Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 696

FIGURE 11.31
The characteristic equation.

We find

2 EI
4 L 2

ε< 2 < π for L 1 > 0: or P cr

2 EI
L 2

2 . 05 π

for L 1 = 0: επ = 1 . 4303 π or P cr =

2 EI
L 2

2 EI
L 2

π 2 . 05 π

for − L < L 1 < 0: π<ε< 1 . 4303 π or < P cr <

(8)

2 EI
L 2

= π for L 1 =− L : ε = π or P cr

2 EI
4 L 2

2 EI
L 2

2 <ε<π π

P cr < π for L 1 < − L : or <

2 EI
4 L 2
The buckling load is plotted in Fig. 11.32a as a function of L 1 .
The buckled configurations are displayed in Fig. 11.32b. The buckled configurations of
columns with hinged ends, which have the same buckling forces as those of the original
structure of Fig. 11.30, are shown.

ε = 2 = π

for L 1 =±∞ : or P cr

EXAMPLE 11.9 Column of Variable Cross-Section
Find the buckling load for the stepped column of Fig. 11.33.
Since the transfer matrix for a column element has been established (Eq. 11.47), columns
formed of elements of different geometries are readily analyzed for the conditions of in-
stability. The transfer matrix method of analysis developed in Chapter 5 can be applied to
such columns as that of Fig. 11.33.