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

Phân tích động mờ khung thép phẳng được giằng<br /> sử dụng thuật toán tiến hóa vi phân<br /> Fuzzy dynamic analysis of 2d-braced steel frame using differential evolution optimization<br /> Viet T. Tran, Anh Q. Vu, Huynh X. Le<br /> <br /> <br /> Tóm tắt 1. Introduction<br /> <br /> Bài báo nghiên cứu áp dụng thủ tục phần tử hữu In the dynamic analysis of steel frame structures with semi-rigid<br /> connections, rigidity of the connection (or fixity factor of the connection),<br /> hạn mờ phân tích động kết cấu khung thép phẳng<br /> loads, mass per unit volume, damping ratio … have a significant influence<br /> với các đại lượng đầu vào mờ. Các hệ số liên kết<br /> on the time – history response of steel frame structure [4]. In practice,<br /> giữa dầm – cột, cột – móng, tải trọng, khối lượng<br /> however, many parameters like worker skill, quality of welds, properties<br /> riêng và hệ số cản được mô tả dưới dạng các số mờ of material and type of the connecting elements affect the behavior of a<br /> tam giác. Phương pháp tích phân số β – Newmark connection, and this fixity factor is difficult to determine exactly. Therefore, in<br /> được áp dụng xác định chuyển vị trong hệ phương a practical analysis of structures, a systematic approach need to include the<br /> trình cân bằng động tuyến tính. Phương pháp tối uncertainty in the connection behavior, and the fixity factor of a connection<br /> ưu mức – α sử dụng thuật toán tiến hóa vi phân modeled as the fuzzy number is reasonable [5]. In addition, the uncertainty<br /> được tích hợp với mô hình phần tử hữu hạn để of input parameters is also described in form of fuzzy numbers, such as<br /> phân tích động kết cấu mờ. Hiệu quả của phương external forces, mass per unit volume and damping ratio. In this paper, the<br /> pháp đề xuất được minh họa thông qua ví dụ liên fuzzy displacement - time dependency of a planar steel frame structure<br /> quan đến khung thép phẳng hai mươi lăm tầng, is determined in which the fixity factor, loads, mass per unit volume, and<br /> ba nhịp được giằng tập trung. damping ratio are described in the form of any triangular fuzzy numbers.<br /> Từ khóa: Khung thép giằng, liên kết mờ, động lực kết cấu A procedure is based on finite element model by combining the α – level<br /> mờ, thuật toán tiến hóa vi phân optimization with the Differential Evolution algorithm (DEa). The Newmark-β<br /> average acceleration numerical integration method is applied to determine<br /> the displacements from the linear dynamic equilibrium equation system of<br /> Abstract the finite element model.<br /> This paper studies the application of the fuzzy finite 2. Finite element with linear semi-rigid connection<br /> element procedure for dynamic analysis of the planar<br /> The linear dynamic equilibrium equation system is given as following<br /> semi-rigid steel frame structures with fuzzy input<br /> parameters. The fixity factors of beam – column and [ M ]{u} + [C ]{u} + [ K ]{u} = {P ( t )} (1)<br /> column – base connections, loads, mass per unit<br /> volume and damping ratio are modeled as triangular where {u} , {u} , and {u } are the vectors of acceleration, velocity, and<br /> fuzzy numbers. The Newmark-β numerical integration displacement respectively; [M], [C], and [K] are the mass, damping, and<br /> method is applied to determine the displacement of stiffness matrices respectively; {P(t)} is the external load vector. The viscous<br /> the linear dynamic equilibrium equation system. The damping matrix [C] can be defined as<br /> α – level optimization using the Differential Evolution = [C ] α M [ M ] + β K [ K ] (2)<br /> (DE) integrated finite element modeling to analyse<br /> dynamic of fuzzy structures. The efficiency of proposed where αM and βK are the proportional damping factors which defined as<br /> methodology is demonstrated through the example 2ω1ω2 2<br /> problem relating to the twenty-five – story, three – bay = α M ξ= ;β ξ<br /> ω1 + ω2 K ω1 + ω2 (3)<br /> concentrically braced frame.