Dominant mode selection and modal coupling analysis for wind-induced response of long-span roofs

Abstract

Mode selection and modal coupling analysis are important to estimate wind-induced structural response of long-span roof structures. This article presents a framework for predicting wind-induced structural response of long-span roof structures based on modal analysis. This framework first identifies the dominant modes according to the correlation between the mode shape and the wind load spatial distribution on the structure as well as a proposed “modal participation coefficient.” Second, the concept of modal strain energy is introduced and a modal coupling coefficient is defined, based on which the dominant coupling modes are determined. A modified square root of the sum of the squares methodology is then developed to account for the modal coupling effects of the background and the resonant response components. The total responses can be obtained by combining the contributions of the dominant coupling modes and the square root of the sum of the squares results from the dominant modes. This avoids the use of the computation expensive complete quadratic combination method. Finally, an illustrative example of wind-induced response analysis of the China National Stadium roof structure is provided to demonstrate the effectiveness of the proposed framework.

Keywords

background response dominant mode long-span roofs modal coupling effects resonant response wind-induced response

Introduction

The length of long-span roofs increases with the use of more flexible and light construction, and the structure becomes more sensitive to wind load, which is a key factor in the design of the structure (Fabio et al., 2012; Holmes, 2002; Katsumura et al., 2007). It is therefore important to analyze the wind-induced response of the long-span roofs efficiently and accurately in both theoretical research and engineering design. The modal superposition approach in frequency domain is often employed to analyze the wind-induced response of structures. These roofs are different from tall buildings as they exhibit dense vibration modes in a narrow frequency range. Multiple modal responses as well as their coupling effects should be considered in the wind-induced response analysis with possibly the inclusion of some higher-order modes (Timothy and David, 1999; Uematsu et al., 1999, 2001). The existing practices showed an under-estimation of the responses because of the omission of contributions from the higher-order modes as well as the modal coupling effects (Chen et al., 2000; Fu et al., 2008). Therefore, it is important to find an effective way for identifying the dominant modes and to take into account the modal coupling effects in the modal superposition analysis with less computational effort.

Nakayama et al. (1998) proposed a criterion for selecting the dominant modes from their contributions to the targeted modal strain energy of the system. Higher-order modes with dominant contribution to the response were determined. This method indirectly selected modes which were representative of the responses excited by the mean wind load. However, the contributions of vibration modes to the responses are also influenced by the dynamic amplification effects of the wind loads, which could not be ignored. Other effective methods were proposed based on the loading distribution of wind load and the structural strain energy; participation factors were defined to select dominant eigenmodes in the background response and dominant vibration modes in the resonant response (Chen et al., 2012). Mataki et al. (1988) referred to select the dominant modes according to the magnitude of the modal participation factors but with no details on the selection process.

Once the dominant modes are selected, the total responses can be obtained by combining their contributions to the responses using the square root of the sum of the squares (SRSS) approach or the complete quadratic combination (CQC) method. The SRSS approach does not take into account the modal coupling effects, and it is not applicable to the long-span roofs. Nevertheless, the CQC method involves double summation calculation, and it is time consuming when multiple modes are considered with a large number of nodes in the structural model of the long-span roof. Gu and Zhou (2009) proposed an approximate yet practical method, the modified SRSS method, to compute the multiple modal responses. The modal coupling effects of the resonant response were taken into account. Then, based on the modified SRSS method, a new method was proposed for computing the resonant component equivalent static wind loads by Zhou and Gu (2010).

In this article, a new framework to consider the contributions of multiple modes to the total responses with the modal coupling effects is proposed for estimating the wind-induced responses of long-span roofs. The dominant modes are first identified on the basis of correlation between the associated mode shape and the wind load spatial distribution on structure as well as a “modal participation coefficient” defined in this article. Then, a modified SRSS methodology is developed to account for the modal coupling effects for the background and resonant response components. Finally, the wind-induced response of the China National Stadium roof structure is analyzed to illustrate the accuracy and effectiveness of this proposed framework with comparison to the results from SRSS and CQC methods.

Dominant mode selection

Basic theory

The response induced by the fluctuating wind loads $P (t)$ can be obtained from equation (1)

M \overset{\cdot\cdot}{X} (t) + C \overset{\cdot}{X} (t) + KX (t) = L P (t)

(1)

where $M$ , $C$ , and $K$ are the mass, damping, and stiffness matrices of the structure, respectively. $\overset{\cdot\cdot}{X} (t)$ , $\overset{\cdot}{X} (t)$ , and $X (t)$ are the acceleration, velocity, and displacement vectors of the structure, respectively. $L$ is the wind load transition matrix, which relates the wind loads obtained from the measured locations to the structural nodes.

According to the principle of modal superposition, the displacement response can be expressed as

X (t) = \sum_{j = 1}^{n} ϕ_{j} q_{j} (t)

(2)

where $ϕ_{j}$ and $q_{j} (t)$ are the mode shape vector and the generalized modal coordinate of the jth mode, respectively. n is the number of modes of the structure.

Substituting equation (2) into equation (1) and applying the orthogonality of the vibration modes, equation (1) can be rewritten in terms of the generalized modal coordinate as

{\overset{\cdot\cdot}{q}}_{j} (t) + 2 ζ_{j} ω_{j} {\overset{\cdot}{q}}_{j} (t) + ω_{j}^{2} q_{j} (t) = ϕ_{j}^{T} LP (t) = F_{j} (t)

(3)

where $ζ_{j}$ , $ω_{j}$ , and $F_{j} (t)$ are the damping ratio, natural frequency, and generalized modal force of the jth mode, respectively.

Based on random vibration theory, the cross power spectral density function between the jth and kth generalized modal coordinates can be written as

S_{q_{j} q_{k}} (ω) = H_{j} (i ω) S_{F_{j} F_{k}} (ω) H_{k}^{* T} (i ω)

(4)

where $H_{j} (i ω) = (1 / (ω_{j}^{2} [1 - {(ω / ω_{j})}^{2} + 2 i ζ_{j} ω / ω_{j}]))$ is the jth frequency response function and $i$ denotes the imaginary unit. $S_{F_{j} F_{k}} (ω) = {ϕ_{j}}^{T} L S_{pp} (ω) L^{T} ϕ_{k}$ is the cross power spectral density function between forces $F_{j} (t)$ and $F_{k} (t)$ . $S_{pp} (ω)$ is the power spectral density function matrix of $P (t)$ .

The power spectral density function matrix of $X (t)$ can be obtained from equations (2) and (4) as

\begin{matrix} S_{XX} (ω) = \sum_{j = 1}^{n} \sum_{k = 1}^{n} ϕ_{j} S_{q_{j} q_{k}} (ω) {ϕ_{k}}^{T} \\ = \sum_{j = 1}^{n} \sum_{k = 1}^{n} ϕ_{j} H_{j} (i ω) S_{F_{j} F_{k}} (ω) H_{k}^{* T} (i ω) {ϕ_{k}}^{T} \end{matrix}

(5)

Taking the mth diagonal element of equation (5) and integrating with respect to $ω$ , the standard deviation of $X (t)$ at the mth node can be obtained as

σ_{x, m} = \sqrt{\int_{- \infty}^{+ \infty} S_{X_{m} X_{m}} (ω) d ω}

(6)

Equations (2) and (5) indicate that the displacement response calculation includes the contributions of n modes to the responses. However, dominant vibration modes can be selected to reduce the computation effort.

Equations (2) to (4) indicate that the degree of contribution of a mode to the response depends on two conditions: (1) the correlation between the mode shape and the wind loads spatial distribution on the structure, shown as $F_{j} (t) = ϕ_{j}^{T} Lp (t)$ and (2) the proximity of the natural frequency of the mode to the fluctuating wind loads’ predominant frequency, shown as $H_{j} (i ω)$ and $S_{pp} (ω)$ .

Then, the dominant modes can be selected based on these two conditions.

Mode selection according to the correlation between mode shapes and wind loads

In engineering practice, a number of lower-order modes, usually up to around 10–20 modes (denoted by l), are selected in the wind-induced dynamic response analysis (Nakayama et al., 1998), and the higher-order modes are neglected. This approximate approach, in general, can satisfy the engineering requirements to certain extent, because most of the lower-order modes are dominant modes approximately satisfying the two conditions in section “Basic theory.” However, as mentioned previously, calculation error may arise with the omission of the contributions of higher-order modes.

Owing to the complexity of wind loads’ spatial distribution on long-span roofs, it is difficult to directly estimate the correlation between the wind loads and the mode shapes. However, according to the previous paragraph analysis, there is a good way to select the higher-order modes that are strongly correlated with the wind loads by means of the lower-order mode shapes.

The main computation steps are as follows:

1. The higher-order modes are selected by projecting the associated mode shape vector onto the first mode shape vector $ϕ_{1}$ :

Projecting the ( $l + 1$ )th mode shape vector $ϕ_{l + 1}$ onto the first mode shape vector $ϕ_{1}$ to form a new vector $α_{1, l + 1}$ as

α_{1, l + 1} = k_{1, l + 1} ϕ_{1}

(7)

The vector $(ϕ_{l + 1} - α_{1, l + 1})$ is orthogonal to the first mode shape vector $ϕ_{1}$ with

〈 (ϕ_{l + 1} - α_{1, l + 1}) \cdot ϕ_{1} 〉 = 0

(8)

where “·” means the dot-product of vectors. Substituting equation (7) into equation (8), the vector of correlation coefficient $k_{1, l + 1}$ between $ϕ_{1}$ and $ϕ_{l + 1}$ can be obtained as

k_{1, l + 1} = \frac{〈 ϕ_{l + 1} \cdot ϕ_{1} 〉}{〈 ϕ_{1} \cdot ϕ_{1} 〉}

(9)

where $k_{1, l + 1}$ approximately indicates the extent of correlation between mode shape vector $ϕ_{l + 1}$ and the wind loads.

(ii) Correlation coefficients $k_{1, l + 2}, k_{1, l + 3}, \dots, k_{1, n}$ can similarly be obtained by repeating Step i.

(iii) Sorting coefficients $k_{1, l + 1}, k_{1, l + 2}, k_{1, l + 3}, \dots, k_{1, n}$ in ascending order, and selecting those modes with the coefficients $k_{1, i}$ larger than a threshold, and the threshold can be decided according to the requirement of engineering practices. The corresponding higher-order modes can approximately be regarded as having strong correlation with the wind loads.

2. The higher-order modes which are strongly correlated with the wind loads are selected by projecting these modes onto the second mode shape vector $ϕ_{2}$ . The computation procedure is similar to Step 1. It is noted that if some of the higher-order modes have been selected in Step 1, they are no longer projected onto the rest of the low-order modes.

3. Similarly, more higher-order modes are selected by projecting them onto $ϕ_{3}, ϕ_{4}, \dots, ϕ_{l}$ in turn.

4. The mode matrix $Φ_{N \times s}$ is then combined from the sets of selected higher-order modes in Steps 1 to 3, where N is the number of nodes and s is the number of selected higher-order modes.

Simplification of modal coordinate variance matrix

The higher-order modes which are strongly correlated with the wind loads have been selected in section “Mode selection according to the correlation between mode shapes and wind loads.” However, whether these selected modes can be as the dominant modes ultimately, they need to go through another selection with reference to the second condition in section “Basic theory.”

According to equation (4), the variance matrix of the generalized modal coordinate is obtained as

\begin{matrix} σ_{q}^{2} = \int_{- \infty}^{+ \infty} H (i ω) S_{FF} (ω) {H^{*}}^{T} (i ω) d ω \\ = \int_{- \infty}^{+ \infty} H (i ω) Φ^{T} L^{T} S_{pp} (ω) L Φ H^{* T} (i ω) d ω \end{matrix}

(10)

where $H (i ω) = diag [H_{1} (i ω), H_{2} (i ω), \dots, H_{n} (i ω)]$ is the frequency response function matrix; $diag [\cdot]$ denotes a diagonal matrix. $Φ = [ϕ_{1}, ϕ_{2}, \dots, ϕ_{n}]$ is the mode shape matrix.

The relative magnitude of the diagonal element of $σ_{q}^{2}$ indicates the contribution of the corresponding mode to the wind-induced response as noted in equation (2). Therefore, it can be used to decide on whether the selected higher-order modes are dominant modes or not.

Substituting the l number of lower-order modes and the s number of selected higher-order modes into equation (10), the $R = l + s$ dimension matrix $σ_{q R \times R}^{2}$ can be obtained. The matrix $σ_{q R \times R}^{2}$ can also be expressed in a simple form as

σ_{q R \times R}^{2} = \int_{- \infty}^{+ \infty} S_{FF} {(ω)}_{R \times R} \otimes {HH}_{R \times R}^{* T} d ω

(11)

where $S_{FF} {(ω)}_{R \times R}$ and ${HH}_{R \times R}^{* T}$ are the power spectral density function matrix of the generalized modal force and transfer function matrix of the frequency response, respectively. $H_{i} (i ω) {H_{j}}^{* T} (i ω)$ is an element of ${HH}_{R \times R}^{* T}$ . “⊗” denotes multiplication of the corresponding elements of two matrices.

The computation of $σ_{q m \times m}^{2}$ in the above form is time consuming and the matrix is therefore simplified in the following paragraphs.

According to the relationship between the wind velocity and wind pressure, the fluctuating wind loads can be obtained as

p (t) = Bv (t)

(12)

where $B$ is the diagonal matrix; $B_{i} = μ_{si} μ_{zi} ρ A_{i} {\bar{V}}_{i}$ is an element of $B$ ; and $μ_{si}$ , $μ_{zi}$ , $ρ$ , $A_{i}$ , and ${\bar{V}}_{i}$ are the shape coefficient of the structure, wind pressure height coefficient, air density, equivalent area of the ith monitored area, and mean wind velocity, respectively. $v (t)$ is the fluctuating wind velocity.

The power spectral density function matrix of the fluctuating wind loads can be obtained according to equation (12) as

S_{pp} (ω) = B S_{vv} (ω) B^{T}

(13)

where $S_{vv} (ω)$ is the power spectral density function matrix of the fluctuating wind velocity, and the elements of $S_{vv} (ω)$ can be expressed as

S_{vivj} (ω) = ρ_{ij} \sqrt{S_{vi} (ω) S_{vj} (ω)}

(14)

where $ρ_{ij}$ is the coherence function of fluctuating wind velocity between the ith and jth nodes, which can be obtained by Shiotani empirical formula (Simiu and Scanlan, 1996). $S_{vi} (ω)$ and $S_{vj} (ω)$ are the power spectral density functions of fluctuating wind velocity at the ith and jth nodes, respectively. Since the height difference is relatively little in long-span roof, the $S_{v} (ω)$ at every node may be considered equal (Davenport, 1961).

The power spectral density function matrix of generalized modal force can be obtained from equations (10), (13), and (14) as follows

S_{FF} {(ω)}_{R \times R} = Φ_{N \times R}^{T} L^{T} B ρ B^{T} L Φ_{N \times R} S_{v} (ω)

(15)

where $ρ$ is the coherence function matrix of the fluctuating wind velocity with its elements $ρ_{ij}$ .

It is noteworthy because the quasi-static hypothesis is applied in simplifying modal coordinate variance matrix; the characteristics of power spectral density functions of fluctuating wind velocity and that of generalized modal force are assumed to be similar, which is noted in equation (15).

According to Simiu and Scanlan (1996), $H_{j} (i ω) {H_{j}}^{* T} (i ω) = {| H_{j} (i ω) |}^{2}$ can be approximately expressed as follows

{\begin{matrix} {| H_{j} (i ω) |}^{2} \approx \frac{1}{{(2 {ω_{j}}^{2} ζ_{j})}^{2}} ω_{j} - \frac{Δ_{j}}{2} < ω < ω_{j} + \frac{Δ_{j}}{2} \\ {| H_{j} (i ω) |}^{2} \approx \frac{1}{{ω_{j}}^{4}} ω < ω_{j} - \frac{Δ_{j}}{2}, ω > ω_{j} + \frac{Δ_{j}}{2} \end{matrix}

(16)

where $Δ_{j}$ is the jth resonant frequency range.

Substituting equations (15) and (16) into equation (11), and considering only the diagonal elements, the alternate form of matrix $σ_{q R \times R}^{2}$ can be written as

{\tilde{} σ}_{qR \times R}^{2} = (2 {W Φ}_{N \times R}^{T} L^{T} B ρ B^{T} L Φ_{N \times R} W) \otimes β

(17)

where $W = diag [1 / {ω_{1}}^{2}, 1 / {ω_{2}}^{2}, \dots, 1 / {ω_{R}}^{2}]$ . $β = diag [β_{1}, β_{2}, \dots, β_{R}]$ , and the elements $β_{j}$ can be expressed as

\begin{matrix} β_{j} = \int_{0}^{+ \infty} S_{v} (ω) d ω + \frac{1}{4 {ζ_{j}}^{2}} \\ \int_{ω_{j} - Δ_{j} / 2}^{ω_{j} + Δ_{j} / 2} S_{v} (ω) d ω (j = 1, 2, \dots, R) \end{matrix}

(18)

According to the characteristic of $S_{v} (ω)$ in the frequency domain (Figure 1), there is a peak value around the predominant frequency of the fluctuating wind velocity. The value is relatively small in the higher frequency range, and it decreases rapidly with increasing frequency. Meanwhile, the fundamental frequency of most long-span roofs is around 1.0 Hz and is larger than the predominant frequency of the fluctuating wind velocity (round 0.05 Hz) (Davenport, 1961). Therefore, $\int_{ω_{j} - Δ_{j} / 2}^{ω_{j} + Δ_{j} / 2} S_{v} (ω) d ω (j = 1, 2, \dots, R)$ on the right-hand side of equation (18) can be assumed to be the same (Figure 1) for all j.

