Remember me and also my parents in your pray friends

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Assala mu alykum My Name is saqib imran and I am the
student of b.tech (civil) in sarhad univeristy of
science and technology peshawer.
I have written this notes by different websites and
some by self and prepare it for the student and also
for engineer who work on field to get some knowledge
from it.
I hope you all students may like it.
Remember me in your pray, allah bless me and all of
you friends.
If u have any confusion in this notes contact me on my
gmail id: Saqibimran43@gmail.com
or text me on 0341-7549889.
Saqib imran.

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Mechanics of Solids
What is Solid Mechanics | Strength of Materials |
Purpose of Studying SOM
Solid mechanics or Mechanics of solids
Solid mechanics or Mechanics of solids is the branch of Mechanics, Physics,
Mathematics that concerns the behavior of solid matter under external actions (e.g.,
external forces, temperature changes, applied displacements, etc.).
Or
It is a branch of science which deals with the internal effects of the forces on the bodies
when they are loaded, from initial point to rupture or break.
Strength of Materials
Strength of materials deals with the relations between externally applied loads and their
internal effects on the bodies.
Purpose of studying strength of materials
To ensure that the structure used will be safe against the maximum internal effects that
may be produced by any combination of external loading.
Types of forces
Forces that are acting on the body, may be of the following types. Effects of these
forces are studied in solid mechanics or mechanics of solids.
Normal forces
The forces that are acting perpendicular to the body are known as normal forces.
Normal forces are of two types:
1. Tensile
2. Compressive.

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Tangential forces
The forces that are acting parallel to the body are known as tangential forces. Shear
forces and frictional forces are example of tangential forces.
Independent forces
Active forces are independent forces.
Dependent forces
Reactive forces are dependent forces.
Concentrated forces
If forces are acting on an area of a body which is negligible as compared to the body,
then these are known as concentrated forces.
Distributed forces
If forces are acting on an area of a body which is not negligible as compared to the
body, then these are known as distributed forces. These may be of uniformly distributed
load (UDL) or Uniformly varying load (UVL) or any kind of general loading.
Effects of forces
The effects of forces which are studied in solid mechanics or mechanics of solids are
1. Translation
2. Rotation (moment and torque)
What is Stress | Yield stress | Compressive
stress | Strain
Stress is a feeling that’s created when we react to particular events. Stress may refer to:
Mechanical, Biological, Music, other.
Stress (physics):
The average amount of force exerted per unit area.
Yield stress:
The stress at which a material begins to deform plastically.

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Compressive stress:
The stress applied to materials resulting in their compaction.
Strain:
It is the deformation of materials caused by the action of stress.
Strain is calculated by first assuming a change between two body states: the beginning state and
the final state. Then the difference in placement of two points in this body in those two states
expresses the numerical value of strain.
Strain therefore expresses itself as a change in size and/or shape.
If strain is equal over all parts of a body, it is referred to as homogeneous strain;
otherwise, it is in homogeneous strain
Stress Strain Curve:
During testing of a material sample, the stress–strain curve is a graphical representation of the
relationship between stress, and strain, i.e. elongation, compression, or distortion. The nature of
the curve varies from material to material.
Tensile Strength | Yield Strength | Ultimate
Strength | Breaking Strength
Tensile Strength
Tensile strength is the stress at the maximum on the stress-strain curve.
Or
The greatest longitudinal or axial stress a material can bear without breaking.
Tensile strength is the maximum amount of tensile stress that a material can take before
failure. There are three definitions of tensile strength
 Yield strength
 Ultimate strength
 Breaking strength
Yield Strength
The stress at which material strain changes from elastic deformation to plastic
deformation, causing it to deform permanently. This point is not a well defined point. We

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can say that yield strength is a strength which a material can bear without permanent
deformation.
Yield strength of Structural steel A36 is 250 MPa.
Ultimate Strength
The maximum stress a material can withstand. Ultimate tensile strength of Structural steel
A36 is 400 MPa.
Breaking Strength
The stress coordinate on the stress-strain curve at the point of rupture.
Ductile and Brittle Materials | Difference Between
Ductile and Brittle Material
Ductile Materials:
 Ductile materials will withstand large strains before the specimen ruptures.
 Ductile materials often have relatively small Young’s moduli and ultimate stresses.
 Ductile materials exhibit large strains and yielding before they fail.
 Steel and aluminum usually fall in the class of Ductile Materials
Brittle Materials:
 Brittle materials fracture at much lower strains.
 Brittle materials often have relatively large Young’s moduli and ultimate stresses.
 Brittle materials fail suddenly and without much warning.
 Glass and cast iron fall in the class of Brittle Materials.
Youngs Modulus and its Units | Proportional
Limit | Ultimate Stress | Rupture Stress
Modulus of Elasticity:
In solid mechanics, Young’s modulus (E) is a measure of the stiffness of a material.It is
defined as the ratio of stress over strain in the region in which Hooke’s Law is obeyed
for the material.

