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Optimization of bent composite cylinders
C‘omposifeSrrucrures 30 (1995) 103-108 Elsevier Science Limited Printed in Great Britain. All rights reserved 0263~8223/95/$9.50 ELSEVIER
0263-8223(94)00031-X
Optimization of bent composite cylinders Joachim L. Grenestedt Department of Lightweight Structures, Royal Institute of Technology S-100 44 Stockholm, Sweden
A circular thin-walled cylinder subject to bending is studied. The optimal radius, wall thickness, and lay-up is obtained under a buckling constraint similar to that of the NASA Space Vehicle Design Criteria, and a global bending stiffness constraint. The lay-up is restricted to symmetric laminates but otherwise with complete freedom in the lay-up angle variation through the thickness, which is obtained by using lamination parameters as design variables. By neglecting the influence of geometry and stiffnesses on the knock-down factor for buckling, the optimal lay-up is found to be independent of the requirements of buckling strength and stiffness, i.e. all cylinders of a specific material will have the same optimal lay-up.
NOTATION A
B, C&
4; q E
E, h
L M
m
Cross-sectional area of the cylinder In-plane stiffnesses Normalized in-plane stiffnesses (Au/h) Coupling stiffnesses Functions of the lay-up only Bending stiffnesses of the cylinder wall Normalized bending stiffnesses ( 12 D,/h “) axial Average Young’s modulus of the cylinder wall Longitudinal modulus of a PlY Wall thickness of the cylinder Lamina thicknesses Second area moment of inertia of the cylinder Normalized buckling according to Ref. 1, function of the lay-up only Length of the cylinder Applied bending moment on the cylinder Bending moments in the cylinder wall Number of half-waves in the axial direction
Number of waves in the circumferential direction Buckling load of uniaxially compressed cylinder Maximum axial compressive stress in the bent cylinder Membrane forces in the cylinder wall Object function for the optimization nL/(mnR) (mnR/L )Z Radius of the cylinder Bending stiffness of the cylinder Required bending stiffness of the cylinder Material constants, linear combinations of the onaxis moduli of a ply Deformations in the x, y, and z directions, respectively Amplitudes of the deformations Axial, circumferential, and through-the-thickness coordinates, respectively Normalized through-thethickness coordinate (2z/h)
Arc tan (&)
Joachim L. Grenestedt
Correlation or knockdown factor Strains in the middle surface of the cylinder wall Lay-up angle as function of z* Lay-up angle Change of curvatures of the middle surface of the cylinder wall Lamination parameters for in-plane stiffnesses A, Lamination parameters for bending stiffnesss D,
INTRODUCTION
The incentive for this investigation was the research at Japan National Aerospace Laboratory (NAL) concerning a high altitude long endurance (HALE) vehicle, whose main wing beam and aft fuselage, in the current design, are circular carbon-fiber cylinders (see Fig. 1). The buckling constraint in the present study requires that the maximum compressive stress in the bent cylinder does not exceed the bifurcation stress of a uniaxially compressed cylinder according to Donnell theory, but with a different correlation factor than used for uniaxial compression. Further, there is a constraint on the bending stiffness. Without the stiffness constraint the optimal lay-up for a bent cylinder is the same as for a
Fig. 1.
uniaxially compressed cylinder, for which Onoda’ showed that the optimal lay-up is quasi-isotropic. ZitzEvancih* considered (among other tasks) maximizing the axial buckling load and the axial stiffness of a cylinder, a problem which is similar to the present one. The lay-up angles were restricted to O”rf:45”, and 90”. Relaxing this restriction leads to some improvements. Use of lamination parameters for the optimization is efficient because large nonlinearities of the object function versus design parameters are avoided, a fact due to the linearity of the stiffnesses in the lamination parameters. Further, the number of design parameters is kept to a minimum while maintaining more design freedom than discrete ply laminates. The main problem with lamination parameters is that the feasible region is not yet fully determined. Fukunaga” derived an inner boundary of the feasible region of the four parameters needed to describe an orthotropic laminate. It should be noted that contrary to what is stated in Ref. 3, the region derived is not the true one, but an inner boundary. It was obtained using a small number of discrete plies in the layup. Grenestedt and Gudmundson4 showed that this region was too small, and derived the true feasible region. A simple outer boundary of the feasible region of the same parameters was also obtained by Grenestedt and Gudmundson4 by using, for example, Schwarz inequality. In the present study, both the outer and inner boundaries of the feasible region are used. The optimal design found using the outer boundary, which
An early design sketch of the wing of the HALE vehicle of NAL, Japan.
Optimizationof bent composite cylinders
might correspond to an unfeasible design, was only 0.1% better than the optimal design obtained using the inner boundary. It was therefore not necessary to use the rather complicated true feasible region for the optimization, but the design obtained by the inner boundary was considered sufficient. The inner boundary, which was obtained by using a discrete lay-up, naturally guarantees feasibility. A fairly accurate representation of the optimal lay-up is [( f 37*5)4, O,, 902,
RJS.
carried out with p as variable instead of p, where p= tan*/3 since the buckling load versus /3 is very smooth, and both very small and very large values of p are included easily by varying /? between 0 and n/2. The use of real instead of integer values of m and n for minimizing the buckling load yields a conservative approximation. With this simplification the optimal lay-up will be independent of the R/h and the L/R ratios. The buckling stress is expressed as K=
GOVERNING EQUATIONS In accordance with the NASA Space Vehicle Design Criteria5 the maximum buckling stress of the bent cylinder was limited by that of a uniformly compressed cylinder, but with a different knock-down factor. The knock-down factor is dependent on the cylinder radius-to-wall thickness ratio, R/h, as well as the elastic stiffnesses, a fact which for the present optimization task was neglected. This is not a severe restriction, since for cylinders with the R/h ratio varying between 200 and 1000 the knock-down factor only varies between approximately 0.8 and 05. By neglecting these dependencies an optimal lay-up independent of the R/h ratio could be determined. The bifurcation load was calculated using Donnell kinematics and lamination theory.6,7 It can be shown that for symmetry reasons A 16,Az6, D 16,and Dz6 should vanish for an optimal lay-up.’ This reduces the number of design parameters to four, &‘, Ef, ljf’, cf, as will be seen later. The resulting equations are fulfilled by the deformation
u(x,y)= u,, cos( mnzx/L
) cos( ny/R)
~(~,y)=v,,sin(mnx/l)
sin(ny/R)
w(x,y)= w,, sin( mJtx/L
) cos( ny/R)
105
(&h*/R)K(c)
(2)
where E, is the longitudinal modulus of a ply, and K is a function of the lay-up, i.e. the lamination parameters c, according to eqns ( 16) and (17) of Ref. 1. Observe that the current definition of stiffnesses and lamination parameters are different from those of Ref. 1. The global bending stiffness of the cylinder is calculated using Euler-Bernoulli beam theory. The cross-sectional area of the thin-walled cylinder is A= 2nRh
(3)
the second area moment of inertia is I= nR”h
(4)
and the average Young’s modulus direction is E=AT, -(AT2)2/A;2
in the axial (9
where A ; = A,/h, see also eqn (7), so the global bending stiffness of the cylinder, S, is S=EI=(AT,
-(AT2)2/A;2)nR”h
(6)
Effects of the order h/R are neglected compared to unity.
(1)
where m and n are integer values which should be selected so that the bifurcation load is minimized and the boundary conditions, if possible, fulfilled. For simplicity, m and n are in this study selected as the real values that minimize the bifurcation load. By introducing the parameters p = nL/ (mzR) and 4 = (mnR/L )” the buckling load was simply minimized analytically versus q,’ while minimization versus p was performed numerically by bracketing initially the minimum between zero and infinity, and then using the Golden Section method. The bracketing and minimization was
CONSTITUTIVE
PARAMETERS
Assuming symmetric equations are
Nx
Nxy
the constitutive
4, AI, AI,
O( NY =
laminates,
(7)
106
Joachim L. Grenestedt
The lamination parameters, derived by Tsai and Pagano,x needed to describe an orthotropic lay-up are defined as I +$,=;
[cos
219,cos 481 dz*
i I (8)
I tf;.2,=; i
_ [cos I
20,cos 48]z** dz*
where the lay-up angle 0 is a function of z*, the normalized through-the-thickness coordinate. Four parameters suffice to describe a general laminated orthotropic material. The in-plane stiffnesses are AT, = 6 + W?+
Q&3’
A;?= U, - U&‘+
U&’
A&=
U,-
(9)
U,{$
A,“,=$ (U, - U,)-
times a correlation factor, y (knock-down factor); - the global bending stiffness, S, exceeds a . . . _ nummum required stiffness, Seq. The cross-sectional area represents the structural weight which is to be minimized under buckling and stiffness constraints. In mathematical terms the optimization problem can be expressed as: minimize A = 2nRh, under the constraints that Nmax5 rlv,, for a specified bending moment, M, and Sk Seq. The maximum compressive stress is
(11) Eliminating h, using eqns (2), (3), (6), and ( 11), the constraints can be rewritten as 47cM R --_-and YE,_ K
Ai2
U,t$
(12)
where A $ = A ii/h, and the bending stiffnesses are DT1= u, + L$t:)+
U&f
D& = U, - U&)+
U&f
D&=
U,-
2 Sreq A’ 2 (A;, -A;;/A;2)R2 (10)
iJ,t”
where Dz = 12D,/h” and U, are material constants. For definitions and examples, see Tsai and Hahn.” Other stiffnesses, i.e. Alh, Az6, Dlh, D26, and all B,, are set to zero. The material used for the current study was graphite/epoxy, see Table 1.
THE OPTIMIZATION TASK The task is to find the radius, wall-thickness and lay-up of a composite material circular thinwalled cylinder with specified applied bending moment, M, such that:
where A,, is the minimum cross-sectional area required not to violate the buckling constraint, and A, is the minimum cross-sectional area required not to violate the global bending stiffness constraint. Accordingly A 2=
Max[A t, A 51= Max[ C, R, C,/R “]
(13)
should be minimized. C, and C, are functions of the lay-up but independent of geometric quantities. If C, R is larger than C,/R 4 then A can be reduced by reducing R until C, R is equal to C,/ R4, and vice versa. Thus the optimal configuration is obtained when both constraints - buckling and bending stiffness - are active. From this requirement R is eliminated, and the crosssectional area thus is expressed as
- the cross-sectional area, A, is minimized; - the maximum compressive stress in the cylinder, N,,,,,, is smaller than the bifurcation load, N,, of a uniaxially compressed cylinder
1 [(A;, -A;2,/A;2)K2]2’5 (14)
Table 1. Material constants’ Material
&_ (GPa)
u, (GPa)
Graphite/epoxy T300-5208
181.0
76.4
U,
(GPa)
85.7
6 (GPa)
u, (GW
19.7
22.6
107
Optimizationof bent composite cylinders
which is a function of the lay-up only, provided the knock-down factor, y, is independent of geometric parameters, which was assumed. The cross-sectional area A is minimized when Obj= $0
I
(AT, -A$A;JK2
(15)
is maximized. This is taken as the objective function for the optimization. The factor 100/U, makes the expression dimensionless and of order unity. When the optimal lay-up has been found, the knock-down factor, y, is determined, for example, according to Ref. 5, the cylinder area A by eqn ( 14), the radius R by eqn (12) and the wall thickness h by eqn (3).
OPTIMAL RESULTS Using the outer boundary of the feasible region of the lamination parameters according to Grenestedt and Gudmundson4 numerical maximization of Obj yields Obj= 6.773
c:‘= 0.3250
&‘=0*3418;
&$I= - 0.3961
(16)
One of the constraints is active, namely El?,”
4
-&+llO
(17)
{:‘= 0.1324
ef = 0.2; &? = 0.04;
tf=
(19)
- 0.7188
while the others are zero. This results in Obj=4*652 which should be compared with Obj= 6.740 for eqn (18). By infinite repetition of the lay-up sequence above, the lamination parameters approach
(20) while
(f = o-3941;
(1+
meters. The result obtained by the inner boundary according to Fukunaga3 is guaranteed to be physically meaningful, so this will be adopted as the optimum. A discrete lay-up corresponding to eqn (18) is [(* &J,,, 0h2, 90,,, 0h41s, where: hI = O-36356, 19~= 37.160”, h2 = O-18395, h, = O-17725, and h,=0*27524 and a good approximation with 22 plies is [( * 37*5)4, 02, 902, 031s,for which Obj= 6.727. The lay-up obtained by ZitzEvancih2 for optimal axial buckling load and axial stiffness, [45,,, 0 ,8, 90,],, results in the lamination parameters
the
others
are
zero,
which results in area and thus the structural weight is proportional to ( 0bj)-1’5. With moment and bending stiffness specified, a 7% weight reduction in favour of the current design compared with that of eqn ( 19) is obtained, and 3% compared with eqn (20) (with A, R and h chosen optimally according to eqns ( 14), ( 12) and Obj= 5.828. The cross-sectional
which was derived in Ref. 4. The optimal design according to eqn ( 16) is outside the inner boundary of the feasible region and the design accordingly may be non-physical, i.e. may not correspond to a true laminate. Maximization of Obj using the inner boundary of the lamination parameters according to Fukunaga3 yields Obj= 6.740
cf = O-3802;
cp= 0.3148
6; = 0.3260;
c$) = - 0.3760
(18)
