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

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1 Journal of Engineering Volume 2013, Article ID , 1 pages Research Article Dnamic Characters of Stiffened Composite Conoidal Shell Roofs with Cutouts: Design Aids and Selection Guidelines Sarmila Sahoo Department of Civil Engineering, Heritage Institute of Technolog, Kolkata , India Correspondence should be addressed to Sarmila Sahoo; sarmila ju@ahoo.com Received 11 December 2012; Revised 3 Ma 2013; Accepted 14 Ma 2013 Academic Editor: Jun Li Copright 2013 Sarmila Sahoo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in an medium, provided the original work is properl cited. Dnamic characteristics of stiffened composite conoidal shells with cutout are analzed in terms of the natural frequenc and mode shapes. A finite element code is developed for the purpose b combining an eight-noded curved shell element with a three-noded curved beam element. The code is validated b 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-pl and angle-pl laminated composite conoids. The effects of these parametric variations on the fundamental frequencies and mode shapes are considered in details. The results furnished here ma be readil used b practicing engineers dealing with stiffened composite conoids with cutouts central or eccentric. 1. Introduction Laminated composite structures are gaining wide importance in various fields of aerospace and civil engineering. Shell roof structures can be convenientl built with composite materials that have man attributes, besides high specific strength and stiffness. Among the different shell panels which are commonl 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, the provide ease of casting and also allow north light in. Hence, this shell is preferred in man places, particularl in medical, chemical, and food processing industries where entr 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 frequenc. In practice, the margin of the cutouts must be stiffened to take account of stress concentration effects. An in-depth stud including bending, buckling, vibration, and impact is required to eploit the possibilities of these curved forms. The present investigation is, however, restricted onl to the free vibration behaviour. A generalized formulation for the doubl 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 briefl reviewed here. The research on conoidal shell started about four decades ago. In 1964, Hadid [1] analsed static characteristics of conoidal shells using the variational method. The research was carried forward and improved b researchers like Brebbia and Hadid [2], Choi [3], Ghosh and Bandopadha [4, 5], De et al. [6], and Das and Bandopadha [7]. De et al. [6] provided a significant contribution on static analsis of conoidal shell. Chakravort et al. [] appliedthefiniteelementtechniquetoeplorethefreevibrationcharacteristicsof shallow isotropic conoids and also observed the effects of ecluding some of the inertia terms from the mass matri on the first four natural frequencies. Chakravort et al. [9 11] publishedaseriesofpaperswherethereportedonfreeand forced vibration characteristics of graphite-epo composite conoidal shells with regular boundar conditions. Later, Naak and Bandopadha [12 15] reported free vibration of stiffened isotropic and composite conoidal shells. Das and Chakravort [16, 17] considered bending and free vibration characteristics of unpunctured and unstiffened composite conoids. Hota and Chakravort [1] studied isotropic punctured conoidal shells with complicated boundar conditions along the four edges, but no such stud about composite

