Refined beam elements with only displacement variables and plate/shell capabilities

Transcription

1 Meccanica (2012) 47: DOI /s Refined beam eements with ony dispacement variabes and pate/she capabiities Erasmo Carrera Marco Petroo Received: 18 June 2010 / Accepted: 14 Juy 2011 / Pubished onine: 25 August 2011 Springer Science+Business Media B.V Abstract This paper proposes a refined beam formuation with dispacement variabes ony. Lagrangetype poynomias, in fact, are used to interpoate the dispacement fied over the beam cross-section. Three- (L3), four- (L4), and nine-point (L9) poynomias are considered which ead to inear, quasi-inear (biinear), and quadratic dispacement fied approximations over the beam cross-section. Finite eements are obtained by empoying the principe of virtua dispacements in conjunction with the Unified Formuation (UF). With UF appication the finite eement matrices and vectors are expressed in terms of fundamenta nucei whose forms do not depend on the assumptions made (L3, L4, or L9). Additiona refined beam modes are impemented by introducing further discretizations over the beam cross-section in terms of the impemented L3, L4, and L9 eements. A number of numerica probems have been soved and compared with resuts given by cassica beam theories (Euer-Bernoui and Timoshenko), refined beam theories based on the use of Tayor-type expansions in the neighborhood of the beam axis, and soid eement modes from commercia codes. Poisson ocking correction is anayzed. Appi- E. Carrera ( ) M. Petroo Department of Aeronautic and Space Engineering, Poitecnico di Torino, Corso Duca degi Abruzzi 24, Torino, Itay e-mai: erasmo.carrera@poito.it M. Petroo e-mai: marco.petroo@poito.it cations to compact, thin-waed open/cosed sections are discussed. The investigation conducted shows that: (1) the proposed formuation is very suitabe to increase accuracy when ocaized effects have to be detected; (2) it eads to she-ike resuts in case of thinwaed cosed cross-section anaysis as we as in open cross-section anaysis; (3) it aows us to modify the boundary conditions over the cross-section easiy by introducing ocaized constraints; (4) it aows us to introduce geometrica boundary conditions aong the beam axis which ead to pate/she-ike cases. Keywords Finite eement method Higher-order beam She-ike capabiities Carrera unified formuation 1 Introduction Beam theories are important toos for structura anaysts. Interest in beam modes is mainy due to their simpicity and their ow computationa costs when compared to 2D (pate/she) or 3D (soid) modes. The use of a beam mode is of particuar interest with sender bodies such as aircraft wings, heicopter rotor bades, and sender bridges. The cassica and bestknown beam theories are those by Euer [16] and hereinafter referred as EBBM, and Timoshenko [30, 31] and hereinafter referred as TBM. The former does not account for transverse shear deformations. The atter foresees a uniform shear distribution aong the

2 538 Meccanica (2012) 47: cross-section of the beam. These modes work propery when sender compact homogeneous structures are considered in bending. Conversey, the anaysis of deep, thin-waed, open beams requires more sophisticated methods, see [22]. Higher-order beam theories that enhance the dispacement fieds of EBBM and TBM can be deveoped to overcome these imits. Severa refined beam modes can be found in open iterature. Attention is herein given to works which are mainy devoted to the anaysis of isotropic, thin-waed, and open cross-section beams. A comprehensive review of beam (incuding pate) theories for vibration, wave propagations, bucking and post-bucking was presented by Kapania and Raciti [20, 21]. An overview of existing beam finite eements was made by Reddy [25], where beam eements based on cassica and higher-order theories were described, and the probems of shear ocking and ocking-free beam eements were discussed. Vinayak et a. [33] used a higher-order beam eement based on the Lo-Christensen-Wu theory to study isotropic and composite, thin and deep compact beams, where axia and transverse dispacements were modeed via cubic and paraboic expansions, respectivey. Severa works reated to the shear correction factor evauation were presented by Gruttmann et a. [18], Gruttmann and Wagner [17], and Wagner and Gruttmann [34]. Here, severa structura probems were addressed: torsiona and fexura shearing stresses in prismatic beams, arbitrary shaped cross-sections, wide and thin-waed structures, and the infuence of Poisson s ratio on the shear correction factor was highighted. A distortiona theory for thin-waed beams was proposed by Jönsson [19], where the distortiona dispacement mode was embedded in the cassica kinematic assumptions of Vasov theory, and the roe of cross-section distortion was investigated for different open and cosed cross-sections undergoing static oads. Petroito [24] and Eisenberger [13] deat with the exact stiffness matrix anaysis of a high-order beam eement: the refined dispacement fied was based on a cubic variation of the axia dispacement over the cross-section of the beam whereas the atera dispacement was kept constant, and comparisons with cassica modes were made, whie the importance of higher order terms in case of short beams was underined. Works by Dinis et a. [12] and Sivestre [29] deat with the bucking anaysis of thin waed open/cosed cross-section beams: the Generaized Beam Theory (GBT) was used to impement beam theories accounting for the in-pane cross-section deformations, and she-type resuts were obtained by using appropriate cross-section shape functions describing the beam dispacement fied; the choice of those functions depends on the geometry of the considered structures. GBT was aso used by Rendek and Baáž [27], for the static anaysis of thin waed beams and comparisons with experimenta resuts. E Fatmi [14, 15] proposed a beam theory with a non-uniform warping distribution. He adopted a kinematic mode that keeps the cross-section shape constant; warping effects were investigated on compact and hoow beams with cosed and open cross-sections, and particuar attention was given to the anaysis of shear and axia stresses. The proposed refined theory proved usefu especiay with short open cross-section beams. Another interesting work deaing with non-uniform warping distributions was presented by Sadé [28]; a detaied overview of the warping probem was given, and static and bucking anayses were performed. A different approach to refine a structura mode is based on the asymptotic method where a characteristic parameter (e.g. the cross-section thickness for a beam) is expoited to buid an asymptotic series, and those terms which exhibit the same order of magnitude as the parameter when it vanishes are retained. Significant exampes of asymptotic buit beam modes were given by Yu et a. [36] and Yu and Hodges [35]. The above works show a cear interest in investigating refined beam theories. The present work fas in the framework of the Carrera Unified Formuation, CUF, which has been deveoped during the ast decade by the first author and his co-workers. CUF was initiay devoted to the deveopment of refined pate and she theories, see [2, 3]. Recenty, it has been extended to beam modeing by Carrera and Giunta [6]. CUF is a hierarchica formuation which considers the order of the theory as an input of the anaysis. This permits us to dea with a wide variety of probems with no need of ad hoc formuations. Non-cassica effects (e.g. warping, in-pane deformations, shear effects, bending-torsion couping) are accounted for by opportuney increasing the order of the adopted mode. Finite eement formuation is adopted to dea with arbitrary geometries, boundary conditions, and oadings. Previous works have been based on the use of Tayor type poynomias to define the dispacement fied above the beam cross-section; each fied consists of

3 Meccanica (2012) 47: a direct extension to higher-order expansions of the Timoshenko beam theory. Static anayses, see [8, 11], showed the strength of CUF in deaing with warping, in-pane deformations, and shear effects. Free vibration anayses, see [9, 10], underined the possibiity of detecting she-ike vibration modes by means of refined beam theories with no need of more cumbersome 2D or 3D modes. An effectiveness anaysis by Carrera and Petroo [7] was aso conducted to highight the roe of each higher-order term in given structura probems. The use of Tayor-type expansions has some intrinsic imitations: the introduced variabes have a mathematica meaning (derivatives at the beam axes); higher order terms cannot have a oca meaning, they can have cross-section properties ony; the extension to arge rotation formuation coud experience difficuties. To overcome these probems, this work proposes new beam theories whose cross-section dispacement fied is described by Lagrange-type poynomias. The choice of this kind of expansion functions eads us to have dispacement variabes ony. This aspect is of particuar interest because: 1. each variabe has a precise physica meaning (the probem unknowns are ony transationa dispacements); 2. unknown variabes can be put in fixed zones (subdomains) of the cross-section area (e.g. cose to oadings); 3. geometrica boundary conditions can be appied in sub-domains of the cross-section (and not ony to the whoe cross-section); 4. geometrica boundary conditions can aso be appied aong the beam-axis; 5. cross-sections can be divided into further beam sections and easiy assembed since the dispacements at each boundary are used as probem unknowns; 6. the extension to geometricay non-inear probems appears more suitabe than in the case of Tayortype higher-order theories. Three- (L3), four- (L4), and nine-point (L9) poynomias are considered in the framework of CUF; this eads to inear, quasi-inear (biinear), and quadratic dispacement fied approximations over the beam cross-section. More refined beam modes are impemented by introducing further discretizations over the beam cross-section in terms of impemented eements. A number of significant probems are treated. Compact, thin-waed, open cross-section, and she-ike Fig. 1 Coordinate frame of the beam mode structures are anayzed. Homogenous isotropic modes are used. The possibiities of deaing with ocaized oadings and to treat in a new manner the geometrica boundary conditions are shown. The paper is organized as foows: brief descriptions of the adopted beam theories and the finite eement formuation are furnished in Sects. 2 and 3; structura probems addressed, together with resuts and discussion, are provided in Sect. 4. The main concusions are then outined in Sect Preiminaries The adopted coordinate frame is presented in Fig. 1. The beam boundaries over y are 0 y L. The dispacement vector is: u(x,y,z)= { u x u y u z } T (1) The superscript T represents the transposition operator. Stress, σ, and strain, ɛ, components are grouped as foows: σ p = { } T σ zz σ xx σ zx, ɛp = { } T ɛ zz ɛ xx ɛ zx σ n = { } T σ zy σ xy σ yy, ɛn = { } T (2) ɛ zy ɛ xy ɛ yy The subscript n stands for terms ying on the crosssection, whie p stands for terms ying on panes which are orthogona to. Linear strain-dispacement reations are used: ɛ p = D p u ɛ n = D n u = (D n + D ny )u (3)

4 540 Meccanica (2012) 47: with: 0 0 z D p = x 0 0, D n = 0 x 0 z 0 0 x z 0 (4) 0 y 0 D ny = y y Tabe 1 L3 cross-section eement point natura coordinates Point r τ s τ The Hooke aw is expoited: σ = Cɛ (5) According to (2), the previous equation becomes: σ p = C pp ɛ p + C pn ɛ n σ n = C np ɛ p + C nn ɛ n (6) In the case of isotropic materia the matrices C pp, C nn, C pn, and C np are: C 11 C 12 0 C pp = C 12 C C 66 C C nn = 0 C 44 0 (7) 0 0 C C 13 C pn = C T np = 0 0 C For the sake of brevity, the dependence of coefficients [ C] ij versus Young s moduus and Poisson s ratio is not reported here. It can be found in the books by Tsai [32] or Reddy [26]. 3 Unified FE formuation In the framework of the Carrera Unified Formuation (CUF), the dispacement fied is the expansion of generic functions, F τ : u = F τ u τ, τ = 1, 2,...,M (8) where F τ vary above the cross-section. u τ is the dispacement vector and M stands for the number of Fig. 2 Cross-section eements in actua geometry terms of the expansion. According to the Einstein notation, the repeated subscript, τ, indicates summation. Tayor-type expansions have been expoited in previous works by Carrera and Giunta [6], Carrera and Petroo [7], Carrera et a. [8 11]. The Euer-Bernoui (EBBM) and Timoshenko (TBM) cassica theories are derived from the inear Tayor-type expansion. Lagrange poynomias are herein used to describe the cross-section dispacement fied. Three-, L3, four-, L4, and nine-point, L9, poynomias are adopted. L3 poynomias are defined on a trianguar domain which is identified by three points. These points define the eement that is used to mode the dispacement fied above the cross-section. Simiary, L4 and L9 crosssection eements are defined on quadriatera domains. The isoparametric formuation is expoited. In the case of the L3 eement, the interpoation functions are given by [23]: F 1 = 1 r s, F 2 = r, F 3 = s (9)

5 Meccanica (2012) 47: Tabe 2 L4 cross-section eement point natura coordinates Point r τ s τ Tabe 3 L9 cross-section eement point natura coordinates Point r τ s τ where r and s beong to the trianguar domain defined by the points in Tabe 1. Figure 2a shows the point ocations in actua coordinates. The L4 eement interpoation functions are given by: F τ = 1 4 (1 + rr τ )(1 + ss τ ), τ = 1, 2, 3, 4 (10) where r and s vary from 1 to+1. Figure 2b shows the point ocations and Tabe 2 reports the point natura coordinates. In the case of a L9 eement the interpoation functions are given by: F τ = 1 4 (r2 + rr τ )(s 2 + ss τ ), τ = 1, 3, 5, 7 F τ = 1 2 s2 τ (s2 ss τ )(1 r 2 ) r2 τ (r2 rr τ )(1 s 2 ) τ = 2, 4, 6, 8 F τ = (1 r 2 )(1 s 2 ), τ = 9 (11) where r and s from 1 to+1. Figure 2c showsthe point ocations and Tabe 3 reports the point natura coordinates. The dispacement fied given by an L4 eement is: Fig. 3 Two assembed L9 eements u x = F 1 u x1 + F 2 u x2 + F 3 u x3 + F 4 u x4 u y = F 1 u y1 + F 2 u y2 + F 3 u y3 + F 4 u y4 u z = F 1 u z1 + F 2 u z2 + F 3 u z3 + F 4 u z4 (12) where u x1,...,u z4 are the dispacement variabes of the probem and they represent the transationa dispacement components of each of the four points of the L4 eement. The cross-section can be discretized by means of severa L-eements. Figure 3 shows the assemby of 2 L9 which share a common edge and three points. The discretization aong the beam axis is conducted via a cassica finite eement approach. The dispacement vector is given by: u = N i F τ q τi (13) where N i stands for the shape functions and q τi for the noda dispacement vector: q τi = { q uxτi q uyτi q uzτi } T (14) For the sake of brevity, the shape functions are not reported here. They can be found in many books, for instance in Bathe [1]. Eements with four nodes (B4) are herein formuated, that is, a cubic approximation aong the y axis is adopted. It has to be highighted that the adopted cross-section dispacement fied mode defines the beam theory. It is therefore possibe to dea with inear (L3), biinear (L4), and quadratic (L9) beam theories. Further refinements can be obtained by adding cross-section eements, in this case the beam mode wi be defined by the number of cross-section eements used. The choice of the cross-section discretization (i.e. the choice of the type, the number and the distribution of cross-section eements) is competey independent of the choice of the beam finite eement to be used aong the beam axis.

6 542 Meccanica (2012) 47: The present formuation has to be considered as an + C 44 F τ,x F s,x d N i N j dy (17) 1D mode since the unknowns of the probem, i.e. the noda unknowns, vary aong the beam axis whereas the dispacement fied of the beam is axiomaticay modeed above the cross-section domain. The introduction + C 33 F τ F s d N i,y N j,y dy of the Lagrange-ike discretization above the crosssection aows us to dea with ocay refinabe 1D Kyz ij τ s = C 55 F τ,z F s d N i N j,y dy modes having ony dispacement variabes. This modeing choice represents the novety of the present work 13 F τ F s,z d N i,y N j dy + C since Tayor-ike poynomias were expoited previousy above the cross-section domain. Kzx ij τ s = C 12 F τ,z F s,x d N i N j dy The stiffness matrix of the eements and the externa oadings, which are consistent with the mode, are + C 66 F τ,x F s,z d N i N j dy obtained via the Principe of Virtua Dispacements: δl int = (δɛ T p σ p + δɛ T n σ n)dv = δl ext (15) Kzy ij τ s = C 13 F τ,z F s d N i N j,y dy V where L int stands for the strain energy, and L ext is the + C 55 F τ F s,z d N i,y N j dy work of the externa oadings. δ stands for the virtua variation. The virtua variation of the strain energy is Kzz ij τ s = C rewritten using (3), (6) and (13): 11 F τ,z F s,z d N i N j dy δl int = δq T τi Kij τ s q sj (16) + C 66 F τ,x F s,x d N i N j dy where K ij τ s is the stiffness matrix in the form of the fundamenta nuceus. Its components are: + C 55 F τ F s d N i,y N j,y dy Kxx ij τ s = C 22 F τ,x F s,x d N i N j dy It shoud be noted that no assumptions on the approximation order have been made. It is therefore possibe + C 66 F τ,z F s,z d N i N j dy to obtain refined beam modes without changing the forma expression of the nuceus components. This is the key-point of CUF which permits, with ony nine + C 44 F τ F s d N i,y N j,y dy FORTRAN statements, to impement any-order beam theories. The shear ocking is corrected through the seective integration (see [1]). The ine and surface inte- Kxy ij τ s = C 23 F τ,x F s d N i N j,y dy gra computation is numericay performed by means + C 44 F τ F s,x d N i,y N j dy of the Gauss method. The assemby procedure of the Lagrange-type eements is anaogous to the one foowed in the case of 2D eements. The procedure key- Kxz ij τ s = C 12 F τ,x F s,z d N i N j dy points are briefy isted: 1. The fundamenta nuceus is expoited to compute + C 66 F τ,z F s,x d N i N j dy the stiffness matrix of each cross-section eement Kyx ij τ s of a structura node. If an L4 eement is considered, = C 44 F τ,x F s d N i N j,y dy this matrix wi have terms. 2. The stiffness matrix of the structura node is then + C 23 F τ F s,x d N i,y N j dy assembed by considering a the cross-section eements and expoiting their connectivity. Kyy ij τ s 3. The stiffness matrix of each beam eement is computed and assembed in the goba stiffness = C 55 F τ,z F s,z d N i N j dy matrix.

7 Meccanica (2012) 47: The variationay coherent oadings vector is derived in the case of a generic concentrated oad P: P = { P ux P uy P uz } T (18) Any other oading condition can be simiary treated. The virtua work due to P is: δl ext = Pδu T (19) The virtua variation of u in the framework of CUF is: δl ext = F τ Pδu T τ (20) By introducing the noda dispacements and the shape functions, the previous equation becomes: δl ext = F τ N i Pδq T τi (21) This ast equation permits us to identify the components of the nuceus which have to be oaded, that is, it eads to the proper assembing of the oading vector by detecting the dispacement variabes that have to be oaded. The imposition of constraints can be carried out by considering each of the three degrees of freedom of cross-section eement points independenty. In other words, a constraint can be either imposed on the whoe cross-section or on an arbitrary number of cross-section points. 4 Resuts and discussion The proposed beam formuation is herein evauated and compared with different modes: cassica beam theories, refined beam eements based on Tayor-type assumptions, and, in some cases, soid eements of the commercia code MSC Nastran. MSC Nastran modes are based on HEX8 eements having approximativey a unitary aspect ratio. Various homogeneous crosssection geometries made of isotropic materias are anayzed. The materia data is: the Young moduus, E,is equa to 75 [GPa]; the Poisson ratio, ν, is equa to Particuar attention is given to probems that show the capabiity of the present beam eement to dea with ocaized boundary conditions over the cross section as we as with pate/she-ike anayses. A the graphic resuts are opportuney scaed. Fig. 4 Rectanguar cross-section Tabe 4 Effect of the number of eements on u z for different beam modes. L/h = 100. Compact square cross-section No. Eem u z 10 2 [m], u zb 10 2 = [m] EBBM TBM N = N = L L Square and rectanguar cross-sections A cantievered beam is used for the preiminary assessment of the present beam mode. The geometry of the cross-section is shown in Fig. 4. The height of the cross-section, h, is0.2 [m], with b as high as h. Two senderness ratios, L/h, are considered: 100 and 10. Sender and moderatey short beams are considered in order to highight the importance of refined modes in case of short structures. A point oad, F z, is appied at [0, L, 0]. The magnitude of F z is equa to 50 [N]. The oaded point vertica dispacement, u z, is evauated. The Euer-Bernoui theory is used for comparison purposes, u zb = F zl 3 3EI, where I is the cross-section moment of inertia. Tabe 4 shows the dispacement vaues for different meshes and beam modes in the case of a sender beam (L/h = 100). Cassica theories, EBBM and TBM, are accounted for (2nd and 3rd rows). Resuts by Tayor-type inear and paraboic modes, N = 1 and N = 2, are reported in the 4th and 5th rows. Lagrange poynomia beam mode resuts are shown in the ast two rows. Four-point, L4, and nine-point, L9, cross-section eements are considered. Tabe 5 reports the resuts of a moderatey

8 544 Meccanica (2012) 47: Tabe 5 Effect of the number of eements on u z for different beam modes. L/h = 10. Compact square cross-section No. Eem u z 10 5 [m], u zb 10 5 = [m] EBBM TBM N = N = N = N = L L Tabe 6 Effect of the number of L4 eements on u z. L/h = 100. Compact square cross-section No. Eem. 1 L4 2 1L4 1 2L4 2 2L4 u z 10 2 [m], u zb 10 2 = [m] thick beam (L/h = 10). Mutipe L4 eements above the cross-section are used in Tabe 6. Each coumn refers to a different L4 discretization. The second coumn indicates the resuts obtained by using one L4, the third is reated to the case of two L4 aong the x-direction, the fourth to the case of two L4 aong the z-direction, and the fifth to the case of two L4 eements aong both cross-section directions. The resuts from a these anayses suggest the foowing considerations. 1. The use of an L9 permits us to obtain good accuracy. This eement gives resuts which are equivaent to those of a Tayor-type paraboic, N = 2, mode. This means that the cubic and quartic poynomia terms (s 2 r, sr 2, and s 2 r 2 ) do not pay a very significant roe in the considered probem, in fact, these terms are not considered in the Tayor case N = L4 have sower convergence rates than L9. However, the subdivision of the cross-section in more than one L4 is very effective. 3. The improvement given by the cross-section discretization in L4 eements is reated to the tota number of eements as we as their distribution Tabe 7 Roe of the biinear term and of the Poisson ocking correction on u z. L/h = 100. Compact square cross-section Correction Tayor biinear 1 L4 u z 10 2 [m], u zb 10 2 = [m] Activated Deactivated above the cross-section. The refinement aong the z-direction is more effective than the one aong the x-direction when a F z oad is appied. The L4 is characterized by the presence of a biinear term in the dispacement fied expression. This term is responsibe for the sow convergence rate. Tabe 7 presents the investigation on the roe of the biinear term together with the Poisson ocking correction. Locking is corrected as in Carrera and Brischetto [4, 5]. The foowing Tayor-type mode is used for comparison purposes: u x = u x1 + xu x2 + zu x3 + xzu x5 u y = u y1 + xu y2 + zu y3 + xzu y5 u z = u z1 + xu z2 + zu z3 + xzu z5 (22) This mode has been obtained as in Carrera and Petroo [7]. The foowing considerations arise from Tabe The L4 is equivaent to the biinear Tayor expansion. 2. The Poisson ocking correction corrupts the effectiveness of both modes because of the presence of the biinear term, but, at the same time, the biinear term is not enough to eiminate the Poisson ocking. 3. As a genera remark it can be stated that a beam mode based on a biinear dispacement fied shoud not be used because of its convergence issues. Linear modes (e.g. TBM or N = 1) with a Poisson ocking correction or at east a second-order mode shoud be preferred to predict the bending behavior of a compact beam. An equiatera trianguar cross-section is now considered to conduct a further investigation on the roe of the Poisson ocking and its correction. Figure 5 shows the geometry of the triange, b is equa to 1 [m] and L/b is as high as 20. A vertica force, F z, is appied at the center point. F z is equa to 30 [N]. Two crosssection discretizations are used: 1 L3 and 2 L3. The

9 Meccanica (2012) 47: Fig. 5 Trianguar cross-section Fig L9 discretization of the rectanguar cross-section Tabe 9 Dispacement and stress vaues of the rectanguar beam SOLID 1 L9 3 3L9 [x, y, z] DOF s 4941 DOF s 8967 DOF s Fig. 6 2 L3 discretization of the trianguar cross-section Tabe 8 Roe of the Poisson ocking correction on u z in case of L3 eements. Compact trianguar cross-section u z 10 7 [m] [0, L, h/2] σ yy 10 4 [Pa] [0, L/10, +h/2] σ yz 10 2 [Pa] [b/2, L/10, 0] Correction 1 L3 2 L3 u z 10 5 [m], u zb 10 5 = [m] Activated Deactivated atter one is shown in Fig. 6. Tabe 8 shows the vertica center point dispacement vaues for both beam modes and the effects of the Poisson ocking correction. The resut anaysis suggests the foowing. 1. The Poisson ocking correction is beneficia in the case of 1 L3 because a inear description of the cross-section dispacement fied is given. This confirms what has been previousy mentioned about the roe of the biinear term. 2. The correction is detrimenta in the case of 2 L3 because the dispacement fied is step-wise inear, therefore is overa higher than the first-order. However, more than 2 L3 eements are needed to nuify the Poisson ocking. That expains why Poisson ocking correction is not effective in L4 beam theories. A rectanguar cantievered beam is now considered. The geometry of the cross-section is shown in Fig. 4. The height of the cross-section, h, is0.1 [m], with b as high as h/4 and L/h equa to six. A point oad, F z, is appied at [0, L, h/2]. The magnitude of F z is equa to 1 [N]. Two cross-section L9 distributions are adopted: a 1 L9 and a 3 3 L9. The atter is shown in Fig. 7. A 20 B4 mesh is used aong the y-direction. Tabe 9 presents vertica dispacements and stress vaues in different points; comparisons with a soid mode are reported together with the computationa cost of each mode. Shear stress distributions above the crosssectionareshowninfig.8. These resuts suggest the foowing. 1. A genera good match is found between the present formuation and the soid mode soution. A sight difference is observed in the vertica dispacement because the oading point is considered where severe oca effects undergo. 2. The cross-section discretization refinement is an effective method that eads to the 3D soid soution.

10 546 Meccanica (2012) 47: Fig. 8 σ yz distributions above the rectanguar cross-section Fig. 10 Cross-section eement distributions for the hoow square beam Fig. 9 Hoow square cross-section Shear stress distributions are particuary improved by the adoption of a refined cross-section mode. 3. The present formuation requires significanty ower computationa efforts than a soid mode. 4.2 Hoow cross-section A hoow square cross-section is considered. Both ends are camped. The cross-section geometry is showninfig.9. The ength-to-height ratio, L/h, is equa to 20. The height-to-thickness ratio, h/t, is equa to 10 with h as high as 1 [m]. A point oad, F z,is first considered and appied at [0, L/2, h/2]. Its magnitude is equa to 1 [N]. Three cross-section discretizations have been used, as shown in Fig. 10. The 8 L9 distribution is symmetric, whereas the 9 L9, and the 11 L9 ones have been refined in the proximity of the oaded point. Tabe 10 shows the dispacement, u z, of the oaded point together with the indication of the number of degrees of freedom of each considered mode. The first row shows the soid mode resut obtained by buiding a FE mode in MSC Nastran. The increasing order Tayor-type modes are considered in rows 2nd to 5th. The present Lagrange mode resuts are shown in the ast three rows. The foowing statements hod. 1. Refined beam theories aows us to obtain the soid mode resuts. 2. The computationa cost of the beam modes is significanty smaer than the one requested by the 3D mode.

11 Meccanica (2012) 47: Tabe 10 u z of the oaded point of the hoow square beam DOF s u z 10 8 [m] SOLID Tayor EBBM N = N = N = Lagrange 8 L9, Fig. 2a L9, Fig. 2b L9, Fig. 2c Tabe 11 Effect of the cross-section eement distribution on the dispacement of the oaded point. Hoow square beam DOF s u ztop 10 9 [m] u zbot 10 9 [m] SOLID Tayor EBBM N = N = N = Lagrange 8 L9, Fig. 2a L9, Fig. 2c An appropriate distribution of the L9 eements above the cross-section is effective in improving the accuracy of the soution. In other words, the oca refinement of a beam mode is possibe and eads to the adaptation of the Lagrange point distribution to the given probem. 4. Lagrange-based modes are abe to detect a more accurate soution than Tayor-based ones with reduced computationa costs. This is due to the possibiity of ocay refining the beam mode which is offered by the use of Lagrange poynomias, whereas a Tayor mode uniformy spreads the refinement above the cross-section with no distinction between owy and highy deformed zones. A second oad case is considered in order to better highight the oca refinement capabiities of the present beam formuation. Two point oadings (F z =±1 [N]) are appied at [0, L/2, h/2]. The adopted L9 distributions are those in Figs. 10a and 10c, that is, symmetric and asymmetric distributions are invoved. The atter has a refined distribution just in the proximity of the bottom side oad point. Tabe 11 shows the dispacements of the two oaded point u ztop and u zbot, respectivey. Soid, as we as Tayor-type modes, are used for comparison purposes. Figure 11 shows the deformed cross-section for each of the considered L9 eement distributions. The foowing considerations are highighted by this ast exampe. 1. Due to the symmetry of the geometry and of the oad, the oaded points shoud be affected by the same vertica dispacements (in magnitude). This resut is obtained in a the considered cases uness the asymmetric L9 distribution is adopted. The o- Fig. 11 Effect of the cross-section eement distribution on the dispacement fied. Hoow square beam cay refined mode eads to higher vaues for the dispacements ony in the proximity of the refinement. 2. The soution improvement offered by Lagrangebased modes is higher and computationay cheaper than the one offered by Tayor-type modes. 3. It has been shown that cassica beam modes, such as EBBM, are not capabe of detecting the dispacements of the oaded points at a. 4.3 Open cross-sections A cantievered C-section beam is considered. The cross-section geometry is shown in Fig. 12. The ength-to-height ratio, L/h, is equa to 20. The heightto-thickness ratio, h/t, is as high as 10 with h and b 2

12 548 Meccanica (2012) 47: Tabe 12 Vertica dispacement, u z, of the top oaded point. C-section beam DOF s u z 10 8 [m] SOLID Tayor EBBM N = N = N = Lagrange 6 L9, Fig. 13a L9, Fig. 13b Fig. 12 C-section geometry presents refinements in the proximity of the oading points. Tabe 12 shows the vertica dispacement, u z, of the point at [0, L, +0.4]. Soid modes as we as Tayor-type beam modes are considered together with the presentbeam formuation. Figure14 show the free tip deformed cross-section for both the adopted L9 distributions. The soid mode soution is reported as we. The foowing statements hod. 1. The 9 L9 mode perfecty detects the soid soution with a significant reduction of the computationa cost. 2. Tayor-type modes require higher than eeventhorder expansions to match the soid mode soution, consequenty, the difference of computationa cost between Tayor- and Lagrange-based beam modes appears to be higher in the case of open, as opposed to cosed cross-sections. 3. As seen previousy, the cassica mode is totay inadequate to detect the dispacement fied of the considered structura probem. Fig. 13 Cross-section L9 distributions for the C-section beam equa to 1 [m], and b 1 as high as b 2 /2. Two point oads are appied at [0, L, ±0.4], and their magnitudes are as high as 1 [N]. Two L9 distributions are adopted and shown in Fig. 13. The 9 L9 distribution A second oading condition is now considered: a fexura-torsiona oad is obtained by means of a point force appied at [b 1, L, h/2], its magnitude is equa to 1 [N]. In this case, two L/h vaues are considered: 20 and 10. The L9 distribution shown in Fig. 13b is adopted. Dispacement and stress vaues at different ocations are presented in Tabes 13 and 14, whereas stress distributions above the cross-section and the 3D deformed configuration are shown in Figs. 15, 16, and 17. These resuts suggest what foows. 1. The fexura-torsiona behavior of a moderatey short open cross-section beam is correcty predicted by the present formuation.

13 Meccanica (2012) 47: Tabe 14 Dispacements and stresses of the C-beam, L/h = 10 SOLID 9 L9 [x, y, z] DOF s 5301 DOF s u z 10 7 [m] [ b 2 /2, L, +h/2] σ yy 10 2 [Pa] [b 1, L/10, +h/2] σ yz 10 2 [Pa] [b 1, L, h/2] σ yz 10 1 [Pa] [0.4, L/10, 0] Fig. 14 Deformed C-sections for different L9 distributions Tabe 13 Dispacements and stresses of the C-beam, L/h = 20 SOLID 9 L9 [x, y, z] DOF s 5301 DOF s u z 10 6 [m] [ b 2 /2, L, +h/2] σ yy 10 2 [Pa] [b 1, L/10, +h/2] σ xy 10 2 [Pa] [b 1, L, h/2] σ yz 10 1 [Pa] [0.4, L/10, 0] 2. As far as stress distributions are considered, a good match with the soid mode soution was found both for axia and shear components. Fig. 15 Stress distributions above the C cross-section, L/h = 20 An open square cross-section is now considered. The cross-section geometry is shown in Fig. 18. The dimensions are the same of those seen in Sect Two opposite unit point oads, ±F x, are appied at [0, L, 0.45]. Three L9 distributions are adopted as shown in Fig. 19. Tabe 15 reports the horizonta dispacement of the right-hand side oaded point which undergoes a positive horizonta force. A soid mode is used to vaidate the resuts. The free-tip deformed crosssection is shown in Fig. 20. A the considered L9 distributions together with the soid mode soution are reported. Figure 21 shows the 3D deformed configuration of the considered structure. The anaysis of

14 550 Meccanica (2012) 47: Fig. 16 3D deformed configuration of the C-section beam, L/h = 10 Fig. 18 Open square cross-section Fig. 17 Stress distributions above the C cross-section, L/h = 10 the open hoow square beam highights the foowing considerations. 1. The Lagrange-based beam mode is abe to dea with cut cross-sections. 2. This type of probem cannot be anayzed with Tayor-type beam modes since the appication of two opposite forces at the same point woud impy nu dispacements. 3. The most appropriate refined L9 distribution does not necessariy ie in the proximity of oad points. Fig. 19 Cross-section L9 distributions for the hoow square beam In this case, the most effective refinement was the one paced above the vertica braces of the crosssection which undergo severe bending deformation.

15 Meccanica (2012) 47: Tabe 15 Horizonta dispacement, u x,at[0,l, h/2]. Open hoow square beam DOF s u x 10 8 [m] SOLID L9, Fig. 20a L9 a,fig.20b L9 b,fig.20c Tabe 16 Vertica dispacement, u z,at[0,l/2, 0] of the rectanguar cross-section beam with new constraints DOF s u z 10 7 [m] SOLID L9, Fig. 20a L9 a,fig.20b Locaized constraints over the cross-section The present Lagrange-based beam formuation offers the important possibiity of deaing with constraints that cannot be considered within cassica and refined beam theories that make use of Tayor-type expansions. Beam mode constraints usuay act above the whoe cross-section as shown in Fig. 22a (the beam ongitudina axis coincides with the y-axis). In the framework of the present approach, each of the three degrees of freedom of every Lagrange point of the beam can be constrained independenty. This means that the cross-section can be partiay constrained. Figure 22b shows a possibe structura probem that can be faced where ony the atera edges of the crosssection are camped. Figure 23 shows the x z view in the case of a rectanguar cross-section. A compact rectanguar beam is first considered. The cross-section geometry and the cross-section constraint distribution is shown in Figs. 4 and 23, respectivey. The ength-to-height ratio, L/h, is equa to 100 with b/h as high as 10 and h equa to 0.01 [m]. A set of 21 unitary point oads is appied aong the mid-span cross-section at z = h/2 with constant x-steps starting from the edge of the cross-section. Two L9 distributions are adopted as shown in Fig. 24. Tabe 16 presents the center-point vertica dispacement, u z,of the considered beam modes and that of soid eements. The deformed mid-span cross-section is shown in Fig. 25. Fig. 20 Deformed cross-sections of the hoow square beam A circuar arch cross-section beam is then anayzed to dea with a she-ike structure. The cross-section geometry and the constraint distribution is shown in Fig. 26. The ength of the beam, L, is equa to 2 [m]. Outer, r 1, and inner, r 2, radii are equa to 1 and 0.9 [m], respectivey. The ange of the arch, θ, is equa

16 552 Meccanica (2012) 47: Fig. 21 3D deformed configuration of the hoow square beam. 11L9 b Fig. 24 Rectanguar cross-section L9 distributions Tabe 17 Vertica dispacement, u z, at the externa surface of the arch cross-section beam, L = L/2, θ = θ/2 DOF s u z [m] SOLID L9 a,fig.20b Tabe 18 Dispacement of the oaded point of the C-section beam DOF s u z 10 8 [m] SOLID L9 a,fig.20b Fig. 22 Comparison of cassica and new constraint imposition approaches Fig. 23 Boundary conditions above the rectanguar cross-section to π/4 [rad]. Three unitary point oads are appied at y = 0, y = L/2, and y = L. Each oad acts aong the radia direction (from the inner to the outer direc- tion). The poar coordinates of the oading points are [r 2, θ/2]. The L9 cross-section discretization is shown in Fig. 27. Tabe 17 shows the vertica dispacement, u z, of a point of the mid-span cross-section. The soid mode soution is aso reported. Figures 28 and 29 show the 2D and 3D deformed configurations, respectivey. The C-section beam is reconsidered to give a fina assessment of this paper. The geometry is as in Sect Constraints are distributed aong the bottom portions of the free-tip cross-sections as shown in Fig. 30. Two unitary point oads, F z, are appied at [0,0,0.4] and [0, L,0.4], respectivey. Both forces act aong the negative direction. The L9 cross-section distribution is shown in Fig. 31. The oaded point vertica dispacement, u z, is reported in Tabe 18 and com-

17 Meccanica (2012) 47: Fig. 26 Circuar arch cross-section Fig. 25 Mid-span deformed rectanguar cross-section for different L9 distributions and comparison with a soid eement mode Fig. 27 L9 distribution above the arch cross-section, 12 L9 pared with the vaue obtained from the soid mode. Figures 32 and 33 show 2D and 3D deformed configurations, respectivey. The foowing considerations can be made. 1. The resuts are in perfect agreement with those from soid modes in a the considered cases. 2. The anaysis of the rectanguar cross-section beam has confirmed the possibiity of deaing with partiay constrained cross-section beams that is offered by the present formuation. 3. The constraints can be arbitrariy distributed in the 3D directions as shown by the anaysis of the C-section beam. 4. The arch beam has shown the strength of the present beam mode in deaing with beams that have she-ike characteristics. The oca effects due to point oadings have aso been detected. 5 Concusions This paper presents a nove beam formuation in the framework of the Carrera Unified Formuation (CUF). Lagrange poynomias have been used to define the dispacement fied above the cross-section of the beam. This choice has ed to a beam formuation with soe dispacement variabes, that is, the unknowns

18 554 Meccanica (2012) 47: Fig. 31 L9 distribution above the C-section, 13 L9 Fig. 28 Free-tip deformed cross-section of the arch cross-section beam Fig. 29 3D deformed configuration of the arch cross-section beam Fig. 32 Deformed cross-section of the C-section beam. y = L Fig. 30 3D camped point distribution on the C-section beam of the probem are the three transationa dispacement components of each Lagrange point above the crosssection. Three-point (L3), four-point (L4), and ninepoint (L9) cross-section eements have been impemented. Mutipe eement distributions have been assembed as we. Severa assessments have been considered: compact cross-sections, hoow cosed and Fig. 33 3D deformed configuration of the C-section beam open beams, and she-ike structures. Point oads have been appied. Tayor-type beam and soid modes have been expoited for comparison purposes. The construction of a refined beam mode by means of the present Lagrange-based formuation has been achieved in two different ways.

19 Meccanica (2012) 47: By increasing the order of each Lagrange-type eement (i.e. using a arger number of interpoation points per eement). 2. By discretizing the cross-section and by using beam eements in sub-domains. This ast option offers the possibiity of adapting the eement distribution to the considered probem with a consequent optimization of the computationa cost. The oca refinement pays a fundamenta roe in deaing with point oads in the presence of open thinwaed cross-sections. In these cases, the Lagrangebased formuation has shown enhanced capabiities compared to the Tayor-based modeing. Other important capabiities of the present formuation are the foowing. 1. The Poisson ocking correction is needed ony in the case of singe L3 eements. The assemby of two L3 or the use of L4 coud make the correction detrimenta. 2. The cassica beam constraining approach has been overcome since a 3D distribution of the boundary conditions is possibe. This impies the possibiity of deaing with partiay constrained cross-section beams, that is, the possibiity of considering boundary conditions which are obtainabe by means of pate/she and soid modes ony. The foowing considerations arise from the comparison with soid eement modes. 1. The resuts compy with a the assessed probems. 2. 3D soutions are obtained by mode refining. 3. The computationa cost of the present beam formuation is consideraby ower than those incurred for 3D modes. The presence of ony transationa degrees of freedom appears to be attractive in the future perspective of the impementation of geometricay non-inear probems. The extension to the ayer-wise anaysis of composite structures coud aso be considered. Acknowedgements The financia support from the Regione Piemonte project MICROCOST is gratefuy acknowedged. References 1. Bathe K (1996) Finite eement procedure. Prentice Ha, New York 2. Carrera E (2002) Theories and finite eements for mutiayered, anisotropic, composite pates and shes. Arch Comput Methods Eng 9(2): Carrera E (2003) Theories and finite eements for mutiayered pates and shes: a unified compact formuation with numerica assessment and benchmarking. Arch Comput Methods Eng 10(3): Carrera E, Brischetto S (2008) Anaysis of thickness ocking in cassica, refined and mixed mutiayered pate theories. Compos Struct 82(4): Carrera E, Brischetto S (2008) Anaysis of thickness ocking in cassica, refined and mixed theories for ayered shes. Compos Struct 85(1): Carrera E, Giunta G (2010) Refined beam theories based on Carrera s unified formuation. Int J App Mech 2(1): Carrera E, Petroo M (2011) On the effectiveness of higherorder terms in refined beam theories. J App Mech 78(2). doi: / Carrera E, Giunta G, Nai P, Petroo M (2010) Refined beam eements with arbitrary cross-section geometries. Comput Struct 88(5 6): doi: /j.compstruc Carrera E, Petroo M, Nai P (2011) Unified formuation appied to free vibrations finite eement anaysis of beams with arbitrary section. Shock Vib 18(3): doi: /sav Carrera E, Petroo M, Vareo A (2011) Advanced beam formuations for free vibration anaysis of conventiona and joined wings. J Aerosp Eng. doi: /(asce)as Carrera E, Petroo M, Zappino E (2011) Performance of CUF approach to anayze the structura behavior of sender bodies. J Struct Eng. doi: /(asce)st X Dinis P, Camotim D, Sivestre N (2006) GBT formuation to anayse the bucking behaviour of thin-waed members with arbitrariy branched open cross-sections. Thin- Waed Struct 44: Eisenberger M (2003) An exact high order beam eement. Comput Struct 81: E Fatmi R (2007) Non-uniform warping incuding the effects of torsion and shear forces. Part I: a genera beam theory. Int J Soids Struct 44: E Fatmi R (2007) Non-uniform warping incuding the effects of torsion and shear forces. Part II: anaytica and numerica appications. Int J Soids Struct 44: Euer L (1744) De curvis easticis. Bousquet, Lausanne 17. Gruttmann F, Wagner W (2001) Shear correction factors in Timoshenko s beam theory for arbitrary shaped crosssections. Comput Mech 27: Gruttmann F, Sauer R, Wagner W (1999) Shear stresses in prismatic beams with arbitrary cross sections. Int J Numer Methods Eng 45: Jönsson J (1999) Distortiona theory of thin-waed beams. Thin-Waed Struct 33: Kapania K, Raciti S (1989) Recent advances in anaysis of aminated beams and pates, part I: shear effects and bucking. AIAA J 27(7): Kapania K, Raciti S (1989) Recent advances in anaysis of aminated beams and pates, part II: vibrations and wave propagation. AIAA J 27(7):

20 556 Meccanica (2012) 47: Novozhiov VV (1961) Theory of easticity. Pergamon, Emsford 23. Oñate E (2009) Structura anaysis with the finite eement method: inear statics, vo 1. Springer, Berin 24. Petroito J (1995) Stiffness anaysis of beams using a higher-order theory. Comput Struct 55(1): Reddy JN (1997) On ocking-free shear deformabe beam finite eements. Comput Methods App Mech Eng 149: Reddy JN (2004) Mechanics of aminated composite pates and shes. Theory and anaysis, 2nd edn. CRC Press, Boca Raton 27. Rendek S, Baáž I (2004) Distortion of thin-waed beams. Thin-Waed Struct 42: Saadé K, Espion B, Warzée G (2004) Non-uniform torsiona behavior and stabiity of thin-waed eastic beams with arbitrary cross sections. Thin-Waed Struct 42: Sivestre N (2007) Generaised beam theory to anayse the bucking behaviour of circuar cyindrica shes and tubes. Thin-Waed Struct 45: Timoshenko SP (1921) On the corrections for shear of the differentia equation for transverse vibrations of prismatic bars. Phios Mag 41: Timoshenko SP (1922) On the transverse vibrations of bars of uniform cross section. Phios Mag 43: Tsai SW (1988) Composites design, 4th edn. Think Composites, Dayton 33. Vinayak RU, Prathap G, Naganarayana BP (1996) Beam eements based on a higher order theory I. Formuation and anaysis of performance. Comput Struct 58(4): Wagner W, Gruttmann F (2002) A dispacement method for the anaysis of fexura shear stresses in thin waed isotropic composite beams. Comput Struct 80: Yu W, Hodges DH (2004) Easticity soutions versus asymptotic sectiona anaysis of homogeneous, isotropic, prismatic beams. J App Mech 71: Yu W, Voovoi VV, Hodges DH, Hong X (2002) Vaidation of the variationa asymptotic beam sectiona anaysis (VABS). AIAA J 40: