# Dynamic Characters of Stiffened Composite Conoidal Shell Roofs with Cutouts: Design Aids and Selection Guidelines

Authors: Sarmila Sahoo
Journal: Journal of Engineering
 (2013)

---

## Abstract

Dynamic characteristics of stiffened composite conoidal shells with cutout are analyzed in terms of the natural frequency and mode shapes. A finite element code is developed for the purpose by combining an eight-noded curved shell element with a three-noded curved beam element. The code is validated by solving benchmark problems available in the literature and comparing the results. The size of the cutouts and their positions with respect to the shell centre are varied for different edge constraints of cross-ply and angle-ply laminated composite conoids. The effects of these parametric variations on the fundamental frequencies and mode shapes are considered in details. The results furnished here may be readily used by practicing engineers dealing with stiffened composite conoids with cutouts central or eccentric.


---

## Body

## 1. Introduction

Laminated composite structures are gaining wide importance in various fields of aerospace and civil engineering. Shell roof structures can be conveniently built with composite materials that have many attributes, besides high specific strength and stiffness. Among the different shell panels which are commonly used as roofing units in civil engineering practice, the conoidal shell has a special position due to a number of advantages it offers. Conoidal shells are often used to cover large column-free areas. Being ruled surfaces, they provide ease of casting and also allow north light in. Hence, this shell is preferred in many places, particularly in medical, chemical, and food processing industries where entry of north light is desirable. Application of conoids in these industries often necessitates cutouts for the passage of light, service lines, and also sometimes for alteration of resonant frequency. In practice, the margin of the cutouts must be stiffened to take account of stress concentration effects. An in-depth study including bending, buckling, vibration, and impact is required to exploit the possibilities of these curved forms. The present investigation is, however, restricted only to the free vibration behaviour. A generalized formulation for the doubly curved laminated composite shell has been presented using the eight-noded curved quadratic isoparametric finite element including three radii of curvature. Some of the important contributions on the investigation of conoidal shells are briefly reviewed here.The research on conoidal shell started about four decades ago. In 1964, Hadid [1] analysed static characteristics of conoidal shells using the variational method. The research was carried forward and improved by researchers like Brebbia and Hadid [2], Choi [3], Ghosh and Bandyopadhyay [4, 5], Dey et al. [6], and Das and Bandyopadhyay [7]. Dey et al. [6] provided a significant contribution on static analysis of conoidal shell. Chakravorty et al. [8] applied the finite element technique to explore the free vibration characteristics of shallow isotropic conoids and also observed the effects of excluding some of the inertia terms from the mass matrix on the first four natural frequencies. Chakravorty et al. [9–11] published a series of papers where they reported on free and forced vibration characteristics of graphite-epoxy composite conoidal shells with regular boundary conditions. Later, Nayak and Bandyopadhyay [12–15] reported free vibration of stiffened isotropic and composite conoidal shells. Das and Chakravorty [16, 17] considered bending and free vibration characteristics of unpunctured and unstiffened composite conoids. Hota and Chakravorty [18] studied isotropic punctured conoidal shells with complicated boundary conditions along the four edges, but no such study about composite conoidal shells is available in the literature. Also, they did not furnish any information on vibration mode shapes. It is also seen from the recent reviews [19, 20] that dynamic characteristics of stiffened conoidal shells with cutout are still missing in the literature. The present study thus focuses on the free vibrations of graphite-epoxy laminated composite stiffened conoids with cutout both in terms of the natural frequencies and mode shapes. The results so obtained may be readily used by practicing engineers dealing with stiffened composite conoids with cutouts. The novelty of the present study lies in the consideration of vibration mode shapes of stiffened composite conoids in presence of cutouts.


## 2. Mathematical Formulation

A laminated composite conoidal shell of uniform thicknessh (Figure 1), and radius of curvature Ry, and radius of cross curvature Rxy is considered. Keeping the total thickness the same, the thickness may consist of any number of thin laminae each of which may be arbitrarily oriented at an angle θ with reference to the x-axis of the coordinate system. The constitutive equations for the shell are given by (a list of notations is separately given)
(1){F}=[E]{ε},
where
(2){F}={Nx,Ny,Nxy,Mx,My,Mxy,Qx,Qy}T,[E]=[[A][B][0][B][D][0][0][0][S]],{ε}={εx0,εy0,γxy0,kx,kxy,γxz0,γyz0}T.
The force and moment resultants are expressed as
(3){Nx,Ny,Nxy,Mx,My,Mxy,Qx,Qy}T=∫-h/2h/2{σx,σy,τxy,σz·z,σy·z,τxy·z,τxz,τyz}Tdz.
The submatrices [A], [B], [D], and [S] of the elasticity matrix [E] are functions of Young’s moduli, shear moduli, and Poisson’s ratio of the laminates. They also depend on the angle which the individual lamina of a laminate makes with the global x-axis. The detailed expressions of the elements of the elasticity matrix are available in several references including Vasiliev et al. [21] and Qatu [22].Figure 1
Conoidal shell with a concentric cutout stiffened along the margins.The strain-displacement relations on the basis of improved first-order approximation theory for thin shell (Dey et al. [6]) are established as
(4){εx,εy,γxy,γxz,γyz}T={εx0,εy0,γxy0,γxz0,γyz0}T+z{kx,ky,kxy,kxz,kyz}T,
where the first vector is the midsurface strain for a conoidal shell and the second vector is the curvature.


## 3. Finite Element Formulation

### 3.1. Finite Element Formulation for Shell

An eight-noded curved quadratic isoparametric finite element is used for conoidal shell analysis. The five degrees of freedom taken into consideration at each node areu,  v,  w,  α,  β. The following expressions establish the relations between the displacement at any point with respect to the coordinates ξ and η and the nodal degrees of freedom
(5)u=∑i=18Niui,v=∑i=18Nivi,w=∑i=18Niwi,α=∑i=18Niαi,β=∑i=18Niβi,
where the shape functions derived from a cubic interpolation polynomial [6] are
(6)Ni=(1+ξξi)(1+ηηi)(ξξi+ηηi-1)4,fori=1,2,3,4,Ni=(1+ξξi)(1-η2)2,fori=5,7,Ni=(1+ηηi)(1-ξ2)2,fori=6,8.
The generalized displacement vector of an element is expressed in terms of the shape functions and nodal degrees of freedom as
(7)[u]=[N]{de},
that is,
(8){u}={uvwαβ}=∑i=18[NiNiNiNiNi]{uiviwiαiβi}.


#### 3.1.1. Element Stiffness Matrix

The strain-displacement relation is given by(9){ε}=[B]{de},
where
(10)[B]=∑i=18[Ni,x00000Ni,y-NiRy00Ni,yNi,x-2NiRxy00000Ni,x00000Ni,y000Ni,yNi,x00Ni,xNi000Ni,y0Ni].
The element stiffness matrix is
(11)[Ke]=∬[B]T[E][B]dxdy.


#### 3.1.2. Element Mass Matrix

The element mass matrix is obtained from the integral
(12)
[
M
e
]
=
∬
[
N
]
T
[
P
]
[
N
]
d
x
d
y
,
where
(13)[N]=∑i=18[Ni00000Ni00000Ni00000Ni00000Ni],[P]=∑i=18[P00000P00000P00000I00000I],
in which
(14)P=∑k=1np∫zk-1zkρdz,I=∑k=1np∫zk-1zkzρdz.


### 3.2. Finite Element Formulation for Stiffener of the Shell

Three-noded curved isoparametric beam element (Figure2) are used to model the stiffeners, which are taken to run only along the boundaries of the shell elements. In the stiffener element, each node has four degrees of freedom, that is, usx, wsx, αsx, and βsx for X-stiffener and vsy, wsy, αsy, and βsy for Y-stiffener. The generalized force-displacement relation of stiffeners can be expressed as
(15)X-stiffener:{Fsx}=[Dsx]{εsx}=[Dsx][Bsx]{δsxi},Y-stiffener:{Fsy}=[Dsy]{εsy}=[Dsy][Bsy]{δsyi},
where
(16){Fsx}=[NsxxMsxxTsxxQsxxz]T,{εsx}=[usx·xαsx·xβsx·x(αsx+wsx·x)]T,{Fsy}=[NsyyMsyyTsyyQsyyz]T,{εsy}=[vsy·yβsy·yαsy·y(βsy+wsy·y)]T.
The generalized displacements of the Y-stiffener and the shell are related by the transformation matrix {δsyi}=[T]{δ}, where
(17)[T]=[1+eRysymmetric010010001].
This transformation is required due to curvature of Y-stiffener, and {δ} is the appropriate portion of the displacement vector of the shell excluding the displacement component along the x-axis.(a) Eight-noded shell element with isoparametric coordinates. (b) Three-noded stiffener element: (i)X-stiffener (ii) Y-stiffener.
(a)
(b)Elasticity matrices are as follows:(18)[Dsx]=[A11bsxB11′bsxB12′bsx0B11′bsxD11′bsxD12′bsx0B12′bsxD12′bsx16(Q44+Q66)dsxbsx30000bsxS11],[Dsy]=[A22bsyB22′bsyB12′bsy0B22′bsy16(Q44+Q66)bsyD12′bsy0B12′bsyD12′bsyD11′dsybsy30000bsyS22],
where
(19)Dij′=Dij+2eBij+e2Aij,Bij′=Bij+eAij,
and Aij, Bij, Dij, and Sij are explained in an earlier paper by Sahoo and Chakravorty [23].Here, the shear correction factor is taken as 5/6. The sectional parameters are calculated with respect to the midsurface of the shell by which the effect of eccentricities of stiffeners is automatically included. The element stiffness matrices are of the following forms:(20)forX-stiffener:[Kxe]=∫[Bsx]T[Dsx][Bsx]dx,forY-stiffener:[Kye]=∫[Bsy]T[Dsy][Bsy]dy.
The integrals are converted to isoparametric coordinates and are carried out by 2-point Gauss quadrature. Finally, the element stiffness matrix of the stiffened shell is obtained by appropriate matching of the nodes of the stiffener and shell elements through the connectivity matrix and is given as
(21)[Ke]=[Kshe]+[Kxe]+[Kye].
The element stiffness matrices are assembled to get the global matrices.


#### 3.2.1. Element Mass Matrix

The element mass matrix for shell is obtained from the integral
(22)
[
M
e
]
=
∬
[
N
]
T
[
P
]
[
N
]
d
x
d
y
,
where
(23)[N]=∑i=18[Ni00000Ni00000Ni00000Ni00000Ni],[P]=∑i=18[P00000P00000P00000I00000I],
in which
(24)P=∑k=1np∫zk-1zkρdz,I=∑k=1np∫zk-1zkzρdz.
Element mass matrix for stiffener element
(25)[Msx]=∬[N]T[P][N]dxforX-stiffener,[Msy]=∬[N]T[P][N]dyforY-stiffener.
Here, [N] is a 3 × 3 diagonal matrix.Consider(26)[P]=∑i=13[ρ·bsxdsx0000ρ·bsxdsx0000ρ·bsxdsx2120000ρ(bsx·dsx3+bsx3·dsx)12]forX-stiffener,[P]=∑i=13[ρ·bsydsy0000ρ·bsydsy0000ρ·bsydsy2120000ρ(bsy·dsy3+bsy3·dsy)12]forY-stiffener.
The mass matrix of the stiffened shell element is the sum of the matrices of the shell and the stiffeners matched at the appropriate nodes
(27)[Me]=[Mshe]+[Mxe]+[Mye].
The element mass matrices are assembled to get the global matrices.


### 3.3. Modeling the Cutout

The code developed can take the position and size of cutout as input. The program is capable of generating nonuniform finite element mesh all over the shell surface. So, the element size is gradually decreased near the cutout margins. One such typical mesh arrangement is shown in Figure3. Such finite element mesh is redefined in steps, and a particular grid is chosen to obtain the fundamental frequency when the result does not improve by more than one percent on further refining. Convergence of results is ensured in all the problems taken up here.Figure 3
Typical 10 × 10 nonuniform mesh arrangements drawn to scale.


### 3.4. Solution Procedure for Free Vibration Analysis

The free vibration analysis involves determination of natural frequencies from the condition(28)|[K]-ω2[M]|=0.This is a generalized eigen value problem and is solved by the subspace iteration algorithm.


## 3.1. Finite Element Formulation for Shell

An eight-noded curved quadratic isoparametric finite element is used for conoidal shell analysis. The five degrees of freedom taken into consideration at each node areu,  v,  w,  α,  β. The following expressions establish the relations between the displacement at any point with respect to the coordinates ξ and η and the nodal degrees of freedom
(5)u=∑i=18Niui,v=∑i=18Nivi,w=∑i=18Niwi,α=∑i=18Niαi,β=∑i=18Niβi,
where the shape functions derived from a cubic interpolation polynomial [6] are
(6)Ni=(1+ξξi)(1+ηηi)(ξξi+ηηi-1)4,fori=1,2,3,4,Ni=(1+ξξi)(1-η2)2,fori=5,7,Ni=(1+ηηi)(1-ξ2)2,fori=6,8.
The generalized displacement vector of an element is expressed in terms of the shape functions and nodal degrees of freedom as
(7)[u]=[N]{de},
that is,
(8){u}={uvwαβ}=∑i=18[NiNiNiNiNi]{uiviwiαiβi}.


### 3.1.1. Element Stiffness Matrix

The strain-displacement relation is given by(9){ε}=[B]{de},
where
(10)[B]=∑i=18[Ni,x00000Ni,y-NiRy00Ni,yNi,x-2NiRxy00000Ni,x00000Ni,y000Ni,yNi,x00Ni,xNi000Ni,y0Ni].
The element stiffness matrix is
(11)[Ke]=∬[B]T[E][B]dxdy.


### 3.1.2. Element Mass Matrix

The element mass matrix is obtained from the integral
(12)
[
M
e
]
=
∬
[
N
]
T
[
P
]
[
N
]
d
x
d
y
,
where
(13)[N]=∑i=18[Ni00000Ni00000Ni00000Ni00000Ni],[P]=∑i=18[P00000P00000P00000I00000I],
in which
(14)P=∑k=1np∫zk-1zkρdz,I=∑k=1np∫zk-1zkzρdz.


## 3.1.1. Element Stiffness Matrix

The strain-displacement relation is given by(9){ε}=[B]{de},
where
(10)[B]=∑i=18[Ni,x00000Ni,y-NiRy00Ni,yNi,x-2NiRxy00000Ni,x00000Ni,y000Ni,yNi,x00Ni,xNi000Ni,y0Ni].
The element stiffness matrix is
(11)[Ke]=∬[B]T[E][B]dxdy.


## 3.1.2. Element Mass Matrix

The element mass matrix is obtained from the integral
(12)
[
M
e
]
=
∬
[
N
]
T
[
P
]
[
N
]
d
x
d
y
,
where
(13)[N]=∑i=18[Ni00000Ni00000Ni00000Ni00000Ni],[P]=∑i=18[P00000P00000P00000I00000I],
in which
(14)P=∑k=1np∫zk-1zkρdz,I=∑k=1np∫zk-1zkzρdz.


## 3.2. Finite Element Formulation for Stiffener of the Shell

Three-noded curved isoparametric beam element (Figure2) are used to model the stiffeners, which are taken to run only along the boundaries of the shell elements. In the stiffener element, each node has four degrees of freedom, that is, usx, wsx, αsx, and βsx for X-stiffener and vsy, wsy, αsy, and βsy for Y-stiffener. The generalized force-displacement relation of stiffeners can be expressed as
(15)X-stiffener:{Fsx}=[Dsx]{εsx}=[Dsx][Bsx]{δsxi},Y-stiffener:{Fsy}=[Dsy]{εsy}=[Dsy][Bsy]{δsyi},
where
(16){Fsx}=[NsxxMsxxTsxxQsxxz]T,{εsx}=[usx·xαsx·xβsx·x(αsx+wsx·x)]T,{Fsy}=[NsyyMsyyTsyyQsyyz]T,{εsy}=[vsy·yβsy·yαsy·y(βsy+wsy·y)]T.
The generalized displacements of the Y-stiffener and the shell are related by the transformation matrix {δsyi}=[T]{δ}, where
(17)[T]=[1+eRysymmetric010010001].
This transformation is required due to curvature of Y-stiffener, and {δ} is the appropriate portion of the displacement vector of the shell excluding the displacement component along the x-axis.(a) Eight-noded shell element with isoparametric coordinates. (b) Three-noded stiffener element: (i)X-stiffener (ii) Y-stiffener.
(a)
(b)Elasticity matrices are as follows:(18)[Dsx]=[A11bsxB11′bsxB12′bsx0B11′bsxD11′bsxD12′bsx0B12′bsxD12′bsx16(Q44+Q66)dsxbsx30000bsxS11],[Dsy]=[A22bsyB22′bsyB12′bsy0B22′bsy16(Q44+Q66)bsyD12′bsy0B12′bsyD12′bsyD11′dsybsy30000bsyS22],
where
(19)Dij′=Dij+2eBij+e2Aij,Bij′=Bij+eAij,
and Aij, Bij, Dij, and Sij are explained in an earlier paper by Sahoo and Chakravorty [23].Here, the shear correction factor is taken as 5/6. The sectional parameters are calculated with respect to the midsurface of the shell by which the effect of eccentricities of stiffeners is automatically included. The element stiffness matrices are of the following forms:(20)forX-stiffener:[Kxe]=∫[Bsx]T[Dsx][Bsx]dx,forY-stiffener:[Kye]=∫[Bsy]T[Dsy][Bsy]dy.
The integrals are converted to isoparametric coordinates and are carried out by 2-point Gauss quadrature. Finally, the element stiffness matrix of the stiffened shell is obtained by appropriate matching of the nodes of the stiffener and shell elements through the connectivity matrix and is given as
(21)[Ke]=[Kshe]+[Kxe]+[Kye].
The element stiffness matrices are assembled to get the global matrices.


### 3.2.1. Element Mass Matrix

The element mass matrix for shell is obtained from the integral
(22)
[
M
e
]
=
∬
[
N
]
T
[
P
]
[
N
]
d
x
d
y
,
where
(23)[N]=∑i=18[Ni00000Ni00000Ni00000Ni00000Ni],[P]=∑i=18[P00000P00000P00000I00000I],
in which
(24)P=∑k=1np∫zk-1zkρdz,I=∑k=1np∫zk-1zkzρdz.
Element mass matrix for stiffener element
(25)[Msx]=∬[N]T[P][N]dxforX-stiffener,[Msy]=∬[N]T[P][N]dyforY-stiffener.
Here, [N] is a 3 × 3 diagonal matrix.Consider(26)[P]=∑i=13[ρ·bsxdsx0000ρ·bsxdsx0000ρ·bsxdsx2120000ρ(bsx·dsx3+bsx3·dsx)12]forX-stiffener,[P]=∑i=13[ρ·bsydsy0000ρ·bsydsy0000ρ·bsydsy2120000ρ(bsy·dsy3+bsy3·dsy)12]forY-stiffener.
The mass matrix of the stiffened shell element is the sum of the matrices of the shell and the stiffeners matched at the appropriate nodes
(27)[Me]=[Mshe]+[Mxe]+[Mye].
The element mass matrices are assembled to get the global matrices.


## 3.2.1. Element Mass Matrix

The element mass matrix for shell is obtained from the integral
(22)
[
M
e
]
=
∬
[
N
]
T
[
P
]
[
N
]
d
x
d
y
,
where
(23)[N]=∑i=18[Ni00000Ni00000Ni00000Ni00000Ni],[P]=∑i=18[P00000P00000P00000I00000I],
in which
(24)P=∑k=1np∫zk-1zkρdz,I=∑k=1np∫zk-1zkzρdz.
Element mass matrix for stiffener element
(25)[Msx]=∬[N]T[P][N]dxforX-stiffener,[Msy]=∬[N]T[P][N]dyforY-stiffener.
Here, [N] is a 3 × 3 diagonal matrix.Consider(26)[P]=∑i=13[ρ·bsxdsx0000ρ·bsxdsx0000ρ·bsxdsx2120000ρ(bsx·dsx3+bsx3·dsx)12]forX-stiffener,[P]=∑i=13[ρ·bsydsy0000ρ·bsydsy0000ρ·bsydsy2120000ρ(bsy·dsy3+bsy3·dsy)12]forY-stiffener.
The mass matrix of the stiffened shell element is the sum of the matrices of the shell and the stiffeners matched at the appropriate nodes
(27)[Me]=[Mshe]+[Mxe]+[Mye].
The element mass matrices are assembled to get the global matrices.


## 3.3. Modeling the Cutout

The code developed can take the position and size of cutout as input. The program is capable of generating nonuniform finite element mesh all over the shell surface. So, the element size is gradually decreased near the cutout margins. One such typical mesh arrangement is shown in Figure3. Such finite element mesh is redefined in steps, and a particular grid is chosen to obtain the fundamental frequency when the result does not improve by more than one percent on further refining. Convergence of results is ensured in all the problems taken up here.Figure 3
Typical 10 × 10 nonuniform mesh arrangements drawn to scale.


## 3.4. Solution Procedure for Free Vibration Analysis

The free vibration analysis involves determination of natural frequencies from the condition(28)|[K]-ω2[M]|=0.This is a generalized eigen value problem and is solved by the subspace iteration algorithm.


## 4. Numerical Examples

The validity of the present approach is checked through solution of benchmark problems. The first problem, free vibration of stiffened clamped conoid, was solved earlier by Nayak and Bandyopadhyay [13]. The second is the free vibration of composite conoid with cutouts solved by Chakravorty et al. [11]. The results obtained by the present method, along with the published results, are presented in Tables 1 and 2, respectively.Table 1
Fundamental frequencies (rad/sec) of clamped conoidal shell with central stiffeners.
Stiffener position
Stiffener alongx-direction
Stiffener alongy-direction
Stiffener along bothx- and y-directions
Nayak and Bandyopadhyay [13]
Present model
Nayak and Bandyopadhyay [13]
Present model
Nayak and Bandyopadhyay [13]
Present model
8
×
8
10
×
10
12
×
12
8
×
8
10
×
10
12
×
12
8
×
8
10
×
10
12
×
12
Concentric
17.28
17.50
17.37
17.31
20.83
21.15
20.85
20.76
20.90
21.24
21.01
20.84
Eccentric at top
17.83
17.92
17.78
17.73
21.57
21.76
21.53
21.45
22.39
22.86
22.51
22.32
Eccentric at bottom
17.55
17.80
17.61
17.52
22.31
22.66
22.40
22.24
22.92
23.25
23.00
22.87
a
=
50 m, b=50 m, h=0.2 m, hh = 10 m, hl = 2.5 m, E=25.4910×109, ν=0.15, ρ=2500 kg/m3, ws=0.3 m, and hs=1 m.Table 2
Nondimensional fundamental frequencies(ω-) for laminated composite conoidal shell with cutout.
a
′
/
a
Corner point supported
Simply supported
Clamped
Chakravorty et al. [11]
Present model
Chakravorty et al. [11]
Present model
Chakravorty et al. [11]
Present model
0.0
23.863
23.494
75.450
74.892
124.736
123.306
0.1
23.554
23.872
75.098
75.278
123.811
123.987
0.2
23.746
23.485
73.668
73.324
122.074
120.588
0.3
23.510
23.768
69.979
69.763
120.515
119.101
0.4
23.205
23.101
61.824
61.524
116.924
115.924
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=2.5, and hl/hh=0.25.Additional problems for conoids with cutouts are solved, varying the size and position of cutout along both of the plan directions of the shell for different practical boundary conditions. In order to study the effect of cutout size on the free vibration response, results for unpunctured conoids are also included in the study.


## 5. Results and Discussion

It is found from Table1 that the fundamental frequencies of stiffened conoids obtained by the present method agree well with those reported by Nayak and Bandyopadhyay [13]. Here, monotonic convergence is noted as the mesh is made progressively finer. Thus, the correctness of the stiffened shell element used here is established. It is evident from Table 2 that the present results agree with those of Chakravorty et al. [11], and the fact that the cutouts are properly modeled in the present formulation is thus established.


### 5.1. Free Vibration Behaviour of Shells with Concentric Cutouts

Tables3 and 4 show the results for the non-dimensional fundamental frequency ω¯ of composite cross-ply and angle-ply  stiffened conoidal shells for different cutout sizes and various combinations of boundary conditions along the four edges. The shells considered are of square planform (a=b), and the cutouts are also taken to be square in plan (a′=b′). The cutout sizes (i.e., a′/a) are varied from 0 to 0.4, and boundary conditions are varied along the four edges. Cutouts are concentric on shell surface. The stiffeners are placed along the cutout periphery and extended up to the edge of the shell. The boundary conditions are designated by describing the support clamped or simply supported as C or S taken in an anticlockwise order from the edge x=0. This means that a shell with CSCS boundary is clamped along x=0, simply supported along y=0, clamped along x=a, and simply supported along y=b. The material and geometric properties of shells and cutouts are mentioned along with the figures.Table 3
Non-dimensional fundamental frequencies(ω-) for laminated composite (0/90/0/90) stiffened conoidal shell for different sizes of the central square cutout and different boundary conditions.
Boundary conditions
Cutout size (a′/a)
0
0.1
0.2
0.3
0.4
CCCC
105.7679
118.9136
124.386
127.4786
124.4929
CSCC
79.3544
87.8992
91.4718
96.4778
97.0499
CCSC
104.0259
117.3182
120.8486
122.5918
119.586
CCCS
79.0009
86.457
91.1477
95.8592
97.0069
CSSC
76.7919
84.417
87.445
91.1546
90.359
CCSS
76.4645
83.126
87.1684
90.6361
90.3194
CSCS
70.8733
76.0322
80.215
86.5974
91.9972
SCSC
96.4607
106.1886
112.2644
116.5409
114.8718
CSSS
65.7562
69.5302
72.8659
77.5964
81.1403
SSSC
65.7398
70.72
75.0802
79.9808
82.2204
SSCS
62.3308
65.2487
69.3811
74.7187
80.4246
SSSS
54.7734
56.6793
59.276
62.3683
65.2097
Point supported
21.1813
21.686
22.6166
24.2309
25.5361
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=5, hl/hh=0.25, E11/E22=25, G23=0.2E22, G13=G12=0.5E22, and ν12=ν21=0.25.Table 4
Non-dimensional fundamental frequencies(ω-) for laminated composite (+45/−45/+45/−45) stiffened conoidal shell for different sizes of the central square cutout and different boundary conditions.
Boundary conditions
Cutout size (a′/a)
0
0.1
0.2
0.3
0.4
CCCC
137.5363
149.748
156.932
154.6134
147.0345
CSCC
127.6366
139.9845
142.6213
142.2069
133.7437
CCSC
133.1847
157.494
165.9908
157.6214
149.8912
CCCS
123.9139
136.05
139.2721
140.9553
134.9193
CSSC
119.3132
129.2799
129.5345
119.8123
107.4374
CCSS
114.1784
124.2418
126.9562
119.4147
107.3405
CSCS
119.1939
129.2601
133.1178
136.9644
128.4247
SCSC
117.6296
129.2476
135.318
141.1146
143.7385
CSSS
102.7485
106.9623
109.5824
110.3696
103.1902
SSSC
105.2055
111.0251
113.482
110.7912
101.8345
SSCS
103.0713
107.6771
111.8604
118.8297
123.5415
SSSS
89.8159
91.6626
94.3912
89.284
83.3021
Point supported
26.4022
26.8267
27.3666
28.542
30.1836
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=5, hl/hh=0.25, E11/E22=25, G23=0.2E22, G13=G12=0.5E22, and ν12=ν21=0.25.


#### 5.1.1. Effect of Cutout Size on Fundamental Frequency

From Tables3 and 4, it is seen that when a cutout is introduced to a stiffened shell, the fundamental frequencies increase. This increasing trend continues up to a′/a=0.4 for both cross- and angle ply shells except some angle ply shells with a′/a>0.2. The initial increase in frequency may be explained by the fact that when a cutout is introduced to an unpunctured surface, the number of stiffeners increases from two to four in the present study. When the cutout size is further increased, the number and dimensions of the stiffeners do not change, but the shell surface undergoes loss of both mass and stiffness. As the cutout grows in size, the loss of mass is more significant than that of stiffness, and hence the frequency increases. But for some angle ply shells with further increase in the size of the cutout, the loss of stiffness gradually becomes more important than that of mass, resulting in decrease in fundamental frequency. This leads to the engineering conclusion that cutouts with stiffened margins may always safely be provided on shell surfaces for functional requirements.


#### 5.1.2. Effect of Boundary Conditions on Fundamental Frequency

The boundary conditions may be divided into six groups, considering number of boundary constraints. The combinations in a particular group have equal number of boundary reactions. These groups areGroup I: CCCC shells,Group II: CSCC, CCSC, and SCCC shells,Group III: CSSC, SSCC, CSCS, and SCSC shells,Group IV: CSSS, SSSC, and SSCS shells,Group V: SSSS shells,Group VI: Corner point supported shell.It is seen from Tables3 and 4 that fundamental frequencies of members belonging to the same groups of boundary combinations may not have close values. So, the different boundary conditions may be regrouped according to performance. According to the values of ω-, the following groups may be identified.For cross ply shellsGroup 1: Contains CCCC, CCSC, and SCSC boundaries which exhibit relatively high frequencies.Group 2: Contains CSCC, CCCS, CSSC, CCSS, CSCS, SSSC, CSSS, and SSCS which exhibit intermediate values of frequencies.Group 3: Contains SSSS and corner point supported boundaries which exhibit relatively low values of frequencies.Similarly for angle ply shells:Group 1: Contains CCCC, CCSC, CSCC, CCCS, CSSC, CCSS, SCSC, and CSCS boundaries which exhibit relatively high frequencies.Group 2: Contains SSSC, CSSS, SSCS, and SSSS boundaries which exhibit intermediate values of frequencies.Group 3: Contains corner point supported shells which exhibit relatively low values of frequencies.It is evident from the present study that the free vibration characteristics mostly depend on the arrangement of boundary constraints rather than their actual number. It can be seen from the present study that if the higher parabolic edge alongx=a is released from clamped to simply supported, there is hardly any change of frequency for cross ply shell. But for angle ply shells, if the edge along higher parabolic edge is released, fundamental frequency even increases more than that of a clamped shell. For cross ply shells, if the edge along y=0 or y=b is released, that is, along the straight edges, frequency values undergo marked decrease. The results indicate that the edge along y=0 or y=b should preferably be clamped in order to achieve higher frequency values, and if the edge has to be released for functional reason, the edge along x=0 and x=a of a conoid must be clamped to make up for the loss of frequency. But for angle ply shells, if any two edges are released, the change in fundamental frequency is not so significant.Tables5 and 6 show the efficiency of a particular clamping option in improving the fundamental frequency of a shell with minimum number of boundary constraints relative to that of a clamped shell. Marks are assigned to each boundary combination in a scale assigning a value of 0 to the frequency of a corner point supported shell and 100 to that of a fully clamped shell. These marks are furnished for cutouts with a′/a=0.2 These tables will enable a practicing engineer to realize at a glance the efficiency of a particular boundary condition in improving the frequency of a shell, taking that of clamped shell as the upper limit.Table 5
Clamping options for 0/90/0/90 conoidal shells with central cutouts havinga′/a ratio 0.2.
Number of sides to be clamped
Clamped edges
Improvement of frequencies with respect to point supported shells
Marks indicating the efficiencies of number of restraints
0
Corner point supported
—
0
0
Simply supported no edges clamped (SSSS)
Good improvement
35
1
(a) Higher parabolic edge alongx=a (SSCS)
Marked improvement
46
(b) Lower parabolic edge alongx=0 (CSSS)
Marked improvement
49
(c) One straight edge alongy=b (SSSC)
Marked improvement
52
2
(a) Two alternate edges  including the higher and lower parabolic edgesx=0 and x=a (CSCS)
Marked improvement
57
(b) 2 straight edges alongy=0 and y=b (SCSC)
Remarkable improvement
88
(c) Any two edges except for the above option (CSSC, CCSS)
Marked improvement
63
3
3 edges including the two parabolic edges  (CSCC, CCCS)
Marked improvement
45
3 edges excluding the higher parabolic edge alongx=a  (CCSC)
Remarkable improvement
96
4
All sides (CCCC)
Frequency attains the highest value
100Table 6
Clamping options for +45/−45/+45/−45 conoidal shells with central cutouts havinga′/a ratio 0.2.
Number of sides to be clamped
Clamped edges
Improvement of frequencies with respect to point supported shells
Marks indicating the efficiencies of number of restraints
0
Corner point supported
—
0
0
Simply supported no edges clamped (SSSS)
Marked improvement
52
1
(a) Higher parabolic edge alongx=a  (SSCS)
Marked improvement
65
(b) Lower parabolic edge alongx=0 (CSSS)
Marked improvement
64
(c) One straight edge alongy=b (SSSC)
Marked improvement
66
2
(a) Two alternate edges  including the higher and lower parabolic edgesx=0 and x=a (CSCS)
Remarkable improvement
81
(b) 2 straight edges alongy=0 and y=b (SCSC)
Remarkable improvement
83
(c) Any two edges except for the above option (CSSC, CCSS)
Remarkable improvement
77–79
3
3 edges including the two parabolic edges (CSCC, CCCS)
Remarkable improvement
86–89
3 edges excluding the higher parabolic edge alongx=a (CCSC)
Frequency attains more than a fully clamped shell.
107
4
All sides (CCCC)
Remarkable improvement
100


#### 5.1.3. Mode Shapes

The mode shapes corresponding to the fundamental modes of vibration are plotted in Figures4 and 5 for cross-ply and angle ply shells, respectively. The normalized displacements are drawn with the shell midsurface as the reference for all the support conditions and for all the laminations used here. The fundamental mode is clearly a bending mode for all the boundary conditions for cross-ply and angle-ply shells, except for corner point supported shell. For corner point supported shells, the fundamental mode shapes are complicated. With the introduction of cutout, mode shapes remain almost similar. When the size of the cutout is increased from 0.2 to 0.4, the fundamental modes of vibration do not change to an appreciable amount.Figure 4
First mode shapes of laminated composite (0/90/0/90) stiffened conoidal shell for different sizes of the central square cutout and boundary conditions.Figure 5
First mode shapes of laminated composite (+45/−45/+45/−45) stiffened conoidal shell for different sizes of the central square cutout and boundary conditions.


