sure distributions causing largest load effects obtained by a conditional sampling (CS) technique. Thus, LRC gives a realistic ESWL distribution. By t

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1 The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 212 Universal equivalent static wind load for structures Yukio Tamura a, Akira Katsumura b a Professor, Tokyo Polytechnic University, Iiyama 1583, Atsugi, Kanagawa, Japan b Vice-President, Wind Engineering Institute, Kawaguchi 395, Fuji Kawaguchiko-cho, Minami-tsuru, Yamanashi, Japan ABSTRACT: Studies on equivalent static wind load (ESWL) distributions are reviewed, and essential matters and necessary conditions of the ESWL are discussed. Then, the efficiency and advantages of multiple-target ESWLs are demonstrated. As an example of a multiple-target ESWL, the universal ESWL (U-ESWL) proposed by Katsumura, Tamura et al. (24, 27) is explained in detail. Reproduced largest load effects by GLF-ESWL, LRC-ESWL and U-ESWL are compared with the real largest load effects obtained by time-domain response analysis for some different types of roof systems including an actual airport terminal roof structure. The results demonstrate the feasibility and usefulness of the U-ESWL. KEYWORDS: Universal Equivalent Static Wind Load, Gust Loading Factor (GLF), Load- Response Correlation (LRC) method, Wind Resistant Design, Long-span Roof, Load Effect 1 INTRODUCTION Structural design of a building is generally based on largest load effects, such as internal forces (bending moments, shear forces, axial forces) and stresses in structural members, and in some cases on the largest story deformation or displacement of a particular part of the building. In structural design of buildings, wind load is not applied separately. We have to consider the combined effects of wind load and many other loads such as dead load, snow load, live load and so on. Therefore, it is convenient and necessary to determine a so-called Equivalent Static Wind Load (ESWL) for combinations with other loads. The ESWL is the static load providing the largest load effects due to dynamic wind excitations. Many studies have been made on ESWL. The ESWL reproducing the largest response of a structure was first introduced by Davenport [1] as that produced by the Gust Loading Factor (GLF) method. Accurate evaluation of the maximum load effect is very important in the structural design of a building, and many studies on the ESWL followed Davenport [1], e.g. Simiu [2], Solari [3], Holmes [4], Kasperski [5], Davenport [6], Piccardo & Solari [7], [8], Zhou & Kareem [9], Chen & Kareem [1], [11], Tamura et al. [12], Repetto & Solari [13], and Kwon & Kareem [14]. The GLF method proposed by Davenport [1] was a milestone in the history of wind resistant design of buildings and structures. It has been widely adopted in many modern wind load codes and standards, including some major codes. The ESWL distribution given by the GLF method is proportional to the mean wind pressure/force distribution. The mean wind pressure/force distribution is a temporarily averaged distribution and such a distribution never happens in any instance. Therefore, the ESWL given by the GLF is an unrealistic distribution. The Load-Response Correlation (LRC) method proposed by Kasperski [5] should be raised as another milestone. The LRC approach is based on a very sophisticated idea with an insight into the physical mechanism of the wind force and response relation. LRC can reproduce the most probable wind load distribution causing a particular maximum (minimum) wind load effect, as Tamura et al. [12] reported by comparing the LRC-ESWLs and ensemble averaged actual pres- 383

2 sure distributions causing largest load effects obtained by a conditional sampling (CS) technique. Thus, LRC gives a realistic ESWL distribution. By the way, when we combine the wind load with dead load or snow load in order to estimate the maximum resultant load effects in a roof beam as an example, not only the up-lift wind effect but also the down-lift wind effect should be considered in determining the critical situation for the targeted beam. Therefore, we should consider both of the largest positive side wind load effect (maximum wind load effect) and the largest negative side wind load effect ( minimum wind load effect), and two ESWLs reproducing both the maximum and minimum load effects should be determined (Tamura et al. [12], Kasperski [15], Katsumura, Tamura et al. [16]). In this paper, the term largest load effect is used for either the maximum or minimum load effect when it is not necessary to clearly indicate one or the other, or when it is intended to mean both of them. 2 EQUIVALENT STATIC WIND LOAD (ESWL) REPRODUCING LARGEST LOAD EFFECTS There are various wind load distributions that can reproduce the largest load effect targeted in structural design, if the target is only one load effect. For example, if the target load effect is the maximum tip displacement (Load Effect 1 in Fig.1), any ESWL distribution can be set to reproduce the same tip displacement as shown in Fig.1, i.e. the GLF method, the LRC method, the CS technique, and even a concentrated force at any particular point. They can all reproduce the same maximum tip displacement if the value of each ESWL is appropriately set. In this context, any wind load distribution will work for the ESWL, if there is only one target load effect. 