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Large deformation finite element analysis of laminated circular composite plates
00457949(94)00300-9
LARGE
Cumpurers & Snucrures Vol. 54. No. I. pp. 5964. 1995 Copyright p 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045.7949195 $9.50 + 0.00
DEFORMATION FINITE ELEMENT LAMINATED CIRCULAR COMPOSITE
ANALYSIS PLATES
OF
C. Sridhar and K. P. Raot Department of Aerospace Engineering, Indian Institute of Science, Bangalore-
012, India
(Received 17 Ortober 1993)
Abstract-A 48 d.o.f., four-noded quadrilateral laminated composite shell finite element is particularised to a sector finite element and is used for the large deformation analysis of circular composite laminated plates. The straindisplacement relationships for the sector element are obtained by reducing those of the quadrilateral shell finite element by substituting proper values for the geometric parameters. Subsequently, the linear and tangent stiffness matrices are formulated using conventional methods. The Newton Raphson method is employed as the nonlinear solution technique. The computer code developed is validated by solving an isotropic case for which results are available in the literature. The method is then applied to solve problems of cylindrically orthotropic circular plates. Some of the results of cylindrically orthotropic case are compared with those available in the literature. Subsequently, application is made to the case of laminated composite circular plates having different lay-up schemes. The computer code can handle symmetric/unsymmetric lay-up schemes. The large displacement analysis is useful in estimating the damage in composite plates caused by low-velocity impact.
1. INTRODUCTION
using a four-noded 48 d.o.f. sector element. The computer code developed is validated by solving an isotropic case for which results are available in the literature. The method is then applied to solve the problems of cylindrically orthotropic circular plates. Some of the results of cylindrically orthotropic case are compared with those available in the literature. Subsequently, application is made to the case of laminated composite circular plates having different lay-up schemes. The computer code can handle symmetric/unsymmetric lay-up schemes.
Fibre-reinforced composite laminates are being increasingly used in the aerospace industry. Most of these structural systems respond with large displacements exhibiting geometric nonlinearity. With increasing emphasis on structural efficiency, there is a necessity for nonlinear analysis as, in practice, a linear analysis may not be accurate for design purposes (for example, a low-velocity impact problem). The manifestation of geometric nonlinearity is a result of the change in geometry and the geometric coupling between the inplane and out of plane deformations (interaction between the inplane stretching and bending modes). The effect of geometric nonlinearity is either to stiffen the system by virtue of the membrane stresses developed, which considerably reduce the displacements of the structure as compared to the linear solution, or to weaken the structure to an extent where deflections increase more rapidly than predicted by a linear solution, the latter resulting in a state where load carrying capacity decreases with continuing deformation. However, the nature of the behaviour is highly dependent on the direction of the loading and the geometry of the structure. They may also have great influence in predicting the strength and stability of structural systems. Thus the effect of geometric nonlinearity on the response of structures is of great importance and interest. In this paper, geometric nonlinear finite element analysis of composite circular plates is performed
tAuthor to whom correspondence
2. SECTOR FINITE ELEMENT
A four-noded 48 d.o.f. doubly curved quadrilateral laminated anisotropic shell finite element was developed by Venkatesh and Rao [I]. The element is bound by two parallel circles and two meridians of a shell of revolution (Fig. 1). Each node has 12 degrees of freedom. The various d.o.f. are
W=
au au azu U'Z'iZ'dSdO~v' { a~ dv a% aw aM, aJw -,a~ rae'raSaO' w’K’Z’rasa6 -1
First order Hermite interpolation polynomials are used to describe the three displacements u, v and w. The capability of the shell element has been proved in linear static analysis [2] and buckling analysis [3,4] of composite shells. In this paper, this element is particularised to obtain a sector element and is used to solve the large deformation problem of laminated composite circular plates.
should be addressed. 59
Sridhar
and K. P. Rao
-22% +!aw
K,r,,=
Fig. 1. Geometry element
of doubly curved quadrilateral for thin shells of revolution.
Strain-displacement
finite
rasae
r
rae
-( ;+; )( &+3-
Q=
relationships
Strain-displacement relationships for axisymmetric shell geometry are given in [I] and [4]. These relations are particularised to flat sector plate geometry. The strain at any point in the plate can be written as
nonlinear
{t y}, (6:) are the strains and (K,}, {K~} are the changes in curvatures of the shell middle surface.
