Free vibrational characteristics of grid-stiffened truncated composite conical shells

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Free vibrational characteristics of grid-stiffened truncated composite conical shells

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a

M. Zarei , G.H. Rahimi

a,∗

, M. Hemmatnezhad

b

a

Faculty of Mechanical Engineering, Tarbiat Modares University, P.O. Box 14115-143, Tehran, Iran b Faculty of Mechanical Engineering, Takestan Branch, Islamic Azad University, Takestan, Iran

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a r t i c l e

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Article history: Received 26 May 2019 Received in revised form 2 December 2019 Accepted 16 January 2020 Available online xxxx Communicated by Grigorios Dimitriadis Keywords: Truncated conical shell Free vibration Stiffened shells Experimental modal analysis Ritz method

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a b s t r a c t In this paper, the free vibrational behavior of composite conical shells stiffened by bevel stiffeners is investigated using experimental, analytical and numerical techniques. The smeared method is employed to superimpose the stiffness contribution of the stiffeners with those of shell in order to obtain the equivalent stiffness parameters of the whole structure. Due to the specific geometry of the conical shell, the whole structure is converted to a conical shell with variable stiffness and thickness. The stiffeners are considered to be of beam-type which support shear load and bending moments in addition to the axial loads. The geodesic path is applied to the stiffeners. The governing equations have been derived based on the first-order shear deformation theory and using the Ritz method. In order to validate the analytical achievements, the experimental modal test is conducted on a stiffened cone. The specimen has been fabricated by a specially-designed filament winding setup. A 3-D finite element model was also built using ABAQUS software to further validate the analytical results and help with parametric study. Comparison of the results obtained from the three approaches revealed good agreements. The effects of the shell geometrical parameters and variations in the cross stiffeners angle on the natural frequencies have been discussed and investigated. The present achievements are novel and can be used as a benchmark for further studies. © 2020 Published by Elsevier Masson SAS.

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1. Introduction

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Grid-stiffened structures are similar to the lattice ones in appearance. These structures simultaneously have both advantages of the conventional composite structures and lattice ones. They are considered as one of the functional and economical structures in various engineering disciplines as reflected by their massive applications in various industries. Grid-stiffened shells which are nowadays well known in many engineering fields, especially in the aerospace industry, are shell structures reinforced by different types of the stiffening structures either on the inner, outer or both sides of the shell. These stiffeners can significantly increase the load bearing capacity of the shell without much increase in its weight. The promising future of these structures have caused a great number of attractions from the side of the global researchers to get involved in the mechanical behavior investigation of the stiffened shells with reinforcing grids or stiffeners. Several publications can be found in the literature which studied the effects of the stiffeners on the vibrational behavior of the orthogonally stiffened cylindrical shells [1–5]. In order to obtain the equivalent stiffness of the structure, some authors have used the discrete technique [6,7], while several others implemented the smeared method [8,9]. However, the number of researches concerning with the mechanical behavior investigation of the stiffened cylindrical shells with helical stiffeners are rare and not as much as those investigating the effects of longitudinal and circumferential stiffeners. In this regard, Kidane et al. [10] obtained the buckling load of grid-stiffened cylindrical shells with helical stiffeners. They improved an analytical model and smeared the forces and moments by the stiffeners onto the shell considering a unit cell. They compared the analytical results with experimental data and reached a good agreement. Parametric analysis of some of the important design variables was performed and several conclusions were drawn as well. Yazdani and Rahimi [11] performed an extensive experimental study on the grid-stiffened composite cylindrical shell. They also examined the effect of the number of stiffeners and grid type on the buckling characteristic of these structures [12,13]. Rahimi et al. [14] also examined the effect of stiffener profile

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Corresponding author. E-mail address: [email protected] (G.H. Rahimi).

https://doi.org/10.1016/j.ast.2020.105717 1270-9638/© 2020 Published by Elsevier Masson SAS.

