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SPIRALS 
 
 for 
 
 Street Railway Curves and 
 Easement Curves 
 
 for 
 
 treet Railway Branch-Off s 
 
 OF THE 
 
 UNIVERSITY 
 
 Complete Formulas and Tables 
 
 The Pennsylvania Steel Company 
 
 a I . 
 
 Steelton, Pa. 
 FROG AND SWITCH DEPARTMENT 
 
 Copyright 1906 by The Pennsylvania Steel Company, SteeUon, Pa. 
 
THE PENNSYLVANIA STEEL COMPANY 
 
 LIST OF OFFICES 
 
 1 ne Pennsylvania kteel (Company 
 
 GENERAL SALES OFFICE 
 Philadelphia, Pa., Girard Trust Building 
 
 Cable .A/Jress, Penns/ee/, Philadelphia. (l/se Lieber's Code.) 
 
 BALTIMORE, MD. . . Continental Trust Building 
 
 BOSTON, MASS. . . . . . . . 70 Kilby Street 
 
 CHICAGO, ILL. . ' . . . Western Union Building 
 LONDON, ENG ..... 110 Cannon Street, E. C. 
 
 MEXICO, MEX. ... 76 Calle de Cinco de Mayo 
 
 NEW YORK CITY, N. Y. ... . . 71 Broadway 
 
 PHILADELPHIA, PA. . . . Girard Trust Building 
 
 SAN FRANCISCO, CAL. . - 
 
 ST. Louis, Mo. . Commonwealth Trust Building 
 STEELTON, PA ........... Works 
 
THE PENNSYLVANIA STEEL COMPANY 5 
 
 JHE Spiral System presented 
 in the following pages has 
 been in use by The Penn- 
 sylvania Steel Company for the last 
 decade, and has given general satis- 
 faction. It consists of a transition 
 formed of arcs of diminishing radii, 
 and is calculated upon the center line 
 of track, since that is substantially the 
 path followed by the center of gravity 
 of the cars. The purpose of a spiral 
 ending on a short radius street railway 
 curve is not as in steam railroad prac- 
 tice, to enable the super-elevation of 
 the outer rail of curve to be gradually 
 attained, but to reduce the shock due 
 to the change in direction of the car. 
 The switch easements are so designed 
 that a plain curve can be converted 
 
 1 
 
6 THE PENNSYLVANIA STEEL COMPANY 
 
 into a branch-off with the minimum 
 disturbance of existing work while re- 
 taining standard switches and mates, 
 and some simplification of computation 
 is also obtained. The solutions of the 
 various problems given present suffi- 
 cient information to enable any data 
 required for the laying out or con- 
 struction of track to be readily secured. 
 
THE PENNSYLVANIA STEEL COMPANY 
 
 PROBLEM 1. To select a spiral. 
 
 (a) The radius of the main curve must be less 
 than the preceding branch of the spiral, must be 
 more than the next branch would be were it pro- 
 duced, and should nearly equal the latter. 
 
 (b] The longer the spiral, the easier the entrance 
 will be. But bear in mind that the main body of the 
 curve should be circular, the spiral simply acting as 
 an entrance to it. 
 
 (c} A spiral of less than three branches should 
 not be used. 
 
 Problem 
 
 PROBLEM 2. Given: A circular curve with sym- 
 metrical spirals, to find the tangent and external 
 
 distances. 
 
 O G R+X versine S R; 
 GS=Y sineSR; 
 
THE PENNSYLVANIA STEEL COMPANY 
 
 Tangent distance O G tan y 2 A +G S ; 
 External distance O G ex sec V 2 A +X- 
 S R. 
 
 PROBLEM 3. Given : The tangent distance V S, 
 the intersection angle A, and the desired length of 
 spiral, to find the radius of the curve. Approximate 
 R=cotangent y z A(V S ^ length of spiral). Hav- 
 ing selected a spiral by this radius, the exact radius 
 may be found, if required, by the following formula : 
 p _cos y 2 A (V S Y X tan y 2 A ) 
 sine (y 2 A S) 
 
 Caution If the result is enough different from 
 the original radius to require a change in the spiral 
 by Problem 1, a second trial must be made. 
 
 PROBLEM 4. Given : The intersection angle A 
 
THE PENNSYLVANIA STEEL COMPANY 
 
 and the external distance V H, to find the radius. 
 Approximate to the radius by finding that for a 
 simple curve passing through the point H, and select 
 a spiral for a radius somewhat smaller. 
 Then R= VHcosKA-X 
 cos S cos y 2 A 
 
 Caution If the result is enough different from 
 the original radius to require a change in the spiral 
 by Problem 1, a second trial must be made. 
 
 Problems 5 and 6 
 
 PROBLEM 5. Given : The X and Y for any point 
 on the spiral ; to find the deflection from the tangent 
 at the point of spiral. 
 
 Tangent deflection angle^^- 
 
 PROBLEM 6. Given : The X and Y for any point 
 on the spiral, to find the long chord. 
 
 Y 
 
 (a) Long chords : -j -. . 
 
 cosine det angle 
 
 or (b] Long chord ^^/xa-i-Y 2 . 
 PROBLEM 7. Given : X and Y for a point on the 
 spiral, -to find X' and Y' on a line parallel to the 
 spiral, and offset the distance S S' inside the spiral. 
 X'=X S S' versine S; 
 Y'=Y SS'sineS . 
 
