# Effect of Geometric Nonlinearity on Membrane Roof Stability in Air Flow

Authors: Weiju Song; Jun Xu; Xiaowei Wang; Changjiang Liu
Journal: Shock and Vibration
 (2020)

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## Abstract

Membrane materials are widely used in construction engineering with small mass and high flexibility, which presents strong geometric nonlinearity in vibration. In this paper, an improved multiscale perturbation method is used to solve the aerostatics stability of membrane roofs on closed and open structures by quantifying the effect of geometric nonlinearity on the single-mode aeroelastic instability wind velocity. Results show that the critical wind velocities of two models are smaller when the geometrical nonlinearity of the membrane material is neglected. In addition, under normal wind load, the influence of geometrical nonlinearity of the membrane on the aerodynamic stability of the roof can be neglected. However, under strong wind load, when the roof deformation reaches 3% of the span, the influence of geometric nonlinearity should be considered and the influence increases with the decrease of transverse and downwind span of the membrane roof. The results obtained in this paper have an important theoretical reference value for the design membrane structures.


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## Body

## 1. Introduction

Fabric membrane is a widely used membrane material in construction engineering. It has the characteristics of high tensile strength and good flexibility. Fabric membranes are mainly composed of substrates and coatings. The substrates are usually braided by orthogonal fibers, which results in the orthotropic properties of the membranes. That is to say, the elastic modulus and Poisson’s ratio in the two orthogonal directions are different. The building, which is made up of membrane material covered on the structural skeleton or tensioned as a whole, has beautiful appearance, good transparency, environmental protection, and energy saving [1, 2]. Therefore, it is widely used in large-scale stadiums, exhibition venues, and other public buildings. Because of the small mass and flexibility, it is easy for vibration under external disturbance, and the stiffness of the membrane material is small, which results in the large vibration deformation of the membrane structure under wind load, showing strong geometric nonlinearity [3, 4]. Many research results show that the single-mode aeroelastic instability can easily occur in membrane structures when the pretension of membrane materials is small [5, 6].In the mathematical analysis of aeroelastic instability of flexible membrane structures, Yang and Liu [7, 8] established the wind-induced dynamic coupling equation of hyperbolic parabolic membrane roof with small sag by using elastic shallow shell theory and ideal fluid potential flow theory in 2006 and determined the critical wind velocity of aeroelastic instability according to Routh–Hurwitz stability criterion. The influence of geometric nonlinearity of membranes was not considered when establishing the mathematical model. In 2011, Xu et al. [9, 10] studied the nonlinear aerodynamic stability of orthotropic-tensioned membrane structures in a rectangular plane and hyperbolic paraboloid, respectively. The critical wind velocities of single-mode instability of the two membrane structures were determined by assuming the solution of the vibration equation. In 2017, Liu et al. [11] studied the aerodynamic stability of closed-tensioned membrane structures by the Galerkin method. The geometric nonlinearity of membrane vibration is weakened, and the critical wind velocity of the instability is obtained by using the weak nonlinearity solution method.In order to investigate the influence of the geometric nonlinearity on the aeroelastic stability of membrane materials, the nonlinear wind-induced dynamic equations of membrane roofs are established based on Von Kamen’s large deflection theory and Darumbel’s principle, taking the flat rectangular orthotropic tensioned membrane roofs with fixed supports on the four sides of open and closed structures as analytical models and considering the effects of geometric nonlinearity and air damping of membrane materials. An improved multiscale method which is suitable for strong geometric nonlinearity is used to solve the vibration equation. The critical wind velocity of instability obtained is compared with the results without considering geometric nonlinearity. The effect of geometric nonlinearity on the wind velocity of single-mode aeroelastic instability of membrane material is obtained quantitatively.


## 2. Analytical Deduction of Single-Mode Instability of Orthotropic Membrane Roofs

### 2.1. Establishment of Basic Equations

Let the length and width of the orthotropic rectangular flexible membrane with four sides fixed bea and b, respectively; the pretension along the length direction is N0x and the width direction is N0y. The wind blows parallel to the roof and toward the membrane surface, which makes the membrane surface vibrate. For flexible membranes, the research results show that the shear stress has little influence on the vibration process of the membranes and can be considered as zero [9–13]. Assuming that the planar membrane is in the XOY plane when in equilibrium and the pretension in X direction is Nx and in the Y direction is Ny, when the membrane is disturbed by external forces to the XOY plane, it will deform and then produce transverse vibration perpendicular to the membrane surface under the action of tension. Select a microunit on the vibrating membrane surface, as shown in Figure 1.Figure 1
Vibration microunits of membrane.Select a unit part dxdy on the membrane surface. When the microunits are deformed, the edges of the microunits are subjected to the tension of the adjacent facets. InX direction, we can regard the surface element as composed of countless chord elements with length dx and width of one unit. The tension acting on the chord element is consistent with its tangent direction. The tension Nx is at an angle α with X coordinate axis. Therefore, the vertical component of the tension acting on the chord element at one end of X is Nxsinα. Because α is small, sinα≈tanα. Let w be the vertical displacement of a point on the membrane away from the equilibrium position. Therefore,(1)Nxsinα=Nxtanα=Nx∂w∂xx.The vertical force acting on the dy edge is Nx∂w/∂xxdy, and the vertical force at the x edge should be Nx∂w/∂xx+dxdy. Thus, the resultant force in Z direction on the x and x + dx sides of the panel is as follows:(2)Nx∂w∂xx+dxdy−Nx∂w∂xxdy=Nx∂2w∂x2dxdy.Similarly, the resultant force ofZ direction component of the tension acting on Y direction can be obtained as follows:(3)Ny∂w∂yy+dydx−Ny∂w∂yydx=Ny∂2w∂y2dxdy.So the total force inZ direction acting on the whole panel is(4)Fz=Nx∂2w∂x2dxdy+Ny∂2w∂y2dxdy+qx,ydxdy,where Nx is the tension in X direction (longitude), Ny is the tension in Y direction (latitude), w is the deflection of the membrane, and qx,y is the external load acting on the unit area of the projection surface of the membrane. According to the force balance, we can obtain that(5)Nx∂2w∂x2dxdy+Ny∂2w∂y2dxdy+qx,ydxdy=0,(6)qx,y+Nx∂2w∂x2+Ny∂2w∂y2=0.The generalized external loads of the flexible membrane roof under wind load include the wind load acting on the membrane surface, structural damping force, and inertial force [11]. If the aerodynamic term is defined as p, then the generalized external load per unit area qx,y is(7)qx,y=px,y,t−2ρc∂wx,y,t∂t−ρ∂2wx,y,t∂t2.For membrane material, the stiffness of the membrane surface comes from the initial pretension of membrane material, so the initial pretension should be added in equation (6). Finally, the differential equations of motion of flexible membranes are obtained as follows:(8)px,y,t−2ρc∂wx,y,t∂t−ρ∂2wx,y,t∂t2+N0x+Nxt∂2w∂x2+N0y+Nyt∂2w∂y2=0.Introducing the stress functionφx,y, Nx=h∂2φ/∂y2 and Ny=h∂2φ/∂x2. Then, equation (6) can be transformed as(9)N0x+h∂2φ∂y2∂2w∂x2+N0y+h∂2φ∂x2∂2w∂y2+p−2ρc∂w∂t−ρ∂2w∂t2=0.After deformation, the membrane surface strain is composed of linear and nonlinear parts. The linear strain is caused by in-plane displacementu and v, and the nonlinear strain is caused by deflection w. After ignoring shear stress, the total strain is as follows:(10)εx=∂u∂x+12∂w∂x2εy=∂v∂y+12∂w∂y2,where εx is the strain in X direction and εy is the strain in Y direction.By eliminatingu and v in equation (10), the continuous deformation conditions satisfying the strain and deflection of the membrane surface can be obtained:(11)∂2εx∂y2+∂2εy∂x2=∂2w∂x∂y2−∂2w∂x2∂2w∂y2.The membrane is orthotropic, and the direction of the fiber is the main direction of elasticity so that it is consistent with the direction of coordinate systemX and Y. Assuming that the direction of fiber is the same as the direction of coordinate system X and Y, Young’s modulus of elasticity in X and Y directions is E1 and E2, respectively. The longitudinal Poisson’s ratio and the latitudinal Poisson’s ratio is μ1 and μ2, respectively. The relationship between elastic modulus and Poisson’s ratio is as follows:(12)μ1E1=μ2E2.The stress-strain relationship is as follows:(13)σxσy=E11−μ1μ2μ1E21−μ1μ2μ2E11−μ1μ2E21−μ1μ2εxεy,where σx and σy is the normal stresses in X and Y directions, respectively. h is the thickness of membrane.LettingNx=h⋅σx and Ny=h⋅σy and substituting equation (13) into equation (11), the compatibility equation is obtained as(14)1E1h∂2Nx∂y2−μ2E2h∂2Ny∂y2−μ1E1h∂2Nx∂x2+1E2h∂2Ny∂x2=∂2w∂x∂y2−∂2w∂x2∂2w∂y2.By substituting the stress function into equation (14), then (14) can be transformed as(15)1E1∂4φ∂y4−μ2E2∂4φ∂x2∂y2−μ1E1∂4φ∂x2∂y2+1E2∂4φ∂x4=∂2w∂x∂y2−∂2w∂x2∂2w∂y2.


