Mechanics of Thin Structure Lecture 15 Wrapping Up the Course Shunji Kanie

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

Example Pure bending problem Solve the problem Equilibrium Compatibility Energy Governing Equation

Example Pure bending problem Solve the problem Equilibrium Compatibility Vertical Moment Equilibrium

Example Pure bending problem Solve the problem Equilibrium Compatibility X direction Equilibrium

Example Pure bending problem Solve the problem Equilibrium Compatibility Compatibility

Example Pure bending problem Solve the problem Equilibrium Compatibility Governing Equation

Example Pure bending problem Energy should be Minimum, so that Energy is calcualted such as; Compatibility Energy

Example Pure bending problem Energy for section

Potential Energy You have the Functional !!

You have the Functional !! Apply the Euler’s Equation

Governing Equation for pure bending beam from the Euler’s Equation

Shearing force Boundary Condition See eq. (6-31) Bending Moment

Example Pure bending problem Mechanical Boundary Condition Geometrical Boundary Condition or

Example Solution 1 : Direct Integration

Example Solution 2 : Virtual Work Equilibrium No Energy produced if the Equilibrium is satisfied Virtual Displacement Virtual Work What is the requirement for the virtual displacement?

Example Solution 2 : Virtual Work Virtual Displacement Assumption

Example Solution 2 : Virtual Work Virtual Displacement Assumption

Example Solution 2 : Virtual Work Virtual Displacement Assumption

Example Solution 2 modified Virtual Displacement Assumption

Example Solution 2 modified It’s same as the solution introduced for Plate Theory Fourier Transform

Example Solution 3 :Point Collocation Dirac Delta Function Assume you would like to have the value at the center

Example Solution 3 :Point Collocation Assumption

Example Solution 4 :Least Square Method Error should be minimum Assumption Equilibrium Estimated

Squared Error Error In order to expect the Error minimized, an appropriate value for a must be

Example Solution 4 :Least Square Method

Example Solution 2 : Virtual Work Virtual Displacement Assumption

Example Solution 4 :Least Square Method

Example Solution 5 :Galerkin Method 1 Starting Equation is Same Weighting Function Assumption

Example Solution 5 :Galerkin Method 2 Starting Equation is Same

Example Solution 5 :Galerkin Method 3

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