A comprehensive review on estimation of equivalent static wind loads on long-span roofs

Abstract

Equivalent static wind load (ESWL) is helpful information for designing wind-sensitive structures since it converts complex wind-induced structural dynamic analysis into a simplified static analysis and facilitates combination of other loads or actions in structural designs. In recent years, there have been a growing number of long-span roof structures, most of which are sensitive to wind actions. Indeed, the accurate determination of ESWLs has been a primary concern in their wind-resistant design. Although some significant progress has been made on this topic, there are various issues to be addressed or improved due to a variety of reasons. With the further development of long-span roofs, there will be higher requirements for their ESWLs. This paper first combs through the existing representative methods depending on wind-induced structural responses (WISRs) and wind-induced structural stability (WISS), where the ESWLs linked with the former are the mainstream and the ones concerned with the latter are still somewhat in their initial stages of development. In each broad category, some momentous tactics of mathematics and mechanics are utilized to derive different methods. In the conclusion, this paper finally summarizes the existing achievements, highlights the technical challenges that hinder the current methods from being widely adopted and proposes some latent solutions for the future research. This review paper is anticipated to be a comprehensive reference for researchers and professionals in this field of study.

Keywords

long-span roof equivalent static wind loads (ESWLs)wind-induced structural responses (WISRs)wind-induced structural stability (WISS)

Introduction

Equivalence thought, which can effectively change a complex process or system into its simple substitution in conjunction with transformation theory under certain principles, is a topic of much attention in various fields. In structural wind engineering, this thought can be embodied by equivalent static wind loads (ESWLs). Stochastic dynamic analysis of structures is often much more intricate than deterministic static analysis, the current structural designs, thereby, often require complicated random dynamic wind loads on structures to be modeled as their ESWLs for simplifying structural designs or analyses-the ESWLs can be combined with other static loads or actions and easily used by engineers during the design of structures (Frontini et al., 2022). Because the treatment is available for evaluating structural dynamic performance indirectly, an admirably deft bond between structural static and dynamic analyses is bridged to reflect the dynamic-static combination idea, one of important ideas in structural engineering.

Because of their architectural, structural, and economical superiorities, long-span roofs are often highly preferred for some large-scale public buildings such as gymnasia, museums, city halls, airports, cinemas, theatres and so on. It can be firmly believed from massive reports that there has been a growing trend in the use of long-span roofs worldwide. In terms of these structures themselves, they are generally featured by light mass, great flexibility, little damping, low and dense natural frequency, and the coupling effect between different structural modes, as well as their varied and irregular structural shapes (Fu et al., 2015). Meanwhile, as far as the external environment is concerned, such roofs are commonly low rise and in the atmospheric boundary layer with relatively large wind speed change and high turbulence, which will cause the structures to experience complicated flows, wind pressure distributions and aerodynamic actions (Fu et al., 2015, Holmes, 2007). Based on the two characteristics, these structures are inherently sensitized to wind excitations which usually dominate their structural designs, their resulting dynamic wind effects are significant but changeable without lucid rules to follow (Holmes, 2007). In the structural designs, the accurate ESWLs determination is believed to play a crucial role (Chen and Zhou, 2007, Holmes, 2007, Uematsu et al., 1997a, 1997b, Zhou et al., 1999). The ESWLs research of long-span roofs has been a hot spot in structural wind engineering for quite a while, since new challenges keep pouring out despite some encouraging achievements achieved.

This paper intends to present a critical review of the representative ESWLs methods for long-span roofs, fixing attention on their computational contents, advantages and disadvantages. At the time of designing structural elements, it can be known from mechanics that when they work properly, their strength, stiffness and stability should be ensured to accommodate to different requirements. Most of the ESWLs are estimated based on wind-induced structural responses (WISRs)-internal forces (or stresses) and displacements. Put differently, they direct at the strength and stiffness questions under wind loads to a certain extent in the structural designs. On the other hand, the ESWLs of some specific roofs are mainly subject to wind-induced structural stability (WISS), for example, single-layer space shells. In the existing methods, the WISRs-based ESWLs will predefine one or multiple WISRs, instead the WISS-based ESWLs require one instability mode instead of WISRs, which will need different mathematical and mechanical principles. The computational results of the ESWLs hailing from WISRs and WISS for a same structure can be considerably distinct from each other (Li and Tamura, 2004, 2005). According to the above three aspects, the ESWLs are categorized into the ones based on the structural WISRs and WISS in this paper. Finally, what have been achieved are summarized, and some beneficial suggestions driving the future research attempts are provided in the conclusion. Consequently, this review paper is expected to be a comprehensive reference for researchers and professionals involved in the wind-resistant analyses or designs of long-span roof structures and even other wind-sensitive structures.

The methods based on WISRs

It is often the case that ESWLs are correlated with WISRs. In this paper, the ESWLs based on WISRs are the static loads, under which structural responses (i.e., ESWLs-induced structural responses or ESWLs-ISRs for short.) are theoretically required to equal the preselected target WISRs (normally the peak dynamic responses under the complicated random dynamic wind loads on structures) (Chen and Zhou, 2007, Frontini et al., 2022). The validity of WISRs-based ESWLs needs to be identified, one admitted way is to rate the errors between ESWLs-ISRs and actual target WISRs which will be mentioned many times later. There is no doubt that the smaller errors will bring the more satisfying ESWLs.

In structural engineering, there are two generic ways to make the loads on structures equivalent: One is to replace complex force systems with their simple ones according to the three elements of vectors (magnitude, direction, and action location). The other is to first compute the structural responses under complex force systems, then get the structural simple force systems reversely based on specially aimed known responses.

From mechanics, it is often required to solve structural responses under given loads, but less work is done to determine their loads by known responses. If the former is mathematically taken as a direct problem, the latter can be considered as an inverse problem. In mathematics, a unique solution will be solved to a direct problem. Nonetheless, an inverse problem is likely to own many and even infinitely many solutions, increasing its complexity. For applicability, an inverse problem is desired to have a unique solution, this can be done with some specific constraints-the initial conditions in mathematics or the boundary conditions in mechanics, the solutions differ from constraint to constraint. Finding the WISRs-based ESWLs can be conceived of as an inverse question with the WISRs being the initial conditions. This will give rise to a bulk of ESWLs for different WISRs or a same WISR, for which extra care should be taken.

So far, the majority of ESWLs are based on WISRs. Several methods are applicable for estimating the ESWLs of high-rise buildings whose WISRs are usually dominated by the first structural mode, with the corresponding ESWLs being aimed at a single target WISR in general. For most of long-span roofs, there is a need to involve multiple structural modes. However, a single monitored target WISR may be insufficient to characterize their dynamic effects and ESWLs. It is thus necessary to consider how to realize the equivalence of multiple peak WISRs. ESWLs can be divided into two categories: Single target and multiple target ESWLs (Frontini et al., 2022), this section will enlarge on their applications in long-span roofs.

The single target equivalent methods

The single target equivalent methods mean that a whole structure’s ESWLs are solved using one preselected target WISR. According to the evolution of these methods, this subsection includes: (1) The ESWLs based on the gust response factor (GRF); and (2) The ESWLS based on the load-response-correlation (LRC) method.

The methods based on the GRF method

It was not until the 1960s that ESWLs were budded in the wind-resistant studies of high-rise buildings. Davenport (Davenport, 1961, 1967) pioneered the idea of gust loading factor (GLF), and leant towards the GRF for more precision (The GLF and GRF are nominated as GRF without a deliberate discrimination in this paper), marking the advent of ESWLs:

{\hat{r}}_{i} = G \cdot {\bar{r}}_{i}

(1a)

G = 1 + g \cdot σ_{r_{i}} / {\bar{r}}_{i}

(1b)

where

{\hat{r}}_{i}

{\bar{r}}_{i}

are the peak and mean values of a target WISR time history-r_i(t) in a time duration, respectively; G and g denote the GRF and the statical peak factor with a given reliability level in this paper, respectively; and

σ_{r_{i}}

is the standard deviation or the root-mean-square (RMS) value of r_i(t).

In order to predict the structural dynamic wind loads, the ESWLs on a high rise building are expressed as the product of the mean wind loads and a specific G, yielding the along-wind structural buffeting responses. In particular, the structural top displacement GRF is defined as one entire structure’s GRF under the premise that the wind loads and/or the WISRs abide by the Gaussian probability distribution.

The approach is handy for a structure with dominant wind-induced drag, in which it is assumed that the wind-induced resonance is in a single mode of vibration (the first structural mode) whose shape varies linearly above ground (Frontini et al., 2022, Isyumov, 2012). This assumption catches the essence of the dynamic wind process for most buildings, and provides a manageable framework for extending the procedure to higher structural modes kindled by buffeting-induced drag loads (Isyumov, 2012). The GRF allows a designer to estimate the ESWLs through a constant amplification factor- G, evaluated on the basis of the mean and peak displacements, and compulsively assumed equal to the ratio between the peak and mean values of an internal force (Frontini et al., 2022).

Due to its simplicity and convenience in application-the primary advantage, this highly profiled method was also acclaimed in determining the ESWLs on long-span roofs at one time. Several researchers (Uematsu et al., 1996, 1997b, 1997a, 1999, 2008, Marukawa et al., 1993, Uematsu and Yamada, 2002) investigated the GRFs for the structural frames of some flat roofs with regular, symmetrical, and simple geometries, as well as the closed forms regardless of interior pressure. From structural point of view, these roofs can be hypothetically supported by a series of simple beams arranged in parallel and connected with one another collar beams, or act like an elastic plate under wind loads. The wind-induced oscillations of these structural frames (beams and plates), thus, may be governed by their fundamental mode and frequency. Building upon the wind-tunnel measurements for the first modal forces acting on the roofs, and upon a statistical analysis over the dynamic WISRs of these structural frames, they first conceptualized a unified GRF (Marukawa et al., 1993, Uematsu et al., 1996, 1997b, 1997a, 1999, 2008, Uematsu and Yamada, 2002):

G = 1 + g \cdot r_{F} \cdot R_{m}

(2a)

g = \sqrt{2 \ln (f_{1} T)} + 0.5772 / \sqrt{2 \ln (f_{1} T)}

(2b)

r_{F} = {\tilde{C}}_{F 1} / | {\bar{C}}_{F 1} |

(2c)

where

{\bar{C}}_{F 1}

and

{\tilde{C}}_{F 1}

stand for the mean and RMS values of the first modal force coefficient-C_F1(t), with their associated ratio γ_F; and R_m is the resonant magnification factor, which can be solved by (Uematsu et al., 1996, 1997b, 1997a, 1999, 2008, Uematsu and Yamada, 2002):

R_{m} = {1 + \frac{π}{4 η_{1}} \frac{f_{1} \cdot S_{F 1} (f_{1})}{σ_{F 1}^{2}}}^{\frac{1}{2}}

(2d)

where σ_F1 is the RMS value of the first modal force F₁(t); S_F1(f₁) represents the power spectrum of F₁(t) relative to the first natural frequency f₁; η₁ is the critical damping ratio of the first mode; and T is time duration of 600s.

For different structures, the studies of (Marukawa et al., 1993, Uematsu et al., 1997a, 1997b, 1999, Uematsu and Yamada, 2002) crystallized r_F and R_m into a series of empirical equations relying on the wind parameters (turbulence intensity of approaching flows, wind velocity, and wind directions) and structural parameters (roof geometry, structural frame location, critical damping ratio, fundamental frequency and mode shape), which are made into Table 1.

Table 1.

The Parameters in equation (2a–2d).

Roof model and coordinate system	Structural frames	Empirical equations
Roof model and coordinate system	Structural frames	r _F	$\frac{f_{1} \cdot S_{F 1} (f_{1})}{σ_{F 1}^{2}}$ or R_m	Explanations
(B=L) (Uematsu et al., 1997a)	A series of simple beams arranged in parallel and connected with one another collar beams	$\begin{array}{l} r_{F} = L_{B} K_{l} \exp (- γ \frac{L}{H}) \\ K_{l} = k_{0} + k_{1} I_{u H}^{2} \\ L_{B} = l_{0} + l_{1} {(ξ / H)}^{2} \end{array}$	$\begin{array}{l} \frac{f_{1} \cdot S_{F 1} (f_{1})}{σ_{F 1}^{2}} \\ = A {(\frac{f H}{U_{H}})}^{- β} \end{array}$	1. 1) the edge region (ξ/H=0.5 and θ=0^o): γ=0.1; L/H≤(L/H)_cr=6; k₀=0.15; k₁=5.0; l₀=0.95; l₁=0.3 2) The interior region (ξ/H=0.5 and θ=90^o): γ=0.15; L/H≤(L/H)_cr=4; k₀=0.2; k₁=3.5; l₀=1; l₁=0 2. f^=f₁H/U_H≥0.4 and L/H=3∼6: 1) ξ/H=0.5 and θ=0^o: A=0.012, β=3; 2) ξ/H=0.5 and θ=90^o: A=0.036, β=2.3 3. U*_H and I_uH =the wind speed and the turbulance intensity at roof height H, respectively
(Uematsu et al., 1997b)	A thin flat plate simply supported on 4 edges	$\begin{array}{l} r_{F} = [K_{l} + 0.043 {(\frac{D}{H})}^{2}] \\ \cdot \exp (- 0.12 \frac{B}{H}) \\ K_{l} = 0.09 + 3.8 I_{u H}^{2} \end{array}$	$\begin{array}{l} \frac{f_{1} \cdot S_{F 1} (f_{1})}{σ_{F 1}^{2}} \\ = A {(\frac{f H}{U_{H}})}^{- β} \end{array}$	1. B/H=width-to-height ratio, and D/H=depth-to-height ratio, where B and D shoulder the roof spans in the across-wind and along-wind directions, respectively. B/H≤6 and D/H≥2 2. f^*=f₁H/U_H≥0.4 and I_uH≥0.3: A=0.014, β=3.5
(Marukawa et al., 1993)	A series of simple beams in L direction	$r_{F} = 2.2 I_{u H}^{2} + 0.19$	$R_{m} = \sqrt{r_{B} + r_{R}}$	1. f^=f₁H/U_H≥0.4 and L/H>1 2. r*_B and r_R represent the quasi-static and resonant WISRs, R_m is only raised conceptually
Circular flat roofs with height H and span D (Uematsu et al., 1999)	A thin elastic plate simply supported on edge	$\begin{array}{l} r_{F} = 1.71 I_{u H}^{2} \\ \cdot \exp (0.2 \frac{D}{H}) + 0.15 \end{array}$	$\begin{array}{l} \frac{f_{1} \cdot S_{F 1} (f_{1})}{σ_{F 1}^{2}} \\ = A {(\frac{f H}{U_{H}})}^{- β} \end{array}$	1. The span-to-height ratio (D/H) ranges from 1 to 8 2. η₁=0.02 3. f^*=f₁D/U_H>1 and I_uH=0.12∼0.26: A=0.11I_uHln(D/H+0.22)+0.01, β=2.7
(Uematsu and Yamada, 2002)	A thin elastic plate simply supported on edge	$\begin{array}{l} r_{F} = 3.4 I_{u H}^{2} \\ \cdot \exp (0.04 \frac{D}{H}) + 0.12 \end{array}$	$\begin{array}{l} \frac{f_{1} \cdot S_{F 1} (f_{1})}{σ_{F 1}^{2}} \\ = A {(\frac{f \sqrt{D H}}{U_{H}})}^{- β} \end{array}$	1. The span to height ratio D/H ranges from 1 to 5.33 2. η₁=0.02 3. $f^{*} = \sqrt{D H} / U_{H} > 0.7$ 4. 1) Flow I: A=0.012, β=3.7; 2) Flow II: A=0.015, β=3.7. Flows I and II are for typical open country and urban terrains, respectively

The above studies condense the ESWLs as regards a big class of roofs into the parametric analysis method. The most striking feature of the parametric analysis methods is that on the basis of wind tunnel test, the ESWLs expressions of a big class of roofs fall back on some structural and wind parameters to facilitate actual engineering applications, without having to do a lot of tedious calculations for every structure. Nevertheless, they suggest that the ESWLs only target at the displacement GRF. In practice, long-span roofs have multiple kinds of WISRs that matter as much as displacement in most cases, such as macroscopic internal forces (bending moment, shear force, axial force and torque). This method is originally capable of suiting any WISRs, but it might bring an inaccurate estimation if the displacement GRF is imposed on any category of WISRs, because the ESWLs for any other WISRs are doomed to be identical to the displacement-related ones (Zhou et al., 1999, Zhou and Kareem, 2001, Kareem and Zhou, 2003). However, the relationship between displacement and other WISRs is not necessarily linear. Thus, the ESWLs for other WISRs save the ESWLs of displacement should be considered.

The ESWLs of beam-supporting flat roofs are supposed to vary with their various maximum WISRs (Tamura et al., 1992), because of the foregoing inverse question’s characteristics. The various maximum WISRs will offer different constraints to result in fickle ESWLs. Tamura et al. (1992) derived the GRFs of some simply supported rigid and elastic beams based on the maximum bending moment, shear force and displacement, showing the diversity between the different GRFs for one same structure.

Due to the disparity in cross-sectional dimensions of structural members, it is demanding to judge the most unfavorable members only by the internal forces in view of the strength design theory in structures (Ma et al., 2022). Among structural responses, microscopic stresses are also important in addition to the above macroscopic internal forces, the corresponding ESWLs are worthy of study. Having no regard for internal pressure, the ESWLs of a rigidly jointed single-layer high-rise dome can be estimated by its GRFs of the axial and bending stresses, the same regime, however, would be thwarted for a low-rise dome (Uematsu et al., 2001). Shortly after (Uematsu et al., 2001), Uematsu et al. (2002) further renovated the WISRs and the ESWLs of these domes. In their work, the dynamic WISRs can be evaluated by quasi-static analysis, because the resonance effect is relatively small. In the common procedure for structural designs, the extreme fiber stress on the member cross section is the most important response for the roofs. Since the GRFs for the single-layer latticed domes in Figure 1 have some in common with the ones (see Table 1) in Uematsu and Yamada (2002), Uematsu et al. (2002), after some modification for Uematsu and Yamada (2002), threw out the stress GRF empirical formulas in parameterization for the structural frames:

\frac{G_{d o m e}}{G_{f l a t}} = {\begin{array}{c} 4 (C_{\max} - 1) \frac{h}{D} + 1.0 (f o r 0 \leq \frac{h}{D} \leq 0.25) \\ C_{\max} (f o r 0.25 \leq \frac{h}{D} \leq 1.0) \end{array}

(3a)

C_{\max} = {\begin{array}{c} 1.0 + 4 \frac{f}{D} (f o r 0 \leq \frac{h}{D} \leq 0.1) \\ 1.8 - 4 \frac{f}{D} (f o r 0.1 < \frac{h}{D} \leq 0.2) \end{array}

(3b)

where G_flat is the GRFs in Uematsu and Yamada (2002). D/H≤7 is asked here to compute γ_F in Uematsu and Yamada (2002); for the quasi-static response, R_m=1 in Uematsu and Yamada (2002); and g=4 in equation (2b) for these roofs.

Figure 1.

The roof geometry (side view) (Uematsu et al., 2002). Note: f/D=0.05, 0.10, 0.20; h/D=0, 1/16, …, 16/16.

An exploration regarding the stress GRFs of an openable and a closed truss roofs was also done by Ma et al. (2022). The GRFs of the structural bars and beams are clearly found to fluctuate greatly for different target stresses, which raises a confusing problem for the GRF method to select one single equivalent target considering the combination of different load effects. In the above-mentioned scattered GRFs, the largest absolute bar stress of each roof is taken as their own equivalent target to calculate their GRFs and corresponding ESWLs, in which (1) the two ESWLs-ISRs corresponding to the target WISRs match well with their respective target WISRs; and (2) the most remaining bar ESWLs-ISRs are generally comparable to their corresponding target WISRs, especially for the larger bar stresses; but (3) the ESWLs screw up in several beams with larger stress, whose ESWLs-ISRs bear a big departure from their target WISRs, requiring additional analyses.

The method of Ma et al. (2022) taps the GRF method in the specified roofs, ushering in a hope to find a suitable target WISR in complicated long-span roofs. However, the method will disable itself when selecting one target WISR morbidly. Therefore, it needs specific analysis on determining the target WISR from the perspective of overall evaluation of this method in Ma et al. (2022).

Lou et al. (2000) managed to explore the WISRs of a flexible square flat roof supported on a closed building, a discussion was made of the nodal GRF at node i:

β_{z i} = \frac{P_{t i}}{P_{s i}} = \frac{P_{d i} + P_{s i}}{P_{s i}} = 1 + \frac{P_{d i}}{P_{s i}}

(4a)

P_{d i} = {\hat{G}}_{a} m_{i} σ_{a i}

(4b)

P_{s i} = \frac{{\bar{C}}_{i} A_{i} {\bar{V}}^{2}}{2}

(4c)

where P_ti, P_di and P_si are the total wind-induced force, the dynamic wind force and the static wind force, respectively;

{\hat{G}}_{a}

= the peak acceleration coefficient approximately set to 3.0; σ_ai and m_i are the acceleration RMS value and the roof mass, respectively;

{\bar{C}}_{i}

and A_i equal to the local mean wind pressure coefficient given by wind tunnel test and the roof surface area, respectively; and

\bar{V}

is the mean wind speed at the roof’s height.

The result of Lou et al. (2000) indicated that the GRF distributions on the whole surface made up a curved surface resembling the first structural mode. From equation (4b), this method entrusts itself to nodal acceleration rather than internal forces or stresses, affirming that the variability of WISRs results in variable ESWLs.

It should be noted that the WISRs in the above long-span roofs are generally dominated by their first structural modes. In this circumstance, every node in the structural frames almost synchronizes with each other to its maximum value as a structure vibrates. Because the fluctuating and mean displacements in the first structural modes are merely included in deriving the formulas, the displacement GRF is constant for an entire structural frame (Zhou et al., 1999, Kareem and Zhou, 2003). The GRF of an isolated WISR in the traditional GRF method of Davenport (Davenport, 1961, 1967) is marketed to that of an entire structural frame, saving the trouble of determining a hand-picked and representative WISR to reflect the ESWLs on an overall roof. In this sense, the narrow definition of the traditional GRF method is gifted with a broader meaning. In effect, many long-span roofs are closely spaced in natural frequencies, so their WISRs will be affected by multi-order structural modes (Uematsu et al., 1997). At this point, the multi-mode participation perhaps paralyzes all nodes in the structural frames from arriving at their maxima synchronously during structural motion, and it will become rather stiff to preselect an accurate displacement and even one other WISR.

Among long-span roofs, the free roofs usually supported only by columns also gain widespread popularity. They seem to be more vulnerable to wind loads than those enclosed buildings, since wind loads will attack their top and bottom surfaces directly (Uematsu et al., 2007, 2013, 2014, 2015, Uematsu and Yamamura, 2019). For some gable, troughed and mono-sloped free-standing canopy roofs, cylindrical free-standing canopy roofs and hyperbolic paraboloid free roofs, if they are assumed to be rigid (quasi-static) and simply propped against four corner columns, the wind-induced column axial forces can thus be regarded as the most important load effect to discuss their design wind loads (Uematsu et al., 2007, 2013, 2014, 2015, Uematsu and Yamamura, 2019).

For the three kinds of roofs which are symmetrical in appearance and solid without porosity: (1) The hyperbolic paraboloid free roofs; (2) The gable, troughed and mono-sloped free-standing canopy roofs; and (3) The cylindrical free-standing canopy roofs, Uematsu et al. (Uematsu et al. 2007, 2014) and Uematsu and Yamamura (2019) sketched their design wind force coefficients in different zones that can be easily converted to the structural ESWLs:

{C_{N W}}^{*} = \frac{γ C_{N W 0}}{G_{f}}

(5a)

{C_{N C}}^{*} = \frac{γ C_{N C 0}}{G_{f}}

(5b)

{C_{N L}}^{*} = \frac{γ C_{N L 0}}{G_{f}}

(5c)

where

{C_{N W}}^{*}

and

{C_{N L}}^{*}

are the design wind force coefficients over the windward and leeward halves for the roofs (1) and (2) or the zones for the roof (3) (see Figure 2);

{C_{N C}}^{*}

is the value in the central zone of the roof (3); C_NW0, C_NC0 and C_NL0 specify the basic values, and are uniformly distributed on the above-mentioned halves or zones. C_NW0, C_NC0 and C_NL0 linearly combine the aerodynamic lift and moment coefficients at the specific wind directions producing the maximum tension and compression in the columns (the specific wind directions are schematically illustrated in Figure 2.), and the aerodynamic lift and moment coefficients result from the even distribution of their corresponding concentrated forces within their respective zones; γ, the correction factor to consider that the axial force in the column may become the maximum for an oblique wind different from the azimuths in Figure 2, is defined as the ratio of the actual peak value to the predicted value from C_NW0, C_NC0 and C_NL0; and G_f is the gust effect factor investigating the dynamic effect of flow turbulence on the column axial forces, and defined as the ratio of the maximum or minimum axial force to the mean value induced in the column.

Figure 2.

The specific wind directions that can produce the maximum load effect and the zoning of three kinds of roofs (θ=0^o or 90^o for the roof (1), θ=0^o or 180^o for the roof (2), and θ=0^o or 45^o for the roof (3). (a) The roof (1) (Uematsu et al., 2014). (b) The roof (2) (Uematsu et al., 2007). (c) The roof (3) (Uematsu and Yamamura, 2019)

For the roofs with the same shapes as those of (Uematsu et al., 2007, 2014) but with holes, the structural porosity will decrease their wind loads significantly (Uematsu et al., 2013, 2015). When determining the structural ESWLs, a reduction factor R, which is defined by the ratio of the ${C_{N W}}^{*}$ or ${C_{N L}}^{*}$ for the porous roofs to that for the corresponding solid roofs and formulated as one simple exponential function of porosity, will be supplemented to consider the porosity’s effect. That is, the ${C_{N W}}^{*}$ and ${C_{N L}}^{*}$ of the porous roofs can be provided by the counterparts for the solid roofs multiplied by the R.

The comparisons of the proposed methods of (Uematsu et al., 2007, 2013, 2014, 2015, Uematsu and Yamamura, 2019) with the relevant standards were made to confirm their validity.

If C_NW0 (and/or C_NC0, C_NL0) and γ/G_f (or Rγ/G_f) take over the mean wind loads and the GRF in the traditional GRF method, respectively, the method of (Uematsu et al., 2007, 2013, 2014, 2015, Uematsu and Yamamura, 2019) can be roughly incorporated into the GRF method. It is still fed with the maximum tension and compression WISRs, but avert selecting one right WISR on the roofs painstakingly that is handed over to the WISR in columns, which widen the traditional GRF method. In the meantime, it considers the influences of the structure geometry (the rise to span ratios, the roof slops, the porosity and the structural symmetry) and the wind fields (the wind directions and the turbulence), the simple parametric expressions are put into full use. Nonetheless, it is necessary to discuss whether the ESWLs derived from the axial force of the columns can secure the roofs against failure.

The above methods are riveted on linear structures, failing to consider the effect of structural nonlinearity. With the increase of complexity and lightness of structures, moderate nonlinearities in some specific structures need to be incorporated, otherwise, the GRFs that exclusively take into account linear effects will discount the safety of these structures (Kasperski, 1992, Kasperski and Niemann, 1992). At some positions of several cable-suspended and cable-supported membrane roofs-two sorts of typical nonlinear tensioning structures, Suzuki et al. (1997), Shen and Yang (1999) and Zhou et al. (2013) probed the GRFs of multiple kinds of control WISRs, and conceived some empirical and parametric equations or values relevant to weak geometrical nonlinearities manifested in Tables 2 and 3 and Figure 3, which broaden the GRF’s use scope.

Table 2.

The GRFs for the Cable-suspended Roof (Suzuki et al., 1997).

The GRFs
	1/4 span	1/2 span	3/4 span
The axial forces	1.8	2.0	1.8
The displacements	5.6	3.6	6.5

Table 3.

The GRFs for the Hyperbolic Paraboloid Cable nets (Shen and Yang, 1999).

The roofs	The GRFs
The roofs		The maximum WISRs	The minimum WISRs
With elliptical plans	The node displacements & the inner forces of pre-tensioning cables	2.3	-0.3
With elliptical plans	The inner forces of main cables	$2.2 + \frac{0.4}{400} (μ_{z} W_{0} - 300)$	$- 0.4 + \frac{0.4}{400} (μ_{z} W_{0} - 300)$
With rhomboic plans	The node displacements	2.2	-0.2
	The inner forces of pre-tensioning cables	2.3	-0.4
	The inner forces of main cables	2.8	-1.6

Note: W₀(N/m²)=the reference wind pressure; μ_z=the exposure factor for wind load.

Figure 3.

The GRFs for the cable-supported membrane roof (Zhou et al., 2013). (a) The nodes and the links. (b) The displacement, stress, and cable force GRFs.

Being an established approach, the GRF method still suffers from the other limitations: (1) The ESWLs are implied to display the same shapes as mean wind loads if a structure only has a GRF, which is not true in some cases (Holmes, 2002, Li and Tamura, 2004, 2005); (2) It falls short in the cases of a zero mean load or WISR, where the ESWLs may not be appropriately defined (Chen and Zhou, 2007, Zhou et al., 1999, 2000, Kareem and Zhou, 2003); (3) It will stumble over WISS analysis, a bigger GRF may pose a lower safety (Li and Tamura, 2004, 2005, Gu and Huang, 2015); (4) If the ESWLs are intended for a particular WISR of interest, the GRFs may be quite distinct from different WISRs for the structures with similar geometric configurations but disparate structural systems, which indicates that the ESWLs given by a single GRF is incapable of providing the adequate predictions of all peak WISRs (Chen and Zhou, 2007); and (5) Under this method, the wind speed, pressure and resulting WISR have been generally treated as stationary random processes, which is appropriate for the large-scale wind systems whose wind process is reasonably stationary, but for the small-scale non-stationary wind events, such as thunderstorm winds, the GRF WISR will greatly under-estimate the actual WISR (Edmund et al., 2002). It should also be recognized that the stationary assumption nearly exists in the following methods. As such, the method may fall into disfavour in long-span roofs, and call for the improved measures.

The methods based on the LRC method

In the ESWLs on long-span roofs, the LRC-based methods boost their developments. This section will introduce the LRC method, and then followed by the ESWLs based on the LRC method.

The LRC method

The WISRs-based ESWLs are often bound up with the makeup of WISRs. By means of the characteristics of along-wind WISR spectra of structures, structural dynamic WISRs are generally made up of mean, background, and resonant components (Davenport, 1967, 1995, Dyrbye and Hansen, 1997), first shedding light on an opportunity to fragment structural ESWLs into static (i.e., mean), quasi-static (i.e., background) and resonant contributions (Repetto and Solari, 2004).

By appealing to the WISR spectra and to the hypothesis that every WISR complies with Gaussian probability distribution, Kasperski (1992) and Kasperski and Niemann (1992) initiated a whole structure’s ESWLs for one special peak target WISR ${\hat{r}}_{i}$ , in which the kth ESWL value is expressed as:

P_{E k} = {\bar{P}}_{k} + g ρ_{r_{i} P_{k}} \cdot σ_{P_{k}}

(6)

where

{\bar{P}}_{k}

denotes the mean ESWL value which can be directly taken from averaging the kth wind load time history P_k(t);

σ_{P_{k}}

takes the RMS value of P_k(t); and

ρ_{r_{i} P_{k}}

arises from the correlation coefficient between the WISR time history of r_i(t) corresponding to

{\hat{r}}_{i}

and P_k(t).

For being more understandable, the column vector {P_E,f} fashioned from all the fluctuating ESWL values in P_Ek, taking discrete structural systems as examples, will bend towards a quotient:

{P_{E, f}} = \frac{g [C_{f}] {(I_{i})}^{T}}{{\hat{r}}_{i, B}}

(7)

where [C_f] takes on the covariance matrix of fluctuating wind loads; (I_i) takes the row vector corresponding to

{\hat{r}}_{i}

out of structural influence function matrix in this paper; (⋅)^T produces a matrix’s or vector’s transpose in this paper; and

{\hat{r}}_{i, B}

is in charge of the background component in

{\hat{r}}_{i}

(the background WISR for short in this paper).

The method (Kasperski, 1992, Kasperski and Niemann, 1992) is commonly known as the load response correlation (LRC) method hereafter referred to as the traditional LRC (T-LRC) method.

Being developed in wind tunnel tests of rigid low-rise buildings (Holmes, 1992, Holmes et al., 1995, Ginger et al., 2000), the T-LRC method can determine the relevant ESWLs from the mean and background WISRs, even without regard to the possibility of resonant amplification (Chen and Zhou, 2007, Holmes, 2002, Ginger et al., 2000). It opens a brand-new research era grounding the anticipated background ESWLs of fluctuating wind loads on solid mathematical and mechanical theories (Holmes, 2002). To be specific, for a desirable peak WISR, it moulds a most probable load distribution with a distinct physical meaning (Kasperski and Niemann, 1992, Chen and Kareem, 2004).

However, when there are a vast quantity of wind load time histories on a structure to consider, all the correlation coefficients will be computationally time-consuming and challenging, it is not required to calculate them directly (Holmes, 1992, Fu et al., 2008). Furthermore, the method just tosses out a basic orientation, how to find out some more feasible, accurate and targeted methods is noteworthy. Consequently, some improved LRC methods with an emphasis on the fluctuating ESWLs, which can afford a good reference for the ESWLs on long-span roofs, should need an exquisite carving.

For rigid low-rise buildings, with the aid of the POD method in (Holmes, 1990), Holmes (1992) and Chen and Zhou (2007) extended the T-LRC method to mainly institute the fluctuating ESWLs for the RMS component of one ${\hat{r}}_{i}$ (i.e., the RMS WISR for short in this paper). Because the correlation coefficients in the T-LRC method can be devolved to the covariance matrix of fluctuating wind loads, the POD method shines here. The fluctuating ESWLs in this paper-{P_E,f} is, hence, calculated as (Chen and Zhou, 2007, Holmes, 1992):

{P_{E, f}} = g \sum_{k = 1}^{K} c_{k} {ϕ_{k}} \sqrt{λ_{k}}

(8a)

c_{k} = \frac{r_{k_σ_{r_{i}}}}{σ_{r_{i}}} = \frac{(I_{i}) {ϕ_{k}} \sqrt{λ_{k}}}{σ_{r_{i}}}

(8b)

where {ϕ_k} and λ_k refer to the kth POD mode and eigenvalue, respectively;

r_{k_σ_{r_{i}}}

means the response corresponding to

σ_{r_{i}}

under the ESWLs (i.e.,

{ϕ_{k}} \sqrt{λ_{k}}

) associated with {ϕ_k}; and c_k is the kth linear combination or weighting or contribution factor, actually signifying the contribution of the ESWLs-ISR furnished by {ϕ_k} to the total

σ_{r_{i}}

This proposition on the subtle use of POD modes as the ESWLs has not only helped mirror the reduced-order aptitude in mathematics, because a small section of lower-order POD modes with larger eigenvalues have a flair for generating high-precision ESWLs, WISRs and wind-field reconstruction (Chen and Zhou, 2007), but has also guided it in grasping how structures respond to spatially varying wind loads (Chen and Zhou, 2007, Holmes, 1992), as the ESWLs only hinge on the aerodynamic characteristics of approaching flows and buildings, getting rid of structural behaviors and systems. Plus, if a linear space is comprised of all $g {ϕ_{k}} \sqrt{λ_{k}}$ , in which $g {ϕ_{k}} \sqrt{λ_{k}}$ is identified as the kth coordinate system, {P_E,f} can be thought of as an expansion in the space with c_k being the kth coordinate, equation (8a) and (8b) is equipped with a clear mathematical and physical meaning.

For {P_E,f}, though the above improved LRC method advances the T-LRC method to a certain extent, it still focuses on the background ESWLs, neglecting the resonant ones and even their cross ones. But without a proper consideration of the fluctuating ESWLs, the method will find itself less employable in engineering structures because noticeable errors will arise (Ma et al., 2022). Consequently, the methods relating to the background, resonant, and even their cross WISRs, as well as the corresponding ESWLs are indispensable.

For long-span bridges, the dynamic displacements based on structural mode decomposition can be viewed as the quasi-static responses under the spring resuming forces composed of multiple-order structural mode inertial forces (SMIFs). Thus, once the dynamic displacements are obtainable, any arbitrary dynamic responses can be calculated in the name of static analysis with the related structural influence function (Chen and Kareem, 2001). The quasi-static property can be grafted onto structural WISRs and ESWLs, the WISRs under the spring resuming forces, at this juncture, match the background displacements. Therefore, the ESWLs for one interesting peak WISR could be accessible by the T-LRC method, in which {P_E,f} was styled into the linear combination of a series of SMIFs (Chen and Kareem, 2001):

{P_{E, f}} = g \sum_{k = 1}^{K} {P_{e, k}} W_{k}

(9a)

{P_{e, k}} = [M] {φ_{k}} ω_{k}^{2} σ_{q_{k}}

(9b)

W_{k} = \frac{ó_{r r_{k}}^{2}}{σ_{r} σ_{r_{k}}}

(9c)

where {P_e,k}=the kth SMIF originating from structural mass matrix-[M], the kth structural mode-{φ_k}, the kth circular frequency-ω_k and the RMS value of kth modal coordinate-

σ_{q_{k}}

; σ_r functions as the RMS value of the total dynamic WISR time history r(t) against the concerned WISR;

σ_{r_{k}}

contributes the RMS value of the component r_k(t) from the kth structural mode in r(t);

σ_{r r_{k}}^{2}

is the covariance between r(t) and r_k(t); and W_k, the weighting/combination/contribution factor of {P_e,k}, serves as the correlation coefficient in physical meaning.

Assuming the dynamic WISR and its every component are jointly the zero-mean Gaussian process, the ESWLs of every structural mode will provide the corresponding most probable peak load distributions (Chen and Kareem, 2001). When multi-mode coupled responses and closely spaced natural frequencies are required, the widely-known CQC method in structural dynamics can be adopted to obtain the fluctuating WISR (Chen and Kareem, 2001).

Above all, this method streamlines computation by leaving out the above separations in the fluctuating WISR and its corresponding ESWLs whose contributions are considered in whole. Meanwhile, the most possible probability level is imparted to the ESWLs. Likewise, {P_E,f} can be expanded in the linear space consisting of g{P_e,k}, W_k is its kth coordinate on the coordinate system-g{P_e,k}, which gives a better insight into the mathematical and physical meanings. Moreover, this method breaks through the hardships of modal response coupling and correlation in multi-mode vibration to raise the accuracy.

It can be summarized here that the above improved LRC methods indicate two significant basic avenues (the POD and structural modes) flourishing to this day for proposing the fluctuating ESWLs, and the POD modes give the enlightenment to the principal static wind loads (PSWLs) modes mentioned later. Additionally, the two methods can be practically used for all wind-sensitive structures, because the mathematical and mechanical equations have nothing to do with structural types.

The methods associated with structural modes and the LRC methods

There is a boom of the LRC method being used in long-span roofs, in which the methods associated with structural modes and the LRC methods prevail in the ESWLs. This section is going to detail these methods according to Three-Component Method and Two-Component Method.

(1) The three-component methods

When considering the dynamic WISRs of fluctuating wind loads for any structure, it is necessary to draw a line between resonant and background components (Holmes, 2007), as the two varying components will bear individual ESWLs. If a total WISR is divided into its mean, background and resonant components whose corresponding ESWLs are calculated, and then the three ESWLs are superposed linearly to form the total ESWLs, in which every combination/weighting/contribution factor indicates its ESWLs contribution proportion to the total ESWLs, this is labeled as “Three-Component Method in ESWLs” (Sun et al., 2015). In general, for a linear structure, because of the linear superposition/decomposition principle in mathematics, the mean ESWLs can be directly taken from the mean wind loads. So, it is unnecessary to recalculate them by the mean WISR, the fluctuating ESWLs are deemed to be significant (Sun et al., 2015, Sun and Zhang, 2020). This paper will mainly elaborate the fluctuating ESWLs in the following.

Holmes (2002, 2007) and Chen et al. (2006) applied the Three-Component-Method to compute the ESWLs of long-span roofs based on a total maximum WISR of interest. In this method, before structuring {P_E,f}, the RMS values for the background WISR and every resonant WISR are amended via multiplying by their own peak factors, every weighting factor for its ESWLs is the proportion of the corresponding amended WISR in the total amended WISR, with the total amended WISR evaluated by the frequently-used square root of the sum of squares (SRSS) method in structural dynamics (Holmes, 2002, 2007). For simplicity, every peak factor for the resonant ESWLs could be an average for all structural modes, leading to {P_E,f} (Holmes, 2002, 2007, Chen et al., 2006):

{P_{E, f}} = W_{B} {P_{E, B}} + W_{R} {P_{E, R}}

(10a)

W_{B} = \frac{g_{B} σ_{r, B}}{\sqrt{{(g_{B} σ_{r, B})}^{2} + {(g_{R} σ_{r, R})}^{2}}}

(10b)

W_{R} = \frac{g_{R} σ_{r, R}}{\sqrt{{(g_{B} σ_{r, B})}^{2} + {(g_{R} σ_{r, R})}^{2}}}

(10c)

σ_{r, R} = \sqrt{\sum_{j = 1}^{N} {(σ_{r, R j})}^{2}}

(10d)

where {P_E,B} and {P_E,R} denote the background and resonant ESWLs, respectively; W_B and W_R are their weighting/combination/contribution factors when {P_E,B} and {P_E,R} are combined; g_B and g_R signal the peak factors for the background WISR-σ_r,B and the resonant WISR-σ_r,R in the total fluctuating WISR-σ_r, respectively; N is the number of all required structural modes; and σ_r,Rj is the resonant WISR from jth structural mode.

In the specific calculation, {P_E,B} is calculated by the POD method like Holmes (1990, 1992) and Chen and Zhou (2007), i.e., equation (8a) and (8b), implying the dedication of all structural modes and having no relation with structural dynamic characteristics. {P_E,R} is acquired by equation (9a)–(9c) (Chen et al., 2006) or W_k=1 in equation (9c) (Holmes, 2002, 2007) from more than one dominant structural mode.

Because pertaining to the fundamental characteristics of wind loads (mean wind loads and the POD modes of fluctuating wind loads) and the structural dynamic characteristic (structural modes or Ritz vectors) (Chen et al., 2006), the method is endowed with a clear physical meaning. The articles of (Holmes, 2002, 2007, Chen et al., 2006), however, took advantage of the SRSS method for computing the fluctuating WISR with an assumption of the uncoupled structural modes, and then ignoring the modal coupling effect for the resonant ESWLs. Unfortunately, such a hypothesis is not always valid. The effect of multi-mode coupling should be considered for some flexible structures with low damping and dense modes and frequencies (Gu and Zhou, 2009, Zhou and Gu, 2010).