### 2.2. Improved Multiscale Solutions of Governing Equations

The initial surface function of the rectangular planar membranez0 (x, y) = 0. Therefore, the surface equation of the flexible membrane under wind load is as follows:(16)zx,y,t=wx,y,t.According to the Bubnov–Galerkin method, assume the solution of the governing equation is [9–12](17)wx,y,t=∑i=1nTitWix,y,φx,y,t=∑i=1nUitϕix,y,where Wix,y is the mode function, ϕix,y is the unknown stress function, and Tit and Uit are time-dependent functions.Because the membrane is fixed on four sides, the vertical deflection at the boundary of the membrane is zero, and the vibration mode function satisfying the conditions is assumed to be [11, 12](18)Wx,y=sinmπxasinnπyb,where m and n are positive integer.Substituting equation (18) into equation (17), the following equation can be obtained:(19)wx,y,t=Ttsinmπxasinnπyb.Substituting equation (19) into equation (15) yields(20)1E1∂4φ∂y4+1E2∂4φ∂x4=m2n2π42a2b2Tt2cos2mπxa+cos2nπyb.Assume that the solution of the stress function in equation (20) is [12](21)φx,y,t=Tt2ϕx,y,ϕx,y=αcos2mπxa+βcos2nπyb.Substituting equation (21) into equation (20) yields(22)α=E2a2n232b2m2,β=E1b2m232a2n2.


#### 2.2.1. Solution of Flexible Membrane Roof on Closed Structure

Flexible membrane covers the top of the closed structure as the roof and the rigid wall around the structure as the vertical bearing member. Because its stiffness is far greater than the membrane’s stiffness, it is assumed that the stiffness of the vertical component is infinite in the process of theoretical derivation in this paper. The membrane roof on closed structure is shown in Figure2.Figure 2
The membrane roof on closed structure.For the membrane roof on closed structure, the aerodynamic force acting on the unit area of the membrane projection surface is expressed as [10](23)p=−ρ02π−V∬RaV∂w/∂x+∂w/∂tx=ξy=ηx−ξx−ξ2+y−η23dξdη+∬RaV∂2w/∂x∂t+∂2w/∂t2x=ξy=ηx−ξ2+y−η2dξdη.Substituting equation (23) into equation (9) yields(24)h∂2φ∂y2+N0x∂2w∂x2+h∂2φ∂x2+N0y∂2w∂y2−2ρc∂w∂t−ρ∂2w∂t2−ρ0π∬Ra1r∂2w∂t2x=ξy=ηdξdη−ρ0V2π∬Ra1r∂2w∂x∂tx=ξy=ηdξdη+ρ0V22π∬Ra1r3∂w∂xx=ξy=ηx−ξdξdη+ρ0V2π∬Ra1r3∂w∂tx=ξy=ηx−ξdξdη=0,where r=x−ξ2+y−η2 and the integral region Ra∈0≤ξ≤a,0≤η≤b.Substituting equations (19), (21), and (22) into equation (24) yields(25)γ1=∬Ra1rWx=ξy=ηdξdη=∬Ra1rsinmπξasinnπηbdξdη,γ2=∬Ra1r∂W∂xx=ξy=ηdξdη=mπa∬Ra1rcosmπξasinnπηbdξdη,γ3=∬Ra1r3∂W∂xx=ξy=ηx−ξdξdη=mπa∬Ra1r3x−ξcosmπξasinnπηbdξdη,γ4=∬Ra1r3Wx=ξy=ηx−ξdξdη=∬Ra1r3x−ξsinmπξasinnπηbdξdη.By using the Bubnov–Galerkin method, (25) can be integrated as(26)∬SρW+ρ0πγ1d2Ttdt2+ρ0V2πγ2-γ4+2ρcWdTtdt-N0x∂2W∂x2+N0y∂2W∂y2+ρ0V22πγ3Tt-h∂2Φ∂y2∂2W∂x2+∂2Φ∂x2∂2W∂y2T3tWx,ydxdy=0,where S∈0≤x≤a,0≤y≤b.Simplifying equation (26) yields(27)Ad2Ttdt2+BdTtdt−CTt−DT3t=0,where(28)A=∬SρW+ρ0πγ1Wdxdy=ρ∬Ssinmπxasinnπyb2dxdy+ρ0π∬Ra∬Ra1rsinmπξasinnπηbdξdηsinmπxasinnπybdxdy=ρab4+ρ0πα1,α1=∬S∬Ra1rsinmπξasinnπηbdξdηsinmπxasinnπybdxdy,B=ρ0V2π∬Sγ2−γ4Wdxdy+2ρc∬SW2dxdy=ρ0mV2a∬S∬Ra1rcosmπξasinnπηbdξdηsinmπxasinnπybdxdy−ρ0V2π∬S∬Ra1r3x−ξsinmπξasinnπηbdξdηsinmπxasinnπybdxdy+2ρc∬Ssinmπxasinnπyb2dxdy=ρ0mV2aα2−ρ0V2πα4+ρcab2,α2=∬S∬Ra1rcosmπξasinnπηbdξdηsinmπxasinnπybdxdy,α4=∬S∬Ra1r3x−ξsinmπξasinnπηbdξdηsinmπxasinnπybdxdy,C=∬SN0x∂2W∂x2+N0y∂2W∂y2+ρ0V22πγ3Wdxdy=∬SN0x∂2W∂x2Wdxdy+∬RaN0y∂2W∂y2Wdxdy+ρ0V22π∬Raγ3Wdxdy=−m2π2bN0x4a−n2π2aN0y4b+ρ0mV22aα3,α3=∬S∬Ra1r3x−ξcosmπξasinnπηbdξdηsinmπxasinnπybdxdy,D=∬Sh∂2Φ∂y2∂2W∂x2+∂2Φ∂x2∂2W∂y2Wdxdy=−hm2n2π4α+β2ab.It can be obtained from numerical calculation thatA≤0 occurs only when b/a≪1, and this will not happen in practical engineering. Equation (27) is a strongly nonlinear vibration equation. For different equations of strongly nonlinear vibration, many scholars have studied the corresponding solutions [13–17]. In this paper, we propose the improved multiscale method based on the principle of modified Lindestedt–Poincaré method to solve equation (27). Letting u=B/εA, ω02=−C/A, ε=−D/A, then equation (27) can be transformed as(29)T¨+ω02T+εμT˙+T3=0.Lettingω be the vibration frequencies of the membrane material and expanding ω2 to the power series of ε near ω02 we obtain the following:(30)ω2=ω02+εω1+ε2ω2+⋯.The transformation parameters are introduced as follows:(31)α=εω1ω02+εω1,(32)ε=ω02αω11−α,ω02+εω1=ω021−α.Expandω2 to the power series of ε as follows:(33)ω2=ω02+εω11+1ω02+εω1ε2ω2+ε3ω3+⋯=ω021−α1+δ2α2+δ3α3+⋯,(34)ω=ω01+12α+38+δ22α2+⋯.The form of the perturbation solution of equation (29) can be(35)Tt,α=T0t0,t1+αT1t0,t1+α2T2t0,t1+⋯,where t0=t and t1=αt.The differential operators are obtained as follows:(36)ddt=D0+αD1+α2D2+⋯,d2dt2=D20+2αD0D1+α2D12+2D0D2+⋯.Substituting equations (31)–(33) and (35) into equation (29) yields(37)1−αD20+2αD0D1+α2D12+2D0D2T0+αT1+α2T2+⋯+1−αω02T0+αT1+α2T2+⋯+αω02ω12T0+αT1+α2T2+⋯3+D0+αD1+α2D2⋅T0+αT1+α2T2+⋯=0,(38)α0,D02T0+ω02T0=0,α1,D02T1+ω02T1+2D0D1T0+ω02ω1D0T0+T03=0,α2,D02T2+T2+2D0D1T1+D12+2D0D2T0+ω02ω13T02T1=0.The solution of the first equation in the system of equation (38) can be obtained as(39)T0=At1eiω0t0+A¯t1e−iω0t0.Substituting equation (39) into the second equation of (38) yields(40)D02T1+ω02T1+2iω0D1A+3ω02ω1A2A¯+iμω03ω1Aeit0+1ω1A3e3it0+cc=0,where cc is the complex conjugate term. Eliminating the term of immortality in equation (40) yields(41)2D1A+3ω0ω1A2A¯+iμω03ω1A=0.Solving equation (41), we can obtain(42)T1=18ω1A3e3it0+A¯3e−3it0.LettingA=1/2feiϕ, bring it into equation (42) and separat the imaginary part from the real part:(43)dfdt1=−12μω02ω1f,fdϕdt1=−3ω08ω1f3.SubstitutingA=1/2feiϕ into equation (39) yields(44)T0=fcosω0t0+ϕ=fcosωt+ϕ0.Comparing the angular frequency in equation (34) with equation (44), we can get the equation under the first-order approximation condition as(45)dϕdt1=ω02.According to equations (43) and (44) yields(46)ω1=34f2.Substituting equation (46) into equation (43) and omitting higher order terms yields(47)ω=ω02+34εf2.The results show that when the critical wind velocity of single-mode instability is reached, the frequency of the characteristic equation of the system approaches zero, which is equivalent to static equilibrium instability [11]. That is,(48)ω02+34εf2=0,where f is the vibration amplitude of the membrane,(49)ω02=2πρabπ+4ρ0α1⋅m2π2b2N0x+n2π2a2N0y2ab−ρ0mα3aV2,α1=∬S∬Ra1rsinmπξasinnπηbdξdηsinmπxasinnπybdxdy,α3=∬S∬Ra1r3x−ξcosmπξasinnπηbdξdηsinmπxasinnπybdxdy,ε=2hm2n2π5α+βabρabπ+4ρ0α1.Solving the equation, the critical wind velocity for single-mode instability of the closed membrane roof is obtained as follows:(50)Vcr=4πm2π2b2N0x+n2π2a2N0y+3εf2ρπa2b2+4ρ0abα18πbρ0mα3,where f is the vibration amplitude corresponding to the instability critical wind velocity. It shows that the critical wind velocity is related to the vibration amplitude. With the consideration of the effect of geometric nonlinearity, the stiffness of the membrane varies with the amplitude in the process of vibration, which will affect the aerodynamic stability of the membrane roof to a certain extent. This conclusion is consistent with that of previous research studies [9, 10, 18]. When f⟶0, the critical wind velocity of instability can be obtained according to the theory of small deflection [9, 10].


