Mechanics of Materials and Structures

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1 Journa of Mechanics of Materias and Structures A NEW MODELING APPROACH FOR PLANAR BEAMS: FINITE-ELEMENT SOLUTIONS BASED ON MIXED VARIATIONAL DERIVATIONS Ferdinando Auricchio, Giuseppe Baduzzi and Caro Lovadina Voume 5, No. 5 May 21 mathematica sciences pubishers

2 JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURES Vo. 5, No. 5, 21 A NEW MODELING APPROACH FOR PLANAR BEAMS: FINITE-ELEMENT SOLUTIONS BASED ON MIXED VARIATIONAL DERIVATIONS FERDINANDO AURICCHIO, GIUSEPPE BALDUZZI AND CARLO LOVADINA 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 soutions. 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 of reproducing the behavior of the Timoshenko beam, with the advantage that the shear correction factor appears naturay from the variationa derivation; the mutiayered beam is capabe of capturing the oca effects produced by boundary constraints and oad distributions; the finite eement is capabe of predicting the cross-section stress distribution with high accuracy, and more generay the behavior of panar structura eements. 1. Introduction To reproduce the mechanica behavior of structura eements characterized by having one dimension predominant with respect to the other two (that is, a beam) the cassica approaches are the we-known Euer Bernoui (EB) and Timoshenko theories [Timoshenko 1955; Hjemstad 25]. Generay adopted for very sender eements, the EB beam mode assumes the cross-section axia and transverse dispacements as 1D unknown fieds, converting the 3D eastic probem, formuated through a set of partia differentia equations, into a simper system of two ordinary differentia equations (ODEs). On the other hand, the Timoshenko beam mode, adopted for ess sender structura eements, assumes the crosssection axia and transverse dispacements, together with rotations, as 1D fieds, eading to a set of three ODEs. As a consequence of the fied assumptions, these modes are often referred to in the iterature as first-order modes. Moreover, we denote these modes as dispacement-based to underine that usuay these modes consider ony the dispacements as independent variabes and the stresses are evauated by postprocessing. Despite their simpicity (aso reated to the possibiity of computing anaytica soutions) and their exceent first-design abiity, the EB and Timoshenko modes have some we-known imitations: oss of accuracy for beams with a ow ratio between the ength and cross-sectiona characteristic dimension, Keywords: aminated inear eastic beam, anaytica soution, finite eement modeing, mixed variationa formuation. The authors woud ike to thank professors Eio Sacco, Aessandro Reai, and Giancaro Sangai for severa hepfu suggestions during the competion of this work. 771

3 772 FERDINANDO AURICCHIO, GIUSEPPE BALDUZZI AND CARLO LOVADINA oss of accuracy in modeing nonhomogeneous beams (such as composites) as we as in modeing noninear materia behavior, and 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 for more accurate 1D modes, researchers have deveoped new approaches that can generay be cassified as higher-order dispacement-based modes, which introduce more sophisticated cross-section kinematics, and mixed or hybrid modes, which aso introduce stresses and sometimes even strains as independent variabes. To give a few exampes, an often cited mode faing in the first category is the one proposed in [Reddy 1984] that introduces section warping in addition to Timoshenko dispacements. As discussed in [Sheinman 21], this mode shows inconsistencies, since transverse dispacement generates a constant cross-section shear distribution, whereas axia dispacements yied a quadratic one. More compete kinematics are adopted in the mode proposed by Lo et a. [1977a] where, considering both section warping and striction, authors eiminate such inconsistencies. For a genera treatment of higher-order, panar, kinematic beam modes and discussion of their anaytica soutions readers may refer to the work of Sheinman [21]. 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, and aso study the convergence of the modes under consideration. In terms of numerica modeing, the number of finite eement (FE) impementations proposed in the iterature is truy vast, since many different choices are possibe, depending on functiona formuation and independent variabes: the simpest functionas consider ony kinematic variabes, as in dispacement based beam modes, whie the most compicated aso consider stresses and strains; order of fied approximation: in addition to the first- versus higher-order cassification, for the case of mutiayered beams, ayerwise modes adopt piecewise ayer-defined functions whie goba modes use gobay defined functions to describe the fied distribution on a cross-section. For both items, intermediate choices are possibe, for exampe, ony some stress components coud be seected as independent variabes, or some fieds described ayerwise, with the remaining ones assumed as goba functions. We note that in the avaiabe review papers (for exampe, [Carrera 2; Wanji and Zhen 28]), modes are usuay compared by considering different fied approximations derived from the same principe. Therefore, it is possibe to understand the effect of basis accuracy, but not improvements arising from the use of a more sophisticated variationa principe. In Section 3 of this paper we review different variationa principes that can be empoyed, briefy discussing their features and possibe drawbacks. The main aim of this paper is to propose a simpe anaytica mode and an FE capabe of capturing the section stress distribution with high accuracy as we as in a mutiayered body. We opt for the dimension reduction variationa approach, cassica in beam modeing and used in Aessandrini et a. [1999] and

4 PLANAR BEAMS: MIXED VARIATIONAL DERIVATION AND FE SOLUTION 773 esewhere for pate probems. This method might aow us to generaize the mode to more compex situations, such as noninear constitutive aws. 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 Section 4 we derive some 1D modes. 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 FE schemes, and 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 imposing the pane stress state hypotheses to a 3D body or to stating 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 x [, ]}, A = y y h 2, h, 2 in which h is the beam thickness and is the beam ength. Obviousy h, so the ongitudina axis is the predominant dimension of the body. Figure 1 represents the domain and the adopted Cartesian coordinate system. As iustrated in Figure 1 we define the ower and upper boundaries as S and S n, respectivey, and the atera cross-sections as A and A. 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 dispacement constrained boundaries, respectivey. The externa oad is defined as a ine force density t : t 2 whie the body oad is defined as an area force density f : 2 ; we specify a sufficienty smooth boundary dispacement function s : s 2. Finay, we suppose that the beam is made of ayers whose thickness is constant aong the axia coordinate x, though 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, that is, E : A and ν : A. y O A S n A h x S Figure 1. Panar beam geometry, coordinate system, dimensions, and adopted notations.

5 774 FERDINANDO AURICCHIO, GIUSEPPE BALDUZZI AND CARLO LOVADINA Introducing the independent variabe fieds σ : 2 2, ε : 2 2, and s : 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, σ = D : ε, σ + f = in ; σ n = t on t ; s = s on s. (1) Here ε = s s is the compatibiity reation, σ = D : ε is the materia constitutive reation, in which D is the appropriate fourth-order inear eastic tensor, and σ + f = represents the equiibrium condition. The ast two equations in (1) represent the boundary equiibrium and the boundary compatibiity condition. 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 and Heinger Reissner and Hu Washizu functionas, considering for each different possibe stationarity conditions. Restricting ourseves to the framework of Section 2, we emphasize that a variationa beam mode can be considered as the outcome of the foowing genera procedure: Step 1: 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. Step 2: For each fied invoved, 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 Step 1, at east if a conforming mode is considered. Step 3: 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 (that is, the beam mode) Tota potentia energy approach. Tota potentia energy (TPE) is the functiona most frequenty used in continuum mechanics and in the standard iterature (for exampe, [Hjemstad 25]); severa authors use it as a starting point to derive first-order beam modes and the most popuar corresponding FE formuations. The TPE functiona can be expressed as foows: J TPE (s) = 1 s s : D : s sd s f d. (2) 2 Boundary conditions wi be suitaby imposed as foows. The critica point of the functiona above corresponds to finding the energy minimizer, which is unique and stabe in the usua framework of admissibe dispacement space. Requiring stationarity of the TPE (2), we obtain the foowing weak probem: Find s W s such that, for a δs W s, δ JTPE s = s (δs) : D : s sd δs f d δs t ds =, (3) t where W s := {s H 1 () : s s = s} and W s := {δs H 1 () : δs s = }.

6 PLANAR BEAMS: MIXED VARIATIONAL DERIVATION AND FE SOLUTION 775 Equation (3) is caed symmetric TPE stationarity. It is often used as a basis for FE deveopment, eading obviousy to a symmetric stiffness matrix. Wanji and Zhen [28] give a review of mutiayered, eastic, dispacement-based (that is, TPE derived) pate FE. Increasing orders of fied approximation are considered, from the simpest modes, in which dispacements are gobay defined aong the cross-section, to the most sophisticated, in which dispacements are defined ayerwise. In the same reference it is noted that amost a the FEs presented perform ony for specific probems, such as thick aminated pates or soft-core sandwiches; 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 [Vinayak et a. 1996a; 1996b], in which the authors deveop a mutiayered panar beam FE starting from (3), using the fied approximation proposed in [Lo et a. 1977a; 1977b] and appropriatey treating the thickness heterogeneity. They propose two ways to evauate the axia and transverse stresses: the first uses the compatibiity and the constitutive reations (1) 1,2, whie the second refines the shear and out-of-pane stress distributions using the equiibrium reation (1) 3. 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 [Rohwer and Rofes 1998; Rohwer et a. 25], in TPE-based modes the critica step is the postprocessing stress evauation which coud compromise the effectiveness of the method Heinger Reissner approach. The Heinger Reissner (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 are enforced depends on the specific variationa formuation empoyed to express the stationarity of the functiona (see 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 the HR functiona requires particuar care in the dispacement and stress fied assumptions, otherwise the mode risks eading to a probem which is not we-posed HR grad-grad stationarity. Requiring stationarity of the HR functiona (4), we obtain the foowing weak probem: Find s W gg s and σ S gg such that, for a δs W gg and a δσ S gg, δ J gg HR = s δs : σ d + δσ : s s d δσ : D 1 : σ d δs f d δs t ds =, (5) t where W gg s := {s H 1 () : s s = s}, W gg := {δs H 1 () : δs s = }, and S gg := {σ L 2 ()}. Equation (5) is caed 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 (an essentia boundary condition), whie σ n t = t turns out to be a natura boundary condition. As mentioned 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 from (5) ead to dispacement fieds that 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 symmetric TPE stationarity (3).

7 776 FERDINANDO AURICCHIO, GIUSEPPE BALDUZZI AND CARLO LOVADINA Many researchers have derived mutiayered pate and beam FEs using HR grad-grad stationarity. [Spiker 1982; Feng and Hoa 1998; Huang et a. 22; Icardi and Atzori 24] are among the most significant exampes, since the computed soutions are generay satisfactory. The main drawback of these schemes, especiay for the case of ayerwise beams and pates, is the high number of degrees of freedom (DOFs), which eads to a heavy FE formuation. Spiker [1982] aeviates this probem by assuming the 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 [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. In [Carrera 2; 21; Carrera and Demasi 22; Demasi 29a; 29b; 29c; 29d; 29e] various pate FEs are derived appying this viewpoint, with different choices of basis function, obtaining reasonabe accuracy in the stress description. Unfortunatey these authors 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 (5) become: s δs : σ d = δs σ nds δs σ d, δσ : s sd = δσ n sds δσ sd. (6) Hence, substituting (6) into (5), the weak formuation becomes: Find s W dd and σ St dd a δs W dd and a δσ S dd, δ J dd HR = δs σ d such that, for δσ sd δσ : D 1 : σ d δs f d + δσ n sds=, (7) s where W dd := {s L 2 ()}, St dd := {σ H(div, ) : σ n t = t}, S dd := {δσ H(div, ) : δσ n t = }. Here and in what foows, H(div, ) denotes the space of square-integrabe vector functions whose divergence is sti square-integrabe. Equation (7) is caed 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. [1999] have derived some homogeneous pate modes, which are more interesting than the ones stemming from the HR grad-grad stationarity (5). However, the techniques deveoped in that work cannot be directy appied to genera heterogeneous pates, because the resuting modes may be divergent; see [Auricchio et a. 24] Hu Washizu approach. The Hu Washizu (HW) functiona may be expressed as J HW (σ, ε, s) = σ : ( s s ε)d + 1 ε : D : εd s f d. (8) 2 Again, how boundary conditions are enforced depends on the specific variationa formuation empoyed to express the stationarity of the functiona. 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.

8 PLANAR BEAMS: MIXED VARIATIONAL DERIVATION AND FE SOLUTION HW grad-grad stationarity. Stationarity of (8) can be expressed as δ J gg HW = s δs : σ d δε : σ d + δσ : ( s s ε)d + δε : D : εd δs f d δs t ds=, (9) t where s satisfies s s = s, whie σ n t = t is a natura boundary condition. Equation (9) is caed HW grad-grad stationarity; an exampe of its use in mutiayered pate modeing is presented in [Auricchio and Sacco 1999] HW div-div stationarity. A second formuation of HW stationarity can be found by introducing (6) into (9), obtaining δ JHW dd = δs σ d δε : σ d δσ sd δσ : εd + δε : D : εd δs f d + (δσ n) sds =, (1) s where σ satisfies σ n = t on t, whie s s = s is a natura boundary condition. This is caed HW div-div stationarity Concusions on weak formuations. We summarize the previous discussion in a few observations (see aso Tabe 1): A the weak formuations considered are symmetric. Each 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 require considering a priori smooth dispacement fieds and ess reguar stress fieds (s H 1 () and σ L 2 ()), whie the atter 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 (see Step 2 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 and sef-equiibrated modes for the div-div formuations. In our view, one of the major imitations of the avaiabe eementary beam modes is that equiibrium equations are not sufficienty enforced within the cross-section. For this reason we wi focus on HR formuations, with some emphasis on the div-div form (7). Principe Dispacement-based TPE Equation (3) Mixed HR grad-grad (5) div-div (7) HW grad-grad (9) div-div (1) Tabe 1. Functiona stationarities cassified in terms of the functionas from which they are derived and equation formats.

9 778 FERDINANDO AURICCHIO, GIUSEPPE BALDUZZI AND CARLO LOVADINA 4. Mode derivations In this section we deveop some beam modes. We wi start from the HR variationa formuations of Section 3.2 (this is Step 1 of the procedure described at the beginning of Section 3), then define the approximated fieds (Step 2), and perform the integration aong the thickness (Step 3). To simpify further discussion, we wi switch to engineering notation, the equivaence between tensor and engineering notation being summarized as foows, where E 1 = 1 1 and E2 = 1 1 : Tensor notation σ S s σ n Engineering notation d dx E 1 + d dy E 2 P σ ˆσ d dx ET 1 + d dy ET 2 P s ŝ (n x E 1 + n y E 2 )P σ ˆσ 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. We first define a profie vector function p γ (y): p γ : A 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), that is, γ(x, y) = pγ T (y) ˆγ (x), (11) where superscript T denotes transposition and ˆγ : 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), the computation of partia derivatives is straightforward, since: x γ = x ( pt γ ˆγ ) = d pt γ dx ˆγ = pt γ ˆγ, y γ = y ( pt γ ˆγ ) = d dy pt γ ˆγ = pt γ ˆγ, where we use a prime to indicate derivatives aong both x and y, as there is no risk of confusion. Adopting the notation introduced in (11) and switching now to an engineering notation we set û(x) su (x, y) p T s(x, y) = = u (y) s v (x, y) pv T (y) = P ˆv(x) s ŝ, (12) σ xx (x, y) p T σ σ (x, y) = σ yy (x, y) = x (y) ˆσ x (x) pσ T y (y) ˆσ τ xy (x, y) pτ T (y) y (x) = P σ ˆσ, (13) ˆτ(x) where, for P s and P σ (ŝ and ˆσ, respectivey), we drop the expicit dependence on y (or x, respectivey), for notationa simpicity. The virtua fieds are anaogousy defined as δs = P s δŝ and δσ = P σ δ ˆσ.

10 PLANAR BEAMS: MIXED VARIATIONAL DERIVATION AND FE SOLUTION 779 In Section 3, D 1 denotes the fourth-order eastic tensor; from now on, we use the same notation to indicate the corresponding square matrix obtained foowing engineering notation. Therefore, we have D 1 = 1 1 ν ν 1. E 2(1+ν) 4.2. Formuation of the probems. We consider the specia case of a beam for which s = on A = s, t = on A, f = in, t = on S S n. Hence, t = A S S n ; the beam is camped on the eft-hand side A, and is subjected to a nonvanishing 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 σxˆt x t = pτ T ˆt. (14) τ Therefore, the boundary condition σ n A = t may be written as (see (13)) ˆσx () ˆt = x. (15) ˆτ() ˆt τ A these assumptions can be modified to cover more genera cases; nonetheess, this simpe mode is aready adequate to iustrate the method s capabiities HR grad-grad approach. In the notation of Section 4.1, the HR grad-grad stationarity (5) becomes δ J gg d HR = dx ET 1 + d T dy ET 2 P s δŝ Pσ ˆσ d + δ ˆσ T Pσ T d dx ET 1 + d dy ET 2 P s ŝ d δ ˆσ T Pσ T D 1 P σ ˆσ d δŝ T Ps T t dy =. (16) t Expanding (16) we obtain δ J gg HR = (δŝ T Ps T E 1 P σ ˆσ + δŝ T Ps T E 2 P σ ˆσ )d + (δ ˆσ T Pσ T ET 1 P sŝ + δ ˆσ T Pσ T ET 2 P s ŝ)d δ ˆσ T Pσ T D 1 P σ ˆσ d δŝ T Ps T t dy =. (17) A Using the Fubini Tonei theorem, (17) can be written as δ J gg HR = (δŝ T G sσ ˆσ + δŝ T H s σ ˆσ + δ ˆσ T G σ s ŝ + δ ˆσ T H σ s ŝ δ ˆσ T H σσ ˆσ )dx δŝ T x= T x =, (18) where G sσ = G T σ s = H s σ = H T σ s = P T A A s E 1 P σ dy, H σσ = Ps T E 2 P σ dy, T x = A A P T σ D 1 P σ dy, P T s t dy. (19)

11 78 FERDINANDO AURICCHIO, GIUSEPPE BALDUZZI AND CARLO LOVADINA 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): δŝ T G sσ ˆσ dx = δŝ T G sσ ˆσ x= x= δŝ T G sσ ˆσ dx. (2) Substituting (2) into (18), recaing that δŝ = on A, and coecting the variabes in a vector we obtain in which [δŝ; δ ˆσ ] G T G = ŝ ˆσ + H gg ŝ dx + δŝ T (G ˆσ sσ ˆσ T x ) x= =, (21) Gsσ G σ s, H gg = H σ s Hs σ H σσ. (22) Requiring that (21) is satisfied for a the possibe variations and imposing the essentia boundary condition ŝ we finay obtain ŝ G ˆσ + H gg ŝ = in, G ˆσ sσ ˆσ = T x at x =, ŝ = at x =. (23) 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, the we-posedness of (23) is guaranteed if the approximated fieds are seected in accordance with the approximation theory of sadde-point probem (5) [Aessandrini et a. 1999] HR div-div approach. Using the notation introduced in Section 4.1 in (7), the HR div-div functiona stationarity becomes δ JHR dd = δŝ T Ps T d dx E 1 + d d dy E 2 P σ ˆσ d dx E 1 + d T dy E 2 P σ δ ˆσ Ps ŝd δ ˆσ T Pσ T D 1 P σ ˆσ d =. (24) Expanding (24), the weak formuation becomes δ JHR dd = (δŝ T Ps T E 1 P σ ˆσ + δŝ T Ps T E 2 Pσ ˆσ )d (δ ˆσ T Pσ T ET 1 P sŝ + δ ˆσ T Pσ T ET 2 P sŝ)d δ ˆσ T Pσ T D 1 P σ ˆσ d =. (25) Again by the Fubini Tonei theorem, (25) becomes δ JHR dd = δŝ T G sσ ˆσ δŝ T H sσ ˆσ δ ˆσ T G σ s ŝ δ ˆσ T H σ sŝ δ ˆσ T H σσ ˆσ dx =, (26) where whie the other matrices are as in (19). H σ s = H T sσ = A P T σ E 2 P s dy

12 PLANAR BEAMS: MIXED VARIATIONAL DERIVATION AND FE SOLUTION 781 Equation (26) represents the weak form of the 1D beam mode. To obtain the corresponding boundary vaue probem, we integrate the third term by parts: δ ˆσ T G σ s ŝdx = δˆσ T G σ s ŝ x= x= + δ ˆσ T G σ s ŝ dx. (27) Substituting (27) into (26), recaing that G sσ δ ˆσ = at x =, and coecting the unknowns in a vector we obtain [δŝ; δ ˆσ ] G T ŝ ˆσ + H dd ŝ dx + δ ˆσ T G ˆσ σ s ŝ x= =, (28) where G is defined as in (22) and H dd is defined as H dd Hsσ =. (29) H σ s H σσ Requiring that (28) be satisfied for a possibe variations, and imposing the essentia boundary condition (15), we finay obtain ŝ G ˆσ + H dd ŝ = in, ˆσ ˆσ x = ˆt x at x =, ˆτ = ˆt τ at x =, G σ s ŝ = at x =. (3) 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 fieds invoved. However, (3) is guaranteed to be weposed if the approximated fieds are seected according to the approximation theory of the sadde-point probem (7); see [Aessandrini et a. 1999] Concusions on the derived beam modes. From the deveopment in this section, we can make the foowing remarks. Starting from different versions of the HR stationarity condition grad-grad (5) and div-div (7) and introducing hypothesis (11), we obtain two different casses of 1D beam modes. Both casses may be described by a boundary vaue probem of the foowing type: G ŝ ˆσ + H ŝ ˆσ = + suitabe boundary conditions. (31) The difference between modes based on the HR grad-grad and HR div-div formuations ies 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 the resuting modes most ikey strongy satisfy the compatibiity aw, though not necessariy the equiibrium equation. H = H dd for the HR div-div formuation. In this case, derivatives are appied to the stress fieds through the divergence operator, and the resuting modes most ikey strongy satisfy the equiibrium equation, though not the necessariy the compatibiity equation. The boundary vaue probem (3) can be expicity written as G sσ ˆσ H sσ ˆσ =, G σ s ŝ H σ sŝ H σσ ˆσ = + suitabe boundary conditions.

13 782 FERDINANDO AURICCHIO, GIUSEPPE BALDUZZI AND CARLO LOVADINA We can compute ˆσ from the second equation and substitute it into the first, obtaining a dispacementike formuation of the probem: where Aŝ + Bŝ + Cŝ = + suitabe boundary conditions, (32) A = G sσ H 1 σσ G σ s, Simiar considerations appy to probem (23). B = G sσ H 1 σσ H σ s + H sσ H 1 σσ G σ s, C = H sσ H 1 σσ H σ s. 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 in Equation (3), we derive a singe ayer beam mode in which we use a first-order dispacement fied, thus showing that the approach under discussion is abe to reproduce the cassica modes; and a mutiayer beam mode, in which we consider aso higher-order kinematic and stress fieds, thus iustrating how the approach can produce a refined mode with a reasonabe soution Singe ayer beam. Considering a homogeneous beam, we assume a first-order kinematic (as in the Timoshenko mode) and the usua cross-section stress distributions (obtained from Jourawsky theory). In other words, we make the foowing hypotheses: u = u (x) + yu 1 (x), that is, p u = 1 y T, û = u u 1 T, v = v(x), that is, p v = 1, ˆv = v, σ xx = σ x (x) + yσ x1 (x), that is, p σx = 1 y T, ˆσx = σ x σ x1 T, σ yy =, that is, p σy =, ˆσ y =, τ = (1 4y 2 /h 2 )τ(x), that is, p τ = 1 4y 2 /h 2, ˆτ = τ. The matrices G and H dd defined in (22) and (29), and entering into the beam mode (3), are expicity given by h h G = 3 h 3 h, H dd = h h. (33) E h3 h E 3 h 2 3 h 8 15 h 2(1+ν) E Since the probem (3) is governed by an ODE system with constant coefficients, the homogeneous

14 PLANAR BEAMS: MIXED VARIATIONAL DERIVATION AND FE SOLUTION 783 soution can be anayticay computed. For exampe, choosing h = 1 mm, = 1 mm, E = 1 5 MPa, and 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, in which the C i are arbitrary constants, which 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 σ s of (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 (see (32)), we obtain the ODE system heu =, h 3 12 Eu Eh 2(1+ν) (v + u 1 ) =, + suitabe boundary conditions. 5 6 Eh 2(1 + ν) (v + u 1 ) = Not surprisingy, we again recover the cassica Timoshenko equations, in which, however, the exact shear correction factor 5 6 automaticay appears. The same resut hods true in the framework of variationa pate modeing proposed in [Aessandrini et a. 1999] Mutiayered beam. We now consider a beam composed of n ayers, as iustrated in Figure 2. The geometric and materia parameters for the generic i-th ayer are 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. y (34) (35) S n S n 1 h n h n 1 S 2 O z h h 3 S 1 S b = 1 h 2 h 1 Figure 2. Cross-section geometry, coordinate system, dimensions, and adopted notations.

15 784 FERDINANDO AURICCHIO, GIUSEPPE BALDUZZI AND CARLO LOVADINA To design a mutiayered beam mode, we foow a two-step procedure: (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 we use here, 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, denoted by deg p γ, and reguarity across each interayer: continuity may or may not be required. 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 the we-posedness of the resuting mode, we seect the fieds according to Tabe 2, where we aso show the numbers of ayer and goba DOFs. Furthermore, we notice that we have to impose σ y = τ = at the top and bottom of the beam. Remark 5.1. More generay, to design a we-posed beam mode one coud choose deg p σx = deg p u = deg p τ 1, deg p τ = deg p v = deg p σy 1, (36) together with the H(div) reguarity for the stress fied A test case. We now present an easy case to test the robustness of the mode: We consider a homogeneous beam, but treat it as if it were formed by three ayers. The geometric and materia properties are described by: h =.367 mm, E = MPa, ν = As a consequence, the tota number of cross-section variabes is 34. The boundary vaue probem (3) is uniquey sovabe in this case. However, now the rank of G is 22, which means that (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 that the boundary conditions in (3) actuay ead to 22 independent constraints, since rank(g σ s ) = 11. deg p γ Interayer continuity Layer DOFs Goba DOFs p u 1 no 2 2n p v 2 no 3 3n p σx 1 no 2 2n p σy 3 yes 4 3n 1 p τ 2 yes 3 2n 1 Tabe 2. Poynomia degrees of the profie vectors, continuity properties, and number of DOFs for a mutiayered beam.

16 PLANAR BEAMS: MIXED VARIATIONAL DERIVATION AND FE SOLUTION 785 We now give the soution of the generaized eigenvaue probem det(λg + H dd ) =, which goes into the construction of the homogeneous soution of (3): [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 the numbers on the right are the eigenvaue mutipicities and the notation denotes eigenvaues that vanish up to the machine precision. We notice that the number of eigenvaues (22 in tota) corresponds exacty to the rank of G. It is aso possibe to evauate the homogeneous soution but, given the compexity of the probem, it is huge and we wi not report it. Nevertheess it is possibe to discuss its structure and make 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 happens in severa other beam modes [Ladevèze 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 introducing a dimension reduction aso aong the beam axis, which eads to a purey agebraic system. The discretization of the axia fieds ŝ and ˆσ can be generay described by ũ N Nu (x) σx (x) σ x ŝ(x) N s s =, ˆσ (x) N N v (x) ṽ σ σ = N σy (x) σ y, N τ (x) τ ũ ŝ N (x) B s s = u (x) Nv (x), ˆσ (x) B ṽ σ σ = Nσ x (x) Nσ y (x) Nτ (x) σ x σ y. τ From now on, for notationa simpicity, we wi drop the expicit dependency on x for fieds 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 third terms of (26). We thus (37)

17 786 FERDINANDO AURICCHIO, GIUSEPPE BALDUZZI AND CARLO LOVADINA obtain: Find ŝ W and ˆσ S such that, for every δŝ W and for every δ ˆσ S, δ J HR = (δŝ T G sσ ˆσ δŝ T H sσ ˆσ + δ ˆσ T G σ s ŝ δ ˆσ T H σ sŝ δ ˆσ T H σσ ˆσ )dx δŝ T T x x= =, (38) where W := {ŝ H 1 () : ŝ x= = } and S := L 2 (). Notice that a the derivatives with respect to x are appied to dispacement variabes, whereas the derivatives with respect to y (incorporated into the H matrices) are appied to cross-section 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 (see Section 3.2.2) most ikey ead to a formuation which accuratey soves the equiibrium equation in the y direction, that is, in the cross-section. x derivatives appied to dispacements and the essentia condition in W most ikey ead to a formuation which accuratey soves the compatibiity equation aong the beam axis. The FE discretization simpy foows from the appication of (37) to the variationa formuation (38): δ J HR = δ s T Bs T G sσ N σ σ δ s T Ns T H sσ N σ σ + δ σ T Nσ T G σ s B s s dx δ σ T Nσ T H σ s N s s + δ σ T Nσ T H σσn σ σ dx δ s T Ns T T x x= =. (39) Coecting unknown coefficients in a vector and requiring (39) to be satisfied for a possibe virtua fieds we obtain Ksσ s T =, (4) σ where K sσ = K T σ s = K σ s K σσ (Bs T G sσ N σ Ns T H sσ N σ )dx, K σσ = Nσ T H σσn σ dx, T = Ns T T x x=. In what foows we wi focus, for a variabes invoved, on the finite eement spaces shown in Tabe 3; we aso reca the profie properties which have ed to the mutiayered beam mode. For the poynomia degrees and continuity requirements we 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 no 2 yes v 2 no 3 yes σ x 1 no 1 no σ y 3 yes 3 no τ 2 yes 2 no Tabe 3. Degree and continuity properties of shape functions with respect to the y and x directions.

18 PLANAR BEAMS: MIXED VARIATIONAL DERIVATION AND FE SOLUTION 787 We remark that this choice of the FE shape functions assures the stabiity and convergence of the resuting discrete scheme. We aso notice that the 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 Mutiayered homogeneous beam. We now consider the same three ayer homogeneous beam of Section Together with the camping condition in A, we assume that = 1 mm and the beam is oaded aong A by the quadratic shear stress distribution t A =[, 3/2(1 4y 2 )] T MPa Convergence. In Tabe 4 we report on the mean vaue of the transverse dispacement aong A, as obtained by empoying (a) the cassica Euer Bernoui beam mode; (b) the cassica Timoshenko beam mode; (c) the numerica mode under investigation, in which the soution is computed using a mesh of 64 eements; (d) a 2D FE scheme in 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 4 we aso report the reative error, defined by e re = v v ex, (41) v ex where v is the mean vaue aong A computed by the various procedures. This e re gives an indication of the accuracy of the mode, even though it is not the usua error measure in terms of the natura norms. Tabe 4 shows the superior performance of the three-ayered mixed FE with respect to the other 1D modes considered. In Figure 3 we study the convergence of our numerica mode. More precisey, we pot the reative quantity defined in (41), evauated considering different mesh sizes δ in the x direction, and i ayers of thickness h i = 1/i, for i = 1, 3, 5, 7. From Figure 3a 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 a highy refined mesh, the reative error e re increases, even as it apparenty converges to a constant cose to 1 3. This behavior can be expained by recaing that a modeing error necessariy arises. Indeed, the soution of the 2D eastic probem and the soution of the mutiayered beam mixed mode do differ from each other, for a fixed ength and thickness of the beam. Beam theory v [1 2 mm] e re [1 3 ] Euer Bernoui Timoshenko Three-ayered mixed FE D soution Tabe 4. Transverse dispacements and reative errors of the free edge of a cantiever ( = 1 mm and h = 1 mm) obtained by different beam theories.

19 788 FERDINANDO AURICCHIO, GIUSEPPE BALDUZZI AND CARLO LOVADINA In Figure 3b we consider another reative quantity, ere, simiar to that defined in (41), but using as the reference soution the one obtained by FE anaysis of the mutiayered mode using the most refined x-direction mesh. We notice that: the sequences of errors ere converge monotonicay to zero; and for a fixed number of ayers, the og-og pots of errors suggest a convergence rate of the order of α, with α 1. Finay, in Figure 3c we pot the reative error e re versus the number of ayers (with the resuts obtained using a mesh of 32 eements). It is evident that the reative error decreases when incrementing the number of ayers even if the succession is not inear Boundary effects. As aready noticed in Section 5.2.1, the mode under investigation is capabe of capturing some oca effects 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 s potentia. In what foows the computations are performed using a mesh of 64 eements. To highight the boundary ayer 3 ayer 5 ayer 7 ayer e re [ ] e * re ayer 3 ayer 5 ayer 7 ayer 1 4 (a) δ [mm] (b) δ [mm] e re [ ] 1 5 (c) h i [mm] 1 1 Figure 3. Reative errors on free-edge and transverse dispacement for different mesh sizes δ and different numbers of ayers. (a) Reative error e re, evauated assuming v ex is the 2D numerica soution. (b) Reative error e re, evauated assuming v ex is the 1D numerica soution obtained using the most refined x-direction mesh. (c) Reative error, potted as a function of the ayer thickness h i, with mesh size δ =.3125 mm.

20 PLANAR BEAMS: MIXED VARIATIONAL DERIVATION AND FE SOLUTION 789 effects we consider a beam anaogous to the one introduced at the beginning of Section 6.2 in which we set = 2.5 mm. The resuts are reported in Figures 4, 5, and 6. 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, boundary effects decay as damped 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 of capturing these kind of boundary effects u 1 u 2 v 1, v 9 v 3, v 4, v 6, v 7 v 2, v 5, v 8 u 3 u [mm x 1 4 ] u 2, u 3 u 4, u 5 u 4 u 5 u 6 v [mm x 1 4 ] 2 4 v 1 v 2 v 3 v 4 v v 6 v v x [mm] v x [mm] Figure 4. Axia dispacements û (eft) and transverse dispacements ˆv (right) as functions of x for a three-ayer homogeneous cantiever, camped at x =, oaded at x = 2.5 by a quadratic shear distribution, and modeed with 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 at x =, oaded at x = 2.5 by a quadratic shear distribution, and modeed with 64 eements.

21 79 FERDINANDO AURICCHIO, GIUSEPPE BALDUZZI AND CARLO LOVADINA.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] t t t t t x [mm] Figure 6. Transverse compressive stress ˆσ y (eft) and shear stress ˆτ (right) as functions of x for a three-ayer homogeneous cantiever, camped at x =, oaded at x = 2.5 by a quadratic shear distribution, and modeed with 64 eements. Section striction, described by the quadratic dispacement terms v 2, v 5, and v 8, is negigibe, as assumed in first-order theories. As specified in Tabe 3, 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 interayer surfaces S 1 and S 2. Figure 7, right, highights that the transverse dispacement jump rapidy decays far from the camped boundary. On the other hand, from the eft haf of the figure 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 u + u [mm x 1 7 ] S 1 S 2 v + v [mm x 1 7 ] S 1 S x [mm] x [mm] Figure 7. Compatibiity mismatches on interayer surfaces y = 1 5 (abeed S 1) and y = 1 6 (S 2): axia dispacement jump u(x) + u(x) (eft) and transverse dispacement jump v(x) + v(x) (right).

22 PLANAR BEAMS: MIXED VARIATIONAL DERIVATION AND FE SOLUTION Mutiayered nonhomogeneous beams. We now consider two nonhomogeneous beams. focusing particuary on the cross-section stress distributions, since such quantities are ony sedom accuratey captured in cassica beam modes Symmetric section. Figure 8 pots the stress distributions at different sections (x = 2.5,.5, and.125) cacuated for a cantiever composed of three ayers, camped at A, for which = 5 mm and.5 2D so 1D so.5 2D so 1D so.5 2D so 1D so y [mm] σ [MPa] x σ [MPa] y.5 x = τ [MPa].5 y [mm] 2D so 1D so y [mm] 2D so 1D so y [mm] 2D so 1D so σ [MPa] x σ [MPa] y.5 x = τ [MPa].5 y [mm] 2D so 1D so 2D so 1D so 2D so 1D so σ [MPa] x σ [MPa] y x = τ [MPa] Figure 8. Cross-section stresses σ x (eft), σ y (midde), and τ (right), as functions of y: one- and two-dimensiona soutions for the symmetric section. Three cross-sections are considered, at various distances from the camped boundary: far (x = 2.5, top), cose (x =.5, midde), and very cose (x =.125, bottom).

23 792 FERDINANDO AURICCHIO, GIUSEPPE BALDUZZI AND CARLO LOVADINA t A =, 3 2 (1 4y2 ) T MPa, and whose geometry and mechanica properties are specified by h =.5 mm, E = MPa, ν = The soution of the 1D mode was evauated with a mesh size of δ = mm (32 eements), and the 2D soution using ABAQUS with a mesh of 2 1 square eements. The abiity of the numerica mode to reproduce the stress distribution very accuratey, far from the camped boundary, is ceary seen from the top row of Figure 8. A simiar feature is aso maintained cose to the camped boundary (midde and bottom rows), 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 exhibits a worse performance (maybe aso because some kind of instabiity arises; see Figure 6, eft) Nonsymmetric section. Figure 9 pots the stress distributions at the section x = 2.5 cacuated for a cantiever composed of four ayers, camped at A, for which = 5 mm and t =... and whose geometric and mechanica properties of the section are specified by h = mm, E = MPa, ν = We evauate the 1D and 2D soutions using the same meshes as for the symmetric case just discussed. As in the symmetric case, there is no significant difference between the 1D and the 2D cross-section stress distributions. We ony notice a sma deviation (beow 1%) in the pot of σ y..5 2D so 1D so.5 2D so 1D so.5 2D so 1D so y [mm] y [mm] y [mm] σ x [MPa] σ y [MPa] x = τ [MPa] Figure 9. Cross-section stresses σ x (eft), σ y (midde), and τ (right), as functions of y: one- and two-dimensiona soutions for the nonsymmetric section. 7. Concusions In this paper we deveoped panar beam modes based on dimension reduction by a variationa approach. More precisey, we seected a suitabe variationa formuation among the ones avaiabe in the iterature.

24 PLANAR BEAMS: MIXED VARIATIONAL DERIVATION AND FE SOLUTION 793 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 mutiayered 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 rigorous mathematica study of the modes, generaization to 3D beams with variabe cross-section, and the treatment of more sophisticated constitutive aws. References [Aessandrini et a. 1999] S. M. Aessandrini, D. N. Arnod, R. S. Fak, and A. L. Madureira, Derivation and justification of pate modes by variationa methods, pp. 1 2 in Pates and shes (Quebec, 1996), edited by M. Fortin, CRM Proc. Lecture Notes 21, Amer. Math. Soc., Providence, RI, [Aix and Dupeix-Couderc 29] O. Aix and C. Dupeix-Couderc, A pate theory as a mean to compute precise 3D soutions incuding edge effects and reated issues, New Trends in Thin Structures 519 (29), [Auricchio and Sacco 1999] F. Auricchio and E. Sacco, A mixed-enhanced finite-eement for the anaysis of aminated composite pates, Int. J. Numer. Methods Eng. 44 (1999), [Auricchio et a. 24] F. Auricchio, C. Lovadina, and A. L. Madureira, An asymptoticay optima mode for isotropic heterogeneous ineary eastic pates, M2AN Math. Mode. Numer. Ana. 38:5 (24), [Carrera 2] E. Carrera, Assessment of mixed and cassica theories on goba and oca response of mutiayered orthotropic pates, Compos. Struct. 5 (2), [Carrera 21] E. Carrera, Deveopments, ideas and evauations based upon Reissner s mixed variationa theorem in the modeing of mutiayered pates and shes, Appied Mechanics Review 54 (21), [Carrera and Demasi 22] E. Carrera and L. Demasi, Cassica and advanced mutiayered pate eements based upon PVD and RMVT, 1: derivation of finite eement matrices, Int. J. Numer. Methods Eng. 55:2 (22), [Demasi 29a] L. Demasi, Mixed pate theories based on the generaized unified formuation, I: governing equations, Compos. Struct. 87 (29), [Demasi 29b] L. Demasi, Mixed pate theories based on the generaized unified formuation, II: ayerwise theories, Compos. Struct. 87 (29), [Demasi 29c] L. Demasi, Mixed pate theories based on the generaized unified formuation, III: advanced mixed high order shear deformation theories, Compos. Struct. 87 (29), [Demasi 29d] L. Demasi, Mixed pate theories based on the generaized unified formuation, IV: zig-zag theories, Compos. Struct. 87 (29), [Demasi 29e] L. Demasi, Mixed pate theories based on the generaized unified formuation, V: resuts, Compos. Struct. 88 (29), [Feng and Hoa 1998] W. Feng and S. V. Hoa, Partia hybrid finite eements for composite aminates, Finite Eem. Ana. Des. 3 (1998), [Hjemstad 25] K. D. Hjemstad, Fundamentas of structura mechanics, Springer, 25. [Huang et a. 22] Y. Huang, S. Di, C. Wu, and H. Sun, Bending anaysis of composite aminated pates using a partiay hybrid stress eement with interaminar continuity, Comput. Struct. 8 (22), [Icardi and Atzori 24] U. Icardi and A. Atzori, Simpe, efficient mixed soid eement for accurate anaysis of oca effects in aminated and sandwich composites, Adv. Engng. Software 35 (24), [Ladevèze and Simmonds 1998] P. Ladevèze and J. Simmonds, New concepts for inear beam theory with arbitrary geometry and oading, Eur. J. Mech. A Soids 17:3 (1998), [Lo et a. 1977a] K. H. Lo, R. M. Christensen, and E. M. Wu, A high order theory for pate deformations, I: homogeneous pates, J. App. Mech. (ASME) 44 (1977),

