Civil Engineering Reference
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more members to it. A structural round which is not constructed using a
tetrahedron but is just stable is a higher round. The algorithm to identify
these rounds is quite complex and has been described elsewhere (Agarwal
et al.
2001a). Three members arranged orthogonally at a rigid joint produce
a well formed structural round (Fig. 8.2b). This round consists of three open
rings. A beam is the simplest space structure forming a structural round.
Such a round is called a leaf round.
The quality of the form of a ring or round is a function of the member
property, orientation, and connectivity. This quality is defi ned as the well-
formedness
Q
and can be obtained using the stiffness sub-matrices associ-
ated with the joints in a structural ring or round:
1
N
=
∑
(
)
Q
=
det
K
ii
N
i
1
where
K
ii
is the stiffness sub-matrix associated with any joint
i
and
N
is the
total number of joints in the ring/round. This measure is independent of the
chosen coordinate system and increases with increasing resistance to loads
along the principal directions. Conceptually, principal directions for a struc-
tural ring are analogous to principal axes for an I section or an unsymmetric
section. Then the measure of well-formedness is the product of principal
stiffnesses analogous to second moment of areas about the principal axes.
It increases with member stiffness and with joint stiffness. Any reduction
of the well-formedness indicates damage to the structure.
8.3.2 Clusters and hierarchy
The purpose of clustering is to successively bring together well-connected
members thus leading to a hierarchical representation of the structure. A
ring or a round itself is a cluster which can grow by adding more members
into it. When a cluster at a higher level is formed, it captures certain attri-
butes from members at lower level as well as new members joining it. The
attributes of a structural member (or cluster) used for cluster formation
include well-formedness of the structural cluster, minimum damage demand
of the structural cluster, nodal connectivity of the structural cluster, and
distance from the reference cluster. The well-formedness of a structural
cluster is calculated similarly to that for a structural round using all of the
joints in the cluster. It is assumed that the damage demand is directly pro-
portional to the loss of the principal stiffness caused by a deteriorating
event. Nodal connectivity is the total number of members connecting to the
joints in the cluster. For structures, the reference cluster is usually the
ground.
The clustering process starts with the identifi cation of seed clusters. A
ring in 2D or a round in 3D with the highest well-formedness is chosen as
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