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Statics by algebraic and graphic methods
STATICS BY ALGEBRAIC AND GRAPHIC METHODS
Intended primarily for students of engineering and architecture
BY LEWIS JEROME JOHNSON
PROFESSOR OF CIVIL ENGINEERING IN HARVARD UNIVERSITY, MEMBER OF THE AMERICAN SOCIETY OF CIVIL ENGINEERS
NEW YORK: JOHN WILEY & SONS; 1908
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Statics by algebraic and graphic methods
PREFACE
This book has been written in the hope of helping students of engineering and architecture to acquire a knowledge of Statics which will include the power to apply it correctly in professional work. To this end an attempt has been made to carry out several specific purposes, prominent among which may be mentioned the following:
1. To give much attention to the starting points of the science, and to make as clear as possible the course of deduction there from.
2. To point out the inherent mathematical limitations of pure Statics, and to show how all its important problems are solved.
3. To develop algebraic and graphic methods of solution, or, if one prefer the terms, Analytical and Graphical Statics, side by side and with equal thoroughness.
4. To present a graded set of problems illustrating not only universal principles but also how Statics is used in engineering practice.
5. Finally, to keep the book of a size commensurate with the small amount of new matter which the reader, versed in the simplest operations of elementary mathematics, need master to gain the desired end.
It may be pointed out that the phrases Analytical Statics and Graphical Statics are avoided. The ground for this is that there seems to be no necessity for using them, in this work at least, and that the terms seem objectionable as tending to obscure the unity of Statics, and to produce the impression that two merely alternative methods of procedure from identical premises and of identical mathematical significance are loosely connected if not actually distinct branches of the science.
The subject will be found developed in the following pages in such a way as to make it possible to solve problems from the outset by both methods in parallel (as in the plates), and the practice of making such double solution is believed to be of great value not only for the drill of checking one's own work, but also for the clearer light in which each method is seen by being kept in close relation to the other. Moreover, as the student checks the correctness of his own work, it is possible, even with classes of upwards of one hundred, to assign different problem data to each student, and still keep the labor of inspecting students' work at a minimum. After some experience with the two methods side by side, it is believed that practice should be had in the rapid solution of a large number of varied problems such as can be found in the familiar works on Statics, in which a single solution by either method is accepted and in which the student judges the correctness of his results not only by examination of his work, but by reflection upon their reasonableness under the conditions.
PREFACE TO THE SECOND EDITION.
The new matter in the Second Edition, although slight in amount, is intended to increase materially the effectiveness of the book. It is believed that it will be welcomed by both instructors and students.
The most important additions are a detailed and time-tested scheme for assigning individual data for the exercises in the text, and a copious collection of practice problems with answers. This makes it possible for the first time to carry out the suggestions of the Preface with regard to these matters without re- course to other publications.
Other additions are a short syllabus intended to facilitate review and to enforce certain important points and seven new double-page plates giving solutions of exercises which have been found especially to need such complete elucidation. Several pages of the text have been slightly rewritten, a few of the cuts have been redrawn, and all known typographical errors have been eliminated.
The new matter appears almost wholly in appendices, and the page and section numbering of the first edition remain unchanged.
The collection of practice problems is regarded as of so much importance as to merit a few lines of special comment. These problems have been gathered from a great variety of sources, but most of them have appeared in a couple of pamphlets published several years ago for the author's classes. Thirteen of them which appear now in this list for the first time have been taken by permission-from Cox's Mechanics.
It will be observed that this collection of problems is massed in an isolated group, is mainly unclassified, that a number of repetitions occur, and that little or no specific aid is offered for the solution of particular problems. A student thus acquires practice in meeting and solving problems upon their individual merits, with a minimum of suggestion from neighboring matter, much as he will have to do in engineering practice. The repetitions afford a drill which rarely proves amiss. It will be noted further that the problems are almost wholly of a sort in which the graphic methods offer no special advantage, the exercises in the text having been found to furnish ample drill in the use of the graphic methods.
