Advanced Methods of Structural Analysis

13.4 Stability of Continuous Beams and Frames 473

The expressions for unit reactions for standard members subjected to compressed

forcePand special types of displacementscan be derived similarly; for typical

uniform beams they are presented in Table A.22. In all cases, the length of the

beam isl, the flexural stiffness isEI, bending stiffness per unit length isiDEI=l.

Corresponding elastic curve is shown by dotted line; the graphs present the real

direction of reactions; the bending moment diagrams are plotted on the tensile fibers.

The row 3 of the Table A.22 presents the bending moment diagrams when a

clamped support rotates and switches in transversal direction simultaneously. While

the angle of rotation is fixed asZD 1 , the displacementdoes not require indi-

cation of its value. These cases may be used for analysis of compressed beams with

elastic supports. Thus, in case of frames with sidesway, the primary system is ob-

tained by introducing constraints, which prevents only angular displacements, and

bending moment diagramsshould be traced as for member with elastic supports.

It is recommended to show elastic curves and remember that bending moment has

one-sign ordinates.

Expressions of special functions in two forms are presented in Table A.24; the

more preferable is form 2. Also this table contains approximate presentation of these

special functions in the form of Maclaurin series. Numerical values of these func-

tions in terms of dimensionless parameterare presented in Table A.25.

13.4.2 Displacement Method........................................

Canonical equations of the displacement method for structure withnunknowns

Zj;.jD1;2;:::;n/are

r 11 Z 1 Cr 12 Z 2 CCr1nZnD0;

r 21 Z 1 Cr 22 Z 2 CCr2nZnD0;



rn1Z 1 Crn2Z 2 CCrnnZnD0:

(13.11)

Features of (13.11):

1.Since the forcesPiare applied only at the joints, then the canonical equations
are homogeneous ones.
2.Bending moment diagram caused by unit displacements of introduced constrains
within thecompressedmembers arecurvilinear. Reactions of constraints depend
on axial forces in the members of the frame, i.e., contain parameterof critical
load. If a frame is subjected to different forcesPi, then critical parameters should
be formulated for each compressed memberi^2 D
Pili^2
.EI/i and after that all of
these parameters should be expressed in terms of parameterfor specified basic
member. Thus, the unit reactions are functions of parameter, i.e.,rik. /.
The trivial solution.ZiD0/of (13.11) corresponds to the initial nondeformable
design diagram. Nontrivial solution.Zi ¤ 0/corresponds to the new form of