Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

4 CHAPTER 1 Basic Elasticity

Fig.1.

Internal force components at the point O.

forcesδPmaybeconsidereduniformlydistributedoverasmallareaδAofeachfaceoftheplaneatthe
correspondingpointO,asinFig.1.2.ThestressatOisthendefinedbytheequation

Stress= lim

δA→ 0

δP

δA

(1.1)

ThedirectionsoftheforcesδPinFig.1.2aresuchthattheyproducetensilestressesonthefaces
oftheplanenn.ItmustberealizedherethatwhilethedirectionofδPisabsolute,thechoiceofplane
is arbitrary so that although the direction of the stress at O will always be in the direction ofδP, its
magnitudedependsontheactualplanechosen,sinceadifferentplanewillhaveadifferentinclination
andthereforeadifferentvaluefortheareaδA.Thismaybemoreeasilyunderstoodbyreferencetothe
barinsimpletensioninFig.1.3.Onthecross-sectionalplanemm,theuniformstressisgivenbyP/A,
whileontheinclinedplanem′m′,thestressisofmagnitudeP/A′.Inbothcases,thestressesareparallel
tothedirectionofP.
Generally,thedirectionofδPisnotnormaltotheareaδA,inwhichcaseitisusualtoresolveδP
intotwocomponents:one,δPn,normaltotheplaneandtheother,δPs,actingintheplaneitself(see
Fig.1.2).NotethatinFig.1.2theplanecontainingδPisperpendiculartoδA.Thestressesassociated
withthesecomponentsareanormalordirectstressdefinedas

σ=lim

δA→ 0

δPn

δA

(1.2)

andashearstressdefinedas

τ= lim

δA→ 0

δPs

δA

(1.3)