Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

16 CHAPTER 1 Basic Elasticity

Itfollowsthat

sin2θ=

−(σx−σy)

√

(σx−σy)^2 + 4 τxy^2

cos2θ=

2 τxy

√

(σx−σy)^2 + 4 τxy^2

sin2(θ+π/ 2 )=

(σx−σy)

√

(σx−σy)^2 + 4 τxy^2

cos2(θ+π/ 2 )=

− 2 τxy

√

(σx−σy)^2 + 4 τxy^2

SubstitutingthesevaluesinEq.(1.9)gives

τmax,min=±

1

2

√

(σx−σy)^2 + 4 τxy^2 (1.14)

Here, as in the case of principal stresses, we take the maximum value as being the greater algebraic
value.
ComparingEq.(1.14)withEqs.(1.11)and(1.12),weseethat

τmax=

σI−σII

2

(1.15)

Equations(1.14)and(1.15)givethemaximumshearstressatthepointinthebodyintheplaneof
thegivenstresses.Forathree-dimensionalbodysupportingatwo-dimensionalstresssystem,thisisnot
necessarilythemaximumshearstressatthepoint.
Since Eq. (1.13) is the negative reciprocal of Eq. (1.10), then the angles 2θgivenbythesetwo
equations differ by 90◦, or the planes of maximum shear stress are inclined at 45◦to the principal
planes.

1.8 Mohr’sCircleofStress..............................................................................

ThestateofstressatapointinadeformablebodymaybedeterminedgraphicallybyMohr’scircleof
stress.
InSection1.6,thedirectandshearstressesonaninclinedplaneweregivenby

σn=σxcos^2 θ+σysin^2 θ+τxysin2θ (Eq.(1.8))

and

τ=

(σx−σy)

2

sin2θ−τxycos2θ (Eq.(1.9))

respectively. The positive directions of these stresses and the angleθare defined in Fig. 1.12(a).
Equation(1.8)mayberewrittenintheform

σn=

σx

2

( 1 +cos2θ)+

σy

2

( 1 −cos2θ)+τxysin2θ