SHEAR FORCE AND BENDING MOMENT DIAGRAMS USING SM EQUATIONS

STRESS STRAIN DIAGRAM

​Stress Strain Diagram

Initially the metal piece is to be placed in between the tension compression testing machine. Later, load is applied on the metal piece with the help of the machine.  One can notice the increment in the size along the axial direction. The increment of the load is measured along with the work piece elongation. The process is repeated until it reaches to failure of the work piece material. Before starting the process note the original values of the work piece material. With the help of the original values of area and length by using them one can find the normal stress and normal strain. For the stress and strain value one needs to draw a graph.

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By taking values of stress on y-axis and values of strain on x-axis, this is known as stress strain diagram. But now we are not giving the standard diagram. The stress strain diagram differs from metal to metal. Now a detail concept of medium carbon steel stress strain diagram is taken into consideration. Metallic materials are categorized as brittle or ductile material. Brittle materials have comparatively small strain rupture point like concrete and cast iron. In case of ductile materials they have high tensile strain ruptures like aluminum and steel.

Parameters we seen in the stress strain curve are

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Proportional limit:
In the above diagram from the origin to the point O is known as Proportional limit. It represents a straight line. Hooks law states that with the Proportional limit the stress is directly proportional to the strain.

Stress is proportional to the strain
Stress ∝ Strain
σ ∝ ε
σ= Stress
ε = Strain
σ = k ε
K = modulus of elasticity or Young’s Modulus.
σ = E ε

Modulus of rigidity:
Modulus of rigidity is also known as shear modulus. Within the elastic limit the ratio between shear stress and shear strain is known as the modulus of rigidity. It is denoted by C, G or N. within the elastic limit ‘q’ is the shear stress and ‘f’ is the shear strain.

G = q / f.

Bulk Modulus:
Within the elastic limit the ratio of the volumetric stress to the volumetric strain is known as Bulk modulus. Within the elastic limit ‘pv’ is the volumetric stress and ‘ev’ is the volumetric strain. The bulk modulus is denoted by K

K = pv / ev.

Elasticity:
It is defined as one of the properties of the material. Elasticity is virtue of the body at which it regains its original shape and size after the deformation forces are removed where we can find the elastic materials very rarely. So the maximum stress exhibits the property of the edacity on the material which is known as elastic limit. If the applied deformed forces cause stress in the material, it leads to excess of the elastic limit. So in this case the body does not regain its original shape and size when the deformed forces are completely removed. Then some residual strains are left in it.

Elastic range:
In the above diagram the area from O to P is known as elastic range

Plastic Range:
In the same diagram the region from P to R is known as Plastic region.

Yield Point:
Yield point is well-defined as the point at which the material will have a considerable yielding or elongation without having any increase in the load.

Ultimate Strength:
Ultimate strength is known as when the maximum stress is applied on the material, and it can withstand by stretching or pulling before failing or breaking. It is also known as tensile strength.

Rupture strength:
Normal stress which are developed in the material at the time of rupture. But the rupture strength is not equal to the ultimate strength. When rupture strength is determining at that case necking is not taken in to the account.  In rare cases true stress are indicated at the rapture. It is also known as breaking strength.

Modulus of resilience:
If the force is increased gradually then the work done on a unit volume is defined as Modulus of resilience. The forces are increases from the point origin to point P. the units are Nm/. We can calculate the area under the curve from the origin to elastic limit. Without creating any permanent distortion the material is having the ability to absorb the energy is known as resilience.

Modulus of Toughness:
If the force is increased gradually then the work done on a unit volume of the material is defined as Modulus of toughness. The forces increases from the point origin to point R the units are Nm/. We can calculate the area under entire curve from the origin to R. This is done without causing any break to the material, as it has the ability to absorb the energy this process is known as Modulus toughness.

Working stress:
Working stress is known as the actual stress of the material under given loads. The maximum stress which is safe and that is carried by a material is named as allowable stress.

Allowable stress: 
Allowable stress has certain a limit and it does not exceed the proportional limit. It is difficult to determine the accurate value of the proportional limit. The allowable stress is taken from the ultimate strength or yield strength which is divided by a safety factor.

Safety factor:
The ratio between the ultimate strength (yield strength) and allowable strength is known as safety factor.