STOCHASTIC WIND-EXCITED RESPONSE OF TALL BUILDINGS INFLUENCE OF MODE SHAPES

STOCHASTIC WIND-EXCITED RESPONSE OF TALL BUILDINGS INFLUENCE OF MODE SHAPES The influence of structural mode shapes on structural response estimates has led to the introduction of several correction procedures. Each procedure focuses on a particular aspect of the overall response analysis framework. This paper first derives expressions for the wind-induced response by separately utilizing an arbitrary and an ideal mode shape. This facilitates in delineating the effect of a non-ideal mode shape on building response. It is noted that the non-ideal mode shapes have different level of influence on various response components. For example, the influence is negligible for the displacement and the base bending moment as noted in the literature. However, the influence on some other response quantities such as the base shear and the mode generalized wind loads is not that significant. This paper also presents a correction procedure for the influence of mode shape on the equivalent static wind load (ESWL), which are very attractive for incorporation in codes and standards.

8 th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability PMC2000-304 STOCHASTIC WIND-EXCITED RESPONSE OF TALL BUILDINGS: INFLUENCE OF M ODE SHAPES Y Zhou, A Kareem, M ASCE University of Notre Dame, Notre Dame, IN 46556 yzhou@nd.edu, kareem@nd.edu M Gu, M ASCE Tongji University, Shanghai, 200092, P R China Abstract The influence of structural mode shapes on structural response estimates has led to the introduction of several correction procedures Each procedure focuses on a particular aspect of the overall response analysis framework This paper first derives expressions for the wind-induced response by separately utilizing an arbitrary and an ideal mode shape This facilitates in delineating the effect of a non-ideal mode shape on building response It is noted that the non-ideal mode shapes have different level of influence on various response components For example, the influence is negligible for the displacement and the base bending moment as noted in the literature However, the influence on some other response quantities such as the base shear and the mode generalized wind loads is not that significant This paper also presents a correction procedure for the influence of mode shape on the equivalent static wind load (ESWL), which are very attractive for incorporation in codes and standards Introduction An ideal mode shape, linear in sway mode, or uniform in torsion, has traditionally been assumed when developing the current analysis procedures, for example, the “gust loading factor” (GLF) approach (Davenport, 1967), and experimental techniques, involving the high frequency base balance (HFBB) and the “stick” type aeroelastic tests Accordingly, these analysis procedures and experimental techniques work best for structures that have mode shapes close to ideal However, the mode shapes of many structures may depart from the ideal Correction procedures are then needed to consider the influence of nonideal mode shapes on the measured or calculated load effects corresponding to the structural mode shapes In this context, several correction procedures have ever been developed in the last few decades A correction formula for the GLF was given by Vickery (1970) in which the error for the actual mode shape deviating from a straight line was within 1%~3% Similar observation was made by Tamura et al (1996), who reported that “the effect of mode shape is within 4% when β = 0.5 ~ ” ( β is the exponent of the first mode shape) In light of this, the effect of a non-linear mode shape has been excluded in the GLF-based analysis procedures given in the major codes (e.g., AS1170.2, 1989; RLB-AIJ, 1994) Vickery’s observation (1970) was also referred to by several investigators in the early development Zhou, Kareem and Gu of the HFBB technique Tschanz and Davenport (1983) identified a need to correct the torsional component measured using the HFBB They suggested an approximate correction based on the base shear component measured by the balance But no theoretical basis was provided As the HFBB technique gained more acceptance, the topic of mode shape correction was focused on the generalized wind loads (e.g., Kareem, 1984; Kijewski and Kareem, 1998; Holmes, 1987) Besides the generalized wind loads, Vickery et al (1985) discussed further the effect of mode shape on the top floor acceleration and the base bending moment Recognizing that the generalized wind load is arbitrary in magnitude which depends only on the normalization scheme of the mode shape, Boggs and Peterka (1989) suggested that the mode shape correction should be applied to actual quantities They introduced a load correction factor, which is essentially the factor for the base moments The Australian Standards, AS1170.2 (1989), provides two correction factors, (1.06-0.06 β ) and (0.76+0.24 β ), for adjusting the base bending moment and the acceleration response in the