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Stability of thick angle-ply composite plates
0045-7949188 S3.00 + 0.00 @ 1988 Per~mon Rw ptc
Compum a SrrucruresVol. 29, No. 2, pp. 317-322, 1988 Printed in Great Britain.
STABILITY OF THICK ANGLE-PLY COMPOSITE PLATES GAJBIR SINGH~ and Y. V. K. SADASIVA RAO$ tStructura1 Engineering Group, Vikram Sarabhai Space Centre, Trivandrum 695 022, India $Stage En~n~~ng and Analysis Division, Liquid Propulsion systems Centre, T~van~m 695 547, India (Received 20 July 1987)
Abstract-An eight-noded quadratic rectangular element is developed in order to study the stability aspects of a layered rectangular composite plate under biaxial loading. The element has five degrees of freedom per node. The formulation includes the Mindlin-TV nonlinear strain~ispla~ent relations. The adoptability of the element is demonstrated by solving a number of problems for which results exist in the literature. The effects of fibre orientation, material properties, layering and boundary conditions on the buckling load are studied in detail. The deficiency of earlier reported results is also pointed out.
NOTATION in-plane, bending-in-plane coupling and bending coefficients twisting and thickness-shear coefficients length of the plate matrix relating the strains and displacements linear and non-linear parts of the matrix B breadth of the plate matrix relating the stress to strain Young’s moduli along and transverse directions of the fibre in-plane and transverse shear moduli fibre directions matrix denoting the serendipity polynominals in-plane loads in x and y directions respectively number of degrees of freedom per each variable total thickness of the plate components of displacements in x, y and z directions respectively variable in space vector denoting fhe sum of internal and external forces generalised displacement vector interpolation functions curvatures Poisson’s ratio of the material stress vector non-dimensional buckling parameter
[A,], [B,], [BJ
=
Nxbz
&mm (unless otherwise defined) total slopes strain vector
The principal advantage of composites is the possibility of tailoring a laminate to suit the structural requirements. By using the directional nature of the material to advantage, highly efficient structures can
be designed. The advent of advanced fibre-reinform composite materials such as glass/epoxy, kevlar/ epoxy and graphite/epoxy with their high potential weight savings has resulted in a significant increase in the use of laminated plates and shells. These structures are subjected to different types of loadings such as bending, tensile, compression etc. Among all the possible loadings their resistance to buckling is of greatest interest to most of the researchers. Several theoretical and experimental papers have been published on the buckling of anisotropic plates under different types of loadings. Jones [l] evaluated the buckling and frequency parameters of unsymmetrically laminated cross-ply rectangular thin plates. Subsequently, Jones et al. [2] extended the above work to uns~met~cally laminated angle-ply plates. Noor [3] calculated the buckling loads of thick laminated cross-ply composite plates by using different shear correction factors for different stacking sequences. Hirano 14, S] attempted, at the optimum stacking sequence, to maximise the uni-axial buckling load of a thin angle-ply laminate. Wang [6] showed that a thin angle-ply laminate with orientation [O/-f?], is superior to a thin cross-ply with [0/9OL, when subjected to axial loads. Very little literature is available on the stability analysis of thick rectangular angle-ply composite plates using finite element method. The present paper is an extension of the finite element formulation [7,8] to the buckling analysis of thick angle-ply plates. Mindlin type non-linear straindisplacement relations are assumed for constructing the geometric stiffness matrix. The accuracy of the element is validated by comparing the available results in the literature. The effects of fibre orientation, material properties and layering are studied in detail. FORMULATION
Let us consider a laminated plate with ‘n’ number of layers having thicknesses I,, t2.. . t,,, each pos317
318
Y _i
19
-_-*
-*-•
IQI
20
14
! 9
9 li --. *‘II
‘5*,i _I
$I
I6 I 13
-u
i
“1
121 -E
I 0
‘1
I 1”
0
l Node Fig. 1. Geometry and finite ekment idealisation uf a layered plate.
