Development of a Multiobjective Optimization Procedure for ...

Pergamon Computers Math. Applic. Vol. 29, No. 6, pp. 81-97, 1995

Copyright©1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0898-1221/95 $9.50 ÷ 0.00 0898-1221(95)00010-0

Deve lopment of a Mult iobject ive Opt imizat ion Procedure for Reducing Edge De laminat ion

Stresses in Compos i te Plates

J . M . F E R R E I R A A N D A . C H A T T O P A D H Y A Y Depar tmen t of Mechanical and Aerospace Engineer ing

Arizona Sta te University, Tempe, AZ 85287, U.S.A.

S . J . P R I N G N I T Z Allied Signal Auxil iary Power

Phoenix, AZ 85010, U.S.A.

(Received September 1993; accepted January 1994)

A b s t r a c t - - T h i s paper addresses the development of an optimization procedure for the reduction of interlaminar stresses in composite plates. The goal is to reduce the interlaminar stress trends near the free edges of composite plates subjected to single and combined loadings. A simplified analytical approximation is used for predicting the free edge delamination stresses. Ply orientations are used as design variables and constraints are imposed on the in-plane material-axis ply stresses, the buckling load and the interlaminar normal stress interior to the free edge of the plate. The problem is formulated with multiple design objectives and the Kreisselmeier-Steinhauser multiobjective optimization technique is used to solve the nonlinear problem. The procedure yields significant reductions in the interlaminar stresses. Results are presented and compared with a reference or baseline design for four Gr /Ep composite plates with symmetric ply arrangements.

N O M E N C L A T U R E

a,b gk(¢) h {k) m

n

S

{s} X, y , Z

Z

[A] [B] [D]

plate dimensions, in

constraint functions

lamina thickness, in

laminate curvature vector, in-1

number of buckle half-waves in the longitudinal direction of the plate

number of buckle half-waves in the transverse direction of the plate

laminate thickness, in

off-axis stress vector, lb/in 2

reference axes

coordinate distance from laminate mid-plane to ply surfaces, in

extensional stiffness matrix, lb/in

coupling stiffness matrix, lb-in/in

bending stiffness matrix, lb-in

E1

E2

EriC) F*(~) F°(~)

FKS(~) G12

{M}

(N) N

NCON

NDV

longitudinal elastic modulus of composite, lb/in2

transverse elastic modulus of composite, lb/in2

objective functions

reduced objective functions

value of ith objective function at the beginning of an iteration

composite K-S objective function

in-plane shear elastic modulus of composite, lb/in 2

resultant moment vector, lb-in/in

resultant force vector, lb/in

total number of constraints and objective functions

number of constraints

number of design variables

The authors acknowledge partial support of this research Number NSF MSS9209961.

by the National Science Foundation through a grant,

Typeset by ~4A~S-TEX

CAHWA 29-6-G 81

82 J . M . FERREIRA et al.

NOBJ number of objective functions

Nx buckling load, Ib/in

Nzcr critical buckling load, lb/in

[Q] off-axis stiffness matrix, lb/in 2

[T] transformation matrix

{e °} laminate mid-plane strain vector

tJ12 major Poisson ratio

q) design variable vector

p K-S function scalar multiplying factor C

{a} material-axis stress vector, lb/in 2 L

O ' I T longitudinal tensile strength, lb/in 2 T

alC longitudinal compressive strength, U lb/in 2

Cr2T

a2C

az , ' r z z , Tyz

T12S

o

SUBSCRIPTS

1, 2

transveme tensile strength, lb/in 2

transveme compressive strength, lb/in 2

interlaminar stresses, lb/in 2

in-plane shear strength, lb/in

ply angle

longitudinal and transverse fiber directions

compressive

lower bound

tensile

upper bound

I N T R O D U C T I O N

The role of composite materials in structural applications is rapidly increasing due to their reduced-weight and high-strength capabilities. The aerospace and automotive industries are the most frequent users of composite materials in their vehicles. To safely utilize these materials in an effective and efficient manner, the designer must be able to predict the stress configurations that occur for a particular loading condition. Classical lamination theory provides the necessary tool for characterizing the planar strength and stiffnesses of composites, however, it lacks the capability of predicting interlaminar stresses. The ability to predict and control out-of-plane stresses is a critical issue with composites because these stresses are directly attributed to causing delamination; a primary failure mechanism unique to composites. Significant reductions in failure strength can occur when delamination is present in a composite. This is very detrimental because it leads to failures which occur at a level below that predicted by classical lamination theory.

To date, a practical method for evaluating free edge stresses, mainly az, has been difficult at best. The only way to truly quantify this stress is with a 3-D finite element model utilizing a very fine mesh at the free edge. Such an approach is impractical during the design phase where numerous ply lay-up configurations are being considered. Consequently, the effect of ply lay-up on free edge stresses is often ignored and methods such as edge trimming or stitching are employed. Also, a 3-D finite element-based procedure would be computationally prohibitive within a design optimization procedure where several calls to the analysis routines are involved and both the function values and the sensitivities are necessary in the gradient-based technique.