 Note Problems 5 and 6 can then be applied to X' 
 and Y' if it is desired to use deflection angles to lay 
 out the curve. 
 
10 THE PENNSYLVANIA STEEL COMPANY 
 
 it. 
 
 Problem 7 
 
 As these curves will almost invariably be laid out 
 on an offset varying with the gage of the road, the 
 deflections are not figured in the table. 
 
 Problem 8 
 
 PROBLEMS. Given: 
 A circular curve with spi- 
 rals to find the distance 
 V V, in order to lay out a 
 
THE PENNSYLVANIA STEEL COMPANY 11 
 
 tangent to the circular curve, from which the latter 
 may be laid out in the usual manner. 
 
 V H = see Problem 2; 
 
 V H=R ex secant (^ A S) ; 
 
 V V'=VH V H. 
 
 -Jt-L 
 
 ------s-- 
 
 PROBLEM 9. General solution for unsymmetrical 
 curves. 
 
 O G=R+X R versine S ; 
 -G S=Y R sine S; 
 O G' R+X' R versine S' ; 
 GS'=Y RsineS '; 
 
 V S=tan y 2 AO G+G S G '~ 
 
 V S'=tan y z AO G+G' S' 
 
 sine A 
 Q G' O G 
 
 tan A 
 
12 
 
 THE PENNSYLVANIA STEEL COMPANY 
 
 Note in above; + if A is more than 90, 
 and if A is less than 90. 
 
 A'= A (S+S 0/ ); 
 V Lor V L'=tan #A' R; 
 
 V B=X+V LsineS ; 
 VB=VS (Y+V L VLversine S); 
 
 VV' = VB if C =45 -; 
 cosine C 
 
 VV'=-X^if C=45+. 
 sine C 
 
 PROBLEM 10. Given : The middle ordinate for a 
 chord of length a b for R and R', to find the middle 
 ordinate at the P C C. From the figure it is evident 
 that d' c' bisects c d. 
 
 .-. ef= 
 
 cf+df 
 2 
 
 Therefore, the middle ordinate at any P C C in the 
 spiral equals one-half the sum of the middle* ordi- 
 nates for the radii on each side for the same chord. 
 
THE PENNSYLVANIA STEEL COMPANY 13 
 
 D' 
 
 v. 
 
 A F B 
 
 Problem 11 
 
 PROBLEM 11. Given: That portion of a spiral 
 with equal chords L, L' and L" and angles a b, a, 
 and a+b, to find the middle ordinate at the center of 
 the chord L in the length D' C'. 
 
 CF=C' AandDF=D'B. 
 
 From the figure it is evident that D' C' bisects C D. 
 CF+DF 
 
 .-. E F=- 
 
 2 
 
 Then C' A=% L tan tfa+L' sine (% a 
 D' B y 2 L tan X+ L " sine (K a 
 
 and since the sines of small angles are proportional 
 to the angles, 
 
 C F+D F 
 
 =E F=X L tan % a+L, sine a. 
 
 But this last equation equals the middle ordinate in 
 the length A B for the radius of the central arc ; and 
 since the increment to the angle b would be equal if 
 L' and L" were equal, the middle ordinate at the 
 center of any arc of the spiral, for any length of 
 chord, is equal to the middle ordinate of the radius 
 of that arc in the same length. 
 
THE PENNSYLVANIA STEEL COMPANY 15 
 
 Standard Spirals. 
 The Pennsylvania Steel Co 
 
 Center Line Data. 
 
16 THE PENNSYLVANIA STEEL COMPANY 
 
 SPIRAL No. 2 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 O 
 
 Versine 
 
 Sine 
 
 300 
 150 
 100 
 
 75 
 60 
 50 
 42^ 
 37^ 
 
 0-30' 
 1-00' 
 1-30' 
 2-00' 
 2-30' 
 3-00' 
 3-30' 
 4-00' 
 
 0.011 
 0.057 
 0.160 
 0.342 
 0.627 
 1.036 
 1.587 
 2.309 
 
 2.618 
 5.235 
 7.851 
 10.463 
 13.065 
 15.651 
 18.187 
 20.703 
 
 0-30' 
 1-30' 
 3-00' 
 5-00' 
 7-30' 
 10-30' 
 14-00' 
 18-00' 
 
 .00004 
 .00034 
 .00137 
 .00381 
 .00856 
 .01675 
 .02970 
 .04894 
 
 .00873 
 .02618 
 05234 
 .08716 
 . 13053 
 .18224 
 .24192 
 .30902 
 
 SWITCH EASEMENT S 2-75 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 s 
 
 This Ease- 
 ment gives an 
 O G equal to 
 and a G S 
 3. 346 less than 
 vSpiral No. 2. 
 
 75 
 
 45K 
 42^ 
 37^ 
 
 7-50' 
 
 2-40' 
 3-30' 
 4-00' 
 
 0.700 
 1.036 
 1.587 
 2.309 
 
 10.222 
 12.305 
 14.841 
 17.357 
 
 7-50' 
 10-30' 
 14-00' 
 18-00' 
 
 SPIRAL No. 2^ 
 
 Rad. 
 
