- Email: [email protected]
Vibration and buckling of hybrid laminated curved panels
Composite Structures 21 (1992) 15-27
Vibration and buckling of hybrid laminated curved panels A. Barai & S. Durvasula Department of Aerospace Engineering, Indian Instituteof Science, Bangalore -- 560 012, India Vibration and buckling of curved plates, made of hybrid laminated composite materials, are studied using first-order shear deformation theory and Reissner's shallow shell theory. For an initial study, only simply-supported boundary conditions are considered. The natural frequencies and critical buckling loads are calculated using the energy method (Lagrangian approach) by assuming a combination of sine and cosine functions in the form of double Fourier series. The effects of curvature, aspect ratio, stacking sequence and ply-orientation are studied. The non-dimensional frequencies and critical buckling load of a hybrid laminate lie in between the values for laminates made of all plies of higher strength and lower strength fibres. Curvature enhances natural frequencies and it is more predominant for a thin panel than a thick one.
NOTATION Panel dimensions along x- and yaxes, respectively A ~/B ~/,D i/ Elements of in-plane, in-plane/ flexural coupling, and flexural stiffness matrices, respectively, Eigenvector or coefficient of Am,l, Ars assumed modes Young's modulus of the material Ell, E22 of the panel in ply principal axes eq Matrix defined in eqn (9) GI2, Gl3, G23 Ply shear modulus in 1-2, 1-3 and 2-3 planes, respectively h Panel thickness i,j Index It, I2, I3 Mass and moments of inertia k Total number of laminae/2 k44, k45, k55,/¢ Shear correction factor K Stiffness matrix Kg, K~, K~ K~Y Geometric stiffness matrices m, n, r, s Indices in the deflection series M Mass matrix {M} Moment resultant vector Mx, My, Mxy Moment resultants N Maximum value of indices n, s {N} Stress resultant vector N_x, N.y, N_x, Stress resultants Nx, Ny, Nxy In-plane applied loads q(t), qi( t) Generalisednormal coordinate a, b
qll|tl
c)!kl ~_ij Qx, Qy r0
Rx, Ry, Rxy S t T u, v, w
u0, v0, w0 U 1/1 V2 x, y, z Zp(X, y) 6 {e} 0
Coefficients in the series expansion of deflection Reduced stiffness of kth ply with fibre orientation 0 Transverse shear force per unit length of laminate Curvature, defined in eqn (3) Non-dimensional in-plane applied load parameters (ATx,/Vy, 1Qxy) aE/ gT h E, respectively Region of integration Time Kinetic energy Displacement vector in rectangular or curvilinear coordinates In-plane displacement components Total potential energy ( VI + 112) Elastic strain energy Potential energy due to in-plane load Cartesian coordinates Equation of the mid-plane of curved panel Rise height of the curved panel mid-plane from flat configuration Mid-plane strain components defined in eqn (4) Ply orientation angle in degrees
15 Composite Structures0263-8223/92/S05.00 © 1992 Elsevier Science Publishers Ltd, England. Printed in Great Britain
16
2
A. Barai, S. Durvasula Mid-plane curvature components defined in eqn (4) Non-dimensional frequency parameter [~oaZ(pmh/ETh3)] 1/2 Poisson's ratio for orthotropic lamina such that v~2E22 = v21 Ell Transverse shear strain vector defined in eqn (7b) Non-dimensional coordinate for shell element Non-dimensional coordinates x/a and y/b, respectively Mid-plane stress components Summation symbol Average values of rotations of line, normal to the middle surface, over thickness Frequency of oscillation (rad/s) =
1"12, '!'21
{,,}
{o} Z
~P,, ~P, O)
1 INTRODUCTION Hybrid composites are produced by combining two or more types of fibres in a common matrix. These materials exhibit some of the desirable characteristics of the unmixed composites and simultaneously attempt to remove some of their unacceptable qualities. Motivation for the use of hybrid composites is well known and it arises from their improved specific strength and stiffness characteristics as well as reduced weight and cost. Hybrid composite construction has shown a potential for increased use in different applications. Applications have been attempted in a variety of areas like aerospace, mechanical and marine engineering.~ Since weight is an important factor in the design of aerospace structural components, hybrid composites provide extra benefit over the usual composites using a single fibre-matrix combination. Alternatively, the addition of cheaper fibres could help to reduce the cost of a composite made from expensive carbon or boron fibres, without seriously affecting the performance of the composite. Although the research and development work on hybrid composites began around 1970, no common names seem to have been given to them so far, in spite of the enormous amount of published data on material characterisation and fabrication techniques. Reference 1 reviews the various aspects of fibre reinforced hybrid composites. Kretsis 2 reviewed the present status of mechanical properties prediction methods mostly based on the rule of mixtures. He also listed a set
of hybrid laminates, used by several investigators for strength or stiffness improvements. Chamis and Lark 3 pointed out that the methods developed for single material composite laminates can be extended for hybrid laminates. However, only a limited number of results seems to be available in the literature on the vibration of hybrid laminated plates. Crawley and Dugundji 4 studied the free vibration characteristics of a cantilever configuration. They developed a scheme based on a partial Ritz (Kantorovich) analysis, which reduces the problem to a set of uncoupled ordinary differential equations. The solution of these equations yields the eigenvalues of free vibration. They also used the finite element method for comparison of some results. They considered a hybrid model with aluminium core and glass/epoxy face layers. An analytical and experimental study on the vibration of rectangular plates with free edges, using hybrid carbon-glass/fibre sandwich composites is reported in Ref. 5. The analysis is based on the finite element method with shear deformation and damping, lyengar and Umaretiya ~ considered hybrid composite plates with two opposite edges simply-supported and the other two edges clamped. In a number of applications, it is well known that thin plates are used as structural members. Some of these plates may have inherent curvature or develop small curvature under various in-plane loadings. Plates with small deviations from the initial flat configuration are known as curved plates in the literature. A major portion of aerospace structures fall into the latter category. Initial curvature has a beneficial effect because it raises the natural frequencies of oscillation. Free vibration behaviour studies of curved panels have been made using linear shell theory as well as nonlinear thin plate theory. They are reviewed by Leissa and Kadi. v All these studies are limited to isotropic materials. Within the scope of linear theory, free vibration behaviour of laminated curved panels is also studied in Refs 8-15. In most of these studies, closed-form solutions could be obtained for antisymmetric cross-ply laminates. Fortier 9 used the Rayleigh-Ritz method for the analysis of some angle-ply laminated curved panels without shear deformation. Only recently, Librescu et al.~4 developed a simple shear-deformable theory for doubly curved shallow cross-ply composite shells and used state-space concepts in conjunction with the Levy method to evaluate the static and dynamic response of panels for various boundary
Vibration and buckling of hybrid laminated curved panels conditions. Chandrashekhara ~5 studied the free vibration characteristics of curved panels using an isoparametric doubly curved quadrilateral shear flexible element. He used first-order shear deformation theory in conjunction with Sander's shell theory. As far as the authors are aware, there are no results on the free vibration behaviour of hybrid laminated curved plates. The buckling problem of laminated composite curved panels was first studied by Viswanathan et al.16 using linear shallow shell theory. They considered long panels with direct and shear loads. Sinha and Rath "~ studied the buckling characteristics of laminated cylindrical curved panel using Reissner's shallow-shell theory including shear deformation. They obtained closed-form solutions of the governing differential equations for simply-supported panels. Zhang and Matthews ~7and Whitney ]~ studied the buckling of cylindrical panels using Galerkin's technique. Rao and Tripathy j9 studied the buckling behaviour of a cylindrically curved panel using sublaminates. For the present study, Reissner's shallow-shell t h e o r y 2°-23 is used for the analysis of curved panels. A general formulation is developed for hybrid laminated curved plates including shear deformation and rotatory inertia. The expression for kinetic energy is obtained by neglecting the energy associated with small curvature. This is essentially the kinetic energy expression for the flat plate as could be directly obtained from YNS (Yang-Norris-Stavsky) theory.24 To obtain governing equations by applying Lagrange's equation, the displacements in the x-, y- and zdirections and rotation about the x- and y-axes are represented by double Fourier sine and cosine series.
2 FORMULATION
quite lengthy; therefore, they are omitted here (see Ref. 23). However, it is important to understand some of the assumptions used for the development of shallow-shell theory because they are important when applying them to a particular problem. Reissner and Wan 2~ developed the shallowshell theory on the basis of the following assumptions. A shell segment will be called shallow if the ratio of its height to base diameter is less than 1/8. The results obtained on the basis of this assumption will often also be applicable to shells which are not shallow, when the displacements are small with respect to undeformed configuration. Under these assumptions, the middle surface can be represented in the folowing form:
x = ~,, y = ~2, Zp = Zp(X, y)
(1)
so that the projections of the ~ - and ~2-curves on the xy-plane form straight lines parallel to the xand y-axes, respectively (see Fig. 1). Therefore, the coordinate curves on the middle surface will, in general, be neither orthogonal nor correspond to the lines of curvature. The slopes of the surface in the x- and y-directions are such that
maxlzo, iZp./] ~. 1
( i , j = x , y)
(2)
3
Fig. la.
Shell element in orthogonal coordinate system.
2.1 Strain-displacement relations for a shallow shell
The undeformed surface of the shell is taken as the reference plane. An orthogonal coordinate system (~l, ~2, ~3)is assumed, in which ~l and ~2 lie in the reference plane and ~3 is perpendicular to the ~ - and ~2-axes. The deformation geometry of a general shell element is complex and the mathematical description is tedious. However, for a shallow shell (or curved panel) it can be simplified considerably. Even so, the derivation of shallow-shell strain-displacement relationships is
17
Y
Fig. lb.
Curved plate geometry.
18
A. Barai, S. Durvasula
where zp., and Zpj are slopes of the mid-plane in the x- and y-directions, respectively. The midplane radii of curvature are given by:
If
h
r~j= - - -
1
-
n
- %--t-
:
-
On-1 'n
(3)
( i , j = x , y)
Z p, ij !
Within the scope of shallow-shell theory, the strain-displacement expressions are given by:2°,21 Ex =gxO +Zl('x;
o I
(a)
Ey = E y 0 "~-ZKy;
g xy = Cx~O + Z K.O,; Exz = W x -- ~).r;
Eyz = W,y -- lffy
where
~X
ExO = Uo,x -- WZp,xx;
(b)
EyO = Uo,y -- WZp, yy;
Laminate definition: (a) ply stacking sequence, (b) ply orientation.
Fig. 2.
e o~ = vo.x + uo,y - 2 wz,xy; K:r =
-- 1])x,x;
~), = -- ~[dy,y;
or
Kv = -('q-'x.y + ~y,x)
(4)
{M}=[B]{e}+[D]{K}
where uo and vo are in-plane displacements and exo, e~ and exg~. are in-plane strain components, respectively, on the middle surface.
ii
k45A45
(6b)
k55As~J[Gzl
or
2.2 Constitutive relations
(7b)
{Q}=[F]{v}
The constitutive relations of a laminate made of orthotropic lamina (see Fig. 2) with shear deformation are given by:
NI,
N~,,
fA,, [A,~
1 Aql ;,, I A2~
A6~Jteo,,j
B22
BI2
B 16
where k44 , k45 and k55 are the shear correction factors. For numerical calculations, they are assumed to be equal to/~, where (A;j,B OD~j)=
B26
B26]/~y ~ //
l
B66 j [ K.,J
(i,j= 1, 2,6)
(8)
h/2
(Sa)
or
(Sb)
{N} =[A]{ e} + [B]{ K}
[h/2 -~.(k)~l. z 2 Q~j ~ , z, )dz d - h/2
B,, B,2 B, l/ x / +
(7a)
Fii =
r d
k i jt0 ~ i(kl j dz
(i,j= 4, 5)
(9)
- h/2
The ~t.~ ~_,j are functions of six elastic constants and the ply angle 0, and they are different for different laminae in hybrid laminates. 2.3 Potential and kinetic energy expressions
Mxy
[B,6
+
B26
For a linear elastic solid, the elastic strain energy is given by:2s
B66_ItG~,J
D,,
D,2
D,6][r~ 1
Jl2
D22
D 16
D26
D26//Ky? // / D66 J [ K . J
lII
~=-~
[{e}r[A]leI+{e}r[BllrI+lrIT[Blle}
(6a) +{r}r[D]{r}+{v}r[F]{v}]dxdy
(10)
Vibration and buckling of hybrid laminated curved panels Expanding the A, B, D and F matrices and substituting for e, r and vfrom eqn (4), we obtain the strain energy expression. The potential energy of the in-plane loads due to transverse deflection is: 1
where (t~,/2,13) =
(1, z, z2)p/k~ dz J - h/2
and Plk) is the material density of kth layer of the hybrid laminate. The average density of a hybrid laminate is given by:
ff
II [&/z0 + wL + &/z0 + w)5 Ja
+ 2.~xy(Zp +W).x(Z p +w).y]dxdy
19
(11)
1 N
pro=--h Z plk)(h,-hk-,)
(17)
k=l
Now the expression for total potential energy can be given by: U= Vt + Vz
and should be used for natural frequency calculations.
(12)
The expression for kinetic energy is given by:25
pIk)[ft2 +#2 + ~2]dxdydz
T=-~
(13)
J J s , / - h/2
Within the scope of linear shell theory, the displacement expressions are given by:
u(x, y, z, t)=(1 +R)uo(x,y, t) + z~p,(x, y, t)
(z)
v(x, y, z, t) = 1 +-~ Vo(X, y, t) JrZ~Dy(X,y, t) (14)
w(x, y, Z, t) = w(x, y, t) Substituting these displacement expressions into eqn (13), the kinetic energy expression for the shell can be obtained. In addition to the flat plate terms, the expression contains terms due to curvature. These extra terms are in the form of:
h/2
dz ptkl(z, z 2) - j-h/2 R
2.4
Assumed deflection functions
To obtain governing differential equations using Lagrange's equations, the displacements (u, v and w) and rotations (~p, and ~py)in the potential and kinetic energy expressions are expressed in terms of admissible functions satisfying the geometric boundary conditions. The following simplysupported boundary conditions are considered for the present study (also known as S2 in the literature): v = w = 7J~ = 0
at x = 0, a
u=w=~Ox=O
aty=0, b
The geometric boundary conditions in eqns (18) are satisfied by the following combination of sine and cosine functions: M
F,.. cos (marx/a) sin(nary/b)
m = ] n = l
M
N
v=ZZ
Gmn sin(marx/a) cos(tory/b)
m=l
n=l
M
N
w=ZE m=l
M
(16)
N
n=l
M
N
m~l
+I2{(~x,t)(U,)+(lpy,t)(Vt)}
(19)
Z Xm,,cos(marx/a)sin(nary/b)
m=l
hV/y = ~ 2.5
H,.. sin(marx~a) sin(nary/b)
n=l
hVdx= ~
T=lfls[II{(U,t)2+(v,t)2+(Wt)2} + 13{(~dx.,)z + (7)y.,)2}]dxdy
N
.=ZZ
(15)
For a shallow shell, 1/R is small, and the above integral, taken over the plate thickness, works out to be negligibly small compared to the corresponding flat plate terms. Therefore, the fiat plate kinetic energy expression only is used in the present study which is given by:
(18)
Z Y,.. sin(marx/a ) cos(nary/b ) n=l
E q u a t i o n for t h e c u r v e d s u r f a c e
The shape of flight structure skin panels may be cylindrical, doubly curved or with curvature and
20
A. Barai, S. Durvasula
twist. For the present study, the equation of the mid-plane of the curved panel, with curvature in both directions but without twist is assumed in the following form: >
Z_p__2 [x-(a~2)] 2 6 (a/2) 2
[Y-(b~2)]2 (b/2) 2
(20)
form: 1
p,,,h[M]{4~,}+--s[K +Kg]{q~,}=Q(t) a
where [Kg] is the geometric stiffness matrix and is given by: X
where 26 is the maximum initial height and 26/a is small. The curvatures are given by: z""' = - 8(b/a2)' Zp,,:,,= - 8(6/b2),
(21)
Zp,xy = 0
For curvature only in one direction (say, the xdirection), the curvature expressions can be written as:
~-1
[x-(a/2)]2
6-
(a/2) 2
(22)
Then the curvatures are given by:
zp.~x= -8(6/a2),
Zp.vy= 0,
U
,%
M
2a a , , t = l
N
M
~ ~ • q,,,,qr, K,,,,,,, n=l
r=l
(24)
g=l
and A/
T=I,.I Pm h E /
m=l
N
M
N
E E E Clmn4rsmmnr,• i1=1 r = l
(28)
where R,, R v and R~v are the non-dimensional mid-plane force parameters. The vector qm. is of the order (5MN x 1) and contains the elements of F,,,,,, G,,,,,, H,,,,,, X,,,,, and Y,,,,, respectively. The mass matrix M and stiffness matrix K are of the order (5 MN x 5 MN).
3.1 Vibration analysis For free vibration, no external force acts on the system, so Q(t)=0; further, for an unstressed panel, the geometric stiffness matrix [Kg]=0. Therefore, eqn (27) reduces to:
p,,,hEM]{i],.,} + 4a [K]{q,.,} = 0
[K]{A,.~} = 22[M]{A,:,}
Substituting the assumed displacement functions eqns (19)and curvature expressions eqns (21)[or eqns (23)] into the potential energy expression eqn (12) and kinetic energy expression eqn (16), we obtain:
~
XV
For harmonic motion, substituting = {A~.,}eiw', in the above equation, we obtain:
3 THE GOVERNING DIFFERENTIAL EQUATIONS
l
y
[Kg] = - Rx[Kg]- RyEKg]- R.yEKg ]
"~p,xy ~ 0
(23)
=
(27)
(25)
s=l
where the qm,, represent, in general, the unknown coefficients Fro,,, etc., of eqns (19). Substituting the potential energy (U) and kinetic energy (T) expressions into Lagrange's equations, 25
(29) {qr~,} (30)
where (31)
~2 = p , , , h w 2 a 4
Equation (30) is a generalised eigenvalue problem. The eigenvalues ~2 and eigenvectors {A,,} are evaluated using standard procedures.
3.2 Buckling analysis For buckling, the term associated with the mass matrix in eqn (27) is set to zero and the combination of in-plane loads for which the initial configuration is no longer stable is calculated. Then, from eqn (27) we can write:
1 --~ [K + Kg]{Ar.,.}= {0} a
(32)
Alternatively, it may be written as: x
[K]{A~} = Rx[Kg]{Ar,} dt
0q =Q(t)
(26)
we obtain the governing equation in the following
+ Ry[K~]{Ar,} + Rxy[KgYl{A~}
(33)
The eigenvalue character of the problem is immediately seen.
Vibration and buckling of hybrid laminated curvedpanels If only Ny is acting, then: [K]{A,~}= Ry[K~g']{Ar~}
(34)
where Ry is the eigenvalue, similar to ,;[2 in eqn (31). This problem is also solved by the same procedures used to solve eqn (30). For biaxial loading, or combined biaxial and shear loading, the procedure is essentially similar by assuming two of the parameters and treating the third as the eigenvalue to be determined.
21
the hybrid construction works out to be lower. Therefore, for the present numerical calculations, for hybrid laminated curved panels, the abovementioned models only are considered. The hybrid laminate model designation scheme is shown in Table 2. The results are presented in non-dimensional form and they are non-dimensionalised with respect to E~.2 (or E r) of the graphite/epoxy material.
4.2 Free vibration of flat and curved panels 4 NUMERICAL CALCULATIONS
4.1 Hybrid laminate modelling Normally, in the literature, the elastic properties of fibrous composites are non-dimensionalised with respect to ET [or E22] for various numerical calculations. They are also used later for the initial comparison of some results. For hybrid laminates, dimensional elastic properties as shown in Table 1 are used. The elastic properties of kevlar/epoxy and glass/epoxy are taken from Ref. 28. The shear correction factor is assumed to be 5/6 for all calculations. However, for the comparison of free vibration and buckling results with the existing results, different sets of material properties and shear correction factors are considered which are taken directly from appropriate references. Although the anisotropic behaviour of a composite laminate reduces as the number of laminae increases, here eight layers are considered sufficient to show the effect of hybridisation. It was shown by the authors 27 that the natural frequencies of eight-layer laminates with 2-4 layers of higher strength material as face plies and the rest as core plies of lower strength material are nearly equal to the laminates with all eight layers of higher strength material. It is also possible that the frequencies of such laminates with four layers of higher strength material as face plies could be higher than those with eight layers of higher strength material, whenever the average density of
E22 v|2 G|2 GI3 G23
/9
Table 2. Hybrid laminate model designation Layer
Graphite/epoxy
Kevlar/epoxy
Glass/epoxy
128"0 GPa 11.0 GPa 0"25 4"48 GPa 4"48 GPa 1"53 GPa 1500"0 kg/m 3
76.0 GPa 5-5 GPa 0"34 2"30 GPa 2"30 GPa 2"30 GPa 1460.0 kg/m 3
38.6 GPa 8.27 GPa 0.26 4.14 GPa 4.14 GPa 4.14 GPa 1800.0 kg/m 3
Model no.
no.
1 2 3 4
HI
H~
H3
H4
H~
H~,
H7
Gr Gr Gr Gr
Kv Kv Kv Kv
GI GI GI GI
Gr Gr Kv Kv
Gr Kv Kv Kv
Gr Gr Gi GI
Gr GI GI GI
Gr, Graphite; Kv, kevlar; GI, glass.
