Fuzzy Analysis of Serviceability Limit State of Slender Beams

Transcription

1 Fuzz Analsis of Serviceabilit Limit State of Slender Beams JAN VALEŠ Department of Structural Mechanics Brno Universit of Technolog, Facult of Civil Engineering Veveří 331/95, Brno CZECH REPUBLIC Abstract: - The maimum permissible deflection values in design standards are determined subjectivel and can thus be considered as fuzz numbers. Using the finite element method, it has been found that the lateral deflection of a slender beam under major ais bending can be relativel high. Nonlinear load-deflection analsis was applied for the compilation of the presented numerical studies. The serviceabilit limit state of beams with initial imperfections was studied. The limit moment, which ields the deflection that is equal to the maimum permissible value, was defined. Fuzz analsis of the limit moment was evaluated using the general etension principle. Slendernesses of members for which the attainment of the maimum permissible value of deflection is the deciding limit state for design were determined. Ke-Words: - Fuzz logic, resistance, imperfections, stabilit, serviceabilit, limit state, beam, bending 1 Introduction Building structures should be designed for strength and serviceabilit during their service life, namel without an significant functionalit loss or necessit of ecessive, and/or unforeseen maintenance. The basis of the design of steel structures is the safeguarding of their safet and reliabilit. The steel structures usuall are slender, and require paing increased attention to stabilit problems. The analsis of ultimate limit states of slender steel members is based on calculation models in closed form [1, ] or on finite element method [3, 4]. The influence of material and geometrical characteristics on the model output can be assessed b means of the advanced sensitivit analsis method, see, e.g., [1, 5]. Advanced approaches analse the reliabilit b appling numerical simulation methods of the Monte Carlo tpe [6]. The safeguarding of serviceabilit represents the second equall important part of the design. The evaluation of the serviceabilit limit state is important in particular in recent time when advanced nonlinear calculation methods enable to analse the deformations of light steel structures [7]. Serviceabilit limit state design of structures includes factors such as deflection, cracking and ecessive vibration, durabilit, overall stabilit, and fire resistance. Basic design criteria of applicabilit of staticall stressed structures are connected with larger deflections, above all the occurrence of which in current service should not eceed the maimum permissible value of deflection (limits) specified b design standards [8, 9]. The limit values of deflection are determined in a ver subjective manner because the are connected with appearance of structure, contentment of users and serviceabilit but also with damage of surfaces or non-supporting members. The limit values of deflections are thus of subjective character, and the can be perceived as fuzz numbers rather than as random numbers or fields. Tpical eemplars are buckling and lateraltorsional buckling of slender beams with initial imperfections. The design of slender beam according to ultimate limit states [8, 9] is based on buckling curves. In design standards, the assessment according to serviceabilit limit state has not the background as strong as ultimate limit state. The maimum permissible values of deflection associated with buckling are not specified at all b design standards for stabilit problems of buckling and lateral torsional buckling. Eperimental research [10] have shown that the lateral deflection of slender imperfect beams under major ais bending ma, in some cases, reach relativel high values. This problem will be studied, in the present article, using the software Anss [11] and the nonlinear load-deflection analsis of the FEM [1]. The paper is divided into five sections including the introduction and conclusion. In the second E-ISSN: Volume 11, 016

