13.3 Stability of Columns with Rigid andElastic Supports 465
presents the columns with specified supports and corresponding stability equations.
The stability equations for cases 4–6 contain dimensionless parameter ( ̨orˇ); the
roots for these cases can be calculated numerically for specified ̨(orˇ).
Table 13.1 Limiting cases for columns with elastic supports (thelength of columnland flexural
stiffnessEI)
Case
k 1 =0
1k 2 =∞k 1 =∞k 2 =∞(^2) k 1 =∞
k 2 =0
3
Stability equation cosnlD 0 tannlDnl tannlD 0
Root.nl /min =2 4.493
Case k
1 =0
k 2
(^4) k 1 =∞
k 2
5
k 1
k 2 =∞
6
Stability equation
tannlD
1
nl ̨
̨D
EI
k 2 l
tannlD nl
n^2 l^2 ̨C 1
;
̨D
EI
k 2 l
tannlDnl.1n^2 l^2 ˇ/
ˇD
EI
k 1 l^3
Limiting case.For case 4, the stability equation
tannlD
1
nl ̨
; ̨D
EI
k 2 l
may be presented as
tannl
nl
D
k 2
Pl
:
For absolutely rigid bodyEID1,and
nD
r
P
EI
!0:
Since
lim
nl!o
tannl
nl
D 1
then a critical force becomes
PcrD
k 2
l
:
This result has been obtained in Sect. 13.2.1.