Civil Engineering Reference
In-Depth Information
Thus, the strains and displacements relationship (Equation 3.28) becomes
(
)
[ ]
Ba
B
B a a
ε =
=
+
(3.36)
0
L
where
B
0
is the same matrix as when geometric nonlinearity is not consid-
ered and
B a
L
( )
is due to the second order of displacement derivatives and
relates to current displacements.
When nonlinearities are considered, the solution of Equation 3.32 has to
be approached by incremental method, in which changes of ψ
( )
respective
to a small increment of
a
are to be noted.
T
d
d
B
a
d
d
σ
∫
∫
d
dv
B
T
dv d
a K a
d
ψ
=
σ
+
=
(3.37)
T
a
In Equation 3.37,
K
T
is the tangential stiffness, respective to small incre-
ment of displacements. Taking the geometric nonlinearity as an example,
the tangential stiffness can be derived as
(3.38)
K
=
K
+
K
+
K
T
0
σ
L
where:
K
=
∫
T
represents the usual stiffness when displacements are small
K
σ
is the first term in Equation 3.37, which reflects the stiffness due
to the existence of stresses, that is, the initial stress or geometric
matrix:
B DB
0
0
0
T
T
d
d
B
a
d
d
B
a
∫
∫
L
K
σ
=
σ
dv
=
σ
dv
(3.39)
K
L
is the stiffness due to large displacements:
(
)
∫
T
T
T
K
=
B DB
+
B DB
+
B DB
dv
(3.40)
L
0
L
L
L
L
0
When material nonlinearity is considered as well, the elastic matrix
D
should be evaluated at strains due to current displacements.
The solutions of nonlinear problems can be reached by iterations on
Equations 3.33 and 3.37. Given initial estimated displacements
a
0
, which
are obtained as linear solution, their corresponding internal strains can be
computed. Furthermore, the internal stresses can be obtained by either linear
or nonlinear stress and strain relationship. As shown in Equation 3.33, the
initial unbalanced general forces ψ
a
0
(
)
can be determined. The unbalanced
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