Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

15.2 Unsymmetrical Bending 437

The moment resultants of the internal direct stress distribution have the same sense as the applied
momentsMxandMy.Therefore,

Mx=

∫

A

σzydA, My=

∫

A

σzxdA (15.17)

SubstitutingforσzfromEq.(15.16)in(15.17)anddefiningthesecondmomentsofareaofthesection
abouttheaxesCx,Cyas

Ixx=

∫

A

y^2 dA, Iyy=

∫

A

x^2 dA, Ixy=

∫

A

xydA

gives

Mx=

Esinα

ρ

Ixy+

Ecosα

ρ

Ixx, My=

Esinα

ρ

Iyy+

Ecosα

ρ

Ixy

or,inmatrixform
{
Mx
My

}

=

E

ρ

[

Ixy Ixx

Iyy Ixy

]{

sinα

cosα

}

fromwhich

E

ρ

{

sinα

cosα

}

=

[

Ixy Ixx

Iyy Ixy

]− 1 {

Mx

My

}

thatis,

E

ρ

{

sinα

cosα

}

=

1

IxxIyy−Ixy^2

[

−Ixy Ixx

Iyy −Ixy

]{

Mx

My

}

sothat,fromEq.(15.16),

σz=

(

MyIxx−MxIxy

IxxIyy−Ixy^2

)

x+

(

MxIyy−MyIxy

IxxIyy−Ixy^2

)

y (15.18)

Alternatively,Eq.(15.18)mayberearrangedintheform

σz=

Mx(Iyyy−Ixyx)

IxxIyy−Ixy^2

+

My(Ixxx−Ixyy)

IxxIyy−Ixy^2

(15.19)

FromEq.(15.19)itcanbeseenthatif,say,My=0,themomentMxproducesastresswhichvarieswith
bothxandy;similarlyforMyifMx=0.
Inthecasewherethebeamcrosssectionhaseither(orboth)CxorCyasanaxisofsymmetry,the
productsecondmomentofareaIxyiszeroandCxyareprincipalaxes.Equation(15.19)thenreducesto

σz=

Mx

Ixx

y+

My

Iyy

x (15.20)