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on the buckling behavior of stiffened cylindrical shell and introduced an optimum profile. Perhaps for the first time, Rahimi et al. [15] investigated the vibrational behavior of grid-stiffened cylinders through analytical approach and also implemented the smeared stiffness approach developed in [10]. Theoretical formulation was established based on Sanders’ thin shell theory. They obtained the vibrational frequencies of the structure for different supporting conditions and concluded that the angle of the stiffeners has a significant effect on the frequencies associated with higher vibrating modes. Hemmatnezhad et al. [16] improved their previous study and studied the free vibration of a composite stiffened cylindrical shell based on the first-order shear deformation theory (FSDT) through analytical and numerical methods. An improved smeared method was employed to superimpose the stiffness contribution of the stiffeners with those of the shell to obtain the equivalent stiffness parameters of the whole panel. In their study, the stiffeners were modeled as a beam and considered to support shear loads and bending moments in addition to the axial loads. Their results showed a good agreement between the results of the two applied approaches. Also, they indicated that the shell thickness has a significant effect on the vibrational frequencies of grid shells, and this effect is more prominent for the unstiffened shell rather than the stiffened one. Furthermore, for the shell thicknesses greater than a specific value, the presence of the grid structure does not have any effect on the natural frequency values and the frequency curves associated with both stiffened and unstiffened shells. Recently, Hemmatnezhad et al. [17] extensively studied the free vibration of stiffened composite cylindrical shells with helical stiffeners using the experimental, numerical and analytical techniques. They performed experimental modal analysis to obtain the modal parameters (i.e. natural frequencies, mode shapes and damping ratios) of these structures and further validate the numerical and analytical methods. The influence of varying shell thickness and also end conditions have been investigated as well. Wang et al. [18] used an effective hybrid model in order to optimize the stiffened shells. Sheng and Wang [19] studied the stability of stiffened functionally graded (FGM) cylindrical shells. The free vibration problem of the composite conical shells has been widely studied by many researchers. Irie et al. [20] used the transfer matrix and Tong [21] applied the power series method for investigating the vibrational behavior of conical shells. Shu [22] investigated the problem of free vibrational behavior of composite laminated conical shells using the generalized differential quadrature (GDQ) method. In this research, he showed that a high order of accuracy and convergence will be achieved by using this method. Lam and Hua [23] studied the free vibration of rotating truncated conical shells using Ritz method. Using the GDQ method, Ma et al. [24] studied the free and forced vibration of a cylindrical-conical shell for arbitrary boundary conditions using Ritz method. The number of studies dealing with the free vibration of stiffened conical shells is not as much as those addressed the cylindrical shells and most of the relative studies are limited to the buckling problem of these structures. Vasilev et al. [25] provided a finite element model for conical lattice structures and proved that the grid structure provides the required buckling load strength by reducing the structural weight about 15 percent with cost saving as well. Jabareen and Sheinman [26] performed the analytical stability of an imperfect conical shell reinforced with longitudinal stiffeners. They used the Donnell’s shell theory and Galerkin method to derive the governing equations. They also included the imperfection effect and found that a classical eigenvalue does not necessarily determine the stable criterion. Furthermore, the nonlinear effects were also considered to determine the actual values of the buckling loads. It was also shown that stiffening may change the imperfection sensitivity and the external stiffeners tend to increase both the sensitivity and the buckling load properties. Shi et al. [27] studied the behavior of a conical lattice structure under the external pressure via the empirical and analytical approaches. They introduced a method for equalization of the stiffeners and calculated the buckling load using energy method. It was concluded that the local buckling is an important factor in optimizing the weight of the lattice structures. Recently, Talebitooti et al. [28] investigated the vibrational behavior of rotating conical shells reinforced with longitudinal and circumferential stiffeners. Using the discrete method for making equal stiffness and energy method, Dung et al. [29] investigated the instability of eccentrically stiffened FGM truncated conical shells under mechanical loads. Daneshjou et al. [30,31] investigated the vibrational behavior of rotating orthogonally stiffened conical shells using GDQ and energy method. They concluded that applying stringer is not recommended unless the buckling phenomenon is significant. They also developed a 3D finite element model using ABAQUS in order to further validate their analytical solutions. Liu et al. [32] examined the linear vibration of the liquid-containing conical shell with variable thickness and reinforced by circumferential stiffeners. They came to the fact that the distance between the rings, shell thickness and stiffness of the rings, have significant effects on the natural frequencies, especially for the higher vibration modes. By studying the effect of the shell thickness variation, they illustrated that the natural frequencies increase with an increment in the shell thickness. Duc et al. [33] investigated the thermal and mechanical stability of reinforced FGM truncated conical shell resting on elastic foundations. In another work, Duc et al. [34] studied the thermal buckling of FGM sandwich truncated conical shells stiffened by orthogonal stiffeners and surrounded on elastic foundations using FSDT. They implemented Galerkin method in order to obtain the stability equations. They concluded that the number of stiffeners highly affects the critical thermal load. Dung and Chan [35] and Chan et al. [36] investigated buckling behavior of FGM truncated conical shells reinforced by orthogonal stiffeners based on FSDT. Dung et al. [37] studied the vibrational behavior of the rotating FGM conical shells stiffened by orthogonal stiffeners. They used Galerkin method and Donnell’s shell theory for extracting the vibration frequencies. They concluded that stiffeners and input parameters have important effects on the natural frequencies of the shell. Bagheri et al. [38] investigated the effect of ring support on the vibrational behavior of conical shells. Totaro [39] analytically formulated the flexural, torsional and axial global stiffness properties of the anisogrid lattice conical shells made of composite materials. Smeared elastic properties have been introduced along with the classical membrane theory of conical shells to determine the global stiffness parameters. They also developed a finite element model for comparing the results and concluded that the results of both methods are in close agreement. Belardi et al. [40] used an optimization method in order to deduce the minimum mass configuration of the lattice structure. Recently, the nonlinear behavior of stiffened structures has been investigated by a number of researchers [41–44]. To the best of the authors’ knowledge, no work has been carried out to study the vibrational behavior of composite conical shells with helical stiffeners based on the FSDT and considering the torsional effects of the stiffeners. The aim of this investigation is to provide a mathematical model capable of predicting the vibrational behavior of grid-stiffened composite conical shells without performing costly numerical simulation and experimental tests. In the analytical procedure, first, the smeared technique is implemented to superpose the stiffness of the stiffeners with those of shell and consequently, the equivalent stiffness of the whole structure will be achieved. Due to the special geometry of the cone, adding the stiffener to the conical shell, creates a variable stiffness in the structure. Therefore, the Ritz method seems to be an appropriate method for such conditions. Using Beam modal functions and Ritz method, the effects of different geometrical parameters, angle of the fibers and angle between stiffeners are examined and novel achievements are given as well. A 3D

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Fig. 1. Conical and tangential coordinates along with the stiffener parameters.

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finite element model is also built using ABAQUS software which takes into consideration the exact geometric configuration of the conical shell and stiffeners. To validate the accuracy of the analytical and numerical approaches, experimental modal analysis is carried out on the fabricated specimens. Several numerical results are given to examine the effects of important design parameters on the vibrational characteristics of composite stiffened conical shells. The present achievements are novel and can be used as a benchmark for further studies. 2. Theoretical formulation

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Fig. 1 depicts a conical lattice structure reinforced with helical stiffeners in which n g , E g , and G lt represent the number of geodesic stiffeners, Young’s modulus along the stiffener direction and shear modulus, respectively. Also, ϕ defines the stiffeners angle with respect to the longitudinal direction whose value at the large radius is ϕ2 and at the smaller one is ϕ1 . Also, h, b g , and A g are the width, height and area of the stiffeners, respectively. It is assumed that the helical stiffeners bear the axial load and the stress uniformly varies over the cross-section. The average stressstrain relations of a stiffener in the normal-tangential coordinate system on the stiffeners considering the torsion and shear effects are given as

⎡ σl ⎤extension 1 0 ⎢ ⎢ σt ⎥ E gbg ⎢ ⎢ ⎥ = ⎢0 ⎢ σlt ⎥ ⎣σ ⎦ a g (x) ⎣ 0 lz 0 σtz ⎡

⎡ σl ⎤torsion 0 0 ⎢ ⎢ σt ⎥ G gbg ⎢ ⎢ ⎥ = ⎢0 ⎢ σlt ⎥ ⎣σ ⎦ a g (x) ⎣ 0 lz 0 σtz ⎡

⎡ σl ⎤shear 0 0 ⎢ ⎢ σt ⎥ G gbg ⎢ ⎢ ⎥ = ⎢0 ⎢ σlt ⎥ ⎣σ ⎦ a g (x) ⎣ 0 lz 0 σtz ⎡

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

⎤⎡

0 εl 0 ⎥ ⎢ εt ⎥⎢ 0 ⎥ ⎢ εlt ⎦⎣ ε 0 lz 0 εtz

⎤⎡

0 εl 0 ⎥ ⎢ εt ⎥⎢ 0 ⎥ ⎢ εlt ⎦⎣ ε 0 lz 0 εtz

⎤⎡

0 εl 0 ⎥ ⎢ εt ⎥⎢ 0 ⎥ ⎢ εlt ⎦⎣ ε 0 lz 1 εtz

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⎥ ⎥ ⎥ = [ Q ]extension ε g ⎦

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(1)

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⎤ ⎥ ⎥ ⎥ = [ Q ]torsion ε g ⎦

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(2)

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⎤ ⎥ ⎥ ⎥ = [ Q ]shear ε g ⎦

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(3)

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a g (x) =

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⎤

where, the index “g” stands for the geodesic stiffener and a g is the vertical distance between two stiffeners which varies along the longitudinal direction and defined as below

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2.1. Smeared approach

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2π R (x)cosϕ ng

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⎡

Here, T is defined as

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(4)

The relationship between the stress components of helical stiffeners in the tangential and conical coordinates is given as follows

⎡ σ x ⎤i σl ⎤i ⎢ σθ ⎥ ⎢ σt ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ σxθ ⎥ = [T ]−1 ⎢ σlt ⎥ , i = extension, torsion, shear ⎣σ ⎦ ⎣σ ⎦ xz lz σθ z σtz

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(5)

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⎡

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c2 ⎢ s2 ⎢ T =⎢ ⎢ −sc ⎣ 0 0

s2 c2 sc 0 0

2sc −2sc c 2 − s2 0 0

0 0 0 c s

⎤

0 0 ⎥ ⎥ 0 ⎥ ⎥, −s ⎦ c

1

s = sin ϕ (x) =

R 2 sin ϕ2 R (x)

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, c = cos ϕ (x)

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The strain components in the conical coordinate system and those in the tangent one are related to each other as

⎡ ⎡ εl ⎤ εxx ⎤ ⎢ ⎢ εt ⎥ ⎢ εθ θ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ γlt ⎥ = [R] [T ] [R]−1 ⎢ γxθ ⎥ , [R] = ⎢ ⎣ ⎣γ ⎦ ⎣γ ⎦ xz lz γtz γθ z ⎡

0 0 0 0 2

0 0 0 2 0

0 0 2 0 0

0 1 0 0 0

1 0 0 0 0

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⎤ ⎥ ⎥ ⎥ ⎦

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(6)

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Using Eqs. (1)-(6), (3), (5) the stress-strain relation due to the stiffeners in the conical coordinates is given as

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⎡

⎡ σ x ⎤i εxx ⎤ ⎢ σθ ⎥ ⎢ εθ θ ⎥ ⎢ ⎥ ⎢ ⎥ [σ ] = ⎢ σxθ ⎥ = [T ]−1 [ Q ]i [R] [T ] [R]−1 ⎢ γxθ ⎥ = [ Q (x)]i [ε ] , i = extension, torsion, shear ⎣σ ⎦ ⎣γ ⎦ xz xz σθ z γθ z

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(7)

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It is assumed that stiffeners and the shell have continuity; also, the strain field is continuous along the thickness of the stiffeners and the shell. For each point of an equivalent shell, the strain components can be expressed as below

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εxθ = εx0θ + zκxθ

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εθ z = εθ0z

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0 xz

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εxz = ε

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(8)

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where, the mid-surface strain and curvature relations in terms of the displacement components based on the Donnell’s theory are given as

∂u ε = , ∂x 0 x

0

εθ =

u sin α − w cos α R

⎡

⎤extension

g

Nx

⎢ g ⎥ ⎣ Nθ ⎦

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ε =

1 ∂v R ∂θ

−

v sin a R

∂v + ∂x

(9)

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⎡

g N xθ

g Mx g

−h−t /2

⎤bending,torsion

⎢ ⎥ ⎣ Mθ ⎦ g

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[ Q (x)]extension {ε } dz

− ˆt /2

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¨ {σ }

extension

−h−t /2

=

zdz +

{σ }

torsion

g Qx g

Qθ

shear

−h−t /2

(11)

[ Q (x)]extension {ε } zdz +

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[ Q (x)]torsion {ε } zd A

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Ag

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− ˆt /2

{σ }shear dz =

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¨

− ˆt /2

=

zd A

Ag

−h−t /2

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− ˆt /2



(10)

−h−t /2

=

M xθ

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− ˆt /2

{σ }extension dz =

=

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,

R ∂θ v cos a 1 ∂w

− ˆt /2

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+

0 xθ

The force and moment resultants due to the stiffeners can be obtained from

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1 ∂v

∂w ε = βx + , εθ0z = βθ − + ∂x R R ∂θ  ∂βx ∂βθ 1 κx = , κθ = + βθ sin α ∂x R (x) ∂θ ∂βθ βθ sin α 1 ∂βx κxθ = − + ∂x R (x) R (x) ∂θ 0 xz

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εθ = εθ0 + zκθ

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εxx = εx0 + zκx

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[ Q (x)]shear {ε } dz

−h−t /2

After integration, the recent relations can be rewritten in the following matrix

(12)

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⎡

⎢ g ⎢ Nθ ⎢ g ⎢N ⎢ xθ ⎢ g ⎢ Mx ⎢ ⎢ Mg ⎢ θ ⎢ g ⎢M ⎢ xθ ⎢ g ⎣ Qx

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⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎡ ⎥ [ A]g ⎥ ⎥ = ⎣ [B ]g ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ ⎦

16

[B ]g [D ]g 0

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⎡ Eg Ag

g

[ A (x)]ϕ =

a g (x)

c

sc 3

Eg Ag g

[D (x)]ϕ =

27

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Gg Ag

g

[H (x)]ϕ =

a g (x)

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2 3 4 5

(13)

⎡

s c s4 cs3

h+t 2

2

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+

c2 cs

cs s2

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sc cs3 ⎦ s2 c 2

h2 12

s2 c 2 s4 cs3



⎡

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⎤

17

sc 3 cs3 ⎦ s2 c 2 c

4

2 2

⎣s c

2 2

sc 3



s c s4 cs3

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⎤

⎡

21

⎤

22

sc 4s c −4s c 2(cs − sc ) JGg ⎣ −4s2 c 2 cs3 ⎦ + 4s2 c 2 2(sc 3 − cs3 ) ⎦ 4a g (x) 2 2 3 3 3 3 s c 2(cs − sc ) 2(sc − cs ) (c 2 − s2 )2 3

2 2

2 2

3

3

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Using the same procedure for circumferential stiffeners, the reduced stiffness matrix would be obtained. Finally by considering all geodesic and hoop stiffeners, the total stiffness matrices due to the stiffeners are expressed as



st

st

st

[A] , [B] , [D] , [H]

st





g

g

g

g



g

g

g

g



= [ A]ϕ , [B]ϕ , [D]ϕ , [H]ϕ + [ A]−ϕ , [B]−ϕ , [D]−ϕ , [H]−ϕ +



[ A]c0 , [B]c0 , [D]c0 , [H]c0

(14)



[A]sh , [B]sh , [D]sh =

46 47

5 n

[ H sh ] =

43 45

n  k=1t

40

44

4

ˆtk



k, [ Q i j ]sh 1, z, z2 dz,



k,

39

[ Q i j ]sh z(k+1) − zk −

k =1

3 3 ( z − z ) k 2 (k+1)

4 3t

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(i , j = 4, 5 only)

(15)

⎡



N M



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⎡

⎤

⎡

N +N [ A] + [ A] = ⎣ M sh + M st ⎦ = ⎣ [ B ]st + [ B ]sh 0 Q sh + Q st sh

st

st

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sh

[B] + [B] [ D ]st + [ D ]sh st

sh

0

⎤

εx0 ⎢ ε0 ⎥ ⎢ θ ⎥ ⎤⎢ γ0 ⎥ ⎡ ⎢ xθ ⎥ 0 [ A] ⎥ ⎢ ⎦ ⎢ κx ⎥ = ⎣ [ B ] 0 ⎥ ⎢ κθ ⎥ 0 [ H ]st + [ H ]sh ⎢ ⎢ κxθ ⎥ ⎥ ⎢ ⎣ γxz ⎦ γxθ

⎡

[B] [D] 0

⎤

εx0 ⎢ ε0 ⎥ ⎢ θ ⎥ ⎤⎢ γ0 ⎥ ⎢ xθ ⎥ 0 ⎢ κx ⎥ ⎥ 0 ⎦⎢ ⎢ κθ ⎥ ⎥ [H ] ⎢ ⎢ κxθ ⎥ ⎥ ⎢ ⎣ γxz ⎦ γxθ

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(16)

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2.2. Linear vibrational analysis of a conical shell

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A 3-D schematic and coordinate system of a conical shell with the vertex angle of “α ” is shown in Fig. 2. The strain and kinetic energies of the conical shell are given as [46]

