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

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 = ∑F_x + ∑F_y

The resultant moment vector acts perpendicular to that plane.

∑M_O = ∑|r × F| = ∑F_d

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:

∑F_x = 0

∑F_y = 0

∑M_O = 0