2 section the model, loading, boundar conditions and initial geometric imperfections are described. The third section discusses the calculation of limit deflection. In the fourth section the fuzz analsis is presented. Problem Formulation The computational model is presented b a simpl supported beam of a hot-rolled profile IPE0 and steel grade S35. The non-dimensional slenderness in lateral-torsional buckling LT [8] was chosen within the interval LT 0., 4. Since the beam length L is a function of this value, it was changed depending on the non-dimensional slenderness. The beam is loaded on both ends b equal bending moments M of the same magnitude and of opposite direction. This represents the case of pure bending. The load is applied b means of compression (tension) on the web p 1 and flange p to determinate the sizes of loading effects p 1 and p. Equations (1) to (4) describe how the sizes of loading effects p 1 and p are calculated. Normal stress has a linear distribution along the cross section due to bending; see Fig.1. The highest absolute value is reached at the outer edges., ma t F, A,ma 1 bt h 1 M h tbt I (3) Substituting Eq. (3) into Eq. (), the force F 1 and pressures p 1 and p [Nm -1 ] can then be subsequentl determined as F 1 3M 1 h t bt 1 h t I 6M 1 h t bt p1 1, (4) h t I M h tt p. I Since the beam is hinged at both ends, the boundar conditions are considered according to [, 13] as follows,, (5) 0 0 L 0 0 L 0 0 L 0 0 L, 0 0 L., Fig.1: Stress distribution and application of compression and tension at the end sections of the beam M z,,ma I Mh (1) I The resultant moment M is given as the sum of moments M 1 and M from the force couples F 1 and F. M M1 M F1 h0 Fh 0 F1 h t 3 3 F h t () The force F, the point of action of which is on the central line of the flange, is given b multipling the respective stress and flange area A. The beam was modelled in finite element software Anss [11]. The model was created using shell element SHELL181 which is a 4-node structural element suitable for analzing thin to moderatel-thick shell structures. The pressure loads were applied using element SURF156. The geometric nonlinear solution was applied using the incremental Newton-Raphson iterative method. The linear elastic stress-strain relationship was used for steel. Residual stress was not considered..1 Initial Imperfections In calculation models, the influence of imperfections on the behavior of structures is considered in various was, see, e.g., [14-15]. In general, imperfections can be considered as deterministic or random quantities [16]. In this work, the initial imperfections are considered as deterministic quantities. E-ISSN: Volume 11, 016

3 The initial out-of-plain imperfection of the beam ais (in the plane, see Fig.), is described b function e sin 0, (6) L where e 0 is the value of amplitude of initial curvature of the beam ais and given as e 0 = L/1000 [1]. In the case of an ideal straight beam, deformation in the plane does not occur as long as the bending moment M does not reach the critical value M cr. Then the beam starts to deflect even in this plane and to rotate. When the ais of the beam has an initial curvature, it starts to deflect in the plane and rotate immediatel after the application of loading M. An eample of a finite element model created in Anss software is depicted in Fig.3. 3 Beam deflections During loading the maimal deflection occurs in the middle of the span length = L/. The deflection value (or the vector deflection) u is calculated as the maimum of lengths AA, BB, CC, DD, see Fig.4. It is a function of the length L and the dimensions of cross section b, h and it is sought during calculation. Fig.4: Deflections of the cross section The deformation of an cross section in 0, L can be described b the components of displacement on each aes u, u a u z, and b rotation of cross sections φ, φ a φ z around these aes. For the cross section in the middle of span length the rotations φ a φ z are negligible 0, 0, 0. (7) 0, L z L Fig.: Initial imperfection But the deflection in the -direction u is not negligible because one end of the beam is free to deflect in the direction of ais. Therefore, the cross sections are shifted on this ais as well. For the vector deflection u it can be written u u u e, 0 sin u L. (8) u z u z, Deflections u, and u, z are calculated as Fig.3: Finite element model from Anss u sgn( z) b h sin, b cos sgn( )sgn( z)arctan h b b h h for :, and z : z,, (9) E-ISSN: Volume 11, 016