2 2 Journal of Engineering conoidal shells is available in the literature. Also, the did not furnish an information on vibration mode shapes. It is also seen from the recent reviews [19, 20] that dnamic characteristics of stiffened conoidal shells with cutout are still missing in the literature. The present stud thus focuses on the free vibrations of graphite-epo laminated composite stiffened conoids with cutout both in terms of the natural frequencies and mode shapes. The results so obtained ma be readil used b practicing engineers dealing with stiffened composite conoidswithcutouts.thenoveltofthepresentstudliesin the consideration of vibration mode shapes of stiffened composite conoids in presence of cutouts. 2. Mathematical Formulation A laminated composite conoidal shell of uniform thickness h (Figure 1), and radius of curvature R,andradiusofcross curvature R is considered. Keeping the total thickness the same, the thickness ma consist of an number of thin laminae each of which ma be arbitraril oriented at an angle θ with reference to the -ais of the coordinate sstem. The constitutive equations for the shell are given b (a list of notations is separatel given) {F} = [E] {ε}, (1) where {F} ={N,N,N,M,M,M,Q,Q } T, [A] [B] [E] = [[B] [D] [0]], [[0] [0] [S]] {ε} ={ε 0,ε0,γ0,k,k,γ 0 z,γ0 z }T. The force and moment resultants are epressed as {N,N,N,M,M,M,Q,Q } T h/2 = {σ,σ,τ,σ z z,σ z,τ z,τ z,τ z } T dz. h/2 (3) The submatrices [A], [B], [D], and[s] of the elasticit matri [E] are functions of Young s moduli, shear moduli, and Poisson s ratio of the laminates. The also depend on the angle which the individual lamina of a laminate makes with the global -ais. The detailed epressions of the elements of the elasticit matri are available in several references including Vasiliev et al. [21]andQatu[22]. The strain-displacement relations on the basis of improved first-order approimation theor for thin shell (De et al. [6]) are established as {ε,ε,γ,γ z,γ z } T ={ε 0,ε0,γ0,γ0 z,γ0 z }T +z{k,k,k,k z,k z } T, (2) (4) Y h b w s a h s Z Figure 1: Conoidal shell with a concentric cutout stiffened along the margins. 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 analsis. The five degrees of freedom taken into considerationateachnodeareu, V, w, α, β. The following epressions establish the relations between the displacement at an point with respect to the coordinates ξ and η and the nodal degrees of freedom u= i=1 N i u i, V = α= i=1 i=1 b h l N i α i, β = a N i V i, w = i=1 X N i β i, i=1 h h N i w i, where the shape functions derived from a cubic interpolation polnomial [6]are N i = (1 + ξξ i)(1+ηη i )(ξξ i +ηη i 1), for i = 1, 2, 3, 4, 4 N i = (1 + ξξ i)(1 η 2 ), for i = 5, 7, 2 N i = (1 + ηη i)(1 ξ 2 ), for i = 6,. 2 The generalized displacement vector of an element is epressed in terms of the shape functions and nodal degrees of freedom as [u] = [N] {d e }, (7) that is, u { V } {u} = w = { α i=1 [ } { β} [ N i N i N i N i N i { ] { ] { u i V i w i α i β i (5) (6) }. () } }

3 Journal of Engineering Element Stiffness Matri. The strain-displacement relation is given b where {ε} = [B] {d e }, (9) N i, N i, N i 0 0 R N [B] = i, N i, 2N i 0 0 R. (10) i= N i, N i, N i, N i, [ 0 0 N i, N i 0 ] [ 0 0 N i, 0 N i ] The element stiffness matri is [K e ]= [B] T [E][B] d d. (11) Element Mass Matri. The element mass matri is obtained from the integral where in which P= [M e ]= [N] T [P][N] d d, (12) N i N i [N] = 0 0 N i 0 0, i=1 [ N i 0 ] [ N i ] np k=1 P P [P] = 0 0 P 0 0, i=1 [ I 0] [ I] z k ρdz, z k 1 I= np k=1 (13) z k zρ dz. (14) z k Finite Element Formulation for Stiffener of the Shell. Three-noded curved isoparametric beam element (Figure 2) areusedtomodelthestiffeners,whicharetakentorunonl along the boundaries of the shell elements. In the stiffener element, each node has four degrees of freedom, that is, u s, w s, α s,andβ s for X-stiffener and V s, w s, α s,andβ s for Y-stiffener. The generalized force-displacement relation of stiffeners can be epressed as X-stiffener: {F s }=[D s ]{ε s }=[D s ][B s ]{δ si }, Y-stiffener: {F s }=[D s ]{ε s }=[D s ][B s ]{δ si }, (15) η 6 (a) ξ Figure 2: (a) Eight-noded shell element with isoparametric coordinates. (b) Three-noded stiffener element: (i) X-stiffener (ii) Ystiffener. where (i) (b) {F s }=[N s M s T s Q sz ] T, {ε s }=[u s α s β s (α s +w s )] T, {F s }=[N s M s T s Q sz ] T, {ε s }=[V s β s α s (β s +w s )] T. ξ η 3 1 (ii) 2 (16) The generalized displacements of the Y-stiffener and the shell are related b the transformation matri {δ si } = [T]{δ}, where 1+ e smmetric [T] = [ [ R ] ] ]. (17) This transformation is required due to curvature of Y-stiffener, and {δ} is the appropriate portion of the displacement vector of the shell ecluding the displacement component along the -ais. Elasticit matrices are as follows: A 11 b s B 11 b s B 12 b s 0 B 11 [D s ]= s D 11 b s D 12 b s 0 [ B 12 b s D 12 b 1, s 6 (Q 44 +Q 66 )d s b 3 s 0 ] [ b s S 11 ] A 22 b s B 22 b s B 12 b s 0 B [D s ]= 22 b 1 s 6 (Q 44 +Q 66 )b s D 12 b s 0 [ B 12 b s D 12 b s D 11 d, sb 3 s 0 ] [ b s S 22 ] (1) where D ij =D ij +2eB ij +e 2 A ij, B ij =B ij +ea ij, (19)

4 4 Journal of Engineering and A ij, B ij, D ij,ands ij are eplained in an earlier paper b Sahoo and Chakravort [23]. Here, the shear correction factor is taken as 5/6. The sectional parameters are calculated with respect to the midsurface of the shell b which the effect of eccentricities of stiffeners is automaticall included. The element stiffness matrices are of the following forms: for X-stiffener: [K e ]= [B s ] T [D s ][B s ]d, for Y-stiffener: [K e ]= [B s ] T [D s ][B s ]d. (20) The integrals are converted to isoparametric coordinates and are carried out b 2-point Gauss quadrature. Finall, the element stiffness matri of the stiffened shell is obtained b appropriate matching of the nodes of the stiffener and shell elements through the connectivit matri and is given as [K e ]=[K she ]+[K e ]+[K e ]. (21) The element stiffness matrices are assembled to get the global matrices Element Mass Matri. The element mass matri for shell is obtained from the integral where in which P= [M e ]= [N] T [P][N] d d, (22) N i N i [N] = 0 0 N i 0 0, i=1 [ N i 0 ] [ N i ] np k=1 P P [P] = 0 0 P 0 0, i=1 [ I 0] [ I] z k ρdz, z k 1 I= np k=1 Element mass matri for stiffener element [M s ]= [N] T [P][N] d [M s ]= [N] T [P][N] d Here, [N] is a 3 3diagonalmatri. (23) z k zρ dz. (24) z k 1 for X-stiffener, for Y-stiffener. (25) Consider ρ b s d s ρ b 3 s d s 0 0 ρ b [P] = s d 2 s [ 12 ] i=1 ρ(b s ds 3 [ b3 s d s) 12 ] for X-stiffener, ρ b s d s ρ b s d s 0 0 ρ b [P] = s d 2 s i=1 [ 12 ρ(b s ds b3 s d ] s) [ 12 ] for Y-stiffener. (26) The mass matri of the stiffened shell element is the sum of thematricesoftheshellandthestiffenersmatchedatthe appropriate nodes [M e ]=[M she ]+[M e ]+[M e ]. (27) 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 graduall decreased near the cutout margins. One such tpical 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 frequenc when the result does not improve b more than one percent on further refining. Convergence of results is ensured in all the problems taken up here Solution Procedure for Free Vibration Analsis. The free vibration analsis involves determination of natural frequencies from the condition [K] ω2 [M] =0. (2) Thisisageneralizedeigenvalueproblemandissolvedb the subspace iteration algorithm. 4. Numerical Eamples The validit of the present approach is checked through solution of benchmark problems. The first problem, free vibration of stiffened clamped conoid, was solved earlier b Naak and Bandopadha [13]. The second is the free vibration of composite conoid with cutouts solved b Chakravort et al. [11]. The results obtained b the present method, along with thepublishedresults,arepresentedintables1 and 2, respectivel. Additional problems for conoids with cutouts are solved, varing the size and position of cutout along both of the plan

