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Applied Elasticity & Plasticity 1. Basic Equations Of APPLIED ELASTICITY & PLASTICITY 1. BASIC EQUATIONS OF ELASTICITY Introduction, The State of Stress at a Point, The State of Strain at a Point, Basic Equations of Elasticity, Methods of Solution of Elasticity Problems, Plane Stress, Plane Strain, Spherical Co-ordinates, Computer Program for Principal Stresses and Principal Planes. 2. TWO-DIMENSIONAL PROBLEMS IN CARTESIAN CO-ORDINATES Introduction, Airy’s Stress Function – Polynomials : Bending of a cantilever loaded at the end ; Bending of a beam by uniform load, Direct method for determining Airy polynomial : Cantilever having Udl and concentrated load of the free end; Simply supported rectangular beam under a triangular load, Fourier Series, Complex Potentials, Cauchy Integral Method , Fourier Transform Method, Real Potential Methods. 3. TWO-DIMENSIONAL PROBLEMS IN POLAR CO-ORDINATES Basic equations, Biharmonic equation, Solution of Biharmonic Equation for Axial Symmetry, General Solution of Biharmonic Equation, Saint Venant’s Principle, Thick Cylinder, Rotating Disc on cylinder, Stress-concentration due to a Circular Hole in a Stressed Plate (Kirsch Problem), Saint Venant’s Principle, Bending of a Curved Bar by a Force at the End. 4. TORSION OF PRISMATIC BARS Introduction, St. Venant’s Theory, Torsion of Hollow Cross-sections, Torsion of thin- walled tubes, Torsion of Hollow Bars, Analogous Methods, Torsion of Bars of Variable Diameter. 5. BENDING OF PRISMATIC BASE Introduction, Simple Bending, Unsymmetrical Bending, Shear Centre, Solution of Bending of Bars by Harmonic Functions, Solution of Bending Problems by Soap-Film Method. 6. BENDING OF PLATES Introduction, Cylindrical Bending of Rectangular Plates, Slope and Curvatures, Lagrange Equilibrium Equation, Determination of Bending and Twisting Moments on any plane, Membrane Analogy for Bending of a Plate, Symmetrical Bending of a Circular Plate, Navier’s Solution for simply supported Rectangular Plates, Combined Bending and Stretching of Rectangular Plates. 7. THIN SHELLS Introduction, The Equilibrium Equations, Membrane Theory of Shells, Geometry of Shells of Revolution. 8. NUMERICAL AND ENERGY METHODS Rayleigh’s Method, Rayleigh – Ritz Method, Finite Difference Method, Finite Element Method. 9. HERTZ’S CONTACT STRESSES Introduction, Pressure between Two-Bodies in contact, Pressure between two-Spherical Bodies in contact, Contact Pressure between two parallel cylinders, Stresses along the load axis, Stresses for two Bodies in line contact Exercises. 10. STRESS CONCENTRATION PROBLEMS Introduction, Stress-Concentration Factor, Fatigue Stress-Concentration Factors. http://books.google.co.in/books?id=KzunZOFUWnoC&lpg=PP1&ots=PrfjDf51Uj&dq=a dvanced%20mechanics%20of%20solids%20by%20ls%20srinath&pg=PP1#v=onepage& q&f=false Unit 1 BASIC EQUATIONS OF ELASTICITY Structure 1.1.Introduction 1.2.Objectives 1.3.The State of Stress at a Point 1.4.The State of Strain at a Point 1.5.Basic Equations of Elasticity 1.6.Methods of Solution of Elasticity Problems 1.7.Plane Stress 1.8.Plane Strain 1.9.Spherical Co-ordinates 1.10. Summary 1.11. Keywords 1.12. Exercise 1.1. Introduction Elasticity: All structural materials possess to a certain extent the property of elasticity, i.e., if external forces, producing deformation of a structure, do not exceed a certain limit, the deformation disappears with the removal of the forces. Throughout this book it will be assumed that the bodies undergoing the action of external forces are perfectly elastic, i.e., that they resume their initial form completely after removal of forces. The molecular structure of elastic bodies will not be considered here. It will be assumed that the matter of an elastic body is homogeneous and continuously distributed over its volume so that the smallest element cut from the body possesses the same specific physical properties as the body. To simplify the discussion it will also be assumed that the body is isotropic, i.e., that the elastic properties are the same in all directions. Structural materials usually do not satisfy the above assumptions. Such an important material as steel, for instance, when studied with a microscope, is seen to consist of crystals of various kinds and various orientations. The material is very far from being homogeneous, but experience shows that solutions of the theory of elasticity based on the assumptions of homogeneity and isotropy can be applied to steel structures with very great accuracy. The explanation of this is that the crystals are very small; usually there are millions of them in one cubic inch of steel. While the elastic properties of a single crystal may be very different in different directions, the crystals arc ordinarily distributed at random and the elastic properties of larger pieces of metal represent averages of properties of the crystals. So long as the geometrical dimensions defining the form of a body are large in comparison with the dimensions of a single crystal the assumption of homogeneity can be used with great accuracy, and if the crystals are orientated at random the material can be treated as isotropic. When, due to certain technological processes such as rolling, a certain orientation of the crystals in a metal prevails, the elastic properties of the metal become different in different directions and the condition of anisotropy must be considered. We have such a condition, for instance, in the case of cold-rolled copper. 1.2. Objectives After studying this unit we are able to understand The State of Stress at a Point The State of Strain at a Point Basic Equations of Elasticity Methods of Solution of Elasticity Problems Plane Stress Plane Strain Spherical Co-ordinates 1.3. The State of Stress at a Point Knowing the stress components σx, σy, τxy at any point of a plate in a condition of plane stress or plane strain, thestress acting on any plane through this point perpendicular to the plateand inclined to the x- and y-axes can be calculated from the equations of statics. Let O be a point of the stressed plate and suppose the stress components σx, σy, τxy are known (Fig. 1). Fig. 1 To find the stress for anyplane through the z-axis and inclined to the x- and y-axes, we take a plane BC parallel to it, at a small distancefrom O, so that this latter planetogether with the coordinate planescuts out from the plate a very small triangular prism OBC. Since thestresses vary continuously over the volume of the body the stress actingon the plane BC will approach the stress on the parallel plane throughO as the element is made smaller. In discussing the conditions of equilibrium of the small triangular prism, the body force can be neglected as a small quantity of a higher order. Likewise, if the element is very small, we can neglect the variation of the stresses over the sides and assume that the stresses are uniformly distributed. The forces acting on the triangular prism can therefore be determined by multiplying the stress components by the areas of the sides. Let N be the direction of the normal to the plane BC, and denote the cosines of the angles between the normal N and the axes x and y by Then, if A denotes the area of the side BC of the element, the areas of the other two sides are Al and Am. If we denote by X and the components of stress acting on the side BC, the equations of equilibrium of the prismatical element give (1) Thus the components of stress on any plane defined by direction cosinesl and m can easily be calculated from Eqs. (1), provided thethree components of stress σx, σy, τxyat the point O are known. Letting α be the angle between the normal N and the x-axis, so that l = cos α and m = sin α, the normal and shearing components of stress on the plane BC are (from Eqs. 1) (2) It may be seen that the angle α can be chosen in such a manner that the shearing stress τbccomes equal to zero. For this case we have or (3) From this equation two perpendicular directions can be found for which the shearing stress is zero. These directions are called principal directions and the corresponding normal stresses principal stresses. If the principal directions are taken as the x- and y-axes, τxy is zero and Eqs. (2) are simplified to (4) The variation of the stress components σ and τ, as we vary the angle α, can be easily represented graphically by making a diagram in which σ and τ are taken as coordinates. For each plane there will correspond a point on this diagram, the coordinates of which represent the values of σ and τ for this plane. Fig. 2 represents such a diagram. For the planes perpendicular to the principal directions we obtain points Aand B with abscissas σx and σy respectively. Now it can be proved that the stress components for any plane BC with an angle α (Fig. 2)will be represented by coordinates of a point on the circle havingABas a diameter. To find this point it is only necessary to measure from the point A in the same direction as α is measured in Fig. 2 an arc subtending an angle equal to 2α. If Dis the point obtained in this manner, then, from the figure, Comparing with Eqs. (4) it is seen that the coordinates of point D give the numerical values of stress components on the plane BC at The angle α. To bring into coincidence the sign of the shearing component we take τ positive in the upward direction (Fig. 2) and consider shearing stresses as positive when they give a couple in the clockwise direction, as on the sides bc and ad of the element abcd (Fig. 2b). Shearingstresses of opposite direction, as on the sidesab and dc of the element,are considered as negative.
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