A semi-analytical solution on static analysis of circular plate exposed to non-uniform axisymmetric transverse loading resting on Winkler elastic foundation_3.pdf

Original Research Article A semi-analytical solution on static analysis of circular plate exposed to non-uniform axisymmetric transverse loading resting on Winkler elastic foundation S. Abbasi, F. Farhatnia *, S.R. Jazi Department of Mechanical Engineering, Islamic Azad University, Khomeinishahr branch, Boulevard Manzariye, Khomeinishahr, Esfahan, Iran 1. Introduction Extensive use of circular plates in particular purposes such as bridge decks, turbine disk, thrust
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  OriginalResearchArticle Asemi-analytical   solutiononstaticanalysisof circularplateexposedtonon-uniformaxisymmetric   transverseloadingrestingonWinklerelasticfoundation S.Abbasi,F.Farhatnia* ,S.R.Jazi Department   of    Mechanical   Engineering,   Islamic    Azad   University,   Khomeinishahr   branch,   Boulevard   Manzariye,Khomeinishahr,   Esfahan,   Iran 1.   Introduction Extensive   use   of    circular    plates   in   particular    purposes   such   asbridge   decks,   turbine   disk,   thrust   bearing    plates   and   clutches,tanks,   structural   components   for    diaphragms,   and   deck   platesin   launch   vehicles,   engineering    and   spatial   structures   re 󿬂 ectsthe   importance   of    circular    plates.   Since   scientists   focus   onfunctionally   graded   material   (FGM)   to   such   an   extent   inengineering    󿬁 eld   recently,   in   this   paper,   FG   circular    plate   isconsidered.   FGMs   are   new   materials,   microscopically   inho-mogeneous   continua,   where   continuous   variation   of    themechanical   properties,   from   metal   to   ceramic,   happensgradually   without   any   sudden   changes.   For    the   󿬁 rst   time   inan   industrial   application,    Japanese   scientists   proposed   FGMfor    thermal   barriers   in   aerospace   structures   [1].This   kind   of new   composites   can   be   found   in   aerospace   structures,   nuclear reactors,   chemical   plants,   semiconductors   and   biomedical archivesofcivilandmechanicalengineeringxxx(2014)xxx–xxx a   r   t   i   c   l   e   i   n   f   o  Articlehistory: Received5October2012Accepted29September2013Availableonlinexxx Keywords: StaticanalysisCircularplatesFunctionallygradedmaterial(FGM)WinklerelasticfoundationDifferentialtransformsmethod(DTM) a   b   s   t   r   a   c   t Thispaperisconcernedwithstaticanalysisoffunctionallygraded(FG)circularplatesresting onWinklerelasticfoundation.Thematerialpropertiesvaryacrossthethicknessdirectionsothepower-lawdistributionisusedtodescribetheconstituentcomponents.Thedifferentialtransformsmethod(DTM)isutilizedtosolvethegoverningdifferentialequationsofbending ofthethincircularplateundervariousboundaryconditions.Byemployingthissolutionmethod,governingdifferentialequationsaretransformedintorecurrencerelationsandboundary/regularityconditionsarechangedintoalgebraicequations.Inthisstudy,theplateissubjectedtouniform/non-uniformtransverseloadintwocasesofboundaryconditions(clampedandsimply-supported).Somenumericalexamplesarepresentedtoshowthein 󿬂 uenceoffunctionallygradedvariation,differentelasticfoundationmodulus,andvaria-tionofthesymmetricaltransverseloadsonthestressanddisplacement 󿬁 elds.Basedontheresults,theobtainedout-planedisplacementcoincidewiththeavailablesolutionforahomogenouscircularplate.Itcanbeconcludedthattheappliedmethodprovidesaccurateresultsanditiseasilyusedforstaticanalysisofcircularplatesonanelasticfoundation. # 2013PolitechnikaWroc ł awska.PublishedbyElsevierUrban&PartnerSp.zo.o.Allrightsreserved.*Correspondingauthor.Tel.:+983113660011x14;, (F.Farhatnia). ACME-152;   No.   of    Pages   13 Pleasecitethisarticleinpressas:S.Abbasietal.,Asemi-analyticalsolutiononstaticanalysisofcircularplateexposedtonon-uniformaxisymmetrictransverseloadingrestingonWinklerelasticfoundation,ArchivesofCivilandMechanicalEngineering(2014),  Available   online   at ScienceDirect  journalhomepage: 1644-9665/$ – seefrontmatter  # 2013PolitechnikaWroc ł awska.PublishedbyElsevierUrban&PartnerSp.zo.o.Allrightsreserved.  industries.   Comprehensive   works   on   static   and   dynamicresponses   of    FG   plates   are   available   in   the   literature.   Next,we   brie 󿬂 y   concentrate   on   some   recent   works   related   to   thestatic   behavior    of    FG   circular    plates.Reddy   et   al.   [2]   investigated   the   axisymmetric   bending    of functionally   graded   circular    and   annular    plates.   They   studiedthe   bending    behavior    of    plate   based   on   the   󿬁 rst   order    shear deformation   Mindlin   plate   theory.   In   their    study,   the   Mindlinsolution   of    FG   circular    plate   was   obtained   for    the   conditionswhere   the   Kirchhoff    solution   for    thin   plate   was   formerlyknown.   Li   et   al.   [3]   developed   the   incremental   load   techniquefor    solving    the   governing    differential   equation   of    thin   circular plate   bending    with   large   deformations.   In   this   technique,   totalapplied   load   was   divided   into   different   small   steps   so   thatlinear    stress   analysis   for    the   plate   was   reasonable.   Civalek   [4]employed   differential   quadrature   method   (DQM)   and   harmon-ic   differential   quadrature   method   (HDQM)   in   analyzing    staticand   vibration   of    columns   as   well   as   circular    and   rectangular plates.   He   compared   accuracy   of    the   two   methods   in   structuralanalysis   and   showed   that   HDQM   needs   less   grid   points   thanDQM   to   achieve   accurate   results.   Li   and   Ding    [5],investigatedbending    of    transversely   isotropic   circular    plates,   whose   elasticcompliance   coef  󿬁 cients   are   arbitrary   functions   of    the   thick-ness   coordinate,   exposed   to   a   transverse   load   as   a   power function   of    radius.   Zheng    and   Zhong    [6]   investigated   axisym-metric   bending    problem   of    FG   circular    plates   under    twoboundary   conditions,   rigid   slipping    and   elastically   supported,subjected   to   transverse   normal   and   shear    loadings.   Theyutilized   Fourier  – Bessel   series   as   the   displacement   function.Civalek   and   Ersoy   [7]   studied   free   vibration   and   bending    of Mindlin   circular    plates   based   on   the   discrete   singular convolution   method   (DSCM)   with   the   use   of    regularizedShannon's   delta   kernel.   They   obtained   the   frequency   param-eters,   de 󿬂 ections,   and   bending    moments   and   showed   that   thesingular    convolution   method   is   an   exact   method.   Sahraee   andSaidi   [8]   investigated   axisymmetric   bending    of    functionallygraded   circular    plates   under    uniform   transverse   loadingsusing    the   fourth-order    shear    deformation   plate   theory.   Theystudied   the   effect   of    various   percentages   of    ceramic – metalvolume   fractions   on   maximum   out-plane   displacement   andshear    stress.   Their    results   were   compared   with   those   obtainedbased   on   the   󿬁 rst-order    shear    deformation   plate   theory,   thethird-order    shear    deformation   plate   theory   of    Reddy   and   theexact   three-dimensional   elasticity   solution   and   found   goodagreement   between   them.   Sahraee   et   al.   [9]   analyzed   bending and   buckling    of    thick   circular    FG   plate   based   third-order    shear deformation   plate   theories.   They   applied   the   shear  – freeconstraint   on   the   top   and   bottom   of    the   plate   and   obtainedthe   static   response   and   critical   buckling    loads   in   bending    andbucking    analysis   of    functionally   graded   circular    plates   using unconstrained   shear    stress   theory   in   terms   of    the   correspond-ing    quantities   of    the   homogeneous   plates   based   on   theclassical   plate   theory.   Yun   et   al.   [10]   carried   out   bending analysis   of    transversely   isotropic   circular    plates   under    arbi-trary   symmetric   transverse   loads.   They   expanded   the   trans-verse   loading    as   Fourier  – Bessel   series.   In   their    work,   thematerial   properties   varied   arbitrarily   along    the   thickness   of    theplate.   They   used   the   direct   displacement   method   for    obtaining the   analytical   solution.   Chen   [11]   suggested   an   innovativetechnique   for    solving    nonlinear    differential   equations   for bending    problem   of    a   circular    plate.   He   used   a   type   of    pseudo-linearization   to   obtain   the   󿬁 nal   solution   for    large   deformationsof    the   circular    plate.   Alipour    and   Shariat   [12]   proposed   stressanalysis   for    axisymmetric   bending    of    circular    FG   sandwichplates   subjected   to   transversely   distributed   loads.   Theyderived   the   governing    equations   based   on   elasticity-equilibri-um-based   zigzag    theory.   They   employed   a   semi-analyticalMaclaurin-type   power-series   solution.In   numerous   engineering    applications,   the   plate   is   contin-uously   supported   within   the   span.   In   the   case   where   thesupport   is   linear    elastic,   its   reaction   is   proportional   to   the   localde 󿬂 ection   of    the   structure   (so-called   Winkler's   elastic   founda-tion).   Accordingly,   if    the   plate   is   supported   by   an   elasticfoundation,   it   experiences   a   local   de 󿬂 ection   w ,   and   thereaction   (counter-pressure)   applied   by   the   foundation   to   theplate   is   kw   where   k   is   a   proportionality   coef  󿬁 cient   called   themodulus   of    the   foundation   [13].In   other    words,   Winkler'selastic   foundation   is   assumed   to   behave   linearly.   It   should   benoted   that   interaction   between   plate   and   elastic   foundation   isa   complicated   issue   which   is   not   easy   to   be   explored.   In   manypractical   engineering    applications,   this   kind   of    model   providessatis 󿬁 ed   results.   It   is   worth   mentioning    that   the   plates   resting on   an   elastic   foundation   have   been   greatly   used   in   modernengineering    structures   such   as   building    footings,   reinforcedconcrete   pavements   of    high   runways,   foundation   of    deepwells,   storage   tanks,   base   of    machines,   aerospace,   biome-chanics,   petrochemical,   civil,   mechanical,   electronic,   nuclear and   foundation   engineering.   Providing    the   exact   solution   for governing    equations   of    static   behavior    and   dynamic   responseof    any   kind   of    plate   in   shape   under    various   form   of    loading    isnot   always   feasible.   So   the   researchers   attempt   to   employ   thesemi-analytical   and   numerical   methods   when   involved   theproblems   in   this   󿬁 eld   of    study.   For    the   󿬁 rst   time,   differentialtransformation   method   (DTM)   was   introduced   by   Zhou   [14]   for solving    linear    and   nonlinear    initial   value   problems   in   electriccircuit   analysis.   This   method   is   a   semi-analytical-numericaltechnique   based   on   Taylor    series   expansion   developed   for various   types   of    differential   equations.   Differential   transformsmethod   solves   a   series   extremely   shorter    and   faster    than   highorder    Taylor    series   method.   It   also   signi 󿬁 cantly   reduces   thecomputation   cost   of    linear    and   nonlinear    problems   and   iseasily   applicable.   By   using    DTM,   governing    differentialequations   are   reduced   to   the   recursive   relations   together    withassociated   boundary   conditions   which   can   be   transformed   to   aset   of    algebraic   equations.   Furthermore,   this   method   reducesthe   computational   dif  󿬁 culties   of    the   other    methods   since   allthe   calculations   can   be   made   with   a   simple   iterative   process[15].   