S. Dey and A. Karmakar / Journal of Mechanical Science and Technology 27 (4) (2013) 963~972 965
The dynamic equilibrium equation for moderate rotational
speeds neglecting Coriolis effect is derived employing La-
grange’s equation of motion and equation in global form is
expressed as [16],
[M] δ ´´ + ([K] + [Kσ]) δ = F(Ω2) + F (11)
where F(Ω2) is the nodal equivalent centrifugal forces and δ is the global displacement vector. [Kσ] depends on the initial stress distribution and is obtained by the iterative proce-dure [17] upon solving,
([K] + [Kσ]) δ = F(Ω2). (12)
The natural frequencies (ωn) are determined from the stan-
dard eigenvalue problem [14] which is represented below and is solved by the QR iteration algorithm,
[A] δ = λ δ (13)
where [A] = ([K] + [Kσ]) - 1 [M] (14)
and λ = 1 / ωn2. (15)
3. Multi-point constraint
Fig. 2 represents the cross-sectional view of a typical de-lamination crack tip where nodes of three plate elements meet together to form a common node. The undelaminated region is modelled by plate element 1 of thickness h, and the delami-nated region is modelled by plate elements 2 and 3 whose interface contains the delamination (h2 and h3 are the thick-nesses of the elements 2 and 3 respectively). The elements 1, 2 and 3 are freely allowed to deform prior to imposition of the constraints conditions. The nodal displacements of elements 2 and 3 at crack tip are expressed as [13]
where z′j is the z-coordinate of mid-plane of element j and the above equation also holds good for element 1 and z′1 equal to zero. The transverse displacements and rotations at a common node have values expressed as,
w1 = w2 = w3 = w (19) θx1 = θx2 = θx3 = θx (20) θy1 = θy2 = θy3 = θy. (21) In-plane displacements of all three elements at crack tip are
where u′1 is the mid-plane displacement of element 1. Eqs. (19)-(25) relating the nodal displacements and rotations of elements 1-3 at the delamination crack tip, are the multipoint constraint equations used in finite element formulation to sat-isfy the compatibility of displacements and rotations. Mid-plane strains between elements 2 and 3 are related as,
ε′j = ε′1 + z′j k (26)
where ε represents the strain vector and k is the curvature vector being identical at the crack tip for elements 1-3. This equation can be considered as a special case for element 1 and z′1 is equal to zero. In-plane stress-resultants, N and mo-ment resultants, M of elements 2 and 3 can be expressed as,
Nj = [A]j ε′1 + (z′j [A]j + [B]j) k (27) Mj = [B]j ε′1 + (z′j [B]j + [D]j) k (where j = 2, 3). (28) The elasticity matrix for the t-th sublaminate is then modi-
fied and is expressed in the form [9]
0
[ ] 0
0 0
oA z A Btij ij ij
oD B z A Dt tij ij ij
Sij
t
+ = +
(29)
/2[ ] [ ]
/2
oh ZtA Q dzotij h Zt
+= ∫
− + (30)
/ 2[ ] [ ] ( )/2
/2[ ] [ ]
/ 2
oh Z otB Q z z dztij o th Ztoh Z ot Q z dz z Ao t tijh Zt
+= −∫− +
+= −∫
− +
(31)
/2 2[ ] [ ]/2
/2 22 [ ] ( ) [ ]/ 2
oh ZtD Q z dztij oh Ztoh Zo otz Q z dz z Aot t tijh Zt
+= −∫
− +
++∫
− +
(32)
where , 1,2,6i j =
Fig. 2. Plate elements at a delamination crack tip.
