Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

8 CHAPTER 1 Basic Elasticity

Fig.1.

Stresses on the faces of an element at a point in an elastic body.

whichsimplifiesto

τxyδyδzδx+

∂τxy

∂x

δyδz

(δx)^2

2

−τyxδxδzδy−

∂τyx

∂y

δxδz

(δy)^2

2

= 0

Dividingbyδxδyδzandtakingthelimitasδxandδyapproachzero.

Similarly,

τxy=τyx

τxz=τzx

τyz=τzy

⎫

⎬

⎭

(1.4)

Wesee,therefore,thatashearstressactingonagivenplane(τxy,τxz,τyz)isalwaysaccompaniedby
anequalcomplementary shearstress(τyx,τzx,τzy)actingonaplaneperpendiculartothegivenplane
andintheoppositesense.
Nowconsideringtheequilibriumoftheelementinthexdirection
(
σx+

∂σx

∂x

δx

)

δyδz−σxδyδz+

(

τyx+

∂τyx

∂y

δy

)

δxδz

−τyxδxδz+

(

τzx+

∂τzx

∂z

δz

)

δxδy

−τzxδxδy+Xδxδyδz= 0