The Fukunaga3 constraint is active, see Fig. 2. The small difference between the two values obtained for Obj, less than 0.5% which corresponds to a 0.1% weight difference for the structure, does not necessitate any further investigation of the feasible region of the lamination Ioara-
Fig. 2. The inner boundary of the feasible region of tp and Ljp (thin line) when 54 and t$’ are fixed, according to Fukunaga.3 The thick line marks the feasible region of E;’ and [t or, equivalently, that of ep and [f. The values of the four . lamination parameters for the optimal design are marked with plus-sigflfj the upper corresponding to (E;” , E$’) IL,, cn\ ana tne lower to (gy, 4 yj.
108
Joachim L. Grenestedt
(3) for each lay-up individually). Compared to a quasi-isotropic lay-up, for which Obj= 4.940, the current design yields a 6% lighter structure. For a unidirectional lay-up with all fibers in the axial direction, Obj= l-600. The current design yields a 25% lighter structure. Thus, more important than the lay-up is proper selection of R and h.
ACKNOWLEDGEMENTS Most of the present work was carried out at Japan National Aerospace Laboratory (NAL), during 1990-91. Special thanks are due to, in alphabetical order, Fukunaga, Ishikawa, Kai, Kasai, Kibe, Shimokawa, and Toda.
SUMMARY AND CONCLUSIONS REFERENCES Optimization of composite circular cylinders subject to bending was performed. The structural weight was minimized when the structure was constrained by requirements on bifurcation buckling and bending stiffness. Lay-up, cylinder radius, and wall thickness were the design variables. Radius and wall thickness could be eliminated from the optimization scheme by simple analytical means. The optimal lay-up thus became independent of all geometric properties as well as load and stiffness requirements. Any physically possible lay-up was included using lamination parameters to describe the lay-up. Geometric properties, i.e. cylinder radius and wall thickness, were determined by requiring that both the buckling and stiffness constraints were active, which was shown to be necessary for an optimal design. A 3% structural weight reduction compared to published lay-ups was obtained, assuming that geometric properties were selected optimally for each lay-up individually. Proper selection of geometric properties exceeds by far the importance of proper lay-up selection.
1. Onoda, J., Optimal laminate configurations of cylindrical shells for axial buckling. AZAA J., 23 (7) (1985) 1093-g. 2. ZitzEvancih, L. D., Designing graphite cylinders to resist buckling. AIAAISAEIASMEIASEE 21st Joint Propulsion Conf., AIAA-851101, 8-10 July 1985, Monterey, California. 3. Fukunaga, H., Netting theory and its application to optimum design of laminated composite shells and plates. AIAAIASMEIASCEIAHS 29th Structures, Dynamics and Materials Conf., Williamsburg, 983-91.
Structural
1988, pp.
4. Grenestedt, J. L. & Gudmundson, P., Lay-up optimization of composite material structures. In Proc. ZUTAM Symp., Lyugby, Denmark, 1992, pp. 31 l-36. 5. Buckling of thin-walled circular cylinders, NASA SP8007, NASA Space Vehicle Design Criteria (Structures), revised Aug. 1968. 6. Dong, S. B., Pister, K. S. & Taylor, R. L., On the theory of laminated anisotropic shells and plates. JAS, 29 (8) (1962) 969-75.
7. Tasi, J., Effect of heterogeneity on the stability of composite cylindrical shells under axial compression. AZAA J., 4 (6) (1966)
1058-62.
8. Tsai, S. W. & Pagano, N. J., Invariant properties of composite materials. In Composite Materials Workshop, Technomic, Westport, CT, 1968, pp. 233-53. 9. Tsai, S. W. & Hahn, H. T., Introduction to Composite Materials, Technomic, Stamford, CT, 1980.
0263-8223(94)00031-X
Optimization of bent composite cylinders Joachim L. Grenestedt Department of Lightweight Structures, Royal Institute of Technology S-100 44 Stockholm, Sweden
A circular thin-walled cylinder subject to bending is studied. The optimal radius, wall thickness, and lay-up is obtained under a buckling constraint similar to that of the NASA Space Vehicle Design Criteria, and a global bending stiffness constraint. The lay-up is restricted to symmetric laminates but otherwise with complete freedom in the lay-up angle variation through the thickness, which is obtained by using lamination parameters as design variables. By neglecting the influence of geometry and stiffnesses on the knock-down factor for buckling, the optimal lay-up is found to be independent of the requirements of buckling strength and stiffness, i.e. all cylinders of a specific material will have the same optimal lay-up.
NOTATION A
B, C&
4; q E
E, h
L M
m
Cross-sectional area of the cylinder In-plane stiffnesses Normalized in-plane stiffnesses (Au/h) Coupling stiffnesses Functions of the lay-up only Bending stiffnesses of the cylinder wall Normalized bending stiffnesses ( 12 D,/h “) axial Average Young’s modulus of the cylinder wall Longitudinal modulus of a PlY Wall thickness of the cylinder Lamina thicknesses Second area moment of inertia of the cylinder Normalized buckling according to Ref. 1, function of the lay-up only Length of the cylinder Applied bending moment on the cylinder Bending moments in the cylinder wall Number of half-waves in the axial direction
Number of waves in the circumferential direction Buckling load of uniaxially compressed cylinder Maximum axial compressive stress in the bent cylinder Membrane forces in the cylinder wall Object function for the optimization nL/(mnR) (mnR/L )Z Radius of the cylinder Bending stiffness of the cylinder Required bending stiffness of the cylinder Material constants, linear combinations of the onaxis moduli of a ply Deformations in the x, y, and z directions, respectively Amplitudes of the deformations Axial, circumferential, and through-the-thickness coordinates, respectively Normalized through-thethickness coordinate (2z/h)
Arc tan (&)
Joachim L. Grenestedt
Correlation or knockdown factor Strains in the middle surface of the cylinder wall Lay-up angle as function of z* Lay-up angle Change of curvatures of the middle surface of the cylinder wall Lamination parameters for in-plane stiffnesses A, Lamination parameters for bending stiffnesss D,
INTRODUCTION
The incentive for this investigation was the research at Japan National Aerospace Laboratory (NAL) concerning a high altitude long endurance (HALE) vehicle, whose main wing beam and aft fuselage, in the current design, are circular carbon-fiber cylinders (see Fig. 1). The buckling constraint in the present study requires that the maximum compressive stress in the bent cylinder does not exceed the bifurcation stress of a uniaxially compressed cylinder according to Donnell theory, but with a different correlation factor than used for uniaxial compression. Further, there is a constraint on the bending stiffness. Without the stiffness constraint the optimal lay-up for a bent cylinder is the same as for a
Fig. 1.
uniaxially compressed cylinder, for which Onoda’ showed that the optimal lay-up is quasi-isotropic. ZitzEvancih* considered (among other tasks) maximizing the axial buckling load and the axial stiffness of a cylinder, a problem which is similar to the present one. The lay-up angles were restricted to O”rf:45”, and 90”. Relaxing this restriction leads to some improvements. Use of lamination parameters for the optimization is efficient because large nonlinearities of the object function versus design parameters are avoided, a fact due to the linearity of the stiffnesses in the lamination parameters. Further, the number of design parameters is kept to a minimum while maintaining more design freedom than discrete ply laminates. The main problem with lamination parameters is that the feasible region is not yet fully determined. Fukunaga” derived an inner boundary of the feasible region of the four parameters needed to describe an orthotropic laminate. It should be noted that contrary to what is stated in Ref. 3, the region derived is not the true one, but an inner boundary. It was obtained using a small number of discrete plies in the layup. Grenestedt and Gudmundson4 showed that this region was too small, and derived the true feasible region. A simple outer boundary of the feasible region of the same parameters was also obtained by Grenestedt and Gudmundson4 by using, for example, Schwarz inequality. In the present study, both the outer and inner boundaries of the feasible region are used. The optimal design found using the outer boundary, which
An early design sketch of the wing of the HALE vehicle of NAL, Japan.
