Higher-order Vlasov torsion theory for thin-walled box beams
Graphical abstract
Introduction
Due to non-rigid cross-sectional deformations, a thin-walled beam subjected to a twisting moment exhibits complex non-uniform torsion responses with a twist rate varying along the longitudinal direction. To reflect these non-negligible sectional flexibilities not considered in the classical St. Venant torsion theory (e.g., [1]), higher-order (or generalized) beam theories [2], [3], [4], [5] for torsion that include torsion-related sectional deformation shapes as additional degrees of freedom have been proposed based on the pioneering Vlasov torsion theory [6]. Numerical and systematic approaches [7], [8], [9], [10], [11], [12], [13], [14], [15], [16] to calculate enriched sectional deformation mode shapes have extended the types of beam section shapes to which higher-order beam theories for torsion can be applied. The resulting enriched mode shapes also allow these theories to handle more advanced analyses such as buckling analysis. However, to the best of authors’ knowledge, no higher-order beam theory for torsion appears to establish explicit F-U and σ-F relations (σ: stresses, U: kinematic variables, F: generalized forces (or work conjugates of U)) that are consistent with those established by the Vlasov torsion theory shown in Fig. 1. If the F-U and σ-F relations can be explicitly described in a manner consistent with those of the Vlasov torsion theory, the physical significance of the generalized forces, especially self-equilibrated forces, can be directly understood [17,18]. Moreover, these relations can be critically useful in deriving explicit equilibrium conditions among the generalized forces including self-equilibrated forces at a joint of multiple thin-walled beams [17]. Motivated by this observation, this study newly proposes a higher-order Vlasov torsion theory (or higher-order beam theory for torsion consistent with the Vlasov torsion theory) that not only includes as many torsion-related sectional deformation modes as desired but also provides the desired explicit F-U and σ-F relations. Along with the Vlasov torsion modes, orthogonal and hierarchical sets of higher-order torsion modes, which are essential for advanced analyses but not considered in the Vlasov theory, are additionally included. Fig. 1 suggests that the developed higher-order torsion theory is fully consistent with the Vlasov torsion theory in that the explicit relations by the Vlasov theory naturally carry over into the proposed theory considering more sectional deformation modes.
Before presenting the essence of the present theory, we review the Vlasov torsion theory first. To address the non-uniform torsion of a thin-walled beam involving restrained warping and distortional deformations, the Vlasov torsion theory [6] which reduces complex two-dimensional (2D) problems of shell theory to one-dimensional (1D) problems was proposed. In the theory, the cross-section of a thin-walled beam is regarded as a hinged beam system so that its edges (or walls) are treated as non-extensible and membrane-like shells. Under these assumptions, the Vlasov method yields two types of sectional deformation modes (i.e., warping and distortion) and their mode shapes. The Vlasov torsion theory incorporates these sectional deformation modes as additional kinematic variables (or degrees of freedom) in addition to typical torsional rotation associated with uniform torsion. The validity of the theory has been demonstrated by subsequent studies on thin-walled beams with different section shapes [2,19,20] and profiles [21], [22], [23]. Further investigations into the bending responses of the edges shown in the distortion [2,3,24] and secondary (or thickness) warping [25,26] have contributed to improve the accuracy of the theory. However, due to the assumptions used in the hinged beam systems, the sectional deformation modes calculated by the Vlasov method are insufficient to fully capture the complicated membrane and bending responses of the edges shown in advanced analyses (e.g., stress analysis) of a thin-walled beam.
Meanwhile, the kinematic variables of the Vlasov torsion theory lead to the definition of generalized 1D forces consisting of the twisting moment, longitudinal bimoment, and transverse bimoment. Here, the longitudinal and transverse bimoments represent the work conjugates of the warping and distortion, respectively, and they are self-equilibrated forces not producing any non-zero resultant force/moment. Although they are self-equilibrated, their physical significance can be readily understood if the F-U and σ-F relations are explicitly given, as suggested in Fig. 1 [17]. Taking the transverse bimoment () for example, one can see that the quantity GJQ is the rigidity corresponding to from the F-U relation and that the stress component is the shear stress σzs induced by from the σ-F relation. Moreover, one can express the edge resultants or the stress resultants at each edge of a beam section [17] which correspond to the bimoments, only when explicit F-U and σ-F relations are available. It has been shown in [17,18] that the concept of edge resultants is critical in establishing explicit relations among 1D field variables at a joint of a multiply-connected thin-walled beam system. This argument suggests that if the explicit F-U and σ-F relations are possible for a higher-order theory including more sectional deformation modes than used in the Vlasov theory, the physical significance of the generalized forces can also be readily available. However, no such relations have yet been established for thin-walled beams undergoing torsional deformation.
