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Buckling and vibrations of unsymmetric laminates resting on elastic foundations under inplane and shear forces
COMPOSITE STRUCTURES ELSEVIER
Composite Structures 44 (1999) 3141
Buckling and vibrations of unsymmetric laminates resting on elastic foundations under in-plane and shear forces M.A. Aiello *, L. Ombres Department
of Materials
Science, University of Lecce, Via per Amesano,
73100 Lecce, Italy
Abstract Buckling loads, free vibrations and vibrations with initial in-plane stresses for moderately thick, simply supported rectangular laminates resting on elastic foundations such as the Pasternak type, are examined in this paper. The analysis is carried out considering unsymmetric flat laminates subjected to biaxial compression and shear loads. First-order shear deformation theory, in conjunction with the Rayleigh-Ritz method is used for the evaluation of buckling loads and vibrations by means of a standard eigenvalue procedure. A large numerical investigation is intended to demonstrate the influence of laminate parameters (fiber orientations, degree of anisotropy, number of layers) and foundation parameters (modulus of subgrade reaction and shear modulus); the results obtained are reported and discussed in the paper. 0 1999 Elsevier Science Ltd. All rights reserved.
Notation width and thickness, respectively Extensional, coupling and bending laminate stiffnesses Young’s moduli along the fiber direction and normal to the fibers, respectively In-plane shear moduli Poisson ratio Displacements in the x-, y- and z-directions, respectively In-plane displacements of the middle plane of the laminate Laminate strains Shear rotations In-plane applied loads Laminate curvatures Elastic strain energy Work of internal stresses Work of external stresses Total potential energy Kinetic energy of the laminate Loads multiplier
a, b, h
Laminate
A,, Bij, Dij EII,
J522
G2,
G3,
G23
1’12 u,
v,
w
uo, 00, wo
l
Gb
Frequency of vibrations Density of materials Modulus of subgrade reaction Shear modulus of the subgrade
n
Nxb2/E22
5 i
NJNx
0
Corresponding author
0263-8223/99/$
length,
Pi ks
h3
N,JNx Wb2/h(plE22)1’2 k,b4/E22 Gb2/E22
h3 h3
1. Introduction Plates supported by elastic foundations represent a very common model in structural engineering; as a consequence the number of technical problems connected with this model, is very large. Generally the analysis is developed on the assumption that the reaction forces of foundation are proportional at every point to the deflection at that point (Winkler model). However, to represent the characteristic behavior of practical foundations, the Winkler model is not adequate and can be substituted by other models such as the Filonenko-Borodhic, Pasternak or Vlasov models [14]. In many cases the two-parameter Pasternak model has proved to be the most effective.
- see front ma.tter 0 1999 Elsevier Science Ltd. All rights reserved
32
M.A. Aiello. L. Ombres I Composite
In the field of composite materials, the model of laminated flat plates resting on elastic foundation is used for the analysis of several structural elements such as, for example, sandwich laminated panels. For this reason, a knowledge of the behavior of laminated plates supported by an elastic foundation of the Pasternak type, is very useful mainly from a design perspective. The evaluation of buckling loads, free vibrations and vibrations with in-plane pre-loads in the field of composite laminates represents a classical subject to many researchers [5]; results of these studies, regarding also laminates resting on elastic foundations, are numerous in the literature [6-121. However, particularly for laminated plates resting on Pasternak foundations, the analysis of vibrations considering the combined effects of foundations and inplane initial stresses is not complete. This problem, in fact, has been analysed only with reference to isotropic thin and, recently, moderately thick symmetric cross-ply laminates subjected to uniformly distributed uniaxial or biaxial forces [l]. The results obtained showed that combined effects of foundation parameters and in-plane stresses can greatly influence the value of plate vibrations [l]. The analysis of this problem becomes, then, essential in order to determine the behavior of laminates under service conditions. On the basis of these considerations, the paper is devoted to the analysis of laminate vibrations in the presence of in-plane stresses under loads smaller than buckling loads. The analysis is carried out with reference to unsymmetric flat simply supported laminates subjected to combined biaxial compression and shear loads. Buckling loads and vibration are determined as solutions of a standard eigenvalue problem by an energetic approach based on the first-order shear deformation theory in conjunction with the Rayleigh-Ritz method. In the case of unsymmetric laminates, because of the bending-stretching coupling, the evaluation of buckling loads as bifurcation load needs to consider added conditions called ‘flatness conditions’ [13] that enable us to take into account the initial curvature of laminates. Flatness conditions furnish a set of restrictions on edge moments and transverse edge shear, depending on the loading and boundary conditions. In order to demonstrate the influence of geometrical and mechanical parameters of laminates (aspect ratio, degree of orthotropy of layers, fiber orientation) and foundation parameters (modulus of subgrade reaction, shear modulus of subgrade) on the buckling and vibrations of laminates, a parametric numerical investigation for simply supported rectangular moderately thick plates subjected to in-plane
Structures
44 (1999) 31-41
compressive conducted.
forces
2. Mathematical
combined
with shear forces,
is
formulations
2.1. Constitutive law of laminates Considering a flat rectangular laminate made with orthotropic layers subjected to in-plane and shear forces the constitutive law is expressed by the following relationships (Fig. 1):
P{K) {MI= PI{&)+ Pl{~) IN] = M{c] +
{Ql = PI{vI being
h/2 (Aij, B,, Dij) = / Qij(l,~>~)
dz (i,j=
1,276)
-h/2 h/2 &j =
kijQjj dz (i,j = s -h/2
4,5)
where k, are shear correction factors, qii, functions of elastic constants and the ply angle 8, differs from one layer to another, while:
Nx i
Fig. 1. Laminate loads.
c
definition:
ply orientation,
ply stacking
and in-plane
MA. Aiello, L. Ombres I Composite Structures 44 (1999) 3141
cS~E=II+A(L;-L,)
33
(1)
where IT = W + U is the strain energy of the laminate, L; the work of internal stresses in the initial configuration for the second-order strain components, L2 the work corresponding to the external loads for the second order displacements components, 62E the second variation of total potential energy, and ;1 the loads multiplier. For a flat rectangular laminate, the work L; is expressed as:
with
~0, ug, EA, Q, and .sXyO being displacements and strains at the middle plane of the laminate. +
2.2. Energy expression:: The total potential energy of a laminate resting on an elastic foundation is expressed as: sZ= W-T+U with W the strain energy of the laminate, T the kinetic energy of the laminate, and U the strain energy due to the Pasternak foundation. Considering a flat rectangular laminate of a and b dimensions along the X- and y-axis, respectively, the strain energy W and the kinetic energy T are expressed as: w+ ss
Gii’lC4 + W’[W4 hdv
+
+A($: being
+ @I dx(dY
&=Jpwdz. &; I2 =/rp(k)z >1, , Jp(k) h/2
I, =
[{s]%I{el. + {s]r[BI{~) + {~]r[B]{s)
-h/2
h/2
-hi2
h/2
-h/2
where pck)is the mass density per unit volume of the k-th layer. The strain energy due to the Pasternak foundation is given by: a b u=;
bt+G(d,x +4,) 1dxdv JJ 00
where k, is the modulus of subgrade reaction and Gt, is the shear modulus of the subgrade.
W&o,xuoy+ uoxvoy+ yxyy)] dx dy.
The work L2 is zero when external loads are constant or independent from U, v and w displacements. For unsymmetric laminates in which the geometric middle plane is different from the neutral plane, the buckling load can be determined as a bifurcation load only if added conditions, which guarantee the flatness of laminates, are imposed. This problem has been investigated by Leissa [13] who defines conditions for unsymmetric laminates to remain flat during the pre-buckling stage when subjected to in-plane loads. Leissa separates the pre-buckling response equations from the buckling equations and works out the conditions for the curvatures to vanish during the pre-buckling stage (flatness conditions): if these conditions are satisfied, the bifurcation phenomenon will occur. For a simply supported laminate subjected to in-plane loading, the flatness conditions involve the introduction of bending moments @ and q acting along the edges of laminates, in the initial configuration. The values of q and q are determined using constitutive laws of the laminate with first-order components of displacements and shear rotations. Bending moments M” and MO furnish a value of L2 that must be consfdered fey the evaluation of d2E. Substituting II, L; and L2 expressions in Eq. (l), the condition 62E = 0 gives the relations that enable determination of the buckling loads of laminates. The solution of the problem is obtained by the Rayleigh-Ritz method assuming linear variation of in-plane displacements u and v and constant transverse deflection w over the panel thickness: U(?Y,Z) = Uo(X,Y) +zIc/,(x,v)
2.3. Buckling load analysis The buckling load of a flat laminate is determined by an energetic approach founded on the Dirichlet principle, expressed as follows [14]:
hY,Z)
= vo(x,r)
W(X,Y,Z)
= W(X,Y).
