Stability Analysis of a Geometrically Imperfect Structure using a Random Field Model

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1 Stabilit Analsis of a Geometricall Imperfect Structure using a Random Field Model JAN VALEŠ, ZDENĚK KALA Department of Structural Mechanics Brno Universit of Technolog, Facult of Civil Engineering Veveří 331/95, Brno CZECH REPUBLIC vales.j@fce.vutbr.c Abstract: - There eist two main categories of structural uncertainties which include spatial correlation, and require the application of random fields. These categories are material and geometrical characteristics. In the presented studies, the random field is applied to the modelling of initial curvature of a slender member ais. Load-carring capacit of the compress member is calculated b using geometricall nonlinear solution in the programme ANSYS. The solution was carried out b using the beam element BEAM188. The specimen application includes the static random response of an imperfect sstem generated b geometrical imperfections of member aes with open and hollow cross-sections. The Latin Hpercube Sampling Method was applied. Ke-Words: Load carring capacit, Initial curvature, Incision, Random field, Limit state, Stress, Warping 1 Introduction One half wave of the function sinus is a frequentl applied idealiation of real curvature of the ais of a slender hot-rolled steel member, although the real geometr can be substantiall more complicated. Model approaches based on one half-wave make possible to calculate the solution of the state of stress in an analtical form b a close-form formula [1,]. On these assumptions, it is not difficult to calculate the load carring capacit, if the amplitude of initial curvature e 0 is known. The amplitude of initial curvature is a random quantit for the identification of which the source materials can be found in literature [, 3]. A more detailed method assumes a spatial distribution of geometrical or material properties and models of this random distribution b a continuous field called a random field [4, 5]. In combination with the finite element method, this approach is generall called the stochastic finite element approach [6]. Modelling of the initial curvature with appling the random field is more complicated, and requires more input parameters to be identified [7]. However, it is not the rule that the initial ais curvature must have the shape of sinus function half-wave or of random field. Both cases are the idealiation of real aial generall spatial curvature. An eplicit solution cannot be obtained for the general spatial curvature. The present paper deals with modelling the member aial curvature based on random fields. The random spatial curvature is considered as the initial geometrical imperfection which influences the load carring capacit of the member. Two smmetrical thin-walled cross sections were chosen for computer modelling, namel the open and the hollow ones. The stress state of an imperfect member under compression is given b pressure, bending, and b torsion. The torsion can significantl contribute to the increase of the stress state, and to the decrease of load carring capacit for slender members with open cross section in particular [8]. The warping torsion the influence of which on stress state, and load carring capacit are studied, is connected with the stress of members having open cross sections. The load carring capacit of compress members was calculated b geometricall nonlinear solution in the ANSYS software. Computational Model A two-hinge member was considered, see Fig. 1. The length of the member is L =.798 m and nondimensional slenderness λ = 1.0. Non-dimensional slenderness λ is the parameter applied to design and dimensioning of member, defined in the standard EUROCODE 3. The parameter λ is ISSN: Volume 1, 016

