Advanced Methods of Structural Analysis

13.4 Stability of Continuous Beams and Frames 481

Stability equation becomes
1 . /D 0 and the lowest root isD=2.The
critical load is equal to

PcrD

2 EI

4h^2

I

This case corresponds to single fixed-free column. For second form of buckling
we getPcrD 2 EI=.0:7h/^2 as for simple fixed-pinned column.
For case of absolutely rigid cross bar, the lowest critical load also corresponds
to the first form. This form is characterized by horizontal displacement of cross bar,
while the fixed joint have not angular displacements because for cross-barEID1.
In this case the unit horizontal reaction equals

r 11 Dk

12i

h^2

2 . / :

Stability equation becomes
2 . /D 0 and the lowest root isD. The critical
load is equal to

PcrD

2 EI

h^2

I

This case corresponds to single column with fixed ends while the one fixed sup-
port permit the horizontal displacement. For second form of buckling, we get
PcrD 42 EI=h^2 as for fixed-fixed column.
We can see that for these regular frames, a critical loadPdoes not depend of the
number of columnskand lengthlof each span.

13.4.3 Modified Approach of the Displacement Method

In general case, the displacement method requires introducing constraints, which
prevent to angular displacement of rigid joints and independent linear displacements
of joints. However, in stability problems of a frame with sidesway, it is possible
some modification of the classical displacement method. Using modified approach,
we can introduce a new type of constraint, mainly the constraint, which prevents
to angular displacement, butsimultaneouslyhas a linear displacement. This type
of constraint is presented for pinned-clamped and clamped-clamped members in
Table A.22, line 3.
Example below presents analysisof the frame with sidesway (Fig.13.18a) us-
ing two approaches. The first approach corresponds to classical primary system of
the displacement method; the primary system containstwointroduced constraints,
one of which prevents angular and another prevents linear displacements. The sec-
ond approach corresponds to modified primary system of the displacement method;
the primary system containsoneintroduced constraint, which prevents angular dis-
placement and allows linear displacement. We will derive the stability equation
using both approaches and calculate the critical load.

First approach. The primary unknowns are angle of rotation of rigid joint and linear
displacement of a cross-bar. Figure13.18b shows the primary system, elastic curve,