Dynamic Response Analyses of Plastic Greenhouse Structure considering Fluctuating Wind Load

Wind load is one of the main factors of plastic greenhouse collapse. To solve the dynamic response problem of greenhouses under wind load and determine the dangerous section of a skeleton structure, the investigated lumpmethod is presented for the dynamic response analysis of a plastic greenhouse, considering pulsating wind on the basis of Timoshenko beam theory. First, the investigated lump is designed according to the Timoshenko beammicrobody concept. On the basis of Timoshenko beam theory, the governing equations of the skeleton structure of the greenhouse are derived, and the realization process of the algorithm is also provided. Second, the accuracy and effectiveness of the proposed numerical method are verified by an example in which the bending wave of a variable cross section beam with free ends propagates. Finally, the dynamic response of the steel skeletons of plastic greenhouses is analyzed under the effect of the simulation wind speed, and the spatial distribution of the maximum node displacement and the section maximum stress of the steel skeleton are obtained. Computational results show that the displacement peak is near the top of the plastic greenhouse. )e most dangerous section of the top chord in the steel skeleton is near the leeward bottom, which has a maximum stress of 219.4MPa, and the most dangerous section of the bottom chord is near the 1 m height on the leeward side of the plastic greenhouse, which has a maximum stress of 248.5MPa. Bending stress is the main factor of the rapid increase of stress at the bottom of the skeleton. )e maximum node displacement and cross-sectional stress caused by fluctuating wind loads are higher than those caused by average wind loads. )e fluctuating wind load should be considered in the wind-induced response analyses of plastic greenhouses.


Introduction
1.1. Literature Review. Greenhouses are an important facility for modern agriculture and have made outstanding contributions to the development of modern agriculture. Greenhouses are also a special form of agricultural construction. In addition to providing animals and plants with the basic space for growth and production and an environment with suitable temperature and humidity, greenhouses must also bear many kinds of loads caused by abnormal operating conditions and extreme natural disasters (e.g., blizzard, gale, and hail). Abnormal weather conditions, such as strong winds and heavy snow, can deform plastic greenhouse structures severely and cause collapses [1,2]. e safety of greenhouse structures and their bearing capacity has gradually warranted attention with the continuous development of greenhouses. e steel skeleton of greenhouses and plastic greenhouses is a light-weight structure, with features, such as light dead weight, large span, and thin and long rods, that make the greenhouse structure sensitive to wind load. e influence of wind load on the skeleton structure is an important consideration in greenhouse structure design [3]. Performing time-history analysis under fluctuating wind load can lead to the accurate grasp of the stress characteristics of greenhouse and plastic greenhouse structures and provide a scientific reference for the structural analysis and wind resistance design of greenhouses and plastic greenhouses. Experimental and numerical simulation methods are used to study the wind load and bearing capacity of greenhouse structures. e data obtained from the test is important for studying the bearing capacity of greenhouse structures [4] and the wind pressure coefficient [2,[5][6][7]. However, due to the high cost of testing and the long cycle, testing cannot be performed for each structure, and a structure with an optimized wind resistance performance is difficult to obtain. e numerical simulation of greenhouse structures has been developed rapidly with the development of computers and numerical calculation methods. e wind pressure coefficients and their distribution on the surfaces of a single-span plastic greenhouse and a solar greenhouse have been investigated [8].
e critical wind speeds at which damage occurs on the surfaces of single-span plastic and solar greenhouses have been determined. e wind direction, spacing, and span of a greenhouse can influence the wind pressure distribution on the surface of greenhouses [6,9,10]. Maximum positive or negative pressure may cause the collapse of the greenhouse frame. By simulating the wind speed of the airflow around and through a greenhouse structure, the influence of strong lateral wind on the leakage rate and internal pressure coefficient of the greenhouse can be studied [11,12]. e mechanical properties of various greenhouse frame structures have been studied under the action of wind load, snow load, and different load combinations [13,14]. In a study, the safety wind speeds for singlespan greenhouses were calculated and compared with the actual wind speeds and snow depths over a period of 8 years in different regions to analyze the structural safety of singlespan greenhouses [15]. e calculation method for the critical load of a circular arch with out-plane bifurcation buckling was proposed [16] for the theoretical analysis for the wind resistance design of arched greenhouses. e stress of the greenhouse covering material and the failure behavior of the film can affect the stress performance of different greenhouse frame structures [17,18]. Unfavorable load combination, such as wind and snow, is one of the reasons for the collapse of arched multispan greenhouse structures [19]. An obvious stress concentration exists at the connection between the truss arch and longitudinal beam of plastic greenhouses. e equivalent stress and displacement of some components exceed the allowable range, and the safety of greenhouse structures is mainly controlled by their strength and rigidity [20]. e maximum bending moment of single-span arched greenhouses appears at the bottom of the windward side wall [21]. e connection modelling condition of the skeleton structure of greenhouses and the addition of the support modelling condition can affect the vibration modes of multispan greenhouses [22]. e fatigue life of greenhouses can be predicted accurately by using a fatigue stress model, considering the dead weight and the wind speed [23]. Complex stress state caused by combination of dead load and wind load was modeled via CFD to examine its effect on the variation of stress distribution in the greenhouse structure [24]. An improved model was established to predict the permeability evolution by the fractal dimension, and strategies for overburden aquifer protection were proposed [25]. A damage-based model of coupled thermal-flowing-mechanical effects was adopted to illustrate the mechanical principle of gas fracturing and simulate the coupled process during the gas fracturing [26].