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Units
Young’s modulus is the ratio of stress, which has units of pressure to strain, which is
dimensionless; therefore Young’s modulus itself has units of pressure.
Proportional Limit :
The point up to which the stress and strain are linearly related is called the proportional
limit.
Ultimate Stress :
The largest stress in the stress strain curve is called the ultimate stress.
Rupture Stress :
The stress at the point of rupture is called the fracture or rupture stress.
Elastic Region | Plastic Region | Yield Point |
Plastic Strain | Strain Hardening | Necking
Elastic Region:
The region of the stress-strain curve in which the material returns to the undeformed state
when applied forces are removed is called the elastic region.
Plastic Region:
The region in which the material deforms permanently is called the plastic region.
Yield Point:
The point demarcating the elastic from the plastic region is called the yield point. The
stress at yield point is called the yield stress.
Plastic Strain:
The permanent strain when stresses are zero is called the plastic strain.
The off-set yield stress is a stress that would produce a plastic strain corresponding to
the specified off-set strain.

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A material that can undergo large plastic deformation before fracture is called a ductile
material.
A material that exhibits little or no plastic deformation at failure is called a brittle
material.
Hardness is the resistance to indentation.
The raising of the yield point with increasing strain is called strain hardening.
Necking Phenomenon:
The sudden decrease in the area of cross-section after ultimate stress is called necking.
Modulus of Resilience | Modulus of Toughness |
Poisson’s Ratio
Modulus of Resilience
The work done on a unit volume of material, as a simple tensile force is gradually
increased from zero to such a value that the proportional limit of the material is reached,
is defined as the modulus of resilience.
This may be calculated as the area under the stress-strain curve from the origin up to
the proportional limit and is represented as the shaded area in the figure below.
Modulus of Toughness
The work done on a unit volume of material as a simple tensile force is gradually
increased from zero to the value causing rupture is defined as the modulus of
toughness.
Poisson’s Ratio
When a bar is subject to a simple tensile loading there is an increase in length of the bar
in the direction of the load, but a decrease in the lateral dimensions perpendicular to the
load. The ratio of the strain in the lateral direction to that in the axial direction is defined
as Poisson’s ratio.

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Classification of Materials | Isotropic Material |
Homogeneous Material | Specific Strength and
Its Units
Classification of Materials
An isotropic material:
Has a stress-strain relationship that is independent of the orientation of the
coordinate system at a point. An isotropic material is the one having the same elastic
properties (E,µ) in all directions at any one point of the body.
A Homogeneous Material:
A material is said to be homogeneous if the material elastic properties (E,µ) are the
same at all points in the body.
Specific Strength:
This quantity is defined as the ratio of the ultimate (or tensile) strength to specific
weight, i.e., weight per unit volume. we have
And,
In the SI system, we have
So that in either system specific strength has units of length. This parameter is useful
for comparisons of material efficiencies.
Definition of Shear Force and Bending Moment
Shear Force:
The Shear Force at the cross section of the beam may be defined as the
unbalanced vertical force to the right or left of the section.
Bending Moment:

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The Bending Moment at the cross section of the beam may be defined as the algebraic
sum of the moments of the forces, to the right or left of the section.
Important Note: While considering the shear force and bending moment at the section ,
the end reactions must also be considered along with other external loads.
Direct Stresses | Shearing Stresses | Tensile
Stresses | Compressive Stresses
Direct Stresses :
The Stresses which are acting normal to the plane of the body are called as normal or
direct stresses.
These are called normal because these are acting perpendicular to the plane of the body.
Shearing Stresses :
The Stresses which act parallel to the stressed surfaces are called as shearing stresses.
Tensile Stresses :
If a straight bar is subjected to a pair of collinear forces acting in opposite direction and
coinciding with the ends of the bar and directed away from the bar, then the bar tends to
increase in length and the stresses developed in the bar will be tensile.
Compressive Stresses :
If a straight bar is subjected to a pair of collinear forces acting in opposite direction and
coinciding with the ends of the bar and directed towards the bar, then the bar tends to
shorter in length and the stresses developed will be compressive stresses.
Definition of Column | Difference between strut
and column | Type of Failure in Columns
Definition of Column
A long slender bar subjected to axial compression is called a column.

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The term is frequently use to describe a vertical member. Sometimes direct stresses
dominate and sometimes flexural stresses dominate.
Axial Compression means the compressive forces act at the two ends of the member in
the opposite direction are along the same axis .
Difference between strut and column
The difference between column and strut is that former is used to describe a vertical
member whereas latter is used for the inclined members.
Strut is any inclined member subjected to compression loads.
Types of Columns
Based upon mode of failure, there are three types.
Short Column
The failure initiates due to crushing of material and direct stresses are dominant. For
short column, if
L < 4d and KL/r min < 30
Where
d = least lateral dimension.
L = Unbraced length of the column.
k = effective length factor depends upon the end conditions of the column.
r min = least radius of gyration.
Slender or long Column
In these, failure initiates due to lateral buckling and flexural stresses are dominant. If
L > 30d
or

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KL/r min > critical slenderness ratio.
Intermediate Column
Failure initiates due to lateral buckling as well as crushing of the material. Direct and
flexural stresses both are dominant.
Column is intermediate when
4d < L < 30d
and
30 < KL /r min < 100 or Critical slenderness ratio. There is no agreed equation for
intermediate column. There are many approaches that are available for design and
analysis of intermediate columns such as:
1. Rankine Gordon approach.
2. Tangent Modulus formula.
3. Straight line formula.
4. Parabolic formula.
5. AISC equations.
Remember all these are empirical approaches.
Type of Failure in Columns
Long compression members fails due to buckling whereas short compression member
may fails due to yielding of material. Buckling of a compression member may occur even
the maximum stresses in the material are less than the yield point of the material . By
Buckling of a compression member it means lateral deflection of the bar.
Definition of critical load of a Column |
Slenderness ratio of a Column
Critical load :The critical load of a column is that value of axial force that is just sufficient
to keep the bar in a slightly deflected shape.
Slenderness ratio of a column : The ratio of effective length of the column (an axial
compression member) to the minimum radius of gyration of the cross-sectional area is
termed as slenderness ratio of the bar.

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As it is a ratio so it has no units and of course it is dimensionless.
Note :
Radius of gyration is always the property of cross-section that is why the word cross-
section is used with radius of gyration.
Definition of a Beam | Cantilever Beams | Simple
Beams
Definition of a Beam: A bar subjected to forces and couples that lie in a plane containing
its longitudinal axis is called a beam.
The Forces will act perpendicular to the longitudinal axis of the bar.
Cantilever Beams :
If a beam is rigidly supported at only one end in such a manner that the beam cannot
rotate at that point, it is called as cantilever beam.
The other end of the beam is free to deflect but from the end where it is fixed, it cannot
rotate. The end which is fixed is said to be ” restrained.” The reaction of the beam at the
fixed end consists of a vertical force and a couple acting in the plane of the applied loads.
Simple Beams :
A beam which is freely supported at both ends is known as simple beam.
The term “freely supported” implies that the end supports of the beam are capable of
exerting forces on the beam not any moment. So in simple beams no “restrained” is offer
to the angular rotation as the beam deflects under the loads.
Overhanging Beams | Statically determinate
Beams | Statically Indeterminate Beams
Overhanging Beam :
A beam which is freely supported at two points and having one or both the ends
extending beyond these supports is known as overhanging beam.

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Statically Determinate Beams :
All the beams in which reactions of the supports may be determined by the use of
equations of static equilibrium are known as statically determinate beams.
The value of these reactions are independent of the deformation of the beam.
Statically indeterminate Beams :
All the beams in which the number of reactions exceeds than the total number of
equations of static equilibrium are known as statically indeterminate beams.
In these beams the static equations must be supplemented by the equation based upon
the deformation of the beam.
Deflection of a Beam | Slope of a Beam | Flexural
Rigidity of Beam
Deflection of a Beam : The deflection at any point on the axis of the beam is the distance
between its position before and after loading.
Slope of a Beam : Slope at any section in a deflected beam is defined as the angle in
radians which the tangent at the section makes with the original axis of the beam.
Flexural Rigidity of Beam : The Product ” EI” is called flexural rigidity of the beam and
is usually constant along the beam.
Theorem 1 of Moment Area Method
Theorem 1 is stated as follows :
” The change in slope between tangents drawn to the elastic curve at any two points A
and B is equal to the product of 1/EI and the area of the moment diagram between these
two points.”

