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Stacking sequence optimization of laminated plates
Composite Structures Vol. 39, Nos 3-4. pp. 283-2X8, 1997 0 1998 Published by Elsevicr Science Ltd. All rights resewed Printed in Great Britain 0263-8223/98/$19.00 + 0.00 ELSEVIER
PlI:SO263-8223(97)00120-7
Stacking sequence optimization of laminated plates C. W. Kim, W. Hwang, H. C. Park & K. S. Han Department of Mechanical Engineering,
Pohang University of Science and Technology, Pohang 790-784, South Korea
Optimum fiber orientations of laminated composite plates for the maximum strength are found under multiple inplane loading conditions. Tsai-Wu failure criterion is taken as objective function. Based on the state space method, effective optimal design formulation is developed and solution procedure is described with the emphasis on the method of calculations of the design sensitivities. Numerical results are presented for the several test problems. 0 1998 Published by Elsevier Science Ltd. All rights reserved.
The present paper treats the optimal stacking sequence design of symmetrically laminated plates under in-plane loadings for maximum strength. Tsai-Wu failure criterion is taken as objective function. Design variables are ply orientation angles. The adjoint variable method is used to formulate the optimal design problem. To treat worst-case design, the optimal design problem is formulated as a min-max problem considering environmental parameters. Based on the adjoint variable method, accurate design sensitivities are calculated using state equation and its results are verified with finite difference method. Optimal stacking sequence design as well as worst stacking sequence design are presented. To compare optimum design with worst case, load carrying ability factor (FL,-) is defined by square root of the ratio of worst design result to the optimum design result. Numerical results are given for various loading conditions and aspect ratios.
INTRODUCTION Composite materials in mechanical, aerospace, and other branches of engineering are increasingly used due to their excellent weight saving and the ease of tailoring. In spite of tremendous progress in analytical capability to analyze the behavior of composite materials and structures, there is a lack of design models which may allow efficient tailoring of their properties to specific requirements for structural components. To improve this long-pending problem, the optimum design of composite materials has been a subject of research for many years. The usual object of optimum design is to design layer thickness or layer orientation which will give the minimum weight [l-3] or the maximum stiffness P-61 under in-plane or transverse loadings. Strength was considered as constraints in many problems [l-3,6-8]. There are, however, a few studies which consider the . strength criterion as an object function of optimum design. Quadratic failure criteria such as Tsai-Wu theory have been used widely for predicting failure of composite materials subject to combined stresses [9]. Recently, the quadratic failure criteria are applied to the optimal stacking sequence design of laminated plates having maximum strength [lo-121.
ANALYSIS The plate stresses are calculated using classical lamination theory (CLT). The constitutive 283
284
C. N Kim, W Hwang, H, C. Park, K. S. Han
equations for a symmetrically site plate are
[$(@=[$
$;
laminated
compo-
i;;T[iJ)
where Qij is transformed can be written as follows:
(1)
stiffness
matrix and
Q,, = Q, ,cos 40+Q,,sin 48 +2(Q, *+2Q&sin *0 cos20 Q22= Q, , sin40+Q,,cos40 +2(Q ,2+2QsG)sin20 cos20 Q,* = (Q, l+Q22-4Q66)sin20 cos20 +Q, ,(cos 40+sin 48) Q,,=(Q,,+Q,,-2Q,,)cos30sin
19
- ( Q2* = Q ,2 - 2QG6)cos 0 sin 38 Q,,=(Q,,
-Q,,-2Q,,)cos
I?,=@-f(e)=0
&=(Q,,+Q,,-2Q,,-2Q,,)sin26,cos
h,= {D) -.f@>=O 2H
+Q,,(cos 0+ sin 48)
(2)
Hooke’s law in conjunction with the straindisplacement relation and the stress and moment resultant definition yields the following relations for a laminated composite plate:
‘A,, Au
4 Nxv
M, =
E:
A22
A26
B,2
B22
B26
E;
A,6
A26
A66
B,6
B26
B66
B,2
B,6
D,I
D,2
D16
Y:, ' Kc,
B,,
B22
B26
D,2
D22
O26
'cv
.B,,
B26
B66
D,6
D26
D663x.v
MY -K,-
A,, B,, B,, 816
A12
14,
s r12
Dij =
s
@i”, dz
7;’ _ r,2 pjr’z” dz
__,* @z
h5 = (a} -f(&?,rc)
=0
