9762

Navier Solution for a Singularly Loaded Kirchhoff Rectangular Plate

This Demonstration illustrates the approximate Navier solution for the bending and stress analysis of a simply supported Kirchhoff rectangular plate subject to a concentrated singular load applied at given coordinates. The deformation of the middle surface is represented on a 3D plot, colored according to its von Mises equivalent stress.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Start from the classical equation for pure bending of a simply supported Kirchhoff plate with dimensions , Young modulus , and Poisson modulus subject to a load :
(flexural rigidity).
Assume a general sinusoidal load of the form
.
Assume a solution for the deformation of the form
.
Consider the boundary conditions of the simply supported plate:
at ,
at .
Then you get the following general solution:
.
In this case, the load is a singular one of value applied at coordinates :
.
Follow Navier's procedure by expanding the load as a Fourier series:
.
Solving for (using orthogonality) you get
.
Introducing the coefficients in the prior general solution and applying the principle of superposition, you can derive a general expression for the deformation under unitary load () applied at coordinates :
.
You can then calculate the deformation , multiplying by the load value :
.
Now you can compute the stresses at the top of the plate ), following the classic theory of Kirchhoff plates:
,
,
.
You can then find the two principal stresses (Kirchhoff theory assumes plane stress):
Finally, you can obtain the von Mises equivalent stress:
Reference
[1] S. P. Timoshenko, Theory of Plates and Shells, New York: McGraw–Hill, 1959.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+