Fundamentals of Engineering Mechanics PDF

# Fundamentals of Engineering Mechanics ## Engineering Mechanics The subject of Engineering Mechanics is the branch of Applied Science that deals with the laws and principles of Mechanics, along with their applications to engineering problems. The subject of Engineering Mechanics is divided into t...

# Fundamentals of Engineering Mechanics ## Engineering Mechanics The subject of Engineering Mechanics is the branch of Applied Science that deals with the laws and principles of Mechanics, along with their applications to engineering problems. The subject of Engineering Mechanics is divided into the following two main groups: 1. Statics 2. Dynamics ## Statics This is the branch of Engineering Mechanics that deals with the forces and their effects while acting upon the bodies at rest. ## Dynamics This is the branch of Engineering Mechanics that deals with the forces and their effects while acting upon the bodies in motion. The subject of Dynamics may be further subdivided into the following two branches: 1. Kinetics 2. Kinematics ## Kinematics This branch of Dynamics deals with the bodies in motion, without any reference to the forces which are responsible for the motion. ## Kinetics This branch of Dynamics deals with the bodies in motion due to the application of forces. ## Rigid Body A rigid body is a solid body in which deformation is so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external forces exerted on it. A rigid body is usually considered as a continuous distribution of mass. ## Force This is defined as an agent which produces, destroys or tends to destroy motion. For example, a horse applies force to pull a cart and to set it in motion. Force is also required to work on a bicycle pump; in this case, the force is supplied by the muscular power of our arms and shoulders. ## System of Forces When two or more forces act on a body, they are called to form a system of forces. The following systems of forces are important from the subject point of view: 1. **Coplanar forces:** The forces whose lines of action lie on the same plane. 2. **Collinear forces:** The forces whose lines of action lie on the same line. 3. **Concurrent forces:** The forces which meet at one point. These forces may or may not be collinear. 4. **Coplanar concurrent forces:** The forces which meet at one point and their lines of action also lie on the same plane. 5. **Coplanar non-concurrent forces:** The forces which do not meet at one point, but their lines of action lie on the same plane. 6. **Non-coplanar concurrent forces:** The forces which meet at one point, but their lines of action do not lie on the same plane. 7. **Non-coplanar non-concurrent forces:** The forces which do not meet at one point and their lines of action do not lie on the same plane. ## Characteristic of a Force We must know the following characteristics of a force to determine the effects of a force, acting on a body: 1. **Magnitude of the force** 2. **The direction of the line along which the force acts** 3. **Nature of the force** ## Effects of a Force A force may produce the following effects in a body, on which it acts: 1. It may change the motion of a body. 2. It may retard the motion of a body. 3. It may retard the forces already acting on a body, thus bringing it to rest or in equilibrium. 4. It may give rise to the internal stresses in the body, on which it acts. ## Principle of Transmissibility If a force acts at any point on a rigid body, it may also be considered to act at any other point on its line of action, provided this point is rigidly connected with the body. ## Principle of Superposition This principle states that the combined effect of force system acting on a particle or a rigid body is the sum of effects of individual forces. ## Action and Reaction Force Forces always act in pairs and always act in opposite directions. When you push on an object, the object pushes back with an equal force. Think of a pile of books on a table. The weight of the books exerts a downward force on the table. This is the action force. The table exerts an equal upward force on the books. This is the reaction force. ## Free Body Diagram A free body diagram is a graphical illustration used to visualize the applied forces, moments, and resulting reactions on a body in a given condition. They depict a body or connected bodies with all the applied forces and moments, and reactions, which act on the body. The body may consist of multiple internal members (such as a truss), or be a compact body (such as a beam). A series of free bodies and other diagrams may be necessary to solve complex problems. ## Resolution of a Force The process of splitting up the given force into a number of components, without changing its effect on the body is called resolution of a force. A force is, generally, resolved along two mutually perpendicular directions. In fact, the resolution of a force is the reverse action of the addition of the component vectors. ## Principle of Resolution It states, "The algebraic sum of the resolved parts of a no. of forces, in a given direction, is equal to the resolved part of their resultant in the same direction." ## Method of Resolution 1. Resolve all the forces horizontally and find the algebraic sum of all the horizontal components. 2. Resolve all the forces vertically and find the algebraic sum of all the vertical components. 3. The resultant R of the given forces will be given by the equation; $R = \sqrt{(\sum H)^2 + (\sum V)^2}$ 4. The resultant force will be inclined at an angle, with the horizontal, such that $tan \theta = \frac{\sum V}{\sum H}$ ## Composition of Forces The process of finding out the resultant force, of a number of given forces, is called composition of forces or compounding of forces. ## Resultant Force If a number of forces are acting simultaneously on a particle, then it is possible to find out a single force which could replace them i.e., which would produce the same effect as produced by all the given forces. This single force is called resultant force and the given forces are called component forces. ## Methods for the Resultant Force Though there are many methods for finding out the resultant force of a number of given forces, yet the following are important from the subject point of view: 1. Analytical method 2. Method of resolution ## Analytical Method for Resultant Force The resultant force, of a given system of forces, may be found out analytically by the following methods : 1. Parallelogram law of forces. 2. Method of resolution. ## Parallelogram Law of Forces It states, "If two forces acting simultaneously on a particle, be represented in magnitude and direction by the two adjacent sides of a parallelogram, their resultant may be represented in magnitude and direction by the diagonal of the parallelogram which passes through their point of intersection." Mathematically, resultant force, $R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 cos \theta}$ and $tan \alpha = \dfrac{F_2 sin \theta}{F_1 + F_2 cos \theta}$ where * $F_1$ and $F_2$ = forces whose resultant is required to be found out, * $\theta$ = angle between the forces $F_1$ and $F_2$, and * $\alpha$ = angle which the resultant force makes with one of the forces (say $F_1$). ## Moment of a Force It is the turning effect produced by a force, on the body, on which it acts. The moment of a force is equal to the product of the force and the perpendicular distance of the point, about which the moment is required and the line of action of the force. Mathematically, moment, $M=PxI$ where * P = Force acting on the body, and * I = Perpendicular distance between the point, about which the moment is required and the line of action of the force. ## Graphical Representation of a Moment Consider a force P represented, in magnitude and direction, by the line AB. Let O be a point, about which the moment of this force is required to be found out, as shown in Fig. From O, draw OC perpendicular to AB. Join OA and OB: Now moment of the force P about O = P x OC = AB × OC But AB × OC is equal to twice the area of triangle ABO. Thus the moment of a force, about any point, is equal to twice the area of the triangle, whose base is the line to some scale representing the force and whose vertex is the point about which the moment is taken. ## Units of Moment Since the moment of a force is the product of force and distance, therefore the units of the moment will depend upon the units of force and distance. Thus, if the force is in Newton and the distance is in meters, then the units of moment will be Newton-meter (briefly written as N-m). Similarly, the units of moment may be kN-m (i.e. kN × m), N-mm (i.e. N x mm) etc. ## Types of Moments 1. **Clockwise moments.** 2. **Anticlockwise moments.** ## Clockwise Moment It is the moment of a force, whose effect is to turn or rotate the body, about the point in the same direction in which hands of a clock move as shown in Fig. ## Anticlockwise Moment It is the moment of a force, whose effect is to turn or rotate the body, about the point in the opposite direction in which the hands of a clock move as shown in Fig. Note. The general convention is to take clockwise moment as positive and anticlockwise moment as negative. ## Varignon's Principle or Law of Moments It states, "If a number of coplanar forces are acting simultaneously on a particle, the algebraic sum of the moments of all the forces about any point is equal to the moment of their resultant force about the same point." ## Laws for the Resultant Force The resultant force, of a given system of forces, may also be found out by the following laws 1. Triangle law of forces 2. Polygon law of forces ## Triangle Law of Forces It states, "If two forces acting simultaneously on a particle, be represented in magnitude and direction by the two sides of a triangle, taken in order; their resultant may be represented in magnitude and direction by the third side of the triangle, taken in opposite order." ## Polygon Law of Forces It is an extension of Triangle Law of Forces for more than two forces, which states, "If a number of forces acting simultaneously on a particle, be represented in magnitude and direction, by the sides of a polygon taken in order; then the resultant of all these forces may be represented, in magnitude and direction, by the closing side of the polygon, taken in opposite order." ## Graphical (Vector) Method for the Resultant Force It is another name for finding out the magnitude and direction of the resultant force by the polygon law of forces. It is done as discussed below: **Construction of space diagram (position diagram):** It means the construction of a diagram showing the various forces (or loads) along with their magnitude and lines of action. **Use of Bow's notations:** All the forces in the space diagram are named by using the Bow's notations. It is a convenient method in which every force (or load) is named by two capital letters, placed on its either side in the space diagram. **Construction of vector diagram (force diagram):** It means the construction of a diagram starting from a convenient point and then go on adding all the forces vectorially one by one (keeping in view the directions of the forces) to some suitable scale. Now the closing side of the polygon, taken in opposite order, will give the magnitude of the resultant force (to the scale) and its direction. ## Couple A pair of two equal and unlike parallel forces (i.e.forces equal in magnitude, with lines of action parallel to each other and acting in opposite directions) is known as a couple. As a matter of fact, a couple is unable to produce any translatory motion (i.e., motion in a straight line). But it produces a motion of rotation in the body, on which it acts. The simplest example of a couple is the forces applied to the key of a lock, while locking or unlocking it. ## Arm of a Couple The perpendicular distance (a), between the lines of action of the two equal and opposite parallel forces, is known as arm of the couple as shown in Fig. ## Moment of a Couple The moment of a couple is the product of the force (i.e., one of the forces of the two equal and opposite parallel forces) and the arm of the couple. Mathematically: Moment of a couple = P x a where * P = Magnitude of the force, and * a = Arm of the couple. ## Classification of Couples The couples may be, broadly, classified into the following two categories, depending upon their direction, in which the couple tends to rotate the body, on which it acts: 1. Clockwise couple 2. Anticlockwise couple ## Clockwise Couple A couple, whose tendency is to rotate the body, on which it acts, in a clockwise direction, is known as a clockwise couple as shown in Fig. (a). Such a couple is also called positive couple. ## Anticlockwise Couple A couple, whose tendency is to rotate the body, on which it acts, in an anticlockwise direction, is known as an anticlockwise couple as shown in Fig. (b). Such a couple is also called a negative couple. ## Characteristics of a Couple A couple (whether clockwise or anticlockwise) has the following characteristics: 1. The algebraic sum of the forces, constituting the couple, is zero. 2. The algebraic sum of the moments of the forces, constituting the couple, about any point is the same, and equal to the moment of the couple itself. 3. A couple cannot be balanced by a single force. But it can be balanced only by a couple of opposite sense. 4. Any no. of co-planer couples can be reduced to a single couple, whose magnitude will be equal to the algebraic sum of the moments of all the couples.