Gu and Zhou (2009) and Zhou and Gu (2010), by fixing some terms in the method of (Holmes, 2002, 2007, Chen et al., 2006), crafted the improved Three-Component Method to compute a peak WISR and its corresponding ESWLs of long-span roof. In their method, the total resonant WISR coming from all the selected structural modes is still obtained by the SRSS method, but the resonant WISR of every structural mode of Holmes (2002, 2007) consider the coupling effect exerted by the other structural modes via a modal coupling factor in equation (11a) and (11b), that is the so-called modified SRSS (MSRSS) method:

c_{j} = \sum_{\begin{array}{l} k = 1 \\ k \neq j \end{array}}^{N} c_{j, k}

(11a)

σ_{r, M R j} = \sqrt{1 + c_{j}} \cdot σ_{r, R j}

(11b)

where N= the number for the selected structural modes; c_j=the jth modal coupling factor allowing for N modal coupling effects on the jth resonant WISR; c_j,k =the modal coupling factor considering the kth modal coupling effect on the jth resonant WISR; σ_r,MRj is viewed as the modified resonant WISR of the jth mode and σ_r,Rj the un-modified resonant WISR of the jth mode in equation (10d).

When calculating the fluctuating ESWLs, the background ESWLs result from the T-LRC method, and the resonant ESWLs of every structural mode are the ones of (Holmes, 2002, 2007) multiplied by the corresponding coupling modified factor, that is, $σ_{q_{k}}$ in equation (9b) is replaced by $\sqrt{1 + c_{k}} \cdot σ_{q_{k}}$ . W_B and W_R are similarly achieved by equation (10b)–(10c) after using the background WISR and σ_r,MRj(j=1,2,⋯N).

Zhou and Gu (2010) went about an actual roof’s displacements and ESWLs based on one specified displacement, in which the MSRSS method outperforms the SRSS method distinctly, the coupling validation is done by comparing the errors between their WISRs and the WISRs from the CQC method.

The long-span roofs with closely spaced natural frequencies may undergo coupled motions when exposed to spatiotemporally varying dynamic wind loads. In solving their WISRs and ESWLs, the modal coupling effects may include the ones between each background structural mode and the ones between each resonant structural mode, as well as the cross correlation between the background and resonance, in which the covariance matrix of the total dynamic displacements, according to the mode superposition method, is casted as (Luo et al., 2018):

\begin{array}{l} [C_{y y}] = {[C_{y y}]}_{B} + {[C_{y y}]}_{R} + 2 {[C_{y y}]}_{R B} \\ = [φ] {[C_{q q}]}_{B} {[φ]}^{T} + [φ] {[C_{q q}]}_{R} {[φ]}^{T} + 2 [φ] {[C_{q q}]}_{R B} {[φ]}^{T} \end{array}

(12)

where [C_yy]_B, [C_yy]_R and [C_yy]_RB are the background, resonant and cross displacement covariance matrices, respectively; [φ] is the structural mode matrix in this paper; [C_qq]_B, [C_qq]_R and [C_qq]_RB are the background, resonant and cross covariance matrices of the generalized displacements, respectively.

As the background components can be calculated more conveniently by the quasi-static analysis, equation (12) is altered into (Luo et al., 2018):

[φ] {[C_{q q}]}_{B} {[φ]}^{T} = [I] {[C_{p p}]}_{B} {[I]}^{T}

(13a)

[C_{y y}] = [I] {[C_{p p}]}_{B} {[I]}^{T} + [φ] {[C_{q q}]}_{R} {[φ]}^{T} + 2 [φ] {[C_{q q}]}_{R B} {[φ]}^{T}

(13b)

where [I] is the known structural influence function matrix in this paper; and [C_pp]_B is the covariance matrix of external wind loads.

Equation (13b) is the complete equation, in which [φ][C_qq]_RB[φ]^T is too small to be counted in many cases, this will spawn the simple equation (Luo et al., 2018):

[C_{y y}] = [I] {[C_{p p}]}_{B} {[I]}^{T} + [φ] {[C_{q q}]}_{R} {[φ]}^{T}

(14)

The modal coupling effects are implicitly encompassed in these covariance matrices. Luo et al. (2018) make it clear that the SRSS method underperforms in computing the RMS displacements as against the complete and simplified equations by reason of the errors.

Luo et al. (2018) launched into the fluctuating ESWLs for an appointed WISR. In advance of computing the ESWLs, the generalized resuming forces composed of fluctuating wind loads and multiple-order SMIFs in quasi-static pattern are defined, under which the so-called structural background WISRs take on the mantle of the actual WISRs, creating a sensible prerequisite to widen applying the T-LRC method. Once [C_f] and ${\hat{r}}_{i, B}$ in equation (7) are superseded by the covariance matrix of the generalized resuming forces-[C_pp] and the RMS WISR- $σ_{r_{i}}$ , respectively, the ESWLs of r_i ${P_{E, r_{i}}}$ are accomplished (Luo et al., 2018):

{P_{E, r_{i}}} = \frac{g [C_{p p}] {(I_{i})}^{T}}{σ_{r_{i}}}

(15)

[C_pp] includes the complete form with the background, resonance, and their cross components, as well as the simplified form with the first two terms, so are the ESWLs (Luo et al., 2018):

\begin{array}{l} {P_{E, r_{i}}} = \frac{g ([K] {[C_{y y}]}_{B} {[K]}^{T} + [K] {[C_{y y}]}_{R} {[K]}^{T} + 2 [K] {[C_{y y}]}_{R B} {[K]}^{T}) {(I_{i})}^{T}}{σ_{r_{i}}} \\ = \frac{g ({[C_{p p}]}_{B} + {[C_{p p}]}_{R} + {[C_{p p}]}_{R B}) {(I_{i})}^{T}}{σ_{r_{i}}} \end{array}

(16a)

{P_{E, r_{i}}} = \frac{g ({[C_{p p}]}_{B} + {[C_{p p}]}_{R}) {(I_{i})}^{T}}{σ_{r_{i}}}

(16b)

where [C_pp]_B, [C_pp]_R and [C_pp]_RB are the background, resonance, and their cross components of the generalized resuming forces, respectively. More specifically, [C_pp]_B comes from equation (13a) and (13b), and [C_pp]_R is the covariance matrix of the SMIFs.

It can be learned from equation (16a) and (16b) that all the three items could have been deduced from their respective covariance matrixes of the general displacements, in which the aforementioned modal coupling and cross correlation are fused. But the background component, for convenience, is substituted by [C_pp]_B to show the real distributions of fluctuating wind loads. The resonant component is the actual distributions of multiple-order SMIFs. The cross component is small enough to be overlooked compared to the resonant one, because the long-span roofs, in many cases, own small damping and natural frequencies (there is no mention of the cross ESWLs later in this paper.). The ESWLs, at this moment, are evolved into (Luo et al., 2018):

{P_{E, r_{i}}} \approx \frac{g ({[C_{p p}]}_{B} + [Q_{0}] {[C_{q q}]}_{R} {[Q_{0}]}^{T}) {(I_{i})}^{T}}{σ_{r_{i}}}

(17a)

[Q_{0}] = [M] [φ] [Λ]

(17b)

where [Λ] forms a diagonal matrix in reference to the squares of the selected circular frequencies (or characteristic values) in this paper.

By comparing one node’s peak displacements between the CQC method and the proposed ESWLs under different wind directions, Luo et al. (2018) confirm the robustness of the proposed methods.

The framework of (Luo et al., 2018) can consider the inter-modal coupling of modal response components and the cross correlation between the background and resonance, bringing sufficient precision and efficiency under the needed structural modes, by which the T-LRC method is enriched.

These Three-Component-Methods in (Holmes, 2002, 2007, Chen et al., 2006, Gu and Zhou, 2009, Zhou and Gu, 2010, Luo et al., 2018) can remedy the main negativity in the T-LRC method and the improved LRC method of (Chen and Zhou, 2007, Holmes, 1992) for one thing, and propose the differing ESWLs in response to wind load and structure features for another. Nevertheless, they must split a fluctuating WISR and its ESWLs into the background and resonant components. No consensus has been reached with respect to how to discriminate both accurately at present yet, because the dynamic WISRs, in the strictest sense, are inseparable. Furthermore, getting the WISR and its associated fluctuating ESWLs separated appears to be needless (Chen and Kareem, 2001, Fu et al., 2008).

(2) The two-component methods

Expressing a total peak WISR as the linear combination of its mean and dynamic fluctuating WISRs is possible, and the corresponding ESWLs are thus obtained. The ESWLs consisting of the mean and dynamic fluctuating components are titled “Two-Component-Method” in this paper, in which the latter amounts to the composite result of the background and resonant ESWLs in the Three-Component-Method (Sun et al., 2015).

As stated earlier, it is through structural modes that the ESWLs can be constructed. In long-span roofs, a fluctuating WISR and its ESWLs can also be completed just by structural mode decomposition and the combination of multiple-order SMIFs, respectively. Fu et al. (2008) also enabled the ESWLs of a peak WISR to be figured by the Two-Component Method, in which the fluctuating ESWLs are designated as a linear combination of a series of the SMIFs made up of each concerned structural mode, paralleling the method of (Chen and Kareem, 2001). The effectiveness of this ESWL method is borne out by Fu et al. (2008) on an actual roof, since the peak displacements by the CQC method under different wind directions are in excellent agreement with the corresponding peak ESWLs-ISRs.

Whilst being viable at some level, the above Three- and Two-Component-Methods rely on the dominant structural modes which count for the WISR and its ESWLs for a complex long-span roof, how to choose them precisely is still either in debate or cumbersome. So, it may be a decent move to compute the ESWLs by shunning structural modes.

The methods independent of structural modes but based on the LRC method

For long-span roofs, Zhou et al. (Zhou et al. 2012, 2013) came up with a modified LRC method to compute the entire structure’s ESWLs for a peak WISR by Two-Component-Method:

{P_{E, {\hat{r}}_{i}}} = {{\bar{P}}_{E}} + {P_{E, f}} = {{\bar{P}}_{E}} + d i a g (P_{E B, {\hat{r}}_{i}}) \cdot {k_{E B, {\hat{r}}_{i}}}

(18)

where

{\hat{r}}_{i}

is the peak WISR determined in the same manner as the T-LRC method;

{{\bar{P}}_{E}}

is the column vector of mean wind loads in this paper; and the fluctuating ESWLs, {P_E,f}, emanate from the diagonal matrix of the background ESWLs in the T-LRC method-

d i a g (P_{E B, {\hat{r}}_{i}})

times a modified coefficient column vector-

{k_{E B, {\hat{r}}_{i}}}

${k_{E B, {\hat{r}}_{i}}}$ is an unknown vector which is dispatched to compensate the background ESWLs for the resonant ones, its solution is the key to this method. Ideally, the following equation holds (Zhou et al., 2012, 2013):

(I_{i}) {P_{E, {\hat{r}}_{i}}} = (I_{i}) {{\bar{P}}_{E}} + (I_{i}) \cdot d i a g (P_{E B, {\hat{r}}_{i}}) \cdot {k_{E B, {\hat{r}}_{i}}} = {\hat{r}}_{i}

(19)

According to equation (19), ${k_{E B, {\hat{r}}_{i}}}$ can be evaluated as (Zhou et al., 2012, 2013):

{k_{E B, {\hat{r}}_{i}}} = p i n v [(I_{i}) \cdot d i a g (P_{E B, {\hat{r}}_{i}})] \cdot ({\hat{r}}_{i} - (I_{i}) \cdot {{\bar{P}}_{E}})

(20)

where the operator pinv(⋅) is to obtain the Moore-Penrose general inverse in this paper.

In equation (20), $(I_{i}) \cdot {{\bar{P}}_{E}}$ is the mean WISR in ${\hat{r}}_{i}$ , so ${\hat{r}}_{i} - (I_{i}) \cdot {{\bar{P}}_{E}}$ can be regarded as the fluctuating WISR. Every element in ${k_{E B, {\hat{r}}_{i}}}$ virtually enacts the adjustment value for every corresponding background ESWL value in $d i a g (P_{E B, {\hat{r}}_{i}})$ .

This method of (Zhou et al., 2012, 2013) blazes a way for the ESWLs independent of structural modes, easing computation. The inverse-question thought underlies the method, which cements the method’s mathematical and physical meanings. More importantly, the fluctuating WISR and its ESWLs contain the background and resonant parts which are not artificially parted from each other, and so is rewarded with less calculation expense, too.

The above methods endeavor to find some plausible ESWLs, in fact, they tend to concentrate on a special but elusive WISR. One long-span roof usually has multiple WISRs, all the WISRs under the ESWLs of a special WISR are very difficult to be consistent with the accurate WISRs, which will impair their reasonability. In the meantime, an inverse question can produce a wide range of solutions for one initial condition. The ESWLs distributions resting on a given WISR are not necessarily unique simply because multiple ESWLs distributions can result in an identical WISR (Chen and Zhou, 2007), which will increase the complexity. Hence, there is a need to seek the ESWLs of multiple target WISRs.

The multiple target equivalent methods

The multiple target equivalent methods try to find a whole structure’s ESWLs by multiple preselected target WISRs. In this subsection, there are: (1) The ESWLs independent of structural modes but based on the LRC method; (2) The ESWLs based on the POD modes of wind loads; (3) The ESWLs of fluctuating wind loads depending only on structural modes and frequencies; (4) The ESWLs based on the POD modes of fluctuating wind loads and structural modes/frequencies; and (5) The method resorting to the RMS wind loads.

The methods independent of structural modes but based on the LRC method

In the wake of the above-indicated modified LRC method in equation (18)–(20), the grouping response method was then suggested by Zhou et al. (Zhou et al. 2012, 2013) to erect the ESWLs for a subset of WISRs (i.e., the grouped WISRs) in a set of pre-selected WISRs.

