Civil Engineering Reference
In-Depth Information
Applying d'Alambert's principle at a position of external and internal equilibrium
defined by
(
)
r
x t
,
, and let the system be subject to an incremental virtual
el
tot
displacement
T
⎡
r
r
r
r
θ
⎤
δ
r
=
⎣
δδδδ
(9.6)
⎦
el
x
y
z
compatible with
T
⎧
[
]
δ
d
=
δ
dd dd d d
δ
δ
δ
δ
δ
δ
d
⎡ ⎤
⎪
1
1
1
2
3
4
5
6
δ
d
=
where
(9.7)
⎨
⎢
⎣ ⎦
δ
d
[
]
T
⎪
⎩
δ
d
=
δ
d
δ
d
δ
d
δ
d
δ
d
δ
d
2
2
7
8
9
10
11
12
such that
()
r
N
x
d
(9.8)
δ
=
⋅
δ
el
Then the external and internal works performed during this motion are given by:
L
T
T
(
)
W
∫
dx
=
δ
dF
⋅
+
δ
r
⋅ −
mr
(9.9)
ext
tot
el
0
el
0
and
L
L
T
T
(
)
∫∫
∫
c r
W
=
δ
ε s
⋅
⋅
(
dxdA
)
+
δ
r
⋅
dx
(9.10)
int
0
el
0
A
0
where (assuming shear centre axis)
⎫
diag m
⎡
m
m
m
⎤
m
=
⎣
⎦
⎬
0
x
y
z
θ
(9.11)
c
=
diag c
⎡
c
c
c
⎤
⎪
⎣
⎦
⎭
0
x
y
z
θ
are diagonal matrices containing the distributed mass and damping properties of the
element, and where
T
⎫
s=
⎡
ssss
⎤
⎣
⎦
x
y
z
yz
⎪
⎬
⎪
el
tot
(9.12)
T
ε
⎡
⎤
δ
=
ε ε ε γ
⎣
⎦ ⎭
x
y
z
θ
s
,
s
and
s
are cross sectional stress contributions from elastic beam stretching in
the
x
direction and bending in the
y
and
z
directions, while
s
is the cross sectional
yz
shear stress due to torsion.
δε
,
δε
,
δε
and
δγ
are the corresponding virtual strain
x
y
z
θ
