Theory Of Plates

This course deals with the theory of plate bending; rectangular and circular plates; energy methods and numerical method

What you’ll learn
Understand the differential equation of equilibrium and the boundary conditions for rectangular and circular plates under pure and cylindrical bending
Calculate the critical loads and the buckling modes of simply supported rectangular plates under different edge conditions and compression directions
Solve the differential equation of equilibrium for simply supported rectangular plates under various loading conditions using Navier’s and Levy’s methods
Apply the finite difference method and the Rayleigh-Ritz method to approximate the deflections and stresses of rectangular plates with different boundaries

Requirements
Basic knowledge of elasticity, energy principles, and classification of various plate theories. Familiarity with variational calculus and differential equations.

Description
The course covers the following topics:Bending of Rectangular Plates: Pure and Cylindrical bending, differential equation, cylindrical bending of uniformly loaded rectangular plates with simply supported and built-in edges. Relations between slope and curvature of slightly bent plates, Moment-curvature relations in pure bending. Strain energy in pure bending.Bending of circular plates: Symmetrical bending, differential equation of equilibrium, uniformly loaded plates at center, Circular plates with circular holes at the center.Buckling of Plates: Differential equation for bending of plate under the combined action of in-plane loading and lateral loading, Calculation of critical loads, buckling of simply supported rectangular plates uniformly compressed in one and two directions with different edge conditions.Small deflections of laterally loaded plates: Differential equation of equilibrium, Boundary conditions, Solution of simply supported rectangular plates under various loading conditions viz. uniformly distributed load (full or partial), concentrated load by Navier’s approach, Levy type solution for rectangular plates under U.D.L with all four edges simply supported or two opposite edges simply supported and other two fixed.Approximate methods for Rectangular Plates: Finite difference method for simply supported or fixed rectangular plates carrying UDL (full or partial) or central point load, Strain energy approaches Rayleigh-Ritz method.

Overview
Section 1: Bending of Rectangular Plates

Lecture 1 Introduction, Differential equation for cylindrical bending of plates

Lecture 2 Cylindrical bending of a uniformly loaded rectangular plate

Lecture 3 Slope, curvature & moment-curvature relations, Strain energy in pure bending

Section 2: Bending of Circular Plates

Lecture 4 Circular plates – basic relations, differential equation of equillibrium

Lecture 5 Deflections and bending moments of uniformly loaded circular plates

Lecture 6 Annular plates with simply supported outer edges

Section 3: Buckling of Plates

Lecture 7 Governing equation, Deflection of plate under combined loading

Lecture 8 Critical buckling load and stress in a plate under uniaxial and biaxial compress

Section 4: Small deflections of laterally loaded plates

Lecture 9 Differential equation for small deflection of plates, boundary conditions

Lecture 10 Deflection of plates due to sinusoidal loading, Navier’s solution for deflection

Lecture 11 Navier’s solution for patch and point loads, Green’s function, Levy’s solution

Lecture 12 Rectangular plate with two opposite edges clamped

Section 5: Approximate methods for Rectangular Plates

Lecture 13 Solution of a rectangular plate using Ritz method

Lecture 14 Finite Difference Method – Introduction, solution for deflection of a plate

Lecture 15 Graphical representation of finite difference equations, deflection of a plate

Intended for ME or MTech Structural Engineering students