A New Modeling Approach for Planar Beams: Finite-Element Solutions based on Mixed Variational Derivations

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1 A New Modeing Approach for Panar Beams: Finite-Eement Soutions based on Mixed Variationa Derivations Ferdinando Auricchio,a,b,c,d, Giuseppe Baduzzi a,e, Caro Lovadina c,d,e May 31, 2 a Dipartimento di Meccanica Strutturae (DMS), Università degi Studi di Pavia, via Ferrata 1, 271 Pavia Itay b European Centre for Training and Research in Earthquake Engineering (EUCENTRE), Pavia, Itay c Istituto di Matematica Appicata e Tecnoogie Informatiche (IMATI), CNR, Pavia, Itay d Centro di Simuazione Numerica Avanzata (CeSNA), IUSS, Pavia, Itay e Dipartimento di Matematica Feice Casorati, Università degi Studi di Pavia, via Ferrata 1, 271 Pavia Itay Abstract This paper iustrates a new modeing approach for panar inear eastic beams. Referring to existing modes, we first introduce the variationa principes that coud be adopted for the beam mode derivation, discussing their reative advantages and disadvantages. Then, starting from the Heinger-Reissner functiona we derive some homogeneous and mutiayered beam modes, discussing some properties of their anaytica soution. Finay, we deveop a panar beam finite eement, foowing an innovative approach that coud be seen as the imposition of equiibrium in the cross-section and compatibiity aong the axis. The homogeneous mode is capabe to reproduce the behavior of Timoshenko beam, with the advantage that the shear correction factor appears naturay from the variationa derivation; the mutiayered beam is capabe to capture the oca effects produced by boundary constraints and oad distributions; the finite eement is capabe to predict the cross-section stress distribution with high accuracy, and more generay the behavior of panar structura eements. Keyword: aminated inear eastic beam; anaytica soution; finite eement modeing; mixed variationa formuation. 1 Introduction To reproduce the mechanica behavior of structura eements characterized by one dimension predominant with respect to the other two (i.e. a beam) the most cassica approaches are the we-known Euer Bernoui (shorty EB) and Timoshenko theories iustrated in standard iterature (e.g. see Timoshenko (1955) and Hjemstad (25)). Generay adopted for very sender eements, the EB beam mode assumes cross-section axia and transverse dispacements as one dimensiona unknown fieds, converting the 3D eastic probem, formuated through a set of partia differentia equations (PDE), into a simper system of two ordinary differentia equations (ODE). On the other hand, adopted for ess sender structura eements, Timoshenko beam mode assumes cross-section axia and transverse dispacements, together with rotations as one-dimensiona fieds, eading to a set of three ODE. As a consequence of the fied assumptions, these modes are often cited in iterature as first-order modes. Moreover, we denote these modes as dispacement-based Emai address: auricchio@unipv.it 1

2 to underine that usuay these modes consider ony dispacements as independent variabes and stresses are evauated by post processing. Despite their simpicity (aso reated to the possibiity of computing anaytica soutions) and their exceent first-design abiity, EB and Timoshenko modes have some we-known imitations, which we wish to cite: oss of accuracy for beams with ow ratio between ength and cross-section characteristic dimension, oss of accuracy in modeing non-homogeneous beams (such as composites) as we as in modeing non-inear materia behavior, difficuty in computing accurate stress profies, connected to the fact that ony mean-vaue dispacements are assumed as fied variabes. Hence, due to the need of more accurate one-dimensiona modes, researchers have deveoped new approaches that can generay be cassified as: high-order, dispacement-based modes, which introduce more sophisticated crosssection kinematics, mixed or hybrid modes, which introduce aso stresses and sometimes even strains as independent variabes. Just to give a few exampes, an often cited mode faing in the first category is the one proposed by Reddy (1984) that introduces section warping in addition to Timoshenko dispacements. As discussed by Sheinman (2), this mode shows an inconsistency, since transverse dispacement generates a constant cross-section shear distribution, whereas axia dispacements give a quadratic one. A more compete kinematics is adopted in the mode proposed by Lo et a. (1977a) where, considering both section warping and striction, authors eiminate such an inconsistency. For a genera treatment of high-order, panar, kinematic beam modes and discussion of their anaytica soutions the readers may refer to the work of Sheinman (2). With respect to mixed modes, a very interesting approach is the one proposed by Aessandrini et a. (1999), where, starting from 3D easticity, the authors iustrate a cear derivation of some pate modes, studying aso the convergence of the modes under consideration. As far as the numerica modeing is concerned, a iterature review shows that the number of proposed finite eement (FE) impementations is neary uncountabe. This aso depends on the fact that many different choices are possibe depending on: functiona formuation and independent variabes: the simpest functionas consider ony kinematic variabes, as in the dispacement based beam modes whie, on the contrary, the most compicated ones consider aso stresses and strains; order of fied approximation: in addition to the first- versus high-order cassification, for the case of mutiayered beams, ayer-wise modes adopt piecewise ayer-defined functions whie goba modes use gobay defined functions to describe the fied distribution on crosssection. For both items, intermediate choices are possibe, e.g. ony some stress components coud be seected as independent variabes, or some fieds are described ayer-wise, whie the remaining ones are assumed as goba functions. It is interesting to observe that in avaiabe review-papers (e.g., see Carrera (2) and Wanji and Zhen (28)), the authors usuay compare the deveoped modes by considering different fied approximations but derived from the same principe. Therefore, it is possibe to understand ony the effect of basis accuracy, but not the improvements introduced considering a more sophisticated variationa principe. With this respect, in Section 3 we present different variationa principes that can be empoyed, briefy discussing their respective features and possibe drawbacks. 2

