53
case 8	13.75	9.42

. The difference between the maximum stress of the air ribs is smaller than that of the tarpaulin, demonstrating that the stress of the air ribs is less affected by the setting angle of the wind ropes.

5.3.2. Structural Displacement Analysis

Figure 16 shows the displacement of the tarpaulin under different wind rope setting angles. The displacement of the tarpaulin is similar in case 5 and case 6. The maximum displacement occurs at the center of the tarpaulin’s end face. Specifically, the maximum displacement is 325.9 mm in case 5 and 318.6 mm in case 6, indicating that the displacement of the end face is not significantly affected by the change in the setting angle of the wind rope. The deformation of the leeward side and the top of the tarpaulin in case 6 closely resembles that of case 5. The wind ropes at 30° are better than those set at 45° in terms of reducing the deformation area range and the influence of the combination of snow and wind loads on the normal use of the membrane structure.

The displacements of tarpaulin in case 7 and case 8 are similar, with the maximum displacement occurring at the end face of the tarpaulin. Specifically, case 7 exhibits a maximum displacement of 622 mm, while case 8 has a maximum displacement of 467 mm. These values indicate that the structural deformation is relatively small when the cable wind rope is set at 30° under a wind angle of 0°. This finding aligns with the results obtained when the wind angle is 90°.

The displacements of the air ribs in case 5 and case 6 are shown in Figure 17. The maximum displacement of the air ribs occurs in the bending zone of the leeward side. The maximum displacement of the air ribs in case 5 is 99.24 mm, and the value in case 6 is 105.8 mm. There is a slight difference in the results of the two cases, primarily due to the similar reduction effect of the wind ropes on the windward side and the equal load applied to the leeward side and top of the air ribs. Consequently, the setting angle of the wind ropes has minimal influence on the displacement of air ribs.

The maximum displacements of the air ribs in case 7 and case 8 both occur at the center of the end face, and the maximum displacement of the air ribs in case 7 is about 435.9 mm. The displacements at other positions of the air ribs in case 8 are minimal and the average value is less than 89.69 mm. This shows that when the wind rope is arranged at 30°, the displacement of the air ribs is small. Therefore, it can be concluded that a wind rope angle of 30° is preferable for all cases.

6. Conclusions

In this paper, a new air-ribbed skeleton membrane structure is proposed, which can be extended rapidly for emergency response. The bearing capacity of the structure under wind and snow loads is analyzed by FE method, and the optimal setting angle of the wind rope is explored. The conclusions are as follows:

(1)

A new air-ribbed skeleton membrane structure of vertical-arch air ribs is constructed. The structure can withstand the action of level 8 wind load and the snow load of a 50-year return period. The maximum displacement (439.4 mm) and maximum stress (29.94 MPa) are both within the safe standard. The structure is reliable and has the advantages of rapid layout and flexible space.

(2)

The maximum stress and displacement of the membrane structure in case 1 are larger than those in case 2. According to the prevailing wind direction, it is advised that the site layout of the membrane structure aligns with a 90° wind angle while avoiding the 0° wind angle.

(3)

The maximum stress of the air ribs appears at the bending area, and the angle of the point to the ground is 45°. The maximum stress of the air ribs in case 8 is lower than that in case 7 by 36%. The maximum stress of the tarpaulin appears at the joint area of the tarpaulin on the end face and the air ribs. The stress of the tarpaulin in case 5 is 35% less than that in case 6. The maximum displacement of the tarpaulin is largest in case 7. It is recommended that the wind rope is installed at 30° to reinforce the membrane structure.