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Keywords: second moment method, random variable, reliability index, distribution function, failure probability.

Summary

Structural members having web openings are frequently used in both commercial and residential constructions. Web openings in beams take place quite often in practice to provide convenient passage of installation such as electrical wiring, plumbing, utility ducts and piping. As a result, story heights in buildings can be reduced, and thus the economy of the structure is affected.

Much research effort concerned with experimental and analytical studies are focused on the behaviour of structural I-beams with circular, elliptical, or rectangular openings located at mid-depth of the section. Tests in reference [1] prove that twelve castellated beams are loaded to failure in order to study the buckling of the web post between the openings. In references [2,3], the tests are carried out in order to investigate the behaviour and load-carrying capacity of the full-size built-up I-beam specimens manufactured such as Vierendeel beams at the ultimate and serviceability limit states. These experimental and theoretical research mentioned above have not been analyzed according to the probabilistic methods.

The probabilistic analysis methods in structural design may be classified in three main groups: analytical methods, numerical integration methods and the Monte Carlo simulation method. Hasofer and Lind [4] have developed the second moment theory for multiple random variables. The Hasofer-Lind method defines the reliability index as the shortest distance from the centre of the density function to the safety boundary, such as a linear or a curved line, a surface or a hyper surface depending on the number of the variables. In reference [5], the methods are improved for computing system reliability, involving first-order, second-order and third-order joint probabilities. In the numerical method developed by Song [6], the open boundaries of the integration domain have been changed to closed ones, then the integration domain has been divided into grids, and then the density function with multiple random variables has been integrated over this domain by the Gaussian numerical integration method. In the Monte Carlo simulation method, a very large sample size is usually required in order to obtain a reasonably precise estimation of failure probabilities. In order to increase the effects of the simulation, some reduction techniques of the variance can be used [7].

In this paper, the reliability of the tested full-size beams in reference [2] is determined according to the probabilistic methods at the ultimate and serviceability limit states. The methods of the partial safety factors generally permit that the internal forces and moments are computed according to the elastic or plastic analysis. When the internal forces and moments are calculated as elastic and the resistance of the element sections plastic, the interaction equations of the element sections can be used as boundary criteria; and when they are calculated as plastic, the equations of mechanism can be used as boundary criteria. The functions of the boundary criteria are constituted in linear and non-linear forms.

Considering the experimental ultimate strength as the resistance of the beams, the reliability of the beams at the ultimate limit state is determined under a certain randomly distributed load. The reliability of the beams at the serviceability limit states is also determined for a certain deflection limit and randomly distributed load. Choosing the various values of the applied load as parameters, the coefficient of variation-reliability index and the deflection limit-reliability index curves are plotted. By means of these curves, the reliability of the beams can be assessed for a given randomly distributed load and deflection limit, the ultimate strengths and service loads can also be predicted for a given failure probability.

References

1: W. Zaarour, R.G. Redwood, "Web Buckling in Thin Webbed Castellated Beams", Journal of the Structural Engineering, ASCE, 8, 860-866, USA, 1996. doi:10.1061/(ASCE)0733-9445(1996)122:8(860)
2: G. Bayramoglu, "Experimental Study of the Rectangular Web Openings with 1/2 NPI Chords", Ph.D Thesis, Institute of Science and Technology, Istanbul Technical University, Turkey, 1991 (in Turkish).
3: G. Bayramoglu, "Deflection Behaviour of Steel I-beams having Rectangular Web Openings", Bulletin of the Technical University of Istanbul, 49(3-4), 653-661, Turkey, 1996.
4: A.M. Hasofer, N.C. Lind, "An Exact and Invariant First-order Reliability Format", Journal of the Engineering Mechanics Division, ASCE, 100(EM1), 111-121, USA, 1974.
5: O. Ditlevsen, "Narrow Reliability Bounds for Structural System", Journal of the Structural Mechanics, 7, 453-472, USA, 1979.
6: B.F. Song, "A Numerical Integration Method for Computing Structural System Reliability", Computers and Structures, 36, 65-70, UK, 1990. doi:10.1080/03601217908905329
7: R.Y. Rubinstein, "Simulation of the Monte Carlo Method", John Wiley & Sons, New York, USA, 1981. doi:10.1016/0045-7949(90)90175-2

