Composite Structures 94 (2012) 2188–2196

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Composite Structures

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Free vibration analyses of multiple delaminated angle-ply compositeconical shells – A finite element approach

Sudip Dey ⇑, Amit KarmakarMechanical Engineering Department, Jadavpur University, Kolkata 700 032, India

a r t i c l e i n f o

Article history:Available online 14 February 2012

Keywords:Angle-plyConical shellMultiple delaminationFinite element

0263-8223/$ - see front matter � 2012 Elsevier Ltd. Adoi:10.1016/j.compstruct.2012.01.006

⇑ Corresponding author.E-mail address: infosudip@gmail.com (S. Dey).

a b s t r a c t

In this paper, the finite element method is employed to investigate the effects of delamination on freevibration characteristics of graphite–epoxy pretwisted shallow angle-ply composite conical shells. Thegeneralized dynamic equilibrium equation is derived from Lagrange’s equation of motion neglecting Cori-olis effect for moderate rotational speeds. The theoretical formulation is based on the Mindlin’s theoryand the multi-point constraint algorithm is considered for an eight noded isoparametric plate bendingelement. The standard eigenvalue problem is solved by applying the QR iteration algorithm. The modeshapes are also depicted for a typical laminate configuration. Non-dimensional natural frequenciesobtained are the first known results for the type of analyses carried out here.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Composite materials are advantageous in weight sensitiveapplications because of their high specific stiffness and strength.The increasing demand of composite materials is due to their highstrength to weight ratio with high compliance to design aspects aswell as their cost-effectiveness. Delamination or inter-laminardebonding is the most common feared mode of damage inthe composite structures. This damage can cause degradation ofstrength leading to structural instability and failure. Rotatingpretwisted conical shells with low aspect ratio can be idealizedas turbo-machinery blades (Fig 1). In general, all vibrations ofblades are closely related to their natural frequencies. In order toensure the safety of operation, a profound knowledge of these nat-ural frequencies and the dynamic behavior of turbomachineryblades are essential for the designers.

The pioneering work on pretwisted composite plates was car-ried out by Qatu and Leissa [1] to determine the natural frequen-cies of stationary plates using laminated shallow shell theoryusing Ritz method. Liew et al. [2] investigated on pretwisted coni-cal shell to find out the vibratory characteristics of stationary con-ical shell by using Ritz procedure and by using the same method,the first known three dimensional continuum vibration analysisincluding full geometric non-linearities and centrifugal accelera-tions in composite blades was carried out by McGee and Chu [3].Regarding delamination model, two worth mentioning investiga-tions were carried out. It included analytical and experimentaldetermination of natural frequencies of delaminated composite

ll rights reserved.

beam by Shen and Grady [4] and the second one dealt with finiteelement treatment of the delaminated composite cantilever beamand plate by Krawczuk et al. [5] for free vibration analyses. On theother hand, Lee et al. [6] exhibited the vibration analysis of twistedcantilevered conical composite shell by using finite element meth-od based on the Hellinger–Reissner principle. Later on, Karmakarand Kishimoto [7] analyzed the free vibration characteristics ofdelaminated composite cylindrical shells. There exists a good num-ber of references on numerical models and experimental investiga-tions of turbomachinery blades idealized according to theTimoshenko theory of beam. It has been found that the applicationof beam theory is far from straight forward and extracts limitedinformation only. Nabi and Ganesean [8] summarized the quanti-tative comparison of natural frequencies of metal matrix compos-ite pretwisted blades in stationary condition using beam and platetheories. Although delamination is one of the most feared damagemodes in laminated composites, the impact behavior of delaminat-ed structures has been addressed only in two investigations bySekine et al. [9] and Hu et al. [10] wherein simply supported plateswith single and multiple delamination were considered for theanalyses. Besides investigation on single delamination, significantwork also incurred on multiple delaminations. Considering multi-ple delaminations, failure analysis of composite plate due to bend-ing and impact was numerically investigated by Parhi et al. [11]using finite element method. Using the same tool, Aymerichet al. [12] simulated impacted cross-ply laminates based oncohesive interface elements. Of late, there are two worth mention-ing paper on free vibration characteristics of angle-ply. The firstone by Viswanathan and Kim [13] is on free vibration of anti-symmetric angle-ply-laminated plates including transverse sheardeformation by Spline method while the second one was carried

Fig. 1. Geometry of (a) Twisted plate and (b) Untwisted shallow conical shell model.