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The overall forces and moments on the stiffened conical shell is the superposition of those due to the stiffener and shell. These quantities can be directly superimposed, as the stiffener’s force and moment contributions have been developed based on the midplane strains and curvatures. Using the principle of superposition, the stiffness of the whole structure can be obtained as

50

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35

38

k −1

49

52

34

37

k = 1, 2, ..., i , j = 1, 2, 6

48

51

32

36

39

42

30

33

where, the upper index “st” stands for all stiffeners. The stiffness matrix of shell based on the plain stress in the conical coordinate is given as follows [45]

38

41

29 31



36 37

6 7

⎤

3

a g (x)

25 26

2 2

⎣ s2 c 2



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4

c4 E g A g (h + t ) g 2 2 ⎣ x = ( ) [B ]ϕ s c 2a g (x) sc 3

21 23

1

where the stiffness matrices are obtained as

17 18

εx0 ⎢ 0 ⎥ ⎢ εθ ⎥ ⎥ ⎢ ⎢ 0 ⎥ ⎤ ⎢ γxθ ⎥ ⎥ ⎢ 0 ⎢ κx ⎥ ⎥ ⎢ ⎦ 0 ⎢κ ⎥ θ ⎥ [H ]g ⎢ ⎥ ⎢ ⎢ κxθ ⎥ ⎥ ⎢ ⎥ ⎢ ⎣ γxz ⎦ γxθ

Qθ

14 15

⎤

⎡

g

11 12

g

Nx

5

U=

1 2

ˆx2 ˆ2π 0 0 ( N x εxx + N θ εθ0θ + N xθ γx0θ + M x κx + M θ κθ + M xθ κxθ + Q x γxz + Q θ γθ0z ) Rdxdθ

k=1 x1 0

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(17)

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Fig. 2. The geometry of the grid-stiffened conical shell and displacement components.

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ˆx2 ˆ2π  1

22

T=

23 24

2

29

32

35

I1 =

38 39 40

−t /2

u=

∞  ∞ 

v=

55 56 57 58 59 60 61 62 63 64 65 66

29 30 31

(19)

2

I sh =

z dz,

amn

w=

z dz,

ρ g = ρg

− 2t

2Ag a g (x)h g

,

ρ c = ρc

Ac

34

ac (x)hc

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bmn χm (x) cos(nθ + ωt )

∞  ∞ 

45 46 47

cmn χm (x) sin(nθ + ωt )

∞  ∞  m =1 n =1

βθ =

37

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∂ χm (x) sin(nθ + ωt ) ∂x

(20)

m =1 n =1

βx =

32 33

2

m =1 n =1

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ac (x)

in which, ρ g , ρ c denote the equivalent density of the stiffeners. The Ritz method is used to calculate the vibration frequencies. The advantage of this method is that the test functions satisfy the boundary conditions and, therefore, this method is applicable to a wide range of problems. The displacement functions for different boundary conditions are defined as

50 52

24

t

− 2t −h g ,hc

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Ac

ˆ2

I g , Ic =

∞  ∞ 

49

(18)



t

44

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+ +ρc

ˆ− 2

m =1 n =1

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a g (x)

ρsh I sh + ρ g I g + ρ c I c

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2Ag



I2 =

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Rdxdθ

27

ρsh dz + ρ g

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ˆt /2

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2

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30 31

2

Here,

27 28



k=1 x1 0

25 26



˙ 2 + I 2 β˙x + β˙θ I 1 u˙ 2 + v˙ 2 + w

∞  ∞ 

48 49 50

∂ χm (x) dmn cos(nθ + ωt ) ∂x

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emn χm (x) sin(nθ + ωt )

55

m =1 n =1

56

where amn , bmn , cmn , dmn and emn are the constants and as



χm (x) = α1 cosh

βm (x − x1 ) L





+ α2 cos

χ (x) is the beam function that satisfies the boundary conditions and is defined

βm (x − x1 ) L



 − ξm α3 sinh



βm (x − x1 ) L



 + α4 sin

βm (x − x1 ) L

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59

(21)

For different boundary conditions the above constants are available in [47]. In the case of the free-free (F-F) boundary condition, the situation is different as displacement fields considered in Eq. (20) are not able to estimate the natural frequencies for m = 0 and m = 1 which are related to the rigid body translational and rotational modes. Ip et al. [48] and Yang et al. [49] considered a useful function for the cylindrical shell but this is not applicable to a conical shell due to the linear variation in the radius and stiffness which leads to the linear variation in the displacement as well. Therefore, the displacement functions for the F-F boundary conditions are considered as

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[m5G; v1.261; Prn:22/01/2020; 10:10] P.7 (1-17)

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Fig. 3. Stiffened specimens with different thicknesses fabricated by specially-designed filament winding setup.

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⎧ ∞ ⎪ u = n=1 a0n sin(nθ + ωt ) ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ v = n∞=1 b0n (x−Lx1 ) cos(nθ + ωt ) ⎪ ⎨ ∞ (x−x w = n=1 c 0n L 1 ) sin(nθ + ωt ) , ⎪ ⎪  ⎪ ∞ ⎪ ⎪ ⎪ βx = n=1 e 0n sin(nθ + ωt ) ⎪ ⎪  ⎩ βθ = n∞=1 d0n (x−Lx1 ) cos(nθ + ωt ) ⎧ ∞ ⎪ u = n=1 a1n sin(nθ + ωt ) ⎪ ⎪ ⎪ ∞ ⎪ (x−x ) ⎪ ⎪ v = n=1 b1n L 2 cos(nθ + ωt ) ⎪ ⎨ ∞ (x−x ) w = n=1 c 1n L 2 sin(nθ + ωt ) , ⎪ ⎪  ⎪ ⎪ ⎪ βx = n∞=1 e 1n sin(nθ + ωt ) ⎪ ⎪ ⎪  ⎩ βθ = n∞=1 d1n (x−Lx2 ) cos(nθ + ωt )