3 The main aim of this paper is to propose a simpe anaytica mode and a FE capabe of capturing the section stress distribution with high accuracy, aso in a mutiayered body. We opt for the dimension reduction variationa approach, cassica in beam modeing, and used among others by Aessandrini et a. (1999) for pate probems. This method coud aow to generaize the mode to other more compex situations, such as non inear constitutive aws, for instance. A summary of the paper is as foows. In Section 2 we define the probem we are going to tacke, in Section 3 we introduce the functionas which may be used for the mode derivation, isting for each one properties and deveoped modes presented in the iterature; by means of this discussion we are abe to choose an appropriate variationa principe as the starting point for the mode derivations. In Sections 4 we derive some 1D modes, whie in Section 5 we present a few exampes, giving some insight on the structure of the corresponding anaytica soution. In Section 6 we deveop suitabe finite eement schemes, and we present some resuts to iustrate their actua computationa performances. 2 Probem definition The object of our study is a panar beam, not necessariy homogeneous aong the thickness and modeed as a 2D body: this is equivaent to impose pane stress state hypotheses to a threedimensiona body or to state that the beam width is negigibe. Moreover, we imit the discussion to inear eastic isotropic materias and sma dispacements. We define the probem domain as: = A where the beam ongitudina axis and the cross-section A are defined as: = { x R x [, ]} { [ A = y R y h 2, h ]} 2 in which h is the beam thickness and is the beam ength. Obviousy >> h so that the ongitudina axis is the predominant dimension of the body. Figure 1 represents the domain and the adopted Cartesian coordinate system. y O A S n A h x S Figure 1: Panar beam geometry, coordinate system, dimensions and adopted notations. As iustrated in Figure 1 we define the atera cross-sections as A and A, whie the bottom and upper boundaries as S and S n, respectivey. We denote the domain boundary as, so that = (A A ) (S S n ). For, we consider the partition { t ; s }, where t and s are the externay oaded and the dispacement constrained boundaries, respectivey. The externa oad is defined as a ine force density t : t R 2 whereas the body oad is defined as an area force density f : R 2 ; moreover we specify a sufficienty smooth boundary dispacement function s : s R 2. Finay, we suppose that the beam is made by ayers whose thickness is constant aong the axia coordinate x, but not necessariy the same at each ayer. Consequenty, the Young s moduus E and the Poisson s ratio ν are scaar fieds depending on the thickness coordinate y, i.e. E : A R and ν : A R. 3

4 S n S n O S 2 S S y b = 1 z h h n h n 1 h 3 h 2 h 1 Figure 2: Cross-section geometry, coordinate system, dimensions and adopted notations. Introducing the foowing independent variabe fieds: σ : R 2 2 ε : R 2 2 s : R 2 where the stress σ and the strain ε are symmetric tensors whie the dispacement s is a vector, the strong formuation of the probem under investigation corresponds to the foowing boundary vaue probem: ε = s s (1a) σ = D : ε in (1b) σ + f = (1c) σ n = t on t (1d) s = s on s (1e) where Equation (1a) is the compatibiity reation, Equation (1b) is the materia constitutive reation in which D is the appropriate fourth order inear eastic tensor and Equation (1c) represents the equiibrium condition. Equations (1d) and (1e) are the boundary equiibrium and the boundary compatibiity condition, respectivey. 3 Variationa probem formuations and modeing methods In this section we are going to present different possibe variationa formuations for Probem (1). In particuar, we introduce approaches based on tota potentia energy, Heinger-Reissner and Hu-Washizu functionas, considering for each one different possibe stationarity conditions. Restricting to the framework of Section 2, we emphasize that a variationa beam mode can be considered as the outcome of the foowing genera procedure. First step. A variationa principe is seected for the 2D panar easticity probem. In particuar, the functiona spaces for the invoved fieds has to be appropriatey chosen. 4

5 Second step. For each invoved fied, an approximation profie is seected. Typicay, to deveop a beam mode, one chooses poynomia or piecewise poynomia shapes aong the thickness direction, whie no profie restrictions are imposed aong the axia direction. However, the approximation fieds shoud fit the functiona framework of the First step above, at east if a conforming mode is considered. Third step. Integration aong the thickness direction is performed. This way, the 2D variationa probem is reduced to a 1D variationa probem, which corresponds to a system of ODEs equipped with boundary conditions (i.e. the beam mode). 3.1 Tota Potentia Energy approach Tota Potentia Energy (shorty TPE) is the functiona most frequenty used in continuum mechanics and in standard iterature (e.g. Hjemstad, 25); severa authors use it as a starting point to derive both first-order beam modes as we as the most popuar corresponding FE formuations. The TPE functiona can be expressed as foows: J TPE (s) = 1 2 s s : D : s sd s f d (2) Boundary conditions wi be suitaby imposed in what foows. The critica point of the functiona above corresponds to find the energy minimizer, which is unique and stabe in the usua framework of admissibe dispacement space. Requiring stationarity of TPE (2), we obtain the foowing weak probem: Find s W s such that δs W s : δjtpe s = s (δs) : D : s s d δs f d δs t ds = (3) t where W s := { s H 1 () : s s = s } W s := { δs H 1 () : δs s = } We define Equation (3) as the TPE symmetric stationarity that is often used as a basis for FE deveopment, eading obviousy to a symmetric stiffness matrix. Wanji and Zhen (28) give a review on mutiayered, eastic, dispacement-based (i.e., TPE derived) pate FE. Increasing orders of fied approximation are considered, from the simpest mode in which dispacements are gobay defined aong the cross-section, to the most sophisticated ones in which dispacements are defined ayer-wise. Wanji and Zhen aso notice that amost a the presented FE perform ony for some specific probems (thick aminated pates, soft-core sandwich, etc.), but they are not abe to accuratey describe the shear distribution aong the thickness in the genera case. An accurate evauation of shear distribution is one of the aims of the work by Vinayak et a. (1996a,b), in which the authors deveop a mutiayered panar beam FE starting from Equation (3), using the fied approximation proposed by Lo et a. (1977a,b) and appropriatey treating the thickness heterogeneity. They propose two ways to evauate the axia and transverse stresses: the first one uses the compatibiity and the constitutive reations (Equations (1a) and (1b)), whie the second one refines the shear and the out-of-pane stress distributions using the equiibrium reation (1c). The resuting numerica schemes are generay satisfactory, but the computed soutions might exhibit instabiities near the boundary, and the stress distributions are not aways sufficienty accurate. As a genera remark (see Rohwer and Rofes (1998) and Rohwer et a. (25)), in TPE-based modes the critica step is the post processing stress evauation that coud compromise the effectiveness of the method. 5

6 3.2 Heinger-Reissner approach The Heinger-Reissner (shorty HR) functiona can be expressed as: J HR (σ, s) = σ : s sd 1 σ : D 1 : σ d s f d (4) 2 How boundary conditions wi be enforced depends on the specific variationa formuation empoyed to express the stationarity of the functiona (cf. Sections and 3.2.2). We aso wish to remark that stationarity of this functiona corresponds to a sadde point probem. Therefore, the mode derivation based on HR functiona requires a particuar care on the dispacement and stress fied assumptions, otherwise the mode risks to ead to a probem acking we-posedness HR grad-grad stationarity Requiring stationarity of HR functiona (4), we obtain the foowing weak probem: Find s W gg s and σ S gg such that δs W gg and δσ S gg : δj gg HR = s δs : σ d+ δσ : s s d δσ : D 1 : σ d δs f d δs tds = (5) t where W gg s := { s H 1 () : s s = s } ; W gg := { δs H 1 () : δs s = } ; S gg := { σ L 2 () } We ca Equation (5) HR grad-grad stationarity because two gradient operators appear in the formuation. We remark that the kinematic boundary condition s s = s is directy enforced in the tria space W gg s (essentia boundary condition), whie σ n t = t turns out to be a natura boundary condition. As anticipated in Section 1, Aessandrini et a. (1999) derived some homogeneous pate modes starting from HR grad-grad stationarity (5). They noticed that in many situations, modes derived by HR grad-grad stationarity (5) ead to mode dispacement fieds which minimize the potentia energy in the cass of the same kinematic assumptions. Therefore, in those cases HR grad-grad modes are essentiay equivaent to the corresponding modes obtained by the TPE symmetric stationarity (3). Many researchers have derived mutiayered pate and beam FE using HR grad-grad stationarity (5). The works by Spiker (1982), Feng and Hoa (1998), Icardi and Atzori (24) and Huang et a. (22) are among the most significant exampes, since the computed soutions are generay satisfactory. The main drawback of these schemes, especiay for the case of ayer-wise beams and pates, is the high number of degrees of freedom (DOFs) that eads to an heavy FE formuation. Spiker (1982) aeviates this probem by assuming stress variabes to be discontinuous aong the pate extension so that they can be condensed at the eement eve reducing the mixed oca stiffness matrix to a dispacement-ike one. An aternative is to consider some stress components as dependent variabes, expressing them a-priori in terms of the dispacements. Reissner s Mixed Variationa Theorem, introduced by Reissner (1986), foows this approach: the out of pane stresses τ xy and σ yy are considered as independent variabes whie the axia stress σ xx is expressed as a function of dispacements; Carrera (2, 2), Carrera and Demasi (22) and Demasi (29a,b,c,d,e) derived different pate FE appying this viewpoint, with different choice of basis function and obtaining a reasonabe accuracy in stress description; unfortunatey they are forced to use a high number of kinematic DOFs, drasticay increasing the computationa effort HR div-div stationarity Integrating by parts, the first and the second terms of Equation (5) become: s δs : σ d = δs σ n ds δs σ d δσ : s s d = δσ n sds δσ sd (6) 6

7 Hence, substituting Expression (6) in (5), the weak formuation becomes: Find s W dd and σ St dd such that δs W dd and δσ S dd δj dd HR = δs σ d δσ : D 1 : σ d δs f d δσ sd + δσ n s ds = s (7) where W dd := { s L 2 () } S dd t := { σ H(div, ) : σ n t = t } ; S dd := { δσ H(div, ) : δσ n t = } Here above and in what foows, H(div, ) denotes the space of vectoria square-integrabe functions, whose divergence is sti square-integrabe. We define Equation (7) as HR div-div stationarity because two divergence operators appear in it. We remark that s s = s is now a natura boundary condition, whie σ n t = t becomes an essentia boundary condition, as it is directy incorporated in the space St dd. Considering the HR div-div stationarity approach (7), Aessandrini et a. have derived some homogeneous pate modes, more interesting than the ones stemming from the HR grad-grad stationarity (5). However, the same techniques deveoped in Aessandrini et a. cannot be directy appied to genera heterogeneous pates, because the resuting modes may be divergent (cf. Auricchio et a. (24)). 3.3 Hu-Washizu approach The Hu-Washizu (shorty HW) functiona may be expressed as foows: J HW (σ, ε, s) = σ : ( s s ε) d + 1 ε : D : ε d 2 s f d (8) Again, how boundary conditions wi be enforced depends on the specific variationa formuation empoyed to express the stationarity of the functiona. We aso remark that, here, aso the strain fied is a prima variabe. In the foowing variationa formuations, we wi not specify the functiona frameworks for the invoved fieds, since they are simiar to the ones of the HR-based corresponding variationa formuations HW grad-grad stationarity Stationarity of Equation (8) can be expressed as: δj gg HW = s δs : σ d δε : σ d + δσ : ( s s ε) d + δε : D : εd δs f d δs tds = t (9) where s satisfies s s = s, whie σ n t = t is a natura boundary condition. As in the discussion of HR stationarities we ca Equation (9) HW grad-grad stationarity because two gradient operators appear in the formuation. An exampe of the use of HW grad-grad stationarity (9) in mutiayered pate modeing is presented by Auricchio and Sacco (1999). 7

8 3.3.2 HW div-div stationarity A second HW stationarity formuation can be found introducing Equations (6) in Equation (9) obtaining: δjhw dd = δs σ d δε : σ d δσ s d δσ : εd + δε : D : ε d (1) δs f d + (δσ n) s ds = s where σ satisfies σ n = t on t, whie s s = s is a natura boundary condition. We ca Equation (1) HW div-div stationarity because two divergence operators appear in the formuation. 3.4 Concusions on probem weak formuations From the previous discussion (briefy summarized in Tabe 1 in terms of principes and equation formats) it is possibe to draw some concuding remarks, isted in the foowing. A the considered weak probem formuations are symmetric. Every mixed weak formuation can be expressed in different formats: the grad-grad formuations (5) and (9), and the div-div formuations (7) and (1). The former ones require to consider a-priori smooth dispacement fieds and ess reguar stress fieds (s H 1 () and σ L 2 ()), whie the atter ones demand a-priori ess reguar dispacement fieds and smooth stress fieds (s L 2 () and σ H(div, )). When seecting the approximation fieds for the mixed mode design (cf. Second step of the procedure described at the beginning of this Section), the combination of the reguarity requirements and the we-posedness of the corresponding sadde point probems, typicay eads to congruent modes for the grad-grad formuations sef-equiibrated modes for the div-div formuations. Since we think that one of the major imitations of the eementary avaiabe beam modes is connected to the fact that equiibrium equations are not sufficienty enforced within the crosssection, in the foowing we focus on HR formuations but with some emphasis on the div-div form (7). Principe dispacement-based TPE eq (3) mixed HR grad-grad eq (5) div-div eq (7) HW grad-grad eq (9) div-div eq (1) Tabe 1: Obtained functiona stationarities cassified in terms of functionas from which are derived and equation formats. 4 Mode derivations In this section we deveop some beam modes. Referring to the procedure described at the beginning of Section 3, we wi start from the HR variationa formuations of Section 3.2 (cf. First step), we wi define the approximated fieds (cf. Second step), and perform the integration aong the thickness (cf. Third step). Moreover, to simpify the further discussions, we wi switch to an engineering notation. 8

