34 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[February, 



the accuracy and trustworthiness of one. It mie^ht he thought 

 that the professed admirers of "t)ie great Sir Christopher AVren" 

 would h)ng ere this have suggested, — and not only suggested, but 

 e;irnestly promoted, some work which should liave for its object 

 the satisfactory illustration of his master-piece. Indeed, a kind of 

 fatality, untoward fate, or destiny seems to hang over St. Paul's; 

 for to what else than fatality, excej>t it be to the most unaccount- 

 able ])er;'erseness, can we ascribe its truly-wretched emplacenient, 

 in which everything stands askew! It is not so much too confined, 

 as it is confusedly huddled-up. Were the same space equalised, 

 and reduced into some regular sha|)e, it would perhaps be suffi- 

 cient, and preferable to a more extended one; inasmuch as too 

 great space around it tends to diminish to the eye the apparent 

 bulk of a building. For the meanness of the surrounding houses 

 it is not at all difficult to account ; but most unaccountable it is 

 that they should have been permitted to grow up quite capri- 

 ciously, — zigzaggedly, and at all sorts of angles, without the 

 slightest regard "to that alignement vvhich is observed for ordinary 

 streets, and which ought most assuredly to have been enforced 

 there. At present, St. Paul's Churchyard is a reproach to the 

 City and its " powers that be ;" the more so as there is no spot in 

 the whole metropolis that holds out greater opportunity for archi- 

 tectural display, while at the same time such display could not 

 but be generally beneficial to the City itself, by serving as a 

 counterpoise to attractions at the west-end of the town. No 

 doubt the value of property just around St. Paul's is so very great 

 as to render any systematic plan of improvement a formidable un- 

 dertaking ; still, were any scheme of the kind carried out, a con- 

 siderable rise in the value of the property might reasonably be 

 looked for. If, however, the outlay required for improvement ac- 

 counts for the actual deformity of the wh(de area not being cor- 

 rected, it does not account for the deformity itself, which appears 

 to have been established perfectly for the nonce. Nothing less 

 than inexplicable is it, that whatever irregularity was permitted 

 elsewhere, some stringent measures should not have been enforced 

 to ensure at least a decent locale for the new cathedral, if only by 

 making the lines of the surrounding houses parallel to its plan, 

 and equidistant from the edifice on every side. Schemes of im- 

 provement have been put forth : one of them, nearly fifty years 

 ago, by the late George Dance, which, besides greatly extending 

 and sjinmetrizing the area immediately around the church, 

 planned a new street carried from the east end of the Churchyard, 

 in a straight line to the Monument, — now more wanted than it 

 then was, in order to relieve the excessive traffic through Cheap- 

 side, increased as it now is by that to the railways on the other side 

 of London-bridge. There was certainly something happy, too, in 

 the idea of approximating, as it were, two of Wren's works, by 

 forming a vista, one end of which would have been terminated by 

 the Monument, and the other by St. Paul's. Little more than 

 twenty years afterwards, Mr. James Elmes brought forward another 

 scheme, confined to the improvement of the "Churchyard," which 

 was ingeniously shaped to follow the outline of the plan of the 

 Cathedral, there being a small crescent facing each of the tran- 

 septs and its semicircular portico. The scheme was to have been 

 promoted by the Duke of York, hut he died before any steps could 

 be taken in it, and it dropped at once. Now, there exists an ob- 

 stacle to such complete improvement which did not at that time, 

 the present St. Paul's School not being then erected. 



NOTES ON ENGINEERING.— No. XII. 

 By HoMERSHAM Cox, B.A. 



The Centrifugal Strains of Wlieels of Railway Carriages. 



The investigation of the strains of the tyres of wheels of rail- 

 way carriages, produced by rotation, is interesting, not only on 

 account of its importance with respect to public safety, but also on 

 account of the very instructive example which it affords of the 

 application of dynamical pi'inciples. 



