Nonlinear discrete analysis method for random vibration of guyed masts under wind load

Journal of Wind Engineering and Industrial Aerodynamics 91 (2003) 513–525

Nonlinear discrete analysis method for random vibration of guyed masts under wind load He Yan-lia,*, Ma Xingb, Wang Zhao-minb a

Space Structures Research Center, Shanghai Jiaotong University, Room 2304, Haoran High Tech., Building, 1954 Huashan Road, Shanghai 200030, China b Department of Civil Engineering, Tongji University, Shanghai 200092, China Received 9 March 2001; received in revised form 17 October 2002; accepted 22 October 2002

Abstract Based on the idea of the discrete analysis method of random vibration, the paper studied the wind-induced response of guyed masts, the Gaussian close assumption is adopted to close the mean square equations when taking into account the nonlinearity of cables. The wind load is generated from spatially correlated filtered white noise. The discrete random vibration method can calculate accurate mean and mean square responses, the wind responses analysis for a numerical example demonstrate this point, and the results of discrete random vibration method is close to the results of experiment through wind-induced response analysis of a wind tunnel experiment model. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Guyed mast; Discrete random vibration; Gaussian close assumption

1. Introduction Guyed masts’ bend, shear and twist deformations are easy to couple, and their natural frequencies are close. The frequency-domain method used in engineering computation cannot efficiently carry out the random response analysis for the guyed mast. Similarly, the time domain method needs large calculation and large numbers of samples, and cannot give a specific reliability for the global structure. Combining the advantages of frequency-domain method and time domain method, the paper

*Corresponding author. E-mail address: [email protected] (H. Yan-li). 0167-6105/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 0 2 ) 0 0 4 5 1 - 8

514

H. Yan-li et al. / J. Wind Eng. Ind. Aerodyn. 91 (2003) 513–525

uses the discrete analysis method of random vibration to analyze the wind-induced responses of guyed masts. The discrete analysis method of random vibration is a numerical method for calculating the stationary or nonstationary random vibration responses. It depends on the recurrence formulae obtained by discretizing the vibration equation in the time domain, and uses the mean process and correlation characteristics of the excitations to calculate the mean and mean square random vibration responses directly [1]. Because this method is only fit for white noise load, the wind load must be generated from spatially correlated filtered white noise; for linear models, the method is absolutely convergent. For nonlinear guyed masts, the high-order terms of displacement should not be ignored; the mean square equations derived through the theory of discrete random vibration are not close. After the Gaussian close assumption is used, the high-order moment of response can be expressed by mean and mean square difference equations. The recurrence mean square equations become nonlinear equations about the mean and mean square difference, which can be solved through an iteration method. With this idea, the paper discusses random vibration analysis for guyed mast which is excited by spatially correlated wind loads. The linear and nonlinear recurrent discrete random vibration equations are derived, and numerical analyses have been carried out for a real guyed mast structure and a wind tunnel experiment model.

2. Assumption and basic theory of calculation (1) The random wind speed is a smooth Gaussian process, the mean of fluctuating wind is zero. (2) Only longitudinal wind load is considered, the horizontal overall wind direction is ignored. (3) Damping is proportional to velocity. (4) The distributed wind load which acts on the mast is transmitted to adjacent nodes, and the wind loads are spatially correlated. (5) The two ends of cables are pin-jointed, the performance of cable conforms to Hooke’s law. The structural vibration equation can be expressed as ½Mfx00 ðtÞg þ ½Cfx0 ðtÞg þ ½KfxðtÞg ¼ fqgsðtÞ;

ð1Þ

where ½M; ½C; ½K are the structural mass, damping and stiffness matrix, respectively. fxðtÞg; fxðtÞg; fxðtÞg are the structural displacement, velocity and ’ . acceleration vectors. sðtÞ is the excitation vector {q} is a matrix which describes the position of the excitations. Transform Eq. (1) into state equation as follows: * 1 ½BfyðtÞg * 1 fugsðtÞ; * fy0 ðtÞg ¼ ½A þ ½A

ð2Þ

H. Yan-li et al. / J. Wind Eng. Ind. Aerodyn. 91 (2003) 513–525

where * ¼ ½A

"

#

I ½M

fug ¼ ½ 0T

;

* ¼ ½B

"

I

½K

½C

515

# ;

fyðtÞg ¼ ½ xT ðtÞ x0T ðtÞ T ;

qT  T :

