STRENGTH OF MATERIALS

1 STRENGTH OF MATERIALS For MECHANICAL ENGINEERING CIVIL ENGINEERING SYLLABUS Stress and strain, elastic constants, Poisson's ratio; Mohr s circle for plane stress and plane strain; thin cylinders; shear force and bending moment diagrams; bending and shear stresses; deflection of beams; torsion of circular shafts; Euler s theory of columns; energy methods; thermal stresses; strain gauges and rosettes; testing of MATERIALS with universal testing machine; testing of hardness and impact STRENGTH . ANALYSIS OF GATE PAPERS MECHANICAL Exam Year 1 Mark Ques. 2 Mark Ques. Total 2003 6 4 14 2004 3 4 11 2005 2 6 14 2006 2 5 12 2007 2 6 14 2008 4 8 20 2009 2 2 6 2010 1 2 5 2011 4 3 10 2012 3 3 9 2013 2 1 4 2014 Set-1 2 3 8 2014 Set-2 3 2 7 2014 Set-3 2 2 6 2014 Set-4 2 2 6 2015 Set-1 3 2 5 2015 Set-2 2 5 12 2015 Set-3 1 1 3 2016 Set-1 3 4 11 2016 Set-2 2 4 10 2016 Set-3 2 3 8 2017 Set-1 5 4 13 2017 Set-2 3 4 7 2018 Set-1 4 5 14 2018 Set-2 2 4 10 CIVIL Exam Year 1 Mark Ques.

2 2 Mark Ques. Total 2003 2 5 12 2004 1 6 13 2005 2 3 8 2006 3 9 21 2007 3 5 13 2008 1 8 17 2009 2 5 12 2010 5 2 9 2011 1 3 7 2012 3 3 9 2013 2 2 9 2014 Set-1 1 3 7 2014 Set-2 1 5 11 2015 Set-1 2 2 6 2015 Set-2 2 2 6 2016 Set-1 3 6 2016 Set-2 4 8 2017 Set-1 2 2 6 2017 Set-2 1 4 9 2018 Set-1 1 3 7 2018 Set-2 1 2 5 STRENGTH OF MATERIALS Copyright Reserved by No part of this material should be copied or reproduced without permission Topics Page No 1. SIMPLE STRESS AND STRAIN Introduction 1 Stresses and Strains 1 Types of stress and strain 2 Hook s law and Elastic Property 5 True Stress and True Strain 5 Elastic Constants 6 Types of MATERIALS 7 Strain Under Tri Axial Loading 7 Axial Stress in Compound Bars 7 Elongation of the Taper Beam under Axial Load 8 Elongation Due to Self-Weight of the Body 9 Thermal Stress 9 Gate Questions 11 2.

3 STRAIN ENERGY AND IMPACT LOAD Strain Energy 29 Gate Questions 31 3. SHEAR FORCE AND BENDING MOMENT DIAGRAM Shear Force and Bending Moment 34 Beam 35 Free Body Diagram 37 Method of Analysis 37 Equations of Equilibrium 37 Example Problems 37 Special Case of Shear Force Diagram and Bending Moment 39 Gate Questions 46 4. BENDING STRESS AND SHEAR STRESS Pure Bending Moment 49 Beam under Uniform STRENGTH 50 Direct Shear Stress 51 Gate Questions 54 5. PURE TORSION CONTENTS Copyright Reserved by No part of this material should be copied or reproduced without permission Pure Torsion 70 Composite Shafts 71 Spring 73 Connection of Springs 76 Gate Questions 77 6.

4 COMBINED STRESS AND STRAIN Introduction 88 Normal Stress and Shear Stress 88 Principal Planes and Principal Stress 88 Maximum Shear Stress Plane and Maximum Shear Stress 88 Principal Strain and Shear strain 89 Combined Bending & Torsion 89 Mohr s circle 90 Gate Questions 91 7. COLUMN AND PRESSURE VESSEL Introduction 95 Types of Column 95 Slenderness ratio 95 Load analysis of column 95 Pressure vessel 96 Gate Questions 100 8. SLOPE AND DEFLECTION Introduction 107 General Expression 107 Methods for Slope &Deflection 107 Special Case of Slope and Deflection 109 9. ASSIGNMENT QUESTIONS 114 10.

5 CIVIL GATE QUESTIONS 126 Copyright Reserved by No part of this material should be copied or reproduced without INTRODUCTION 1. ENGINEERING MECHANICSThe engineering mechanics may be defined as a branch of applied mechanics that deals with behaviors of solid bodies subjected to various types of loadings. This is usually subdivided into further two streams Mechanics of rigid bodies or simply Mechanics and Mechanics of deformable solids. The mechanics of deformable solids is known by several names STRENGTH of MATERIALS , mechanics of MATERIALS etc. OF RIGID BODIESThe mechanics of rigid bodies is concerned with the static and dynamic behavior under external forces of engineering components and systems which are treated as infinitely strong and under formable primarily we deal here with the forces and motions associated with particles and rigid bodies.

6 Mechanics of rigid bodies is further divided in to two subdivision kinematics of mechanics and dynamics of mechanics. OF MATERIALSS trength of material is branch of mechanics that concerns with study of the forces and its effect on the properties of the deformed body. It is concerned with the internal forces and associated changes in the geometry of the components. Particular objective of SOM is that whether the components fail by breaking in service, and 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.

7 STRESSES AND STRAINS we know that in mechanics of deformable solids, externally applied forces acts on a body and body suffers a deformation . From equilibrium point of view, this action should be opposed or reacted by internal forces which are set up within the particles of material due to cohesion. These internal forces give a concept of stress. Therefore, let us define a stress. Let us consider a rectangular bar of some cross sectional area and subjected to some load or force. Let us imagine that the same rectangular bar is assumed to be cut into two halves at section XX. The each portion of this rectangular bar is in equilibrium under the action of load P and the internal forces acting at the section XX has been shown The stress can be defined as total internal resistance per unit area of the component.

8 But after removal of the load, body regains its original position, so total internal resistance should be equal to applied load. So stress can be given by, ( )PStress =AThe basic units of stress are N / m2 (or Pa), N/mm2, MPa, GPa. 1 SIMPLE STRESS AND STRAIN Copyright Reserved by No part of this material should be copied or reproduced without permission693 MPa 10 PaGPa 10 PaKPa 10 Pa=== Some times N / mm2 units are also used, because this is an equivalent to MPa. For direct stresses, if area under consideration is original area, then it is known as Engineering stress or nominal stress or simply stress. But, if the area taken is actual area at the time of considering the stress, then stress is known as true stress.

9 ;oNominal Str s =PAesTrue Stress =PA Where, Ao = Original area of specimen. A = Actual area of specimen. Proof Stress: When a material such as Aluminum does not have an obvious yield point and yet undergoes large strains after the proportional limit is exceeded, an arbitrary yield stress may be determined by the offset method. A line parallel to initial linear part is drawn, which is offset by some standard amount of Strain such as The intersection of the off set point (A) defines the yield stress or off set yield stress, which is slightly above the proportional limit and is called proof stress. Sign convention for stress Tensile Stresses or pulling Stresses are considered as positive whereas compressive stresses are considered as negative.

10 2. STRAIN If a bar is subjected to a direct load and body is deformed body there will be deformation under action of load. If the bar has an original length L and changes by an amount L, The strain produce is defined as follows ( )Change of length LStrain ==Original lengthL Unit of Strain: Strain is a measure of the deformation of the material and is a non dimensional Quantity. It is simply a ratio of two quantities with the same unit. So strain is dimension less quantity. Sign convention for strain Tensile strains are positive whereas compressive strains are negative. TYPES OF STRESS AND STRAIN 1. TYPES OF STRESS According to direction of loading, Stress can be divided into following way: Copyright Reserved by No part of this material should be copied or reproduced without permission Normal stress ( ) We have defined stress as force per unit area.