-
azw rasae
( I(
__-_-
i aw r
rae 1
_
&j-g($) -
and
=
is
(1.1)
{c) = {cl1 + @213 where {L,} and {c2} are the linear strains respectively. Again,
au
K 2so
-au as
alw -+;z r2ae2
( >(
i aw
>
-
(%l+s)(~).
(1.5)
The linear elastic and tangent stiffness matrices [K,] and [K,] are obtained following the conventional procedures. 3. RESULTS AND DISCUSSION
L:,, = g
+u r
a0 u au E 01.10= - + - - rae as r azW ‘G.7 = ---i asah KIO = - ---r2ae2
i aw r
as
The standard Newton-Raphson method is employed as a nonlinear solution technique. First, linear static finite element analysis is performed. The deflection vector is obtained by solving equilibrium equation {P} = [K,]{q} for {q}, where {P} is the nodal load vector and [K,] is linear elastic stiffness matrix. This linear deflection vector (q}, is used as an initial guess for the subsequent Newton-Raphson iterations. In every iteration, the deflection vector {q} computed in the previous iteration is used to calculate new tangent stiffness matrix [K,] and the new equilibrium equation {P} = [KT]{q} is solved for {q}. The iterations are carried out until the convergence is attained on a characteristic element in the deflection vector. In the present work, the characteristic element in the deflection vector is chosen to be the central vertical displacement of the plate. In the case of an
Finite element analysis of composite plates
w
Typical
61
finite
element
Fig. 2. Typical finite element mesh adopted for the circular plate.
annular plate, it is the largest vertical displacement of the inner edge of the plate. Full details regarding the
formulation of [Kr] and the application of the Newton-Raphson method are given in [S]. Convergence study with respect to the mesh size, prior to the analysis of composite circular plates, is carried out. As a result, the mesh size of the plate (Fig. 2) is decided as 5 x 10 (radial x circumferential) for all the cases considered here. First, we consider the cases of isotropic and cylindrically orthotropic circular plates for validating the computer code. The geometric properties, material properties and boundary conditions are assumed as follows: Thickness of the plate(h) = 13.748 mm. Radius of the plate(u) = 1.01 m. Isotropic case: E = 2.9 GPa; v = 0.25. Orthotropic case: E, = 2.9 GPa; E. = BE,; G,,, = E&V
+ v,,).
at r = a for a clamped plate
Boundary conditions are:
0 0
k WJ, % =
v, VI), V#
=
w, w,, WY, w,u= 0 and for a simply supported plate are: &r+,,u,,=O 0, u/j1u,o= 0 w, WC), w,u= 0. Figure 3 shows nondimensionalized load-central deflection curves obtained by [6] and the present analysis for the case of a clamped, uniformly loaded, isotropic circular plate. Figure 4 shows the nondimensionalized stress-central displacement relationship obtained by [6] and the present study. In both the cases, a very good agreement can be observed. The case of a clamped, isotropic circular plate with a concentrated load at the centre is analysed next. Table 1 shows a comparison of the results obtained here with those of[6]; good agreement is seen.
IO
f * l
05
0 0
2
4
6
8
IO
qo’/Eh’
Fig. 3. Comparison of maximum deflection in an isotropic circular plate subjected to a uniformly distributed load.
0
04
08
I2
w/h
Fig. 4. Comparison of stresses at clamped edge and centreclamped circular isotropic plates subjected to a uniformly distributed load.
C. Sridhar
62
and K. P. Rao
Table I. Comparison of central deflection of a clamped isotropic circular plate under a central concentrated load with the results of Timoshenko 161 Central deflection (mm) Error
Load (N)
Present work
250 500 750 1000
6.650 11.300 14.650 17.270
Timoshenko 6.653 II.303 14.654 17.281
W) 0.04 0.026 0.027 0.04
Using continuum mechanics, Alwar and Reddy [7] solved the large displacement problem of orthotropic circular plates with and without a central hole. They have assumed the plate to be cylindrically orthotropic, i.e. material properties of the plate are axisymmetric about the axis transversely passing through the centre of the plate. Typical cases of annular cylindrically orthotropic plates are presently considered for the purpose of comparison. Figure 5 shows the comparison of present results with those of Alwar and Reddy [7] for the case of an orthotropic circular plate for /3 = 0.5, 0.75 and v,(, = 0.3. Figure 6 shows the comparison of results for : = 0.25; /r = 0.5, 1.0. The present study shows slightly larger deflections compared to [7]. However, the value of v,,, used in their work is not available. In order to study the effect of Poisson’s ratio, we consider a few examples. Figures 7 and 8 show the effect of Poisson’s ratio on the load-central displacement relationship for a simply supported plate and a clamped plate, respectively, under a uniformly distributed load. The value of p is assumed to be 0.5 and 5 is 0.25. It can be inferred from the figures that an increase in Poisson‘s ratio causes stiffening of the plate. Similar behaviour is also observed in isotropic plates; this can be explained as follows. Consider any small sector element with radial stresses on the plate. The tangential strains produced in the element are proportional to Poisson’s ratio. Since the plate is cylindrically orthotropic (i.e. the material properties are invariant
0
20
40
60
qo4/Eeh*
Fig. 6. Comparison of results of a clamped annular cylindrically orthotropic circular plate with those of [7].
about the transverse axis passing through the centre of the plate), a restraint exists in the tangential direction; this causes tangential membrane stresses, which stiffen the plate. Finally, composite circular plates made of T300/N5208 laminae are analysed for large displacements under a uniformly distributed load. Keeping the boundary conditions the same as given above for a clamped plate, the effect of lay-up sequences on the nonlinear behaviour of plates is studied. The material properties of T300/N5208 are E, = 13 1.O GPa, E, = 13.0 GPa, G,, = 6.4 GPa and v,, = 0.34. The geometric properties of the plate are a = 1.O m and h = 12.0mm. The different lay-up schemes considered are: [O’] (Orthotropic&Lay-up A [O”/450/900/- 45’1, (Quasi-isotropic)-Lay-up B [0~/45~/90~/-45:)-Lay-up C [O”/450/900/ - 45°/Oo/450/900/ - 45’]-Lay-up D [O”/600/- 60’1, (Quasi-isotropic)-Lay-up E [0;/60!/ - 60:]-Lay-up F [O”/600/ - 60°/Oo/600/ - 600]-Lay-up G.
05-
0
2
4
6
8
IO
qa ’ / Eeh4
Fig. 5. Comparison of results of a clamped circular cylindrically orthotropic plate with those of [7].
qa4/Eh4
Fig. 7. Loaddisplacement relationship with respect to ‘v’ for a simply supported annular cylindrically orthotropic plate under a uniformly distributed load.
Finite element
analysis
of composite
0 40
20 qo4/E
60
63
plates
0
Lwd
Fig. 1 I. Variation nated composite
Lay-up
scheme
Lay-up
scheme
F
l
Loy
up scheme
G
‘+
L
h4
Fig. 8. Load-displacement relationship with respect to ‘v’ for a clamped annular cylindrically orthotropic plate under a uniformly distributed load.
l
.
0
E
tl
lkN/m’l
of central deflection with load for lamicircular plates with different lay-ups.
(orthotropic) laminate. Figure 10 compares the stiffness of a laminates with different lay-up sequences. It can be inferred that lay-up scheme B, i.e. the quasi isotropic lay-up scheme, is the stiffest of the three. A similar graph is plotted for 5 laminates (Fig. 11).
4.
6
4
Load (kN/rr2)
Fig. 9. Loaddisplacement thotropic plate under
relationship for a clamped ora uniformly distributed load.
q
I
Lay-up
scheme
B
.
Lay-up
scheme
C
.
Lay-up
scheme
D
CONCLUSIONS
In this paper, a four-noded 48 d.o.f. laminated anisotropic sector element is used to study the large deformation behaviour of laminated composite circular plates. The computer code developed based on this finite element is validated by comparison of the results available in the literature for both isotropic and cylindrically orthotropic circular plates. Application is subsequently made to typical 3 and f laminates made of T300/N5208 plies. It is found that the quasi-isotropic lay-up gives the maximum stiffness for uniformly distributed loading. Application of the present nonlinear finite element analysis for the estimation and growth of damage due to low-velocity impact in laminated composite circular plates will be reported in a follow-up paper. Acknowledgements-The authors wish to thank Dr Biswajit Tripathy, Scientist, Aeronautical Development Establishment, Bangalore, for his help during the formulation of nonlinear stiffness matrices and preparation of the computer code.