5 Journal of Engineering 5 Table 1: Fundamental frequencies (rad/sec) of clamped conoidal shell with central stiffeners. Stiffener along -direction Stiffener along-direction Stiffener along both-and-directions Stiffener Naak and Naak and Naak and position Present model Present model Present model Bandopadha Bandopadha Bandopadha [13] [13] [13] Concentric Eccentric at top Eccentric at bottom a=50m, b=50m, h=0.2m, h h =10m,h l =2.5m,E = , ] =0.15, ρ = 2500 kg/m 3, w s =0.3m, and h s =1m. Table 2: Nondimensional fundamental frequencies (ω) for laminated composite conoidal shell with cutout. a Corner point supported Simpl supported Clamped /a Chakravortet al. [11] Present model Chakravortet al.[11] Present model Chakravortet al.[11] Present model a/b = 1, a/h = 100, a /b =1, a/h h =2.5,andh l /h h =0.25. Figure 3: Tpical nonuniform mesh arrangements drawn to scale. directions of the shell for different practical boundar conditions. In order to stud the effect of cutout size on the free vibration response, results for unpunctured conoids are also included in the stud. 5. Results and Discussion It is found from Table 1 that the fundamental frequencies of stiffened conoids obtained b the present method agree well with those reported b Naak and Bandopadha [13]. Here, monotonic convergence is noted as the mesh is made progressivel 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 Chakravort et al. [11],and the fact that the cutouts are properl modeled in the present formulationisthusestablished Free Vibration Behaviour of Shells with Concentric Cutouts. Tables 3 and 4 show the results for the non-dimensional fundamental frequenc ω of composite cross-pl and angle-pl stiffened conoidal shells for different cutout sizes and various combinations of boundar conditions along the four edges. The shells considered are of square planform (a = b),andthe cutouts are also taken to be square in plan (a = b ).The cutout sizes (i.e., a /a)arevariedfrom0to0.4,andboundar conditions are varied along the four edges. Cutouts are concentric on shell surface. The stiffeners are placed along the cutout peripher and etended up to the edge of the shell. The boundar conditions are designated b describing the support clamped or simpl supported as C or S taken in an anticlockwise order from the edge = 0.Thismeansthat a shell with CSCS boundar is clamped along =0,simpl supported along = 0,clampedalong = a,andsimpl supported along = b. The material and geometric properties of shells and cutouts are mentioned along with the figures Effect of Cutout Size on Fundamental Frequenc. From Tables 3 and 4, itisseenthatwhenacutoutisintroduced to a stiffened shell, the fundamental frequencies increase. This increasing trend continues up to a /a = 0.4 for both cross- and angle pl shells ecept some angle pl shells with a /a > 0.2. The initial increase in frequenc ma be eplained b the fact that when a cutout is introduced to an unpunctured surface, the number of stiffeners increases fromtwotofourinthepresentstud.whenthecutoutsize 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

6 6 Journal of Engineering Boundar condition a a CCCC 0 CCSC CCCS CSSC CCSS CSCS SCSC CSSS SSSC SSCS SSSS Point supported CSCC Figure 4: First mode shapes of laminated composite (0/90/0/90) stiffened conoidal shell for different sizes of the central square cutout and boundar conditions. the frequenc increases. But for some angle pl shells with further increase in the size of the cutout, the loss of stiffness graduall becomes more important than that of mass, resulting in decrease in fundamental frequenc. This leads to the engineering conclusion that cutouts with stiffened margins ma alwas safel be provided on shell surfaces for functional requirements Effect of Boundar Conditions on Fundamental Frequenc. The boundar conditions ma be divided into si groups, considering number of boundar constraints. The combinations in a particular group have equal number of boundar reactions. These groups are Group I: CCCC shells, Group II: CSCC, CCSC, and SCCC shells,

7 Journal of Engineering CCCC CSCC CCSC CCCS CSSC CCSS CSCS SCSC CSSS SSSC SSCS SSSS Point supported Boundar a a condition Figure 5: First mode shapes of laminated composite (+45/ 45/+45/ 45) stiffened conoidal shell for different sizes of the central square cutout and boundar conditions. 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 Tables 3 and 4 that fundamental frequencies of members belonging to the same groups of boundar combinations ma not have close values. So, the different boundar conditions ma be regrouped according to performance. According to the values of ω, thefollowinggroups ma be identified.

8 Journal of Engineering Figure 6: First mode shapes of laminated composite (0/90/0/90) stiffened conoidal shell for different positions of the square cutout with CCCC boundar 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 boundar condition.

9 Journal of Engineering Figure : First mode shapes of laminated composite (+45/ 45/+45/ 45) stiffened conoidal shell for different positions of the square cutout with CCCC boundar 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 boundar condition.

