GBT-based finite element to assess the buckling behaviour of steel–concrete composite beams

Thin-Walled Structures 107 (2016) 207–220

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GBT-based finite element to assess the buckling behaviour of steel–concrete composite beams David Henriques, Rodrigo Gonçalves a,n, Dinar Camotim b a b

CERIS, ICIST and Departamento de Engenharia Civil, Faculdade de Ciências e Tecnologia, Universidade NOVA de Lisboa, 2829–516 Caparica, Portugal CERIS, ICIST, DECivil, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049–001 Lisboa, Portugal

art ic l e i nf o

a b s t r a c t

Article history: Received 5 June 2016 Accepted 8 June 2016

In this paper, an accurate and computationally efficient Generalised Beam Theory (GBT) finite element is proposed, which makes it possible to calculate buckling (bifurcation) loads of steel–concrete composite beams subjected to negative (hogging) bending. Two types of buckling modes are considered, namely (i) local (plate-like) buckling of the web, possibly involving a torsional rotation of the lower flange, and (ii) distortional buckling, combining a lateral displacement/rotation of the lower flange with cross-section transverse bending. The determination of the buckling loads is performed in two stages: (i) a geometrically linear pre-buckling analysis is first carried out, accounting for shear lag and concrete cracking effects, and (ii) an eigenvalue buckling analysis is subsequently performed, using the calculated pre-buckling stresses and allowing for cross-section in-plane and out-of-plane (warping) deformation. The intrinsic versatility of the GBT approach, allowing the incorporation of a relatively wide range of assumptions, is used to obtain a finite element with a reasonably small number of DOFs and, in particular, able to comply with the principles of the “inverted U-frame” model prescribed in Eurocode 4 [1]. Several numerical examples are presented, to illustrate the application of the proposed GBT-based finite element and provide clear evidence of its capabilities and potential. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Steel-concrete composite beams Generalised Beam Theory (GBT) Distortional buckling Local buckling Shear lag

1. Introduction It is well known that steel–concrete composite beams with a slender steel I-section and subjected to negative (hogging) bending can experience distortional buckling, where the lower (compressed) flange undergoes warping and a lateral displacement, whereas the web (mainly) and the concrete slab exhibit transverse bending, as shown in Fig. 1. In Eurocode 4 (EC4 [1]), which concerns the design of steel–concrete composite structures, it is stated in Section 6.4.1 that the design methods for lateral-torsional buckling prescribed in Eurocode 3 [2] (design of steel structures) may be applied, and that the beneficial restraint due to cross-section distortion may be taken into account. For the particular case of beams in buildings with class 1–3 cross-sections and uniform steel section, this restraint can be calculated according to the so-called “continuous inverted U-frame” model (see Fig. 1 and, e.g., [3]), as detailed in Section 6.4.2 of EC4. Nevertheless, the calculation of the critical moment requires taking into account various other complex effects, such as shear lag, concrete cracking, creep, construction sequence and non-uniform bending—clearly, a non-trivial task. n

Corresponding author. E-mail address: [email protected] (R. Gonçalves).

http://dx.doi.org/10.1016/j.tws.2016.06.005 0263-8231/& 2016 Elsevier Ltd. All rights reserved.

The calculation of elastic distortional buckling loads has deserved the attention of several researchers in the past. A beam finite element including cross-section deformation has been proposed in [4], where the upper flange is assumed fixed and the lower flange can undergo a lateral displacement and an independent in-plane rotation, causing the web to distort into a cubic curve.1 Other authors have proposed beam-on-elasticfoundation models, which make it possible to derive rather simple and practical solutions involving coefficients provided in tables or charts. This is the case of the column-like approach developed by Svensson [8] and the general formulation for composite beams proposed by Hanswille et al., [9,10], which accounts for crosssection distortion in accordance with the U-frame model of EC4. The simple spring-type model developed in [11] also deserves to be mentioned. Generalised Beam Theory (GBT) is a thin-walled prismatic bar theory that includes cross-section in-plane and out-of-plane (warping) deformation through the consideration of so-called 1 This model was later extended to inelastic buckling in [5]. It is also worth mentioning early inelastic models that account for web deformation, such as the plate finite element approach employed in [6] and the finite strip model proposed in [7].

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2. GBT-based buckling analysis of steel–concrete beams 2.1. Fundamental aspects

Fig. 1. Distortional buckling of steel–concrete composite beams (the “inverted U-frame”).

“cross-section deformation modes”. GBT was introduced by Schardt [12,13], 50 years ago, and has since been considerably developed, being presently well established as an efficient and valuable tool to analyse the linear, buckling, post-buckling, vibration and dynamic behaviour of thin-walled members (e.g., [14– 16]). The first applications of GBT in the field of steel–concrete composite I-beams and box-girder bridges were reported in [17]. In this paper, it was demonstrated that GBT can handle, efficiently and accurately, cross-section distortion (including constraints due to diaphragms), shear lag and shear connection flexibility. Several illustrative examples were provided, concerning linear elastostatic and undamped free vibration analyses, and semi-analytical buckling solutions for simply supported members under constant hogging moment were also presented. Partial cracking and other types of material non-linearity were not taken into consideration. More recently, in [18], the authors have proposed a very accurate and computationally efficient physically non-linear beam element that is able to capture the (geometrically linear) behaviour of wide-flange steel and steel–concrete composite beams up to collapse. This element takes into consideration concrete cracking/crushing, shear lag effects and steel plasticity. Moreover, analytical solutions for the elastic shear lag effect were also presented. The above investigations have shown that GBT can offer significant advantages in the field of steel–concrete composite beams, namely with respect to standard shell/solid finite element and finite strip approaches. Indeed, (i) semi-analytical solutions can be retrieved in particular cases, (ii) much lower DOF numbers are generally required to achieve accurate results, (iii) the computation times are greatly reduced in physically non-linear problems and (iv) the “trademark” modal decomposition of the GBT solution into hierarchic and structurally meaningful cross-section deformation modes provides in-depth insight into the mechanics of the problem under consideration. In this paper, a new GBT-based finite element is proposed, which can accurately and efficiently calculate elastic local-distortional bucking (bifurcation) loads2 of steel–concrete composite beams. In particular, the element can handle discrete changes of the cross-section and comply (not exclusively) with the principles of the “inverted U-frame” model of EC4. The calculation involves two steps: (i) a geometrically linear analysis is first carried out, to obtain the pre-buckling stresses, taking into account shear lag and concrete cracking effects—the construction sequence and creep effects can be also considered in a simplified manner —, and (ii) a buckling eigenvalue analysis is performed, allowing for crosssection distortion and local buckling of the web. To illustrate the capabilities of the proposed finite element, several numerical examples are presented and discussed throughout the paper. For validation and comparison purposes, results obtained with finite strip and shell finite element models are provided.

2 Note that plastic bifurcation problems have also been investigated using GBT, see [19].

The notation employed in this paper follows closely that introduced in [20], together with the matrix forms presented in [21,22], although the geometric non-linear terms are simplified in accordance with the usual assumptions pertaining to a linearised buckling analysis. As in [18], the parameters associated with concrete, steel I-section and rebars are identified by the subscripts c, a and s, respectively. It is worth recalling that, in the GBT approach, the member deformed configuration is expressed in terms of k = 1, … , D predetermined cross-section deformation modes (D is the total number of modes included in the analysis), each one multiplied by the corresponding longitudinal amplitude function ϕk (the problem unknowns). Therefore, the member configuration is unequivocally defined by ϕ, the column vector collecting the individual mode amplitude functions. In accordance with the inverted U-frame model, it is assumed that the beam cross-section is of the type shown in Fig. 2(a), comprising a reinforced concrete slab—a perfect bond between the rebars and concrete is assumed—and a steel I-section. The relevant geometric parameters are indicated in the figure, where it should be noted that hw is the distance between the mid-lines of the I-section flanges and As is the rebar area per unit length. Fig. 2 (b) displays the wall mid-lines and local axes, which constitute the basis of the GBT kinematic description and “crosssection analysis”, i.e., the procedure that retrieves the cross-section deformation modes (see Sections 2.3 and 2.4). The local axes are defined as follows: x, y correspond to the beam axis and wall midline directions, respectively, and z is the through-thickness coordinate. As in the usual GBT procedure, the cross-section is subdivided into several walls that are connected through so-called “natural” nodes: (i) two flanges and one web, for the steel I-section, and (ii) two reinforced concrete flanges that include rebar layers. The mid-lines of the I-section top flange and concrete slab are connected through an in-plane rigid link, even if longitudinal slip can be allowed, as proposed in [17]. Note also that symmetry boundary conditions are included in the cross-section.3 A plane stress state is assumed in the walls and equilibrium is written in terms of Green-Lagrange strains E and second Piola– Kirchhoff stresses S , through the virtual work statement in Voigt notation ( ET = [Exx Eyy 2Exy ], ST = [Sxx Syy Sxy ]) , yielding

δW = −

∫V

δE T S dV +

∫Ω δU¯ T q¯ dΩ = 0,

(1)

where V and Ω are the beam volume and wall mid-surfaces at the initial configuration, q¯ are mid-surface forces (for simplicity, volume forces are converted into mid-surface ones) and U¯ are the work-conjugate (mid-surface) displacements. For the pre-buckling analysis, geometric linearity is assumed. Small strains thus hold ( E = ε) and the equilibrium equation and its incremental/iterative linearisation, in the direction of a variation of the configuration Δϕ, are given by

−

∫V

δεT S dV +

∫Ω δU¯ T q¯ dΩ = 0,

(2)

3 If the slab is continuous over several I-beams, the boundary conditions should be changed to reflect the appropriate kinematics of the buckling mode (the pre-buckling analysis is not affected by these boundary conditions, since crosssection in-plane deformation is not allowed, see Section 2.2). Although this can be easily handled with GBT, it was decided to adopt the single inverted U-frame model because it falls on the conservative side.

D. Henriques et al. / Thin-Walled Structures 107 (2016) 207–220

209

Fig. 2. Steel–concrete composite beam (a) cross-section geometry, (b) wall mid-lines and boundary conditions, (c) pre-buckling analysis deformation modes, (d) buckling analysis in-plane independent DOFs and (e) associated mode shapes, (f) orthogonal distortional deformation mode and (g) local modes in the web. The mode shapes displayed were obtained using linear amplitude functions ( ϕk = x ).

DδW [Δϕ] = −

∫V

δεT Ct Δε dV ,

(3)

where Ct is the tangent constitutive matrix, accounting for concrete cracking (the material behaviours are otherwise assumed linear). With a finite element interpolation of the amplitude functions in ϕ, Eq. (2) yields the element out-of-balance force vector and Eq. (3) leads to the element tangent stiffness matrix. For the bucking analysis, the pre-buckling stresses S are retained and define a reference loading, which is multiplied by the load parameter λ. The bifurcation equation corresponds to the linearisation of Eq. (1) in the direction of a buckling mode Δϕ and discarding pre-buckling displacements ( ϕ = 0 , hence δE = δε, ΔE = Δε). This leads to a standard eigenvalue problem of the form

−

⎛

2.2. Kinematic relations Following the standard GBT kinematic description, using the wall mid-surface local axes displayed in Fig. 1(b) and Kirchhoff's thin plate assumption, the displacement vector for each wall U is expressed as

⎡ Ux ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ϕ (x) ⎥ ¯ , U (x, y , z ) = ⎢ Uy ⎥ = (ΞU (y) + zΞU (y) ) ⎢ ϕ (x)⎥ ⎢⎣ U ⎥⎦ ⎢⎣ , x ⎥⎦ z

(5)

⎞

∫V ⎝ δεT Ct Δε + λΔδE T S⎠ dV = 0, ⎜

tangent stiffness matrix, whereas the second term leads to the element geometric matrix.

⎟

(4)

where the eigenvalues λ are the bifurcation load parameters. Resorting once again to a finite element interpolation of the amplitude functions in ϕ, the first term of Eq. (4) yields the element

⎡ ¯T⎤ ⎢ 0 u ⎥ Ξ¯ U (y) = ⎢ v¯ T 0 ⎥, ⎢⎣ w ¯ T 0 ⎥⎦

⎡ 0 w ¯ T⎤ ⎥ ⎢ T ΞU (y) = − ⎢ w ¯ , y 0 ⎥, ⎥ ⎢⎣ 0 0 ⎦

(6)

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where the commas indicate differentiations (e.g., a, y = da/dy ), ϕ(x ) is a column vector containing the k = 1, … , D deformation mode amplitude functions (as previously mentioned) and u¯ (y ), v¯ (y ), ¯ (y ) are k-element column vectors collecting the wall mid-line w displacement functions along x, y, z, respectively. In this paper, the calculation of these displacement functions (explained in Sections 2.3 and 2.4) differs considerably from the usual GBT cross-section analysis, which for arbitrary cross-sections is described in detail in [20,23,24] and is implemented in the GBTUL program (release 2.05), freely available at www.civil.ist.utl.pt/gbt. The Green-Lagrange strain components are subdivided into membrane (M) and bending (B) components. The bending components are assumed to be small. For the membrane components, M only EM xx and Exy need to be retained, since the problems addressed in this paper only concern buckling triggered by longitudinal normal stresses Sxx and shear stresses Sxy. In addition, the nonlinear terms associated with warping displacements are discarded, as in [19]. Therefore one has, according to the notation in [22],

⎡ ε M⎤ B⎤ ⎡ E M, NL ⎤ ⎡ Exx ⎤ ⎡ εxx ⎢ xx ⎥ ⎢ xx ⎥ ⎥ ⎢ B⎥ ⎢ M⎥ E = ⎢ Eyy ⎥ = ⎢ εyy ⎥ + ⎢ εyy + ⎢ 0 ⎥, ⎢ ⎥ ⎢ M, NL ⎥ ⎢ 2E ⎥ ⎢ B ⎥ M⎥ ⎢⎣ γxy ⎣ xy ⎦ ⎢⎣ γxy ⎥⎦ ⎣ 2Exy ⎦ ⎦     

⏟ εB

⎡ ϕ⎤ ⎥ ⎢ ε B = zΞεB (y) ⎢ ϕ, x ⎥, ⎥ ⎢ ⎢⎣ ϕ, xx ⎥⎦

εM

=

⎡ ϕ⎤ ⎥ ⎢ ϕ, x ⎥, ⎥ ⎢ ⎣ ϕ, xx ⎦

ΞεM (y) ⎢

E M , NL

εM

(7)

⎡ 0 ¯ T⎤ 0 w ⎥ ⎢ T ¯ , yy 0 0 ⎥, ΞεB (y) = − ⎢ w ⎥ ⎢ ¯ ,Ty 0 ⎥⎦ ⎢⎣ 0 2w

(8)

⎡ 0 0 u¯ T ⎤ ⎥ ⎢ T 0 0 ⎥, = ⎢ v¯ , y ⎥ ⎢ ⎣ 0 (u¯ , y + v¯ )T 0 ⎦

(9)

ΞεM (y)

M, NL Exx =

⎞ 1 T⎛ T ¯ ¯ + ww ¯ ¯ T ⎟ ϕ, x , ϕ ⎜ vv ⎠ 2 ,x ⎝

(10)

M, NL Exy =

⎞ 1 T⎛ ¯ ,y w ¯ T ⎟ ϕ, x . ϕ ⎜ v¯ , y v¯ T + w ⎠ 2 ⎝

(11)

One final note to stress that the efficiency of the proposed approach stems from a combination of the GBT intrinsic versatility to model the behaviour of thin-walled members and the use of specific assumptions concerning the stresses and strains, which are discussed next. These assumptions are essential to comply with the principles of the “inverted U-frame” model of EC4 and, simultaneously, keep the number of deformation modes (hence DOFs) as low as possible, without sacrificing accuracy. 2.3. Pre-buckling analysis The pre-buckling analysis is similar to that proposed in [18], although concrete cracking is the only source of non-linearity considered here. It is assumed that the cross-section is in-plane M B B undeformable ( εyy = εyy = γxy = 0) and that Vlasov's null memM brane shear strain assumption ( γxy = 0) holds in the steel flanges.

M B These assumptions lead to Sxy = 0 in the = 0 in all walls and Sxy

M B steel flanges. In addition, it is assumed that Syy = Syy = 0, otherwise over-stiff solutions are retrieved due to Poisson effects. The rebars behave uniaxially only. Under these assumptions, the relevant cross-section deformation modes are those displayed in Fig. 2(c) (only the wall mid-lines are represented, the reinforcement layers are omitted), comprising: axial extension (E), Euler-Bernoulli bending (B), web uniform shear (S),

longitudinal slip in the shear connection (LS) and linear/quadratic warping in each concrete wall (LW/QW), to capture shear lag effects. Attention is called to the following aspects concerning these modes: (i) Even if no axial force is applied to the beam, the E mode must be included in the analysis, to capture the neutral axis shift due to cracking, shear lag and longitudinal slip effects. (ii) For uniform members, the B mode is calculated assuming that the concrete is uncracked. For non-uniform members (discretely varying), see the example in Section 3.3.2. (iii) Since a single web shear mode (S) is employed, constant prebuckling shear stresses Sxy are always retrieved in the web, which is generally acceptable in wide-flange sections. (iv) Although additional shear lag modes can be added for improved accuracy, it was shown in [18] that the LW/QW modes already lead to excellent results, even in the presence of steel plasticity and concrete crushing effects. (v) The LS mode was proposed in [17]. When this mode is considered, the virtual work Eq. (1) must include an addition term due to the work done by the longitudinal shear force along the steel–concrete interface. Since this mode essentially influences the pre-buckling results and does not participate in the buckling mode, it is not employed in the examples presented in Section 3—linear examples including this mode are already available in [25,17]. It should be noted that assuming Syy = 0 in the web means that the buckling analysis does not take into account the destabilizing effect of these stresses, which can be important near concentrated loads, such as support reactions. It is therefore assumed that appropriate web transverse stiffeners are provided, to mitigate these effects. With the adopted assumptions, the only non-null stress comM M ponents are SBxx, SM xx and Sxy, although Sxy = 0 holds in the walls where Vlasov's assumption is enforced. Therefore, the tangent constitutive matrix is of the form:

⎡ Et 0 0 ⎤ ⎥ ⎢ Ct = ⎢ 0 0 0 ⎥ , ⎢⎣ 0 0 Gt ⎥⎦

(12)

where Et and Gt are the tangent uniaxial and shear moduli. For the steel beam and longitudinal rebars, a linear elastic constitutive relation is adopted (but recall that the rebars behave uniaxially only). The transverse rebars are not taken into consideration in the pre-buckling analysis, as εyy = 0 is assumed. For concrete, a simplified version of the constitutive law proposed in [18] is adopted, with zero tensile strength and a linear compressive branch. In this case, Et ¼Ec for εxx < 0 (otherwise Et ¼ 0) and Gt = βGc , where β ≤ 1 is a reduction factor for cracked concrete.4 As in [18], a standard GBT finite element interpolation of the amplitude functions is employed, with ϕ = Ψd , where matrix Ψ contains the interpolation functions and vector d contains the unknowns, i.e. the nodal values of the amplitude functions. Hermite cubic polynomials are employed for the B and S modes, whereas Lagrange quadratic polynomials are used for the remaining modes (those involving only warping displacements). This leads to a 26 DOF element if all eight deformation modes are considered. For symmetric sections, the shear lag modes may be paired (LW1þLW2 and QW1þQW2), leading to a reduced 20 DOF element. The element out-of-balance force vector g , tangent stiffness 4 Note that, since no tensile strength is considered and Syy = 0 is assumed, cracking occurs if εxx > 0 or Sxy ≠ 0 . Because the latter condition is generally satisfied due to the shear lag modes, the β factor can be considered constant throughout the whole slab.

D. Henriques et al. / Thin-Walled Structures 107 (2016) 207–220

matrix Kt and incremental load vector Δf are obtained from the numerical integration of the expressions

∫V

g=

Kt =

Δf =

⎡ Ψ ⎤T ⎢Ψ ⎥ T ⎢ , x ⎥ Ξε S dV − ⎢⎣ Ψ, xx ⎥⎦

∫V

⎡ Ψ ⎤T T ⎢ ⎥ Ξ¯U q¯ dΩ, Ω ⎣ Ψ, x ⎦

∫

(13)

⎡ Ψ ⎤ ⎡ Ψ ⎤T ⎥ ⎢ ⎢Ψ ⎥ T Ψ ⎢ , x ⎥ Ξε Ct Ξε ⎢ , x ⎥ dV , ⎢ Ψ, xx ⎥ ⎢⎣ Ψ, xx ⎥⎦ ⎦ ⎣

⎡

⎤T

∫Ω ⎢⎣ ΨΨ,x ⎥⎦

(14)

Ξ¯ UT Δq¯ dΩ,

(15)

where Ξε = ΞεB + zΞεM . Numerical integration is carried out using Gauss quadrature, with 3 points along x and a variable number of points along y and z. In the through-thickness direction z, an odd number of points is adopted to enable retrieving the mid-surface stresses, subsequently employed in the buckling analyses: a single point for the rebar layers, 3 points for the steel beam walls and 5 points for the concrete, in order to capture cracking accurately. In the mid-line direction y, the number of points depends on the modes included in the buckling analysis (see Section 2.4). The prebuckling analysis is carried out with Newton-Raphson iterations until convergence is achieved, for a single load step corresponding to the reference loading adopted for the subsequent calculation of the buckling loads. The procedure was implemented in MATLAB [26]. In certain cases, the construction sequence and/or creep effects must be taken into consideration in the calculation of the prebuckling stresses. With the present approach, this can be achieved in an approximate manner by performing a separate pre-buckling analysis for each stage, considering the appropriate static system and concrete modular ratio. The final pre-buckling stresses and concrete longitudinal strains are calculated by adding the values obtained in each analysis, which is acceptable if the cracked zone due to εxx is approximately the same in all stages. 2.4. Buckling analysis The buckling analysis is carried out by solving the discretised form of the eigenvalue problem given by Eq. (4). Although it is still M M assumed that Eyy = Syy = 0, as in the pre-buckling stage, transverse bending

and

B Sxy ,

B εxy )

B Exy (

torsional

stresses/strains

B B B Syy , Eyy ( = εyy )

and

are now developed due to cross-section in-plane = deformation. Concerning the deformation modes, the set displayed in Fig. 2 (c) must be supplemented with additional modes to enable capturing cross-section deformation, namely local and distortional buckling. These additional deformation modes are obtained on the basis of the structural model of Fig. 2(b) and a special cross-section analysis, where nodal in-plane rotational DOFs are used5 and no mode orthogonalisation is performed. With this approach, the deformation modes become independent of the wall stiffness properties, a feature that facilitates enforcing compatibility between elements with different cross-sections (see the example in Section 3.3.2). However, since the solution is expressed in terms of non-orthogonal modes, a post-processing procedure is performed to obtain the participation of the most relevant orthogonal modes, 5 In the standard GBT approach, the cross-section in-plane nodal rotations are condensed out, see [13].

211

as explained next. M With the Eyy = 0 constraint and assuming that the left concrete B wall does not undergo transverse bending ( εyy = 0), only five independent DOFs exist in the structural model of Fig. 2(b) and those identified in Fig. 2(d) are selected. In addition, Vlasov's assumption (for small strains) is deemed valid in all walls, which means that displacements along y are only possible if warping displacements also occur, satisfying u¯ , y = − v¯ . Recalling that the web vertical displacement, together with Vlasov's assumption, is already accounted for by the B mode (see Fig. 2(c)), only four independent DOFs remain, which generate the four modes in Fig. 2(e). All these modes correspond to the imposition of unit displacements at each DOF, separately, except for mode L2, which is obtained by condensing the DOF associated with mode L1—i.e., mode L1 can be viewed as if following the standard hierarchic shape function approach. Mode ND in Fig. 2(e) is termed “distortional”, as it involves warping in the lower flange, due to its lateral displacement,6 and also “non-orthogonal”, since it is not orthogonalised in the usual GBT sense. The standard GBT orthogonal distortional mode (D) is shown in Fig. 2(f) and, together with modes E and B, completes the so-called “Vlasov natural mode set” [20,17]. The D mode corresponds to a combination of the ND, L2 and L3 modes,7 whose coefficients depend on the transverse bending stiffness of the various walls. Instead of following the standard GBT approach (see [17]), the D mode can be easily obtained by applying a lateral displacement at the lower flange, while restraining the web vertical displacement and condensing out all other DOFs in Fig. 2(d). The warping functions of the lower and upper flange of the I-section are then found by ensuring that no axial force exists in each one and that u¯ , y = − v¯ . It should be noted that, if the slab is transversally cracked, the transverse bending stiffness of the D mode is quite similar to that obtained using Section 6.4.2 of EC4, although the gap between the steel top flange and concrete is discarded.8 Further details on the application of the EC4 approach are provided by Hanswille et al. [9,10], where the critical moment is obtained using a beam-onelastic-foundation model that includes the EC4 U-frame transverse bending stiffness, plus the warping stiffness of the lower flange and the St. Venant torsional stiffness of the steel section (assuming that no transverse web bending occurs). Therefore, if only the D mode is included in the buckling analysis, the proposed GBT approach can be considered roughly equivalent to the EC4-Hanswille approach.9 Since the GBT analysis is carried out with the ND mode, instead of the D mode, the results are post-processed to retrieve the participation of the latter, thus providing further insight into the mechanics of the problem. This is achieved by calculating the D mode and projecting the solution in the new deformation mode space. A similar procedure could be carried out for the L1-L3 modes, which can also be replaced by orthogonal ones (in the GBT sense), but this has not been done because these modes already enable a clear interpretation of the results—each one is associated with a specific localised deformation.

6 Actually, mode L2 also involves minute warping in the steel top flange, as the rotation is applied at the node connecting the two concrete walls and, therefore, a lateral displacement is produced in this flange. 7 Note that mode L1 is not necessary to define the D mode. This fact was precisely the motivation for adopting the hierarchical approach to define modes L1 and L2. 8 Note that using the full gap, as in the proposed GBT approach, constitutes an approximation when the slab is cracked, since the neutral axis for transverse bending is not located at the slab mid-height. 9 However, note that the GBT approach includes a pre-buckling analysis that takes into account, automatically and quite accurately, shear lag, cracking and shear connection stiffness effects.

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In addition, a set of web local modes is also included in the analyses, in order to capture local buckling. These modes, designated L4–L7, are given by polynomials of increasing degree and exhibit null displacements and rotations at the web/flange junc¯ k = fk /max (fk ), with tions, given by w

f4 = y2 (y − h w )2 ,

(16)

f5 = y2 (y − h w )2 (y − h w /2),

(17)

(ii) In the transverse direction, the slab is assumed either fully cracked (as prescribed by EC4) or uncracked. Since null membrane transverse extensions are enforced, direct throughthickness integration leads to over-stiff solutions, as the neutral axis is not located at z ¼0. This difficulty is circumvented by using directly the appropriate bending stiffness Dfcy, which includes the contribution of the transverse rebars. (iii) β is assumed constant throughout the whole slab, as in the pre-buckling analysis. One therefore has

f6 = y2 (y − h w )2 (y − h w /3)(y − 2h w /3),

(18)

f7 = y2 (y − h w )2 (y − h w /4)(y − h w /2)(y − 3h w /4).

(19)

Fig. 2 (g) displays these four mode shapes. It should be mentioned that it has been assumed that the lower flange is stocky enough to preclude the occurrence of transverse bending in this wall. Although such effect can be easily incorporated, by including appropriate deformation modes, this has not been done in the present work. For steel, an elastic material law is adopted, making it possible to uncouple membrane and bending terms, as well as perform analytical integration in the trough-thickness direction. For the steel beam, the constitutive matrix for the membrane terms is as given in Eq. (12), whereas for the bending terms a plane stress matrix is employed, leading to

⎡ Ea t 0 0 ⎤ ⎥ ⎢ M C¯ ta = ⎢ 0 0 0 ⎥, ⎢⎣ 0 0 Ga t ⎥⎦

(20)

⎡ 1 νa 0 ⎤ ⎥ ⎢ B ν 1 0 ⎥ C¯ ta = Dfa ⎢ a , ⎢ 1 − νa ⎥ ⎥ ⎢0 0 ⎣ 2 ⎦

⎤ ⎡ E h3 ⎢ tcx c 0 0 ⎥ ⎥ ⎢ 12 B Dfcy 0 ⎥, C¯ tc = ⎢ 0 ⎥ ⎢ ⎢ βGc hc3 ⎥ 0 0 ⎥ ⎢⎣ 12 ⎦

(26)

and recall that Etcx is calculated at z¼0 only. The geometrically non-linear term in Eq. (4) is also integrated in the through-thickness direction. The steel beam and longitudinal rebars behave elastically and therefore the throughthickness integration of the pre-buckling stress bending components vanishes. Note that the displacements of the rebars must be obtained from Eq. (6), substituting z by the value corresponding to M each layer. For the concrete, it is assumed that Sxx ≈ Sxx and recall that, necessarily, Sxx ≤ 0. One therefore has, for the longitudinal pre-buckling stresses, M ¯ ¯ T + ww ¯ ¯ T ) tλSxx Δϕ, x . ∫ ΔδExx λSxx dz = δϕ,Tx ( vv

(21)

(22)

For each longitudinal rebar layer, only the membrane component is considered and integration along z provides B C¯ tsx ≈ 0.

(23)

The transverse rebars are included in the concrete slab, as explained next. For concrete, Poisson coupling is discarded due to cracking and, at Gauss point level, the tangent constitutive matrix reads

⎡ Etcx 0 0 ⎤ ⎥ ⎢ Ctc = ⎢ 0 Etcy 0 ⎥, ⎢⎣ 0 0 βGc ⎥⎦

(25)

(27)

For the shear stresses, only the contribution of the web is conM sidered. Taking into account that v¯k, y = 0, due to the εyy = 0 assumption, one has M ¯ ¯ ,Ty Δϕ) tλSxy . ∫ 2ΔδExy λSxy dz = ( δϕT w¯ ,y w¯ T Δϕ,x + δϕ,Tx ww

Ea t 3 Dfa = . 12 (1 − νa2 )

⎡ Es Asx 0 0⎤ M ⎢ ⎥ C¯ tsx = ⎢ 0 0 0 ⎥, ⎢⎣ 0 0 0⎥⎦

⎡ Etcx hc 0 0 ⎤ ⎥ ⎢ M C¯ tc = ⎢ 0 0 0 ⎥, ⎢⎣ 0 0 βGc hc ⎥⎦

(28)

For a finite element solution, the interpolation scheme of the pre-buckling analysis is followed, although a higher number of deformation modes are included in the analysis. For the additional modes, displayed in Fig. 2(e) and (g), Hermite polynomials are employed and therefore 4 × 8 = 32 DOFs are added to the finite element of the pre-buckling analysis. The discretised eigenvalue problem is given by

( Kt + λG ) Δd = 0,

(29)

where the element tangent stiffness and geometric matrices read

Kt =

⎡ Ψ ⎤T ⎢Ψ ⎥ ⎢ ,x ⎥ Ω ⎢⎣ Ψ, xx ⎥⎦

∫

((

T M ΞεM C¯ t ΞεM

)

+

T B ΞεB C¯ t ΞεB

( )

)

⎡ Ψ ⎤ ⎥ ⎢ ⎢ Ψ , x ⎥ dΩ , ⎢ Ψ, xx ⎥ ⎦ ⎣

(30)

(24)

where Etcx is zero if εxx > 0 from the pre-buckling analysis. It is possible to eliminate membrane/bending coupling and integrate in the through-thickness direction if the following assumptions are adopted: (i) Etcx is assumed constant throughout the slab depth, equalling its mid-line value (at z¼ 0).

G=

+

⎛

⎞

¯ ¯ T + ww ¯ ¯ T Ψ , x dΩ ∫Ω tSxxM Ψ ,Tx ⎝ vv ⎠ ⎜

⎛

⎟

(31)

⎞

¯ ¯ ,Ty Ψ dΩ, ∫Ω tSxyM ⎝ ΨT w¯ ,y w¯ T Ψ,x + Ψ,Tx ww ⎠ ⎜

⎟

(32)

D. Henriques et al. / Thin-Walled Structures 107 (2016) 207–220

and it should be noted that the integrations are performed along the member mid-surface only. Numerical integration is carried out with 3 Gauss points along x (as in the pre-buckling analysis) and M ¯2 the number of points along y required to integrate Sxx wk in each wall—e.g., for the web, due to the L7 mode and the fact that SM xx is linear, 8 Gauss points should be adopted. The finite element procedure, including the calculation of the eigenvalues and eigenvectors, was implemented in MATLAB.

3. Numerical examples 3.1. Simply supported beam under hogging bending The first example concerns simply supported beams subjected to uniform negative bending moments. The relevant geometrical and material parameters, as well as the results obtained, are provided in Fig. 3. Two steel web height (hw) values are considered and the span length L is varied between 1 and 17 m. The longitudinal rebars are assumed located at z¼0 and the concrete is deemed fully uncracked in the transverse direction (the contribution of transverse reinforcement is not considered). Due to the uniform moment distribution, no shear effects are present in the pre-buckling stage and the deck is fully cracked in the longitudinal direction. Therefore, the GBT pre-buckling analyses are carried out without the S, LS, LW and QW modes. For the buckling analyses, the ND and L1–L7 modes are added to the previously employed mode set. The beam is discretized using 10 equal-length finite elements.

213

In this particular case, sinusoidal amplitude functions constitute exact solutions and the CUFSM finite strip program [27] provides accurate solutions, which are used for comparison purposes. The top graph in Fig. 3 plots the critical moment vs. the span L and the two bottom diagrams provide the corresponding GBT modal participations, obtained by means of a strain energy criterion (see, e.g., [28]). Furthermore, a representative sample of the GBT buckling modes obtained is included in the figure. First of all, it is observed that the GBT-based critical moments are in excellent agreement with the finite strip ones throughout the whole L range considered, indicating that the selected GBT deformation mode set is appropriate for the problem under consideration. These results show that, as expected, local buckling is characterised by very small half-wave lengths, whereas distortional buckling is associated with moderate half-wave lengths and is critical for span values of practical interest. The modal participation diagrams make it possible to conclude that local buckling has major contributions from modes L4 þL5 (web local deformation) and noteworthy participations of modes L3þL6 (rotation of the lower flange and web local deformation with 3 half-waves). On the other hand, the distortional mode corresponds essentially to the D mode, with a very small participation of mode L3—note that this last deformation mode is the only one participating in both buckling modes. The dashed curves in the top graph in Fig. 3 correspond to GBTbased buckling analyses performed with the D mode alone. It is observed that they practically match those obtained with all the modes, for the span range associated with critical distortional buckling. This shows that, in this particular case, the D mode alone

Fig. 3. Buckling of simply supported steel–concrete beams subjected to uniform negative moment.

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a

E = 210 GPa = 0.3 a = 2400 mm b = 800 mm t = 15 mm

p L = 20 m

b

Modal participation

100% 80%

L = 20 m

Steel beam

Reinforcement

Concrete

Ea = 210 GPa = 0.3 hw = 800 mm tw = 15, 30 mm bf = 300 mm tf = 30 mm

Es = 200 GPa Asx = 1.5 % Asy = 1.5 %

Ec = 37 GPa =0 hc = 0.20 m bc1 = 1.50 m bc2 = 2.00 m

60%

Fig. 5. Two-span beam geometrical and material parameters. 40% 20% 0% L2

L3

L4

L5

L6

L7

Fig. 4. Buckling of a simply supported plate subjected to pure shear.

is capable of providing very accurate distortional buckling loads. It is still worth mentioning that the results obtained show that increasing the web height hw influences significantly the local buckling load (as expected), but has a small impact on the distortional buckling load concerning the larger spans, although the transition between modes with 1 and 2 half-waves occurs for considerably different lengths (11 m for hw ¼0.8 m and 15 m for hw ¼1.2 m). 3.2. Shear buckling of a web panel This example aims at assessing the accuracy of the proposed finite element in providing the solution of a classical plate shear buckling problem. The plate is simply supported along all edges and subjected to pure shear, as shown in Fig. 4. The critical shear stress can be generally expressed as

τcr = k

⎛ t ⎞2 π 2E ⎜ ⎟ , 2 12 (1 − ν ) ⎝ b ⎠

(33)

where b , t are the plate height and thickness, respectively, and k is the buckling coefficient. For an aspect ratio a/b = 3, the EBPlate program [29] retrieves k¼5.84. For the GBT analysis, only the web of the I-section is considered, with h w = b, tw = t and L ¼a. In addition, only modes L2– L7 in Fig. 2 are included in the buckling analyses. The results obtained with only two finite elements fall only 4.4% above the EBPlate value and differences below 1% are obtained if five or more elements are employed. Fig. 4 displays the critical buckling mode obtained with the proposed finite element and the corresponding modal participations, showing that the first two deformation modes (L2, L3) are the most relevant, although significant participations of the L4 and L5 modes are also observed. The last two modes (L6–L7) have very small participations. 3.3. Two-span beam subjected to an uniformly distributed load In this section, the two-span composite beam shown in Fig. 5 is analysed by adopting several assumptions. The beam is subjected to an uniformly distributed vertical load acting in the plane of the web. All the relevant material and geometric parameters are provided in the figure and note that two web thickness values are considered.10 The three supports fully restrain all cross-section inplane displacements (including the concrete slab ones), but allow warping. 10 Note that, for tw = 30 mm , the web and flange have the same thickness, which is unrealistic. This case was considered because it constitutes a “limit case”, when the web is very stocky.

It should be noted that this problem is significantly more complex than the previous ones, since the pre-buckling stresses vary quite rapidly along the beam length and buckling is localised near the intermediate support region. The GBT pre-buckling analyses are carried out using a symmetry simplification, whereas the buckling analyses assume antisymmetry of the buckling modes with respect to the intermediate support. Such strategy makes it possible to analyse a single span member, thus achieving a significant DOF economy. For comparison purposes, results obtained with refined meshes of 4-node MITC shell finite elements (over 53 000 DOFs) are provided. The ADINA program [30] was used for the calculations. In these models, the top steel flange and the concrete slab are connected through rigid links (no slip is allowed) and the two spans are modelled. At the supports, the cross-section in-plane displacements are prevented with the exception of the horizontal displacements of the concrete slab and lower flange (except the web-flange node), to avoid stress concentrations. Nevertheless, since the web is restrained, Syy stresses inevitably occur near the supports. 3.3.1. Uncracked analysis First, the concrete slab is assumed fully uncracked and the contribution of the reinforcement is discarded. With these assumptions, the GBT results can be straightforwardly compared with those obtained with a standard elastic isotropic shell finite element model. The results obtained are shown in Figs. 6 ( tw = 15 mm ) and 7 (tw = 30 mm ). In each one, the top-left graph displays the variation of the GBT-based first two buckling loads (pb) with the number of equal-length finite elements, obtained considering (i) only Sxx prebuckling stresses and all deformation modes, (ii) Sxx, Sxy and all deformation modes, and (iii) Sxx and all deformation modes in Fig. 2(c) (pre-buckling analysis) plus the distortional (D) mode in Fig. 2(f). For comparison purposes, the shell model results are also provided in these graphs (the thick horizontal lines). The tables below these graphs show the shell- and GBT-based buckling loads (the latter obtained with 100 elements). In addition, each figure shows the modal participations for each buckling mode, obtained also with 100 elements, and the corresponding buckling mode shapes—note that the GBT buckling modes concern a single span, whereas the shell model ones include the two spans, as already mentioned. First of all, the influence of the GBT element length on the buckling loads is analysed. The top-left graphs in Figs. 6 and 7 show that 20–30 elements already provide quite good results, although more elements are required in some cases (namely the second mode when tw = 30 mm ). These discretisation levels are naturally higher than those adopted in Example 3.1, as the present problem is clearly more complex (as already mentioned). For tw = 15 mm (Fig. 6), the first two buckling loads are quite close— pb = 250.5, 270.3 kN/m for the shell model—and the critical buckling mode combines cross-section distortion with web local buckling. The presence of the latter is somewhat surprising,

D. Henriques et al. / Thin-Walled Structures 107 (2016) 207–220

215

Fig. 6. Buckling of a two-span elastic beam subjected to an uniformly distributed load ( tw = 15 mm ).

as the web is not particularly slender.11 Concerning the GBT results, the following conclusions are drawn: (i) The GBT buckling loads obtained with 100 finite elements and both pre-buckling stresses (256.8 and 284.3 kN/m) lie only slightly above the shell model ones (2.5% and 5.2%, respectively). The corresponding buckling modes confirm this similarity: some differences are observed in the first mode (the GBT solution predicts a more pronounced web local deformation), but an

11 According to EC4 and EC3, if a low steel grade is adopted and the neutral axis is located near the top steel flange, with hw /tw = 53.3, no allowance for shear buckling is necessary and the web is class 3. Also note that the lower flange is rather stocky, as bf /2t f = 5.

excellent agreement is found for the second mode. The associated GBT modal participation diagrams show that both buckling modes have high participations of the web local deformation modes (namely L4), with the second mode exhibiting also an important contribution of the D mode (23%). (ii) Consider now the GBT results with Sxx only and including all deformation modes. The first buckling load is somewhat higher than the shell model one, even when 100 elements are used (8.1%), although the modal participations and buckling mode configuration are rather similar to those obtained with both Sxx and Sxy. This shows that, in this case, both pre-buckling stresses should be considered in the analyses. Conversely, the second buckling load is much closer to the shell model value (only 1.6% above for 100 elements), which contradicts

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D. Henriques et al. / Thin-Walled Structures 107 (2016) 207–220

Fig. 7. Buckling of a two-span elastic beam subjected to an uniformly distributed load ( tw = 30 mm ).

the previous assertion.12 However, a close inspection of the buckling mode shape and modal participations reveals that the web local modes L4–L5 now almost vanish and that the buckling mode becomes essentially distortional—thus, although the buckling load is in better agreement with the shell model value, the buckling mode is not. (iii) If the GBT analyses are carried out with Sxx and the D mode

12 The fact that excluding Sxy leads to lower buckling loads is not surprising, as the “true” stress state also includes Syy stresses, which are being discarded in the proposed finite element.

alone (plus the pre-buckling modes), the first buckling load obtained with 100 elements is 13.6% above the shell model value, a difference that could be considered acceptable in some circumstances (e.g., pre-design stage). Nevertheless, this result shows that the distortional deformation mode alone may not provide accurate buckling loads (in this case, the L2 mode is also required). In addition, the second buckling mode is strongly overestimated (pb ¼997.8 kN/m), a value outside the graph range. For tw = 30 mm , the shell model results shown in Fig. 7 make it

D. Henriques et al. / Thin-Walled Structures 107 (2016) 207–220

217

Fig. 8. Critical buckling loads and modes for a two-span beam with non-uniform web thickness.

end support

intermediate support Fig. 9. Orientation of the cracks at each mid-surface integration point, for one span of the continuous beam (5 integration points are employed in each cross-section wall).

possible to conclude that the two buckling loads are quite far apart ( pb = 625.1, 1046 kN/m ) and that the first buckling mode is distortional, whereas the second mode is local. The observation of the GBT results leads to the following conclusions: (i) With both pre-buckling stresses, the first/second buckling loads for 100 elements are 5.5/4.6% above the shell model ones and a very good match between the buckling modes is again observed. The modal participations confirm that the first buckling mode is distortional (with an additional rotation of the top flange, evidenced by the presence of mode L2), whereas the second is local in the web with a very small contribution from distortion (3%). (ii) Using Sxx only, all deformation modes and 100 elements, the buckling loads are quite accurately captured (2.8 and 5.4% above the shell model values), as well as the buckling mode configurations. The modal participations are very similar to those obtained with the two pre-buckling stresses. Although there are no significant differences between the buckling loads obtained with/without Sxy, it should be noted that, like for the tw = 15 mm case, a slightly lower buckling load is retrieved without Sxy when the buckling mode has a noteworthy participation of the D mode (the second buckling mode for tw = 15 mm ; the first one for tw = 30 mm ). Even if the differences are rather small, it is therefore advisable to carry out both types of analyses (with/without Sxy). (iii) With Sxx and the D mode alone, the first buckling load with 100 elements is almost 30% above the shell value, which is not acceptable. This difference is due to the fact that the L2 mode has a significant participation. The second buckling load is once more outside the graph range.

3.3.2. Non-uniform cross-sections In this example, it is demonstrated how the proposed formulation handles a change in cross-section geometry. Consider that the web thickness equals 20 mm within 3 m of the intermediate support and is 15 mm elsewhere. As in the previous case, the slab is assumed fully uncracked and reinforcement is not considered. In the GBT analyses, compatibility between finite elements with different web thickness is ensured by the modes in Fig. 1(c), (e), (g), with the exception of the B mode, which is calculated using the particular cross-section stiffness properties. In order to avoid using compatibility equations, the B mode is calculated for tw = 20 mm (also for the elements with tw = 15 mm ) and the solution is subsequently projected in the correct mode space, as in the case of the D mode. A total of 80 equal-length elements are employed in a single span (12 elements in the thicker web length). Fig. 8 displays the critical buckling loads and modes obtained. A comparison between these results and those concerning the first (critical) mode with tw = 15 mm (Fig. 6) reveals that (i) a 30% increase of the critical load occurs and (ii) the buckling mode becomes essentially distortional. Once more, the GBT-based results are in very good agreement with the shell model ones, particularly if Sxy = 0.13 3.3.3. Cracked analysis Consider now the effects of cracking. In this case, the length of the negative moment region decreases and buckling becomes even more localised and affected by stress concentrations (near the intermediate support) than in the previous examples—thus, the problem is even more complex. The reinforcement is taken into account, assumed concentrated at the concrete slab mid-line, and β = 0.2 is adopted. The slab transverse bending stiffness Dfcy is calculated for either cracked or uncracked states—recall that the former case is prescribed by EC4. In the GBT analyses, 30 equal-length elements were found to be sufficient. Due to cracking, the pre-buckling analysis is carried out with Newton–Raphson iterations, as was described in Section 2.3. For illustrative purposes, Fig. 9 displays a typical cracking pattern. For comparison purposes, results obtained with refined shell finite element models (ADINA) are also provided, following an approximate method to reproduce the assumptions of the GBT model concerning the concrete slab. An orthotropic shell element is used for concrete, with the material parameters in accordance M with Eq. (24), with Etcx = 0 or Ec (if εxx > 0 or <0, respectively) and 3 Etcy = 12Dfcy /hc . The longitudinal reinforcement is included through additional orthotropic shell elements, with the appropriate longitudinal stiffness and zero stiffness otherwise, which 13 Note that this is in agreement with the conclusions of the previous example, as the buckling mode has a significant participation of the D deformation mode.

218

D. Henriques et al. / Thin-Walled Structures 107 (2016) 207–220

Fig. 10. Buckling of a two-span cracked beam subjected to an uniformly distributed load.

are laid over the concrete shell elements, matching the corresponding nodes. Since Etcx depends on the longitudinal strains, a quite cumbersome and time-consuming iterative pre-buckling analysis must be carried out: at the end of each iteration, the M concrete elements with εxx > 0 are identified and Etcx is changed accordingly, until no significant changes are observed in consecutive iterations.

The critical buckling loads and corresponding buckling modes are displayed in Fig. 10, together with the GBT modal participations. These results prompt the following remarks: (i) With respect to the shell buckling loads concerning the uncracked slab case (Section 3.3.1), a slight increase is observed, up to 12% for tw = 15 mm and a transversally uncracked slab.

D. Henriques et al. / Thin-Walled Structures 107 (2016) 207–220

700 600

t w = 30 mm

pcr (kN/m)

500 400

t w = 15 mm

300 200 100 0 0%

20%

40%

pL/pcr,S

60%

80%

100%

Fig. 11. Influence of the long/short-term loading ratio.

the Sxy stresses are is discarded. The graph in Fig. 11 plots the buckling loads as a function of the ratio between pL and the critical load for short-term loading ( pcr , S ), which equals 295.7 kN/m for tw = 15 mm and 628.2 kN/m for tw = 30 mm (see Fig. 10). These results show that the buckling load decreases slightly as pL /pcr , S increases, with a maximum drop of 6% and 7%, for tw = 15 mm and tw = 30 mm , respectively. No results for ratios beyond pL /pcr , S = 90% are provided, because buckling is triggered by the long-term load alone. It should also be mentioned that it was found that the modal participations do not change with pL /pcr , S and therefore the corresponding graphs in Fig. 10, for the transversally cracked cases, remain valid.

4. Concluding remarks

This increase is not so pronounced when the slab is transversally cracked and, in fact, for tw = 30 mm , only a 2% drop is obtained. (ii) The buckling mode shapes show that distortion plays a major role in all cases (as attested by the modal participation graphs). It is also worth noting that significant transverse bending of the slab is observed for tw = 30 mm . (iii) The GBT buckling loads are in very good agreement with the shell model ones, particularly when Sxy is discarded. This is in agreement with the previous observations for the buckling modes involving significant distortion. It is also noted that the differences with respect to the shell models are higher for the slender web case ( tw = 15 mm ), revealing that web localised deformation14 and/or stress concentrations15 may play a significant role in this case. (iv) The GBT modal participation diagrams show that the flange rotation modes L2 and L3 are present in some cases, particularly for the transversally cracked case. This means that, like in the case of Fig. 7, using the D mode alone will not lead to satisfactory results. 3.3.4. Effect of long/short-term loading This last example aims at illustrating the application of the proposed finite element when several pre-buckling analyses need to be performed. Consider a short-term live load pS added to a long-term (dead) load pL and that the former is increased until the critical buckling load is reached, i.e., the total load is pL + λpS . The stresses due to the long-term loading are calculated using a concrete modular ratio EL = Ec /2, to account for creep, whereas for the short-term loading one has ES ¼ Ec. It is assumed that these stresses can be calculated independently, which constitutes a rough approximation if the cracked length changes significantly, as mentioned in Section 2.3—in this particular case, this assumption is acceptable. The corresponding discretised buckling eigenvalue problem reads

( Kt + GL + λGS ) Δd = 0,

219

(34)

where the loading parameter λ only affects the short-term loading geometric matrix. Both web thickness values are considered and the slab is assumed cracked in the longitudinal and transverse directions, with β = 0.2. In all cases, discretisations with 50 equal-length finite elements are adopted. Note that the results concerning the shortterm loading alone are given in Fig. 10, for the transversally cracked cases. Following the conclusions drawn in Section 3.3.3, 14 Note that, in the GBT modal participation diagrams, the web local modes L4– L5 are involved only in the tw = 15 mm case. 15 Recall that, with the proposed GBT-based finite element, SM yy is discarded and SM xy is assumed constant in the web.

A new GBT-based beam finite element was proposed in this paper, which is capable of calculating, accurately and efficiently, elastic bifurcation loads of steel–concrete composite beams. The element includes only a few deformation modes (upto 8 for the pre-buckling analysis and 16 for the buckling analysis) and accounts for shear lag, concrete cracking, shear connection stiffness, cross-section distortion, local (plate-type) deformation in the web and discrete changes of the cross-section. The construction sequence and creep can also be taken into consideration, although in a simplified manner. The numerical examples presented in the paper provide ample evidence of the capabilities of the proposed element, as the buckling loads and modes obtained match quite accurately those provided by finite strip and shell finite element models, while involving a much lower DOF number. Moreover, the GBT modal decomposition features make it possible to draw meaningful conclusions concerning the buckling mode nature. For example, it was observed that it generally involves a combination of the D and L modes, and that local-distortional interaction can occur, even in beams with stocky webs. It should be stressed that the key features of the proposed element stem from the fact that GBT allows incorporating a wide range of assumptions concerning the stresses and strains, without sacrificing accuracy. In particular, the proposed element is designed to enable attributing independent values to the various bending/membrane stiffness terms of the concrete slab, a feature that allows, for instance, complying with the principles of the U-frame model prescribed in EC4—note that this is not easily achieved using shell elements.

References [1] EN 1994-1-1:2004, Eurocode 4: Design of Composite Steel and Concrete Structures – Part 1-1: General Rules and Rules for Buildings, CEN, Brussels, Belgium, 2004. [2] EN 1993-1-1:2005, Eurocode 3: Design of Steel Structures – Part 1-1: General Rules and Rules for Buildings, CEN, Brussels, Belgium, 2005. [3] R. Johnson, D. Anderson, Designers’ Guide to EN 1994-1-1: Eurocode 4: Design of Composite Steel and Concrete Structures. General rules and rules for buildings, Thomas Telford, 2004. [4] M. Bradford, Z. Gao, Distortional buckling solutions for continuous composite beams, J. Struct. Eng. (ASCE) 118 (1) (1992) 73–89. [5] M. Bradford, Inelastic buckling of i-beams with continuous elastic tension flange restraint, J. Constr. Steel Res. 48 (2–3) (1998) 63–77. [6] G. Weston, D. Nethercot, M. Crisfield, Lateral buckling in continuous composite bridge girders, Struct. Eng. 5 (1991) 79–87. [7] M. Bradford, R. Johnson, Inelastic buckling of composite bridge girders near internal supports, Proc. Inst. Civil Eng. Part 2 83 (1987) 143–159. [8] S. Svensson, Lateral buckling of beams analyzed as elastically supported columns subjected to a varying axial force, J. Construct. Steel Res. 5 (3) (1985) 179–193. [9] G. Hanswille, J. Lindner, D. Münch, Zum Biegedrillknicken von Verbundträgern, Stahlbau 7 (1998) 525–535. [10] G. Hanswille, Lateral torsional buckling of composite beams. Comparison of

220

[11]

[12] [13] [14]

[15]

[16]

[17] [18]

[19]

[20]

D. Henriques et al. / Thin-Walled Structures 107 (2016) 207–220

more accurate methods with Eurocode 4, in: J. Hajjar, M. Hosain, W. Easterling, B. Shahrooz (Eds.), Composite Construction in Steel and Concrete IV, ASCE, 2002, pp. 105–116. N. Dekker, A. Kemp, P. Trinchero, Factors influencing the strength of continuous composite beams in negative bending, J. Construct. Steel Res. 34 (2–3) (1995) 161–185. R. Schardt, Eine erweiterung der technischen biegetheorie zur berechnung prismatischer faltwerke, Stahlbau 35 (1966) 161–171 (German). R. Schardt, Verallgemeinerte Technische Biegetheorie, Springer Verlag, Berlin, Germany, 1989 (German). D. Camotim, C. Basaglia, R. Bebiano, R. Gonçalves, N. Silvestre, Latest developments in the GBT analysis of thin-walled steel structures, in: E. Batista, P. Vellasco, L. Lima (Eds.), Proc. Int. Coll. Stability and Ductility of Steel Struct., Rio de Janeiro, Brazil, 2010, pp. 33–58. D. Camotim, C. Basaglia, N. Silva, N. Silvestre, Numerical analysis of thinwalled structures using generalised beam theory (GBT): recent and future developments, in: B. Topping, J. Adam, F. Pallarés, R. Bru, M. Romero (Eds.), Computational Technology Reviews, vol. 1, Saxe-Coburg, Stirlingshire, 2010, pp. 315–354. D. Camotim, C. Basaglia, Buckling analysis of thin-walled steel structures using generalized beam theory (gbt): state-of-the-art report, Steel Constr. 6 (2) (2013) 117–131. R. Gonçalves, D. Camotim, Steel–concrete composite bridge analysis using generalised beam theory, Steel Compos. Struct. 10 (3) (2010) 223–243. D. Henriques, R. Gonçalves, D. Camotim, A physically non-linear GBT-based finite element for steel and steel–concrete beams including shear lag effects, Thin-Walled Struct. 90 (2015) 202–215. R. Gonçalves, P. Le Grognec, D. Camotim, GBT-based semi-analytical solutions for the plastic bifurcation of thin-walled members, Int. J. Solids Struct. 47 (1) (2010) 34–50. R. Gonçalves, M. Ritto-Corrêa, D. Camotim, A new approach to the calculation

[21]

[22]

[23]

[24]

[25]

[26] [27]

[28]

[29] [30]

of cross-section deformation modes in the framework of Generalized Beam Theory, Comput. Mech. 46 (5) (2010) 759–781. R. Gonçalves, D. Camotim, Generalised beam theory-based finite elements for elastoplastic thin-walled metal members, Thin-Walled Struct. 49 (10) (2011) 1237–1245. R. Gonçalves, D. Camotim, Geometrically non-linear generalised beam theory for elastoplastic thin-walled metal members, Thin-Walled Struct. 51 (2012) 121–129. R. Gonçalves, R. Bebiano, D. Camotim, On the shear deformation modes in the framework of generalized beam theory, Thin-Walled Struct. 84 (2014) 325–334. R. Bebiano, R. Gonçalves, D. Camotim, A cross-section analysis procedure to rationalise and automate the performance of GBT-based structural analyses, Thin-Walled Struct. 92 (2015) 29–47. R. Gonçalves, D. Camotim, Steel–concrete composite beam analysis using Generalised Beam Theory, in: D. Lam (Ed.), Proceedings of the 9th International Conference on Steel Concrete Composite and Hybrid Structures (ASCCS), 2009, pp. 657–662. MATLAB, version 7.10.0 (R2010a), The MathWorks Inc., Massachusetts, 2010. Z. Li, B. Schafer, Buckling analysis of cold-formed steel members with general boundary conditions using CUFSM: conventional and constrained finite strip methods, in: Proceedings of 20th International Specialty Conference on ColdFormed Steel Structures (St. Louis, 3-4/11), 2010, pp. 17–31. G. Garcea, R. Gonçalves, A. Bilotta, D. Manta, R. Bebiano, L. Leonetti, D. Magisano, D. Camotim, Deformation modes of thin-walled members: a comparison between the method of generalized eigenvectors and generalized beam theory, Thin-Walled Struct. 100 (2016) 192–212. Y. Galéa, P. Martin, Contrainte critiques de voilement de plaques rectangulaires - Présentation du logiciel libre EBPlate, Constr. Métall. 44 (3) (2007) 33–74. K.J. Bathe, ADINA System, ADINA R&D Inc., 2012.