Mechanical Science

LESSON PLAN MECHANICAL SCIECES ME 101 : MECHANICAL SCIENCES CONTACT : 3L+ 1T = 4 CREDIT : 4 1. CHARTER 1 – FRICTION : Concept of friction , Laws of Coulumb Friction , Angle of repose. 4L 2. CHAPTER 2 – CENTROID AND CENTER OF GRAVITY : Moment of inertia of plane figures, moment of inertia of plane figures with respect to an in its plane, moment of inertia of plane figure with respect to an axis perpendicular to the pane of figure, Parallel axis theorem , Mass moment of inertia of symmetrical bodies , e.g. cylinder, sphere rod . 9 L 3. CHAPTER 3 – STRENGTH OF MATERIALS : Concept of simple stresses and strains , normal stress, shear stress , bearing stress , normal strain , shearing strain , Hookes law, Poissons ratio, Examples.4L 4. CHAPTER 4 – STRENGTH OF MATERIALS : Stress and strainsunder axial loading stress – strain diagram of ductile materials, Working stress , Factor of safety, Proportional limit , Elastic limit , Ultimate stress, Yielding, Modulus of elasticity, definitions of malleability , ductility, toughness and resilience. Concept of thermal stress.5L

LECTURE 1 

FRICTION 1.

Whenever the surfaces of two bodies are in contact there will be a limited amount of resistance to sliding between them, which is called friction.

2. LAWS OF FRICTION



The total friction that can be developed is independent of the magnitude of the area of contact.

 The total friction that can be developed is proportional to the normal force. 

For low velocities of sliding, the total friction that can be developed is practically independent of the velocity, although the experiments show that the force F necessary to start sliding is greater than that necessary to maintain sliding.



The angle of limiting friction may be defined as the angle between the resultant reaction and the normal to the plane on which the motion of the body is impending.



The maximum inclination called of the plane on which a body , free from external forces, can repose (sleep) is called angle of repose.

3. In the solution of problems where friction is involved, we may deal either with the total reaction R or with its rectangular components F and N will be more convenient when we are using the algebraic method of projections and the single force R when we are working graphically.

STRENGTH OF MATERIALS

CONCEPT OF STRESS AND STRAIN 1. Force per unit area is known as stress. The stress may be tensile, compressive, shear or internal pressure. 2. When a structure is subjected to stress, it undergoes deformation and is said to be strained. The ratio of change in length in the direction of application of stress to the original length is called strain. 3. When a bar is subjected to a force P normal to the surface then the stress produced is called normal stress. The force parallel to the cross section of bar is shear force and the stress produced is called shear stress. 4. Normal strain is produced under the action of direct or normal stresses. Shear strain is produced under the action of shear stress and is measured by change in angle. 5. FACTOR OF SAFETY AND WORKING STRESS In order to utilize completely the mechanical properties of a machine as a whole , the maximum stress to which any member is designed is much less than the ultimate stress or critical stress , and this stress is called working stress or allowable stress . Thus, Factor Of Safety = Critical stress/ allowable stress The factor of safety depends on many considerations when the material is subjected to varying stresses , or it is non – homogeneous , or subject to corrosion, the factor of safety is high . The factor of safety must always be such that the working stress is below the elastic limit. LECTURE 3

STRENGTH OF MATERIALS

1. Elasticity is the property by which a body returns to its original shape after the removal of external load is called elasticity. Plasticity is a term meant opposite to elasticity.

HOOKES LAW Hooke’s law states that stress is directly proportional to strain up to the elastic limit. 2. Definition of Young’s Modulus: Young’s Modulus or modulus of elasticity is defined as the ratio of the direct stress to direct strain . The stress and the strain may be compressive or tensile in nature. Ratio of modulus of elasticity of two materials is called modular ratio. 3. Definition of Poisson’s Ratio: The ratio of lateral strain to the longitudinal strain is a constant quantity and is called the Poisson’s ratio and is denoted by 1/m.

EXPLANATION OF DIFFERENT POINTS OBTAINED UNDER STRESSSTRAIN DIAGRAM (a) Limit of proportionality: It is the limiting value of the stress is proportional to strain. (b) Elastic limit is the limiting value of stress up to which stress is proportional to strain. (c) Upper yield point is the stress at which, the load starts reducing and the extention increases. (d) Lower yield point is the point at which the stress remains same but strain increases for some time. (e) Ultimate stress: It is the maximum stress the material can resist. At this stage cross –sectional area at a particular section starts reducing very fast. This is called neck formation. (f) Breaking point: The stress at which finally the specimen fails is called the breaking point.

LECTURE 4

STRENGTH OF MATERIALS CONCEPT OF THERMAL STRESS A bar when subjected to a change in temperature it undergoes an elongation or shorting depending on whether the bar is heated or cooled .If the end conditions of the bar are such that so as not to allow the deformation to take place, some stresses shall be produced in the bar . A bar of length L and made of material having co- efficient of thermal expansion alpha,when subjected to a raise in temperature of T degree celcius undergoes an elongation COMPOUND BARS Bars having components of different materials are known as compound bars or composite bars . A compound bar consisting of two or more components , when loaded , shall have same strain for different components , and stresses in different components shall be proportional to their respective moduli of elasticity. The sum of loads carried by different components will expand shall be equal to the applied load. A compound bar consisting of components made of different materials when subjected to a rise in temperature, shall develop different stresses in different components depending upon the co- efficient of thermal expansion of different components . The bar having higher thermal co- efficient of expansion will expand more and shall try to pull the bar having lower co- efficient of thermal expansion thereby causing tensile stresses in it and compressive stresses in itself.

LECTURE 5

CENTROID AND MOMENT OF INERTIA

1.The centre of gravity can be defined as the point through which resultant of force of gravity (weight) of the body acts. Center of gravity is a misnomer for the area . It is to be called as centroid. 2. Difference between centre of gravity and centroid (i)

The term centre of gravity applies to the bodies with mass and weight, and centroid applies to plan areas.

(ii)

Centre of gravity of a body is a point through which the resultant gravitational force (weight) acts for any orientation of the body whereas centroid is a point in a plane area such that the moment of area about any axis through that point is zero.

3. USE OF AXIS OF SYMMETRY Centroid of an area lies on the axis of symmetry if it exists. This is useful theorem to locate the centroid of an area. Making use of the symmetry we can conclude that (i) (ii)

Centroid of a circle is its centre. Centroid of rectangle of sides b and d is at a distance b/2 and d/2 from either sides.

DETERMINATION OF CENTROID OF SIMPLE FIGURES FROM FIRST PRINCIPLE For Simple figures like triangle and semicircle , we can write general expression for the elemental area and its distance from an axis . The location of the centroid using the above equations may be considered as finding centroid from first principles. Now, we can find out the centroid of the following figures , (i) (ii) (iii) (iv)

Centroid of triangle of base b and height h is h/3. Centroid of a semicircle of diameter d is 4R/3 where R is the radius of the circle. Centroid of sector of a circle is 2R /3 Centroid of a parabolic spandrel of base a and height h is (3a/4, 3h/10).

CENTROID OF COMPOSITE SECTIONS So far, the discussions was confined to locating the centroid of simple figures like rectangle , triangle, circle ,semicircle, etc. In engineering practice , use of sections which are built up of many simple sections is very common. Such sections may be

called as built –up sections or composite sections.In order to find out the centroid of each composite sction ,the given composite section can be split into suitable simple figures and then the centroid of each simple figure can be found by inspection or using the standard formulae. LECTURE 6 MOMENT OF INERTIA 1. Consider an elemental area dA with co-ordinates x and y. The term is called moment of inertia of the area about x axis and is denoted as I.Similarly , the moment of inertia about y axis is I = xdA 2. The moment of inertia is a purely mathematicl term . The moment of inertia is a fourth dimensional term since it is a term obtained by multiplying area by the square of the distance.Hence, in SI units , if meter (m) is the unit for linear measurements used then m is the unit of moment of inertia. 3. POLR MOMENT OF INERTIA Moment of inertia about an axis is perpendicular to the plane of an area is known as polar moment of inertia.It may be denoted as J . 4. RADIUS OF GYRATION Radius of gyration is a mathematical term defined by the relation K = I/A Where K = radius of gyration I = moment of inertia And A = the cross – sectional area. 5. THEOREMS OF MOMENTS OF INERTIA There are two theorems of moment of inertia : (1) Perpendicular axis theorem , and (2) Parallel axis theorem PERPENDICULAR AXIS THEOREM The moment of inertia of an area about an axis perpendicular to its plane (polar moment of inertia) at any point O is equal to the sum of moments of inertia about any two mutually perpendicular axis through the same point O and lying in the plane of the area. PARALLEL AXIS THEOREM Moment of inertia about any axis in the plane of an area is equal to the sum of moment of inertia about a parallel centroidal axis and the product of area and square of the distance between the two parallel axes.

LECTURE 7 MOMENT OF INERTIA 1.For simple figures , moment of inertia can be obtained by writing down the general expression for an element and then carrying out integration so as to cover the entire . This procedure is illustrated with the following three cases (1) Moment of inertia of rectangle about the centroidal axis (2) Moment of inertia of a triangle about the base (3) Moment of inertia of a circle about a dimetral axis of width b and depth d .Moment of inertia about the centroidal axis x-x parallel to the short side is given by the formulae I = bd / 12 3. Moment of inertia of a triangle about its base – Consider a triangle of base b and height h . Moment of inertia of the triangle is about an axis AB parallel to the base is given by the formulae I = bh / 12. 4. Moment of inertia of circle about its dimetral axis: Consider a circle of diameter D . Moment of inertia of the circle about the dimetral axis is given by the formulae I= 5. MOMENT OF INERTIA OF COMPOSITE SECTIONS Beams and columns having composite sections are commonly used in structures .Moment of inertia of these sections about an axis can be found by the following 2. Moment of inertia of a rectangle about the centroidal axis : consider a rectangle steps: (1) Divide the given figure into a number of simple figures . (2) Locate the centroid of each simple figure by inspection or using standard expressions. (3) Find the moment of inertia of each simple figure about its centroidal axis . Add the term Ay where A is the area of the simple figure and y is the distance of the centroid of the simple figure from the refrence axis This gives moment of inertia of the simple figure about the reference axis. (4) Sum up moments of inertia of all simple figures to get the moment of inertia of the composite section