To better represent complex global/local twisting responses of thin-walled beams, higher-order (or generalized) beam theories for torsion incorporating enriched torsion-related sectional deformation modes have been proposed based on the Vlasov torsion theory. The abundant sectional mode shapes introduced in these higher-order theories have been numerically obtained by using various cross-section analysis methods. In the generalized beam theory (GBT) [7,[27], [28], [29]], a plane beam frame model that represents the 1D discretization of a beam section has been proposed with kinematic concepts to calculate GBT deformation modes. Other relevant works employing the plane beam frame model can also be found in [8,9,12,30,31]. Methods of 2D cross-section analysis extracted from three-dimensional (3D) continuum theory have also been proposed. The variational asymptotic beam section analysis (VABS) [11,32,33] has been developed from 3D energy functional by applying the variational asymptotic method [34]. The method of generalized eigenvectors (GE) [13,35] has been constructed from 3D equilibrium equations based on an extension of the St. Venant rod theory. To reflect the sectional flexibilities without cross-section analysis, the Carrera Unified Formulation (CUF) [10,[36], [37], [38]] has been established in which the complex 3D beam behavior is represented by generic expansion of 1D kinematic variables defined at specific section points. In a similar sense, a nod-based degree-of-freedom approach [39] utilizing GBT elementary deformation modes was also proposed. The review of the aforementioned studies shows that higher-order torsion modes representing membrane or bending deformations of section edges, which were not considered in the traditional procedure of the Vlasov method, are critical for more advanced analyses [12,[40], [41], [42], [43], [44], [45], [46]] such as torsional buckling or stress analysis. However, it appears that no higher-order beam theory for torsion has been proposed in which explicit F-U and σ-F relations consistent with those established by the Vlasov theory are established.
As far as the F-U and σ-F relations are concerned, it may be possible to write these relations with a currently available higher-order beam theory for torsion [13,47], but the resulting relations appear to be difficult to be written in a manner consistent with those of the Vlasov theory. The main reason is that the derivatives of the torsion-related sectional mode shapes may be involved in the σ-U relation; because the appearance of the derivative terms does not generally allow orthogonality among the sectional functions (i.e., sectional mode shapes and their derivatives) included in the σ-U relation, the resulting F-U and σ-F relations are usually written in a complex form rather than in a form consistent with the Vlasov theory. Here, it is worth emphasizing that in the Vlasov theory [17], the σ-U relation can be described only by the orthogonal sectional mode shapes themselves (without any derivative term) because the derivative of the warping mode shape can be replaced by a linear combination of other mode shapes of the torsional rotation and distortion. Along the same argument, the F-U and σ-F relations can be explicitly written in the Vlasov theory. If these observations can be utilized and extended for a higher-order theory involving more sectional modes, it may be possible to resolve the aforementioned difficulty. So, we propose to derive explicit relations that express each of the derivatives involved in the σ-U relation as a linear combination of other sectional mode shapes.
It may be worth reiterating the advantages to establish explicit F-U and σ-F relations for a higher-order beam theory for torsion. First, the concept of the Vlasov generalized forces, such as the twisting moment and two bimoments, can be directly extended to the higher-order theory. Thereby, the physical significance of newly added self-equilibrated forces (i.e., the work conjugates of the higher-order torsion modes) can be clearly interpreted. Second, each of the newly added self-equilibrated forces can be explicitly decomposed into edge resultants so that explicit equilibrium conditions among the generalized forces of a higher-order theory at a joint of multiple thin-walled beams can be established. Using the explicit equilibrium conditions together with the principle of virtual work, one can derive explicit matching conditions among 1D kinematic variables at a joint that are consistently useful regardless of the number of beam members connected at a joint, as shown in [17] for the analysis of box beam joints based on the Vlasov kinematics. The significance of establishing a higher-order theory with these consistent joint matching conditions can be found in other joint matching approaches [43,[48], [49], [50], [51], [52]] such as those using the displacement continuity at specific points or lines around a joint.
