Civil Engineering Reference
In-Depth Information
12
EI
L
6
EI
L
y
y
−
∆
q
=
P
M
3
2
iz
iz
6
EI
L
4
EI
L
iy
iy
y
y
−
2
12
EI
L
6
EI
L
y
y
−
3
2
[
=
K
(4.19)
ii
6
EI
L
4
EI
L
y
y
−
2
The transmission matrix derived in Section 4.3 for the X-Z system can be
used to find the forces at the
j
-end.
10
1
[]
=
T
L
From statics equilibrium, the forces at the
i
i-end transmitted to the
j
-end
plus the forces at the
j
-end are equal to zero. The forces at the
j
-end due to
motions at the
i
i-end are denoted as stiffness matrix [
K
ji
].
[]
[
+
=
TK K
0
ii
ji
=−
[]
[]
K
T K
ji
ii
12
EI
6
EI
L
12
EI
L
6
EI
L
y
y
y
y
−
−
10
1
−
−−
10
1
=−
3
2
3
2
L
K
=
ji
L
6
EI
L
4
EI
L
L
6
EI
L
4
EI
L
y
y
y
y
−
−
2
2
12
EI
L
6
EI
L
y
y
−
3
2
=
K
(4.20)
ji
6
EI
L
2
EI
L
y
y
−
2
For a prismatic member, it can be shown that the stiffness matrix is sym-
metric about the main diagonal. This results in stiffness [
K
ij
] being equal
to the transpose of [
K
ji
], where the forces at the
j
-end due to motions at the
i
i-end are denoted as stiffness matrix [
K
ji
].
