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
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