2.1 Wind-induced response of long-span roof structures Time domain dynamic response analyses using fluctuating pressure data obtained by a simultaneous multi-channel pressure measuring system (SMPMS) have been conducted in the practical design of buildings and structures such as long-span roof structures, structures with non-linear dynamic characteristics, and buildings adopting non-linear damping devices since the 198 s in Japan (Tamura, [17]). For tall buildings with complicated sectional shapes changing along the vertical axis, as for many recent tall buildings, eccentricity causes complicated coupled motions of translational and torsional vibration components, and three-dimensional time-domain dynamic response analysis using those resultant fluctuating wind force components or by directly applying pressures obtained by the SMPMS is required to appropriately estimate the combined effects of the various wind force components and the resultant three-dimensional motions. For other buildings, a simpler but sufficiently accurate method is required for wind resistant design. ESWLs reproducing largest load effects have been examined for this purpose. The wind-induced response behavior of long-span roof structures is very complicated, showing significant contributions of multiple vibration modes of up to 1 or more. The largest load effects such as bending moments, shear forces, and axial forces in a huge number of members, e.g. 1, or more, should be considered and their largest values should never happen simultaneously. It is also necessary to consider the effects of wind directions. Different members reach their largest load effects for different wind directions. The structural designer should design all structural members based on the largest load effects found in each member under such complicated conditions. Thus, it is obviously difficult to design such structures using traditional building codes or standards based on the GLF or LRC method, which basically reproduce one particular load effect for a specifically targeted member. 384

3 The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 212 As mentioned before, if there is only one target load effect, say Load Effect 1 in Fig.1, any one of the ESWL distributions shown in Fig.1 would do to reproduce the target load effect. However, if there are two target load effects, say Load Effect 1 and Load Effect 2 in Fig.1, all of the ESWLs except for the U-ESWL would be needed to determine another ESWL distribution to reproduce Load Effect 2 as shown in Fig.1. Thus, if there are N target load effects, we need to determine N different ESWL distributions to reproduce them. The only exception is the U-ESWL, which can reproduce any number of multiple largest load effects for all the targeted members. Figure 1. Various ESWL distributions reproducing target load effects 2.2 Largest load effects and single-target ESWL distributions and multiple-target ESWL distributions As an example of single-target ESWLs like those produced by the GLF and LRC methods, let s assume the target largest load effect, rˆ, is an internal force at a particular point of a beam excited by dynamic wind load. The target load effect is expressed as a scalar value. T rˆ I Fˆ I Fˆ (1) i ri i r Here, the ESWL distribution { Fˆ } reproducing the largest load effect rˆ is given in a vector expression, and { I r } T is an influence function vector. Once the ESWL { Fˆ } is obtained, the single target largest load effect rˆ can be given as the product of the influence function vector { I r } T and the ESWL vector { Fˆ }. Thus, a static analysis can reproduce the load effect equivalent to the largest dynamic load effect. As an example of the multiple-target U-ESWLs, let s assume target largest load effects at various points of the same beam. The largest load effects should be expressed in a vector form as: Rˆ I Fˆ (2) R Here, { Rˆ } is the largest load effect vector, in which elements are the target load effects at various points, and [ I R ] is the influence function matrix. The target largest load effects are given as the product of the influence function matrix [ I R ] and the ESWL vector { Fˆ }. A scalar value of the target load effect rˆ on the left hand side of Eq.(1) is replaced by a target load effect vector { Rˆ } in Eq.(2), and the influence function vector { I r } T in Eq.(1) is replaced by the influence function matrix [ I R ] in Eq.(2). These are differences between mathematical expressions of a single-target ESWL and a multiple-target ESWL. 385

4 2.3 U-ESWL reproducing multiple load effects As a multiple-target ESWL distribution, Katsumura, Tamura et al. [16], [18], [19] proposed a U- ESWL method to reproduce all targeted largest load effects { Rˆ } based on Eq.(2). For the U-ESWL, the targeted largest load effects can theoretically be any load effects, and the number of targeted load effects is not necessarily the same as the number of structural members. An U-ESWL distribution { Fˆ } estimated for the target largest load effects { Rˆ } is assumed to be expressed by a linear combination of a set of basic wind load distributions (BWLD) { f i } as: Fˆ c f c f c f F C (3) M M Here, [ F ] is the BWLD matrix consisting of the BWLD vectors { f i }, and { C } is the combination factor vector. It is also very important that any wind load distributions, even a set of concentrated forces, are theoretically available for the BWLDs, { f i }. It is of course better to use the most efficient distributions for the BWLDs. In Katsumura, Tamura et al. [16], [18], [19], the POD eigenvectors of the wind pressure field acting on the building/structure model obtained by SMPMS are used, but it should be emphasized again that the POD eigenvectors are not necessarily required for the BWLDs in Eq.(3). Combining Eqs.(2) and (3), the target largest load effect vector { Rˆ } is given as: Rˆ I Fˆ I F C R C (4) R R Once the BWLD matrix [ F ] is given, as the influence function matrix [ I R ] can be obtained accordingly for the structural system, the product matrix [ R ] = [ I R ][ F ] can be a known matrix. Then, the problem is only to solve the combination factor vector { C } based on the target largest load effect vector { Rˆ }. The target largest load effects { Rˆ } can be obtained by any response analysis, but the time-domain response analysis applying the SMPMS pressure field to an FEM model may be generally made. If the number of target largest load effects, N t, is equal to the number of BWLD terms, M, in Eq.(3), i.e., N t = M, Eq.(4) can be solved uniquely. If the number of target largest load effects N t is less than M, i.e., N t < M, the number of BWLD terms, M, can be appropriately reduced to N t ; thus Eq.(4) can also be solved uniquely. However, in many cases, the number of target largest load effects, N t, is more than the number of BWLD terms M, i.e., N t > M, because of the huge number of structural members. Of course, you can increase the number of BWLD terms, M, by increasing the number of loading points for the FEM model or by any other way to equalize N t and M. In Katsumura, Tamura, et al. [16], [18], [19], the Singular Value Decomposition (SVD) technique has been used to find the most appropriate solutions approximating the simultaneous equations given by Eq.(4). It is a matter of course that any mathematical technique can be used to obtain appropriate solutions. 2.4 Simple cantilever roof model The simple cantilever roof model shown in Fig.3 was discussed in Katsumura, Tamura et al. [16], [19]. Here, the number of structural members N m is 9, and the number of loading points is set at 48. The loading points are set on the nodes of the grid members, and the number of BWLD terms M is also set at 48. All structural members are targeted, and one largest load effect is considered for each member. Then, the number of target largest load effects is set at N t = N m = 9. In this case, N t > M. As unique solutions cannot be obtained for Eq.(4), the SVD technique is applied. A wind tunnel experiment was performed to obtain fluctuating pressures acting on the upper and lower roof surfaces at the same 48 positions on the cantilever model. The net wind forces acting at these positions were applied in the FEM model for time domain dynamic response analysis. 386