This grouping response method commences with grouping the WISRs and ends up calculating the ESWLs. In accordance with the modified LRC method, the ESWLs of every peak WISR in the pre-selected WISRs can be handily obtained, under which the structural WISRs (the ESWLs-ISRs) corresponding to the pre-selected WISRs are reproduced, if some of the ESWLs-ISRs get close to each other, their corresponding pre-selected WISRs are classified into a group, i.e., the grouped WISRs. The ESWLs of the grouped WISRs are a linear combination of equation (18) for every peak WISR in this group (Zhou et al., 2012, 2013):

\begin{array}{l} {P_{E, M}} = \sum_{i = 1}^{N} k_{E, M - i} \cdot {P_{E, M - {\hat{r}}_{i}}} = [\begin{array}{c} {P_{E, M - {\hat{r}}_{1}}} & {P_{E, M - {\hat{r}}_{2}}} & \dots & {P_{E, M - {\hat{r}}_{N}}} \end{array}] {\begin{array}{c} k_{E, M - 1} \\ k_{E, M - 2} \\ ⋮ \\ k_{E, M - N} \end{array}} \\ = [P_{E, M - \hat{r}}] {k_{E, M}} \end{array}

(21)

where {P_E,M} is the total ESWLs for the Mth group with N peak WISRs;

{P_{E, M - {\hat{r}}_{i}}}

is the ESWLs aiming at

{\hat{r}}_{i}

in the Mth group; and k_E,M-i is the weighting/combination/contribution factor indicating the contribution from

{P_{E, M - {\hat{r}}_{i}}}

to {P_E,M}.

If {P_E,M} can be mathematically described as an expansion in the linear space of the vectors in $[P_{E, M - \hat{r}}]$ , these linearly independent column vectors play the roles of bases (The bases are the basic wind load distributions (BWLDs)), which conduces to understanding the physical meaning. Through the linear least-square method without any constraints, {k_E,M} can be granted its optimal solution, herein the objective function is (Zhou et al., 2012, 2013):

T ({k_{E, M}}) = \min {‖ [I_{M}] ([P_{E, M - \hat{r}}] {k_{E, M}}) - {{\hat{r}}_{M}} ‖}_{2}

(22a)

{{\hat{r}}_{M}} = {\begin{array}{c} {\hat{r}}_{1} \\ {\hat{r}}_{2} \\ ⋮ \\ {\hat{r}}_{N} \end{array}}

(22b)

[I_{M}] = [\begin{array}{c} (I_{1}) \\ (I_{2}) \\ ⋮ \\ (I_{N}) \end{array}]

(22c)

where ‖⋅‖₂ denotes a vector’s Euclid norm;

{{\hat{r}}_{M}}

is the column vector of the peak WISRs in the Mth group; and [I_M] is the structural influence function matrix in connection with

{{\hat{r}}_{M}}

Then the solution to {k_E,M} is (Zhou et al., 2012, 2013):

{k_{E, M}} = p i n v ([I_{M}] [P_{E, M - \hat{r}}]) {{\hat{r}}_{M}}

(23)

The authors of (Zhou et al., 2012, 2013) exemplified the grouping ESWL method on a space truss roof with node displacements and axial forces, in which the grouped WISRs agree well with their corresponding ESWLs-ISRs, the un-grouped WISRs deviate from their corresponding ESWLs-ISRs significantly, however, showing that the method is partial toward the grouped WISRs.

While the grouped WISRs are appropriately chosen, the range of the ESWLs magnitudes will be like that of natural wind loads and so within reason. Instead, if the grouped selection is improper, the ESWLs may be beset by some erratic and irrational ESWLs with extremely large values. It should be noted that when the grouped WISRs increases in number, this individuality is more pronounced, which also exists in (Katsumura et al., 1994, 2005a, 2005b, 2007, Tamura and Katsumura, 2012, Sun et al., 2015) inevitably.

To obtain the ESWLs with a reasonable value range, Zhou et al. (2014) thought up the constrained least-square method to compute the ESWLs aiming at all peak WISRs in a certain group of WISRs. In the method, the ESWLs are regarded as a linear combination of two kinds of pre-defined BWLDs, one of which is still from $[P_{E, M - \hat{r}}]$ . For steering the ESWLs away from the above-mentioned abnormality, it is much-needed to restrain the value ranges of the ESWLs by controlling the upper and lower bounds of {k_E,M}. Then the solution for {k_E,M} is, in actuality, a constrained linear least-square problem. The objective function in equation (22a) is combined with the constraint condition in equation (24) to search for {k_E,M} (Zhou et al., 2014):

k_{E, \min} \leq k_{E, M - i} \leq k_{E, \max}

(24)

where k_E,min and k_E,max are the upper and lower limits for any k_E,M-i of {k_E,M}, respectively, their optimal values can be found by trial and error.

Meanwhile, a few typical WISRs rather than all WISRs in the certain group can be given more significance, their pre-established weighting factors are imported together with the constraint to lift the accuracy of these typical WISRs signally. The solution to {k_E,M} is turned into a weighted and constrained linear least-square problem, the objective function in equation (25) and the constraint condition in equation (24) are empowered to find the solution to {k_E,M} (Zhou et al., 2014):

T ({k_{E, M}}) = \min {‖ {W} * {[I_{M}] ([P_{E, M - \hat{r}}] {k_{E, M}}) - {{\hat{r}}_{M}}} ‖}_{2}

(25)

where {W} works as the weighting factor column vector; and the notation * in this paper is just a multiplication in form, meaning that the elements at the same position of two same-sized vectors are multiplied.

Zhou et al. (2014) still took the structure in Zhou et al. (2012) as an instance to uphold their ESWLs. In search of the credible ESWLs which bear the obvious similarity to the natural wind pressure in range, k_E,min=-2 and k_E,max=2, the weighting factors for the typical WISRs are set as 10 and those for the non-typical WISRs as one to ameliorate the accuracy of the typical WISRs.

These methods of Zhou et al. (Zhou et al. 2012, 2014) eliminate the dependence on structural modes and frequencies, thus giving some relief from computing the WISRs and the ESWLs. Meanwhile, they can accommodate multiple WISRs simultaneously rather than one WISR. Furthermore, the authors spearhead the way to find a balance between accuracy and reasonability via these constraints. Finally, because of inheriting the T-LRC method, the ESWLs are analogous to the ones with the characteristics of the T-LRC method and demonstrate certain physical meanings (Zhou et al., 2012, 2014).

The following downsides to these methods, however, still exist: (1) These methods trade the errors for the reasonability, even if the accuracies can be better ensured in the ESWLs of the selected WISR group, they are unsatisfactory for the other WISRs. (2) The ESWLs connect themselves to the grouped patterns which can be seen as the initial conditions in the inverse question, so they will vary with the grouped WISRs, seeming illogical. (3) The selected WISRs are all peak values. But different WISRs can rarely peak at the same time, and their correlations should be considered, or the ESWLs of numerous WISRs will potentially be too large to be used, although the above constraints can cushion the problem at the sacrifice of accuracy. (4) When more WISRs require the above constraints, or the constraints become more rigorous, such a way of proceeding is no longer applicable by virtue of the greater errors as compared to the results without the constraints.

The methods based on the POD modes of wind loads

In mathematics, based on the variable separation method, POD method is credited with analyzing a complex random field which can be expanded into the linear combination of a series of orthogonal basis functions (i.e., POD modes), where the combination factors are their principal coordinates (Solari and Carassale, 2000). This method also gains favor with spatiotemporally changing wind fields, some significant physical distributions, such as the POD modes of wind fields, may express ESWLs availably (Katsumura et al., 2007).

The method based on the POD modes of total wind loads

Davenport and Surry (1984) also made use of Two-Component-Method to calculate the maximum and minimum ESWLs of a saddle-shaped hyperbolic paraboloid roof with a nearly circular plan form, herein the linear combination of the multiple POD modes of total (mean and fluctuating) wind loads are passing for the mean and fluctuating ESWLs. The maximum and minimum ESWLs at the polar coordinates-(r,θ) are expressed as (Davenport and Surry, 1984):

P_{D} (r, θ) = q {\sum_{i} [{\bar{C}}_{F_{i}} \pm g_{i} \cdot {\tilde{C}}_{F_{i}} \cdot {(R M F)}_{i}]} ϕ_{i} (r, θ)

(26)

in which, q is the mean reference velocity pressure at the saddle roof height; ϕ_i(r,θ) is the ith POD mode defining the ith shape of the structural wind loads at the position (r,θ); g_i is the statistical peak factor for the ith mode who is related to the number of POD modes contributing to a critical loading case. When more than one POD mode act, g_i will be systematically reduced by an empirical load combination factor;

{\bar{C}}_{F_{i}}

and

{\tilde{C}}_{F_{i}}

are the mean and RMS values of the ith modal force coefficient time history-

C_{F_{i}} (t)

; (RMF)_i is the resonant magnification factor of

C_{F_{i}} (t)

which can be given by an established random noise theory.

In equation (26), the force coefficients stipulating the amplitudes of these orthogonal loading patterns are each summarized by ${\bar{C}}_{F_{i}}$ , ${\tilde{C}}_{F_{i}}$ and (RMF)_i (Davenport and Surry, 1984). $C_{F_{i}} (t)$ assumes the role of the POD principal coordinate in the POD decomposition.

In practical computation, the POD modes are set as some simple mathematical shape functions (harmonic Fourier functions) coinciding with the structural modes of a circular membrane and drawing close to the ones of the hyperbolic paraboloid surface. Meanwhile, the shape functions relate intimately to the characteristics of the structural WISRs.

Unlike most methods, Davenport and Surry (1984) did not settle the WISRs-based ESWLs, but first obtained the ESWLs, following any one peak WISR for design- ${\hat{r}}_{D}$ as long as the influence function for the WISR took the place of the corresponding POD mode:

{\hat{r}}_{D} = q {\sum_{i} [{\bar{C}}_{F_{i}} \pm g_{i} \cdot {\tilde{C}}_{F_{i}} \cdot {(R M F)}_{i}]} I_{i} (r)

(27)

where I_i(r) is the influence function for the WISR-r_i due to the unit action of ϕ_i(r,θ).

In addition to avoiding involving the influence function directly in the load description, the method directly makes for the representatives of highly complicated load patterns (Davenport and Surry, 1984). The ESWLs also start diverging from structural modes and verge on the POD modes of wind loads, which cut back the computation. However, the above-mentioned WISRs for design stem from the ESWLs, so they may be in disaccord with the actual WISRs. What is more, the mean wind loads are embraced in the total wind loads to pass off as the POD computation, they ought to be excluded from the POD analysis, and their contribution is processed separately (Tamura et al., 1999).

The methods based on the POD modes of fluctuating wind loads

Most of the preceding methods tie the ESWLs of an overall structure to one WISR or some suitable WISRs. This only accounts for a small proportion in all WISRs, thus mistakenly using one part to represent the whole. The selected WISR or WISRs can be said to be subjective and even inappropriate, and the resultant pattern tailored to the different WISR or WISRs, as mentioned earlier, will beget inconsistent ESWLs for the same structure, which does not seem to make sense. Besides, it is often tough to pinpoint the WISR or WISRs in advance, especially for a complex structure with a shower of WISRs, since the WISR or WISRs may not stand out from the others (Li and Tamura, 2005). Such being the case, it may be expedient to select the entirety in one kind or different sorts of WISRs as much as possible.

As WISRs vary spatiotemporally, their largest values for all structural members do not occur simultaneously. The universal ESWLs (UESWLs), which can synchronously reproduce these WISRs in theory by using an inverse-analysis technique and be seen as a linear combination of several known arbitrary BWLDs, will be practical especially in the early design stage. This holds true even though they may have small changes in structural design (Katsumura et al., 1994, 2005a, 2005b, 2007, Tamura and Katsumura, 2012).

For the fluctuating UESWLs on long-span roofs, scilicet the uncompensated ESWLs {P_E,f-u} called by Sun et al. (2015), the articles of (Katsumura et al., 1994, 2005a, 2005b, 2007, Tamura and Katsumura, 2012) advocated the known intrinsic POD modes of fluctuating wind loads as the BWLDs:

{P_{E, f - u}} = [ϕ] {C}

(28)

where [ϕ] is the selective POD mode matrix of fluctuating wind loads; and {C} is the undetermined column vector of weighting/combination/contribution factors.

Here, {P_E,f-u} can be perceived as one expansion in the linear place of [ϕ] with {C} being its coordinates in the coordinate system.

In preparation for obtaining {C}, the known WISRs-{r_e} can be formulated as (Katsumura et al., 1994, 2005a, 2005b, 2007, Tamura and Katsumura, 2012):

{r_{e}} = [I] {P_{E, f - u}} = [I] [ϕ] {C}

(29)

Equation (29) forms a homogeneous linear equation set, whereupon {C} can be solved by the least-square method (Katsumura et al., 1994, 2005a, 2005b, 2007, Tamura and Katsumura, 2012):

{C} = p i n v ([I] [ϕ]) {r_{e}}

(30)

This method gains the uncompensated ESWLs straightforward by the POD modes, simplifying the computation on account of ending the relationship with structural modes and frequencies. Indeed, not only can it take care of the ESWLs for one category of WISRs such as the ESWLs of displacement or axial force, but it also provides the ESWLs for different sorts of WISRs like the ESWLs of displacement and axial force, making a welcome contribution to broaden its application scope. Moreover, it can think of the target WISRs for multiple and even all wind directions, which also opens a new window into its broader application field (Ma et al., 2022). Finally, it is invested with a widespread applicability holding for all linear engineering structures, because the computational processes do not lean on structural styles at all.

If the wind loads on structures must be obtained experimentally, the total number of the POD modes of fluctuating wind loads will be equal to that of the independent measuring points (Sun et al., 2015). Owing to the limitations of test equipment and wind tunnel model, it usually happens that the number of independent measuring points is far fewer than the nodal number of structural finite-element model (Sun et al., 2015). The method turns out to be more mathematical in nature and is a better fit for the simple structures with fewer WISRs in the case of fewer POD modes. However, it may nullify itself if all the POD modes are greatly outnumbered by a wealth of WISRs in the complex linear structures for the greater errors between the accurate {r_e} and the approximate WISRs subjected to {P_E,f-u} (Sun et al., 2015), that is, the POD reduced-order talent will be even more eclipsed here. Another problem with the method is that only the background ESWLs can be computed in the articles of (Holmes, 1992, 2002, Ginger et al., 2000, Chen et al., 2006, 2012, 2014, Yang et al., 2013, Blaise and Denoel, 2013), and it doesn’t consider structural mechanical behavior and a possible resonant contribution in WISRs (Blaise and Denoel, 2013).

As to these defects, Sun et al. (2015) theorized a modified UESWL method of the fluctuating wind loads on the complex linear long-span roofs by incorporating the above UESWL method with the POD compensation strategy. In this strategy, based on the response differences (the compensated objective) between {r_e} and the approximate ESWLs-ISRs in the UESWL method, the compensated POD mode and factor are inversely coined through the least-square method. Then their product is the compensated ESWLs, and is merged with {P_E,f-u} to form the total ESWLs.