(5)
In the adjoint variable method, the design sensitivity vector li of a function ‘Pi (z,b) can be written [ 131: ah’
ab
(3
ab
(9
The adjoint variable vector I’ is calculated from the solution of following adjoint equation
r/2
Bij=
h4= ” - f(A,D) = 0 11ic
I’= -_----_ PI’;
where A,, B, and D, are extensional stiffness matrix, coupling matrix, and bending stiffness matrix, respectively. The matrices can be written as follows: Aij =
h2={A}-f(Q)=0
8 sin ‘8
- (Q22 - Q I2- 2Q&cos ‘8 sin 6,
NX
The accuracy of optimal solutions depends on the design sensitivity analysis. Most engineering optimization problems use the finite difference method for its convenience in spite of its truncation error and round off error. To get more accurate design sensitivity, the state space method [13] is adopted to formulate an optimization problem and the adjoint variable method is used. In the state space method, the governing equation is defined as a state equation and state variables represent the behavior of the system. The analysis procedures of laminated plates are considered as a set of state equations. The transformed stiffness matrix Q, extensional stiffness matrix A, bending stiffness matrix D, strain E, and stress 0 are state variables, z, while stacking sequence 8 is design variables, b. The resulting state equations are
JTA'=
84’7
2
az
(7)
where J denotes the Jacobian matrix and
dz
s
(4)
J= g
[z”,b”l
PROBLEM FORMULATION
The design sensitivities can be calculated above eqns (6) and (7).
by the
Sensitivity analysis
Optimization procedure
Every iterative algorithm for solving an optimization problem requires a gradient of functions.
The optimization problem that we consider here is to maximize the strength of composite
285
Stacking sequence optimization of laminated plates
Optimization procedure The optimization problem that we consider here is to maximize the strength of composite laminates for a given layer thickness. Each layer
0.02
I
angle is taken as a design variable. To consider strength, the Tsai-Wu quadratic failure criterion is used. The state space method is used to formulate the optimal design problem. To treat worst-case design, the optimal design problem is formulated as a min-max problem considering the environmental parameter, CC In this problem, maximum failure index through thickness is treated as an environmental parameter. The optimization problem can be expressed in mathematical form as follows: Minimize Maximum tmin
<
r < trnax
+
2 02
2 01
XX’
+
(8)
YY’ +s2
ClC’2
-
2,XX’YY’
(9)
where F(o,, (TV,T,~) is the Tsai-Wu failure criterion and defined as failure index (H). Min-max design formulation can be reduced to a parametric optimal design formulation using a dummy design variable b(j + 1) and new constraints. The reduced parametric formulation can be written as follows [13]: Minimize Y’, = b(j+ 1) subject to Y, =Maximum F(a,,o,,z,,)-_(j+l)
<
z < tmax
with the state equation
values occurs
0
10
20
30
40
50
60
70
80
90
Angie
Fig. 1.
h(z,b,cc)= 0
2 212
F.I.
-0.01
F(o,,o,,z,,)
with the state equation
--
minus
NUMERICAL RESULTS AND DISCUSSION Some examples are presented to demonstrate the effect of optimization of stacking sequence on the strength of composite plates. The optimization problem stated in eqn (10) is solved by gradient projection algorithm with the design sensitivity information. A conceptual flow chart for solving the process of optimal design problem is given in Fig. 2. In calculating numerical solutions, the following material properties of T300/5208 graphite/epoxy are applied: E,, = 181 GPa G,, = 7.17 GPa
Y=Y’=43$MPa
E22= 10.30 GPa
u12=O.28 X = X’ = 1500 MPa S=86.9MPa
(“‘4”’ (101
I
h(z,b,cx)= 0
In some cases, the above Tsai-Wu failure criterion is not applicable, especially when the unidirectional angle ply laminates are subjected to axial force or torsion [14]. For instance, minus failure index values occur near 20-30” when the unidirectional angle ply laminates are subjected to tensile loading N, as shown in Fig. 1. As a result, the optimum angle, which gives minimum failure index, is found to be around 20”. This result is not only far from reality but also physically meaningless. To avoid this inconsistency, it is assumed that transverse compressive strength is same as tensile strength, i.e. Y’ = Y.