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Mathematically, it can be written as
For simplification we can say first draw the elastic curve diagram of the beam, then draw
tangents at those two points for which change in slope has to be determined ( in the above
diagram these points are A and B ) . Then Θab is the change in slope between tangents
drawn to the elastic curve at these two points A and B.
Theorem 2 of Moment Area Method
It is stated as follows :
” The deviation of any point B relative to a tangent drawn to the elastic curve at any other
point A in a direction perpendicular to the original position of the beam, is equal to the
product of 1/EI and the moment of area about b of that part of the moment diagram
between points A and B.”
Mathematically it can be written as
Similarly
Where Āb and Āa are the distances from center of gravity of the moment diagram to the
point B and A respectively.
For simplification we can say that Ā is the distance from center of gravity of the moment
diagram to the point where we have to find deviation.
For instance we are finding the deviation at point B then Ā will be the distance from center
of gravity of the moment diagram to the point B, because we are finding deflection at B.
Same is the case with any other point such as A.

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Definition of Thin-walled open sections
Thin-walled open sections
Thin-walled open sections may be defined as
The sections in which the thickness of material is small as compared with the overall
geometry of the material and there is only one or no axis of symmetry.
These sections are widely used. Their main use is in air craft industry. Common examples
are channel section, top hat, zed, angle and circular sections.
Within the section, shear stress variation is almost constant and we can assume shear
stress constant.
Definition of shear Center
Shear Center
Shear center is defined as the point on the beam section where load is
applied and no twisting is produced.
At shear center, resultant of internal forces passes. On symmetrical sections, shear
center is the center of gravity of that section.
In unsymmetrical sections, if the external applied forces act through the centroid of the
section, then in addition to bending, twisting is also produced. To avoid twisting, and
cause only bending, it is necessary for the forces to act through the particular point, which
may not coincide with the centroid. The position of the this point is a function only of the
geometry of the beam section. It is termed as shear center.
Such sections in which there is a sliding problem, we place loads at the shear center.
Strain measuring instruments | Strain Gauges
Strain measuring instruments :
These are of the following types :

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 Mechanical.
 Electrical. (Strain gauges)
 Photoelectrical.
 Basic working principle of strain gauges :
In 1856, Kelvin observed that resistance of the conductor is directly proportional to the
length and inversely proportional to the area of the conductor.
This principle was used in 1933 by defense department of US. They form gauges of
alloys of
 Copper + Nickle ( 55 : 45 ).
 Nickle + Chromium + Iron.
Advantages of strain gauges :
 We can use strain gauges after cementing them to structural components. We can
use them for a longer time.
 Remote reading is possible.
 The same strain meter can be used to measure the strains at 20 to 30 different
places.
 It can measure static and dynamic strains.
 Strain can be measured in any required direction.
 Micro strain measurement is possible.
Disadvantages :
 It can not be used immediately after cementing it as curing is needed. Curing :It is
to put something in favourable conditions to gain strength.
 It measures only surface strain.
 We can not use them beyond the elastic limit of the material of the strain gauge.
 It is very much sensitive to change in temperature.
 Sensitive to change in humidity
Construction of strain gauges :
 Length of strain gauges varies from 2-25 mm.
 Gauges for compression members varies from 2-4 mm.
 Whereas gauges for tension members ranges from 10-25 mm. Diameter of strain
gauge wire is 0.025 mm.
 Range of strain gauge resistance is 60-10000 ohm.
 Range of safe current is 25-50 mAmp.
 Range of safe voltage is 35-50 volts