sessing individual properties, acted upon by axial compressive loads N, and NYin the x and y directions as shown in Fig. 1. The dis~ia~ments at any paint in the plate are given by u k Y, r) = ~%, Y) - stlY.lx, Y) D~~~Y,~~=~“f~,Y~-zJlytx,Y~
where a comma denotes diffe~ntiation. can be written as
Equation (2)
cc> = @Lf + &.It where {e&t)and {tNL) are the matrices containing linear and non-linear terms. The stress and moment resultants are determined in the standard manner by considering all the ‘n’ layers, yielding
and
where u, v and w are the displacements in x, y and z directions, uO, v” and w” are the associated mid-plane displa~ments and JI, anf $,, are the total slopes along the x and y directions respectively. For the case of a Mindlin plate, the mid-plane strains and curvatures are given by
and K2K
QI Q2 i~C
=
&k&
Ec,&& K&8,
1’
(3)
where A,, 3, ttii (i, j = 1,2,6) and A# (i, j = 4,s) are
319
Stability of thick angle-ply composite plates
where {cr} = [D](C) and {E} is the vector containing. the respective in-plane, bending-in-plane coupling, external forces due to imposed loads bending and twisting, and thickness-shear coefficients respectively; K4 and KS are the shear correction factors which are taken as 0.8333 [8]. N, d(y) = 0, and Mi are the in-plane stress and moment resultants. where FINITE ELEMENT FORMULATION
d(y} =
Let the region of the plate be divided into a number of quadratic elements with eight nodes per element and five degrees of freedom (a, v, w, $, and $,) per node as shown in Fig. 1. The displacements at any point in the element can be taken in terms of the surrounding nodal displacements as
s”
d[81r(a}
do +
s”
[B]‘d{b} do
where
where N is the matrix containing the standard serendipity polynomial functions and {u}, represents the nodal displacements. The generalised displacements
(6) = [u,0, w, J/m$yl’
Kld{~I =
I”
4Kd-~~I da
in each element can be represented by
[KJ,[KJ and [K,] are the small displacement, large displacement and geometric stiffness matrices respectively. Assuming [KNL] = 0,the initial stability problem of eqn (7) reduces to
(8)
d(y)=([K,l+I[K,l}d(~}=0.
[K,]can be obtained by carrying out stress analysis as reported in (81. The eigenvalues, 1, of eqn (8) give rise to the buckling loads of the laminated plate. Gaussian quadrature formula adopting reduced integration wherever necessary is used in obtaining the element stiffness and geometric stiffness matrices.
where dy(a = 1,2,3) is the interpolation function corresponding to the ith node in the element and r, s and p are the number of degrees of freedom per variable. For simplicity, we take 4; = #J: = 4: and r=s=JJ[7j. Substitution of (5) in (2) gives rise to
The matrix [B] contains linear and non-linear which can be written as
terms
The matrices [BJ and BNJ are given in the Appendix. The vector sum of the internal and external forces is
1~1=
[Wa)
dv - {RI,
NUMERICAL RESULTS AND DISCUSSION
In order to study the applicability and accuracy of the element developed, a number of numerical examples are investigated. Some of these results are compared with those existing in the literature and are presented in Tables 1-3. The values of the nondimensional buckling parameter, 1, for an isotropic Table 1. Simply supported square isotropic plate Non-dimensional buckling parameters, thick plate theory Exact theory Mindlin’s theory [9] [91 3.924 3.909 3.741 3.729 5 3.150 3.115
b/t ti
y CO.3, 6 =N,b2
n=D'
Present theory
Thin plate theory [IO]
3.941 3.745 3.162
4.0 4.0
4.0
GAJBIRSINGHand Y. V. K. SADASIVARAO
320
Table 2. Simply supported, cross-ply square plate
(ii) Material 2: EL/E,= 13.82 E,= 5.50Gpa G,,/Er=0.418 GLZ=GTZ=GLT VLT= 0.34
Non-dimensional buckling parameter, N,bZ
*=a)
No.
