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Bending and vibration of symmetric cross-ply laminated plates using ply dependent shear deformation model
Journal of Sound and Vibration (1992) 158(2), 257-265
B E N D I N G A N D V I B R A T I O N OF S Y M M E T R I C CKOSS-PLY L A M I N A T E D PLATES USING PLY D E P E N D E N T SHEAR. D E F O R M A T I O N M O D E L K. JAWAHAR REDDY AND K. VIJAYAKUMAR
Department of Aerospace Engineering, Indian Institute of Science, Bangaiore 560 012, India (Received 4 February 1991, and in final form ! July 1991)
A new ply dependent shear deformation model based on a priori assumed slopes of transverse shear stresses is developed. The performance of the present model relative to that of an earlier model [ 1, 2] is assessed by a comparison with the exact elasticitysolutions. The present model is found to be in good agreement with the exact elasticity distributions of the displacements and stresses. The flexural frequenciespredicted by the present model are lower than the corresponding estimates from the CPT, HSDT and the earlier model [ 1, 2]. In the case of thickness twist mode frequencies, the estimates provided by the present model are found to be significantlydifferentfrom those obtained by using the other models. 1. INTRODUCTION Recent advances made in the analysis of laminated beams and plates with emphasis on transverse shear effects and development of several 2-D approximate models due to complexity of 3-D elasticity approach have been extensively reviewed by Noor and Burton [3], Kapania and Raciti [4, 5], and Reddy [6]. In displacement-based models for the analysis of laminated plates, the displacements are assumed to have a priori assigned thicknesswise distributions either plywise or in the total laminate. In the latter case, it is useful and becomes necessary for thick plates to incorporate a realistic ply-dependent character of the assumed displacements in order to account for the thicknesswise variation of the stress-strain relations while maintaining continuity across the interfaces. With the aim of incorporating such a ply-dependent character of the assumed displacements, we have recently proposed a new approach [1] to account for transverse shear effects in the analysis of flexure of symmetric laminates. This approach is based on assumed plywise distributions in terms of reference plane variables for transverse normal strain (ez, generally assumed to be zero) and slopes of transverse shear stresses (o'4,z and os.~). In the bending of symmetric laminates, the simplest shear deformation model in the above approach is derived from assumed zero values for cr4,z and crs.~,and the resulting model (model I) can be shown to be equivalent to an earlier model corresponding to theory II presented in their work by Sun and Whitney [7]. A refined model is obtained by assuming thickness-wise linear variation of cr4.z and ors..- in the laminate (model II). An equivalent model has been recently proposed independently by Lee et al. [2] for bending of unsymmetric laminates. In the above models, thickness-wise distributions of transverse shear stresses are independent of ply material properties. However, the transverse shear strains are ply dependent, having discontinuities across the interfaces. On integration of the transverse shear strain-displacement relations along with ~: equal to zero and maintaining continuity across interfaces, the derived in-plane displacement distributions also become 257 0022-460x/92/200257 + 09 $08.00/0 © 1992AcademicPress Limited
258
K. J. R E D D Y
AND
K. V I J A Y A K U M A R
dependent on transverse shear compliances. In order to make t~4 and '~5 distributions dependent on ply material properties, in-plane equilibrium equations in each ply are considered from which or4.: and ~5.~ are assumed to be plywise linear with slope changing from ply to ply according to the change in the corresponding direct modulus (model IIl), with the Poisson and in-plane shear effects ignored. It has been observed that model I is not adequate to provide realistic response characteristics of laminates [1 ]. In this paper, a detailed numerical study with regard to the performance of models II and III is presented by comparing results with those from exact elasticity solutions for flexure of simply supported square plates subjected to sinusoidal loading. These models are also used for obtaining flexural and thickness twist mode frequencies of a simply supported [0/90]s square plate and these results are compared with those obtained from HSDT [8]. Rotary inertia is included in the present investigations.
