Advanced Methods of Structural Analysis

472 13 Stability of Elastic Systems

As earlier, we need to have the reactions of standard elements, which are subjected
to unit settlements of constraints and axial compressed load. Calculation of reactions
for typical member is shown below.

P

l

P

Q 0 =RA

EC^1 M

A

B

j 0

Fig. 13.12 Reaction of compressed beam subjected to unit angular displacement at supportB

The pinned-clamped beam is subjectedto unit angular displacement of supportB
and axial compressed forceP. The length of the beam isl;EIDconst. Figure13.12
presents the elastic curve (EC) and positive unknown reactionsRandM.Fortheir
determination we can use the initial parameters method.
The origin is placed at supportA.Sincey 0 D 0 andM 0 D 0 ,then(13.10)
becomes

y.x/D' 0

sinnx

n

Q 0

nxsinnx

nP

;

y^0 .x/D' 0 cosnxQ 0

1 cosnx

P

:

For calculation of two unknown initial parameters' 0 andQ 0 DRwe have two
boundary conditions:y.l/D 0 andy^0 .l /D 1 , therefore

y.l/D' 0

sinnl

n

Q 0

nlsinnl

nP

D0;

y^0 .l /D' 0 cosnlQ 0

1 cosnl

P

D1:

Solution of these equations is

Q 0 D

3i

l

^2 tan

3.tan/

;

whereiDEI=l; the dimensionless parameter of critical force is

DnlDl

r

P

EI

:

SinceQ 0 is negative, then reactionRAis directed downward. Reactive moment at
supportBis

MDRAlD3i

^2 tan

3.tan/

:

The negative sign means that the reactive moment acts clockwise; indeed direction
of moment coincides with direction of angular displacement of clamped support.