Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

54 CHAPTER 2 Two-Dimensional Problems in Elasticity

Fig.2.4

Stress distributions illustrating St. Venant’s principle.

We may, therefore, apply the theory to sections of bodies away from points of applied loading
or constraint. The determination of stresses in these regions requires, for some problems, separate
calculation.

2.5 Displacements........................................................................................

Havingfoundthecomponentsofstress,Eq.(1.47)(forthecaseofplanestress)isusedtodetermine
the components of strain. The displacements follow from Eqs. (1.27) and (1.28). The integration of
Eqs.(1.27)yieldssolutionsoftheform

u=εxx+a−by (2.10)

v=εyy+c+bx (2.11)

in whicha,b,andcare constants representing movement of the body as a whole orrigid bodydis-
placements.Ofthese,aandcrepresentpuretranslatorymotionsofthebody,whilebisasmallangular
rotationofthebodyinthexyplane.Ifweassumethatbispositiveinananticlockwisesense,thenin
Fig.2.5thedisplacementv′duetotherotationisgivenby

v′=P′Q′−PQ

=OPsin(θ+b)−OPsinθ

which,sincebisasmallangle,reducesto

v′=bx

Similarly,

u′=−byasstated