Motivation.
When studying the bearing capacity and stress characteristics of a greenhouse structure, wind is usually applied to the greenhouse structure in the form of a quasistatic load. Wind speed can be composed of the mean and fluctuating winds. Instantaneous wind force is the most disadvantageous to light-weight greenhouse structures, and a few seconds of instantaneous wind can destroy an entire skeleton structure.
Using a 10 min time-distance is unsafe for greenhouse structures [27]. However, few studies on the pulsating wind effect on the mechanical properties of greenhouses have been conducted. e time-history analysis of greenhouse structures under wind loads can lead to an accurate grasp of the stress characteristics of the structures. For example, Hur and Kwon [23] did not perform time-history analysis on wind speed when they studied the safety of greenhouse structures under dynamic wind loads. erefore, the effect of fluctuating wind load on the dynamic response of greenhouse structures should be studied further through timehistory analysis.
In this study, from the perspective of the wave propagation of greenhouse structures and on the basis of Timoshenko beams theory, considering shear deformation and rotational inertia, a calculation method for the dynamic response time-history analysis of greenhouse structures under wind load is proposed, and the dynamic response of the steel skeleton structure of a greenhouse is analyzed. e time-history analysis of greenhouses can lead to the accurate grasp of the stress characteristics of skeleton structures under dynamic action, providing meaningful results for the dynamic analysis and wind-resistant design of greenhouse structures.

Principle of Calculation
According to the equilibrium conditions of a microunit, the dynamic equilibrium equation of the Timoshenko beam [28], considering shear deformation and rotational inertia, is obtained as follows: ρI where ρ is the material density, kg/m 3 ; A is the cross-sectional area, m 2 ; u is the transverse displacement of the neutral axis of the beam, m; t is the time, s; V is the shear force of any cross section, N; x is the any cross-sectional position, m; q(x) is the transverse force per unit length on the beam, N/m; I is the second moment of the area, m 4 ; θ is the section angle caused by bending deformation, rad; and M the bending moment of any cross section, N·m. In Timoshenko's theory, the total rotation angle of the neutral axis of the beam can be composed of two parts: the section angle caused by bending deformation and the rotation angle of the neutral axis caused by shear deformation: where (zu/zx) is the total rotation angle of the beam's neutral axis, rad; and c is the rotation angle of the neutral axis caused by shear deformation, rad. e shear force and strain relationship is e bending moment and curvature relationship is where E is the elastic modulus, N/mm 2 ; G is the shear modulus, N/mm 2 ; and k is the section shear correction coefficient.