Fig. 2. Plate elements at a delamination crack tip.

S. Dey, A. Karmakar / Composite Structures 94 (2012) 2188–2196 2189

out by Khare et al. [14] for composite and sandwich laminates witha higher-order facet shell element. Later on, Patel et al. [15] carriedout a study to determine the post-buckling characteristics of angle-ply laminated truncated circular conical shells. In contrast, staticand vibration analysis of axi-symmetric angle-ply laminated cylin-drical shell using state space differential quadrature method wasinvestigated by Alibeigloo [16].

To the best of the authors’ knowledge, there is no literatureavailable which deals with multiple delaminated pretwisted rotat-ing composite angle-ply conical shells. To fill up this apparent void,the present analyses employed a finite-element based approach tostudy the free vibration characteristics of pretwisted multiple del-aminated graphite–epoxy angle-ply composite rotating shallowconical shells. The analyses are carried out using an eight-nodedisoparametric plate bending element considering the effects oftransverse shear deformation and rotary inertia based Mindlin’stheory. The undelaminated region is modeled by a single layer ofplate elements while the delaminated region is modeled usingtwo layers of plate elements whose interface contains the delami-nation. To ensure the compatibility of deformation and equilibriumof resultant forces and moments at the delamination crack front amulti-point constraint algorithm [17] is incorporated which leadsto anti-symmetric element stiffness matrices. The QR iterationalgorithm [18] is utilized to solve the standard eigenvalue problem.This paper presents a finite element based numerical approach todetermine the non-dimensional natural frequencies of multipledelaminated composite conical shells neglecting effect of dynamiccontact between delaminated layers.

2. Theoretical formulation

A shallow shell is characterized by its middle surface which isdefined by the equation [19],

z ¼ �12

x2

Rxþ 2

xyRxyþ y2

Ry

� �ð1Þ

where Rx and Ry denote the radii of curvature in the x and y direc-tions, respectively. The radius of twist (Rxy), length (L) of shell andtwist angle (W) are related as,

tan w ¼ � LRxy

ð2Þ

The dynamic equilibrium equation for moderate rotationalspeeds neglecting Coriolis effect is derived from Lagrange’s equa-tion of motion and equation in global form is expressed as [20],

½M�½€d� þ ð½K� þ ½Kr�Þfdg ¼ fFðX2Þg ð3Þ

where [M], [K], [Kr] are global mass, elastic stiffness and geometricstiffness matrices [21], respectively. {d} and {F(X2)} are global dis-placement vector and nodal equivalent centrifugal forces. [Kr] de-

pends on the initial stress distribution and is obtained by theiterative procedure [22] upon solving,

ð½K� þ ½Kr�Þd ¼ FðX2Þ ð4Þ

The natural frequencies (xn) are determined from the standardeigenvalue problem [18] which is represented below and is solvedby the QR iteration algorithm,

½A�fdg ¼ kfdg ð5Þwhere ½A� ¼ ð½K� þ ½Kr�Þ � 1½M� ð6Þk ¼ 1=x2

n ð7Þ

3. Multi-point constraint

Fig. 2 represents the cross-sectional view of a typical delamina-tion crack tip where nodes of three plate elements meet togetherto form a common node. The undelaminated region is modeled byplate element 1 of thickness h, and the delaminated region ismodeled by plate elements 2 and 3 whose interface contains thedelamination (h2 and h3 are the thicknesses of elements 2 and 3respectively). The elements 1, 2 and 3 are freely allowed to deformprior to imposition of constraints conditions. Nodal displacementsof elements 2, 3 at crack tip [17],

uj ¼ u0j � ðz� z0jÞhxj ð8Þ

v j ¼ v 0j � ðz� z0jÞhyj ð9Þ

wj ¼ w0j ðwhere; j ¼ 2;3Þ ð10Þ

where u0j, v 0j and w0j are the mid-plane displacements and z0j is the z-coordinate of mid-plane of element j and hx, hy are the rotationsabout x and y axes, respectively. The above equation also holdsgood for element 1 and z01 equal to zero. The transverse displace-ments and rotations at a common node have values expressed as,

w1 ¼ w2 ¼ w3 ¼ w ð11Þ

hx1 ¼ hx2 ¼ hx3 ¼ hx ð12Þ

hy1 ¼ hy2 ¼ hy3 ¼ hy ð13Þ

Table 2Non-dimensional fundamental frequencies [x = xn L2 p(qh/D)] of graphite–epoxycomposite rotating cantilever plate, L/bo = 1, h/L = 0.12, D = Eh3/{12(1 � m2)}, m = 0.3.

