research article estimation of local delamination...

Research ArticleEstimation of Local Delamination Buckling in OrthotropicComposite Plates Using Kirchhoff Plate Finite Elements

Zoltaacuten Juhaacutesz and Andraacutes Szekreacutenyes

Department of Applied Mechanics Budapest University of Technology and Economics Muegyetem rkp 5Building MM 1111 Budapest Hungary

Correspondence should be addressed to Andras Szekrenyes szekimmbmehu

Received 3 May 2015 Accepted 13 July 2015

Academic Editor Paolo Lonetti

Copyright copy 2015 Z Juhasz and A Szekrenyes This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We analyse the buckling process of composite plateswith through-the-width delamination and straight crack front applying uniaxialcompressionWe are focusing on themixedmode buckling case where the non-uniform distribution of the in-plane forces controlsthe occurence of the buckling of the delaminated layers For the analysis semi-discrete finite elements will be derived based on theLevy-type method The method of harmonic balance is used for taking into account the force distribution that is generally nonuniform in-plane

1 Introduction

Nowadays fiber-reinforced composite materials are appliedeverywhere in the industrial products such as cars airplanes ships space vehicles and sport equipment [1 2]The heterogeneous behaviour of these materials makes theirdamage analysis very important [3ndash6] One of the mainfailure modes of layered composites is the delaminationfracture which can occur because ofmachining errors or lowvelocity impact during the service life [7ndash16] Buckling causedby uniaxial compression is one of the critical failures for thesematerials [17ndash24]

Many researchers have studied the field of delaminatedcomposite plate buckling experimentally analytically usingfinite element method (FEM) Chai et al [25] have developedan analytical one-dimensional model for simulating bucklingof delaminated beam or plate specimens The delaminationwas assumed to be close to the surface (thin-film approach)and mixed buckling modes have been determined On abeam-plate model it was shown that the transverse sheareffect reduces the critical buckling loads [26]The effect of theboundary conditions was investigated on a 1D model takinginto account the large deformations and the results werecompared with the thin-film approximation and experiments[27] Anastasiadis and Simitses [28] presented a modified 1D

model for improving the critical buckling loads They usedartificial springs along the crack line for preventing crackopening which would result in inadmissible mode shapes[29] Ovesy et al [30] developed a layerwise theory basedon the First-Order Shear Deformation Theory (FSDT) foranalysing the postbuckling behaviour of multiple delami-nated laminates In order to prevent the inadmissible modeshapes they used contact constraints on the delaminated areaFor the double delaminated beam-plate model an analyticalsolution was introduced by Shu using a constrained modelfor the global buckling calculation [31] Kim and Kedward[32] used the Classical Laminate Plate Theory (CLPT) formodelling a delaminated rectangular plate and analysed itsbehaviour with respect to the buckling using the Naviermethod For the global stability analysis the delaminatedregion was treated as a reduced stiffness zone The localstability analysis was carried out on a clamped plate assumingthe same load along the local part as for the global modelThedelamination buckling and delamination growth of cross-plylaminates were examined induced by low velocity impact in[33] The results of the optical measurement were comparedwith X-ray nondestructive evaluation results Wang and Lu[34] examined the failure mechanism of near the surfaceembedded delaminations under compression load using theenergymethod and experimental testingThrough-the-width

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 749607 14 pageshttpdxdoiorg1011552015749607

2 Mathematical Problems in Engineering

delaminated composite laminates subjected to compressiveloads were investigated using the Rayleigh-Ritz method andCLPT theory by Kharazi and Ovesy [35] Experiments onmultiple delaminated composite plates and a FEM studyusing ANSYS were carried out by Aslan and Sahin [36]Kharazi developed a layerwise theory based on the FSDTwhere the prevention of the penetration of the delaminatedlayers was solved by using the penalty method and artificialsprings [37] The mixed mode buckling was captured with acontact model in the paper of Hwang and Liu [38] A layer-wise FEMsolution usingAbaqus andVICONOPT is given forcomposite plates with embedded rectangular delamination inDamghani et al [39] Marjanovic and Vuksanovic [40] usedthe Generalized Layerwise Plate Theory of Reddy and builta finite element (FE) model which was capable of handlingmore embedded delaminations in the composite structureA 3D FEM model was developed for analysing the bucklingbehaviour of embedded and through-the-width delaminatedplates in [41]

For the global model an analytical solution was presentedin Juhasz and Szekrenyes [42] In that paper it was shown thatthe in-plane force distribution greatly influences the bucklingbehaviour of the plate In this paper a FEmodel is introducedwhich concerns a plate with two simply supported oppositeedges so the plate problem cannot be solved using beamtheories The other two boundary conditions (BCs) can bearbitrarily chosen The plate is loaded with uniaxial com-pression The critical in-plane loads for the global bucklingare calculated using a constrained model [29 31] Using thisapproach there is no need for using constraints along thedelamination which results in a very efficient calculationmethod contrary to the widely used contact models Thecontinuity between the delaminated andundelaminated partsis maintained using special transitional elementsThis modelis capable of predicting the global critical buckling loads anddetermines the corresponding in-plane force distributionsUsing the solution of the global model the local stability isanalysed separately using a local FE model for each delami-nated part Because of the nonuniformity in the distributionof axial forces the stability analysis was carried out usingthe method of harmonic balance [43] It was shown thatthese effects greatly influence the local buckling loads In thispaper the effects of different type of boundary conditionsare analysed in a numerical example where the length ofthe delamination is varied and the local and global criticalamplitudes are calculated which results in a stability mapfor the different BCs It is shown that the different BCsgreatly influence the buckling loads and the correspondingmixed mode buckling shapes Using our method and theresulting stability maps the failure process of a compresseddelaminated plate can be estimated and it can be determinedat which point will the plate buckle globally or locally or inmixed mode case

2 Model Creation

Let us consider a layered plate with several orthotropic pliesand a closed through-the-width delamination (see Figure 1)

z

y

x

b aL

2

3

1

Simply supported

DelaminationNload

x

tt tb

L1

L3

t

Figure 1 Simply supported layered plate with delamination sub-jected to uniaxial compression

The plate length is 119871 the length of the delamination is 119886and the thickness of the plate is 119905 The plate is composed oftwo equivalent single layers (ESL) with thicknesses 119905

119905and 119905119887

respectivelyThe edges parallel to the 119909-axis are simply supported and

the constraint of the other two can be arbitrarily chosen Theplate is subjected to uniaxial compression in the midplaneWe are considering thin structures therefore the solution isbased on CLPT [44ndash47]

119906 (119909 119910 119911) = 0 (119909 119910) + 119906120575

0 (119909 119910) minus 1199111205971199080 (119909 119910)

120597119909

V (119909 119910 119911) = V0 (119909 119910) + V120575

0 (119909 119910) minus 119911120597119908120575

0 (119909 119910)

120597119910

119908 (119909 119910 119911) = 1199080 (119909 119910)

(1)

where 0 and V0 are constant parts in the in-plane displace-ment functions of the undelaminated portions which arisebecause of the kinematic coupling on the interface as Figure 2shows 1199061205750 and V

120575

0 are constant through the thickness in the topand bottom parts where 120575 takes either ldquo119905rdquo or ldquo119887rdquo respectively(see Figure 2) Moreover 1199080 is the deflection of the plateparts respectively Along the undelaminated parts we assumeperfect adherence and no crack propagation at the cracktips Therefore the continuity of the undelaminated portionsis ensured using the system of exact kinematic conditions(SEKC) [48]

11990611990510038161003816100381610038161003816119911=minus119905

1199052 = 119906

11988710038161003816100381610038161003816119911=1199051198872

V11990510038161003816100381610038161003816119911=minus119905

1199052 = V119887

10038161003816100381610038161003816119911=1199051198872

(2)

And the in-plane displacements in the global reference planeare

119905119905+ 119905119887

2

le 119905119887

11990611988710038161003816100381610038161003816119911=119905

1199052 = 0 V119887

10038161003816100381610038161003816119911=1199051199052 = V0

ge 119905119887

1199061199051003816100381610038161003816119911=minus119905

1198872 = 0 V1199051003816100381610038161003816119911=minus119905

1198872 = V0

(3)

From the literature it is well known that a free modelallows the intersection of the delaminated portions into

Mathematical Problems in Engineering 3

Xx

x

InterfaceUndelaminated portions

Yy

y

Undelaminated portion

Delamination

Delaminated portion

Bottom platereference plane

Top platereference plane

tt

tb

0 u0

t(x y z)

ut(x y z)

ub(x y z)b(x y z)

Z z Z z

Figure 2 Cross-sections and deformations of the top and bottom elements of an unsymmetrically delaminated orthotropic plate

each other which results in kinematically inadmissible modeshapes and wrong critical loads [29 37 49ndash51] Avoidingthis a constrained model is used where the deflection ofthe delaminated top and bottom portions is common Usingthese assumptions the strain fields of the undelaminatedportions can be given as

1205980119909

1205980119910

1205740119909119910

119905

=

0119909 minus119905119887

21199080119909119909

V0119910 minus119905119887

21199080119910119910

0119910 + V0119909 minus 1199051198871199080119909119909

119905

1205981119909

1205981119910

1205741119909119910

119905

=

minus1199080119909119909

minus1199080119910119910

minus21199080119909119910

119905

1205980119909

1205980119910

1205740119909119910

119887

=

0119909 +119905119905

21199080119909119909

V0119910 +119905119905

21199080119910119910

0119910 + V0119909 + 1199051199051199080119909119909

119887

1205981119909

1205981119910

1205741119909119910

119887

=

minus1199080119909119909

minus1199080119910119910

minus21199080119909119910

119887

(4)

And for the delaminated portions the strain field will be

1205980119909

1205980119910

1205740119909119910

=

119906120575

0119909

V12057501199101199060119910 + V1205750119909

1205981119909

1205981119910

1205741119909119910

=

minus1199080119909119909

minus1199080119910119910

minus21199080119909119910

(5)

where 0 means the constant and 1 means the linear partof the strain field in terms of the 119911 coordinate [52 53]For expressing the stresses of the composite laminate withthe strain fields we have to use the constitutive equation ofcomposite laminates (CEL) [52ndash55]

NM

120575

= [

A BB D

]

120575

1205980

1205981

120575

(6)

where the matrices A B and D can be calculated based onthe literature

Using these the strain energy density of the delaminatedand undelaminated portions can be given as [17 52 56]

119906 =

119873119909

119873119910

119873119909119910

sdot

1205980119909

1205980119910

1205740119909119910

+

119872119909

119872119910

119872119909119910

sdot

1205981119909

1205981119910

1205741119909119910

(7)

where the in-plane force and moment resultants are depend-ing on the strain fields based on (6)

3 Finite Element Discretization

The finite element discretization concept can be seen inFigure 3

As we have written in the previous section the in-planedisplacement fields are continuous on the undelaminatedportions and independent of each other in the delaminatedtop and bottom portions (see Figure 3) Therefore betweenthe delaminated and undelaminated portions special tran-sition elements were used on sections 2119871 and 2119877 whichcapture the crack tips These elements ensure the continuityof the displacements [57] Because of the opposite simplysupported edges the Levy-type method is applicable and thedisplacements fields in the 119910 direction can be given by theterms of Fourier seriesHosseini-Hashemi et al [58] Bodaghiand Saidi [55] Thai and Kim [59] Szekrenyes [48 60] Thaiand Kim [61] and Nguyen et al [62]

1199060

V01199080

(119909 119910) =

infin

sum

119899=1

1198800119899 (119909) sin1205731199101198810119899 (119909) cos1205731199101198820119899 (119909) sin120573119910

(8)


Crack tip Crack tipDelamination

X

Z2L 2R

Transitional elementNode

1 32

1205791

120579i 120579nwn

un

w1 u1

ui

wi

120579(i+1)

120579(i+1)

w(i+1)

w(i+1)

ub(i+1)

ut(i+1)

w(nminusjminus2)

120579(nminusjminus2)

120579(nminusj) u(nminusj)

w(nminusj)ut(nminusjminus2)

ub(nminusjminus2)

Figure 3 Finite element discretization and nodal DOFs of the layered plate with delamination

where 120573 = 119899120587119887 and 1198800119899(119909) 1198810119899(119909) 1198820119899(119909) are theamplitudes in the 119909 direction As the displacement in the 119910direction is given it is enough to discretize the model along119909 so we can write the nodal displacements as

U119890= R119890u119890 (9)

whereu119890is the nodal displacement vector andR

119890is a diagonal

matrix containing the corresponding trigonometric termsfrom (8) This results in a semidiscrete finite element whichis capable of modelling plate-like buckling For the analysiswe need to derive the material and the geometric stiffnessmatrices of the different elements of Figure 3

Integrating the strain energy density thematerial stiffnessmatrix of an element can be derived based on the Hamiltonprinciple [63ndash66]

119880119890=12intΩ119890

119906 119889119881 =12U119879119890K119872119890

U119890 (10)

where K119872119890

is the material or general stiffness matrix and Ω119890

is the element domain Based on the literature the geometricstiffness matrix with respect to the axial compression can begiven as [63 67]

K119866119890= 1198730119897119890 int

1

0B119908otimesB119908119889120585 (11)

As the in-plane119873119909119910

forces along the delaminated region willnot be zero for the local stability analysis we also need thegeometric stiffness matrix with respect to the119873

119909119910load

K119909119910119866119890= 1198730119897119890

120573

2int

1

0B119908

119879otimesN119908119889120585 (12)

In ((11)-(12)) the1198730 is the applied load on the element N119908is

the vector of interpolation functions of the element and B119908

is the derivative of the vector of interpolation functions

31 Element of the Nondelaminated Parts For the undelami-nated parts an 8DoF element is used

u119879119890= 1199061 V1 1199081 1205791 1199062 V2 1199082 1205792 (13)

where 120579119894is the rotation of the cross-section in the node The

matrix of the trigonometric coefficients can be given as

R119890=

[[[[[[

[

sin120573119910 0 0 sdot sdot sdot

0 cos120573119910 0 sdot sdot sdot

0 0 sin120573119910 sdot sdot sdot

d

]]]]]]

]

(14)

For the in-plane displacements linear interpolation functionswere used whereas for the transverse deflection a third-orderfunction was applied

119906 (120585) = 1198860 + 1198861120585 =8sum

119894=1119873119906119894119906119890119894 (15)

V (120585) = 1198870 + 1198871120585 =8sum

119894=1119873V119894119906119890119894 (16)

119908 (120585) = 1198880 + 1198881120585 + 11988821205852+ 1198883120585

3=

8sum

119894=1119873119908119894119906119890119894 (17)

where 119886119894 119887119894 and 119888

119894are constant coefficients and 120585 varies

between 0 and 1 Their value can be calculated using theCLPT from the following equations

119906 (0) = 1199061

119906 (1) = 1199062(18)

V (0) = V1

V (1) = V2(19)

119908 (0) = 1199081

119908 (1) = 1199082(20)

1119897119890

1199081015840(0) = 1205791

1119897119890

1199081015840(1) = 1205792

(21)


where the derivation is carried out with respect to thedimensionless 120585 coordinate By solving ((15)ndash(21)) the vectorinterpolation functions can be obtained [63]

N120581

119879= 1198731205811 1198731205818 (22)

where 120581 can be 119906 V or 119908Substituting the discretized displacement fields into (10)

the strain energy can be given as

119880119890= int

119887

0(U119890

119879int

1

0

12(Kint119872119890

+Kint119872119890

119879

) 119889120585U119890)119889119910 (23)

Carrying out the integration over 119889120585 the element materialstiffness matrix can be obtained

Using the interpolation functions from (11) the geometricstiffness matrix can be derived

32 Element of the Delaminated Parts Because of the con-strained model the transverse deflection is common but thein-plane displacements are independent in the delaminatedportion which results in a 12DoF element

u119879119890= 119906119905

1 V119905

1 119906119887

1 V119887

1 1199081 1205791 119906119905

2 V119905

2 119906119887

2 V119887

2 1199082 1205792 (24)

The R119890matrix can be composed based on the nodal displace-

ment vector For the in-plane displacement the same linearinterpolation functions were used given by ((15)-(16)) andfor the transverse deflection the third-order function wasapplied in accordance with (17)

As the in-plane displacements are independent thepotential energy has to be evaluated for both the top andbottom parts and the material stiffness matrix can be derivedfrom the sum of the potential energies based on (23)

33TheTransition Elements In accordancewith Figure 3 theelements of sections 2119871 and 2119877 ensure the kinematic conti-nuity between the delaminated and undelaminated portionsThe vector of displacements of the 2119871 element is

u119879119890= 1199061 V1 1199081 1205791 119906

119905

2 V119905

2 119906119887

2 V119887

2 1199082 1205792 (25)

And for the element denoted by 2119877 we have

u119879119890= 119906119905

1 V119905

1 119906119887

1 V119887

1 1199081 1205791 1199062 V2 1199082 1205792 (26)

Based on the kinematic continuity the following 4 equa-tions can be written based on the applied plate theory for the2119871 section

119906119905(0) = 1199061 minus

119905119887

21205791

119906119905(1) = 119906

119905

2

119906119887(0) = 1199061 +

119905119905

21205791

119906119887(1) = 119906

119887

2

V119905 (0) = V1 minus119905119887

21205731199081

V119905 (1) = V1199052

V119887 (0) = V1 +119905119905

21205731199081

V119887 (1) = V1198872

(27)

The equations take similar form for the 2119877 section

119906119905(0) = 119906

119905

2

119906119905(1) = 1199061 minus

119905119887

21205791

119906119887(0) = 119906

119887

2

119906119887(1) = 1199061 +

119905119905

21205791

V119905 (0) = V1199052

V119905 (1) = V1 minus119905119887

21205731199081

V119887 (0) = V1198872

V119887 (1) = V1 +119905119905

21205731199081

(28)

Using the equations above and (20) and (21) the vectorof interpolation functions can be obtained The stiffnessmatrices can be calculated on the same way as it was shownbefore

4 Stability Analysis

Based on Section 3 the structuralmatrices of the globalmodelcan be obtained After applying the selected BCs on the 119909 =

0 and 119909 = 119871 edges the critical loads and the correspondingeigenvectors can be calculated as

(K+119873Load119909

K119909119866)U119879 = 0 (29)

The corresponding global mode shapes and the resultant in-plane force distributions can be obtained using the vector ofinterpolation functions by (22) and the CEL given by (6)Because of the in-plane resultant forces at the crack tips the


z

y

x

a

Simply supported

Simply supportedBuilt-in end

Built-in endSine distribution

Nx

(a)

L x

Nx

L1 + aL1

Nbx

Ntx

(b)

Figure 4 Model of the local stability analysis (top or bottom part) (a) Example for the distribution of the in-plane normal force along the 119909direction (b)

plate is able to buckle locally along the delamination Thelocal stability is analysed individually for the top and bottomdelaminated plate portions assuming plates with built-in endBCs along the crack tips The local stability is affected by thedistribution of the in-plane forces (see Figure 4) For the localFE model we derived the elements of the individual top andbottom layers using the same method as for the elements ofthe global model The nonuniform resultant in-plane forcesof the global model are evaluated for every element at themiddle These values were normed with the value at thecrack tip and the element geometric stiffness matrices weremultiplied with these values taking into consideration thedistribution along 119909 Because of the simply supported edges

the plate will have a half wave shape along the width whichresults in the fact that the load along the crack tip will not beuniform (see Figure 4) Taking this aspect of the problem intoconsideration we applied the method of harmonic balanceand wrote the Fourier series of the nodal displacements [43]

U119879 = d0 +infin

sum

119894=1d119894cos

119894120587119910

119887 d119894= 119889119894120601 (30)

where 119889119894are constant coefficients and 120601 is the vector of

displacement values Taking this back into (29) and applyingsome trigonometric identities we can obtain a system ofequations in matrix form

[[[[[[[[[

[

K120575

12119873

Load119909119897

K119909119866120575

0 0 sdot sdot sdot

12119873

Load119909119897

K119909119866120575

K120575

12119873

Load119909119897

K119909119866120575

0 sdot sdot sdot

0 12119873

Load119909119897

K119909119866120575

K120575

12119873

Load119909119897

K119909119866120575

sdot sdot sdot

d

]]]]]]]]]

]

[[[[[[[[[

[

1198890

1198891

1198892

1198893

]]]]]]]]]

]

120601 = 0 (31)

The critical values and the corresponding mode shapes canbe calculated from (31) and (29)

For validation purposes the model was solved usingAbaqus The plate is made by carbonepoxy material usingthe following layup order [plusmn45∘119891 0∘ plusmn45∘119891] Engineeringconstants of the layers are detailed in Table 1 The seriesexpansion in (30) was carried out for two terms Along the119909 direction the plate was discretized using 14 elements tocapture the higher order mode shapes The obtained criticalvalues from (31) are sim40 higher than the loads of theproblem with constant distribution along 119910 The top ESLof the example in Section 6 was checked assuming constantforce distribution along 119909Thewidth of the plate was 100mmand the length of the plate was 105mmThe S4R shell elementwas used for the analysis with 1mm element size The resultsshow good agreement with the present calculations (seeTable 2)

5 Boundary and Continuity Conditions

In this paper the process of loss of stability is determined byusing a displacement controlled model based on Section 2For solution the Levy-type method is used with the state-space approach [52] From (7) using Hamiltonrsquos principle thegoverning PDEs of each section can be derived [52] Applying(8) the obtained ODEs can be rearranged into the state-spacemodel [52 68]

Z1015840 = TZ (32)

where Z is the state vector The general solution of (32) is [5268 69]

Z120572(119909) = 119890

(T119909)K120572 (33)

where K120572is the vector of constants 119870

120572119894 At the crack

tips we have to define 10-10 continuity conditions (CCs)


Table 1 Elastic properties of single carbonepoxy composite laminates

1198641 [GPa] 1198642 [GPa] 1198643 [GPa] 11986612 [GPa] 11986613 [GPa] 11986623 [GPa] ]12 [mdash] ]13 [mdash] ]23 [mdash]plusmn45∘119891 1639 1639 164 164 546 546 03 05 050∘ 148 965 965 371 466 491 03 025 027

Table 2 Difference between the critical amplitudes of the constantand sine loaded plate and the difference between the two types ofloads of each method (Δ) and the difference between the results ofthe ABAQUS model with sinusoidal loading and the results of thepresent method (Δ sin)The dimensions of the results are in Nmmminus1

Present method FEM119873

Crit1119909Const 119873

Crit1119909 Sine Δ [] 119873

Abaqus 1119909 Sine 119873

Abaqus 1119909Const Δ [] Δ sin []

1105 15685 4194 10014 14608 4588 737119873

Crit2119909Const 119873

Crit2119909 Sine Δ [] 119873

Abaqus 2119909 Sine 119873


15918 22638 4221 15129 22104 4610 241

between the plate portions A B and B C Because of theclosed delamination (see Figure 1) the so-called Mujumdarconditions have to be used for fitting the 119872

119909moment and

the Kirchhoff equivalent shear force [29]

11988001198992119887 (119909) = 1198800119899120572 (119909) +119905119887

21198821015840

0119899120572 (119909)

11988101198992119887 (119909) = 1198810119899120572 (119909) +119905119887

21205731198820119899120572 (119909)

11988001198992119905 (119909) = 1198800119899120572 (119909) minus119905119905

21198821015840

0119899120572 (119909)

11988101198992119905 (119909) = 1198810119899120572 (119909) minus119905119905

21205731198820119899120572 (119909)

11988201198992 (119909) = 1198820119899120572 (119909)

1198821015840

01198992 (119909) = 1198821015840

0119899120572 (119909)

1198991199091198992119905 (119909) + 1198991199091198992119887 (119909) = 119899

119909119899120572119905(119909) + 119899

119909119899120572119887(119909)

1198991199091199101198992119905 (119909) + 1198991199091199101198992119887 (119909) = 119899

119909119910119899120572119905(119909) + 119899

119909119910119899120572119887(119909)

1198981199091198992119905 (119909) +

119905119887

21198991199091198992119905 (119909) +1198981199091198992119887 (119909)

minus119905119905

21198991199091198992119887 (119909) = 119898

119909119899120572119905(119909) +

119905119887

2119899119909119899120572119905

(119909)

+119898119909119899120572119887

(119909) minus119905119905

2119899119909119899120572119887

(119909)

1198981015840

1199091198992119905 (119909) +119905119887

21198991015840

1199091198992119905 (119909)

minus 2(1205731198981199091199101198992119905 (119909) +

119905119887

21198991199091199101198992119905 (119909)) +119898

1015840

1199091198992119887 (119909)

minus119905119905

21198991015840

1199091198992119887 (119909)

minus 2(1205731198981199091199101198992119887 (119909) minus

119905119905

21198991199091199101198992119887 (119909)) = 119898

1015840

119909119899120572119905(119909)

+119905119887

21198991015840

119909119899120572119905(119909) minus 2(120573119898

119909119910119899120572119905(119909) +

119905119887

2119899119909119910119899120572119905

(119909))

+1198981015840

119909119899120572119887(119909) minus

119905119905

21198991015840

119909119899120572119887(119909)

minus 2(120573119898119909119910119899120572119887

(119909) minus119905119905

2119899119909119910119899120572119887

(119909))

(34)

where 119909 can take either 1198711 or (1198711 + 119886) respectivelyand 119899

119909119899120572120575 119899119910119899120572120575

119899119909119910119899120572120575

119898119909119899120572120575

119898119910119899120572120575

119898119909119910119899120572120575

depends on1198800119899120572120575 11988101198991205721205751198820119899120572 and their derivatives 120572 can take 1 2 or3 depending on the sections which will be fit In the BCs an1198800 axial displacement at 119909 = 0 has to be prescribed and thereis no other load Substituting the solution of the state-spacemodel into the BCs and CCs a system of inhomogeneousequations can be obtained

MKall = 1198800 0 0 0119879 (35)

which can be solved for the119870all119894

constants Using (33) we canget the displacement functions and the in-plane forces canbe calculated using (6)

Using this model we calculated the arising forces at theedge of the plate and at the crack tips with respect to the axialdisplacement 1198800 The critical values of the global and localstability analysis were compared with these results

51 Criterion of Constant Arc Length All of the mode shapeswere calculated with a maximum amplitude of 1mm andscaled to fit the physical requirements The amplitudes ofthe global and local modes were controlled using an arclength criterion [57] This means that the arc length of thesuperimposed eigenshapes minus the axial displacement hasto be equal to the length of the plate or the delamination

intradic1 + (120597sum119899

119894(119891120574

119894119882120574

119894(119909))

120597119909)

2

119889119909minusΔ119906

=

119871 if 120574 = 119892

119886 if 120574 = 119897120575

(36)

where 119891120574119894is the scale factor for the mode shapes and119882120574

119894(119909) is

the buckled shape of the 119894th buckling mode For global modeshapesΔ119906 = 1198800 For local mode shapes it is the signed sum of


Table 3 Geometric parameters of the plate modelled for thenumerical examples

119871 [mm] 119887 [mm] 119905 [mm] ℎ [mm]200 100 45 05

the axial displacements at the left and right crack tips In caseof mixed mode buckling Δ119906 is

Δ119906

= 119882Amp120574

(1198800)(119899

sum

119894

119880119892

119894(1198711) minus119880119892

119894(1198711+ 119886))

+ (119880Static120575

(119880Crit10119892 1198711) minus119880Static

120575(119880

Crit10119892 1198711+ 119886))

(37)

where 119880119892119894(119909) is the axial displacement of the global model

for the 119894th mode scaled with the 119882Amp120574

(1198800) amplitude and119880

Static120575

(1198800 119909) is the static axial displacement

52 Superposition of the Mode Shapes Based on the linearrelationship between the in-plane normal force and the axialdisplacement of the displacement controlled model criticalaxial displacements can be calculated for the critical loadsFrom these values the amplitudes are rising linearily

119891Amp

(1198800) =

0 0 le 1198800 ge 119880Crit1198940120574

1198800 minus 119880Crit1198940120574

119880Max0 minus 119880

Crit1198940120574

119880Crit1198940120574 lt 1198800 ge 119880

Max0

(38)

We assume that the dominant in-plane force distributionis determined by the first global mode Therefore localbuckling inmixedmode case occurs only if the critical valuesof the local modes which belong to the first global mode arereached

6 Numerical Example

In this section we adopt the method on a carbonepoxylayered plate The ply order of the plate is [plusmn45∘119891 0∘ plusmn45∘119891]The plate consists of 9 layers The corresponding materialdata can be found in Table 1 The plate is symmetricallydelaminated and its geometric data is presented in Table 3The stiffness matrices of each single layer were determinedbased on the elastic properties given by Table 1 The analysiswas carried out with 119899 = 1 condition in (8) The order ofthe matrix in (31) was set to 2 The plate was discretizedusing 12 elements in all sections and 1-1 additional transitionalelements were used at the crack tips The position of thedelamination was set above the 5th layer At the edges 119909 = 0and 119909 = 119871 the same simply supported (S-S) or built-in (B-B) BCs were used The length of the delamination was variedfrom 10mm to 100mmThe global critical forces with respectto the delamination length can be seen in Figure 5

It can be seen that the obtained critical loads of the built-in plate are higher but as the length of the delamination

20 40 60 80 100

600

800

1000

1200

S-SB-B

a

Nx

Figure 5 The global critical amplitudes with respect to the delami-nation length

0 20 40 60 80 100

1000

2000

3000

4000

5000

Nx

S-SB-B

a

Figure 6 The local critical amplitudes of the top plate portion withrespect to the delamination length

increases the effect of BCs gets less significant The criticalamplitudes of the local top and bottom delaminated portionscan be seen in Figures 6-7

As it can be seen the local critical values are higher in thesimply supported cases This is because different eigenshapebelongs to the different BCs which results in different in-plane force distribution Again as the delamination lengthincreases the effect of the BCs gets less significant Usingthe displacement controlled model the critical axial displace-ments can be calculated for each critical amplitude Basedon this calculation stability diagrams can be obtained withrespect to the axial displacement and the delamination length(see Figures 8-9)