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m=0

(22)

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m=1

29

(23)

32 33 34

⎢ ⎢ ⎢ ⎣

C 13 C 23 C 33 C 34 C 35

C 14 C 24 C 34 C 44 C 45

C 15 C 25 C 35 C 45 C 55

⎤⎧ ⎪ ⎪ ⎥⎪ ⎨ ⎥ ⎥ ⎦⎪ ⎪ ⎪ ⎩

amn bmn cmn dmn emn

⎫ ⎪ ⎪ ⎪ ⎬

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⎪ ⎪ ⎪ ⎭

(25)

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For a nontrivial solution of this, the determinant of the coefficient matrix should be zero, resulting in an eigenvalue problem as

λ0 ω10 + λ1 ω8 + λ2 ω6 + λ3 ω4 + λ4 ω2 + λ5 = 0

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49

(26)

By solving the above equation and extracting the frequency values, the smallest frequency is considered as the fundamental natural frequency for the grid-stiffened conical shell.

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3. Experimental modal analysis (EMA)

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=0

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38

(24)

For the state of equilibrium, the total potential energy should be least. By applying the strain-displacement relations and minimizing the above functional with respect to the unknown constants, one arrives at the following square matrix

C 12 C 22 C 23 C 24 C 25

30 31

 = T max − U max

C 11 C 12 C 13 C 14 C 15

22 23

For m > 1, the beam function of Eq. (21) is capable of providing a solution. The circumferential, longitudinal wave numbers and the frequency of stiffened conical shell are represented by n, m and ω respectively. Finally, the energy functional for the stiffened conical shell is given by the Lagrangian function as follows

⎡

21

The specimens were composed of E-glass fiber, woven glass fibers and epoxy resin. Silicon dies and specially-designed filament winding setup were used in order to fabricate the specimens. The schematic view of the fabricated specimens is represented in Fig. 3. Three specimens with three different thicknesses were provided. The nominal material properties for the skin and stiffeners are given in Tables 1 and 2, respectively. The rule of mixture was implemented to obtain the material properties of the specimens [50]. For the experimental modal testing, the F-F boundary condition was provided by hanging the structure by two flexible strings. Fig. 4 shows the experimental modal analysis setup. A 4-channel 24-bit data analyzer with 50 g accelerometer were used for the measurement of the frequency response. Roving excitation was performed by exciting all 77 grid points of the structure using a hammer. The mass of the accelerometer (5 g) is negligible as compared to that of the structure (475 g). Three specimens with different thickness were considered for present analysis. Finally, the frequency response functions (FRFs) were processed by MODAL VIEW software to extract the natural frequencies of composite stiffened conical shells corresponding to each vibration mode through curve fitting (see Fig. 5).

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[m5G; v1.261; Prn:22/01/2020; 10:10] P.8 (1-17)

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Table 1 Mechanical properties of the shell.

2 3

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4

Young’s modulus Shear modulus Poisson’s ratio Density

5 6 7

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ν12 ρ (kg/m3 )

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Table 2 Mechanical properties of the stiffeners.

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Properties

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E 11 , E 22 , E 33 (Gpa) G 12 , G 13 , G 23 (Gpa)

ν12 , ν13 , ν23 ρ (kg/m3 )

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Fig. 4. Experimental modal analysis setup.

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Table 3 Geometrical dimensions of the shell and stiffeners.

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Geometry

value

Large radius (R 2 ) Small radius (R 1 ) Semi-vertex angle (α ) Thickness (t) Stiffener angle (ϕ ) Stiffeners cross section area (bh × h) Number of stiffeners

147 mm 77 mm 14.03 1.3 mm 11.5 4 mm×4 mm 22

4. Finite element analysis

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A 3-D model was created for the inner grid-stiffened conical shell using the ABAQUS CAE software (Fig. 6). The current model consists of 22 stiffeners based on the geodesic path and the stiffener angles with respect to the longitudinal direction of the conical shell is considered as 11.5◦ . The skin (shell) is modeled as a laminate. The stiffeners are also modeled as a 3-D beam. To assign the properties to the stiffener, a local coordinate system is defined in such a way that its main axis is along the helical stiffener. Also, the rotation of the stiffener is accompanied with the axial rotation of the coordinate system. The stiffeners are tied firm to the conical shell and therefore the stiffener and the shell acts as a unique structure. The quadratic planar elements with 8 nodes (S8R) and a three-dimensional quadratic cubic elements with 20 nodes (C3D20R) were used to mesh the shell and stiffeners, respectively. The skin was modeled as eight ply laminate with stacking sequence of [0, 90], each layer having a thickness of 0.17 mm. The geometrical parameters of the shell and stiffeners are given in Table 3.

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[m5G; v1.261; Prn:22/01/2020; 10:10] P.9 (1-17)

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Fig. 6. 3-D models used in the finite element analysis.