#### 2.2.2. Solution of Flexible Membrane Roof on Open Structure

For membrane roofs on open structures, air flows on both sides of the membrane surface due to the smaller thickness of the membrane, which can be substantially determined by the thin-airfoil theory. The membrane roof on open structure is shown in Figure3.Figure 3
The membrane roof on open structure.For the membrane roof on open structure, the aerodynamic force acting on the unit area of the membrane projection surface is expressed as [11](51)p=ρ0∂∂t∫0xγcξ,y,tdξ+ρ0Vγc,where γc is the density of vortices:(52)γc=aVγj=aVa1jTt+a2jT′tV,j=1,2,…,M×N.Substituting equation (52) into equation (9) yields(53)h∂2φ∂y2+N0x∂2w∂x2+h∂2φ∂x2+N0y∂2w∂y2−2ρξ0∂w∂t+ρ0Vγc+ρ0∫0y∂γc∂tdη=ρ∂2w∂t2.Solutions of stress functions in compatible equations such as equation (21). Substituting equations (19) and (21) into equation (53) yields(54)ρWd2Ttdt2+2ρξ0WdTtdt−N0x∂2W∂x2+N0y∂2W∂y2Tt−h∂2ϕ∂y2∂2W∂x2+∂2ϕ∂x2∂2W∂y2T3t−ρ0Vγc−ρ0∫0y∂γc∂tdη=0.By using the Bubnov–Galerkin method, (54) can be integrated as(55)∬SρWd2Ttdt2+2ρξ0WdTtdt−N0x∂2W∂x2+N0y∂2W∂y2Tt−h∂2ϕ∂y2∂2W∂x2+∂2ϕ∂x2∂2W∂y2T3t−ρ0Vγc−ρ0∫0y∂γc∂tdηWx,ydxdy=0,where S∈0≤x≤a,0≤y≤b.Integrating and simplifying equation (55) yields(56)Ad2Ttdt2+BdTtdt−CTt−DT3t−E=0,where(57)A=∬SρW2dxdy=ρab4,B=2ρξ0∬SW2dxdy=2ρξ0∬Ssinmπxasinnπyb2dxdy=ρξ0ab2,C=∬SN0y∂2Wx,y∂y2+N0x∂2Wx,y∂x2Wx,ydxdy=∬SN0x∂2W∂x2Wdxdy+∬SN0y∂2W∂y2Wdxdy=−m2π2bN0x4a−n2π2aN0y4b,D=∬Sh∂2Φ∂y2∂2W∂x2+∂2Φ∂x2∂2W∂y2Wdxdy=−hm2n2π4α+β2ab,E=∬Sρ0Vγ+ρ0∫0y∂γ∂tdηWx,ydxdy.Substituting equation (52) into equation (56) yields(58)A1d2Ttdt2+B1dTtdt−C1Tt−DT3t=0,where(59)A1=A−ρ0a∬S∫0ya2jdηWx,ydxdy,B1=B−ρ0aV∬S∫0ya1jdηWx,ydxdy−ρ0aV∬Sa2jWx,ydxdy,C1=C+ρ0aV2∬Sa1jWx,ydxdy.The composition of equation (58) is consistent with the vibration control equation of closed membrane roof, and the solving process is consistent with the above, which is not discussed here. Thus, the expression of critical wind velocity for single-mode instability of open flexible membrane roof can be obtained as(60)Vcr=πm2bN0x/4a+n2aN0y/4b+3hm2n2π2α+βf2/8abρ0aab/MN∑j=1M×Na1jsinmπxj/asinnπyj/b.


