1.5 Boundary Conditions 9whichgives
∂σx
∂x+
∂τyx
∂y+
∂τzx
∂z+X= 0
Or,writingτxy=τyxandτxz=τzxfromEq.(1.4).
Similarly,∂σx
∂x+
∂τxy
∂y+
∂τxz
∂z+X= 0
∂σy
∂y+
∂τyx
∂x+
∂τyz
∂z+Y= 0
∂σz
∂z+
∂τzx
∂x+
∂τzy
∂y+Z= 0
⎫
⎪⎪
⎪⎪
⎪⎪
⎪⎬
⎪⎪
⎪⎪
⎪⎪
⎪⎭
(1.5)
Theequationsofequilibriummustbesatisfiedatallinteriorpointsinadeformablebodyundera
three-dimensionalforcesystem.
1.4 PlaneStress...........................................................................................
Most aircraft structural components are fabricated from thin metal sheet so that stresses across the
thickness of the sheet are usually negligible. Assuming, say, that thezaxis is in the direction of the
thickness,thenthethree-dimensionalcaseofSection1.3reducestoatwo-dimensionalcaseinwhich
σz,τxz,andτyzare all zero. This condition is known asplane stress; the equilibrium equations then
simplifyto
∂σx
∂x+
∂τxy
∂y+X= 0
∂σy
∂y+
∂τyx
∂x+Y= 0
⎫
⎪⎪
⎬
⎪⎪
⎭
(1.6)
1.5 BoundaryConditions................................................................................
Theequationsofequilibrium(1.5)(andalso(1.6)foratwo-dimensionalsystem)satisfytherequirements
ofequilibriumatallinternalpointsofthebody.Equilibriummustalsobesatisfiedatallpositionson
theboundaryofthebodywherethecomponentsofthesurfaceforceperunitareaareX,Y,andZ.The
triangularelementofFig.1.7attheboundaryofatwo-dimensionalbodyofunitthicknessisthenin
equilibriumundertheactionofsurfaceforcesontheelementallengthABoftheboundaryandinternal
forcesoninternalfacesACandCB.
Summationofforcesinthexdirectiongives
Xδs−σxδy−τyxδx+X1
2
δxδy= 0