Analysis of Thin-Walled Beams

This book presents an advanced beam theory for accurate and efficient analyses of thin-walled beam structures, focusing primarily on thin-walled closed beams but including other types. Beam members exhibit non-negligible sectional deformations such as warping and distortion if they consist of thin-walled sections. Because classical beam theories, such as the Euler and Timoshenko beam theories [see, e.g., Gere and Timoshenko (1997)], use only six degrees of freedom (DOFs) representing three rigid-body translations and three rigid-body rotations of a beam cross-section, the aforementioned non-rigid sectional deformations cannot be depicted at all by them. Therefore, additional DOFs corresponding to non-rigid sectional deformations must be incorporated for an accurate analysis of a thin-walled beam, even when a beam theory is used. However, it is difficult to derive the sectional deformations systematically, and it is much more difficult to establish matching conditions among the corresponding degrees of freedom at a joint of multiply-connected thin-walled beams. It may be apparent that the standard field matching conditions established for classical beam theories are no longer useful if the field variables include DOFs representing non-rigid sectional deformations in addition to conventional six DOFs. In this case, therefore, an alternative field matching approach for these field variables should be established for an analysis of a thin-walled beam-joint structure.

As can be observed from the T-joint problem considered in Fig. 1.8, the effects of section deformations, i.e., torsional warping and torsional distortion, on the overall stiffness of a thin-walled beam are significant due to their coupling behavior with torsional rotation when the beam is subjected to a torsional load or a more general load. The torsional warping and torsional distortion modes involve deformation of the cross-section of a beam, but they will be treated as modes belonging to the fundamental mode set in the HoBT; these two modes as well as the six rigid-body section modes are treated as fundamental modes in this book. As shall be shown later, none of the higher-order section-deformable modes, such as torsional warping and torsional distortion modes (see Table 1.1), produce net non-zero resultant force or moment. Therefore, the stress and displacement of these modes should decay along the axis of a beam under a static load. However, their decay rates can be quite different depending on the mode type and order. For instance, the stress and displacement of torsional warping and torsional distortion modes can survive even several times longer than the cross-sectional width, while those of other higher-order modes decay rapidly. It should also be noted that due to geometric characteristics, the effects of the torsional warping mode for open-section beams generally survive longer than those for closed-sectioned beams. Because the warping effect in open-section beams is generally significant, a structural analysis of such beams using field variables having non-zero resultant forces and moments only produces unacceptably inaccurate results. For this reason, Vlasov (1961) mostly focused his analyses on the torsional warping mode in open-section beams in his beam theory. In the case of closed thin-walled cross-sections, the torsional distortion mode can be induced by torsional warping (Kim and Kim 1999a, b, 2000, 2003; Choi and Kim 2021; Camotim et al. 2010; Goncalves et al. 2010; Yu et al. 2012). Therefore, both torsional warping and torsional distortion modes should be used simultaneously to yield accurate results. (Note that torsion warping is generally coupled with torsion in thin-walled closed-section beams.)

Chapter 1 suggests that the three-dimensional displacements on the midline of a cross-sectional wall can be expressed as products of the generalized 1D displacements and sectional shape functions. The displacements at a general point on a cross-section are calculated using those on the midline based on the kinematic assumption Kirchhoff’s thin plate theory, i.e., the assumption of no shear and normal strains in the thickness direction \({(\varepsilon_{nn} = \gamma_{ns} = \gamma_{zn} = 0)}\). The local coordinates \((z,n,s)\) defined on the midline of a cross-sectional wall will be used to facilitate the expression of the strain field from the three-dimensional displacements. A two-dimensional stress field can be derived assuming a plane-stress state \((\text{i.e.,}\,\sigma_{nn} = \tau_{ns} = \tau_{zn} = 0)\) for thin walls.

