Composite Structures 118 (2014) 351–366

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Stress concentration phenomena in the vicinity of compositeplate-doubler junctions by a layerwise analysis approach

http://dx.doi.org/10.1016/j.compstruct.2014.07.0410263-8223/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: christian.mittelstedt@sogeti-hightech.de (C. Mittelstedt).

1 Current address: Institute of Materials Research, Helmholtz-Zentrum Geesthacht,Materials Mechanics, Max-Planck-Str. 1, D-21052 Geesthacht, Germany.

Jan Eike Butzke 1, Christian Mittelstedt ⇑SOGETI High Tech GmbH, Hein-Sass-Weg 30, D-21129 Hamburg, Germany

a r t i c l e i n f o

Article history:Available online 7 August 2014

Keywords:DoublerPatchStress singularitySemi-analytical methodInterlaminar stressesLayerwise analysis

a b s t r a c t

This paper presents an analysis method for the determination of displacements, strains and stresses in alaminated composite plate subjected to tensile load that is reinforced by a doubler. The analysis approachconsists of two parts. Firstly, a ‘global’ solution that is based on Classical Laminated Plate Theory (CLPT) isintroduced. Secondly, a ‘local’ model is derived that enables the assessment of the three-dimensionalstress state near the plate-doubler junction. The local solution employs a discretization of the physicallayers into a number of mathematical layers which necessitates the numerical treatment of a quadraticeigenvalue problem. Consequently, the current approach can be classified as being a semi-analyticallayerwise analysis method. Results are generated for several different plate-doubler configurations,and it is found that the current analysis model delivers excellent results when compared to finite elementsimulations, however with only a fraction of the computational time and effort that are needed for theFEM analyses.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Composite materials in the form of fiber-reinforced plastics(FRP) – i.e. a matrix material (for instance epoxy) which is beingreinforced by unidirectional fibers (e.g. carbon fibers) – have foundan increasing use in many engineering branches due to their supe-rior properties in terms of high strength-to-weight and stiffness-to-weight ratios. Naturally, such FRP which often appear in theform of laminated composite plates or shells (i.e. thin-walledstructures that consist of a number of FRP-layers) are very attrac-tive for such applications where the structural weight plays animportant role. Of course, this holds true especially in lightweightengineering branches such as aeronautics and astronautics, butalso in automotive engineering or in the wind energy sector.

In many cases it is necessary to reinforce a laminated plate orshell at certain locations in order to ensure their structural integ-rity. As an example, such reinforcements may be the result of astructural repair at locations where a laminate has been damagedand where after the repair measures have been completed a patch/doubler is being attached in order to additionally strengthen theplate or shell. In certain cases reinforcements may also be

necessary due to static requirements whenever a stronger cross-section is required locally so that the applied loads can be carriedby the structure without failure, or when a hole through the thick-ness of the laminate requires reinforcements. Two general over-view papers concerning repair concepts for composite structuresare available with Myhre and Beck [20] and Myhre and Labor [21].

In Fig. 1 a typical structural situation consisting of a laminatedplate (‘basic structure’) and a rectangular doubler (i.e. an externalrepair solution) is depicted, where it should be noted that the plan-form of doublers in composites engineering can take up virtuallyany form (i.e. circular, quadratic, rectangular, elliptic, polygonial),depending on the application case and the specific requirements.Further, a distinction has to be made between tapered repairpatches (i.e. such patches where there is no abrupt change of thick-ness at the plate-doubler junction but rather a smooth transitionfrom the reinforced structure to the basic structure is ensured,see for instance He et al. [7]) and such patches where no taperingis used and where the change in thickness at the plate-doublerjunction is discontinuous and abrupt. The present contribution isconcerned with the stress analysis of the latter type of patchreinforcements.

Naturally, the presence of the doubler will lead to significantstress concentrations in the vicinity of the plate-doubler junctiondue to the geometrical discontinuity and the abrupt change inthe thickness of the structure which needs to be taken into accountcarefully whenever such patches/doublers are applied. These stress

Fig. 1. An example for a laminated plate reinforced by a rectangular doubler.

352 J.E. Butzke, C. Mittelstedt / Composite Structures 118 (2014) 351–366

concentrations will manifest themselves in the form of significantthree-dimensional stress states at the edges and corners of theinterface between plate and doubler, even though the far-fieldloading might consist of a simple load case such as uniaxial tensileload as exemplarily shown in Fig. 1. Further, due to the presence ofthe free edges of the doubler, it can be expected that free-edgestress fields will arise in this area as well which must be taken intoconsideration when designing and analyzing reinforcements in theform of doublers.

Free-edge effects are mainly triggered by the inherent anisot-ropy and thus the general mismatch of the elastic properties ofadjacent layers in a composite laminate and are characterized byvery localized and potentially singular interlaminar stress fieldsin the vicinity of the traction free edges of composite laminates.The free-edge effect has been under investigation by the scientificcommunity since the pioneering publication by Pipes and Pagano[24] in which a plane laminated specimen under uniaxial tensionhas been treated by a finite difference formulation, and a virtuallyuncountable number of papers employing a broad variety ofclosed-form analytical, semi-analytical, purely numerical, but alsoexperimental approaches is available today. An encompassing lit-erature survey is well beyond the scope of this paper, and a goodnumber of survey papers on this specific topic is available a selec-tion of which is cited with Salamon [29], Pagano [22], Herakovich[9], Reddy and Robbins [25], Kant and Swaminathan [12], Mittel-stedt and Becker [16], Mittelstedt and Becker [18]. The interestedreader is advised to consult these references in order to gain a dee-per insight into the state of the art concerning the modeling andanalysis of free-edge effects in composite laminates.

The closed-form analytical, semi-analytical and numerical mod-eling as well as the experimental study of composite structuresthat are reinforced by doublers has been the topic of a good num-ber of publications in the last decades. In the following a shortselective overview of relevant references is given where it mustbe noted that no claim of completeness is made here.

A rather simple and straightforward approach for assessing thestrength of a repair solution of a thin-walled composite laminatedstructure based on a stress and strain calculation in the frameworkof Classical Laminated Plate Theory (short: CLPT) in conjunctionwith stress-based failure criteria was reported by Robson et al. in[28]. Soutis and Hu [30] presented a closed-form analytical methodfor the determination of interlaminar stresses in plate-doublerinterfaces based on a rather simple shear-lag analysis approach.Further, they employed a refined three-dimensional finite elementmesh for the determination of stress concentration factors andcritical locations at round external patch repairs and studied theinfluence of the thickness of the repair patch on the stress fields.In a follow-up paper, Soutis et al. [31] performed experiments oncomposite laminates under compressive load that were repairedby circular patches. Furthermore, by using a full-scale three-dimensional finite element model Soutis et al., performed a stressanalysis of the repaired region in order to predict the location, typeand initiation of damage in the vicinity of the plate-doublerinterface. Hu and Soutis [10] again investigated the compressive

behavior of laminated composite specimens under compressiveload that were strengthened by a circular patch and used a shearlag analysis approach in order to derive design guidelines for anoptimized choice of patch geometry, size and inplane stiffness. Athree-dimensional finite element model in conjunction with acohesive zone approach delivered results for the stress fields inthe vicinity of the plate-doubler junction and enabled an estimateof the ultimate compressive loads of such repaired specimens. Leeand Knauss [13] performed experimental and numerical three-point and four-point bending studies on composite laminatedspecimens which included abrupt thickness changes and deter-mined the load level at which damage initiation takes place forcombined loading conditions including tension, bending and inter-laminar shear. Tsamasphyros et al., [34] performed elasticity-basedanalytical calculations and numerical computations of isotropicplates that contain cracks and that are reinforced by compositepatches, wherein patches on one side of the plate as well aspatches attached to both sides were considered. A closed-formanalysis of the inplane stress fields in the vicinity of patch repairsof composite laminated plates was presented by Engels and Becker[6] who investigated composite containing an elliptical hole whichis reinforced by an elliptical doubler and used the complex poten-tial method for deriving a closed-form analytical analysis method.However, their analysis was restricted to the pure inplane stressstate, and interlaminar stresses were not part of the investigations.A similar analysis approach allowing for arbitrarily shaped holesand patches was presented by Zemlyanova [38]. Duong and Yu[5] investigated the thermal stresses that arise in a compositebonded repair where an octagonal patch is applied on a crackedplate by means of an elasticity based approach. Again, the analysisapproach only captured inplane stress fields, the typical interlam-inar stresses at plate-doubler junctions were not part of the inves-tigations. A numerical model for the assessment of the stress fieldsin the vicinity of cracked plates that are being repaired and rein-forced by composite patches was developed by Bachir Bouiadjraet al. [2]. Specifically, the cracks were considered under pure modeI as well as under mixed mode loading, and the stress intensity fac-tors occurring in this region were determined. Mathias et al. [15]reported the application of genetic algorithms to the optimizationof composite patches applied to aluminum plates containing circu-lar holes. The aim of the optimizations was to reduce the stressesin the vicinity of the holes considering constraints in terms ofthe patch shape, size and location by altering the stackingsequence and size of the patch. In the same way Brighenti [3] opti-mized patches attached to cracked plates under mode I and mode IIloading and determined optimum topologies in order to maximizethe fracture resistance and fatigue life behavior. Wigger and Becker[36,37] performed asymptotic investigations of the inplane stresssingularities in the vicinity of corners of doublers attached to com-posite plates and used the complex variables method and theboundary finite element method, respectively. They showed thateven though the orders of the occurring stress singularities are gen-erally far lower than for typical singular stress problems in the vicin-ity of crack tips, they may become significant nonetheless, and thecorners of the plate-doubler junctions are generally prone to delam-ination failure. Note that in these investigations the complete three-dimensional stress field and the according stress singularities werenot considered. Papanikos et al. [23] developed a numerical modelfor the progressive failure analysis of isotropic plates containingthrough-the-thickness cracks that were repaired by tapered com-posite patches attached to both sides of the plates. A study from adifferent application field and yet related to the scope of the presentpaper was published by Carpinteri et al. [4] who investigated thestructural behavior of beams with CFRP reinforcements. Specifically,Carpinteri et al., considered the debonding behavior of the CFRPreinforcements due to the stress concentrations at the interface

Fig. 2. Nomenclature.

