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Mechanics of materials - Church
MECHANICS OF MATERIALS
A Treatise on the Elasticity and Strength of Beams, Columns, Arches, Etc., for students of engineering.
BY IRVING P. CHURCH,
NEW YORK; JOHN WILEY & SONS; 1887
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Mechanics of materials
PREFACE.
For the engineering student, pursuing the study of Applied Mechanics as part of his professional training, and not as additional mathematical culture, not only is a thoroughly systematic, clear, and consistent treatment of the subject quite essential, but one which presents the quantities and conceptions involved in as practical and concrete a form as possible, with all the aids of the printer's and engraver's arts; and especially one which, besides showing the derivation of formulae from principles, illustrates, inculcates, and lays stress on correct numerical substitution and the consistent and proper use of units of measurement; for without this no reliable results can be reached, and the principal object of these formulae is frustrated.
With these requirements in view, and aided by the experience of ten years in teaching the Mechanics of Engineering at this institution, the writer has been led to prepare the present work, in which attention is called to the following features:
The diagrams are very numerous (about one to every page; an appeal to the eye is often worth a page of verbal description).
The symbols for distances, angles, forces, etc., used in the algebraic work are, as far as possible, inserted directly in the diagrams, rendering the latter full and explicit, and thus saving time and mental effort to the student. In problems in Dynamics three kinds of arrows are used to distinguish forces, velocities, and accelerations, respectively, and thus to prevent confusion of ideas.
Illustrations and examples of a practical nature, both algebraic and numerical, are of frequent occurrence.
Formulae are divided into two classes; those (homogeneous) admitting of the use of any system of units whatever for measurements of force, space, mass, and time, in numerical substitution; and those which are true for specified units only. Attention is repeatedly directed to the matter of correct numerical substitution, especially in Dynamics, where time and mass, as well as force and space, are among the quantities considered. The importance, in this connection, of frequent mention of the quality of the various kinds of quantity employed, is also recognized, and a corresponding phraseology adopted.
The definition of force (§8) is made to include and illustrate Newton's law of action and reaction, the misconception of which leads to such lengthy discussions in technical journals every few years.
In the matter of "Centrifugal force," the artificial method, so commonly adopted, of regarding a particle moving uniformly in a circle as in equilibrium, i. e., acted on by a balanced system of forces, one of which is the "Centrifugal force," has been avoided, as being at variance with a system of Mechanics founded on Newton's laws, according to the first of which a particle moving in any other than a straight line cannot be in equilibrium. In such a system of Mechanics nothing can be recognized as a force which is not a definite poll, posh, pressure, rub, attraction or repulsion, of one body upon, or against, another. '
It is true that the artificial nature of the method referred to is in some text-books fully explained in the context, (in Goodeve's Steam Engine, for instance, in treating the governor ball,) but is too often not mentioned at all, so that the student risks being led into error in attempting kindred problems by what would then seem to him correct methods.
The general theorem of Work and Energy in machines is developed gradually by definite and limited steps, in preference to giving a single demonstration which, from its generality, might be too vague and abstruse to be readily grasped by the student.
In the use of the Calculus, (in the elements of which the student is supposed to have had the training usually given in technical schools by the end of the second year) the integral sign is always used to indicate summation (except on p. 857) while the name of anti-derivative of & given function (of one variable) has been adopted for that function whose derivative, or differential co-efficient, is the given function.
The signs and are used for perpendicular and parallel, respectively. In Torsion and Flexure of Beams, the well worn and simple theories of Navier have been thought sufficient for establishing practical formulae for safe loads and deflections of beams and shafts; and prominence has been given to the methods of designing the cross-sections and riveting of built-beams and plate-girders, forming the basis of the tables and rules usually given in the pocket-books of our iron and steel manufacturers.
The graphics of curved beams or arch ribs is made to precede that of the straight girder, since the treatment of the latter as a particular case of the former is then a comparatively simple matter. Hence Prof. Eddy's methods* (inserted by his kind permission) for the arch rib of hinged ends, and also that of fixed ends, are presented as special geometrical devices, instead of being based on Prof. Eddy's general theorem (involving a straight girder of the same section and mode of support).
With these requirements in view, and aided by the experience of ten years in teaching the Mechanics of Engineering at this institution, the writer has been led to prepare the present work, in which attention is called to the following features:
The diagrams are very numerous (about one to every page; an appeal to the eye is often worth a page of verbal description).
The symbols for distances, angles, forces, etc., used in the algebraic work are, as far as possible, inserted directly in the diagrams, rendering the latter full and explicit, and thus saving time and mental effort to the student. In problems in Dynamics three kinds of arrows are used to distinguish forces, velocities, and accelerations, respectively, and thus to prevent confusion of ideas.
