Numerical Simulation of Structural Performance in a Single-Tube Frame for 12 m-Span Chinese Solar Greenhouses Subjected to Snow Loads

Abstract

To address the structural concerns of a 12.0 m-span landing assembled single-tube frame (LASF) for Chinese solar greenhouses subjected to snow loads, the internal forces and deformations of LASF and its reinforced counterpart (RLASF) were numerically simulated to determine the ultimate bearing capacities (L_u) and the failure loads (L_f). During the simulations, steel tubes were modeled as beam188 elements and cables as link180 elements. The frame constraints and the connections were assumed to be fixed supports and rigid, respectively. The loads were determined according to the Chinese standard (GB51183-2016). Simulations revealed that the LASF and RLASF primarily withstand bending moments and are prone to strength failures under snow loads. Both exhibited lower L_u and L_f under non-uniform snow loads than under uniform snow loads. The results also indicated that crop loads could deteriorate the structural safety of the LASF and RLASF. L_u and L_f were found to be proportional to the section modulus of the tubes. The effects of wind loads and initial geometry imperfections on L_f of the LASF and RLASF can be neglected. Furthermore, the RLASF exhibited higher L_f compared to the LASF. Steel usage of the RLASF could be further reduced by replacing circular tubes with rectangular tubes, making the RLASF a feasible option for constructing Chinese solar greenhouses.

1. Introduction

The Chinese solar greenhouse (hereafter referred to as ‘solar greenhouse’) is widely used for vegetable production in northern China with little or no additional heating during the winter months [1,2]. It plays a crucial role in ensuring a consistent vegetable supply throughout this critical period [3,4,5,6,7]. Moreover, heavy snowfall is a major cause of structural damage and failure in solar greenhouse frames [8,9]. Thus, evaluating the structural safety of solar greenhouse frames under snow loads has become an important aspect of structural design for solar greenhouses in northern China.

Traditionally, solar greenhouses have been predominantly constructed using truss structures to address the above problem. However, in recent years, there has been an observable trend toward replacing these truss structure frames with landing-assembled single-tube frames (hereafter referred to as ‘LASF’) with a span up to 12.0 m [10]. The LASF is known for its low construction cost and large interior space for agricultural machinery. Furthermore, it allows for the application of flexible walls made of insulation blankets, which are also increasingly favored recently over conventional north walls typically built with bricks or rammed earth [11,12]. Thereby, the LASF has become an increasingly popular choice for solar greenhouse construction. However, concerns regarding potential structural safety hazards associated with single-tube frames under heavy snow have been raised simultaneously and impact the application of the LASF [13].

Previous studies have demonstrated that single-tube frames predominantly experience bending moments under snow loads and are particularly susceptible to non-uniform snow distribution [14,15]. The factors affecting the snow load bearing capacity of single-tube frames include the cross-section shape of tubes, the overall frame design, the column support conditions, and the load combinations. Liu et al. [16] conducted a comparative analysis on the stress and deformation responses of a 10.0 m span solar greenhouse single-tube frame constructed from tubes of round, Ω, elliptical, and square cross-sections under snow load conditions. Their findings revealed that frames utilizing elliptical tubes exhibited the least cross-section displacement, whereas those with round tubes showed the lowest levels of normal stress. Furthermore, Bai et al. identified that the maximum bending moment experienced by an 8.0 m span LASF with elliptical tubes could be modified by adjusting the incline angle of the north column [12]. Moriyama et al. observed that the maximum permissible snow load on a greenhouse frame diminishes as ridge heights increase [17]. Wang et al. concluded that the ultimate load-bearing capacity of an LASF decreases as the span increases, based on the simulation results [18]. They also determined that there is a positive correlation between ultimate bearing capacity and the section modulus and size of the frame. It is reported in their study that the frame’s support condition at the soil level could shift from a fixed type to a hinge type due to soil saturation, thereby reducing the snow load-bearing capacity. These observations were corroborated by Kang et al. [19] and Yan et al. [20]. Briassoulis et al. [21], through numerical analysis of a collapsed multi-span steel greenhouse, concluded that the failure under snow conditions resulted from an unfavorable combination of snow and wind actions, as well as the heating system not operating.

Reinforcement provides an efficient and economical method to enhance the structural safety of greenhouse frames and mitigate damages caused by heavy snowfalls. Yang et al. [22] observed the structural instability failure of a solar greenhouse frame under snow and wind loads. They proposed to enhance the frame’s structural safety by installing columns and nylon cables on the front roof. Qi et al. [23] recommended that the use of temporary support columns beneath the front roof could reduce the frame deformation and improve their structural safety under wind and snow loads. Moriyama et al. [17] demonstrated that the inclusion of diagonal braces on the roof significantly increased the maximum allowable snow load of a plastic greenhouse. Kang et al. also reported that the lateral sway of arch-type greenhouse frames could be effectively diminished by the temporary installation of diagonal braces [19]. Xie et al. [24], through force analyses of a three-span solar greenhouse frame, found that adding a vertical suspension rod onto the existing frame could decrease the maximum equivalent stress and deformation under wind loads by 89.6% and 59.0%, respectively. Moreover, Yan et al. [20] studied the internal forces of a 12 m-span elliptical tube single-tube arch under the load combinations of self-weight, crop, and wind. They recognized that local stress concentration occurred at the north wall columns and proposed to enhance these north wall columns with lattice columns to alleviate these stresses.