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5. Results and discussion 5.1. Comparative study

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57 58

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In order to evaluate the results, a comparison has been made between the natural frequencies of the grid-stiffened conical shell obtained via the analytical approach with those of numerical ones via finite element method (FEM) and illustrated in Fig. 7. It can be seen that the results of both methods are in good agreement and the differences between the analyses is due to the simplification assumptions that were considered in the analysis as well as FE simulation, such as the effect of the initial curve of the stiffeners. The different boundary conditions considered here are the combinations of simply supported (S), clamped (C) and free (F) ones. In the case of F-F boundary condition, the modal beam functions are unable to estimate the natural frequencies associated with the longitude mode numbers of m = 0 and 1. As mentioned earlier, two types of the modal functions were defined and analytical solutions obtained as well for the rigid body modes. Figs. 8 and 9 plots the natural frequencies associated with two rigid body modes for three

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[m5G; v1.261; Prn:22/01/2020; 10:10] P.10 (1-17)

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Table 4 Natural frequencies and mode shapes for a F-F grid stiffened composite conical shell (t = 1.3 mm).

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different values of skin thickness. It can be seen that the results agree well with the experimental ones and this agreement is better for higher thickness values. As would be observed, the effect of thickness variation on the natural frequencies is more significant for higher circumferential wave numbers. However, the modal function can predict the natural frequencies for m > 1 (see Fig. 10). Table 4 lists the first six natural frequencies and corresponding mode shapes of the F-F composite stiffened conical shells achieved via EMA and FEM methods. As can be seen, the natural frequencies obtained from the two types of analysis meet a good agreement and also the corresponding mode shapes are very much alike. The sensible difference between the two types of analyses in some vibration modes is mainly due to the complexity of the structure which leads to the possible inaccuracies while fabricating the specimens. In addition, the use of simplification assumptions while building the structure’s finite element model may cause the predictive models to become distant from the real specimens. In general, this comparison verifies that the finite element model is qualified enough for predicting the vibrational behavior of stiffened composite conical shells.

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[m5G; v1.261; Prn:22/01/2020; 10:10] P.11 (1-17)

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Fig. 9. Comparison of the analytical and experimental results for the stiffened shell with different thicknesses and F-F boundary condition (m = 1).

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Fig. 10. Comparison of the analytical and experimental results for the stiffened shell with different thicknesses and F-F boundary condition (m > 1).

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5.2. Effect of shell parameters on the natural frequency

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5.2.1. Effect of the semi-vertex angle on the vibration frequency As discussed before, the stiffeners behave like beam structures. Having solved the governing equations of a beam, its natural frequency for the different boundary conditions will be obtained as



64 65 66

ωbeam =

Ki 2π

L2

61 62 63 64

EI

ρA

(27)

65 66

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12

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[m5G; v1.261; Prn:22/01/2020; 10:10] P.12 (1-17)

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Fig. 11. Effect of the semi-vertex angle on the vibration frequencies of stiffened conical shells for the fixed radii and S-S boundary condition.

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Fig. 12. Effect of shell’s semi-vertex angle on the frequency of stiffened and unstiffened conical shells with the same mass for fixed length and fixed small radius (n = 2).

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

where I and L denote the moment of inertia and length of the beam, respectively. K i , is a variable constant, due to the different boundary conditions. There are two approaches for studying the effect of semi-vertex angle of a stiffened cone. In the first approach, the large and small radii are kept fixed and only the length of a conical shell changes. Second approach expresses assumes the small radius and length of the conical shell to be fixed and only the large radius changes. Fig. 11 indicates the frequency variation with this angle for the first approach. It can be observed that by increasing the semi-vertex angle, the natural frequency of the unstiffened shell is greater than the stiffened one, but from a specific value, the trend become reverse and the frequency of grid-stiffened conical shell is greater than the unstiffened one. It is due to this fact that increase in the semi-vertex angle with the fixed radii leads to decrease in length of the shell and stiffener. From Eq. (27), it is clear that by reducing the stiffener’s length, the natural frequency increases with the power of two. It is clear that as the angle reaches 90◦ , the difference between the frequency values of stiffened and unstiffened conical shells meets its peak. In this situation, the conical shell is converted to a circular plate with a cutout for which the natural frequency is completely dependent on the bending stiffness of the stiffened shell. Stiffeners have a high bending stiffness due to their high moment of inertia compared to the shell. In this situation, stiffeners exhibit maximum effect on the structure. In other words, the increase rate of stiffness is greater than that of inertia. Therefore, the moment of inertia acts as a positive parameter for the increasing frequency of the structure. It can be observed that at higher modes, the stiffness contribution of the stiffener increases at the smaller semi-vertex angles because the deflections at higher modes are much more than the lower ones therefore, the effect of stiffener on the stiffness is much prominent. In the second approach, the small radius and length of the conical shell are fixed but the large radius varies (see Fig. 12). In this situation, the conical shell’s length is more increased as compared to the previous situation. Hence, the natural frequency of the stiffened shell is higher than the unstiffened one for the larger values of semi-vertex angles. However, for lower semi-vertex angles, the natural frequencies of unstiffened shells are greater than the stiffened ones. This is mainly due to the fact that at lower angles, the increasing rate of the stiffener inertia is more than its stiffness contribution. The effects of boundary conditions are also exhibited in Fig. 12. According to preliminary texts, it is proved that K C −C > K S − S > K C − F . As expected before, the maximum and minimum natural frequencies are associated with C-C and F-F boundary conditions, respectively.

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40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

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5.2.2. Effect of shell thickness The natural frequency variations versus the circumferential wave number for the stiffened conical shell are shown in Fig. 13 and compared with those of unstiffened one. As can be observed, at lower thicknesses, the natural frequencies of the stiffened shell are greater than those of unstiffened one, but at higher values of the shell thickness, the trend becomes reverse. This is due to this fact that at

63 64 65 66

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[m5G; v1.261; Prn:22/01/2020; 10:10] P.13 (1-17)

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Fig. 13. Effect of shell thickness on the natural frequencies of the stiffened and unstiffened conical shell under C-C end conditions.

