Chapter
STRESS
1.1
1
STRAIN—
AXIAL L OADING
AND
INTRODUCTION
The purpose to study the mechanics of materials is to acquire analytical tools necessary for
analysis and design of load taking members of practical utility structures. These members
could be parts of some machine or members of a building. The role of a design engineer is
to ensure strength and rigidity of a member considering its physical dimensions, mechanical
properties of the used material and the economy so that the member and the structure as
a whole should neither break nor deform excessively in the life span of the structure while
in service.
The analytical tools are developed based on the laws of physics and application of
mathematical methods. The properties of materials are known from systematically carried
out experiments on the materials in the laboratories. In the study of mechanics of materials,
we determine the stresses, strains and displacements in structures and their components
caused by different loads acting on them. Further, we may compute these quantities for
different combination and magnitude of loads until the failure is caused. Such a study gives
a complete picture of behaviour of these structures.
1.2
STRESS
The first fundamental concept in mechanics of materials is stress. The strength against an
axially loaded member of a structure is assessed on the magnitude of stress, the intensity
of the axial force, i.e., the force per unit area and is denoted by the Greek letter σ (sigma).
P
σ =
...(1.1)
A
where,
P = Axial load (a load that acts along the axis of the member)
and
A = Area of cross-section measured perpendicular to the load direction.
The stress may be assumed to be uniform over the area when the cross-sectional area
is small and load acts on the centroid of the geometry of the cross-section.
The stress that acts in the direction perpendicular to the area, is called normal stress.
Sign convention: A positive sign is assigned to indicate a tensile stress (the member is in
tension) and a negative sign to indicate a compressive stress (the member is in compression).
If a bar is pulled outward by the force P, the stress is tensile stress, if the force is
inward (into the area) causing the bar to be compressed, the stress is called compressive
1
Evisdom
2
MECHANICS OF MATERIALS
stress (Refer Fig. T-1.1). Elementary form of a member is a prismatic bar, a structural
member having same cross-section throughout its length.
In S.I. units, the force (P) is expressed in newton (N) and area (A) in square meter (m2),
the stress in (N/m2), that is, pascals (Pa). A large value of stress is sometimes indicated as:
1 kPa (kilopascal) = 1 × 103 N/m2
1 MPa (megapascal) = 1 × 106 N/m2
1 GPa (gigapascal) = 1 × 109 N/m2
A
P(+)
P(+)
A
P(–)
P(+)
(Bar under tension)
P(–)
P(+)
(Bar under compression)
Fig. T-1.1
1.3 SHEARING STRESS
There could be a situation where a body is subjected to a transverse force in the direction
normal to its axis. If the transverse force is quite large, the body may split into two parts.
This phenomenon may be well understood with the making of slices out of a bread using
a knife. The force on the knife makes it to pass through the cross-section of the bread. The
transverse force is along/tangential to the plane of area of the body. While split takes place
one part slides over its other part.
The transverse or tangential force divided by the associated area of cross-section, we
get average shear stress. Denoting shearing stress by the Greek letter τ (tau), we write
P
τ =
...(1.2)
A
This is known as direct shear or simple shear as well. In a situation when a member
is split at one plane, the member is in single shear. There could be another situation where
split occurs at two planes, the member is said to be in double shear, (determine shear stress
considering the area twice of the area in single shear), (Refer Fig. T-1.2). Direct shear arises
in the design of bolts, pins, welds and glued joints.
P
A
P
P
P
τ=
P
A
τ=
P
A
P
P
(Single shear)
(Double shear)
Fig. T-1.2
P
2A
3
Shear stresses also arise in an indirect manner when members are subjected to tension,
torsion, and bending.
1.4
BEARING STRESS
The bearing stress occurs at the surface of contact of two members experiencing external
load. Consider two plates of thickness t each is joined together with a pin of diameter d. The
two plates, when are pulled by force P, the outer surface of the pin comes in contact with
the plate surface. The pin surface will tend to crush the plate, while in turn, the plate
through its surface in contact with the pin would crush the pin. The intensity of stress thus
produced would depend on the contact area which is rectangular and its value is t d. Thus,
the stress called bearing stress is given by:
P
td
σb =
...(1.3)
Pin
t
P
P
Plate
P
t
Bearing area (Rectangle)
d
Fig. T-1.3
1.5 PUNCHING STRESS
We know about a paper punching machine. When pressed, a solid pin pierces through
the paper in making a hole in the paper and a piece of paper circular in shape drops
down. Such a process is adopted to mint coins from a metallic plate.