2. FORMULATION Consider a rectangular plate of plan-form dimensions a and b and thickness 2h. The plate is composed of 2N layers of orthotropic material, and the axes of material symmetry are parallel to the plate axes x, y and z. 2.1. CONSTITUTIVE EQUATIONS The constitutive equations in the kth ply, orthotropic in the laminate co-ordinate system, are, in the usual contracted notation,
tri=Q~ei,
i=1,2,6,
e~=S~tr~,
a=4,5,
e3=S33tr3+S31trl,
(la-c)
where Ca are plane stress reduced elastic constants and $3:, S ~ are elastic compliances. In these relations, a repeated indexj indicates summation over its range of specified integer values l, 2, 6. (Suffix k is generally not indicated, but used whenever it is required for clarity.) 2.2.
IN-PLANE
DISPLACEMENTS
As mentioned in the introduction, the thickness-wise distributions for slopes of transverse shear stresses in model II are assumed as o-~: = -2z¢,¢O)(x, y)
(model II).
(2)
In model II1, 3-D equilibrium equations are considered in each ply and the in-plane stresses are expressed in terms of in-plane strains with the use of reduced stiffness coefficients. In these equations, we retain only the terms associated with direct stiffnesses QII and Q22, ignoring the influence of Poisson and shear coupling material constants, and express the slopes of transverse shear stresses as
aa,~/~z = - Q(")zv°~)(x, y)
(model III),
(3)
where 0<4) = 022,
0 °) = Q,,.
(4)
Model II is a special case of model III, with each of QI~ and 022 replaced by a constant value 2 for all plies.
PLY D E P E N D E N T SHEAR D E F O R M A T I O N M O D E L
259
Transverse shear stresses are obtained by integrating equation (3) with integration constants determined from interface continuity conditions and shear free conditions. These transverse shear stresses are substituted in constitutive relation (1 b) to obtain the transverse shear strains e,. Using the transverse shear strains thus obtained along with zero normal strain ~3 in the strain-displacement relations e3 = w.~ ,
e4 = w,y + v.~ ,
e5 = W,x + u.~ ,
(5)
and integrating, one obtains the following expressions for displacements in each ply, maintaining continuity across interfaces: w = wo(x, y),
(6a)
u = -ZWo,x + [G (s) - {((BtS))A,--(B~S))k)Zk + O(S)Z~(Zk/3 -- hk)/2}Sss] u/ts),
(6b)
V = --ZWo,y+ [ G f4) - { ((B(4)),v - (B(4))k)Zk'Jl- O(4)Z~(Zk/3 -- hk)/Z}S44]llt (4).
(6c)
Here k
{B~)}k= E OJ~)(t~/2-hth),
zk=hk -z,
(7, 8)
I=1 k
{G~a)}k= E Sa~[({Bt~)}N- {B{~)}k)h+ O ~ " ) t ~ ( h / 3 - h 3 / 2 ] .
(9)
I--I
The unknown variables w0, ~4) and ~5) are obtained by using the energy method (principle of virtual displacements) for the static case and Hamilton's principle for the free vibration response. Details of this algebra are not presented here.
4. NUMERICAL RESULTS AND DISCUSSION 4.1. BENDING A simply supported square symmetric cross-ply laminate subjected to sinusoidal load, (0-3)~=+h= 4-(q0/2) sin (fix~a) sin (fly~a),
(10)
is considered for numerical investigations. The solutions for the displacements take the forms w = ff sin (Trx/a) sin (fiy/a),
u = t~(z) cos (fix~a) sin (fly~a),
v = ~(z) sin ( f i x / a ) cos (fly/a).
(1 l a,b) (llc)
Numerical data have been generated both by the present procedure and by the exact analysis given by Pagano [9], for the following material properties, with material axes L and T along and perpendicular to the fiber direction, respectively: EL = 25Er, GLr = O"5Er, G r r = O . 2 E r , VLr = Vrr=0-25. For a [0/90], laminate, a/2h ratios of 4, 5, 6, 7.5, 10, 20, 50 and 100 have been considered. For a thick plate, a/2h = 4, the thicknesswise distributions of displacements and stresses are presented in the following non-dimensionalized forms: W= lO0~,Er/2hqoS 4, (el,
0"2, 0-6) = ( e t ,
02,
(ti, t3)= 100(if, O)Er/2hqoS 3,
O6)/qoS 2,
(04, 6"5)=(04, Os)/qoS.
(12a, b) (12c, d)
260
K.J.
REDDY
AND
K. V I J A Y A K U M A R
Here S = a/2h.
(13)
Thickness-wise distributions of displacements and stresses obtained by the models II and III are compared with those given by the exact solutions in Figures 1-7. The estimates of transverse shear stresses correspond to statically equivalent distributions. It can be seen that all the above mentioned distributions obtained by model III are in good agreement with the exact solutions, better than those obtained by model II, and reflect all essential features of the exact solutions. Variations of percentage errors in the estimates for the maximum values of displacements and stresses with respect to 2h/a are shown in Figures 8-10. Errors in these estimates by the present models are generally found to increase with increasing values of 2h/a. It is observed that the present model gives a maximum error in the estimates of t~ which is less than 5-2% even for S = 5. 4.2. FREE VIBRATION In Table 1 are shown the non-dimensionalized fundamental frequencies cb = (coa2/2h)x/p-/Er of flexural and thickness twist modes for square plates with span to 0"5
~,.~"~-e~.'
\
"'~"
~
'
\\
04
\\\ !'~/i. \\\\~i
0"3
\
0.2
O.I
0"0 -I.0
",
,L -0"5
,
~
-~.
'\\ VI/t' 0
Figure 1. T h i c k n e s s - w i s e d i s t r i b u t i o n of t~ in a [0/90], square laminated plate ......
, model II; -----,
05
(EL/GLr=50).
model III; - - -, CPT.
0"5
0.4
0"3
0.2
0-1
0.0 -3"0
I
-20
-I.0
F i g u r e 2. A s F i g u r e 1, b u t d i s t r i b u t i o n o f 0.
0
- -
-, E x a c t ;
PLY DEPENDENT
SHEAR
DEFORMATION
MODEL
0"5
0'4
0.3
Z
/
/
0,2
0.1
0"0
_ _ . 1 . ~ -0.25
0,25
J 050
J.__
0-75
Figure 3. As Figure 1, but distribution of 6",.
0.4
0"3f ///~ /
........
,,/,..'"'"
0 2,
o.,p
.....
0.0 ~ / ' ' ~
1
I
I
0.5
G
I 1.0
Figure 4. As Figure !, but distribution of
o.5
d2.
-~.....! ~ \ , .~ -~................ i~\\ \
\~'%X
~ ~ . ."
0.4
".~,,,,,. \ Oq,
0.2
0'I
O-C -
I
0.05
[
- O. 0;)5
I
I
0
Figure 5. As Figure I, but distribution of a6.
-0.025
261
262
K, J. R E D D Y 05
----t
......
I
AND
K. V I J A Y A K U M A R
I
~
...... T -
t--
l
......
\'~ '
.....:
0.4
i. 0.0 ~
L
[
\ I
'
Ou
I
02
j
0"3
e4 F i g u r e 6. A s F i g u r e 1, b u t distribution of d 4 .
0.5
0.4
0.5
0.2
0.1
I I
0-0
I 0"1
I
I
I
I 0"3
0"2
F i g u r e 7. A s F i g u r e 1, b u t distribution of ~5.
,
,
4
r~
/ r / - / (v) / / / / { u ~/ / /
2
. fJ
(u)
j/l" I.I
.~o
"
0
I 0.05
I
\
010
L 015
I 0'20
0.25
2h/a
Figure 8. Percentage errors in the estimates of displacements (w, u, v) for [0/90], laminate ., Model II; - - - , model IIl.
vs. 2h/a
(EL/GLr
=
50).
depth thickness ratios (a/2h) equal to 2, 5, 10, 20 and 50. The frequency estimates from model II and model III are compared with those from HSDT. The flexural frequency estimate from model III is lower and hence more accurate than the corresponding estimates from each of the models CPT, HSDT and model II. In the case of thickness twist mode frequencies, the estimates from the present model are significantly different from those
PLY DEPENDENT
SHEAR
DEFORMATION
263
MODEL
// // 0.2 / / / / ° " 6 / / / ° " 1
4
8_
////
2
./"/'/'/"
0.1
ii ° -4
0.05
0"10
0,15
0'20
0"25
2h/o
Figure 9. As Figure 8, but errors of in-planestresses (tr,, trz, o6).
