24 Questions
What is the contribution of the twisting moment per unit length due to?
A couple of vertical forces of intensity Mxy
What happens to the vertical forces in the internal side of the element?
They balance each other
What is the equilibrium equation for Mxy?
Mxy + ∂Mxy/∂y dy = Mxy'
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
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)
What is the condition for the rectangular plate?
The plate is simply-supported along the sides
What is the solution method used for the rectangular plate?
Navier Solution
What is the load on the rectangular plate?
A transverse load p(x,y) across the plate
What is the primary approach used to solve the bending of a rectangular plate?
Fourier Series Approach
What is the purpose of expanding the load p(x,y) into a Fourier series?
To decompose the load into its harmonic components
What is the property of the Fourier series terms that allows us to compute the coefficient 𝑝 m n?
Orthogonality
What is the governing equation for the bending of a rectangular plate?
The biharmonic equation
What is the purpose of the boundary conditions in the plate bending problem?
To describe the physical constraints on the plate
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
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
What is the benefit of using the Fourier series approach to solve the plate bending problem?
It provides an exact solution
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
What is the assumption made about p(x,y) in the analysis?
p(x,y) is a known function
What is the purpose of stopping the expansion at certain terms in the Fourier series approach?
To evaluate the convergence of the series
What is the equation that must be satisfied for every (x,y)?
The field equation
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,…
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
What is the equation that describes the equilibrium of the plate in deformed configuration?
The equilibrium equation
What is the result of substituting the expression of w(x,y) into the field equation?
The desired solution is obtained
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.
Solve rectangular plate bending problems using a Fourier series approach. Learn how to multiply and integrate equations to obtain solutions.
Make Your Own Quizzes and Flashcards
Convert your notes into interactive study material.
Get started for free
