202 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



fSEPTEMDER, 



equal to the centrifiicral force, plus the normal component of the 

 weifrht. The curvature of a deflected girder is in fjeneral so ex- 

 ceedinjily small, tliat it will he (piite safe to assume the pressure 

 equal to the centrifutral force, ])lus the weight itself. The curve 

 assumed hy the surface of tlie heam depends on the forces acting; 

 on it ; and we here supjjose the beam to be at rest, althoutth the 

 load upon it is in motion. Hence, the elastic forces of tlie beam 

 are in statiml equilibrium with tlie pressures on the curve. 



The origin of co-ordinates beinff at one end of the beam, and the 

 axis of .r, measured vertically downwards, at any point (.r, y) ot 

 tlie curve, the tangent of the angle of horizontal inclination is 



all. Hence, neglecting the square of 



— ^, which is always very smii 

 a y 



quantity as inconsiderable compared with unity, we may 



the inverse of the radius of curvature at the point 



tliat 

 put 



(P- X 



' dy'- 

 where concave upwards.) 



(■'■, .'/) 



(The sign never changes, as the curve is every- 



From the theory of perfectly elastic beams, it appears that 



dir- 



is, at every part of the beam, proportional to the moment about 

 (j', y) of all the pressure acting between that point and either ex- 

 tremity of the beam. Here, the pressures between (,r, y) and tlie 

 origin are the centrifugal forces and the weights acting downwards, 

 and the pressure of the abutment acting upwards. The moment 

 of all the centrifugal forces may be first ascertained. 



Each small portion of the load may be supposed to act inde- 

 pendently of the rest, or to press on the curve with its own weight 

 and its own centrifugal force. Let m be the unit of mass; and 

 therefore, at any point (.t' y') intermediate between (,r, y) and the 

 end of the beam, m d y' the mass of an element of the load. 

 Calling V its linear velocity, it appears from what has been 



already said about tho radius of curvature, that V' -— ^ mdy' is 



the centrifugal force of that element. The moment of this 



centrifugal force is 



■V2 — . iy-y') mdy'. 



way between (,r, y) and the origin, and therefore equals 



The moment of all the centrifugal forces about point (.r,j/) will 

 be found by integrating this expression between the limits;/ := 

 and y' = y. So it may be ascertained that the moment of these 

 forces is 



m V' {y tan ^—x), 

 where /3 is the horizontal inclination of the curve at the origin. 

 y tan $ — .v is the length of a vertical line drawn downwards from 

 the point {>\y) to meet the tangent drawn from the origin; and 

 is very small. 



The weight of the portion of the load upon the horizontal 

 length of the heam y, is m g y ; and its moment about the point 

 (,c, J/) is the same as if the weight were collected at a point half 



m g y^ 



Also, if P be the pressure of the abutment, Vy is its moment; and 

 representing the constant, by which tlie radius of the curvature has 

 to be multiplied to render it equal to the sum of the moments, 

 by E I, we have — 



EI^ = mo ^' - P y+ mV^iyUnB-x). 

 dy' "2 " ' ^* ' 



This equation is integrable in its present form ; but as the last 



term of it is very small, we may make an alteration which will 



tend very much to the simplicity of the results. The centrifugal 



jiressures cannot under any circumstances be great compared with 



the other forces, as may readily be foreseen by considering that in 



all cases of actual practice, the curvature is very small on account 



of the very small proportion which the central deflection bears to 



the length of the beam. F'or any'central defection preinoiisly assigned, 



the curve would be very little altered if the centrifugal pressure 



were uniformly distributed. Therefore, in the above equation, 



the small term jn V- (_;/ tan 8 — .r) is neglected in estimating the 



radius of the curvature merely. Now, it appears from the " iVIe- 



chanical Principles of Engineering," Art. 374, that when the beam 



is subject to any uniformly distributed pressures whatever, 



d'o! 24.D ,, . 



where a is half the length of the beam, and D is its central de- 

 flection. The curvature of the beam, when it assumes its perma- 

 nent form under the influeaoe of a passing load, will not greatly 



diff'er from that which this equation indicates. Of course, this 

 hypothesis does not suppose the distribution of the centrifugal 

 pressures to be actually uniform — it merely presumes that the 

 curvature of tlie boani,y«)' a given deflection, is nearly the same as 

 if the pressures were so distributed. 