Figure 1.

Sketch of $β_{j}$ calculation.

For long-span roofs, it can be assumed that the damping ratio $ζ_{j}$ of every mode is identical. Therefore, the following result can be approximately obtained as

β_{1} = β_{2} \dots = β_{j} \dots = β_{R} = β

(19)

Substituting equation (19), equation (17) can be rewritten as

{\tilde{} σ}_{qR \times R}^{2} = (2 {W Φ}_{N \times R}^{T} L^{T} B ρ B^{T} L Φ_{N \times R} W) β = D \cdot β

(20)

It is clear that the proportional relation between the diagonal elements of matrices $D$ , ${\tilde{} σ}_{qR \times R}^{2}$ and $σ_{q R \times R}^{2}$ is identical, but the computational expense of $D$ is less than ${\tilde{} σ}_{qR \times R}^{2}$ and $σ_{q R \times R}^{2}$ .

Modal participation coefficient

The modal participation coefficient $α_{j}$ is introduced to estimate the contribution fraction of jth mode to the total response, and it is expressed as equation (21). The dominant modes can therefore be selected according to the value of $α_{j}$

α_{j} = \frac{D_{jj}}{\sum_{jj = 1}^{R} D_{jj}}

(21)

where $D_{jj}$ is the diagonal elements of matrix $D$ .

Moreover, a threshold value on the total fraction of participation $\sum α_{j}$ to form the total response can be selected to determine the number of dominant modes to be selected.

In engineering practice, if it takes the threshold value as $\sum α_{j} \geq 90 %$ , the precise of the corresponding displacement response can reach at more than 95%, which satisfies engineering requirement. As a result, take $\sum α_{j} \geq 90 %$ as the threshold value.

Modal coupling effect analysis

After selecting the dominant modes, the total response can be obtained by combining the contributions of dominant modes to the responses. The wind-induced response is often divided into background and resonant components. Davenport (1967) proposed the concept of the background response and resonant response with definitions based on the relationship between the power spectrum of fluctuating wind loads and that of the structural response. The background response describes the quasi-static effect of fluctuating wind loads with a power spectral density function similar to that of the fluctuating wind loads. The resonant response describes the dynamic effect of the fluctuating loads, which is related to the peaks of the power spectrum of structural response. The variance of the total response can be obtained as (Chen et al., 2012; Davenport, 1967)

σ_{x}^{2} = σ_{x, b}^{2} + σ_{x, r}^{2}

(22)

where $σ_{x, b}^{2}$ and $σ_{x, r}^{2}$ are the variances of the background and resonant responses, respectively, and they are obtained by combining the contributions of dominant modes to the responses. As mentioned previously, the individual coupling effects of the background and resonant component should be considered.

Modal coupling effects of background response

Based on the concept of background response (Davenport, 1967), the terms on the inertial loads and the damping loads in equation (3) are neglected, that is, the first and second terms on the left-hand side of equation (3) are set equal to 0, and the generalized modal coordinate of the background responses can be obtained as

q_{j, b} (t) = \frac{F_{j} (t)}{ω_{j}^{2}}

(23)

Substituting equation (23) into equation (3), the background displacement response can be expressed as

X_{b} (t) = \sum_{j = 1}^{n} ϕ_{j} q_{j, b} (t)

(24)

The cross power spectral density function between $q_{j, b} (t)$ and $q_{k, b} (t)$ can be written from equations (4) and (23) as

\begin{matrix} S_{q_{j, b} q_{k, b}} (ω) = \frac{1}{ω_{j}^{2}} S_{F_{j} F_{k}} (ω) \frac{1}{ω_{k}^{2}} \\ = H_{j, b} (ω) S_{F_{j} F_{k}} (ω) H_{k, b}^{T} (ω) \end{matrix}

(25)

where $H_{j, b} (ω) = 1 / ω_{j}^{2}$ is the jth background frequency response function.

The power spectral density function matrix of $X_{b} (t)$ can be obtained from equations (24) and (25) as

\begin{matrix} S_{X_{b} X_{b}} (ω) = \sum_{j = 1}^{n} \sum_{k = 1}^{n} ϕ_{j} S_{q_{j, b} q_{k, b}} (ω) {ϕ_{k}}^{T} \\ = \sum_{j = 1}^{n} \sum_{k = 1}^{n} ϕ_{j} H_{j, b} (ω) S_{F_{j} F_{k}} (ω) H_{k, b}^{T} (ω) {ϕ_{k}}^{T} \end{matrix}

(26)

The variance of $X_{b} (t)$ of the jth mode at the mth node can be obtained from equations (5), (6), and (26) as

\begin{matrix} σ_{x, j, b, m}^{2} = \int_{- \infty}^{+ \infty} S_{X_{b, m} X_{b, m}} (ω) d ω \\ = \int_{- \infty}^{+ \infty} ϕ_{jm} H_{j, b} (ω) S_{F_{j} F_{j}} (ω) H_{j, b}^{T} (ω) ϕ_{jm} d ω \\ = \frac{ϕ_{jm}^{2} σ_{F_{j} F_{j}}^{2}}{ω_{j}^{4}} \end{matrix}

(27)

where $σ_{F_{j} F_{j}}^{2}$ is the variance of jth generalized modal force. $ϕ_{jm}$ is the element of jth mode shape vector at the mth node.

