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The stresses in framed structures
THE STRESSES IN FRAMED STRUCTURES
Including the strength of materials and theory of flexure, also the determination of dimensions and designing of details, specifications and complete designs and working drawings.
BY JAY DU BOIS,
PROFESSOR OF CIVIL ENGINEERING IN THE SHEFFIELD SCIENTIFIC SCHOOL OF VALE UNIVERSITY.
NEW YORK; JOHN WILEY & SONS; 1896.
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The stresses in framed structures
PREFACE TO THE TENTH EDITION.
The present edition is essentially a new work. It appears not only in a new form, greatly reduced in size and weight, but with additions and changes so extensive and important that it is practically rewritten and reset.
In the change of title from "Strains" to Stresses we have at last conformed to the established usage of all physicists and the large and increasing majority of engineers. In changing the signs for compression, tension, and rotation we conform to the universal notation of analytical mechanics. These minor changes would have been made by the author long since had it not been for the expense, necessitating as they do an almost entire resetting of the type. The present work is thus in harmony as regards notation and nomenclature with modern practice.
As to more important changes, in Part I, Section I, page 37, we have replaced the articles upon “Graphic Representation of Moments for any Number of Forces” by new and correct matter. These articles have until now appeared in successive editions, erroneous in statement and demonstration. Later writers have not called the attention of the author to this blemish. Indeed, some have done him the favor of giving the same erroneous presentation without calling attention to its source. It is, however, hoped that in their future editions they may give him the credit of its correction.
Examination of the rest of this section will reveal many other minor changes, too numerous for mention here.
In Part I, Section II, many of the changes are of a more important character.
On page 97 et seq. we give a revised treatment of the “Method of Calculation by Concentrated Load Systems;” also by “equivalent uniform load,” by “one locomotive excess and equivalent uniform load,” and by “two locomotive excesses and actual uniform train load.” We also give a comparison of the results of these various methods, and point out the reasons why the last method is now and always has been preferred by the author, and used in all the illustrative examples. At the same time the matter is so presented that any method may be used, the illustrations of the application of the statical principles being so full and complete that the reader can have no difficulty in using any system of loading he may prefer.
In Chapter VI, page 148, we have given the principle of least work, and illustrated its application to redundant members and to the deflection of a framed girder. We have there given our reasons for considering such applications of little practical value, although in recent works much stress has been laid upon them. This principle of least work is, however, capable of applications of great value in directions where it has not hitherto been applied, and we believe that in the present work several such applications will be found made for the first time, and the way opened for others.
Thus in Chapter VII, page 155, we have applied it to Swing Bridges, and have obtained new and general methods and formulas which apply to varying depth and chord section. We have given an example fully worked out, for a centre-bearing pivot-span, also a comparison of results with those obtained by the formulas hitherto in use. It is also shown that these formulas are but special cases of our more general method, when depth and chord-sections are constant. They do not therefore apply to most practical cases, although heretofore they have been the best in use. The new formulas of this chapter are deduced by themselves at the end. They follow from the principle of least work so directly and simply, that the value of this principle is well brought out.
Again, in Chapter IX, page 190, we have applied this principle to the Braced Arch, with the same results. Here also we obtain new and general methods and formulas which apply to varying depth and chord sections and to all forms, flat or full centre. Again we find that the formulas heretofore in use are but special cases for a flat parabolic arch of constant depth. They do not therefore apply to many practical cases. Here again the formulas of the chapter are deduced at the end, and those familiar with the usual mathematical treatment of the braced arch will appreciate the directness and simplicity of that afforded by the principle of least work.
Again, in Chapter X, page 217, this principle gives a new solution for the Suspension System. The new formulas thus obtained are recapitulated on page 228 for ready reference, in shape for practical application, and an example is worked out and results given both by the old and new methods. The old method assumes that the cable carries the entire load, dead and live, and that the truss acts merely to distribute a partial loading. The error of such an assumption is pointed out, and by the application of the principle of least work we determine the share of the loading, dead and live, which each system must carry. The new formulas obtained are simple and easy of application.
In the appendix to Part I, Chapter II, page 270, the theory of Flexure and Mechanics of Materials is rewritten and given in greater fulness, clearness, and with improved notation. Here again the principle of least work is applied, giving a direct and simpler determination of the reactions in the cases of beam fixed at one end and supported at the other, and fixed at both ends. On page 289 the fundamental formulas are recapitulated for reference, and these formulas are derived anew from the principle of work. We have also treated the topics of combined stresses and secondary stresses.
With a view to complete treatment, we have given in Chapter III, page 328, the application of the Theory of Flexure to Torsion, although the subject finds no application in Framed Structures. Thus the student need not go outside of the present work for his course on Mechanics of Materials.
In Chapter IV., page 333, we have given a new treatment of the “ideal column” and of long struts. The new formula of Mr. Prichard, M. Am. Soc. C. E., is here given for the first time. This formula is probably the most important on this subject since the derivation of Euler's well-known formula. We have also given the practical column formulas in general use. For the reasons stated in this chapter, we consider the formula of Prof. Merriman as the best practical formula thus far proposed.
As to Part II., which deals with practical designing and structural principles, the changes and additions are numerous, mainly relating to improved method of presentation or to present practice. This part of the subject is in process of change and development. The present work was one of the first to give due prominence to it, and the first to give anything like a complete and practical presentation. In subsequent editions this portion has been greatly enlarged. A point has been reached, however, where it would be impossible to try to keep pace with changing details and minutiae of practice. Nor is such an attempt desirable in a work of this character. We have therefore sought in the present edition to give clearly and logically those permanent principles and practices, never out of date, upon which design must always rest, in such a manner as to indicate their application to constructions in general, whether framed structures or machines, together with such illustrations, special details, and applications to framed structures in especial, as may serve to give the student ability to use and apply those principles intelligently to other cases. The teacher must always supplement such an endeavor by more detailed instruction and by as great a variety of examples as the time allotted to the subject may permit; while the student and designer must seek in other works than this, in specifications, and in the records of the most recent practice for those changing rules and details which constitute the latest usage.
GENERAL CONTENTS.
PART I.
SECTION I. DIFFERENT METHODS OF CALCULATION.
General Principles. Introductory
CHAPTER I. Graphic Resolution of Forces. General Principles, Forces in the Same Plane, Common Point of Application
CHAPTER II. Analytic Resolution of Forces. General Principles. Forces in the Same Plane, Common Point of Application
CHAPTER III. Method of Moments - Algebraic Solution. General Principles. Forces in the Same Plane Different Points of Application
CHAPTER IV. Method of Moments - Graphic Solution. General Principles. Forces in the Same Plane, Different Points of Application
SECTION II. PRACTICAL APPLICATION OF PRECEDING METHODS TO VARIOUS STRUCTURES.
Introductory - Classification of Structures
CHAPTER I. Structures which Sustain a Dead Load Only - Roof Trusses
CHAPTER II. Structures which Sustain a Live as well as a Dead Load - Bridge Trusses
CHAPTER III. Bridge Girders with Parallel Chords - Triangular Girder
CHAPTER IV. Bridge Girders with Parallel Chords
CHAPTER V. Bridge Girders with Inclined Chords
CHAPTER VI. Principle of Least Work - Redundant Members - Deflection of a Framed Girder
CHAPTER VII. The Pivot or Swing Bridge
CHAPTER VIII. The Continuous Girder
CHAPTER IX. The Braced Arch
CHAPTER X. Composite Structures - Suspension System with Stiffening Truss
APPENDIX.
CHAPTER I. Concentrated Load Systems
CHAPTER IL Strength of Materials and Theory of Flexure
CHAPTER III. Torsion
CHAPTER IV. Column Formulas
CHAPTER V. The Continuous Girder
PART II. DETERMINATION OF DIMENSIONS AND DESIGNING OF DETAILS.
CHAPTER I. Ultimate Strength. Elastic Limit. Methods of Dimensioning.
CHAPTER II. Cross-sectioning. Determination of Dimensions - Tension Members
CHAPTER III. Cross-sectioning. Determination of Dimensions – Compression Members
CHAPTER IV. Pins and Eye-bars
CHAPTER V. Riveting
CHAPTER VI. Wind Bracing - Miscellaneous Details.
CHAPTER VII. Floor System. Cross-Girders. Stringers. Floor.
CHAPTER VIII. Roof and Bridge Trusses. Dead Weight. Economic Depth.
CHAPTER IX. Specifications. List of Bridge Members.
CHAPTER X. Complete Design for a Railway Bridge.
CHAPTER XI.
CHAPTER XII. The Order Book. Shipping and Inspecting.
CHAPTER XIII. The Erection of Engineering Structures.
CHAPTER XIV. Modern High-Building Construction.
INTRODUCTORY.
Definition of Framed Structures. - A framed structure, or ”truss” is a collection of straight “members” so joined together by pins or rivets as to form a rigid framework.
The office of such a structure may be either to transmit or transform motion or work, in which case it may form part or whole of a mechanism or machine ; or to resist the action of external forces tending to cause motion, in which case it is a structure of stability, or a statical construction. The principles which govern the discussion of the first case are therefore dynamical, and belong to the science of kinetics; while in the second they are those of statical equilibrium, and belong to the science of statics. The latter class of structures alone is discussed in this work.
The simplest kind of truss is a triangle, because that is the only figure whose shape cannot alter without changing the length of its sides. The triangle is thus the truss element, and all framed structures, no matter how complicated, which contain no superfluous members, may be considered as assemblages of triangles.
The members are always straight, because if a member is curved its axis does not coincide with the force at each end which it is designed to resist. The consequence is a tendency to deformation.
External Forces. - Every structure which we shall consider is acted upon by external forces, such as the loads applied at various points, the reactions of the supports, the weight of the structure itself, the force of the wind, the weight of snow, shocks, etc. These external forces act to distort the structure and the various members of which it is composed. As we shall see later, they can all be resolved into forces applied at the ends of each member. These forces we may distinguish by the effect they produce in the member.
We thus distinguish:
Force of tension, or tensile force, which acts to elongate a member in the direction of its length.
Force of compression, or compressive force, which acts to compress a member in the direction of its length.
Force of shear, or shearing force, which acts upon a member at right angles to its length.
Stress. - Let a member be acted upon at its ends by two equal and opposite external forces in the direction of its length, so that it is compressed or extended. Then, if equilibrium exists, it follows that at any imaginary section through the member there must exist two internal forces equal and opposite to the external forces at each end. These internal forces are called stresses.
We thus distinguish:
Stress of tension, or tensile stress, due to attraction between the particles, which resists a tensile force.
Stress of compression, or compressive stress, due to repulsion between the particles, which resists a compressive force.
Stress of shearing, or shearing stress, which resists a shearing force.
Stress, then, is always internal. We speak of the force on a member, and the resulting stress in the member.
Strain. - When a member is acted upon by two equal and opposite forces in the direction of its length it is compressed or elongated. This compression or elongation in opposition to existing stress is called strain.
If the resisting stress is compressive, the strain is a compression.
If the resisting stress is tensile, the strain is an extension.
If the resisting stress is shearing stress, the strain is a shearing strain.
Stress, then, is measured in units of force, as, for instance, in pounds; while strain is measured in units of length, as, for instance, in inches, and it must always be opposite in direction to coexisting stress.
Strut, Tie, Brace, Counterbrace, etc. — The word "member," which we have used already so many times, signifies a body whose length is generally great in comparison to its other dimensions. It is always straight. By the union of such members the structure is formed, and the whole combination is termed z framework. The member has different names according to the stress it is designed to resist. When it resists a compressive stress in general it is called a Strut, and when the strut is vertical it becomes a Post. When the stress is tensile the member is called a Tie, The term Brace is used to denote both struts and ties. When a brace is rendered capable of acting either as a strut or as a tie indifferently it is said to be counterbraced.
Beam, Girder. - In the case of a bending stress the member is called a Beam. When the beam is of considerable length and subjected to transverse stresses only it is called a Girder, and may be either solid or flanged. The cross-section of a solid girder is either rectangular, triangular, or round, or some modification of these forms. The flanged girder consists of one or two flanges of any desirable cross-section .united to a thin vertical web. The office of the flanges is to resist the compressive and tensile stresses. That of the web is to resist the shearing stress. The web may be continuous, as in plate girders, or open-work as in framed girders. It is with the latter only that we have to do in this work. The intersection of a brace with a flange is called an Apex. That portion of a flange between two adjacent apices is called a Bay or Panel.
Fundamental Principles. - All the various methods of investigating the conditions of stability of framed structures are based upon one of two principles - the so-called “principles of statical equilibrium”.
The first of these is as follows:
If any number of forces, all in the same plane, and acting at a common point of application or at different points of the same rigid body are in equilibrium, the algebraic sum of all their components in any given direction is zero. That is, the sum of all the components tending to cause motion in any one given direction is exactly equal to the sum of all those tending to cause motion in the precisely opposite direction.
This we shall call the “principle of the resolution of forces”.
The second principle is as follows:
If any number of forces all in the same plane and acting at a common point of application or at different points of the same rigid body, are in equilibrium, the algebraic sum of the moments of these forces, taken with reference to any point whatever in the plane of the forces is zero. That is, the sum of the moments tending to cause rotation in one direction is balanced by the sum of the moments tending to cause rotation in the other direction.
This we shall call the “principle of the equality of moments.”
Definition of “Moment.” - The “moment” of a force is the product of the force into its “lever arm,” The lever arm of a force with respect to any point, which is called the “centre of moments,” is the shortest distance of that point from the direction of the force, that is, it is the length of the perpendicular let fall from the point upon the force, prolonged in direction if necessary.
The office of such a structure may be either to transmit or transform motion or work, in which case it may form part or whole of a mechanism or machine ; or to resist the action of external forces tending to cause motion, in which case it is a structure of stability, or a statical construction. The principles which govern the discussion of the first case are therefore dynamical, and belong to the science of kinetics; while in the second they are those of statical equilibrium, and belong to the science of statics. The latter class of structures alone is discussed in this work.
The simplest kind of truss is a triangle, because that is the only figure whose shape cannot alter without changing the length of its sides. The triangle is thus the truss element, and all framed structures, no matter how complicated, which contain no superfluous members, may be considered as assemblages of triangles.
The members are always straight, because if a member is curved its axis does not coincide with the force at each end which it is designed to resist. The consequence is a tendency to deformation.
External Forces. - Every structure which we shall consider is acted upon by external forces, such as the loads applied at various points, the reactions of the supports, the weight of the structure itself, the force of the wind, the weight of snow, shocks, etc. These external forces act to distort the structure and the various members of which it is composed. As we shall see later, they can all be resolved into forces applied at the ends of each member. These forces we may distinguish by the effect they produce in the member.
We thus distinguish:
Force of tension, or tensile force, which acts to elongate a member in the direction of its length.
Force of compression, or compressive force, which acts to compress a member in the direction of its length.
Force of shear, or shearing force, which acts upon a member at right angles to its length.
Stress. - Let a member be acted upon at its ends by two equal and opposite external forces in the direction of its length, so that it is compressed or extended. Then, if equilibrium exists, it follows that at any imaginary section through the member there must exist two internal forces equal and opposite to the external forces at each end. These internal forces are called stresses.
We thus distinguish:
Stress of tension, or tensile stress, due to attraction between the particles, which resists a tensile force.
Stress of compression, or compressive stress, due to repulsion between the particles, which resists a compressive force.
Stress of shearing, or shearing stress, which resists a shearing force.
Stress, then, is always internal. We speak of the force on a member, and the resulting stress in the member.
Strain. - When a member is acted upon by two equal and opposite forces in the direction of its length it is compressed or elongated. This compression or elongation in opposition to existing stress is called strain.
If the resisting stress is compressive, the strain is a compression.
If the resisting stress is tensile, the strain is an extension.
If the resisting stress is shearing stress, the strain is a shearing strain.
Stress, then, is measured in units of force, as, for instance, in pounds; while strain is measured in units of length, as, for instance, in inches, and it must always be opposite in direction to coexisting stress.
Strut, Tie, Brace, Counterbrace, etc. — The word "member," which we have used already so many times, signifies a body whose length is generally great in comparison to its other dimensions. It is always straight. By the union of such members the structure is formed, and the whole combination is termed z framework. The member has different names according to the stress it is designed to resist. When it resists a compressive stress in general it is called a Strut, and when the strut is vertical it becomes a Post. When the stress is tensile the member is called a Tie, The term Brace is used to denote both struts and ties. When a brace is rendered capable of acting either as a strut or as a tie indifferently it is said to be counterbraced.
Beam, Girder. - In the case of a bending stress the member is called a Beam. When the beam is of considerable length and subjected to transverse stresses only it is called a Girder, and may be either solid or flanged. The cross-section of a solid girder is either rectangular, triangular, or round, or some modification of these forms. The flanged girder consists of one or two flanges of any desirable cross-section .united to a thin vertical web. The office of the flanges is to resist the compressive and tensile stresses. That of the web is to resist the shearing stress. The web may be continuous, as in plate girders, or open-work as in framed girders. It is with the latter only that we have to do in this work. The intersection of a brace with a flange is called an Apex. That portion of a flange between two adjacent apices is called a Bay or Panel.
Fundamental Principles. - All the various methods of investigating the conditions of stability of framed structures are based upon one of two principles - the so-called “principles of statical equilibrium”.
The first of these is as follows:
If any number of forces, all in the same plane, and acting at a common point of application or at different points of the same rigid body are in equilibrium, the algebraic sum of all their components in any given direction is zero. That is, the sum of all the components tending to cause motion in any one given direction is exactly equal to the sum of all those tending to cause motion in the precisely opposite direction.
This we shall call the “principle of the resolution of forces”.
The second principle is as follows:
If any number of forces all in the same plane and acting at a common point of application or at different points of the same rigid body, are in equilibrium, the algebraic sum of the moments of these forces, taken with reference to any point whatever in the plane of the forces is zero. That is, the sum of the moments tending to cause rotation in one direction is balanced by the sum of the moments tending to cause rotation in the other direction.
This we shall call the “principle of the equality of moments.”
Definition of “Moment.” - The “moment” of a force is the product of the force into its “lever arm,” The lever arm of a force with respect to any point, which is called the “centre of moments,” is the shortest distance of that point from the direction of the force, that is, it is the length of the perpendicular let fall from the point upon the force, prolonged in direction if necessary.
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