Factor of Safety, Thermal Expansion.

1. Mechanics of
materials
(AE-106)
Lecture: 03
Engr. Abdullah Khan
Visiting Lecturer,
Department of Agricultural Engineering.

2. Factor of Safety
If some member of a structure be exposed to a
stress somewhat less than the ultimate stress of the
material of which the member is composed, it may
be supposed that the member is quite safe to carry
the load imposed upon it.
Failure could not occur if the actual
stress never reached the ultimate stress.
In actual practice, however, great care is always taken to
ensure that the working stress shall never be nearly so great
as the ultimate stress.
Thus it is customary for the designer of a machine to
proportion the various parts in such a manner that the working
stress will only be a certain fraction of the ultimate stress.
In this way a certain margin of
safety in working is ensured.
The ratio of the ultimate stress to the working stress is
termed the factor of safety.

3. Reasons for adopting Factor of Safety
1
As already pointed out, different samples of the
same material may vary considerably in structure
and composition, and consequently in strength.
2
Again, in the construction of the machine or
structure, there is always the possibility of bad
workmanship entering into the question, in
consequence of which the actual strength of the
parts may be more or less seriously reduced.
3
The factor to be considered is the deterioration or
weakening of the materials which may take place
as time goes on from various causes.
A further factor, and one of the greatest
importance, is the manner in which the loads to be
sustained act on the machine.
4
If the loads be steady and uniform we can
generally calculate the stresses due to
them, but otherwise it may be impossible
to determine them with any degree of
certainty.
For instance, we have seen that the tensile strength of
cast iron varies from as little as 5 tons to as much as 15
tons per square inch, whilst that of mild steel varies
from 27 tons to 33 tons per square inch.

4. How the type of load affects F.S?
It is customary to class the loads acting upon a
machine or structure as dead loads or live loads.
Live Load
which is applied more or less suddenly, or one
which varies from instant to instant.
Dead Load
which produces a constant stress or a stress
which increases or decreases very gradually.
The load on a road bridge, due to the weight of the roadway, is
an example of a dead load, whilst the load due to the weight of a
train running rapidly over a railway bridge is a live load.
The value of the factor of safety to be
adopted in any given case naturally
varies in accordance with the
conditions.
Thus
If the materials used in the construction
are reliable and the loads steady,
A lower factor of safety may be used than
would be required if the materials were
not reliable and the loads not steady.

5. In actual practice the factor of safety is seldom less than three, i.e. the working
stress is seldom more ultimate stress ; in some cases it is as much as twelve or
even more.
• As an example of a structure where a comparatively low factor of safety may be
adopted we may mention the steam boiler.
Here the material from which the structure is made, viz. mild steel, is very
reliable, its strength being known with certainty, the forces acting on the
structure may also be calculated accurately, and the forces are extremely steady.
Further, if the conditions of working be satisfactory, deterioration may be
almost entirely prevented.
• As an example of a structure where a high factor of safety is necessary a crane
may be cited.
Here, although the material may be of the best quality, the actual loads may be
applied in the nature of shocks, so that they are, strictly speaking, indeterminate
• Further, repeated application of unsteady loads may impair the quality of the
material as time goes on, and it is most important therefore that a high factor of
safety should be adopted.

6. Stresses due to the Forces of Expansion and contraction caused by
Heating and Cooling.
• It is a well-known fact that most metals expand when heated and contract when
cooled.
• The forces of expansion and contraction are practically irresistible, and in the case
of certain structures exposed to changes of temperature these forces must be duly
provided against.
• In a range of steam pipes, for example, it is found that the amount of expansion
which occurs when steam is turned into the cold pipes is approximately 2.5 inches
per 100 feet of length, the actual amount depending, of course, on the temperature
of the steam, the material of which the pipes are made, etc.

7. • Unless suitable provision be made for taking up the expansive movements, the
forces of expansion are such that the pipes may be seriously strained and even
fractured, particularly if of cast iron, by the stresses set up in the material.
• Very dangerous stresses may be, and sometimes are, set up in steam boilers by the
forces of contraction which result from sudden cooling of the hot plates.
• Thus a fireman will frequently blow off the water and steam under pressure, and
then, with the object of cooling down the boiler quickly for cleaning or inspection
purposes, will play over the plates with cold water from a hose pipe.
• The sudden contraction which results from this cannot take place freely, the boiler
being more or less a rigid structure, and, in consequence, the plates become very
highly stressed, so much so in fact that fracture may occur.

8. • Fortunately, in most cases, the engineer is not only able to make suitable provision
against the forces of expansion and contraction, but he is frequently able to put
them to practical advantage.
• Thus, the walls of a building which have become bulged have frequently been
straightened by passing bolts, provided with large nuts and washers, through them,
heating the bolts through a certain range of temperature, screwing up the nuts until
the washers bear against the walls, allowing the bolts to contract, and repeating the
operation one or more times as required.
• As contraction takes place an enormous force is exerted upon the walls, which are
in consequence eventually straightened.
• It is a simple matter to calculate how much a bar of any particular metal expands
or contracts on being heated or cooled through a given range of temperature.

9. • From experiment, we know with considerable accuracy what fraction of its length
at 32 degrees Fahrenheit a bar of any metal will expand for every degree through
which it is heated. This fraction is termed the coefficient of expansion of the
metal.
• For steel, the coefficient is 0.00000672, which means to say that a bar of steel of
any uniform section expands 0.00000672 of its length at 32 degrees F.
• For every degree through which it is heated ; the bar would contract exactly the
same amount for every degree it was cooled down. It is to be noted that the length
of the bar referred to is the length at a certain temperature, viz. 32 degrees F.
• Knowing the range of temperature through which the bar is heated or cooled, then
the expansion or contraction is obtained by multiplying the length by the
coefficient and by the range of temperature.

10. • Thus, let
α = coefficient of expansion,
ϒ = range of temperature.
Δx = amount of expansion or contraction.
Then
Δx = L * α * ϒ where, δ = Δx / L
• The stress which may be set up in a metal bar when the bar is so fixed that the expansion or contraction
cannot take place may be calculated in the following manner :
let
ϭ = stress in the material.
E= modulus of elasticity.
Then
E = ϭ / δ
from which
ϭ = E * δ = E * α * ϒ.
• It will be noticed that the length cancels out, so that in finding the stress in any particular case the length
need not enter into the question.