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

: 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.


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 Agronomy 2024, 14, 1122 3 of 19

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.

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.

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 (sk) 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: where sk is the standard value of snow load, kN•m −2 ; and µr is the distribution coefficient of snow loads on roof.ct 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 .µr was taken according to Figure 2. It is noted that in case of non-uniformly distributed snow, the maximum value of µr (µr,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].

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: 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 s 0 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 (wk) are calculated using the following equation: where wk is the standard value of the wind loads, kN•m −2 ; µs is the wind pressure coefficient; and µz 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 µs 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; µz 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): The wind loads (w k ) are calculated using the following equation: 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.w 0 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.The wind loads (wk) are calculated using the following equation: where wk is the standard value of the wind loads, kN•m −2 ; µs is the wind pressure coefficient; and µz 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 µs 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; µz 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): 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): 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.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: where D is the span of the LASF, 12.0 m.

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] : where Sd is the design value of actions; γG is the partial factor of G; SGk is the characteristic value of G; γQ1 is the partial factor of dominant variable loads; SQ1k is the characteristic value of dominant variable loads; γQi is the partial factor of non-dominant variable loads; φQi is the combination coefficient of non-dominant variable load; SQik 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.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 µr,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 nonuniform pattern 1 and 2, respectively.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: where D is the span of the LASF, 12.0 m.

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]: 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.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.

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.

Finite Element Model and Simulations
Numerical simulation has become a prevalent method to assess the structural sa greenhouse frames.It is also demonstrated by Briassoulis et al. that the safety assessm a greenhouse can be effectively carried out using a 2D analysis, despite greenhouse a 3D structure [21].In this study, a 2D finite element model of the frames was cons for analysis using ANSYS Workbench 2021a software.Then, the tubes for const frames and the cable are modeled as BEAM188 elements and LINK180 elements, tively.Fixed boundary conditions are assumed considering that the ends of the LAS embedded in concrete foundations in practical scenarios.The connections were as to be rigid.In addition, the impact of geometric imperfections on the structural sa the LASF was investigated.The initial geometric imperfections were achieved acc to the first-order buckling mode, derived from the buckling analysis of the ideal m The frame is made of Q235 steel with a yield strength of 235 MPa, Young's mod 206 GPa, Poisson's ratio of 0.3, and a density of 7850 kg•m −3 .The design strength v the frame tubes is 205 MPa.Both the Bilinear Kinematic (BKIN) model and the la formation are considered to characterize the nonlinear properties of the steel mater the geometric non-linearity of the frame, respectively.
During the simulations, the loading process was methodically divided into thre applying G in the first step, applying wk or C in the second step, and applying sk in th step.Furthermore, the non-uniformly distributed snow loads, acting on the front roo loads, were discretized into concentrated forces and subsequently applied to the later.Firstly, the projection of the front roof covered with snow is segmented into tions with n + 1 nodes (Figure 5).Node 0 denotes the projection point of the ridg node n marks the projection of the point on the front roof where the tangent angle i to 60°.Subsequently, the concentrated force (Qi, kN) at each node i (0 < i < n + 1) is lated and exerted to the respective node.When i = 1, Q1 is calculated according to the following equation: where Dm represents the distance from the point projection of maximum non-unifo ear snow load occurrence on the front roof to node 0, taken as 6.119 m.Dr,s deno horizontal projection length of the front roof, which has a tangent angle less th taken as 9.157 m.
When i > 1 and i•Δx ≤ Dm, Qi is calculated as follows: When i = n, Qn is calculated as follows: When i = 1, Q 1 is calculated according to the following equation: 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•∆x ≤ D m , Q i is calculated as follows: When i > 1 and i•∆x ≥ D m , Q i is calculated as follows: When i = n, Q n is calculated as follows: 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: where W is the section modulus (mm 3 ); and A is the cross-sectional area of tubes (m 2 ).

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.Through simulation, the vertical displacement of the front roof mid-point and the maximum combined stresses (δc,max) under different snow loads were simulated to determine the ultimate bearing capacity (Lu) and the failure load (Lf) according to the methods introduced by Zhou and Ju [26].The maximum total deformation (dm) of the frames, the maximum bending moment(M), shear (Fs), and normal forces (Fa) at the snow loads of Lf were simulated as well.Then, the maximum bending (δb,max), shear (δs,max), and normal stresses (δn,max) were calculated according to the following equations: where W is the section modulus (mm 3 ); and A is the cross-sectional area of tubes (m 2 ).

