ELEMENTS OF MECHANICAL ENGINEERING

Transcription

1 ELEMENTS OF MECHANICAL ENGINEERING (For B.E./B.Tech Engineering Students) As per New Revised Syllabus of Dr. APJ Abdul Kalam Technical University - UP Dr. S. Ramachandran, M.E., Ph.D., Saswath Kumar Das, HOD - Mechanical Engineering Dr. D.R. Somashekar, M.E., Ph.D Director IIMT College of Engineering, GREATER NOIDA (UP) AIRWALK PUBLICATIONS (Near All India Radio) 80, Karneeshwarar Koil Street Mylapore, Chennai Ph.: , aishram2006@gmail.com, airwalk800@gmail.com

2 th First Edition: 9 August 2017 ISBN: Price : ISBN :

3 ELEMENTS OF MECHANICAL ENGINEERING Syllabus S.1 UNIT-I: Force System: Force, Parallelogram Law, Lami s theorem, Principle of Transmissibility of forces. Moment of a force, Couple, Varignon s theorem, Resolution of a force into a force and a couple. Resultant of coplanar force system. Equilibrium of coplanar force system, Free body diagrams, Determination of reactions. Concept of Centre of Gravity and Centroidand Area Moment of Inertia, Perpendicular axis theorem and Parallel axis theorem UNIT-II: Plane Truss: Perfect and imperfect truss, Assumptions and Analysis of Plane Truss by Method of joints and Method of section. Beams: Types of beams, Statically Determinate Beams, Shear force and bending moment in beams, Shear force and bending moment diagrams, Relationships between load, shear and bending moment. UNIT-III: Simple stress and strain: Normal and shear stresses. One Dimensional Loading; members of varying cross section, bars in series. Tensile Test diagram for ductile and brittle materials, Elastic constants, Strain energy. Bending (Flexural) Stresses: Theory of pure bending, neutral surface and neutral axis, stresses in beams of different cross sections. Engineering Materials: Importance of engineering materials, classification, mechanical properties and applications of Ferrous, Nonferrous and composite materials. UNI-IV: Basic Concepts and Definitions of Thermodynamics: Introduction and definition of thermodynamics, Microscopic and Macroscopic approaches, System, surrounding and universe, Concept of continuum, Thermodynamic equilibrium, Thermodynamic properties, path, process and cycle, Quasi static process, Energy and its forms, Work and heat. Thermodynamic definition of work.

4 S.2 Elements of Mechanical Engineering Zeroth law of thermodynamics: Temperature and its measurement. First law of thermodynamics: First law of thermodynamics, Internal energy and enthalpy. First law analysis for non-flow processes. Non-flow work Steady flow energy equation; Boilers, Condensers, Turbine, Throttling process, Pumps etc. UNIT-V: Second law: Thermal reservoir, Kelvin Planck statement, Heat engines, Efficiency; Clausius statement Heat pump, refrigerator, Coefficient of Performance. Carnot cycle, Carnot theorem and it s corollaries.clausius inequality, Concept of Entropy. Properties of pure substances: P-v, T-s and h-s diagram, dryness fraction and steam tables. Rankine Cycle. Internal Combustion Engines: Classification of I.C. Engines and their parts, working principle and comparison between 2 Stroke and 4 stroke engine, difference between SI and CI engines. Pv and T-s diagramsof Otto and Diesel cycles, comparison of efficiency.

5 Contents C.1 Contents Chapter 1 Force System - Centroid and Moment of Inertia 1.1 Engineering Mechanics Classification of Engineering Mechanics Fundamental Concepts of Mechanics Scalar and Vector Quantities Laws of Mechanics Newton s Three Fundamental Laws Newton s Law of Gravitation The Parallelogram Law of Forces Triangular Law of Forces Polygon Law of Forces Law (or) Principle of Transmissibility of Forces Force and Force System Types of forces Types of force system (I) Coplanar force system (a) Concurrent forces (b) Coplanar - concurrent force system (c) Non concurrent and non-parallel forces (d) Coplanar - Non concurrent forces (e) Collinear forces (f) Parallel forces II Non coplanar force system (a) Non coplanar concurrent forces (b) Non coplanar Non concurrent forces (c) Non coplanar parallel forces (d) Non coplanar Non concurrent and Non parallel forces (skew forces) Composition of Forces Resultant of two coplanar concurrent forces

6 C.2 Elements of Mechanical Engineering I. Analytical method - Parallelogram law of forces II Analytical Method Triangle Law of Forces III Graphical method - Parallelogram law of forces IV Graphical method - Triangle law of forces Resolution of Forces Resultant of Coplanar Force System - (Method of projections) Summary Equilibrium of Coplanar Force System Equilibrant Equations of Equilibrium of a particle Free Body Diagram Free body diagram in simple words Equilibrium Of A Three Force Body Condition For Three Forces In Equilibrium Lami s Theorem Graphical Method Moment of a Force Principle of Moments (or) Varignon s theorem General Case of Parallel Forces in a Plane Solved Problems in Varignon s theorem Couple Moment of a Couple Equivalent couples Resolution of a Given Force Into a Force and a Couple : Force - Couple System Support and Determination of Reactions Types of Supports Problems for determination of reactions Equivalent Force System Resultant of Force and Couple System Centre of Gravity or Centroid

7 Contents C First Moment of Area Centroid of a Uniform Lamina Centre of Gravity Problems on Centroids of composite plane figures and curves Theorems of Pappus and Guldinus Surface of Revolution Volume of Revolution Pappus-Guldinus Theorem I Pappus-Guldinus Theorem II Area Moment of Inertia (Second Moment of Area) Perpendicular Axis theorem Polar Moment of Inertia and Perpendicular Axis Theorem Perpendicular Axis theorem Radius of Gyration of an Area Parallel Axis Theorem Problems on Moment of Inertia Moment of Inertia of Composite Section Mass Moment of Inertia of Cylinder and thin Disc Parallel Axis Theorem Problems on Mass Moment of Inertia Chapter - 2 PLANE TRUSSES AND BEAMS 2.1 Introduction The Classifications of Structure Frame Truss Classification of Dimensional configuration Classification of Determinacy Classification of supports Perfect Truss

8 C.4 Elements of Mechanical Engineering Mathematical equation for perfect truss Imperfect truss Assumptions and Analysis of Plane Truss Trusses For Various Applications Analysis of Forces in a Truss Determination of the reactions at the supports Determination of the internal forces in the members of the frame Analytical method Method of joints Problems on Method of Joints Method of sections Beams Types of Beams (i) Simply supported Beam: (ii) Cantilever Beam: (iii) Overhanging Beam: (iv) Fixed Beam: (v) Continuous Beam: (vi) Propped Cantilever Beam: Diagrammatic Conventions for Supports and Loading Supports and Support Reactions Types of Supports and their Reactions (i) Simple support or Knife Edge support (ii) Roller Support (iii) Hinge or Pin-Jointed support (iv) Fixed or built-in support (v) Smooth surface support or Frictionless support Static Equilibrium Equations Determinate and Indeterminate Beams Types of Loading in Beams (a) Point or Concentrated load (b) Uniformly Distributed Load (UDL)

9 Contents C.5 (c) Uniformly Varying Load (UVL) Shear Force in Beams (S.F) Sign Convention for Shear Force In Beam Couple or Moment Bending Moment in Beams Sign Convention for Bending Moment In Beams Shear Force (S.F) And Bending Moment (B.M) diagrams Relationship Between Load, Shear force and Bending Moment Method of Drawing Shear Force and Bending Moment Diagrams by Summation Approach Points to be Remembered for Drawing S.F.D and B.M.D Chapter 3 Simple Stress and Strain, Bending Stresses and Engineering Materials 3.1 Simple Stress and Strain Stress Unit of Stress Strain Normal and Shear Stresses Normal Stress: Axially Loaded Bar Tensile Stress and Tensile Strain Compressive Stress and Compressive Strain Shear Stress and Shear Strain BEaring Stress (crushing Stress) In Connections Stress-strain Diagram Stress-Strain Curve (Tensile Test diagram) for Ductile Materials Stress-Strain Curves (Tensile test diagram) for Brittle Materials Stress Strain Curves (Compression) Hooke s Law - Axial & Shear Deformation