9 4.1 Profie approximation and notations We now introduce the notation we use for a generic approximated fied γ (x, y) invoved in the beam modes. e first define a profie vectoria function p γ (y): p γ : A R m We insist that the m components of p γ (y) are a set of ineary independent functions of the cross-section coordinate y. They may be goba poynomias, as we as piece-wise poynomias or other shape functions. The approximate fied γ (x, y) is expressed as a inear combination of the components of p γ (y), with arbitrary functions of the axia coordinate x as coefficients, coected in a vector ˆγ(x). Therefore, γ (x, y) can be written as a scaar product between p γ (y) and ˆγ(x), i.e. where the superscript T denotes the transposition operation, and γ (x, y) = p T γ (y)ˆγ(x) (11) ˆγ : R m contains the functiona coefficients as components. We emphasize that, once the profie vector p γ (y) has been assigned, the fied γ (x, y) is uniquey determined by the components of ˆγ(x). Such components, for a the invoved fieds, are indeed the unknowns of the beam modes we wi deveop. We aso remark that, due to assumption (11), computation of partia derivatives is straightforward, since it hods: x γ = ( p T γ x ˆγ ) = p T d γ dx ˆγ = p T γ ˆγ y γ = ( p T γ y ˆγ ) = d dy pt γ ˆγ = p T γ ˆγ where we use a prime to indicate derivatives both aong x and aong y, because there is no risk of confusion. Adopting notation introduced in (11) and switching now to an engineering notation we set: { } su (x, y) s(x, y) = = s v (x, y) σ xx (x, y) σ(x, y) = σ yy (x, y) = τ xy (x, y) [ p T u (y) ] { ûu(x) p T v (y) ˆv(x) p T σ x (y) p T σ y (y) p T τ (y) } = P s ŝs (12) = P σˆσ (13) ˆσ x (x) ˆσ y (x) ˆτ(x) where, for P s and P σ (ŝs and ˆσ, respectivey), we drop the expicit dependence on y (x, respectivey), for notationa simpicity. Virtua fieds are anaogousy defined as: δs = P s δŝs δσ = P σ δˆσ Coherenty with the engineering notation just introduced, in Tabe 2 we re-define the differentia operators and the norma versor product where [ ] [ ] 1 1 E 1 = E 1 2 = 1 In Section 3, with D 1 we denoted the fourth order eastic tensor whie from now on, we use the same notation to indicate the corresponding square matrix obtained foowing engineering notation. Therefore, we have: D 1 = 1 E 1 ν ν 1 2 (1 + ν) 9

10 Tensoria notation σ S s σ n ( Engineering notation d dx E 1 + d ) dy E 2 P σˆσ ( d dx ET 1 + d ) dy ET 2 P s ŝs (n x E 1 + n y E 2 ) P σˆσ Tabe 2: Tensoria and Engineering equivaent notations. 4.2 Probem formuations In the foowing, we consider the specia case of a beam for which it hods: s = on A = s, t on A f = in, t = on S S n Hence, t = A S S n ; moreover, the beam is camped on the eft-hand side A, and it is subjected to a non-vanishing traction fied on the right-hand side A. Furthermore, we suppose that t A can be exacty represented using the chosen profies for σ n. Recaing (13), and noting that n A = (1, ) T, this means that there exist suitabe vectors ˆt x and ˆt τ such that { } P T σ t = xˆt x P Tˆt (14) τ τ Therefore, the boundary condition σ n A = t may be written as (cf. (13)) { } } ˆσ x () {ˆt x = ˆτ() ˆt τ (15) We aso remark that a these assumptions can be modified to cover more genera cases; nonetheess, this simpe mode is aready adequate to iustrate the method capabiities HR grad-grad approach Using the notations introduced in Subsection 4.1 the HR grad-grad stationarity (5) becomes: δj gg HR = [( d dx E T 1 + d dy E 2 E 2 T ) δˆσ T P T σ D 1 P σˆσ d T P s δŝs] P σˆσ d + δŝs T P T s tdy = t δˆσ T P T σ [( d dx ET 1 + d ) ] dy ET 2 P s ŝs d Expanding Equation (16) we obtain: ( ) ( ) δj gg HR = δŝs T P T s E 1 P σˆσ + δŝs T P T s E 2 P σˆσ d + δˆσ T P T σ ET 1 P s ŝs + δˆσ T P T σ ET 2 P sŝs d δˆσ T P T σ D 1 P σˆσ d δŝs T P T s t dy = A By using Fubini-Tonei theorem, Equation (17) can be written as: ( ) δj gg HR = δŝs T G sσˆσ + δŝs T H s σˆσ + δˆσ T G σs ŝs + δˆσ T H σs ŝs δˆσ T H σσˆσ dx δŝs T x= T x = (18) 1 (16) (17)

