Civil Engineering Reference
In-Depth Information
y
3
45
x
1
2
L
W
F
IGURE
17.4
Truss of Ex. 17.1
The external forces are applied at node 2 such that
F
x
,2
=
0,
F
y
,2
=−
W
; the nodal
forces at 1 and 3 are then unknown reactions.
The first step in the solution is to assemble the stiffness matrix for the complete frame-
work by writing down the member stiffness matrices referred to the global axes using
Eq. (17.23). The direction cosines
λ
and
µ
take different values for each of the three
members; therefore, remembering that the angle
θ
is measured anticlockwise from
the positive direction of the
x
axis we have the following:
Member
θ (deg)
λ
µ
12
0
1
0
13
90
0
1
−
23
135
0.707
0.707
The member stiffness matrices are therefore
10
10
00 00
−
0000
010
AE
L
AE
L
1
0000
0
−
[
K
12
]
=
[
K
13
]
=
−
10 10
00 00
−
10 1
0
.
5
−
0
.
5
−
0
.
5
.
5
AE
−
0
.
5
.
5
.
5
−
0
.
5
[
K
23
]
=
(i)
1
·
414
L
−
0
.
5
.
5
.
5
−
0
.
5
0
.
5
−
0
.
5
−
0
.
5
.
5
The complete stiffness matrix is now assembled using the method suggested in the
discussion of Eq. (17.14). The matrix will be a 6
×
6 matrix since there are six nodal