The two-dimensional stress state assumed in classical theory is quite accurate for predicting stresses at regions away from free edges (e.g., holes, exterior edges of a plate). In the vicinity of a free edge, the stress state becomes three-dimensional [1-3] and classical theory is unable to predict out-of-plane stresses. The three-dimensional equations that govern the free edge stress configuration are coupled second-order partial differential equations that do not have a simple closed-form solution. Therefore, detailed finite element analysis is typically used to evaluate these stresses [2-6]. These methods are computationally prohibitive and are not easy to incorporate in a design study. As a result, some studies have been performed on developing simpler approaches that allow the relative quantification of interlaminar stresses near free edges [7-12]. One of these methods, recently explored by Pringnitz [12], is quite capable of predicting trends in the free edge delamination stresses of composites and is used in this research. The purpose of the present paper is to develop an optimization procedure, for reducing the edge delamination stresses in composite plates using the analysis methodology presented by Pringnitz.

Multiobjective Optimization 83

P R O B L E M S T A T E M E N T

This paper addresses the development of a formal optimization procedure to minimize the free edge interlaminar stresses in composite plates subjected to uniaxial compression, bending and combined compression and bending. A practical approach is used to evaluate the relative free edge stresses for a given ply lay-up under various loading conditions. The objective is to reduce the interlaminar stresses that occur near the free edges of each lamina by using geometric design variables. To maintain structural integrity of the plate, constraints are imposed on the individual ply longitudinal, transverse and in-plane shear stresses, the interlaminar normal stress interior to the free edge of the plate and the buckling load. Plates made of symmetric orthotropic laminates are considered. By varying the individual ply orientations, specific combinations are obtained that reduce the interlaminar stresses with respect to a baseline or reference design. A structural analysis procedure, based upon classical laminate theory, is used for the analysis. The problem is formulated with multiple design objectives so a multiobjective formulation technique is used to solve the optimization problem.

S T R U C T U R A L MODEL

A geometric illustration of a typical orthotropic composite plate used in this study is presented in Figure 1, along with a three-dimensional positive stress state. Each plate consists of a number of laminated plies made of Graphite/Epoxy. The total number of plies used to make up the plate thickness, s, are numbered starting from I through the final ply in the laminate, with the first ply being the top most one. Symmetric laminates are used to reduce the number of design variables. The plates all have a length, a, of 10in. and width, b, of 10in. (a/b = 1). A representative value of 0.01 in. is used for the thickness of each ply. Three different baseline or reference lay-ups consisting of eight plies with ply orientations of [(15)2/(30)21s, [(q-45)2]s and [(-}-30)/(=i=45)]8 are optimized. The properties of Gr/Ep that are used in this study are presented in Table 1.

Table 1. Material properties of Gr/Ep.

Longitudinal modulus, E1 (psi) Transverse modulus, E2 (psi) Shear modulus, G12 (psi) Major Poisson ratio, v12 Longitudinal tensile strength, O'lT (psi) Longitudinal compressive strength, ale (psi) Transverse tensile strength, Cr2w (psi) Transverse compressive strength, a2c (psi) Shear strength, ~12s (psi)

22,200,000 1,580,000 810,000

0.3 100,000 110,000 4,000 13,900 22,200

ANALYSIS

In this section, a brief description of the structural analysis procedures is provided. The laminate analysis theory, which includes interlaminar stress calculations, is presented first, followed by the failure and the buckling analysis theory.

Laminate Analysis

IN-PLANE. The individual material-axis ply stresses of the orthotropic composite plates are analyzed using classical laminate theory [13,14]. The constitutive relations provided by this theory, relating ply strains and curvatures to the resultant forces and moments, are as follows:

{N} = [A] {¢0} + [B]{k},

{M} = [B] {¢0} + [D] {k}.

(1) (2)

84 J.M. FERREIRA et al.

The relations simplify, in the case of uniaxiai compression, with all elements of the matr ix {M} being zero and the only nonzero element in matr ix {N} is the axial compressive stress Nz. In the case of a bending moment, all elements of the matr ix {N} are zero and the only nonzero element in matrix {M} is the applied bending moment Mx. The constitutive equations

are solved to obtain the mid-plane strains and the curvatures of the plate. The individual off-axis and material-axis ply stresses are evaluated from the strain and curvature values as follows:

{s} = [0] { : } + z [0] {k}, {u} = [Tl{s},

(3) (4)

where the positive direction of the z-axis is as shown in Figure 1.

w I ~ b ~---I

I/' !

(Y Z

XZ

cy

(y x "C

xy

Figure 1. Typical composite plate mad stress configuration.

(y Z

Z

Approximate ~ / / / 7

/ ~ Y

115 ~z,

-~ 213 s ..~ !i-.~-- I]3 s ----~', Compression i

(b/2) - s b/2

Figure 2. FEM and approximate interlaminar normal stress distribution.


0 yl

Ozl T. ~vzl V, Ozl

Y

o y2

o y3

Oy4

Oz2 T "Cvz2 T Oz2

Oz3 z3 ~yz3 O

0 ~ 0 z4 yz4 z4

Figure 3. Free body diagram of laminate at the free edge.

OUT-OF-PLANE. The distribution of stresses along the free edge of a composite plate has been evaluated in numerous works and validated with finite element analyses. Pagano and Pipes [1] evaluated the interlaminar stresses and showed that az and Txz exhibit singularity, possibly approaching infinity at the free edge. Therefore, a: and Txz are the delamination stresses of concern in composites with free edges. The stress distribution of az near the free edge has been characterized by finite element analysis. Additionally, Pagano and Pipes [1] developed a linear approximation to the interlaminar normal stress distribution, which is overlaying the finite element distribution in Figure 2. This approximation assumes that when a tensile value of interlaminar normal stress, az, exists at a free edge, a compressive interlaminar stress of magnitude 1/5az exists at points interior to the free edge and vice versa. Numerous studies have also shown that the extent of these edge effect stresses is confined to a distance from the edge equal to the thickness of the laminate, s.