Although progress has been made on the structural safety of greenhouse frames under snow loads, information on the stress and deformation responses of LASFs to these conditions is scant. The impacts of crop loads, wind loads, tube cross-sectional shapes, and initial geometric imperfections on these responses remain unexplored. Furthermore, there are currently no reinforcement measures available that enhance the structural integrity of the LASF without significantly increasing steel usage. Therefore, the objectives of this paper are, by using numerical analysis methods, to investigate the stress and deformation responses of the LASF under snow loads, evaluate the effects of crop loads, wind loads, and tube cross-sectional shapes on these stresses, and develop reinforcement measures to ensure the structural safety of this type of solar greenhouse frame without increased steel usage.

2. Materials and Methods

2.1. LASF of Solar Greenhouse

The solar greenhouse is constructed with the LAST covered with film and insulation blanket. The schematic diagram of the 12.0 m span LASF is made with bent single steel tubes with a round cross-section (Φ 60.3 mm × 3.5 mm) and installed at an interval of 1.0 m. The ridge height of the LASF is 5.525 m. The projection widths of the back roof and front roof are 2.32 m and 9.68 m, respectively. The incline angles of the north wall column and the back roof are 85° and 45°, respectively. The remaining parameters are shown in Figure 1.

2.2. Loads

The loads acting on the solar greenhouse arch include permanent loads (G), snow loads, wind loads, and crop loads, etc. G is generated by the self-weights of the frame, plastic film, insulation blankets, and permanently fixed equipment. The self-weight of the frame can be included in the calculation by considering the gravitational acceleration and the steel density. The self-weights of 1 mm thick polyethylene film and 20 mm thick foam insulation blankets, which are typically used in solar greenhouses are 0.001 kN∙m⁻² and 0.015 kN∙m⁻², respectively, according to the Chinese standard GB/T 51183-2016 [25]. The weights of permanently fixed equipment are considered negligible in this study, as they were seldom used in solar greenhouses. In case of daytime when the thermal blanket is rolled up, G acting on the front roof consists of self-weight of the frame and plastic film. At night, when the insulation blanket is extended, G acting on the front roof includes the self-weights of the frame, plastic film, and insulation blanket. G acting on the remaining parts consists of the self-weights of the frame, plastic film, and insulation blanket.

According to the Chinese standard GB/T 51183-2016 [25], snow load (s_k) is distributed on the front roof in areas where the tangent angle is less than 60°, and can be calculated according to the following equation:

s_k = μ_r∙c_t∙s₀

(1)

where s_k is the standard value of snow load, kN∙m⁻²; and μ_r is the distribution coefficient of snow loads on roof. c_t is the heating influence coefficient, which can be taken as 1.0 for solar greenhouses without heating measures [25]; and s₀ is the basic snow pressure, kN∙m⁻².

μ_r was taken according to Figure 2. It is noted that in case of non-uniformly distributed snow, the maximum value of μ_r (μ_r,m) is 1.0 with the film only on the front roof, and 2.0 with both the film and insulation blanket according to the Chinese standard GB/T 51183-2016 [25].

The wind loads (w_k) are calculated using the following equation:

w_k = μ_s∙μ_z∙w₀

(2)

where w_k is the standard value of the wind loads, kN∙m⁻²; μ_s is the wind pressure coefficient; and μ_z is the wind pressure variation coefficient. w₀ is the basic wind pressure, kN∙m⁻².

According to the Chinese standard GB/T 51183-2016 [25], the values of μ_s under northward winds and southward winds are determined following Figure 3. Additionally, solar greenhouses are often constructed in B class areas, such as fields, rural areas, forests, and hilly terrains; μ_z can be taken as 0.75.

In solar greenhouses used for cultivating fruit vegetables, crop loads (C) exert concentrated forces on the frames via hanging wires directly affixed to them (Figure 4). These concentrated forces, resulting from the tension in the horizontal hanging wires, are divided into both horizontal (H) and vertical components (N). Following the provisions of the Chinese standard GB/T 51183-2016 [25], the values of H and N were determined as 1752 N and 234 N, respectively, according to Equations (3) and (4):

H = q∙d∙l²/(8f)

(3)

N = q∙d∙l/2

(4)

where q is the crop loads, 0.15 kN∙m⁻¹; d is the distance between frames, m; and l is distance between adjacent support points of the horizontal wire, m. f is the sag of the horizontal wire, m. f is taken as l/30 from a conservative perspective.

The concentrated forces (V) resulting from the tension in the vertical hanging wires are the same in magnitude and direction. It was calculated as 756 N according to the following equation:

V = q∙d∙D/2

(5)

where D is the span of the LASF, 12.0 m.

