Hindawi Publishing CorporationISRN Civil EngineeringVolume 2013 Article ID 562482 8 pageshttpdxdoiorg1011552013562482

Research ArticlePoissonrsquos Theory for Analysis of Bending of Isotropic andAnisotropic Plates

K Vijayakumar

Department of Aerospace Engineering Indian Institute of Science Bangalore 560 012 India

Correspondence should be addressed to K Vijayakumar kazavijayakumargmailcom

Received 30 May 2013 Accepted 19 June 2013

Academic Editors M Garg and D Huang

Copyright copy 2013 K Vijayakumar This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Sixteen-decade-old problemof Poisson-Kirchhoff rsquos boundary conditions paradox is resolved in the case of isotropic plates through atheory designated as ldquoPoissonrsquos theory of plates in bendingrdquo It is based on ldquoassumingrdquo zero transverse shear stresses instead of strainsReactive (statically equivalent) transverse shear stresses are gradients of a function (in place of in-plane displacements as gradients ofvertical deflection) so that reactive transverse stresses are independent of material constants in the preliminary solution Equationsgoverning in-plane displacements are independent of the vertical (transverse) deflection 119908

0(119909 119910) Coupling of these equations

with 1199080is the root cause for the boundary conditions paradox Edge support condition on 119908

0does not play any role in obtaining

in-plane displacements Normally solutions to the displacements are obtained from governing equations based on the stationaryproperty of relevant total potential and reactive transverse shear stresses are expressed in terms of these displacements In the presentstudy a reverse process in obtaining preliminary solution is adapted in which reactive transverse stresses are determined first anddisplacements are obtained in terms of these stresses Equations governing second-order corrections to preliminary solutions ofbending of anisotropic plates are derived through application of an iterative method used earlier for the analysis of bending ofisotropic plates

1 Introduction

Kirchhoff rsquos theory [1] and first-order shear deformationtheory based on Henckyrsquos work [2] abbreviated as FSDT ofplates in bending are simple theories and continuously usedto obtain design information Kirchhoff rsquos theory consistsof a single variable model in which in-plane displacementsare expressed in terms of gradients of vertical deflection1199080(119909 119910) so that zero face shear conditions are satisfied 119908

0

is governed by a fourth-order equation associated with twoedge conditions instead of three edge conditions requiredin a 3D problem Consequence of this lacuna is the well-known Poisson-Kirchhoff boundary conditions paradox (seeReissnerrsquos article [3])

Assumption of zero transverse shear strains is dis-carded in FSDT forming a three-variable model Verticaldeflection 119908

0(119909 119910) and in-plane displacements [119906 V] =

119911[1199061(119909 119910) V

1(119909 119910)] are coupled in the governing differen-

tial equations and boundary conditions Reactive (statically

equivalent) transverse shears are combined with in-planeshear resulting in approximation of associated torsion prob-lem instead of flexure problem In Kirchhoff rsquos theory in-plane shear is combined with transverse shear implied inKelvin and Taitrsquos physical interpretation of contracted bound-ary condition [4] In fact torsion problem is associated withflexure problem whereas flexure problem (unlike directlyor indirectly implied in energy methods) is independent oftorsion problem Reissner [3] in his article felt that inclusionof transverse shear deformation effects was presumed to bethe key in resolving Poisson-Kirchhoff rsquos boundary condi-tions paradox Recently it is shown that the second-ordercorrection to 119908

0(119909 119910) by either Reissnerrsquos theory or FSDT

corresponds to approximate solution of a torsion problem [5]Coupling between bending and associated torsion problemsis eliminated earlier [6] by using zero rotation120596

119911= (V119909minus119906119910)

about the vertical axis The proposal of zero 120596119911does not

however resolve the paradox in a satisfactory manner Inthe reference [6] face deflection 119908

0in (8) is from zero face

2 ISRN Civil Engineering

shear conditions whereas neutral plane deflectio n 1199080from

edge support condition is associated with self-equilibratingtransverse shear stresses

The condition120596119911= 0 decoupling the bending and torsion

problems is satisfied in Kirchhoff rsquos theory If this condition isimposed in FSDT sum of the strains (120576

119909+ 120576119910) in the isotropic

plate is governed by a second-order equation due to appliedtransverse loads Reactive transverse shear stresses are interms of gradients of (120576

119909+120576119910) uncoupled from119908

0Thickness-

wise linear distribution of 120590119911is zero from the equilibrium

equation governing transverse stresses Normal strain 120598119911from

constitutive relation is linear in 119911 in terms of (120576119909+120576119910) Reactive

transverse shear stresses and thickness-wise linear strain 120576119911

form the basis for resolving the paradox and for obtaininghigher order corrections to the displacements The theorythus developed is designated as ldquoPoissonrsquos theory of plates inbendingrdquo

The previously mentioned Poissonrsquos theory is applied tothe analysis of anisotropic plates using reactive transverseshear stresses as gradients of a function Normally solutionsto the displacements are obtained from governing equationsbased on stationary property of relevant total potential andthe reactive transverse shear stresses are expressed in termsof these displacements In the present work reverse process isadapted in which reactive transverse stresses are determinedfirst and the displacements are obtained in terms of thesestresses Equations governing second-order corrections to thepreliminary solutions are derived through the application ofan iterativemethod used earlier [7] for the analysis of bendingof isotropic plates

2 Equations of Equilibrium andEdge Conditions

For simplicity in presentation a rectilinear domain boundedby 0 le 119883 le 119886 0 le 119884 le 119887 119885 = plusmnℎ with referenceto Cartesian coordinate system (119883119884 119885) is considered Forconvenience coordinates 119883119884 119885 and displacements 119880119881119882in nondimensional form 119909 = 119883119871 119910 = 119884119871 119911 =

119885ℎ 119906 = 119880ℎ V = 119881ℎ 119908 = 119882ℎ and half-thicknessratio 120572 = ℎ119871 with reference to a characteristic length 119871 in119883-119884 plane are used (119871 is defined such that mod of 119909 and 119910

are equal to or less than 1) With the previous notation andhArr indicating interchange equilibrium equations in terms ofstress components are

120572 (120590119909119909

+ 120591119909119910119910

) + 120591119909119911119911

= 0 lArrrArr (119909 119910) (1)

120572 (120591119909119911119909

+ 120591119910119911119910

) + 120590119911119911

= 0 (2)

in which suffix after ldquordquo denotes partial derivative operatorEdge conditions are prescribed such that (119908 120591

119909119911 120591119910119911

120574119909119911 120574119910119911) are even in 119911 and (119906 V 120590

119909 120590119910 120591119909119910 120576119909 120576119910 120574119909119910 120590119911 120576119911)

are odd in 119911 In the primary flexure problem the plate issubjected to asymmetric load 120590

119911= plusmn119902(119909 119910)2 and zero shear

stresses along 119911 = plusmn1 faces Three conditions to be satisfiedalong 119909 (and 119910) constant edges are prescribed in the form

119906 = 0 or 120590119909= 119911119879119909(119910) lArrrArr (119909 119910) (119906 V) (3a)

V = 0 or 120591119909119910

= 119911119879119909119910(119910) lArrrArr (119909 119910) (119906 V) (3b)

119908 = 0 or 120591119909119911

= (

1

2

) (1 minus 1199112) 119879119909119911(119910) lArrrArr (119909 119910) (3c)

3 Stress-Strain andStrain-Displacement Relations

In displacement based models stress components areexpressed in terms of displacements via six stress-strainconstitutive relations and six strain-displacement relations Inthe present study these relations are confined to the classicalsmall deformation theory of elasticity

It is convenient to denote displacement and stress com-ponents of anisotropic plates as

[119906 V 119908] = [119906119894] [120590

119909 120590119910 120591119909119910] = [120590

119894] (119894 = 1 2 3) (4a)

[120591119909119911 120591119910119911 120590119911] = [120590

119894] (119894 = 4 5 6) (4b)

Strain-stress relations in terms of compliances [119878119894119895] with the

usual summation convention are120576119894= 119878119894119895120590119895 120576119903= 119878119903119904120590119904 (119894 119895 = 1 2 3 6) (119903 119904 = 4 5) (5)

We have from semi-inverted strain-stress relations with [119876119894119895]

denoting inverse of [119878119894119895]

120590119894= 119876119894119895(120576119895minus 1198786119895120590119911) (119894 119895 = 1 2 3) (6)

Normal strain 120576119911is given by120576119911= 1198786119895120590119895 119895 = 1 2 3 6 (7)

Transverse shear strains with [119876119903119904] denoting inverse of [119878

119903119904]

are120576119903= 119876119903119904120590119904 (119903 119904 = 4 5) (8)

Strains 120576119894from strain-displacement relations are

[1205761 1205762 1205763] = 120572 [119906

119909 V119910 119906119910+ V119909

]

[1205764 1205765 1205766] = [119906 + 120572119908

119909 V + 120572119908

119910 119908119911]

(9)

4 119891119899(119911) Functions and Their Use

We use thickness-wise distribution functions119891119899(119911) generated

from recurrence relations [7] with 1198910

= 1 1198912119899+1119911

= 1198912119899

1198912119899+2119911

= minus1198912119899+1

such that 1198912119899+2

(plusmn1) = 0 They are (up to119899 = 5)

[1198911 1198912 1198913] = [119911

(1 minus 1199112)

2

(119911 minus 11991133)

2

]

[1198914 1198915] = [

(5 minus 61199112+ 1199114)

24

(25119911 minus 101199113+ 1199115)

120

]

(10)

ISRN Civil Engineering 3

Displacements strains and stresses are expressed in the form(sum 119899 = 0 1 2 3 )

[119908 119906 V] = [11989121198991199082119899 1198912119899+1

1199062119899+1

1198912119899+1

V2119899+1

]

[120576119909 120576119910 120574119909119910 120576119911] = 1198912119899+1

[120576119909 120576119910 120574119909119910 120576119911]2119899+1

(11)

(Note that 1199082119899

= minus1205761199112119899minus1

119899 ge 1)

[120590119909 120590119910 120591119909119910 120590119911] = 1198912119899+1

[120590119909 120590119910 120591119909119910 120590119911]2119899+1

[120574119909119911 120574119910119911 120591119909119911 120591119910119911] = 1198912119899[120574119909119911 120574119910119911 120591119909119911 120591119910119911]

(12)

In the previous equations variables associated with 119891(119911)

functions are functions of (119909 119910) only

5 Sequence of Trivially Known Steps withoutAny Assumptions in Preliminary Analysis

(a) Due to prescribed zero (120591119909119911 120591119910119911) along 119911 = plusmn1 faces of the

plate (1205911199091199110

1205911199101199110

) equiv 0 in the plate(b) (1205741199091199110

1205741199101199110

) equiv 0 from constitutive relations (5)(c) Static equilibrium equation (2) gives 120590

1199111equiv 0

(d) In the absence of (1199061 V1) (1205761199091 1205761199101 1205741199091199101

) equiv 0 fromstrain-displacement relations (8)

(e) (1205901199091 1205901199101 1205911199091199101

) equiv 0 from constitutive relations (6)(f) 1205761199111

equiv 0 from relation (7)(g) 119908 = 119908

0(119909 119910) from thickness-wise integration of 120576

119911=

0(h) From steps (b) and (g) integration of [119906

119911+120572119908119909 V119911+

120572119908119910] = [0 0] in the thickness direction gives [119906 V] =

minus119911120572[1199080119909

1199080119910]

In the present study as in the earlier work [6] [119906 V] linearin 119911 are unknown functions as in FSDT From integration of[119906119911+ 120572119908119909 V119911+ 120572119908119910] = [0 0] in the face plane instead of

thickness direction one gets vertical deflection1199080(119909 119910) in the

form

1205721199080= minusint [119906

1119889119909 + V

1119889119910] (13)

Since 1199080is from satisfaction of zero face shear conditions it

can be considered as face deflection1199080119865

though it is same forall face parallel planes (note that prescribed zero 119908

0along

a segment of the neutral plane implies zero 1199080along the

corresponding segment of the intersection of face plane withwall of the plate since 119908

0is independent of 119911 and vice versa)

It is analytic in the domain of the plate if 120596119911is zero In such a

case we note from strain-displacement relations (8) that

[1205761119910 1205762119909 1205763119910 1205763119909] = 1205722[V119909119909

119906119910119910

2119906119910119910

2V119909119909

] (14)

(i) In the absence of applied transverse loads integrationsof (1) (2) using Kirchhoff rsquos displacements give a homoge-neous fourth-order equation governing 119908

0 In the present

analysis as in FSDT static equations governing [1199061 V1] are

however given by

1198761119894120576119894119909

+ 1198763119894120576119894119910

= 0 1198762119894120576119894119910

+ 11987631198941205763119894119909

= 0 (119894 = 1 2 3)

(15)

They are subjected to the edge conditions (3a) (3b) along 119909-(and 119910-) constant edges

Equations (15) governing [1199061 V1] are uncoupled due to

relations (14) in the case of isotropic plates whereas theyare coupled without cross-derivatives [119906

1119909119910 V1119909119910

] in the caseof anisotropic plates In the absence of prescribed bendingstress all along the closed boundary of the plate in-plane120591119909119910

distribution corresponds to pure torsion where as itis different in the presence of bending load These 120591

119909119910

distributions are discussed later in Section 62(j) In FSDT (15) correspond to neglecting shear energy

from transverse shear deformations Vertical deflection 1199080

has to be obtained from (14) using the solutions for [1199061 V1]

from (15) Reactive transverse stresses are expressed in termsof [1199061 V1]They are dependent onmaterial constants different

from prescribed nonzero transverse shear stresses along theedges of the plate

(k) In the development of Poissonrsquos theory (15) arecoupled with reactive transverse stresses

6 Poissonrsquos Theory

In the preliminary solution steps (a)ndash(g) in Section 5 areunaltered That is transverse stresses and strains are zeroand 119908 = 119908

0(119909 119910) In the absence of higher order in-plane

displacement terms reactive transverse shear stresses areparabolic and gradients of a function 120595(119909 119910) that is in-plane distributions of transverse shear stresses [120591

119909119911 120591119910119911] =

120572[1205952119909 1205952119910] Equation governing 120595(= 120595

2) from thickness-

wise integration of (2) is

1205722Δ120595 + 120590

1199113= 0 (16)

In the previous equation 1205901199113

is coefficient of 1198913(119911) Satisfac-

tion of the load condition 120590119911= plusmn1199022 along 119911 = plusmn1 faces gives

1205901199113

= (32)119902 so that

1205722Δ120595 + (

3

2

) 119902 = 0 (17)

Edge condition on 120595 is either 120595 = 0 or its outwardnormal gradient is equal to the prescribed shear stress alongeach segment of the edge Note that the transverse stressesthus obtained are independent of material constants In theisotropic plate 120595(119909 119910) = 119864

10158401198901in which 119864

1015840= 119864(1 minus ]2) and

1198901= (1205761199091

+ 1205761199101)

Integrated equilibrium equations (1) in terms of in-planestrains are (sum 119895 = 1 2 3)

[1198761119895120576119895119909

+ 1198763119895120576119895119910] minus 1205911199091199112

= 0

[1198762119895120576119895119910

+ 1198763119895120576119895119909] minus 1205911199101199112

= 0

(18)

By substituting strains in terms of [1199061 V1] (18) are two

equations governing (1199061 V1) and have to be solved with

conditions along 119909- (and 119910-) constant edge

119906 = 0 or 120590119909= 119911119879119909(119910) lArrrArr (119909 119910) (119906 V) (19a)

V = 0 or 120591119909119910

= 119911119879119909119910(119910) lArrrArr (119909 119910) (119906 V) (19b)

4 ISRN Civil Engineering

Vertical deflection 1199080is given by (13) As mentioned earlier

cross-derivatives of [1199061 V1] do not exist in (18) (19a) and

(19b) due to (14) In the isotropic case (18) are uncoupled andare simply given by

1205722Δ1199061= 1205721198901119909

lArrrArr (119909 119910) (119906 119910) (20)

Edge conditions (19a) and (19b) are also uncoupled and theyare given by

1199061= 0 or 1198641015840 (1 minus 120592) 120572119906

1119909= 119879119909(119910) lArrrArr (119909 119910) (119906 V)

V1= 0 or 2119866120572V

1119909= 119879119909119910(119910) lArrrArr (119909 119910) (119906 V)

(21)

The condition1205950= 0 in solving (17) is different from the usual

condition 1199080= 0 in the Kirchhoff rsquos theory and FSDT The

function 1205950is related to the normal strain 120576

119911 In the isotropic

case 1205950is proportional to 119890

1(119909 119910) = (120576

1199091+ 1205761199101) Gradients of

1205950are proportional to gradients of transverse strains in FSDT

If the plate is free of applied transverse stresses that is theplate is subjected to bending and twisting moments only 119890

1

is proportional toΔ1199080The function120595

0(119909 119910) thereby Δ119908

0is

identically zero from (2) (in Kirchhoff rsquos theory the tangentialgradient ofΔ119908

0is proportional to the corresponding gradient

of applied 120591119909119910

along the edge of the plate implied from Kelvinand Taitrsquos physical interpretation of the contracted transverseshear condition) This Laplace equation is not adequate tosatisfy two in-plane edge conditions One needs its conjugateharmonic functions to express in-plane displacements in theform

[1199061 V1] = minus120572119911 [119908

0119909+ 1205930119910 1199080119910

minus 1205930119909] (22)

The function 1205930was introduced earlier by Reissner [8] as a

stress function in satisfying (2) Note that the two variables1199061and V1are expressed in terms of gradients of two functions

1199080and 120593

0

After finding1199080and1205930from solving in-plane equilibrium

equations (1) correction 1199080119888

to 1199080from zero face shear

conditions is given by

1199080119888= int [120601

0119910119889119909 + 120601

0119909119889119910] (23)

One should note here that119908(119909 119910) = 1199080+1199080119888does not satisfy

prescribed edge condition on 119908In FSDT [119906

1 V1] are obtained from solving (1) There is

no provision to find 1199080and it has to be obtained from zero

face shear conditions in the form

1205721199080= minusint [119906

1119889119909 + V

1119889119910] (24)

If the applied transverse stresses are nonhomogeneous alonga segment of the edge 119908

0is coupled with [119906

1 V1] in (1) and

(2) and edge conditions (3a) (3b) and (3c)With reference to the present analysis it is relevant to

note the following observation in the preliminary solution120590119911 120572[120591119909119911 120591119910119911] 1205722[120590

119909 120590119910 120591119909119910] are of119874(1) As such estimation

of in-plane stresses thereby in-plane displacements is notdependent on 119908

0 The support condition 119908

0= 0 along the

edge of the plate does not play any role in determining thein-plane displacements

61 Comparison with Analysis of Extension Problems It isinteresting to compare the previous analysis with that ofextension problem In extension problem applied shearstresses along the faces of the plate are gradients of a potentialfunction and they are equivalent to linearly varying bodyforces independent of elastic constants in the in-planeequilibrium equations Rotation 120596

119911= 0 (but Δ120596

119911= 0) and

equations governing displacements [1199060 V0] are coupled Also

vertical deflection is given by thickness-wise integration of1205761199110from constitutive relation and higher order corrections are

dependent on this vertical deflection In the present analysisof bending problems in-plane distributions of transverseshear stresses are gradients of a function related to appliededge transverse shear and face normal load through equi-librium equation (2) They are also independent of elasticconstants and derivative 119891

2119911of parabolic distribution of each

of them is equivalent to body force in linearly varying in-plane equilibrium equations Vertical deflection 119908

0is purely

from satisfaction of zero shear stress conditions along faces ofthe plate Higher order corrections are dependent on normalstrains and independent of119908

0 It can be seen that steps in the

analysis of bending problem are complementary to those inthe classical theory of extension problem

62 Associated Torsion Problems First-order shear deforma-tion theory and higher order shear deformation theoriesdo not provide proper corrections to initial solution (fromKirchhoff rsquos theory) of primary flexure problems In thesetheories corrections are due to approximate solutions ofassociated torsion problems In fact one has to have a relookat the use of shear deformation theories based on plate(instead of 3D) element equilibrium equations other thanKirchhoff rsquos theory in the analysis of flexure problems Inthe earlier work [5] we have mentioned that this couplingof bending and torsion problems is nullified in the limit ofsatisfying all equations in the 3D problems However thesehigher order theories in the case of primary flexure problemdefined from Kirchhoff rsquos theory are with reference to findingthe exact solution of associated torsion problem only In factone can obtain exact solution of the torsion problem (insteadof using higher order polynomials in 119911 by expanding 119911 and1198913(119911) in sine series and119891

2(119911) in cosine series In a pure torsion

problem normal stresses and strains are zero implying that

[119906 V 119908] = [119906 (119910 119911) V (119909 119911) 119908 (119909 119910)] (25)

Rotation 120596119911

= 0 and warping displacements (119906 V) are inde-pendent of vertical displacement 119908 In FSDT 120596

119911= 0 and

corrections to Kirchhoff rsquos displacements are due to thesolution of approximate torsion problem In the isotropicrectangular plate warping function 119906(119910 119911) is harmonic in 119910-119911-plane It has to satisfy 119866120572119906

119910= 119879119906(119910 119911) with prescribed 119879

119906

along an 119909 = constant edge By expressing 119906 in product form119906 = 119891119906(119910)119892119906(119911) we have

1205722119891119906119910119910

119892119906+ 119891119906119892119906119911119911

= 0 (26)

ISRN Civil Engineering 5

By taking 119892119906119899

= sin 120582119899119911 where 120582

119899= (2119899 + 1)1205872 satisfying

zero face shear conditions we get

119891119906119899

= 119860119906119899cosh(

120582119899

120572

)119910 + 119861119906119899sinh(

120582119899

120572

)119910 (27)

If prescribed 119879119906is 119891119906119899

with specified constants (119860119906119899 119861119906119899)

clearly there is no provision to satisfy zero in-plane shearcondition along 119910 = constant edges It has to be nullifiedwith corresponding solution from bending problem That iswhy a torsion problem is associated with bending problembut not vice versa In a corresponding bending problem allstress components are zero except 120591

119909119910= minus119891

119906119899(119910)119892119906119899(119911)

This solution is used only in satisfying edge condition along119910 = constant edges in the presence of specified 119879

119906119899along

119909 = constant edges It shows that 120591119909119910

distribution in flexureproblem is nullified in the limit in shear deformation theoriesdue to torsion

It is interesting to note that the previous mentioneddeficiency due to coupling with 119908

0in the plate element

equilibrium equations in FSDT and higher order sheardeformation theories does not exist if applied 120591

119909119910is zero all

along the closed boundary of the plate It is complementaryto the fact that boundary condition paradox in Kirchhoff rsquostheory does not exist if tangential displacement and 119908

0are

zero all along the boundary of the plate

63 Illustrative Example Simply Supported Isotropic SquarePlate Consider a simply supported isotropic square platesubjected to vertical load

119902 = 1199020sin(120587119909

119886

) sin(120587119910

119886

) (28)

Exact values for neutral plane and face deflections obtainedearlier [7] with ] = 03 120572 = (16) are

(

119864

21199020

)119908(

119886

2

119886

2

0) = 4487 (29)

(

119864

21199020

)119908(

119886

2

119886

2

1) = 4166 (30)

In this example Poisson-Kirchhoff rsquos boundary conditionsparadox does not exist Hence Poissonrsquos theory gives thesame solution obtained in the authorrsquos previous mentionedwork Concerning face value in (30) Kirchhoff rsquos theory givesa value of 227 that is same for all face parallel planes Poissonrsquostheory gives an additional value of 0267 attributable to 120576

1199111

and it is same for neutral plane deflection since119908(119909 119910 119911) canbe expressed as

119908 (119909 119910 119911) = 1199080119865

(119909 119910) + 1198912(119911) 1199082

= 1199080119873

(119909 119910) minus (

1

2

) 11991121205761199111

(31)

In the previous expression 1199080119873

is deflection of the neutralplane Higher order correction to119908

0uncoupled from torsion

is 1262 so that total correction to the value from Kirchhoff rsquostheory is about 153 [7] Correction due to coupling with

torsion is 145 [5] whereas it is 1423 from FSDT and othersixth order theories [9] With reference to numerical valuesreported by Lewinski [9] the previous correction is 1412 inReddyrsquos 8th-order theory and less than 123 in Reissnerrsquos 12th-order and other higher order theories It clearly shows thatthese shear deformation theories do not lead to solutions ofbending problems (note that the value (227 + 1423) fromFSDT corresponds to neutral plane deflection and it is inerror by about 177 from the exact value)

In view of the previous observations it is relevant tomakethe following remarks physical validity of 3D equations isconstrained by limitations of small deformation theory For120572 = 1 this constraint is dependent on material constantsgeometry of the domain and applied loads Range of validityof 2D plate theories arises for small values of 120572 Kirchhoff rsquostheory gives lower bound for this 120572 For this value of 120572 and forslightly higher values of 120572 than this lower bound Kirchhoff rsquostheory is used in spite of its known deficiencies due to itssimplicity to obtain design information Similarly FSDT andother sixth-order shear deformation theories are used forsome additional range of values of 120572 due to the correctionsover Kirchhoff rsquos theory though these corrections are due tocoupling with approximate torsion problemsThey serve onlyin giving guidance values for applicable design parametersfor small range of values of 120572 beyond its lower boundThey do not provide initial set of equations in formulatingproper sequence of sets of 2D problems converging to the 3Dproblem As such comparison of solutions from Kirchhofftheory and FSDT with the corresponding solutions in thepresent analysis does not serve much purpose

Since 1199080119873

has to be higher than 1199080119865 one has to obtain

the correction to 1199080119873

before finding correction to internaldistribution of119908(119909 119910 119911) along with correction to119908

0119865 Initial

correction to 1199080119873

is obtained earlier [7] in the form

119908 = 1199080minus [1198912(0) 1205761199111+ 1198914(0) 1205761199113] (32)

It is the same for face deflection (note that 1199082in the

previouslymentioned authorrsquos work is required only to obtain1205761199113) As such 119908

0119873is further corrected from solution of

a supplementary problem This total correction over facedeflection is about 0658 giving a value of 4458 which is veryclose to the exact value 4487 in (29) However we note thatthe error in the estimated value (=38) of 119908

0119865is relatively

high compared to the accuracy achieved in the neutral planedeflection It is possible to improve estimation of 119908

0119865by

including 1205761199111in (1199063 V3) such that (120591

1199091199112 1205911199101199112

) are independentof 1205761199111 In the present example correction to face deflection

changes to

(

119864

21199020

)119908face max =((65) ((1 + 120592) (1 minus 120592))) 120573

2

((25) ((4 minus 120592) (1 minus 120592))) 1205732minus 1

(33)

In the previous equation 1205732 = 2119908011986512057221205872 Correction 120576

1199113(=

1262) to face value changes to 1431 giving 397 (= 227 +

0267 + 1431) for face deflection which is under 47 fromexact value

6 ISRN Civil Engineering

7 Anisotropic PlateSecond-Order Corrections

We note that 1205761199111does not participate in the determination of

(1199061 V1) and it is obtained from the constitutive relation (6) in

the interior of the plate It is zero along an edge of the plateif 119908 = 0 is specified condition With zero transverse shearstrains specification of 120576

1199111= 0 instead of 119908

0= 0 is more

appropriate along a supported edge since 1199080

= 0 implieszero tangential displacement along a straight edge which isthe root cause of Poisson-Kirchhoff rsquos boundary conditionsparadox As such edge support condition on w does not playany role in obtaining the in-plane displacements and reactivetransverse stresses

In the first stage of iterative procedure second-orderreactive transverse stresses have to be obtained by consid-ering higher order in-plane displacement terms 119906

3and V

3

However one has to consider limitation in the previousPoissonrsquos theory like in Kirchhoff rsquos theory namely reactive120590119911is zero at locations of zeros of 119902 It is identically zero in

higher order approximations since 1198912119899+1

(119911) functions are notzero along 119911 = plusmn1 faces Moreover it is necessary to accountfor its dependence on material constants To overcome theselimitations it is necessary to keep 120590

1199115as a free variable by

modifying 1198915in the form

119891lowast

5(119911) = 119891

5 (119911) minus 120573

31198913 (119911) (34)

In the previous equation 1205733= [1198915(1)1198913(1)] so that119891lowast

5(plusmn1) =

0 Denoting coefficient of 1198913in 120590119911by 120590lowast1199113 it becomes

120590lowast

1199113= 1205901199113minus 12057331205901199115 (35)

Displacements (1199063 V3) are modified such that they are cor-

rections to face parallel plane distributions of the preliminarysolution and are free to obtain reactive stresses 120591

1199091199114 1205911199101199114

1205901199115

and normal strain 1205761199113

We have from constitutive relations

1205741199091199112

= 119878441205911199091199112

+ 119878451205911199101199112

lArrrArr (119909 119910) (4 5) (36)

Modified displacements and the corresponding derivedquantities denoted with lowast are

119906lowast

3= 1199063minus 1205721199082119909

+ 1205741199091199112

lArrrArr (119909 119910) (119906 V) (37)

Strain-displacement relations give

120576lowast

1199093= 1205761199093

minus 12057221199082119909119909

+ 1205721205741199091199112119909

lArrrArr (119909 119910)

120574lowast

1199091199103= 1205741199091199103

minus 212057221199082119909119910

+ 120572 (1205741199091199112119910

+ 1205741199101199112119909

)

120574lowast

1199091199113= 1199063+ 1205741199091199112

lArrrArr (119909 119910) (119906 V)

(38)

In-plane and transverse shear stresses from constitutiverelations are

120590lowast

119894(3)= 119876119894119895120576lowast

119895(3)(119894 119895 = 1 2 3) (39)

120591lowast

1199091199112= 119876441199063+ 11987645V3+ 1205911199091199112

lArrrArr (119909 119910) (4 5) (40)

Suffix (3) in (39) is to indicate that the stresses and strainscorrespond to 119906

3and V3

We get from (2) (15) (35) (37) and (40) with referenceto coefficient of 119891

3(119911)

120572 [(119876441199063+ 11987645V3)119909

+ (119876541199063+ 11987655V3)119910

] = 12057331205901199115 (41)

Note that 1199082is not present in the previous equation

From integration of equilibrium equations we have withsum on 119895 = 1 2 3

1205911199091199114

= 120572[1198761119895120576lowast

119895119909+ 1198763119895120576lowast

119895119910](3)

1205911199101199114

= 120572[1198762119895120576lowast

119895119910+ 1198763119895120576lowast

119895119909](3)

(42)

Note that 1199082is present in the previous expression

120572 (1205911199091199114119909

+ 1205911199101199114119910

) + 1205901199115

= 0 (Coefficient of 1198915) (43)

1205761199113

= [1198786119895120590119895+ 11987866120590119911](3) (44)

One gets one equation governing in-plane displacements(1199063 V3) from (41) (43) noting that 119891

5119911119911+1198913= 0 in the form

1205733(1205911199091199114119909

+ 1205911199101199114119910

)

= 1205722[(119876441199063+ 11987645V3)119909119909

+ (119876541199063+ 11987655V3)119910119910

]

(45)

The second equation governing these variables is from thecondition 120596

119911= 0 Here it is more convenient to express

(1199063 V3) as

[1199063 V3] = minus120572 [(120595

3119909minus 1206013119910) (1205953119910

+ 1206013119909)] (46)

so that Δ1205933= 0 and (45) becomes a fourth-order equation

governing 1205953 This sixth-order system is to be solved for 120595

3

and 1205933with associated edge conditions

119906lowast

3= 0 or 120590

lowast

3= 0

Vlowast

3= 0 or 120591

lowast

1199091199103= 0

1205953= 0 or 120591

lowast

1199091199113= 0

(47)

along an 119909 = constant edge and analogous conditions along a119910 = constant edge

In (45) expressed in terms of 1205953and 120593

3 we replace 119908

2by

1205953for obtaining neutral plane deflection since contributions

of 1199082and 120595

3are one and the same in finding [119906

3 V3]

71 Supplementary Problem Because of the use of integrationconstant to satisfy face shear conditions vertical deflection ofneutral plane is equal to that of the face plane deflectionThisis physically incorrect since face plane is bounded by elasticmaterial on one side whereas neutral plane is bounded onboth sidesThis lack in interior solution is rectified by addingthe solution of a supplementary problem based on leadingcosine term that is enough in view of 119891

2(119911) distribution

of reactive shear stresses This is based on the expansion

ISRN Civil Engineering 7

of 120590119911= 1198913(119911)(32)119902 in sine series Here load condition is

satisfied by the leading sine term and coefficients of all othersine terms are zero

Corrective displacements in the supplementary problemare assumed in the form

119908 = 1199082119904

120587

2

cos(1205872

119911) 119906119904= 1199063119904sin(120587

2

119911) lArrrArr (119906 V)

1205903119904119894

= 1198761198941198951205763119904119895

(119894 119895 = 1 2 3)

(48)

We have from integration of equilibrium equations

1205912119904119904119909119911

= minus(

2

120587

)120572 [11987611198951205763119904119895119909

+ 11987631198951205763119904119895119910

] (49a)

1205912119904119910119911

= minus(

2

120587

)120572 [11987621198951205763119904119895119910

+ 11987631198951205763119904119895119909

] (49b)

In-plane distributions 1199063119904

and V3119904

are added as correctionsto the known 119906

lowast

3and Vlowast

3so that (119906 V) in the supplementary

problem are

119906 = (119906lowast

3+ 1199063119904) sin(120587

2

119911) lArrrArr (119906 V) (50)

We note that 120590119911from integration of equilibrium equations

with 119904 variables is the same as 120590119911from static equilibrium

equations with lowast variables Hence

(

2

120587

)

2

1205722[11987611198951205763119904119895119909119909

+ 211987631198951205763119904119895119909119910

+ 11987621198951205763119904119895119910119910

] + 12057331205901199115

= 0

(51)

It is convenient to use here also

[1199063119904 V3119904] = minus120572 [(120595

3119904119909minus 1206013119904119910

) (1205953119904119910

+ 1205933119904119909

)] (52)

Correction to neutral plane deflection due to solution ofsupplementary problem is from

1205761199113119904

= [11987861198951205903119904119895

+ 119878661205901199113119904

] (sum 119895 = 1 2 3) (53)

72 Summary of Results from the First Stage of IterativeProcedure The previous analysis gives displacements [119906V] =1198913[1199063 V3] consistent with [120591

119909119911 120591119910119911] = 120572119891

2[120595119909 120595119910] and

corrective transverse stresses from the first stage of iteration

119908 = 1199080minus [11989121205761199111+ 11989141205761199113] minus 1205761199111199043

120587

2

cos(1205872

119911)

119906 = 11989111199061+ 1198913119906lowast

3+ (119906lowast

3+ 1199063119904) sin(120587

2

119911) lArrrArr (119906 V)

120591119909119911

= 11989121205912119909119911

+ 1205912119904119909119911

120587

2

cos(1205872

119911) lArrrArr (119909 119910)

120590119911= 1198913(119911) 120590lowast

1199113(From (35))

(54)

It is to be noted that the dependence of transverse stresses onmaterial constants is through the solution of supplementaryproblem One may add 119891

4terms in shear components and 119891

5

term in normal stress componentThese components are alsodependent on material constants but they need correctionfrom the solution of a supplementary problem Successiveapplication of the previous iterative procedure leads to thesolution of the 3D problem in the limit

8 Concluding Remarks

Poissonrsquos theory developed in the present study for the analy-sis of bending of anisotropic plates within small deformationtheory forms the basis for generation of proper sequence of2D problems Analysis for obtaining displacements therebybending stresses along faces of the plate is different fromsolution of a supplementary problem in the interior ofthe plate A sequence of higher order shear deformationtheories lead to solution of associated torsion problem onlyIn the preliminary solution reactive transverse stresses areindependent of material constants In view of layer-wisetheory of symmetric laminated plates proposed by the presentauthor [10] Poissonrsquos theory is useful since analysis of faceplies is independent of lamination which is in confirmationof (8)-(9) in a recent NASA technical publication by Tessleret al [11]

One significant observation is that sequence of 2D prob-lems converging to 3D problems in the analysis of extensionbending and torsion problems are mutually exclusive to oneother

Highlights

(i) Poisson-Kirchhoff boundary conditions paradox isresolved

(ii) Transverse stresses are independent of material con-stants in primary solution

(iii) Edge support condition on vertical deflection has norole in the analysis

(iv) Finding neutral plane deflection requires solution ofa supplementary problem

(v) Twisting stress distribution in pure torsion nullifies itsdistribution in bending

Appendix

119886 = side length of a square plate2ℎ = plate thickness119891119899(119911)= through-thickness distribution functions 119899 =

0 1 2

119871 = characteristic length of the plate in119883-119884-plane119902(119909 119910) = applied face load density119876119894119895= stiffness coefficients

119878119894119895= elastic compliances

[119879119909 119879119909119910 119879119909119911] = prescribed stress distributions along

an 119909 = constant edge

8 ISRN Civil Engineering

[119880 119881119882] = displacements in 119883119884 119885-directionsrespectively[119906 V 119908] = [119880 119881119882]ℎ

119874 minus 119883119884119885 = Cartesian coordinate system[119909 119910 119911] = nondimensional coordinates [119883119871 119884119871119885ℎ]

120572 = ℎ119871

1205732119899+1

= [1198912119899+3

(1)12057221198912119899+1

(1)]

Δ = Laplace operator (12059721205971199092 + 12059721205971199102)

[120576119909 120576119910 120574119909119910] = in-plane strains

[120574119909119911 120574119910119911 120576119911] = transverse strains

120596119911= 120572(V119909minus 119906119910) rotation about 119911-axis

[120590119909 120590119910 120591119909119910] = bending stresses

[120591119909119911 120591119910119911 120590119911] = transverse stresses

References

[1] G Kirchhoff ldquoUber das Gleichgewicht und die Bewegung einerelastischen Scheiberdquo Journal fur die Reine und AngewandteMathematik vol 40 pp 51ndash58 1850

[2] HHencky ldquoUber die Berucksichtigung der Schubverzerrung inebenen Plattenrdquo Ingenieur-Archiv vol 16 no 1 pp 72ndash76 1947

[3] E Reissner ldquoReflections on the theory of elastic platesrdquoAppliedMechanics Reviews vol 38 no 11 pp 1453ndash1464 1985

[4] A E H Love A Treatise on Mathematical Theory of ElasticityCambridge University Press Cambridge UK 4th edition 1934

[5] K Vijayakumar ldquoModified Kirchhoff rsquos theory of plates includ-ing transverse shear deformationsrdquo Mechanics Research Com-munications vol 38 pp 211ndash213 2011

[6] K Vijayakumar ldquoNew look at kirchhoff rsquos theory of platesrdquoAIAA Journal vol 47 no 4 pp 1045ndash1046 2009

[7] K Vijayakumar ldquoA relook at Reissnerrsquos theory of plates inbendingrdquo Archive of Applied Mechanics vol 81 no 11 pp 1717ndash1724 2011

[8] E Reissner ldquoThe effect of transverse shear deformations on thebending of elastic platesrdquo Journal of Applied Mechanics vol 12pp A69ndashA77 1945

[9] T Lewinski ldquoOn the twelfth-order theory of elastic platesrdquoMechanics Research Communications vol 17 no 6 pp 375ndash3821990

[10] K Vijayakumar ldquoLayer-wise theory of bending of symmetriclaminates with isotropic pliesrdquo AIAA Journal vol 49 no 9 pp2073ndash2076 2011

[11] A Tessler M Di Sciuva and M Gherlone ldquoRefined zigzagtheory for homogeneous laminated composite and sandwichplates a homogeneous limit methodology for zigzag functionselectionrdquo Tech Rep NASATP-2920102162141 2010

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International Journal of

2 ISRN Civil Engineering

shear conditions whereas neutral plane deflectio n 1199080from

edge support condition is associated with self-equilibratingtransverse shear stresses

The condition120596119911= 0 decoupling the bending and torsion

problems is satisfied in Kirchhoff rsquos theory If this condition isimposed in FSDT sum of the strains (120576

119909+ 120576119910) in the isotropic

plate is governed by a second-order equation due to appliedtransverse loads Reactive transverse shear stresses are interms of gradients of (120576

119909+120576119910) uncoupled from119908

0Thickness-

wise linear distribution of 120590119911is zero from the equilibrium

equation governing transverse stresses Normal strain 120598119911from

constitutive relation is linear in 119911 in terms of (120576119909+120576119910) Reactive

transverse shear stresses and thickness-wise linear strain 120576119911

form the basis for resolving the paradox and for obtaininghigher order corrections to the displacements The theorythus developed is designated as ldquoPoissonrsquos theory of plates inbendingrdquo

The previously mentioned Poissonrsquos theory is applied tothe analysis of anisotropic plates using reactive transverseshear stresses as gradients of a function Normally solutionsto the displacements are obtained from governing equationsbased on stationary property of relevant total potential andthe reactive transverse shear stresses are expressed in termsof these displacements In the present work reverse process isadapted in which reactive transverse stresses are determinedfirst and the displacements are obtained in terms of thesestresses Equations governing second-order corrections to thepreliminary solutions are derived through the application ofan iterativemethod used earlier [7] for the analysis of bendingof isotropic plates

2 Equations of Equilibrium andEdge Conditions

For simplicity in presentation a rectilinear domain boundedby 0 le 119883 le 119886 0 le 119884 le 119887 119885 = plusmnℎ with referenceto Cartesian coordinate system (119883119884 119885) is considered Forconvenience coordinates 119883119884 119885 and displacements 119880119881119882in nondimensional form 119909 = 119883119871 119910 = 119884119871 119911 =

119885ℎ 119906 = 119880ℎ V = 119881ℎ 119908 = 119882ℎ and half-thicknessratio 120572 = ℎ119871 with reference to a characteristic length 119871 in119883-119884 plane are used (119871 is defined such that mod of 119909 and 119910

are equal to or less than 1) With the previous notation andhArr indicating interchange equilibrium equations in terms ofstress components are

120572 (120590119909119909

+ 120591119909119910119910

) + 120591119909119911119911

= 0 lArrrArr (119909 119910) (1)

120572 (120591119909119911119909

+ 120591119910119911119910

) + 120590119911119911

= 0 (2)

in which suffix after ldquordquo denotes partial derivative operatorEdge conditions are prescribed such that (119908 120591

119909119911 120591119910119911

120574119909119911 120574119910119911) are even in 119911 and (119906 V 120590

119909 120590119910 120591119909119910 120576119909 120576119910 120574119909119910 120590119911 120576119911)

are odd in 119911 In the primary flexure problem the plate issubjected to asymmetric load 120590

119911= plusmn119902(119909 119910)2 and zero shear

stresses along 119911 = plusmn1 faces Three conditions to be satisfiedalong 119909 (and 119910) constant edges are prescribed in the form

119906 = 0 or 120590119909= 119911119879119909(119910) lArrrArr (119909 119910) (119906 V) (3a)

V = 0 or 120591119909119910

= 119911119879119909119910(119910) lArrrArr (119909 119910) (119906 V) (3b)

119908 = 0 or 120591119909119911

= (

1

2

) (1 minus 1199112) 119879119909119911(119910) lArrrArr (119909 119910) (3c)

3 Stress-Strain andStrain-Displacement Relations

In displacement based models stress components areexpressed in terms of displacements via six stress-strainconstitutive relations and six strain-displacement relations Inthe present study these relations are confined to the classicalsmall deformation theory of elasticity

It is convenient to denote displacement and stress com-ponents of anisotropic plates as

[119906 V 119908] = [119906119894] [120590

119909 120590119910 120591119909119910] = [120590

119894] (119894 = 1 2 3) (4a)

[120591119909119911 120591119910119911 120590119911] = [120590

119894] (119894 = 4 5 6) (4b)

Strain-stress relations in terms of compliances [119878119894119895] with the

usual summation convention are120576119894= 119878119894119895120590119895 120576119903= 119878119903119904120590119904 (119894 119895 = 1 2 3 6) (119903 119904 = 4 5) (5)

We have from semi-inverted strain-stress relations with [119876119894119895]

denoting inverse of [119878119894119895]

120590119894= 119876119894119895(120576119895minus 1198786119895120590119911) (119894 119895 = 1 2 3) (6)

Normal strain 120576119911is given by120576119911= 1198786119895120590119895 119895 = 1 2 3 6 (7)

Transverse shear strains with [119876119903119904] denoting inverse of [119878

119903119904]

are120576119903= 119876119903119904120590119904 (119903 119904 = 4 5) (8)

Strains 120576119894from strain-displacement relations are

[1205761 1205762 1205763] = 120572 [119906

119909 V119910 119906119910+ V119909

]

[1205764 1205765 1205766] = [119906 + 120572119908

119909 V + 120572119908

119910 119908119911]

(9)

4 119891119899(119911) Functions and Their Use

We use thickness-wise distribution functions119891119899(119911) generated

from recurrence relations [7] with 1198910

= 1 1198912119899+1119911

= 1198912119899

1198912119899+2119911

= minus1198912119899+1

such that 1198912119899+2

(plusmn1) = 0 They are (up to119899 = 5)

[1198911 1198912 1198913] = [119911

(1 minus 1199112)

2

(119911 minus 11991133)

2

]

[1198914 1198915] = [

(5 minus 61199112+ 1199114)

24

(25119911 minus 101199113+ 1199115)

120

]

(10)

ISRN Civil Engineering 3

Displacements strains and stresses are expressed in the form(sum 119899 = 0 1 2 3 )

[119908 119906 V] = [11989121198991199082119899 1198912119899+1

1199062119899+1

1198912119899+1

V2119899+1

]

[120576119909 120576119910 120574119909119910 120576119911] = 1198912119899+1

[120576119909 120576119910 120574119909119910 120576119911]2119899+1

(11)

(Note that 1199082119899

= minus1205761199112119899minus1

119899 ge 1)

[120590119909 120590119910 120591119909119910 120590119911] = 1198912119899+1

[120590119909 120590119910 120591119909119910 120590119911]2119899+1

[120574119909119911 120574119910119911 120591119909119911 120591119910119911] = 1198912119899[120574119909119911 120574119910119911 120591119909119911 120591119910119911]

(12)

In the previous equations variables associated with 119891(119911)

functions are functions of (119909 119910) only

5 Sequence of Trivially Known Steps withoutAny Assumptions in Preliminary Analysis

(a) Due to prescribed zero (120591119909119911 120591119910119911) along 119911 = plusmn1 faces of the

plate (1205911199091199110

1205911199101199110

) equiv 0 in the plate(b) (1205741199091199110

1205741199101199110

) equiv 0 from constitutive relations (5)(c) Static equilibrium equation (2) gives 120590

1199111equiv 0

(d) In the absence of (1199061 V1) (1205761199091 1205761199101 1205741199091199101

) equiv 0 fromstrain-displacement relations (8)

(e) (1205901199091 1205901199101 1205911199091199101

) equiv 0 from constitutive relations (6)(f) 1205761199111

equiv 0 from relation (7)(g) 119908 = 119908

0(119909 119910) from thickness-wise integration of 120576

119911=

0(h) From steps (b) and (g) integration of [119906

119911+120572119908119909 V119911+

120572119908119910] = [0 0] in the thickness direction gives [119906 V] =

minus119911120572[1199080119909

1199080119910]

In the present study as in the earlier work [6] [119906 V] linearin 119911 are unknown functions as in FSDT From integration of[119906119911+ 120572119908119909 V119911+ 120572119908119910] = [0 0] in the face plane instead of

thickness direction one gets vertical deflection1199080(119909 119910) in the

form

1205721199080= minusint [119906

1119889119909 + V

1119889119910] (13)

Since 1199080is from satisfaction of zero face shear conditions it

can be considered as face deflection1199080119865

though it is same forall face parallel planes (note that prescribed zero 119908

0along

a segment of the neutral plane implies zero 1199080along the

corresponding segment of the intersection of face plane withwall of the plate since 119908

0is independent of 119911 and vice versa)

It is analytic in the domain of the plate if 120596119911is zero In such a

case we note from strain-displacement relations (8) that

[1205761119910 1205762119909 1205763119910 1205763119909] = 1205722[V119909119909

119906119910119910

2119906119910119910

2V119909119909

] (14)

(i) In the absence of applied transverse loads integrationsof (1) (2) using Kirchhoff rsquos displacements give a homoge-neous fourth-order equation governing 119908

0 In the present

analysis as in FSDT static equations governing [1199061 V1] are

however given by

1198761119894120576119894119909

+ 1198763119894120576119894119910

= 0 1198762119894120576119894119910

+ 11987631198941205763119894119909

= 0 (119894 = 1 2 3)

(15)

They are subjected to the edge conditions (3a) (3b) along 119909-(and 119910-) constant edges

Equations (15) governing [1199061 V1] are uncoupled due to

relations (14) in the case of isotropic plates whereas theyare coupled without cross-derivatives [119906

1119909119910 V1119909119910

] in the caseof anisotropic plates In the absence of prescribed bendingstress all along the closed boundary of the plate in-plane120591119909119910

distribution corresponds to pure torsion where as itis different in the presence of bending load These 120591

119909119910

distributions are discussed later in Section 62(j) In FSDT (15) correspond to neglecting shear energy

from transverse shear deformations Vertical deflection 1199080

has to be obtained from (14) using the solutions for [1199061 V1]

from (15) Reactive transverse stresses are expressed in termsof [1199061 V1]They are dependent onmaterial constants different

from prescribed nonzero transverse shear stresses along theedges of the plate

(k) In the development of Poissonrsquos theory (15) arecoupled with reactive transverse stresses

6 Poissonrsquos Theory

In the preliminary solution steps (a)ndash(g) in Section 5 areunaltered That is transverse stresses and strains are zeroand 119908 = 119908

0(119909 119910) In the absence of higher order in-plane

displacement terms reactive transverse shear stresses areparabolic and gradients of a function 120595(119909 119910) that is in-plane distributions of transverse shear stresses [120591

119909119911 120591119910119911] =

120572[1205952119909 1205952119910] Equation governing 120595(= 120595

2) from thickness-

wise integration of (2) is

1205722Δ120595 + 120590

1199113= 0 (16)

In the previous equation 1205901199113

is coefficient of 1198913(119911) Satisfac-

tion of the load condition 120590119911= plusmn1199022 along 119911 = plusmn1 faces gives

1205901199113

= (32)119902 so that

1205722Δ120595 + (

3

2

) 119902 = 0 (17)

Edge condition on 120595 is either 120595 = 0 or its outwardnormal gradient is equal to the prescribed shear stress alongeach segment of the edge Note that the transverse stressesthus obtained are independent of material constants In theisotropic plate 120595(119909 119910) = 119864

10158401198901in which 119864

1015840= 119864(1 minus ]2) and

1198901= (1205761199091

+ 1205761199101)

Integrated equilibrium equations (1) in terms of in-planestrains are (sum 119895 = 1 2 3)

[1198761119895120576119895119909

+ 1198763119895120576119895119910] minus 1205911199091199112

= 0

[1198762119895120576119895119910

+ 1198763119895120576119895119909] minus 1205911199101199112

= 0

(18)

By substituting strains in terms of [1199061 V1] (18) are two

equations governing (1199061 V1) and have to be solved with

conditions along 119909- (and 119910-) constant edge

119906 = 0 or 120590119909= 119911119879119909(119910) lArrrArr (119909 119910) (119906 V) (19a)

V = 0 or 120591119909119910

= 119911119879119909119910(119910) lArrrArr (119909 119910) (119906 V) (19b)

4 ISRN Civil Engineering

Vertical deflection 1199080is given by (13) As mentioned earlier

cross-derivatives of [1199061 V1] do not exist in (18) (19a) and

(19b) due to (14) In the isotropic case (18) are uncoupled andare simply given by

1205722Δ1199061= 1205721198901119909

lArrrArr (119909 119910) (119906 119910) (20)

Edge conditions (19a) and (19b) are also uncoupled and theyare given by

1199061= 0 or 1198641015840 (1 minus 120592) 120572119906

1119909= 119879119909(119910) lArrrArr (119909 119910) (119906 V)

V1= 0 or 2119866120572V

1119909= 119879119909119910(119910) lArrrArr (119909 119910) (119906 V)

(21)

The condition1205950= 0 in solving (17) is different from the usual

condition 1199080= 0 in the Kirchhoff rsquos theory and FSDT The

function 1205950is related to the normal strain 120576

119911 In the isotropic

case 1205950is proportional to 119890

1(119909 119910) = (120576

1199091+ 1205761199101) Gradients of

1205950are proportional to gradients of transverse strains in FSDT

If the plate is free of applied transverse stresses that is theplate is subjected to bending and twisting moments only 119890

1

is proportional toΔ1199080The function120595

0(119909 119910) thereby Δ119908

0is

identically zero from (2) (in Kirchhoff rsquos theory the tangentialgradient ofΔ119908

0is proportional to the corresponding gradient

of applied 120591119909119910

along the edge of the plate implied from Kelvinand Taitrsquos physical interpretation of the contracted transverseshear condition) This Laplace equation is not adequate tosatisfy two in-plane edge conditions One needs its conjugateharmonic functions to express in-plane displacements in theform

[1199061 V1] = minus120572119911 [119908

0119909+ 1205930119910 1199080119910

minus 1205930119909] (22)

The function 1205930was introduced earlier by Reissner [8] as a

stress function in satisfying (2) Note that the two variables1199061and V1are expressed in terms of gradients of two functions

1199080and 120593

0

After finding1199080and1205930from solving in-plane equilibrium

equations (1) correction 1199080119888

to 1199080from zero face shear

conditions is given by

1199080119888= int [120601

0119910119889119909 + 120601

0119909119889119910] (23)

One should note here that119908(119909 119910) = 1199080+1199080119888does not satisfy

prescribed edge condition on 119908In FSDT [119906

1 V1] are obtained from solving (1) There is

no provision to find 1199080and it has to be obtained from zero

face shear conditions in the form

1205721199080= minusint [119906

1119889119909 + V

1119889119910] (24)

If the applied transverse stresses are nonhomogeneous alonga segment of the edge 119908

0is coupled with [119906

1 V1] in (1) and

(2) and edge conditions (3a) (3b) and (3c)With reference to the present analysis it is relevant to

note the following observation in the preliminary solution120590119911 120572[120591119909119911 120591119910119911] 1205722[120590

119909 120590119910 120591119909119910] are of119874(1) As such estimation

of in-plane stresses thereby in-plane displacements is notdependent on 119908

0 The support condition 119908

0= 0 along the

edge of the plate does not play any role in determining thein-plane displacements

61 Comparison with Analysis of Extension Problems It isinteresting to compare the previous analysis with that ofextension problem In extension problem applied shearstresses along the faces of the plate are gradients of a potentialfunction and they are equivalent to linearly varying bodyforces independent of elastic constants in the in-planeequilibrium equations Rotation 120596

119911= 0 (but Δ120596

119911= 0) and

equations governing displacements [1199060 V0] are coupled Also

vertical deflection is given by thickness-wise integration of1205761199110from constitutive relation and higher order corrections are

dependent on this vertical deflection In the present analysisof bending problems in-plane distributions of transverseshear stresses are gradients of a function related to appliededge transverse shear and face normal load through equi-librium equation (2) They are also independent of elasticconstants and derivative 119891

2119911of parabolic distribution of each

of them is equivalent to body force in linearly varying in-plane equilibrium equations Vertical deflection 119908

0is purely

from satisfaction of zero shear stress conditions along faces ofthe plate Higher order corrections are dependent on normalstrains and independent of119908

0 It can be seen that steps in the

analysis of bending problem are complementary to those inthe classical theory of extension problem

62 Associated Torsion Problems First-order shear deforma-tion theory and higher order shear deformation theoriesdo not provide proper corrections to initial solution (fromKirchhoff rsquos theory) of primary flexure problems In thesetheories corrections are due to approximate solutions ofassociated torsion problems In fact one has to have a relookat the use of shear deformation theories based on plate(instead of 3D) element equilibrium equations other thanKirchhoff rsquos theory in the analysis of flexure problems Inthe earlier work [5] we have mentioned that this couplingof bending and torsion problems is nullified in the limit ofsatisfying all equations in the 3D problems However thesehigher order theories in the case of primary flexure problemdefined from Kirchhoff rsquos theory are with reference to findingthe exact solution of associated torsion problem only In factone can obtain exact solution of the torsion problem (insteadof using higher order polynomials in 119911 by expanding 119911 and1198913(119911) in sine series and119891

2(119911) in cosine series In a pure torsion

problem normal stresses and strains are zero implying that

[119906 V 119908] = [119906 (119910 119911) V (119909 119911) 119908 (119909 119910)] (25)

Rotation 120596119911

= 0 and warping displacements (119906 V) are inde-pendent of vertical displacement 119908 In FSDT 120596

119911= 0 and

corrections to Kirchhoff rsquos displacements are due to thesolution of approximate torsion problem In the isotropicrectangular plate warping function 119906(119910 119911) is harmonic in 119910-119911-plane It has to satisfy 119866120572119906

119910= 119879119906(119910 119911) with prescribed 119879

119906

along an 119909 = constant edge By expressing 119906 in product form119906 = 119891119906(119910)119892119906(119911) we have

1205722119891119906119910119910

119892119906+ 119891119906119892119906119911119911

= 0 (26)

ISRN Civil Engineering 5

By taking 119892119906119899

= sin 120582119899119911 where 120582

119899= (2119899 + 1)1205872 satisfying

zero face shear conditions we get

119891119906119899

= 119860119906119899cosh(

120582119899

120572

)119910 + 119861119906119899sinh(

120582119899

120572

)119910 (27)

If prescribed 119879119906is 119891119906119899

with specified constants (119860119906119899 119861119906119899)

clearly there is no provision to satisfy zero in-plane shearcondition along 119910 = constant edges It has to be nullifiedwith corresponding solution from bending problem That iswhy a torsion problem is associated with bending problembut not vice versa In a corresponding bending problem allstress components are zero except 120591

119909119910= minus119891

119906119899(119910)119892119906119899(119911)

This solution is used only in satisfying edge condition along119910 = constant edges in the presence of specified 119879

119906119899along

119909 = constant edges It shows that 120591119909119910

distribution in flexureproblem is nullified in the limit in shear deformation theoriesdue to torsion

It is interesting to note that the previous mentioneddeficiency due to coupling with 119908

0in the plate element

equilibrium equations in FSDT and higher order sheardeformation theories does not exist if applied 120591

119909119910is zero all

along the closed boundary of the plate It is complementaryto the fact that boundary condition paradox in Kirchhoff rsquostheory does not exist if tangential displacement and 119908

0are

zero all along the boundary of the plate

63 Illustrative Example Simply Supported Isotropic SquarePlate Consider a simply supported isotropic square platesubjected to vertical load

119902 = 1199020sin(120587119909

119886

) sin(120587119910

119886

) (28)

Exact values for neutral plane and face deflections obtainedearlier [7] with ] = 03 120572 = (16) are

(

119864

21199020

)119908(

119886

2

119886

2

0) = 4487 (29)

(

119864

21199020

)119908(

119886

2

119886

2

1) = 4166 (30)

In this example Poisson-Kirchhoff rsquos boundary conditionsparadox does not exist Hence Poissonrsquos theory gives thesame solution obtained in the authorrsquos previous mentionedwork Concerning face value in (30) Kirchhoff rsquos theory givesa value of 227 that is same for all face parallel planes Poissonrsquostheory gives an additional value of 0267 attributable to 120576

1199111

and it is same for neutral plane deflection since119908(119909 119910 119911) canbe expressed as

119908 (119909 119910 119911) = 1199080119865

(119909 119910) + 1198912(119911) 1199082

= 1199080119873

(119909 119910) minus (

1

2

) 11991121205761199111

(31)

In the previous expression 1199080119873

is deflection of the neutralplane Higher order correction to119908

0uncoupled from torsion

is 1262 so that total correction to the value from Kirchhoff rsquostheory is about 153 [7] Correction due to coupling with

torsion is 145 [5] whereas it is 1423 from FSDT and othersixth order theories [9] With reference to numerical valuesreported by Lewinski [9] the previous correction is 1412 inReddyrsquos 8th-order theory and less than 123 in Reissnerrsquos 12th-order and other higher order theories It clearly shows thatthese shear deformation theories do not lead to solutions ofbending problems (note that the value (227 + 1423) fromFSDT corresponds to neutral plane deflection and it is inerror by about 177 from the exact value)

In view of the previous observations it is relevant tomakethe following remarks physical validity of 3D equations isconstrained by limitations of small deformation theory For120572 = 1 this constraint is dependent on material constantsgeometry of the domain and applied loads Range of validityof 2D plate theories arises for small values of 120572 Kirchhoff rsquostheory gives lower bound for this 120572 For this value of 120572 and forslightly higher values of 120572 than this lower bound Kirchhoff rsquostheory is used in spite of its known deficiencies due to itssimplicity to obtain design information Similarly FSDT andother sixth-order shear deformation theories are used forsome additional range of values of 120572 due to the correctionsover Kirchhoff rsquos theory though these corrections are due tocoupling with approximate torsion problemsThey serve onlyin giving guidance values for applicable design parametersfor small range of values of 120572 beyond its lower boundThey do not provide initial set of equations in formulatingproper sequence of sets of 2D problems converging to the 3Dproblem As such comparison of solutions from Kirchhofftheory and FSDT with the corresponding solutions in thepresent analysis does not serve much purpose

Since 1199080119873

has to be higher than 1199080119865 one has to obtain

the correction to 1199080119873

before finding correction to internaldistribution of119908(119909 119910 119911) along with correction to119908

0119865 Initial

correction to 1199080119873

is obtained earlier [7] in the form

119908 = 1199080minus [1198912(0) 1205761199111+ 1198914(0) 1205761199113] (32)

It is the same for face deflection (note that 1199082in the

previouslymentioned authorrsquos work is required only to obtain1205761199113) As such 119908

0119873is further corrected from solution of

a supplementary problem This total correction over facedeflection is about 0658 giving a value of 4458 which is veryclose to the exact value 4487 in (29) However we note thatthe error in the estimated value (=38) of 119908

0119865is relatively

high compared to the accuracy achieved in the neutral planedeflection It is possible to improve estimation of 119908

0119865by

including 1205761199111in (1199063 V3) such that (120591

1199091199112 1205911199101199112

) are independentof 1205761199111 In the present example correction to face deflection

changes to

(

119864

21199020

)119908face max =((65) ((1 + 120592) (1 minus 120592))) 120573

2

((25) ((4 minus 120592) (1 minus 120592))) 1205732minus 1

(33)

In the previous equation 1205732 = 2119908011986512057221205872 Correction 120576

1199113(=

1262) to face value changes to 1431 giving 397 (= 227 +

0267 + 1431) for face deflection which is under 47 fromexact value

6 ISRN Civil Engineering

7 Anisotropic PlateSecond-Order Corrections

We note that 1205761199111does not participate in the determination of

(1199061 V1) and it is obtained from the constitutive relation (6) in

the interior of the plate It is zero along an edge of the plateif 119908 = 0 is specified condition With zero transverse shearstrains specification of 120576

1199111= 0 instead of 119908

0= 0 is more

appropriate along a supported edge since 1199080

= 0 implieszero tangential displacement along a straight edge which isthe root cause of Poisson-Kirchhoff rsquos boundary conditionsparadox As such edge support condition on w does not playany role in obtaining the in-plane displacements and reactivetransverse stresses

In the first stage of iterative procedure second-orderreactive transverse stresses have to be obtained by consid-ering higher order in-plane displacement terms 119906

3and V

3

However one has to consider limitation in the previousPoissonrsquos theory like in Kirchhoff rsquos theory namely reactive120590119911is zero at locations of zeros of 119902 It is identically zero in

higher order approximations since 1198912119899+1

(119911) functions are notzero along 119911 = plusmn1 faces Moreover it is necessary to accountfor its dependence on material constants To overcome theselimitations it is necessary to keep 120590

1199115as a free variable by

modifying 1198915in the form

119891lowast

5(119911) = 119891

5 (119911) minus 120573

31198913 (119911) (34)

In the previous equation 1205733= [1198915(1)1198913(1)] so that119891lowast

5(plusmn1) =

0 Denoting coefficient of 1198913in 120590119911by 120590lowast1199113 it becomes

120590lowast

1199113= 1205901199113minus 12057331205901199115 (35)

Displacements (1199063 V3) are modified such that they are cor-

rections to face parallel plane distributions of the preliminarysolution and are free to obtain reactive stresses 120591

1199091199114 1205911199101199114

1205901199115

and normal strain 1205761199113

We have from constitutive relations

1205741199091199112

= 119878441205911199091199112

+ 119878451205911199101199112

lArrrArr (119909 119910) (4 5) (36)

Modified displacements and the corresponding derivedquantities denoted with lowast are

119906lowast

3= 1199063minus 1205721199082119909

+ 1205741199091199112

lArrrArr (119909 119910) (119906 V) (37)

Strain-displacement relations give

120576lowast

1199093= 1205761199093

minus 12057221199082119909119909

+ 1205721205741199091199112119909

lArrrArr (119909 119910)

120574lowast

1199091199103= 1205741199091199103

minus 212057221199082119909119910

+ 120572 (1205741199091199112119910

+ 1205741199101199112119909

)

120574lowast

1199091199113= 1199063+ 1205741199091199112

lArrrArr (119909 119910) (119906 V)

(38)

In-plane and transverse shear stresses from constitutiverelations are

120590lowast

119894(3)= 119876119894119895120576lowast

119895(3)(119894 119895 = 1 2 3) (39)

120591lowast

1199091199112= 119876441199063+ 11987645V3+ 1205911199091199112

lArrrArr (119909 119910) (4 5) (40)

Suffix (3) in (39) is to indicate that the stresses and strainscorrespond to 119906

3and V3

We get from (2) (15) (35) (37) and (40) with referenceto coefficient of 119891

3(119911)

120572 [(119876441199063+ 11987645V3)119909

+ (119876541199063+ 11987655V3)119910

] = 12057331205901199115 (41)

Note that 1199082is not present in the previous equation

From integration of equilibrium equations we have withsum on 119895 = 1 2 3

1205911199091199114

= 120572[1198761119895120576lowast

119895119909+ 1198763119895120576lowast

119895119910](3)

1205911199101199114

= 120572[1198762119895120576lowast

119895119910+ 1198763119895120576lowast

119895119909](3)

(42)

Note that 1199082is present in the previous expression

120572 (1205911199091199114119909

+ 1205911199101199114119910

) + 1205901199115

= 0 (Coefficient of 1198915) (43)

1205761199113

= [1198786119895120590119895+ 11987866120590119911](3) (44)

One gets one equation governing in-plane displacements(1199063 V3) from (41) (43) noting that 119891

5119911119911+1198913= 0 in the form

1205733(1205911199091199114119909

+ 1205911199101199114119910

)

= 1205722[(119876441199063+ 11987645V3)119909119909

+ (119876541199063+ 11987655V3)119910119910

]

(45)

The second equation governing these variables is from thecondition 120596

119911= 0 Here it is more convenient to express

(1199063 V3) as

[1199063 V3] = minus120572 [(120595

3119909minus 1206013119910) (1205953119910

+ 1206013119909)] (46)

so that Δ1205933= 0 and (45) becomes a fourth-order equation

governing 1205953 This sixth-order system is to be solved for 120595

3

and 1205933with associated edge conditions

119906lowast

3= 0 or 120590

lowast

3= 0

Vlowast

3= 0 or 120591

lowast

1199091199103= 0

1205953= 0 or 120591

lowast

1199091199113= 0

(47)

along an 119909 = constant edge and analogous conditions along a119910 = constant edge

In (45) expressed in terms of 1205953and 120593

3 we replace 119908

2by

1205953for obtaining neutral plane deflection since contributions

of 1199082and 120595

3are one and the same in finding [119906

3 V3]

71 Supplementary Problem Because of the use of integrationconstant to satisfy face shear conditions vertical deflection ofneutral plane is equal to that of the face plane deflectionThisis physically incorrect since face plane is bounded by elasticmaterial on one side whereas neutral plane is bounded onboth sidesThis lack in interior solution is rectified by addingthe solution of a supplementary problem based on leadingcosine term that is enough in view of 119891

2(119911) distribution

of reactive shear stresses This is based on the expansion

ISRN Civil Engineering 7

of 120590119911= 1198913(119911)(32)119902 in sine series Here load condition is

satisfied by the leading sine term and coefficients of all othersine terms are zero

Corrective displacements in the supplementary problemare assumed in the form

119908 = 1199082119904

120587

2

cos(1205872

119911) 119906119904= 1199063119904sin(120587

2

119911) lArrrArr (119906 V)

1205903119904119894

= 1198761198941198951205763119904119895

(119894 119895 = 1 2 3)

(48)

We have from integration of equilibrium equations

1205912119904119904119909119911

= minus(

2

120587

)120572 [11987611198951205763119904119895119909

+ 11987631198951205763119904119895119910

] (49a)

1205912119904119910119911

= minus(

2

120587

)120572 [11987621198951205763119904119895119910

+ 11987631198951205763119904119895119909

] (49b)

In-plane distributions 1199063119904

and V3119904

are added as correctionsto the known 119906

lowast

3and Vlowast

3so that (119906 V) in the supplementary

problem are

119906 = (119906lowast

3+ 1199063119904) sin(120587

2

119911) lArrrArr (119906 V) (50)

We note that 120590119911from integration of equilibrium equations

with 119904 variables is the same as 120590119911from static equilibrium

equations with lowast variables Hence

(

2

120587

)

2

1205722[11987611198951205763119904119895119909119909

+ 211987631198951205763119904119895119909119910

+ 11987621198951205763119904119895119910119910

] + 12057331205901199115

= 0

(51)

It is convenient to use here also

[1199063119904 V3119904] = minus120572 [(120595

3119904119909minus 1206013119904119910

) (1205953119904119910

+ 1205933119904119909

)] (52)

Correction to neutral plane deflection due to solution ofsupplementary problem is from

1205761199113119904

= [11987861198951205903119904119895

+ 119878661205901199113119904

] (sum 119895 = 1 2 3) (53)

72 Summary of Results from the First Stage of IterativeProcedure The previous analysis gives displacements [119906V] =1198913[1199063 V3] consistent with [120591

119909119911 120591119910119911] = 120572119891

2[120595119909 120595119910] and

corrective transverse stresses from the first stage of iteration

119908 = 1199080minus [11989121205761199111+ 11989141205761199113] minus 1205761199111199043

120587

2

cos(1205872

119911)

119906 = 11989111199061+ 1198913119906lowast

3+ (119906lowast

3+ 1199063119904) sin(120587

2

119911) lArrrArr (119906 V)

120591119909119911

= 11989121205912119909119911

+ 1205912119904119909119911

120587

2

cos(1205872

119911) lArrrArr (119909 119910)

120590119911= 1198913(119911) 120590lowast

1199113(From (35))

(54)

It is to be noted that the dependence of transverse stresses onmaterial constants is through the solution of supplementaryproblem One may add 119891

4terms in shear components and 119891

5

term in normal stress componentThese components are alsodependent on material constants but they need correctionfrom the solution of a supplementary problem Successiveapplication of the previous iterative procedure leads to thesolution of the 3D problem in the limit

8 Concluding Remarks

Poissonrsquos theory developed in the present study for the analy-sis of bending of anisotropic plates within small deformationtheory forms the basis for generation of proper sequence of2D problems Analysis for obtaining displacements therebybending stresses along faces of the plate is different fromsolution of a supplementary problem in the interior ofthe plate A sequence of higher order shear deformationtheories lead to solution of associated torsion problem onlyIn the preliminary solution reactive transverse stresses areindependent of material constants In view of layer-wisetheory of symmetric laminated plates proposed by the presentauthor [10] Poissonrsquos theory is useful since analysis of faceplies is independent of lamination which is in confirmationof (8)-(9) in a recent NASA technical publication by Tessleret al [11]

One significant observation is that sequence of 2D prob-lems converging to 3D problems in the analysis of extensionbending and torsion problems are mutually exclusive to oneother

Highlights

(i) Poisson-Kirchhoff boundary conditions paradox isresolved

(ii) Transverse stresses are independent of material con-stants in primary solution

(iii) Edge support condition on vertical deflection has norole in the analysis

(iv) Finding neutral plane deflection requires solution ofa supplementary problem

(v) Twisting stress distribution in pure torsion nullifies itsdistribution in bending

Appendix

119886 = side length of a square plate2ℎ = plate thickness119891119899(119911)= through-thickness distribution functions 119899 =

0 1 2

119871 = characteristic length of the plate in119883-119884-plane119902(119909 119910) = applied face load density119876119894119895= stiffness coefficients

119878119894119895= elastic compliances

[119879119909 119879119909119910 119879119909119911] = prescribed stress distributions along

an 119909 = constant edge

8 ISRN Civil Engineering

[119880 119881119882] = displacements in 119883119884 119885-directionsrespectively[119906 V 119908] = [119880 119881119882]ℎ

119874 minus 119883119884119885 = Cartesian coordinate system[119909 119910 119911] = nondimensional coordinates [119883119871 119884119871119885ℎ]

120572 = ℎ119871

1205732119899+1

= [1198912119899+3

(1)12057221198912119899+1

(1)]

Δ = Laplace operator (12059721205971199092 + 12059721205971199102)

[120576119909 120576119910 120574119909119910] = in-plane strains

[120574119909119911 120574119910119911 120576119911] = transverse strains

120596119911= 120572(V119909minus 119906119910) rotation about 119911-axis

[120590119909 120590119910 120591119909119910] = bending stresses

[120591119909119911 120591119910119911 120590119911] = transverse stresses

References

[1] G Kirchhoff ldquoUber das Gleichgewicht und die Bewegung einerelastischen Scheiberdquo Journal fur die Reine und AngewandteMathematik vol 40 pp 51ndash58 1850

[2] HHencky ldquoUber die Berucksichtigung der Schubverzerrung inebenen Plattenrdquo Ingenieur-Archiv vol 16 no 1 pp 72ndash76 1947

[3] E Reissner ldquoReflections on the theory of elastic platesrdquoAppliedMechanics Reviews vol 38 no 11 pp 1453ndash1464 1985

[4] A E H Love A Treatise on Mathematical Theory of ElasticityCambridge University Press Cambridge UK 4th edition 1934

[5] K Vijayakumar ldquoModified Kirchhoff rsquos theory of plates includ-ing transverse shear deformationsrdquo Mechanics Research Com-munications vol 38 pp 211ndash213 2011

[6] K Vijayakumar ldquoNew look at kirchhoff rsquos theory of platesrdquoAIAA Journal vol 47 no 4 pp 1045ndash1046 2009

[7] K Vijayakumar ldquoA relook at Reissnerrsquos theory of plates inbendingrdquo Archive of Applied Mechanics vol 81 no 11 pp 1717ndash1724 2011

[8] E Reissner ldquoThe effect of transverse shear deformations on thebending of elastic platesrdquo Journal of Applied Mechanics vol 12pp A69ndashA77 1945

[9] T Lewinski ldquoOn the twelfth-order theory of elastic platesrdquoMechanics Research Communications vol 17 no 6 pp 375ndash3821990

[10] K Vijayakumar ldquoLayer-wise theory of bending of symmetriclaminates with isotropic pliesrdquo AIAA Journal vol 49 no 9 pp2073ndash2076 2011

[11] A Tessler M Di Sciuva and M Gherlone ldquoRefined zigzagtheory for homogeneous laminated composite and sandwichplates a homogeneous limit methodology for zigzag functionselectionrdquo Tech Rep NASATP-2920102162141 2010

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ISRN Civil Engineering 3

Displacements strains and stresses are expressed in the form(sum 119899 = 0 1 2 3 )

[119908 119906 V] = [11989121198991199082119899 1198912119899+1

1199062119899+1

1198912119899+1

V2119899+1

]

[120576119909 120576119910 120574119909119910 120576119911] = 1198912119899+1

[120576119909 120576119910 120574119909119910 120576119911]2119899+1

(11)

(Note that 1199082119899

= minus1205761199112119899minus1

119899 ge 1)

[120590119909 120590119910 120591119909119910 120590119911] = 1198912119899+1

[120590119909 120590119910 120591119909119910 120590119911]2119899+1

[120574119909119911 120574119910119911 120591119909119911 120591119910119911] = 1198912119899[120574119909119911 120574119910119911 120591119909119911 120591119910119911]

(12)

In the previous equations variables associated with 119891(119911)

functions are functions of (119909 119910) only

5 Sequence of Trivially Known Steps withoutAny Assumptions in Preliminary Analysis

(a) Due to prescribed zero (120591119909119911 120591119910119911) along 119911 = plusmn1 faces of the

plate (1205911199091199110

1205911199101199110

) equiv 0 in the plate(b) (1205741199091199110

1205741199101199110

) equiv 0 from constitutive relations (5)(c) Static equilibrium equation (2) gives 120590

1199111equiv 0

(d) In the absence of (1199061 V1) (1205761199091 1205761199101 1205741199091199101

) equiv 0 fromstrain-displacement relations (8)

(e) (1205901199091 1205901199101 1205911199091199101

) equiv 0 from constitutive relations (6)(f) 1205761199111

equiv 0 from relation (7)(g) 119908 = 119908

0(119909 119910) from thickness-wise integration of 120576

119911=

0(h) From steps (b) and (g) integration of [119906

119911+120572119908119909 V119911+

120572119908119910] = [0 0] in the thickness direction gives [119906 V] =

minus119911120572[1199080119909

1199080119910]

In the present study as in the earlier work [6] [119906 V] linearin 119911 are unknown functions as in FSDT From integration of[119906119911+ 120572119908119909 V119911+ 120572119908119910] = [0 0] in the face plane instead of

thickness direction one gets vertical deflection1199080(119909 119910) in the

form

1205721199080= minusint [119906

1119889119909 + V

1119889119910] (13)

Since 1199080is from satisfaction of zero face shear conditions it

can be considered as face deflection1199080119865

though it is same forall face parallel planes (note that prescribed zero 119908

0along

a segment of the neutral plane implies zero 1199080along the

corresponding segment of the intersection of face plane withwall of the plate since 119908

0is independent of 119911 and vice versa)

It is analytic in the domain of the plate if 120596119911is zero In such a

case we note from strain-displacement relations (8) that

[1205761119910 1205762119909 1205763119910 1205763119909] = 1205722[V119909119909

119906119910119910

2119906119910119910

2V119909119909

] (14)

(i) In the absence of applied transverse loads integrationsof (1) (2) using Kirchhoff rsquos displacements give a homoge-neous fourth-order equation governing 119908

0 In the present

analysis as in FSDT static equations governing [1199061 V1] are

however given by

1198761119894120576119894119909

+ 1198763119894120576119894119910

= 0 1198762119894120576119894119910

+ 11987631198941205763119894119909

= 0 (119894 = 1 2 3)

(15)

They are subjected to the edge conditions (3a) (3b) along 119909-(and 119910-) constant edges

Equations (15) governing [1199061 V1] are uncoupled due to

relations (14) in the case of isotropic plates whereas theyare coupled without cross-derivatives [119906

1119909119910 V1119909119910

] in the caseof anisotropic plates In the absence of prescribed bendingstress all along the closed boundary of the plate in-plane120591119909119910

distribution corresponds to pure torsion where as itis different in the presence of bending load These 120591

119909119910

distributions are discussed later in Section 62(j) In FSDT (15) correspond to neglecting shear energy

from transverse shear deformations Vertical deflection 1199080

has to be obtained from (14) using the solutions for [1199061 V1]

from (15) Reactive transverse stresses are expressed in termsof [1199061 V1]They are dependent onmaterial constants different

from prescribed nonzero transverse shear stresses along theedges of the plate

(k) In the development of Poissonrsquos theory (15) arecoupled with reactive transverse stresses

6 Poissonrsquos Theory

In the preliminary solution steps (a)ndash(g) in Section 5 areunaltered That is transverse stresses and strains are zeroand 119908 = 119908

0(119909 119910) In the absence of higher order in-plane

displacement terms reactive transverse shear stresses areparabolic and gradients of a function 120595(119909 119910) that is in-plane distributions of transverse shear stresses [120591

119909119911 120591119910119911] =

120572[1205952119909 1205952119910] Equation governing 120595(= 120595

2) from thickness-

wise integration of (2) is

1205722Δ120595 + 120590

1199113= 0 (16)

In the previous equation 1205901199113

is coefficient of 1198913(119911) Satisfac-

tion of the load condition 120590119911= plusmn1199022 along 119911 = plusmn1 faces gives

1205901199113

= (32)119902 so that

1205722Δ120595 + (

3

2

) 119902 = 0 (17)

Edge condition on 120595 is either 120595 = 0 or its outwardnormal gradient is equal to the prescribed shear stress alongeach segment of the edge Note that the transverse stressesthus obtained are independent of material constants In theisotropic plate 120595(119909 119910) = 119864

10158401198901in which 119864

1015840= 119864(1 minus ]2) and

1198901= (1205761199091

+ 1205761199101)

Integrated equilibrium equations (1) in terms of in-planestrains are (sum 119895 = 1 2 3)

[1198761119895120576119895119909

+ 1198763119895120576119895119910] minus 1205911199091199112

= 0

[1198762119895120576119895119910

+ 1198763119895120576119895119909] minus 1205911199101199112

= 0

(18)

By substituting strains in terms of [1199061 V1] (18) are two

equations governing (1199061 V1) and have to be solved with

conditions along 119909- (and 119910-) constant edge

119906 = 0 or 120590119909= 119911119879119909(119910) lArrrArr (119909 119910) (119906 V) (19a)

V = 0 or 120591119909119910

= 119911119879119909119910(119910) lArrrArr (119909 119910) (119906 V) (19b)

4 ISRN Civil Engineering

Vertical deflection 1199080is given by (13) As mentioned earlier

cross-derivatives of [1199061 V1] do not exist in (18) (19a) and

(19b) due to (14) In the isotropic case (18) are uncoupled andare simply given by

1205722Δ1199061= 1205721198901119909

lArrrArr (119909 119910) (119906 119910) (20)

Edge conditions (19a) and (19b) are also uncoupled and theyare given by

1199061= 0 or 1198641015840 (1 minus 120592) 120572119906

1119909= 119879119909(119910) lArrrArr (119909 119910) (119906 V)

V1= 0 or 2119866120572V

1119909= 119879119909119910(119910) lArrrArr (119909 119910) (119906 V)

(21)

The condition1205950= 0 in solving (17) is different from the usual

condition 1199080= 0 in the Kirchhoff rsquos theory and FSDT The

function 1205950is related to the normal strain 120576

119911 In the isotropic

case 1205950is proportional to 119890

1(119909 119910) = (120576

1199091+ 1205761199101) Gradients of

1205950are proportional to gradients of transverse strains in FSDT

If the plate is free of applied transverse stresses that is theplate is subjected to bending and twisting moments only 119890

1

is proportional toΔ1199080The function120595

0(119909 119910) thereby Δ119908

0is

identically zero from (2) (in Kirchhoff rsquos theory the tangentialgradient ofΔ119908

0is proportional to the corresponding gradient

of applied 120591119909119910

along the edge of the plate implied from Kelvinand Taitrsquos physical interpretation of the contracted transverseshear condition) This Laplace equation is not adequate tosatisfy two in-plane edge conditions One needs its conjugateharmonic functions to express in-plane displacements in theform

[1199061 V1] = minus120572119911 [119908

0119909+ 1205930119910 1199080119910

minus 1205930119909] (22)

The function 1205930was introduced earlier by Reissner [8] as a

stress function in satisfying (2) Note that the two variables1199061and V1are expressed in terms of gradients of two functions

1199080and 120593

0

After finding1199080and1205930from solving in-plane equilibrium

equations (1) correction 1199080119888

to 1199080from zero face shear

conditions is given by

1199080119888= int [120601

0119910119889119909 + 120601

0119909119889119910] (23)

One should note here that119908(119909 119910) = 1199080+1199080119888does not satisfy

prescribed edge condition on 119908In FSDT [119906

1 V1] are obtained from solving (1) There is

no provision to find 1199080and it has to be obtained from zero

face shear conditions in the form

1205721199080= minusint [119906

1119889119909 + V

1119889119910] (24)

If the applied transverse stresses are nonhomogeneous alonga segment of the edge 119908

0is coupled with [119906

1 V1] in (1) and

(2) and edge conditions (3a) (3b) and (3c)With reference to the present analysis it is relevant to

note the following observation in the preliminary solution120590119911 120572[120591119909119911 120591119910119911] 1205722[120590

119909 120590119910 120591119909119910] are of119874(1) As such estimation

of in-plane stresses thereby in-plane displacements is notdependent on 119908

0 The support condition 119908

0= 0 along the

edge of the plate does not play any role in determining thein-plane displacements

61 Comparison with Analysis of Extension Problems It isinteresting to compare the previous analysis with that ofextension problem In extension problem applied shearstresses along the faces of the plate are gradients of a potentialfunction and they are equivalent to linearly varying bodyforces independent of elastic constants in the in-planeequilibrium equations Rotation 120596

119911= 0 (but Δ120596

119911= 0) and

equations governing displacements [1199060 V0] are coupled Also

vertical deflection is given by thickness-wise integration of1205761199110from constitutive relation and higher order corrections are

dependent on this vertical deflection In the present analysisof bending problems in-plane distributions of transverseshear stresses are gradients of a function related to appliededge transverse shear and face normal load through equi-librium equation (2) They are also independent of elasticconstants and derivative 119891

2119911of parabolic distribution of each

of them is equivalent to body force in linearly varying in-plane equilibrium equations Vertical deflection 119908

0is purely

from satisfaction of zero shear stress conditions along faces ofthe plate Higher order corrections are dependent on normalstrains and independent of119908

0 It can be seen that steps in the

analysis of bending problem are complementary to those inthe classical theory of extension problem

62 Associated Torsion Problems First-order shear deforma-tion theory and higher order shear deformation theoriesdo not provide proper corrections to initial solution (fromKirchhoff rsquos theory) of primary flexure problems In thesetheories corrections are due to approximate solutions ofassociated torsion problems In fact one has to have a relookat the use of shear deformation theories based on plate(instead of 3D) element equilibrium equations other thanKirchhoff rsquos theory in the analysis of flexure problems Inthe earlier work [5] we have mentioned that this couplingof bending and torsion problems is nullified in the limit ofsatisfying all equations in the 3D problems However thesehigher order theories in the case of primary flexure problemdefined from Kirchhoff rsquos theory are with reference to findingthe exact solution of associated torsion problem only In factone can obtain exact solution of the torsion problem (insteadof using higher order polynomials in 119911 by expanding 119911 and1198913(119911) in sine series and119891

2(119911) in cosine series In a pure torsion

problem normal stresses and strains are zero implying that

[119906 V 119908] = [119906 (119910 119911) V (119909 119911) 119908 (119909 119910)] (25)

Rotation 120596119911

= 0 and warping displacements (119906 V) are inde-pendent of vertical displacement 119908 In FSDT 120596

119911= 0 and

corrections to Kirchhoff rsquos displacements are due to thesolution of approximate torsion problem In the isotropicrectangular plate warping function 119906(119910 119911) is harmonic in 119910-119911-plane It has to satisfy 119866120572119906

119910= 119879119906(119910 119911) with prescribed 119879

119906

along an 119909 = constant edge By expressing 119906 in product form119906 = 119891119906(119910)119892119906(119911) we have

1205722119891119906119910119910

119892119906+ 119891119906119892119906119911119911

= 0 (26)

ISRN Civil Engineering 5

By taking 119892119906119899

= sin 120582119899119911 where 120582

119899= (2119899 + 1)1205872 satisfying

zero face shear conditions we get

119891119906119899

= 119860119906119899cosh(

120582119899

120572

)119910 + 119861119906119899sinh(

120582119899

120572

)119910 (27)

If prescribed 119879119906is 119891119906119899

with specified constants (119860119906119899 119861119906119899)

clearly there is no provision to satisfy zero in-plane shearcondition along 119910 = constant edges It has to be nullifiedwith corresponding solution from bending problem That iswhy a torsion problem is associated with bending problembut not vice versa In a corresponding bending problem allstress components are zero except 120591

119909119910= minus119891

119906119899(119910)119892119906119899(119911)

This solution is used only in satisfying edge condition along119910 = constant edges in the presence of specified 119879

119906119899along

119909 = constant edges It shows that 120591119909119910

distribution in flexureproblem is nullified in the limit in shear deformation theoriesdue to torsion

It is interesting to note that the previous mentioneddeficiency due to coupling with 119908

0in the plate element

equilibrium equations in FSDT and higher order sheardeformation theories does not exist if applied 120591

119909119910is zero all

along the closed boundary of the plate It is complementaryto the fact that boundary condition paradox in Kirchhoff rsquostheory does not exist if tangential displacement and 119908

0are

zero all along the boundary of the plate

63 Illustrative Example Simply Supported Isotropic SquarePlate Consider a simply supported isotropic square platesubjected to vertical load

119902 = 1199020sin(120587119909

119886

) sin(120587119910

119886

) (28)

Exact values for neutral plane and face deflections obtainedearlier [7] with ] = 03 120572 = (16) are

(

119864

21199020

)119908(

119886

2

119886

2

0) = 4487 (29)

(

119864

21199020

)119908(

119886

2

119886

2

1) = 4166 (30)

In this example Poisson-Kirchhoff rsquos boundary conditionsparadox does not exist Hence Poissonrsquos theory gives thesame solution obtained in the authorrsquos previous mentionedwork Concerning face value in (30) Kirchhoff rsquos theory givesa value of 227 that is same for all face parallel planes Poissonrsquostheory gives an additional value of 0267 attributable to 120576

1199111

and it is same for neutral plane deflection since119908(119909 119910 119911) canbe expressed as

119908 (119909 119910 119911) = 1199080119865

(119909 119910) + 1198912(119911) 1199082

= 1199080119873

(119909 119910) minus (

1

2

) 11991121205761199111

(31)

In the previous expression 1199080119873

is deflection of the neutralplane Higher order correction to119908

0uncoupled from torsion

is 1262 so that total correction to the value from Kirchhoff rsquostheory is about 153 [7] Correction due to coupling with

torsion is 145 [5] whereas it is 1423 from FSDT and othersixth order theories [9] With reference to numerical valuesreported by Lewinski [9] the previous correction is 1412 inReddyrsquos 8th-order theory and less than 123 in Reissnerrsquos 12th-order and other higher order theories It clearly shows thatthese shear deformation theories do not lead to solutions ofbending problems (note that the value (227 + 1423) fromFSDT corresponds to neutral plane deflection and it is inerror by about 177 from the exact value)

In view of the previous observations it is relevant tomakethe following remarks physical validity of 3D equations isconstrained by limitations of small deformation theory For120572 = 1 this constraint is dependent on material constantsgeometry of the domain and applied loads Range of validityof 2D plate theories arises for small values of 120572 Kirchhoff rsquostheory gives lower bound for this 120572 For this value of 120572 and forslightly higher values of 120572 than this lower bound Kirchhoff rsquostheory is used in spite of its known deficiencies due to itssimplicity to obtain design information Similarly FSDT andother sixth-order shear deformation theories are used forsome additional range of values of 120572 due to the correctionsover Kirchhoff rsquos theory though these corrections are due tocoupling with approximate torsion problemsThey serve onlyin giving guidance values for applicable design parametersfor small range of values of 120572 beyond its lower boundThey do not provide initial set of equations in formulatingproper sequence of sets of 2D problems converging to the 3Dproblem As such comparison of solutions from Kirchhofftheory and FSDT with the corresponding solutions in thepresent analysis does not serve much purpose

Since 1199080119873

has to be higher than 1199080119865 one has to obtain

the correction to 1199080119873

before finding correction to internaldistribution of119908(119909 119910 119911) along with correction to119908

0119865 Initial

correction to 1199080119873

is obtained earlier [7] in the form

119908 = 1199080minus [1198912(0) 1205761199111+ 1198914(0) 1205761199113] (32)

It is the same for face deflection (note that 1199082in the

previouslymentioned authorrsquos work is required only to obtain1205761199113) As such 119908

0119873is further corrected from solution of

a supplementary problem This total correction over facedeflection is about 0658 giving a value of 4458 which is veryclose to the exact value 4487 in (29) However we note thatthe error in the estimated value (=38) of 119908

0119865is relatively

high compared to the accuracy achieved in the neutral planedeflection It is possible to improve estimation of 119908

0119865by

including 1205761199111in (1199063 V3) such that (120591

1199091199112 1205911199101199112

) are independentof 1205761199111 In the present example correction to face deflection

changes to

(

119864

21199020

)119908face max =((65) ((1 + 120592) (1 minus 120592))) 120573

2

((25) ((4 minus 120592) (1 minus 120592))) 1205732minus 1

(33)

In the previous equation 1205732 = 2119908011986512057221205872 Correction 120576

1199113(=

1262) to face value changes to 1431 giving 397 (= 227 +

0267 + 1431) for face deflection which is under 47 fromexact value

6 ISRN Civil Engineering

7 Anisotropic PlateSecond-Order Corrections

We note that 1205761199111does not participate in the determination of

(1199061 V1) and it is obtained from the constitutive relation (6) in

the interior of the plate It is zero along an edge of the plateif 119908 = 0 is specified condition With zero transverse shearstrains specification of 120576

1199111= 0 instead of 119908

0= 0 is more

appropriate along a supported edge since 1199080

= 0 implieszero tangential displacement along a straight edge which isthe root cause of Poisson-Kirchhoff rsquos boundary conditionsparadox As such edge support condition on w does not playany role in obtaining the in-plane displacements and reactivetransverse stresses

In the first stage of iterative procedure second-orderreactive transverse stresses have to be obtained by consid-ering higher order in-plane displacement terms 119906

3and V

3

However one has to consider limitation in the previousPoissonrsquos theory like in Kirchhoff rsquos theory namely reactive120590119911is zero at locations of zeros of 119902 It is identically zero in

higher order approximations since 1198912119899+1

(119911) functions are notzero along 119911 = plusmn1 faces Moreover it is necessary to accountfor its dependence on material constants To overcome theselimitations it is necessary to keep 120590

1199115as a free variable by

modifying 1198915in the form

119891lowast

5(119911) = 119891

5 (119911) minus 120573

31198913 (119911) (34)

In the previous equation 1205733= [1198915(1)1198913(1)] so that119891lowast

5(plusmn1) =

0 Denoting coefficient of 1198913in 120590119911by 120590lowast1199113 it becomes

120590lowast

1199113= 1205901199113minus 12057331205901199115 (35)

Displacements (1199063 V3) are modified such that they are cor-

rections to face parallel plane distributions of the preliminarysolution and are free to obtain reactive stresses 120591

1199091199114 1205911199101199114

1205901199115

and normal strain 1205761199113

We have from constitutive relations

1205741199091199112

= 119878441205911199091199112

+ 119878451205911199101199112

lArrrArr (119909 119910) (4 5) (36)

Modified displacements and the corresponding derivedquantities denoted with lowast are

119906lowast

3= 1199063minus 1205721199082119909

+ 1205741199091199112

lArrrArr (119909 119910) (119906 V) (37)

Strain-displacement relations give

120576lowast

1199093= 1205761199093

minus 12057221199082119909119909

+ 1205721205741199091199112119909

lArrrArr (119909 119910)

120574lowast

1199091199103= 1205741199091199103

minus 212057221199082119909119910

+ 120572 (1205741199091199112119910

+ 1205741199101199112119909

)

120574lowast

1199091199113= 1199063+ 1205741199091199112

lArrrArr (119909 119910) (119906 V)

(38)

In-plane and transverse shear stresses from constitutiverelations are

120590lowast

119894(3)= 119876119894119895120576lowast

119895(3)(119894 119895 = 1 2 3) (39)

120591lowast

1199091199112= 119876441199063+ 11987645V3+ 1205911199091199112

lArrrArr (119909 119910) (4 5) (40)

Suffix (3) in (39) is to indicate that the stresses and strainscorrespond to 119906

3and V3

We get from (2) (15) (35) (37) and (40) with referenceto coefficient of 119891

3(119911)

120572 [(119876441199063+ 11987645V3)119909

+ (119876541199063+ 11987655V3)119910

] = 12057331205901199115 (41)

Note that 1199082is not present in the previous equation

From integration of equilibrium equations we have withsum on 119895 = 1 2 3

1205911199091199114

= 120572[1198761119895120576lowast

119895119909+ 1198763119895120576lowast

119895119910](3)

1205911199101199114

= 120572[1198762119895120576lowast

119895119910+ 1198763119895120576lowast

119895119909](3)

(42)

Note that 1199082is present in the previous expression

120572 (1205911199091199114119909

+ 1205911199101199114119910

) + 1205901199115

= 0 (Coefficient of 1198915) (43)

1205761199113

= [1198786119895120590119895+ 11987866120590119911](3) (44)

One gets one equation governing in-plane displacements(1199063 V3) from (41) (43) noting that 119891

5119911119911+1198913= 0 in the form

1205733(1205911199091199114119909

+ 1205911199101199114119910

)

= 1205722[(119876441199063+ 11987645V3)119909119909

+ (119876541199063+ 11987655V3)119910119910

]

(45)

The second equation governing these variables is from thecondition 120596

119911= 0 Here it is more convenient to express

(1199063 V3) as

[1199063 V3] = minus120572 [(120595

3119909minus 1206013119910) (1205953119910

+ 1206013119909)] (46)

so that Δ1205933= 0 and (45) becomes a fourth-order equation

governing 1205953 This sixth-order system is to be solved for 120595

3

and 1205933with associated edge conditions

119906lowast

3= 0 or 120590

lowast

3= 0

Vlowast

3= 0 or 120591

lowast

1199091199103= 0

1205953= 0 or 120591

lowast

1199091199113= 0

(47)

along an 119909 = constant edge and analogous conditions along a119910 = constant edge

In (45) expressed in terms of 1205953and 120593

3 we replace 119908

2by

1205953for obtaining neutral plane deflection since contributions

of 1199082and 120595

3are one and the same in finding [119906

3 V3]

71 Supplementary Problem Because of the use of integrationconstant to satisfy face shear conditions vertical deflection ofneutral plane is equal to that of the face plane deflectionThisis physically incorrect since face plane is bounded by elasticmaterial on one side whereas neutral plane is bounded onboth sidesThis lack in interior solution is rectified by addingthe solution of a supplementary problem based on leadingcosine term that is enough in view of 119891

2(119911) distribution

of reactive shear stresses This is based on the expansion

ISRN Civil Engineering 7

of 120590119911= 1198913(119911)(32)119902 in sine series Here load condition is

satisfied by the leading sine term and coefficients of all othersine terms are zero

Corrective displacements in the supplementary problemare assumed in the form

119908 = 1199082119904

120587

2

cos(1205872

119911) 119906119904= 1199063119904sin(120587

2

119911) lArrrArr (119906 V)

1205903119904119894

= 1198761198941198951205763119904119895

(119894 119895 = 1 2 3)

(48)

We have from integration of equilibrium equations

1205912119904119904119909119911

= minus(

2

120587

)120572 [11987611198951205763119904119895119909

+ 11987631198951205763119904119895119910

] (49a)

1205912119904119910119911

= minus(

2

120587

)120572 [11987621198951205763119904119895119910

+ 11987631198951205763119904119895119909

] (49b)

In-plane distributions 1199063119904

and V3119904

are added as correctionsto the known 119906

lowast

3and Vlowast

3so that (119906 V) in the supplementary

problem are

119906 = (119906lowast

3+ 1199063119904) sin(120587

2

119911) lArrrArr (119906 V) (50)

We note that 120590119911from integration of equilibrium equations

with 119904 variables is the same as 120590119911from static equilibrium

equations with lowast variables Hence

(

2

120587

)

2

1205722[11987611198951205763119904119895119909119909

+ 211987631198951205763119904119895119909119910

+ 11987621198951205763119904119895119910119910

] + 12057331205901199115

= 0

(51)

It is convenient to use here also

[1199063119904 V3119904] = minus120572 [(120595

3119904119909minus 1206013119904119910

) (1205953119904119910

+ 1205933119904119909

)] (52)

Correction to neutral plane deflection due to solution ofsupplementary problem is from

1205761199113119904

= [11987861198951205903119904119895

+ 119878661205901199113119904

] (sum 119895 = 1 2 3) (53)

72 Summary of Results from the First Stage of IterativeProcedure The previous analysis gives displacements [119906V] =1198913[1199063 V3] consistent with [120591

119909119911 120591119910119911] = 120572119891

2[120595119909 120595119910] and

corrective transverse stresses from the first stage of iteration

119908 = 1199080minus [11989121205761199111+ 11989141205761199113] minus 1205761199111199043

120587

2

cos(1205872

119911)

119906 = 11989111199061+ 1198913119906lowast

3+ (119906lowast

3+ 1199063119904) sin(120587

2

119911) lArrrArr (119906 V)

120591119909119911

= 11989121205912119909119911

+ 1205912119904119909119911

120587

2

cos(1205872

119911) lArrrArr (119909 119910)

120590119911= 1198913(119911) 120590lowast

1199113(From (35))

(54)

It is to be noted that the dependence of transverse stresses onmaterial constants is through the solution of supplementaryproblem One may add 119891

4terms in shear components and 119891

5

term in normal stress componentThese components are alsodependent on material constants but they need correctionfrom the solution of a supplementary problem Successiveapplication of the previous iterative procedure leads to thesolution of the 3D problem in the limit

8 Concluding Remarks

Poissonrsquos theory developed in the present study for the analy-sis of bending of anisotropic plates within small deformationtheory forms the basis for generation of proper sequence of2D problems Analysis for obtaining displacements therebybending stresses along faces of the plate is different fromsolution of a supplementary problem in the interior ofthe plate A sequence of higher order shear deformationtheories lead to solution of associated torsion problem onlyIn the preliminary solution reactive transverse stresses areindependent of material constants In view of layer-wisetheory of symmetric laminated plates proposed by the presentauthor [10] Poissonrsquos theory is useful since analysis of faceplies is independent of lamination which is in confirmationof (8)-(9) in a recent NASA technical publication by Tessleret al [11]

One significant observation is that sequence of 2D prob-lems converging to 3D problems in the analysis of extensionbending and torsion problems are mutually exclusive to oneother

Highlights

(i) Poisson-Kirchhoff boundary conditions paradox isresolved

(ii) Transverse stresses are independent of material con-stants in primary solution

(iii) Edge support condition on vertical deflection has norole in the analysis

(iv) Finding neutral plane deflection requires solution ofa supplementary problem

(v) Twisting stress distribution in pure torsion nullifies itsdistribution in bending

Appendix

119886 = side length of a square plate2ℎ = plate thickness119891119899(119911)= through-thickness distribution functions 119899 =

0 1 2

119871 = characteristic length of the plate in119883-119884-plane119902(119909 119910) = applied face load density119876119894119895= stiffness coefficients

119878119894119895= elastic compliances

[119879119909 119879119909119910 119879119909119911] = prescribed stress distributions along

an 119909 = constant edge

8 ISRN Civil Engineering

[119880 119881119882] = displacements in 119883119884 119885-directionsrespectively[119906 V 119908] = [119880 119881119882]ℎ

119874 minus 119883119884119885 = Cartesian coordinate system[119909 119910 119911] = nondimensional coordinates [119883119871 119884119871119885ℎ]

120572 = ℎ119871

1205732119899+1

= [1198912119899+3

(1)12057221198912119899+1

(1)]

Δ = Laplace operator (12059721205971199092 + 12059721205971199102)

[120576119909 120576119910 120574119909119910] = in-plane strains

[120574119909119911 120574119910119911 120576119911] = transverse strains

120596119911= 120572(V119909minus 119906119910) rotation about 119911-axis

[120590119909 120590119910 120591119909119910] = bending stresses

[120591119909119911 120591119910119911 120590119911] = transverse stresses

References

[1] G Kirchhoff ldquoUber das Gleichgewicht und die Bewegung einerelastischen Scheiberdquo Journal fur die Reine und AngewandteMathematik vol 40 pp 51ndash58 1850

[2] HHencky ldquoUber die Berucksichtigung der Schubverzerrung inebenen Plattenrdquo Ingenieur-Archiv vol 16 no 1 pp 72ndash76 1947

[3] E Reissner ldquoReflections on the theory of elastic platesrdquoAppliedMechanics Reviews vol 38 no 11 pp 1453ndash1464 1985

[4] A E H Love A Treatise on Mathematical Theory of ElasticityCambridge University Press Cambridge UK 4th edition 1934

[5] K Vijayakumar ldquoModified Kirchhoff rsquos theory of plates includ-ing transverse shear deformationsrdquo Mechanics Research Com-munications vol 38 pp 211ndash213 2011

[6] K Vijayakumar ldquoNew look at kirchhoff rsquos theory of platesrdquoAIAA Journal vol 47 no 4 pp 1045ndash1046 2009

[7] K Vijayakumar ldquoA relook at Reissnerrsquos theory of plates inbendingrdquo Archive of Applied Mechanics vol 81 no 11 pp 1717ndash1724 2011

[8] E Reissner ldquoThe effect of transverse shear deformations on thebending of elastic platesrdquo Journal of Applied Mechanics vol 12pp A69ndashA77 1945

[9] T Lewinski ldquoOn the twelfth-order theory of elastic platesrdquoMechanics Research Communications vol 17 no 6 pp 375ndash3821990

[10] K Vijayakumar ldquoLayer-wise theory of bending of symmetriclaminates with isotropic pliesrdquo AIAA Journal vol 49 no 9 pp2073ndash2076 2011

[11] A Tessler M Di Sciuva and M Gherlone ldquoRefined zigzagtheory for homogeneous laminated composite and sandwichplates a homogeneous limit methodology for zigzag functionselectionrdquo Tech Rep NASATP-2920102162141 2010

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International Journal of

4 ISRN Civil Engineering

Vertical deflection 1199080is given by (13) As mentioned earlier

cross-derivatives of [1199061 V1] do not exist in (18) (19a) and

(19b) due to (14) In the isotropic case (18) are uncoupled andare simply given by

1205722Δ1199061= 1205721198901119909

lArrrArr (119909 119910) (119906 119910) (20)

Edge conditions (19a) and (19b) are also uncoupled and theyare given by

1199061= 0 or 1198641015840 (1 minus 120592) 120572119906

1119909= 119879119909(119910) lArrrArr (119909 119910) (119906 V)

V1= 0 or 2119866120572V

1119909= 119879119909119910(119910) lArrrArr (119909 119910) (119906 V)

(21)

The condition1205950= 0 in solving (17) is different from the usual

condition 1199080= 0 in the Kirchhoff rsquos theory and FSDT The

function 1205950is related to the normal strain 120576

119911 In the isotropic

case 1205950is proportional to 119890

1(119909 119910) = (120576

1199091+ 1205761199101) Gradients of

1205950are proportional to gradients of transverse strains in FSDT

If the plate is free of applied transverse stresses that is theplate is subjected to bending and twisting moments only 119890

1

is proportional toΔ1199080The function120595

0(119909 119910) thereby Δ119908

0is

identically zero from (2) (in Kirchhoff rsquos theory the tangentialgradient ofΔ119908

0is proportional to the corresponding gradient

of applied 120591119909119910

along the edge of the plate implied from Kelvinand Taitrsquos physical interpretation of the contracted transverseshear condition) This Laplace equation is not adequate tosatisfy two in-plane edge conditions One needs its conjugateharmonic functions to express in-plane displacements in theform

[1199061 V1] = minus120572119911 [119908

0119909+ 1205930119910 1199080119910

minus 1205930119909] (22)

The function 1205930was introduced earlier by Reissner [8] as a

stress function in satisfying (2) Note that the two variables1199061and V1are expressed in terms of gradients of two functions

1199080and 120593

0

After finding1199080and1205930from solving in-plane equilibrium

equations (1) correction 1199080119888

to 1199080from zero face shear

conditions is given by

1199080119888= int [120601

0119910119889119909 + 120601

0119909119889119910] (23)

One should note here that119908(119909 119910) = 1199080+1199080119888does not satisfy

prescribed edge condition on 119908In FSDT [119906

1 V1] are obtained from solving (1) There is

no provision to find 1199080and it has to be obtained from zero

face shear conditions in the form

1205721199080= minusint [119906

1119889119909 + V

1119889119910] (24)

If the applied transverse stresses are nonhomogeneous alonga segment of the edge 119908

0is coupled with [119906

1 V1] in (1) and

(2) and edge conditions (3a) (3b) and (3c)With reference to the present analysis it is relevant to

note the following observation in the preliminary solution120590119911 120572[120591119909119911 120591119910119911] 1205722[120590

119909 120590119910 120591119909119910] are of119874(1) As such estimation

of in-plane stresses thereby in-plane displacements is notdependent on 119908

0 The support condition 119908

0= 0 along the

edge of the plate does not play any role in determining thein-plane displacements

61 Comparison with Analysis of Extension Problems It isinteresting to compare the previous analysis with that ofextension problem In extension problem applied shearstresses along the faces of the plate are gradients of a potentialfunction and they are equivalent to linearly varying bodyforces independent of elastic constants in the in-planeequilibrium equations Rotation 120596

119911= 0 (but Δ120596

119911= 0) and

equations governing displacements [1199060 V0] are coupled Also

vertical deflection is given by thickness-wise integration of1205761199110from constitutive relation and higher order corrections are

dependent on this vertical deflection In the present analysisof bending problems in-plane distributions of transverseshear stresses are gradients of a function related to appliededge transverse shear and face normal load through equi-librium equation (2) They are also independent of elasticconstants and derivative 119891

2119911of parabolic distribution of each

of them is equivalent to body force in linearly varying in-plane equilibrium equations Vertical deflection 119908

0is purely

from satisfaction of zero shear stress conditions along faces ofthe plate Higher order corrections are dependent on normalstrains and independent of119908

0 It can be seen that steps in the

analysis of bending problem are complementary to those inthe classical theory of extension problem

62 Associated Torsion Problems First-order shear deforma-tion theory and higher order shear deformation theoriesdo not provide proper corrections to initial solution (fromKirchhoff rsquos theory) of primary flexure problems In thesetheories corrections are due to approximate solutions ofassociated torsion problems In fact one has to have a relookat the use of shear deformation theories based on plate(instead of 3D) element equilibrium equations other thanKirchhoff rsquos theory in the analysis of flexure problems Inthe earlier work [5] we have mentioned that this couplingof bending and torsion problems is nullified in the limit ofsatisfying all equations in the 3D problems However thesehigher order theories in the case of primary flexure problemdefined from Kirchhoff rsquos theory are with reference to findingthe exact solution of associated torsion problem only In factone can obtain exact solution of the torsion problem (insteadof using higher order polynomials in 119911 by expanding 119911 and1198913(119911) in sine series and119891

2(119911) in cosine series In a pure torsion

problem normal stresses and strains are zero implying that

[119906 V 119908] = [119906 (119910 119911) V (119909 119911) 119908 (119909 119910)] (25)

Rotation 120596119911

= 0 and warping displacements (119906 V) are inde-pendent of vertical displacement 119908 In FSDT 120596

119911= 0 and

corrections to Kirchhoff rsquos displacements are due to thesolution of approximate torsion problem In the isotropicrectangular plate warping function 119906(119910 119911) is harmonic in 119910-119911-plane It has to satisfy 119866120572119906

119910= 119879119906(119910 119911) with prescribed 119879

119906

along an 119909 = constant edge By expressing 119906 in product form119906 = 119891119906(119910)119892119906(119911) we have

1205722119891119906119910119910

119892119906+ 119891119906119892119906119911119911

= 0 (26)

ISRN Civil Engineering 5

By taking 119892119906119899

= sin 120582119899119911 where 120582

119899= (2119899 + 1)1205872 satisfying

zero face shear conditions we get

119891119906119899

= 119860119906119899cosh(

120582119899

120572

)119910 + 119861119906119899sinh(

120582119899

120572

)119910 (27)

If prescribed 119879119906is 119891119906119899

with specified constants (119860119906119899 119861119906119899)

clearly there is no provision to satisfy zero in-plane shearcondition along 119910 = constant edges It has to be nullifiedwith corresponding solution from bending problem That iswhy a torsion problem is associated with bending problembut not vice versa In a corresponding bending problem allstress components are zero except 120591

119909119910= minus119891

119906119899(119910)119892119906119899(119911)

This solution is used only in satisfying edge condition along119910 = constant edges in the presence of specified 119879

119906119899along

119909 = constant edges It shows that 120591119909119910

distribution in flexureproblem is nullified in the limit in shear deformation theoriesdue to torsion

It is interesting to note that the previous mentioneddeficiency due to coupling with 119908

0in the plate element

equilibrium equations in FSDT and higher order sheardeformation theories does not exist if applied 120591

119909119910is zero all

along the closed boundary of the plate It is complementaryto the fact that boundary condition paradox in Kirchhoff rsquostheory does not exist if tangential displacement and 119908

0are

zero all along the boundary of the plate

63 Illustrative Example Simply Supported Isotropic SquarePlate Consider a simply supported isotropic square platesubjected to vertical load

119902 = 1199020sin(120587119909

119886

) sin(120587119910

119886

) (28)

Exact values for neutral plane and face deflections obtainedearlier [7] with ] = 03 120572 = (16) are

(

119864

21199020

)119908(

119886

2

119886

2

0) = 4487 (29)

(

119864

21199020

)119908(

119886

2

119886

2

1) = 4166 (30)

In this example Poisson-Kirchhoff rsquos boundary conditionsparadox does not exist Hence Poissonrsquos theory gives thesame solution obtained in the authorrsquos previous mentionedwork Concerning face value in (30) Kirchhoff rsquos theory givesa value of 227 that is same for all face parallel planes Poissonrsquostheory gives an additional value of 0267 attributable to 120576

1199111

and it is same for neutral plane deflection since119908(119909 119910 119911) canbe expressed as

119908 (119909 119910 119911) = 1199080119865

(119909 119910) + 1198912(119911) 1199082

= 1199080119873

(119909 119910) minus (

1

2

) 11991121205761199111

(31)

In the previous expression 1199080119873

is deflection of the neutralplane Higher order correction to119908

0uncoupled from torsion

is 1262 so that total correction to the value from Kirchhoff rsquostheory is about 153 [7] Correction due to coupling with

torsion is 145 [5] whereas it is 1423 from FSDT and othersixth order theories [9] With reference to numerical valuesreported by Lewinski [9] the previous correction is 1412 inReddyrsquos 8th-order theory and less than 123 in Reissnerrsquos 12th-order and other higher order theories It clearly shows thatthese shear deformation theories do not lead to solutions ofbending problems (note that the value (227 + 1423) fromFSDT corresponds to neutral plane deflection and it is inerror by about 177 from the exact value)

In view of the previous observations it is relevant tomakethe following remarks physical validity of 3D equations isconstrained by limitations of small deformation theory For120572 = 1 this constraint is dependent on material constantsgeometry of the domain and applied loads Range of validityof 2D plate theories arises for small values of 120572 Kirchhoff rsquostheory gives lower bound for this 120572 For this value of 120572 and forslightly higher values of 120572 than this lower bound Kirchhoff rsquostheory is used in spite of its known deficiencies due to itssimplicity to obtain design information Similarly FSDT andother sixth-order shear deformation theories are used forsome additional range of values of 120572 due to the correctionsover Kirchhoff rsquos theory though these corrections are due tocoupling with approximate torsion problemsThey serve onlyin giving guidance values for applicable design parametersfor small range of values of 120572 beyond its lower boundThey do not provide initial set of equations in formulatingproper sequence of sets of 2D problems converging to the 3Dproblem As such comparison of solutions from Kirchhofftheory and FSDT with the corresponding solutions in thepresent analysis does not serve much purpose

Since 1199080119873

has to be higher than 1199080119865 one has to obtain

the correction to 1199080119873

before finding correction to internaldistribution of119908(119909 119910 119911) along with correction to119908

0119865 Initial

correction to 1199080119873

is obtained earlier [7] in the form

119908 = 1199080minus [1198912(0) 1205761199111+ 1198914(0) 1205761199113] (32)

It is the same for face deflection (note that 1199082in the

previouslymentioned authorrsquos work is required only to obtain1205761199113) As such 119908

0119873is further corrected from solution of

a supplementary problem This total correction over facedeflection is about 0658 giving a value of 4458 which is veryclose to the exact value 4487 in (29) However we note thatthe error in the estimated value (=38) of 119908

0119865is relatively

high compared to the accuracy achieved in the neutral planedeflection It is possible to improve estimation of 119908

0119865by

including 1205761199111in (1199063 V3) such that (120591

1199091199112 1205911199101199112

) are independentof 1205761199111 In the present example correction to face deflection

changes to

(

119864

21199020

)119908face max =((65) ((1 + 120592) (1 minus 120592))) 120573

2

((25) ((4 minus 120592) (1 minus 120592))) 1205732minus 1

(33)

In the previous equation 1205732 = 2119908011986512057221205872 Correction 120576

1199113(=

1262) to face value changes to 1431 giving 397 (= 227 +

0267 + 1431) for face deflection which is under 47 fromexact value

6 ISRN Civil Engineering

7 Anisotropic PlateSecond-Order Corrections

We note that 1205761199111does not participate in the determination of

(1199061 V1) and it is obtained from the constitutive relation (6) in

the interior of the plate It is zero along an edge of the plateif 119908 = 0 is specified condition With zero transverse shearstrains specification of 120576

1199111= 0 instead of 119908

0= 0 is more

appropriate along a supported edge since 1199080

= 0 implieszero tangential displacement along a straight edge which isthe root cause of Poisson-Kirchhoff rsquos boundary conditionsparadox As such edge support condition on w does not playany role in obtaining the in-plane displacements and reactivetransverse stresses

In the first stage of iterative procedure second-orderreactive transverse stresses have to be obtained by consid-ering higher order in-plane displacement terms 119906

3and V

3

However one has to consider limitation in the previousPoissonrsquos theory like in Kirchhoff rsquos theory namely reactive120590119911is zero at locations of zeros of 119902 It is identically zero in

higher order approximations since 1198912119899+1

(119911) functions are notzero along 119911 = plusmn1 faces Moreover it is necessary to accountfor its dependence on material constants To overcome theselimitations it is necessary to keep 120590

1199115as a free variable by

modifying 1198915in the form

119891lowast

5(119911) = 119891

5 (119911) minus 120573

31198913 (119911) (34)

In the previous equation 1205733= [1198915(1)1198913(1)] so that119891lowast

5(plusmn1) =

0 Denoting coefficient of 1198913in 120590119911by 120590lowast1199113 it becomes

120590lowast

1199113= 1205901199113minus 12057331205901199115 (35)

Displacements (1199063 V3) are modified such that they are cor-

rections to face parallel plane distributions of the preliminarysolution and are free to obtain reactive stresses 120591

1199091199114 1205911199101199114

1205901199115

and normal strain 1205761199113

We have from constitutive relations

1205741199091199112

= 119878441205911199091199112

+ 119878451205911199101199112

lArrrArr (119909 119910) (4 5) (36)

Modified displacements and the corresponding derivedquantities denoted with lowast are

119906lowast

3= 1199063minus 1205721199082119909

+ 1205741199091199112

lArrrArr (119909 119910) (119906 V) (37)

Strain-displacement relations give

120576lowast

1199093= 1205761199093

minus 12057221199082119909119909

+ 1205721205741199091199112119909

lArrrArr (119909 119910)

120574lowast

1199091199103= 1205741199091199103

minus 212057221199082119909119910

+ 120572 (1205741199091199112119910

+ 1205741199101199112119909

)

120574lowast

1199091199113= 1199063+ 1205741199091199112

lArrrArr (119909 119910) (119906 V)

(38)

In-plane and transverse shear stresses from constitutiverelations are

120590lowast

119894(3)= 119876119894119895120576lowast

119895(3)(119894 119895 = 1 2 3) (39)

120591lowast

1199091199112= 119876441199063+ 11987645V3+ 1205911199091199112

lArrrArr (119909 119910) (4 5) (40)

Suffix (3) in (39) is to indicate that the stresses and strainscorrespond to 119906

3and V3

We get from (2) (15) (35) (37) and (40) with referenceto coefficient of 119891

3(119911)

120572 [(119876441199063+ 11987645V3)119909

+ (119876541199063+ 11987655V3)119910

] = 12057331205901199115 (41)

Note that 1199082is not present in the previous equation

From integration of equilibrium equations we have withsum on 119895 = 1 2 3

1205911199091199114

= 120572[1198761119895120576lowast

119895119909+ 1198763119895120576lowast

119895119910](3)

1205911199101199114

= 120572[1198762119895120576lowast

119895119910+ 1198763119895120576lowast

119895119909](3)

(42)

Note that 1199082is present in the previous expression

120572 (1205911199091199114119909

+ 1205911199101199114119910

) + 1205901199115

= 0 (Coefficient of 1198915) (43)

1205761199113

= [1198786119895120590119895+ 11987866120590119911](3) (44)

One gets one equation governing in-plane displacements(1199063 V3) from (41) (43) noting that 119891

5119911119911+1198913= 0 in the form

1205733(1205911199091199114119909

+ 1205911199101199114119910

)

= 1205722[(119876441199063+ 11987645V3)119909119909

+ (119876541199063+ 11987655V3)119910119910

]

(45)

The second equation governing these variables is from thecondition 120596

119911= 0 Here it is more convenient to express

(1199063 V3) as

[1199063 V3] = minus120572 [(120595

3119909minus 1206013119910) (1205953119910

+ 1206013119909)] (46)

so that Δ1205933= 0 and (45) becomes a fourth-order equation

governing 1205953 This sixth-order system is to be solved for 120595

3

and 1205933with associated edge conditions

119906lowast

3= 0 or 120590

lowast

3= 0

Vlowast

3= 0 or 120591

lowast

1199091199103= 0

1205953= 0 or 120591

lowast

1199091199113= 0

(47)

along an 119909 = constant edge and analogous conditions along a119910 = constant edge

In (45) expressed in terms of 1205953and 120593

3 we replace 119908

2by

1205953for obtaining neutral plane deflection since contributions

of 1199082and 120595

3are one and the same in finding [119906

3 V3]

71 Supplementary Problem Because of the use of integrationconstant to satisfy face shear conditions vertical deflection ofneutral plane is equal to that of the face plane deflectionThisis physically incorrect since face plane is bounded by elasticmaterial on one side whereas neutral plane is bounded onboth sidesThis lack in interior solution is rectified by addingthe solution of a supplementary problem based on leadingcosine term that is enough in view of 119891

2(119911) distribution

of reactive shear stresses This is based on the expansion

ISRN Civil Engineering 7

of 120590119911= 1198913(119911)(32)119902 in sine series Here load condition is

satisfied by the leading sine term and coefficients of all othersine terms are zero

Corrective displacements in the supplementary problemare assumed in the form

119908 = 1199082119904

120587

2

cos(1205872

119911) 119906119904= 1199063119904sin(120587

2

119911) lArrrArr (119906 V)

1205903119904119894

= 1198761198941198951205763119904119895

(119894 119895 = 1 2 3)

(48)

We have from integration of equilibrium equations

1205912119904119904119909119911

= minus(

2

120587

)120572 [11987611198951205763119904119895119909

+ 11987631198951205763119904119895119910

] (49a)

1205912119904119910119911

= minus(

2

120587

)120572 [11987621198951205763119904119895119910

+ 11987631198951205763119904119895119909

] (49b)

In-plane distributions 1199063119904

and V3119904

are added as correctionsto the known 119906

lowast

3and Vlowast

3so that (119906 V) in the supplementary

problem are

119906 = (119906lowast

3+ 1199063119904) sin(120587

2

119911) lArrrArr (119906 V) (50)

We note that 120590119911from integration of equilibrium equations

with 119904 variables is the same as 120590119911from static equilibrium

equations with lowast variables Hence

(

2

120587

)

2

1205722[11987611198951205763119904119895119909119909

+ 211987631198951205763119904119895119909119910

+ 11987621198951205763119904119895119910119910

] + 12057331205901199115

= 0

(51)

It is convenient to use here also

[1199063119904 V3119904] = minus120572 [(120595

3119904119909minus 1206013119904119910

) (1205953119904119910

+ 1205933119904119909

)] (52)

Correction to neutral plane deflection due to solution ofsupplementary problem is from

1205761199113119904

= [11987861198951205903119904119895

+ 119878661205901199113119904

] (sum 119895 = 1 2 3) (53)

72 Summary of Results from the First Stage of IterativeProcedure The previous analysis gives displacements [119906V] =1198913[1199063 V3] consistent with [120591

119909119911 120591119910119911] = 120572119891

2[120595119909 120595119910] and

corrective transverse stresses from the first stage of iteration

119908 = 1199080minus [11989121205761199111+ 11989141205761199113] minus 1205761199111199043

120587

2

cos(1205872

119911)

119906 = 11989111199061+ 1198913119906lowast

3+ (119906lowast

3+ 1199063119904) sin(120587

2

119911) lArrrArr (119906 V)

120591119909119911

= 11989121205912119909119911

+ 1205912119904119909119911

120587

2

cos(1205872

119911) lArrrArr (119909 119910)

120590119911= 1198913(119911) 120590lowast

1199113(From (35))

(54)

It is to be noted that the dependence of transverse stresses onmaterial constants is through the solution of supplementaryproblem One may add 119891

4terms in shear components and 119891

5

term in normal stress componentThese components are alsodependent on material constants but they need correctionfrom the solution of a supplementary problem Successiveapplication of the previous iterative procedure leads to thesolution of the 3D problem in the limit

8 Concluding Remarks

Poissonrsquos theory developed in the present study for the analy-sis of bending of anisotropic plates within small deformationtheory forms the basis for generation of proper sequence of2D problems Analysis for obtaining displacements therebybending stresses along faces of the plate is different fromsolution of a supplementary problem in the interior ofthe plate A sequence of higher order shear deformationtheories lead to solution of associated torsion problem onlyIn the preliminary solution reactive transverse stresses areindependent of material constants In view of layer-wisetheory of symmetric laminated plates proposed by the presentauthor [10] Poissonrsquos theory is useful since analysis of faceplies is independent of lamination which is in confirmationof (8)-(9) in a recent NASA technical publication by Tessleret al [11]

One significant observation is that sequence of 2D prob-lems converging to 3D problems in the analysis of extensionbending and torsion problems are mutually exclusive to oneother

Highlights

(i) Poisson-Kirchhoff boundary conditions paradox isresolved

(ii) Transverse stresses are independent of material con-stants in primary solution

(iii) Edge support condition on vertical deflection has norole in the analysis

(iv) Finding neutral plane deflection requires solution ofa supplementary problem

(v) Twisting stress distribution in pure torsion nullifies itsdistribution in bending

Appendix

119886 = side length of a square plate2ℎ = plate thickness119891119899(119911)= through-thickness distribution functions 119899 =

0 1 2

119871 = characteristic length of the plate in119883-119884-plane119902(119909 119910) = applied face load density119876119894119895= stiffness coefficients

119878119894119895= elastic compliances

[119879119909 119879119909119910 119879119909119911] = prescribed stress distributions along

an 119909 = constant edge

8 ISRN Civil Engineering

[119880 119881119882] = displacements in 119883119884 119885-directionsrespectively[119906 V 119908] = [119880 119881119882]ℎ

119874 minus 119883119884119885 = Cartesian coordinate system[119909 119910 119911] = nondimensional coordinates [119883119871 119884119871119885ℎ]

120572 = ℎ119871

1205732119899+1

= [1198912119899+3

(1)12057221198912119899+1

(1)]

Δ = Laplace operator (12059721205971199092 + 12059721205971199102)

[120576119909 120576119910 120574119909119910] = in-plane strains

[120574119909119911 120574119910119911 120576119911] = transverse strains

120596119911= 120572(V119909minus 119906119910) rotation about 119911-axis

[120590119909 120590119910 120591119909119910] = bending stresses

[120591119909119911 120591119910119911 120590119911] = transverse stresses

References

[1] G Kirchhoff ldquoUber das Gleichgewicht und die Bewegung einerelastischen Scheiberdquo Journal fur die Reine und AngewandteMathematik vol 40 pp 51ndash58 1850

[2] HHencky ldquoUber die Berucksichtigung der Schubverzerrung inebenen Plattenrdquo Ingenieur-Archiv vol 16 no 1 pp 72ndash76 1947

