Page 271 - Marine Structural Design

Chapter 12 A Theory of Nonlinear Finite Element Analysis 247

Eliminating (do) from Eqs. (A.40), (A.41), Ail is obtained from the following equation,

(A.42)

Substituting Eq. (A.42) into Eq. (A.40), we obtain,

{An}= [De]-
I -{&r{$}+{gr (A.43)
[Del{g}
= (De]+ [DPD(d&)
J
here, [DP is expressed as,

(A.44)

and this must be considered when the material is in the plastic condition.
Substituting Eq. (A.42) into Eq. (A.37), the plastic strain increment is expressed in the
following equation.

{A&P ] = (A.45)

In the process of plastic deformation, Ail in Eq. (A37) must have a positive value. Therefore,
by checking the sign of A2 in Eq. (A.42), the unloading condition can be detected.

12.6 Appendix B: Deformation Matrix
The deformation matrix [k,] is symmetric, nonzero terms are given below:

0
k, (1,2) = k, (7,s) = -k, (1,8) = -k, (2,7) = - pf (EA/LXB,, + Qzz)/l
0
k, (1,3) = k, (7,9) = -k, (1,9) = -k, (3,7) = p: (EA/L)(6,,, + OYZ)/1
k, (1,5) = -k, (5,7) = *, + p:EA (- 46,, + 6,,)/30
k, (1,6) = -k, (6,7) = -a, + pf EA (- 4e,, + e,,)/30

k, (i,i 1) = -k, (7,l I) = a,, + p;m(e,, - 4eYz)/30