Linear response of a planar FGM beam with non-linear variation of the
mechanical properties
Giuseppe Balduzzi*, Mehdi Aminbaghai*, and Josef Füssl*
*
Institute for Mechanics of Materials and Structures (IMWS)
Vienna University of Technology
Karlsplatz 13/202 A-1040 Vienna, Austria
e-mails: [email protected], [email protected],
[email protected]
ABSTRACT
Functionally Graded Material (FGM) beams and plates are more and more used in several engineering
fields since they allow to optimize the distribution of strength and stiffness within the structure leading,
as an example, to the maximal material exploitation and significant cost saving. The most natural
choice for the modelling of these structures is the usage of 1D (beams) or 2D (plates) models.
Unfortunately, models developed for bodies with constant mechanical properties along the axis/midplane could be inadequate and too coarse for the engineering practice. As an example, a non-symmetric
distribution of the mechanical properties within the cross-section induces significant transversal
displacement in a beam subjected to axial load. Since standard beams model separately the axial and
the shear bending problem, they are not able to catch the so far described effect and, therefore, ad hoc
models must be developed. Furthermore, the shear stress distribution within FGM beams can vary
significantly from one cross-section to the other, according to the changes of mechanical properties
distribution [1]. Consequently, also the shear correction factor seems to vary along the beam axis,
leading the shear stiffness to have a non-trivial distribution.
This contribution aims at (i) highlighting the non-trivial effects that a continuous variation of the
mechanical properties induces on stresses, displacement and stiffness distributions within a planar
beam body, (ii) proposing a linear model capable to tackle the effects of mechanical properties
variation, and (iii) demonstrating the proposed model capabilities. In greater detail, starting from the
remarks discussed in previous paragraph and following a derivation path recently proposed in [2, 3],
this contribution develops a suitable planar beam model capable to account the non-trivial effects so
far introduced. In greater detail, the beam model assumes the Timoshenko beam kinematics and it
results naturally expressed by several linear Ordinary Differential Equations (ODEs) considering both
displacement and internal forces as unknowns.
Few numerical examples will demonstrate that the proposed beam model can catch with high accuracy
all the so far listed effects, allowing for a simple and effective modelling of a large class of structures.
Specifically, the non-linear variation of the mechanical properties both within the thickness and along
the beam axis induces (i) variations of the cross-section geometric-centre position (introducing some
analogies with curved and non-prismatic beams), (ii) non trivial stress distribution (e.g., in a FGM
beam subjected to axial load non-vanishing shear stresses occur), (iii) complex beam constitutive
relations (i.e., the stiffness matrix is full, whereas for homogeneous beams it is diagonal), and (iv) nontrivial stiffness and displacements.
REFERENCES
[1] J. Murin, M. Aminbaghai, J. Hrabovský, V. Kutiš, and S. Kugler, “Modal analysis of the FGM
beams with effect of the shear correction function.” Composites Part B: Engineering, Vol. 45(1),
pp.1575-1582, (2013).
[2] G. Balduzzi, M. Aminbaghai, E. Sacco, J. Füssl, J. Eberhardsteiner, and F. Auricchio, “Nonprismatic beams: a simple and effective Timoshenko-like model”. International Journal of Solids
and Structures Vol. 90, pp.236-250, (2016).
[3] G. Balduzzi, M. Aminbaghai, F. Auricchio, and J. Füssl, “Planar Timoshenko-like model for
multilayer mon-prismatic beams”. Submitted, pp. 1-17, (2016).