CffiCULAR CYLINDERS UNDER CERTAIN PRACTICAL SYSTEMS OF LOAD. 207 
§ 22. Principal Stresses at each Point. Lines of Principal Stress. 
Now, when we have a distribution of stress 
rr, zz, rz, 
which is symmetrical aliout an axis, then the principal stresses are 
Ell, c^, ZZ; 
where is the same as before, and EE, ZZ are two tractions in the meridian plane, 
EPt making an angle 6 with rr, where 
tan 26 = 
izr 
rr — zz 
and the values of EPt and ZZ are given by 
. ( 110 ), 
RR = + i- ' 
rr — 
+ 4 {zrf 
2Z = - 4 V - zzf + 4 [zr)~ 
111 ). 
Whence, using the tables in § 21, we find the following values for EE and ZZ, 
compared with the mean pressure over the ends. 
Table of Princijjal Stresses. 
RR/Q. 
r. 
z = 
2 ; = cf. 
0 
II 
2 = 3t-/6. 
0 
• 
II 
2 = 5c/6. 
r. 
0 
- -00274 
- -01414 
- -05134 
- -12416 
- -25325 
- -48151 
- -89668 
ajZ 
- -00181 
- -01113 
- -04199 
- -10178 
- -20063 
- -33690 
- -34800 
2ap 
- -00110 
- -00457 
- -01758 
- -04405 
- -08433 
- -09804 
-09139 
a 
-00000 
-00000 
-00000 
- 00000 
-00000 
-00000 
-00000 
ZZ/Q. 
r. 
z = 0. 
0 
II 
M 
1 
^ = 2c/Q. 
^ = 3c/Q. 
z = 4c/6. 
z = 5c/6. 
= c. 
0 
- 1-13382 
- 1-13436 
- 1-13322 
- 1-12146 
- 1-08000 
- 1-03372 
- 0 - 68576 
a/3 
- 1-09971 
- 1-10057 
- 1-10018 
- 1-09039 
- 1-05849 
-0-99907 
- 1-06087 
2a/3 
-1-00724 
- 1-00898 
- 1-01314 
- 1-01628 
- 1-01467 
- 1-01602 
- 1-11961 
a 
-0-89430 
-0-88809 
-0-87216 
-0-85845 
-0-88177 
- 1-04077 
- 1-68635 