Another    advantage   of    this   method   is   exact   results   whichcan   be   obtained   with   a   rapid   convergence.DTM   has   recently   attracted   the   attention   of    scientists   invarious   󿬁 elds   of    engineering.   Yalcin   et   al.   [16]   represented   freevibration   analysis   of    circular    plates   by   differential   transfor-mation   method.   Özdemir    and   Kaya   [17]   investigated   󿬂 ap   wisebending    vibration   of    a   rotating    tapered   cantilever    Bernoulli – Euler    beam   by   differential   transforms   method.   Balkaya   andKaya   [18]   employed   differential   transforms   method   to   predictthe   vibrating    behavior    of    Euler  – Bernoulli   and   Timoshenkobeams   resting    on   an   elastic   foundation   (elastic   soil).   Theyshowed   that   it   is   a   useful   tool   for    analytical   and   numericalsolutions   and   that   the   solution   procedure   can   be   easily   applied archivesofcivilandmechanicalengineeringxxx(2014)xxx–xxx 2 ACME-152;   No.   of    Pages   13 Pleasecitethisarticleinpressas:S.Abbasietal.,Asemi-analyticalsolutiononstaticanalysisofcircularplateexposedtonon-uniformaxisymmetrictransverseloadingrestingonWinklerelasticfoundation,ArchivesofCivilandMechanicalEngineering(2014),  to   governing    equation   of    beam   vibration   and.   Attarnejad   et   al.[19]   utilized   DTM   for    calculating    natural   frequency   of    aTimoshenko   beam   resting    on   two-parameter    elastic   founda-tion.   Soltanizadeh   [20]   utilized   two   dimensional   DTM   for solving    the   hyperbolic   telegraph   equation.   He   carried   out   somenumerical   tests   to   show   the   advantages   and   disadvantages   of the   proposed   method.As   seen   in   the   literature   above,   the   differential   transformsmethod   (DTM)   has   been   used   for    solving    a   vast   range   of problems   in   different   󿬁 elds   of    engineering.   To   the   bestknowledge   of    the   authors,   no   research   effort   has   beendevotedso   far    to   󿬁 nd   the   solution   of    bending    of    a   functionally   gradedcircular    plate   resting    on   an   elastic   foundation   by   employing DTM.   In   our    work,   bending    analysis   of    functionally   gradedcircular    plates   resting    on   Winkler    elastic   foundation   is   carriedout   usingdifferential   transforms   method   (DTM).   The   plate   issubjected   to   axisymmetric   transverse   load   that   is   assumed   to   berepresented   by   a   power    law   distribution   along    the   radialdirection   of    the   plate.   This   study   attempts   to   incorporate   boththe   effect   of    elastic   foundation   modulus   and   FG   power    index   onout-plane   displacement   of    the   plate   resting    on   elastic   founda-tion.   Also,the   distributions   of    radial   and   circumferentialstresses   along    the   radius   and   across   the   thickness   are   obtained.The   results   are   compared   with   the   published   literature   andFinite   Element   Method   to   demonstrate   the   applicability   and   thecomputational   ef  󿬁 ciency   of    the   proposed   method. 2.   Governing   equation   of     bending   of    FGcircular    plate   subjected   to   symmetric   transverseload Consider    a   circular    plate   subjected   to   uniform/non-uniformtransverse   loading    while   resting    on   an   elastic   foundation   asshown   in   Fig.   1.   Geometric   parameters   R and   h   are   radius   andthickness   of    the   plate,   respectively,   and   k w  is   the   foundationelastic   modulus.Differential   equation   of    a   circular    plate   subjected   to   thesymmetric   transverse   load   in   the   form   of    q 0 ( r  / R )  p ,   where    p  0and   is   a   󿬁 nite   even   number,   rested   on   Winkler    elasticfoundation,   according    to   the   classical   plate   theory   (CPT)   [13],is   given   as   follows: r 4 w   ¼ q 0 ð r = R Þ  p    k w wD  (1)where   w   stands   for    the   out-plane   displacement   (de 󿬂 ection)   of any   point   of    the   plate   mid-surface,   the   radial   coordinate   isdenoted   by   r ,   k w  is   the   Winkler    foundation   modulus,   q 0  is   aconstant   value,   D   is   the   󿬂 exural   rigidity   of    plate,   and   r 4 is   thebi-harmonic   operator,   which   is   de 󿬁 ned   as   follows   in   a   polar cylindrical   coordinate   system: r 4 w   ¼  d 2 dr 2 þ 1 rddrd 2 wdr 2  þ 1 rdwdr !!  (2)Upon   substituting    r 4 w   into   Eq.   (2),the   simpli 󿬁 ed   form   isobtained   as d 4 wdr 4  þ 2 rd 3 wdr 3    1 r 2 d 2 wdr 2  þ  1 r 3 dwdr  ¼ q 0 ð r = R Þ  p    k w wD  (3)By   de 󿬁 nition   the   following    non-dimensionless   parameters: ’   ¼  rR ;   W    ¼ wR  ;   K w  ¼ k w R 4 D  ;   q   ¼ q 0 R 3 D The   governing    Eq.   (1)   can   be   rearranged   in   the   non-dimensional   form   below: ’ 3  d 4 W d ’ 4  þ   2 ’ 2  d 3 W d ’ 3     ’ d 2 W d ’ 2  þ dW d ’ þ   K w ’ 3 W       q ’ m þ 3 ¼   0   (4)Young    modulus   of    a   functionally   graded   plate,   E ( z ),smoothly   changes   based   on   the   power-law   distribution   acrossthe   thickness   direction   from   metal   to   ceramic,   i.e., E ð z Þ   ¼   ð E c    E m Þ  zh þ 12  g þ   E m  (5)where   E m and   E c are   the   Young    modulus   of    metal   and   ceramics,respectively,   and   ( g   is   volume   fraction   index   in   which   g =0( g !1 )   represents   a   fully   ceramic   (homogeneous   metal)   plate.Poisson's   ratio   is   considered   as   a   constant   ratio   throughout   thethickness.ThedifferentialEq.(1)   is   utilized   for    an   isotropic   homoge-nous   platein   whichphysicalneutral   surfaceand   geometricmiddlesurfacearethe   same.Basedon   the   asymmetricmechanicalpropertiesof    FGM   plateswithrespect   to   themiddle   plane,   the   positionof    the   physicalneutral   plane(wherethestrain   and   stressarezero),is   notlocatedon   themiddle   plane.Also,   there   is   stretching  – bending    coupling effect   in   thegoverningequations   of    functionally   gradedplatefor    static   behavioranddynamicresponse.   Byselecting    theproperreference   plane   (whichis   called   physical   neutralsurface),the   governing    differentialequationsof    FG   thinplateshavethesimpleform   asthose   of    classicalthinplate   theoryfor homogeneous   isotropicmaterials.Thepositionof    thisplane( z 0 )   from   the   middlesurfaceis   introduced   asfollows   [21]: z 0  ¼ R  h = 2  h = 2  zE ð z Þ dz R  h = 2  h = 2  E ð z Þ dz (6)where   z   is   the   direction   along    the   thickness.Consequently,   the   elastic   󿬂 exural   rigidity   is   determined   asfollows: D   ¼ Z   h = 2  h = 2 ð z      z 0 Þ 2 E ð z Þ 1      n 2  dz   (7) D   can   be   derived   in   terms   of    thickness,   Poisson   ratio,   neutralposition   from   middle   plane,   ceramic   and   metal   Young    modu-lus,   as   follows   [22]: Fig.   1   –   Geometric   and   foundation   parameters   of    FG   circular plate. archivesofcivilandmechanicalengineeringxxx(2014)xxx–xxx 3 ACME-152;   No.   of    Pages   13 Pleasecitethisarticleinpressas:S.Abbasietal.,Asemi-analyticalsolutiononstaticanalysisofcircularplateexposedtonon-uniformaxisymmetrictransverseloadingrestingonWinklerelasticfoundation,ArchivesofCivilandMechanicalEngineering(2014),  D   ¼  h ð 1      n 2 Þ ( E m h 2 12  1   þ  9 ð g   þ   1 Þð g   þ   2 Þð g   þ   3 Þ  þð E c    E m Þ h 2 4  1     6 g ð g   þ   1 Þð g   þ   2 Þð g   þ   3 Þ )    z 0 h g ð E c    E m Þð g   þ   1 Þð g   þ   2 Þþ   z 20  E m þð E c    E m Þ g   þ   1   (8) 3.   Boundary   and   regularity   conditions Out-plane   displacement   (de 󿬂 ection),   w ,   must   satisfy   theboundary   conditions   at   the   outer    edge   of    the   circular    plate( r = R )   for    clamped,   simply   supported   plate   and   regularitycondition   at   the   center    of    the   circular    plate.   The   boundary/regularity   conditions   can   be   written   in   terms   of    dimensionlessde 󿬂 ection   as   shown   in   Table   1. 