5 The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 212 Then, a set of largest bending moments and largest shear forces in all 9 members were extracted. Targeting these largest load effects, U-ESWL distributions were obtained. In this case, the POD eigenvectors { i } were adopted for the BWLD vectors { f i }. Figures 4(a) and 4(b) show the up-lift side U-ESWLs for the largest bending moments and the largest shear forces. No No mm 6 18mm mm mm (a) Elevation (b) One-way type frame model (c) Two-way type frame model Figure 3. Simple cantilever model (a) U-ESWL (Shear Fs.) (b) U-ESWL (Bending Ms.) (c) GLF (Shear F., GLF=2.4) (d) LRC (Shear F., g Q =4.3) Figure 4. Up-lift side ESWLs for largest load effects on all 9 members ((a) and (b)) and for a single (maximum) largest shear force ((c) and (d)) for a one-way type model shown in Fig.3(b) Two U-ESWLs for the largest shear forces and the largest bending moments are similar. This suggests that either Fig.4(a) or 4(b) is enough for both load effects in this case. Figures 4(c) and 4(d) show ESWLs obtained by the GLF and LRC methods, respectively, for the largest shear forces. The GLF-ESWL distribution shown in Fig.4(c) is proportional to the mean wind force distribution, which is an unrealistic distribution never happening at any instance as are the U- ESWLs shown in Figs.4(a) and 4(b). However, the LRC-ESWL shown in Fig.4(d) is a realistic distribution, which most probably generates the quasi-static component of the largest shear force of a target member. Figure 5 compares the largest shear forces in members obtained by the time domain response analysis and those reproduced by the ESWLs (Katsumura, Tamura et al., [16]). The abscissa indicates 48 main members numbered 1 to 48. The maximum shear force in Member No.1 obtained by time-domain dynamic response analysis showed the largest value of all members, and it was selected as the target largest load effect in the calculations of ESWLs for both the GLF and LRC methods. Therefore, the largest shear forces in Member No.1 reproduced by GLF- ESWL and LRC-ESWL are exactly the same as the actual largest shear force obtained by the time-domain dynamic response analysis as indicated in the figure. However, the reproduced largest shear forces in the other members are underestimated by both GLF-ESWL and LRC- ESWL as shown in Figs.5(a) and 5(b), respectively. It is interesting that GLF-ESWL, propor- 387

6 tional to the unrealistic mean wind force distribution, gives better estimation than LRC-ESWL, based on the realistic instantaneous wind force distribution. This is understandable because the internal forces in the other members do not necessarily reach their maximum values at the same time that the maximum internal force in the targeted Member No.1 appears. (a) GLF method (G f =2.4) (b) LRC method (g Q =4.3) (c) U-ESWL Figure 5. Comparisons of largest load effects obtained by time domain response analysis and ESWLs [kn] 8 6 by U-ESWL(+) Time-domain Maximum(positive) 4 2 SF Along-span member Along-ridge member Member's number Response Analysis by Minimum(negative) U-ESWL( ) (a) U-ESWL for (+) Shear Fs. (b) U-ESWL for ( ) Shear Fs. (c) Reproduced largest shear forces Figure 7. U-ESWLs for positive side (a) and negative side (b) largest shear forces (without mean components) for a two-way type model (see Fig.3(c)), and (c) comparisons of largest shear forces obtained by time domain response analysis and U-ESWLs 3 REQUIREMENTS FOR ESWL As seen in Fig.5, the realistic LRC-ESWL distribution is not necessarily better than an unrealistic GLF-ESWL or U-ESWL distribution in terms of reproduction of maximum load effects for non-targeted members. The LRC approach can catch the typical load distribution condition causing the largest load effect in a target member (or at a target position). However, this is too typical (or too good) for the target load effect. For example, a typical LRC-ESWL distribution causing the largest bending moment at the end of a roof beam and that causing the largest bending moment at the center of a roof beam are quite different (Tamura et al., [12]). The LRC- ESWL for the beam end bending moment is obviously inappropriate for reproducing the largest beam center bending moment. Superimposition of the two ESWL distributions for these two different targets is of course also inappropriate. For such a multiple-target estimation, an U-ESWLlike approach is required. Anyway, it is strongly suggested that the LRC-ESWL is good only for the target load effect, but not for other load effects. Of course, a realistic LRC-like approach is useful for some particular purposes and has other advantages. In order to clarify the essential matters in ESWLs, let s go back to the GLF by Davenport [1]. He assumed the following conditions for estimating the ESWL for a structure: 388