+z+y/,(x,Y)
For a simply supported flat laminate the displacements that satisfy the geometric boundary conditions:
34
M.A. Aiello, L. Ombres I Composite
v=w=*,=o
at x = 0, a
u=w=*X=o
at y = 0,b
ypIn” m
cos(mrc~/a) sin(nny/b)
n
vO(x,y) = yyIV& m n
sin(mm/a) cos(nrry/b)
$Jx, Y) = TX&B m n
cos(m7-la)
W(X,Y) =
sin(n~ylb)
(2)
WI+ WlHq~= 0
W and L2
(4)
(5) in which [L] is the matrix containing the terms corresponding to the in-plane pre-loads.
Numerical analysis is carried out with reference to flat rectangular simply supported laminates resting on elastic foundations. Results are obtained for symmetric and
(3)
which represents a system of linear equations of 5mn x 5mn order. The solution of Eq. (3) is obtained as the solution of a standard eigenvalue problem that can be solved by available standard procedures. [L] and [Kj are matrices of 5mn x 5mn order that contain terms relative to strain energy and external work; the vector {q} contains series coefficients U,,, I’,,, W,,, X,, and The buckling load coincides with the smallest eigenvalue of Eq. (3). In the presence of biaxial compression and shear the solution is obtained considering ratios:
and evaluating the critical value of N,. 2.4. Vibration analysis The analysis of laminate vibrations is carried out by the same procedure described for the buckling analysis. In this case the total potential energy R must be derived considering the maximum value of the kinetic energy besides the strain energy. Evaluating free vibrations, external loads are equal to zero then [L] = 0. In accordance with the Rayleigh-Ritz method, the analysis is carried out assuming periodic displacement functions with time of the following form: _ U(X,y,
WI + ~2MHql = (01
where [n/rl is the mass matrix of the laminate and {q} the vector of unknowns coefficients. Free vibrations correspond to eigenvalues of Eq. (4). For vibrations in the presence of in-plane stresses, Eq. (4) becomes:
3. Numerical investigations
sin(mrc.x/a) sin(nrcy/b). ~y?+Kln
Substituting Eq. (2) into the expressions Eq. (1) becomes:
w
44 (1999) 31-41
Minimization of the total potential energy furnishes the following governing equation:
are chosen in the form of the following trigonometric series:
U&Y) =
Structures
t) = u(x, y) sin tit
Table 1 Comparison of uniaxial buckling loads for square symmetric laminates resting on Pastemak foundation 5
where u(x,y), v(x,y) and w(x,y) are the displacement functions expressed by Eq. (2) and o is the frequency of vibration.
b/h = 1000
b/h= 10
Ref. [l]
Present
Ref. [I]
Present
50.7515 49.2666
50.7511 49.2666
66.2895 66.2895
66.2888 66.2888
22.2288 19.5904
22.2288 19.5902
26.7300 20.9930
26.7300 20.9928
(n =~%$b’/E~~h’; Sk = 100; 6, = 10). Lay-up: 0”/90”10”; EillEz2 =40; Glz= f&=0.6 E2*; &~,=0.5 E,,; ~,~=0.25.
Table 2 Comparison of non-dimensional svmmetric laminates
frequency parameters
for square
Mode
Present Ref. [lo]
1
2
3
4
5
19.058 18.950
27.659 27.390
46.977 46.670
69.439 69.380
74.268 73.950
o = Wb2(plE22) ‘i2., Lay-up: 0”/90”/0”; a/h = 50; El l/E22 = 40; ($2 = G,3 =0.6 E,,; G2) = 0.5 E22; v,~ = 0.25.
Table 3 Comparison of non-dimensional fundamental frequency parameters for square antisymmetric laminates
e
Number of layers (n)
Present
Ref. [I l]
30” 45” 45”
10 10 4
18.585 19.454 18.960
18.510 19.380 18.460
u(x,y, t) = v(x,y) sin Gt
w(x, y, t) = w(x, y) sin tit
alb
w = tib2(plE&/2; Lay-up: 0l-WV-01. J-8; alh = 10; E111E22= 40; G,~=G,3=0.6 E,,; G23=0.5 Ez2; ~,~=0.25.
MA. Table 4 Comparison unsymmetric
of non-dimemional laminates
frequency
Aiello, L. Ombres I Composite
parameters
for
square
Mode
Present Ref. [S]
1
2
3
4
11.835 11.700
32.064 31.900
32.064 31.900
46.947 46.400
ru = 8b2(p/E22)“2; Lay-up: 0”/90”; a/h = 50; G,J = 1.0 E22; G,, =0..5 Ez2; v,~ = 0.25.
-
EII/Ez2 = 40;
G,2 =
Table 5 Comparison of non-dimensional frequency parameters for square symmetric laminates resting on elastic foundation in the presence of innlane stresses
5
b/h
Ref. [l]
Present.
Ref. [l]
Present
Ref. [l]
Present
(1,l)
(1,l)
(1,2)
(1,2)
(2,l)
(2.1)
0
10 1000
264.51 327.12
264.53 327.14
825.38 991.94
825.39 991.96
992.70 4413.0
992.70 4413.20
1
10 1000
295.26 390.43
295.26 390.45
534.85 659.53
534.86 659.53
1441.8 5062.0
1441.8 5061.8
(9 = wb2(p/Ez2) “‘; iy = 0.5; & = 100; 6, = 10. Lay-up: G,z=G,s=0.6 E22; G2,=0.5 E22; ~,~=0.25.
0”/90”/O”; El,/
E,,=40;
unsymmetric laminates assuming the shear correction factor equal to kij = 5/6. 3.1. Comparison with available data Initially the analysis is carried out considering symmetric laminates resting on elastic foundations: results are then compared with those available in the literature and reported in Tables l-5 for buckling loads, free vibrations and vibratio:ns in the presence of in-plane stresses. The results obtained demonstrate good agreement between the present model and models defined by other authors [8,10,11].
Structures
44 (1999) 3141
3.5
In order to determine the influence of mechanical parameters of foundations, in the following figures and tables, results obtained for unsymmetric laminates made with four layers of constant thickness and lay-up B/-8/ &/8, are shown. The 6, value is assumed equal to O”, 45” and 90”, while values of foundation parameters have been chosen in order to represent the Winkler model (& = 100, 6, = 0), the Filonenko-Borodhic model (6k = 0, 6, = 10) and the Pasternak model (& = 100, 6, = 10): results corresponding to each model are compared with those relative to unrestrained laminates (& = 0, 6, = 0). Figures 2 and 3 show the variation of uniaxial buckling loads versus lamination angle varying foundation parameters for unsymmetric simply supported laminates with (!A, = 45” and & = 90”, respectively. The results obtained confirm the very sensible influence of foundation on the behavior of laminates: buckling loads are higher than ones corresponding to unrestrained laminates and their maximum values are obtained for different lamination angles varying foundation parameters. For laminates resting on elastic foundations the maximum value of buckling loads corresponds to 6’ nearly equal to 20”. The best response has been found for a two-parameter foundation (Pasternak model). For unrestrained laminates maximum buckling loads are obtained with 8 nearly equal to 40”. In the presence of shear forces, as shown in Figs. 4 and 5, buckling load values are lower than those corresponding to uniaxial compression; however, in this case the behavior of laminates, varying foundation pa-
n
3.2. Parametric analysis Results of a numerical parametric analysis, carried out with reference to unsymmetric laminates, are shown in following tables and figures. Materials properties used in the analysis are: Material I: E,,lEzz = 40; G,z = G,3 = 0.6 E22; G23= 0.5 E22; VI2= 0.25 Material II: E,,lEzz = 10; G,z = G,3 = 0.5 E22; G23= 0.5 E22; 1112 = 0.30 Results are shown considering the following non-dimensional quantities: & = ksb4/Ez2h3; 6, = Gbb2/Ez2h3
Fig. 2. Non-dimensional unsymmetric laminates terial II.
buckling load versus lamination angle for W-8/45/8 (b/h = 50; u/b = 2; < = 0; [ = 0). Ma-
M.A. Aiello, L. Ombres I Composite Structures 44 (1999) 31-41
36
Fig. 3. Non-dimensional unsymmetric laminates terial II.
buckling load versus lamination angle for 8/-0/90/O (b/h = 50; u/b = 2; 5 = 0; ( = 0). Ma-
rameters, is similar. Maximum values of buckling loads, in fact, are obtained for 8 = 0” or &’= 90” in all case In Fig. 6 n-8 diagrams for unsymmetric lamin; subject to biaxial compression are compared; res show that curves corresponding to unrestrained la____ nates and laminates resting on Filonenko-Borodhic foundation are almost parallel and the maximum value of the buckling load is attained with the same 0 value.
Fig. 5. Non-dimensional unsymmetric laminates terial II.
buckling load versus lamination angle for f3/-0/45&l (b/h = 50; u/b = 2; 5 = 0; < = 1). Ma-
Analogous behavior can be seen for laminates resting ‘asternak and Winkler foundations. lurves drawn in Fig. 7 relate to laminates under 14 .ial compression and shear forces: it is possible to see _-._._the behavior corresponds to that of laminates under uniaxial compression and shear forces. In this case, however, the best response of laminates corresponds to the value 0 = 90”.
n
n
20 15 10 5
e 0
15
Fig. 4. Non-dimensional unsymmetric laminates terial II.
30
45
60
75
90
buckling load versus lamination angle for 8/-0/O/0 (b/h = 50; u/b = 2; c = 0; { = 1). Ma-
8
0 0
15
Fig. 6. Non-dimensional unsymmetric laminates terial II.
30
45
60
75
90
buckling load versus lamination angle for 0/-0/90/Q (b/h = 50; u/b = 2; 5 = 1; [ = 0). Ma-
M.A. Aiello, L. Ombres I Composite
n
Structures
Table 6 Buckling loads and foundation
20
e 0 15 24 25 30 37 45 60 75 90 Lay-up:
15
0
Fig. 7. Non-dimensional unsymmetric laminates terial II.
30
45
60
75
90
buckling load versus lamination angle for 8/-Q/O/0 (b/h = 50; a/b = 2; < = 1; [ = 1). Ma-
n
Fig. 8. Non-dimensional buckling load versus lamination angle for unsymmetric laminates 6+&f&/0 (b/h = 50; a/b = 2; Sk = 100; 6, = 10; &, = 0”, 45”, 90’). Material Il.
In Fig. 8, n-8 curves for all load combinations examined for laminates resting on Pasternak foundations, varying the O0 values, are drawn. The results obtained show the influence of loading combinations on the buckling load and also that 80 variation has little effect. Results of buckling analysis are also shown in Tables 69. Vibrations of laminates resting on elastic foundations in the presence of in-plane stresses, are shown in
for unsymmetric parameters
laminates
varying
lamination
&=O
61; = 100
s,=o
6,=0
&=O a,=10
61, = 100 6,=10
7.523 9.702 12.637 13.003 13.553 13.782 13.551 11.327 8.957 7.883
21.317 22.104 22.945 22.804 22.047 20.772 19.007 15.437 12.053 10.747
31.185 32.296 33.560 33.685 33.123 31.916 30.102 26.065 22.200 20.713
39.761 39.147 38.322 38.208 37.548 36.306 34.407 28.659 24.742 23.078
0/-0/0”/0.
Table 7 Buckling loads and foundation
Material
for unsymmetric parameters
laminates
varying
lamination
&=O s,=o
61,= 100 6, =o
&=O 6, = 10
6k = 100 6,=10
0 15 30 45 51 60 68 75 90
1.628 2.059 3.123 4.307 4.760 5.371 5.821 5.673 5.537
6.506 7.092 8.650 9.939 10.127 9.743 9.341 8.990 8.491
11.628 12.060 13.123 14.307 14.759 15.371 15.820 15.670 15.536
16.506 17.091 18.650 19.938 20.126 19.742 19.340 19.000 18.490
Table 8 Buckling loads and foundation
Material
for unsymmetric parameters
laminates
varying
lamination
&=O 6,=0
& = 100 s,=o
&=O a,=10
61, = 100 a,=10
0 15 30 45 60 75 90
5.536 5.896 6.976 7.191 6.778 6.235 6.768
11.650 10.366 10.141 9.684 9.000 8.285 8.815
18.286 16.898 16.611 16.041 15.358 15.200 16.478
21.525 19.495 18.525 17.900 17.173 16.863 18.253
9/-@/90”/0. Material
angle
II; (5 = 1, [ = 0; u/b = 2; b/h = 50).
0
Lay-up:
angle
II; (5 = [ = 0; u/b = 2; b/h = 50).
8
Lay-up:tWtV45”/8.
25
37
44 (1999) 31-41
angle
II; (5 = 0, [ = 1; u/b = 2; b/h = 50).
Figs. 9-15: in-plane stresses are assumed equal to a fraction of buckling loads CI= N/N,,. Results are obtained considering in the Rayleigh-Ritz procedure an adequate number of waves (m,n) defined on the basis of a convergence test: for the laminates examined convergence of results is attained for different m,n values varying both laminate configurations and foundation parameters. In particular, for unrestrained laminates convergence has been obtained for m = n = 10, for laminates resting on Winkler foundations m = n = 8,
M.A. Aiello, L. Ombres I Composite
38 Table 9 Buckling loads and foundation
44 (1999) 31-41 0
for unsymmetric parameters
laminates
varying
lamination
0
&=O 6,=0
& = 100 6, =o
&=O a,=10
& = 100 a,=10
0 15 30
1.570 1.882 2.713
5.585 5.448 5.805
9.712 9.669 10.050
12.493 11.813 11.866
45 60
3.536 4.182
6.132 6.167
10.356 10.667
11.951 12.124
75 90
4.490 5.084
6.456 7.444
11.402 12.995
12.904 14.826
Lay-up:
Structures
Ol-0/45”/0.
Material
II; (5 = 1,
angle
[ = 1; alb = 2; blh = 50).
35
30-
I 1
25-
0,2
0,4
0,6
0,8
Fig. 10. Non-dimensional frequency parameter (Mode plane loads ratios for square unsymmetric laminates. Lay-up: 30”/-30”/0”; b/h = 10; Material II.
8 0.2
I 0.4
1 0.6
I 0.8
Fig. 9. Non-dimensional fundamental frequency parameter plane loads ratios for square unsymmetric laminates. Lay-up: 30”/-30”/0”; b/h = 10; Material II.
1,0 2) versus
in-
40-
1.0 I
versus in-
for laminates resting on Filonenko-Borodhic foundations m = n = 6, while for laminates resting on Pasternak foundations m = n = 5. In Figs. 9-l 1 diagrams ~+a relative to various vibration modes for an unsymmetric thick laminate, varying foundation parameters, are shown. Analysing the results shows that vibrations of laminates increase as the foundation parameters increase, whilst increasing the ratio of in-plane loads has the opposite effect. With reference to a Pasternak foundation, in Fig. 12, cc>-acurves, varying the mode number of vibrations for a square unsymmetric laminate, are shown: it is possible to observe that increasing in-plane stresses, fundamental frequencies of laminate decrease more sensibly than vibrations of modes 2 and 3.
30-
20: 0,2
a I
0,4
I
0,6
I
0,8
Fig. 11. Non-dimensional frequency parameter (Mode plane loads ratios for square unsymmetric laminates. Lay-up: 30%30”/0”; b/h = 10; Material II.
1,0 3) versus
in-
Non-dimensional values of vibrations in the presence of in-plane initial stresses for a 90”/-90”/90”/90” laminate resting on elastic foundations are shown in Table 10. Values of vibrations shown in Table 10, relative to a thin laminate, confirm the influence of foundation parameters and in-plane loads as previously described.
39
M.A. Aiello, L. Ombres I Composite Structures 44 (1999) 3141
26 24
40-
22 20
30-
20-
p
Mode A0
18
3 -a-
16
-+
14
10-l----la
0,2
0,4
0,6
0,8
1,0
Fig. 12. Non-dimensional frequency parameter versus in-plane loads ratios for square unsymmetric laminates resting on Pasternak foundation. o = 6b2(p/E22)‘/2. Lay-up: 30”/-30”/0”; b/h = 10; & = 100; 6, = 10; Material II.
J-l-7-I 25
SO 75
%
100 125 150 175 200
Fig. 13. Non-dimensional fundamental frequency parameter versus foundation parameters for unsymmetric UJ= ob*(p/E#* laminates. Lay-up: 45”/-45”/0”/45”; b/h = 10; a/b = 2; c(= 0.4. Material II.
This shows that the two-parameter foundation values are nearly twice the others. The variation in vibrations of unsymmetric laminates resting on a Pasternak foundation varying foundation parameters, is shown in Fig. 13. The analysis of results confirms that frequencies of laminates increase when foundation parameters increase; in particular, the in-
10
0
20
30
40
Fig. 14. Non-dimensional frequency parameter w = Wb2(p/E22)‘/2 versus anisotropy ratios b = Ell/Et2 for unsymmetric laminates. Lay-up: 30”/-3OV45”; b/h = 10; a/b = 1; a = 0.40; 6, = 10. Material II.
104 0
a/b 1
2
3
4
5
Fig. 15. Non-dimensional frequency parameter o = Wb*(p/E22)‘/* versus ratios u/b for unsymmetric laminates. Lay-up: 90”/-90”/0”; b/ h = 10; G(= 0.50; & = 200. Material I.
crease is more relevant, increasing the shear modulus of the subgrade. The influence of laminate anisotropy on the vibrations in the presence of in-plane stresses of an unsymmetric laminate resting on Pasternak foundation, is shown in Fig. 14. The results obtained, relative to fundamental frequencies of vibrations, show that the increase of the degree of anisotropy with higher foun-
40
M.A. Aiello, L. Ombres 1 Composite
Table 10 Non-dimensional frequency parameters presence of initial in-plane loads
for unsymmetric
laminates
a
Mode
bl,=O 6, =o
6k = 100 6,=0
&=O 6,=10
SI, = 100 a,=10
0.00
1 2 3
9.326 10.503 13.020
13.673 14.500 16.414
14.502 17.538 22.135
17.614 20.190 24.281
0.25
1 2 3
9.090 9.650 11.444
13.514 13.894 15.194
14.354 17.044 21.250
17.492 19.759 23.482
0.50
1 2 3
8.652 8.836 9.503
13.221 13.343 13.791
14.197 16.510 20.287
17.363 19.300 22.615
0.75
1 2 3
6.843 7.433 8.556
12.133 12.460 13.161
14.030 15.933 19.231
17.228 18.809 21.672
o = tib2(p/E22)‘/2. Lay-up: 90”/-90”/90”/90”. a/b = 2; b/h = 50; Sk = 100; 6, = 10).
Material
in
I; (5 = 0, i = 1;
Table 11 Non-dimensional fundamental frequencies versus load combinations varying u/b for four layers thick unsymmetric laminates alb t=o; t=o; 5=1; E=l:
[=O [=l [=O i=l
0.5
1.0
2.0
5.0
49.503 49.480 49.453 49.430
25.846 25.819 25.751 25.724
18.359 18.342 18.226 18.208
16.026 16.021 15.874 15.870
(x = 0.5; b/h = 20; dk = 100; 6, = 5). Lay-up: II.
450/-450/900/45”;
Structures
44 (1999) 31-ll
The results obtained for simply supported laminates, mainly on Pasternak foundations, lead to the following general conclusions: 1. The presence of foundations increases both the buckling load and vibration response of laminates when compared with unrestrained conditions. This increase is especially marked in the case of two-parameter Pasternak foundations. 2. The maximum buckling loads of laminates resting on elastic foundations are clearly different from equivalent unrestrained ones for all load combinations. In addition, the optimum laminate configuration for maximum buckling resistance is strongly influenced by the foundation parameters. 3. Initial in-plane stresses cause a decrease in laminate vibration frequencies as the ratio of in-plane loads increases. 4. An increase in shear modulus of the subgrade produces a beneficial laminate response from varying both geometrical and mechanical parameters. 5. Biaxial compression and shear forces reduce laminate buckling loads without influencing the fundamental frequencies of laminates with in-plane loads. The combined effect of in-plane stresses and foundation restraints produces a remarkable increase in laminate performance in terms of both stability and vibration. This can be extremely useful from a design perspective in order to optimize structural performance.
Material
References
dation parameters, produces an increase of vibration values. Results corresponding to vibrations with in-plane stresses for square unsymmetric laminates, varying ratio a/b are shown in Fig. 15. It is evident that as the a/b ratio increases vibrations decrease. However, increasing subgrade shear modulus has the opposite effect. Values of non-dimensional fundamental frequencies parameters varying the a/b ratio and load combinations are shown in Table 11. Analysing the results, it is possible to observe that the presence of biaxial compression and shear forces does not modify vibrations of laminates; their values, in fact, are almost constant.
4. Concluding remarks In this paper, buckling loads, free vibrations and vibrations in the presence of initial in-plane stresses of unsymmetric flat composite laminates resting on elastic foundations, have been analysed.
111Xiang
Y, Kitipornchai S, Liew KM. Buckling and vibration of thick laminates on Pasternak foundation, Journal of Engineering Mechanics 1996;122:5463. 121Turvey GJ. Uniformly loaded, simply supported, antisymmetritally laminated, rectangular plate on a Winkler-Pastemak foundation, Int. Journal Solids Structures 1977:13:437-44. J. The Vlasov foundation model, Int. J. [31 Jones R, Xenophontos Mech. Science 1977;19:317-23. of orthotropic plates on two-parameter 141 Shen H-S. Postbuckling elastic foundation, Journal of Eng. Mechanics 1995;121:5&56. [51 Leissa AW. Advances in vibration, buckling and postbuckling studies on composite plates. In: Composite Structures, Proceedings of the 1st International Conference on Composite Structures, 1981:312-334. PI Dawe DJ, Craig TJ. The vibration and stability of symmetricallylaminated composite rectangular plates subjected to in-plane stresses, Composite Structures 1986;5:281-307. [71 Noor AK, Scott Burton W. Stress and free vibration analyses of composite plates, Composite Structures multilayered 1989;11:183-204. S. Vibration and buckling for hybrid 181 Barai A, Durvasula laminated curved panels, Composite Structures 1992;21:15-27. 191 Khdeir AA, Librescu L. Analysis of symmetric cross-ply laminated elastic plates using higher-order theory: Part II - Buckling and free vibration, Composite Structures 1988;9:259-77.
MA.
Aiello, L. Ombres I Composite
[lo] Khdeir AA, Comparison between shear deformable and Kirchhoff theories for bending, buckling and vibration of antisymmetric angle-ply laminated plates, Composite Structures 1989;13: 159-72. [l l] Srinivas S, Rao AK. Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates, Int. Journal Solids Structures 1970;6:1463-81.
Structures
44 (1999) 31-41
41
[12] Jones RM. Buckling and vibration of unsymmetrically laminated cross-ply rectangular plates, AIAA Journal 1973;ll: 162632. [13] Leissa AW. Conditions for laminated plates to remain flat under inplane loading, Composite Structures 1986;6:261-70. [14] Aiello MA, Ombres L. Maximum buckling loads for unsymmetric thin hybrid laminates under in-plane and shear forces, Composite Structures 1997;36:l-l 1.
Composite Structures 44 (1999) 3141
Buckling and vibrations of unsymmetric laminates resting on elastic foundations under in-plane and shear forces M.A. Aiello *, L. Ombres Department
of Materials
Science, University of Lecce, Via per Amesano,
73100 Lecce, Italy
Abstract Buckling loads, free vibrations and vibrations with initial in-plane stresses for moderately thick, simply supported rectangular laminates resting on elastic foundations such as the Pasternak type, are examined in this paper. The analysis is carried out considering unsymmetric flat laminates subjected to biaxial compression and shear loads. First-order shear deformation theory, in conjunction with the Rayleigh-Ritz method is used for the evaluation of buckling loads and vibrations by means of a standard eigenvalue procedure. A large numerical investigation is intended to demonstrate the influence of laminate parameters (fiber orientations, degree of anisotropy, number of layers) and foundation parameters (modulus of subgrade reaction and shear modulus); the results obtained are reported and discussed in the paper. 0 1999 Elsevier Science Ltd. All rights reserved.
Notation width and thickness, respectively Extensional, coupling and bending laminate stiffnesses Young’s moduli along the fiber direction and normal to the fibers, respectively In-plane shear moduli Poisson ratio Displacements in the x-, y- and z-directions, respectively In-plane displacements of the middle plane of the laminate Laminate strains Shear rotations In-plane applied loads Laminate curvatures Elastic strain energy Work of internal stresses Work of external stresses Total potential energy Kinetic energy of the laminate Loads multiplier
a, b, h
Laminate
A,, Bij, Dij EII,
J522
G2,
G3,
G23
1’12 u,
v,
w
uo, 00, wo
l
Gb
Frequency of vibrations Density of materials Modulus of subgrade reaction Shear modulus of the subgrade
n
Nxb2/E22
5 i
NJNx
0
Corresponding author
0263-8223/99/$
length,
Pi ks
h3
N,JNx Wb2/h(plE22)1’2 k,b4/E22 Gb2/E22
h3 h3
1. Introduction Plates supported by elastic foundations represent a very common model in structural engineering; as a consequence the number of technical problems connected with this model, is very large. Generally the analysis is developed on the assumption that the reaction forces of foundation are proportional at every point to the deflection at that point (Winkler model). However, to represent the characteristic behavior of practical foundations, the Winkler model is not adequate and can be substituted by other models such as the Filonenko-Borodhic, Pasternak or Vlasov models [14]. In many cases the two-parameter Pasternak model has proved to be the most effective.
- see front ma.tter 0 1999 Elsevier Science Ltd. All rights reserved
32
M.A. Aiello. L. Ombres I Composite
In the field of composite materials, the model of laminated flat plates resting on elastic foundation is used for the analysis of several structural elements such as, for example, sandwich laminated panels. For this reason, a knowledge of the behavior of laminated plates supported by an elastic foundation of the Pasternak type, is very useful mainly from a design perspective. The evaluation of buckling loads, free vibrations and vibrations with in-plane pre-loads in the field of composite laminates represents a classical subject to many researchers [5]; results of these studies, regarding also laminates resting on elastic foundations, are numerous in the literature [6-121. However, particularly for laminated plates resting on Pasternak foundations, the analysis of vibrations considering the combined effects of foundations and inplane initial stresses is not complete. This problem, in fact, has been analysed only with reference to isotropic thin and, recently, moderately thick symmetric cross-ply laminates subjected to uniformly distributed uniaxial or biaxial forces [l]. The results obtained showed that combined effects of foundation parameters and in-plane stresses can greatly influence the value of plate vibrations [l]. The analysis of this problem becomes, then, essential in order to determine the behavior of laminates under service conditions. On the basis of these considerations, the paper is devoted to the analysis of laminate vibrations in the presence of in-plane stresses under loads smaller than buckling loads. The analysis is carried out with reference to unsymmetric flat simply supported laminates subjected to combined biaxial compression and shear loads. Buckling loads and vibration are determined as solutions of a standard eigenvalue problem by an energetic approach based on the first-order shear deformation theory in conjunction with the Rayleigh-Ritz method. In the case of unsymmetric laminates, because of the bending-stretching coupling, the evaluation of buckling loads as bifurcation load needs to consider added conditions called ‘flatness conditions’ [13] that enable us to take into account the initial curvature of laminates. Flatness conditions furnish a set of restrictions on edge moments and transverse edge shear, depending on the loading and boundary conditions. In order to demonstrate the influence of geometrical and mechanical parameters of laminates (aspect ratio, degree of orthotropy of layers, fiber orientation) and foundation parameters (modulus of subgrade reaction, shear modulus of subgrade) on the buckling and vibrations of laminates, a parametric numerical investigation for simply supported rectangular moderately thick plates subjected to in-plane
Structures
44 (1999) 31-41
compressive conducted.
forces
2. Mathematical
combined
with shear forces,
is
formulations
2.1. Constitutive law of laminates Considering a flat rectangular laminate made with orthotropic layers subjected to in-plane and shear forces the constitutive law is expressed by the following relationships (Fig. 1):
P{K) {MI= PI{&)+ Pl{~) IN] = M{c] +
{Ql = PI{vI being
h/2 (Aij, B,, Dij) = / Qij(l,~>~)
dz (i,j=
1,276)
-h/2 h/2 &j =
kijQjj dz (i,j = s -h/2
4,5)
where k, are shear correction factors, qii, functions of elastic constants and the ply angle 8, differs from one layer to another, while:
Nx i
Fig. 1. Laminate loads.
c
definition:
ply orientation,
ply stacking
and in-plane
MA. Aiello, L. Ombres I Composite Structures 44 (1999) 3141
cS~E=II+A(L;-L,)
33
(1)
where IT = W + U is the strain energy of the laminate, L; the work of internal stresses in the initial configuration for the second-order strain components, L2 the work corresponding to the external loads for the second order displacements components, 62E the second variation of total potential energy, and ;1 the loads multiplier. For a flat rectangular laminate, the work L; is expressed as:
with
~0, ug, EA, Q, and .sXyO being displacements and strains at the middle plane of the laminate. +
2.2. Energy expression:: The total potential energy of a laminate resting on an elastic foundation is expressed as: sZ= W-T+U with W the strain energy of the laminate, T the kinetic energy of the laminate, and U the strain energy due to the Pasternak foundation. Considering a flat rectangular laminate of a and b dimensions along the X- and y-axis, respectively, the strain energy W and the kinetic energy T are expressed as: w+ ss
Gii’lC4 + W’[W4 hdv
+
+A($: being
+ @I dx(dY
&=Jpwdz. &; I2 =/rp(k)z >1, , Jp(k) h/2
I, =
[{s]%I{el. + {s]r[BI{~) + {~]r[B]{s)
-h/2
h/2
-hi2
h/2
-h/2
where pck)is the mass density per unit volume of the k-th layer. The strain energy due to the Pasternak foundation is given by: a b u=;
bt+G(d,x +4,) 1dxdv JJ 00
where k, is the modulus of subgrade reaction and Gt, is the shear modulus of the subgrade.
W&o,xuoy+ uoxvoy+ yxyy)] dx dy.
The work L2 is zero when external loads are constant or independent from U, v and w displacements. For unsymmetric laminates in which the geometric middle plane is different from the neutral plane, the buckling load can be determined as a bifurcation load only if added conditions, which guarantee the flatness of laminates, are imposed. This problem has been investigated by Leissa [13] who defines conditions for unsymmetric laminates to remain flat during the pre-buckling stage when subjected to in-plane loads. Leissa separates the pre-buckling response equations from the buckling equations and works out the conditions for the curvatures to vanish during the pre-buckling stage (flatness conditions): if these conditions are satisfied, the bifurcation phenomenon will occur. For a simply supported laminate subjected to in-plane loading, the flatness conditions involve the introduction of bending moments @ and q acting along the edges of laminates, in the initial configuration. The values of q and q are determined using constitutive laws of the laminate with first-order components of displacements and shear rotations. Bending moments M” and MO furnish a value of L2 that must be consfdered fey the evaluation of d2E. Substituting II, L; and L2 expressions in Eq. (l), the condition 62E = 0 gives the relations that enable determination of the buckling loads of laminates. The solution of the problem is obtained by the Rayleigh-Ritz method assuming linear variation of in-plane displacements u and v and constant transverse deflection w over the panel thickness: U(?Y,Z) = Uo(X,Y) +zIc/,(x,v)
2.3. Buckling load analysis The buckling load of a flat laminate is determined by an energetic approach founded on the Dirichlet principle, expressed as follows [14]:
hY,Z)
= vo(x,r)
W(X,Y,Z)
= W(X,Y).
+z+y/,(x,Y)
For a simply supported flat laminate the displacements that satisfy the geometric boundary conditions:
34
M.A. Aiello, L. Ombres I Composite
v=w=*,=o
at x = 0, a
u=w=*X=o
at y = 0,b
ypIn” m
cos(mrc~/a) sin(nny/b)
n
vO(x,y) = yyIV& m n
sin(mm/a) cos(nrry/b)
$Jx, Y) = TX&B m n
cos(m7-la)
W(X,Y) =
sin(n~ylb)
(2)
WI+ WlHq~= 0
W and L2
(4)
(5) in which [L] is the matrix containing the terms corresponding to the in-plane pre-loads.
Numerical analysis is carried out with reference to flat rectangular simply supported laminates resting on elastic foundations. Results are obtained for symmetric and
(3)
which represents a system of linear equations of 5mn x 5mn order. The solution of Eq. (3) is obtained as the solution of a standard eigenvalue problem that can be solved by available standard procedures. [L] and [Kj are matrices of 5mn x 5mn order that contain terms relative to strain energy and external work; the vector {q} contains series coefficients U,,, I’,,, W,,, X,, and The buckling load coincides with the smallest eigenvalue of Eq. (3). In the presence of biaxial compression and shear the solution is obtained considering ratios:
and evaluating the critical value of N,. 2.4. Vibration analysis The analysis of laminate vibrations is carried out by the same procedure described for the buckling analysis. In this case the total potential energy R must be derived considering the maximum value of the kinetic energy besides the strain energy. Evaluating free vibrations, external loads are equal to zero then [L] = 0. In accordance with the Rayleigh-Ritz method, the analysis is carried out assuming periodic displacement functions with time of the following form: _ U(X,y,
WI + ~2MHql = (01
where [n/rl is the mass matrix of the laminate and {q} the vector of unknowns coefficients. Free vibrations correspond to eigenvalues of Eq. (4). For vibrations in the presence of in-plane stresses, Eq. (4) becomes:
3. Numerical investigations
sin(mrc.x/a) sin(nrcy/b). ~y?+Kln
Substituting Eq. (2) into the expressions Eq. (1) becomes:
w
44 (1999) 31-41
Minimization of the total potential energy furnishes the following governing equation:
are chosen in the form of the following trigonometric series:
U&Y) =
Structures
t) = u(x, y) sin tit
Table 1 Comparison of uniaxial buckling loads for square symmetric laminates resting on Pastemak foundation 5
where u(x,y), v(x,y) and w(x,y) are the displacement functions expressed by Eq. (2) and o is the frequency of vibration.
b/h = 1000
b/h= 10
Ref. [l]
Present
Ref. [I]
Present
50.7515 49.2666
50.7511 49.2666
66.2895 66.2895
66.2888 66.2888
22.2288 19.5904
22.2288 19.5902
26.7300 20.9930
26.7300 20.9928
(n =~%$b’/E~~h’; Sk = 100; 6, = 10). Lay-up: 0”/90”10”; EillEz2 =40; Glz= f&=0.6 E2*; &~,=0.5 E,,; ~,~=0.25.
Table 2 Comparison of non-dimensional svmmetric laminates
frequency parameters
for square
Mode
Present Ref. [lo]
1
2
3
4
5
19.058 18.950
27.659 27.390
46.977 46.670
69.439 69.380
74.268 73.950
o = Wb2(plE22) ‘i2., Lay-up: 0”/90”/0”; a/h = 50; El l/E22 = 40; ($2 = G,3 =0.6 E,,; G2) = 0.5 E22; v,~ = 0.25.
Table 3 Comparison of non-dimensional fundamental frequency parameters for square antisymmetric laminates
e
Number of layers (n)
Present
Ref. [I l]
30” 45” 45”
10 10 4
18.585 19.454 18.960
18.510 19.380 18.460
u(x,y, t) = v(x,y) sin Gt
w(x, y, t) = w(x, y) sin tit
alb
w = tib2(plE&/2; Lay-up: 0l-WV-01. J-8; alh = 10; E111E22= 40; G,~=G,3=0.6 E,,; G23=0.5 Ez2; ~,~=0.25.
MA. Table 4 Comparison unsymmetric
of non-dimemional laminates
frequency
Aiello, L. Ombres I Composite
parameters
for
square
Mode
Present Ref. [S]
1
2
3
4
11.835 11.700
32.064 31.900
32.064 31.900
46.947 46.400
ru = 8b2(p/E22)“2; Lay-up: 0”/90”; a/h = 50; G,J = 1.0 E22; G,, =0..5 Ez2; v,~ = 0.25.
-
EII/Ez2 = 40;
G,2 =
Table 5 Comparison of non-dimensional frequency parameters for square symmetric laminates resting on elastic foundation in the presence of innlane stresses
5
b/h
Ref. [l]
Present.
Ref. [l]
Present
Ref. [l]
Present
(1,l)
(1,l)
(1,2)
(1,2)
(2,l)
(2.1)
0
10 1000
264.51 327.12
264.53 327.14
825.38 991.94
825.39 991.96
992.70 4413.0
992.70 4413.20
1
10 1000
295.26 390.43
295.26 390.45
534.85 659.53
534.86 659.53
1441.8 5062.0
1441.8 5061.8
(9 = wb2(p/Ez2) “‘; iy = 0.5; & = 100; 6, = 10. Lay-up: G,z=G,s=0.6 E22; G2,=0.5 E22; ~,~=0.25.
0”/90”/O”; El,/
E,,=40;
unsymmetric laminates assuming the shear correction factor equal to kij = 5/6. 3.1. Comparison with available data Initially the analysis is carried out considering symmetric laminates resting on elastic foundations: results are then compared with those available in the literature and reported in Tables l-5 for buckling loads, free vibrations and vibratio:ns in the presence of in-plane stresses. The results obtained demonstrate good agreement between the present model and models defined by other authors [8,10,11].
Structures
44 (1999) 3141
3.5
In order to determine the influence of mechanical parameters of foundations, in the following figures and tables, results obtained for unsymmetric laminates made with four layers of constant thickness and lay-up B/-8/ &/8, are shown. The 6, value is assumed equal to O”, 45” and 90”, while values of foundation parameters have been chosen in order to represent the Winkler model (& = 100, 6, = 0), the Filonenko-Borodhic model (6k = 0, 6, = 10) and the Pasternak model (& = 100, 6, = 10): results corresponding to each model are compared with those relative to unrestrained laminates (& = 0, 6, = 0). Figures 2 and 3 show the variation of uniaxial buckling loads versus lamination angle varying foundation parameters for unsymmetric simply supported laminates with (!A, = 45” and & = 90”, respectively. The results obtained confirm the very sensible influence of foundation on the behavior of laminates: buckling loads are higher than ones corresponding to unrestrained laminates and their maximum values are obtained for different lamination angles varying foundation parameters. For laminates resting on elastic foundations the maximum value of buckling loads corresponds to 6’ nearly equal to 20”. The best response has been found for a two-parameter foundation (Pasternak model). For unrestrained laminates maximum buckling loads are obtained with 8 nearly equal to 40”. In the presence of shear forces, as shown in Figs. 4 and 5, buckling load values are lower than those corresponding to uniaxial compression; however, in this case the behavior of laminates, varying foundation pa-
n
3.2. Parametric analysis Results of a numerical parametric analysis, carried out with reference to unsymmetric laminates, are shown in following tables and figures. Materials properties used in the analysis are: Material I: E,,lEzz = 40; G,z = G,3 = 0.6 E22; G23= 0.5 E22; VI2= 0.25 Material II: E,,lEzz = 10; G,z = G,3 = 0.5 E22; G23= 0.5 E22; 1112 = 0.30 Results are shown considering the following non-dimensional quantities: & = ksb4/Ez2h3; 6, = Gbb2/Ez2h3
Fig. 2. Non-dimensional unsymmetric laminates terial II.
buckling load versus lamination angle for W-8/45/8 (b/h = 50; u/b = 2; < = 0; [ = 0). Ma-
M.A. Aiello, L. Ombres I Composite Structures 44 (1999) 31-41
36
Fig. 3. Non-dimensional unsymmetric laminates terial II.
buckling load versus lamination angle for 8/-0/90/O (b/h = 50; u/b = 2; 5 = 0; ( = 0). Ma-
rameters, is similar. Maximum values of buckling loads, in fact, are obtained for 8 = 0” or &’= 90” in all case In Fig. 6 n-8 diagrams for unsymmetric lamin; subject to biaxial compression are compared; res show that curves corresponding to unrestrained la____ nates and laminates resting on Filonenko-Borodhic foundation are almost parallel and the maximum value of the buckling load is attained with the same 0 value.
Fig. 5. Non-dimensional unsymmetric laminates terial II.
buckling load versus lamination angle for f3/-0/45&l (b/h = 50; u/b = 2; 5 = 0; < = 1). Ma-
Analogous behavior can be seen for laminates resting ‘asternak and Winkler foundations. lurves drawn in Fig. 7 relate to laminates under 14 .ial compression and shear forces: it is possible to see _-._._the behavior corresponds to that of laminates under uniaxial compression and shear forces. In this case, however, the best response of laminates corresponds to the value 0 = 90”.
n
n
20 15 10 5
e 0
15
Fig. 4. Non-dimensional unsymmetric laminates terial II.
30
45
60
75
90
buckling load versus lamination angle for 8/-0/O/0 (b/h = 50; u/b = 2; c = 0; { = 1). Ma-
8
0 0
15
Fig. 6. Non-dimensional unsymmetric laminates terial II.
30
45
60
75
90
buckling load versus lamination angle for 0/-0/90/Q (b/h = 50; u/b = 2; 5 = 1; [ = 0). Ma-
M.A. Aiello, L. Ombres I Composite
n
Structures
Table 6 Buckling loads and foundation
20
e 0 15 24 25 30 37 45 60 75 90 Lay-up:
15
0
Fig. 7. Non-dimensional unsymmetric laminates terial II.
30
45
60
75
90
buckling load versus lamination angle for 8/-Q/O/0 (b/h = 50; a/b = 2; < = 1; [ = 1). Ma-
n
Fig. 8. Non-dimensional buckling load versus lamination angle for unsymmetric laminates 6+&f&/0 (b/h = 50; a/b = 2; Sk = 100; 6, = 10; &, = 0”, 45”, 90’). Material Il.
In Fig. 8, n-8 curves for all load combinations examined for laminates resting on Pasternak foundations, varying the O0 values, are drawn. The results obtained show the influence of loading combinations on the buckling load and also that 80 variation has little effect. Results of buckling analysis are also shown in Tables 69. Vibrations of laminates resting on elastic foundations in the presence of in-plane stresses, are shown in
for unsymmetric parameters
laminates
varying
lamination
&=O
61; = 100
s,=o
6,=0
&=O a,=10
61, = 100 6,=10
7.523 9.702 12.637 13.003 13.553 13.782 13.551 11.327 8.957 7.883
21.317 22.104 22.945 22.804 22.047 20.772 19.007 15.437 12.053 10.747
31.185 32.296 33.560 33.685 33.123 31.916 30.102 26.065 22.200 20.713
39.761 39.147 38.322 38.208 37.548 36.306 34.407 28.659 24.742 23.078
0/-0/0”/0.
Table 7 Buckling loads and foundation
Material
for unsymmetric parameters
laminates
varying
lamination
&=O s,=o
61,= 100 6, =o
&=O 6, = 10
6k = 100 6,=10
0 15 30 45 51 60 68 75 90
1.628 2.059 3.123 4.307 4.760 5.371 5.821 5.673 5.537
6.506 7.092 8.650 9.939 10.127 9.743 9.341 8.990 8.491
11.628 12.060 13.123 14.307 14.759 15.371 15.820 15.670 15.536
16.506 17.091 18.650 19.938 20.126 19.742 19.340 19.000 18.490
Table 8 Buckling loads and foundation
Material
for unsymmetric parameters
laminates
varying
lamination
&=O 6,=0
& = 100 s,=o
&=O a,=10
61, = 100 a,=10
0 15 30 45 60 75 90
5.536 5.896 6.976 7.191 6.778 6.235 6.768
11.650 10.366 10.141 9.684 9.000 8.285 8.815
18.286 16.898 16.611 16.041 15.358 15.200 16.478
21.525 19.495 18.525 17.900 17.173 16.863 18.253
9/-@/90”/0. Material
angle
II; (5 = 1, [ = 0; u/b = 2; b/h = 50).
0
Lay-up:
angle
II; (5 = [ = 0; u/b = 2; b/h = 50).
8
Lay-up:tWtV45”/8.
25
37
44 (1999) 31-41
angle
II; (5 = 0, [ = 1; u/b = 2; b/h = 50).
Figs. 9-15: in-plane stresses are assumed equal to a fraction of buckling loads CI= N/N,,. Results are obtained considering in the Rayleigh-Ritz procedure an adequate number of waves (m,n) defined on the basis of a convergence test: for the laminates examined convergence of results is attained for different m,n values varying both laminate configurations and foundation parameters. In particular, for unrestrained laminates convergence has been obtained for m = n = 10, for laminates resting on Winkler foundations m = n = 8,
M.A. Aiello, L. Ombres I Composite
38 Table 9 Buckling loads and foundation
44 (1999) 31-41 0
for unsymmetric parameters
laminates
varying
lamination
0
&=O 6,=0
& = 100 6, =o
&=O a,=10
& = 100 a,=10
0 15 30
1.570 1.882 2.713
5.585 5.448 5.805
9.712 9.669 10.050
12.493 11.813 11.866
45 60
3.536 4.182
6.132 6.167
10.356 10.667
11.951 12.124
75 90
4.490 5.084
6.456 7.444
11.402 12.995
12.904 14.826
Lay-up:
Structures
Ol-0/45”/0.
Material
II; (5 = 1,
angle
[ = 1; alb = 2; blh = 50).
35
30-
I 1
25-
0,2
0,4
0,6
0,8
Fig. 10. Non-dimensional frequency parameter (Mode plane loads ratios for square unsymmetric laminates. Lay-up: 30”/-30”/0”; b/h = 10; Material II.
8 0.2
I 0.4
1 0.6
I 0.8
Fig. 9. Non-dimensional fundamental frequency parameter plane loads ratios for square unsymmetric laminates. Lay-up: 30”/-30”/0”; b/h = 10; Material II.
1,0 2) versus
in-
40-
1.0 I
versus in-
for laminates resting on Filonenko-Borodhic foundations m = n = 6, while for laminates resting on Pasternak foundations m = n = 5. In Figs. 9-l 1 diagrams ~+a relative to various vibration modes for an unsymmetric thick laminate, varying foundation parameters, are shown. Analysing the results shows that vibrations of laminates increase as the foundation parameters increase, whilst increasing the ratio of in-plane loads has the opposite effect. With reference to a Pasternak foundation, in Fig. 12, cc>-acurves, varying the mode number of vibrations for a square unsymmetric laminate, are shown: it is possible to observe that increasing in-plane stresses, fundamental frequencies of laminate decrease more sensibly than vibrations of modes 2 and 3.
30-
20: 0,2
a I
0,4
I
0,6
I
0,8
Fig. 11. Non-dimensional frequency parameter (Mode plane loads ratios for square unsymmetric laminates. Lay-up: 30%30”/0”; b/h = 10; Material II.
1,0 3) versus
in-
Non-dimensional values of vibrations in the presence of in-plane initial stresses for a 90”/-90”/90”/90” laminate resting on elastic foundations are shown in Table 10. Values of vibrations shown in Table 10, relative to a thin laminate, confirm the influence of foundation parameters and in-plane loads as previously described.
39
M.A. Aiello, L. Ombres I Composite Structures 44 (1999) 3141
26 24
40-
22 20
30-
20-
p
Mode A0
18
3 -a-
16
-+
14
10-l----la
0,2
0,4
0,6
0,8
1,0
Fig. 12. Non-dimensional frequency parameter versus in-plane loads ratios for square unsymmetric laminates resting on Pasternak foundation. o = 6b2(p/E22)‘/2. Lay-up: 30”/-30”/0”; b/h = 10; & = 100; 6, = 10; Material II.
J-l-7-I 25
SO 75
%
100 125 150 175 200
Fig. 13. Non-dimensional fundamental frequency parameter versus foundation parameters for unsymmetric UJ= ob*(p/E#* laminates. Lay-up: 45”/-45”/0”/45”; b/h = 10; a/b = 2; c(= 0.4. Material II.
This shows that the two-parameter foundation values are nearly twice the others. The variation in vibrations of unsymmetric laminates resting on a Pasternak foundation varying foundation parameters, is shown in Fig. 13. The analysis of results confirms that frequencies of laminates increase when foundation parameters increase; in particular, the in-
10
0
20
30
40
Fig. 14. Non-dimensional frequency parameter w = Wb2(p/E22)‘/2 versus anisotropy ratios b = Ell/Et2 for unsymmetric laminates. Lay-up: 30”/-3OV45”; b/h = 10; a/b = 1; a = 0.40; 6, = 10. Material II.
104 0
a/b 1
2
3
4
5
Fig. 15. Non-dimensional frequency parameter o = Wb*(p/E22)‘/* versus ratios u/b for unsymmetric laminates. Lay-up: 90”/-90”/0”; b/ h = 10; G(= 0.50; & = 200. Material I.
crease is more relevant, increasing the shear modulus of the subgrade. The influence of laminate anisotropy on the vibrations in the presence of in-plane stresses of an unsymmetric laminate resting on Pasternak foundation, is shown in Fig. 14. The results obtained, relative to fundamental frequencies of vibrations, show that the increase of the degree of anisotropy with higher foun-
40
M.A. Aiello, L. Ombres 1 Composite
Table 10 Non-dimensional frequency parameters presence of initial in-plane loads
for unsymmetric
laminates
a
Mode
bl,=O 6, =o
6k = 100 6,=0
&=O 6,=10
SI, = 100 a,=10
0.00
1 2 3
9.326 10.503 13.020
13.673 14.500 16.414
14.502 17.538 22.135
17.614 20.190 24.281
0.25
1 2 3
9.090 9.650 11.444
13.514 13.894 15.194
14.354 17.044 21.250
17.492 19.759 23.482
0.50
1 2 3
8.652 8.836 9.503
13.221 13.343 13.791
14.197 16.510 20.287
17.363 19.300 22.615
0.75
1 2 3
6.843 7.433 8.556
12.133 12.460 13.161
14.030 15.933 19.231
17.228 18.809 21.672
o = tib2(p/E22)‘/2. Lay-up: 90”/-90”/90”/90”. a/b = 2; b/h = 50; Sk = 100; 6, = 10).
Material
in
I; (5 = 0, i = 1;
Table 11 Non-dimensional fundamental frequencies versus load combinations varying u/b for four layers thick unsymmetric laminates alb t=o; t=o; 5=1; E=l:
[=O [=l [=O i=l
0.5
1.0
2.0
5.0
49.503 49.480 49.453 49.430
25.846 25.819 25.751 25.724
18.359 18.342 18.226 18.208
16.026 16.021 15.874 15.870
(x = 0.5; b/h = 20; dk = 100; 6, = 5). Lay-up: II.
450/-450/900/45”;
Structures
44 (1999) 31-ll
The results obtained for simply supported laminates, mainly on Pasternak foundations, lead to the following general conclusions: 1. The presence of foundations increases both the buckling load and vibration response of laminates when compared with unrestrained conditions. This increase is especially marked in the case of two-parameter Pasternak foundations. 2. The maximum buckling loads of laminates resting on elastic foundations are clearly different from equivalent unrestrained ones for all load combinations. In addition, the optimum laminate configuration for maximum buckling resistance is strongly influenced by the foundation parameters. 3. Initial in-plane stresses cause a decrease in laminate vibration frequencies as the ratio of in-plane loads increases. 4. An increase in shear modulus of the subgrade produces a beneficial laminate response from varying both geometrical and mechanical parameters. 5. Biaxial compression and shear forces reduce laminate buckling loads without influencing the fundamental frequencies of laminates with in-plane loads. The combined effect of in-plane stresses and foundation restraints produces a remarkable increase in laminate performance in terms of both stability and vibration. This can be extremely useful from a design perspective in order to optimize structural performance.
Material
References
dation parameters, produces an increase of vibration values. Results corresponding to vibrations with in-plane stresses for square unsymmetric laminates, varying ratio a/b are shown in Fig. 15. It is evident that as the a/b ratio increases vibrations decrease. However, increasing subgrade shear modulus has the opposite effect. Values of non-dimensional fundamental frequencies parameters varying the a/b ratio and load combinations are shown in Table 11. Analysing the results, it is possible to observe that the presence of biaxial compression and shear forces does not modify vibrations of laminates; their values, in fact, are almost constant.
4. Concluding remarks In this paper, buckling loads, free vibrations and vibrations in the presence of initial in-plane stresses of unsymmetric flat composite laminates resting on elastic foundations, have been analysed.
111Xiang
Y, Kitipornchai S, Liew KM. Buckling and vibration of thick laminates on Pasternak foundation, Journal of Engineering Mechanics 1996;122:5463. 121Turvey GJ. Uniformly loaded, simply supported, antisymmetritally laminated, rectangular plate on a Winkler-Pastemak foundation, Int. Journal Solids Structures 1977:13:437-44. J. The Vlasov foundation model, Int. J. [31 Jones R, Xenophontos Mech. Science 1977;19:317-23. of orthotropic plates on two-parameter 141 Shen H-S. Postbuckling elastic foundation, Journal of Eng. Mechanics 1995;121:5&56. [51 Leissa AW. Advances in vibration, buckling and postbuckling studies on composite plates. In: Composite Structures, Proceedings of the 1st International Conference on Composite Structures, 1981:312-334. PI Dawe DJ, Craig TJ. The vibration and stability of symmetricallylaminated composite rectangular plates subjected to in-plane stresses, Composite Structures 1986;5:281-307. [71 Noor AK, Scott Burton W. Stress and free vibration analyses of composite plates, Composite Structures multilayered 1989;11:183-204. S. Vibration and buckling for hybrid 181 Barai A, Durvasula laminated curved panels, Composite Structures 1992;21:15-27. 191 Khdeir AA, Librescu L. Analysis of symmetric cross-ply laminated elastic plates using higher-order theory: Part II - Buckling and free vibration, Composite Structures 1988;9:259-77.
MA.
Aiello, L. Ombres I Composite
[lo] Khdeir AA, Comparison between shear deformable and Kirchhoff theories for bending, buckling and vibration of antisymmetric angle-ply laminated plates, Composite Structures 1989;13: 159-72. [l l] Srinivas S, Rao AK. Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates, Int. Journal Solids Structures 1970;6:1463-81.
Structures
44 (1999) 31-41
41
[12] Jones RM. Buckling and vibration of unsymmetrically laminated cross-ply rectangular plates, AIAA Journal 1973;ll: 162632. [13] Leissa AW. Conditions for laminated plates to remain flat under inplane loading, Composite Structures 1986;6:261-70. [14] Aiello MA, Ombres L. Maximum buckling loads for unsymmetric thin hybrid laminates under in-plane and shear forces, Composite Structures 1997;36:l-l 1.