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Strength of Materials P STRENGTH OF MATERIALS L S A T N R T E N D G E T S H I G O N F M & A T E E C R O I N A O L M S ©This book is protected by law under the Copyright Act of India. This book can only be used by the I student to whom the book was provided by Career Avenues GATE Coaching as a part of its GATE course. Any other use of the book such as reselling, copying, photocopying, etc is a legal offense. C S © CAREER AVENUES /SOM 1 C H STRENGTH OF MATERIALS STRENGTH OF MATERIAL 0 INTRODUCTION 6 Introduction On the basis of time of action of load On the basis of direction of load On the basis of area of acting the load 9 1 Couple L0ADS Pure Bending torsion Free Body Diagram Introduction Strength Classification of stresses Normal stresses Shear stress 22 2 Stress tensor STRESSES Effects of various loads acting on the body Introduction Classification of strain Normal strain Longitudinal and lateral strain 44 Volumetric strain 3 Shear strains STRAINS Sign conventions for shear strains Properties of materials Young‟s modulus of elasticity Modulus of rigidity Bulk modulus 55 4 Poisson‟s ratio ELASTIC State of simple shear CONSTANTS Relationships between various constants © CAREER AVENUES /SOM 2 STRENGTH OF MATERIALS Stress and strain diagram Limit of proportionality Elasticity Plasticity 73 5 Ductility, Brittleness, Malleability MECHANICAL Yield strength, ultimate strength, rapture PROPERTIES strength OF MATERIALS Work done by load Strain energy due to torsion Strain energy due to bending Resilience 81 6 Toughness STRAIN Effect of carbon percentage on properties ENERGY, RESIELENCE AND TOUGHNESS Principle of superposition Elongation of bar due to axial load Bar of varying cross-section 7 Uniformly tapering circular bar 98 NORMAL Uniformly tapering rectangular bar STRESSES Elongation of bar of uniform cross-section AND STRAIN due to self- weight Compound bars or parallel bars Statically indeterminate problems Thermal effects Free expansion of bar Temperature stresses in bar fixed at the both 8 ends 108 THERMAL Temperature stresses in composite bars STRESSES © CAREER AVENUES /SOM 3 STRENGTH OF MATERIALS Pure bending Theory of simple bending Moment of resistance 9 Bending equation 129 PURE Assumptions BENDING AND Design criteria BENDING Analysis of bending equation STRESSES Distribution of shear stresses Assumptions Shear stress distribution –rectangular section 10 Circular section 144 SHEAR I-section STRESSES IN BEAMS Pure torsion Moment of resistance Torsion equation 11 Assumptions 151 TORSION Shear stress distribution in shafts Analysis of torque equation Compound shafts Introduction Stresses on inclined section pq State of stress at a point due to biaxial stress 12 State of stress due to simple shear 167 PRINCIPAL State of stress due to normal and shear stress Normal and shear stress on a plane STRESSES AND STRAINS perpendicular to oblique plane Mohr Circle Type of support Types of beams Sign conventions 13 SFD and BMD 192 S.F.D Relationship b/w load, force and B.M. AND Cantilever beam B.M.D Simply supported beams © CAREER AVENUES /SOM 4 STRENGTH OF MATERIALS Thin cylindrical shell subjected to internal pressure 14 Maximum shear stress in cylindrical shell 244 Volumetric strain of thin cylindrical shell THIN Design of thin cylinder CYLINDER Thin spherical shells subjected to internal pressure Volumetric strain in spherical shell Cylindrical shell with hemispherical ends Lame‟s theory When only external pressure is zero 15 When only internal pressure is zero 261 When internal pressure is pr and external THICK pressure pR CYLINDER For solid circular shaft, subjected only to external pressure pr Graphical representation of lame‟s theory Compound cylinders Shrinking another cylinder over the cylinder Shrink fit allowance Thick spherical shell Differential equation of the deflection curve of beam 16 Sign conventions Double integration method 268 DEFLECTION Steps for solving the problems OF BEAMS Macaulay‟s method Moment area method Conjugate beam method Strain energy method Introduction Close coiled helical spring: axial pull Closed – coiled helical springs: axial couple or torque 307 Open – coiled helical spring: axial force 17 Open coiled helical spring: axial torque SPRINGS Series and parallel arrangement of springs Leaf or carriage springs Flat spiral springs © CAREER AVENUES /SOM 5 STRENGTH OF MATERIALS Introduction Equilibrium of elastic body Buckling stress 18 Slenderess ratio 314 COLUMNS Euler‟s theory End conditions Rankine theory Introduction Maximum principal stress theory : Rankines‟s theory Maximum principal strain theory: St. 332 Venant‟s theory 19 Maximum shear stress theory: guest‟s THEORIES OF theory ELASTIC Maximum strain energy theory or haigh‟s FAILURE theory Maximum shear strain energy (or distortion energy) theory Mises- Henky theory Tests 20 Answer Key 352 TESTS © CAREER AVENUES /SOM 6 STRENGTH OF MATERIALS INTRODUCTION There are three fundamental areas of engineering mechanics: i. Statics ii. Dynamics iii. Strength of materials or mechanics of materials Statics and dynamics deals with the effect of forces on rigid bodies i.e. the bodies in which change in shape can be neglected. Strength of material deals with the relation between externally applied loads and their internal effects on sold bodies. “Strength of material is a branch of applied mechanics that deals with the behaviour of solid bodies subjected to various types of loading.” The principal objective of strength of materials is to determine the stresses, strains, and displacements in structures and their components due to the loads acting on them. An understanding of mechanical behaviour is essential for the safe design of all types of structures, whether airplanes and antennas, buildings and bridges, machines and motors, or ships and spacecraft. In designing, engineer must consider both dimensions and material properties to satisfy the requirements of strength and rigidity. Mechanics of Rigid Bodies The mechanics of rigid bodies is primarily concerned with the static and dynamic behavior under external forces of engineering components and systems which are treated as infinitely strong and undeformable. Primarily, we deal here with the forces and motions associated with particles and rigid bodies. A basic requirement for the study of the mechanics of deformable bodies and the mechanics of fluids is essential for the design and analysis of many types of structural members, mechanical components, electrical devices, etc, encountered in engineering. A rigid body does not deform under load © CAREER AVENUES /SOM 7 STRENGTH OF MATERIALS Mechanics of deformable solids Mechanics of Solids The mechanics of deformable solids is more concerned with the internal forces and associated changes in the geometry of the components involved. Of particular importance are the properties of the materials used, the strength of which will determine whether the components fail by breaking in service, and the stiffness of which will determine whether the amount of deformation they suffer is acceptable. Therefore, the subject of mechanics of materials or strength of materials is central to the whole activity of engineering design. Usually the objectives in analysis here will be the determination of the stresses, strains, and deflections produced by loads. Theoretical analyses and experimental results have an equal role in this field. In short, Mechanics of Solids deals with the relation between the loads applied to a solid (non-rigid) body and the resulting internal forces and deformations induced in the body. Principle Objective = determine the stresses, strains, and displacements in structures and their components due to loads acting on them. Alternate Names = Strength of Materials or Mechanics of Deformable Bodies These notes will provide a basis to determine: The materials to be used in constructing a machine or structure to perform a given function. The optimal sizes and proportions of various elements of a machine or structure. If a given design is adequate and economical. The actual load carrying capacity of a structure or machine. (structure may have been design for a purpose other than one being considered). Guru Gyan Mass is a property of matter that does not change from one location to another. Weight refers to the gravitational attraction of the earth on a body or quantity of mass. Its magnitude depends upon the elevation at which the mass is located Weight of a body is the gravitational force acting on it. © CAREER AVENUES /SOM 8 STRENGTH OF MATERIALS CHAPTER 1 LOADS INTRODUCTION . Load may be defined as the external force or couple to which a component is subjected during its functioning. Load is a vector quantity. All the external forces acting on bodies are SURFACE forces. Externally applied forces may be due to Working environment Service conditions Contact with other members Fluid pressure Gravity or inertia forces . The forces acting on the body due to volume of the body is called BODY force. Loads may be classified on following basis: On the basis of time On basis of direction of load On the basis of area ON THE BASIS OF TIME OF ACTION OF LOAD . On the basis of time of action of load, load may be classified as Static load Dynamic load STATIC LOAD may be Dead load Gradually applied load © CAREER AVENUES /SOM 9 STRENGTH OF MATERIALS DEAD LOAD . Dead load includes loads that are relatively constant over time, including the weight of the structure itself. GRADUALLY APPLIED LOAD . Gradually applied load may be defined as the load whose magnitude increases gradually with the time. Gradually applied loads become dead load after a certain period of time. FIGURE 1.1 DYNAMIC LOAD may be Impact Fatigue IMPACT LOAD . The load which are acting for short interval time are said to be impact load. If t = time of application of load T = time period of vibration. Then 푇 푡 < 푖푚푝푎푐푡 푙표푎푑 2 푡 ≥ 3푇 푠푡푎푡푖푐 푙표푎푑 푡 = 2푇 푎푠푠푢푚푒푑 푖푚푝푎푐푡 푙표푎푑 © CAREER AVENUES /SOM 10 STRENGTH OF MATERIALS Since stress produced is greater than the static load.
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