 Angle 
 
 x 
 
 Y 
 
 s 
 
 Versine 
 
 Sine 
 
 444 
 222 
 
 148 
 111 
 89 
 
 74 
 63^ 
 55^ 
 49 
 44X 
 40^ 
 
 0-20' 
 0-40' 
 1-00' 
 1-20' 
 1-40' 
 2-00' 
 2-20' 
 2-40' 
 3-00' 
 3-20' 
 3-40' 
 
 0.007 
 0.038 
 0.105 
 0.226 
 0.414 
 0.684 
 1.051 
 1.529 
 2.128 
 2.870 
 3.763 
 
 2.583 
 5.166 
 
 7.748 
 10.328 
 12.910 
 15.478 
 18.038 
 20.576 
 23.070 
 25.550 
 27.983 
 
 0-20' 
 1-00' 
 2-00' 
 3-20' 
 5-00' 
 7-00' 
 9-20' 
 12-00' 
 15-00' 
 18-20' 
 22-00' 
 
 .00002 
 00015 
 .00061 
 .00169 
 00381 
 .00745 
 .01324 
 .02185 
 .03407 
 .05076 
 .07282 
 
 .00582 
 .01745 
 .03490 
 .05814 
 .08716, 
 .12187 
 .16218 
 .20791 
 .25882 
 .31454 
 .37461 
 
 
THE PENNSYLVANIA STEEL COMPANY 17 
 
 SWITCH EASEMENT S 2^-100 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 o 
 
 This 
 
 Ease- 
 
 
 
 
 
 
 ment gives an 
 
 102^ 
 
 6-30' 
 
 0.658 
 
 11.584 
 
 6-30' 
 
 O G equal to 
 
 56X 
 
 5-30' 
 
 1.529 
 
 16.936 
 
 12-00' 
 
 and a 
 
 G S 
 
 49 
 
 44^ 
 40^ 
 
 3-00' 
 3-20' 
 3-40' 
 
 2.128 
 2.870 
 3.763 
 
 19.430 
 21.910 
 24.343 
 
 15-00' 
 18-20' 
 22-00' 
 
 3. 640 less than 
 Spiral No. 2|-. 
 
 SWITCH EASEMENT S 2^-200 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 s 
 
 Tvjiic 
 
 
 200 
 
 4-00' 
 
 0.487 
 
 13.951 
 
 4-00' 
 
 ment gives an 
 
 129 
 
 1-00' 
 
 0.664 
 
 16.196 
 
 5-00' 
 
 O G 
 
 0.250 
 
 74 
 63^ 
 55^ 
 49 
 
 2-00' 
 2-20' 
 2-40' 
 3-00' 
 
 0.934 
 1.301 
 1.779 
 
 2.378 
 
 18.764 
 21.324 
 23.862 
 26.356 
 
 7-00' 
 9-20' 
 12-00' 
 15-00' 
 
 greater 
 G S 
 greater 
 
 and a 
 3.286 
 
 than 
 
 44^ 
 
 3-20' 
 
 3.120 
 
 28.836 
 
 18-20' 
 
 Spiral No. 2. 
 
 40^ 
 
 3-40' 
 
 4.013 
 
 31.269 
 
 22-00' 
 
 
 
 SPIRAL No. 3 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 s 
 
 Versine 
 
 Sine 
 
 300 
 
 1-00' 
 
 0.046 
 
 5.236 
 
 1-00' 
 
 .00015 
 
 .01745 
 
 150 
 
 2-00' 
 
 0.229 
 
 10.468 
 
 3-00' 
 
 .00137 
 
 .05234 
 
 100 
 
 3-00' 
 
 0.639 
 
 15.688 
 
 6-00' 
 
 .00548 
 
 .10453 
 
 75 
 
 4-00' 
 
 1.368 
 
 20.871 
 
 10-00' 
 
 .01519 
 
 .17365 
 
 60 
 
 5-00' 
 
 2.501 
 
 25.982 
 
 15-00' 
 
 .03407 
 
 .25882 
 
 50 
 
 6-00' 
 
 4.118 
 
 30.959 
 
 21-00' 
 
 .06642 
 
 .35837 
 
 40 
 
 7-00' 
 
 6.143 
 
 35.403 
 
 28-00' 
 
 .11705 
 
 .46947 
 
 
 
18 THE PENNSYLVANIA STEEL COMPANY 
 
 SWITCH EASEMENT S 3-100 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 S 
 
 This Ease- 
 ment gives an 
 O G equal to 
 and a G S 
 4. 391 less than 
 Spiral No. 3. 
 
 102X 
 81 
 60 
 50 
 
 40 
 
 6-30' 
 3-30' 
 5-00' 
 6-00' 
 7-00' 
 
 0.658 
 1.368 
 2.501 
 4.118 
 6.143 
 
 11.584 
 16.480 
 21.591 
 26.568 
 31.012 
 
 6-30 
 10-00' 
 15-00' 
 21-00' 
 28-00' 
 
 SWITCH EASEMENT S 3-200 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 S 
 
 This Ease- 
 ment gives an 
 O G 0.250 
 greater and a 
 G S 2.853 
 greater than 
 Spiral No. 3. 
 