2.3. Load Combination Method

The load combinations controlled by variable load effects are used as follows according to the Chinese standard GB/T 51183-2016 [25]:

$$S_d={\gamma }_G{S}_{Gk}+{\gamma }_{Q1}{SQ}_{Q1k}+{\displaystyle \sum _{i=2}^n{\gamma }_{Qi}{\phi }_{Qi}{S}_{Qik}}$$

(6)

where S_d is the design value of actions; γ_G is the partial factor of G; S_Gk is the characteristic value of G; γ_Q1 is the partial factor of dominant variable loads; S_Q1k is the characteristic value of dominant variable loads; γ_Qi is the partial factor of non-dominant variable loads; φ_Qi is the combination coefficient of non-dominant variable load; S_Qik is the characteristic value of non-dominant variable loads; and n is the number of variable negative loads involved in the combination.

Solar greenhouses are typically subjected to a variety of loads simultaneously. Consequently, four load combinations taking the snow loads as the dominant loads were considered in this study and are showed in

Table 1. Values of partial factors and combination coefficients for three load combinations.

Load Cases	Partial Factors/Combination Coefficients for Various Loads
	G	sk	C	wk
1	1.0	1.2	-	-
2	1.0	1.2	1.2/0.7	-
3	1.0	1.2	0.0	1.2/0.6
4	1.0	-	-	1.0

. The partial factors and the combination coefficients for each load are determined according to the Chinese standard GB/T 51183-2016 [25]. The non-uniform distributions of the snow loads with µ_r,m values of 1 and 2 were defined as non-uniform pattern 1 and 2, respectively. Alphabets of ‘a’, ‘b’, and ‘c’ are added after the load combination numbers to denote that the snow is distributed in a uniform and non-uniform pattern 1 and 2, respectively.

2.4. Finite Element Model and Simulations

Numerical simulation has become a prevalent method to assess the structural safety of greenhouse frames. It is also demonstrated by Briassoulis et al. that the safety assessment of a greenhouse can be effectively carried out using a 2D analysis, despite greenhouses being a 3D structure [21]. In this study, a 2D finite element model of the frames was constructed for analysis using ANSYS Workbench 2021a software. Then, the tubes for constructing frames and the cable are modeled as BEAM188 elements and LINK180 elements, respectively. Fixed boundary conditions are assumed considering that the ends of the LASF were embedded in concrete foundations in practical scenarios. The connections were assumed to be rigid. In addition, the impact of geometric imperfections on the structural safety of the LASF was investigated. The initial geometric imperfections were achieved according to the first-order buckling mode, derived from the buckling analysis of the ideal model.

The frame is made of Q235 steel with a yield strength of 235 MPa, Young’s modulus of 206 GPa, Poisson’s ratio of 0.3, and a density of 7850 kg∙m⁻³. The design strength value of the frame tubes is 205 MPa. Both the Bilinear Kinematic (BKIN) model and the large deformation are considered to characterize the nonlinear properties of the steel material and the geometric non-linearity of the frame, respectively.

During the simulations, the loading process was methodically divided into three steps: applying G in the first step, applying w_k or C in the second step, and applying s_k in the third step. Furthermore, the non-uniformly distributed snow loads, acting on the front roof as line loads, were discretized into concentrated forces and subsequently applied to the frames later. Firstly, the projection of the front roof covered with snow is segmented into n sections with n + 1 nodes (Figure 5). Node 0 denotes the projection point of the ridge, and node n marks the projection of the point on the front roof where the tangent angle is equal to 60°. Subsequently, the concentrated force (Q_i, kN) at each node i (0 < i < n + 1) is calculated and exerted to the respective node.

When i = 1, Q₁ is calculated according to the following equation:

Q₁ = (3s₀∙d∙Δx/2)∙((μ_r,m − μ_b)∙Δx/D_r,s + μ_b)

(7)

where D_m represents the distance from the point projection of maximum non-uniform linear snow load occurrence on the front roof to node 0, taken as 6.119 m. D_r,s denotes the horizontal projection length of the front roof, which has a tangent angle less than 60°, taken as 9.157 m.

When i > 1 and i∙Δx ≤ D_m, Q_i is calculated as follows:

Q_i = s₀∙d∙Δx∙(i∙Δx∙(μ_r,m − μ_b)/D_m) + μ_b

(8)

When i > 1 and i∙Δx ≥ D_m, Q_i is calculated as follows:

Q_i = s₀∙d∙Δx∙μ_r,m∙(D_r,s − i∙Δx)/(D_r,s − D_m)

(9)

When i = n, Q_n is calculated as follows:

Q_n = s₀∙d∙(D_r,s − (n − 1/2)∙Δx)∙μ_r,m∙(D_r,s − i∙Δx)/(D_r,s − D_m)

(10)

Through simulation, the vertical displacement of the front roof mid-point and the maximum combined stresses (δ_c,max) under different snow loads were simulated to determine the ultimate bearing capacity (L_u) and the failure load (L_f) according to the methods introduced by Zhou and Ju [26]. The maximum total deformation (d_m) of the frames, the maximum bending moment(M), shear (F_s), and normal forces (F_a) at the snow loads of L_f were simulated as well. Then, the maximum bending (δ_b,max), shear (δ_s,max), and normal stresses (δ_n,max) were calculated according to the following equations:

δ_b = M·10³/W

(11)

δ_s,max = F_s/A

(12)

δ_n = F_a/A

(13)

where W is the section modulus (mm³); and A is the cross-sectional area of tubes (m²).

3. Results

3.1. Validation of the Finite Element Model

To validate the finite element method, the modeling approach used in this research was verified using the results of a single-span greenhouse covered with different snow depth obtained by Lee et al. (2020) [27]. As shown in Figure 6, the two results showed a good agreement and indicate that the proposed modeling method can be used for assessing the structural behavior of solar greenhouse frames under snow loads.

3.2. Influence of Uniform Snow Loads on Structural Safety of the LASF

Under load combination 1a in which the snow loads are uniformly distributed, the LASF is only subjected to G and s_k. The load–displacement curve, as illustrated in Figure 7, demonstrates the relationship between s_k and the vertical displacement of the front roof mid-point. It indicates that the LASF may encounter instability failure when reaching L_u of 0.70 kN∙m⁻². On the other hand, L_f, which was defined as the s_k when δ_c,max is equal to 205 MPa, the design strength value of the tube, is calculated to be 0.33 kN∙m⁻² and lower than L_u. These results indicate that the LASF is prone to strength failure under the aforementioned conditions before it experiences structural instability.

When s_k equals L_f under the aforementioned conditions, the deformation of the LASF is shown in Figure 8. The deformation has been magnified five times for easier observation. The front roof exhibits inward sagging, whereas the back roof and north wall columns show outward deflection. d_m of the LASF is 247 mm, occurring at the midpoint of the front roof. Furthermore, δ_b,max, δ_s,max, and δ_n,max are 207.5, 1.5, and −2.9 MPa (the negative value indicating compressive stresses), respectively (

Table 2. Maximum stresses of the LASF and RLASF under load combination 1.

Load Combination	LASF			RLASF
	δb,max (MPa)	δs,max (MPa)	δn,max (MPa)	δb,max (MPa)	δs,max (MPa)	δn,max (MPa)
Load combination 1a	207.5	1.5	−2.9	190.3	3.3	−21.4
Load combination 1b	207.8	1.6	−2.6	194.0	3.3	−16.2
Load combination 1c	208.0	1.6	−2.6	192.7	3.3	−16.7

). Notably, δ_b,max exceeds both δ_s,max and δ_n,max. This result indicates that the LASF is primarily subjected to a bending moment under the aforementioned conditions, thereby increasing its susceptibility to bending failure. Additionally, δ_b,max happened at the base of the north wall columns, indicating that this point is the critical section.

3.3. Influence of Non-Uniform Snow Loads on Structural Safety of the LASF

In practice, wind can cause snow drift, resulting in a non-uniform distribution of snow across the roof. To assess the impact of non-uniform snow loads on the deformation and stresses of the LASF, this study converts the non-uniform snow load into concentrated forces and applies them to the back roof and the front roof. Figure 6 illustrates the relationship between Δx and both d_m and δ_c,max, under a non-uniform snow load of 0.25 kN∙m⁻², which is distributed in pattern 1. It was observed that when Δx is less than 350 mm, the variations remain below 0.5% and are considered negligible (Figure 9). Consequently, a Δx of 350 mm has been adopted for calculations to balance computational efficiency with precision.

The load–displacement curves achieved under load combinations 1b and 1c, wherein the non-uniform snow loads were distributed in pattern 1 and 2, are similar as that achieved under load combination 1a (Figure 6). The results revealed that L_u of the LASF are 0.48 and 0.26 kN∙m⁻² for load combination 1b and 1c, respectively. Additionally, L_f of the LASF is 0.24 and 0.13 kN∙m⁻², respectively, under the above conditions. Both L_u and L_f are lower than those observed under load combination 1a. Based on those results, it can be inferred that the LASF is sensitive to non-uniform snow load and also prone to experiencing strength failure under the aforementioned conditions. Furthermore, the LASF exhibits a high susceptibility to damage under non-uniform snow loads distributed in pattern 2 compared to pattern 1.

The deformation characteristics of the LASF at the s_k of L_f under load combination 1b and 1c are consistent with those observed under load combination 1a. Thus, the deformation figures were not presented here. Under this condition, the values of d_m are 257 and 265 mm for the LASF under load combination 1b and 1c, respectively. The deformation characteristics of the LASF are consistent with those observed under load combination 1a. Notably, δ_b,max is more than 98% higher than both δ_s,max and δ_n,max. Both δ_c,max and δ_b,max occur at the base of the north wall column, identifying this point as the critical section of the LASF under non-uniform snow loads. These findings confirm that the LASF primarily experiences bending moments, and it is recommended to select tubes with greater bending resistance to mitigate bending stresses.

