Seismic risk models for aging and deteriorating buildings and civil infrastructure - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 383

respect to a physical parameter; for instance, in structures it can be the
stiffness or the inter-story drift. In order to be general, the parameter that
will be used to describe the system performance in this chapter is the struc-
tural capacity/resistence.

15.2.2 Progressive degradation
Progressive (graceful) degradation results in capacity/resistance continu-
ously being removed from the structure at a rate that may, or may not,
change (randomly) over time. Most progressive degradation models avail-
able in the literature assume that the form of the degradation process is
known, but the parameters are uncertain. The solution to this problem
conveys to a parameter estimation problem. Thus, if V p ( t ) is the state of the
system at a given time t , which in practice, may be expressed in terms of,
for example, remaining capacity, reliability, safety and durability, then these
type of models have the following general form:

v 0 ≤≤ t t

⎩ ⎭

0 e

() = ,

[15.1]

Vt

p

( )

vh t − p , − t t > t

0 e e

where 0 and t e is a time

before degradation initiation (e.g., time to corrosion initiation). The func-
tion h may take a linear, non-linear, exponential of any other form based
on the appropriate selection of the vector parameter p , which depends upon
the problem at hand.
Progressive degradation can be handled also by evaluating variations in
the degradation rate, i.e.,

ν 0 is the remaining life of the system at time t =

0. Thus, the system state (e.g., remaining
structual capacity) at time t can be expressed as:

δ p ( t ), t ≥

t
() =−

∫

()

Vt v

0 δτ τ

d [15.2]

p 0 p

where the rate may be dependent or independent of time. Finally, the
degradation function can be modeled also as a discrete process with
random changes at fi xed/random time intervals. If the distribution or the
time between these changes is selected appropriately, it is possible to fi nd
a good approximation to the continuous solution. These models are
used for shock-based deterioration processes and will be discussed in
Section 15.4.

15.2.3 Degradation caused by shocks

Shocks can be defi ned as events that cause a signifi cant change in a system's
physical property in a small time interval. Then, shock-based degradation