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The action of materials under stress or structural mechanics
THE ACTION OF MATERIALS UNDER STRESS OR STRUCTURAL MECHANICS
Comprising the strength and resistance of materials and elements of structural design with examples and problems.
BY CHARLES E. GREENE,
PROFESSOR OF CIVIL ENGINEERING, UNIVERSITY OF MICHIGAN
ANN ARBOR, MICH.; Printed for the Author; 1897
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The action of materials under stress or structural mechanics
PREFACE
The author, in teaching for many years the subjects embraced in the following pages, has found it advantageous to take at first but a portion of what is included in the several chapters, and, after a general survey of the field, to return and extend the investigation more in detail. Some of the sections, therefore, are printed in smaller type and can be omitted at first reading. A few of the special investigations may become of interest only when the problems to which they relate occur in actual practice.
It is hoped that this book will be serviceable after the class-room work is concluded, and reference is facilitated by a more compact arrangement of the several matters than the course suggested above would give. The attempt has been made to deal with practicable cases, and the examples for the most part are shaped with that end in view. A full index will enable one to find any desired topic.
The treatment of the subject of internal stress is largely graphical. All the constructions are simple, and the results, besides being useful in themselves, shed much light on various problems. The time devoted to a careful study of the chapter in question will be well expended.
The notation is practically uniform throughout the book, and is that used by several standard authors. Forces and moments are expressed by capital letters, and unit loads and stresses by small letters. The co-ordinate x is measured along the length of a piece, the co-ordinate y in the direction of variation of stress in a section, and z is the line of no variation of stress, that is, the line parallel to the moment axis.
One who has mastered the subjects discussed here can use the current formulas, the pocket-book rules, and tables, not blindly, but with discrimination, and ought to be prepared to design intelligently.
Mr Albert E. Greene has rendered much assistance in preparing the material for publication.
TABLE OF CONTENTS
Introduction. Chapter I. Action of a Piece under Direct Force
Chapter II. Materials
Chapter III. Beams
Chapter IV. Torsion
Chapter V. Moments of Inertia
Chapter VI. Flexure and Deflection of Simple Beams
Chapter VII. Restrained Beams: Continuous Beams
Chapter VIII. Pieces under Tension
Chapter IX. Compression Pieces: - Columns, Posts and Struts
Chapter X. Safe Working Stresses
Chapter XI. Internal Stress: Change of Form
Chapter XII. Rivets: Pins
Chapter XIII. Envelopes: - Boilers, Pipes, Dome
Chapter XIV. Plate Girder
Chapter XV. Earth Pressure: Retaining Wall: Springs: Plates
Chapter XVI. Details in Wood and Iron
INTRODUCTION.
1. External Forces. - The engineer, in designing a new structure, or critically examining one already built, determines from the conditions of the case the actual or probable external forces which the structure is called upon to resist. He may then prepare, either by mathematical calculations or by graphical methods, a sheet which shows the maximum and minimum direct forces of tension and compression which the several pieces or parts of the structure are liable to experience, as well as the bending moments on such parts as are subjected to them.
These forces and moments are determined from the requirements of equilibrium, if the pieces are at rest. For forces acting in one plane, a condition which suffices for the analysis of most cases, it is necessary that, for the structure as a whole, as well as for each piece, there shall be no tendency to move up or down, to move to the right or left, or to rotate. These limitations are usually expressed in Mechanics as, that the sum of the X forces, the sum of the Y forces, and the sum of the moments shall each equal zero.
If the structure is a machine, the forces and moments in action at any time, and their respective magnitudes, call for a consideration of the question of acceleration or retardation of the several parts and the additional maximum forces and moments called into action by the greatest rate of change of motion at any instant. Hence the weight or mass of the moving part or parts is necessarily taken into account.
