Civil Engineering Reference
In-Depth Information
2.4.9 Dividing an edge into segments
In FE MG, it is often required to add more nodes to a line segment. The method employed
to generate more nodes on an edge depends on the required node distribution and the data
available. The minimum required information is the specified node spacing at the ending
nodes of the given edge. In this case, for a smooth transition of element size, nodes are gener-
ated along the edge following a geometric progression.
2.4.9.1 Element size is specified at nodal points
Let ρ
1
and ρ
2
be the specified spacings at the ending nodes and d be the length of the edge.
a. ρ
1
= ρ
2
The edge is to be divided following a uniform nodal distribution. Number of segment
(division), n = NINT(d/ρ
1
). Adjusted spacing between nodes, s = d/n.
b. ρ
2
≠ ρ
1
Without loss of generality, assume ρ
2
> ρ
1
. Suppose that the edge is to be divided into
n segments with geometric progression ratio r > 1. The length of the first segment s
=
ρ
1
and we have for the last (n
th
) segment ρ
2
= ρ
1
r
n−1
. The length of the edge as a sum of
all the n segments is given by
n
r
r
−
−
1
1
r
ρρ ρ
ρ
−
−
d
d
−
−
2
1
1
d
=
ρ
=
r
=
1
r
1
2
Knowing the ratio r between adjacent segments, the number of segments n is given
by
ρ
ρ
log
2
1
ρ
ρ
n
−
1
1
2
nNINT
=
+
1
andr
=
log( )
r
1
The geometric progression ratio r has to be adjusted as n is rounded up to a whole
number, and the adjusted length of the first segment is given by
n
r
r
−
−
1
1
ss
r
r
−
−
1
1
d
=
=
d
n
Remark:
These formulas are applicable for edges with a length greater than a threshold
value, i.e.
d >
++
+
2
2
ρρρρ
ρρ
1
12
2
, and for edges of shorter lengths, no subdivision is needed.
1
2
Consider the division of the edges of a rectangle (100 × 50) into segments according to
the element size specified at the corner points, as shown in Figure 2.19. Given the specified
size of 2, 5, 16 and 8 at the corner points, the edge division formulas are applied to each
edge in turn. Starting from the bottom edge following the counter-clockwise direction, the
four edges are divided into 30, 5, 9 and 11 segments, respectively. It is noted that nodes are
automatically clustered near the bottom-left corner where a smaller spacing of two units has
been prescribed.
