Rectangular Plate Bending: Fourier Series Approach

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Questions and Answers

What is the contribution of the twisting moment per unit length due to?

A couple of axial forces of intensity Mxy

A couple of horizontal forces of intensity Mxy

A couple of moment forces of intensity Mxy

A couple of vertical forces of intensity Mxy (correct)

What happens to the vertical forces in the internal side of the element?

They add up to a concentrated load

They balance each other (correct)

They cancel each other out

They create a twisting moment

What is the equilibrium equation for Mxy?

Mxy - ∂Mxy/∂y dy = Mxy'

Mxy + ∂Mxy/∂y dy = Mxy' (correct)

Mxy + ∂Mxy/∂y dy = 0

Mxy - ∂Mxy/∂y dy = 0

What is the result of considering the contribution of Mxy as produced by vertical forces?

Two concentrated loads will be generated at the end of the side Signup and view all the answers

What is the resultant force at the corners of the rectangular plate?

R = Mxy(x=0,a;y=0,b) + Myx(x=0,a;y=0,b) = 2Mxy(x=0,a;y=0,b) Signup and view all the answers

What is the condition for the rectangular plate?

The plate is simply-supported along the sides Signup and view all the answers

What is the solution method used for the rectangular plate?

Navier Solution Signup and view all the answers

What is the load on the rectangular plate?

A transverse load p(x,y) across the plate Signup and view all the answers

What is the primary approach used to solve the bending of a rectangular plate?

Fourier Series Approach Signup and view all the answers

What is the purpose of expanding the load p(x,y) into a Fourier series?

To decompose the load into its harmonic components Signup and view all the answers

What is the property of the Fourier series terms that allows us to compute the coefficient 𝑝 m n?

Orthogonality Signup and view all the answers

What is the governing equation for the bending of a rectangular plate?

The biharmonic equation Signup and view all the answers

What is the purpose of the boundary conditions in the plate bending problem?

To describe the physical constraints on the plate Signup and view all the answers

What is the significance of the terms 𝜕2 𝑤/𝜕𝑦 2 and 𝜕2 𝑤/𝜕𝑥 2 in the boundary conditions?

They represent the curvature of the plate Signup and view all the answers

What is the assumption made about the solution and the load p(x,y) in the Fourier series approach?

They can be expanded into a Fourier series Signup and view all the answers

What is the benefit of using the Fourier series approach to solve the plate bending problem?

It provides an exact solution Signup and view all the answers

What is the result of multiplying both terms of the equation by sin(m'πx/a)sin(n'πy/b) and integrating it on the domain 0 ≤ x ≤ a; 0 ≤ y ≤ b?

Only the terms having m = m', n = n' will survive Signup and view all the answers

What is the assumption made about p(x,y) in the analysis?

p(x,y) is a known function Signup and view all the answers

What is the purpose of stopping the expansion at certain terms in the Fourier series approach?

To evaluate the convergence of the series Signup and view all the answers

What is the equation that must be satisfied for every (x,y)?

The field equation Signup and view all the answers

What is the expression of p_mn in the case p(x,y) = p0?

p_mn = p0/(mn) with m, n = 1,3,5,7,… Signup and view all the answers

What is the purpose of the Fourier series approach in the solution of the rectangular plate in bending problem?

To find the solution in terms of a series Signup and view all the answers

What is the equation that describes the equilibrium of the plate in deformed configuration?

The equilibrium equation Signup and view all the answers

What is the result of substituting the expression of w(x,y) into the field equation?

The desired solution is obtained Signup and view all the answers

Study Notes

Rectangular Plate in Bending - Solution by a Fourier Series Approach

The rectangular plate in bending solution involves expressing the load p(x, y) as a Fourier series:
The solution involves multiplying both terms of the equation by sin(m'πx/a) sin(n'πy/b) and integrating over the domain 0 ≤ x ≤ a, 0 ≤ y ≤ b.
This yields the coefficient p mn, which can be obtained if p(x, y) is known.
In the case of p(x, y) = p0, the expression for p mn involves m, n = 1, 3, 5, 7,…

Equilibrium Equations of the Plate in Deformed Configuration

The twisting moment per unit length can be considered as due to a couple of vertical forces of intensity M xy and a brace dy.
This gives rise to an additional shearing component Q x'.

Bending/Twisting Equations of Motion - Boundary Conditions

By considering the contribution of M xy as produced by vertical forces, two concentrated loads will be generated at the end of the side in rectangular plates.
The boundary conditions reduce to:

Solution for a Rectangular Simply-Supported Plate under Transverse Loading (Navier Solution)

The solution involves using the governing equations and boundary conditions:
The boundary conditions simplify to:
The solution can be obtained using a Fourier series approach.

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Description

Solve rectangular plate bending problems using a Fourier series approach. Learn how to multiply and integrate equations to obtain solutions.