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Finite Element Method

Finite Element Method

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Finite Element Method

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  1. TS 4466 2 Credits Finite Element Method Instructor: Wong Foek Tjong, Ph.D.

  2. Course description • The course aims to enable the students to understand the basic concepts andproceduresof the finite element method (FEM) and to apply the FEM by using a commercial software • Teaches understanding of how finite element methods work rather than how to use a software

  3. The instructor • Graduated from Universitas Parahyangan, Bandung in March1994 Final project: Dynamic Analysis of Multi-degree-of- Freedom Structures Subjected to Ergodic Random Excitation • Graduated from Institut Teknologi Bandung in April1998 Master thesis:Active Vibration Control of Structures by Using Artificial Neural Network Observer • Graduated from Asian Institute of Technology, Thailand in May 2009 Dissertation: Kriging-based Finite Element Method for Plates and Shells • Contact: P Building, Room P402B Tel. 62-31-298-3391

  4. Course outline • Overview of the FEM • The direct stiffness method • Spring and bar systems • Truss structures • One-dimensional elements • Bar, beam, torsional bar elements • Frame element in 3D space • Two-dimensional elements for plane-strain/plane-stress problems • Constant strain triangle element • Bilinear isoparametric quadrilateral element • Introduction to plate and shell elements • Applications of the FEM using SAP2000

  5. References • D.L. Logan (2007) A First Course in the Finite Element Method the 4th Ed., Toronto, Nelson • D.V. Hutton (2004) Fundamentals of Finite Element Analysis New York, McGraw-Hill • R. D. Cook, D.S. Malkus, M.E. Plesha and R.J. Witt (2002) Concepts and Applications of Finite Element Analysis 4th Ed., John Wiley and Sons • W. Weaver, Jr. and P.R. Johnston (1984) Finite Elements for Structural Analysis New Jersey, Prentice-Hall

  6. References (cont’d) • Computers and Structures, Inc. (2006) CSI Analysis Reference Manual, Berkeley, CSI • C. Felippa (2008) Introduction to Finite Element Methods • R. Krisnakumar (2010) Introduction to Finite Element Methods (Video of lecture series on FEMs)

  7. Softwares • MATLAB Ver. 6.5 Strongly recommended software for matrix computation and programming • SAP 2000 Ver. 11.0.0 For applications

  8. Grading weights • Homework assignments 15% • Mid-semester exam 35% • Take home test 15% • Final exam-- project 35%

  9. Late coming to the class The tolerance for coming late to the class is 20 minutes. Those who come late more than 20 minutes are NOT allowed to attend the class. Please refer to the “FEM Lecture Plan” for more academic norms

  10. Any question about the course before we begin withtheOverview of the FEM?

  11. Discussion: the task of a structural engineer • Let take a look on a typical job vacancy announcement that you may read once you graduate from your study

  12. Discussion (cont’d) • Why do you think a design engineer is required to master a structural analysis and design software? • An engineer needs to understand the behavior of a structure so that he/she can make judicious decisions in design, retrofitting, or rehabilitation of the structure

  13. Behavior of a Real Structure Simulation Experiment Simplifications and assumptions of the real structure Replicate conditions of the structure (possibly on a smaller scale) and observe the behavior of the model Mathematical Model Physical Model

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

  15. The need for modeling • A real structure cannot be analyzed, it can only be “load tested” to determine the responses • We can only analyze a “model” of the structure (perform simulation) • We need to model the structure as close as possible to represent the behavior of the real structure Source: W. Kanok-Nukulchai

  16. The idealization process for a simple structure Source: C. Felippa

  17. Mathematical Models Analytical Solution Techniques Numerical Solution Techniques Closed-form Solutions • Finite difference methods • Finite element methods • Boundary element methods • Mesh-free methods • etc. Only possible for simple geometries and boundary conditions

  18. Finite element method (1) • It is a computational technique used to obtain approximate solutions of engineering problems. • In the context of structural analyzes, it may be regarded as a generalized direct stiffness method. • The direct stiffness method you studied in MK 4215 Structural Analysis III is actually the application of the FEM to frame structures • 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.

