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Structural Design for Cold Region Engineering

Structural Design for Cold Region Engineering. Lecture 14 Thory of Plates Shunji Kanie. Theory of Plates Kirchhoff Plate. Kirchhoff Plate. Pure Bending. Such as Bernoulli Euler. Assumptions. Isotropic and homogeneous The thickness of the plate is thin

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Structural Design for Cold Region Engineering

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  1. Structural Design forCold Region Engineering Lecture 14 Thory of Plates Shunji Kanie

  2. Theory of PlatesKirchhoff Plate Kirchhoff Plate Pure Bending Such as Bernoulli Euler Assumptions Isotropic and homogeneous The thickness of the plate is thin (Comparatively to the length and width ) Linear filaments of the plate (Even after the deformation) Kirchhoff hypothesis

  3. Theory of PlatesKirchhoff Plate Kirchhoff Plate Length : a in x direction Width : b in y direction Thickness : h in z direction Deflection Rotation angle

  4. Theory of PlatesKirchhoff Plate Kirchhoff Plate Rotation angle Displacement due to deflection

  5. Theory of PlatesKirchhoff Plate Kirchhoff Plate Stress-Strain Relation Plane Stress !

  6. Theory of PlatesKirchhoff Plate Sectional Force Bending Moments and Torsional Moment are calculated at least

  7. Theory of PlatesKirchhoff Plate X direction y direction z direction

  8. Theory of PlatesKirchhoff Plate z direction

  9. Theory of PlatesKirchhoff Plate Governing Equation

  10. Theory of PlatesKirchhoff Plate Governing Equation

  11. Theory of PlatesKirchhoff Plate Introducing Laplacian

  12. Theory of PlatesKirchhoff Plate Boundary Conditions Simple support Fixed support Free support Effective transverse shear Kirchhoff force

  13. Theory of PlatesSolution Simply Supported Plate Assuming Deformation Boundary Condition?

  14. Theory of PlatesSolution Simply Supported Plate Governing Equation Assumed Deformation

  15. Theory of PlatesSolution Simply Supported Plate Assumed Deformation Deformation

  16. Theory of PlatesSolution Simply Supported Plate Deformation Bending & Twisting Moments

  17. Theory of PlatesSolution Simply Supported Plate Bending & Twisting Moments If we are very LUCKY enough

  18. Theory of PlatesSolution Simply Supported Plate If qmn is successfully calculated and we can have the solution Is there any good idea if q is uniformly distributed load?

  19. Theory of PlatesSolution Simply Supported Plate Apply Double Fourier Expansion for q

  20. Theory of PlatesSolution Simply Supported Plate When q is constant as q0 You can solve the problem for any shape of load distribution

  21. Theory of PlatesSolution Plate supported like Assuming Deformation Single Fourier Expansion

  22. Theory of PlatesSolution Plate supported like If q is constant in x direction m=1,3,5,…….

  23. Theory of PlatesSolution Plate supported like If q is linear in x direction m=1,2,3,4,…….

  24. Theory of PlatesSolution Plate supported like If q is constant in x direction m=1,3,5,……. If q is linear in x direction m=1,2,3,4,……. Solve

  25. Theory of PlatesSolution Plate supported like Solve General Solution Characteristic Equation

  26. Theory of PlatesSolution Plate supported like General Solution Singular Solution should be constant such as

  27. Theory of PlatesSolution Plate supported like General Solution Singular Solution Solution

  28. Theory of PlatesSolution Difference Method

  29. Theory of PlatesSolution Difference Method Governing Equation Simple support Fixed support

  30. Theory of PlatesSolution Galerkin Method Assume Approximation Governing Equation Weighted Residual Same with Double Fourier

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