Based on equation (30), the compensated objective is (Sun et al., 2015):

{r_{e, c}} = {r_{e}} - [I] [ϕ] (p i n v ([I] [ϕ]) {r_{e}})

(31)

The unit compensated POD mode {ϕ_c} and corresponding compensated contribution factor C_c are constructed (Sun et al., 2015):

{ϕ_{c}} = \frac{p i n v ([I]) \cdot {r_{e, c}}}{C_{c}}

(32a)

C_{c} = n o r m (p i n v ([I]) \cdot {r_{e, c}})

(32b)

The total fluctuating ESWLs are (Sun et al., 2015):

{P_{E, f}} = [ϕ] \cdot p i n v ([I] [ϕ]) {r_{e}} + {ϕ_{c}} \cdot C_{c}

(33)

The authors of (Sun et al., 2015) compared the relevant responses, and computed the errors between the accurate WISRs {r_e} and the ESWLs-ISRs under {P_E,f-u} or {P_E,f}, demonstrating the need for compensation.

More important than just helping with the previous assets, the modified method can exterminate the errors that grow out of all modes or a portion of lower-order modes considered in the UESWL method itself, and only leaves the innate errors caused by the Moore-Penrose generalized inverse of the non-square [I], so it can obtain the accurate displacement-based ESWLs, as well as the highest-fidelity ESWLs based on internal force and internal force/displacement (The precise ESWLs based on internal force, as Wu (2007) has claimed, do not usually exist.), revealing its clear physical meaning and high accuracy (Sun et al., 2015). Whereas it should be spelled out that all WISRs are also hypothesized to reach their maxima simultaneously in the two UESWL methods (this point is expressly termed the common issue in the UESWLs in this paper), which may raise some erratic and irrational ESWLs with huge absolute values to erode their wide availability indisputably (Sun et al., 2015). In practice, the possibility for all WISRs to arrive at their maxima concurrently is slim (Katsumura et al., 1994, 2005a, 2005b, 2007, Tamura and Katsumura, 2012). It is a must to consider their correlation, little headway with this consideration has been made currently, however (Sun et al., 2015).

The method of fluctuating wind loads depending only on structural modes and frequencies

In the Two-Component-Method, it is fruitful to compute the fluctuating UESWLs only through structural modes and frequencies.

The method based on multiple-order SMIFs

The only distinction between the method based on the multiple-order SMIFs and the methods based on the POD modes of fluctuating wind loads is that the multiple-order SMIFs can be appropriated in place of the POD modes.

For a linear complex long-span roof with quite a few degrees of freedom (DOFs), a joint consideration for the frequency characteristics of structures and wind loads is often encouraged to compute structural WISRs based on dynamics of structures (Sun and Zhang, 2020). It is quite possible that most of the structural intermediate and higher-order frequencies exceed far beyond the frequency scope of frequently encountered wind loads, and that very few of them and their associated structural modes can be excited by wind loads (Holmes, 2007; Macdonald, 1975). A minority of low-order frequencies and modes, correspondingly, most often predominate over the WISRs. Based on the linear combination of their multiple-order SMIFs, the corresponding UESWLs of fluctuating wind loads for these low-order frequencies and modes, the uncompensated ESWLs, were built by (Sun and Zhang, 2020):

{P_{E, f - u}} = [M] [φ] [Λ] {C}

(34)

where [M], [φ], [Λ] and {C} hold the same meanings as before; {P_E,f-u}can also be seen as a vector to be unfolded in the linear space of [M][φ][Λ]; and {C} is still envisioned as the coordinates whose solution is (Sun and Zhang, 2020):

{C} = p i n v ([I] [M] [φ] [Λ]) {r_{e}}

(35)

It’s not hard to know from mathematics that equation (34) will incur considerable errors in the uncompensated ESWLs, despite the usage of the least-square method. Sun and Zhang (2020) invented another compensated tactic to surmount the negative consequence in the absence of the medium and higher modes and frequencies for the ESWLs. In this method, considering the response differences between {r_e} and the uncompensated ESWLs-ISRs, the authors turn to the least-square method for inversely setting up one compensated characteristic value, mode and contribution factor whose product constitutes the compensated ESWLs. The method ends with the compensated ESWLs being superimposed into the uncompensated ones to forge the total ESWLs.

The compensated objective becomes (Sun and Zhang, 2020):

\begin{array}{l} {r_{e, c}} = {r_{e}} - ([I] [M] [φ] [Λ]) \cdot (p i n v ([I] [M] [φ] [Λ]) \cdot {r_{e}}) \\ = [I] [M] {φ_{c}} Λ_{c} C_{c} \end{array}

(36)

where {φ_c}, Λ_c and C_c are the compensated mode, characteristic value and contribution factor, respectively, which can be coined (Sun and Zhang, 2020):

{φ_{c}} = \frac{{Φ}}{Λ_{c} C_{c}} = \frac{{Φ}}{\sqrt{{Φ}^{T} [M] {Φ}}}

(37a)

Λ_{c} = \frac{{φ_{c}}^{T} [K] {φ_{c}}}{{φ_{c}}^{T} [M] {φ_{c}}}

(37b)

C_{c} = \frac{\sqrt{{Φ}^{T} [M] {Φ}}}{Λ_{c}}

(37c)

{Φ} = p i n v ([I] [M]) \cdot {r_{e, c}}

(37d)

where [K] is the structural matrix; and {Ф} is looked on as the compensated vector.

The total fluctuating ESWLs are (Sun and Zhang, 2020):

{P_{E, f}} = ([M] [φ] [Λ]) \cdot (p i n v ([I] [M] [φ] [Λ]) {r_{e}}) + [M] {φ_{c}} Λ_{c} C_{c}

(38)

Sun and Zhang (2020) compared the relevant responses and calculated their errors, which substantiate the necessity for this compensation.

In this method, the errors born out of {P_E,f-u} themselves are eradicated, and the unavoidable ones triggered by the Moore-Penrose generalized inverse of [I][M] are only preserved, which refine the accuracy and exhibit clear mathematical and physical meanings.

The parametric analysis methods

Most of the current UESWLs methods are derived from pure numerical algorithms, and there are no general practical parametrical expressions for the engineering applications, bringing the inconveniency to the designers (Chen et al., 2018, Wang et al., 2023). The parametric analysis methods provide an alternative approach.

Single-layer cylindrical shells, plate-like flat roofs and railway station canopies are also in the spotlight. In most cases, sufficient structural modes are recruited to calculate their WISRs, but if the wind and structural parameters of these structures can cover the common engineering application ranges of practical interest, few dominant structural modes with the maximum strain energy will be prioritized to calculate their dynamic WISRs and UESWLs in the parametric study (Chen et al., 2018, Wang et al., 2020, 2023).

For the single-layer cylindrical steel shells located in open terrains and supported by hinged constraints along four edges (Chen et al., 2018): (1) Their structural members along the arch direction are the primary load-resistant members, with the response peculiarity similar to those of the plane archs; (2) Their bending stress and and vertical displacement are the unfavorable responses, in which the background WISRs far outweigh the resonant ones; and (3) The two unfavorable wind directions for the WISRs and the UESWLs are 0^o and 45^o. Chen et al. (2018) found two most dominant structural modes in the shells, and gave the UESWLs for all nodal displacements under the parametric analysis in a manner that balances precision and convenience:

{P_{E, t}} = {{\bar{P}}_{E}} + {P_{E, f}} = \frac{1}{2} ρ {({\bar{V}}_{0})}^{2} ({{\bar{C}}_{p} (x, y)} \pm [C_{1, f} {φ_{1} (x, y)} + C_{2, f} {φ_{2} (x, y)}])

(39)

where (x,y) sets one coordinate on the surface of the shells; {P_E,t},

{{\bar{P}}_{E}}

and {P_E,f} mean the total, mean and fluctuating UESWLs, respectively;

{{\bar{C}}_{p} (x, y)}

is on behalf of the known mean wind pressure coefficients; ρ and

{\bar{V}}_{0}

are air density and the mean reference wind velocity in this paper, respectively; and {φ_k(x,y)}(k=1,2) and C_k,f(k=1,2) are the dominant structural modes and their corresponding combination factors in this paper, respectively.

Many structural modes are necessary to compute the dynamic WISRs, it is difficult to obtain the UESWLs for both all nodal displacements and all member stress with {φ_k(x,y)}(k=1,2). Therefore, equation (39) is multiplied by an adjustment factor β to express the total UESWLs for all member stress (Chen et al., 2018):

{P_{E, t}} = β \frac{1}{2} ρ {({\bar{V}}_{0})}^{2} ({{\bar{C}}_{p} (x, y)} \pm [C_{1, f} {φ_{1} (x, y)} + C_{2, f} {φ_{2} (x, y)}])

(40)

In equations (39) and (40), the approximate and simple expressions of {φ_k(x,y)}can be found in Chen et al. (2018). C_k,f can be obtained like Chen et al. (Chen et al. 2012, 2014) and Yang et al. (2013) in principle. However, to facilitate engineering application, the parametric results of the UESWLs can be simply expressed. After investigating the dependence of C_k,f and β on the wind and structural parameters (the unfavorable wind direction θ, the structural span S, the rise-span ratio f/S, the length-span ratio L/S, the roof distributed mass, and the reduced frequency f^*=f₁S/V₀ being f₁ the basic frequency in this paper and V₀ the mean wind velocity at 10 m height.), a serious of conservative empirical equations and values for both are fitted. The actual distributions of

{{\bar{C}}_{p} (x, y)}

obtained from the wind tunnel tests are rather complex, to make their simplified forms into full use in engineering, the roofs are separated into several rectangular zones, wherein their portioned values with reference to L/S, f/S and θ are experimentally provided in Chen et al. (2018).

For the plate-like flat roofs supported on four sides (Wang et al., 2020): (1) The bending stress and and vertical displacement are the unfavorable responses; (2) In view of the structural WISRs and UESWLs, the two unfavorable wind directions are the wind directions along the short (0^o) and long (90^o) building sides; and (3) The first structural mode, a double sinusoidal surface, is the most dominant mode. Wang et al. (2020), in combination with parametric analysis, studied the structural UESWLs for all nodal displacements and member stress:

{P_{E, t}} = {{\bar{P}}_{E}} + {P_{E, f}} = \frac{1}{2} ρ {({\bar{V}}_{0})}^{2} ({{\bar{C}}_{p} (x, y)} \pm C_{1, f} {φ_{1} (x, y)})

(41)

where (x,y) prescribes one coordinate on the roofs; {φ₁(x,y)} is recorded in Wang et al. (2020). If the structural mass is evenly distributed, C_1,f can be simplified as (Wang et al., 2020):

C_{1, f} = 4 g σ \sqrt{1 + \frac{π}{4 ζ_{1}} \frac{A (f_{1} H / V)}{{[1 + B {(f_{1} H / V)}^{2}]}^{1.2}}}

(42)

where σ =the RMS of the first modal generalized wind force coefficient; ζ₁ =the damping ratio; and f₁H/V=the reduced frequency corresponding to the mean wind velocity V at the roof eave height H.

Using the test pressures, the generalized force spectrum curve $\frac{A (f_{1} H / V)}{{[1 + B {(f_{1} H / V)}^{2}]}^{1.2}}$ and σ in equation (42) can be fitted by the least-squares method for engineering convenience. The fitted values of A, B and σ for different terrain types, length-width ratios and unfavorable wind directions are well documented in Wang et al. (2020).

The roofs are also parceled out into several rectangular domains, whose simple expressions of ${{\bar{C}}_{p} (x, y)}$ depending on length, width, terrain types, length-width ratios and unfavorable wind directions are offered in Wang et al. (2020).

For the regular canopies with single- or three-span frames in the small or medium-sized railway stations (Wang et al., 2023): (1) The structural systems can be roughly modeled as plane frames, and the wind loads on the frame columns are too small to be ignored, therefore the UESWLs on the frame beams represent the UESWLs of the canopies; (2) In the plane frames, the first structural mode rules over the horizontal displacements and the frame column stresses, and the second structural mode dominates the vertical displacements and the frame beam stresses of the canopy frames; (3) The unfavorable wind directions are determined for the center, near-center and end frames; (4) The canopies of railway stations are fully open without walls or other obstacles between the columns; and (5) The interference caused by other buildings on the railway stations are omitted. Wang et al. (2023) devised the total UESWLs of the frame beam for all nodal displacements and member stress in the frame:

{P_{E, t}} = \frac{1}{2} ρ {({\bar{V}}_{0})}^{2} ({{\bar{C}}_{p} (x)} n \pm [{C_{1, f} (x)} * {φ_{1} (x)} + {C_{2, f} (x)} * {φ_{2} (x)}])

(43)

where x is the coordinate on the beam; n is the normal vector projecting the vertical

{{\bar{C}}_{p} (x)}

onto the roof’s normal direction thanks to the beam slope (i.e., the roof slope); {φ_k(x)}(k=1,2) are the beam components in the frame’s dominant structural modes, which can be obtained from either the classical finite element analyses or the simplified functions parametrically defined in Wang et al. (2023); The practical

{{\bar{C}}_{p} (x)}

in engineering demand are experimentally gained, whose simplified parametrical expressions and distributions relying on the structural and wind parameters can be discovered in Wang et al. (2023); and {C_k,f(x)}(k=1,2) are the combination factor vectors (or the vectors containing the equivalent fluctuating wind pressure coefficient C_k,f(x)) on the frame beam, in which the element C_k,f(x) reads (Wang et al., 2023):

C_{k, f} (x) = \frac{m (x)}{\frac{1}{2} ρ {({\bar{V}}_{0})}^{2} W M_{k}^{*}} g \sqrt{{(\frac{1}{2} ρ {({\bar{V}}_{0})}^{2} σ_{k} A_{s})}^{2} + \frac{π f_{k}}{4 ξ} S_{F F k} (f_{k})}

(44)

where m(x)=the distributed mass per unit length of the frame beam; A_s=LW with L=the distance between the frame columns and W=the longitudinal spacing of the frames;

M_{k}^{*}

=the kth structural mode’s generalized mass; f_k=the kth structural Hz frequency; σ_k=the RMS of the kth structural mode generalized force coefficient; and S_FFk( f_k)=the kth structural mode generalized force spectrum. After the least-square fitting, the intuitive parametrical expressions of S_FFk(f_k) and σ_k banking on the wind and structural parameters are given in Wang et al. (2023), which are conducive to the structure design.

The overall good match between the ESWLs-ISRs and the target WISRs and the deviation from the traditional GRF method both reveal the adequacy of these methods of (Chen et al., 2018, Wang et al., 2020, 2023).

Most of the UESWL methods will summon numerous BWLDs for complicated structures, and fit single case analysis due to their incompetence in expressing the results of a certain kind of long-span roofs, all of which can be overcome by (Chen et al., 2018, Wang et al., 2020, 2023), however. These methods individuate the UESWLs as the parametric expressions, pursuing a novel, simple, high-precision and usable route for a serious of roofs via a small amount of structural modes and the characteristics of the WISRs. Nevertheless, these methods are enslaved to the assumption that there is a simple linear relation between the UESWLs of all node displacements and those of all element stress, which is not always the case in fact for many long-span roofs. Furthermore, if the UESWLs need more structural modes definitely, the effectiveness of these methods demands further investigation.

The methods of combining multiple-order SMIFs and single value decomposition

The methods of combining multiple-order SMIFs and single value decomposition (SVD) are also stated as PSWLs in the papers of (Blaise et al., 2012, Blaise and Denoel, 2013; Frontini et al., 2022). SVD is a high-powered technique solving the sets of matrix-form equations classically, and is adoptable for the complicated conditions in which some classical methods such as LU decompositions and Gaussian elimination feel incompetent to provide satisfactory results, or even in the other cases where the number of unknowns is more or less than that of the equations, it can assist in identifying the number of modes in each frequency response band of the corresponding transfer function in a stable way (Biglieri et al., 1989). One of the main benefits of SVD is its capability of separating and distinguishing the different modes produced in a single transfer function that corresponds to each mode without residual effects (Rizzo et al., 2023).

Blaise et al. (2012) and Blaise and Denoel (2013) were still keen on Two-Component-Method. In their method, a ESWLs matrix, $[P_{E, f}^{L R C}]$ , is first computed for each fluctuating buffeting WISR of interest using equation (9a)–(9c) (Blaise et al., 2012, Blaise and Denoel, 2013):

[P_{E, f}^{L R C}] = [\begin{array}{c} {P_{E, f, 1}} & {P_{E, f, 2}} & \dots & {P_{E, f, M}} \end{array}]

(45)

where {P_E,f,i}(i=1,2,⋯,M) extracts the ESWLs for ith fluctuating WISR of interest.

Then equation (45) is factorized by SVD as (Blaise et al., 2012, Blaise and Denoel, 2013):

[P_{E, f}^{L R C}] = [P_{E, f}^{P S W L}] [Σ] [V^{T}]

(46)

where

[P_{E, f}^{P S W L}] = [\begin{array}{c} {P_{E, f, 1}^{P S W L}} & {P_{E, f, 2}^{P S W L}} & \dots & {P_{E, f, N}^{P S W L}} \end{array}]

is the square matrix of PSWLs, in which the orthogonal vector

{P_{E, f, i}^{P S W L}}

(or the load modes) are called PSWLs; [Σ] collects, along its main diagonal, the singular values (i.e., the principal coordinates) ordered by descending magnitude; and [V^T] contains the combination coefficients to rebuild

[P_{E, f}^{L R C}]

from the linear combination of

[P_{E, f}^{P S W L}]

, weighed by the singular values.

Inspired by common POD applications, only the first few load modes determined by the corresponding dominant principal coordinates may be kept for representing $[P_{E, f}^{L R C}]$ sufficiently accurately (Blaise et al., 2012, Blaise and Denoel, 2013). A first main advantage is that the selection is straightforward since they are ranked by decreasing importance. Moreover, each load mode (principal load) is no longer associated with a specific WISR, but rather aims at a global reconstruction of the set of ESWLs and, as a corollary, of the envelope of WISRs (Blaise and Denoel, 2013).

The way to define PSWLs suggests that linear combinations may be considered, any combination of the PSWLs can produce the new fluctuating UESWLs-{P_E,f} of long-span roofs and the corresponding ESWLs-ISRs- ${r_{e, f}^{I S R s}}$ (Blaise et al., 2012, Blaise and Denoel, 2013):

{P_{E, f}} = [P_{E, f}^{P S W L}] {C}

(47a)

{r_{e, f}^{I S R s}} = [I] {P_{E, f}}

(47b)

where {C} is an undetermined vector of arbitrary combination coefficients. The only limitation on solving {C} is that

{r_{e, f}^{I S R s}}

(the reconstructed envelope) is enforced to never overestimate its corresponding target envelopes (i.e., the fluctuating target WISRs), and that

{r_{e, f}^{I S R s}}

is constrained to lie inside the target envelopes and to be at least in a point tangent to its boundary

[P_{E, f}^{P S W L}]

(Blaise and Denoel, 2013, Frontini et al., 2022). Consequently, in the multi-dimensional space related to the envelopes,

{r_{e, f}^{I S R s}}

is a parametric representation of the subspace spanned by the principal static wind loads collected in

[P_{E, f}^{P S W L}]

(Blaise and Denoel, 2013).

The method allows appreciating the gain of a desired reconstructed envelope accuracy by combining a finite number of ${P_{E, f, i}^{P S W L}}$ , fully reflecting the reduced-order capacity. The SVD operation puts a new slant on constructing the BWLDs in the UESWLs. The method’s main advantage is that the principal bases used to define $[P_{E, f}^{P S W L}]$ are optimal to solve envelope reconstruction problem (Frontini et al., 2022). Regrettably, this method doesn’t point out evaluating {C} clearly. Even through a Monte Carlo optimization used in Blaise and Denoel (2013), the high computation costs are unavoidable for general structures, especially for the complex cases for which just a few ${P_{E, f, i}^{P S W L}}$ might be unable to accurately reproduce the envelopes. Moreover, the error limit is relatively loose because ${r_{e, f}^{I S R s}}$ is confined a wide band.

Following this PSWLs thread, Frontini et al. (2022) pushed forward with the method. Thanks to the non-Gaussian distribution of WISRs, each peak WISR (namely the target envelop) can be derived from the Gumbel method (Frontini et al., 2022). For nodal displacements, when the kth WISR peaks, the corresponding total (mean plus fluctuating) ESWLs are (Frontini et al., 2022):

{P_{E, t, k}} = [K] {d_{{\hat{d}}_{k}}} = [K] [φ] {q_{{\hat{q}}_{k}}} = [M] [φ] [Λ] {q_{{\hat{q}}_{k}}}

(48)

where

{d_{{\hat{d}}_{k}}}

and

{q_{{\hat{q}}_{k}}}

are the column vectors of the displacements and the modal coordinates in which the kth values reach their peak WISRs.

The global matrix containing all {P_E,t,k}(k=1,2,⋯,M) is (Frontini et al., 2022):

[P_{E, t}] = [\begin{array}{c} {P_{E, t, 1}} & {P_{E, t, 2}} & \dots & {P_{E, t, M}} \end{array}]

(49)

Following Blaise et al. (2012) and Blaise and Denoel (2013), a SVD of [P_E,t] is performed to obtain the load mode ${P_{E, t, i}^{P S W L}}$ , whose PSWLs matrix is $[P_{E, t}^{P S W L}] = [\begin{array}{c} {P_{E, t, 1}^{P S W L}} & {P_{E, t, 2}^{P S W L}} & \dots & {P_{E, t, N}^{P S W L}} \end{array}]$ . The total UESWLs-{P_E,t} and the corresponding ESWLs-ISRs- ${r_{e, t}^{I S R s}}$ are (Frontini et al., 2022):

{P_{E, t}} = [P_{E, t}^{P S W L}] {C}

(50a)

{r_{e, t}^{I S R s}} = ([I] [φ]) \cdot p i n v ([M] [φ] [Λ]) \cdot [P_{E, t}^{P S W L}] {C}

(50b)

Under selecting a small percentage of ${P_{E, t, i}^{P S W L}}$ to build {P_E,t} for a sizeable number of WISRs, and then reproduce ${r_{e, t}^{I S R s}}$ , an optimization procedure must be used to find {C}, with ${r_{e, t}^{I S R s}}$ matching the target envelops approximately. To this end, the method in Blaise et al. (2012) and Blaise and Denoel (2013) can be ameliorated introducing a tolerance band in the reconstruction of the target envelopes and a genetic algorithm optimization for the optimal combination of ${P_{E, t, i}^{P S W L}}$ .

A comparison with the simple target ESWLs method has shown the method’s optimality.

First, the approach is based on both relaxing the constraint requirement enforced on the reconstructed envelope in Blaise et al. (2012) and Blaise and Denoel (2013) and defining a global optimization parameter used for automated optimization algorithms (Frontini et al., 2022). Instead of a perfect match between the target and reconstructed envelopes, an approximated envelope provides a reasonably accurate reconstruction after the balance of precision and simplicity, in which a bounded overcoming of the target envelopes should be allowed for: this approach therefore implies an overestimation of the acting forces in designing the structural components (Frontini et al., 2022). Second, the method can consider a large amount of structural parameters, removing the need to torturously identify critical structural elements in the verification of the structure, and consequently the designer’s subjectivity (Frontini et al., 2022).

The methods of (Blaise et al., 2012, Blaise and Denoel, 2013, Frontini et al., 2022) recast the UESWLs into a combination of some lower-order new BWLDs. However, they are mathematical by essence, because of the looser error, the ESWLs-ISRs may have serious deviation from their original equivalent targets seriously, misleading designers.

It should be accented here that these advocated methods of fluctuating wind loads depending only on structural modes and frequencies still shrivel for tackling the common issue in the UESWLs.

The methods based on the POD modes of fluctuating wind loads and structural modes/frequencies

{P_E,f-u} in equation (28) is justified in coping with the background ESWLs only, it’s well worth developing the other alternatives to consider the impact of the resonant ESWLs directly or indirectly except for the compensated channel proposed by Sun et al. (2015). In the methods based on the POD modes of wind loads and the methods of fluctuating wind loads depending only on structural modes and frequencies, the BWLDs are reliant on either POD modes or multiple-order SMIFs, the union of both into one may be problem-solving for the resonant ESWLs.

Enlightened by Katsumura et al. (1994, 2005a, 2005b, 2007), Tamura and Katsumura (2012) and Chen and Yang (2019), Chen et al. (Chen et al. 2012, 2014) and Yang et al. (2013) also ran the UESWLs on long-span roofs via the Two-Component-Method, in which the fluctuating ESWLs-{P_E,f} is a linear combination of some dominant POD modes and [M][φ][Λ] from the prominent structural modes/frequencies, i.e., both these POD modes and [M][φ][Λ] are excellent vehicles for the BWLDs. These POD modes and structural modes/frequencies can be specifically selected by the background response participation factor and Ritz-POD method, respectively. Solving the column vector-{C} in the least square is not unlike that in Katsumura et al. (1994, 2005a, 2005b, 2007) and Tamura and Katsumura (2012).

The studies of Chen et al. (Chen et al. 2012, 2014) and Yang et al. (2013) selected some vertical node displacements and support reactions in a roof to analyze their fluctuating ESWLs and verified the superiority of the admired method by comparing these responses.

Apart from considering the background ESWLs, the method also contains the resonant ESWLs moderately, which surpasses Katsumura et al. (Katsumura et al. 1994, 2005a, 2005b, 2007) and Tamura and Katsumura (2012). As a matter of fact, when solving {C}, the method can be boiled down to a least-square solution without any constricts, thereupon, the above-mentioned mathematical weaknesses in Katsumura et al. (Katsumura et al. 1994, 2005a, 2005b, 2007) and Tamura and Katsumura (2012) will come to pass.

Luo et al. (2017) also took interest in the fluctuating UESWLs on long-span roofs. In this method, the generalized resuming force of fluctuating wind loads and multiple-order SMIFs is exploited for getting access to its covariance matrix and then the useful POD modes, namely the BWLDs which are so unlike those of Chen et al. (Chen et al. 2012, 2014) and Yang et al. (2013). Then, much like Katsumura et al. (Katsumura et al. 1994, 2005a, 2005b, 2007) and Tamura and Katsumura (2012), the UESWLs {P_E,f} is linearly superposed by these POD modes. To moderate those erratic and irrational ESWLs, every UESWL’s absolute value will be constrained within a reasonable range by the constraint factor and the extreme generalized restoring forces (Luo et al., 2017):

| {P_{E, f}} | \leq α {{\hat{p}}_{E, f}}

(51)

where α is the pre-determined constraint factor; and

{{\hat{p}}_{E, f}}

is the column vector of the extreme generalized restoring forces, in which the ith value can be expressed as (Luo et al., 2017):

{\hat{p}}_{E, f - i} = g σ_{p e, i}

(52a)

{σ_{p e}} = {\begin{array}{c} σ_{p e, 1} \\ σ_{p e, 2} \\ ⋮ \\ σ_{p e, N} \end{array}} = \sqrt{d i a g ([C_{p p}])}

(52b)

where {σ_pe} is the RMS value column vector of the generalized restoring forces; diag(⋅) is a mathematical operator for taking the diagonal elements in a matrix to a column vector; and [C_pp] matches equation (15).

When solving {C}, the equation set comes down to a least-square solution with constraint condition of equation (51), the looser constraint condition, apparently, will breed the more favorable solution. Luo et al. (2017) studied a roof’s ESWLs based on some largest displacements, in which α is recommended to take the range from 1 to 2. The ESWLs with constraints become better in controlling their value range, although there is a slight discrepancy between the accurate WISRs and the approximate ESWLs-ISRs.

Although equation (51) can relieve those erratic and irrational ESWLs in some degree, it is more likely that the ESWLs-ISRs are unequal to the actual WISRs under the rigorous constraint condition and a lot of WISRs, which will devalue the constraint effect. So, this method basically inherits the same traits of Katsumura et al. (Katsumura et al. 1994, 2005a, 2005b, 2007) and Tamura and Katsumura (2012).

The method resorting to the RMS wind loads

With the help of the gust response envelop (GRE) method, Cao et al. (2018) and Su et al. (2018) also adopted the Two-Component-Method to put forward the UESWLs on cantilevered grandstand roofs, in which the fluctuating ESWLs take the form of:

{P_{E, f}} = \pm g [K] {σ_{d}} \approx \pm g μ_{r} μ_{b} {σ_{P}} = \pm g μ {σ_{P}}

(53)

where {σ_d} fetches the column vector of displacement RMS WISRs; μ_r is the resonant factor reflecting dynamic effect; μ_b is the background factor to portray the correlation of wind loads on structures; μ=μ_rμ_b forms the GRE factor; and {σ_p} takes the known RMS wind loads corresponding to the RMS WISRs.

