A method for optimally designing laminated plates subject to fatigueloads for minimum weight using a cumulative damage constraint

M. Walker *

Centre for Advanced Materials, Design and Prototyping Research (Cadence), Technikon Natal, P.O. Box 953, Durban 4000, South Africa

Abstract

A procedure to optimally design laminated plates for a speci®c cyclic life using a cumulative damage constraint is described. The

objective is minimum weight, and the design variables are the ®ber orientation, and the plate thickness. The plates are subjected to

cyclic bending loads, and the ®nite element method, in conjunction with the Golden Section method, is used to determine the design

variables optimally. The FE formulation is based on Mindlin theory for moderately thick laminated plates and shells, and the

formulation includes bending±twisting coupling. In order to demonstrate the procedure, several plates with di�ering events, load

magnitudes and type, aspect ratios, boundary conditions and cyclic lives are optimised, and compared. Ó 1999 Elsevier Science

Ltd. All rights reserved.

Keywords: Optimal design; Composite structures; Cyclic loads

1. Introduction

The use of laminated composite materials in struc-tural applications has increased dramatically in the lastquarter of the twentieth century, particularly in themarine and aerospace industries [1]. This is mainly dueto the high strength-to-weight and sti�ness-to-weightratios these materials a�ord. Many composite structuresare routinely subjected to cyclic loading regimes, andthough the applied stresses may be low, failure mayoccur due to fatigue.

Damage occurs during each cycle of fatigue loading,and worsens as the number of cycles increases, becauseit is cumulative. Eventually the damage may exceed thelimit a material can sustain. At this point, fatiguefailure, one of the most common forms of failure, takesplace.

Various researchers have investigated the behavior oflaminated composites under static and dynamic loads(eg., Refs. [2,3]), and cyclic loads (eg., Refs. [4±7]). Mostof these fatigue studies are experimental, and haveadded to the understanding of the mechanisms involvedin the fatiguing of composites. Some have even devel-oped design methodologies, and an extensive list of thesecan be found in a paper by Nyman [8] which also de-

scribes a simpli®ed fatigue design approach. None theless the amount of work reported in the literature con-cerning procedures for optimally designing laminatedcomposite structures under fatigue loads is small.

In terms of design optimisation, Adali [9] used anapproach described by Rotem and Hashin [10] to opti-mally design laminated plates subjected to cyclic in-plane loads, with the objective of minimising the weight.The formulation used is analytical, and thus limited inits application.

One phenomenon associated with symmetric angle-ply con®gurations is the occurrence of bending±twistingcoupling which may cause signi®cantly di�erent resultscompared to cases in which this coupling is exactly zero[11]. The e�ect of bending±twisting coupling becomeseven more pronounced for laminates with few layers.Due to this coupling, closed-form solutions cannot beobtained for any boundary conditions and this situationled to the neglection of bending±twisting coupling indesign studies. In actual fact, closed-form solutions forsymmetric laminates are not available even for thesimpli®ed models where this coupling is neglected exceptif the boundary conditions are simply supported allaround. The present study adopts a numerical approachto include the e�ect of bending±twisting coupling and toobtain the optimal design solutions for a variety ofboundary conditions.

Optimisation of composite plates for a given fatiguelife is necessary to realise the full potential of ®ber-re-

Composite Structures 48 (2000) 213±218

* Tel.: +27-31-204-2116; fax: +27-31-204-2139.

E-mail address: walker@umfolozi.ntech.ac.za (M. Walker)

0263-8223/00/$ - see front matter Ó 1999 Elsevier Science Ltd. All rights reserved.

PII: S 0 2 6 3 - 8 2 2 3 ( 9 9 ) 0 0 0 9 7 - 5

inforced materials. The procedure described here is usedto optimally design laminated plates which are subjectedto fatigue loading for a speci®c cyclic life using a damagerule constraint. The objective is minimum weight, and thedesign variables are the ®ber orientation, and the platethickness. The plates are subjected to cyclic bendingloads, and the ®nite element method, in conjunction withthe Golden Section method [12], is used to determine thedesign variables optimally. The ®nite element formula-tion is based on Mindlin theory for moderately thicklaminated plates and shells, and the formulation includesbending±twisting coupling. In order to demonstrate theprocedure, several plates with di�ering events, loadmagnitudes and type, aspect ratios, boundary conditionsand cyclic lives are optimised, and compared.

