Page 205 - Marine Structural Design

Chapter 9 Buckling and Local Buckling of Tubular Members 181

Mi = Bending moment at nodal point i at the end of the n-th step
Q = Bending moment due to distributed lateral load
M = Bending moment given by Eq. (9.1 8)
Xi = Axial force at the end of the n-th step
AX, = Increment of axial force during the (n+l)-th step
Mi = Increment of bending moment at nodal point i during the (n+l)-th step
AQ = Bending moment increment due to distributed lateral load during (n+l)-th
step and
em = dMi/dxi e,, =AQ/AXi (9.68)
Xi, Mi, Mi, Mi, AQ are known variables after the (n+l)-th step has ended.
Q,
and
Considering the equilibrium condition of forces in the axial direction, geometrical conditions
regarding the slope, and Eq. (9.77), the following equations are obtained:
for Case A Stress Distribution:

(9.69)
(9.70)
(9.71)

(9.72)
PW+4% +%)=A +h, +k -4 +(A +h4hI/(t7+fi)+fs (9.73)
(9.74)
c2/(77 + fi) = K
17 = R(cos~, -cosc~,)/~ (9.75)
After the initial yielding, elasto-plastic analysis by the simplified method is performed using
Eqs. (9.69) thru (9.71) or Eqs. (9.72) thru (9.77) at each step of the Idealized Structural Unit
analysis until the ultimate strength is attained at a certain step.
Here, a more accurate method is introduced to determine the length of plastic zoneZ, . If the
axial force P and bending moment M are given, the parameters 17 and a, (and a*), which
determine the axial strain€ and curvature 4(x) are obtained from Eqs. (9.17) and (9.18).
Then, the increment of the curvature d&c) from the former step is evaluated. With this
increment, the length of plastic zone is given as

1, = pC,(x)qd4, (9.76)

d+, (x) = d((x)- dM(x)/EI (9.77)
where d4, represents the maximum plastic curvature increment in the plastic region.