Engineering Mechanics & Design of Steel Structures: Equilibrium of Forces & Law of motion &friction

By Ashutosh Yadav|Updated : October 21st, 2021

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Basic Concept

 Mechanics can be defined as the branch of physics concerned with the state of rest or motion of bodies that subjected to the action of forces. OR It may be defined as the study of forces acting on body when it is at rest or in motion is called mechanics.

Mechanics can be divided into two branches. 

  • Statics It is the branch of mechanics that deals with the study of forces acting on a body in equilibrium. Either the body at rest or in uniform motion is called statics
  • Dynamics: It is the branch of mechanics that deals with the study of forces on body in motion is called dynamics. It is further divided into two branches.
    • Kinetics It is the branch of the dynamics which deals the study of body in motion under the influence of force i.e. is the relationship between force and motion are considered or the effect of the force are studied
    • Kinematics: It is the branch of the dynamics that deals with the study of body in motion without considering the force.

Force

  • Force In general force is a Push or Pull, which creates motion or tends to create motion, destroy or tends to destroys motion.
  • In Engineering mechanics force is the action of one body on another.
  • A force tends to move a body in the direction of its action, A force is characterized by its point of application, magnitude, and direction, i.e. a force is a vector quantity.

Units of force

The following force units are frequently used.

  • Newton
    • The S.I unit of force is Newton and denoted by N. which may be defined as 1N = 1 kg. 1 m/s2 
  • Dynes
    • Dyne is the C.G.S unit of force. 1 Dyne = 1 g. 1 cm/s2 One Newton force = 105 dyne
  • Pounds
    • The FPS unit of force is the pound. 1 lbf = 1 lbm. 1ft/s2 One pound force = 4.448 N One dyne force = 2.248 x 10ˉ6 lbs

Principle of Transmissibility of forces

  • The state of rest of motion of a rigid body is unaltered if a force acting in the body is replaced by another force of the same magnitude and direction but acting anywhere on the body along the line of action of the replaced force.
  • For example the force F acting on a rigid body at point A. According to the principle of transmissibility of forces, this force has the same effect on the body as a force F applied at point B.

 

 

 

Free-Body Diagram: 

  • A diagram or sketch of the body in which the body under consideration is freed from the contact surface (surrounding) and all the forces acting on it (including reactions at contact surface) are drawn is called free body diagram. Free body diagram for few cases are shown in below

 

Steps to draw a free-body diagram:

  1. Select the body (or part of a body) that you want to analyze, and draw it.
  2. Identify all the forces and couples that are applied onto the body and draw them on the body. Place each force and couple at the point that it is applied.
  3. Label all the forces and couples with unique labels for use during the solution process.
  4. Add any relevant dimensions onto your picture.

Equilibrium: The concept of equilibrium is introduced to describe a body which is stationary or which is moving with a constant velocity. In statics, the concept of equilibrium is usually used in the analysis of a body which is stationary, or is said to be in the state of static equilibrium.

Particles: A particle is a body whose size does not have any effect on the results of mechanical analyses on it and, therefore, its dimensions can be neglected.

Rigid body: A body is formed by a group of particles. The size of a body affects the results of any mechanical analysis on it. A body is said to be rigid when the relative positions of its particles are always fixed and do not change when the body is acted upon by any load (whether a force or a couple).

Force System:

  • When a member of forces simultaneously acting on the body, it is known as force system. A force system is a collection of forces acting at specified locations. Thus, the set of forces can be shown on any free body diagram makes-up a force system.

Types of system of forces

  • Collinear forces :
    • In this system, line of action of forces act along the same line is called collinear forces. For example consider a rope is being pulled by two players as shown in figure.
  • Coplanar forces
    • When all forces acting on the body are in the same plane the forces are coplanar
  • Coplanar Concurrent force system
    • A concurrent force system contains forces whose lines of action meet at same one point. Forces may be tensile (pulling) or Forces may be compressive (pushing)

  • Non-Concurrent Co-Planar Forces
    • A system of forces acting on the same plane but whose line of action does not pass through the same point is known as non concurrent coplanar forces or system, for example, a ladder resting against a wall and a man is standing on the rung but not on the center of gravity. 
  • Coplanar parallel forces
    • When the forces acting on the body are in the same plane but their line of actions are parallel to each other known as coplanar parallel forces for example forces acting on the beams and two boys are sitting on the sea saw.
  • Non-coplanar parallel forces
    • In this case all the forces are parallel to each other but not in the same plane, for example the force acting on the table when a book is kept on it.