0
0"05
O'lO
0'15
0"20
0"25
2hla
Figure I0. As Figure 8, but errors of transverse shear stresses (04, its).
of the other models. This is, perhaps, due to the better thicknesswise distributions for displacements obtained by the present model in the static case. 5. CONCLUDING REMARKS A new ply dependent shear deformation model based on a priori assumed slopes of the transverse shear stresses has been presented. The displacements in the model are expressed in terms of reference plane variables but account is taken of the ply to ply variation of direct stiffnesses and transverse shear compliances. Its performance relative to that of an earlier model II [ l, 2] has been studied by comparing results with those from exact elasticity solutions for the flexure of simply supported [0/90Is square laminate under sinusoidal load. Errors in the estimates for displacements and stresses from the present model are generally found to increase with increasing values of 2h/a. The error in the estimate of 0 is maximum for the present model but it is within 5.2% for S = 5 and EL/GLz=50. The thickness-wise distributions obtained from the present model are found to be better than those obtained from the earlier model II. As such, the estimates of the thickness twist mode frequencies from the present model and model II are found to be significantly different from each other. The present model, like FSDT and HSDT, is quite suitable for finite element implementation.
I
2 15-90 5-51 5 18.30 10-79 10 18-74 15.11 20 18-85 17.65 50 18.89 18.67 t Results obtained by using HSDT [8].
a/2h
CPT I 20.76 75-67 256.70 974-62 5745.12
II
HSDTt
37.69 107.72 291.86 979.52 5999-94
III
. 5.61 10-85 15.14 17.66 18.67
. I 20.25 74.80 255.56 971.12 5716.56
Model II . . II 37.43 107.04 290.20 976.22 5994.48
III
Non-dimensionalized flexural and thickness-twist mode frequencies, cb = ( coa2/ 2h ) ~
TABLE 1
5-42 10-68 15.06 17-63 18.67
I
25.74 88.79 290-75 983-10 5768.22
II
Model III III 37.92 108.35 294.84 1097.49 6726-57
of [0/90]s square plate
>
> .< >
> Z
70
y..
7~
.Ix
t,~
PLY DEPENDENT SHEAR DEFORMATION MODEL
265
ACKNOWLEDGMENTS The authors express their thanks to Professor A. V. Krishna Murty for his valuable comments and keen interest shown during the course of the present investigation. The authors are grateful to the Aeronautics Research and Development Board, Ministry of Defence, Government of India, for providing financial assistance under grant No. 481.
REFERENCES 1. K. VIJAVAKUMAR,A. V. KRISHNA MURTV, K. S. NAOARAJA and K. JAWAHAR REDOV 1990 Department of Aerospace Engineering, Indian Institute of Science, Bangalore-12, Report No. ARDB/STR/481/90/01, 23 pp. +55 tables. Higher order theories for estimation of interlaminar stresses in laminates. 2. K. H. LEE, N. R. SENTHILNATHAN, S. P. ElM and S. T. CHOW 1990 Composite Structures 15, 137-148. An improved zig-zag model for the bending of laminated composite plates. 3. A. K. NooR and W. S. BURTON 1989 Applied Mechanics Reviews 42, 1-13. Assessment of shear deformation theories for multilayered composite plates. 4. R. K. KAPANIA and S. RACITI 1989 American Institute of Aeronautics and Astronautics Journal 27(7), 923-934. ReCent advances in analysis of laminated beams and plates, part I: shear effects and buckling. 5. R. K. KAPANIA and S. RACITI 1989 American Institute of Aeronautics and Astronautics Journal 27(7), 935-946. Recent advances in analysis of laminated beams and plates, part II: vibration and wave propagation. 6. J. N. REDDY 1990 Shock and Vibration Digest 22(7), 3-17. A review of refined theories of laminated composite plates. 7. C. T. SUN and J. M. WHITNEY 1973 American Institute of Aeronautics and Astronautics Journal 11, 178-183. Theories for the dynamic response of laminated plates. 8. J. N. REODY and N. D. PHAN 1985 Journal of Sound and Vibration 98, 157-170. Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory. 9. N. J. PA~ANO 1970 Journal of Composite Materials 4, 20-34. Exact solutions for rectangular bidirectional 'composites and sandwich plates.
