Page 199 - Marine Structural Design

Chapter 9 Buckling and Local Buckling of Tubular Members 175

= 0-7 iW) (9.38)
G=1.21&

The axial strain in a tube wall fiber where local buckling has occurred may be expressed as:

E = E, + (1/2S)f(dWb/drfdr = + (7?/416/3)2 (9.39)
On the other hand, considering the equilibrium condition of a bending moment in a strip with
its unit width cut out fiom the tube wall, the following equation is obtained. (See Figure 9.16
(b)):
AF,6-2dM, =o (9.40)
The interaction between the strips is not considered when Eq. (9.40) is derived. According to
the assumptions previously mentioned, local buckling takes place in the plastic region.
Consequently, AF, and AM, should satisfy the hlly plastic interaction relationships, which
are expressed as:
A M,/M~ = 1 -(A F,/F,)~ (9.41)
where,

Fo =tu,
M, = t2cy /4 (9.42)
Using Eqs. (9.39), (9.40), and (9.41), the stress-strain and local lateral deflection stress
relationships may be obtained as follows:
c/o1. = pi2 - $2 (9.43)

&/t = (1 - +Y Y /(2+, ) (9.44)
where,

p = (4s/rdh/G (9.45)
The stress-strain relationship represented by Eq. (9.43) is schematically illustrated in Figure
9.16 (c). Applying this model, the stress distributions for tube cross-section after the
occurrence of local buckling are represented as in Figure 9.17. For a case A' stress distribution,
the following relationships are derived in place of Eqs. (9.22) and (9.23).
&I+ A)= f2 + f; +(c +c;h (9.46)
P(@+eo)=f, +f:+V4 +f,v)/Cr, +v)+fs (9.47)

where,
f; = 2ayR2t(g, -a)cosa, (9.48)
f: = 20, Rt(g, - R sin a)