Page 170 - Marine Structural Design

146 Part II Ultimate Strength

1
M, = Pel - (8.25)
kl
cos -
2
(8.26)

Eq. (8.26) is called the secant formula. Taking the first two terms of the formula expansion:

(8.27)

and substituting Eq. (8.27) into Eq. (8.28), we obtain:

n2 e, P
(8.28)

8.2.2 Beam-Column with Initial Deflection and Eccentric Load
The deflection for a beam-column in Figure 8.4 may be obtained easily by superposition of Eq.
(8.9) and Eq. (8.23), the total deflection is:
[ (:
e
w=- Woma sin-+A cos --kx ) :] (8.29)
-cos-
1 -a kl
cos-
2
The maximum deflection occurs at the center of the beam-column:
(8.30)

The bending moment at any section x of the beam-column is:
M = P(e, + w)
r kl (: )I 1
m e (8.31)
=p w,,, sin-+Lcos --kr
1
cos -
2
and the maximum moment at the center of the beam-column is:

(8.32)

From Eq. (8.15) and Eq. (8.28), the maximum stress at the center of the beam-column is:

(8.33)