Free Vibration of Laminated Composite Hypar Shell Roofs with Cutouts

Abstract

Use of laminated composites in civil engineering structural components including shell roofs is increasing day by day due to their light weight, high specific strength, and stiffness properties. In the present paper, laminated composite hypar shell (hyperbolic paraboloidal shells bounded by straight edges) roofs with cutouts are analyzed for their free vibration characteristics using finite element method. An eight-noded curved shell element is used for modeling the shell. Specific numerical problems of earlier investigators are solved to compare their results with the present formulation. A number of problems are further solved where the size of the cutouts and their positions with respect to the shell centre are varied for different edge constraints. The results are furnished in the form of figures and tables. The results are examined thoroughly to arrive at some meaningful conclusions useful to designers.

1. Introduction

Different shell forms are common in civil engineering applications. Among the shell forms used as roofing units, the skewed hypars (hyperbolic paraboloidal shells bounded by straight edges) have a special position because these architecturally pleasant forms may be cast and fabricated conveniently being doubly ruled surfaces. Hypar shell is preferred in many places, particularly in medical, chemical and food processing industries where entry of north light is desirable. Application of hypars in these industries often necessitates provision of cutouts for the passage of light, to provide accessibility to other parts of the structure, for venting and also sometimes for alteration of the resonant frequency. A comprehensive idea about the static and free vibration characteristics of such shell roofs with cutouts is essential for a designer for successfully applying these forms. Moreover, nowadays researchers are emphasizing more on laminated composite shells realizing the strength and stiffness potentials of this advanced material.

The free vibration of composite as well as isotropic plate with cutout was studied by different researchers from time to time. Reddy [2] investigated large amplitude flexural vibration of composite plate with cutout. Later in 1989, Malhotra et al. [3] studied free vibration of composite plate with cutout for different boundary conditions. One of the early reports on free vibration of curved panels with cutout was due to Sivasubramonian et al. [4]. They analysed the effect of cutouts on the natural frequencies of plates with some classical boundary conditions. The plate had a curvature in one direction and was straight in the other. The effect of fibre orientation and size of cutout on natural frequency on orthotropic square plates with square cutout was studied using Rayleigh-Ritz method. Later Sivakumar et al. [5], Rossi [6], Huang and Sakiyama [7], and Hota and Padhi [8] studied free vibration of plate with various cutout geometries. Chakravorty et al. [1] analyzed the effect of concentric cutout on different shell options. Again in 1999, Sivasubramonian et al. [9] studied the effect of curvature and cutouts on square panels with different boundary conditions. The size of the cutout (symmetrically located) as well as curvature of the panels is varied. Hota and Chakravorty [10] published useful information about free vibration of stiffened conoidal shell roofs with cutout. Later Nanda and Bandyopadhyay [11] studied the effect of different parametric variation on free vibration of cylindrical shell with cutout.

A scrutiny of the literature on vibration of shell panels with cutout indicates that there is enough scope of research to be carried out on the hypar shell. Although Chakravorty et al. [1] did a few studies about hypar shell with concentric cutout, many practically important aspects are yet to be addressed. In the present paper the free vibration of hypar shell with cutouts (Figure 1) is studied considering different boundary conditions. The variation of fundamental frequency due to change in eccentricity of cutout along 𝑥 and 𝑦 direction is also considered.

Figure 1 

Surface of a hypar shell with cutout. Surface equation 𝑧=(4𝑐/𝑎𝑏)(𝑥−𝑎/2)(𝑦−𝑏/2)

