THE MECHANICAL INTERPRETATION OF JOINTS 9 
Thus in Fig. 13 we lay off OA, =0;, OA.= —o., and draw the princi- 
pal circlesonOA,andOA,. For states in which the principal stresses 
do not considerably exceed o, and —o», we may regard the common 
external tangents to these circles as representing approximately a 
portion of the envelope. For the region in which this approxima- 
tion is allowable, the angle between the shearing planes is constant 
and given by 
Un Un Is Fil O02 
acie,  jalbe’ aes (9) 
cos = 
The angle 6 is thus independent of the particular state of stress 
and depends only upon the ratio of ultimate compressive to 
ultimate tensile stress. According as o.> = <o;, 6 will be an 
acute, right, or obtuse angle. For most materials ¢,>0,; values of 
6 corresponding to different values of x in this case are given in the 
following table: 
O02 
Ke aaillo ak 5) 2 3 4 SLO ZO 
Or 
SOs 7 Ge Oo” Be AO” BS AS” 
The principal circle for the state in which o,=—c,, that is, a 
circle about O as center and tangent to the envelope tangents, 
determines the ultimate torsional strength: p,=O7. The ulti- 
mate shearing strength is given by 7,=OS; for this ordinate 
represents the maximum shear for zero normal stress. From the 
geometry of the figure, 
o10> ey 
So T,=3V o.O2. 
P3 Gi ee ae 102 
When a; and a, are nearly equal we have p,=7,=40, a result in 
concordance with many old and modern experiments and not 
satisfactorily explained by the earlier theories of rupture. 
II. MOHR’S THEORY APPLIED TO EXPERIMENTAL DATA 
Opi Os 
=, connects the angle of shearing 
O20; 
with two important physical constants, the ultimate tensile strength 
and the crushing strength. 
Within certain limits, that is to say, inso far as the curve T,, =f(c) 
between the points G, and G, approaches a straight line, it confirms 
Mohr’s formula, cos @= 
