smart technologies for safety engineering

226 Smart Technologies for Safety Engineering
Note, however, that although the elements of the full principal matrix F S do not depend on
time and can thus be computed only once, its size is time-dependent, since the third equation of
(28) is valid only for the currently yielding elements and the extent of the plastic zone changes
in time. Thus, the principal matrix F S and the system of Equation (28) have to be updated
each time the plastic zone changes. The algorithm for simulation of material redistribution in
elastoplastic structures is shown in Table 6.3.
Table 6.3 Algorithm for material redistribution in elastoplastic structures
Data and initial calculations
Input data
Construction under external load
Initially empty plastic zone B =∅
Yield stress limits σ ⋆
i , hardening parameters γi , cross-section modification parameters μ A i
Calculations
Linear response εL i (t) and üL N (t)
ε Dynamic influence matrices Dij (t), Df iN (t), ¨B ε Nj (t) and ¨B f NM (t)
Principal matrix FS by Equation (28)
Calculations in each time step t
Strains ε =t
i (t) and accelerations ü =t
N (t)
Estimate the trial stresses σ TR
i (t) = Ei ε =t
i (t) − β0
i (t − 1)
Estimate the extent of the plastic zone B by the condition |σ TR
i (t)| >σ⋆ i + γi Ei
1−γi i(t)
NONEMPTY PLASTIC ZONE
(a) Update the principal matrix FS according to the plastic zone B and sign(σ TR
i )
(b) Virtual distortions β 0 k (t), ε0 j (t) and f 0 M (t) by Equation (28)
(c) Plastic distortions: β0 k (t) := β0 k (t − 1) + β 0 k (t)
(d) Total plastic strain for isotropic hardening: k(t) := k(t − 1) +|β 0 k (t)|
(e) Actual strains by Equation (20) and accelerations
(f) Stresses by Equation (26)
(g) Verify the sign of β 0 k (t)σk(t). If negative, reestimate the extent of the plastic zone
B using the computed stresses σi(t) instead of the trial stresses σ TR
i (t) and go back
to (a)
EMPTY PLASTIC ZONE
(a) Update the principal matrix FS according to the empty plastic zone B =∅
(b) Virtual distortions ε0 j (t) and f 0 M (t)
(c) Plastic distortions β0 k (t) := β0 k (t − 1)
(d) Actual strains by Equation (20) and accelerations
(e) Stresses by Equation (26)
If necessary, by the corresponding influence matrices compute the response: the displacements
u N (t), the velocities ˙u N (t), etc.
If necessary, calculate the derivatives of the response (Section 6.2.2)
t := t + 1