2. Governing Equations

A laminated composite hypar shell (Figure 2) of uniform thickness ℎ and twist radius of curvature 𝑅_𝑥𝑦 is considered. Keeping the total thickness 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 co-ordinate system. The constitutive equations for the shell are given by (a list of notations is already given) {𝐹}=[𝐷]{𝜀}, (1) where {𝐹}=𝑁_𝑥𝑁_𝑦𝑁_𝑥𝑦𝑀_𝑥𝑀_𝑦𝑀_𝑥𝑦𝑄_𝑥𝑄_𝑦^𝑇,[𝐷]=𝐴₁₁𝐴₁₂𝐴₁₆𝐵₁₁𝐵₁₂𝐵₁₆𝐴00₁₂𝐴₂₂𝐴₂₆𝐵₁₂𝐵₂₂𝐵₂₆𝐴00₁₆𝐴₂₆𝐴₆₆𝐵₁₆𝐵₂₆𝐵₆₆𝐵00₁₁𝐵₁₂𝐵₁₆𝐷₁₁𝐷₁₂𝐷₁₆𝐵00₁₂𝐵₂₂𝐵₂₆𝐷₁₂𝐷₂₂𝐷₂₆𝐵00₁₆𝐵₂₆𝐵₆₆𝐷₁₆𝐷₂₆𝐷₆₆00000000𝑆₁₁𝑆₁₂000000𝑆₁₂𝑆₂₂,{𝜀}=𝜀⁰_𝑥𝜀⁰_𝑦𝛾⁰_𝑥𝑦𝑘_𝑥𝑘_𝑦𝑘_𝑥𝑦𝛾⁰_𝑥𝑧𝛾⁰_𝑦𝑧^𝑇. (2) The force and moment resultants are expressed as𝑁_𝑥𝑁_𝑦𝑁_𝑥𝑦𝑀_𝑥𝑀_𝑦𝑀_𝑥𝑦𝑄_𝑥𝑄_𝑦=^ℎ/2_−ℎ/2𝜎_𝑥𝜎𝑑𝑧_𝑦𝜏𝑑𝑧_𝑥𝑦𝜎𝑑𝑧_𝑥𝜎𝑧𝑑𝑧_𝑦𝜏𝑧𝑑𝑧_𝑥𝑦𝜏𝑧𝑑𝑧_𝑥𝑧𝜏𝑑𝑧_𝑦𝑧𝑑𝑧. (3) The stiffness coefficients are functions of Young’s moduli, shear moduli, and the 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 Qatu [12] and Vasiliev et al. [13].

Figure 2 

Laminations in skewed hypar shell.

The stiffness coefficients are defined as𝐴_𝑖𝑗=^np_𝑘=1𝑄_𝑖𝑗𝑘𝑧_𝑘−𝑧_𝑘−1,𝐵_𝑖𝑗=12^np𝑘=1𝑄_𝑖𝑗𝑘𝑧²_𝑘−𝑧²_𝑘−1,𝐷_𝑖𝑗=13^np𝑘=1𝑄_𝑖𝑗𝑘𝑧³_𝑘−𝑧³_𝑘−1𝑆𝑖,𝑗=1,2,6,_𝑖𝑗=^np_𝑘=1𝐹_𝑖𝐹_𝑗𝐺_𝑖𝑗𝑘𝑧_𝑘−𝑧_𝑘−1𝑖,𝑗=1,2, (4) where 𝑄_𝑖𝑗 are elements of the off-axis elastic constant matrix which is given by 𝑄_𝑖𝑗off=[𝑇]^𝑇𝑄_𝑖𝑗on[𝑇] (5) in which 𝑄_𝑖𝑗on=𝑄₁₁𝑄₁₂0𝑄₁₂𝑄₂₂000𝑄₆₆,[𝑇]=𝑚²𝑛²𝑛𝑚𝑛₂𝑚²−𝑚𝑛−2𝑚𝑛2𝑚𝑛𝑚²−𝑛² (6) with 𝑚=cos𝜃,𝑛=sin𝜃. (7) The elements of the [𝑄_𝑖𝑗]_on matrix [13] are 𝑄₁₁=1−𝜈₁₂𝜈₂₁⁻¹𝐸₁₁,𝑄₂₂=1−𝜈₁₂𝜈₂₁⁻¹𝐸₂₂,𝑄₁₂=1−𝜈₁₂𝜈₂₁⁻¹𝐸₁₁𝜈₂₁,𝑄₆₆=𝐺₁₂. (8)𝐹_𝑖 and 𝐹_𝑗 of (4) are two shear correction factors presently taken as unity for thin shells and the elements of the 𝐺_𝑖𝑗 matrix are given by 𝐺_𝑥𝑥=𝐺₁₃cos²𝜃+𝐺₂₃sin²𝐺𝜃,_𝑥𝑦=𝐺₁₃−𝐺₂₃𝐺cos𝜃sin𝜃,_𝑦𝑦=𝐺₁₃sin²𝜃+𝐺₂₃cos²𝜃. (9) The 𝐺_𝑖𝑗 matrix has the form 𝐺_𝑖𝑗=𝐺_𝑥𝑥𝐺_𝑥𝑦𝐺_𝑥𝑦𝐺_𝑦𝑦. (10) The strain-displacement relations on the basis of improved first-order approximation theory for thin shell [14] are established as 𝜀_𝑥,𝜀_𝑦,𝛾_𝑥𝑦,𝛾_𝑥𝑧,𝛾_𝑦𝑧^𝑇=𝜀⁰_𝑥,𝜀⁰_𝑦,𝛾⁰_𝑥𝑦,𝛾⁰_𝑥𝑧,𝛾⁰_𝑦𝑧^𝑇+𝑧𝑘_𝑥,𝑘_𝑦,𝑘_𝑥𝑦,𝑘_𝑥𝑧,𝑘_𝑦𝑧^𝑇, (11) where, the first vector is the midsurface strain for a hypar shell and the second vector is the curvature. These are given, respectively, by 𝜀⁰_𝑥𝜀⁰_𝑦𝛾⁰_𝑥𝑦𝛾⁰_𝑥𝑧𝛾⁰_𝑦𝑧=𝜕𝑢𝜕𝑥𝜕𝑣𝜕𝑦𝜕𝑢+𝜕𝑦𝜕𝑣−𝜕𝑥2𝑤𝑅_𝑥𝑦𝛼+𝜕𝑤𝜕𝑥𝛽+𝜕𝑤,𝑘𝜕𝑦^𝑥𝑘_𝑦𝑘_𝑥𝑦𝑘_𝑥𝑧𝑘_𝑦𝑧=𝜕𝛼𝜕𝑥𝜕𝛽𝜕𝑦𝜕𝛼+𝜕𝑦𝜕𝛽00𝜕𝑥. (12) The radius of cross curvature may be evaluated by differentiating the surface equation of shell in the form 𝑧=𝑓(𝑥,𝑦) and for shallow shells which are taken up for the present study the same may be expressed as 1/𝑅_𝑥𝑦=𝑑²𝑧/𝑑𝑥𝑑𝑦.

2.1. Finite Element Formulation

An eight-noded curved quadratic isoparametric finite element (as shown in Figure 3) is used for hypar shell analysis. The five degrees of freedom taken into consideration at each node are 𝑢,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: 𝑢=⁸_𝑖=1𝑁_𝑖𝑢_𝑖,𝑣=⁸_𝑖=1𝑁_𝑖𝑣_𝑖,𝑤=⁸_𝑖=1𝑁_𝑖𝑤_𝑖,𝛼=⁸_𝑖=1𝑁_𝑖𝛼_𝑖,𝛽=⁸_𝑖=1𝑁_𝑖𝛽_𝑖, (13) where the shape functions derived from a cubic interpolation polynomial [14] are 𝑁_𝑖=1+𝜉𝜉_𝑖1+𝜂𝜂_𝑖𝜉𝜉_𝑖+𝜂𝜂_𝑖−14𝑁,for𝑖=1,2,3,4,_𝑖=1+𝜉𝜉_𝑖1−𝜂²2𝑁,for𝑖=5,7,_𝑖=1+𝜂𝜂_𝑖1−𝜉²2,for𝑖=6,8. (14)

Figure 3 

Eight noded shell element with isoparametric coordinates.

The generalized displacement vector of an element is expressed in terms of the shape functions and nodal degrees of freedom as [𝑢]=[𝑁]𝑑_𝑒, (15) that is, {𝑢}=𝑢𝑣𝑤𝛼𝛽=⁸_𝑖=1𝑁_𝑖𝑁_𝑖𝑁_𝑖𝑁_𝑖𝑁_𝑖𝑢_𝑖𝑣_𝑖𝑤_𝑖𝛼_𝑖𝛽_𝑖. (16)

2.2. Element Stiffness Matrix

The strain-displacement relation is given by {𝜀}=[𝐵]𝑑_𝑒, (17) where [𝐵]=⁸_𝑖=1𝑁_𝑖,𝑥00000𝑁_𝑖,𝑦𝑁000_𝑖,𝑦𝑁_𝑖,𝑥−2𝑁_𝑖𝑅_𝑥𝑦00000𝑁_𝑖,𝑥00000𝑁_𝑖,𝑦000𝑁_𝑖,𝑦𝑁_𝑖,𝑥00𝑁_𝑖,𝑥𝑁_𝑖000𝑁_𝑖,𝑦0𝑁_𝑖. (18) The element stiffness matrix is 𝐾_𝑒=[𝐵]^𝑇[𝐸][𝐵]𝑑𝑥𝑑𝑦. (19)

2.3. Element Mass Matrix

The element mass matrix is obtained from the integral 𝑀_𝑒=[𝑁]^𝑇[𝑃][𝑁]𝑑𝑥𝑑𝑦, (20) where[𝑁]=⁸_𝑖=1𝑁_𝑖00000𝑁_𝑖00000𝑁_𝑖00000𝑁_𝑖00000𝑁_𝑖,[𝑃]=⁸_𝑖=1,𝑃00000𝑃00000𝑃00000𝐼00000𝐼 (21) in which 𝑃=^np_𝑘=1^𝑧𝑘_𝑧𝑘−1𝜌𝑑𝑧,𝐼=^np_𝑘=1^𝑧𝑘_𝑧𝑘−1𝑧𝜌𝑑𝑧. (22)

2.4. 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 4. 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 4 

Typical 10 × 10 nonuniform mesh arrangements drawn to scale.

2.5. Solution Procedure for Dynamic Analysis

The element stiffness and mass matrices are evaluated first by expressing the integrals in the local natural coordinates 𝝃 and 𝜼 of the 2×2 Gauss quadrature because the shape functions are derived from a cubic interpolation polynomial (Dey et al. [14]) and it is an established fact that a polynomial of degree 2𝑛−1 is integrated exactly by 𝑛 point Gauss quadrature [15]. Then the element matrices are assembled after performing appropriate transformations due to the curved shell surface to obtain the respective global matrices [𝐾] and [𝑀].

The free vibration analysis involves determination of natural frequencies from the condition [𝐾]−𝜔²[𝑀]=0. (23) This is a generalized eigen value problem and is solved by the subspace iteration algorithm.

3. Validation Study

The accuracy of the present formulation is first validated by comparing the results of the following problem available in the existing literature.

Free vibration of simply supported and clamped hypar shell with (0/90)₄ shell of aspect ratio 𝑎/𝑏=1, 𝑐/𝑎=0.2, and side to thickness ratio 𝑎/ℎ=100 with cutouts [1] is considered as the benchmark problem. The nondimensional frequency parameter is 𝜔=𝜔𝑎²(𝜌/𝐸₂₂ℎ²)^1/2. Material properties are as follows: 𝐸₁₁/𝐸₂₂=25, 𝐺₂₃=0.2𝐸₂₂, 𝐺₁₃=𝐺₁₂=0.5𝐸₂₂, 𝜈₁₂=𝜈₂₁=0.25.

The fundamental frequencies of hypar shell with cutout obtained by the present method agree well with those reported by Chakravorty et al. [1] as evident from Table 1, establishing the correctness of the present results. The fact that the cutouts are properly modeled in the present formulation is thus also established. The present approach uses the improved first-order approximation theory for thin shells [13] considering the radius of cross curvature. For this class of thin shells a shear correction factor of unity is found to yield good results. It is observed that the results remain the same when analysis is repeated with the commonly used shear correction factor of 𝜋/√12.

Table 1 

Non-dimensional fundamental frequencies (𝜔) for hypar shells (lamination (0/90)₄) with concentric cutouts.