## 2.1. Establishment of Basic Equations

Let the length and width of the orthotropic rectangular flexible membrane with four sides fixed bea and b, respectively; the pretension along the length direction is N0x and the width direction is N0y. The wind blows parallel to the roof and toward the membrane surface, which makes the membrane surface vibrate. For flexible membranes, the research results show that the shear stress has little influence on the vibration process of the membranes and can be considered as zero [9–13]. Assuming that the planar membrane is in the XOY plane when in equilibrium and the pretension in X direction is Nx and in the Y direction is Ny, when the membrane is disturbed by external forces to the XOY plane, it will deform and then produce transverse vibration perpendicular to the membrane surface under the action of tension. Select a microunit on the vibrating membrane surface, as shown in Figure 1.Figure 1
Vibration microunits of membrane.Select a unit part dxdy on the membrane surface. When the microunits are deformed, the edges of the microunits are subjected to the tension of the adjacent facets. InX direction, we can regard the surface element as composed of countless chord elements with length dx and width of one unit. The tension acting on the chord element is consistent with its tangent direction. The tension Nx is at an angle α with X coordinate axis. Therefore, the vertical component of the tension acting on the chord element at one end of X is Nxsinα. Because α is small, sinα≈tanα. Let w be the vertical displacement of a point on the membrane away from the equilibrium position. Therefore,(1)Nxsinα=Nxtanα=Nx∂w∂xx.The vertical force acting on the dy edge is Nx∂w/∂xxdy, and the vertical force at the x edge should be Nx∂w/∂xx+dxdy. Thus, the resultant force in Z direction on the x and x + dx sides of the panel is as follows:(2)Nx∂w∂xx+dxdy−Nx∂w∂xxdy=Nx∂2w∂x2dxdy.Similarly, the resultant force ofZ direction component of the tension acting on Y direction can be obtained as follows:(3)Ny∂w∂yy+dydx−Ny∂w∂yydx=Ny∂2w∂y2dxdy.So the total force inZ direction acting on the whole panel is(4)Fz=Nx∂2w∂x2dxdy+Ny∂2w∂y2dxdy+qx,ydxdy,where Nx is the tension in X direction (longitude), Ny is the tension in Y direction (latitude), w is the deflection of the membrane, and qx,y is the external load acting on the unit area of the projection surface of the membrane. According to the force balance, we can obtain that(5)Nx∂2w∂x2dxdy+Ny∂2w∂y2dxdy+qx,ydxdy=0,(6)qx,y+Nx∂2w∂x2+Ny∂2w∂y2=0.The generalized external loads of the flexible membrane roof under wind load include the wind load acting on the membrane surface, structural damping force, and inertial force [11]. If the aerodynamic term is defined as p, then the generalized external load per unit area qx,y is(7)qx,y=px,y,t−2ρc∂wx,y,t∂t−ρ∂2wx,y,t∂t2.For membrane material, the stiffness of the membrane surface comes from the initial pretension of membrane material, so the initial pretension should be added in equation (6). Finally, the differential equations of motion of flexible membranes are obtained as follows:(8)px,y,t−2ρc∂wx,y,t∂t−ρ∂2wx,y,t∂t2+N0x+Nxt∂2w∂x2+N0y+Nyt∂2w∂y2=0.Introducing the stress functionφx,y, Nx=h∂2φ/∂y2 and Ny=h∂2φ/∂x2. Then, equation (6) can be transformed as(9)N0x+h∂2φ∂y2∂2w∂x2+N0y+h∂2φ∂x2∂2w∂y2+p−2ρc∂w∂t−ρ∂2w∂t2=0.After deformation, the membrane surface strain is composed of linear and nonlinear parts. The linear strain is caused by in-plane displacementu and v, and the nonlinear strain is caused by deflection w. After ignoring shear stress, the total strain is as follows:(10)εx=∂u∂x+12∂w∂x2εy=∂v∂y+12∂w∂y2,where εx is the strain in X direction and εy is the strain in Y direction.By eliminatingu and v in equation (10), the continuous deformation conditions satisfying the strain and deflection of the membrane surface can be obtained:(11)∂2εx∂y2+∂2εy∂x2=∂2w∂x∂y2−∂2w∂x2∂2w∂y2.The membrane is orthotropic, and the direction of the fiber is the main direction of elasticity so that it is consistent with the direction of coordinate systemX and Y. Assuming that the direction of fiber is the same as the direction of coordinate system X and Y, Young’s modulus of elasticity in X and Y directions is E1 and E2, respectively. The longitudinal Poisson’s ratio and the latitudinal Poisson’s ratio is μ1 and μ2, respectively. The relationship between elastic modulus and Poisson’s ratio is as follows:(12)μ1E1=μ2E2.The stress-strain relationship is as follows:(13)σxσy=E11−μ1μ2μ1E21−μ1μ2μ2E11−μ1μ2E21−μ1μ2εxεy,where σx and σy is the normal stresses in X and Y directions, respectively. h is the thickness of membrane.LettingNx=h⋅σx and Ny=h⋅σy and substituting equation (13) into equation (11), the compatibility equation is obtained as(14)1E1h∂2Nx∂y2−μ2E2h∂2Ny∂y2−μ1E1h∂2Nx∂x2+1E2h∂2Ny∂x2=∂2w∂x∂y2−∂2w∂x2∂2w∂y2.By substituting the stress function into equation (14), then (14) can be transformed as(15)1E1∂4φ∂y4−μ2E2∂4φ∂x2∂y2−μ1E1∂4φ∂x2∂y2+1E2∂4φ∂x4=∂2w∂x∂y2−∂2w∂x2∂2w∂y2.


## 2.2. Improved Multiscale Solutions of Governing Equations

The initial surface function of the rectangular planar membranez0 (x, y) = 0. Therefore, the surface equation of the flexible membrane under wind load is as follows:(16)zx,y,t=wx,y,t.According to the Bubnov–Galerkin method, assume the solution of the governing equation is [9–12](17)wx,y,t=∑i=1nTitWix,y,φx,y,t=∑i=1nUitϕix,y,where Wix,y is the mode function, ϕix,y is the unknown stress function, and Tit and Uit are time-dependent functions.Because the membrane is fixed on four sides, the vertical deflection at the boundary of the membrane is zero, and the vibration mode function satisfying the conditions is assumed to be [11, 12](18)Wx,y=sinmπxasinnπyb,where m and n are positive integer.Substituting equation (18) into equation (17), the following equation can be obtained:(19)wx,y,t=Ttsinmπxasinnπyb.Substituting equation (19) into equation (15) yields(20)1E1∂4φ∂y4+1E2∂4φ∂x4=m2n2π42a2b2Tt2cos2mπxa+cos2nπyb.Assume that the solution of the stress function in equation (20) is [12](21)φx,y,t=Tt2ϕx,y,ϕx,y=αcos2mπxa+βcos2nπyb.Substituting equation (21) into equation (20) yields(22)α=E2a2n232b2m2,β=E1b2m232a2n2.


