A new Method to determine Stress Cycle Histograms of Structures under Gusty Wind Load

Transcription

1 new Method to determine Stress Cycle Histograms of Structures under Gusty Wind Load Fran H. Kemper Dept. for Wind Engineering, Institute for Steel Structures, RWTH achen University, 574 achen, Germany. bstract Based on spectral computations with a subsequent frequency based Rainflow count algorithm (Dirli method), a thorough investigation of the response behaviour and the associated Rainflow histograms has been performed for different strcutures under wind load. With respect to stress histograms over the lifetime, it has been identified that mainly the structural frequency and damping, as well as the shape coefficient of the Weibull distribution for the wind speed on site are relevant. Based on this conclusion, concerted calculations of various structural and site-related parameters have been performed. For each case, the Rainflow shape has been fitted to a quadratic logarithmic function. The associated quadratic and linear coefficients could finally be expressed based on the four main influence parameters dependent on structure and location. Summarized in surface plots, an easy to use method to determine the shape of the stress cycle count histogram for any structure and any location has been developed. Introduction For the structural safety of modern steel structures, a realistic consideration all quasi-static or dynamic cyclic actions, a correct interpretation of the resulting structural stresses, a meaningful damage cumulation approach and a realistic fatigue-related material resistance is needed. For structures under gusty wind load, the description of the stress cycles during the foreseen lifetime is often estimated based on simplified closed-form approaches. In most cases, these methods overestimate the Damage value significantly with increasing bandwith of the response process. In earlier investigations it has been shown, that detailed spectral structural computations in combination with a spectral Rainflow cycle count approach lead to reliable stress range distributions for arbitrary response bandwiths (Kemper & Feldmann, b). Spectral Computations of Stress Range Spectra The most common and accepted counting algorithm for cyclic stressed structures is the Rainflow method from Matsuishi & Endo (968). s this algorithm is based on a transient formulation a usage for stochastic wind load processes is limited to the field of Monte-Carlo Simulations with artificial time-series of the wind field. lot of efforts have been made to estimate Rainflow-lie distributions based on spectral densities of response processes, which is much more convinient for the treatment of stochastic fields. n overview of some current methods has been given from Halfpenny (). Own calculations showed, that Rainflow stress range spectra of gust excited structures can be determined in good agreement to

2 6 th European and frican Wind Engineering Conference transient methods based on an emprical formula from Dirli (985). lthough the computation succeeds much quicer than transient methods, for a general rating of structures or an implication into codes it still seems too complex. Especially in the field of mechanical engineering an empirical approach for estimating the Rainflow distribution based on the power spectral density is meanwhile well established. Dirli derived this method based on continuous random processes and subsequent numerical Rainflow counting Dirli (985). He discovered an empirical relationship between the frequency range description of the signal and the Rainflow distribution. His studies were directed to stationary, ergodic random processes with arbitrary bandwidth. Consistently, the Dirli probability density function of pea values of a process with any bandwidth is a weighted sum of a normal distribution and a Rayleigh distribution. In previous studies of Wirsching and Shehata?, a similar approach was chosen, but there an attempt was made to describe the Rainflow distributions by Weibull functions. The simulations of Dirli, as well as own simulations showed, that the Rainflow distributions of peas often differs significantly from a Weibull distribution. For this reason, the Wirsching-Factor, which has been used for instance from Holmes Holmes (), is not realistic in case of higher response bandwidth Kemper & Feldmann (a). With Dirli s formula, the density of the number of cycle counts N of a given stress level can be computed to: whereat: N () = E P T f D () () f D () = D Q e Z() Q + D Z() R e Z() R + D 3 Z() e Z() m () x m = m m R = γ x m D m m 4 γ D + D (3a) D = (x m γ ) + γ D = γ D + D R (3b) D 3 = D D Z() = m (3c) Q =.5 (γ D 3 D R) D are auxillary values. The accuracy of the method has been verified with respect to gust excited structures by comparing the results to corresponding transient simulations. In all cases, the discrepancies between transient Rainflow count and Dirli formula where negligible Kemper & Feldmann (b). Even geometrical nonlinear structures can be analyzed based on the presented spectral approach. The necessary stochastic description of the mechanical system has been presented in Kemper & Feldmann (c). The method presented in this paper is aimed to clarify the influence parameters on the stress range distribution for structures under gusty wind loads and to deduce a simplified approach which taes into account the most relevant influences. The stress range density over the lifetime T Life of a structure can be expressed as a convolution of the stress range spectra within a short reference period N(, u), e.g. T ref = min. at a given wind speed and the frequency of occurence of wind speeds f(u) at the considered site (Kemper & Feldmann, ): (3d)

