Table Of Content“Frontmatter”
Structural Engineering Handbook
Ed. Chen Wai-Fah
Boca Raton: CRC Press LLC, 1999
Structural Engineering Contents
1 BasicTheoryofPlatesandElasticStability EikiYamaguchi
2 StructuralAnalysis J.Y.RichardLiew, N.E.Shanmugam,and C.H.Yu
3 StructuralSteelDesign1 E.M.Lui
4 StructuralConcreteDesign2 AmyGriderandJulioA.RamirezandYoungMookYun
5 EarthquakeEngineering CharlesScawthorn
6 CompositeConstruction EdoardoCosenzaandRiccardoZandonini
7 Cold-FormedSteelStructures Wei-WenYu
8 AluminumStructures MauriceL.Sharp
9 TimberStructures KennethJ.Fridley
10 BridgeStructures ShoujiToma,LianDuan,andWai-FahChen
11 ShellStructures ClarenceD.Miller
12 MultistoryFrameStructures J.Y.RichardLiewandT.BalendraandW.F.Chen
13 SpaceFrameStructures TienT.Lan
14 CoolingTowerStructures PhillipL.GouldandWilfriedB.Krätzig
15 TransmissionStructures Shu-jinFang,SubirRoy,andJacobKramer
15B Tunnel Structures Birger Schmidt, Christian Ingerslev, Brian Brenner, and J.-N. Wang
16 Performance-Based Seismic Design Criteria For Bridges LianDuanandMarkReno
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17 EffectiveLengthFactorsofCompressionMembers LianDuanandW.F.Chen
18 StubGirderFloorSystems ReidarBjorhovde
19 PlateandBoxGirders MohamedElgaaly
20 SteelBridgeConstruction JacksonDurkee
21 BasicPrinciplesofShockLoading O.W.BlodgettandD.K.Miller
22 WeldedConnections O.W.BlodgettandD.K.Miller
23 CompositeConnections RobertoLeon
24 FatigueandFracture RobertJ.DexterandJohnW.Fisher
25 UndergroundPipe J.M.DoyleandS.J.Fang
26 StructuralReliability3 D.V.Rosowsky
27 PassiveEnergyDissipationandActiveControl T.T.SoongandG.F.Dargush
28 AnInnnovativeDesignForSteelFrameUsingAdvancedAnalysis4 Seung-EockKimand
W.F.Chen
29 WeldedTubularConnections—CHSTrusses PeterW.Marshall
30 Earthquake Damage to Structures Mark Yashinsky
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Yamaguchi, E. “Basic Theory of Plates and Elastic Stability”
Structural Engineering Handbook
Ed. Chen Wai-Fah
Boca Raton: CRC Press LLC, 1999
Basic Theory of Plates and Elastic
Stability
1.1 Introduction
1.2 Plates
BasicAssumptions(cid:15)GoverningEquations(cid:15)BoundaryCon-
ditions(cid:15)CircularPlate(cid:15)ExamplesofBendingProblems
1.3 Stability
Basic Concepts (cid:15) Structural Instability (cid:15) Columns (cid:15) Thin-
WalledMembers(cid:15)Plates
EikiYamaguchi
1.4 DefiningTerms
DepartmentofCivilEngineering,
References
KyushuInstituteofTechnology,
Kitakyusha,Japan FurtherReading
1.1 Introduction
Thischapterisconcernedwithbasicassumptionsandequationsofplatesandbasicconceptsofelastic
stability. Herein,weshallillustratetheconceptsandtheapplicationsoftheseequationsbymeansof
relativelysimpleexamples;morecomplexapplicationswillbetakenupinthefollowingchapters.
1.2 Plates
1.2.1 Basic Assumptions
WeconsideracontinuumshowninFigure1.1. Afeatureofthebodyisthatonedimensionismuch
smallerthantheothertwodimensions:
t <<Lx;Ly (1.1)
wheret;Lx,andLyarerepresentativedimensionsinthreedirections(Figure1.1). Ifthecontinuum
hasthisgeometricalcharacteristicofEquation1.1andisflatbeforeloading,itiscalledaplate. Note
thatashellpossessesasimilargeometricalcharacteristicbutiscurvedevenbeforeloading.
ThecharacteristicofEquation1.1lendsitselftothefollowingassumptionsregardingsomestress
andstraincomponents:
(cid:27)z D 0 (1.2)
"z D "xz D"yz D0 (1.3)
WecanderivethefollowingdisplacementfieldfromEquation1.3:
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FIGURE1.1: Plate.
@w
u.x;y;z/ D u .x;y/−z 0
0 @x
@w
(cid:23).x;y;z/ D (cid:23)0.x;y/−z @y0 (1.4)
w.x;y;z/ D w .x;y/
0
whereu; (cid:23); andwaredisplacementcomponentsinthedirectionsofx-,y-,andz-axes,respectively.
AscanberealizedinEquation1.4,u and(cid:23) aredisplacementcomponentsassociatedwiththeplane
0 0
ofzD0. Physically,Equation1.4impliesthatthelinearfilamentsoftheplateinitiallyperpendicular
to the middle surface remain straight and perpendicular to the deformed middle surface. This is
knownastheKirchhoffhypothesis. AlthoughwehavederivedEquation1.4fromEquation1.3inthe
above,onecanarriveatEquation1.4startingwiththeKirchhoffhypothesis: theKirchhoffhypothesis
isequivalenttotheassumptionsofEquation1.3.
1.2.2 Governing Equations
Strain-DisplacementRelationships
Usingthestrain-displacementrelationshipsinthecontinuummechanics, wecanobtainthe
followingstrainfieldassociatedwithEquation1.4:
@u @2w
" D 0 −z 0
x
@x @x2
@(cid:23) @2w
"y D 0 −z 0 (1.5)
@y @y2
(cid:18) (cid:19)
1 @u @(cid:23) @2w
" D 0 C 0 −z 0
xy
2 @y @x @x@y
Thisconstitutesthestrain-displacementrelationshipsfortheplatetheory.
EquilibriumEquations
In the plate theory, equilibrium conditions are considered in terms of resultant forces and
moments. This is derived by integrating the equilibrium equations over the thickness of a plate.
BecauseofEquation1.2,weobtaintheequilibriumequationsasfollows:
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@N @N
x xy
C Cqx D 0 (1.6a)
@x @y
@N @N
xy y
C Cqy D 0 (1.6b)
@x @y
@V @V
x y
C Cqz D 0 (1.6c)
@x @y
whereNx; Ny, andNxy arein-planestressresultants; Vx andVy areshearingforces; andqx; qy,
andqzaredistributedloadsperunitarea. Thetermsassociatedwith(cid:28)xzand(cid:28)yzvanish,sinceinthe
plateproblemsthetopandthebottomsurfacesofaplatearesubjectedtoonlyverticalloads.
Wemustalsoconsiderthemomentequilibriumofaninfinitelysmallregionoftheplate,which
leadsto
@M @M
x xy
C −Vx D 0
@x @y
@M @M
xy y
C −Vy D 0 (1.7)
@x @y
whereMx andMy arebendingmomentsandMxy isatwistingmoment.
Theresultantforcesandthemomentsaredefinedmathematicallyas
Z
Nx D (cid:27)xdz (1.8a)
Zz
Ny D (cid:27)ydz (1.8b)
z Z
Nxy D Nyx D (cid:28)xydz (1.8c)
Z z
Vx D (cid:28)xzdz (1.8d)
Zz
Vy D (cid:28)yzdz (1.8e)
Zz
Mx D (cid:27)xzdz (1.8f)
Zz
My D (cid:27)yzdz (1.8g)
z Z
Mxy D Myx D (cid:28)xyzdz (1.8h)
z
TheresultantforcesandthemomentsareillustratedinFigure1.2.
ConstitutiveEquations
Sincethethicknessofaplateissmallincomparisonwiththeotherdimensions,itisusually
acceptedthattheconstitutiverelationsforastateofplanestressareapplicable. Hence,thestress-strain
relationshipsforanisotropicplatearegivenby
8 9 2 38 9
< (cid:27)x = E 1 (cid:23) 0 < "x =
: (cid:28)(cid:27)xyy ;D 1−(cid:23)2 4 (cid:23)0 01 .1−0(cid:23)/=2 5: γ"xyy ; (1.9)
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FIGURE1.2: Resultantforcesandmoments.
whereEand(cid:23)areYoung’smodulusandPoisson’sratio,respectively. UsingEquations1.5,1.8,and1.9,
theconstitutiverelationshipsforanisotropicplateintermsofstressresultantsanddisplacementsare
describedby
(cid:18) (cid:19)
Et @u @(cid:23)
Nx D 0 C(cid:23) 0 (1.10a)
1−(cid:23)2 @x @y
(cid:18) (cid:19)
Et @(cid:23) @u
Ny D 0 C(cid:23) 0 (1.10b)
1−(cid:23)2 @y @x
(cid:18) (cid:19)
Et @(cid:23) @u
Nxy D Nyx 0 C 0 (1.10c)
2.1C(cid:23)/ @x @y
(cid:18) (cid:19)
@2w @2w
Mx D −D 0 C(cid:23) 0 (1.10d)
@x2 @y2
(cid:18) (cid:19)
@2w @2w
My D −D 0 C(cid:23) 0 (1.10e)
@y2 @x2
@2w
Mxy D Myx D−.1−(cid:23)/D 0 (1.10f)
@x@y
wheret isthethicknessofaplateandDistheflexuralrigiditydefinedby
Et3
D D (1.11)
12.1−(cid:23)2/
InthederivationofEquation1.10,wehaveassumedthattheplatethicknesstisconstantandthatthe
initialmiddlesurfaceliesintheplaneofZ D0. ThroughEquation1.7,wecanrelatetheshearing
forcestothedisplacement.
Equations 1.6, 1.7, and 1.10 constitute the framework of a plate problem: 11 equations for 11
unknowns,i.e.,Nx; Ny; Nxy; Mx; My; Mxy; Vx; Vy; u0;(cid:23)0,andw0. Inthesubsequentsections,
weshalldropthesubscript0thathasbeenassociatedwiththedisplacementsforthesakeofbrevity.
In-PlaneandOut-Of-PlaneProblems
Asmayberealizedintheequationsderivedintheprevioussection, theproblemcanbede-
composedintotwosetsofproblemswhichareuncoupledwitheachother.
1. In-planeproblems
Theproblemmaybealsocalledastretchingproblemofaplateandisgovernedby
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@N @N
x xy
C Cqx D0
@x @y
@N @N
xy y
C Cqy D0 (1.6a,b)
@x @y
(cid:18) (cid:19)
Et @u @(cid:23)
N D C(cid:23)
x
1−(cid:23)2 @x @y
(cid:18) (cid:19)
Et @(cid:23) @u
N D C(cid:23)
y
1−(cid:23)2 @y @x
(cid:18) (cid:19)
Et @(cid:23) @u
Nxy D Nyx D C (1.10a(cid:24)c)
2.1C(cid:23)/ @x @y
Herewehavefiveequationsforfiveunknowns. Thisproblemcanbeviewedandtreated
inthesamewayasforaplane-stressprobleminthetheoryoftwo-dimensionalelasticity.
2. Out-of-planeproblems
Thisproblemisregardedasabendingproblemandisgovernedby
@V @V
x y
C Cqz D0 (1.6c)
@x @y
@M @M
x xy
C −Vx D0
@x @y
@M @M
xy y
C −Vy D0 (1.7)
@x @y
(cid:18) (cid:19)
@2w @2w
M D −D C
x
@x2 @y2
(cid:18) (cid:19)
@2w @2w
M D −D C
y
@y2 @x2
@2w
Mxy D Myx D−.1−(cid:23)/D (1.10d(cid:24)f)
@x@y
Herearesixequationsforsixunknowns.
EliminatingVx andVy fromEquations1.6cand1.7,weobtain
@2M @2M @2M
x xy y
C2 C Cqz D0 (1.12)
@x2 @x@y @y2
SubstitutingEquations1.10d(cid:24)fintotheabove, weobtainthegoverningequationintermsofdis-
placementas
(cid:18) (cid:19)
@4w @4w @4w
D C2 C Dqz (1.13)
@x4 @x2@y2 @y4
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or
q
z
r4w D (1.14)
D
wheretheoperatorisdefinedas
r4 D r2r2
@2 @2
r2 D C (1.15)
@x2 @y2
1.2.3 Boundary Conditions
Since the in-plane problem of a plate can be treated as a plane-stress problem in the theory of
two-dimensionalelasticity,thepresentsectionisfocusedsolelyonabendingproblem.
Introducingthen-s-zcoordinatesystemalongsideboundariesasshowninFigure1.3,wedefine
themomentsandtheshearingforceas
Z
M D (cid:27) zdz
n n
z Z
Mns D Msn D (cid:28)nszdz (1.16)
Z z
V D (cid:28) dz
n nz
z
Intheplatetheory,insteadofconsideringthesethreequantities,wecombinethetwistingmoment
andtheshearingforcebyreplacingtheactionofthetwistingmomentMns withthatoftheshearing
force,ascanbeseeninFigure1.4. Wethendefinethejointverticalas
@M
ns
Sn DVnC (1.17)
@s
Theboundaryconditionsarethereforegiveningeneralby
w DworSn DSn (1.18)
@w
− D(cid:21)norMn DMn (1.19)
@n
wherethequantitieswithabarareprescribedvaluesandareillustratedinFigure1.5. Thesetwosets
ofboundaryconditionsensuretheuniquesolutionofabendingproblemofaplate.
FIGURE1.3: n-s-zcoordinatesystem.
Theboundaryconditionsforsomepracticalcasesareasfollows:
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