Page 197 - Marine Structural Design

Chapter 9 Buckling and Local Buckling of Tubular Members 173

&7+f,)=fz+h,+(c,-hz)rl (9.28)
P(w+eCJ=f, +h4 +dr, -h, +cr5 +h,hV(V+fi)+f6 (9.29)

=
CZ/h + f,) K (9.30)
77 = R(cosa, -cosa,)/2 (9.31)
where,
h, = 2a,R2t(sina, -a, cosa,)

h, = 4a,Rta,
h, = a,R’t(a, +- sina, cosa,) (9.32)

h, = 2a,R2tsina,
Solving Eqs. (9.22) thru (9.24) for case A and Eqs. (9.28) thru (9.31) for case B, with respect
to P, 7, and a, (and a, ), respectively, the relationship between axial load and lateral
deflection may be obtained.
The mean compressive axial strain in the elasto-plastic range may be given as:

(9.33)

The second term in the right-hand side of Eq. (9.33) represents the plastic component of the
axial strain. It is assumed that the plastic strain of qa,/(77 + Rcosa,) is uniformly distributed
within a region 2R.

Critical Condition for Local Buckling
According to the classical theory of elastic stability, critical buckling strain in a cylindrical
shell under axial compression is given as follows (Timoshenko and Gere, 1961):
I t t
E, = -- = 0.61- (9.34)
347R R
On the other hand, the critical strain for plastic shell buckling is given by Gerard (1962),
Batterman (1965) and others. Here, Reddy (1979) concluded that the critical buckling strain of
a shell occurs within the limits represented below including the pure bending case:
t t
0.2- ( E,~ ( 0.4- (9.35)
R R
In general, axial force and bending moment exist at the cross-section of tubular members.
Consequently, the strain at a cross-section is not uniform. This chapter proposes an empirical
formula, which represents the critical buckling strain in terms of the ratio of the maximum
bending strain to the axial strain E~/E, , and the wall thickness to radius ratio t/R, as follows:
E, = 0.155(0.25(~,/~,)2 +l.O}(t/R) for (2.5 (9.36)