Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

3.1Prandtl Stress Function Solution 71

ItisconvenienttointroduceatorsionconstantJdefinedbythegeneraltorsionequation

T=GJ

dθ

dz

(3.12)

The productGJis known as thetorsional rigidityof the bar and may be written, from Eqs. (3.8)
and(3.11),

GJ=−

4 G

∇^2 φ

∫∫

φdxdy (3.13)

ConsidernowthelineofconstantφinFig.3.5.Ifsisthedistancemeasuredalongthislinefrom
somearbitrarypoint,then

∂φ

∂s

= 0 =

∂φ

∂y

dy

ds

+

∂φ

∂x

dx

ds

UsingEqs.(3.2)and(3.6),wemayrewritethisequationas

∂φ

∂s

=τzxl+τzym= 0 (3.14)

FromFig.3.5thenormalandtangentialcomponentsofshearstressare

τzn=τzxl+τzym τzs=τzyl−τzxm (3.15)

ComparingthefirstofEqs.(3.15)withEq.(3.14),weseethatthenormalshearstressiszerosothat
theresultantshearstressatanypointistangentialtoalineofconstantφ.Theseareknownaslinesof
shearstressorshearlines.
SubstitutingφinthesecondofEqs.(3.15),wehave

τzs=−

∂φ

∂x

l−

∂φ

∂y

m

Fig.3.5

Lines of shear stress.