4-72 
ME. W. H. BAELOW ON THE EESISTANCE OE ELESUEE 
Referring again to fig, 3, if J represents any point in the upper surface of a solid 
beam, before strain is applied, and g the same point when loaded, hr* will Tary directly 
as 7'g. But rg represents the difference between the extension of the fibre, at or nearest 
the neutral axis, and that at the outer portion of the beam ; therefore the resistance to 
flexure will vaiy directly as this difference. 
In the case of the open beam, the resistance to flexure being only due to that of the 
bar deflected, whereas the ultimate deflection of the beam is equal to that of a sohd beam 
of the same total depth, the resistance of flexm-e in the open beam will be to that of 
the solid beam, at the moment of rupture, as the depth of the bar to the half-depth of 
the beam ; and this is also proportional to the difference between the extension of the 
fibres nearest the neutral axis, and those at the outer portion of the beam. 
The foregoing consideration of the subject, therefore, points out the following proper- 
ties as belonging to the resistance of flexure : — 
1st. That it is a resistance acting in addition to the chrect extension and compression. 
2nd. That it is evenly distributed over the smTace, and consequently (within the limits 
of its operation) its points of action will be at the centres of grarity of the half-section. 
3rd. That this uniform resistance is due to the lateral cohesion of the adjacent sur- 
faces of the fibres or particles, and to the elastic reaction which thus ensues between the 
portions of a beam unequally strained. 
4th. That it is proportional to, and varies with, the mequality of strain between the 
fibres or particles nearest the neutral axis and those most remote. 
We are enabled, under the above-mentioned conditions, to arrive at the relation 
between the straining and resisting forces in any of the forms of section experimented 
upon, as resulting from the combined effect of the resistances of tension, compression 
and flexure. 
Using the same letters as before to represent the tension, weight, length, depth, &c., 
let Ip = the resistance of flexure acting as a force evenly spread over the siuTace of the 
section. 
1 
Then, instead of the expression y, as representing the resistance at the distance x, 
we shall have, according to the preceding view, the expression 
and these forces acting as before, the moment will be 
The sum of these moments, including those above and below the neutral axis, uill be 
which, taken between the limits ^=0 and x—d, becomes 
* r, wliieh is aot represented in tlie figure, is the intersection of the lines hn, pg. 
