Advanced Methods of Structural Analysis

474 13 Stability of Elastic Systems

equilibrium. This occurs if the determinant, which is consisting of coefficients of
unknowns, equals zero, i.e.,

det

2

6

6

4

r 11 . / r 12 . /  r1n. /

r 21 . / r 22 . /  r2n. /

   

rn1. / rn2. /  rnn. /

3

7

7

5 D^0 (13.12)

Condition (13.12) is called the stability equation of a structure in the form of
displacement method. For practical engineering, it is necessary to calculate the min-
imum root of the above equation. This root defines the smallest parameterof
critical force or smallest critical force.
It is obvious that condition (13.12) leads to transcendental equation with respect
to parameter. Since the functions'./and
./are tabulated (Tables A.24 and
A.25), then a solution of stability equation may be obtained by graphical method.
Since the determinant is very sensitive with respect to parameter, it is recom-
mended to solve the equation (13.12) using a graphing calculator or computer.
The displacement method is effective forstability analysis of stepped continuous
beams on rigid supports with several axial compressed forces along the beam and
for frames with/without sidesway.
Let us derive the stability equation and determine the critical load for frame
shown in Fig.13.13a. This frame has one unknown of the displacement method.
The primary unknown is the angle of rotation of rigid joint. Figure13.13bshows
the primary system, elastic curve, and bending moment diagram caused by unit
rotation of introduced constrain. The bending moments diagram for compressed
vertical member of the frame is curvilinear. The ordinate for this member is taken
from Table A.22, row 1.

P

l 1

l 2

EI 1

EI 2

Elastic curve

r 11

4 i 1 j 2 (u 1 ) 4 i

2

Z= 1

ab

Fig. 13.13 (a) Design diagram; (b) Primary system of the displacement method and unit bending
moment diagram

The bending moment diagram yieldsr 11 D4i 1 ' 2 . 1 /C4i 2 , where parameter
of critical load

 1 Dl 1

s

P

EI 1

: