## 1 Introduction  
Thin-walled beams are widely used as structural components in weight-sensitive industrial applications, largely due to the advantage of high strength to weight ratio. However, they are also prone to cross-section deformation due to \"thin walls\", which may cast significant influences on the mechanical performance [1]. Therefore, it is essential to consider the phenomenon in the design process. Classic beam theories fail to achieve this, since they preclude the detection of cross-section warping and distortion. To overcome this, many attempts have been made to develop new beam theories for thin-walled structures over the years. Among them, one normal strategy adopted is to reduce the three-dimensional (3D) elasticity formulation to a one-dimensional (1D) model. Such models are usually quite exact since no further assumptions are added. Typically, the Saint-Venant solution (SVS) has been the theoretical basis of several thin-walled beam models. For example, see [PERSON] [2] and [PERSON] et al. [3]. By adding enrichment terms to the Saint Venant solution, various non-Classic effects can be taken into account in beam models. Detailed implementations can be found in [PERSON] and [PERSON] [4]. However, the derived solutions are always problem-dependent [5]. Variational asymptotic methods (VAM) represent another powerful tool, which allows the decomposition of a 3D elasticity problem into a 1D formulation [6]. Now, it has stepped toward the refined beam model with a transcendental accuracy [7]. However, the solution is in closed form for only a few cross-section geometries and materials. In order to deal with arbitrary cross-sections, the variational asymptotic beam sectional analysis (VABS) was developed based on VAM, exploiting the finite element method over the cross-section to solve the variational problem and deriving beam-like equations for higher order effects [8]. Related work can be found in studies by [PERSON] et al. [9] and [PERSON] [10]. A recent contribution to refined beam theories is the Carrera unified formulation (CUF) models, where the order of the theory is optional and related to the convergence analysis. As a hierarchical formulation, it can reduce 3D problems to 2D or 1D by considering the definition of the displacement field through arbitrary expansions [11]. Over the past decade, CUF has been developed for the higher order analysis of thin-walled structures. Some valuable developments were reported by [PERSON] et al. [12] and [PERSON] et al. [13]. It should be noted that it allows special improvements on the accuracy of representing higher order effects [14]. Moreover, CUF is not dependent of the structural problem and thus no extra assumptions should be made to represent a particular structural effect. Therefore, this formulation has been successfully adopted in the modelling of thin-walled beams with arbitrary cross-sections [15]. Some refined formulations have also been developed by considering the addition of representative deformation modes. The cross-section nonuniform warping defined by [PERSON] [16] was the first higher order mode introduced to refine the displacement field. From then on, various warping functions have been formulated. The work reported in [PERSON] et al. [17] and [PERSON] et al. [18] are typical examples. Some approaches have focused on the use of shear correction factors to enhance classic beam theories. These shear correction factors provide a more appropriate shear distribution over the beam cross-section and thus accommodate the stress-free boundary conditions. Shear correlation factors can be defined in various ways, but are dependent on problem characteristics [19]. In the field of bridge engineering, some higher order deformation modes have been introduced to capture special structural behaviours or load conditions [20]. The analyses of the multi-cell distortion [21], torsional warping [22] and distortional warping [23] due to shear-lag effects in box girders are successful examples.