smart technologies for safety engineering

Dynamic Load Monitoring 143
Single-Stage Load Reconstruction
Instead of separate successive reconstructions of the reconstructible and unreconstructible
loading components, the loading can be reconstructed in one stage only, but at the cost of
accuracy. The objective functions h1(m1) of Equation (53) and h2(m2) of Equation (55) resemble
the components of the general objective function h(f) in Equation (39), which is used
in the overdetermined case. Hence, instead of successive separate optimizations of h1(m1) and
h2(m2), both functions may be optimized simultaneously, as in h(f), weighted by an appropriate
coefficient δ>0:
hδ(f) := ε M − D f f 2 + δ Bf 2
As the second term also plays the role of a regularization term, there is no need to use the
regularized system matrix D fδ , and hence no need for numerically costly (although one-time
only) singular value decomposition. The minimization of the compound objective function hδ
can be performed relatively quickly by the conjugate gradient technique described earlier.
This one-stage approach makes no distinction between the reconstructible and the unreconstructible
loading components and retrieves them simultaneously. Therefore, it is generally less
accurate, since the heuristic assumptions influence both components of the reconstructed loading,
while the two-stage approach described in the preceding subsection properly reconstructs
the reconstructible component on the basis of the measurements ε M only.
4.3.2.3 Elastoplastic Systems
The approaches described before rely heavily on the linearity of the system equation (37).
The elastoplastic case of Equation (38) has to be treated separately. In general, three cases are
possible:
1. Strongly overdetermined system, possible in the case of a very limited loading area. The
number of sensors exceeds or equals the total number of potentially loading-exposed DOFs
and potentially plastified truss elements. The approach described before for overdetermined
linear systems is straightforwardly applicable (unless the system is singular), with both
loads fN (t) and plastic distortions β 0 i
(t) treated as independent unknowns.
2. Overdetermined system. The number of sensors exceeds or equals the number of potentially
loading-exposed DOFs. The unique evolution of the load can be reconstructed from the
measurements, unless the system is singular. However, the system is not linear and thus the
approaches of the preceding sections are not applicable.
3. Underdetermined system. The number of potentially loading-exposed DOFs exceeds the
number of sensors. In general, the gradient-based optimization approach presented below
reconstructs a nonunique evolution of loading, which is observationally indistinguishable
from the actual loading.
In the overdetermined elastoplastic case, Equation (38) can be, in general, uniquely solved
by minimizing the objective function (40) with any gradient-based optimization algorithm.
However, the modeled system response ε(t) is no longer a linear function of the loading f(t)
and the derivatives, instead of in Equation (41), take the following form:
∂h(f) ∂ Bf2
= δ − 2
∂ fN (t) ∂ fN (t)
i
T
τ=
max(0,t)
ε M i (τ) − εi(τ) ∂εi(τ)
∂ fN (t)
(58)
(59)