DISCUSSIONS AND CLOSURES
Discussion of “On the Comparison
of Timoshenko and Shear Models
in Beam Dynamics” by Noël Challamel
October 2006, Vol. 132, No. 10, pp. 1141–1145.
DOI: 10.106/ASCE0733-93992006132:101141
J. Dario Aristizabal-Ochoa
1
1
125-Year Generation Professor, National Univ., School of Mines,
Medellín, Colombia. E-mail: jdaristi@unalmed.edu.co
This technical note questions the validity of a general approach
developed by Aristizabal-Ochoa 2004 for the dynamic analysis
of a Timoshenko beam-column with generalized boundary condi-
tions and the nonclassical modes of vibration of shear beams, in
particular the theoretical relationship between both models for
large values of bending to shear stiffness parameter. A simply
supported beam is analytically studied for both models by the
author of the discussed paper, and asymptotic solutions are ob-
tained for large values of bending to shear stiffness parameter.
Using the particular case of a simply supported beam with zero
rotational inertia i.e., r =0, the author claims that, “In the gen-
eral case, it is proven that the shear beam model cannot be de-
duced from the Timoshenko model, by considering large values
of bending to shear stiffness parameter. This is only achieved for
specific geometrical parameter in the present example.” Finally,
the author concludes based on his asymptotic solutions of the
simply supported beam with zero rotational inertia that “the ca-
pability of the shear model to approximate Timoshenko model for
large values of bending to shear stiffness parameter is firmly de-
pendent on the material and geometrical characteristics of the
beam section and on the boundary conditions.”
The author is congratulated for studying the free vibration of
beams. However, it is unfortunate that the author uses a degener-
ate beam model to prove that the general approach developed by
Aristizabal-Ochoa 2004 for the dynamic analysis of a Timosh-
enko beam-column with generalized boundary conditions and the
nonclassical modes of vibration of shear beams is in some way
incorrect. The main objective of this discussion is to show that 1
it is incorrect to assume a Timoshenko beam with r =0 and still
expect that it is capable of predicting the free vibration of the
Timoshenko shear model for large values of bending to shear
stiffness parameter; and 2 the conclusions based on the
asymptotic solutions presented by the author are not correct be-
cause they are based on a free-vibration analysis of a degenerate
Timoshenko beam.
What follows is the proof that it is incorrect to assume that the
rotational inertia can be simply ignored in the free-vibration
analysis of shear beams and to expect that the natural frequencies
and modal shapes are correct. Kausel 2002 elegantly proved this
point in shear beams with unrestrained or partially restrained ends
against rotation of the cross sections, showing that the classical
solutions which are based on r =0 violate the principle of angu-
lar moment.
This argument can be easily proved using the governing equa-
tions of the Timoshenko beam model presented by the author
m ¯
2
y
t
2
- GA
s
2
y
x
2
+ GA
s
x
=0 1
m ¯ r
2
2
t
2
- GA
s
y
x
- - EI
2
x
2
=0 2
Eqs. 1 and 2 are the transverse motion and moment equi-
librium equations of a differential element of a Timoshenko
beam, respectively. The two continuous degrees of freedom y and
are always coupled together in a Timoshenko beam as long as
both inertias m ¯ and m ¯ r
2
are different from zero. However, in a
Timoshenko shear beam / x = 0, and consequently Eq. 1 is
reduced to
m ¯
2
y
t
2
- GA
s
2
y
x
2
= 0 or
2
y
x
2
=
1
C
s
2
2
y
t
2
3
where C
s
=transverse shear wave velocity.
However, in the classical analysis of shear beams it is assumed
that y / x = ;
2
/ x
2
=0 since / x =0 and r =0. As a conse-
quence, Eq. 2 is totally ignored because each one of the three
terms of Eq. 2 is equal to zero. This explains the claims by
Kausel 2002. Shear beams based on the classical Bernoulli and
Euler theories are definitely degenerate systems becoming evi-
dent mostly when the member is free-free and pinned-free at
very low values of r, see Figs. 3 and 5 of Kausel 2002 and also
Figs. 2–6 of Aristizabal-Ochoa 2004.
On the other hand, in a Timoshenko shear beam, Eq. 2 re-
mains unchanged. Now, knowing that M =-EI / x, Eq. 2 can
be written as follows:
m ¯ r
2
2
t
2
- GA
s
y
x
- +
M
x
= 0 or
Mx = M
x=0
+
0
x
GA
s
y
x
- - m ¯ r
2
2
t
2
dx 4
Eqs. 3 and 4 are identical to those developed by Kausel
2002, p. 664 for Timoshenko shear beams when the effects of
body forces are neglected. Kausel shows that 1 Eq. 4 must be
satisfied, and, together with Eq. 3 and the boundary conditions,
the natural frequencies and corresponding modes of vibration can
be determined directly; and 2 the effects of the rotational inertia
m ¯ r
2
become significant for low values of r and vanish when r
tends to infinity preventing the rotation .
Based on this analysis, it is wrong to assume r =0 for two
reasons:
1. Since the term
0
x
m ¯ r
2
2
/ t
2
dx cannot be equal to zero
unless Mx = M
x=0
= 0 and y / x = , then rotational accel-
eration
2
/ t
2
must become infinity. This situation is par-
ticularly evident in the case of a simply supported shear
beam used by the author as proof, since the bending mo-
ments at both ends are zero i.e., M
x=L
= M
x=0
=0. This
conflict is similar to that described by Panovko and
Gubanova 1964, pp. 127–130 in the “half degree of free-
dom” system; and
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