Little more than a year ago, a fatal accident occurred on one of 

 the principal railways of this kingdom, by the tyre of a railway 

 carriage iu motion being thrown off by its centrifugal force, and 

 striking a carriage of another train. Many other cases have 

 occurred of the similar disruption and violent projection of the 

 ])onderous masses of metal of which the tyres of railway wheels 

 are composed. The practical importance of the question, there- 

 fore, becomes very great, when it is considered that the centri- 

 fugal strains upon tyres may be so great as to seriously atfect their 



strength, and that the momentum which they acquire when pro- 

 jected may be so great as to render them most destructive agents. 



When a material substance is moving in a curve, the total ex- 

 ternal force acting on it in a direction normal to the curve may he 

 estimated from a knowledge of the actual velocity and the radius 

 of curvature. This normal force is usually called centrifugal ; 

 and the principal value of the theory of centrifugal force consists 

 in this — that it leads to a determination of the normally-resolved 

 part of the external forces acting on a moving body, when the 

 magnitudes and directions of the external forces themselves can- 

 not be ascertained. 



By a principle which need not be here demonstrated, since it is 

 to be found iu numerous mechanical treatises. The centrifiigiU force 

 (in ])ounds) of a small body moving in a curve = the weight (in 

 pounds) X the square of the number of feet described per second 

 -^ 32j times the radius of curvature (in feet). For instance — if 

 the weight of a body be 10 lb., and its velocity 8 feet per second, 

 in a curve of which the radius is 5 feet, the product of the weight 

 and square of the number of feet per second is 10 X square of 8 

 = 10 X C* = etO. This divided by 32i times the radius (= 32^ 

 X 5, or 1()1) gives S-J^ lb., or nearly 4 lb. for the amount of the 

 centrifugal force. 



The rule above enunciated, when expressed by a mathematical 

 formula, gives the value of the centrifugal force equal to 



mv- W V- 



r ~ g'r^ 

 where m is the mass of the body, v its velocity, W its weight, 

 r the radius of curvature of its path, and g the force of gravity. 

 AVhen the velocity is expressed in feet per second, the force g 

 must be similarly expressed, and therefore ^ 32j, since that is the 

 velocity, in feet, generated during one second in a body falling 

 freely by the action of gravity. 



The formula just given will now be applied to determine the 

 tension due to centrifugal force of a circular ring revolving uni- 

 formly about a fixed centre. 



Since every part of the ring revolves with the same velocity 

 about the same centre, it is acted on by the same centrifugal force. 

 It is easily seen, then, that if the ring were perfectly flexible, its 

 circular form would not be altered by the centrifugal forces. 

 Hence, it follows that at every point its tension is tangential, or 

 in the direction of its length, as it would be if the ring were a 

 flexible string. If the tension were in any other than the tangen- 

 tial direction, it would tend to bend the ring. It is also clear that 

 the tension is the same in every part of the ring : let this tension 

 be called T. 



This being premised, let us consider the forces acting on a quad- 

 rant of the ring. The quadrant at its two extremities is acted on 

 by two tangential forces, T, which are evidently at right angles to 

 each other; and also at every point by its centrifugal forces nor- 

 mally. If rf.v be an element of the arc, y^ds its mass, v its linear 

 velocity, and r tlie radius of the ring, the centrifugal force of any 

 element of it, by the principle above laid down, is 



^a« . — . 

 r 



If the radius at any point of the quadrant be inclined to the 



radius at one extremity at an angle 6, the part of the centrifugal 



force resolved parallel to that radius is 



. t»- , , 



lids . - cos 6; or, uafl ti' cos fl ; 

 r 



since ds := rde. 



Now, the sum of the resolved parts of the centrifugal force 



parallel to T must equal T, since there is eciuilibrium, and all the 



other forces acting on the quadrant are perpendicular to these. 



Hence, ,^ /" , „ 



T = I lids V cos e = IXV-, 



integrating between limits and 90°. Let / be the length of the 

 whole circle, and therefore /</ its mass. Therefore, putting its 



weight = W, jti ^ - ,• Also, if t be the time of revolution, or 



that in which any point of the ring moves through the space /, we 



have vt = I, and v = - . Substituting these values of v and ft iu 



the above expression for T, 



Wl 

 tension of ring = — — ; 



which formula gives the following rule for finding the tension due 



