10.4 Dynamic Stiffness Matrix in Bending
403
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
Y
Y
Y
Y
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
=
⎡
⎢
⎢
⎣
sin λx
cos λx
sinh λx
cosh λx
λ cos λx −λ sin λx λ cosh λx λ sinh λx
−λ
2 sin λx −λ
2 cos λx λ
2 sinh λx λ
2 cosh λx
−λ
3 cos λx λ
3 sin λx λ
3 cosh λx λ
3 sinh λx
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
C 1
C 2
C 3
C 4
⎤
⎥
⎥
⎦ (10.39)
The end deformations of the beam segment can be expressed in terms of four
constants
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
Y i
Y j
θ i
θ j
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
=
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
Y at x = 0
Y at x = L
−Y
at x = 0
Y
at x = L
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
=
⎡
⎢
⎢
⎣
0
1
0
1
sin λL cos λL sinh λL
cosh λL
−λ
0
−λ
0
−λ cos λL λ sin λL −λ cosh λL −λ sinh λL
⎤
⎥
⎥
⎦
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
C 1
C 2
C 3
C 4
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
(10.40)
In compact form, Eq. (10.40) can be written as
{δ} e = [G]{C}
(10.41)
The end forces acting on the member can be expressed in terms of constants
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
V i
V j
M i
M j
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
=
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
E I Y
at x = 0
−E I Y
at x = L
E I Y
at x = 0
E I Y
at x = L
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
= E I λ
2
⎡
⎢
⎢
⎣
−λ
0
λ
0
λ cos λL −λ sin λL −λ cosh λL −λ sinh λL
0
−1
0
1
sin λL cos λL − sinh λL − cosh λL
⎤
⎥
⎥
⎦
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
C 1
C 2
C 3
C 4
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
(10.42)
In compact form, Eq. (10.42) can be written as
{P e } = [H ]{C}
(10.43)
From Eq. (10.41), we get
{C} = [G]
−1
{δ} e
(10.44)
Combining Eqs. (10.43) and (10.44) leads to
{P e } = [H ][G]
−1
{δ} e
(10.45)