294 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[Septeubbr, 



ELASTICITY OF SOLIDS. 



On the LniBS nfthe Elasticity of Solids. By W. J. Macquorn Ran- 

 KiNE, C.K., F.K.S.E. 



This paper is intended to form the foundation of the theoretical 

 ])art of a series of researches on the strength of materials. Its 

 immediate object is to investigate the relations which must exist 

 between the elasticities of different kinds possessed by a given sub- 

 stance, and between the different values of these elasticities in dif- 

 ferent directions. 



The different kinds of elasticity possessed by a solid substance 

 are distinguished into tlirce, viz. : — First, longitudinal ehuticity, 

 representing the forces called into play in a given direction by 

 condensation or dilatation of the particles of the body in the same 

 direction ; Secondly, lateral elantU-itij, representing those called into 

 play in a given direction by condensation or dilatation of the par- 

 ticles of the body in a direction at right angles to that of the force; 

 and thirdly, transverse ehmticitij or riyiditi/, being the force by which 

 solid substances resist distortion or change of figure, and the pro- 

 perty which distinguishes solids from fluids. The author's re- 

 searches refer chiefly to substances whose elasticity varies in dif- 

 ferent directions. His first endeavour is, to determine the laws 

 of elasticity of such substances, so far as they are independent of 

 liypotheses respecting the constitution of matter ; a course which 

 has not hitherto been followed. 



The first Theorem or law states the existence of three axes of 

 elasticity at right angles to each other at each point of each sub- 

 stance possessing a certain degree of symmetry of molecular action. 

 The elasticity of a body, as referred to these three axes, is expressed 

 by twelve coefficients, three of longitudinal elasticity, six of lateral 

 elasticity, and three of rigidity, which are connected by the follow- 

 ing laws. 



Theorem II. Tlie coefficient of rigidity is the same for all direc- 

 tions of distortion in a given plane. 



Theorem III. In each of the co-ordinate planes of elasticity, the 

 coefficient of rigidity is equal to one-fourtli part of the sum of the 

 two coefficients of longitudinal elasticity, diminished by one-fourth 

 part of the sum of the two coefficients of lateral elasticity in the 

 same plane. 



The investigation having now been carried as far as is possible 

 without the aid of hypotheses, the author determines in the first 

 place the consequences of the supposition of Boscovich, that elas- 

 ticity arises solely from the mutual action of atomic centres of 

 force. In the following theorems a perfect solid means a body so 

 constituted. 



Theorem IV. In each of the co-ordinate planes of elasticity of a 

 perfect solid, the two coefficients of lateral elasticity, and the co- 

 efficient of rigidity, are all equal to each other. 



Theorem V. For each axis of elasticity of a perfect solid the co- 

 efficient of longitudinal elasticity is equal to three times the sum 

 of the two coefficients of rigidity for the co-ordinate planes which 

 pass through that axis, diminished by three times the coefficient of 

 rigidity for the plane normal to that axis. 



Thus in perfect solids all the coefficients of elasticity are functions 

 of three independent coefficients — those of rigidity. In no pre- 

 vious investigation has the number of independent co-efficients 

 been reduced below six. 



To represent the phenomena of imperfect solids, there is intro- 

 duced the hypothesis of molecular i^ortices, in addition to that of 

 atomic centres ; that is to say, each atomic centre is su|>posed to 

 be surrounded by a fluid atmosphere, retained round the centre 

 by attraction, and diffused from it by the centrifugal force of 

 revolutions constituting heat. The author has already applied 

 this hypothesis to the theory of the elasticity of gases and vapours. 

 (Trans. Hoy. Sue. Edin., Vol. XX. Part I.) Applied to solids, it 

 leads to the following conclusions : — 



Theorem VI. In an imperfect solid, each of the coefficients of 

 longituilinal and lateral elasticity is equal to the same function of 

 th(f coefficients of rigidity wliich would have been its value in a 

 jierfect solid, added to a coefficient oi fluid elasticity which is the 

 same in all directions. 



Thus the number of independent coefficients for such substances 

 is. /bur. 



The rest of the paper is occupied by the deduction from these 

 principles of some important consequences, relative to coefficients 

 of compressibility and extensibility, and to elasticities correspond- 

 ing to directions not coinciding with either of the three axes. 



FOBCE OF WAVES. 



