Civil Engineering Reference
In-Depth Information
It is taken for granted that the material behaviour is linear elastic and that
displacements are small (i.e. there is no geometric non-linearity). The relationship
between cross sectional stress resultants and the derivatives of the corresponding
displacements are then given by the following differential equations (see e.g. Chen &
Atsuta [27]):
(
)
(
)
(
)
M xt GI r xt EI r xt
Mxt EIrxt
Mxt EIrxt
Vxt
,
,
,
⎫
=⋅
′
− ⋅
′′′
xr
t
r
w
r
θ
θ
⎪
(
)
(
)
,
=−
⋅
′′
,
⎪
⎪
yr
yzr
(
)
(
)
,
′′
,
=⋅
(7.64)
⎬
⎪
zr
zyr
(
)
(
)
(
)
,
Mxt
′
,
EIrxt
′′′
,
=−
=−
⋅
⎪
⎪
yr
zr
zyr
(
)
(
)
(
)
Vxt M xt
,
,
EI rxt
,
=
′
= −
⋅
′′′
⎭
zr
yr
yzr
where the prime behind symbols indicate derivation with respect to
x
. Defining the
cross sectional property matrix
()
T
x
r
0
EI
0
0
0
0
−
⎡
⎤
z
⎢
⎥
0
0
0
EI
0
0
−
⎢
y
⎥
⎢
⎥
()
x
0
0
0
0
GI
EI
T
=
−
(7.65)
r
t
w
⎢
⎥
0
0
EI
0
0
0
−
⎢
⎥
y
⎢
⎥
EI
0
0
0
0
0
⎣
⎦
z
then
(
)
x
,
t
as defined in Eq. 7.63 is given by
F
r
T
(
)
F
xt
,
T
⎡
rrrrrr
θθ
′′
′′′
′′
′′′
′
′′′
⎤
=⋅
(7.66)
r
⎣
y
y
z
z
⎦
Multi mode approach
Introducing the six by
N
mode shape derivative matrix
mod
φ
′′
""
φ
′′
φ
′′
⎡
⎤
y
y
y
1
i
N
mod
⎢
⎥
′′′
′′′
′′′
⎢
φ
""
φ
φ
⎥
y
y
y
1
i
N
⎢
mod
⎥
⎢
φ
′′
""
φ
′′
φ
′′
⎥
z
z
z
1
i
N
⎢
mod
⎥
()
x
β
=
⎢
(7.67)
r
′′′
′′′
′′′
⎥
φ
""
φ
φ
z
z
z
1
i
N
⎢
⎥
mod
⎢
⎥
φ
′
""
φ
′
φ
′
θ
θ
θ
1
i
N
⎢
⎥
mod
⎢
⎥
′′′
′′′
′′′
φ
""
φ
φ
θ
θ
θ
⎣
1
i
N
⎦
mod