 200 
 132 
 75 
 60 
 50 
 40 
 
 4-00' 
 2-00' 
 4-00' 
 5-00' 
 6-00' 
 7-00' 
 
 0.487 
 0.889 
 1.618 
 2.751 
 4.368 
 6.393 
 
 13.951 
 18.541 
 23.724 
 28.835 
 33.812 
 38.256 
 
 4-00' 
 6-00' 
 10-00' 
 15-00' 
 21-00' 
 28-00' 
 
 SPIRAL No. 4 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 Co 
 
 Versine 
 
 Sine 
 
 420 
 210 
 140 
 105 
 84 
 70 
 60 
 
 0-42' 
 l-24' 
 2-06' 
 2-48' 
 3-30' 
 4-12' 
 4-54' 
 
 0.031 
 0.157 
 0.439 
 0.939 
 1.720 
 2.839 
 4.352 
 
 5.131 
 10.261 
 15.384 
 20.490 
 25.561 
 30.567 
 35.469 
 
 0-42' 
 2-06' 
 4-12' 
 7-00' 
 10-30' 
 14-42' 
 19-36' 
 
 .00007 
 .00067 
 .00269 
 .00745 
 .01675 
 .03273 
 .05794 
 
 .01222 
 .03664 
 .07324 
 .12187 
 . 18224 
 .25376 
 .33545 
 
 SWITCH EASEMENT S 4-200 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 s 
 
 This Ease- 
 ment gives an 
 O G 0.178 
 greater and a 
 G S equal to 
 Spiral No. 4. 
 
 200 
 125^ 
 84 
 70 
 60 
 
 4-00' 
 3-00' 
 3-30' 
 4-12' 
 4-54' 
 
 0.487 
 1.117 
 1.898 
 3.017 
 4.530 
 
 13.951 
 20.490 
 25.561 
 30.567 
 35.469 
 
 4-00' 
 7-00' 
 10-30' 
 14-42' 
 19-36' 
 
 
THE PENNSYLVANIA STEEL COMPANY 19 
 
 SPIRAL No. 5 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 S 
 
 Versine 
 
 Sine 
 
 600 
 300 
 200 
 150 
 120 
 100 
 85 
 
 0-30' 
 1-00' 
 1-30' 
 2-00' 
 2-30' 
 3-00' 
 3-30' 
 
 0.023 
 0.114 
 0.320 
 0.685 
 1.255 
 2.073 
 3.175 
 
 5.236 
 10.471 
 15.703 
 20.926 
 26.130 
 31.302 
 36.374 
 
 0-30' 
 1-30' 
 3-00' 
 5-00' 
 7-30' 
 10-30' 
 14-00' 
 
 .00004 
 .00034 
 .00137 
 .00381 
 .00856 
 .01675 
 .02970 
 
 .00873 
 .02618 
 .05234 
 .08716 
 .13053 
 .18224 
 .24192 
 
 SWITCH EASEMENT S 5-200 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 S 
 
 This Ease- 
 ment gives an 
 O G equal to 
 and a G S 
 
 4. 468 less than 
 Spiral No. 5. 
 
 200 
 144 
 120 
 100 
 
 85 
 
 4-00' 
 1-00' 
 2-30' 
 3-00' 
 3-30' 
 
 0.487 
 0.685 
 1.255 
 2.073 
 3.175 
 
 13.951 
 16.458 
 21.662 
 26.834 
 31.906 
 
 4-00' 
 5-00' 
 7-30' 
 10-30' 
 14-00' 
 
 SPIRAL No. 6 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 S 
 
 Versine 
 
 Sine 
 
 900 
 450 
 300 
 225 
 180 
 150 
 128 
 
 0-20' 
 0-40' 
 1-00' 
 1-20' 
 l-4Q' 
 2-00' 
 2-20' 
 
 0.015 
 0.076 
 0.213 
 0.457 
 0.837 
 1.385 
 2.125 
 
 5.236 
 10.472 
 15.706 
 20.936 
 26.158 
 31.365 
 36.524 
 
 0-20' 
 1-00' 
 2-00' 
 3-20' 
 5-00' 
 7-00' 
 9-20' 
 
 .00002 
 .00015 
 .00061 
 .00169 
 .00381 
 .00745 
 .01324 
 
 .00582 
 .01745 
 .03490 
 .05814 
 .08716 
 .12187 
 .16218 
 
 SWITCH EASEMENT S 6-200 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 S 
 
 This Ease- 
 ment gives an 
 O G equal to 
 and a G S 
 7. 770 less than 
 Spiral No. 6. 
 
 200 
 255 
 150 
 
 128 
 
 4-00' 
 1-00' 
 2-00' 
 
 2-20' 
 
 0.487 
 0.837 
 1.385 
 2.125 
 
 13.951 
 18.388 
 23.595 
 
 28.754 
 
 4-00' 
 5-00' 
 7-00' 
 9-20' 
 
 
20 THE PENNSYLVANIA STEEL COMPANY 
 
 SPIRAL No. 7 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 S 
 
 Versine 
 
 Sine 
 
 1260 
 
 0-15' 
 
 0.012 
 
 5.498 
 
 0-15' 
 
 .00001 
 
 .00436 
 
 630 
 
 0-30' 
 
 0.060 
 
 10.995 
 
 0-45' 
 
 .00009 
 
 .01309 
 
 420 
 
 0-45' 
 
 0.168 
 
 16.492 
 
 1-30' 
 
 .00034 
 
 .02618 
 
 315 
 
 1-00' 
 
 0.360 
 
 21.987 
 
 2-30' 
 
 .00095 
 
 .04362 
 
 252 
 
 1-15' 
 
 0.660 
 
 27.475 
 
 3-45' 
 
 .00214 
 
 .06540 
 
 210 
 
 1-30' 
 
