Planar Timoshenko-like model for multilayer non-prismatic beams
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Planar Timoshenko-like model for multilayer non-prismatic beams Giuseppe Balduzzi
. Mehdi Aminbaghai . Ferdinando Auricchio . Josef Fu¨ssl
Received: 12 August 2016 / Accepted: 20 December 2016 The Author(s) 2017. This article is published with open access at Springerlink.com
Abstract This paper aims at proposing a Timoshenkolike model for planar multilayer (i.e., non-homogeneous) non-prismatic beams. The main peculiarity of multilayer non-prismatic beams is a non-trivial stress distribution within the cross-section that, therefore, needs a more careful treatment. In greater detail, the axial stress distribution is similar to the one of prismatic beams and can be determined through homogenization whereas the shear distribution is completely different from prismatic beams and depends on all the internal forces. The problem of the representation of the shear stress distribution is overcame by an accurate procedure that is devised on the basis of the Jourawsky theory. The paper demonstrates that the proposed representation of cross-section stress distribution and the rigorous procedure adopted for the derivation of constitutive, equilibrium, and compatibility equations lead to Ordinary Differential Equations that couple the axial and the shear bending problems, but allow practitioners to calculate both analytical and numerical solutions for almost arbitrary beam geometries. Specifically, the numerical
G. Balduzzi (&) M. Aminbaghai J. Fu¨ssl Institute for Mechanics of Materials and Structures (IMWS), Vienna University of Technology, Karlsplatz 13/202, 1040 Vienna, Austria e-mail: [email protected] F. Auricchio Department of Civil Engineering and Architecture (DICAr), University of Pavia, Via Ferrata 3, 27100 Pavia, Italy
examples demonstrate that the proposed beam model is able to predict displacements, internal forces, and stresses very accurately and with moderate computational costs. This is also valid for highly heterogeneous beams characterized by thin and extremely stiff layers. Keywords Non-homogeneous non-prismatic beam Tapered beam Beam of variable cross-section First order beam model Arch shaped beam
1 Introduction According to the terminology introduced by Balduzzi et al. (2016), the definition multilayer non-prismatic beam refers to a continuous body made of layers of different homogeneous materials, in which the geometry of each layer can vary arbitrarily along the prevailing dimension of the beam. Both researchers and practitioners are interested in non-prismatic beams since they allow to reach extremely important optimization goals such as the desired strength with the least material usage. Furthermore, multilayer nonprismatic beams are nowadays more and more employed in different engineering fields since the workability of materials (like steel, aluminum, composites, wooden or plastic products) and modern production technologies (e.g., automatic welding machines, 3D printers) allow to manufacture elements with complex geometry without a significant increase of productio
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