Similarly, the cross covariance of $X_{b} (t)$ between the jth and kth modes at mth node can be obtained from equations (5), (6), and (26) as

\begin{matrix} σ_{x, jk, b, m}^{2} = \int_{- \infty}^{+ \infty} ϕ_{jm} H_{j, b} (ω) S_{F_{j} F_{k}} (ω) H_{k, b}^{T} (ω) ϕ_{km} d ω \\ = \frac{ϕ_{jm} ϕ_{km} σ_{F_{j} F_{k}}^{2}}{ω_{j}^{2} ω_{k}^{2}} \end{matrix}

(28)

where $σ_{F_{j} F_{k}}^{2}$ is the cross covariance between the jth and kth generalized modal force. $ϕ_{km}$ is the element of the kth mode shape vector at the mth node.

The strain energy of the structure induced by the background response of the jth mode can be expressed as

W_{j, b} (t) = \frac{1}{2} X_{j, b}^{T} (t) K X_{j, b} (t)

(29)

Similarly, the strain energy of the structure induced by the coupling background responses of the jth and kth modes can be expressed as

W_{jk, b} (t) = \frac{1}{2} X_{jk, b}^{T} (t) K X_{jk, b} (t)

(30)

where $X_{j, b} (t)$ is the background response of jth mode. $X_{jk, b} (t)$ is the background response of the jth and kth modal coupling effects.

Taking time average of equations (29) and (30), and substituting equations (27) and (28) into them, respectively, the following equations can be obtained

{\bar{W}}_{j, b} = \frac{1}{2} ω_{j}^{2} σ_{x, j, b}^{2} = \frac{σ_{F_{j} F_{j}}^{2}}{2 ω_{j}^{2}}

(31)

{\bar{W}}_{jk, b} = \frac{1}{2} ω_{j} ω_{k} σ_{x, jk, b}^{2} = \frac{σ_{F_{j} F_{k}}^{2}}{2 ω_{j} ω_{k}}

(32)

where overhead (-) denotes the time-averaged value. $σ_{x, j, b}^{2}$ is the variance of background response of the jth mode. $σ_{x, jk, b}^{2}$ is the cross variance of background response between the jth and kth modes, that is, the modal coupling effects between the jth and kth modes.

The background response modal coupling coefficient can then be defined as

η_{jk, b} = \frac{{\bar{W}}_{jk, b}}{{\bar{W}}_{j, b} + {\bar{W}}_{k, b}}

(33)

It indicates the degree of modal coupling effects from the background response of the jth and kth modes.

Substituting equations (31) and (32) into equation (33), and taking $ρ_{jk} = ω_{j} / ω_{k}$ and $θ_{jk} = σ_{F_{j} F_{j}}^{2} / σ_{F_{k} F_{k}}^{2}$ , equation (33) can further be written as

η_{jk, b} = \frac{ω_{j} ω_{k} σ_{F_{j} F_{k}}^{2}}{ω_{k}^{2} σ_{F_{j} F_{j}}^{2} + ω_{j}^{2} σ_{F_{k} F_{k}}^{2}} = \frac{ρ_{jk} \sqrt{θ_{jk}}}{θ_{jk} + ρ_{jk}^{2}}

(34)

According to equation (34), the selection criteria of the background response dominant coupling modes are taken as

η_{jk, b} > ε_{b}

(35)

where $ε_{b}$ is the threshold value.

Once the value of $ε_{b}$ is decided, the background response dominant coupling modes can be selected by means of equation (35). The total background response can be obtained by combining the contributions of background response dominant coupling modes and the SRSS results from the dominant modes as

σ_{x, b}^{2} = \sum_{j = 1}^{R'} σ_{x, j, b}^{2} + \sum_{j = 1}^{l_{b} - 1} \sum_{k = j + 1}^{l_{b}} σ_{x, jk, b}^{2}

(36)

where $R'$ and $l_{b}$ are the number of dominant modes and background response dominant coupling modes, respectively.

Modal coupling effects of resonant response

Based on the concept of background response (Davenport, 1967), the variance of resonant response of the jth mode at the mth node can be obtained from equations (5) and (6) as

\begin{matrix} σ_{x, j, r, m}^{2} = 2 \int_{ω_{j} - Δ_{j} / 2}^{ω_{j} + Δ_{j} / 2} ϕ_{jm}^{2} H_{j} (i ω) H_{j}^{* T} (i ω) S_{F_{j} F_{j}} (ω) d ω \\ = ϕ_{jm}^{2} \frac{π}{2 ω_{j}^{3} ξ_{j}} S_{F_{j} F_{j}} (ω_{j}) \end{matrix}

(37)

Similarly, the cross variance of resonant response between the jth and kth modes at the mth node can be obtained from equations (5) and (6) as follows

\begin{matrix} σ_{x, jk, r, m}^{2} = 4 \int_{ω_{j} - Δ_{j} / 2}^{ω_{j} + Δ_{j} / 2} ϕ_{jm} H_{j} (i ω) H_{k}^{* T} (i ω) S_{F_{j} F_{k}} (ω) ϕ_{km} d ω \\ + 4 \int_{ω_{k} - Δ_{k} / 2}^{ω_{k} + Δ_{k} / 2} ϕ_{km} H_{k} (i ω) H_{j}^{* T} (i ω) S_{F_{j} F_{k}} (ω) ϕ_{jm} d ω \end{matrix}

(38)

where $Δ_{k}$ is the kth resonant frequency range.

