Summary: Rigid Body | Engineering Mechanics - Civil Engineering (CE) PDF Download

Mechanics

Mechanics is the branch of physics concerned with the state of rest or motion of bodies subjected to the action of forces. It explains how and why bodies move or remain at rest when acted upon by forces.

Many scientists contributed to the development of mechanics: Archimedes (287-212 BC), Galileo (1564-1642), Sir Isaac Newton (1642-1727), Einstein (1878-1955), and others including Varignon, Euler and D'Alembert.

Mechanics may be grouped broadly as:

• Classical mechanics / Newtonian mechanics
• Relativistic mechanics
• Quantum mechanics / Wave mechanics

Sir Isaac Newton consolidated laws and experimental findings on motion and rest and stated the three laws of motion and the law of universal gravitation; mechanics based on these is called Classical (Newtonian) mechanics.

Albert Einstein showed that Newtonian mechanics does not correctly describe bodies moving at speeds near the speed of light and proposed Relativistic mechanics.

Schrödinger and de Broglie showed that Newtonian mechanics fails at atomic scales and proposed Quantum mechanics.

Classifications of Classical Mechanics

Rigid Body

Definition: A rigid body is an idealised body in which the relative positions of any two particles remain unchanged under the action of forces. Equivalently, the distance between any two points in a rigid body remains the same before and after applying external forces.

In practice, physical bodies deform slightly when loaded. If that deformation is negligible compared to the body's size for the problem under consideration, the body is treated as rigid.

Branches of Rigid Body Mechanics

• Statics - studies forces acting on bodies in equilibrium (bodies at rest or moving uniformly).
• Dynamics - studies bodies in motion under action of forces; dynamics is further subdivided into kinematics (description of motion without regard to forces) and kinetics (relation between motion and the forces causing it).

Types of Rigid Body Motion

• Translation - every particle of the body moves along parallel paths; no change of orientation.
• Rotation about a fixed axis - every particle moves in a circle about the fixed axis; distances from the axis remain constant.
• General plane motion - combination of translation and rotation in a plane (e.g., body moving while also rotating).

Force

Force is any action that tends to change the state of rest or motion of a body. A force is a vector quantity; its SI unit is the Newton (N).

Three quantities required to fully specify a force are:

• Magnitude
• Point of application
• Direction / Line of action

Types of Quantities

• Scalar - quantity with magnitude only. Examples: time, volume, density, speed, energy, mass.
• Vector - quantity with magnitude and direction, obeys vector addition (parallelogram law). Examples: displacement, velocity, acceleration, force, moment, momentum.

Effects of a Force on a Body

• A force may change the state of motion of a body (cause translation).
• A force may deform the body (internal effect producing strain and stress).
• A force may induce rotation when applied at a point not coincident with the body's centre of mass (creates a moment or couple).

Line of Action of a Force

The line of action of a force is the straight line passing through the point of application in the direction of the force; it determines the direction along which the force tends to move the body.

Graphical Representation of Force

A force may be represented graphically by a directed line segment whose length is proportional to the magnitude and whose orientation gives the direction. The point where the segment meets the body is the point of application.

Laws of Mechanics

The fundamental laws used in rigid body mechanics include:

• Newton's laws of motion
• Newton's law of universal gravitation
• Law of transmissibility of forces
• Parallelogram law of forces

Newton's Laws of Motion

First law (Law of Inertia): A particle remains at rest or continues to move with uniform velocity unless acted upon by an unbalanced external force.

Second law: The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction of the applied force.

For a constant-mass body this gives the familiar form: F = ma, where F is the vector sum of forces, m is mass and a is acceleration.

Third law: To every action there is an equal and opposite reaction; the mutual forces of action and reaction between two bodies are equal in magnitude, opposite in direction and collinear.

Newton's Law of Universal Gravitation

Any two point masses attract each other with a force directed along the line joining them. The magnitude of the force is directly proportional to the product of the masses and inversely proportional to the square of the distance between them:

F = G m1 m2 / r², where G is the gravitational constant, m1 and m2 are the masses, and r is the separation.

Triangular and Polygon Laws of Forces

Triangular law of forces: If two forces acting on a body are represented in magnitude and direction by two sides of a triangle taken in order, then the third side (closing side) of the triangle taken in the opposite order represents the resultant in magnitude and direction.

Polygon law of forces: If several concurrent forces acting on a body are represented in magnitude and direction by the sides of a polygon taken in order, then their resultant is represented by the closing side of the polygon taken in the reverse order.

Law of Transmissibility of Force

The principle of transmissibility states that the external effect of a force on a rigid body is the same if the force is applied at any other point on the same line of action. In other words, a force can be moved along its line of action without changing the external effect on a rigid body.

Note: This principle does not hold if deformation of the body is significant for the problem.

Examples Fig : Rigid Bodies (valid) Fig : Deformable Bodies (not valid) Note : In engineering mechanics we deal with rigid bodies. If deformation is considered in a problem, the law of transmissibility will not be valid.

Parallelogram Law of Forces

If two forces acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through the point.

Analytical proof (outline)

Given OA = P and OB = Q represented as vector segments. Construct parallelogram OBCA. Drop perpendicular CD on extension of OA.

From the geometry,

AD = Q cos θ

CD = Q sin θ

Using right triangle relationships, the magnitude of resultant R is found by combining components of P and Q and gives the same result as the diagonal of the parallelogram.

Note : The parallelogram law is valid for all vectors and thus for forces.

Force Systems

A force system is when several forces of different magnitudes and directions act on a body or particle. The resultant and the net effect of the system depend on their magnitudes, directions and points of application.

Types of Force Systems

1. Collinear Force System

Collinear forces act along the same straight line.

2. Coplanar and Non-coplanar Force Systems

Coplanar force system: All forces lie in the same plane.

Non-coplanar force system: Forces do not lie in a single common plane.

3. Concurrent and Non-concurrent Force Systems

Concurrent forces are forces whose lines of action meet at a common point. They may be coplanar or non-coplanar.

Non-concurrent forces have lines of action that do not all meet at a single point.

4. Parallel Force Systems

Forces with lines of action parallel to one another form a parallel force system. They are classified as:

• Like parallel forces - forces parallel and acting in the same direction.
• Unlike parallel forces - forces parallel but acting in opposite directions.

Resultants and Equilibrium of Rigid Bodies

To analyse a rigid body under several forces, we find the resultant force and the resultant moment. For a rigid body to be in equilibrium, the following conditions must be satisfied:

• The resultant of all forces must be zero. In a plane this gives ΣFx = 0 and ΣFy = 0.
• The sum of moments about any point must be zero: ΣM = 0.

For three-dimensional problems the vector form of equilibrium is used: ΣF = 0 and ΣM = 0 (vector equations).

Moments and Couples

Moment (or torque) of a force about a point is the tendency of the force to cause rotation about that point. For a force F with position vector r, the moment is the vector cross-product M = r × F.

A couple consists of two equal, parallel and opposite forces whose lines of action do not coincide; a couple produces pure rotation and its moment is free vector (independent of the reference point).

Centre of Mass and Centroid

Centre of mass of a rigid body is the point at which the entire mass may be considered to be concentrated for the purpose of translational motion analysis. For discrete masses, the coordinates of the centre of mass are given by the weighted average of particle positions. For continuous bodies, integrals over the mass distribution are used.

Centroid is the geometric centre of an area or volume and is used when the area or volume has uniform density; for uniform thickness bodies centroid and centre of mass coincide.

Moment of Inertia (Second Moment of Area and Mass)

Mass moment of inertia (for dynamics) is a scalar or tensor quantity that measures a body's resistance to angular acceleration about an axis. For a rigid body rotating about a fixed axis, the kinetic energy and dynamics depend on the mass moment of inertia.

Second moment of area (often called area moment of inertia) is used in strength of materials to relate bending stress and deflection of beams.

Free Body Diagram (FBD)

To apply equilibrium equations, first draw a free body diagram: isolate the body or part, show all external forces and moments, including reactions at supports, weights, and applied loads. Indicate coordinate axes and the points about which moments are taken.

Simple Worked Example (qualitative)

Consider a beam supported at two ends with a central load. The procedure to find support reactions is:

• Draw the free body diagram showing the beam, support reactions at the ends and the applied central load.
• Apply equilibrium equations: ΣFy = 0 to relate vertical reactions to the applied load.
• Apply moment equilibrium about one support to find the other reaction.
• Check all equilibrium equations are satisfied.

Applications of Rigid Body Mechanics

• Analysis and design of structures: beams, trusses, frames.
• Machine design: shafts, gears, linkages.
• Stability and support reactions for buildings and bridges.
• Dynamics of rigid bodies in vehicles, robotics and mechanisms.

Final summary: Rigid body mechanics provides the foundations to analyse forces, moments and equilibrium for bodies idealised as non-deformable. Statics determines load distributions and reactions under equilibrium; dynamics links forces to motion. Understanding force systems, resultants, moments, free body diagrams, and equilibrium equations is essential for engineering analysis and design.