Structural Design for Cold Region Engineering Lecture 14 Thory of Plates Shunji Kanie

Theory of Plates Kirchhoff Plate Pure Bending Kirchhoff Plate Such as Bernoulli Euler 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 Assumptions

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

Theory of Plates Kirchhoff Plate Kirchhoff Plate Displacement due to deflection Rotation angle

Theory of Plates Kirchhoff Plate Kirchhoff Plate Stress-Strain Relation Plane Stress !

Theory of Plates Kirchhoff Plate Bending Moments and Torsional Moment are calculated at least Sectional Force

Theory of Plates Kirchhoff Plate X direction y direction z direction

Theory of Plates Kirchhoff Plate z direction

Theory of Plates Kirchhoff Plate Governing Equation

Theory of Plates Kirchhoff Plate Governing Equation

Theory of Plates Kirchhoff Plate Introducing Laplacian

Theory of Plates Kirchhoff Plate Boundary Conditions Simple support Fixed support Free support Effective transverse shear Kirchhoff force

Theory of Plates Solution Simply Supported Plate Assuming Deformation Boundary Condition?

Theory of Plates Solution Simply Supported Plate Governing Equation Assumed Deformation

Theory of Plates Solution Simply Supported Plate Assumed Deformation Deformation

Theory of Plates Solution Simply Supported Plate Deformation Bending & Twisting Moments

Theory of Plates Solution Simply Supported Plate Bending & Twisting Moments If we are very LUCKY enough

Theory of Plates Solution 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?

Theory of Plates Solution Simply Supported Plate Apply Double Fourier Expansion for q

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

Theory of Plates Solution Plate supported like Assuming Deformation Single Fourier Expansion

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

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

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

Theory of Plates Solution Plate supported like Solve General Solution Characteristic Equation

Theory of Plates Solution Plate supported like General Solution Singular Solution should be constant such as

Theory of Plates Solution Plate supported like General Solution Singular Solution Solution

Theory of Plates Solution Difference Method

Theory of Plates Solution Difference Method Governing Equation Simple support Fixed support

Theory of Plates Solution Galerkin Method Governing Equation Assume Approximation Weighted Residual Same with Double Fourier