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Where
un
is the vector of virtual displacements,
n
is the associated virtual
strains, b
n
is the vector of applied body forces,
tractions, is the vector of stresses, is the mass density,
n
parameter and refers to differentiation with respect to time.
interest, has two boundaries:
specified and boundary on which displacements
u
t
n
24
t
n
is the vector of surface
c
n
is the damping
is the domain of
boundary on which boundary conditions
u
n
are specified.
n
t
n
are
For a finite-element representation, the displacements and strains and their
virtual counterparts are given by the expressions as [15]
n
m
id
i
n
, u
n
Nid
i
n
u N
i1
m
m
i1
n
Bid
i
n
,
n
Bid
i
n
i1
where at the time
n
, and the vector
of virtual nodal variables is , N
i
is the global shape functions matrix
m
i1
t for node I vector of nodal displacement is d
d i
n
and B
i
is the global stain-displacement matrix. The total number of nodes is m.
After substituting these equations into Eqn. (1), the resulting equation is true for
any set of virtual displacements, then we can obtain for each node i the equations
[11] as,
p f f f f
(4)
0
i n
Bi n
where the internal resisting forces are
Ii
n
Di
T
p
B
d
i
n i n
(5)
the consistent forces for the applied body forces are
the inertial forces are
T
f
N
b d
n
Ti
n
n i
(2)
(3)
Bi n i n
(6)
T
f
N
N , N ,..., N
Ii
n
i
n
1
2
m
n
d
1
d2
n
.
d
.
.
dm
n
(7)