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Cementing procedure of strain gauges :
 Clean oil, grease or dust from the surface usually from sand paper. The paint
should be removed through sand paper and then paste.
 Make the surface rough for good bond. Structural material and strain gauge
should have proper bond. If there will be slippage then some of the strain will not
be determined.
 Clean oil, grease and dust from the backside of the strain gauge with tissue paper.
 Apply paste/cementing material on the structural component e.g nitrocellulose and
epoxy resins.
 Paste strain gauge very carefully in proper direction and proper location by
pressing strain gauge from one end to other end using finger. So that the surplus
cementing material should be taken out.
 Apply four coats of water proofing agent to avoid from temperature and humidity
effects.
 Give sufficient time for curing, normally 1-2 days.
 Connect strain gauge with wheat stone bridge circuit.
Definition of Principal stresses | Principal plane |
Principal Axis
 Principal stresses :
Principal stresses may be defined as
” The extreme values of the normal stresses possible in the material.”
These are the maximum normal stress and the minimum normal stress. Maximum normal
stress is called major principal stress while minimum normal stress is called minor
principal stress.
 Principal plane :
Principal plane may be defined as
” The plane on which normal stress attains its maximum and minimum value.”
So these planes are also called as major principal plane and minor principal plane. The
shear stress on principal plane is zero.
 Principal Axis :

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Principal axis may be defined as
” The axis where moment of inertia is maximum or minimum and product of inertia
is zero.”
Note :
 Principal plane is the property of stress whereas principal axis is the property of
moment of inertia. There is no relation between principal axis and principal plane.
Maximum Principal strain theory or Saint Venant
theory with equation
Maximum principal strain theory :
This theory states that “ Yielding in a member subjected to complex/compound stress
system commences when maximum principal strain of the system becomes equal to the
strain corresponding to the yield point under simple tension test of the same material.”
For a 3 – dimensional state of stress system

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For a 2 – dimensional state of stress system

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Where
 є1 = maximum principal strain.
 υ = Poisson’s ratio.
 σ1= Major principal stress in x direction.
 σ2 = Minor principal stress in y direction.
 σ3 = Minor principal stress in z direction.
 σyt= Maximum Tensile strength in yielding.
 σyc =Maximum Compressive strength in yielding.
Merits :
 It computes the failure load based on all 3 values of principal stresses.
De merits :
 This theory gives unsatisfactory results for various materials.
 This theory is not much used for design purposes.
 This is not a very popular theory.
Maximum principal stress theory of failure with
equations
February 5, 2012 by admin Leave a Comment
 Maximum principal stress theory :
This theory states that “A structural component will fail when maximum principal stress
of the system will become equal to the yield strength of same material in a simple tension
test.”
 3-d equations for ductile materials:
σ1=σyt

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σ3=σyc
 2-d equations for ductile materials:
σ1=σyt
σ2=σyc
 3-d Equation for brittle materials:
σ1= σut
σ3= σ uc
 2-d equation for brittle materials:
σ1= σ ut
σ2= σ uc
where
 σ1 = Major principal stress in x direction.
 σ2 = Minor principal stress in y direction.
 σ3 = Minor principal stress in z direction.
 σyt= Maximum Tensile strength in yielding.
 σyc =Maximum Compressive strength in yielding.
 σut= Ultimate tensile strength.
 σut = Ultimate compressive strength.
This theory gives satisfactory results for brittle materials because brittle materials fails in
tension. It gives unsatisfactory results for ductile materials because ductile materials fail
in maximum shear stress.
This hypothesis, proposed by Rankine, which was also intended for use to predict yielding
of a ductile material.
Merits :

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 Calculations are easy.
 Gives satisfactory results for brittle materials.
De-merits :
 It computes failure load based only on values of principal stress.
 Gives unsatisfactory results for ductile materials.
Shear stress is zero at principal plane
Proof of the statement “Shear stress is zero at principal plane.”
At principal plane shear stress is zero. There is maximum direct or normal stress but no
shear stress at principal plane.

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Derivation of the angle at which shear stress is maximum:
Shear stress is maximum at 45 degree from the principal plane. Below is the derivation
of the angle at which shear stress is maximum.
Principal plane:
At principal plane direct or normal stress is maximum and there is no shear stress on
principal plane.
Derivation:
 Shear stress equation on an oblique section of body:

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Below is the derivation of the shear stress equation on an oblique section of a body in
terms of co-ordinate stresses.

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Derivation of maximum normal stress at principal plane
To find the maximum value of normal stress at some angle, we have to differentiate the
normal stress transformation equation, (which we derived in the previous article) with
respect to dθ.
Principal Plane :
It is the plane at which normal stress attains the maximum value.
Below is the derivation of maximum normal stress at principal plane.

