Large Deflection of a Non-Linear, Elastic, Asymmetric Ludwick Cantilever Beam

The investigated cantilever beam is characterized by a
constant rectangular cross-section and is subjected to a
concentrated vertical constant load at the free end. The same
beam is made by an elastic non-linear asymmetric Ludwick
type material with different behavior in tension and
compression. Namely the constitutive law of the proposed
material is characterized by two different elastic moduli and
two different strain exponential coefficients. The aim of this
study is to describe the deformation of the beam neutral surface
and particularly the horizontal and vertical displacements of the
free end cross-section. The analysis of large deflection is based
on the Euler-Bernoulli bending beam theory, for which crosssections,
after the deformation, remain plain and perpendicular
to the neutral surface; furthermore their shape and area do not
change. On the stress viewpoint, the shear stress effect and the
axial force effect are considered negligible in comparison with
the bending effect. The mechanical model deduced from the
identified hypotheses includes two kind of non-linearity: the
first due to the material and the latter due to large deformations.
The mathematical problem associated with the mechanical
model, i.e. to compute the bending deformations, consists in
solving a non-linear algebraic system and a non-liner second
order ordinary differential equation. Thus a numerical
algorithm is developed and some examples of specific results
are shown in this paper. Precisely, the proposed problem is a
generalization of similar cases in literature, consequently
numerical comparisons are performed with these previous
works, i.e. assuming linear elastic materials or assuming
symmetric Ludwick type material with same behavior in
tension and compression like aluminum alloy and annealed
copper. After verifying a proper agreeing with the literature, in
order to investigate the effect of the different material behavior
on the horizontal and vertical displacements of the free end
cross-section, numerical results are obtained for different
values of elastic moduli and strain exponential coefficients. The
arising conclusions are coherent with the assumed hypotheses
and with similar works in literature.