4.   The   de 󿬁 nition   and   operation   of    differentialtransforms   method The   differential   transforms   method   (DTM)   provides   ananalytical   solution   procedure   in   the   form   of    polynomials   tosolve   ordinary   and   partial   differential   equations.   In   thismethod,   differential   transformation   of    m th   derivative   function  f  ( r )   and   differential   inverse   transformation   (DT)   of    F [ m ]arerespectively   de 󿬁 ned   as   follows: F ½ m ¼  1 m !  d m  f    ð r Þ dr m ! r ¼ r 0 (9)where  f  ð r Þ   ¼ X 1 m ¼ 0 F ½ m ð r    r 0 Þ m (10)In   Eq.   (9), F [ m ]is   denoted   as   the   tranformed   function(T-function).   The   lower    case   and   upper    case   letters   representthe   srcinal   and   transformed   functions,   respectively,   and   r = r 0 represent   any   point   in   the   domain.   The   function    f  ( r )isconsidered   analytic   in   a   domain   R   and   is   intruduced   as   a 󿬁 nite   power    series   whose   center    is   located   at   r 0 .   Therefore,Eq.   (10)   can   be   expressed   as  f  ð r Þ   ¼ X Nm ¼ 0 F ½ m ð r    r 0 Þ m (11)in   which   N   determines   the   convergence   of    non-dimensionalde 󿬂 ection   which   implies   that    f    ð r Þ   ¼ P 1 m ¼ N þ 1  F ½ m ð r    r 0 Þ m isnegligible.   By   combining    Eqs.   (9)   and   (10),   the   following    relationis   obtained:  f  ð r Þ   ¼ X 1 m ¼ 0 ð r      r 0 Þ m m !  d m  f    ð r Þ dr m ! r ¼ r 0 (12)As   is   seen,   the   concept   of    differential   transformation   isbased   upon   the   Taylor    series   expansion.   From   the   de 󿬁 nitionsof    DTM   in   Eqs.   (9)   and   (10),fundamental   theorems   of differential   transforms   method   [23 – 25]   can   be   performed   thatare   listed   in   Table   2. 5.   Application   of    DTM   in   the   governingequations   and    boundary/regularity   conditions 5.1.   Differential   transformation   of    the   governing   equation Thenon-dimensional   form   of    the   differentialequationof    circularplatesrestingon   Winklerelastic   foundation(Eq.(5))   canbe   solved   using    theabovedifferentialtransformstheorems   at r 0 =0.Let   W  [ m ]   to   be   the   differentialtransformof    w ( r );   Applying Table1,   thedifferentialtransform   versionof    Eq.(3)   canbe   exploited   from   thesolution   approach   below: X mm 1 ¼ 0 d ð m 1  3 Þð 4 þ m  m 1 Þð 3 þ m  m 1 Þð 2 þ m  m 1 Þð 1 þ m  m 1 Þ W  ½ 4   þ m  m 1 þ 2 X mm 1 ¼ 0 d ð m 1  2 Þð 3 þ m  m 1 Þð 2   þ m  m 1 Þð 1 þ m  m 1 Þ W  ½ 3   þ m  m 1  X mm 1 ¼ 0 d ð m 1  1 Þð 2 þ m  m 1 Þð 1 þ m  m 1 Þ W  ½ 2   þ m  m 1 þð 1 þ m Þ W  ½ m þ 1 þ K w X mm 1 ¼ 0 d ð m 1  3 Þ W  ½ m  m 1  q d ð m ð  p þ 3 ÞÞ¼ 0for  m  0 (13) Table   1   –   The   dimensional/    non-dimensional    boundary/regularity   conditions. Type   of    boundaryconditionDimensionalboundary   conditionNon-dimensionalboundary   conditionRegularity   condition   (R.C.)Dimensional   R.C.   Non-Dimensional   R.C. Clampededge w | r = R =0 W  | w =1 =0  dwdr  j r ¼ 0  ¼ 0  dW d ’  j ’ ¼ 0  ¼ 0 dwdr  j r ¼ R  ¼ 0  dW d ’  j ’ ¼ 1  ¼ 0 SimplySupported w | r = R =0 W  | w =1 =0 M r j r ¼ R  ¼ D ð d 2 wdr 2  þ n rdwdr Þ¼ 0  D ð ’ d 2 W d ’ 2  þ y dW d ’ Þ¼ 0 Table   2   –   Fundamental   theorems   of    one-dimensionalDTM. Original   function   Transformed   function  f  ( r )=  y ( r )    z ( r )   F [ m ]= Y  [ m ]  Z [ m ]  f  ( r )= l  y ( r )   F [ m ]= l Y  [ m ]  f  ( r )=  y ( r )   . z ( r ) F ½ m    ¼ X mm 1 ¼ 0 Y  ½ m 1  Z ½ m  m 1   f    ð r Þ¼ d  p  y ð r Þ dr  p  F ½ m    ¼ ð    p þ m Þ ! m !  Y  ½ m þ  p   f  ( r )   = r  p F ½ m    ¼ d ð m     p Þ   ¼ ( 0if  m   6¼  p 1if  m   ¼  p archivesofcivilandmechanicalengineeringxxx(2014)xxx–xxx 4  ACME-152;   No.   of    Pages   13 Pleasecitethisarticleinpressas:S.Abbasietal.,Asemi-analyticalsolutiononstaticanalysisofcircularplateexposedtonon-uniformaxisymmetrictransverseloadingrestingonWinklerelasticfoundation,ArchivesofCivilandMechanicalEngineering(2014),
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