of layers
Ref. [3]
Present element
2 3 4 5 6 10
9.3146 19.3040 17.7829 20.4663 19.6394 20.6347
9.3152 19.4276 17.8371 20.4841 19.7339 20.7354
(iii) Material 3: EL/E,=40 ET= 5.172 Gpa G,,IE,= 0.5 Gz= GLZ= GLT VLT= 0.25,
WE, = 30, G,,IE, = 0.6, G,/E,= 0.5,G,IE,= 0.5, vLr= 0.25, b/t= 10. material under uni-axial loads obtained from the present formulation is compared with the exact and Mindlin’s results in Table 1. The present element is found to yield an upper bound in thick and thin regions of the plate and is closer to the thin plate value for b/t = 20, thus indicating a good comparison. Table 2 shows the comparison of nondimensional buckling parameter values for a crossply plate under uniaxial load and varying the number of layers. Apparently, the results indicate a good agreement. Next, the critical load parameters for a two-layered (angle-ply) anisotropic simply supported plate under uniaxial compression with different fibre orientations are compared with the existing results and are tabulated in Table 3. The results from the present theory compare well, although these yield both upper and lower values. Thus, the capability of the present element to solve problems involving isotropic, orthotropic and anisotropic plates is fully established. Further results in this paper are confined to square plates with the following material properties: (i) Material 1: EL/ET= 11.512 ET = 10.30 Gpa G,,jE,= 0.696 G,JE,= 1.0 G&E,= 1.0 vLr = 0.28 Table 3. Simply supported angle-ply square plate (+ 0 / - 0) 6 (degrees) 0
15 30 45 60 15 90
Non-dimensional buckling parameter Ref. [2] Present theory 35.831 21.734 20.441 21.709 19.392 12.915 13.132
34.654 20.716 21.3422 22.9731 20.381 14.665 12.895
ELET= 40, G,,IE,=0.5, vLr= 0.25, G,IE,= 0.2, G,IE,= 0.2, b/t= 25,
where E, G and v denote the Young’s and shear moduli and Poisson’s ratio respectively. During the stress analysis computation, the transverse displacement and slopes are assumed to vanish in the entire plate for determining the initial stresses. The other boundary conditions used in this paper are: (i) Simply supported (SS): Stress analysis u = 0 along Y-axis v = 0 along X-axis Stability w = tjx = 0 along X-axis w = $J = 0 along Y-axis (ii) Clamped (CC): Stress analysis u = 0 along Y-axis v = 0 along X-axis Stability u = w = $, = I+?,= 0 along X-axis u = w = J/, = $y = 0 along Y-axis. The effect of fibre orientation on the buckling load of a square unidirectional laminate under SS boundary is shown in Fig. 2 for two materials. The nondimensional buckling parameter, 1, is consistently lower for thick plates at all the angles considered. The lowest i occurs at different angles for each material which can be mainly attributed to the material properties. However, this load occurs at the same angle of orientation for thick and thin plates of the same material. The computed values of I for a thin plate of material 1 with 0 and 90” fibre orientation are 4.6104 and 3.4173146 respectively. These can be calculated as 4.62498 and 3.347123 from [12], indicating a good comparison. Figure 2 also shows the numerical results from [ 11, Table 51 corresponding to the above plate. These indicate a completely different behaviour owing to the fact that the formulation of [1 1] is suitable only for orthotropic cases and does not include the effects of anisotropic coefficients, viz. A,,,Az6,D,,and D,,,and hence is not applicable for analysis of plates with fibre orientation angles other
Stability of thick angle-ply composite plates
6
ET = 0.75 x IO6 MPo, G,/Ev0.5,
4 50,
Moteriol
-*---
e=lO,
Moteriot-l
-----
t=lO,
Moterial-2
-v-x-
REF. CI II SS Condition
q
321
____I__ -*___
00 15’ L$y
-r-l_
75’
~~~-0.25, GLTGLz .GLT
900 SS Condition /
0
I
I
30
60
I 90
Angle of orientation
(8)
Fig. 2. Variation of buckling parameter with fibre orientation (square plate under uniaxial compression). than 0 and 90”. Moreover, the above mentioned results do not compare with those of [12], even at orientation angles of 0 and 90”, due to erroneous computation. However, it is found that this error is in the definition of D, and D, in [I I] which should be corrected as D, = E&12(1
- vi=)
and D, = Ert3/12(1 -v:,). Figure 3 shows the variation of ;I with the fibre orientation for a biaxially loaded plate with different
__-----
+0.5 $0.25 +=I,,
Moteriol-I
and -2
SS Condition
I
I
I
I
I
I
15
30
45
60
75
so
Angle of orlentotion (8)
Fig. 3. Variation of buckling parameter with fibre orientation (square plate under biaxial compression). CAS 29/2-J
I
I
5
IO
i 20
I
I
30
40
E, /ET
Fig. 4. Effect of modulus ratio on the buckling load of a square plate.
