Stress: Stress is defined as the internal resistance force What do you mean by polar moment of iner a ?

What do you mean by polar moment of iner a ? Difference between direct stress and bending stress :
(Opposite to applied force) developed by the body over It is s measure of a circular beams ‘s ability to resist Direct tensile and compressive stress is set up due to load
unit area of its cross section. Unit of stress is N/m2 (S.I). *As the magnitude of radial stress is very small, therefore torsion when some amount of torque is applied to in in a applied parallel to the length of the object and direct
Strain: Strain is the ratio of change in length to original they are neglected while analysing thin cylinder. specified axis .It is required to calculate the twist of beam shear stress is set up in the sec on which is parallel to the
length . Unit 2 subjected to a torque . it is called second polar moment of line of• ac on of the shear load. But bending stress is
Hooke's Law: Within proportional limit, stress is directly Moment of Iner a: It is the mass property of a rigid body area etupdue to load at right angles to the
proportional to strain. that defines the torque needed for a desired change in length of the object subjected to bending.
Stress ~ strain
angular velocity about an axis of rota on. It is also known Write short note on radius of gyra on : In case of direct stress, nature and intensity of stress is the
Stress=constant x strain
as mass moment of Iner a. It is defined as the distance from the axis of rota on to a same at any layer in the sec on of the object subjected to
E = Young's modulus
Elasticity: When a body is subjected to external forces, it It depends on the shape of the body and may be different point where the total mass of the direct stress, but in case of bending stress nature of
undergoes deformations and develops internal resisting around different axes of rota on. body is supposed to be concentrated, so that the moment stress is opposite on opposite side of the neutral axis, and
force to balance the externally applied forces. When the The moment of Iner a of the mass (m) about the point at of iner a about the axis may remain the same. intensity of stress is different at different layers of the
external forces are removed, it comes back to its original a distance (r)is, I=mr2 . What is moment if iner a. What iS the difference sec on of the object subjected to bending.
shape and size. Materials when exhibit this property is Area moment of Iner a: The area moment of iner a is a between mass moment of iner a and area moment of In case of direct stress, intensity of stress is the same in a
known as elastic materials and the property itself is called property of a two dimensional plane shape which iner a: A quan ty expressing a body's tendency to resist sec on taken through any point of the object. But in case
elasticity. characterizes its deflec on under loading. It is also Known angular accelera on which is the sum of the products of of bending stress, intensity of stress is different at the
Plasticity: Plasticity. is the property because of which a as the second moment of iner a. The area moment of the mass of each par cle in the body with the square of same layer of the sec on taken through different points of
material subjected to forces undergo deformations which iner a has dimensions of length of the fourth power. its distance from the axis of rota on. the object
do not disappear on the removal or external forces. Mathema cally, I=∑AK2 or, I= ∑MK2 Pure bending is that bending where it is assumed that
Ductility: Ductility is the property because of which it is Theorem of Perpendicular Axis: The basic difference between mass moment of iner a and effect of shear stress is nil
possible to draw thin wire of a metal. Copper and mild It states, If Ixx and Iyy be the moments of iner a of a area moment of iner a is that mass M.I is conceded with Pure bending: Pure bending occurred in three cases given
steel exhibit this property. plane sec on about two perpendicular axis mee ng at O, the iner a of rota ng body whereas area M.I is concerned below.
Hardness: Hardness is the resistance of a material to the moment of iner a I about the axis ZZ perpendicular to with the trending stresses developed in the body. (a) The BM is constant along the length of the beam and
indentation, scratching cutting or wear by abrasion.
the plane and Describe the method of finding out the moment of the SF is zero
Fatigue: Fatigue is the phenomenon of a material failing
passing through the intersec on of X.X and Y-Y is given by: iner a of a composite sec on. (b) The BM is constant between the loads and the SF in
under very little sue due to repeated cycle of loading.
Proof : The moment of iner a o fa composite sec on may be zero
Creep: Creep is the property by which a material
undergoes deformation constant stress over a period of Consider a small lamina (P) of area da having co-ordinates found out by the following steps: (c) The BM is maximum where the S.F is zero or change
time. . as x and y along OX and OY two mutually perpendicular i) Split up the given sec on into plane areas sign.
