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Lecture 1 (40)

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Concept of Particles and Free Body Diagram

Why FBD diagrams are used during the analysis?

It enables us to check the body for equilibrium.
By considering the FBD, we can clearly define the exact system of forces which we must use in the investigation of any constrained body.
It helps to identify the forces and ensures the correct use of equation of equilibrium.

Note:
Reactions on two contacting bodies are equal and opposite on account of Newton's III Law.
The type of reactions produced depends on the nature of contact between the bodies as well as that of the surfaces.
Sometimes it is necessary to consider internal free bodies such that the contacting surfaces lie within the given body. Such a free body needs to be analyzed when the body is deformable.

Physical Meaning of Equilibrium and its essence in Structural Application
The state of rest (in appropriate inertial frame) of a system particles and/or rigid bodies is called equilibrium.

A particle is said to be in equilibrium if it is in rest. A rigid body is said to be in equilibrium if the constituent particles contained on it are in equilibrium.
The rigid body in equilibrium means the body is stable.
Equilibrium means net force and net moment acting on the body is zero.

Essence in Structural Engineering
To find the unknown parameters such as reaction forces and moments induced by the body.
In Structural Engineering, the major problem is to identify the external reactions, internal forces and stresses on the body which are produced during the loading. For the identification of such parameters, we should assume a body in equilibrium. This assumption provides the necessary equations to determine the unknown parameters.
For the equilibrium body, the number of unknown parameters must be equal to number of available parameters provided by static equilibrium condition.

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Lecture 1 (40)

  1. 1. Engineering Mechanics: Statics Chapter 1: General Principles
  2. 2. • Statics will build upon what you were supposed to learn in your basic physics and mathematics courses. • We will talk about forces – vector forces – about moments and torques, reactions and the requirements of static equilibrium of a particle or a rigid body. WHY Statics?
  3. 3. • You have seen a good bit of the basic stuff of this course before, but we will not assume you know the way to talk about, or work with, these concepts, principles, and methods so fundamental to our subject. • So we will recast the basics in our own language, the language of engineering mechanics. WHY Statics?
  4. 4. • Think of this course as a language text; of yourself as a language student beginning the study of Engineering Mechanics (Statics). • You must learn the language if you aspire to be an engineer. • But this is a difficult language to learn, unlike any other foreign language you have learned. WHY Statics?
  5. 5. • It is difficult because, on the surface, it appears to be a language you already know. But you will have to be on guard, careful, not to presume the word you have heard before bears the same meaning. Words and phrases you have already used now take on a more special and, in most cases, narrower meaning; a couple of forces is more than just two forces. WHY Statics?
  6. 6. • The best way to learn a new language is to use it – speak it, read it, listen to it on audio tapes, watch it on television; better yet, go to the land where it is the language in use and use it to buy a loaf of bread, get a hotel room for the night, ask to find the nearest post office. WHY Statics?
  7. 7. • So too in statics, we insist you begin to use the language. Doing problems and exercises, taking quizzes and the final, is using the language. • Statics course contains exercises explained, as well as exercises for you to tackle such as homeworks... WHY Statics?
  8. 8. Chapter Outline • Mechanics (an introduction) • Fundamental Concepts (Newton’s Laws of Motion) • The International System of Units (principles for applying the SI) • Units of Measurement • Numerical Calculations (procedures for performing numerical calculations) • General Procedure for Analysis (solving problems)
  9. 9. Mechanics • Mechanics can be divided into 2 branches: 1- Rigid-body Mechanics 2- Deformable-body Mechanics - Solid bodies - Fluids (liquids and gases • Rigid-body Mechanics deals with - Statics - Dynamics
  10. 10. MechanicsMechanics Rigid-Body Mechanics Rigid-Body Mechanics Deformable-Body Mechanics Deformable-Body Mechanics StaticsStatics DynamicsDynamics Accelerated motion of bodies ‘a = constant’. Equilibrium of Bodies (restrest ‘v=0’ or movemove with constant velocitywith constant velocity ‘v=constant’) Mechanics
  11. 11. Mechanics • Statics – Equilibrium of bodies At rest Move with constant velocity • Dynamics – Non-equilibrium of bodies Accelerated / Deccelerated motion of bodies
  12. 12. What may happen if statics is not applied properly
  13. 13. Force – “push” or “pull” exerted by one body on another – Occur due to direct contact between bodies Eg: Person pushing against the wall – Occur through a distance without direct contact Eg: Gravitational, electrical and magnetic forces Fundamentals Concepts
  14. 14. Fundamentals Concepts • Particles – Consider mass but neglect size Eg: Size of Earth insignificant compared to its size of orbit • Rigid Body – Combination of large number of particles – Neglect material properties Eg: Deformations in structures, machines and mechanism
  15. 15. Fundamentals Concepts • Concentrated Force – Effect of loading, assumed to act at a point on a body – Represented by a concentrated force, provided loading area is small compared to overall size Eg: Contact force between wheel and ground
  16. 16. Fundamentals Concepts Newton’s Three Laws of Motion • First Law “A particle originally at rest, or moving in a straight line with constant velocity, will remain in this state provided that the particle is not subjected to an unbalanced force” First Law: ΣF = 0
  17. 17. Fundamentals Concepts Newton’s Three Laws of Motion • Second Law “A particle acted upon by an unbalanced force F experiences an acceleration ‘a’ that has the same direction as the force and a magnitude that is directly proportional to the force” Second Law: ΣF =
  18. 18. Fundamentals Concepts Newton’s Three Laws of Motion • Third Law “The mutual forces of action and reaction between two particles are equal and opposite and collinear” Third law: Faction= Freaction
  19. 19. Fundamentals Concepts WEIGHT • m is the mass (kg) • g is the acceleration due to gravity (m/s2 ) • Most engineering calculations, g at sea level and at a latitude of 45° is enough. g=9.81 m/s2 W mg=
  20. 20. Fundamentals Concepts • At the standard location, g = 9.806 65 m/s2 • For calculations, we use g = 9.81 m/s2 • Thus, W = mg (g = 9.81m/s2 ) • Hence, a body of mass 1 kg has a weight of 9.81 N, a 2 kg body weighs 19.62 N
  21. 21. Units of Measurement Basic Quantities • Length – Locate position and describe size of physical system – Define distance and geometric properties of a body
  22. 22. Units of Measurement Basic Quantities • Mass – Comparison of action of one body against another – Measure of resistance of matter to a change in velocity
  23. 23. Basic Quantities • Time – Period between two succession of events Units of Measurement
  24. 24. Units of Measurement SI Units • Système International d’Unités • F = ma is maintained only if – Three of the units, called basic units, are arbitrarily defined – Fourth unit is derived from the equation • SI units specifies:  length in meters (m),  time in seconds (s) and  mass in kilograms (kg) • Unit of force, called Newton (N) is derived from F = ma
  25. 25. Units of Measurement       2 . s mkg Name Length Time Mass Force International Systems of Units (SI) Meter (m) Second (s) Kilogram (kg) Newton (N)
  26. 26. The International System of Units Prefixes • For a very large or very small numerical quantity, the units can be modified by using a prefix • Each represent a multiple or sub-multiple of a unit Eg: 4 000 000 N = 4 000 kN (kilo-Newton) = 4 MN (Mega- Newton) 0.005 m = 5 mm (milli-meter)
  27. 27. International System of Units Exponential Form Prefix SI Symbol Multiple 1 000 000 000 109 Giga G 1 000 000 106 Mega M 1 000 103 Kilo k Sub-Multiple 0.001 10-3 Milli m 0.000 001 10-6 Micro μ 0.000 000 001 10-9 nano n
  28. 28. International System of Units Rules for Use • Never write a symbol with a plural “s”. Easily confused with second (s) • Symbols are always written in lowercase letters, except the 2 largest prefixes, mega (M) and giga (G) • Symbols named after an individual are capitalized Eg: Newton (N)
  29. 29. International System of Units Rules for Use • Quantities defined by several units which are multiples, are separated by a dot Eg: N = kg.m/s2 = kg.m.s-2 • The exponential power represented for a unit having a prefix refer to both the unit and its prefix Eg: μN2 = (μN)2 = μN. μN
  30. 30. • If a derived unit is obtained by dividing a basic unit to an other basic unit, the prefix should always be used for the unit at the numerator and never for the unit at the denominator. • Ex: If under 100 N load, a spring elongates 20 mm, the elongation constant ‘k’ of the spring is: �=(100 N)/(20 mm)=(100 N)/(0.02 m) = 5000 N/m or = 5 kN/m Never k = 5 N/mm International System of Units
  31. 31. International System of Units Rules for Use • Physical constants with several digits on either side should be written with a space between 3 digits rather than a comma (,) Eg: 73 569.213 427
  32. 32. Numerical Calculations Significant Figures - The accuracy of a number is specified by the number of significant figures it contains. - A significant figure is any digit including even zero. The location of the decimal point of a number is not important Eg: 5 604 and 34.52 both have four significant numbers
  33. 33. Numerical Calculations Significant Figures - When numbers begin or end with zero, we make use of prefixes to clarify the number of significant figures Eg: 400 → as 1 significant figure would be 0.4 (103 ) or 4 (102 ) 2 500 → as 3 significant figures would be 2.50 (103 )
  34. 34. Numerical Calculations Rounding Off Numbers For numerical calculations, the accuracy obtained from the solution of a problem would never be better than the accuracy of the problem data! Often handheld calculators or computers involve more figures in the answer than the number of significant figures in the data.
  35. 35. Numerical Calculations Rounding Off Numbers Calculated results should always be “rounded off” to an appropriate number of significant figures.
  36. 36. Numerical Calculations To ensure the accuracy of the final results, always give your answers 3 digits after the decimal point. Eg: 45.703 101.007 1 398.400
  37. 37. Numerical Calculations For plane angles used in trigonometry, in this course, please give your answers in 4 digits after the decimal point both for the angles and their trigonometric equivalencies. Eg: Sin 35.0000˚ = 0.5736 Cos 45.0380˚ = 0.7066 Tan-1 1.3459 = 53.3878˚
  38. 38. General Procedure for Analysis • Most efficient way of learning is to solve problems: • To be successful at this, it is important to present the work in a logical and orderly way as suggested: 1) Read the problem carefully and try to correlate actual physical situation with theory; 2) Draw any necessary diagrams and tabulate the problem data;
  39. 39. General Procedure for Analysis 3) Apply relevant principles, generally in mathematical forms; 4) Solve the necessary equations, algebraically as far as practical, making sure that they are dimensionally homogenous, using a consistent set of units and complete the solution numerically;
  40. 40. General Procedure for Analysis 5) Report the answer with no more significance figures than accuracy of the given data; 6) Study the answer with technical judgment and common sense to determine whether or not it seems reasonable.

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