4 u sgn( z) b h sin z, b sin sgn( )sgn( z)arctan h b b h h for :, and z : z,. (10) The non-dimensional slenderness LT of the beams was considered within the interval LT 0., 4. Due to the relation between slenderness and length, the lengths are within the interval L 0.5, 4. Together, 416 beams with the length increase of 0.5 m were created. For each value L, there is a corresponding value Eurocode 3. LT, see 4 Fuzz Analsis The application of fuzz sets in the construction industr/civil engineering is undergoing significant development, particularl in areas involving multiple criteria decision making [17]. The application of fuzz analsis to the verification of the reliabilit of the design of structures is most frequentl focused on the serviceabilit [18] or the safet of structures [19]. As the moment M increases from zero to M ma, the deflection u increases from the initial value of e 0 = L/1000 to u ma. The limit moment M lim will occur when maimal deflection u ma is equal to the maimum permissible value of deflection u lim. As u lim is a fuzz number, the limit moment M lim is also a fuzz number. u lim has a membership function u = (u, L/450, L/50, L/173), see Fig.5. u lim analsis of membership function of fuzz number M lim [0, 1]. Mlim M sup ( u) (11) ulim f um The fuzz analsis of two selected slendernesses is presented in Fig.6 and Fig.7. The support of M lim is bounded b deflection values L/450 and L/173. These are considered to be the maimum vector deflections. The value of the centre of gravit (COG) of the area below the membership function is the defuzzified value. Fig.6: Fuzz analsis M lim of slenderness 0.4 Fig.7: Fuzz analsis M lim of slenderness 1. The fuzz analsis M lim for other nondimensional slendernesses is presented in Fig. 8. Fig.5: Membership function of u lim The relationship (11) is based on the general etension principle and it was applied for the fuzz E-ISSN: Volume 11, 016

5 Fig.8: Fuzz analsis M lim vs slendernesses The elastic resistance R f, see, e.g., [], is plotted in Fig.8 as a black dash line. Its values correspond to the moment at which the ield strength is reached in the most stressed point. 5 Conclusion The article was focused on a fuzz analsis of the maimum moment M lim at which the maimum permissible deflection u lim has been reached. A nonlinear dependence between the deformation and the loading moment was discovered. u lim is a subjectivel chosen value. Thus it can be considered as a fuzz number. Although u lim was considered as a linear smmetrical fuzz number Λ, M lim is not linear and smmetric anmore, see Fig.7 and Fig.8. The defuzzified values (COG) are greater than the kernel for LT (L.7m). The biggest difference between COG and the kernel is reached for the lowest values of slenderness. From the slenderness LT the COG values are lower than the kernel, and the difference becomes more significant with increasing value of LT. It has been found out from the numerical studies that the lateral deflections ( ais) are substantiall smaller than those in the vertical plane (z ais) for lower slenderness LT With increasing value of slenderness, the lateral deformations increase as well. Lateral and vertical deformations are approimatel the same for LT. 0. The ield strength is a random variable that is evaluated eperimentall []. The elastic resistance R f is the phsical limit of the moment M above which there is an objective reduction in safet due to initialization of permanent deformations. The curve of R f intersects the curve of the COG at the point LT (L = 3.3 m). The difference between COG and R f decreases with increasing slenderness, and the difference is negligible for slenderness LT It is not possible to make a general conclusion of comparison of COG and R f because R f was calculated from the characteristic value of ield strength f = 35 MPa, and the initial curvature of the beam e 0 was considered as a non-random value L/1000. However, R f can be more accuratel considered as a fuzz random variable because the ield strength f and initial curvature of the beam e 0 are real random variables. The article provides topics intended for elaboration in further numerical studies of reliabilit analsis focused on the design of slender steel beam structures using stochastic methods and fuzz logic. It also contributes to the discussions on the appropriateness of using fuzz or stochastic approaches depending on the tpe of task. Studing serviceabilit limit state using stochastic methods cannot be considered sufficient, because the limit deformation in design standards are determined subjectivel b humans. Fuzz analsis is suitable in these cases. Acknowledgment: The paper has been worked out under the project No. LO1408 AdMAs UP. References: [1] Z. Kala, Sensitivit and Reliabilit Analses of Lateral-torsional Buckling Resistance of Steel Beams, Archives of Civil and Mechanical Engineering, Vol.15, No.4, 015, pp E-ISSN: Volume 11, 016

