som final.docx

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STRENGTH OF MATERIALS
STRESS (σ-sigma, N/mm2):
The intensity of force (i.e. force per unit area is) is called stress. The force of resistance
offered by a body against the deformation for a unit area is called stress.
STRESS
NORMAL STRESS (PERPENDICULAR) SHEAR STRESS (PARALLEL)
TENSILE (+VE) COMPRESSIVE (-VE)
STRAIN (ε-Epsilon, No unit):
The elongation or contraction per unit length is called strain.
WHY STRESS – STRAIN DIAGRAM?
Usual procedure to find the mechanical behaviour is to place small specimens of material in
testing machines, apply the loads and then measure the resulting deformations (load -
displacement curves). This test results generally depend upon the dimension of specimen
being tested. To express the test results in the form that can be applied to members of any
size we convert the test results to stresses and strains.
WHEN TO APPLY FORMULA FOR STRESS? (σ = P / A)
The principal requirement is that deformation of the bar be uniform throughout its volume,
which in turn requires that the bar be prismatic, loads act at centroids of the cross sections
and the material be homogenous.
TRUE STRESS AND ENGINEERING STRESS;

When the initial area of the specimen is used for calculating stress, the stress is called
engineering stress (i.e. nominal or conventional stress).
A more exact value of the axial stress, called true stress (σact) can be calculated by using the
actual area at the cross section where the failure occurs.
STRESS-STRAIN DIAGRAM:
Mild steel (Not to scale)
Mild steel (to scale)
PROPOTIONAL LIMIT: The point A up to which SSD is linear and proportional (the ratio
remains constant)
YIELD POINT: From point B considerable increase in elongation without increase in stress.
The corresponding stress value is called Yield strength.
ULTIMATE STRENGTH:
The maximum value of stress (D) the material can bear is called ultimate strength. Further
stretching resulting in reduction in load.
SSD OF OTHER MATERIALS:
RUBBER CAST IRON(BRITTLE) ALUMINIUM(NO YIELD POINT)
DIFF BETWEEN STRENGTH AND STRESS:
Strength is a general term that refers to the capacity of a structure to resist loads. It is a
property of a material (yield strength, ultimate tensile strength etc).

Stress varies when the load applied varies. It’s not a property of a material.
YOUNG’S MODULUS;
Up to proportional limit, the ratio of axial stress and axial strain is called young’s modulus.
𝐸 =
𝜎
𝜀
RIGIDITY MODULUS:
Up to proportional limit, the ratio of shear stress and shear strain is called rigidity modulus.
𝐺 =
𝜏
𝛾
HOW TO FIND YIELD STRESS OF ALUMINIUM?
As aluminium is not having defined yield point we have to measure it by OFFSET METHOD. A
straight line drawn parallel to initial linear part of the curve with a offset of
0.2%strain(0.002) intersect the SSD gives the Yield strength(Proof, offset yield strength).This
yield strength is not an inherent property of the material. Its calculated arbitrarily.
PHYSICAL SIGNIFICANCE OF YOUNG’S MODULUS?
The value signifies the resistance to elastic deformation. (Erubber=0.1GPa,Esteel=210Gpa).
The greater the modulus the stiffer the material. 𝐾 =
𝐸𝐴
𝐿
EXPLAIN STRAIN HARDENING?
Due to strain hardening the material undergoes change in crystalline structure, resulting in
increased resistance to further deformation. Therefore further increase in strain needs
increase in stress so SSD so you get positive slope .

WHAT IS GAGE LENGTH (L0)?
The length of specimen which taken for consideration during testing within the
extensometer arms. ASTM standard gage length-50mm. It’s used to measure strain and % of
elongation. %EL=
𝐿1−𝐿0
𝐿0
× 100,L1-final length
WHEN TO GO FOR TRUE STRESS:
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
WHAT IS BAUSCHINGER EFFECT (STRAIN SOFTENING, WORK SOFTENING)?
When a metal with tensile yield stress Y, is subjected to tension into the plastic range and
then the load is released and applied in compression, the yield stress in compression is lower
than that in tension. This phenomenon is known as Bauschinger effect. This effect is also felt
when compression is followed by tension.
WHICH REPRESENTS TOUGHNESS AND RESILIENCE REGION IN SSD?
TOUGHNESS RESILIENCE

