Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

10 CHAPTER 1 Basic Elasticity

Fig.1.

Stresses on the faces of an element at the boundary of a two-dimensional body.

which,bytakingthelimitasδxapproacheszero,becomes

X=σx

dy

ds

+τyx

dx
ds
Thederivativesdy/dsanddx/dsarethedirectioncosineslandmoftheanglesthatanormaltoAB
makeswiththexandyaxes,respectively.Itfollowsthat

X=σxl+τyxm

andinasimilarmanner,

Y=σym+τxyl

A relatively simple extension of this analysis produces the boundary conditions for a three-
dimensionalbody,namely

X=σxl+τyxm+τzxn

Y=σym+τxyl+τzyn

Z=σzn+τyzm+τxzl

⎫

⎪⎬

⎪⎭

(1.7)

wherel,m,andnbecomethedirectioncosinesoftheanglesthatanormaltothesurfaceofthebody
makeswiththex,y,andzaxes,respectively.

1.6 DeterminationofStressesonInclinedPlanes....................................................

ThecomplexstresssystemofFig.1.6isderivedfromaconsiderationoftheactualloadsappliedtoa
bodyandisreferredtoapredetermined,thougharbitrary,systemofaxes.Thevaluesofthesestresses
may not give a true picture of the severity of stress at that point, so it is necessary to investigate the