ADDITION OF FORCES

  • ADDITION OF (FORCES) BY HEAD TO TAIL RULE
    • To add two or more than two vectors (forces), join the head of the first vector with the tail of the second vector, and join the head of the second vector with the tail of the third vector and so on.
    • Then the resultant vector is obtained by joining the tail of the first vector with the head of the last vector. The magnitude and the direction of the resultant vector (Force) are found graphically and analytically.
  • RESULTANT FORCE
    • A resultant force is a single force, which produces same effect so that of number of forces can produce is called resultant force

COMPOSITION OF FORCES

  • The process of finding out the resultant Force of given forces (components vector) is called composition of forces. A resultant force may be determined by following methods 
    • PARALLELOGRAM METHOD
      • According to parallelogram method ‘If two forces (vectors) are acting simultaneously on a particle be represented (in magnitude and direction) by two adjacent sides of a parallelogram, their resultant may represent (in magnitude and direction) by the diagonal of the parallelogram passing through the point.
      • The magnitude and the direction of the resultant can be determined either graphically or analytically as explained below.
      • Graphical method Let us suppose that two forces F1 and F2 acting simultaneously on a particle as shown in the figure (a) the force F2 makes an angle θ with force F1

  • First of all we will draw a side OA of the parallelogram in magnitude and direction equal to force F1 with some suitable scale. Similarly draw the side OB of parallelogram of same scale equal to force F2, which makes an angle θ with force F1. Now draw sides BC and AC parallel to the sides OA and BC. Connect the point O to Point C which is the diagonal of the parallelogram passes through the same point O and hence it is the resultant of the given two forces. By measurement the length of diagonal gives the magnitude of resultant and angle α gives the direction of the resultant as shown in fig (A)

Analytical method

  • In the paralleogram OABC, from point C drop a perpendicular CD to meet OA at D as shown in fig (B)
  • In parallelogram OABC, OA = F1 OB = F2 Angle AOB = θ
  • Now consider the ∆CAD in which Angle CAD = θ AC = F2
  • By resolving the vector F2 we have, ,CD = F2 Sin θ and AD = F2 Cosine θ
  • Now consider ∆OCD ,Angle DOC = α. Angle ODC = 90º

According to Pythagoras theorem ,(Hyp) ² = (per) ² + (base) ²

  • OC² = DC² + OD², OC² = DC² + (OA + AD) ²
  • FR ² = F² Sin²θ + (F1 + F2 Cosine θ) ²
  • FR ² = F²2 Sin²θ + F²1 + F²2 Cos²θ + 2 F1 F2 Cosine θ.
  • FR ² = F²2 Sin²θ + F²2 Cos²θ +F²1 + 2 F1 F2 Cosine θ.
  • FR ² = F²2 (Sin²θ + Cos²θ) + F²1+ 2 F1 F2 Cosine θ.
  • FR ² = F²2 (1) + F²1+ 2 F1 F2 Cosine θ.
  • FR ² = F²2 + F²1+ 2 F1 F2 Cosine θ.
  • FR ² = F²1+F²2 + 2 F1 F2 Cosine θ
  • FR =√ F²1+F²2 + 2 F1 F2 Cosine θ.

TRIANGLE METHOD OR TRIANGLE LAW OF FORCES

  • According to triangle law or method” If two forces acting simultaneously on a particle by represented (in magnitude and direction) by the two sides of a triangle taken in order their resultant is represented (in magnitude and direction) by the third side of the triangle taken in opposite order. OR If two forces are acting on a body such that they can be represented by the two adjacent sides of a triangle taken in the same order, then their resultant will be equal to the third side (enclosing side) of that triangle taken in the opposite order. The resultant force (vector) can be obtained graphically and analytically or trigonometry.
  • Graphically, Now draw lines OA and AB to some convenient scale in magnitude equal to F1 and F2.
  • Join point O to point B the line OB will be the third side of triangle, passes through the same point O and hence it is the resultant of the given two forces.
  • By measuring the length of OB gives the magnitude of resultant and angle α gives the direction of the resultant as shown in fig (B).


 

  • Now draw a line OC to represent the vector in magnitude, which makes an angle θ with x-axis with some convenient scale.
  • Drop a perpendicular CD at point C which meet x axis at point D, now join point O to point D, the line OD is called horizontal component of resultant vector and represents by Fx in magnitude in same scale.
  • Similarly draw perpendicular CE at point C, which will meet y-axis at point E now join O to E. The line OE is called vertical component of resultant vector and represents by Fy in magnitude of same scale

Analytically or trigonometry

In ∆COD, Angle COD = θ , Angle ODC = 90°

  • OC = F
  • OD = Fx
  • OE = CD = Fy
  • We know that Cosine θ = OD/OC. Cosine θ = Fx/F And Fx = F Cosine θ
  • Similarly we have Sin θ = DC /OC, Sin θ = Fy /F And Fy = F Sine θ

Equilibrium Equations for a rigid body:

A rigid acted upon by any applied load will tend to translate dan rotate about a particular axis. The tendency to translate is due to the action of the resultant force on the body and the tendency to rotate is due to the action of the resultant couple.

Equilibrium will occur on the body if the resultant force, as well as the resultant couple, are both zero.

  • For equilibrium, the sum of all forces acting on the body is zero.

Resultant Force = ∑F = 0

  • The sum of the moment about any axis must be zero.

Resultant Moment = ∑M = 0

Equilibrium Equations in 2D:

The resultant force vector for a planar force system acts on the plane of action of the original force system.

∑F = ∑Fx + ∑Fy

The resultant moment vector acts perpendicular to that plane.

∑MO = ∑|r × F| = ∑Fd

where d is the perpendicular distance between any moment centre O and the line of action of F.

The equilibrium equations for the two-dimensional case can be written in the scalar form as follows:

∑Fx = 0

∑Fy = 0

∑MO = 0

 

free-body-diagrams-and-equilibrium

 

 

 

Law of Motion

Law 1:

Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it. Projectiles continue in their motions, so far as they are not retarded by the resistance of air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are continually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by air. The greater bodies of planets and comets, meeting with less resistance in freer spaces, preserve their motions both progressive and circular for a much longer time.

Law 2:

The change of motion is proportional to the motive force impressed, and is made in the direction of the right line in which that force is impressed.
If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force is impressed altogether and at once or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

Law 3:

To every action there is always opposed an equal reaction: or, the mutual action of two bodies upon each other are always equal, and directed to contrary parts.
Whatever draws or presses another is as much drawn or pressed by that other. If you press on a stone with your finger, the finger is also pressed by the stone. The Third Law, commonly known as the “action-reaction” law, is the most surprising of the three laws. Newton’s great discovery was that when two objects interact, they each exert the same magnitude of force on each other. We shall refer to objects that
interact as an interaction pair.

Friction Force

Frictional forces also exist when there is a thin film of liquid between two surfaces or within a liquid itself. This is known as the viscous force. We will not be talking about such forces and will focus our attention on Coulomb friction i.e., frictional forces between two dry surfaces only. Frictional force always opposes the motion or tendency of an object to move against another object or against a surface. We distinguish between two kinds of frictional forces - static and kinetic - because it is observed that kinetic frictional force is slightly less than maximum static frictional force.

     

The block does not move until the applied force reaches a maximum value Fmax. Thus from F = 0 up to F = Fmax, the frictional force adjusts itself so that it is just sufficient to stop the motion. It was observed by Coulombs that F max is proportional to the normal reaction of the surface on the object. You can observe all this while trying to push a table across the room; heavier the table, larger the push required to move it. Thus we can write

where µs is known as the coefficient of static friction. It should be emphasize again that is the maximum possible value of frictional force, applicable when the object is about to stop, otherwise frictional force could be less than, just sufficient to prevent motion. We also note that frictional force is independent of the area of contact and depends only on .

As the applied force goes beyond , the body starts moving now experience slightly less force compound to. This force is seem to be when is known as the coefficient of kinetic friction. At low velocities it is a constant but decrease slightly at high velocities. A schematic plot of frictional force as a function of the applied force is as shown in figure 2.

Values of frictional coefficients for different materials vary from almost zero (ice on ice) to as large as 0.9 (rubber tire on cemented road) always remaining less than 1.

A quick way of estimating the value of static friction is to look at the motion an object on an inclined plane. Its free-body diagram is given in figure 3.

1

Since the block has a tendency to slide down, the frictional force points up the inclined plane. As long as the block is in equilibrium

3

As θ is increased, mgsinθ increases and when it goes past the maximum possible value of friction fmax the block starts sliding down. Thus at the angle at which it slides down we have,

2

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Ashutosh YadavAshutosh YadavMember since Jan 2020
ESE, GATE, SSC-JE and other Civil Engineering exams.
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