Example
A
steel bolt 160 mm long undergoes an elongation
=
0,12 mm during tightening. The modulus of elasticity of the material
is E =
MPa.
Determine the stress in the bolt.
Solution.
The unit elongation is
.
The stress in the bolt is determined from formula
II. Lateral Strain23 in Tension and Compression
Experiments
show that even if a rod undergoes very small deformations in the
longitudinal direction its lateral dimensions change. An elongation
in the longitudinal direction produces a contraction in the
transverse direction and conversely the shortening in the
longitudinal direction is accompanied by a lateral expansion.
Consequently, a body under tension lengthens and becomes thinner and
under compression it shortens and becomes thicker. Lateral strains in
tension or compression are proportional to longitudinal strains. If
the longitudinal strain is denoted by
(longitudinal
compressive strain) and the lateral strain by
(lateral
tensile strain), then, as is found from experiments,
is only a fraction of
,
i. e.
.
The
factor
is known as Poisson’s ratio.
Poisson’s ratio in tension is defined as and in compression
.
Poisson
thought that the ratio
was the same and equal to 0,25 for all materials. However, subsequent
experiments showed that Poisson’s ratio is different for different
materials,
ranging from 0 to 0,5. Average numerical values of this ratio for
some materials are given in table 2. In design practice
is taken as 0,3 for steel, beyond the elastic limit
increases to 0,5.
Table 2
Poisson’s Ratio for Some Materials
|
Material |
|
Material |
|
|
Cork |
0 |
Copper |
0,34 |
|
Carbon steel |
0,24 to 0,28 |
Bronze |
0,35 |
|
Chrome-nickel steels |
0,25 to 0,30 |
Rubber |
0,47 |
|
Aluminium |
0,26 to 0,36 |
Paraffine wax |
0,50 |
Using this ratio, it is possible to determine the change in volume of a rod under tension or compression. Let us first solve this problem in the general form. The volume of a rod of square cross section before extension is
.
After
extension each unit of the original length becomes equal to
(1+
),
consequently, the new length of the rod becomes equal to
(1+
).
The unit of length in the transverse direction shortens and becomes
equal to (1–
)
or (1–
).
There fore, the cross-sectional area after extension is
.
The volume of the rod after extension is
.
Neglecting
terms containing the factors
and
as
small quantities of higher order, we obtain
.
III. Experimental Study of Materials in Tension
The design of structures calls for a knowledge of the properties of materials of which these structures are made. The mechanical properties of materials are revealed by testing them under load.
The test most commonly used is a tension test. The reason for this is that the mechanical characteristics obtained from a tension test make it possible in many cases to predict sufficiently accurately the behaviour of the material under other types of deformation, such as compression, shear,24 torsion and bending. Besides, a tension test is easiest to perform. Materials which have to withstand primarily compressive loads (stone, concrete, etc.) are tested in compression as well.
Tension tests are carried out on special specimens of materials in specially designed tension testing machines. Specimens are usually of circular section, less frequently of rectangular section. At the ends of a specimen there are heads of heavier section. The heads are inserted into special grips of the testing machine. The transition from the specimen head to the middle (gauge) length is made smooth, in the form of a cone in circular specimens and a fillet in flat specimens. Uniform extension of a specimen occurs over a distance where the specimen section is constant; therefore elongations are measured only over this distance, called the gauge length.25 The gauge length of the specimen is designated as L.
As experiments show, only geometrically similar specimens of the same material give identical results. Consequently, in comparing mechanical qualities of different materials the absolute dimensions of specimens may by different provided that the law of geometric similarity is maintained.
In the case of brittle materials comparison is made by testing specimens of the same dimensions.
The
shapes and dimensions of specimens are standardized, if however, for
some reason or other «normal» specimens cannot be prepared,
comparable results may be obtained on specimens of circular or
rectangular cross section similar to normal ones with the ratio
,
where
is the gauge length of the specimen and A its cross-sectional area.
This value of the ratio
for a circular specimen is obtained when
=10.
In order that a tensile force acts precisely along the specimen axis, the gripping devices of the machine should be built with self-centering spherical seats.
Tension testing machines subject a specimen to a load increasing gradually from zero to a value causing fracture, and provide the so-called static loading. The load is measured by load-measuring instruments (dynamometers).
Tension testing machines vary in construction, shows a schematic diagram of a machine widely used in materials testing laboratories. The heads A of a test specimen are held in the grips of the machine. The lower grip remains stationary during testing. It is raised or lowered only when the specimen is being mounted. The raising or lowering of the lower grip is affected by means of a screw turning the handle. The tensile force is produced by gradually pumping oil into a cylinder mounted on the machine frame. Piston moves up and lifts the upper grip through a system of pin-connected rods. Since the lower grip remains stationary during testing and the upper grip moves up, the specimen is stretched.
Tension testing machines are usually provided with recording instruments which trace a curve showing the relation between the tensile load and the resulting elongation of the specimen. As mentioned, direct measurement of deformations is made with special instruments – strain gauges.