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Strength of materials - Erik Oberg
STRENGTH OF MATERIALS
PRINCIPLES OF THE THEORY METHODS FOR CALCULATING STRENGTH AND DETERMINING DIMENSIONS OF MACHINE PARTS
BY ERIK OBERG
MACHINERY'S REFERENCE BOOK NO. 136
PUBLISHED BY MACHINERY, NEW YORK
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CONTENTS
- Principles of the Strength of Materials
- Tension, Compression and Shearing Stresses
- Beams and Bending Stresses
- Torsional Stresses Shafting and Springs
- Miscellaneous Applications
PREFACE
In the design of machinery, there is nothing more important than to be able to determine the stresses to which a machine member is subjected and to give it the adequate strength for the purpose for which it is intended. It is the object of this book to give in as simple a manner as possible the principles of the methods used in calculating the strength of parts met with in machine design and engineering work generally, and to give the formulas used in such a shape that they can be directly employed by the practical man. No attempt has been made to show how the formulas are derived mathematically, as this would be impossible in a book of limited size. However, the book will be all the more acceptable to the great number of practical men, because it avoids involved mathematical treatment and presents every formula in its simplest shape. Numerous examples have also been given to show the application of the rules and formulas. While thus the subject is treated in as simple a manner as possible, the book is still comprehensive and covers all the more important questions connected with the subject.
A special effort has been made to indicate the use of standard engineering handbooks in connection with the calculations of strength of materials, and numerous references are made to tables and formulas that are to be found in works of that kind.
A special effort has been made to indicate the use of standard engineering handbooks in connection with the calculations of strength of materials, and numerous references are made to tables and formulas that are to be found in works of that kind.
PRINCIPLES OF THE STRENGTH OF MATERIALS
It may be said, in a general way, that when designing machinery the designer must take into consideration two main factors. One is to so design the mechanism mechanically that the various motions required can be obtained by means of the machine. The other is to so proportion the parts that they will be strong enough to do the work for which they are intended without breakage and, in most cases, without distortion. A third factor also enters prominently into the design, that of so designing a machine that the various parts can be easily manufactured, but this last factor, while commercially fully as important as the other two, has not a direct bearing upon the actual working of the completed mechanism.
The subject that will be dealt with in this Reference Book relates to the second of the two main questions affecting the design, that of proportioning the parts so that they will be strong enough to properly do the work for which they are intended. This problem, in turn, may be divided into two sections, one of which deals with the determining of the forces which act upon a machine part, tending to break or distort it, and the determination of the actual proportions necessary to resist these forces. The science of determining the forces acting upon a machine part or, in general, upon an engineering structure, is termed mechanics, this word being used in its more limited sense, often referred to as theoretical mechanics. The problem of actually determining the dimensions of details of machines or structures with regard to their strength is covered by that part of mechanical knowledge which is known as the strength of materials. It should be understood, however, that the question of determining the forces acting upon an engineering structure or machine and the determination of the actual dimensions and materials required to resist these forces, are so closely connected that in dealing with the strength of materials we must also deal, to a very large extent, with the mechanical theory of forces.
When the forces acting upon a machine part are definitely known, it is, as a rule, comparatively easy to determine the actual proportions required to resist the action of these forces. There are, however, a number of instances in which the forces are applied in such a manner that it has so far proved impossible for the mathematician to determine with exact preciseness formulas that would cover each individual case, and many of the formulas used in calculating the strength of materials are based on results obtained by experiments and on practical experience. Of course, it must be understood at the outset that the actual strength of any material, such as steel, brass, copper, etc., must have been experimentally determined before any calculations at all can be made that would give definite results. Hence, what is known as the testing of the strength of materials lies at the basis of all calculations of strength and endurance.
It may be said, in a general way, that when designing machinery the designer must take into consideration two main factors. One is to so design the mechanism mechanically that the various motions required can be obtained by means of the machine. The other is to so proportion the parts that they will be strong enough to do the work for which they are intended without breakage and, in most cases, without distortion. A third factor also enters prominently into the design, that of so designing a machine that the various parts can be easily manufactured, but this last factor, while commercially fully as important as the other two, has not a direct bearing upon the actual working of the completed mechanism.
The subject that will be dealt with in this Reference Book relates to the second of the two main questions affecting the design, that of proportioning the parts so that they will be strong enough to properly do the work for which they are intended. This problem, in turn, may be divided into two sections, one of which deals with the determining of the forces which act upon a machine part, tending to break or distort it, and the determination of the actual proportions necessary to resist these forces. The science of determining the forces acting upon a machine part or, in general, upon an engineering structure, is termed mechanics, this word being used in its more limited sense, often referred to as theoretical mechanics. The problem of actually determining the dimensions of details of machines or structures with regard to their strength is covered by that part of mechanical knowledge which is known as the strength of materials. It should be understood, however, that the question of determining the forces acting upon an engineering structure or machine and the determination of the actual dimensions and materials required to resist these forces, are so closely connected that in dealing with the strength of materials we must also deal, to a very large extent, with the mechanical theory of forces.
When the forces acting upon a machine part are definitely known, it is, as a rule, comparatively easy to determine the actual proportions required to resist the action of these forces. There are, however, a number of instances in which the forces are applied in such a manner that it has so far proved impossible for the mathematician to determine with exact preciseness formulas that would cover each individual case, and many of the formulas used in calculating the strength of materials are based on results obtained by experiments and on practical experience. Of course, it must be understood at the outset that the actual strength of any material, such as steel, brass, copper, etc., must have been experimentally determined before any calculations at all can be made that would give definite results. Hence, what is known as the testing of the strength of materials lies at the basis of all calculations of strength and endurance.
Important Definitions
There are a number of expressions used in connection with calculations of strength of materials that must be explained at the outset. A stress is a force acting within a material or machine part resisting deformation. A load is a force applied from without to a material. The load tends to produce deformation and is resisted by the stress which it creates within the body.
A working load is the maximum load applied to a material under ordinary working conditions. A working stress is the stress produced in the material by this working load. A safe working stress is the maximum permissible working stress under given conditions, as, for example, for a certain material.
The ultimate strength of a material is its breaking strength in pounds per square inch, in tension, compression or shearing, as the case may be.
The total stress is the sum of all the stresses caused at one section of the body, irrespective of its area in square inches; while the expression stress, working stress, or intensity of stress generally means the number of pounds stress per each square inch of section.
Analysis of the Forces that Act upon a Machine Member
As mentioned, it is necessary to analyze or determine the forces that act upon a machine member, in order to be able to determine the actual dimensions necessary to insure its strength to resist distortion or breaking. Ordinarily, only the actual load resting upon or transmitted through the machine element need to be considered, but, in many instances, as in the case of bridges, elevator ropes for deep shafts, beams, etc., the weight of the part itself must be taken into account. In other instances, frictional resistance and forces due to inertia caused by change of velocity, as well as centrifugal forces must be considered. This latter consideration is especially necessary in the case of flywheels or pulleys moving at high velocities. In some instances, stresses are caused by forces due to changes in the temperature, as when a metal part is constrained between other metal surfaces.
These loads may be applied in three different ways. They may be applied steadily in one direction, in which case we speak of a steady or dead load. They may be alternately applied and removed, the load being constantly in motion, in which case we speak of a live load. A live load may be applied first in one direction and then in the re- verse direction, or it may be applied intermittently in one direction. It may be gradually applied and gradually relieved, or it may be suddenly applied, in which case we speak of the material as being subjected to shock or impact.