load ratios, ~~1~~. The value of NYalone is varied in this computation. For both the materials considered, the buckling parameter increases at all angles of orientation as the load ratio decreases. We again see that the angle at which the lowest buckling load occurs is a function of the material properties. The variation of A with the modulus ratio EL/ET for a unidirectional square laminate under uniaxial compression is shown in Fig. 4, in which the value of EL is varied keeping ET constant. As expected, the laminate with 0” fibre orientation gives the highest L as Et/E, is increased, while that with 45” yields the lowest values. In order to bring out the effect of layering, an angle-ply square laminate under uniaxial loads with two and four layers is taken for analysis. The numerical results for such a laminate under SS and CC conditions are presented in Fig. 5. The effects of anisotropy and symmetric and antisymmetric layering can be studied from these results. While the anisotropic effects bring down the buckling load, the four-layered antisymmetric laminate gives a higher buckling load than the symmetric one. The vanishing of the D,, and D,,terms in the case of the antisymmetric laminate yields a higher buckling load when compared to a symmetric one in which the B,, and Bz6 terms are zero. The thickness effects are found to be more predominant for a plate with a clamped boundary when compared with one which is simply supported at all edges. CONCLUSION
A finite element fo~ulation amounting for the transverse shear effects is given in this paper for the
GAJBIR SINGH and Y.
322 -.---x-I-
+45/-45 +45/-45/+45/-45 +45/-45/-45/+45
b/t
Fig. 5. Effect of layering, stacking sequence and thickness on the buckling load of a square laminated plate. buckling analysis of layered thick composite plates under biaxial loading. The accuracy of the element is demonstrated by comparing the existing results in the literature for isotropic, orthotropic and anisotropic cases. The effects of fibre orientation, material properties and layering are studied in detail. Results are presented for the cases of plates with different materials and under various boundary conditions.
1. R. M. Jones, Buckling and vibration of unsymmetrically laminated cross-ply rectangular plates. AIAA Jnl 11, 16261632 (1973). 2. R. M. Jones, S. Morgan and J. M. Whitney, Buckling and vibration of antisymmetrically laminated angle-ply rectangular plates. J. uppl. Mech. 1143-1144 (1973). A. K. Noor, Stability of multilayered composite plates. Fibre Sci. Technol. 8, 81-89 (1975). Y. Hirano, Optimum design of laminated plates under axial compression. AZAA Jnl 17, 1017-1019 (1979). Y. Hirano, Optimum design of laminated plates under shear. J. Comn. Mat. 13. 329-334 (1973). J. T. S. Wang, Best angles against buckhng of rectangular laminates. Proceedings of the 4th international conference on composite materials, pp. 575-582 (1982). 7. S. V. Rajgopal, G. Singh and Y. V. K. Sadasiva Rao, Large deflection and non-linear vibration of multilayered sandwich plates. AIAA Jnl (to appear). 8. G. Singh and Y. V. K. Sadasiva Rao, Large deflection behaviour of thick composite plates. (communicated to Composite Structures). 9. S. Srinivas and A. K. Rao, Buckling of thick rectangular plates. AIAA Jnl 7, 1645 (196%. 1o S. P. Timoshenko and J. M. Gere. Theory of Elastic ’ Stability. McGraw-Hill, New York’ (1961): ” 11. T. Chelladurai, B. P. Shastry and G. V. Rao, Effect of fibre orientation on the stability of orthotropic rectangular plates. Fibre Sci. Technol. 20, 121-134 (1984). 12 S. G. Lekhintskii, Anisotropic Plates. Gordon & Breach, New York (1956).
