Advanced Methods of Structural Analysis

13.3 Stability of Columns with Rigid andElastic Supports 467

P

Q 0

φ 0

y 0

M 0

Q 0

Y

x

Fig. 13.9 Initial parameters of a beam

Constants Ci may be expressed in terms of kinematical initial parameter
(displacementy 0 and slope' 0 Dy^0 D ddyx) and static initial parameters (bending
momentM 0 DEIy 000 and shearQ 0 DEIy 0000 ) as follows

C 1 D

M 0

n^2 EI

D

M 0

P

;C 2 D

Q 0

n^3 EI

D

Q 0

nP

;C 3 D' 0 

Q 0

P

;C 4 Dy 0 

M 0

P

Substitution of these constantsCiin (13.8a) leads to the following expressions in
terms of initial parameters

y.x/ Dy 0 C' 0 xM 0

1 cosnx

P

Q 0

nxsinnx

nP

;

y^0 .x/D' 0 M 0

nsinnx

P

Q 0

1 cosnx

P

;

M.x/DM 0 cosnxCQ 0

sinnx

n

;

Q.x/DM 0 nsinnxCQ 0 cosnx:

(13.9)

These equations present the first form of the initial parameter method for com-
pressed columns. It can be seen that, in spite of the external lateral load being absent,
the shearQ.x/is variable along the column. It happens because (13.9)ispresented
in terms ofQ 0 , which is directed as perpendicular to the tangent of anelasticline
of the beam.
Let us calculate the critical load for uniform clamped-free column (Fig.13.7);
EIDconstant. The origin is placed at the clamped support. The geometrical initial
parameters arey 0 D 0 and' 0 D 0. The third equation of system (13.9) becomes

M.x/DM 0 cosnxCQ 0

sinnx

n

:

Since the bending moment for free end of the column.xDl/is zero, then

MlDM 0 cosnlCQ 0

sinnl

n

D0:

It is obvious thatQ 0 D 0 andM 0 ¤ 0 , therefore the stability equation becomes
cosnlD 0. This result had been obtained using integration of the differential equa-
tion (13.4).