### 5.2. Effect of Eccentricity of Cutout Position

#### 5.2.1. Fundamental Frequency

The effect of eccentricity of cutout positions on fundamental frequencies is studied from the results obtained for different locations of a cutout witha′/a=0.2. The non-dimensional coordinates of the cutout centre (x-=x/a,y-=y/a) were varied from 0.2 to 0.8 along each direction, so that the distance of a cutout margin from the shell boundary was not less than one tenth of the plan dimension of the shell. The margins of cutouts were stiffened with four stiffeners. The study was carried out for all the thirteen boundary conditions for both cross ply and angle ply shells. The fundamental frequency of a shell with an eccentric cutout is expressed as a percentage of fundamental frequency of a shell with a concentric cutout. This percentage is denoted by r. In Tables 7 and 8, such results are furnished.Table 7
Values of “r” for 0/90/0/90 conoidal shells.
Edge condition
y
-
x
-
0.2
0.3
0.4
0.5
0.6
0.7
0.8
CCCC
0.2
82.951
90.621
99.503
101.707
93.088
85.111
79.437
0.3
82.557
90.021
99.158
103.034
93.743
85.175
79.190
0.4
81.946
89.117
97.773
101.280
92.554
84.271
78.405
0.5
81.955
88.983
97.220
100.015
91.853
83.955
78.247
0.6
81.946
89.115
97.771
101.280
92.555
84.271
78.405
0.7
82.557
90.020
99.159
103.036
93.743
85.175
79.189
0.8
82.887
90.511
99.407
101.694
93.070
85.088
79.424
CSCC
0.2
93.862
100.890
106.419
103.740
95.957
89.329
85.169
0.3
96.374
104.672
110.505
108.656
101.815
94.918
89.726
0.4
93.858
102.085
108.275
106.920
100.353
93.653
88.508
0.5
90.589
97.867
102.329
100
93.876
88.419
84.453
0.6
88.150
94.257
97.223
94.591
89.294
84.754
81.462
0.7
87.105
92.080
94.299
92.0577
87.583
83.555
80.475
0.8
87.065
91.375
93.317
91.458
87.383
83.527
80.519
CCSC
0.2
80.918
87.869
96.510
101.953
95.020
87.039
81.087
0.3
80.426
87.123
95.768
102.726
95.727
87.269
81.107
0.4
79.615
85.977
94.270
101.072
94.674
86.601
80.675
0.5
79.448
85.705
93.791
100
93.987
86.365
80.627
0.6
79.612
85.977
94.271
101.073
94.674
86.601
80.675
0.7
80.426
87.122
95.770
102.726
95.727
87.268
81.106
0.8
80.837
87.743
96.390
101.907
95.007
87.017
81.075
CCCS
0.2
87.175
91.751
93.546
91.679
87.294
83.462
80.475
0.3
87.180
92.361
94.480
92.063
87.489
83.463
80.416
0.4
88.309
94.562
97.427
94.587
89.193
84.660
81.409
0.5
90.816
98.202
102.571
100
93.771
88.341
84.427
0.6
94.105
102.438
108.564
106.999
100.349
93.673
88.565
0.7
96.611
105.025
110.815
108.872
101.974
95.062
89.875
0.8
93.762
101.038
106.545
103.958
96.168
89.383
85.123
CSSC
0.2
91.560
99.045
106.094
105.722
98.527
91.427
86.004
0.3
91.466
99.489
107.457
108.698
103.151
96.316
90.410
0.4
89.314
96.955
104.290
105.667
101.025
94.902
89.407
0.5
88.004
95.094
100.516
100
95.159
90.036
85.737
0.6
87.238
93.625
97.473
95.882
91.217
86.821
83.288
0.7
87.033
92.615
95.568
94.032
89.939
85.975
82.704
0.8
87.165
92.303
94.864
93.620
89.886
86.065
82.823
CCSS
0.2
87.339
92.674
95.098
93.607
89.769
85.968
82.749
0.3
87.146
92.875
95.730
94.015
89.811
85.845
82.608
0.4
87.407
93.903
97.657
95.859
91.084
86.689
83.196
0.5
88.217
95.390
100.738
100
95.034
89.925
85.675
0.6
89.541
97.258
104.563
105.762
101.012
94.895
89.435
0.7
91.697
99.797
107.751
108.900
103.289
96.435
90.531
0.8
91.033
98.862
106.134
105.856
98.729
91.447
85.903
CSCS
0.2
87.176
90.165
91.941
90.278
87.528
86.028
85.511
0.3
90.236
93.667
95.133
93.168
90.295
88.544
87.299
0.4
93.274
97.392
98.931
97.165
94.295
92.109
89.886
0.5
95.713
99.837
101.259
100
97.767
96.416
93.011
0.6
93.271
97.392
98.930
97.154
94.291
92.103
89.883
0.7
90.232
93.666
95.133
96.907
90.294
88.544
87.296
0.8
87.045
90.057
91.800
90.237
87.504
86.015
85.510
SCSC
0.2
86.114
93.707
101.983
100.984
91.907
84.864
79.915
0.3
85.570
92.996
101.670
102.014
92.324
84.978
79.910
0.4
84.341
91.527
100.008
100.904
91.661
84.571
79.679
0.5
83.782
90.945
99.224
100
91.321
84.472
79.663
0.6
84.342
91.527
100.007
100.904
91.662
84.571
79.680
0.7
85.570
92.996
101.673
102.014
92.325
84.979
79.911
0.8
86.038
93.588
101.881
100.967
91.891
84.845
79.907
CSSS
0.2
90.957
94.019
96.237
95.319
93.007
91.535
90.574
0.3
93.329
96.650
98.463
97.318
95.163
93.907
92.929
0.4
94.987
98.154
99.743
99.094
97.663
96.789
95.576
0.5
96.157
98.431
100.035
100
99.376
99.252
98.900
0.6
94.984
98.155
99.743
99.092
97.661
96.784
95.572
0.7
93.325
96.649
98.463
97.317
95.162
93.907
92.926
0.8
90.826
93.898
96.032
95.227
92.974
91.510
90.575
SSSC
0.2
97.801
107.461
112.236
106.669
97.842
91.027
86.339
0.3
100.320
109.209
113.744
109.522
101.475
94.421
89.126
0.4
99.538
107.571
110.641
105.969
98.634
92.339
87.472
0.5
97.804
105.127
105.975
100
93.276
88.232
84.425
0.6
96.806
103.135
102.653
96.372
90.208
85.948
82.902
0.7
96.657
101.965
101.055
95.197
89.518
85.604
82.855
0.8
96.784
101.605
100.554
94.955
89.461
85.654
82.981
SSCS
0.2
93.160
96.470
95.571
91.662
88.790
87.812
87.727
0.3
96.515
99.494
98.334
94.512
91.801
90.728
90.027
0.4
99.292
102.150
101.314
97.900
95.242
93.801
92.363
0.5
100.559
103.324
102.905
100
97.629
96.284
94.946
0.6
99.290
102.155
101.314
97.896
95.240
93.802
92.362
0.7
96.514
99.481
98.332
94.512
91.80
90.726
90.024
0.8
93.059
96.355
95.442
91.626
88.753
87.798
87.733
SSSS
0.2
91.510
96.363
98.295
97.493
95.587
93.691
91.976
0.3
95.252
98.790
99.925
98.895
97.114
95.560
94.407
0.4
97.747
100.113
100.764
99.692
98.064
96.681
95.608
0.5
98.751
100.497
101.006
100
98.510
97.233
96.154
0.6
97.742
100.114
100.764
99.690
98.063
96.682
95.608
0.7
95.251
98.786
99.922
98.897
97.113
95.560
94.405
0.8
91.451
96.286
98.154
97.412
95.567
93.666
91.951
CS
0.2
137.892
128.050
114.295
104.647
100.791
101.361
109.077
0.3
135.205
125.573
111.227
101.093
97.006
97.834
106.312
0.4
132.315
124.337
110.519
100.187
95.990
97.045
105.585
0.5
131.075
123.720
110.365
100
95.813
96.912
105.499
0.6
132.292
124.356
110.550
100.211
96.017
97.088
105.603
0.7
135.228
125.582
111.219
101.090
96.984
97.817
106.303
0.8
135.480
125.754
112.435
102.528
98.801
99.577
107.915
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=5, hl/hh=0.25, E11/E22=25, G23=0.2E22, G13=G12=0.5E22, and ν12=ν21=0.25.Table 8
Values of “r” for +45/−45/+45/−45 conoidal shells.
Edge condition
y
-
x
-
0.2
0.3
0.4
0.5
0.6
0.7
0.8
CCCC
0.2
72.823
78.236
82.758
83.828
81.568
76.803
71.923
0.3
74.884
80.770
86.548
88.893
86.815
81.249
75.551
0.4
76.701
83.522
91.226
95.703
93.788
87.111
80.040
0.5
77.112
84.768
94.005
100
97.403
89.411
81.517
0.6
76.805
83.652
91.325
95.680
93.763
87.160
80.108
0.7
75.008
80.906
86.626
88.855
86.795
81.298
75.617
0.8
72.647
77.908
82.266
83.759
81.579
76.633
71.755
CSCC
0.2
81.077
91.056
97.803
91.446
86.032
83.235
78.655
0.3
83.525
93.130
102.170
99.590
94.922
89.192
82.211
0.4
84.074
93.824
101.696
103.807
101.498
94.596
86.412
0.5
83.003
90.427
97.733
100
99.474
96.299
87.547
0.6
81.276
87.968
94.605
96.334
95.817
92.081
84.895
0.7
79.646
86.287
92.866
94.104
92.136
86.701
80.220
0.8
78.630
85.619
92.366
92.585
89.172
83.321
77.139
CCSC
0.2
70.077
77.727
85.937
83.800
79.229
74.361
69.326
0.3
71.072
78.703
87.768
88.368
83.336
77.766
72.302
0.4
71.984
80.017
89.895
95.290
89.759
82.905
76.264
0.5
72.468
81.214
91.396
100
93.470
85.277
77.810
0.6
71.937
79.893
89.673
95.311
89.871
83.044
76.355
0.7
71.086
78.669
87.710
88.508
83.481
77.891
72.371
0.8
70.006
77.569
85.788
83.973
79.340
74.231
69.123
CCCS
0.2
78.059
85.085
91.568
92.620
90.166
84.849
78.646
0.3
78.792
85.576
91.983
93.758
92.489
87.571
81.169
0.4
80.502
87.433
93.909
95.951
95.851
92.337
85.557
0.5
82.841
90.481
97.657
100
99.876
96.861
88.986
0.6
85.401
94.488
102.330
104.306
102.177
96.765
88.819
0.7
85.313
94.803
103.136
100.952
96.520
91.365
84.617
0.8
81.938
91.745
98.552
93.014
87.371
84.653
80.319
CSSC
0.2
80.251
90.347
102.001
98.147
92.559
90.103
85.813
0.3
81.688
90.911
103.714
105.159
101.546
97.766
88.928
0.4
81.340
89.735
99.850
103.865
102.503
98.783
91.054
0.5
81.528
89.780
96.778
100
99.635
97.691
92.230
0.6
81.914
89.289
95.251
97.877
98.401
97.867
92.317
0.7
81.213
88.071
94.478
97.330
98.425
94.980
87.252
0.8
80.276
87.186
94.253
97.479
97.826
90.750
83.637
CCSS
0.2
79.271
85.734
92.215
96.177
97.750
92.440
85.311
0.3
80.024
86.471
92.663
96.147
97.781
95.350
88.242
0.4
81.065
87.896
94.083
97.107
98.178
97.531
92.578
0.5
82.087
90.360
97.023
100
100.117
98.606
94.251
0.6
82.521
90.825
101.803
104.549
103.368
100.351
93.755
0.7
82.941
91.821
103.719
105.727
102.486
99.509
91.399
0.8
80.346
90.089
100.955
99.195
93.707
91.345
87.507
CSCS
0.2
78.163
86.022
92.995
92.450
88.854
85.808
80.401
0.3
79.871
86.979
93.768
95.528
94.670
90.304
83.255
0.4
81.380
88.475
95.483
97.758
98.510
95.335
87.720
0.5
83.417
91.209
98.288
100
101.457
99.522
90.636
0.6
83.285
90.593
97.462
99.337
99.875
96.908
89.038
0.7
81.807
89.179
96.019
97.206
95.825
91.360
84.354
0.8
79.311
87.551
94.710
93.330
88.907
85.864
80.620
SCSC
0.2
85.127
94.329
95.136
88.093
85.184
82.426
78.466
0.3
86.016
95.544
99.993
92.933
89.052
85.588
80.885
0.4
86.853
97.202
105.501
97.941
92.882
88.930
83.293
0.5
87.280
98.396
107.888
100
94.353
90.351
84.256
0.6
86.654
96.782
104.675
97.456
92.684
88.887
83.209
0.7
85.821
95.213
99.433
92.519
89.003
85.631
80.778
0.8
84.798
93.899
94.867
87.932
85.323
82.385
78.165
CSSS
0.2
84.944
92.468
99.487
102.083
102.496
102.211
95.697
0.3
87.001
94.415
100.612
101.921
103.359
105.519
99.083
0.4
87.482
95.487
101.492
100.522
100.612
102.250
100.822
0.5
87.931
97.254
102.629
100
99.420
100.620
99.680
0.6
88.552
96.975
102.732
101.723
101.674
102.885
100.813
0.7
88.608
96.436
102.875
103.969
105.120
106.825
99.475
0.8
85.793
93.995
101.843
104.209
103.551
102.657
95.378
SSSC
0.2
90.301
101.510
103.781
96.224
91.653
89.271
87.137
0.3
91.100
102.460
108.098
102.693
98.673
95.766
90.177
0.4
90.295
101.404
106.028
102.757
98.698
95.253
90.038
0.5
89.523
99.110
102.720
100
96.670
94.017
90.034
0.6
88.793
96.857
100.664
98.472
96.351
94.739
91.563
0.7
88.172
95.669
99.702
97.818
96.797
96.041
90.722
0.8
87.395
94.930
98.931
96.571
95.997
95.427
88.365
SSCS
0.2
88.795
97.759
99.936
94.981
92.168
90.694
87.596
0.3
90.150
98.974
101.928
98.665
97.216
96.201
89.404
0.4
91.309
100.310
103.409
99.753
97.841
97.300
92.291
0.5
92.288
101.662
104.357
100
97.487
96.678
93.643
0.6
92.516
101.539
104.514
100.363
98.033
97.305
92.917
0.7
91.859
100.918
103.929
99.899
98.067
97.444
90.291
0.8
89.927
99.404
101.683
95.907
92.707
91.701
87.789
SSSS
0.2
88.805
93.985
97.537
99.443
99.34104
97.458
93.660
0.3
90.116
95.853
98.238
99.300
99.55631
98.869
96.461
0.4
89.799
95.332
98.006
97.770
96.847
96.012
94.848
0.5
89.665
94.925
97.577
100
95.462
94.360
93.375
0.6
90.247
95.512
98.509
98.722
97.792
96.672
95.013
0.7
90.943
96.728
99.725
101.290
101.508
100.518
97.141
0.8
88.912
94.653
99.280
101.663
101.056
99.261
95.529
CS
0.2
124.611
120.401
110.994
104.153
102.644
103.581
108.893
0.3
120.487
117.591
108.489
101.849
100.596
101.659
107.324
0.4
117.053
116.177
107.584
100.426
99.256
100.691
105.789
0.5
115.794
115.786
107.429
100
98.788
100.425
105.330
0.6
118.003
116.844
107.793
100.623
99.239
100.341
105.584
0.7
122.490
118.779
109.260
102.384
100.470
100.978
106.534
0.8
122.076
118.091
109.558
102.712
101.196
101.869
107.386
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=5, hl/hh=0.25, E11/E22=25, G23=0.2E22, G13=G12=0.5E22, and ν12=ν21=0.25.It can be seen that eccentricity of the cutout along the length of the shell towards the parabolic edges makes it more flexible. It is also seen that towards the lower parabolic edge,r value is greater than that of the higher parabolic edge. This means that if a designer has to provide an eccentric cutout along the length, he should preferably place it towards the lower height boundary. The exception is there in some cases of cross ply shells. For cross ply shells with three edges simply supported when cutout shifts towards the lower parabolic edge (including the lower parabolic edge) the shell becomes stiffer. Again for corner point supported shells, r values increase towards the parabolic edges and are maximum along lower parabolic edge. For cross ply shells, four out of thirteen boundary conditions yield the maximum value of r along x-=0.5, four  yield maximum values of r along x-=0.4, and others show maximum values within x-=0.4  to 0.5.It is observed from Table7 that if the eccentricity of a cutout is varied along the width, the shell becomes stiffer when the cutout shifts towards clamped edges. So, for functional purposes, if a shift of central cutout is required, eccentricity of a cutout along the width should preferably be towards the clamped straight edge. For shells having two straight edges of identical boundary condition, the maximum fundamental frequency occurs along y-=0.5. For corner point supported shells, the maximum fundamental frequency always occurs along the boundary of the shell. All these are true for cross ply shells only. For an angle ply shell, such unified trend is not observed, and the boundary conditions and the fundamental frequency behave in a complex manner as evident from Table 8. But for corner point supported angle-ply shells also, the maximum values of r are along the boundary.Tables9 and 10 provide the maximum values of r together with the position of the cutout. These tables also show the rectangular zones within which r is always greater than or equal to 95 and 90. It is to be noted that at some other points, r values may have similar values, but only the zone rectangular in plan has been identified. This study identifies the specific zones within which the cutout centre may be moved so that the loss of frequency is less than 5% or 10%, respectively, with respect to a shell with a central cutout. This will help a practicing engineer to make a decision regarding the eccentricity of the cutout centre that can be allowed.Table 9
Maximum values ofr with corresponding coordinates of cutout centre and zones where r≥90 and r≥95 for 0/90/0/90 conoidal shells.
Boundary condition
Maximum values ofr
Co-ordinate of cutout centre
Area in which the value ofr≥90
Area in which the value ofr≥95
CCCC
103.036
x
-
=
0.5, y-=0.7
x
-
=
0.6, 0.2≤y-≤0.8
0.4
≤
x
-
0.5, 0.2≤y-≤0.8
CSCC
110.505
x
-
=
0.4, y-=0.3
0.3
≤
x
-
0.5, 0.6≤y-≤0.8
0.3
≤
x
-
0.6, 0.2≤y-≤0.5
CCSC
102.726
x
-
=
0.5, y-=0.3   x-=0.5, y-=0.7
x
-
=
0.4, 0.6, 0.2≤y-≤0.8
x
-
=
0.5, 0.2≤y-≤0.8
CCCS
110.815
x
-
=
0.4,  y-=0.7
0.3
≤
x
-
0.5, 0.2≤y-≤0.4
0.3
≤
x
-
0.6, 0.5≤y-≤0.8
CSSC
108.698
x
-
=
0.5, y-=0.3
0.3
≤
x
-
0.5, 0.6≤y-≤0.8
0.3
≤
x
-
0.6, 0.2≤y-≤0.5
CCSS
108.900
x
-
=
0.5,  y-=0.7
0.3
≤
x
-
0.5, 0.2≤y-≤0.4
0.3
≤
x
-
0.6, 0.5≤y-≤0.8
CSCS
101.259
x
-
=
0.4, y-=0.5
0.3
≤
x
-
0.5, 0.2≤y-≤0.3   0.7≤y-≤0.8
0.3
≤
x
-
0.5, 0.4≤y-≤0.6
SCSC
102.014
x
-
=
0.5, y-=0.7
x
-
=
0.3, 0.5, 0.2≤y-≤0.8
0.4
≤
x
-
0.5, 0.2≤y-≤0.8
CSSS
100.035
x
-
=
0.4, y-=0.5
0.2
≤
x
-
0.8, 0.2≤y-≤0.8
0.3
≤
x
-
0.6, 0.3≤y-≤0.7
SSSC
113.744
x
-
=
0.4, y-=0.3
0.6
≤
x
-
0.7, 0.2≤y-≤0.4
0.2
≤
x
-
0.5, 0.2≤y-≤0.8
SSCS
103.324
x
-
=
0.3, y-=0.5
0.5
≤
x
-
0.8, 0.3≤y-≤0.7
0.2
≤
x
-
0.4, 0.3≤y-≤0.7
SSSS
101.006
x
-
=
0.4, y¯=0.5
x
-
=
0.8, 0.2≤y-≤0.8
0.2
≤
x
-
0.7, 0.3≤y-≤0.7
CS
137.892
x
-
=
0.2, y-=0.2
nil
0.2
≤
x
-
0.8, 0.2≤y-≤0.8
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=5, hl/hh=0.25, E11/E22=25, G23=0.2E22, G13=G12=0.5E22, and ν12=ν21=0.25.Table 10
Maximum values ofr with corresponding coordinates of cutout centre and zones where r≥90 and r≥95 for +45/−45/+45/−45 conoidal shells.
Boundary condition
Maximum values ofr
Co-ordinate of cutout centre
Area in which the value ofr≥90
Area in which the value ofr≥95
CCCC
100.000
x
-
=
0.5,  y-=0.5
x
-
=
0.4, 0.6 0.4≤y-≤0.6
x
-
=
0.5, 0.4≤y-≤0.6
CSCC
103.807
x
-
=
0.5,  y-=0.4
x
-
=
0.3, 0.2≤y-≤0.5; 0.6≤x-0.7, 0.3≤y-≤0.6
0.4
≤
x
-
≤
0.5, 0.3≤y-≤0.5
CCSC
100.000
x
-
=
0.5,  y-=0.5
0.4
≤
x
-
≤
0.6, y-=0.5
x
-
=
0.5   0.4≤y-≤0.6
CCCS
104.306
x
-
=
0.5,  y-=0.6
0.4
≤
x
-
≤
0.6, 0.2≤y-≤0.4
0.4
≤
x
-
≤
0.7, 0.5≤y-≤0.6
CSSC
105.159
x
-
=
0.5, y-=0.3
0.4
≤
x
-
≤
0.7, 0.7≤y-≤0.8; x-=0.8, 0.4≤y-≤0.6
0.4
≤
x
-
≤
0.7, 0.3≤y-≤0.6
CCSS
105.727
x
-
=
0.5,  y-=0.7
x
-
=
0.5, 0.7 0.2≤y-≤0.4   x-=0.3, 0.8 0.5≤y-≤0.7
0.5
≤
x
-
≤
0.6, 0.2≤y-≤0.4   0.4≤x-≤0.7, 0.5≤y-≤0.7
CSCS
101.457
x
-
=
0.6,  y-=0.5
0.4
≤
x
-
≤
0.7, y-=0.3
0.4
≤
x
-
≤
0.7, 0.4≤y-≤0.7
SCSC
107.888
x
-
=
0.4,  y-=0.5
0.5
≤
x
-
≤
0.6, 0.4≤y-≤0.6
0.3
≤
x
-
≤
0.4, 0.3≤y-≤0.7
CSSS
106.825
x
-
=
0.7,  y-=0.7
x
-
=
0.3, 0.2≤y-≤0.8
0.4
≤
x
-
≤
0.8, 0.2≤y-≤0.8
SSSC
108.098
x
-
=
0.4,  y-=0.3
0.7
≤
x
-
≤
0.8, 0.3≤y-≤0.7
0.3
≤
x
-
≤
0.6, 0.2≤y-≤0.7
SSCS
104.514
x
-
=
0.4,  y-=0.6
x
-
=
0.2, 0.8 0.3≤y-≤0.7
0.3
≤
x
-
≤
0.7, 0.3≤y-≤0.7
SSSS
101.663
x
-
=
0.5,  y-=0.8
x
-
=
0.3, 0.8 0.2≤y-≤0.8
0.4
≤
x
-
≤
0.7, 0.2≤y-≤0.8
CS
124.611
x
-
=
0.2,  y-=0.2
nil
0.2
≤
x
-
0.8, 0.2≤y-≤0.8
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=5, hl/hh=0.25, E11/E22=25, G23=0.2E22, G13=G12=0.5E22, and ν12=ν21=0.25.


#### 5.2.2. Mode Shapes

The mode shapes corresponding to the fundamental modes of vibration are plotted in Figures6, 7, 8, and 9 for cross-ply and angle-ply shells of CCCC and CCSC shells for different eccentric positions of the cutout. As CCCC and CCSC shells are most efficient with respect to number of restraints, the mode shapes of these shells are shown as typical results. All the mode shapes are bending modes. It is found that for different position of cutout, mode shapes are somewhat similar, only the crest and trough positions change.Figure 6
First mode shapes of laminated composite (0/90/0/90) stiffened conoidal shell for different positions of the square cutout with CCCC boundary condition.Figure 7
First mode shapes of laminated composite (0/90/0/90) stiffened conoidal shell for different positions of the square cutout with CCSC boundary condition.Figure 8
First mode shapes of laminated composite (+45/−45/+45/−45) stiffened conoidal shell for different positions of the square cutout with CCCC boundary condition.Figure 9
First mode shapes of laminated composite (+45/−45/+45/−45) stiffened conoidal shell for different positions of the square cutout with CCSC boundary condition.The present study considers the dynamic characteristics of stiffened composite conoidal shells with square cutout in terms of the natural frequency and mode shapes. The size of the cutouts and their positions with respect to the shell centre are varied for different edge constraints of cross-ply and angle-ply laminated composite conoids. The effects of these parametric variations on the fundamental frequencies and mode shapes are considered in details. However, the effect of the shape and orientation of the cutout on the dynamic characters of the conoid has not been considered in the present study. Future studies will evaluate these aspects.


## 5.1. Free Vibration Behaviour of Shells with Concentric Cutouts

Tables3 and 4 show the results for the non-dimensional fundamental frequency ω¯ of composite cross-ply and angle-ply  stiffened conoidal shells for different cutout sizes and various combinations of boundary conditions along the four edges. The shells considered are of square planform (a=b), and the cutouts are also taken to be square in plan (a′=b′). The cutout sizes (i.e., a′/a) are varied from 0 to 0.4, and boundary conditions are varied along the four edges. Cutouts are concentric on shell surface. The stiffeners are placed along the cutout periphery and extended up to the edge of the shell. The boundary conditions are designated by describing the support clamped or simply supported as C or S taken in an anticlockwise order from the edge x=0. This means that a shell with CSCS boundary is clamped along x=0, simply supported along y=0, clamped along x=a, and simply supported along y=b. The material and geometric properties of shells and cutouts are mentioned along with the figures.Table 3
Non-dimensional fundamental frequencies(ω-) for laminated composite (0/90/0/90) stiffened conoidal shell for different sizes of the central square cutout and different boundary conditions.
Boundary conditions
Cutout size (a′/a)
0
0.1
0.2
0.3
0.4
CCCC
105.7679
118.9136
124.386
127.4786
124.4929
CSCC
79.3544
87.8992
91.4718
96.4778
97.0499
CCSC
104.0259
117.3182
120.8486
122.5918
119.586
CCCS
79.0009
86.457
91.1477
95.8592
97.0069
CSSC
76.7919
84.417
87.445
91.1546
90.359
CCSS
76.4645
83.126
87.1684
90.6361
90.3194
CSCS
70.8733
76.0322
80.215
86.5974
91.9972
SCSC
96.4607
106.1886
112.2644
116.5409
114.8718
CSSS
65.7562
69.5302
72.8659
77.5964
81.1403
SSSC
65.7398
70.72
75.0802
79.9808
82.2204
SSCS
62.3308
65.2487
69.3811
74.7187
80.4246
SSSS
54.7734
56.6793
59.276
62.3683
65.2097
Point supported
21.1813
21.686
22.6166
24.2309
25.5361
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=5, hl/hh=0.25, E11/E22=25, G23=0.2E22, G13=G12=0.5E22, and ν12=ν21=0.25.Table 4
Non-dimensional fundamental frequencies(ω-) for laminated composite (+45/−45/+45/−45) stiffened conoidal shell for different sizes of the central square cutout and different boundary conditions.
Boundary conditions
Cutout size (a′/a)
0
0.1
0.2
0.3
0.4
CCCC
137.5363
149.748
156.932
154.6134
147.0345
CSCC
127.6366
139.9845
142.6213
142.2069
133.7437
CCSC
133.1847
157.494
165.9908
157.6214
149.8912
CCCS
123.9139
136.05
139.2721
140.9553
134.9193
CSSC
119.3132
129.2799
129.5345
119.8123
107.4374
CCSS
114.1784
124.2418
126.9562
119.4147
107.3405
CSCS
119.1939
129.2601
133.1178
136.9644
128.4247
SCSC
117.6296
129.2476
135.318
141.1146
143.7385
CSSS
102.7485
106.9623
109.5824
110.3696
103.1902
SSSC
105.2055
111.0251
113.482
110.7912
101.8345
SSCS
103.0713
107.6771
111.8604
118.8297
123.5415
SSSS
89.8159
91.6626
94.3912
89.284
83.3021
Point supported
26.4022
26.8267
27.3666
28.542
30.1836
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=5, hl/hh=0.25, E11/E22=25, G23=0.2E22, G13=G12=0.5E22, and ν12=ν21=0.25.


### 5.1.1. Effect of Cutout Size on Fundamental Frequency

From Tables3 and 4, it is seen that when a cutout is introduced to a stiffened shell, the fundamental frequencies increase. This increasing trend continues up to a′/a=0.4 for both cross- and angle ply shells except some angle ply shells with a′/a>0.2. The initial increase in frequency may be explained by the fact that when a cutout is introduced to an unpunctured surface, the number of stiffeners increases from two to four in the present study. When the cutout size is further increased, the number and dimensions of the stiffeners do not change, but the shell surface undergoes loss of both mass and stiffness. As the cutout grows in size, the loss of mass is more significant than that of stiffness, and hence the frequency increases. But for some angle ply shells with further increase in the size of the cutout, the loss of stiffness gradually becomes more important than that of mass, resulting in decrease in fundamental frequency. This leads to the engineering conclusion that cutouts with stiffened margins may always safely be provided on shell surfaces for functional requirements.