Brittleness: Brittleness is the tendency of a material to axes on a plane sec on. ii) Find the centre of gravity of the sec on n pure bending, the beam bends in the shape of an arc of
shatter on receiving shock. This happens due to lack of NOW consider a plane OZ perpendicular to OX and OY- iii) Find the moments of iner a of these areas about their a circle
ductility. Let (r) be the distance of the lamina (P) from Z-Z axis such respec ve centres of gravity Assump ons:
Elastic limit: Elastic limit is the stress up to which the that OP=r iv) Calculate moments of iner a about the required axis i) The material of the beam is perfectly homogenous and
material recovers to its original length on removal of the From the geometry of the figure, we find that z by theorem of parallel axis. isotropic
load . r2=x2+y2 v) For a given sec on, moment of iner a may now be ii) the material of beam obeys the Hooke's Law.
proportional limit: The stress up to which stress is We know that the moment of iner a of the lamina P obtained by the algebraic sum of moment of iner a about iii) Elas city constant (E) is the same in tension and
proportional to strain is known as proportional limit. about X-X axis, the required axis compression
Rigid Body: A rigid body is an idealization of a solid body Ixx =da.y2 …. [ “. l=Area x (Distance)2] . iv) The beam is in equilibrium, there is no resultant pull or
in which deformation is neglected. Similarly, Iyy =da.x2 push in the beam sec on
Malleability: Malleability is the ability of a metal to be Izz =da.r2 = da (x2+y2) …… ( r2 = x2 + y2) v) The transverse sec on which were plane before
hammered into thin sheets. Gold and silver are highly = da.x2 + da.y2 = Iyy + Ixx bending, remain plane a er
malleable.
bending
Machinability: Machinability has no direct definition, like
Modulus of Rupture
grades or number. In a broad sense it includes the ability of
the work piece material to be machined, the calculated at rupture using the bending equa on is known
wear it creates on the cutting edge and the chip formation as the modulus of The term modulus of rupture is almost
that can be obtained. exclusively confined to cast iron and
Weldability: The ability of a material to be welded under the rectangular sec on ...
imposed conditions into specific, suitable structure and to Neutral axis:
perform satisfactorily for its intended use. the neutral is an axis in the cross-sec on of a beam (a
Young's Modulus: It is the ratio between tensile stress and member resis ng bending ) sha along which there are
tensile strain or compressive stress. It is denoted by E. no longitudinal stresses or strain.
Modulus of righty: The ratio between shear stress to shear Neutral Plane
strain is called Modulus of rigidity. \ Since the beam is undergoing uniform bending, a plane on
Poisson's Ratio: The ratio between lateral strains to the beam remains same. This called neutral plane.
longitudinal strain is called Poisson's ratio. Moment of resistance :
Bulk Modulus: It is the ratio of hydrostatic stress and The moment of resistance at any sec on of a beam is
volumetric strain. defined as the moment of couple, fomed by the
Factor of safety: The ratio of the ultimate stress and the
longitudinal internal forces of opposite nature and of
working stress for a material is called factor of safety.
equal magnitude, set up at that sec on on either side of
Law of transmissibility: The principle of transmissibility
states that the point of application of a force can be moved the neutral axis due to bending. In magnitude moment of
anywhere along its line of action without changing the Types of beam : resistance at any sec on of a beam is equal to the bending
external reaction forces on a rigid body. Consider a strip of a)A simply supported beam with a hinge at one end and a at that sec on of it so long as the beam does not bend.
roller support at the other. The moment of resistance0fa beam sec on is also known

Theorem of Parallel Axis: b)can lever which has a fixed support at one end and is
free at the other.
as its "flexural strength
"..
c) An overhanging beam where one support of a simply
It states, 1f the moment of iner a of a plane area about
supported beam is shi ed to the interior
an axis through its centre of gravity is denoted by IG then
d) Doubly over hung beam with over hanging segments
moment of iner a of the area about any other axis AB.
at both the ends.
parallel to the first and at a distance h from the centre of
e) A hinged beam where a hinge is used to connect two
gravity is given by:
segments O fa beam. Where,
IAB= IG + ah2
f) A fixed beam which have a fixed support at both end. M= moment of resistance
Where, IAB = Moment of iner a of the area about
Types of load: I= Moment or iner a of the sec on about the neutral
an axis AB,
(i) Point load which is ac ng at a point. axis.
IG= Moment of Iner a of the area about its centre of
(ii) Uniformly distributed load which is uniform en re F= Stress set up in the layer EF at a distance set up in
gravity
length. the layer EF at a distance y' from NA due to bending.
a = Area of the sec on, and
(iii) Gradually varying load which vary with respect to E = Young's modulus of the material of the beam.
h = Distance between centre of gravity of the sec on and
length. Neutral axis always passes through the C.G of the sec on
axis AB.
Shear force : of the beam. Bending stress nil at the neutral axis.is
a circle, whose moment of inertia is required to be found Proof: Shear force is an unbalance ver cal load. Shear force has
two modes
52. Describe bending stress.:
Bhai a beam is loaded with external loads in the
out about a line AB as shown in Fig. 11.7. Consider a strip of a circle, whose moment of iner a is Downward+ Right------shear force (+ ve) transverse direc on of the beam, all the will experience a
Let Oa = Area of the strip required to be found out about a line AB . Upward + le --------shear force(-ve) bending moment. The resistance, offered by the in mal
y= Distance of the strip from the Let Da= Area of the strip
centre of gravity the section and G Bending moment : bending moment is a moment which resistor the bending is called bending stress. The nature Of
y= Distance of the strip from the created bending in beam or sha or bar etc . bending stress is tensile Ed compressive both. Bending
h
centre of gravity the sec on and Point contra flexure: stress increases from zero at neural axis to the extreme
Jlar
h = Distance between centre of In a bending beam, a point is known as a point of contra resec on unlike normal stress which remains same all
h = Distance between centre of
gravity of the sec on and the flexure if it is a loca on at which no bending occurs. A over the sec ons.
gravity of the section and the
axis AB. A- B axis AB. sta onary point on a curve at which the tangent is S3. What is Pure Bending?:
Fig. 11.7. Theorem of parallel We know that moment of iner a of the whole sec on horizontal or bending is a condi on of stress where a bending moment
axis. about an axis passing through the centre of gravity of the A point at which the direc on of bending changes. is applied to a beam simultaneous presence of axial, shear
CI We know that moment of inertia of the whole section sec on =Da .Y.2 or torsional forces. Pure bending occurs under a constant
about and moment of iner a of the whole sec on about an axis Define SFD and BMD. bending moment where the shear force is zero.
an axis passing through the centre of gravity of the section passing through its centre of gravity, Shear force diagram (SFD) and bending moment diagrams 5. What is simple bending? What are the assump ons
= da. y2 I G= sigma Da .y2 (BMD) are analy cal tools in conjunc on with structural made in the simple: bending?
and moment of inertia of the whole section about an axis Moment of iner a of the sec on about the axis AB, analysis to help perform structural design by determining Bending is usually associated with shear. However, for
passing through its centre of gravity, IAB =ah2 + IG+0 value of shear force and bending moments at a given simplicity we neglect the effect of Shea and consider
Shear Stress: *This stresses are always act parallel to the
point of a structural element such a beam. SFD shows bending moment alone to find the stresses due to
stressed surface under consideration.
varia on in shear force along the length of the barn. BMD bending. Such a theory which deals with finding stresses
*Equal and opposite forces act tangentially on any cross -
sectional plane of a body. shows varia on of bending moment along the length. due to pure bending moment at a sec on is
Shear Strain: It is a measure of the angle through which a Called simple bending theory.
body is deformed by the applied force. Assump ons in simple theory of bending:
The value of Poisson’s ratio: l. The beam is ini ally straight and uniform cross-sec on
Cork= 0, Steel=0.25 to 0.33, Rubber=0.5 and every layer of it is free to
Longitudinal Strain: The deformation/strain produced in expand or contract independently of the layer above or
a material body in the longitudinal direction due to below it.
longitudinal force applied to the body is known as 2• The material is homogeneous and isotropic (i.e., of
longitudinal strain. equal elas c proper es in all
Isotropic Materials: Material properties are same in the direc on).
every direction at a point. 3. Young's modulus is same in tension and compression.
