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The elements of graphic statics - Hoskins
THE ELEMENTS OF GRAPHIC STATICS
A TEXT BOOK FOR STUDENTS OF ENGINEERING
BY L. M. HOSKINS
Professor of Applied Mechanics in the Leland Stanford Junior University
THE MACMILLAN COMPANY; 1899
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The elements of graphic statics
PREFACE TO REVISED EDITION
The method of treatment adopted in this work is designed to meet the needs of the beginner. To this end the endeavor has been made to secure simplicity of presentation without sacrifice of logical rigor.
In scope, the work has been planned with reference to the requirements of students of engineering. This limits the development of the general theory to such principles and methods as are practically useful. It also excludes many applications which, though leading to practical results, are likely to prove useful and to save labor only in the hands of the expert in graphical methods. Graphic Statics is treated as a branch of Mechanics rather than of Geometry, and those beautiful developments whose chief interest is geometrical have not been included.
Although graphical methods are especially useful to the structural engineer, it is believed that students in all departments of engineering will find it profitable to become familiar with the general theory of complanar forces from the graphical side, as well as with the simpler applications to the determination of stresses in framed structures, and of centroids and moments of inertia of plane areas. The application to trusses carrying moving loads is of less general interest, its practical by utility being limited mainly to the solution of problems in bridge design. The methods developed in Chapter VII will therefore be of practical value mainly to students giving special attention to this branch of engineering.
In the present revised edition no change has been made in general plan, and few changes in the treatment adopted, except in the portions relating to beams and trusses carrying moving loads (Chapters VI and VII). These portions have been wholly re-written. It is believed that a substantial improvement has been made upon the methods hitherto used, particularly in the criterion for determining the position of a given load-series which causes maximum stress in any member of a truss. The improvement consists in generalization, which is believed to be gained without sacrifice of simplicity. The graphical method of applying the criterion in the case of trusses with parallel chords has been fully treated by Professor H. T. Eddy. The method here given applies without the restriction to parallel chords. The algebraic statement of the same criterion, as given in Art. 152, is also believed to be a useful generalization of the methods heretofore used. Whether the algebraic or the graphical treatment is preferred, a method is useful in proportion to its generality, provided this does not involve a loss of simplicity. There is a decided ad- vantage in the use of a single general equation, applicable to any member of any truss, instead of several particular equations, each applicable to a special member or to a special form of truss.
In scope, the work has been planned with reference to the requirements of students of engineering. This limits the development of the general theory to such principles and methods as are practically useful. It also excludes many applications which, though leading to practical results, are likely to prove useful and to save labor only in the hands of the expert in graphical methods. Graphic Statics is treated as a branch of Mechanics rather than of Geometry, and those beautiful developments whose chief interest is geometrical have not been included.
Although graphical methods are especially useful to the structural engineer, it is believed that students in all departments of engineering will find it profitable to become familiar with the general theory of complanar forces from the graphical side, as well as with the simpler applications to the determination of stresses in framed structures, and of centroids and moments of inertia of plane areas. The application to trusses carrying moving loads is of less general interest, its practical by utility being limited mainly to the solution of problems in bridge design. The methods developed in Chapter VII will therefore be of practical value mainly to students giving special attention to this branch of engineering.
In the present revised edition no change has been made in general plan, and few changes in the treatment adopted, except in the portions relating to beams and trusses carrying moving loads (Chapters VI and VII). These portions have been wholly re-written. It is believed that a substantial improvement has been made upon the methods hitherto used, particularly in the criterion for determining the position of a given load-series which causes maximum stress in any member of a truss. The improvement consists in generalization, which is believed to be gained without sacrifice of simplicity. The graphical method of applying the criterion in the case of trusses with parallel chords has been fully treated by Professor H. T. Eddy. The method here given applies without the restriction to parallel chords. The algebraic statement of the same criterion, as given in Art. 152, is also believed to be a useful generalization of the methods heretofore used. Whether the algebraic or the graphical treatment is preferred, a method is useful in proportion to its generality, provided this does not involve a loss of simplicity. There is a decided ad- vantage in the use of a single general equation, applicable to any member of any truss, instead of several particular equations, each applicable to a special member or to a special form of truss.
CONTENTS.
PART I. - GENERAL THEORY.
Chapter I. Definitions. - Concurrent Forces.
1. Preliminary Definitions
2. Composition of Concurrent Forces
3. Equilibrium of Concurrent Forces
4. Resolution of Concurrent Forces
Chapter II Non-concurrent Forces.
1. Composition of Non-concurrent Forces Acting on the Same Rigid Body
2. Equilibrium of Non-concurrent Forces
3. Resolution into Non-concurrent Systems
4. Moments of Forces and of Couples
5. Graphic Determination of Moments
6. Summary of Conditions of Equilibrium
Chapter III. Internal Forces and Stresses
1. External and Internal Forces
2. External and Internal Stresses
3. Determination of Internal Stresses
PART II. - STRESSES IN SIMPLE STRUCTURES.
Chapter IV. Introductory.
1. Outline of Principles and Methods
Chapter V. Roof Trusses. - Framed Structures Sustaining - Stationary Loads.
1. Loads on Roof Trusses
2. Roof Truss with Vertical Loads
3. Stresses Due to Wind Pressure
4. Maximum Stresses
5. Cases Apparently Indeterminate
6. Three-hinged Arch
7. Counterbracing
Chapter VI Simple Beams.
1. General Principles
2. Beam Sustaining Fixed Loads
3. Beam Sustaining Moving Loads
Chapter VII Trusses Sustaining Moving Loads.
1. Bridge Loads
2. Truss Regarded as a Beam
3. Truss Sustaining Any Series of Moving Loads
4. Truss with Subordinate Bracing
5. Uniformly Distributed Moving Load
PART III - CENTROIDS AND MOMENTS OF INERTIA.
Chapter VIII. Centroids.
1. Centroid of Parallel Forces
2. Center of Gravity - Definitions and General Principles
3. Centroids of Lines and of Areas
Chapter IX. Moments of Inertia.
1. Moments of Inertia of Forces
2. Moments of Inertia of Plane Areas
Chapter X. Curves of Inertia.
1. General Principles
2. Inertia-Ellipses for Systems of Forces
3. Inertia-Curves for Plane Areas
CHAPTER VI - SIMPLE BEAMS
General Principles.
Classification of Beams. A beam has been defined in Art. 79. Beams may be treated in two main classes, the basis of classification being that described in Art. 75. These two classes will be called simple and non-simple beams respectively. The present chapter deals only with simple beams, the definition of which may be stated as follows:
A simple beam is one so supported that it may be regarded as a rigid body in determining the reactions.
A simple beam may rest on two supports at the ends; or it may overhang one or both supports.
A cantilever is any beam projecting beyond its supports. Such a beam may be either simple or non-simple.
A continuous beam is one resting on more than two supports. Such a beam is non-simple.
A beam may be supported in several ways. It is simply supported at a point when it rests against the support so that the reaction has a fixed direction. It is constrained at a point if so held that the tangent to the axis of the beam at that point must maintain a fixed direction. If hinged at a support, the reaction may have any direction. We shall deal mainly with the case of simple support.
In what follows it will be assumed that the beam rests in a horizontal position, since this is the usual case.
External Shear, Resisting Shear, and Shearing Stress.
- The external shear at any section of a beam is the algebraic sum of the external vertical forces acting on the portion of the beam to the left of the section.
The resisting shear at any section is the algebraic sum of the internal vertical forces in the section acting on the portion of the beam to the left, and exerted by the portion to the right of the section.
The shearing stress at any section is the stress which consists of the internal vertical forces in the section, exerted by the two portions of the beam upon each other. It consists of the resisting shear and the reaction to it. (See Art. 63.)
Let AB (Fig. 43) be a beam in equilibrium under the action of any external forces. At any point in its length, as C, conceive a plane to be passed perpendicular to the axis of the beam, and consider the portion AC, to the left of the section. The principles of equilibrium apply to the body AC, and the external forces acting upon it include, besides those forces to the left of C that are external to the whole bar, certain forces acting across the section at C that are internal to AB, but external to AC. (Art. 61.) These latter forces comprise that constituent of the internal stress between AC and CB which is exerted by CB upon AC.
Represent by V the algebraic sum of the resolved parts in the vertical direction of all forces acting on AB to the left of the section at C, upward forces being called positive. V is the external shear at the given section as above denned.
Since the body AC is in equilibrium, condition (i), Art. 58, requires that the algebraic sum of the resolved parts in the vertical direction of all forces acting on it must equal zero. Hence the forces acting on AC m the section at C must have a vertical component equal to V. This vertical component is called the resisting shear in the given section. This resisting shear is one of the forces of a stress of which the other is an equal and opposite force exerted by A C upon CB. This stress is called the shearing stress in the section, and is called positive when it resists a tendency of AC to move upward, and of CB to move downward.
Bending Moment, Resisting Moment, and Stress Moment,
The bending moment at any section of a beam is the algebraic sum of the moments of all the external forces acting on the portion of the beam to the left of the section; the origin of moments being taken in the section.
The resisting moment at any section is the algebraic sum of the moments of the internal forces in the section acting on the portion of the beam to the left, and exerted by the portion to the right of the section; the origin of moments being the same as for bending moment.
The stress moment or moment of internal stress at any section consists of the two equal and opposite moments of the forces exerted across the section by the two portions of the beam upon each other.
Referring again to Fig. 43, let us analyze further the forces in the section at C. Applying to the body AC the second condition of equilibrium ((2) of Art. 58), and taking an origin at a point in the section, we see that the algebraic sum of the moments of all the external forces acting on the beam to the left of the section plus the sum of the moments of the internal forces acting on AC in the section must equal zero. The former sum is denned as the bending moment at the given section. Represent it by M. The latter sum is denned as the resisting moment at the section, and must be equal to M, by the above principle.
We have thus far referred only to the internal forces exerted by CB upon AC; but evidently the equal and opposite forces exerted by AC upon CB have a moment numerically equal to M. The equal and opposite moments of the equal and opposite forces of the stress in the section together constitute the stress-moment in the section.
If the external forces applied to the beam are all vertical, the value of M will be the same at whatever point of the section the origin is taken; since the arm of each force will be the same for all origins in the same vertical line. If the loads and reactions are not all vertical, the value of M will generally depend upon what point in the section is taken as the origin of moments.
117. Curves of Shear and Bending Moment. The curve of shear for a beam is a curve whose abscissas are parallel to the axis of the beam, and whose ordinate at any point represents the external shear at the corresponding section of the beam.
Classification of Beams. A beam has been defined in Art. 79. Beams may be treated in two main classes, the basis of classification being that described in Art. 75. These two classes will be called simple and non-simple beams respectively. The present chapter deals only with simple beams, the definition of which may be stated as follows:
A simple beam is one so supported that it may be regarded as a rigid body in determining the reactions.
A simple beam may rest on two supports at the ends; or it may overhang one or both supports.
A cantilever is any beam projecting beyond its supports. Such a beam may be either simple or non-simple.
A continuous beam is one resting on more than two supports. Such a beam is non-simple.
A beam may be supported in several ways. It is simply supported at a point when it rests against the support so that the reaction has a fixed direction. It is constrained at a point if so held that the tangent to the axis of the beam at that point must maintain a fixed direction. If hinged at a support, the reaction may have any direction. We shall deal mainly with the case of simple support.
In what follows it will be assumed that the beam rests in a horizontal position, since this is the usual case.
External Shear, Resisting Shear, and Shearing Stress.
- The external shear at any section of a beam is the algebraic sum of the external vertical forces acting on the portion of the beam to the left of the section.
The resisting shear at any section is the algebraic sum of the internal vertical forces in the section acting on the portion of the beam to the left, and exerted by the portion to the right of the section.
The shearing stress at any section is the stress which consists of the internal vertical forces in the section, exerted by the two portions of the beam upon each other. It consists of the resisting shear and the reaction to it. (See Art. 63.)
Let AB (Fig. 43) be a beam in equilibrium under the action of any external forces. At any point in its length, as C, conceive a plane to be passed perpendicular to the axis of the beam, and consider the portion AC, to the left of the section. The principles of equilibrium apply to the body AC, and the external forces acting upon it include, besides those forces to the left of C that are external to the whole bar, certain forces acting across the section at C that are internal to AB, but external to AC. (Art. 61.) These latter forces comprise that constituent of the internal stress between AC and CB which is exerted by CB upon AC.
Represent by V the algebraic sum of the resolved parts in the vertical direction of all forces acting on AB to the left of the section at C, upward forces being called positive. V is the external shear at the given section as above denned.
Since the body AC is in equilibrium, condition (i), Art. 58, requires that the algebraic sum of the resolved parts in the vertical direction of all forces acting on it must equal zero. Hence the forces acting on AC m the section at C must have a vertical component equal to V. This vertical component is called the resisting shear in the given section. This resisting shear is one of the forces of a stress of which the other is an equal and opposite force exerted by A C upon CB. This stress is called the shearing stress in the section, and is called positive when it resists a tendency of AC to move upward, and of CB to move downward.
Bending Moment, Resisting Moment, and Stress Moment,
The bending moment at any section of a beam is the algebraic sum of the moments of all the external forces acting on the portion of the beam to the left of the section; the origin of moments being taken in the section.
The resisting moment at any section is the algebraic sum of the moments of the internal forces in the section acting on the portion of the beam to the left, and exerted by the portion to the right of the section; the origin of moments being the same as for bending moment.
The stress moment or moment of internal stress at any section consists of the two equal and opposite moments of the forces exerted across the section by the two portions of the beam upon each other.
Referring again to Fig. 43, let us analyze further the forces in the section at C. Applying to the body AC the second condition of equilibrium ((2) of Art. 58), and taking an origin at a point in the section, we see that the algebraic sum of the moments of all the external forces acting on the beam to the left of the section plus the sum of the moments of the internal forces acting on AC in the section must equal zero. The former sum is denned as the bending moment at the given section. Represent it by M. The latter sum is denned as the resisting moment at the section, and must be equal to M, by the above principle.
We have thus far referred only to the internal forces exerted by CB upon AC; but evidently the equal and opposite forces exerted by AC upon CB have a moment numerically equal to M. The equal and opposite moments of the equal and opposite forces of the stress in the section together constitute the stress-moment in the section.
If the external forces applied to the beam are all vertical, the value of M will be the same at whatever point of the section the origin is taken; since the arm of each force will be the same for all origins in the same vertical line. If the loads and reactions are not all vertical, the value of M will generally depend upon what point in the section is taken as the origin of moments.
117. Curves of Shear and Bending Moment. The curve of shear for a beam is a curve whose abscissas are parallel to the axis of the beam, and whose ordinate at any point represents the external shear at the corresponding section of the beam.
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