In equation (53), μ_r and μ_b turn to (Cao et al., 2018, Su et al., 2018):

{σ_{r, B}} = μ_{b} {σ_{r, 0}}

(54a)

{σ_{r}} = μ_{r} {σ_{r, B}}

(54b)

where {σ_r,B} and {σ_r} request the RMS vectors from the known background and total dynamic WISRs, respectively; and {σ_r,0} is composed of the basic WISRs particularly induced by the completely correlated loads (Cao et al., 2018, Su et al., 2018):

{σ_{r, 0}} = a b s ([I] {[K]}^{- 1} {σ_{P}})

(55)

where [K] is the known structural stiffness matrix in this paper; [⋅]⁻¹ is to get a square matrix’s inverse matrix; abs(⋅) means finding the absolute values.

The solutions of μ_r and μ_b are (Cao et al., 2018, Su et al., 2018):

μ_{b} = p i n v ({σ_{r, 0}}) {σ_{r, B}}

(56a)

μ_{r} = p i n v ({σ_{r, B}}) {σ_{r}}

(56b)

In consideration of engineering convenience, μ is empirically provided (Cao et al., 2018):

μ = [1 + \frac{a}{\sqrt{ξ}} \exp (- b f_{n})] μ_{b}

(57)

where a and b are two parameters to be statistically determined together with μ_b. a is recommended to be 0.3 and 0.2 for the rectangular and arc-shaped plane roofs, respectively; b is recommended as 1.5; μ_b=0.85 and μ_b=0.9 are suggested for the rectangular and arc-shaped plane roofs, respectively; ξ=the structural damping ratio in this paper; and f_n=the first structural Hz frequency in general.

By contrasting the ESWLs-ISRs with the target WISRs and confronting the ESWLs with some national standards, the method is proved to lead an advantageous approximation.

What distinguishes the method from the previous UESWLs methods is its ingenious ability to discard the vital problem of constructing multiple well-behaved BWLDs elaborately, while replace them with one synthetic value of {σ_p} for the other target WISRs, which dramatically reduces the computation for application convenience. However, this method’s accuracy is highly dependent on pinv({σ_r,0}) and pinv({σ_r,B}) just from a mathematical point of view, it is evident that the more elements in {σ_r,0} and {σ_r,B} and the bigger difference of the special distribution characteristics between {σ_r,0} and {σ_r,B} will lead to the worse errors, so the compromise between the determination of the target WISRs and the accuracy, an intractable issue, pops up again. Moreover, the fluctuating ESWLs for the other target WISRs excepting the target displacement are proportional to {σ_p}, it is somewhat short of clear physical meaning to bracket together the ESWLs shape with {σ_p}, especially for the complicated-shaped roofs. The method still specializes in the maximum target WISRs with its inability to consider their correlation. For the structures as simple as cantilevered plate-like roofs, it seems precise, because the first structural mode is dominant, leading to the strong response synchronization. For the other complicated-shaped structures, the same adversities in the previous UESWLs methods may occur, which requires further discussion.

In this section, if there are plenty of WISRs in a long-span roof, but some of which are allocated to solve the whole structure’s ESWLs, the methods of (Katsumura et al., 1994, 2005a, 2005b, 2007; Tamura and Katsumura, 2012, Chen et al., 2012, 2014; Yang et al., 2013, Luo et al., 2017), the uncompensated and compensated methods in Sun and Zhang (2020) and the POD compensated method in Sun et al. (2015) can undertake the job. In such context, the terms in [I] corresponding to the selected WISRs are merely retained to solve the least-square solution of the associated linear equation set. For the solutions without constraints or with the constraint in equation (51), the errors between the selected WISRs and their corresponding ESWLs-ISRs are smaller than the ones between the unselected WISRs and their corresponding ESWLs-ISRs.

In addition, provided that a long-span roof possesses a great mass of WISRs, all of which will take part in the structural ESWLs, however, only a handful of them are privileged to equal their corresponding ESWLs-ISRs, and the rest of the WISRs approximate to their corresponding ESWLs-ISRs. The ESWLs catering to the above needs belong to the least-square method with equality-constraint when the least-square solution is applied (Sun and Zhang, 2020). Keeping to the premises for one solution in Sun and Zhang (2020), the methods of (Tamura and Katsumura, 2012, Chen et al., 2012, 2014; Yang et al., 2013, Luo et al., 2017), the uncompensated and compensated methods in Sun and Zhang (2020) and the POD compensated method in Sun et al. (2015) can fulfill the task. The equality constraint, as compared with the least-square solution irrespective of any constraints, will deteriorate the errors between all the WISRs and their corresponding ESWLs-ISRs, albeit no errors come up in the selected WISRs (Sun and Zhang, 2020).

The Methods based on WISS

Strength, stiffness, and stability are three important concerns in the designs of structures and structural components. In trying to see into the WISRs-related ESWLs which are counted as the strength- and stiffness-aimed questions, the WISS-dependent ESWLs of long-span roofs should also gain attention. The ESWLs concerned with WISRs are ill-suited to the WISS analyses for some long-span roofs whose stability is of significance, too (Li and Tamura, 2004, 2005).

By implementing the Two-Component-Method, Li and Tamura (Li and Tamura, 2004, 2005) embarked on the most unfavorable ESWLs of a single-layer reticulated shell. In this approach, the mean ESWLs- ${{\bar{P}}_{E}}$ , based on the load code in effect, is obtained from the reference wind pressure (Li and Tamura, 2004, 2005):

{{\bar{P}}_{E}} = μ_{z} μ_{s} w_{0} {A}

(58)

where μ_z is the height variation factor; μ_s is the shape factor; w₀ is the reference wind pressure at the site; and {A} is the corresponding tributary area vector of a shell.

Then, to gain the fluctuating ESWLs, a stability analysis is conducted under the load combination of the dead load {P_D}, the live load {P_L} and ${{\bar{P}}_{E}}$ (Li and Tamura, 2004, 2005):

{P_{t e m p}} = C_{D} {P_{D}} + C_{L} {P_{L}} + C_{W} {{\bar{P}}_{E}}

(59)

where C_D, C_L and C_W are the corresponding combination coefficients, respectively. Usually the vectors {P_D}, {P_L} and C_D, C_L and C_W can be determined according to the load code.

Just prior to the occurrence of the instability point in the equilibrium path, an eigenvalue analysis of the current tangent stiffness matrix-[K_T] of the structure in static nonlinear iteration is carried out to obtain the current possible instability mode-{υ}, the first eigenvector is utilized as a rule. Next, a most unfavorable distribution of fluctuating wind loads {ε} is (Li and Tamura, 2004, 2005):

{ε} = [K_{T}] {υ}

(60)

In the end, after normalizing {ε}, the structural estimated fluctuating ESWLs {P_E,f} can be obtained (Li and Tamura, 2004, 2005):

{P_{E, f}} = {ε_{i} A_{i}} g ρ {\bar{V}}_{0} σ_{v}

(61)

where {ε_i} is the normalized vector of {ε}, and can be taken as the most unfavorable distribution estimation of the fluctuating wind loads; {A_i} is the vector of the tributary area at node i; and σ_v is the RMS value of the reference wind speed.

Since the fluctuating wind loads are random, the possible instability mode is used as a most unfavorable estimation of their ESWLs. Therefore, this method can supply a conservative estimation for the effects of the fluctuating wind loads on the structural deformation and stability (Li and Tamura, 2004, 2005). It can also determine a suitable reference displacement WISR for using the Holmes’s method of (Holmes, 2002, 2007). The method in combination with Holmes’s method can judge the ESWLs for structural deformation and stability analyses efficiently (Li and Tamura, 2004, 2005). However, it is powerless to consider the dynamic instability since it only considers the quasi-static stability under ${{\bar{P}}_{E}}$ in essence. Accordingly, the instability mode for the ESWLs may be not the actual one under total wind loads, lacking an explicit physical meaning.

Inspired by the GRF method, Gu and Huang (2015) followed a similar pattern to delve into a roof’s dynamic instability-led ESWLs, which is equal to the mean wind loads multiplied by a dynamic instability factor:

\hat{p} (x, y, z) = φ_{D} \cdot \bar{p} (x, y, z)

(62)

where

\hat{p} (x, y, z)

is the ESWL at (x,y,z);

\bar{p} (x, y, z)

is the mean wind load corresponding to the structural design wind velocity; and φ_D is the dynamic instability factor which is defined by (Gu and Huang, 2015):

φ_{D} = f_{S} / f_{D}

(63)

where f_S and f_D are the critical wind load incremental factors, as determined by the nonlinear static and dynamic stability analyses of the structure, respectively.

φ_D indicates the influence of the dynamic wind loads acted on structural stability. f_S is determined by the static stability analysis under proportionally increasing load of $f \cdot \bar{p} (x, y, z)$ , in which f is the wind load incremental factor. When determining f_D and the dynamic failure types, the well-received Budiansky-Roth criterion and a bilinear kinematic hardening elastoplastic model are adopted, respectively.

Wang et al. (2020) took charge of the method of Gu and Huang (2015) but considered three kinds of wind-induced failure modes under different rise-span ratios: (1) the elastic dynamic instability failure for the elastic state of the whole structure, (2) the elastoplastic dynamic instability failure for the plastic state of some members in the structure, and (3) the plastic dynamic strength failure for the entire structure’s plastic state. During the determination of the critical wind load incremental factors, they take account of the full combination of the Budiansky-Roth and Hsu S. C. criteria aiming at the stable or unstable post-failure load-displacement curves.

The ESWLs of Gu and Huang (2015) and Wang et al. (2020) become simple by borrowing the GRF measure. Furthermore, because the critical static wind load incremental factor under the ESWLs equals to the critical dynamic wind load incremental factor, the static stability design under the ESWLs can produce the real dynamic instability factor in the dynamic wind loads (Gu and Huang, 2015) which expressly validates its physical meaning. In Davenport’s GRF, both the mean and maximum WISRs come from a same peak WISR. But the static wind load incremental factors issue from the maximum static displacements of all nodes, when the mean wind loads mount up proportionally, these displacements might not be in a same DOF, which agrees with the dynamic wind load incremental factors. Even then, the two kinds of incremental factors perhaps also do not show up on the same DOF. If a critical wind load incremental factor can characterize an instability mode indirectly, the instability mode under the ESWLs, for the above-analyzed reason, does not necessarily reproduce the actual dynamic instability mode at least for their maximum displacements, which will thus dilute their practicability.

Conclusions

Generating the ESWLs on long-span roofs is an important subject for structural engineering because it can short-circuit structural design through bypassing complex computation. At present, the studies concerning the ESWLs on long-span roofs run into some bottlenecks, there is a strong need to tease out these existing methods and point out their future research directions. To advance this topic, this paper presents a comprehensively retrospective look at the existing ESWLs methods in long-span roofs. They can be subdivided into two broad categories, namely, the methods related to the WISRs and the WISS, in which most methods center on WISRs and fewer methods are based on WISS.

In order to characterize the features of the preselected WISRs of different structures better, the WISRs-relied ESWLs are further divided into the single and multiple target equivalent methods, in which some important mathematical and mechanical means are employed to derive different methods. The single target equivalent methods are conditioned by the target WISR. There are the following achievements in the GRF-based methods: (1) For some special structures whose responses are very much dictated by their first structural modes, the synchronization characteristics of the responses can result in a constant GRF for one whole roof whose concise parametrical expression is useful to design an actual engineering; (2) The ESWLs for internal forces, stresses and nodal accelerations are required to satisfy the different needs of structural designs; and (3) For the sake of safety, some structures also need to consider the effect of weak nonlinearity on the ESWLs. The LRC-based methods own the outputs below: (1) The Two- or Three-Component Methods are well established according to one target WISR’s different components; and (2) When the ESWLs correlate with structural modes, the mode coupling and the cross correlation can be computed to improve the accuracy. In the single target equivalent methods, the target WISR’s determination will make a great difference on the ESWLs. Because selecting one proper target WISR is often knotty, the multiple target equivalent methods probably become wiser, in which: (1) Multiple BWLDs constructed by some mathematical and mechanical tools are linearly combined to obtain the ESWLs, whose values can be rationalized and optimized by the compensation tools and the linear least-square methods with certain constraints; and (2) Analogously to the single target equivalent methods, for some special structures with few dominant structural modes to their WISRs, the well-received parametrical expressions can be channeled into the UESWLs.

The WISS of long-span roofs mainly occurs in single-layer shells, and other long-span structures are not prone to WISS. Their WISS-based ESWLs are equally imperative because they are very different from the WISRs-related ESWLs. Different mechanisms will introduce different instability modes, by which the existing methods try to map out limited ESWLs models.

By exploring all these existing approaches, it can be concluded that they are heading for simple computation, plain physical meaning, high accuracy, and convenience for the ongoing application within the engineering field. While some achievements have been made, their applications are still restricted. For a couple of promising methods to be viably employed in engineering practices soon, the subsequent research can double down on a few pivotal questions:

(1) When the ESWLs are WISRs-conscious, the number and values of the targeted WISRs should be precise beforehand, for which there is currently no cure in the existing methods.

(2) When the ESWLs of single-layer shells are WISS-oriented, the instability mode under the ESWLs should agree with the one under the actual total wind loads at least for the maximum displacement, which is not be efficiently disposed by some silver bullets. Currently, there are few WISS studies of single-layer shells under actual wind loads, and their resulting WISS mechanism or instability mode, one more meaningful topic, is unclear, retarding the studies on the customized ESWLs.

(3) The current methods only take aim at WISRs and WISS separately, if both are expected to be considered in a structure, how to find such ESWLs will be worthwhile.

(4) The methods are geared toward the common wind loads, especially the stationary Gaussian-distribution ones. The structural ESWLs for the little-seen wind loads (like tornado and thunderstorm winds) or the multiple disasters (for instance, wind and rain) are a blank, and merit profound consideration.

By resolving these knots, the ESWLs will be raised to a new height, and will speed up the development of long-span roofs. At the same time, the other engineering structures will profit from them, because many methods are versatile.

Footnotes

Acknowledgements

This project is fully supported by the Fundamental Research Funds for the Central Universities (CUG2013059013), the 2021 First-class course “Principle of Reinforced Concrete Structure” of Hubei Province, and City University of Hong Kong (Project No.: 7005770 and 9667237), which is gratefully acknowledged.

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 City University of Hong Kong (Project No: 7005770 and 9667237) and Fundamental Research Funds for the Central Universities (CUG2013059013).

ORCID iDs

Wuyi Sun

Meixia Zhang

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