Finite Element Analysis Lecture 1. Dr./ Ahmed Nagib

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1 Finite Element Analysis Lecture 1 Dr./ Ahmed Nagib April 30, 2016

2 Research and Development

3 Mathematical Model

4

5 Mathematical Model

6 Mathematical Model

7 Finite Element Analysis The linear equation of motion for vibration is M x ሷ + C x ሶ + K x = F x, which is the displacement vector x ሷ, which is the acceleration vector M, which is the Mass matrix K, which is the stiffness matrix C, which is the damping matrix F, which is the load vector

8 Finite Element Softwares

9 Finite Element Analysis

10 Finite Element Analysis

11 Finite Element Analysis

12 Static Analysis For a linear static structural analysis, the global displacement vector x is solved for in the matrix equation below: K x = F Assumptions made for linear static structural analysis are: K, which is the global stiffness matrix, is constant Linear elastic material behavior is assumed Small deflection theory is used F, which is the global load vector, is statically applied No time-varying forces are considered No damping effects

13 Axial Stress 13

14 Beam under the action of two tensile forces 14

15 Beam under the action of two tensile forces 15

16 Torsion Stress 16

17 Torsion Stress 17

18 Torsion Stress 18

19 Angle of Twist 19

20 Torsion of a Shaft with Circular Cross-Section 20

21 Torsion of a Shaft with Circular Cross-Section 21

22 Torsion of a Beam with the Square Cross-Section 22

23 Torsion of a Beam with the Square Cross-Section 23

24 Bending Stress 24

25 Bending Stress 25

26 Bending a Cantilever Beam under a Concentrated Load 26

27 Bending a Cantilever Beam under a Concentrated Load 27

28 Bending Stress 28

29 Bending Stress 29

30 Bending Stress 30

31 Bending Stress 31

32 Bending Stress 32

33 Bending Stress 33

34 Bending Stress 34

35 Bending of Curved beam Displacement Stress in x direction 35

36 Finite Element Analysis

37 Finite Element Analysis

38 Static Analysis

39 Static Analysis

40 Static Analysis

41 Static Analysis

42 Static Analysis

43 Static Analysis Linear vs Non Linear solve In a linear analysis, the matrix equation [K]{x}={F} is solved in one iteration. That means the model stiffness does not change during solve : [K] is constant. A non linear solve allow stiffness changes and uses an iterative process to solve the problem. In a static structural analysis, ANSYS runs a non linear solve automatically when the model contains : - Non linear material laws : Plasticity, Creep, Gasket, Viscoelasticity - Non linear contact : Frictionless, Rough, Frictional - Large deflection turned <<ON>> - Joints - Bolt pretension

44 ሷ ሷ Modal Analysis The linear equation of motion for free, un-damped vibration is M x + K x = 0 Assume harmonic motion: x = φ i sin ω i t + θ i x = ω 2 i φ i sin ω i t + θ i Substituting x and eigenvalue equation: xሷ in the governing equation gives an ω 2 i M + K where ω i : Natural Frequencies φ i : Mode Shapes φ i = 0

45 Modal Analysis

46 Modal Analysis

47 Modal Analysis

48 Modal Analysis Assumptions for Modal Analysis [K] and [M] are constant: Linear elastic material behavior is assumed Small deflection theory is used, and no nonlinearities included [C] is not present, so damping is not included {F} is not present, so no excitation of the structure is assumed Mode shapes φ i are relative values, not absolute

49 Modal Analysis Modal Results: Because there is no excitation applied to the structure the mode shapes are relative values not actual ones. Because a modal result is based on the model s properties and not a particular input, we can interpret where the maximum or minimum results will occur for a particular mode shape but not the actual value.

50 Modal Analysis

51 Modal Analysis

52 Modal Analysis

53 Modal Analysis

54 Modal Analysis

55 Modal Analysis

56 Modal Analysis

57 Dynamic Analysis

58 Dynamic Analysis

59 Dynamic Analysis

60 Dynamic Analysis

61 Dynamic Analysis

62 Dynamic Analysis

63 Dynamic Analysis

64 Dynamic Analysis

65 Dynamic Analysis

66 Dynamic Analysis

67 Dynamic Analysis

68 Dynamic Analysis

69 Dynamic Analysis

70 Dynamic Analysis

71 Dynamic Analysis

72 Dynamic Analysis

73 Dynamic Analysis

74 Dynamic Analysis

75 Dynamic Analysis

76 Dynamic Analysis

77 Dynamic Analysis

78 Dynamic Analysis

79 Fluid-Structure Interaction Solid Mechanics-Structural Analysis Fluid Dynamics Solved by Finite Element Analysis Computational Fluid Dynamics (CFD) 79

80 Recent Computational Methodology Commercial Software Finite Element Analysis Ansys Mechanical, Abaqus Computational Fluid Dynamics (CFD) Ansys Fluent, Ansys CFX, Open-foam 80

81 Recent Computational Methodology 81

82 Recent Computational Methodology 82

83 Fluid-Structure Interaction

84 Recent Computational Methodology 1 way FSI vs Two way FSI 84

85 Fluid-Structure Interaction

86 Fluid Structure Interaction

Table of Contents. Preface... 13

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