Advanced Methods of Structural Analysis

468 13 Stability of Elastic Systems

The initial parameters equations (13.9) may be presented in equivalent form (sec-
ond form), i.e., in terms ofQ 0 , which is directed as perpendicular to theinitial
straight line of the rod (Fig.13.9).
According to equilibrium equation

P
Y D 0 (axisY is directed alongQ 0 )
we have
Q 0 DQ 0 cos' 0 CPsin' 0 ŠQ 0 CP' 0 :

Then, substitution of this expression in (13.9) gives the initial parameters equations
in terms ofQ 0

y.x/ Dy 0 C' 0

sinnx

n

M 0

1 cosnx

P

Q 0

nxsinnx

nP

;

y^0 .x/D' 0 cosnxM 0

nsinnx

P

Q 0

1 cosnx

P

;

M.x/D' 0 nEIsinnxCM 0 cosnxCQ 0

sinnx

n

;

Q.x/DQ 0 :

(13.10)

In second form (13.1), the shear along the beam is constant, since an external lateral
load is absent. Both forms (13.9)and(13.10) are equivalent.
Now we illustrate the application of (13.10) for stability analysis of the stepped
fixed-free column shown in Fig.13.10; the upper and lower portions of the column
bel 1 andl 2 , respectively. The bending stiffness for both portions areEI 1 andEI 2.
The column is loaded by two compressed axial forcesP 1 DPandP 2 DˇP,where
ˇis any positive number. It means that growth of all loads up to critical condition of
a structure occurs in such way that relationships between all loads remain constant
(simple loading).

P 1 =P

P 2 =bP

l 2

l

EI 1

EI 2

P

P 2

Initial parameters for first portion

j 0

x

(^01)
l 1
(^02)
j 1 ≠ 0, M 1 ≠ 0, Q 1 = 0
j 0 ≠ 0, M 0 = 0, Q 0 = 0
Initial parameters for second portion
Fig. 13.10 Stepped clamped-free column
Let the origin 01 is located at the free end (Fig.13.10). Initial parameters (at free
end) for upper portion are' 0 Dy 00 ¤0; M 0 D 0 IQ 0 D 0. The slope and bending
moment at the end of the first portion (at thexDl 1 ) according to (13.10)are
' 1 .xDl 1 /D' 0 cosn 1 l 1 ;
M 1 .xDl 1 /D' 0 n 1 EI 1 sinn 1 l 1 ;
n 1 D
r
P 1
EI 1
:
(a)