Microsoft Word - ETASR_V13_N1_pp9888-9892 Engineering, Technology & Applied Science Research Vol. 13, No. 1, 2023, 9888-9892 9888 www.etasr.com Fahmi & Al-Zaidee: Buckling and Vibration Estimation of Girder Steel Portal Frames using the … Buckling and Vibration Estimation of Girder Steel Portal Frames using the Bayesian Updating Methods Fahmi H. Fahmi Department of Civil Engineering, College of Engineering, University of Baghdad, Iraq f.fahmi1901m@coeng.uobaghdad.edu.iq (corresponding author) Salah Al-Zaidee Department of Civil Engineering, College of Engineering, University of Baghdad, Iraq salah.r.al.zaidee@coeng.uobaghdad.edu.iq Received: 26 October 2022 | Revised: 5 November 2022 | Accepted: 7 November 2022 ABSTRACT In practice, structural systems are complicated. When dealing with such systems, the use of analytical simulations is or may be impossible to apply, since the problem includes many variables with partial differential formulations. The stability and dynamic response of structures is an important aspect that must be paid particular attention to in order to ensure safety against collapse. A physical phenomenon of a reasonably straight, slender member bending laterally (usually abruptly) from its longitudinal position due to compression is referred to as buckling. Two kinds of buckling can be distinguished: (1) bifurcation-type buckling and (2) deflection-amplification-type buckling. In vibration, frequencies and mode shapes must be known in order to use some methods of dynamic response analysis. To calculate frequencies and modes, an eigenvalue problem is solved in algebraic form. In structural dynamic problems, only the lowermost eigenpairs are of important interest. The highest eigenpairs are not needed and are not accurate due to discretization errors. Keywords-interior supporting frame; Bayesian method; sample data; prior data; posterior data; first order; MCs I. INTRODUCTION The majority of engineering designs are based on deterministic variables and often do not consider the variations in the material properties and the geometry of the structure [1]. Uncertainty is vital to the analysis and design of an engineering system and the effects of uncertainties must be quantified and propagated. Increasing amounts of data in engineering systems are collected and stored. This information can and should be used to decrease the uncertainty in engineering models and optimize the management of these systems. A coherent and effective way for merging new with existing information is provided by the Bayesian method, in which prior probabilistic information is updated with new data and observations. The Bayesian framework enables the combination of uncertain and incomplete information with models from different sources and provides probabilistic information on the accuracy of the updated model [2]. The current study emphasizes the analysis of the buckling and vibration of steel portal frames, where new data have been acquired for the independent variables of the system. Subsequently, the Bayesian method presents an efficient way to update the prior knowledge regarding the stability and natural frequency of the frame. The first-order approximation method and Monte Carlo (MC) simulation have been adopted to investigate the statistical properties of the buckling and vibration. II. THE BAYESIAN METHOD In engineering, one often needs to use whatever information is available in formulating a sound basis for making decisions. This may include observed data (field or experimental), information derived from theoretical models, and judgments based on experience. Moreover, the available information may need to be updated as new information or data are acquired [3]. The proper tool for combining and updating the available information is embodied in the Bayesian approach. Parameter estimation