X Present FEM Sreenivasamurthy et al. [22]

2190 S. Dey, A. Karmakar / Composite Structures 94 (2012) 2188–2196

In-plane displacements of all three elements at crack tip areequal and they are related as [17],

u02 ¼ u01 � z02hx ð14Þ

0.0 3.4174 3.43680.2 3.4933 3.5185 v 02 ¼ v 01 � z02hy ð15Þ 0.4 3.7110 3.75280.6 4.0458 4.12870.8 4.4690 4.5678 u03 ¼ u01 � z03hx ð16Þ 1.0 4.9549 5.0916

Table 3Non-dimensional fundamental frequencies [x = xn b2

o

p(qh/D), D = Eh3/12(1 � m2)]

for the pretwisted shallow conical shell with m = 0.3, s/h = 1000, hv = 15� and ho = 30�.

w Aspect ratio(L/s)

Present FEM(8 � 8)

Present FEM(6 � 6)

Liew et al.[2]

0� 0.6 0.3524 0.3552 0.35990.7 0.2991 0.3013 0.30600.8 0.2715 0.2741 0.2783

30� 0.6 0.2805 0.2834 0.28820.7 0.2507 0.2528 0.25750.8 0.2364 0.2389 0.2417

Table 4NDFF [x = xn L2 p(q/E1h2)] of graphite–epoxy composite rotating cylindrical shellswith 25% delamination located at several positions [for 2 delamination: 0�/0�/30�//

v 03 ¼ v 01 � z03hy ð17Þ

where u01 is the mid-plane displacement of element 1. Equations of(11)–(17) relating the nodal displacements and rotations of ele-ments 1, 2 and 3 at the delamination crack tip, are the multipointconstraint equations used in finite element formulation to satisfythe compatibility of displacements and rotations. Mid-plane strainsbetween elements 2 and 3 related as [17],

fe0gj ¼ fe0g1 þ z0jfkg ð18Þ

where {e} and {k} represent the strain vector and the curvature vec-tor being identical at the crack tip for elements 1, 2 and 3, respec-tively. This equation can be considered as a special case forelement 1 and z01 is equal to zero. In-plane stress-resultants, {N}and moment resultants, {M} of elements 2 and 3 can be expressedas,

fNgj ¼ ½A�jfe0g1 þ ðz0j½A�j þ ½B�jÞfkg ðwhere j ¼ 2;3Þ ð19Þ

�30�/�30�/30�/0�//0� and for 3 delamination: 0�//0�/30�/�30�//�30�/30�//0�/0�,where // indicates the location of delamination] across the thickness, a/b = 1, b/h = 100, b/Ry = 0.5.

X Two delaminations Three delaminations

PresentFEM

Karmakar et al.[24]

PresentFEM

Karmakar et al.[24]

0.0 1.9316 1.9332 1.9048 1.89940.5 2.1080 2.1921 2.0469 2.14951.0 2.6735 2.7475 2.5996 2.6874

fMgj ¼ ½B�jfe0g1 þ ðz0j½B�j þ ½D�jÞfkg ðwhere j ¼ 2;3Þ ð20Þ

where [A], [B] and [D] are the extension, bending-extension cou-pling and bending stiffness coefficients, respectively. Thus the for-mulation based on the multi-point constraints condition leads tounsymmetric stiffness matrix. The resultant forces and momentsat the delamination front for the elements 1, 2 and 3 satisfy the fol-lowing equilibrium conditions,

fNg ¼ fNg1 ¼ fNg2 þ fNg3 ð21Þ

fMg ¼ fMg1 ¼ fMg2 þ fMg3 þ z02fNg2 þ z03fNg3 ð22Þ

fQg ¼ fQg1 ¼ fQg2 þ fQg3 ð23Þ

where {Q} denotes the transverse shear resultants. An eight nodedisoparametric quadratic plate bending element with five degreesof freedom at each node (three translations and two rotations) isemployed wherein the shape functions are as follows [18],

Ni ¼ ð1þ eeiÞð1þ ggiÞðeei þ ggi � 1Þ=4 ðfor i ¼ 1;2;3;4Þ ð24Þ

Ni ¼ ð1� e2Þð1þ ggiÞ=2 ðfor i ¼ 5;6Þ ð25Þ

Ni ¼ ð1� g2Þð1þ eeiÞ=2 ðfor i ¼ 6;8Þ ð26Þ

where g and e are the local natural coordinates of the element.