)
Guess initial design
I
,I r
Solve the state equation Calculate design sensitivities and the other design information if NO Calculate design improvement and modify the current design Do the design and the design information satisfy all design requirements ?
I
(-G--j Fig. 2.
286
C. W Kim, W Hwang, H. C. Park, K. S. Han
Table 1. Accuracy of design sensitivity
([O/45/901, laminate,
i
bi
1 :
0 45 90
4
- 1~0000
46370 25673 - 4.6370 x 10 -’
4.6371 7.2049 - 4.6369 x 10 - ’
4.6369 - 7.1998 - 4.6371 x low
- l%nMI
- 1+000
- 1~0000
lj, adjoint variable method; lZw,forward finite difference difference scheme.
NW = 100 MN/mm)
scheme; li(‘) , backward finite difference
Sensitivity analysis The design sensitivities of the 2nd layer of [O/45/90], laminated plate subjected to shear load Nq = 100 MN/mm are presented in Table 1. The sensitivities evaluated by F.D.M. with various step size are also compared with the present design sensitivity in Fig. 3. The comparison shows that the present design sensitivity calculated using the adjoint variable method is very accurate. It can be seen that all the results evaluated by F.D.M. give reliable design sensitivities for a relatively wide range of step size from lo-’ to 10-3. However, in certain ranges of step size large numerical errors are observed when the design sensitivities are evaluated by F.D.M. Illustrative examples
4.6370 25781 - 4.6370 x lo-*
scheme; lp), central finite
the ratio of loading capabilities between the worst designed laminate and optimum designed laminate under Tsai-Wu failure condition. It can be written as follows:
F,, =
/x
(11)
To verify the present method, optimum solutions of [0], and [0,/6& laminates subjected to uniaxial tensile loading are sought. It is obvious that the maximum strength of the laminates occurs when the all layer angles are 0” and the minimum strength of the laminates occurs when the all layer angles are 90”. The same results are obtained in this study as shown in Fig. 4 and Table 2 (case 1). Figure 4 shows three dimensional plot and contours of the objective
To demonstrate the validity of the method developed, the worst solutions as well as optimal solutions are presented. The load carrying ability factor, F,, is defined by the square root of the ratio of worst failure index to optimal failure index. The load carrying factor implies
- - Q4.641
F.D.Y(f) F.D.M.(b) ---O-- F.D.M.(c) _ adjolnt variable
-90
r” > = f $
-90 4.637
-45 0
45 4.633 l(-J-'o1o-~lo-* lO~'lO~
lo"lo~lo~lo"lo"lo"
Step Size Fig. 3.
90 -90
-45
An&
45
Fig. 4.
90
Angle
Stacking
287
sequence optimization of laminated plates
Table 2. Optimum and worst results of symmetric laminated plates under in-plane loading Case
No. of layers
1 2 3 4 5
2 4 2 4 2 4 2 4 2 4
Loading ratio (N,, N,, N,)
Optimum
Worst angle
angle
Wls
(LO, 0)
‘9K01s y$
(1, 2,O) (0, (1, 1)
[45/45\3,
Pls WOIS
(1, 4, 0)
[-31.71, [-31.71-31.71,
(1, 2, 1)
function with respect to design variables under uniaxial loading. It can be seen in the figure that minimum failure index is found in [Q],, laminate while [O,/@& laminate has maximum failure index. As it is expected, F,, is 375 which is exactly same as the ratio of transverse strength to longitudinal strength. Other calculations are carried out in-plane biaxially loaded plates using (N,:N, = 1:2). This loading condition is the same as the cylindrical pressure vessel under internal pressure. The worst and optimal stacking sequences are presented in Table 2 (case 2). It can be seen that the optimum solutions are close to the [ +54*7], laminate which is the optimum result from the netting analysis of cylindrical pressure vessel subjected to internal pressure. In comparison with the worst solution, the optimally designed plate can carry as much as 13.3 times greater external loads. The worst and optimum solutions under various loading conditions [(N,, NY, NT) = (0, 0, l), (1, 4, 0), and (1, 2, l)] are evaluated and presented in Table 2. It is found that the optimized laminate can carry external loads more than ten times greater than the worst designed laminate under a shear loading (case 3). In cases 4 and 5, the load carrying ability factors are found to be 21. 7 and 28.6, respectively. It is very interesting that optimum solutions vary with number of plies while worst solutions do not change.