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.

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 sk.The load-displacement curve, as illustrated in Figure 7, demonstrates the relationship between sk and the vertical displacement of the front roof mid-point.It indicates that the LASF may encounter instability failure when reaching Lu of 0.70 kN•m −2 .On the other hand, Lf, which was defined as the sk when δc,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 Lu.These results indicate that the LASF is prone to strength failure under the aforementioned conditions before it experiences structural instability.

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 sk equals Lf 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.dm of the LASF is 247 mm, occurring at the midpoint of the front roof.Furthermore, δb,max, δs,max, and δn,max are 207.5, 1.5, and −2.9 MPa (the negative value indicating compressive stresses), respectively (Table 2).Notably, δb,max exceeds both δs,max and δn,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, δb,max happened at the base of the north wall columns, indicating that this point is the critical section.

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 dm and δc,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.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).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.When sk equals Lf 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.dm of the LASF is 247 mm, occurring at the midpoint of the front roof.Furthermore, δb,max, δs,max, and δn,max are 207.5, 1.5, and −2.9 MPa (the negative value indicating compressive stresses), respectively (Table 2).Notably, δb,max exceeds both δs,max and δn,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, δb,max happened at the base of the north wall columns, indicating that this point is the critical section.

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 dm and δc,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.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 ∆x 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 ∆x is less than 350 mm, the variations remain below 0.5% and are considered negligible (Figure 9).Consequently, a ∆x 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, w the non-uniform snow loads were distributed in pattern 1 and 2, are similar as that ac under load combination 1a (Figure 6).The results revealed that Lu of the LASF are 0. 0.26 kN•m −2 for load combination 1b and 1c, respectively.Additionally, Lf of the LASF and 0.13 kN•m −2 , respectively, under the above conditions.Both Lu and Lf are lower than observed under load combination 1a.Based on those results, it can be inferred that the is sensitive to non-uniform snow load and also prone to experiencing strength failure the aforementioned conditions.Furthermore, the LASF exhibits a high susceptibility to age under non-uniform snow loads distributed in pattern 2 compared to pattern 1.
The deformation characteristics of the LASF at the sk of Lf under load combinat and 1c are consistent with those observed under load combination 1a.Thus, the deform figures were not presented here.Under this condition, the values of dm are 257 and 26 for the LASF under load combination 1b and 1c, respectively.The deformation chara tics of the LASF are consistent with those observed under load combination 1a.No δb,max is more than 98% higher than both δs,max and δn,max.Both δc,max and δb,max occur at th of the north wall column, identifying this point as the critical section of the LASF non-uniform snow loads.These findings confirm that the LASF primarily exper bending moments, and it is recommended to select tubes with greater bending resi to mitigate bending stresses.

Influence of Snow Loads on Structural Safety of the RLASF
Increasing the cross-section area of tubes is a commonly used method to enhan structural safety of solar greenhouse frames.However, this strategy often leads to costs and introduces complications in terms of processing and installation.Therefor important to explore reinforcement measures in controlling the cost of the LASF an proving their structural safety.
Based on the deformation and stress characteristics of the LASF subjected to the loads, the reinforcement measures shown in Figure 10 were adopted.These measu clude the addition of a diagonal brace to restrict the outward deflection of the back ro the north wall columns, a diagonal strut to prevent the sag in the front roof, and a positioned beneath the diagonal strut to limit deformations of the lower front roo lowing these enhancements, this reinforced LASF is referred to as the RLASF.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.