<br /> Keywords: braced steel frame, fuzzy connection, fuzzy where ξ is the damping ratio; ω 1 and ω 2 are the natural radian frequencies<br /> of the first and second modes of the considered frame, respectively.<br /> structural dynamic, differential evolution algorithm<br /> In this study, the frame element with linear semi – rigid connection is<br /> shown in Fig. 1, with E - the elastic modulus, A – the section area, I – the<br /> inertia moment, m - the mass per unit volume, k1 and k2 – rotation resistance<br /> MS. Viet T. Tran<br /> stiffness at connections.<br /> Faculty of civil engineering, Duy Tan University<br /> Email: The element stiffness matrix - [Kel] and the mass matrix - [Mel] of the<br /> Ass. Prof. Anh Q. Vu frame are given by [4], with si = Lki / (3EI + Lki) denote the fixity factor of<br /> Faculty of civil engineering semi – rigid connection at the boundaries (i = 1,2). In Eq. (1), when fixity<br /> Hanoi Architectural University factors of connections, external loads, mass per unit volume and damping<br /> Email: ratio are given by fuzzy numbers, the displacements of joints are also fuzzy<br /> Prof. Huynh X. Le numbers. In steel structures, the common fuzzy connections can be defined<br /> Faculty of civil engineering by linguistic terms as shown in Fig. 2. Eleven linguistic terms are assigned<br /> National University of Civil Engineering numbers from 0 to 10 ( si = 0,1,...10 ) [5].<br /> Email: In the classical finite element method (FEM), in Eq. (1), the displacement<br /> – time dependency of the joints is determined by solving the linear dynamic<br /> equilibrium equation system. The Newmark-β method has been chosen for<br /> <br /> <br /> S¬ 27 - 2017 45<br />  KHOA H“C & C«NG NGHª<br /> <br /> <br /> <br /> <br /> Figure 1. Frame element with Figure 2. Membership functions of fuzzy fixity factors<br /> linear semi-rigid connection<br /> <br /> <br /> <br /> <br /> Figure 4. Fuzzy displacement-time response at joint<br /> 26 in x direction<br /> <br /> <br /> <br /> <br /> Figure 3. Concentrically braced steel frame with Figure 5. The membership functions of fuzzy<br /> fuzzy input parameters displacement at joint 26<br /> <br /> <br /> <br /> the numerical integration of this equation system because of The displacement of the joint at each time step is<br /> its simplicity [1]. The fuzzy displacement is determined by the determined by this algorithm of linear elastic dynamic<br /> fuzzy finite element method (FFEM) using the α-cut strategy analysis.<br /> with the optimization approaches. FFEM is an extension of<br /> FEM in the case that the input quantities in the FEM are 3.2. α – level optimization using Differential Evolution<br /> modeled as fuzzy numbers. In this study, an optimization algorithm (DEa)<br /> approach is presented in the next sections: the differential For fuzzy structural analysis, the α-level optimization is<br /> evolution algorithm (DEa). known as a general approach in which all the fuzzy inputs<br /> are discretized by the intervals that are equal α-levels.<br /> 3. Proceduce for fuzzy structural dynamic analysis The output intervals are then searched by the optimization<br /> 3.1. Linear elastic dynamic analysis algorithm algorithms. The optimization process is implemented directly<br /> The Newmark-β method is based on the solution of an by the finite element model and the goal function is evaluated<br /> incremental form of the equations of motion. For the equations many times in order to reach to an acceptable value. In this<br /> of motion (1), the incremental equilibrium equation is: study, the output intervals are the displacement intervals at<br /> each time step, and the solution procedure is proposed by<br /> [ M ]{∆u} + [C ]{∆u} + [ K ]{∆u} ={∆P} (4) combining the Differential Evolution algorithm (DEa) with<br /> where {u} , {u} , and {u} are the vectors of incremental the α-level optimization. The DEa has shown better than the<br /> acceleration, velocity, and displacement respectively; {∆P} is genetic algorithm (GA) and is simple and easy to use. Basic<br /> the external load increment vector. procedure of DEa is described as [6].