 1.091 
 
 32.957 
 
 5-15' 
 
 .00420 
 
 .09150 
 
 180 
 
 l-45' 
 
 1.678 
 
 38.424 
 
 7-00' 
 
 .00745 
 
 .12187 
 
 157 
 
 
 
 
 
 
 
 SPIRAL No. 8 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 S 
 
 Versine 
 
 Sine 
 
 1890 
 
 0-10' 
 
 0.008 
 
 5.498 
 
 0-10' 
 
 .00000 
 
 .00291 
 
 945 
 
 0-20' 
 
 0.040 
 
 10.996 
 
 0-30' 
 
 .00004 
 
 .00873 
 
 630 
 
 0-30' 
 
 0.112 
 
 16.493 
 
 1-00' 
 
 .00015 
 
 .01745 
 
 472^ 
 
 0^0' 
 
 0.241 
 
 21.990 
 
 1-40' 
 
 .00042 
 
 .02908 
 
 378 
 
 0-50' 
 
 0.441 
 
 27.483 
 
 2-30' 
 
 .00095 
 
 .04362 
 
 315 
 
 1-00' 
 
 0.729 
 
 32.973 
 
 3-30' 
 
 .00187 
 
 .06105 
 
 270 
 
 1-10' 
 
 1.120 
 
 38.457 
 
 4-40 
 
 .00332 
 
 .08136 
 
 236 
 
 
 
 
 
 
 
 SPIRAL No. 9 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 S 
 
 Versine 
 
 Sine 
 
 2730 
 
 0- 7' 
 
 0.006 
 
 5.559 
 
 o- r 
 
 .00000 
 
 .00204 
 
 1365 
 
 0-14' 
 
 0.028 
 
 11.118 
 
 0-21' 
 
 .00002 
 
 .00611 
 
 910 
 
 0-21' 
 
 0.079 
 
 16.677 
 
 0-42' 
 
 .00007 
 
 .01222 
 
 682^ 
 
 0-28' 
 
 0.170 
 
 22.234 
 
 1-10 
 
 .00021 
 
 .02036 
 
 546 
 
 0-35' 
 
 0.311 
 
 27.791 
 
 l-45' 
 
 .00047 
 
 .03054 
 
 455 
 
 0-42' 
 
 0.515 
 
 33.346 
 
 2-27' 
 
 .00091 
 
 .04275 
 
 390 
 
 0-49' 
 
 0.792 
 
 38.899 
 
 3-16' 
 
 .00162 
 
 .05698 
 
 341 
 
 
 
 
 
 
 
 
THE PENNSYLVANIA STEEL COMPANY 21 
 
 SPIRAL No. 10 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 S 
 
 Versine 
 
 Sine 
 
 3780 
 
 0-05' 
 
 0.004 
 
 5.498 
 
 0-05' 
 
 .00000 
 
 .00145 
 
 1890 
 
 0-10' 
 
 0.020 
 
 10.996 
 
 0-15' 
 
 .00001 
 
 .00436 
 
 1260 
 
 0-15' 
 
 0.056 
 
 16.493 
 
 0-30' 
 
 .00004 
 
 .00873 
 
 945 
 
 0-20' 
 
 0.120 
 
 21.991 
 
 0-50' 
 
 .00011 
 
 .01454 
 
 756 
 
 0-25' 
 
 0.220 
 
 27.488 
 
 1-15' 
 
 .00024 
 
 .02181 
 
 630 
 
 0-30' 
 
 0.364 
 
 32.983 
 
 l-45' 
 
 .00047 
 
 .03054 
 
 540 
 
 0-35' 
 
 0.560 
 
 38.478 
 
 2-20' 
 
 .00083 
 
 .04071 
 
 472 
 
 
 
 
 
 
 
 SPIRAL No. 11 
 
 Rad. 
 
 Angle 
 
 X 
 
 Y 
 
 o 
 
 Versine 
 
 Sine 
 
 5250 
 
 0-04' 
 
 .0035 
 
 6.109 
 
 0-04' 
 
 .00000 
 
 .00116 
 
 2625 
 
 0-08' 
 
 .0178 
 
 12.217 
 
 0-12' 
 
 .00001 
 
 .00349 
 
 1750 
 
 0-12' 
 
 .0498 
 
 18.326 
 
 0-24' 
 
 .00002 
 
 .00698 
 
 1312^ 
 
 0-16' 
 
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 1020 
 
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THE PENNSYLVANIA STEEL COMPANY 61 
 
 
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62 THE PENNSYLVANIA STEEL COMPANY 
 
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THE PENNSYLVANIA STEEL COMPANY 63 
 
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64 THE PENNSYLVANIA STEEL COMPANY 
 
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THE PENNSYLVANIA STEEL COMPANY 65 
 
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66 THE PENNSYLVANIA STEEL COMPANY 
 
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THE PENNSYLVANIA STEEL COMPANY 
 
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68 THE PENNSYLVANIA STEEL COMPANY 
 
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THE PENNSYLVANIA STEEL COMPANY 69 
 
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70 THE PENNSYLVANIA STEEL COMPANY 
 
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THE PENNSYLVANIA STEEL COMPANY 71 
 
 
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THE PENNSYLVANIA STEEL COMPANY 78 
 