P
Punch
d
d
t
Plate
τ
t
Metal piece punched out
Fig. T-1.4
CHAPTER 1
STRESS AND STRAIN—AXIAL LOADING
4
MECHANICS OF MATERIALS
Here the cylindrical surface of the metal piece bears the stress, the corresponding
punching stress is given by:
P
...(1.4)
π dt
where π dt is the area of the curved surface of the metal piece that offers resistance
to the punching load P.
τ =
y
1.6 SHEAR STRESS
a
This occurs when each of two equal forces is
applied in opposite direction along two parallel
planes of a body (here we consider a rectangular
parallelopiped). The shear stress:
τ =
F
a× c
1.7 ALLOWABLE LOAD AND
ALLOWABLE STRESS;
FACTOR OF SAFETY
c
F
b
x
...(1.5)
F
z
Fig. T-1.5 An element of
From safety consideration, that is avoiding
material subjected to shear stress
structural failure, a structural member or a
machine part is loaded less than its strength. Here, the strength is meant as the maximum
load that can be sustained by a member. The maximum load is termed as ultimate load as
well. While a smaller load than this is termed as allowable load (also called the permissible
load or the safe load). The stress corresponding to allowable load is called as working stress
or allowable stress. Then we have,
Ultimate load
Allowable load
An alternative definition based on stress could be
Factor of safety (F.S.) =
...(1.6)
Ultimate stress or Yield stress
...(1.7)
Allowable stress
Selection of magnitude of factor of safety depends on many factors, such as properties
of a material, loading conditions, uncertainity factors related to manufacturing process,
construction techniques, the perfectness in the analysis and computational techniques and
importance of the structure under consideration.
There are design codes (In India these are known as IS codes) available for use, these
are prepared by experienced engineers. They recommend appropriate values of factor of
safety which could be any thing more than 1.
Factor of safety (F.S.) =
1.8 STRESSES ON INCLINED SECTIONS
On an axially loaded prismatic bar, we easily determine the normal stress, dividing the axial
load by the area of bar normal to the load. For complete analysis, we need to determine the
stresses that act on inclined sections of the bar.
Consider an axial load P acting along x-axis, passing through the centroid of the bar.
The direct-normal stress on x-plane of the bar is:
P
σx =
...(1.8)
A
5
An inclined plane mn passes through the bar Fig. T-1.6, its inclination is angle θ,
measured in counter-clockwise direction with respect to y-axis. The interest, here, is to
investigate stress values on this plane. We consider the free body diagram and condition of
static equilibrium for the cut-out part containing the plane mn of the bar (say the left part).
On the inclined plane, a force P has to act in opposite direction to that it acts on the negative
x-plane. Resolving this force P, into two components, a normal force N that is perpendicular
to the inclined plane mn and a shear force V that is tangential to it. These are:
N = P cos θ, V = P sin θ
The corresponding normal and shear stresses are obtained if dividing them by area Aθ,
of the inclined section. Thus,
σθ =
where
Aθ =
or,
σθ =
V
N
, τθ = –
Aθ
Aθ
A
cos θ
ÊPˆ
P
. cos2 θ, τθ = – ÁË ˜¯ sin θ cos θ
A
A
1 + cos 2 θ
, sin 2θ = 2 sin θ cos θ,
2
σx
σx
we get
σθ =
(1 + cos 2θ), τθ = –
(sin 2θ)
...(1.9)
2
2
Sign convention: The subscript θ is used to indicate that the stresses act on a plane
(section) inclined at an angle θ. Normal stresses σθ are positive in tension and shear stresses
τθ are positive when they tend to rotate the part in counter-clockwise direction.
Writing,
cos2 θ =
y
Positive x-plane
+θ
m
•
P
Negative x-plane
•
x
P
Centroid
of
area
n
Area = A
τθ
Clockwise rotation
σθ
•
P
(–)
V
N
θ
P
Area =
A
cos θ
τθ
Fig. T-1.6 Stresses acting on an inclined plane mn (prismatic bar in tension)
From Equation (1.9), we may conclude:
the maximum normal stress occurs at θ = 0 and is
σmax = σx
...(1.10)
CHAPTER 1
STRESS AND STRAIN—AXIAL LOADING
6
MECHANICS OF MATERIALS
and maximum shear stress by magnitude
σx
τmax =
...(1.11)
2
The largest positive shear stress is at θ = – 45° and the largest negative value of shear
stress is at θ = 45°.
