CACULATING AND DESIGN FOR a RAILWAY BRIDGE USING FINITE ELEMENT METHOD

HỘI NGHỊ KHCN TỒN QUỐC VỀ CƠ KHÍ - ĐỘNG LỰC NĂM 2017 Ngày 14 tháng 10 năm 2017 Trường ĐH Bách Khoa – ĐHQG TP HCM CACULATING AND DESIGN FOR A RAILWAY BRIDGE USING FINITE ELEMENT METHOD The Van Tran, Trong Nghia Hoang, Anh Tuan Do Hung Yen University of Technology and Education ABSTRACT: Finite element method is a popular and efficient method for mumerical solving the different technical problems as stress analysis and strain analysis in mechanical structures of the car elements, high building structures and bridge bars, etc In this study, a structure of the railway bridge is modelling for calculating stress, strain and displacements of bars The calculated results obtained base on finite element method is compared with that of Matlab It shows that the built railway bridge model is statified with the real model Keywords: railway bridge, finite element method, stress, strain, displacement INTRODUCTION A railway bridge is a structure designed to carry freight and passenger trains across an obstacle in the landscape These bridges represent complex feats of engineering and design, and often require the cooperation of a team of engineers and builders While many railway bridges are designed to cross bodies of water, others span valleys, canyons, or other obstacles that once prevented rail travel within the area A railway bridge often has a major impact on travel, allowing for shorter trips and faster freight delivery, as the train no longer needs to take a longer route around the obstacle As the popularity of train travel declines, railway bridges are often preserved or reconfigured for other uses, such as hiking or bike trails As rail travel is replaced by other forms of travel, rail bridges continue to play an important role in society Many are celebrated for their beauty or structure, while others are adopted by historic preservation groups In the US, "rails to MODEL OF STRUCTURE The basic railway bridge consists of a simple beam or girder, and is designed to cross short spans, such as a small creek The addition of triangular trusses allowed for longer, stronger railway bridges Railway engineers also took advantage of the natural strength of the arch to trails" programs are particularly popular As part of these programs, communities transform old railway paths and bridges into scenic trails for recreation and hiking Compared to road bridges, railway bridges are different, because the trains that pass bridges bring about different requirements When trains pass a bridge, the traffic loads are higher, which means that the relation between dead load and live load is a different one as compared to road bridges Translated into the language of the engineer, higher forces move relatively fast over a structure, having implications on the design of the bridge itself as well as for the protection system of the bridge The maximum deflection of a railway bridge is dependent on speed of the train, span length, mass, stiffness and damping of the structure, axle loads of the train Up to now railway bridges have been designed only due to a static analysis design bridges with an arch-shaped support Suspension bridges rely on high-tension cables for support, which allows them to span even greater distances than earlier bridge designs The most advanced units featured things like doubledecker construction, allowing railcars to share the same bridge as vehicles or pedestrians Trang 261 HỘI NGHỊ KHCN TỒN QUỐC VỀ CƠ KHÍ - ĐỘNG LỰC NĂM 2017 Ngày 14 tháng 10 năm 2017 Trường ĐH Bách Khoa – ĐHQG TP HCM Because A railway bridge must be equipped to handle the extreme loads of a train and its cargo, as well as the additional forces generated by the speed of the train However, this topic we assume there is one train stop on the train, since the railway bridge is applied constant load The railway bridge is modeled as following figure: Figure Some types of railway bridges (a) (b) Figure Some example about applying load on the railway bridge THEORETICAL CALCULATING MODEL BY FINITE ELEMENT METHOD A railway bridge must be equipped to handle the extreme loads of a train and its cargo, as