7 The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 212 (1) Mean displacement caused by mean wind load is almost proportional to the 1st vibration mode. (2) The contribution of the 1st vibration mode is predominant in the dynamic response and the contributions of other higher modes are negligible. Under the above conditions, the distribution of the maximum displacement along the vertical axis of a tall building can be proportional to that of the mean displacement caused by the mean wind load. Therefore, even if the targeted load effect of the GLF is only the maximum tip displacement, the ESWL can also reproduce the maximum displacement at any other story. The temporary and spatially fluctuating pressure field acting on a building never shows the temporary averaged (mean) pressure distribution at any instance. Thus, the mean wind force distribution is a completely unrealistic distribution, and in this regard, there is no difference from a single unrealistic concentrated load. Thus, the original ESWL proposed by Davenport [1] was intended to reproduce not only the target largest load effect, i.e. the largest tip displacement, but also the other largest load effects simultaneously, i.e. the largest displacements of all other stories. By the way, the reason why Zhou & Kareem [9] adopted the largest base-bending moment rather than the largest tip displacement was to reproduce the largest internal forces, which are more important than the largest displacement in the design of tall buildings. In general, the 1st-mode contribution is predominant for the resonant component of the windinduced building response, but the dynamic response consists not only of the resonant component but also of the background component. The latter is not necessarily proportional to the 1st vibration mode, and the 2nd condition is not acceptable in some cases. Especially in the case of long-span roof structures, the higher mode contributions are very significant for the maximum internal forces in the structural members, and they never appear simultaneously. Thus, in general, the largest internal forces appear at different instances, and the LRC-ESWL by Kasperski [5] does not guarantee the largest internal forces in other non-targeted members. Anyway, it is obvious that the main purpose of the ESWL is to reproduce the targeted largest load effect, and it is also essential to reproduce other non-targeted largest load effects. These two are very important requirements for ESWLs in practical applications, and it is also true that a realistic wind force distribution essentially does not satisfy the latter requirement except in some special cases and only unrealistic wind force distributions are possible. Then, if these two requirements are the most important, it is very natural to directly seek an unrealistic wind force distribution reproducing the largest internal forces simultaneously in all structural members. In answer to this question, Katsumura and Tamura et al. [16], [18], [19] proposed the U-ESWL to reproduce the largest load effects in all structural members. 4 REQUIRED POSITIVE-SIDE AND NEGATIVE-SIDE U-ESWL DISTRIBUTIONS The U-ESWL distributions are derived by an inverse analysis based on the actual largest load effects in all or some important targeted members obtained by dynamic time domain response analyses applying fluctuating pressure fields. Figure 5(c) compares the actual largest shear forces reproduced by the U-ESWL shown in Fig.4(a) and the actual largest shear forces obtained by the dynamic time domain response analysis in 48 targeted members. By applying the simple U- ESWL distribution shown in Fig.4(a), all largest load effects were almost perfectly reproduced. This is quite different from the results shown in Figs.5(a) and 5(b) for GLF-ESWL and LRC- ESWL. The efficiency of the U-ESWL might be understood from Fig.5(c). It should be noted that two U-ESWL distributions reproducing positive-side and negativeside largest load effects are necessary in building design, considering the combined load effects 389

8 with other loads such as dead load, snow load and so on. If we include the mean component of the pressure field, both the obtained maximum and the minimum wind-induced load effects in all members can be all positive, all negative, or a mixture of positive and negative. However, if the mean components of pressures are subtracted and only fluctuating components are applied, the maximum load effects can be all positive and the minimum load effects can be all negative. For simplicity but not losing generality, we only apply the fluctuating components of pressures in the time domain analyses. The effects of the mean components can be taken into account separately or superimposed finally. Figures 7(a) - 7(c) show positive-side and negative-side U- ESWLs and the largest load effects reproduced by these two U-ESWL distributions for the twoway type cantilever frame model shown in Fig.3(c). Here, the red solid lines shown in Fig.7(c) indicate the maximum and the minimum shear forces in all 9 members obtained by timedomain response analysis. If the target load effects are the maximum and minimum shear forces in these 9 (= N t ) members, we should consider N t positive shear forces Q j + ( j = 1, 2,... N t ) and N t negative shear forces Q j ( j = 1, 2,... N t ) as the target load effects. In this case, two sets of largest