Civil Engineering Reference
In-Depth Information
V
min
Theory
... Analysis
Figure 16.8
Distribution of maximum compressive stress, comparison with theory
16.4
DYNAMICS
Here we extend the coupling method to dynamics. The dynamic equilibrium equations
which arise from finite element discretisation (see Bathe
4
) can be written as
> @
^`
> @
^`
> @
^` ^`
Mu
Cu
Ku
F
(16.19)
where
>
M
,
>
C
,
>
K
are the assembled mass, damping and stiffness matrices and
^
u
,
^
u
,
^
u
are the acceleration, velocity and displacement vectors. The time may be
discretised into
n
time steps of size ' . Assuming an average acceleration within the
time step the system of differential equations can be transformed into a system of
algebraic equations (Newmark method
4
)
^`
^`
ªº
(16.20)
¬
Ku
()
t
F
where
tnt
'and
t
'
( )
n t
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