# Mechanics: Statics and Dynamics

Oct 08, 1999
1607 Words

Mechanics: Statics and Dynamics

TABLE OF CONTENTS

INTRODUCTION.........................................................1

Chapter

I. General Principles........................................2 I. Systems of Force.........................................4 II. Stress..................................................6 III. Properties of Material.................................7 IV. Bolted and Welded Joints................................10 V. Beams -- A Practical Application.........................13 VI. Beam Design.............................................17 VII. Torsional Loading: Shafts, Couplings, and Keys........19 VIII. Conclusion............................................20

BIBLIOGRAPHY.........................................................21

INTRODUCTION

Mechanics is the physical science concerned with the dynamic behavior of bodies that are acted on by mechanical disturbances. Since such behavior is involved in virtually all the situations that confront an engineer, mechanics lie at the core of much engineering analysis. In fact, no physical science plays a greater role in engineering than does mechanics, and it is the oldest of all physical sciences. The writings of Archimedes covering buoyancy and the

lever were recorded before 200 B.C. Our modern knowledge of gravity and motion was established by Isaac Newton (1642-1727).

Mechanics can be divided into two parts: (1) Statics, which relate to bodies at rest, and (2) dynamics, which deal with bodies in motion. In this paper we will explore the static dimension of mechanics and discuss the various types of force on an object and the different strength of materials.

The term strength of materials refers to the ability of the individual parts of a machine or structure to resist loads. It also permits the selection of materials and the determination of dimensions to ensure the sufficient strength of the various parts.

General Principles

Before we can venture to explain statics, one must have a firm grasp on classical mechanics. This is the study of Newton's laws and their extensions. Newton's three laws were originally stated as follows:

1. Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed on it.

2. The change of motion is proportional to the motive force impressed and is made in the direction in which that force is impressed.

3. To every action there is always opposed an equal reaction; or the mutual actions of two bodies on each other are equal and direct to contrary parts.

Newton's law of gravitational attraction pertains to celestrial bodies or any object onto which gravity is a force and states: "Two particles will be attracted toward each other along their connecting line with a force whose magnitude is directly proportional to the product of the masses and inversely proportional to the distance squared between the particles.

When one of the two objects is the earth and the other object is near the surface of the earth (where r is about 6400 km) / is essentially constant, then the attraction law becomes f = mg.

Another essential law to consider is the Parallelogram Law. Stevinius (1548-1620) was the first to demonstrate that forces could be combined by representing them by arrows to some suitable scale, and then forming a parallelogram in which the diagonal represents the sum of the two forces. All vectors must combine in this manner.

When solving static problems as represented as a triangle of force, three common theorems are as follows:

1. Pythagorean theorem. In any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs

2. Law of sines. In any triangle, the sides are to each other as the sines of the opposite angle

3. Law of cosines. In any triangle, the square of any side is equal to the sum of the squares of the other two sides minus twice the product of the sides and the cosine of their included angle

By possessing an understanding of Newton's Laws, following these three laws of graphical solutions, and understanding vector algebra you can solve most engineering static problems.

Systems of Force

Systems of force acting on objects in equilibrium can be classified as either concurrent or nonconcurrent and as either coplanar or noncoplanar. This gives us four general categories of systems.

The first category, concurrent-coplanar forces occur when the lines of action of all forces lie in the same plane and pass through a common point. Figure 1 illustrates a concurrent-coplanar force in such that F1, F2, and W all lie in the same plane (the paper) and all their lines of action have point O in common. To determine the resultant of concurrent force systems, you can use the Pythagorean theorem, the law of sines, or the law of cosines as outlined in the previous chapter.

Nonconcurrent-coplanar force is when the lines of action of all forces lie in the same plane but do not pass through a common point as illustrated in figure 2. The magnitude and direction of the resultant force can be determined by the rectangular component method using the first two equations in figure 2, and the perpendicular distance of the line of action of R from the axis of rotation of the body can be found using the third equation in figure 2.

Concurrent-noncoplanar forces are when Application the lines of action of all forces pass through a common point and are not in the same plane. To find the resultant of these forces it is best to resolve each force into components along three axes that make angles of 90 degrees with each other.

Nonconcurrent-noncoplanar forces are when the lines of action of all forces do not pass through a common point and the forces do not all lie in the same plane.

Stress

When a restrained body is subject to external forces, there is a tendency for the shape of the body that is subject to the external force to be deformed or changed. Since materials are not perfectly rigid, the applied forces will cause the body to deform. The internal resistance to deformation of the fibers of a body is called stress. Stress can be classified as either simple stress, sometimes referred to as direct stress, or indirect stress.

The various types of direct stress are tension, compression, shear, and bearing. The various types of indirect stress are bending and torsion. A third variety of stress is categorized as any combination of direct and indirect stress.

Simple stress is developed under direct loading conditions. That is, simple tension and simple compression occur when the applied force is in line with the axis of the member and simple shear occurs when equal, parallel, and opposite forces tend to cause a surface to slide relative to the adjacent surface. When any type of simple stress develops we can calculate the magnitude of the stress by the formula, where:

· s = average unit stress;

· F = external force causing stress to develop; · A = area over which stress develops.

Indirect stress, or stress due to bending should be properly classified under statics of rigid bodies and not under strength of materials. The bending moment in a beam depends only on the loads on the beam and on its consequent support reactions. Torsion is when a shaft is acted upon by two equal and opposite twisting moments in parallel planes. Torsion can be either stationary or rotating uniformly. Indirect stress will be discussed in detail in later sections.

Properties of Material

In order for the engineer to effectively design any item, whether it is a frame which holds an object or a complicated piece of automated machinery, it is very important to have a strong knowledge of the mechanical and physical properties of metals, wood, concrete, plastics and composites, and any other material an engineer is considering using to construct an object. The rest of this paper will deal with strength of materials and how to best choose a material and construction technique to effectively accomplish what was set out without "over-engineering."

Strength of materials deals with the relationship between the external forces applied to elastic bodies and the resulting deformations and stresses. In the design of structures and machines, the application of the principles of strength of materials is necessary if satisfactory materials are to be utilized and adequate proportions obtained to resist functional forces.

In today's global economy is crucial for success to be able to build the "biggest and best" while spending the least. To do that successfully it is imperative to have a firm understanding of different materials and their correct uses. The load per unit area, called stress, and the deformation per unit length, called strain, must be understood. The formula for stress is:

The formula for strain is:

The amount of stress and strain a material can endure before deformation occurs is known as the proportional limit. Up to this point, any stress or strain induced into the material will allow the material to return to its original shape. When stress and strain exceed the proportional limit of the material and a permanent deformation, or set, occurs the object is said to have reached its elastic limit. Modulus of elasticity, also called Young's modulus, is the ratio of unit stress to unit strain within the proportional limit of a material in tension or compression. Some representatives values of Young's modulus (in 10^6 psi) are as follows:

· Aluminum, cast, pure...................................9 · Aluminum, wrought, 2014-T6............................10.6 · Beryllium copper......................................19 · Brass, naval..........................................15 · Titanium, alloy, 5 Al, 2.5 Sn.........................17 · Steel for buildings and bridges, ASTM A7-61T...29

Once the elastic limit of a material is reached, the material will elongate rather easily without a significant increase in the load. This is known as the yield point of the material. Not all materials have a yield point. Some repre

TABLE OF CONTENTS

INTRODUCTION.........................................................1

Chapter

I. General Principles........................................2 I. Systems of Force.........................................4 II. Stress..................................................6 III. Properties of Material.................................7 IV. Bolted and Welded Joints................................10 V. Beams -- A Practical Application.........................13 VI. Beam Design.............................................17 VII. Torsional Loading: Shafts, Couplings, and Keys........19 VIII. Conclusion............................................20

BIBLIOGRAPHY.........................................................21

INTRODUCTION

Mechanics is the physical science concerned with the dynamic behavior of bodies that are acted on by mechanical disturbances. Since such behavior is involved in virtually all the situations that confront an engineer, mechanics lie at the core of much engineering analysis. In fact, no physical science plays a greater role in engineering than does mechanics, and it is the oldest of all physical sciences. The writings of Archimedes covering buoyancy and the

lever were recorded before 200 B.C. Our modern knowledge of gravity and motion was established by Isaac Newton (1642-1727).

Mechanics can be divided into two parts: (1) Statics, which relate to bodies at rest, and (2) dynamics, which deal with bodies in motion. In this paper we will explore the static dimension of mechanics and discuss the various types of force on an object and the different strength of materials.

The term strength of materials refers to the ability of the individual parts of a machine or structure to resist loads. It also permits the selection of materials and the determination of dimensions to ensure the sufficient strength of the various parts.

General Principles

Before we can venture to explain statics, one must have a firm grasp on classical mechanics. This is the study of Newton's laws and their extensions. Newton's three laws were originally stated as follows:

1. Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed on it.

2. The change of motion is proportional to the motive force impressed and is made in the direction in which that force is impressed.

3. To every action there is always opposed an equal reaction; or the mutual actions of two bodies on each other are equal and direct to contrary parts.

Newton's law of gravitational attraction pertains to celestrial bodies or any object onto which gravity is a force and states: "Two particles will be attracted toward each other along their connecting line with a force whose magnitude is directly proportional to the product of the masses and inversely proportional to the distance squared between the particles.

When one of the two objects is the earth and the other object is near the surface of the earth (where r is about 6400 km) / is essentially constant, then the attraction law becomes f = mg.

Another essential law to consider is the Parallelogram Law. Stevinius (1548-1620) was the first to demonstrate that forces could be combined by representing them by arrows to some suitable scale, and then forming a parallelogram in which the diagonal represents the sum of the two forces. All vectors must combine in this manner.

When solving static problems as represented as a triangle of force, three common theorems are as follows:

1. Pythagorean theorem. In any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs

2. Law of sines. In any triangle, the sides are to each other as the sines of the opposite angle

3. Law of cosines. In any triangle, the square of any side is equal to the sum of the squares of the other two sides minus twice the product of the sides and the cosine of their included angle

By possessing an understanding of Newton's Laws, following these three laws of graphical solutions, and understanding vector algebra you can solve most engineering static problems.

Systems of Force

Systems of force acting on objects in equilibrium can be classified as either concurrent or nonconcurrent and as either coplanar or noncoplanar. This gives us four general categories of systems.

The first category, concurrent-coplanar forces occur when the lines of action of all forces lie in the same plane and pass through a common point. Figure 1 illustrates a concurrent-coplanar force in such that F1, F2, and W all lie in the same plane (the paper) and all their lines of action have point O in common. To determine the resultant of concurrent force systems, you can use the Pythagorean theorem, the law of sines, or the law of cosines as outlined in the previous chapter.

Nonconcurrent-coplanar force is when the lines of action of all forces lie in the same plane but do not pass through a common point as illustrated in figure 2. The magnitude and direction of the resultant force can be determined by the rectangular component method using the first two equations in figure 2, and the perpendicular distance of the line of action of R from the axis of rotation of the body can be found using the third equation in figure 2.

Concurrent-noncoplanar forces are when Application the lines of action of all forces pass through a common point and are not in the same plane. To find the resultant of these forces it is best to resolve each force into components along three axes that make angles of 90 degrees with each other.

Nonconcurrent-noncoplanar forces are when the lines of action of all forces do not pass through a common point and the forces do not all lie in the same plane.

Stress

When a restrained body is subject to external forces, there is a tendency for the shape of the body that is subject to the external force to be deformed or changed. Since materials are not perfectly rigid, the applied forces will cause the body to deform. The internal resistance to deformation of the fibers of a body is called stress. Stress can be classified as either simple stress, sometimes referred to as direct stress, or indirect stress.

The various types of direct stress are tension, compression, shear, and bearing. The various types of indirect stress are bending and torsion. A third variety of stress is categorized as any combination of direct and indirect stress.

Simple stress is developed under direct loading conditions. That is, simple tension and simple compression occur when the applied force is in line with the axis of the member and simple shear occurs when equal, parallel, and opposite forces tend to cause a surface to slide relative to the adjacent surface. When any type of simple stress develops we can calculate the magnitude of the stress by the formula, where:

· s = average unit stress;

· F = external force causing stress to develop; · A = area over which stress develops.

Indirect stress, or stress due to bending should be properly classified under statics of rigid bodies and not under strength of materials. The bending moment in a beam depends only on the loads on the beam and on its consequent support reactions. Torsion is when a shaft is acted upon by two equal and opposite twisting moments in parallel planes. Torsion can be either stationary or rotating uniformly. Indirect stress will be discussed in detail in later sections.

Properties of Material

In order for the engineer to effectively design any item, whether it is a frame which holds an object or a complicated piece of automated machinery, it is very important to have a strong knowledge of the mechanical and physical properties of metals, wood, concrete, plastics and composites, and any other material an engineer is considering using to construct an object. The rest of this paper will deal with strength of materials and how to best choose a material and construction technique to effectively accomplish what was set out without "over-engineering."

Strength of materials deals with the relationship between the external forces applied to elastic bodies and the resulting deformations and stresses. In the design of structures and machines, the application of the principles of strength of materials is necessary if satisfactory materials are to be utilized and adequate proportions obtained to resist functional forces.

In today's global economy is crucial for success to be able to build the "biggest and best" while spending the least. To do that successfully it is imperative to have a firm understanding of different materials and their correct uses. The load per unit area, called stress, and the deformation per unit length, called strain, must be understood. The formula for stress is:

The formula for strain is:

The amount of stress and strain a material can endure before deformation occurs is known as the proportional limit. Up to this point, any stress or strain induced into the material will allow the material to return to its original shape. When stress and strain exceed the proportional limit of the material and a permanent deformation, or set, occurs the object is said to have reached its elastic limit. Modulus of elasticity, also called Young's modulus, is the ratio of unit stress to unit strain within the proportional limit of a material in tension or compression. Some representatives values of Young's modulus (in 10^6 psi) are as follows:

· Aluminum, cast, pure...................................9 · Aluminum, wrought, 2014-T6............................10.6 · Beryllium copper......................................19 · Brass, naval..........................................15 · Titanium, alloy, 5 Al, 2.5 Sn.........................17 · Steel for buildings and bridges, ASTM A7-61T...29

Once the elastic limit of a material is reached, the material will elongate rather easily without a significant increase in the load. This is known as the yield point of the material. Not all materials have a yield point. Some repre