Shallow Shells

Abstract

Shallow shells are open shells that have small curvatures (i.e. large radii of curvatures compared with other shell parameters such as length and width).

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Shallow shells are open shells that have small curvatures (i.e. large radii of curvatures compared with other shell parameters such as length and width). Vlasov (1951) and Leissa (1973) describe a shallow shell as follows:

Consider a shell outlined in part by some surface and which is a thin-walled spatial structure with a comparatively small rise above the plane covered by this structure. We call such shells shallow. If, for example, a building which has a rectangular floor plan is covered by a shell with a rise of not more than 1/5 of the smallest side of the rectangle lying in the plane of the supporting points of the structure, then we class such a spatial structure in the category of shallow shells.

These shells are sometimes referred to as curved plates. The general shell theory given in Chap. 1 can be readily applied to shallow shells. However, the general shell equations are rare complicated because the bending of a general shell is coupling with its stretching. When a shell is shallow as previous described, certain additional assumptions can be made to reduce the complexity in general shell equations considerably. The resulting set of equations is referred to as shallow shell theory. The development of the shallow shell theory is principally credited to, Leissa (1973), Leissa et al. (1984) and Qatu (2004), etc. The classical shallow shell theories (CSST) and shear deformation shallow shell theory (SDSST) are obtained by making following additional assumptions to the general shell theories (Qatu 2004):

• 1. The radii of curvature are very large compared to the inplane displacements (i.e., the curvature changes caused by the tangential displacement components u and v are small in a shallow shell, in comparison with changes caused by the normal component w). Also, the transverse shear forces are much smaller than the term R _i ∂N _i /∂i:

  $$ \frac{{u_{i} }}{{R_{i} }} \ll 1\quad \frac{{Q_{i} }}{{R_{i} }} \ll \frac{{\partial N_{i} }}{\partial i} $$

  (8.1)

  where u _i is either of the inplane displacement components u and v; Q _i is either of the shear forces Q _α and Q _β ; N _i is N _α , N _β or N _αβ ; and R _i is R _α , R _β or R _αβ ; The term ∂i indicates derivative with respect to either α or β;

• 2. The deepness term (1 + z/R _i ) is close to 1; where R _i can be R _α , R _β or R _αβ ;

• 3. The shell is shallow enough to be represented by the plane coordinate systems. For the case of rectangular orthotropy, this leads to constant Lamé parameters.

It should be stressed that the shallow shell theories should be used for maximum span to minimum radius ratio of 1/2 or less.

With the progress of composite materials, shallow shells constructed by composite laminas are extensively used in many fields of modern engineering practices requiring high strength-weight and stiffness-weight ratios such as aircraft structures, space vehicles. A complete understanding of the bucking, bending, vibration and other characteristics of these shells is of particular importance. Laminated shallow shells can be formed as rectangular, triangular, trapezoidal, circular or any other planforms and various types of curvatures such as singly-curved (e.g., cylindrical), double-curved (e.g., spherical, hyperbolic paraboloidal) or other complex shapes such as turbomachinery blades. In the context of this chapter, we consider laminated shallow shells formed in rectangular planform with rectangular orthotropy, in which the fibers in each layer to the planform being straight.

In recent decades, a huge amount of research efforts have been devoted to the vibration analysis of laminated shallow shells. So far, some of the static and dynamic behaviors of these shells with classical boundary conditions had being presented precisely. Some research papers and articles oriented to such contributions may be found in following enumeration. Fazzolari and Carrera (2013) developed a hierarchical trigonometric Ritz formulation for free vibration and dynamic response analysis of doubly-curved anisotropic laminated shallow and deep shells. Reddy and Asce (1984) presented the exact solutions of the equations and fundamental frequencies for simply supported, doubly-curved, cross-ply laminated shells. Khdeir and Reddy (1997) predicted free and force vibration of cross-ply laminated composite shallow arches by a generalized modal approach. Qatu (1995a, b) studied natural vibration of completely free laminated composite triangular and trapezoidal shallow shells. Soldatos and Shu (1999) used the five-degrees-of-freedom shallow shell theory in the stress analysis of cross-ply laminated plates and shallow shell panels having a rectangular plan-form. Some other contributors in this subject are Dogan and Arslan (2009), Ghavanloo and Fazelzadeh (2013), Kurpa et al. (2010), Leissa and Chang (1996), Librescu et al. (1989a, b), Qatu (1995a, b, 1996, 2011), Qatu and Leissa (1991), Singh and Kumar (1996). More detailed and systematic summarizations can be seen in the excellent monographs by Leissa (1973), Qatu (2002a, b, 2004), Qatu et al. (2010), and Reddy (2003).

In this chapter, we consider vibration of thin and moderately thick laminated shallow shells with general boundary conditions. Fundamental equations of thin and thick shallow shells are presented in the first and second sections, respectively. Then, numerous vibration results of thin and thick laminated shallow shells with different boundary conditions, lamination schemes and geometry parameters are given in the third section by using the SDSST and the modified Fourier series. The results are obtained by applying the weak form solution procedure.

Consider a laminated shallow shell in rectangular planform as shown in Fig. 8.1, the length, width and thickness of the shell are represented by a, b and h, respectively. The shell is shallow enough so that is can be represented by the orthogonal Cartesian coordinate system (x, y and z). The laminated shallow shell is characterized by its middle surface, which can be defined by (Qatu 2004):

$$ z = - \frac{1}{2}\left( {\frac{{x^{2} }}{{R_{x} }} + \frac{xy}{{R_{xy} }} + \frac{{y^{2} }}{{R_{y} }}} \right) $$

(8.2)

where R _x and R _y represent the radii of curvature in the x, y directions as depicted in Fig. 8.1. R _xy is the corresponding radius of twist. In the present chapter, we focus on the cases when R _x , R _y and R _xy are constants. In addition, the x and y coordinates are conveniently oriented to be parallel to boundaries so that R _xy  = ∞. The displacements of the shell in the x, y and z directions are denoted by u, v and w, respectively.

Fig. 8.1

Schematic diagram of laminated shallow shells with rectangular planform

In Fig. 8.2, shallow shells constructed as various types of curvatures are plotted. It can be flat (i.e. R _x  = R _y  = R _xy  = ∞), spherical (e.g., R _x /R _y  > 0, R _xy  = ∞), circular cylindrical (e.g., R _y  = R _xy  = ∞) and hyperbolic paraboloidal (e.g., R _x /R _y  < 0, R _xy  = ∞).

Fig. 8.2

Rectangular planform shallow shells with various types of curvatures

Considering the shallow shell in Fig. 8.1 and its Cartesian coordinate system, the coordinates, characteristics of the Lamé parameters and radius of curvatures are:

$$ \alpha = x\quad \beta = y\quad A = B = 1\quad R_{\alpha } = R_{x} \quad R_{\beta } = R_{y} $$

(8.3)

The equations of shallow shells are a reduction of the general shell equations given in Chap. 1 by substituting Eq. (8.3) into such shell equations.

8.1 Fundamental Equations of Thin Laminated Shallow Shells

Fundamental equations of thin laminated shallow shells are given here by substituting Eq. (8.3) into the general thin shell equations developed in Sect. 1.2. Similarly, the equations are given for the general dynamic analysis.

8.1.1 Kinematic Relations

Substituting Eq. (8.3) into Eq. (1.7), the middle surface strains and curvature changes of thin laminated shallow shells can be written in terms of middle surface displacements. Taking Eq. (8.1) into consideration, they are given as:

$$ \begin{aligned} \varepsilon_{x}^{0} & = \frac{\partial u}{\partial x} + \frac{w}{{R_{x} }},\quad \chi_{x} = - \frac{{\partial^{2} w}}{{\partial x^{2} }} \\ \varepsilon_{y}^{0} & = \frac{\partial v}{\partial y} + \frac{w}{{R_{y} }},\quad \chi_{y} = - \frac{{\partial^{2} w}}{{\partial y^{2} }} \\ \gamma_{xy}^{0} & = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x},\quad \chi_{xy} = - 2\frac{{\partial^{2} w}}{\partial x\partial y} \\ \end{aligned} $$

(8.4)

where $\varepsilon_{x}^{0}$, $\varepsilon_{y}^{0}$ and $\gamma_{xy}^{0}$ indicate the strains in middle surface; χ _x , χ _y and χ _xy are the curvature changes. Thus, the liner strains in the kth layer space of a laminated shallow shell can be defined as:

$$ \begin{aligned} \varepsilon_{x} & = \varepsilon_{x}^{0} + z\chi_{x} \\ \varepsilon_{y} & = \varepsilon_{y}^{0} + z\chi_{y} \\ \gamma_{xy} & = \gamma_{xy}^{0} + z\chi_{xy} \\ \end{aligned} $$

(8.5)

where Z _k  < z < Z _k+1. Z _k and Z _k+1 denote the distances from the top surface and the bottom surface of the layer to the referenced middle surface, respectively.