 Middle Ordinates 
 for 
 
 10-Foot Chords 
 
74 THE PENNSYLVANIA STEEL COMPANY 
 
 MIDDLE ORDINATES, 10 FT. CHORDS 
 
 M. O. 
 
 Radius 
 
 M. 0. 
 
 Radius 
 
 M. O. 
 
 Radius 
 
 " 
 
 infinity 
 
 1" 
 
 150- Q l / 2 " 
 
 2" 
 
 75'- 1" 
 
 *x 
 
 4807'- 8" 
 
 1 i " 
 
 145'- 6" 
 
 gi" 
 
 73' HT?T" 
 
 
 2399'- 3" 
 
 W 
 
 141'- 2H" 
 
 ^yV" 
 
 72'- 9X" 
 
 A* 
 
 1600'- 6" 
 
 1A" 
 
 137'- 2/8" 
 
 2 3 ^" 
 
 71'- 8X" 
 
 H" 
 
 1200'- 9" 
 
 W 
 
 133'- 4X" 
 
 2>|" 
 
 70'- 8y 8 " 
 
 A; 
 
 960'- 1" 
 
 1A" 
 
 129'- 9 T y 
 
 2A" 
 
 69'- 7^" 
 
 
 800'- 3 T y 
 
 1A 
 
 126'- 4^" 
 
 2 T y 
 
 68'- 7f|" 
 
 f" 
 
 682- 3 T y 
 
 IFF 
 
 123'- 2^" 
 
 37." 
 
 67'- 8y&" 
 
 
 
 600'- 0^" 
 
 IX" 
 
 120'- 0^" 
 
 sx;; 
 
 66'- 9>|" 
 
 a 9 / 
 
 533'- 4^" 
 
 1A" 
 
 117'- IX" 
 
 
 65-10^" 
 
 " 
 
 480'- T y 
 
 1A" 
 
 114'- 4^" 
 
 2 T B / 
 
 64'-ll T y 
 
 " 
 
 436'- 4 T y 
 
 lii" 
 
 in'- 8 T y 
 
 2**[ 
 
 64- i T y 
 
 " 
 
 400'- T y 
 
 1H" 
 
 109'- IX" 
 
 
 63'- 8A" 
 
 H; 
 
 369'- 3" 
 
 HI;; 
 
 106'- 8|*" 
 
 s^r 
 
 62'- 5#" 
 
 
 342'-10 T y 
 
 
 104- 4H" 
 
 
 61'- 7H" 
 
 ; 
 
 320'- OX" 
 300'- OX" 
 
 i>l" 
 
 102'- 2X" 
 100'- OX" 
 
 *f" 
 
 2 ^f" 
 
 60'- IX" 
 
 if 
 
 282'- 4 T y 
 
 HI" 
 
 98- OX" 
 
 
 59'- 4^" 
 
 : 
 
 266'- 8 T y 
 253'- Oif" 
 
 iA' 
 
 HI" 
 
 96'- OX" 
 94'- 2 T 3 ff " 
 
 S : 
 
 58'- 7X" 
 57-11X" 
 
 #" 
 
 240'- OA" 
 
 W 
 
 92'- 4^" 
 
 2^" 
 
 57'- 3" 
 
 21" 
 
 228'- 7 T y 
 
 if*' 
 
 90'- 7 T y 
 
 2|i" 
 
 56'- 7" 
 
 11" 
 
 218'- 1H" 
 
 iff' 
 
 88-11 X" 
 
 2ii" 
 
 55-11 H" 
 
 ft* 
 
 209'- l^f" 
 
 jB: 
 
 87'- 4>6" 
 
 2ft" 
 
 55'- 3 T y 
 
 'X* 
 
 200'- 2 T y 
 
 
 85'- 9 T y 
 
 2X" 
 
 54'- 7||" 
 
 ir 
 
 192'- 0^" 
 
 it!" 
 
 Q/l ' Q 7 " 
 
 84- d T % 
 
 2ff" 
 
 54'- T y 
 
 it: 
 
 184'- 7X" 
 177- 9ii" 
 
 ifi; 
 
 82-10" 
 81'- 5X" 
 
 8t : 
 
 53'- 5 T y 
 
 52-10^" 
 
 |I" 
 
 171'- 5/ 2 " 
 
 
 80'- OH" 
 
 2^1 " 
 
 52'- 3^" 
 
 29" 
 
 165'- 6^" 
 
 ill" 
 
 78'- 9X" 
 
 2f-|" 
 
 51'- 8{f * 
 
 15" 
 
 160'- OX" 
 
 i**" 
 
 77'- 6" 
 
 2|f" 
 
 51'- 2X" 
 
 ' 
 
 
 if*" 
 
 76'- 3X" 
 
 3fi" 
 
 50'- 71f 
 
 
THE PENNSYLVANIA STEEL COMPANY 
 
 75 
 
 MIDDLE ORDINATES, 10 FT. CHORDS 
 
 M. O. 
 
 Radius 
 
 M. O. 
 
 Radius 
 
 M. 0. 
 
 Radius 
 
 3" 
 
 
 
 3ff 
 
 50'- !#' 
 49'- 7 T y 
 49'- IX" 
 48'- 7^" 
 48'- 1 T V 
 
 47'- 7^" 
 47'- 2 T y 
 46- 8}f 
 46'- 3^" 
 45'-10 T y 
 
 45'- 5 T V' 
 45'- 0" 
 44'- 7" 
 44'- 2y 8 " 
 43'- 9^i" 
 
 43'- 4ft' 
 43'- T y 
 
 42'- iy 2 n 
 
 42'- 3 T y 
 
 41'- 6; 
 41'- ft< 
 40-10" 
 40'- 5j 
 40- 1; 
 
 39'- 6 r y 
 39- 2A" 
 
 38'- 3}f 
 37-11^' 
 
 4" 
 
 4/8 
 
 37'- 8" 
 37'- 4; 
 37'- Ij 
 36'- 9; 
 36'- 6, 
 
 35'- 
 35- 5ft" 
 
 OK' O9" 
 
 OO ^y% 
 
 84'-llA' 
 
 Jt": 
 
 34'- 2ft' 
 33'-!!^" 
 
 33'- 6X' 
 
 QO' O T/ ' 
 
 OO ^5/2 
 
 33'- Olf ' 
 32-10^' 
 
 32- iy 2 ' 
 
 32'- 4^' 
 32'- 2^' 
 
 31'- 
 31'- 
 
 31'- 
 31'- 
 30'-11 
 30'- 9 
 30'- 7 
 30'- 4 
 
 30'- 2}4' 
 30'- OX" 
 
 29'-10 T V" 
 
 29- 5lf" 
 
 29'- 3H" 
 29'- 1 T V" 
 
 28'- 9>^" 
 28'- lyz" 
 
 28- 5>^" 
 28'- %y 2 " 
 28'- 1 T 9 ^' 
 27'-!!^" 
 27- 9X" 
 
 27'- 7^" 
 27'- 6" 
 27^- 4^ 
 
 27'- T 9 ^" 
 
 26'-101f" 
 26'- 9 T y 
 26'- 7 T 5 ff " 
 26'- 5^" 
 26'- 31f" 
 26'- 2X" 
 
 26^- 0^ 
 
 25;- 7ft; 
 
 25'- 4A' 
 
76 THE PENNSYLVANIA STEEL COMPANY 
 
 \Vneel Contours 
 
 GREAT variety of wheel con- 
 tours are used on street rail- 
 ways, and often the extremes 
 are used on the same track sys- 
 tem to the great detriment of both special 
 work and wheels. 
 
 Since there are now no standard wheel 
 contours except the M. C. B. (steam rail- 
 road), it seems proper to offer some de- 
 signs which will meet most of the difficul- 
 ties found in our practice. 
 
 The contour of wheel fixes the charac- 
 ter of the special work as to whether the 
 frog work is to be " flange bearing." This 
 term means that at the ''waist of frog," 
 i. e., just in advance of the point of frog, 
 the floor of the throat is raised to carry the 
 wheel through the waist and past the point 
 upon its flange. The latter cannot be of a 
 shape or character of metal well adapted to 
 this service, and more or less chipped 
 flanges are thereby caused. It is, however, 
 inevitable that this support be provided, 
 
THE PENNSYLVANIA STEEL COMPANY 77 
 
 unless the tread is wide enough to carry 
 the wheel past this critical point. 
 
 Contour "A" is the M. C. B. standard 
 wheel, and is adapted to open track and 
 streets where rails with deep and wide 
 flangeways are provided, and the paving is 
 such that the tread will not be seriously 
 chipped by contact with the same. 
 
 Contour "B" is a compromise wheel, 
 for use where a portion of the track system 
 is used by wheels having Contour "A" and 
 the remainder has not the deep flangeway, 
 but the width of flangeway is provided. If 
 the paving conditions of this .portion are 
 such that the width of tread must be re- 
 duced from that shown, all frog work on 
 the entire portion of system used by Con- 
 tour "B " must be made "flange bearing." 
 
 Contour "C" is adapted to track sys- 
 tems where wide and deep flangeways can- 
 not be provided but the paving conditions 
 are such that the width of the tread shown 
 can be used. 
 
 Contour "D" is adapted to track sys- 
 tems where a narrower tread is required by 
 
78 THE PENNSYLVANIA STEEL COMPANY 
 
 the paving conditions, the shape and size 
 of flange being the same as " C." As wide 
 a tread as possible should be used, as the 
 life of special work will thereby be mate- 
 rially prolonged besides increasing the fac- 
 tor of safety on any open track in the 
 system. " Flange-bearing " frogs will be 
 required for this contour. 
 
 Contours "A" and "C" will give the 
 best results where conditions permit their 
 use and do not require "flange-bearing" 
 frogs. 
 
 Contour "B" will not give entire sat- 
 isfaction, since the flange is not of the best 
 shape, and should only be used when com- 
 pelled by the conditions stated. 
 
 Contours "C" and " D " can be used 
 on the same track system if "flange-bear- 
 ing " frogs are provided for that portion on 
 which Contour "D" is used. 
 
 Contours "C" and "D" cannot be 
 satisfactorily used on track systems de- 
 signed for Contours "A" or "B," or vice 
 versa. 
 
