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Prepared by Dr.A.Vinoth Jebaraj
Solid Mechanics
Mechanics of Rigid
Bodies
Mechanics of
Deformable Bodies
Studying the external effects of
forces on machine components and
structures
Studying the internal effects of
forces on machine components and
structures
Types of Loading
Tensile loading
Compressive
loading
Shear loading
(In plane stress)
Bending load
(out of plane stress)
Torsional shear
Transverse shear
Normal loading : Tensile, compressive (out of plane)
Shear loading: tangential loading or parallel loading (in plane
loading)
Axial or normal load
Direct shear
Torsional shear
Bending stresses ( out of
plane stresses)
Direct Shear
Torsion
An effect that produces
shifting of horizontal
planes of a material
The twisting and distortion
of a material’s fibers in
response to an applied load
Torsional shear
Direct Shear Stress - Examples
Single shear
Double shear
= =
= =
( )
Uniaxial loading Biaxial loading Triaxial loading
Different types of Loadings
Types of strain (plane strain ,
volumetric strain)
Longitudinal (or) linear strain
Transverse (or) lateral strain
Types of stresses ( plane stress,
volumetric stress)
Normal stresses
Shear stresses (or) tangential (or)
parallel stresses [τ] Internal resistance
Within the elastic limit, lateral strain is directly proportional to longitudinal
strain.
This constant is known as Poisson’s ratio which is denoted by μ (or) 1/m.
Note: Linear strain = - μ × Lateral strain
Elastic Region Yielding
Plastic
Region
Fracture
Behavior of
material during
loading
Different Stages in Loading
Ductile materials Brittle materials
Stress Strain Curve
Hooke’s Law
In 1678, Robert Hooke’s found that for many structural materials,
within certain limits, elongation of a bar is directly proportional to
the tensile force acting on it. (Experimental finding)
Statement : Within the elastic limit, stress is directly
proportional to strain.
Where E = Young’s Modulus or Modulus of Elasticity (MPa)
Its for tensile, compressive and shear stresses
Modulus of Rigidity [or] Shear Modulus
Statement : Within the elastic limit, shear stress is directly
proportional to shear strain.
Where G = Shear Modulus or Modulus of Rigidity (MPa)
Bulk Modulus
When a body is subjected to three mutually perpendicular stresses,
of equal intensity, then the ratio of the direct stress to the
corresponding volumetric strain is known as bulk modulus.
Total stress [σ] = σ x + σy + σ z
Total strain [ε] = ε x + εy + ε z
σ xσ x
σ y
σ y
σ z
σ z
Analysis of Stresses in varying cross section bar
Stress in an Inclined Plane
 The plane perpendicular to the line of action of the load is a principal plane.
[Because, It is having the maximum stress value and shear stress in this plane
is zero.]
 The plane which is at an angle of 90° will have no normal and tangential
stress.
Max. shear stress = ½ Normal stress
Note to Remember:
 When a component is subjected to a tensile loading, then
both tensile and shear stresses will be induced. Similarly,
when it is subjected to a compressive loading, then both
compressive and shear stresses induced.
 At one particular plane the normal stress (either tensile or
compressive) will be maximum and the shear stress in that
plane will be zero. That plane is known as principal plane.
That particular magnitude of normal stress is known as
principal stress.
 The magnitude of maximum principal stress will be higher
when compare with the loads acting.
Elastic Constants ( E, G, K and μ)
Where
E = Young’s Modulus
K = Bulk Modulus
G [or] C = Shear Modulus
μ = Poisson’s Ratio
Principal Stresses
When a combination of axial, bending and shear loading
acts in a machine member , then identifying the plane of
maximum normal stress is difficult . Such a plane is known as
principal plane and the stresses induced in a principal plane is
known as principal stresses. Principal stress is an equivalent
of all stresses acting in a member.
Combined loading
Max principal
normal stresses
Max principal
shear stresses
Biaxial and shear
loading
Mohr’s Circle Diagram
Thermal Stresses
When a body is subjected to heating or cooling, it will try to
expand or contract. If this expansion or contraction is restricted
by some external FORCES, then thermal stresses will be
induced.
Actual Thermal Stresses
Strength  ability to resist external forces
Stiffness  ability to resist deformation under stress
Elasticity  property to regain its original shape
Plasticity  property which retains the deformation produced
under load after removing load
Ductility  property of a material to be drawn into wire form
with using tensile force
Brittleness  property of breaking a material without any
deformation
Mechanical properties of materials
Malleability  property of a material to be rolled or
hammered into thin sheets
Toughness  property to resist fracture under impact load
Machinability  property of a material to be cut
Resilience  property of a material to absorb energy
Creep  material undergoes slow and permanent
deformation when subjected to constant stress with high
temperature
Fatigue  failure of material due to cyclic loading
Hardness  resistant to indentation, scratch
Mechanical Properties of Materials
Contd. . .
Bending Moment
“Bending moment = Force × Perpendicular distance
between the line of the action of the force and the point
about which the moment produced”
“Vertical force in a section which is in transverse direction
perpendicular to the axis of a beam is known as transverse shear
force”
Cantilever Beam
Transverse Shear Force
Applied Load
Reaction Force
Shear Force (S.F)
Applied force
Reaction force
Shear force
 For drawing shear force diagram, consider left or right portion of the
section in a beam.
 Add the forces (including reaction) normal to the beam on one of the
portion. If the right portion is chosen, a force on the right portion acting
downwards is positive while a force acting upwards is negative and vice
versa for left portion.
 Positive value of shear force are plotted above the base line.
 S.F. Diagram will increase or decrease suddenly by a vertical straight line
at a section where there is point vertical load.
 Shear force between two vertical loads will be constant.
Points to remember
“The resultant force acting on any one portion (left or right) normal to the axis of
the beam is called shear force at the section X-X. This shear force is known as
transverse shear force”
Simply Supported Beam carrying 1000 N at its midpoint. The
reactions at the supports will be equal to RA = RB = 500 N
Pure Bending
Assumptions in the Evaluation of Bending stress
R = Radius of shaft, L = Length of the shaft
T = Torque applied at the free end
C = Modulus of Rigidity of a shaft material
τ = torsional shear stress induced at the cross section
Ø = shear strain, θ = Angle of twist
Torsional Equation
Design for Bending
Design for Bending & Twisting
 When a shaft is subjected to pure rotation, then it has to be designed for
bending stress which is induced due to bending moment caused by self
weight of the shaft.
Example: Rotating shaft between two bearings.
 When a gear or pulley is mounted on a shaft by means of a key, then it
has to be designed for bending stress (induced due to bending moment)
and also torsional shear stress which is caused due to torque induced
by the resistance offered by the key .
Example: gearbox shaft (splines)
Shear stress distribution in solid & hollow shafts
Hoop stress (or) Circumferential stress
Circumferential or hoop Stress (σ H) = (p .d)/ 2t
Longitudinal stress = (σ L) = (p .d)/ 4t
Euler’s Formula for Buckling
Both ends pinned Both ends fixed
One end fixed, other free One end fixed and other pinned

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Strength of materials by A.Vinoth Jebaraj

  • 2. Solid Mechanics Mechanics of Rigid Bodies Mechanics of Deformable Bodies Studying the external effects of forces on machine components and structures Studying the internal effects of forces on machine components and structures
  • 3. Types of Loading Tensile loading Compressive loading Shear loading (In plane stress) Bending load (out of plane stress) Torsional shear Transverse shear Normal loading : Tensile, compressive (out of plane) Shear loading: tangential loading or parallel loading (in plane loading) Axial or normal load
  • 4. Direct shear Torsional shear Bending stresses ( out of plane stresses)
  • 5. Direct Shear Torsion An effect that produces shifting of horizontal planes of a material The twisting and distortion of a material’s fibers in response to an applied load Torsional shear
  • 6. Direct Shear Stress - Examples Single shear Double shear = = = = ( )
  • 7. Uniaxial loading Biaxial loading Triaxial loading Different types of Loadings
  • 8. Types of strain (plane strain , volumetric strain) Longitudinal (or) linear strain Transverse (or) lateral strain Types of stresses ( plane stress, volumetric stress) Normal stresses Shear stresses (or) tangential (or) parallel stresses [τ] Internal resistance
  • 9. Within the elastic limit, lateral strain is directly proportional to longitudinal strain. This constant is known as Poisson’s ratio which is denoted by μ (or) 1/m. Note: Linear strain = - μ × Lateral strain
  • 10. Elastic Region Yielding Plastic Region Fracture Behavior of material during loading Different Stages in Loading
  • 11. Ductile materials Brittle materials Stress Strain Curve
  • 12. Hooke’s Law In 1678, Robert Hooke’s found that for many structural materials, within certain limits, elongation of a bar is directly proportional to the tensile force acting on it. (Experimental finding) Statement : Within the elastic limit, stress is directly proportional to strain. Where E = Young’s Modulus or Modulus of Elasticity (MPa) Its for tensile, compressive and shear stresses
  • 13. Modulus of Rigidity [or] Shear Modulus Statement : Within the elastic limit, shear stress is directly proportional to shear strain. Where G = Shear Modulus or Modulus of Rigidity (MPa)
  • 14. Bulk Modulus When a body is subjected to three mutually perpendicular stresses, of equal intensity, then the ratio of the direct stress to the corresponding volumetric strain is known as bulk modulus. Total stress [σ] = σ x + σy + σ z Total strain [ε] = ε x + εy + ε z σ xσ x σ y σ y σ z σ z
  • 15. Analysis of Stresses in varying cross section bar
  • 16. Stress in an Inclined Plane  The plane perpendicular to the line of action of the load is a principal plane. [Because, It is having the maximum stress value and shear stress in this plane is zero.]  The plane which is at an angle of 90° will have no normal and tangential stress.