Research performed by Pringnitz [12] expanded on that of Conti and De Paulis [7] to develop simplified governing equations for the interlaminar stresses at a free edge. Figure 3 shows a free body diagram of the top four plies of a laminate at a cross section removed from the loading ends where the stress is essentially independent of the axial coordinate, x. It includes interlaminar stresses and the stress predicted by classical lamination theory. By assuming the interlaminar normal stress approximation as shown in Figure 2, cz can be represented as a distribution of rectangular and triangular areas along the y-direction, with equivalent forces expressed as the stress times the area acted upon. Figure 4 shows a more detailed free body diagram of the edge of the top ply in Figure 3, which includes the stresses T=y and T=~. Summing the moments of all the forces in the first lamina about point A, allows az to be defined as a function of ay at the first ply interface as follows:

~--~M~ = 0:

O'yl -[- ~'~ O'zl 8 --'i'~(7z18 "~8 -- -i--~ O'zl S 8 : 0 , (5)

az l = y ayl , (6)

where h is the thickness of the lamina (ply) and s is the thickness of the laminate. The shear stress Tuz is zero at locations away from the free edge and at the edge itself, but has

some finite value in the area near the free edge. Determination of the magnitude of %z is not the focus of this study. Therefore, for evaluating a~, %~ is assumed to be constant. Likewise, Txz is zero everywhere except in the vicinity of the free edge and for simplicity is assumed to have

86

%1 /

xyl

J. M. FERREmA et al.

s

(~z] / "C "~xzl yzI

Figure 4. Free body diagram of top lamina at the free edge.

pt. A

a constant distribution. Summing the forces in the y-direction (Figure 4) defines an expression for T~z at the first ply interface as follows:

~-~ F~ = O :

- zl(S) = 0 , ( 7 )

Likewise, summing the forces in the x-direction (Figure 4) defines an expression for Wxz at the first ply interface as shown below:

~--'~ Fx = 0 :

~'xyl(h) - Txzl(8) = O, (9)

If this same procedure is applied to the subsequent lamina, from the outer plies to the laminate center, it is observed that the governing equations for the interlaminar normal stress, az, and the interlaminar shear stresses, Tyz and Txz, simplify to:

Crz~ = -~- ~ [(2i - 2j + 1)~u~], (11) j = l

i

j = l

i

j = l

where the subscript 'i ' refers to the interface between the ith and the i + I th laminae (i.e., i = 1 refers to interlaminar stresses occurring between plies 1 and 2). These interlaminar stresses are functions of geometric parameters and in-plane ply stresses, which are evaluated using classical lamination theory. The relations are formulated such that they apply to a free edge on a com- ponent under any loading case. Equation (11) shows that the lamina interface normal stress, az, is a function of transverse normal stress, ay, in all the laminae outboard of the interface of interest. In addition, the effect of each lamina is weighted such that the outer laminae have significantly greater influence on the interlaminar normal stress. This is very critical in the design of a laminate which controls the delaminating normal stress.

Failure Analys is

Material failure will occur in a composite plate if the longitudinal, transverse or in-plane shear stress of an individual ply exceeds the allowable strengths of the material. Therefore, to avoid this,


the individual ply stresses must be constrained against failure. The interaction failure criterion presented by Tsai and Wu [15] is used in this research. The general form of this relation, for a plane stress condition, is*as follows:

( 1 1 ) ( 1 1 ) cr12 ffla2 ~2 ./.122 a~T a~'C a l + a2"T o';c a2-l I- - - - - < 1. (14) tilT ¢rlC ~/6rlT O'IC 6r2T O'2C O'2T ff2T -}- T122S

Material-axis ply stresses along with their respective strengths are used in this relation. A value of less than one indicates a safe ply stress configuration.

Buckling Analysis

The critical buckling load of the plate represents the value of load amplitude, N~, which is a minimum with respect to variations in buckling modes. For plates subjected to uniaxial compression, the applied axial load must not exceed this critical value if structural stability is to be maintained. The values of the critical buckling load obtained in this paper are based on the form of the stability criterion as presented by [13]:

(15)

where,

D1 = Dn, (16)

D2 = D22, (17)

D3 = 2 (Dr2 + 2D68) • (18)

The critical buckling load, Nxcr, is obtained by varying m and n (the number of buckle half- waves in the longitudinal and transverse directions of the plate, respectively) until a combination is found that produces the lowest load, Nx. The boundary conditions for the case of a simply supported plate are assumed in this analysis.

OPTIMIZATION F O R M U L A T I O N

The formal optimization problem can be mathematically stated as follows:

Minimize: Fi (¢j) ,

subject to: gk (¢j) _< 0

¢iL < ¢i <_ ¢~u

i = 1 , . . . ,NOBJ

j = 1 , . . . ,NDV

k = 1 , . . . ,NCON

(objective functions)

(inequality constraints)

(side constraints),

where • is the design variable vector, NOBJ is the number of objective functions, NDV is the number of design variables, NCON is the total number of constraints and the subscripts L and U correspond to the lower and upper bounds imposed on the design variables. Side constraints are imposed on the design variables to avoid unrealistic designs.

Kreisselmeier-Steinhauser (K-S) Function Approach

To solve the multiobjective nonlinear optimization problem, the Kreisselmeier-Steinhanser multiobjective optimization technique [16] is employed. This method was chosen because it was found to be efficient in practical and highly nonlinear design environments [17].

