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A treatise on analytical statics - Routh
A TREATISE ON ANALYTICAL STATICS - VOLUME I AND II
WITH NUMEROUS EXAMPLES
BY EDWARD JOHN ROUTH,
Cambridge: at the University Press; 1909
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A treatise on analytical statics VOL I
A treatise on analytical statics VOL II
PREFACE (VOL I)
During many years it has been my duty and pleasure to give courses of lectures on various Mathematical subjects to successive generations of students. The course on Statics has been made the groundwork of the present treatise. It has however been necessary to make many additions; for in a treatise all parts of the subject must be discussed in a connected form while in a series of lectures a suitable choice has to be made. A portion only of the science of Statics has been included in this volume. It is felt that such subjects as Attractions, Astatics, and the Bending of rods could not be adequately treated at the end of a treatise without either making the volume too bulky or requiring the other parts to be unduly curtailed. These remaining portions appear in the second volume.
In order to learn Statics it is essential to the student to work numerous examples. Besides some of my own construction, I have collected a large number from the University and College Examination papers. Some of these are so good as to deserve to rank among the theorems of the science rather than among the examples. Solutions have been given to many of the examples, sometimes at length and in other cases in the form of hints when these appeared sufficient.
I have endeavoured to refer each result to its original author. I have however found that it is a very difficult task to effect this with any completeness. The references will show that I have searched many of the older books and memoirs as well as some of those of recent date to discover the first mention of a theorem.
In this edition I have made many additions and have also omitted several things which on after consideration appeared to be of minor importance. The explanations also have been simplified wherever there appeared to be any obscurity. For the convenience of reference I have retained the order of the articles as far as that was possible.
The latter part of the chapter on forces in three dimensions has been enlarged by the addition of several theorems and the portions on five and six forces re-arranged. The chapter on graphical statics also has been almost entirely rewritten.
An index has been added which it is hoped will be found useful.
In order to learn Statics it is essential to the student to work numerous examples. Besides some of my own construction, I have collected a large number from the University and College Examination papers. Some of these are so good as to deserve to rank among the theorems of the science rather than among the examples. Solutions have been given to many of the examples, sometimes at length and in other cases in the form of hints when these appeared sufficient.
I have endeavoured to refer each result to its original author. I have however found that it is a very difficult task to effect this with any completeness. The references will show that I have searched many of the older books and memoirs as well as some of those of recent date to discover the first mention of a theorem.
In this edition I have made many additions and have also omitted several things which on after consideration appeared to be of minor importance. The explanations also have been simplified wherever there appeared to be any obscurity. For the convenience of reference I have retained the order of the articles as far as that was possible.
The latter part of the chapter on forces in three dimensions has been enlarged by the addition of several theorems and the portions on five and six forces re-arranged. The chapter on graphical statics also has been almost entirely rewritten.
An index has been added which it is hoped will be found useful.
CONTENTS (VOL I)
CHAPTER I. THE PARALLELOGRAM OF FORCES.
- Elementary considerations on forces
- Dynamical and statical laws
- Rigid bodies
- Resultant forces
- Parallelogram of forces
- Historical summary
CHAPTER II. FORCES ACTING AT A POINT.
- Geometrical Method. Triangle, Polygon and Parallelepiped of forces. Three methods of resolution
- Method of analysis. Resultant of any number of forces
- Forces which act normally to the faces of a polyhedron
- Theorems on Determinants
- Oblique axes
- The mean centre. Its use in resolving and compounding forces
- Equilibrium of a particle. Smooth curves and surfaces
- The principle of Work
- Astatic equilibrium. The astatic triangle of forces
- Stable and unstable equilibrium. A free body under two nd three forces
CHAPTER III. PARALLEL FORCES.
- Resultant of two and any number of parallel forces
- The centre of parallel forces
- Conditions of equilibrium
- A body suspended from a fixed point
- A body resting on a plane
- Theory of couples
CHAPTER IV. FORCES IN TWO DIMENSIONS.
