Mechanics of Thin Structure

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Mechanics of Thin Structure

• Lecture 15 Wrapping Up the Course
• Shunji Kanie

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Mechanics of Thin Structure
What you learned are
Introduction for Linear Elasticity Stress and
Strain with 3D General Expressions Plane Stress
and Plane Strain Principle of Energy Principle of
Virtual Work Calculus of Variations Theory of
Beams Theory of Plates
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ExamplePure bending problem
Solve the problem
Equilibrium
Compatibility
Governing Equation
Energy
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ExamplePure bending problem
Solve the problem
Equilibrium
Equilibrium
Compatibility
Vertical
Moment
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ExamplePure bending problem
Solve the problem
Equilibrium
Equilibrium
Compatibility
X direction
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ExamplePure bending problem
Solve the problem
Equilibrium
Compatibility
Compatibility
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ExamplePure bending problem
Solve the problem
Equilibrium
Compatibility
Governing Equation
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ExamplePure bending problem
Compatibility
Energy
Energy should be Minimum, so that Energy is
calcualted such as
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ExamplePure bending problem
Energy for section
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ExamplePure bending problem
Potential Energy
You have the Functional !!
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ExamplePure bending problem
You have the Functional !!
Apply the Eulers Equation
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ExamplePure bending problem
Governing Equation for pure bending beam
from the Eulers Equation
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ExamplePure bending problem
Boundary Condition
See eq. (6-31)
Shearing force
Bending Moment
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ExamplePure bending problem
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ExampleSolution 1 Direct Integration
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ExampleSolution 2 Virtual Work
Equilibrium
No Energy produced if the Equilibrium is satisfied
Virtual Displacement
Virtual Work
What is the requirement for the virtual
displacement?
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Example Solution 2 Virtual Work
Assumption
Virtual Displacement
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Example Solution 2 Virtual Work
Assumption
Virtual Displacement
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Example Solution 2 Virtual Work
Assumption
Virtual Displacement
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ExampleSolution 2 modified
Assumption
Virtual Displacement
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ExampleSolution 2 modified
Fourier Transform
Its same as the solution introduced for Plate
Theory
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ExampleSolution 3 Point Collocation
Assume you would like to have the value at the
center
Dirac Delta Function
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ExampleSolution 3 Point Collocation
Assumption
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ExampleSolution 4 Least Square Method
Equilibrium
Error should be minimum
Assumption
Estimated
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ExampleSolution 4 Least Square Method
Error
Squared Error
In order to expect the Error minimized, an
appropriate value for a must be
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ExampleSolution 4 Least Square Method
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Example Solution 2 Virtual Work
Assumption
Virtual Displacement
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ExampleSolution 4 Least Square Method
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ExampleSolution 5 Galerkin Method 1
Starting Equation is Same
Assumption
Weighting Function
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ExampleSolution 5 Galerkin Method 2
Starting Equation is Same
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ExampleSolution 5 Galerkin Method 2
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ExampleSolution 5 Galerkin Method 2
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ExampleSolution 5 Galerkin Method 2
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ExampleSolution 5 Galerkin Method 2
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ExampleSolution 5 Galerkin Method 3
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ExampleSolution 5 Galerkin Method 3
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ExampleSolution 5 Galerkin Method 3
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ExampleSolution 5 Galerkin Method 3
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Mechanics of Thin Structure
What you learned are
Introduction for Linear Elasticity Stress and
Strain with 3D General Expressions Plane Stress
and Plane Strain Principle of Energy Principle of
Virtual Work Calculus of Variations Theory of
Beams Theory of Plates