THE PENNSYLVANIA STEEL COMPANY 79 
 
THE PENNSYLVANIA STEEL COMPANY 81 
 
THE PENNSYLVANIA STEEL COMPANY 
 
 Angles 
 
 Bulb 
 
 Reinforcing 
 
 Switch 
 
 Equal Legs 
 
 Unequal Legs 
 
 Braces 
 
 Acme 
 
 Switch 
 
 Guard 
 
 Interlocking 
 
 Tie Plate 
 
 Bridges 
 Buildings 
 
 Chairs 
 
 Rail chairs for steam railroads in paved streets 
 
 Channels 
 
 Circles 
 
 Turntables 
 
 Coal storage tracks 
 
 Clamps 
 
 Guard rail 
 
 Crossings 
 
 Bolted 
 
 Bolted Plate 
 Keyed 
 
 " Manard" Steel 
 
 Movable Point 
 Riveted 
 
 Street Railway 
 "Manard" Renewable Centre 
 Double Slip 
 
 Single Slip 
 
 Three Rail 
 
THE PENNSYLVANIA STEEL COMPANY 
 
 DROP FORGED SOCKET 
 
 
 ACME BRACE 
 
84 THE PENNSYLVANIA STEEL COMPANY 
 
 MANARD ANVIL FACE FROG, DESIGN 153 
 
THE PENNSYLVANIA STEEL COMPANY 85 
 
 MANARD ANVIL FACE SPRING RAIL FROG 
 
THE PENNSYLVANIA STEEL COMPANY 
 
 Crossovers 
 
 Single Crossovers 
 
 Double Crossovers 
 
 Portable Crossovers 
 Forgings 
 
 Frogs, Rigid 
 
 Bolted 
 
 Bolted Plate 
 
 Keyed 
 
 "Manard" Steel 
 
 "Manard" Renewable Centre 
 
 Riveted 
 Frogs, Spring 
 Bolted 
 
 Bolted Plate 
 
 Double 
 
 Hinged 
 
 Keyed 
 "Manard" Steel 
 
 Riveted Plate 
 Twin 
 
 Vaughan Hinged 
 
 Vaughan Sliding 
 Joints 
 
 Angle 
 
 Channel 
 
 Compromise 
 
 Deep Girder Rail 
 
 Plain 
 Knees 
 
 Mates ' 
 
 Built 
 
 "Manard" Bolted 
 
 "Manard" Key Fast 
 Plates 
 
 . Frog 
 
 Slide 
 
 Switch 
 
 Tie 
 Rails, Guard 
 
THE PENNSYLVANIA STEEL COMPANY 
 
 MANARD GROOVED TONGUE SWITCH FOR STEAM 
 RAILROADS IN PAVED STREETS 
 
88 
 
 THE PENNSYLVANIA STEEL COMPANY 
 
THE PENNSYLVANIA STEEL COMPANY 89 
 
 RAPID RENEWABLE MANARD CENTRE FROG 
 
90 THE PENNSYLVANIA STEEL COMPANY 
 
 Rails 
 
 A. S. C. E. Sections 
 
 Cast Weld Compromise 
 Check 
 
 Flat 
 
 Girder 
 Girder Guard 
 
 Slot Rails and Conductor Bars 
 Tee 
 
 High Tee 
 
 Tram 
 Rails, Renewable Guard 
 
 Rods, Switch Connecting 
 
 Splice Bars 
 
 Spikes 
 
 Spiral Curves 
 
 Steam Railroad Track Equipment 
 
 Street Railway Special Work 
 
 Street Railway Track Fastenings 
 
 Steel 
 
 Bessemer 
 Billets 
 
 Blooms 
 
 Castings 
 
 Forgings 
 
 Flats 
 
 Basic Open Hearth Acid Open Hearth 
 
 " Manard " 
 
 Merchant 
 
 Round 
 
 Shapes 
 
 Slabs 
 
 Special 
 
 Square 
 
 Tool 
 Structural Steel 
 
THE PENNSYLVANIA STEEL COMPANY 
 
 91 
 
92 THE PENNSYLVANIA STEEL COMPANY 
 
 GROUND LEVER, MODEL 16 
 
 Low NEW CENTURY ADJUSTABLE SWITCH STAND 
 MODEL 51 A 
 
THE PENNSYLVANIA STEEL COMPANY 
 
 INTERMEDIATE MAIN LINE SWITCH STAND 
 MODEL 47 B 
 
94 THE PENNSYLVANIA STEEL COMPANY 
 
 Switches 
 
 Adjustable 
 Angle 
 
 Challenge 
 
 Lorenz 
 
 " Manard " Big Pin Grooved Tongue 
 
 " Manard " Big Pin Tongue 
 "Manard" Steel 
 
 Plain 
 
 Reinforced 
 Socket 
 
 Stub 
 
 Three Way 
 Switch Stands 
 Automatic 
 Banner 
 
 Steelton Detective 
 Ground Levers 
 
 Upright Levers 
 
 Long Safety 
 Main Line Adjustable 
 Positive 
 Automatic 
 Mine 
 
 Mine Kickover 
 New Century 
 
 New Century Adjustable 
 New Era 
 
 Pet 
 Semaphore with disappearing blade 
 
 Spring Ground Throw 
 
 Tie Plates Yard Stands 
 
 Tie Rods 
 Track Bolts 
 Track Fastenings 
 Track Girders 
 
 Track Material of All Kinds 
 Trough Floors 
 Trough Sections 
 Viaducts 
 "Z" Bars 
 
THE PENNSYLVANIA STEEL COMPANY 95 
 
 INTERMEDIATE SEMAPHORE SWITCH STAND, MODEL 50 E 
 
THE PENNSYLVANIA STEEL COMPANY 
 
 OF THE 
 
 UNIVERSITY 
 
 OF 
 
 Chasmar-Winchell Press New York and Pittsburgh 
 
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