- Resultant of any number of forces
- Conditions of equilibrium
- Varignon's theorem
- The single resultant force or couple
- The solution of problems. Constrained rods, discs, spheres
- Equilibrium of four repelling or attracting particles
- Reactions at joints. Theorems of Euler, Fuss, &c. A note on a property of triangles
- Polygon of heavy rods. Funicular polygon. See also Arts
- Examples
- Reactions at rigid connections. Bending moment. Beam with weights, diagram of stress. Various examples
- Indeterminate problems
- Frameworks Conditions of stiffness
- Equations of equilibrium of a body or framework derived from those of a single particle. See foot-note
- Sufficiency of the equations to find the reactions when the frame is just stiff. Exceptional cases.
- Supplementary equations when the frame is over stiff
- Asiatics. Astatic resultant. Centre of forces
CHAPTER V. ON FRICTION.
- Laws of friction. The friction force and friction couple. The angle of friction and limiting friction
- Particle constrained by a rough curve, surface. The cone of friction
- Problems on friction when the direction is known. Different methods of solution. Various examples
- Wheel and axle with friction
- Friction in unknown directions. Two methods
- Examples. Rotation of bodies on a finite and an infinite number of supports. String of particles
CHAPTER VI. THE PRINCIPLE OF WORK.
- Proof of the principle
- The forces which do not appear in the equation of work
- Work of a bent string.
- Infinite forces
- Converse of the principle of work
- Initial motion
- Equations of equilibrium derived from work
- Examples on the principle
- The work function
- Stable and unstable equilibrium. Analytical method
- Attracting or repelling atoms
- Determination of stress in a simply stiff frame. Examples
- Abnormal deformations
- Indeterminate tensions. Theorems of Crofton and Levy.
- Geometrical method of determining the stability of a body. The circle of stability
- Rocking stones. Examples
- Rocking stones, spherical and not spherical to a second approximation in two and three dimensions
- Lagrange's proof of the principle of virtual work
CHAPTER VII. FORCES IN THREE DIMENSIONS.
- Resultants of a system at forces. Conditions of equilibrium.
- Components of a force
- Moment of a force
- Problems on equilibrium. Pressures on an axis, rods, spheres, rod on wall, curtain ring,
- Poinsot's central axis
- The equivalent wrench. Analytical method
- The Invariants
- Equivalent wrench of two wrenches or forces
- The Cylindroid
- Work of a wrench
- Reciprocal screws
- The Null plane
- Conjugate forces
- Theorems on three forces
- Four forces. The hyperboloid. Forces acting normally to the faces of a tetrahedron. Forces normal to a surface
- Five forces. The two directors
- Six Forces, analytical view. The polar plane. Lines in involution and the two determinants. Resolution of a force along six arbitrary lines
- The sufficiency of the general conditions of equilibrium
- Six Forces, geometrical view. The transversals. Theorems on the fifth and sixth force
- Tetrahedral coordinates
CHAPTER VIII. GRAPHICAL STATICS.
- Analytical view. Reciprocal figures
- Theorems of Maxwell and Cremona
- Example of Maxwell's theorem
- Mechanical property of reciprocal figures. Figures which have no reciprocals
- Methods of lettering and drawing reciprocal figures
- Euler's theorems
- Statical view. Construction of the resultant force. Conditions of equilibrium
- Parallel forces. Funicular polygon
- Various graphical constructions
- Graphical construction of stress problems
- Frameworks
- Method of sections
- Indeterminate tensions.
- Line of pressure. Theorems
- Examples
CHAPTER IX. CENTRE OF GRAVITY.
- Definitions and fundamental equations
- Working rule. Examples
- Tetrahedron. Volume, faces, and edges. Pyramid, cone and double tetrahedra. Isosceles tetrahedron
- Centres of gravity of arcs. Circle, catenary, cycloid,
- Centres of gravity of circular areas
- Geometrical and analytical projection of areas
- Centre of gravity of any area
- Theorems of Pappus
- Areas on the surface of a right cone
- Areas on a sphere. Theory of maps
- Surfaces and solids of revolution. Moments and products of inertia
- Ellipsoidal volumes and shells
- Any surfaces and volumes
- Heterogeneous bodies. Octant of ellipsoid. Triangular area, &c 285287
- Lagrange's two theorems
- Applications to pure geometry
CHAPTER X. ON STRINGS.
