Universidade do Porto, Porto, Portugal c Department of Mathematics, Faculty of Science, King Abdulaziz University, Saudi Arabia

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1 This article was downloaded by: [RMIT University] On: 05 June 2014, At: 21:36 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Journal of Thermal Stresses Publication details, including instructions for authors and subscription information: A Thermo-Mechanical Analysis of Isotropic and Composite Beams via Collocation with Radial Basis Functions G. Giunta a, N. Metla a, S. Belouettar a, A. J. M. Ferreira b c & E. Carrera d a Centre de Recherche Public Henri Tudor, Luxembourg-Kirchberg, Luxembourg b Departamento de Engenharia Mecânica e Gestão Industrial, Faculdade de Engenharia, Universidade do Porto, Porto, Portugal c Department of Mathematics, Faculty of Science, King Abdulaziz University, Saudi Arabia d Departament of Mechanical and Aerospace Engineering, Politecnico di Torino, Turin, Italy Published online: 12 Sep To cite this article: G. Giunta, N. Metla, S. Belouettar, A. J. M. Ferreira & E. Carrera (2013) A Thermo-Mechanical Analysis of Isotropic and Composite Beams via Collocation with Radial Basis Functions, Journal of Thermal Stresses, 36:11, , DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 Journal of Thermal Stresses, 36: , 2013 Copyright Taylor & Francis Group, LLC ISSN: print/ x online DOI: / A THERMO-MECHANICAL ANALYSIS OF ISOTROPIC AND COMPOSITE BEAMS VIA COLLOCATION WITH RADIAL BASIS FUNCTIONS G. Giunta 1, N. Metla 1, S. Belouettar 1, A. J. M. Ferreira 23, and E. Carrera 4 1 Centre de Recherche Public Henri Tudor, Luxembourg-Kirchberg, Luxembourg 2 Departamento de Engenharia Mecânica e Gestão Industrial, Faculdade de Engenharia, Universidade do Porto, Porto, Portugal 3 Department of Mathematics, Faculty of Science, King Abdulaziz University, Saudi Arabia 4 Departament of Mechanical and Aerospace Engineering, Politecnico di Torino, Turin, Italy In this article, the mechanical behavior of beams subjected to thermal loads is investigated. The temperature field is obtained by exactly solving Fourier s heat conduction equation and it is considered as an external load within the mechanical analysis. Several higher-order beam models as well as Timoshenko s classical theory are derived thanks to a compact notation for the a priori approximation of the displacement field upon the cross-section. The governing differential equations and boundary conditions are obtained in a compact nucleal form that does not depend upon the displacements expansion order. The latter can be regarded as a free parameter of the formulation. A meshless strong-form solution based upon collocation with Wendland s radial basis functions is adopted. Isotropic and laminated orthotropic beams are investigated. Results are validated toward an analytical Navier-type solution and three-dimensional FEM results. It is shown that good accuracy can be obtained. Keywords: Beam structures; Hierarchical modeling; Radial basis functions; Thermo-mechanical response INTRODUCTION Several examples of isotropic and composite beam-like structures operating in complex environments can be found in many engineering fields. A severe temperature field, which is very common in many engineering applications, can be a cause of structural failure. It, therefore, calls for refined beam models. Efforts devoted toward the improvement of the structural models for an accurate and Received 6 November 2012; accepted 26 November The present research has been supported by the Fonds National de la Recherche Luxembourg via the CORE project C09/MS/05 FUNCTIONALLY. Address correspondence to Gaetano Giunta, Research Scientist, Department of Advanced Materials and Structures, Centre de Recherche Public Henri Tudor, 29, av. John F. Kennedy, Luxembourg-Kirchberg L-1855, Luxembourg. gaetano.giunta@tudor.lu 1169

3 1170 G. GIUNTA ET AL. efficient prediction of the thermo-mechanical response of isotropic and composite beams are, therefore, important and up-to-date. A global overview of thermo-elasticity can be found in Nowiski [1] and Hetnarski and Eslami [2]. Ghiringhelli [3] presented a finite element semidiscretization for composite beams in which the temperature distribution within the beam cross-section was computed by a two-dimensional finite element analysis. Comparison with three-dimensional finite element analysis were presented. Kapuria et al. [4] presented a higher-order zig-zag theory for thermal stress analysis of laminated simply supported beams. The thermal field was assumed a priori as a piecewise linear function along the through-the-thickness direction. The governing equations are derived using the principle of virtual work and solved via a Navier-type solution. Vidal and Polit [5] derived a conforming three-node thermomechanical beam finite element for the analysis of laminated beams. A sinus-refined model was used for the displacements field. Carpinteri and Poggi [6] carried out a thermo-elastic analytical stress analysis of multi-layered beams by means of Euler Bernoulli s kinematic hypotheses. Variable thickness beams were investigated by Xu and Zhou [7]. A non-linear temperature profile along the beam s thickness was computed solving the heat conduction equation. The solution was assessed towards results obtained by the commercial finite element software ANSYS. Despite relevant improvement in meshing generation, refinement and updating (see, e.g., Alliez et al. [8 10], Zhang and Bajaj [11] and Ito et al. [12]), some problems still need to be addressed, see Shepherd and Johnson [13]. Meshless methods have been proposed over the past years to overcome meshing-related issues. Very comprehensive and thorough reviews also addressing computer implementation can be found in Liu and Gu [14] and Nguyen et al. [15]. In particular, the collocation method was also addressed there. This latter method together with Wendland s Radial Basis Functions (RBFs) is used in the present article. RBFs were first proposed by Hardy [16] in topography for the analytical approximation of irregular surfaces using scattered data by means of multiquadric functions. Kansa [17, 18] extended the approximation via RBFs first to estimate partial derivatives and, then, to the solution of partial differential equations (PDEs). Wu [19, 20] proofed the convergence of collocation with RBFs for both data interpolation and solution of PDEs. In the latter case, it was shown that convergence is of the order of Oh d+1 where h is the density of the collocation points and d is a problem spatial dimension. Within the literature, several solution examples of structural problems by means of collocation with RBFs can be found. To the best of the authors knowledge, Ferreira [21, 22] was among the first to study composite plate and beam structures via collocation with RBFs. The work by Liu and Gu [23] and Liu and Wang [24] are also worth mentioning. The advantages of collocation with RBFs reside in the fact that it is easy to implement; it provides higher-order smoothness of the solution; new collocation points can be easily added to the solution space; and, since it is based on a strong form solution, RBFs are evaluated only at nodes and not at the integration points, they have to be differentiated but not integrated (this last task might be a problem in non-polynomial weak form meshless methods). Essential boundary conditions are also easily imposed. Disadvantages consist in the loss of symmetry of the stiffness matrix that is also dense in the case of globally supported RBFs. Another major disadvantage is the experience required for a correct choice

4 ISOTROPIC AND COMPOSITE BEAMS THERMO-MECHANICAL ANALYSIS 1171 of RBFs shape parameter. More details on this topic can be found in Davydov and Oanh [25]. A thermo-mechanical analysis of beams made of isotropic and laminated orthotropic materials is presented in this article. According to Nowiski [1], an uncoupled solution procedure is used. The temperature field is first obtained by solving the Fourier heat conduction equation and a mechanical analysis is, then, carried out accounting for the temperature as an external loading. Several refined models as well as Timoshenko s beam theory (TBT) are used for the latter analysis. These models are derived via a Unified Formulation (UF) that has been previously proposed for plates and shells (see Carrera [26] and Carrera and Giunta [27, 28]) and then extended to the analysis of beam structures via finite element or Navier-type solutions, see Carrera and Giunta [29], Carrera et al. [30, 31] and Giunta et al. [32 35]. A general and compact form for the approximation of the displacement components over the beam cross-section is assumed. This results in a nucleal form of the governing differential equations and boundary conditions that does not depend upon the order of expansion of the displacements. The fundamental nucleo has been obtained through the Principle of Virtual Displacements. Refined models accounting for cross-section in- and out-of-plane warping can be straightforwardly obtained. A strong form solution of problem governing equations is obtained by collocation with Wendland s RBFs. The proposed models are validated through comparison with the corresponding UF closed-form Navier-type models and three-dimensional finite element solutions. Numerical results show that accurate results can be obtained with a reduced computational effort. PRELIMINARIES A beam is a structure whose axial extension (l) is predominant when compared with any other dimension orthogonal to it. The beam cross-section () is identified by intersecting the beam with planes that are orthogonal to its axis. A Cartesian reference system is adopted: y- and z-axis are two orthogonal directions laying on. The x coordinate is coincident to the axis of the beam. It is bounded such that 0 x l. Beam geometry and reference system are presented in Figure 1. The displacement field is: u T x y z = u x x y z u y x y z u z x y z (1) in which u x, u y and u z are the displacement components along x-, y- and z-axes. Superscript T represents the transposition operator. Stress,, and total strain, t, vectors are grouped into vectors n and tn that lay on the cross-section: T n = xx xy xz T tn = txx txy txz and p and tp laying on planes orthogonal to : p T = { } yy zz yz T tp = { } tyy tzz tyz (2) (3)

5 1172 G. GIUNTA ET AL. Figure 1 Beam geometry and reference system. Under the hypothesis of linear analysis, the following strain-displacement geometrical relations hold: T tn = { } u xx u xy + u yx u xz + u zx (4) T tp = { } u yy u zz u yz + u zy Subscripts x, y and z, when preceded by comma, represent derivation versus the corresponding spatial coordinate. A compact vectorial notation can be adopted for Eqs. (4): tn = D np u + D nx u tp = D p u (5) where D np, D nx, and D p are the following differential matrix operators: D np = 0 0 y 0 0 z D nx = I x D p = 0 y z 0 z y (6) and I is the unit matrix. In the case of thermo-mechanical problems, Hooke s law reads: = C e = C t = C t T = C t T (7)

