Page 202 - Marine Structural Design

178 Part II Ultimate Strength

Integrating AF, and dM,respectively, the force F, and the bending moment M, acting at the
bottom of a dent are obtained as:
Fb = 21AFbd8 (9.56)

M, =2[dMbdB (9.57)
where a, represents a half dent angle, and has a limiting value, a, as mentioned in Chapter
9.2. After a, is attained, two other dents are introduced as illustrated in Figure 9.10 (c). For
the specimen tested in this chapter a, = z/4, which coincides with the calculated results by
Toi et.al. (1983).
Applying this model, the stress distributions after local buckling may be represented as shown
in Figure 9.19. In this figure, the case with one dent is indicated as case A" distribution, and
that with three dents is a case B" distribution. For a case A" stress distribution, Eqs. (9.22) and
(9.23) are replaced with:
(9.58)

(9.59)
where,
f,"= Fbi (9.60)
f: = M, + Fbi R COS pi (9.61)

pi is the angle of the center of the i-th dent measured from the vertical centerline, as shown in
Figure 9.19.
For a case B" stress distribution, Eqs. (9.28) and (9.29) are replaced with:
(~-fiXv+fi)=f* +(c, -h2h (9.62)
+hl
W+e,)=f3 +f:+h +v4 -h, +dr, +h4h}/(V+fi)+fa (9.63)

Procedure of Numerical Analysis
Until initial yielding is detected, Eq. (9.3) gives the relationship between axial compressive
loads and lateral deflection. The mean compressive axial strain is evaluated by Eq. (9.8).
After plastification has started, the analysis is performed in an incremental manner using the
plastic component of deflection shown in Figure 9.13. This deflection mode expressed by Eqs.
(9.10) thru (9.12) gives a constant plastic curvature increment in the region Z, . If the actual
plastic region length ld in Figure 9.20 (a) is taken as 1, , it reduces to prescribe excess plastic
curvature especially near the ends of the plastic region. To avoid this, a bi-linear distribution
of plastic curvature increments is assumed in the region Id, as indicated in Figure 9.20 (b).
Then, the change of the plastic slope increment along the plastic region Id, may be expressed
as:
de, = 1,d K~ /2 (9.64)