well as the additional forces generated by the speed of the train These bridges should also be capable of withstanding extreme wind and weather In this paper, the train stopped on the bridge and the applied load of train is static load (Figure 3(a)) 3.1 Node numbering scheme Figure 3(b) shows a node numbering scheme The bandwidth of the overall or global Trang 262 characteristic matrix depends on the node numbering scheme and the number of degrees of freedom considered per node If the bandwidth can minimize, the storage requirements as well as solution time can also be minimized The bandwidth (B) is defined: B  D  1 f (1) Where D is the maximum largest difference in the node numbers occurring for all elements of the assemblage, and f is the number of degrees of freedom at each node The previous equation indicates that D has to be minimized in order to minimize the bandwidth Thus, a shorter HỘI NGHỊ KHCN TOÀN QUỐC VỀ CƠ KHÍ - ĐỘNG LỰC NĂM 2017 Ngày 14 tháng 10 năm 2017 Trường ĐH Bách Khoa – ĐHQG TP HCM Table Node index of the element bandwidth can be obtained simply by numbering the nodes across the shortest dimension of the body Node Node i Node j 1 3 5 4 6 Element 10 11 (a) (b) Figure The model calculation (a) and Node numbering scheme (b) 3.2 Determine element stiffness matrix The review a one general element: u  ui  ui cos  vi sin   cos sin    i   vi  (2) u  vi  ui sin   vi cos  -sin cos   i   vi  (3) In matrix form:  ui   cos  v     sin   i  sin    ui  cos   vi  For the two nodes of a bar element: (4)  ui   cos  v    i     sin  u j       vj   sin  cos 0 0 cos  sin    ui      vi  sin   u j    cos   v j  (5) The nodal forces are transformed in the same way:  f xi   cos f   yi     sin   f xj       f yj   sin  cos 0 0 cos  sin    f xi      f yi  (6) sin    f xj    cos   f yj  where f' and f are the force in the local and global coordinate system, respectively Trang 263 HỘI NGHỊ KHCN TỒN QUỐC VỀ CƠ KHÍ - ĐỘNG LỰC NĂM 2017 Ngày 14 tháng 10 năm 2017 Trường ĐH Bách Khoa – ĐHQG TP HCM In the local coordinate system, the displacements can be obtained: 1  EA  L  1  0   u 'i   fi '        v 'i        u ' j   f j'     v ' j    1 0 0 (7) Element stiffness matrix for elements e2, e4, e6, e8, e10: K 2,4,6,8,10 1 0 0 1  EA   L  1  0 0  0  0 (10) Element stiffness matrix for elements e3, e7, e11:      EA  K 3,7,11  L        Figure Coordinate system for a node Table Node in local and global systems coordin ate 4   4      3    (11) 3       Apply load and boundary conditions: u1  v1  0, v7   3  P3 y  210.10 N , P5 y  280.10 N (12) 3.3 Calculating process for stress, strain and displacement of bars by Matab software The block diagram of program is constructed as following: Element stiffness matrix:  C2  EA  CS K L  C   CS CS S2 CS S CS   S  CS   S  C CS C2 CS where C=cos , S=sin between two elements and  Reading input data: materials, geometric structure, meshing control, load, connecting elements, bound conditions (8) is the angle Table Angle of elements in global coordinate system elements Angle C2 S2 CS e1, e5, e9 600 4 e2, e4, e6, e8, e10 00 0 e3, e7, e11 120 4  Element stiffness matrix for elements e1, e5, e9: K1,5,9      EA   L       Trang 264 4   4  4   3   3          (9) Calculating element stiffness matrix [k]e = node and calculating element load vector[f]e = node Defining global stiffness matrix [K] and load vector[F] Setting bound conditions Determining displacement vector of nodes by solving equations [K][u] = [F] Calculating stress, strain, reaction force,… Printing results: Displacement, stress, strain, reaction force Figure The block diagram of Matlab program HỘI NGHỊ KHCN TOÀN QUỐC VỀ CƠ KHÍ - ĐỘNG LỰC NĂM 2017 Ngày 14 tháng 10 năm 2017 Trường ĐH Bách Khoa – ĐHQG TP HCM - The first: input data of the structure as elastic module (E), length of bars (l1,l2,l3,…), load (P1,P2,…), bound conditions is setup - The second: element stiffness matrix and element load vector is calculated - The third: global stiffness matrix and global load vector are defined from element stiffness matrix and element load vector based on connecting algorithm - The fourth: bound conditions are setup as Eq (12) - The fifth: displacements of