May io, 1894] 



NATURE 



application of mathematics is more familiar than in that which 

 relates to the calculation of the strength and rigidity of slnictures 

 of various kinds. It is impossible to take up any book dealing 

 with the subject wiihout finding that it is crammed either wiih 

 mathematical formulas, or with geometrical figures. The question 

 is not whether mathematics is necessary to an adequate com- 

 prehension of the subject, but whether analytical or purely 

 geometrical methods are more convenient. Of course one 

 might occupy many lectures in discussing the practical appli- 

 cation of mathematics to the question of bridge building, roofs, 

 guns, shafting, and the like. Our object must be to illustrate 

 by various e-xamples rather than to attempt anything like a 

 complete discussion. 



Consider the ease of a long strut, so long that its transverse 

 dimensions can be regarded as insignificant in comparison with 

 its length. Whilst the strut remains perfectly symmetrical about 

 its middle line, its strength will depend only upon the resistance 

 of the material to crushing. Everyone knows that this would 

 be an inadequate conclusion ; we have to consider another 

 element, namely, its stability, that is, we must examine what 

 will happen to the strut if from any cause it is displaced some- 

 what from the direct line between its extremities. A mathe- 

 matical discussion of the question results in a differential equa- 

 tion of the second order with one independent variable. Upon 

 consideration of this we are enabled to see that if the thrust 

 upon the strut be less than a certain critical value, a slightly 

 bent strut will tend to return to its straight condition ; but 

 that if the thrust upon the strut be greater than this critical 

 value the displacement will tend to increase, and the strut will 

 give way. Further, that the critical value will depend upon 

 whether one or both of the two extremities of the strut are held 

 free, or whether they are rigidly attached liy flanges or other- 

 wise, so that the direction of the axis of the stiut at this point 

 must remain unaltered. Again, we infer that if the ends are 

 held rigidly fixed, the length of the strut may be twice as great 

 for a given critical value of the thrust as if the two ends ate free to 

 turn. We can also infer what the critical value will be for 

 struts of various lengths and of varying cross sections. This 

 critical value depends not upon the resistance of the material to 

 crushing, but upon its rigidity. 



Another example, having a certain degree of similarity with 

 the case of struts, is that of a shaft running at a high number of 

 revolutions per minute, and with a substantial distance between 

 its bearings ; for simplicity, we will suppose that there are no 

 additional weights, such as pulleys, upon the shaft. How will 

 the shaft behave itself in regard to centrifugal force as the 

 speed increases? In this case, so long as the shaft remains 

 absolutely straight it will not tend to be in any way affected by 

 the centrifugal force, but suppose the shaft becomes slightly 

 bent, it is obvious to anyone that if the speed be enormously 

 high this bending will increase, and go on increasing until the 

 shaft breaks. In this case also we may use mathematical 

 treatment ; we find that the condition of the shaft is expressed 

 by a diliferential equation of the fourth order, and from con- 

 sideration of the solution of this equation we can say that if the 

 speed of any particular shalt be less than a certain critical 

 speed, the shaft will tend to straighten itself if it be momentarily 

 bent, but that, on the other hand, if the speed exceeds this 

 critical value, the bending will tend to increase with the probable 

 destruction of the shaft. I do not know that either of these 

 two questions can be properly understood wiihout some know- ; 

 ledge of differential equation-. 



A problem having a certain analogy to those to which I have 

 just referred is that of hollow cylinders under compression from 

 without, such as boiler tubes. Whether the tubes be thick or 

 thin, so long as they are perfect circular cylinders, they should 

 stand until the material was crushed. But if the tubes are thin, 

 what will happen if the tube from any cause deviate ever so 

 little from the cylindrical form? The solution cannot be 

 obtained without a substantial quantity of mathematics. 



The next illustration shows how a mathematical conclusion, 

 correc. within the limits to which it applies, may mislead if ap- 

 plied beyond those limits, and how a more thorough mathenn- 

 tical discussion will give a correct result. Considering a case of 

 shafting in torsion it was shown by Coulomb that the stillfness 

 and strength of a shaft having the form of a complete circular 

 cylinder could be readily calculated if the transverse elasticity 

 of the material and its resistance to shearing were known. From 

 the com|ilcte symmetry about the axis it is evident that points 

 which lie in a plane perpendicular to the axis before twisting will 



still be in that plane when the shaft is twisted ; it is also clear 

 that the angle through which all points in the same plane move 

 will be the same ; hence the problem was as simple as problem 

 could be. But many who had occasion to make use of Coulomb's 

 results gave them an application which was wholly unwarranted. 

 They assumed that they were equally applicable to other cases 

 than complete circular cylinders ; they assumed in fact that every 

 point of the material which lay in a plane perpendicular to the 

 axis would remain in that plane when the shaft was twisted, 

 whether the shaft was symmetrical about its axis or not, and they 

 consequently arrived at very erroneous results. That the assump- 

 tion was erroneous is obvious enough from a consideration of an 

 extreme case. In Fig. i is shown in cross- section a hollow cylin- 

 drical shaft, which is not complete, but divided by a plane passing 

 through its axis. In this case the shaft when twisted will be as 

 illustrated in the side elevation ; two points, a and i;, were in 

 one plane perpendicular to the axis when the shaft was free from 

 twist ; they cease to be in one plane when the shaft is twisted. 

 St. Venant' in 1855 investigated the question of shafts without 

 making incorrect assumptions ; he expressed the condition of 

 the material by a partial dififerential equation of the second 

 order, and gave suitable surface conditions. A general solution 

 of the problem for all forms of shafts has not been obtained, but 

 St. Venant gives a number of solutions for particular forms, and 

 he obtains some general results of interest. In all cases the 

 stififness of the shaft is less than would be inferred from an 

 erroneous application of Coulomb's theory. Fig. 2 shows 

 diagrammatically the strain in a shaft of triangular section ; the 

 full lines indicate that the parts of the shaft which lay in one 



Fig. 



Fig. 2, 



plane before twisting when twisted rise above the plane : the 

 dotted lines indicate that they lie behind the plane of the 

 paper. The shearing stress is least at the angles of the triangle, 

 and is greatest at the middles of the sides. At this point then 

 the shaft will begin to break under torsion. The fact is prob- 

 ably well known to men of practical experience, but it is 

 directly contradictory to the conclusion at which one would 

 arrive by a careless use of Coulomb's theory beyond the narrow 

 limits within which it is applicable. The longitudinal ribs 

 which one often sees on old cast-iron shafts are useful enough to 

 give stiffness to the shafts against bending, but are good for 

 very little if torsional stiffne.ss or torsional strength is desired. 



Another application of mathematical theory which has been 

 carried somewhat further than the premises warrant is found in 

 the case of girders. It is almost invariably customary to ticat a 

 girder as though the sections retain when the girder is bent the 

 form and size which they had before bending. Making this 

 assumption, it is very easy to calculate the strength and stiffness 

 of a girder of any section. Unfortunately, the assumption is 

 untrue ; but, fortunately, it is approximately true in the case of 

 most girders with which engineers in practice have to deal. 

 That it is untrue can be readily seen from consideration of a 

 girder of exaggerated form, the section of which is .shown in 

 jTig 3. Any practical man would at once see that the outer 

 parts of the flanges would add little to the strength of the girder, 

 but according to the usual mathematical theory the outer parts of 

 this flange should be as useful as the parts which are nearer to 



' " Mcmoircs des S.ivanls Ku.ingers," i355; and Thomson and Tail, 

 *' Treatise on Natural Philosophy."' 



NO. 1280, VOL. 50] 



