ASSIGNMNET

 ASSIGNMENT  NO-1

LAST DATE OF SUBMISSION-5/10/2020

SUB NAME -STRENGTH OF MATERIAL

Q1 Write   Difference Between  Elasticity  and plasticity

Q2 Write  a short note  with diagram  a) types of stresses    b)  types of strain 

Q3  Explain stress -strain diagram  with  digaram  and also  Hooke's   Law  

Q4  Write  a short note  with  diagram a)  working  stresss  b) factor  of safety   c)lateral  strain  d) volumetric strain e) Poisson's ratio   

Q5  Write a short note  on a) young'  modulus    b)  bulk  modulus   c) modulus of rigidity    

 

Q6 Derive relationship between  elastic moduli?

Q7 Derive Torsion equation with assumption  ?

Q8 what is strain energy? Derive strain energy for gradual, sudden and impact loading

                          Assignment No-2

               Last date of submission-16/11/2020

Q1 explain different types of load and types of supports in beam

Q2 Draw sfd and bmd of udl of intesity wkn/m over a span of  L metre  Beam is simply supported

Q3 Draw sfd and bmd of uvl  over a beam of L m beam is simply supported

Q4 Draw sfd and bmd of cantilever carrying a udl over a span of L m. Beam is simply supported 

Q5 Derive relation between load  shear and bending moment 

Q6 derive bending equation with assumption

Q7 Define section modulus  ? Derive section modulus of a) rectangular section b) hollow rectangular section c) circular section d) hollow circular section 

INTRODUCTION AND REVIEW

GO THROUGH THESE VIDEOS FIRST

https://www.youtube.com/watch?v=YkdQB0JnJD4&list=PL9RcWoqXmzaLlfmNg2Ku1SdZtvXnYrLbc&index=4

https://www.youtube.com/watch?v=AzICg5-Whlo&list=PL9RcWoqXmzaLlfmNg2Ku1SdZtvXnYrLbc&index=8

https://www.youtube.com/watch?v=c6woVjcC-q8&list=PL9RcWoqXmzaLlfmNg2Ku1SdZtvXnYrLbc&index=9

https://www.youtube.com/watch?v=c6woVjcC-q8&list=PL9RcWoqXmzaLlfmNg2Ku1SdZtvXnYrLbc&index=9

https://www.youtube.com/watch?v=Gtd-eLxtP7Y

UNIT  -2

Stress: stress is defined as the force intensity or force per unit area. Here we use a symbol s to represent the stress.

                                                                           

Units :

The basic units of stress in S.I units i.e. (International system) are N / m² (or Pa)

MPa = 10⁶ Pa

GPa = 10⁹ Pa

KPa = 10³ Pa

 TYPES OF STRESSES :

only two basic stresses exists : 

(1) normal stress and

 (2) shear shear stress. 

Normal stresses : We have defined stress as force per unit area. If the stresses are normal to the areas concerned, then these are termed as normal stresses. The normal stresses are generally denoted by a Greek letter ( s )

Tensile or compressive stresses :

The normal stresses can be either tensile or compressive whether the stresses acts out of the area or into the area

Shear stresses :Let us consider now the situation, where the cross – sectional area of a block of material is subject to a distribution of forces which are parallel, rather than normal, to the area concerned. Such forces are associated with a shearing of the material, and are referred to as shear forces. The resulting force interistes are known as shear stresses

The resulting force intensities are known as shear stresses, the mean shear stress being equal to

Where P is the total force and A the area over which it acts.

The greek symbol t ( tau ) ( suggesting tangential ) is used to denote shear stress

Hydrostatic stress : The term Hydrostatic stress is used to describe a state of tensile or compressive stress equal in all directions within or external to a body. Hydrostatic stress causes a change in volume of a material, which if expressed per unit of original volume gives a volumetric strain denoted by Î_v. So let us determine the expression for the volumetric strain.

CONCEPT OF STRAIN:

Concept of strain : if a bar is subjected to a direct load, and hence a stress the bar will change in length. If the bar has an original length L and changes by an amount dL, the strain produce is defined as follows:

Strain is thus, a measure of the deformation of the material and is a non

dimensional Quantity i.e. it has no units. It is simply a ratio of two quantities with the same unit.

Shear strain is measured as the displacement of the surface that is in direct contact with the applied shear stress from its original position.

Volumetric Strain:

lateral strain, also known as transverse strain, is defined as the ratio of the change in diameter of a circular bar of a material to its diameter due to deformation in the longitudinal direction. 

