10.8 Beam Segments with Distributed Mass Having Shear Deformation…
419
V
M
=
⎡
⎣
( ¯
u − μAGT ) ( ¯
u − μAGT )
−η + μAGU
η − μAGU
sinh bαξ
cosh bαξ
sin bβξ
cos bβξ
− cosh bαξ − sinh bαξ
τ
cos bβξ
τ
sin bβξ
⎤
⎦
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
C 1
C 2
C 3
C 4
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
(10.95)
in which
¯
u = μAG
bα
L
(10.96)
η = μAG
bβ
L
(10.97)
= E I
bα
L
T
(10.98)
τ
= E I
bβ
L
U
(10.99)
Let the displacements at the ith node of the beam segment be Y i and ψ i , and
at the jth node, they are Y j and ψ j (Fig. 10.13). Similarly, the shear force and the
bending moment at the ith end and jth end are V i , M i and V j , M j , respectively. The
displacement and the force matrices are given by
Fig. 10.13 A beam element
419
V
M
=
⎡
⎣
( ¯
u − μAGT ) ( ¯
u − μAGT )
−η + μAGU
η − μAGU
sinh bαξ
cosh bαξ
sin bβξ
cos bβξ
− cosh bαξ − sinh bαξ
τ
cos bβξ
τ
sin bβξ
⎤
⎦
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
C 1
C 2
C 3
C 4
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
(10.95)
in which
¯
u = μAG
bα
L
(10.96)
η = μAG
bβ
L
(10.97)
= E I
bα
L
T
(10.98)
τ
= E I
bβ
L
U
(10.99)
Let the displacements at the ith node of the beam segment be Y i and ψ i , and
at the jth node, they are Y j and ψ j (Fig. 10.13). Similarly, the shear force and the
bending moment at the ith end and jth end are V i , M i and V j , M j , respectively. The
displacement and the force matrices are given by
Fig. 10.13 A beam element