REFERENCES
2
0
4
6
8
Load lkN/m’l
Fig. 10. Variation nated composite
Figures for seven the lay-up the plate. CA.9 54. I--t
of central deflection with load for lamicircular plates with different lay-ups.
9 to 11 show load-central deflection curves different lay-up schemes. It can be seen that scheme considerably affects the stiffness of Figure 9 shows the stiffness curve of the O”
A. Venkatesh
and K. P. Rao, A doubly curved quadrilateral finite element for the analysis of laminated anisotropic thin shells of revolution. Compur. Struct. 12, 8255832 (1980). A. Venkatesh and K. P. Rao, Analysis of laminated shells of revolution with laminated stiffeners using a doubly curved quadrilateral finite element. Compur. Swucr. 20, 669-682 (1985). K. P. Rao and B. Tripathy, Composite cylindrical panels-optimisation of lay-up for buckling by ranking. Cornput. Srruct. 38, 217-225 (1991).
64
C. Sridhar
4. B. Tripathy and K. P. Rao, Stiffened composite axisymmetric shells-optimum lay-up for buckling by ranking. Comput. Struct. 46, 299-309 (1993). 5. C. Sridhar, Damage and its growth in laminated composite circular plates undergoing large deformations, M.Sc. (Engng) Thesis, Indian Institute of Science, July 1993.
and K. P. Rao 6. S. Timoshenko and S. W. Kreiger, Theory of Plates and Shells, Int. Students Edition, 2nd Edn, pp. 407-408. McGraw-Hill, New York (1983). 7. R. S. Alwar and B. S. Reddy, Large deflection static and dynamic analysis of isotropic and orthotropic annular plates. Inr. J. Nonlinear Mech. 14, 347-359 (1979).
LARGE
Cumpurers & Snucrures Vol. 54. No. I. pp. 5964. 1995 Copyright p 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045.7949195 $9.50 + 0.00
DEFORMATION FINITE ELEMENT LAMINATED CIRCULAR COMPOSITE
ANALYSIS PLATES
OF
C. Sridhar and K. P. Raot Department of Aerospace Engineering, Indian Institute of Science, Bangalore-
012, India
(Received 17 Ortober 1993)
Abstract-A 48 d.o.f., four-noded quadrilateral laminated composite shell finite element is particularised to a sector finite element and is used for the large deformation analysis of circular composite laminated plates. The straindisplacement relationships for the sector element are obtained by reducing those of the quadrilateral shell finite element by substituting proper values for the geometric parameters. Subsequently, the linear and tangent stiffness matrices are formulated using conventional methods. The Newton Raphson method is employed as the nonlinear solution technique. The computer code developed is validated by solving an isotropic case for which results are available in the literature. The method is then applied to solve problems of cylindrically orthotropic circular plates. Some of the results of cylindrically orthotropic case are compared with those available in the literature. Subsequently, application is made to the case of laminated composite circular plates having different lay-up schemes. The computer code can handle symmetric/unsymmetric lay-up schemes. The large displacement analysis is useful in estimating the damage in composite plates caused by low-velocity impact.
1. INTRODUCTION
using a four-noded 48 d.o.f. sector element. The computer code developed is validated by solving an isotropic case for which results are available in the literature. The method is then applied to solve the problems of cylindrically orthotropic circular plates. Some of the results of cylindrically orthotropic case are compared with those available in the literature. Subsequently, application is made to the case of laminated composite circular plates having different lay-up schemes. The computer code can handle symmetric/unsymmetric lay-up schemes.
Fibre-reinforced composite laminates are being increasingly used in the aerospace industry. Most of these structural systems respond with large displacements exhibiting geometric nonlinearity. With increasing emphasis on structural efficiency, there is a necessity for nonlinear analysis as, in practice, a linear analysis may not be accurate for design purposes (for example, a low-velocity impact problem). The manifestation of geometric nonlinearity is a result of the change in geometry and the geometric coupling between the inplane and out of plane deformations (interaction between the inplane stretching and bending modes). The effect of geometric nonlinearity is either to stiffen the system by virtue of the membrane stresses developed, which considerably reduce the displacements of the structure as compared to the linear solution, or to weaken the structure to an extent where deflections increase more rapidly than predicted by a linear solution, the latter resulting in a state where load carrying capacity decreases with continuing deformation. However, the nature of the behaviour is highly dependent on the direction of the loading and the geometry of the structure. They may also have great influence in predicting the strength and stability of structural systems. Thus the effect of geometric nonlinearity on the response of structures is of great importance and interest. In this paper, geometric nonlinear finite element analysis of composite circular plates is performed
tAuthor to whom correspondence
2. SECTOR FINITE ELEMENT
A four-noded 48 d.o.f. doubly curved quadrilateral laminated anisotropic shell finite element was developed by Venkatesh and Rao [I]. The element is bound by two parallel circles and two meridians of a shell of revolution (Fig. 1). Each node has 12 degrees of freedom. The various d.o.f. are
W=
au au azu U'Z'iZ'dSdO~v' { a~ dv a% aw aM, aJw -,a~ rae'raSaO' w’K’Z’rasa6 -1
First order Hermite interpolation polynomials are used to describe the three displacements u, v and w. The capability of the shell element has been proved in linear static analysis [2] and buckling analysis [3,4] of composite shells. In this paper, this element is particularised to obtain a sector element and is used to solve the large deformation problem of laminated composite circular plates.
should be addressed. 59
Sridhar
and K. P. Rao
-22% +!aw
K,r,,=
Fig. 1. Geometry element
of doubly curved quadrilateral for thin shells of revolution.
Strain-displacement
finite
rasae
r
rae
-( ;+; )( &+3-
Q=
relationships
Strain-displacement relationships for axisymmetric shell geometry are given in [I] and [4]. These relations are particularised to flat sector plate geometry. The strain at any point in the plate can be written as
nonlinear
{t y}, (6:) are the strains and (K,}, {K~} are the changes in curvatures of the shell middle surface.
-
azw rasae
( I(
__-_-
i aw r
rae 1
_
&j-g($) -
and
=
is
(1.1)
{c) = {cl1 + @213 where {L,} and {c2} are the linear strains respectively. Again,
au
K 2so
-au as
alw -+;z r2ae2
( >(
i aw
>
-
(%l+s)(~).
(1.5)
The linear elastic and tangent stiffness matrices [K,] and [K,] are obtained following the conventional procedures. 3. RESULTS AND DISCUSSION
L:,, = g
+u r
a0 u au E 01.10= - + - - rae as r azW ‘G.7 = ---i asah KIO = - ---r2ae2
i aw r
as
The standard Newton-Raphson method is employed as a nonlinear solution technique. First, linear static finite element analysis is performed. The deflection vector is obtained by solving equilibrium equation {P} = [K,]{q} for {q}, where {P} is the nodal load vector and [K,] is linear elastic stiffness matrix. This linear deflection vector (q}, is used as an initial guess for the subsequent Newton-Raphson iterations. In every iteration, the deflection vector {q} computed in the previous iteration is used to calculate new tangent stiffness matrix [K,] and the new equilibrium equation {P} = [KT]{q} is solved for {q}. The iterations are carried out until the convergence is attained on a characteristic element in the deflection vector. In the present work, the characteristic element in the deflection vector is chosen to be the central vertical displacement of the plate. In the case of an
Finite element analysis of composite plates
w
Typical
61
finite
element
Fig. 2. Typical finite element mesh adopted for the circular plate.
annular plate, it is the largest vertical displacement of the inner edge of the plate. Full details regarding the
formulation of [Kr] and the application of the Newton-Raphson method are given in [S]. Convergence study with respect to the mesh size, prior to the analysis of composite circular plates, is carried out. As a result, the mesh size of the plate (Fig. 2) is decided as 5 x 10 (radial x circumferential) for all the cases considered here. First, we consider the cases of isotropic and cylindrically orthotropic circular plates for validating the computer code. The geometric properties, material properties and boundary conditions are assumed as follows: Thickness of the plate(h) = 13.748 mm. Radius of the plate(u) = 1.01 m. Isotropic case: E = 2.9 GPa; v = 0.25. Orthotropic case: E, = 2.9 GPa; E. = BE,; G,,, = E&V
+ v,,).