2 sometimes considered to be a quantit more transparent than the member length, because it (the parameter) is the function also of the radius of gration about the relevant ais, determined using the properties of the gross cross section. To the load carring capacit analsis of the compress member, the member element BEAM188 with seven degrees of freedom was applied which is a part of the offer, and is accessible in the ANSYS software. Seven degrees of freedom include three degrees of freedom corresponding to translations along aes,,, other three ones, to rotations around these aes, and the 7 th degree of freedom corresponds to warp. The member scheme is in Fig. 1. In the node a, translations in directions of all three aes, and the rotation about ais are prevented. In the node b, the translations in the direction of aes and and rotation around ais are prevented. Further on, it is assumed that both end cross sections in nodes a and b cannot warp; b this, the model approaches a real laborator eperiment. The model is subsequentl loaded b translation in the node b in the direction of ais. In node a, the value of the reaction in direction is followed to determine the load carring capacit. The hollow cross section is smmetrical along both aes, and thus the position of the centre of gravit C g corresponds to the position of shear centre S s. In open cross section, the position of shear centre was shifted to the right, according to Fig. b). Cross-section characteristics of both cross sections are given in Table 1. Table 1: Cross section characteristics. Characteristic Hollow Open Area A Second moment of area I Second moment of area I Second warping moment I ω These cross sections are then applied to each realiation of spatial curvature of member ais (see Chapter.). In view of the fact that the initial curvature is here independent of the applied profile, the open cross section is; rotated into four positions. The load carring capacit of member of the given realiation is then calculated for five cross sections according to Fig. 3. L a b I) II) III) Fig.1: Model of steel member..1 Cross Section In the given problem, two smmetrical square thinwalled cross sections were used, the first being hollow, and the second, open. In the middle of one side of open cross section, there is an incision. Both cross sections are presented in Fig.. The member is modelled so that all the points of the centre of gravit of cross section lie on the member ais C g =S s [mm] 6.0 C g Fig.: Cross sections: a) hollow, b) open S s IV) V) Fig.3: Hollow and open cross sections.. Random Input Variables, Random Field The Gauss probabilit densit distribution with mean value 97.3 MPa, and with standard deviation 16.8 MPa was considered for ield point f [9]. The initial curvature of member ais was modelled using eleven nodes through which the spline was interlaid, see Fig. 4. Each of these nodes had the Gauss probabilit densit distribution with ero mean value, and with standard deviation sin(π i /L) m, where i is the position on the member ais. The value was calculated, based on the assumption that 95 % realiations of initial spatial deformation la within the tolerance limits ISSN: Volume 1, 016

3 ±0,15 % L. The values of coordinates in each of the two planes are mutuall correlated b means of the correlation matri. The correlation matri defines the random field with correlation length L cor = 1.44 m. The correlation is considered among the values of coordinates of nodes ling on one plane. It means that the curvature on one plane is independent of the curvature on the other one. Random realiations of ield strengths and initial curvature were simulated b appling the method Latin Hpercube Sampling [10, 11] using Freet software, see L=.798 m 9 10 Fig.4: Definition of spatial curvature of the ais. The other parameters of the model were considered as deterministic ones, and were taken b their nominal values. The Young s modulus of steel E=10GPa and geometrical characteristics of the cross section, e.g., are concerned. B loading using the method step-b-step, the strength was searched for which the maimum value of the von Mises stress of the member would be equal to ield strength. 60 random realiations of aial curvatures of members and 60 random realiations of ield strength were simulated. Each realiation of aial curvature of the member forms a pair with one realiation of ield point. An eample of one realiation of initial curvature of member ais is illustrated in Fig. 5 and Fig Stress State and Limit State Building structures are usuall designed so that maimum epected stresses were within the limits of linear elasticit, it means that deformation caused b internal stress is directl proportional to them. At loading, the real imperfect member is in the state of spatial stress. For the assessment, it is necessar to know when the stress state is approaching the limit state of stress in a material. Yield strength f is usuall considered to be the limit state of stress for structural steel. Combined stresses cannot be described b a single vector. For the limit condition, there was used the von Mises (Huber, Henck) condition of plasticit in the form. f = σ = 0 (1) 0 where σ is equivalent stress [Pa], which is ( σ ) ( ) ( ) ( ) + σ + σ τ + τ + τ + 1 σ = () σ 0 corresponds to f, elements σ, σ, σ are called orthogonal normal stresses (relative to the chosen coordinate sstem), and τ, τ, τ, orthogonal shear stresses. σ is longitudinal stress in direction of ais, σ is longitudinal stress in direction of ais, σ is longitudinal stress in direction of ais. τ is shear stress on the plane in direction of, τ is shear stress on the plane in direction, τ is shear stress on the plane in direction. The analsis of stress can be considerabl simplified for members that are subjected to moderate compressing, bending and twisting. When the solution is carried out b member model, it is assumed that σ =σ =τ =0. The formula () is so reduced to the form ( τ τ ) σ = σ + + (3) 3 Fig.5: Ais curvature of 1 random run - plane. So, the limit state occurs, when ield point f is reached in the maimum stressed point of member. σ = (4) f Fig.6: Ais curvature of 1 random run - plane. 4 Load Carring Capacit The load carring capacit values of all sit random realiations of the member are presented in Table. B Chi-quadrate test of good agreement on the significance level 5 %, the hpothesis on ISSN: Volume 1, 016

