Page 166 - Marine Structural Design

142 Part XI Ultimate Strength

Then the deflection w, due to the initial deformation is determined from the differential
equation:

EI- - (8.4)

Substituting Eq. (8.1) into Eq. (8.4), we may obtain:
d2wl
m
-+ k2w, = -k2w,,, sin-
dX2 I
where,

The general solution of Eq. (8.5) is:
1 .m
wI = Asinkx+Bcoskx+- w,,, sin -
?r2
-- 1 I
k212
To satisfy the boundary condition (w, = 0 for x = 0 and x = 1)for any value of k, A =
B = 0. Also, by using the notationa for the ratio of the longitudinal force to its critical value:

where,
Z’EI
PE =-
I2
we obtain the following:
a m
w, =- wonax sin -
1-a I
The final form of the deflection curve is:

m
a
.m
w=wo+wl =w,,,sin-++w,,,sin-=~sin- m
I 1-a I 1-a I (8.9)
This equation shows that the initial deflection w,,, at the middle of the column is magnified
a
at the ratio - the action of the longitudinal compressive force. When the compressive
by
1-a
force P approaches its critical value, a approaches 8.0, the deflection w increases infinitely.
Substituting Eq. (8.9) into Eq. (8.3), we obtain:
(8.10)