### 2.2.1. Solution of Flexible Membrane Roof on Closed Structure

Flexible membrane covers the top of the closed structure as the roof and the rigid wall around the structure as the vertical bearing member. Because its stiffness is far greater than the membrane’s stiffness, it is assumed that the stiffness of the vertical component is infinite in the process of theoretical derivation in this paper. The membrane roof on closed structure is shown in Figure2.Figure 2
The membrane roof on closed structure.For the membrane roof on closed structure, the aerodynamic force acting on the unit area of the membrane projection surface is expressed as [10](23)p=−ρ02π−V∬RaV∂w/∂x+∂w/∂tx=ξy=ηx−ξx−ξ2+y−η23dξdη+∬RaV∂2w/∂x∂t+∂2w/∂t2x=ξy=ηx−ξ2+y−η2dξdη.Substituting equation (23) into equation (9) yields(24)h∂2φ∂y2+N0x∂2w∂x2+h∂2φ∂x2+N0y∂2w∂y2−2ρc∂w∂t−ρ∂2w∂t2−ρ0π∬Ra1r∂2w∂t2x=ξy=ηdξdη−ρ0V2π∬Ra1r∂2w∂x∂tx=ξy=ηdξdη+ρ0V22π∬Ra1r3∂w∂xx=ξy=ηx−ξdξdη+ρ0V2π∬Ra1r3∂w∂tx=ξy=ηx−ξdξdη=0,where r=x−ξ2+y−η2 and the integral region Ra∈0≤ξ≤a,0≤η≤b.Substituting equations (19), (21), and (22) into equation (24) yields(25)γ1=∬Ra1rWx=ξy=ηdξdη=∬Ra1rsinmπξasinnπηbdξdη,γ2=∬Ra1r∂W∂xx=ξy=ηdξdη=mπa∬Ra1rcosmπξasinnπηbdξdη,γ3=∬Ra1r3∂W∂xx=ξy=ηx−ξdξdη=mπa∬Ra1r3x−ξcosmπξasinnπηbdξdη,γ4=∬Ra1r3Wx=ξy=ηx−ξdξdη=∬Ra1r3x−ξsinmπξasinnπηbdξdη.By using the Bubnov–Galerkin method, (25) can be integrated as(26)∬SρW+ρ0πγ1d2Ttdt2+ρ0V2πγ2-γ4+2ρcWdTtdt-N0x∂2W∂x2+N0y∂2W∂y2+ρ0V22πγ3Tt-h∂2Φ∂y2∂2W∂x2+∂2Φ∂x2∂2W∂y2T3tWx,ydxdy=0,where S∈0≤x≤a,0≤y≤b.Simplifying equation (26) yields(27)Ad2Ttdt2+BdTtdt−CTt−DT3t=0,where(28)A=∬SρW+ρ0πγ1Wdxdy=ρ∬Ssinmπxasinnπyb2dxdy+ρ0π∬Ra∬Ra1rsinmπξasinnπηbdξdηsinmπxasinnπybdxdy=ρab4+ρ0πα1,α1=∬S∬Ra1rsinmπξasinnπηbdξdηsinmπxasinnπybdxdy,B=ρ0V2π∬Sγ2−γ4Wdxdy+2ρc∬SW2dxdy=ρ0mV2a∬S∬Ra1rcosmπξasinnπηbdξdηsinmπxasinnπybdxdy−ρ0V2π∬S∬Ra1r3x−ξsinmπξasinnπηbdξdηsinmπxasinnπybdxdy+2ρc∬Ssinmπxasinnπyb2dxdy=ρ0mV2aα2−ρ0V2πα4+ρcab2,α2=∬S∬Ra1rcosmπξasinnπηbdξdηsinmπxasinnπybdxdy,α4=∬S∬Ra1r3x−ξsinmπξasinnπηbdξdηsinmπxasinnπybdxdy,C=∬SN0x∂2W∂x2+N0y∂2W∂y2+ρ0V22πγ3Wdxdy=∬SN0x∂2W∂x2Wdxdy+∬RaN0y∂2W∂y2Wdxdy+ρ0V22π∬Raγ3Wdxdy=−m2π2bN0x4a−n2π2aN0y4b+ρ0mV22aα3,α3=∬S∬Ra1r3x−ξcosmπξasinnπηbdξdηsinmπxasinnπybdxdy,D=∬Sh∂2Φ∂y2∂2W∂x2+∂2Φ∂x2∂2W∂y2Wdxdy=−hm2n2π4α+β2ab.It can be obtained from numerical calculation thatA≤0 occurs only when b/a≪1, and this will not happen in practical engineering. Equation (27) is a strongly nonlinear vibration equation. For different equations of strongly nonlinear vibration, many scholars have studied the corresponding solutions [13–17]. In this paper, we propose the improved multiscale method based on the principle of modified Lindestedt–Poincaré method to solve equation (27). Letting u=B/εA, ω02=−C/A, ε=−D/A, then equation (27) can be transformed as(29)T¨+ω02T+εμT˙+T3=0.Lettingω be the vibration frequencies of the membrane material and expanding ω2 to the power series of ε near ω02 we obtain the following:(30)ω2=ω02+εω1+ε2ω2+⋯.The transformation parameters are introduced as follows:(31)α=εω1ω02+εω1,(32)ε=ω02αω11−α,ω02+εω1=ω021−α.Expandω2 to the power series of ε as follows:(33)ω2=ω02+εω11+1ω02+εω1ε2ω2+ε3ω3+⋯=ω021−α1+δ2α2+δ3α3+⋯,(34)ω=ω01+12α+38+δ22α2+⋯.The form of the perturbation solution of equation (29) can be(35)Tt,α=T0t0,t1+αT1t0,t1+α2T2t0,t1+⋯,where t0=t and t1=αt.The differential operators are obtained as follows:(36)ddt=D0+αD1+α2D2+⋯,d2dt2=D20+2αD0D1+α2D12+2D0D2+⋯.Substituting equations (31)–(33) and (35) into equation (29) yields(37)1−αD20+2αD0D1+α2D12+2D0D2T0+αT1+α2T2+⋯+1−αω02T0+αT1+α2T2+⋯+αω02ω12T0+αT1+α2T2+⋯3+D0+αD1+α2D2⋅T0+αT1+α2T2+⋯=0,(38)α0,D02T0+ω02T0=0,α1,D02T1+ω02T1+2D0D1T0+ω02ω1D0T0+T03=0,α2,D02T2+T2+2D0D1T1+D12+2D0D2T0+ω02ω13T02T1=0.The solution of the first equation in the system of equation (38) can be obtained as(39)T0=At1eiω0t0+A¯t1e−iω0t0.Substituting equation (39) into the second equation of (38) yields(40)D02T1+ω02T1+2iω0D1A+3ω02ω1A2A¯+iμω03ω1Aeit0+1ω1A3e3it0+cc=0,where cc is the complex conjugate term. Eliminating the term of immortality in equation (40) yields(41)2D1A+3ω0ω1A2A¯+iμω03ω1A=0.Solving equation (41), we can obtain(42)T1=18ω1A3e3it0+A¯3e−3it0.LettingA=1/2feiϕ, bring it into equation (42) and separat the imaginary part from the real part:(43)dfdt1=−12μω02ω1f,fdϕdt1=−3ω08ω1f3.SubstitutingA=1/2feiϕ into equation (39) yields(44)T0=fcosω0t0+ϕ=fcosωt+ϕ0.Comparing the angular frequency in equation (34) with equation (44), we can get the equation under the first-order approximation condition as(45)dϕdt1=ω02.According to equations (43) and (44) yields(46)ω1=34f2.Substituting equation (46) into equation (43) and omitting higher order terms yields(47)ω=ω02+34εf2.The results show that when the critical wind velocity of single-mode instability is reached, the frequency of the characteristic equation of the system approaches zero, which is equivalent to static equilibrium instability [11]. That is,(48)ω02+34εf2=0,where f is the vibration amplitude of the membrane,(49)ω02=2πρabπ+4ρ0α1⋅m2π2b2N0x+n2π2a2N0y2ab−ρ0mα3aV2,α1=∬S∬Ra1rsinmπξasinnπηbdξdηsinmπxasinnπybdxdy,α3=∬S∬Ra1r3x−ξcosmπξasinnπηbdξdηsinmπxasinnπybdxdy,ε=2hm2n2π5α+βabρabπ+4ρ0α1.Solving the equation, the critical wind velocity for single-mode instability of the closed membrane roof is obtained as follows:(50)Vcr=4πm2π2b2N0x+n2π2a2N0y+3εf2ρπa2b2+4ρ0abα18πbρ0mα3,where f is the vibration amplitude corresponding to the instability critical wind velocity. It shows that the critical wind velocity is related to the vibration amplitude. With the consideration of the effect of geometric nonlinearity, the stiffness of the membrane varies with the amplitude in the process of vibration, which will affect the aerodynamic stability of the membrane roof to a certain extent. This conclusion is consistent with that of previous research studies [9, 10, 18]. When f⟶0, the critical wind velocity of instability can be obtained according to the theory of small deflection [9, 10].


### 2.2.2. Solution of Flexible Membrane Roof on Open Structure

For membrane roofs on open structures, air flows on both sides of the membrane surface due to the smaller thickness of the membrane, which can be substantially determined by the thin-airfoil theory. The membrane roof on open structure is shown in Figure3.Figure 3
The membrane roof on open structure.For the membrane roof on open structure, the aerodynamic force acting on the unit area of the membrane projection surface is expressed as [11](51)p=ρ0∂∂t∫0xγcξ,y,tdξ+ρ0Vγc,where γc is the density of vortices:(52)γc=aVγj=aVa1jTt+a2jT′tV,j=1,2,…,M×N.Substituting equation (52) into equation (9) yields(53)h∂2φ∂y2+N0x∂2w∂x2+h∂2φ∂x2+N0y∂2w∂y2−2ρξ0∂w∂t+ρ0Vγc+ρ0∫0y∂γc∂tdη=ρ∂2w∂t2.Solutions of stress functions in compatible equations such as equation (21). Substituting equations (19) and (21) into equation (53) yields(54)ρWd2Ttdt2+2ρξ0WdTtdt−N0x∂2W∂x2+N0y∂2W∂y2Tt−h∂2ϕ∂y2∂2W∂x2+∂2ϕ∂x2∂2W∂y2T3t−ρ0Vγc−ρ0∫0y∂γc∂tdη=0.By using the Bubnov–Galerkin method, (54) can be integrated as(55)∬SρWd2Ttdt2+2ρξ0WdTtdt−N0x∂2W∂x2+N0y∂2W∂y2Tt−h∂2ϕ∂y2∂2W∂x2+∂2ϕ∂x2∂2W∂y2T3t−ρ0Vγc−ρ0∫0y∂γc∂tdηWx,ydxdy=0,where S∈0≤x≤a,0≤y≤b.Integrating and simplifying equation (55) yields(56)Ad2Ttdt2+BdTtdt−CTt−DT3t−E=0,where(57)A=∬SρW2dxdy=ρab4,B=2ρξ0∬SW2dxdy=2ρξ0∬Ssinmπxasinnπyb2dxdy=ρξ0ab2,C=∬SN0y∂2Wx,y∂y2+N0x∂2Wx,y∂x2Wx,ydxdy=∬SN0x∂2W∂x2Wdxdy+∬SN0y∂2W∂y2Wdxdy=−m2π2bN0x4a−n2π2aN0y4b,D=∬Sh∂2Φ∂y2∂2W∂x2+∂2Φ∂x2∂2W∂y2Wdxdy=−hm2n2π4α+β2ab,E=∬Sρ0Vγ+ρ0∫0y∂γ∂tdηWx,ydxdy.Substituting equation (52) into equation (56) yields(58)A1d2Ttdt2+B1dTtdt−C1Tt−DT3t=0,where(59)A1=A−ρ0a∬S∫0ya2jdηWx,ydxdy,B1=B−ρ0aV∬S∫0ya1jdηWx,ydxdy−ρ0aV∬Sa2jWx,ydxdy,C1=C+ρ0aV2∬Sa1jWx,ydxdy.The composition of equation (58) is consistent with the vibration control equation of closed membrane roof, and the solving process is consistent with the above, which is not discussed here. Thus, the expression of critical wind velocity for single-mode instability of open flexible membrane roof can be obtained as(60)Vcr=πm2bN0x/4a+n2aN0y/4b+3hm2n2π2α+βf2/8abρ0aab/MN∑j=1M×Na1jsinmπxj/asinnπyj/b.