J.E. Butzke, C. Mittelstedt / Composite Structures 118 (2014) 351–366 353

edge between the beam and the reinforcement by using two differ-ent failure criteria, one based on the assessment of the interfacialshear stresses, and the other based on fracture mechanics. Leeet al. [14] investigated the free-edge stress fields that arise in acomposite patch that is attached to a composite plate under bend-ing load and used a stress-function based equivalent single-layertheory approach. Weißgraeber and Becker [35] and Hell et al. [8]used a finite fracture mechanics analysis approach (i.e. a combinedstress and energy fracture criterion) for the determination of thecrack onset in the interfaces of adhesively bonded joints. Whilethe authors used this coupled fracture criterion for the analysisof crack onset in bonded joints, the analysis approach is alsoapplicable for the assessment of the fracture criticality ofplate-doubler junctions.

This contribution is devoted to the development of an efficientand accurate analysis method for the determination of displace-ments, strains and stresses in a tensile specimen as depicted inFig. 1. For this purpose, a semi-analytical analysis method for themodeling of a composite laminated plate under tensile load thatis reinforced by a rectangular doubler will be presented, whereinboth plate and doubler are supposed to consist of symmetriccross-ply layups. In the vicinity of the junction between plateand doubler significant three-dimensional and potentially evensingular stress fields are to be expected, and the developed analysismethod specifically aims at capturing the special circumstancesthat arise in this region of interest. The analysis approach assumesthat as long as the area of investigation is sufficiently remote fromthe corners of the doubler, a plane state of strain can be assumed,and consequently the analysis will be performed using a two-dimensional strip idealization. The analysis itself consists of twoparts. Firstly, a ’global’ model will be established in Section 2 thatis based on the assumptions of Classical Laminate Plate Theory (seee.g. Jones [11], Reddy [26]) and reduces the model to a representa-tive plane of reference. The global approach thus enables an esti-mation of displacements and stresses in such regions where theanticipated fully three-dimensional stress fields and singularitiesin the vicinity of the plate-doubler junction do not have an influ-ence. Secondly, a ‘local’ model will be established in Section 3 bywhich the ‘global’ model is upgraded so that also the very localizedstress concentrations in the near-field of the plate-doubler vicinitycan be assessed properly. For this purpose, all physical layers of theplate and doubler laminates are being discretized into a number ofmathematical layers with respect to the thickness direction (so-called layerwise theory, see Reddy [26] for a general overview).Within each mathematical layer, a separate displacement formula-tion is used wherein in each interface a priori unknown displace-ment functions with respect to the longitudinal direction arepostulated, while in the thickness direction a linear interpolationbetween the interfacial displacement functions is used. Theunknown displacement functions are determined by virtue of theprinciple of minimum elastic potential which leads to a set ofdifferential equations that can be solved in an exact closed-formanalytical manner. The solution approach eventually leads to aquadratic eigenvalue problem that requires numerical treatment.According layerwise approaches have been employed by e.g. Rob-bins and Reddy [27], Tahani and Nosier [32], Mittelstedt andBecker [17], Mittelstedt and Becker [19], Tahani and Andakhshideh[33], Andakhshideh and Tahani [1] with great success for the anal-ysis of free-edge effects in composite laminates, and consequentlythis paper is an extension of such approaches for the simultaneousanalysis of the stress concentrations in the vicinity of the plate-doubler junction as well as at the free edges of the applied doubler.In the framework of the present analysis approach, all quantities –i.e. displacements, strains and stresses – at any point of thereinforced laminated plate can be determined in a closed-formanalytical manner, however a numerical solution of an eigenvalue

problem is required. Consequently, it is adequate to speak of asemi-analytical analysis method. A comparison with finite elementsimulations will be presented in Section 4 of this paper, and theresults will show that the present analysis approach works witha very high accuracy with comparatively low computational effortwhich makes it very suitable for engineering practice for all pur-poses where such stress concentration phenomena need to beassessed and where computational time is a critical aspect. Finally,the paper closes with a summary and an outlook on futureinvestigation in Section 5.

2. The global model

2.1. Displacements, strains and stresses in the framework of CLPT

2.1.1. Modeling approachConsider the structural situation as given in Fig. 1. The employed

nomenclature for the geometric properties of the specimen isexplained in Fig. 2. Herein, l1; b1 and h1 are the length, width andheight of the plate, while l2; b2 and h2 are the geometric propertiesof the doubler. Apparently, this structural situation exhibits certainsymmetry properties that will be taken advantage of in the follow-ing. Both the plate and the doubler are manufactured of symmetriccross-ply laminates so that none of the typical coupling effects (i.e.extension–shear-coupling, extension–bending-coupling, bending–twisting-coupling) occur due to the employed lamination schemes.However, as will be shown later on, due to the discontinuity of theheight of the structure, some secondary bending effects will occurnevertheless, even though the structure is subjected to a plane ten-sile load.

The analysis method that is going to be developed in the follow-ing makes use of the fact that at the free doubler edge remote fromthe free corners, a state of plane strain holds so that for the devel-opment of an efficient analysis method it is sufficient to consider atwo-dimensional strip of the plate-doubler-structure that is cutfrom the structure in the direction of the applied tensile load(Fig. 3) and that only covers one half of the length, i.e. l1

2, of theplate. This two-dimensional strip is then assumed to be in a stateof plane strain with respect to the global y-direction. The employedboundary conditions are depicted in Fig. 4, upper part, and stemdirectly from the above mentioned symmetry properties of theconsidered structural situation. Firstly, at the cutting edge atx ¼ 0 for all z, it can be stated that due to the symmetry of thestructural situation, the applied load case in the form of uniaxialtension, and the chosen laminate layups, no displacements u0 inthe x-direction are possible. In order to ensure that no rigid-bodytranslations occur, it is further assumed that at x ¼ 0 and z ¼ 0the displacement w0 is prevented. The exact location of the originof the coordinate system x; z will be explained in detail later on.Lastly, the load introduction at the free end needs to be discussed.In the current analysis approach, the load introduction is modeled

Fig. 3. Two-dimensional plate-doubler strip.

354 J.E. Butzke, C. Mittelstedt / Composite Structures 118 (2014) 351–366

as being displacement controlled, so that at x ¼ l12 a displacement

value uinit is prescribed.In order to facilitate the computations, the model is decom-

posed into the two segments I and II (see Fig. 4, middle and lowerpart) with the thicknesses h1 (Segment I) and h1 þ h2 (Segment II).In the following, two local coordinate systems xI; z and xII; z areintroduced as indicated in Fig. 4, middle part.

The displacement-controlled load introduction leads to the nor-mal force N0ðIÞ

xx in the middle plane of segment I in positive x1-direc-tion (see Fig. 4, middle). It can be directly deduced that this normalforce will transfer itself into segment II due to equilibrium require-ments. Note that the normal force N0ðIIÞ

xx does not act in the middleplane of segment II, but rather has an offset d ¼ h1

2 (see Fig. 4).Further, it has been assumed that the load introduction in the anal-ysis model is such that the section of segment I remains plane andperpendicular to the x-direction under any given tensile load. Inorder to enforce this requirement, a bending moment M0ðIÞ

xx is intro-duced at the free end of segment I that runs counter to the rotationof the section of segment I. This bending moment is also present insegment II, denoted as M0ðIIÞ

xx . The resultant analysis model is shownin Fig. 4, lower part. From equilibrium at the junction between seg-ments I and II it can be deduced that N0ðIÞ

xx ¼ N0ðIIÞxx and M0ðIÞ

xx ¼ M0ðIIÞxx

hold.

2.1.2. Constitutive behaviorIt is assumed that both the plate and the doubler have a sym-

metric cross-ply layup. Hence, the constitutive law according toCLPT for segment I can be written as:

Fig. 4. Analys

N0ðIÞxx

N0ðIÞyy

N0ðIÞxy

M0ðIÞxx

M0ðIÞyy

M0ðIÞxy

0BBBBBBBBBBB@

1CCCCCCCCCCCA¼

AI11 AI

12 0 0 0 0

AI12 AI

22 0 0 0 0

0 0 AI66 0 0 0

0 0 0 DI11 DI

12 0

0 0 0 DI12 DI

22 0

0 0 0 0 0 DI66

266666666664

377777777775

e0ðIÞxx

e0ðIÞyy

e0ðIÞxy

j0ðIÞxx

j0ðIÞyy

j0ðIÞxy

0BBBBBBBBBB@

1CCCCCCCCCCA; ð1Þ

wherein AIij are the membrane stiffness components and DI

ij are theflexural stiffnesses (i; j ¼ 1;2;6). Note that since segment I is sym-metric, no coupling stiffness components BI

ij need to be taken intoaccount.