2. Basic equations

2.1. Cumulative damage theory

Service operation at any given cyclic stress amplitudeproduces fatigue damage (which is assumed to be per-manent), the seriousness of which will be related to thetotal number of cycles that would be required to pro-duce failure of an undamaged component at that stressamplitude. The approach adopted in this study is alinear cumulative damage method based on Miner'srule. To better understand the general theory, considerthe following example.

Assume that operation at several di�erent stress am-plitudes S1; S2; . . . ; St in sequence for a number of cyclesn1; n2; . . . ; nt will result in an accumulation of totaldamage equal to the sum of the damage incrementsaccrued at each individual stress level. Then if operationat a stress amplitude S1 produces complete damage (orfailure) in N1 cycles, operation at stress amplitude S1

(event 1 ± in the following development each one of thedi�erent load level operations, which may consist of anumber of cycles, is called an event) for a number ofcycles n1 smaller than N1 will produce a smaller fractionof damage, say D1: Factor D1 is termed a damage frac-tion (or usage factor). Operation over a spectrum ofdi�erent stress levels results in a usage factor Di for eachof the di�erent stress levels Si in the spectrum. Whenthese factors sum to unity, failure is predicted; viz.

D1 � D2 � � � � � Di P 1: �1�The linear damage rule states that the damage fraction,Di, at stress level Si is equal to the cycle ratio ni=Ni: Thus,the damage fraction D due to one cycle of loading is 1/N.In other words, the application of one cycle of loadingconsumes 1/N of the fatigue life. The failure criterion forvariable amplitude loading can now be stated as

n1

N 1� n2

N2

� � � � � ni

NiP 1: �2�

2.2. Bending of rectangular laminates

Consider a symmetrically laminated rectangular plateof length a; width b and thickness h under a transversebending load q�x; y�. The plate is located in the x; y; zplane and constructed of an arbitrary number K oforthotropic layers of equal thickness hk and ®bre ori-entation hk where k � 1; 2; . . . ;K (Fig. 1). The dis-placement of a point (x0; y0; z0� on the reference surfaceis denoted by �u0; v0;w0�:

The governing equation for the de¯ection w in the zdirection under a transverse load q is given by [13]

D11w;xxxx�4D16w;xxxy �2�D12 � 2D66�w;xxyy

� 4D26w;xyyy �D22w;yyyy � q; �3�where variables after the comma denote di�erentiationwith respect to that variable, and

Dij �Z h=2

ÿh=2

�Q�k�ij z2 dz; �4�

are the bending sti�nesses (and should not be confusedwith the damage fraction Di) and �Q�k�ij are components ofthe transformed reduced sti�ness matrix for the kthlayer.

As no simpli®cations are assumed, the �D� matrix isfully populated and bending±twisting coupling is thusincluded due to the terms D16 and D26.

2.3. Finite element formulation

We now consider the ®nite element formulation of theproblem [14]. Let the region S of the plate be dividedinto n sub-regions Sr�Sr 2 S; r � 1; 2; . . . ; n� such that

P�u� �Xn

r�1

PSr�u�; �5�

Fig. 1. Geometry and loading of laminated plate with loading q�x; y�.

214 M. Walker / Composite Structures 48 (2000) 213±218

where P and PSr are potential energies of the plate andthe element, respectively, and u is the displacementvector. Using the same shape functions associated withnode j �j � 1; 2; . . . ; n�; Sj�x; y�; for interpolating thevariables in each element, we can write

u �Xn

j�1

Sj�x; y�uj; �6�

where uj is the value of the displacement vector corre-sponding to node j; and is given by

u � fu�j�; v�j�;w�j�;/�j�1 ;/�j�2 gT

: �7�The displacements fu; v;w;/1;/2g are approximated as

u �Xn

j�1

ujwj�x; y�; v �Xn

j�1

vjwj�x; y�;

w �Xn

j�1

wjwj�x; y�;