Motivated by the aforementioned observations, we newly propose a higher-order Vlasov torsion theory (HoVTT) that not only includes as many torsion-related sectional deformation modes as desired but also provides explicit F-U and σ-F relations fully consistent with those of the Vlasov theory (see Fig. 1). Along with the Vlasov torsion modes , orthogonal and hierarchical N sets (N≥ 1) of higher-order torsion modes (ts: torsion) are additionally included as the kinematic variables to fully describe both membrane and bending responses of the edges (or walls) in a thin-walled beam. Specifically, among the membrane modes , describe extension of the edges in the contour direction while describe complex longitudinal deformations of the edges, which cannot be represented by . The edge-bending modes describe pure bending deformations of the edges in the thickness direction. As the 1D generalized forces of the HoVTT, self-equilibrated forces are newly defined in addition to the Vlasov generalized forces . As illustrated in Fig. 1, the explicit F-U and σ-F relations established by the proposed HoVTT are fully consistent with those by the Vlasov theory in that each term of F in the F-U relation is explicitly related to the terms of U or their derivatives through a single proportional constant (i.e., rigidity) and that stress caused by β (β ∈ F) in the σ-F relation is explicitly written by the force β, the generalized moment of inertia (Jβ) for β, and the sectional mode shape (ψα) of the mode α (or the work conjugate of β). Moreover, the explicit F-U and σ-F relations for established by the Vlasov theory naturally carry over into the proposed HoVTT. To provide a detailed description for the HoVTT, we consider a thin-walled box beam depicted in Fig. 1 (the extension to beams with general sections is given in [53]). The box beam dimensions are denoted by length L, sectional width b, sectional height h, and edge thickness t. In addition to the global Cartesian coordinate system (x, y, z), the local rectangular coordinate system (nj, sj, z) with origin at the center of Edge j (j=1, 2, 3, 4) is introduced where (nj, sj, z) represent the through-thickness, contour, and longitudinal direction, respectively.
To establish the explicit F-U and σ-F relations, we newly introduce recursive relations and orthogonality conditions among the sectional mode shapes of the higher-order torsion modes. By doing so, the σ-U relation in the HoVTT can be expressed only in terms of the sectional functions (i.e., sectional mode shapes or their derivatives) that satisfy the orthogonality conditions. Because membrane stress in the σ-U relation are related to both the membrane deformation shapes of the modes (see Fig. 2) and their derivatives, we derive recursive relations that can be used to express each of the derivatives of the membrane deformation shapes for the Nth (N≥ 1) higher-order membrane modes as a linear combination of the membrane deformation shapes for the lower-order modes . In addition, the orthogonality conditions among the membrane deformation shapes of the modes are also employed. Meanwhile, edge-bending shear stress in the σ-U relation is expressed only in terms of the derivatives of edge-bending deformation shapes of the modes (see Fig. 2). Thereby, we newly establish orthogonality conditions with respect to these derivatives where no similar relation has been considered in earlier studies. More details on this aspect are presented in Section 2 along with the kinematic and generalized force concepts established in the proposed HoVTT.
An important aspect in establishing the recursive relations and orthogonality conditions is that sectional mode shapes for the higher-order torsion modes must be compatible with these establishments. To this end, the membrane deformation shapes of the higher-order membrane modes are newly derived in closed form in Section 3. Section 3 also discusses why the proposed recursive relations can be justified from the mechanics perspective. For example, the recursive relation corresponding to the Nth (N≥ 1) edge-extension mode will be shown to be theoretically derived using a normal strain in the contour direction of a thin-walled beam subjected to longitudinal membrane stress by (where the Poisson effect is considered). Meanwhile, along with the longitudinal membrane stress, longitudinal edge-bending stress varying linearly along the thickness direction is also induced by the bimoment due to the secondary (or thickness) warping effect [25,26]. For this reason, the edge-extension modes also include edge-bending deformation shapes as well as the membrane deformation shapes derived in Section 3 (see Fig. 2). Accordingly, edge-bending deformation shapes of the modes are analytically derived in Section 4 by considering an edge-bending normal strain in the contour direction caused by the longitudinal edge-bending stress of . While the modes are represented by both membrane and edge-bending deformation shapes, the higher-order edge-bending modes are represented only by the edge-bending deformation shapes as shown in Fig. 2. In Section 5, these edge-bending mode shapes are analytically derived by using the newly proposed orthogonality conditions concerning their derivatives. In addition, continuity conditions at section corners are also considered so that these edge-bending mode shapes satisfy the displacement, slope, and moment continuities at the corners [2,54].
To check the validity and accuracy of the proposed HoVTT, several numerical examples are considered in Section 6. We examine both membrane and edge-bending stresses along the section centerline and off-centerline, respectively, together with displacement distributions. Free vibration analysis is also performed. The results by the HoVTT are compared with those by the Vlasov torsion theory, Abaqus shell analysis [55], and GBTUL [56].
Section snippets
Higher-order Vlasov torsion theory
This section presents the fundamental concepts in the proposed higher-order Vlasov torsion theory (HoVTT) and establishes the field relations of the proposed HoVTT, such as the u-U, ε-u, σ-ε, F-U, and σ-F relations (u: displacements, ε: strains). The membrane and bending fields of the section edges are presented in Sections 2.1 and 2.2, respectively. For example, the membrane displacement field um (s, z) which is uniform in the thickness (n) direction will be considered in Section 2.1 while the
Derivation of membrane mode shapes
This section is devoted to the derivation of the sectional membrane mode shapes of the higher-order membrane modes that are consistent with the recursive relations in Eq. (10a) and satisfy the orthogonality conditions in Eq. (10b). Before deriving these mode shapes, it may be worth explaining why the proposed recursive relations are physically reasonable. To this end, the recursive analysis method is extended to deal with torsion-related membrane mode
Derivation of involved in the mode shapes of
If the mode shape of χts is represented only by the membrane deformation shape derived in Section 3.1, the displacement continuity at the corners cannot be satisfied in the mode shape of χts as shown in Fig. 5(b). This shows why edge-bending deformation representing the n-directional displacement of section edges is additionally required in the mode shape of χts. Hence, this section will derive the edge-bending deformation shapes involved in the mode shapes of
Derivation of edge-bending mode shapes
This section derives the sectional mode shapes of the higher-order edge-bending modes that satisfy the newly proposed orthogonality conditions in Eq. (25). Unlike the mode shapes of derived in Sections 3 and 4, the mode shapes of are represented only by the edge-bending deformation shapes (i.e., no membrane deformation is included) as shown in Fig. 2. Thus, in these mode shapes, no
Numerical analysis
This section presents the procedure to set up the 1D finite element equation of the HoVTT. Numerical results by the 1D equation will then be provided to demonstrate the validity and accuracy of the HoVTT.
Conclusions
This study newly developed a higher-order Vlasov torsion theory (HoVTT) for thin-walled box beams in which the F-U and σ-F relations (Eqs. (13), (14), (26), and (27)) fully consistent with those of the Vlasov torsion theory are explicitly established. For kinematic variables, hierarchical N sets (N≥ 1) of higher-order torsion modes are additionally included together with the Vlasov torsion modes. It was shown that the recursive relations and orthogonality conditions newly established (Eqs. (10a)
CRediT authorship contribution statement
Soomin Choi: Conceptualization, Methodology, Software, Validation, Writing - original draft, Writing - review & editing, Visualization. Yoon Young Kim: Supervision, Writing - original draft, Writing - review & editing, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This research was supported by grants (2016R1A5A1938472, 2016R1A6A3A01007359, and 2014M3A6B3063711) from the National Research Foundation of Korea (NRF) funded by the Korea government (MOE, MSIT) contracted through IAMD and SRRC at Seoul National University.
References (61)
- et al.
The effect of distortion in thin-walled box-spine beams
Int J Solids Struct
(1984) - et al.
First-order generalised beam theory for arbitrary orthotropic materials
Thin-Walled Struct
(2002) - et al.
A cross-section analysis procedure to rationalise and automate the performance of GBT-based structural analyses
Thin-Walled Struct
(2015) - et al.
A higher order thin-walled beam model including warping and shear modes
Int J Mech Sci
(2013) - et al.
A new beam element with transversal and warping eigenmodes
Comput Struct
(2014) - et al.
Higher-order thin-walled beam analysis for axially varying generally shaped cross sections with straight cross-section edges
Comput Struct
(2017) - et al.
A generalized model for heterogeneous and anisotropic beams including section distortions
Thin-Walled Struct
(2014) - et al.
Global-distortional buckling mode influence on post-buckling behaviour of lip-channel beams
Int J Mech Sci
(2020) - et al.
Data-driven approach for a one-dimensional thin-walled beam analysis
Comput Struct
(2020) - et al.
Exact matching at a joint of multiply-connected box beams under out-of-plane bending and torsion
Eng Struct
(2016)