Numerical free vibration analysis of homogeneous or composite beam using a refined beam theory built on Saint Venant’s solution Fares Naccache, Rached El Fatmi ⇑ Université de Tunis El Manar, Ecole Nationale d’Ingénieurs de Tunis, LGC, BP 37, Le Belvédère, 1002 Tunis, Tunisia article info Article history: Received 8 March 2018 Accepted 19 August 2018 Available online xxxx Keywords: Refined beam theory Saint-Venant’s solution Composite section Vibration Warping Distortion Finite element method Matlab tool abstract Free vibration problem of an arbitrary cross-sectional homogeneous or composite beam is investigated using a refined 1D beam theory (RBT). This theory includes a set of 3D displacement modes of the cross-section (CS) which reflects its mechanical behavior: the main part of these sectional modes is extracted from the 3D Saint Venant’s solution and another part is related to the CS dynamic behavior. These sectional modes, which are first derived from a CS analysis, lead to a consistent 1D beam model which really fits the section nature (shape and materials), and hence the beam problem. The numerical strategy to apply such general approach, is based on a first set of CS problems solved by 2D-FEM computations to get the sectional modes, and then the dynamic beam problem is solved by 1D- FEM computation according to RBT displacement model to provide (in fine) the first natural frequencies and 3D vibration mode shapes of the beam. To do so and in order to easily apply such method, a user friendly Matlab numerical tool named CSB (Cross-Section and Beam analysis) has been developed. To illustrate the capabilities and the accuracy of the method to catch the main 3D-effects, such as elas- tic/inertial coupling effects and 3D local/global mode shapes, a significant set of beam cross-section con- figurations with isotropic and anisotropic materials are analyzed. The first ten natural frequencies and 3D mode shapes are systematically compared to those obtained by full 3D-FEM computations, and some of them to literature. Ó 2018 Elsevier Ltd. All rights reserved. 1. Introduction Beam-like structures are one of the most used elements in structural engineering. High performance requirements make an increasing use of composite materials and a design of non- conventional cross-sections (shapes and materials). These opti- mized structures exhibit a complex mechanical behavior. Lami- nated composite beams are known to present phenomena such as coupled deformations arising from the anisotropic nature of the layers and from the stacking sequences, and the situation is more complex when, to reduce cost and weight, thin-walled open/- closed composite sections are involved. Detailed structural models are then essential in order to understand and predict such specific 3D effects due to the cross-section shape and materials. The use of 3D finite element (3D-FEM) analysis to help design is computationally costly, especially during the design phase when a lot of analyzes have to be carried out. This calls for the develop- ment of general and realistic beam theories and efficient numerical tools, designed for engineering practice, and suitable for the anal- ysis of beams exhibiting important 3D effects, for which the classi- cal beam theory assumptions are obviously no longer valid. The present work is a contribution to that expectation. Moreover, it is worth noting that the distinction about the kind of section (compact, thin walled and open/closed section) is no longer relevant for arbitrary composite cross-section as shown by El Fatmi [1]. Indeed, the mechanical behavior of a composite beam is the result of the whole nature of its cross-section: shape and material(s). Besides, as rightly emphasized by Silvestre and Camo- tim [2], it is necessary to beware of an intuition-based reasoning developed in the context of isotropic and homogeneous beams. An extensive research has been dedicated in the last decades to the dynamic analysis of homogeneous and composite beams focus- ing on thin-walled members because of their great advantages in many engineering applications. Different critical points have been investigated such as the effect of the length-to-thickness ratio, the material coupling effects, the shear deformation, the sectional in and out of plane warpings and the boundary conditions. All these points may have a significant influence on the dynamic structural behavior of a beam, as it can be drawn from the overview of some recent works that follows; this brief overview will also show a https://doi.org/10.1016/j.compstruc.2018.08.005 0045-7949/Ó 2018 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: fares.naccache@enit.utm.tn (F. Naccache), rached.elfatmi@e- nit.utm.tn (R. El Fatmi). Computers and Structures xxx (2018) xxx–xxx Contents lists available at ScienceDirect Computers and Structures journal homepage: www.elsevier.com/locate/compstruc Please cite this article in press as: Naccache F, El Fatmi R. Numerical free vibration analysis of homogeneous or composite beam using a refined beam the- ory built on Saint Venant’s solution. Comput Struct (2018), https://doi.org/10.1016/j.compstruc.2018.08.005