According to the results of [1] and [2], the unconditionally stable recurrence formulae which are able to calculate the accurate structural stationary mean and mean square responses are [1]: * 1 fugs%; fyg % ¼ ½B

ð3Þ

* 0:5Dt½B * I½Ryy ðn 1Þ½2ð½A * 0:5Dt½BÞ * IT * 1 Þ½A * 1 ½A ½Ryy  ¼ ½2ð½A * 0:5Dt½BÞ * 0:5Dt½BÞ * 1 ½Ruu ½ð½A * 1 ; þ 2pDtð½A ð4Þ where the random excitations are assumed to be stationary white noises. s% and ½Ruu  are their mean processes and auto-correlation matrix, respectively. fyg % and ½Ryy  are structural mean and mean square responses, respectively. The power spectrum and auto-correlation function of white noise are used during the course of deriving the Eqs. (3) and (4). But, the random fluctuating wind speed is not white noises, it should be generated from filtered white noises first, and Eqs. (3) and (4) can then be used.

3. Generation of spatially correlated wind vector The wind load is composed of two parts, static mean wind load and fluctuating wind load: % þ fpg ¼ fPg % þ ½efvðtÞg; fPg ¼ fPg

ð5Þ

P% i ¼ Ai cpi mzi W0 ; pi ¼ ei vðtÞ; ei ¼ A i cp i

pffiffiffiffiffiffi mzi rV0

% is static wind load, fpg is the fluctuating wind where fPg is the wind load vector, {P} vector, A is the area exposed to wind, cp is the wind pressure coefficient, mz is the height coefficient, W0 is basic wind pressure, V0 is the mean wind speed at 10 m height, r is the density of air, vðtÞ is the fluctuating component of wind speed. The paper only discusses how the longitudinal wind speed is generated from filtered white noises, because the guyed mast is little affected by vertical wind speed. The Davenport spectrum is adopted as longitudinal wind power spectrum, vðtÞ can be expressed as fvðtÞg ¼ sv ff ðtÞg;

ð6Þ

H. Yan-li et al. / J. Wind Eng. Ind. Aerodyn. 91 (2003) 513–525

516

where sv is the standard deviation of wind velocity, ff ðtÞg is the normalized wind velocity by sv : The power spectrum density function of fvðtÞg is expressed as [2] Sf ðoÞ ¼

2a2 3oð1 þ

a2 Þ4=3

;

a¼

600o ; pV0

ð7Þ

oX0;

where Sf ðoÞ is the power spectrum density of fvðtÞg; o is the fluctuating wind frequency. It is assumed that a filtered white noise process ff  ðtÞg having power spectrum  Sf ðoÞ can be generated from a white noise process fwðtÞg by using an appropriate filter. The filter is usually described as follows fz00 g þ afz0 g þ bfzg ¼ fwðtÞg;

ð8Þ

ff  ðtÞg ¼ gfz0 ðtÞg;

ð9Þ 00

0

where fwðtÞg is the white noise whose power spectrum is equal to 1.0, fz g; fz g; fzg are the state vectors of the filter, a; b; g are parameters of the filter, then, the power spectrum Sf ðoÞ of the generated filtered white noise process ff  ðtÞg can be expressed as [3] Sf ðoÞ ¼ S0 jHf ðoÞj2 ¼

g2 o2 S0 : ðb o2 Þ2 þ a2 o2

ð10Þ

When the parameters are selected as a ¼ 1:208 10 2 V0 ;

b ¼ 1:645 10 5 V02 ;

g¼

pffiffiffiffiffiffi pffiffiffiffiffiffiffiffi a=p ¼ 6:2 10 2 V0 ;

the two sides power spectrum Sf ðoÞ of f  ðtÞ approach the one side power spectrum Sf ðoÞ; and the filtered white noise process ff  ðtÞg approches to ff ðtÞg: Sf ðoÞ ¼ Sf ð oÞD12Sf ðoÞ;

ð11Þ

f  ðtÞDf ðtÞ:

ð12Þ

Combining Eqs. (6), (9) and (12), the fluctuating component of wind load can be expressed as fpg ¼ ½efvðtÞg ¼ ½esv ff ðtÞg ¼ ½esv gfz0 ðtÞg: Disturbance effects can be considered for two points in the wind field. pffiffiffiffiffiffiffiffiffiffiffiffiffi Sfij ðoÞ ¼ Rc Sfii Sfjj ¼ Rc Sf ðoÞ;

ð13Þ

ð14Þ

where, Rc is coherent factor, depending only on the geometric position [4,5].