1.9 STRAIN AND STRESS-STRAIN DIAGRAM
The other important concept in mechanics of materials is strain. A body subjected to a change
in temperature or to an external load undergoes some change in shape, termed as deformation.
A specific relationship between stress and deformation is characteristic of the material and
helps a design engineer in selecting a proper material to meet the requirement of the structural
member. Consider a prismatic bar of a homogeneous material is subjected to an axial force
P, the bar of length L shows a change in length by δ (deformation). The change in length per
unit length is defined a strain, and is denoted by the Greek letter ε (epsilon).
δ
ε =
...(1.12)
L
If the bar is in tension, the strain is called a tensile strain representing an elongation of
the material. If the bar is in compression, the strain is a compressive strain, the bar shortens.
Sign convention: Tensile strain is taken as positive and compressive strain as negative.
The strain ε is called a normal strain as it is associated with normal stresses. It is a
dimensionless quantity.
In the design and construction of structures, the knowledge of mechanical behaviour of
materials in use is important. One such behaviour of the material is stress-strain relationship.
The best way to obtain this is to conduct experiments on the material in the laboratory and
to derive its stress-strain diagram. A specimen of the material under study is taken and put
on the special machine. By applying loads in a specified manner, parameters like change in
length, diameters, and nature of breaking are recorded and the results are interpreted.
12 mm
P
Gauge length 50 mm
•
L
P
An extensometer or
a strain gauge is
attached to measure
elongation
A
δ
P
Fig. T-1.7 Test specimen with tensile load
The specimen is cylindrical in shape of diameter 12 mm and a gauge length of 50 mm
between gauge marks, which are the points where the extensometer arms are attached
to the specimen Fig. T-1.7. The test specimen is placed in a testing machine (tensile-test
machine or universal-test machine) which is used to apply an axial load P. As the load P
is increased the gauge length L is measured, and the elongation δ is determined for each
7
value of P. For each pair of readings P and δ, the stress σ is computed by dividing P by the
original cross-sectional area of the specimen, and the strain ε by dividing the elongation δ
by the original distance between the two gauge marks. The stress-strain diagram may then
be obtained by plotting ε as an abscissa and σ as an ordinate. A stress-strain diagram is a
characteristic of the particular material being tested and provides important informations
about the mechanical properties and type of behaviour.
Stress-strain diagram (Fig. T-1.8): We shall discuss results of the material known as
structural steel (also known as mild steel or low carbon steel). Structural steel is one of the
most commonly used materials in the construction of buildings, bridges, cranes, ships
automobiles, locomotives and in many other constructions.
1.9.1 Proportional Limit
The diagram begins from the origin O, OA is a straight line. This shows a linear relationship
between stress and strain and the two have proportionality relationship. Beyond point A, the
proportionality no longer exists, hence the stress at A, is called the proportional limit. For
structural steels, this limit is in the range of 210–350 MPa and for high tensile steels the
value is around 550 MPa. In many of designs, the stress corresponding to proportional limit
is the maximum stress that is allowed in the structural component.
1.9.2 Elastic Limit
This is a stress value at F beyond which if the load is removed the material will not return
to its original shape. At this state of material a permanent deformation in the structural
member sets in. This state of material is known as permanent set.
1.9.3 Yield Point
This is a point B on the graph from where excessive elongation begins in the material. The
elongation of the mild-steel in the region B to C is about 10–15 times elongation that takes
place in the linear portion. From B to C, the behaviour of the material is that of a plastic,
the slope of the curve is nearly zero, which means that it deforms without an increase in
the applied load (in some case, the load may decrease while yielding occurs).
1.9.4 Yield Stress
This is the stress at yield point. The stress at yield point is also known yield strength of the
material denoted by σY. This is a very important value of the material, on many instances
it helps in deciding the working stress in the designs.
1.9.5 Strain Hardening
When the loading continues to increase beyond yield, the restructuring of crystalline
arrangement begins within the material and it starts resisting the impressed load and
deformation rate is slowed and load carrying capacity increases. This process is known as
strain-hardening. In the graph the line from C to D has positive slope and is about 1/50th
of the slope of line between the origin, O and A.
1.9.6 Ultimate Stress
In the process of increasing load on the specimen, the load bearing capacity of the material
eventually reaches its maximum value at point D. It is highest ordinate of the curve. The
corresponding stress is ultimate stress, σU or ultimate strength.
CHAPTER 1
STRESS AND STRAIN—AXIAL LOADING
8
MECHANICS OF MATERIALS
1.9.7 The Rupture Strength
After having reached the point D, the load starts falling (this will be observed as the back
motion of the pointer on the dial of the machine) while elongation continues to increase.
Eventually the specimen breaks at point E on the graph. The stress σB corresponding to
rupture is called the breaking strength. This value is smaller than the ultimate strength.
Stress
σ=
P
A
D
Ultimate
stress
E
Yield
stress
B
F
A
C
Elastic
limit
σB Fracture
strength
σU
σY
Proportional
limit
Strain
δ
ε=
L
O
Yielding
Elastic
behaviour
Fracture
point
Perfect
plasticity
Strain
hardening
Necking
(a) Stress-strain diagram for a typical structural steel in tension (not to scale)
P
σ
σU= σB
Cup
Proportional
limit
Fracture
A
Cone
O
P
(b) Necking of steel-specimen at failure
ε
(c) Stress-strain diagram for a typical brittle material
Fig. T-1.8
1.9.8 Necking
At loading condition corresponding to point D on the graph, the diameter of the test
specimen gets progressively reduced (thinning or narrowing). The phenomenon is called
9
necking. Thereafter the material breaks into two parts. The shape of two parts have special
appearance, one is of cup shape and the other is of a cone. The cup-shaped surface forms
an angle of approximately 45° with the original surface of the specimen.
1.9.9 Ductility
This is a special property of the material. The ductility is a measure to the extent a material
undergoes plastic deformation (a permanent strain) during increasing load. The material that
undergoes greater deformation more ductile it is. Ductility is measured either by %age reduction
in area or by the %age elongation of a known measured length of specimen.
Ai − Af
× 100
...(1.13)
Percent reduction in area =
Ai
Li − L f
× 100
or,
Percent elongation =
...(1.14)
Li
subscript i is for initial value and f for the final value at the point of fracture.
Some of the known ductile materials are: structural steel, aluminium, copper,
magnesium, lead, molybdenum, nickel, brass, bronze, monel metal, nylon. For structural
steel the per cent reduction in area is of the order 60 to 70%. The property of ductility
of the material enables to give some arbitrary shape to suit the construction requirement.
1.9.10
Brittle Property
There are materials which fail at low strain when subjected to a load to cause tension,
at elongation little beyond its proportional limit, A. The reduction in area at point of
failure is insignificant absence of any necking. Such materials are known as brittle. A
typical stress-strain curve for a brittle material has been shown in the Fig. T-1.8. Highcarbon steels (high-strength steels) have high value of proportional limit about 550 MPa
very high yield stresses about 700 MPa but they behave in a brittle manner. Ordinary
glass is an ideal brittle material, because it exhibits almost no ductility. Its ultimate
stress is about 70 MPa. Glass fibres have ultimate stress upto about 7 GPa. The
prominent brittle materials are: concrete, stone, cast iron, glass, ceramics.
1.9.11 Comparative Stress-Strain Curves for Different Materials
The relative strength and behaviour of different materials under axial loading (tension)
can best be understood by plotting, stress-strain values on the same σ-ε graph as shown
in the Fig. T-1.9.
s
High carbon steel
s
Tempered alloy steel
High-strength,
low-alloy steel
Cast iron
Carbon steel
Aluminium
Concrete
Pure iron
e
(b)
(a)
(a) Comparative stress-strain diagrams for different materials
(b) Stress-strain diagrams for iron and different grades of steel
Fig. T-1.9
e
CHAPTER 1
STRESS AND STRAIN—AXIAL LOADING
10
MECHANICS OF MATERIALS
The structural steel has about 0.2% carbon. With increasing carbon content, steel
becomes less ductile but possesses higher yield stress and higher ultimate stress. That
is, some of the physical properties of structural metals, such as strength, ductility, and
corrosion resistance, can be greatly affected by alloying (adding other metals), heat
treatment (tempering), and the manufacturing process (rolling, forging) used. We may
find from the Fig. T-1.9 (b) that the pure iron and three different grades of steel have
large variations in the yield strength, ultimate strength and final strain (ductility).
1.9.12
Proof Stress
Aluminium and many other ductile materials do not
exhibit a distinct yield point on stress-strain diagram,
instead the stress keep increasing-not linearly-until
the ultimate strength is reached and eventually
rupture takes place. The yield strength is determined
by the offset method. A straight line is drawn on the
stress-strain diagram parallel to the initial part of
the curve but offset by some standard strain, such
as ε = 0.002 on ε-axis. The intersection of the offset
line and the stress-strain curve, the point A defines the
yield stress. This value is distinguished from a true yield
stress by referring to it as the proof stress.
1.9.13
Compression Test
σ
A
0.002 offset
O
ε
Fig. T-1.10 Arbitrary yield stress
determined by the offset method
For compression tests, the specimens of the metals
are in the shape of cubes—50 mm a side or circular cylinders-diameter 25 mm and 25
mm to 300 mm. Both the load applied by the machine and the shortening of the specimen
are measured. In order to eliminate end effects, the gauge length is selected to be
smaller than the total length of the specimen.
Concrete is tested in compression on every important construction project to ensure
that required strength has been achieved. The test specimen could be a cube of 150 mm
a side or a cylinder 150 mm dia. and 30 cm long and is subjected to curing for 28 days.
Smaller specimens are used when performing compression test on stones.
A concrete of M20 grade has compressive
σ
strength of 20 N/mm2 and that of sand stone is
2
65 N/mm .
Stress-strain curves for materials, in general,
in compression differ from those in tension.
Ductile metals-steel, aluminum, and copper
have proportional limits nearly same to those in
tension and the stress-strain diagrams are also
same. Beyond yield point, the behaviour is quite
different when in compression. With compressive
loading, the specimen bulges outward (Fig. T-1.11).
ε
O
Brittle materials loaded in compression, iniFig. T-1.11 Stress-strain diagram
tially have σ-ε relationship—a straight line, furfor a ductile material in compression
ther the shortening increases at a higher rate.
The stress-strain curves for compression and
tension often have similar shapes. The ultimate stresses are higher than those in tension.
1.10
11
ELASTICITY AND PLASTICITY—PROPERTIES OF A MATERIAL
The elasticity and plasticity are the properties exhibited by a material during loading,
unloading and then loading in a sequence and in an extended manner. The entire behaviour
of the material is best understood on the σ-ε diagram developed through experimental
results of a specimen.
σ
σ
F
B
F
E
E
A
A
ε
O
Elastic
O
C
D
Residual
strain
Plastic
(a )
ε
Elastic
recovery
(b)
Fig. T-1.12
Consider a specimen subjected to tensile load (increasing). The stress and strain go from
the origin O to point A on the stress-strain curve of Fig. T-1.12(a). On removal of load at this
point, the material traces back to the origin O. This property of a material, by which it
returns to its original dimensions/shape during unloading, is called elasticity. The material
is known as elastic. The largest value of the stress at point E for which the material behaves
elastically is called the elastic limit of the material.
Suppose the load is increased from point E and taken to point B on the curve and the
unloading takes place, the stress and strain decrease along a straight line BC parallel to the
initial portion of the loading process. The point C represents on the curve where the load
is no more on the specimen and ε is not zero-Fig. T-1.12(b). The magnitude of OC indicates
a residual strain, or permanent strain or plastic deformation of the material. When the
material is subjected to loads beyond elastic limit the strain values are inelastic the strain
is known as plasticity.
Suppose the specimen is loaded again with tensile load from the point C on the graph,
the new curve is along the old line CB and further, it follows the stress-strain curve of the
original curve toward point F.
For reloading, B is proportional limit, which is at a higher stress than the original
elastic limit, E. The point of rupture, F remains same. The ductility is reduced—it is
measured as BF which is smaller than that of original–one way loading measured as EF.
1.11 HOOKE’S LAW, MODULUS OF ELASTICITY
Most of engineering structures are designed for deformation which corresponds to a
point on the proportionality line, the stress is normally not allowed to exceed the limit
of proportionality, where the stress σ is directly proportional to the strain ε, we may
write
σ = E . ε
CHAPTER 1
STRESS AND STRAIN—AXIAL LOADING