### 5.1.2. Effect of Boundary Conditions on Fundamental Frequency

The boundary conditions may be divided into six groups, considering number of boundary constraints. The combinations in a particular group have equal number of boundary reactions. These groups areGroup I: CCCC shells,Group II: CSCC, CCSC, and SCCC shells,Group III: CSSC, SSCC, CSCS, and SCSC shells,Group IV: CSSS, SSSC, and SSCS shells,Group V: SSSS shells,Group VI: Corner point supported shell.It is seen from Tables3 and 4 that fundamental frequencies of members belonging to the same groups of boundary combinations may not have close values. So, the different boundary conditions may be regrouped according to performance. According to the values of ω-, the following groups may be identified.For cross ply shellsGroup 1: Contains CCCC, CCSC, and SCSC boundaries which exhibit relatively high frequencies.Group 2: Contains CSCC, CCCS, CSSC, CCSS, CSCS, SSSC, CSSS, and SSCS which exhibit intermediate values of frequencies.Group 3: Contains SSSS and corner point supported boundaries which exhibit relatively low values of frequencies.Similarly for angle ply shells:Group 1: Contains CCCC, CCSC, CSCC, CCCS, CSSC, CCSS, SCSC, and CSCS boundaries which exhibit relatively high frequencies.Group 2: Contains SSSC, CSSS, SSCS, and SSSS boundaries which exhibit intermediate values of frequencies.Group 3: Contains corner point supported shells which exhibit relatively low values of frequencies.It is evident from the present study that the free vibration characteristics mostly depend on the arrangement of boundary constraints rather than their actual number. It can be seen from the present study that if the higher parabolic edge alongx=a is released from clamped to simply supported, there is hardly any change of frequency for cross ply shell. But for angle ply shells, if the edge along higher parabolic edge is released, fundamental frequency even increases more than that of a clamped shell. For cross ply shells, if the edge along y=0 or y=b is released, that is, along the straight edges, frequency values undergo marked decrease. The results indicate that the edge along y=0 or y=b should preferably be clamped in order to achieve higher frequency values, and if the edge has to be released for functional reason, the edge along x=0 and x=a of a conoid must be clamped to make up for the loss of frequency. But for angle ply shells, if any two edges are released, the change in fundamental frequency is not so significant.Tables5 and 6 show the efficiency of a particular clamping option in improving the fundamental frequency of a shell with minimum number of boundary constraints relative to that of a clamped shell. Marks are assigned to each boundary combination in a scale assigning a value of 0 to the frequency of a corner point supported shell and 100 to that of a fully clamped shell. These marks are furnished for cutouts with a′/a=0.2 These tables will enable a practicing engineer to realize at a glance the efficiency of a particular boundary condition in improving the frequency of a shell, taking that of clamped shell as the upper limit.Table 5
Clamping options for 0/90/0/90 conoidal shells with central cutouts havinga′/a ratio 0.2.
Number of sides to be clamped
Clamped edges
Improvement of frequencies with respect to point supported shells
Marks indicating the efficiencies of number of restraints
0
Corner point supported
—
0
0
Simply supported no edges clamped (SSSS)
Good improvement
35
1
(a) Higher parabolic edge alongx=a (SSCS)
Marked improvement
46
(b) Lower parabolic edge alongx=0 (CSSS)
Marked improvement
49
(c) One straight edge alongy=b (SSSC)
Marked improvement
52
2
(a) Two alternate edges  including the higher and lower parabolic edgesx=0 and x=a (CSCS)
Marked improvement
57
(b) 2 straight edges alongy=0 and y=b (SCSC)
Remarkable improvement
88
(c) Any two edges except for the above option (CSSC, CCSS)
Marked improvement
63
3
3 edges including the two parabolic edges  (CSCC, CCCS)
Marked improvement
45
3 edges excluding the higher parabolic edge alongx=a  (CCSC)
Remarkable improvement
96
4
All sides (CCCC)
Frequency attains the highest value
100Table 6
Clamping options for +45/−45/+45/−45 conoidal shells with central cutouts havinga′/a ratio 0.2.
Number of sides to be clamped
Clamped edges
Improvement of frequencies with respect to point supported shells
Marks indicating the efficiencies of number of restraints
0
Corner point supported
—
0
0
Simply supported no edges clamped (SSSS)
Marked improvement
52
1
(a) Higher parabolic edge alongx=a  (SSCS)
Marked improvement
65
(b) Lower parabolic edge alongx=0 (CSSS)
Marked improvement
64
(c) One straight edge alongy=b (SSSC)
Marked improvement
66
2
(a) Two alternate edges  including the higher and lower parabolic edgesx=0 and x=a (CSCS)
Remarkable improvement
81
(b) 2 straight edges alongy=0 and y=b (SCSC)
Remarkable improvement
83
(c) Any two edges except for the above option (CSSC, CCSS)
Remarkable improvement
77–79
3
3 edges including the two parabolic edges (CSCC, CCCS)
Remarkable improvement
86–89
3 edges excluding the higher parabolic edge alongx=a (CCSC)
Frequency attains more than a fully clamped shell.
107
4
All sides (CCCC)
Remarkable improvement
100


### 5.1.3. Mode Shapes

The mode shapes corresponding to the fundamental modes of vibration are plotted in Figures4 and 5 for cross-ply and angle ply shells, respectively. The normalized displacements are drawn with the shell midsurface as the reference for all the support conditions and for all the laminations used here. The fundamental mode is clearly a bending mode for all the boundary conditions for cross-ply and angle-ply shells, except for corner point supported shell. For corner point supported shells, the fundamental mode shapes are complicated. With the introduction of cutout, mode shapes remain almost similar. When the size of the cutout is increased from 0.2 to 0.4, the fundamental modes of vibration do not change to an appreciable amount.Figure 4
First mode shapes of laminated composite (0/90/0/90) stiffened conoidal shell for different sizes of the central square cutout and boundary conditions.Figure 5
First mode shapes of laminated composite (+45/−45/+45/−45) stiffened conoidal shell for different sizes of the central square cutout and boundary conditions.


## 5.1.1. Effect of Cutout Size on Fundamental Frequency

From Tables3 and 4, it is seen that when a cutout is introduced to a stiffened shell, the fundamental frequencies increase. This increasing trend continues up to a′/a=0.4 for both cross- and angle ply shells except some angle ply shells with a′/a>0.2. The initial increase in frequency may be explained by the fact that when a cutout is introduced to an unpunctured surface, the number of stiffeners increases from two to four in the present study. When the cutout size is further increased, the number and dimensions of the stiffeners do not change, but the shell surface undergoes loss of both mass and stiffness. As the cutout grows in size, the loss of mass is more significant than that of stiffness, and hence the frequency increases. But for some angle ply shells with further increase in the size of the cutout, the loss of stiffness gradually becomes more important than that of mass, resulting in decrease in fundamental frequency. This leads to the engineering conclusion that cutouts with stiffened margins may always safely be provided on shell surfaces for functional requirements.


## 5.1.2. Effect of Boundary Conditions on Fundamental Frequency

The boundary conditions may be divided into six groups, considering number of boundary constraints. The combinations in a particular group have equal number of boundary reactions. These groups areGroup I: CCCC shells,Group II: CSCC, CCSC, and SCCC shells,Group III: CSSC, SSCC, CSCS, and SCSC shells,Group IV: CSSS, SSSC, and SSCS shells,Group V: SSSS shells,Group VI: Corner point supported shell.It is seen from Tables3 and 4 that fundamental frequencies of members belonging to the same groups of boundary combinations may not have close values. So, the different boundary conditions may be regrouped according to performance. According to the values of ω-, the following groups may be identified.For cross ply shellsGroup 1: Contains CCCC, CCSC, and SCSC boundaries which exhibit relatively high frequencies.Group 2: Contains CSCC, CCCS, CSSC, CCSS, CSCS, SSSC, CSSS, and SSCS which exhibit intermediate values of frequencies.Group 3: Contains SSSS and corner point supported boundaries which exhibit relatively low values of frequencies.Similarly for angle ply shells:Group 1: Contains CCCC, CCSC, CSCC, CCCS, CSSC, CCSS, SCSC, and CSCS boundaries which exhibit relatively high frequencies.Group 2: Contains SSSC, CSSS, SSCS, and SSSS boundaries which exhibit intermediate values of frequencies.Group 3: Contains corner point supported shells which exhibit relatively low values of frequencies.It is evident from the present study that the free vibration characteristics mostly depend on the arrangement of boundary constraints rather than their actual number. It can be seen from the present study that if the higher parabolic edge alongx=a is released from clamped to simply supported, there is hardly any change of frequency for cross ply shell. But for angle ply shells, if the edge along higher parabolic edge is released, fundamental frequency even increases more than that of a clamped shell. For cross ply shells, if the edge along y=0 or y=b is released, that is, along the straight edges, frequency values undergo marked decrease. The results indicate that the edge along y=0 or y=b should preferably be clamped in order to achieve higher frequency values, and if the edge has to be released for functional reason, the edge along x=0 and x=a of a conoid must be clamped to make up for the loss of frequency. But for angle ply shells, if any two edges are released, the change in fundamental frequency is not so significant.Tables5 and 6 show the efficiency of a particular clamping option in improving the fundamental frequency of a shell with minimum number of boundary constraints relative to that of a clamped shell. Marks are assigned to each boundary combination in a scale assigning a value of 0 to the frequency of a corner point supported shell and 100 to that of a fully clamped shell. These marks are furnished for cutouts with a′/a=0.2 These tables will enable a practicing engineer to realize at a glance the efficiency of a particular boundary condition in improving the frequency of a shell, taking that of clamped shell as the upper limit.Table 5
Clamping options for 0/90/0/90 conoidal shells with central cutouts havinga′/a ratio 0.2.
Number of sides to be clamped
Clamped edges
Improvement of frequencies with respect to point supported shells
Marks indicating the efficiencies of number of restraints
0
Corner point supported
—
0
0
Simply supported no edges clamped (SSSS)
Good improvement
35
1
(a) Higher parabolic edge alongx=a (SSCS)
Marked improvement
46
(b) Lower parabolic edge alongx=0 (CSSS)
Marked improvement
49
(c) One straight edge alongy=b (SSSC)
Marked improvement
52
2
(a) Two alternate edges  including the higher and lower parabolic edgesx=0 and x=a (CSCS)
Marked improvement
57
(b) 2 straight edges alongy=0 and y=b (SCSC)
Remarkable improvement
88
(c) Any two edges except for the above option (CSSC, CCSS)
Marked improvement
63
3
3 edges including the two parabolic edges  (CSCC, CCCS)
Marked improvement
45
3 edges excluding the higher parabolic edge alongx=a  (CCSC)
Remarkable improvement
96
4
All sides (CCCC)
Frequency attains the highest value
100Table 6
Clamping options for +45/−45/+45/−45 conoidal shells with central cutouts havinga′/a ratio 0.2.
Number of sides to be clamped
Clamped edges
Improvement of frequencies with respect to point supported shells
Marks indicating the efficiencies of number of restraints
0
Corner point supported
—
0
0
Simply supported no edges clamped (SSSS)
Marked improvement
52
1
(a) Higher parabolic edge alongx=a  (SSCS)
Marked improvement
65
(b) Lower parabolic edge alongx=0 (CSSS)
Marked improvement
64
(c) One straight edge alongy=b (SSSC)
Marked improvement
66
2
(a) Two alternate edges  including the higher and lower parabolic edgesx=0 and x=a (CSCS)
Remarkable improvement
81
(b) 2 straight edges alongy=0 and y=b (SCSC)
Remarkable improvement
83
(c) Any two edges except for the above option (CSSC, CCSS)
Remarkable improvement
77–79
3
3 edges including the two parabolic edges (CSCC, CCCS)
Remarkable improvement
86–89
3 edges excluding the higher parabolic edge alongx=a (CCSC)
Frequency attains more than a fully clamped shell.
107
4
All sides (CCCC)
Remarkable improvement
100


## 5.1.3. Mode Shapes

The mode shapes corresponding to the fundamental modes of vibration are plotted in Figures4 and 5 for cross-ply and angle ply shells, respectively. The normalized displacements are drawn with the shell midsurface as the reference for all the support conditions and for all the laminations used here. The fundamental mode is clearly a bending mode for all the boundary conditions for cross-ply and angle-ply shells, except for corner point supported shell. For corner point supported shells, the fundamental mode shapes are complicated. With the introduction of cutout, mode shapes remain almost similar. When the size of the cutout is increased from 0.2 to 0.4, the fundamental modes of vibration do not change to an appreciable amount.Figure 4
First mode shapes of laminated composite (0/90/0/90) stiffened conoidal shell for different sizes of the central square cutout and boundary conditions.Figure 5
First mode shapes of laminated composite (+45/−45/+45/−45) stiffened conoidal shell for different sizes of the central square cutout and boundary conditions.


