10.7 Dynamic Stiffness Matrix of a Bar Subjected to Axial …
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10.7 Dynamic Stiffness Matrix of a Bar Subjected to Axial
and Bending Deformations
When both axial and flexural deformations of a bar are considered, there are three
degrees of freedom at each node, and they are u i , y i and θ i at node i and u j , y j and
θ j at node j (Fig.10.11). Two types of analyses are presented in the following. In the
first case, it is assumed that the dynamic force effects are small enough, so that the
axial and bending terms are not coupled. In the second category, we will consider the
case where axial force has significant effect on bending, and as such the combined
effect of axial force and bending is to be considered as coupled.
10.7.1 Combined Uncoupled Axial and Bending Stiffness
As the dynamic bending and axial stiffness terms are uncoupled, the dynamic stiffness
matrix for this case is simply a combination of axial stiffness matrix and bending
stiffness matrix of Eqs. (10.58) and (10.47), respectively. It may be written as follows:
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
V i
V j
M i
M j
N i
N j
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
=
E I
L
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
α
L 3 −
α
L 3 −
β
L 2 −
β
L 2
0
0
−
α
L 3
α
L 3
β
L 2
β
L 2
0
0
−
β
L 2
β
L 2
γ
L
¯
γ
L
0
0
−
β
L 2
β
L 2
¯
γ
L
γ
L
0
0
0
0
0
0
ψ 1
L 2
L
r
2 ψ 2
L 2
L
r
2
0
0
0
0
ψ 2
L 2
L
r
2 ψ 1
L 2
L
r
2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
y i
y j
θ i
θ j
u i
u j
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(10.59)
Fig. 10.11 A bar with axial force and bending