Page 209 - Marine Structural Design

Chapter 9 Buckling and Local Buckling of Tubular Members 185

As described in 9.3.2.3, there exists no one-to-one correspondence between plastic nodal
displacements and plastic strains at a nodal point. However, plastic strains may be
concentrated near the cross-section where local buckling occurs. So, the axial strain and
curvature at this cross-section are approximated by:
e=PfEA+e,, +(up -uFr)flp (9.87)
K=M/EI+KFr+(Op -OFr)/lP (9.88)

lp in the above equations represents the length of plastic zone, and is taken to be equal to the
diameter D(=2R ) as in the case of a simplified method. Considering Eqs. (9.87) and (9.88),
the filly plastic interaction relationship reduces to:

The elasto-plastic stiffness matrix after local buckling occurs, is derived based on the filly
plastic interaction relationship expressed by Eq. (9.89). The condition to maintain the plastic
state is written as:
ar ar ar ar
dT=-dP+-dM+-du, +-de, = 0 (9.90)
ap a~ auP a@,
or in the matrix form as:

(9.91)

where, {dR) and { dh,} are the increment of nodal forces and plastic nodal displacements,
respectively, see Figure 9.12 and the following Equations:
(bj = {ar/ax,, ar/az,, aqa!}

4j = (ar/ax,, arjaz, , aqaM,} (9.92)
vi = @q/hpi, ar/aYpi. arpPi}
vj = brp, j, aqh, I aqas, \ (9.93)

Here, considering ras a plastic potential, the increments of plastic nodal displacements are
given as

(9.94)

When only nodal point j is plastic, d;li = 0. Contrary to this, dAj = 0 when only node point i is
plastic.
On the other hand, the increments of nodal forces are expressed in terms of the elastic stiffness
matrix and the elastic components of nodal displacement increments as follows: