Phân tích động mờ khung thép phẳng được giằng sử dụng thuật toán tiến hóa vi phân

45 S¬ 27 - 2017 Phân tích động mờ khung thép phẳng được giằng sử dụng thuật toán tiến hóa vi phân Fuzzy dynamic analysis of 2d-braced steel frame using differential evolution optimization Viet T. Tran, Anh Q. Vu, Huynh X. Le Tóm tắt Bài báo nghiên cứu áp dụng thủ tục phần tử hữu hạn mờ phân tích động kết cấu khung thép phẳng với các đại lượng đầu vào mờ. Các hệ số liên kết giữa dầm – cột, cột – móng, tải trọng, khối lượng riêng và hệ số cản được mô tả dưới dạng các số mờ tam giác. Phương pháp tích phân số β – Newmark được áp dụng xác định chuyển vị trong hệ phương trình cân bằng động tuyến tính. Phương pháp tối ưu mức – α sử dụng thuật toán tiến hóa vi phân được tích hợp với mô hình phần tử hữu hạn để phân tích động kết cấu mờ. Hiệu quả của phương pháp đề xuất được minh họa thông qua ví dụ liên quan đến khung thép phẳng hai mươi lăm tầng, ba nhịp được giằng tập trung. Từ khóa: Khung thép giằng, liên kết mờ, động lực kết cấu mờ, thuật toán tiến hóa vi phân Abstract This paper studies the application of the fuzzy finite element procedure for dynamic analysis of the planar semi-rigid steel frame structures with fuzzy input parameters. The fixity factors of beam – column and column – base connections, loads, mass per unit volume and damping ratio are modeled as triangular fuzzy numbers. The Newmark-β numerical integration method is applied to determine the displacement of the linear dynamic equilibrium equation system. The α – level optimization using the Differential Evolution (DE) integrated finite element modeling to analyse dynamic of fuzzy structures. The efficiency of proposed methodology is demonstrated through the example problem relating to the twenty-five – story, three – bay concentrically braced frame. Keywords: braced steel frame, fuzzy connection, fuzzy structural dynamic, differential evolution algorithm MS. Viet T. Tran Faculty of civil engineering, Duy Tan University Email: Ass. Prof. Anh Q. Vu Faculty of civil engineering Hanoi Architectural University Email: Prof. Huynh X. Le Faculty of civil engineering National University of Civil Engineering Email: 1. Introduction In the dynamic analysis of steel frame structures with semi-rigid connections, rigidity of the connection (or fixity factor of the connection), loads, mass per unit volume, damping ratio have a significant influence on the time – history response of steel frame structure [4]. In practice, however, many parameters like worker skill, quality of welds, properties of material and type of the connecting elements affect the behavior of a connection, and this fixity factor is difficult to determine exactly. Therefore, in a practical analysis of structures, a systematic approach need to include the uncertainty in the connection behavior, and the fixity factor of a connection modeled as the fuzzy number is reasonable [5]. In addition, the uncertainty of input parameters is also described in form of fuzzy numbers, such as external forces, mass per unit volume and damping ratio. In this paper, the fuzzy displacement - time dependency of a planar steel frame structure is determined in which the fixity factor, loads, mass per unit volume, and damping ratio are described in the form of any triangular fuzzy numbers. A procedure is based on finite element model by combining the α – level optimization with the Differential Evolution algorithm (DEa). The Newmark-β average acceleration numerical integration method is applied to determine the displacements from the linear dynamic equilibrium equation system of the finite element model. 2. Finite element with linear semi-rigid connection The linear dynamic equilibrium equation system is given as following [ ]{ } [ ]{ } [ ]{ } ( ){ }M u C u K u P t+ + =  (1) where { }u , { }u , and { }u are the vectors of acceleration, velocity, and displacement respectively; [M], [C], and [K] are the mass, damping, and stiffness matrices respectively; {P(t)} is the external load vector. The viscous damping matrix [C] can be defined as [ ] [ ] [ ]M KC M Kα β= + (2) where αM and βK are the proportional damping factors which defined as 1 2 1 2 1 2 2 2;M K ω ω α ξ β ξ ω ω ω ω = = + + (3) where ξ is the damping ratio; ω1 and ω2 are the natural radian frequencies of the first and second modes of the considered frame, respectively. In this study, the frame element with linear semi – rigid connection is shown in Fig. 1, with E - the elastic modulus, A – the section area, I – the inertia moment, m - the mass per unit volume, k1 and k2 – rotation resistance stiffness at connections. The element stiffness matrix - [Kel] and the mass matrix - [Mel] of the frame are given by [4], with si = Lki / (3EI + Lki) denote the fixity factor of semi – rigid connection at the boundaries (i = 1,2). In Eq. (1), when fixity factors of connections, external loads, mass per unit volume and damping ratio are given by fuzzy numbers, the displacements of joints are also fuzzy numbers. In