  19. 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

  20. Finite element method (3) • The name “finite element method” was coined by R.W. Clough in 1960. It is called “finite” in order to distinguish with “infinitesimal element” in Calculus. • 1967: First FEM book by O.C. Zienkiewicz

  21. Finite element method (4) • The computation is carried out automatically using a computer or a network of computers. • The results are generally not exact.

  22. Example of applications in structural engineering 1. Framed structures (b) Grid (a) Truss Source: Weaver and Johnston, 1984

  23. Example of applications in structural engineering (cont’d) 1. Framed structures (cont’d) (c) Frame (d) Arch Source: Weaver and Johnston, 1984

  24. Example of applications in structural engineering (cont’d) 2. Two-dimensional continua (b) Plane strain (a) Plane stress Source: Weaver and Johnston, 1984

  25. Example of applications in structural engineering (cont’d) 3. Three-dimensional continua (a) General solid (b) Axisymmetric solid Source: Weaver and Johnston, 1984

  26. Example of applications in structural engineering (cont’d) 4. Plate in bending Source: Weaver and Johnston, 1984

  27. Example of applications in structural engineering (cont’d) 5. Shells (a) General shell (b) Axis symmetric shell Source: Weaver and Johnston, 1984

  28. Example of applications in structural engineering (cont’d) The analysis of a double curvature dam taking into account soil-structure interactions effects The analysis and design of buildings Source:

  29. 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:

  30. Discretization (1) • 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. • Mesh generation programs, called preprocessors, help the user in doing this work • GiD, a software for pre and post processor

  31. Some basic element shapes

  32. Some 1st order (linear) elements

  33. Some 2nd order (quadratic) elements

  34. Discretization (2) • One suggestion on performing discretization is to divide structural regions with high stress concentration into finer division e.g. in the vicinity of the support and around the hole(s). • The accuracy of the results can be improved by using a finer mesh (h-refinement) or using a higher order elements (p-refinement).

  35. Examples of discretization (1)

  36. z y x h D Examples of discretization (2) Clamped D=100, D/h =100 E = 2 x 106 ; ν = 0.3; k = 5/6 Load: uniform q = -1E-6 76 nodes, 119 elements 172 active DOF

  37. 150 m Examples of discretization (3) Cooling Tower– Nuclear Power Plant (taken from a FEM Course Project of Doddy and Andre, Dec 2008)

  38. Structural Model and Its Example of the Analysis Results Membrane force contour in the circumferential direction The structure is divided into smaller parts called “element”

  39. The FE Model with a Finer Mesh The result is now better The structure is modeled with a finer mesh

  40. Examples of FEM software • For General purposes: NASTRAN, ANSYS, ADINA, ABAQUS, etc. • For structural analysis, particularly in Civil Engineering: SANS, SAP, STAAD, GT STRUDL, etc. • For building structures: ETABS, BATS etc. • For geotechnical design: PLAXIS • For conducting researches on earthquake engineering: DRAIN-2D, DRAIN-3D, RUAOMOKO, OpenSees etc.

  41. Typical capabilities of a FE program • Data generation • Automatic generation of nodes, elements, and restraints • Element types • E.g. SAP2000: Frame, Cable, Shell, Plane, Asolid, Solid, etc. • Material behavior • Linear-elastic, nonlinear • Load types • Force, displacement, thermal, time-varying excitation • Plotting results • Original and deformed geometry, stress contours

  42. Why do we need to study the basic theory of FEM? • Cook, Malkus, and Plesha (1989, pp.6) Concepts and assumptions behind the computer codes (FEM software) should be mastered. Engineers are expected to be able to use the software to gain better advantages and will less likely misuse them. • SAP2000 disclaimer The user accepts and understands that no warranty is expressed or implied by the developers or the distributors on the accuracy or reliability of the program. The user must explicitly understand the assumptions of the program and must independently verify the results.

  43. Any question before we proceed to computational steps of the FEM?

  44. 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.

  45. 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.

  46. 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).

  47. Any question before we continue to a brief introduction to MATLAB?

  48. Example • Suppose you want to calculate the natural frequency (Hz) of a SDOF system with the mass m=100 kg and stiffness k=5 KN/m • The formula is • Type in the Command Window: >>m=100 >>k=5*1000 >> f=1/(2*pi)*sqrt(k/m)