4. Discrete method of random vibration for guyed masts The difference equation of guyed masts under random wind load can be expressed as   % þ ½efvðtÞg; ½Mfx00 ðtÞg þ ½C x0 ðtÞ þ ½KfxðtÞg ¼ fPg ð15Þ

H. Yan-li et al. / J. Wind Eng. Ind. Aerodyn. 91 (2003) 513–525

517

% is the mean wind load, the response under fPg % is the mean response which where fPg can be obtained using the static method. The mean square response of a guyed mast under total wind load is the mean square response under fluctuating wind load which can be obtained using the following discrete analysis method. (I) Linear calculation model: According to Eqs. (13) and (15), the dynamic differential equation of a guyed mast under fluctuating wind load can be written as ½Mfx00 ðtÞg þ ½Cfx0 ðtÞg þ ½KfxðtÞg ¼ ½esv gfz0 ðtÞg

ð16Þ

transforming Eq. (16) into a state equation as follows: 38 9 8 9 2 38 90 2 x> >0> x> 0 I 0 0 I > > > > > > > > > > > > > = < < 6 7 6 7 0> 0 ½esg 7 x0 = < 0 = ½M 6 ½K ½C 6 7 x þ sðtÞ: ¼6 7 6 7 4 5> 0 0 I 5> z> z> I > 4 0 > > > > > >0> > > ; ; > ; : 0> : 0> : > I0 0 0

bI aI z z I ð17Þ Simplifying Eq. (17) as fy0 ðtÞg ¼ ½A* L  1 ½B* L fyL ðtÞg þ ½A* L  1 fuL gsðtÞ;

ð18Þ

L

where 2

3

I

6 6 ½A* L  ¼ 6 4

7 7 7; 5

½M I

2

fuL g ¼ ½ 0T

0T

I

6 ½K ½C 6 ½B* L  ¼ 6 4 0 0

I fyL ðtÞg ¼ ½ xT ðtÞ

0

0

0

0 0 0

bI

0

3

½esv g 7 7 7; I 5

aI

x0T ðtÞ V T ðtÞ V 0T ðtÞ T ; 0T

qT  T :

According to the solution of mean square responses of structures which are subjected to white noises given in Eq. (4), the formula for calculating the mean square is ½Ry y ðnÞ ¼ ½2ð½A* L  0:5Dt½B* L  1 Þ½A* L  I½Ry y ðn 1Þ½2ð½A* L  0:5Dt½B* L Þ 1 L L

L L

½A* L  IT ; þ2pDtð½A* L  0:5Dt½B* L Þ 1 ½Ruu ½ð½A* L  0:5Dt½B* L Þ 1 ; ð19Þ where 2 6 6 Ruu ¼ 6 4

3

0

7 7 7 5

0 0 Rc

is the spatial correlation matrix of the wind load. ½RyL yL  is the auto-correlation matrix of the responses, the square roots of its diagonal elements are the mean square of response.

518

H. Yan-li et al. / J. Wind Eng. Ind. Aerodyn. 91 (2003) 513–525

The mean square of internal force of element can be calculated after the mean square of displacement is obtained. ½RFeFe  ¼ ½Ke ½Ryeye ½Ke T ;

ð20Þ

where ½RFeFe  is the internal force covariance matrix of the element, ½Ryeye  is the displacement covariance matrix of the element, ½Ke  is the stiffness matrix of the element. The maximum responses of structures can be expressed as frg ¼ fr%g þ mfsr gsig nðr%Þ

ð21Þ

frg is the maximum responses of the structure, displacement or internal force. fr%g is the mean responses, i.e. the responses under mean wind load. m is a peak factor which limits vibration amplitude within threshold for the specific reliability[6]. sr are r.m.s responses. (II) Nonlinear calculation model: Guyed masts are nonlinear structures, which embody the second moment of mast and nonlinear performance of cables. This paper derived the nonlinear recurrence equations in discrete time domain taking account the high-order terms of displacement of the cables. It is satisfactory to consider the effects of second moment of the mast through depreciating the stiffness of the mast [6]. Owing to high-order terms of displacement, the recurrence equation dose not close; the paper adopted Gaussian truncation theory to close it [3]. The nonlinear vibration equation of the cable is ½Mc fQ00 g þ ½Cc fQ0 g þ ½Kc fQg þ ½E½TfUg ¼ fFc g þ fRg;

ð22Þ

where ½Mc ; ½Cc ; ½Kc  are the mass, damping and stiffness matrix of the cable, respectively. fQg; fQ0 Þg; fQ00 g are the displacement, velocity and accelaration vector of cable, respectively, fUg is the displacement vector of the cables’ nodes under global coordinate system, ½T is a coordinate transform matrix, ½E is coefficient matrix [7], fFc g is the external excitation vector of the cable, fRg is a nonlinear term, subscript (c) refers to the cable. The dynamic tension of the cable can be expressed as fHg ¼ ½DfUg þ ½gfQg þ fR g:

ð23Þ

fHg is the dynamic tension of the cable, ½D and ½g are coefficient matrix [7], fR g is the nonlinear part of tension under global coordinate system. Taking into account the dynamic tension of the cables, the vibration equation of guyed masts can be expressed as ½Ms fU 00 g þ ½Cs fU 0 g þ ½Ks fUg þ ½Tg fHg ¼ fF g;

ð24Þ

where ½Ms ; ½Cs ; ½Ks  are, respectively, the mass, damping and stiffness matrixes of structure when cables are taken as tension bars. fUg; fU 0 Þg; fU 00 g are the displacement, velocity and accelaration vectors, respectively. fTg g is a distribution matrix, it distributes the dynamic tension at the nodes of the mast. Subscript (s) refers to the structure when the cables are taken as tension bars.

H. Yan-li et al. / J. Wind Eng. Ind. Aerodyn. 91 (2003) 513–525

519

Combining Eqs. (22) and (24), the coupling vibration equation of the guyed mast can be expressed as " #( ) " #( ) ½Ms  fU 00 g ½Cs  fU 0 g þ ½Mc  fQ00 g ½Cc  fQ0 g " #( ) ( ) ½Ks  þ ½Tg ½D ½Tg ½g fF g fUg þ þ fFR g ¼ : ð25Þ fFc g ½E½T ½Kc  fQg Eq. (25) can be simplified as ½Mfx00 g þ ½Cfx0 g þ ½Kfxg þ fFR g ¼ fqgsðtÞ;

ð26Þ

where fFR g is a nonlinear term which is composed of high-order terms of displacement, other paraments in Eq. (25) are the same as those in Eq. (1). Transform Eq. (26) into state equation as follows: * 1 fuL gsðtÞ; fy0L ðtÞg ¼ ½A* L  1 ½B* L fyL ðtÞg þ ½A* L  1 fGðtÞg þ ½A

ð27Þ

where fGðtÞg ¼ ½ 0T FRT T ; is a nonlinear term. The other parameters in Eq. (26) are same as those in Eq. (16). According to the discrete method of random vibration, the formula for calculating the mean square is * A* L  IÞ½Ry y ðn 1Þð2½J½ * A* L  IÞT þ Dtð2½J½ * A* L  IÞ ½RyN yN ðnÞ ¼ ð2½J½ N N T * þ Dt½J½R * Gy ðn 1Þð2½J½ * A* L  IÞT

½RyG ðn 1Þ½J * GG ½J * T þ 2pDt½J½R * uu ½J *T þDt2 ½J½R * ¼ ð½A* L  0:5Dt½B* L Þ 1 ½J

ð28Þ

½RyN yN  is an auto-correlation matrix, ½RyG  and ½RGy  are mutual correlation matrixes between linear and nonlinear terms, ½RGG  is an auto correlation matrix of nonlinear term. The other parameters in Eq. (28) are same as those in Eq. (19). ½RyG ; ½RGy  and ½RGG  contain high-order moment of responses, the paper adopted Gaussian truncation theory, then the high-order moment can be expressed using mean and mean square [3]. The third moment of response is X E½y1 y2 y3  ¼ E½y1 y2 E½y3  2E½y1 E½y2 E½y3 ; ð29Þ the fourth moment of response is X X E½y1 y2 y3 E½y4  þ E½y1 y2 E½y3 y4  E½y1 y2 y3 y4  ¼ X

2 E½y1 y2 E½y3 E½y4  þ 6E½y1 E½y2 E½y3 E½y4 

ð30Þ

substituting Eqs. (29) and (30) into Eqs. (28), the closure recurrence equation of mean square of response can be obtained, the equation can be solved using an iteration method.

520

H. Yan-li et al. / J. Wind Eng. Ind. Aerodyn. 91 (2003) 513–525

5. Numerical analysis The example is a 150 m tall radio guyed mast with two levels of guys and a ground exposure B. Every level has three guys, they are presented in Fig. 1. The mast is made of grid steel tubes, the cross-section of the mast is an equilateral triangle with 1 m width, the web bar is f102=6 steel tube, the chord bar is f54=4 steel tube, the elastic modulus of the steel tube is 2.06 105 N/mm2. The cables are galvanized wire cables, the diameters of upper and lower layers are, respectively, 18.5 and 14.5 mm, the initial stress of cables of the two layer are 250 N/mm2, the elastic modulus is 1.20 105 N/mm2. The basic wind pressure acting on the guyed mast is w0 ¼ 450 N/m2, the wind direction is f ¼ 01: (1) Mean responses of the guyed mast: For the nonlinear guyed mast, the mean response of the structure under wind load (mean) is not similar to the response of the structure under mean wind load (M-wind). In order to compare them, the paper calculated mean responses of the example using the time domain method (TD). Tables 1–3 list comparisons of nodal displacement, stress of the cable and moment of the mast for the guyed mast, respectively. It can be seen that the difference between mean responses and responses under mean wind is inferior to 5% from Table 1 to Table 3. So, the mean response can be substituted with response under mean wind in practical calculation. (2) Mean square response of the guyed mast: The paper analyzed the mean square response of the guyed mast using the time domain method (TD), the linear discrete random vibration method (LDRV) and the nonlinear discrete random vibration method (NDRV), respectively. Tables 4–6 list a comparision of the nodal

Fig. 1. Guyed mast structure.

H. Yan-li et al. / J. Wind Eng. Ind. Aerodyn. 91 (2003) 513–525

521

Table 1 Displacement comparison U% y (m) Node

A

B

C

D

E

Mean M-wind

0.206 0.208

0.150 0.143

0.267 0.259

0.440 0.427

1.231 1.222

Table 2 Stress of cable comparison (N/mm2) Cable

1i

1j

2i

2j

Mean M-wind

396 387

187 185

435 427

167 168

Table 3 % x (kN m) Moment of mast comparison M Node

A

B

C

D

Mean M-wind

84

87

118 117

33

32

243 243

Table 4 Standard deviation of displacement comparison sUy (m) Node Method

A

B

C

D

E

TD LDRV NDRV

0.132 0.135 0.134

0.071 0.068 0.061

0.151 0.175 0.165

0.112 0.161 0.141

0.558 0.654 0.608

Table 5 Standard deviation of stress comparison (N/mm2) Cable Method

1i

1j

2i

2j

TD LDRV NDRV

71 66 68

44 49 51

50 60 63

40 38 41

522

H. Yan-li et al. / J. Wind Eng. Ind. Aerodyn. 91 (2003) 513–525

Table 6 Standard deviation of moment comparison sMx (KN m) Node Method

A

B

C

D

TD LDRV NDRV

64 67 68

60 58 59

68 78 74

94 105 101

Table 7 Max displacement comparison Uy ¼ U% y þ msUy (m) Node Method

A

B

C

D

E

TD LDRV NDRV

0.496 0.505 0.503

0.306 0.293 0.277

0.600 0.636 0.614

0.687 0.781 0.737

2.459 2.661 2.560

Table 8 % x þ msMx (KN m) Max moment comparision Mx ¼ M Node Method

A

B

C

D

TD LDRV NDRV

225

231

234

250 246 248

183

205

197

450 474 464

displacement, stress of cable and moment of the mast for the guyed mast, respectively. The results of TD, LDRV and NDRV are very close as seen from Table 4 to Table 6 Comparing NDRV with LDRV, the displacement of the structure is smaller and the stress of cable is bigger using NDRV. The reason is that the NDRV considered the nonlinearity of the cable, the stiffness of the cable is enhanced using NDRV, namely the damping of the structure is increased (Tables 7 and 8). (3) The maximum responses of the guyed mast: The maximum displacement and moment of mast comparison of TD and NDRV are shown in Fig. 2. Fig. 2 shows that the results of TD andNDRV are very close. Of course, the results of TD cannot give statistical results on the whole time axis, because the samples are not enough and the time of the sample is not long enough. In addition, the results of LDRV are rather accurate when the guyed mast is weakly nonlinear, i.e. the tensions of the cables are strong. At that time, the LDRV method can satisfy the precision in engineering field.

H. Yan-li et al. / J. Wind Eng. Ind. Aerodyn. 91 (2003) 513–525

523

TD NDRV

Fig. 2. Displacement and moment of guyed mast.

The calculation results of the numerical example present that the discrete random vibration method has good convergence and stability, and it can calculate accurate stationary mean and mean square responses.