From the equation last given, the value of -; — at the centre, 



d y'^ 



St 1) T,, . 



IS —. — -• Ihcreiorc, at the centre, a 



10 u-^ 



weight moving with 



24 D 



10~a= 



times its mass. 



velocity V, has a centrifugal pressure = V' 



To ascertain the whole effect of centrifugal pressure, we have 

 evidently the expression 



—JmV ^ dy = -m V= ^i^ fi^'f-ay) dy, 



integrating between limits y — and )/ = 2 a. 



From this, it appears that the total centrifugal pressure 



32 D 1 V- 



— — rr, 10 a, which becomes ?H w a- — D, if? — 32 



— TO V- ' ^ 10 a2 ' 



Now, if T be the number of seconds in w'lich either end of tie 



V V 4 



load traverses the beam, ,., = 2a, and -- r= ,-. Substituting 



this value, and remembering that the total weight on the beam 

 is 2 mqa, we find ultimately, 



^ ' ID 



Centrifugal pressure on whole heam ■ 



5 T- 



X the weight. 



What very strongly confirms this conclusion, and shows that no 

 niatei-ially great error is contained in it, is the consideration that 

 if the curve had been supposed to be a circular arc passing through 

 the middle and two ends of the curve, tlie effect of centrifugal 

 pressure would be almost the same as the above formula gives it. 

 The only difference (as will be seen hereafter) would be, that in 

 the formula we must substitute J for tL. ^V'hen it is considered 

 how exceedingly small the curvature of the beam must necessarily 

 be in all practical cases, it becomes clear that a circular arc of 

 large radius would represent the curve with at least tolerable ac- 

 curacy. At all events, that assumption furnishes a safe test of 

 the foregoing conclusions. 



Rule for Calculating the Prcsstire. 

 The formula then gives all the information that can be generally 

 required, respecting the influence of the velocity of the train, or 

 its pressure on the deflected girder, when the mass of the former 

 is not small compared with that of the latter. Put into ordinary 

 language, the formula amounts to this — that u-hen a long uniform 

 load moves over a girder ivhich is perfectly elastic, originally horizontal, 

 the greatest pressure on the girder is that of the weight on it at any 

 time -\- a small fraction of that weight, which fraction is found by 

 dividing one-fifth the deflection Cin parts of a foot) by the square of 

 the number of seconds in which either end of the load traverses the 

 girder. 



In order to give a clear idea of the value of the formula, and to 

 show how small the influence of velocity generally is, one or two 

 practical applications may be given. 



A heavy train moves over a girder 88 feet long, at the rate of a 

 mile a minute, and the observed deflection is one-third of a foot. 

 To find the pressure on the girder. 



In this case, either end of the train moves over the girder in 

 one second. The square of the number of seconds is therefore 1. 

 The deflection is i. Therefore, the fraction of the weight is ■^, 

 Or the extreme pressure on the girder is one-fifteenth more than 

 the weight on it at any time. 



A train moves off a girder in three-fourths of a second, and the 

 observed deflection is one-fourtli of a foot. 



Here the square of £ is ^. One-fifth the deflection is jV; and 

 ^ divided by -ps g^ives ,^3 ; or the pressure is not quite one-twelfth 

 more than the weight at any time on the girder. 



These instances give the dynamical pressure as large as it is 

 ever likely to be with a properly-constructed girder-bridge. They 

 consequently show that the dynamical pressure of heavy loads, 

 even at high velocities, verj; little e.xceeds the statical ; and at low 

 velocities, differs from it only in an inappreciable degree. 



It will be observed, that if the velocity be indefinitely increased, 

 or T in the formula indefinitely diminished, the dynamical pres- 

 sure is indefinitely increased. But the formula virtually excludes 

 these hypothetical cases ; for the investigation proceeded on the 

 assumption that the centrifugal pressures are comparatively smaU, 

 and that the whole pressure produces but a small deflectiOD. 