APPENDIX
Matrices [BL] and [BNL]from eqn (6) are given by:
with
SADASIVA RAO
REFERENCES
Materiol-3
12 -
V. K.
Compum a SrrucruresVol. 29, No. 2, pp. 317-322, 1988 Printed in Great Britain.
STABILITY OF THICK ANGLE-PLY COMPOSITE PLATES GAJBIR SINGH~ and Y. V. K. SADASIVA RAO$ tStructura1 Engineering Group, Vikram Sarabhai Space Centre, Trivandrum 695 022, India $Stage En~n~~ng and Analysis Division, Liquid Propulsion systems Centre, T~van~m 695 547, India (Received 20 July 1987)
Abstract-An eight-noded quadratic rectangular element is developed in order to study the stability aspects of a layered rectangular composite plate under biaxial loading. The element has five degrees of freedom per node. The formulation includes the Mindlin-TV nonlinear strain~ispla~ent relations. The adoptability of the element is demonstrated by solving a number of problems for which results exist in the literature. The effects of fibre orientation, material properties, layering and boundary conditions on the buckling load are studied in detail. The deficiency of earlier reported results is also pointed out.
NOTATION in-plane, bending-in-plane coupling and bending coefficients twisting and thickness-shear coefficients length of the plate matrix relating the strains and displacements linear and non-linear parts of the matrix B breadth of the plate matrix relating the stress to strain Young’s moduli along and transverse directions of the fibre in-plane and transverse shear moduli fibre directions matrix denoting the serendipity polynominals in-plane loads in x and y directions respectively number of degrees of freedom per each variable total thickness of the plate components of displacements in x, y and z directions respectively variable in space vector denoting fhe sum of internal and external forces generalised displacement vector interpolation functions curvatures Poisson’s ratio of the material stress vector non-dimensional buckling parameter
[A,], [B,], [BJ
=
Nxbz
&mm (unless otherwise defined) total slopes strain vector
The principal advantage of composites is the possibility of tailoring a laminate to suit the structural requirements. By using the directional nature of the material to advantage, highly efficient structures can
be designed. The advent of advanced fibre-reinform composite materials such as glass/epoxy, kevlar/ epoxy and graphite/epoxy with their high potential weight savings has resulted in a significant increase in the use of laminated plates and shells. These structures are subjected to different types of loadings such as bending, tensile, compression etc. Among all the possible loadings their resistance to buckling is of greatest interest to most of the researchers. Several theoretical and experimental papers have been published on the buckling of anisotropic plates under different types of loadings. Jones [l] evaluated the buckling and frequency parameters of unsymmetrically laminated cross-ply rectangular thin plates. Subsequently, Jones et al. [2] extended the above work to uns~met~cally laminated angle-ply plates. Noor [3] calculated the buckling loads of thick laminated cross-ply composite plates by using different shear correction factors for different stacking sequences. Hirano 14, S] attempted, at the optimum stacking sequence, to maximise the uni-axial buckling load of a thin angle-ply laminate. Wang [6] showed that a thin angle-ply laminate with orientation [O/-f?], is superior to a thin cross-ply with [0/9OL, when subjected to axial loads. Very little literature is available on the stability analysis of thick rectangular angle-ply composite plates using finite element method. The present paper is an extension of the finite element formulation [7,8] to the buckling analysis of thick angle-ply plates. Mindlin type non-linear straindisplacement relations are assumed for constructing the geometric stiffness matrix. The accuracy of the element is validated by comparing the available results in the literature. The effects of fibre orientation, material properties and layering are studied in detail. FORMULATION
Let us consider a laminated plate with ‘n’ number of layers having thicknesses I,, t2.. . t,,, each pos317
318
Y _i
19
-_-*
-*-•
IQI
20
14
! 9
9 li --. *‘II
‘5*,i _I
$I
I6 I 13
-u
i
“1
121 -E
I 0
‘1
I 1”
0
l Node Fig. 1. Geometry and finite ekment idealisation uf a layered plate.
sessing individual properties, acted upon by axial compressive loads N, and NYin the x and y directions as shown in Fig. 1. The dis~ia~ments at any paint in the plate are given by u k Y, r) = ~%, Y) - stlY.lx, Y) D~~~Y,~~=~“f~,Y~-zJlytx,Y~
where a comma denotes diffe~ntiation. can be written as
Equation (2)
cc> = @Lf + &.It where {e&t)and {tNL) are the matrices containing linear and non-linear terms. The stress and moment resultants are determined in the standard manner by considering all the ‘n’ layers, yielding
and
where u, v and w are the displacements in x, y and z directions, uO, v” and w” are the associated mid-plane displa~ments and JI, anf $,, are the total slopes along the x and y directions respectively. For the case of a Mindlin plate, the mid-plane strains and curvatures are given by
and K2K
QI Q2 i~C
=
&k&
Ec,&& K&8,
1’
(3)
where A,, 3, ttii (i, j = 1,2,6) and A# (i, j = 4,s) are
319
Stability of thick angle-ply composite plates
where {cr} = [D](C) and {E} is the vector containing. the respective in-plane, bending-in-plane coupling, external forces due to imposed loads bending and twisting, and thickness-shear coefficients respectively; K4 and KS are the shear correction factors which are taken as 0.8333 [8]. N, d(y) = 0, and Mi are the in-plane stress and moment resultants. where FINITE ELEMENT FORMULATION
d(y} =
Let the region of the plate be divided into a number of quadratic elements with eight nodes per element and five degrees of freedom (a, v, w, $, and $,) per node as shown in Fig. 1. The displacements at any point in the element can be taken in terms of the surrounding nodal displacements as
s”
d[81r(a}
do +
s”
[B]‘d{b} do
where
where N is the matrix containing the standard serendipity polynomial functions and {u}, represents the nodal displacements. The generalised displacements
(6) = [u,0, w, J/m$yl’
Kld{~I =
I”
4Kd-~~I da
in each element can be represented by
[KJ,[KJ and [K,] are the small displacement, large displacement and geometric stiffness matrices respectively. Assuming [KNL] = 0,the initial stability problem of eqn (7) reduces to
(8)
d(y)=([K,l+I[K,l}d(~}=0.
[K,]can be obtained by carrying out stress analysis as reported in (81. The eigenvalues, 1, of eqn (8) give rise to the buckling loads of the laminated plate. Gaussian quadrature formula adopting reduced integration wherever necessary is used in obtaining the element stiffness and geometric stiffness matrices.
where dy(a = 1,2,3) is the interpolation function corresponding to the ith node in the element and r, s and p are the number of degrees of freedom per variable. For simplicity, we take 4; = #J: = 4: and r=s=JJ[7j. Substitution of (5) in (2) gives rise to
The matrix [B] contains linear and non-linear which can be written as
terms
The matrices [BJ and BNJ are given in the Appendix. The vector sum of the internal and external forces is
1~1=
[Wa)
dv - {RI,
NUMERICAL RESULTS AND DISCUSSION
In order to study the applicability and accuracy of the element developed, a number of numerical examples are investigated. Some of these results are compared with those existing in the literature and are presented in Tables 1-3. The values of the nondimensional buckling parameter, 1, for an isotropic Table 1. Simply supported square isotropic plate Non-dimensional buckling parameters, thick plate theory Exact theory Mindlin’s theory [9] [91 3.924 3.909 3.741 3.729 5 3.150 3.115
b/t ti
y CO.3, 6 =N,b2
n=D'
Present theory
Thin plate theory [IO]
3.941 3.745 3.162
4.0 4.0
4.0
GAJBIRSINGHand Y. V. K. SADASIVARAO
320
Table 2. Simply supported, cross-ply square plate
(ii) Material 2: EL/E,= 13.82 E,= 5.50Gpa G,,/Er=0.418 GLZ=GTZ=GLT VLT= 0.34
Non-dimensional buckling parameter, N,bZ
*=a)
No.
of layers
Ref. [3]
Present element
2 3 4 5 6 10
9.3146 19.3040 17.7829 20.4663 19.6394 20.6347
9.3152 19.4276 17.8371 20.4841 19.7339 20.7354
(iii) Material 3: EL/E,=40 ET= 5.172 Gpa G,,IE,= 0.5 Gz= GLZ= GLT VLT= 0.25,
WE, = 30, G,,IE, = 0.6, G,/E,= 0.5,G,IE,= 0.5, vLr= 0.25, b/t= 10. material under uni-axial loads obtained from the present formulation is compared with the exact and Mindlin’s results in Table 1. The present element is found to yield an upper bound in thick and thin regions of the plate and is closer to the thin plate value for b/t = 20, thus indicating a good comparison. Table 2 shows the comparison of nondimensional buckling parameter values for a crossply plate under uniaxial load and varying the number of layers. Apparently, the results indicate a good agreement. Next, the critical load parameters for a two-layered (angle-ply) anisotropic simply supported plate under uniaxial compression with different fibre orientations are compared with the existing results and are tabulated in Table 3. The results from the present theory compare well, although these yield both upper and lower values. Thus, the capability of the present element to solve problems involving isotropic, orthotropic and anisotropic plates is fully established. Further results in this paper are confined to square plates with the following material properties: (i) Material 1: EL/ET= 11.512 ET = 10.30 Gpa G,,jE,= 0.696 G,JE,= 1.0 G&E,= 1.0 vLr = 0.28 Table 3. Simply supported angle-ply square plate (+ 0 / - 0) 6 (degrees) 0
15 30 45 60 15 90
Non-dimensional buckling parameter Ref. [2] Present theory 35.831 21.734 20.441 21.709 19.392 12.915 13.132
34.654 20.716 21.3422 22.9731 20.381 14.665 12.895
ELET= 40, G,,IE,=0.5, vLr= 0.25, G,IE,= 0.2, G,IE,= 0.2, b/t= 25,
where E, G and v denote the Young’s and shear moduli and Poisson’s ratio respectively. During the stress analysis computation, the transverse displacement and slopes are assumed to vanish in the entire plate for determining the initial stresses. The other boundary conditions used in this paper are: (i) Simply supported (SS): Stress analysis u = 0 along Y-axis v = 0 along X-axis Stability w = tjx = 0 along X-axis w = $J = 0 along Y-axis (ii) Clamped (CC): Stress analysis u = 0 along Y-axis v = 0 along X-axis Stability u = w = $, = I+?,= 0 along X-axis u = w = J/, = $y = 0 along Y-axis. The effect of fibre orientation on the buckling load of a square unidirectional laminate under SS boundary is shown in Fig. 2 for two materials. The nondimensional buckling parameter, 1, is consistently lower for thick plates at all the angles considered. The lowest i occurs at different angles for each material which can be mainly attributed to the material properties. However, this load occurs at the same angle of orientation for thick and thin plates of the same material. The computed values of I for a thin plate of material 1 with 0 and 90” fibre orientation are 4.6104 and 3.4173146 respectively. These can be calculated as 4.62498 and 3.347123 from [12], indicating a good comparison. Figure 2 also shows the numerical results from [ 11, Table 51 corresponding to the above plate. These indicate a completely different behaviour owing to the fact that the formulation of [1 1] is suitable only for orthotropic cases and does not include the effects of anisotropic coefficients, viz. A,,,Az6,D,,and D,,,and hence is not applicable for analysis of plates with fibre orientation angles other
Stability of thick angle-ply composite plates
6
ET = 0.75 x IO6 MPo, G,/Ev0.5,
4 50,
Moteriol
-*---
e=lO,
Moteriot-l
-----
t=lO,
Moterial-2
-v-x-
REF. CI II SS Condition
q
321
____I__ -*___
00 15’ L$y
-r-l_
75’
~~~-0.25, GLTGLz .GLT
900 SS Condition /
0
I
I
30
60
I 90
Angle of orientation
(8)
Fig. 2. Variation of buckling parameter with fibre orientation (square plate under uniaxial compression). than 0 and 90”. Moreover, the above mentioned results do not compare with those of [12], even at orientation angles of 0 and 90”, due to erroneous computation. However, it is found that this error is in the definition of D, and D, in [I I] which should be corrected as D, = E&12(1
- vi=)
and D, = Ert3/12(1 -v:,). Figure 3 shows the variation of ;I with the fibre orientation for a biaxially loaded plate with different
__-----
+0.5 $0.25 +=I,,
Moteriol-I
and -2
SS Condition
I
I
I
I
I
I
15
30
45
60
75
so
Angle of orlentotion (8)
Fig. 3. Variation of buckling parameter with fibre orientation (square plate under biaxial compression). CAS 29/2-J
I
I
5
IO
i 20
I
I
30
40
E, /ET
Fig. 4. Effect of modulus ratio on the buckling load of a square plate.
load ratios, ~~1~~. The value of NYalone is varied in this computation. For both the materials considered, the buckling parameter increases at all angles of orientation as the load ratio decreases. We again see that the angle at which the lowest buckling load occurs is a function of the material properties. The variation of A with the modulus ratio EL/ET for a unidirectional square laminate under uniaxial compression is shown in Fig. 4, in which the value of EL is varied keeping ET constant. As expected, the laminate with 0” fibre orientation gives the highest L as Et/E, is increased, while that with 45” yields the lowest values. In order to bring out the effect of layering, an angle-ply square laminate under uniaxial loads with two and four layers is taken for analysis. The numerical results for such a laminate under SS and CC conditions are presented in Fig. 5. The effects of anisotropy and symmetric and antisymmetric layering can be studied from these results. While the anisotropic effects bring down the buckling load, the four-layered antisymmetric laminate gives a higher buckling load than the symmetric one. The vanishing of the D,, and D,,terms in the case of the antisymmetric laminate yields a higher buckling load when compared to a symmetric one in which the B,, and Bz6 terms are zero. The thickness effects are found to be more predominant for a plate with a clamped boundary when compared with one which is simply supported at all edges. CONCLUSION
A finite element fo~ulation amounting for the transverse shear effects is given in this paper for the
GAJBIR SINGH and Y.
322 -.---x-I-
+45/-45 +45/-45/+45/-45 +45/-45/-45/+45
b/t
Fig. 5. Effect of layering, stacking sequence and thickness on the buckling load of a square laminated plate. buckling analysis of layered thick composite plates under biaxial loading. The accuracy of the element is demonstrated by comparing the existing results in the literature for isotropic, orthotropic and anisotropic cases. The effects of fibre orientation, material properties and layering are studied in detail. Results are presented for the cases of plates with different materials and under various boundary conditions.
1. R. M. Jones, Buckling and vibration of unsymmetrically laminated cross-ply rectangular plates. AIAA Jnl 11, 16261632 (1973). 2. R. M. Jones, S. Morgan and J. M. Whitney, Buckling and vibration of antisymmetrically laminated angle-ply rectangular plates. J. uppl. Mech. 1143-1144 (1973). A. K. Noor, Stability of multilayered composite plates. Fibre Sci. Technol. 8, 81-89 (1975). Y. Hirano, Optimum design of laminated plates under axial compression. AZAA Jnl 17, 1017-1019 (1979). Y. Hirano, Optimum design of laminated plates under shear. J. Comn. Mat. 13. 329-334 (1973). J. T. S. Wang, Best angles against buckhng of rectangular laminates. Proceedings of the 4th international conference on composite materials, pp. 575-582 (1982). 7. S. V. Rajgopal, G. Singh and Y. V. K. Sadasiva Rao, Large deflection and non-linear vibration of multilayered sandwich plates. AIAA Jnl (to appear). 8. G. Singh and Y. V. K. Sadasiva Rao, Large deflection behaviour of thick composite plates. (communicated to Composite Structures). 9. S. Srinivas and A. K. Rao, Buckling of thick rectangular plates. AIAA Jnl 7, 1645 (196%. 1o S. P. Timoshenko and J. M. Gere. Theory of Elastic ’ Stability. McGraw-Hill, New York’ (1961): ” 11. T. Chelladurai, B. P. Shastry and G. V. Rao, Effect of fibre orientation on the stability of orthotropic rectangular plates. Fibre Sci. Technol. 20, 121-134 (1984). 12 S. G. Lekhintskii, Anisotropic Plates. Gordon & Breach, New York (1956).
APPENDIX
Matrices [BL] and [BNL]from eqn (6) are given by:
with
SADASIVA RAO
REFERENCES
Materiol-3
12 -
V. K.