Table 1Non-dimensional fundamental natural frequencies [x = xn L2 p(q/E1h2)] of threelayered [h, �h, h] graphite–epoxy twisted plates, L/b = 1, b/h = 20, w = 30�.

Fiber orientation angle, h Present FEM Qatu and Leissa [1]

15� 0.8618 0.875930� 0.6790 0.692345� 0.4732 0.483160� 0.3234 0.3283

Fig. 3. Influence of the relative position of delamination on the first naturalfrequency of the composite cantilever beam [5].

4. Results and discussion

Non-dimensional natural frequencies for conical shells (Rx =1)having a square plan-form (L/bo = 1), curvature ratio (bo/Ry = 0.5)

Table 5Comparisons between NDFF and NDSF [x = xn L2 p(q/E1h2)] of graphite–epoxy angle-ply composite conical shells with delamination for various twist angles and fiberorientation angle at different rotational speeds (X) considering n = 8, nd = 1, h = 0.004, s/h = 1000, a/L = 0.33, d/L = 0.5, L/s = 0.7, ho = 45�, hv = 20�.

W h NDFF at nd = 4 NDSF at nd = 1

NDa Mid-plane delamination ND Mid-plane delamination

X = 0.0 X = 0.5 X = 1.0 X = 0.0 X = 0.5 X = 1.0

0� 0� 0.7956 0.7443 0.8238 1.0261 2.1804 1.7205 1.7991 2.015415� 0.7242 0.6750 0.7451 0.9267 1.9848 1.5587 1.6282 1.822430� 0.5244 0.4865 0.5262 0.6440 1.4540 1.1569 1.2020 1.332245� 0.3473 0.3229 0.3007 1.0006 0.9687 0.7802 0.7984 1.248360� 0.2541 0.2382 0.3023 0.2089 0.7073 0.5701 0.6002 0.636775� 0.2157 0.2032 0.1868 0.5295 0.5993 0.4820 0.4966 1.022690� 0.2059 0.1940 0.1584 0.5052 0.5718 0.4598 0.4723 0.9782

15� 0� 0.7776 0.7181 0.7908 0.9827 2.1545 1.6922 1.7658 1.969315� 0.7025 0.6431 0.7006 0.8637 1.9522 1.5056 1.5635 1.734930� 0.5146 0.4732 0.4888 0.5336 1.4346 1.1289 1.1575 1.234645� 0.3428 0.3176 0.6882 0.3095 0.9581 0.7674 1.5530 0.834760� 0.2512 0.2351 0.2510 0.4506 0.7002 0.5625 0.5835 0.687275� 0.2133 0.2008 0.1638 0.5022 0.5935 0.4762 0.4874 1.016990� 0.2037 0.1920 0.4490 0.5033 0.5666 0.4551 0.7936 0.9718

30� 0� 0.7225 0.6490 0.7067 0.8747 2.0781 1.6055 1.6666 1.840415� 0.6558 0.5841 0.6149 0.7349 1.8748 1.4071 1.4487 1.583330� 0.4909 0.4460 0.3275 0.4866 1.3856 1.0716 1.0501 1.172545� 0.3300 0.3048 0.3245 1.8691 0.9290 0.7371 0.7620 3.283460� 0.2422 0.2267 0.1399 0.1208 0.6797 0.5424 0.5412 0.583475� 0.2055 0.1935 0.0777 0.4643 0.5761 0.4597 0.4600 0.994390� 0.1962 0.1851 0.4358 0.1389 0.5500 0.4398 0.9128 0.5129

45� 0� 0.6265 0.5509 0.5927 0.7345 1.9430 1.4539 1.4986 1.634515� 0.5746 0.4984 0.4886 1.2243 1.7429 1.2550 1.2751 2.694430� 0.4439 0.3972 0.4149 0.9391 1.2988 0.9738 0.9993 2.108945� 0.3032 0.2790 0.0680 0.4651 0.8760 0.6824 0.6602 0.812160� 0.2229 0.2089 0.4625 0.4601 0.6422 0.5055 1.0797 1.114875� 0.1887 0.1778 0.4053 0.4043 0.5442 0.4289 0.9195 0.951690� 0.1798 0.1696 0.4000 0.3962 0.5195 0.4108 0.8809 0.9104

a No delamination.

Table 6Variation of NDFF and NDSF [x = xn L2 p(q/E1h2)] for delaminated angle-ply composite conical shells for various twist angles and fiber orientation at different rotational speedsconsidering n = 8, nd = 4, h = 0.004, s/h = 1000, a/L = 0.33, d/L = 0.5, L/s = 0.7, ho = 45�, hv = 20�.

W h NDFF at nd = 4 NDSF at nd = 4

ND Mid-plane delamination ND Mid-plane delamination

X = 0.0 X = 0.5 X = 1.0 X = 0.0 X = 0.5 X = 1.0

0� 0� 0.7956 0.6989 0.7776 0.9745 2.1804 1.1482 1.1593 1.325915� 0.7242 0.5892 0.6421 0.8231 1.9848 0.7576 0.7879 0.956830� 0.5244 0.4219 0.4514 0.5555 1.4540 0.8432 0.8503 0.907545� 0.3473 0.3201 0.2927 0.6910 0.9687 0.7480 0.7461 0.906160� 0.2541 0.2249 0.2661 0.1884 0.7073 0.4937 0.5148 0.553675� 0.2157 0.1945 0.1740 0.4593 0.5993 0.4193 0.4331 0.725390� 0.2059 0.1855 0.1462 0.4476 0.5718 0.4018 0.4141 0.8537

15� 0� 0.7776 0.6258 0.6839 0.8537 2.1545 1.2931 1.3082 1.425015� 0.7025 0.6163 0.6649 0.8139 1.9522 1.1342 1.1713 1.312530� 0.5146 0.4469 0.4856 0.5641 1.4346 0.5190 0.6719 1.016045� 0.3428 0.3198 0.2896 0.3473 0.9581 0.8403 0.7352 0.788360� 0.2512 0.2251 0.2461 0.1269 0.7002 0.4898 0.5142 0.562575� 0.2133 0.1933 0.1674 0.2952 0.5935 0.4179 0.4321 0.492390� 0.2037 0.1823 0.2692 0.4414 0.5666 0.3967 0.4156 0.8353

30� 0� 0.7225 0.5163 0.5532 0.6830 2.0781 1.2528 1.2952 1.425815� 0.6558 0.5469 0.5632 0.6492 1.8748 1.2563 1.3036 1.422030� 0.4909 0.4542 0.4493 0.4736 1.3856 0.9129 1.0169 1.253045� 0.3300 0.2986 0.3111 0.7767 0.9290 0.5508 0.6024 1.295160� 0.2422 0.2209 0.1904 0.5530 0.6797 0.4756 0.4954 1.041675� 0.2055 0.1909 0.1448 0.6309 0.5761 0.4052 0.4202 0.818190� 0.1962 0.1718 0.1009 0.4247 0.5500 0.3800 0.3794 0.7892

45� 0� 0.6265 0.4222 0.4424 0.5407 1.9430 1.0923 1.1229 1.224715� 0.5746 0.4634 0.4380 0.6576 1.7429 1.1288 1.1530 1.303830� 0.4439 0.6719 0.6350 0.7773 1.2988 0.9446 1.2724 1.772745� 0.3032 0.3374 0.3496 0.0232 0.8760 0.5887 0.6426 0.543660� 0.2229 0.1872 0.3502 0.2911 0.6422 0.3937 0.4628 0.518575� 0.1887 0.1877 0.0799 0.4962 0.5442 0.3865 0.4091 0.700590� 0.1798 0.1515 0.1556 0.0288 0.5195 0.3471 0.3595 0.4157

S. Dey, A. Karmakar / Composite Structures 94 (2012) 2188–2196 2191

Fig. 4. Arrangement of layers of eight layered angle-ply composite withdelamination.