Since the contribution of the imaginary part in $H_{j} (i ω) H_{k}^{* T} (i ω)$ to the response is relatively little (Simiu and Scanlan, 1996), only the real part is analyzed, and it can be expressed as

Re [H_{j} (i ω) H_{k}^{* T} (i ω)] = \frac{(ω_{j}^{2} - ω^{2}) (ω_{k}^{2} - ω^{2}) + 4 ξ_{j} ξ_{k} ω_{j} ω_{k} ω^{2}}{π_{j} π_{k}}

(39)

where $π_{j} = {(ω_{j}^{2} - ω^{2})}^{2} + {(2 ξ_{j} ω_{j} ω)}^{2}$ , and $π_{k}$ is similarly defined.

Substituting equation (39) into equation (38), and rewriting the integral calculation into discrete sum calculation around $ω_{j}$ and $ω_{k}$ , equation (38) can be rewritten as

\begin{matrix} σ_{x, jk, r, m}^{2} = 4 ϕ_{jm} Δ_{j} Re [H_{j} (i ω) H_{k}^{* T} (i ω)] |_{ω} = ω_{j} S_{F_{j} F_{k}} (ω_{j}) ϕ_{km} \\ + 4 ϕ_{jm} Δ_{k} Re [H_{j} (i ω) H_{k}^{* T} (i ω)] |_{ω = ω_{k}} S_{F_{j} F_{k}} (ω_{k}) ϕ_{km} \\ = 4 ϕ_{jm} K_{jk} S_{F_{j} F_{k}} (ω_{j}) ϕ_{km} + 4 ϕ_{jm} K_{kj} S_{F_{j} F_{k}} (ω_{k}) ϕ_{km} \end{matrix}

(40)

where $K_{jk} = (π ξ_{k} ω_{k}) / ({(ω_{k}^{2} - ω_{j}^{2})}^{2} + (2 ξ_{k} ω_{k} {ω_{j}}^{2}))$ , and $K_{kj}$ is similarly defined.

The strain energy of the structure induced by the resonant response of the jth mode can be expressed as

W_{j, r} = \frac{1}{2} {\overset{\cdot}{X}}_{j, r}^{T} (t) M {\overset{\cdot}{X}}_{j, r} (t) + \frac{1}{2} X_{j, r}^{T} (t) K X_{j, r} (t)

(41)

where ${\overset{\cdot}{X}}_{j, r} (t)$ and $X_{j, r} (t)$ are the velocity and displacement resonant responses of the jth mode, respectively

Similarly, the strain energy of the structure due to the coupling resonant responses of the jth and kth modes can be expressed as

W_{jk, r} = \frac{1}{2} {\overset{\cdot}{X}}_{jk, r}^{T} (t) M {\overset{\cdot}{X}}_{jk, r} (t) + \frac{1}{2} X_{jk, r}^{T} (t) K X_{jk, r} (t)

(42)

where ${\overset{\cdot}{X}}_{jk, r} (t)$ and $X_{jk, r} (t)$ are the coupling velocity and displacement resonant responses of both the jth and kth modes, respectively.

Taking the time average of equations (41) and (42), and substituting equations (37) and (40) into them, respectively, the following equations can be obtained

{\bar{W}}_{j, r} = \frac{1}{2} σ_{v, j, r}^{2} + \frac{1}{2} ω_{j}^{2} σ_{x, j, r}^{2} = \frac{π S_{F_{j} F_{j}} (ω_{j})}{2 ω_{j} ξ_{j}}

(43)

\begin{matrix} {\bar{W}}_{jk, r} = \frac{1}{2} σ_{v, jk, r}^{2} + \frac{1}{2} ω_{j} ω_{k} σ_{x, jk, r}^{2} \\ = \frac{2 π ξ_{k} ω_{k}^{2} ω_{j} S_{F_{j} F_{k}} (ω_{j})}{{(ω_{k}^{2} - ω_{j}^{2})}^{2} + {(2 ξ_{k} ω_{k} ω_{j})}^{2}} \\ + \frac{2 π ξ_{j} ω_{j}^{2} ω_{k} S_{F_{j} F_{k}} (ω_{k})}{{(ω_{j}^{2} - ω_{k}^{2})}^{2} + {(2 ξ_{j} ω_{j} ω_{k})}^{2}} \end{matrix}

(44)

where $σ_{v, j, r}^{2}$ and $σ_{x, j, r}^{2}$ are the variance of the velocity and displacement resonant responses of the jth mode, respectively. $σ_{v, jk, r}^{2}$ and $σ_{x, jk, r}^{2}$ are the cross variance of the velocity and displacement resonant responses of the jth and kth modes, that is, with the modal coupling effects between the two modes.

The resonant response modal coupling coefficient is defined as

η_{jk, r} = \frac{{\bar{W}}_{jk, r}}{{\bar{W}}_{j, r} + {\bar{W}}_{k, r}}

(45)

Substituting equations (43) and (44) into equation (45), and taking $ρ_{jk} = ω_{j} / ω_{k}$ , $λ_{jk, j} = S_{F_{j} F_{j}} (ω_{j}) / S_{F_{k} F_{k}} (ω_{j})$ , $λ_{kj, k} = S_{F_{k} F_{k}} (ω_{k}) / S_{F_{j} F_{j}} (ω_{k})$ , $λ_{jk} = S_{F_{j} F_{j}} (ω_{j}) / S_{F_{k} F_{k}} (ω_{k})$ , and assuming $ξ_{j} = ξ_{k} = ξ$ , then equation (45) can be rewritten as

η_{jk, r} = \frac{4 ρ_{jk}^{2} ξ^{2} (λ_{jk} \sqrt{\frac{1}{λ_{jk, j}}} + ρ_{jk} \sqrt{\frac{1}{λ_{kj, k}}})}{[{(1 - ρ_{jk}^{2})}^{2} + 4 ξ^{2} ρ_{jk}^{2}] (λ_{jk} + ρ_{jk})}

(46)

According to equation (46), the selection criteria for the resonant response dominant coupling modes are then defined as

η_{jk, r} > ε_{r}

(47)

where $ε_{r}$ is the threshold value.

