174 APPLIED MECHANICS 
centre is sufficient to counteract the tendency to break at the centre, as 
explained above. 
In the case of brittle materials, where the contraction is negligitil 
the position of the fracture is either at the weakest section, or at the 
section on which the distribution of stress is most variable. The fracture 
is therefore at the weakest section, or near one end. 
164. Long versus Short Tension Test Bars.—Every np is 
more or less variable in quality, and test pieces taken from the same 
piece will probably have different tenacities, In a long bar there will be 
one part weaker than the remainder, and if this bar is tested as a whole, 
it will probably fracture at the weakest part. But if a number of test 
bars be cut from the long bar, only one of them will contain the original 
weakest part. Hence the short bars will generally show a higher tenacity 
than long ones. A familiar illustration of the above is found in boot- 
laces. In tightening the lace it is more likely to break when the pull is 
applied at the end than when the pull is applied locally at the boot. 
165. Effect of Notches and Perforations on. Tenacity of Test — 
Bars.—Reducing the cross section of a bar by notching it, as shown at — 
(a) and (0), Fig. 246, or by perforating it, as shown at (c) and (d), will 
evidently fix the position of the frac- 
ture, and as the probability is in favour a Be 
of this not being at the section where the | 
material is weakest, the tenacity of the Oo oO 
notched or perforated bar will for this (b d)\ 
reason probably be higher than that of (a) ) (c) (d) ; 
the unaltered bar. Again, the notching imi Ee 
or perforating will evidently disturb the Fra. 246. 
distribution of the stress at the reduced 
section, making it less uniform, with the result that the contraction of 
area will be reduced. On this account, therefore, the tenacity would be ~ 
increased. On the other hand, however, the notch or perforation may — 
cause such an unequal distribution of stress that fracture may take 
place in consequence and the tenacity be reduced. The effect of the 
notch or perforation in reducing the tenacity will evidently be greater — 
the sharper the re-entrant angle formed by the notch or perforation, 
and it will also be greater the more brittle the material, because a brittle 
material does not yield sufficiently where the stress is greatest to throw 
part of the stress on to the part of the bar where the stress is least. 
Notching or perforating a bar of mild steel raises its tenacity, but 
notching a piece of cast-iron lowers its tenacity very considerably. 
Another important point to consider is the effect of the notch or 
perforation on the resilience of the bar, which is a measure of its power — 
to resist shocks. Let 7 be the length of a bar of uniform cross section @ — 
Let A be the area of the cross section of another bar of the same length, 
but having an indefinitely narrow notch in it, the area of the section at 
the bottom of the notch being a. Let fbe the maximum tensile stress on — 
each bar. Then the stress on the second bar, except at the notch, is ; i. 
The resilience of the first bar is ~ (see Art. 88, page 69). The 