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.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.How ever, Lu 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.Lf 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 time 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 structura 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 equals Lf.The deformation characteristics of the RLASF under non-uniform snow loads are consistent with those observed under load combination 1a.Thus, the deformation figure were not presented.Under these conditions, the deformation patterns reveal that the uppe 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.dm is 54 mm under load combination 1a, and 61 and 59 mm under both load combina tion 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, δb,max of the RLASF under the aforementioned conditions exceeds both δs,max and δn,max by over 98% and ove 89%, respectively (Table 1).Notably, both δc,max and δb,max occur at the junction where the cable connects with the north wall column.Thus, the RLASF primarily experiences bend ing moments under these conditions, and the critical section is changed to the junction  The load-displacement curves of the RLASF under load combination 1a, shown in Figure 11.The trends of these curves are similar to those of the origin ever, Lu for the RLASF under load combination 1a, 1b, and 1c is 1.96, 1.20, an respectively, which is 2.3, 1.6, and 1.6 times higher than those of the LASF.Lf under the above conditions is 0.93, 0.54, and 0.29 kN•m −2 , which are also 2.3, 1.6 greater than the LASF.The results indicated that the RLASF is also prone to un failure under the snow loads and is sensitive to non-uniform snow loads, bu safety under such conditions is significantly improved compared to the LA Figure 12 illustrates the deformation of the RLASF under load combinat equals Lf.The deformation characteristics of the RLASF under non-uniform s consistent with those observed under load combination 1a.Thus, the deform were not presented.Under these conditions, the deformation patterns reveal front roof exhibits inward sagging, while the lower front roof deforms out Concurrently, the north wall experiences inward bending due to the tension rope.dm is 54 mm under load combination 1a, and 61 and 59 mm under both tion 1b and 1c.These values represent only 23.8%, 22.8%, and 22.4% of those o LASF under load combination 1a, 1b, and 1c, respectively.Consequentl demonstrates smaller deformations compared to the LASF.Moreover, δb,max under the aforementioned conditions exceeds both δs,max and δn,max by over 89%, respectively (Table 1).Notably, both δc,max and δb,max occur at the junct cable connects with the north wall column.Thus, the RLASF primarily expe ing moments under these conditions, and the critical section is changed t where the cable connects with the north wall column.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.

Influences of Crop Loads on Structural Safety of the LASF and RLASF
The simulation findings indicate that the LASF equipped with a crop hanging fails to satisfy the structural safety requirements of solar greenhouses under load co tion 2, as δc,max surpasses the design strength value of the frame tubes.Conversely, L RLASF under load combinations 2a, 2b, and 2c, where in the snow loads are distrib uniform, non-uniform pattern 1, and non-uniform pattern 2 are 18.1%, 21.4%, and lower than those observed under load combinations 1a, 1b, and 1c, according to Similarly, Lf of the RLASF under the same load combinations experienced reduc 37.0%, 38.1%, and 38.2%, respectively, compared to those under load combinations and 1c, according to Table 3.These results suggest that the presence of crop loads ad affects the snow-load-bearing capacity of both the LASF and RLASF.Therefore, in th production scenarios, it is preferable to suspend crops using brackets fixed to the and avoid the high construction costs associated with them.In subsequent studie load will be excluded from the subsequent structural analyses of the LASF and the

Influences of Wind Loads on Structural Safety of the LASF and RLASF
Wind and snow can occur simultaneously, making special challenges for the st safety of frames under snow.This study investigated the effects of wind load on the st safety of the LASF under snow conditions, using the wind pressure in Beijing (0.37 kN a case study.When the wind blows from the north, Lu under load combinations 3a, 3b were found to be 23%, 20%, and 15% times greater, respectively, than those recorded load combinations 1a, 1b, and 1c (Table 4).Similarly, Lf under load combinations 3a, 3c, were 82%, 63%, and 54% times greater than those recorded under load combi 1a, 1b, and 1c (Table 4).On the other hand, when the wind blows from the south, L under load combination 3 were slightly higher than those recorded under load co tions 1 (Table 4).

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.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.

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).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).The deformation characteristics of the LASF at the sk of Lf 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 Lu and Lf 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, Lu 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, Lf 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 sk of Lf 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 Lf 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.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.The deformation characteristics of the LASF at the sk of Lf 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 Lu and Lf 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, Lu 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, Lf 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 sk of Lf 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 Lf 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.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.

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.

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 Lu and Lf.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 Figure16, 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 Lu and Lf of the LASF were not impacted by the initial geometric imperfections.However, for the RLASF, Lf of the RLASF with initial geometric imperfections the under load combinations were close to those observed on the flawless RLASF.It is observed that dm 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 Lf.On the other hand, Lu of the RLASF under load 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.