The author has found that an acceptable way to use such a set of problems with his classes is to assign lessons from them upon which written recitations or tests are held at frequent intervals. Heavy emphasis is laid upon successful work in these tests, success being understood to involve numerical accuracy as well as correct use of statical principles.
The faithful solution of a large number of such problems contributes to a vigor and likelihood of success in applying the subject which is not only indispensable, but is hardly to be obtained in any other way.
The most important additions are a detailed and time-tested scheme for assigning individual data for the exercises in the text, and a copious collection of practice problems with answers. This makes it possible for the first time to carry out the suggestions of the Preface with regard to these matters without re- course to other publications.
Other additions are a short syllabus intended to facilitate review and to enforce certain important points and seven new double-page plates giving solutions of exercises which have been found especially to need such complete elucidation. Several pages of the text have been slightly rewritten, a few of the cuts have been redrawn, and all known typographical errors have been eliminated.
The new matter appears almost wholly in appendices, and the page and section numbering of the first edition remain unchanged.
The collection of practice problems is regarded as of so much importance as to merit a few lines of special comment. These problems have been gathered from a great variety of sources, but most of them have appeared in a couple of pamphlets published several years ago for the author's classes. Thirteen of them which appear now in this list for the first time have been taken by permission-from Cox's Mechanics.
It will be observed that this collection of problems is massed in an isolated group, is mainly unclassified, that a number of repetitions occur, and that little or no specific aid is offered for the solution of particular problems. A student thus acquires practice in meeting and solving problems upon their individual merits, with a minimum of suggestion from neighboring matter, much as he will have to do in engineering practice. The repetitions afford a drill which rarely proves amiss. It will be noted further that the problems are almost wholly of a sort in which the graphic methods offer no special advantage, the exercises in the text having been found to furnish ample drill in the use of the graphic methods.
The author has found that an acceptable way to use such a set of problems with his classes is to assign lessons from them upon which written recitations or tests are held at frequent intervals. Heavy emphasis is laid upon successful work in these tests, success being understood to involve numerical accuracy as well as correct use of statical principles.
The faithful solution of a large number of such problems contributes to a vigor and likelihood of success in applying the subject which is not only indispensable, but is hardly to be obtained in any other way.
CONTENTS.
PART I.
GENERAL PRINCIPLES AND METHODS.
1. Algebraic and Graphic Methods Compared
2. History of Graphic Methods in Statics
CHAPTER I. DEFINITIONS AND PRELIMINARIES.
3. Mechanics and its Subdivisions
4. Rigid Body. Particle
5. Rest and Motion
6. Force
7. Elements of Force
8. Magnitude
9. Direction and Sense
10. Point of Application
11. Coplanar and Non-Coplanar Forces
12. Concurrent and Non- Concurrent Forces
13. Equilibrium
14. Equivalence
15. Resultant and Equilibrant
16. Components
17. Relation between Two Components and their Resultant. Parallelogram of Forces
Exercise 1
18. Couples and their Moments
19. Center of Rotation for a Couple
20. Single Force Always Replaceable by a Single Force and a Couple
21. Moment of a Force
Exercise 2
22. Propositions Regarding the Moment of a Force
23. Proposition Regarding the Moment of a Couple
24. Sets of Forces Classified
25. Conditions of Equivalence
CHAPTER II. NOTATION AND CONVENTIONS.
26. Notation. Conventions Regarding Elements of Forces
27. Space Diagram and Magnitude Diagram
28. Illustration of Scheme of Notation
CHAPTER III. PARALLELOGRAM OF FORCES AND ITS DERIVATIVES, THE TRIANGLE OF FORCES, THE MAGNITUDE POLYGON, AND THE STRING POLYGON.
29. Demonstration of the Parallelogram of Forces
30. Triangle of Forces
31. Polygon of Forces. Location of Resultant of Inclined Forces. Magnitude Polygon
32. String Polygon
33. Additional Remarks on the String Polygon
CHAPTER IV. ALGEBRAIC AND GRAPHIC STATEMENTS OF THE CONDITIONS OF EQUILIBRIUM WITH APPLICATIONS.
34. Nature of Statical Problems
35. Condition of Equilibrium
36. Algebraic Statement of (A) and (B] (the Conditions of Equilibrium) for Coplanar Forces
37. Graphic Interpretation of (A) and (B)
38. Exercises in the Composition of Forces
Exercises 3-7
39. Generalization of the Three Classes of Resultants
40. Establishment of Equilibrium by Use of Moments Alone
41. Determination of Magnitudes by Single Moment Equations Alone
42. Convention as to Algebraic Signs in Moment Equations
43. Six Methods of Stating the Conditions of Equilibrium
CHAPTER V. SCOPE OF PURE STATICS.
44. General Survey of the Scope of Pure Statics
45. The Four Cases
CHAPTER VI SOLUTIONS OF STATICAL PROBLEMS, WITH SPECIAL REFERENCE TO THE FOUR MOST IMPORTANT CASES.
46. The Solution of Statical Problems
47. Solution of Case 1
48. Solution of Case 2a
48 a. Solution of Case 2b
49. Solution of Case 3
50. Solution of Case 4
50 a. Remarks on Cases 3 and 4
Exercises 8-15
CHAPTER VII. ADDITIONAL GENERAL TOPICS AND PROCESSES.
51. Graphic Representation of the Moment of a Force
52. String Polygon for Parallel Forces a Diagram of Moments
53. Remarks on the String Polygon as a Diagram of Moments
54. To Pass a String Polygon Through Three Given Points
Exercise 16
55. An Alternative View oi the Rays and the String Polygon
PART II
CHAPTER VIII. CENTERS OF GRAVITY.
56. Center of Gravity
Exercise 17
CHAPTER IX. STRESS.
57. External and Internal Forces
58. Stress
59. Kinds of Stress
60. Combined Stresses
61. Further Particulars Relating to Stress
CHAPTER X. STRUCTURES
62. Definitions
63. Extent of Approximation to True Frames in Practice
64. Loads Applied Elsewhere than at Joints
65. Frames in General
66. Loads
67. Stresses in Structures
CHAPTER XI. STRESSES IN NON-FRAMED STRUCTURES.
68. Stresses in Non-Framed Structures
Exercise
69. Shear Diagrams
70. Flexure Diagrams
Exercise 19
71. Connection Between Shear and Change in Flexure
Exercise 20
CHAPTER XII. STRESSES IN FRAMED STRUCTURES.
72. Stresses in Framed Structures
73. Method of Sections
74. Method for Determining all the Stresses in a Frame under a Given Load
75. Example
76. Stress Diagrams
77. General Instructions Regarding Exercises Involving Stress Diagrams
78. Special Instructions Regarding Exercises 21-23
Exercises 21-23
CHAPTER XIII. ADDITIONAL TOPICS AND EXAMPLES.
79. Complications in Connection with the Analysis of Frames
80. Reactions due to Non-vertical Forces
81. The" Fink Truss
Exercise 24
82. Triangular Frame with Trussed Top Chord 108
Exercise 25
83. Counters
Exercise 26
84. Bent of a Mill Building no
Exercise 27
85. Cantilever Bridge
Exercise 28
86. Three-Hinged Arch
87. Line of Pressure
Exercises 29 and 30
88. Hammer Beam Truss
Exercise 31
89. Stresses Due to Moving Loads
90. Stability of a Masonry Dam
Exercise 32
91. Action and Reaction Not Necessarily Normal to the Surface of Contact
92. Friction
APPENDICES
APPENDIX I. Additional Regarding the Scope of Pure Statics
APPENDIX II. Syllabus
APPENDIX III. Individual Data for Exercises
APPENDIX IV. Practice Problems with Answers
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