- Catenary. Equations and properties
- Problems on free and constrained catenaries
- Stability of catenaries
- Heterogeneous chains. Cycloid, parabola, and the catenary of equal strength
- String under any forces. General intrinsic equations
- General Cartesian equations
- Constrained strings. Light string on a smooth curve. Problem of Bernoulli
- Heavy string on a smooth curve. The anti-centre and statical directrix.
- Light string on a rough curve. Bough pegs
- Heavy string on a rough curve
- Endless strings. Strings which just fit a curve
- Central forces. Various laws of force. Kinetic analogy. Two centres
- String on a smooth surface. Cartesian and Intrinsic equations. Various theorems. Case of no forces
- String on a surface of revolution
- Spherical catenary
- String on a cylinder
- String on a right cone
- String on a rough surface
- Minimum force to move a heavy circular string on a rough plane, and other problems
- Calculus of variations
- Elastic string. Hooke's law
- Heavy elastic string suspended by one end
- Work of an elastic string
- Heavy elastic string on a smooth curve. The statical directrix
- Light elastic string on a rough curve. Rough pegs
- Elastic string under any forces
- Elastic catenary
CHAPTER XI. THE MACHINES.
- Mechanical advantage. Efficiency
- The Lever. Conditions of equilibrium. Pressure on axis. Various kinds of Levers.
- Roberval's balance
- The common balance
- The common and Danish Steelyard
- The single pulley
- The system of pulleys with one rope
- Rigidity of cords
- Systems of pulleys with several ropes
- The inclined plane
- The wheel and axle. The differential axle
- Toothed wheels
- The wedge
- The screw
PREFACE (VOL II)
In the first edition of this treatise the subject of Attractions was presented only in its gravitational aspect. This limitation was formerly customary, when electricity was less studied than now, but the result has become somewhat unsatisfactory. When lecturing on the subject the Author found that some of the most striking examples of Attraction were those derived from the theory of electricity. While it was impossible wholly to pass these over, it appeared that the interest in them was sensibly diminished if they were discussed without explanations of their meaning. Examples on the attractions of thin layers of matter, subject to what appeared to be arbitrary laws, seemed to have no real applications.
For these reasons a selection has been made of those propositions in Magnetism and Electricity which appeared most forcibly to illustrate the theory of Attraction. These have been joined together, with brief introductions, so as to form a continuous story which could be understood without reference to any other book.
These illustrations have been so far separated from the rest of the volume that any portion of them may be omitted by a reader who desires to confine his attention chiefly to gravitational problems.
Some theorems, which it was not deemed expedient to include in the text, have been shortly discussed in the notes at the end of the volume. These are not always closely connected with the theory of attractions, yet, being natural developments of the text, will probably assist the reader in following the argument
The general arrangement of the gravitational part of "At- traction" has been only slightly altered. New theorems have, however, been introduced and the demonstrations of some of the old ones simplified.
The second part of this volume is on the stretching and bending of rods. The investigation of the stretching, and consequent thinning, of a rod is founded on Hooke's law. The fact that (with certain restrictions) the stress couple is proportional to the bending is assumed as an experimental result and applied to determine the bending of rods and springs under various circumstances. The problem, when put into this form, is properly included in a treatise on Statics. Although this chapter is not a treatise on the theory of Elasticity, it did not seem proper wholly to omit the theoretical considerations by which the truth of the fundamental law is confirmed. Accordingly some simple examples which had been briefly discussed in the last edition have been retained.
The theory of Astatics occupies the third part of this volume. It was discussed with sufficient fullness in the first edition and only very slight alterations have now been made.
A separate index to each of the three chapters has been given. So many results are included under the head of Attraction that it was found impossible to mention them all without unduly lengthening the list. It was also necessary to classify some theorems only under one heading.
Finally, I desire to express my thanks to Mr J. D. H. Dickson of Peter house for the very great assistance he has given me in correcting most of the proof-sheets and for his many valuable suggestions.
For these reasons a selection has been made of those propositions in Magnetism and Electricity which appeared most forcibly to illustrate the theory of Attraction. These have been joined together, with brief introductions, so as to form a continuous story which could be understood without reference to any other book.
These illustrations have been so far separated from the rest of the volume that any portion of them may be omitted by a reader who desires to confine his attention chiefly to gravitational problems.
Some theorems, which it was not deemed expedient to include in the text, have been shortly discussed in the notes at the end of the volume. These are not always closely connected with the theory of attractions, yet, being natural developments of the text, will probably assist the reader in following the argument
The general arrangement of the gravitational part of "At- traction" has been only slightly altered. New theorems have, however, been introduced and the demonstrations of some of the old ones simplified.
The second part of this volume is on the stretching and bending of rods. The investigation of the stretching, and consequent thinning, of a rod is founded on Hooke's law. The fact that (with certain restrictions) the stress couple is proportional to the bending is assumed as an experimental result and applied to determine the bending of rods and springs under various circumstances. The problem, when put into this form, is properly included in a treatise on Statics. Although this chapter is not a treatise on the theory of Elasticity, it did not seem proper wholly to omit the theoretical considerations by which the truth of the fundamental law is confirmed. Accordingly some simple examples which had been briefly discussed in the last edition have been retained.
The theory of Astatics occupies the third part of this volume. It was discussed with sufficient fullness in the first edition and only very slight alterations have now been made.
A separate index to each of the three chapters has been given. So many results are included under the head of Attraction that it was found impossible to mention them all without unduly lengthening the list. It was also necessary to classify some theorems only under one heading.
Finally, I desire to express my thanks to Mr J. D. H. Dickson of Peter house for the very great assistance he has given me in correcting most of the proof-sheets and for his many valuable suggestions.
CONTENTS (VOL II)
ATTRACTIONS
INTRODUCTORY REMARKS
- Law of attraction, units and magnitude
- Law of the direct distance
- Attraction of Rods, Discs
- Components and resultant attraction
- Attraction of an infinite cylinder
- Examples and theorems on rods
- Curvilinear rods. Examples and theorems
- Some inverse problems
- Circular disc. Infinite plate. Young's rule
- Attraction of cylinders
- Attraction of a lamina. Playfair's rule
- Solid of greatest attraction
- Attraction of mountains; density of the earth
- The Potential
- Work and potential
- Level surfaces, various theorems
- Potentials of rods. Rectilinear lamina, with confocal level
- Discs and cylinders
- Attraction of a homogeneous cylindrical shell at (1) an internal,
- Spherical Surface
- Potential and attraction of a thin shell
- Ellipsoidal and cylindrical elliptic shells
- Various theorems
- Solid sphere
- Other laws of force
- Eccentric shells
- Gauss and Poisson on mean potential
- Find the law of force that the attraction may be the same as that of a particle
LAPLACE'S, POISSON'S AND GAUSS' THEOREMS.
- Laplace's theorem. Jellett's theorem,
- Potential constant through a finite space
- Invariant property of Laplace's operator
- Potential at internal points
- Poisson's theorem
- Gauss' theorem
- Polar, cylindrical, orthogonal and elliptic coordinates
THEOREMS ON THE POTENTIAL
- Potential constant through any space
- Points of equilibrium. Stability. Separating cone
- Intersections of a level surface. Eankine's theorem
- Tubes of force, &c. Theorems
- Bodies with equal potentials
- Extensions of these theorems
- To trace level curves and lines of force
- Potential at a distant point. Maccullagh's theorem
- Theorem on equipotential bodies
- Centrobaric bodies
- Potential constant through a finite space
ATTRACTION OF A THIN STRATUM.