88 J . M . FERREIRA et al.

The use of this technique requires that all of the original objective functions must first be converted to reduced objective functions of the following form:

FO 1 - gmax _< 0, i = 1 , . . . , N O B J, (19)

where • is the design variable vector, F ° is the value of the ith objective function, Fi, at the beginning of an iteration and gmax is the value of the largest violated constraint. These reduced objective functions are now additional constraints in the problem and, therefore, a new constraint vector must be defined as fn(¢) , n = 1 , . . . ,N, where N = NCON + NOBJ. This new vector includes the reduced objective functions equation (19) and the original inequality constraints.

The reduced objective functions, along with the original problem constraints, are now combined to form an envelope of the entire function set using the K-S function as follows:

N

FKS('~) = fmax + I log e E eP(/~(~)-Ym~x)' (20) P n = l

where fmax is the maximum value of the new constraint vector, fn(O), and p is analogous to a draw down factor tha t controls the distance from the K-S function surface to the surface of maximum function value. When p is large, the K-S function will closely follow the surface of the largest constraint function. When p is small, the K-S function will include contributions from all violated constraints. This envelope function is then searched for a minimum using an unconstrained optimization technique.

Objective Functions

The overall objective of the optimization study is to minimize the interlaminar stress trends near the free edge of a composite plate, subjected to various loading conditions. Therefore, the interlaminar stresses, az, T~z, and ~-~, that occur at each ply interface are the individual objective functions to be minimized. The total number of objective functions for a particular laminate will be three times the number of ply interfaces in the plate.

Constraints

To maintain structural and material integrity of the plate the following constraints are imposed

on the problem:

(i) Failure occurs when at least one unidirectional layer looses its strength by exceeding the allowable stress limit. Therefore, the Tsai-Wu interaction failure criterion is used to constrain the in-plane material-axis stresses, a l , a2, and 7-12 , of each ply in the laminate. Since symmetric laminates are used, only the stresses in half the plies are constrained.

(ii) Since the optimization problem is formulated for minimizing az at the free edge, theoreti- cally, the procedure can drive this value to negative infinity, in which case the stresses just inboard of the free edge would be positive infinity. To preclude this situation, constraints are necessary on the stresses inboard of the free edge. When the interlaminar normal stress near the free edge of the plate is in a state of compression a delaminating tensile stress of 1/5az occurs between plies, at points interior to the free edge (Figure 2). To prevent interior ply delamination a constraint is imposed so that the magnitude of 1/5az remains less than the transverse tensile strength of the composite, a2W.

(iii) In the case of a plate subject to a uniaxial compressive load, an instability occurs if the applied load exceeds the critical buckling load of the plate. To prevent buckling, a constraint is placed upon the critical buckling load, Nxcr, of the plate so that it remains higher than the applied load, N~.


Design Variables

The design variables selected in this problem are the individual ply orientations in the laminate. Since symmetric laminates are used, only orientations of half the plies are used as design variables.

O P T I M I Z A T I O N I M P L E M E N T A T I O N

The optimization process is initiated by defining all the necessary preassigned parameters (e.g., plate dimensions and ply thicknesses) of the problem. Next, the design variables and the optimization parameters are initialized and the structural analysis is performed. The structural analysis consists of an in-house code that calculates the in-plane stresses, the interlaminar stresses and the buckling load based on classical lamination theory. The objective functions and constraints are then evaluated followed by a sensitivity analysis, in which the method of finite differences is used to compute all gradients. The multiobjective optimization problem is formulated using the K-S function method. A sequential unconstrained minimization technique, as implemented in the computer-code KSOPT (see [16]), is used for the optimization. Convergence is based upon the K-S objective function value over three consecutive iterations, where an iteration comprises a complete analysis and optimization. A convergence tolerance of 0.005 is used.

R E S U L T S A N D D I S C U S S I O N

Results obtained using the above optimization procedure are presented in this section. A total of three Gr/Ep plates consisting of eight plies with orientations of [(15)2/(30)2]s, [(i45)21s, and [(±30)/(=t=45)] s are optimized. The first plate is subjected to a uniaxial compressive load of 1201b/in, the second plate is subjected to a bending moment of 501b-in/in and the third plate is subjected to a combined uniaxial compressive load and bending moment of 120 lb/in and 50 lb-in/in, respectively. The above configurations are chosen because they produce high interlaminar stress values and the three different load conditions are selected to test the ability of the optimization procedure under single, as well as, combined loading conditions. All the optimized results are compared against the reference or baseline designs. In each case, optimum configurations for minimum interlaminar stresses are obtained in 6-33 cycles.

All three optimized plate configurations satisfy the in-plane ultimate strength, the buckling and the interior interlaminar strength requirements and have significantly reduced interlaminar stress magnitudes at the end of each optimization process. The results of the optimization are summarized in Table 2 and Figures 5-13. The normal strength data are used as upper bounds due to the lack of available data. This should not have any effect on the relative ply angle comparison. Table 2 presents the interlaminar stresses (az, 7yz and rxz) at the ply interfaces of both the reference and the optimum plates for the three different loading conditions. Since each interlaminar stress represents an objective function, all three plates have a total of 21 individual objective functions that are to be minimized. In all three cases, the interlaminar stresses at each interface are reduced, except for Ty~ at the 3-4 and 5-6 ply interfaces in the case of the plate subjected to a bending moment. The nature of the interlaminar stresses change, in some cases, from tensile to compressive and vice versa after optimization. For example, in the case of the plate subjected to a combined loading the value of Tzz at the interface between plies 3 and 4 is tensile in the reference plate and compressive in the optimum plate. This is of advantage since compressive interlaminar stresses help to prevent delamination. However, not all the interlaminar stresses are compressive after optimization, and therefore, by reducing the magnitude of the interlaminar stresses the onset of delamination can be prevented. By examining Table 2, it is also very clear that az and Txz are the dominating stresses (in magnitude) in each plate, as suggested by previous studies [1-3].