11 where: G sσ = G T σs = P T s E 1 P σ dy H σσ = P T σd 1 P σ dy A A H s σ = H T σs = P T s E 2 P σ dy T x = P T s t dy A A Equation (18) represents the weak form of the 1D beam mode. To obtain the corresponding boundary vaue probem, we integrate by parts the first term of (18): δŝs T G sσˆσdx = δŝs T G sσˆσ x= x= (19) δŝs T G sσˆσ dx (2) Substituting Equation (2) in (18), recaing that δŝs = on A and coecting variabes in a vector we obtain: { [δŝs; δˆσ] (G T ŝs } { }) ŝs ˆσ + H gg dx + δŝs T (G ˆσ sσˆσ T x ) x= = (21) in which G = [ G sσ G σs ] [ H gg = H s σ H σs H σσ Requiring to satisfy Equation (21) for a the possibe variations, and imposing the essentia boundary condition ŝs we finay obtain: { ŝs } { } { } G ˆσ + H gg ŝs = in ˆσ (23) G sσˆσ = T x at x = ŝs = at x = We remark that boundary vaue probem (23) is not necessariy we-posed. This depends on how the profie vectors have been chosen for a the invoved fieds. However, we-posedness of (23) is guaranteed if the approximated fieds are seected according with the approximation theory of sadde-point probem (5) (see Aessandrini et a. (1999)) HR div-div approach Using the notation introduced in Section 4.1 in Equation (7), the HR div-div functiona stationarity becomes: [( δjhr dd = δŝs T d P T s dx E 1 + d ] [( d dy E 2 )P σˆσ d dx E 1 + d ) T dy E 2 P σ δˆσ] P s ŝsd (24) δˆσ T P T σd 1 P σˆσ d = Expanding Equation (24), the weak formuation becomes: ( ) ( ) δjhr dd = δŝs T P T s E 1 P σˆσ + δŝs T P T s E 2 P σˆσ d δˆσ T P T σ ET 1 P s ŝs + δˆσ T P T σ ET 2 P s ŝs d δˆσ T P T σd 1 P σˆσ d = By using Fubini-Tonei theorem, Equation (25) becomes: ( ) δjhr dd = δŝs T G sσˆσ δŝs T H sσ ˆσ δˆσ T G σs ŝs δˆσ T H σ sŝs δˆσ T H σσˆσ dx = (26) 11 ] (22) (25)

12 where H σ s = A P T σ E 2 P s dy whie the other matrices are defined as (19). Equation (26) represents the weak form of the 1D beam mode. To obtain the corresponding boundary vaue probem, we integrate by parts the third term of (26): δˆσ T G σs ŝsdx = δˆσ T G σs ŝs x= + x= δˆσ T G σs ŝs dx (27) Substituting Equation (27) in Equation (26), recaing that G sσ δˆσ = at x =, and coecting the unknowns in a vector we obtain: { [δŝs; δˆσ] (G T ŝs } { }) ŝs ˆσ + H dd dx + δˆσ T G ˆσ σs ŝs = (28) x= where G is defined as in (22) and H dd is defined as: [ ] H dd H = sσ H σ s H σσ (29) Requiring to satisfy Equation (28) for a the possibe variations, and imposing the essentia boundary condition (15), we finay obtain: } } { { ŝs } { G ˆσ + H dd ŝs = in ˆσ ˆσ x = ˆt x at x = ˆτ = ˆt τ at x = G σs ŝs = at x = We remark that boundary vaue probem (3) is not necessariy we-posed. This depends on how the profie vectors have been chosen for a the invoved fieds. However, we-posedness of (3) is guaranteed if the approximated fieds are seected according with the approximation theory of sadde-point probem (7) (see Aessandrini et a. (1999)). 4.3 Concusions on the derived beam modes From what we have deveoped in this section, we can make the foowing remarks. Starting from two different versions of the HR stationarity condition (i.e., grad-grad (5) and div-div (7)) and introducing hypothesis (11), we obtain two different casses of onedimensiona beam modes. Both casses may be described by a boundary vaue probem of the foowing type: { ŝs } { } { } ŝs G ˆσ + H = ˆσ (31) + suitabe boundary conditions The difference between modes based on the HR grad-grad formuation and the HR div-div one stands in the H matrix. More precisey: H = H gg for the HR grad-grad formuation. In this case, derivatives are appied to the dispacement fieds through the symmetric gradient operator, and most ikey the resuting modes strongy satisfy the compatibiity aw, but not necessariy the equiibrium equation. (3) 12

13 H = H dd for the HR div-div formuation. In this case, derivatives are appied to the stress fieds through the divergence operator, and most ikey the resuting modes strongy satisfy the equiibrium equation, but not the necessariy the compatibiity one. Boundary vaue probem (3) can be expicity written as: G sσˆσ H sσ ˆσ = G σs ŝs H σ sŝs H σσˆσ = + suitabe boundary conditions We can compute ˆσ from the second equation and substitute it in the first one, obtaining a dispacement-ike formuation of the probem: { } Aŝs + Bŝs + Cŝs = (32) + suitabe boundary conditions where A = G sσ H 1 σσg σs B = G sσ H 1 σσh σ s + H sσ H 1 σσg σs C = H sσ H 1 σσh σ s Anaogous considerations appy to probem (23). 5 Exampes of beam modes In this section we give two exampes of beam modes deveoped using the strategies of Section 4. More precisey, starting from the HR div-div approach (Equation (3)), we derive: 1. a singe ayer beam mode in which we use a first order dispacement fied; by means of this exampe we wi show that the approach under discussion is abe to reproduce the cassica modes; 2. a mutiayer beam mode, in which we consider aso higher order kinematic and stress fieds; by means of this exampe we wi iustrate how the approach coud produce a refined mode with a reasonabe soution. 5.1 Singe ayer beam Considering a homogeneous beam, we assume a first-order kinematic (as in Timoshenko mode) and the usua cross-section stress distributions (obtained from the Jourawsky theory). In other words we make the foowing hypotheses: { } { } 1 u u = u (x) + yu 1 (x) i.e. p u = ûu = y v = v(x) i.e. p v = {1} ˆv = {v} { } { } 1 σx σ xx = σ x (x) + yσ x1 (x) i.e. p σx = ˆσ y x = σ x1 σ yy = i.e. p σy = {} ˆσ y = {} τ = ( 1 4y 2 /h 2) τ(x) i.e. p τ = { 1 4y 2 /h 2} ˆτ = {τ} u 1 13