The determination of the stiffness components AIIij , BII

ij and DIIij for

segment II is done analogously. However, in order to use the samereference plane for both segments I and II, the offset d is taken intoaccount for segment II. This eventually leads to a coupling of themembrane and the flexural behavior as follows:

ABDðþdÞ ¼A Bþ dA

� �Bþ dA� �

Dþ 2dBþ d2A� �

264

375; ð2Þ

and specifically, for segment II the following constitutive lawresults:

N0ðIIÞxx

N0ðIIÞyy

N0ðIIÞxy

M0ðIIÞxx

M0ðIIÞyy

M0ðIIÞxy

0BBBBBBBBBBB@

1CCCCCCCCCCCA¼

AII11 AII

12 0 dAII11 dAII

12 0

AII12 AII

22 0 dAII12 dAII

22 0

0 0 AII66 0 0 dAII

66

dAII11 dAII

12 0 DII11þd2AII

11 DII12þd2AII

12 0

dAII12 dAII

22 0 DII12þd2AII

12 DII22þd2AII

22 0

0 0 dAII66 0 0 D2

66þd2AII66

266666666664

377777777775

e0ðIIÞxx

e0ðIIÞyy

e0ðIIÞxy

j0ðIIÞxx

j0ðIIÞyy

j0ðIIÞxy

0BBBBBBBBBB@

1CCCCCCCCCCA:

ð3Þ

In all that follows, the stiffness components describing segment IIwill be written with a superposed index ðþdÞ as BIIðþdÞ

ij ¼ dAIIij and

DIIðþdÞij ¼ DII

ij þ d2AIIij .

2.1.3. Simplification of the modelDue to the assumption of a plane state of strain of the consid-

ered two-dimensional strip with respect to the global y-axis, sev-eral simplifications can be implemented in the constitutive lawfor both segments I and II in order to facilitate the analysis

is model.

J.E. Butzke, C. Mittelstedt / Composite Structures 118 (2014) 351–366 355

approach. Firstly, due to the assumed state of plane strain, thekinematic quantities e0

yy; c0xy;j0

yy, and j0xy can be neglected in both

segments I and II:

e0ðIÞyy ¼ e0ðIIÞ

yy ¼ 0; c0ðIÞxy ¼ c0ðIIÞ

xy ¼ 0;

j0ðIÞyy ¼ j0ðIIÞ

yy ¼ 0; j0ðIÞxy ¼ j0ðIIÞ

xy ¼ 0: ð4Þ

Further, the force and moment fluxes N0yy;N

0xy;M

0yy, and M0

xy disap-pear as well due to the given load case:

N0ðIÞyy ¼ N0ðIIÞ

yy ¼ 0; N0ðIÞxy ¼ N0ðIIÞ

xy ¼ 0;

M0ðIÞyy ¼ M0ðIIÞ

yy ¼ 0; M0ðIÞxy ¼ M0ðIIÞ

xy ¼ 0: ð5Þ

2.2. Determination of axial and out-of-plane deformations

Since the current analysis model assumes a plane state of strain,the out-of-plane displacements w0 are uncoupled from the widthcoordinate y, i.e. wI

0 ¼ f ðxIÞ and wII0 ¼ f ðxIIÞ. From the fourth rows

in (1) and (3), the curvatures j0ðIÞxx and j0ðIIÞ

xx can be readily deter-mined as:

j0ðIÞxx ¼ �

@2wI0ðxIÞ

@x2I

¼ M0ðIÞxx

DI11

; ð6Þ

j0ðIIÞxx ¼ � @

2wII0ðxIIÞ@x2

I

¼ M0ðIÞxx � BIIðþdÞ

11 e0ðIIÞxx

DIIðþdÞ11

: ð7Þ

Twofold integration leads to the following expressions for the out-of-plane displacements wI

0 and wII0 in segments I and II:

wI0ðxIÞ ¼ �

ZZj0ðIÞ

xx dxIdxI ¼ �M0ðIÞ

xx

DI11

x2I

2þ C1xI þ C2; ð8Þ

wII0ðxIIÞ ¼ �

ZZj0ðIIÞ

xx dxIIdxII ¼ �M0ðIIÞ

xx � BIIðþdÞ11 e0ðIIÞ

xx

� �DIIðþdÞ

11

x2II

2þ C3xII þ C4:

ð9Þ

Therein, C1; . . . ;C4 are integration constants. The constants C3 andC4 can be determined from the requirement of vanishing displace-

ments wII0ðxIIÞ and a vanishing slope @wII

0 ðxIIÞ@xII

at the supported end of

segment II at xII ¼ l22. This immediately leads to:

C3 ¼M0ðIIÞ

xx � BIIðþdÞ11 e0ðIIÞ

xx

DIIðþdÞ11

l2

2; ð10Þ

C4 ¼ �M0ðIIÞ

xx � BIIðþdÞ11 e0ðIIÞ

xx

� �DIIðþdÞ

11

l228: ð11Þ

The remaining constants C1 and C2 can be computed from the com-patibility requirements between segment I and II, i.e.

@wI0ðxI ¼ 0Þ@xI

¼ � @wII0ðxII ¼ 0Þ@xII

; wI0ðxI ¼ 0Þ ¼ wII

0ðxII ¼ 0Þ: ð12Þ

Evaluation then directly yields C1 ¼ �C3 and C2 ¼ C4.A further requirement arises at the loaded end of segment I at

xI ¼ l1�l22 where it has to be enforced that the cross-section of seg-

ment I remains normal to the global x-axis also in the loaded state.Since the analysis approach operates within the perimeters of

CLPT, this requirement is fulfilled if the slope@wI

0 xI¼l1�l2

2

� �@xI

vanishes.

In this way, the bending moment M0ðIÞxx can be determined as:

M0ðIÞxx ¼

DI11BIIðþdÞ

11 e0ðIIÞxx l2

DIIðþdÞ11 ðl1 � l2Þ þ l2DI

11

: ð13Þ

Upon insertion of the now determined constants C1; . . . ;C4 into theEqs. (8) and (9) it becomes clear that the axial strain e0ðIIÞ

xx is the only

remaining unknown quantity in the expressions for wI0ðxIÞ and

wII0ðxIIÞ. From the first line in (1), the normal force N0ðIÞ

xx can beextracted as a function of the normal strain e0ðIÞ

xx :

N0ðIÞxx ¼ AI

11e0ðIÞxx : ð14Þ

The first line of (3) in conjunction with (14) then allows for a com-putation of e0ðIIÞ

xx as:

e0ðIIÞxx ¼ AI

11DIIðþdÞ11 e0ðIÞ

xx

AII11DIIðþdÞ

11 þ BIIðþdÞ11

� �2 DI11 l2

l1�l2ð ÞDIIðþdÞ11 þl2DI

11

� 1� � : ð15Þ

Note that this formulation for e0ðIIÞxx includes the axial strain e0ðIÞ

xx . Theaxial displacement uII

0ðxIIÞ of segment II can be determined byintegration of (15) with respect to xII:

uII0ðxIIÞ ¼

Ze0ðIIÞ

xx dxII

¼ AI11DIIðþdÞ

11 e0ðIÞxx

AII11DIIðþdÞ

11 þ BIIðþdÞ11

� �2 DI11 l2

l1�l2ð ÞDIIðþdÞ11

þl2DI11

� 1� � xII þ C5: ð16Þ

The integration constant C5 is determined from the requirementthat the axial displacement uII

0ðxIIÞ has to vanish at xII ¼ l22, leading

to:

C5 ¼ �l2

2AI

11DIIðþdÞ11 e0ðIÞ

xx

AII11DIIðþdÞ

11 þ BIIðþdÞ11

� �2 DI11 l2

l1�l2ð ÞDIIðþdÞ11 þl2DI

11

� 1� � : ð17Þ

With this result, also the axial displacement uII0ðxIIÞ is known as a

function of the axial strain e0ðIÞxx . The axial displacement uI

0ðxIÞ isdetermined by integration of e0ðIÞ

xx with respect to xI:

uI0ðxIÞ ¼

Ze0ðIÞ

xx dxI ¼ e0ðIÞxx xI þ C6: ð18Þ

The integration constant C6 is found from the requirement that theaxial displacement uI

0ðxIÞ at the loaded edge equals the prescribeddisplacement uinit:

C6 ¼ uinit � e0ðIÞxx

l1 � l22

: ð19Þ

At this point the axial strain e0ðIÞxx is the only remaining unknown

quantity in all displacement formulations. If we examine the axialdisplacement uI;II at the junction between segment I and II closer,we can determine e0ðIÞ

xx as follows:

e0ðIÞxx ¼

uinit � uI;IIðl1�l2Þ

2

: ð20Þ

Therein, the axial displacement uI;II at the junction of segments I andII is determined from the remaining continuity condition:

uII0ðxII ¼ 0Þ ¼ �uI;II: ð21Þ

At this point, the displacement functions uI0; u

II0 , wI

0 and wII0 are fully

known, and the total displacement field can be establishedaccording to the kinematical assumptions of CLPT as:

uICLPTðxI; zÞ ¼ uI

0ðxIÞ � z@wI

0ðxIÞ@xI

; ð22Þ

wICLPTðxIÞ ¼ wI

0ðxIÞ; ð23Þ

uIICLPTðxII; zÞ ¼ uII

0ðxIIÞ � z@wII

0ðxIIÞ@xII

; ð24Þ

wIICLPTðxIIÞ ¼ wII

0ðxIIÞ: ð25Þ

Once the displacements are known, the strains and accordinglythe stresses in each laminate layer can be computed straightfor-wardly. Since the according kinematic and constitutive relationsarewell-known and can be found in any textbook on the

Table 2Geometry parameters of the considered structural situation.

Geometry parameters

l1 200 [mm] Plate lengthl2 100 [mm] Doubler lengthb1 100 [mm] Plate widthb2 50 [mm] Doubler widthd 0.125 [mm] Layer thicknessn 4 [–] Number of layers in plate and doublerh1 0.5 [mm] Plate thicknessh2 0.5 [mm] Doubler thickness

Fig. 5. The finite element model.

356 J.E. Butzke, C. Mittelstedt / Composite Structures 118 (2014) 351–366

mechanics of composite structures (see e.g. Reddy [26 or Jones[11]), this will not be elaborated on in more detail at this point.

2.3. Comparison of the global model with FEM computations

In order to verify the accuracy of the ‘global’ model, a compar-ison is made with the results of a finite element model of thesame structural situation. Since the determined displacementfunctions uI

0;uII0 , wI

0 and wII0 refer to displacements of a specified

reference plane (see Fig. 4), the displacements results accordingto the global model and the FEM are compared in this plane.The results refer to the global coordinate system as indicated inFig. 4. Results will be shown for the case in which both the plateand the doubler have the same layup ½0�=90��S. Results have beengenerated for a number of other layups as well with a similarresults quality which, however, will not be presented here dueto reasons of brevity. The employed material parameters for alaminate layer are given in Table 1. The employed geometry dataare given in Table 2.

A sketch of a section of the employed finite element model ispresented in Fig. 5. As can be seen, the finite element model is veryrefined especially in the regions around the local plate-doublerjunction which is important for the simulation of the interlaminarstress fields that occur around the tip of the junction as well as atthe free edges of the doubler.

Fig. 6 includes results for the displacements u0 and w0 withrespect to the reference plane as indicated in Fig. 4. Obviouslythe agreement between both the FEM results and the results bythe global model is excellent, and the results comply with whatwould be expected from engineering intuition. Accordingly, theaxial displacements u0 exhibit a change in the slope at the plate-doubler junction at x ¼ l2

2 which can be explained by the abruptchange of the extensional stiffness of the considered structure. Asfor the bending deformations, in segment II for 0 < x < l2

2 a negativebending deformation takes place, and the curvature of the bendingline attains a reversed sign for segment I. While the former obser-vation is due to the fact that the considered structure exhibits asudden thickness change which is not symmetric with respect tothe x-axis, the latter observation stems from the applied bendingmoment at the loaded end of segment I in order to enforce thatthe cross-section remains perpendicular to the global x-axis alsoin the loaded state.

A further comparison is made for the intralaminar stresses asthey arise at a certain distance from the plate-doubler junctionwhere the anticipated very local interlaminar stress concentrationsdo not have an influence on the stress field, and where the FEMresults should be matching the analytical CLPT results. A compar-ison of the stress results for rxx and ryy is made at the locationsx ¼ l2

2 þl1�l2

4 and x ¼ l24, i.e. the middle points of segments II and I,

respectively, which is shown in Figs. 7 and 8. Apparently, the com-parison contained in Figs. 7 and 8 shows an excellent agreementbetween both the numerical and the closed-form analytical resultswhich lends credibility to the global model.

Table 1Linear elastic orthotropic material parameters for a laminate layer.

Material parameters

E1 135000 [MPa] Longitudinal Young’s modulusE2 ¼ E3 10000 [MPa] Transverse Young’s modulusG12 ¼ G13 5000 [MPa] Shear modulusG23 3937 [MPa] Shear modulusm12 ¼ m13 ¼ m23 ¼ m32 0.27 [–] Poisson’s ratiom21 ¼ m31 0.02 [–] Poisson’s ratio

Fig. 6. Comparison of the results for the displacements u0 (upper) and w0 (lower)according to FEM and the global model for a plate-doubler structure with the layups½0�=90��S for both plate and doubler.

3. The local model

The global model is based on the assumptions of CLPT which apriori excludes the occurrence of interlaminar stresses at any pointin the laminate. However, at the plate-doubler junction in the

Fig. 7. Comparison of the results for the intralaminar stresses rxx (upper) and ryy

(lower) according to FEM and the global model for a plate-doubler structure withthe layups ½0�=90��S for both plate and doubler. Results were extracted atx ¼ l2

2 þl1�l2

4 through the thickness.

Fig. 8. Comparison of the results for the intralaminar stresses rxx (upper) and ryy

(lower) according to FEM and the global model for a plate-doubler structure withthe layups ½0�=90��S for both plate and doubler. Results were extracted at x ¼ l2

4through the thickness.

J.E. Butzke, C. Mittelstedt / Composite Structures 118 (2014) 351–366 357

vicinity of the location x ¼ l22, significant interlaminar stress fields are

to be expected which of course cannot be assessed with the globalmodel, even though the results of the preceding section have shownthat the global model is highly accurate at those locations where arather uniform state of stress prevails that is in line with the assump-tions of CLPT. The global model is thus upgraded in order to enablethe analysis of interlaminar stress fields that may potentiallybecome even singular at the plate-doubler junction and which of

course need to be taken into account carefully whenever such com-posite structures need to be designed and analyzed.

3.1. Layerwise analysis approach

In order to capture the non-homogeneous stress state in thevicinity of the plate-doubler junction, each physical layer of bothplate and doubler is discretized into a number m of mathematicallayers (Fig. 9). Hence, if both plate and doubler contain n physicallayers, segment I and segment II will consist of nL ¼ mn andnL ¼ 2mn layers, respectively. Note that the discretization onlytakes place with respect to the thickness coordinate.

In each mathematical layer, a separate displacement field ispostulated that consists of the displacements according to the glo-bal solution (22)–(25), and additional displacement functions thatare supposed to capture the special circumstances that arise in thevicinity of the plate-doubler junction and which further exhibit apronounced localized characteristic and a rapidly decaying behav-ior with increasing distance from the plate-doubler junction sothat in regions remote from the junction, the stress field accordingto CLPT is recovered. It is important to note that since in the vicin-ity of the plate-doubler junction the interlaminar stresses are ofinterest it is necessary to consider the transverse displacementsw that arise from the global model due to the transverse straineC

zz (the letter C signifies the contraction of the laminate in thethickness direction) which was not considered in the global model.Considering that in the framework of the assumptions made in theglobal model we have rzz ¼ 0 (plane stress state with respect to thethickness direction) and due to the plane state of strain withrespect to the y-direction we further have eyy ¼ 0, the strain eCðkÞ

zz

in layer ðkÞ can be determined directly from Hooke’s generalizedlaw for an orthotropic material:

rðkÞxx

rðkÞyy

rðkÞzz

sðkÞyz

sðkÞxz

sðkÞxy

0BBBBBBBBB@

1CCCCCCCCCA

|fflfflfflfflffl{zfflfflfflfflffl}rðkÞ

¼

CðkÞ11 CðkÞ12 CðkÞ13 0 0 0

CðkÞ12 CðkÞ22 CðkÞ23 0 0 0

CðkÞ13 CðkÞ23 CðkÞ33 0 0 0

0 0 0 CðkÞ44 0 0

0 0 0 0 CðkÞ55 0

0 0 0 0 0 CðkÞ66

266666666664

377777777775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}CðkÞ

eðkÞxx

eðkÞyy

eðkÞzz

cðkÞyz

cðkÞxz

cðkÞxy

0BBBBBBBBBB@

1CCCCCCCCCCA

|fflfflfflfflffl{zfflfflfflfflffl}eðkÞ

:ð26Þ

Hence:

eCðkÞzz ¼ �CðkÞ13

CðkÞ33

eCLPTðkÞxx : ð27Þ

Therein, the quantities CðkÞ13 and CðkÞ33 are stiffness components in layerðkÞ, and the overbar indicates transformed quantities according tothe respective fiber orientation of the layer. Integration with respectto z leads to:

wðkÞe ðzÞ ¼Z

eCðkÞzz dz ¼ �CðkÞ13

CðkÞ33

ZeCLPTðkÞ

xx dz: ð28Þ

In total, the following displacement field in mathematical layer ðkÞcan be written which consists of the solution according to the globalmodel (uCLPTðx; zÞ;wCLPTðxÞ, see Eqs. (22)–(25)), the transverse dis-placements weðzÞ, and the additional layerwise terms UVARðx; zÞbzw. WVARðx; zÞ that are taken to be functions of both x and z:

uIðkÞ ¼ uIðkÞCLPTðxI; zÞ þ UIðkÞ

VARðxI; zÞ;uIIðkÞ ¼ uIIðkÞ

CLPTðxII; zÞ þ UIIðkÞVAR ðxII; zÞ;

wIðkÞ ¼ wIðkÞCLPTðxIÞ þwIðkÞ

e ðzÞ þWIðkÞVARðxI; zÞ;

wIIðkÞ ¼ wIIðkÞCLPTðxIIÞ þwIIðkÞ

e ðzÞ þWIIðkÞVAR ðxII; zÞ: ð29Þ

Fig. 9. Discretization of the physical layers into a number of mathematical layers at the example of a ½0�=90��S laminate.

