Advanced Methods of Structural Analysis

13.3 Stability of Columns with Rigid andElastic Supports 463

13.3.1.2 Uniform Columns with Elastic Supports

Now let us consider the compressed column with elastic supports at the both ends
(Fig.13.8a). The flexural stiffness of the column isEIDconstant; the stiffness co-
efficients of elastic supports arek 1 (kN/m) andk 2 (kN m/rad) for linear and angular
displacements, respectively.

y

x

P

l EI

A

B

k 1

k 2

x

y

f

y

x

P

RB=k 1 f

j

MA=k 2 j

RA

P

ab

Fig. 13.8 Column with elastic supports

The cross section at supportAcan rotate through angle'while the supportB
has a linear displacementf. Thus reactions of elastic supports are moment at the
supportsAand force at supportB. They are equalMADk 2 'andRBDk 1 f.
Deformable state, elastic curve,and all reactions are shown in Fig.13.8b.
The bending moment at any sectionxequals

M.x/DP.fCy/Ck 1 f.lx/ :

Substituting this expression into (13.4) leads to the buckling differential equation of
the column
EI

d^2 y

dx^2

CPyDfŒk 1 .lx/P: (13.7)

This is the second-order nonhomogeneous differential equation. In case ofk 1 D1,
we get homogeneous differential equation.
The partial solution of (13.7) isy. Substituting this constant into (13.7) leads to
the following expression:

yDf



k 1

P

.lx/ 1

:

The general solution of differential equation (13.7) and corresponding slope are

yDC 1 cosnxCC 2 sinnxCy;n^2 D

P

EI

;

dy

dx

Dy^0 DC 1 nsinnxCC 2 ncosnx

fk 1

P

: