Non uniform warping for composite beams. Theory and numerical applications

Transcription

1 19 ème Congrès Français de Mécanique Marseille, août 29 Non uniform warping for composite beams. Theor and numerical applications R. EL FATMI, N. GHAZOUANI Laboratoire de Génit Civil, Ecole Nationale d Ingénieurs de Tunis B.P. 37, Campus universitaire, Le Belvédère, 12, Tunis, Tunisie Abstract : Started from the 3D solution of the original St Venant problem, a general non-uniform warping beam theor (NUW- BT) has recentl been established for an arbitrar homogeneous isotropic and elastic cross-section 1. Using now the solution of the etended St Venant problem for composite beam, NUW-BT ma be rewritten in order to be etended for an composite cross-section. For the present work, which constitutes a first step of this etension, the cross-section is assumed to be smmetric and made b orthotropic materials ; however, Poisson s effects (called here in-plane warping) are also taken into account. Theoretical results are given for the structural behavior of the composite beam and for the epressions of the 3D stresses ; these ones, eas to compare with 3D St Venant stresses, make clear the additional contribution of the new internal forces induced b the non-uniformit of the (in and out) warpings. Numerical results are given for a torsion and a shear-bending of a cantilever sandwich beam. Résumé : Basée sur la solution 3D du problème de St Venant, une théorie gérérale du gauchissement non uniforme a été récemment établie pour une poutre homogène isotrope et de section quelconque 1. Cette théorie peut être étendue au sections composites en s appuant sur la solution du problème de St Venant étendu au sections composites. Cette etension se restreint, dans le présent papier, au sections smétriques à phases orthotropes, mais introduit aussi la prise en compte des effets Poisson dans la déformation des sections. Les résultats théoriques concernent le comportement généralisé de la poutre et l epression 3D des contraintes ; celles-ci, faciles à comparer à celles de St Venant, montrent eplicitement la contribution de chacun des nouveau efforts intérieurs. Les résultats numériques présentés analsent la torsion et la fleion-simple d une poutre console de section sandwich. Mots clefs : non-uniform warping, Poisson s effect, torsion, shear-bending, composite beam, orthotropic material, finite elements method 1 Introduction For an homogeneous and isotropic beam, it is well known that (out-of-plane) warpings are due to the torsional moment and the shear forces. For an arbitrar composite cross-section, the situation is much more comple. Indeed, warpings ma also be induced b aial force and bending moments, and several elastic couplings between etensional, fleural and torsional deformations ma occur, even if the cross-section is smmetric. Laminated composite beams are known to ehibit such coupled deformations arising from the anisotropic nature of the laminae and from the stacking sequences. Further, in general cases of loading and boundar condition, these warpings are non-uniform along the beam ais. This leads to a beam mechanical behavior that ma be enough different from that predicted b (original or etended) St Venant (SV) beam theor 2, or other simplified theories which are restricted to uniform warping. To detect and describe the warping effects, different non-uniform warping theories have been proposed. The most part of these theories, called high order beam theories are based on kinematics including a non-uniform (out-of-plane) warping w with respect to the beam ais of the following shape w(, X) = η() ψ(x), where is the beam ais coordinate, X the in-section vector position, η the warping parameter and ψ a warping mode associated to the torsion or to one of the shear forces. In each case, the mode ψ is supposed to represent the corresponding SV-warping-function, which is considered as the reference to describe the natural warping of a cross-section. Various assumptions are present in the literature, the concern principall the shape of the crosssectional warping and the shear distribution (which are two connected points) and the choice of the warping parameter η as independent or linked to the cross-sectional strains (e.g. the twisting rate is used instead η in the non-uniform torsion theor of Vlasov). An another assumption, commonl made, is to maintain the shape of the cross-section ; this assumption, acceptable for an homogeneous section, should be reconsidered for a composite one, speciall when the gap between the rigidities of the materials is important (indeed, Poisson s effects in 1

