A complete dynamic approach to the Generalized Beam Theory cross-section analysis including extension and shear modes

Abstract

A simple and efficient ‘complete dynamic approach’ is proposed, and named GBT-D, to evaluate a suitable basis of modes for the elastic analysis of thin-walled members in the framework of Generalized Beam Theory (GBT). The basis includes conventional and non-conventional modes, the latter accounting for transverse extension and membrane shear strain of the plate elements forming the cross-section, which identically vanish in the former set. The method relies on the solution of two distinct eigenvalue problems, governing the in-plane and the out-of-plane free oscillations of a segment of a thin-walled beam. Both the eigenvalue problems, differential in origin, and defined on a one-dimensional spatial domain, are transformed into an algebraic problem by means of a discretization carried out at the cross-section middle line. Numerical examples are then presented to outline the ease of use of the proposed method considering a single plate, an open cross-section and a partially closed one. Member analyses are also performed for the simplest boundary conditions, to validate the accuracy of the proposed GBT-D approach against finite element method results and analytical solutions, highlighting the importance in including the non-conventional modes.

Keywords

Generalized Beam Theory thin-walled beams cross-section analysis dynamic approach extensional modes shear modes

1. Introduction

Thin-walled beams (TWBs) behave in a substantially different way from compact beams, since their response to loads is remarkably influenced, even in the linear field, by the deformation of the cross-section, both in-plane and out-of-plane (warping). The classical Vlasov theory [1] accounts for warping effects only, while it ignores any in-plane deformation. In recent decades, many researchers have proposed different extensions to Vlasov theory in order to account for in-plane deformations in refined beam models.

Among the different approaches presented in the literature, the Generalized Beam Theory (GBT) seems to have attracted the attention of several researchers (e.g. [2 –9] for a wide overview). The theory is based on a semi-variational approach, in which the displacement field is expressed as a linear combination of assumed cross-sectional deformation modes and unknown amplitude functions, depending on the beam abscissa. This approach has been described from a general point of view in several books and papers, for example [10 –13], and successfully applied in many different contexts, including the formulation of reduced plate and beam models in piezoelasticity (e.g. [14 –17]) and poroelasticity problems (e.g. [18 –20]). From a methodological point of view, it may serve not only to the reduction from three-dimensional (3D) to one-dimensional (1D) problems, but also for a simplification of the geometry while remaining on 3D models, for example [21,22]. Concerning beams with deformable cross-sections, it has to be remarked that a GBT solution may profit from any ‘a priori’ knowledge about the deformation of the beam section. In this context the formal expansions obtained in [23 –26] may suggest interesting extensions of Timoshenko and Bernoulli–Euler models, by also including second- or higher-order effects of the thickness-to-diameter ratio. Furthermore, to the best knowledge of the authors, few references exist, for example [27], on the consequences at the macro-mechanical level of higher displacement gradients, which can lead to interesting developments in the theory of TWBs. In the framework of GBT, the 3D elasticity problem is transformed into a simpler 1D problem, as in the compact beam theory. Differently from that, however, the unknowns are not the classical translations and rotations of the rigid cross-section, but the amplitude of the deformation modes (the rigid modes being a subset of them). Moreover, the GBT approach refers to cross-sections formed with flat plate segments that deform according to the Kirchhoff model, unlike the direct approach that considers beams as deformable curves endowed with a certain microstructure (e.g. [28,29]). The use of the GBT relies on two fundamental steps being performed: (a) selection of the deformation modes for the cross-section (usually referred to as the cross-section analysis); and (b) solution of the 1D problem (member analysis). While the second step, except for very simple boundary conditions of the beam, is usually carried out by means of the finite element (FE) approach, the first step has received extensive attention in the research community over the years, as briefly outlined below. From a mathematical point of view, this procedure consists of a sort of separation of variables (since the cross-section deformation is expressed as a function of s, the curvilinear abscissa along the middle line of the transverse profile C, whereas the member analysis depends on z, the abscissa along the beam axis). In this sense it presents strong formal similarities with the technique of the Proper Generalized Decomposition, which is proven to be very suitable for solving highly multi-dimensional models [30], although, at the present state of research, the interest of the GBT formulation has not yet specifically addressed the construction of reduced-order models.

The classical GBT approach was first introduced by Schardt [2,3] and disseminated in English by Davies and co-workers, for example [4,31,32]. It was then applied and extended by Camotim and co-worker to a wide range of structural problems, for example [5,33 –39]. The classical method is based on the two fundamental Vlasov hypotheses: (V1) the walls of the beam are unshearable; and (V2) they are inextensible in the cross-section plane. Due to unshearability, tangential in-plane displacements and warping displacements are linked to each other so that, once one of the two fields is known, the other can be consequently evaluated. The classical GBT procedure consists of the following two steps: (a) a piecewise linear warping distribution is assumed to occur along the cross-section middle line, so that the warping is identified from its nodal values; (b) based on assumptions V1 and V2, the corresponding in-plane displacements are calculated performing a linear elastic analyses of the cross-section, considered as an inextensible planar frame. These obtained modes are supplemented by purely in-plane (no-warping) modes, still evaluated by means of a linear elastic analysis of the planar frame.

Most of these modes, however, have a localized character, since they include the deformation of a limited part of the cross-section. Therefore, in order to deal with modes possessing a global character, in which deformations are extended to the whole section, a change of basis is performed to produce a mutually orthogonal set of modes, which diagonalize some of (but not all) the matrices governing the structural problem. In the resultant, final, set of modes it is possible to distinguish: (a) rigid modes (in which the cross-section remains undeformed, that is, the classical Vlasov modes); (b) distortional modes (in which the ‘natural’ nodes, at which the plates of the assembly join each other, undergo in-plane displacements); (c) local modes (in which the natural nodes remains virtually at rest), still evaluated by linear elastic analysis of the planar frame. The set of the three types of modes is usually referred to as conventional, in accordance with the definition provided in [8], to stress the fact that these describe the deformation patterns that are believed to be most relevant in the representation of the mechanical behaviour of a TWB.

Still limited to conventional modes, a completely different approach was proposed in [40], where the classical methodology of GBT cross-section analysis was reversed. An in-plane analysis was first carried out by solving a dynamic eigenvalue problem relevant to an inextensible (V2 hypothesis) planar frame, having the shape of the cross-section middle line. Successively, the warping was evaluated by enforcing the V1 hypothesis. The procedure permits one to avoid laborious elastic analyses, as well the orthogonalization among the modes, which turn out to be directly of global type, although slightly different from that of the classical approach.

An alternative procedure to evaluate the conventional modes was also proposed in [9,41], where a complex eigenvalue problem, formulated in terms of the elastic potential energy, was used to select the deformation modes. A variant of this approach was recently proposed in [42], which consists of solving a real eigenvalue problem, suitably defined. In particular, a set of orthogonal modes is directly determined, identical to that produced by the two-steps procedure of the classical approach.

Classical modes, however, are not exhaustive of all the deformation modes that a TWB can experience. The topic has very recently been discussed in [8,39], where shear modes and extensional modes were additionally introduced by releasing the V1 and V2 assumptions. These non-conventional modes are expected to play an important role in some particular linear problems. For example, extensional modes are considered to be essential in a non-linear analysis, where coupling of the flexural and membrane behaviours of plate elements occurs. The construction of the non-conventional modes were carried out in [8,39] using the classical methodology, with the addition of selected orthogonalizations of modal matrices considered in pairs.

In this paper, a new ‘complete dynamic approach’ is presented: it is able to evaluate the whole set of conventional and non-conventional modes, thus extending the approach proposed in [40]. The method, referred to as GBT-D throughout this paper, is based on the formulation of two distinct eigenvalue problems, relevant to a segment of TWB of unitary length, namely: (i) a Planar Eigenvalue Problem (PEP), governing the in-plane free oscillations of the segment, behaving as an extensible planar frame; and (ii) a Warping Eigenvalue Problem (WEP), governing the out-of-plane free oscillation of the segment, behaving as a purely shear beam. Both eigenvalue problems are differential in origin, and can be transformed in an algebraic problem applying a suitable discretization to the cross-section middle line.

With this approach, the evaluation of the deformation modes is achieved carrying out the following steps: (1) a constrained PEP is formulated and solved, in which the members are assumed to be inextensible (V2 hypothesis), thus leading to planar inextensional eigenmodes; (2) a second constrained PEP is considered and solved, in which the solutions are required to be orthogonal to the space spanned by the vectors determined in the previous step, so that planar extensional eigenmodes are evaluated; (3) a (unconstrained) WEP is defined and solved, from which purely warping shear modes are determined. The three sets of modes are exhaustive of all the modes that the TWB can experience, and therefore constitute a complete basis for the TWB. It is worth pointing out that any non-singular linear combination of these modes also forms a complete basis for the problem space. Consistent with the idea of capturing the main aspects of the structural mechanical behaviour using the smallest number of modes, it is still convenient to associate warping components to the planar inextensional modes relying on assumption V1. In this way, the conventional modes are included in the selected basis. In summary, the basis obtained in this study is formed by: (a) conventional modes (which include in-plane and out-of-plane components); (b) extensional modes (being of purely planar type); and (c) shear modes (being of purely warping type).

The paper is organized as follows. In Section 2 a short overview of the GBT analysis, extended to non-conventional modes, is provided. In Section 3 the cross-section analysis is carried out by means of the proposed new dynamic approach. In Section 4 an exact member analysis for simply supported TWBs is developed. Section 5 is devoted to numerical results, which include comparison with a benchmark membrane problem, for which an exact solution is also presented. Section 6 reports some conclusions. Four appendices close the paper.

2. Generalized Beam Theory formulation

A brief overview of the GBT is presented in this section. A TWB is considered as a prismatic shell, made of flat plates connected along edges, and possibly constrained at the end cross-sections. The displacement field of the middle surface $S$ of the shell is expressed as (Figure 1)

u (s, z) = u (s, z) e_{s} (s) + v (s, z) e_{y} (s) + w (s, z) e_{z} (s)

(1)

where $s$ is the curvilinear abscissa along the middle line of the transverse profile $C$ ; $z$ is the abscissa along the beam axis; $e_{s} (s), e_{y} (s), e_{z} (s)$ are the tangential, normal and binormal unit-vectors, respectively; and $u (s, z), v (s, z), w (s, z)$ are the relevant scalar components of displacement.

The displacement field at a point $P = (s, y, z)$ external to $S$ is expressed by the Kirchhoff hypothesis as

d (s, y, z) = [\begin{matrix} d_{s} (s, y, z) \\ d_{y} (s, y, z) \\ d_{z} (s, y, z) \end{matrix}] = [\begin{matrix} u - yv,_{s} \\ v \\ w - yv,_{z} \end{matrix}]

(2)

where a comma denotes differentiation with respect the following variable. The corresponding infinitesimal strains, $ε_{s} = d_{s, s}, ε_{z} = d_{z, z,} γ_{zs} = d_{z, s} + d_{s, z,}$ are formed by a membrane component $(y = 0)$ and a flexural one $(y \neq 0)$ , denoted by superscripts m and f, respectively. These are defined as follows:

ε_{s}^{m} = u_{, s}, ε_{z}^{m} = w_{, z}, ε_{s}^{f} = - {yv,}_{ss}, ε_{z}^{f} = - {yv,}_{zz}, γ_{zs}^{m} = u_{, z} + w_{, s}, γ_{zs}^{f} = - 2 yv,_{sz}

(3)

According to the semi-variational approach, the displacement components are expressed as a linear combination of several known deformation modes $U_{k} (s), V_{k} (s), Ω_{k} (s)$ and $W_{j} (s)$ , defined on $C$ , and unknown amplitude functions $φ_{k} (z), ψ_{j} (z)$ , defined along the beam axis, as

\begin{matrix} u (s, z) = \sum_{k = 1}^{K} U_{k} (s) φ_{k} (z) = U^{T} φ \\ v (s, z) = \sum_{k = 1}^{K} V_{k} (s) φ_{k} (z) = V^{T} φ \\ w (s, z) = \sum_{k = 1}^{K} Ω_{k} (s) {φ'}_{k} (z) + \sum_{j = 1}^{J} W_{j} (s) ψ_{j} (z) = Ω^{T} φ' + W^{T} ψ \end{matrix}

(4)

where a dash denotes differentiation with respect the (sole) independent variable, and $U, V, Ω, W$ and $φ, ψ$ are column vectors collecting the homonymous scalar functions. In this representation, a physical dimension of length is assigned to $φ$ and $ψ$ , from which U, V and W are pure numbers and $Ω$ has the dimension of a length. It should be noted that, similarly to [43], the warping displacement is made of two independent contributions. The first one $φ$ is the classical term related to the conventional modes, which depend on z via the $z$ -derivatives of the in-plane components. This choice allows the condition of membrane unshearability $γ_{zs}^{m} = 0$ (assumption V1) to be enforced satisfying $Ω'_{k} (s) + U_{k} (s) = 0$ . The condition of inextensibility (assumption V2) requires $U_{k} (s)$ to remain constant over each plate element, from which $Ω_{k} (s)$ varies linearly; therefore, both of them are (constant or linear) stepwise on each plate segments of $C$ . The second, non-conventional, contribution $ψ$ accounts for additional warping caused by the shear modes, in which the membrane shear strains $γ_{zs}^{m}$ are no longer zero.

Substituting Equations (4) into Equations (3), the strain field can be rewritten as

θ = [\begin{matrix} ε_{s}^{m} \\ ε_{z}^{m} \\ γ_{zs}^{m} \\ ε_{s}^{f} \\ ε_{s}^{f} \\ γ_{zs}^{f} \end{matrix}] = [\begin{matrix} {U'}^{T} φ \\ Ω^{T} φ ″ + W^{T} ψ' \\ (U^{T} + {Ω'}^{T}) φ' + {W'}^{T} ψ \\ - y {V ″}^{T} φ \\ - y V^{T} φ ″ \\ - 2 y {V'}^{T} φ' \end{matrix}]

(5)

Assuming a linear elastic law, the (membrane and flexural) stresses $σ$ are expressed as $σ = E θ + σ_{r}$ . Here $σ_{r}$ are the reactive part of the stresses, related to the internal constraints, whose evaluation is beyond the scope of this work. $E$ is the elastic matrix, which is defined as

E = [\begin{matrix} E & 0 & 0 & 0 & 0 & 0 \\ 0 & E & 0 & 0 & 0 & 0 \\ 0 & 0 & G & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{E}{1 - ν^{2}} & \frac{ν E}{1 - ν^{2}} & 0 \\ 0 & 0 & 0 & \frac{ν E}{1 - ν^{2}} & \frac{E}{1 - ν^{2}} & 0 \\ 0 & 0 & 0 & 0 & 0 & G \end{matrix}]

(6)

where $E, G, ν$ are the longitudinal, tangential and Poisson moduli, respectively. Based on the specified material properties, the (active) stresses can be evaluated using

\begin{matrix} σ_{s}^{m} (s, z) = E {U'}^{T} φ \\ σ_{z}^{m} (s, z) = E (Ω^{T} φ ″ + W^{T} ψ') \\ τ_{zs}^{m} (s, z) = G [(U^{T} + {Ω'}^{T}) φ' + {W'}^{T} ψ] \\ σ_{s}^{f} (s, y, z) = - y \frac{E}{1 - ν^{2}} [{V ″}^{T} φ + ν V^{T} φ ″] \\ σ_{z}^{f} (s, y, z) = - y \frac{E}{1 - ν^{2}} [ν {V ″}^{T} φ + V^{T} φ ″] \\ τ_{zs}^{f} (s, z) = - 2 yG {V'}^{T} φ' \end{matrix}

(7)

The weak formulation of the elasticity problem is derived by means of the Principle of Virtual Work. This can be expressed, assuming external forces to be constant (or averaged) over the plate thickness, as

\int_{V} σ^{T} δ θ d V = \int_{S} f^{T} δ u d S + \int_{C} \sum_{H} F_{H}^{T} δ u_{H} d s