Ultimate Strength of Materials
The materials used in machine building are mainly metals, whereas, in civil engineering, wood, natural and artificial stones, as well as metals, are used. These materials may be subjected to a stress either in tension, that is when the forces acting upon the material are trying to pull it apart, or in compression, when the forces acting are trying to crush the material, or in shear, a stress which results either from a direct shearing action or a twisting or turning action. The strength of materials is usually measured in pounds per square inch.
For example, when we say that structural steel has a strength of 60,000 pounds per square inch in tension, we mean that a bar of structural steel, the cross-section of which is one square inch, will, on an average, not break before subjected to a load trying to pull it apart, of 60,000 pounds.
Structural steel has also a strength of 60,000 pounds per square inch in compression, but many metals have not the same strength in tension as in compression. Cast iron, for example, has only a strength of 15,000 pounds per square inch in tension, whereas its strength in compression, on an average, is 80,000 pounds per square inch. These values are the ultimate strength of the metals. When used in structures or machine members, the metals must never be subjected to so severe a strain, but the actual load must be much less per square inch in order to provide for a factor of safety.
In calculations of the strength of materials, one of the first considerations, after an analysis of the forces acting upon the material has been made and the problem thus been mechanically determined, is to assume the average ultimate strength of the material used in the construction in pounds per square inch. As already mentioned, this assumption is based upon the experiments made on these materials by many investigators in the past, and a table is given herewith which shows the strength of the metals most commonly used. A table is also given showing the average ultimate strength of common materials other than metals. In all calculations the ultimate strength of the materials may be taken from these tables, but in order to make sure that there is a margin for safety, a suitable factor of safety must, of course, be assumed.
The Factor of Safety
If the ultimate strength of a material like machine steel is 60,000 pounds per square inch in tension or compression, and we subject it to a load of 10,000 pounds per square inch, a factor of safety of 6 is used; that is, the ultimate strength of the material is six times as great as the load to which the material is subjected. The factor of safety must be greater for moving loads than for dead or steady loads. It must also be greater if the load is applied suddenly and suddenly removed than if the load is constant at all times. In other words, when the load varies from zero to maximum in one direction, as shown in the accompanying table, "Factors of Safety," that is, when the load varies from no-load to full-load, the factor of safety must increase as compared with that for a steady load. Now, if the load varies from a maximum in one direction to no-load, and then to a maximum in the other direction, as, for example, in a rod which is first submitted to a pulling or tensional stress, then to no load at all, and then to a crushing or compression stress, the factor of safety must again be increased. If the loads vary very suddenly, taking the nature of shocks, a very high factor of safety is required, even with the most reliable materials. This will be understood by studying the figures in the table "Factors of Safety."* The factor of safety required with various materials also differs. It must, for example, be greater for cast iron than for wrought iron because cast iron is not so dependable a material.
Influence of Temperature on the Strength of Metals
The degree of temperature to which a machine or a structural member made from metal is subjected has a considerable influence upon its strength. If we assume that metals have what we might call a "normal strength" at 70 degrees P., we will find upon investigation that this strength often increases with an increase in temperature up to a certain degree, and then rapidly decreases with further increase in temperature.
Elasticity and Elastic Limit
When external forces act upon a material they produce stresses within it as mentioned. These stresses are fundamentally tension, compression or shearing stresses, although we sometimes speak of bending or torsional stresses. Bending stresses, however, are only a combination of tension and compression stresses, and possibly also of shearing stresses, as will be explained later. Torsional stresses are merely shearing stresses. In most instances, a combination of two or more of these stresses is produced, especially in machine parts. In structural designs, such as bridges, for example, it is quite common that members are subjected to tension only or compression only, but in machine parts simple stresses of this kind are not as frequently met with, especially if the parts enter into the moving mechanism. All stresses to which a material is subjected tend to cause a deformation in it. If the stress is not too great, however, the material will return to its original shape and dimensions when the external load is removed. The property which enables a material to return to its original shape and dimensions is called elasticity and differs greatly in different materials. Of the metals, lead, for example, has little or no elasticity, whereas the elasticity of steel is great by comparison.
