Page 259 - Marine Structural Design

Chapter 12 A Theory of Nonlinear Finite Element Analysis 235

where,
H,i = [H;k + H;i + H:, /(&t)], hi (12.50)

12.3.4 Elastic-Plastic Stiffness Equation for Elements
When both nodes 1 and 2 are plastic, the following matrix equation may be established from
Eq. (1 2.49):
[@I' (dnj - [H']{dd) + {A) = 0 (12.51)
where,
[@I = [{4 1 I42 11
[H'] = [H:, H:,] (2x2 diagonal matrix)
(4 d4 f'
=
and

(4 = a2 Y
From Eq. (12.38), the increments of the plastic nodal displacement {dup ]are given as:
{~f= [@Ida) (12.52)
The increments of the total nodal displacement @}are expressed by the summation of the
elastic and plastic components as:
{du) = hue}+ @up} (12.53)
Substitution of Eqs. (12.52) and (12.53) into Eq. (12.4) gives:
[kE Xb1 - [@lt4) = b) (12.54)

Solving Eqs. (12.51) and (12.54) with respect to (dd) we obtain:
w = uff 1 + [@I' [kE I@Ir [kE 14 + M) (12.55)

substituting (dl} into Eq. (12.54) gives the elastic plastic stiffness equation:
[kpl(dul={4+Idr 1 (12.56)

(12.57)
&')= [k,I[@lb'l+ [@1'[kEI[@lrM (12.58)

If the sign of (dl,}or (dA2}is found to be negative, unloading occurs at the plastic node and
the node should then be treated as elastic. It is noted that the effects of large displacements and
strain hardening as well as strain-rate hardening have been taken into account in the derived
elastic-plastic stiffness equation.