6 ISOTROPIC AND COMPOSITE BEAMS THERMO-MECHANICAL ANALYSIS 1173 where subscripts e and refer to the elastic and the thermal contributions, respectively. C is the material elastic stiffness, the vector of the thermal expansion coefficients and their product. T stands for temperature. According to the stress and strain vectors splitting, Eq. (7) becomes: p = C pp tp + C pn tn p T n = C np tp + C nn tn n T (8) In the general case of orthotropic materials with fibers laying on planes parallel to the xy one, matrices C pp, C pn, C np and C nn are: Cpp = C 22 C 23 0 C C 23 C 33 0 C pn = C 12 C 26 0 C T np = C 13 C 36 0 C 11 C 16 0 nn = C 16 C C C C 55 For the sake of brevity, coefficients C ij in Eqs. (9) as function of the engineering material constants and fiber rotation angle are not reported here. They can be found in Reddy [36]. The coefficients n and p : T n = { } T p = { } (10) are related to the thermal expansion coefficients n and p : through the following equations: T n = { 1 00 } T p = { } (11) (9) p = C pp p + C pn n n = C np p + C nn n (12) FOURIER S HEAT CONDUCTION EQUATION The beam models are derived considering the temperature as an external loading resulting from the internal thermal stresses. This requires the temperature profile to be known over the whole beam domain. Toward this end, Fourier s heat conduction equation for a multi-layered FGM beam: ( K k 1 x ) T k + ( K k 2 x y ) T k + ( K k 3 y z ) T k = 0 (13) z is solved via a Navier-type solution in which the temperature is written as follows: Tx y z = n x y z (14)

7 1174 G. GIUNTA ET AL. by ideally dividing the cross-section into N k non-overlapping sub-domains (or layers) along the through-the-thickness direction z: = N k For a kth layer, the Fourier differential equation becomes: k=1 k (15) K k 2 T k 1 + K k 2 T k x K k 2 T k y 2 3 = 0 (16) z 2 where K i k are the thermal conductivity coefficients. In order to obtain a closed form analytical solution, it is further assumed that the temperature does not depend upon the through-the-width coordinate y. The continuity of the temperature and the through-the-thickness heat flux q z hold at each interface between two consecutive sub-domains: T k = T k+1 q k z = qk+1 z with q k z = K k T k 3 z Subscripts and stand for sub-domain s top and bottom, respectively. The following temperatures are also imposed at cross-section through-the-thickness top and bottom: where T N k T N k = T N k sinx T 1 = T 1 sinx and T 1 are maximal amplitudes and is: = m l (17) (18) (19) with m N + representing the half-wave number along the beam axis. The following temperature field: T k x z = k z sin nx = T k e skz sinx (20) represents a solution of the considered heat conduction problem. T k is an unknown constant obtained by imposing the boundary conditions, whereas s is obtained by replacing Eq. (20) into Eq. (16): s k 12 =± K1 k K3 k (21)

8 ISOTROPIC AND COMPOSITE BEAMS THERMO-MECHANICAL ANALYSIS 1175 k z, therefore, becomes: k z = T k 1 es 1z + T k 2 es 2z (22) or, equivalently: k z = Ck 1 cosh ( K1 k K3 k z ) + C k2 sinh ( K1 k K3 k z ) (23) For a cross-section division into N k sub-domains, 2 N k unknowns C k j are present. The problem is mathematically well posed since the boundary conditions in Eqs. (17) and (18) yield a linear algebraic system of 2 N k equations in C k j. HIERARCHICAL BEAM THEORIES The variation of the displacement field over the cross-section can be postulated a priori. Several displacement-based theories can be formulated on the basis of the following generic kinematic field: ux y z = F y zu x with = 1 2N u (24) N u stands for the number of unknowns. It depends on the approximation order N that is a free parameter of the formulation. The compact expression is based on Einstein s notation: repeated indexes implicitly indicate summation. Thanks to this notation, problems governing differential equations and boundary conditions can be derived in terms of a single fundamental nucleo. The complexity related to higher than classical approximation terms is tackled and the theoretical formulation is valid for the generic approximation order and approximating functions F y z. The approximating functions F are Mac Laurin s polynomials. This choice is inspired by the classical beam models. N u and F as functions of N can be obtained through Pascal s triangle as shown in Table 1. The actual governing differential equations and boundary conditions due to a fixed approximation order are obtained straightforwardly via summation of the nucleo corresponding to each term of the Table 1 Mac Laurin s polynomials terms via Pascal s triangle N N u F 0 1 F 1 = F 2 = yf 3 = z 2 6 F 4 = y 2 F 5 = yz F 6 = z F 7 = y 3 F 8 = y 2 zf 9 = yz 2 F 10 = z 3 N N +1N +2 2 F N 2 +N +2 2 = y N F N 2 +N +4 2 = y N 1 z F NN +3 2 = yz N 1 F N +1N +2 2 = z N

9 1176 G. GIUNTA ET AL. expansion. According to the previous choice of polynomial function, the generic, N -order displacement field is: u x = u x1 + u x2 y + u x3 z + +u x N 2 +N +2 2 u y = u y1 + u y2 y + u y3 z + +u y N 2 +N +2 2 u z = u z1 + u z2 y + u z3 z + +u z N 2 +N +2 2 y N + +u x N +1N +2 z N 2 y N + +u y N +1N +2 z N 2 y N + +u z N +1N +2 2 z N (25) A detailed investigation of the effectiveness of each expansion term in the a priori kinematic field can be found in Carrera et al. [31] and Carrera and Petrolo [37]. The kinematic field of Timoshenko s beam theory is u x = u x1 + u x2 y + u x3 z u y = u y1 (26) u z = u z1 The latter is straightforwardly derived from the first-order approximation model by penalizing numerically the degrees of freedom u ij i= y z j = 2 3 and using reduced material stiffness coefficients. The latter is required in order to contrast the Poisson locking (see, Carrera and Brischetto [38, 39]). Poisson s ratio couples the normal elastic deformations along the spatial directions. Because of this, a constant approximation of the displacement components u y and u z does not yield accurate results, even in the case of slender beams. A reduced version of the material s constitutive equations, in which the stiffness coefficients are opportunely modified, is obtained imposing yy and zz equal to zero in Hooke s law. An linear algebraic system in the elastic normal stress components eyy and ezz is obtained. By substituting its solution into Hooke s equations regarding xx and xy the reduced stiffness coefficients Q 11, Q 16 and Q 66 are obtained: Q 11 = C 11 + C 12 C 12 C 33 C 13 C 23 C 2 23 C 22 C 33 + C 13 C 22 C 13 C 12 C 23 C 2 23 C 22 C 33 Q 16 = C 16 + C 26 C 12 C 33 C 13 C 23 C 2 23 C 22 C 33 + C 36 C 22 C 13 C 12 C 23 C 2 23 C 22 C 33 (27) Q 66 = C 66 + C 26 C 26 C 33 C 36 C 23 C 2 23 C 22 C 33 + C 36 C 22 C 36 C 26 C 23 C 2 23 C 22 C 33 The new constitutive relations in the case of TBT read: xx = Q 11 txx + Q 16 txy xy = Q 16 txx + Q 66 txy (28) xz = C 55 txz yz = C 45 txz

10 ISOTROPIC AND COMPOSITE BEAMS THERMO-MECHANICAL ANALYSIS 1177 It should be noted that in the case of isotropic materials Q 11 reduced to the Young modulus, Q 66 to the shear one and Q 16 is equal to zero. No shear correction coefficient is considered, since it depends upon several parameters, such as the geometry of the cross-section (see, for instance, Cowper [40] and Murty [41]). GOVERNING EQUATIONS The governing equations and the boundary conditions are derived through the Principle of Virtual Displacements (PVD): L i = 0 (29) where stands for a virtual variation and L i represents the strain energy. According to the grouping of the stress and strain components in Eqs. (2) and (3), the virtual variation of the strain energy for a thermo-mechanical problem is: L i = T tn n + T tp pddx (30) l By substitution of the geometrical relations in Eqs. (5), the constitutive equations in Eqs. (8) and the unified hierarchical approximation of the displacements in Eq. (24), Eq. (30) becomes: L i = D np F T C np D p F s + C nn D np F s + C nn F s D nx u T l + D p F T C pp D p F s + C pn D np F s + C pn F s D nx + D T nx C np F D p F s + C nn F D np F s + C nn F F s D nx d u s dx D np F T n I + D p F T p I + D T nx F n Id n dx u T l After integration by parts, Eq. (31) reads: L i = D np F T C np D p F s + C nn D np F s + C nn F s D nx u T l + D p F T C pp D p F s + C pn D np F s + C pn F s D nx (31) D T nx C np F D p F s + C nn F D np F s + C nn F F s D nx du s dx u T D np F T n I + D p F T p I D T nx F n Id n dx l + u T F C np D p F s + C nn D np F s + C nn F s D nx d u s x=l x=0 u T F n Id n x=l x=0 (32)

11 1178 G. GIUNTA ET AL. This latter can be written in the following compact vectorial form: L i = u T Ks uu u s dx u T K u n dx + u T s uu u s x=l x=0 ut u n x=l x=0 (33) l l The components of the differential stiffness matrix K s uu are: ( ) K s uuxx = J 66 y s y + J 55 z s z + J 16 y s J 16 s y x J 11 s ( ) K s uuxy = J 66 y s J 12 s y x + J 26 y s y + J 45 z s z J 16 s K s uuxz = J 45 z s y + J 36 y s z + ( ) J 55 z s J 13 s z x ( ) K s uuyx = J 26 y s y + J 45 z s z + J 12 y s J 66 s y K s uuyy = J 22 y s y + J 44 z s z ( J 26 y s J 26 s y ) ( ) K s uuyz = J 44 z s y + J 23 y s z + J 45 z s J 36 s z x ( ) K s uuzx = J 36 z s y + J 45 y s z + J 13 z s J 55 s z x ( ) K s uuzy = J 23 z s y + J 44 y s z + J 36 z s J 45 s z x K s uuzz = J 44 y s y + J 33 z s z ( J 45 y s J 45 s y ) The generic term J gh s is a cross-section moment: 2 x 2 2 x 2 2 x J 16 s x 2 (34) 2 x J 66 s x 2 2 x J 55 s x 2 J gh s = C gh F F s d (35) The components of the differential thermo-mechanical coupling matrix K u are: K uxx = J 6 y J 1 x K uyy = J 2 y J 6 x (36) K uzz = J 3 z The generic term J g is: J g = F g d (37)

12 ISOTROPIC AND COMPOSITE BEAMS THERMO-MECHANICAL ANALYSIS 1179 As far as the boundary conditions are concerned, the components of s uu are: s uuxx = J 16 s y + J 11 s x s uuyy = J 26 s y + J 66 s x s uuzz = J 45 s y + J 55 s x s uuxy = J 12 s y + J 16 s x s uuyx = J 66 s y + J 16 s x s uuzx = J 55 s z s uuzy = J 45 s z s uuxz = J 13 s z s uuyz = J 36 s z (38) and the components of u are: uxx = J 1 uyy = J 6 uzz = 0 (39) The fundamental nucleo of the governing equations is obtained by properly choosing the virtual displacement components: Its explicit form reads: u T Ks uu u s = K u n (40) u x J 11 s u xs xx + J 16 y s J 16 s y u xsx + J 66 y s y + J 55 z s z u xs J 16 s u ys xx + J 66 y s J 12 s y u ysx + J 26 y s y + J 45 z s z u ys + J 55 z s J 13 s z u zsx + J 45 z s y + J 36 y s z u zs = J 6 y n J 1 n x u y J 16 s u xs xx + J 12 y s J 66 s y u xsx + J 26 y s y + J 45 z s z u xs J 66 s u ys xx + J 26 y s J 26 s y u ysx + J 22 y s y + J 44 z s z u ys + J 45 z s J 36 s z u zsx + J 23 y s z + J 44 z s y u zs = J 2 y n J 6 n x u z J 13 z s J 55 s z u xsx + J 36 z s y + J 45 y s z u xs + J 36 z s J 45 s z u ysx + J 23 z s y + J 44 y s z u ys J 55 s u zs xx + J 45 y s J 45 s y u zsx + J 44 y s y + J 33 z s z u zs = J 3 z n (41) The fundamental nucleo of the natural and mechanical boundary conditions at x = 0 and l are: either u x = u x or J 16 s y u xs + J 11 s u xs x + J 12 s y u ys + J 16 s u ys x + J 13 s z u zs = J 1 n either u y = u y or J 26 s y u xs + J 66 s u xs x + J 66 s y u ys + J 16 s u ys x + J 36 s z u zs = J 6 n (42) either u z = u z or J 45 s y u xs + J 55 s u xs x + J 55 s z u ys + J 45 s z u zs = 0 where an overlined symbol stands for an imposed value. For a fixed approximation order, the nuclei have to be expanded versus the indexes and s in order to obtain the governing equations and the boundary conditions of the desired model. COLLOCATION WITH RADIAL BASIS FUNCTIONS Collocation solution procedure stems from weighted residual methods in which the governing equations and boundary conditions are not satisfied in an

13 1180 G. GIUNTA ET AL. Figure 2 Effect of the shape parameter on the normalized difference between the RBF solution and the corresponding Navier-type solution, Chebyshev s nodes distribution, N = 2, N n = 23 and l/b = 10. average sense but, using Dirac delta function as weight functions, on a set 0l of N n distinct nodes x i along the beam axis: = I E = x i i= 2 3N n 1 x 1 = 0x Nn = l (43) I and E are two sub-sets containing the internal and the external nodes, respectively. The axial variation of the displacement field and its derivatives are approximated via a linear combination of RBFs i x x i c: u x = U i i x x i c u x x = U i ix x x i c with i = 1 2N n (44) u xx x = U i ixx x x i c where U i are the unknown linear combination coefficients and i are Wendland s C 6 functions: [ ] i x x i c= 1 cx x i c 3 x x i 3 +25c 2 x x i 2 +8cx x i +1 (45) being c a shape parameter that has to be opportunely chosen, see Roque and Ferreira [42], as it will be discussed next. Stiffness Matrix Algebraic Fundamental Nucleo The algebraic fundamental nucleo of the governing equation is obtained by replacing Eq. (44) into Eq. (41) and computing them for an internal node x j I : J 55 z s z + J 66 y s y ij + J 16 y s J 16 s y ijx J 11 s ijxxu xsi + J 26 y s y + J 45 z s z ij + J 66 y s J 12 s y ijx J 16 s ijxxu ysi

14 ISOTROPIC AND COMPOSITE BEAMS THERMO-MECHANICAL ANALYSIS 1181 Figure 3 Effect of the number of nodes on the normalized difference between the RBF solution and the corresponding Navier-type solution for (a) uniform and (b) Chebyshev s nodes distributions, N = 2, c = c and l/b = J 45 z s y + J 36 y s z ij + J 55 z s J 13 s z ijx U zsi = J 6 y nj J 1 n x j J 26 y s y + J 45 z s z ij + J 12 y s J 66 s y ijx J 16 s ijxxu xsi + J 22 y s y + J 44 z s z ij + J 26 y s J 26 s y ijx J 66 s ijxxu ysi + J 23 y s z + J 44 z s y ij + J 45 z s J 36 s z ijx U zsi = J 2 y nj J 6 n x j (46) J 36 z s y + J 45 y s z ij + J 13 z s J 55 s z ijx U xsi + J 23 z s y + J 44 y s z ij + J 36 z s J 45 s z ijx U ysi + J 44 y s y + J 33 z s z ij + J 45 y s J 45 s y ijx J 55 s ijxxu zsi = J 3 z nj

15 1182 G. GIUNTA ET AL. Figure 4 Effect of the shape parameter in the convergence of u z versus the nodes number for a uniform nodes distribution, N = 2 and l/b = 10. where, for the sake of brevity, the following notation has been introduced: ijxxx = ixxx x j x i c nxj = nx x j Eqs. (46) can be rewritten into the following compact matrix form: (47) K sij U si = P j (48) For a fixed approximation order N, the nucleo in Eqs. (48) has to be expanded versus the indexes and s. Table 2 Displacement components [m] for slender and short isotropic beams l/b = 100 l/b = ũ x 10 3 ũ y ũ z 10 2 ũ x 10 3 ũ y 10 2 ũ z FEM 3D a FEM 3D b N = 10 (Navier) N = N = N = N = N = N = N = N = N = TBT a: Elements number b: Elements number

16 ISOTROPIC AND COMPOSITE BEAMS THERMO-MECHANICAL ANALYSIS 1183 Figure 5 Variation of u x [m] versus z at x/l = 0 and 2y/a = 1, isotropic beam, l/b = 10. Boundary Conditions Algebraic Fundamental Nucleo The boundary conditions are obtained replacing Eq. (44) within Eq. (42): either U xi ij = u x or J 16 s y ij + J 11 s ijxu xsi + J 12 s y ij + J 16 s ijxu ysi + J 13 s z ij U zsi = J 1 n either U yi ij = u y or J 26 s y ij + J 66 s ijxu xsi + J 66 s y ij + J 16 s ijxu ysi + J 36 s z ij U zsi = J 6 n either U zi ij = u z or J 45 s y ij + J 55 s ijxu xsi + J 55 s z ij U ysi + J 45 s z ij U zsi = 0 (49) Figure 6 Variation of u z [m] versus z at 2x/l = 1 and y/a = 0, isotropic beam, l/b = 10.

17 1184 G. GIUNTA ET AL. Table 3 Stress components [Pa] for isotropic beams, l/b = xx 10 6 xz 10 6 xy 10 5 zz 10 6 yy 10 5 yz FEM 3D a FEM 3D b N = 13 (Navier) N = N = N = N = N = N = N = N = N = N = N = N = TBT c a: Elements number b: Elements number c: Result not provided by the theory. The compact matrix form of the boundary conditions algebraic fundamental nucleo is: sij U si = B j (50) Also this nucleo has to be expanded versus the indexes and s. A clamped edge is obtained imposing natural or Dirichlet-type boundary conditions at once with u x = u y = u z = 0. A free edge is obtained imposing Robin-type boundary conditions at a time. A simply supported edge is obtained considering the natural boundary conditions for displacement components on the cross-section and the Robin-type one for the axial one. Resulting Algebraic System An algebraic linear system in U si is obtained computing the nucleo in Eq. (48) for each node in I and the boundary conditions nucleo in Eq. (50) for each node in E and assembling them as follows: K s21 K s22 K s2n n 2 K s2n n 1 K s2n n U s1 K s31 K s32 K s3n n 2 K s3n n 1 K s3n n U s2 K sn n 11 K sn n 12 K sn n 1N n 2 K sn n 1N n 1 K sn n 1N n U snn s11 s12 s1n n 2 s1n n 1 s1n 2 n U sn n1 sn n2 sn nn n 2 sn nn n 1 sn nn n snn 1 U snn

18 ISOTROPIC AND COMPOSITE BEAMS THERMO-MECHANICAL ANALYSIS 1185 Figure 7 Variation of xx [Pa] over the cross-section at 2x/l = 1 via (a) FEM 3D a, (b) N = 9 and (c) TBT, isotropic beam, l/b = 10. (Color figure available online.)

19 1186 G. GIUNTA ET AL. Figure 8 Variation of xz [Pa] versus y at x/l = z/b = 0, isotropic beam, l/b = 10. = P 2 P 3 P Nn 1 B 1 B Nn Hon and Shaback [43] showed that a general proof of non-singularity of a linear system obtained via collocation with RBFs is impossible. Nevertheless, on the basis of the numerical evidence and experience, they concluded that cases in which (51) Figure 9 Variation of xy [m] versus z at x/l = 0 and y/a = 1/4, isotropic beam, l/b = 10.

20 ISOTROPIC AND COMPOSITE BEAMS THERMO-MECHANICAL ANALYSIS 1187 Figure 10 Variation of zz [Pa] versus y at 2x/l = 1 and z/b = 0, isotropic beam, l/b = 10. the linear system is singular are rare and they have to be appositely constructed. Increasing the order of the problem, the coefficients matrix, which is a dense matrix, can become ill-conditioned. Kansa and Hon [44] proposed several techniques, such as domain decomposition, variable shape parameter and truncated multiquadric basis functions, to improve the conditioning of the coefficients matrix. Toward this end, Zhang et al. [45] introduced some auxiliary points, besides the collocation ones, and the problem was solved in a least-square sense. In the present work, it has been found that, increasing the expansion order N, the problem can be severely ill-conditioned. Nevertheless, equilibration of the coefficients matrix via row and column scaling was sufficient to obtain a well-conditioned problem for each considered case. Figure 11 Variation of yy [m] versus z at 2x/l = 1 and y/a = 0, isotropic beam, l/b = 10.

21 1188 G. GIUNTA ET AL. Figure 12 Variation of yz [m] versus z at 2x/l = 1 and y/a = 1/4, isotropic beam, l/b = 10. NUMERICAL RESULTS AND DISCUSSION The beam support is 0l a/2a/2 b/2b/2. Square cross-section with a = b = 1 m are considered. The length-to-side ratio l/b is equal to 100 and 10. Slender and thick beams are, therefore, investigated. The thermal boundary conditions, see Eq. (18), are: T = 400 K and T = 300 K. A half-wave is considered for the temperature variation along the beam axis. Simply supported beams are considered for which a closed-form Navier-type analytical solution is present, see Giunta et al. [35]. It should be noted that within the framework of each one-dimensional theory, this latter solution represents the exact solution. Threedimensional FEM models are also developed within the commercial code ANSYS. The three-dimensional quadratic element Solid90 is used for the thermal analysis, whereas the 20-node element Solid186 is considered for the mechanical problem. In order to present the convergence of the three-dimensional reference solution, two different meshes are considered for each analysis. The acronym FEM 3D a stands for a refined elements mesh , whereas a coarser solution is addressed by FEM 3D b. As far as the computational effort is concerned, the degrees of freedom (DOFs) of the three-dimensional FEM mechanical models are about 10 5 and for the FEM 3D a and FEM 3D b models, respectively. The number of DOFs (N DOFs ) for the present meshless method are function of the expansion order N and the number of nodes N n : N + 1N + 2 N DOFs = 3 N 2 n (52) In the case of a 13th-order 23-node solution, N DOFs is equal to Isotropic Beams Isotropic beams made of an aluminium alloy are first considered. The mechanical properties are: E = 72 GPa, = 03, K = 121 W/mK, = K 1.

22 ISOTROPIC AND COMPOSITE BEAMS THERMO-MECHANICAL ANALYSIS 1189 Figure 13 Temperature profile along the through-the-thickness direction for different length-to-side ratio values for (a) 0/90, (b) 90 2 and (c) 0 2 laminations.

23 1190 G. GIUNTA ET AL. Table 4 Displacements [m] and stresses [Pa] for a 0/90 laminated beam, l/b = ũ x 10 3 ũ y 10 2 ũ z 10 8 xx 10 6 xz 10 6 xy FEM 3D a FEM 3D b N = 14 (Navier) N = N = N = N = N = N = N = N = N = N = N = N = N = a: Elements number b: Elements number Unless differently stated, the displacements and the stresses evaluated at the following points are considered: ( ũ x = u x 0 a 2 b ) 2 ( l xx = xx 2 a 2 b ) 2 ) ( l ũ y = u y 2 a 2 b ( l ũ 2 z = u z 2 0 b 2 ( xz = xz 0 a ) ( 2 0 xy = xy 0 a 4 b ) 2 Table 5 Displacements [m] and stresses [Pa] for a 90 2 laminated beam, l/b = 10 ) (53) 10 2 ũ x 10 4 ũ y 10 2 ũ z 10 4 xx 10 3 xz 10 3 xy FEM 3D a FEM 3D b N = 13 (Navier) N = N = N = N = N = N = N = N = N = N = N = N = a: Elements number b: Elements number

24 ISOTROPIC AND COMPOSITE BEAMS THERMO-MECHANICAL ANALYSIS 1191 Table 6 Displacements [m] and stresses [Pa] for a 0 2 laminated beam, l/b = ũ x 10 3 ũ y 10 3 ũ z 10 7 xx 10 4 xz 10 6 xy FEM 3D a FEM 3D b N = 13 (Navier) N = N = N = N = N = N = N = N = N = N = N = N = a: Elements number b: Elements number zz = zz ( l ) yy = yy ( l 2 0 b 2 ) yz = yz ( l 2 a 4 b 4 The influence of the shape parameter c is investigated first. It is well known that it plays a very important role in collocation with RBFs for approximating functions (see Press et al. [46]) and solving partial differential equations (see Roque and Ferreira [42]). According to a good or a poor choice of it, the solution accuracy can differ by several orders of magnitude. In literature, several values of c have been proposed depending upon the number of nodes, the distance between the nodes and the type of RBFs. For instance, Fasshauer [47] proposed a shape parameter inversely proportional to the square root of the number of nodes for data fitting and solution of non-linear partial differential equations over a constant unit square domain. In this work, the length-to-thickness ratio explicitly appears in the shape parameter: ) c = b 2l (54) Convergence analysis versus c has been carried out with 23 nodes distributed according to a Chebyshev grid, see Philips [48]: x m = l 2 [ ( ) ] m 1 cos N n with m = 1 2N n (55) A short beam and a second-order theory are considered. Results for slender beams and higher-order models are very similar and they are not here presented for

25 1192 G. GIUNTA ET AL. Figure 14 Variation of u x [m] versus z at x/l = y/a = 0, 0/90 beam, l/b = 10. the sake of brevity. Figure 2 presents the normalised difference between the RBF solution and the corresponding Navier-type one in the case of the transverse displacement u z and the normal and shear stresses xx and xz. Plotted quantities are defined as follows: u e uz = zrbf u znavier e xx = xxrbf xxnavier e xz = xzrbf xznavier (56) u znavier xxnavier xznavier The displacement and the stresses are computed at the same points as in Eqs. (53). For small values of the shape parameter, the errors are high. For the considered cases, a shape parameter that does not explicitly account for the beam length yields inaccurate results for both slender and short beams, whereas results computed for c = c are accurate. This is believed to be due to the fact that a parameter representative of the leading dimension of the structure should be also considered. A more regular convergence is observed in the case of short beams than for slender ones. In the remaining part of the article, the shape parameter as in Eqs. (54) will be used. As far as the convergence versus the nodes number is concerned, the variation of the normalised differences e uz, e xx and e xz versus N n is presented in Figure 3. The analysis is carried out considering both a uniform and a Chebyshev s node distribution. As also shown in Fasshauer [47], they yield a non-monotonic rate of convergence, being the latter faster than the former one. In the case of Chebyshev s node distribution, the solution becomes unstable for N n 50. As shown in Eq. (55), this is due to the fact that at the neighbourhood of the beam ends the distance between two consecutive nodes becomes too small. In the remaining of the numerical investigations, 23 nodes will be used and they will be positioned according to Chebyshev s distribution. The cusp in e uz in the case of uniform nodes distribution is due to the chosen value for the shape parameter as presented in Figure 4. The cusp disappears for c 007.

26 ISOTROPIC AND COMPOSITE BEAMS THERMO-MECHANICAL ANALYSIS 1193 Figure 15 Variation of u y [m] versus y at 2x/l = 2z/b = 1, 0/90 beam, l/b = 10. Table 2 presents the displacement components u x, u y and u z in case of both slender and short beams. TBT yields accurate results for the axial and the throughthe-thickness displacement components, the absolute error versus the reference three-dimensional FEM solution being about 15%. A nil displacement component u y is obtained since TBT accounts for a rigid cross-section on its plane and the problem is symmetric versus the plane Oxz. The variation of the displacement components u x and u z along the z-axis is presented in Figures 5 and 6. They are plot over the cross-section where they present the maximum value. According to the Navier-type solution, see Giunta et al. [35], the axial position 2x/l is either equal to zero or one. TBT predicts a though-the-width constant axial displacement, whereas it actually varies also versus y. This latter variation is not presented here for the sake of brevity. A second-order model matches the FEM 3D a solution. Figure 16 Variation of u z [m] versus z at 2x/l = 1 and y/a = 0, 0/90 beam, l/b = 10.

27 1194 G. GIUNTA ET AL. Figure 17 Variation of xx [Pa] over the cross-section at x/l = 1/2 via (a) FEM 3D a and (b) N = 14, orthotropic beam, l/b = 10. (Color figure available online.) The six stress components are presented in Table 3 for l/b = 10. Poor results are obtained through lower-order models (N 3) and TBT, whereas highorder theories for N 9 match the reference solution. The difference between the reference solution and N = 9 is about 08%, at worst. It decreases to about 05% for N = 13. Figures 7 to 12 show the variation over the cross-section of the stress

28 ISOTROPIC AND COMPOSITE BEAMS THERMO-MECHANICAL ANALYSIS 1195 Figure 18 Variation of xz [Pa] over the cross-section at x/l = 0 via (a) FEM 3D a and (b) N = 14, orthotropic beam, l/b = 10. (Color figure available online.) components. A ninth-order model matches the reference three-dimensional FEM solution. For the axial stress, the TBT solution is also shown. It presents a throughthe-thickness bi-linear variation that is very different from the reference one. Due to this marked diversity, a different scale is used.

29 1196 G. GIUNTA ET AL. Orthotropic Beams A two-layer composite beam is considered. Three stacking sequences (0/90, 0 2 and 90 2 ) are investigated. Lamination starts from cross-section top; laminae lay on planes parallel to the Oxy one and fibers orientation angle is measured towards the axial axis. The material elastic and thermal properties are: E L = GPa, E T = 691 GPa, G LT = 345 GPa, G TT = 138 GPa, LT = TT = 025, K L = 3642 W/mK, K T = 096 W/mK, L = K 1 and T = K 1. Subscripts L and T stand for a direction parallel and perpendicular to the fibers, respectively. Figure 13 presents the temperature profile Tz at mid-span cross-section. Several values of the length-to-thickness ratio are considered. For slender beams, a linear variation from T to T is observed, regardless the stacking sequence. For short beams presenting fibers along the axial direction, the temperature can be lower than T. The thermal boundary conditions at the beam ends influence the temperature profile at mid-span. This is due to the high difference in value of the thermal conductivity coefficients and it is not present in the 90 2 beam. The solution of Fourier s heat conduction equation matches the reference three-dimensional FEM one. Tables 4 to 6 present the displacement and stress components for all the considered stacking sequences. For the sake of brevity only short beams are considered. Results are computed at the same point as presented in Eqs. (53) except for the shear stress component xz that is evaluated at 0 a b. As far as 2 4 displacements are concerned, lower-order theories yield good results for the 90 2 lamination only. This case presents a temperature profile that is similar to the case of the isotropic material. For the other two stacking sequences, the temperature profile is more complex (as shown in Figures 13) resulting in a complex mechanical behavior. The stress field is actually three-dimensional and, therefore, very difficult to be accurately predicted by a one-dimensional approach. Higher-order models are mandatory. Figures 14 to 16 present the variation along z- or y-axes of the displacement components for a 0/90 beam. The color maps of the stress components xx and xz over the cross-section are presented in Figures 17 and 18. Although displacements are accurately predicted, the proposed models yield a fair description of the stresses. The latter present a high gradient at layers interface due to the difference in material thermal properties. Further improvements of the proposed models can be obtained by accounting for two peculiar characteristics of the mechanics of composite structures: the continuity of the transverse shear stresses and the change in slope of displacements at layers interface. CONCLUSIONS A unified formulation of one-dimensional beam models has been proposed for the thermo-mechanical analysis of isotropic and laminated beams. The temperature field has been computed by solving Fourier s heat conduction equation and it has been accounted for in the mechanical analysis as an external load. Several higherorder beam theories as well as Timoshenko s classical model have been derived straightforwardly. As a first endeavor, a strong form solution of the governing differential equations has been obtained via collocation with Wendland s locally

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