in the Bayesian approach is based on the updating formula: ���� = ��������� (1) where ���� is the prior Probability Density Function (PDF) representing the initial state of knowledge about the unknown parameters �, ���� is the likelihood function representing the Engineering, Technology & Applied Science Research Vol. 13, No. 1, 2023, 9888-9892 9889 www.etasr.com Fahmi & Al-Zaidee: Buckling and Vibration Estimation of Girder Steel Portal Frames using the … knowledge gained from a set of observations, and the constant � is a normalizing factor. The posterior PDF ���� represents the updated state of knowledge regarding the parameters �. A. Prior Distribution The prior distribution represents the distribution of possible parameter values from which the parameter has been drawn. The selection of the prior distribution is an important matter in Bayesian modeling. Since the choice of the prior distribution has a major effect on the resulting inference, this choice must be conducted with the utmost care [4]. B. Posterior Distribution The current information about the parameter is contained in the posterior probability distribution. It merges the information of the prior distribution and the likelihood. This results in a strong representation of the information than separate sources of information [5]. The prior and posterior densities are presented in Table I and the mean and variance of the posterior parameters in Table II. TABLE I. RANDOM VARIABLES AND THEIR PRIOR AND POSTERIOR DENSITIY DISTRIBUTIONS [3] Basic random variables Parameter Prior and posterior distribution Normal � � � = 1√2�� � � �− 12 � − �� ��� � Normal ���� = ��� �� � �⁄ � + ̅� ���� � �⁄ + ���� �� Lognormal � � �= 1√2� � � !− 12 "ln − % & �' % Normal �(�%�= 1√2�� � � !− 12 "% − �� & �' TABLE II. MEAN AND VARIANCE OF THE PARAMETERS AND THEIR POSTERIOR STATISTICS [3] Mean and variance of parameters Posterior statistics )��� = �� ���� = ��� �� � �⁄ � + ̅� ����� �⁄ + ���� �� *+,��� = ��� ���� = - ���� ����� �⁄ ����� �� + �� �⁄ )�%� = � ��� = .�� � �⁄ � + � � ln ̅ � �⁄ + � � *+,�%� = � � � �� = / � �� � �⁄ ��� + � �⁄ III. STATISTICAL CHARACTERISTICS OF THE INDEPENDENT VARIABLES This study considers the applied loads, cross-section dimensions, modulus of elasticity, girder span, plastic section modulus, and yield strength as random variables. The sample data of the independent variables were summarized and are presented in Table III. The prior statistical characteristics of these variables have been collected from the literature and can be seen in Table IV. The posterior statistical data of the input variables have been calculated from the updating process of the prior data by the Bayesian method. The modulus of elasticity of steel )0 has been considered as an example of this process. Updating mean, standard deviation, and coefficient of variance have been estimated based on the relations in (1)-(3). The posterior statistical characteristics of the independent variables are summarized in Table V. ��� = 1 23" 4√5&678! 9:�4:�6' 1 <" 4√5&678! <�4:�6' = =̅�>:�68�:"465 &�>:�68"465 & = ��?@��AB�8��??�� :�6�465 ��>:�68�465 � = / �AB�� ::�:: = P.�Q�?� = 0.016 (3) TABLE III. SAMPLE STATISTICAL DATA OF THE VARIABLES Variable STUVWXYZVU[ COV Distribution type Reference Dead load \] 1.03 0.08 Normal [6] Live load \̂ 1.00 0.1 Gumbel [7] Moment of inertia _= 0.96 0.03 Normal Local survey Modulus of elasticity )0 1.025 0.05 Normal [8] Modulus of elasticity )` 0.98 0.07 Lognormal [9] Member span � 1.00 0.07 Lognormal [10] Plastic section modulus a= 1.04 0.05 Lognormal [11] Yield strength of steel bc 1.10 0.07 Lognormal [11] TABLE IV. PRIOR STATISTICAL DATA OF THE VARIABLES Variable STUVWXYZVU[ COV Distribution type Reference Dead load \] 1.03 0.08 Normal [6] Live load \̂ 1.00 0.1 Gumbel [7] Moment of inertia _= 0.96 0.05 Normal [12] Modulus