Civil Engineering Reference
In-Depth Information
Figure 4.32.
Example 4.18 Δ
iy
stiffness.
Solving the first equation for
P
iy
in terms of
M
iz
and then substituting into
the second equation, the stiffness value can be found.
2
M
P
=
iz
iy
L
2
M
L
EI
EI
L
∆
=
iz
iy
6
6
z
z
M
=
∆
(4.26)
iz
iy
2
This can be substituted back into the equation for
P
iy
to obtain the last
stiffness value.
=
12
EI
P
z
∆
(4.27)
iy
iy
3
L
The four terms given in Equations 4.24 through 4.27 are the flexural stiff-
ness terms for the forces at the
i
i-end due to motions at the
i
i-end. This is
denoted as stiffness matrix [
K
ii
] in Equation 4.28. The stiffness equation
and matrix form of this are as follows:
[[]
=
[]
K
d
F
ii
i
i
12
EI
L
6
EI
L
z
z
∆
θ
P
iy
=
3
2
iy
6
EI
L
4
EI
L
M
z
z
iz
iz
2
12
EI
L
6
EI
L
z
z
3
2
[
=
(4.28)
K
ii
6
EI
L
4
EI
L
z
z
2
The transmission matrix derived in Section 4.3 for the X-Y system can be
used to find the forces at the
j
-end.
10
1
[]
=
T
−
L