6 [] Z. Kala, Elastic lateral-torsional buckling of simpl supported hot-rolled steel I-beams with random imperfections, Procedia Engineering, Vol.57, 013, pp [3] Z. Kala, J. Kala, Sensitivit Analsis of Stabilit Problems of Steel Structures using Shell Finite Elements and Nonlinear Computation Methods, WSEAS Transactions on Applied and Theoretical Mechanics, Vol.4, No.3, 009, pp [4] Z. Kala, J. Kala, A. Omishore, Probabilistic Buckling Analsis of Thin-walled Steel Columns using Shell Finite Elements, International Journal of Mechanics, Vol.10, 016, pp [5] Z. Kala, Fuzz Sensitivit analsis for Reliabilit Assessment of Building Structures, AIP Conference Proceedings, Vol.1738, Article number , 016. [6] Z. Kala, Reliabilit Analsis of the Lateral Torsional Buckling Resistance and the Ultimate Limit State of Steel Beams with Random Imperfections, Journal of Civil Engineering and Management, Vol.1, No.7, 015, pp [7] Z. Kala, G. Sandovič, Fuzz Stochastic Analsis of Serviceabilit and Ultimate Limit States of Two-span Pedestrian Steel Bridge, AIP Conference Proceedings, Vol.1479, No.1, 01, pp [8] EN : Eurocode 3: Design of Steel Structures, Part 1 1, General Rules and Rules for Buildings, Brussels, CEN, 004. [9] EN 1993-: Eurocode 3 Design of Steel Structures - Part : Steel Bridges, Brussels, CEN, 006. [10] J. Melcher, M. Karmazínová, Lateral buckling of steel sigma beams with web holes. Proceedings SDSS Rio 010 International Colloquium Stabilit and Ductilit of Steel Structures held in Rio de Janeiro, Federal Universit of Rio de Janeiro, Vol., 010. pp ISBN: [11] Anss, Inc.: Release 1.1 Documentation for Anss. SAS IP, Inc 010. [1] Z. Kala, J. Valeš, Modelling and statistical approaches to lateral-torsional buckling. Advanced Materials Research 969, pp (014). DOI / [13] Z. Kala, J. Kala, Lateral-torsional buckling analsis of I-beams using shell finite elements and nonlinear computation methods. TE. Simos (ed.) Proceedings of International Conference of Numerical Analsis and Applied Mathematics ICNAAM 01, Vol in ICNAAM, pp American Institute of Phsics, Rhodes (01). DOI / [14] Z. Kala, Sensitivit analsis of carring capacit of steel plane frames to imperfections. TE. Simos (ed.) Proceedings of International Conference of Numerical Analsis and Applied Mathematics ICNAAM 008, Vol in ICNAAM, pp , American Institute of Phsics, Rhodes (008). [15] J. Valeš, Analsis of load-carring capacit of thin-walled closed beams with different slenderness, Advanced Materials Research 969, pp (014) / [16] Z. Bažant, L. Cedolin, Stabilit of Structures, Oford Universit Press, New York-Oford, 003. [17] A. Mardani, A. Jusoh, E. K. Zavadskas, Fuzz multiple criteria decision-making techniques and applications Two decades review from 1994 to 014. Epert Sstems with Applications 4(8), pp (015). DOI /j.eswa [18] Z. Kala, J. Valeš, Fuzz Analsis of serviceabilit limit state of slender steel beam under bending. TE. Simos (ed.) Proceedings of International Conference of Numerical Analsis and Applied Mathematics ICNAAM 014, American Institute of Phsics, Rhodes (014). [19] Z. Kala, Stabilit problems of steel structures in the presence of stochastic and fuzz uncertaint. Thin-Walled Structures 45(10-11), pp (007). DOI /j.tws [0] L. A. Zadeh, Fuzz sets. Information and Control 8(3), pp (1965). [1] D. Dubois, Fuzz Sets and Sstems Theor and Applications, Academic Press, pp. 394 (1981), ISBN [] Z. Kala, J. Melcher, L. Puklický, Material and geometrical characteristics of structural steels based on statistical analsis of metallurgical products. Journal of Civil Engineering and Management 15(3), pp (009). DOI / E-ISSN: Volume 11, 016