- Attraction of a stratum on an element of itself
- Changes of force and potential at the surface
- Green's theorem deduced from Gauss'
- Relation between normal force and constant potential
GREEN'S THEOREM
- A volume integral replaced by a surface integral. Also a surface integral by a line integral
- Green's equivalent layer
- Points at which V is infinite
- Multiple-valued functions
GIVEN THE POTENTIAL, FIND THE BODY
- Solid and superficial matter
METHOD OF INVERSION
- Inversion from a line
- Extension of the theory
CIRCULAR RINGS AND ANCHOR RINGS
- Potential found by the use of a conic
- Law of the inverse k th power
- Anchor rings
ATTRACTION OF ELLIPSOIDS
- Two kinds of shells
- Homoeoid, internal point
- Thin homoeoids, attractions and lines of force
- Homoeoid, external point
- Solid homogeneous ellipsoid
- Level surfaces
- Other laws of force
- Spheroids. Nearly spherical ellipsoids
- Ivory's theorem
- Maclaurin's theorem
- Mutual attraction
- Ellipsoidal quadratic layers. Heterogeneous shells
- Elliptic cylinders, attraction and potential
- Heterogeneous ellipsoid, similar strata
- Ellipsoids with any law of density
- Elliptic discs. Confocal level surfaces
- Rectilinear Figures
- Potential of a lamina found in terms of those of the sides and that of a solid in terms of those of the faces
- Application to a tetrahedron
LAPLACE'S FUNCTIONS AND SPHERICAL HARMONICS
- Legendre's functions. Elementary theorems
- Expression for the potential of a body
- Laplace's second equation
- Three fundamental theorems
- Examples and theorems
- Heterogeneous spherical strata
- Nearly spherical bodies
- Potential found when given on spheres
- Solid of revolution
- Potential of a ring, &c. Oblate spheroid
- Clairaut's theorem, first and second approximations
- Figure of Saturn acted on by the ring
- Jacobi's ellipsoid. Nearly spherical earth and Haughton's problem on a fluid nucleus
MAGNETIC ATTRACTIONS
- Potentials, couples, forces and energy
- Lines of force of a rectilinear row of particles and of parallel rods in a plane
- Examples and applications
- Magnetic body. Elementary rule, rod, sphere, ellipsoid, cylinder
- Infinite cylinder in a field of force
- Terrestrial magnetism
- Poisson's general theorem
- Magnetic force and Magnetic induction
- Solenoids 180181
- Lamellar shells. Potential force, mutual energy. Circular shells
ELECTRICAL ATTRACTIONS
- Introductory statement and elementary examples
- Ellipsoids. Quantity on a portion of the surface. Elliptic discs, potentials
- Conductor with a cavity
- Two methods of solution. Green's theorem
- Sphere acted on by a point charge. Lines of force, &c. Quantity on a portion of the sphere and its potential.
- Electrified ring
- Cylinders with parallel axes
- Plane conductors. Intersecting at right angles or at an angle
- Condensers. Electric cables
- Nearly spherical bodies, &c. Point charge
- Sphere with electrified ring
- Two orthogonal spheres. Capacity
- Spheres intersecting. Three orthogonal
- Theory of a system of conductors. Energy. Junction and introduction of conductors
- Circular disc with point charge
- Spherical bowl
- Two separate spheres
MAGNETIC INDUCTION
- Induced magnetism. The boundary condition
- Specific Inductive Capacity. Dielectrics. Kelvin's theorem.