On both pictures below the blue line the plate is stableIn the orange region the plate buckles globally in thegreen region it buckles globally and the crack opens as thelocal top plate loses its stability and above the green linethe delaminated bottom portion buckles too It has to beremarked that in the B-B case the bottom part buckles only at


0 20 40 60 80 100

2000

4000

6000

8000

10000

12000

14000

Nx

S-SB-B

a

Figure 7 The local critical amplitudes of the bottom plate portionwith respect to the delamination length

0 20 40 60 80 10000

05

10

15

20

GlobalLocal topLocal bottom

Stable

Unstable

a

U0

Figure 8 The stability diagram of the simply supported plate

0 20 40 60 80 10000

05

10

15

20

Stable

Unstable

a


U0

Figure 9 The stability diagram of the built-in end plate

017 0

30

29

xg

xg

xg

Am

p (N

mm

)

Ngx

Ntx

Nbx

Ntxy

Nbxy

0150

0

U0 (mm)

NCrit3

NCrit2

NCrit1

Figure 10 The static 119873119909and 119873

119909119910curves and global critical forces

and the corresponding axial displacements of the simply supportedcase

Table 4 The global critical buckling loads in Nmmminus1

Modes BCS-S B-B

I 4546 5021II 4898 5805III 8521 11912

higher axial compression therefore the green line is outsidethe range shown in Figure 9 The maximal critical amplitudewas set to 2mm It can be seen that the built-in end plate ismore stable and its bottom part does not lose its stabilityup to the maximal axial displacement whereas the simplysupported plate loses its stability on smaller amplitudes Itcan be noticed that as the delamination length increases thepoint of the global and local stability loss of the top plate getsclose to each other The presented critical loads are the firstcritical amplitudes But if the plate is weak against uniaxialcompression higher order mode shapes are also feasibleThesemode shapes can be superimposed using the arc lengthcriterion In the followingwewill show the process of stabilityloss of the simply supported and built-in end plates with100mm delamination length The global critical amplitudesfor the two types of BCs are listed in Table 4 For these valuesthe critical axial compressions can be determined based onthe displacement controlled model The resulting forces withrespect to the axial displacement for the simply supportedcase are shown in Figure 10 On the sameway the critical axialdisplacements of the built-in end plate can be determinedWhereas the critical loads are higher than in case of simplysupported BCs the critical axial displacements of the first 2modes are smaller and only the third mode appears at higherdisplacement 014mm 015mm and 033mmThe maximalaxial compression was chosen in both cases for the 120of the third mode The critical values of the delaminated


(mm

)W

Am

pW

Am

pW

Am

p(m

m)

minus10

minus10

minus08

minus06

minus04

minus02

(mm

)

minus10

minus08

minus06

minus04

minus02

minus30

minus20

minus10

10

20

minus20

minus60

minus40

minus20

20

40

60

minus10

10

20

30

minus05

50

05

10

100 150

50 100 150

50 100 200150

50 100 200150

W

W

W

x(N

mm

)x

(Nm

m)

x(N

mm

)

Ntx

Nbx

Ntx

Nbx

Ntx

Nbx

x (mm)

x (mm)

50 100 200150x (mm)

x (mm)

x (mm)

50 100 150x (mm)

NA

mp

NA

mp

NA

mp

Figure 11The globalmode shapes and the corresponding in-plane force distributions of the simply supported case Note that the distributionsinvolve a half sine wave in the 119910 direction

portions were calculated for the local buckling case wherethe nonuniform distribution of the in-plane forces does notcount but the calculated critical axial displacements werehigher than the critical axial displacement of the first globalmode therefore the plate loses its stability first globally

From Figures 11 and 12 it can be seen that because of thedifferent BCs different mode shapes appear For the mixedmode buckling the local critical values were calculated forboth cases using the nonuniform force distribution of theglobal modes Here we present only the critical loads of therealizing local modes (see Table 5) As it can be seen onlythe first two local modes appear in both cases The third

mode would only appear at higher axial compression Atthe built-in end case the local modes calculated with theforce distribution of the second global mode are not presentduring the stability loss because the critical values of thesemodes are much higher The plate was also examined for the119873119909119910forces but according to the results no stability loss occurs

with respect to the119873119909119910

forces at the crack tip In accordancewith Figures 8 and 9 the delaminated bottom part does notlose its stability at the selected maximal axial displacement

The shapes of these modes were calculated with the in-plane force distribution resulting from the correspondingglobal modes and were superimposed using the arc length


(mm

)

(mm

)

(mm

)

minus06

minus08

minus10 minus10

minus05

minus02

minus04

minus06

minus08

minus10

05

10

minus04

minus02

50 100 150 200x (mm)

50 100 150 200x (mm)

50 100 150 200

x (mm)

W(x) W(x)

W(x)

WA

mp

WA

mp

WA

mp

Figure 12 The global mode shapes of the built-in end case Note that the distributions involve a half sine wave in the 119910 direction

Table 5 The local critical119873119909amplitudes in Nmmminus1

Cases Modes1st global 2nd global 3rd global

S-S Corresponding 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rdlocal top 1148 1609 mdash 147 2164 mdash 1218 1773 mdash

B-B Corresponding 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rdlocal top 1537 2273 mdash mdash mdash mdash 1621 mdash mdash

criterion Figures 13 and 14 show the buckled shapes at119880Crit10119892

119880Crit20119892 119880Crit3

0119892 and 119880Max0 On the superimposed shapes it can

be seen that the dominant part of the solution is always theglobal and local first modes but the higher order modesinfluence the shape slightly

7 Conclusion

In this paper the buckling process of a delaminated layeredplate was investigated The formulation of the problem isbased on the system of exact kinematic conditions (SEKC)by cutting the plate in the plane of the delamination andforming the continuity conditions The problem was solvedusing FEM with self-developed semidiscrete finite elementsThe model contains special transitional elements whichensure the kinematic continuity between the delaminated and

undelaminated portions The delaminated region was mod-elled as a constrained section in the global model thereforethere is no need for using contact along the delaminated areawhich results in a calculation efficient and simple methodfor the estimation of the global critical buckling loads andthe corresponding shapes The local behaviour of the delam-inated portion was analysed by a separate FE model For theconsideration of the nonuniform in-plane force distributionthe method of harmonic balance was used On a numericalexample the effects of the simply supported and built-inend BCs were determined with respect to the delaminationlength It was shown that the BCs are influencing not onlythe critical loads but also the corresponding global modeshapes Because of the different global mode shapes the localbehaviour of the delaminated portions is different as the in-plane force distributions differ significantlyThis results in thefact that whereas the simply supported plate buckles globally


(mm

)(m

m)

(mm

)

minus01

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

01

00

minus2

2

0

x (mm)50 100 150

x (mm)50 100 150 200

x (mm)50 100 150 200x (mm)

50 100 150 200

WA

mp

WA

mp

WA

mp

WA

mp

Figure 13 The buckled shapes of the simply supported case at 119880Crit10119892 119880Crit2

0119892 119880Crit30119892 and 119880Max

0

(mm

)

minus01

01

00

x (mm)50 100 150

(mm

)

minus2

2

0

x (mm)50 100 150 200

x (mm)50 100 150 200

x (mm)50 100 150 200

(mm

)

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

WA

mp

WA

mp

WA

mp

WA

mp

Figure 14 The buckled shapes of the built-in end case at 119880Crit10119892 119880Crit2

0119892 119880Crit30119892 and 119880Max

0

at lower values this configuration is more stable locally thanthe built-in end configuration It was also shown that thiseffect is more significant if the delamination length is smallStability diagrams with respect to the axial displacementand the delamination length were given where the globaland mixed mode stability loss cases were shown At onedelamination length the process of stability losswas presentedfor both BCs Here the effect of the BCs and the nonuniformin-plane force distribution can be seen This nonuniformdistribution was not observed with respect to the differenttype of BCs in the literature and we can state that it greatlyalerts the buckled shape of the delaminated layers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the Hungarian National Scien-tific Research Fund (OTKA) under Grant no 44615-066-15(108414)

References

[1] O Gohardani and D W Hammond ldquoIce adhesion to pristineand eroded polymer matrix composites reinforced with carbonnanotubes for potential usage on future aircraftrdquo Cold RegionsScience and Technology vol 96 pp 8ndash16 2013

[2] S Giannis and K Hansen ldquoInvestigation on the joining ofCFRP-to-CFRP and CFRP-to-aluminium for a small aircraftstructural applicationrdquo in ProceedingsmdashAmerican Society forComposites 25th Technical Conference of the American Societyfor Composites and 14th US-Japan Conference on Composite


Materials 20-22 September 2010 Dayton Ohio USA J B LantzEd vol 1 pp 333ndash346 CurranAssociates RedHook NY USA2011

[3] D A Hills P A Kelly D N Dai and A M Korsunsky Solu-tion of Crack Problems The Distributed Dislocation TechniqueKluwer Academic Dordrecht The Netherlands 1996

[4] D F Adams L A Carlsson and R B Pipes ExperimentalCharacterization of Advanced Composite Materials CRC PressBoca Raton Fla USA 3rd edition 2000

[5] B D Davidson F O Sediles and K D Humphreys ldquoA shear-torsion-bending test for mixed-mode I-II-III delaminationtoughness determinationrdquo in Proceedings of the 25th TechnicalConference of the American Society for Composites and 14thUS-Japan Conference on Composite Materials pp 1001ndash1020Dayton Ohio USA September 2010

[6] M F S F De Moura R M Guedes and L Nicolais ldquoFractureinterlaminarrdquo in Wiley Encyclopedia of Composites pp 60ndash78John Wiley amp Sons 2011

[7] L N Phillips Ed Design with Advanced Composite MaterialsSpringer The Design Council Berlin Germany 1989

[8] V Rizov A Shipsha andD Zenkert ldquoIndentation study of foamcore sandwich composite panelsrdquo Composite Structures vol 69no 1 pp 95ndash102 2005

[9] V I Rizov ldquoNon-linear indentation behavior of foam coresandwich compositematerialsmdasha 2DapproachrdquoComputationalMaterials Science vol 35 no 2 pp 107ndash115 2006

[10] A D Zammit S Feih and A C Orifici ldquo2D numericalinvestigation of pre-tension on low velocity impact damage ofsandwich structuresrdquo in Proceedings of the 18th InternationalConference on Composite Materials (ICCM18 rsquo11) pp 1ndash6Jeju International Convention Center Jeju Republic of KoreaAugust 2011

[11] R A Chaudhuri and K Balaraman ldquoA novel method for fab-rication of fiber reinforced plastic laminated platesrdquo CompositeStructures vol 77 no 2 pp 160ndash170 2007

[12] N Carrere T Vandellos and E Martin ldquoMultilevel analysis ofdelamination initiated near the edges of composite structuresrdquoin Proceedings of the 17th International Conference on CompositeMaterials (ICCM rsquo09) pp 1ndash10 Edinburgh UK July 2009

[13] V N Burlayenko and T Sadowski ldquoA numerical study of thedynamic response of sandwich plates initially damaged by low-velocity impactrdquo Computational Materials Science vol 52 no 1pp 212ndash216 2012

[14] J Rhymer H Kim and D Roach ldquoThe damage resistanceof quasi-isotropic carbonepoxy composite tape laminatesimpacted by high velocity icerdquo Composites Part A AppliedScience and Manufacturing vol 43 no 7 pp 1134ndash1144 2012

[15] G Goodmiller and S TerMaath ldquoInvestigation of compositepatch performance under low-velocity impact loadingrdquo inProceedings of the 55th AIAAASMEASCEAHSSC StructuresStructural Dynamics and Materials Conference National Har-bor Md USA 2014

[16] C Elanchezhian B V Ramnath and J Hemalatha ldquoMechanicalbehaviour of glass and carbon fibre reinforced compositesat varying strain rates and temperaturesrdquo Procedia MaterialsScience vol 6 pp 1405ndash1418 2014 Proceedings of the 3rdInternational Conference on Materials Processing and Charac-terisation (ICMPC rsquo14)

[17] R Guo and A Chattopadhyay ldquoDevelopment of a finite-element-based design sensitivity analysis for buckling andpostbuckling of composite platesrdquo Mathematical Problems inEngineering vol 1 no 3 pp 255ndash274 1995

[18] L P Kollar ldquoBuckling of rectangular composite plates withrestrained edges subjected to axial loadsrdquo Journal of ReinforcedPlastics and Composites vol 33 no 23 pp 2174ndash2182 2014

[19] G Tarjan A Sapkas and L P Kollar ldquoStability analysis oflong composite plates with restrained edges subjected to shearand linearly varying loadsrdquo Journal of Reinforced Plastics andComposites vol 29 no 9 pp 1386ndash1398 2010

[20] H-TThai and D-H Choi ldquoAnalytical solutions of refined platetheory for bending buckling and vibration analyses of thickplatesrdquo Applied Mathematical Modelling vol 37 no 18-19 pp8310ndash8323 2013

[21] H-T Thai M Park and D-H Choi ldquoA simple refined theoryfor bending buckling and vibration of thick plates resting onelastic foundationrdquo International Journal ofMechanical Sciencesvol 73 pp 40ndash52 2013

[22] C Klobedanz A study of the effect of delamination size on thecritical sublaminate buckling load in a composite plate usingthe Ritz method [PhD thesis] Rensselaer Polytechnic InstituteTroy NY USA 2014

[23] S A M Ghannadpour H R Ovesy and E Zia-DehkordildquoBuckling and post-buckling behaviour of moderately thickplates using an exact finite striprdquo Computers amp Structures vol147 pp 172ndash180 2015

[24] H R Ovesy A Totounferoush and S A M GhannadpourldquoDynamic buckling analysis of delaminated composite platesusing semi-analytical finite strip methodrdquo Journal of Sound andVibration vol 343 pp 131ndash143 2015

[25] H Chai C D Babcock and W G Knauss ldquoOne dimensionalmodelling of failure in laminated plates by delamination buck-lingrdquo International Journal of Solids and Structures vol 17 no11 pp 1069ndash1083 1981

[26] G A Kardomateas and D W Schmueser ldquoBuckling andpostbuckling of delaminated composites under compressiveloads including transverse shear effectsrdquo AIAA Journal vol 26no 3 pp 337ndash343 1988

[27] G A Kardomateas ldquoLarge deformation effects in the postbuck-ling behavior of composites with thin delaminationsrdquo AIAAJournal vol 27 no 5 pp 624ndash631 1989

[28] J S Anastasiadis and G J Simitses ldquoSpring simulated delam-ination of axially-loaded flat laminatesrdquo Composite Structuresvol 17 no 1 pp 67ndash85 1991

[29] P M Mujumdar and S Suryanarayan ldquoFlexural vibrations ofbeams with delaminationsrdquo Journal of Sound and Vibration vol125 no 3 pp 441ndash461 1988

[30] H R Ovesy M A Mooneghi and M Kharazi ldquoPost-bucklinganalysis of delaminated composite laminates with multiplethrough-the-width delaminations using a novel layerwise the-oryrdquoThin-Walled Structures vol 94 pp 98ndash106 2015

[31] D Shu ldquoBuckling ofmultiple delaminated beamsrdquo InternationalJournal of Solids and Structures vol 35 no 13 pp 1451ndash14651998

[32] H Kim and K T Kedward ldquoA method for modeling thelocal and global buckling of delaminated composite platesrdquoComposite Structures vol 44 no 1 pp 43ndash53 1999

[33] J T Ruan F Aymerich J W Tong and Z Y Wang ldquoOpticalevaluation on delamination buckling of composite laminatewith impact damagerdquo Advances in Materials Science and Engi-neering vol 2014 Article ID 390965 9 pages 2014

[34] XWang andG Lu ldquoLocal buckling of composite laminar plateswith various delaminated shapesrdquo Thin-Walled Structures vol41 no 6 pp 493ndash506 2003


[35] MKharazi andHROvesy ldquoPostbuckling behavior of compos-ite plates with through-the-width delaminationsrdquo Thin-WalledStructures vol 46 no 7ndash9 pp 939ndash946 2008

[36] Z Aslan and M Sahin ldquoBuckling behavior and compressivefailure of composite laminates containing multiple large delam-inationsrdquoComposite Structures vol 89 no 3 pp 382ndash390 2009

[37] M Kharazi H R Ovesy and M Asghari Mooneghi ldquoBucklinganalysis of delaminated composite plates using a novel layerwisetheoryrdquoThin-Walled Structures vol 74 pp 246ndash254 2014

[38] S-F Hwang and G-H Liu ldquoBuckling behavior of compositelaminates withmultiple delaminations under uniaxial compres-sionrdquo Composite Structures vol 53 no 2 pp 235ndash243 2001

[39] M Damghani D Kennedy and C Featherston ldquoGlobal buck-ling of composite plates containing rectangular delaminationsusing exact stiffness analysis and smearing methodrdquo Computersamp Structures vol 134 pp 32ndash47 2014

[40] M Marjanovic and D Vuksanovic ldquoLayerwise solution of freevibrations and buckling of laminated composite and sandwichplates with embedded delaminationsrdquo Composite Structuresvol 108 no 1 pp 9ndash20 2014

[41] J D Whitcomb ldquoMechanics of instability-related delaminationgrowthrdquo in Composite Materials Testing and Design vol 9 pp215ndash230 ASTM 1990

[42] Z Juhasz and A Szekrenyes ldquoProgressive buckling of a sim-ply supported delaminated orthotropic rectangular compositeplaterdquo International Journal of Solids and Structures 2015

[43] W W Bolotin Kinetische Stabilitat Elastischer Systeme VEBDeutscher Verlag der Wissenschaften Berlin Germany 1961

[44] A Szekrenyes ldquoAnalysis of classical and first-order sheardeformable cracked orthotropic platesrdquo Journal of CompositeMaterials vol 48 no 12 pp 1441ndash1457 2014

[45] L S Ma and T J Wang ldquoRelationships between axisymmetricbending and buckling solutions of FGMcircular plates based onthird-order plate theory and classical plate theoryrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 85ndash101 2004

[46] M Amabili and S Farhadi ldquoShear deformable versus classicaltheories for nonlinear vibrations of rectangular isotropic andlaminated composite platesrdquo Journal of Sound and Vibrationvol 320 no 3 pp 649ndash667 2009

[47] AM Zenkour ldquoExactmixed-classical solutions for the bendinganalysis of shear deformable rectangular platesrdquo Applied Math-ematical Modelling vol 27 no 7 pp 515ndash534 2003

[48] A Szekrenyes ldquoThe system of exact kinematic conditionsand application to delaminated first-order shear deformablecomposite platesrdquo International Journal of Mechanical Sciencesvol 77 pp 17ndash29 2013

[49] C N Della and D Shu ldquoVibration of delaminated multilayerbeamsrdquoComposites Part B Engineering vol 37 no 2-3 pp 227ndash236 2006

[50] Y Guo M Ruess and Z Gurdal ldquoA contact extended iso-geometric layerwise approach for the buckling analysis ofdelaminated compositesrdquoComposite Structures vol 116 pp 55ndash66 2014

[51] J Wang and L Tong ldquoA study of the vibration of delami-nated beams using a nonlinear anti-interpenetration constraintmodelrdquoComposite Structures vol 57 no 1ndash4 pp 483ndash488 2002

[52] J N Reddy Mechanics of Laminated Composite Plates andShellsmdashTheory and Analysis CRC Press Boca Raton Fla USA2004

[53] L Kollar and G Springer Mechanics of Composite StructuresCambridge University Press Cambridge UK 2002

[54] J Ye Laminated Composite Plates and Shellsmdash3D modellingSpringer London UK 2003

[55] M Bodaghi and A R Saidi ldquoLevy-type solution for bucklinganalysis of thick functionally graded rectangular plates basedon the higher-order shear deformation plate theoryrdquo AppliedMathematical Modelling vol 34 no 11 pp 3659ndash3673 2010

[56] S W Tsai Theory of Composites Design Think CompositesDayton Ohio USA 1992

[57] A Szekrenyes ldquoA special case of parametrically excited systemsfree vibration of delaminated composite beamsrdquo EuropeanJournal of MechanicsmdashASolids vol 49 pp 82ndash105 2015

[58] S Hosseini-Hashemi M Fadaee and H Rokni DamavandiTaher ldquoExact solutions for free flexural vibration of Levy-typerectangular thick plates via third-order shear deformationrdquoAppliedMathematicalModelling vol 35 no 2 pp 708ndash727 2011

[59] H-T Thai and S-E Kim ldquoLevy-type solution for bucklinganalysis of orthotropic plates based on two variable refined platetheoryrdquo Composite Structures vol 93 no 7 pp 1738ndash1746 2011

[60] A Szekrenyes ldquoApplication of Reddyrsquos third-order theory todelaminated orthotropic composite platesrdquo European Journal ofMechanics A Solids vol 43 pp 9ndash24 2014

[61] H-TThai and S-E Kim ldquoLevy-type solution for free vibrationanalysis of orthotropic plates based on two variable refinedplate theoryrdquoAppliedMathematical Modelling vol 36 no 8 pp3870ndash3882 2012

[62] Q-H Nguyen E Martinelli and M Hjiaj ldquoDerivation of theexact stiffnessmatrix for a two-layer Timoshenko beamelementwith partial interactionrdquo Engineering Structures vol 33 no 2pp 298ndash307 2011

[63] K-J Bathe Finite Element Procedures Prentice Hall UpperSaddle River NJ USA 1996

[64] M Petyt Introduction to Finite Element Vibration AnalysisCambridgeUniversity Press Cambridge UK 2nd edition 2010

[65] E Ventsel and T KrauthammerThin Plates and ShellsmdashTheoryAnalysis and Applications Marcel Dekker New York NY USA2001

[66] T Ozben and N Arslan ldquoFEM analysis of laminated compositeplate with rectangular hole and various elastic modulus undertransverse loadsrdquo Applied Mathematical Modelling vol 34 no7 pp 1746ndash1762 2010

[67] R Szilard Theories and Applications of Plate Analysis JohnWiley amp Sons Hoboken NJ USA 2004

[68] W Q Chen Y FWu and R Q Xu ldquoState space formulation forcomposite beam-columns with partial interactionrdquo CompositesScience and Technology vol 67 no 11-12 pp 2500ndash2512 2007

[69] K Xu A K Noor and Y Y Tang ldquoThree-dimensional solu-tions for coupled thermoelectroelastic response of multilayeredplatesrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 126 no 3-4 pp 355ndash371 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


delaminated composite laminates subjected to compressiveloads were investigated using the Rayleigh-Ritz method andCLPT theory by Kharazi and Ovesy [35] Experiments onmultiple delaminated composite plates and a FEM studyusing ANSYS were carried out by Aslan and Sahin [36]Kharazi developed a layerwise theory based on the FSDTwhere the prevention of the penetration of the delaminatedlayers was solved by using the penalty method and artificialsprings [37] The mixed mode buckling was captured with acontact model in the paper of Hwang and Liu [38] A layer-wise FEMsolution usingAbaqus andVICONOPT is given forcomposite plates with embedded rectangular delamination inDamghani et al [39] Marjanovic and Vuksanovic [40] usedthe Generalized Layerwise Plate Theory of Reddy and builta finite element (FE) model which was capable of handlingmore embedded delaminations in the composite structureA 3D FEM model was developed for analysing the bucklingbehaviour of embedded and through-the-width delaminatedplates in [41]

For the global model an analytical solution was presentedin Juhasz and Szekrenyes [42] In that paper it was shown thatthe in-plane force distribution greatly influences the bucklingbehaviour of the plate In this paper a FEmodel is introducedwhich concerns a plate with two simply supported oppositeedges so the plate problem cannot be solved using beamtheories The other two boundary conditions (BCs) can bearbitrarily chosen The plate is loaded with uniaxial com-pression The critical in-plane loads for the global bucklingare calculated using a constrained model [29 31] Using thisapproach there is no need for using constraints along thedelamination which results in a very efficient calculationmethod contrary to the widely used contact models Thecontinuity between the delaminated andundelaminated partsis maintained using special transitional elementsThis modelis capable of predicting the global critical buckling loads anddetermines the corresponding in-plane force distributionsUsing the solution of the global model the local stability isanalysed separately using a local FE model for each delami-nated part Because of the nonuniformity in the distributionof axial forces the stability analysis was carried out usingthe method of harmonic balance [43] It was shown thatthese effects greatly influence the local buckling loads In thispaper the effects of different type of boundary conditionsare analysed in a numerical example where the length ofthe delamination is varied and the local and global criticalamplitudes are calculated which results in a stability mapfor the different BCs It is shown that the different BCsgreatly influence the buckling loads and the correspondingmixed mode buckling shapes Using our method and theresulting stability maps the failure process of a compresseddelaminated plate can be estimated and it can be determinedat which point will the plate buckle globally or locally or inmixed mode case

2 Model Creation

Let us consider a layered plate with several orthotropic pliesand a closed through-the-width delamination (see Figure 1)

z

y

x

b aL

2

3

1

Simply supported

DelaminationNload

x

tt tb

L1

L3

t

Figure 1 Simply supported layered plate with delamination sub-jected to uniaxial compression

The plate length is 119871 the length of the delamination is 119886and the thickness of the plate is 119905 The plate is composed oftwo equivalent single layers (ESL) with thicknesses 119905

119905and 119905119887

respectivelyThe edges parallel to the 119909-axis are simply supported and

the constraint of the other two can be arbitrarily chosen Theplate is subjected to uniaxial compression in the midplaneWe are considering thin structures therefore the solution isbased on CLPT [44ndash47]

119906 (119909 119910 119911) = 0 (119909 119910) + 119906120575

0 (119909 119910) minus 1199111205971199080 (119909 119910)

120597119909

V (119909 119910 119911) = V0 (119909 119910) + V120575

0 (119909 119910) minus 119911120597119908120575

0 (119909 119910)

120597119910

119908 (119909 119910 119911) = 1199080 (119909 119910)

(1)

where 0 and V0 are constant parts in the in-plane displace-ment functions of the undelaminated portions which arisebecause of the kinematic coupling on the interface as Figure 2shows 1199061205750 and V

120575

0 are constant through the thickness in the topand bottom parts where 120575 takes either ldquo119905rdquo or ldquo119887rdquo respectively(see Figure 2) Moreover 1199080 is the deflection of the plateparts respectively Along the undelaminated parts we assumeperfect adherence and no crack propagation at the cracktips Therefore the continuity of the undelaminated portionsis ensured using the system of exact kinematic conditions(SEKC) [48]

11990611990510038161003816100381610038161003816119911=minus119905

1199052 = 119906

11988710038161003816100381610038161003816119911=1199051198872

V11990510038161003816100381610038161003816119911=minus119905

1199052 = V119887

10038161003816100381610038161003816119911=1199051198872

(2)

And the in-plane displacements in the global reference planeare

119905119905+ 119905119887

2

le 119905119887

11990611988710038161003816100381610038161003816119911=119905

1199052 = 0 V119887

10038161003816100381610038161003816119911=1199051199052 = V0

ge 119905119887

1199061199051003816100381610038161003816119911=minus119905

1198872 = 0 V1199051003816100381610038161003816119911=minus119905

1198872 = V0

(3)

From the literature it is well known that a free modelallows the intersection of the delaminated portions into


Xx

x


Yy

y


Delamination

Delaminated portion



tt

tb

0 u0

t(x y z)

ut(x y z)

ub(x y z)b(x y z)

Z z Z z



1205980119909

1205980119910

1205740119909119910

119905

=

0119909 minus119905119887

21199080119909119909

V0119910 minus119905119887

21199080119910119910

0119910 + V0119909 minus 1199051198871199080119909119909

119905

1205981119909

1205981119910

1205741119909119910

119905

=

minus1199080119909119909

minus1199080119910119910

minus21199080119909119910

119905

1205980119909

1205980119910

1205740119909119910

119887

=

0119909 +119905119905

21199080119909119909

V0119910 +119905119905

21199080119910119910

0119910 + V0119909 + 1199051199051199080119909119909

119887

1205981119909

1205981119910

1205741119909119910

119887

=

minus1199080119909119909

minus1199080119910119910

minus21199080119909119910

119887

(4)


1205980119909

1205980119910

1205740119909119910

=

119906120575

0119909

V12057501199101199060119910 + V1205750119909

1205981119909

1205981119910

1205741119909119910

=

minus1199080119909119909

minus1199080119910119910

minus21199080119909119910

(5)


NM

120575

= [

A BB D

]

120575

1205980

1205981

120575

(6)



119906 =

119873119909

119873119910

119873119909119910

sdot

1205980119909

1205980119910

1205740119909119910

+

119872119909

119872119910

119872119909119910

sdot

1205981119909

1205981119910

1205741119909119910

(7)





1199060

V01199080

(119909 119910) =

infin

sum

119899=1

1198800119899 (119909) sin1205731199101198810119899 (119909) cos1205731199101198820119899 (119909) sin120573119910

(8)



X

Z2L 2R


1 32

1205791

120579i 120579nwn

un

w1 u1

ui

wi

120579(i+1)

120579(i+1)

w(i+1)

w(i+1)

ub(i+1)

ut(i+1)

w(nminusjminus2)




ub(nminusjminus2)



U119890= R119890u119890 (9)


119890is a diagonal



119880119890=12intΩ119890

119906 119889119881 =12U119879119890K119872119890

U119890 (10)

where K119872119890



K119866119890= 1198730119897119890 int

1

0B119908otimesB119908119889120585 (11)

As the in-plane119873119909119910


119909119910load

K119909119910119866119890= 1198730119897119890

120573

2int

1

0B119908

119879otimesN119908119889120585 (12)





u119879119890= 1199061 V1 1199081 1205791 1199062 V2 1199082 1205792 (13)



R119890=

[[[[[[

[




d

]]]]]]

]

(14)


119906 (120585) = 1198860 + 1198861120585 =8sum

119894=1119873119906119894119906119890119894 (15)

V (120585) = 1198870 + 1198871120585 =8sum

119894=1119873V119894119906119890119894 (16)

119908 (120585) = 1198880 + 1198881120585 + 11988821205852+ 1198883120585

3=

8sum

119894=1119873119908119894119906119890119894 (17)

where 119886119894 119887119894 and 119888



119906 (0) = 1199061

119906 (1) = 1199062(18)

V (0) = V1

V (1) = V2(19)

119908 (0) = 1199081

119908 (1) = 1199082(20)

1119897119890

1199081015840(0) = 1205791

1119897119890

1199081015840(1) = 1205792

(21)



N120581

119879= 1198731205811 1198731205818 (22)



119880119890= int

119887

0(U119890

119879int

1

0

12(Kint119872119890

+Kint119872119890

119879

) 119889120585U119890)119889119910 (23)




u119879119890= 119906119905

1 V119905

1 119906119887

1 V119887

1 1199081 1205791 119906119905

2 V119905

2 119906119887

2 V119887

2 1199082 1205792 (24)





u119879119890= 1199061 V1 1199081 1205791 119906

119905

2 V119905

2 119906119887

2 V119887

2 1199082 1205792 (25)


u119879119890= 119906119905

1 V119905

1 119906119887

1 V119887

1 1199081 1205791 1199062 V2 1199082 1205792 (26)


119906119905(0) = 1199061 minus

119905119887

21205791

119906119905(1) = 119906

119905

2

119906119887(0) = 1199061 +

119905119905

21205791

119906119887(1) = 119906

119887

2

V119905 (0) = V1 minus119905119887

21205731199081

V119905 (1) = V1199052

V119887 (0) = V1 +119905119905

21205731199081

V119887 (1) = V1198872

(27)


119906119905(0) = 119906

119905

2

119906119905(1) = 1199061 minus

119905119887

21205791

119906119887(0) = 119906

119887

2

119906119887(1) = 1199061 +

119905119905

21205791

V119905 (0) = V1199052

V119905 (1) = V1 minus119905119887

21205731199081

V119887 (0) = V1198872

V119887 (1) = V1 +119905119905

21205731199081

(28)





(K+119873Load119909

K119909119866)U119879 = 0 (29)



z

y

x

a

Simply supported



Nx

(a)

L x

Nx

L1 + aL1

Nbx

Ntx

(b)




U119879 = d0 +infin

sum

119894=1d119894cos

119894120587119910

119887 d119894= 119889119894120601 (30)



[[[[[[[[[

[

K120575

12119873

Load119909119897

K119909119866120575

0 0 sdot sdot sdot

12119873

Load119909119897

K119909119866120575

K120575

12119873

Load119909119897

K119909119866120575

0 sdot sdot sdot

0 12119873

Load119909119897

K119909119866120575

K120575

12119873

Load119909119897

K119909119866120575

sdot sdot sdot

d

]]]]]]]]]

]

[[[[[[[[[

[

1198890

1198891

1198892

1198893

]]]]]]]]]

]

120601 = 0 (31)





Z1015840 = TZ (32)


Z120572(119909) = 119890

(T119909)K120572 (33)









Crit1119909Const 119873

Crit1119909 Sine Δ [] 119873

Abaqus 1119909 Sine 119873


1105 15685 4194 10014 14608 4588 737119873

Crit2119909Const 119873

Crit2119909 Sine Δ [] 119873

Abaqus 2119909 Sine 119873


15918 22638 4221 15129 22104 4610 241


119909moment and


11988001198992119887 (119909) = 1198800119899120572 (119909) +119905119887

21198821015840

0119899120572 (119909)

11988101198992119887 (119909) = 1198810119899120572 (119909) +119905119887

21205731198820119899120572 (119909)

11988001198992119905 (119909) = 1198800119899120572 (119909) minus119905119905

21198821015840

0119899120572 (119909)

11988101198992119905 (119909) = 1198810119899120572 (119909) minus119905119905

21205731198820119899120572 (119909)

11988201198992 (119909) = 1198820119899120572 (119909)

1198821015840

01198992 (119909) = 1198821015840

0119899120572 (119909)

1198991199091198992119905 (119909) + 1198991199091198992119887 (119909) = 119899

119909119899120572119905(119909) + 119899

119909119899120572119887(119909)

1198991199091199101198992119905 (119909) + 1198991199091199101198992119887 (119909) = 119899

119909119910119899120572119905(119909) + 119899

119909119910119899120572119887(119909)

1198981199091198992119905 (119909) +

119905119887

21198991199091198992119905 (119909) +1198981199091198992119887 (119909)

minus119905119905

21198991199091198992119887 (119909) = 119898

119909119899120572119905(119909) +

119905119887

2119899119909119899120572119905

(119909)

+119898119909119899120572119887

(119909) minus119905119905

2119899119909119899120572119887

(119909)

1198981015840

1199091198992119905 (119909) +119905119887

21198991015840

1199091198992119905 (119909)

minus 2(1205731198981199091199101198992119905 (119909) +

119905119887

21198991199091199101198992119905 (119909)) +119898

1015840

1199091198992119887 (119909)

minus119905119905

21198991015840

1199091198992119887 (119909)

minus 2(1205731198981199091199101198992119887 (119909) minus

119905119905

21198991199091199101198992119887 (119909)) = 119898

1015840

119909119899120572119905(119909)

+119905119887

21198991015840

119909119899120572119905(119909) minus 2(120573119898

119909119910119899120572119905(119909) +

119905119887

2119899119909119910119899120572119905

(119909))

+1198981015840

119909119899120572119887(119909) minus

119905119905

21198991015840

119909119899120572119887(119909)

minus 2(120573119898119909119910119899120572119887

(119909) minus119905119905

2119899119909119910119899120572119887

(119909))

(34)


119909119899120572120575 119899119910119899120572120575

119899119909119910119899120572120575

119898119909119899120572120575

119898119910119899120572120575

119898119909119910119899120572120575


MKall = 1198800 0 0 0119879 (35)





intradic1 + (120597sum119899

119894(119891120574

119894119882120574

119894(119909))

120597119909)

2

119889119909minusΔ119906

=

119871 if 120574 = 119892

119886 if 120574 = 119897120575

(36)


119894(119909) is




119871 [mm] 119887 [mm] 119905 [mm] ℎ [mm]200 100 45 05


Δ119906

= 119882Amp120574

(1198800)(119899

sum

119894

119880119892

119894(1198711) minus119880119892

119894(1198711+ 119886))

+ (119880Static120575


120575(119880

Crit10119892 1198711+ 119886))

(37)




Static120575



119891Amp

(1198800) =

0 0 le 1198800 ge 119880Crit1198940120574

1198800 minus 119880Crit1198940120574

119880Max0 minus 119880

Crit1198940120574

119880Crit1198940120574 lt 1198800 ge 119880

Max0

(38)


6 Numerical Example



20 40 60 80 100

600

800

1000

1200

S-SB-B

a

Nx


0 20 40 60 80 100

1000

2000

3000

4000

5000

Nx

S-SB-B

a






0 20 40 60 80 100

2000

4000

6000

8000

10000

12000

14000

Nx

S-SB-B

a


0 20 40 60 80 10000

05

10

15

20


Stable

Unstable

a

U0


0 20 40 60 80 10000

05

10

15

20

Stable

Unstable

a


U0


017 0

30

29

xg

xg

xg

Am

p (N

mm

)

Ngx

Ntx

Nbx

Ntxy

Nbxy

0150

0

U0 (mm)

NCrit3

NCrit2

NCrit1





Modes BCS-S B-B

I 4546 5021II 4898 5805III 8521 11912



(mm

)W

Am

pW

Am

pW

Am

p(m

m)

minus10

minus10

minus08

minus06

minus04

minus02

(mm

)

minus10

minus08

minus06

minus04

minus02

minus30

minus20

minus10

10

20

minus20

minus60

minus40

minus20

20

40

60

minus10

10

20

30

minus05

50

05

10

100 150

50 100 150

50 100 200150

50 100 200150

W

W

W

x(N

mm

)x

(Nm

m)

x(N

mm

)

Ntx

Nbx

Ntx

Nbx

Ntx

Nbx

x (mm)

x (mm)

50 100 200150x (mm)

x (mm)

x (mm)

50 100 150x (mm)

NA

mp

NA

mp

NA

mp









(mm

)

(mm

)

(mm

)

minus06

minus08

minus10 minus10

minus05

minus02

minus04

minus06

minus08

minus10

05

10

minus04

minus02

50 100 150 200x (mm)

50 100 150 200x (mm)

50 100 150 200

x (mm)

W(x) W(x)

W(x)

WA

mp

WA

mp

WA

mp







119880Crit20119892 119880Crit3



7 Conclusion




(mm

)(m

m)

(mm

)

minus01

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

01

00

minus2

2

0

x (mm)50 100 150

x (mm)50 100 150 200

x (mm)50 100 150 200x (mm)

50 100 150 200

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0

(mm

)

minus01

01

00

x (mm)50 100 150

(mm

)

minus2

2

0

x (mm)50 100 150 200

x (mm)50 100 150 200

x (mm)50 100 150 200

(mm

)

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0




Acknowledgment


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Xx

x


Yy

y


Delamination

Delaminated portion



tt

tb

0 u0

t(x y z)

ut(x y z)

ub(x y z)b(x y z)

Z z Z z



1205980119909

1205980119910

1205740119909119910

119905

=

0119909 minus119905119887

21199080119909119909

V0119910 minus119905119887

21199080119910119910

0119910 + V0119909 minus 1199051198871199080119909119909

119905

1205981119909

1205981119910

1205741119909119910

119905

=

minus1199080119909119909

minus1199080119910119910

minus21199080119909119910

119905

1205980119909

1205980119910

1205740119909119910

119887

=

0119909 +119905119905

21199080119909119909

V0119910 +119905119905

21199080119910119910

0119910 + V0119909 + 1199051199051199080119909119909

119887

1205981119909

1205981119910

1205741119909119910

119887

=

minus1199080119909119909

minus1199080119910119910

minus21199080119909119910

119887

(4)


1205980119909

1205980119910

1205740119909119910

=

119906120575

0119909

V12057501199101199060119910 + V1205750119909

1205981119909

1205981119910

1205741119909119910

=

minus1199080119909119909

minus1199080119910119910

minus21199080119909119910

(5)


NM

120575

= [

A BB D

]

120575

1205980

1205981

120575

(6)



119906 =

119873119909

119873119910

119873119909119910

sdot

1205980119909

1205980119910

1205740119909119910

+

119872119909

119872119910

119872119909119910

sdot

1205981119909

1205981119910

1205741119909119910

(7)





1199060

V01199080

(119909 119910) =

infin

sum

119899=1

1198800119899 (119909) sin1205731199101198810119899 (119909) cos1205731199101198820119899 (119909) sin120573119910

(8)



X

Z2L 2R


1 32

1205791

120579i 120579nwn

un

w1 u1

ui

wi

120579(i+1)

120579(i+1)

w(i+1)

w(i+1)

ub(i+1)

ut(i+1)

w(nminusjminus2)




ub(nminusjminus2)



U119890= R119890u119890 (9)


119890is a diagonal



119880119890=12intΩ119890

119906 119889119881 =12U119879119890K119872119890

U119890 (10)

where K119872119890



K119866119890= 1198730119897119890 int

1

0B119908otimesB119908119889120585 (11)

As the in-plane119873119909119910


119909119910load

K119909119910119866119890= 1198730119897119890

120573

2int

1

0B119908

119879otimesN119908119889120585 (12)





u119879119890= 1199061 V1 1199081 1205791 1199062 V2 1199082 1205792 (13)



R119890=

[[[[[[

[




d

]]]]]]

]

(14)


119906 (120585) = 1198860 + 1198861120585 =8sum

119894=1119873119906119894119906119890119894 (15)

V (120585) = 1198870 + 1198871120585 =8sum

119894=1119873V119894119906119890119894 (16)

119908 (120585) = 1198880 + 1198881120585 + 11988821205852+ 1198883120585

3=

8sum

119894=1119873119908119894119906119890119894 (17)

where 119886119894 119887119894 and 119888



119906 (0) = 1199061

119906 (1) = 1199062(18)

V (0) = V1

V (1) = V2(19)

119908 (0) = 1199081

119908 (1) = 1199082(20)

1119897119890

1199081015840(0) = 1205791

1119897119890

1199081015840(1) = 1205792

(21)



N120581

119879= 1198731205811 1198731205818 (22)



119880119890= int

119887

0(U119890

119879int

1

0

12(Kint119872119890

+Kint119872119890

119879

) 119889120585U119890)119889119910 (23)




u119879119890= 119906119905

1 V119905

1 119906119887

1 V119887

1 1199081 1205791 119906119905

2 V119905

2 119906119887

2 V119887

2 1199082 1205792 (24)





u119879119890= 1199061 V1 1199081 1205791 119906

119905

2 V119905

2 119906119887

2 V119887

2 1199082 1205792 (25)


u119879119890= 119906119905

1 V119905

1 119906119887

1 V119887

1 1199081 1205791 1199062 V2 1199082 1205792 (26)


119906119905(0) = 1199061 minus

119905119887

21205791

119906119905(1) = 119906

119905

2

119906119887(0) = 1199061 +

119905119905

21205791

119906119887(1) = 119906

119887

2

V119905 (0) = V1 minus119905119887

21205731199081

V119905 (1) = V1199052

V119887 (0) = V1 +119905119905

21205731199081

V119887 (1) = V1198872

(27)


119906119905(0) = 119906

119905

2

119906119905(1) = 1199061 minus

119905119887

21205791

119906119887(0) = 119906

119887

2

119906119887(1) = 1199061 +

119905119905

21205791

V119905 (0) = V1199052

V119905 (1) = V1 minus119905119887

21205731199081

V119887 (0) = V1198872

V119887 (1) = V1 +119905119905

21205731199081

(28)





(K+119873Load119909

K119909119866)U119879 = 0 (29)



z

y

x

a

Simply supported



Nx

(a)

L x

Nx

L1 + aL1

Nbx

Ntx

(b)




U119879 = d0 +infin

sum

119894=1d119894cos

119894120587119910

119887 d119894= 119889119894120601 (30)



[[[[[[[[[

[

K120575

12119873

Load119909119897

K119909119866120575

0 0 sdot sdot sdot

12119873

Load119909119897

K119909119866120575

K120575

12119873

Load119909119897

K119909119866120575

0 sdot sdot sdot

0 12119873

Load119909119897

K119909119866120575

K120575

12119873

Load119909119897

K119909119866120575

sdot sdot sdot

d

]]]]]]]]]

]

[[[[[[[[[

[

1198890

1198891

1198892

1198893

]]]]]]]]]

]

120601 = 0 (31)





Z1015840 = TZ (32)


Z120572(119909) = 119890

(T119909)K120572 (33)









Crit1119909Const 119873

Crit1119909 Sine Δ [] 119873

Abaqus 1119909 Sine 119873


1105 15685 4194 10014 14608 4588 737119873

Crit2119909Const 119873

Crit2119909 Sine Δ [] 119873

Abaqus 2119909 Sine 119873


15918 22638 4221 15129 22104 4610 241


119909moment and


11988001198992119887 (119909) = 1198800119899120572 (119909) +119905119887

21198821015840

0119899120572 (119909)