of elasticity )0 1.00 0.02 Normal [11] Modulus of elasticity )` 1.00 0.03 Lognormal [13] Member span � 1.00 0.004 Lognormal [14] Plastic section modulus a= 1.00 0.04 Lognormal [15] Yield strength of steel bc 1.10 0.06 Lognormal [12] TABLE V. POSTERIOR STATISTICAL DATA OF THE VARIABLES Variable STUVWXYZVU[ COV Distribution type Dead load \] 1.03 0.08 Normal Live load \̂ 1.00 0.1 Gumbel Moment of inertia _= 0.96 0.005 Normal Modulus of elasticity )0 1.01 0.016 Normal Modulus of elasticity )` 1.00 0.02 Lognormal Member span � 1.00 0.003 Lognormal Plastic section modulus a= 1.00 0.03 Lognormal Yield strength of steel bc 1.10 0.04 Lognormal IV. PARAMETRIC STUDY A steel single-story warehouse, as shown in Figure 1, with 45m total length and 15m width, consisting of 5 bays, has been considered in this parametric study. The superimposed dead Engineering, Technology & Applied Science Research Vol. 13, No. 1, 2023, 9888-9892 9890 www.etasr.com Fahmi & Al-Zaidee: Buckling and Vibration Estimation of Girder Steel Portal Frames using the … load has been determined based on the assumption of a flooring system of a concrete slab with a thickness of 100mm and a metal deck of 7.5mm, supported on IPE 300 floor beams. The first inner frame shown in Figure 2 has been considered the most critical, as it includes the exterior face of the first interior support where the shear force is about 15% greater than the average value. The frame consists of IPE 600 steel girder and columns. The applied live loads on the girder have been determined based on the ASCE 7 specifications. The self- weight of the girder and columns have been determined depending on the cross-section dimensions and material density. Fig. 1. Steel single-story warehouse building. Fig. 2. Summary of the dead and live load on the interior supporting frame. V. DERIVED AND SIMULATED STATISTICAL PROPERTIES FOR THE BUCKLING AND VIBRATION OF THE GIRDER Two stages have been considered in the first-order approximation method and the MC simulation method. The randomness of the dependent variables, in the first stage, has been deduced based on the prior knowledge of Table IV, while in the second stage, it has been deduced based on the posterior knowledge of Table V. A. First-Order Approximation Method It is possible to expand the original model into an infinite Taylor Series (TS) around the mean values: d�e� = d�� � + �e − � � fgf + A� �e − � �� f6gf 6 + ⋯ +Ai! �e − � �i f5gf 5 (4) where the function and derivatives are evaluated at � . It is common to include only linear terms, assuming that random input variables are independent. A function d �e� of N independent random variables can be approximated by linear terms of the TS, which are as follows [16]: )�k� l d��= � (5) and *+,�k� ≃ *+,�e − � � �fgf �� = *+,� � �fgf �� (6) 1) Buckling Analysis The buckling \̀ n is a function of the span �, the modulus of elasticity ), and the moment of inertia _. By refereeing to (4) and (5) the mean and variance of the buckling can be estimated by: )�\̀ n � = Ao6 p6qrsqtu̅vwxy� ŝ �6 (7) *+,�\̀ n � = *+,�_� �z{|yzt �� + *+,��� �z{|yz^ �� +*+,�)� �z{|yzr �� (8) Based on the statistical characteristics of the independent variables illustrated in Tables IV and V, the prior and posterior statistical characteristics for \̀ n have been estimated and are presented in Tables VI and VII. TABLE VI. PRIOR STATISTICAL CHARACTERISTICS OF THE BUCKLING OF THE GIRDER Random variable Nominal (kN) Mean (kN) Standard deviation (kN) COV \̀ n 44198 42072 2172 0.051 TABLE VII. POSTERIOR STATISTICAL CHARACTERISTICS OF THE BUCKLING OF THE GIRDER Random variable Nominal (kN) Mean (kN) Standard deviation (kN) COV \̀ n 44198 42493 750 0.017 2) Vibration Analysis The vibration �i is a function of the load P, span L, modulus of elasticity E, and moment of inertia I. The mean and variance of