B E N D I N G A N D V I B R A T I O N OF S Y M M E T R I C CKOSS-PLY L A M I N A T E D PLATES USING PLY D E P E N D E N T SHEAR. D E F O R M A T I O N M O D E L K. JAWAHAR REDDY AND K. VIJAYAKUMAR
Department of Aerospace Engineering, Indian Institute of Science, Bangaiore 560 012, India (Received 4 February 1991, and in final form ! July 1991)
A new ply dependent shear deformation model based on a priori assumed slopes of transverse shear stresses is developed. The performance of the present model relative to that of an earlier model [ 1, 2] is assessed by a comparison with the exact elasticitysolutions. The present model is found to be in good agreement with the exact elasticity distributions of the displacements and stresses. The flexural frequenciespredicted by the present model are lower than the corresponding estimates from the CPT, HSDT and the earlier model [ 1, 2]. In the case of thickness twist mode frequencies, the estimates provided by the present model are found to be significantlydifferentfrom those obtained by using the other models. 1. INTRODUCTION Recent advances made in the analysis of laminated beams and plates with emphasis on transverse shear effects and development of several 2-D approximate models due to complexity of 3-D elasticity approach have been extensively reviewed by Noor and Burton [3], Kapania and Raciti [4, 5], and Reddy [6]. In displacement-based models for the analysis of laminated plates, the displacements are assumed to have a priori assigned thicknesswise distributions either plywise or in the total laminate. In the latter case, it is useful and becomes necessary for thick plates to incorporate a realistic ply-dependent character of the assumed displacements in order to account for the thicknesswise variation of the stress-strain relations while maintaining continuity across the interfaces. With the aim of incorporating such a ply-dependent character of the assumed displacements, we have recently proposed a new approach [1] to account for transverse shear effects in the analysis of flexure of symmetric laminates. This approach is based on assumed plywise distributions in terms of reference plane variables for transverse normal strain (ez, generally assumed to be zero) and slopes of transverse shear stresses (o'4,z and os.~). In the bending of symmetric laminates, the simplest shear deformation model in the above approach is derived from assumed zero values for cr4,z and crs.~,and the resulting model (model I) can be shown to be equivalent to an earlier model corresponding to theory II presented in their work by Sun and Whitney [7]. A refined model is obtained by assuming thickness-wise linear variation of cr4.z and ors..- in the laminate (model II). An equivalent model has been recently proposed independently by Lee et al. [2] for bending of unsymmetric laminates. In the above models, thickness-wise distributions of transverse shear stresses are independent of ply material properties. However, the transverse shear strains are ply dependent, having discontinuities across the interfaces. On integration of the transverse shear strain-displacement relations along with ~: equal to zero and maintaining continuity across interfaces, the derived in-plane displacement distributions also become 257 0022-460x/92/200257 + 09 $08.00/0 © 1992AcademicPress Limited
258
K. J. R E D D Y
AND
K. V I J A Y A K U M A R
dependent on transverse shear compliances. In order to make t~4 and '~5 distributions dependent on ply material properties, in-plane equilibrium equations in each ply are considered from which or4.: and ~5.~ are assumed to be plywise linear with slope changing from ply to ply according to the change in the corresponding direct modulus (model IIl), with the Poisson and in-plane shear effects ignored. It has been observed that model I is not adequate to provide realistic response characteristics of laminates [1 ]. In this paper, a detailed numerical study with regard to the performance of models II and III is presented by comparing results with those from exact elasticity solutions for flexure of simply supported square plates subjected to sinusoidal loading. These models are also used for obtaining flexural and thickness twist mode frequencies of a simply supported [0/90]s square plate and these results are compared with those obtained from HSDT [8]. Rotary inertia is included in the present investigations.