2192 S. Dey, A. Karmakar / Composite Structures 94 (2012) 2188–2196

and thickness ratio (s/h = 1000) are obtained corresponding to dif-ferent speeds of rotation X = 0.0, 0.5 and 1.0 (where X = X0/xo)and relative distance, d/L = 0.33, 0.5 and 0.66, considering three dif-ferent angles of twist, namely w = 15�, 30� and 45�, in addition to theuntwisted one (w = 0�). Parametric studies are carried out with re-spect to twist angles, location of delamination and rotational speedson natural frequencies of angle-ply composite shallow conicalshells. The parameters n, nd, X0, xo, q, L, bo, h, d, a, hv and ho representthe number of layers, number of delaminations, actual angularspeed of rotation, fundamental natural frequency of a non-rotatingshell, density, length, reference width, thickness, distance of thecenterline of delamination from the clamped (fixed) end, cracklength, vertex angle and base subtended angle of cone, respectively.The finite element formulation employs eight noded plate bendingelement with five degrees of freedom at each node. Material proper-ties of graphite–epoxy composite [23] considered as E1 = 138.0 GPa,E2 = 8.96 GPa, m = 0.3, G12 = 7.1 GPa, G13 = 7.1 GPa, G23 = 2.84 GPa.Convergence studies are performed using uniform mesh divisionof (6 � 6) and (8 � 8) and the results are found to be nearly equal,with the difference being around 1%. and the results also corrobo-rate monotonic downward convergence. The slight differences be-tween the value of present solution and those of Liew et al. [2] canbe attributed to consideration of transverse shear deformationand rotary inertia in the present FEM and also to the fact that Ritzmethod overestimates the structural stiffness of the conical shell([2]). Moreover, increasing the size of matrix because of highermesh size increases the ill-conditioning of the numerical eigenvalueproblem ([1]). Hence, the lower mesh size (6 � 6) consisting of 36elements and 133 nodes, has been used for the analysis due tocomputational efficiency.

4.1. Validation of results

Based on the present finite element modeling, computer codesare developed and the results obtained are compared and validatedwith those published in open literature [1,22,2,24,5] as furnishedin Tables 1–4 and in Fig 3. The comparative study depicts an excel-lent agreement with the previously published results and hence itdemonstrates the capability of the computer codes developed andinsures the accuracy of analyses. Table 1 presents the non-dimen-sional fundamental frequencies of graphite–epoxy compositetwisted plates with different fiber-orientation angle [1], while Ta-ble 2 furnishes the non-dimensional fundamental natural frequen-cies of graphite–epoxy composite rotating cantilever plate [22].Table 3 presents non-dimensional fundamental frequencies oftwisted conical shells [2], while Table 4 furnishes the non-dimen-sional fundamental natural frequencies of graphite–epoxy

Table 7NDFF and NDSF [x = xn L2 p(q/E1h2)] of angle-ply [45�/�45�/45�/�45�]s composite coniL = 0.33, d/L = 0.33 and 0.66, h = 0.004, s/h = 1000, L/s = 0.7, ho = 45�, hv = 20�.

W X NDFF

d/L = 0.33

0� 0 0.26550.5 0.11431 0.2782

15� 0 0.27930.5 0.72501 0.2926

30� 0 0.27160.5 0.30741 0.1855

45� 0 0.26760.5 0.26591 0.2347

composite with eight layered bending stiff [0�/0�/30�/�30�]s rotat-ing cylindrical shells with 25% double and triple delaminations[24]. The span-wise variation of fundamental frequency of com-posite cantilever beam with relative position of delamination [5]is illustrated in Fig 3.

4.2. Effect of stacking sequence

Non-dimensional fundamental natural frequencies (NDFF) andnon-dimensional second natural frequencies (NDSF) of eight lay-ered graphite–epoxy composite rotating conical shells for a partic-ular size and number of the delamination is presented in Tables 5and 6 for different twist angles. Some aspects of the results are alsopresented in graphical form in Figs. 5. The shells with 33% mid-plane delamination centered at a relative distance, d/L = 0.33 fromthe fixed end are considered for the analyses. In general, the fre-quency parameters at stationary conditions are found to decreasewith increase in the twist angle for all fiber orientations. At station-ary condition, delaminated non-dimensional fundamental fre-quencies are found lower than the corresponding undelaminatedNDFF irrespective of twist angle and fiber orientation. In all thecases of both twisted and untwisted non-rotating delaminatedconical shells, non-dimensional fundamental frequencies are foundto be minimum at h = 90�. For untwisted delaminated conical shellsat stationary condition, the non-dimensional fundamental fre-quencies are observed to attain the maximum value for h = 15�and gradually decrease to a minimum value for h = 90�.

cal shells by varying relative position of delamination along the span considering a/

NDSF

d/L = 0.66 d/L = 0.33 d/L = 0.66

0.3271 0.7380 0.60580.3084 0.6907 0.62390.4578 0.7867 1.3828

0.3239 0.7399 0.60320.2374 1.0525 0.60350.4633 0.8488 0.7561

0.3153 0.6582 0.59300.4956 0.6920 0.75560.3573 0.7410 0.6998

0.2976 0.6589 0.57510.3077 0.6857 0.59990.2472 0.7644 0.6888

Ω nd 1= nd 4=

0.5

1.0

Fig. 5. Variation of relative frequencies (NDFF) of graphite–epoxy angle-ply composite conical shells at X = 0.5 and 1.0, with respect to fiber orientation, considering n = 8,h = 0.004, s/h = 1000, a/L = 0.33, L/s = 0.7, ho = 45�, hv = 20�.

