Analysis of laminated composite skew shells using higher order shear deformation theory

Static analysis of skew composite shells is presented by developing a C 0 finite element (FE) model based on higher order shear deformation theory (HSDT). In this theory the transverse shear stresses are taken as zero at the shell top and bottom. A realistic parabolic variation of transverse shear strains through the shell thickness is assumed and the use of shear correction factor is avoided. Sander’s approximations are considered to include the effect of three curvature terms in the strain components of composite shells. The C 0 finite element formulation has been done quite efficiently to overcome the problem of C 1 continuity associated with the HSDT. The isoparametric FE used in the present model consists of nine nodes with seven nodal unknowns per node. Since there is no result available in the literature on the problem of skew composite shell based on HSDT, present results are validated with few results available on composite plates/shells. Many new results are presented on the static response of laminated composite skew shells considering different geometry, boundary conditions, ply orientation, loadings and skew angles. Shell forms considered in this study include spherical, conical, cylindrical and hypar shells.


INTRODUCTION
Laminated composite shell structures are widely used in civil, mechanical, aerospace and other engineering applications.Laminated composites materials are becoming popular because of their high strength to weight and strength to stiffness ratios.The most important feature for the analysis of composite structures is that the material (composite) is weak in shear compared to extensional rigidity.Due to this reason transverse shear deformation of the composite shell has to be modeled very efficiently.L = Total number of layers in a lamination scheme S = Ratio of length to total height of shell panel (a/h) Classical theories originally developed for thin elastic shells are based on the Love-Kirchhoff assumptions.These theories neglect the effect of transverse shear deformations.However, application of such theories to laminated composite shells where shear deformation is very significant, may lead to errors in calculating deflections, stresses and frequencies.
In subsequent development of shell theories transverse shear deformation was included in a manner where the shear strain is uniform throughout the thickness of the shell.These theories are known as first order shear deformation theory (FSDT).In this theory a shear correction factors are required for the analysis and these factors should be calculated based on the orientations of different layers in different directions.
The effects of transverse shear and normal stresses in shells were considered by Hildebrand, Reissner and Thomas [18], Lure [25], and Reissner [35].The effect of transverse shear deformation and transverse isotropy, as well as thermal expansion through the thickness of cylindrical shells were considered by Gulati and Essenberg [16], Zukas and Vinson [43], Dong et al. [12,13], Hsu and Wang [19], and Whitney and Sun [37,38].The higher-order shell theories presented in [37,38] are based on a displacement field in which the displacements in the surface of the shell are expanded as linear functions of the thickness coordinate and the transverse displacement is expanded as a quadratic function of the thickness coordinate.These higher-order shell theories are cumbersome and computationally more demanding, because, with each additional power of the thickness co-ordinates, an additional dependent unknown is introduced into the theory.
Reddy and Liu [33] presented a simple higher-order shear deformation theory (HSDT) for the analysis of laminated shells.It contains the same dependent unknowns as in the first-order shear deformation theory (FSDT) in which the displacements of the middle surface are expanded as linear functions of the thickness coordinate and the transverse deflection is assumed to be constant through the thickness.The theory is based on a displacement field in which the displacements of the middle surface are expanded as cubic functions of the thickness coordinate and the transverse displacement is assumed to be constant through the thickness.The additional dependent unknowns introduced with the quadratic and cubic powers of the thickness coordinate are evaluated in terms of the derivatives of the transverse displacement and the rotations of the normals at the middle surface.This displacement field leads to the parabolic distribution of the transverse shear stresses (and zero transverse normal strain) and therefore no shear correction factors are used.Huang [20] presented modified Reddy's theory and further improved the accuracy.Xiao-ping [39] presented a shell theory based on Love's first-order geometric approximation and Donnell's simplification for shallow shells.The theory improves the in-plane displacement (u, v) distribution through thickness by ensuring the continuity of interlaminar transverse shear stresses and zero transverse shear strains on the surface.The theory contains then same dependent unknown and the same order of governing equations as in the first-order shear deformation theory.Without the need for shear correction factors, the theory predicts more accurate responses than first-order theory and some higher-order theories, and the solutions are very close to the elasticity solutions.However, these theories [33,39] demand C1 continuity of transverse displacements during finite element implementations.
Yang [41] developed a higher-order shell element with three constant radii of curvature, two principal radii, orthogonal to each other and one twist radius.The displacement functions u, v and w are composed of products of one-dimensional Hermite interpolation formulae.Shu and Sun [40] developed an improved higher-order theory for laminated composite plates.This theory satisfies the stress continuity across each layer interface and also includes the influence of different materials and ply-up patterns on the displacement field.Liew and Lim [24] proposed a higher-order theory by considering the Lame´ parameter (1+z/Rx) and (1+z/Ry) for the transverse strains, which were neglected by Reddy and Liu [33].This theory accounts for cubic distribution (non-even terms) of the transverse shear strains through the shell thickness in contrast with the parabolic shear distribution (even-terms) of Reddy and Liu [33].Kant and Khare [21] presented a higher-order facet quadrilateral composite shell element.Bhimaraddi [4], Mallikarjuna and Kant [26], Cho et al. [8] are among the others to develop higher-order shear deformable shell theory.It is observed that except for the theory of Yang [41], remaining higher-order theories do not account for twist curvature (1/Rxy), which is essential while analyzing shell forms like hypar and conoid shells.
However, application of higher-order theory for studying the behavior of laminated composite shells with the combination of all three radii of curvature is very limited in literature.Pradyumna and Bandyopadhyay [31] studied the behavior of laminated composite shells based on a higher-order shear deformation theory (HSDT) developed by Kant and Khare [21].They also [21] also extended the theory to the shells to include all three radii of curvature.However, this theory [21] contains some nodal unknowns which are not having any physical significance and therefore, incorporation of appropriate boundary conditions becomes a problem.
It is also observed that there is no literature available on the analysis of composite skew shell using HSDT while very few publications are available on the problem using FSDT [17] for isotropic materials only.
In view of the above, a new finite element model has been developed in the present study for static analysis of composite skew shell panels using a simple higher order shear deformation theory [33].The problem of C1 continuity associated with theory has been overcome in this model and an existing C0 isoparametric finite element has been utilized for this purpose.The element contains nine nodes with seven nodal unknowns at each node.The analysis has been performed considering shallow shell assumptions.The effect of all the three radii of curvature is also included in the formulation.The present finite element model based on HSDT is applied to solve many problems of composite skew shells considering different shell geometries, boundary conditions, loadings and other parameters.The present results are also validated with some published results.

THEORY AND FORMULATION
A laminated shell element made of a finite number of uniformly thick orthotropic layers oriented arbitrarily with respect to the shell co-ordinates (x, y, z) is shown in Figure 1.The reference plane (i.e., mid plane) is defined at at z = 0.In the present formulation the in-plane displacement components u (x, y, z) and v (x, y, z) at any point in the shell space are expanded in Taylor's series in terms of the thickness co-ordinates while the transverse displacement, w(x, y) is taken as constant throughout the shell thickness.As the transverse shear stresses actually vary parabolically through the shell thickness the in-plane displacement fields are expanded as cubic functions of the thickness co-ordinate.The displacement fields are assumed in the form as given by Reddy [32] which satisfy the abovecriteria and may be expressed as below: where u, v and w are the displacements of a general point (x, y, z) in an element of the laminate along x, y and z directions, respectively.The parameters u 0 , v 0 , and w 0 denote the displacements of a point (x ,y) on the mid plane, and and are the rotations of normals with respect to the mid plane about the y and x axes, respectively.The functions ξ x ,ζ x , ξ y and ζ y can be determined using the conditions that the transverse shear strains, γ xz and γ yz vanish at the top and bottom surfaces of the shell.We have The displacement field in equation ( 1) becomes The linear strain-displacement relations according to Sanders' approximation are: Substituting equation (4) in equation ( 5): Latin American Journal of Solids and Structures 10(2013) 891 -919 where, A 1 is a tracer by which the analysis can be reduced to that of shear deformable Love's first approximation.A 1 is important factor as it helps to incorporate the shear part which plays a critical role in failure analysis of composite laminates.Hence, A 1 must be chosen cautiously accordingly to shear deformation theory used to analyze the composites.
The constitutive relations for a typical lamina (k-th) with reference to the material axis may be written as: or, in matrix form: ! !After that, we need to transform the lamina stiffness matrix into a global form using the transformed coefficient.In global form, lamina with different angles and thickness of each layer are computed.Hence, the transformed stiffness matrix Q ij can be calculated.The stress-strain relations for a lamina about the structure axis system (x, y, z) may be written as below after doing the necessary transformation, Integrating the stresses through the laminate thickness, the resultant forces and moments acting on the laminate may be obtained as below, and  is the rigidity matrix of size 17x17., , , , , , , , , , , , , ,

Finite element formulation
A nine-noded isoparametric C 0 element with seven unknowns per node developed by Singh et.al [34] is used in the present finite element model.The nodal unknown vector {d} on the midsurface of a typical element is given by: where {d i } is the nodal unknown vector corresponding to node i and N i is the interpolating shape function associated with the node i.The problem of a skew shell as shown in Figure [2] is studied by using the proposed finite element model.

Figure 2 Mappedskewshell panel
As the sides AB and DC are inclined to global axis by an angle, α, necessary transformation is made to express the degrees of freedom of the nodes on these two sides.
After getting the generalized nodal unknown vector {d} within an element, the generalized midsurface strains at any point of the shell (Eq.: 6), can be expressed in terms of global displacements as shown below, (12) where [B i ] is the differential operator matrix of interpolation functions which may be obtained from Equation (6).
The element stiffness matrix for an element (say, e-th), which includes membrane, flexure, thetransverse shear effects and their couplings may be expressed by the following equation: A 3 × 3 Gaussian Quadrature scheme has been used in all numerical integrations.The element matrices are then assembled to obtain the global stiffness matrix, [K] by following the standard procedure of finite element method [10].

RESULTS AND DISCUSSION
In this section many problems of laminated composite shells are solved for normal as well as skew configurations using the present finite element model based on HSDT.A computer code is developed based on the above formulation for the implementation of the above model.As there is no result available in the literature in the present problem of laminated composite skew shell, the results obtained by the proposed model are validated by some published results on laminated composite shells of normal geometry.A mapped skew shell is shown in Figure 2. To validate the results for skew geometry, the present finite element results are first compared with some results of isotropic and composite skew plates.Then the results obtained by the present model are compared with some limited published results of skew shells using isotropic material.Many new results are generated for the benefit of the future researchers.The shell forms mainly considered for validation are spherical, conical, cylindrical and hypar shells while the problems of skew shells are restricted to cylindrical and hypar geometry.
The elastic properties of the lamina with respective to the material axes has been taken as

Validation of the Present Formulation
In order to validate the present formulation, some problems are solved which are already reported in the literature.

Convergence Study
Convergence study is carried out in order to determine the required mesh size N × N at which the displacement values converge.Table 1 shows the convergence of the results of a simply supported cross-ply spherical shell subjected to uniform loading with a/b=1 and R x =R y =R with the 0, at = 0, a orthotropic elastic properties as mentioned earlier.Three different lamination schemes (0 0 /90 0 , 0 0 /90 0 /90 0 and 0 0 /90 0 /90 0 /0 0 ) are considered.

Cross-ply laminated spherical shell
A cross-ply laminated spherical shell with a/b=1 and R x =R y =R with simply supported boundaries subjected to sinusoidal loading q = q 0 sin π x a sin π y b ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ is considered for the analysis.This problem is earlier solved by Reddy and Liu [33].The lamination schemes are taken as in the previous example.It is found that the present results are in close agreement with the results of Reddy and Liu [33] based on HSDT.The non-dimensional central displacements are obtained as, A laminated (0 0 /90 0 /0 0 ) conical shell panel is considered in this example with a slant edge L, radii at top and bottom R t and R b , respectively.The shell is subjected to uniform lateral pressure p. 12 X 12mesh is used to discretise the shell.The material properties considered are as follows: The central deflections (Table 3) based on the present theory were found to be in good agreement with those obtained by Bhaskar and Varadan [3] based on HSDT for clamped boundary conditions.New results for boundary conditions like SSSS, SSCC and SCSC are also presented in Table 3.It can be observed that as the ratio E L /E T increases the deflection values decrease.On the other hand as the ratio R b /h is increased from 10 to 100 there is considerable decrease in deflection.The geometry of conical shell is as shown in Figure [3].Simply supported three-ply (0 0 /90 0 /0 0 ), four-ply (0 0 /90 0 /0 0 /90 0 ) and five-ply (0 0 /90 0 /0 0 /90 0 /0 0 ) laminated cylindrical shells with R 1 /a = 4 and b/a = 3 is considered in this example.Taking sinusoidal variation of loading q = q 0 sin π x a sin π y b ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ on the shells, the central deflections obtai- ned by thepresent model are compared with the results of Reddy and Liu [33].It is observed in Table 4 that for lower values of thickness ratio (a/h) present results slightly differ from the results presented by Reddy and Liu [33].This difference may be due to the fact that the deflection results of Reddy and Liu [33] shown in Table 4 are multiplied by a factor (1 + h/2R 1 ) (1 + h/2R 2 ,).Moreover, Reddy and Liu [33] have not included the effect of twist curvature (1/R xy ) in their formulation.The geometry of cylindrical shell is as shown in Figure [4].It can also be observed that with increasing values of R 1 /a and a/h the deflections decrease.The lamination scheme (0 0 /90 0 /0 0 ) was found to give lesser deflection compared to other schemes for a/h =10, 100 while for a/h=5, lamination scheme (0 0 /90 0 /0 0 /90 0 /0 0 ) is found give lesser values for the deflections.It is also observed that the anti-symmetric lamination scheme (0 0 /90 0 /0 0 /90 0 ) gives higher results in all the cases.

Angle-ply cylindrical shell
In this example, an angle-ply (/- /  /- / ) cylindrical shell panel of square plan form, simply supported at all edges and subjected to uniform transverse pressure p is considered.Nondimensional central deflections obtained by using the present model are compared with the corresponding results of Bhaskar and Varadan [3].It may be observed in Table 5 that there are differences between the two results for higher values of ply angles ().This may be due to the reasons that the loading considered in the two models are slightly different; and also Bhaskar and Varadan [3] have not considered the twist curvature (1/R xy ) in their formulation which may be significant for angle ply lamination schemes.Some new results are also presented in Table 5.It is observed minimum deflection values corresponds to =45 for all the cases.

Hypar shell
In this example cross-ply laminated hypar shells having four lamination schemes of (0 0 /90 0 ), (0 0 /90 0 /90 0 /0 0 ), (0 0 /90 0 /0 0 /90 0 ) and (0 0 /90 0 /0 0 /90 0 /0 0 )with simply supported as well as clamped boundary conditions with varying c/a ratios (0 to 0.2) are considered.The shell is subjected to uniform as well as sinusoidal loadings.It may be noted that the c/a ratio is an indicator of the twist curvature of the hypar shell.The results are compared with those obtained by Pradyumna and Bandyopadhyay [31] and found to match very well with each other.Some new results are also presented.Lamination scheme 0 0 /90 0 /90 0 /0 0 is found to give lesser deflections in all cases especially in case of clamped boundary (CCCC) conditions.The geometry of hypar shell is as shown in Figure [5].

Analysis of laminated composite skew shell
In this section some examples of laminated composite skew hypar and cylindrical shells are studied using the present FE model based on HSDT.
In Table 9 the values of non-dimensional central deflections w ( ) of three different lamination schemes are presented for different skew angles.It can be observed that the deflection values decrease with increase in skew angle.Also there is not much effect of R/a ratio on the deflection values.The clamped boundary condition is more effective in reduction of deflection values as compared to the simply supported boundary condition.The reduction in deflection with increase in a/h ratio from 10 to 100 makes thin shells more effective.The superiority of (0 0 /90 0 /0 0 ) lamination scheme is observed in all the cases mentioned in Table 9.
Present results for stresses of cylindrical shell panel (b/a=3, R/a=4) subjected to sinusoidal loading are shown in Table 10.Some of the present results are compared with those of Xiao-ping [39] for zero skew angle and found to be in good agreement.The lamination scheme (0 0 /90 0 /0 0 ) is found to be superior.Normal stresses decrease as the value of skew angle increases.Stresses are least in the case of clamped boundary condition.In all the cases listed; values of shear stress are much lesser compared to other stresses.

Angle-ply laminated skew cylindrical shell
Angle-ply laminated (45 0 /-45 0 /45 0 /-45 0 ) skew cylindrical shell (b/a =3) is considered in this example.The shell is subjected to uniform loading with different boundary conditions such as simply supported and clamped.The skew angle is varied from 0 0 to 45 0 .The R/a (3, 10, 100) as well as a/h (10, 100) ratios are also varied.In this paper, a new finite element model has been developed for the analysis of skew composite shells on higher order shear deformation theory (HSDT) using a C 0 formulation.In this model there is no need to include any shear correction factor.Three radii of curvatures including the cross curvature effects are also considered in the FE formulation which accounts for twisting effect of the geometry.Different shell forms considered in this study include spherical, conical, cylindrical and hypar shells.It is observed there is no result available in the literature on the present problem of skew composite shell.Therefore, many new results are generated on the static response of laminated composite skew shells considering different geometry, boundary conditions, ply orientation, loadings and skew angles which should be useful for future research.

Variation of non-dimensional central deflection (Figures 6a-6e
The following general conclusions are made: 1. Central deflection values are lesser for thin shells (a/h=100) as compared to thick shells (a/h=10) for all the types of shells considered in this paper.
3. In cylindrical and spherical shells, as the R/a ratio increases central deflection values decrease.4. Laminated composite angle-ply ( /- /  /- /) cylindrical shell gives minimum deflection values corresponds to  =45 for all the cases.5. Laminated composite skew hypar shell give minimum central deflection for lamination scheme (0 0 /90 0 /0 0 /90 0 ).6. In-plane normal stresses decrease as the skew angle increases in case of laminated composite skew cylindrical shell.7. In-plane shear stresses are much lesser compared to in-plane normal stresses for both cylindrical and hypar shells.

Latin
American Journal of Solids and Structures 10(2013) 891 -919 Nomenclature a = Length of the mappedshell panel parallel to x-axis in plan α = Angle between skewed edge of mapped shell panel with respect to y-axis measured in plan b = Width of the mapped shell panel parallel to y-axis in plan h = Total height of the shell panel N x N = FE mesh divison denoted as (Number of divisions in X-direction) x (Number of divisions in Y-direction).

Figure 3
Figure 3 Conical Shell panels

Figure 4
Figure 4 Perspective view of cylindrical shell panel

Figure 5
Figure 5 Perspective view of hypar shell panel ) and in-plane normal stress (Figures 7a-7e) with skew angle is presented [Figures (a, b): Simply supported, Figures (c, d): Clamped and Figures (e, f):-Simply supported-clamped boundary conditions].As in the case of clamped boundary condition, the in-plane shear stress was found to be zero for all the cases; variation of in-plane shear stress with skew angle is shown (Figures 8a-8d) only for two boundary conditions [Figure 8(a, b): Simply supported, 8(c, d): Simply supported-clamped boundary conditions].From the figures it can be observed that the deflection and stresses follow the expected general trend.With increase in R/a ratio, central deflection values differ significantly in case of thin shells (a/h = 100) as compared to those for thick shells (a/h = 10).Deflection and stresses tend to decrease as the skew angle increases.In Figures 8 it can be observed that in-plane shear stresses tend to attain minimum value approximately at skew angle of 30 0 .

Figure 6 Figure 7 Figure
Figure 6 Variation of non-dimensional central deflections w ( ) of laminated angle ply skew composite cylindrical shell With skew angles

Table 2
Non-dimensional central deflections of simply supported cross-ply laminated spherical shells under sinusoidally distributed load (a/b=1, Rx=Ry=R)

Table 7
Non-dimensional central deflections of a laminated composite hypar shell panel subjected to uniform loading (b/a =1)

Table 8
Non-dimensional stresses of a laminated composite hypar shell panel subjected to uniform loading (b/a =1, a/h = 100)

Table 9
Non-dimensional central deflections of a laminated composite cylindrical shell panel subjected to uniform loading (b/a =3)