3.4. Influence of Snow Loads on Structural Safety of the RLASF

Increasing the cross-section area of tubes is a commonly used method to enhance the structural safety of solar greenhouse frames. However, this strategy often leads to higher costs and introduces complications in terms of processing and installation. Therefore, it is important to explore reinforcement measures in controlling the cost of the LASF and improving their structural safety.

Based on the deformation and stress characteristics of the LASF subjected to the snow loads, the reinforcement measures shown in Figure 10 were adopted. These measures include the addition of a diagonal brace to restrict the outward deflection of the back roof and the north wall columns, a diagonal strut to prevent the sag in the front roof, and a cable positioned beneath the diagonal strut to limit deformations of the lower front roof. Following these enhancements, this reinforced LASF is referred to as the RLASF.

The load–displacement curves of the RLASF under load combination 1a, 1b, and 1c are shown in Figure 11. The trends of these curves are similar to those of the original LASF. However, L_u for the RLASF under load combination 1a, 1b, and 1c is 1.96, 1.20, and 0.64 kN∙m⁻², respectively, which is 2.3, 1.6, and 1.6 times higher than those of the LASF. L_f for the RLASF under the above conditions is 0.93, 0.54, and 0.29 kN∙m⁻², which are also 2.3, 1.6, and 1.6 times greater than the LASF. The results indicated that the RLASF is also prone to undergo strength failure under the snow loads and is sensitive to non-uniform snow loads, but its structural safety under such conditions is significantly improved compared to the LASF.

Figure 12 illustrates the deformation of the RLASF under load combination 1a when s_k equals L_f. The deformation characteristics of the RLASF under non-uniform snow loads are consistent with those observed under load combination 1a. Thus, the deformation figures were not presented. Under these conditions, the deformation patterns reveal that the upper front roof exhibits inward sagging, while the lower front roof deforms outward sagging. Concurrently, the north wall experiences inward bending due to the tension in the tension rope. d_m is 54 mm under load combination 1a, and 61 and 59 mm under both load combination 1b and 1c. These values represent only 23.8%, 22.8%, and 22.4% of those observed in the LASF under load combination 1a, 1b, and 1c, respectively. Consequently, the RLASF demonstrates smaller deformations compared to the LASF. Moreover, δ_b,max of the RLASF under the aforementioned conditions exceeds both δ_s,max and δ_n,max by over 98% and over 89%, respectively (Table 1). Notably, both δ_c,max and δ_b,max occur at the junction where the cable connects with the north wall column. Thus, the RLASF primarily experiences bending moments under these conditions, and the critical section is changed to the junction where the cable connects with the north wall column.

3.5. Influences of Crop Loads on Structural Safety of the LASF and RLASF

The simulation findings indicate that the LASF equipped with a crop hanging system fails to satisfy the structural safety requirements of solar greenhouses under load combination 2, as δ_c,max surpasses the design strength value of the frame tubes. Conversely, L_u of the RLASF under load combinations 2a, 2b, and 2c, where in the snow loads are distributed in uniform, non-uniform pattern 1, and non-uniform pattern 2 are 18.1%, 21.4%, and 22.9% lower than those observed under load combinations 1a, 1b, and 1c, according to

Table 3. L_u and L_f of the RLASF under load combination 2.

Load Combination	Lu (kN∙m−2)	Lf (kN∙m−2)
Load combination 2a	1.63	0.68
Load combination 2b	1.03	0.39
Load combination 2c	0.54	0.21

. Similarly, L_f of the RLASF under the same load combinations experienced reductions of 37.0%, 38.1%, and 38.2%, respectively, compared to those under load combinations 1a, 1b, and 1c, according to Table 3. These results suggest that the presence of crop loads adversely affects the snow-load-bearing capacity of both the LASF and RLASF. Therefore, in the actual production scenarios, it is preferable to suspend crops using brackets fixed to the ground and avoid the high construction costs associated with them. In subsequent studies, crop load will be excluded from the subsequent structural analyses of the LASF and the RLASF.

3.6. Influences of Wind Loads on Structural Safety of the LASF and RLASF

Wind and snow can occur simultaneously, making special challenges for the structural safety of frames under snow. This study investigated the effects of wind load on the structural safety of the LASF under snow conditions, using the wind pressure in Beijing (0.37 kN·m⁻²) as a case study. When the wind blows from the north, L_u under load combinations 3a, 3b, and 3c, were found to be 23%, 20%, and 15% times greater, respectively, than those recorded under load combinations 1a, 1b, and 1c (

Table 4. L_u and L_f of the LASF and RLASF under load combination 3.

Load Combination	LASF				RLASF
	Northward Wind		Southward Wind		Northward Wind		Southward Wind
	Lu (kN∙m−2)	Lf (kN∙m−2)	Lu (kN∙m−2)	Lf (kN∙m−2)	Lu (kN∙m−2)	Lf (kN∙m−2)	Lu (kN∙m−2)	Lf (kN∙m−2)
Load combination 3a	0.86	0.60	0.72	0.33	2.03	1.06	1.85	1.28
Load combination 3b	0.58	0.39	0.50	0.24	1.26	0.58	1.39	0.74
Load combination 3c	0.30	0.20	0.27	0.13	0.67	0.31	0.74	0.40

). Similarly, L_f under load combinations 3a, 3b, and 3c, were 82%, 63%, and 54% times greater than those recorded under load combinations 1a, 1b, and 1c (Table 4). On the other hand, when the wind blows from the south, L_u and L_f under load combination 3 were slightly higher than those recorded under load combinations 1 (Table 4).

