An investigation of the free vibration behavior of thin-walled composite box-beams is carried out by considering different assumptions in the constitutive equations. Within the present model some non-classical effects, such as restrained warping and transverse shear, are incorporated. Free vibration results are validated against experimental and numerical results, which are available in the literature. The natural frequencies obtained based on the different assumptions of constitutive equations are compared, and it is revealed that these assumptions play an important role in the proper treatment of the free vibration behavior of torsion-bending coupled composite beams. The results obtained based on the proposed constitutive equations are demonstrated to have a good agreement with the finite element results as far as the lower natural frequencies of the beams are concerned.
The enhancement of the theory of thin-walled composite beams has notably drawn the attention of many researchers in the past two decades. It has been revealed that in order to properly treat the behavior of thin-walled composite beams, a number of nonclassical effects, such as restrained warping, transverse shear, and also the mechanisms of structural couplings, have to be carefully considered.
Song and Librescu  have proposed a model in which the effects of restrained warping and transverse shear on the vibration behavior of composite beams are studied. By including these effects, the predicted values of the natural frequencies have become lower than the corresponding values obtained by neglecting restrained warping effects. In a model developed by Chandra and Chopra , it is suggested that the variation of shear stiffness along the contour of a crosssection has a significant influence on the warping and twist characteristic of a composite beam. Volovoi and Hodges  have used an asymptotically correct linear theory for com posite thin-walled beams. They have revealed that the inclusion of hoop bending moment and shell bending is important for a proper prediction of torsional stiffness of a number of thin-walled composite beams. The latter finding is used by Jung et al.  in order to subsequently suggest a so-called mixed method, which combines the force and displacement approaches in a unified form. They have then achieved similar results to those of Volovoi and Hodges . Qin and Librescu  have validated the model developed earlier in Song and Librescu  against experimental data. They have shown that the static and dynamic behaviors predicted by this refined model are in good agreement with experimental data and other analytical models. Suresh and Nagaraj , in a comprehensive comparison between experimental and analytical results, have revealed that their proposed model, in which the refined warping function and higher shear deformation theory are considered, can predict the static and dynamic behavior of thin-walled composite beams efficiently. They suggested that the way the warping is modeled has a significant role in a correct treatment of thin-walled composite beams.
The model that is used in the current study is mainly based on the works by Song and Librescu  and Qin and Librescu  with some modifications in the constitutive equations. It will be shown that these modifications can significantly influence the free vibration behavior of single-celled thin-walled composite beams.
2. Theoretical Developments
For completeness, the previous model by Song and Librescu  and Qin and Librescu , on which the current study is mainly based, is presented below.
The geometric configuration and the chosen coordinate system are depicted in Figures 1 and 2.
In order to model a single-celled cross-section fiberreinforced thin-walled beam, the following assumptions are adopted :
(1) The cross-sections do not deform in their own planes.
(2) Transverse shear effects are incorporated. In addition, it is stipulated that the transverse shear strains, xz and yz, are uniform over the cross-sections.
(3) In addition to the warping displacement along the midline contour (referred to as primary warping), the off midline contour warping (referred to as the secondary warping) is also incorporated.
(4) It is assumed that over the cross-section, nn and Nsn are negligibly small when deriving the stress–strain constitutive law.
(5) The deformations are small and linear elasticity theory is used.
3. Results and Discussions
Initially, the validation of the theoretical developments is attempted to ensure the proper implementation of the methodology and the corresponding assumptions in the constitutive equations. This is achieved by obtaining the results for a composite box-beam based on the third set of assumptions given above, and comparing the results with those already available in the literature, which are obtained by an experimental test  or other theoretical studies [5, 21, 22]. Two cases with CAS lay-up configurations, which are denoted by CAS1 and CAS2 (see Table 1), are considered. Table 2 shows the geometric dimensions and material properties used in the corresponding box-beams.
In order to study the effects of three different sets of constitutive equations on the free vibration behavior of thin-walled composite box-beams, two cases of lay-up configurations referred to as CAS3 and CAS4 with varying ply angles are considered. These lay-ups have initially been introduced by Volovoi and Hodges . The stacking sequences of these layups are given in Table 3 and the material properties and geometric dimensions used in the analysis are listed in Table 2.
The predicted results by different sets of assumptions are compared with finite element method (FEM) results that are obtained by implementing an ANSYS code using Shell99 element type.