Civil Engineering Reference
In-Depth Information
as in Eq. (17.61), whence
[
B
]
T
[
D
][
B
] d(vol)
[
K
e
]
=
vol
[
C
][
A
−
1
] where [
A
] is defined in Eq. (17.72) and [
C
]inEq.
(17.76). The elasticity matrix [
D
] is defined in Eq. (17.79) for plane stress problems
or in Eq. (17.80) for plane strain problems. We note that the [
C
], [
A
] (therefore [
B
])
and [
D
] matrices contain only constant terms and may therefore be taken outside the
integration in the expression for [
K
e
], leaving only
d(vol) which is simply the area,
A
, of the triangle times its thickness
t
. Thus
In this expression [
B
]
=
[
K
e
]
[[
B
]
T
[
D
][
B
]
At
]
=
(17.81)
Finally the element stresses follow from Eq. (17.66), i.e.
δ
e
{
σ
}=
[
H
]
{
}
where [
H
]
[
D
][
B
] and [
D
] and [
B
] have previously been defined. It is usually found
convenient to plot the stresses at the centroid of the element.
=
Of all the finite elements in use the triangular element is probably the most versatile.
It may be used to solve a variety of problems ranging from two-dimensional flat plate
structures to three-dimensional folded plates and shells. For three-dimensional appli-
cations the element stiffness matrix [
K
e
] is transformed from an in-plane
xy
coordinate
system to a three-dimensional system of global coordinates by the use of a transform-
ation matrix similar to those developed for the matrix analysis of skeletal structures.
In addition to the above, triangular elements may be adapted for use in plate flexure
problems and for the analysis of bodies of revolution.
E
XAMPLE
17.3
A constant strain triangular element has corners 1(0, 0), 2(4, 0)
and 3(2, 2) referred to a Cartesian O
xy
axes system and is 1 unit thick. If the elasticity
matrix [
D
] has elements
D
11
=
D
22
=
a
,
D
12
=
D
21
=
b
,
D
13
=
D
23
=
D
31
=
D
32
=
0 and
D
33
=
c
, derive the stiffness matrix for the element.
From Eq. (17.69)
w
1
=
α
1
+
+
α
2
(0)
α
3
(0)
i.e.
w
1
=
α
1
(i)
w
2
=
α
1
+
α
2
(4)
+
α
3
(0)
i.e.
w
2
=
α
1
+
4
α
2
(ii)
w
3
=
α
1
+
α
2
(2)
+
α
3
(2)
