Mr. W. H. Barlow on an Element of Strength in Beams. 4973 
strength of the metal as obtained by direct experiment. From this 
comparison applied to the different forms of beam, it would follow 
that the resistance at the outer fibre varies from 25,271 lbs. to 
53,966 lbs., while the tensile strength of the metal, obtained by ex- 
periments on direct tension, averages only 18,750 lbs.; and the dis- 
crepancy and variation will be found to arise from the received 
theory not taking into account the resistance consequent on the 
molecular disturbance accompanying curvature. 
In his former paper the author gave a formula by which the dif- 
ference between the tensile strength and the apparent resistance at 
the outer fibre could be computed, approximatively, in solid rectan- 
gular beams and open girders; and he now proposes to trace the 
operation of the resistance of flexure, considered as a separate ele- 
ment of strength, and to show its effect, in each of the forms of 
section above indicated. Observing that the usual supposition of only 
two resistances in a beam, tension and compression, fails to account 
either for the strength, or for the visible changes of figure which 
take place under transverse strain, he proceeds to discuss the effects 
involved in such change of figure, and thence arrives at the follow- 
ing conclusions applicable:to the resistance of flexure :— 
1. That it is a resistance acting in addition to the direct exten- 
sion and to compression. 
2. That it is evenly distributed over the surface, and consequently 
(within the limits of its operation) its points of action will be at the 
centres of gravity of the half-section. 
3. That this uniform resistance is due to the lateral cohesion of 
the adjacent surfaces of the fibres or particles, and to the elastic 
reaction which thus ensues between the portions of a beam unequally 
strained. 
4. That it is proportional to and varies with the inequality of 
strain, as between the fibres or particles nearest the neutral axis and 
those most remote. 
Formule are then given, according to these principles, exhibiting 
the relation between the straining and resisting forces in the several 
forms of section experimented on, as resulting from the joint effect™ 
of the resistances of tension, compression and flexure. The appli- 
cation of these formulz to the actual experiments yields a series of 
equations with numerical coefficients, in which, were the metal of 
uniform strength, the tensile strength f, and the resistance of flexure 
¢, would be constant quantities, and their value might be obtained 
from any two of the equations ; but as the strength varies even in 
castings of the same dimensions, and as a reduction of strength per 
unit of section takes place when the thickness is increased, the values 
of f and @ will necessarily vary, and can only be ascertained in each 
experiment by first establishing the ratio they bear to each other. 
For this purpose the first ten experiments are used, in all of which 
the metal was from ? to 1 inch in thickness, and its mean tensile 
strength ascertained by direct experiment to be 18,750 Ibs. per inch. 
The resulting mean value of » is=16,573 lbs., and the ratio of f to » 
as 1 to *847. 
