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axial force
N
0
is
v
T
N
0
N
T
N
dx
v
N
T
EI
N
dx
−
δ
W
=
δ
−
v
T
G
T
EI
N
u
dx
G
N
0
G
T
dx
G
v
N
T
u
N
u
N
u
=
δ
−
(11.60)
Define for element
i
G
T
k
lin
=
N
T
EI
N
dx
N
T
u
EI
N
u
dx
G
=
N
0
N
0
G
T
(11.61)
k
i
geo
=−
N
T
N
dx
N
u
N
u
dx
G
=−
Matrix
k
lin
is the same element stiffness matrix that was discussed in Chapters 4 and 5. This
stiffness matrix is used in the static analysis of structures. If the same (static) shape function
N
is used in forming
k
i
geo
as in forming
k
lin
, then
k
i
geo
,
which is called the
geometric stiffness
matrix,
is said to be
consistent.
The matrix
k
i
geo
is also known as the
initial stress stiffness
matrix, stability coefficient matrix,
or simply the
stress stiffness matrix
.
In this section, we will study the development of the element matrices
k
lin
and
k
i
geo
and applications to simple beam models. The assembly into system matrices of these ele-
ment matrices will be discussed in Section 11.4, along with stability analyses of complex
structures.
One of the most common assumed displacements is a trial function formed of Hermitian
polynomials. The Hermitian polynomial shape functions of Chapter 4, Eq. (4.47b)
w
=
w
a
H
1
+
θ
a
H
2
+
w
b
H
3
+
θ
b
H
4
led to the exact element stiffness matrix for the static response
in Chapter 4. As mentioned above, the use of the same (static response) polynomial to form
the stability matrix
k
i
geo
provides a “consistent” matrix. Introduce (Chapter 4, Eq. (4.47a))
1000
0
w
a
θ
−
100
a
2
3
]
w
=
Nv
=
[1
ξξ
ξ
(11.62)
−
3231
2
w
b
−
1
−
2
−
1
θ
b
N
u
G
v
with
2
]
1
N
u
=
[012
ξ
3
ξ
1
N
u
=
[0026
ξ
]
2
into Eq. (11.61). This gives the stiffness matrix
k
lin
[
w
a
θ
a
w
b
θ
b
]
12
−
6
−
12
−
6
EI
−
64 62
k
lin
=
(11.63)
−
26 26
−
3
62 64
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