F LC
375 37.5 2.0 13.3 :,;7, 4.2 21.7 7.4 28.6
an objective function. The design sensitivity with respect to layer angle is analyzed based on the adjoint variable method. It is found that this method provides more precise design sensitivity than F.D.M. In conclusion, the numerical results show that the failure index is minimized successfully and the reasonable optimal stacking sequence design is sought.
REFERENCES 1. Schmit, L. A. Jr. and Farshi, B., Optimum laminated design for strength and stiffness. ht. J. for Numerical Methods in Engineering
1977, 7, 519-536.
2. Schmit, L. A. Jr. and Farshi, B., Optimum design of laminated fiber composite plates. Znt. J. for Numerical Methods in Engineering 1977,11, 623-640. 3. Fukunaga, H. and Vanderplaats, G. N., Strength optimization of laminated composites with respect to layer thickness and/or layer orientation angle. Computers and Structures 1991, 40, 1429-1439.
4. Tauchert, T. R. and Adibhatla, S., Design of laminated plates for maximum stiffness. Journal of Composite Materials 1984, l&58-69. 5. Kam, T. Y. and Snyman, J. A., Optimal design of laminated composite plates using a global optimization technique. Composite Structure 1991, 19, 351-370. 6. Fukunaga, H. and Sekine, H., Optimum design of composite structures for shape, layer angle and layer thickness distributions. Journal of Composite Materials 1993,27,
(15)
1479-1492.
7. Liu, I.-W. and Lin, C.-C., Optimum design of composite wing structures by a refined optimal@ criterion. Composite Structures 1991, 17, 51-65.
8. Adali, S., Summers, E. B. and Verijenko, V. E., Optimization of laminated cylindrical pressure vessels under strength criterion. Composite Structures 1993, 25,305-312.
CONCLUSIONS An efficient approach for strength optimization of composite laminates subjected to in-plane loadings is presented. The layer angles could be regarded as design variables without difficulty by taking Tsai-Wu quadratic failure criterion as
9. Rowlands, R. E., Strength (failure) theories and their experimental correlation. In Failure Mechanics of Composites, ed. G. C. Sih and A. M. Sudra. Elsevier, 1985. 10. Chao, C. C., Koh, S. L. and Sun, C. T., Optimization of buckling and yield strengths of laminated composites.AIAA Journal 1975, 13, 1131-1132. 11. Graesser, D. L., Zabinsky, Z. B., Tuttle, M. E. and Kim, G. I., Optimal design of a composite structure. Composite Structures 1993, 24, 273-281.
288
C. W Kim, U? Hwang, H. C. Park, K. S. Han
12. Kim, C. W., Hwang, W., Park, H. C. & Han, K. S., An optimal stacking sequence design of laminated composite cylinder. ICCM-9, Madrid, Spain, 1993. 13. Haug, E. J. & Arora, J. S., Applied Optimal Design. John Wiley, 1979.
14. Kim, C. W., Song, S. R., Hwang, W., Park, H. C. and Han, K. S., On the failure indices of quadratic failure criteria for optimal stacking sequence design of laminated plate. Applied Composite Materials 1994, 1, 81-85.
PlI:SO263-8223(97)00120-7
Stacking sequence optimization of laminated plates C. W. Kim, W. Hwang, H. C. Park & K. S. Han Department of Mechanical Engineering,
Pohang University of Science and Technology, Pohang 790-784, South Korea
Optimum fiber orientations of laminated composite plates for the maximum strength are found under multiple inplane loading conditions. Tsai-Wu failure criterion is taken as objective function. Based on the state space method, effective optimal design formulation is developed and solution procedure is described with the emphasis on the method of calculations of the design sensitivities. Numerical results are presented for the several test problems. 0 1998 Published by Elsevier Science Ltd. All rights reserved.
