RESISTANCE OF MATEUI U.S. 



-TANCE OF MATERIALS. 



36 



to the VMM!, the immersed part being less, the pressure of the wate 

 gain* the front will akw be Ion. 



Mr. Barlow observe* that, with small velocities, tho force of traction 

 on canal* U leu than oo railways ; and when the velocity it equal to 



..!.- ] r h-.nr. the force* are equal. Beyond this velocity tlir 

 advantage i in favour <>f the railway. 



)K MATKlUAl".S. AVh, n solid bodies are exposed 

 to the action of external force*, they are capable of minting those 

 force* by rea*oo of the cohesion or of the elasticity they may posses*, 

 until their own power* are exceeded, when the particle* of which the 

 olid body in question U composed begin fintly to change tli.'ir 

 respective positions, and finally aeparate from one another. \Vithin 

 certain limit*, the solid bodies in question are susceptible of ramming 

 their original form after their particles hare begun thus to change 

 their positions (under condition* depending upon tho nature of the 

 external force) when the force U withdrawn ; and it U according to tho 

 energy and the mode of exhibition of this power that bodies ore ranged 

 in the classei of elastic or non-elastic bodies, of highly elastic or 

 perfectly eUatio bodies. When the force is exercised in a direction 

 parallel to the longitudinal axis of the body in a manner to pull or to 

 extend it, the force is said to be one of traction, or extension ; when it 

 act* in the direction able to bind the particle* of the body into closer 

 contact, it U said to be an effort of comprtttion ; when it acts in a 

 direction able to cause one part of the body to slide over the other, 

 or, in other words, to split it, the force is said to produce an effort of 

 dttrtuiuH ; and when it acts so as to twist the particles or fibres of the 

 material one over the other, the effort is said to be one of torsion. 

 Tho terms elattic, higUy elastic, and won-dattic bodies Hiifliciently 

 explain themselves ; but it may be desirable to add that bodies are 

 aid to be perfectly tlattic when they resist, with equal energy, efforts 

 of compression and of extension. The best wrought iron is on illus- 

 tration of this property ; In it in the case of good cost iron, the resist- 

 ance to compression is equal to nearly 64 times the resistance to 

 extension : the former is nearly a perfectly elastic body, the latter is 

 only imperfectly such. It was formerly considered that the com- 

 pression of solid bodies took place equally, or, in the words of Hooke, 

 that / rnufo, tie rit ; but more recent experiments have shown that, 

 beyond certain limits, the compression and extension take place with 

 a greater degree of rapidity than would be proportional to the increase 

 of the effort. The limits of equable resistance thus alluded to corre- 

 spond with the range of the unimpaired elastic powers of the body ; 

 for if the effort should be such as to cause the body to compress or to 

 extend with an accelerated velocity, the original dimensions and form 

 will not be rausumed when tho effort is withdrawn. Bodies so 

 affected are said to have had their permanent elasticity interfered 

 with ; or their elasticity has been changed, so as to produce either a 

 permanent elongation, contraction, or flexure, as tho case may be. 



There is a very important consideration which must always be borne 

 in mind in determining the effort to be applied to any body, namely, 

 that the length of time during which it is so applied has a material 

 influence upon the resistance ; or, in other words, bodies will resist 

 instantaneous efforts of far greater value than they can resist perma- 

 nently, without alteration in their elasticity. It therefore becomes a 

 matter of necessary precaution (in all building or mechanical operations), 

 to keep the forces to which the various materials are exposed consider- 

 ably within the limits of what would bo able to produce instantaneous 

 changes of their natural elastic states ; and this is the more necessary 

 because tho materials alluded to are, in practice, subject to shocks, 

 jars, or accidental efforts, which may be of a serious nature. 



Mridnlut of Eltuticity. If a prismatic, or cylindrical body of a 

 given length, L, and an area A, be exposed to an effort of longitudinal 

 traction in the direction of its axis /', it would extend under this 

 action by a quantity we may call 7; and if this quantity should 



be proportional to the total length, in such a manner as that - 



Ll 



should be a constant quantity, it may be represented by i, and will 

 represent the elongation for every unit of length. Now, so long as 

 this quantity does not exceed the limits of the perfect elasticity of 

 the material, i increase* proportionally to the load and tho area, or 



to the ratio -; so that in fact jj is a constant quantity, which 



i* called the rn-rffinenl, or modultu of tltuticily, and is usually expressed 

 If then, the transverse section were equal to the unity of 

 surface, r.nd the elongation i, for every unit of length, were equal to 

 that unit of length, A i = 1, and r = E woul.l be the effort supported 

 by the unity of surface, and able to produce for the unity of length an 

 elastic elongation equal to that unity. The same remarks -nil! apply 

 to efforts of compression, and it U generally admitted that the co- 

 efficient of elasticity has the same value in the two cases, although in 

 certain granular bodies this law does not appear always to hold. As 



j, 

 the relation p = A r. i become* P = E i and from thence E T, when 



the load r U supported by every unity of the section A, it is easy to 

 determine the value of H for every such unity of section, and thence to 

 calculate the value of the load able to produce a given elongation of 

 the body presenting that section, or to calculate the elongation pro- 

 duced by a given load. 



