8th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability
PMC2000-016
STRUCTURAL RELIABILITY ANALYSIS OF STEEL PORTAL FRAMES UNDER DYNAMIC WIND LOAD USING WIND TUNNEL DATA
M. Kasperski Ruhr-University Bochum, 44780 Bochum, Germany michael.kasperski@aib.ruhr-uni-bochum.de
Abstract The analysis of the structural reliability of steel portal frames under dynamic wind load requires non-linear calculations in the time domain to evaluate the limit state function. In the present study, the limit state function is obtained with the software DRAFS which models the opening and closing of plastic hinges for vibrating frame structures. The required time histories of the wind loads have been obtained in an accurately scaled boundary layer wind tunnel experiment. To obtain a high statistical stability, 240 individual runs have been performed which can be understood as 240 independent storms. The set of basic variables contest random deviations from the geometry, random material properties and random load amplitudes. A sensitivity analysis allows to reduce the set of basic variables to only three decisive contributions: the plastic carrying capacity of the cross section Mpl, the sum of all vertical loads q and the mean wind speed v. As example of application, a lighter roof system is analysed. This type of roof is typical for building sites where no considerable snow loads occur (e.g., Australia or Southern U.S.A.).
Introduction In the ultimate limit state, the structural behavior of steel portal frames is governed by considerable geometric and physical non-linearities which are induced by larger deformations and the forming of plastic hinges. The limit state function therefore has to be obtained from a non-linear structural analysis. The exceedence probability of a specific limit state is obtained from a multi-dimensional integral over the joint probability density of the contributing random variables. In the following, the object of the study is briefly introduced, followed by a discussion of the basic variables and possibilities of further reduction. The estimation of the limit state function is based on a special software-package named DRAFS© (Dynamic Reliability Analysis of Frame Structures) which has been developed in Bochum (Koss, 1999). As most important feature on the structural side, this program models the opening and closing of plastic hinges for a vibrating structure. The dynamic analysis is based on time histories of the wind loads which have been obtained in an accurately scaled boundary layer wind tunnel experiment. A sensitivity analysis finally allows to identify three decisive basic variables which influence the position of the traces of the exceedence probabilities. An example of application demonstrates that, with the nowadays computer capacity, a full reliability analysis for steel portal frames becomes practicable. Object of the Study Object of the study is a typical portal frame of an industrial low-rise building. The basic dimensions of the building are: eaves height h = 11 m, width d = 27.5 m, length l = 45 m, roof pitch α = 5°. The distance of the frames is 5 m. In Europe, designing engineers prefer to use standardized rolled sections. In many cases, columns and rafters have the
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same stiffness. The study is based on the group of the HEA-series (DIN 1025, 1995). As static system, a hinged support is introduced. Basic Variables The basic variables can be sorted in three groups: geometry, properties of the material and loads. Geometry includes the geometry of the structural system itself and the geometry of the cross-section. DIN 18203 specifies the quality demands for steel structures in general. Deviations from the nominal values of the basic geometry have to be extremely small, therefore it is reasonable to handle this aspect as deterministic. The geometry of the cross-section on the other hand has to be treated as partially random. DIN 1025 specifies allowable tolerances for the height and width of the profile and for the thickness of the web and flange. From the fabrication process, it is reasonable to assume independent uniform distributions for each of the random dimensions. A MonteCarlo simulation helps to reduce the number of independent basic variables. For the structural analysis, the section area A, the plastic section modulus wpl and the moment of inertia I have to be introduced. All three variables can be approximated by a normal distribution with a coefficient of variation in the order of 2% to 5%, depending on the actual size of the rolled section. It is worth mentioning that of course the distributions are limited to both ends. While the probability densities of the cross section areas are overlapping for adjacent profiles in the HEA-series (e.g., HEA 450 and HEA 500), the respective curves for the plastic section modulus show a clear separation for different profiles. For the joint-probabilities of the area and the plastic section modulus, narrow distributions are obtained indicating a strong correlation between wpl and A and only small variations of A for a specific value of wpl. The respective coefficient of variation of A for a specific value wpl is in the range of 1% to 0.5%. The next group of basic variables are the material properties, i.e., the density ρ, the modulus of elasticity E (Young's modulus) and the yielding stress σy. The variation of the density of rolled steel sections is smaller then 1% and may be neglected. The Young's modulus for steel has typically a coefficient of variation of 5%. The corresponding probability distribution has a considerable skewness and cannot be described by a normal distribution. The most decisive basic variable on the material side is the yielding stress. Several years of testing rolled steel section have led to the following values (Petersen, 1990): mean value 277 N/mm² and standard deviation 17.79 N/mm². The last group of random variables are the load amplitudes of the dead load, snow load and wind load. The dead load consists of two parts, the dead load of the structure and the dead load of the secondary construction, i.e. purlins, cladding, fasteners and thermal insulation. The dead load of the structure gstruc is obtained by the cross-section area A times the steel weight γ, i.e., it is obtained as a dependent variable. The dead load of the secondary construction gsec. has to be treated as an independent basic variable and can be described by a normal distribution with a coefficient of variation between 5% to 10%.