Thin Cylinder: 4. Stresses are within elas c limit.
When the thickness of the wall of the cylinder is less than The transverse sec ons which were plane before bending
1/10 to 1/15 of the diameter of the cylinder, then the remain plane a er bending.
cylinder is considered as thin cylinder.
6' The radius of curvature is large compared to the depth
𝑡<(1/10to1/15)𝑑
of beam.
Example: Oil tanks, steam boilers, gas pipes, water supply
7' The beam is subjected to pure bending and therefore
mains etc.
Assumption in Thin Cylinders: bends in the arc of a circle.
*It is assumed that the stresses are uniformly distributed
throughout the thickness of the wall.
*The type of stresses developed in thin cylinders are hoop
stress and longitudinal stress.
 difference between deflection & bending moment: ** What is torsional rigidity?
Deflection is a measure of bending caused due to a Answer: Torsional rigidity is the object's torsional
load or force in an element. Deflection is the distance resistance to twisting as a torque is applied to the
measured from its original position. Bending moment is component and is dependent on a component's
the moment created by the external load or force geometry. As torsional Rigidity increases, the torque
which causes the deflection. required to produce a twist of one-unit angular
**Struts: A structural member subjected to an axial measurement per unit length of the shaft, increases.
compressive force, is called a strut. ** What is meant by section modulus?
**Column: As per definition, struts may be horizontal, Answer: Section modulus is a geometric property for a
inclined or even vertical. But an Erical strut used in given cross-section used in the design beams of beam
buildings or frames is called a column. or flexural members. Section modulus is defined as the
**Difference between column and strut: ratio of moment of inertia of a beam (I) about its C.G
>> A vertical member subjected to axial compressive to the maximum distance of extreme outermost
load is called column (Example: Pillars of a building. section of the beam (Ymax).
>>A strut is a structural member which is subjected to Mathematically, Z=I/ Y max
an axial compressive force. The more the section modulus of a section the more
>>A strut may be horizontal, vertical or inclined with the ability to resist bending load.
any end fixity condition. ** Define angle of twist.
**Classification of Columns: According to the nature of Answer: The angle between the plane of the anchoring
failure – short, medium and long columns. ring and the square plane containing the metal and its
*Short column – whose length is so related to its c/s four directly bonded atoms.
area that failure occurs mainly due to direct **Tortional resilience: It is the strain energy stored in
compressive stress only and the role of bending stress the shaft due to torque.
is negligible. Torsional resilience=½ *T*0 Where T is the mean
*Medium Column - whose length is so related to its c/s torque,0 is the angle twist.
area that failure occurs by a combination of direct **Torsional stiffness: Also called torsional rigidity.It is
compressive stress and bending stress. **Modulus of Rigidity: It is defined as the ratio
* Long Column - whose length is so related to its c/s shearing stress to shearing strain.
area that failure occurs mainly due to bending stress **What are the effects if a bar is subjected to
and the role of direct compressive stress is negligible. torsional load?
**Critical or Crippling or Buckling load: The critical Answer: Effects are: Twisting moment and shear stress
load of a column or strut is the load at which buckling produced on the bar.
of a column or strut will start. At the critical load, the
To transmit energy by rotation, it is necessary to apply
column is said to have developed an elastic instability.
a turning force. In the case of a bar, if the force is
**Safe load: In actual practice, the column is subjected
to a load much less than critical load. This load is called applied tangentially and in the plane of transverse
a safe load or working load. Mathematically, Safe load cross-section, the torque or twisting moment can be
= 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑙𝑜𝑎𝑑/ 𝐹𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑠𝑎𝑓𝑒𝑡𝑦. calculated by multiplying the force with the radius of
**Euler’s Theory >>Columns and struts which fail by the shaft (bar). If the bar is subjected to two opposite
buckling may be analyzed by Euler’s theory. turning moments it is said to be in pure torsion and it is
**Assumptions made in Euler’s theory: exhibit the tendency of shearing-off at every cross-
1. Initially the column or strut is perfectly straight, and section which is perpendicular to longitudinal axis.
load is axial.
