Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

56 CHAPTER 2 Two-Dimensional Problems in Elasticity

Fig.2.6

Bending of an end-loaded cantilever.

giving

A=−

Bb^2

8

Fromthesecond

−

∫b/ 2

−b/ 2

τxydy=P (seesignconventionforτxy)

or

−

∫b/ 2

−b/ 2

(

Bb^2

8

−

By^2

2

)

dy=P

fromwhich

B=−

12 P

b^3

ThestressesfollowfromEqs.(ii)

σx=−

12 Pxy

b^3

=−

Px

I

y

σy= 0

τxy=−

12 P

8 b^3

(b^2 − 4 y^2 )=−

P

8 I

(b^2 − 4 y^2 )

⎫

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎭

(iii)

whereI=b^3 /12thesecondmomentofareaofthebeamcrosssection.
WenotefromthediscussionofSection2.4thatEqs.(iii)representsanexactsolutionsubjecttothe
followingconditionsthat:

(1) theshearforcePisdistributedoverthefreeendinthesamemannerastheshearstressτxygiven

byEqs.(iii)