Civil Engineering Reference
In-Depth Information
()
()
RRR
. This also applies at a global structural level, i.e.
t
r
r
r
t
=+
=+
n
n
n
tot
tot
()
t
and
RRR
. Before proceeding it is necessary to define the connectivity matrix
=+
tot
A
describing the relationship between element degrees of freedom
d
and global
n
n
tot
degrees of freedom
tot
r
, i.e.:
d
=⋅
Ar
(9.67)
n
n
tot
tot
Applying a set of virtual displacements to the system
δ
r
, then
δ
d
=⋅
Ar
δ
, and
tot
n
n
tot
tot
since the virtual work exerted by the external forces (at global as well as at element
level) must be equal to the sum of the virtual work of the internal forces, then
N
N
T
∑
T
∑
T
Rr
⋅
δ
=
R d
⋅
δ
=
F d
⋅
δ
(9.68)
tot
n
n
n
n
tot
tot
tot
tot
n
1
n
1
=
=
where
N
is the total number of elements in the system. Introducing Eq. 9.67, then
T
T
N
N
⎛
⎞
⎛
⎞
∑
T
∑
T
RA
⋅
δ
r
=
FA
⋅
δ
r
(9.69)
⎜
⎟
⎜
⎟
n
n
n
n
n
n
tot
tot
tot
tot
⎝
⎠
⎝
⎠
n
1
n
1
=
=
and thus, it is seen that the equilibrium condition at a global structural level is given by
N
N
∑
T
∑
T
AR
=
AF
(9.70)
nn
nn
tot
tot
n
1
n
1
=
=
where
N
is the total number of elements in the system. As shown in Eq. 9.24
Fmdc dkd
=
+
+
(9.71)
n
n
n
n
n
n
tot
tot
and assuming that Eq 9.62 applies (i.e. that indicial functions are not in use), then
()
()
()
t
t
t
RRR R
=+
+
=+
RR c
+
d
+
k
d
(9.72)
n
n
n
ae
n
n
ae
tot
ae
tot
tot
n
n
n
Introducing Eqs. 9.67, 9.71 and 9.72 into the equilibrium condition in Eq. 9.70 (and
acknowledging that the time derivative of mean values are zero) then the following is
obtained