10 10 Journal of Engineering 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 boundar conditions. Boundar Cutout size (a /a) conditions CCCC CSCC CCSC CCCS CSSC CCSS CSCS SCSC CSSS SSSC SSCS SSSS Point supported a/b = 1, a/h = 100, a /b =1, a/h h =5, h l /h h =0.25, E 11 /E 22 =25, G 23 = 0.2E 22, G 13 =G 12 =0.5E 22,and] 12 = ] 21 =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 boundar conditions. Boundar Cutout size (a /a) conditions CCCC CSCC CCSC CCCS CSSC CCSS CSCS SCSC CSSS SSSC SSCS SSSS Point supported a/b = 1, a/h = 100, a /b =1, a/h h =5, h l /h h =0.25, E 11 /E 22 =25, G 23 = 0.2E 22, G 13 =G 12 =0.5E 22,and] 12 = ] 21 =0.25. For cross pl shells Group 1: Contains CCCC, CCSC, and SCSC boundaries which ehibit relativel high frequencies. Group 2: Contains CSCC, CCCS, CSSC, CCSS, CSCS, SSSC, CSSS, and SSCS which ehibit intermediate values of frequencies. Group 3: Contains SSSS and corner point supported boundaries which ehibit relativel low values of frequencies. Similarl for angle pl shells: Group 1: Contains CCCC, CCSC, CSCC, CCCS, CSSC, CCSS, SCSC, and CSCS boundaries which ehibit relativel high frequencies. Group 2: Contains SSSC, CSSS, SSCS, and SSSS boundaries which ehibit intermediate values of frequencies. Group 3: Contains corner point supported shells which ehibit relativel low values of frequencies. It is evident from the present stud that the free vibration characteristics mostl depend on the arrangement of boundar constraints rather than their actual number. It can be seen fromthepresentstudthatifthehigherparabolicedgealong =ais released from clamped to simpl supported, there is hardl an change of frequenc for cross pl shell. But for angle pl shells, if the edge along higher parabolic edge is released, fundamental frequenc even increases more than that of a clamped shell. For cross pl shells, if the edge along =0or =bis released, that is, along the straight edges, frequenc values undergo marked decrease. The results indicate that the edge along =0or =bshould preferabl be clamped in order to achieve higher frequenc values, and if the edge has to be released for functional reason, the edge along =0and =aof a conoid must be clamped to make up for the loss of frequenc. But for angle pl shells, if an twoedgesarereleased,thechangeinfundamentalfrequenc is not so significant. Tables 5 and 6 show the efficienc of a particular clamping option in improving the fundamental frequenc of a shell with minimum number of boundar constraints relative to that of a clamped shell. Marks are assigned to each boundar combination in a scale assigning a value of 0 to the frequenc of a corner point supported shell and 100 to that of a full 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 efficienc of a particular boundar condition in improving the frequenc 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-pl and angle pl shells, respectivel. 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 clearl a bending mode for all the boundar conditions for cross-pl and angle-pl shells, ecept 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 donotchangetoanappreciableamount.

11 Journal of Engineering 11 Number of sides to be clamped Table 5: Clamping options for 0/90/0/90 conoidal shells with central cutouts having a /a ratio 0.2. 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 Simpl supported no edges clamped (SSSS) Good improvement 35 (a) Higher parabolicedgealong =a(sscs) Marked improvement 46 1 (b)lowerparabolicedgealong=0(csss) Marked improvement 49 (c) Onestraightedgealong =b(sssc) Marked improvement 52 2 (a) Two alternate edges including the higher and lower parabolic edges =0and =a(cscs) Marked improvement 57 (b) 2 straight edges along =0and =b(scsc) Remarkable improvement (c) An two edges ecept for the above option (CSSC, CCSS) Marked improvement 63 3 edges including the two parabolic edges (CSCC, 3 CCCS) Marked improvement 45 3 edges ecluding the higher parabolic edge along Remarkable =a(ccsc) improvement 96 4 All sides (CCCC) Frequenc attains the highest value 100 Number of sides to be clamped Table 6: Clamping options for +45/ 45/+45/ 45 conoidal shells with central cutouts having a /a ratio 0.2. 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 Simpl supported no edges clamped (SSSS) Marked improvement 52 (a) Higher parabolic edge along =a(sscs) Marked improvement 65 1 (b)lowerparabolicedgealong=0(csss) Marked improvement 64 (c) One straight edge along =b(sssc) Marked improvement 66 2 (a) Two alternate edges including the higher and lower parabolic edges =0and =a(cscs) Remarkable improvement 1 (b) 2straight edges along =0and =b(scsc) Remarkable improvement 3 (c) An two edges ecept for the above option (CSSC, CCSS) Remarkable improvement edges including the two parabolic edges (CSCC, 3 CCCS) Remarkable improvement edges ecluding the higher parabolic edge along Frequenc attains more than a =a(ccsc) full clamped shell All sides (CCCC) Remarkable improvement Effect of Eccentricit of Cutout Position Fundamental Frequenc. The effect of eccentricit of cutout positions on fundamental frequencies is studied from the results obtained for different locations of a cutout with a /a = 0.2. The non-dimensional coordinates of the cutout centre ( =/a,=/a)were varied from 0.2 to 0. along each direction, so that the distance of a cutout margin from the shell boundar was not less than one tenth of the plan dimension of the shell. The margins of cutouts were stiffened with four stiffeners. The stud was carried out for all the thirteen boundar conditions for both cross pl and angle pl shells. The fundamental frequenc of a shell with an eccentric cutout is epressed as a percentage of fundamental frequenc of a shell with a concentric cutout. This percentage is denoted b r. In Tables 7 and, such results are furnished. Itcanbeseenthateccentricitofthecutoutalongthe length of the shell towards the parabolic edges makes it more fleible. It is also seen that towards the lower parabolic edge, r value is greater than that of the higher parabolic edge. This meansthatifadesignerhastoprovideaneccentriccutout along the length, he should preferabl place it towards the lower height boundar. The eception is there in some cases of cross pl shells. For cross pl shells with three edges simpl

12 12 Journal of Engineering Table 7: Values of r for 0/90/0/90 conoidal shells. Edge condition CCCC CSCC CCSC CCCS CSSC CCSS CSCS

13 Journal of Engineering 13 Table 7: Continued. Edge condition SCSC CSSS SSSC SSCS SSSS CS a/b = 1, a/h = 100, a /b =1, a/h h =5, h l /h h =0.25, E 11 /E 22 =25, G 23 =0.2E 22, G 13 =G 12 =0.5E 22,and] 12 = ] 21 =0.25. supportedwhencutoutshiftstowardsthelowerparabolic 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 maimum along lower parabolic edge. For cross pl shells, four out of thirteen boundar conditions ield the maimum value of r along = 0.5, four ield maimum values of r along = 0.4, and others show maimum values within = 0.4 to 0.5. It is observed from Table 7 that if the eccentricit 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, eccentricitofacutoutalongthewidthshouldpreferablbe towards the clamped straight edge. For shells having two straight edges of identical boundar condition, the maimum fundamental frequenc occurs along = 0.5. For corner

14 14 Journal of Engineering Table : Values of r for +45/ 45/+45/ 45 conoidal shells. Edge condition CCCC CSCC CCSC CCCS CSSC CCSS CSCS

15 Journal of Engineering 15 Table : Continued. Edge condition SCSC CSSS SSSC SSCS SSSS CS a/b = 1, a/h = 100, a /b =1, a/h h =5, h l /h h =0.25, E 11 /E 22 =25, G 23 =0.2E 22, G 13 =G 12 =0.5E 22,and] 12 = ] 21 =0.25. point supported shells, the maimum fundamental frequenc alwasoccursalongtheboundaroftheshell.alltheseare trueforcrossplshellsonl.foranangleplshell,such unified trend is not observed, and the boundar conditions and the fundamental frequenc behave in a comple manner as evident from Table. But for corner point supported angle-pl shells also, the maimum values of r are along the boundar. Tables 9 and 10 provide the maimum values of r together with the position of the cutout. These tables also show the rectangular zones within which r is alwas greater than or equalto95and90.itistobenotedthatatsomeotherpoints,