10 C.6 Elements of Mechanical Engineering Factor of Safety Deformation of a body due to force acting on it Significance of percentage of Elongation & Reduction in Area One Dimensional Loading Deformation In Simple Bar Subjected to Axial Load Deformation for a Bar of Varying Cross Section and Bars in Series Principle of Superposition Stress in Bars of Uniformly Tapering Cross Section Deformation of Uniformly Tapering Rectangular Bar Deformation in Compound or Composite Bars Strain Energy Strain Energy In Pure Shearing Expression for strain energy stored in a body due to shear stress Elastic Constants Modulus of Elasticity Rigidity Modulus (or) Shear Modulus Bulk Modulus Linear Strain and Lateral Strain Poisson s ratio Biaxial And Triaxial Deformations Volumetric Strain Rectangular Body Subjected to Axial Loading Rectangular Bar Subjected to 3 Mutually perpendicular Forces Cylindrical Rod Subjected to Axial Load Bulk Modulus Relationship Between Elastic Constants Relation between Bulk Modulus and Young s Modulus Shear Stress and Strain

11 Contents C Shear Modulus or Modulus of Rigidity Relation between Modulus of Elasticity and Modulus of Rigidity Bending (flexural) Stresses In Beams - Theory of Pure Bending Simple Bending (or) Pure Bending Assumption in Theory of Simple Bending Theory of simple bending - Bending stress equation (or) Flexural Formula Derivation Limitations in Theory of Simple Bending: Stresses In Beams of Different Cross Sections Section modulus or Modulus of section Flexural Strength of A Section Engineering Materials Importance of Engineering Materials Classification of Engineering materials Metals Ferrous Metals Non-Ferrous metals Mechanical Properties of Materials Mechanical Properties And Applications of Ferrous, Non Ferrous And Composite Materials Steel Effect of alloying additions on steel Stainless Steels Properties of Stainless Steels Applications Tool Steels Properties of tool steels High Strength Low Alloy Steels (hsla Steels) Applications of HSLA Maraging Steels (ultra High Strength Steels)

12 C.8 Elements of Mechanical Engineering Properties of Maraging Steels Applications Cast Iron Composition of cast iron Types of Cast Iron Grey cast iron White cast iron Malleable Cast Iron Spheroidal Graphite Cast Iron: (Ductile iron (or) Nodular iron) Alloy Cast Iron Non Ferrous Metals Aluminium And Aluminium Alloys Aluminium Aluminium alloys Classification Copper And Copper Alloys Classification of Copper alloys Bearing Alloys Non - Metallic Materials Polymers Classification of Polymers Thermoplastic polymers Thermosetting Polymers Commodity and Engineering Polymers Plastics Properties and Applications of Some Common Thermoplastics Properties and Applications of Some Common Thermosetting Polymers Elastomer (or) Rubber Natural Rubber Synthetic Rubbers Ceramics

13 Contents C Classification of Ceramic Materials on the Basis of Applications Glasses Clay Products Refractory Materials Cements Composite Materials (or) Composites Classification of Composites Chapter 4 Basic Concepts, Zeroth Law and First Law of Thermodynamics 4.1 Introduction Role of Thermodynamics in Engineering and Science Applications of Thermodynamics Basic Concepts Macroscopic And Microscopic Approaches Macroscopic approach Microscopic approach Concept of Continuum Thermodynamic System, Surroundings and Universe Types of System Closed System Open System Isolated System Homogeneous and Heterogeneous Systems Property State, Path and Process Cycle Thermodynamic Equilibrium Quasi-static (or) Quasi equilibrium Process Energy

14 C.10 Elements of Mechanical Engineering Different forms of energy Work Work Transfer - A Path Function Different Modes of Work Electrical work Shaft Work Spring Work Paddle wheel work (or) Stirring work Flow Work Workdone on Elastic solid bars Work Associated with the stretching of a Liquid Film Workdone per unit volume on a magnetic material Heat Heat Transfer A Path Function Thermodynamic Definition Of Work Zeroth Law Of Thermodynamics Equality of temperature Temperature and its Measurements Thermometry Applications of thermometry First Law of Thermodynamics Joules experiment Perpetual motion of machine of first kind-i Internal Energy Enthalpy Application of First Law To Non-flow Process (or) Closed System Constant Volume Process Constant Pressure Process (or) isobaric Process Constant Temperature Process (or) isothermal Process Reversible Adiabatic Process (or) isentropic Process

15 Contents C Polytropic process: PV n constant n n Steady Flow Energy Equation: Application of First Law to Steady Flow Process (Open System) Applications of first law Nozzle: Diffusor: Throttling Device: Turbine: Compressor: Heat Exchanger: Chapter 5 Second Law, Properties of Pure substances and IC Engines 5.1 Introduction to the Second Law of Thermodynamics Thermal Energy Reservoirs Heat Engines Heat Engine Cycle The Second Law of Thermodynamics: Kelvin-planck Statement The Second Law of Thermodynamics: Clausius Statement Refrigerators and Heat Pumps Refrigerator Heat Pump Coefficient Of Performance - COP Carnot Cycle Process 4-1 Isentropic Compression Process Reversed Carnot Cycle The Carnot Principles (or) Carnot Theorem Reversed Heat Engines Corollaries of Carnot s Theorem Clausius Inequality Concept of Entropy Characteristics of entropy

16 C.12 Elements of Mechanical Engineering Entropy Transfer with Heat Flow Properties of Pure Substances Thermodynamic properties of pure substances in solid, liquid and vapour phases Property Diagrams for Phase-Change Processes T-v Diagram P v Diagram P-T Diagram T-s Diagram h s Diagram or Mollier Diagram Thermodynamic Properties of Steam Standard Rankine Cycle The Ideal Cycle for Vapour Power Cycles Efficiency of Standard Rankine Cycle Introduction to IC Engines Basic Terms Connected with I.C. Engines Classification of IC Engines Four Stroke SI (Petrol) Engine Four Stroke CI (diesel) Engine Working of Two Stroke Cycle Engine Working of Two Stroke Petrol Engine Two Stroke Cycle Compression Ignition Engine IC Engine Components Functions and Materials Comparison Of Four-stroke and Two Stroke Cycle engines Otto Cycle (or) Constant Volume Cycle Diesel Cycle or Constant Pressure Cycle Comparison of Efficiency

17 Force System-Centroid and Moment of Inertia 1.1 Chapter 1 Force System - Centroid and Moment of Inertia Force System: Force, Parallelogram Law, Lami s theorem, Principle of Transmissibility of forces. Moment of a force, Couple, Varignon s theorem, Resolution of a force into a force and a couple. Resultant of coplanar force system. Equilibrium of coplanar force system, Free body diagrams, Determination of reactions. Concept of Centre of Gravity and Centroid and Area Moment of Inertia, Perpendicular axis theorem and Parallel axis theorem. 1.1 ENGINEERING MECHANICS The different motions that we notice, everyday, like balls bouncing or wheels rolling, are interaction of different bodies and effect of forces acting on them - the study is called mechanics. It can be defined as that science which describes and predicts the condition of rest or motion of bodies under the action of forces. Mechanics is divided into three major parts. 1. Mechanics of rigid bodies 2. Mechanics of deformable bodies 3. Mechanics of fluids Mechanics, when applied in engineering is called Engineering mechanics which concerns itself mainly with the application of the principles of mechanics to the solution of problems commonly encountered in the field of engineering practice. Thus, Engineering mechanics is the study of forces and motion of bodies in mechanisms.

18 1.2 Elements of Mechanical Engineering CLASSIFICATION OF ENGINEERING MECHANICS Engineering mechanics may be classified based upon the type or nature of the body involved and is shown in Fig Engineering Mechanics Mechanics of solids Mechanics of fluid Rigid bodies Deformable bodies Ideal fluid Viscous fluid Compressible fluid Statics Dynamics Strength of Materials Theory of Elasticity Theory of Plasticity Kinetics Kinematics Fig. 1.1 Classification of Engineering Mechanics A particle is defined as an idealized model of actual physical body of real world, as an entity having only mass and location in space. Its dimensions are negligible when compared with the distances involved in the discussion of its motion. Rigid body Rigid body is one which can retain its shape, size or one which does not undergo any deformation under the loads. It is a combination of a large number of particles which occupy fixed positions with respect to each other both before and after applying a load. A rigid body is one which does not deform under load. However, actual bodies are not absolutely rigid and deform slightly. Since such slight deformations do not affect the conditions of equilibrium or motion, they are neglected in the study of rigid bodies.

19 Force System-Centroid and Moment of Inertia 1.3 Rigid body mechanisms found a suitable basis for the analysis and design of structural, mechanical or electrical engineering devices and are divided into two areas - Statics and Dynamics. Dynamics is further divided into Kinetics and Kinematics. Statics is branch of mechanics which deals with the force and its effects on bodies at rest. The configuration of different forces is such that the resultant force on the system is zero. Dynamics is branch of mechanics which deals with the force and its effects on bodies in motion. Kinetics is the branch of mechanics which deals with the body in motion when the forces which cause the motion are considered. Kinematics is the branch of mechanics which deals with the body in motion, when the forces causing the motion are not considered. Deformable bodies are one which undergoes deformation under application of forces. The branch of mechanics which deals with the study of internal force distribution, stress and strain developed in the bodies is called mechanics of deformable bodies or mechanics of materials. Fluid mechanics is the branch of mechanics which deals with study of fluids both liquids and gases at rest or in motion. 1.3 FUNDAMENTAL CONCEPTS OF MECHANICS The basic concepts used in everyday mechanics are based on newtonian mechanics. The basic concepts used in newtonian mechanics are space, time, mass and force. These are absolute concepts because they are independent of each other. Space is a geometric region in which events involving bodies occur. Space is associated with the position of a point P. The position of P can be defined by three lengths measured from a certain reference point or origin in three given directions. These lengths are known as coordinates of P. Time is a measure of the succession of events. The standard unit used for its measurement is the second s, which is based on duration. Mass is used to characterize and compare bodies on the basis of certain fundamental mechanical experiments. Two bodies of the same mass, will be attracted by the earth in the same manner and, they will also offer same resistance to a change in translation motion. Mass is quantitative measure of inertia. Mass of a body is constant regardless of its location.

20 1.4 Elements of Mechanical Engineering Force is the action of one body on another. Each body tends to move in the direction of external force acting on it. A force is characterized by its point of application, its magnitude and its direction. A force is represented by a vector. Simply force is push or pull, which by acting on a body changes or tends to change its state of rest or motion. Weight is the force with which a body is attracted towards the centre of earth by the gravitational pull. The relation between the mass and weight of a body is given in the equation 1.1. Weight (W) mass (m) gravity (g)... (1.1) where W weight in Newton m mass of body in kg g acceleration due to gravity i.e 9.81 m/s SCALAR AND VECTOR QUANTITIES Scalar quantities are those physical quantities that have only magnitude but no direction. eg. mass, time, volume, etc. Vector quantities are those physical quantities that have both magnitude and direction. eg. Displacement, velocity, acceleration, momentum, force, etc. A vector is represented graphically as shown in Fig Magnitude (length) Sense (arrow) Direction Line of Action Head of Vector Reference axis Tail of Vector Fig.1.2 Graphical representation of a vector For mathematical operations involving vectors, the rules of vector algebra should be applied.

21 Force System-Centroid and Moment of Inertia LAWS OF MECHANICS The study of mechanics rests on the following fundamental principles based on the experimental evidences. (i) Newton s three fundamental laws (First law, Second law, and Third law) (ii) Newton s Law of Gravitation (iii) The principle of transmissibility of forces (iv) The parallelogram law of addition of forces (v) The triangular law of forces (vi) The law of conservation of Energy (vii) The principle of work and Energy Newton s Three Fundamental Laws (a) Newton s First Law Newton s first law states that If the net force or the resultant force acting on a particle is zero, the particle will remain at rest (if originally at rest) or will move with constant speed in a straight line (if originally in motion). Or in other words, Every particle continues in its state of rest or of motion in a straight line unless it is compelled to change that state by an external force imposed on the body. So the first law is used to define the forces. (b) Newton s Second Law Newton s second law states that If the resultant force acting on a particle is not zero, the particle will have an acceleration proportional to the magnitude of the resultant force and will move in the direction of this resultant force. F Resultant force or Net force F ma where m Mass of the body a Acceleration of the body This law is used to measure a force.

22 1.6 Elements of Mechanical Engineering Newton s Second Law in other words: The rate of change of momentum of a body is directly proportional to the imposed force and it takes place in the direction in which the force acts. i.e., Force Rate of change of momentum Now, Momentum Mass Velocity Rate of change of momentum mass rate of change of velocity Mass Acceleration Force mass acceleration i.e., F d dt F K ma mv m dv dt m a K Constant of proportionality 1 in S.I. units (c) Newton s Third Law Each and every action (FORCE) has equal and opposite reaction. This means that the forces of action and reaction between two bodies are equal in magnitude but opposite in direction and have the same line of action Newton s Law of Gravitation This states that two particles of mass M and m are mutually attracted with equal and opposite forces F and F. This force F is defined as follows: F G Mm r 2 where r Distance between the two particles. G Universal constant, also called the Constant of Gravitation. Refer Fig 1.3(a).

23 Force System-Centroid and Moment of Inertia 1.7 Example The attraction of the earth on a particle located on its surface. The force F exerted by the earth on the particle is known as weight W. By the law of gravitation we have, W GMm 2, Introducing the constant g GM r 2, we have M -F Fig.1.3 (a) r F m W mg where W weight of the body in Newton N m mass of the body in kg g 9.81 m/sec 2 acceleration due to gravity The Parallelogram Law of Forces Construct a parallelogram using a force P and a force Q as two sides of the parallelogram. Now the diagonal P passing through A represents the resultant force. This is called as parallelogram law for addition of two forces. Refer Fig 1.3 (b). Or in other words A Q Fig.1.3(b) If two forces P and Q acting at a point A are represented by the adjacent sides of a parallelogram, then the diagonal of the parallelogram passing through that point of intersection represents the resultant. Resultant R P 2 Q 2 2PQ cos

24 1.8 Elements of Mechanical Engineering Triangular Law of Forces If two forces acting simultaneously on a body are represented by the sides of a triangle taken in order, then their resultant is represented by the closing side of the triangle taken in the opposite order. Refer the Fig. 1.4 (a). The sum of two forces P and Q may be obtained by arranging P and Q in tip to tail fashion and then connecting the tail of P with tip of Q. Addition of Two Vectors (Forces) P and Q [Refer Fig. 1.4 (a) and (b)] P Q Even if we P Resultant P+Q change the order we get same resultant Resultant Q+P (a) Fig.1.4. Q (b) Now P Q Q P. So addition of two forces is commutative. The subtraction of a force is obtained by the addition of corresponding negative force. The force P Q is obtained by adding to P with the negative force Q as shown in Fig. 1.5 (a) & (b). So, P Q P Q P P Q P - Q -Q subtraction (a) (b) Fig.1.5. Resultant P- Q

25 Force System-Centroid and Moment of Inertia Polygon Law of Forces If a number of concurrent forces acting simultaneously on a body, are represented in magnitude and direction by the sides of a polygon taken in order, then the resultant is represented in magnitude and direction by the closing side of the polygon, taken in the opposite order. Graphically arrange all the given forces in the tip-to-tail fashion in any order. Join the tail of the second force with the tip of the first force and so on. To obtain the resultant, draw vector joining tail of first force and tip of last force as shown in Fig. 1.6 (b). Concurrent Forces acting on a body. Refer Fig. 1.6 (a) F 5 F 2 F 5 By using Polygon Law of Forces Vectorial Addition of F 4 Fig.1.6.(a) F 3 F 1, F 2, F 3, F 4, and F 5 Resultant This means: We can add the forces as in Fig 1.6 (b) We can also add the forces as in Fig. 1.6 (c). F 2 F 1 F 4 F 3 F 5 Resultant. Consider Fig. 1.6 (b) and 1.6 (c). Here order is changed. Even then, we get the same resultant. Hence, the resultant R does not depend upon the order in which the forces are chosen to draw the polygon.

26 1.10 Elements of Mechanical Engineering R e sultan t = F + F +F +F +F o By using Polygon law of forces F 1 Both Resultant are equal F 2 Change the order : o F 4 R es ulta nt = F +F + F + F + F F 2 F 5 F 4 F 5 F 3 F 1 F 3 Fig.1.6.(b) Fig.1.6.( c ) Law (or) Principle of Transmissibility of Forces The condition of equilibrium or motion of a rigid body will remain unchanged if the point of application of a force acting on the rigid body is transferred to any other point along its line of action. Refer Fig. 1.7 and 1.8 Line of action a F A B b = a Same Effect A B F b (Push) Fig.1.7. F (Pull) A line of action Fig.1.8. = line of action B F

27 Force System-Centroid and Moment of Inertia 1.11 A force F acting on a body at point A is transferred to point B along the same line of action without changing its net effect on the rigid body. Principle of Transmissibility in other words (Refer Fig. 1.8) Alternatively, the principle of transmissibility states that the conditions of equilibrium or the motion of a rigid body will remain unchanged if a force F acting at a given point A of the rigid body is replaced by a force F of the same magnitude and same direction, but acting at a different point B, provided that the two forces have the same line of action. The two forces F and F have the same effect on the rigid body and are said to be equivalent. 1.6 FORCE AND FORCE SYSTEM Force Force is an action that changes or tends to change the state of the body on which it acts. Simply, force is a push or pull on a body. Force cannot be seen and only its effect on the object upon which it acts can be felt. A force can also change or tend to change the size and shape of deformable bodies. The characteristics of forces are (i) Magnitude (ii) Direction (slope) (iii) Sense (iv) Point of application. These characteristics distinguish one force from another. Representation of a force is shown in the Fig Space diagram is the diagram in which the forces are represented and their magnitudes are written along their lines of action. (Fig. 1.10). Force diagram is the diagram in which the forces are drawn to scale, parallel to their respective line of action. O M a gn itu de (F ) point of action F = inclination of force w ith horizontal Fig. 1.9 Force lin e of a ctio n

28 1.12 Elements of Mechanical Engineering C C=20N B B=30N C B B A=40N A (a) Space diag ram Fig 1.10 C A (b) Force diag ram A Types of forces Forces may be classified as (i) External forces (contacting or applied forces) (ii) Body forces (non-contacting or non applied forces) (a) External forces Forces which act on a body due to external causes or agency are termed as external forces. The external forces acting on the surface of a body are called surface forces. The equilibrium of the body is affected by all the external forces (body forces and contact forces) which act on the body. A body pressing against another is subjected to both normal and tangential force. F 1 Normal force is perpendicular to the surface of contact. The tangential force of the contact surfaces is called frictional force. Body The applied forces are further classified F 2 as F (i) concentrated or point forces 3 Fig 1.11 Point force (ii) Distributed forces Point forces are those forces which act at specific points on the body. Example - force exerted by wheels of vehicle on road. Fig

29 Force System-Centroid and Moment of Inertia 1.13 W/m Simply supported beam X a b Fig 1.12 Distributed force Y Distributed forces are those forces which are distributed over definite areas of surface of the body. Example - a beam of a ceiling or a floor Fig (b) Body forces The external forces acting on each and every particle of a body are termed as body force or volume force. Examples are gravitational pull on all physical bodies, magnetic force, etc. If the force acting on an area of specific size and if the area is significant, the resultant force per unit area is termed as pressure or stress and is described as force per unit area N/m Types of force system When several forces are acting upon a body simultaneously, they constitute a system of forces. Force system is classified based upon the two dimensional or three dimensional space of forces and the orientation of line of action of forces. The classification is given below. Force system Coplanar Non-coplanar Concurrent Parallel Non-concurrent, Non-parallel Collinear Concurrent Parallel Non-concurrent, Non-parallel I. Coplanar force system (a) Concurrent (b) Parallel (c) Non-concurrent (d) Collinear II. Non coplanar force system (a) Concurrent (b) Parallel (c) Non-concurrent

30 1.14 Elements of Mechanical Engineering (I) Coplanar force system A system of forces that are contained in a single plane or system of forces having their line of actions in the single plane is called coplanar force system. (Fig. 1.14) Forces F 1, F 2, F 3 are coplanar forces. F 1 F 2 F 3 Plane Forces (a) Concurrent forces When the lines of action of all the forces of a system intersect at a common point, the system of forces are said to be concurrent. (Refer Fig. 1.15) Fig 1.14 Coplanar forces F 1 F 2 O (b) Coplanar - concurrent force system When the system of forces lie in the same plane and the line of action of forces pass through the common point, then the system of forces is called coplanar - concurrent force system (Fig. 1.16) Forces F 1, F 2, F 3 are in same plane and pass through common point O. F 3 Fig Concurrent Forces F 3 F 2 O 2 1 Plane F 1 (c) Non concurrent and non-parallel forces When the system of forces whose line of action does not pass through a common point and the forces are not parallel is called non concurrent and non-parallel system. (Refer Fig.1.17) Fig 1.16 Coplanarconcurrent forces Fig: 1.17 Non-Concurrent Non-Parallel

31 Force System-Centroid and Moment of Inertia 1.15 (d) Coplanar - Non concurrent forces When the system of forces is acting in a single plane and does not have their line of action passing through a common point are called coplanar non concurrent forces. Fig is an example of coplanar - non concurrent forces. Plane F 1 F 3 forces F 2 Fig 1.18 Coplanar unlike parallel forces (e) Collinear forces When the system of forces acting in a single plane with a common line of action are called collinear forces. Fig shows collinear force system. O F 2 F 1 F 3 F 4 F 5 Plane Common line of action Forces (f) Parallel forces When the lines of action of all the forces of the system are parallel, then the system is called parallel force system. When the lines of action of all the forces are parallel and all of them act in the same direction, then the force system is called like parallel forces. Fig 1.19 Collinear forces Plane F 1 F 2 F 3 F 4 Fig 1.20 Coplanar like parallel forces

32 1.16 Elements of Mechanical Engineering When the line of action of all the forces are parallel and some forces act in one direction while others in opposite direction, then the force system is called unlike parallel force system. Fig shows an example of coplanar like parallel force system and Fig shows an example of coplanar unlike parallel force system. II Non coplanar force system A system of forces in which the forces lie in different planes or in a three dimensional space is called a Non F 1 F 2 F 3 Plane Fig 1.21 Coplanar unlike parallel forces forces Z Planes Z Planes F 1 F 1 F 3 O O X O X F 2 F 2 Y Fig 1.22 Non Coplaner Concurrent forces Y Fig 1.23 Non Coplanar non concurrent forces coplanar force system. (a) Non coplanar concurrent forces A system of forces lying in different planes with their line of action intersecting at a common point are called Non coplanar concurrent force system. Fig shows such an example.

33 Force System-Centroid and Moment of Inertia 1.17 (b) Non coplanar Non concurrent forces A system of forces lying in different planes with different lines of actions are called non coplanar non concurrent forces (Fig. 1.23) (c) Non coplanar parallel forces A system of forces lying in different plane but their line of action parallel to each other are called non coplanar parallel forces. In Fig forces F 1 and F 2 are non coplanar like parallel force and F 3 and F 4 are non coplanar unlike parallel forces. (d) Non coplanar Non concurrent and Non parallel forces (skew forces) Z Z F 1 F 2 F 1 F 3 F 4 O X F 3 O X F 2 Y Fig 1.24 Non Coplanar parallel forces Y Fig 1.25 Skew forces A system of forces that are not lying in same plane with different line of action and not passing through same common point and are not parallel to each other are called Non coplanar Non concurrent and non parallel forces. These forces are also called as skew forces. F 3 F 2 R (Resultant ) Particle F 1 Fig 1.26 Resultant force

34 1.18 Elements of Mechanical Engineering COMPOSITION OF FORCES Composition of a force system is a process of finding a single force, known as resultant, that can produce the same effect on the particle as that of the system of forces. For example Fig shows a system of three forces acting on a particle with the resultant R which can produce the same effect. A particle means that the size and shape of body does not significantly affect the solution of problems and all the forces are assumed to act at the same point. Thus, the resultant is a representative force which has the same effect on the particle as the group of forces it replaces Resultant of two coplanar concurrent forces Resultant of two coplanar concurrent forces can be obtained by the following methods. 1. Analytical method (a) Parallelogram law of forces (b) Triangular law of forces 2. Graphical method (a) Parallelogram law of forces, (b) Triangular law of forces I. Analytical method - Parallelogram law of forces F 2 F 2 R R A F 1 A F 1 A Fig 1.27(a) Parallelogram law When two forces F 1, F 2 acting on a particle are represented by two adjacent sides of a parallelogram, the diagonal connecting the two sides represents the Resultant force R in magnitude and direction.[fig (a)] Hence, the relationship between F 1, F 2 and R can be derived as follows [Fig (b)].

35 Consider the parallelogram OACB. Let OA and OB represent the forces F 1 and F2 acting at a point O. The diagonal OC represents the resultant R which can be expressed as, OC 2 OA AD 2 CD 2 OA 2 2 OA AD AD 2 CD 2 OA 2 2 OA AD AC 2 [... AC 2 AD 2 CD 2 ] R 2 F F 1 F 2 cos F 2 2 [... AC OB F 2 ] 2 Hence R F 1 2 F2 2F1 F 2 cos Also tan CD OA AD Consider the following special cases. F 2 sin F 1 F 2 cos [... AD AC cos F 2 cos ] Note (1): If F 1, F 2 are at right angles, then 90, cos , R F 1 2 F2 tan F 2 F 1 Note (2): If F 1, F 2 are collinear and are in the same direction, then 0, cos 1 Force System-Centroid and Moment of Inertia 1.19 F 2 B R O F 1 A D Fig. 1.27(b) Resultant of two forces C R 2 2 F 1 2 F2 2F1 F 2 Resultant R F 1 F 2, tan 0 or 0 Case (3): If F 1 F 2 are collinear and are in opposite directions F 1 F 2, then 180