Static bending and free vibration of cross ply laminated composite plates using NURBS based finite element method and unified formulation
aa r X i v : . [ m a t h . NA ] D ec Static bending and free vibration of cross ply laminated composite plates using NURBSbased finite element method and unified formulation
S Natarajan a,1, , Hung Nguyen-Xuan b , AJM Ferreira c,e , E Carrera d,e a School of Civil & Environmental Engineering, The University of New South Wales, Sydney, Australia b Department of Mechanics, Faculty of Mathematics & Computer Science, University of Science, HCMC, Vietnam c Faculdade de Engenharia da Universidade do Porto, Porto, Portugal. d Department of Aeronautics and Aerospace Engineering, Politecnico di Torino, Italy. e Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia.
Abstract
This paper presents an effective formulation to study the response of laminated composites based on isogeomet-ric approach (IGA) and Carrera unified formulation (CUF). The IGA utilizes the non-uniform rational B-spline(NURBS) functions which allows to construct higher order smooth functions with less computational effort. Thestatic bending and the free vibration of thin and moderately thick laminates plates are studied. The present ap-proach also suffers from shear locking when lower order functions are employed and the shear locking is suppressedby introducing a modification factor. The combination of the IGA with the CUF allows a very accurate predictionof the field variables. The effectiveness of the formulation is demonstrated through numerical examples.
Keywords:
A. Lamina/ply; A. Layered Structures; B. Vibration; C. Computational Modelling; C. Finiteelement analysis; C. Numerical analysis
1. Introduction
The need for high strength-to and high stiffness-to-weight ratio materials has led to the development of laminatedcomposite materials. This class of material has seen increasing utilization as structural elements, because of thepossibility to tailor the properties to optimize the structural response. Since, its inception, different approacheshave been employed to study the response of such laminated composite plates, ranging from complete 3D analysisto two-dimensional theories. A brief overview on the development of different plate theories is given in [1, 2]. Thevarious two dimensional plate theories can be further classified into three different approaches: (a) equivalent singlelayer theories [3]; (b) discrete layer theories [4] and (c) mixed plate theory. Among these, the equivalent singlelayer theories, viz., first order shear deformation theory [5], second and higher order accurate theory [3, 6] are themost popular theories employed to describe the plate kinematics. Existing approaches in the literature to studyplate and shell structures made up of laminated composites uses the finite element method based on Lagrange basisfunctions [7] or non-uniform rational B splines (NURBS) [8] or meshfree methods [9]. Not only these approachessuffer from shear locking when applied to thin plates, these techniques does not provide a single platform totest the performance of various theories. Thanks to the recent derivation of series of axiomatic approaches byCarrera [10], coined as Carrera Unified Formulation [11] for the general description of two-dimensional formulationsfor multilayered plates and shells. With this unified formulation, it is possible to implement in a single software aseries of hierarchical formulations, thus affording a systematic assessment of different theories ranging from simpleequivalent single layer models up to higher order layerwise descriptions.Recent interest in the unified formulation has led to the development of discrete models such as those based on finiteelement method [12, 13] and more recently meshless methods [9]. Nevertheless, even with the unified framework,there is an important shortcoming. With lower order basis functions within the finite element framework, whenapplied to thin plates, the formulation suffers from shear locking. Intensive research over the the past decadeshas to led to some of the robust methods to suppress the shear locking syndrome. This includes: (a) reducedintegration [14]; (b) use of assumed strain method [15]; (c) using field redistributed shape functions [16]; (d)
Preprint submitted to Elsevier June 5, 2018 ixed interpolation tensorial components (MITC) technique with strain smoothing [17] and (e) very recently,twist Kirchhoff plate element [18] and the 3D consistent formulation based on the scaled boundary finite elementmethod [19].The main objective of this manuscript is to investigate the potential application of the NURBS based isogeometricfinite element method within the Carrera Unified Formulation (CUF) to study the global response of cross-plylaminated composites. The present formulation also suffers from shear locking when lower order basis functionsare employed to thin plates. To address lower-order NURBS elements for plates, the introduction of a stabilizationtechnique into shear locking has been studied in [20]. The other approach to suppress is to employ higher orderbasis functions [21]. In this study, to alleviate shear locking, a simple modification is done to the shear term whenlower order NURBS basis functions are used. However, the draw back of this approach is that the shear correctionfactor is problem dependent. The influence of various parameters, viz., the ply thickness, the ply orientation, theplate geometry, the material property and the boundary conditions on the global response is numerically studied.The paper commences with a brief discussion on the unified formulation for plates and the finite element discretiza-tion. Section 3 describes the isogeometric approach employed in this study, followed by a technique to addressshear locking when lower order NURBS functions are used to discretize the field variables. The efficiency of thepresent formulation, numerical results and parametric studies are presented in Section 4, followed by concludingremarks in the last section.
2. Carrera Unified Formulation
Let us consider a laminated plate composed of perfectly bonded layers with coordinates x, y along the in-planedirections and z along the thickness direction of the whole plate, while z k is the thickness of the k th layer.The CUF is a useful tool to implement a large number of two-dimensional models with the description at thelayer level as the starting point. By following the axiomatic modelling approach, the displacements u ( x, y, z ) =( u ( x, y, z ) , v ( x, y, z ) , w ( x, y, z )) are written according to the general expansion as: u ( x, y, z ) = N X τ =0 F τ ( z ) u τ ( x, y ) (1)where F ( z ) are known functions to model the thickness distribution of the unknowns, N is the order of theexpansion assumed for the through-thickness behaviour. By varying the free parameter N , a hierarchical seriesof two-dimensional models can be obtained. The strains are related to the displacement field via the geometricalrelations: ε pG = (cid:2) ε xx ε yy γ xy (cid:3) T = D p u ε nG = (cid:2) γ xz γ yz ε zz (cid:3) T = ( D np + D nz ) u (2)where the subscript G indicate the geometrical equations, D p , D np and D nz are differential operators given by: D p = ∂ x ∂ y ∂ y ∂ x , D np = ∂ x ∂ y , D nz = ∂ z ∂ z
00 0 ∂ z . (3)The 3D constitutive equations are given as: σ pC = C pp ε pG + C pn ε nG σ nC = C np ε pG + C nn ε nG (4)2ith C pp = C C C C C C C C C C pn = C C C C np = C C C C nn = C C C C
00 0 C (5)where the subscript C indicate the constitutive equations. The Principle of Virtual Displacements (PVD) in caseof multilayered plate subjected to mechanical loads is written as: N k X k =1 Z Ω k Z A k n ( δ ε kpG ) T σ kpC + ( δ ε knG ) T σ knC o dΩ k d z = N k X k =1 Z Ω k Z A k ρ k δ u k T s ¨ u k dΩ k d z + N k X k =1 δ L ke (6)where ρ k is the mass density of the k th layer, Ω k , A k are the integration domain in the ( x, y ) and the z direction,respectively. Upon substituting the geometric relations (Equation (2)), the constitutive relations (Equation (4))and the unified formulation into the PVD statement, we have: Z Ω k Z A k (cid:26)(cid:16) D kp F s δ u ks (cid:17) T n C kpp D kp F τ u kτ + C kpn ( D kn Ω + D knz ) F τ u kτ o + h ( D kn Ω + D knz ) f x δ u ks ) T ( C knp D kp F τ u kτ + C knn ( D kn Ω + D knz ) F τ u kτ ) io dΩ k d z = N k X k =1 Z Ω k Z A k ρ k δ u k T s ¨ u k dΩ k d z + N k X k =1 δ L ke (7)After integration by parts, the governing equations for the plate are obtained: K kτsuu u kτ = P kuτ (8)and in the case of free vibrations, we have: K kτsuu u kτ = M kτs ¨ u kτ (9)where the fundamental nucleus K kτsuu is: K kτsuu = h ( − D kp ) T ( C kpp D kp + C kpn ( D kn Ω + D nz ) + ( − D kn Ω + D knz ) T ( C knp D kp + C knn ( D kn Ω + D knz )) i F τ F s (10)and M kτs is the fundamental nucleus for the inertial term given by: M kτsij = (cid:26) ρ k F τ F s if i = j i = j (11)where P kuτ are variationally consistent loads with applied pressure. For more detailed derivation and for theexplicit form of the fundamental nuclei, interested readers are referred to [11, 22].
3. Non-uniform rational B-splines
In this study, the finite element approximation uses NURBS basis function. We give here only a brief introductionto NURBS. More details on their use in FEM are given in [23]. The key ingredients in the construction of NURBSbasis functions are: the knot vector (a non decreasing sequence of parameter values, ξ i ≤ ξ i +1 , i = 0 , , · · · , m − P i , the degree of the curve p and the weight associated to a control point, w . The i th B-splinebasis function of degree p , denoted by N i,p is defined as: N i, ( ξ ) = (cid:26) ξ i ≤ ξ ≤ ξ i +1 N i,p ( ξ ) = ξ − ξ i ξ i + p − ξ i N i,p − ( ξ ) + ξ i + p +1 − ξξ i + p +1 − ξ i +1 N i +1 ,p − ( ξ ) (12)A p th degree NURBS curve is defined as follows: C ( ξ ) = m P i =0 N i,p ( ξ ) w i P im P i =0 N i,p ( ξ ) w i (13) Ξ B as i s f un c t i on s Figure 1: non-uniform rational B-splines, order of the curve = 3 where P i are the control points and w i are the associated weights. Figure (1) shows the third order non-uniformrational B-splines for a knot vector, Ξ = { , , , , / , / , / , / , / , , , , } . NURBS basis functionshas the following properties: (i) non-negativity, (ii) partition of unity, P i N i,p = 1; (iii) interpolatory at the endpoints. As the same function is also used to represent the geometry, the exact representation of the geometry ispreserved. It should be noted that the continuity of the NURBS functions can be tailored to the needs of theproblem. The B-spline surfaces are defined by the tensor product of basis functions in two parametric dimensions ξ and η with two knot vectors, one in each dimension as: C ( ξ, η ) = n X i =1 m X j =1 N i,p ( ξ ) M j,q ( η ) P i,j (14)where P i,j is the bidirectional control net and N i,p and M j,q are the B-spline basis functions defined on the knotvectors over an m × n net of control points P i,j . The NURBS surface is then defined by: C ( ξ, η ) = P ni =1 P mj =1 N i,p ( ξ ) M j,q ( η ) P i,j w i w j w ( ξ, η ) (15)4here w ( ξ, η ) is the weighting function. The displacement field within the control mesh is approximated by: u τ ( x, y ) = C ( ξ, η ) q τ ( x, y ) , (16)where q τ ( x, y ) are the nodal variables and C ( ξ, η ) are the basis functions given by Equation (15). Similar to the finite element based on Lagrange basis functions, locking appears when lower order NURBS basisfunctions are employed [21, 24], for example with quadratic, cubic and quartic elements . One approach to alleviatethe shear locking is to employ interpolation functions of order 5 or higher [21], but this inevitably increases thecomputational cost. A stabilization technique for several lower-order NURBS elements for plates was reported in[20]. In this paper, we adopt a stabilization technique proposed in [25] and later used in [24] to study the responseof Reissner-Mindlin plates. In this approach, the material matrix related to the shear terms are multiplied by thefollowing factor: shearFactor = h h + α ℓ (17)where ℓ is the longest length of the edges of the NURBS element and α is a positive constant given in the interval0 . ≤ α ≤ .
15. It is found from numerical experiments of NURBS-based isogeometric plate elements that α canbe fixed at 0.1, which provide reasonably accurate solutions.
4. Numerical Results
In this section, we present the static response and the natural frequencies of laminated composite plates usingthe combined IGA and CUF framework. In this study we use a hybrid displacement assumption, where the in-plane displacements u and v are expressed as sinusoidal expansion in the thickness direction, and the transversedisplacement, w is quadratic in the thickness direction. We refer to this theory as SINUS-W2. The displacementsare expressed as: u ( x, y, z, t ) = u o ( x, y, t ) + zu ( x, y, t ) + sin (cid:16) πzh (cid:17) u ( x, y, t ) v ( x, y, z, t ) = v o ( x, y, t ) + zv ( x, y, t ) + sin (cid:16) πzh (cid:17) v ( x, y, t ) w ( x, y, z, t ) = w o ( x, y, t ) + zw ( x, y, t ) + z w ( x, y, t ) (18)where u o , v o and w o are translations of a point at the middle-surface of the plate, w is higher order translation,and u , v , u and v denote rotations [26] and considers a quadratic variation of the transverse displacement w allowing for the through-the-thickness deformations. The effect of the plate aspect ratio, the ply angle and theratio of Young’s modulus E /E on the static bending and free vibration is numerically studied. The static analysis is conducted for cross-ply laminated plates with three and four layers under the followingsinusoidal load: p z ( x, y ) = P o sin (cid:16) πxa (cid:17) sin (cid:16) πya (cid:17) (19)where P o is the amplitude of the mechanical load. The origin of the coordinate system is located at the lower-left corner on the midplane. The physical quantities are non-dimensionalized by the following relations, unless Linear NURBS basis functions are same as the linear Lagrange basis functions and are not discussed here. Approaches employedfor Lagrange basis functions can readily be applied to NURBS basis functions with order 1 w = w ( a/ , a/ ,
0) 100 h E P a ; σ xx = σ xx ( a/ , a/ , h/ h P a ; σ yy = σ yy ( a/ , a/ , h/ h P a ; τ xz = τ xz (0 , a/ , hP a ; (20) Validation.
Before proceeding with a detailed numerical study on the effect of various parameters on the globalresponse of cross-ply laminated composites, the results from the proposed formulation are compared againstavailable results pertaining to static bending of laminated plates. In this study, we consider three orders ofNURBS basis functions, viz., quadratic, cubic and quartic. It is noted that, in this study, we do not considerfirst order NURBS basis functions. This is because, the first order NURBS basis functions are similar to theconventional bilinear shape functions. The performance of which is discussed in detail in [12, 13]. In this study,the results from the present formulation are denoted by Quadratic, Cubic and Quartic, which corresponds to theorder of shape functions employed, which is referred to as p − refinement. Three different mesh discretizations,viz., 5 ×
5, 7 × × h − refinement. Table 1 shows the convergence of thecentral deflection and stresses of a simply supported cross-ply laminated square plate. It is seen that with both h − and p − refinement, the results from the present formulation converge. It is seen that highly accurate resultsare obtained from the present formulation even with a coarse mesh. A comparison with other approaches and anelasticity solution is given in Table 2. Table 1: Convergence of the central deflection w = w ( a/ , a/ , E h Pa of a simply supported cross-ply laminated square plate[0 ◦ / ◦ / ◦ / ◦ ] with E = 25 E , G = G = 0.5 E , G = 0.2 E , ν =0.25. Method Meshes5 × × × w Quadratic 1.9207 1.9100 1.9058Cubic 1.9076 1.9038 1.9021Quartic 1.9045 1.9020 1.9010HSDT [3] 1.8937Elasticity [27] 1.9540 σ xx Quadratic 0.6966 0.7009 0.7029Cubic 0.7074 0.7063 0.7061Quartic 0.7062 0.7060 0.7058HSDT [3] 0.6651Elasticity [27] 0.7200 σ yy Quadratic 0.6179 0.6221 0.6239Cubic 0.6277 0.6270 0.6268Quartic 0.6268 0.6267 0.6266HSDT [3] 0.6322Elasticity [27] 0.6660 τ xz Quadratic 0.2293 0.2246 0.2227Cubic 0.2210 0.2205 0.2202Quartic 0.2205 0.2202 0.2201HSDT [3] 0.2064Elasticity [27] 0.27006 .1.1. Four layer (0 ◦ /90 ◦ ) s square cross-ply laminated plate under sinusoidal load A square simply supported laminate of side a and thickness h , composed of four equally thick layers orientedat (0 ◦ /90 ◦ ) s is considered. The plate is subjected to a sinusoidal vertical pressure given by Equation (19). Thematerial properties are as follows: E = 25 E ; G = G = 0.5 E ; G = 0.2 E ; ν = 0.25. For this example, athree-dimensional exact solution by Pagano [27] is available. The central deflection and the corresponding stressesfor the SINUS-W2 theory with isogeometric approach are presented in Table 2. We compare the results withhigher order plate theories [3, 28], first order theory [29], an exact solution [27] and also with the strain smoothingapproach with SINUS-W2. The effect of plate the thickness is also shown in Table 2. It is clear that the firstorder shear deformation theories (FSDT) cannot be used for thick laminates. It can be seen that the resultsfrom the present formulation are in very good agreement with those in the literature and very precise transversedisplacements and stresses are obtained. Table 2: The normalized central deflection w = w ( a/ , a/ , E h Pa , stresses, σ xx = σ xx ( a/ , a/ , h/ h Pa , σ yy = σ yy ( a/ , a/ , h/ h Pa and τ xz = τ xz (0 , /a , hPa of a simply supported cross-ply laminated square plate [0 ◦ / ◦ / ◦ / ◦ ], with E =25 E , G = G = 0.5 E , G = 0.2 E , ν =0.25. a/h Method w σ xx σ yy τ xz
10 HSDT [3] 0.7147 0.5456 0.3888 0.2640FSDT [29] 0.6628 0.4989 0.3615 0.1667Elasticity [27] 0.7430 0.5590 0.4030 0.3010RBF [28] 0.7325 0.5627 0.3908 0.3321CS-FEM Q4 (4 subcells) [13] 0.7195 0.5597 0.3905 0.2952Present (Quadratic 9 ×
9) 0.7250 0.5571 0.3908 0.2985Present (Cubic 9 ×
9) 0.7203 0.5596 0.3913 0.2983Present (Quartic 9 ×
9) 0.7187 0.5594 0.3907 0.2967100 HSDT [3] 0.4343 0.5387 0.2708 0.2897FSDT [29] 04337 0.5382 0.2705 0.1780Elasticity [27] 0.4347 0.5390 0.2710 0.3390RBF [28] 0.4307 0.5431 0.2730 0.3768CS-FEM Q4 (4 subcells) [13] 0.4304 0.5368 - 0.3285Present (Quadratic 9 ×
9) 0.4383 0.5334 - 0.4069Present (Cubic 9 ×
9) 0.4336 0.5368 - 0.3271Present (Quartic 9 ×
9) 0.4317 0.5366 - 0.3275 ◦ /90 ◦ /0 ◦ ) square cross ply laminated plate under sinusoidal load In this case, a square laminate of side a and thickness h , composed of three equally thick layers oriented at(0 ◦ /90 ◦ /0 ◦ ) is considered. It is simply supported on all edges and subjected to a sinusoidal vertical pressure of theform given by Equation (19). The material properties for this example are: E =132.38 GPa, E = E =10.756GPa, G =3.606 GPa, G = G = 5.6537 GPa, ν = ν = 0.24, ν = 0.49. In Table 3, we present results forthe SINUS-W2 theory with isogeometric approach with quadratic, cubic and quartic NURBS basis functions witha 9 × Table 3: Transverse displacement w = w ( a/ , a/ , h/
2) at the center of a multilayered plate [0 ◦ / ◦ / ◦ ] with E = 132.38 GPa, E = E = 10.756 GPa, G = 3.606 GPa, G = G = 5.6537 GPa, ν = ν = 0.24, ν = 0.49. w a/h
10 50 100 500 1000Analytical (ESL-2) [30, 10] 0.9249 0.7767 0.7720 0.7705 0.7704MITC4 [12] 0.9195 0.7713 0.7666 0.7650 0.7650CS-FEM Q4 (4 subcells) [13] 0.9235 0.7703 0.7655 0.7639 0.7639Present (Quadratic 9 ×
9) 0.9252 0.7713 0.7650 0.7624 0.7624Present (Cubic 9 ×
9) 0.9226 0.7704 0.7656 0.7640 0.7639Present (Quartic 9 ×
9) 0.9217 0.7695 0.7646 0.7631 0.7630
In this example, all layers of the laminate are assumed to be of the same thickness, density and made up of thesame linear elastic material. The following material parameters are considered for each layer E E = 10,20,30, or 40; G = G = 0.6 E ; G = 0.5 E ; ν = 0.25 . The subscripts 1 and 2 denote the directions normal and the transverse to the fiber direction in a lamina, whichmay be oriented at an angle to the plate axes. The ply angle of each layer is measure from the global x − axis to thefiber direction. The example considered is a simply supported square plate of the cross-ply lamination (0 ◦ /90 ◦ ) s .The thickness and the length of the plate are denoted by h and a , respectively. The thickness-to-span ratio h/a =0.2 is employed in the computations. In this study, we present the non dimensionalized free flexural frequenciesas, unless specified otherwise: Ω = ω a h r ρE Table 4 shows the convergence of the normalized fundamental frequency of a simply supported cross-ply laminatedsquare plate based on the current isogeometric approach. The performance of various basis functions with NURBSmesh refinement is studied. It is seen that with h − refinement, the solutions converge and with p − refinement,the accuracy increases for the same mesh size, as expected. Table 5 lists the fundamental frequency for a simplysupported cross-ply laminated square plate with h/a = 0.2 and for different Young’s modulus ratios, E /E . Itcan be seen that the results from the present formulation are in very close agreement with the values of [32] basedon higher order theory, the meshfree results of Liew et al., [31] and Ferreira et al., based on FSDT and higher ordertheories with radial basis functions [33]. The effect of plate thickness on the fundamental frequency is shown inTable 6. It can be seen that the results agree with the results available in the literature. The present formulationis insensitive to shear locking. 8 able 4: Convergence of the normalized fundamental frequency Ω = ωa /h p ρ/E of a simply supported cross-ply laminated squareplate (0 ◦ /90 ◦ ) s with h/a = 0 . E E = 40, G = G = 0.6 E , G = 0.5 E , ν = 0.25.. Method Meshes5 × × × Table 5: The normalized fundamental frequency Ω = ωa /h p ρ/E of a simply supported cross-ply laminated square plate (0 ◦ /90 ◦ ) s with h/a = 0 . E E = 10, 20, 30 or 40, G = G = 0.6 E , G = 0.5 E , ν = 0.25. Method E /E
10 20 30 40Liew [31] 8.2924 9.5613 10.3200 10.8490Reddy, Khdeir [32] 8.2982 9.5671 10.3260 10.8540HSDT [33] ( ν = 0 .
18) 8.2999 9.5411 10.2687 10.7652CS-FEM Q4 (4 subcells) [13] 8.3642 9.5793 10.2973 10.7887Present (Quadratic 9 ×
9) 8.3358 9.5437 10.2572 10.7454Present (Cubic 9 ×
9) 8.3417 9.5532 10.2691 10.7590Present (Quartic 9 ×
9) 8.3439 9.5566 10.2734 10.7640
Table 6: Variation of fundamental frequencies, Ω = ωa /h p ρ/E with a/h for a simply supported square laminated plate[0 ◦ / ◦ / ◦ / ◦ ], Ω = ωa /h p ρ/E , with E /E = 40, G = G =0.6 E , G =0.5 E , ν = ν = ν = 0.25. Method a/h ×
9) 5.3931 9.2701 15.0660 17.5781 18.5913 18.7579Present (Cubic 9 ×
9) 5.3945 9.2785 15.1086 17.649 18.6711 18.8343Present (Quartic 9 ×
9) 5.3951 9.2815 15.1239 17.6749 18.7024 18.86659 .3. Circular plates
In this example, consider a circular four layer [ θ/ − θ/ − θ/θ ] laminated plate with fully clamped boundaryconditions. The influence of the fiber orientations on the free vibration of clamped circular laminated plate isstudied. The following material properties are used: E E = 40; G = G = 0.6 E ; G = 0.5 E ; ν = 0.25 . The subscripts 1 and 2 denote the directions normal and the transverse to the fiber direction in a lamina. Thecircular plate has a radius-to-thickness of 5 (
R/h =5). For this problem, a NURBS quadratic basis functionis enough to model exactly the circular geometry. Any further refinement, if done, will only improve the accu-racy of the solution. The following knot vectors for the coarsest mesh with one element are defined as follows:Ξ =[0,0,0,1,1,1]; and H =[0,0,0,1,1,1]. The data for the circular plate is given in Table 7. In this study, 13 × Table 7: Control points and weights for a circular plate with radius R = 0.5. i 1 2 3 4 5 6 7 8 9 x i - √ - √ √ √ √ √ y i √ √ √ √ √ √ w i √ √ √ √ Table 8: Influence of fiber orientations on the fundamental frequencies, Ω = ωa /h p ρ/E for clamped circular laminated plates. θ Method Ω1 2 30 MLSDQ-FSDT [31] 22.2110 29.651 41.1010IGA [36] 23.5781 30.7459 42.0042Present 22.6663 30.3485 41.7294 π/
12 MLSDQ-FSDT [31] 22.7740 31.4550 43.350IGA [36] 23.6090 31.7743 43.9569Present 23.0024 31.5752 43.7671 π/ π/
5. Conclusions
In this article, the isogeometric approach was combined with the unified formulation to study the static bendingand the free vibration of laminated composites. The present approach allows us to achieve smooth approximationof the unknown fields with arbitrary continuity. When employing lower order elements, the method suffers from10hear locking syndrome, which is alleviated by multiplying the shear term with a correction factor. The resultsfrom the present formulations are in very good agreement with the solutions available in the literature. It isbelieved that the present formulation is definitely a effective computational formulation for practical problems.On one hand, the unified formulation allows the user to test different theories within a single framework, whilst,the isogeometric approach not only provides flexibility in constructing higher smooth basis functions, but thegeometry is accurately described.
Acknowledgements
S Natarajan would like to acknowledge the financial support of the School of Civil and Environmental Engineering,The University of New South Wales for his research fellowship for the period September 2012 onwards.
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