966 S. Dey and A. Karmakar / Journal of Mechanical Science and Technology 27 (4) (2013) 963~972
/ 2 2[ ] [ ] ( )/2
oh Z otS Q z z dztij o th Zt
+= −∫− +
(33)
where [ ]Q is the transformed reduced stiffness as defined in Ref. [18] while o
tz is the z-co-ordinate of mid-plane of t-th
sublaminate and h is the thickness of the t-th sub-laminate. Thus, the formulation based on the multi-point constraints condition leads to unsymmetric stiffness matrix. The resultant forces and moments at the delamination front for the elements 1-3 satisfy the following equilibrium conditions,
where Q denotes the transverse shear resultants. An eight noded isoparametric quadratic plate bending element of five degrees of freedom at each node (three translation and two rotations) is employed where shape functions are as follows [14],
Ni = (1 + ξ ξi) (1 + η ηi) (ξ ξi + η η i - 1) / 4 (37) (for i=1, 2, 3, 4) Ni = (1 - ξ2) (1 + η ηi) / 2 (for i = 5, 7) (38) Ni = (1 - η2) (1 + ξ ξi) / 2 (for i = 6, 8). (39)
4. Results and discussion
Non-dimensional natural frequencies for conical shells (Rx = ∞) having a square plan-form (L/bo = 1), curvature ratio (bo/Ry) of 0.5 and thickness ratio (s/h) of 1000 are obtained corresponding to different speeds of rotation Ω = 0.0, 0.5 and 1.0 (where Ω = Ω´/ωo) and relative distance, d/L = 0.33, 0.5 and 0.66, considering three different angles of twist, namely ψ = 15°, 30° and 45°, in addition to the untwisted one (ψ = 0°). The finite element formulation employed an eight-noded isoparametric plate bending element with five degrees of free-dom at each node. Material properties of graphite-epoxy com-posite [19, 20] are considered as E1 = 138.0 GPa, E2 = 8.96 GPa, ν12 = 0.3, G12 = 7.1 GPa, G13 = 7.1 GPa, G23 = 2.84 GPa. Convergence studies are also performed to determine the con-verged mesh size (Table 3). It is observed from the conver-gence study that uniform mesh divisions of (6 x 6) and (8 x 8) considering the complete planform of the shell provide nearly equal results the difference being around one percent (1%) and the results also corroborate monotonic downward conver-gence. The slight differences between the value of present solution and those of Liew et al. [5] can be attributed to con-sideration of transverse shear deformation and rotary inertia in the present FEM and also to the fact that Ritz method always overestimates the structural stiffness. Moreover, increasing the size of matrix because of higher mesh size increases the ill-conditioning of the numerical eigen value problem. Hence, the lower mesh size (6 x 6) consisting of 36 elements and 133 nodes, has been used for the analysis due to computational
efficiency. The total number of degrees of freedom involved in the computation is 665 as each node of the isoparametric plate bending element is having five degrees of freedom com-prising of three translations and two rotations. A comparative study is also conducted between bending stiff and torsion stiff composite configuration.
4.1 Validation
Based on finite element, the computer codes are developed. The results obtained are validated with the results of published literatures [4, 5, 8, 17] as furnished in Tables 1-3 and Fig. 3, respectively. The numerical results show an excellent agree-ment with the previously published results and hence, it dem-onstrates the capability of the computer codes developed and proves the accuracy of the analyses. Table 1 furnishes non-dimensional fundamental frequencies of graphite-epoxy com-posite twisted plates with different fibre-orientation angles [4], while Table 2 presents the convergence study for NDFF of graphite-epoxy composite pretwisted shallow conical shells [5]. Fig. 3 represents the validation of fundamental natural
Fibre orientation angle, θ Present FEM Qatu and Leissa [4]
15° 0.8618 0.8759
30° 0.6790 0.6923
45° 0.4732 0.4831
60° 0.3234 0.3283
Table 2. Convergence study for NDFF [ω = ωn bo
2 √(ρh/D), where D = Eh3/12(1-ν2)] of the pretwisted shallow conical shells, considering ν = 0.3, s/h = 1000, θv = 15°, θo = 30°.
Ψ L/s Present FEM
(8 x 8) Present FEM
(6 x 6) Liew et al. [5]
0.6 0.3524 0.3552 0.3599
0.7 0.2991 0.3013 0.3060 0°
0.8 0.2715 0.2741 0.2783
0.6 0.2805 0.2834 0.2882
0.7 0.2507 0.2528 0.2575 30°
0.8 0.2364 0.2389 0.2417
Table 3. NDFF of graphite-epoxy composite isotropic rotating cantile-ver plate with L/b = 1, h/L = 0.12, D = Eh3/12(1-ν2), ν12 = 0.3.
