STATIC AND DYNAMIC ANALYSIS OF AIRCRAFT STRUCTURES BY COMPONENT-WISE APPROACH. E. Carrera A. Pagani, M. Petrolo

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1 Proceedings of the ASME 213 Internationa Mechanica Engineering Congress & Eposition IMECE 213 November 13-21, 213, San Diego, Caifornia, USA IMECE STATIC AND DYNAMIC ANALYSIS OF AIRCRAFT STRUCTURES BY COMPONENT-WISE APPROACH E. Carrera A. Pagani, M. Petroo Department of Mechanica and Aerospace Engineering Poitecnico di Torino Corso Duca degi Abrui 29, 1129 Torino, Ita ABSTRACT This paper proposes an advanced approach to the anasis of reinforced-she aircraft structures. This approach, denoted as Component-Wise (), is deveoped b using the Carrera Unified Formuation (CUF). CUF is a hierarchica formuation aowing for the straightforward impementation of an-order onedimensiona (1D) beam theories. Lagrange-ike ponomias are used to discretie the dispacement fied on the cross-section of each component of the structure. Depending on the geometrica and materia characteristics of the component, the capabiities of the mode can be enhanced and the computationa costs can be kept ow through smart discretiation strategies. The goba mathematica mode of compe structures (e.g. wings or fuseages) is obtained b assembing each component mode at the cross-section eve. Net, a cassica 1D finite eement (FE) formuation is used to deveop numerica appications. It is shown that MSC/PATRAN can be used as pre- and post-processor for the modes, whereas MSC/ DMAP aters can be used to sove both static and dnamic probems. A number of tpica aeronautica structures are anaed and resuts are compared to cassica beam theories (Euer-Bernoui and Timoshenko), refined modes and cassica soid/she FE soutions from the commercia code MSC/. The resuts highight the enhanced capabiities of the proposed formuation. In fact, the approach is cear the natura too to anae wing structures, since it eads to resuts that can be on obtained Corresponding author, emai: afonso.pagani@poito.it through three-dimensiona easticit (soid) eements whose computationa costs are at east one-order of magnitude higher than modes. INTRODUCTION Primar aircraft structures are essentia reinforced thin shes [1]. These are so-caed semimonocoque constructions which are obtained b assembing three main components: skins (or panes), ongitudina stiffening members (incuding spar caps) and transversa stiffeners (ribs). The free vibration anasis and the determination of stress/strain fieds in these structura components are of prime interest for structura anasts. Man different approaches were deveoped in the first haf of the ast centur. These are discussed in major reference books [1, 2] and more recent in [3]. Among these approaches the so-caed Pure Semimonocoque (PS) (or ideaied semimonocoque ) is the most popuar, since it assumes constant shear into panes and shear webs. The main advantage of PS is that it eads to a sstem of inear agebraica equations. However the number of such equations rapid increases for muti-ba bo structures with high redundanc. The number of resuting equations (and redundanc) can be strong reduced b couping PS with assumptions from Euer-Bernoui (Euer-Bernoui Beam Mode, EBBM) or Timoshenko (Timoshenko Beam Mode, TBM) theories. This ead to the so-caed Beam Semimonocoque (BS) approach. Man works are known to overcome imitations reated 1 Copright c 213 b ASME

2 to constant shear hpotheses, see [4 8] as eampes. Due to the advent of computationa methods, most FEM, the anasis of compe aircraft structures continued to be made using a combination of soids (3D), pates/shes (2D) and beams (1D). These were impemented first in codes. Man others commercia FE codes have been deveoped and used in aerospace industries. Nowadas FEM modes with a number of unknowns (degrees of freedom, DOFs) cose to 1 6 are wide used in common practise. The possibe manner in which stringers, spar caps, spar webs, panes, ribs are introduced into FE mathematica modes is part of the knowedge of structura anasts. A short discussion of this foows. A number of works have shown the necessit for a proper simuation of the stiffeners-pane inkage. Koi and Chandrashekhara [9] formuated an FE mode with 9-node pate and 3-node beam eements. Gangadhara [1] carried out inear static anaes of composite aminated shes using a combination of 8-node pate eements and 3-node beam eements. As far as dnamic anasis is concerned, Samanta and Mukhopadha [11] deveoped a new stiffened she eement and, subsequent, the used this eement to determine natura frequencies and mode shapes of different stiffened structures. In [12], Thinh and Khoa deveoped a new 9-node rectanguar stiffened pate eement for the free vibration anasis of aminated stiffened pates based on Mindin s deformation pate theor. Recent, Vörös [13] formuated a new pate/she stiffener eement. In Vörös theor, the stiffener eement is deveoped b means of a genera beam theor, which incudes the constraint torsiona warping effect and the second order terms of finite rotations. The works mentioned so far show a cear interest in investigating FEM appications to reinforced-she structures. In most of the artices in iterature, such as those cited above, pates/shes and stiffeners are modeed as separate eements and a simuation of the stiffener-pane inkage is often necessar. Usua, beam nodes are connected to the she eement nodes via rigid fictitious inks. This methodoog presents some inconsistencies. The main probem is that the out-of-pane warping dispacements in the stiffener section are negected and the beam torsiona rigidit is not correct predicted. The main aim of the present work is to introduce a new 1D formuation which is abe to mode reinforced-she aircraft structures. The present component-wise () approach deas with shes and stiffeners b means of a unique 1D formuation, with no need to introduce fictitious inks to connect beam and she eements. The approach has recent been epoited for the anasis of aminated composites [14] and it has proven to be abe to mode singe fibers and reated matrices, entire aers and whoe mutiaers. Furthermore, the modes have shown their enhanced capabiities in deaing with both static [15] and free vibration [16] anasis of wing structures. The present work is part of the framework of the onedimensiona Carrera Unified Formuation, CUF, which was recent proposed b the first author and his co-workers [17, 18]. Two casses of CUF 1D modes were formuated: the Taorepansion cass, hereafter referred to as TE, and the Lagrangeepansion cass, hereafter referred to as LE. TE modes epoit N-order Taor-ike ponomias to define the dispacement fied above the cross-section with N as a free parameter of the formuation. The strength of CUF TE modes in deaing with arbitrar geometries, thin-waed structures and oca effects were evident in static [19, 2] and free-vibration anasis [21, 22]. An important feature of TE modes is that EBBM and TBM cassica beam theories can be derived as degenerate cases of the inear Taor-tpe epansion. Converse, the LE cass is based on Lagrange-ike ponomias to discretie the cross-section dispacement fied and LE modes have on pure dispacement variabes. Recent, static anases on isotropic [18] and composite structures [23] have reveaed the strength of LE modes in deaing with open cross-sections, arbitrar boundar conditions and obtaining Laer-Wise descriptions of the 1D mode. In the foowing a brief overview on CUF is proposed and the approach is described. Net, some eampes are addresses. Fina, the main concusions are outined. PRELIMINARIES The notation assumed in this paper is hereafter introduced. The adopted coordinate frame is presented in Fig. 1. Let us introduce the transposed dispacement vector, u(,,) = { u u u } T (1) The cross-section of the structure is Ω, and the beam boundaries FIGURE 1. COORDINATE FRAME OF THE BEAM MODEL over are L. The stress, σ, and strain, ε, components are grouped as foows: σ p = { σ σ σ } T, ε p = { ε ε ε } T σ n = { σ σ σ } T, εn = { ε ε ε } T (2) 2 Copright c 213 b ASME