Dynamic Response Analysis Method for Steel Skeleton Structure of Greenhouse
e dynamic response of the steel skeleton structure is obtained to simulate the propagation process of the bending wave in the steel skeleton structure of a greenhouse. First, the composition of the investigated lump of the greenhouse skeleton structure is provided based on Timoshenko beam microunits.
e dynamic equilibrium and rotation equations of the investigated lump are established based on the Timoshenko beam's dynamic equation. On the basis of the shear force and strain, bending moment and curvature, and axial force and strain relationships of beam theory, the relationship between the internal force and displacement in a discrete segment is obtained. en, by applying the equations provided above and computing alternately in the time domain, the numerical method is presented for the dynamic response analyses of a greenhouse structure. Finally, the effectiveness of the investigated lump method is verified. e strategic structure of the dynamic response analysis method for greenhouse structures is shown in Figure 1. e steel skeleton structure of the greenhouse is divided into discrete spaces by nodes ( Figure 2). Discrete segment l exists between nodes i and j. e investigated lumps are composed of half of the spatial discrete segments associated with the nodes, and the endpoint number of discrete segments can be used to represent the number of investigated lumps. Investigated lump i is composed of half of the top chord and the web member connected to node i, that is, the area surrounded by the dotted ellipse on the left. Investigated lump k is composed of half of the bottom chord and the web member connected to node k, that is, the area surrounded by the dotted ellipse on the right. Figure 3 shows the spatial position and central internal force diagram of discrete segment l of investigated lump i. Let n be the number of components associated with investigated lump i, and the force of the investigated lump comes from the wind pressure and the internal force of the central section of the discrete members of the structure. In Figure 3, x and z are the global coordinate axes, and x' and z' are the local coordinate system axes. φ l is the angle between the x ′ axis of the local coordinate system and the x axis of the global coordinate system of discrete segment l, rad. N l and V l , respectively, represent the axial and shear forces acting on the central section of discrete segment l, N. M l is the bending moment of the central section of discrete segment l, N·m, and N i and V i , respectively, represent the axial and shear forces of the i-end section of discrete section l, N. M i is the bending moment of the i-end section of discrete section l, N·m. N j and V j , respectively, denote the axial and shear forces of the j-end section of discrete section l, N. M j is the bending moment of the j-end section of discrete section l, N·m. L l is the length of discrete segment l, m. q(x ′ )is the external load of discrete segment l, N/m.
According to equation (1), the discrete form of the dynamic equilibrium equation for the translational motion of the investigated lump can be established using Figure 3.
at is, where m i � n l�1 (m l /2) is the mass of investigated lump i, and m l is the mass of discrete section l, kg.
€ u i and € w i are the accelerations of investigated lump i along the x-axis and z-axis directions, respectively, m/s 2 . F ql is the resultant force of transverse distributed load q(x ′ ) on discrete segment l, that is, According to equation (1), the discrete form of the dynamic equilibrium equation for the rotational motion of the investigated lump can be established using Figure 3. at is, where J i � n l�1 (1/3) · (m l /2) · (L l /2) 2 is the moment of inertia of investigated lump i, kg·m 2 . € θ i is the angular acceleration of investigated lump i around the endpoint axis, rad/s 2 . M l is the bending moment of the central section of Advances in Civil Engineering discrete segment l, N·m. M ql is the moment of distributed load q(x ′ ) on discrete segment l to point i; that is,

Relationship between Internal Force and Displacement
in Discrete Segments of Skeleton Structure. By the spatial discretization of equations (4) and (5), the shear force (V l ) and bending moment (M l ) of the central section of discrete segment l are obtained as follows: where E l is the elastic modulus of discrete segment l, N/ mm 2 . I l is the second moment of the area, m 4 . G l is the shear modulus, N/mm 2 . A l is the cross-sectional area of discrete segment l, m 2 . k l is the section shear correction coefficient of discrete segment l.w i ′ and w j ′ are the respective transverse displacements of the i-and j-ends of discrete section l, m.    Advances in Civil Engineering and θ j ′ are the respective rotation angles of the i-and j-ends of discrete section l, m. According to the relationship between the axial force and strain of beam theory, the axial force of the central section of discrete segment l can be obtained as follows: where u i ′ and u j ′ are the respective axial displacements of the i-and j-ends of discrete section l, m. e total normal stress of the cross section is the sum of the bending and axial normal stresses. From formulas (10) and (11), the total normal stress of the cross section can be obtained as where W y is the section modulus in bending, mm 3 .

