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Introduction to FE Analysis

Tutorial CD

Introduction to FE Analysis

Tutorial CD

Buckling of a simply supported isotropic panel, subject to an end load.

be part of a vehicle outer skin, a panel in a piece of furniture or a section of aircraft

wing skin. A schematic of the structure, applied loading and boundary conditions is

shown below.

There are various classical solutions for this problem, based on assumed displacement

of the plate. Roark, Megson are typical references.

The solution to the actual buckled shape is controlled by the number of assumed

wavelengths in each direction. By convention we have

the load is assumed to be applied and

picture below shows an FE analysis in which a buckled shape is showing 3 half

wavelengths in the loaded direction and 1 half wave length in the transverse direction.

The classical solutions establish an equation for the critical buckling load in terms of

and

1, 2, 3 etc.) to see what is the lowest critical buckling load that can be found.

In the case of the square panel above, the lowest solution is with m = 1. M = 3 would be

achieved if the panel were snubbed or guided in some way such that m = 1 or 2 could

not be achieved. We will see shortly how the FE analysis is able to find a full range of

buckling mode shapes.

The Roark equation for the critical buckling load is:

The parameter k, is a factor to allow for various (m,n) corrections to be made based on

the aspect ratio a/b of the plate. In this case the user looks up the term k from a table

as below.

The dimensions of the plate used in this example a re a = 24 inches, b = 24 inches.

Thickness is 0.1 inches.

Young's Modulus, E is 1.07E7 lbs/in^2

Poisson's raitio is 0.3

The aspect ratio in this case is 2.0, so the parameter is 3.29.

This gives a critical buckling load of 1646 lbf for the plate.

The first buckled mode shape from the FE analysis shown below. It matches the Roark

solution with a critical buckling load of 1645.8 lbf.

The method that an FE program such as Nastran uses to establish the critical buckling

loads and mode shapes is via an Eigen-Value analysis. This is usually associated with a

normal modes solution, wher we use the structural stiffness and mass matrices. It will be

seen later that the buckling problem can be set up as a very similar eigen-value

problem, where the traditional stiffness matrix and the geometric stiffness matrix used.

To establish the geometric stiffness matrix terms we need a perturbation of the basic

structural analysis. The FE solution is therefore in two parts. A static solution, with

known loading is applied. An eigen value solution is then used to establish the scaling

factor on the applied load that will occur at critical buckling.

In the first case above a load of 10,000 lbs was applied and the first eigen value was

0.16458. This scales the load to the expected critical buckling load of 1545.8 lbs.

It is useful to look at the family of eigen values ( load scaling factors) and eigen vectors

( buckled mode shapes).

Megson provides an expanded expression for the critical buckling load, which allows the

user to input any combination of m and n to find a solution.

where

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