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Fig. 14. Effect of laminate fiber angle on the natural frequency of the stiffened composite conical shell with C-C boundary conditions.

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lower thicknesses of the shell, the skin acts as a membranous structure therefore, the major stiffness of the stiffened structure is provided by the stiffener and the stiffness contribution of the stiffener is higher than its inertial one. However, at higher thicknesses, the stiffness contribution of the shell exceeds that of the stiffener, resulting into a reverse behavior.

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5.2.3. Effect of fiber angle on the natural frequency Fig. 14 depicts the variation of the natural frequency versus circumferential wave number for different cross-ply lamination sequences of the skin and C-C boundary conditions. It is seen that the difference between frequency curves increases with an increment in the mode number. 5.3. Effect of the stiffener orientation angle on the natural frequency

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Fig. 15 illustrates the fundamental frequency variation with laminate stacking sequence for a four-layered laminated conical shell ([θ /–θ /θ /–θ ]) under S-S conditions. It is clear that the maximum value of the natural frequency is obtained when the stiffener orientation angle meets its peak (the maximum stiffener angle allowed in the present truncated conical shell is 30◦ ) and θ = 55◦ . Also, the minimum fundamental natural frequency is related to the lowest values of the stiffener orientation and fiber angles. As it is expected, an increment in the stiffener orientation angle results into the increasing stiffness in the hoop direction which leads to the natural frequency increment as a consequence. Fig. 16 shows the natural frequency variation of a conical shell with different stiffener angles and circumferential mode numbers. It can be seen that the natural frequency increases with an increment in the stiffener orientation angle and this increase is more prominent for higher modes. This is mainly due to the fact that the stiffness in the circumferential direction increases by increasing the stiffener angle, which leads to the increasing bending stiffness and therefore natural frequencies. Besides, the deflection increases at higher vibration modes and hence the effect of stiffeners will be more prominent. Fig. 17 shows the fundamental frequency variation versus semi-vertex angle for different stiffener angles and mode numbers. As would be observed, at semi-vertex angles lower than 60◦ (α < 60◦ ), an increment in the stiffener angle results in the natural frequency increase. However, for higher values of the angle (α > 60◦ ), the natural frequency decreases as the stiffener angle increases. This can be justified knowing the fact that at lower semi-vertex angles, the stiffener angle increase leads to an increment in the circumferential stiffness

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AID:105717 /FLA

[m5G; v1.261; Prn:22/01/2020; 10:10] P.14 (1-17)

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Fig. 16. Effect of the stiffener orientation angle on the natural frequency of the S-S conical shell.

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but at higher ones, it causes the bending stiffness to be reduced and as a result, the natural frequency decreases. The variation of the fundamental frequency of a grid-stiffened conical shell with stiffener orientation angle is plotted in Fig. 18. It is clear that the maximum value of the natural frequency is obtained for the greatest stiffener orientation angles and α = 53◦ . Furthermore, a comparison is made by Fig. 19 between the two different types of the grid structure including angle grid and triangle grid. As can be seen for lower values of the semi-vertex angle, the natural frequency of stiffened shell with triangle grid is greater than those with angle grid. However, the trend becomes reverse for higher semi-vertex angles with a negligible difference between the two curves. This is mainly due to the fact that the ring’s stiffness contribution in bending vibration decreases by increasing semi-vertex angle. 6. Conclusion

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In this study, the free vibrational behavior of grid-stiffened composite conical shells have been thoroughly investigated using experimental, analytical and numerical techniques probably for the first time. A specially-designed filament-winding setup was utilized for fabricating the stiffened composite conical shells. The results achieved via the three analyses were in good agreement. The minor differences refer to the complexity of the structure and possible errors may happen during the specimen production. Comparison of the results indicated that the analytical and numerical models could well predict the natural frequencies of vibration. Since, fabricating several conical shell specimens with different structures and configurations and performing experimental modal tests on them with different boundary conditions is an expensive work requiring more advance setups, an extensive parametric study was conducted in order to examine the effects of boundary conditions, geometrical parameters of the shell and stiffeners on the vibrational characteristics. These results can be used as a benchmark for future investigations on this subject. For the lower skin thicknesses and higher mode numbers, the natural frequency of the grid-stiffened conical shell was higher than that of the unstiffened one. This is mainly due to the fact that for the lower thicknesses, the stiffness contribution of the stiffener is more than its mass. The orientation angle of the stiffeners was also shown to be more effective at higher vibration modes. For higher values of

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Fig. 17. Effect of the semi-vertex angle on the fundamental frequency of the S-S conical shell with fixed radii.

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Fig. 18. Contour plot of the fundamental frequency with varying semi-vertex and stiffener orientation angles for the S-S conical shell with fixed radii.

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Fig. 19. Effect of grid structure on the fundamental natural frequency of a conical shell with fixed radii and S-S boundary conditions.

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the semi-vertex angle, the natural frequencies of the grid-stiffened conical shell are greater compared to the unstiffened one because the bending stiffness of the stiffeners became more prominent in these cases. Further to these, the significant influence of the shell’s boundary conditions on the natural frequencies was illustrated. The quality of the present experimental results may be improved in the future by using more advanced equipment. Declaration of competing interest

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This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue. The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript. References

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