Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

16.1 General Stress, Strain, and Displacement Relationships 481

Fig.16.2

Axial, tangential, and normal components of displacement of a point in the beam wall.

Fig.16.3

Determination of shear strainγin terms of tangential and axial components of displacement.

Itispossibletoderiveasimpleexpressionforthedirectstrainεsintermsofvt,vn,s,andthecurvature
1/rinthexyplaneofthebeamwall.However,aswedonotrequireεsinthesubsequentanalysis,we
shall,forbrevity,merelyquotetheexpression

εs=

∂vt

∂s

+

vn

r

(16.5)

Theshearstrainγisfoundintermsofthedisplacementswandvtbyconsideringthesheardistortion
ofanelementδs×δzofthebeamwall.FromFig.16.3,weseethattheshearstrainisgivenby

γ=φ 1 +φ 2

or,inthelimitasbothδsandδztendtozero

γ=

∂w

∂s

+

∂vt

∂z

(16.6)