Environmental Engineering Reference
In-Depth Information
elastic properties [8-11]. The problem is formulated with respect to the remaining
third direction defi ning the axis of the beam also called elastic axis.
Beam theory considers combined bending, tension and torsion. There are sev-
eral beam models of varying complexity. The simplest is the fi rst order or Euler-
Bernoulli model in which the elastic axis is considered rectilinear while cross
sections originally normal to it remain so in the deformed state. As a consequence
shear is eliminated. Shear will be included in Section 7.1 while in Section 7.2,
second order theory will be briefl y presented.
There are three steps to take: (a) defi ne the deformation kinematics, (b) intro-
duce the stress-strain relations, (c) apply dynamic equilibrium to a differential
volume of the structure. To this end, consider a beam with its elastic axis along the
y
-axis of the co-ordinate system [O;
xyz
] which for this reason is called
beam
sys-
tem. Bending takes place in the
x
(lead-lag) and
z
(fl ap) directions while tension
and torsion both take place in the
y
direction. For any point of the structure, let
T
=
denote its position in the un-deformed and
deformed state, respectively. Assuming small rotations (Fig. 1 shows the situation in
the
xz
plane):
r
=
(,,)
x
y
z
T
(,,)
and
r
xyz
0
0
0
0
⎛⎞
U
100
u
⎡
0
z
0
⎤
q
q
q
⎛⎞⎡
⎤⎛⎞
0
x
⎢
⎥
⎢
⎥
⎜⎟
⎜⎟
⎜⎟
Urr
=− =
V
=
010
v
+−
z
0
x
( 3)
⎜⎟
⎢
⎥
⎢
⎥
0
⎜⎟
⎜⎟
0
0
y
⎜⎟
⎢
⎥
⎢
⎥
⎝⎠
W
001
⎝⎠
w
0
−
x
0
⎣
⎦
⎝⎠
⎣
⎦
0
z
are the displacement and rota-
tion vectors of the section that defi ne its deformation state. In the Euler-Bernoulli
model,
T
(,) (,, )
uyt
=
uvw
T
(,) ( , , )
xyz
where
and
q
yt
=
q
q
q
so there are only four independent deformation
variables
u
(
y
,
t
) = (
u, v, w, q
y
)
T
:
q
=∂
w
,
q
=−∂
u
,
x
y
z
y
0
1
T
rr
=+
S
()
r
⋅ +
u
S
()
r
⋅
u
′
,
u
(,) (,,, ), )
yt
=
uvw
q
′
=∂ ∂
)/
y
0
0
0
y
100
z
0
00
0
⎡
⎤
⎡
⎤
0
⎢
⎥
⎢
⎥
( 4 )
0
1
S
()
r
=
0 1 0 0
,
S
()
r
= −
z
0
−
x
0
⎢
⎥
⎢
⎥
0
0
0
0
⎢
⎥
⎢
⎥
001
−
x
0
00
0
⎣
⎦
⎣
⎦
0
Equation ( 4 ) defi nes the deformation kinematics with respect to the beam system.
z
y
y
z
z
P
P
0
w
q
y
u
u
x
x
x
Figure 1: Deformation a beam cross section.
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