11988101198992119887 (119909) = 1198810119899120572 (119909) +119905119887

21205731198820119899120572 (119909)

11988001198992119905 (119909) = 1198800119899120572 (119909) minus119905119905

21198821015840

0119899120572 (119909)

11988101198992119905 (119909) = 1198810119899120572 (119909) minus119905119905

21205731198820119899120572 (119909)

11988201198992 (119909) = 1198820119899120572 (119909)

1198821015840

01198992 (119909) = 1198821015840

0119899120572 (119909)

1198991199091198992119905 (119909) + 1198991199091198992119887 (119909) = 119899

119909119899120572119905(119909) + 119899

119909119899120572119887(119909)

1198991199091199101198992119905 (119909) + 1198991199091199101198992119887 (119909) = 119899

119909119910119899120572119905(119909) + 119899

119909119910119899120572119887(119909)

1198981199091198992119905 (119909) +

119905119887

21198991199091198992119905 (119909) +1198981199091198992119887 (119909)

minus119905119905

21198991199091198992119887 (119909) = 119898

119909119899120572119905(119909) +

119905119887

2119899119909119899120572119905

(119909)

+119898119909119899120572119887

(119909) minus119905119905

2119899119909119899120572119887

(119909)

1198981015840

1199091198992119905 (119909) +119905119887

21198991015840

1199091198992119905 (119909)

minus 2(1205731198981199091199101198992119905 (119909) +

119905119887

21198991199091199101198992119905 (119909)) +119898

1015840

1199091198992119887 (119909)

minus119905119905

21198991015840

1199091198992119887 (119909)

minus 2(1205731198981199091199101198992119887 (119909) minus

119905119905

21198991199091199101198992119887 (119909)) = 119898

1015840

119909119899120572119905(119909)

+119905119887

21198991015840

119909119899120572119905(119909) minus 2(120573119898

119909119910119899120572119905(119909) +

119905119887

2119899119909119910119899120572119905

(119909))

+1198981015840

119909119899120572119887(119909) minus

119905119905

21198991015840

119909119899120572119887(119909)

minus 2(120573119898119909119910119899120572119887

(119909) minus119905119905

2119899119909119910119899120572119887

(119909))

(34)


119909119899120572120575 119899119910119899120572120575

119899119909119910119899120572120575

119898119909119899120572120575

119898119910119899120572120575

119898119909119910119899120572120575


MKall = 1198800 0 0 0119879 (35)





intradic1 + (120597sum119899

119894(119891120574

119894119882120574

119894(119909))

120597119909)

2

119889119909minusΔ119906

=

119871 if 120574 = 119892

119886 if 120574 = 119897120575

(36)


119894(119909) is




119871 [mm] 119887 [mm] 119905 [mm] ℎ [mm]200 100 45 05


Δ119906

= 119882Amp120574

(1198800)(119899

sum

119894

119880119892

119894(1198711) minus119880119892

119894(1198711+ 119886))

+ (119880Static120575


120575(119880

Crit10119892 1198711+ 119886))

(37)




Static120575



119891Amp

(1198800) =

0 0 le 1198800 ge 119880Crit1198940120574

1198800 minus 119880Crit1198940120574

119880Max0 minus 119880

Crit1198940120574

119880Crit1198940120574 lt 1198800 ge 119880

Max0

(38)


6 Numerical Example



20 40 60 80 100

600

800

1000

1200

S-SB-B

a

Nx


0 20 40 60 80 100

1000

2000

3000

4000

5000

Nx

S-SB-B

a






0 20 40 60 80 100

2000

4000

6000

8000

10000

12000

14000

Nx

S-SB-B

a


0 20 40 60 80 10000

05

10

15

20


Stable

Unstable

a

U0


0 20 40 60 80 10000

05

10

15

20

Stable

Unstable

a


U0


017 0

30

29

xg

xg

xg

Am

p (N

mm

)

Ngx

Ntx

Nbx

Ntxy

Nbxy

0150

0

U0 (mm)

NCrit3

NCrit2

NCrit1





Modes BCS-S B-B

I 4546 5021II 4898 5805III 8521 11912



(mm

)W

Am

pW

Am

pW

Am

p(m

m)

minus10

minus10

minus08

minus06

minus04

minus02

(mm

)

minus10

minus08

minus06

minus04

minus02

minus30

minus20

minus10

10

20

minus20

minus60

minus40

minus20

20

40

60

minus10

10

20

30

minus05

50

05

10

100 150

50 100 150

50 100 200150

50 100 200150

W

W

W

x(N

mm

)x

(Nm

m)

x(N

mm

)

Ntx

Nbx

Ntx

Nbx

Ntx

Nbx

x (mm)

x (mm)

50 100 200150x (mm)

x (mm)

x (mm)

50 100 150x (mm)

NA

mp

NA

mp

NA

mp









(mm

)

(mm

)

(mm

)

minus06

minus08

minus10 minus10

minus05

minus02

minus04

minus06

minus08

minus10

05

10

minus04

minus02

50 100 150 200x (mm)

50 100 150 200x (mm)

50 100 150 200

x (mm)

W(x) W(x)

W(x)

WA

mp

WA

mp

WA

mp







119880Crit20119892 119880Crit3



7 Conclusion




(mm

)(m

m)

(mm

)

minus01

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

01

00

minus2

2

0

x (mm)50 100 150

x (mm)50 100 150 200

x (mm)50 100 150 200x (mm)

50 100 150 200

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0

(mm

)

minus01

01

00

x (mm)50 100 150

(mm

)

minus2

2

0

x (mm)50 100 150 200

x (mm)50 100 150 200

x (mm)50 100 150 200

(mm

)

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0




Acknowledgment


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











X

Z2L 2R


1 32

1205791

120579i 120579nwn

un

w1 u1

ui

wi

120579(i+1)

120579(i+1)

w(i+1)

w(i+1)

ub(i+1)

ut(i+1)

w(nminusjminus2)




ub(nminusjminus2)



U119890= R119890u119890 (9)


119890is a diagonal



119880119890=12intΩ119890

119906 119889119881 =12U119879119890K119872119890

U119890 (10)

where K119872119890



K119866119890= 1198730119897119890 int

1

0B119908otimesB119908119889120585 (11)

As the in-plane119873119909119910


119909119910load

K119909119910119866119890= 1198730119897119890

120573

2int

1

0B119908

119879otimesN119908119889120585 (12)





u119879119890= 1199061 V1 1199081 1205791 1199062 V2 1199082 1205792 (13)



R119890=

[[[[[[

[




d

]]]]]]

]

(14)


119906 (120585) = 1198860 + 1198861120585 =8sum

119894=1119873119906119894119906119890119894 (15)

V (120585) = 1198870 + 1198871120585 =8sum

119894=1119873V119894119906119890119894 (16)

119908 (120585) = 1198880 + 1198881120585 + 11988821205852+ 1198883120585

3=

8sum

119894=1119873119908119894119906119890119894 (17)

where 119886119894 119887119894 and 119888



119906 (0) = 1199061

119906 (1) = 1199062(18)

V (0) = V1

V (1) = V2(19)

119908 (0) = 1199081

119908 (1) = 1199082(20)

1119897119890

1199081015840(0) = 1205791

1119897119890

1199081015840(1) = 1205792

(21)



N120581

119879= 1198731205811 1198731205818 (22)



119880119890= int

119887

0(U119890

119879int

1

0

12(Kint119872119890

+Kint119872119890

119879

) 119889120585U119890)119889119910 (23)




u119879119890= 119906119905

1 V119905

1 119906119887

1 V119887

1 1199081 1205791 119906119905

2 V119905

2 119906119887

2 V119887

2 1199082 1205792 (24)





u119879119890= 1199061 V1 1199081 1205791 119906

119905

2 V119905

2 119906119887

2 V119887

2 1199082 1205792 (25)


u119879119890= 119906119905

1 V119905

1 119906119887

1 V119887

1 1199081 1205791 1199062 V2 1199082 1205792 (26)


119906119905(0) = 1199061 minus

119905119887

21205791

119906119905(1) = 119906

119905

2

119906119887(0) = 1199061 +

119905119905

21205791

119906119887(1) = 119906

119887

2

V119905 (0) = V1 minus119905119887

21205731199081

V119905 (1) = V1199052

V119887 (0) = V1 +119905119905

21205731199081

V119887 (1) = V1198872

(27)


119906119905(0) = 119906

119905

2

119906119905(1) = 1199061 minus

119905119887

21205791

119906119887(0) = 119906

119887

2

119906119887(1) = 1199061 +

119905119905

21205791

V119905 (0) = V1199052

V119905 (1) = V1 minus119905119887

21205731199081

V119887 (0) = V1198872

V119887 (1) = V1 +119905119905

21205731199081

(28)





(K+119873Load119909

K119909119866)U119879 = 0 (29)



z

y

x

a

Simply supported



Nx

(a)

L x

Nx

L1 + aL1

Nbx

Ntx

(b)




U119879 = d0 +infin

sum

119894=1d119894cos

119894120587119910

119887 d119894= 119889119894120601 (30)



[[[[[[[[[

[

K120575

12119873

Load119909119897

K119909119866120575

0 0 sdot sdot sdot

12119873

Load119909119897

K119909119866120575

K120575

12119873

Load119909119897

K119909119866120575

0 sdot sdot sdot

0 12119873

Load119909119897

K119909119866120575

K120575

12119873

Load119909119897

K119909119866120575

sdot sdot sdot

d

]]]]]]]]]

]

[[[[[[[[[

[

1198890

1198891

1198892

1198893

]]]]]]]]]

]

120601 = 0 (31)





Z1015840 = TZ (32)


Z120572(119909) = 119890

(T119909)K120572 (33)









Crit1119909Const 119873

Crit1119909 Sine Δ [] 119873

Abaqus 1119909 Sine 119873


1105 15685 4194 10014 14608 4588 737119873

Crit2119909Const 119873

Crit2119909 Sine Δ [] 119873

Abaqus 2119909 Sine 119873


15918 22638 4221 15129 22104 4610 241


119909moment and


11988001198992119887 (119909) = 1198800119899120572 (119909) +119905119887

21198821015840

0119899120572 (119909)

11988101198992119887 (119909) = 1198810119899120572 (119909) +119905119887

21205731198820119899120572 (119909)

11988001198992119905 (119909) = 1198800119899120572 (119909) minus119905119905

21198821015840

0119899120572 (119909)

11988101198992119905 (119909) = 1198810119899120572 (119909) minus119905119905

21205731198820119899120572 (119909)

11988201198992 (119909) = 1198820119899120572 (119909)

1198821015840

01198992 (119909) = 1198821015840

0119899120572 (119909)

1198991199091198992119905 (119909) + 1198991199091198992119887 (119909) = 119899

119909119899120572119905(119909) + 119899

119909119899120572119887(119909)

1198991199091199101198992119905 (119909) + 1198991199091199101198992119887 (119909) = 119899

119909119910119899120572119905(119909) + 119899

119909119910119899120572119887(119909)

1198981199091198992119905 (119909) +

119905119887

21198991199091198992119905 (119909) +1198981199091198992119887 (119909)

minus119905119905

21198991199091198992119887 (119909) = 119898

119909119899120572119905(119909) +

119905119887

2119899119909119899120572119905

(119909)

+119898119909119899120572119887

(119909) minus119905119905

2119899119909119899120572119887

(119909)

1198981015840

1199091198992119905 (119909) +119905119887

21198991015840

1199091198992119905 (119909)

minus 2(1205731198981199091199101198992119905 (119909) +

119905119887

21198991199091199101198992119905 (119909)) +119898

1015840

1199091198992119887 (119909)

minus119905119905

21198991015840

1199091198992119887 (119909)

minus 2(1205731198981199091199101198992119887 (119909) minus

119905119905

21198991199091199101198992119887 (119909)) = 119898

1015840

119909119899120572119905(119909)

+119905119887

21198991015840

119909119899120572119905(119909) minus 2(120573119898

119909119910119899120572119905(119909) +

119905119887

2119899119909119910119899120572119905

(119909))

+1198981015840

119909119899120572119887(119909) minus

119905119905

21198991015840

119909119899120572119887(119909)

minus 2(120573119898119909119910119899120572119887

(119909) minus119905119905

2119899119909119910119899120572119887

(119909))

(34)


119909119899120572120575 119899119910119899120572120575

119899119909119910119899120572120575

119898119909119899120572120575

119898119910119899120572120575

119898119909119910119899120572120575


MKall = 1198800 0 0 0119879 (35)





intradic1 + (120597sum119899

119894(119891120574

119894119882120574

119894(119909))

120597119909)

2

119889119909minusΔ119906

=

119871 if 120574 = 119892

119886 if 120574 = 119897120575

(36)


119894(119909) is




119871 [mm] 119887 [mm] 119905 [mm] ℎ [mm]200 100 45 05


Δ119906

= 119882Amp120574

(1198800)(119899

sum

119894

119880119892

119894(1198711) minus119880119892

119894(1198711+ 119886))

+ (119880Static120575


120575(119880

Crit10119892 1198711+ 119886))

(37)




Static120575



119891Amp

(1198800) =

0 0 le 1198800 ge 119880Crit1198940120574

1198800 minus 119880Crit1198940120574

119880Max0 minus 119880

Crit1198940120574

119880Crit1198940120574 lt 1198800 ge 119880

Max0

(38)


6 Numerical Example



20 40 60 80 100

600

800

1000

1200

S-SB-B

a

Nx


0 20 40 60 80 100

1000

2000

3000

4000

5000

Nx

S-SB-B

a






0 20 40 60 80 100

2000

4000

6000

8000

10000

12000

14000

Nx

S-SB-B

a


0 20 40 60 80 10000

05

10

15

20


Stable

Unstable

a

U0


0 20 40 60 80 10000

05

10

15

20

Stable

Unstable

a


U0


017 0

30

29

xg

xg

xg

Am

p (N

mm

)

Ngx

Ntx

Nbx

Ntxy

Nbxy

0150

0

U0 (mm)

NCrit3

NCrit2

NCrit1





Modes BCS-S B-B

I 4546 5021II 4898 5805III 8521 11912



(mm

)W

Am

pW

Am

pW

Am

p(m

m)

minus10

minus10

minus08

minus06

minus04

minus02

(mm

)

minus10

minus08

minus06

minus04

minus02

minus30

minus20

minus10

10

20

minus20

minus60

minus40

minus20

20

40

60

minus10

10

20

30

minus05

50

05

10

100 150

50 100 150

50 100 200150

50 100 200150

W

W

W

x(N

mm

)x

(Nm

m)

x(N

mm

)

Ntx

Nbx

Ntx

Nbx

Ntx

Nbx

x (mm)

x (mm)

50 100 200150x (mm)

x (mm)

x (mm)

50 100 150x (mm)

NA

mp

NA

mp

NA

mp









(mm

)

(mm

)

(mm

)

minus06

minus08

minus10 minus10

minus05

minus02

minus04

minus06

minus08

minus10

05

10

minus04

minus02

50 100 150 200x (mm)

50 100 150 200x (mm)

50 100 150 200

x (mm)

W(x) W(x)

W(x)

WA

mp

WA

mp

WA

mp







119880Crit20119892 119880Crit3



7 Conclusion




(mm

)(m

m)

(mm

)

minus01

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

01

00

minus2

2

0

x (mm)50 100 150

x (mm)50 100 150 200

x (mm)50 100 150 200x (mm)

50 100 150 200

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0

(mm

)

minus01

01

00

x (mm)50 100 150

(mm

)

minus2

2

0

x (mm)50 100 150 200

x (mm)50 100 150 200

x (mm)50 100 150 200

(mm

)

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0




Acknowledgment


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











N120581

119879= 1198731205811 1198731205818 (22)



119880119890= int

119887

0(U119890

119879int

1

0

12(Kint119872119890

+Kint119872119890

119879

) 119889120585U119890)119889119910 (23)




u119879119890= 119906119905

1 V119905

1 119906119887

1 V119887

1 1199081 1205791 119906119905

2 V119905

2 119906119887

2 V119887

2 1199082 1205792 (24)





u119879119890= 1199061 V1 1199081 1205791 119906

119905

2 V119905

2 119906119887

2 V119887

2 1199082 1205792 (25)


u119879119890= 119906119905

1 V119905

1 119906119887

1 V119887

1 1199081 1205791 1199062 V2 1199082 1205792 (26)


119906119905(0) = 1199061 minus

119905119887

21205791

119906119905(1) = 119906

119905

2

119906119887(0) = 1199061 +

119905119905

21205791

119906119887(1) = 119906

119887

2

V119905 (0) = V1 minus119905119887

21205731199081

V119905 (1) = V1199052

V119887 (0) = V1 +119905119905

21205731199081

V119887 (1) = V1198872

(27)


119906119905(0) = 119906

119905

2

119906119905(1) = 1199061 minus

119905119887

21205791

119906119887(0) = 119906

119887

2

119906119887(1) = 1199061 +

119905119905

21205791

V119905 (0) = V1199052

V119905 (1) = V1 minus119905119887

21205731199081

V119887 (0) = V1198872

V119887 (1) = V1 +119905119905

21205731199081

(28)





(K+119873Load119909

K119909119866)U119879 = 0 (29)



z

y

x

a

Simply supported



Nx

(a)

L x

Nx

L1 + aL1

Nbx

Ntx

(b)




U119879 = d0 +infin

sum

119894=1d119894cos

119894120587119910

119887 d119894= 119889119894120601 (30)



[[[[[[[[[

[

K120575

12119873

Load119909119897

K119909119866120575

0 0 sdot sdot sdot

12119873

Load119909119897

K119909119866120575

K120575

12119873

Load119909119897

K119909119866120575

0 sdot sdot sdot

0 12119873

Load119909119897

K119909119866120575

K120575

12119873

Load119909119897

K119909119866120575

sdot sdot sdot

d

]]]]]]]]]

]

[[[[[[[[[

[

1198890

1198891

1198892

1198893

]]]]]]]]]

]

120601 = 0 (31)





Z1015840 = TZ (32)


Z120572(119909) = 119890

(T119909)K120572 (33)









Crit1119909Const 119873

Crit1119909 Sine Δ [] 119873

Abaqus 1119909 Sine 119873


1105 15685 4194 10014 14608 4588 737119873

Crit2119909Const 119873

Crit2119909 Sine Δ [] 119873

Abaqus 2119909 Sine 119873


15918 22638 4221 15129 22104 4610 241


119909moment and


11988001198992119887 (119909) = 1198800119899120572 (119909) +119905119887

21198821015840

0119899120572 (119909)

11988101198992119887 (119909) = 1198810119899120572 (119909) +119905119887

21205731198820119899120572 (119909)

11988001198992119905 (119909) = 1198800119899120572 (119909) minus119905119905

21198821015840

0119899120572 (119909)

11988101198992119905 (119909) = 1198810119899120572 (119909) minus119905119905

21205731198820119899120572 (119909)

11988201198992 (119909) = 1198820119899120572 (119909)

1198821015840

01198992 (119909) = 1198821015840

0119899120572 (119909)

1198991199091198992119905 (119909) + 1198991199091198992119887 (119909) = 119899

119909119899120572119905(119909) + 119899

119909119899120572119887(119909)

1198991199091199101198992119905 (119909) + 1198991199091199101198992119887 (119909) = 119899

119909119910119899120572119905(119909) + 119899

119909119910119899120572119887(119909)

1198981199091198992119905 (119909) +

119905119887

21198991199091198992119905 (119909) +1198981199091198992119887 (119909)

minus119905119905

21198991199091198992119887 (119909) = 119898

119909119899120572119905(119909) +

119905119887

2119899119909119899120572119905

(119909)

+119898119909119899120572119887

(119909) minus119905119905

2119899119909119899120572119887

(119909)

1198981015840

1199091198992119905 (119909) +119905119887

21198991015840

1199091198992119905 (119909)

minus 2(1205731198981199091199101198992119905 (119909) +

119905119887

21198991199091199101198992119905 (119909)) +119898

1015840

1199091198992119887 (119909)

minus119905119905

21198991015840

1199091198992119887 (119909)

minus 2(1205731198981199091199101198992119887 (119909) minus

119905119905

21198991199091199101198992119887 (119909)) = 119898

1015840

119909119899120572119905(119909)

+119905119887

21198991015840

119909119899120572119905(119909) minus 2(120573119898

119909119910119899120572119905(119909) +

119905119887

2119899119909119910119899120572119905

(119909))

+1198981015840

119909119899120572119887(119909) minus

119905119905

21198991015840

119909119899120572119887(119909)

minus 2(120573119898119909119910119899120572119887

(119909) minus119905119905

2119899119909119910119899120572119887

(119909))

(34)


119909119899120572120575 119899119910119899120572120575

119899119909119910119899120572120575

119898119909119899120572120575

119898119910119899120572120575

119898119909119910119899120572120575


MKall = 1198800 0 0 0119879 (35)





intradic1 + (120597sum119899

119894(119891120574

119894119882120574

119894(119909))

120597119909)

2

119889119909minusΔ119906

=

119871 if 120574 = 119892

119886 if 120574 = 119897120575

(36)


119894(119909) is




119871 [mm] 119887 [mm] 119905 [mm] ℎ [mm]200 100 45 05


Δ119906

= 119882Amp120574

(1198800)(119899

sum

119894

119880119892

119894(1198711) minus119880119892

119894(1198711+ 119886))

+ (119880Static120575


120575(119880

Crit10119892 1198711+ 119886))

(37)




Static120575



119891Amp

(1198800) =

0 0 le 1198800 ge 119880Crit1198940120574

1198800 minus 119880Crit1198940120574

119880Max0 minus 119880

Crit1198940120574

119880Crit1198940120574 lt 1198800 ge 119880

Max0

(38)


6 Numerical Example



20 40 60 80 100

600

800

1000

1200

S-SB-B

a

Nx


0 20 40 60 80 100

1000

2000

3000

4000

5000

Nx

S-SB-B

a






0 20 40 60 80 100

2000

4000

6000

8000

10000

12000

14000

Nx

S-SB-B

a


0 20 40 60 80 10000

05

10

15

20


Stable

Unstable

a

U0


0 20 40 60 80 10000

05

10

15

20

Stable

Unstable

a


U0


017 0

30

29

xg

xg

xg

Am

p (N

mm

)

Ngx

Ntx

Nbx

Ntxy

Nbxy

0150

0

U0 (mm)

NCrit3

NCrit2

NCrit1





Modes BCS-S B-B

I 4546 5021II 4898 5805III 8521 11912



(mm

)W

Am

pW

Am

pW

Am

p(m

m)

minus10

minus10

minus08

minus06

minus04

minus02

(mm

)

minus10

minus08

minus06

minus04

minus02

minus30

minus20

minus10

10

20

minus20

minus60

minus40

minus20

20

40

60

minus10

10

20

30

minus05

50

05

10

100 150

50 100 150

50 100 200150

50 100 200150

W

W

W

x(N

mm

)x

(Nm

m)

x(N

mm

)

Ntx

Nbx

Ntx

Nbx

Ntx

Nbx

x (mm)

x (mm)

50 100 200150x (mm)

x (mm)

x (mm)

50 100 150x (mm)

NA

mp

NA

mp

NA

mp









(mm

)

(mm

)

(mm

)

minus06

minus08

minus10 minus10

minus05

minus02

minus04

minus06

minus08

minus10

05

10

minus04

minus02

50 100 150 200x (mm)

50 100 150 200x (mm)

50 100 150 200

x (mm)

W(x) W(x)

W(x)

WA

mp

WA

mp

WA

mp







119880Crit20119892 119880Crit3



7 Conclusion




(mm

)(m

m)

(mm

)

minus01

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

01

00

minus2

2

0

x (mm)50 100 150

x (mm)50 100 150 200

x (mm)50 100 150 200x (mm)

50 100 150 200

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0

(mm

)

minus01

01

00

x (mm)50 100 150

(mm

)

minus2

2

0

x (mm)50 100 150 200

x (mm)50 100 150 200

x (mm)50 100 150 200

(mm

)

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0




Acknowledgment


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










z

y

x

a

Simply supported



Nx

(a)

L x

Nx

L1 + aL1

Nbx

Ntx

(b)




U119879 = d0 +infin

sum

119894=1d119894cos

119894120587119910

119887 d119894= 119889119894120601 (30)



[[[[[[[[[

[

K120575

12119873

Load119909119897

K119909119866120575

0 0 sdot sdot sdot

12119873

Load119909119897

K119909119866120575

K120575

12119873

Load119909119897

K119909119866120575

0 sdot sdot sdot

0 12119873

Load119909119897

K119909119866120575

K120575

12119873

Load119909119897

K119909119866120575

sdot sdot sdot

d

]]]]]]]]]

]

[[[[[[[[[

[

1198890

1198891

1198892

1198893

]]]]]]]]]

]

120601 = 0 (31)





Z1015840 = TZ (32)


Z120572(119909) = 119890

(T119909)K120572 (33)









Crit1119909Const 119873

Crit1119909 Sine Δ [] 119873

Abaqus 1119909 Sine 119873


1105 15685 4194 10014 14608 4588 737119873

Crit2119909Const 119873

Crit2119909 Sine Δ [] 119873

Abaqus 2119909 Sine 119873


15918 22638 4221 15129 22104 4610 241


119909moment and


11988001198992119887 (119909) = 1198800119899120572 (119909) +119905119887

21198821015840

0119899120572 (119909)

11988101198992119887 (119909) = 1198810119899120572 (119909) +119905119887

21205731198820119899120572 (119909)

11988001198992119905 (119909) = 1198800119899120572 (119909) minus119905119905

21198821015840

0119899120572 (119909)

11988101198992119905 (119909) = 1198810119899120572 (119909) minus119905119905

21205731198820119899120572 (119909)

11988201198992 (119909) = 1198820119899120572 (119909)

1198821015840

01198992 (119909) = 1198821015840

0119899120572 (119909)

1198991199091198992119905 (119909) + 1198991199091198992119887 (119909) = 119899

119909119899120572119905(119909) + 119899

119909119899120572119887(119909)

1198991199091199101198992119905 (119909) + 1198991199091199101198992119887 (119909) = 119899

119909119910119899120572119905(119909) + 119899

119909119910119899120572119887(119909)

1198981199091198992119905 (119909) +

119905119887

21198991199091198992119905 (119909) +1198981199091198992119887 (119909)

minus119905119905

21198991199091198992119887 (119909) = 119898

119909119899120572119905(119909) +

119905119887

2119899119909119899120572119905

(119909)

+119898119909119899120572119887

(119909) minus119905119905

2119899119909119899120572119887

(119909)

1198981015840

1199091198992119905 (119909) +119905119887

21198991015840

1199091198992119905 (119909)

minus 2(1205731198981199091199101198992119905 (119909) +

119905119887

21198991199091199101198992119905 (119909)) +119898

1015840

1199091198992119887 (119909)

minus119905119905

21198991015840

1199091198992119887 (119909)

minus 2(1205731198981199091199101198992119887 (119909) minus

119905119905

21198991199091199101198992119887 (119909)) = 119898

1015840

119909119899120572119905(119909)

+119905119887

21198991015840

119909119899120572119905(119909) minus 2(120573119898

119909119910119899120572119905(119909) +

119905119887

2119899119909119910119899120572119905

(119909))

+1198981015840

119909119899120572119887(119909) minus

119905119905

21198991015840

119909119899120572119887(119909)

minus 2(120573119898119909119910119899120572119887

(119909) minus119905119905

2119899119909119910119899120572119887

(119909))

(34)


119909119899120572120575 119899119910119899120572120575

119899119909119910119899120572120575

119898119909119899120572120575

119898119910119899120572120575

119898119909119910119899120572120575


MKall = 1198800 0 0 0119879 (35)





intradic1 + (120597sum119899

119894(119891120574

119894119882120574

119894(119909))

120597119909)

2

119889119909minusΔ119906

=

119871 if 120574 = 119892

119886 if 120574 = 119897120575

(36)


119894(119909) is




119871 [mm] 119887 [mm] 119905 [mm] ℎ [mm]200 100 45 05


Δ119906

= 119882Amp120574

(1198800)(119899

sum

119894

119880119892

119894(1198711) minus119880119892

119894(1198711+ 119886))

+ (119880Static120575


120575(119880

Crit10119892 1198711+ 119886))

(37)




Static120575



119891Amp

(1198800) =

0 0 le 1198800 ge 119880Crit1198940120574

1198800 minus 119880Crit1198940120574

119880Max0 minus 119880

Crit1198940120574

119880Crit1198940120574 lt 1198800 ge 119880

Max0

(38)


6 Numerical Example



20 40 60 80 100

600

800

1000

1200

S-SB-B

a

Nx


0 20 40 60 80 100

1000

2000

3000

4000

5000

Nx

S-SB-B

a






0 20 40 60 80 100

2000

4000

6000

8000

10000

12000

14000

Nx

S-SB-B

a


0 20 40 60 80 10000

05

10

15

20


Stable

Unstable

a

U0


0 20 40 60 80 10000

05

10

15

20

Stable

Unstable

a


U0


017 0

30

29

xg

xg

xg

Am

p (N

mm

)

Ngx

Ntx

Nbx

Ntxy

Nbxy

0150

0

U0 (mm)

NCrit3

NCrit2

NCrit1





Modes BCS-S B-B

I 4546 5021II 4898 5805III 8521 11912



(mm

)W

Am

pW

Am

pW

Am

p(m

m)

minus10

minus10

minus08

minus06

minus04

minus02

(mm

)

minus10

minus08

minus06

minus04

minus02

minus30

minus20

minus10

10

20

minus20

minus60

minus40

minus20

20

40

60

minus10

10

20

30

minus05

50

05

10

100 150

50 100 150

50 100 200150

50 100 200150

W

W

W

x(N

mm

)x

(Nm

m)

x(N

mm

)

Ntx

Nbx

Ntx

Nbx

Ntx

Nbx

x (mm)

x (mm)

50 100 200150x (mm)

x (mm)

x (mm)

50 100 150x (mm)

NA

mp

NA

mp

NA

mp









(mm

)

(mm

)

(mm

)

minus06

minus08

minus10 minus10

minus05

minus02

minus04

minus06

minus08

minus10

05

10

minus04

minus02

50 100 150 200x (mm)

50 100 150 200x (mm)

50 100 150 200

x (mm)

W(x) W(x)

W(x)

WA

mp

WA

mp

WA

mp







119880Crit20119892 119880Crit3



7 Conclusion




(mm

)(m

m)

(mm

)

minus01

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

01

00

minus2

2

0

x (mm)50 100 150

x (mm)50 100 150 200

x (mm)50 100 150 200x (mm)

50 100 150 200

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0

(mm

)

minus01

01

00

x (mm)50 100 150

(mm

)

minus2

2

0

x (mm)50 100 150 200

x (mm)50 100 150 200

x (mm)50 100 150 200

(mm

)

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0




Acknowledgment


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Crit1119909Const 119873

Crit1119909 Sine Δ [] 119873

Abaqus 1119909 Sine 119873


1105 15685 4194 10014 14608 4588 737119873

Crit2119909Const 119873

Crit2119909 Sine Δ [] 119873

Abaqus 2119909 Sine 119873


15918 22638 4221 15129 22104 4610 241


119909moment and


11988001198992119887 (119909) = 1198800119899120572 (119909) +119905119887

21198821015840

0119899120572 (119909)

11988101198992119887 (119909) = 1198810119899120572 (119909) +119905119887

21205731198820119899120572 (119909)

11988001198992119905 (119909) = 1198800119899120572 (119909) minus119905119905

21198821015840

0119899120572 (119909)

11988101198992119905 (119909) = 1198810119899120572 (119909) minus119905119905

21205731198820119899120572 (119909)

11988201198992 (119909) = 1198820119899120572 (119909)

1198821015840

01198992 (119909) = 1198821015840

0119899120572 (119909)

1198991199091198992119905 (119909) + 1198991199091198992119887 (119909) = 119899

119909119899120572119905(119909) + 119899

119909119899120572119887(119909)

1198991199091199101198992119905 (119909) + 1198991199091199101198992119887 (119909) = 119899

119909119910119899120572119905(119909) + 119899

119909119910119899120572119887(119909)

1198981199091198992119905 (119909) +

119905119887

21198991199091198992119905 (119909) +1198981199091198992119887 (119909)

minus119905119905

21198991199091198992119887 (119909) = 119898

119909119899120572119905(119909) +

119905119887

2119899119909119899120572119905

(119909)

+119898119909119899120572119887

(119909) minus119905119905

2119899119909119899120572119887

(119909)

1198981015840

1199091198992119905 (119909) +119905119887

21198991015840

1199091198992119905 (119909)

minus 2(1205731198981199091199101198992119905 (119909) +

119905119887

21198991199091199101198992119905 (119909)) +119898

1015840

1199091198992119887 (119909)

minus119905119905

21198991015840

1199091198992119887 (119909)

minus 2(1205731198981199091199101198992119887 (119909) minus

119905119905

21198991199091199101198992119887 (119909)) = 119898

1015840

119909119899120572119905(119909)

+119905119887

21198991015840

119909119899120572119905(119909) minus 2(120573119898

119909119910119899120572119905(119909) +

119905119887

2119899119909119910119899120572119905

(119909))

+1198981015840

119909119899120572119887(119909) minus

119905119905

21198991015840

119909119899120572119887(119909)

minus 2(120573119898119909119910119899120572119887

(119909) minus119905119905

2119899119909119910119899120572119887

(119909))

(34)


119909119899120572120575 119899119910119899120572120575

119899119909119910119899120572120575

119898119909119899120572120575

119898119910119899120572120575

119898119909119910119899120572120575


MKall = 1198800 0 0 0119879 (35)





intradic1 + (120597sum119899

119894(119891120574

119894119882120574

119894(119909))

120597119909)

2

119889119909minusΔ119906

=

119871 if 120574 = 119892

119886 if 120574 = 119897120575

(36)


119894(119909) is




119871 [mm] 119887 [mm] 119905 [mm] ℎ [mm]200 100 45 05


Δ119906

= 119882Amp120574

(1198800)(119899

sum

119894

119880119892

119894(1198711) minus119880119892

119894(1198711+ 119886))

+ (119880Static120575


120575(119880

Crit10119892 1198711+ 119886))

(37)




Static120575



119891Amp

(1198800) =

0 0 le 1198800 ge 119880Crit1198940120574

1198800 minus 119880Crit1198940120574

119880Max0 minus 119880

Crit1198940120574

119880Crit1198940120574 lt 1198800 ge 119880

Max0

(38)


6 Numerical Example



20 40 60 80 100

600

800

1000

1200

S-SB-B

a

Nx


0 20 40 60 80 100

1000

2000

3000

4000

5000

Nx

S-SB-B

a






0 20 40 60 80 100

2000

4000

6000

8000

10000

12000

14000

Nx

S-SB-B

a


0 20 40 60 80 10000

05

10

15

20


Stable

Unstable

a

U0


0 20 40 60 80 10000

05

10

15

20

Stable

Unstable

a


U0


017 0

30

29

xg

xg

xg

Am

p (N

mm

)

Ngx

Ntx

Nbx

Ntxy

Nbxy

0150

0

U0 (mm)

NCrit3

NCrit2

NCrit1





Modes BCS-S B-B

I 4546 5021II 4898 5805III 8521 11912



(mm

)W

Am

pW

Am

pW

Am

p(m

m)

minus10

minus10

minus08

minus06

minus04

minus02

(mm

)

minus10

minus08

minus06

minus04

minus02

minus30

minus20

minus10

10

20

minus20

minus60

minus40

minus20

20

40

60

minus10

10

20

30

minus05

50

05

10

100 150

50 100 150

50 100 200150

50 100 200150

W

W

W

x(N

mm

)x

(Nm

m)

x(N

mm

)

Ntx

Nbx

Ntx

Nbx

Ntx

Nbx

x (mm)

x (mm)

50 100 200150x (mm)

x (mm)

x (mm)

50 100 150x (mm)

NA

mp

NA

mp

NA

mp









(mm

)

(mm

)

(mm

)

minus06

minus08

minus10 minus10

minus05

minus02

minus04

minus06

minus08

minus10

05

10

minus04

minus02

50 100 150 200x (mm)

50 100 150 200x (mm)

50 100 150 200

x (mm)

W(x) W(x)

W(x)

WA

mp

WA

mp

WA

mp







119880Crit20119892 119880Crit3



7 Conclusion




(mm

)(m

m)

(mm

)

minus01

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

01

00

minus2

2

0

x (mm)50 100 150

x (mm)50 100 150 200

x (mm)50 100 150 200x (mm)

50 100 150 200

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0

(mm

)

minus01

01

00

x (mm)50 100 150

(mm

)

minus2

2

0

x (mm)50 100 150 200

x (mm)50 100 150 200

x (mm)50 100 150 200

(mm

)

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0




Acknowledgment


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119871 [mm] 119887 [mm] 119905 [mm] ℎ [mm]200 100 45 05


Δ119906

= 119882Amp120574

(1198800)(119899

sum

119894

119880119892

119894(1198711) minus119880119892

119894(1198711+ 119886))

+ (119880Static120575


120575(119880

Crit10119892 1198711+ 119886))

(37)




Static120575



119891Amp

(1198800) =

0 0 le 1198800 ge 119880Crit1198940120574

1198800 minus 119880Crit1198940120574

119880Max0 minus 119880

Crit1198940120574

119880Crit1198940120574 lt 1198800 ge 119880

Max0

(38)


6 Numerical Example



20 40 60 80 100

600

800

1000

1200

S-SB-B

a

Nx


0 20 40 60 80 100

1000

2000

3000

4000

5000

Nx

S-SB-B

a






0 20 40 60 80 100

2000

4000

6000

8000

10000

12000

14000

Nx

S-SB-B

a


0 20 40 60 80 10000

05

10

15

20


Stable

Unstable

a

U0


0 20 40 60 80 10000

05

10

15

20

Stable

Unstable

a


U0


017 0

30

29

xg

xg

xg

Am

p (N

mm

)

Ngx

Ntx

Nbx

Ntxy

Nbxy

0150

0

U0 (mm)

NCrit3

NCrit2

NCrit1





Modes BCS-S B-B

I 4546 5021II 4898 5805III 8521 11912



(mm

)W

Am

pW

Am

pW

Am

p(m

m)

minus10

minus10

minus08

minus06

minus04

minus02

(mm

)

minus10

minus08

minus06

minus04

minus02

minus30

minus20

minus10

10

20

minus20

minus60

minus40

minus20

20

40

60

minus10

10

20

30

minus05

50

05

10

100 150

50 100 150

50 100 200150

50 100 200150

W

W

W

x(N

mm

)x

(Nm

m)

x(N

mm

)

Ntx

Nbx

Ntx

Nbx

Ntx

Nbx

x (mm)

x (mm)

50 100 200150x (mm)

x (mm)

x (mm)

50 100 150x (mm)

NA

mp

NA

mp

NA

mp









(mm

)

(mm

)

(mm

)

minus06

minus08

minus10 minus10

minus05

minus02

minus04

minus06

minus08

minus10

05

10

minus04

minus02

50 100 150 200x (mm)

50 100 150 200x (mm)

50 100 150 200

x (mm)

W(x) W(x)

W(x)

WA

mp

WA

mp

WA

mp







119880Crit20119892 119880Crit3



7 Conclusion




(mm

)(m

m)

(mm

)

minus01

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

01

00

minus2

2

0

x (mm)50 100 150

x (mm)50 100 150 200

x (mm)50 100 150 200x (mm)

50 100 150 200

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0

(mm

)

minus01

01

00

x (mm)50 100 150

(mm

)

minus2

2

0

x (mm)50 100 150 200

x (mm)50 100 150 200

x (mm)50 100 150 200

(mm

)

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0




Acknowledgment


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100

2000

4000

6000

8000

10000

12000

14000

Nx

S-SB-B

a


0 20 40 60 80 10000

05

10

15

20


Stable

Unstable

a

U0


0 20 40 60 80 10000

05

10

15

20

Stable

Unstable

a


U0


017 0

30

29

xg

xg

xg

Am

p (N

mm

)

Ngx

Ntx

Nbx

Ntxy

Nbxy

0150

0

U0 (mm)

NCrit3

NCrit2

NCrit1





Modes BCS-S B-B

I 4546 5021II 4898 5805III 8521 11912



(mm

)W

Am

pW

Am

pW

Am

p(m

m)

minus10

minus10

minus08

minus06

minus04

minus02

(mm

)

minus10

minus08

minus06

minus04

minus02

minus30

minus20

minus10

10

20

minus20

minus60

minus40

minus20

20

40

60

minus10

10

20

30

minus05

50

05

10

100 150

50 100 150

50 100 200150

50 100 200150

W

W

W

x(N

mm

)x

(Nm

m)

x(N

mm

)

Ntx

Nbx

Ntx

Nbx

Ntx

Nbx

x (mm)

x (mm)

50 100 200150x (mm)

x (mm)

x (mm)

50 100 150x (mm)

NA

mp

NA

mp

NA

mp









(mm

)

(mm

)

(mm

)

minus06

minus08

minus10 minus10

minus05

minus02

minus04

minus06

minus08

minus10

05

10

minus04

minus02

50 100 150 200x (mm)

50 100 150 200x (mm)

50 100 150 200

x (mm)

W(x) W(x)

W(x)

WA

mp

WA

mp

WA

mp







119880Crit20119892 119880Crit3



7 Conclusion




(mm

)(m

m)

(mm

)

minus01

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

01

00

minus2

2

0

x (mm)50 100 150

x (mm)50 100 150 200

x (mm)50 100 150 200x (mm)

50 100 150 200

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0

(mm

)

minus01

01

00

x (mm)50 100 150

(mm

)

minus2

2

0

x (mm)50 100 150 200

x (mm)50 100 150 200

x (mm)50 100 150 200

(mm

)

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0




Acknowledgment


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(mm

)W

Am

pW

Am

pW

Am

p(m

m)

minus10

minus10

minus08

minus06

minus04

minus02

(mm

)

minus10

minus08

minus06

minus04

minus02

minus30

minus20

minus10

10

20

minus20

minus60

minus40

minus20

20

40

60

minus10

10

20

30

minus05

50

05

10

100 150

50 100 150

50 100 200150

50 100 200150

W

W

W

x(N

mm

)x

(Nm

m)

x(N

mm

)

Ntx

Nbx

Ntx

Nbx

Ntx

Nbx

x (mm)

x (mm)

50 100 200150x (mm)

x (mm)

x (mm)

50 100 150x (mm)

NA

mp

NA

mp

NA

mp









(mm

)

(mm

)

(mm

)

minus06

minus08

minus10 minus10

minus05

minus02

minus04

minus06

minus08

minus10

05

10

minus04

minus02

50 100 150 200x (mm)

50 100 150 200x (mm)

50 100 150 200

x (mm)

W(x) W(x)

W(x)

WA

mp

WA

mp

WA

mp







119880Crit20119892 119880Crit3



7 Conclusion




(mm

)(m

m)

(mm

)

minus01

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

01

00

minus2

2

0

x (mm)50 100 150

x (mm)50 100 150 200

x (mm)50 100 150 200x (mm)

50 100 150 200

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0

(mm

)

minus01

01

00

x (mm)50 100 150

(mm

)

minus2

2

0

x (mm)50 100 150 200

x (mm)50 100 150 200

x (mm)50 100 150 200

(mm

)

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0




Acknowledgment


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(mm

)

(mm

)

(mm

)

minus06

minus08

minus10 minus10

minus05

minus02

minus04

minus06

minus08

minus10

05

10

minus04

minus02

50 100 150 200x (mm)

50 100 150 200x (mm)

50 100 150 200

x (mm)

W(x) W(x)

W(x)

WA

mp

WA

mp

WA

mp







119880Crit20119892 119880Crit3



7 Conclusion




(mm

)(m

m)

(mm

)

minus01

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

01

00

minus2

2

0

x (mm)50 100 150

x (mm)50 100 150 200

x (mm)50 100 150 200x (mm)

50 100 150 200

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0

(mm

)

minus01

01

00

x (mm)50 100 150

(mm

)

minus2

2

0

x (mm)50 100 150 200

x (mm)50 100 150 200

x (mm)50 100 150 200

(mm

)

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0




Acknowledgment


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(mm

)(m

m)

(mm

)

minus01

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

01

00

minus2

2

0

x (mm)50 100 150

x (mm)50 100 150 200

x (mm)50 100 150 200x (mm)

50 100 150 200

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0

(mm

)

minus01

01

00

x (mm)50 100 150

(mm

)

minus2

2

0

x (mm)50 100 150 200

x (mm)50 100 150 200

x (mm)50 100 150 200

(mm

)

minus4

minus2

0

2

4

(mm

)

minus4

minus2

0

2

4

WA

mp

WA

mp

WA

mp

WA

mp


0119892 119880Crit30119892 and 119880Max

0




Acknowledgment


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article estimation of local delamination...

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