<br /> <br /> <br /> <br /> 46 T„P CHŠ KHOA H“C KI¦N TR”C - XŸY D¼NG<br /> Table 1: Section properties used for analysis of the portal steel frame<br /> Member Section Cross – section area, A (m2) Moment of inertia, I (m4)<br /> Column (1st to 4th story) W30x391 7.35E-02 8.616E-03<br /> Column (5th to 8th story) W30x326 6.17E-02 6.993E-03<br /> Column (9th to 14th story) W27x307 5.82E-02 5.453E-03<br /> Column (15th to 20th story) W24x306 5.79E-02 4.454E-03<br /> Beam (1st to 20th story) W24x250 4.74E-02 3.534E-03<br /> <br /> 4. Numerical illustration central value) from the SAP2000 software, with t = 1.10, 2.05,<br /> A twenty-five – story, three – bay concentrically braced 2.90, 4.05, 5.10, and 6.00 seconds.<br /> frame subjected to fuzzy impulse force as shown in Fig. 3 5. Conclusion<br /> is considered. The fuzzy input parameters are: m  1 = (7.85,<br /> 0.785, 0.785), m  2 = (50, 5, 5), s1 =9, s2 = 8, s3 = 7, s4 = 6, A fuzzy finite element analysis based on the Differential<br /> s5 =5, s6 =1. The fuzzy damping ratio is ξ = (0.05, 0.005, Evolution (DE) in combination with the α – level optimization,<br /> 0.005). The fuzzy impulse force is: P ( t ) = P (0 ≤ t ≤ 3 s), and in which the Newmark-β average acceleration method is<br /> P ( t ) = 0 (t > 3 s), with P = (40, 4, 4). These fuzzy terms applied to determine the deterministic displacement. The<br /> are considered to be triangular fuzzy numbers with 20% fuzzy input parameters such as fixity factors of connections,<br /> absolute spread. A time step Δt of 0.05 second is chosen in external forces, mass per unit volume, and damping ratio<br /> the dynamic analysis. The output intervals of displacement have a significant influence on the time dependency of the<br /> are calculated by using DE programmed by MATLAB. The fuzzy displacement. With the example is considered, fuzzy<br /> section properties used for analysis of the frame are shown displacments show more different shapes of membership<br /> in Table I. functions at different times. Moreover, these fuzzy<br /> displacements have absolute spreads from 40% to 150%. In<br /> Fig. 4 shows the fuzzy displacement-time response and adition, the determinant results are also compared with ones<br /> the membership functions of fuzzy displacement at different of the SAP2000 software and give a good agreement./.<br /> times in 3D – axis. Fig. 5 shows the membership functions<br /> of fuzzy displacement and the deterministic displacement (at<br /> <br /> <br /> Tài liệu tham khảo 5. A. Keyhani, S. M. R. Shahabi. Fuzzy connections in structural<br /> analysis. ISSN 1392 – 1207 MECHANIKA, 18(4) (2012) 380-<br /> 1. N. M. Newmark. A method of computation for structural<br /> 386.<br /> dynamic. Journal of the Engineering Mechanics Division, ASCE,<br /> vol. 85 (1959) 67-94. 6. M. M. Efrén, R. S. Margarita, A. C. Carlos. Multi-Objective<br /> Optimization using Differential Evolution: A Survey of the State-<br /> 2. R. Storn, and K. Price. Differential Evolution – A Simple and<br /> of-the-Art. Soft Computing with Applications (SCA), 1(1) (2013).<br /> Efficient Heuristic for Global Optimization over Continuous<br /> Spaces. Journal of Global Optimization 11, Netherlands, (1997) 7. P. H Anh, N. X. Thanh, N. V. Hung. Fuzzy Structural Analysis<br /> 341-359. Using Improved Differential Evolution Optimization.<br /> International Conference on Engineering Mechanic and<br /> 3. M. Hanss. The transformation method for the simulation and<br /> Automation (ICEMA 3), Hanoi, October 15-16 (2014) 492-498.<br /> analysis of systems with uncertain parameters. Fuzzy Sets and<br /> Systems 130(3) (2002) 277-289. 8. T. T. Viet, V. Q. Anh, L. X. Huynh. Fuzzy analysis for stability of<br /> steel frame with fixity factor modeled as triangular fuzzy number.<br /> 4. V. Q. Anh, N. M. Hien. Geometric nonlinear vibration analysis<br /> Advances in Computational Design 2(1) (2017) 29-42.<br /> of steel frames with semi-rigid connections and rigid zones.<br /> Vietnam Journal of Mechanics, VAST 25 (2) (2003) 122-128.<br /> <br /> <br /> <br /> <br /> S¬ 27 - 2017 47<br />