at r = a for a clamped plate
Boundary conditions are:
0 0
k WJ, % =
v, VI), V#
=
w, w,, WY, w,u= 0 and for a simply supported plate are: &r+,,u,,=O 0, u/j1u,o= 0 w, WC), w,u= 0. Figure 3 shows nondimensionalized load-central deflection curves obtained by [6] and the present analysis for the case of a clamped, uniformly loaded, isotropic circular plate. Figure 4 shows the nondimensionalized stress-central displacement relationship obtained by [6] and the present study. In both the cases, a very good agreement can be observed. The case of a clamped, isotropic circular plate with a concentrated load at the centre is analysed next. Table 1 shows a comparison of the results obtained here with those of[6]; good agreement is seen.
IO
f * l
05
0 0
2
4
6
8
IO
qo’/Eh’
Fig. 3. Comparison of maximum deflection in an isotropic circular plate subjected to a uniformly distributed load.
0
04
08
I2
w/h
Fig. 4. Comparison of stresses at clamped edge and centreclamped circular isotropic plates subjected to a uniformly distributed load.
C. Sridhar
62
and K. P. Rao
Table I. Comparison of central deflection of a clamped isotropic circular plate under a central concentrated load with the results of Timoshenko 161 Central deflection (mm) Error
Load (N)
Present work
250 500 750 1000
6.650 11.300 14.650 17.270
Timoshenko 6.653 II.303 14.654 17.281
W) 0.04 0.026 0.027 0.04
Using continuum mechanics, Alwar and Reddy [7] solved the large displacement problem of orthotropic circular plates with and without a central hole. They have assumed the plate to be cylindrically orthotropic, i.e. material properties of the plate are axisymmetric about the axis transversely passing through the centre of the plate. Typical cases of annular cylindrically orthotropic plates are presently considered for the purpose of comparison. Figure 5 shows the comparison of present results with those of Alwar and Reddy [7] for the case of an orthotropic circular plate for /3 = 0.5, 0.75 and v,(, = 0.3. Figure 6 shows the comparison of results for : = 0.25; /r = 0.5, 1.0. The present study shows slightly larger deflections compared to [7]. However, the value of v,,, used in their work is not available. In order to study the effect of Poisson’s ratio, we consider a few examples. Figures 7 and 8 show the effect of Poisson’s ratio on the load-central displacement relationship for a simply supported plate and a clamped plate, respectively, under a uniformly distributed load. The value of p is assumed to be 0.5 and 5 is 0.25. It can be inferred from the figures that an increase in Poisson‘s ratio causes stiffening of the plate. Similar behaviour is also observed in isotropic plates; this can be explained as follows. Consider any small sector element with radial stresses on the plate. The tangential strains produced in the element are proportional to Poisson’s ratio. Since the plate is cylindrically orthotropic (i.e. the material properties are invariant
0
20
40
60
qo4/Eeh*
Fig. 6. Comparison of results of a clamped annular cylindrically orthotropic circular plate with those of [7].
about the transverse axis passing through the centre of the plate), a restraint exists in the tangential direction; this causes tangential membrane stresses, which stiffen the plate. Finally, composite circular plates made of T300/N5208 laminae are analysed for large displacements under a uniformly distributed load. Keeping the boundary conditions the same as given above for a clamped plate, the effect of lay-up sequences on the nonlinear behaviour of plates is studied. The material properties of T300/N5208 are E, = 13 1.O GPa, E, = 13.0 GPa, G,, = 6.4 GPa and v,, = 0.34. The geometric properties of the plate are a = 1.O m and h = 12.0mm. The different lay-up schemes considered are: [O’] (Orthotropic&Lay-up A [O”/450/900/- 45’1, (Quasi-isotropic)-Lay-up B [0~/45~/90~/-45:)-Lay-up C [O”/450/900/ - 45°/Oo/450/900/ - 45’]-Lay-up D [O”/600/- 60’1, (Quasi-isotropic)-Lay-up E [0;/60!/ - 60:]-Lay-up F [O”/600/ - 60°/Oo/600/ - 600]-Lay-up G.
