Seismic vulnerability analysis of a complex interconnected civil infrastructure - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 470

have fragility models in the form of a set of fragility functions, each describ-
ing the probability of attaining or exceeding a pre-defi ned limit-state or
performance-level as a function of a scalar or vector IM. In contrast,
line-like components commonly have fragility models in the form of a
Poisson distribution of damage per unit length, with rate that is again a
function of a scalar or vector IM. Within a WSS, pipes have a line-like fra-
gility model, while other components, such as dams (Lupoi and Callari,
2011), tanks or pumping stations, have point-like fragility models. For long
pipes, going through different soil conditions and with varying distance
from the epicentre, whether the IM and hence the rate of damages (ruptures
and/or leakages) is sampled at a single point (the centroid) or at many
points along the line (the centroids of the automatically generated smaller
segments), clearly makes a difference in the predicted damage distribution.
Other attributes of pipes include the depth (derived from that of the con-
nected nodes), the diameter, the roughness, the numbers of breaks and
leaks, and the leakage area (equal to the total area if at least a break is
present).
The functional model for this network consists of the N + E nonlinear

fl ow equations (Houghtalen et al ., 2009):

( =

AqQh 0
Rqq A h Ah 0

T

−

⎩

N N

[18.1]

( ) =

+ +

NN SS

where N , E and S are the number of internal (non-source) nodes, of edges
and of sources, respectively. The fi rst N equations express fl ow balance at
the internal nodes (sum of incoming and outgoing fl ows equal either zero
or the demands in the end-user nodes), while the next E equations express
the fl ow resistance of the edges.
The E

× N and E × S matrices A N and A S are sub-matrices of the E × ( N

+ S ) matrix A which contains 0, 1 and − 1 terms as a function of the network

connectivity. The N × 1 and S × 1 vectors h N and h S are the corresponding

1 vector h collecting the N unknown heads in
the internal nodes and the S known heads in the water-source nodes. The
E

partitions of the ( N + S ) ×

× 1 vector q collects the unknown fl ows in the E links, and R is the E ×

D −5
(according to Darcy's law) and L i is the i th link length. Figure 18.6 (top)
shows a sample WSS, with three nodes and three edges, and the correspond-
ing equations. As shown in the fi gure there are fi ve equations in the fi ve
unknowns. Two equations express the balance in demand nodes 2 and 3.
The next three equations express the dissipation of energy during fl ow
through pipes 1 to 3. The interested reader should consult with recent text-
books such as Houghtalen et al . (2009) or Swame and Sharma (2008), for
an in-depth treatment of water systems analysis.

E diagonal matrix of resistance, with terms r i = u i L i , where u i = β