4. Results and Discussion

In order to study the effect of cutout size on the free vibration response additional problems for hypar shells with 0/90/0/90 and +45/−45/+45/−45 lamination and different boundary conditions have been solved. The selection of the 0/90/0/90 and +45/−45/+45/−45 lamination is based on an earlier study by Sahoo and Chakravorty [16] which revealed that repeating 0/90 unit and +45/−45 unit more than once and keeping the total shell thickness constant do not improve the fundamental frequency to an appreciable extent. The positions of the cutouts are varied along both of the plan directions of the shell for different practical boundary conditions to study the effect of eccentricity of cutout on the fundamental frequency.

4.1. Free Vibration Behaviour of Shells with Concentric Cutouts

Figures 5 and 6 furnish the results of non-dimensional frequency (𝜔) of 0/90/0/90 and +45/−45/+45/−45 hypar shells. The shells considered are of square plan form (𝑎=𝑏), and the cutouts are also taken to be square in plan (𝑎^′=𝑏^′). The cutouts are placed concentrically on the shell surface. The cutout sizes (i.e., 𝑎^′/𝑎) are varied from 0 to 0.4, and boundary conditions are varied along the four edges. 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 a shell with CSCS boundary is clamped along 𝑥=0, simply supported along 𝑦=0 and 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.

Figure 5 

Values of non-dimensional fundamental frequency (𝜔) of 0/90/0/90 hypar shell with cutout for different sizes of central cutout and boundary conditions. 𝑎/𝑏=1, 𝑎/ℎ=100, 𝑎′/𝑎=0.1, 𝑎′/𝑏′=1, 𝑐/𝑎=0.2; 𝐸11/𝐸22=25, 𝐺23=0.2𝐸22, 𝐺13=𝐺12=0.5𝐸22, 𝜈12=𝜈21=0.25.

Figure 6 

Values of non-dimensional fundamental frequency (𝜔) of +45/−45/+45/−45 hypar shell with cutout for different sizes of central cutout and boundary conditions. 𝑎/𝑏=1, 𝑎/ℎ=100, 𝑎′/𝑎=0.1, 𝑎′/𝑏′=1, 𝑐/𝑎=0.2; 𝐸11/𝐸22=25, 𝐺23=0.2𝐸22, 𝐺13=𝐺12=0.5𝐸22, 𝜈12=𝜈21=0.25.

4.1.1. Effect of Cutout Sizes

From the figures it is seen that when a cutout is introduced to a shell the fundamental frequency increases in 6 out of 12 cases in case of symmetric crossply shell. But in case of symmetric angle ply one the fundamental frequency increases in 10 out of 12 cases. In order to study the effect of cutout size, in more detail, the ratio of the fundamental frequency of a concentric punctured shell to that of a shell without cutout is expressed in percentage. The increase or decrease in percentage of fundamental frequency from the full shell is denoted by 𝑝. Tables 2 and 3 contain such 𝑝 values for 0/90/0/90 and +45/−45/+45/−45 shells, respectively. Negative sign indicates decrease in frequency. It is evident from Tables 2 and 3 that in all the cases with the introduction of cutout with 𝑎^′/𝑎=0.1 the increase or decrease in frequency is not more than 1%. But with further increase in cutout size, that is, when 𝑎^′/𝑎=0.2, fundamental frequency decreases in all the cases, except for simply supported symmetric angle ply shells. Because in such cases loss of stiffness is more significant than loss of mass. Here also the decrease in fundamental frequency is between 1 and 10%. But with further increase in cutout sizes 𝑎^′/𝑎=0.3 and 0.4 fundamental frequency decreases to an appreciable extent (up to 30%). For some cases of symmetric angle ply shell no such unified trend is observed. This leads to the engineering conclusion that concentric cutouts may be provided safely on shell surfaces for functional requirements up to 𝑎^′/𝑎=0.2.

Table 2 

Values of “𝑝” for 0/90/0/90 hypar shell.

Table 3 

Values of “𝑝” for +45/−45/+45/−45 hypar shell.

4.1.2. Effect of Boundary Conditions

The boundary conditions have been divided into four groups, so that the combinations in a particular group have equal number of boundary reactions. The groups are of the following forms:Group I: contains CCCC shells.Group II: contains CSCC, CCSC, and CCCS shells.Group III: contains CSSC, CCSS, CSCS, and SCSC shells.Group IV: contains SSSS shells.

As evident from Figures 5 and 6, fundamental frequencies of members belonging to different boundary combinations may be grouped according to performance.

According to the values of (𝜔), the group III may be subdivided into Group IIIa and Group IIIb for both 0/90/0/90 and +45/−45/+45/−45 shells.Group I: contains CCCC shells.Group II: contains CSCC, CCSC, and CCCS shells.Group IIIa: contains CSSC, CCSS shells.Group IIIb: contains CSCS and SCSC shells.Group IV: contains SSSS shells.

This observation indicates that the impact of arrangement of boundary constraints is far more important than their actual number in determining the free vibration characteristics.

The frequencies are further studied, and marks are given to the options of clamping the edges of a simply supported shell in order to gradually improve performances. Tables 4 and 5 furnish such clamping options for crossply and angle ply shells, respectively. The scale is chosen like this: 0 is assigned to a simply supported shell and 100 to a clamped shell. These marks are furnished for cutouts with 𝑎^′/𝑎=0.2. These tables will help a practicing engineer. If one takes the frequency of a clamped shell as upper limit and that of the simply supported as lower limit, one can easily realize the efficiency of a particular boundary condition.

Table 4 

Clamping options for 0/90/0/90 hypar shells with central cutouts having 𝑎^′/𝑎 ratio 0.2.

Table 5 

Clamping options for +45/−45/+45/−45 hypar shells with central cutouts having a′/a ratio 0.2.

4.2. Effect of Eccentricity of Cutout Position on Fundamental Frequency

To study the effect of cutout positions on fundamental frequencies, results are obtained for different locations of a cutout with 𝑎^′/𝑎=0.2. As with the introduction of cutout with 𝑎^′/𝑎=0.2, the change in fundamental frequency with that of an unpunctured shell is within 1–10%, so 𝑎^′/𝑎=0.2 is chosen for the further study. Each of the non-dimensional coordinates of the cutout centre (𝑥=𝑥/𝑎, 𝑦=𝑦/𝑎) is varied from 0.2 to 0.8 along both the plan directions so that the distance of a cutout margin from the shell boundary is not less than one-tenth of the plan dimension of the shell. The study is carried out for all the twelve boundary conditions for both 0/90/0/90 and +45/−45/+45/−45 hypar shells. The ratio of the fundamental frequency of a shell with an eccentric puncture to that of a shell with concentric puncture (obtainable from Figures 5 and 6) expressed in percentage is denoted by 𝑟. Tables 6 and 7 contain the value of 𝑟 for 0/90/0/90 and +45/−45/+45/−45 hypar shells.

Table 6 

Values of “𝑟” for 0/90/0/90 hypar shells.

Table 7 

Values of “𝑟” for +45/−45/+45/−45 hypar shells.

From the 𝑟 values it is observed that a shell with eccentric cutout has a fundamental frequency above 90% to that of a shell having concentric cutout in almost all the cases of 0/90/0/90 shells. So cutout centre may be moved along any directions, resulting in loss of frequency of not more than 5–10% with respect to a shell with concentric cutout. But in case of a +45/−45/+45/−45 shell, the 𝑟 value is between 84.02 and 102.10.

For 0/90/0/90 shell when the four edges are clamped the stiffness of shell decreases more when the cutout centre is nearer to the shell centre line whereas when the cutout centre goes towards the boundary shell stiffness increases resulting in increased frequency. For such a shell 𝑟 value is minimum along centre line and increases towards the boundary. But such increase or decrease is marginal in each case. When one simply supported edge is introduced along the shell boundary 𝑟 value is maximum along the simply supported boundary in almost all the cases. Similarly when two adjacent edges are simply supported the values of 𝑟 have comparatively greater value towards simply supported edges. But exactly reverse trend is observed when two opposite edges are simply supported. In such cases the maximum 𝑟 values are obtained along the centre line of the shell which is equidistant from each simply supported edge and 𝑟 value decreases towards the simply supported edges. It is further noticed that when three edges are simply supported the 𝑟 value is maximum along the line which is equidistant from two simply supported edges and 𝑟 value is minimum along the centre which is equidistant from one simply supported edge and another clamped edge. When four edges are simply supported 𝑟 value is maximum along the diagonal.