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A full probabilistic model of snow loads on roofs does not exist. If the climatic conditions at the building site require to take into account a considerable snow load, usually only the snow load on the ground is treated as an independent basic variable. The snow load on the roof s is obtained by multiplying this snow load on the ground with a 'deterministic' dimensionless shape coefficient. A more sophisticated model uses an additional error term which has a log-normal distribution (O'Rourke, 1982). For the probability distribution of the snow load on the ground, different models are in use, varying from the three extreme value distributions (I - Gumbel, II - Frechet, III - Weibull) to the lognormal distribution (Sack, 1997; Soukhov, 1998). The wind load has two contributing random parts: the extreme wind speed of a storm and the extreme action during a single storm. It is the second contribution which requires an extensive time domain analysis based on time histories of the wind induced actions. These time series have been obtained in a wind tunnel experiment in an appropriately scaled boundary layer flow. For the actual study, 240 independent runs have been performed, each corresponding to 10 minutes storm duration in full scale (Kasperski et al., 1996). The high number of independent runs is required for a sufficient high statistical stability of the non-linear limit state function. The analysis of the local wind climate in regard to extreme wind conditions has to be based on independent storm mechanisms like, e.g., frontal depressions, thunderstorms and tropical cyclones (Gomes and Vickery, 1978). Basic variables are therefore the extreme wind speeds v of the occurring storm phenomena. In many practical cases, only one storm phenomenon has to be taken into account. Usual models for the extremes of v are the three extreme value distributions, some researchers prefer to describe the extremes by a Gumbel distribution for v² (Kasperski, 2000; Cook, 1985). In general, the further reliability analysis requires the knowledge of the joint-probability of snow and wind. As a matter of fact, this information has to be obtained from an appropriate statistical analysis of the relevant meteorological date. However, in many cases, a simplified model can be used, if the extremes of snow and the extremes of wind are mutually excluding. This is for instance the case for most building sites in Germany. Calculation of the Limit State Function The calculation of the limit state function is performed for discrete combinations of basic variables (wpl, A(wpl), I(wpl), E, σy, gstruc(A), gsec., s, v). Two limit states are of major interest: the occurrence of a first plastic hinge and the collapse of the structure induced by a complete chain of plastic hinges. A non-linear analysis for each of 240 independent 'storms' leads for each of the storms to a result in binary form: '0' if the specific limit state is not exceeded and '1' if the limit state occurs. The number of '1's divided by the total number of storms (240) leads to a good estimate for the exceedence probability of the specific limit state. Varying in the next step the mean wind speed level v allows to produce a trace of exceedence probabilities covering the range from 0 to 1. Especially for the collapse of the structure, a multi-modal shape is obtained which reflects the fact that more than one failure path exists and contributes to the total exceedence probability.
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Sensitivity Analysis and Final Definition of the Required Family of Traces So far, nine values have to be set to perform the non-linear analysis. The question arises, if a further reduction of required information is possible. A sensitivity analysis for varying the Young's modulus E and the section moment of inertia I shows, that the traces are not affected at all for a variation of E·I from 0.9 to 1.1, where 1.0 marks the product of the respective mean values. Therefore, E and I can be handled as deterministic values and are introduced in the following with there mean values. For a chosen profile, the structural failure probability finally is obtained as:
pf =
w pl σy g sec . s v
∫∫ ∫∫∫p
l.s. ( x ) ⋅p( w pl ) ⋅p(σy ) ⋅p(g sec . ) ⋅p(s, v ) dw pl
dσy ⋅dg sec . ⋅ds ⋅dv
( x ) = ( w pl , A(w pl ), E ⋅I(w pl ), σy , g struc (A ), g sec . , s, v)
The above equation defines the exceedence probability of a specific limit state pl.s. to be dependent on five independent and three dependent variables. The position of a specific trace on the other hand explicitly depends on only three variables: • • • the amplitude of the plastic carrying capacity of the cross-section Mpl which is the product of the plastic section modulus and the yielding stress, the total vertical load which is the sum of the dead load of the structure, the dead load of the secondary construction and the snow load on the roof, and finally the mean wind speed.
The respective variations lead to families of traces as shown in Figure 1 for a variation of the vertical load amplitude and the carrying capacity of the cross-section.
1.9 0.9 rel. frequency of obtaining a plastic hinge 0.8 0.7 1.4 0.6 1.3 0.5 0.4 0.3 0.2 1.1 0.1 0.0 0.0 1.2
1.8 1.7 1.6 1.5 0.0 0.8
probability of occurrence of a chain of plastic hinges
1.0
1.0
0.8
250 350
0.6 450 0.4
0.2
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.0
0.5
1.0 mean velocity pressure [kN/m²]
1.5
2.0
mean wind velocity pressure [kN/m²]
variation of the vertical load amplitude q
variation of the plastic carrying capacity Mpl
Figure 1. Family of traces for the exceedence probabilities
The information on the exceedence probability of a specific limit state pl.s. for any combination of q , Mpl and v is obtained by interpolation between the respective curves.