Once the value of $ε_{r}$ is decided, the resonant response dominant coupling modes can be selected by means of equation (47). The total resonant response can be obtained by combining the contributions of the resonant response dominant coupling modes with the results from the SRSS analysis on the dominant modes to give

σ_{x, r}^{2} = \sum_{j = 1}^{R'} σ_{x, j, r}^{2} + \sum_{j = 1}^{l_{r} - 1} \sum_{k = j + 1}^{l_{r}} σ_{x, jk, r}^{2}

(48)

where $R'$ and $l_{r}$ are the number of dominant modes and resonant response dominant coupling modes, respectively.

Case study

The wind-induced response of the roof of the China National Stadium is analyzed with the proposed method. The Stadium is the main stadium of the 2008 Olympic Games in Beijing with 332.3-m-long axis and 296.4-m-short axis. The height of the roof varies from 40.1 to 68.5 m. The main roof consists of 48 groups of lattice rigid frames and a middle ring girder. An empty area of 185.3 m × 127.5 m locates in the middle of the roof. The lattice rigid frames are supported by 24 groups of columns.

Wind tunnel test and parameters for computation

A wind tunnel test was conducted in the boundary layer wind tunnel. The geometric scale was 1:300. The boundary layer was simulated with a power law exponent of $α = 0.16$ , and the surrounding buildings were also modeled. Figure 2 shows the photograph of the model. A total of 509 pressure taps were installed on the roof. The full-scale structure basic wind pressure was 0.5 kN/m² and the sampling frequency for full-scale roof was 5.0 Hz. Measurements were taken for 36 wind directions at equally spaced 10° intervals (Figure 3).

Figure 2.

Model of the structure in wind tunnel test.

Figure 3.

Wind position and number of key nodes.

In this example, the wind load data from the wind tunnel test at 340° wind angle are utilized, and the contour plots of net mean and root mean square (RMS) pressure coefficient distribution over the roof are shown in Figure 4. They are related to the total pressure on the topside and underside of the roof, and that at the highest point on the roof serves as the reference for the pressure coefficient.

Figure 4.

(a) Net mean pressure coefficient distribution and (b) RMS pressure coefficient distribution of the roof.

The dynamic characteristic analysis is performed with the finite element program software ANSYS. The first 500 mode shapes and the corresponding natural frequencies are obtained. Figure 5 gives the first two mode shapes of the structure, which are vibrating mainly in the vertical direction. Table 1 gives the natural frequencies of the first 30 modes, which is noted to have a dense distribution. Therefore, multiple modes and their coupling effects in the wind-induced response analysis should be considered.

Figure 5.

Distribution of the first 2 mode shapes: (a) the first mode and (b) the second mode.

Table 1.

Natural frequencies of the first 30 modes.

Mode no.	Value (Hz)	Mode no.	Value (Hz)	Mode no.	Value (Hz)
1	0.939	11	2.139	21	2.613
2	1.016	12	2.421	22	2.636
3	1.093	13	2.498	23	2.653
4	1.164	14	2.512	24	2.668
5	1.366	15	2.528	25	2.679
6	1.423	16	2.544	26	2.685
7	1.705	17	2.564	27	2.731
8	1.877	18	2.567	28	2.747
9	1.894	19	2.572	29	2.820
10	1.910	20	2.583	30	2.832

Results of dominant mode selection

According to Yang and Tian (2011), the first $l = 10$ modes are selected as the lower-order modes from engineering judgment, and the remaining 11th to 500th modes are regarded as the higher-order modes for subsequent selection. It is noteworthy, according to engineering practice, l is taken up to around 10–20 modes in general; if l is less than 10, some dominant modes may be lost, and if l is greater than 20, the computational efficiency will be decreased. The correlation coefficient $k_{i, j}$ ( $i = 1, 2, \dots, 10$ ; $j = 11, 12, \dots, 500$ ) are computed by means of the method proposed in section “Mode selection according to the correlation between mode shapes and wind loads.”Figure 6 gives the results of $k_{1, j}$ and $k_{3, j}$ for illustration, and the number in the figure are the mode numbers.

Figure 6.

Correlation coefficient distribution: (a) distribution of $k_{1, j}$ and (b) distribution of $k_{3, j}$ .

According to the value of $k_{i, j}$ , 57 modes are preliminarily selected with their correlation coefficient larger than 0.1. On this basis, the mode participation matrix $Φ_{N \times 67}$ is formed from these 57 selected modes and the $l = 10$ lower-order modes.

The modal participation coefficient $α_{j}$ of the 67 modes are calculated according to the method proposed in section “Simplification of modal coordinate variance matrix.” The criterion $α_{j} > 0.05 %$ is adopted with finally 33 dominant modes selected. The modal accumulation participation coefficient from these modes satisfies $\sum α_{j} > 98 %$ . Some of the larger $α_{j}$ and the corresponding mode numbers are shown in Table 2.

Table 2.

Results of mode participation coefficients.

Mode no.	$α_{j}$	Mode no.	$α_{j}$	Mode no.	$α_{j}$
1	38.33	6	1.764	16	0.186
2	29.53	5	1.049	49	0.155
4	9.253	10	0.813	56	0.126
3	7.507	9	0.644	26	0.124
7	3.920	21	0.348	23	0.108
8	2.612	22	0.236	72	0.096

It is noted in Table 2 that the $α_{j}$ of the lower-order modes are much bigger than those of the higher-order modes, in general. For example, with the largest value of 38.33% for the first mode, the $α_{j}$ of the higher-order modes are small and close except those of the 21st and 22nd modes. This indicates the need to include the contributions of the higher-order modes to the responses and multi-mode responses should be taken into consideration in the wind-induced response analysis.

Wind-induced response analysis

Background response analysis

The background response modal coupling coefficients $η_{jk, b}$ are calculated by equation (34) from the 33 dominant modes selected in section “Results of dominant mode selection,” and 27 groups of background response dominant coupling modes are selected according to the criterion $η_{jk, b} \geq 0.1$ . The largest 12 modal coupling coefficients are shown in Table 3.

