Introduction to Finite Element Method

Transcription

1 Guest Lecture in Prodi Teknik Sipil Introduction to Finite Element Method Wong Foek Tjong, Ph.D. Petra Christian University Surabaya

2 Lecture Outline 1. Overview of the FEM 2. Computational steps of the FEM 3. Learning the direct stiffness method using ta29.petra.ac.id 2014/6/7 2

3 Behavior of a Real Structure Simulation Experiment Simplifications and assumptions of the real structure Mathematical Model Replicate conditions of the structure (possibly on a smaller scale) and observe the behavior of the model Physical Model 2014/6/7 3

4 An example of the FEM applications Real experiment It is often expensive or dangerous FE simulation It replicates conditions of the real experiment Source: W.J. Barry (2003), FEM Lecture Slides, AIT Thailand 2014/6/7 4

5 Mathematical Models Analytical Solution Techniques Numerical Solution Techniques Closed-form Solutions Only possible for simple geometries and boundary conditions Finite difference methods Finite element methods Boundary element methods Mesh-free methods etc. 2014/6/7 5

6 What is FEM? It is a computational technique used to obtain approximate solutions of engineering problems. The results are generally not exact. However, the accuracy of the results can be improved either using finer mesh (hrefinement) or higher degree elements (prefinement) 2014/6/7 6

7 Solution refinements in FEM h-refinement h=1 h=1/2 h=1/4

8 Solution refinements in FEM (cont d) p-refinement u=a+bx+cy h=1 h=1/2 u=a+bx+cy+ dx 2 +exy+fy 2

9 Finite element method (1) In the context of structural analyzes, it may be regarded as a generalized matrix method of structural analysis. It is originated as a method of structural analysis but is now widely used in various disciplines such as heat transfer, fluid flow, seepage, electricity and magnetism, and others. 2014/6/7 9

10 Finite element method (2) Modern FEM were first developed and applied by aeronautical engineers, i.e. M.J. Turner et al., at Boeing company in the period 1950s. 1956: The first engineering FEM paper 2014/6/7 10

11 Finite element method (3) The name finite element method was coined by R.W. Clough in It is called finite in order to distinguish with infinitesimal element in Calculus. 1967: First FEM book by O.C. Zienkiewicz 2014/6/7 11

12 Examples of FEM software For General purposes: NASTRAN, ANSYS, ADINA, ABAQUS, etc. For structural analysis, particularly in Civil Engineering: SANS, SAP2000, STAAD, GT STRUDL, MIDAS, DIANA, STRAND 7, etc. For building structures: ETABS, BATS etc. 2014/6/7 12

13 Example of applications in structural engineering 2014/6/7 Beijing National Stadium in China (Bird s Nest Stadium) Source: MidasGen project apllications 13

14 Example of applications in structural engineering (cont d) The structural analysis of an F-16 aircraft The analysis of the Cathedral of Barcelona using 3D solid elements. (courtesy of Barcelona Cathedral) Source: /6/7 14

15 Simulation of Wave Propagation Passing a Solid Barrier Source: G. B. Wijaya (2005), a Petra-AIT Thesis 7-Jun-14 15

16 Lecture Outline 1. Overview of the FEM 2. Computational steps of the FEM 3. Learning the direct stiffness method using ta29.petra.ac.id 2014/6/7 16

17 Discretization Fundamental concept is discretization, i.e. dividing a continuum (continuous body, structural system) into a finite number of smaller and simple elements whose union approximates the geometry of the continuum. 2014/6/7 17

18 Some basic element shapes 2014/6/7 18

19 Examples of discretization (3) Cooling Tower Nuclear Power Plant (taken from a FEM Course Project of Doddy and Andre, Dec 2008) 150 m 2014/6/7 19

20 Structural Model and Its Example of the Analysis Results 2014/6/7 The structure is divided into smaller parts called element Membrane force contour in the circumferential direction 20

21 The FE Model with a Finer Mesh 2014/6/7 The structure is modeled with a finer mesh The result is now better 21

22 Computational steps of the FEM- the direct stiffness method Discretize the structure (problem domain) Divide the structure or continuum into finite elements Once the structure has been discretized, the computational steps faithfully follow the steps in the direct stiffness method. The direct stiffness method: The global stiffness matrix of the discrete structure are obtained by superimposing (assembling) the stiffness matrices of the element in a direct manner. 2014/6/7 22

23 Computational steps (cont d) Generate element stiffness matrix and element force matrix for each element. Assemble the element matrices to obtain the global stiffness equation of the structure. Apply the known nodal loads. Specify how the structure is supported: Set several nodal displacements to known values. 2014/6/7 23

24 General steps of the FEM (cont d) Solve simultaneous linear algebraic equation. The nodal parameters (displacements) are obtained. Calculate element stresses or stress resultants (internal forces). 2014/6/7 24

25 Discretization of Plane Frame Structures p 2 p 3 Joint CONTINUOUS MODEL p 1 D 8 D 5 D 9 D 7 D 11 x 2 D 6 D 4 D 12 D 10 D 2 DISCRETE MODEL D 3 D 1 {D} = Joint Degree-of-freedom = Free DOF = Locked DOF Source: ACECOMS, AIT-Thailand

26 Plane Frame Element The stiffness equation for the plane frame element can be obtained by superposing the bar and beam equations y 4 z x Local coordinate 7-Jun-14 26

27 The discretized equation: kd=f d = d 1 d 2 d 3 d 4 d 5 d 6 = u 1 v 1 θ 1 u 2 v 2 θ 2 ; f = f 1 f 2 f 3 f 4 f 5 f 6 = N 1 V 1 M 1 N 2 V 2 M 2 7-Jun-14 27

28 The stiffness matrix: k = C C C 2 6C 2 L 0 12C 2 6C 2 L 0 6C 2 L (4 + φ)c 2 L 2 0 6C 2 L 2 φ C 2 L 2 C C C 2 6C 2 L 0 12C 2 6C 2 L 0 6C 2 L (2 φ)c 2 L 2 0 6C 2 L (4 + φ)c 2 L 2 C 1 = EA L ; C 2 = EI z L 3 (1+φ y ) ; φ y = 12EI z A s GL 2 7-Jun-14 28

29 Source: ACECOMS, AIT-Thailand =

30 Source: ACECOMS, AIT-Thailand =

31 Source: ACECOMS, AIT-Thailand =

32 Source: ACECOMS, AIT-Thailand =

33 r2 = r = -

34 Final System r2 = r = -

35 Lecture Outline 1. Overview of the FEM 2. Computational steps of the FEM 3. Learning the direct stiffness method using ta29.petra.ac.id 2014/6/7 35

36 Example (SAP2000 Verification Manual, 2007) Find the internal forces, support reactions and the deflection at the middle of the beam (node 5) 36

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