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

Dynamic characteristics of stiffened composite conoidal shells with cutout are analyzed in terms of the natural frequency andmode 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 andmode 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.


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][10][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][13][14][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.

Mathematical Formulation
A laminated composite conoidal shell of uniform thickness ℎ (Figure 1), and radius of curvature   , and radius of cross curvature   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  with reference to the -axis of the coordinate system.The constitutive equations for the shell are given by (a list of notations is separately given) where ( The force and moment resultants are expressed as The submatrices  [21] and Qatu [22]. The strain-displacement relations on the basis of improved first-order approximation theory for thin shell (Dey et al. [6]) are established as where the first vector is the midsurface strain for a conoidal shell and the second vector is the curvature.

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 are , V, , , .The following expressions establish the relations between the displacement at any point with respect to the coordinates  and  and the nodal degrees of freedom where the shape functions derived from a cubic interpolation polynomial [6] are

2
, for  = 5, 7, The generalized displacement vector of an element is expressed in terms of the shape functions and nodal degrees of freedom as that is,

Element Stiffness Matrix.
The strain-displacement relation is given by where The element stiffness matrix is

Element Mass Matrix. The element mass matrix is obtained from the integral
where in which

Finite Element Formulation for Stiffener of the Shell.
Three-noded curved isoparametric beam element (Figure 2) 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,   ,   ,   , and   for -stiffener and V  ,   ,   , and   for -stiffener.The generalized force-displacement relation of stiffeners can be expressed as where The generalized displacements of the -stiffener and the shell are related by the transformation matrix {  } = []{}, where This transformation is required due to curvature of -stiffener, and {} is the appropriate portion of the displacement vector of the shell excluding the displacement component along the -axis.Elasticity matrices are as follows: where and   ,   ,   , and   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: for -stiffener: 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 The element stiffness matrices are assembled to get the global matrices.

Element Mass Matrix. The element mass matrix for shell is obtained from the integral
where in which Element mass matrix for stiffener element Here, [] is a 3 × 3 diagonal matrix. Consider 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 The element mass matrices are assembled to get the global matrices.

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 Figure 3.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.

Solution Procedure for Free Vibration Analysis. The free vibration analysis involves determination of natural frequencies from the condition
This is a generalized eigen value problem and is solved by the subspace iteration algorithm.

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.
Additional problems for conoids with cutouts are solved, varying the size and position of cutout along both of the plan

Stiffener along 𝑥-direction
Stiffener along -direction Stiffener along both and -directions Nayak and Bandyopadhyay [13] Present model Nayak and Bandyopadhyay [13] Present model Nayak and Bandyopadhyay [13] Present model   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.

Results and Discussion
It is found from Table 1 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.

Free Vibration Behaviour of Shells with Concentric Cutouts.
Tables 3 and 4 show the results for the non-dimensional fundamental frequency  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 ( = ), and the cutouts are also taken to be square in plan (  =   ).The cutout sizes (i.e.,   /) 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  = 0.This means that a shell with CSCS boundary is clamped along  = 0, simply supported along  = 0, clamped along  = , and simply supported along  = .The material and geometric properties of shells and cutouts are mentioned along with the figures.

Effect of Cutout Size on Fundamental
Frequency.From Tables 3 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   / = 0.4 for both cross-and angle ply shells except some angle ply shells with   / > 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.It is seen from Tables 3 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 , the following groups may be identified.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 along  =  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  = 0 or  =  is released, that is, along the straight edges, frequency values undergo marked decrease.The results indicate that the edge along  = 0 or  =  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  = 0 and  =  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.

Effect of Boundary Conditions on
Tables 5 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   / = 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.

Mode Shapes.
The mode shapes corresponding to the fundamental modes of vibration are plotted in Figures 4 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.

Fundamental Frequency.
The effect of eccentricity of cutout positions on fundamental frequencies is studied from the results obtained for different locations of a cutout with   / = 0.2.The non-dimensional coordinates of the cutout centre ( = /,  = /) 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 .In Tables 7 and 8, such results are furnished.
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,  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  It is observed from Table 7 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  = 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  are along the boundary.Tables 9 and 10 provide the maximum values of  together with the position of the cutout.These tables also show the rectangular zones within which  is always greater than or equal to 95 and 90.It is to be noted that at some other points,

Figure 1 :
Figure 1: Conoidal shell with a concentric cutout stiffened along the margins.

Figure 4 :
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.
,   : Inplane strain component : Angle of twist   ,   ,   : Shearing strain components ] 12 , ] 21 : Poisson's ratios , , : Isoparametric coordinates : D e n s i t yo fm a t e r i a l   ,   : Inplane stress components   ,   ,   : Shearing stress components : Natural frequency : N o n d i m e n s i o n a ln a t u r a l frequency =  2 (/ 22 ℎ 2 ) 1/2 .