Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 710

11.5.4 Determination of the Buckling Loads Using Other Approximate Methods

There are several approximate methods of interest for finding a critical load. These include
a simple iteration procedure, finite differences, and a weighted-residual approach.

An Iteration Technique (Stodola's, 2 Picard's or Stodola-Vianello's Method)
The fourth order di ff erential equation for a beam with a compressive axial force P , but no
transverse loading p z , is given by [Eq. (11.31)]

v

w =

i

EI w + P 0

For constant EI and P , integration gives

w + EI P w = C 1 + C 2 x

With the introduction of redefined constants C 1 and C 2 this expression can be rewritten as

P
EI (w +

w =−

C 1 + C 2 x )

Integration gives

x x

0 (w +

C 4

P
EI

w =− C 1 + C 2 x ) dx dx + C 3 x + (11.70)

0

An iteration scheme can be designed, which converges after a number of iterations to
a “good” approximation, beginning with an assumed w on the right-hand side. From

Eq. (11.70) for the ( n − 1)th iteration

x x

0 (w

C 4

P cr
EI

n − 1 n − 1

w =− + C 1 + C 2 x ) dx dx + C 3 x + (11.71)

0

Define

x x

0 (w

C 4

n n − 1

w =− + C 1 + C 2 x ) dx dx + C 3 x + (11.72)

0

From Eqs. (11.71) and (11.72)

P cr
EI w

n − 1 n

w

= (11.73)

which provides an approximation for the eigenvalue P cr , i.e.,

EI w

n − 1

P cr = (11.74)

w

n

the constants C i ,i = 1 , 2 , 3 , 4 are determined at each iteration step using the boundary

conditions.

2 Aurel Stodola (1859-1942) was a Hungarian-born engineer, specializing in the development of steam and gas
turbines. He served on the faculty of the University of Zurich, Switzerland, for more than 35 years.