2 19 ème Congrès Français de Mécanique Marseille, août 29 the plane of the section ma generate significant stresses speciall in the interfaces between the materials (e.g. the free edge effect in a laminated beam, which ma induce the delamination)). Beside, there is a serious lack (particularl for composite sections) of eperimental results and benchmark problems making us able to assess the validit or the relative importance of these various assumptions, which can obscure differences among theories 3. The results provided b high order beam theories for non uniform warping are more or less in agreement for the structural behavior of the beam, but the situation is not so clear for the 3D stress field, speciall close to the regions where the warping is restrained. The alternative to eperimental validations is a full 3D finite elements computation (3DFEM) of the beam ; however, this resort is not so used. Recentl, a general non-uniform warping beam theor (denoted b NUW-BT) for homogeneous elastic and isotropic beam, including torsional and shear forces effects, has been proposed b El Fatmi 1, 4. The theoretical development of this NUW-BT is completel based on the knowledge, a priori, of the 3D solution of the original SV-problem which provides, in particular, the SV-warping functions of the cross-section and their properties. This makes this theor free from the classical assumptions and valid for an arbitrar shape of cross-section 5. According to the NUW-BT approach, and using now the 3D etended SV-solution established for straight composite beams 2, NUW-BT ma be rewritten for an elastic composite cross-section. However, the problem being much more comple than for homogeneous and isotropic section, this etension will be, as a first step, restricted in this paper to the case of a smmetric cross-section (denoted b spo-cs) made b orthotropic materials for which the principal material coordinates coincide with those of the beam. This etended NUW-BT is based on a kinematics including the out-of-plane warpings due to the torsion and the shear forces and the in-plane warpings (i. e. Poisson s effect) due to the aial force and the bending moments. The stud is carried out within the framework of small perturbations and for a straight elastic composite beam with an spo-cs. Starting from a displacement model including si independent warping parameters and using as (out or in) warping modes the SV-warping-functions of the composite section, the corresponding nonuniform warping beam theor is derived. It should be noted that the theoretical development of this theor is based on the knowledge of the SV-functions and their properties, but it is not possible, to specif these properties in this present short paper. Thus we just recall in the first section the epression of the 3D SVsolution for the particular case of a spo-cs. 2 The etended St Venant problem and its solution for a spo-cs This etended problem is a 3D equilibrium elastic problem. The beam is along the ais and is occuping a prismatic domain Ω of a constant cross-section S and length L. S lat is the lateral surface and S and S L are the etremit cross-sections. The beam is in equilibrium onl under surface force densities H and H L acting on S and S L, respectivel. The cross-section is an spo-cs (bod-cs, open/closed thin-walled-cs), where the materials are ---orthotropic and perfectl bonded together. We call SV-solution the solution that satisfies FIG. 1 The 3D St Venant s problem for composite beam all the equations of the linearied equilibrium problem, ecept the boundar conditions on (S, S L ) which are satisfied onl in terms of resultant (force and moment). Let (R = (N, T, T ), M = M, M, M )) denoting the cross-sectional stresses where (N, T, T, M, M, M ) are the aial force the shear forces the torsional moment and the the bending moments, respectivel. For the case of a spo-cs, St Venant displacement ξ ( the upperbar ( ) denotes SV-quantities) is given b : V W V W V W ξ=u+ω X+ Ñ + M + M + M + T + T EA ẼI ẼI GJ GA GA (1) where (u, ω) are the cross-sectional displacements, and (ẼA, ẼI, ẼI, GJ, GA, GA ) the cross-sectional constants that depend onl on the cross-section nature (shape and materials) and govern the structural elastic behavior of the beam. V, W, V, W V, W ) are the in plane SV-warping functions (related to Poisson s effects) due to N, M and M, respectivel ; (φ, φ, φ ) are the out of plane SV-warping functions due to M, T and T, respectivel. All these SV-warpings are functions of (, ) and are characteristics of the cross-section. For an ---orthotropic material, Hooke s law σ= Kε ma be splitted to σ= K 1 ε 1 and τ = K 2 ε 2 where ψ ψ ψ 2

3 19 ème Congrès Français de Mécanique Marseille, août 29 ( σ, τ, ε 1, ε 2, K 1, K 2 ) are defined b K = σ σ= τ K1 O O K 2 σ= σ σ σ τ K 1 = τ = K 11 K 12 K 13 K 12 K 22 K 23 K 13 K 23 K 33 G τ τ ε= ε1 ε 2 K 2 = ε 1 = ε ε ε 2ε G G ε 2 = 2ε 2ε The 3D SV-stresses are linear with respect to the internal forces and their epressions ma be splitted and written in the following compact form N M σ = σ N + σ M + σ M = σ M τ = τ M + τ T + τ T = τ T (4) M T 3 Non uniform warping beam theor (NUW-BT) For the sake of simplicit the beam reference problem is taken similar to the 3D SV-problem defined in section- 2. The kinematical modeling is the following St Venant-like displacement field (Eq.1) : ψ ψ ψ ξ(u, ω, η, α) = u+ω X+α V +α V +α V +η +η +η W W W (5) where, with i {,, }, (η i, α i ) are the out and in warping parameters and (ψ i, V i, W i ) are the out and in plane SV-warping functions, respectivel. At the abscissa, each warping parameter measures the magnitude of the corresponding warping. The beam theor associated with this displacement, parametried b (v, θ, η, α), will be derived b the principle of virtual work. Let us denote b ξ = ξ(û, ω, η, α) a virtual displacement and ε = ε( ξ) the corresponding strain tensor. The internal virtual work is W i = σ : ε d. Using the displacement model Eq.(5), W i takes the form : L W i = ( L R γ + M ω + M ψ η + M s η + A ν α + B s α ) d = L R û + ( M + R ) ω + ( M ψ M) η + ( A ν B ) α d (6) R û + M ω + M ψ η + A ν α+ L O where ( ) is the derivative with respect to, (R, M) are the classical cross-sectional stresses and (M ψ, M s, A ν, B s ) are the new (or additional) ones defined b M M ψ = (Mψ, M ψ, M ψ ψ = σ ψ ) M ψ = σ ψ Mψ = σ ψ M M s = (Ms, Ts, Ts s = τ ψ, + τ ψ, ) Ts = τ ψ, + τ ψ, Ts = τ ψ, + τ ψ {, A = τ V + τ W (7) A ν = (A ν, A ν, A ν) A = τ V + τ W A = τ V + τ W N s = σ V B s = (N s, Ms, Ms, + σ W, + τ (V, + W,) ) Ms = σ V, + σ W, + τ (V, + W,) Ms = σ V, + σ W, + τ (V, + W,) where ( ) denotes S ( )ds. M ψ, M s introduced in 1 are called the bimoment vector and the secondar internal force vector, and the are both related to the out of plane warping. Similarl, it corresponds to the in plane warping two internal forces (A ν, B s ) related to the Poisson s effects (the subscript (.) s as secondar is chosen to indicate that the components (N s, M s, M s ) of B s ma be seen as secondar aial force and secondar bending moments. The eternal virtual work is W e = H O ξ O + H L ξ L takes the form W e = P O û O + C O ω O + Q O η O + S O α O + P L û L + C L ω L + Q L η L + S L α L (8) 3 (2) (3)

4 19 ème Congrès Français de Mécanique Marseille, août 29 where the onedimensional eternal forces (P, C, Q, S) are defined b P = H Q = H ψ i i C = GM H S = H V i + H W i i i {,, } (9) Thanks to the principle of virtual work, Eqs.6-8 are used to provide the equilibrium equations and the boundar conditions R = M + R = M ψ M = A ν B = (1) = (R, M, M ψ, A ν ) = (P O, C O, Q O, S O ) = L (R, M, M ψ, A ν ) = (P L, C L, Q L, S L ) Beam structural behavior. Let D=( D 1, D 2 )=(( D 1, D w 1 ), ( D 2, D w 2 )) denote the generalied strain vector and T =( T 1, T 2 )=(( T p 1, T w 1 ), ( T p 2, T w 2 )) the corresponding generalied force vector defined b : γ Np χ M p D 1 = χ T p 1 = Mp D 2 = γ T p 2 = Tp D w 1 = χ η η η α γ α χ α χ T w 1 = M p M ψ M ψ M ψ N s M s M s D w 2 = γ η χ η γ η γ α α R γ α + R γ } T w 2 = T p M s T s T s A ν A ν A ν (11) (12) where (N p, M p, M p, M p, T p, T p ), called the primar internal forces are defined b N p = N + N s Mp = M + Ms Mp = M + Ms Tp = T + Ts R A ν Mp = M + Ms Tp = T + Ts + R A ν (13) where R GA = g ; R GA = g. Using all the properties of the SV-solution and in particular the mathematical properties of the SV-functions (V i, W i, ψ i ), we can show that the generalied 1D behavior, defined b T = Γ D, gei gei is given b the operator Γ which has the following form : Γ= Γ w 1 = Γ 1 O O O O Γ w 1 O O O O Γ 2 O O O O Γ w 2 ; Γ 1 = ẼA ẼI ẼI ; Γ 2 = GJ GA GA KI ψ KI ψ a KI ψ a K 11 ẼA a 2 K 11 ẼI a + 2 K 11 ẼI GI GJ G GA R (f e ) e Γ w 2 = G GA + R (f e ) e g e g e g where KI ψ = K 11(ψ ) 2 g = G (V ) 2 +G (W ) 2 a = K 11 ψ KI ψ = K 11(ψ ) 2 g = G (V ) 2 +G (W ) 2 a = K 11 ψ KI ψ = K 11(ψ ) 2 g = G (V ) 2 +G (W ) 2 f = G V e = G ψ,v +G ψ,w e = G ψ,v +G ψ,w f = G W 4 (14) (15) (16) (17)

5 19 ème Congrès Français de Mécanique Marseille, août 29 Comments. The components of Γ define the cross-sectional contants of the composite beam. The si constants (ẼA, ẼI, ẼI, GJ, GA, GA ) are known, and the new ones are : si constants K 11, K 11 2, K 11 2, GI = G 2 +G 2, G, G related to the shape and the materials of the section ; si warping rigidities ( KI ψ, KI ψ, KI ψ,g, g, g ) related to the out and in-plane warpings ; and si constants (a, a, e, e, f, f ) which epress the coupling between the in and out warpings related to the bending moments (M, M ) and the shear forces (T, T ). We can also deduce from Γ the following constitutive relation for the primar internal forces N p = ẼAγ M p = ẼI χ M p = ẼI χ M p = GJχ T p = GA γ T p = GA γ (18) Note that these constitutive relations coincide with those of St Venant beam theor. Stresses. Using Hooke s law, we can establish that the stresses σ and τ ma be written in the following compact form M ψ ψ ψ ψ N Ns σ= σ M + σ M V, V, V M ψ, M s + K 1 Ms W, W, W, (Γ w 1 ) 1 M ψ N }{{} (V, +W,) (V,+W,) (V,+W,) s Ms sv Ms τ = τ M T T } {{ } sv + τ M s T s - R A ν T s + R A ν ψ + K, ψ, ψ, V V V 2 ψ, ψ, ψ, W W W (Γ w 2 ) 1 (19) Remark. Compared with SV-stresses, the epression of the stresses make clear the additional contribution of the new internal forces induced b the non-uniformit of the warpings. 4 Numerical applications In order to appl this NUW-BT, it is imperative to previousl know, for an given cross-section, all its constants and in particular its SV-warping-functions. This is achieved b using the software tool designated b SECOPE which is available within the finite elements code CASTEM. SECOPE has been developped conforming to the numerical method proposed b 6 for the computation of the 3D St Venant solution within the framework of the eact beam theor 2. This NUW-BT is applied to analse the elastic behavior of a cantilever sandwich beam subjected to a tip torsional moment (C = 1) or a tip transversal force (F = 1). The geometr of the cross-section and the mechanical constants of the materials (isotropic) are given in Fig-2. The SV-warpingfunctions are depicted in Fig-3. We present some numerical results for the 1D-behavior and the 3D stresses. For the stresses comparisons are made with a full 3DFEM computations. Torsion. Fig-4 compares the variations of ω and η along the span with those of SV-BT and, starting from the built-in section, Fig-6 compares, for the point B 1, the -variation of the aial stress σ with that obtained b a 3DFEM computation. Shear-bending. Fig-5 compares the variations of α, η and u along the span with those of SV-BT and, starting from the built-in section, Fig-7 compares, for the point A 2, the -variation of the aial stress σ with that obtained b a 3DFEM computation. Références 1 El Fatmi R. Non-uniform warping including the effects of torsion and shear forces. part-i : A general beam theor. International Journal of Solids and Structures, 44, , Ladevèe P. and Simmonds J. New concepts for linear beam theor with arbitrar geometr and loading. European Journal of Mechanics, 17(3), , Volovoi V. V., Hodges D. H., Cesnik C. E. S., and Popescu B. Assessment of beam modeling methods for rotor blade applications. Mathematical and computer modelling, 33, , El Fatmi R. Non-uniform warping theor for beams. Compte Rendu de l Académie des Sciences, C. R. Mécanique, 335, , El Fatmi R. Non-uniform warping including the effects of torsion and shear forces. part-ii : Analtical and numerical applications. International Journal of Solids and Structures, 44, , 27. M s T s T s A ν A ν A ν 5

6 19 ème Congrès Français de Mécanique Marseille, août 29 6 El Fatmi R. and Zenri H. A numerical method for the eact elastic beam theor. Applications to homogeneous and composite beams. International Journal of Solids and Structures, 41, , 24. 6