8.1.2 Stress-Strain Relations and Stress Resultants

The stress-strain relations, force and moment resultants for thin laminated shallow shells formed in rectangular planform are the same as those derived earlier for thin laminated rectangular plates, see Eqs. (4.4) and (4.6).

8.1.3 Energy Functions

The strain energy (U _s ) of thin laminated shallow shells during vibration can be defined in terms of middle surface strains, curvature changes and stress resultants as:

$$ U_{s} = \frac{1}{2}\int\limits_{0}^{a} {\int\limits_{0}^{b} {\left\{ \begin{aligned} & N_{x} \varepsilon_{x}^{0} + N_{y} \varepsilon_{y}^{0} + N_{xy} \gamma_{xy}^{0} \hfill \\ & + M_{x} \chi_{x} + M_{y} \chi_{y} + M_{xy} \chi_{xy} \hfill \\ \end{aligned} \right\}} } dxdy $$

(8.6)

And the kinetic energy (T) of thin laminated shallow shells during vibration is:

$$ T = \frac{1}{2}\int\limits_{0}^{a} {\int\limits_{0}^{b} {I_{0} \left\{ {\left( {\frac{\partial u}{\partial t}} \right)^{2} + \left( {\frac{\partial v}{\partial t}} \right)^{2} + \left( {\frac{\partial w}{\partial t}} \right)^{2} } \right\}} } dxdy $$

(8.7)

where the inertia term I ₀ is given in Eq. (1.19). Suppose q _x , q _y and q _z are the external loads in the x, y and z directions, respectively. Thus, the external work can be expressed as:

$$ W_{e} = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left\{ {q_{x} u + q_{y} v + q_{z} w} \right\}} } dxdy $$

(8.8)

The artificial spring boundary technique is adopted here to realize the general boundary conditions of a thin shallow shell. Under the current framework, symbols $k_{\psi }^{u}$, $k_{\psi }^{v}$, $k_{\psi }^{w}$ and $K_{\psi }^{w}$ (ψ = x0, y0, x1 and y1) are used to indicate the stiffness of the boundary spring components at the boundaries x = 0, y = 0, x = a and y = b, respectively, see Fig. 8.3. Thus, the deformation strain energy about the boundary springs (U _sp ) can be defined as:

$$ \begin{aligned} U_{sp} & = \frac{1}{2}\int\limits_{0}^{b} {\left\{ \begin{aligned} & \left[ {k_{x0}^{u} u^{2} + k_{x0}^{v} v^{2} + k_{x0}^{w} w^{2} + K_{x0}^{w} \left( {\partial w/\partial x} \right)} \right]\left| {_{x = 0} } \right. \hfill \\ & + \left[ {k_{x1}^{u} u^{2} + k_{x1}^{v} v^{2} + k_{x1}^{w} w^{2} + K_{x1}^{w} \left( {\partial w/\partial x} \right)} \right]\left| {_{x = a} } \right. \hfill \\ \end{aligned} \right\}} dy \\ & \quad + \frac{1}{2}\int\limits_{0}^{a} {\left\{ \begin{aligned} & \left[ {k_{y0}^{u} u^{2} + k_{y0}^{v} v^{2} + k_{y0}^{w} w^{2} + K_{y0}^{w} \left( {\partial w/\partial y} \right)} \right]\left| {_{y = 0} } \right. \hfill \\ & + \left[ {k_{y1}^{u} u^{2} + k_{y1}^{v} v^{2} + k_{y1}^{w} w^{2} + K_{y1}^{w} \left( {\partial w/\partial y} \right)} \right]\left| {_{y = b} } \right. \hfill \\ \end{aligned} \right\}dx} \\ \end{aligned} $$

(8.9)

Fig. 8.3

Boundary conditions of thin laminated shallow shells

8.1.4 Governing Equations and Boundary Conditions

Substituting Eq. (8.3) into Eq. (1.28) and then simplifying the expressions and taking Eq. (8.1) into consideration, the reduced governing equations of thin laminated shallow shells are:

$$ \begin{array}{l} \frac{{\partial N_{x} }}{\partial x} + \frac{{\partial N_{xy} }}{\partial y} + q_{x} = I_{0} \frac{{\partial^{2} u}}{{\partial t^{2} }} \\ \frac{{\partial N_{xy} }}{\partial x} + \frac{{\partial N_{y} }}{\partial y} + q_{y} = I_{0} \frac{{\partial^{2} v}}{{\partial t^{2} }} \\ - \left( {\frac{{N_{x} }}{{R_{x} }} + \frac{{N_{y} }}{{R_{y} }}} \right) + \frac{{\partial^{2} M_{x} }}{{\partial x^{2} }} + 2\frac{{\partial^{2} M_{xy} }}{\partial x\partial y} + \frac{{\partial^{2} M_{y} }}{{\partial y^{2} }} + q_{z} = I_{0} \frac{{\partial^{2} w}}{{\partial t^{2} }} \end{array} $$

(8.10)

Substituting Eq. (8.4) and the stress resultants equations Eq. (4.6) into Eq. (8.10), the governing equations of thin laminated shallow shells can be writing in matrix form as:

$$ \left( {\left[ {\begin{array}{*{20}c} {L_{11} } & {L_{12} } & {L_{13} } \\ {L_{21} } & {L_{22} } & {L_{23} } \\ {L_{31} } & {L_{32} } & {L_{33} } \\ \end{array} } \right] - \omega^{2} \left[ {\begin{array}{*{20}c} { - I_{0} } & 0 & 0 \\ 0 & { - I_{0} } & 0 \\ 0 & 0 & { - I_{0} } \\ \end{array} } \right]} \right)\left[ {\begin{array}{*{20}c} u \\ v \\ w \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - p_{x} } \\ { - p_{y} } \\ { - p_{z} } \\ \end{array} } \right] $$

(8.11)

The coefficients of the linear operator L _ij are written as

$$ \begin{aligned} L_{11} & = \frac{\partial }{\partial x}\left( {A_{11} \frac{\partial }{\partial x} + A_{16} \frac{\partial }{\partial y}} \right) + \frac{\partial }{\partial y}\left( {A_{16} \frac{\partial }{\partial x} + A_{66} \frac{\partial }{\partial y}} \right) \\ L_{12} & = L_{21} = \frac{\partial }{\partial x}\left( {A_{12} \frac{\partial }{\partial y} + A_{16} \frac{\partial }{\partial x}} \right) + \frac{\partial }{\partial y}\left( {A_{26} \frac{\partial }{\partial y} + A_{66} \frac{\partial }{\partial x}} \right) \\ L_{13} & = \frac{\partial }{\partial x}\left( {\frac{{A_{11} }}{{R_{x} }} + \frac{{A_{12} }}{{R_{y} }} - B_{11} \frac{{\partial^{2} }}{{\partial x^{2} }} - B_{12} \frac{{\partial^{2} }}{{\partial y^{2} }} - 2B_{16} \frac{{\partial^{2} }}{\partial x\partial y}} \right) \\ & \quad + \frac{\partial }{\partial y}\left( {\frac{{A_{16} }}{{R_{x} }} + \frac{{A_{26} }}{{R_{y} }} - B_{16} \frac{{\partial^{2} }}{{\partial x^{2} }} - B_{26} \frac{{\partial^{2} }}{{\partial y^{2} }} - 2B_{66} \frac{{\partial^{2} }}{\partial x\partial y}} \right) \\ L_{22} & = \frac{\partial }{\partial x}\left( {A_{26} \frac{\partial }{\partial y} + A_{66} \frac{\partial }{\partial x}} \right) + \frac{\partial }{\partial y}\left( {A_{22} \frac{\partial }{\partial y} + A_{26} \frac{\partial }{\partial x}} \right) \\ L_{23} & = \frac{\partial }{\partial x}\left( {\frac{{A_{16} }}{{R_{x} }} + \frac{{A_{26} }}{{R_{y} }} - B_{16} \frac{{\partial^{2} }}{{\partial x^{2} }} - B_{26} \frac{{\partial^{2} }}{{\partial y^{2} }} - 2B_{66} \frac{{\partial^{2} }}{\partial x\partial y}} \right) \\ & \quad + \frac{\partial }{\partial y}\left( {\frac{{A_{12} }}{{R_{x} }} + \frac{{A_{22} }}{{R_{y} }} - B_{12} \frac{{\partial^{2} }}{{\partial x^{2} }} - B_{22} \frac{{\partial^{2} }}{{\partial y^{2} }} - 2B_{26} \frac{{\partial^{2} }}{\partial x\partial y}} \right) \\ L_{31} & = - \frac{1}{{R_{x} }}\left( {A_{11} \frac{\partial }{\partial x} + A_{16} \frac{\partial }{\partial y}} \right) - \frac{1}{{R_{y} }}\left( {A_{12} \frac{\partial }{\partial x} + A_{26} \frac{\partial }{\partial y}} \right) \\ & \quad + \frac{{\partial^{2} }}{{\partial x^{2} }}\left( {B_{11} \frac{\partial }{\partial x} + B_{16} \frac{\partial }{\partial y}} \right) + 2\frac{{\partial^{2} }}{\partial x\partial y}\left( {B_{16} \frac{\partial }{\partial x} + B_{66} \frac{\partial }{\partial y}} \right) \\ & \quad + \frac{{\partial^{2} }}{{\partial y^{2} }}\left( {B_{12} \frac{\partial }{\partial x} + B_{26} \frac{\partial }{\partial y}} \right) \\ L_{32} & = - \frac{1}{{R_{x} }}\left( {A_{12} \frac{\partial }{\partial y} + A_{16} \frac{\partial }{\partial x}} \right) - \frac{1}{{R_{y} }}\left( {A_{22} \frac{\partial }{\partial y} + A_{26} \frac{\partial }{\partial x}} \right) \\ & \quad + \frac{{\partial^{2} }}{{\partial x^{2} }}\left( {B_{12} \frac{\partial }{\partial y} + B_{16} \frac{\partial }{\partial x}} \right) + 2\frac{{\partial^{2} }}{\partial x\partial y}\left( {B_{26} \frac{\partial }{\partial y} + B_{66} \frac{\partial }{\partial x}} \right) \\ & \quad + \frac{{\partial^{2} }}{{\partial y^{2} }}\left( {B_{22} \frac{\partial }{\partial y} + B_{26} \frac{\partial }{\partial x}} \right) \\ L_{33} & = - \frac{1}{{R_{x} }}\left( {\frac{{A_{11} }}{{R_{x} }} + \frac{{A_{12} }}{{R_{y} }} - B_{11} \frac{{\partial^{2} }}{{\partial x^{2} }} - B_{12} \frac{{\partial^{2} }}{{\partial y^{2} }} - 2B_{16} \frac{{\partial^{2} }}{\partial x\partial y}} \right) \\ & \quad - \frac{1}{{R_{y} }}\left( {\frac{{A_{12} }}{{R_{x} }} + \frac{{A_{22} }}{{R_{y} }} - B_{12} \frac{{\partial^{2} }}{{\partial x^{2} }} - B_{22} \frac{{\partial^{2} }}{{\partial y^{2} }} - 2B_{26} \frac{{\partial^{2} }}{\partial x\partial y}} \right) \\ & \quad + \frac{{\partial^{2} }}{{\partial x^{2} }}\left( {\frac{{B_{11} }}{{R_{x} }} + \frac{{B_{12} }}{{R_{y} }} - D_{11} \frac{{\partial^{2} }}{{\partial x^{2} }} - D_{12} \frac{{\partial^{2} }}{{\partial y^{2} }} - 2D_{16} \frac{{\partial^{2} }}{\partial x\partial y}} \right) \\ & \quad + 2\frac{{\partial^{2} }}{\partial x\partial y}\left( {\frac{{B_{16} }}{{R_{x} }} + \frac{{B_{26} }}{{R_{y} }} - D_{16} \frac{{\partial^{2} }}{{\partial x^{2} }} - D_{26} \frac{{\partial^{2} }}{{\partial y^{2} }} - 2D_{66} \frac{{\partial^{2} }}{\partial x\partial y}} \right) \\ & \quad + \frac{{\partial^{2} }}{{\partial y^{2} }}\left( {\frac{{B_{12} }}{{R_{x} }} + \frac{{B_{22} }}{{R_{y} }} - D_{12} \frac{{\partial^{2} }}{{\partial x^{2} }} - D_{22} \frac{{\partial^{2} }}{{\partial y^{2} }} - 2D_{26} \frac{{\partial^{2} }}{\partial x\partial y}} \right) \\ \end{aligned} $$

(8.12)

And according to Eqs. (1.29) and (1.30), the boundary conditions of thin shallow shells are:

$$ \begin{aligned} x & =0:\left\{ {\begin{array}{*{20}l} {N_{x} + \frac{{M_{x} }}{{R_{x} }} - k_{x0}^{u} u = 0} \hfill \\ {N_{xy} + \frac{{M_{xy} }}{{R_{y} }} - k_{x0}^{v} v = 0} \hfill \\ {Q_{x} + \frac{{\partial M_{xy} }}{\partial y} - k_{x0}^{w} w = 0} \hfill \\ { - M_{x} - K_{x0}^{w} \frac{\partial w}{\partial x} = 0} \hfill \\ \end{array} } \right.\quad x=a:\left\{ {\begin{array}{*{20}l} {N_{x} + \frac{{M_{x} }}{{R_{x} }} + k_{x1}^{u} u = 0} \hfill \\ {N_{xy} + \frac{{M_{xy} }}{{R_{y} }} + k_{x1}^{v} v = 0} \hfill \\ {Q_{x} + \frac{{\partial M_{xy} }}{\partial y} + k_{x1}^{w} w = 0} \hfill \\ { - M_{x} + K_{x1}^{w} \frac{\partial w}{\partial x} = 0} \hfill \\ \end{array} } \right. \\ y & =0:\left\{ {\begin{array}{*{20}l} {N_{xy} + \frac{{M_{xy} }}{{R_{x} }} - k_{y0}^{u} u=0} \hfill \\ {N_{y} + \frac{{M_{y} }}{{R_{y} }} - k_{y0}^{v} v=0} \hfill \\ {Q_{y} + \frac{{\partial M_{xy} }}{\partial x} - k_{y0}^{w} w=0} \hfill \\ { - M_{y} - K_{y0}^{w} \frac{\partial w}{\partial y}=0} \hfill \\ \end{array} } \right.\quad y=0:\left\{ {\begin{array}{*{20}l} {N_{xy} + \frac{{M_{xy} }}{{R_{x} }} + k_{y1}^{u} u=0} \hfill \\ {N_{y} + \frac{{M_{y} }}{{R_{y} }} + k_{y1}^{v} v=0} \hfill \\ {Q_{y} + \frac{{\partial M_{xy} }}{\partial x} + k_{y1}^{w} w=0} \hfill \\ { - M_{y} + K_{y1}^{w} \frac{\partial w}{\partial y}=0} \hfill \\ \end{array} } \right. \\ \end{aligned} $$

(8.13)

In each boundary of thin laminated shallow shells, there exists 12 possible classical boundary conditions. Taking boundaries x = constant for example, the possible classical boundary conditions are given in Table 8.1, similar boundary conditions can be obtained for boundaries y = constant.

Table 8.1 Possible classical boundary conditions for thin shallow shells at each boundary of x = constant

8.2 Fundamental Equations of Thick Laminated Shallow Shells

Like general thin shell theory, the classical shallow shell theory (CSST) is applicable where the thickness is smaller than 1/20 of the smallest of the wave length and/or radii of curvature (Qatu 2004). For thick shallow shells, the Kirchhoff hypothesis should be relaxed and the shear deformation and rotary inertia should be included in the formulation. Fundamental equations of thick laminated shallow shells will be derived in this section. The following equations are derived from the general shear deformation shell theory (SDST) described in Sect. 1.3 by imposing Eq. (8.3) into those of general shells.

8.2.1 Kinematic Relations

As was done for general shells and plates, we assume that normals to the undeformed middle surface remain straight but do not normal to the deformed middle surface and the shell inplane displacements are expanded in terms of shell thickness of first order expansion. Thus, the displacement field of a thick shallow shell can be expressed as

$$ \begin{aligned} U(x,y,z) & = u(x,y) + z\phi_{x} \\ V(x,y,z) & = v(x,y) + z\phi_{y} \\ W(x,y,z) & = w(x,y) \\ \end{aligned} $$

(8.14)

where u, v and w are the middle surface displacements of the shallow shell in the x, y and z directions, respectively, and ϕ _x and ϕ _y represent the rotations of transverse normal respect to y- and x-axes. Substituting α = x, β = y into Eq. (1.33) and deleting the z/R _α and z/R _β terms, the normal and shear strains at any point in the shallow shell then can be written in terms of middle surface strains and curvature changes as:

$$ \begin{aligned} \varepsilon_{x} & = \varepsilon_{x}^{0} + z\chi_{x} ,\quad \gamma_{xz} = \gamma_{xz}^{0} \\ \varepsilon_{y} & = \varepsilon_{y}^{0} + z\chi_{y} , \quad \gamma_{yz} = \gamma_{yz}^{0} \\ \gamma_{xy} & = \gamma_{xy}^{0} + z\chi_{xy} \\ \end{aligned} $$

(8.15)

Substituting Eq. (8.3) into Eq. (1.34) and taking Eq. (8.1) into consideration, the middle surface strains and curvature changes are:

$$ \begin{aligned} \varepsilon_{x}^{0} & = \frac{\partial u}{\partial x} + \frac{w}{{R_{x} }}, \quad \chi_{x} = \frac{{\partial \phi_{x} }}{\partial x} \\ \varepsilon_{y}^{0} & = \frac{\partial v}{\partial y} + \frac{w}{{R_{y} }}, \quad \chi_{y} = \frac{{\partial \phi_{y} }}{\partial y} \\ \gamma_{xy}^{0} & = \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y},\quad \chi_{xy} = \frac{{\partial \phi_{y} }}{\partial x} + \frac{{\partial \phi_{x} }}{\partial y} \\ \gamma_{xz}^{0} & = \frac{\partial w}{\partial x} + \phi_{x} ,\quad \gamma_{yz}^{0} = \frac{\partial w}{\partial y} + \phi_{y} \\ \end{aligned} $$

(8.16)

The above equations constitute the fundamental strain-displacement relations of a thick shallow shell formed in rectangular planform.

8.2.2 Stress-Strain Relations and Stress Resultants

The stress-strain relations and stress resultants for thick laminated shallow shells formed in rectangular planform are the same as those derived earlier for thick laminated rectangular plates, see Eqs. (4.19), (4.21) and (4.22).

8.2.3 Energy Functions

The strain energy (U _s ) of thick shallow shells during vibration can be defined as

$$ U_{s} = \frac{1}{2}\int\limits_{0}^{a} {\int\limits_{0}^{b} {\left\{ \begin{aligned} & N_{x} \varepsilon_{x}^{0} + N_{y} \varepsilon_{y}^{0} + N_{xy} \varepsilon_{xy}^{0} + N_{yx} \varepsilon_{yx}^{0} + M_{x} \chi_{x} \hfill \\ & + M_{y} \chi_{y} + M_{xy} \chi_{xy} + M_{yx} \chi_{yx} + Q_{y} \gamma_{yz} + Q_{x} \gamma_{xz} \hfill \\ \end{aligned} \right\}} } dydx $$

(8.17)

And the corresponding kinetic energy function (T) is:

$$ T\begin{array}{*{20}c} = \\ \end{array} \frac{1}{2}\int\limits_{0}^{a} {\int\limits_{0}^{b} {\left\{ {\begin{array}{*{20}l} {I_{0} \left( {\frac{\partial u}{\partial t}} \right)^{2} + 2I_{1} \frac{\partial u}{\partial t}\frac{{\partial \phi_{x} }}{\partial t} + I_{2} \left( {\frac{{\partial \phi_{x} }}{\partial t}} \right)^{2} + I_{0} \left( {\frac{\partial v}{\partial t}} \right)^{2} } \hfill \\ { + 2I_{1} \frac{\partial v}{\partial t}\frac{{\partial \phi_{y} }}{\partial t} + I_{2} \left( {\frac{{\partial \phi_{y} }}{\partial t}} \right)^{2} + I_{0} \left( {\frac{\partial w}{\partial t}} \right)^{2} } \hfill \\ \end{array} } \right\}dydx} } $$

(8.18)

The inertia terms I _i (i = 0, 1, 2) are written as in Eq. (1.52). Assuming the distributed external forces q _x , q _y and q _z are in the x, y and z directions, respectively and m _x and m _y represent the external couples in the middle surface, thus, the work done by the external forces and moments is

$$ W_{e} = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left\{ {q_{x} u + q_{y} v + q_{z} w + m_{x} \phi_{x} + m_{y} \phi_{y} } \right\}} } dydx $$

(8.19)

As was done for thin laminated shallow shells, the artificial spring boundary technique is adopted here to realize the general boundary conditions of thick laminated shallow shells, in which three groups of linear springs (k _u , k _v , k _w ) and two groups of rotational springs (K _x , K _y ) are distributed uniformly along each shell boundary artificially, see Fig. 8.4. Therefore, the deformation strain energy (U _sp ) of the boundary springs during vibration can be written as:

$$ \begin{aligned} U_{sp} & = \frac{1}{2}\int\limits_{0}^{b} {\left\{ \begin{aligned} & \left[ {k_{x0}^{u} u^{2} + k_{x0}^{v} v^{2} + k_{x0}^{w} w^{2} + K_{x0}^{x} \phi_{x}^{2} + K_{x0}^{y} \phi_{y}^{2} } \right]\left| {_{x = 0} } \right. \hfill \\ & + \left[ {k_{x1}^{u} u^{2} + k_{x1}^{v} v^{2} + k_{x1}^{w} w^{2} + K_{x1}^{x} \phi_{x}^{2} + K_{x1}^{y} \phi_{y}^{2} } \right]\left| {_{x = a} } \right. \hfill \\ \end{aligned} \right\}} dy \\ & \quad + \frac{1}{2}\int\limits_{0}^{a} {\left\{ \begin{aligned} & \left[ {k_{y0}^{u} u^{2} + k_{y0}^{v} v^{2} + k_{y0}^{w} w^{2} + K_{y0}^{x} \phi_{x}^{2} + K_{y0}^{y} \phi_{y}^{2} } \right]\left| {_{y = 0} } \right. \hfill \\ & + \left[ {k_{y1}^{u} u^{2} + k_{y1}^{v} v^{2} + k_{y1}^{w} w^{2} + K_{y0}^{x} \phi_{x}^{2} + K_{y0}^{y} \phi_{y}^{2} } \right]\left| {_{y = b} } \right. \hfill \\ \end{aligned} \right\}dx} \\ \end{aligned} $$

(8.20)

where $k_{\psi }^{u}$, $k_{\psi }^{v}$, $k_{\psi }^{w}$, $K_{\psi }^{x}$ and $K_{\psi }^{y}$ (ψ = x0, y0, x1 and y1) represent the rigidities (per unit length) of the boundary springs at the boundaries x = 0, y = 0, x = a and y = b, respectively. In such case, the stiffness of the boundary springs can take any value from zero to infinity to better model many real-world boundary conditions including all the classical boundaries and the uniform elastic ones. For instance, the clamped restraint condition is essentially obtained by setting the spring stiffness substantially larger than the bending rigidity of the involved shallow shell (10⁷ × D).

Fig. 8.4

Boundary conditions of thick laminated shallow shells

8.2.4 Governing Equations and Boundary Conditions

The governing equations and boundary conditions of thick laminated shallow shells can be derived by the Hamilton’s principle in the same manner as described in Sect. 1.2.4 or specializing from those of general thick shells. Substituting Eq. (8.3) into Eq. (1.59) and taking Eq. (8.1) into consideration, the governing equations for thick shallow shells are

$$ \begin{array}{l} \frac{{\partial N_{x} }}{\partial x} + \frac{{\partial N_{xy} }}{\partial y} + q_{x} =I_{0} \frac{{\partial^{2} u}}{{\partial t^{2} }} + I_{1} \frac{{\partial^{2} \phi_{x} }}{{\partial t^{2} }} \\ \frac{{\partial N_{xy} }}{\partial x} + \frac{{\partial N_{y} }}{\partial y} + q_{y} =I_{0} \frac{{\partial^{2} v}}{{\partial t^{2} }} + I_{1} \frac{{\partial^{2} \phi_{y} }}{{\partial t^{2} }} \\ - \left( {\frac{{N_{x} }}{{R_{x} }} + \frac{{N_{y} }}{{R_{y} }}} \right) + \frac{{\partial Q_{x} }}{\partial x} + \frac{{\partial Q_{y} }}{\partial y} + q_{z} =I_{0} \frac{{\partial^{2} w}}{{\partial t^{2} }} \\ \frac{{\partial M_{x} }}{\partial x} + \frac{{\partial M_{xy} }}{\partial y} - Q_{x} + m_{x} = I_{1} \frac{{\partial^{2} u}}{{\partial t^{2} }} + I_{2} \frac{{\partial^{2} \phi_{x} }}{{\partial t^{2} }} \\ \frac{{\partial M_{xy} }}{\partial x} + \frac{{\partial M_{y} }}{\partial y} - Q_{y} + m_{y} =I_{1} \frac{{\partial^{2} v}}{{\partial t^{2} }} + I_{2} \frac{{\partial^{2} \phi_{y} }}{{\partial t^{2} }} \\ \end{array} $$

(8.21)

Furthermore, the above equations can be written in terms of displacements and rotation components as

$$ \left( {\left[ {\begin{array}{*{20}c} {L_{11} } & {L_{12} } & {L_{13} } & {L_{14} } & {L_{15} } \\ {L_{21} } & {L_{22} } & {L_{23} } & {L_{24} } & {L_{25} } \\ {L_{31} } & {L_{32} } & {L_{33} } & {L_{34} } & {L_{35} } \\ {L_{41} } & {L_{42} } & {L_{43} } & {L_{44} } & {L_{45} } \\ {L_{51} } & {L_{52} } & {L_{53} } & {L_{54} } & {L_{55} } \\ \end{array} } \right] - \omega^{2} \left[ {\begin{array}{*{20}c} {M_{11} } \\ 0 \\ 0 \\ {M_{41} } \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ {M_{22} } \\ 0 \\ 0 \\ {M_{52} } \\ \end{array} \begin{array}{*{20}c} 0 \\ 0 \\ {M_{33} } \\ 0 \\ 0 \\ \end{array} \begin{array}{*{20}c} {M_{14} } \\ 0 \\ 0 \\ {M_{44} } \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ {M_{25} } \\ 0 \\ 0 \\ {M_{55} } \\ \end{array} } \right]} \right)\left[ {\begin{array}{*{20}c} u \\ v \\ w \\ {\phi_{x} } \\ {\phi_{y} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - p_{x} } \\ { - p_{y} } \\ { - p_{z} } \\ { - m_{x} } \\ { - m_{y} } \\ \end{array} } \right] $$

(8.22)

The coefficients of the linear operator L _ij are listed below

$$ \begin{aligned} L_{11} & = \frac{\partial }{\partial x}\left( {A_{11} \frac{\partial }{\partial x} + A_{16} \frac{\partial }{\partial y}} \right) + \frac{\partial }{\partial y}\left( {A_{16} \frac{\partial }{\partial x} + A_{66} \frac{\partial }{\partial y}} \right) \\ L_{12} & = \frac{\partial }{\partial x}\left( {A_{12} \frac{\partial }{\partial y} + A_{16} \frac{\partial }{\partial x}} \right) + \frac{\partial }{\partial y}\left( {A_{26} \frac{\partial }{\partial y} + A_{66} \frac{\partial }{\partial x}} \right) \\ L_{13} & = \frac{\partial }{\partial x}\left( {\frac{{A_{11} }}{{R_{x} }} + \frac{{A_{12} }}{{R_{y} }}} \right) + \frac{\partial }{\partial y}\left( {\frac{{A_{16} }}{{R_{x} }} + \frac{{A_{26} }}{{R_{y} }}} \right) \\ L_{14} & = \frac{\partial }{\partial x}\left( {B_{11} \frac{\partial }{\partial x} + B_{16} \frac{\partial }{\partial y}} \right) + \frac{\partial }{\partial y}\left( {B_{16} \frac{\partial }{\partial x} + B_{66} \frac{\partial }{\partial y}} \right) \\ L_{15} & = \frac{\partial }{\partial x}\left( {B_{12} \frac{\partial }{\partial y} + B_{16} \frac{\partial }{\partial x}} \right) + \frac{\partial }{\partial y}\left( {B_{26} \frac{\partial }{\partial y} + B_{66} \frac{\partial }{\partial x}} \right) \\ L_{21} & = \frac{\partial }{\partial x}\left( {A_{16} \frac{\partial }{\partial x} + A_{66} \frac{\partial }{\partial y}} \right) + \frac{\partial }{\partial y}\left( {A_{12} \frac{\partial }{\partial x} + A_{26} \frac{\partial }{\partial y}} \right) \\ L_{22} & = \frac{\partial }{\partial x}\left( {A_{26} \frac{\partial }{\partial y} + A_{66} \frac{\partial }{\partial x}} \right) + \frac{\partial }{\partial y}\left( {A_{22} \frac{\partial }{\partial y} + A_{26} \frac{\partial }{\partial x}} \right) \\ L_{23} & = \frac{\partial }{\partial x}\left( {\frac{{A_{16} }}{{R_{x} }} + \frac{{A_{26} }}{{R_{y} }}} \right) + \frac{\partial }{\partial y}\left( {\frac{{A_{12} }}{{R_{x} }} + \frac{{A_{22} }}{{R_{y} }}} \right) \\ L_{24} & = \frac{\partial }{\partial x}\left( {B_{16} \frac{\partial }{\partial x} + B_{66} \frac{\partial }{\partial y}} \right) + \frac{\partial }{\partial y}\left( {B_{12} \frac{\partial }{\partial x} + B_{26} \frac{\partial }{\partial y}} \right) \\ L_{25} & = \frac{\partial }{\partial x}\left( {B_{26} \frac{\partial }{\partial y} + B_{66} \frac{\partial }{\partial x}} \right) + \frac{\partial }{\partial y}\left( {B_{22} \frac{\partial }{\partial y} + B_{26} \frac{\partial }{\partial x}} \right) \\ L_{31} & = - \frac{1}{{R_{x} }}\left( {A_{11} \frac{\partial }{\partial x} + A_{16} \frac{\partial }{\partial y}} \right) - \frac{1}{{R_{y} }}\left( {A_{12} \frac{\partial }{\partial x} + A_{26} \frac{\partial }{\partial y}} \right) \\ L_{32} & = - \frac{1}{{R_{x} }}\left( {A_{12} \frac{\partial }{\partial y} + A_{16} \frac{\partial }{\partial x}} \right) - \frac{1}{{R_{y} }}\left( {A_{22} \frac{\partial }{\partial y} + A_{26} \frac{\partial }{\partial x}} \right) \\ L_{33} & = - \frac{1}{{R_{x} }}\left( {\frac{{A_{11} }}{{R_{x} }} + \frac{{A_{12} }}{{R_{y} }}} \right) - \frac{1}{{R_{y} }}\left( {\frac{{A_{11} }}{{R_{x} }} + \frac{{A_{12} }}{{R_{y} }}} \right) + \frac{\partial }{\partial x}\left( {A_{45} \frac{\partial }{\partial y} + A_{55} \frac{\partial }{\partial x}} \right) \\ & \quad + \frac{\partial }{\partial y}\left( {A_{44} \frac{\partial }{\partial y} + A_{45} \frac{\partial }{\partial x}} \right) \\ L_{34} & = - \frac{1}{{R_{x} }}\left( {B_{11} \frac{\partial }{\partial x} + B_{16} \frac{\partial }{\partial y}} \right) - \frac{1}{{R_{y} }}\left( {B_{12} \frac{\partial }{\partial x} + B_{26} \frac{\partial }{\partial y}} \right) + A_{55} \frac{\partial }{\partial x} + A_{45} \frac{\partial }{\partial y} \\ L_{35} & = - \frac{1}{{R_{x} }}\left( {B_{12} \frac{\partial }{\partial y} + B_{16} \frac{\partial }{\partial x}} \right) - \frac{1}{{R_{y} }}\left( {B_{22} \frac{\partial }{\partial y} + B_{26} \frac{\partial }{\partial x}} \right) + A_{45} \frac{\partial }{\partial x} + A_{44} \frac{\partial }{\partial y} \\ L_{41} & = \frac{\partial }{\partial x}\left( {B_{11} \frac{\partial }{\partial x} + B_{16} \frac{\partial }{\partial y}} \right) + \frac{\partial }{\partial y}\left( {B_{16} \frac{\partial }{\partial x} + B_{66} \frac{\partial }{\partial y}} \right) \\ L_{42} & = \frac{\partial }{\partial x}\left( {B_{12} \frac{\partial }{\partial y} + B_{16} \frac{\partial }{\partial x}} \right) + \frac{\partial }{\partial y}\left( {B_{26} \frac{\partial }{\partial y} + B_{66} \frac{\partial }{\partial x}} \right) \\ L_{43} & = \frac{\partial }{\partial x}\left( {\frac{{B_{11} }}{{R_{x} }} + \frac{{B_{12} }}{{R_{y} }}} \right) + \frac{\partial }{\partial y}\left( {\frac{{B_{16} }}{{R_{x} }} + \frac{{B_{26} }}{{R_{y} }}} \right) - \left( {A_{45} \frac{\partial }{\partial y} + A_{55} \frac{\partial }{\partial x}} \right) \\ L_{44} & = \frac{\partial }{\partial x}\left( {D_{11} \frac{\partial }{\partial x} + D_{16} \frac{\partial }{\partial y}} \right) + \frac{\partial }{\partial y}\left( {D_{16} \frac{\partial }{\partial x} + D_{66} \frac{\partial }{\partial y}} \right) - A_{55} \\ L_{45} & = \frac{\partial }{\partial x}\left( {D_{12} \frac{\partial }{\partial y} + D_{16} \frac{\partial }{\partial x}} \right) + \frac{\partial }{\partial y}\left( {D_{26} \frac{\partial }{\partial y} + D_{66} \frac{\partial }{\partial x}} \right) - A_{45} \\ L_{51} & = \frac{\partial }{\partial x}\left( {B_{16} \frac{\partial }{\partial x} + B_{66} \frac{\partial }{\partial y}} \right) + \frac{\partial }{\partial y}\left( {B_{12} \frac{\partial }{\partial x} + B_{26} \frac{\partial }{\partial y}} \right) \\ L_{52} & = \frac{\partial }{\partial x}\left( {B_{26} \frac{\partial }{\partial y} + B_{66} \frac{\partial }{\partial x}} \right) + \frac{\partial }{\partial y}\left( {B_{22} \frac{\partial }{\partial y} + B_{26} \frac{\partial }{\partial x}} \right) \\ L_{53} & = \frac{\partial }{\partial x}\left( {\frac{{B_{16} }}{{R_{x} }} + \frac{{B_{26} }}{{R_{y} }}} \right) + \frac{\partial }{\partial y}\left( {\frac{{B_{12} }}{{R_{x} }} + \frac{{B_{22} }}{{R_{y} }}} \right) - \left( {A_{44} \frac{\partial }{\partial y} + A_{45} \frac{\partial }{\partial x}} \right) \\ L_{54} & = \frac{\partial }{\partial x}\left( {D_{16} \frac{\partial }{\partial x} + D_{66} \frac{\partial }{\partial y}} \right) + \frac{\partial }{\partial y}\left( {D_{12} \frac{\partial }{\partial x} + D_{26} \frac{\partial }{\partial y}} \right) - A_{45} \\ L_{55} & = \frac{\partial }{\partial x}\left( {D_{26} \frac{\partial }{\partial y} + D_{66} \frac{\partial }{\partial x}} \right) + \frac{\partial }{\partial y}\left( {D_{22} \frac{\partial }{\partial y} + D_{26} \frac{\partial }{\partial x}} \right) - A_{44} \\ M_{11} & = M_{22} = M_{33} = - I_{0} \\ M_{14} & = M_{41} = M_{15} = M_{51} = - I_{1} \\ M_{44} & = M_{55} = - I_{2} \\ \end{aligned} $$

(8.23)

And according to Eqs. (1.60), (1.61) and (8.3), the general boundary conditions of thick shallow shells are

$$ \begin{aligned} x & =0:\left\{ {\begin{array}{*{20}l} {N_{x} - k_{x0}^{u} u = 0} \hfill \\ {N_{xy} - k_{x0}^{v} v = 0} \hfill \\ {Q_{x} - k_{x0}^{w} w = 0} \hfill \\ M_{x} - K_{x0}^{x} \phi_{x} = 0 \hfill \\ M_{xy} - K_{x0}^{y} \phi_{y} =0 \hfill \\ \end{array} } \right. \quad x=a:\left\{ {\begin{array}{*{20}l} {N_{x} + k_{x1}^{u} u = 0} \hfill \\ {N_{xy} + k_{x1}^{v} v = 0} \hfill \\ {Q_{x} + k_{x1}^{w} w = 0} \hfill \\ M_{x} + K_{x1}^{x} \phi_{x} = 0 \hfill \\ M_{xy} + K_{x1}^{y} \phi_{y} =0 \hfill \\ \end{array} } \right. \\ y & = 0:\left\{ {\begin{array}{*{20}l} N_{yx} - k_{y0}^{u} u=0 \hfill \\ N_{y} - k_{y0}^{v} v=0 \hfill \\ {Q_{y} - k_{y0}^{w} w=0} \hfill \\ {M_{yx} - K_{y0}^{x} \phi_{x} =0} \hfill \\ {M_{y} - K_{y0}^{y} \phi_{y} =0} \hfill \\ \end{array} } \right. \quad y=0:\left\{ {\begin{array}{*{20}l} N_{yx} + k_{y1}^{u} u=0 \hfill \\ N_{y} + k_{y1}^{v} v=0 \hfill \\ {Q_{y} + k_{y1}^{w} w=0} \hfill \\ {M_{yx} + K_{y1}^{x} \phi_{x} =0} \hfill \\ {M_{y} + K_{y1}^{y} \phi_{y} =0} \hfill \\ \end{array} } \right. \\ \end{aligned} $$

(8.24)

For thick shallow shells, there exists 24 possible classical boundary conditions in each boundary. The classification of the classical boundary conditions shown in Table 4.2 for thick rectangular plates is applicable for thick shallow shells in rectangular planform.

The classical boundary conditions of laminated shallow shells can be readily realized by applying the artificial spring boundary technique. In this chapter, we mainly consider the F, SD, S and C boundary conditions. Taking edge x = 0 for example, the corresponding spring rigidities for the four classical boundary conditions are given as in Eq. (4.30).

8.3 Vibration of Laminated Shallow Shells

Free vibration of laminated shallow shells formed on rectangular planforms with general boundary conditions, arbitrary lamination schemes and various types of curvatures will be considered, including shallow cylindrical, spherical and hyperbolic paraboloidal shells, see Fig. 8.2. Solutions in the framework of shear deformation shallow shell theory (SDSST) will be presented.

From Eq. (8.23), it is obvious that each displacements/rotation component of thick laminated shallow shells is required to have up to the second derivatives. Therefore, regardless of boundary conditions, each displacements/rotation component of a laminated shallow shell is expressed as the proposed modified Fourier series expansion in which several auxiliary functions are introduced to ensure and accelerate the convergence of the series expansion:

$$ \begin{aligned} u(x,y) & = \sum\limits_{m = 0}^{M} {\sum\limits_{n = 0}^{N} {A_{mn} } } \cos \lambda_{m} x\cos \lambda_{n} y + \sum\limits_{l = 1}^{2} {\sum\limits_{n = 0}^{N} {a_{ln} P_{l} (x)} } \cos \lambda_{n} y \\ & \quad + \sum\limits_{l = 1}^{2} {\sum\limits_{m = 0}^{M} {b_{lm} P_{l} (y)} } \cos \lambda_{m} x \\ v(x,y) & = \sum\limits_{m = 0}^{M} {\sum\limits_{n = 0}^{N} {B_{mn} } } \cos \lambda_{m} x\cos \lambda_{n} y + \sum\limits_{l = 1}^{2} {\sum\limits_{n = 0}^{N} {c_{ln} P_{l} (x)} } \cos \lambda_{n} y \\ & \quad + \sum\limits_{l = 1}^{2} {\sum\limits_{m = 0}^{M} {d_{lm} P_{l} (y)} } \cos \lambda_{m} x \\ w(x,y) & = \sum\limits_{m = 0}^{M} {\sum\limits_{n = 0}^{N} {C_{mn} } } \cos \lambda_{m} x\cos \lambda_{n} y + \sum\limits_{l = 1}^{2} {\sum\limits_{n = 0}^{N} {e_{ln} P_{l} (x)} } \cos \lambda_{n} y \\ & \quad + \sum\limits_{l = 1}^{2} {\sum\limits_{m = 0}^{M} {f_{lm} P_{l} (y)} } \cos \lambda_{m} x \\ \phi_{x} (x,y) & = \sum\limits_{m = 0}^{M} {\sum\limits_{n = 0}^{N} {D_{mn} } } \cos \lambda_{m} x\cos \lambda_{n} y + \sum\limits_{l = 1}^{2} {\sum\limits_{n = 0}^{N} {g_{ln} P_{l} (x)} } \cos \lambda_{n} y \\ & \quad + \sum\limits_{l = 1}^{2} {\sum\limits_{m = 0}^{M} {h_{lm} P_{l} (y)} } \cos \lambda_{m} x \\ \phi_{y} (x,y) & = \sum\limits_{m = 0}^{M} {\sum\limits_{n = 0}^{N} {E_{mn} } } \cos \lambda_{m} x\cos \lambda_{n} y + \sum\limits_{l = 1}^{2} {\sum\limits_{n = 0}^{N} {i_{ln} P_{l} (x)} } \cos \lambda_{n} y \\ & \quad + \sum\limits_{l = 1}^{2} {\sum\limits_{m = 0}^{M} {j_{lm} P_{l} (y)} } \cos \lambda_{m} x \\ \end{aligned} $$

(8.25)

where λ _m  = mπ/a and λ _n  = nπ/b. The auxiliary functions P _l (x) and P _l (y) are given in Eqs. (4.32) and (4.33).

Like rectangular plates, each shallow shell formed on rectangular planforms has four boundaries. Thus, for the sake of simplicity, a four-letter string is employed to represent the boundary condition of a shallow shell, such as FCSSD identifies the shell having F, C, S and SD boundary conditions at boundaries x = 0, y = 0, x = a, y = b, respectively. Furthermore, for all numerical examples, unless otherwise stated, the layers of the considered laminated shallow shells are of equal thickness and made of the same composite material: E ₂ = 10 GPa, E ₁/E ₂ = open, μ ₁₂ = 0.25, G ₁₂ = 0.6E ₂, G ₁₃ = 0.6E ₂, G ₂₃ = 0.5E ₂ and ρ = 1,450 kg/m³. In addition, the natural frequencies of the considered shells will be expressed by the non-dimensional parameters as $\Omega = \omega a^{2} \sqrt {\rho /E_{2} h^{2} }$.

8.3.1 Convergence Studies and Result Verification

Table 8.2 shows the first six frequency parameters $\Omega = \omega a^{2} \sqrt {\rho /E_{1} h^{2} }$ for completely free, [30°/−30°/−30°/30°] laminated graphite/epoxy shallow cylindrical, spherical and hyperbolic paraboloidal shells with different truncation schemes (i.e. M = N = 11, 12, 14, 15), respectively, together with results presented by Qatu (2004). The FFFF and CCCC boundary conditions are performed in the study. The geometry and material constants of the layers of the considered shells are given as: b/a = 1, h/a = 0.01, R _y /a = 2, E ₁ = 138 GPa, E ₂ = 8.96 GPa μ ₁₂ = 0.3, G ₁₂ = G ₁₃ = 7.1 GPa, G ₂₃ = 3.9 GPa. It is obvious that the modified Fourier series solution converges quickly. The maximum difference between the 11 × 11-term solutions and those of 15 × 15-term is less than 0.24 %. Based on this analysis, the truncated number of the displacement expressions will be uniformly selected as M = N = 15 in all subsequent calculations. Furthermore, by comparing with results published by Qatu (2004), we can find a well agreement between the two results. The differences between these two results are attributed a different shallow theory was used by Qatu (2004).

Table 8.2 Convergence of frequency parameters $\Omega = \omega a^{2} \sqrt {\rho /E_{1} h^{2} }$ for [30°/−30°/−30°/30°] laminated graphite/epoxy shallow cylindrical, spherical and hyperbolic paraboloidal shells

Table 8.3 shows the comparison of the fundamental parameters Ω for cross-ply cylindrical, spherical and hyperbolic paraboloidal shallow shells with SDSDSDSD boundary conditions and different curvature ratios (a/R). The shells are composed of graphite/epoxy (E ₁/E ₂ = 15, μ ₁₂ = 0.25, G ₁₂ = G ₁₃ = G ₂₃ = 0.5E ₂) and having geometry properties: a/b = 1, h/a = 0.1. The symmetric lamination [0°/90°/90°/0°] and two asymmetric laminations [0°/90°] and [90°/0°] are used in the comparison. “–” represents results that were not considered in the referential work. Curvature ratios are varied from 0 (flat plate) to 0.5 (limit of shallow shell theory). The table shows that the present solutions match very well with those reported by Qatu (2004). Except the case of shallow cylindrical shell with [90°/0°] and curvature ratio a/R = 0.5, the maximum differences between the two results are less than 1.12, 0.012 and 0.012 % for the shallow cylindrical, spherical and hyperbolic paraboloidal shells, respectively.

Table 8.3 Comparison of the fundamental frequency parameters Ω for cross-ply cylindrical, spherical and hyperbolic paraboloidal shallow shells with SDSDSDSD boundary conditions and different curvature ratios

Table 8.4 shows the comparison of the fundamental parameters Ω for a three-layered, [0°/90°/0°] shallow spherical shell with different boundary conditions and curvature ratios (a/R). The shell having SD boundary conditions at the edges y = 0, y = b and the other two edges having arbitrary boundary conditions. The shell is formed on square planform and composed of composite layers having following material and geometry parameters: a/b = 1, h/a = 0.1, E ₁/E ₂ = 25, μ ₁₂ = 0.25, G ₁₂ = G ₁₃ = 0.5E ₂, G ₂₃ = 0.2E ₂. The state space solutions provided by Librescu et al. (1989a) are selected as the benchmark solutions. Comparing the two results, we can find that the current solutions match very well with the referential data. The maximum difference between the two results is very small and less than 0.17 %.

Table 8.4 Comparison of the fundamental frequency parameters Ω for a three-layered, [0°/90°/0°] shallow spherical shell with different boundary conditions and curvature ratios

8.3.2 Laminated Shallow Shells with General Boundary Conditions

Table 8.5 shows the first four non-dimension frequency parameters Ω of a three-layered, [0°/90°/0°] shallow cylindrical shell with different boundary conditions and thickness-length ratios. Four different thickness-length ratios i.e. h/a = 0.01, 0.05, 0.1 and 0.15, corresponding to thin to thick shallow shells are performed in the analysis. The shell parameters used are: a/b = 1, R _x  = ∞, a/R _y  = 0.1, E ₁/E ₂ = 15. It can be seen from the table that the augmentation of the thickness-length ratio leads to the decrease of the frequency parameters. Tables 8.6 and 8.7 show the similar results for shallow spherical (a/R _x  = a/R _y  = 0.1) and hyperbolic paraboloidal (a/R _x  = 0.1, a/R _y  = −0.1) shells, respectively. The similar observation can be seen in the tables as well. In addition, comparing the three tables, it can be found that the frequency parameters of the hyperbolic paraboloidal shell are higher than the cylindrical and spherical ones. And the results of the shallow cylindrical shell are the smallest. Furthermore, from Table 8.5, we can see that frequency parameters of cylindrical shell with curved edge cantilevered (CFFF) are higher than those of straight edge cantilevered (FFFC).

Table 8.5 Frequency parameters Ω for a three-layered, [0°/90°/0°] shallow cylindrical shell with different boundary conditions and thickness-length ratios (a/b = 1, R _x  = ∞, a/R _y  = 0.1, E ₁/E ₂ = 15)

Table 8.6 Frequency parameters Ω for a three-layered, [0°/90°/0°] shallow spherical shell with different boundary conditions and thickness-length ratios (a/b = 1, a/R _x  = a/R _y  = 0.1, E ₁/E ₂ = 15)

Table 8.7 Frequency parameters Ω for a three-layered, [0°/90°/0°] shallow hyperbolic paraboloidal shell with different boundary conditions and thickness-length ratios (a/b = 1, a/R _x  = 0.1, a/R _y  = −0.1, E ₁/E ₂ = 15)

Influence of the fiber orientations on the modal frequencies of shallow shells is also investigated. In Fig. 8.5, variations of the lowest three frequency parameters Ω of certain single-layered ([ϑ]) shallow shells with various fiber orientations are given. Three types of curvature considered in the study are: cylindrical (R _x  = ∞, a/R _y  = 0.1), spherical (a/R _x  = a/R _y  = 0.1) and hyperbolic paraboloidal (a/R _x  = 0.1, a/R _y  = −0.1). In each case, the fiber orientation ϑ is varies from 0° to 180° by an increment of 10°. The shallow shells under consideration are assumed to be with SDSDSDSD boundary conditions. The material constants and geometry parameters of the laminas of the shallow shells are: b/a = 1, h/a = 0.05, E ₁/E ₂ = 15. Many interesting characteristics can be observed from the figure. The first observation is that all the subfigures are symmetrical about central line (i.e., ϑ = 90°). The second observation is that the fundamental frequency parameter traces climb up and then decline, and may reach their crests around ϑ = 45° (when ϑ is increased from 0° to 90°). The similar characteristics can be observed in the subfigure of the second mode. However, for the third mode, the maximum frequency parameters may occur around ϑ = 30° or ϑ = 60°. In addition, it is obvious that frequency parameters of the shallow spherical shell are larger than the shallow cylindrical and hyperbolic paraboloidal ones. Furthermore, since the length-to-radius ratio used in each case in this investigation is small (a/R = 0.1), thus, the differences between these results are small as well. The maximum differences between the results of cylindrical and spherical shallow shells, cylindrical and hyperbolic paraboloidal shallow shells, spherical and hyperbolic paraboloidal shallow shells are less than 1.14, 0.61, 6.24 %, respectively.

Fig. 8.5

The lowest three frequency parameters Ω of single-layered ([ϑ]) shallow shells with SDSDSDSD boundary condition and various fiber orientations (b/a = 1, h/a = 0.05, a/R = 0.1)

Figure 8.6 performs the similar study for the shallow shells with CCCC boundary conditions. The different observation is that the variation tendencies of the fundamental modes decline firstly and then climb up. The minimum frequency parameters may occur around ϑ = 45°. For the second mode, the frequency parameters increase when ϑ increases from 0° to 40° and decrease when ϑ increases from 50° to 90°. The variation tendencies of the third mode are the same as those in Fig. 8.5. The figure also shows that the differences between the three results are small due to the fact that a low length-to-radius ratio is used in this investigation.

Fig. 8.6

The lowest three frequency parameters Ω of single-layered ([ϑ]) shallow shells with CCCC boundary condition and different fiber orientations (b/a = 1, h/a = 0.05, a/R = 0.1)

Figure 8.7 shows the similar study for the shallow shells with SDSDSDSD boundary conditions and a higher length-to-radius ratio, i.e., a/R = 0.2 (limit of the shallow theory). The material and other geometry properties are the same as those used in Figs. 8.6 and 8.7. The observations from this figure are the same as those of Fig. 8.5. However, the maximum differences between the results of cylindrical and spherical shallow shells, cylindrical and hyperbolic paraboloidal shallow shells, spherical and hyperbolic paraboloidal shallow shells can be as many as 5.29, 6.50, 22.9 %, respectively. It is attributed to the effects of curvatures on the frequency parameters of shallow shells are more significant when the length-to-radius ratio is higher. The figure also shows that the variation tendency of each frequency parameter of the cylindrical shallow shell is the same as the corresponding ones of the spherical and hyperbolic paraboloidal shells. Comparing Figs. 8.5, 8.6 and 8.7, it can be found that the influence of fiber orientations on the modal frequencies of shallow shells vary with mode sequence and boundary conditions.

Fig. 8.7

The lowest three frequency parameters Ω of single-layered ([ϑ]) shallow shells with SDSDSDSD boundary condition and different fiber orientations (b/a = 1, h/a = 0.05, a/R = 0.2)

Effects of the fiber orientations on the frequency parameters and mode shapes of laminated shallow shells are further reported. In Tables 8.8, 8.9 and 8.10, the lowest four frequency parameters Ω of certain three-layered, [0°/ϑ/0°] shallow shells with various boundary conditions and fiber orientations are presented. The aspect ratio is chosen to be b/a = 1. The thickness-to-width ratio h/b = 0.1 is used in the calculation. As usual, the shallow cylindrical (R _x  = ∞, a/R _y  = 0.1), spherical (a/R _x  = a/R _y  = 0.1) and hyperbolic paraboloidal (a/R _x  = 0.1, a/R _y  = −0.1) shells are considered in the study. The fiber direction angle ϑ is varied from 0° to 90° with an increment of 30°. The layers of the shells are made of composite materials with orthotropy ratio E ₁/E ₂ = 15. From Table 8.8, it can be noticed that the results of the shell with FFCC boundary conditions equal to those of CCFF boundary conditions. The similar observation can be seen from Tables 8.9 and 8.10 as well.

Table 8.8 Frequency parameters Ω for a three-layered, [0°/ϑ/0°] shallow cylindrical shell with different boundary conditions and fiber orientations (a/b = 1, h/a = 0.1, R _x  = ∞, a/R _y  = 0.1, E ₁/E ₂ = 15)

Table 8.9 Frequency parameters Ω for a three-layered, [0°/ϑ/0°] shallow spherical shell with different boundary conditions and fiber orientations (a/b = 1, h/a = 0.1, a/R _x  = a/R _y  = 0.1, E ₁/E ₂ = 15)

Table 8.10 Frequency parameters Ω for a three-layered, [0°/ϑ/0°] shallow hyperbolic paraboloidal shell with different boundary conditions and fiber orientations (a/b = 1, h/a = 0.1, a/R _x  = 0.1, a/R _y  = −0.1, E ₁/E ₂ = 15)

Figures 8.8, 8.9 and 8.10 give the first three contour plots of the mode shapes and frequency parameters with various lamination angles for CFFF, single-layered [ϑ] shallow cylindrical, spherical and hyperbolic paraboloidal shells, respectively. The aspect ratio a/b = 1, thickness-to-length ratio h/a = 0.05 and orthotropy ratio E ₁/E ₂ = 25 are used in the presentation. Mode shapes and frequency parameters are shown for ϑ = 0, 30°, 60° and 90°. From the figures, we can find that when ϑ = 0, the mode shapes of the shallow cylindrical, spherical and hyperbolic paraboloidal shells are symmetrical with respect to the longitudinal center line. It is attributed to the boundary conditions, geometry and material properties of the shallow shells under consideration are symmetrical with respect to the longitudinal center line as well. Furthermore, it is obviously that the maximum frequency parameters for the first, second and third modes in each case occur at ϑ = 0, ϑ = 0 and ϑ = 30°, respectively. The figures also show that that the node lines (i.e., lines with zero displacements) of the mode shapes vary with respect to the fiber orientation. In addition, since the length-to-radius ratio used in this investigation is small (a/R = 0.1), thus, the differences between the frequency parameters are small as well.

Fig. 8.8

Mode shapes and frequency parameters for CFFF shallow cylindrical shells ([ϑ], a/b = 1, h/a = 0.05, R _x  = ∞, a/R _y  = 0.1, E ₁/E ₂ = 25)

Fig. 8.9

Mode shapes and frequency parameters for CFFF shallow spherical shells ([ϑ], a/b = 1, h/a = 0.05, a/R _x  = a/R _y  = 0.1, E ₁/E ₂ = 25)

Fig. 8.10

Mode shapes and frequency parameters for CFFF hyperbolic paraboloidal shallow shells ([ϑ], a/b = 1, h/a = 0.05, a/R _x  = 0.1, a/R _y  = −0.1, E ₁/E ₂ = 25)

Tables 8.11, 8.12 and 8.13 present the first four non-dimension frequency parameters Ω of certain two-layered, angle-ply [45°/−45°] shallow cylindrical, spherical and hyperbolic paraboloidal shells with different boundary conditions and length-to-radius ratios, respectively. Four different length-to-radius ratios i.e. a/R = 0.05, 0.1, 0.15 and 0.2, corresponding to considerably to slightly shallow shells are performed in the analysis. The shells are formed in rectangular planform with following geometry and material properties: a/b = 1/2, h/a = 0.1, E ₁/E ₂ = 15, R _x  = ∞ for shallow cylindrical shells, R _y  = R _x for the spherical shells and R _y  = −R _x for the hyperbolic paraboloidal ones. Seven different boundary conditions, i.e., FFFC, FFCC, FCCC, CCCC, CCCF, CCFF and CFFF are performed in the calculation. It can be seen from the table that the augmentation of the length-to-radius ratio leads to the increases of the frequency parameters. Table 8.11 also shows that the results of shallow cylindrical shells cantilevered in the curved edge (CFFF) are higher than those of straight edge cantilevered (FFFC). From Table 8.12, it can be noticed that the results for the spherical shells with FFCC boundary conditions equal to those of CCFF boundary conditions. The similar observation can be seen from Tables 8.10 and 8.13 as well.

Table 8.11 Frequency parameters Ω for two-layered, angle-ply [45°/−45°] shallow cylindrical shells with different boundary conditions and length-to-radius ratios (a/b = 1/2, h/a = 0.1, R _x  = ∞, E ₁/E ₂ = 15)

Table 8.12 Frequency parameters Ω for two-layered, angle-ply [45°/−45°] shallow spherical shells with different boundary conditions and length-to-radius ratios (a/b = 1/2, h/a = 0.1, R _y  = R _x , E ₁/E ₂ = 15)

Table 8.13 Frequency parameters Ω for two-layered, angle-ply [45°/−45°] shallow hyperbolic paraboloidal shells with different boundary conditions and length-to-radius ratios (a/b = 1/2, h/a = 0.1, R _y  = −R _x , E ₁/E ₂ = 15)

As the last case, parameter studies are carried out in Figs. 8.11 and 8.12 to further investigate the effects of length-to-radius ratios a/R _x and a/R _y on the frequency parameters Ω of laminated shallow shells. Figure 8.11 depicts the lowest three frequency parameters Ω versus length-to-radius ratios a/R _x and a/R _y for a [0°/90°] layered shallow shell with aspect ratio of a/b = 5 and CCCC boundary conditions. The other shell parameters used in the study are: h/b = 0.1, E ₁/E ₂ = 15. It is seen that each frequency parameter Ω of the shell increases with the increments of length-to-radius ratios a/R _x or a/R _y . The figure also shows that the effects of length-to-radius ratios a/R _x and a/R _y on the frequency parameters Ω of laminated shallow shells vary with mode sequence.

Fig. 8.11

The lowest three frequency parameters Ω versus length-to-radius ratios a/R _x and a/R _y for a [0°/90°] layered shallow shell with aspect ratio of a/b = 5

Fig. 8.12

The lowest three frequency parameters Ω versus length-to-radius ratios a/R _x and a/R _y for a [0°/90°] layered shallow shell with aspect ratio of a/b = 1

Figure 8.12 shows the similar study for the shallow shell with aspect ratio of a/b = 1. The figure shows that the fundamental frequency parameter Ω of the shell increases with the increase of length-to-radius ratios a/R _x or a/R _y . However, for the second mode, the maximum and minimum frequency parameters occur at a/R _x  = 0.2, a/R _y  = −0.2 and a/R _x  = 0, a/R _y  = 0.2, respectively. And for the third mode, the minimum and maximum frequency parameters occur at a/R _x  = a/R _y  = 0 (plate) and a/R _x  = a/R _y  = 0.2 (spherical curvature), respectively. Figures 8.11 and 8.12 also reveal that the effects of length-to-radius ratios on the vibration characteristics of shallow shells vary with aspect ratios.