Higher-order Vlasov torsion theory for thin-walled box beams

https://doi.org/10.1016/j.ijmecsci.2020.106231Get rights and content

Highlights

  • newly develops a higher-order beam theory for torsion consistent with the Vlasov torsion theory.

  • includes hierarchical N (N ≥ 1) sets of higher-order torsion modes together with the Vlasov modes.

  • establishes explicit F-U and σ-F relations (U: kinematic variables, F: generalized forces, σ: stresses).

  • shows that the explicit relations are fully consistent with those by the Vlasov torsion theory.

  • found that stresses calculated directly using the explicit σ-F relation are sufficiently accurate.

Abstract

Non-negligible sectional deformations, such as warping and distortion, occur in thin-walled beams under a twisting moment. For accurate analysis, these deformations need to be considered as additional kinematic degrees besides the degrees of freedom used in the classical St. Venant torsion theory. Vlasov pioneered to develop a higher-order beam theory for torsion that incorporates warping and distortion, but more sectional deformation modes than those considered in the Vlasov theory are needed to improve solution accuracy. Several theories were developed towards this direction, but no higher-order beam theory for torsion appears to allow explicit F-U and σ-F relations (U: kinematic variables, F: generalized forces, σ: stresses) as established by the Vlasov theory. In that the explicit relations are useful to interpret the physical significance of the generalized forces and can be critical in deriving explicit equilibrium conditions among the generalized forces at a joint of multiple thin-walled beams, a theory allowing the explicit relations needs to be developed. In this study, we newly propose a higher-order Vlasov torsion theory that not only includes as many torsion-related modes as desired but also provides the explicit F-U and σ-F relations that are fully consistent with those by the Vlasov theory. Towards this direction, we show that expressing the σ-U relation only with sectional mode shapes orthogonal to each other is critical in establishing explicit F-U and σ-F relations. We then establish new recursive relations that can be used to express each of derivatives for the sectional mode shapes involved in the σ-U relation as a linear combination of other orthogonal sectional mode shapes. In the developed theory, even stresses at off-centerline positions of the beam cross-section are explicitly related to F. The validity and accuracy of the proposed theory are confirmed by examining displacements, stresses, and eigenfrequencies for several torsion problems. The numerical results by the proposed theory are in good agreement with those by the shell analysis.

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 (Q0ts) for example, one can see that the quantity GJQ is the rigidity corresponding to Q0ts from the F-U relation and that the stress component (Q0ts/JQ)·ψχ0ts is the shear stress σzs induced by Q0ts 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 {θz,χ0ts,W0ts}, orthogonal and hierarchical N sets (N≥ 1) of higher-order torsion modes {χkts,Wkts,η¯kts,η^kts}k=1,,N (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 {χkts,Wkts}k=1,,N, {χkts}k=1,,N describe extension of the edges in the contour direction while {Wkts}k=1,,N describe complex longitudinal deformations of the edges, which cannot be represented by W0ts. The edge-bending modes {η¯kts,η^kts}k=1,,N describe pure bending deformations of the edges in the thickness direction. As the 1D generalized forces of the HoVTT, self-equilibrated forces {Qkts,Bkts,R¯kts,R^kts}k=1,,N are newly defined in addition to the Vlasov generalized forces {Mz,Q0ts,B0ts}. 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 {Mz,Q0ts,B0ts} 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 {θz,χ0ts,W0ts,{χkts,Wkts}k=1,,N} (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 {χNts,WNts} as a linear combination of the membrane deformation shapes for the lower-order modes {θz,χ0ts,W0ts,{χkts,Wkts}k=1,,N1}. In addition, the orthogonality conditions among the membrane deformation shapes of the modes {θz,χ0ts,W0ts,{χkts,Wkts}k=1,,N} 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 {θz,χ0ts,{η¯kts,η^kts}k=1,,N} (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 {χkts,Wkts}k=1,,N 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 χNts 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 {B0ts,{Bkts}k=1,,N} (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 B0ts due to the secondary (or thickness) warping effect [25,26]. For this reason, the edge-extension modes {χkts}k=1,,N 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 {χkts}k=1,,N 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 B0ts. While the modes {χkts}k=1,,N are represented by both membrane and edge-bending deformation shapes, the higher-order edge-bending modes {η¯kts,η^kts}k=1,,N 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 {ψsχkts,ψzWkts}k=1,,N