If a material has been subjected to such a load that upon its removal the material cannot fully return to its original shape and dimensions, it is said that it has been stressed beyond its elastic limit. Up to the elastic limit deformation is directly proportional to the load, but when the elastic limit has been reached, and the load is still increasing, the deformation will cease to be proportional to the stress, although the material will not actually break before a much greater load has been applied The elastic limit is difficult to determine with accuracy, although in iron and steel which has not been heat-treated, it is frequently about one-half of the ultimate strength of the material. In all engineering designs, the loads applied to the material must never be so great that the elastic limit is ever exceeded. If it is, there will be a permanent set in the material which naturally interferes both with the action of a machine and with its safe operation.
Modulus of Elasticity
The modulus of elasticity is another expression used to determine a certain quality in materials of engineering which is of great importance in the calculations of strength. The modulus of elasticity of a material may be denned as the quotient obtained by dividing the stress per square inch by the elongation in the length of one inch caused by this stress. The modulus of elasticity is quite generally denoted by E.
A working load is the maximum load applied to a material under ordinary working conditions. A working stress is the stress produced in the material by this working load. A safe working stress is the maximum permissible working stress under given conditions, as, for example, for a certain material.
The ultimate strength of a material is its breaking strength in pounds per square inch, in tension, compression or shearing, as the case may be.
The total stress is the sum of all the stresses caused at one section of the body, irrespective of its area in square inches; while the expression stress, working stress, or intensity of stress generally means the number of pounds stress per each square inch of section.
Analysis of the Forces that Act upon a Machine Member
As mentioned, it is necessary to analyze or determine the forces that act upon a machine member, in order to be able to determine the actual dimensions necessary to insure its strength to resist distortion or breaking. Ordinarily, only the actual load resting upon or transmitted through the machine element need to be considered, but, in many instances, as in the case of bridges, elevator ropes for deep shafts, beams, etc., the weight of the part itself must be taken into account. In other instances, frictional resistance and forces due to inertia caused by change of velocity, as well as centrifugal forces must be considered. This latter consideration is especially necessary in the case of flywheels or pulleys moving at high velocities. In some instances, stresses are caused by forces due to changes in the temperature, as when a metal part is constrained between other metal surfaces.
These loads may be applied in three different ways. They may be applied steadily in one direction, in which case we speak of a steady or dead load. They may be alternately applied and removed, the load being constantly in motion, in which case we speak of a live load. A live load may be applied first in one direction and then in the re- verse direction, or it may be applied intermittently in one direction. It may be gradually applied and gradually relieved, or it may be suddenly applied, in which case we speak of the material as being subjected to shock or impact.
Ultimate Strength of Materials
The materials used in machine building are mainly metals, whereas, in civil engineering, wood, natural and artificial stones, as well as metals, are used. These materials may be subjected to a stress either in tension, that is when the forces acting upon the material are trying to pull it apart, or in compression, when the forces acting are trying to crush the material, or in shear, a stress which results either from a direct shearing action or a twisting or turning action. The strength of materials is usually measured in pounds per square inch.
For example, when we say that structural steel has a strength of 60,000 pounds per square inch in tension, we mean that a bar of structural steel, the cross-section of which is one square inch, will, on an average, not break before subjected to a load trying to pull it apart, of 60,000 pounds.
Structural steel has also a strength of 60,000 pounds per square inch in compression, but many metals have not the same strength in tension as in compression. Cast iron, for example, has only a strength of 15,000 pounds per square inch in tension, whereas its strength in compression, on an average, is 80,000 pounds per square inch. These values are the ultimate strength of the metals. When used in structures or machine members, the metals must never be subjected to so severe a strain, but the actual load must be much less per square inch in order to provide for a factor of safety.
In calculations of the strength of materials, one of the first considerations, after an analysis of the forces acting upon the material has been made and the problem thus been mechanically determined, is to assume the average ultimate strength of the material used in the construction in pounds per square inch. As already mentioned, this assumption is based upon the experiments made on these materials by many investigators in the past, and a table is given herewith which shows the strength of the metals most commonly used. A table is also given showing the average ultimate strength of common materials other than metals. In all calculations the ultimate strength of the materials may be taken from these tables, but in order to make sure that there is a margin for safety, a suitable factor of safety must, of course, be assumed.
The Factor of Safety
If the ultimate strength of a material like machine steel is 60,000 pounds per square inch in tension or compression, and we subject it to a load of 10,000 pounds per square inch, a factor of safety of 6 is used; that is, the ultimate strength of the material is six times as great as the load to which the material is subjected. The factor of safety must be greater for moving loads than for dead or steady loads. It must also be greater if the load is applied suddenly and suddenly removed than if the load is constant at all times. In other words, when the load varies from zero to maximum in one direction, as shown in the accompanying table, "Factors of Safety," that is, when the load varies from no-load to full-load, the factor of safety must increase as compared with that for a steady load. Now, if the load varies from a maximum in one direction to no-load, and then to a maximum in the other direction, as, for example, in a rod which is first submitted to a pulling or tensional stress, then to no load at all, and then to a crushing or compression stress, the factor of safety must again be increased. If the loads vary very suddenly, taking the nature of shocks, a very high factor of safety is required, even with the most reliable materials. This will be understood by studying the figures in the table "Factors of Safety."* The factor of safety required with various materials also differs. It must, for example, be greater for cast iron than for wrought iron because cast iron is not so dependable a material.
Influence of Temperature on the Strength of Metals
The degree of temperature to which a machine or a structural member made from metal is subjected has a considerable influence upon its strength. If we assume that metals have what we might call a "normal strength" at 70 degrees P., we will find upon investigation that this strength often increases with an increase in temperature up to a certain degree, and then rapidly decreases with further increase in temperature.
Elasticity and Elastic Limit
When external forces act upon a material they produce stresses within it as mentioned. These stresses are fundamentally tension, compression or shearing stresses, although we sometimes speak of bending or torsional stresses. Bending stresses, however, are only a combination of tension and compression stresses, and possibly also of shearing stresses, as will be explained later. Torsional stresses are merely shearing stresses. In most instances, a combination of two or more of these stresses is produced, especially in machine parts. In structural designs, such as bridges, for example, it is quite common that members are subjected to tension only or compression only, but in machine parts simple stresses of this kind are not as frequently met with, especially if the parts enter into the moving mechanism. All stresses to which a material is subjected tend to cause a deformation in it. If the stress is not too great, however, the material will return to its original shape and dimensions when the external load is removed. The property which enables a material to return to its original shape and dimensions is called elasticity and differs greatly in different materials. Of the metals, lead, for example, has little or no elasticity, whereas the elasticity of steel is great by comparison.
If a material has been subjected to such a load that upon its removal the material cannot fully return to its original shape and dimensions, it is said that it has been stressed beyond its elastic limit. Up to the elastic limit deformation is directly proportional to the load, but when the elastic limit has been reached, and the load is still increasing, the deformation will cease to be proportional to the stress, although the material will not actually break before a much greater load has been applied The elastic limit is difficult to determine with accuracy, although in iron and steel which has not been heat-treated, it is frequently about one-half of the ultimate strength of the material. In all engineering designs, the loads applied to the material must never be so great that the elastic limit is ever exceeded. If it is, there will be a permanent set in the material which naturally interferes both with the action of a machine and with its safe operation.
Modulus of Elasticity
The modulus of elasticity is another expression used to determine a certain quality in materials of engineering which is of great importance in the calculations of strength. The modulus of elasticity of a material may be denned as the quotient obtained by dividing the stress per square inch by the elongation in the length of one inch caused by this stress. The modulus of elasticity is quite generally denoted by E.
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