The present paper treats the optimal stacking sequence design of symmetrically laminated plates under in-plane loadings for maximum strength. Tsai-Wu failure criterion is taken as objective function. Design variables are ply orientation angles. The adjoint variable method is used to formulate the optimal design problem. To treat worst-case design, the optimal design problem is formulated as a min-max problem considering environmental parameters. Based on the adjoint variable method, accurate design sensitivities are calculated using state equation and its results are verified with finite difference method. Optimal stacking sequence design as well as worst stacking sequence design are presented. To compare optimum design with worst case, load carrying ability factor (FL,-) is defined by square root of the ratio of worst design result to the optimum design result. Numerical results are given for various loading conditions and aspect ratios.
INTRODUCTION Composite materials in mechanical, aerospace, and other branches of engineering are increasingly used due to their excellent weight saving and the ease of tailoring. In spite of tremendous progress in analytical capability to analyze the behavior of composite materials and structures, there is a lack of design models which may allow efficient tailoring of their properties to specific requirements for structural components. To improve this long-pending problem, the optimum design of composite materials has been a subject of research for many years. The usual object of optimum design is to design layer thickness or layer orientation which will give the minimum weight [l-3] or the maximum stiffness P-61 under in-plane or transverse loadings. Strength was considered as constraints in many problems [l-3,6-8]. There are, however, a few studies which consider the . strength criterion as an object function of optimum design. Quadratic failure criteria such as Tsai-Wu theory have been used widely for predicting failure of composite materials subject to combined stresses [9]. Recently, the quadratic failure criteria are applied to the optimal stacking sequence design of laminated plates having maximum strength [lo-121.
ANALYSIS The plate stresses are calculated using classical lamination theory (CLT). The constitutive 283
284
C. N Kim, W Hwang, H, C. Park, K. S. Han
equations for a symmetrically site plate are
[$(@=[$
$;
laminated
compo-
i;;T[iJ)
where Qij is transformed can be written as follows:
(1)
stiffness
matrix and
Q,, = Q, ,cos 40+Q,,sin 48 +2(Q, *+2Q&sin *0 cos20 Q22= Q, , sin40+Q,,cos40 +2(Q ,2+2QsG)sin20 cos20 Q,* = (Q, l+Q22-4Q66)sin20 cos20 +Q, ,(cos 40+sin 48) Q,,=(Q,,+Q,,-2Q,,)cos30sin
19
- ( Q2* = Q ,2 - 2QG6)cos 0 sin 38 Q,,=(Q,,
-Q,,-2Q,,)cos
I?,=@-f(e)=0
&=(Q,,+Q,,-2Q,,-2Q,,)sin26,cos
h,= {D) -.f@>=O 2H
+Q,,(cos 0+ sin 48)
(2)
Hooke’s law in conjunction with the straindisplacement relation and the stress and moment resultant definition yields the following relations for a laminated composite plate:
‘A,, Au
4 Nxv
M, =
E:
A22
A26
B,2
B22
B26
E;
A,6
A26
A66
B,6
B26
B66
B,2
B,6
D,I
D,2
D16
Y:, ' Kc,
B,,
B22
B26
D,2
D22
O26
'cv
.B,,
B26
B66
D,6
D26
D663x.v
MY -K,-
A,, B,, B,, 816
A12
14,
s r12
Dij =
s
@i”, dz
7;’ _ r,2 pjr’z” dz
__,* @z
h5 = (a} -f(&?,rc)
=0
(5)
In the adjoint variable method, the design sensitivity vector li of a function ‘Pi (z,b) can be written [ 131: ah’
ab
(3
ab
(9
The adjoint variable vector I’ is calculated from the solution of following adjoint equation
r/2
Bij=
h4= ” - f(A,D) = 0 11ic
I’= -_----_ PI’;
where A,, B, and D, are extensional stiffness matrix, coupling matrix, and bending stiffness matrix, respectively. The matrices can be written as follows: Aij =
h2={A}-f(Q)=0
8 sin ‘8
- (Q22 - Q I2- 2Q&cos ‘8 sin 6,
NX
The accuracy of optimal solutions depends on the design sensitivity analysis. Most engineering optimization problems use the finite difference method for its convenience in spite of its truncation error and round off error. To get more accurate design sensitivity, the state space method [13] is adopted to formulate an optimization problem and the adjoint variable method is used. In the state space method, the governing equation is defined as a state equation and state variables represent the behavior of the system. The analysis procedures of laminated plates are considered as a set of state equations. The transformed stiffness matrix Q, extensional stiffness matrix A, bending stiffness matrix D, strain E, and stress 0 are state variables, z, while stacking sequence 8 is design variables, b. The resulting state equations are
JTA'=
84’7
2
az
(7)
where J denotes the Jacobian matrix and
dz
s
(4)
J= g
[z”,b”l
PROBLEM FORMULATION
The design sensitivities can be calculated above eqns (6) and (7).
by the
Sensitivity analysis
Optimization procedure
Every iterative algorithm for solving an optimization problem requires a gradient of functions.
The optimization problem that we consider here is to maximize the strength of composite
285
Stacking sequence optimization of laminated plates
Optimization procedure The optimization problem that we consider here is to maximize the strength of composite laminates for a given layer thickness. Each layer
0.02
I
angle is taken as a design variable. To consider strength, the Tsai-Wu quadratic failure criterion is used. The state space method is used to formulate the optimal design problem. To treat worst-case design, the optimal design problem is formulated as a min-max problem considering the environmental parameter, CC In this problem, maximum failure index through thickness is treated as an environmental parameter. The optimization problem can be expressed in mathematical form as follows: Minimize Maximum tmin
<
r < trnax
+
2 02
2 01
XX’
+
(8)
YY’ +s2
ClC’2
-
2,XX’YY’
(9)
where F(o,, (TV,T,~) is the Tsai-Wu failure criterion and defined as failure index (H). Min-max design formulation can be reduced to a parametric optimal design formulation using a dummy design variable b(j + 1) and new constraints. The reduced parametric formulation can be written as follows [13]: Minimize Y’, = b(j+ 1) subject to Y, =Maximum F(a,,o,,z,,)-_(j+l)
<
z < tmax
with the state equation
values occurs
0
10
20
30
40
50
60
70
80
90
Angie
Fig. 1.
h(z,b,cc)= 0
2 212
F.I.
-0.01
F(o,,o,,z,,)
with the state equation
--
minus
NUMERICAL RESULTS AND DISCUSSION Some examples are presented to demonstrate the effect of optimization of stacking sequence on the strength of composite plates. The optimization problem stated in eqn (10) is solved by gradient projection algorithm with the design sensitivity information. A conceptual flow chart for solving the process of optimal design problem is given in Fig. 2. In calculating numerical solutions, the following material properties of T300/5208 graphite/epoxy are applied: E,, = 181 GPa G,, = 7.17 GPa
Y=Y’=43$MPa
E22= 10.30 GPa
u12=O.28 X = X’ = 1500 MPa S=86.9MPa
(“‘4”’ (101
I
h(z,b,cx)= 0
In some cases, the above Tsai-Wu failure criterion is not applicable, especially when the unidirectional angle ply laminates are subjected to axial force or torsion [14]. For instance, minus failure index values occur near 20-30” when the unidirectional angle ply laminates are subjected to tensile loading N, as shown in Fig. 1. As a result, the optimum angle, which gives minimum failure index, is found to be around 20”. This result is not only far from reality but also physically meaningless. To avoid this inconsistency, it is assumed that transverse compressive strength is same as tensile strength, i.e. Y’ = Y.
)
Guess initial design
I
,I r
Solve the state equation Calculate design sensitivities and the other design information if NO Calculate design improvement and modify the current design Do the design and the design information satisfy all design requirements ?
I
(-G--j Fig. 2.
286
C. W Kim, W Hwang, H. C. Park, K. S. Han
Table 1. Accuracy of design sensitivity
([O/45/901, laminate,
i
bi
1 :
0 45 90
4
- 1~0000
46370 25673 - 4.6370 x 10 -’
4.6371 7.2049 - 4.6369 x 10 - ’
4.6369 - 7.1998 - 4.6371 x low
- l%nMI
- 1+000
- 1~0000
lj, adjoint variable method; lZw,forward finite difference difference scheme.
NW = 100 MN/mm)
scheme; li(‘) , backward finite difference
Sensitivity analysis The design sensitivities of the 2nd layer of [O/45/90], laminated plate subjected to shear load Nq = 100 MN/mm are presented in Table 1. The sensitivities evaluated by F.D.M. with various step size are also compared with the present design sensitivity in Fig. 3. The comparison shows that the present design sensitivity calculated using the adjoint variable method is very accurate. It can be seen that all the results evaluated by F.D.M. give reliable design sensitivities for a relatively wide range of step size from lo-’ to 10-3. However, in certain ranges of step size large numerical errors are observed when the design sensitivities are evaluated by F.D.M. Illustrative examples
4.6370 25781 - 4.6370 x lo-*
scheme; lp), central finite
the ratio of loading capabilities between the worst designed laminate and optimum designed laminate under Tsai-Wu failure condition. It can be written as follows:
F,, =
/x
(11)
To verify the present method, optimum solutions of [0], and [0,/6& laminates subjected to uniaxial tensile loading are sought. It is obvious that the maximum strength of the laminates occurs when the all layer angles are 0” and the minimum strength of the laminates occurs when the all layer angles are 90”. The same results are obtained in this study as shown in Fig. 4 and Table 2 (case 1). Figure 4 shows three dimensional plot and contours of the objective
To demonstrate the validity of the method developed, the worst solutions as well as optimal solutions are presented. The load carrying ability factor, F,, is defined by the square root of the ratio of worst failure index to optimal failure index. The load carrying factor implies
- - Q4.641
F.D.Y(f) F.D.M.(b) ---O-- F.D.M.(c) _ adjolnt variable
-90
r” > = f $
-90 4.637
-45 0
45 4.633 l(-J-'o1o-~lo-* lO~'lO~
lo"lo~lo~lo"lo"lo"
Step Size Fig. 3.
90 -90
-45
An&
45
Fig. 4.
90
Angle
Stacking
287
sequence optimization of laminated plates
Table 2. Optimum and worst results of symmetric laminated plates under in-plane loading Case
No. of layers
1 2 3 4 5
2 4 2 4 2 4 2 4 2 4
Loading ratio (N,, N,, N,)
Optimum
Worst angle
angle
Wls
(LO, 0)
‘9K01s y$
(1, 2,O) (0, (1, 1)
[45/45\3,
Pls WOIS
(1, 4, 0)
[-31.71, [-31.71-31.71,
(1, 2, 1)
function with respect to design variables under uniaxial loading. It can be seen in the figure that minimum failure index is found in [Q],, laminate while [O,/@& laminate has maximum failure index. As it is expected, F,, is 375 which is exactly same as the ratio of transverse strength to longitudinal strength. Other calculations are carried out in-plane biaxially loaded plates using (N,:N, = 1:2). This loading condition is the same as the cylindrical pressure vessel under internal pressure. The worst and optimal stacking sequences are presented in Table 2 (case 2). It can be seen that the optimum solutions are close to the [ +54*7], laminate which is the optimum result from the netting analysis of cylindrical pressure vessel subjected to internal pressure. In comparison with the worst solution, the optimally designed plate can carry as much as 13.3 times greater external loads. The worst and optimum solutions under various loading conditions [(N,, NY, NT) = (0, 0, l), (1, 4, 0), and (1, 2, l)] are evaluated and presented in Table 2. It is found that the optimized laminate can carry external loads more than ten times greater than the worst designed laminate under a shear loading (case 3). In cases 4 and 5, the load carrying ability factors are found to be 21. 7 and 28.6, respectively. It is very interesting that optimum solutions vary with number of plies while worst solutions do not change.
F LC
375 37.5 2.0 13.3 :,;7, 4.2 21.7 7.4 28.6
an objective function. The design sensitivity with respect to layer angle is analyzed based on the adjoint variable method. It is found that this method provides more precise design sensitivity than F.D.M. In conclusion, the numerical results show that the failure index is minimized successfully and the reasonable optimal stacking sequence design is sought.
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