smart technologies for safety engineering

VDM-Based Remodeling 219
Finally, Equations (7) and (10) yield together the following linear system of equations,
which can serve in successive time steps to determine the virtual strain distortions and force
distortions, which model the modifications μA i of the element cross-sectional areas:
⎡
⎣ δij −
1 − μA
i Dε
i j (0) − 1 − μA
i Df i K (0)
MNM ¨B ε Mj (0) δNK + MNM ¨B f MK (0)
−MNMü =t
M
⎤
0 ε j (t)
⎦
f 0 K (t)
⎡
1 − μ
= ⎣
A
i ε =t
⎤
i (t)
⎦ (12a)
which can be stated in the following shorter form:
Fx 0 = b (12b)
where the vector x 0 collects the virtual distortions. The principal matrix F is time-independent
and hence it is determined and decomposed only once. Moreover, it is also indispensable in
the sensitivity analysis.
The algorithm for the material redistribution analysis is shown in Table 6.1.
Table 6.1 Algorithm for material redistribution in elastic structures
Data and initial calculations
Input data:
Construction under external load
Cross-section modification parameters μA i
Calculations:
Linear response εL i (t) and üL N
ε Dynamic influence matrices Dij , Df iN , ¨B ε Nj and ¨B f NM
Principal time-independent matrix F by Equation (12).
Calculations in each time step t
(a) Strains ε =t
i (t) and accelerations ü =t
N by Equations (8) and (11)
(b) Virtual distortions ε0 i (t) and f 0 N (t) by Equation (12)
(c) Actual strains and accelerations:
εi(t) = ε =t
i (t) + D ε ij (0)ε0 j (t) + Df iM (0) f 0 M (t)
ü N (t) = ü =t
N (t) + ¨B ε Nj ε0 j (t) + ¨B f NM f 0 M (t)
(d) If necessary, by the corresponding influence matrices compute the response:
the displacements u N (t), the velocities ˙u N (t), the stresses σi(t), etc.
(e) If necessary, calculate the derivatives of the response (Section 6.1.2)
(f) t := t + 1