90 J. M. FERREIRA et al.

Table 2. Interlaminar ply stresses of the reference and optimum plates.

Compression Bending Combined

Ref. Opt . Ref. Opt . Ref. Opt .

a z (psi) -11 .1 0.2 406.7 66.1 -114 .2 - 3 . 1

1-2" T~z (psi) --13.8 0.2 506.1 82.2 --142.1 --3.9

• rxz (psi) --37.1 2.0 --1,494.3 --160.3 --2,240.1 338.7

crz (psi) --44.2 --0.4 580.9 221.3 --863.7 --22.1

2--3* r~z (psi) --27.5 --1.0 --289.2 110.9 --790.8 --19.9

Tzz (psi) --74.2 --6.3 853.9 199.0 426.4 --98.4

a z (psi) --77.4 --1.3 290.5 242.9 --1,174.7 --21.9

3--4* Tuz (psi) --13.8 --0.2 --72.3 --84.0 403.9 19.7

T z z (psi) --37.1 --1.4 213.5 --77.3 1,033.5 --159.3

a z (psi) --88.4 --1.4 0.0 0.0 --266.5 12.8

4--5* VVz (psi) 0.0 0.0 0.0 0.0 0.0 0.0

~'xz (psi) 0.0 0.0 0.0 0.0 0.0 0.0

az (psi) --77.4 --1.3 --290.5 --242.9 708.3 30.2

5--6* Tuz (psi) -- 13.8 --0.2 72.3 84.0 --486.8 1.6

~'xz (psi) --37.1 --1.4 --213.5 77.3 --1,051.0 133.6

Crz (psi) --44.2 --0.4 --580.9 --221.3 597.3 16.5

6--7* "ruz (psi) --27.5 --1.0 289.2 --110.9 624.9 15.5

Tzz (psi) --74.2 --6.3 --853.9 --199.0 --426.4 77.3

a z (psi) --11.1 0.2 --406.7 --66.1 47.5 2.0

7--8* r y z (psi) --13.8 0.2 --506.1 --82.2 59.2 2.5

~'xz (psi) --37.1 2.0 1,494.3 160.3 1,950.6 --297.4

• laminate interface between these number plies.

Ref. Opt. Ref. Opt. Ref. Opt. Ref. [ Opt. Ref. Opt. Ref. Opt. Ref. Opt.

-90

Referetw.e orientation: [(15)]00)2] s

-80 -70 -60 -50 -40 -30 -20 -10 0 10

Op6.murn orientation: [3/7/28/4]s

• Interface 1-2

[ ] htcffacc 2-3

[ ] Imcffacc 3-4

[ ] Interface 4-5

[ ] Interface 5-6

[ ] Interface 6-7

!~ Interface 7-8

Figure 5. Inter laminar normal stress, az, magni tudes of 8-ply plate subjec t to compression.

Ref.

opt Ref. Opt Ref.

Opt Ref.

Opt Ref. Opt. Rcf. opt Rcf.

OPt

Multiobjective Optimization

Reference orientation: [(15)2/(30)21 s Optimum orientation: [3/7/28/4] s

[ ] Interface

[ ] Interface

[ ] Interface

[ ] Interface

[ ] Interface

[ ] Interface

[ ] Interface

-30 -25 -20 -15 -10 -5 0 5

(Ibfm 2) yz

Figure 6. Interlaminar shear stress, TVz , magnitudes of 8-ply plate subject to compression.

1-2

2-3

3-4

4-5

5-6

6-7

7-8

91

Ref. Opt Ref. Opt. Ref. Op:. Ref.

OP~ Ref. OPt Ref. Opt Ref.

Op~ m

-80

Reference orientation: [(15)2/(30)2] s

-70 -60 -50 -40 -30 -20 -10 0 10

Optimum orientation: [3/7/28/4] s

[ ] Interface 1-2

[ ] Inmrface 2-3

[ ] Inmrface 3-4

[ ] Inmrface 4-5

[ ] Interface 5-6

[ ] Inmrface 6-7

[ ] In,trace 7-8

Zxz 0bfm2)

Figure 7. Interlaminar shear stress, rxz, magnitudes of 8-ply plate subject to compression,

The percent change in the magnitudes and the distribution of the interlaminar stresses through the laminate, from reference to optimum, are presented in Figures 5-13. Figures 5-7 present the results for the plate subjected to a uniaxial compressive load of 120 lb/in. The ply orientation of the reference plate is [(15)2/(30)218 and the optimized plate has a ply orientation of [3/7/28/4]8. It can be seen that the distribution of the interlaminar normal and shear stresses are symmetric with respect to the mid-plane of the laminate before, as well as, after optimization. This trend is due to the fact that a symmetric orientation is used for the composite layup and an axial load is applied to the plate. The largest interlaminar normal stress occurs at the 4-5 ply interface of the


reference and the optimized plate (Figure 5) and is a result of the weighted nature of the governing equation equation (11) used to calculate az. The largest interlaminar shear stresses occur at both the 2-3 and the 6-7 ply interfaces (Figures 6 and 7) in the reference and the optimized plate. Also, the reference and the optimum plates exhibit no shear stresses at the 4-5 ply interface. Under this loading condition, significant reductions are obtained in all the interlaminar stresses. Reductions in magnitudes in the range of 98% to 99% (Figure 5) are obtained for the interlaminar normal stress, az. The interlaminar shear stresses, Tyz and Tzz, are reduced in magnitude in the range of 96% to 98% and 91% to 96%, respectively, as shown in Figures 6 and 7.

Ref. Opt_ Re'f. Or,r- Ref. op'.. Re*:.. Or,-.- RE.

R ~ . [~ Op:- Ref. orr-

-600

Reference orientation: [(+45)2] s optimum orientation: [2/-6145/..42]s

0~ i

62%

~ / / / / / / / / / / / / / / / / / / / A : 62%

X)OZKXXX~ 16%

84~ -400 -200 0 200 400 600

• Interface 1-2

[ ] Inr.erfaee 2-3

[ ] Interface 3-4

[ ] Interface 4-5

[ ] Intzrface 5-6

[ ] Interface 6-7

[ ] Interface 7-8

Oz Obfm-)

Figure 8. Interlaminar normal stress, O'z, magnitudes of 8-ply plate subject to bending.

Ref. Opt. Ref. Opt. Ref. Opt. Ref. Opt. Ref. Opt. Ref. Op,. Ref. opt.

-600

Reference orientation: [(+45)2] s Optimum orientation: [2/-6145/-42] s

÷16%

0%

62~ KK'K'

~ r - ~ -46o -zbo

62%

m

m

\ \ \ \ \ \ \ \ \ ' 4

I 0 2D0 400 600

• Interfaee~

[ ] Interfac,e~

[ ] Interfac-e

[ ] Interface,.

[ ] Interface

[ ] Interface

[ ] Inteffa~

,1

"~yz tlb/in-)

Figure 9. Interlaminar shear stress, ruz, magnitudes of 8-ply plate subject to bending.

1-2

2-3

3-4

4-5

5-6

6-7

7-8


Refi o~t. Refi

Ref. Opt. Refl Optl Ref.

Ref. Opt. Ref. opt.

-1500

Reference orientation: [(+45)2] s Optimum orientation: [2/-6/45/-42] s

89% [] 7 / / i 11 / / / / / i / / / / 111 ,

64~

O~ l l m m

77% k '~

77%

lll] 89%

- 600 -sbo 0 5 6 0 500

• Interface I-2

[] Interface 2-3

[] Interface 3-4

[] Interface 4-5

[] Interface 5-6

[] Interface 6-7

[] Interface 7-8

~xz (Ibym2) Figure 10. Interlaminar shear stress, T z z , magnitudes of 8-ply plate subject to bending.

Figures 8-10 present the results for the plate subjected to a bending moment of 50 lb-in/in with a reference ply orientation of [(~45)218. The optimized plate has a ply orientation of [ 2 / - 6 / 4 5 / - 42]s. The distribution of the interlaminar stresses are still symmetric (in magnitude) with respect to the mid-plane of the plate, before and after optimization, as with the axial compressive case. However, due to the compressive state that exists in the top half of the laminate and the tensile state that exists in the lower half of the laminate, the interlaminar stresses are opposite in sign. The highest interlaminar normal stress in the reference plate now occurs at the 2-3 and 6-7 ply interfaces (Figure 8) because of the alternating stress states through the laminate. Once optimized, the largest stress occurs at the 3-4 and 5-6 ply interfaces in the plate. The highest interlamina~ shear stresses occur at both of the outer ply interfaces in the reference plate (Figures 9 and 10) and after optimization occur at the 2-3 and 6-7 ply interfaces. No interlaminar stresses are exhibited at the mid-plane interface of the laminate, before and after optimization, because in-plane stresses do not exist on the neutral axis of a plate in pure bending. Again, significant reductions of the interlaminar stresses are obtained in all but one instance. The interlaminar normal stress is reduced in magnitude in the range of 16% to 84% (Figure 8). This trend in the reduction of the interlaminar normal stress results due to the weighted nature of the governing equation. The interlaminar shear stress, Tyz, is reduced by 62% and 84%, as shown in Figure 9. At the 3-4 and 5-6 ply interfaces, however, a 16% increase occurs. This is perhaps due to the use of the K-S function formulation technique where an envelope function, which has contributions from each of the individual objective functions, is minimized. The optimizer has possibly reached a local minimum of the envelope function which does not guarantee minimization of all the individual objective functions, but rather the best weighted solution. The magnitudes of ~-~zare reduced in the range of 64% to 89% (see Figure 10).

To examine the effects of multiple loading on interlaminar stresses, a combined case is also examined where a compressive load of 1201b/in and a bending moment of 50 lb-in/in are si- multaneously applied to the plate. The ply orientation of the reference plate is [+30)/(±45)] 8 and the optimized plate has a ply orientation of [ - 5 / 5 / 6 7 / - 49]~. Results from this case are presented in Figures 11-13. Combining the compressive load with the bending moment produces a similar type of interlaminar normal stress distribution trend that occurs in the pure bending


Ref. Op~ Ref.

OP~ Ref. OP~ Ref. Op~ Ref. OPT. Ref.

OP~ Ref.