K.iifn(m and fomprtaion. A very great number of observations 

 have been made for the purpose of . ascertaining, experimentally, the 

 law* which regulate the extension of nnlid bodies, the results of which 

 may be briefly stated as follows. The load which is capable of producing 

 rupture by extension, is directly proportional to the transverse section 

 of the body considered ; and the load has no reference to the length, 

 provided the material be homogeneous, and the weight of the body 

 be taken into account. The substances which have the highest co- 

 efficient of elasticity, are those which resist rupture by extension in tho 

 most energetic manner ; that is to say, they are the most tenacious. 

 ipuraturu of the bodies considered must be taken into con- 

 -i'lrraiioii ; for the expansion produced by an increase of temperature 

 in many substances acts in such a manner as to produce a longitudinal 

 extension, whereas in others it produces a change in volume ; and this 

 remark may be extended also to the changes which follow upon any 

 alteration in the molecular structure of a body ; because, in the first 

 place, every new crystalline arrangement is liable to produce some 

 change in the volume of the substance, and in the second, the relative 

 positions of the axes of the crystals may, and often do, singularly affect 

 the powers- of resistance to extension. In a great number of the 

 materials used in the arts the resistances to extension and to com- 

 pression are nearly equal ; but there are cases (as for instance, building 

 stones and cast iron) in which there are marked differences in these 

 respective powers. 



As a general rule, it is more important in the arts to know the 

 limits of the resistance of the substances employed to efforts of com- 

 pression, and it thus happens that the majority of the experiments, 

 made upon the physical properties of those bodies, have been made 

 with a view to the solution of that class of investigations. The law 

 before stated, as applying to the resistance to extension, sensibly holds 

 with respect to compression, and it is expressed by the formula N = AR, 

 in which N = the total effort exercised normally to the direction of the 

 base, A = the sectional area of the prism, and R = the resistance for 

 every unity of the section. When, however, the bodies pass certain 

 lengths, compared with the dimensions of their sides, the resistances 

 to compression cease to follow the ordinary law, and it becomes 

 necessary to divide the theoretical results, obtained by the application 

 of the tables of resistance, by a co-efficient varying with the propor- 

 tionate length of the prism to the diameter of the polygon circum- 

 scribed upon its base. Thus, when the diameter of the polygon is in a 

 lesser ration than 1 to the following tabular numbers, representing the 

 height, the co-efficient becomes, in each case, as under : 



Ratio 1 . 10 15 20 25 30 35 I 42 46 $0 



Co -efficient 



1-J l-i, 



I- 



it 



Again, the form of the body experimented upon, and the relative 

 positions of its molecules, have an important influence on iti 

 resistance, for Vicat has shown (' Annales des Fonts et Chaussees,' 

 1833) that the resistance of a cylinder of stone to compression, in a 

 direction at right angles to its bedding, is rather greater in proportion 

 than that of the circumscribed square ; a cylinder laid fl;it crushes 

 under a weight which does not exceed } of tho one required to crush 

 it when loaded on end ; and the inscribed sphere will crush under J of 

 the load of the cylinder. Compound substances were found to crush 

 more readily than homogeneous ones; that is to say, that cubes 

 of stone built up of several pieces were found to crush more 

 easily than monoliths did ; but the general conditions of their 

 resistance to compression were, after due allowance for this law, pre- 

 cisely analogous to those of solid bodies. In fact, in large masses of 

 masonry, the resistance to efforts of compression is regulated by the 

 resistance of their weakest parts, that is to say, by the resistance of 

 the mortar used ; and if it were required to calculate within safe 

 limits the condition of the stability of a lofty pier, it would also be 

 necessary to apply the co-efficient for the relative heights and bases 

 just quoted. Vicat observes that in many cases loads, which for as 

 much as 95 days were not able to produce any perceptible effect upon 

 the bodies exposed to their compressive action, were able ultimately to 

 destroy them ; and he thence inferred that it was not safe to employ 

 any materials, under efforts of permanent compression exceeding ^ of 

 the effort required to produce instantaneous rupture. In the case of 

 substances possessing very imperfect elasticity a diminution of volume 

 may frequently be produced by an effort of compression, which 

 would not be recovered if that effort were withdrawn, even 

 though the substance had not begun to disintegrate, nor its molecules 

 had lost their cohesion. The clays and loams, so frequently met 

 with in foundation works, are exposed to this peculiar action ; and it 

 requires to be taken seriously into account in building operations. 

 Water is one of the imperfectly elastic bodies, but it resists compres- 

 sion with very great energy. 



Temperature has a decided influence upon the powers of resistance 

 of bodies to efforts of compression. For an incn-aso of temperature, 

 liryond the atmospheric average, diminishes in a gradually accelerating 

 ratio the solidity of the bodies, whilst a decrease of temperature, 

 below the freezing point, by affecting the powers of cohesion (or in 

 common phrase by its rendering the bodies more brittle), causes the 

 bodies to break up more rapidly. In materials obtained from stratified 

 deposits, such as the decidedly laminated building stones, &c., Vicat 



