Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

68 CHAPTER 3 Torsion of Solid Sections

Thus,φisconstantonthesurfaceofthebar,andsincetheactualvalueofthisconstantdoesnotaffect
thestressesofEq.(3.2),wemayconvenientlytaketheconstanttobezero.Hence,onthecylindrical
surfaceofthebar,wehavetheboundarycondition

φ= 0 (3.7)

Ontheendsofthebar,thedirectioncosinesofthenormaltothesurfacehavethevaluesl=0,m=0,
andn=1.Therelatedboundaryconditions,fromEqs.(1.7),arethen

X=τzx

Y=τzy

Z= 0

Wenowobservethattheforcesoneachendofthebarareshearforceswhicharedistributedoverthe
ends of the bar in the same manner as the shear stresses are distributed over the cross section. The
resultantshearforceinthepositivedirectionofthexaxis,whichweshallcallSx,isthen

Sx=

∫∫

Xdxdy=

∫∫

τzxdxdy

or,usingtherelationshipofEqs.(3.2),

Sx=

∫∫

∂φ

∂y

dxdy=

∫

dx

∫

∂φ

∂y

dy= 0

asφ=0attheboundary.Inasimilarmanner,Sy,theresultantshearforceintheydirection,is

Sy=−

∫

dy

∫

∂φ

∂x

dx= 0

Itfollowsthatthereisnoresultantshearforceontheendsofthebarandtheforcesrepresentatorque
ofmagnitude,referringtoFig.3.3

T=

∫∫

(τzyx−τzxy)dxdy

inwhichwetakethesignofTasbeingpositiveintheanticlockwisesense.
Rewritingthisequationintermsofthestressfunctionφ

T=−

∫∫

∂φ

∂x

xdxdy−

∫∫

∂φ

∂y

ydxdy

Integratingeachtermontheright-handsideofthisequationbyparts,andnotingagainthatφ=0atall
pointsontheboundary,wehave

T= 2

∫∫

φdxdy (3.8)