Discussion on ‘A comprehensive analysis
of hardening/softening behavior of shearable planar beams
with whatever axial boundary constraint’, S. Lenci et al.,
Received: 12 November 2016 / Accepted: 30 December 2016 / Published online: 18 January 2017
Ó Springer Science+Business Media Dordrecht 2017
Abstract Two inconsistencies in the beam model
presented in the paper cited in the title are shown.
Keywords Non-linear elasticity Á Helmholtz
conditions Á Beam theory
This is a brief comment on the choice of strain
measures, equilibrium equations and mass matrix for
studying a shear deformable beam in a paper published
in this Journal , as well as in the related studies
[2–4]. In the following note, equation numbers are
from , if not stated differently.
In those papers, Authors investigate the non-linear
dynamic behavior of a planar shear deformable beam,
neglecting damping. The derivation of the theoretical
model starts from the deﬁnition of the kinematics (i.e.
a strains–displacements relationship), which, together
with the equilibrium equations and the constitutive
law, deﬁnes the elastic behavior of the beam.
As I shall discuss below, the developed model is
mechanically inconsistent: the Principle of Virtual
Work (PVW) does not hold because the internal elastic
energy of the beam cannot be deﬁned. Furthermore it
does not satisfy objectivity (i.e. frame invariance).
An approach to get the elastica of a beam is to
deﬁne strains through the kinematics, together with the
stresses as energy-conjugate of such strains. Then, it is
postulated that a relationship (the constitutive law)
exists between such conjugate pairs. In this approach,
the kinematics is usually assumed or justiﬁed by
geometrical arguments, but the equilibrium equations
follow analytically upon the application of the PVW.
The inverse approach, which for the non-linear
planar Cosserat rod has been pioneered by Reissner
, is to write the equilibrium equations of the beam.
If these equations are consistent, then one ﬁnds the
kinematics through the PVW. I remark, using Reiss-
ner’s words after writing the equilibrium equations,
that ‘We are ignorant, at this point,[…] in regard to
deﬁnitions for the components of strain’. In the same
way, if we follow the previous inverse approach, we
should be ignorant in regard to the deﬁnitions for the
components of stress.
A third (semi-inverse) approach could be to assume
both the kinematics and the equilibrium equations, as
done by Authors, but for such an approach it is
mandatory to ﬁnally check if the PVW holds.
The beam kinematics deﬁned by the Authors is non-
standard because curvature strain is chosen as the
derivative of the rotation angle with respect to the
deformed, rather than reference, arc-length. They
This Commentary refers to the Article available at: 10.1007/
D. Genovese (&)
Meccanica (2017) 52:3003–3004