nodes determined - Last one: The stresses, strains and reaction forces are determined from displacement The displacements at each nodes stresses and strains in each bar, reactive load is presented in Table and Table It shows that the results are obtained from Matlab program is approximately with that of the results obtained by directly solving equations Table Deplacement and load of nodes on elements nodes stresses and strains in each bar, reactive load is determined By solving above simultaneous equation (Eq (7)), the displacements can be obtained After that, the stress in each bar is calculated as following: u,    i  E  EB  i,   E    L u j   ui    vi E  C  S C S    uj  L    v j   C  L   S 0 C (N ) Matlab Hand Matlab Hand 0 7.805 0 2.937 2.937 0 -3.336 -3.337 0 0.711 0.711 0 -6.263 -6.263 -210000 -210000 1.516 1.516 0 -6.892 -6.892 0 2.2026 2.20281 0 -6.659 -6.660 -280000 -280000 -0.047 -0.047 0 -3.555 -3.556 0 2.984 2.985 0 0 (13) (14) From the global FE equation, the reaction forces is calculated  u  u  P1x  K1, j   , P1 y  K 2, j   u  u  ,  ui    P7 y  K14, j u ,    j  , i , j    (15) 256670 256659.968 Table Strain and stress of the elements Elements  i  i Ei 233333 233326.547 0  S From stresses in each bar, the strain is calculated as: , i , j Load of nodes (mm) Nodes NUMERICAL RESULTS A railway bridge is assembled from steel, the same cross-section of steel bars with each other and equal 3250 mm2 The train stopped on the bridge, which have to apply the load of the train (as Figure 3(a)) The displacements at each Deplacement of nodes Strain of elements Stress of elements 103 (mm) ( N / mm2 ) Matlab -0.3948 Hand -0.3948 Matlab Hand -82.8995 -82.9016 0.1974 0.1974 41.4467 41.4508 82.9020 0.3948 0.3948 82.8995 -0.3947 -0.3948 -82.8934 -82.9016 -0.0395 -0.0395 -8.29 -8.2903 0.4145 0.4145 87.038 87.0465 0.0395 0.0395 8.29 8.2901 -0.4342 -0.4343 -91.1827 -91.1915 0.4342 0.4343 91.1895 91.1917 45.5963 10 0.2171 0.2171 45.5914 11 -0.4342 -0.4343 -91.1895 -91.1919 Trang 265 HỘI NGHỊ KHCN TOÀN QUỐC VỀ CƠ KHÍ - ĐỘNG LỰC NĂM 2017 Ngày 14 tháng 10 năm 2017 Trường ĐH Bách Khoa – ĐHQG TP HCM CONCLUSIONS This paper has proposed a highly practical model for calculating and design railway bridge using finite element method This method can be used to solve more 1–D and 2-D problems The algorithm of program is designed exactly Matlab program runs stability and gets good result the same calculating result The program is highly flexible REFERENCES [1] V C Nguyễn, V H Trần, T B Mạc, Phân tích thiết kế khí, Nhà xuất Khoa học Kỹ thuật (2016) [4] H S Govinda Rao, Finite element method and classical methods, New age international publishers (2007) [2] X L Nguyễn, Phương pháp phần tử hữu hạn, NXB GTVT (2007) [5] S S Rao, The Finite Element Method in Engineering, 4nd ed Elsevier Butterworth– Heinemann, USA (2005) [3] I T Trần, N K Ngô, Phương pháp phần tử hữu hạn, NXB Hà Nội (2007) TÍNH TOÁN THIẾT KẾ CẦU ĐƯỜNG SẮT XE LỬA ỨNG DỤNG PHƯƠNG PHÁP PHẦN TỬ HỮU HẠN TÓM TẮT: Phương pháp phần tử hữu hạn (PTHH) phương pháp phổ biến hữu hiệu cho lời giải số tốn kỹ thuật khác phân tích trạng thái ứng suất, biến dạng kết cấu khí, chi tiết tơ, khung nhà cao tầng, dầm cầu, Trong nghiên cứu này, kết cấu cầu xe lửa đường sắt mơ hình hóa cho việc tích tốn trạng thái ứng suất, biến dạng chuyển vị Kết thu từ việc tính tốn dựa lý thuyết phần tử hữu hạn so sánh với kết từ phần mềm Matlab Từ cho thấy, mơ hình xây dựng phù hợp với thiết kế thực tế cho kết cấu cầu đường sắt Từ khóa: cầu đường sắt, phương pháp phần tử hữu hạn, ứng suất, biến dạng, chuyển vị Trang 266 ... Some example about applying load on the railway bridge THEORETICAL CALCULATING MODEL BY FINITE ELEMENT METHOD A railway bridge must be equipped to handle the extreme loads of a train and its cargo,... calculating and design railway bridge using finite element method This method can be used to solve more 1–D and 2-D problems The algorithm of program is designed exactly Matlab program runs stability... Bách Khoa – ĐHQG TP HCM Because A railway bridge must be equipped to handle the extreme loads of a train and its cargo, as well as the additional forces generated by the speed of the train However,