05-
0
2
4
6
8
IO
qa ’ / Eeh4
Fig. 5. Comparison of results of a clamped circular cylindrically orthotropic plate with those of [7].
qa4/Eh4
Fig. 7. Loaddisplacement relationship with respect to ‘v’ for a simply supported annular cylindrically orthotropic plate under a uniformly distributed load.
Finite element
analysis
of composite
0 40
20 qo4/E
60
63
plates
0
Lwd
Fig. 1 I. Variation nated composite
Lay-up
scheme
Lay-up
scheme
F
l
Loy
up scheme
G
‘+
L
h4
Fig. 8. Load-displacement relationship with respect to ‘v’ for a clamped annular cylindrically orthotropic plate under a uniformly distributed load.
l
.
0
E
tl
lkN/m’l
of central deflection with load for lamicircular plates with different lay-ups.
(orthotropic) laminate. Figure 10 compares the stiffness of a laminates with different lay-up sequences. It can be inferred that lay-up scheme B, i.e. the quasi isotropic lay-up scheme, is the stiffest of the three. A similar graph is plotted for 5 laminates (Fig. 11).
4.
6
4
Load (kN/rr2)
Fig. 9. Loaddisplacement thotropic plate under
relationship for a clamped ora uniformly distributed load.
q
I
Lay-up
scheme
B
.
Lay-up
scheme
C
.
Lay-up
scheme
D
CONCLUSIONS
In this paper, a four-noded 48 d.o.f. laminated anisotropic sector element is used to study the large deformation behaviour of laminated composite circular plates. The computer code developed based on this finite element is validated by comparison of the results available in the literature for both isotropic and cylindrically orthotropic circular plates. Application is subsequently made to typical 3 and f laminates made of T300/N5208 plies. It is found that the quasi-isotropic lay-up gives the maximum stiffness for uniformly distributed loading. Application of the present nonlinear finite element analysis for the estimation and growth of damage due to low-velocity impact in laminated composite circular plates will be reported in a follow-up paper. Acknowledgements-The authors wish to thank Dr Biswajit Tripathy, Scientist, Aeronautical Development Establishment, Bangalore, for his help during the formulation of nonlinear stiffness matrices and preparation of the computer code.
REFERENCES
2
0
4
6
8
Load lkN/m’l
Fig. 10. Variation nated composite
Figures for seven the lay-up the plate. CA.9 54. I--t
of central deflection with load for lamicircular plates with different lay-ups.
9 to 11 show load-central deflection curves different lay-up schemes. It can be seen that scheme considerably affects the stiffness of Figure 9 shows the stiffness curve of the O”
A. Venkatesh
and K. P. Rao, A doubly curved quadrilateral finite element for the analysis of laminated anisotropic thin shells of revolution. Compur. Struct. 12, 8255832 (1980). A. Venkatesh and K. P. Rao, Analysis of laminated shells of revolution with laminated stiffeners using a doubly curved quadrilateral finite element. Compur. Swucr. 20, 669-682 (1985). K. P. Rao and B. Tripathy, Composite cylindrical panels-optimisation of lay-up for buckling by ranking. Cornput. Srruct. 38, 217-225 (1991).
64
C. Sridhar
4. B. Tripathy and K. P. Rao, Stiffened composite axisymmetric shells-optimum lay-up for buckling by ranking. Comput. Struct. 46, 299-309 (1993). 5. C. Sridhar, Damage and its growth in laminated composite circular plates undergoing large deformations, M.Sc. (Engng) Thesis, Indian Institute of Science, July 1993.
and K. P. Rao 6. S. Timoshenko and S. W. Kreiger, Theory of Plates and Shells, Int. Students Edition, 2nd Edn, pp. 407-408. McGraw-Hill, New York (1983). 7. R. S. Alwar and B. S. Reddy, Large deflection static and dynamic analysis of isotropic and orthotropic annular plates. Inr. J. Nonlinear Mech. 14, 347-359 (1979).