Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

2.4 St. Venant’s Principle 53

FromEq.(vi)

B+ 15 Dh^2 =0(ix)

sothat,subtractingEq.(viii)fromEq.(ix)

D=−

q

40 h^3

Then

B=

3 q

8 h

A=−

q

4

C=−

q
20 h
and

φ=

q

40 h^3

[− 10 h^3 x^2 + 15 h^2 x^2 y− 2 h^2 y^3 −( 5 x^2 y^3 −y^5 )]

Theobviousdisadvantageoftheinversemethodisthatwearedeterminingproblemstofitassumed
solutions,whereasinstructuralanalysisthereverseisthecase.However,insomeproblemstheshapeof
thebodyandtheappliedloadingallowsimplifyingassumptionstobemade,therebyenablingasolution
to be obtained. St. Venant suggested asemi-inverse methodfor the solution of this type of problem
inwhichassumptionsaremadeastostressordisplacementcomponents.Theseassumptionsmaybe
basedonexperimentalevidenceorintuition.St.Venantfirstappliedthemethodtothetorsionofsolid
sections(Chapter3)andtotheproblemofabeamsupportingshearloads(Section2.6).

2.4 St.Venant’sPrinciple................................................................................

IntheexamplesofSection2.3,wehaveseenthataparticularstressfunctionformmaybeapplicable
toavarietyofproblems.Differentproblemsarededucedfromagivenstressfunctionbyspecifying,in
thefirstinstance,theshapeofthebodyandthenassigningavarietyofvaluestothecoefficients.The
resultingstressfunctionsgivestresses,whichsatisfytheequationsofequilibriumandcompatibilityat
allpointswithinandontheboundaryofthebody.Itfollowsthattheappliedloadsmustbedistributed
aroundtheboundaryofthebodyinthesamemannerastheinternalstressesattheboundary.Inthecase
ofpurebending,forexample(Fig.2.2(a)),theappliedbendingmomentmustbeproducedbytensileand
compressiveforcesontheendsoftheplate,theirmagnitudesbeingdependentontheirdistancefrom
theneutralaxis.Ifthisconditionisinvalidatedbytheapplicationofloadsinanarbitraryfashionorby
preventingthefreedistortionofanysectionofthebody,thenthesolutionoftheproblemisnolonger
exact.Asthisisthecaseinpracticallyeverystructuralproblem,itwouldappearthattheusefulnessof
thetheoryisstrictlylimited.Tosurmountthisobstacle,weturntotheimportantprincipleofSt.Venant,
whichmaybesummarizedasstating:

that while statically equivalent systems of forces acting on a body produce substantially different

localeffectsthestressesatsectionsdistantfromthesurfaceofloadingareessentiallythesame.

Therefore, at a section AA close to the end of a beam supporting two point loadsP, the stress
distributionvariesasshowninFig.2.4,whileatthesectionBB,adistanceusuallytakentobegreater
thanthedimensionofthesurfacetowhichtheloadisapplied,thestressdistributionisuniform.