across-wind direction, respectively Similarly, the AIJ standard, RLB-AIJ (1994) provides (0.73+0.27 β ) as a correction for displacement In this paper, the effect of non-ideal mode shapes is examined in the overall analysis framework for the wind-induced response Two formulations for the wind-induced response are derived, one is based on an arbitrary non-ideal mode shape and the second involves an ideal mode shape Through a comparison between these two formulations, mode shape corrections are evaluated at each step of the overall analysis process for a range of wind-induced response components A parameter study is carried out to clarify the effect of some involved parameters on the CFs The results are compared to those reported in the literature Since the CFs presented here are dependent on the response component of interest, a correction procedure for the ESWL is also proposed Analysis of Wind-induced Response The RMS displacement response in the first mode is given by σD ( z) = 1/2  ∞  H ( f ) S ( f ) df ⋅ ϕ( z )   P*  k *  ∫0 (1) where H ( f ) is the structural transfer function; k * = (2πf ) m * is the generalized stiffness in the first mode; f is the natural frequency of the first mode; m( z ) = m0 (1 − λ( z / H ) ) is the mass distribution which is assumed to be linearly distributed; S P * ( f ) is the generalized wind load and the first mode shape may be approximated by ϕ( z ) = c1 ( z / H ) β To estimate the generalized wind load, the coherence of the externally applied wind force is approximated by P( z1 , f ) P * ( z2 , f ) = P( z1 ) P( z ) S p ( f )Q( z1 , z , f ); P ( z ) = PH ( z / H )γ Q( z1 , z , f ) = exp( −CK f / V ⋅ z1 − z / H ) (2) where S p ( f ) is the unit fluctuating wind force or torque spectrum; PH is the amplitude of the fluctuating wind force evaluated at the building top; Q is the coherence of the Zhou, Kareem and Gu fluctuating wind pressures; γ is the fluctuating wind profile exponent and C K is a coefficient The wind-induced dynamic response is customarily divided into the resonant and the background components For the sake of brevity, the derivation in this paper involves only the resonant component in the sway mode Other details can be found in a paper by the authors (Zhou et al, 2000) With the resonant displacement response, the ESWL and an arbitrary load effect, can be computed by PH (1 + β)( + β) σ FR ( z ) = ⋅ H [( + β) − λ(1 + 2β )] πf z  z   J (γ , β, f ) S p ( f ) 1 − λ   4ζ H  H   β (3) σrR = ∫ σ FR ( z )i ( z )dz H (4) PH ic ⋅ (1 + β)( + 2β ) [( + β + β0 ) − λ(1 + β + β0 ) ] πf = ⋅ J (γ , β, f ) S (f ) [( + β ) − λ(1 + β)] ( + β + β0 ) ⋅ (1 + β + β0 ) 4ζ p where J (γ , β, f ) is usually referred to as the joint acceptance function; ζ is the damping ratio; and i ( z ) = ic ( z / H ) β is the influence function For the base shear force and base moment, these coefficients are ic = , β0 = and i c = H , β0 = , respectively Simplified Procedure for Structure with Ideal Mode Shape The analysis procedure in the preceding section is accurate while, in practice, can give meaningful solution only for a few limited situations, such as some buffeting problems where quasi-steady and strip theories hold However, in most engineering applications, the information about the aerodynamic loads is not always available On the other hand, when a building has a linear mode shape in sway modes, there is a very expedient way to evaluate the generalized wind load on the building S P* ( f ) = S M ( f ) / H (5) where S P * and S M ( f ) are the PSD of the generalized wind load and applied base bending moment, respectively Given the generalized wind load, the wind-induced response can be computed using the procedure provided in the last section, which is usually simpler in expressions as provided in the following: σ ′FR ( z) = ′ = σ rR 12 PH z  z  πf  ⋅ J (γ ,1, f1 ) S p ( f ) 1 − λ   H ( − 3λ ) 4ζ H  H   12 ⋅ PH i c ⋅ [( + β ) − λ (2 + β )] π f1 J (γ ,1, f1 ) S p ( f1 ) (4 − 3λ ) ⋅ ( + β ) ⋅ ( + β ) 4ζ (6) (7) where the superscript prime indicates linear mode shape It is noted that the applied moment in the simplified formulation is directly measured using a HFBB Zhou, Kareem and Gu Mode Shape Corrections for Generalized Wind Loads When the mode shape of an actual building deviates from a straight line, the following mode shape correction is introduced to modify the base bending moment to the generalized wind load (e.g., Kijewski and Kareem, 1998) φX ( β) = σP * ( β ) / (σM / H ) (8) φX where σ is the RMS value The variation in the generalized wind load CF with the mode shapes 2.0 Full cor Delta cor and other involved parameters is shown in Fig.1 γ =0.00 1.8 γ =0.15 As expected, when the mode shape of the γ =0.25 1.6 γ =0.35 structure is linear the CF is unity regardless of 1.4 other involved parameters However, significant 1.2 corrections are needed when the actual mode shapes deviate from this assumption since the 1.0 CF is very much dependent on the mode shape 0.8 exponent Although the mode shape correction 0.6 for the generalized