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Example of Application For a first example of application, a light roof system is chosen as it is typical for climates where no considerable snow loads occur. The dead load of the secondary construction then can be introduced with a mean value of 0.2 kN/m² and a coefficient of variation of 5%. The wind climate is assumed to follow a Gumbel distribution with a coefficient of variation of 12.5%. This is more or less typical for the wind climate of Western and Central Europe. The mean value of the extreme wind speeds is varied from 20 m/s to 35 m/s. Figure 2 summarizes the respective results. The final step is to specify the target reliability. In the Eurocode 1, this target value is given as 10-6. Then, the required cross-section can be obtained from the plot. Table 1 presents the required plastic section modulus for two different target reliabilities and the different wind climates, respectively.
1 0.1 0.01 1E-3 1E-4 1E-5 1E-6 1E-7 1E-8 1E-9 1E-10 1E-11 1E-12 1E-13 1E-14 1E-15 1E-16 1E-17 1E-18 1E-19 0
failure probability - chain of plastic hinges
v = 20 m/s v = 25 m/s v = 30 m/s v = 40 m/s
1000 2000 3000 4000 5000 6000 7000
plastic section modulus [cm³]
Figure 2. Failure probability for a lighter roof system for different wind climates Table 1. Required plastic section modulus in [cm3] for a lighter roof system for different wind climates (v - mean value of the extreme wind speeds, coefficient of variation 12.5%, extreme value distribution type I)
v pf = 10-5 pf = 10-6
20 m/s 1449 1819
25 m/s 2285 2766
30 m/s 3209 3800
35 m/s 4249 4995
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Conclusions In the ultimate limit state, steel portal frames of industrial low-rise buildings will experience larger plastifications. Therefore, a reliability analysis has to be based on a non-linear structural analysis. The loads induced by wind may cause considerable dynamic effects, i.e., the analysis has to be performed in the time domain. Based on 240 independent runs in a wind tunnel experiment, a sufficient large ensemble is sampled which allows to estimate the limit state function very accurately. In the initial stage, a set of eight basic variables has been identified to be possibly contributing to the failure probability. A sensitivity analysis allowed to reduce this set to only three basic variables which are the plastic carrying capacity of the cross-section, the vertical loads and the wind speed. Then, it becomes possible from a practical point of view to estimate the failure probability, i.e., the convolution of the limit state function and the probability densities of the contributing basic variables requires not too much computing time. A first example is presented for a lighter roof construction which is typical for building sites with no snow load.
Acknowledgements The above study is based on results which have been obtained in the diploma thesis of Jörg Waltring whose task was to calculate the 'families of traces' and the diploma thesis of Anne Hartmann who developed the computer code for the efficient convolution. Both studies have been supervised with the help of Holger Koss. The development of DRAFS has been mainly sponsored by the Deutsche Forschungsgemeinschaft. All these supports are gratefully acknowledged. References DIN 1025-3, (1994), "Warmgewalzte I-Träger", (hot rolled I-section) DIN 18202, (1986), " Maßtoleranzen im Hochbau", (tolerances for offsizes in building construction) Cook, N.J. (1985), "The designer's guide to wind loading of building structures, Part 1", Butterworth, London, ISBN 0 408 00870 9 ENV1991-1, (1993), "Basis of Design and Action on Structures" Gomes, L. and Vickery, B.J. (1978), "Extreme Wind Speeds in Mixed Climates", Journal of Wind Engineering and Industrial Aerodynamics 2, 331-344 Kasperski, M., Koss, H. and Sahlmen, J. (1996), "BEATRICE Joint-Project: Wind Action on Low-Rise Buildings, Part I: Basic Information and First Results", Journal of Wind Engineering and Industrial Aerodynamics 64, 101-125 Kasperski, M. (2000), "Specification and Codification of Design Wind Loads", Habilitation-Thesis, Faculty of Civil Engineering, Ruhr-University Bochum Koss, H. (1999), DRAFS, Dynamic Reliability Analysis of Frame Structures, Version 1.3, Internal Report Aerodynamik im Bauwesen, Ruhr-University Bochum O'Rourke, M.J., Redfield, R. and Bradsky, P.v. (1982), "Uniform Snow Loads on Structures", ASCE Journal of Structural Engineering, 109(7), 2781–2798 Sack, R.L. (1997) "Perspectives on the Science Engineering Effects of Snow", 3rd International Conference on Snow Engineering, Sendai, Japan, May 26-31, 1996, Proceedings, 3-10, Balkema ISBN 90 5410 8657 Soukhov, D. (1998), “The Probability Distribution Function for Snow Loads in Germany", Leipzig Annual Engineering Report, ISSN 1432-6590
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