Observations on the Force of the Waves. By Thomas Ste\xnsox, 

 F.R.S.E., Civil Engineer. 



The author, after some introductory remarks, described the 

 action of the Marine Dynamometer, the self-registering instrument 

 with which the observations were made, and one of the instruments 

 was exhibited. He stated, that a theoretical objection might, 

 perhaps, be started to referring the action of the sea to a statical 

 value, but contended, that in designing sea works the attempt of 

 the engineer is to oppose the dynamical action of the sea by the 

 dead weight or inertia of the masonry, so that the indications of 

 the Marine Dynamometer furnish exactly the kind of information 

 which the engineer requires. The greatest result registered in 

 the Atlantic Ocean was at Skerryvore, during the westerly gale of 

 the 29th of March, 1845, when the force was C083 lb., or 3 /otm 

 fier square foot. The greatest result registered in the German 

 Ocean was 30131b., or about 1^ ton per square foot. It further 

 appeared, fnmi taking an average result for five of the summer 

 montlis during the years 1843 and 1844, that the force in the At- 

 lantic Ocean was 6111b. jier square foot, while the corresponding 

 averag^e for six of the winter months was 2086 lb., or three times as 

 great as in summer. These ol)servations he liad communicated in 

 1845 to the Royal Society of Edinburgh, and were printed in the 

 twelfth volume of the 'Transactions' of that body. 



The author then stated, that tlie greatness of those results had 

 excited surprise in almost all to wliom they had been communi- 

 cated, and positive doubts were expressed by many as to the 

 correctness of the indications. Three classes of facts, essentially 

 different from each other, may be appealed to, as proving that if 

 the indications of the Dynamometer are incorrect, tlie error must be 

 in defect, and not in excess. The first fact to which reference was 

 made was the elevation of spray caused by waves meeting with an 

 obstruction to their onward motion. Most persons are familiar 

 with the frontispiece representations of the Eddystone and Bell 

 Rock Lighthouses during storms, which are attached to the 

 descriptive accounts of the erection of those works; and although 

 some deduction may be allowed for the fancy of the artists, still 

 there can be no doubt that they are, in the main, faithful repre- 

 sentations of a natural phenomenon. On the 20th of November, 

 1827, in a heavy ground swell after a storm, solid water rose at 

 the Bell Rock, 106 feet above the level of the sea, irrespective of 

 the depth of the trough of the wave. Such an elevation is due to 

 a head of water of the same height. The force, then, which urges 

 the lower courses of the Bell Rock must have been nearly three 

 tons per square foot, while the highest indication of the Marine 

 Dynamometer at the same place, since the observations were 

 commenced hardly equalled 1-| ton. The second class of facts to 

 which the author alluced was the fracture of materials of known 

 strength. The instance adduced was a small harbour in Argyll- 

 shire, where, in order to preserve the tranquility of the tide basin, 

 a contrivance, called '■booms,' well known in harbour architecture, 

 had been resorted to. The booms are logs of timber, which are 

 placed across the entrance to a harbour, and fit into checks or 

 grooves, which are made in the masonry on either side. The booms, 

 therefore, act as a temporary wall or barrier against the waves. 

 The set of booms referred to have been in use for about five years, 

 and in that time the waves have broken no less than four Memel 

 logs, measuring each one foot square in the middle, and spanning 

 an entrance of 20 feet. From the known strength of the material 

 it will be found, that on these four occasions a force must have 

 been exerted equivalent to the uniform distribution of a dead 

 weight of 30 tons, or at the rate of \\ ton per square foot, while the 

 highest result that had been recorded at the same place during the 

 short period that observations were made, was about 1^ ton per 

 square foot. 



The last class of effects to which the author alluded was the 

 movement of heavy blocks of stone. The information derived 

 from s\ich observations was not so certain or satisfactory as from 

 the other instances. The only record he could adduce was the 

 movement of a block of stone weighing about Ij ton, to which a 

 Marine Dynamometer had been bolted. The stone was turned 

 upside down, and the dynamometer indicated a pressure of little 

 more than one ton. 



The autlior then referred to the overturning of the Carr Rock 

 Beacon by the sea in 1817, during a heavy gale, but stated that, as 

 we do not know the manner in which waves act when encountering 

 obstacles, it was impossible to calculate what force had in this 

 instance been exerted. The part of the column which was over- 



