648 REPORT—1883. 
may be made the fresh air inlet tunnel; so that the gangs of men working in it 
may have perfectly fresh air to work in, and be free from all danger from the run- 
ning trains. Ifa breakdown of a train occurs in any one tunnel, that can be at 
once converted into the fresh-air inlet tunnel, whilst the traffic is carried on through 
the other two, thereby avoiding delay. It was explained that in the event of motors 
being invented requiring no ventilation, an additional line of rails would be available 
for traffic without further delay. 
A permanent way was described, by the adoption of which much economy of 
labour for repairs would result. 
6. On the Resistance of Beams when strained beyond the Elastic Limit. 
By Watrer R. Browne, M.A., MInst.0.E. 
It is well known that the ordinary theory of the resistance of beams to trans- 
verse strain depends on the following assumptions :— 
(1) All straight lines normal to the axis of the beam in its unstrained condition 
remain straight and normal to the axis in its strained condition. 
(2) Hooke’s law holds; that is, the strain on each layer or fibre is proportional 
to the stress causing it, 
(3) The modulus of elasticity is the same on both sides of the neutral axis; 
ze, the extension and compression produced by equal stresses are themselves equal. 
It is not generally pointed out that the second of these assumptions tacitly in- 
volves another, which is as follows:— 
(4) The shearing stress acting between the successive layers or fibres may be 
neglected; in other words, the resistance offered by each fibre to the tensile stress 
is the same as if it were not connected to the fibres above and below it in any way. 
Let M be the bending moment at any given section of a rectangular beam, 
y the distance of any fibre from the neutral axis, T the unit stress on that fibre, 
R the radius of curvature; then the above assumptions lead to the equations— 
E 
Tay 
u=| Tdy x y= R| vay 
Now if the shearing stress between any two fibres is to be neglected, it follows 
that the shearing strain, or the amount by which one surface has shifted over to 
the other, must be small. For cases below the elastic limit, in which the original 
normal sections still continue normal, two successive layers are strained so nearly 
by the same amount that their difference in length—in other words, the distance by 
which they have shifted over each other—will be excessively small. Hence, so 
long as the tensile stress, or T, is within the elastic limit of the material, this con- 
dition holds; but when T passes this limit, and especially when it approaches the 
ultimate tensile strength, the case is different. 
We may refer to the actual extension of a bar of mild Siemens steel, as de- 
termined by Professor Kennedy (‘ Proceedings Inst. Mech. Engineers,’ 1881, plate 30), 
under stresses varying from 0 to 60,000 lbs. per square inch. The same figure will 
represent the actual extension in the successive layers of the extension side of a steel 
bar of the length and of the depth shown, provided we assume that the stress on 
vnese layers increases uniformly from the neutral axis at P to the outside at A, as 
in the ordinary theory of elasticity it is supposed todo. This assumption, as we 
have seen, involves the hypothesis that the shearing stress between the different 
layers may be neglected, and for this it is necessary that the extension of any one 
layer beyond that next to it should be small. Is this thecase? On looking at the 
figure, we see that the difference in successive extensions is very small up to a point 
L, where the stress is about 41,000 Ibs. per square inch. At this point, however, 
the extension increases by about 2 inch (in 40 inches), without any further 
increase of stress; and it then goes on increasing rapidly up to fracture. 
Let us consider the behaviour of the fibre at L (taken to be the outside fibre of 
