Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

70 CHAPTER 3 Torsion of Solid Sections

Fig.3.4

Rigid body displacement in the cross section of the bar.

or

u=−θyv=θx (3.9)

ReferringtoEqs.(1.20)and(1.46)

γzx=

∂u

∂z

+

∂w

∂x

=

τzx

G

γzy=

∂w

∂y

+

∂v

∂z

=

τzy

G

RearrangingandsubstitutingforuandvfromEqs.(3.9)

∂w

∂x

=

τzx

G

+

dθ

dz

y

∂w

∂y

=

τzy

G

−

dθ

dz

x (3.10)

For a particular torsion problem Eqs. (3.10) enable the warping displacementwof the originally
planecrosssectiontobedetermined.Notethatsinceeachcrosssectionrotatesasarigidbody,θisa
functionofzonly.
DifferentiatingthefirstofEqs.(3.10)withrespecttoy,thesecondwithrespecttox,andsubtracting,
wehave

0 =

1

G

(

∂τzx

∂y

−

∂τzy

∂x

)

+ 2

dθ

dz

Expressingτzxandτzyintermsofφgives

∂^2 φ

∂x^2

+

∂^2 φ

∂y^2

=− 2 G

dθ

dz

or,fromEq.(3.4)

− 2 G

dθ

dz

=∇^2 φ=F(constant) (3.11)