Optimizationof bent composite cylinders
might correspond to an unfeasible design, was only 0.1% better than the optimal design obtained using the inner boundary. It was therefore not necessary to use the rather complicated true feasible region for the optimization, but the design obtained by the inner boundary was considered sufficient. The inner boundary, which was obtained by using a discrete lay-up, naturally guarantees feasibility. A fairly accurate representation of the optimal lay-up is [( f 37*5)4, O,, 902,
RJS.
carried out with p as variable instead of p, where p= tan*/3 since the buckling load versus /3 is very smooth, and both very small and very large values of p are included easily by varying /? between 0 and n/2. The use of real instead of integer values of m and n for minimizing the buckling load yields a conservative approximation. With this simplification the optimal lay-up will be independent of the R/h and the L/R ratios. The buckling stress is expressed as K=
GOVERNING EQUATIONS In accordance with the NASA Space Vehicle Design Criteria5 the maximum buckling stress of the bent cylinder was limited by that of a uniformly compressed cylinder, but with a different knock-down factor. The knock-down factor is dependent on the cylinder radius-to-wall thickness ratio, R/h, as well as the elastic stiffnesses, a fact which for the present optimization task was neglected. This is not a severe restriction, since for cylinders with the R/h ratio varying between 200 and 1000 the knock-down factor only varies between approximately 0.8 and 05. By neglecting these dependencies an optimal lay-up independent of the R/h ratio could be determined. The bifurcation load was calculated using Donnell kinematics and lamination theory.6,7 It can be shown that for symmetry reasons A 16,Az6, D 16,and Dz6 should vanish for an optimal lay-up.’ This reduces the number of design parameters to four, &‘, Ef, ljf’, cf, as will be seen later. The resulting equations are fulfilled by the deformation
u(x,y)= u,, cos( mnzx/L
) cos( ny/R)
~(~,y)=v,,sin(mnx/l)
sin(ny/R)
w(x,y)= w,, sin( mJtx/L
) cos( ny/R)
105
(&h*/R)K(c)
(2)
where E, is the longitudinal modulus of a ply, and K is a function of the lay-up, i.e. the lamination parameters c, according to eqns ( 16) and (17) of Ref. 1. Observe that the current definition of stiffnesses and lamination parameters are different from those of Ref. 1. The global bending stiffness of the cylinder is calculated using Euler-Bernoulli beam theory. The cross-sectional area of the thin-walled cylinder is A= 2nRh
(3)
the second area moment of inertia is I= nR”h
(4)
and the average Young’s modulus direction is E=AT, -(AT2)2/A;2
in the axial (9
where A ; = A,/h, see also eqn (7), so the global bending stiffness of the cylinder, S, is S=EI=(AT,
-(AT2)2/A;2)nR”h
(6)
Effects of the order h/R are neglected compared to unity.
(1)
where m and n are integer values which should be selected so that the bifurcation load is minimized and the boundary conditions, if possible, fulfilled. For simplicity, m and n are in this study selected as the real values that minimize the bifurcation load. By introducing the parameters p = nL/ (mzR) and 4 = (mnR/L )” the buckling load was simply minimized analytically versus q,’ while minimization versus p was performed numerically by bracketing initially the minimum between zero and infinity, and then using the Golden Section method. The bracketing and minimization was
CONSTITUTIVE
PARAMETERS
Assuming symmetric equations are
Nx
Nxy
the constitutive
4, AI, AI,
O( NY =
laminates,
(7)
106
Joachim L. Grenestedt
The lamination parameters, derived by Tsai and Pagano,x needed to describe an orthotropic lay-up are defined as I +$,=;
[cos
219,cos 481 dz*
i I (8)
I tf;.2,=; i
_ [cos I
20,cos 48]z** dz*
where the lay-up angle 0 is a function of z*, the normalized through-the-thickness coordinate. Four parameters suffice to describe a general laminated orthotropic material. The in-plane stiffnesses are AT, = 6 + W?+
Q&3’
A;?= U, - U&‘+
U&’
A&=
U,-
(9)
U,{$
A,“,=$ (U, - U,)-
times a correlation factor, y (knock-down factor); - the global bending stiffness, S, exceeds a . . . _ nummum required stiffness, Seq. The cross-sectional area represents the structural weight which is to be minimized under buckling and stiffness constraints. In mathematical terms the optimization problem can be expressed as: minimize A = 2nRh, under the constraints that Nmax5 rlv,, for a specified bending moment, M, and Sk Seq. The maximum compressive stress is
(11) Eliminating h, using eqns (2), (3), (6), and ( 11), the constraints can be rewritten as 47cM R --_-and YE,_ K
Ai2
U,t$
(12)
where A $ = A ii/h, and the bending stiffnesses are DT1= u, + L$t:)+
U&f
D& = U, - U&)+
U&f
D&=
U,-
2 Sreq A’ 2 (A;, -A;;/A;2)R2 (10)
iJ,t”
where Dz = 12D,/h” and U, are material constants. For definitions and examples, see Tsai and Hahn.” Other stiffnesses, i.e. Alh, Az6, Dlh, D26, and all B,, are set to zero. The material used for the current study was graphite/epoxy, see Table 1.