S. Dey, A. Karmakar / Composite Structures 94 (2012) 2188–2196 2193

For both single and quadruple delamination at stationary condi-tion, maximum values of NDFF and NDSF of undelaminated angle-ply composite conical shells are identified at h = 0� (for w = 0�, 15�,30� and 45�), while minimum values invariably observed at h = 90�irrespective of twist angles, wherein the differences betweenmaximum and minimum values of non-dimensional naturalfrequencies are found 74.1%, 73.8%, 72.8% and 71.3% (for NDFF)and 73.8%, 73.7%, 73.5% and 73.3%, (for NDSF) corresponding tow = 0�, 15�, 30� and 45� respectively. But in contrast, both NDFFand NDSF for delaminated cases at stationary condition, maximumvalues of non-dimensional natural frequencies are identified ath = 0� invariant of twist angle except at h = 15� corresponding tow = 30� and 45� for nd = 4, while minimum values consistently de-picted at h = 90� irrespective of twist angles, wherein the differ-ences between maximum and minimum values of NDFF arefound 73.9%, 73.3%, 71.5% and 69.2% (for nd = 1) and 73.5%, 70.9%,68.6% and 77.5% (for nd = 4) corresponding to w = 0�, 15�, 30� and45� respectively. Similarly, for delaminated cases at stationarycondition, the difference between maximum and minimum valuesof NDSF are identified are obtained as 73.3%, 73.1%, 72.6% and71.7% (for nd = 1) and 65.0%, 69.3%, 69.8% and 69.3% (for nd = 4) cor-responding to w = 0�, 15�, 30� and 45� respectively. From Tables 5and 6, it is observed that the effect of centrifugal stiffening (i.e.,increase of structural stiffness with increase of rotational speed)is predominantly found with reference to NDFF in case of h = 0�irrespective of twist angles. The trend of relative frequencies ofNDFF (ratio of delaminated frequency at X = 0.5 or 1.0 and delami-nated frequency at X = 0.0) is shown in Fig. 5. A specific trendagainst the variation of fiber orientation is depicted in respect of

the deviation in non-dimensional fundamental frequency valuesof delaminated non-rotating shells compared to undelaminatedone and maximum deviation for NDFF is identified at h = 0� irre-spective of twist angles except at W = 45� (nd = 1) and at W = 0�(nd = 4). At rotating condition, non-dimensional fundamental andsecond natural frequencies have a typical trend for both delami-nated and undelaminated cases. At lower rotational speed(X = 0.5), the difference between maximum and minimum valuesof NDFF with single mid-plane delamination are observed 80.8%,79.3%, 89.0% and 88.5% corresponds to W = 0�, 15�, 30� and 45�respectively, while the same for NDFF with quadruple delamina-tions found to be 81.2%, 75.5%, 82.1% and 87.4%, corresponding toW = 0�, 15�, 30� and 45� respectively. In the contrast, at higher rota-tional speed (X = 1.0), the difference between maximum and min-imum values of NDFF at nd = 1 are observed 79.6%, 68.5%, 93.5% and67.6% corresponds to W = 0�, 15�, 30� and 45� respectively, whilefor NDFF at nd = 4, the difference found to be 80.7%, 85.1%, 45.3%and 97.0%, corresponding to W = 0�, 15�, 30� and 45� respectively.

4.3. Delamination along span

A delamination of relative length a/L = 0.33 is considered andthis is centered at a relative distances of d/L = 0.33 and 0.66 fromthe fixed end (Table 7). The delamination is considered at the inter-face of 45� and �45� layers (nd = 4 of Fig. 4). At stationary condi-tion, NDFF are observed to decrease with the increase of twistangle. At stationary condition, NDFF are predominantly found toreduce as the delamination moves towards the free end except atX = 0.5 for w = 15� while the reverse trend is substantially noted

Ψ = 0° Ψ = 15°

Ψ = 30° Ψ = 45°

Fig. 6. Variation of NDFF of angle-ply [45�/�45�/45�/�45�]s composite conical shells by varying relative position of delamination across thickness (h0/h) considering n = 8,nd = 1, h = 0.004, s/h = 1000, a/L = 0.33, d/L = 0.5, L/s = 0.7, ho = 45�, hv = 20�.