Ω Present FEM Sreenivasamurthy and
Ramamurti [17]
0.0 3.4174 3.4368
0.2 3.4933 3.5185
0.4 3.7110 3.7528
0.6 4.0458 4.1287
0.8 4.4690 4.5678
1.0 4.9549 5.0916
S. Dey and A. Karmakar / Journal of Mechanical Science and Technology 27 (4) (2013) 963~972 967
frequencies at different relative positions (spanwise) of de-lamination of graphite-epoxy composite cantilever beam [8]. On the other hand, Table 3 presents the non-dimensional fun-damental natural frequencies of isotropic flat rotating cantile-ver plate [17].
4.2 Effect of stacking sequence
In general, at stationary condition, non-dimensional funda-mental frequencies decrease with the increase of twist angle for both delaminated and undelaminated bending and torsion stiff configuration. From Table 4, it is observed that at station-ary condition, non-dimensional fundamental natural frequen-cies (NDFF) of both bending stiff and torsion stiff configura-tions attained maximum value for twist angle ψ = 0° and gradually decreased to a minimum value for twist angle ψ = 45° in case of no delamination (ND) and with delamination. It is also observed that delaminated non-dimensional fundamen-tal frequencies are found lower than undelaminated one at stationary condition irrespective of twist angle. At rotating condition, non-dimensional fundamental frequencies of de-laminated composite laminates with bending stiff configura-tion found drooping trend with the increase of twist angle, while a typical trend is exhibited with increase of twist angle for torsion stiff configuration. The centrifugal stiffening effect (i.e., increase of structural stiffness with increase of rotational speed because of greater contribution of geometric stiffness arising out of centrifugal rotation) is predominantly found with reference to non-dimensional fundamental frequencies of bending stiff composites irrespective of twist angle, while the same is not identified for torsion stiff except for ψ = 15°. Ta-ble 5 presents the non-dimensional second natural frequencies (NDSF) at stationary condition for bending stiff and torsion stiff configurations wherein the same attained maximum value for twist angle ψ = 15° and gradually decreased to a minimum value at ψ = 45° for both no delamination (ND) and with de-lamination cases. At stationary condition, the delaminated NDSF are found lower than undelaminated one irrespective of twist angle. At rotating condition, delaminated NDSF of bend-ing stiff configuration at Ω = 0.5 shows similar trend that of
undelaminated case, while there is no significant trend found for torsion stiff configuration. In case of NDSF of bending stiff, the centrifugal stiffening effect is observed at ψ = 30° and ψ = 45°, while the same is not identified for torsion stiff laminates.
4.3 Effect of delamination across thickness
In all three laminates under single delamination across thickness, the mid-plane (i.e., at h´/h = 0.5) is identified as the plane of symmetry depicting the mirror image values of NDFF corresponding to relative position of delamination at stationary condition as also observed by Krawczuk et al. [12]. The variations of NDFF due to single delamination considered across the thickness corresponding to torsion stiff [ ± 45°/ m 45°]s and bending stiff [0°2/ ± 30°]s graphite-epoxy com-posite conical shells at stationary condition are furnished in Fig. 4 wherein NDFF is found to attain a minimum value at relative position, h´/h = 0.5 for twisted cases of torsion stiff and bending stiff laminates. It is also noted that for untwisted cases of torsion stiff laminate, the value of NDFF is found to be almost invariant with a lower value compared to untwisted cases of bending stiff. The maximum values of NDFF as noted at two free ends for both twisted and untwisted cases of all three laminates establishes the fact that undelaminated values are higher than the delaminated cases as expected. For a particular value of relative position of the delamination across the thickness, NDFF are found to decrease with the increase in the angle of twist from 0° to 45° for both bending
Fig. 3. Influence of the relative position of delamination on the first natural frequency of the composite cantilever beam [8].
Table 4. NDFF [ω = ωnL2√(ρ/E1h2] of bending stiff and torsion stiff composite conical shells with mid-plane delamination, considering n = 8, h = 0.0004, s/h = 1000, d/L = 0.5, L/s = 0.7, θo = 45°, θv = 20°.
968 S. Dey and A. Karmakar / Journal of Mechanical Science and Technology 27 (4) (2013) 963~972
stiff and torsion stiff laminates. A specific common trend is observed for bending stiff and torsion stiff laminates for all twist angles that the frequency value gradually reduces from the top free surface towards the mid-plane and then rises to-wards the bottom free surface as the location of delamination changes.