The higher-order sectional deformations of a box beam differ depending on the type of an applied load (see Vlasov (1961), Schardt (1994), Cesnik and Hodges (1997), Kim and Kim (1999), Carrera et al. (2011), Genoese et al. (2014), Bebiano et al. (2018)). Different section deformation modes, which may be classified as torsional, extensional, and bending modes, are needed to deal with a thin-walled beam subjected to different load types. This chapter and Chap. 5 are devoted to detailed derivations of higher-order sectional shape functions corresponding to torsional modes (Table 4.1).

In Chap. 4, the sectional shape functions \((\psi_{z}^{W}\), \(\psi_{s}^{\chi })\) for a box beam under torsion, which correspond to the wall-membrane field, were derived. This section is devoted to the derivation of the sectional shape functions \((\psi_{n}^{\chi }\), \(\psi_{n}^{{\overline{\eta }}}\), \(\psi_{n}^{{\hat{\eta }}})\) corresponding to the wall-bending field. (See Ferradi and Cespedes (2014), Bebiano et al. (2015), and Choi et al. (2017) for earlier developments.) To argue for the co-existence of \(\psi_{n}^{\chi }\) with \(\psi_{s}^{\chi }\), we observe that if the distortion mode \(\chi\) has a non-zero \(\psi_{s}^{\chi }\) (the s-directional displacement component) only (see Fig. 5.1a) without its n-directional counterpart, \(\psi_{n}^{\chi }\), two adjacent sectional edges cannot remain connected at the corners. Therefore, \(\psi_{n}^{\chi }\) cannot be zero. It was shown in Chap. 2 that the zeroth-order distortion mode \(\chi_{0}\) has a non-zero \(\psi_{n}^{{\chi_{0} }}\), as given by Eq. (2.50). If \(\psi_{n}^{\chi } (z,s)\) does not vanish, it will induce a non-zero \(\tilde{u}_{s} (z,n,s)\) for \(n \ne 0\), as expressed by Eq. (3.3c) and thus causes the bending of cross-sectional walls. The sectional shape functions \(\psi_{n}^{{\chi_{k} }}\) for \(k \ge 1\) will be derived in Sect. 5.3 identically to how \(\psi_{s}^{{\chi_{k} }}\) was derived in Chap. 4; \(\psi_{n}^{{\chi_{k} }}\) can be obtained as the secondary deformation of the axial stress through Poisson’s effect.

In this chapter, we will derive the sectional shape functions for a box beam subjected to an extensional (or axial) load using a recursive and hierarchical method similar to that used to derive sectional shape functions for a box beam under torsion, as presented in Chaps. 4 and 5. In this chapter, the shape functions \((\psi_{z}^{W}\), \(\psi_{s}^{\chi })\) corresponding to the wall-membrane field and those (\(\psi_{n}^{\chi }\), \(\psi_{n}^{{\overline{\eta }}}\), \(\psi_{n}^{{\hat{\eta }}}\)) corresponding to the wall-bending field will be derived altogether. Although the applied load type considered in this chapter differs from the torsional loads considered in Chaps. 4 and 5, the characteristics of the sectional shape functions derived for extensional loads are identical to those derived for torsional loads. Therefore, the derivation procedures for both sets of shape functions are nearly identical. Accordingly, we will not present the details of the procedure used to derive the shape functions needed to deal with extensional loads.

The sectional shape functions of a box beam subjected to a flexural load are derived in this chapter using a procedure similar to that presented in Chaps. 4–6. (Other approaches may be found in Ferradi and Cespedes (2014) and Bebiano et al. (2015)). As in the cases for torsional or extensional loads, three types of deformable section modes are considered in addition to rigid-body section modes: (1) warping modes \(\{ W_{k} \}_{k = 1,2, \ldots }\), (2) unconstrained distortion modes \(\{ \chi_{k} \}_{k = 1,2, \ldots }\), and (3) constrained distortion modes \(\{ \overline{\eta }_{k} ,\hat{\eta }_{k} \}_{k = 1,2, \ldots }\). The warping mode \(W_{k}\) has the z-directional shape function \(\psi_{z}^{{W_{k} }} (s)\) only, which depicts the wall-membrane deformations of a beam section. On the other hand, the unconstrained distortion mode \(\chi_{k}\) has both the s-directional shape function \(\psi_{s}^{{\chi_{k} }} (s)\) representing wall-membrane deformation and the n-directional shape function \(\psi_{n}^{{\chi_{k} }} (s)\) representing wall-bending deformation. The constrained distortional modes \(\overline{\eta }_{k}\) and \(\hat{\eta }_{k}\) have only the n-directional shape functions \(\psi_{n}^{{\overline{\eta }_{k} }} (s)\) and \(\psi_{n}^{{\hat{\eta }_{k} }} (s)\), respectively, representing wall-bending deformations. The shape functions \(\psi_{s}^{{\chi_{k} }} (s)\) and \(\psi_{z}^{{W_{k} }} (s)\) representing wall-membrane deformations are derived in Sect. 7.2 while \(\psi_{n}^{{\chi_{k} }} (s)\), \(\psi_{n}^{{\overline{\eta }_{k} }} (s)\) and \(\psi_{n}^{{\hat{\eta }_{k} }} (s)\) representing wall-bending deformations are derived in Sects. 7.3–7.5. Section 7.6 presents numerical results using the derived modes.

Chapters 4, 5, 6, 7 presented procedures for deriving the shape functions of higher-order section-deformable modes of thin-walled rectangular cross-sections (or box beam cross-sections) recursively and hierarchically. To extend the procedure to thin-walled beams of generally shaped cross-sections, including open, closed, or open–closed sections, it is important to summarize the types of higher-order modes of a rectangular cross-section and the recursive relationships of their sectional shape functions. This summary will guide us to develop recursive relationships for the sectional shape functions of generally shaped cross-sections.

In Chaps. 4, 5, 6, 7, the shape functions of the deformable section modes of a box beam were derived in an approach with three key steps. Recursive equations derived in a differential form were integrated edgewise to find sectional shape functions. To determine the unknown coefficients and integration constants of the sectional shape functions, the geometric symmetry of a rectangular cross-section, continuity of field quantities at every corner of the section, and orthogonality conditions were used. However, as the conditions of geometric symmetry are mostly not available for a cross-section with a general shape, we need to find new conditions to determine the unknowns.

This chapter presents a higher-order beam analysis of a joint structure in which multiple straight box beam members are connected at a joint, as shown in Fig. 10.1a. Owing to extensive section deformation occurring near the joint, the overall structural behavior of the joint structure becomes considerably more flexible than predicted by the classical beam theory (Donders et al. (2009); Mundo et al. (2009)). One can certainly expect improved or more accurate predictions with a higher-order beam theory, but the field variables of multiple box beams at the joint are difficult to match (Basaglia et al. (2012); Basaglia et al. (2018); Choi et al. (2012); Choi and Kim (2016a, 2016b); Jang et al. (2008); Jang and Kim (2009); Jang et al. (2013). Unless they are matched accurately, there is no way to make the use of the advantages of a higher-order beam theory which is shown to be accurate for straight box beams without any joints.

In Chap. 10, we derived joint-matching conditions for a box beam by considering the equilibrium conditions of sectional and edge resultants, which are represented in terms of generalized forces. The principle of virtual work was used to derive the matching conditions in analytic form for the field variables. For a cross-section with a general shape, edge resultants may also be defined by following a procedure similar to that of a box beam. However, it is difficult to establish the matching conditions in analytic form because the relative positions of the matching edges of joining beams can be neither parallel nor intersecting. Furthermore, sectional edges are not always one-to-one matched when beams of different sectional shapes are connected at a joint. Therefore, an alternative method which does not explicitly deal with edge resultants should be developed.