THE OPTIMIZATION TASK The task is to find the radius, wall-thickness and lay-up of a composite material circular thinwalled cylinder with specified applied bending moment, M, such that:
where A,, is the minimum cross-sectional area required not to violate the buckling constraint, and A, is the minimum cross-sectional area required not to violate the global bending stiffness constraint. Accordingly A 2=
Max[A t, A 51= Max[ C, R, C,/R “]
(13)
should be minimized. C, and C, are functions of the lay-up but independent of geometric quantities. If C, R is larger than C,/R 4 then A can be reduced by reducing R until C, R is equal to C,/ R4, and vice versa. Thus the optimal configuration is obtained when both constraints - buckling and bending stiffness - are active. From this requirement R is eliminated, and the crosssectional area thus is expressed as
- the cross-sectional area, A, is minimized; - the maximum compressive stress in the cylinder, N,,,,,, is smaller than the bifurcation load, N,, of a uniaxially compressed cylinder
1 [(A;, -A;2,/A;2)K2]2’5 (14)
Table 1. Material constants’ Material
&_ (GPa)
u, (GPa)
Graphite/epoxy T300-5208
181.0
76.4
U,
(GPa)
85.7
6 (GPa)
u, (GW
19.7
22.6
107
Optimizationof bent composite cylinders
which is a function of the lay-up only, provided the knock-down factor, y, is independent of geometric parameters, which was assumed. The cross-sectional area A is minimized when Obj= $0
I
(AT, -A$A;JK2
(15)
is maximized. This is taken as the objective function for the optimization. The factor 100/U, makes the expression dimensionless and of order unity. When the optimal lay-up has been found, the knock-down factor, y, is determined, for example, according to Ref. 5, the cylinder area A by eqn ( 14), the radius R by eqn (12) and the wall thickness h by eqn (3).
OPTIMAL RESULTS Using the outer boundary of the feasible region of the lamination parameters according to Grenestedt and Gudmundson4 numerical maximization of Obj yields Obj= 6.773
c:‘= 0.3250
&‘=0*3418;
&$I= - 0.3961
(16)
One of the constraints is active, namely El?,”
4
-&+llO
(17)
{:‘= 0.1324
ef = 0.2; &? = 0.04;
tf=
(19)
- 0.7188
while the others are zero. This results in Obj=4*652 which should be compared with Obj= 6.740 for eqn (18). By infinite repetition of the lay-up sequence above, the lamination parameters approach
(20) while
(f = o-3941;
(1+
meters. The result obtained by the inner boundary according to Fukunaga3 is guaranteed to be physically meaningful, so this will be adopted as the optimum. A discrete lay-up corresponding to eqn (18) is [(* &J,,, 0h2, 90,,, 0h41s, where: hI = O-36356, 19~= 37.160”, h2 = O-18395, h, = O-17725, and h,=0*27524 and a good approximation with 22 plies is [( * 37*5)4, 02, 902, 031s,for which Obj= 6.727. The lay-up obtained by ZitzEvancih2 for optimal axial buckling load and axial stiffness, [45,,, 0 ,8, 90,],, results in the lamination parameters
the
others
are
zero,
which results in area and thus the structural weight is proportional to ( 0bj)-1’5. With moment and bending stiffness specified, a 7% weight reduction in favour of the current design compared with that of eqn ( 19) is obtained, and 3% compared with eqn (20) (with A, R and h chosen optimally according to eqns ( 14), ( 12) and Obj= 5.828. The cross-sectional
which was derived in Ref. 4. The optimal design according to eqn ( 16) is outside the inner boundary of the feasible region and the design accordingly may be non-physical, i.e. may not correspond to a true laminate. Maximization of Obj using the inner boundary of the lamination parameters according to Fukunaga3 yields Obj= 6.740
cf = O-3802;
cp= 0.3148
6; = 0.3260;
c$) = - 0.3760
(18)
The Fukunaga3 constraint is active, see Fig. 2. The small difference between the two values obtained for Obj, less than 0.5% which corresponds to a 0.1% weight difference for the structure, does not necessitate any further investigation of the feasible region of the lamination Ioara-
Fig. 2. The inner boundary of the feasible region of tp and Ljp (thin line) when 54 and t$’ are fixed, according to Fukunaga.3 The thick line marks the feasible region of E;’ and [t or, equivalently, that of ep and [f. The values of the four . lamination parameters for the optimal design are marked with plus-sigflfj the upper corresponding to (E;” , E$’) IL,, cn\ ana tne lower to (gy, 4 yj.
