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Elastic curve transmitting force and couple-Kirchhoff's ki-

netic comparison-Common pendulum and plane elastic

curve-Graphic construction of elastic curve transmitting

force in one plane-Equation of the plane elastic curve-

Bow slightly bent-Plane elastic curve and common pen-

dulum

Wire of any shape disturbed by forces and couples applied

through its length-Longitudinal tension-Equations of

torsion-flexure-Torsion, and two components of curva-

ture, of wire (or component angular velocities of rotating

solid)-Terminal conditions--Straight beam infinitely

little bent-Case of independent flexure in two planes-

Plank bent by its own weight-Plank supported by its

ends; by its middle-Droops compared-Plank supported

by its ends or middle; by three or more points-Plank

supported by its ends and middle-Rotation of a wire

round its elastic central line-Elastic universal flexure

joint-Equable elastic rotating joint-Practical inequali-

ties-Elastic rotating joint-Rotation round its elastic

central circle, of a straight wire made into a hoop-Rota-

tion round its elastic central circle, of a hoop of wire

equally flexible in all directions, but circular when un-

strained-Wire unequally flexible in different directions,

and circular when unstrained, bent to another circle by

balancing-couples applied to its ends-Conical bendings of

developable surface

Flexure of a plane elastic plate-Definitions-Geometrical

preliminaries-Limitation of flexure not to imply a stretch-

ing of middle surface comparable with that of either side—

Stretching of a plane by synclastic or anticlastic flexure-

Stretching of a curved surface by flexure not fulfilling

Gauss's condition-Gauss's theorem regarding flexure-

Limitations as to the forces and flexures to be admitted in

elementary theory of elastic plate-Results of general

theory stated in advance-Laws for flexure of elastic plate

assumed in advance-Stress-couple acting across a normal

609-613

614-626

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627-642

649-651

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654-657

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face in terms of rectangular specification of stress-Stress-

quadric-Principal planes and axes of a stress-Varieties

of stress-quadric-Composition of stresses-Laws of strain

and stress compared-Rectangular elements of strains and

stresses-Work done by a stress within a varying solid-

Work done on the surface of a varying solid-Strain-

components in terms of displacement-Work done through

interior; agrees with work done on surface-Differential

equation of work done by a stress- Physical application—
Perfectly elastic body defined, in abstract dynamics--Its
conditional fulfilment in nature-Potential energy of an
elastic solid held strained-Stress-components expressed
in terms of strain-Strain-components expressed in terms
of stress-Average stress through any changing strain

Homogeneousness defined-Molecular hypothesis assumes a

very fine grained texture in crystals, but no ultimate

homogeneousness-Scales of average homogeneousness—
Isotropic and eolotropic substances defined-Isotropy and
eolotropy of different sets of properties-Practical limita-
tion of isotropy, and homogeneousness of eolotropy, to the
average in the aggregate of molecules-Conditions fulfilled
in elastic isotropy-Measures of resistance to compression
and resistance to distortion-Bulk-modulus or modulus of
compression-Compressibility-Rigidity, or elasticity of

figure, defined

9nk

3k+n

-Ratio of lateral contraction to

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658-674

Young's modulus defined-Same as longitudinal rigidity-
Weight-modulus and length of modulus-Velocity of
transmission of a simple longitudinal stress through a rod
-Specific Young's modulus of an isotropic body in terms
of the absolute unit; or of the force of gravity on the unit
of mass in any particular locality-Metrical denomina-
tions of moduluses of elasticity in general

Practical rules for velocities of waves; distortional without
change of bulk; compressional, in an elastic solid; com-
pressional in liquid; compressional in gas; gravitational
in liquid; transversal vibration of stretched cord-Digres-
sion on Resilience, from Art. Elasticity, Encyc. Brit.

VOL. II.

с

Stress required to maintain a simple longitudinal strain-
Stress components in terms of strain for isotropic body-
Equation of energy for the same

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Fundamental problems of mathematical theory-Conditions of

internal equilibrium, expressed by three equations-

General equations of interior equilibrium-Being suffi-

cient, they imply that the forces on any part supposed

rigid fulfil the six equations of equilibrium in a rigid

body-Verification of equations of equilibrium for any

part supposed rigid-Simplified equations for isotropic

solid

St Venant's application to torsion problems-Torsion pro-

blem stated-Lemma - Torsional rigidity of circular

cylinder-Prism of any shape constrained to a simple

twist requires tractions on its sides-Traction on sides of

prism constrained to a simple twist-St Venant's correc-

tion to give the strain produced by mere twisting couples

applied to the ends-Hydro-kinetic analogue to torsion

problem-Solution of torsion problem-Equations of

strain, stress, and internal equilibrium-Surface traction

to be made zero-Couple resultant of traction in normal

section-Hydro-kinetic application of torsional equation-

St Venant's invention of solvable cases-Solution for

elliptic cylinder, for equilateral triangle, for curvilinear

squares, for star with four rounded points-St Venant's

reduction to Green's problem-Solution for rectangular

prism, found by Fourier's analysis-Extension to a class
of curvilinear rectangles-Lamé's transformation to plane
isothermal co-ordinates-Theorem of Stokes and Lamé-
Solution for rectangle of plane isothermals-Example.
Rectangle bounded by two concentric arcs and two radii—
Contour lines of normal section of elliptic cylinder, as
warped by torsion: equilateral hyperbolas-Contour lines
of normal section of triangular prism, as warped by tor-
sion-Diagram of St Venant's curvilinear squares for
which torsion problem is solvable-Contour lines for St
Venant's "étoile à quatre points arrondis"-Contour lines
of normal section of square prism, as warped by torsion-
Elliptic square, and flat rectangular bars twisted

Torsional rigidity less in proportion to sum of principal

flexural rigidities than according to false extension (§ 703)

of Coulomb's law-Ratios of torsional rigidities to those of

solid circular rods (a) of same moment of inertia, (b) of

same quantity of material-Places of greatest distortion in

twisted prisms-Solid of any shape having edges or pyra-

midal or conical angles, under stress-Strain at projecting

angles evanescent, at re-entrant angles infinite-Liability

692-695

.

699-708

to cracks proceeding from re-entrant angles, or any places

of too sharp concave curvature-Cases of curvilinear rect-

angles for which torsion problem has been solved—Distor-

tion zero at central angle of sector (4), infinite at central

angle of sector (6); zero at all the other angles-Problem of

flexure-Forced condition of no distortion in normal sec-

tions-Surface traction (P, Q), required to prevent distor-

tion in normal section-Correction to do away with lateral

traction, and bodily force-St Venant's solution of flexure

problem-Flexure of a bar-Line through centres of in-

ertia of normal sections remains unchanged in length-

Flexure through finite angle in one plane; must be in either

of two principal planes, if produced simply by balancing

couples on the two ends-Principal flexural rigidities and

axes-Geometrical interpretation of distortion in normal

plane-Anticlastic and conical curvatures produced in the

four sides of a rectangular prism by flexure in a principal

plane-Experimental illustration - Uncalculated effects

of ordinary bendings of a thin flat spring-Hence neces-

sity for stricter limitation, § 628, of curvature than § 588

when a thin flat spring is bent in a plane perpendicular

to its breadth

Flexure of a plate: by a single bending stress; by simultaneous

bending stresses in two planes at right angles to one an-

other-Stress in cylindrical curvature: in spherical curva-

ture in anticlastic curvature-Flexural rigidities of a

plate: (4) cylindrical, (h) synclastic, (k) anticlastic—

Same result for anticlastic flexure of a plate arrived at also

by transition from simple torsion of rectangular prism-

Analysis of traction in normal section of twisted rectangu-

lar prism-Composition of action in normal section of a

long rectangular lamina under torsion-Uniform distribu-

tion of couple applied to its edges to render the stress uni-

form from the edges inwards-Algebraic solution express-

ing displacement, strain, and stress, through a plate bent

to uniform anticlastic curvature-Thin rectangular plate

subjected to the edge-traction of § 647-Transition to plate

without corners subjected to edge-traction of § 647-Origin

shifted from middle plane to one side of plate-Displace.

ment of substance produced by edge-traction of § 647

Case of § 647 independently investigated—Rapid decrease of

disturbance from edge inwards

Problems to be solved-General problem of infinite solid:
solved for isotropic substance-General equations for
infinite isotropic solid integrated-Force applied uni-
formly to spherical portion of infinite homogeneous solid-
Dilatation produced by it-Investigation of displacement---

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