Hook's Law :

A material is said to be elastic if it returns to its original, unloaded dimensions when load is removed.

Hook's law therefore states that

Stress ( s ) a strain( Î )

Working Stress and Factor of Safety

Working stress is the safe stress taken within the elastic range of the material. For brittle materials, it is taken equal to the ultimate strength divided by suitable factor of safety. However, for materials possessing well defined yield point, it is equal to yield stress divided by a factor of safety. 

Factor of safety is a number used to determine the working stress. It is fixed based on the experimental works on the material. It accounts all uncertainties such as, material defects, unforeseen loads, manufacturing defects, unskilled workmanship, temperature effects etc. Factor of safety is a dimensionless number. It is fixed based on experimental works on each material. 

Poisson's ratio: If a bar is subjected to a longitudinal stress there will be a strain in this direction equal to s / E . There will also be a strain in all directions at right angles to s . The final shape being shown by the dotted lines.

It has been observed that for an elastic materials, the lateral strain is proportional to the longitudinal strain. The ratio of the lateral strain to longitudinal strain is known as the poison's ratio .

Poison's ratio ( m ) = - lateral strain / longitudinal strain

STRESS STRAIN DIAGRAM 

 Proportional Limit (Hooke's Law)

From the origin O to the point called proportional limit, the stress-strain curve is a straight line.This linear relation between elongation and the axial force causing   is called Hooke's Law that within the proportional limit, the stress is directly proportional to strain or

 

σ∝ε$\sigma \propto \epsilon$   or   σ=kε

The constant of proportionality k is called the Modulus of Elasticity E or Young's Modulus and is equal to the slope of the stress-strain diagram from O to P. Then

σ=Eε

Elastic Limit
The elastic limit is the limit beyond which the material will no longer go back to its original shape when the load is removed, or it is the maximum stress that may e developed such that there is no permanent or residual deformation when the load is entirely removed.

Yield Point
Yield point is the point at which the material will have an appreciable elongation or yielding without any increase in load.
 

Ultimate Strength

The maximum ordinate in the stress-strain diagram is the ultimate strength or tensile strength.
 

Rapture Strength
Rapture strength is the strength of the material at rupture. This is also known as the breaking strength.

Modulus of elasticity(E) : Within the elastic limits of materials i.e. within the limits in which Hook's law applies, it has been shown that

Stress / strain = constant

This constant is given by the symbol E and is termed as the modulus of elasticity or Young's modulus of elasticity

Thus 

Shear modulus also known as Modulus of rigidity(G)  :It is the measure of the rigidity of the body, given by the ratio of shear stress to shear strain. Often denoted by G sometimes by S or μ.

Definition	Shear modulus is the ratio of shear stress to shear strain in a body.
Symbol	G or S or μ
SI unit	Pascal (Pa), N/m2
Formula	Shear stress/shear strain
Dimension formula	M1L-1T-2

.The bulk modulus (K) -describes volumetric elasticity, or the tendency of an object to deform in all directions when uniformly loaded in all directions.

It is defined as volumetric stress over volumetric strain, and is the inverse of compressibility.

• G is the Shear Modulus
• E is the Young’s Modulus
• K is the Bulk Modulus
• υ is Poisson’s Ratio

Elastic moduli & the relationship between them

Relation between E, G and u :

Let us establish a relation among the elastic constants E,G and u. Consider a cube of material of side ‘a' subjected to the action of the shear and complementary shear stresses as shown in the figure and producing the strained shape as shown in the figure below.

Assuming that the strains are small and the angle A C B may be taken as 45⁰.

Therefore strain on the diagonal OA

= Change in length / original length

Since angle between OA and OB is very small hence OA @ OB therefore BC, is the change in the length of the diagonal OA

Now this shear stress system is equivalent or can be replaced by a system of direct stresses at 45⁰ as shown below. One set will be compressive, the other tensile, and both will be equal in value to the applied shear strain.

Thus, for the direct state of stress system which applies along the diagonals:

Relation between E, K and u :

Consider a cube subjected to three equal stresses s as shown in the figure below

The total strain in one direction or along one edge due to the application of hydrostatic stress or volumetric stress s is given as

Relation between E, G and K :

The relationship between E, G and K can be easily determained by eliminating u from the already derived relations

E = 2 G ( 1 + u ) and E = 3 K ( 1 - u )

Thus, the following relationship may be obtained