4.4 Effect of frequency ratio and relative frequency
The trend of frequency ratio (FR) (ratio of delaminated and undelaminated fundamental natural frequency) and relative frequencies (RF) (ratio of rotating natural frequency and sta-tionary natural frequency) at Ω = 0.5, 1.0 are furnished in Fig. 5. At stationary condition, it is noted that the percentage dif-ferences between maximum and minimum frequency ratios (NDFF) are found to be 2.6% and 2.1% for bending stiff and torsion stiff configuration, respectively. The frequency ratio (NDFF) of torsion stiff laminate is always found higher than that of bending stiff laminate at stationary condition. The dif-ferences in the value of frequency ratio between torsion stiff and bending stiff laminates are found maximum at Ψ = 0° and identified to be converged with the increase of twist angles and observed to be of minimum value at Ψ = 45°. Similarly, the frequency ratio (NDSF) for bending stiff configuration is found lower than torsion stiff laminates for both untwisted and other twisted cases, except at Ψ = 15°. Almost similar trend is observed for both relative frequencies (NDFF) and relative frequencies (NDSF) at lower rotating speeds, while an excep-tion is identified at higher rotating speeds for relative fre-quency (NDSF) at Ψ = 30°. In both the cases, relative fre-quencies at Ψ = 0° and 45°, are found higher in torsion stiff laminate compared to bending stiff configuration while the reverse trend is observed for Ψ = 15° and 30°.
At lower rotating speeds (Ω = 0.5), it is identified that the percentage differences between maximum and minimum rela-
(a)
(b)
(c)
(d)
Fig. 4. Variation of NDFF for single delaminated torsion stiff [ ± 45°/ m 45°]s, and bending stiff [0°2/ ± 30°]s graphite-epoxy com-posite conical shells by varying relative position of delamination across thickness (h´/h) considering n = 8, h = 0.004, s/h = 1000, a/L = 0.33, d/L = 0.5, L/s = 0.7, θo = 45°, θv = 20°.
(a)
(b)
Fig. 5(a,b). Variation of frequency ratio (FR) at Ω = 0.0 for NDFF and NDSF at different twist angles for delaminated bending stiff ([0°2/ ± 30°]s) and torsion stiff ([ ± 45°/ + 45°]s) composite conical shells, considering n = 8, h = 0.0004, s/h = 1000, d/L = 0.5, L/s = 0.7, θo = 45°, θv = 20°.
S. Dey and A. Karmakar / Journal of Mechanical Science and Technology 27 (4) (2013) 963~972 969
tive frequencies (NDFF) are obtained as 38.6% and 63.2% for bending stiff and torsion stiff configuration, respectively. The percentage differences between maximum and minimum rela-tive frequencies (NDFF) at higher rotational speeds (Ω = 1.0) are found 60.1% and 85.0% for bending stiff and torsion stiff
configuration, respectively. Hence, it leads to the fact that for bending stiff configuration, relative frequencies (NDFF) have pronounced effect for higher rotating speeds while the same is also observed for torsion stiff configuration with higher per-centage difference compared to bending stiff configuration. On the other hand, the percentage differences between maxi-mum and minimum frequency ratio (NDFF) at stationary con-dition are found to be 8.6% and 3.4% for bending stiff and torsion stiff configuration, respectively. At lower rotating speeds (Ω = 0.5), the percentage differences between maxi-mum and minimum relative frequencies (NDSF) are obtained as 25.9% and 52.2% while the same at higher rotating speeds (Ω = 1.0) are obtained as 52.4% and 58.4% for bending stiff and torsion stiff configuration, respectively. Hence it leads to the fact that for bending stiff configuration, relative frequen-cies (NDSF) have pronounced effect at higher rotating speeds compared to torsion stiff configuration.
4.5 Effect of twist and rotation along span
A delamination of relative length a/L = 0.33 is considered and this is centered at a relative distances of d/L = 0.33 and 0.66 from the fixed end as indicated in Table 6 and Table 7. The delamination is considered at the interface of each layer. At stationary condition, NDFF are observed to decrease with the increase of twist angle for both bending stiff and torsion stiff configuration. For both the cases at stationary condition, NDFF for both twisted and untwisted conical shells are found to increase as the delamination moves towards the free end. It is observed that with d/L = 0.33, 0.67 at rotating condition (Ω = 0.5 and 1.0), NDFF decreases with increase of twist angle for bending stiff configuration except for ψ = 30° at higher rotating speeds (Ω = 1.0). In the contrast, for torsion stiff con-
(c)
(d)
(e)
(f)
Fig. 5. Variation of relative frequencies (RF) at Ω = 0.5, 1.0 for NDFF and NDSF at different twist angles for delaminated bending stiff ([0°2/ ± 30°]s) and torsion stiff ([ ± 45°/ + 45°]s) composite conical shells, considering n = 8, h = 0.0004, s/h = 1000, d/L = 0.5, L/s = 0.7, θo = 45°, θv = 20°.
Table 6. NDFF [ω=ωnL2√(ρ/E1h2] of graphite-epoxy bending stiff ([0°2/ ± 30°]s) composite conical shells with delamination along span, considering n = 8, h = 0.0004, s/h = 1000, L/s = 0.7, θo = 45°, θv = 20°.
NDFF at d/L = 0.33 NDFF at d/L = 0.67 Ψ
Ω = 0.0 Ω = 0.5 Ω = 1.0 Ω = 0.0 Ω = 0.5 Ω = 1.0
0° 0.3902 0.5223 0.6004 0.4015 0.5243 0.6041
15° 0.2339 0.4760 0.5136 0.2418 0.4310 0.4470
30° 0.1390 0.3689 0.1826 0.1450 0.1929 0.5854
45° 0.0911 0.2227 0.3786 0.0947 0.1799 0.3431
Table 7. NDFF [ω = ωnL2√(ρ/E1h2] of graphite-epoxy torsion stiff ([ ± 45°/ m 45°]s) composite conical shells with delamination along span, considering n = 8, h = 0.0004, s/h = 1000, L/s = 0.7, θo = 45°, θv = 20°.
NDFF at d/L = 0.33 NDFF at d/L = 0.67 Ψ
Ω = 0.0 Ω = 0.5 Ω = 1.0 Ω = 0.0 Ω = 0.5 Ω = 1.0
0° 0.2451 0.4180 0.2806 0.2453 0.4096 0.2761
15° 0.2082 0.3202 0.3582 0.2117 0.2663 0.3524
30° 0.1472 0.2288 0.2430 0.1521 0.1650 0.2816
45° 0.0941 0.3173 0.2876 0.0972 0.2430 0.2604
970 S. Dey and A. Karmakar / Journal of Mechanical Science and Technology 27 (4) (2013) 963~972
figuration, the same is found to decrease with increase of twist angle, except for ψ = 30° (at d/L = 0.33 with Ω = 0.5 and 1.0 and at d/L = 0.67 with Ω = 0.5). From Tables 4 and 5, it is observed that the effect of centrifugal stiffening is predomi-nantly found with reference to non-dimensional fundamental frequency in case of ψ = 15° for both bending stiff and torsion stiff configuration.
5. Mode shapes
The mode shapes of both two-dimensional and three-dimensional view corresponding to NDFF and NDSF are fur-
nished in Fig. 6 (bending stiff) and 7 (torsion stiff), respec-tively for various twist angles (ψ = 0°, 15°, 30°, 45°), consid-ering eight layered graphite-epoxy delaminated composite twisted conical shells. The fundamental natural frequency corresponds to the first torsion for both the cases under twisted condition. The first span wise bending is observed in torsion stiff configuration at stationary condition corresponding to untwisted fundamental and second natural frequencies and also in bending stiff corresponding to untwisted second natu-ral frequency.
Torsion stiff (2-D view) ψ
Mode 1 Mode 2
0°
15°
30°
45°
(a)
Torsion stiff (3-D view)
ψ Mode 1 Mode 2
0°
15°
30°
45°
(b) Fig. 7. Effect of twist on mode shapes: (a) 2D View; (b) 3D View of graphite-epoxy torsion stiff ([ ± 45°/ m 45°]s) composite conical shells with mid-plane delamination, considering n = 8, Ω = 0.0, d/L = 0.5, h = 0.0004, s/h = 1000, L/s = 0.7, θo = 45°, θv = 20°.
Bending stiff (2-D view) ψ
Mode 1 Mode 2
0°
15°
30°
45°
(a)
Bending stiff (3-D view) ψ
Mode 1 Mode 2
0°
15°
30°
45°
(b) Fig. 6. Effect of twist on mode shapes: (a) 2D View; (b) 3D View of graphite-epoxy bending stiff ([0°2/ ± 30°]s) composite conical shells with mid-plane delamination, considering n = 8, Ω = 0.0, d/L = 0.5, h = 0.0004, s/h = 1000, L/s = 0.7, θo = 45°, θv = 20°.
S. Dey and A. Karmakar / Journal of Mechanical Science and Technology 27 (4) (2013) 963~972 971
6. Conclusions
The finite element formulation presented in this paper can be successfully applied to analyse the free vibration character-istics of undelaminated and delaminated composite conical shells for any particular laminate configuration. In general, at stationary condition, NDFF is found to decrease with the in-crease of twist angle for both delaminated and undelaminated bending stiff and torsion stiff configurations. At rotating con-dition, non-dimensional fundamental frequencies of delami-nated composite laminates with bending stiff configuration are found to decrease with the increase of twist angle, while there is no significant trend observed for the same in case of torsion stiff. As the location of single delamination changes its posi-tion across thickness, the mid-plane (i.e., at h´/h = 0.5) is iden-tified as the plane of symmetry depicting the mirror image values of NDFF for torsion stiff and bending stiff conical shells at stationary condition. NDFF is found to attain a mini-mum value at relative position, h´/h = 0.5 for twisted cases of torsion stiff and bending stiff laminates. For both the cases at stationary condition, NDFF for both twisted and untwisted conical shells are identified to increase as the delamination moves towards the free end. The fundamental frequency cor-responds to the first torsion for both bending stiff and torsion stiff configuration under twisted condition. The first span wise bending is observed in torsion stiff configuration at stationary condition corresponding to untwisted fundamental and second natural frequencies and also in bending stiff corresponding to untwisted second natural frequency. The non-dimensional frequencies obtained are the first known results which could serve as reference solutions for the future investigators.
Rx : Radius of curvature in x-direction Ry : Radius of curvature in y-direction Rxy : Radius of twist Ψ : Twist angle Ω : Non-dimensional speed of rotation (Ω′/ ωo) Ω′ : Actual angular speed of rotation ωo : Fundamental frequency of a non-rotating
shell ρ : Mass density L : Length a : Crack length bo : Reference width ν : Poisson’s ratio h : Thickness d : Distance of centerline of delamination from
clamped (fixed) end θv : Vertex angle θo : Base subtended angle of cone E1, E2 : Elastic moduli along 1 and 2 axes G12, G13, G23 : Shear moduli along 1-2, 1-3 and 2-3 planes [M] : Global mass matrix
[K] : Elastic stiffness matrix [Kσ] : Geometric stiffness matrix δ : Global displacement vector λ : Non-dimensional frequency uj, vj, wj : Nodal displacements u´j, v´j, w´j : Mid-plane displacements θx, θy : Rotation about x and y axes Q : Transverse shear resultants [A] : Extension coefficient [B] : Bending-extension coupling coefficient [D] : Bending stiffness coefficients k : Curvature vector N : In-plane stress-resultants M : Moment resultants ε : Strains vector η, ξ : Local natural coordinates of the element x, y, z : Local coordinate axes (shell coordinate sys-
tem) n : Number of layers L/s : Aspect ratio ω : Natural frequency of rotating shell ND : No delamionation or undelaminated case NDFF : Non-dimensional fundamental natural fre-
quency NDSF : Non-dimensional second natural frequency
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Sudip Dey is a research scholar in Me-chanical Engineering Department, Jadavpur University, Kolkata, India. His basic research interests are mechanics of composites, structural health monitoring, computational mechanics and modelling, functionally graded materials and impact mechanics. He has more than 10 years
industry and research experience. He is also actively engaged in various industrial and research projects. He has published several papers in reputed international journals.
