mechanics of material

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MECHANICS OF MATERIAL By: - Prof. K.G.SONTAKKEDept. of Mechanical Engg.B.D.C.O.E. Sevagram : 

MECHANICS OF MATERIAL By: - Prof. K.G.SONTAKKEDept. of Mechanical Engg.B.D.C.O.E. Sevagram

Course Objectives Expected Outcomes: -To understand the basic concepts of stress, strain and their variations under different types of loading.It includes the basic concepts involved in mechanics of materials, bending moment, shear force, stresses in beams, slope and deflection in beams under different loading and support conditions, understanding of torsional shear stress in shaft, crippling load (Buckling load) in struts and columns. At the end of this course, students will be able to analyze different stresses, strains and deflections in a simple mechanical element under various loading and support conditions. : 

Course Objectives Expected Outcomes: -To understand the basic concepts of stress, strain and their variations under different types of loading.It includes the basic concepts involved in mechanics of materials, bending moment, shear force, stresses in beams, slope and deflection in beams under different loading and support conditions, understanding of torsional shear stress in shaft, crippling load (Buckling load) in struts and columns. At the end of this course, students will be able to analyze different stresses, strains and deflections in a simple mechanical element under various loading and support conditions.

SYLLABUSUNIT – I [ 8 Hrs.]Concept of simple stresses and strains: Introduction, stress, strain, types of stresses, stress and strain diagram for brittle & ductile material, elastic limit, Hooks law, modulus of elasticity, modulus of rigidity, factor of safety, analysis of tapered rod, analysis of composite section, thermal stress and strain. Longitudinal strain & stress, lateral stresses and strains, Poisson’s ratio, volumetric stresses and strain with uni-axial, bi-axial & tri-axial loading, bulk modulus, relation between Young’s modulus and modulus of rigidity, Poisson’s ratio and bulk modulus. : 

SYLLABUSUNIT – I [ 8 Hrs.]Concept of simple stresses and strains: Introduction, stress, strain, types of stresses, stress and strain diagram for brittle & ductile material, elastic limit, Hooks law, modulus of elasticity, modulus of rigidity, factor of safety, analysis of tapered rod, analysis of composite section, thermal stress and strain. Longitudinal strain & stress, lateral stresses and strains, Poisson’s ratio, volumetric stresses and strain with uni-axial, bi-axial & tri-axial loading, bulk modulus, relation between Young’s modulus and modulus of rigidity, Poisson’s ratio and bulk modulus.

UNIT – II [ 8 Hrs.]Shear force and bending moment: - Types of beam (cantilever beam, simply supported beam, overhung beam etc.), Types of loads (Concentrated and UDL), shear force and bending moment diagrams for different types of beams subjected to different types of loads, sign conventions for bending moment and shear force, shear force and bending moment diagrams for beams subjected to couple, Relation between load, shear force and bending moment. Stresses in beams: - Pure bending, theory of simple bending with assumptions & expressions for bending stress, derivation of bending equation, bending stresses in symmetrical sections, section modulus for various shapes of beam sections. Shear stresses in beams: - Concept, derivation of shear stress distribution formula , shear stress distribution diagram for common symmetrical sections, maximum and average shear stress. : 

UNIT – II [ 8 Hrs.]Shear force and bending moment: - Types of beam (cantilever beam, simply supported beam, overhung beam etc.), Types of loads (Concentrated and UDL), shear force and bending moment diagrams for different types of beams subjected to different types of loads, sign conventions for bending moment and shear force, shear force and bending moment diagrams for beams subjected to couple, Relation between load, shear force and bending moment. Stresses in beams: - Pure bending, theory of simple bending with assumptions & expressions for bending stress, derivation of bending equation, bending stresses in symmetrical sections, section modulus for various shapes of beam sections. Shear stresses in beams: - Concept, derivation of shear stress distribution formula , shear stress distribution diagram for common symmetrical sections, maximum and average shear stress.

UNIT–III [ 8 Hrs.]Deflection of beams:- Deflection & slope of cantilever,simply supported, overhung beams subjected to concentrated load, UDL, Relation between slope, deflection & radius curvature Macaulay’s method to determine deflection of beam. Principal stresses and strains:- Definition of principal planes & principal stresses, analytical method of determining stresses on oblique section when member is subjected to direct stresses in one plane in mutually perpendicular two planes, when member is subjected to shear stress and direct stresses in two mutually perpendicular planes, Mohr’s circle for representation of principal stresses. : 

UNIT–III [ 8 Hrs.]Deflection of beams:- Deflection & slope of cantilever,simply supported, overhung beams subjected to concentrated load, UDL, Relation between slope, deflection & radius curvature Macaulay’s method to determine deflection of beam. Principal stresses and strains:- Definition of principal planes & principal stresses, analytical method of determining stresses on oblique section when member is subjected to direct stresses in one plane in mutually perpendicular two planes, when member is subjected to shear stress and direct stresses in two mutually perpendicular planes, Mohr’s circle for representation of principal stresses.

UNIT-IV [ 8 Hrs.]Torsion of circular shafts: - Derivation of torsion equation with the assumptions made in it. Torsion shear stress induced in the shaft, when it is subjected to torque. Strength and rigidity criterion for design of shaft. Torque transmitted by solid & hollow circular shaft. Equivalent twisting and bending moment in shaft when it is subjected to bending moment, torque & axial load. Column & Struts: - Failure of long & short column, slenderness ratio, assumptions made in Euler’s column theory, end conditions for column. Expression for crippling load for various end conditions of column and derivation on column with both ends hinged. Effective length of column, limitations of Euler’s formula, Rankine formula. : 

UNIT-IV [ 8 Hrs.]Torsion of circular shafts: - Derivation of torsion equation with the assumptions made in it. Torsion shear stress induced in the shaft, when it is subjected to torque. Strength and rigidity criterion for design of shaft. Torque transmitted by solid & hollow circular shaft. Equivalent twisting and bending moment in shaft when it is subjected to bending moment, torque & axial load. Column & Struts: - Failure of long & short column, slenderness ratio, assumptions made in Euler’s column theory, end conditions for column. Expression for crippling load for various end conditions of column and derivation on column with both ends hinged. Effective length of column, limitations of Euler’s formula, Rankine formula.

UNIT-V [ 8 Hrs.]Introduction to fracture mechanics: - Modes of fracture, stress intensity factors, crack propagation, creep phenomenon. Strain energy & impact loading: - Definition of strain energy stored in a body when it is subjected to gradually applied load, suddenly applied loads & impact loads. Strain energy stored in bending & torsion. : 

UNIT-V [ 8 Hrs.]Introduction to fracture mechanics: - Modes of fracture, stress intensity factors, crack propagation, creep phenomenon. Strain energy & impact loading: - Definition of strain energy stored in a body when it is subjected to gradually applied load, suddenly applied loads & impact loads. Strain energy stored in bending & torsion.

UNIT-VI [ 8 Hrs.]Factor of safety: Statistical methods in determining factor of safety. Theories of failure, modes of fa ilure, compound stresses, eccentric axial loading, variable stresses in machine parts, Endurance, S-N Curve, stress concentration & stress raisers, notch sensitivity, stress concentration factor, methods for reducing stress concentration. Goodmans criteria, Soderberg criteria, Gerber’s criteria, fatigue design for finite and infinite life of the parts subjected to variable loads with uniform cross section. : 

UNIT-VI [ 8 Hrs.]Factor of safety: Statistical methods in determining factor of safety. Theories of failure, modes of fa ilure, compound stresses, eccentric axial loading, variable stresses in machine parts, Endurance, S-N Curve, stress concentration & stress raisers, notch sensitivity, stress concentration factor, methods for reducing stress concentration. Goodmans criteria, Soderberg criteria, Gerber’s criteria, fatigue design for finite and infinite life of the parts subjected to variable loads with uniform cross section.