2. The cross section of the column is uniform ** State the assumptions made in the theory of pure
throughout its length. torsion.
3. The column is perfectly elastic, homogenous and Answer: The following assumptions are made in
isotropic.
4. The shortening of the column due to direct deducing torsion equation:
1) The shaft material is uniform throughout its length.
compressive stress is neglected.
5. The column is very long as compared to cross 2) The cross-section of a shaft remains plane after
sectional area. application of torque
6. Failure of the column occurs only due to buckling. 3) The cross-section of a circular shaft remains circular
Euler’s formula for critical load, 𝑃𝑒 = 𝜋 2𝐸𝐼/ 𝑙𝑒𝑞 2 after application of torque
**Slenderness Ratio: Slenderness ratio is the ratio of 4) The twist along the length of the shaft is uniform
the equivalent/effective length of a column and the throughout.
radius of gyration of its cross section. Slenderness 5) The maximum shear stress induced in the shaft is
Ratio = 𝑙𝑒𝑞/𝑘. within elastic limit of the material of the shaft.
**Limitations of Euler’s Formula:
• There is always some eccentricity and initial Torque:It is It is the turning moment or twisting
curvature present.
moment tends to produce rotation of an object.
• In practice a strut suffers a deflection before the
Torgue is measured by multiplying the applied force
Crippling load.
**Rankine’s Formula: and its perpendicular distance from the axis of rotation
>> Euler’s formula is applicable to long columns only
for which (𝑙𝑒𝑞 𝑘) ratio is larger than a particular value.
>>Also doesn’t take into account the direct compressive
stress.
>> Thus for columns of medium length it doesn’t
provide accurate results.
>> Rankine forwarded an empirical relation
**Flexural rigidity: It is defined as the force couple
required to bend a non-rigid structure to a unit
curvature or it can be defined as the resistance offered
by a structure while undergoing bending.
**Why Euler's Euler's theory is not applicable for
short columns? Answer –Euler’s theory is not
applicable for short column due to-Euler's long columns
fail only by bucking but in short column crushing failure
is also a factor.
*What do you mean by effective length of a column?
The Elective length of a column is defined as the
distance between successive inflection points of zero
moments.
** What do you mean by polar modulus?
Answer: Polar modulus is the ratio of polar moment of
inertia (J) to the outer radius of a circular shaft(R).
Mathematically, Polar modulus, Zp=J/R
It is also called torsional section modulus.
Stiffness of a spring: The define stiffness of a spri ng:
laad required to produce a unit The stiffness of a body is a
deflection in a spring is called measure of the resistance
spring stiffness or stiffness of otfered by an elastic body ta
a spring. deformation. Generally, for
Types of spring: spring, the spring stiffness
i)Bending springs or leaf is the force required to
spring ii) Torsion springs cause unt deformation
Bending spring : A spring Therefore, Stiffness K
which is subjected to bending =F/Y
only and the resilience is also where deflection Y=1
due to it, is known as spring. Stiffness (K) =F (force)
Laminated spring or leaf What are the applacation
spring are also called bending areas of spring?
spring. Spring change their shape
Torsional spring :A spring, when force is applied and
which is subjected to torsion regains their original
it, is known as a torsion position when we remove it.
spring. subjected to torsion or We use many products in
twisting moment only and the our day-to-day life that use
resilience is also due to it is springs to achieve to
known as torsion spring. required function. For
Torsional strain energy in a example, in analogue
close coil helical spring watches, bicycles, cars and
subjected to axial load is given toys etc. to store mechanical
by energy and realese it.

U=
What is the helix angle of
helica lspring?
The shape formed by the
wires of the helical spring is
called helix angle. The angle
between the coils and the base
of the spring is called pitch
angle or helix angle.