## 5.2. Effect of Eccentricity of Cutout Position

### 5.2.1. Fundamental Frequency

The effect of eccentricity of cutout positions on fundamental frequencies is studied from the results obtained for different locations of a cutout witha′/a=0.2. The non-dimensional coordinates of the cutout centre (x-=x/a,y-=y/a) were varied from 0.2 to 0.8 along each direction, so that the distance of a cutout margin from the shell boundary was not less than one tenth of the plan dimension of the shell. The margins of cutouts were stiffened with four stiffeners. The study was carried out for all the thirteen boundary conditions for both cross ply and angle ply shells. The fundamental frequency of a shell with an eccentric cutout is expressed as a percentage of fundamental frequency of a shell with a concentric cutout. This percentage is denoted by r. In Tables 7 and 8, such results are furnished.Table 7
Values of “r” for 0/90/0/90 conoidal shells.
Edge condition
y
-
x
-
0.2
0.3
0.4
0.5
0.6
0.7
0.8
CCCC
0.2
82.951
90.621
99.503
101.707
93.088
85.111
79.437
0.3
82.557
90.021
99.158
103.034
93.743
85.175
79.190
0.4
81.946
89.117
97.773
101.280
92.554
84.271
78.405
0.5
81.955
88.983
97.220
100.015
91.853
83.955
78.247
0.6
81.946
89.115
97.771
101.280
92.555
84.271
78.405
0.7
82.557
90.020
99.159
103.036
93.743
85.175
79.189
0.8
82.887
90.511
99.407
101.694
93.070
85.088
79.424
CSCC
0.2
93.862
100.890
106.419
103.740
95.957
89.329
85.169
0.3
96.374
104.672
110.505
108.656
101.815
94.918
89.726
0.4
93.858
102.085
108.275
106.920
100.353
93.653
88.508
0.5
90.589
97.867
102.329
100
93.876
88.419
84.453
0.6
88.150
94.257
97.223
94.591
89.294
84.754
81.462
0.7
87.105
92.080
94.299
92.0577
87.583
83.555
80.475
0.8
87.065
91.375
93.317
91.458
87.383
83.527
80.519
CCSC
0.2
80.918
87.869
96.510
101.953
95.020
87.039
81.087
0.3
80.426
87.123
95.768
102.726
95.727
87.269
81.107
0.4
79.615
85.977
94.270
101.072
94.674
86.601
80.675
0.5
79.448
85.705
93.791
100
93.987
86.365
80.627
0.6
79.612
85.977
94.271
101.073
94.674
86.601
80.675
0.7
80.426
87.122
95.770
102.726
95.727
87.268
81.106
0.8
80.837
87.743
96.390
101.907
95.007
87.017
81.075
CCCS
0.2
87.175
91.751
93.546
91.679
87.294
83.462
80.475
0.3
87.180
92.361
94.480
92.063
87.489
83.463
80.416
0.4
88.309
94.562
97.427
94.587
89.193
84.660
81.409
0.5
90.816
98.202
102.571
100
93.771
88.341
84.427
0.6
94.105
102.438
108.564
106.999
100.349
93.673
88.565
0.7
96.611
105.025
110.815
108.872
101.974
95.062
89.875
0.8
93.762
101.038
106.545
103.958
96.168
89.383
85.123
CSSC
0.2
91.560
99.045
106.094
105.722
98.527
91.427
86.004
0.3
91.466
99.489
107.457
108.698
103.151
96.316
90.410
0.4
89.314
96.955
104.290
105.667
101.025
94.902
89.407
0.5
88.004
95.094
100.516
100
95.159
90.036
85.737
0.6
87.238
93.625
97.473
95.882
91.217
86.821
83.288
0.7
87.033
92.615
95.568
94.032
89.939
85.975
82.704
0.8
87.165
92.303
94.864
93.620
89.886
86.065
82.823
CCSS
0.2
87.339
92.674
95.098
93.607
89.769
85.968
82.749
0.3
87.146
92.875
95.730
94.015
89.811
85.845
82.608
0.4
87.407
93.903
97.657
95.859
91.084
86.689
83.196
0.5
88.217
95.390
100.738
100
95.034
89.925
85.675
0.6
89.541
97.258
104.563
105.762
101.012
94.895
89.435
0.7
91.697
99.797
107.751
108.900
103.289
96.435
90.531
0.8
91.033
98.862
106.134
105.856
98.729
91.447
85.903
CSCS
0.2
87.176
90.165
91.941
90.278
87.528
86.028
85.511
0.3
90.236
93.667
95.133
93.168
90.295
88.544
87.299
0.4
93.274
97.392
98.931
97.165
94.295
92.109
89.886
0.5
95.713
99.837
101.259
100
97.767
96.416
93.011
0.6
93.271
97.392
98.930
97.154
94.291
92.103
89.883
0.7
90.232
93.666
95.133
96.907
90.294
88.544
87.296
0.8
87.045
90.057
91.800
90.237
87.504
86.015
85.510
SCSC
0.2
86.114
93.707
101.983
100.984
91.907
84.864
79.915
0.3
85.570
92.996
101.670
102.014
92.324
84.978
79.910
0.4
84.341
91.527
100.008
100.904
91.661
84.571
79.679
0.5
83.782
90.945
99.224
100
91.321
84.472
79.663
0.6
84.342
91.527
100.007
100.904
91.662
84.571
79.680
0.7
85.570
92.996
101.673
102.014
92.325
84.979
79.911
0.8
86.038
93.588
101.881
100.967
91.891
84.845
79.907
CSSS
0.2
90.957
94.019
96.237
95.319
93.007
91.535
90.574
0.3
93.329
96.650
98.463
97.318
95.163
93.907
92.929
0.4
94.987
98.154
99.743
99.094
97.663
96.789
95.576
0.5
96.157
98.431
100.035
100
99.376
99.252
98.900
0.6
94.984
98.155
99.743
99.092
97.661
96.784
95.572
0.7
93.325
96.649
98.463
97.317
95.162
93.907
92.926
0.8
90.826
93.898
96.032
95.227
92.974
91.510
90.575
SSSC
0.2
97.801
107.461
112.236
106.669
97.842
91.027
86.339
0.3
100.320
109.209
113.744
109.522
101.475
94.421
89.126
0.4
99.538
107.571
110.641
105.969
98.634
92.339
87.472
0.5
97.804
105.127
105.975
100
93.276
88.232
84.425
0.6
96.806
103.135
102.653
96.372
90.208
85.948
82.902
0.7
96.657
101.965
101.055
95.197
89.518
85.604
82.855
0.8
96.784
101.605
100.554
94.955
89.461
85.654
82.981
SSCS
0.2
93.160
96.470
95.571
91.662
88.790
87.812
87.727
0.3
96.515
99.494
98.334
94.512
91.801
90.728
90.027
0.4
99.292
102.150
101.314
97.900
95.242
93.801
92.363
0.5
100.559
103.324
102.905
100
97.629
96.284
94.946
0.6
99.290
102.155
101.314
97.896
95.240
93.802
92.362
0.7
96.514
99.481
98.332
94.512
91.80
90.726
90.024
0.8
93.059
96.355
95.442
91.626
88.753
87.798
87.733
SSSS
0.2
91.510
96.363
98.295
97.493
95.587
93.691
91.976
0.3
95.252
98.790
99.925
98.895
97.114
95.560
94.407
0.4
97.747
100.113
100.764
99.692
98.064
96.681
95.608
0.5
98.751
100.497
101.006
100
98.510
97.233
96.154
0.6
97.742
100.114
100.764
99.690
98.063
96.682
95.608
0.7
95.251
98.786
99.922
98.897
97.113
95.560
94.405
0.8
91.451
96.286
98.154
97.412
95.567
93.666
91.951
CS
0.2
137.892
128.050
114.295
104.647
100.791
101.361
109.077
0.3
135.205
125.573
111.227
101.093
97.006
97.834
106.312
0.4
132.315
124.337
110.519
100.187
95.990
97.045
105.585
0.5
131.075
123.720
110.365
100
95.813
96.912
105.499
0.6
132.292
124.356
110.550
100.211
96.017
97.088
105.603
0.7
135.228
125.582
111.219
101.090
96.984
97.817
106.303
0.8
135.480
125.754
112.435
102.528
98.801
99.577
107.915
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=5, hl/hh=0.25, E11/E22=25, G23=0.2E22, G13=G12=0.5E22, and ν12=ν21=0.25.Table 8
Values of “r” for +45/−45/+45/−45 conoidal shells.
Edge condition
y
-
x
-
0.2
0.3
0.4
0.5
0.6
0.7
0.8
CCCC
0.2
72.823
78.236
82.758
83.828
81.568
76.803
71.923
0.3
74.884
80.770
86.548
88.893
86.815
81.249
75.551
0.4
76.701
83.522
91.226
95.703
93.788
87.111
80.040
0.5
77.112
84.768
94.005
100
97.403
89.411
81.517
0.6
76.805
83.652
91.325
95.680
93.763
87.160
80.108
0.7
75.008
80.906
86.626
88.855
86.795
81.298
75.617
0.8
72.647
77.908
82.266
83.759
81.579
76.633
71.755
CSCC
0.2
81.077
91.056
97.803
91.446
86.032
83.235
78.655
0.3
83.525
93.130
102.170
99.590
94.922
89.192
82.211
0.4
84.074
93.824
101.696
103.807
101.498
94.596
86.412
0.5
83.003
90.427
97.733
100
99.474
96.299
87.547
0.6
81.276
87.968
94.605
96.334
95.817
92.081
84.895
0.7
79.646
86.287
92.866
94.104
92.136
86.701
80.220
0.8
78.630
85.619
92.366
92.585
89.172
83.321
77.139
CCSC
0.2
70.077
77.727
85.937
83.800
79.229
74.361
69.326
0.3
71.072
78.703
87.768
88.368
83.336
77.766
72.302
0.4
71.984
80.017
89.895
95.290
89.759
82.905
76.264
0.5
72.468
81.214
91.396
100
93.470
85.277
77.810
0.6
71.937
79.893
89.673
95.311
89.871
83.044
76.355
0.7
71.086
78.669
87.710
88.508
83.481
77.891
72.371
0.8
70.006
77.569
85.788
83.973
79.340
74.231
69.123
CCCS
0.2
78.059
85.085
91.568
92.620
90.166
84.849
78.646
0.3
78.792
85.576
91.983
93.758
92.489
87.571
81.169
0.4
80.502
87.433
93.909
95.951
95.851
92.337
85.557
0.5
82.841
90.481
97.657
100
99.876
96.861
88.986
0.6
85.401
94.488
102.330
104.306
102.177
96.765
88.819
0.7
85.313
94.803
103.136
100.952
96.520
91.365
84.617
0.8
81.938
91.745
98.552
93.014
87.371
84.653
80.319
CSSC
0.2
80.251
90.347
102.001
98.147
92.559
90.103
85.813
0.3
81.688
90.911
103.714
105.159
101.546
97.766
88.928
0.4
81.340
89.735
99.850
103.865
102.503
98.783
91.054
0.5
81.528
89.780
96.778
100
99.635
97.691
92.230
0.6
81.914
89.289
95.251
97.877
98.401
97.867
92.317
0.7
81.213
88.071
94.478
97.330
98.425
94.980
87.252
0.8
80.276
87.186
94.253
97.479
97.826
90.750
83.637
CCSS
0.2
79.271
85.734
92.215
96.177
97.750
92.440
85.311
0.3
80.024
86.471
92.663
96.147
97.781
95.350
88.242
0.4
81.065
87.896
94.083
97.107
98.178
97.531
92.578
0.5
82.087
90.360
97.023
100
100.117
98.606
94.251
0.6
82.521
90.825
101.803
104.549
103.368
100.351
93.755
0.7
82.941
91.821
103.719
105.727
102.486
99.509
91.399
0.8
80.346
90.089
100.955
99.195
93.707
91.345
87.507
CSCS
0.2
78.163
86.022
92.995
92.450
88.854
85.808
80.401
0.3
79.871
86.979
93.768
95.528
94.670
90.304
83.255
0.4
81.380
88.475
95.483
97.758
98.510
95.335
87.720
0.5
83.417
91.209
98.288
100
101.457
99.522
90.636
0.6
83.285
90.593
97.462
99.337
99.875
96.908
89.038
0.7
81.807
89.179
96.019
97.206
95.825
91.360
84.354
0.8
79.311
87.551
94.710
93.330
88.907
85.864
80.620
SCSC
0.2
85.127
94.329
95.136
88.093
85.184
82.426
78.466
0.3
86.016
95.544
99.993
92.933
89.052
85.588
80.885
0.4
86.853
97.202
105.501
97.941
92.882
88.930
83.293
0.5
87.280
98.396
107.888
100
94.353
90.351
84.256
0.6
86.654
96.782
104.675
97.456
92.684
88.887
83.209
0.7
85.821
95.213
99.433
92.519
89.003
85.631
80.778
0.8
84.798
93.899
94.867
87.932
85.323
82.385
78.165
CSSS
0.2
84.944
92.468
99.487
102.083
102.496
102.211
95.697
0.3
87.001
94.415
100.612
101.921
103.359
105.519
99.083
0.4
87.482
95.487
101.492
100.522
100.612
102.250
100.822
0.5
87.931
97.254
102.629
100
99.420
100.620
99.680
0.6
88.552
96.975
102.732
101.723
101.674
102.885
100.813
0.7
88.608
96.436
102.875
103.969
105.120
106.825
99.475
0.8
85.793
93.995
101.843
104.209
103.551
102.657
95.378
SSSC
0.2
90.301
101.510
103.781
96.224
91.653
89.271
87.137
0.3
91.100
102.460
108.098
102.693
98.673
95.766
90.177
0.4
90.295
101.404
106.028
102.757
98.698
95.253
90.038
0.5
89.523
99.110
102.720
100
96.670
94.017
90.034
0.6
88.793
96.857
100.664
98.472
96.351
94.739
91.563
0.7
88.172
95.669
99.702
97.818
96.797
96.041
90.722
0.8
87.395
94.930
98.931
96.571
95.997
95.427
88.365
SSCS
0.2
88.795
97.759
99.936
94.981
92.168
90.694
87.596
0.3
90.150
98.974
101.928
98.665
97.216
96.201
89.404
0.4
91.309
100.310
103.409
99.753
97.841
97.300
92.291
0.5
92.288
101.662
104.357
100
97.487
96.678
93.643
0.6
92.516
101.539
104.514
100.363
98.033
97.305
92.917
0.7
91.859
100.918
103.929
99.899
98.067
97.444
90.291
0.8
89.927
99.404
101.683
95.907
92.707
91.701
87.789
SSSS
0.2
88.805
93.985
97.537
99.443
99.34104
97.458
93.660
0.3
90.116
95.853
98.238
99.300
99.55631
98.869
96.461
0.4
89.799
95.332
98.006
97.770
96.847
96.012
94.848
0.5
89.665
94.925
97.577
100
95.462
94.360
93.375
0.6
90.247
95.512
98.509
98.722
97.792
96.672
95.013
0.7
90.943
96.728
99.725
101.290
101.508
100.518
97.141
0.8
88.912
94.653
99.280
101.663
101.056
99.261
95.529
CS
0.2
124.611
120.401
110.994
104.153
102.644
103.581
108.893
0.3
120.487
117.591
108.489
101.849
100.596
101.659
107.324
0.4
117.053
116.177
107.584
100.426
99.256
100.691
105.789
0.5
115.794
115.786
107.429
100
98.788
100.425
105.330
0.6
118.003
116.844
107.793
100.623
99.239
100.341
105.584
0.7
122.490
118.779
109.260
102.384
100.470
100.978
106.534
0.8
122.076
118.091
109.558
102.712
101.196
101.869
107.386
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=5, hl/hh=0.25, E11/E22=25, G23=0.2E22, G13=G12=0.5E22, and ν12=ν21=0.25.It can be seen that eccentricity of the cutout along the length of the shell towards the parabolic edges makes it more flexible. It is also seen that towards the lower parabolic edge,r value is greater than that of the higher parabolic edge. This means that if a designer has to provide an eccentric cutout along the length, he should preferably place it towards the lower height boundary. The exception is there in some cases of cross ply shells. For cross ply shells with three edges simply supported when cutout shifts towards the lower parabolic edge (including the lower parabolic edge) the shell becomes stiffer. Again for corner point supported shells, r values increase towards the parabolic edges and are maximum along lower parabolic edge. For cross ply shells, four out of thirteen boundary conditions yield the maximum value of r along x-=0.5, four  yield maximum values of r along x-=0.4, and others show maximum values within x-=0.4  to 0.5.It is observed from Table7 that if the eccentricity of a cutout is varied along the width, the shell becomes stiffer when the cutout shifts towards clamped edges. So, for functional purposes, if a shift of central cutout is required, eccentricity of a cutout along the width should preferably be towards the clamped straight edge. For shells having two straight edges of identical boundary condition, the maximum fundamental frequency occurs along y-=0.5. For corner point supported shells, the maximum fundamental frequency always occurs along the boundary of the shell. All these are true for cross ply shells only. For an angle ply shell, such unified trend is not observed, and the boundary conditions and the fundamental frequency behave in a complex manner as evident from Table 8. But for corner point supported angle-ply shells also, the maximum values of r are along the boundary.Tables9 and 10 provide the maximum values of r together with the position of the cutout. These tables also show the rectangular zones within which r is always greater than or equal to 95 and 90. It is to be noted that at some other points, r values may have similar values, but only the zone rectangular in plan has been identified. This study identifies the specific zones within which the cutout centre may be moved so that the loss of frequency is less than 5% or 10%, respectively, with respect to a shell with a central cutout. This will help a practicing engineer to make a decision regarding the eccentricity of the cutout centre that can be allowed.Table 9
Maximum values ofr with corresponding coordinates of cutout centre and zones where r≥90 and r≥95 for 0/90/0/90 conoidal shells.
Boundary condition
Maximum values ofr
Co-ordinate of cutout centre
Area in which the value ofr≥90
Area in which the value ofr≥95
CCCC
103.036
x
-
=
0.5, y-=0.7
x
-
=
0.6, 0.2≤y-≤0.8
0.4
≤
x
-
0.5, 0.2≤y-≤0.8
CSCC
110.505
x
-
=
0.4, y-=0.3
0.3
≤
x
-
0.5, 0.6≤y-≤0.8
0.3
≤
x
-
0.6, 0.2≤y-≤0.5
CCSC
102.726
x
-
=
0.5, y-=0.3   x-=0.5, y-=0.7
x
-
=
0.4, 0.6, 0.2≤y-≤0.8
x
-
=
0.5, 0.2≤y-≤0.8
CCCS
110.815
x
-
=
0.4,  y-=0.7
0.3
≤
x
-
0.5, 0.2≤y-≤0.4
0.3
≤
x
-
0.6, 0.5≤y-≤0.8
CSSC
108.698
x
-
=
0.5, y-=0.3
0.3
≤
x
-
0.5, 0.6≤y-≤0.8
0.3
≤
x
-
0.6, 0.2≤y-≤0.5
CCSS
108.900
x
-
=
0.5,  y-=0.7
0.3
≤
x
-
0.5, 0.2≤y-≤0.4
0.3
≤
x
-
0.6, 0.5≤y-≤0.8
CSCS
101.259
x
-
=
0.4, y-=0.5
0.3
≤
x
-
0.5, 0.2≤y-≤0.3   0.7≤y-≤0.8
0.3
≤
x
-
0.5, 0.4≤y-≤0.6
SCSC
102.014
x
-
=
0.5, y-=0.7
x
-
=
0.3, 0.5, 0.2≤y-≤0.8
0.4
≤
x
-
0.5, 0.2≤y-≤0.8
CSSS
100.035
x
-
=
0.4, y-=0.5
0.2
≤
x
-
0.8, 0.2≤y-≤0.8
0.3
≤
x
-
0.6, 0.3≤y-≤0.7
SSSC
113.744
x
-
=
0.4, y-=0.3
0.6
≤
x
-
0.7, 0.2≤y-≤0.4
0.2
≤
x
-
0.5, 0.2≤y-≤0.8
SSCS
103.324
x
-
=
0.3, y-=0.5
0.5
≤
x
-
0.8, 0.3≤y-≤0.7
0.2
≤
x
-
0.4, 0.3≤y-≤0.7
SSSS
101.006
x
-
=
0.4, y¯=0.5
x
-
=
0.8, 0.2≤y-≤0.8
0.2
≤
x
-
0.7, 0.3≤y-≤0.7
CS
137.892
x
-
=
0.2, y-=0.2
nil
0.2
≤
x
-
0.8, 0.2≤y-≤0.8
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=5, hl/hh=0.25, E11/E22=25, G23=0.2E22, G13=G12=0.5E22, and ν12=ν21=0.25.Table 10
Maximum values ofr with corresponding coordinates of cutout centre and zones where r≥90 and r≥95 for +45/−45/+45/−45 conoidal shells.
Boundary condition
Maximum values ofr
Co-ordinate of cutout centre
Area in which the value ofr≥90
Area in which the value ofr≥95
CCCC
100.000
x
-
=
0.5,  y-=0.5
x
-
=
0.4, 0.6 0.4≤y-≤0.6
x
-
=
0.5, 0.4≤y-≤0.6
CSCC
103.807
x
-
=
0.5,  y-=0.4
x
-
=
0.3, 0.2≤y-≤0.5; 0.6≤x-0.7, 0.3≤y-≤0.6
0.4
≤
x
-
≤
0.5, 0.3≤y-≤0.5
CCSC
100.000
x
-
=
0.5,  y-=0.5
0.4
≤
x
-
≤
0.6, y-=0.5
x
-
=
0.5   0.4≤y-≤0.6
CCCS
104.306
x
-
=
0.5,  y-=0.6
0.4
≤
x
-
≤
0.6, 0.2≤y-≤0.4
0.4
≤
x
-
≤
0.7, 0.5≤y-≤0.6
CSSC
105.159
x
-
=
0.5, y-=0.3
0.4
≤
x
-
≤
0.7, 0.7≤y-≤0.8; x-=0.8, 0.4≤y-≤0.6
0.4
≤
x
-
≤
0.7, 0.3≤y-≤0.6
CCSS
105.727
x
-
=
0.5,  y-=0.7
x
-
=
0.5, 0.7 0.2≤y-≤0.4   x-=0.3, 0.8 0.5≤y-≤0.7
0.5
≤
x
-
≤
0.6, 0.2≤y-≤0.4   0.4≤x-≤0.7, 0.5≤y-≤0.7
CSCS
101.457
x
-
=
0.6,  y-=0.5
0.4
≤
x
-
≤
0.7, y-=0.3
0.4
≤
x
-
≤
0.7, 0.4≤y-≤0.7
SCSC
107.888
x
-
=
0.4,  y-=0.5
0.5
≤
x
-
≤
0.6, 0.4≤y-≤0.6
0.3
≤
x
-
≤
0.4, 0.3≤y-≤0.7
CSSS
106.825
x
-
=
0.7,  y-=0.7
x
-
=
0.3, 0.2≤y-≤0.8
0.4
≤
x
-
≤
0.8, 0.2≤y-≤0.8
SSSC
108.098
x
-
=
0.4,  y-=0.3
0.7
≤
x
-
≤
0.8, 0.3≤y-≤0.7
0.3
≤
x
-
≤
0.6, 0.2≤y-≤0.7
SSCS
104.514
x
-
=
0.4,  y-=0.6
x
-
=
0.2, 0.8 0.3≤y-≤0.7
0.3
≤
x
-
≤
0.7, 0.3≤y-≤0.7
SSSS
101.663
x
-
=
0.5,  y-=0.8
x
-
=
0.3, 0.8 0.2≤y-≤0.8
0.4
≤
x
-
≤
0.7, 0.2≤y-≤0.8
CS
124.611
x
-
=
0.2,  y-=0.2
nil
0.2
≤
x
-
0.8, 0.2≤y-≤0.8
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=5, hl/hh=0.25, E11/E22=25, G23=0.2E22, G13=G12=0.5E22, and ν12=ν21=0.25.


### 5.2.2. Mode Shapes

The mode shapes corresponding to the fundamental modes of vibration are plotted in Figures6, 7, 8, and 9 for cross-ply and angle-ply shells of CCCC and CCSC shells for different eccentric positions of the cutout. As CCCC and CCSC shells are most efficient with respect to number of restraints, the mode shapes of these shells are shown as typical results. All the mode shapes are bending modes. It is found that for different position of cutout, mode shapes are somewhat similar, only the crest and trough positions change.Figure 6
First mode shapes of laminated composite (0/90/0/90) stiffened conoidal shell for different positions of the square cutout with CCCC boundary condition.Figure 7
First mode shapes of laminated composite (0/90/0/90) stiffened conoidal shell for different positions of the square cutout with CCSC boundary condition.Figure 8
First mode shapes of laminated composite (+45/−45/+45/−45) stiffened conoidal shell for different positions of the square cutout with CCCC boundary condition.Figure 9
First mode shapes of laminated composite (+45/−45/+45/−45) stiffened conoidal shell for different positions of the square cutout with CCSC boundary condition.The present study considers the dynamic characteristics of stiffened composite conoidal shells with square cutout in terms of the natural frequency and mode shapes. The size of the cutouts and their positions with respect to the shell centre are varied for different edge constraints of cross-ply and angle-ply laminated composite conoids. The effects of these parametric variations on the fundamental frequencies and mode shapes are considered in details. However, the effect of the shape and orientation of the cutout on the dynamic characters of the conoid has not been considered in the present study. Future studies will evaluate these aspects.


## 5.2.1. Fundamental Frequency

The effect of eccentricity of cutout positions on fundamental frequencies is studied from the results obtained for different locations of a cutout witha′/a=0.2. The non-dimensional coordinates of the cutout centre (x-=x/a,y-=y/a) were varied from 0.2 to 0.8 along each direction, so that the distance of a cutout margin from the shell boundary was not less than one tenth of the plan dimension of the shell. The margins of cutouts were stiffened with four stiffeners. The study was carried out for all the thirteen boundary conditions for both cross ply and angle ply shells. The fundamental frequency of a shell with an eccentric cutout is expressed as a percentage of fundamental frequency of a shell with a concentric cutout. This percentage is denoted by r. In Tables 7 and 8, such results are furnished.Table 7
Values of “r” for 0/90/0/90 conoidal shells.
Edge condition
y
-
x
-
0.2
0.3
0.4
0.5
0.6
0.7
0.8
CCCC
0.2
82.951
90.621
99.503
101.707
93.088
85.111
79.437
0.3
82.557
90.021
99.158
103.034
93.743
85.175
79.190
0.4
81.946
89.117
97.773
101.280
92.554
84.271
78.405
0.5
81.955
88.983
97.220
100.015
91.853
83.955
78.247
0.6
81.946
89.115
97.771
101.280
92.555
84.271
78.405
0.7
82.557
90.020
99.159
103.036
93.743
85.175
79.189
0.8
82.887
90.511
99.407
101.694
93.070
85.088
79.424
CSCC
0.2
93.862
100.890
106.419
103.740
95.957
89.329
85.169
0.3
96.374
104.672
110.505
108.656
101.815
94.918
89.726
0.4
93.858
102.085
108.275
106.920
100.353
93.653
88.508
0.5
90.589
97.867
102.329
100
93.876
88.419
84.453
0.6
88.150
94.257
97.223
94.591
89.294
84.754
81.462
0.7
87.105
92.080
94.299
92.0577
87.583
83.555
80.475
0.8
87.065
91.375
93.317
91.458
87.383
83.527
80.519
CCSC
0.2
80.918
87.869
96.510
101.953
95.020
87.039
81.087
0.3
80.426
87.123
95.768
102.726
95.727
87.269
81.107
0.4
79.615
85.977
94.270
101.072
94.674
86.601
80.675
0.5
79.448
85.705
93.791
100
93.987
86.365
80.627
0.6
79.612
85.977
94.271
101.073
94.674
86.601
80.675
0.7
80.426
87.122
95.770
102.726
95.727
87.268
81.106
0.8
80.837
87.743
96.390
101.907
95.007
87.017
81.075
CCCS
0.2
87.175
91.751
93.546
91.679
87.294
83.462
80.475
0.3
87.180
92.361
94.480
92.063
87.489
83.463
80.416
0.4
88.309
94.562
97.427
94.587
89.193
84.660
81.409
0.5
90.816
98.202
102.571
100
93.771
88.341
84.427
0.6
94.105
102.438
108.564
106.999
100.349
93.673
88.565
0.7
96.611
105.025
110.815
108.872
101.974
95.062
89.875
0.8
93.762
101.038
106.545
103.958
96.168
89.383
85.123
CSSC
0.2
91.560
99.045
106.094
105.722
98.527
91.427
86.004
0.3
91.466
99.489
107.457
108.698
103.151
96.316
90.410
0.4
89.314
96.955
104.290
105.667
101.025
94.902
89.407
0.5
88.004
95.094
100.516
100
95.159
90.036
85.737
0.6
87.238
93.625
97.473
95.882
91.217
86.821
83.288
0.7
87.033
92.615
95.568
94.032
89.939
85.975
82.704
0.8
87.165
92.303
94.864
93.620
89.886
86.065
82.823
CCSS
0.2
87.339
92.674
95.098
93.607
89.769
85.968
82.749
0.3
87.146
92.875
95.730
94.015
89.811
85.845
82.608
0.4
87.407
93.903
97.657
95.859
91.084
86.689
83.196
0.5
88.217
95.390
100.738
100
95.034
89.925
85.675
0.6
89.541
97.258
104.563
105.762
101.012
94.895
89.435
0.7
91.697
99.797
107.751
108.900
103.289
96.435
90.531
0.8
91.033
98.862
106.134
105.856
98.729
91.447
85.903
CSCS
0.2
87.176
90.165
91.941
90.278
87.528
86.028
85.511
0.3
90.236
93.667
95.133
93.168
90.295
88.544
87.299
0.4
93.274
97.392
98.931
97.165
94.295
92.109
89.886
0.5
95.713
99.837
101.259
100
97.767
96.416
93.011
0.6
93.271
97.392
98.930
97.154
94.291
92.103
89.883
0.7
90.232
93.666
95.133
96.907
90.294
88.544
87.296
0.8
87.045
90.057
91.800
90.237
87.504
86.015
85.510
SCSC
0.2
86.114
93.707
101.983
100.984
91.907
84.864
79.915
0.3
85.570
92.996
101.670
102.014
92.324
84.978
79.910
0.4
84.341
91.527
100.008
100.904
91.661
84.571
79.679
0.5
83.782
90.945
99.224
100
91.321
84.472
79.663
0.6
84.342
91.527
100.007
100.904
91.662
84.571
79.680
0.7
85.570
92.996
101.673
102.014
92.325
84.979
79.911
0.8
86.038
93.588
101.881
100.967
91.891
84.845
79.907
CSSS
0.2
90.957
94.019
96.237
95.319
93.007
91.535
90.574
0.3
93.329
96.650
98.463
97.318
95.163
93.907
92.929
0.4
94.987
98.154
99.743
99.094
97.663
96.789
95.576
0.5
96.157
98.431
100.035
100
99.376
99.252
98.900
0.6
94.984
98.155
99.743
99.092
97.661
96.784
95.572
0.7
93.325
96.649
98.463
97.317
95.162
93.907
92.926
0.8
90.826
93.898
96.032
95.227
92.974
91.510
90.575
SSSC
0.2
97.801
107.461
112.236
106.669
97.842
91.027
86.339
0.3
100.320
109.209
113.744
109.522
101.475
94.421
89.126
0.4
99.538
107.571
110.641
105.969
98.634
92.339
87.472
0.5
97.804
105.127
105.975
100
93.276
88.232
84.425
0.6
96.806
103.135
102.653
96.372
90.208
85.948
82.902
0.7
96.657
101.965
101.055
95.197
89.518
85.604
82.855
0.8
96.784
101.605
100.554
94.955
89.461
85.654
82.981
SSCS
0.2
93.160
96.470
95.571
91.662
88.790
87.812
87.727
0.3
96.515
99.494
98.334
94.512
91.801
90.728
90.027
0.4
99.292
102.150
101.314
97.900
95.242
93.801
92.363
0.5
100.559
103.324
102.905
100
97.629
96.284
94.946
0.6
99.290
102.155
101.314
97.896
95.240
93.802
92.362
0.7
96.514
99.481
98.332
94.512
91.80
90.726
90.024
0.8
93.059
96.355
95.442
91.626
88.753
87.798
87.733
SSSS
0.2
91.510
96.363
98.295
97.493
95.587
93.691
91.976
0.3
95.252
98.790
99.925
98.895
97.114
95.560
94.407
0.4
97.747
100.113
100.764
99.692
98.064
96.681
95.608
0.5
98.751
100.497
101.006
100
98.510
97.233
96.154
0.6
97.742
100.114
100.764
99.690
98.063
96.682
95.608
0.7
95.251
98.786
99.922
98.897
97.113
95.560
94.405
0.8
91.451
96.286
98.154
97.412
95.567
93.666
91.951
CS
0.2
137.892
128.050
114.295
104.647
100.791
101.361
109.077
0.3
135.205
125.573
111.227
101.093
97.006
97.834
106.312
0.4
132.315
124.337
110.519
100.187
95.990
97.045
105.585
0.5
131.075
123.720
110.365
100
95.813
96.912
105.499
0.6
132.292
124.356
110.550
100.211
96.017
97.088
105.603
0.7
135.228
125.582
111.219
101.090
96.984
97.817
106.303
0.8
135.480
125.754
112.435
102.528
98.801
99.577
107.915
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=5, hl/hh=0.25, E11/E22=25, G23=0.2E22, G13=G12=0.5E22, and ν12=ν21=0.25.Table 8
Values of “r” for +45/−45/+45/−45 conoidal shells.
Edge condition
y
-
x
-
0.2
0.3
0.4
0.5
0.6
0.7
0.8
CCCC
0.2
72.823
78.236
82.758
83.828
81.568
76.803
71.923
0.3
74.884
80.770
86.548
88.893
86.815
81.249
75.551
0.4
76.701
83.522
91.226
95.703
93.788
87.111
80.040
0.5
77.112
84.768
94.005
100
97.403
89.411
81.517
0.6
76.805
83.652
91.325
95.680
93.763
87.160
80.108
0.7
75.008
80.906
86.626
88.855
86.795
81.298
75.617
0.8
72.647
77.908
82.266
83.759
81.579
76.633
71.755
CSCC
0.2
81.077
91.056
97.803
91.446
86.032
83.235
78.655
0.3
83.525
93.130
102.170
99.590
94.922
89.192
82.211
0.4
84.074
93.824
101.696
103.807
101.498
94.596
86.412
0.5
83.003
90.427
97.733
100
99.474
96.299
87.547
0.6
81.276
87.968
94.605
96.334
95.817
92.081
84.895
0.7
79.646
86.287
92.866
94.104
92.136
86.701
80.220
0.8
78.630
85.619
92.366
92.585
89.172
83.321
77.139
CCSC
0.2
70.077
77.727
85.937
83.800
79.229
74.361
69.326
0.3
71.072
78.703
87.768
88.368
83.336
77.766
72.302
0.4
71.984
80.017
89.895
95.290
89.759
82.905
76.264
0.5
72.468
81.214
91.396
100
93.470
85.277
77.810
0.6
71.937
79.893
89.673
95.311
89.871
83.044
76.355
0.7
71.086
78.669
87.710
88.508
83.481
77.891
72.371
0.8
70.006
77.569
85.788
83.973
79.340
74.231
69.123
CCCS
0.2
78.059
85.085
91.568
92.620
90.166
84.849
78.646
0.3
78.792
85.576
91.983
93.758
92.489
87.571
81.169
0.4
80.502
87.433
93.909
95.951
95.851
92.337
85.557
0.5
82.841
90.481
97.657
100
99.876
96.861
88.986
0.6
85.401
94.488
102.330
104.306
102.177
96.765
88.819
0.7
85.313
94.803
103.136
100.952
96.520
91.365
84.617
0.8
81.938
91.745
98.552
93.014
87.371
84.653
80.319
CSSC
0.2
80.251
90.347
102.001
98.147
92.559
90.103
85.813
0.3
81.688
90.911
103.714
105.159
101.546
97.766
88.928
0.4
81.340
89.735
99.850
103.865
102.503
98.783
91.054
0.5
81.528
89.780
96.778
100
99.635
97.691
92.230
0.6
81.914
89.289
95.251
97.877
98.401
97.867
92.317
0.7
81.213
88.071
94.478
97.330
98.425
94.980
87.252
0.8
80.276
87.186
94.253
97.479
97.826
90.750
83.637
CCSS
0.2
79.271
85.734
92.215
96.177
97.750
92.440
85.311
0.3
80.024
86.471
92.663
96.147
97.781
95.350
88.242
0.4
81.065
87.896
94.083
97.107
98.178
97.531
92.578
0.5
82.087
90.360
97.023
100
100.117
98.606
94.251
0.6
82.521
90.825
101.803
104.549
103.368
100.351
93.755
0.7
82.941
91.821
103.719
105.727
102.486
99.509
91.399
0.8
80.346
90.089
100.955
99.195
93.707
91.345
87.507
CSCS
0.2
78.163
86.022
92.995
92.450
88.854
85.808
80.401
0.3
79.871
86.979
93.768
95.528
94.670
90.304
83.255
0.4
81.380
88.475
95.483
97.758
98.510
95.335
87.720
0.5
83.417
91.209
98.288
100
101.457
99.522
90.636
0.6
83.285
90.593
97.462
99.337
99.875
96.908
89.038
0.7
81.807
89.179
96.019
97.206
95.825
91.360
84.354
0.8
79.311
87.551
94.710
93.330
88.907
85.864
80.620
SCSC
0.2
85.127
94.329
95.136
88.093
85.184
82.426
78.466
0.3
86.016
95.544
99.993
92.933
89.052
85.588
80.885
0.4
86.853
97.202
105.501
97.941
92.882
88.930
83.293
0.5
87.280
98.396
107.888
100
94.353
90.351
84.256
0.6
86.654
96.782
104.675
97.456
92.684
88.887
83.209
0.7
85.821
95.213
99.433
92.519
89.003
85.631
80.778
0.8
84.798
93.899
94.867
87.932
85.323
82.385
78.165
CSSS
0.2
84.944
92.468
99.487
102.083
102.496
102.211
95.697
0.3
87.001
94.415
100.612
101.921
103.359
105.519
99.083
0.4
87.482
95.487
101.492
100.522
100.612
102.250
100.822
0.5
87.931
97.254
102.629
100
99.420
100.620
99.680
0.6
88.552
96.975
102.732
101.723
101.674
102.885
100.813
0.7
88.608
96.436
102.875
103.969
105.120
106.825
99.475
0.8
85.793
93.995
101.843
104.209
103.551
102.657
95.378
SSSC
0.2
90.301
101.510
103.781
96.224
91.653
89.271
87.137
0.3
91.100
102.460
108.098
102.693
98.673
95.766
90.177
0.4
90.295
101.404
106.028
102.757
98.698
95.253
90.038
0.5
89.523
99.110
102.720
100
96.670
94.017
90.034
0.6
88.793
96.857
100.664
98.472
96.351
94.739
91.563
0.7
88.172
95.669
99.702
97.818
96.797
96.041
90.722
0.8
87.395
94.930
98.931
96.571
95.997
95.427
88.365
SSCS
0.2
88.795
97.759
99.936
94.981
92.168
90.694
87.596
0.3
90.150
98.974
101.928
98.665
97.216
96.201
89.404
0.4
91.309
100.310
103.409
99.753
97.841
97.300
92.291
0.5
92.288
101.662
104.357
100
97.487
96.678
93.643
0.6
92.516
101.539
104.514
100.363
98.033
97.305
92.917
0.7
91.859
100.918
103.929
99.899
98.067
97.444
90.291
0.8
89.927
99.404
101.683
95.907
92.707
91.701
87.789
SSSS
0.2
88.805
93.985
97.537
99.443
99.34104
97.458
93.660
0.3
90.116
95.853
98.238
99.300
99.55631
98.869
96.461
0.4
89.799
95.332
98.006
97.770
96.847
96.012
94.848
0.5
89.665
94.925
97.577
100
95.462
94.360
93.375
0.6
90.247
95.512
98.509
98.722
97.792
96.672
95.013
0.7
90.943
96.728
99.725
101.290
101.508
100.518
97.141
0.8
88.912
94.653
99.280
101.663
101.056
99.261
95.529
CS
0.2
124.611
120.401
110.994
104.153
102.644
103.581
108.893
0.3
120.487
117.591
108.489
101.849
100.596
101.659
107.324
0.4
117.053
116.177
107.584
100.426
99.256
100.691
105.789
0.5
115.794
115.786
107.429
100
98.788
100.425
105.330
0.6
118.003
116.844
107.793
100.623
99.239
100.341
105.584
0.7
122.490
118.779
109.260
102.384
100.470
100.978
106.534
0.8
122.076
118.091
109.558
102.712
101.196
101.869
107.386
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=5, hl/hh=0.25, E11/E22=25, G23=0.2E22, G13=G12=0.5E22, and ν12=ν21=0.25.It can be seen that eccentricity of the cutout along the length of the shell towards the parabolic edges makes it more flexible. It is also seen that towards the lower parabolic edge,r value is greater than that of the higher parabolic edge. This means that if a designer has to provide an eccentric cutout along the length, he should preferably place it towards the lower height boundary. The exception is there in some cases of cross ply shells. For cross ply shells with three edges simply supported when cutout shifts towards the lower parabolic edge (including the lower parabolic edge) the shell becomes stiffer. Again for corner point supported shells, r values increase towards the parabolic edges and are maximum along lower parabolic edge. For cross ply shells, four out of thirteen boundary conditions yield the maximum value of r along x-=0.5, four  yield maximum values of r along x-=0.4, and others show maximum values within x-=0.4  to 0.5.It is observed from Table7 that if the eccentricity of a cutout is varied along the width, the shell becomes stiffer when the cutout shifts towards clamped edges. So, for functional purposes, if a shift of central cutout is required, eccentricity of a cutout along the width should preferably be towards the clamped straight edge. For shells having two straight edges of identical boundary condition, the maximum fundamental frequency occurs along y-=0.5. For corner point supported shells, the maximum fundamental frequency always occurs along the boundary of the shell. All these are true for cross ply shells only. For an angle ply shell, such unified trend is not observed, and the boundary conditions and the fundamental frequency behave in a complex manner as evident from Table 8. But for corner point supported angle-ply shells also, the maximum values of r are along the boundary.Tables9 and 10 provide the maximum values of r together with the position of the cutout. These tables also show the rectangular zones within which r is always greater than or equal to 95 and 90. It is to be noted that at some other points, r values may have similar values, but only the zone rectangular in plan has been identified. This study identifies the specific zones within which the cutout centre may be moved so that the loss of frequency is less than 5% or 10%, respectively, with respect to a shell with a central cutout. This will help a practicing engineer to make a decision regarding the eccentricity of the cutout centre that can be allowed.Table 9
Maximum values ofr with corresponding coordinates of cutout centre and zones where r≥90 and r≥95 for 0/90/0/90 conoidal shells.
Boundary condition
Maximum values ofr
Co-ordinate of cutout centre
Area in which the value ofr≥90
Area in which the value ofr≥95
CCCC
103.036
x
-
=
0.5, y-=0.7
x
-
=
0.6, 0.2≤y-≤0.8
0.4
≤
x
-
0.5, 0.2≤y-≤0.8
CSCC
110.505
x
-
=
0.4, y-=0.3
0.3
≤
x
-
0.5, 0.6≤y-≤0.8
0.3
≤
x
-
0.6, 0.2≤y-≤0.5
CCSC
102.726
x
-
=
0.5, y-=0.3   x-=0.5, y-=0.7
x
-
=
0.4, 0.6, 0.2≤y-≤0.8
x
-
=
0.5, 0.2≤y-≤0.8
CCCS
110.815
x
-
=
0.4,  y-=0.7
0.3
≤
x
-
0.5, 0.2≤y-≤0.4
0.3
≤
x
-
0.6, 0.5≤y-≤0.8
CSSC
108.698
x
-
=
0.5, y-=0.3
0.3
≤
x
-
0.5, 0.6≤y-≤0.8
0.3
≤
x
-
0.6, 0.2≤y-≤0.5
CCSS
108.900
x
-
=
0.5,  y-=0.7
0.3
≤
x
-
0.5, 0.2≤y-≤0.4
0.3
≤
x
-
0.6, 0.5≤y-≤0.8
CSCS
101.259
x
-
=
0.4, y-=0.5
0.3
≤
x
-
0.5, 0.2≤y-≤0.3   0.7≤y-≤0.8
0.3
≤
x
-
0.5, 0.4≤y-≤0.6
SCSC
102.014
x
-
=
0.5, y-=0.7
x
-
=
0.3, 0.5, 0.2≤y-≤0.8
0.4
≤
x
-
0.5, 0.2≤y-≤0.8
CSSS
100.035
x
-
=
0.4, y-=0.5
0.2
≤
x
-
0.8, 0.2≤y-≤0.8
0.3
≤
x
-
0.6, 0.3≤y-≤0.7
SSSC
113.744
x
-
=
0.4, y-=0.3
0.6
≤
x
-
0.7, 0.2≤y-≤0.4
0.2
≤
x
-
0.5, 0.2≤y-≤0.8
SSCS
103.324
x
-
=
0.3, y-=0.5
0.5
≤
x
-
0.8, 0.3≤y-≤0.7
0.2
≤
x
-
0.4, 0.3≤y-≤0.7
SSSS
101.006
x
-
=
0.4, y¯=0.5
x
-
=
0.8, 0.2≤y-≤0.8
0.2
≤
x
-
0.7, 0.3≤y-≤0.7
CS
137.892
x
-
=
0.2, y-=0.2
nil
0.2
≤
x
-
0.8, 0.2≤y-≤0.8
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=5, hl/hh=0.25, E11/E22=25, G23=0.2E22, G13=G12=0.5E22, and ν12=ν21=0.25.Table 10
Maximum values ofr with corresponding coordinates of cutout centre and zones where r≥90 and r≥95 for +45/−45/+45/−45 conoidal shells.
Boundary condition
Maximum values ofr
Co-ordinate of cutout centre
Area in which the value ofr≥90
Area in which the value ofr≥95
CCCC
100.000
x
-
=
0.5,  y-=0.5
x
-
=
0.4, 0.6 0.4≤y-≤0.6
x
-
=
0.5, 0.4≤y-≤0.6
CSCC
103.807
x
-
=
0.5,  y-=0.4
x
-
=
0.3, 0.2≤y-≤0.5; 0.6≤x-0.7, 0.3≤y-≤0.6
0.4
≤
x
-
≤
0.5, 0.3≤y-≤0.5
CCSC
100.000
x
-
=
0.5,  y-=0.5
0.4
≤
x
-
≤
0.6, y-=0.5
x
-
=
0.5   0.4≤y-≤0.6
CCCS
104.306
x
-
=
0.5,  y-=0.6
0.4
≤
x
-
≤
0.6, 0.2≤y-≤0.4
0.4
≤
x
-
≤
0.7, 0.5≤y-≤0.6
CSSC
105.159
x
-
=
0.5, y-=0.3
0.4
≤
x
-
≤
0.7, 0.7≤y-≤0.8; x-=0.8, 0.4≤y-≤0.6
0.4
≤
x
-
≤
0.7, 0.3≤y-≤0.6
CCSS
105.727
x
-
=
0.5,  y-=0.7
x
-
=
0.5, 0.7 0.2≤y-≤0.4   x-=0.3, 0.8 0.5≤y-≤0.7
0.5
≤
x
-
≤
0.6, 0.2≤y-≤0.4   0.4≤x-≤0.7, 0.5≤y-≤0.7
CSCS
101.457
x
-
=
0.6,  y-=0.5
0.4
≤
x
-
≤
0.7, y-=0.3
0.4
≤
x
-
≤
0.7, 0.4≤y-≤0.7
SCSC
107.888
x
-
=
0.4,  y-=0.5
0.5
≤
x
-
≤
0.6, 0.4≤y-≤0.6
0.3
≤
x
-
≤
0.4, 0.3≤y-≤0.7
CSSS
106.825
x
-
=
0.7,  y-=0.7
x
-
=
0.3, 0.2≤y-≤0.8
0.4
≤
x
-
≤
0.8, 0.2≤y-≤0.8
SSSC
108.098
x
-
=
0.4,  y-=0.3
0.7
≤
x
-
≤
0.8, 0.3≤y-≤0.7
0.3
≤
x
-
≤
0.6, 0.2≤y-≤0.7
SSCS
104.514
x
-
=
0.4,  y-=0.6
x
-
=
0.2, 0.8 0.3≤y-≤0.7
0.3
≤
x
-
≤
0.7, 0.3≤y-≤0.7
SSSS
101.663
x
-
=
0.5,  y-=0.8
x
-
=
0.3, 0.8 0.2≤y-≤0.8
0.4
≤
x
-
≤
0.7, 0.2≤y-≤0.8
CS
124.611
x
-
=
0.2,  y-=0.2
nil
0.2
≤
x
-
0.8, 0.2≤y-≤0.8
a
/
b
=
1, a/h=100, a′/b′=1, a/hh=5, hl/hh=0.25, E11/E22=25, G23=0.2E22, G13=G12=0.5E22, and ν12=ν21=0.25.


## 5.2.2. Mode Shapes

The mode shapes corresponding to the fundamental modes of vibration are plotted in Figures6, 7, 8, and 9 for cross-ply and angle-ply shells of CCCC and CCSC shells for different eccentric positions of the cutout. As CCCC and CCSC shells are most efficient with respect to number of restraints, the mode shapes of these shells are shown as typical results. All the mode shapes are bending modes. It is found that for different position of cutout, mode shapes are somewhat similar, only the crest and trough positions change.Figure 6
First mode shapes of laminated composite (0/90/0/90) stiffened conoidal shell for different positions of the square cutout with CCCC boundary condition.Figure 7
First mode shapes of laminated composite (0/90/0/90) stiffened conoidal shell for different positions of the square cutout with CCSC boundary condition.Figure 8
First mode shapes of laminated composite (+45/−45/+45/−45) stiffened conoidal shell for different positions of the square cutout with CCCC boundary condition.Figure 9
First mode shapes of laminated composite (+45/−45/+45/−45) stiffened conoidal shell for different positions of the square cutout with CCSC boundary condition.The present study considers the dynamic characteristics of stiffened composite conoidal shells with square cutout in terms of the natural frequency and mode shapes. The size of the cutouts and their positions with respect to the shell centre are varied for different edge constraints of cross-ply and angle-ply laminated composite conoids. The effects of these parametric variations on the fundamental frequencies and mode shapes are considered in details. However, the effect of the shape and orientation of the cutout on the dynamic characters of the conoid has not been considered in the present study. Future studies will evaluate these aspects.


## 6. Conclusions

The following conclusions are drawn from the present study.(1)
As this approach produces results in close agreement with those of the benchmark problems, the finite element code used here is suitable for analyzing free vibration problems of stiffened conoidal roof panels with cutouts. The present study reveals that cutouts with stiffened margins may always safely be provided on shell surfaces for functional requirements.(2)
The arrangement of boundary constraints along the four edges is far more important than their actual number; so far the free vibration is concerned. The relative-free vibration performances of shells for different combinations of edge conditions along the four sides are expected to be very useful in decision making for practicing engineers.(3)
The information regarding the behaviour of stiffened conoids with eccentric cutouts for a wide spectrum of eccentricity and boundary conditions for cross ply and angle ply shells may also be used as design aids for structural engineers.



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*Source: 230120-2013-06-05.xml*