(8)

where $V$ is the volume of the TWB, $f (s, z) = f_{s} (s, z) e_{s} (s) + f_{y} (s, z) e_{y} (s) + f_{z} (s, z) e_{z} (s)$ are forces per unit area applied to the middle surface $S$ , $F_{H} (s) = F_{Hs} (s) e_{s} (s) + F_{Hy} (s) e_{y} (s) + F_{Hz} (s) e_{z} (s)$ are forces per unit length acting on the middle line $C$ of the end cross-sections $H = 0, L$ (i.e. at z=0,L, respectively), and the $δ$ -operator denotes a virtual quantity. Substituting Equations (4) and (7) into the variational principle of Equation (8), the following field equations are obtained based on standard steps of variational calculus:

\begin{matrix} (C^{f} + C_{Ω Ω}^{a}) φ^{IV} + C_{Ω W}^{a} ψ ‴ + (D^{f} + D^{f^{T}} - D_{Ω Ω}^{s} - D^{t}) φ ″ - D_{Ω W}^{s} ψ' + (B^{f} + B^{e}) φ = p_{Ω} \\ - C_{W Ω}^{a} φ ‴ - C_{WW}^{a} ψ ″ + D_{W Ω}^{s} φ' + D_{WW}^{s} ψ = p_{W} \end{matrix}

(9)

with the corresponding boundary condition terms applied at $z = 0, L$ :

\begin{matrix} δ {φ'}^{T} [(C^{f} + C_{Ω Ω}^{a}) φ ″ + C_{Ω W}^{a} ψ' + D^{f} φ - P_{M}] & = 0 \\ δ φ^{T} [(C^{f} + C_{Ω Ω}^{a}) φ ‴ + C_{Ω W}^{a} ψ ″ + (D^{f} - D_{Ω Ω}^{s} - D^{t}) φ' - D_{Ω W}^{s} ψ + P_{Ω}] & = 0 \\ δ ψ^{T} [C_{W Ω}^{a} φ ″ + C_{WW}^{a} ψ' - P_{W}] & = 0 \end{matrix}

(10)

Equations (9) and (10) are here referred as GBT non-conventional equations. The former are a set of coupled differential equations in the amplitude functions $φ$ , $ψ$ . The latter lead to the mechanical boundary conditions at the ends of the TWB, once the geometric constraints have been prescribed there. It is worth pointing out that, specifying ψ = 0, the first of Equations (9) and the first two of Equations (10) lead to the classical form of the GBT problem that enables the evaluation of conventional modes, e.g. [40]. In Equations (9) and (10) the following positions hold for the structural matrices:

\begin{matrix} C^{f} = \int_{c} \frac{E t^{3}}{12 (1 - ν^{2})} V V^{T} d s \\ C^{a} = [\begin{matrix} C_{Ω Ω}^{a} & C_{Ω W}^{a} \\ C_{W Ω}^{a} & C_{WW}^{a} \end{matrix}] = [\begin{matrix} \int_{c} Et Ω Ω^{T} d s \int_{c} Et Ω W^{T} d s \\ \int_{c} Et W Ω^{T} d s \int_{c} Et W W^{T} d s \end{matrix}] \\ D^{f} = \int_{c} \frac{ν E t^{3}}{12 (1 - ν^{2})} V {V ″}^{T} d s, D^{t} = \int_{c} \frac{G t^{3}}{3} V' {V'}^{T} d s \\ D^{s} = [\begin{matrix} D_{Ω Ω}^{s} & D_{Ω W}^{s} \\ D_{W Ω}^{s} & D_{WW}^{s} \end{matrix}] = [\begin{matrix} \int_{c} Gt {({Ω'}^{T} + U^{T})}^{T} ({Ω'}^{T} + U^{T}) d s \int_{c} Gt {({Ω'}^{T} + U^{T})}^{T} {W'}^{T} d s \\ \int_{c} Gt W' ({Ω'}^{T} + U^{T}) d s \int_{c} Gt {W' W'}^{T} d s \end{matrix}] \\ B^{f} = \int_{c} \frac{E t^{3}}{12 (1 - ν^{2})} V ″ {V ″}^{T} d s, B^{e} = \int_{c} Et U' {U'}^{T} d s \end{matrix}

(11)

with superscripts $f, t, s, e$ and a referring to the flexural, torsional, shear, (transverse) extensional and axial nature of the underlying energy terms, respectively. Subscripts $Ω$ and $W$ refer to the conventional and non-conventional part of the warping modes, respectively. The relevant loading vectors included in both field and boundary equations are

\begin{array}{l} p_{Ω} = \int_{c} f_{s} U d s + \int_{c} f_{y} V d s - \int_{c} {f'}_{z} Ω d s, p_{W} = \int_{c} f_{z} W d s \\ P_{M} = \int_{c} \sum_{H} F_{H z} Ω d s, P_{Ω} = \int_{c} \sum_{H} (F_{H s} U + F_{H y} V) d s + \int_{c} f_{z} Ω d s, P_{W} = \int_{c} \sum_{H} F_{H z} W d s \end{array}

(12)

In summary, GBT requires two different steps to be carried out: (a) a cross-section analysis, in which a suitable set of deformation modes (described by $U, V, Ω$ and W) is determined; (b) a member analysis, in which matrices and vectors (11), (12) are first evaluated, and the GBT equations (9) and (10) are solved in terms of unknowns $φ$ and $ψ$ , either analytically or numerically. In the post-processing, other variables, including the stress distributions, can be computed.

3. Dynamic models for cross-section analysis

The fundamental step of the GBT relies on an adequate selection of the deformation modes. The analysis should capture the beam response in the most accurate way, while including the smallest number of modes and minimizing the complexity of the beam model.

According to the spirit of the semi-variational methods, and similar to what is usually done in applying the Ritz–Galerkin variational methods, it seems quite natural to assume for the deformation modes of the TWB the eigenfunctions of one or more linear eigenvalue problems defined on the 1D domain $C$ . Of course, the mathematical problems used for the determination of the deformation modes do not need to have any relation with the real problem considered, so that their choice is quite arbitrary in the wide field of mathematical physics. Among these, the dynamic eigenvalue problem, which governs the free oscillations of a TWB segment, lends itself for the GBT cross-sectional analysis because it (a) relies on a simple mathematical formulation, (b) is easy to implement in a computer code, and (c) is expected to be efficient because, for its own nature, it provides ‘global’ deformation modes, already ordered according to their importance.

In this context, a ‘complete dynamic approach’ is proposed, leading to a GBT-D version of the classical GBT analysis. A segment of TWB of length $dz$ is considered (see Figure 2(a), where a channel section is considered as an example) to be free in the space, weightless, with mass proportional to the local thickness. The segment is susceptible to two types of motions.

In-plane oscillations, in which all points move transversally to the axis (Figure 2(b)). Since $dz$ is infinitesimal, it can be assumed that the in-plane motion does not depend on $z$ , so that the only strains involved are the flexural ones in the cross-section plane and the transverse extension. Under these conditions, the segment behaves as an unconstrained planar extensible frame.

Out-of-plane (or warping) oscillations, in which all points move parallel to the $z$ -axis (Figure 2(c)). Under the assumption that warping is independent from $z$ , the only non-zero strain is the membrane shear. In this case, the segment behaves as an unconstrained pure shear beam.

The previous two problems are governed by a set of uncoupled 1D differential equations referred to in the following as the PEP and the WEP, respectively. These are now outlined below separately for clarity.

3.1 The Planar Eigenvalue Problem

3.1.1 The boundary value problem

The PEP is governed by the well-known equations for axial and transverse vibrations of a beam, which are expressed as

\begin{matrix} U ″ (s) + α^{2} U (s) = 0 \\ V^{IV} (s) - β^{4} V (s) = 0 \end{matrix}

(13)

where $α^{2} = m λ / EA, β^{4} = m λ / EI$ , m is the mass per unit length and $t$ is the plate thickness, while $A = t dz$ and $I = t^{3} dz / 12$ represent the cross-sectional area and second moment of area, respectively. Variable $λ$ depicts the unknown eigenvalue, defined as the squared natural circular frequency of the frame, that is, $λ = ω^{2}$ .

Equations (13) represent the field equations applicable to each of the $M_{b}$ beam elements used in to discretize the cross-section as a planar frame. Parameters $m, E, A, I$ can change from one beam element to another, but are assumed to remain constant for each beam. The general solution of Equations (13) can be written as

\begin{matrix} U = A_{1} \cos (α s) + A_{2} \sin (α s) \\ V = B_{1} \cos (β s) + B_{2} \sin (β s) + B_{3} \cosh (β s) + B_{4} \sinh (β s) \end{matrix}

(14)

where $A_{i}$ and $B_{j}$ are $6 M_{b}$ arbitrary constants, to be evaluated applying the boundary conditions at the ‘natural nodes’ (i.e. at the corners of the profile $C$ ), namely: (a) geometrical conditions, establishing continuity of displacements and rotations; and (b) mechanical conditions, expressing local equilibrium of forces and moments. Enforcing to a nil value the determinant of the matrix of the coefficients, a transcendent characteristic equation of the type $f (λ) = 0$ is obtained, which admits an infinite number of eigenvalues $λ_{k}$ . Associated with these, an infinite number of deformation modes $(U_{k} (s), V_{k} (s)), k = 1, 2, \dots$ is determined via Equations (14).

Since the frame is unconstrained, the problem admits the triple eigenvalue $λ = 0$ , to which three eigenvectors, describing rigid motions of the cross-section, are associated. These can be freely and conveniently chosen as two translations along the principal inertia axes of the cross-section, and a rotation around its shear centre.

3.1.2 The discretized problem

Although the continuous problem (Equations (13) plus boundary conditions) is in principle simple and suitable for a matrix formulation (e.g. see [44]), its discrete version is often preferred in the literature. In this approach, indeed, simpler polynomial functions are used in place of the circular and hyperbolic functions of Equations (14); moreover, as discussed in [44], inextensibility conditions are easier to implement.

The discretization is carried out by subdividing the frame into FEs, obtained by interposing additional nodes between the natural ones. By taking $n$ nodes, and assuming three degrees of freedom (dofs) per node (two translations and one rotation), the frame is reduced to $N = 3 n$ dofs. The tangential displacement $U (s)$ is interpolated between adjacent nodes using a linear polynomial, while the transverse displacement field $V (s)$ is depicted by a cubic function. Following standard FE steps [45,46], the stiffness and mass matrices $K_{p}^{e}$ and $M_{p}^{e}$ (with $e = 1, 2, \dots, M$ and subscript ‘p’ identifying the planar problem) are evaluated for each of the $M$ elements, and then assembled. Details of $K_{p}^{e}$ and $M_{p}^{e}$ are provided in Appendix A. The boundary value problem is thus transformed in the following algebraic eigenvalue problem:

(K_{p} - λ M_{p}) q = 0

(15)

where $K_{p}, M_{p}$ are $N \times N$ global stiffness and mass matrices and $q$ is an $N -$ vector collecting the nodal displacements. The solutions $(λ_{k}, q_{k}), k = 1, 2, \dots, N$ of Equations (15) are discrete approximations of the exact eigensolutions. Unlike the latter ones, these are limited in number as they depend on the dofs of the adopted discretization.

3.1.3 The inextensional and extensional planar modes

Since the matrices $K_{p}$ and $M_{p}$ are symmetric, their eigenvectors $q_{k} (k = 1, 2, \dots, N)$ form a complete set that spans the entire space of the admissible configurations of the problem. Obviously, any non-singular change of basis leads to a set of vectors that still represents a complete basis for the space considered. In view of further developments, it is convenient to divide the space into two supplementary subspaces, in each of which a basis is sought, namely: (a) a space in which the frame undergoes flexure without extension; and (b) its orthogonal supplement, in which the system undergoes extension and flexure (pure extension is not possible, due to the connection among non-parallel beams). In this way, the classical inextensional behaviour assumed by Vlasov is enucleated from the totality of the planar mechanisms that the cross-section can experience, and the remaining ‘essentially different’ mechanisms highlighted. The eigenvectors $q$ belonging to the former subspace, and the relevant deformation modes, are referred to as inextensional eigenvectors, and denoted by $q_{i}$ ; the eigenvectors belonging to the supplementary subspace, with the relevant deformation mode, are referred to as extensional eigenvectors, and denoted by $q_{e}$ . The determination of $q_{i}, q_{e}$ requires the solution of two constrained eigenvalue problems, derived by the original Equations (15).

The inextensibility of the $M$ elements is enforced by a linear condition of the type $A q_{i} = 0$ , where $A$ is an $M \times N$ matrix, which can be solved with respect to $N_{i} = N - M$ master variables $q_{iM}$ to produce $q_{i} = R_{i} q_{iM}$ (where $R_{i}$ is an $N \times N_{i}$ constraint matrix). Details related to the implementation of the inextensibility condition are provided in Appendix B or in [40]. Based on this, the constrained eigenvalue problem is rewritten as

(K_{i} - λ_{i} M_{i}) q_{iM} = 0

(16)

where

K_{i} = R_{i}^{T} K_{p} R_{i}, M_{i} = R_{i}^{T} M_{p} R_{i}

(17)

are $N_{i} \times N_{i}$ reduced matrices. Equations (16) provide $N_{i}$ solutions $(λ_{ik}, q_{iMk}), k = 1, 2, \dots, N_{i}$ , from which inextensional deformation modes $(U_{ik} (s), V_{ik} (s)), k = 1, 2, \dots, N_{i}$ are determined, based on the adopted interpolating functions.

The inextensional eigenvectors span a $N_{i}$ -dimensional subspace of the $N$ -dimensional space. They can be collected in the $N \times N_{i}$ modal matrix as

Q_{i} = [q_{i 1} | q_{i 2} | \dots | q_{i N_{i}}]

(18)

Any vector $q_{e}$ that is orthogonal to this subspace must satisfy $N_{i}$ conditions, which can be expressed in matrix form as $Q_{i}^{T} G q_{e} = 0$ , G being an N x N definite positive matrix as the identity matrix $I$ or the mass matrix $M_{p}$ ; in the following, it is assumed that $G \equiv I$ . These are used as constraint conditions to solve the dynamic eigenvalue problem (15) in the extensional space. Once the constraints have been solved with respect $N_{e} = N - N_{i} \equiv M$ master variables $q_{eM}$ , they enable the calculation of $q_{e} = R_{e} q_{eM}$ , where $R_{e}$ is an $N \times N_{e}$ constraint matrix (see Appendix B). Consequently, the eigenvalue problem is re-arranged as

(K_{e} - λ_{e} M_{e}) q_{eM} = 0

(19)

where

K_{e} = R_{e}^{T} K_{p} R_{e}, M_{e} = R_{e}^{T} M_{p} R_{e}

(20)

are $N_{e} \times N_{e}$ reduced matrices. Equations (19) provide $N_{e}$ solutions $(λ_{eh}, q_{eMh}), h = 1, 2, \dots, N_{e}$ , with $q_{eh}$ orthogonal to the inextensional eigenvectors $q_{ih}$ and, because of this, completing the basis sought. The entire profile of the extensional deformation modes $(U_{eh} (s), V_{eh} (s)), h = 1, 2, \dots, N_{e}$ is calculated from $q_{eh}$ , recalling the adopted interpolating functions.

3.2 The Warping Eigenvalue Problem

3.2.1 The boundary value problem

The WEP governs the free oscillations of a pure shear beam. This is an internally constrained Timoshenko beam, in which cross-section rotations are prevented, so that they are only permitted to slide orthogonally to the axis. Therefore, the shear $γ$ is the only strain, and the shear force $T$ the only active stress. The relevant equilibrium equation is $T' - mw,_{θ θ} = 0$ , where $w = w (s, θ)$ is the transverse displacement and $θ$ depicts the time. By considering an elastic material behaviour, the active shear force is expressed as $T = GA γ$ , where $G$ is the tangential modulus, $A = t dz$ is the cross-section area and, from kinematics, $γ = w,_{s}$ . Based on this, the equation of motion is rewritten as $GAw,_{ss} - mw,_{θ θ} = 0$ . A solution of the problem can be sought assuming an harmonic motion for the transverse displacement, that is, $w (s, θ) = W (s) \exp (i ω θ)$ , from which the ordinary differential equation follows:

W ″ (s) + α^{2} W (s) = 0

(21)

which is formally identical to Equation (13), with $α^{2} = m λ / GA$ and $λ = ω^{2}$ . Equation (21) is the field equation of the problem, which must hold for each of the $M_{b}$ sides of the profile $C$ (on which $m, G, A$ are constant and can change from one side to the other). Its general integral is

W = C_{1} \cos (α s) + C_{2} \sin (α s)

(22)

so that there are $2 M_{b}$ arbitrary constants $C_{j}$ , to be determined through (a) continuity of the warping $W$ , and (b) continuity of the shear force $GAW'$ at natural nodes. The zeroing of the determinant of the coefficients supplies a transcendent equation of the type $f (λ) = 0$ , whose roots $λ_{k}$ , infinite in number, represent the warping eigenvalues. The associated eigenfunctions $W_{k} (s), k = 1, 2, \dots$ depict the warping modes.

Since the shear beam is free at the ends, it admits the (simple) zero eigenvalue $λ = 0$ , which is associated to the rigid motion, that is, $W (s) = 1$ ; this represents the uniform axial extension of the Vlasov theory.

It is also worth noticing that there is no need to subdivide the domain $C$ in $M_{b}$ intervals (i.e. as many intervals as the number of plate elements) because, at the corners where just two sides converge (and irrespective of the angle of incidence), the function $W (s)$ is continuous with a continuous slope. Therefore, such two adjacent sides can be merged in a unique sub-domain, provided they have the same thickness. As a limited, but extremely important case, unbranched open sections with uniform thickness can be integrated on a unique interval (i.e. by ‘unrolling’ the segment of the TWB on a plane and, consequently, its profile $C$ on a straight line, see Figure 3(a)). For these unbranched sections, therefore, by requiring that the shear force vanishes at the free end, it is $W_{k} = \cos (k π s / b), k = 0, 1, 2, \dots$ , with $s \in [0, b]$ and $b$ being the total length of $C$ . These considerations remarkably reduce the number of unknowns. For example, for a boxed section with two wings (see Figure 3(b)), four sub-domains are required, instead of six, so that there are eight arbitrary constants. These must be evaluated enforcing (a) two kinematic and one equilibrium conditions at each of the internal nodes $B, C$ , and (b) one kinematic and one equilibrium condition at each of the external nodes $A, D$ .

3.2.2 The discretized problem

The WEP (Equation (21) and relevant boundary conditions) can, of course, be discretized, although it turns out to be much simpler than the PEP. By using the same discretization adopted for this latter, that is, $n$ nodes and $M$ elements, and accounting for the fact that each node now possesses a unique dof (namely the out-of-plane displacement), the system has $n$ dofs. The warping field $W (s)$ is interpolated between adjacent nodes by a linear law and, by standard steps, the stiffness and mass matrices $K_{w}^{e}, M_{w}^{e}$ , $e = 1, 2, \dots, M$ (reported in Appendix A), are evaluated. Their assembly leads to the following algebraic eigenvalue problem:

(K_{w} - λ M_{w}) q_{w} = 0

(23)

where $K_{w}, M_{w}$ are $n \times n$ global stiffness and mass matrices, respectively, $q_{w}$ is an $n$ -vector collecting the nodal displacements, and the index $w$ indicates the warping problem. The solutions $(λ_{j}, q_{wj}), j = 1, 2, \dots, n$ of Equations (23), which are limited in number, discretely approximate the exact eigensolutions. Once the eigenvectors are determined, the warping modes $W_{j} (s), j = 1, 2, \dots, n$ are computed via interpolation.

3.3 Conventional and non-conventional modes

The previous analysis led to determination of three sets of deformation modes: (a) inextensional planar modes $(U_{ik} (s), V_{ik} (s)), k = 1, 2, \dots, N_{i}$ , in number equal to the planar dofs $N = 3 n$ , diminished of the number of elements $M$ ; (b) extensional planar modes $(U_{eh} (s), V_{eh} (s)), h = 1, 2, \dots, N_{e}$ , in number equal to the elements $N_{e} = M$ ; (c) warping modes $W_{j} (s), j = 1, 2, \dots, n$ , in number equal to the nodes; overall, $N + n = 4 n$ modes, that is, the totality of the modes the discretized cross-section can experience. Therefore, these modes could be directly used in the GBT equations (9) and (10) as a basis of independent vectors. However, it immediately turns out that this choice is not convenient, since membrane shear strains are associated with inextensional modes. Thus, if a slender TWB with open profile was considered, for which membrane shear strains are small, a large number of warping modes is expected to be triggered, as the resulting displacement field is shear-free. Similarly, if a boxed TWB was considered, for which the shear strains are expected to satisfy the Bredt’s condition requiring the tangential stress flow to be constant along the loop, a large number of warping modes would be involved. As a limit case, even the De Saint Venant uniform flexure of the beam, occurring in a principal inertia plan, would be described by a pure translation of the cross-section (one rigid inextensional mode) and all the warping modes (except the uniform axial extension); a similar case would occur for the torsion of a boxed section.

For the previous reasons, aimed to build-up a set of deformation modes which, even if taken in a small number, is able to capture the main behaviour of the beam, it is convenient to supplement the inextensional modes by warping components $Ω_{ik} (s)$ , able (a) to identically vanish the membrane shear strains in open branches of $C$ ; (b) to ensure the constancy of the tangential stress flow around each closed branch that composes $C$ . These modes are referred to as conventional in the literature and can be obtained as the solution of the following problem:

given the planar displacement field $u_{ik} (s, z) = U_{ik} (s) φ_{k} (s)$ , with $U_{ik} (s) = U_{ik}^{e}, e = 1, 2, \dots, M$ , stepwise constant on $C$ , the out-of-plane displacement field $w_{ik} (s, z) = Ω_{k} (s) φ'_{k} (z)$ needs to be evaluated so that the associated shear strain $γ_{zs}^{m} = u_{ik, z} + w_{ik, s}$ either (a) is nil along open branches of $C$ , or (b) satisfies $γ_{zs}^{m} t = const$ on each closed branch of $C$ . The solution to this problem has already been detailed in [40,42] and, for completeness, a brief description of this procedure is provided in Appendix C.

In summary, the basis of $4 n$ modes proposed here is formed by: (a) conventional modes $(U_{ik} (s), V_{ik} (s), Ω_{k} (s)), k = 1, 2, \dots, N_{i}$ , generally described by all three components of displacement, in which the Vlasov hypotheses concerning membrane unshearability and (transverse) inextensibility are satisfied, the first one being replaced by the Bredt’s hypothesis along closed loops; (b) extensional modes $(U_{eh} (s), V_{eh} (s), 0), h = 1, 2, \dots, N_{e}$ , which are purely planar; (c) warping or shear modes $(0, 0, W_{j} (s)), j = 1, 2, \dots, n$ (the latter locution being from now on preferable, due to the existence of a ‘dependent’ warping $Ω_{k} (s)$ included in the conventional modes), which are depicted by out-of-plane displacements only. The set of extensional and shear modes can also be referred as non-conventional modes.

4. Member analysis of simply supported thin-walled beams

The member analysis of TWBs is carried out solving the GBT equations (9) and (10). These are a system of differential equations and boundary conditions, often solved in the literature by means of the finite element method (FEM), for example [47]. Such an approach would require a convergence analysis to be performed to estimate possible errors induced for different levels of discretization. This aspect of the problem is outside the scope of this paper, where attention is instead focused on the section analysis. Therefore, it is preferred here to consider special boundary conditions that permit an exact solution of GBT equations.

Based on the above, a TWB is considered and assumed to be simply supported, restrained to torsion and free to warp at the ends. Geometric boundary conditions require $u (s, \bar{z}) = v (s, \bar{z}) = 0$ at $\bar{z} = 0, L$ . Since these must be satisfied for any $s$ , $φ (\bar{z}) = 0$ follows from the first two of Equations (4). However, $φ' (\bar{z})$ and $ψ (\bar{z})$ are arbitrary, so that two mechanical conditions follow from the first and third of Equations (10), enforcing to zero the bracketed terms, both involving even derivatives of $φ (\bar{z})$ and odd derivatives of $ψ$ . Therefore, a solution of the following type exactly satisfies the field equation and the boundary conditions:

φ = a \sin (\frac{m π z}{L}), ψ = b \cos (\frac{m π z}{L}), m = 1, 2, \dots

(24)

with $a = {a_{k}}, k = 1, 2, \dots 3 n$ and $b = {b_{j}}, j = 1, 2, \dots, n$ unknown constants to be suitably chosen, provided the distributed loads $f$ have a sinusoidal/cosinusoidal law such as

f_{s} = f_{sm} \sin (\frac{m π z}{L}), f_{y} = f_{ym} \sin (\frac{m π z}{L}), f_{z} = f_{zm} \cos (\frac{m π z}{L}), m = 1, 2, \dots

(25)

Assuming that no forces $F_{H}$ are applied at the end-sections, the loading vector defined in Equations (12) can be expressed as

p_{Ω} = p_{Ω}^{0} \sin (\frac{m π z}{L}), p_{W} = p_{W}^{0} \cos (\frac{m π z}{L}), m = 1, 2, \dots

(26)

with $P_{M} = P_{Ω} = P_{W} = 0$ .

Substitution of Equations (24) into Equations (9) leads to the following $4 n$ algebraic equations in the $4 n$ unknowns $a, b$ :

[\begin{matrix} K_{Ω a} & K_{Ω b} \\ K_{Wa} & K_{Wb} \end{matrix}] (\begin{matrix} a \\ b \end{matrix}) = (\begin{matrix} p_{Ω}^{0} \\ p_{W}^{0} \end{matrix})

(27)

where

\begin{matrix} K_{Ω a} = (C^{f} + C_{Ω Ω}^{a}) {(\frac{m π}{L})}^{4} - (D^{f} + D^{f^{T}} - D_{Ω Ω}^{s} - D^{t}) {(\frac{m π}{L})}^{2} + (B^{f} + B^{e}) \\ K_{Ω b} = - C_{Ω W}^{a} {(\frac{m π}{L})}^{3} + D_{Ω W}^{s} (\frac{m π}{L}) \\ K_{Wa} = C_{W Ω}^{a} {(\frac{m π}{L})}^{3} + D_{W Ω}^{s} (\frac{m π}{L}) \\ K_{Wb} = C_{WW}^{a} {(\frac{m π}{L})}^{2} + D_{WW}^{s} \end{matrix}

(28)

Once $a$ and $b$ are determined, the displacement field is evaluated with Equations (24) and (4). The active stresses can be post-processed as follows (see Equations (7)):

\begin{matrix} σ_{s}^{m} (s, z) = E {U'}^{T} a \sin (\frac{m π z}{L}) \\ σ_{z}^{m} (s, z) = - E (Ω^{T} \frac{m^{2} π^{2}}{L^{2}} a + W^{T} b \frac{m π}{L}) \sin (\frac{m π z}{L}) \\ τ_{zs}^{m} (s, z) = G [(U^{T} + {Ω'}^{T}) \frac{m π}{L} a + {W'}^{T} b] \cos (\frac{m π z}{L}) \\ σ_{s}^{f} (s, y, z) = - y \frac{E}{1 - ν^{2}} [{V ″}^{T} a - ν \frac{m^{2} π^{2}}{L^{2}} V^{T} a] \sin (\frac{m π z}{L}) \\ σ_{z}^{f} (s, y, z) = y \frac{E}{1 - ν^{2}} [\frac{m^{2} π^{2}}{L^{2}} V^{T} a - ν {V ″}^{T} a] \sin (\frac{m π z}{L}) \\ τ_{zs}^{f} (s, z) = - 2 yG \frac{m π}{L} {V'}^{T} a \cos (\frac{m π z}{L}) \end{matrix}

(29)

5. Numerical results

The accuracy of the GBT-D formulation is checked with a benchmark membrane problem, for which an analytical solution can be derived considering the case of a flat plate, loaded by in-plane forces acting at longitudinal edges. This is followed by the results calculated for two numerical examples to highlight the ease of use of the proposed approach for the analysis of an open-branched section and partially closed one. In all these problems, the role of the shear and extensional modes on the overall response of the beam is investigated and discussed.

5.1 Benchmark membrane problem of a flat plate

A rectangular flat plate of dimensions $b \times L$ , lying in the $(s, z)$ -plane, is assumed to be simply supported at the opposite edges (i.e. $z = 0, L$ ) and subjected to in-plane harmonic forces (per unit area), that is, $f_{s} = f_{y} = 0$ and $f_{z} = f_{0} \cos (π z / L)$ , applied at the remaining free edges $s = 0, b$ , as outlined in Figure 4. The classical phenomenon of the shear-lag is expected to occur in this system when the length-to-width ratio is close to 1. The structural system is considered as a TWB, with its profile $C$ represented by a single plate segment. An analytical solution is easily obtained for this problem, when the plate is assumed to be transversally inextensible, according to the V2 hypothesis (see Appendix D for its derivation). This exact solution is used to benchmark the results of the following GBT-D analysis.

For the purpose of the numerical examples, both width and length of the plate are taken equal to 100 mm and the thickness equal to 1 mm. The plate width is discretized with 20 FEs. The material properties include an elastic modulus of 200 GPa and a zero Poisson ratio. The maximum amplitude of the external load $f_{0}$ is taken as 20 N/mm.

The PEP is first solved in the inextensional space (Equation (16)), leading to modes having $(U (s), V (s))$ -components, illustrated in Figure 5. In particular, only the first 10 modes obtained are displayed, with the first three being the rigid ones. Among them, only mode 1 includes a tangential displacement $U (s)$ , which is associated with a membrane shear strain. The remaining modes are purely normal to the cross-section and are depicted by values for $V (s)$ only, that is, these are shear-free because associated with a nil U(s). To remove shear strain from mode 1, a warping displacement $Ω (s)$ is added, evaluated by the procedure outlined in Section 3.3 and in Appendix C. Therefore, mode 1 (made of $(U (s), 0, Ω (s))$ ), and modes 2,3,… (made of $(0, V (s), 0)$ ), displayed in Figure 5, depict the conventional modes for the single plate considered.

Successively, the PEP is solved in the extensional subspace (Equation (19)), and deformation modes of purely tangential type $(U (s), 0, 0)$ are determined. The diagrams of the first six modes are shown in Figure 6. These modes produce transverse extension and shear strains in the plate. Finally, the WEP is addressed (Equation (23)), and purely warping modes $(0, 0, W (s))$ are evaluated. The first of them is an axial extension of the plate, and the successive are shear modes, all depicted in Figure 7 (first 1+5 modes displayed).

The numerical results presented in the following are calculated considering different subsets of the modes previously introduced to evaluate how these influence the accuracy of the solution. In particular, these subsets consist of (a) conventional modes; (b) conventional and shear modes; (c) all modes (conventional, shear and extensional modes). In all cases, all the modes of the subsets obtained from the cross-sectional analysis are used (not only those displayed in Figures 5 –7). Two additional solutions are also considered for comparative purposes: (d) FE results obtained with the commercial software Abaqus [48] (in which suitable material and element constraints are introduced), and (e) the analytical solution presented in Appendix D. The Abaqus model is developed with shell elements S4R and with the condition of inextensibility enforced by requiring that no displacements occur in the s direction. A nil Poisson ratio is specified, to reflect the typical GBT assumption (Equation (6)) in which the two membrane normal stresses are not coupled.

Figures 8 reports results (warping, longitudinal and shear stresses evaluated at $z = 0, L / 2, 0$ , respectively) obtained by ignoring transverse extension. It appears that GBT-D formulation with the sole conventional modes leads to (transversally) constant warping and, consequently, to constant normal stress $σ_{z}$ and zero shear stress $τ_{zs}$ . Accounting for the contribution of the shear modes, an almost perfect match is found between GBT results, closed-form solutions (Equations (40), (41)) and Abaqus values. In particular, the less satisfactory constant values assumed by the shear stresses $τ_{zs}$ in each FE are due to the linear interpolation function adopted for warping. Obviously, higher-order polynomials (or even the exact solution included in Equation (22)) could be used in the determination of the shear modes, thus improving the accuracy of the solution.

As a final comment, the variation over the width of the section of the normal stress well depicts the occurrence of shear-lag. This, however, is significant because the length-to-width ratio $L / b$ is of the order of 1, becoming negligible for larger spans.

The relevant results (warping, longitudinal and transverse normal stresses, tangential stresses, evaluated at $z = 0, L / 2, L / 2, 0$ , respectively), calculated including the extensional modes in the solution, thus considering the complete set of modes, are reported in Figure 9 and compared with those obtained for the case of the inextensional plate. The GBT-D results are in excellent agreement with those furnished by Abaqus (with $s$ -displacements now free), once the (already commented) limitation on the stepwise representation for the transverse normal and shear stresses has been accepted.

5.2 Numerical examples: T-section and box girder

The GBT-D is applied in this section to the analysis of two simply supported TWBs, that is, a T-section beam and a box girder, to better outline its ease of use and highlight the role of the non-conventional modes when considering the accuracy of the solution.

5.2.1 T-section beam

The geometric layout of the TWB considered in this example is depicted in Figure 10. The elastic modulus is 200 GPa and the Poisson ratio 0.3. The loads (forces per unit length) act in the symmetry plane, and are applied to the top flange; their expressions are: $f_{s} = f_{z} = 0$ , $f_{y} = f_{0} \sin (π z / L)$ , with $f_{0}$ = 20 N/mm.

The deformation modes, obtained by discretizing each of the three plates in 10 FEs, are illustrated in Figure 11 (where the U- and V-components are displayed by the final position of the nodes, while the W-components are diagrammed). The deformation modes consist of: (a) conventional modes (Figure 11(a)); (b) extensional modes (Figure 11(b)); and (c) shear modes (Figure 11(c)), which include the uniform axial extension mode.

Results concerning the member analysis are reported in Figure 12, where displacement and stresses are plotted at the abscissa where they attain the maximum values (i.e. at $z = 0$ or $z = L$ ). Two subsets of modes have been considered in the analyses: (a) conventional and extensional modes (‘C+E’ curves); and (b) the full set of modes (‘all modes’ curves). A shear-lag effect is highlighted by the non-uniform distribution of the longitudinal normal stresses at the top flange, which can only be captured when the shear modes are included in the analysis. Shear modes also significantly affect the distribution of the tangential stresses, particularly over the web. The stepwise distributions of $σ_{s}^{M}$ and $τ_{zs}^{M}$ (already commented) depend on the linearity of the warping interpolation functions.

5.2.2 Box girder

A box girder (Figure 13) is considered to outline the ability of GBT-D to deal with partially closed sections. The applied load consists of an in-plane longitudinal force (per unit area), $f_{s} = f_{y} = 0$ and $f_{z} = f_{0} \cos (π z / L)$ , with $f_{0} = 1$ N/mm², applied over a strip of 20-mm wide. The relevant conventional, extensional and shear modes are depicted in Figure 14.

Similar to the T-section, a comparison between analyses with all modes and no shear modes is presented in Figure 15. These results confirm the need to account for the shearing deformation in order to accurately predict the structural response in terms of both in-plane and warping displacements (Figures 15(a) and (b)), as well as stresses (Figures 15(c)–(e)). This kind of response is particularly important for short members, such as those considered here, for which the length is comparable to the cross-sectional width.

6. Conclusions

A new approach to the cross-section analysis of TWBs, to be performed in the framework of the GBT, has been proposed. It is based on evaluation of the dynamic eigenmodes of beam systems, and therefore named GBT-D. The method permits one to replace the sets of (quite involved) elastic analyses required in the classical approach, by simpler eigenmode analyses. The method calls for the following steps to be performed.

Solving the eigenvalue problem for an inextensional frame of planar beam, having the shape of the middle line of the TWB. Supplementing the inextensional planar modes so obtained by warping displacements that assure (a) shear-indeformability of open branches, and (b) constant shear-stress flow along closed loops; these modes are called conventional.

Solving the eigenvalue problem for an extensional planar frame, of the same shape, under the constraint that modes are orthogonal to the inextensional ones; these planar modes are called extensional.

Solving the eigenvalue problem for a planar shear beam, supplying purely warping modes for the TWB; these are named shear modes.

Performing a member analysis, aimed to solve GBT equations in the unknown amplitudes of conventional, extensional and shear deformation modes.

A benchmark membrane problem has been considered, for which an analytical solution can be obtained under inextensibility conditions. It has been shown that the GBT-D analysis is able to accurately reproduce the exact solution when the shear modes, in addition to the conventional, are taken into account. These, of course, are important when shear-lag effects play an important role in describing the structural response, as is the case for short TWBs. Extensional modes are also significant when the inextensibility condition is released, as shown by the comparisons carried out against FE results. Two additional numerical examples have been presented to highlight the efficiency of GBT-D in predicting the response of open and partially closed TWBs. Although similar results can be obtained with refined FE models (for example, by using shell elements), it is in the opinion of the authors that the GBT technique improves the interpretation of the results, and opens the way to the study of reduced-order models. Therefore, the GBT-D is regarded as a simple and efficient tool for the elastic analysis of TWBs.

Footnotes

Appendix A: Stiffness and mass element matrices

The element matrices used in this study for the two-node planar Euler–Bernoulli beam, based on linear and cubic interpolation functions for the axial and transverse displacements, respectively, and relying on the consistent mass approach, are defined as

(30)

K_{p}^{e} = \frac{EI}{b} [\begin{matrix} \frac{EA}{EI} & 0 & 0 & - \frac{EA}{EI} & 0 & 0 \\ 0 & \frac{12}{b^{2}} & \frac{6}{b} & 0 & - \frac{12}{b^{2}} & \frac{6}{b} \\ 0 & \frac{6}{b} & 4 & 0 & - \frac{6}{b} & 2 \\ - \frac{EA}{EI} & 0 & 0 & \frac{EA}{EI} & 0 & 0 \\ 0 & - \frac{12}{b^{2}} & - \frac{6}{b} & 0 & \frac{12}{b^{2}} & - \frac{6}{b} \\ 0 & \frac{6}{b} & 2 & 0 & - \frac{6}{b} & 4 \end{matrix}], M_{p}^{e} = \frac{mb}{420} [\begin{matrix} 140 & 0 & 0 & 70 & 0 & 0 \\ 0 & 156 & 22 b & 0 & 54 & - 13 b \\ 0 & 22 b & 4 b^{2} & 0 & 13 b & - 3 b^{2} \\ 70 & 0 & 0 & 140 & 0 & 0 \\ 0 & 54 & 13 b & 0 & 156 & - 22 b \\ 0 & - 13 b & - 3 b^{2} & 0 & - 22 b & 4 b^{2} \end{matrix}]

where $K_{p}^{e}$ and $M_{p}^{e}$ are the stiffness and mass element matrices, respectively, and all quantities are referred to the eth element.

The stiffness ( $K_{w}^{e}$ ) and the mass ( $M_{w}^{e}$ ) element matrices used for the two-node planar FE shear beam, based on a linear interpolation of the displacements and a consistent mass approach, are

(31)

K_{w}^{e} = \frac{Gt}{b} [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}], M_{w}^{e} = mb [\begin{matrix} \frac{1}{3} & \frac{1}{6} \\ \frac{1}{6} & \frac{1}{3} \end{matrix}]

Appendix B: Constrained algebraic eigenvalue problems

The constrained dynamic eigenvalue problem can be expressed as

(32)

\begin{matrix} δ q^{T} (K - λ M) q = 0 \\ Aq = 0 \end{matrix}

where the first equation is the Virtual Work Principle and the second equation expresses a linear constraint for $q$ . Partitioning the latter equation as $[\begin{matrix} A_{M} & A_{S} \end{matrix}] [\begin{matrix} q_{M} \\ q_{S} \end{matrix}] = 0$ , in which indexes M and S denote master (free) and slave (dependent) variables, respectively, and assuming $A_{S}$ to be non-singular, vector q can be expressed in terms of the master displacements $q_{M}$ as $q = R q_{M}$ , with $R = [\begin{matrix} I \\ \dots \dots \dots \\ - A_{S}^{- 1} A_{M} \end{matrix}]$ . By substituting $δ q = R δ q_{M}$ in the first of Equation (32), the following eigenvalue problem is derived:

(33)

(K_{R} - λ M_{R}) q_{M} = 0

where $K_{R} = R^{T} KR$ and $M_{R} = R^{T} MR$ .

If the constraints express the inextensibility of a planar frame (inextensional PEP), then $A$ collects the coefficients of the equations $(u_{j} - u_{i}) \cos α_{e} + (v_{j} - v_{i}) \sin α_{e} = 0$ , where u and v are the global displacements of the end nodes $i, j$ of the $e$ th beam, inclined by $α_{e}$ with respect to the x-axis.

In the case the constraints express orthogonality of $q$ to a subspace spanned by the column of a matrix $Q_{i}$ (extensional PEP), then $A = Q_{i}^{T}$ and, therefore, $R = [\begin{matrix} I \\ - Q_{iS}^{- T^{}} Q_{iM}^{T} \end{matrix}]$ .

Appendix C: Warping component of the conventional modes

The warping in the conventional modes is determined by enforcing the membrane shear-strain in each beam element forming the cross-section to be either (a) zero, when dealing with open branches, or (b) equal to a constant shear flow, when considering closed loops included in the cross-section. For the $k$ th mode, this can be expressed as $γ_{zs}^{M} = Γ_{k} (s) {φ'}_{k} (z)$ , with $Γ_{k} (s) = U_{k} (s) + Ω'_{k} (s)$ .

Considering that $U_{k} (s)$ is stepwise constant over the width of each plate element (and assumes known values $U_{k}^{e}$ ), and that $Γ_{k} (s)$ is also stepwise constant (and assumes unknown values $Γ_{k}^{e}$ ), the variation for $Ω_{k} (s)$ is stepwise linear, and can be calculated as $Ω_{k} (s) = (Ω_{k}^{j} - Ω_{k}^{i}) (s_{e} / b_{e}) + Ω_{k}^{i}$ , with $s_{e} \in [0, b_{e}]$ , and $Ω_{k}^{i}, Ω_{k}^{j}$ being the unknown values of warping at the end nodes $i, j$ of element e. For the kth mode, the relationship between $Γ_{k}^{e}$ , $U_{k}^{e}$ , $Ω_{k}^{j}$ and $Ω_{k}^{i}$ is expressed by

(34)

Γ_{k}^{e} = U_{k}^{e} + \frac{Ω_{k}^{j} - Ω_{k}^{i}}{b_{e}}, e = 1, 2, \dots, M

If the beam possesses an open profile, then $Γ_{k}^{e} = 0$ is enforced over the entire cross-section. In these cases, Equations (34) consist of a linear system of $M$ equations in $n = M + 1$ unknowns $Ω_{k}^{i}$ , with the indeterminacy being due to the existence of the axial extension of the member. The remaining (M+1)th condition can be obtained normalizing the warping distribution over the cross-section to have a nil average value over $C$ .

In the presence of closed loops in $C$ , additional unknowns $Γ_{k}^{e} \neq 0$ need to be included in Equations (34), one for each loop an element belongs to. These unknowns, however, can be eliminated by means of equilibrium conditions requiring that the flow of the shear stresses in each element, $G t_{e} Γ_{e}$ , is the algebraic sum of the (unknown) constant flows $Q_{k}^{l}$ acting along each loop $l = 1, 2, \dots, L$ that contains the element $e$ . This condition can be written for the eth element as

(35)

G t_{e} (U_{k}^{e} + \frac{Ω_{k}^{j} - Ω_{k}^{i}}{b_{e}}) = \sum_{l = 1}^{L} δ_{el} Q_{k}^{l}

where $δ_{el} = \pm 1$ , if the element $e$ belongs to the loop $l$ and its orientation is concordant or discordant, respectively, with the positive direction adopted for the lth loop; $δ_{el} = 0$ if the element $e$ does not belong to the loop $l$ .

In summary, the system of equations to be solved consists of one equation for each element, either of type (34) if the element belongs to an open branch, or of type (35) if it belongs to one or more loops. An additional equation is included, which enforces the warping average over the entire cross-section to be nil. The system is then solved for the unknowns, which include the nodal values of warping $Ω_{k}^{i}$ and the shear flows $Q_{k}^{l}$ related to the closed loops. A worked example providing the details of this procedure is presented in [40].

Appendix D: Closed-form solution for the benchmark membrane problem of a flat plate

The closed-form solution describing the response of the flat plate depicted in Figure 4 is derived in this section. By assuming that the plate is inextensible along the transverse $s$ -axis (i.e. assumption V2) and accounting for the symmetry with respect the median axis, the tangential displacement turns out to be identically zero, that is, $u (s, z) = 0$ . For this problem, the only non-zero displacement is the warping $w (s, z)$ with corresponding non-zero strains $ε_{z} = w_{, z}$ and $γ_{zs} = w_{, s}$ . The inextensibility condition implies that the transverse normal stress $σ_{s} (s, z)$ is of reactive type, while the longitudinal normal and tangential stresses $σ_{z} (s, z)$ and $τ_{zs} (s, z)$ are active. In the case of linear elastic material, the active stresses can be expressed as $σ_{z} = E ε_{z} = E w_{, z}$ and $τ_{zs} = G γ_{zs} = {Gw}_{, s}$ and need to satisfy the following equilibrium conditions:

(36)

\begin{matrix} σ_{z, z} (s, z) + τ_{zs, s} (s, z) = 0 \\ τ_{zs} (0, z) = - f_{z} (0, z), τ_{zs} (b, z) = f_{z} (b, z) \\ σ_{z} (s, 0) = 0, σ_{z} (s, L) = 0 \end{matrix}

where the field equation governs equilibrium in the $z$ -direction, and boundary conditions require the equilibrium with the prescribed loads. The equilibrium problem, expressed in terms of displacements, can be rewritten as

(37)

\begin{matrix} {Ew}_{, zz} + {Gw}_{, ss} = 0 \\ {Gw}_{, s} (0, z) = - f_{0} \cos (π z / L), {Gw}_{, s} (b, z) = f_{0} \cos (π z / L) \\ w_{, z} (s, 0) = 0, w_{, z} (s, L) = 0 \end{matrix}

Considering external loads with harmonic distributions, the expression for the warping distribution can be obtained with a separation of variable technique as

(38)

w = W (s) \cos (π z / L)

which, when substituted in Equations (37), leads to

(39)

\begin{matrix} W_{, ss} - \frac{E}{G} \frac{π^{2}}{L^{2}} W = 0 \\ G W_{, s} (0) = - f_{0} \frac{L}{π}, G W_{, s} (b) = f_{0} \frac{L}{π} \end{matrix}

The solution of Equations (39) is then calculated using

(40)

W (s) = \frac{f_{0} L \coth (β b / 2)}{β G π} \cosh β s - \frac{f_{0} L}{β G π} \sinh β s

with $β = (π / L) \sqrt{E / G}$ . Backward substitution leads to the following active stress field:

(41)

σ_{z} (s, z) = - EW (s) {(\frac{π}{L})}^{2} \sin \frac{π z}{L}, τ_{zs} (s, z) = G W_{, s} (s) \frac{π}{L} \cos \frac{π z}{L}

Finally, the distribution of the reactive stresses $σ_{s} (s, z)$ is derived by integrating the equilibrium equation $σ_{s, s} (s, z) + τ_{zs, z} (s, z) = 0$ along s with boundary conditions $σ_{s} (0, z) = σ_{s} (b, z) = 0$ (one of them being identically satisfied from the kinematic compatibility). The following expression is derived:

(42)

σ_{s} (s, z) = - G {(\frac{π}{L})}^{2} [\frac{f_{0} L \coth (β b / 2)}{β G π} (- 1 + \cosh β s) - \frac{f_{0} L}{β G π} \sinh β s] \sin \frac{π z}{L}

Acknowledgements

The contributions of the first and third authors were partially supported by the Italian Ministry of Education, Universities and Research (MIUR), through the PRIN funded program ‘Dynamics, Stability and Control of Flexible Structures’ (grant number 2010MBJK5B). The work of the second author was supported by the Australian Research Council through its Discovery Projects funding scheme (grant number DP1096454).

Conflict of interest

None declared.

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