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Resistance of materials for beginners in engineering
RESISTANCE OF MATERIALS FOR BEGINNERS IN ENGINEERING
BY S. E. SLOCUM,
PROFESSOR OF APPLIED MATHEMATICS IN THE UNIVERSITY OF CINCINNATI
GINN AND COMPANY; NEW YORK; 1914
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Resistance of materials for beginners in engineering
PREFACE
The chief feature which distinguishes this volume from other American textbooks on the same subject is that the Principle of Moments is used consistently throughout in place of the usual calculus processes. By basing the work on this principle it has been found practicable to give a simple and obvious treatment of many topics for which the calculus is usually thought to be indispensable, such as the calculation of moments of inertia, the deflection of beams, the buckling of columns, and the strength of thick cylinders. Experience has shown conclusively that the average engineering graduate, and even the practicing engineer, is deficient in the ability to apply the Principle of Moments readily, but when thus used as the central and coordinating principle, it must necessarily make an indelible impression on the mind of the student and go far toward remedying this deficiency.
The mechanics of materials is of such fundamental importance in all branches of technology that it is important to begin its study as early in the course as possible. Heretofore it has been necessary to defer it - awaiting the completion of the calculus - until junior year, when the curriculum is already crowded with technical subjects requiring its application. This text makes it possible for the course to parallel or even to precede the calculus. In addition, it makes the subject available for trade or architectural schools where no calculus is taught.
Although simple and obvious, the treatment is adequate, and its simplicity in no way limits its range or generality. The text is supplemented by a variety of engineering applications, giving practical information as well as a mastery of the principles involved.
The chief feature which distinguishes this volume from other American textbooks on the same subject is that the Principle of Moments is used consistently throughout in place of the usual calculus processes. By basing the work on this principle it has been found practicable to give a simple and obvious treatment of many topics for which the calculus is usually thought to be indispensable, such as the calculation of moments of inertia, the deflection of beams, the buckling of columns, and the strength of thick cylinders. Experience has shown conclusively that the average engineering graduate, and even the practicing engineer, is deficient in the ability to apply the Principle of Moments readily, but when thus used as the central and coordinating principle, it must necessarily make an indelible impression on the mind of the student and go far toward remedying this deficiency.
The mechanics of materials is of such fundamental importance in all branches of technology that it is important to begin its study as early in the course as possible. Heretofore it has been necessary to defer it - awaiting the completion of the calculus - until junior year, when the curriculum is already crowded with technical subjects requiring its application. This text makes it possible for the course to parallel or even to precede the calculus. In addition, it makes the subject available for trade or architectural schools where no calculus is taught.
Although simple and obvious, the treatment is adequate, and its simplicity in no way limits its range or generality. The text is supplemented by a variety of engineering applications, giving practical information as well as a mastery of the principles involved.
CONTENTS
- STRESS AND DEFORMATION
- FIRST AND SECOND MOMENTS
- BENDING-MOMENT AND SHEAR DIAGRAMS
- STRENGTH OF BEAMS
- DEFLECTION OF CANTILEVER AND SIMPLE BEAMS
- CONTINUOUS BEAMS
- RESTRAINED, OR BUILT-IN, BEAMS
- COLUMNS AND STRUTS
- TORSION
- SPHERES AND CYLINDERS UNDER UNIFORM PRESSURE
- FLAT PLATES
- RIVETED JOINTS AND CONNECTIONS
- REENFORCED CONCRETE
- SIMPLE STRUCTURES
SECTION I - STRESS AND DEFORMATION
1. Elastic resistance, or stress. The effect of an external force acting upon an elastic body is to produce deformation, or change of shape. For example, if a bar is placed in a testing machine and a tensile load applied, it will be found that the length of the bar is increased and the area of its cross section diminished. Similarly, if a compressive load is applied, the length of the bar is diminished and the area of its cross section increased.
All solid bodies offer more or less resistance to the deformation, or change of shape, produced by external force. This internal resistance, when expressed in definite units, is called stress. A body under the action of stress is said to be strained.
In general the stress is not the same at all points of a body, but varies from point to point. The intensity of the stress at any particular point is therefore expressed as the force in pounds which would be exerted if the stress were uniform and acted over an area one square inch in extent. That is to say, whatever the actual extent of the area considered, whether finite or infinitesimal, the stress is expressed in pounds per square inch.