- Problems on Condensers and on Induction
- Magnetic shells
- Surface integral of magnetic induction
- Gauss' and Poisson's theorems applied to dielectrics
- Potential of an electric system
THE BENDING OF RODS
INTRODUCTORY REMARKS
- Definition of central line, thickness of rod, extent of deformation
- The Stretching of Rods
- Hooke's law
- Isotropic and aeolotropic bodies
- Theory of a stretched rod. Change of sectional area
THE BENDING OF RODS
- Equations of equilibrium. Two methods
- The additional experimental law. Flexural rigidity
- Limits to the law of elasticity
- Work of bending an element
- Deflection of a straight heavy rod
- Equation of the three moments
- General method of solving problems. Yielding
- Examples of bent rods. Determination of the constants of bending by experiment
- Problems on the Britannia Bridge
- Supported rods with weights. Relation of the pressures at consecutive points of support. Also the deflection
- A bent bow. Euler's theorem on the strength of columns. Greenhill's problems on columns
- Theory of a bent rod. The displacements of the fibres
- Airy's problem on the proper method of supporting standards of length
- Bending of circular rods ; inextensible and extensible
- Work of bending and stretching a circular rod
- Very flexible rods
RODS IN THREE DIMENSIONS
- Measures of twist
- Resolved curvature
- Relations of stress to strain
- Work of bending
- Helical twisted rods
- Spiral springs
- Equations of equilibrium
ASTATICS
A STATIC COUPLES
- Definitions of the astatic arm, the astatic angle. History
- A couple may be moved parallel to itself
- Resultant of two couples, (1) forces parallel, (2) arms parallel
- Working rule
- Three couples cannot in general be in astatic equilibrium
THE CENTRAL ELLIPSOID
- Reduction to a force and three couples
- Astatic notation. The twelve elements.
- Conditions of equilibrium, &c.
- If a system be equivalent to three forces their points of application lie in a fixed plane
- The central ellipsoid
- Principal axes at any base. The principal couples
- The initial positions
- There are only four positions in which the forces are equivalent to a single resultant passing through the base .
- Other ellipsoids
THE CENTRAL PLANE AND THE CENTRAL POINT
- The central ellipsoids at different points compared
- When the central ellipsoid is a cylinder the locus of the base is the central plane
- Reduction to a force and two couples
- The central point. The arms of the two couples are perpendicular; the forces are perpendicular to each other and to E
- Working rule to find the central point
- Equation to the central plane expressed in terms of the forces and their mutual inclinations
- Summary of reductions 325 326
- Analogy to moments of inertia 326 327
THE CONFOCALS
- Arrangement of central ellipsoids about the central point
- The principal axes are the normals to the three confocals
- The focal conies and the focal lines
- Theorems on focal lines. Distance of a focal line from the centre. Four real focal lines can be drawn through a given point or parallel to a given line. Locus of focal lines passing through a point on a focal conic is a right cone. Locus of the base when the central ellipsoid is a surface of revolution. Other theorems
ARRANGEMENT OF POINSOT'S CENTRAL AXES
- When the body is rotated about R, Poinsot's axis is fixed in space and describes a right cylinder in the body. To so place the body that, when possible, a given straight line is a Poinsot's axis
- Inversion of the body
- Equation to Poinsot's axis referred to the principal axes at the central point. Minding's theorem
- To find the axes when the couple-moment is given. Another construction on Poinsot's axis
- Cylinders which intersect the circular cylinder in four
- Poinsot's axes of given moment
- Minding's theorem
- Relations of Poinsot's axes to the confocals
REDUCTION TO THREE AND TO FOUR FORCES
- Reduction to three forces
- Astatic triangles lie on the central plane
- Hence a rule to find the central plane
- Interpretation of a column of the twelve elements
- Reduction to four forces
- The imaginary focal conic
- Various theorems
- Transformation of astatic tetrahedrons
- A system cannot in general be reduced to fewer than three
- Conditions that it can be reduced to two
- The invariants of two forces
- Conditions that a system can be reduced to a single force
NOTES
A. A surface integral replaced by a line integral
B. Potential of a thin circular ring
C. Attraction of a solid ellipsoid; another proof
D. Potential of a homoeoid for any law of force
E. Heterogeneous ellipsoids
F. The expansion of the potential for other laws of force
G. Theorems on Legendre's functions
H. Laplace's equation $Y m Y n du=Q
I. A magnetic sphere
K. Magnetic forces
M. The vector potential
N. Figure showing the distribution of electricity on a sphere
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