[3] E Reissner ldquoReflections on the theory of elastic platesrdquoAppliedMechanics Reviews vol 38 no 11 pp 1453ndash1464 1985

[4] A E H Love A Treatise on Mathematical Theory of ElasticityCambridge University Press Cambridge UK 4th edition 1934

[5] K Vijayakumar ldquoModified Kirchhoff rsquos theory of plates includ-ing transverse shear deformationsrdquo Mechanics Research Com-munications vol 38 pp 211ndash213 2011

[6] K Vijayakumar ldquoNew look at kirchhoff rsquos theory of platesrdquoAIAA Journal vol 47 no 4 pp 1045ndash1046 2009

[7] K Vijayakumar ldquoA relook at Reissnerrsquos theory of plates inbendingrdquo Archive of Applied Mechanics vol 81 no 11 pp 1717ndash1724 2011

[8] E Reissner ldquoThe effect of transverse shear deformations on thebending of elastic platesrdquo Journal of Applied Mechanics vol 12pp A69ndashA77 1945

[9] T Lewinski ldquoOn the twelfth-order theory of elastic platesrdquoMechanics Research Communications vol 17 no 6 pp 375ndash3821990

[10] K Vijayakumar ldquoLayer-wise theory of bending of symmetriclaminates with isotropic pliesrdquo AIAA Journal vol 49 no 9 pp2073ndash2076 2011

[11] A Tessler M Di Sciuva and M Gherlone ldquoRefined zigzagtheory for homogeneous laminated composite and sandwichplates a homogeneous limit methodology for zigzag functionselectionrdquo Tech Rep NASATP-2920102162141 2010

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International Journal of

ISRN Civil Engineering 5

By taking 119892119906119899

= sin 120582119899119911 where 120582

119899= (2119899 + 1)1205872 satisfying

zero face shear conditions we get

119891119906119899

= 119860119906119899cosh(

120582119899

120572

)119910 + 119861119906119899sinh(

120582119899

120572

)119910 (27)

If prescribed 119879119906is 119891119906119899

with specified constants (119860119906119899 119861119906119899)

clearly there is no provision to satisfy zero in-plane shearcondition along 119910 = constant edges It has to be nullifiedwith corresponding solution from bending problem That iswhy a torsion problem is associated with bending problembut not vice versa In a corresponding bending problem allstress components are zero except 120591

119909119910= minus119891

119906119899(119910)119892119906119899(119911)

This solution is used only in satisfying edge condition along119910 = constant edges in the presence of specified 119879

119906119899along

119909 = constant edges It shows that 120591119909119910

distribution in flexureproblem is nullified in the limit in shear deformation theoriesdue to torsion

It is interesting to note that the previous mentioneddeficiency due to coupling with 119908

0in the plate element

equilibrium equations in FSDT and higher order sheardeformation theories does not exist if applied 120591

119909119910is zero all

along the closed boundary of the plate It is complementaryto the fact that boundary condition paradox in Kirchhoff rsquostheory does not exist if tangential displacement and 119908

0are

zero all along the boundary of the plate

63 Illustrative Example Simply Supported Isotropic SquarePlate Consider a simply supported isotropic square platesubjected to vertical load

119902 = 1199020sin(120587119909

119886

) sin(120587119910

119886

) (28)

Exact values for neutral plane and face deflections obtainedearlier [7] with ] = 03 120572 = (16) are

(

119864

21199020

)119908(

119886

2

119886

2

0) = 4487 (29)

(

119864

21199020

)119908(

119886

2

119886

2

1) = 4166 (30)

In this example Poisson-Kirchhoff rsquos boundary conditionsparadox does not exist Hence Poissonrsquos theory gives thesame solution obtained in the authorrsquos previous mentionedwork Concerning face value in (30) Kirchhoff rsquos theory givesa value of 227 that is same for all face parallel planes Poissonrsquostheory gives an additional value of 0267 attributable to 120576

1199111

and it is same for neutral plane deflection since119908(119909 119910 119911) canbe expressed as

119908 (119909 119910 119911) = 1199080119865

(119909 119910) + 1198912(119911) 1199082

= 1199080119873

(119909 119910) minus (

1

2

) 11991121205761199111

(31)

In the previous expression 1199080119873

is deflection of the neutralplane Higher order correction to119908

0uncoupled from torsion

is 1262 so that total correction to the value from Kirchhoff rsquostheory is about 153 [7] Correction due to coupling with

torsion is 145 [5] whereas it is 1423 from FSDT and othersixth order theories [9] With reference to numerical valuesreported by Lewinski [9] the previous correction is 1412 inReddyrsquos 8th-order theory and less than 123 in Reissnerrsquos 12th-order and other higher order theories It clearly shows thatthese shear deformation theories do not lead to solutions ofbending problems (note that the value (227 + 1423) fromFSDT corresponds to neutral plane deflection and it is inerror by about 177 from the exact value)

In view of the previous observations it is relevant tomakethe following remarks physical validity of 3D equations isconstrained by limitations of small deformation theory For120572 = 1 this constraint is dependent on material constantsgeometry of the domain and applied loads Range of validityof 2D plate theories arises for small values of 120572 Kirchhoff rsquostheory gives lower bound for this 120572 For this value of 120572 and forslightly higher values of 120572 than this lower bound Kirchhoff rsquostheory is used in spite of its known deficiencies due to itssimplicity to obtain design information Similarly FSDT andother sixth-order shear deformation theories are used forsome additional range of values of 120572 due to the correctionsover Kirchhoff rsquos theory though these corrections are due tocoupling with approximate torsion problemsThey serve onlyin giving guidance values for applicable design parametersfor small range of values of 120572 beyond its lower boundThey do not provide initial set of equations in formulatingproper sequence of sets of 2D problems converging to the 3Dproblem As such comparison of solutions from Kirchhofftheory and FSDT with the corresponding solutions in thepresent analysis does not serve much purpose

Since 1199080119873

has to be higher than 1199080119865 one has to obtain

the correction to 1199080119873

before finding correction to internaldistribution of119908(119909 119910 119911) along with correction to119908

0119865 Initial

correction to 1199080119873

is obtained earlier [7] in the form

119908 = 1199080minus [1198912(0) 1205761199111+ 1198914(0) 1205761199113] (32)

It is the same for face deflection (note that 1199082in the

previouslymentioned authorrsquos work is required only to obtain1205761199113) As such 119908

0119873is further corrected from solution of

a supplementary problem This total correction over facedeflection is about 0658 giving a value of 4458 which is veryclose to the exact value 4487 in (29) However we note thatthe error in the estimated value (=38) of 119908

0119865is relatively

high compared to the accuracy achieved in the neutral planedeflection It is possible to improve estimation of 119908

0119865by

including 1205761199111in (1199063 V3) such that (120591

1199091199112 1205911199101199112

) are independentof 1205761199111 In the present example correction to face deflection

changes to

(

119864

21199020

)119908face max =((65) ((1 + 120592) (1 minus 120592))) 120573

2

((25) ((4 minus 120592) (1 minus 120592))) 1205732minus 1

(33)

In the previous equation 1205732 = 2119908011986512057221205872 Correction 120576

1199113(=

1262) to face value changes to 1431 giving 397 (= 227 +

0267 + 1431) for face deflection which is under 47 fromexact value

6 ISRN Civil Engineering

7 Anisotropic PlateSecond-Order Corrections

We note that 1205761199111does not participate in the determination of

(1199061 V1) and it is obtained from the constitutive relation (6) in

the interior of the plate It is zero along an edge of the plateif 119908 = 0 is specified condition With zero transverse shearstrains specification of 120576

1199111= 0 instead of 119908

0= 0 is more

appropriate along a supported edge since 1199080

= 0 implieszero tangential displacement along a straight edge which isthe root cause of Poisson-Kirchhoff rsquos boundary conditionsparadox As such edge support condition on w does not playany role in obtaining the in-plane displacements and reactivetransverse stresses

In the first stage of iterative procedure second-orderreactive transverse stresses have to be obtained by consid-ering higher order in-plane displacement terms 119906

3and V

3

However one has to consider limitation in the previousPoissonrsquos theory like in Kirchhoff rsquos theory namely reactive120590119911is zero at locations of zeros of 119902 It is identically zero in

higher order approximations since 1198912119899+1

(119911) functions are notzero along 119911 = plusmn1 faces Moreover it is necessary to accountfor its dependence on material constants To overcome theselimitations it is necessary to keep 120590

1199115as a free variable by

modifying 1198915in the form

119891lowast

5(119911) = 119891

5 (119911) minus 120573

31198913 (119911) (34)

In the previous equation 1205733= [1198915(1)1198913(1)] so that119891lowast

5(plusmn1) =

0 Denoting coefficient of 1198913in 120590119911by 120590lowast1199113 it becomes

120590lowast

1199113= 1205901199113minus 12057331205901199115 (35)

Displacements (1199063 V3) are modified such that they are cor-

rections to face parallel plane distributions of the preliminarysolution and are free to obtain reactive stresses 120591

1199091199114 1205911199101199114

1205901199115

and normal strain 1205761199113

We have from constitutive relations

1205741199091199112

= 119878441205911199091199112

+ 119878451205911199101199112

lArrrArr (119909 119910) (4 5) (36)

Modified displacements and the corresponding derivedquantities denoted with lowast are

119906lowast

3= 1199063minus 1205721199082119909

+ 1205741199091199112

lArrrArr (119909 119910) (119906 V) (37)

Strain-displacement relations give

120576lowast

1199093= 1205761199093

minus 12057221199082119909119909

+ 1205721205741199091199112119909

lArrrArr (119909 119910)

120574lowast

1199091199103= 1205741199091199103

minus 212057221199082119909119910

+ 120572 (1205741199091199112119910

+ 1205741199101199112119909

)

120574lowast

1199091199113= 1199063+ 1205741199091199112

lArrrArr (119909 119910) (119906 V)

(38)

In-plane and transverse shear stresses from constitutiverelations are

120590lowast

119894(3)= 119876119894119895120576lowast

119895(3)(119894 119895 = 1 2 3) (39)

120591lowast

1199091199112= 119876441199063+ 11987645V3+ 1205911199091199112

lArrrArr (119909 119910) (4 5) (40)

Suffix (3) in (39) is to indicate that the stresses and strainscorrespond to 119906

3and V3

We get from (2) (15) (35) (37) and (40) with referenceto coefficient of 119891

3(119911)

120572 [(119876441199063+ 11987645V3)119909

+ (119876541199063+ 11987655V3)119910

] = 12057331205901199115 (41)

Note that 1199082is not present in the previous equation

From integration of equilibrium equations we have withsum on 119895 = 1 2 3

1205911199091199114

= 120572[1198761119895120576lowast

119895119909+ 1198763119895120576lowast

119895119910](3)

1205911199101199114

= 120572[1198762119895120576lowast

119895119910+ 1198763119895120576lowast

119895119909](3)

(42)

Note that 1199082is present in the previous expression

120572 (1205911199091199114119909

+ 1205911199101199114119910

) + 1205901199115

= 0 (Coefficient of 1198915) (43)

1205761199113

= [1198786119895120590119895+ 11987866120590119911](3) (44)

One gets one equation governing in-plane displacements(1199063 V3) from (41) (43) noting that 119891

5119911119911+1198913= 0 in the form

1205733(1205911199091199114119909

+ 1205911199101199114119910

)

= 1205722[(119876441199063+ 11987645V3)119909119909

+ (119876541199063+ 11987655V3)119910119910

]

(45)

The second equation governing these variables is from thecondition 120596

119911= 0 Here it is more convenient to express

(1199063 V3) as

[1199063 V3] = minus120572 [(120595

3119909minus 1206013119910) (1205953119910

+ 1206013119909)] (46)

so that Δ1205933= 0 and (45) becomes a fourth-order equation

governing 1205953 This sixth-order system is to be solved for 120595

3

and 1205933with associated edge conditions

119906lowast

3= 0 or 120590

lowast

3= 0

Vlowast

3= 0 or 120591

lowast

1199091199103= 0

1205953= 0 or 120591

lowast

1199091199113= 0

(47)

along an 119909 = constant edge and analogous conditions along a119910 = constant edge

In (45) expressed in terms of 1205953and 120593

3 we replace 119908

2by

1205953for obtaining neutral plane deflection since contributions

of 1199082and 120595

3are one and the same in finding [119906

3 V3]

71 Supplementary Problem Because of the use of integrationconstant to satisfy face shear conditions vertical deflection ofneutral plane is equal to that of the face plane deflectionThisis physically incorrect since face plane is bounded by elasticmaterial on one side whereas neutral plane is bounded onboth sidesThis lack in interior solution is rectified by addingthe solution of a supplementary problem based on leadingcosine term that is enough in view of 119891

2(119911) distribution

of reactive shear stresses This is based on the expansion

ISRN Civil Engineering 7

of 120590119911= 1198913(119911)(32)119902 in sine series Here load condition is

satisfied by the leading sine term and coefficients of all othersine terms are zero

Corrective displacements in the supplementary problemare assumed in the form

119908 = 1199082119904

120587

2

cos(1205872

119911) 119906119904= 1199063119904sin(120587

2

119911) lArrrArr (119906 V)

1205903119904119894

= 1198761198941198951205763119904119895

(119894 119895 = 1 2 3)

(48)

We have from integration of equilibrium equations

1205912119904119904119909119911

= minus(

2

120587

)120572 [11987611198951205763119904119895119909

+ 11987631198951205763119904119895119910

] (49a)

1205912119904119910119911

= minus(

2

120587

)120572 [11987621198951205763119904119895119910

+ 11987631198951205763119904119895119909

] (49b)

In-plane distributions 1199063119904

and V3119904

are added as correctionsto the known 119906

lowast

3and Vlowast

3so that (119906 V) in the supplementary

problem are

119906 = (119906lowast

3+ 1199063119904) sin(120587

2

119911) lArrrArr (119906 V) (50)

We note that 120590119911from integration of equilibrium equations

with 119904 variables is the same as 120590119911from static equilibrium

equations with lowast variables Hence

(

2

120587

)

2

1205722[11987611198951205763119904119895119909119909

+ 211987631198951205763119904119895119909119910

+ 11987621198951205763119904119895119910119910

] + 12057331205901199115

= 0

(51)

It is convenient to use here also

[1199063119904 V3119904] = minus120572 [(120595

3119904119909minus 1206013119904119910

) (1205953119904119910

+ 1205933119904119909

)] (52)

Correction to neutral plane deflection due to solution ofsupplementary problem is from

1205761199113119904

= [11987861198951205903119904119895

+ 119878661205901199113119904

] (sum 119895 = 1 2 3) (53)

72 Summary of Results from the First Stage of IterativeProcedure The previous analysis gives displacements [119906V] =1198913[1199063 V3] consistent with [120591

119909119911 120591119910119911] = 120572119891

2[120595119909 120595119910] and

corrective transverse stresses from the first stage of iteration

119908 = 1199080minus [11989121205761199111+ 11989141205761199113] minus 1205761199111199043

120587

2

cos(1205872

119911)

119906 = 11989111199061+ 1198913119906lowast

3+ (119906lowast

3+ 1199063119904) sin(120587

2

119911) lArrrArr (119906 V)

120591119909119911

= 11989121205912119909119911

+ 1205912119904119909119911

120587

2

cos(1205872

119911) lArrrArr (119909 119910)

120590119911= 1198913(119911) 120590lowast

1199113(From (35))

(54)

It is to be noted that the dependence of transverse stresses onmaterial constants is through the solution of supplementaryproblem One may add 119891

4terms in shear components and 119891

5

term in normal stress componentThese components are alsodependent on material constants but they need correctionfrom the solution of a supplementary problem Successiveapplication of the previous iterative procedure leads to thesolution of the 3D problem in the limit

8 Concluding Remarks

Poissonrsquos theory developed in the present study for the analy-sis of bending of anisotropic plates within small deformationtheory forms the basis for generation of proper sequence of2D problems Analysis for obtaining displacements therebybending stresses along faces of the plate is different fromsolution of a supplementary problem in the interior ofthe plate A sequence of higher order shear deformationtheories lead to solution of associated torsion problem onlyIn the preliminary solution reactive transverse stresses areindependent of material constants In view of layer-wisetheory of symmetric laminated plates proposed by the presentauthor [10] Poissonrsquos theory is useful since analysis of faceplies is independent of lamination which is in confirmationof (8)-(9) in a recent NASA technical publication by Tessleret al [11]

One significant observation is that sequence of 2D prob-lems converging to 3D problems in the analysis of extensionbending and torsion problems are mutually exclusive to oneother

Highlights

(i) Poisson-Kirchhoff boundary conditions paradox isresolved

(ii) Transverse stresses are independent of material con-stants in primary solution

(iii) Edge support condition on vertical deflection has norole in the analysis

(iv) Finding neutral plane deflection requires solution ofa supplementary problem

(v) Twisting stress distribution in pure torsion nullifies itsdistribution in bending

Appendix

119886 = side length of a square plate2ℎ = plate thickness119891119899(119911)= through-thickness distribution functions 119899 =

0 1 2

119871 = characteristic length of the plate in119883-119884-plane119902(119909 119910) = applied face load density119876119894119895= stiffness coefficients

119878119894119895= elastic compliances

[119879119909 119879119909119910 119879119909119911] = prescribed stress distributions along

an 119909 = constant edge

8 ISRN Civil Engineering

[119880 119881119882] = displacements in 119883119884 119885-directionsrespectively[119906 V 119908] = [119880 119881119882]ℎ

119874 minus 119883119884119885 = Cartesian coordinate system[119909 119910 119911] = nondimensional coordinates [119883119871 119884119871119885ℎ]

120572 = ℎ119871

1205732119899+1

= [1198912119899+3

(1)12057221198912119899+1

(1)]

Δ = Laplace operator (12059721205971199092 + 12059721205971199102)

[120576119909 120576119910 120574119909119910] = in-plane strains

[120574119909119911 120574119910119911 120576119911] = transverse strains

120596119911= 120572(V119909minus 119906119910) rotation about 119911-axis

[120590119909 120590119910 120591119909119910] = bending stresses

[120591119909119911 120591119910119911 120590119911] = transverse stresses

References

[1] G Kirchhoff ldquoUber das Gleichgewicht und die Bewegung einerelastischen Scheiberdquo Journal fur die Reine und AngewandteMathematik vol 40 pp 51ndash58 1850

[2] HHencky ldquoUber die Berucksichtigung der Schubverzerrung inebenen Plattenrdquo Ingenieur-Archiv vol 16 no 1 pp 72ndash76 1947

[3] E Reissner ldquoReflections on the theory of elastic platesrdquoAppliedMechanics Reviews vol 38 no 11 pp 1453ndash1464 1985

[4] A E H Love A Treatise on Mathematical Theory of ElasticityCambridge University Press Cambridge UK 4th edition 1934

[5] K Vijayakumar ldquoModified Kirchhoff rsquos theory of plates includ-ing transverse shear deformationsrdquo Mechanics Research Com-munications vol 38 pp 211ndash213 2011

[6] K Vijayakumar ldquoNew look at kirchhoff rsquos theory of platesrdquoAIAA Journal vol 47 no 4 pp 1045ndash1046 2009

[7] K Vijayakumar ldquoA relook at Reissnerrsquos theory of plates inbendingrdquo Archive of Applied Mechanics vol 81 no 11 pp 1717ndash1724 2011

[8] E Reissner ldquoThe effect of transverse shear deformations on thebending of elastic platesrdquo Journal of Applied Mechanics vol 12pp A69ndashA77 1945

[9] T Lewinski ldquoOn the twelfth-order theory of elastic platesrdquoMechanics Research Communications vol 17 no 6 pp 375ndash3821990

[10] K Vijayakumar ldquoLayer-wise theory of bending of symmetriclaminates with isotropic pliesrdquo AIAA Journal vol 49 no 9 pp2073ndash2076 2011

[11] A Tessler M Di Sciuva and M Gherlone ldquoRefined zigzagtheory for homogeneous laminated composite and sandwichplates a homogeneous limit methodology for zigzag functionselectionrdquo Tech Rep NASATP-2920102162141 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 ISRN Civil Engineering

7 Anisotropic PlateSecond-Order Corrections

We note that 1205761199111does not participate in the determination of

(1199061 V1) and it is obtained from the constitutive relation (6) in

the interior of the plate It is zero along an edge of the plateif 119908 = 0 is specified condition With zero transverse shearstrains specification of 120576

1199111= 0 instead of 119908

0= 0 is more

appropriate along a supported edge since 1199080

= 0 implieszero tangential displacement along a straight edge which isthe root cause of Poisson-Kirchhoff rsquos boundary conditionsparadox As such edge support condition on w does not playany role in obtaining the in-plane displacements and reactivetransverse stresses

In the first stage of iterative procedure second-orderreactive transverse stresses have to be obtained by consid-ering higher order in-plane displacement terms 119906

3and V

3

However one has to consider limitation in the previousPoissonrsquos theory like in Kirchhoff rsquos theory namely reactive120590119911is zero at locations of zeros of 119902 It is identically zero in

higher order approximations since 1198912119899+1

(119911) functions are notzero along 119911 = plusmn1 faces Moreover it is necessary to accountfor its dependence on material constants To overcome theselimitations it is necessary to keep 120590

1199115as a free variable by

modifying 1198915in the form

119891lowast

5(119911) = 119891

5 (119911) minus 120573

31198913 (119911) (34)

In the previous equation 1205733= [1198915(1)1198913(1)] so that119891lowast

5(plusmn1) =

0 Denoting coefficient of 1198913in 120590119911by 120590lowast1199113 it becomes

120590lowast

1199113= 1205901199113minus 12057331205901199115 (35)

Displacements (1199063 V3) are modified such that they are cor-

rections to face parallel plane distributions of the preliminarysolution and are free to obtain reactive stresses 120591

1199091199114 1205911199101199114

1205901199115

and normal strain 1205761199113

We have from constitutive relations

1205741199091199112

= 119878441205911199091199112

+ 119878451205911199101199112

lArrrArr (119909 119910) (4 5) (36)

Modified displacements and the corresponding derivedquantities denoted with lowast are

119906lowast

3= 1199063minus 1205721199082119909

+ 1205741199091199112

lArrrArr (119909 119910) (119906 V) (37)

Strain-displacement relations give

120576lowast

1199093= 1205761199093

minus 12057221199082119909119909

+ 1205721205741199091199112119909

lArrrArr (119909 119910)

120574lowast

1199091199103= 1205741199091199103

minus 212057221199082119909119910

+ 120572 (1205741199091199112119910

+ 1205741199101199112119909

)

120574lowast

1199091199113= 1199063+ 1205741199091199112

lArrrArr (119909 119910) (119906 V)

(38)

In-plane and transverse shear stresses from constitutiverelations are

120590lowast

119894(3)= 119876119894119895120576lowast

119895(3)(119894 119895 = 1 2 3) (39)

120591lowast

1199091199112= 119876441199063+ 11987645V3+ 1205911199091199112

lArrrArr (119909 119910) (4 5) (40)

Suffix (3) in (39) is to indicate that the stresses and strainscorrespond to 119906

3and V3

We get from (2) (15) (35) (37) and (40) with referenceto coefficient of 119891

3(119911)

120572 [(119876441199063+ 11987645V3)119909

+ (119876541199063+ 11987655V3)119910

] = 12057331205901199115 (41)

Note that 1199082is not present in the previous equation

From integration of equilibrium equations we have withsum on 119895 = 1 2 3

1205911199091199114

= 120572[1198761119895120576lowast

119895119909+ 1198763119895120576lowast

119895119910](3)

1205911199101199114

= 120572[1198762119895120576lowast

119895119910+ 1198763119895120576lowast

119895119909](3)

(42)

Note that 1199082is present in the previous expression

120572 (1205911199091199114119909

+ 1205911199101199114119910

) + 1205901199115

= 0 (Coefficient of 1198915) (43)

1205761199113

= [1198786119895120590119895+ 11987866120590119911](3) (44)

One gets one equation governing in-plane displacements(1199063 V3) from (41) (43) noting that 119891

5119911119911+1198913= 0 in the form

1205733(1205911199091199114119909

+ 1205911199101199114119910

)

= 1205722[(119876441199063+ 11987645V3)119909119909

+ (119876541199063+ 11987655V3)119910119910

]

(45)

The second equation governing these variables is from thecondition 120596

119911= 0 Here it is more convenient to express

(1199063 V3) as

[1199063 V3] = minus120572 [(120595

3119909minus 1206013119910) (1205953119910

+ 1206013119909)] (46)

so that Δ1205933= 0 and (45) becomes a fourth-order equation

governing 1205953 This sixth-order system is to be solved for 120595

3

and 1205933with associated edge conditions

119906lowast

3= 0 or 120590

lowast

3= 0

Vlowast

3= 0 or 120591

lowast

1199091199103= 0

1205953= 0 or 120591

lowast

1199091199113= 0

(47)

along an 119909 = constant edge and analogous conditions along a119910 = constant edge

In (45) expressed in terms of 1205953and 120593

3 we replace 119908

2by

1205953for obtaining neutral plane deflection since contributions

of 1199082and 120595

3are one and the same in finding [119906

3 V3]

71 Supplementary Problem Because of the use of integrationconstant to satisfy face shear conditions vertical deflection ofneutral plane is equal to that of the face plane deflectionThisis physically incorrect since face plane is bounded by elasticmaterial on one side whereas neutral plane is bounded onboth sidesThis lack in interior solution is rectified by addingthe solution of a supplementary problem based on leadingcosine term that is enough in view of 119891

2(119911) distribution

of reactive shear stresses This is based on the expansion

ISRN Civil Engineering 7

of 120590119911= 1198913(119911)(32)119902 in sine series Here load condition is

satisfied by the leading sine term and coefficients of all othersine terms are zero

Corrective displacements in the supplementary problemare assumed in the form

119908 = 1199082119904

120587

2

cos(1205872

119911) 119906119904= 1199063119904sin(120587

2

119911) lArrrArr (119906 V)

1205903119904119894

= 1198761198941198951205763119904119895

(119894 119895 = 1 2 3)

(48)

We have from integration of equilibrium equations

1205912119904119904119909119911

= minus(

2

120587

)120572 [11987611198951205763119904119895119909

+ 11987631198951205763119904119895119910

] (49a)

1205912119904119910119911

= minus(

2

120587

)120572 [11987621198951205763119904119895119910

+ 11987631198951205763119904119895119909

] (49b)

In-plane distributions 1199063119904

and V3119904

are added as correctionsto the known 119906

lowast

3and Vlowast

3so that (119906 V) in the supplementary

problem are

119906 = (119906lowast

3+ 1199063119904) sin(120587

2

119911) lArrrArr (119906 V) (50)

We note that 120590119911from integration of equilibrium equations

with 119904 variables is the same as 120590119911from static equilibrium

equations with lowast variables Hence

(

2

120587

)

2

1205722[11987611198951205763119904119895119909119909

+ 211987631198951205763119904119895119909119910

+ 11987621198951205763119904119895119910119910

] + 12057331205901199115

= 0

(51)

It is convenient to use here also

[1199063119904 V3119904] = minus120572 [(120595

3119904119909minus 1206013119904119910

) (1205953119904119910

+ 1205933119904119909

)] (52)

Correction to neutral plane deflection due to solution ofsupplementary problem is from

1205761199113119904

= [11987861198951205903119904119895

+ 119878661205901199113119904

] (sum 119895 = 1 2 3) (53)

72 Summary of Results from the First Stage of IterativeProcedure The previous analysis gives displacements [119906V] =1198913[1199063 V3] consistent with [120591

119909119911 120591119910119911] = 120572119891

2[120595119909 120595119910] and

corrective transverse stresses from the first stage of iteration

119908 = 1199080minus [11989121205761199111+ 11989141205761199113] minus 1205761199111199043

120587

2

cos(1205872

119911)

119906 = 11989111199061+ 1198913119906lowast

3+ (119906lowast

3+ 1199063119904) sin(120587

2

119911) lArrrArr (119906 V)

120591119909119911

= 11989121205912119909119911

+ 1205912119904119909119911

120587

2

cos(1205872

119911) lArrrArr (119909 119910)

120590119911= 1198913(119911) 120590lowast

1199113(From (35))

(54)

It is to be noted that the dependence of transverse stresses onmaterial constants is through the solution of supplementaryproblem One may add 119891

4terms in shear components and 119891

5

term in normal stress componentThese components are alsodependent on material constants but they need correctionfrom the solution of a supplementary problem Successiveapplication of the previous iterative procedure leads to thesolution of the 3D problem in the limit

8 Concluding Remarks

Poissonrsquos theory developed in the present study for the analy-sis of bending of anisotropic plates within small deformationtheory forms the basis for generation of proper sequence of2D problems Analysis for obtaining displacements therebybending stresses along faces of the plate is different fromsolution of a supplementary problem in the interior ofthe plate A sequence of higher order shear deformationtheories lead to solution of associated torsion problem onlyIn the preliminary solution reactive transverse stresses areindependent of material constants In view of layer-wisetheory of symmetric laminated plates proposed by the presentauthor [10] Poissonrsquos theory is useful since analysis of faceplies is independent of lamination which is in confirmationof (8)-(9) in a recent NASA technical publication by Tessleret al [11]

One significant observation is that sequence of 2D prob-lems converging to 3D problems in the analysis of extensionbending and torsion problems are mutually exclusive to oneother

Highlights

(i) Poisson-Kirchhoff boundary conditions paradox isresolved

(ii) Transverse stresses are independent of material con-stants in primary solution

(iii) Edge support condition on vertical deflection has norole in the analysis

(iv) Finding neutral plane deflection requires solution ofa supplementary problem

(v) Twisting stress distribution in pure torsion nullifies itsdistribution in bending

Appendix

119886 = side length of a square plate2ℎ = plate thickness119891119899(119911)= through-thickness distribution functions 119899 =

0 1 2

119871 = characteristic length of the plate in119883-119884-plane119902(119909 119910) = applied face load density119876119894119895= stiffness coefficients

119878119894119895= elastic compliances

[119879119909 119879119909119910 119879119909119911] = prescribed stress distributions along

an 119909 = constant edge

8 ISRN Civil Engineering

[119880 119881119882] = displacements in 119883119884 119885-directionsrespectively[119906 V 119908] = [119880 119881119882]ℎ

119874 minus 119883119884119885 = Cartesian coordinate system[119909 119910 119911] = nondimensional coordinates [119883119871 119884119871119885ℎ]

120572 = ℎ119871

1205732119899+1

= [1198912119899+3

(1)12057221198912119899+1

(1)]

Δ = Laplace operator (12059721205971199092 + 12059721205971199102)

[120576119909 120576119910 120574119909119910] = in-plane strains

[120574119909119911 120574119910119911 120576119911] = transverse strains

120596119911= 120572(V119909minus 119906119910) rotation about 119911-axis

[120590119909 120590119910 120591119909119910] = bending stresses

[120591119909119911 120591119910119911 120590119911] = transverse stresses

References

[1] G Kirchhoff ldquoUber das Gleichgewicht und die Bewegung einerelastischen Scheiberdquo Journal fur die Reine und AngewandteMathematik vol 40 pp 51ndash58 1850

[2] HHencky ldquoUber die Berucksichtigung der Schubverzerrung inebenen Plattenrdquo Ingenieur-Archiv vol 16 no 1 pp 72ndash76 1947

[3] E Reissner ldquoReflections on the theory of elastic platesrdquoAppliedMechanics Reviews vol 38 no 11 pp 1453ndash1464 1985

[4] A E H Love A Treatise on Mathematical Theory of ElasticityCambridge University Press Cambridge UK 4th edition 1934

[5] K Vijayakumar ldquoModified Kirchhoff rsquos theory of plates includ-ing transverse shear deformationsrdquo Mechanics Research Com-munications vol 38 pp 211ndash213 2011

[6] K Vijayakumar ldquoNew look at kirchhoff rsquos theory of platesrdquoAIAA Journal vol 47 no 4 pp 1045ndash1046 2009

[7] K Vijayakumar ldquoA relook at Reissnerrsquos theory of plates inbendingrdquo Archive of Applied Mechanics vol 81 no 11 pp 1717ndash1724 2011

[8] E Reissner ldquoThe effect of transverse shear deformations on thebending of elastic platesrdquo Journal of Applied Mechanics vol 12pp A69ndashA77 1945

[9] T Lewinski ldquoOn the twelfth-order theory of elastic platesrdquoMechanics Research Communications vol 17 no 6 pp 375ndash3821990

[10] K Vijayakumar ldquoLayer-wise theory of bending of symmetriclaminates with isotropic pliesrdquo AIAA Journal vol 49 no 9 pp2073ndash2076 2011

[11] A Tessler M Di Sciuva and M Gherlone ldquoRefined zigzagtheory for homogeneous laminated composite and sandwichplates a homogeneous limit methodology for zigzag functionselectionrdquo Tech Rep NASATP-2920102162141 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

ISRN Civil Engineering 7

of 120590119911= 1198913(119911)(32)119902 in sine series Here load condition is

satisfied by the leading sine term and coefficients of all othersine terms are zero

Corrective displacements in the supplementary problemare assumed in the form

119908 = 1199082119904

120587

2

cos(1205872

119911) 119906119904= 1199063119904sin(120587

2

119911) lArrrArr (119906 V)

1205903119904119894

= 1198761198941198951205763119904119895

(119894 119895 = 1 2 3)

(48)

We have from integration of equilibrium equations

1205912119904119904119909119911

= minus(

2

120587

)120572 [11987611198951205763119904119895119909

+ 11987631198951205763119904119895119910

] (49a)

1205912119904119910119911

= minus(

2

120587

)120572 [11987621198951205763119904119895119910

+ 11987631198951205763119904119895119909

] (49b)

In-plane distributions 1199063119904

and V3119904

are added as correctionsto the known 119906

lowast

3and Vlowast

3so that (119906 V) in the supplementary

problem are

119906 = (119906lowast

3+ 1199063119904) sin(120587

2

119911) lArrrArr (119906 V) (50)

We note that 120590119911from integration of equilibrium equations

with 119904 variables is the same as 120590119911from static equilibrium

equations with lowast variables Hence

(

2

120587

)

2

1205722[11987611198951205763119904119895119909119909

+ 211987631198951205763119904119895119909119910

+ 11987621198951205763119904119895119910119910

] + 12057331205901199115

= 0

(51)

It is convenient to use here also

[1199063119904 V3119904] = minus120572 [(120595

3119904119909minus 1206013119904119910

) (1205953119904119910

+ 1205933119904119909

)] (52)

Correction to neutral plane deflection due to solution ofsupplementary problem is from

1205761199113119904

= [11987861198951205903119904119895

+ 119878661205901199113119904

] (sum 119895 = 1 2 3) (53)

72 Summary of Results from the First Stage of IterativeProcedure The previous analysis gives displacements [119906V] =1198913[1199063 V3] consistent with [120591

119909119911 120591119910119911] = 120572119891

2[120595119909 120595119910] and

corrective transverse stresses from the first stage of iteration

119908 = 1199080minus [11989121205761199111+ 11989141205761199113] minus 1205761199111199043

120587

2

cos(1205872

119911)

119906 = 11989111199061+ 1198913119906lowast

3+ (119906lowast

3+ 1199063119904) sin(120587

2

119911) lArrrArr (119906 V)

120591119909119911

= 11989121205912119909119911

+ 1205912119904119909119911

120587

2

cos(1205872

119911) lArrrArr (119909 119910)

120590119911= 1198913(119911) 120590lowast

1199113(From (35))

(54)

It is to be noted that the dependence of transverse stresses onmaterial constants is through the solution of supplementaryproblem One may add 119891

4terms in shear components and 119891

5

term in normal stress componentThese components are alsodependent on material constants but they need correctionfrom the solution of a supplementary problem Successiveapplication of the previous iterative procedure leads to thesolution of the 3D problem in the limit

8 Concluding Remarks

Poissonrsquos theory developed in the present study for the analy-sis of bending of anisotropic plates within small deformationtheory forms the basis for generation of proper sequence of2D problems Analysis for obtaining displacements therebybending stresses along faces of the plate is different fromsolution of a supplementary problem in the interior ofthe plate A sequence of higher order shear deformationtheories lead to solution of associated torsion problem onlyIn the preliminary solution reactive transverse stresses areindependent of material constants In view of layer-wisetheory of symmetric laminated plates proposed by the presentauthor [10] Poissonrsquos theory is useful since analysis of faceplies is independent of lamination which is in confirmationof (8)-(9) in a recent NASA technical publication by Tessleret al [11]

One significant observation is that sequence of 2D prob-lems converging to 3D problems in the analysis of extensionbending and torsion problems are mutually exclusive to oneother

Highlights

(i) Poisson-Kirchhoff boundary conditions paradox isresolved

(ii) Transverse stresses are independent of material con-stants in primary solution

(iii) Edge support condition on vertical deflection has norole in the analysis

(iv) Finding neutral plane deflection requires solution ofa supplementary problem

(v) Twisting stress distribution in pure torsion nullifies itsdistribution in bending

Appendix

119886 = side length of a square plate2ℎ = plate thickness119891119899(119911)= through-thickness distribution functions 119899 =

0 1 2

119871 = characteristic length of the plate in119883-119884-plane119902(119909 119910) = applied face load density119876119894119895= stiffness coefficients

119878119894119895= elastic compliances

[119879119909 119879119909119910 119879119909119911] = prescribed stress distributions along

an 119909 = constant edge

8 ISRN Civil Engineering

[119880 119881119882] = displacements in 119883119884 119885-directionsrespectively[119906 V 119908] = [119880 119881119882]ℎ

119874 minus 119883119884119885 = Cartesian coordinate system[119909 119910 119911] = nondimensional coordinates [119883119871 119884119871119885ℎ]

120572 = ℎ119871

1205732119899+1

= [1198912119899+3

(1)12057221198912119899+1

(1)]

Δ = Laplace operator (12059721205971199092 + 12059721205971199102)

[120576119909 120576119910 120574119909119910] = in-plane strains

[120574119909119911 120574119910119911 120576119911] = transverse strains

120596119911= 120572(V119909minus 119906119910) rotation about 119911-axis

[120590119909 120590119910 120591119909119910] = bending stresses

[120591119909119911 120591119910119911 120590119911] = transverse stresses

References

[1] G Kirchhoff ldquoUber das Gleichgewicht und die Bewegung einerelastischen Scheiberdquo Journal fur die Reine und AngewandteMathematik vol 40 pp 51ndash58 1850

[2] HHencky ldquoUber die Berucksichtigung der Schubverzerrung inebenen Plattenrdquo Ingenieur-Archiv vol 16 no 1 pp 72ndash76 1947

[3] E Reissner ldquoReflections on the theory of elastic platesrdquoAppliedMechanics Reviews vol 38 no 11 pp 1453ndash1464 1985

[4] A E H Love A Treatise on Mathematical Theory of ElasticityCambridge University Press Cambridge UK 4th edition 1934

[5] K Vijayakumar ldquoModified Kirchhoff rsquos theory of plates includ-ing transverse shear deformationsrdquo Mechanics Research Com-munications vol 38 pp 211ndash213 2011

[6] K Vijayakumar ldquoNew look at kirchhoff rsquos theory of platesrdquoAIAA Journal vol 47 no 4 pp 1045ndash1046 2009

[7] K Vijayakumar ldquoA relook at Reissnerrsquos theory of plates inbendingrdquo Archive of Applied Mechanics vol 81 no 11 pp 1717ndash1724 2011

[8] E Reissner ldquoThe effect of transverse shear deformations on thebending of elastic platesrdquo Journal of Applied Mechanics vol 12pp A69ndashA77 1945

[9] T Lewinski ldquoOn the twelfth-order theory of elastic platesrdquoMechanics Research Communications vol 17 no 6 pp 375ndash3821990

[10] K Vijayakumar ldquoLayer-wise theory of bending of symmetriclaminates with isotropic pliesrdquo AIAA Journal vol 49 no 9 pp2073ndash2076 2011

[11] A Tessler M Di Sciuva and M Gherlone ldquoRefined zigzagtheory for homogeneous laminated composite and sandwichplates a homogeneous limit methodology for zigzag functionselectionrdquo Tech Rep NASATP-2920102162141 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 ISRN Civil Engineering

[119880 119881119882] = displacements in 119883119884 119885-directionsrespectively[119906 V 119908] = [119880 119881119882]ℎ

119874 minus 119883119884119885 = Cartesian coordinate system[119909 119910 119911] = nondimensional coordinates [119883119871 119884119871119885ℎ]

120572 = ℎ119871

1205732119899+1

= [1198912119899+3

(1)12057221198912119899+1

(1)]

Δ = Laplace operator (12059721205971199092 + 12059721205971199102)

[120576119909 120576119910 120574119909119910] = in-plane strains

[120574119909119911 120574119910119911 120576119911] = transverse strains

120596119911= 120572(V119909minus 119906119910) rotation about 119911-axis

[120590119909 120590119910 120591119909119910] = bending stresses

[120591119909119911 120591119910119911 120590119911] = transverse stresses

References

[1] G Kirchhoff ldquoUber das Gleichgewicht und die Bewegung einerelastischen Scheiberdquo Journal fur die Reine und AngewandteMathematik vol 40 pp 51ndash58 1850

[2] HHencky ldquoUber die Berucksichtigung der Schubverzerrung inebenen Plattenrdquo Ingenieur-Archiv vol 16 no 1 pp 72ndash76 1947

[3] E Reissner ldquoReflections on the theory of elastic platesrdquoAppliedMechanics Reviews vol 38 no 11 pp 1453ndash1464 1985

[4] A E H Love A Treatise on Mathematical Theory of ElasticityCambridge University Press Cambridge UK 4th edition 1934

[5] K Vijayakumar ldquoModified Kirchhoff rsquos theory of plates includ-ing transverse shear deformationsrdquo Mechanics Research Com-munications vol 38 pp 211ndash213 2011

[6] K Vijayakumar ldquoNew look at kirchhoff rsquos theory of platesrdquoAIAA Journal vol 47 no 4 pp 1045ndash1046 2009

[7] K Vijayakumar ldquoA relook at Reissnerrsquos theory of plates inbendingrdquo Archive of Applied Mechanics vol 81 no 11 pp 1717ndash1724 2011

[8] E Reissner ldquoThe effect of transverse shear deformations on thebending of elastic platesrdquo Journal of Applied Mechanics vol 12pp A69ndashA77 1945

[9] T Lewinski ldquoOn the twelfth-order theory of elastic platesrdquoMechanics Research Communications vol 17 no 6 pp 375ndash3821990

[10] K Vijayakumar ldquoLayer-wise theory of bending of symmetriclaminates with isotropic pliesrdquo AIAA Journal vol 49 no 9 pp2073ndash2076 2011

[11] A Tessler M Di Sciuva and M Gherlone ldquoRefined zigzagtheory for homogeneous laminated composite and sandwichplates a homogeneous limit methodology for zigzag functionselectionrdquo Tech Rep NASATP-2920102162141 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of