The deformation characteristics of the LASF at the s_k of L_f under load case combination 3 align with those observed under load case 1. However, when the LASF was subjected to load case combination 4 with northward winds, the back roof and north wall columns of the LASF deflect inward, whereas the front roof columns deflect outward (Figure 13). These configurations are the opposite to those exhibited under snow loads and help to mitigate the impacts of snow loads on the LASF. Furthermore, when the LASF was subjected to load case combination 4 with southward winds, the upper half of the front roof protrudes outward, which is also contrary to that observed under snow loads, thereby counteracting the impact of snow loads on the LASF as well (Figure 13).

Comparable findings were also observed for the RLASF. Both L_u and L_f for load combination 3 with northward winds rose slightly, compared to those for load combinations 1 (Table 4). However, when the wind blows from the south, L_u under load combination 3a decreased by 4.6% compared to that observed under load combination 1a, but were 15.8% and 15.6% higher than those observed under load combinations 1b and 1c, respectively (Table 4). Further research should be conducted to clarify this phenomenon. On the other hand, L_f for load combinations 3a, 3b, and 3c experienced increases of 37.6%, 18.5%, and 37.9%, respectively. On the other hand, the deformation characteristics of the RLASF at the s_k of L_f are consistent with those observed under load case 1. But the RLASF showed deformation patterns with the upper part of the roof deflecting outward and the lower part deflecting inward under load combination 4 (Figure 14). These patterns contrast with those observed under snow loads, and help mitigate the impact of snow loads on the RLASF.

In summary, applying wind load did not reduce L_f of the LASF and the RLASF. Due to the variability of wind speeds and the lack of synchronicity between peak wind and snow loads in practical situations, wind loads were omitted from consideration in subsequent studies. Therefore, only load combination 1 was utilized in subsequent studies.

3.7. Influences of Initial Geometric Imperfections on Structural Safety of the LASF and RLASF

Initial geometric imperfections in frames can originate from tube processing and installation, which could impact both L_u and L_f. According to the Chinese standard GB 50017-2017 [28], the maximum allowable geometric displacement defect for steel structures is limited to 1/400 of the span. In this research, the initial geometric imperfection was obtained from the first-order mode of the linear buckling analysis, featuring a 30 mm displacement of the ridge deflecting outward from the X–Z plane (equivalent to 1/400 of the LASF span) (Figure 15). Subsequently, these initial geometric imperfections were used to analyze the impacts of the geometric imperfections on the structural safety of the LASF and the RLASF under snow loads.

Figure 16 shows the deformation of the LASF and RLASF under load combination 1a. The deformation characteristics of the frames under load combination 1b and 1c were consistent with that under load combination 1a. Thus, those deformations are not presented. According to Figure 16, both the LASF and the RLASF with initial geometric imperfections showed that the deformation characteristics within the X–Z plane are similar to those observed in the flawless LASF and the RLASF, along with a deflection X–Z plane.

According to the simulations, both L_u and L_f of the LASF were not impacted by the initial geometric imperfections. However, for the RLASF, L_f of the RLASF with initial geometric imperfections the under load combinations were close to those observed on the flawless RLASF. It is observed that d_m of the RLASF with initial geometry imperfections were 107, 76, and 74 mm under load combinations 1a, 1b, and 1c, which were larger than those observed on the flawless RLASF. The larger deformation may alleviate the bending moments in the frame and slightly increase L_f. On the other hand, L_u of the RLASF under load combinations 1a, 1b, and 1c showed decreases of 38.9%, 12.4%, and 29.7% compared to the flawless RLASF (

Table 5. L_u and L_f of the RLASF with initial geometry imperfection under load combination 1.

Load Combination	Lu (kN∙m−2)	Lf (kN∙m−2)
Load combination 1a	1.16	0.94
Load combination 1b	0.87	0.57
Load combination 1c	0.46	0.30

). This may be because high s_k enhanced the deflection of the RLASF with initial geometry imperfection from the X–Z plane. Moreover, L_u values were consistently higher than initial values, suggesting that the RLASF is more prone to bending failure before it would fail due to structural safety. As a result, in structural design, it is acceptable to overlook the initial structural imperfections and focus solely on checking the design strength value of the frame tubes. Thus, these imperfections were not considered in subsequent studies.

3.8. Influences of Tube Cross Section on Structural Safety of the LASF and RLASF

To assess the influences of tube cross-section shapes on the structural safety of the LASF and the RLASF, a comparison was conducted among the tubes with different cross-section shapes but similar cross-section areas. The cross-section shapes included I-shaped and II-shaped square tubes, I-shaped and II-shaped rectangular tubes, and circular tubes (

Table 6. Dimensions and geometric parameters of tube cross-section shapes.

Cross-Section Shapes	Size (mm)	Cross-Sectional Area (mm2)	Section Modulus (mm3)
Round	Φ 60.3 × 3.5	675	8967
Square I	45 × 4	656	8246
Square II	40 × 5	700	7292
Rectangular I	60 × 40 × 3.5	651	10,353
Rectangular II	55 × 38 × 4	680	9720

Note: dimensions of circular, square, and rectangular tubes are represented with Φ outer diameter × thickness, side length × thickness, length × width × thickness.

). The calculations showed a direct positive relationship between the section modulus and both L_u and L_f for both the LASF and the RLASF (Figure 17 and Figure 18). Given that both the LASF and the RLASF primarily experience bending moments under the aforementioned conditions, it is advisable to select a cross-section shape with a larger section modulus, given the same cross-section area. This strategy helps mitigate bending stresses, thereby enhancing the snow bearing capacity and reducing construction costs.

3.9. Cross-Section Optimization of the LASF and RLASF

Based on the aforementioned analyses, it is crucial to investigate the steel usage of the frames to assess the economical feasibility of substituting the LASF with RLASF, in addition to considering a structural safety perspective. Using the snow load standards for Beijing as a reference, a solar greenhouse frame needs to withstand a basic snow pressure of 0.25 kN∙m⁻². Since both the LASF and the RLASF are sensitive to non-uniform snow loads, the analyses were conducted specifically for load combinations 1b and 1c. For the frames employing circular tubes, tubes of Φ 76.1 mm × 3.5 mm and Φ 88.9 mm × 4.0 mm should be used for the LASF to satisfy the aforementioned load requirements under load combinations 1b and 1c, respectively (

Table 7. Suitable cross-section dimensions for the LASF and RLASF in Beijing. (Unit: mm).

Frame	Components	Load Case 1b		Load Case 1c
		Round Tube	Rectangular Tube	Round Tube	Rectangular Tube
LASF	north wall, back- and front roof	Φ 76.1 × 3.5	60 × 40 × 4.0	Φ 88.9 × 4.0	70 × 50 × 4.0
RLASF	north wall, back- and front roof	Φ 48.3 × 3.5	45 × 30 × 2.5	Φ 60.3 × 3.5	55 × 38 × 2.5
	bracing column1 and 2	Φ 26.9 × 2.8	32 × 13 × 2	Φ 33.7 × 3.5	35 × 14 × 2

). By using the RLASF instead, the steel usage can be reduced by 16.1% for load combination 1b and 19.6% for load combination 1c.

It is worth noting that rectangular tubes offer the potential to further reduce steel usage. Considering the aforementioned conditions, the LASF constructed with 60 mm × 40 mm× 4.0 mm rectangular tubes can meet the load requirements for load combination 1b, while 70 mm × 50 mm× 4.0 mm rectangular tubes are sufficient for load combination 1c. These changes lead to a 16.0% reduction in steel usage for load combination 1b and a 16.1% reduction for load combination 1c compared to the LASF employing circular tubes. Moreover, if the RLASF were constructed using rectangular tubes, the steel usage could be reduced by 45.3% for load combination 1b and 47.5% for load combination 1c, in comparison to the LASF constructed with circular tubes.

4. Discussion

Multiple studies have highlighted the critical role of boundary conditions at frame ends in the structural safety of greenhouse frames. For frames installed directly on the ground, Japanese code provisions state that the boundary condition of the frame can be classified as a fixed boundary condition if the frame end is positioned 500 mm below ground level [29]. However, ground conditions may change, potentially leading to hinged boundary conditions. This alteration typically results in increased frame deformations and stresses [27]. On the other hand, in practical scenarios, the LASF are commonly fixed in concrete foundations, which could restrict the rotations and displacements of the LASF ends. Thus, fixed boundary conditions are used for the structural analyses of the LASF and RLASF.

In this study, the load–displacement curves of the LASF and RLASF under snow loads vary in the form of a logarithmic function. Furthermore, L_u under non-uniform snow loads is higher than that under uniform loads. These results align with the research conducted by Wang et al. [14] and Ding et al. [30], indicating that the LASF and RLASF have a high possibility of stable failure under snow loads, especially under non-uniform snow loads. It is also found that when the greenhouse roof is covered with an insulation blanket, the uneven distribution of snow on the roof due to wind could be more pronounced, leading to a further reduction in L_u. Moreover, L_f was calculated as well in this study. L_f under non-uniform snow loads are also lower than that under uniform load. These findings support the conclusion that non-uniform snow loads are more detrimental to the structural safety of the greenhouse frame compared to uniform snow loads. It is also worth noting that L_f is lower than L_u. This result further indicated that the frame is more likely to experience strength failure before undergoing stable failure under snow loads. Therefore, it is necessary to assess the structural safety of single-tube frames under non-uniform snow load conditions.