25 794 FERDINANDO AURICCHIO, GIUSEPPE BALDUZZI AND CARLO LOVADINA [Lo et a. 1977b] K. H. Lo, R. M. Christensen, and E. M. Wu, A high order theory for pate deformations, II: aminated pates, J. App. Mech. (ASME) 44 (1977), [Reddy 1984] J. N. Reddy, A simpe higher-order theory of aminated composite pates, ASME, Journa of Appied Mechanics of Composite Materias 51 (1984), [Reissner 1986] E. Reissner, On a variationa theorem and on shear deformabe pate theory, Int. J. Numer. Methods Eng. 23 (1986), [Rohwer and Rofes 1998] K. Rohwer and R. Rofes, Cacuating 3D stresses in ayered composite pates and shes, Mech. Compos. Mater. 34 (1998), [Rohwer et a. 25] K. Rohwer, S. Friedrichs, and C. Wehmeyer, Anayzing aminated structures from fiber-reinforced composite materia an assessment, Tech. Mech. 25 (25), [Sheinman 21] I. Sheinman, On the anaytica cosed-form soution of high-order kinematic modes in aminated beam theory, Int. J. Numer. Methods Eng. 5 (21), [Spiker 1982] R. L. Spiker, Hybrid-stress eight-node eements for thin and thick mutiayer aminated pates, Int. J. Numer. Methods Eng. 18 (1982), [Timoshenko 1955] S. Timoshenko, Strength of materias, I: eementary theory and probems, 3rd ed., Van Nostrand, New York, [Vinayak et a. 1996a] R. U. Vinayak, G. Prathap, and B. P. Naganarayana, Beam eements based on a higher order theory, I: formuation and anaysis of performance, Comput. Struct. 58 (1996), [Vinayak et a. 1996b] R. U. Vinayak, G. Prathap, and B. P. Naganarayana, Beam eements based on a higher order theory, II: boundary ayer sensitivity and stress osciations, Comput. Struct. 58 (1996), [Wanji and Zhen 28] C. Wanji and W. Zhen, A seective review on recent deveopment of dispacement-based aminated pate theories, Recent Patents Mech. Engng. 1 (28), Received 3 Dec 29. Revised 1 Jun 21. Accepted 16 Jun 21. FERDINANDO AURICCHIO: auricchio@unipv.it Dipartimento di Meccanica Strutturae, Università degi Studi di Pavia, Via Ferrata 1, 271 Pavia, Itay GIUSEPPE BALDUZZI: giuseppe.baduzzi@unipv.it Dipartimento di Meccanica Strutturae / Dipartimento di Matematica, Università degi Studi di Pavia, Via Ferrata 1, 271 Pavia, Itay CARLO LOVADINA: caro.ovadina@unipv.it Dipartimento di Matematica, Università degi Studi di Pavia, Via Ferrata 1, 271 Pavia, Itay

26 JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURES Founded by Chares R. Steee and Marie-Louise Steee EDITORS CHARLES R. STEELE DAVIDE BIGONI IWONA JASIUK YASUHIDE SHINDO Stanford University, U.S.A. University of Trento, Itay University of Iinois at Urbana-Champaign, U.S.A. Tohoku University, Japan EDITORIAL BOARD H. D. BUI Écoe Poytechnique, France J. P. CARTER University of Sydney, Austraia R. M. CHRISTENSEN Stanford University, U.S.A. G. M. L. GLADWELL University of Wateroo, Canada D. H. HODGES Georgia Institute of Technoogy, U.S.A. J. HUTCHINSON Harvard University, U.S.A. C. HWU Nationa Cheng Kung University, R.O. China B. L. KARIHALOO University of Waes, U.K. Y. Y. KIM Seou Nationa University, Repubic of Korea Z. MROZ Academy of Science, Poand D. PAMPLONA Universidade Catóica do Rio de Janeiro, Brazi M. B. RUBIN Technion, Haifa, Israe A. N. SHUPIKOV Ukrainian Academy of Sciences, Ukraine T. TARNAI University Budapest, Hungary F. Y. M. WAN University of Caifornia, Irvine, U.S.A. P. WRIGGERS Universität Hannover, Germany W. YANG Tsinghua University, P.R. China F. ZIEGLER Technische Universität Wien, Austria PRODUCTION PAULO NEY DE SOUZA SHEILA NEWBERY SILVIO LEVY Production Manager Senior Production Editor Scientific Editor Cover design: Aex Scorpan See inside back cover or for submission guideines. Cover photo: Wikimedia Commons JoMMS (ISSN ) is pubished in 1 issues a year. The subscription price for 21 is US $5/year for the eectronic version, and $66/year (+$6 shipping outside the US) for print and eectronic. Subscriptions, requests for back issues, and changes of address shoud be sent to Mathematica Sciences Pubishers, Department of Mathematics, University of Caifornia, Berkeey, CA JoMMS peer-review and production is managed by EditFLOW from Mathematica Sciences Pubishers. PUBLISHED BY mathematica sciences pubishers A NON-PROFIT CORPORATION Typeset in LATEX Copyright 21. Journa of Mechanics of Materias and Structures. A rights reserved.

27 Journa of Mechanics of Materias and Structures Voume 5, No. 5 May 21 Axia compression of hoow eastic spheres ROBERT SHORTER, JOHN D. SMITH, VINCENT A. COVENEY and JAMES J. C. BUSFIELD 693 Couping of peridynamic theory and the finite eement method BAHATTIN KILIC and ERDOGAN MADENCI 77 Genetic programming and orthogona east squares: a hybrid approach to modeing the compressive strength of CFRP-confined concrete cyinders AMIR HOSSEIN GANDOMI, AMIR HOSSEIN ALAVI, PARVIN ARJMANDI, ALIREZA AGHAEIFAR and REZA SEYEDNOUR 735 Appication of the Kirchhoff hypothesis to bending thin pates with different modui in tension and compression XIAO-TING HE, QIANG CHEN, JUN-YI SUN, ZHOU-LIAN ZHENG and SHAN-LIN CHEN 755 A new modeing approach for panar beams: finite-eement soutions based on mixed variationa derivations FERDINANDO AURICCHIO, GIUSEPPE BALDUZZI and CARLO LOVADINA 771 SIFs of rectanguar tensie sheets with symmetric doube edge defects XIANGQIAO YAN, BAOLIANG LIU and ZHAOHUI HU 795 A noninear mode of thermoeastic beams with voids, with appications YING LI and CHANG-JUN CHENG 85 Dynamic stiffness vibration anaysis of thick spherica she segments with variabe thickness ELIA EFRAIM and MOSHE EISENBERGER 821 Appication of a matrix operator method to the thermoviscoeastic anaysis of composite structures ANDREY V. PYATIGORETS, MIHAI O. MARASTEANU, LEV KHAZANOVICH and HENRYK K. STOLARSKI 837