Seismic risk models for aging and deteriorating buildings and civil infrastructure - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 384

occurs when a fi xed amount of capacity/resistance is removed from the
system at discrete points in time (Sánchez-Silva et al. , 2011). If Y i is a
random variable describing the degradation caused by shock i , the total
degradation by time t is: Dt

() =∑ =1 ; where N t represents the number of
shocks by time t . Note that in many practical applications the time between
shocks is also random; therefore, N t is also a random variable. Then, the
remaining structural capacity/resistance by time t can be computed as:

N

Y

t

s i

i

N t
() =− () =− = ∑ Vt v Dt v Y [15.3]

s 0 s 0 i

i 1

Extensive research has been carried out on developing mathematical
models for shock degradation. Additional details about these methods can
be found in Barlow and Proshan (1965), Aven and Jansen (1999), Frangopol
et al. (2004) and Sánchez-Silva et al. (2011).

15.2.4 General degradation model

Frequently, progressive and shock-based deterioration occur simultane-
ously (Fig. 15.1). Thus, for a structural component with initial capacity,

ψ 0 ,
subject to both continuous and sudden damaging events acting indepen-
dently, the remaining capacity/resistance by time t can be computed as
(Sánchez-Silva et al. , 2011):

N t

t

=

() =−

∫

()

Vt v δ u u d − Y [15.4]

0 p i

0

i 1

It is important to stress that in some cases both events are not independent
and, therefore, the coupled effect should be taken into consideration.

15.3 Shock-based damage accumulation models

15.3.1 Damage accumulation due to
successive earthquakes
Some of the early work on seismic damage accumulation of structures as a
result of successive earthquakes can be found in Elnashai et al. (1998).
Elnashai et al. showed that the ductility demand imposed on a structure,
following multiple earthquake ground motions, is often several times higher
than the ductility demand required by a single earthquake occurrence.
Additionally, studies by Murià-Vila and Jaramillo (1998) revealed a signifi -
cant reduction in lateral stiffness of a building founded in soft soil under
repeated low magnitude earthquake excitations. The work by Amadio et al.
(2003) focused primarily on the behavior of inelastic single degree of