3 6 th European and frican Wind Engineering Conference 3 Δσ Δσ max in % 5 β α Regression of Inverse N Δσ (N Life) = α log (N Life)+ β log(n Life)+ Δσ max Life -Function: N Life Δσ Δσ max ( ) Figure : Quadratic pproximation of the Inverse N Life -Function Δσ Δσ max Δσ m,8 8 Δσ uref Δσ m,i +.5 Δσ b m,7 7 Δσ m,6 6 N T Life () Life = N= f(u) i Δσ N N(, ( Δσ) dδσ u) du (4) life m,5 5 T ref Δσ m,i -.5 Δσ b Δσ m,4 4 It is meaningful Δσ b Δσ m,3 to decouple parameters which mainly affect the scale of the stress range amplitudes 3 (e.g. the reference Δσ m, wind speed on site) and those which affect the distribution as well (e.g. the structural frequency). s N () Life describes the density of cycle counts over the structural effects, a Δσ m, direct relation to the maximum amplitude 3 succeeds 4 based 5 on 6 the cumulative 7 8 stress N Life ( Δσ range ) distribution: σ σ max n example of,the N Life -function is given in Fig.. s a result of the cumulative representation,,5 N Life (/ max ) = / max σ σ Considering all parameters of the lan G. Davenport Wind Loading Chain, it can be stated, max N that mainly N Life the natural frequency of the structure and its damping behavior as well as the shape Life coefficient of the probability density function of the wind speed on site influence the distibution of stress range spectra. Based on systematic spectral computations, the regression parameter α and β of the general N Life - Curve (cp. Fig. ) have been determined, to identify the relations between the mentioned parameters and the shape of the Rainflow stress range cycle count. In order to extract the indiviual results of the performed computations to a simplified model, for each considered location (expressed by its Weibull parameters and ), a two-dimensional fit has been performed (cp. Fig. ). Finally, the surface regression coefficients (3 in case of a bilinear fit) have been determined. s a result, the shape of the Inverse N Life -Function, expressed by its coefficients α and β can be described related to the dynamical properties of the structure for any location: N Life (t) dt (5) the maximum stress range amplitude is located at N Life =, which corresponds to a probability of exceedance of p = /T Life. For a general usage, the amplitudes may be normalized to the maximum. 3 Influence Parameters and Model Simplification max (N g ) = (κ log f + κ log δ + κ 3 ) log N g... (λ log f + λ log δ + λ 3 ) log N g + (6)