4 normalit of distribution for no solved cross-section tpes is rejected. Based on the standard EN1990, the design load carring capacit is calculated as 0.1 percentile. It corresponds to design reliabilit inde β d = percentile was calculated on assumption that the load carring capacit had the Gauss probabilit densit function. Table 1: Cross section characteristics. Cross section I II III IV V Mean value [kn] Variation coefficient percentile [kn] The design load carring capacit of the member with hollow cross section (cross section I) is according to EUROCODE kn. The EUROCODE 3 does not state the design load carring capacit of open cross sections (cross sections II, III, IV). 5 Conclusion The statistical modeling of a structure is usuall carried out for the purpose of finding the probabilistic response, or for reliabilit assessment. In the paper, there was presented the application of a random field for modelling initial curvature of the member ais. Statistical characteristics of load carring capacit evaluated for two cross-section tpes are the result of the stud. The Table 1 contains load carring capacities of members with several variants of open cross sections, completed b the variant with hollow cross section. Mean values and standard deviations of members with open cross sections are similar. The design load carring capacit of these cross sections is not stated b the standard EUROCODE 3 epressl, but in compliance with standard proceedings, the curve of buckling strength b can be used, and the value of design load carring capacit 35 kn can be obtained. The value 35 kn 0.1 is higher than all 0.1 percentiles of Gauss probabilit densit function, b which the values of load carring capacit of members with open cross sections were approimated. The design load carring capacit of members with hollow cross section according to EUROCODE 3 is 6.9 kn. This value is higher than 0.1% quantile of normal distribution, as well. The comparison of design value according to EUROCODE 3 with the design value obtained as 0.1 percentile represents the basic principle for verification of design reliabilit. The statistical analsis points out that it is necessar to introduce more strict standard design values. However, it must be verified b eperimental research and a further probabilistic analsis according to EN1990. Random fields are an appropriate instrument for research into this phenomenon. Acknowledgement The article was elaborated within the framework of project GAČR S. References: [1] S. Timoshenko, J. Gere, Theor of Elastic Stabilit, McGraw-Hill, New York, [] Z. Kala, Sensitivit Assessment of Steel Members under Compression, Engineering Structures, Vol.31, 009, pp [3] Z. Kala, Elastic Lateral-torsional Buckling of Simpl Supported Hot-rolled Steel I-beams with Random Imperfections, Procedia Engineering, Vol.57, 013, pp [4] C. Bucher, Applications of Random Field Models in Stochastic Structural Mechanics, Solid Mechanics and its Applications, Vol.140, 006, pp [5] J. Valeš, Modeling and Simulation of Random Fields in Stabilit Problems of Compressed Members, In Proc. of the 1st International Conference on Soft Computing MENDEL015, Brno (Cech Republic), 015, pp [6] H.G. Matthies, Ch.E. Brenner, Ch.G. Bucher, C.G. Soares, Uncertainties in Probabilistic Numerical Analsis of Structures and Solids - Stochastic Finite Elements, Structural Safet, Vol.19, No. 3, 1997, pp [7] J. Valeš, The Influence of Random Initial Ais Curvature of Compression Steel Slender Member on its Load Carring Capacit, In Proc. of the 18th International Conference on Soft Computing MENDEL01, Brno (Cech Republic), 01, pp [8] J. Freund, A. Karako, Warping Displacement of Timoshenko Beam Model, International Journal of Solids and Structures, Vol.9-93, 016, pp [9] J. Melcher, Z. Kala, M. Holický, M. Fajkus, L. Rolívka, Design Characteristics of Structural Steels Based on Statistical Analsis of ISSN: Volume 1, 016

5 Metallurgical Products, Journal of Constructional Steel Research, Vol.60, No.3-5, 004, pp [10] M. D. McKe, R. J. Beckman, W. J. Conover, A Comparison of the Three Methods of Selecting Values of Input Variables in the Analsis of Output from a Computer Code, Technometrics, Vol. 1, 1979, pp [11] R. C. Iman, W. J. Conover, Small Sample Sensitivit Analsis Techniques for Computer Models with an Application to Risk Assessment, Communications in Statistics Theor and Methods, Vol.9, No.17, 1980, pp ISSN: Volume 1, 016