DUCTILITY:
It is a measure of plastic deformation that has been sustained at fracture. It’s measured as
%EL= (
𝑳𝟏−𝑳𝟎
𝑳𝟎
) × 𝟏𝟎𝟎 (The value of %EL depends upon gage length. So always we should
specify gage length with %EL)
%AR =(
𝑨𝟎−𝑨𝟏
𝑨𝟎
) ×100 (For a material value of %EL, %AR will be different)
BRITTLE:
A material with very little or no plastic deformation at fracture is termed as BRITTLE.
A material having less than 5% fracture strain is considered as BRITTLE. (EG: Cast iron,
Stone, Concrete, and Ceramics)
FATIGUE:
Fatigue is defined as the deterioration of a material under repeated cycles of stress and strain,
resulting in progressive cracking that produce fracture at loads lesser than failure load in
static condition. The fatigue strength is obtained by plotting S-N curve or ENDURANCE curve.
S-N CURVE (S-STRESS,N-NO OF CYCLES TO FAILURE)
FATIGUE LIMIT OR ENDURANCE LIMIT:
The limit of stress below which fatigue failure will not occur regardless of how many times
the cycle is repeated.
RELATE THE 3 ELASTIC CONSTANTS OF A MATERIAL (E, G, K):
𝑬 =
𝟗𝑲𝑮
𝟑𝑲+𝑮
𝑮 =
𝑬
𝟐(𝟏+𝜸)
, 𝑲 =
𝑬
𝟑(𝟏−𝟐𝜸)
Where E- YOUNGS MODULUS, G- RIGIDITY MODULUS , K-BULK MODULUS , γ- POISSON
RATIO.

FACTOR OF SAFETY (n-value range 1-10):
𝐹𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑠𝑎𝑓𝑒𝑡𝑦, 𝑛 =
𝑎𝑐𝑡𝑢𝑎𝑙 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ
𝑎𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ
𝑦𝑖𝑒𝑙𝑑 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ
𝑤𝑜𝑟𝑘𝑖𝑛𝑔 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ
(For ductile material)
𝑢𝑙𝑡𝑖𝑚𝑎𝑡𝑒 𝑡𝑒𝑛𝑠𝑖𝑙𝑒 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ
𝑎𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ
(For brittle material)
POISSONS RATIO (ν-NU)(VALUE RANGE: 0-0.5)
The ratio of lateral strain at any point of the bar to the axial strain
at the same point is known as poisons ratio.
=
−𝜀′
𝜀
Since the
axial and lateral strains are opposite in sign, a negative sign is
introduced in the equation to make ν positive
HOOP‘S STRESS (circumferential): 𝜎1 =
𝑝𝑑
2𝑡
If the direction of tensile stress is along the circumference of the shell, the
stress so induced is called hoop stress or circumferential stress.
MOMENT EQUATION:
𝑀
𝐼
=
𝜎
𝑦
=
𝐸
𝑅
M Bending moment at Section
I Moment of Inertia of Section about Neutral
axis
Σ Bending Stress
Y Distance of extreme fibre from neutral axis
E Young’s Modulus of Material
R Radius of Curvature of beam
TORSION EQUATION:
MATERIAL RATIO(NO UNIT)
CORK 0
CONCRETE 0.1 OR 0.2
MILD STEEL 0-3
RUBBER 0.5

𝑇
𝐽
=
𝜏
𝑟
=
𝐺. 𝜃
𝑙
𝜏 Torsional Shear Stress
𝑟 Radius of Shaft
𝑇 Torque or Twisting Moment
𝐽 Polar Moment of Inertia
𝐺 Modulus of Rigidity
𝜃 Twist Angle
𝑙 Shaft Length
CANTILEVER BEAMS:
A beam which is fixed or built in at one end while the other end is free is called a cantilever.
SIMPLY SUPPORTED BEAMS:
A beam with pin support at one end and roller support at other end is called a simply
support beam or simple beam.
SECTION MODULUS:
It is defined as the ratio of the moment of inertia about the neutral axis and the distance of
the most distant point from the neutral axis
𝒁 =
𝑴𝑶𝑴𝑬𝑵𝑻 𝑶𝑭 𝑰𝑵𝑬𝑹𝑻𝑰𝑨 𝑨𝑩𝑶𝑼𝑻 𝑻𝑯𝑬 𝑵𝑬𝑼𝑻𝑹𝑨𝑳 𝑨𝑿𝑰𝑺
𝑫𝑰𝑺𝑻𝑨𝑵𝑪𝑬 𝑶𝑭 𝑻𝑯𝑬 𝑴𝑶𝑺𝑻 𝑫𝑰𝑺𝑻𝑨𝑵𝑪𝑬 𝑭𝑹𝑶𝑴 𝑵𝑬𝑼𝑻𝑹𝑨𝑳 𝑨𝑿𝑰𝑺
=
𝑰
𝒀𝒎𝒂𝒙
COLUMNS:
Column is the name given to vertical members used in building frames subjected to
compressive loads.
STRUT:
Strut is the name given to a compression member of a truss.
BUCKLING:
When an axial compressive load reaches a certain critical value the member begins to bend
or buckle.
SLENDERNESS RATIO:

Slenderness ratio depends only on the dimensions of the member. It is the ratio of length
and width. i.e. l/r for bars , l/ b for rectangular cross section .
PARALLEL AXIS THEOREM:
The moment of inertia of an area with respect to any axis in its plane is equal to the
moment of inertia with respect to a parallel centroidal axis plus the product of the area and
square of the distance between two axes.
PERPENDICULAR AXES THEOREM:
If Iox and Ioy be the moments of inertia of a lamina about mutually perpendicular axes OX
and OY in the plane of the lamina and Ioz be the moment of inertia of the lamina about an
axis normal to the lamina and passing through the point of intersection of the axes OX and
OY ,
I oz = I ox + I oy
MOHR S CIRCLE:
The plot of Transformation Equation for plane stress in graphical form is known as Mohr’s
circle. It is used to visualize the relationship between normal and shear stress acting on
various inclined planes at a point in stressed body. It’s used to calculate
 Principal stress
 Maximum shear stress
 Stress on inclined plane
Draw a set of coordinate’s axes with σx1 as abscissa (positive to the right) and τx1y1 as
ordinate (positive). COUNTER CLOCKWISE

2. Locate the centre C of the circle at the point having coordinates σx1 = σavg and τx1y1 = 0.
3. Locate point A, representing the stress conditions on the x face of the element by plotting
its coordinates σx1 = σx and τx1y1 = τxy . Note that point A on the circle corresponds to θ = 0 .
Also, note that x face of the element is noted A to show its correspondence with point A on
the circle.
4. Locate point B, representing the stress conditions on the y face of the element by plotting
its coordinates σx1 = σy and τx1y1 =- τxy. Note that
point B on the circle corresponds to θ =90. Also note that x face of the element is noted B
to show its correspondence with point B on the circle.
5. Draw a line from point A to point B. This line is a diameter of the circle and passes
through the centre C. Points A and B, representing the stresses on planes at 90 to each
other, are at opposite ends of the diameter (and therefore are 180 apart on the circle ) .
Using C as centre, draw Mohr’s circle through points A and B . The circle drawn in this
manner has radius as shown below .
Note: Angle 2θ in Mohrs circle corresponds to an angle θ on a stress element.
INFERENCE:
I. If the angle θ is known we can measure σx1, σy1 and τx1y1.(by measuring stress 2θ
counterclockwise from radius CA, because A corresponds to θ=0)
II. The maximum shear stress are located at angle 2 ϴ=90 from points P1,P2(which is
equal to radius of mohrs circle).
III. The shear stresses are zero on the principal planes.
IV. Here, θp has two values θp1, and θp2 that differ by 90° with one value between 0°
and 90° and the other between 90° and 180°. These two values define the principal
planes that contain maximum and minimum stresses.

SHEAR FORCE ABD BENDING MOMENT:
SIGN CONVENTION USED:
1. For shear force: A S.F having an upward direction to the left hand side of the section or
downward direction to the right of section is considered as positive. Similarly negative stress
when downward direction to the left and upward direction to the right of the section
considered.
2. For bending moment: A B.M causing concavity upwards is taken as positive and B.M causing
convexity upwards is taken as negative.
3. The problems are analysed from left hand side for the beams.
IMPORTANT NOTE: You can take sign convention just opposite to what given above. But you will
get completely reversed S.F and B.M diagrams.
Simply supported with point load: Simply supported with UDL Simply supported with UVL

CANTILIVER WITH POINT LOAD UDL UDL & POINT LOAD
OVERHANGING:

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