2. FORMULATION Consider a rectangular plate of plan-form dimensions a and b and thickness 2h. The plate is composed of 2N layers of orthotropic material, and the axes of material symmetry are parallel to the plate axes x, y and z. 2.1. CONSTITUTIVE EQUATIONS The constitutive equations in the kth ply, orthotropic in the laminate co-ordinate system, are, in the usual contracted notation,
tri=Q~ei,
i=1,2,6,
e~=S~tr~,
a=4,5,
e3=S33tr3+S31trl,
(la-c)
where Ca are plane stress reduced elastic constants and $3:, S ~ are elastic compliances. In these relations, a repeated indexj indicates summation over its range of specified integer values l, 2, 6. (Suffix k is generally not indicated, but used whenever it is required for clarity.) 2.2.
IN-PLANE
DISPLACEMENTS
As mentioned in the introduction, the thickness-wise distributions for slopes of transverse shear stresses in model II are assumed as o-~: = -2z¢,¢O)(x, y)
(model II).
(2)
In model II1, 3-D equilibrium equations are considered in each ply and the in-plane stresses are expressed in terms of in-plane strains with the use of reduced stiffness coefficients. In these equations, we retain only the terms associated with direct stiffnesses QII and Q22, ignoring the influence of Poisson and shear coupling material constants, and express the slopes of transverse shear stresses as
aa,~/~z = - Q(")zv°~)(x, y)
(model III),
(3)
where 0<4) = 022,
0 °) = Q,,.
(4)
Model II is a special case of model III, with each of QI~ and 022 replaced by a constant value 2 for all plies.
PLY D E P E N D E N T SHEAR D E F O R M A T I O N M O D E L
259
Transverse shear stresses are obtained by integrating equation (3) with integration constants determined from interface continuity conditions and shear free conditions. These transverse shear stresses are substituted in constitutive relation (1 b) to obtain the transverse shear strains e,. Using the transverse shear strains thus obtained along with zero normal strain ~3 in the strain-displacement relations e3 = w.~ ,
e4 = w,y + v.~ ,
e5 = W,x + u.~ ,
(5)
and integrating, one obtains the following expressions for displacements in each ply, maintaining continuity across interfaces: w = wo(x, y),
(6a)
u = -ZWo,x + [G (s) - {((BtS))A,--(B~S))k)Zk + O(S)Z~(Zk/3 -- hk)/2}Sss] u/ts),
(6b)
V = --ZWo,y+ [ G f4) - { ((B(4)),v - (B(4))k)Zk'Jl- O(4)Z~(Zk/3 -- hk)/Z}S44]llt (4).
(6c)
Here k
{B~)}k= E OJ~)(t~/2-hth),
zk=hk -z,
(7, 8)
I=1 k
{G~a)}k= E Sa~[({Bt~)}N- {B{~)}k)h+ O ~ " ) t ~ ( h / 3 - h 3 / 2 ] .
(9)
I--I
The unknown variables w0, ~4) and ~5) are obtained by using the energy method (principle of virtual displacements) for the static case and Hamilton's principle for the free vibration response. Details of this algebra are not presented here.
4. NUMERICAL RESULTS AND DISCUSSION 4.1. BENDING A simply supported square symmetric cross-ply laminate subjected to sinusoidal load, (0-3)~=+h= 4-(q0/2) sin (fix~a) sin (fly~a),
(10)
is considered for numerical investigations. The solutions for the displacements take the forms w = ff sin (Trx/a) sin (fiy/a),
u = t~(z) cos (fix~a) sin (fly~a),
v = ~(z) sin ( f i x / a ) cos (fly/a).
(1 l a,b) (llc)
Numerical data have been generated both by the present procedure and by the exact analysis given by Pagano [9], for the following material properties, with material axes L and T along and perpendicular to the fiber direction, respectively: EL = 25Er, GLr = O"5Er, G r r = O . 2 E r , VLr = Vrr=0-25. For a [0/90], laminate, a/2h ratios of 4, 5, 6, 7.5, 10, 20, 50 and 100 have been considered. For a thick plate, a/2h = 4, the thicknesswise distributions of displacements and stresses are presented in the following non-dimensionalized forms: W= lO0~,Er/2hqoS 4, (el,
0"2, 0-6) = ( e t ,
02,
(ti, t3)= 100(if, O)Er/2hqoS 3,
O6)/qoS 2,
(04, 6"5)=(04, Os)/qoS.
(12a, b) (12c, d)
260
K.J.
REDDY
AND
K. V I J A Y A K U M A R
Here S = a/2h.
