True stress and true strain

True stress and true strain: 


The stress was calculated by
dividing the load P by the initial cross section of the specimen.
But it is clear that as the specimen elongates its diameter decreases and the decrease in
cross section is apparent during necking phase.
Hence, the actual stress which is obtained by dividing the load by the actual cross
sectional area in the deformed specimen is different from that of the engineering stress
that is obtained using undeformed cross sectional area

                                          
                                σact = P/Aact



Though the difference between the true stress and the engineering stress is negligible for
smaller loads, the former is always higher than the latter for larger loads.
Similarly, if the initial length of the specimen is used to calculate the strain, it is called
engineering strain as obtained in equation 1.9
But some engineering applications like metal forming process involve large deformations
and they require actual or true strains that are obtained using the successive recorded
lengths to calculate the strain.

                                            

True strain

True strain is also called as actual strain or natural strain and it plays an important role in
theories of viscosity.
The difference in using engineering stress-strain and the true stress-strain is noticeable
after the proportional limit is crossed

Read More

Mechanical properties of materials

Mechanical properties of materials:


A tensile test is generally conducted on a standard specimen to obtain the relationship
between the stress and the strain which is an important characteristic of the material.
In the test, the uniaxial load is applied to the specimen and increased gradually. The
corresponding deformations are recorded throughout the loading.
Stress-strain diagrams of materials vary widely depending upon whether the material is
ductile or brittle in nature.
If the material undergoes a large deformation before failure, it is referred to as ductile
material or else brittle material.


Initial part of the loading indicates a linear relationship between stress and strain, and the
deformation is completely recoverable in this region for both ductile and brittle materials.
This linear relationship, i.e., stress is directly proportional to strain, is popularly known as
Hooke's law.

                                            σ = Eε




The co-efficient E is called the modulus of elasticity or Young's modulus.
Most of the engineering structures are designed to function within their linear elastic region
only.


After the stress reaches a critical value, the deformation becomes irrecoverable. The
corresponding stress is called the yield stress or yield strength of the material beyond
which the material is said to start yielding.
In some of the ductile materials like low carbon steels, as the material reaches the yield
strength it starts yielding continuously even though there is no increment in external
load/stress. This flat curve in stress strain diagram is referred as perfectly plastic region.
The load required to yield the material beyond its yield strength increases appreciably and
this is referred to strain hardening of the material.
In other ductile materials like aluminum alloys, the strain hardening occurs immediately
after the linear elastic region without perfectly elastic region.
After the stress in the specimen reaches a maximum value, called ultimate strength, upon
further stretching, the diameter of the specimen starts decreasing fast due to local
instability and this phenomenon is called necking.
The load required for further elongation of the material in the necking region decreases
with decrease in diameter and the stress value at which the material fails is called the
breaking strength.
In case of brittle materials like cast iron and concrete, the material experiences smaller deformation before rupture and there is no necking.

Read More

Strain

Strain:


The structural member and machine components undergo deformation as they are brought
under loads.
To ensure that the deformation is within the permissible limits and do not affect the
performance of the members, a detailed study on the deformation assumes significance.
A quantity called strain defines the deformation of the members and structures in a better
way than the deformation itself and is an indication on the state of the material.

Consider a rod of uniform cross section with initial length.
Application of a tensile load P at one end of the rod results in elongation of the rod by
L0


Application of a tensile load P at one end of the rod results in elongation of the rod by
L0
δ .
After elongation, the length of the rod is L. As the cross section of the rod is uniform, it is
appropriate to assume that the elongation is uniform throughout the volume of the rod. If
the tensile load is replaced by a compressive load, then the deformation of the rod will be a
contraction. The deformation per unit length of the rod along its axis is defined as the
normal strain. It is denoted by ε


                                               ε =   δ/L = L-L0/L




Though the strain is a dimensionless quantity, units are often given in mm/mm, μm/m.

Read More

Stress on inclined planes under axial loading:

Stress on inclined planes under axial loading:


When a body is under an axial load, the plane normal to the axis contains only the normal
stress as discussed in section.

However, if we consider an oblique plane that forms an angle with normal plane, it
consists shear stress in addition to normal stress.
Consider such an oblique plane in a bar. The resultant force P acting on that plane will

keep the bar in equilibrium against the external load P'


The resultant force P on the oblique plane can be resolved into two components Fn and Fs
that are acting normal and tangent to that plane, respectively.
If A is the area of cross section of the bar, A/cos is the area of the oblique plane. Normal
and shear stresses acting on that plane can be obtained as follows.
Fn= Pcosθ
Fs = -Psinθ (Assuming shear causing clockwise rotation negative).

σ = Pcos

θ/A/cos

θ = P/A cos

2

θ

τ = -Psin

θ/A/cos

θ = -P/Asin

θcos

θ

 It define the normal and shear stress values on an inclined plane that
makes an angle θ with the vertical plane on which the axial load acts.
From above equations, it is understandable that the normal stress reaches its maximum
when θ = 0o and becomes zero when θ = 90o.


But, the shear stress assumes zero value at θ = 0o and θ = 90o and reaches its maximum
when θ = 45o.
The magnitude of maximum shear stress occurring at θ = 45o plane is half of the maximum
normal stress that occurs at θ = 0o for a material under a uniaxial loading.


  τ =

max

P/

2A =

σmax/2

Now consider a cubic element A in the rod which is represented in two dimension

 such that one of its sides makes an angle with the vertical plane

To determine the stresses acting on the plane mn, equations 1.6 and 1.7 are used as such
and to knows the stresses on plane om, θ is replaced by θ + 90o.
Maximum shear stress occurs on both om and mn planes with equal magnitude and
opposite signs, when mn forms 45o angle with vertical plane.

Read More

Bearing Stress:

Bearing Stress:


In the bolted connection  a highly irregular pressure gets developed on the
contact surface between the bolt and the plates.

The average intensity of this pressure can be found out by dividing the load P by the
projected area of the contact surface. This is referred to as the bearing stress.



The projected area of the contact surface is calculated as the product of the diameter of
the bolt and the thickness of the plate.
Bearing stress

σb = P/A= P / t×d

Read More

Shear Stress

Shear Stress:


The stresses acting perpendicular to the surfaces considered are normal stresses and
were discussed in the preceding section.
Now consider a bolted connection in which two plates are connected by a bolt with cross
section A.



The tensile loads applied on the plates will tend to shear the bolt at the section AA.
Hence, it can be easily concluded from the free body diagram of the bolt that the internal
resistance force V must act in the plane of the section AA and it should be equal to the
external load P.
These internal forces are called shear forces and when they are divided by the
corresponding section area, we obtain the shear stress on that section.

τ = V/A


It defines the average value of the shear stress on the cross section and the
distribution of them over the area is not uniform.

In general, the shear stress is found to be maximum at the centre and zero at certain
locations on the edge. This will be dealt in detail in shear stresses in beams (module 6).

 In It the bolt experiences shear stresses on a single plane in its body and hence it

the bolt experiences shear on two sections AA and BB. Hence, the bolt is
said to be under double shear and the shear stress on each section is


                  τ = V/A = P/2A


Assuming that the same bolt is used in the assembly  and
the same load P is applied on the plates, we can conclude that the shear stress is reduced
by half in double shear when compared to a single shear.

Shear stresses are generally found in bolts, pins and rivets that are used to connect
various structural members and machine components

Read More

Saint - Venant's Principle:

Saint - Venant's Principle:

Consider a slender bar with point loads at its ends.

The normal stress distribution across sections located at distances b/4 and b from one and
of the bar.
It is found that the stress varies appreciably across the cross section in the
immediate vicinity of the application of loads.

The points very near the application of the loads experience a larger stress value whereas,
the points far away from it on the same section has lower stress value.

The variation of stress across the cross section is negligible when the section considered
is far away, about equal to the width of the bar, from the application of point loads.

Thus, except in the immediate vicinity of the points where the load is applied, the stress
distribution may be assumed to be uniform and is independent of the mode of application
of loads. This principle is called Saint-Venant's principle.

Read More

NORMAL STRESS

Normal Stress:

When a structural member is under load, predicting its ability to withstand that load is not
possible merely from the reaction force in the member.
It depends upon the internal force, cross sectional area of the element and its material
properties.
Thus, a quantity that gives the ratio of the internal force to the cross sectional area will
define the ability of the material in with standing the loads in a better way.
That quantity, i.e., the intensity of force distributed over the given area or simply the force
per unit area is called the stress.
P
A
σ = 1.1
In SI units, force is expressed in newtons (N) and area in square meters. Consequently,
the stress has units of newtons per square meter (N/m2) or Pascals (Pa).
In figure 1.2, the stresses are acting normal to the section XX that is perpendicular to the
axis of the bar. These stresses are called normal stresses.
The stress  is obtained by dividing the force by the cross sectional area and hence it represents the average value of the stress over the entire cross section.


Consider a small area ΔA on the cross section with the force acting on it ΔF

. Let the area contain a point C.

Now, the stress at the point C can be defined as,
A 0
lim F
Δ → A
Δ
σ =
Δ 1.2
The aver

age stress values obtained using equation 1.1 and the stress value at a point from

equation 1.2 may not be the same for all cross sections and for all loading conditions

Read More

STRENGTH OF MATERIALS

INTRODUCTION:

An important aspect of the analysis and design of  structures relates to the deformations
caused by the loads applied to a structure. Clearly it is important to avoid deformations so
large that they may prevent the structure from fulfilling the purpose for which it is intended.
But the analysis of deformations may also help us in the determination of stresses. It is not
always possible to determine the forces in the members of a structure by applying only the
principle of statics. This is because statics is based on the assumption of undeformable,
rigid structures. By considering engineering structures as deformable and analyzing the
deformations in their various members, it will be possible to compute forces which are
statically indeterminate. Also the distribution of stresses in a given member is
indeterminate, even when the force in that member is known. To determine the actual
distribution of stresses within a member, it is necessary to analyze the deformations which
take place in that member. This chapter deals with the deformations of a structural
member such as a rod, bar or a plate under axial loading.

Read More

Shear Stress:

Shear Stress:

The stresses acting perpendicular to the surfaces considered are normal stresses and

were discussed in the preceding section.

Now consider a bolted connection in which two plates are connected by a bolt with cross

section A 

The tensile loads applied on the plates will tend to shear the bolt at the section AA.

Hence, it can be easily concluded from the free body diagram of the bolt that the internal

resistance force V must act in the plane of the section AA and it should be equal to the

external load P.

These internal forces are called shear forces and when they are divided by the

corresponding section area, we obtain the shear stress on that section

   

                                          

τ = V/A

 The average value of the shear stress on the cross section and the

distribution of them over the area is not uniform.

In general, the shear stress is found to be maximum at the centre and zero at certain

locations on the edge. This will be dealt in detail in shear stresses in beams  the bolt experiences shear stresses on a single plane in its body and hence it

is said to be under single shear

the bolt experiences shear on two sections AA and BB. Hence, the bolt is

said to be under double shear and the shear stress on each section is

V P

A 2A

τ = V/A = P/2A

Assuming that the same bolt is used in the assembly as shown in figure 1.5 and 1.6 and

the same load P is applied on the plates, we can conclude that the shear stress is reduced

by half in double shear when compared to a single shear.

Shear stresses are generally found in bolts, pins and rivets that are used to connect

various structural members and machine components.

Read More