108
Joachim L. Grenestedt
(3) for each lay-up individually). Compared to a quasi-isotropic lay-up, for which Obj= 4.940, the current design yields a 6% lighter structure. For a unidirectional lay-up with all fibers in the axial direction, Obj= l-600. The current design yields a 25% lighter structure. Thus, more important than the lay-up is proper selection of R and h.
ACKNOWLEDGEMENTS Most of the present work was carried out at Japan National Aerospace Laboratory (NAL), during 1990-91. Special thanks are due to, in alphabetical order, Fukunaga, Ishikawa, Kai, Kasai, Kibe, Shimokawa, and Toda.
SUMMARY AND CONCLUSIONS REFERENCES Optimization of composite circular cylinders subject to bending was performed. The structural weight was minimized when the structure was constrained by requirements on bifurcation buckling and bending stiffness. Lay-up, cylinder radius, and wall thickness were the design variables. Radius and wall thickness could be eliminated from the optimization scheme by simple analytical means. The optimal lay-up thus became independent of all geometric properties as well as load and stiffness requirements. Any physically possible lay-up was included using lamination parameters to describe the lay-up. Geometric properties, i.e. cylinder radius and wall thickness, were determined by requiring that both the buckling and stiffness constraints were active, which was shown to be necessary for an optimal design. A 3% structural weight reduction compared to published lay-ups was obtained, assuming that geometric properties were selected optimally for each lay-up individually. Proper selection of geometric properties exceeds by far the importance of proper lay-up selection.
1. Onoda, J., Optimal laminate configurations of cylindrical shells for axial buckling. AZAA J., 23 (7) (1985) 1093-g. 2. ZitzEvancih, L. D., Designing graphite cylinders to resist buckling. AIAAISAEIASMEIASEE 21st Joint Propulsion Conf., AIAA-851101, 8-10 July 1985, Monterey, California. 3. Fukunaga, H., Netting theory and its application to optimum design of laminated composite shells and plates. AIAAIASMEIASCEIAHS 29th Structures, Dynamics and Materials Conf., Williamsburg, 983-91.
Structural
1988, pp.
4. Grenestedt, J. L. & Gudmundson, P., Lay-up optimization of composite material structures. In Proc. ZUTAM Symp., Lyugby, Denmark, 1992, pp. 31 l-36. 5. Buckling of thin-walled circular cylinders, NASA SP8007, NASA Space Vehicle Design Criteria (Structures), revised Aug. 1968. 6. Dong, S. B., Pister, K. S. & Taylor, R. L., On the theory of laminated anisotropic shells and plates. JAS, 29 (8) (1962) 969-75.
7. Tasi, J., Effect of heterogeneity on the stability of composite cylindrical shells under axial compression. AZAA J., 4 (6) (1966)
1058-62.
8. Tsai, S. W. & Pagano, N. J., Invariant properties of composite materials. In Composite Materials Workshop, Technomic, Westport, CT, 1968, pp. 233-53. 9. Tsai, S. W. & Hahn, H. T., Introduction to Composite Materials, Technomic, Stamford, CT, 1980.