Table3. Convergence study for the non-dimensional frequency parameter of a square fiat panel Order of matrix (mxn)
Table 1. Material properties
Ell
First, a convergence study for the non-dimensional frequencies of an unsymmetrically laminated cross-ply, cylindrically curved panel is made by increasing the number of terms in the assumed displacement functions of eqns (19) as shown in Table 3. The numerical convergence calculations for a cylindrically curved panel are first carried out for a composite made of single fibre construction in order to be able to compare results with those available in the literature: Table 4 shows the non-dimensional frequencies of unsymmetrically laminated curved panels with various curvatures compared with the results reported in Ref. 8. The results show good agreement. To study the effect of stacking sequence, symmetric laminates [0°/90°/0°] are considered. The non-dimensional frequency parameter, for the
2 3 4
2 3 4
No. of terms
Mode no.
1 20 45 80
19"0 19"0 19"0 (1,1) a
2
3
4
53"5 53"5 74"8 53"5 53'5 74-8 53"3 53"3 74-8 (2,1)(1,2)(2,2)
5 -112"8 112"8 (1,3)
a/h=50, Ell/E22=40.0, G|2/E22=1"0, G23/E22= 0"5, shear correction factor k = 5/6, vl2 -- 0"25, 2 = [toa2(pmh/E2~h3)l/2].
Lay-up: [0°/90°],
GI3/E22=
aModes.
1"0,
A. BaraL S. Durvasula
22
panel configuration and material properties used in Table 3, are shown in Table 5. Comparison of the results in Tables 4 and 5 shows that bending-extensional coupling for unsymmetric laminates significantly reduces natural frequencies. Table 4. Comparison of non-dimensional frequency parameters for square, fiat and curved panels
5/a
Coupling ignored B 0 = 0 (2 layers)
With coupling
B,j# 0 (2 layers)
Closed-form Present Closed-form Present solution (R-R method)" solution (R-R (ref. 8) (ref. 8) method) 0"0
19'0 ~ (1, I)' 52.9 (2,1) 52.9 (1,2) 74.3 (2,2)
19'0 (1,1) 53.3 (2,1) 53.3 (1,2) 74.8 (2,2)
11"6" (1,1) --
24.4" (1,1) 53"4 (2,1) 60.4 (1,2) 75"9 (2, 2)
24.4 (1,1) 53"8 (2,1) 60.8 (1,2) 76.3 (2, 2)
19.2" (1,1) --
36.0" (1,1) 54.9 (2, 1) 78.7 (1,2) 80.3 (2,2)
36.0 (1,1) 55.3 (2, 1) 79.0 (1,2) 80.8 (2,2)
32.6" (1,1) --
0.02
0.04
---
---
---
11"7 (1,1) 31.9 (2,1) 31.9 (1,2) 46.4 (2,2) 19.2 (1,1) 32"7 (2,1) 43.0 (1,2) 48.8 (2, 2) 32.7 (1,1) 35-1 (2, 1) 55.5 (2,2) 66.2 (1,2)
Lay-up: [0°/90°], material properties and panel geometry are the same as in Table 3, )1.= [~oa2(Pmh/E22h3p/2]. "Rayleigh-Ritz method. hTaken from graph. 'Modes. Table 5. Non-dimensional frequency parameter symmetrically laminated, square, curved panels
6/a
0.0 0"02 0.04 0.06
for
Table 6. Non-dimensional frequency parameter symmetrically laminated, square, curved panels
6/a
Mode no. 1
2
3
4
5
18.95 (1,1) 24.30 (1, 1) 35.84 (1,1) 49.40 (1,1)
27.39 (1,2) 40.37 (1, 2) 65.32 (1,2) 72.52 (2,1)
46.67 (1,3) 62.44 (1, 3) 70-79 (2,1) 86.89 (2,2)
69"38 (2,1) 69"73 (2, 1) 79.96 (2,2) 93-09 (1,2)
73.95 (2,2) 75.50 (2, 2) 95.18 (1,3) 109.49 (2,3)
Lay-up: [0°/90°/0°], (a/h = 50), material properties and panel geometry are the same as in Table 3, 2=[e~a2(pmh/
E22 h3)1/2].
Hence, the majority of the calculations in the various studies are made only for symmetric laminates. Tables 4 and 5 also confirm that the frequencies increase with curvature. Next, the free vibration behaviour of a square thick panel (a/h= 10) is studied and the results are presented in Table 6. The geometry and material properties are similar to those used in Table 3. Since this is a thick panel, and first-order shear deformation theory is used in the present formulation, there will be some error in the nondimensional frequencies. The results are presented to show the effect of thickness ratio on the free vibration characteristics of laminated curved panels. It is clear from Tables 5 and 6 that the relative increase in the frequencies for thin panels is more than that for the corresponding thick panels, for the same change in curvature. The increase in frequency for curved panels is due to the existence of coupling between bending and extension. This is a kind of geometric coupling which is well known for shell configurations and different from the coupling that exists in unsymmetric lay-ups which is due to anisotropic material behaviour. The extensional stiffness of a panel is several orders of magnitude greater than its bending stiffness. Therefore, for a small amount of lateral deflection, the extensional stiffness of a thin panel is much more effective than for a thick panel where deflection is resisted by bending stiffness. Before studying the free vibration characteristics of hybrid laminated curved panels, the behaviour of a hybrid laminated flat panel is studied first. For this purpose, a symmetrically laminated ([ + 0]2,) square plate of a/h = 40 is considered. The variation of the first three fre-
0-0 0.02 0.04 0.06
for
Mode no. 1
2
3
4
5
16.13 (1,1) 16.42 (1,1) 17.25 (1,1) 18.55 (1,1)
29.05 (1,2) 29-62 (1,2) 31.30 (1,2) 33.90 (1,2)
41.19 (2,1) 41.22 (2,1) 41.30 (2,1) 41.42 (2,1)
48.35 (2,2) 48.45 (2,2) 48.74 (2,2) 49.21 (2,2)
50.83 (1,3) 51.46 (1,3) 53.29 (1,3) 56-20 (1,3)
Lay-up: [0°/90°/0°], (a/h = 10), material properties and panel geometry are the same as in Table 3, £=[~oae(p,,h/
E22h3)1/2].
Vibration and buckling of hybrid laminated curved panels quencies in non-dimensional form [wa2(pmh/ E.rh3)] 1/2 with play angle (0) is plotted in Figs 3a, 3b and 3c, respectively, for all seven models. The first and second natural frequencies of angle-ply laminates are higher than laminates with all 0 ° or 90 ° layers. All models exhibit maximum frequencies at 45 °, except for the third mode. The 2 - 0 variation is symmetric about 0 = 4 5 °. As in the previous study (Ref. 27, for boron and kevlar materials), models H 4 and H 6 with four graphite/ epoxy layers as face plies and the remaining four layers of kevlar or glass as core plies exhibit frequencies quite close to model H~ with eight layers of graphite/epoxy. The third mode generally shows frequency maxima between 20 ° and 25 °, and 65 ° and 70 °, and a local minimum at 45 ° which is more than the value at 0 ° or 90 °. Model H 3 (all glass/epoxy) exhibits some peculiar characteristics in its third mode; the minimum frequency occurs at 45 ° and the maximum values at 0 ° and 90 °. The models H5 and H 7 with two graphite face plies and six kevlar or glass core plies exhibit intermediate frequencies with respect to models H~, H 2 and H 3. The behaviour of the fourth and fifth modes for all of the models are similar to that of the first two modes and they are not presented here. It is apparent that the hybrid construction under consideration raises the free vibration frequencies of flat plates in comparison with those of flat plates of single material construction. Now, the same square plate as mentioned above will be used as a reference for the study of curved plate behaviour. First, a small curvature in the xdirection only is introduced. For these studies only models H t -- a single material construction -- and H 4 and H 7 - - which are of hybrid construction -- will be considered. These three laminates are expected to demonstrate the salient features of the curvature effects as well as the effects of hybridisation. Figures 4a, 4b and 4c show the variation of the first three non-dimensional frequencies with ply angle for cylindrically curved panels with small initial curvature (6/a=0.01). The 2 - 0 variation curves for first three frequencies are similar to those of flat panels, presented in Figs 3a-3c. To study further the effect of curvature, the same square plate configuration (a/h = 50), made of graphite/epoxy, is considered. Figure 5a shows the variation of non-dimensional frequencies with ply angle for different values of curvature only in the x-direction. The non-dimensional frequency parameter increases with curvature and for small
a/b =1 a/h = 40
23
;~=¢oaZOOh/E.rh'~ ff'~ 141
12
1C
h
H3
:
I
I
0
I
20
I
I
40 O (deg)
I
I
60
I
I
80
Variation of fundamental frequency of a flat plate with ply angle;lay-up: [ _+012,.
Fig. 3a.
!
28}/
a/b = 1 o/h = ~0
)_ ,,o :o
2O
16
1 0 Fig. 3b.
20
40 e (deg)
60
80
Variation of second mode frequency o f a flat plate
with ply angle;lay-up: [ + 012,. 36
alb = I
32
28
24
20 I
0
Fig. 3c.
I
20
I
I
40 0 (deg}
I
I
I
60
I
80
Variation of third mode frequency o f a fiat plate
with ply angle; lay-up: [ _+0]2s.
A. Barai, S. Durvasula
24 a/b = 1 a/h = 50
16
40
H1
~.J2,11 / '~
6/a = 0.01
a/b=l, a/h = 50 GrGphite/Epoxy
/
14
X
12
10
~ 0
Fig. 4a.
I
20
I
I
40 e (deg)
1
60
80
Variation of fundamental frequency of a curved
0
plate with ply angle; lay-up: [ + 0]_,,. 30
H1
20
40 (3 [deg)
80
Fig. 5a. Variation of fundamental frequency of a cylindrically curved (in the x-direction) panel with ply angle; lay-up:
a/b = 1
[ + o]_~,.
alh = 50 =
60
.
26 a/b=1 , a / h = 5 0 AJJ,z) (z,i/~ 8 0 _ Graphite/Epoxy
f
22
"L
-
(,,2)
. . . .
60 18 40 I"_
0
Fig. 4b.
i
i
~
20
i
L
40 e (deg)
J
60
t
80
20
Variation of second mode frequency of a curved plate with ply angle; lay-up: [ + 0]=,.
38[
Q/b =1 QIh =50 6]a =0.1
•
Fig. 5b. ~'~
H
30
M6 H5
2 i
0
Fig. 4c.
20
i
1
40 O (deg)
1
20
]
I
I
40 0 (deg)
1
60
i
[
80
Variation of fundamental frequency of a doubly curved panel with ply angle: lay-up: [ + 0]2,.