14 The matrices G and H dd defined in (22) and (29), and entering into the beam mode (3), are expicity given by: h h h G= 3 h ; H h dd = h h 3 E h E 3 h 2 3 h 8 (1+ν) h2 15 E (33) Since probem (3) is governed by an ODE system with constant coefficients, the homogeneous soution can be anayticay computed. For exampe, choosing h = 1 [mm] = 1 [mm] E = 1 5 [MPa] ν =.25 [ ] the homogeneous soution is given by u = C 4 x + C 1 u 1 = C 6 x C 5 x + C 2 v = ( C 6 x C 5 x C 2 x + C 3 ) C 6 x σ x = C 4 σ x1 = 4. C 6 x + C 5 τ = C 6 (34) in which C i are arbitrary constants. The six constants of (34) may be determined by imposing the boundary conditions specified in (3). Indeed, for the beam mode under consideration, the boundary conditions in (3) ead to a set of six ineary independent equations, since the matrix G σs (cf. (19) and (22)) is invertibe. We remark that the soution (34) is compatibe with the one obtained by the Timoshenko beam mode. However, we underine that the stress distributions aong the beam axis are obtained directy from the mode soution, and not by means of the dispacement derivatives, as happens in cassica formuations. We aso notice that, reducing the mode to the dispacement formuation (cf. (32)), we obtain the foowing ODE system: heu = h 3 12 Eu 1 5 Eh 62(1 + ν) (v + u 1 ) = (35) 5 Eh 62(1 + ν) (v + u 1) = + suitabe boundary conditions Therefore, not surprisingy, we again recover the cassica Timoshenko equations, where, however, the exact shear correction factor (5/6) automaticay appears. The same resut hods true in the framework of variationa pate modeing proposed by Aessandrini et a. (1999). 5.2 Mutiayered beam We now consider a beam composed by n ayers, as iustrated in Figure 2. The geometric and materia parameters for the generic i-th ayer are defined by the thickness h i, the Young s moduus E i and the Poisson ratio ν i, coected in the n-dimensiona vectors h, E and ν, respectivey. To design a mutiayered beam mode, we foow the two-step procedure described beow. 14

15 1. In each ayer, we choose suitabe profies for every fied invoved in the modeing. Of course, given a generic fied, the simpest choice, which is the one we use in this paper, is to use the same profies for every ayer. 2. Across each interayer, we impose the necessary continuity to ensure that the stresses beong to H(div). As a consequence, given a generic fied γ, it is possibe to define its profie vector p γ, which is characterized by: The highest poynomia degree, with respect to y, used in a generic ayer. This number is denoted by deg (p γ ) in what foows. The reguarity across each interayer: in the foowing, C 1 stands for no continuity requirement, C for standard continuity requirement. Since a main aim of this paper is to deveop a mode with an accurate stress description, a natura choice is to assume deg (p τ ) = 2, as in Jourawsky theory. To ensure we-posedness of the resuting mode, we thus seect the invoved fieds according to Tabe 3, where we aso show the number of ayer and goba DOFs. Furthermore, we notice that we have to impose σ y = τ = at the top and bottom of the beam. deg (p γ ) inter-ayer continuity ayer DOF goba DOF p u 1 C 1 2 2n p v 2 C 1 3 3n p σx 1 C 1 2 2n p σy 3 C 4 3n-1 p τ 2 C 3 2n-1 Tabe 3: Poynomia degrees of the profies vectors, continuity properties and number of DOFs for a mutiayered beam. Remark 5.1. More generay, to design a we-posed beam mode one coud choose together with the H(div) reguarity for the stress fied A test case deg (p σx ) = deg (p u ) = deg (p τ ) 1 deg (p τ ) = deg (p v ) = deg ( p σy ) 1 (36) We now present an easy case to test the mode robustness. More precisey, we consider a homogeneous beam, but we treat it as if it were formed by three ayers. Thus, the geometric and materia properties are described by: h = [mm] E = [MPa] ν = [ ] As a consequence, the tota number of cross-section variabes is 34. We notice that the boundary vaue probem (3) is uniquey sovabe aso in this case. However, we now have rank(g) = 22, which means that probem (3) is actuay a differentia-agebraic boundary vaue probem. Thus, 22 variabes are soutions of a differentia probem, whie the remaining 12 unknowns are agebraicay determined by the former ones. We aso remark the the boundary conditions in (3) actuay ead to 22 independent constraints, since rank(g σs ) =

16 We now give the soution of the generaized eigenvaue probem det ( λg + H dd) =, which enters in the construction of the homogeneous soution of (3). [12] [6] ± 3.87i [1] ± 3.87i [1] ± 2.585i [1] λ = ± 2.585i [1] 4.23 ± 2.52i [1] 4.23 ± 2.52i [1] ± 6.21i [1] ± 6.21i [1] where in square brackets we show the eigenvaue mutipicities, and we used the symbo to denote eigenvaues which vanish up to the machine precision. It is possibe to evauate aso the homogeneous soution but, given the probem compexity, it is huge and we wi not report it. Nevertheess it is possibe to discuss its structure and made some important remarks. The zero eigenvaues ead to poynomia terms anaogous to the Timoshenko homogeneous soution described in Section 5.1. The compex conjugate eigenvaues (a ± ib) ead to functions ike C i e ax sin(bx + C j ), which describe oca effects near the boundaries, as it happens in severa other beam modes (see Ladeveze and Simmonds, 1998; Aix and Dupeix-Couderc, 29). 6 Numerica mutiayered beam mode In this section we deveop the FE corresponding to the mutiayered beam mode introduced in Section 5.2. This is equivaent to introduce a dimension reduction aso aong the beam axis, eading, therefore, to a purey agebraic system. The discretization of axia fieds ŝs and ˆσ can be generay described as foows: ŝs(x) N s s = ŝs (x) B s s = [ Nu (x) N v (x) [ N u (x) N v(x) ] { ũu ṽv ] { ũu ṽv } ; ˆσ(x) N σ σ = } ; ˆσ (x) B σ σ = N σx (x) N σy (x) N τ (x) N σ x (x) N σ y (x) N τ (x) σ x σ y ; τ σ x σ y τ (37) From now on, for the sake of notationa simpicity, we wi drop the expicit dependency on x for the invoved fieds. 6.1 Weak probem formuation We now make expicit the weak formuation we wi use as a starting point for the FE discretization. To this end, we first reca the beam mode variationa formuation (26). Then, we integrate by parts with respect to the x direction both the first and the third terms of Equation (26). We thus obtain: Find ŝs W and ˆσ S such that for every δŝs W and for every δˆσ S ( ) δj HR = δŝs T G sσˆσ δŝs T H sσ ˆσ + δˆσ T G σs ŝs δˆσ T H σ sŝs δˆσ T H σσˆσ dx δŝs T x= T x = (38) 16