shear forces should be considered for two U-ESWLs, but elements of the two sets are arbitrarily selected. It is not necessary that one set should consist of all positive shear forces and the other should consist of all negative shear forces. Any combination of the largest positive and negative shear forces is theoretically fine, if Q j + and Q j appear in either of the two sets as follows: Set A: Q + 1, Q 2, Q + 3, Q + 4, Q 5,..., Q + Nt-1, Q Nt Set B: Q 1, Q + 2, Q 3, Q 4, Q + 5,..., Q Nt-1, Q Nt + However, for general use of U-ESWL distributions and its desired robustness with various structural systems, a moderate wind load distribution like those shown in Figs.4(a) or 4(b) is better. If the combination of the signs of the largest load effects in a set was not adequate, even an obtained U-ESWL can reproduce all targeted largest load effects appropriately; the obtained ESWL distribution is too specific for the given structural system and cannot show robustness for different structural systems. In order to overcome this difficulty, Katsumura, Tamura et al. [16] proposed to use some POD eigenvectors of fluctuating load effects in all members to decide the combination of signs of load effects for the two sets, Set A and Set B. It should be noted that the POD eigenvectors are different from those of the fluctuating pressure field used for the BWLD in Eq.(3). The POD technique is applied for a time series of internal forces in all members obtained by the time-domain response analysis. 4.1 Two-way type cantilever roof model Katsumura, Tamura et al. [16] discussed the contributions of the 1st and 2nd POD modes to the fluctuating shear forces in the 9 members of the two-way frame shown in Fig.3(c). The 1st mode contribution was very significant for the along-span members, while the 2nd mode contribution was very significant for the along-ridge members. Based on these, the signs of the largest shear forces in the along-span members are set the same as the 1st POD eigenvector of the fluctuating shear forces in all members, and those of the largest shear forces in along-ridge members are set the same as the 2nd POD eigenvector. In this way, the two U-ESWL distributions are obtained as shown in Figs.7(a) and 7(b). By applying these two U-ESWL distributions, positiveside and negative-side largest wind load effects can be obtained for all members as shown in Fig.7(c). 39

9 The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, Flat roof models Following the procedure explained before, two sets of U-ESWL distributions are obtained for a truss-beam flat-roof model, as shown in Figs.1(a) and 1(b) (a) U-ESWL for Set A (b). U-ESWL for Set B (c) Reproduced largest shear forces Figure 1. U-ESWL distributions for a flat-roof model and comparison of largest load effects obtained by timedomain response analysis and U-ESWLs Figure 1(c) compares the largest axial forces, bending moments and shear forces in all members of the truss-beam model obtained by the time domain response analysis and those reproduced by the U-ESWLs. In this case, various different kinds of internal forces are simultaneously targeted, and all of them are appropriately reproduced by only two U-ESWL distributions, as shown in Figs.1(a) and 1(b). Here, the efficiency and the feasibility of the U-ESWL are clearly demonstrated. 5 CONCLUDING REMARKS The essential matters required for ESWLs are discussed and some advantages of the multipletarget ESWL, especially U-ESWL, are demonstrated for long-span roof structures. There is no significant difference between the procedures for obtaining ESWL by the various approaches. GLF-ESWL, LRC-ESWL, U-ESWL and so on all require the following procedures: - Getting largest load effects to be targeted, probably by time-domain response analysis - Conducting inverse analysis to get ESWL based on given target load effect(s) - Accumulation of ESWL distributions to create database for codification The GLF method has been widely adopted in many wind load codes and standards, and the aerodynamic database for the GLF methods. The advantage of the simplicity of GLF is also obvious, and can be reasonably applied for tall buildings. It is also well known that many novel ideas on ESWL have been proposed and recent improvement of the GLF method is significant. The Gust Front Factor has been proposed by Kwon & Kareem [14] to envelope the GLF method and to reflect the effects of an unsteady flow field. However, it is also true that a traditional single-target approach such as GLF or LRC has a limitation for roof structures, and the multiple-target approach such as U-ESWL is required and has the possibility of further development in its application. More studies on U-ESWL are desirable. 391

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