  • 17. Max. shear stress = ½ Normal stress
  • 18. Note to Remember:  When a component is subjected to a tensile loading, then both tensile and shear stresses will be induced. Similarly, when it is subjected to a compressive loading, then both compressive and shear stresses induced.  At one particular plane the normal stress (either tensile or compressive) will be maximum and the shear stress in that plane will be zero. That plane is known as principal plane. That particular magnitude of normal stress is known as principal stress.  The magnitude of maximum principal stress will be higher when compare with the loads acting.
  • 19. Elastic Constants ( E, G, K and μ) Where E = Young’s Modulus K = Bulk Modulus G [or] C = Shear Modulus μ = Poisson’s Ratio
  • 20. Principal Stresses When a combination of axial, bending and shear loading acts in a machine member , then identifying the plane of maximum normal stress is difficult . Such a plane is known as principal plane and the stresses induced in a principal plane is known as principal stresses. Principal stress is an equivalent of all stresses acting in a member. Combined loading
  • 21. Max principal normal stresses Max principal shear stresses Biaxial and shear loading
  • 23. Thermal Stresses When a body is subjected to heating or cooling, it will try to expand or contract. If this expansion or contraction is restricted by some external FORCES, then thermal stresses will be induced.
  • 25. Strength  ability to resist external forces Stiffness  ability to resist deformation under stress Elasticity  property to regain its original shape Plasticity  property which retains the deformation produced under load after removing load Ductility  property of a material to be drawn into wire form with using tensile force Brittleness  property of breaking a material without any deformation Mechanical properties of materials
  • 26. Malleability  property of a material to be rolled or hammered into thin sheets Toughness  property to resist fracture under impact load Machinability  property of a material to be cut Resilience  property of a material to absorb energy Creep  material undergoes slow and permanent deformation when subjected to constant stress with high temperature Fatigue  failure of material due to cyclic loading Hardness  resistant to indentation, scratch Mechanical Properties of Materials Contd. . .
  • 27. Bending Moment “Bending moment = Force × Perpendicular distance between the line of the action of the force and the point about which the moment produced”
  • 28. “Vertical force in a section which is in transverse direction perpendicular to the axis of a beam is known as transverse shear force” Cantilever Beam Transverse Shear Force Applied Load Reaction Force
  • 29. Shear Force (S.F) Applied force Reaction force Shear force
  • 30.  For drawing shear force diagram, consider left or right portion of the section in a beam.  Add the forces (including reaction) normal to the beam on one of the portion. If the right portion is chosen, a force on the right portion acting downwards is positive while a force acting upwards is negative and vice versa for left portion.  Positive value of shear force are plotted above the base line.  S.F. Diagram will increase or decrease suddenly by a vertical straight line at a section where there is point vertical load.  Shear force between two vertical loads will be constant. Points to remember
  • 31. “The resultant force acting on any one portion (left or right) normal to the axis of the beam is called shear force at the section X-X. This shear force is known as transverse shear force”
  • 32. Simply Supported Beam carrying 1000 N at its midpoint. The reactions at the supports will be equal to RA = RB = 500 N
  • 33.
  • 34.
  • 35.
  • 36.
  • 37.
  • 38.
  • 40. Assumptions in the Evaluation of Bending stress
  • 41.
  • 42. R = Radius of shaft, L = Length of the shaft T = Torque applied at the free end C = Modulus of Rigidity of a shaft material τ = torsional shear stress induced at the cross section Ø = shear strain, θ = Angle of twist Torsional Equation
  • 43.
  • 44. Design for Bending Design for Bending & Twisting  When a shaft is subjected to pure rotation, then it has to be designed for bending stress which is induced due to bending moment caused by self weight of the shaft. Example: Rotating shaft between two bearings.  When a gear or pulley is mounted on a shaft by means of a key, then it has to be designed for bending stress (induced due to bending moment) and also torsional shear stress which is caused due to torque induced by the resistance offered by the key . Example: gearbox shaft (splines)
  • 45.
  • 46. Shear stress distribution in solid & hollow shafts
  • 47. Hoop stress (or) Circumferential stress Circumferential or hoop Stress (σ H) = (p .d)/ 2t
  • 48. Longitudinal stress = (σ L) = (p .d)/ 4t
  • 50. Both ends pinned Both ends fixed One end fixed, other free One end fixed and other pinned