Advanced Methods of Structural Analysis

476 13 Stability of Elastic Systems

The axial compressed forces in both spans are equal, so the dimensionless pa-

rameters 1 and 2 for both spans are

 1 Dl 1

s

P

EI 1

I  2 Dl 2

s

P

EI 2

:

Let the left span is considered as the basic member, so 1 D. In this case, the

parameter 2 can be presented in term of the basic parameteras follows

 2 D 1

l 2

l 1

s

EI 1

EI 2

D ̨; ̨D

l 2

l 1

s

EI 1

EI 2

:

The primary unknown of the displacement method is the angle of rotation at the

intermediate support. The primary system and bending moment diagram due to unit

rotation of the introduced constrain are presented in Fig.13.14b. Since both spans

are compressed, then the bending moment diagrams are curvilinear. According to

Table A.22, the moment for clamped-pinned beam in case of angular displacement

has the multiple' 1 . /. Unit reaction, which arises in introduced constraint, is

r 11 . /D

3 EI 1

l 1

' 1 . 1 /C

3 EI 2

l 2

' 1 . 2 /D

3 EI 1

l 1

' 1 . /C

3 EI 2

l 2

' 1. ̨ / :

As before, the subscript 1 at the function' 1 . /reflects thetype of the beamandtype

of displacement(Table A.22), while subscripts 1 and 2 at the parameterdenote

the number of the span.

Canonical equation of the displacement method isr 11 . /ZD 0. Nontrivial so-

lution of this equation leads to equation of a critical force

r 11 . /D

3 EI 1

l 1

' 1 . /C

3 EI 2

l 2

' 1. ̨/D0:

Special cases:

1.Assumel 2! 0. In this case, intermediate pinned support is transformed
intoclampedsupport and initial beam becomes one-span pinned-clamped beam
lengthl 1. Stability equation becomes' 1 . 1 /D1. Root of this equation is
 1 D4:488and critical force

PcrD

 12 EI

l^21

D

4:488^2 EI

l 12

D

2 EI

.0:7l 1 /^2

;D0:7

2.Assumel 1 Dl 2 DlandEI 1 DEI 2 DEI. In this case, parameter ̨D 1 ,the
stability equation becomes' 1 . /D 0 , and parameter of critical loadD.
So the critical loadPcrD
2 EI
l^2

. This critical load corresponds to column with
pinned-rolled supports.