Advanced Methods of Structural Analysis

13.6 Compressed Rods with Lateral Loading 491

DisplacementZ 1 D 1 and corresponding bending moment diagram is shown in
Fig.13.21d. The unit reaction

r 11 D3i 1 C4i 2 D0:5EIC0:5EID1:0EI:

The primary system subjected to external unknown coupleM is presented in
Fig.13.21e, soR1PDM. The canonical equation becomes 1 EIZ 1 MD 0.
If the angle of rotation Z 1 D 1 ,thenMDkD1:0EI. For given parametersRand
̨the stability equation (13.28) of the structure becomes

tann

6

D

n

cot

6

C



n^2  1



20

or tan.0:5236n/D

20n

33:64Cn^2

:

The root of this equationnD7:955. The critical load is

qcrD



7:955^2  1

EI

R^3

D62:28

EI

R^3

: (13.29)

According to Table13.6, the critical load for arch with fixed supports and for two-
hinged arch (the central angle in both cases is2 ̨D 60 ı)areqcrD73:3REI 3 and

qcrD (^35) REI 3 , respectively. Above calculated critical load (13.29) is located between
two limiting cases.

13.6 Compressed Rods with Lateral Loading

In the previous chapters, the bending structure had been analyzed on the basis of a
nondeformable scheme. However, the axial compressed forcePcreates additional
moment on the displacementsıdue to lateral load. Even if a small lateral load leads
to small lateral displacements, the compressed load on these small displacements
leads to additional moments and displacements. Influence of the compressed axial
force becomes especially significant for tall structures.
A straight member that is simultaneously subjected to axial compression and
lateral bending is called a beam-column. Often analysis for such structures is refers
asP-delta analysis. Other title of such analysis is analysis of a structure on the basis
of deformable scheme. This analysis is nonlinear one and it allows finding a more
real distribution of internal forces and deflections, while the analysis on the basis of
the nondeformable design diagram leads to theirs underestimating.
Two different methods for beam-columns analysis are presented below. They are
the double integration and initial parameters methods.