The fruit vegetable crops grown in solar greenhouses are typically suspended using wires supported by frames, resulting in concentrated forces exerted on the frames due to the weight of the crops. Although the crop loads were considered in the research conducted by Liu et al. [7], Ding et al. [30], and Wang et al. [31], the influence of crop loads on L_u and L_u of solar greenhouse frames were not explored. In this study, it was found that the crop loads decreased both L_u and L_f under snow. Thus, the presence of crop loads impacts the structural safety of the LASF and the RLASF. It is not recommended to exert crop loads on either the LASF or RLASF.

It is pertinent to acknowledge that wind loads may occur concurrently with snow loads. Maraveas et al. [32] incorporated both wind and snow loads in their methodology for the ultimate limit state design of steel structures. Briassoulis et al. [21] observed that the wind loads were a contributing factor to the structural failure of a greenhouse with a vaulted roof under snow conditions. Regarding the greenhouse frames, Yang et al. [22] demonstrated that wind loads decreased L_u. Nevertheless, the conclusions of this study are inconsistent with the aforementioned research. This inconsistency may be attributed to differences in frame shape, which could affect the deformations and stresses of the frames. According to the simulation, both the LASF and the RLASF experience strength failure under snow loads, and the application of wind loads could counteract part of the snow load’s effects on frames. Consequently, it is reasonable to assess the structural safety of the LASF and RLASF under snow conditions without incorporating wind loads.

The initial geometric imperfections did not significantly affect both L_u and L_f of the LASF in this study. This observation is consistent with the findings from Wang et al.’s research conducted on the ‘Ω’ shaped skeleton [14]. This could be ascribed to the low Lu and L_f of the LASF, where the out-of-plane deflection was rather small and had little impact on structural behavior of the LASF under load combination 1. Consequently, the LASF demonstrates limited sensitivity towards the initial geometric defect. On the other hand, the deflection of the RLASF with an initial geometry imperfection from the X–Z plane was large under high s_k. As a result, the structural stability of the RLASF could be decreased, leading to lower L_u values under load combination 1, compared to those observed on the flawless RLASF. Moreover, L_u were consistently higher than L_f, suggesting that the RLASF is more prone to bending failure before it would fail due to structural stability. As a result, in structural design, it is acceptable to overlook the initial structural imperfections and focus solely on checking the design strength value of the frame tubes. Thus, these imperfections were not considered in subsequent studies. It also should be noted that this research uses a 2D model for simulation analysis. In future research, 3D frame models could be employed to examine the impact of out-of-plane deformations on the structural safety of the LASF and the RLASF.

The results of this study showed that the frames primarily experience bending moments under snow loads, leading to bending failure. Notably, L_u and L_f were positively correlated with the sectional modulus of the tubes. Wang et al. [31] found the same results on an 8.0 m span solar greenhouse frame. Thus, it is recommended to select the cross-section shape of tubes with the larger section modulus.

The simulation of this study also indicates that the RLASF has higher L_u and L_f than those of the LASF. Then, the RLASF made with DN30 tubes can meet the structural safety requirement of most cities of northern China. Furthermore, the tubes of the RLASF also could be optimized for a specific snow load and achieve low steel usage. Thus, the RLASF helps reducing concerns about the structural safety of the LASF and advancing the evolution of Chinese solar greenhouses. Furthermore, considering that the high temperature and high-humidity environment inside greenhouses in practice would accelerate the corrosion of steel tubes and finally decrease L_f. Thus, it is necessary to conduct research on frames made of aluminum or composite materials to avoid the above problems.

5. Conclusions

The study analyzed the stress and deformation responses of the 12.0 m-span LASF for snow loads, alongside evaluating the impact of crop loads, wind loads, initial geometric imperfections, and variations in tube cross-section shapes on these responses. It also compared the structural safety and the reduction in steel usage of the RLASF compared to the LASF.

The simulations were conducted using ANSYS workbench software. The steel tubes were modeled as beam188 elements and cables as link180 elements. The frame constraints and the connections were assumed to be fixed supports and rigid, respectively. The loads were determined according to the Chinese standard (GB51183-2016). It is observed that both the LASF and RLASF primarily experience bending moments and are susceptible to strength failure under snow load conditions. Moreover, both frames are sensitive to non-uniform snow loads. The application of crop loads could further deteriorate the structural safety of both the LASF and the RLASF. Conversely, the wind loads and the initial geometric imperfections did not decrease L_f of the LASF and the RLASF. Thus, it is reasonable to assess the structural safety of the LASF and RLASF without considering the effects of the wind loads and the initial geometric imperfection. Furthermore, the positive correlations were observed between the cross-section modulus of the tubes with similar cross-sectional area and both L_u and L_f for the LASF and the RLASF.

The RLASF was constructed by adding bracing columns and cables onto the LASF. L_f of the RLASF under load combinations 1a, 1b, and 1c are 1.81, 1.25, and 1.23 times greater than those observed on the LASF. Furthermore, taking an LASF built in Beijing as an example, the steel usage of solar greenhouses could be reduced by 16.1% using the RLASF instead of the LASF. This reduction could be further increased to 45.3% by substituting rectangular tubes with a larger cross-section modulus for the RLASF. Hence, it is possible to employ the RLASF for alleviating the structural safety concerns of the LASF for Chinese solar greenhouses with low cost.