4 6 th European and frican Wind Engineering Conference 4 NL Opt. α [:4.8 :.6] NL Opt. β [:4.8 :.6].5.5. f in Hz..δ. f in Hz..δ Figure : Regression of α and β dependent on the dynamic Properties of all considered Structures where κ i and λ i are the areal regression parameter of a bilinear approach, summarized in Fig. 3. The usage of the presented approach is quite simple:. Computation of the maximum stress amplitude max. Determination of auxiliary values κ i and λ i dependent on the location of the structure 3. Formulation of the cumulative Rainflow spectrum N Life dependent on the structural dynamic parameters with Eq. (6) The stress range density N Life (/ max) can be found to: N (s) = dn(s) ds N(s) = log(e) β 4 α + 4 s α Finally, stress range collectives can be derived for arbitrary class widths and the associated fatigue damage leads do: D = D L N Life ( () ) m + N D D D where: m and m describe the inclination of the structural S-N curve. 4 DMGE EQUIVLENCE FCTOR CONCEPT (7) N Life ( () ) m d (8) N D D In the preceding sections the main computation strategy and a simplified possibility for the determination of realistic cycle count distributions is presented. The latter allows a detailed verification of the actual fatigue life considering appropriate S-N curves and damage accumulation strategies. However, at least for a pre-design stage, the damage assessment might be still too extensive, as the damage integral has still to be solved. For this reason some additional considerations have been made, allowing to reduce the fatigue assessment to a really simple chec of the fatigue endurance. The following items are addressed in order to allow a further simplification: Consideration of a realistic shape of the N Life -curve based on α(f, δ,, ) and β(f, δ,, ) ssumption of a certain shape of the structural S-N curve (trilinear or single slope) Linear damage hypothesis (elementary or modified Miner rule)

5 6 th European and frican Wind Engineering Conference 5 κ κ κ λ λ λ Figure 3: Parameters κ and λ for the determination of realistic Rainflow stress range spectra under consideration of dynamic properties and wind speed probabilities Based on these preconditions, an equivalent stress amplitude E at a given number of cycles can be computed which leads to an identical damage than the original collective. The ratio between the maximum stress amplitude max and E, is therefore denoted as a damage equivalence factor. In order to calculate the damage due to the N Life cycle count distribution in a general form, the density of cycle numbers N Life (s) in accordance to Eq. (4) is needed. The deviation of the cumulative cycle count function is given by: where: N Life (s) = dn Life(s) ds = ln() [ γ(s) exp β + γ(s) ] ln() α (9) γ(s) = β 4 α + 4 s α s = / max The general formulation of the structural damage under consideration of the elementary Miner rule leads to: D = N Life () N c ( c ) m d () where c is the notch case of the structural detail and m ist the slope of the S-N curve. s the N Life curve represends the cumulative cycle counts within 5 years, Eq. () yields to the associated damage within this period.

6 6 th European and frican Wind Engineering Conference 6 4. Elementary Miner-Rule (acc. to Palmgren) In this section the steps are described, which allow the development of damage equivalence factors using the elementary Miner rule. The equivalent stress range E can be calculated implicitly by the definition of the number equivalent of cycles n E = N D = 5 6 and by the demand of an identity of damages: n E = N D! = D Miner,elementary () The associated stress range amplitude E is defined based on the structural S-N curve: E = ( ) D m ND m Under consideration of Eq. (), the equivalent stress amplitude can finally be derived to: E = N D N Life ( s max ) s m ds The damage equivalence factor K F for the consideration of the shape of the stress range collective is given by: m () (3) K F = max / E (4) The fatigue endurance verification can be performed considering characteristic values as follows: max /K F! D (5) realistic damage assessment is enabled using the bearable cycle counts, which are associated with the stress amplitude E : ( ) m D = N D (6) E Finally, the corresponding damage value is defined as: D = n E = N D (7) The verification now includes implicitly the individual shape of the stress and leads to the same collective damage value as the integral of the actual collective consideration based on Eq. (). 4. Damage Equivalence Factors for pplication For an enabling of a preferably simple and general fatigue assessment concept, it was essential to avoid dependencies of the damage equivalence factor K F to other parameters than the shape of the cycle count distribution. Especially an influence of the maximum stress amplitude max and the notch case C would complicate the concept. Considering Eq. (3), it becomes apparent that the equivalent stress range depends on the fatigue endurance limit D in a direct way and indirectly by the damage D. Therefore, the equivalent stress range E, and thus the damage equivalence factor K F, are not dependent on the notch case C. However, as soon as a S-N curve with different inclinations is used, this independency is lost, due to the necessary case distinction.