16 16 Journal of Engineering Table 9: Maimum values of r with corresponding coordinates of cutout centre and zones where r 90and r 95for 0/90/0/90 conoidal shells. Boundar condition Maimum values of r Co-ordinate of cutout centre CCCC = 0.5, = 0.7 CSCC = 0.4, = 0.3 CCSC = 0.5, = 0.3 = 0.5, = 0.7 CCCS = 0.4, = 0.7 CSSC = 0.5, = 0.3 CCSS = 0.5, = 0.7 CSCS = 0.4, = 0.5 SCSC = 0.5, = 0.7 CSSS = 0.4, = 0.5 SSSC = 0.4, = 0.3 SSCS = 0.3, = 0.5 SSSS = 0.4, = 0.5 Area in which the value of r 90 = 0.6, , = 0.4, 0.6, , , , , = 0.3, 0.5, , , , = 0., CS = 0.2, = 0.2 nil a/b = 1, a/h = 100, a /b =1, a/h h =5, h l /h h =0.25, E 11 /E 22 =25, G 23 =0.2E 22, G 13 =G 12 =0.5E 22,and] 12 = ] 21 =0.25. Area in which the value of r , , = 0.5, , , , , , , , , , , r valuesmahavesimilarvalues,butonlthezonerectangular in plan has been identified. This stud identifies the specific zones within which the cutout centre ma be moved so that the loss of frequenc is less than 5% or 10%, respectivel,withrespecttoashellwithacentralcutout.thiswill help a practicing engineer to make a decision regarding the eccentricit of the cutout centre that can be allowed Mode Shapes. The mode shapes corresponding to the fundamental modes of vibration are plotted in Figures 6, 7,, and 9 for cross-pl and angle-pl 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 tpical results. All the mode shapes are bending modes. It is found that for different position of cutout, mode shapes are somewhatsimilar,onlthecrestandtroughpositionschange. The present stud considers the dnamic characteristics of stiffened composite conoidal shells with square cutout in terms of the natural frequenc 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-pl and angle-pl 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 dnamic characters of the conoid has not been considered in the present stud. Future studies will evaluate these aspects. 6. Conclusions The following conclusions are drawn from the present stud. (1) As this approach produces results in close agreement with those of the benchmark problems, the finite element code used here is suitable for analzing free vibration problems of stiffened conoidal roof panels with cutouts. The present stud reveals that cutouts with stiffened margins ma alwas safel be provided on shell surfaces for functional requirements. (2) The arrangement of boundar 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 epected to be ver useful in decision making for practicing engineers.

17 Journal of Engineering 17 Table 10: Maimum values of r with corresponding coordinates of cutout centre and zones where r 90and r 95for +45/ 45/+45/ 45 conoidal shells. Boundar condition Maimum values of r Co-ordinate of cutout centre CCCC = 0.5, = 0.5 CSCC = 0.5, = 0.4 CCSC = 0.5, = 0.5 CCCS = 0.5, = 0.6 CSSC = 0.5, = 0.3 CCSS = 0.5, = 0.7 CSCS = 0.6, = 0.5 SCSC 107. = 0.4, = 0.5 CSSS = 0.7, = 0.7 SSSC = 0.4, = 0.3 SSCS = 0.4, = 0.6 SSSS = 0.5, = 0. Area in which the value of r 90 = 0.4, = 0.3, ; , , = , , ; = 0., = 0.5, = 0.3, , = , = 0.3, , = 0.2, = 0.3, CS = 0.2, = 0.2 nil a/b = 1, a/h = 100, a /b =1, a/h h =5, h l /h h =0.25, E 11 /E 22 =25, G 23 =0.2E 22, G 13 =G 12 =0.5E 22,and] 12 = ] 21 =0.25. Area in which the value of r 95 = 0.5, , = , , , , , , , , , , , (3) The information regarding the behaviour of stiffened conoids with eccentric cutouts for a wide spectrum of eccentricit and boundar conditions for cross pl and angle pl shells ma also be used as design aids for structural engineers. Notations a, b: Lengthandwidthofshellinplan a,b : Length and width of cutout in plan b st : Width of stiffener in general b s,b s : WidthofX-andY-stiffeners, respectivel B s,b s : Strain-displacement matri of stiffener elements d st : Depth of stiffener in general d s,d s : DepthofX-andY-stiffeners, respectivel {d e }: Element displacement e: Eccentricities of both - and-direction stiffeners with respect to shell midsurface E 11,E 22 :Elasticmoduli G 12,G 13,G 23 : Shear moduli of a lamina with respect to 1, 2, and 3 aes of fibre h: Shellthickness h h : Higherheightofconoid h l : Lower height of conoid M,M : Moment resultants M : Torsion resultant np: Number of plies in a laminate N 1 N : Shapefunctions N,N : Inplane force resultants N : Inplane shear resultant Q,Q : Transverse shear resultant R,R : Radii of curvature and cross curvature of shell, respectivel u, V, w: Translational degrees of freedom,, z: Local coordinate aes X, Y, Z: Global coordinate aes z k : Distanceofbottomofthekth pl from midsurface of a laminate α,β: Rotational degrees of freedom

