finite element analysis lecture 1 - ahmed...
TRANSCRIPT
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Finite Element AnalysisLecture 1
Dr./ Ahmed Nagib
April 30, 2016
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Research and Development
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Mathematical Model
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Mathematical Model
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Mathematical Model
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Finite Element Analysis
The linear equation of motion for vibration is
𝑴 ሷ𝒙 + 𝑪 ሶ𝒙 + 𝑲 𝒙 = 𝑭
• 𝒙 , which is the displacement vector• ሷ𝒙 , 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
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Finite Element Softwares
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Finite Element Analysis
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Finite Element Analysis
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Finite Element Analysis
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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
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Axial Stress
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Beam under the action of two tensile forces
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Beam under the action of two tensile forces
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Torsion Stress
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Torsion Stress
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Torsion Stress
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Angle of Twist
19
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Torsion of a Shaft with Circular Cross-Section
20
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Torsion of a Shaft with Circular Cross-Section
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Torsion of a Beam with the Square Cross-Section
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Torsion of a Beam with the Square Cross-Section
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Bending Stress
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Bending Stress
25
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Bending a Cantilever Beam under a Concentrated Load
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Bending a Cantilever Beam under a Concentrated Load
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Bending Stress
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Bending Stress
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Bending Stress
30
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Bending Stress
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Bending Stress
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Bending Stress
33
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Bending Stress
34
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Bending of Curved beam
35
Displacement Stress in x direction
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Finite Element Analysis
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Finite Element Analysis
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Static Analysis
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Static Analysis
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Static Analysis
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Static Analysis
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Static Analysis
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Static Analysis
Linear vs Non Linear solveIn 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
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Modal AnalysisThe linear equation of motion for free, un-damped vibration is
𝑴 ሷ𝒙 + 𝑲 𝒙 = 𝟎Assume harmonic motion:
𝒙 = 𝝓 𝒊 sin 𝜔𝑖𝑡 + 𝜃𝑖ሷ𝒙 = −𝜔𝒊
𝟐 𝝓 𝒊 sin 𝜔𝑖𝑡 + 𝜃𝑖
Substituting 𝒙 and ሷ𝒙 𝐢n the governing equation gives an eigenvalue equation:
−𝜔𝒊𝟐 𝑴 + 𝑲 𝝓 𝒊 = 𝟎
where𝜔𝑖: Natural Frequencies𝝓 𝒊 : Mode Shapes
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Modal Analysis
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Modal Analysis
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Modal Analysis
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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
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Modal AnalysisModal 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.
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Modal Analysis
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Modal Analysis
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Modal Analysis
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Modal Analysis
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Modal Analysis
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Modal Analysis
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Modal Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Fluid-Structure Interaction
Solid Mechanics-Structural Analysis Fluid Dynamics
Solved by Finite Element Analysis Computational Fluid Dynamics (CFD)
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Recent Computational Methodology
Finite Element Analysis Computational Fluid Dynamics (CFD)
CommercialSoftware
Ansys Mechanical, Abaqus Ansys Fluent, Ansys CFX, Open-foam
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Recent Computational Methodology
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Recent Computational Methodology
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Fluid-Structure Interaction
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Recent Computational Methodology
1 way FSI vs Two way FSI
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Fluid-Structure Interaction
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Fluid Structure Interaction