Displacement Relation between Local and Global Coordinate
System. e dynamic equilibrium equation of investigated lump i is established in the global coordinates, and the constitutive equation of internal force at the central section is established in the local coordinate system. erefore, the displacement relationship between the local and global coordinate systems must be established. e transformation equation from the global to the local coordinate system is e transformation equation from the local to the global coordinate system is 3.2. Algorithm Implementation. Wind force is derived from the wind speed calculation, and the time-history of wind force is applied to the skeleton structure. e dynamic equilibrium equation of the investigated lump and the relationship between the central section internal force and the displacement in the discrete segment of the skeleton structure are alternately used, and the displacement relationships between the local and global coordinate systems are combined. Computing alternately in the time domain realizes the dynamic response analysis of the greenhouse steel skeleton structure. e pseudocode of this method is presented in Algorithm 1.

Method Validation.
e flexural wave propagation of a variable cross section beam with two free ends is calculated by using the investigated lump method. e calculation results are compared with those of the finite difference method in reference [29] to verify the correctness and effectiveness of the proposed numerical algorithm method. e elastic modulus (E) of the beam is 209 GPa, the density (ρ) is 8000 kg/m 3 , Poisson's ratio (]) is 0.3, the shear modulus (G) is 80.4 GPa, and the shear cross section correction coefficient (k') is 0.886. e length of the beam (X) is 5 m. e beam section changes at the middle point, the radius of the left half is 0.1 m, and the radius of the right half is 0.08 m. e number of discrete segments is 200 in the length direction for the uniform beam, and the calculation time step of this paper is Δt � 1.25 μs. e length from the left end of the beam is x, and the bending moment at the section is M.
After normalization, time t is (1/X) ����� (E/ρ)t, the length from the left end of the beam is x � (x/X), the lateral displacement is w � (w/X), and the bending moment is M � (MX/EI).
e inclination moment of the left end of the beam is

Analysis of Dynamic Response of Plastic
Greenhouse Steel Skeleton Structure

Calculation Model and Parameters of Plastic Greenhouse
Steel Skeleton. e research object is a plastic greenhouse steel skeleton structure, which adopts a single flat plane frame model ( Figure 5). e diameter of the steel pipe of the top chord is 33 mm, and the wall thickness of the steel pipe is 3.3 mm. e diameter of the bottom chord reinforcement is 12 mm, and that of the web member is 8 mm.
e strength grade of the reinforcement is HPB 300. e skeleton structure has 55 nodes, that is, 55 investigated lumps. e nodes are numbered from left to right, with those at the top chord set to 1-28, as shown in nodes 5, 15, and 24. e example is represented as a solid point. e nodes at the bottom chord are numbered 29-55, as shown in nodes 33, 42, and 52. Each segment between nodes is regarded as a discrete segment, with a total of 107 discrete segments. e discrete segments are numbered from left to right, with the segments at the top chord set to ①- 27 , and those at the bottom chord to 28 -53 . e web members are numbered 54 -107. e calculation time step is 0.05 ms.

Advances in Civil Engineering
Input: Time Step Δt, Total time steps T; Total number of discrete segments n, Discrete length L l , l ∈ n; Total number of investigated lump R, Mass of investigated lump m i , Moment of inertia of investigated lump J i , i ∈ R; Load point R f , R f ∈ R; fluctuating wind speed v i ,i ∈ R f ; Constraint point R c . Output: e displacement of investigated lumps u i and w i , rotation angle θ i , maximum stress σ max , shear forces at ends of discrete segment V i and V j , bending moments at ends of discrete segment M i and M j .
External force in the x direction at loading point n l�1 F ql · sin φ l in equation (6), external force in the y direction − n l�1 F ql · cos φ l in equation (7), Moment n l�1 M ql in equation (8). (9) end do (10) Define constraints: Calculate u i ′ , u j ′ and w i ′ , w j ′ at both ends of segment l by equation (13). Calculate N l by equation (11), Calculate V l by equation (9), Calculate M l by equation (10), Calculate σ max by equation (12); Calculate V i and V j by equation (11), (13) end do (14) do l � 1, n (15) Calculate u i , u j and w i , w j at both ends of segment l by equation (14), Calculate the force and moment of the investigated lump n l�1 (N l cos φ l − V l sin φ l ) in equation (6) n l�1 (N l sin φ l + V l cos φ l ) in equation (7), n l�1 (M l − V l (L l /2)) in equation Calculate acceleration of the investigated lump € u i in equation (6), € w i in equation (7) and angular acceleration € θ i in equation (8). (19) if i ∈ R f then (20) superimposing Step 8, Calculate acceleration of the investigated lump € u i in equation (6), € w i in equation (7), angular acceleration € θ i in equation (8). (21) end if (22) Calculate velocity _  Advances in Civil Engineering

