Formula Sheet: Mechanics of Materials (MOM) | Formula Sheets of Mechanical Engineering PDF Download

 Page 1


 
 
Stress: When a material is subjected to an external force, a resisting force is set up in the component. 
The internal resistance force per unit area acting on a material is called the stress at a point. It is a 
tensor quantity having unit of N/m
2
 or Pascal. 
 
Types of Stresses 
? Normal stress 
 
 
? Shear Stress 
 
 
? Bulk Stress 
 
 
Strain: It is the deformation produced in the material due to simple stress. It usually represents the 
displacement between particles in the body relative to a reference length. 
Types of Strains 
? Normal Strain: The normal strain of a body is generally expressed as the ratio of total 
displacement to the original length. 
 
Page 2


 
 
Stress: When a material is subjected to an external force, a resisting force is set up in the component. 
The internal resistance force per unit area acting on a material is called the stress at a point. It is a 
tensor quantity having unit of N/m
2
 or Pascal. 
 
Types of Stresses 
? Normal stress 
 
 
? Shear Stress 
 
 
? Bulk Stress 
 
 
Strain: It is the deformation produced in the material due to simple stress. It usually represents the 
displacement between particles in the body relative to a reference length. 
Types of Strains 
? Normal Strain: The normal strain of a body is generally expressed as the ratio of total 
displacement to the original length. 
 
 
 
 
Strain is dimensionless quantity. 
It is of two types: Longitudinal strain and Lateral Strain 
Longitudinal strain is the defined as the ratio of the change in length of the body due to the 
deformation to its original length in the direction of the force. 
Lateral Strain is defined as the ratio of the change in length (breadth of a rectangular bar or diameter 
of a circular bar) of the body due to the deformation to its original length (breadth of a rectangular 
bar or diameter of a circular bar) in the direction perpendicular to the force. 
  
? Shear strain 
 
 
? Bulk Strain or Volumetric Strain 
 
 
 
Page 3


 
 
Stress: When a material is subjected to an external force, a resisting force is set up in the component. 
The internal resistance force per unit area acting on a material is called the stress at a point. It is a 
tensor quantity having unit of N/m
2
 or Pascal. 
 
Types of Stresses 
? Normal stress 
 
 
? Shear Stress 
 
 
? Bulk Stress 
 
 
Strain: It is the deformation produced in the material due to simple stress. It usually represents the 
displacement between particles in the body relative to a reference length. 
Types of Strains 
? Normal Strain: The normal strain of a body is generally expressed as the ratio of total 
displacement to the original length. 
 
 
 
 
Strain is dimensionless quantity. 
It is of two types: Longitudinal strain and Lateral Strain 
Longitudinal strain is the defined as the ratio of the change in length of the body due to the 
deformation to its original length in the direction of the force. 
Lateral Strain is defined as the ratio of the change in length (breadth of a rectangular bar or diameter 
of a circular bar) of the body due to the deformation to its original length (breadth of a rectangular 
bar or diameter of a circular bar) in the direction perpendicular to the force. 
  
? Shear strain 
 
 
? Bulk Strain or Volumetric Strain 
 
 
 
 
Stress and Strain both are tensor quantity i.e. it has both change in magnitude as well as 
direction. 
True Stress and True Strain 
? The true stress is defined as the ratio of the load to the cross section area at any instant. 
 
Where s and e is the engineering stress and engineering strain respectively. 
? The true strain is defined as 
 
Lo- original length, L-successive values of the length as it changes 
? The volume of the specimen is assumed to be constant during plastic deformation. 
Stress-Strain Relationship 
? The stress-strain diagram is shown in the figure. In brittle materials, there is no appreciable change 
in the rate of strain. There is no yield point and no necking takes place. 
         
? In figure (a), the specimen is loaded only upto point A, when load is gradually removed the curve 
follows the same path AO and strain completely disappears. Such a behaviour is known as the 
elastic behaviour. 
? In figure (b), the specimen is loaded upto point B beyond the elastic limit E. When the specimen is 
gradually loaded the curve follows path BC, resulting in a residual strain OC or permanent strain. 
Comparison of engineering stress and the true stress-strain curves shown below: 
 
Page 4


 
 
Stress: When a material is subjected to an external force, a resisting force is set up in the component. 
The internal resistance force per unit area acting on a material is called the stress at a point. It is a 
tensor quantity having unit of N/m
2
 or Pascal. 
 
Types of Stresses 
? Normal stress 
 
 
? Shear Stress 
 
 
? Bulk Stress 
 
 
Strain: It is the deformation produced in the material due to simple stress. It usually represents the 
displacement between particles in the body relative to a reference length. 
Types of Strains 
? Normal Strain: The normal strain of a body is generally expressed as the ratio of total 
displacement to the original length. 
 
