Civil Engineering Reference
In-Depth Information
Substituting Equation 5.27 into the following equation repeated from ear-
lier results in Equation 5.28 for the second stiffness value.
MP
L
L
EI
L
=
−
P
q
−
q
z
iz
iy
ix
iz
iz
2
12
4
EI
L
−
2
15
L
(5.28)
M
=
q
z
+
P
q
iz
iz
ix
iz
Take note that the first terms in each of these stiffness equations are the
same as the elastic stiffness values derived in Equations 4.24 and 4.25.
The second term is the geometric component due to the deflected shape
and the axial thrust. All four terms can be written in matrix form.
12
EI
L
6
EI
L
−−
6
1
10
z
z
∆
iy
P
M
=
3
2
5
L
iy
+
P
ix
6
EI
L
4
EI
L
1
10
−−
2
15
L
q
z
z
iz
iz
2
The first matrix on the left side of the equation in the basic elastic stiffness
matrix will be called [
K
]. The second matrix on the left side of the equa-
tion in the geometric stiffness matrix will be called [
G
]. In general terms,
the equation may be written as follows:
(
)
[]
=
[]
[]
+
[]
K G F
ix
P
∆
The same transformation used for previous stiffness matrix derivations can
be applied to find the rest of the geometric stiffness matrix. The geometric
stiffness matrix for the coplanar X-Y system is given as Equation 5.29.
The sign convention on
P
ix
is positive for tension.
00 000
0
6
5
1
10
6
5
1
10
0
0
−
L
L
1
10
2
15
L
1
10
L
0
0
−
−
30
[
]
=
KP
m
ix
00 000
0
6
5
1
10
6
5
1
10
0
−
−
0
−
L
L
1
10
L
1
10
2
15
L
0
−
0
−
30
(5.29)