7 6th European and frican Wind Engineering Conference 7. d.. KF 4 b 6 8 = Determine KF via a or b lternatively a simplified determination via (Weibull), f and d is possible =.9. f. = a..4 Figure 4: Damage Equivalence Factors KF dependent on the regression parameter of the NLif e function and assuming the elementary Miner rule Generally, the maximum value of the stress collective max affects the computed structural damage. In case of a simple linear S-N curve (elementary Miner rule) it follows for different collective maximums max, and max, : max, m D = D (8) max, Consequently, in conjunction with equation (3) it follows, that the equivalent stress range E acts linear to the maximum stress amplitude max of the collective. Thus, the damage equivalence factor KF is also independent of the collective maximum amplitude max. When the modified Miner s rule is used, the damage equivalence factor KF depends on both the collective maximum and the notch case of the S-N curve. For the establishment of an easy concept it is therefore advantageous to use the conservative elementary Miner rule with a unique slope of m = 3. In Fig. 4, the damage equivalence factors are plotted for arbitrary combinations of the regression parameter α and β of the cycle count distribution NLif e. Considering the influence of wind characteristic and structural dynamic properties, the equivalent stress factors can be read directly from the chart. Therefore, the bi-linear regression model has been used. s the plot is restricted to three different values for the shape parameter of the Weibull distribution, the usage is limited to rather approximative calculations, for example as a part of a pre-dimensioning. For the accurate determination of the damage equivalent factors it is recommended to determine the parameter α and β in advance using the equations described in section 3. For the fatigue assessment finally Eq. (5) can be simplified to: γf f max! D KF γm f (9) Due to the implicit form of the elementary Miner rule this proof is generally on the safe side. The corresponding damage value can be calculated as follows: D= D KF max m ()

8 6 th European and frican Wind Engineering Conference 8 5 Conclusions The presented method allows for the determination of realistic Rainflow stress range distributions for structures under gusty wind load. The formulation taes into account the individual location of the structure by considering the frequency of occurrence of wind speeds, as well as the dynamical parameter of the structure. s main influencing parameters of the stress cycle count distribution, mainly the structural frequency, the damping behavior and the wind statistic on site has been identified. Under consideration of appropriate shape function, the four independent parameters have been brought together in order to enable a simplified, graphical description. Finally, the damage equivalence factor concepts has been introduced on the base of the regression parameters which describe the actual shape of the cycle count distribution. In combination with the elementary Miner rule it was possible to calculated divisors, which allow a reduction of the maximum stress level in order to consider the distribution of cycle counts. References Dirli, T pplication of computers to fatigue analysis. Ph.D. thesis, Warwic University, Warwic. Halfpenny, ndrew.. Rainflow Cycle Counting and coustic Fatigue nalysis Techniques for Random Loading. In: RSD. Holmes, J. D.. Wind loading of structures. London and New Yor: Spon Press. Kemper, F. H., & Feldmann, M.. Rating of the fatigue relevance of gust induced vibrations based on spectral methods. Pages of: Proceedings of the 9th UK Conference on Wind Engineering. Lulu.com. Kemper, F. H., & Feldmann, M. a. ppraisement of Fatigue Phenomena due to Gust induced Vibrations based on closed-form pproaches. In: ICWE 3. Kemper, F. H., & Feldmann, M. b. Fatigue life prognosis for structural elements under stochastic wind loading based on spectral methods: Part : Linear structures. In: EURODYN. Kemper, F. H., & Feldmann, M. c. Fatigue life prognosis for structural elements under stochastic wind loading based on spectral methods: Part : Nonlinear structures. In: EURODYN. Matsuishi, M., & Endo, T Fatigue of metals subjected to varying stress. Japan Society of Mechanical Engineers.