 
 
 
Strain is dimensionless quantity. 
It is of two types: Longitudinal strain and Lateral Strain 
Longitudinal strain is the defined as the ratio of the change in length of the body due to the 
deformation to its original length in the direction of the force. 
Lateral Strain is defined as the ratio of the change in length (breadth of a rectangular bar or diameter 
of a circular bar) of the body due to the deformation to its original length (breadth of a rectangular 
bar or diameter of a circular bar) in the direction perpendicular to the force. 
  
? Shear strain 
 
 
? Bulk Strain or Volumetric Strain 
 
 
 
 
Stress and Strain both are tensor quantity i.e. it has both change in magnitude as well as 
direction. 
True Stress and True Strain 
? The true stress is defined as the ratio of the load to the cross section area at any instant. 
 
Where s and e is the engineering stress and engineering strain respectively. 
? The true strain is defined as 
 
Lo- original length, L-successive values of the length as it changes 
? The volume of the specimen is assumed to be constant during plastic deformation. 
Stress-Strain Relationship 
? The stress-strain diagram is shown in the figure. In brittle materials, there is no appreciable change 
in the rate of strain. There is no yield point and no necking takes place. 
         
? In figure (a), the specimen is loaded only upto point A, when load is gradually removed the curve 
follows the same path AO and strain completely disappears. Such a behaviour is known as the 
elastic behaviour. 
? In figure (b), the specimen is loaded upto point B beyond the elastic limit E. When the specimen is 
gradually loaded the curve follows path BC, resulting in a residual strain OC or permanent strain. 
Comparison of engineering stress and the true stress-strain curves shown below: 
 
 
? True stress-strain curve gives a true indication of deformation characteristics because it is based 
on the instantaneous dimension of the specimen. 
? In engineering stress-strain curve, stress drops down after necking since it is based on the original 
area. 
? In true stress-strain curve, the stress however increases after necking since the cross sectional 
area of the specimen decreases rapidly after necking. 
Hooke's Law: 
According to Hooke’s law the stress is directly proportional to strain i.e. normal stress (s) ? normal 
strain (e) 
and shearing stress ( ? ) ? shearing strain ( ? ). 
s = Ee and ? = ?G 
The co-efficient E is called the modulus of elasticity i.e. its resistance to elastic strain. The coefficient 
G is called the shear modulus of elasticity or modulus of rigidity.  
Properties of Materials 
Some properties of materials which judge the strength of materials are given below: 
? Elasticity: Elasticity is the property by virtue of which a material is deformed under the load and 
is enabled to return to its original dimension when the load is removed. 
? Plasticity: Plasticity is the converse of elasticity. A material in the plastic state is permanently 
deformed by the application of load and it has no tendency to recover. The characteristic of the 
material by which it undergoes inelastic strains beyond those at the elastic limit is known as 
plasticity. 
? Ductility: Ductility is the characteristic which permits a material to be drawn out longitudinally to 
a reduced section, under the action of a tensile force (large deformation). 
? Brittleness: Brittleness implies the lack of ductility. A material is said to be brittle when it cannot 
be drawn out by tension to the smaller section. 
? Malleability: Malleability is a property of a material which permits the material to be extended in 
all directions without rapture. A malleable material possesses a high degree of plasticity, but not 
necessarily great strength.Malleability is a physical property of metals that defines their ability to 
be hammered, pressed, or rolled into thin sheets without breaking 
? Toughness: Toughness is the property of a material which enables it to absorb energy without 
fracture 
? Hardness:Hardness is the ability of a material to resist indentation or surface abrasion. Brinell 
hardness test is used to check hardness. 
? Strength: The strength of a material enables it to resist fracture under load. 
Engineering Stress-Strain Curve 
? The stress-strain diagram is shown in the figure. The curve starts from an origin. Showing thereby 
that there is no initial stress of strain in the specimen. 
? The stress-strain curve diagram for a ductile material like mild steel is shown in the figure below. 
Page 5


 
 
Stress: When a material is subjected to an external force, a resisting force is set up in the component. 
The internal resistance force per unit area acting on a material is called the stress at a point. It is a 
tensor quantity having unit of N/m
2
 or Pascal. 
 
Types of Stresses 
? Normal stress 
 
 
? Shear Stress 
 
 
? Bulk Stress 
 
 
Strain: It is the deformation produced in the material due to simple stress. It usually represents the 
displacement between particles in the body relative to a reference length. 
Types of Strains 
? Normal Strain: The normal strain of a body is generally expressed as the ratio of total 
displacement to the original length. 
 
