Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

18 CHAPTER 1 Basic Elasticity

Hence,

σn=

σx+σy

2

+

(

σx−σy

2

)

cos2θ+CP 1 tanβsin2θ

which,onrearranging,becomes

σn=σxcos^2 θ+σysin^2 θ+τxysin2θ

asinEq.(1.8).Similarly,itmaybeshownthat

Q′N=τxycos2θ−

(

σx−σy

2

)

sin2θ=−τ

asinEq.(1.9).NotethattheconstructionofFig.1.12(b)correspondstothestresssystemofFig.1.12(a)
so that any sign reversal must be allowed for. Also, the Oσand Oτaxes must be constructed to the
samescale,ortheequationofthecircleisnotrepresented.
The maximum and minimum values of the direct stress—that is, the major and minor principal
stressesσIandσII—occurwhenNandQ′coincidewithBandA,respectively.Thus,

σ 1 =OC+radiusofcircle

=

(σx+σy)

2

+

√

CP^21 +P 1 Q^21

or

σI=

(σx+σy)

2

+

1

2

√

(σx−σy)^2 + 4 τxy^2

andinthesamefashion

σII=

(σx+σy)

2

−

1

2

√

(σx−σy)^2 + 4 τxy^2

Theprincipalplanesarethengivenby2θ=β(σI)and2θ=β+π(σII).
AlsothemaximumandminimumvaluesofshearstressoccurwhenQ′coincideswithDandEat
theupperandlowerextremitiesofthecircle.
Atthesepoints,Q′Nisequaltotheradiusofthecirclewhichisgivenby

CQ 1 =

√

(σx−σy)^2

4

+τxy^2

Henceτmax,min=±^12

√

(σx−σy)^2 + 4 τxy^2 asbefore.Theplanesofmaximumandminimumshearstresses

aregivenby2θ=β+π/2and2θ=β+ 3 π/2,thesebeinginclinedat45◦totheprincipalplanes.

Example 1.3
Directstressesof160N/mm^2 (tension)and120N/mm^2 (compression)areappliedataparticularpointin
anelasticmaterialontwomutuallyperpendicularplanes.Theprincipalstressinthematerialislimited