Civil Engineering Reference
In-Depth Information
If we solve for time step
N
, the results for the previous time steps are known and can
be put to the right hand side:
N
1
N
1
¦¦
>@
^`
>@
^`
>@
^`
>@
^`
(14.47)
Tu
Uq
Tu
Uq
N
N
N
N
n
n
n
n
n
1
n
1
or
> @
^`
> @
^` ^`
Tu
Uq
F
(14.48)
N
N
N
N
where the vector
^`
contains the effect of the time history. The coefficients of
^`
F
F
are
N
1
E
J
N
1
E
J
¦¦¦
¦¦¦
(14.49)
e
e
F
'
Uq
'
Tu
i
ijNn
nj
ijNn
nj
n
1
e
1
j
1
n
1
e
1
j
1
14.4
ELASTODYNAMICS
We now turn our attention to general problems in elasticity. The differential equation for
dynamics in the frequency domain can be written in matrix form as:
2
(
O
G
)
u
G
u
U
b
UZ
u
(14.50)
where
b
is a body force vector
§
2
2
2
·
w
w
w
¨
¸
2
ww ww
x yxz
w
x
¨
¸
§·
u
u
¨
¸
x
2
2
2
¨¸
w
w
w
¨
¸
(14.51)
u
¨¸
;
y
2
¨
ww
y x
ww
y z
¸
w
y
¨¸
u
¨
¸
©¹
z
2
2
2
w
w
w
¨
¸
¨
¸
2
ww ww
zx
zy
w
z
©
¹
and
§
2
2
2
·
w
w w
0
0
¨
¸
2
2
2
w
x
w
y
w
z
¨
¸
¨
¸
2
2
2
w
w
w
¨
¸
0
0
(14.52)
¨
2
2
2
¸
w
x
w
y
w
z
¨
¸
2
2
2
¨
w
w w
¸
0
0
¨
¸
2
2
2
w
x
w
y
w
z
©
¹
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