17 where W := { ŝs H 1 () : ŝs x= = } ; S := L 2 () We may notice that a the derivatives with respect to x are appied to dispacement variabes, whereas derivatives with respect to y (incorporated into the H matrices) are appied to crosssection stress vectors. The resuting variationa formuation has the foowing features: The obtained weak formuation (38) is symmetric. y-derivatives appied to stresses and the essentia conditions of St dd (cf. Section 3.2.2) most ikey ead to a formuation which accuratey sove the equiibrium equation in the y direction, i.e. in the cross-section. x derivatives appied to dispacements and the essentia condition in W most ikey ead to a formuation which accuratey sove the compatibiity equation aong the beam axis. The FE discretization simpy foows from the appication of (37) into the variationa formuation (38). We get: ( δj HR = δ s T B T s G sσ N σ σ δ s T N T s H sσ N σ σ + δ σ T N T σg σs B ) s s dx+ ( ) (39) δ σ T N T σ H σ sn s s + δ σ T N T σ H σσ N σ σ dx δ s T N T s T x= x = Coecting unknown coefficients in a vector and requiring to satisfy Equation (39) for a the possibe virtua fieds we obtain: [ } } where K sσ = K T σs = ]{ K sσ s K σs K σσ σ = { T ( B T s G sσ N σ N T ) s H sσ N σ dx Kσσ = N T σh σσ N σ dx (4) T = N T s T x x= In what foows we wi focus, for a the invoved variabes, on the finite eement spaces shown in Tabe 4, where we aso reca the profie properties which has ed to the muti-ayered beam mode. For the poynomia degrees and continuity requirements, we here use the same notation as in Section 5.2. Thus, for exampe, the fied v is approximated by means of piecewise cubic poynomias, continuous aong the axia direction. deg (p γ ) y continuity deg (N γ ) x continuity u 1 C 1 2 C v 2 C 1 3 C σ x 1 C 1 1 C 1 σ y 3 C 3 C 1 τ 2 C 2 C 1 Tabe 4: Degree and continuity properties of shape functions with respect to y and x directions. We remark that this choice of the finite eement shape functions assures the stabiity and convergence of the resuting discrete scheme. We aso notice that stresses are discontinuous across eements aong the x direction, so that it is possibe to staticay condensate them out at the eement eve, reducing the dimension of the goba stiffness matrix and improving efficiency. 6.2 Mutiayered homogeneous beam We now consider the same three ayer homogeneous beam introduced in Section Together with the camping condition in A, we assume = 1mm and the beam to be oaded aong A by the quadratic shear stress distribution t A = [, 3/2 ( 1 4y 2)] T [MPa]. 17

18 6.2.1 Convergence In Tabe 5 we report on the mean vaue of the transverse dispacement aong A, as obtained by empoying the foowing different procedures: 1. the cassica Euer-Bernoui beam mode; 2. the cassica Timoshenko beam mode; 3. the numerica mode under investigation, in which the soution is computed using a mesh of 64 eements; 4. a 2D finite eement scheme of the structura anaysis program ABAQUS, using a fine reguar grid of eements. Due to the arge number of eements used, we consider the atter soution as the reference soution, and we denote with v ex its mean vaue aong A. In Tabe 5 we aso report the foowing reative error quantity: e re = v v ex v ex where v is the mean vaue aong A computed by the various procedures. We remark that e re gives an indication of the mode accuracy, even though it is not the usua error measure in terms of the natura norms. Beam theory v (1) [ ] mm 1 2 e re [ ] 1 3 Euer-Bernoui Timoshenko Three-ayered mixed FE D soution Tabe 5: Transverse dispacements and reative errors of the free-edge of a cantiever ( = 1 and h = 1) obtained by different beam theories. Tabe 5 shows the superior performance of the three-ayered mixed FE with respect to the other considered 1D modes. In Figure 3 we study the convergence of our numerica mode. More precisey, we pot the reative quantity defined in Equation (41), evauated considering i ayers of thickness h i = 1/i (i = 1, 3, 5, 7), and different mesh sizes δ in the x direction. From Figure 3(a)) we notice that: using even a few eements the quantity e re is under 1%; the error e re decreases as the number of ayers increases; using highy-refined mesh, the reative error e re increases, even if it apparenty converges to a constant cose to 1 3. This behaviour can be expained by recaing that a modeing error does necessariy arise. Indeed, the soution of the 2D eastic probem and the one of the muti-ayered beam mixed mode do differ from each others, for a fixed ength and thickness of the beam. In Figure 3(b) we consider another reative quantity, namey e re. Such a quantity is simiar that defined in (41), but here the reference soution is the one obtained by the FE anaysis of the muti-ayered mode using the most refined mesh. We notice that: the sequences of errors e re are monotonicay convergent to zero; fixing the number of ayers, the og-og pots of errors suggest a convergence rate of the order of α, with α 1. Finay, in Figure 3(c) we pot the reative error e re versus the number of ayers (resuts obtained using a mesh of 32 eements). It is evident that the reative error decreases incrementing the number of ayers even if the succession is not inear. (41) 18

19 e re [ ] e * re ayer 3 ayer 5 ayer 7 ayer δ [mm] ayer 3 ayer 5 ayer 7 ayer (a) Reative error e re, evauated assuming v ex = 2D numerica soution δ [mm] 1 1 (b) Reative error e re, evauated assuming vex = 1D numerica soution obtained using the most refined x-direction mesh e re [ ] h i [mm] 1 1 (c) Reative error, potted as a function of the ayer thickness h i, with mesh size δ =.3125mm. Figure 3: Reative errors on free-edge, transverse dispacement for different mesh sizes δ and different number of ayers. 19

20 6.2.2 Boundary effects As aready noticed in Section 5.2.1, the mode under investigation is capabe to capture some oca effect near the camped boundary; we here present some resuts focused on that issue. Even though this study is far from being exhaustive, it gives an indication of the mode potentias. In what foows the computations are performed using a mesh of 64 eements. To highight the boundary effects we consider a beam anaogous to the one introduced at the beginning of Section6.2 in which we set = 2.5mm. The resuts are reported in Figures 4, 5 and 6. 2 u 1 v 3, v 4, v 6, v 7 v 2, v 5, v u 2 v 1, v 9 u 3 1 u 4 u 5 2 v 1 u [mm x 1 4 ].5.5 u 2, u 3 u 4, u 5 u 6 v [mm x 1 4 ] 4 v 2 v 3 v 4 v v 6 v v x [mm] (a) Axia dispacements ûu(x). v x [mm] (b) Transverse dispacements ˆv(x). Figure 4: Axia and transverse dispacements of a three-ayer, homogeneous cantiever, camped in x =, oaded at x = 2.5 by a quadratic shear distribution and modeed by means of 64 eements σ x [MPa] s x1 s x2 s x3 s x4 s x5 s x x [mm] Figure 5: Axia stress ˆσ x(x) of a three ayer, homogeneous cantiever, camped in x =, oaded at x = 2.5 by a quadratic shear distribution and modeed with 64 eements. We can appreciate the foowing. Far from the boundary, it is possibe to recognize the cassica beam soution (constant shear stress τ(x), inear axia stress σ x (x), quadratic horizonta dispacement u(x) and cubic transverse dispacement v(x)). As expected, the boundary effects decay as a dumped harmonic functions. The oca effects decay very rapidy, so that ony the first osciation is significant. This resut is consistent with the other modes capabe to capture this kind of boundary effects. 2