LIST OF TUTORIALS:1) Two problems on principle stresses2) Two problems on Mohr’s circle3) Two problems on Thermal stresses with heat flow4) Three problems on S.F. & B.M. diagrams5) Two problems on Stresses in beam bending6) Two problems on shear stresses7) Two problems on Macaulay’s methods8) Two problems on area moment method9) Two problems on shafts10) Two problems on columns & struts11) Two problems on compound loading12) Two problems on fatigue & variable loads : 

LIST OF TUTORIALS:1) Two problems on principle stresses2) Two problems on Mohr’s circle3) Two problems on Thermal stresses with heat flow4) Three problems on S.F. & B.M. diagrams5) Two problems on Stresses in beam bending6) Two problems on shear stresses7) Two problems on Macaulay’s methods8) Two problems on area moment method9) Two problems on shafts10) Two problems on columns & struts11) Two problems on compound loading12) Two problems on fatigue & variable loads

LIST OF PRACTICALS:Minimum Eight Practical's out of following areas shall be performed:1. Study of Universal Testing Machine1. Tension test on metals.2. Compression test on materials.3. Shear test on metals.4. Impact test on metals.5. Hardness test on metals.6. Torsion test on metals.7. Deflection of beams.8. Modulus of rupture test.9. Buckling of columns.10. Deflection of springs. : 

LIST OF PRACTICALS:Minimum Eight Practical's out of following areas shall be performed:1. Study of Universal Testing Machine1. Tension test on metals.2. Compression test on materials.3. Shear test on metals.4. Impact test on metals.5. Hardness test on metals.6. Torsion test on metals.7. Deflection of beams.8. Modulus of rupture test.9. Buckling of columns.10. Deflection of springs.

TEXT BOOKS:1. Strength of Materials, Ramamurtham, Dhanapat Rai Publication. 2. Strength of Materials, R K Bansal, Laxmi Publications Strength of Materials, R S Khurmi, S.Chand Publications 3. Elements of Strength of Materials, S. Timoshenko and O.H.Young, East West Press Private Ltd.4. Design Data for Machine Elements, B.D. Shiwalkar, Denett & Company5. Strength of Material, R.K. Rajput, S.Chand Publication6. PSG Data Book.7. Mechanics of Structures Vol.-1:- S.B.Junarkar & H.J. Shah REFERENCE BOOKS:8. Strength of Material, Ferdinard L. Singer, Harper and Row, New York9. Elements of Strength of Materials, V. Natarajan, Oxford & IBH Publishing Company10. Strength of Materials, S S Rattan, Tata McGraw-Hill11. Mechanics of Material, Beer & Johnson, Tata Mc-Graw Hill : 

TEXT BOOKS:1. Strength of Materials, Ramamurtham, Dhanapat Rai Publication. 2. Strength of Materials, R K Bansal, Laxmi Publications Strength of Materials, R S Khurmi, S.Chand Publications 3. Elements of Strength of Materials, S. Timoshenko and O.H.Young, East West Press Private Ltd.4. Design Data for Machine Elements, B.D. Shiwalkar, Denett & Company5. Strength of Material, R.K. Rajput, S.Chand Publication6. PSG Data Book.7. Mechanics of Structures Vol.-1:- S.B.Junarkar & H.J. Shah REFERENCE BOOKS:8. Strength of Material, Ferdinard L. Singer, Harper and Row, New York9. Elements of Strength of Materials, V. Natarajan, Oxford & IBH Publishing Company10. Strength of Materials, S S Rattan, Tata McGraw-Hill11. Mechanics of Material, Beer & Johnson, Tata Mc-Graw Hill

Slide 12: 

REFERENCE PAPERS: 12. Strength and fracture of engineering solids. David K. Felbeck & A. G. Atkins, Prentice Hall (1995). 13. E. J. Hearn,Mechanics of materials (An Introduction to the Mechanics of Elastic And Plastic Deformation of Solids and Structural Materials,.

Strength of a Material : 

Strength of a Material The strength of a material is defined as the ability to resist stress without failure. In engineering the term strength is always defined and is probably one of the following Tensile strength Compressive strength Shear strength depending on the type of loading.