Illustrations and examples of a practical nature, both algebraic and numerical, are of frequent occurrence.
Formulae are divided into two classes; those (homogeneous) admitting of the use of any system of units whatever for measurements of force, space, mass, and time, in numerical substitution; and those which are true for specified units only. Attention is repeatedly directed to the matter of correct numerical substitution, especially in Dynamics, where time and mass, as well as force and space, are among the quantities considered. The importance, in this connection, of frequent mention of the quality of the various kinds of quantity employed, is also recognized, and a corresponding phraseology adopted.
The definition of force (§8) is made to include and illustrate Newton's law of action and reaction, the misconception of which leads to such lengthy discussions in technical journals every few years.
In the matter of "Centrifugal force," the artificial method, so commonly adopted, of regarding a particle moving uniformly in a circle as in equilibrium, i. e., acted on by a balanced system of forces, one of which is the "Centrifugal force," has been avoided, as being at variance with a system of Mechanics founded on Newton's laws, according to the first of which a particle moving in any other than a straight line cannot be in equilibrium. In such a system of Mechanics nothing can be recognized as a force which is not a definite poll, posh, pressure, rub, attraction or repulsion, of one body upon, or against, another. '
It is true that the artificial nature of the method referred to is in some text-books fully explained in the context, (in Goodeve's Steam Engine, for instance, in treating the governor ball,) but is too often not mentioned at all, so that the student risks being led into error in attempting kindred problems by what would then seem to him correct methods.
The general theorem of Work and Energy in machines is developed gradually by definite and limited steps, in preference to giving a single demonstration which, from its generality, might be too vague and abstruse to be readily grasped by the student.
In the use of the Calculus, (in the elements of which the student is supposed to have had the training usually given in technical schools by the end of the second year) the integral sign is always used to indicate summation (except on p. 857) while the name of anti-derivative of & given function (of one variable) has been adopted for that function whose derivative, or differential co-efficient, is the given function.
The signs and are used for perpendicular and parallel, respectively. In Torsion and Flexure of Beams, the well worn and simple theories of Navier have been thought sufficient for establishing practical formulae for safe loads and deflections of beams and shafts; and prominence has been given to the methods of designing the cross-sections and riveting of built-beams and plate-girders, forming the basis of the tables and rules usually given in the pocket-books of our iron and steel manufacturers.
The graphics of curved beams or arch ribs is made to precede that of the straight girder, since the treatment of the latter as a particular case of the former is then a comparatively simple matter. Hence Prof. Eddy's methods* (inserted by his kind permission) for the arch rib of hinged ends, and also that of fixed ends, are presented as special geometrical devices, instead of being based on Prof. Eddy's general theorem (involving a straight girder of the same section and mode of support).
CONTENTS
PART III. STRENGTH OF MATERIALS. (OR MECHANICS OF MATERIALS)
CHAPTER I. ELEMENTARY STRESSES AND STRAINS.
Stress and Strain; of Two Kinds. Oblique Section of Rod in Tension. Hooke's Law. Elasticity. Safe Limit. Elastic Limit. Rupture. Modulus of Elasticity. Isotropes. Resilience. Internal Stress. Temperature Stresses
TENSION.
Hooke's Law by Experiment Strain Diagrams. Lateral Contraction. Modulus of Tenacity. Resilience of Stretched Prism. Load Applied Suddenly. Prism Under Its Own Weight. Solid of Uniform Strength. Temperature Stresses
COMPRESSION OF SHORT BLOCKS.
Short and Long Columns. Remarks on Crushing
EXAMPLES IN TENSION AND COMPRESSION.
Tables. Examples. Factor of Safety. Practical
SHEARING.
Rivets. Shearing Distortion. Table. Punching. Examples
CHAPTER II. TORSION.
Angle and Moment of Torsion. Torsional Strength, Stiffness, and Resilience. Non-Circular Shafts. Transmission of Power. Autographic Testing Ma-
chine. Examples
CHAPTER III. FLEXURE OF HOMOGENEOUS PRISMS UNDER PERPENDICULAR FORCES IN ONE PLANE.
The Common Theory. Elastic Forces. Neutral Axis. The "Shear" and “Moment” Flexural Strength and Stiffness. Radius of Curvature. Resilience
ELASTIC CURVES.
Single Central Load; at Rest, and Applied Suddenly. Eccentric Load. Uniform Load. Cantilever
SAFE LOADS.
Maximum Moment. Shear - x-Derivative of the Moment. Simple Beams With Various Loads. Comparative Strength and Stiffness of Rectangular Beams. Moments of Inertia. "Shape Iron." I- Beams. Etc. Cantilevers. Tables. Numerical Examples
SHEARING STRESSES IN FLEXURE.
Shearing Stress Parallel to Neutral Surface; and in Cross Section. Web of I-Beam. Riveting of Built Beams
SPECIAL PROBLEMS IN FLEXURE.
Designing Sections of Built Beams. Moving Loads. Special Cases of Quiescent Loads. Hydrostatic Load. Derivatives of Ordinate of Elastic Curve. Weight
Falling on Beam. Crank Shaft. Other Shafts. 310
CHAPTER IV. FLEXURE; CONTINUOUS GIRDERS.
Analytical Treatment of Symmetrical Cases of Beams on Three Supports; also, Built in
THE DANGEROUS SECTION IN NON-PRISMATIC BEAMS.
Double Truncated Wedge, Pyramid, and Cone
NON-PRISMATIC BEAM8 OF UNIFORM STRENGTH.
Parabolic and Wedge-Shaped Beams. I-Beam. Elliptical Beam and Cantilevers
DEFLECTION OF BEAMS OF UNIFORM STRENGTH.
Parabolic and Wedge-Shaped Beams. Special Problems
CHAPTER V. FLEXURE OF PRISMATIC BEAMS UNDER OBLIQUE FORCES.
Gravity and Neutral Axes. Shear, Thrust, and Stress-Couple. Strength and Stiffness of Oblique Cantilever. Hooks. Crane
CHAPTER VI. FLEXURE OF LONG COLUMNS.
End-Conditions. Euler's Formula. “Incipient Flexure.” Hodgkinson's Formulae. Rankine's Formula. Radii of Gyration Built Columns. Trussed Girders. Buckling of Web Plates. Examples
CHAPTER VII. LINEAR ARCHES (OF BLOCKWORK.)
Inverted Catenary. Parabolic Arch. Circular Arch. Transformed Catenary as Arch
CHAPTER VIII. ELEMENTS OF GRAPHICAL STATICS.
Force Polygons. Concurrent and Non-Concurrent Forces in a Plane. Force Diagrams. Equilibrium Polygons. Constructions for Resultant, Pier-Reactions, and Stresses in Roof Truss. Bow's Notation. The Special Equilibrium Polygon
CHAPTER IX. GRAPHICAL STATICS OF VERTICAL FORCES.
Jointed Rods. Centre of Gravity. Useful Relations Between Force Diagrams and Their Equilibrium Polygons
CHAPTER X. RIGHT ARCHES OF MASONRY.
Definitions. Mortar and Friction. Pressure in Joints. Conditions of Safe Equilibrium. True Linear Arch. Arrangement of Data for Graphic Treatment. Graphical Treatment of Arch. Symmetrical and Unsymmetrical Cases
CHAPTER XI. ARCH-RIBS.
Mode of Support. Special Equilibrium Polygon and its Force Diagram. Change in Angle Between Rib Tangents. Displacement of Any Point on Rib. Graphical Arithmetic. Summation of Products. Moment of Inertia by Graphics. Classification of Arch-Ribs. Prof. Eddy's Graphical Method for Arch-Ribs of Hinged Ends; and of Fixed Ends. Stress Diagrams. Temperature Stresses. Braced
Arches
HORIZONTAL STRAIGHT GIRDERS.
Prismatic Girder Supported at the Extremities, also with Ends Fixed Horizontally
CHAPTER XII. GRAPHICS OF CONTINUOUS GIRDERS.
The Elastic Curve an Equilibrium Polygon. Mohr's Theorem. Example and Numerical Case. End-Tangents. Re-arrangement of Moment- Area. Positive Moment-Areas; Amount and Gravity Vertical. Construction of the "False Polygons" in Any Given Case. Moment Curves, Shears, and Reactions. Numerical Example in Detail.
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CHAPTER I. ELEMENTARY STRESSES AND STRAINS.
Stress and Strain; of Two Kinds. Oblique Section of Rod in Tension. Hooke's Law. Elasticity. Safe Limit. Elastic Limit. Rupture. Modulus of Elasticity. Isotropes. Resilience. Internal Stress. Temperature Stresses
TENSION.
Hooke's Law by Experiment Strain Diagrams. Lateral Contraction. Modulus of Tenacity. Resilience of Stretched Prism. Load Applied Suddenly. Prism Under Its Own Weight. Solid of Uniform Strength. Temperature Stresses
COMPRESSION OF SHORT BLOCKS.
Short and Long Columns. Remarks on Crushing
EXAMPLES IN TENSION AND COMPRESSION.
Tables. Examples. Factor of Safety. Practical
SHEARING.
Rivets. Shearing Distortion. Table. Punching. Examples
CHAPTER II. TORSION.