OP~

Reference orientation: [(+_30)/-2_45)] s

[ ] 97%

~//////////////////,~/////2 97%1

Optimum orientation: [-5151671..49] s

95%

97~

• 96%

[ ] Interface 1-2

[ ] Interface 2-3

[ ] Interface 3-4

[ ] Interface 4-5

[ ] Interface 5-6

[ ] Interface 6-7

[ ] Interface 7-8

-1200-1000-800-600-400-200 0 200 400 600 800

a z 0b/in 2)

Figure 11. Interlaminar normal stress, az, magnitudes of 8-ply plate subject to compression and bending.

Ref. opt. Ref. opt. Ref. Opt. Ref. Opt. Ref. Opt. Ref. Opt. Ref. opt.

-8oo "

Refe:ence orientation: [(*-30)~-45)] s

J 97%

~ / / I / / / / / / / / / / / / / / / / ~ 98%

9 s ~

0%

~ 9 9 % k \ \ \ \ \ \ x ~ \\.\\\\.-x~

~996 % %

-400 - 2 ~ 0 200 400 660

Optimum orientation: [-5151671-49]s

• Interface 1-2

[ ] Interface 2-3

[ ] Interface 3-4

[ ] Interface 4-5

[ ] Interface 5-6

[ ] Interface 6-7

[ ] Interface 7-8

800

X.v z (lb] in2)

Figure 12. Interlaminar shear stress, Tyz, magnitudes of 8-ply plate subject to compression and bending.

case (i.e., compressive values in one half of the laminate and tensile values in the other). How- ever, the combined loads produce an eccentric stress resultant with respect to the mid-plane of the laminate. As a result, the magnitudes of the stress distributions for both the interlaminar normal and shear stress are not symmetric in nature as with the single loading cases. Also, a value of interlaxainar normal stress of significant magnitude now appears at the mid-plane of the laminate. Interlaminar shear stresses also appear at the mid-plane but their magnitudes are so low in value that they are negligible. The largest interlaminar normal stress occurs in the upper half of the reference laminate, at the 3-4 ply interface, and after optimization it occurs in the


Reference orientation: [(+_30)/'2:45)] Optimum orientation: [-5/5/67/-49] s

Ref. opt. Ref. Opt. Ref. Opt. Ref. opt. Ref. Opt. Ref. Opt. Ref. OpL ss,'. r - '=

-2500.2doo-l~00-IdOO-5bo o

85% i~

0%

m

~ K , : : ' : 9 ' ~

II Z7~

82~

560 l O'OO 5'oo 2000

• Interface I-2

[ ] Interface 2-3

[ ] Interf~,e 3-4

[ ] Imerfa~e 4-5

[ ] Interf .a~e 5-6

[ ] Irtterf .a~e 6-7

[ ] Interface 7-8

~xz (Ibfm2)

Figure 13. Interlaminar shear stress, rxz, magnitudes of 8-ply plate subject to compression and bending.

lower half of the laminate at the 5-6 ply interface, as shown in Figure 11. The highest interlaminar shear stresses, Tyz and Tzz, also occur in the upper half of the reference and optimized laminate at the 2-3 and 1-2 ply interfaces, respectively. As in the uniaxial compressive load case, significant reductions are obtained in all of the interlaminar stresses during optimization. Reductions in magnitude in the range of 95% to 98% are achieved for the interlaminar normal stress (Figure 11). The magnitudes of interlaminar shear stress, ryz, are reduced in the range of 95% to 99% (Figure 12) and rzz are reduced by 77% to 87% from their original values, as shown in Figure 13.

Table 3. Comparison of design variables.

Compression Bending Combined

Ref. Opt. Ref. Opt. Ref. Opt.

01 (degrees) 15 3 45 2 30 -5 02 (degrees) 15 7 -45 -6 -30 5 03 (degrees) 30 28 45 45 45 67 04 (degrees) 30 4 -45 -42 -45 -49

Table 3 presents the design variable values for the reference and the optimum plates. Only half the ply orientations (0i) are presented, due to conditions of symmetry. An interesting trend is observed regarding the ply orientations. For all three plates, optimization significantly reduces the magnitude of the outermost ply orientations from their respective reference values. This suggests tha t smaller ply orientations (magnitude wise) are more suitable for reducing interlaminar stresses. The reason for this can be seen by examining equations (11)-(13). It is noted that the interlaminar stresses are functions of the in-plane ply stresses cry and ~'xu whose magnitudes are governed by ply orientation. In the case of the applied axial compressive load, the interlaminar normal stress is driven by Poisson mismatch. Because of the weight effect placed on the outer plies in equation (11), the optimizer tends to minimize the angle change (i.e., Poisson mismatch) between the outer plies by driving them toward zero. This reduces the magnitude of a u in the outer plies which helps reduce the interlaminar normal stress. Also, the magnitude of

96 J.M. FERrtEZP~ et al.

the dominating bending stiffness term Dll, in the buckling relation equation (15) is increased by having more ply orientations in the laminate closer to zero. This increases the critical buckling load of the plate allowing the buckling load constraint to be well satisfied. Hence, it can be seen, from Table 3, that/94 in the compression case is also reduced to a lower value, in addition to 01 and 02. In the case of the applied moment, the upper plies in the composite are in transverse compression and the lower plies are in transverse tension. Examining equation (11), it must be noted that the interlaminar normal stress, a~, is a function of in-plane transverse stress, au,

and the effect of a u in the outer plies have higher weights. This results in a dominating effect throughout the composite layup. To minimize au (thus reducing az) the outer plies should be at angles closer to zero. In the combined loading case, the same rationale can be applied, and it can be seen that the optimizer adjusts the outer ply orientations in a similar fashion. This confirms that one of the key ways of reducing interlaminar stresses, for these three types of loading conditions, is the use of plies of smaller orientations at the outer surfaces, as shown by Pringnitz [12].

0.9

0.8: ..-~

o

= 07£

6S ~ 0

2 2~ 05 O

0.4:,

• "~ 03- - 2£ o O. Z

0.I

0 0

---B-- Compression

Bending

,L Combined

• A A • A • , A a • A A A • • • A ~ , ~ j , ~

io

Cycle

Figure 14. Kreisselemeier-Steinhauser composite objective function iteration histories.

Figure 14 presents the K-S composite objective function iteration histories for each of the loading cases. Consistent decreases and smooth convergence of the K-S functions demonstrate the ability of the optimization procedure to minimize 21 individual objective functions (interlaminar stresses). The large drops in the K-S function values during the optimization procedure are attributed to the changing value of the scalar multiplying factor, p, which is monitored during optimization and has the effect of drawing the K-S envelop function closer to the surface of maximum function value. The increase in the number of cycles for the combined load case is due to the larger design space in this case.

C O N C L U D I N G R E M A R K S

A procedure has been developed to reduce the interlaminar stresses in composite plates subjected to uniaxial compression, bending and combined compression and bending. An analytical approximation has been used to predict the interlaminar stresses at the ply interfaces. Ply orientations are used as design variables and constraints are placed on the in-plane material-axis ply stresses, the buckling load and the interlaminar normal stress interior to the free edge of the plate. The optimization problem involves multiple design objectives, therefore, the Kreisselmeier- Steinhauser approach has been used to formulate this problem. Results are presented for three


different Gr/Ep plates consisting of eight plies. The procedure yields smooth convergence. The following observations have been made from this study:

(1) Optimization reduced the magnitudes of the interlaminar stresses from the reference values while satisfying all constraints. The uniaxial compressive load case displayed the most significant reductions.

(2) The magnitudes and the nature of the interlaminar stresses in each ply changed significantly from reference to optimum.

(3) Significant changes occurred in the values of the design variables. Ply orientations nearer the outer surface of the plate were reduced the most in magnitude.

(4) The optimization procedure displayed the ability of minimizing a large number of objective functions.

R E F E R E N C E S

1. N.J. Pagano and R.B. Pipes, Interlaminar stresses in composite laminates under uniform axial extension, Journal of Composite Materials 4, 538-548 (1970).

2. N.J. Pagano and R.B. Pipes, The influence of stacking sequence on laminate strength, Journal of Composite Materials 5, 50-57 (1971).

3. N.J. Pagano and R.B. Pipes, Some observations on the interlaminar strength of composite laminates, International Journal of Mechanical Sciences 15, 679-688 (1973).

4. W.E. Howard, T. Gossaxd, Jr. and l:t.M. Jones, Composite laminate free-edge reinforcement with u-shaped caps, Part I: Stress analysis, AIAA Journal 27 (5), 610-616 (1989).

5. W.E. Howard, T. Gossaxd, Jr. and R.M. Jones, Composite laminate free-edge reinforcement with u-shaped caps, Part II: Theoretical-experimental correlation, AIAA Journal 27 (5), 617-623 (1989).

6. W.C. Hwang and C.T. Sun, Failure analysis of laminated composites by using iterative three-dimensional finite element method, Computers and Structures 33 (1), 41-47 (1989).

7. P. Conti and A. De Paulis, A simple model to simulate the interlaminar stresses generated near the free edge of a composite, In Delamination and Debonding of Materials, ASTM STP 876, (Edited by W.S. Johnson), American Society for Testing Materials, Philadelphia, 35-51, (1985).

8. R.R. Valisetty and L.W. Rehfield, A new ply model for interlaminar stress analysis, In Delamination and Debonding of Materials, ASTM STP 876, (Edited by W.S. Johnson), pp. 52-68, American Society for Testing and Materials, Philadelphia, (1985).

9. J.D. Whitcomb, Parametric analytical study of instability-related delamination growth, Composites Science and Technology 25, 19-48 (1986).

10. N.J. Pagano and S.R. Soni, Interlaminar Response of Composite Materials, pp. 1-68, Elsevier Science Publishers B. V., (1989).

11. C. Kassoapoglou, Determination of interlaminar stresses in composite laminates under combined loads, Journal of Reinforced Plastics and Composites 9, 33-58 (1990).

12. S.J. Pringnitz, Prediction of free edge delamination stresses in composite laminates, M.S. Thesis, Arizona State University, (1991).

13. J.l:t. Vinson and R.L. Sierakowski, The Behavior of Structures Composed of Composite Materials, Kluwer, Dordreeht, (1987).

14. B.D. Agarwal and L.J. Broutman, Analysis and Performance of Fiber Composites, John Wiley, USA, (1980).

15. S.W. Tsai and E.M. Wu, A general theory of strength for anisotropic materials, Journal of Composite Materials 5, 58-80 (1971).

16. G.A. Wrenn, An indirect method for numerical optimization using the Kreisselmeier-Steinhauser function, NASA CR-~P20 (1989).

17. A. Chattopadhyay and T.R. McCarthy, Multiobjective design optimization of helicopter rotor blades with multidisciplinary couplings, Presented at the Proc. OPTI 91-Computer Aided Optimum Design of Struc- turcs, Boston, MA, (June 1991).

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