Slide 14: 

DIRECT STRESS: - When a force is applied to an elastic body, the body deforms. The way in which the body deforms depends upon the type of force applied to it. Compression force makes the body shorter. A tensile force makes the body longer Direct Stress = Applied Force (F) Cross Sectional Area (A)

Slide 15: 

Shear Stress This cylinder is in Tension Forces Flexural (bending) stress This cylinder is in compression Compression Tension Bending Shear Forces

Tension and Compression : 

Tension and Compression

Testing for strength : 

Testing for strength Fig:- Strength testing machine

-This is a measure of the internal resistance in a material to an externally applied load. -Stress is the force per unit area upon which it acts. -For direct compressive or tensile loading the stress is designated  and is defined as: : 

-This is a measure of the internal resistance in a material to an externally applied load. -Stress is the force per unit area upon which it acts. -For direct compressive or tensile loading the stress is designated  and is defined as: STRESS Unit of Stress: Pascal = 1 N/m2 , KN/m2 , MN/m2 , GN/m2 1 MPa = 1 N/mm2

Slide 19: 

Fig: 1 Fig: 2

Slide 20: 

STRAIN When loads are applied to a body, some deformation will occur resulting to a change in dimension. It is defined as deformation per unit length. It is the ratio of change in length to original length. P  L Strain has no dimensions. It is expressed as a percentage or in microstrain (s).

Slide 21: 

Tensile strain: -A positive axial strain represents extension. (+ Ve) () Increase in length  Original length L Compressive strain: - A negative axial strain represents a contraction. (- Ve) () Decrease in length  Original length L A strain of 1 s is an extension of one part per million. A strain of 0.2% is equal to 2000 s. = = = =

Units of stress and strain : 

Units of stress and strain The basic unit for Force and Load is the Newton (N) which is equivalent to kg m/s2. One kilogramme (kg) weight is equal to 9.81 N.  In industry the units of stress are normally Newtons per square millimetre (N/mm2) but this is not a base unit for calculations. The MKS unit for pressure is the Pascal. 1 Pascal = 1 N/m2 Pressure and Stress have the same units 1 MPa = 1 N/mm2

Slide 23: 

Shear Stress The shear stress  is a measure of the internal resistance of a material to an externally applied shear load. The shear stress is defined as: Shear stress is the stress tangent to a surface. Shear stresses are produced by equal and opposite parallel forces not in line. The forces tend to make one part of the material slide over the other part. Shear stress is tangential to the area over which it acts.

Slide 24: 

For shear loads the strain is defined as the angle . This is measured in radians. Shear force Shear Force Area resisting shear Shear displacement (x) Shear strain is angle  L Shear Strain

Stress- Strain Curve for Mild Steel (Ductile Material) : 

Stress- Strain Curve for Mild Steel (Ductile Material) Strain Stress Plastic state Of material Elastic State Of material Yield stress Point E = Modulus of elasticity Ultimate stress point Breaking stress point

Slide 26: 

STRESS STRAIN DIAGRAM

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Elastic behaviour The curve is straight line through out most of the region Stress is proportional with strain Material to be linearly elastic Proportional limit The upper limit to linear line The material still respond elastically The curve tend to bend and flatten out Elastic limit Upon reaching this point, if load is remove, the specimen still return to original shape Yielding A Slight increase in stress above the elastic limit will result in breakdown of the material and cause it to deform permanently. This behaviour is called yielding The stress that cause = YIELD STRESS@YIELD POINT Plastic deformation Once yield point is reached, the specimen will elongate (Strain) without any increase in load Material in this state = perfectly plastic

# Strain Hardening: -When yielding has ended, further load applied, resulting in a curve that rises continuouslyBecome flat when reached ULTIMATE STRESSThe rise in the curve = STRAIN HARDENINGWhile specimen is elongating, its cross sectional will decrease The decrease is fairly uniform.# Necking: -At the ultimate stress, the cross sectional area begins its localised region of specimenit is caused by slip planes formed within materialActual strain produced by shear strainAs a result, “neck” tend to formSmaller area can only carry lesser load, hence curve donward Specimen break at FRACTURE STRESS : 

# Strain Hardening: -When yielding has ended, further load applied, resulting in a curve that rises continuouslyBecome flat when reached ULTIMATE STRESSThe rise in the curve = STRAIN HARDENINGWhile specimen is elongating, its cross sectional will decrease The decrease is fairly uniform.# Necking: -At the ultimate stress, the cross sectional area begins its localised region of specimenit is caused by slip planes formed within materialActual strain produced by shear strainAs a result, “neck” tend to formSmaller area can only carry lesser load, hence curve donward Specimen break at FRACTURE STRESS

Elastic vs. Plastic Behavior : 

2 - 30 Elastic vs. Plastic Behavior If the strain disappears when the stress is removed, the material is said to behave elastically. When the strain does not return to zero after the stress is removed, the material is said to behave plastically. The largest stress for which this occurs is called the elastic limit.

Elastic and Plastic deformation : 

Elastic and Plastic deformation Stress Strain Stress Strain Permanent Deformationa Elastic deformation Plastic deformation

Slide 32: 

Temperature Stresses When the temperature of a component is increased or decreased the material respectively expands or contracts. If this expansion or contraction is not resisted in any way then the process takes place free of stress. If however, the changes in dimensions are restricted then stresses termed temperature stresses will be set up. New length:

Slide 33: 

HOOKE’S LAW When a material is loaded , within elastic limit, Stress is directly proportional to Strain. i.e. Where, E=Young’s modulus  = P/A and  =  / L P/A = E ( / L)  =PL /AE E     =E 

Slide 34: 

Modulus of Elasticity OR Young’s Modulus (E) It states that providing the limit of proportionality of a material is not exceeded, the stress is directly proportional to the strain produced. It is the Ratio of Stress () to strain (). Value of E is same in Tension & Compression A   O stress E E =  /  strain slope of stress-strain Curve.

Slide 35: 

If the strain is "elastic“, Hooke's law may be used to define Young's modulus is also called the modulus of elasticity or stiffness and is a measure of how much strain occurs due to a given stress. Because strain is dimensionless, Young's modulus has the units of stress or pressure.

Slide 36: 

Factor of Safety The load which any member of a machine carries is called working load, and stress produced by this load is the working stress. Obviously, the working stress must be less than the yield stress, tensile strength or the ultimate stress. This working stress is also called the permissible stress or the allowable stress or the design stress. Factor of Safety = Ultimate or Yield stress Design or Working stress Permissible stress or allowable stress or working stress = Yield stress or Ultimate stress Factor of safety.

Slide 37: 

The ultimate strength is the measured stress at failure but this is not normally used for design because safety factors are required. Selection of Factor of Safety Variations that may occur in the properties of the member under considerations The number of loading that may be expected during the life of the structural/machine The type of loading that are planned for in the design, or that may occur in the future The type of failure that may occur Uncertainty due to the methods of analysis Deterioration that may occur in the future because of poor maintenance / because of unpreventable natural causes The importance of a given member to the integrity of the whole structure

MODULUS OF RIGIDITY (N): OR SHEARING MODULUS OR MODULUS OF TRANSVERSE ELASTICITY It has been experimentally found, that within the elastic limit, shear stress () directly proportional to shearing strain ().    i.e./=N;  = C  Where, Modulus of Rigidity or Shear Modulus C =  /  : 

MODULUS OF RIGIDITY (N): OR SHEARING MODULUS OR MODULUS OF TRANSVERSE ELASTICITY It has been experimentally found, that within the elastic limit, shear stress () directly proportional to shearing strain ().    i.e./=N;  = C  Where, Modulus of Rigidity or Shear Modulus C =  / 

Slide 39: 

POISSON’S RATIO It has been experimentally found, that if a body is stressed within elastic limit, the lateral strain bears a constant ratio to the linear strain. It is defined as: lateral strain = Radial strain linear strain = axial strain and is dimensionless.

When a body is subjected to the identical stress  in three mutually perpendicular directions, the body undergoes uniform changes in three directions without the distortion of the shape. The ratio of change in volume to original volume has been defined as volumetric strain (v ) . Then the bulk modulus, K is defined as : 

When a body is subjected to the identical stress  in three mutually perpendicular directions, the body undergoes uniform changes in three directions without the distortion of the shape. The ratio of change in volume to original volume has been defined as volumetric strain (v ) . Then the bulk modulus, K is defined as BULK MODULUS (K) K=  / v

Slide 41: 

K=  / v Where, v = V/V Change in volume = Original volume Volumetric Strain = Bulk Modulus (K): -

Slide 42: 

YOUNG’S MODULUS: E =  /  K =  / v BULK MODULUS: MODULUS OF RIGIDITY: C =  /  ELASTIC CONSTANTS

Slide 43: 

2.5 mm 2 m 12 N 0.3 mm (X) Problems

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2

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