## 2.2.1. Solution of Flexible Membrane Roof on Closed Structure

Flexible membrane covers the top of the closed structure as the roof and the rigid wall around the structure as the vertical bearing member. Because its stiffness is far greater than the membrane’s stiffness, it is assumed that the stiffness of the vertical component is infinite in the process of theoretical derivation in this paper. The membrane roof on closed structure is shown in Figure2.Figure 2
The membrane roof on closed structure.For the membrane roof on closed structure, the aerodynamic force acting on the unit area of the membrane projection surface is expressed as [10](23)p=−ρ02π−V∬RaV∂w/∂x+∂w/∂tx=ξy=ηx−ξx−ξ2+y−η23dξdη+∬RaV∂2w/∂x∂t+∂2w/∂t2x=ξy=ηx−ξ2+y−η2dξdη.Substituting equation (23) into equation (9) yields(24)h∂2φ∂y2+N0x∂2w∂x2+h∂2φ∂x2+N0y∂2w∂y2−2ρc∂w∂t−ρ∂2w∂t2−ρ0π∬Ra1r∂2w∂t2x=ξy=ηdξdη−ρ0V2π∬Ra1r∂2w∂x∂tx=ξy=ηdξdη+ρ0V22π∬Ra1r3∂w∂xx=ξy=ηx−ξdξdη+ρ0V2π∬Ra1r3∂w∂tx=ξy=ηx−ξdξdη=0,where r=x−ξ2+y−η2 and the integral region Ra∈0≤ξ≤a,0≤η≤b.Substituting equations (19), (21), and (22) into equation (24) yields(25)γ1=∬Ra1rWx=ξy=ηdξdη=∬Ra1rsinmπξasinnπηbdξdη,γ2=∬Ra1r∂W∂xx=ξy=ηdξdη=mπa∬Ra1rcosmπξasinnπηbdξdη,γ3=∬Ra1r3∂W∂xx=ξy=ηx−ξdξdη=mπa∬Ra1r3x−ξcosmπξasinnπηbdξdη,γ4=∬Ra1r3Wx=ξy=ηx−ξdξdη=∬Ra1r3x−ξsinmπξasinnπηbdξdη.By using the Bubnov–Galerkin method, (25) can be integrated as(26)∬SρW+ρ0πγ1d2Ttdt2+ρ0V2πγ2-γ4+2ρcWdTtdt-N0x∂2W∂x2+N0y∂2W∂y2+ρ0V22πγ3Tt-h∂2Φ∂y2∂2W∂x2+∂2Φ∂x2∂2W∂y2T3tWx,ydxdy=0,where S∈0≤x≤a,0≤y≤b.Simplifying equation (26) yields(27)Ad2Ttdt2+BdTtdt−CTt−DT3t=0,where(28)A=∬SρW+ρ0πγ1Wdxdy=ρ∬Ssinmπxasinnπyb2dxdy+ρ0π∬Ra∬Ra1rsinmπξasinnπηbdξdηsinmπxasinnπybdxdy=ρab4+ρ0πα1,α1=∬S∬Ra1rsinmπξasinnπηbdξdηsinmπxasinnπybdxdy,B=ρ0V2π∬Sγ2−γ4Wdxdy+2ρc∬SW2dxdy=ρ0mV2a∬S∬Ra1rcosmπξasinnπηbdξdηsinmπxasinnπybdxdy−ρ0V2π∬S∬Ra1r3x−ξsinmπξasinnπηbdξdηsinmπxasinnπybdxdy+2ρc∬Ssinmπxasinnπyb2dxdy=ρ0mV2aα2−ρ0V2πα4+ρcab2,α2=∬S∬Ra1rcosmπξasinnπηbdξdηsinmπxasinnπybdxdy,α4=∬S∬Ra1r3x−ξsinmπξasinnπηbdξdηsinmπxasinnπybdxdy,C=∬SN0x∂2W∂x2+N0y∂2W∂y2+ρ0V22πγ3Wdxdy=∬SN0x∂2W∂x2Wdxdy+∬RaN0y∂2W∂y2Wdxdy+ρ0V22π∬Raγ3Wdxdy=−m2π2bN0x4a−n2π2aN0y4b+ρ0mV22aα3,α3=∬S∬Ra1r3x−ξcosmπξasinnπηbdξdηsinmπxasinnπybdxdy,D=∬Sh∂2Φ∂y2∂2W∂x2+∂2Φ∂x2∂2W∂y2Wdxdy=−hm2n2π4α+β2ab.It can be obtained from numerical calculation thatA≤0 occurs only when b/a≪1, and this will not happen in practical engineering. Equation (27) is a strongly nonlinear vibration equation. For different equations of strongly nonlinear vibration, many scholars have studied the corresponding solutions [13–17]. In this paper, we propose the improved multiscale method based on the principle of modified Lindestedt–Poincaré method to solve equation (27). Letting u=B/εA, ω02=−C/A, ε=−D/A, then equation (27) can be transformed as(29)T¨+ω02T+εμT˙+T3=0.Lettingω be the vibration frequencies of the membrane material and expanding ω2 to the power series of ε near ω02 we obtain the following:(30)ω2=ω02+εω1+ε2ω2+⋯.The transformation parameters are introduced as follows:(31)α=εω1ω02+εω1,(32)ε=ω02αω11−α,ω02+εω1=ω021−α.Expandω2 to the power series of ε as follows:(33)ω2=ω02+εω11+1ω02+εω1ε2ω2+ε3ω3+⋯=ω021−α1+δ2α2+δ3α3+⋯,(34)ω=ω01+12α+38+δ22α2+⋯.The form of the perturbation solution of equation (29) can be(35)Tt,α=T0t0,t1+αT1t0,t1+α2T2t0,t1+⋯,where t0=t and t1=αt.The differential operators are obtained as follows:(36)ddt=D0+αD1+α2D2+⋯,d2dt2=D20+2αD0D1+α2D12+2D0D2+⋯.Substituting equations (31)–(33) and (35) into equation (29) yields(37)1−αD20+2αD0D1+α2D12+2D0D2T0+αT1+α2T2+⋯+1−αω02T0+αT1+α2T2+⋯+αω02ω12T0+αT1+α2T2+⋯3+D0+αD1+α2D2⋅T0+αT1+α2T2+⋯=0,(38)α0,D02T0+ω02T0=0,α1,D02T1+ω02T1+2D0D1T0+ω02ω1D0T0+T03=0,α2,D02T2+T2+2D0D1T1+D12+2D0D2T0+ω02ω13T02T1=0.The solution of the first equation in the system of equation (38) can be obtained as(39)T0=At1eiω0t0+A¯t1e−iω0t0.Substituting equation (39) into the second equation of (38) yields(40)D02T1+ω02T1+2iω0D1A+3ω02ω1A2A¯+iμω03ω1Aeit0+1ω1A3e3it0+cc=0,where cc is the complex conjugate term. Eliminating the term of immortality in equation (40) yields(41)2D1A+3ω0ω1A2A¯+iμω03ω1A=0.Solving equation (41), we can obtain(42)T1=18ω1A3e3it0+A¯3e−3it0.LettingA=1/2feiϕ, bring it into equation (42) and separat the imaginary part from the real part:(43)dfdt1=−12μω02ω1f,fdϕdt1=−3ω08ω1f3.SubstitutingA=1/2feiϕ into equation (39) yields(44)T0=fcosω0t0+ϕ=fcosωt+ϕ0.Comparing the angular frequency in equation (34) with equation (44), we can get the equation under the first-order approximation condition as(45)dϕdt1=ω02.According to equations (43) and (44) yields(46)ω1=34f2.Substituting equation (46) into equation (43) and omitting higher order terms yields(47)ω=ω02+34εf2.The results show that when the critical wind velocity of single-mode instability is reached, the frequency of the characteristic equation of the system approaches zero, which is equivalent to static equilibrium instability [11]. That is,(48)ω02+34εf2=0,where f is the vibration amplitude of the membrane,(49)ω02=2πρabπ+4ρ0α1⋅m2π2b2N0x+n2π2a2N0y2ab−ρ0mα3aV2,α1=∬S∬Ra1rsinmπξasinnπηbdξdηsinmπxasinnπybdxdy,α3=∬S∬Ra1r3x−ξcosmπξasinnπηbdξdηsinmπxasinnπybdxdy,ε=2hm2n2π5α+βabρabπ+4ρ0α1.Solving the equation, the critical wind velocity for single-mode instability of the closed membrane roof is obtained as follows:(50)Vcr=4πm2π2b2N0x+n2π2a2N0y+3εf2ρπa2b2+4ρ0abα18πbρ0mα3,where f is the vibration amplitude corresponding to the instability critical wind velocity. It shows that the critical wind velocity is related to the vibration amplitude. With the consideration of the effect of geometric nonlinearity, the stiffness of the membrane varies with the amplitude in the process of vibration, which will affect the aerodynamic stability of the membrane roof to a certain extent. This conclusion is consistent with that of previous research studies [9, 10, 18]. When f⟶0, the critical wind velocity of instability can be obtained according to the theory of small deflection [9, 10].


