Civil Engineering Reference
In-Depth Information
3.3.1 Direction Cosines
In practice, a finite element model is constructed by defining nodes at specified
coordinate locations followed by definition of elements by specification of the
nodes connected by each element. For the case at hand, nodes
i
and
j
are defined
in global coordinates by (
X
i
,
Y
i
) and (
X
j
,
Y
j
). Using the nodal coordinates, element
length is readily computed as
L
X
i
)
2
Y
i
)
2
]
1
/
2
=
[(
X
j
−
+
(
Y
j
−
(3.29)
and the unit vector directed from node
i
to node
j
is
1
L
[(
X
j
−
=
X
i
)
I
+
(
Y
j
−
Y
i
)
J
]
=
cos
X
I
+
cos
Y
J
(3.30)
where
I
and
J
are unit vectors in global coordinate directions
X
and
Y
, respec-
tively. Recalling the definition of the scalar product of two vectors and referring
again to Figure 3.4, the trigonometric values required to construct the element
transformation matrix are also readily determined from the nodal coordinates as
the
direction cosines
in Equation 3.30
X
j
−
X
i
cos
=
cos
X
=
·
I
=
(3.31)
L
Y
j
−
Y
i
sin
=
cos
Y
=
·
J
=
(3.32)
L
Thus, the element stiffness matrix of a bar element in global coordinates can
be completely determined by specification of the nodal coordinates, the cross-
sectional area of the element, and the modulus of elasticity of the element material.
3.4 DIRECT ASSEMBLY OF GLOBAL
STIFFNESS MATRIX
Having addressed the procedure of transforming the element characteristics of
the one-dimensional bar element into the global coordinate system of a two-
dimensional structure, we now address a method of obtaining the global equilib-
rium equations via an element-by-element assembly procedure. The technique of
directly assembling the global stiffness matrix for a finite element model of a
truss is discussed in terms of the simple two-element system depicted in Fig-
ure 3.2. Assuming the geometry and material properties to be completely speci-
fied, the element stiffness matrix in the global frame can be formulated for each
element using Equation 3.28 to obtain
k
(1)
11
k
(1)
12
k
(1)
13
k
(1)
14
k
(1)
21
k
(1)
22
k
(1)
23
k
(1)
24
K
(1)
=
(3.33)
k
(1)
31
k
(1)
32
k
(1)
33
k
(1)
34
k
(1)
41
k
(1)
42
k
(1)
43
k
(1)
44
