In this article we consider the using of finite element method for solving problems of the resistance of materials and compare results solved different methods: analytic solution, hand-calculation solution by finite element method and solution in ANSYS

Pronkevich, S. A.

Senk, I.

English

Repository of Belarusian National Technical University (BNTU)

USING OF FINITE ELEM ENT M ETHOD FOR SOLVING PROBLEMS OF THERESISTANCE OF MATERIALS.Pronkevich S.A., Belarusian National Technical University, Minsk;Senk I, Faculty of Technical Science, Novi Sad.In this article we consider the using o f finite element method for solving problems o f the resistance of materials ami compare results solved different methods: analytic solution, hand-calculation solution by finite element method and solution in ANSYS.The wide distribution of modern computer technic has changed the engineering appreciably. The using of modern computer methods provides capability to research different characteristics of projected objects. It allows changing construction of this objects without prototyping and long and expensive simulations.The finite element method takes up important place among modem computing methods. The finite element method is a numerical procedure that can be used to obtain solution to a wide class of engineering problems involving stress analysis, heat transfer, fluid flow, etc.Consider an example of solution of a simple problem of the resistance of materials by finite element method with hand-calculation, with ANSYS and compare the results with analytical solution.Consider a bar with a variable cross section supporting a load P, as shown in Figure 1. The bar is fixed at one end and carries the load P at the other end. Let us designate the width of the bar at the top by wi at the bottom by the W2, its thickness by t, and its length by L. The bar’s modulus of elasticity will be denoted by E. We are interested in determining how much the bar will deflect at various points along its length when it is subjected to the load P. We will neglect the weight of the bar in the following analysis, assuming that the applied load is considerably larger than the weight of the bar.1' fFigure 1. A bar under axial loadingThe given bar is modeled using four individual segments, with each segment having a uniform cross section. The cross-sectional area of each element is represented by an average area of the cross section at the nodes that make up (define) the element. This model is shown in Figure 2.266w“Т« з Тт»4element 1 2elcmeni 23element 34eletncm 4 t  5^  PFigure 2. Subdividing the bar into elements and nodesIn order to obtain numerical values of the nodal displacement, let us assume thatE - 10.4 1 0 ^ -^  , w, -  2in,W2 = \in ,t -  0Л25 in, L = \0in (I = 2.5 in), P = 1000 Zb inThe variation of the cross-section area of the bar in the у -  direction can be expressed byA(y) = (w, + = (2 + y) • 0.125 = 0.25 -  0.0125y (*)Using (*) w'e can compute the cross-section areas at each node:0.25 Zn^  Ą  = 0.21875.2  Л _  A  1 .- .-2Л3 =0.1875Zn A, =0.15625 Zn"Д  = 0.125 in^Next, the equivalent stiffness coefficient for each element in computed from the equationsЛ+І + A 17k  =■2/Z:, =975-10,lbinZ:, = 845-10 '-^  inlbZ:, = 7 1 5 - 1 0 ' Z:^  =585-10’ — in inThe global stiffness matrix is obtained by assembling, or adding, together each element’s position in the global stiffness matrix:+ [ K f^1^ -K 0 0 0-^1 Z:, +k^ -кг 0 00 -кг -l-Zc, -кг 00 0 -кг Z:, + -к0 0 0 -Zc4 к^Applying the boundary conditions m, = 0  and the load P = IOOO/Z2 we get26710’1 0 0 0 0 «I 0-975 1820 -845 0 0 «2 00 -845 1560 -715 0 ««3 > = . 00 0 -715 1300 -585 «4 00 0 0 -585 585 _ «5 10’If we will have solved this matrix we find displacement;Mi=0, «2=0-001026/«, «3 =0.002210/«, «4=0.003608/«, «5 =0.005317/«Modeling in ANSYS consists of the following steps:1. to select the type of finite element -  “link 1 ”.2. to set material properties. In this task we use elastic isotropic material and input modulus of elasticity and Poisson's ratio.3. because the model consists of 4 finite elements with different cross-sections we have to set up an individual set of real constants for each element and apply it while creating the elements.4. According to problem situation we fix rigidly the upper node and apply load P to the lower node.Model is shown on Figure 3.ELEMEWIS REAL HUMA NРШГ Ю. 15L_.fcFigure 3. Finite elements model created in ANSYS. The results of the nodes displacement are shown on Figure 4.268"ЖЕ FOLLOHING DEGREE OF FREEOOH RESULTS ARE IN GLOBAL COORDINflfiSNODE UV1 D.DODD2 -D.1D256E-023 -D.22D91E-D2  ^ -D.36D77E-D2 5 -D.53171E-D2Figure 4. The results of the nodes displacement.Consider an analytic solution of this problem. Assuming that the material of the bar meets Hook’slaw we can use the following formulae to the calculation of bar displacement □/ = f or inJ EA{y)2.5-iour model for each step □/,. = j 1000„ 10.4 ■ 1 O'’ -(0.25-0.0125 y) Calculating for each step we will find;dy□Zj = 0 in□/2 =0.00102716 in □/3 =0.00221294 Ш □Z, =0.00361541 in □ Z5 =0.0053319 /«Compare results:1 2 3 4 5FEA 0 0.001026 0.002210 0.003608 0.005317ANSYS 0 0.0010256 0.0022091 0.0036077 0.0053171Analytic 0 0.00102761 0.00221294 0.00361541 0.0053319We can see the solution results of different methods are equal.ReferencesМиролюбив И.Н. и др. Пособие к решению задач по сопротивлению материалов. «Высшая школа» 1967 г.Saeed Moaveni. Finite Element Analysis. Theory and Application with ANSYS. Minnesota State University, Mancato.ANSYS User’s Manual.269

Using of finite element method for solving problems of the resistance of materials

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Using of finite element method for solving problems of the resistance of materials

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