An Approximate Analytical Solution of Transversal …
43
2 The Governing Equation
In what follows, we consider a flexible Euler–Bernoulli beam of length l subjected
by a constant axial force P = P 0 (Fig. 1).
The Cartesian coordinate system is adopted in the symmetric plane of the beam,
and w is displacement of a point in the middle plane of the beam in y-direction. In
Fig. 2 is considered an infinitesimal length of the Euler–Bernoulli beam with the
ends F and G.
Applying the Hamilton’s principle, we obtain [9, 10]:
δ
t 2
t 1
Ldt +
t 2
t 1
δW dt = 0
( 1 )
with L the Lagrange function and W the virtual work. The kinetic energy of the beam
is T =
m
2
l
0 (u
2
t + v
2
t )dx, m being the mass per unit length of the beam, and (u, v)
are longitudinal and transversal displacement, respectively, of end F from Fig. 2.
The strain potential energy of the beam is U =
1
2E
v σ
2 dv, where we consider the
linear relation between the stress and strain σ = Eε. The total strain of the point
F is ε = ε 0 + ε 1 where the strain caused by the axial displacement of the beam
is ε 0 =
ds−dx
dx
=
(1 + u x ) 2 + v 2
x − 1, and the strain of the point F located at the
distance z from the middle plane, caused by the rotation of the cross-sectional plane
θ (x, t) is ε 1 = z
dθ
ds
= z
W xx (1 + W
2
x )
−3/2
. By means of Taylor series, the strains ε 0
and ε 1 can be written in the form [9, 10]
Fig. 1 Flexible
Euler–Bernoulli beam
Fig. 2 A segment of an
infinitesimal length