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 83 PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro Paper 123 Probabilistic Evaluation of the Test Results of Steel I-beams with Web Openings G. Bayramoglu and A. Ozgen Civil Engineering Faculty, Istanbul Technical University, Turkey doi:10.4203/ccp.83.123 purchase the full-text of this paper Full Bibliographic Reference for this paper G. Bayramoglu, A. Ozgen, "Probabilistic Evaluation of the Test Results of Steel I-beams with Web Openings", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 123, 2006. doi:10.4203/ccp.83.123 Keywords: second moment method, random variable, reliability index, distribution function, failure probability. Summary Structural members having web openings are frequently used in both commercial and residential constructions. Web openings in beams take place quite often in practice to provide convenient passage of installation such as electrical wiring, plumbing, utility ducts and piping. As a result, story heights in buildings can be reduced, and thus the economy of the structure is affected. Much research effort concerned with experimental and analytical studies are focused on the behaviour of structural I-beams with circular, elliptical, or rectangular openings located at mid-depth of the section. Tests in reference [1] prove that twelve castellated beams are loaded to failure in order to study the buckling of the web post between the openings. In references [2,3], the tests are carried out in order to investigate the behaviour and load-carrying capacity of the full-size built-up I-beam specimens manufactured such as Vierendeel beams at the ultimate and serviceability limit states. These experimental and theoretical research mentioned above have not been analyzed according to the probabilistic methods. The probabilistic analysis methods in structural design may be classified in three main groups: analytical methods, numerical integration methods and the Monte Carlo simulation method. Hasofer and Lind [4] have developed the second moment theory for multiple random variables. The Hasofer-Lind method defines the reliability index as the shortest distance from the centre of the density function to the safety boundary, such as a linear or a curved line, a surface or a hyper surface depending on the number of the variables. In reference [5], the methods are improved for computing system reliability, involving first-order, second-order and third-order joint probabilities. In the numerical method developed by Song [6], the open boundaries of the integration domain have been changed to closed ones, then the integration domain has been divided into grids, and then the density function with multiple random variables has been integrated over this domain by the Gaussian numerical integration method. In the Monte Carlo simulation method, a very large sample size is usually required in order to obtain a reasonably precise estimation of failure probabilities. In order to increase the effects of the simulation, some reduction techniques of the variance can be used [7]. In this paper, the reliability of the tested full-size beams in reference [2] is determined according to the probabilistic methods at the ultimate and serviceability limit states. The methods of the partial safety factors generally permit that the internal forces and moments are computed according to the elastic or plastic analysis. When the internal forces and moments are calculated as elastic and the resistance of the element sections plastic, the interaction equations of the element sections can be used as boundary criteria; and when they are calculated as plastic, the equations of mechanism can be used as boundary criteria. The functions of the boundary criteria are constituted in linear and non-linear forms. Considering the experimental ultimate strength as the resistance of the beams, the reliability of the beams at the ultimate limit state is determined under a certain randomly distributed load. The reliability of the beams at the serviceability limit states is also determined for a certain deflection limit and randomly distributed load. Choosing the various values of the applied load as parameters, the coefficient of variation-reliability index and the deflection limit-reliability index curves are plotted. By means of these curves, the reliability of the beams can be assessed for a given randomly distributed load and deflection limit, the ultimate strengths and service loads can also be predicted for a given failure probability. References 1 W. Zaarour, R.G. Redwood, "Web Buckling in Thin Webbed Castellated Beams", Journal of the Structural Engineering, ASCE, 8, 860-866, USA, 1996. doi:10.1061/(ASCE)0733-9445(1996)122:8(860) 2 G. Bayramoglu, "Experimental Study of the Rectangular Web Openings with 1/2 NPI Chords", Ph.D Thesis, Institute of Science and Technology, Istanbul Technical University, Turkey, 1991 (in Turkish). 3 G. Bayramoglu, "Deflection Behaviour of Steel I-beams having Rectangular Web Openings", Bulletin of the Technical University of Istanbul, 49(3-4), 653-661, Turkey, 1996. 4 A.M. Hasofer, N.C. Lind, "An Exact and Invariant First-order Reliability Format", Journal of the Engineering Mechanics Division, ASCE, 100(EM1), 111-121, USA, 1974. 5 O. Ditlevsen, "Narrow Reliability Bounds for Structural System", Journal of the Structural Mechanics, 7, 453-472, USA, 1979. 6 B.F. Song, "A Numerical Integration Method for Computing Structural System Reliability", Computers and Structures, 36, 65-70, UK, 1990. doi:10.1080/03601217908905329 7 R.Y. Rubinstein, "Simulation of the Monte Carlo Method", John Wiley & Sons, New York, USA, 1981. doi:10.1016/0045-7949(90)90175-2 purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £140 +P&P)
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