1915-16.] Torsional Vibration of Beams of Commercial Section. 33 
of projecting points or corners, the inefficiency of the section as a whole 
becoming more marked as the elementary circular section is departed from. 
From a consideration of the strains produced, consequent upon the 
application of a torque, St Venant has deduced expressions for the effective 
polar moment of inertia for the commoner non-circular sections.* Writing 
formula (1) in the form 
,_TL 
6 CJ'’ 
where J' is the effective value of J, the following table gives values of 
J' for the more important of the simpler sections : — 
Table I. 
Section. 
Remarks. 
Effective Value of 
J( = J'). 
Square .... 
Side = s 
T4 s 4 . 
Rectangle . . j 
Breadth = b . 
Depth = d . 
dbH *66/ ¥ \1 
3 L d\ 
Any symmetrical section 
including rectangles in 
which ratio of outside 
dimensions in any two 
directions in a cross 
section is small. 
A = Area of section. 
J = Geometrical polar 
moment of inertia. 
A 4 
40J ' 
Sections other than the above, i.e. I, channel, and similar commercial 
sections, do not readily yield to mathematical treatment, and resort must 
be made to experimental research to determine the efficiency of such 
sections under torsion. Experiments carried out by the author indicate 
that such sections as are used in engineering practice are particularly 
weak when subjected to the action of a twisting-moment. In the case 
of an I section 8" deep x 5" flange, for instance, the section developed 
only -01 of its theoretical torsional strength, as calculated from the geo- 
metrical properties of the section. The following formula was deduced 
from the result of a large number of experiments carried out on various 
sections : — 
If J' is the effective polar moment of inertia, and if A is the area of 
the section, then 
A n 
J' 
* See Todhunter and Pearson, History of Elasticity , vol. ii. 
VOL. XXXVI. 
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