the vibration are calculated by: )��i� = 0.18 A~�� � g���y��y ⟹ 1.247� g�� � rst̅{s ŝ E (9) *+,��i� � *+,�_� �z�5zt � � � *+,��� �z�5z^ � � � *+,�)� �z�5zr � � � *+,�\� �z�5z{ � � (10) Based on Tables IV and V, the prior and posterior statistical characteristics for vibration are presented in Tables VIII and IX. Engineering, Technology & Applied Science Research Vol. 13, No. 1, 2023, 9888-9892 9891 www.etasr.com Fahmi & Al-Zaidee: Buckling and Vibration Estimation of Girder Steel Portal Frames using the … TABLE VIII. PRIOR STATISTICAL CHARACTERISTICS Random variable Nominal (kN) Mean (kN) Standard deviation (kN) COV �i 3.32 7.23 0.214 0.029 TABLE IX. POSTERIOR STATISTICAL CHARACTERISTICS Random variable Nominal (kN) Mean (kN) Standard deviation (kN) COV �i 3.32 7.267 0.103 0.014 B. Monte Carlo Simulation The MC method is considered one of the most powerful and accurate simulation tools and can be applied to many practical problems allowing the direct consideration of any type of probability distribution for the random variables [17]. The MC method has been adopted to achieve the simulation processes of the buckling, and vibration for the interior supporting frame. MATLAB code has been used to generate pseudo-random sampling with a size of 1000 for each input variable. 1) Buckling Analysis The prior statistical characteristics of the buckling are presented in Figure 3 and are summarized in Table X. \̀ n Fig. 3. Histogram of the prior buckling of a typical interior girder. TABLE X. PRIOR STATISTICAL CHARACTERISTICS OF THE BUCKLING OF THE GIRDER Random variable Nominal (kN) Mean (kN) Standard deviation (kN) COV \̀ n 41999 2279 0.054 Normal The posterior statistical characteristics of the buckling were estimated and are presented in Figure 4 and Table XI. TABLE XI. POSTERIOR STATISTICAL CHARACTERISTICS OF THE BUCKLING OF THE GIRDER Random variable Nominal (kN) Mean (kN) Standard deviation (kN) COV \̀ n 42481 756 0.017 Normal \̀ n Fig. 4. Histogram of the posterior buckling of a typical interior girder. 2) Vibration Analysis The prior statistical characteristics of the natural frequency are presented in Figure 5 and Table XII. �i Fig. 5. Histogram of the prior natural frequency of a typical interior girder. TABLE XII. PRIOR STATISTICAL CHARACTERISTICS OF THE NATURAL FREQUENCY OF THE GIRDER Random variable Nominal (kN) Mean (kN) Standard deviation (kN) COV �i 7.30 0.311 0.042 Normal The estimated posterior statistical characteristics of the natural frequency are presented in Figure 6 and Table XIII. TABLE XIII. POSTERIOR STATISTICAL CHARACTERISTICS OF THE NATURAL FREQUENCY OF THE GIRDER Random variable Nominal (kN) Mean (kN) Standard deviation (kN) COV �i 7.36 0.26 0.035 Normal Engineering, Technology & Applied Science Research Vol. 13, No. 1, 2023, 9888-9892 9892 www.etasr.com Fahmi & Al-Zaidee: Buckling and Vibration Estimation of Girder Steel Portal Frames using the … �i Fig. 6. Histogram of the posterior natural frequency of a typical interior girder. VI. CONCLUSION In this paper, buckling and vibration analysis was performed for the girder of the first inner frame for the prior and posterior statistical characteristics. Through the application of the Bayesian method, and from the results of the prior and posterior analysis, it was shown that the greater the knowledge gained about the independent parameters, the less randomness is in the dependent parameters, and thus the analysis and design of the system is enhanced. In the first order analysis, the covariance of the critical buckling load decreased from 0.051 to 0.017, and for the natural frequency from 0.029 to 0.014, while in Monte Carlo simulation, the covariance results were from 0.054 to 0.017 for the critical buckling load, and from 0.042 to 0.035 for the natural frequency. 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