Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

544 CHAPTER 19 Structural Idealization

Theseshearloadsproducedirectstressesduetobendingintheboomsandskinandshearstressesin
theskin.Supposethattheshearflowsintheskinadjacenttotherthboomofcross-sectionalareaBrare
q 1 andq 2 .Then,fromFig.19.6(b),

(

σz+

∂σz

∂z

δz

)

Br−σzBr+q 2 δz−q 1 δz= 0

whichsimplifiesto

q 2 −q 1 =−

∂σz

∂z

Br (19.4)

SubstitutingforσzinEq.(19.4)from(15.18),wehave

q 2 −q 1 =−

[

(∂My/∂z)Ixx−(∂Mx/∂z)Ixy

IxxIyy−I^2 xy

]

Brxr

−

[

(∂Mx/∂z)Iyy−(∂My/∂z)Ixy

IxxIyy−I^2 xy

]

Bryr

or,usingtherelationshipsofEqs.(15.23)and(15.24),

q 2 −q 1 =−

(

SxIxx−SyIxy

IxxIyy−Ixy^2

)

Brxr−

(

SyIyy−SxIxy

IxxIyy−Ixy^2

)

Bryr (19.5)

Equation(19.5)givesthechangeinshearflowinducedbyaboomwhichitselfissubjectedtoadirect
load(σzBr).Eachtimeaboomisencountered,theshearflowisincrementedbythisamountsothatif,
atanydistancesaroundtheprofileofthesection,nboomshavebeenpassed,theshearflowatthepoint
isgivenby

qs=−

(

SxIxx−SyIxy

IxxIyy−Ixy^2

)⎛

⎝

∫s

0

tDxds+

∑n

r= 1

Brxr

⎞

⎠

−

(

SyIyy−SxIxy

IxxIyy−Ixy^2

)⎛

⎝

∫s

0

tDyds+

∑n

r= 1

Bryr

⎞

⎠

(19.6)

Example 19.3
Calculate the shear flow distribution in the channel section shown in Fig. 19.7 produced by a ver-
tical shear load of 4.8kN acting through its shear center. Assume that the walls of the section are
onlyeffectiveinresistingshearstresses,whilethebooms,eachofarea300mm^2 ,carryallthedirect
stresses.