Strength It is the capacity of the material to withstand the action of external agencies without exhibiting any type of fracture of failure.

Stiffness

It is the capacity of the material to withstand the action of load undergoing lesser deformation.

'SOM' deals with the development of mathematical models, equations to determine strength and stiffness characteristics of materials.

Load

Any external force applied on a body is called 'LOAD' on the body.

Unit: 

         Newton (N) is the SI unit of load

         k N, Kilo Newton - 103N

         MN, Mega Newton - 106N

         GN, Giga Newton - 109N

Load Classification Classification of loads:

I. Based on the way in which load is applied on the structure

   (a) Gradually increasing load

   (b) Suddenly applied load

   (c) Load with impact or Dynamic load

Impact load causes severest effect on the material.

II. Based on the way in which load occupies the structure

    (a) Point load or concentrated load

    (b) Uniformly Distributed Load (UDL)

    (c) Uniformly Varying Load (UVL)

    (d) Trapezoidal load

III. Based on the direction of the load with respect to the plane

under consideration

     (a) Direct load or Normal load (P)

Assumptions made in S.O.M

1. All bodies are deformable.

2. Materials are perfectly elastic.

3. Materials are isotropic and homogenous.

4. Principle of superposition is valid.

5. St. Venant's Principle is valid.

Elasticity:

It is the property by virtue of which material undergoes deformation due to the action of load and regains its original shape and size after removal of load. No material is perfectly elastic.

Plasticity:

Plasticity is the property by virtue of which material undergoes deformation due to the action of load and retains its deformed shape and size even after the removal of load. No material is perfectly plastic.

Isotropy:It is the property by virtue of which elastic behaviour of the material at a point will be same along all the directions.The material which is not isotropic is called anisotropic. Eg: Wood

Homogeneity:It is the property by virtue of which elastic behaviour of material will be same at all the points.

If the material is not homogenous, then it is called heterogonous.

Principle of Superposition:

"The net effect of loads applied in any sequence on a body is given by the algebraic sum of effect of individual forces on the body"

St. Venant's Principle:"At points away from the loading points, the behaviour of material will be independent of gripping forces or type of application of load or local effects." 

Explanation:

Definition of Stress:-

When a bar of certain material is subjected to load it undergoes certain amount of deformation and then attains state of equilibrium due to the resistive force developed inside the material. The resistive force offered by the molecules of the section against the applied load is called 'STRESS'. For the equilibrium, total resistive force developed must be equal to total load applied.

Stress Distribution Stress distribution indicates the variation of stress across the

resisting section. There are 2 types of stress distribution.

1. Uniform Stress Distribution

2. Non-uniform Stress Distribution.

1. Uniform Stress Distribution

If the molecules of the resisting section offer equal amount of

resistance then stress distribution is said to be uniform.

2. Non-uniform Stress Distribution

If the molecules of the section offer unequal amount of resistance,

then the stress distribution is said to be non-uniform.

Intensity of Stress Intensity of Stress or Unit Stress or Stress Intensity:

Intensity of Stress or Unit Stress or Stress Intensity at a point is the

resistive force developed over unit area considered around the point

in the section.

Calculation of intensity of stress:

If the stress distribution is uniform then intensity of stress will be

same at all the points. In such cases intensity of stress at any point

is given by the ratio of total resistive force developed to the total

area of resistance.

If the stress distribution is non-uniform then intensity of stress at a

point is given by,

where  R is the resistive force developed over elemental area  A

N/mm 2 is the convenient unit for intensity of stress.

1 N/mm2 =  = 106 N/m2

1 N/mm2 = 1 MPa

Types of Stresses Depending on the type of load

i. Direct Stress/ Normal Stress: 

This is the stress developed or resistive force developed

inside the material against the direct load applied. It can be

tensile or compressive. 

ii. Tangential or Shear Stress: 

It is the stress developed or resistive force developed

against tangential load or shear force.

Strain It is the measure of deformation caused by the load on the material.

Types of Strain

i. Direct Strain or Normal Strain (e or  ) :

It is the strain caused by a direct load (tensile or

compressive) along its line of action. It is calculated as the

ratio of deformation caused by the load along its line of

action to the corresponding original dimension. 

Case 1: 

Case 2:-

ii. Lateral Strain (elat):

It is the strain produced by a direct load along direction/s perpendicular to its line of action. It

is calculated as the ratio of change in lateral dimension to the original lateral dimension.

iii. Shear Strain (Ø):

It is the strain produced by shear force 

ii. Volumetric Strain / Bulk Strain (ev):

It is the ratio of change in volume caused by the load or system of loads to the initial or

original volume.

Unit of Strain:Any strain is a dimensionless quantity and has no unit.

Hooke's Law "For all elastic materials, stress is directly proportional to strain

within the limit of proportionality"

Explanation:

Stress   Strain (Within limit of proportionality)

Stress = (Constant) x Strain (Within limit of proportionality)

(Y = m x X)

Limit of Proportionality:It is the stress value within which material obeys Hooke's law or Stress-Strain relationship is rectilinear (straight line).

Elastic limit:It is the stress value upto which the material exhibits the property of elasticity. Elastic limit will be more than the limit of proportionality.

Elastic Constants

There are 4 elastic constants; they are:

(a) Young's modulus of elasticity (E)

(b) Rigidity or shear modulus of elasticity ( C or N or G)

(c) Bulk modulus of elasticity (K)

(d) Poisson's ratio

.a Young's modulus of elasticity (E):

For all elastic materials the ratio of direct stress to direct strain is a

constant within the limit of proportionality called 'Young's modulus of

elasticity'

.b Rigidity or shear modulus of elasticity (C):

For all elastic materials, the ratio of shear stress to shear strain is a const

within the limit of proportionality called 'Rigidity or Shear modulus of

elasticity'

C Bulk modulus of elasticity (K): 

For all elastic materials the ratio of stress of equal intensity applied along all the

direction to the corresponding volumetric strain is a constant within the limit of

proportionality called 'Bulk Modulus'.

d. Poisson's ratio (µ):

For all elastic materials, the ratio of lateral strain to the direct strain

produced by a direct load is a constant within the limit of proportionality

called 'Poisson's ratio'.

For all elastic materials, µ lies between 0 and 0.5. 

Relationship among elastic constants:

Types of Bars:-

1. Uniform bar or prismatic bar:

A bar whose cross-sectional dimensions remain same over its entire

length is called 'Uniform or Prismatic bar'. Its volume is given by the

product of cross-sectional area and length.

2. Bar of varying cross sections:

A bar whose cross - section's dimensions vary over its length is called bar

of varying cross sections. Here there are 2 types.

a. Bar with abrupt or sudden changes in cross section:

(Stepped bars)

b. Bar with gradual changes in cross section: (tapering bar)

Example

1. A steel rod 10m long used in a control mechanism must transmit a tensile force of 5 kN

without stretching more than 3 mm, nor exceeding an allowable stress of 150 MPa.Nominal

Stress and nominal Strain:

Stress and strain of material calculated at any instant of time with respect to the original

dimensions of the specimen are called nominal stress and nominal strain. They are also called

Engineer's stress and Engineer's Strain.

2. True stress and true strain:

Stress and strain of a material calculated at any instant of time with respect to the dimensions

prevailing at that instant of time are called "True Stress" and "True Strain".

Stress-Strain Curve for a Typical Ductile Material (Mild Steel) Under Tension:

1. Between 'O' and 'A' the stress-strain relationship is rectilinear and material obeys Hooke's law.

The stress corresponding to 'A' is called 'limit of proportionality'. The slope of the straight line

gives Young's modulus of Elasticity. 

2. Between 'A' and 'B' material is elastic but will not obey Hooke's law. Stress corresponding to 'B'

upto which material is elastic is called 'Elastic limit'. 

3. 'C' is the "upper yield point" and 'D' is the 'lower yield point'. Between yield points material

undergoes large amt. of deflection although there is no increase in stress. Stress

corresponding to yield point is called 'Yield Stress'. 

4. 'E' is the ultimate point and corresponds to the maximum load taken by the material. Stress

corresponding to ultimate point is called 'ultimate Stress' or ultimate Tensile strength or

'Tenacity' of the material. 

5. 'F' corresponds to the point of failure and is called 'Breaking Point'. Stress corresponding to

breaking point is called 'Breaking Stress'. Nominal Breaking stress is less than Nominal

Ultimate stress.

6. Working Stress:To avoid the risk of loading the material upto its maximum capacity, a stress value less than the maximum stress that it can take is adopted as allowable or permissible stress for the material and is called 'Working Stress'.

7.

8.Factor of Safety:It is the ratio of maximum stress that the material can take to the working stress adopted for design.

9.

(a) What is the dia of the rod required? give answer to the nearest mm (E = 210 GPa) 

(b) Does the strength or stiffness control the design?

To det dia required to satisty C2:

                

To determine dia required to satisfy C1: 

(b). Here, stiffness controls the design.