68 APPLIED MECHANICS 
be determined without great error by assuming that the stress is every- 
where uniformly distributed and given by the formula f= P/s, where fis 
the stress at a cross section whose area is a, and P is the load. 
The effect of sudden changes of section on the behaviour of a loaded 
bar is further considered in Article 165, p. 174. 
85. Strength and Factor of Safety.—If the load on a piece which 
is in tension or shear be continuously increased, the piece will ultimately 
fracture or break in two, and the smallest load which will do this is 
called the breaking load, and the corresponding stress or breaking load 
per unit of original section is called the ultimate stress or ultimate 
strength of the piece. All solid materials have an ultimate tensile and 
ultimate shearing strength, and many have an ultimate crushing strength, 
but for certain ductile materials, such as wrought-iron and mild steel, 
there is no definite load which will cause complete fracture when they are 
subjected to compression. 
When a piece is loaded up to the elastic limit, the stress produced is 
the elastic strength of the piece. 
The largest load, repeatedly applied, which a piece will carry without 
taking a permanent set is called the proof load, and the corresponding 
stress is the proof stress or proof strength. 
The proof strength, as above defined, is less than the elastic strength, 
because experiment has shown that a load less than that required to 
produce permanent set may, if repeated a sufficient number of times, 
cause permanent set, and a load just under the elastic load will, after 
one or two applications, generally cause permanent set. This proof 
strength is difficult to determine, and in practice the term proof strength 
is often taken to mean elastic strength. Also, the elastic strength is 
frequently taken to mean the stress when the first decided set has taken 
place, as in mild steel, when the yield point is reached. 
The load put upon a piece in actual use is the working load, and the 
corresponding stress is the working stress or working strength. For 
safety the working stress must be less than the proof stress. The 
working stress is usually determined by dividing the ultimate stress by 
a number called a factor of safety, but it may also be fixed by dividing 
the proof stress by another factor of safety. a 
The value of the factor of safety to be used in any particular case 
must be determined by experience and judgment. Some of the con- 
siderations which influence the value of the factor of safety are—(1) the 
degree of certainty as to the magnitude of the greatest load which is 
likely to act on the piece ; (2) the character of the load, z.e. whether it 
is a fixed or constant load, or a constantly changing ioad; (3) the 
consequences of a breakdown; (4) the reliability of the material used ; 
(5) the amount of deterioration or wear which may take place in the 
piece when in use. 
86. Stress-strain Diagrams.—If the strains and corresponding 
stresses on a loaded bar be plotted in the usual way (Fig. 79), then since 
stress-strain is a constant up to the elastic limit, the diagram up to 
this point will be a sloping straight line OA. After the elastic limit 
is reached the strains increase more rapidly than the stresses, and the 
results are represented by a more or less irregularly curved line AB. 
So long as the cross section of a loaded bar does not sensibly alter, 