Table 3.

Results of background response modal coupling coefficients.

Mode no.		$η_{jk, b}$	Mode no.		$η_{jk, b}$
1	2	0.499	7	8	0.482
3	5	0.498	5	8	0.472
6	9	0.497	3	9	0.465
9	10	0.496	2	7	0.447
4	7	0.495	1	4	0.432
5	10	0.490	4	6	0.402

The 33 dominant modes and 27 groups of background response dominant coupling modes are then considered in the background response analysis. The background displacement responses of the roof are calculated from equation (36). The results are also obtained from the SRSS and CQC methods with all the first 500 modes for comparison with the latter set serving as the reference. Table 4 shows comparison of the results on nine selected nodes on the roof. The locations of these nodes are shown in Figure 3.

Table 4.

Comparison of the background displacement response of representative nodes.

Node no.	CQC (mm)	SRSS		Proposed method
Node no.	CQC (mm)	Response (mm)	Error (%)	Response (mm)	Error (%)
1	7.966	7.098	10.89	7.667	3.756
2	8.311	6.922	16.70	7.822	5.884
3	9.177	6.972	24.02	8.794	4.169
4	9.387	6.847	27.06	8.959	4.560
5	9.010	6.730	25.29	8.601	4.532
6	9.187	6.983	23.99	8.845	3.731
7	8.019	6.921	13.69	7.895	1.546
8	7.968	6.504	18.37	7.636	4.172
9	7.640	6.034	21.02	7.501	1.830

CQC: complete quadratic combination; SRSS: square root of the sum of squares.

Table 4 indicates that the error of the computed background response from the SRSS method is greater than 10% with the largest value of 27.06% at the fourth node. The results from the proposed method have calculation error less than 5%, in general, satisfying the engineering requirement.

Resonant response analysis

Similarly, the resonant response modal coupling coefficients $η_{jk, r}$ are calculated by equation (46) from the 33 dominant modes selected in section “Results of dominant mode selection.” A total of 18 groups of resonant response dominant coupling modes are selected according to the criterion $η_{jk, r} \geq 0.1$ . The largest 12 modal coupling coefficients are shown in Table 5.

Table 5.

Results of resonant response modal coupling coefficients.

Mode no.		$η_{jk, r}$	Mode no.		$η_{jk, r}$
8	9	0.975	1	2	0.421
9	10	0.971	2	3	0.412
8	10	0.925	8	7	0.374
6	9	0.684	11	12	0.291
5	6	0.607	1	3	0.246
3	4	0.484	11	13	0.208

The 33 dominant modes and 18 groups of resonant response dominant coupling modes are considered in the resonant response analysis. The resonant displacement responses of the roof are obtained from equation (48). The results from the SRSS and CQC methods are also obtained for comparison from using all the first 500 modes in the computation. The latter set of results serves as the reference. Table 6 shows the comparison of results at nine selected nodes on the structure.

Table 6.

Comparison of resonant displacement response of representative nodes.

Node no.	CQC (mm)	SRSS		Proposed method
Node no.	CQC (mm)	Response (mm)	Error (%)	Response (mm)	Error (%)
1	2.33	1.86	20.0	2.32	3.05
2	2.34	1.87	20.2	2.33	2.77
3	2.40	1.91	20.4	2.40	2.51
4	2.64	2.10	20.6	2.64	1.38
5	2.76	2.20	20.3	2.76	0.92
6	3.25	2.70	16.9	3.25	0.86
7	3.22	2.80	13.1	3.23	1.74
8	3.13	2.80	10.6	3.14	1.90
9	3.00	2.71	9.41	3.00	1.77

CQC: complete quadratic combination; SRSS: square root of the sum of squares.

The results in Table 6 indicate that the resonant response from the SRSS method has error greater than 10% with the largest value of 20.6% at the fourth node, while the largest error from the proposed method is 3.05% at the first node.

Moreover, according to Tables 4 and 6, the contributions of the resonant component can be found smaller compared with those of the background one. As the stiffness of China National Stadium is relatively greater, which makes the contributions of background component to be dominating, the dynamic characteristic of the structure can be found in Table 1.

Conclusion

A methodology for selecting the dominant modes and for calculating the modal coupling effects in the wind-induced response analysis of long-span roofs is proposed in this article. The wind-induced responses of the China Natural Stadium are calculated with this new approach for illustration of the main features of the proposed approach with the following conclusions:

The contributions of multiple modes to the response should be considered in the wind-induced response analysis of long-span roofs. Based on the lower-order modes shapes, the higher-order mode shapes which are strongly correlated with wind loads could be preliminarily selected. On this basis, the modal participation coefficient is obtained by considering the modal frequency and wind load characteristics. It could be used to effectively select the final dominant modes.

The omission of contributions of modal coupling effects to the wind-induced response of long-span roofs will lead to an under-estimation. According to the “background (resonant) response modal coupling coefficient” defined in this article, background (resonant) response dominant coupling modes can be identified. The wind-induced responses can be obtained by combining the contributions of dominant coupling modes to the response with the results from the SRSS method on the dominant modes. This approach avoids the computation expensive CQC method and effectively takes into account the modal coupling effects.

The study on the wind-induced response of the China National Stadium roof indicates that the proposed method is effective and with sufficient accuracy for engineering application.

Footnotes

Acknowledgements

The authors are grateful to Prof. S. S. Law of the Hong Kong Polytechnic University.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (project nos 51278314 and 51378061), the Natural Science Foundation of Hebei Province (project no. E2012210002), the PhD Programs Foundation of Ministry of Education of China (project no. 20120009110037), and the Research Foundation of Hebei Province for Outstanding Young Teachers in University (project no. YQ2013028).

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