(13)
Thickness-wise distributions of displacements and stresses obtained by the models II and III are compared with those given by the exact solutions in Figures 1-7. The estimates of transverse shear stresses correspond to statically equivalent distributions. It can be seen that all the above mentioned distributions obtained by model III are in good agreement with the exact solutions, better than those obtained by model II, and reflect all essential features of the exact solutions. Variations of percentage errors in the estimates for the maximum values of displacements and stresses with respect to 2h/a are shown in Figures 8-10. Errors in these estimates by the present models are generally found to increase with increasing values of 2h/a. It is observed that the present model gives a maximum error in the estimates of t~ which is less than 5-2% even for S = 5. 4.2. FREE VIBRATION In Table 1 are shown the non-dimensionalized fundamental frequencies cb = (coa2/2h)x/p-/Er of flexural and thickness twist modes for square plates with span to 0"5
~,.~"~-e~.'
\
"'~"
~
'
\\
04
\\\ !'~/i. \\\\~i
0"3
\
0.2
O.I
0"0 -I.0
",
,L -0"5
,
~
-~.
'\\ VI/t' 0
Figure 1. T h i c k n e s s - w i s e d i s t r i b u t i o n of t~ in a [0/90], square laminated plate ......
, model II; -----,
05
(EL/GLr=50).
model III; - - -, CPT.
0"5
0.4
0"3
0.2
0-1
0.0 -3"0
I
-20
-I.0
F i g u r e 2. A s F i g u r e 1, b u t d i s t r i b u t i o n o f 0.
0
- -
-, E x a c t ;
PLY DEPENDENT
SHEAR
DEFORMATION
MODEL
0"5
0'4
0.3
Z
/
/
0,2
0.1
0"0
_ _ . 1 . ~ -0.25
0,25
J 050
J.__
0-75
Figure 3. As Figure 1, but distribution of 6",.
0.4
0"3f ///~ /
........
,,/,..'"'"
0 2,
o.,p
.....
0.0 ~ / ' ' ~
1
I
I
0.5
G
I 1.0
Figure 4. As Figure !, but distribution of
o.5
d2.
-~.....! ~ \ , .~ -~................ i~\\ \
\~'%X
~ ~ . ."
0.4
".~,,,,,. \ Oq,
0.2
0'I
O-C -
I
0.05
[
- O. 0;)5
I
I
0
Figure 5. As Figure I, but distribution of a6.
-0.025
261
262
K, J. R E D D Y 05
----t
......
I
AND
K. V I J A Y A K U M A R
I
~
...... T -
t--
l
......
\'~ '
.....:
0.4
i. 0.0 ~
L
[
\ I
'
Ou
I
02
j
0"3
e4 F i g u r e 6. A s F i g u r e 1, b u t distribution of d 4 .
0.5
0.4
0.5
0.2
0.1
I I
0-0
I 0"1
I
I
I
I 0"3
0"2
F i g u r e 7. A s F i g u r e 1, b u t distribution of ~5.
,
,
4
r~
/ r / - / (v) / / / / { u ~/ / /
2
. fJ
(u)
j/l" I.I
.~o
"
0
I 0.05
I
\
010
L 015
I 0'20
0.25
2h/a
Figure 8. Percentage errors in the estimates of displacements (w, u, v) for [0/90], laminate ., Model II; - - - , model IIl.
vs. 2h/a
(EL/GLr
=
50).
depth thickness ratios (a/2h) equal to 2, 5, 10, 20 and 50. The frequency estimates from model II and model III are compared with those from HSDT. The flexural frequency estimate from model III is lower and hence more accurate than the corresponding estimates from each of the models CPT, HSDT and model II. In the case of thickness twist mode frequencies, the estimates from the present model are significantly different from those
PLY DEPENDENT
SHEAR
DEFORMATION
263
MODEL
// // 0.2 / / / / ° " 6 / / / ° " 1
4
8_
////
2
./"/'/'/"
0.1
ii ° -4
0.05
0"10
0,15
0'20
0"25
2h/o
Figure 9. As Figure 8, but errors of in-planestresses (tr,, trz, o6).