This section is devoted to the derivation of the sectional membrane mode shapes {ψsχkts,ψzWkts}k=1,,N of the higher-order membrane modes {χkts,Wkts}k=1,,N 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 {ψnχkts}k=1,,N involved in the mode shapes of {χkts}k=1,,N

If the mode shape of χts is represented only by the membrane deformation shape ψsχts 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 ψnχts 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 {ψnχkts}k=1,,N involved in the mode shapes of {χkts}k

Derivation of edge-bending mode shapes {ψnη¯kts,ψnη^kts}k=1,,N

This section derives the sectional mode shapes {ψnη¯kts,ψnη^kts}k=1,,N of the higher-order edge-bending modes {η¯kts,η^kts}k=1,,N that satisfy the newly proposed orthogonality conditions in Eq. (25). Unlike the mode shapes of {χkts}k=1,,N derived in Sections 3 and 4, the mode shapes of {η¯kts,η^kts}k=1,,N are represented only by the edge-bending deformation shapes {ψnη¯kts,ψnη^kts}k=1,,N (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)

  • S. Choi et al.

    Analysis of two box beams-joint systems under in-plane bending and axial loads by one-dimensional higher-order beam theory

    Int J Solids Struct

    (2016)
  • S.H. Zhang et al.

    A thin-walled box beam finite element for curved bridge analysis

    Comput Struct

    (1984)
  • Y. Kim et al.

    Analysis of thin-walled curved box beam under in-plane flexure

    Int J Solids Struct

    (2003)
  • L.F. Boswell et al.

    An experimental investigation of the behaviour of thin-walled box beams

    Thin-Walled Struct

    (1985)
  • M.M. Attard et al.
    (1989)
  • A. Martins et al.

    A new modal theory for wrinkling analysis of stretched membranes

    Int J Mech Sci

    (2020)
  • R. Schardt

    Generalized beam theory—an adequate method for coupled stability problems

    Thin-Walled Struct

    (1994)
  • R.F. Vieira et al.

    Definition of warping modes within the context of a higher order thin-walled beam model

    Comput Struct

    (2015)
  • I.S. Choi et al.

    Higher order analysis of thin-walled beams with axially varying quadrilateral cross sections

    Comput Struct

    (2017)
  • W. Yu et al.

    Variational asymptotic beam sectional analysis–an updated version

    Int J Eng Sci

    (2012)
  • G. Garcea et al.

    Deformation modes of thin-walled members: a comparison between the method of generalized eigenvectors and generalized beam theory

    Thin-Walled Struct

    (2016)
  • E. Carrera et al.

    Analysis of reinforced and thin-walled structures by multi-line refined 1D/beam models

    Int J Mech Sci

    (2013)
  • E. Carrera et al.

    Hierarchical theories of structures based on legendre polynomial expansions with finite element applications

    Int J Mech Sci

    (2017)
  • R. Gonçalves et al.

    Improving the efficiency of GBT displacement-based finite elements

    Thin-Walled Struct

    (2017)
  • E. Carrera et al.

    A global/local approach based on CUF for the accurate and efficient analysis of metallic and composite structures

    Eng Struct

    (2019)
  • D. Camotim et al.

    GBT buckling analysis of thin-walled steel frames: a state-of-the-art report

    Thin-Walled Struct

    (2010)
  • V. Mokos et al.

    Secondary torsional moment deformation effect by BEM

    Int J Mech Sci

    (2011)
  • E. Ghafari et al.

    Vibration analysis of rotating composite beams using polynomial based dimensional reduction method

    Int J Mech Sci

    (2016)
  • A. Genoese et al.

    A geometrically exact beam model with non-uniform warping coherently derived from the Saint Venant rod

    Eng Struct

    (2014)
  • S. de Miranda et al.

    A high performance flexibility-based GBT finite element

    Comput Struct

    (2015)
  • Cited by (0)

    View full text