6. Analysis of wind tunnel experiment for a guyed mast model The experiment was carried out in TJ-2 wind tunnel lab of Tongji University, the dimension of this wind tunnel is 15 m 3 m 2.5 m (length width height). A turbulent boundary layer with a power-law exponent of 0.16 was generated on the wind tunnel floor, the simulating height of atmospheric boundary layer was about 1.6 m. The geometric scale of the model is 1:100, the height of the model is 2 m, the mast was tensioned by three levels of cables, every level had three guys, they are presented in Fig. 3. The mast is made of grid tubes, the cross-section of the mast is an equilateral triangle with 15 mm width. The details of the dimensions are labeled in Fig. 3. The material of the mast is f 0.8 mm copper wire, the material of cables is f 0.15 mm constantan wire, springs are tied to cables in order that the stiffness of spring substitute the stiffness of the cable. The initial stress of guy is 150 N/mm2. The positions of acceleration sensors are shown in Fig. 4, Nos. 1,3,4 and 5 acceleration sensors are along the wind direction. Nos. 2 and 6 sensors are perpendicular to wind direction, i.e. horizontal direction. The paper analyzed the mean and standard deviation responses of the wind tunnel model using the discrete method of random vibration (NDRV) when the wind speed is 18 m/s. The comparison between experiment and theory is shown in Tables 9–11, the letter H in parentheses describes that sensors were distributed along horizontal wind direction. The mean and mean square responses of experiment and theoretical calculation are very close, as seen from Table 9 to Table 11. It is proved that the discrete method of random vibration for wind-induced responses of guyed masts is reliable. In

524

H. Yan-li et al. / J. Wind Eng. Ind. Aerodyn. 91 (2003) 513–525

Fig. 3. Gued mast model.

Fig. 4. Distribution of acceleration sensors.

Table 9 Standard deviation of acceleration (m/s2) Measuring point

2(H)

3

4

6(H)

5

Experiment NDRV

0.067 0.066

0.169 0.172

0.124 0.119

0.142 0.150

0.198 0.195

Table 10 Mean displacement (mm) Measuring point

2(H)

3

4

6(H)

5

Experiment NDRV

0.01 0.0

6.3 6.2

5.2 4.9

0.041 0.0

9.1 9.9

addition, the freqencies are distributed densely for guyed masts, strictly speaking, it is not suitable to analyze the wind responses of guyed mast using mode superposition method.

H. Yan-li et al. / J. Wind Eng. Ind. Aerodyn. 91 (2003) 513–525

525

Table 11 Standard deviation of displacement (mm) Measuring point

2(H)

3

4

6(H)

5

experiment NDRV

0.22 0.20

2.85 2.91

1.59 1.61

0.48 0.51

5.0 5.2

7. Conclusions (1) This paper applied the discrete method of random vibration in analyzing the wind-induced responses of a guyed mast. When taking account of nonlinear cables, the paper derived the nonlinear discrete random vibration method for guyed masts adopting Gaussian truncation theory. (2) The discrete method of random vibration has good convergence and stability. The results of this method are very close to those of the time domain method and experiment, which proved that the method is accurate and reliable for guyed masts. (3) In addition, the linear discrete random vibration method can be used for weakly nonlinear structures, which consumes less computational time. (4) This paper derived formulae for the discrete method which can also be applied to structures whose frequencies are close, such as cable roof structures, transmission towers, large-span spatial domes, etc.

References [1] Tan Dongyao, Yang Qingshan, Zhao Chen. Discrete analysis method for random vibration of structures subjected to spatially correlated filtered white noises, Comput. Struct. 43 (6) (1992) 1051–1056. [2] V. Kolousek, M. Pirne, O. Fischer, et al., Wind Effects on Civil Engineering Structures, Academia, Prague, 1984. [3] Zhu Weiqiu, Random Vibration Theorem, Chinese Academic Press, Beijing, 1992. [4] Zhang Xiangting, Structural Design Handbook for Wind Engineering, Chinese Architectural Industrial Publishing House, Beijing, 1998. [5] J.L. Lumley, H.A. Panofsky, The Structure of Atmospheric Turbulence, Wiley, New York, 1964. [6] Wang Zhaomin, U. Peil, Tower and Guyed Mast Structures, Tongji University Publishing House, Shanghai, 1989. [7] Ma Xing, Research on vibration theory and effect coefficient of guyed mast under wind load, Doctoral Dissertation, Tongji University, Shanghai, 1999.