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 Lu and Lf.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 Figure16, 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 Lu and Lf of the LASF were not impacted by the initial geometric imperfections.However, for the RLASF, Lf of the RLASF with initial geometric imperfections the under load combinations were close to those observed on the flawless RLASF.It is observed that dm 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 Lf.On the other hand, Lu of the RLASF under load 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).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.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 IIshaped square tubes, I-shaped and II-shaped rectangular tubes, and circular tubes (Table 6).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 (Figures 17 and 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.Note: dimensions of circular, square, and rectangular tubes are represented with Φ outer diameter × thickness, side length × thickness, length × width × thickness.
Agronomy 2024, 14, x FOR PEER REVIEW 14 of combinations 1a, 1b, and 1c showed decreases of 38.9%, 12.4%, and 29.7% compared to flawless RLASF (Table 5).This may be because high sk enhanced the deflection of the RLA with initial geometry imperfection from the X-Z plane.Moreover, Lu values were consisten higher than initial values, suggesting that the RLASF is more prone to bending failure bef it would fail due to structural safety.As a result, in structural design, it is acceptable to ov look the initial structural imperfections and focus solely on checking the design streng value of the frame tubes.Thus, these imperfections were not considered in subsequ studies.To assess the influences of tube cross-section shapes on the structural safety of LASF and the RLASF, a comparison was conducted among the tubes with different cro section shapes but similar cross-section areas.The cross-section shapes included I-shap and II-shaped square tubes, I-shaped and II-shaped rectangular tubes, and circular tub (Table 6).The calculations showed a direct positive relationship between the section mod lus and both Lu and Lf for both the LASF and the RLASF (Figures 17 and 18).Given that bo the LASF and the RLASF primarily experience bending moments under the aforemention conditions, it is advisable to select a cross-section shape with a larger section modulus, giv the same cross-section area.This strategy helps mitigate bending stresses, thereby enha ing the snow bearing capacity and reducing construction costs.

Cross-Section Optimization of the LASF and RLASF
Based on the aforementioned analyses, it is crucial to investigate the steel usage of frames to assess the economical feasibility of substituting the LASF with RLASF, in addit to considering a structural safety perspective.Using the snow load standards for Beijing a reference, a solar greenhouse frame needs to withstand a basic snow pressure of 0 kN•m −2 .Since both the LASF and the RLASF are sensitive to non-uniform snow loads, analyses were conducted specifically for load combinations 1b and 1c.For the frames e ploying circular tubes, tubes of Φ 76.1 mm × 3.5 mm and Φ 88.9 mm × 4.0 mm should used for the LASF to satisfy the aforementioned load requirements under load combinatio 1b and 1c, respectively (Table 7).By using the RLASF instead, the steel usage can be reduc 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 usa Considering the aforementioned conditions, the LASF constructed with 60 mm × 40 m 4.0 mm rectangular tubes can meet the load requirements for load combination 1b, while mm × 50 mm× 4.0 mm rectangular tubes are sufficient for load combination 1c.These chan lead to a 16.0% reduction in steel usage for load combination 1b and a 16.1% reduction for lo combination 1c compared to the LASF employing circular tubes.Moreover, if the RLASF w constructed using rectangular tubes, the steel usage could be reduced by 45.3% for load co bination 1b and 47.5% for load combination 1c, in comparison to the LASF constructed w circular tubes.

Discussion
Multiple studies have highlighted the critical role of boundary conditions at frame en in the structural safety of greenhouse frames.For frames installed directly on the grou Japanese code provisions state that the boundary condition of the frame can be classified a fixed boundary condition if the frame end is positioned 500 mm below ground level [2 However, ground conditions may change, potentially leading to hinged boundary con tions.This alteration typically results in increased frame deformations and stresses [27].

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).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.

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.

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.

Figure 2 .
Figure 2. Distribution coefficient of snow loads on solar greenhouse roof.(a) Uniform snow load; (b) non-uniform snow load.µr,m is 1.0 with film only covering the front roof, and 2.0 with both film and insulation blanket.µr,b is the µr on the back roof.

Figure 3 .
Figure 3. Wind pressure coefficient of wind loads on solar greenhouse roof.(a) Wind pressure coefficient under northward wind.(b) Wind pressure coefficient under southward wind.

Figure 2 .
Figure 2. Distribution coefficient of snow loads on solar greenhouse roof.(a) Uniform snow load; (b) non-uniform snow load.µ r,m is 1.0 with film only covering the front roof, and 2.0 with both film and insulation blanket.µ r,b is the µ r on the back roof.

Figure 2 .
Figure 2. Distribution coefficient of snow loads on solar greenhouse roof.(a) Uniform snow load; (b) non-uniform snow load.µr,m is 1.0 with film only covering the front roof, and 2.0 with both film and insulation blanket.µr,b is the µr on the back roof.