3 The subscript n stands for terms ing on the cross-section, whie p stands for terms ing on panes which are orthogona to Ω. In the case of sma dispacements with respect to a characteristic dimension of Ω, inear strain - dispacement reations can be used ε p = D p u ε n = D n u = (D nω + D n )u where D p and D n are inear differentia operators. The can be found in [17]. Constitutive aws were epoited to obtain stress components, (3) σ = Cε (4) For LE cass, F τ are Lagrange-ike ponomias. In this work, three tpes of cross-section ponomia sets were adopted: four- (L3), four- (L4), and nine-point (L9) eements. The isoparametric formuation was epoited to dea with arbitrar shaped geometries. The L3, L4 and L9 interpoation functions are given in [25]. For instance, the L4 function is F τ = 1 4 (1 + r r τ)(1 + s s τ ) τ = 1,2,3,4 (8) where r and s var from 1 to +1, whereas r τ and s τ are the coordinates of the four points whose numbering and ocation in the natura coordinate frame are shown in Fig. 2a. Unike TE, According to Eqn.s (2), Eqn. (4) becomes (-1, 1) (1, 1) (-1, 1) (, 1) (1, 1) σ p = C pp ε p + C pn ε n σ n = C np ε p + C nn ε n (5) The matrices C pp, C nn, C pn, and C np contains the materia coefficients. For the sake of brevit the are not reported here. The can be found in [24]. r (-1, ) (1, ) (-1, -1) s (1, -1) (a) Four-point eement, L4 s r (-1, ) (1, ) (, ) (-1, -1) (, -1) (1, -1) (b) Nine-point eement, L9 HIGHER-ORDER FINITE BEAM ELEMENTS In the framework of the CUF, the dispacement fied above the cross-section is the epansion of generic functions, F τ, u(,,) = F τ (,)u τ (), τ = 1,2,...,M (6) where F τ var over the cross-section. u τ is the dispacement vector and M stands for the number of terms of the epansion. According to the Einstein notation, the repeated subscript, τ, indicates summation. The choice of F τ determines the cass of 1D CUF mode that has to be adopted. Two cases are addressed in this paper: TE and /LE. TE 1D modes are based on ponomia epansions, i j, of the dispacement fied above the cross-section of the structure, where i and j are positive integers. For instance, the dispacement fied of the second-order (N = 2) TE mode is epressed b u = u 1 + u 2 + u u 4 + u u 6 u = u 1 + u 2 + u u 4 + u u 6 u = u 1 + u 2 + u u 4 + u u 6 (7) The order N of the epansion is arbitrar and defines the beam theor. N can be set as an input of the anasis. FIGURE 2. OMETRY CROSS-SECTION L-ELEMENTS IN NATURAL GE- one of the most important feature of LE modes is that the have on pure dispacement degrees of freedom. More detais about LE modes can be found in the paper b Carrera and Petroo [18]. For both TE and LE modes, the FE approach was adopted to discretie the structure aong the -ais. This process is conducted via a cassica finite eement technique, where the dispacement vector is given b u(,,) = F τ (,)N i ()q τi (9) N i stands for the shape functions and q τi for the noda dispacement vector. For the sake of brevit, the shape functions are not reported here. The can be found in man books, for instance in [26]. Eements with four nodes (B4) were adopted in this work, that is, a cubic approimation aong the -ais was assumed. The stiffness matri of the eements, the mass matri and the eterna oadings vector can be obtained via the principe of virtua dispacements. For the sake of brevit, the derivation of the eementa matrices and the oading vector is not provided in this paper, but it can be found in [17]. For iustrative purpose, 3 Copright c 213 b ASME

4 TABLE 1. DISPLACEMENT VALUES, u, AT THE LOADED POINT AND NUMBER OF DEGREES OF FREEDOM FOR THE CONSIDERED STRUCTURAL CONFIGURATIONS OF THE THREE-BAY WING BOX Fu Mode No Ribs Case Open Mid-ba Case u 1 2 m DOFs u 1 2 m DOFs u 1 2 m DOFs MSC/ SOLID/SHELL Cassica and Refined TE Beam Modes EBBM TBM N = N = N = N = the stiffness matri in the form of fundamenta nuceus is given in the foowing: K i j τ s =I i j ( Dnp T F τ I )[ ( C np Dp F s I ) ( + C nn Dnp F s I )] + ( D T p F τ I )[ ( C pp Dp F s I ) ( + C pn Dnp F s I )] Ω + [ (D T np F τ I ) C nn + ( Dp T F τ I ) ] C pn F s Ω I Ω + where: ( I i j I i j, I i, j I Ω F τ [ C np ( Dp F s I ) + C nn ( Dnp F s I )] Ω + I i, j, I Ω F τ C nn F s Ω I Ω (1) I Ω = Ω = 1, I i j,, I i, j Ω... dω (11), I i, ) ) j, = (N i N j, N i N j,, N i, N j, N i, N j, d (12) It shoud be noted that K i jτs does not depend either on the epansion order or on the choice of the F τ epansion ponomias. These are the ke-points of CUF which aows, with on nine FORTRAN statements, the impementation of an-order of mutipe cass theories. The component-wise approach The refined TE modes described above are characteried b degrees of freedom (dispacements and N-order derivatives of dispacements) with a correspondence to the ais of the beam. The epansion can aso be made b using on pure dispacement (a) FIGURE 3. (b) (c) APPROACH THROUGH LE ELEMENTS vaues, e.g. b using Lagrange ponomias. The resuting LE can be used for the whoe cross-section or can be introduced b dividing the cross-section into various sub-domains (see [18]). This characteristic aows us to separate mode, for instance, stringers and panes. The LE formuation was used in this paper to impement modes of reinforced-she wing structures, as 4 Copright c 213 b ASME

5 shown in Fig. 3a where a two-stringer spar is considered. Figure 3b shows a possibe mode of the spar where each component was modeed via one 1D LE eement. Each LE eement is then assembed above the cross-section to obtain the goba stiffness matri based on the 1D formuation. Since panes coud not be reasonab modeed via a 1D formuation, 1D modes can be refined b using severa L-eements for one component. This aspect is shown in Fig. 3c where the pane is modeed via two 1D LE eements. B epoiting the present 1D formuation, the anasis capabiities of a structura mode can be enhanced b 1. oca refining the LE discretiation; 2. using higher-order LE eements (e.g. 4-node, 9-node, 16-node, etc.). NUMERICAL RESULTS Some eampes are discussed in this section. First, the static anasis of a three-ba wing bo is addressed. Net, free vibration anases of a fuseage section and a compete wing are introduced. The resuts are compared both with cassica beam theories and soid/she eements of the commercia code MSC/. The attention is focused on the abiit of the present formuation to foresee the effects due to both ongitudina and transverse stiffeners as we as open sections on thin-waed aerospace structures. eampes highight the capabiit of the present advanced 1D modes to accurate describe the effects due to ribs and open sections. The structures consist of three wing boes each with a ength,, equa to.5 m. The cross-section is a trapeium with a height b = 1 m. The two webs of the spars have a thickness of m, whereas h 1 =.16 m and h 2 =.8 m. The top and the bottom panes have a thickness of m, as we as ribs. The area of the stringers is A s = m 2. The wing is compete made of an auminium ao 224, having G/E =.4. The cross-section in = was camped and a point oad, F = [N], was appied at [b,2,h 2 /2]. Tabe 1 shows the vertica dispacement vaues, u and the computationa costs for each mode impemented. Cassica, increasing order TE and modes are reported. The modes were obtained b a combination of L4 and L9 eements. Resuts are vaidated b an MSC/ mode buit both with soid and she FE. Figures 5, 6 and 7 show the span-wise variation of the aia and the shear stress components for the three different configurations. BS and PS soutions are provided for the fu mode of the three-ba wing bo for comparison. The structure has three redundancies. Fina, Fig. 8 shows that the present mode is abe to detect the distribution of transverse stress components on ribs. The foowing remarks can be made: Static anasis of a three-ba wing bo The first anasis case was carried out on the three-ba wing bo [15] for which PS and BS soutions were given in Riveo s book [2]. The considered structure is shown in Fig. 4a [2, chap. 11 p. 31], whereas Fig.s 4b and c show its variations. These Y F (a) = Z h 1 X h 2 (a) Fu mode b Y F Y F (b) = 2 Z Z Open Section X X (b) No ribs case (c) Fu mode with open mid-ba FIGURE 4. DIFFERENT STRUCTURAL CONFIGURATIONS OF THE THREE-BAY WING BOX FIGURE 8. TRANSVERSE STRESSES DISTRIBUTION, σ, ON THE RIBS OF THE FULL MODEL OF THE THREE-BAY WING BOX, MODEL 5 Copright c 213 b ASME

6 PS BS N PS BS N σ [kpa] 6 σ [kpa] [m] (a) Bottom right spar cap; σ at = b, = h [m] (b) Rear spar; σ at = b, = FIGURE 5. STRESS COMPONENTS DISTRIBUTION ALONG THE WING SPAN, FULL MODEL OF THE THREE-BAY WING BOX 25 N N σ [kpa] 1 σ [kpa] [m] (a) Bottom right spar cap; σ at = b, = h [m] (b) Rear spar; σ at = b, = FIGURE 6. STRESS COMPONENTS DISTRIBUTION ALONG THE WING SPAN, THREE-BAY WING BOX WITH NO RIBS 2 N N σ [kpa] σ [kpa] [m] (a) Bottom right spar cap; σ at = b, = h [m] (b) Rear spar; σ at = b, = FIGURE 7. STRESS COMPONENTS DISTRIBUTION ALONG THE WING SPAN, THREE-BAY WING BOX WITH OPEN MID-BAY 6 Copright c 213 b ASME

7 1. /LE modes correct predict ribs and oca effects, as the match the resuts obtained with soid/she modes. 2. Higher than sith-order TE modes are required to correct predict the cross-section deformabiit. Moda anasis of a fuseage section The free vibration anasis of a fuseage section was carried out net. The cross-section of the fuseage is considered to be circuar and it is shown in Fig. 9. The outer diameter, d, was set to 2 m, whereas the thickness, t, was.2 m. The ength-todiameter ratio, L/d, was taken to be equa to 1. The cinder was made of an auminium ao with eastic moduus E = MPa and Poisson ratio ν =.33. The fuseage was camped at both its ends. (a) Bending mode, H t d (b) Torsiona mode, 8.71 H FIGURE 9. CIRCULAR CROSS-SECTION OF THE FUSELAGE In Tab. 2 the main natura frequencies are shown and CUF modes are compared to a MSC/ mode constructed with CQUAD4 she eements. Both cassica and refined TE modes are given. The resuts b the mode are reported in the ast row. The mode was obtained b discretiing the fuseage cross-section with 2 L9 Lagrange eements. Comparison between MSC/ computationa time and the presented method is aso highighted in Tab. 2. Figure 1 shows the first bending, torsiona and she-ike moda shapes b mode. The foowing statements are worth of carefu stud: 1. Cassica and ower-order TE modes are not abe to describe the dnamic behavior of the fuseage structure. 2. Higher-order TE modes are abe to detect both goba (bending, torsiona) and oca (she-ike) modes of the fuseage. 3. The proposed mode of the thin-waed cinder on detects bending and torsiona natura frequencies. Loca she-ike modes are not correct described. This coud be due to the high distortions that infict LE eements for this particuar probem. Improved resuts can be obtained b increasing the number of LE eements above the fuseage cross-section. (c) She-ike mode, H FIGURE 1. MAIN MODAL SHAPES OF THE FUSELAGE SEC- TION, MODEL Moda anasis of a compete aircraft wing The free vibration anasis of a compete aircraft wing was carried out for the fina assessment. The cross-section of the wing is shown in Fig. 11. The NACA 2415 airfoi was used and two spar webs and four spar caps were added. The airfoi has the chord, c, equa to 1 m. The ength, L, aong the span direction is equa to 6 m. The thickness of the panes is m, whereas the thickness of the spar webs is m. The whoe structure is made of the same isotropic materia of the previous anasis case. The wing was camped at the root. For the present wing structure, two different configurations were considered. Configuration A had no transverse stiffening members. In Configuration B the wing was divided into three equa bas, each separated b 7 Copright c 213 b ASME

8 TABLE 2. NATURAL FREQUENCIES (H) OF THE FUSELAGE SECTION AND COMPUTATIONAL TIME IN PERCENTAGE OF THE REF- ERENCE MODEL Mode I Bending II Bending I She-ike II She-ike I Torsiona II Torsiona DOFs Computationa time MSC/ SHELL % Cassica and Refined TE Beam Modes EBBM % TBM % N = % N = % N = % N = % N = % 2 L % *: not provided b the mode Z X c FIGURE 11. CROSS-SECTION OF THE WING a rib with a thickness of m. Tabe 3 shows the main moda frequencies of both structura configurations of the wing. In this tabe, the resuts obtained through the CUF modes are compared to those from cassica beam theories and to those from SOLID modes. In the ast two rows of Tab. 3, the frequencies of the first two she-ike modes are stated. The foowing considerations hod. 1. The bending modes of the wing are correct detected b both the ower-order and higher-order TE modes. 2. As reveaed b the previous numerica eampes, at east a cubic epansion on the dispacement fied (TE N = 3) is necessar to correct detect the torsiona modes. 3. The modes match the SOLID soutions, in fact, sheike modes can be obtained b means of LE beam eements. 4. The computationa effort of a higher-order beam mode is significant ower than the ones requested b soid modes. To dea with compe structures, such as the one consid- (a) Mode 26 ( H) (b) Mode 54 (246.7 H) FIGURE 12. SHELL-LIKE MODES OF THE WING (CONFIGU- RATION A) EVALUATED WITH THE MODEL ered in this section, the modes were incuded into the commercia software MSC/, which was used to sove the eigenvaue probem through DMAP aters, and MSC/PATRAN 8 Copright c 213 b ASME

9 TABLE 3. GLOBAL AND LOCAL MODAL FREQUENCIES OF THE COMPLETE AIRCRAFT WING Configuration A EBBM TBM N = 1 N = 2 N = 3 SOLID DOFs Goba Modes I Bending I Bending II Bending I Torsiona III Bending Loca Modes I She-ike II She-ike Configuration B DOFs Goba Modes I Bending I Bending II Bending I Torsiona III Bending Loca Modes I She-ike II She-ike Bending ξ : bending mode aong the ξ -ais was used for the post-processing of the mode of the wing. Two she-ike modes evauated b means of the mode are shown in Fig. 12 for Configuration A. CONCLUSIONS This paper has considered and compared eisting methods and recent approaches that epoit one-dimensiona structura theories based on the Unified Formuation, which aows for the straightforward impementation of higher-order anasis without the need of etensive revisions of the mode. The main concusion to be drawn is that the present component-wise anasis appears to the authors to be the most convenient wa, in terms of both accurac and computationa costs, in order to capture the goba and oca mechanica behavior of wing structures. However, particuar attention has to be paid when discretiing structures with ow radii of curvature. Additiona, the present approach aows us to buid FE mathematica modes b on using phsica surfaces. This characteristic of modes is a unique feature that makes this approach advantageous in a CAE/CAD scenario. 9 Copright c 213 b ASME

10 REFERENCES [1] Bruhn, E. F., Anasis and Design of Fight Vehice Structures. Tri-State Offset Compan, Cincinnati, USA. [2] Riveo, R. M., Theor and anasis of fight structures. McGraw-Hi, New York, USA. [3] Carrera, E., 211. Fondamenti su Cacoo di Strutture a Guscio Rinforato per Veicoi Aerospaiai. Levrotto & Bea, Torino, Ita. [4] Cicaa, P., Su cacoo dee strutture a guscio. L Aerotecnica, XXVI(3), pp Part 1 of 4. [5] Goode, W. J., A stressed skin probem. Aircraft Engineering and Aerospace Technoog, 1(1), pp [6] Ebner, H., and Koer, H., Zur berechnung des kraftveraufes in versteiften indershaen. Luftfahrtforschung, 14, pp [7] Ebner, H., and Koer, H., Ueber den kraftverauf in ängs und querversteiften scheiben. Luftfahrtforschung, 15, pp [8] Brogio, L., Introduione di un metodo generae per i cacoo dee strutture a guscio. Istituto poigrafico deo Stato, Roma. Monografie scientifiche di aeronautica n. 1. [9] Koi, M., and Chandrashekhara, K., Finite eement anasis of stiffened aminated pates under transverse oading. Composite Science and Technoog, 56, pp [1] Gangadhara Prust, B., 23. Linear static anasis of hat stiffened aminated shes using finite eements. Finite eements in anasis and design, 39, pp [11] Samanta, A., and Mukhopadha, M., 24. Free vibration anasis of stiffened shes b the finite eement technique. European Journa of Mechanics - A/Soids, 23(1), pp DOI: 1.116/j.euromechso [12] Thinh, T. I., and Khoa, N. N., 28. Free vibration anasis of stiffened aminated pates using a new stiffened eement. Technische Mechanik, 28(3 4), pp [13] Vörös, G. M., 27. Finite eement anasis of stiffened pates. Periodica Potechnica, 51(2), pp [14] Carrera, E., Maiarú, M., and Petroo, M., 212. Component-wise anasis of aminated anisotropic composites. Internationa Journa of Soids and Structures, 49, pp DOI: 1.116/j.ijsostr [15] Carrera, E., Pagani, A., and Petroo, M., 213. Cassica, refined and component-wise theories for static anasis of reinforced-she wing structures. AIAA Journa. In Press. [16] Carrera, E., Pagani, A., and Petroo, M., 213. Component-wise method appied to vibration of wing structures. Journa of Appied Mechanics. In Press. [17] Carrera, E., Giunta, G., and Petroo, M., 211. Beam Structures: Cassica and Advanced Theories. John Wie & Sons. DOI: 1.12/ [18] Carrera, E., and Petroo, M., 212. Refined beam eements with on dispacement variabes and pate/she capabiities. Meccanica, 47(3), pp DOI: 1.17/s [19] Carrera, E., Giunta, G., Nai, P., and Petroo, M., 21. Refined beam eements with arbitrar cross-section geometries. Computers and Structures, 88(5 6), pp DOI: 1.116/j.compstruc [2] Carrera, E., Petroo, M., and Zappino, E., 212. Performance of cuf approach to anae the structura behavior of sender bodies. Journa of Structura Engineering, 138(2), pp DOI: 1.161/(ASCE)ST X.42. [21] Carrera, E., Petroo, M., and Nai, P., 211. Unified formuation appied to free vibrations finite eement anasis of beams with arbitrar section. Shock and Vibrations, 18(3), pp DOI: /SAV [22] Carrera, E., Petroo, M., and Vareo, A., 212. Advanced beam formuations for free vibration anasis of conventiona and joined wings. Journa of Aerospace Engineering, 25(2), pp DOI: 1.161/(ASCE)AS [23] Carrera, E., and Petroo, M., 212. Refined onedimensiona formuations for aminated structure anasis. AIAA Journa, 5(1), pp DOI: /1.J [24] Washiu, K., Variationa Methods in Easticit and Pasticit. Pergamon, Oford. [25] Oñate, E., 29. Structura Anasis with the Finite Eement Method: Linear Statics, Voume 1. Springer. [26] Bathe, K. J., Finite eement procedure. Prentice ha. 1 Copright c 213 b ASME