Note for Strength Of Materials - SOM By JNTU Heroes

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LECTURE NOTES ON STRENGTH OF MATERIALS – I (A30107) II B. Tech I semester (JNTUH-R13) CIVIL ENGINEERING

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Contents Unit I: Simple Stress and Strains, Strain Energy ..................................................................................... 1 Introduction to Mechanics of Materials ............................................................................................. 1 Mechanical Properties of Engineering Materials................................................................................ 1 Concept of Stress and Strain ............................................................................................................... 2 Saint Venant’s principle: ................................................................................................................. 2 Types of Stress .................................................................................................................................... 3 A) Normal/Direct Stress............................................................................................................... 3 B) Shear/Tangential Stress .............................................................................................................. 4 Concept of complementary shear stresses......................................................................................... 6 C) Bearing Stress ............................................................................................................................. 7 Strain ................................................................................................................................................... 8 Types of Strain................................................................................................................................. 8 Existence of Normal and Shear Stresses due to Axial Loading ......................................................... 10 Hooke’s Law ...................................................................................................................................... 12 Simple Tension Test for Mild Steel Specimen on Universal Testing Machine (UTM) ....................... 13 Concepts of Permanent Plastic Deformation (Slip, Creep) And Strain-Hardening ........................... 15 Cyclic Loading, Fatigue – Endurance Limit and Fatigue Limit ........................................................... 15 Working Stress and Factor of Safety ................................................................................................. 15 Lateral Strain and Poisson’s Ratio ..................................................................................................... 16 Three Dimensional Stresses and Strains – Stress Tensor – Generalized Hooke’s Law ..................... 16 Volumetric Strain and Bulk Modulus ................................................................................................ 16 Bars of Varying Sections.................................................................................................................... 20 Stress in the bars of uniformly tapering circular sections ................................................................ 22 Bars of Composite section or Composite Bars .................................................................................. 23 Thermal stresses or Temperature stresses ....................................................................................... 24 Strain Energy ..................................................................................................................................... 25 Definitions ..................................................................................................................................... 25 Expression for Strain Energy in a body when the load is gradually applied ................................. 26 Strain energy stored in a body when the load is suddenly applied .............................................. 27 Strain energy stored in a body when the impact load is applied ................................................. 28 UNIT – II................................................................................................................................................. 30 Shear Force and Bending Moment ....................................................................................................... 30 Page | i

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Definition: Beam ............................................................................................................................... 30 Types of beams: According to their support................................................................................. 30 Types of loading ............................................................................................................................ 31 Shear Force (SF) ............................................................................................................................ 31 Bending Moment (BM) ................................................................................................................. 32 Shear Force Diagrams and Bending Moment Diagrams ................................................................... 33 Important points for drawing SFD & BMD: ................................................................................... 33 Cantilever beam with a point load at its free end ........................................................................ 35 Cantilever with Uniformly distributed load .................................................................................. 36 Cantilever with a gradually varying load ....................................................................................... 39 Simply supported beam with a point load at its mid-point .......................................................... 41 Simply supported beam with eccentric point load ....................................................................... 41 Simply Supported beam with a uniformly distributed load.......................................................... 44 Simply supported beam carrying a uniformly varying load (Triangle load) .................................. 48 Simply Supported beam with a gradually varying load from zero at one end to w per unit length at other end .................................................................................................................................. 50 Point of Contraflexure (or Point of Inflexion): .............................................................................. 53 Shear Force and Bending Moment Diagram for overhanging beams: ......................................... 53 Relationship between Load, Shear Force and Bending Moment ..................................................... 54 UNIT – III................................................................................................................................................ 55 FLEXURAL STRESSES, SHEAR STRESSES ................................................................................................. 55 PART A: FLEXURAL STRESSES ............................................................................................................ 55 Introduction ...................................................................................................................................... 55 Theory of Simple Bending ................................................................................................................. 57 Assumptions in the theory of simple of bending .......................................................................... 58 Derivation of bending equation and bending stress..................................................................... 58 Neutral Axis ................................................................................................................................... 60 Moment of Resistance .................................................................................................................. 61 Section modulus................................................................................................................................ 62 Section modulus for various shapes ............................................................................................. 62 Determination of bending stresses................................................................................................... 68 PART B: SHEAR STRESSES .................................................................................................................. 69 Transverse and Longitudinal Shear Stresses in Beam Sections ........................................................ 69 Shear Stress at a Section in a Beam .................................................................................................. 70 Page | ii

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Shearing stress distribution in typical cross-sections ....................................................................... 73 Shear stress distribution in beams of a rectangular section......................................................... 73 Shear stress distribution in beams of circular cross-section......................................................... 75 Shear stress distribution in beams of I - section ........................................................................... 77 UNIT - IV ................................................................................................................................................ 80 Part A: Principal Stresses and Strains ................................................................................................... 80 Normal and shear stresses on inclined sections ............................................................................... 80 State of Stress (Compound Stress) ................................................................................................... 81 Transformation of plane stress ......................................................................................................... 82 Principal stresses and maximum shear stress .................................................................................. 83 Mohr's circle for plane stress ............................................................................................................ 84 Construction of the Mohr’s circle (Plane Stress) .............................................................................. 85 Plane Strain ....................................................................................................................................... 86 Mohr’s Circle for plane strain ........................................................................................................... 90 Part B: Theories of Elastic Failure of Materials ..................................................................................... 91 Introduction ...................................................................................................................................... 91 The Failure Theories.......................................................................................................................... 92 Maximum principal stress theory [Rankine’s Theory] .................................................................. 92 Maximum shear stress theory [Guest’s Theory or Tresca’s Theory] ............................................ 92 Maximum normal strain theory [St. Venant’s Theory] ................................................................. 93 Total strain energy theory [Haigh’s Theory] ................................................................................. 93 Maximum shear strain energy theory [Mises-Henky or Von Mises’ theory] ................................ 93 Industrial Applications of Failure Theories ................................................................................... 95 UNIT-V ................................................................................................................................................... 96 Deflection of Beams .............................................................................................................................. 96 Introduction ...................................................................................................................................... 96 A. Methods of Determining Beam Deflections ......................................................................... 96 B. Superposition Principle ......................................................................................................... 96 C. Nomenclature ....................................................................................................................... 97 D. Review of Simple Bending..................................................................................................... 97 Differential Equation for the Elastic Curve ..................................................................................... 100 Evaluation of deflection by double-integration method ................................................................ 101 A. Example 1- Cantilever beam ............................................................................................... 101 Macaulay’s Method / Singularity Functions ................................................................................... 102 Page | iii