## 2.2.2. Solution of Flexible Membrane Roof on Open Structure

For membrane roofs on open structures, air flows on both sides of the membrane surface due to the smaller thickness of the membrane, which can be substantially determined by the thin-airfoil theory. The membrane roof on open structure is shown in Figure3.Figure 3
The membrane roof on open structure.For the membrane roof on open structure, the aerodynamic force acting on the unit area of the membrane projection surface is expressed as [11](51)p=ρ0∂∂t∫0xγcξ,y,tdξ+ρ0Vγc,where γc is the density of vortices:(52)γc=aVγj=aVa1jTt+a2jT′tV,j=1,2,…,M×N.Substituting equation (52) into equation (9) yields(53)h∂2φ∂y2+N0x∂2w∂x2+h∂2φ∂x2+N0y∂2w∂y2−2ρξ0∂w∂t+ρ0Vγc+ρ0∫0y∂γc∂tdη=ρ∂2w∂t2.Solutions of stress functions in compatible equations such as equation (21). Substituting equations (19) and (21) into equation (53) yields(54)ρWd2Ttdt2+2ρξ0WdTtdt−N0x∂2W∂x2+N0y∂2W∂y2Tt−h∂2ϕ∂y2∂2W∂x2+∂2ϕ∂x2∂2W∂y2T3t−ρ0Vγc−ρ0∫0y∂γc∂tdη=0.By using the Bubnov–Galerkin method, (54) can be integrated as(55)∬SρWd2Ttdt2+2ρξ0WdTtdt−N0x∂2W∂x2+N0y∂2W∂y2Tt−h∂2ϕ∂y2∂2W∂x2+∂2ϕ∂x2∂2W∂y2T3t−ρ0Vγc−ρ0∫0y∂γc∂tdηWx,ydxdy=0,where S∈0≤x≤a,0≤y≤b.Integrating and simplifying equation (55) yields(56)Ad2Ttdt2+BdTtdt−CTt−DT3t−E=0,where(57)A=∬SρW2dxdy=ρab4,B=2ρξ0∬SW2dxdy=2ρξ0∬Ssinmπxasinnπyb2dxdy=ρξ0ab2,C=∬SN0y∂2Wx,y∂y2+N0x∂2Wx,y∂x2Wx,ydxdy=∬SN0x∂2W∂x2Wdxdy+∬SN0y∂2W∂y2Wdxdy=−m2π2bN0x4a−n2π2aN0y4b,D=∬Sh∂2Φ∂y2∂2W∂x2+∂2Φ∂x2∂2W∂y2Wdxdy=−hm2n2π4α+β2ab,E=∬Sρ0Vγ+ρ0∫0y∂γ∂tdηWx,ydxdy.Substituting equation (52) into equation (56) yields(58)A1d2Ttdt2+B1dTtdt−C1Tt−DT3t=0,where(59)A1=A−ρ0a∬S∫0ya2jdηWx,ydxdy,B1=B−ρ0aV∬S∫0ya1jdηWx,ydxdy−ρ0aV∬Sa2jWx,ydxdy,C1=C+ρ0aV2∬Sa1jWx,ydxdy.The composition of equation (58) is consistent with the vibration control equation of closed membrane roof, and the solving process is consistent with the above, which is not discussed here. Thus, the expression of critical wind velocity for single-mode instability of open flexible membrane roof can be obtained as(60)Vcr=πm2bN0x/4a+n2aN0y/4b+3hm2n2π2α+βf2/8abρ0aab/MN∑j=1M×Na1jsinmπxj/asinnπyj/b.


## 3. Validation of Theoretical Model

### 3.1. Verification Statement

Due to the limitation of the existing research funds, we have not carried out the aeroelastic experiment of the membrane roof at present. Therefore, we use the wind tunnel experiment results of the aeroelastic models of membrane structure in reference [19] to verify the theoretical calculation results in this paper. Later, we will do relevant experiments, and the experimental results will be discussed in our next paper.


### 3.2. Verification Results

The experimental model in reference [19] is a hyperbolic membrane structure with a small sag, with length of 416 mm and width of 325 mm, as shown in Figure 4. The wind direction is along the length. Experimental model parameters are shown in Table 1.Figure 4
Experimental system.
(a)
(b)Table 1
Physical parameters of the experimental model.
Model directory
h (mm)
ρ (kg/m3)
E (MPa)
ω (Hz)
Damping ratio (%)
N
0x (N/mm)
N
0y (N/mm)
B1
0.164
0.155
1.55
13.0
1.9
1.31
0.75
B2
0.163
0.152
1.54
15.4
1.1
1.83
2.91
B3
0.161
0.152
1.56
14.3
1.7
2.40
0.98
B4
0.162
0.155
1.54
15.2
1.2
3.1
1.02Uematsu et al. found in experiments that when the wind velocity exceeds a certain threshold, the test model will have a single-frequency vibration. At the same time, the amplitude of the pulsation displacement jumps and then decays. Then, the wind velocity corresponding to this amplitude is the critical wind velocity of a certain order of instability of the membrane structure. Based on this, the critical wind velocities of the first two orders of each model are finally obtained. Finally, the theoretical calculation value of the single-mode method of the open membrane roof is calculated by (60). Because the amplitude of the test jump point is incomplete and f in (60) cannot be obtained, the theoretical result can only obtain the minimum critical wind velocity, as shown in Table 2.Table 2
Comparison of theoretical calculation results with experimental results.
Model directory
Experimental results
Theoretical calculation results
V
c
r
,
t
1 (m/s)
V
c
r
,
t
2 (m/s)
V
c
r
,
s
1 (m/s)
V
c
r
,
s
2 (m/s)
B1
3.0
4.7
>3.43
>4.90
B2
4.1
5.9
>4.72
>6.03
B3
3.5
5.5
>3.91
>5.87
B4
3.7
5.6
>4.02
>5.98It can be obtained from Table2. The theoretical results are close to the experimental results, and the general variation of theoretical calculation results is in line with the experimental results, but the theoretical calculation results of different models are higher than the experimental results. The reason is that there is a curvature difference between the theoretically calculated flat model and the experimental model, which cause the aerodynamic forces on the surface of the membrane roof to be underestimated. Furthermore, the relative difference of the first-order critical wind velocity is greater than the second-order wind velocity, which indicates that the aerodynamic effect has a greater impact on the first-order mode shape of the membrane, which is consistent with existing research conclusions [20–22]. In summary, it shows that the theoretical calculation method in this paper has the good applicability for membrane structures.


## 3.1. Verification Statement

Due to the limitation of the existing research funds, we have not carried out the aeroelastic experiment of the membrane roof at present. Therefore, we use the wind tunnel experiment results of the aeroelastic models of membrane structure in reference [19] to verify the theoretical calculation results in this paper. Later, we will do relevant experiments, and the experimental results will be discussed in our next paper.


