smart technologies for safety engineering

134 Smart Technologies for Safety Engineering
of the system transfer functions ˜D f iN relating the response in the ith sensor location to local
impulse loading in the Nth potentially load-exposed DOF.
In Equation (26) T is the maximum system response time (i.e. the maximum propagation
time of an elastic wave between a loading-exposed DOF and a sensor). Due to the intended
limited number of sensors, the considered system is rarely collocated; thus T > 0. The
measurements and the reconstruction process may be triggered by a strong excitation (e.g. an
impact), which is picked up delayed at most T ; hence the time shift is necessary.
In real-world applications continuous responses are rarely known and so Equation (26)
should be discretized with respect to time. With the simplest quadrature rule it takes the
following form:
εi(t) =
t
τ=−T
D f i (t − τ)T f(τ) (27)
where D f i is the discretized and accordingly rescaled system transfer function Df i = t ˜D f i .
Equation (27), rewritten for each sensor location i and each measurement time instance t, can
be stated in the form of a general linear equation:
ε = D f f (28)
where ε is the vector of system responses in all sensor locations i and measurement time
instances t = 0,...,T , loading vector f represents the loading forces in all loading-exposed
DOFs and in all loading time instances τ =−T,...T , while D f is the system transfer matrix
compound of discretized D f i .
4.3.1.2 Elastoplastic Systems
The description of system dynamics stated above can be extended to include the elastoplastic
system behavior by combining the computationally effective virtual distortion method (VDM)
[18] with the return-mapping algorithm of Simo and Hughes [29]. The small deformation
restriction still applies and the extension is obviously at the cost of the linearity. For notational
simplicity the concept is presented here for trusses only and εi(t) denotes in this subsection
the strain in the ith truss element. Nevertheless, with inessential modifications (which result,
however, in much notational overhead), the concept is readily applicable to other types of
structures and linear sensors.
Equation (26) in the elastoplastic case has to take into account the effect of the plastic
distortions of the truss elements,
εi(t) =
t
τ=−T
D f i (t − τ)T f(τ) +
t
τ=−T
D ε i (t − τ)T β 0 (τ) (29)
where vector β 0 contains the discretized plastic distortions of all truss elements and D ε i is
the vector of the discrete system transfer functions Dε ij , which relate the response in the ith
sensor location (i.e. the strain in the ith element according to the convention of this subsection)
to the unit plastic distortion of the jth truss element. Equation (29), being very similar to
Equation (26), seems to be linear, but it obviously cannot be the case here. One of the reasons
is that the plastic distortions β 0 are nonlinearly dependent on the loading f.