/1 �Xn

j�1

S1j wj�x; y�;

/2 �Xn

j�1

S2j wj�x; y�; �8�

where wj are the Lagrange family of interpolationfunctions. From the equilibrium equations of ®rst-ordertheory, and Eq. (8), we obtain the ®nite element model,X5

b�1

Xn

j�1

Kabij Db

j ÿ F ai � 0; �a � 1; 2; . . . ; 5� �9�

or

�K�fDg ÿ fF g � f0g; �10�where K is the sti�ness matrix, F the nodal force vector,and the variable D denotes the nodal values of w and itsderivatives [15].

3. Analysis procedure for cyclic loading

Firstly, the stresses at the nodes throughout the FEplate model are determined. Next (using an appropriateS±N curve), the partial usage factor Di for the ith al-ternating stress intensity is evaluated in that list byevaluating the cycle ratio ni=Ni (as its equivalence). Theith alternating stress intensity is formed by the combi-nation of loadings AE and BF , where the subscripts Eand F are the corresponding events. Here, ni corre-sponds to the lower number of cycles interpolated fromthe design fatigue curve (S±N curve).

After evaluating the partial usage factor Di, updatesare made to the alternating stress intensity list by re-ducing the number of cycles of both events E and F byni. Consequently, one of the two events E or F will beeliminated (or both if E and F have the same number of

cycles) and the other event will have ni cycles less in thelater calculations. Elimination of an event results inelimination of the corresponding loadings. Once aloading is eliminated the corresponding stress intensities(formed by the combination of that loading with otherloadings) will also be eliminated from the list. Afterupdating the list, the next alternating stress intensity inthe list is checked and the corresponding partial usagefactor evaluated, added that to the cumulative one andthe list updated. This procedure is repeated for the nextalternating stress intensity in the list and continues untilall stress intensities are considered.

For any alternating stress within the stress range S1

and S2 (the ®rst and last points) of an S±N curve, in-terpolation is used to ®nd the corresponding cycles.

4. Optimal design problem

The objective of the design problem is, for a givenspeci®c fatigue life (consisting of several events), tominimise the maximum damage Dmax�x; y� and then theweight W of the plate by minimising the thickness hsubject to a cumulative damage law constraint. Mini-mising Dmax�x; y� is achieved by optimally determiningthe ®ber orientations given by hk � �ÿ1�k�1h for k6K=2and hk � �ÿ1�kh for k P K=2� 1 in order to minimisethe maximum cumulative damage which occurs duringloading events.

Thus, the ®rst part is stated as

Dmin �D minh�Dmax�h��; 0°6 h6 90°; �11�

where

Dmax�h� � maxx;y

D�x; y; h�: �12�

The second part of the design problem involves mini-mising the laminate thickness h within the constraints ofthe damage rule, and may be stated as

hmin � minh

h�hopt�: �13�

The nodal stresses and subsequently the remaining life ni

(which is interpolated from the S±N curve) and thus themaximum damage Dmax are determined from the ®niteelement solution of the problem given by (10) and (2).The ®rst optimisation procedure involves the stages ofdetermining the maximum damage Dmax�x; y� for a givenh and improving the ®ber angle to minimise Dmax. Thesecond optimisation stage involves evaluating D�hopt�for a given h and improving the laminate thickness tominimise the weight. This step may be described ex-plicitly as

minh

D�hopt��� ÿ 1

�� �14�

in order to minimise the thickness. Thus the computa-tional solution consists of successive stages of analysis

M. Walker / Composite Structures 48 (2000) 213±218 215

and optimisation until a convergence is obtained and theoptimal ®bre angle hopt and then hmin is determinedwithin a speci®ed accuracy. In both optimisation stages,the Golden Section method is employed ®rstly to deter-mine hopt and then hmin:

5. Numerical examples

For the purpose of illustrating the method described,numerical results are given for a typical T300/5208graphite/epoxy material with E1 � 181 GPa, E2 � 10:3GPa, G12 � 7:17 GPa and m12 � 0:28 [16]. The symmet-ric plates studied here are constructed of four equalthickness layers with h1 � ÿh2 � ÿh3 � h4 � h and fordesign purposes, the S±N curve of Fig. 2 is speci®ed [17].Di�erent combinations of free (F), simply supported (S)and clamped (C) boundary conditions are implementedat the four edges of the plates. In particular, the di�erentcombinations studied are (F, S, C, S), (S, S, S, S) and (C,C, C, C), where the ®rst letter refers to the ®rst plateedge, and the others follow in an anti-clockwise direc-tion as shown in Fig. 1. Also, the plates are subjected touniformly distributed transverse bending loads of mag-nitude P Pa, which are applied cyclically (and fully re-versed).

The dependence of stresses and thus the damage D onthe ®bre angle h is illustrated in Fig. 3, for a (F, S, C, S)plate and (C, C, C, C) plate, both having a=b � 1: Theplates are subjected to the ®rst loading regime or eventshown in Table 1 (viz. the load of 100,000 Pa is appliedand then removed 50,000 times). The (F, S, C, S) platehas h=b � 0:0074; while the other has h=b � 0:0042: Thisdemonstrates that the damage is indeed dependent onthe ®ber orientation, and it is clear that the minimumdamage for a laminate can be several times lower thanthe damage at other ®bre angles. In addition, the platethickness can be determined optimally, such that the

plate is designed for a particular cyclic life. This factemphasizes the importance of carrying out optimizationin design work of this nature to obtain the best perfor-mance of ®bre composite plates.

Example 1 (Plates with differing events). Consider a (C,C, C, C) plate with a=b � 2: The plate is subjected to the®rst loading regime or event shown in Table 1, andshould fail at the end of the last cycle of that event.Thus, the plate must be designed such that h and h areselected optimally to ensure minimum weight and su�-cient life.

The result is compared to that for a similar plate(plate no. 2) which is subjected to both events shown inTable 1.

As expected, the plate which is subjected to bothevents has a minimum thickness which is approximately40% more than the plate which is designed optimally toendure the ®rst event only (see Table 2).

Example 2 (Plates with differing aspect ratios). In orderto demonstrate the e�ect of a=b on the results, two plateswith di�erent aspect ratios were optimally designed. Theboundary conditions implemented along the edges ofeach plate are (C, C, C, C), with the ®rst havinga=b � 1:25, and the second a=b � 2 (the standard plateof Example 1). The plates are subject to both events.

The plate with the larger aspect ratio ends up with aminimum thickness which is approximately 20% greaterthan the smaller plate after the process of optimisation iscompleted (see Table 3). This is expected, since thesmaller plate is sti�er.

Fig. 2. S±N curve for T300/5208 material.

Fig. 3. Dependence of damage D on the ®bre angle.

Table 1

Loading vs. life for the two events

Event Load (Pa) Cycles

1 100,000 50,000

2 200,000 25,000

216 M. Walker / Composite Structures 48 (2000) 213±218

Example 3 (Plates with differing boundary conditions).In order to demonstrate the e�ect of the boundaryconditions on the results, three plates, each of aspectratio a=b � 1:75 but with di�ering boundary conditionswere optimally designed. The ®rst is clamped along allfour edges, while the second is simply supported alongall four edges. The third plate has a combination of free,simply supported and clamped boundary conditions,viz. (F, S, C, S), and all are subjected to the ®rst eventonly.

As the number of degrees of freedom of a plate iscurtailed along the boundary, so it becomes sti�er. Forthis reason, the (F, S, C, S) plate is 26% thicker than the(S, S, S, S) plate, and 55% thicker than the (C, C, C, C)plate, as can be seen from the results in Table 4.

Example 4 (Plates with differing load types). A rectan-gular (C, C, C, C) laminate with a=b � 2 subjected to acyclic patch UDL (see Fig. 4) of magnitude 105 Pa isoptimally designed, and compared to the standard plate(which is the same geometrically as that with the patchload except that the whole of the plate surface is sub-jected to a UDL of magnitude 105 Pa). The plates aresubjected to the ®rst event only.

As expected, the plate with the patch load has aminimum thickness which is nearly 30% less than theplate subjected to a UDL over its entire surface (seeTable 5).

Example 5 (Plates with differing load magnitudes). Inorder to demonstrate the e�ect of di�ering load magni-tude, the optimal designs of two similar plates withdi�erent loads are compared (see Table 6). The plateseach have a=b � 2; are clamped along all the edges (viz.the standard plate). The ®rst is subjected to the ®rst

event, while the second is also subjected to the ®rst eventonly, except that the load is doubled to a magnitude of2� 105 Pa.

The plate with the larger load has a minimumthickness which is 43% more than the plate with thesmaller load.

Example 6 (Plates with differing cyclic lives). In order todemonstrate the e�ect of di�ering the number of cycles,a (C, C, C, C) plate with a=b � 2 (viz. the standardplate) is subjected to an event which is similar to event 1,except that the number of cycles is increased to 75,000.The result is compared to that for the standard platewhich has a life of 50,000 cycles (see Table 7).

As expected, the plate which has to endure a longercyclic life (viz. a greater number of cycles) is slightlythicker than that which has been designed for a shorterlife.

6. Conclusions

A procedure for optimally designing laminated platesfor a speci®c cyclic life using a damage rule constraint isdescribed. The objective is minimum weight, and thedesign variables are the ®ber orientation, and the platethickness. The plates are subjected to cyclic bendingloads, and the ®nite element method, in conjunctionwith a search routine is used to determine the designvariables optimally. The FEM formulation is based on

Table 4

Optimal designs for plates with di�ering boundary conditions

Boundary conditions hopt hmin=b� 10ÿ3

(C, C, C, C) 90� 4.48

(S, S, S, S) 87.7� 6.02

(F, S, C, S) 26.9� 9.93

Fig. 4. Laminated with UDL patch load of magnitude P. (Note:

c=b � 0:5, d=b � 0:3125, e=b � 0:1875, f=b � 0:3).

Table 3

Optimal designs for plates with di�ering aspect ratios

a=b hopt hmin=b� 10ÿ3

1.25 90� 6.15

2 45� 7.69

Table 2

Optimal designs for plates subjected to di�ering events

Plate hopt hmin=b� 10ÿ3

1 90� 4.50

2 45� 7.69

Table 5

Optimal designs for plates with di�ering load types

Load type hopt hmin=b� 10ÿ3

Patch 69.1� 3.20

Whole surface 90� 4.50

M. Walker / Composite Structures 48 (2000) 213±218 217

Mindlin-type laminated plate theory. The numericalapproach employed in the present study is necessitatedby the fact that bending-twisting coupling and consid-eration of various combinations of free, clamped andsimply supported boundary conditions, rule out an an-alytical approach. It must be emphasised that di�erentdamage rules may be used, and that the one chosen inthis study is used merely to demonstrate the procedure.

The e�ect of partial optimisation on the damage wasinvestigated by plotting D against the ®rst design vari-able h for a given plate thickness (Fig. 3). The resultsshow that the di�erence in the damage of optimal andnon-optimal plates could be substantial, emphasisingthe importance of optimisation for ®ber compositestructures.

In order to demonstrate the complete procedure de-scribed in this paper, several plates are then completelyoptimised, so that both the ®ber angle and plate thick-ness are determined optimally. The plates experiencedi�ering events, load magnitudes and type, aspect ratios,boundary conditions and cyclic lives, and the optimaldesign of each is compared to the others. Such di�er-ences result in di�ering optimal designs, thus empha-sizing the importance of optimisation in every case.

7. Closure

When the plates described by the results depicted inFig. 3 are completely optimised, the results are as fol-lows:

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Boundary condition hopt hmin=b� 10ÿ3

(F, S, C, S) 0° 6.62(C, C, C, C) 45° 4.15

Table 6

Optimal designs for plates with di�ering load magnitudes

Load (Pa) hopt hmin=b� 10ÿ3

100,000 90� 4.50

200,000 45� 7.84

Table 7

Optimal designs for plates with di�ering cyclic lives

Cycles hopt hmin=b� 10ÿ3

75,000 84.3� 4.56

50,000 90� 4.50

218 M. Walker / Composite Structures 48 (2000) 213±218