358 J.E. Butzke, C. Mittelstedt / Composite Structures 118 (2014) 351–366

The additional displacement functions UVARðx; zÞ and WVARðx; zÞ areset up in such a way that in layer ðkÞ in both interfaces ðkÞ andðkþ 1Þ, a set of a priori unknown displacement functions UðkÞ;W ðkÞ

and Uðkþ1Þ;W ðkþ1Þ, respectively, is postulated that are solely depen-dent on x, while within the layer ðkÞ a linear interpolation betweenthese interface functions is being performed by use of the interpo-lation functions wðkÞ1 and wðkÞ2 that are functions of z only. A graphicalillustration is given in Fig. 10 at the example of UðkÞVARðx; zÞ in layer ðkÞ.Hence, the additional displacement functions UðkÞVARðx; zÞ andW ðkÞ

VARðx; zÞ in the framework of the local model can be written forlayer ðkÞ as follows:

UIðkÞVARðxI; zÞ ¼ UIðkÞðxIÞwðkÞ1 ðzÞ þ UIðkþ1ÞðxIÞwðkÞ2 ðzÞ; ð30Þ

UIIðkÞVAR ðxII; zÞ ¼ UIIðkÞðxIIÞwðkÞ1 ðzÞ þ UIIðkþ1ÞðxIIÞwðkÞ2 ðzÞ; ð31Þ

WIðkÞVARðxI; zÞ ¼WIðkÞðxIÞwðkÞ1 ðzÞ þWIðkþ1ÞðxIÞwðkÞ2 ðzÞ; ð32Þ

WIIðkÞVAR ðxII; zÞ ¼WIIðkÞðxIIÞwðkÞ1 ðzÞ þWIIðkþ1ÞðxIIÞwðkÞ2 ðzÞ: ð33Þ

The linear interpolation functions wðkÞ1 and wðkÞ2 have to be postulatedin a way that they fulfill the following requirements:

wðkÞ1 ðz ¼ zðk�1ÞÞ ¼ 1; wðkÞ1 ðz ¼ zðkÞÞ ¼ 0;

wðkÞ2 ðz ¼ zðk�1ÞÞ ¼ 0; wðkÞ2 ðz ¼ zðkÞÞ ¼ 1; ð34Þ

which leads to the following formulations:

wðkÞ1 ðzÞ ¼1

dðkÞzðkÞ � z� �

; wðkÞ2 ðzÞ ¼1

dðkÞz� zðk�1Þ� �

: ð35Þ

With the total displacement field (29) according to the local modelbeing established, the strain field in each mathematical layer ðkÞ canbe determined. Due to the assumption of a plane state of strain with

Fig. 10. Additional layerwise displacement field according to the local model at theexample of UVARðx; zÞ in layer ðkÞ.

respect to the y-direction and consequently eyy ¼ cyz ¼ cxy ¼ 0, thefollowing strain components remain:

eIðkÞxx ¼ eCLPTIðkÞ

xx þ @UIðkÞ

@xIwðkÞ1 þ

@UIðkþ1Þ

@xIwðkÞ2 ; ð36Þ

eIIðkÞxx ¼ eCLPTIIðkÞ

xx þ @UIIðkÞ

@xIIwðkÞ1 þ

@UIIðkþ1Þ

@xIIwðkÞ2 ; ð37Þ

eIðkÞzz ¼ eCIðkÞ

zz þWIðkÞ @wðkÞ1

@zþWIðkþ1Þ @w

ðkÞ2

@z; ð38Þ

eIIðkÞzz ¼ eCIIðkÞ

zz þWIIðkÞ @wðkÞ1

@zþWIIðkþ1Þ @w

ðkÞ2

@z; ð39Þ

cIðkÞxz ¼ UIðkÞ @w

ðkÞ1

@zþ UIðkþ1Þ @w

ðkÞ2

@zþ @WIðkÞ

@xIwðkÞ1 þ

@WIðkþ1Þ

@xIw2ðkÞ; ð40Þ

cIIðkÞxz ¼ UIIðkÞ @w

ðkÞ1

@zþ UIIðkþ1Þ @w

ðkÞ2

@zþ @WIIðkÞ

@xIIwðkÞ1 þ

@WIIðkþ1Þ

@xIIwðkÞ2 : ð41Þ

From the layerwise strain field the stresses in mathematical layerðkÞ can be determined by virtue of (26).

3.2. Variational statement and governing equations

In order to derive the governing equations for the additionaldisplacement functions UVARðx; zÞ and WVARðx; zÞ, the principle ofminimum elastic potential P ¼ Pi þPa is employed, wherein Pi

is the inner potential, and Pa is the potential according to theapplied loadings. The inner potential of segments I and II can bewritten as:

PIi ¼

12

XnL2

r¼1

ZVIðrÞ

eIðrÞT CIðrÞ

eIðrÞ|fflfflfflffl{zfflfflfflffl}rIðrÞ

dVIðrÞ; ð42Þ

PIIi ¼

12

XnL

r¼1

ZVIIðrÞ

eIIðrÞT CIIðrÞ

eIIðrÞ|fflfflfflfflffl{zfflfflfflfflffl}rIIðrÞ

dVIIðrÞ: ð43Þ

An outer potential only exists in segment I (i.e. PIIa ¼ 0) which can

be determined as the product of the normal force N0ðIÞxx and the axial

displacement uinit:

PIa ¼ �N0ðIÞ

xx uinit: ð44Þ

The total elastic potential P then can be postulated as followswhere it is required for the structure to be in equilibrium that aminimum is attained:

J.E. Butzke, C. Mittelstedt / Composite Structures 118 (2014) 351–366 359

P ¼ PI þPII ¼ 12

XnL2

r¼1

ZVIðrÞ

eIðrÞT CIðrÞ

eIðrÞdVIðrÞzfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{PI

i

� N0ðIÞxx uinit

zfflfflfflfflffl}|fflfflfflfflffl{PIa

þ 12

XnL

r¼1

ZVIIðrÞ

eIIðrÞT CIIðrÞ

eIIðrÞdVIIðrÞ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}PII

i

¼ Min: ð45Þ

Note that with a given displacement uinit , the outer potential PIa is a

constant value so that a minimization of the total potential onlyinvolves the two inner potentials PI

i and PIIi in (45).

Since all state variables have been postulated in a variable-sep-arable form, the volume integral in (45) can be disassembled intothree separate integrals as

RV ð. . .ÞdV ¼

Rx

Ry

Rzð. . .Þdzdydx. Since

the current analysis model is a two-dimensional strip idealizationin a plane state of stress so that all state variable are not dependenton y, the integral with respect to the y-direction can be solved witha unity value. Further, since the additional displacement functionshave been formulated in terms of the interpolation functions wðkÞ1

and wðkÞ2 with respect to z, the integrals concerning z can be solvedfundamentally and exactly in a closed-form analytical manner andtake up certain constant values. As a consequence, the integralterms that need to be minimized reduce to:

12

XnL2

r¼1

Z ðl1�l2 Þ2

0

Z zðrÞ

zðr�1Þ

eIðrÞT CIðrÞ

eIðrÞdz|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}FIðrÞ

dxI ¼ Min; ð46Þ

12

XnL

r¼1

Z l22

0

Z zðrÞ

zðr�1Þ

eIIðrÞT CIIðrÞ

eIIðrÞdz|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}FIIðrÞ

dxII ¼ Min: ð47Þ

Hence, only the integrals with respect to xI and xII remain, and theintegrands FIðkÞ and FIIðkÞ take up the following general form:

FIðkÞ ¼ f xI;UIðkÞ;WIðkÞ;

@UIðkÞ

@xI;@WIðkÞ

@xI

!; ð48Þ

FIIðkÞ ¼ f xII;UIIðkÞ;WIIðkÞ;

@UIIðkÞ

@xII;@WIIðkÞ

@xII

!: ð49Þ

Thus, the additional displacement functions have to be determinedin a way that the integrals in (46) and (47) attain a minimumvalue. The according Euler–Lagrange equations read in the currentcase:

@FI

@UIðkÞ �d

dxI

@FI

@UIðkÞ

@xI

0@

1A ¼ 0;

@FI

@WIðkÞ �d

dxI

@FI

@WIðkÞ

@xI

0@

1A ¼ 0;

@FII

@UIIðkÞ �d

dxII

@FII

@UIIðkÞ

@xII

0@

1A ¼ 0;

@FII

@WIIðkÞ �d

dxII

@FII

@WIIðkÞ

@xII

0@

1A ¼ 0: ð50Þ

Therein, FI and FII are defined as the sums of the layerwise inte-grands FIðkÞ and FIIðkÞ, respectively. After some symbolic manipula-tion, the evaluation of (50) eventually yields:

K1d2

dx2I

UI þ K2d

dxIWI þ K3UI ¼ R1; ð51Þ

K4d2

dx2I

WI þ K5d

dxIUI þ K6WI ¼ R2; ð52Þ

K7d2

dx2II

UII þ K8d

dxIIWII þ K9UII ¼ R3; ð53Þ

K10d2

dx2II

WII þ K11d

dxIIUII þ K12WII ¼ R4: ð54Þ

Eqs. (51)–(54) constitute a set of linear inhomogeneous second-order differential equations with constant coefficient matrices Ki

(i ¼ 1;2; . . . ;12). The matrices Ki and the right-hand side vectorsR1; . . . ;R4 are given in detail in the appendix. The vectorsUI;UII;WI and WII include the additional displacement functionsaccording to the global solution as follows:

UI ¼UIð1Þ

..

.

UIðnL2 Þ

0BB@

1CCA; UII ¼

UIIð1Þ

..

.

UIIðnLÞ

0BB@

1CCA;

WI ¼WIð1Þ

..

.

WIðnL2 Þ

0BB@

1CCA; WII ¼

WIIð1Þ

..

.

WIIðnLÞ

0BB@

1CCA: ð55Þ

If we further condense the vectors of the additional displacementfunctions as

UItot ¼

UI

WI

� �; UII

tot ¼UII

WII

� �; ð56Þ

then the equation system (51)–(54) can be written as:

K1 00 K4

" #zfflfflfflfflfflffl}|fflfflfflfflfflffl{K1;tot

d2

dx2I

UItotþ

0 K2

K5 0

" #zfflfflfflfflfflffl}|fflfflfflfflfflffl{K2;tot

ddxI

UItotþ

K3 00 K6

" #zfflfflfflfflfflffl}|fflfflfflfflfflffl{K3;tot

UItot¼

0R2

� �; ð57Þ

K7 00 K10

" #|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

K7;tot

d2

dx2II

UIItotþ

0 K8

K11 0

" #|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

K8;tot

ddxII

UIItotþ

K9 00 K12

" #|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

K9;tot

UIItot¼

0R4

� �: ð58Þ

Therein, the matrices K1;tot , K2;tot and K3;tot are of the dimension2 nL

2 þ 1� �

� 2 nL2 þ 1� �

. The coefficient matrices of segment II K7;tot ,K8;tot and K9;tot have the dimensions 2 nL þ 1ð Þ � 2 nL þ 1ð Þ.

3.3. Solution and boundary conditions

The unknown additional displacement functions can be deter-mined by solving the differential equation systems (57) and (58).Firstly, the homogeneous form of (57) and (58) is treated. A solu-tion for Uhom can be sought in the following general form:

Uhom ¼ Aekx: ð59Þ

Inserting (59) into (57) and (58), the exponential terms cancel outaltogether, and two quadratic eigenvalue problems are achieved:

kI2K1;tot þ kIK2;tot þ K3;tot

� �AI ¼ 0; ð60Þ

kII2K7;tot þ kIIK8;tot þ K9;tot

� �AII ¼ 0: ð61Þ

Therein, A plays the role of the eigenvector, and k is the associatedeigenvalue. In the framework of this contribution, these two eigen-value problems have been solved numerically using a commercialprogramming language. The solution of (60) leads to 4 nL

2 þ 1� �

eigenvalues, while (61) yields 4 nL þ 1ð Þ eigenvalues. It is importantto note that all eigenvalues appear in pairs with opposite signs.Since the additional displacement functions in the framework ofthe local solution should display the typical decaying behavior ofthe state variables in the vicinity of the plate-doubler junction, onlythose eigenvalues with a negative sign are of interest for the solu-tion (59). Hence, only 2 nL

2 þ 1� �

eigenvalues from (60) and2 nL þ 1ð Þ eigenvalues from (61) remain.

It was found that during the solution of (60) and (61) severaleigenvalues with zero values occurred. Such eigenvalues corre-spond to rigid body translations and additional strain states thatare physically meaningless in the framework of the current analy-sis approach. Hence, in order to overcome this problem, a slight

Fig. 11. Structural paths for stress evaluation.

360 J.E. Butzke, C. Mittelstedt / Composite Structures 118 (2014) 351–366

numerical manipulation of the coefficient matrices has been per-formed such that the main and secondary diagonals of Ki are chan-ged by small amounts. Defining the quantities

DðkÞ1 ¼Z zðk�1Þ

zðk�2Þ

wðk�1Þ1 wðk�1Þ

2 dz; ð62Þ

DðkÞ2 ¼Z zðk�1Þ

zðk�2Þ

wðk�1Þ1 wðk�1Þ

2 dzþZ zðkÞ

zðk�1Þ

wðkÞ1 wðkÞ2 dz; ð63Þ

DðkÞ3 ¼Z zðkÞ

zðk�1Þ

wðkÞ1 wðkÞ2 dz; ð64Þ

assembling (62)–(64) for segment I in a matrix as

KID ¼

Dð1Þ2 Dð1Þ3 0 0 0

Dð2Þ1 Dð2Þ2 Dð2Þ3 0 0

..

. ... . .

. ... ..

.

0 0 DðnL

2�1Þ1 D

ðnL2�1Þ

2 DðnL

2�1Þ3

0 0 0 DðnL

2 Þ1 D

ðnL2 Þ

2

26666666664

37777777775; ð65Þ

and in segment II as

KIID ¼

Dð1Þ2 Dð1Þ3 0 0 0

Dð2Þ1 Dð2Þ2 Dð2Þ3 0 0

..

. ... . .

. ... ..

.

0 0 DðnL�1Þ1 DðnL�1Þ

2 DðnL�1Þ3

0 0 0 DðnLÞ1 DðnLÞ

2

2666666664

3777777775; ð66Þ

the original coefficient matrices are manipulated using (65) and(66) as follows:

Kmodi ¼ Ki � kkKD: ð67Þ

Therein, kk is an arbitrary factor that is chosen as small as it is pos-sible in order to avoid eigenvalues with zero values. Since the rele-vant eigenvalues are rather high when compared to the resultantsmall manipulated eigenvalues, this numerical manipulation doesnot have a significant effect on the resultant stress fields. For allcomputations contained in the results section, a value ofkk ¼ �10�3 has been used.

Eventually, the total homogeneous solution can be written as:

UIhom ¼

Xr¼2nL2þ1ð Þ

r¼1

bIrA

Ire

kIr xI ; UII

hom ¼Xr¼2 nLþ1ð Þ

r¼1

bIIr AII

r ekIIr xII : ð68Þ

Herein, bIr and bII

r are free constants that will be adjusted to theboundary and continuity conditions at the plate-doubler junction.

Concerning a particular solution of (57) and (58) it is importantto note that the right-hand side terms are constant quantities.Hence, a solution approach could be sought in the form of constantterms which in turn would again lead to rigid body motions andadditional strain states. For this reason, the particular solution willbe discarded:

Upart ¼ 0: ð69Þ

In order to determine the free constants bIr and bII

r , a total of2 nL

2 þ 1� �

equations for segment I and 2 nL þ 1ð Þ equations for seg-ment II are necessary. Firstly, the additional displacement functionshave to be equal in the interface between segments I and II for allmathematical layers within the plate. Writing the vectors UI

hom

and UIIhom of the homogeneous solutions as

UIhom ¼

UI

WI

� �; UII

hom ¼UII

WII

� �; ð70Þ

we may formulate identical displacements at x1 ¼ 0 and x2 ¼ 0 as:

UIðxI ¼ 0Þ ¼ �UIIðxII ¼ 0Þ; WIðxI ¼ 0Þ ¼WIIðxII ¼ 0Þ: ð71Þ

This gives 2 nL2 þ 1� �

equations.Secondly, the stresses rI

xx and rIIxx as well as sI

xz and sIIxz have to

be continuous across the interface between segment I and II. Thisrequirement cannot be fulfilled in a pointwise fashion so that anintegral formulation employing the linear interpolation functionsis used instead:Z zðk�1Þ

zðk�2Þ

rIðk�1Þxx ðxI¼0Þwðk�1Þ

2 dzþZ zðkÞ

zðk�1Þ

rIðkÞxx ðxI¼0ÞwðkÞ1 dz¼

Z zðk�1Þ

zðk�2Þ

rIIðk�1Þxx ðxII¼0Þwðk�1Þ

2 dzþZ zðkÞ

zðk�1Þ

rIIðkÞxx ðxII¼0ÞwðkÞ1 dz; ð72Þ

Z zðk�1Þ

zðk�2Þ

sIðk�1Þxz ðxI ¼ 0Þwðk�1Þ

2 dzþZ zðkÞ

zðk�1Þ

sIðkÞxz ðxI ¼ 0ÞwðkÞ1 dz ¼

�Z zðk�1Þ

zðk�2Þ

sIIðk�1Þxz ðxII ¼ 0Þwðk�1Þ

2 dz�Z zðkÞ

zðk�1Þ

sIIðkÞxz ðxII ¼ 0ÞwðkÞ1 dz: ð73Þ

This constitutes an additional 2 nL2 þ 1� �

equations.Thirdly and lastly, at the free edge of the doubler the stress

components rIIxx and sII

xz have to vanish. Again using an integral for-mulation as followsZ zðk�1Þ

zðk�2Þ

rIIðk�1Þxx ðxII ¼ 0Þwðk�1Þ

2 dzþZ zðkÞ

zðk�1Þ

rIIðkÞxx ðxII ¼ 0ÞwðkÞ1 dz ¼ 0; ð74Þ

Z zðk�1Þ

zðk�2Þ

sIIðk�1Þxz ðxII ¼ 0Þwðk�1Þ

2 dzþZ zðkÞ

zðk�1Þ

sIIðkÞxz ðxII ¼ 0ÞwðkÞ1 dz ¼ 0 ð75Þ

gives another 2 nL2

� �equations for the determination of the free con-

stants bIr and bII

r .This completes the derivation of the global solution.

4. Verification, results and discussion

In this section, the current analysis model will be checked forthe quality of its results by comparison with accompanying finiteelement simulations. Results will be presented for plate and dou-bler structures with the symmetric cross-ply layups ½0�=90��S and½90�=0��S. Results will be evaluated along the interfaces 0–3 ofthe doubler as shown in Fig. 11, as well as through the thicknessas it is also indicated in Fig. 11.

4.1. Convergence behavior

In order to verify the convergence behavior of the derived anal-ysis model with an increasing number m of mathematical layers,Fig. 12 shows results for the interlaminar stresses rzz and sxz inthe interfaces 3 and 1, respectively, for a ½90�=0��S laminate. Itcan be observed that for all tested degrees of discretization theresults match very well, with only some slight deviations whencomparing the results for m ¼ 5 with those generated usingm ¼ 10 and m ¼ 15. Further layups have been tested which willnot be discussed at this point for reasons of brevity. The outcome

Fig. 12. Interlaminar stresses rzz (upper) and sxz (lower) in the interfaces 3 and 1 ofthe doubler for different numbers m of mathematical layers; plate and doubler bothhave the layup ½0�=90��S .

Fig. 13. Interlaminar normal stress rzz in the interfaces 0; . . . ;3, counted from topto bottom; Plate and doubler: ½0�=90��S layups.

J.E. Butzke, C. Mittelstedt / Composite Structures 118 (2014) 351–366 361

of all convergence studies, however, has been the same in all cases,so that is seems sufficient to use m ¼ 15 mathematical layers perphysical layer for the subsequent discussion.

4.2. Results and discussion

4.2.1. Plate and doubler with ½0�=90��S layupsFig. 13 shows results for the interlaminar normal stress rzz in

the interfaces 0; . . . ;3 and a comparison to the corresponding finiteelement results. Obviously, the agreement between the results byboth analysis methods is excellent which lends credibility to thepresently derived analysis approach. Deviations are observed onlyin interface 0 directly in the vicinity of the expected stress singu-larity at xII ¼ 0 which is an expected and typical outcome. Thestress values given in Fig. 13 show that the most severe interlam-inar normal stresses rzz occur in interface 0, i.e. in the interfacebetween plate and doubler which is also an expected result. Theresults in interfaces 1, 2, and 3 show that rzz also arises at theselocations due to the free-edge that occurs at the doubler’s edges,but that the resultant stress values are much lower than in inter-face 0. Hence, the most probable location for a delamination failurefor this specimen and load case will be directly in the interfacebetween plate and doubler in the vicinity of the plate-doublerjunction.

Fig. 14 includes analogous results for the interlaminar shearstress sxz. In all, the conclusions that were already drawn fromthe discussion of Fig. 13 also hold in the case of the interlaminarshear stress sxz so that a renewed discussion can be omitted at thispoint. However, the excellent agreement between the numericalresults and the current analysis method should be highlighted.

In order to gain some insight about the stress distributionsthrough the thickness of the plate-doubler junction, Fig. 15 includesresults for the normal stresses rxx;ryy, and rzz as well as for theinterlaminar shear stress sxz at xII ¼ 0. For all stress components,

the current analysis method delivers results that match the finiteelement results very closely which is a satisfying outcome. How-ever, in the near field of the plate-doubler junction (i.e. in interface0 and its closer vicinity), certain deviations and further somediscontinuities of the stress distributions can be observed. Thisphenomenon can be explained as follows. The present layerwiseapproach constitutes a C0-continuous displacement field but agenerally discontinuous stress field with respect to the thickness

Fig. 14. Interlaminar shear stress sxz in the interfaces 0; . . . ;3, counted from top tobottom; Plate and doubler: ½0�=90��S layups.

Fig. 15. Stresses through the thickness at the plate-doubler junction; Plate anddoubler: ½0�=90��S layups.

362 J.E. Butzke, C. Mittelstedt / Composite Structures 118 (2014) 351–366

direction due to the discretization procedure that has beenemployed in the z-direction. As a consequence, the present analysismethod predicts two stress values for each stress component ateach interface between two adjacent mathematical layers ðkÞ andðkþ 1Þ, namely one at the upper edge of layer ðkÞ, and one at thelower edge of layer ðkþ 1Þ. It is important to note that a ratherstrong stress singularity exists in the vicinity of the plate-doublerjunction where the gradients of the stress field are very pronounced

as already became clear from the results contained in 13. At suchlocations where the influence of the stress singularity is notsignificant, the two stress values at the interface between twoadjacent mathematical layers ðkÞ and ðkþ 1Þ generally agree verywell and the distribution of the stresses are rather smooth, howeverdiscontinuities in the stress distributions such as they can beobserved in Fig. 15 near the plate-doubler junction are to beexpected, even though their spatial influence usually decreaseswith an increasing number of mathematical layers. It is importantto note that this issue also arises within the finite element methodas long as standard displacement based solid elements areemployed. The reason why this does not become apparent hereis that generally all commercial finite element codes use a

Fig. 16. Interlaminar normal stress rzz in the interfaces 0; . . . ;3, counted from topto bottom; Plate and doubler: ½90�=0��S layups.

Fig. 17. Interlaminar shear stress sxz in the interfaces 0; . . . ;3, counted from top tobottom; Plate and doubler: ½90�=0��S layups.

J.E. Butzke, C. Mittelstedt / Composite Structures 118 (2014) 351–366 363

smoothening of the stress results for graphical display using meanvalues for the stresses which, however, has not been implementedin the present analysis approach. Further, due to the expected steepstress gradients quadratic elements and a strong mesh refinementin the vicinity of the stress singularities have been employed, thusgiving an advantage to the finite element method. In all, the qualityof the results is very satisfying, the observed deviations are to beexpected, and the stress distributions could be smoothened to a

certain degree if mean values of the two stress results at the inter-faces between two mathematical layers were used which, however,has not been implemented in the current approach. Further notethat the stress components rxx and sxy satisfy the boundaryconditions of traction-free doubler edges very well, even thoughthese boundary conditions have been formulated in an integralsense. Deviations again occur in the vicinity of the plate-doublerjunction which is to be expected.

364 J.E. Butzke, C. Mittelstedt / Composite Structures 118 (2014) 351–366

4.2.2. Plate and doubler with ½90�=0��S layupsIn this subsection, the current analysis approach is verified for a

structure where the plate and the doubler both consist of symmet-ric ½90�=0��S cross-ply laminates. Fig. 16 shows results for the inter-laminar normal stress rzz in the interfaces 0; . . . ;3. As in the case ofthe ½0�=90��S laminates, the agreement between the numericalresults and the present analysis method is excellent. Analogously,Fig. 17 includes results for the interlaminar shear stress sxy. Again,the agreement with the numerical results as well as the fulfillmentof the boundary conditions of traction-free doubler edges is verysatisfying. As a closure, Fig. 18 shows results for the stress compo-nents rxx;ryy;rzz and sxz through the thickness of the specimen atxII ¼ 0. As in the case of the ½0�=90��S laminates, the agreement is

Fig. 18. Stresses through the thickness at the plate-doubler junction; Plate anddoubler: ½90�=0��S layups.

generally very satisfying, and in the vicinity of the plate-doublerjunction, the results by the finite element computations and thepresent analysis method match in an averaged sense.

5. Summary and conclusions

In this paper, an analysis method for the determination of dis-placements, strains and stresses in a cross-ply composite plateunder longitudinal tensile load that is padded up by a cross-plydoubler has been presented. The analysis method consists of twoparts, namely a ‘global’ solution that is based on the theoreticalframework of CLPT, and a ‘local’ model that takes the inherentlythree-dimensional stress state in the vicinity of the plate-doublerjunction into account. While the former solution is of a completelyclosed-form analytical nature due to its relative simplicity, the lat-ter solution relies on a discretization of the physical layers into anumber of mathematical layers which eventually leads to aquadratic eigenvalue problem that requires numerical evaluation.Hence, it is adequate to speak of a semi-analytical analysis method.It has been shown that the current analysis model delivers verysatisfying results when compared to full-scale finite element sim-ulations, however with only a fraction of the computational timeand effort that are needed for the FEM analyses. As a consequence,the presented analysis approach can be recommended for allengineering applications where such plate-doubler structuresneed to be designed and analyzed in a rapid and yet reliablemanner.

Further work should be invested into the modeling and analy-sis of similar structural situations where layups different from thecurrently considered cross-ply laminates can be assessed as well.This would also necessitate the explicit consideration of the typ-ical coupling effects on the global scale, wherein bending-twistingcoupling will probably be the one with the highest practical sig-nificance due to the preferred application of quasi-isotropic andsymmetric laminates in aerospace engineering. Apart from that,additional and potentially more complex load cases should beconsidered and introduced into the analysis method as well.Examples would be constant pressure on one of the surfaces ofthe plate-doubler structure (and thus representing a pressurizedaircraft fuselage), pure bending, or transverse single forces atarbitrary positions of the structure. Further, the currently appliedset of boundary conditions would be representative for a tensiletest specimen. However, in a real lightweight structure theboundary conditions can be far more complex, and thus theadaption of the presented analysis method to different boundaryconditions would be an important topic as well. Lastly, it shouldbe mentioned that the currently considered rectangular doublersrepresent only one of many other possible doubler shapes, and itshould be a long-term goal to establish closed-form analysismethods for the investigation of doublers with arbitraryplanforms.

Appendix A. Appendix

Using the abbreviations

EðkÞopcd ¼Z zk

zðk�1Þ

CðkÞop wðkÞc wðkÞd dz; ð76Þ

FðkÞopcd ¼Z zk

zðk�1Þ

CðkÞop wðkÞc@wðkÞd

@zdz; ð77Þ

GðkÞopcd ¼Z zk

zðk�1Þ

CðkÞop

@wðkÞc

dz@wðkÞd

dzdz; ð78Þ

the coefficient matrices K1; . . . ;K12 can be written as:

J.E. Butzke, C. Mittelstedt / Composite Structures 118 (2014) 351–366 365

K1¼�

Eð1Þ1111 Eð1Þ1112 0 0 0

Eð1Þ1112 Eð1Þ1122þEð2Þ1111 Eð2Þ1112 0 0

..

. ... . .

. ... ..

.

0 0 EðnL

2 �1Þ1112 E

ðnL2 �1Þ

1122 þEðnL

2 Þ1111 E

ðnL2 Þ

1112

0 0 0 EðnL

2 Þ1112 E

ðnL2 Þ

1122

2666666664

3777777775; ð79Þ

K2¼

Fð1Þ5511�Fð1Þ1311 Fð1Þ5521�Fð1Þ1312 0 0 0

Fð1Þ5512�Fð1Þ1321 Fð2Þ5511þFð1Þ5522 Fð2Þ5521�Fð2Þ1312 0 0

�Fð2Þ1311�Fð1Þ1322

..

. ... . .

. ... ..

.

0 0 FðnL

2 �1Þ5512 �F

ðnL2 �1Þ

1321 FðnL

2 Þ5511þF

ðnL2 �1Þ

5522 FðnL

2 Þ5521�F

ðnL2 Þ

1312

�FðnL

2 Þ1311�F

ðnL2 �1Þ

1322

0 0 0 FðnL

2 Þ5512�F

ðnL2 Þ

1321 FðnL

2 Þ5522�F

ðnL2 Þ

1322

2666666666666664

3777777777777775; ð80Þ

K3¼

Gð1Þ5511 Gð1Þ5512 0 0 0

Gð1Þ5512 Gð1Þ5522þGð2Þ5511 Gð2Þ5512 0 0

..

. ... . .

. ... ..

.

0 0 GðnL

2 �1Þ5512 G

ðnL2 �1Þ

5522 þGðnL

2 Þ5511 G

ðnL2 Þ

5512

0 0 0 GðnL

2 Þ5512 G

ðnL2 Þ

5522

2666666664

3777777775; ð81Þ

K4¼�

Eð1Þ5511 Eð1Þ5512 0 0 0

Eð1Þ5512 Eð1Þ5522þEð2Þ5511 Eð2Þ5512 0 0

..

. ... . .

. ... ..

.

0 0 EðnL

2 �1Þ5512 E

ðnL2 �1Þ

5522 þEðnL

2 Þ5511 E

ðnL2 Þ

5512

0 0 0 EðnL

2 Þ5512 E

ðnL2 Þ

5522

2666666664

3777777775; ð82Þ

K5¼

Fð1Þ1311�Fð1Þ5511 Fð1Þ1321�Fð1Þ5512 0 0 0

Fð1Þ1312�Fð1Þ5521 Fð1Þ1322þFð2Þ1311 Fð2Þ1321�Fð2Þ5512 0 0

�Fð2Þ5511�Fð1Þ5522

..

. ... . .

. ... ..

.

0 0 FðnL

2 �1Þ1312 �F

ðnL2 �1Þ

5521 FðnL

2 �1Þ1322 þF

ðnL2 Þ

1311 FðnL

2 Þ1321�F

ðnL2 Þ

5512

�FðnL

2 Þ5511�F

ðnL2 �1Þ

5522

0 0 0 FðnL

2 Þ1312�F

ðnL2 Þ

5521 FðnL

2 Þ1322�F

ðnL2 Þ

5522

2666666666666664

3777777777777775; ð83Þ

K6¼

Gð1Þ3311 Gð1Þ3312 0 0 0

Gð1Þ3312 Gð1Þ3322þGð2Þ3311 Gð2Þ3312 0 0

..

. ... . .

. ... ..

.

0 0 GðnL

2 �1Þ3312 G

ðnL2 �1Þ

3322 þGðnL

2 Þ3311 G

ðnL2 Þ

3312

0 0 0 GðnL

2 Þ3312 G

ðnL2 Þ

3322

2666666664

3777777775; ð84Þ

K7¼�

Eð1Þ1111 Eð1Þ1112 0 0 0

Eð1Þ1112 Eð2Þ1111þEð1Þ1122 Eð2Þ1112 0 0

..

. ... . .

. ... ..

.

0 0 EðnL�1Þ1112 EðnL Þ

1111þEðnL�1Þ1122 EðnL Þ

1112

0 0 0 EðnL Þ1112 EðnL Þ

1122

266666664

377777775; ð85Þ

K8¼

Fð1Þ5511�Fð1Þ1311 Fð1Þ5521�Fð1Þ1312 0 0 0

Fð1Þ5512�Fð1Þ1321 Fð1Þ5522þFð2Þ5511 Fð2Þ5521�Fð2Þ1312 0 0

�Fð2Þ1311�Fð1Þ1322

..

. ... . .

. ... ..

.

0 0 FðnL�1Þ5512 �FðnL�1Þ

1321 FðnL�1Þ5522 þFðnL Þ

5511 FðnL Þ5521�FðnL Þ

1312

�FðnL Þ1311�FðnL�1Þ

1322

0 0 0 FðnL Þ5512�FðnL Þ

1321 FðnL Þ5522�FðnL Þ

1322

266666666666664

377777777777775; ð86Þ

K9¼

Gð1Þ5511 Gð1Þ5512 0 0 0

Gð1Þ5512 Gð2Þ5511þGð1Þ5522 Gð2Þ5512 0 0

..

. ... . .

. ... ..

.

0 0 GðnL�1Þ5512 GðnL Þ

5511þGðnL�1Þ5522 GðnL Þ

5512

0 0 0 GðnL Þ5512 GðnL Þ

5522

266666664

377777775; ð87Þ

K10¼�

Eð1Þ5511 Eð1Þ5512 0 0 0

Eð1Þ5512 Eð1Þ5522þEð2Þ5511 Eð2Þ5512 0 0

..

. ... . .

. ... ..

.

0 0 EðnL�1Þ5512 EðnL�1Þ

5522 þEðnL Þ5511 EðnL Þ

5512

0 0 0 EðnL Þ5512 EðnL Þ

5522

2666666664

3777777775; ð88Þ

K11¼

Fð1Þ1311�Fð1Þ5511 Fð1Þ1321�Fð1Þ5512 0 0 0

Fð1Þ1312�Fð1Þ5521 Fð1Þ1322þFð2Þ1311 Fð2Þ1321�Fð2Þ5512 0 0

�Fð2Þ5511�Fð1Þ5522

..

. ... . .

. ... ..

.

0 0 FðnL�1Þ1312 �FðnL�1Þ

5521 FðnL�1Þ1322 þFðnL Þ

1311 FðnL Þ1321�FðnL Þ

5512

�FðnL Þ5511�FðnL�1Þ

5522

0 0 0 FðnL Þ1312�FðnL Þ

5521 FðnL Þ1322�FðnL Þ

5522

2666666666666664

3777777777777775; ð89Þ

K12¼

Gð1Þ3311 Gð1Þ3312 0 0 0

Gð1Þ3312 Gð1Þ3322þGð2Þ3311 Gð2Þ3312 0 0

..

. ... . .

. ... ..

.

0 0 GðnL�1Þ3312 GðnL�1Þ

3322 þGðnL Þ3311 GðnL Þ

3312

0 0 0 GðnL Þ3312 GðnL Þ

3322

2666666664

3777777775: ð90Þ

The non-vanishing right-hand side vectors read:

R2 ¼

CIðkÞ13111

CIðk�1Þ13112 þ CIðkÞ

13111

..

.

CIðnL

2 Þ13112 þ C

ðnL2þ1Þ

13111

CIðnL

2þ1Þ13112

0BBBBBBBBB@

1CCCCCCCCCA; R4 ¼

CIIðkÞ13111

CIIðk�1Þ13112 þ CIIðkÞ

13111

..

.

CIIðnLÞ13112 þ CIIðnLþ1Þ

13111

CIIðnLþ1Þ13112

0BBBBBBBB@

1CCCCCCCCA: ð91Þ

Therein, the following abbreviation has been employed:

CðkÞopijc ¼Z zðkÞ

zðk�1Þ

CðkÞop eCLPTðkÞij

@wðkÞc

@zdz: ð92Þ

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