Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

1.8Mohr’s Circle of Stress 17

Fig.1.12

(a) Stresses on a triangular element; (b) Mohr’s circle of stress for stress system shown in (a).

or

σn−

1

2

(σx+σy)=

1

2

(σx−σy)cos2θ+τxysin2θ

SquaringandaddingthisequationtoEq.(1.9),weobtain
[
σn−

1

2

(σx+σy)

] 2

+τ^2 =

[

1

2

(σx−σy)

] 2

+τxy^2

whichrepresentstheequationofacircleofradius^12

√

(σx−σy)^2 + 4 τxy^2 andhavingitscenteratthepoint

((σx−σy)/2,0).
ThecircleisconstructedbylocatingthepointsQ 1 (σx,τxy)andQ 2 (σy,−τxy)referredtoaxesOστas
showninFig.1.12(b).ThecenterofthecirclethenliesatCtheintersectionofQ 1 Q 2 andtheOσaxis;

clearlyCisthepoint((σx−σy)/2,0),andtheradiusofthecircleis^12

√

(σx−σy)^2 + 4 τxy^2 asrequired.

CQ′is now set off at an angle 2θ(positive clockwise) to CQ 1 , and Q′is then the point (σn,−τ)as
demonstratedinthefollowing.FromFig.1.12(b),weseethat

ON=OC+CN

orsinceOC=(σx+σy)/2,CN=CQ′cos(β− 2 θ),andCQ′=CQ 1 ,wehave

σn=

σx+σy

2

+CQ 1 (cosβcos2θ+sinβsin2θ)

But

CQ 1 =

CP 1

cosβ

and CP 1 =

(σx−σy)

2