Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

3.2St. Venant Warping Function Solution 75

bythesevaluesgives

T=G

dθ

dz

πa^3 b^3

(a^2 +b^2 )

(iv)

fromwhich(seeEq.(3.12))

J=

πa^3 b^3

(a^2 +b^2 )

(v)

The shear stress distribution is obtained in terms of the torque by substituting for the product
G(dθ/dz)inEq.(iii)fromEq.(iv)andthendifferentiatingasindicatedbytherelationshipsofEqs.(3.2).
Thus,

τzx=−

2 Ty

πab^3

τzy=

2 Tx

πa^3 b

(vi)

Sofarwehavesolvedforthestressdistribution,Eqs.(vi),andtherateoftwist,Eq.(iv).Itremains
todeterminethewarpingdistributionwoverthecrosssection.ForthiswereturntoEqs.(3.10)which
become,onsubstitutingfromtheprecedingforτzx,τzy,anddθ/dz

∂w

∂x

=−

2 Ty

πab^3 G

+

T

G

(a^2 +b^2 )

πa^3 b^3

y

∂w

∂y

=

2 Tx

πa^3 bG

−

T

G

(a^2 +b^2 )

πa^3 b^3

x

or

∂w

∂x

=

T

πa^3 b^3 G

(b^2 −a^2 )y

∂w

∂y

=

T

πa^3 b^3 G

(b^2 −a^2 )x (vii)

IntegratingbothofEqs.(vii)

w=

T(b^2 −a^2 )

πa^3 b^3 G

yx+f 1 (y) w=

T(b^2 −a^2 )

πa^3 b^3 G

xy+f 2 (x)

Thewarpingdisplacementgivenbyeachoftheseequationsmusthavethesamevalueatidenticalpoints
(x,y).Itfollowsthatf 1 (y)=f 2 (x)=0.Hence,

w=

T(b^2 −a^2 )

πa^3 b^3 G

xy (viii)

Linesofconstantw,therefore,describehyperbolaswiththemajorandminoraxesoftheellipticalcross
sectionasasymptotes.Further,forapositive(anticlockwise)torquethewarpingisnegativeinthefirst
andthirdquadrants(a>b)andpositiveinthesecondandfourth.

3.2 St.VenantWarpingFunctionSolution............................................................

In formulating his stress function solution, Prandtl made assumptions concerned with the stress dis-
tribution in the bar. The alternative approach presented by St. Venant involves assumptions as to the
mode of displacement of the bar—namely, that cross sections of a bar subjected to torsion maintain