wind load has been most 0.0 0.5 1.0 1.5 2.0 β widely used in research, some inherent shortcomings relevant to this concept still call Figure Mode shape correction for special attention The first is related to the for generalized wind loads definition of this correction since the generalized wind load, which is arbitrary in magnitude depending on the normalization scheme of the mode shape The second is about the correct use of this CF in the wind-induced response analysis procedure, which may introduce additional inconsistencies (Vickery et al, 1985) Mode Shape Corrections for Wind-Induced Response Following the mode shape correction for the generalized wind load, which is actually the ratio between the actual generalized wind load and that based on an ideal mode shape, the mode shape correction for the wind-induced response can be formulated by referring to Eqs and 7, as ηrXR = σ rXR (4 − 3λ ) ⋅ (1 + β )(2 + β ) = × σ ′rXR 12 [( + β ) − λ (1 + β )] (3 + β )(2 + β ) ⋅ [(2 + β + β ) − λ (1 + β + β ) ] ( + β + β )(1 + β + β ) ⋅ [(3 + β ) − λ (2 + β ) ] J (γ , β , f1 ) J (γ ,1, f ) (9) It is noted that the CF in Eq depends not only on the wind exponent, correlation, and the mode shape exponent, but also on the response component of interest Using a full correlation and substituting β0 = in Eq will lead to the base moment CF, which was alluded to in Boggs and Peterka (1989) as the load CF This CF ranges from 0.84 to 1.06 as shown in Fig 2(a) However, this CF is different from that for the base shear, which, as plotted in Fig 2(b), varies in the range of 0.75~1.18 that cannot simply be ignored Zhou, Kareem and Gu 1.2 Full cor Delta cor γ =0.00 γ =0.15 γ =0.25 γ =0.35 1.2 1.1 η XR 1.1 ηXR 1.3 Full cor Delta cor γ =0.00 γ =0.15 γ =0.25 γ =0.35 1.0 1.0 0.9 0.9 0.8 1.06-0.06*β (AS1170.2-89) 0.8 0.7 0.5 1.0 1.5 2.0 0.5 1.0 1.5 β 2.0 β (a) Figure Mode shape corrections for resonant response in sway mode (b) Corrections for Equivalent Static Wind Loads σ FXR ( z ) = η FXRH ⋅ σ ′FXR (H ) ⋅ (z / H ) η FXRH = σ FXR ( H ) (1 + β ) = ⋅ σ ′FXR ( H ) Full cor Delta cor 1.4 η FXRH Since the above mode shape corrections are all dependent on β0 or the response component of interest, it may not be convenient to evaluate the CF for each desired response component in practice To facilitate the use of CFs in design practice, a correction procedure for the ESWL is developed Comparing Eq with Eq 6, when assuming a uniform distribution of the mass as implied in most codes and standards, the ESWL on the actual structure for the sway and the torsion modes can be expressed by 1.2 γ =0.00 γ =0.15 γ =0.25 γ =0.35 1.0 0.76+0.24*β (AS1170.2-89) 0.8 0.5 1.0 1.5 2.0 β β J (γ , β , f ) J (γ ,1, f ) Figure Mode shape correction for (10) resonant ESWL component where σ ′FXR ( H ) = ( / H ) πf S MX ( f ) / 4ζ is the resonant ESWL component evaluated at the top of the building in the sway mode Figure shows the variation in this CF, which can be approximated by ηFXRH = 0.76 + 0.24 × β (11) Conclusions To access the wind-induced response of a particular building structure with non-ideal mode shapes using the base moments input, the mode shape CF can be evaluated at each step of the overall analysis framework through a comparison between the ideal and the Zhou, Kareem and Gu non-ideal mode shape formulations A parameter study shows that the effect of a nonideal mode shape is actually negligible for the displacement and the base bending moment, but is not so small for other load effects, such as the generalized wind load, the base shear and the acceleration This observation is consistent with the comments made by earlier researchers Although the existing correction procedures are effective in their own context, caution must be exercised in utilizing these procedures indiscriminately for other load effects Among several mode shape correction schemes, the ESWL-based correction procedure offers a convenient framework for implementation in codes and standards and correct interpretation of wind tunnel measurements Acknowledgements The authors gratefully acknowledge the partial support from NSF Grants # CMS 95-03779 and CMS 9522145 for this study Authors would like to thank Dr Xinzhong Chen and Ms Tracy Kijewski for their comments on this manuscript References Architectural Institute of Japan (1994) “Recommendations for Loads on Buildings.” (in Japanese) Australian Standards (1989) “SAA Loading code, part 2- wind forces, AS1170.2-89.” Boggs D W and Peterka J A (1989), “Aerodynamic model tests of tall buildings”, ASCE Journal of Engineering Mechanics, 115(3), 618-635 Holmes J D (1987), “Mode shape corrections for dynamic response to wind”, Engineering Structures, 9, 210-212 Kareem A (1984), “Model for predication of the across-wind response of buildings”, Engineering Structures, 6(2), 136-141 Kijewski T and Kareem A (1998), “Dynamic wind effects: a comparative study of provisions in codes and standards with wind tunnel data”, Wind and Structures, 1(1), 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