/ ~.\
H1
X
0
I
60
1
i
80
Variation of third mode frequency of a curved plate with ply angle; lay-up: [ + 0]2.,.
values of curvature it attains a m a x i m u m value of 45 °. However, as the curvature increases (beyond 6/a = 0.04), the m a x i m u m shifts from 45 ° towards 0 °. For small value of curvatures (6/a = 0 - 0 1 and 0.02), the fundamental frequency occurs in c a m -
bination of m = 1 and n = 1. Thus, the curve panel behaviour is similar to that of a flat panel (6/a = 0-0). However, for 6/a=0.04 the curved panel behaves like a flat panel up to 30 ° (the fundamental frequency occurs in a c o m b i n a t i o n of m = 1 and n = 1 ). From 0 = 35 ° to 90 °, the panel behaves like a shell and fundamental frequency occurs in a combination of m = 2 and n = 1. For 6/a=0.06, the variation of fundamental frequency is similar to 6/a = 0.04, but shell m o d e (m = 2 and n = 1) appears at 0 = 20 ° and this trend is continued up to 90 °. For the same plate configuration, if curvature exists in both directions then an increase in frequency is observed c o m pared to the case of curvature in o n e direction only, for all ply angles, as s h o w n in Fig. 5b. It is seen that frequency maxima occur at 45 ° and that
Vibration and buckling of hybrid laminatedcurvedpanels the ;t-O variation is symmetric with respect to 0 = 45 °. In this case, the shell modes are observed for 6/a--0.02. At 0 = 40* and 45", fundamental frequencies occur in a combination of m = 2 and n -- 1, but at 0 = 50* it is in a combination of m = 1 and n---2. For curvatures beyond 6/a = 0.02, the fundamental frequencies for different ply angles occur due to various combinations of m and n, as shown in Fig. 5b.
4.3 Buckling of flat and curved panels First, a comparison for the critical buckling load is made using composite panels with a single fibre-matrix combination, because some results are available in literature for such laminates. The results are presented in Table 7 for unsymmetrically laminated curved panels with a/b 1 and a~ h -- 100. The panel is made of graphite/epoxy and its elastic properties are also presented in Table 7. The results are compared with the closed-form solution of Ref. 10. The results show good agreement. =
Table 7. Comparison of critical buckling loads (Erha)] of curved panels
6/a
Closed-form solution" Ref. 10
0.0 0.5 1.0
9.5 68,0 139,0
[Nya21
Present (R-R method) 9,66 68.34 141.25
Lay-up: [0°/90°], a/h=lO0, EH =172 GN/m 2, E22=6"89 GN/m z, G12=3.45 GN/m 2, G2.~=Gt3=l'38 GN/m 2, Vl2 = 0.25, shear correction factor k = 2/3, "Taken from graph. bRayleigh-Ritz method.
25
The variation of critical buckling loads (Nya2/ E r h 3) of flat panels with eight layers, symmetrically laminated, is shown in Fig. 6. The critical buckling loads for angle-ply laminates are higher than for all zero degree layers, for all models. Hybrid laminates, with four layers of higher strength material as face plies and four layers of lower strength material as core plies, exhibit buckling behaviour similar to all higher strength material, for similar ply orientations. Laminates with two layers of higher strength material as face plies and six layers of lower strength material as core plies exhibit critical buckling load which lies between the critical buckling load of all higher strength material and all lower strength material, for all ply orientation angles. All models show buckling load maxima when 0-- 55 °, except model H 3 (all glass/epoxy) which shows a maximum value at 50 °. Like the free vibration characteristics, the critical buckling load of curved panels increases with curvature. Figure 7 shows the critical buckling load for all models with small initial curvature (6/a = 0.02) while other parameters are similar to the flat panel in Fig. 6. one significant difference in response compared to the flat panel is that all curved panels attain buckling load maxima at two ply angles (35 ° or 40 ° and 60 °) except model H6 which has a maximum value at 60 °. The effect of curvature on the buckling load versus ply angle response is shown in Fig. 8, for model H4 (a/h=60). For small curvature, the maximum buckling load occurs only at one ply angle (50 °) but as the curvature increases, maxima
36
28 N y az
Nya 2
E+h 3 20
2l--
Fig. 6.
I (
./h =SO I
0
o/b =1 o/h = SO
I
20
I
I
I
40 9 (deg)
I 6/0
60 ~
I= o
I
I
80
0
Variation of critical buckling load of a fiat plate with ply angle; lay-tip: [ _+012~.
Fig. 7.
I
I
20
I
I
40 0 (deg)
I
I
60
I
I
I
80
Vafiationofcfiticalbuckling load of a cylindrically curved panel with ply angle; lay-up: [ ± 0]2s.
26
A. Barai, S. Durvasula 80 o/b
6o "Y°' L
/
= 1
5.
\ 6/o=00, m,
ETh3
o.o3
6. 7. 8. 0t
f
i
20
i
I
i
40 e (deg)
I
60
i
i
9.
80
Fig. 8. Variation of fundamental frequency of a cylindrically curved (in the x-direction) panel with ply angle; lay-up: [-+ 0L~.
o c c u r at two d i f f e r e n t ply angles. T h e critical buckling l o a d n o r m a l l y increases with c u r v a t u r e .
10. 11. 12.
13. 5 CONCLUSIONS
Hybrid construction in a laminated composite plate increases the natural frequencies and critical buckling load in comparison with single material construction. The increase in natural frequency and critical buckling load for curved panels is essentially d u e to the p r e s e n c e of b e n d i n g e x t e n s i o n coupling. T h e effect o f c o u p l i n g is g r e a t e r in the case o f thin panels ( a / h = 50) a n d c o m p a r a t i v e l y less f o r thick p a n e l s ( a / h = 10). F o r s o m e values o f c u r v a t u r e a n d ply angle, the p a n e l b e h a v e s m o r e like a shell a n d the f u n d a m e n t a l m o d e o c c u r s for a c o m b i n a t i o n of h i g h e r indices (e.g. m = 2 , n = l o r m = l , n = 2 ) especially for angle-ply laminates. F r e q u e n c i e s of d o u b l y c u r v e d panels are h i g h e r t h a n t h o s e of singly c u r v e d panels.
REFERENCES
1. Hancox, N. L. (ed.), Fibre Composite Hybrid Materials, Applied Science Publishers Ltd, London, 1981. 2. Kretsis, G., A review of the tensile, compressive, flexural and shear properties of hybrid fibre-reinforced plastics. Composites, 18 (1987) 13-23. 3. Chamis, C. C. & Lark, R. E, Non-metallic hybrid composites: Analysis, design, application and fabrication. In Hybrid and Select Metal Matrix Composites -- A State-of-the-Art Review, ed. W. J. Renton. AIAA Education Series, AIAA, New York, 1977, pp. 13-51. 4. Crawley, E. E & Dugundji, J:, Frequency determination
14.
and non-dimensionalization for composite cantilever plates. J. Sound Vibration, 72 ( 1980) 1- 10. Lin, D. X., Ni, R. G. & Adams, R. D., The vibration analysis of carbon fibre glass fibre sandwich hybrid composite plates. AIAA Paper 85-0605,, Proc. AIAA/ ASME/ASCE/AHS 26th Structures, Structural Dynamics and Materials Conf., Orlando, FL, 15-17 April 1985, Part 2, pp. 120-5. Iyengar, N. G. R. & Umaretiya, J. R., Transverse vibration of hybrid laminated plates. J. Sound Vibration, 104 (1986)425-35. Leissa, A. W. & Kadi, A. S., Curvature effects on shallow shell vibrations. J. Sound Vibration, 16 ( 1971 ) 173-87. Fortier, R. C. & Rossettos, J. N., On the vibration of shear deformable curved anisotropic composite plates. J. App. Mech., 40 (1973) 299-301. Fortier, R. C., Transverse vibrations related to stability of curved anisotropic plates. AIAA Journal 11 (1973) 1782-3. Sinha, P. K. & Rath, A. K., Vibration and buckling of cross-ply laminated circular cylindrical panels. AeronauticalQuarterly, 26 (3)(1975) 211-17. Soldatos, K. P. & Tzivanidis, G. J., Buckling and vibration of cross-ply laminated circular cylindrical panels. J. AppL Math. &Phys. (ZAMP), 33 (1982) 230-40. Soldatos, K. P., Influence of thickness shear deformation on free vibrations of rectangular plates, cylindrical panels and cylinders of antisymmetric angle-ply construction. J. Sound Vibration, 119 (1987) 1 l 1-37. Mustafa, B. A. J. & Ali, R., Prediction of natural frequency of vibration of stiffened cylindrical shells and orthogonally stiffened curved panels. J. Sound Vibration, 113 (1987) 317-27. Librescu, L., Khdier, A. A. & Frederich, D., A shear deformable theory of laminated composite shallow shell-type panels and their response analysis. 1: Free vibration and buckling. Acta Mechanica, 76 (1989) 1-33.
15. Chandrashekhara, K., Free vibration of anisotropic laminated doubly curved shells. Computers and Structures, 33, (1989) 435-40. 16. Viswanathan, A. V., Tamekuni, M. & Baker, L. L., Elastic stability of laminated, flat and curved, long rectangular plates subjected to combined in-plane loads. NASA CR-2330, June 1974. 17. Zhang, Y. & Matthews, F. L., Initial buckling of curved panels of generally layered composite materials. Composite Structures, 1 (1983) 3-30. 18. Whitney, J. M., Buckling of anisotropic laminated cylindrical plates. A1AA Journal, 22 (1984) 1641-5. 19. Rao, K. P. & Tripathy, B., On buckling of composite cylindrical panels. J. Aero. Soc. India, 41 (1989) 111-17. 20. Reissner, E. & Wan, F. Y. M., On the equations of linear shallow shell theory. Studies in Applied Mathematics, 48 (1969) 133-45. 21. Reissner, E. & Wan, E Y. M., A note on the linear theory of shallow shear-deformable shells. J. Appl. Math. & Phys. (ZAMP), 33 (1982) 425-7. 22. Reissner, E., On the derivation of the differential equations of linear shallow shell theory. In Flexible Shells: Theory and Applications, ed. E. L. Axelrad & E A. Emmerling. Springer-Verlag, Berlin, 1984. 23. Mollmann, H., Introduction to the Theory of Thin Shells. John Wiley, New York, 1981. 24. Yang, P. C., Norris, C. H. & Stavsky, Y., Elastic wave propagation in heterogeneous plates. Int. J. Solids Struct., 2 (1966) 665-84. 25. Reddy, J. N., Energy and Variational Methods in Applied Mechanics. John Wiley, New York, 1984.
Vibration and buckling of hybrid laminated curved panels 26. Dowell, E. H., Nonlinear flutter of curved plates. AIAA Journal, 7 (1969) 424-31. 27. Barai, A. & Durvasula, S., Free vibration of hybrid laminated plates with simply supported edges. Paper presented at the Technical Session of 39th Annual
27
General Meeting of the Aeronautical Society of India, Delhi, 10-11 March 1988. 28. Tsai, S. W. & Hahn, H. T., Introduction to Composite Materials. Technomic Publishing Co., Westport, CT, 1980.
Vibration and buckling of hybrid laminated curved panels A. Barai & S. Durvasula Department of Aerospace Engineering, Indian Instituteof Science, Bangalore -- 560 012, India Vibration and buckling of curved plates, made of hybrid laminated composite materials, are studied using first-order shear deformation theory and Reissner's shallow shell theory. For an initial study, only simply-supported boundary conditions are considered. The natural frequencies and critical buckling loads are calculated using the energy method (Lagrangian approach) by assuming a combination of sine and cosine functions in the form of double Fourier series. The effects of curvature, aspect ratio, stacking sequence and ply-orientation are studied. The non-dimensional frequencies and critical buckling load of a hybrid laminate lie in between the values for laminates made of all plies of higher strength and lower strength fibres. Curvature enhances natural frequencies and it is more predominant for a thin panel than a thick one.
NOTATION Panel dimensions along x- and yaxes, respectively A ~/B ~/,D i/ Elements of in-plane, in-plane/ flexural coupling, and flexural stiffness matrices, respectively, Eigenvector or coefficient of Am,l, Ars assumed modes Young's modulus of the material Ell, E22 of the panel in ply principal axes eq Matrix defined in eqn (9) GI2, Gl3, G23 Ply shear modulus in 1-2, 1-3 and 2-3 planes, respectively h Panel thickness i,j Index It, I2, I3 Mass and moments of inertia k Total number of laminae/2 k44, k45, k55,/¢ Shear correction factor K Stiffness matrix Kg, K~, K~ K~Y Geometric stiffness matrices m, n, r, s Indices in the deflection series M Mass matrix {M} Moment resultant vector Mx, My, Mxy Moment resultants N Maximum value of indices n, s {N} Stress resultant vector N_x, N.y, N_x, Stress resultants Nx, Ny, Nxy In-plane applied loads q(t), qi( t) Generalisednormal coordinate a, b
qll|tl
c)!kl ~_ij Qx, Qy r0
Rx, Ry, Rxy S t T u, v, w
u0, v0, w0 U 1/1 V2 x, y, z Zp(X, y) 6 {e} 0
Coefficients in the series expansion of deflection Reduced stiffness of kth ply with fibre orientation 0 Transverse shear force per unit length of laminate Curvature, defined in eqn (3) Non-dimensional in-plane applied load parameters (ATx,/Vy, 1Qxy) aE/ gT h E, respectively Region of integration Time Kinetic energy Displacement vector in rectangular or curvilinear coordinates In-plane displacement components Total potential energy ( VI + 112) Elastic strain energy Potential energy due to in-plane load Cartesian coordinates Equation of the mid-plane of curved panel Rise height of the curved panel mid-plane from flat configuration Mid-plane strain components defined in eqn (4) Ply orientation angle in degrees
15 Composite Structures0263-8223/92/S05.00 © 1992 Elsevier Science Publishers Ltd, England. Printed in Great Britain
16
2
A. Barai, S. Durvasula Mid-plane curvature components defined in eqn (4) Non-dimensional frequency parameter [~oaZ(pmh/ETh3)] 1/2 Poisson's ratio for orthotropic lamina such that v~2E22 = v21 Ell Transverse shear strain vector defined in eqn (7b) Non-dimensional coordinate for shell element Non-dimensional coordinates x/a and y/b, respectively Mid-plane stress components Summation symbol Average values of rotations of line, normal to the middle surface, over thickness Frequency of oscillation (rad/s) =
1"12, '!'21
{,,}
{o} Z
~P,, ~P, O)
1 INTRODUCTION Hybrid composites are produced by combining two or more types of fibres in a common matrix. These materials exhibit some of the desirable characteristics of the unmixed composites and simultaneously attempt to remove some of their unacceptable qualities. Motivation for the use of hybrid composites is well known and it arises from their improved specific strength and stiffness characteristics as well as reduced weight and cost. Hybrid composite construction has shown a potential for increased use in different applications. Applications have been attempted in a variety of areas like aerospace, mechanical and marine engineering.~ Since weight is an important factor in the design of aerospace structural components, hybrid composites provide extra benefit over the usual composites using a single fibre-matrix combination. Alternatively, the addition of cheaper fibres could help to reduce the cost of a composite made from expensive carbon or boron fibres, without seriously affecting the performance of the composite. Although the research and development work on hybrid composites began around 1970, no common names seem to have been given to them so far, in spite of the enormous amount of published data on material characterisation and fabrication techniques. Reference 1 reviews the various aspects of fibre reinforced hybrid composites. Kretsis 2 reviewed the present status of mechanical properties prediction methods mostly based on the rule of mixtures. He also listed a set
of hybrid laminates, used by several investigators for strength or stiffness improvements. Chamis and Lark 3 pointed out that the methods developed for single material composite laminates can be extended for hybrid laminates. However, only a limited number of results seems to be available in the literature on the vibration of hybrid laminated plates. Crawley and Dugundji 4 studied the free vibration characteristics of a cantilever configuration. They developed a scheme based on a partial Ritz (Kantorovich) analysis, which reduces the problem to a set of uncoupled ordinary differential equations. The solution of these equations yields the eigenvalues of free vibration. They also used the finite element method for comparison of some results. They considered a hybrid model with aluminium core and glass/epoxy face layers. An analytical and experimental study on the vibration of rectangular plates with free edges, using hybrid carbon-glass/fibre sandwich composites is reported in Ref. 5. The analysis is based on the finite element method with shear deformation and damping, lyengar and Umaretiya ~ considered hybrid composite plates with two opposite edges simply-supported and the other two edges clamped. In a number of applications, it is well known that thin plates are used as structural members. Some of these plates may have inherent curvature or develop small curvature under various in-plane loadings. Plates with small deviations from the initial flat configuration are known as curved plates in the literature. A major portion of aerospace structures fall into the latter category. Initial curvature has a beneficial effect because it raises the natural frequencies of oscillation. Free vibration behaviour studies of curved panels have been made using linear shell theory as well as nonlinear thin plate theory. They are reviewed by Leissa and Kadi. v All these studies are limited to isotropic materials. Within the scope of linear theory, free vibration behaviour of laminated curved panels is also studied in Refs 8-15. In most of these studies, closed-form solutions could be obtained for antisymmetric cross-ply laminates. Fortier 9 used the Rayleigh-Ritz method for the analysis of some angle-ply laminated curved panels without shear deformation. Only recently, Librescu et al.~4 developed a simple shear-deformable theory for doubly curved shallow cross-ply composite shells and used state-space concepts in conjunction with the Levy method to evaluate the static and dynamic response of panels for various boundary
Vibration and buckling of hybrid laminated curved panels conditions. Chandrashekhara ~5 studied the free vibration characteristics of curved panels using an isoparametric doubly curved quadrilateral shear flexible element. He used first-order shear deformation theory in conjunction with Sander's shell theory. As far as the authors are aware, there are no results on the free vibration behaviour of hybrid laminated curved plates. The buckling problem of laminated composite curved panels was first studied by Viswanathan et al.16 using linear shallow shell theory. They considered long panels with direct and shear loads. Sinha and Rath "~ studied the buckling characteristics of laminated cylindrical curved panel using Reissner's shallow-shell theory including shear deformation. They obtained closed-form solutions of the governing differential equations for simply-supported panels. Zhang and Matthews ~7and Whitney ]~ studied the buckling of cylindrical panels using Galerkin's technique. Rao and Tripathy j9 studied the buckling behaviour of a cylindrically curved panel using sublaminates. For the present study, Reissner's shallow-shell t h e o r y 2°-23 is used for the analysis of curved panels. A general formulation is developed for hybrid laminated curved plates including shear deformation and rotatory inertia. The expression for kinetic energy is obtained by neglecting the energy associated with small curvature. This is essentially the kinetic energy expression for the flat plate as could be directly obtained from YNS (Yang-Norris-Stavsky) theory.24 To obtain governing equations by applying Lagrange's equation, the displacements in the x-, y- and zdirections and rotation about the x- and y-axes are represented by double Fourier sine and cosine series.
2 FORMULATION
quite lengthy; therefore, they are omitted here (see Ref. 23). However, it is important to understand some of the assumptions used for the development of shallow-shell theory because they are important when applying them to a particular problem. Reissner and Wan 2~ developed the shallowshell theory on the basis of the following assumptions. A shell segment will be called shallow if the ratio of its height to base diameter is less than 1/8. The results obtained on the basis of this assumption will often also be applicable to shells which are not shallow, when the displacements are small with respect to undeformed configuration. Under these assumptions, the middle surface can be represented in the folowing form:
x = ~,, y = ~2, Zp = Zp(X, y)
(1)
so that the projections of the ~ - and ~2-curves on the xy-plane form straight lines parallel to the xand y-axes, respectively (see Fig. 1). Therefore, the coordinate curves on the middle surface will, in general, be neither orthogonal nor correspond to the lines of curvature. The slopes of the surface in the x- and y-directions are such that
maxlzo, iZp./] ~. 1
( i , j = x , y)
(2)
3
Fig. la.
Shell element in orthogonal coordinate system.
2.1 Strain-displacement relations for a shallow shell
The undeformed surface of the shell is taken as the reference plane. An orthogonal coordinate system (~l, ~2, ~3)is assumed, in which ~l and ~2 lie in the reference plane and ~3 is perpendicular to the ~ - and ~2-axes. The deformation geometry of a general shell element is complex and the mathematical description is tedious. However, for a shallow shell (or curved panel) it can be simplified considerably. Even so, the derivation of shallow-shell strain-displacement relationships is
17
Y
Fig. lb.
Curved plate geometry.
18
A. Barai, S. Durvasula
where zp., and Zpj are slopes of the mid-plane in the x- and y-directions, respectively. The midplane radii of curvature are given by:
If
h
r~j= - - -
1
-
n
- %--t-
:
-
On-1 'n
(3)
( i , j = x , y)
Z p, ij !
Within the scope of shallow-shell theory, the strain-displacement expressions are given by:2°,21 Ex =gxO +Zl('x;
o I
(a)
Ey = E y 0 "~-ZKy;
g xy = Cx~O + Z K.O,; Exz = W x -- ~).r;
Eyz = W,y -- lffy
where
~X
ExO = Uo,x -- WZp,xx;
(b)
EyO = Uo,y -- WZp, yy;
Laminate definition: (a) ply stacking sequence, (b) ply orientation.
Fig. 2.
e o~ = vo.x + uo,y - 2 wz,xy; K:r =
-- 1])x,x;
~), = -- ~[dy,y;
or
Kv = -('q-'x.y + ~y,x)
(4)
{M}=[B]{e}+[D]{K}
where uo and vo are in-plane displacements and exo, e~ and exg~. are in-plane strain components, respectively, on the middle surface.
ii
k45A45
(6b)
k55As~J[Gzl
or
2.2 Constitutive relations
(7b)
{Q}=[F]{v}
The constitutive relations of a laminate made of orthotropic lamina (see Fig. 2) with shear deformation are given by:
NI,
N~,,
fA,, [A,~
1 Aql ;,, I A2~
A6~Jteo,,j
B22
BI2
B 16
where k44 , k45 and k55 are the shear correction factors. For numerical calculations, they are assumed to be equal to/~, where (A;j,B OD~j)=
B26
B26]/~y ~ //
l
B66 j [ K.,J
(i,j= 1, 2,6)
(8)
h/2
(Sa)
or
(Sb)
{N} =[A]{ e} + [B]{ K}
[h/2 -~.(k)~l. z 2 Q~j ~ , z, )dz d - h/2
B,, B,2 B, l/ x / +
(7a)
Fii =
r d
k i jt0 ~ i(kl j dz
(i,j= 4, 5)
(9)
- h/2
The ~t.~ ~_,j are functions of six elastic constants and the ply angle 0, and they are different for different laminae in hybrid laminates. 2.3 Potential and kinetic energy expressions
Mxy
[B,6
+
B26
For a linear elastic solid, the elastic strain energy is given by:2s
B66_ItG~,J
D,,
D,2
D,6][r~ 1
Jl2
D22
D 16
D26
D26//Ky? // / D66 J [ K . J
lII
~=-~
[{e}r[A]leI+{e}r[BllrI+lrIT[Blle}
(6a) +{r}r[D]{r}+{v}r[F]{v}]dxdy
(10)
Vibration and buckling of hybrid laminated curved panels Expanding the A, B, D and F matrices and substituting for e, r and vfrom eqn (4), we obtain the strain energy expression. The potential energy of the in-plane loads due to transverse deflection is: 1
where (t~,/2,13) =
(1, z, z2)p/k~ dz J - h/2
and Plk) is the material density of kth layer of the hybrid laminate. The average density of a hybrid laminate is given by:
ff
II [&/z0 + wL + &/z0 + w)5 Ja
+ 2.~xy(Zp +W).x(Z p +w).y]dxdy
19
(11)
1 N
pro=--h Z plk)(h,-hk-,)
(17)
k=l
Now the expression for total potential energy can be given by: U= Vt + Vz
and should be used for natural frequency calculations.
(12)
The expression for kinetic energy is given by:25
pIk)[ft2 +#2 + ~2]dxdydz
T=-~
(13)
J J s , / - h/2
Within the scope of linear shell theory, the displacement expressions are given by:
u(x, y, z, t)=(1 +R)uo(x,y, t) + z~p,(x, y, t)
(z)
v(x, y, z, t) = 1 +-~ Vo(X, y, t) JrZ~Dy(X,y, t) (14)
w(x, y, Z, t) = w(x, y, t) Substituting these displacement expressions into eqn (13), the kinetic energy expression for the shell can be obtained. In addition to the flat plate terms, the expression contains terms due to curvature. These extra terms are in the form of:
h/2
dz ptkl(z, z 2) - j-h/2 R
2.4
Assumed deflection functions
To obtain governing differential equations using Lagrange's equations, the displacements (u, v and w) and rotations (~p, and ~py)in the potential and kinetic energy expressions are expressed in terms of admissible functions satisfying the geometric boundary conditions. The following simplysupported boundary conditions are considered for the present study (also known as S2 in the literature): v = w = 7J~ = 0
at x = 0, a
u=w=~Ox=O
aty=0, b
The geometric boundary conditions in eqns (18) are satisfied by the following combination of sine and cosine functions: M
F,.. cos (marx/a) sin(nary/b)
m = ] n = l
M
N
v=ZZ
Gmn sin(marx/a) cos(tory/b)
m=l
n=l
M
N
w=ZE m=l
M
(16)
N
n=l
M
N
m~l
+I2{(~x,t)(U,)+(lpy,t)(Vt)}
(19)
Z Xm,,cos(marx/a)sin(nary/b)
m=l
hV/y = ~ 2.5
H,.. sin(marx~a) sin(nary/b)
n=l
hVdx= ~
T=lfls[II{(U,t)2+(v,t)2+(Wt)2} + 13{(~dx.,)z + (7)y.,)2}]dxdy
N
.=ZZ
(15)
For a shallow shell, 1/R is small, and the above integral, taken over the plate thickness, works out to be negligibly small compared to the corresponding flat plate terms. Therefore, the fiat plate kinetic energy expression only is used in the present study which is given by:
(18)
Z Y,.. sin(marx/a ) cos(nary/b ) n=l
E q u a t i o n for t h e c u r v e d s u r f a c e
The shape of flight structure skin panels may be cylindrical, doubly curved or with curvature and
20
A. Barai, S. Durvasula
twist. For the present study, the equation of the mid-plane of the curved panel, with curvature in both directions but without twist is assumed in the following form: >
Z_p__2 [x-(a~2)] 2 6 (a/2) 2
[Y-(b~2)]2 (b/2) 2
(20)
form: 1
p,,,h[M]{4~,}+--s[K +Kg]{q~,}=Q(t) a
where [Kg] is the geometric stiffness matrix and is given by: X
where 26 is the maximum initial height and 26/a is small. The curvatures are given by: z""' = - 8(b/a2)' Zp,,:,,= - 8(6/b2),
(21)
Zp,xy = 0
For curvature only in one direction (say, the xdirection), the curvature expressions can be written as:
~-1
[x-(a/2)]2
6-
(a/2) 2
(22)
Then the curvatures are given by:
zp.~x= -8(6/a2),
Zp.vy= 0,
U
,%
M
2a a , , t = l
N
M
~ ~ • q,,,,qr, K,,,,,,, n=l
r=l
(24)
g=l
and A/
T=I,.I Pm h E /
m=l
N
M
N
E E E Clmn4rsmmnr,• i1=1 r = l
(28)
where R,, R v and R~v are the non-dimensional mid-plane force parameters. The vector qm. is of the order (5MN x 1) and contains the elements of F,,,,,, G,,,,,, H,,,,,, X,,,,, and Y,,,,, respectively. The mass matrix M and stiffness matrix K are of the order (5 MN x 5 MN).
3.1 Vibration analysis For free vibration, no external force acts on the system, so Q(t)=0; further, for an unstressed panel, the geometric stiffness matrix [Kg]=0. Therefore, eqn (27) reduces to:
p,,,hEM]{i],.,} + 4a [K]{q,.,} = 0
[K]{A,.~} = 22[M]{A,:,}
Substituting the assumed displacement functions eqns (19)and curvature expressions eqns (21)[or eqns (23)] into the potential energy expression eqn (12) and kinetic energy expression eqn (16), we obtain:
~
XV
For harmonic motion, substituting = {A~.,}eiw', in the above equation, we obtain:
3 THE GOVERNING DIFFERENTIAL EQUATIONS
l
y
[Kg] = - Rx[Kg]- RyEKg]- R.yEKg ]
"~p,xy ~ 0
(23)
=
(27)
(25)
s=l
where the qm,, represent, in general, the unknown coefficients Fro,,, etc., of eqns (19). Substituting the potential energy (U) and kinetic energy (T) expressions into Lagrange's equations, 25
(29) {qr~,} (30)
where (31)
~2 = p , , , h w 2 a 4
Equation (30) is a generalised eigenvalue problem. The eigenvalues ~2 and eigenvectors {A,,} are evaluated using standard procedures.
3.2 Buckling analysis For buckling, the term associated with the mass matrix in eqn (27) is set to zero and the combination of in-plane loads for which the initial configuration is no longer stable is calculated. Then, from eqn (27) we can write:
1 --~ [K + Kg]{Ar.,.}= {0} a
(32)
Alternatively, it may be written as: x
[K]{A~} = Rx[Kg]{Ar,} dt
0q =Q(t)
(26)
we obtain the governing equation in the following
+ Ry[K~]{Ar,} + Rxy[KgYl{A~}
(33)
The eigenvalue character of the problem is immediately seen.
Vibration and buckling of hybrid laminated curvedpanels If only Ny is acting, then: [K]{A,~}= Ry[K~g']{Ar~}
(34)
where Ry is the eigenvalue, similar to ,;[2 in eqn (31). This problem is also solved by the same procedures used to solve eqn (30). For biaxial loading, or combined biaxial and shear loading, the procedure is essentially similar by assuming two of the parameters and treating the third as the eigenvalue to be determined.
21
the hybrid construction works out to be lower. Therefore, for the present numerical calculations, for hybrid laminated curved panels, the abovementioned models only are considered. The hybrid laminate model designation scheme is shown in Table 2. The results are presented in non-dimensional form and they are non-dimensionalised with respect to E~.2 (or E r) of the graphite/epoxy material.
4.2 Free vibration of flat and curved panels 4 NUMERICAL CALCULATIONS
4.1 Hybrid laminate modelling Normally, in the literature, the elastic properties of fibrous composites are non-dimensionalised with respect to ET [or E22] for various numerical calculations. They are also used later for the initial comparison of some results. For hybrid laminates, dimensional elastic properties as shown in Table 1 are used. The elastic properties of kevlar/epoxy and glass/epoxy are taken from Ref. 28. The shear correction factor is assumed to be 5/6 for all calculations. However, for the comparison of free vibration and buckling results with the existing results, different sets of material properties and shear correction factors are considered which are taken directly from appropriate references. Although the anisotropic behaviour of a composite laminate reduces as the number of laminae increases, here eight layers are considered sufficient to show the effect of hybridisation. It was shown by the authors 27 that the natural frequencies of eight-layer laminates with 2-4 layers of higher strength material as face plies and the rest as core plies of lower strength material are nearly equal to the laminates with all eight layers of higher strength material. It is also possible that the frequencies of such laminates with four layers of higher strength material as face plies could be higher than those with eight layers of higher strength material, whenever the average density of
E22 v|2 G|2 GI3 G23
/9
Table 2. Hybrid laminate model designation Layer
Graphite/epoxy
Kevlar/epoxy
Glass/epoxy
128"0 GPa 11.0 GPa 0"25 4"48 GPa 4"48 GPa 1"53 GPa 1500"0 kg/m 3
76.0 GPa 5-5 GPa 0"34 2"30 GPa 2"30 GPa 2"30 GPa 1460.0 kg/m 3
38.6 GPa 8.27 GPa 0.26 4.14 GPa 4.14 GPa 4.14 GPa 1800.0 kg/m 3
Model no.
no.
1 2 3 4
HI
H~
H3
H4
H~
H~,
H7
Gr Gr Gr Gr
Kv Kv Kv Kv
GI GI GI GI
Gr Gr Kv Kv
Gr Kv Kv Kv
Gr Gr Gi GI
Gr GI GI GI
Gr, Graphite; Kv, kevlar; GI, glass.
Table3. Convergence study for the non-dimensional frequency parameter of a square fiat panel Order of matrix (mxn)
Table 1. Material properties
Ell
First, a convergence study for the non-dimensional frequencies of an unsymmetrically laminated cross-ply, cylindrically curved panel is made by increasing the number of terms in the assumed displacement functions of eqns (19) as shown in Table 3. The numerical convergence calculations for a cylindrically curved panel are first carried out for a composite made of single fibre construction in order to be able to compare results with those available in the literature: Table 4 shows the non-dimensional frequencies of unsymmetrically laminated curved panels with various curvatures compared with the results reported in Ref. 8. The results show good agreement. To study the effect of stacking sequence, symmetric laminates [0°/90°/0°] are considered. The non-dimensional frequency parameter, for the
2 3 4
2 3 4
No. of terms
Mode no.
1 20 45 80
19"0 19"0 19"0 (1,1) a
2
3
4
53"5 53"5 74"8 53"5 53'5 74-8 53"3 53"3 74-8 (2,1)(1,2)(2,2)
5 -112"8 112"8 (1,3)
a/h=50, Ell/E22=40.0, G|2/E22=1"0, G23/E22= 0"5, shear correction factor k = 5/6, vl2 -- 0"25, 2 = [toa2(pmh/E2~h3)l/2].
Lay-up: [0°/90°],
GI3/E22=
aModes.
1"0,
A. BaraL S. Durvasula
22
panel configuration and material properties used in Table 3, are shown in Table 5. Comparison of the results in Tables 4 and 5 shows that bending-extensional coupling for unsymmetric laminates significantly reduces natural frequencies. Table 4. Comparison of non-dimensional frequency parameters for square, fiat and curved panels
5/a
Coupling ignored B 0 = 0 (2 layers)
With coupling
B,j# 0 (2 layers)
Closed-form Present Closed-form Present solution (R-R method)" solution (R-R (ref. 8) (ref. 8) method) 0"0
19'0 ~ (1, I)' 52.9 (2,1) 52.9 (1,2) 74.3 (2,2)
19'0 (1,1) 53.3 (2,1) 53.3 (1,2) 74.8 (2,2)
11"6" (1,1) --
24.4" (1,1) 53"4 (2,1) 60.4 (1,2) 75"9 (2, 2)
24.4 (1,1) 53"8 (2,1) 60.8 (1,2) 76.3 (2, 2)
19.2" (1,1) --
36.0" (1,1) 54.9 (2, 1) 78.7 (1,2) 80.3 (2,2)
36.0 (1,1) 55.3 (2, 1) 79.0 (1,2) 80.8 (2,2)
32.6" (1,1) --
0.02
0.04
---
---
---
11"7 (1,1) 31.9 (2,1) 31.9 (1,2) 46.4 (2,2) 19.2 (1,1) 32"7 (2,1) 43.0 (1,2) 48.8 (2, 2) 32.7 (1,1) 35-1 (2, 1) 55.5 (2,2) 66.2 (1,2)
Lay-up: [0°/90°], material properties and panel geometry are the same as in Table 3, )1.= [~oa2(Pmh/E22h3p/2]. "Rayleigh-Ritz method. hTaken from graph. 'Modes. Table 5. Non-dimensional frequency parameter symmetrically laminated, square, curved panels
6/a
0.0 0"02 0.04 0.06
for
Table 6. Non-dimensional frequency parameter symmetrically laminated, square, curved panels
6/a
Mode no. 1
2
3
4
5
18.95 (1,1) 24.30 (1, 1) 35.84 (1,1) 49.40 (1,1)
27.39 (1,2) 40.37 (1, 2) 65.32 (1,2) 72.52 (2,1)
46.67 (1,3) 62.44 (1, 3) 70-79 (2,1) 86.89 (2,2)
69"38 (2,1) 69"73 (2, 1) 79.96 (2,2) 93-09 (1,2)
73.95 (2,2) 75.50 (2, 2) 95.18 (1,3) 109.49 (2,3)
Lay-up: [0°/90°/0°], (a/h = 50), material properties and panel geometry are the same as in Table 3, 2=[e~a2(pmh/
E22 h3)1/2].
Hence, the majority of the calculations in the various studies are made only for symmetric laminates. Tables 4 and 5 also confirm that the frequencies increase with curvature. Next, the free vibration behaviour of a square thick panel (a/h= 10) is studied and the results are presented in Table 6. The geometry and material properties are similar to those used in Table 3. Since this is a thick panel, and first-order shear deformation theory is used in the present formulation, there will be some error in the nondimensional frequencies. The results are presented to show the effect of thickness ratio on the free vibration characteristics of laminated curved panels. It is clear from Tables 5 and 6 that the relative increase in the frequencies for thin panels is more than that for the corresponding thick panels, for the same change in curvature. The increase in frequency for curved panels is due to the existence of coupling between bending and extension. This is a kind of geometric coupling which is well known for shell configurations and different from the coupling that exists in unsymmetric lay-ups which is due to anisotropic material behaviour. The extensional stiffness of a panel is several orders of magnitude greater than its bending stiffness. Therefore, for a small amount of lateral deflection, the extensional stiffness of a thin panel is much more effective than for a thick panel where deflection is resisted by bending stiffness. Before studying the free vibration characteristics of hybrid laminated curved panels, the behaviour of a hybrid laminated flat panel is studied first. For this purpose, a symmetrically laminated ([ + 0]2,) square plate of a/h = 40 is considered. The variation of the first three fre-
0-0 0.02 0.04 0.06
for
Mode no. 1
2
3
4
5
16.13 (1,1) 16.42 (1,1) 17.25 (1,1) 18.55 (1,1)
29.05 (1,2) 29-62 (1,2) 31.30 (1,2) 33.90 (1,2)
41.19 (2,1) 41.22 (2,1) 41.30 (2,1) 41.42 (2,1)
48.35 (2,2) 48.45 (2,2) 48.74 (2,2) 49.21 (2,2)
50.83 (1,3) 51.46 (1,3) 53.29 (1,3) 56-20 (1,3)
Lay-up: [0°/90°/0°], (a/h = 10), material properties and panel geometry are the same as in Table 3, £=[~oae(p,,h/
E22h3)1/2].
Vibration and buckling of hybrid laminated curved panels quencies in non-dimensional form [wa2(pmh/ E.rh3)] 1/2 with play angle (0) is plotted in Figs 3a, 3b and 3c, respectively, for all seven models. The first and second natural frequencies of angle-ply laminates are higher than laminates with all 0 ° or 90 ° layers. All models exhibit maximum frequencies at 45 °, except for the third mode. The 2 - 0 variation is symmetric about 0 = 4 5 °. As in the previous study (Ref. 27, for boron and kevlar materials), models H 4 and H 6 with four graphite/ epoxy layers as face plies and the remaining four layers of kevlar or glass as core plies exhibit frequencies quite close to model H~ with eight layers of graphite/epoxy. The third mode generally shows frequency maxima between 20 ° and 25 °, and 65 ° and 70 °, and a local minimum at 45 ° which is more than the value at 0 ° or 90 °. Model H 3 (all glass/epoxy) exhibits some peculiar characteristics in its third mode; the minimum frequency occurs at 45 ° and the maximum values at 0 ° and 90 °. The models H5 and H 7 with two graphite face plies and six kevlar or glass core plies exhibit intermediate frequencies with respect to models H~, H 2 and H 3. The behaviour of the fourth and fifth modes for all of the models are similar to that of the first two modes and they are not presented here. It is apparent that the hybrid construction under consideration raises the free vibration frequencies of flat plates in comparison with those of flat plates of single material construction. Now, the same square plate as mentioned above will be used as a reference for the study of curved plate behaviour. First, a small curvature in the xdirection only is introduced. For these studies only models H t -- a single material construction -- and H 4 and H 7 - - which are of hybrid construction -- will be considered. These three laminates are expected to demonstrate the salient features of the curvature effects as well as the effects of hybridisation. Figures 4a, 4b and 4c show the variation of the first three non-dimensional frequencies with ply angle for cylindrically curved panels with small initial curvature (6/a=0.01). The 2 - 0 variation curves for first three frequencies are similar to those of flat panels, presented in Figs 3a-3c. To study further the effect of curvature, the same square plate configuration (a/h = 50), made of graphite/epoxy, is considered. Figure 5a shows the variation of non-dimensional frequencies with ply angle for different values of curvature only in the x-direction. The non-dimensional frequency parameter increases with curvature and for small
a/b =1 a/h = 40
23
;~=¢oaZOOh/E.rh'~ ff'~ 141
12
1C
h
H3
:
I
I
0
I
20
I
I
40 O (deg)
I
I
60
I
I
80
Variation of fundamental frequency of a flat plate with ply angle;lay-up: [ _+012,.
Fig. 3a.
!
28}/
a/b = 1 o/h = ~0
)_ ,,o :o
2O
16
1 0 Fig. 3b.
20
40 e (deg)
60
80
Variation of second mode frequency o f a flat plate
with ply angle;lay-up: [ + 012,. 36
alb = I
32
28
24
20 I
0
Fig. 3c.
I
20
I
I
40 0 (deg}
I
I
I
60
I
80
Variation of third mode frequency o f a fiat plate
with ply angle; lay-up: [ _+0]2s.
A. Barai, S. Durvasula
24 a/b = 1 a/h = 50
16
40
H1
~.J2,11 / '~
6/a = 0.01
a/b=l, a/h = 50 GrGphite/Epoxy
/
14
X
12
10
~ 0
Fig. 4a.
I
20
I
I
40 e (deg)
1
60
80
Variation of fundamental frequency of a curved
0
plate with ply angle; lay-up: [ + 0]_,,. 30
H1
20
40 (3 [deg)
80
Fig. 5a. Variation of fundamental frequency of a cylindrically curved (in the x-direction) panel with ply angle; lay-up:
a/b = 1
[ + o]_~,.
alh = 50 =
60
.
26 a/b=1 , a / h = 5 0 AJJ,z) (z,i/~ 8 0 _ Graphite/Epoxy
f
22
"L
-
(,,2)
. . . .
60 18 40 I"_
0
Fig. 4b.
i
i
~
20
i
L
40 e (deg)
J
60
t
80
20
Variation of second mode frequency of a curved plate with ply angle; lay-up: [ + 0]=,.
38[
Q/b =1 QIh =50 6]a =0.1
•
Fig. 5b. ~'~
H
30
M6 H5
2 i
0
Fig. 4c.
20
i
1
40 O (deg)
1
20
]
I
I
40 0 (deg)
1
60
i
[
80
Variation of fundamental frequency of a doubly curved panel with ply angle: lay-up: [ + 0]2,.
/ ~.\
H1
X
0
I
60
1
i
80
Variation of third mode frequency of a curved plate with ply angle; lay-up: [ + 0]2.,.
values of curvature it attains a m a x i m u m value of 45 °. However, as the curvature increases (beyond 6/a = 0.04), the m a x i m u m shifts from 45 ° towards 0 °. For small value of curvatures (6/a = 0 - 0 1 and 0.02), the fundamental frequency occurs in c a m -
bination of m = 1 and n = 1. Thus, the curve panel behaviour is similar to that of a flat panel (6/a = 0-0). However, for 6/a=0.04 the curved panel behaves like a flat panel up to 30 ° (the fundamental frequency occurs in a c o m b i n a t i o n of m = 1 and n = 1 ). From 0 = 35 ° to 90 °, the panel behaves like a shell and fundamental frequency occurs in a combination of m = 2 and n = 1. For 6/a=0.06, the variation of fundamental frequency is similar to 6/a = 0.04, but shell m o d e (m = 2 and n = 1) appears at 0 = 20 ° and this trend is continued up to 90 °. For the same plate configuration, if curvature exists in both directions then an increase in frequency is observed c o m pared to the case of curvature in o n e direction only, for all ply angles, as s h o w n in Fig. 5b. It is seen that frequency maxima occur at 45 ° and that
Vibration and buckling of hybrid laminatedcurvedpanels the ;t-O variation is symmetric with respect to 0 = 45 °. In this case, the shell modes are observed for 6/a--0.02. At 0 = 40* and 45", fundamental frequencies occur in a combination of m = 2 and n -- 1, but at 0 = 50* it is in a combination of m = 1 and n---2. For curvatures beyond 6/a = 0.02, the fundamental frequencies for different ply angles occur due to various combinations of m and n, as shown in Fig. 5b.
4.3 Buckling of flat and curved panels First, a comparison for the critical buckling load is made using composite panels with a single fibre-matrix combination, because some results are available in literature for such laminates. The results are presented in Table 7 for unsymmetrically laminated curved panels with a/b 1 and a~ h -- 100. The panel is made of graphite/epoxy and its elastic properties are also presented in Table 7. The results are compared with the closed-form solution of Ref. 10. The results show good agreement. =
Table 7. Comparison of critical buckling loads (Erha)] of curved panels
6/a
Closed-form solution" Ref. 10
0.0 0.5 1.0
9.5 68,0 139,0
[Nya21
Present (R-R method) 9,66 68.34 141.25
Lay-up: [0°/90°], a/h=lO0, EH =172 GN/m 2, E22=6"89 GN/m z, G12=3.45 GN/m 2, G2.~=Gt3=l'38 GN/m 2, Vl2 = 0.25, shear correction factor k = 2/3, "Taken from graph. bRayleigh-Ritz method.
25
The variation of critical buckling loads (Nya2/ E r h 3) of flat panels with eight layers, symmetrically laminated, is shown in Fig. 6. The critical buckling loads for angle-ply laminates are higher than for all zero degree layers, for all models. Hybrid laminates, with four layers of higher strength material as face plies and four layers of lower strength material as core plies, exhibit buckling behaviour similar to all higher strength material, for similar ply orientations. Laminates with two layers of higher strength material as face plies and six layers of lower strength material as core plies exhibit critical buckling load which lies between the critical buckling load of all higher strength material and all lower strength material, for all ply orientation angles. All models show buckling load maxima when 0-- 55 °, except model H 3 (all glass/epoxy) which shows a maximum value at 50 °. Like the free vibration characteristics, the critical buckling load of curved panels increases with curvature. Figure 7 shows the critical buckling load for all models with small initial curvature (6/a = 0.02) while other parameters are similar to the flat panel in Fig. 6. one significant difference in response compared to the flat panel is that all curved panels attain buckling load maxima at two ply angles (35 ° or 40 ° and 60 °) except model H6 which has a maximum value at 60 °. The effect of curvature on the buckling load versus ply angle response is shown in Fig. 8, for model H4 (a/h=60). For small curvature, the maximum buckling load occurs only at one ply angle (50 °) but as the curvature increases, maxima
36
28 N y az
Nya 2
E+h 3 20
2l--
Fig. 6.
I (
./h =SO I
0
o/b =1 o/h = SO
I
20
I
I
I
40 9 (deg)
I 6/0
60 ~
I= o
I
I
80
0
Variation of critical buckling load of a fiat plate with ply angle; lay-tip: [ _+012~.
Fig. 7.
I
I
20
I
I
40 0 (deg)
I
I
60
I
I
I
80
Vafiationofcfiticalbuckling load of a cylindrically curved panel with ply angle; lay-up: [ ± 0]2s.
26
A. Barai, S. Durvasula 80 o/b
6o "Y°' L
/
= 1
5.
\ 6/o=00, m,
ETh3
o.o3
6. 7. 8. 0t
f
i
20
i
I
i
40 e (deg)
I
60
i
i
9.
80
Fig. 8. Variation of fundamental frequency of a cylindrically curved (in the x-direction) panel with ply angle; lay-up: [-+ 0L~.
o c c u r at two d i f f e r e n t ply angles. T h e critical buckling l o a d n o r m a l l y increases with c u r v a t u r e .
10. 11. 12.
13. 5 CONCLUSIONS
Hybrid construction in a laminated composite plate increases the natural frequencies and critical buckling load in comparison with single material construction. The increase in natural frequency and critical buckling load for curved panels is essentially d u e to the p r e s e n c e of b e n d i n g e x t e n s i o n coupling. T h e effect o f c o u p l i n g is g r e a t e r in the case o f thin panels ( a / h = 50) a n d c o m p a r a t i v e l y less f o r thick p a n e l s ( a / h = 10). F o r s o m e values o f c u r v a t u r e a n d ply angle, the p a n e l b e h a v e s m o r e like a shell a n d the f u n d a m e n t a l m o d e o c c u r s for a c o m b i n a t i o n of h i g h e r indices (e.g. m = 2 , n = l o r m = l , n = 2 ) especially for angle-ply laminates. F r e q u e n c i e s of d o u b l y c u r v e d panels are h i g h e r t h a n t h o s e of singly c u r v e d panels.
REFERENCES
1. Hancox, N. L. (ed.), Fibre Composite Hybrid Materials, Applied Science Publishers Ltd, London, 1981. 2. Kretsis, G., A review of the tensile, compressive, flexural and shear properties of hybrid fibre-reinforced plastics. Composites, 18 (1987) 13-23. 3. Chamis, C. C. & Lark, R. E, Non-metallic hybrid composites: Analysis, design, application and fabrication. In Hybrid and Select Metal Matrix Composites -- A State-of-the-Art Review, ed. W. J. Renton. AIAA Education Series, AIAA, New York, 1977, pp. 13-51. 4. Crawley, E. E & Dugundji, J:, Frequency determination
14.
and non-dimensionalization for composite cantilever plates. J. Sound Vibration, 72 ( 1980) 1- 10. Lin, D. X., Ni, R. G. & Adams, R. D., The vibration analysis of carbon fibre glass fibre sandwich hybrid composite plates. AIAA Paper 85-0605,, Proc. AIAA/ ASME/ASCE/AHS 26th Structures, Structural Dynamics and Materials Conf., Orlando, FL, 15-17 April 1985, Part 2, pp. 120-5. Iyengar, N. G. R. & Umaretiya, J. R., Transverse vibration of hybrid laminated plates. J. Sound Vibration, 104 (1986)425-35. Leissa, A. W. & Kadi, A. S., Curvature effects on shallow shell vibrations. J. Sound Vibration, 16 ( 1971 ) 173-87. Fortier, R. C. & Rossettos, J. N., On the vibration of shear deformable curved anisotropic composite plates. J. App. Mech., 40 (1973) 299-301. Fortier, R. C., Transverse vibrations related to stability of curved anisotropic plates. AIAA Journal 11 (1973) 1782-3. Sinha, P. K. & Rath, A. K., Vibration and buckling of cross-ply laminated circular cylindrical panels. AeronauticalQuarterly, 26 (3)(1975) 211-17. Soldatos, K. P. & Tzivanidis, G. J., Buckling and vibration of cross-ply laminated circular cylindrical panels. J. AppL Math. &Phys. (ZAMP), 33 (1982) 230-40. Soldatos, K. P., Influence of thickness shear deformation on free vibrations of rectangular plates, cylindrical panels and cylinders of antisymmetric angle-ply construction. J. Sound Vibration, 119 (1987) 1 l 1-37. Mustafa, B. A. J. & Ali, R., Prediction of natural frequency of vibration of stiffened cylindrical shells and orthogonally stiffened curved panels. J. Sound Vibration, 113 (1987) 317-27. Librescu, L., Khdier, A. A. & Frederich, D., A shear deformable theory of laminated composite shallow shell-type panels and their response analysis. 1: Free vibration and buckling. Acta Mechanica, 76 (1989) 1-33.
15. Chandrashekhara, K., Free vibration of anisotropic laminated doubly curved shells. Computers and Structures, 33, (1989) 435-40. 16. Viswanathan, A. V., Tamekuni, M. & Baker, L. L., Elastic stability of laminated, flat and curved, long rectangular plates subjected to combined in-plane loads. NASA CR-2330, June 1974. 17. Zhang, Y. & Matthews, F. L., Initial buckling of curved panels of generally layered composite materials. Composite Structures, 1 (1983) 3-30. 18. Whitney, J. M., Buckling of anisotropic laminated cylindrical plates. A1AA Journal, 22 (1984) 1641-5. 19. Rao, K. P. & Tripathy, B., On buckling of composite cylindrical panels. J. Aero. Soc. India, 41 (1989) 111-17. 20. Reissner, E. & Wan, F. Y. M., On the equations of linear shallow shell theory. Studies in Applied Mathematics, 48 (1969) 133-45. 21. Reissner, E. & Wan, E Y. M., A note on the linear theory of shallow shear-deformable shells. J. Appl. Math. & Phys. (ZAMP), 33 (1982) 425-7. 22. Reissner, E., On the derivation of the differential equations of linear shallow shell theory. In Flexible Shells: Theory and Applications, ed. E. L. Axelrad & E A. Emmerling. Springer-Verlag, Berlin, 1984. 23. Mollmann, H., Introduction to the Theory of Thin Shells. John Wiley, New York, 1981. 24. Yang, P. C., Norris, C. H. & Stavsky, Y., Elastic wave propagation in heterogeneous plates. Int. J. Solids Struct., 2 (1966) 665-84. 25. Reddy, J. N., Energy and Variational Methods in Applied Mechanics. John Wiley, New York, 1984.
Vibration and buckling of hybrid laminated curved panels 26. Dowell, E. H., Nonlinear flutter of curved plates. AIAA Journal, 7 (1969) 424-31. 27. Barai, A. & Durvasula, S., Free vibration of hybrid laminated plates with simply supported edges. Paper presented at the Technical Session of 39th Annual
27
General Meeting of the Aeronautical Society of India, Delhi, 10-11 March 1988. 28. Tsai, S. W. & Hahn, H. T., Introduction to Composite Materials. Technomic Publishing Co., Westport, CT, 1980.