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for NDSF except at X = 1.0 (w = 0�) and X = 0.5 (w = 30�). The cen-trifugal stiffening effects are identified for NDSF at all the twist an-gles of d/L = 0.67 except w = 30� while the same holds good for d/L = 0.33 at w = 30� and 45�. For both d/L = 0.33 and 0.67, centrifugalstiffening effect is found absent corresponding to NDFF.

4.4. Delamination across thickness

From Fig. 6, it is observed that NDFF attains a minimum value atrelative position, h0/h = 0.5 irrespective of twist angle. In otherwords, minimum value of NDFF is identified when delaminationis located at the mid-plane for any values of twist angle. For a par-ticular relative position of the delamination across the thickness, itis also noted that the non-dimensional fundamental natural fre-quencies are found to decrease with the increase in the angle oftwist from 0� to 45�. Irrespective of twist angle, it shows a specifictrend i.e., as the location of delamination changes, the frequencyvalue gradually reduces to minimum at the mid-plane from thetop surface and then rises towards the bottom surface.

4.5. Mode shapes

The mode shapes corresponding to NDFF for a typical laminateconfiguration are shown in Fig. 7. The other parameters consideredare as mentioned in Table 5. It is identified that the symmetrymodes are absent when twist angle is non-zero and the nodal linesindicate with zero displacement amplitude. The first span wisebending is predominantly observed for both untwisted and twistedconical shell at stationary condition corresponding to its funda-mental frequency. The effect of first span wise bending modeincreases with fiber orientation angle for both twisted and un-twisted cases in respect of fundamental frequency. The dominance

of torsion mode is found at h = 0� to 45� irrespective of twist angleexcept at h = 0� and 45� (for w = 0�) and at h = 30� and 45� (forw = 30�) wherein span wise bending modes are identified.

5. Conclusions

In general, the frequency parameters at stationary conditionsare found to decrease with increase in the twist angle for fiber ori-entations. At stationary condition, delaminated non-dimensionalfundamental and second natural frequencies are found to be lowerthan the corresponding undelaminated one irrespective of twistangle and fiber orientation. In all the cases of both twisted and un-twisted non-rotating delaminated conical shells, non-dimensionalfundamental frequencies are found to be minimum at h = 90�.The effect of centrifugal stiffening (i.e., increase of structural stiff-ness with increase of rotational speed) is predominantly foundwith reference to non-dimensional fundamental natural frequencyat h = 0� irrespective of twist angles. Non-dimensional fundamen-tal natural frequencies at stationary condition are observed to de-crease with the increase of twist angle corresponding to locationsof delaminations near the fixed end as well as near the free end.The non-dimensional fundamental natural frequencies for twistedconical shell at stationary condition increases as the delaminationmoves towards the free end while non-dimensional second naturalfrequencies of stationary twisted conical shells decreases as thedelamination moves towards the free end. The non-dimensionalfundamental and second natural frequencies attain a minimum va-lue when the delamination is located at the mid-plane (i.e., be-tween fourth and fifth layers) for all values of the twist angle.The non-dimensional fundamental natural frequencies decreasewith the increase of twist angle for a particular relative positionof the delamination across the thickness.

θ ψ = 0° ψ = 15° ψ = 30° ψ = 45°

0°

15°

30°

45°

60°

75°

90°

Fig. 7. Effect of twist and rotational speeds on mode shapes of NDFF [x = xn L2p(q/E1h2)] of graphite–epoxy symmetric angle-ply (h/�h/h/�h)s composite conical shells withdelaminations for different twist angles, considering n = 8, nd = 4, h = 0.004, a/L = 0.33, d/L = 0.5, s/h = 1000, L/s = 0.7, ho = 45�, hv = 20�.

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The mode shapes predominantly show the effect of first spanwise bending mode for both twisted and untwisted conical shellsin respect of fundamental frequency wherein an exception of tor-sion mode is found at h = 0–45� irrespective of twist angle exceptat h = 0� and 45� for w = 0� and at h = 30� and 45� for w = 30�wherein span wise bending modes are identified. The non-dimen-sional frequencies obtained are the first known results of the typeof analyses carried out here and the results could serve as referencesolutions for future investigators.

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