Advanced Methods of Structural Analysis

470 13 Stability of Elastic Systems

P 1 crD

2:0678EI 1

l^2

:

Summary. Initial parameters method may be applied for stability analysis of the
stepped columns subjected to several forces. In this case, the origin should be shifted
for each following portion. Initial parameters for each following portion coincide
with parameters at the end of the previous portion.

Example 13.3.Design diagram of the beam with elastic support is presented in
Fig.13.11. The flexural stiffness of the beam isEI. Stiffness coefficient of elastic
support isk[kN m/rad]. Derive the stability equation.

x

y(x)

k

x

l

j 0

M 0

P P

RR

Fig. 13.11 Beam with elastic support

Solution.Suppose that the section at the left end of the beam is rotated clockwise.
Corresponding reactionsM 0 (which arise in elastic support) andRof the beam are
shown in Fig.13.11. The lateral displacement of the beam according to the first
equation of the system (13.10)

y.x/D' 0

sinnx

n

M 0

1 cosnx

P

Q 0

nxsinnx

nP

: (a)

Initial parameters areM 0 Dk' 0 , and shear

Q 0 DRD

k' 0

l

:

Their signs are accepted according to Fig.13.9. Thus, (a) may be rewritten as

y.x/D' 0

sinnx

n

Ck' 0

1 cosnx

P



k' 0

l

nxsinnx

nP

: (b)

Boundary condition: atxDlyD 0. Therefore

y.l/D' 0



sinnl

n

Ck

1 cosnl

P



k

l

nlsinnl

nP

D0:

Since' 0 ¤ 0 ,then

sinnl

n

Ck

1 cosnl

P



k

l

nlsinnl

nP

D0:

This expression leads to the following stability equation

tannlD

nl

n^2 l^2 ̨C 1

; (c)