Advanced Methods of Structural Analysis

502 13 Stability of Elastic Systems

It is obvious thatr 12 D 0. The free-body diagram for the cross bar is presented in
Fig.13.25e.
The equilibrium equation

P

XD 0 for the cross bar becomes

r 22 D 2 0:01435EIC0:024EID0:0527EI.kN=m/

It is obvious that the free terms of canonical equations are

R1PD0;andR2PDFD 30 kN:

The canonical equations become

2 EIZ 1 C 0 Z 2 D0;

0 Z 1 C0:0527EIZ 2 FD0:

(b)

The roots of these equations are

Z 1 D0; Z 2 D

F

0:0527EI

D

569:25

EI

:

The bending moment diagram can be constructed using the principle of superposi-
tion:
MPDM 1 Z 1 CM 2 Z 2 CMP^0 (c)

Since Z 1 D 0 and acting loadPdoes not cause bending of the members, then
formula (c) becomesMPDM 2 Z 2. The bending moments at the clamped sup-
ports are

MADMCD0:1118EI

569:25

EI

D63:64 .kNm/;

MBD0:12EI

569:25

EI

D68:31 .kNm/:

:

The resulting bending moment diagram is presented in Fig.13.25f. The number in
parenthesis is bending moments calculated on the basis of the nondeformable design
diagram (Example 8.1,FD 30 kN).
We can see thatP-delta effect is significant. Increasing of the horizontal dis-
placementZ 2 is 36.6%.

13.6.4 Graph Multiplication Method for Beam-Columns

Vereshchagin rule for computation of displacements may be modified for case of the
uniform members subjected to any transversal load and compressive forceP.The
unit state should be created as usual. The bending moment diagrams in the actual
and unit states are plotted without compressive load. However, axial load is taking
into account by factors depending of axial compressive force. Two important cases
are presented below.