21 .5 σ y [MPa] s y1 s y2 s y3 s y4 s y5 s y6 s y7 τ [MPa].5 1 s y x [mm] (a) Transverse compressive stress ˆσ y(x). t 1 t 2 t 3 t 4 t x [mm] (b) Shear stress ˆτ(x). Figure 6: Stresses ˆσ y(x) and ˆτ(x) of a three ayer, homogeneous cantiever, camped in x =, oaded at x = 2.5 by a quadratic shear distribution and modeed by means of 64 eements. The section striction, described by the dispacement quadratic terms v 2, v 5 and v 8, is negigibe, as assumed in first-order theories. As specified in Tabe 4, in the numerica mode under discussion we do not a-priori impose dispacement continuity across ayers. In Figure 7 we pot the jump of the dispacement fied across the inter-ayer surfaces S 1 and S 2. Figure 7(b) highights that the transverse dispacement jump u + u [mm x 1 7 ] S 1 S 2 v + v [mm x 1 7 ] S 1 S x [mm] (a) Axia dispacement jump u(x) + u(x) x [mm] (b) Transverse dispacement jump v(x) + v(x). Figure 7: Reative axia and transverse dispacements (i.e. compatibiity errors) evauated on interayer surfaces S 1 : y = 1/5 and S 2 : y = 1/6. rapidy decay far from the camped boundary. On the other hand, from Figure 7(a) we see that, far from the camped boundary, the axia dispacement jump u(x) + u(x) tends to a vaue different from zero (of the order of 1 7 mm). However, we notice that the dispacement fied is much greater, since it is of the order of 1 4 mm. 6.3 Mutiayered non-homogeneous beams We now consider two non-homogeneous beams. For these cases, we particuary focus on the crosssection stress distributions, since such quantities are ony sedom accuratey captured in cassica beam modes. 21

22 6.3.1 Symmetric section We consider a cantiever composed by three ayers, camped in A, for which = 5mm and t A = [, 3/2 ( 1 4y 2)] T [MPa]; geometry and mechanica properties are specified in the foowing vectors: h = [mm] E = [MPa] ν = [ ] We evauate the soution of the 1D mode assuming a mesh size δ =.15625mm (32 eements) and the 2D soution using ABAQUS software and considering a mesh of 2 1 square eements. For both the modes we evauate the stress distributions at different sections: x = 2.5, x =.5 and x =.125, as reported in Figures 8, 9 and 1 respectivey..5 2D so 1D so.5 2D so 1D so y [mm] y [mm] σ x [MPa] (a) σ x(y) cross-section distribution σ y [MPa] (b) σ y(y) cross-section distribution..5 2D so 1D so y [mm] τ [MPa] (c) τ(y) cross-section distribution. Figure 8: Cross-section stress-distributions evauated at x = 2.5, far from the camped boundary. 1D and 2D soutions for the symmetric section. The capabiity of the numerica mode to reproduce a very accurate stress distribution, far from the camped boundary, is ceary seen from Figures 8. A simiar feature is aso maintained cose the camped boundary (see Figures 9 and 1), even though some error progressivey arises as we approach x =. In particuar, the axia stress σ x and the shear stress τ are very accuratey described, whereas the σ y approximation exhibit a worse performance (maybe aso because some kind of instabiity arises, cf. Figure 6(a)). 22

23 .5.5 y [mm] 2D so 1D so y [mm] 2D so 1D so σ x [MPa] (a) σ x(y) cross-section distribution σ y [MPa] (b) σ y(y) cross-section distribution..5 y [mm] 2D so 1D so τ [MPa] (c) τ(y) cross-section distribution. Figure 9: Cross-section stress-distributions evauated at x =.5, cose to the camped boundary. 1D and 2D soutions for the symmetric section Non-symmetric section We consider a cantiever composed by four ayers, camped in A, and for which = 5mm. The appied traction is: t A = [, 3/2 ( 1 4y 2)] T [MPa].The geometric and mechanica properties of the section are: h = [mm] E = [MPa] ν = [ ] We evauate the 1D and 2D soutions using the same meshes as for the symmetric case of Section The computed stress distributions at x = 2.5 is reported in Figures 11. As in the symmetric case, there is no significant difference between the 1D and the 2D crosssection stress distributions. We ony notice a sma deviation (beow 1%) in the pot of the σ y. 23

24 .5.5 y [mm] 2D so 1D so y [mm] 2D so 1D so σ x [MPa] (a) σ x(y) cross-section distribution σ y [MPa] (b) σ y(y) cross-section distribution..5 y [mm] 2D so 1D so τ [MPa] (c) τ(y) cross-section distribution. Figure 1: Cross-section stress-distributions evauated at x =.125, very cose to the camped boundary. 1D and 2D soutions for the symmetric section. 7 Concusions In this paper we deveoped panar beam modes based on a dimension reduction by variationa approach. More precisey, we seected a suitabe variationa formuation among the ones avaiabe in the iterature. Our choice of the variationa principe was motivated by the need to accuratey describe the stress profies. In Sections 4 and 5 we gave the genera derivation of the beam modes, together with a coupe of exampes. In Section 6 we focused on the muti-ayered mode introduced in Section 5, for which we deveoped an efficient Finite Eement scheme, as assessed by the presented numerica resuts. Future deveopments of this work coud incude 1) the rigorous mathematica study of the modes; 2) the generaization to 3D beams with variabe cross-section; 3) the treatment of more sophisticated constitutive aws. Acknowedgments The authors woud ike to thank professors Eio Sacco, Aessandro Reai and Giancaro Sangai for severa hepfu suggestions during the competion of this work. 24