steel structures, the common fuzzy connections can be defined by linguistic terms as shown in Fig. 2. Eleven linguistic terms are assigned numbers from 0 to 10 ( 0,1,...10is = ) [5]. In the classical finite element method (FEM), in Eq. (1), the displacement – time dependency of the joints is determined by solving the linear dynamic equilibrium equation system. The Newmark-β method has been chosen for 46 T„P CHŠ KHOA H“C KI¦N TR”C - XŸY D¼NG KHOA H“C & C«NG NGHª Figure 2. Membership functions of fuzzy fixity factors Figure 4. Fuzzy displacement-time response at joint 26 in x direction Figure 5. The membership functions of fuzzy displacement at joint 26 Figure 1. Frame element with linear semi-rigid connection the numerical integration of this equation system because of its simplicity [1]. The fuzzy displacement is determined by the fuzzy finite element method (FFEM) using the α-cut strategy with the optimization approaches. FFEM is an extension of FEM in the case that the input quantities in the FEM are modeled as fuzzy numbers. In this study, an optimization approach is presented in the next sections: the differential evolution algorithm (DEa). 3. Proceduce for fuzzy structural dynamic analysis 3.1. Linear elastic dynamic analysis algorithm The Newmark-β method is based on the solution of an incremental form of the equations of motion. For the equations of motion (1), the incremental equilibrium equation is: [ ]{ } [ ]{ } [ ]{ } { }M u C u K u P∆ + ∆ + ∆ = ∆  (4) where { }u , { }u , and { }u are the vectors of incremental acceleration, velocity, and displacement respectively; { }P∆ is the external load increment vector. The displacement of the joint at each time step is determined by this algorithm of linear elastic dynamic analysis. 3.2. α – level optimization using Differential Evolution algorithm (DEa) For fuzzy structural analysis, the α-level optimization is known as a general approach in which all the fuzzy inputs are discretized by the intervals that are equal α-levels. The output intervals are then searched by the optimization algorithms. The optimization process is implemented directly by the finite element model and the goal function is evaluated many times in order to reach to an acceptable value. In this study, the output intervals are the displacement intervals at each time step, and the solution procedure is proposed by combining the Differential Evolution algorithm (DEa) with the α-level optimization. The DEa has shown better than the genetic algorithm (GA) and is simple and easy to use. Basic procedure of DEa is described as [6]. Figure 3. Concentrically braced steel frame with fuzzy input parameters 47 S¬ 27 - 2017 4. Numerical illustration A twenty-five – story, three – bay concentrically braced frame subjected to fuzzy impulse force as shown in Fig. 3 is considered. The fuzzy input parameters are: 1m = (7.85, 0.785, 0.785), 2m = (50, 5, 5), 1s =9, 2s = 8, 3s = 7, 4s = 6, 5s =5, 6s =1. The fuzzy damping ratio is ξ = (0.05, 0.005, 0.005). The fuzzy impulse force is: ( )P t P=  (0 ≤ t ≤ 3 s), and ( ) 0P t = (t > 3 s), with P  = (40, 4, 4). These fuzzy terms are considered to be triangular fuzzy numbers with 20% absolute spread. A time step Δt of 0.05 second is chosen in the dynamic analysis. The output intervals of displacement are calculated by using DE programmed by MATLAB. The section properties used for analysis of the frame are shown in Table I. Fig. 4 shows the fuzzy displacement-time response and the membership functions of fuzzy displacement at different times in 3D – axis. Fig. 5 shows the membership functions of fuzzy displacement and the deterministic displacement (at central value) from the SAP2000 software, with t = 1.10, 2.05, 2.90, 4.05, 5.10, and 6.00 seconds. 5. Conclusion A fuzzy finite element analysis based on the Differential Evolution (DE) in combination with the α – level optimization, in which the Newmark-β average acceleration method is applied to determine the deterministic displacement. The fuzzy input parameters such as fixity factors of connections, external forces, mass per unit volume, and damping ratio have a significant influence on the time dependency of the fuzzy displacement. With the example is considered, fuzzy displacments show more different shapes of membership functions at different times. Moreover, these fuzzy displacements have absolute spreads from 40% to 150%. In adition, the determinant results are also compared with ones of the SAP2000 software and give a good agreement./. Table 1: Section properties used for analysis of the portal steel frame Member Section Cross – section area, A (m2) Moment of inertia, I (m4) Column (1st to 4th story) W30x391 7.35E-02 8.616E-03 Column (5th to 8th story) W30x326 6.17E-02 6.993E-03 Column (9th to 14th story) W27x307 5.82E-02 5.453E-03 Column (15th to 20th story) W24x306 5.79E-02 4.454E-03 Beam (1st to 20th story) W24x250 4.74E-02 3.534E-03 Tài liệu tham khảo 1. N. M. Newmark. 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