Figure 3 .
Figure 3. Wind pressure coefficient of wind loads on solar greenhouse roof.(a) Wind pressure coefficient under northward wind.(b) Wind pressure coefficient under southward wind.

Figure 3 .
Figure 3. Wind pressure coefficient of wind loads on solar greenhouse roof.(a) Wind pressure coefficient under northward wind.(b) Wind pressure coefficient under southward wind.

Figure 5 .
Figure 5. Division of nodes in the horizontal projection of the front roof covered with snow are nodes of the front roof.Δx is the distance between nodes i and i + 1, and Δxn is the d between nodes n − 1 and n.

Figure 5 .
Figure 5. Division of nodes in the horizontal projection of the front roof covered with snow.i and n are nodes of the front roof.∆x is the distance between nodes i and i + 1, and ∆x n is the distance between nodes n − 1 and n.

Figure 8 .
Figure 8. Deformation of the LASF under load combination 1a wherein sk equals Lf (magnified by five times).Black line represents the undeformed LASF.

Figure 8 .
Figure 8. Deformation of the LASF under load combination 1a wherein sk equals Lf (magnified by five times).Black line represents the undeformed LASF.

Figure 8 .
Figure 8. Deformation of the LASF under load combination 1a wherein s k equals L f (magnified by five times).Black line represents the undeformed LASF.

Figure 9 .
Figure 9. Effects of node spacing (Δx) on dm and δc,max of the LASF under snow loads of 0.25 distributed in non-uniform pattern 1.

Figure 9 .
Figure 9. Effects of node spacing (∆x) on d m and δ c,max of the LASF under snow loads of 0.25 kN•m −2 distributed in non-uniform pattern 1.

Figure 15 .
Figure 15.Schematic diagram of initial geometric imperfection (axonometric view).(a) LASF.(b) RLASF.Black line represents the undeformed frames and blue line represents the first-order mode of the linear buckling analysis of the frames.

Figure 16 .
Figure 16.Deformation of the LASF and RLASF with initial geometric imperfection under load combination 1a (magnified by five times).(a) LASF.(b) RLASF.Black line represents the undeformed frames.

Figure 15 .
Figure 15.Schematic diagram of initial geometric imperfection (axonometric view).(a) LASF.(b) RLASF.Black line represents the undeformed frames and blue line represents the first-order mode of the linear buckling analysis of the frames.

Figure 15 .
Figure 15.Schematic diagram of initial geometric imperfection (axonometric view).(a) LASF.(b) RLASF.Black line represents the undeformed frames and blue line represents the first-order mode of the linear buckling analysis of the frames.

Figure 16 .
Figure 16.Deformation of the LASF and RLASF with initial geometric imperfection under load combination 1a (magnified by five times).(a) LASF.(b) RLASF.Black line represents the undeformed frames.

Figure 16 .
Figure 16.Deformation of the LASF and RLASF with initial geometric imperfection under load combination 1a (magnified by five times).(a) LASF.(b) RLASF.Black line represents the undeformed frames.

Figure 17 .
Figure 17.Ultimate bearing capacities of arches under load combination 1 affected by the sect modules of tube cross-section shapes.(a) LASF.(b) RLASF.

Figure 17 .
Figure 17.Ultimate bearing capacities of arches under load combination 1 affected by the section modules of tube cross-section shapes.(a) LASF.(b) RLASF.

Figure 18 .
Figure 18.Failure snow loads of frames under load combination 1 affected by the section modu of tube cross-section shapes.(a) LASF.(b) RLASF.

Figure 18 .
Figure 18.Failure snow loads of frames under load combination 1 affected by the section modules of tube cross-section shapes.(a) LASF.(b) RLASF.

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

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

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

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

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

Table 3 .
Lu and Lf of the RLASF under load combination 2.

Table 4 .
Lu and Lf of the LASF and RLASF under load combination 3.
Figure 12.Deformation of the RLASF under load combination 1a (magnified by five times).Black line represents the undeformed RLASF.

Table 3 .
L u and L f of the RLASF under load combination 2.

Table 4 .
L u and L f of the LASF and RLASF under load combination 3. )

Table 5 .
L u and L f of the RLASF with initial geometry imperfection under load combination 1. Influences of Tube Cross Section on Structural Safety of the LASF and RLASF

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

Table 5 .
Lu and Lf of the RLASF with initial geometry imperfection under load combination 1.

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

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

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