In case of +45/−45/+45/−45 shells when the four edges are clamped the 𝑟 values are maximum along the centre line of shell in both directions but towards the boundary no unified trend is observed. Also along each clamped edge 𝑟 value is maximum at the middle of each edge. When one of the edges is simply supported, the 𝑟 value decreases towards the edges opposite to the simply supported edges and are minimum at the centre of that edge. But along the simply supported edge and other two clamped edges fundamental frequency is maximum at the middle of each edge. When the two adjacent edges are simply supported and other two are clamped minimum 𝑟 values are obtained when cutout centre is near the middle of the clamped edges and maximum when the same is near the middle of the simply supported edges. When two opposite edges are simply supported in more than 50% cases 𝑟 value is greater than 100. In rest of the cases 𝑟 values are more than 95 except one or two cases. When three edges are simply supported in a central rectangular zone the 𝑟 value is greater than 100. But when four edges are simply supported the zone where 𝑟 value is greater than 100 gets reduced.

5. Conclusions

From the present study the following conclusions are drawn.(1)The finite element code used here is suitable for analyzing free vibration problems of hypar shell roofs with cutouts as this approach produces results in close agreement with those of the benchmark problems.(2) Concentric cutouts may be provided safely on hypar shell surfaces for functional requirements up to 𝑎^′/𝑎=0.2.(3) The arrangement of boundary constraint along the four edges is far more important than their actual number so far the free vibration stiffness is concerned.(4) Fundamental frequency undergoes marked improvement when the edge is converted to clamped from simply supported condition.(5) Tables 4 and 5 provide a clear picture about the relative free vibration performances of hypar shells for different combinations of edge conditions along the four sides and are expected to be very useful in decision making for practicing engineers.(6) Tables 6 and 7 provide the information regarding behaviour of hypar shell with eccentric cutouts for wide spectrum of eccentricity and boundary conditions and may be used as design aids by structural engineers.

Notations

𝑎,𝑏:	Length and width of shell in plane
𝑎′,𝑏′:	Length and width of cutout in plane
𝑐:	Rise of hypar shell
{𝑑𝑒}:	Element displacement
𝐸11, 𝐸22:	Elastic moduli
𝐺12, 𝐺13, 𝐺23:	Shear moduli of a lamina with respect to 1, 2 and 3 axes of fibre
ℎ:	Shell thickness
𝑀𝑥, 𝑀𝑦:	Moment resultants
𝑀𝑥𝑦:	Torsion resultant
np:	Number of plies in a laminate
𝑁1-𝑁8:	Shape functions
𝑁𝑥, 𝑁𝑦:	Inplane force resultants
𝑁𝑥𝑦:	Inplane shear resultant
𝑄𝑥, 𝑄𝑦:	Transverse shear resultant
𝑅𝑥𝑦:	Radii of cross curvature of hypar shell
𝑢,𝑣,𝑤:	Translational degrees of freedom
𝑥,𝑦,𝑧:	Local co-ordinate axes
𝑋,𝑌,𝑍:	Global co-ordinate axes
𝑧𝑘:	Distance of bottom of the 𝑘th ply from midsurface of a laminate
𝛼,𝛽:	Rotational degrees of freedom
𝜀𝑥, 𝜀𝑦:	Inplane strain component
𝜙:	Angle of twist
𝛾𝑥𝑦,𝛾𝑥𝑧,𝛾𝑦𝑧:	Shearing strain components
𝜈12,𝜈21:	Poisson’s ratios
𝜉,𝜂,𝜏:	Isoparametric coordinates
𝜌:	Density of material
𝜎𝑥, 𝜎𝑦:	Inplane stress components
𝜏𝑥𝑦,𝜏𝑥𝑧,𝜏𝑦𝑧:	Shearing stress components
𝜔:	Natural frequency
𝜔:	Non-dimensional natural frequency =𝜔𝑎2(𝜌/𝐸22ℎ2)1/2.