Angle and Moment of Torsion. Torsional Strength, Stiffness, and Resilience. Non-Circular Shafts. Transmission of Power. Autographic Testing Ma-
chine. Examples
CHAPTER III. FLEXURE OF HOMOGENEOUS PRISMS UNDER PERPENDICULAR FORCES IN ONE PLANE.
The Common Theory. Elastic Forces. Neutral Axis. The "Shear" and “Moment” Flexural Strength and Stiffness. Radius of Curvature. Resilience
ELASTIC CURVES.
Single Central Load; at Rest, and Applied Suddenly. Eccentric Load. Uniform Load. Cantilever
SAFE LOADS.
Maximum Moment. Shear - x-Derivative of the Moment. Simple Beams With Various Loads. Comparative Strength and Stiffness of Rectangular Beams. Moments of Inertia. "Shape Iron." I- Beams. Etc. Cantilevers. Tables. Numerical Examples
SHEARING STRESSES IN FLEXURE.
Shearing Stress Parallel to Neutral Surface; and in Cross Section. Web of I-Beam. Riveting of Built Beams
SPECIAL PROBLEMS IN FLEXURE.
Designing Sections of Built Beams. Moving Loads. Special Cases of Quiescent Loads. Hydrostatic Load. Derivatives of Ordinate of Elastic Curve. Weight
Falling on Beam. Crank Shaft. Other Shafts. 310
CHAPTER IV. FLEXURE; CONTINUOUS GIRDERS.
Analytical Treatment of Symmetrical Cases of Beams on Three Supports; also, Built in
THE DANGEROUS SECTION IN NON-PRISMATIC BEAMS.
Double Truncated Wedge, Pyramid, and Cone
NON-PRISMATIC BEAM8 OF UNIFORM STRENGTH.
Parabolic and Wedge-Shaped Beams. I-Beam. Elliptical Beam and Cantilevers
DEFLECTION OF BEAMS OF UNIFORM STRENGTH.
Parabolic and Wedge-Shaped Beams. Special Problems
CHAPTER V. FLEXURE OF PRISMATIC BEAMS UNDER OBLIQUE FORCES.
Gravity and Neutral Axes. Shear, Thrust, and Stress-Couple. Strength and Stiffness of Oblique Cantilever. Hooks. Crane
CHAPTER VI. FLEXURE OF LONG COLUMNS.
End-Conditions. Euler's Formula. “Incipient Flexure.” Hodgkinson's Formulae. Rankine's Formula. Radii of Gyration Built Columns. Trussed Girders. Buckling of Web Plates. Examples
CHAPTER VII. LINEAR ARCHES (OF BLOCKWORK.)
Inverted Catenary. Parabolic Arch. Circular Arch. Transformed Catenary as Arch
CHAPTER VIII. ELEMENTS OF GRAPHICAL STATICS.
Force Polygons. Concurrent and Non-Concurrent Forces in a Plane. Force Diagrams. Equilibrium Polygons. Constructions for Resultant, Pier-Reactions, and Stresses in Roof Truss. Bow's Notation. The Special Equilibrium Polygon
CHAPTER IX. GRAPHICAL STATICS OF VERTICAL FORCES.
Jointed Rods. Centre of Gravity. Useful Relations Between Force Diagrams and Their Equilibrium Polygons
CHAPTER X. RIGHT ARCHES OF MASONRY.
Definitions. Mortar and Friction. Pressure in Joints. Conditions of Safe Equilibrium. True Linear Arch. Arrangement of Data for Graphic Treatment. Graphical Treatment of Arch. Symmetrical and Unsymmetrical Cases
CHAPTER XI. ARCH-RIBS.
Mode of Support. Special Equilibrium Polygon and its Force Diagram. Change in Angle Between Rib Tangents. Displacement of Any Point on Rib. Graphical Arithmetic. Summation of Products. Moment of Inertia by Graphics. Classification of Arch-Ribs. Prof. Eddy's Graphical Method for Arch-Ribs of Hinged Ends; and of Fixed Ends. Stress Diagrams. Temperature Stresses. Braced
Arches
HORIZONTAL STRAIGHT GIRDERS.
Prismatic Girder Supported at the Extremities, also with Ends Fixed Horizontally
CHAPTER XII. GRAPHICS OF CONTINUOUS GIRDERS.
The Elastic Curve an Equilibrium Polygon. Mohr's Theorem. Example and Numerical Case. End-Tangents. Re-arrangement of Moment- Area. Positive Moment-Areas; Amount and Gravity Vertical. Construction of the "False Polygons" in Any Given Case. Moment Curves, Shears, and Reactions. Numerical Example in Detail.
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