18 1 Journal of Engineering ε,ε : Inplane strain component φ: Angle of twist γ,γ z,γ z : Shearing strain components ] 12, ] 21 : Poisson s ratios ξ, η, τ: Isoparametric coordinates ρ: Densitofmaterial σ,σ : Inplane stress components τ,τ z,τ z :Shearingstresscomponents ω: Natural frequenc ω: Nondimensionalnatural frequenc =ωa 2 (ρ/e 22 h 2 ) 1/2. References [1] H. A. Hadid, An analtical and eperimental investigation into the bending theor of elastic conoidal shells [Ph.D. thesis], Universit of Southampton, [2] C. Brebbia and H. Hadid, Analsis of plates and shells using rectangular curved elements, CE/5/71 Civil engineering department, Universit of Southampton, [3] C. K. Choi, A conoidal shell analsis b modified isoparametric element, Computers and Structures, vol.1,no.5,pp , 194. [4] B.GhoshandJ.N.Bandopadha, Bendinganalsisofconoidal shells using curved quadratic isoparametric element, Computers and Structures,vol.33,no.3,pp ,199. [5] B.GhoshandJ.N.Bandopadha, Approimatebendinganalsis of conoidal shells using the galerkin method, Computers and Structures,vol.36,no.5,pp.01 05,1990. [6] A. De, J. N. Bandopadha, and P. K. Sinha, Finite element analsis of laminated composite conoidal shell structures, Computers and Structures,vol.43,no.3,pp ,1992. [7] A. K. Das and J. N. Bandopadha, Theoretical and eperimental studies on conoidal shells, Computers and Structures, vol. 49, no. 3, pp , []D.Chakravort,J.N.Bandopadha,andP.K.Sinha, Free vibration analsis of point-supported laminated composite doubl curved shells-a finite element approach, Computers and Structures, vol. 54, no. 2, pp , [9] D. Chakravort, J. N. Bandopadha, and P. K. Sinha, Finite element free vibration analsis of point supported laminated composite clindrical shells, Journal of Sound and Vibration, vol.11,no.1,pp.43 52,1995. [10] D. Chakravort, J. N. Bandopadha, and P. K. Sinha, Finite element free vibration analsis of doubl curved laminated composite shells, Journal of Sound and Vibration, vol. 191, no. 4, pp , [11] D. Chakravort, P. K. Sinha, and J. N. Bandopadha, Applications of FEM on free and forced vibration of laminated shells, Journal of Engineering Mechanics,vol.124,no.1,pp.1,199. [12] A. N. Naak and J. N. Bandopadha, On the free vibration of stiffened shallow shells, Journal of Sound and Vibration, vol. 255, no. 2, pp , [13] A.N.NaakandJ.N.Bandopadha, Freevibrationanalsis and design aids of stiffened conoidal shells, Journal of Engineering Mechanics,vol.12,no.4,pp ,2002. [14] A. N. Naak and J. N. Bandopadha, Free vibration analsis of laminated stiffened shells, Journal of Engineering Mechanics, vol. 131, no. 1, pp , [15] A.N.NaakandJ.N.Bandopadha, Dnamicresponseanalsis of stiffened conoidal shells, Journal of Sound and Vibration, vol. 291, no. 3 5, pp , [16] H. S. Das and D. Chakravort, Design aids and selection guidelines for composite conoidal shell roofs a finite element application, Journal of Reinforced Plastics and Composites, vol.26, no.17,pp ,2007. [17] H. S. Das and D. Chakravort, Natural frequencies and mode shapes of composite conoids with complicated boundar conditions, Journal of Reinforced Plastics and Composites, vol.27, no.13,pp ,200. [1] S. S. Hota and D. Chakravort, Free vibration of stiffened conoidal shell roofs with cutouts, JVC/Journal of Vibration and Control,vol.13,no.3,pp ,2007. [19] M. S. Qatu, E. Asadi, and W. Wang, Review of recent literature on static analses of composite shells: , Open Journal of Composite Materials,vol.2,pp.61 6,2012. [20]M.S.Qatu,R.W.Sullivan,andW.Wang, Recentresearch advances on the dnamic analsis of composite shells: , Composite Structures,vol.93,no.1,pp.14 31,2010. [21] V. V. Vasiliev, R. M. Jones, and L. L. Man, Mechanics of Composite Structures, Talor and Francis, New York, NY, USA, [22] M. S. Qatu, Vibration of Laminated Shells and Plates, Elsevier, London, UK, [23] S. Sahoo and D. Chakravort, Finite element bending behaviour of composite hperbolic paraboloidal shells with various edge conditions, Journal of Strain Analsis for Engineering Design,vol.39,no.5,pp ,2004.

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