Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 708

Use the Hermitian polynomials as assumed displacements and find the stiffness matrix for
this single element system to be

k lin + k geo

K =

. . − .

6 β 2 β 6 β 4 β 12 − − 6 + 12 + − 6 +

βλ . +

. +

βλ .

− 1 . 2 λ + 0 . 6 0 . 1 λ − 0 . 1 βλ 1 . 2 λ − 0 . 6 + 0 . 1 λ

········· · ········· · ········· · ·········

. . .

2 β − β 2 β − β − 6 + 4 6 − 2

βλ . − 30 . − βλ . + λ/

+ 0 . 1 λ − 0 . 1 2 λ/ 15 + βλ/ 0 . 1 λ + 0 . 1 30 − βλ/ 60

EI 0
L 3

= ········· · ········· · ········· · ········· (4)

. . .

6 β 2 β 6 β 4 β − 12 + 6 − 12 − 6 −

βλ . −

. −

βλ .

+ 1 . 2 λ − 0 . 6 0 . 1 λ + 0 . 1 βλ 1 . 2 λ + 0 . 6 − 0 . 1 λ

········· · ········· · ········· · ·········

. . .

4 β − β 4 β 3 β

− 6 + 2 6 − 4 −

. + λ/

60 .

. −

+ 0 . 1 λ 30 − βλ/ − 0 . 1 λ 2 λ/ 15 + 0 . 1 βλ

2

P 0 L 2 where EI 0 .Tofind the buckling load, apply the boundary conditions to

form a reduced K and solve det K

λ = ε = /

0. This, if desired, can be represented for this single
element model in two terms in the form
det k lin +

=

k geo =

0 (5)

where k lin and k geo are matrices reduced by application of the boundary conditions.

EXAMPLE 11.14 A Stepped Column
The stepped column of Fig. 11.42a will be used to illustrate several of the techniques for
computing buckling loads for a structural system. The boundary and in-span conditions
are w a = w c = w d = 0 ,M a = M d = 0.

FIGURE 11.42
A stepped column.