496 13 Stability of Elastic Systems
Each formula of system (13.37) contains of two parts. The first part takes into ac-
count the initial parameters of the structure. The second part depends on the lateral
load; these terms are denoted by symbol./and they are presented in Table13.7.
Table 13.7 Additional terms of (13.36);nD
q
P
EI
qa
xFa
xMa
xy nq (^2) EI
h.xa/ 2
2 C
cosn.xa/ 1
n^2
i
F
n^2 EI
h
.xa/sinn.xna/
i
nM (^2) EIŒ1cosn.xa/
nq (^2) EI
h
.xa/sinn.xna/
i
F
n^2 EIŒ1cosn.xa/
M
nEIsinn.xa/
M nq 2 Œ1cosn.xa/ Fnsinn.xa/ Mcosn.xa/
Q qnsinn.xa/ Fcosn.xa/ Mnsinn.xa/
Superposition principle fordifferent lateralloads is applicable for thesame com-
pressedload. For example, in case of several concentrated forcesFi, which are
applied atx Dai, and uniformly distributed loadqwithin the whole span, the
additional terms are
yD
1
n^3 EI
X
FiŒn .xai/sinn.xai/C
q
n^4 EI
n^2 x^2
2
Ccosnx 1
:
Note, that for given sectionx, only loads located to the left of sectionx, should be
considered.
Let us illustrate the initial parameters method for free-clamped beam subjected to
axial compressed forcePand lateral forceF(Fig.13.24). It is necessary to compute
the vertical and angular displacements atthe free end and bending moment at the
fixed support and to provide the numericalP-delta analysis.
Fig. 13.24 Free-clamped
compressed-bending beam
P
P
l
x
y
x
RB
F
B
y 0
According to Table13.7the terms, which take into account a lateral load are
yD
F
n^2 EI
.xa/
sinn.xa/
n
D
F
n^2 EI
x
sinnx
n
D
F
n^3 EI
.nxsinnx/ ;
D
F
n^2 EI
Œ1cosn.xa/D
F
n^2 EI
.1cosnx/ ; aD0; (13.38)
MD
F
n
sinn.xa/ :