 
 
 
Strain is dimensionless quantity. 
It is of two types: Longitudinal strain and Lateral Strain 
Longitudinal strain is the defined as the ratio of the change in length of the body due to the 
deformation to its original length in the direction of the force. 
Lateral Strain is defined as the ratio of the change in length (breadth of a rectangular bar or diameter 
of a circular bar) of the body due to the deformation to its original length (breadth of a rectangular 
bar or diameter of a circular bar) in the direction perpendicular to the force. 
  
? Shear strain 
 
 
? Bulk Strain or Volumetric Strain 
 
 
 
 
Stress and Strain both are tensor quantity i.e. it has both change in magnitude as well as 
direction. 
True Stress and True Strain 
? The true stress is defined as the ratio of the load to the cross section area at any instant. 
 
Where s and e is the engineering stress and engineering strain respectively. 
? The true strain is defined as 
 
Lo- original length, L-successive values of the length as it changes 
? The volume of the specimen is assumed to be constant during plastic deformation. 
Stress-Strain Relationship 
? The stress-strain diagram is shown in the figure. In brittle materials, there is no appreciable change 
in the rate of strain. There is no yield point and no necking takes place. 
         
? In figure (a), the specimen is loaded only upto point A, when load is gradually removed the curve 
follows the same path AO and strain completely disappears. Such a behaviour is known as the 
elastic behaviour. 
? In figure (b), the specimen is loaded upto point B beyond the elastic limit E. When the specimen is 
gradually loaded the curve follows path BC, resulting in a residual strain OC or permanent strain. 
Comparison of engineering stress and the true stress-strain curves shown below: 
 
 
? True stress-strain curve gives a true indication of deformation characteristics because it is based 
on the instantaneous dimension of the specimen. 
? In engineering stress-strain curve, stress drops down after necking since it is based on the original 
area. 
? In true stress-strain curve, the stress however increases after necking since the cross sectional 
area of the specimen decreases rapidly after necking. 
Hooke's Law: 
According to Hooke’s law the stress is directly proportional to strain i.e. normal stress (s) ? normal 
strain (e) 
and shearing stress ( ? ) ? shearing strain ( ? ). 
s = Ee and ? = ?G 
The co-efficient E is called the modulus of elasticity i.e. its resistance to elastic strain. The coefficient 
G is called the shear modulus of elasticity or modulus of rigidity.  
Properties of Materials 
Some properties of materials which judge the strength of materials are given below: 
? Elasticity: Elasticity is the property by virtue of which a material is deformed under the load and 
is enabled to return to its original dimension when the load is removed. 
? Plasticity: Plasticity is the converse of elasticity. A material in the plastic state is permanently 
deformed by the application of load and it has no tendency to recover. The characteristic of the 
material by which it undergoes inelastic strains beyond those at the elastic limit is known as 
plasticity. 
? Ductility: Ductility is the characteristic which permits a material to be drawn out longitudinally to 
a reduced section, under the action of a tensile force (large deformation). 
? Brittleness: Brittleness implies the lack of ductility. A material is said to be brittle when it cannot 
be drawn out by tension to the smaller section. 
? Malleability: Malleability is a property of a material which permits the material to be extended in 
all directions without rapture. A malleable material possesses a high degree of plasticity, but not 
necessarily great strength.Malleability is a physical property of metals that defines their ability to 
be hammered, pressed, or rolled into thin sheets without breaking 
? Toughness: Toughness is the property of a material which enables it to absorb energy without 
fracture 
? Hardness:Hardness is the ability of a material to resist indentation or surface abrasion. Brinell 
hardness test is used to check hardness. 
? Strength: The strength of a material enables it to resist fracture under load. 
Engineering Stress-Strain Curve 
? The stress-strain diagram is shown in the figure. The curve starts from an origin. Showing thereby 
that there is no initial stress of strain in the specimen. 
? The stress-strain curve diagram for a ductile material like mild steel is shown in the figure below. 
 
? Upto point A, Hooke's Law is obeyed and stress is proportional to strain. Point A is called limit of 
proportionality. 
 
? Point B is called the elastic limit point. 
? At point B the cross-sectional area of the material starts decreasing and the stress decreases to a 
lower value to point D, called the lower yield point. 
? The apparent stress decreases but the actual or true stress goes on increasing until the specimen 
breaks at point C, called the upper yield point 
? From point E onwards, the strain hardening phenomena become predominant and the strength of 
the material increases thereby requiring more stress for deformation, until point F is reached. Point 
F is called the ultimate point. 
Elongation 
A prismatic bar loaded in tension by an axial force  
 
For a prismatic bar loaded in tension by 
an axial force P. The elongation of the bar 
can be determined as 
d=PL/AE 
Elongation of composite body 
Elongation of a bar of varying cross section A1 ,A2 ,----------,An of lengths l1 , l2,--------ln respectively 
 
Elongation of a tapered body