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Normal Stress transformation equation:
Below is the derivation of normal stress transformation equation when the axis is
inclined at some angle.
Sign conventions:
 For normal stresses:
We will show tension with positive sign and compression with negative sign.
 For shear stresses:
If vertical couple is Counter clockwise then (+) sign and if vertical couple is Clockwise (-)
sign will use.
If horizontal couple is Counter clockwise then (-) sign and if horizontal couple is
clockwise then + sign.

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Plane stress | Plain strain | Normal or Direct
stress | Induced stress
Plane stress :

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The plane stress is the case in which stress is zero in one direction. In this case stress is
in 3 directions.
Plane strain:
In plane strain case, strain is in 3 directions and in one direction strain value is zero.
Normal or Direct stress :
The stress which is applied by us intentionally is known as direct or normal stress.
Induced stress :
This is the stress which we do not want to produce but is induced due to direct or normal
stress.
Direct & Shear Stress on a plane inclined to the direction of loading:
If a piece of material is cut along a plane inclined at some angle θ to the direction of
loading, then in order to maintain equilibrium, a system of forces would have to be applied
to the plane.
In figure 1 below, a bar is subjected to an axial tensile force P. The area of cross section
normal to the axis of the bar is A. If the bar is cut along the plane BC, inclined at an angle
θ, then the force P applied to this portion of the bar can be reacted by the component
forces N normal and T tangential to the plane BC.
Below is the derivation of the Direct and normal stress on a plane inclined to the direction
of loading.

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Conclusion:

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From the above we conclude that on the plane, where direct stress is maximum, shear
stress is zero.

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Strength of Materials
What is Beam
A structural member acted upon by a system of external forces at right angle to its
axis is called beam.
The beams may be classified :-
1. Cantilever : A beam whose one end is fixed and other end being free is called
cantilever.
2. Simply supported beam : A beam whose both the ends are resting on
supports,which may be walls or columns is called freely supported beam.
3. Encastre beam : A beam whose ends are built in into the walls is called
encastre or built in beam.
4. Continuous beam : A beam which is continuous over intermediate supports is
called continuous beam.
Define Columns and struts
A member subjected to compressive stress is termed as post, strut or pillar.
Main supporting members of building are generally called columns o pillars.
The load carried by short column is
P = p X A
where, p is the intensity of stress and A is the area of cross-section of the
column.
A column that fails primarily due to direct stress is called a short column and
the one that fails primarily due to the bending stress is called the long column.
Equivalent lengths for various end conditions

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Define Hookes law
Under the elastic limit stress is directly proportional to strain.
Stress ∝ Strain
Stress = E strain
E = Stress / Strain
where, E is a constant and called modulus of elasticity.
What is Modulus of rigidity

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It is defined as the ratio of tangential stress to shearing strain.
Modulus of rigidity G or N
= Stress / Strain
= F/A/ θ
Define Poissons ratio
The ratio of lateral strain to longitudinal strain is called Poisson s ratio and denoted
by 1/m.
Define Rankines formula
For long column with both ends hinged carrying a load W is given by
where, fc is crushing stress
k = Least value of radius of gyration
a = Rankine's constant.
What is Shear centre
It is the point of intersection of bending axis and the plane of transverse section of
a beam. The beams composed of thin sections generally are rolled I-sections,
angles, channels etc. Therefore load must act through the shear centre to avoid
twisting of the beam.
Shear stress
The shear stress is produced when the body is subjected to two equal and
opposite forces P and tends to slide two adjacent layers. The intensity of shear
stress
q = P/A
Explain Springs
Spring is a device in which the material is arranged in such a way that it can
undergo considerable change without being permanently distorted. Spring are of
three types :-
1. Laminated spring
2. Closely coiled helical spring
3. Open coiled helical spring
What is Stress
When a body is loaded, the restoring force per unit area, comes into play
inside the body is called stress. Its unit is Pascal.

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Stress = Force applied / Area of cross-section = P/A.
Define Temperature stress
When expansion or contraction of a body due to a change in temperature is
prevented thermal stress and thermal strain is produced.
Temperature stress = E.α.t
where E = Modulus of elasticity
α = Coefficient of thermal expansion
t = Change in temperature
Testing of Materials
Different tests and methods used for testing materials are :-
1. Tensile test machine : It consists of a mean of applying load to
specimen, a device to measure the applied load and accessories for griping
the test piece, recording the load and power unit etc.
2. Tensile test of ductile metals
3. Percentage elongation
4. Tensile test of brittle metals
5. Compression test : This test is opposite to tensile test with respect to
the direction of the applied stress.
6. Shear test : A force acting parallel to the plane produces shear stress
which is of two types : (a) Direct shear (b) Torsional shear.
7. Hardness test : This test is carried out by pressing a loaded diamond
against the surface under test.
8. Fatigue test : This test is carried out in parts subjected to continuously
varying stresses either in magnitude or from one kind stress to the reversed
kind.
9. Impact test : This load applied suddenly on a machine component is
called impact blow. The impact load may be applied in flexure, tension,
compression or torsion. Test enables to determination of toughness.
Thin walled cylinder
Thin walled cylindrical vessels are used to store fluids under pressure upto less than
2500 kg/cm2
. Beyond this pressure vessels used in a way. The thickness of thin
walled cylinder is about d/20 cm.
Due to internal pressure in the thin walled cylinder the three types of stresses may
be observed
1. Hoop or circumferential stress acting across the longitudinal section.
2. Longitudinal stress acting across the section perpendicular to the longitudinal
section.
3. Radial stress acting radially, being too small and can be ignored.
These three types of stresses are mutually perpendicular to the area.
Define Youngs modulus

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It is defined as the ratio of stress to longitudinal strain.
Young's modulus (E) = Stress / Strain
= F/A / δl/l
Define Bulk modulus
It is defined as the ratio of stress to volume strain.
Bulk modulus = Stress / Strain
= p / Δv / V
Direct stress
When the section is subjected to axial pull or push acting normally across the
section the stress so produced is direct stress.
Euler's Column Formulae
What is Load
A load on a material is the combination of external forces acting on it. When it is
constant then called dead load and when it is variable then called as live load.
Mohr's theorem
(a) The angle in radians between the tangents to the elastic curve at two points on
a straight member under bending is equal to the area between those two points.
(b) The deflection of a point on a straight member under bending in the direction
perpendicular to the original straight axis of the member measured from the

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tangent at another point on the member, is equal to the member between those
two points, where the deflection occurs.
Different Properties of Materials
The following are the main properties of materials :-
1. Hardness : It is the property which enables the material to resist scratching, wear
and abrasion, penetration by hard objects.
2. Malleability : It is the property which enables the material to undergo great
changes in size without rupture under compressive stress.
3. Ductility : It is the property due to which material can be drawn into wires.
4. Plasticity : The property of the material by virtue of which it can not return to its
original position when the load is removed, is known as plasticity. In this case there is
a permanent deformation in the material.
5. Elasticity : It is the property of material by virtue of which material springs back to
its original position with in certain limit (Elastic limit) with the removal of external
load.
6. Brittleness : It is the property of the material due to which it breaks under load
without permanent distortion.
7. Toughness : It is the property of the material enabling it to resist fracture due to
high impact loads like hammer blows.
Relationship between elastic constants
What is Shear flow
The shear flow at a point in the section is the force per unit length, and is
expressed by
 t = VQ / I

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where,
 = Shear stress
t = Thickness
V = Total vertical load
Q = a y analogous
I M.O.I. of cross-section about N.A.
Simple bending
A beam or a cantilever is said to be subjected to simple bending when it bends
under the action of uniform bending moment action of any load.
Assumptions in theory of simple bending :-
1. Plane of transverse sections of the beam, remain plane even after bending.
2. The material of the beam always obeys Hooke's law and is assumed
homogeneous.
3. The beam is initially straight of constant cross-section.
4. The moduli of elasticity for tension and compression are same.
5. The stresses in the beam do not exceed elastic limit.
6. The beam bends to a circular arc.
7. The radius of curvature is large compared to the section dimensions.
8. The loads act perpendicular to the beam axis.
What is Strain
It is defined as the ratio of change in length, volume, or shape to the original
length, volume or shape. It is dimensionless. It is of three types :-
1. Longitudinal strain = Change in length / Original length
2. Volume strain = Change in volume / Original volume
3. Shearing strain = Change in shear
Stress on inclined plane
Stress on a plane inclined through O to the cross-section carrying a load P.
Normal stress Pn = P cos2
θ
Tangential stress Pi = P sin 2θ / 2
Thick walled cylinders
Thick wall cylindrical vessels are used to store fluids under pressure more than
2500 kg/cm2
. A thick walled cylinder is subjected to the following three principal
stresses
1. The radial pressure p
2. The hoop stress sfn
3. The longitudinal tensile stress P

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Ultimate stress
The maximum stress which a material is subjected to before rupture is called
ultimate stress.