## 3.2. Verification Results

The experimental model in reference [19] is a hyperbolic membrane structure with a small sag, with length of 416 mm and width of 325 mm, as shown in Figure 4. The wind direction is along the length. Experimental model parameters are shown in Table 1.Figure 4
Experimental system.
(a)
(b)Table 1
Physical parameters of the experimental model.
Model directory
h (mm)
ρ (kg/m3)
E (MPa)
ω (Hz)
Damping ratio (%)
N
0x (N/mm)
N
0y (N/mm)
B1
0.164
0.155
1.55
13.0
1.9
1.31
0.75
B2
0.163
0.152
1.54
15.4
1.1
1.83
2.91
B3
0.161
0.152
1.56
14.3
1.7
2.40
0.98
B4
0.162
0.155
1.54
15.2
1.2
3.1
1.02Uematsu et al. found in experiments that when the wind velocity exceeds a certain threshold, the test model will have a single-frequency vibration. At the same time, the amplitude of the pulsation displacement jumps and then decays. Then, the wind velocity corresponding to this amplitude is the critical wind velocity of a certain order of instability of the membrane structure. Based on this, the critical wind velocities of the first two orders of each model are finally obtained. Finally, the theoretical calculation value of the single-mode method of the open membrane roof is calculated by (60). Because the amplitude of the test jump point is incomplete and f in (60) cannot be obtained, the theoretical result can only obtain the minimum critical wind velocity, as shown in Table 2.Table 2
Comparison of theoretical calculation results with experimental results.
Model directory
Experimental results
Theoretical calculation results
V
c
r
,
t
1 (m/s)
V
c
r
,
t
2 (m/s)
V
c
r
,
s
1 (m/s)
V
c
r
,
s
2 (m/s)
B1
3.0
4.7
>3.43
>4.90
B2
4.1
5.9
>4.72
>6.03
B3
3.5
5.5
>3.91
>5.87
B4
3.7
5.6
>4.02
>5.98It can be obtained from Table2. The theoretical results are close to the experimental results, and the general variation of theoretical calculation results is in line with the experimental results, but the theoretical calculation results of different models are higher than the experimental results. The reason is that there is a curvature difference between the theoretically calculated flat model and the experimental model, which cause the aerodynamic forces on the surface of the membrane roof to be underestimated. Furthermore, the relative difference of the first-order critical wind velocity is greater than the second-order wind velocity, which indicates that the aerodynamic effect has a greater impact on the first-order mode shape of the membrane, which is consistent with existing research conclusions [20–22]. In summary, it shows that the theoretical calculation method in this paper has the good applicability for membrane structures.


## 4. Analysis of the Effect of Geometric Nonlinearity on Critical Wind Velocity

Assuming that the wind velocity is in theX direction, m=n=1, b=20m, N0x=2kN/m, and N0y=2kN/m. Then, the difference between the critical wind velocity with and without considering the geometric nonlinearity of membranes are discussed. Finally, the necessity of the consideration of the geometrical nonlinearity in the design of such structures is given. In this paper, the expression of critical wind speed is obtained by mathematical derivation, and it is related to the vibration amplitude of the membrane. Because the geometric nonlinearity of the membrane is considered, the stiffness of the membrane varies with the amplitude in the process of vibration, which will affect the aerodynamic stability of the membrane roof to a certain extent.Defineλ is the ratio of cross-wind (Y) span to along-wind (X) span. The variation of critical wind velocity with vibration amplitude for two types of membrane roofs with different span ratios is shown in Figure 5.Figure 5
Curves of critical wind velocity with amplitude for two types roof models. (a)λ = 0.5. (b) λ = 1. (c) λ = 2.
(a)
(b)
(c)Taking the membrane roof on the closed structure as an example and considering the geometric nonlinearity of the membrane, the critical wind velocity for single-mode instability of the closed membrane roof is obtained by solving the equation:(61)Vcr=4πm2π2b2N0x+n2π2a2N0y+3εf2ρπa2b2+4ρ0abα18πbρ0mα3.By substitutingf = 0 into equation (61), the results of critical wind velocity calculation without considering geometric nonlinearity can be obtained as(62)Vcr,L=πam2bN0x/a+n2aN0y/b2ρ0mα3.Comparing the critical wind velocity of the membrane roof on closed structure with and without considering the geometric nonlinearity, the following results are obtained:(63)VcrVcr,Lclosed=1+3εf2ρπa2b2+4ρ0abα14πm2π2b2N0x+n2π2a2N0y.Similarly, we can also get the comparison results of membrane roofs on open structure as(64)VcrVcr,Lopen=1+3εf2ρπa2b2+4ρ0abα14πm2π2b2N0x+n2π2a2N0y.Obviously, whether for the membrane roof on the closed structure or open structure, the ratio of critical wind velocity is the same with and without considering the geometric nonlinearity. Although compared with the open structure, the wall of the closed structure will affect the aerodynamic force of the membrane roof, but for the ratio of critical wind velocity equations (63) and (64), they are based on the same structure. Therefore, it is reasonable that the ratio of critical wind velocity of the closed structure and the open structure is the same.


## 5. Discussion

According to (63) and (64), whether for the membrane roof on the open structure or closed structure, Vcr/Vcr,L>1. So, the critical wind velocity of single-mode instability obtained with considering the geometrical nonlinearity of the membrane is larger than the linear one. The reason for this is that the stress increment, which is acquired from the geometric nonlinearity, can improve the lateral stiffness and enhance the aerodynamic stability of structures. It is conservative for structural design. By substituting the specific values, the ratio of critical wind velocity Vcr/Vcr,L of the membrane roof on the closed structure can be obtained, as shown in Table 3.Table 3
Ratio of critical wind velocity of the membrane roof on the closed structure with and without considering geometric nonlinearity.
f = 0.1 m
f = 0.2 m
f = 0.4 m
f = 0.6 m
f = 0.8 m
f = 1m
f = 1.2 m
f = 1.4 m
λ = 0.5
1.00
1.04
1.14
1.29
1.47
1.68
1.90
2.14
Λ = 1
1.00
1.01
1.05
1.11
1.19
1.29
1.40
1.52
Λ = 2
1.00
1.01
1.05
1.11
1.20
1.29
1.40
1.52From Table3, it can be seen that when the cross-wind span ratio is small. The geometric nonlinearity has a greater influence on the critical wind velocity of single-mode instability of the membrane roof, and the influence increases with the increase of the vibration amplitude. When the span ratio is greater than 1, the influence is relatively small. Article 5.3.4 of China’s Technical Regulations for Membrane Structures (CECS 158:2015) [23] stipulates that for integral tensioned and cable-supported membrane structures, and the deformation of the membrane structure should not be larger than 1/200 of the span when considering the combination of wind load effects. In the calculation example in this paper, the normal displacement is limited to 0.1 m, and the ratio of critical wind velocity is 1.00 when geometric nonlinearity is considered or not. Therefore, the influence of membrane geometric nonlinearity on the aerodynamic stability of roofs can be neglected under normal wind loads. However, under strong wind loads, the deformation of roofs will exceed the norm limit. At this time, the influence of geometric nonlinearity should be considered.


## 6. Conclusions

In this paper, the aerodynamic stability of orthotropic rectangular planar membranes on closed and open structures is investigated by the mathematical analytic method. The governing equations of wind-induced nonlinear vibration of the tensioned membrane roofs are established by using the theory of large deflection of membrane and Darumbell’s principle. The Bubov–Galerkin method is applied to transform the aerodynamically coupled governing equations into second-order constant coefficient nonlinear differential equations. Then, the periodic solution of the equation is calculated by using an improved multiscale method. Finally, critical wind velocity for single-mode instability of the membrane roof considering geometric nonlinearity is obtained by judging the stability of the periodic solutions of the equations. Furthermore, the influence of geometrical nonlinearity on the critical wind velocity of single-mode aeroelastic instability of membrane material is quantitatively obtained by comparing the results with those without consideration. The main conclusions can be summarized as follows:(i)
Considering the geometric nonlinearity of membranes vibration, the critical wind velocity of single-mode instability of membrane roof nonlinearly increases with the increase of transverse vibration displacement of membranes.(ii)
The critical wind velocity of single-mode instability with the geometrical nonlinearity of the membrane considered is larger than those of linear results. For structural design, it is conservative. When the span along the wind direction is small, the geometrical nonlinearity has a great influence on the critical wind velocity of single-mode instability of the membrane roof and with the amplitude of vibration. When the span ratio is greater than 1, there is no difference, as shown in Table3. Only the vibration amplitude of 0.8 shows an insignificant.(iii)
Under normal wind loads, the influence of membrane geometric nonlinearity on the aerodynamic stability of roofs can be neglected. However, under strong wind loads, the deformation of roofs may exceed the norm limit and reach about 3% of the span, the influence of geometric nonlinearity should be considered.



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*Source: 2305145-2020-04-09.xml*