Wind Load Time-History Simulation.
e wind load must be used as the input load parameter for the dynamic response analysis of the plastic greenhouse steel frame of the plane frame model. In this study, the input load parameters needed for calculation are obtained through the numerical simulation of wind speed.
According to reference [30], the input wind load parameters are obtained by calculation. e average wind speed at 10 m above the ground is 29.67 m/s. According to reference [3], the terrain roughness index of the plastic greenhouse structure is 0.16, and the terrain roughness coefficient is 0.03.
Using the Davenport spectrum and harmonic superposition method [31], the fluctuating wind speed of each space point on the surface of the plastic greenhouse shown in Figure 5 is simulated. e fluctuating wind is assumed to be a stationary Gaussian random process with a zero mean. Fluctuating wind speed v j (t) can be expressed as follows: where S is the sampling points of the fluctuating wind frequency, and j is the number of simulation points. H jk (f l ) represents the elements in the lower triangular matrix. Δf � ((f u − f d )/S) is the frequency increment, and f u and f d are the upper and lower limits of the fluctuating wind interception frequency, respectively. β kl represents random numbers that are uniformly distributed between 0 and 2π. e wind pressure (W(z, t)) at height z in the downwind direction can be obtained through the wind speed: where ρ is the air density, v(z)is the mean wind speed at height z, v(x, y, z, t) is the fluctuating wind speed at height z, (1/2)ρv(z) 2 is the mean wind pressure at height z, and (1/2)ρ[2v(z)v(x, y, z, t) + v(x, y, z, t) 2 ] is the fluctuating wind pressure at height z. Figure 6 shows the time-history curve of the fluctuating wind speed at nodes 4 and 10. Figure 7 compares the simulated wind speed power spectrum of nodes 4 and 10 with the Davenport fluctuating wind speed power spectrum. e figure indicates that the change trend of the simulated power spectrum is consistent with the target power spectrum. Figure 8 shows the spatial coherence comparison at different points. e figure shows that the wind speed at two adjacent points (e.g., points 3 and 4) has the strongest coherence. e maximum value of the normalized crosscorrelation function (R max ) is 0.9681, and that of points 3 and 10 is R max � 0.8669. e coherence of wind speeds at two adjacent points is strong. e coherence of wind speed gradually weakens with the increase of the distance between the two points, which conforms to the coherence law of the wind field. e fluctuating wind speed's time-history at different points of the plastic greenhouse can be verified by Figures 7  and 8. Figure 9 shows the time-history curve of the fluctuating wind pressure at points 4 and 10 on the surface of the plastic greenhouse structure obtained by simulation. e input wind pressure time-history of the plastic greenhouse skeleton structure can be obtained by the superposition of the fluctuating and mean wind pressures' time-history [30]. e left side of the plastic greenhouse is the windward side, as shown in Figure 5.