0
0"05
O'lO
0'15
0"20
0"25
2hla
Figure I0. As Figure 8, but errors of transverse shear stresses (04, its).
of the other models. This is, perhaps, due to the better thicknesswise distributions for displacements obtained by the present model in the static case. 5. CONCLUDING REMARKS A new ply dependent shear deformation model based on a priori assumed slopes of the transverse shear stresses has been presented. The displacements in the model are expressed in terms of reference plane variables but account is taken of the ply to ply variation of direct stiffnesses and transverse shear compliances. Its performance relative to that of an earlier model II [ l, 2] has been studied by comparing results with those from exact elasticity solutions for the flexure of simply supported [0/90Is square laminate under sinusoidal load. Errors in the estimates for displacements and stresses from the present model are generally found to increase with increasing values of 2h/a. The error in the estimate of 0 is maximum for the present model but it is within 5.2% for S = 5 and EL/GLz=50. The thickness-wise distributions obtained from the present model are found to be better than those obtained from the earlier model II. As such, the estimates of the thickness twist mode frequencies from the present model and model II are found to be significantly different from each other. The present model, like FSDT and HSDT, is quite suitable for finite element implementation.
I
2 15-90 5-51 5 18.30 10-79 10 18-74 15.11 20 18-85 17.65 50 18.89 18.67 t Results obtained by using HSDT [8].
a/2h
CPT I 20.76 75-67 256.70 974-62 5745.12
II
HSDTt
37.69 107.72 291.86 979.52 5999-94
III
. 5.61 10-85 15.14 17.66 18.67
. I 20.25 74.80 255.56 971.12 5716.56
Model II . . II 37.43 107.04 290.20 976.22 5994.48
III
Non-dimensionalized flexural and thickness-twist mode frequencies, cb = ( coa2/ 2h ) ~
TABLE 1
5-42 10-68 15.06 17-63 18.67
I
25.74 88.79 290-75 983-10 5768.22
II
Model III III 37.92 108.35 294.84 1097.49 6726-57
of [0/90]s square plate
>
> .< >
> Z
70
y..
7~
.Ix
t,~
PLY DEPENDENT SHEAR DEFORMATION MODEL
265
ACKNOWLEDGMENTS The authors express their thanks to Professor A. V. Krishna Murty for his valuable comments and keen interest shown during the course of the present investigation. The authors are grateful to the Aeronautics Research and Development Board, Ministry of Defence, Government of India, for providing financial assistance under grant No. 481.
REFERENCES 1. K. VIJAVAKUMAR,A. V. KRISHNA MURTV, K. S. NAOARAJA and K. JAWAHAR REDOV 1990 Department of Aerospace Engineering, Indian Institute of Science, Bangalore-12, Report No. ARDB/STR/481/90/01, 23 pp. +55 tables. Higher order theories for estimation of interlaminar stresses in laminates. 2. K. H. LEE, N. R. SENTHILNATHAN, S. P. ElM and S. T. CHOW 1990 Composite Structures 15, 137-148. An improved zig-zag model for the bending of laminated composite plates. 3. A. K. NooR and W. S. BURTON 1989 Applied Mechanics Reviews 42, 1-13. Assessment of shear deformation theories for multilayered composite plates. 4. R. K. KAPANIA and S. RACITI 1989 American Institute of Aeronautics and Astronautics Journal 27(7), 923-934. ReCent advances in analysis of laminated beams and plates, part I: shear effects and buckling. 5. R. K. KAPANIA and S. RACITI 1989 American Institute of Aeronautics and Astronautics Journal 27(7), 935-946. Recent advances in analysis of laminated beams and plates, part II: vibration and wave propagation. 6. J. N. REDDY 1990 Shock and Vibration Digest 22(7), 3-17. A review of refined theories of laminated composite plates. 7. C. T. SUN and J. M. WHITNEY 1973 American Institute of Aeronautics and Astronautics Journal 11, 178-183. Theories for the dynamic response of laminated plates. 8. J. N. REODY and N. D. PHAN 1985 Journal of Sound and Vibration 98, 157-170. Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory. 9. N. J. PA~ANO 1970 Journal of Composite Materials 4, 20-34. Exact solutions for rectangular bidirectional 'composites and sandwich plates.