Finally, noting the rapidity and frequency of the change of force and moment at any section of any piece or connection, the engineer selects, as judgment dictates, the allowable stresses of the several kinds per square inch, making allowance for the effect of impact, shock and vibration in intensifying their action, and proceeds to find the necessary cross-sections of the parts and the proportions of the connections between them. As all structures are intended to endure the forces and vicissitudes to which they are usually exposed, the allowable unit-stresses, expressed in pounds per square inch, must be safe stresses.
It is largely with the development of the latter part of this subject, after the forces have been found to which the several parts are liable, that this book is concerned.
2. Ties, Struts and Beams. - There are, in general, three kinds of pieces in a frame or structure; ties or tension members; columns, posts and struts or compression members; and beams, which support a transverse load and are subject to bending and its accompanying shear. A given piece may also be, at the same time, a tie and a beam, or a strut and a beam, and at different times a tie and a strut.
3. Relation of External Forces to Internal Stresses. - The forces and moments which a member is called upon to resist, and which may properly be considered as external to that member, give rise to actions between all the particles of material of which such a member is composed, tending to move adjacent particles from, towards or by one another, and causing change of form. There result internal stresses or resistances to displacement, between the several particles.
These internal stresses, or briefly stresses, must be of such kind, magnitude, distribution and direction, at any imaginary section of a piece or structure, that their resultant force and moment will satisfy the requirements of equilibrium or change of motion with the external resultant force and moment at that section; and no stress per square inch can, for a correct design, be greater than the material will safely bear. Hence may be determined the necessary area and form of the cross-section at the critical points, when the resultant forces and moments are known.
4. Internal Stress. - There are three kinds of stress, or action of adjacent particles one on the other, to which the particles of a body may be subjected, when external forces and its own weight are considered, viz.: tensile stress, tending to remove one particle farther from its neighbor, and manifested by an accompanying stretch or elongation of the body; compressive stress, tending to make a particle approach its neighbor, and manifested by an accompanying shortening or compression of the body; and shearing stress, tending to make a particle move or slide laterally with reference to an adjacent particle, and manifested by an accompanying distortion. Whether the stress produces change of form, or the attempted change of form gives rise to internal stresses as resistances, is of little consequence; the stress between two particles and the change of position of the particles are always associated, and one being given the other must exist,
5. Tension and Shear, or Compression and Shear. - If the direction of the stress is oblique, that is, not normal or perpendicular, on any section of a body, the stress may be resolved into a tensile or compressive stress normal to that section, and a tangential stress along the section,, which, from its tendency to cause sliding of one portion of the body, by or along the section, has been given the name of shear, from the resemblance to the action of a pair of shears, one blade passing by the other along the opposite sides of the plane of section. Draw two oblique and directly opposed arrows, one on either side of a straight line representing the trace of a sectional plane, decompose those oblique stresses normally and tangentially to the plane, and notice the resulting directly opposed tension or compression, and the shear. Hence tension and shear, or compression and shear, may be found on any given plane in a body, but tension and compression cannot simultaneously occur at one point in a given area.
6. Sign of Stress. - Ties are usually slender members; struts have larger lateral dimensions. Longitudinal tension tends to diminish the cross-section of the piece which carries it, and hence may conveniently be represented by - , the negative sign; longitudinal compression tends to increase the cross sectional area and may be called + or positive. Shear, being at right angles to the tension and compression in the preceding illustration, has no-sign; and lies, in significance, between tension and compression. If a rectangular plate is pulled in the direction of two of its opposite sides and compressed in the direction of its other two sides, there will be some shearing stress on every plane of section except those parallel to the sides, and nothing but shear on two certain oblique planes, as will be seen later.
7. Unit Stresses. - These internal stresses are measured by units of pounds and inches by English and American engineers, and are stated as so many pounds of tension, compression or shear per square inch, called unit tension, compression or shear. Thus, in a bar of four square inches cross-section, under a total pull of 36,000 pounds centrally applied, the internal unit tension is 9000 pounds per square inch, provided the pull is uniformly distributed on the particles adjacent to any cross-section. If the pull is not central or the stress not uniformly distributed, the average or mean unit tensile stress is still 9000 pounds.
If an oblique section of the same bar is made, the total force acting on the particles adjacent to the section is the same as before, but the area of section is increased; hence the unit stress, found by dividing the force by the new area, is diminished. The stress will also be oblique to the section, as its direction must be that of the force . When the unit stress is notnormal to the plane of section on which it acts, it can be decomposed into a normal unit tension and a unit shear.
When the stress varies in magnitude from point to point, its amount on any very small area (the infinitesimal area of the Calculus) may be divided by that area, and the quotient will be the unit stress, or the amount which would exist on a square inch, if a square inch had the same stress all over it as the very small area has.
8. Unit Stresses on Different Planes not to be Treated as Forces. - It will be seen, upon inspection of the results of analyses which come later, that unit stresses acting on different planes must not be compounded and resolved as if they were forces. But the entire stress upon a certain area, found by multiplying the unit stress by that area, is a force, and this force may be compounded with other forces or resolved, and the new force may then be divided by the new area of action, and a new unit stress be thus found.
Some persons may be assisted in understanding the analysis of problems by representing in a sketch, or mentally, the unit stresses at different parts of a cross-section by ordinates which make up, in their assemblage, a volume. This volume, whose base is the cross-section, will represent or be proportional to the total force on the section. The position of the resultant force or forces, i. e., traversing the centre of gravity of the volume, the direction and law of distribution of the stress are then quite apparent.
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These forces and moments are determined from the requirements of equilibrium, if the pieces are at rest. For forces acting in one plane, a condition which suffices for the analysis of most cases, it is necessary that, for the structure as a whole, as well as for each piece, there shall be no tendency to move up or down, to move to the right or left, or to rotate. These limitations are usually expressed in Mechanics as, that the sum of the X forces, the sum of the Y forces, and the sum of the moments shall each equal zero.
If the structure is a machine, the forces and moments in action at any time, and their respective magnitudes, call for a consideration of the question of acceleration or retardation of the several parts and the additional maximum forces and moments called into action by the greatest rate of change of motion at any instant. Hence the weight or mass of the moving part or parts is necessarily taken into account.
Finally, noting the rapidity and frequency of the change of force and moment at any section of any piece or connection, the engineer selects, as judgment dictates, the allowable stresses of the several kinds per square inch, making allowance for the effect of impact, shock and vibration in intensifying their action, and proceeds to find the necessary cross-sections of the parts and the proportions of the connections between them. As all structures are intended to endure the forces and vicissitudes to which they are usually exposed, the allowable unit-stresses, expressed in pounds per square inch, must be safe stresses.
It is largely with the development of the latter part of this subject, after the forces have been found to which the several parts are liable, that this book is concerned.
2. Ties, Struts and Beams. - There are, in general, three kinds of pieces in a frame or structure; ties or tension members; columns, posts and struts or compression members; and beams, which support a transverse load and are subject to bending and its accompanying shear. A given piece may also be, at the same time, a tie and a beam, or a strut and a beam, and at different times a tie and a strut.
3. Relation of External Forces to Internal Stresses. - The forces and moments which a member is called upon to resist, and which may properly be considered as external to that member, give rise to actions between all the particles of material of which such a member is composed, tending to move adjacent particles from, towards or by one another, and causing change of form. There result internal stresses or resistances to displacement, between the several particles.
These internal stresses, or briefly stresses, must be of such kind, magnitude, distribution and direction, at any imaginary section of a piece or structure, that their resultant force and moment will satisfy the requirements of equilibrium or change of motion with the external resultant force and moment at that section; and no stress per square inch can, for a correct design, be greater than the material will safely bear. Hence may be determined the necessary area and form of the cross-section at the critical points, when the resultant forces and moments are known.
4. Internal Stress. - There are three kinds of stress, or action of adjacent particles one on the other, to which the particles of a body may be subjected, when external forces and its own weight are considered, viz.: tensile stress, tending to remove one particle farther from its neighbor, and manifested by an accompanying stretch or elongation of the body; compressive stress, tending to make a particle approach its neighbor, and manifested by an accompanying shortening or compression of the body; and shearing stress, tending to make a particle move or slide laterally with reference to an adjacent particle, and manifested by an accompanying distortion. Whether the stress produces change of form, or the attempted change of form gives rise to internal stresses as resistances, is of little consequence; the stress between two particles and the change of position of the particles are always associated, and one being given the other must exist,
5. Tension and Shear, or Compression and Shear. - If the direction of the stress is oblique, that is, not normal or perpendicular, on any section of a body, the stress may be resolved into a tensile or compressive stress normal to that section, and a tangential stress along the section,, which, from its tendency to cause sliding of one portion of the body, by or along the section, has been given the name of shear, from the resemblance to the action of a pair of shears, one blade passing by the other along the opposite sides of the plane of section. Draw two oblique and directly opposed arrows, one on either side of a straight line representing the trace of a sectional plane, decompose those oblique stresses normally and tangentially to the plane, and notice the resulting directly opposed tension or compression, and the shear. Hence tension and shear, or compression and shear, may be found on any given plane in a body, but tension and compression cannot simultaneously occur at one point in a given area.
6. Sign of Stress. - Ties are usually slender members; struts have larger lateral dimensions. Longitudinal tension tends to diminish the cross-section of the piece which carries it, and hence may conveniently be represented by - , the negative sign; longitudinal compression tends to increase the cross sectional area and may be called + or positive. Shear, being at right angles to the tension and compression in the preceding illustration, has no-sign; and lies, in significance, between tension and compression. If a rectangular plate is pulled in the direction of two of its opposite sides and compressed in the direction of its other two sides, there will be some shearing stress on every plane of section except those parallel to the sides, and nothing but shear on two certain oblique planes, as will be seen later.
7. Unit Stresses. - These internal stresses are measured by units of pounds and inches by English and American engineers, and are stated as so many pounds of tension, compression or shear per square inch, called unit tension, compression or shear. Thus, in a bar of four square inches cross-section, under a total pull of 36,000 pounds centrally applied, the internal unit tension is 9000 pounds per square inch, provided the pull is uniformly distributed on the particles adjacent to any cross-section. If the pull is not central or the stress not uniformly distributed, the average or mean unit tensile stress is still 9000 pounds.
If an oblique section of the same bar is made, the total force acting on the particles adjacent to the section is the same as before, but the area of section is increased; hence the unit stress, found by dividing the force by the new area, is diminished. The stress will also be oblique to the section, as its direction must be that of the force . When the unit stress is notnormal to the plane of section on which it acts, it can be decomposed into a normal unit tension and a unit shear.
When the stress varies in magnitude from point to point, its amount on any very small area (the infinitesimal area of the Calculus) may be divided by that area, and the quotient will be the unit stress, or the amount which would exist on a square inch, if a square inch had the same stress all over it as the very small area has.
8. Unit Stresses on Different Planes not to be Treated as Forces. - It will be seen, upon inspection of the results of analyses which come later, that unit stresses acting on different planes must not be compounded and resolved as if they were forces. But the entire stress upon a certain area, found by multiplying the unit stress by that area, is a force, and this force may be compounded with other forces or resolved, and the new force may then be divided by the new area of action, and a new unit stress be thus found.
Some persons may be assisted in understanding the analysis of problems by representing in a sketch, or mentally, the unit stresses at different parts of a cross-section by ordinates which make up, in their assemblage, a volume. This volume, whose base is the cross-section, will represent or be proportional to the total force on the section. The position of the resultant force or forces, i. e., traversing the centre of gravity of the volume, the direction and law of distribution of the stress are then quite apparent.
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