Spatial Distribution of Maximum Displacement of
Node. Figure 10 shows the spatial distribution curve of the maximum displacement of each node of the plastic greenhouse skeleton structure with a span of 10 m. "Mean-+ Fluctuating" in the figure represents the calculated results considering both the mean and fluctuating wind loads. "Fluctuating" represents the calculated results considering only the effect of the fluctuating wind load. "Mean" represents the calculated results considering only the effect of the mean wind load. Figure 10 shows that the displacement of the top chord increases significantly between nodes 10 and 19, the maximum displacement at node 15 is 20.5 mm, and secondary peaks of displacement appear at nodes 5 and 24. e displacement of the bottom chord increases significantly between nodes 38 and 46. e maximum displacement at node 42 is 20.8 mm, and secondary peaks appear at nodes 33 and 52. Figure 5 shows that the displacement value of the top chord of plastic greenhouses above the 2.5 m height increases significantly, the displacement peak appears at the top of the plastic greenhouse, and the second peak is at the 1.5 m height on the windward and leeward sides.      Advances in Civil Engineering position of the maximum displacement can be seen directly in the figure. Table 1 shows the relative peak value of node displacement in the node displacement curve.  Advances in Civil Engineering times that considering only the mean wind load. e fluctuating wind load alone is 2.1-2.5 times that of the mean wind load alone. e comparison results in Figure 10 and Table 1 indicate that the maximum displacement of each node is the largest under "Mean + Fluctuating," followed by that under "Fluctuating," and that under "Mean" is the smallest. e effect of fluctuating wind on node displacement is significant. Figure 12 shows the spatial distribution comparison curve of the maximum value of the total normal stress in the cross section of each discrete segment of the plastic greenhouse skeleton structure. Total normal stress is the sum of the absolute values of the axial and bending normal stresses.

Spatial Distribution of Maximum Total Stress in Cross-Section.
e "Mean + Fluctuating," "Fluctuating," and "Mean" in the figure are the same as those in Figure 10. Figure 12 indicates that, for both the top and bottom chords, the maximum stress is generally the largest under "Mean + Fluctuating," followed by that under "Fluctuating," and that under "Mean" is the smallest, and the variation trend of the stress at each discrete section is basically the same under the three loads.  The original structure After the deformation   Table 2 shows the relative peak stress of the top and bottom chords under different wind loads. e results in Figure 12 and Table 2 indicate that the effect of fluctuating wind on stress is basically consistent with the displacement. Figure 13 shows the comparison curve of the stress components of each discrete section of the top chord under the action of both mean and fluctuating winds. e main component of the total stress is axial stress, but bending stress is the main component at sections 1, 2, 26, and 27 of the discrete section, that is, near the two bottom ends of the skeleton structure. For the plastic greenhouse structure, axial force is the main internal force, and the influence of bending moment on the internal force of both ends of the skeleton structure cannot be ignored.

Conclusion
In this paper, the investigated lump method for the dynamic response analysis of plastic greenhouse structures under the wind load is presented. e accuracy and effectiveness of the proposed numerical method are verified by an example in which the bending wave of a variable cross section beam with free ends propagates. e physical meaning of the investigated lump is clear, and the classical dynamics equations are unnecessary. ree conditions of numerical wind speed simulation (mean and fluctuating winds, fluctuating wind, and mean wind) are used as input load parameters for the dynamic response time-history analysis of a plastic greenhouse steel skeleton structure. e spatial distribution of the maximum value of node displacement and section stress in the dynamic response time-history record of the plastic greenhouse steel skeleton structure is obtained. e displacement of each node and the stress of each section are generally the largest under "Mean + Fluctuating," followed by those under "Fluctuating," and those under "Mean" are the smallest. e maximum values of node displacement and section stress under fluctuating wind load are approximately 2.5 and 2 times those of the mean wind load, indicating that the effect of fluctuating wind on the dynamic response of a plastic greenhouse steel skeleton structure is more important than that of mean wind. e relative peak values of node displacement appear near the top of the skeleton, at the 1.5 m height on the windward side, and at the 1.4 m and at the 1.2 m height on the leeward side. e relative peak values of cross section stress appear near the top of the skeleton, at the 1.3-1.6 m height on the windward side, and at the 1.3-1.4 m height on the leeward side. e most dangerous section of the top chord in the steel skeleton is near the leeward bottom, and the most dangerous section of the bottom chord is near the 1 m height on the leeward side of the plastic greenhouse. Axial stress is the main internal force of the greenhouse steel skeleton structure, and the section near the left end of the structure is the most dangerous. Bending stress is the main reason for the rapid increase of the stress at this location.
Data Availability e analysis result data used to support the findings of this study are included within the article. e calculation data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest.