1 Introduction

Reticulated structures are space frames made by short beam members placed regularly along some curved surfaces. The structures have been widely adopted in architectural engineering, aerospace structures, medical apparatus and instruments, etc. Mechanical behaviors of such structures have drawn considerable attention from the researchers since the last three decades. In general, there are two main strategies to analyze the reticulated structures, i.e., the discrete methods^[1-2] and the equivalent continuum modeling methods^[3-4]. Compared with the discrete methods, the use of equivalent continuum modeling is time saving for some complicated reticulated structures with a large number of beam members. In particular, it can simplify the process of solution for linear and nonlinear problems of various reticulated structures, e.g., plate-type, shell-type, and other space structures. Specially, the equivalent continuum model can be effectively used to predict buckling critical loads and frequencies of entire structures.

So far, there has been much work on the analysis of the buckling and vibration of reticulated shell structures with and without geometric imperfection based on the Bernoulli-Euler beam theory and Kirchhoff-Love assumption^[5-11], and the work has also been extended to analyze the effects of different supporting media^[12]. Fan et al.^[13] investigated the effect of member buckling on the stability of latticed shells. Numerical simulations of the dynamic behaviors of domes subject to impact on the apex were also conducted^[14]. Löpez et al.^[15-16] and Ma et al.^[17-18] studied the influence of the semi-rigid joints on the global behavior of the reticulated shell structure using numerical analysis and experimental tests. The thermal behavior of aluminum reticulated shell structures was investigated by Liu et al.^[19]. Sabzikar and Eslami^[20] presented the snap-through buckling analysis of shallow clamped spherical shells made of functionally graded material and surface-bonded piezoelectric actuators under the thermo-electro-mechanical loading. Kato et al.^[21] and Kumagai and Ogawa^[22] studied dynamic buckling behavior of single-layer domes. Recently, seismic performance of reticulated shells was explored. Furthermore, the effects of substructures and support flexibility on seismic responses were presented by Yu et al.^[23] and Li et al.^[24]. Kato et al.^[25] also proposed a proportioning method for member sections of single layer reticulated domes subject to uniform loads and then extended the method to the structures under non-uniform loads. Since single-layer reticulated cylindrical shells were sensitive to the external load distribution, the stability of the structure under wind loading was demonstrated by Li et al.^[26]. The stability of different types of reticulated shell structures was analyzed, such as pretensioned cylindrical reticulated mega-structures^[27], novel cable-stiffened single-layer latticed shells^[28], and inverted catenary cylindrical reticulated shells^[29]. Meanwhile, the effects of the grid shape, the grid spacing, and the span-to-height ratio on the failure load of the shell were assessed by Malek et al.^[30]. However, among the above studies, little attention has been paid to the influence of the transverse shear on the nonlinear behaviors of the reticulated shells^[31]. The results by Li et al.^[32] and Groh and Weaver^[33] showed that transverse shear deformation has effects on the critical buckling load of the solid continuum shell structures.

This paper aims to study the effect of transverse shear deformation on nonlinear free frequency of the reticulated shallow spherical shells whose beam members are placed in two orthogonal directions. The fundamental governing equations for nonlinear free vibration of the shell are presented for the case of axisymmetrical bending deformation. The asymptotic iteration method is used to construct a solution for frequency of the shell. Numerical examples are given to illustrate the effect of transverse shear on the linear and nonlinear frequencies of reticulated shells and plates with beam members having circular and rectangular cross-sections for various nondimensional apex heights. The softening and hardening nonlinear behaviors are discussed.

2 Mathematical formulation of the problem

Let us consider a reticulated shallow spherical shell whose beam members are assumed to be placed uniformly in two orthogonal directions and along every central angle θ₀ over the entire spherical surface, as shown in Fig. 1. The span and radius of curvature of the shallow shell are 2a and R, respectively. Each beam member has the same geometrical size and material property and its length, cross-sectional area, transverse and lateral bending stiffnesses, and shear stiffness are denoted by L, A, EI, EI₀, and GJ, respectively. For the case of axisymmetrical bending deformation, the fundamental governing equations for free vibration of the shell can be expressed in terms of the deflection w and the rotational angle _r together with the membrane force N_r as^[31]

(1)

(2)

(3)

where m₀ = ρh, and ρ and h are the density and the thickness of the shell, respectively. Introduce the following nondimensional variables and quantities:

(4)

(5)

and

(6)

where α is the modification coefficient of transverse shear deformation. The nondimensional governing equations are written by

(7)

(8)

(9)

where the differential operator .

The clamped boundary condition is

(10a)

(10b)

and the hinged boundary condition is

(10c)

(10d)

where , and . It can be observed that Eq. (10c) will change to Eq. (10a) when m→ ∞. The following analysis will be confined to the case of the hinged boundary condition.

Suppose

(11a)

(11b)

(11c)

where V_r (ρ), V(ρ), T₁ (ρ), and T₂ (ρ) are undetermined functions, and s is an unknown parameter, which are determined by using the following variational relations:

(12a)

(12v)

Notice

Substituting the above equations and Eqs. (11a)-(11c) into Eqs. (12a) and (12b), and considering that δV_r, δV, and δs may take arbitrary values, the two integrations with respect to τ result in

(13)

(14)

(15)

where

and

In addition, substituting Eqs. (11a)-(11c) into Eq. (9) yields

(16)

(17)

The corresponding boundary conditions are

(18a)

(18b)

3 Asymptotic iteration solution 3.1 First iteration

In view of Eqs. (13) and (14), considering a corresponding linear boundary-value problem, the reduced equations in terms of V(ρ) and V_r (ρ) are expressed as follows:

(19)

(20)

and

(21a)

(21b)

where the superscript (1) stands for the first iteration. ω₁ is the reference frequency when s=0. In fact, it is seen from Eqs. (19) and (20) that the frequency corresponds to the natural frequency for linear free vibration of a circular reticulated plate of radius a. From Eqs. (16) and (17), the undetermined functions T₁⁽¹⁾ and T₂⁽¹⁾ directly satisfy the following equations:

(22)

(23)

with

(24a)

(24b)

The linear solutions for V⁽¹⁾(ρ) and V_r⁽¹⁾(ρ) can be written by

(25)

(26)

where

(27)

and

Substituting Eqs. (25) and (26) into Eq. (21a), the use of nontrivial solutions for two coefficients A₁ and A₂ leads to the equation of ω₁,

(28)

where (·)' =d(·)/dρ. Denote V⁽¹⁾|_ρ=0 =W_c. The expressions for A₁ and A₂ are obtained by

Further, using V⁽¹⁾ and V_r⁽¹⁾, the results for T₁⁽¹⁾ and T₂⁽¹⁾ in Eqs. (22) and (23) in connection with Eqs. (24a) and (24b) can be obtained as

(29)

(30)

where the expressions for f₁ (ρ) and f₂ (ρ) are omitted here due to space limitation. Substituting the resulting equations of T₁⁽¹⁾, T₂⁽¹⁾, V⁽¹⁾(ρ), and V_r⁽¹⁾(ρ) into Eq. (15), the value of the parameter s can be determined when W_c is prescribed.

3.2 Second iteration

Based on the above solution for the first iteration, let us consider the following set of equations:

(31)

(32)

where . The superscript (2) represents the second iteration. The corresponding boundary condition is

(33a)

(33b)

and

(34)

Similarly, the unknowns V⁽²⁾(ρ) and V_r⁽²⁾(ρ) have the following form of solutions:

(35)

(36)

in which

(37)

where

and V^(2)* (V_r^(2)*) are particular solutions and can be derived by determination on the coefficient β_k when the solution in the form of series of powers is assumed^[9]. The unknown coefficients A₃ and A₄ in Eq. (37) can be determined by substituting Eqs. (35) and (36) into Eq. (33) and using Eq. (34), and the equation of frequency ω₂ can also be derived,

(38)

where V_r^(2)*' (1)=dV_r^(2)* (ρ)/dρ|_ρ=1, and V_r^(2)*(1)=V_r^(2)*(ρ)|_ρ=1. The above equation indicates that the frequency ω₂ relates to W_c. For a given value of W_c, the parameter s is obtained from Eq. (15), and the nonlinear frequency is thus determined. The characteristic relation between ω and the amplitude W_m =W_c(1+s) is finally established. Especially, if W_c =0 and s=0, the frequency will reduce to the natural frequency for linear free vibration of the shell denoted by ω₀₁. It should be pointed out that if the transverse shear effect is neglected, i.e., the parameter M=(3EI+GJ)/(4a²αGA)→0 in Eq. (6), Eq. (7) reduces to a classical relation ψ_r =dW/dρ. Then, Eqs. (8) and (9) will become the known equations for the reticulated shell based on the classical shallow shell theory. The above asymptotic analysis and form of solutions can be degenerated to the available results for shells and plates for R→ ∞^[9-10].

4 Numerical examples and analysis

In computation, geometrical parameters and material constants of the shells are listed below^{[8, 10]}. The span of the shell is assumed to be 2.0 m, and each beam member has the same length of 0.2 m and circular cross-section with 0.005 8 m diameter. Poisson's ratio is 0.3. The radii of curvature are chosen as 43.109 2 m, 21.563 3 m, and 7.868 9 m corresponding to nondimensional apex heights H/h=2, 4, 11, respectively.

The relation between the parameter s and the amplitude W_m of the hinged shell is presented in Fig. 2. The result shows that a smaller value of H/h causes a smaller value of s. For the case of reticulated plate, i.e., R→∞ or H/h→ 0, there is always s=0.

The characteristic relation between the ratio of nonlinear frequency to linear frequency and the amplitude for various nondimensional apex heights is displayed in Fig. 3. For a smaller value of the apex height, e.g., when H/h=2, the value of the frequency ratio ω/ω₀₁ is always larger than one and increases with the amplitude W_m, which reflects hardening nonlinear behavior of the large-deflection shell. However, when H/h=11, the frequency ratio decreases with the amplitude for smaller amplitudes, but increases when the amplitude becomes large enough. The shell vibrates with the initial softening behavior. In addition, for the case of H/h=4, it is observed that the value of the frequency ratio ω/ω₀₁ is about one for smaller amplitudes, and then increases with the amplitude. From the figure, it is concluded that there exists a critical value of the nondimensional apex height H/h corresponding to a conversion of pure hardening to softening-hardening nonlinear behaviors. The computational result shows that the critical value is four or so.

As mentioned in Subsection 3.1, the frequency ω₁ is the linear frequency for free vibration of a circular reticulated plate with the radius being half a span of the shell. For axisymmetrical vibration of the plate with circular and rectangular cross-sections of beams, nondimensional results of first three order frequencies are listed in Tables 1 and 2, respectively. The computation shows that the frequency of the plate incorporating transverse shear effect is lower than the one without the transverse shear effect. In addition, for an identical shell, a higher order number generates a larger decrease in the frequency when the transverse shear effect is taken into account. The calculation results indicate that such a decrease in the frequency will be more remarkable when the thickness of the shell becomes large. Specifically, as shown in Table 2, the decreases in the first and third frequencies are by 0.44% and 7.79% respectively for the case of h=0.1 m. However, when h=0.5 m, such decreases are by 9.20% and 53.86%, respectively.

For the reticulated spherical shell whose radius of curvature is 43.109 2 m and cross-sections of beam members are rectangular with width b=0.1 m and different heights h=0.1 m, 0.3 m, and 0.5 m, the relation between the nondimensional frequency ω/ω₀ and the amplitude of the shell without and with the transverse shear effects are displayed in Figs. 4 and 5, respectively. In contrast to the above mentioned linear frequency ω₀₁ with the transverse shear effect, the frequency ω₀ represents the linear frequency without the shear effect. The results show that the nonlinear frequency increases with the amplitude. For an identical shell, the frequency with the transverse shear effect is expected to be smaller than one without the effect due to a decrease in the global stiffness of the shell when the transverse shear effect is included. Comparison of Fig. 4 with Fig. 5 indicates that for a thicker shell, i.e., the shell with a larger height of each beam, the influence of transverse shear deformation on the linear frequency responding to W_m=0 is more considerable. When the shell vibrates with larger amplitude, the transverse shear effect may become weak, and geometrical nonlinearity presents dominant influence on the amplitude-frequency relation.

5 Conclusions

Nonlinear vibration of the reticulated shallow spherical shells made by beam members placed in two orthogonal directions is studied in this paper. The transverse shear effect is incorporated. The nondimensional fundamental governing equations for the free vibration of such shell structures are presented for the case of axisymmetrical bending deformation. The amplitude-frequency relation is obtained by the asymptotic iteration method. The resulting theoretical solution can be degenerated to the available results for the shells and the plates if the transverse shear effect is neglected. Numerical examples are given to illustrate the effect of transverse shear deformation on the linear and nonlinear frequencies of reticulated shells and plates. The present model is helpful to predict the frequencies and evaluate the nonlinear vibration behaviors of such space reticulated structures.