Civil Engineering Reference
In-Depth Information
2
4
3
5
12
EI
L
3
6
EI
L
2
12
EI
L
3
6
EI
L
2
2
4
3
5
P
L
P
L
−
−
0
0
6
EI
L
2
4
EI
L
6
EI
L
2
2
EI
L
−
0000
P
L
K
i
=
+
ð7
:
9Þ
12
EI
L
3
6
EI
L
2
12
EI
L
3
6
EI
L
2
P
L
0
−
0
−
−
−
6
EI
L
2
2
EI
L
6
EI
L
2
4
EI
L
−
0000
Because the geometric nonlinear stiffness matrix in Eq. (7.9) is a simple linear function of the
axial force per unit length with other terms being set equal to zero, it has been used extensively
in many commercial software packages (e.g., Perform-3D (CSI 2011) and OpenSees (Regents
of UC 2000)). However, keep in mind that Eq. (7.9) is the stiffness matrix that considers only
the large
P
-
Δ
effect but ignores the small
P
-
δ
effect. This approach for handling geometric
nonlinearity will be difficult to incorporate into the force analogy method, which requires a
detailed expression to capture the moment magnification due to the small
P
-
δ
effect along
any point on the member with material nonlinearity.
7.1.2 The Geometric Stiffness Approach
In this approach, both large
P
-
Δ
and small
P
-
δ
are considered by assuming that the strain at any
point along the member and cross section is second-order with all higher-order terms truncated.
This gives
"
#
2
+
2
dx
−
y
d
2
v
ε
x
, ðÞ=
du
dx
2
+
1
du
dx
dv
dx
ð7
:
10Þ
2
Because
u
and
v
are functions of
x
only, and the strain
ε
(
x
,
y
) has a linear variation of
y
along
the cross section as shown in the second term of Eq. (7.10), the assumption “plane sections
remain plane” is again employed. The minimum energy method can be used to compute the
stiffness matrix:
W
=
ð
V
σ
x
, ð
ε
x
, ð
dV
ð7
:
11Þ
where
W
is the work done due to the internal strain energy,
σ
(
x
,
y
) is the stress at any point along
the member and cross section, and the integration is done over the entire volume of the beam
member. Substituting Eq. (7.10) into Eq. (7.11) and using the interpolation functions for the
displacements in Eqs. (7.1) to (7.4) and Eq. (7.6) rewritten as:
L
u
ðÞ+
x
x
u
ðÞ=1−
L
u
ðÞ
ð7
:
12aÞ
