Civil Engineering Reference
In-Depth Information
■
Solution
Using the specified data, The cross-sectional area is
1
in.
2
A
=
1(1)
=
And the area moment of inertia about the
z
axis is
=
bh
3
083
in.
4
I
z
/
12
=
1
/
12
=
0
.
The characteristic axial stiffness is
10
6
)
10
5
)
lb/in.
AE
/
L
=
1(10
×
/
20
=
(5
×
and the characteristic bending stiffness is
EI
z
/
L
3
10
6
(0
20
3
=
10
×
.
083)
/
=
104
.
2
lb/in.
Denoting member
OB
as element 1 and member
BC
as element 2, the stiffness
matrices in the element coordinate systems are identical and given by
5(10
5
)
−
5(10
5
)
0
0
0
0
.
−
.
0
1, 250
4
12, 504
0
1, 250
4
12, 504
k
(1)
=
k
(2)
=
0
12, 504
166, 720
0
−
12, 504
83, 360
−
5(10
5
)
0
0
5(10
5
)
0
0
0
−
1, 250
.
4
−
12, 504
0
1, 250
.
4
−
12, 504
0
12, 504
83, 360
0
−
12, 504
166, 720
Choosing the global coordinate system and displacement numbering as in Figure 4.14b,
we observe that element 2 requires no transformation, as its element coordinate system is
aligned with the global system. However, as shown in Figure 4.14c, element 1 requires
transformation. Using
=
/
2
, Equations 4.64 and 4.65 are applied to obtain
1, 250
.
4
−
12, 504
1, 250
.
4
−
12, 504
0
0
5(10
5
)
−
5(10
5
)
0
0
0
0
K
(1)
=
−
12, 504
0
166, 720
12, 504
0
83, 360
1, 250
.
4
0
12, 504
1, 250
.
4
0
12, 504
0
−
5(10
5
)
0
0
5(10
5
)
0
−
12, 504
0
83, 360
12, 504
0
166, 720
Note particularly how the stiffness matrix of element 1 changes as a result of the 90°
rotation. The values of individual components in the stiffness matrix are unchanged. The
positions of the terms in the matrix are changed to reflect, quite simply, the directions of
bending and axial displacements of the element when described in the global (system)
coordinate system.
The displacement correspondence table is shown in Table 4.6 and the assembled sys-
tem stiffness matrix, by the direct assembly procedure, is in Table 4.7. Note, as usual, the
“overlap” of the element stiffness matrices at the displacements associated with the com-
mon node. At these positions in the global stiffness matrix, the stiffness terms from the
individual element stiffness matrices are additive.
