456
12 Finite Element Method in Vibration Analysis
Fig. 12.7 Timoshenko beam
element
nodes, it is termed as complex. It has been observed that the simple Timoshenko
beam element is aptly adequate for general purpose use. We present the derivation
of one such element.
The Timoshenko beam element shown in Fig. 12.7 has two nodes at its two ends.
At each end, the degrees of freedom are the total deflection w and the bending rotation
θ .
The deflection function is given by
w = α 0 + α 1 x + α 2 x
2
+ α 3 x
3
(12.59)
The relation between transverse shear strain γ, w
and θ is
w
= θ + γ
(12.60)
Transverse shear strain is assumed to be independent of x, that is
γ = β 0
(12.61)
The bending moment–curvature relationship is
M = − E I
∂θ
∂ x
(12.62)
The shear force V is related to transverse shear strain γ by
V = μAGγ
(12.63)
12 Finite Element Method in Vibration Analysis
Fig. 12.7 Timoshenko beam
element
nodes, it is termed as complex. It has been observed that the simple Timoshenko
beam element is aptly adequate for general purpose use. We present the derivation
of one such element.
The Timoshenko beam element shown in Fig. 12.7 has two nodes at its two ends.
At each end, the degrees of freedom are the total deflection w and the bending rotation
θ .
The deflection function is given by
w = α 0 + α 1 x + α 2 x
2
+ α 3 x
3
(12.59)
The relation between transverse shear strain γ, w
and θ is
w
= θ + γ
(12.60)
Transverse shear strain is assumed to be independent of x, that is
γ = β 0
(12.61)
The bending moment–curvature relationship is
M = − E I
∂θ
∂ x
(12.62)
The shear force V is related to transverse shear strain γ by
V = μAGγ
(12.63)