Taking any plane section of a body under strain, the stress acting on this plane section may in general be resolved, like any force, into two components, one perpendicular to the plane and the other lying in the plane. The perpendicular, or normal, component is called direct stress and is either tension or compression. The tangential component, or that lying in the plane of the cross section, is called shear. In what follows, the letter p will always be used to denote normal, or direct, stress, and q to denote tangential stress, or shear.
The effect of a normal stress is to produce extension or compression, that is, a lengthening or shortening of the fibers, thereby changing the dimensions of the body ; whereas shear tends to slide any given cross section over the one adjacent to it, thus producing angular deformation, or change in shape, of the body without altering its dimensions.
2. Varieties of strain. The nature of the deformation produced by external forces acting on an elastic body depends on where and how these forces are applied. Although only two kinds of stress can occur, namely, normal stress (tension or compression) and shear, these may arise in various ways. In general, five different cases of strain may be distinguished, each of which must be considered separately. These are as follows:
1. If the forces act along the same line, toward or away from one another, the strain is called compression or tension (Fig. 1, a).
2. If the forces tend to slice off a portion of the body by separating it along a surface, the strain is called shear (Fig. 1, ft).
3. If the forces act transverse to the length of the body (usually perpendicular to the long axis of the piece), so as to produce lateral deflection, the strain is called bending, or flexure (Fig. 1, e).
4. If one dimension of the body is large as compared with the other two, and the forces act in the direction of the long dimension and toward one another, the strain is called buckling, or column flexure (Fig. 1, d).
5. If the forces exert a twist on the body, the strain is called torsion (Fig. 1, e).
3. Strain diagram. In the case of tension or compression it is easy to show graphically the chief features of the strain. Thus, suppose that a test bar is placed in a testing machine, and that the total load on the bar at any instant is read on the scale beam of the machine, and its corresponding length in inches is measured with an extensometer. Assuming that the stress is uniformly distributed over any cross section of the bar, the unit stress is obtained by dividing the total load in pounds acting on the bar by the area of its cross section in square inches.
4. Hooke's law. By inspection of the curves in Fig. 2 it is evident that the strain diagram for each material has certain characteristic features. For instance, in the case of wrought iron the strain diagram from to A is a straight line ; this means that for points between and A the stress is proportional to the corresponding deformation.
5. Elastic limit. It is found by experiment that as long as the stress does not pass the elastic limit, the deformation disappears when the external forces are removed. If the unit stress (or, more properly, the unit deformation) exceeds the elastic limit, however, then the deformation does not entirely disappear upon removal of the load, but the body retains a permanent set. At the elastic limit, therefore, the body begins to lose its elastic properties, and hence, in constructions which are intended to last for any length of time, the members should be so designed that the actual stresses lie well below the elastic limit.
It has also been found by experiment that, for iron and steel, if the stress lies well within the elastic limit, it can be removed and repeated indefinitely without causing rupture ; but if the metal is stressed beyond the elastic limit, and the stress is repeated or alternates between tension and compression, it will eventually cause rupture, the number of changes necessary to produce failure decreasing as the difference between the upper and lower limits of the strain increases. This is known as the fatigue of metals, and indicates that in determining the resistance of any material the elastic limit is much more important than the ultimate strength.
Overstrain of any kind results in a gradual hardening of the material. Where this has already occurred, the elastic properties of the material can be partially or wholly restored by annealing ; that is, by heating the metal to a cherry redness and allowing it to cool slowly.
6. Working stress. The stress which can be carried by any material without losing its elastic properties is called the allowable stress or working stress, and must always lie below the elastic limit.
Average values of the ultimate strength, factors of safety, and other elastic constants for the various materials used in construction are given in Table I.
Since for wrought iron and steel the elastic limit can be definitely located, the working stresses for these materials is usually assumed as a certain fraction, say i to |, of the elastic limit.
Materials like cast iron, stone, and concrete have no definite elastic limits; that is, they do not conform perfectly to Hooke's law.
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All solid bodies offer more or less resistance to the deformation, or change of shape, produced by external force. This internal resistance, when expressed in definite units, is called stress. A body under the action of stress is said to be strained.
In general the stress is not the same at all points of a body, but varies from point to point. The intensity of the stress at any particular point is therefore expressed as the force in pounds which would be exerted if the stress were uniform and acted over an area one square inch in extent. That is to say, whatever the actual extent of the area considered, whether finite or infinitesimal, the stress is expressed in pounds per square inch.
Taking any plane section of a body under strain, the stress acting on this plane section may in general be resolved, like any force, into two components, one perpendicular to the plane and the other lying in the plane. The perpendicular, or normal, component is called direct stress and is either tension or compression. The tangential component, or that lying in the plane of the cross section, is called shear. In what follows, the letter p will always be used to denote normal, or direct, stress, and q to denote tangential stress, or shear.
The effect of a normal stress is to produce extension or compression, that is, a lengthening or shortening of the fibers, thereby changing the dimensions of the body ; whereas shear tends to slide any given cross section over the one adjacent to it, thus producing angular deformation, or change in shape, of the body without altering its dimensions.
2. Varieties of strain. The nature of the deformation produced by external forces acting on an elastic body depends on where and how these forces are applied. Although only two kinds of stress can occur, namely, normal stress (tension or compression) and shear, these may arise in various ways. In general, five different cases of strain may be distinguished, each of which must be considered separately. These are as follows:
1. If the forces act along the same line, toward or away from one another, the strain is called compression or tension (Fig. 1, a).
2. If the forces tend to slice off a portion of the body by separating it along a surface, the strain is called shear (Fig. 1, ft).
3. If the forces act transverse to the length of the body (usually perpendicular to the long axis of the piece), so as to produce lateral deflection, the strain is called bending, or flexure (Fig. 1, e).
4. If one dimension of the body is large as compared with the other two, and the forces act in the direction of the long dimension and toward one another, the strain is called buckling, or column flexure (Fig. 1, d).
5. If the forces exert a twist on the body, the strain is called torsion (Fig. 1, e).
3. Strain diagram. In the case of tension or compression it is easy to show graphically the chief features of the strain. Thus, suppose that a test bar is placed in a testing machine, and that the total load on the bar at any instant is read on the scale beam of the machine, and its corresponding length in inches is measured with an extensometer. Assuming that the stress is uniformly distributed over any cross section of the bar, the unit stress is obtained by dividing the total load in pounds acting on the bar by the area of its cross section in square inches.
4. Hooke's law. By inspection of the curves in Fig. 2 it is evident that the strain diagram for each material has certain characteristic features. For instance, in the case of wrought iron the strain diagram from to A is a straight line ; this means that for points between and A the stress is proportional to the corresponding deformation.
5. Elastic limit. It is found by experiment that as long as the stress does not pass the elastic limit, the deformation disappears when the external forces are removed. If the unit stress (or, more properly, the unit deformation) exceeds the elastic limit, however, then the deformation does not entirely disappear upon removal of the load, but the body retains a permanent set. At the elastic limit, therefore, the body begins to lose its elastic properties, and hence, in constructions which are intended to last for any length of time, the members should be so designed that the actual stresses lie well below the elastic limit.
It has also been found by experiment that, for iron and steel, if the stress lies well within the elastic limit, it can be removed and repeated indefinitely without causing rupture ; but if the metal is stressed beyond the elastic limit, and the stress is repeated or alternates between tension and compression, it will eventually cause rupture, the number of changes necessary to produce failure decreasing as the difference between the upper and lower limits of the strain increases. This is known as the fatigue of metals, and indicates that in determining the resistance of any material the elastic limit is much more important than the ultimate strength.
Overstrain of any kind results in a gradual hardening of the material. Where this has already occurred, the elastic properties of the material can be partially or wholly restored by annealing ; that is, by heating the metal to a cherry redness and allowing it to cool slowly.
6. Working stress. The stress which can be carried by any material without losing its elastic properties is called the allowable stress or working stress, and must always lie below the elastic limit.
Average values of the ultimate strength, factors of safety, and other elastic constants for the various materials used in construction are given in Table I.
Since for wrought iron and steel the elastic limit can be definitely located, the working stresses for these materials is usually assumed as a certain fraction, say i to |, of the elastic limit.
Materials like cast iron, stone, and concrete have no definite elastic limits; that is, they do not conform perfectly to Hooke's law.
DOWNLOAD FREE BOOK: Resistance of materials for beginners in engineering
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