Environmental Engineering Reference
In-Depth Information
dynamic formulations of structures, such as the methodology based on the con-
straint of geometrical deformation [
39
], geometric nonlinearities [
40
], and initial
stress [
41
].
The geometric nonlinearities considering the relation between the strain and
deformation are nonlinear because the stiffness matrix of flexible bodies should be
a function of motion status and stress. Therefore, the nonlinear relationship
between the strain and deformation should be retained. For an elastic plane beam,
the relation between the strain and deformation for one arbitrary point on the
nonmidline is given by:
2
2
v
0
e
xx
¼
o
u
0
ox
y
o
ox
2
þ
1
o
v
0
ox
ð
9
:
2
Þ
2
where u
0
and v
0
are the longitudinal and transverse deformations of the corre-
sponding point on the midline, respectively. Thus, the corresponding potential
energy of strain is also nonlinear. Bakr and Shabana reserved cubic terms without
considering quartic ones [
40
] and presented the corresponding potential energy of
strain:
Z
L
2
dx
þ
1
2
Z
L
2
dx
U
¼
1
2
o
u
0
ox
EA
o
u
0
ox
o
v
0
ox
EA
0
0
ð
9
:
3
Þ
2
Z
L
o
2
v
0
ox
2
þ
1
2
EI
dx
0
where E is elastic modulus, A the section area, and I the moment of inertia.
Mayo retained the quartic terms in the expression of the potential energy of
strain [
42
]:
Z
L
2
dx
þ
1
2
Z
L
2
dx
U
¼
1
2
o
u
0
ox
EA
o
u
0
ox
o
v
0
ox
EA
0
0
ð
9
:
4
Þ
Z
L
2
Z
L
4
dx
o
2
v
0
ox
2
þ
1
2
dx
þ
1
2
1
4
EA
o
v
0
ox
EI
0
0
The total stiffness matrix derived from the potential energy of strain can be
written as
K
¼
K
0
þ
K
1
ð
a
Þ
ð
9
:
5
Þ
where K
0
is the normal modal stiffness matrix, which is a constant matrix, and
K
1
(a) is the stiffness matrix of the geometric nonlinearities, which is a function of
the modal coordinate.
