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Laminated composite shells are frequently used in various engineering applications including aerospace, mechanical, marine, and automotive engineering. This article reviews the recent literature on the static analysis of composite shells. It follows up with the previous work published by the first author [1-4] and it is a continuation of another recent article that focused on the dynamics of composite shells [3]. This paper reviews most of the research done in recent years (2000-2010) on the static and buckling behavior (including postbuckling) of composite shells. This review is conducted with an emphasis on the analysis performed (static, buckling, postbuckling, and others), complicating effects in both material (e.g. piezoelectric) and structure (e.g. stiffened shells), and the various shell geometries (cylindrical, conical, spherical and others). Attention is also given to the theory being applied (thin, thick, 3D, nonlinear …). However, more details regarding the theories have been described in previous work [1,3].

The use of laminated composite shells in many engineering applications has been expanding rapidly in the past four decades due to their higher strength and stiffness to weight ratios when compared to most metallic materials. Composite shells now constitute a large percentage of recent aerospace or submarine structures. They are used increasingly in areas such as automotive engineering, biomedical engineering and other applications.

Literature on composite shell research can be found in many national and international conferences and journals. A recent article [

Present article reviews only recent research (2000 through 2010) done on the static and buckling analyses of composite shells. It includes stress, deformation, buckling and post buckling analyses under mechanical, thermal, hygrothermal or electrical loading. Since there are extensive papers on experimental and optimization studies in literature, those topics have not been discussed in this review separately. However, papers in those topics based on their obtained results are classified in the topics of this review.

This article classifies research based upon the typically used shell theories. These include thin (or classical) and thick shell theories (including shear deformation and three dimensional theories), shallow and deep theories, linear and nonlinear theories, and others. Most theories are classified based on the thickness ratio of the shell being treated (defined as the ratio of the thickness of the shell to the shortest of the span lengths and/or radii of curvature), its shallowness ratio (defined as the ratio of the shortest span length to one of the radii of curvature) and the magnitude of deformation (compared mainly to its thickness). Fundamental equations are listed for the types of shells used by most researchers in other publications [1-4].

The literature is reviewed while focusing on various aspects of research. Focus will first be placed on the various shell geometries that are receiving attention in recent years. Among classical shell geometries are the cylindrical, spherical, conical shells and other shells of revolution; other shells like shallow shells are also included in this review. Stress and deformation analyses, in which various boundary conditions and/or shell geometries are considered, buckling and post-buckling problems, and finally research dealing with thermal and/or hygrothermal environments will be reviewed. The third aspect of research will focus on material-related complexities, which include piezoelectric or other complex materials. Structural-related complexities will be the final category that will be addressed. This will include stiffened shells, shells with cut-outs, shells with imperfections or other complexities.

Shells are three dimensional bodies bounded by two, relatively close, curved surfaces. The three dimensional equations of elasticity are complicated when written in curvilinear, or shell, coordinates. Researchers simplify such shell equations by making certain assumptions for particular applications. Almost all shell theories (thin and thick, deep and shallow …) reduce the three-dimensional (3D) elasticity problem into a two dimensional (2D) problem. The accuracy of thin and thick shell theories is established when their results are compared to those of 3D theory of elasticity.

A shell is a three dimensional body confined by two parallel (unless the thickness is varying) surfaces. In general, the distance between those surfaces is small compared with other shell parameters. In this section, the equations from the theory of 3D elasticity in curvilinear coordinates are presented. The literature regarding Mechanics of laminated shells using 3D elasticity theory will then be reviewed.

Consider a shell element of thickness h, radii of curvature R_{a} and R_{b} (a radius of twist R_{a}_{b} is not shown here) (

The laminated composite shells are assumed to be composed of plies of unidirectional long fibers embedded in a matrix material. On a macroscopic level, each layer may be regarded as being homogeneous and orthotropic. However, the fibers of a typical layer may not be parallel to the coordinates in which the shell equations are expressed. The stress-strain relationship for a typical nth lamina in a laminated composite shell made of N laminas as shown in

The positive notations of the stresses are shown in

In order to develop a consistent set of equations, the boundary conditions and the equilibrium equations will be derived using the principle of virtual work, which yields the following equilibrium equations

The principle of virtual work will also yield boundary terms that are consistent with the other equations. The boundary terms for z = constant are:

where s_{0z} , s_{0}_{az} and s_{0}_{bz}_{ }are surface tractions and u^{0}, v^{0} and w^{0} are displacement functions at z = constant. Similar results are obtained for the boundaries a = constant and b = constant. A three dimensional shell element has six surfaces. With three equations at each surface, a total of 18 equations can be obtained for a single-layered shell.

The above equations are valid for single-layered shells. To use 3D elasticity theory for multi-layered shells, each layer must be treated as an individual shell. Both displacements and stresses must be continuous between each layer (layer k to layer k + 1) in a n-ply laminate to insure that there are no free internal surfaces (i.e., delamination) between the layers.

For k = 1,···, N – 1.

Among the recent work that used 3D theory of elasticity is the work of Sheng and Ye [

Thick shells are defined as shells with a thickness smaller by at least one order of magnitude when compared with other shell parameters such as wavelength and/or radii of curvature (thickness is at least 1/10 of the smaller length of the shell). The main differentiation between thick shell and thin shell theories is the inclusion of shear deformation and rotary inertia effects. Theories that include shear deformation are referred to as thick shell theories or shear deformation theories.

Thick shell theories are typically based on either a displacement or stress approach. In the former, the midplane shell displacements are expanded in terms of shell thickness, which can be a first order expansion, referred to as first order shear deformation theories.

The 3D elasticity theory is reduced to a 2D theory using the assumption that the normal strains acting upon the plane parallel to the middle surface are negligible compared with other strain components. This assumption is generally valid except within the vicinity of a highly concentrated force (St. Venant’s principle). In other words, no stretching is assumed in the z-direction (i.e., e_{z} = 0). Assuming that normals to the midsurface strains remain straight during deformation but not normal, the displacements can be written as [

where u_{0}, v_{0} and w_{0} are midsurface displacements of the shell and y_{a} and y_{b} are midsurface rotations. An alternative derivation can be made with the assumption s_{z} = 0. The subscript (0) will refer to the middle surface in subsequent equations. The above equations describe a typical first-order shear deformation shell theory, and will constitute the only assumption made in this analysis when compared with the 3D theory of elasticity. As a result, strains are written as [

where the midsurface strains are:

and the curvature and twist changes are:

The force and moment resultants (Figures 3 and 4) are obtained by integrating the stresses over the shell thickness considering the (1 +) term that appears in the denominator of the stress resultant equations [

where are defined in [

It has been shown [1,5] that the above Equations (9) and (10) yield more accurate results when compared with those of plates and those traditionally used for shells [

The boundary terms for the boundaries with a = constant are

Similar equations can be obtained for b = constant.

Equations (9) and (10) are significantly different from those that cover most of first order shear deformation theories (FSDTs) for shells which neglect the effect of in the stress resultant equations. Asadi et al. [

studied static and free vibration of composite shells using Equations (9) and (10) and compared their results with other FSDTs and 3D elasticity results. They showed that presented FSDT improves the prediction of displacements, force resultants and moment resultants signifycantly.

Shear deformation theories were used by many authors (e.g. Qatu [

Piskunov et al. [

Zhen and Wanji [

In general, layer-wise laminate theories are used to properly represent local effects, such as interlaminar stress distribution, delaminations, etc. These theories are typically employed for cases involving anisotropic materials in which transverse shear effects cannot be ignored. Recent studies include Yuan et al. [

If the shell thickness is less than 1/20 of the other shell dimensions (e.g. length) and/or radii of curvature, a thin shell theory, where shear deformation and rotary inertia are negligible, is generally acceptable. Depending on various assumptions made during the derivation of the strain-displacement relations, stress-strain relations, and the equilibrium equations, various thin shell theories can be derived [

where the midsurface strains, curvature and twist changes are

and where

Applying Kirchhoff hypothesis of neglecting shear deformation and the assumption that e_{z} is negligible, the stress-strain equations for an element of material in the kth lamina may be written as [

where s_{a} and s_{b} are normal stress components, t_{a}_{b} is the in-plane shear stress component [_{a} and e_{b} are the normal strains, and g_{a}_{b} is the in-plane engineering shear strain. The terms Q_{ij} are the elastic stiffness coefficients for the material. If the shell coordinates (a,b) are parallel or perpendicular to the fibers, then the terms Q16 and Q26 are both zero. Stresses over the shell thickness (h) are integrated to get the force and moment resultants as given by

where A_{ij}, B_{ij}, and D_{ij} are the stiffness coefficients arising from the piecewise integration over the shell thickness (Equation (14b)). For shells which are laminated symmetrically with respect to their midsurfaces, all the B_{ij} terms become zero. Note that the above equations are the same as those for laminated plates, which are also valid for thin laminated shells. Using principle of virtual work yields the following equilibrium equations.

where

The following boundary conditions can be obtained for thin shells for a = constant (similar equations can be obtained for b = constant).

where and are, respectively, the start and end points of the shell in direction. Qatu and Asadi [

The magnitude of transverse displacement compared to shell thickness is the third criterion used in classifying shell equations. In many cases, nonlinear terms in the fundamental shell equations are expanded using perturbbation methods, and smaller orders of the rotations are retained. Most frequently, the first order only is retained and occasionally third orders have been included in nonlinear shell theories. In some shell problems, the material used can also be nonlinear (e.g., rubber, plastics and others). Theories that include materials nonlinearity are also referred to as nonlinear shell theories as well. The vast majority of shell theories, however, deal with geometric nonlinearity only.

Galishin and Shevchenko [

Other nonlinear analyses include Chaudhuri [

Shells may have different geometries based mainly on their curvature characteristics. In most shell geometries, the fundamental equations have to be treated at a very basic level. The equations are affected by the choice of the coordinate system, the characteristics of the Lame parameters and curvature [1-4]. Equations for cylindrical, spherical, conical and barrel shells can be derived from the equations of the more general case of shells of revolution. Equations for cylindrical, barrel, twisted and shallow shells can also be derived from the general equations of doubly curved shells. Cylindrical shells, doubly curved shallow shells, spherical and conical shells are the most treated geometries in research.

Bespalova and Urusova [

Shin et al. [

Studies on buckling of cylindrical shells include Wangi and Xiao [

Wang et al. [

Other analyses include those of Sheng and Ye [

Khare et al. [

Conical shells are other special cases of shells of revolution. For these shells, a straight line revolves about an axis to generate the surface. Wu et al. [

Spherical shells are other special cases of shells of revolution. For these shells, a circular arc, rather than a straight line, revolves about an axis to generate the surface. If the circular arc is half a circle and the axis of rotation is the circle’s own diameter, a closed sphere will result. Smithmaitrie and Tzou [

Tzou et al. [

Sai et al. [202,203] investigated shells with and without cut-outs. Other study includes Latifa and Sinha [

Analyses can be dynamic in nature. These include free and transient vibrations, wave propagation, dynamic stability, shock and impact loadings and others. These were covered in another review article [

Pinto Correiaa et al. [

Other static analyses include Alibeigloo and Nouri [

Lee and Lee [

Studies on buckling of cylindrical shells include Wangi and Xiao [

Other buckling analyses include Matsunaga [

Shin et al. [

Other studies on postbuckling analysis include Shen [36,37,43,69,124,127,128,133-135], Li and Shen [21,25, 26], Li [

Galishin [

Studies that treated thermal and/or hygrothermal effects include those of Li and Shen [21,25,26], Ruhi et al. [

Zhang et al. [

Other studies on failure of composite shells include those of Galishin [

Morozov [

Material complexity in composites occurs in various ways. Composite shells can have active or piezoelectric layers. They can also be braided or made of wood or natural fibers or a combination of materials.

Ren and Parvizi-Majidi [

Other studies on piezoelectric shells include Santos et al. [

Picha et al. [

Structural complexity occurs when the geometry or boundary conditions of the shells deviate from the classical shells described earlier. These include stiffened shells, shells with internal boundaries from cracks, imperfect shells as well as other types of complexities.

Ambur and Janunky [

Studies on stiffened composite shells include Prusty [

Several recent studies have focused on various composite shell structures with cutouts. Hillburger and Starnes [

Starnes and Hilburger [

Vasilenko et al. [

It is interesting to see that despite advances made in computational power, researchers avoided in general usage of 3D theory of elasticity. Experience shows that extensive usage of 3D elements in practical problems is not feasible even with advanced computers. Researchers looked for, developed and used thick shell theories to solve engineering problems. Finite element is the most used method in the analysis. Its ability to treat general boundary conditions, loading and geometry have certainly attributed to its popularity.

Cylindrical shells are still the subject of research of most recent articles. Doubly curved shallow shells have also received considerable interest. These shells can be spherical, barrel, cylindrical, or other shape.

Complicating effects of various kinds have received considerable interest. The use of piezoelectric shells necessitated by various applications and certain advanced materials resulted in considerable literature in the field. Other complicating effect of stiffened shells received some attention.

Looking at recent innovations in the area of composite plates, the authors think that it is a matter of time before these composites start making strong presence in research on shells. Areas of innovation include the use of natural fiber, single-walled and multi-walled carbon nanotubes, varying fiber orientation (both short and long fibers) as we as others. Such innovation are becoming more necessary as composite materials are required to deliver simultaneously structural functions (strength, stiffness, damping, toughness…) and non-structural ones (thermal and electrical conductivity). Both modeling and testing of such composites can be a corner-stone of future research on composite shells.

The authors thank Mr. Imran Aslam for his help gathering the papers.