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+The Project Gutenberg EBook of Transactions of the American Society of
+Civil Engineers, Vol. LXX, Dec. 1910, by Beverly S.  Randolph
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: Transactions of the American Society of Civil Engineers, Vol. LXX, Dec. 1910
+       Locomotive Performance On Grades Of Various Lengths, Paper No. 1172
+
+Author: Beverly S.  Randolph
+
+Release Date: July 3, 2006 [EBook #18747]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK AMERICAN SOCIETY OF CIVIL ENGINEERS ***
+
+
+
+
+Produced by Juliet Sutherland, Sigal Alon and the Online
+Distributed Proofreading Team at http://www.pgdp.net
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+
+
+
+AMERICAN SOCIETY OF CIVIL ENGINEERS
+
+INSTITUTED 1852
+
+
+TRANSACTIONS
+
+Paper No. 1172
+
+
+LOCOMOTIVE PERFORMANCE ON GRADES OF
+VARIOUS LENGTHS.
+
+BY BEVERLY S. RANDOLPH, M. AM. SOC. C. E.
+
+WITH DISCUSSION BY MESSRS. C. D. PURDON, JOHN C. TRAUTWINE, JR.,
+AND BEVERLY S. RANDOLPH.
+
+
+
+
+In the location of new railways and the improvement of lines already in
+operation, it is now well recognized that large economies can be
+effected by the careful study of train resistance due to grades and
+alignment, distributing this resistance so as to secure a minimum cost
+of operation with the means available for construction.
+
+While engaged in such studies some years ago, the attention of the
+writer was attracted by the fact that the usual method of calculating
+the traction of a locomotive--by assuming from 20 to 25% of the weight
+on the drivers--was subject to no small modification in practice.
+
+In order to obtain a working basis, for use in relation to this feature,
+he undertook the collection of data from the practical operation of
+various roads. Subsequent engagements in an entirely different direction
+caused this to be laid aside until the present time. The results are
+given in Table 1, from which it will be seen that the percentage of
+driver weight utilized in draft is a function of the length as well as
+the rate of grade encountered in the practical operation of railways.
+
+In this table, performance will be found expressed as the percentage of
+the weight on the drivers which is utilized in draft. This is calculated
+on a basis of 6 lb. per ton of train resistance, for dates prior to
+1880, this being the amount given by the late A. M. Wellington, M. Am.
+Soc. C.. E.,[A] and 4.7 lb. per ton for those of 1908-10, as obtained by
+A. C. Dennis, M. Am. Soc. C. E.,[B] assuming this difference to
+represent the advance in practice from 1880 to the present time. Most of
+the data have been obtained from the "Catalogue of the Baldwin
+Locomotive Works" for 1881, to which have been added some later figures
+from "Record No. 65" of the same establishment, and also some obtained
+by the writer directly from the roads concerned. Being taken thus at
+random, the results may be accepted as fairly representative of American
+practice.
+
+Attention should be directed to the fact that the performance of the
+10-34 E, Consolidation locomotive on the Lehigh Valley Railroad in 1871
+is practically equal to that of the latest Mallet compounds on the Great
+Northern Railway. In other words, in the ratio between the ability to
+produce steam and the weight on the drivers there has been no change in
+the last forty years. This would indicate that the figures are not
+likely to be changed much as long as steam-driven locomotives are in
+use. What will obtain with the introduction of electric traction is
+"another story."
+
+These results have also been platted, and are presented in Fig. 1, with
+the lengths of grade as abscissas and the percentages of weight utilized
+as ordinates. The curve sketched to represent a general average will
+show the conditions at a glance. The results may at first sight seem
+irregular, but the agreement is really remarkable when the variety of
+sources is considered; that in many cases the "reputed" rate of grade is
+doubtless given without actual measurement; that the results also
+include momentum, the ability to utilize which depends on the conditions
+of grade, alignment, and operating practice which obtain about the foot
+of each grade; and that the same amount of energy due to momentum will
+carry a train farther on a light grade than on a heavy one.
+
+There are four items in Table 1 which vary materially from the general
+consensus. For Item 9, the authorities of the road particularly state
+that their loads are light, because, owing to the congested condition of
+their business, their trains must make fast time. Item 10 represents
+very old practice, certainly prior to 1882, and is "second-hand." The
+load consisted of empty coal cars, and the line was very tortuous, so
+that it is quite probable that the resistance assumed in the calculation
+is far below the actual. Items 15 and 17 are both high. To account for
+this, it is to be noted that this road has been recently completed,
+regardless of cost in the matter of both track and rolling stock, and
+doubtless represents the highest development of railroad practice. Its
+rolling stock is all new, and is probably in better condition to offer
+low resistance than it will ever be again, and there were no "foreign"
+cars in the trains considered. The train resistance, therefore, may be
+naturally assumed to be much less than that of roads hauling all classes
+of cars, many of which are barely good enough to pass inspection. As the
+grades are light in both cases, this feature of train resistance is
+larger than in items including heavier grades. Attention should be
+called to the fact that a line connecting the two points representing
+these items on Fig. 1 would make only a small angle with the sketched
+curve, and would be practically parallel to a similar line connecting
+the points represented by Items 13 and 16. There is, therefore, an
+agreement of ratios, which is all that needs consideration in this
+discussion.
+
+[Illustration: FIG. 1.--DIAGRAM SHOWING PERCENTAGE OF WEIGHT ON DRIVERS
+WHICH IS UTILIZED IN TRACTION ON GRADES OF VARIOUS LENGTHS]
+
+Wellington, in his monumental work on railway location, presents a table
+of this character. The percentages of weight on the drivers which is
+utilized in draft show the greatest irregularity. He does not give the
+length of the grades considered, so that it is impossible to say how far
+the introduction of this feature would have contributed to bring order
+out of the chaos. In his discussion of the table he admits the
+unsatisfactory character of the results, and finally decides on 25% as a
+rough average, "very approximately the safe operating load in regular
+service." He further states that a number of results, which he omits for
+want of space, exceeds 33 per cent. The highest shown in Table 1 will be
+found in Item 1 (0.06 mile, 0.066 grade), showing 33 per cent. There is
+no momentum effect here, as the grade is a short incline extending down
+to the river, and the start is necessarily a "dead" one. The reports of
+Item 3, which shows 31%, and Item 5, which shows 27%, state specifically
+that the locomotives will stop and start the loads given at any point on
+the grade.
+
+The results of a series of experiments reported by Mr. A. C. Dennis in
+his paper, "Virtual Grades for Freight Trains," previously referred to,
+indicate a utilization of somewhat more than 23%, decreasing with the
+speed.
+
+All this indicates that the general failure of locomotives to utilize
+more than from 16 to 18% on long grades, as shown by Table 1, can only
+be due to the failure of the boilers to supply the necessary steam.
+While the higher percentage shown for the shorter grades may be ascribed
+largely to momentum present when the foot of the grade is reached, the
+energy due to stored heat is responsible for a large portion of it.
+
+When a locomotive has been standing still, or running with the steam
+consumption materially below the production, the pressure accumulates
+until it reaches the point at which the safety valve is "set." This
+means that the entire machine is heated to a temperature sufficient to
+maintain this pressure in the boiler. When the steam consumption begins
+to exceed the production, this temperature is reduced to a point where
+the consumption and production balance.
+
+The heat represented by this difference in temperature has passed into
+the steam used, thus adding to the energy supplied by the combustion
+going on in the furnace. The engines, therefore, are able to do
+considerably more work during the time the pressure is falling than they
+can do after the fall has ceased.
+
+The curve in Fig. 1 would indicate that the energy derived from the two
+sources just discussed is practically dissipated at 15 miles, though the
+position of the points representing Items 16, 18, 19, 20, and 21 would
+indicate that this takes place more frequently between 10 and 12 miles.
+From this point onward the performance depends on the efficiency of the
+steam production, which does not appear to be able to utilize more than
+16% of the weight on the drivers. The diagrams presented by Mr. Dennis
+in his paper on virtual grades, and by John A. Fulton, M. Am. Soc.
+C. E., in his discussion of that paper, indicate that similar results
+would be shown were they extended to include the distance named.
+
+From this it would appear that a locomotive is capable of hauling a
+larger train on grades less than 10 miles in length than on longer
+grades, and that, even when unexpectedly stopped, it is capable of
+starting again as soon as the steam pressure is sufficiently built up.
+Conversely, it should be practicable to use a higher rate of ascent on
+shorter grades on any given line without decreasing the load which can
+be hauled over it. In other words, what is known as the "ruling grade"
+is a function, strictly speaking, of the length as well as the rate of
+grade.
+
+In any discussions of the practicability of using a higher rate on the
+short grades, which the writer has seen, the most valid objection has
+appeared to be the danger of stalling and consequent delay. As far as
+momentum is relied on, this objection is valid. Within the limits of the
+load which can be handled by the steam, it has small value, as it is
+only a question of waiting a few minutes until the pressure can be built
+up to the point at which the load can be handled. As this need only be
+an occasional occurrence, it is not to be balanced against any material
+saving in cost of construction.
+
+The writer does not know of any experiments which will throw much light
+on the value of heat storage as separated from momentum, though the
+following discussion may prove suggestive:
+
+A train moving at a rate of 60 ft. per sec., and reaching the foot of a
+grade, will have acquired a "velocity head" of 56.7 ft., equivalent to
+stored energy of 56.7 × 2,000 = 113,400 ft-lb. per ton. On a 0.002
+grade (as in Item 15 of Table 1) the resistance would be, gravity
+4 lb. + train 4.7 lb. = 8.7 lb., against which the energy above given
+would carry the train through 113,400 ÷ 8.7 = 13,034 ft., say, 2.5
+miles, leaving 5 miles to be provided for by the steam production.
+Examining the items in the table having grades in excess of 10 miles, it
+will be noted that 16% is about all the weight on drivers which can be
+utilized by the current supply of steam. In Item 15 the energy derived
+from all sources is equivalent to 24.3%; hence the stored heat may be
+considered as responsible for an equivalent of 24.3% - 16% = 8.3% for a
+distance of 5 miles.
+
+
+TABLE 1.
+
+===========================================================================
+Item No.
+  |Length of grade, in miles.
+  |     |Rate of grade.
+  |     |      |Maximum curvature.
+  |     |      |      |Compensation.
+  |     |      |      |    |Gross weight of load, in tons.
+  |     |      |      |    |     |Weight of tender, in tons.
+  |     |      |      |    |     |  |Weight of locomotive, in tons.
+  |     |      |      |    |     |  |     |Weight on drivers, in tons.
+  |     |      |      |    |     |  |     |     |Percentage of weight on
+  |     |      |      |    |     |  |     |     |drivers utilized in draft.
+  |     |      |      |    |     |  |     |     |     |Class.
+  |     |      |      |    |     |  |     |     |     |
+  |     |      |      |    |     |  |     |     |     |
+--+-----+------+------+----+-----+--+-----+-----+-----+--------------------
+ 1| 0.06|0.066 |      |    |  115|  | 37.5| 29  |0.358| 8-28-1/3 C
+ 2| 0.33|0.0203|25°20'|    |  242|25| 35  | 23  |0.285| 8-28     C
+ 3| 1.0 |0.06  |16°   |0.05|  192|22| 57.5| 50  |0.310|10-36     E
+ 4| 1.3 |0.0127|      |    |  600|16| 40  | 32.5|0.300|Mogul.
+ 5| 1.4 |0.0128| 3°12'|    |  750|15| 51  | 44  |0.270|10-34     E
+ 6| 2.0 |0.01  |      |    |1,000|15| 51  | 44  |0.291|10-34     E
+ 7| 2.2 |0.013 | 3°   |    |  725|15| 51  | 44  |0.245|10-34     E
+ 8| 2.5 |0.0144| 6°   |    |  400|27| 42  | 32  |0.237|10-32     E
+ 9| 2.5 |0.004 |      |    |2,700|70| 96.7| 85.8|0.207| H 6 - A
+10| 3.5 |0.033 |14°   |    |  100|25| 35  | 35  |0.160|
+11| 3.6 |0.035 |10°   |0.05|  236|22| 57.5| 50  |0.245|10-36     E
+12| 4.0 |0.0085| 4°   |    |1,020|30| 51  | 44  |0.256|10-34     E
+13| 6.0 |0.0145|      |    |  308|25| 38  | 28  |0.207|10-28     D
+14| 6.0 |0.020 |10°   |0.05|  460|32| 57.5| 50  |0.242|10-34     E
+15| 7.5 |0.002 |      |  C |6,152|86|134.5|109.5|0.243|Mallet.
+16| 9.75|0.018 |      |    |  200|18| 29  | 29  |0.170|
+17|10.0 |0.006 |      |  C |6,173|86|299  |265  |0.203|Mallet.
+18|12.0 |0.018 |10°   |    |  280|30| 51  | 44  |0.160|10-34     E
+19|12.0 |0.022 |      |    |  850|74|175  |156  |0.166|D-D 16
+20|13.0 |0.022 |      |    |  800|74|177  |158  |0.153|D-D  1
+21|13.0 |0.022 |14°   |    |  415|50| 91  | 83  |0.154|Consol.
+22|16.0 |0.0044|      |    |  950|30| 51  | 44  |0.164|10-34     E
+23|20.0 |0.022 |      |    |  500|62| 97.5| 90  |0.170|F 8, Consol.
+24|20.0 |0.022 |      |    |  800|74|177  |158  |0.159|L-1, Mallet.
+===========================================================================
+
+
+============================================================================
+  |Maker.  |Railroad.                   |Reporting Officer.           |Year.
+--+--------+----------------------------+-----------------------------+-----
+ 1|Baldwin.|Morgan's Louisiana & Texas  |Newell Tilton, Asst. Supt.   |1880
+ 2|   "    |Long Island                 |S. Spencer, Gen. Supt.       |1878
+ 3|   "    |Atchison, Topeka & Santa Fe |J. D. Burr,  Asst. Engr.     |1879
+ 4|   "    |Chillan & Talcahuana        |J. E. Martin, Local Supt.    |1879
+ 5|   "    |Chicago, Burlington & Quincy|H. B. Stone                  |1880
+ 6|   "    |Chicago, Burlington & Quincy|      "                      |1880
+ 7|   "    |Chicago, Burlington & Quincy|      "                      |1880
+ 8|   "    |St. Louis & San Francisco   |C. W. Rogers, Gen. Mgr.      |1879
+ 9|Pa. R.R |Cumberland Valley.          |                             |1910
+10|        |                            |                             |1910
+11|Baldwin.|Atchison, Topeka & Santa Fe |J. D. Burr,  Asst. Engr.     |1879
+12|   "    |Missouri Pacific            |John Hewitt, Supt. M. P.     |1880
+13|   "    |Western Maryland            |D. Holtz, M. of Mach'y.      |1878
+14|   "    |Atchison, Topeka & Santa Fe |J. D. Burr,  Asst. Engr.     |1879
+15|   "    |Virginian Ry.               |                             |1910
+16|        |Pennsylvania                |                             |1910
+17|Baldwin.|Virginian Ry.               |                             |1910
+18|   "    |Lehigh Valley, Wyoming Div. |A. Mitchell, Div. Supt.      |1871
+19|   "    |Great Northern              |Grafton Greenough.           |1908
+20|   "    |Great Northern              |Grafton Greenough.           |1908
+21|   "    |Baltimore & Ohio            |F. E. Blaser, Div. Supt.     |1910
+22|   "    |Central of N. J.            |W. W. Stearns, Asst.Gen.Supt.|1880
+23|   "    |Great Northern              |Grafton Greenough.           |1908
+24|   "    |Great Northern              |Grafton Greenough.           |1906
+============================================================================
+
+
+==============================================================================
+  |Source of Data.                          |Remarks.
+--+-----------------------------------------+---------------------------------
+ 1|Baldwin Catalogue, 1881, p. 134          |
+ 2|   "        "      1881, "   72          |10 miles per hour.
+ 3|   "        "      1881, "  115          | 8   "    "   "
+  |                                         |    Stops and starts on grade.
+ 4|   "        "      1881, "  100          |
+ 5|   "        "      1881, "  116          |Stops and starts at any point
+  |                                         |    on grade.
+ 6|   "        "      1881, "  116          |
+ 7|   "        "      1881, "  116          |
+ 8|   "        "      1881, "   87          |
+ 9|                                         |
+10|Trautwine's Pocket Book, Ed. 1882, p. 412|Empty cars; many curves and
+  |                                         |    reversions.
+11|Baldwin Catalogue, 1881, p. 114          |
+12|   "        "      1881, "  112          |
+13|   "        "      1881, "   86          |12 miles per hour.
+14|   "        "      1881, "  114          | 8   "    "   "
+15|_Engineering News_, Jan. 13, 1910.       |
+16|Trautwine's Pocket Book, Ed. 1882, p. 412|
+17|_Engineering News_, Jan. 13, 1910.       |Road locomotive and helper.
+18|Baldwin Catalogue, 1881, p. 112          |
+19|Baldwin Loco. Wks. Record, No. 65, p.  29|
+20|Baldwin Loco. Wks. Record, No. 65, p.  29|
+21|                                         |Very crooked line. Uncompensated.
+22|Baldwin Catalogue, 1881, p. 113          |
+23|Baldwin Loco. Wks. Record, No. 65, p.  29|
+24|Baldwin Loco. Wks. Record, No. 65, p.  29|
+==============================================================================
+
+In proportioning grade resistance for any line, therefore, a locomotive
+may be counted on to utilize 24.3% of the weight on the drivers for a
+distance of 5 miles on a 0.002 grade without any assistance from
+momentum, and, in the event of an unexpected stop, should be able, as
+soon as a full head of steam is built up, to start the train and carry
+it over the grade. This is probably a maximum, considering the condition
+of the equipment of this Virginian Railway, as previously mentioned.
+
+Treating Item 14 in the same way, a distance of 2,310 ft. is accounted
+for by momentum, leaving, say, 5.5 miles for the steam, or the length of
+a 0.02 grade on which a locomotive may be loaded on a basis of tractive
+power equal to 24.2% of the weight on the drivers.
+
+From these figures it may be concluded that on lines having grades from
+12 to 15 or more miles in length, grades of 3 to 5 miles in length may
+be inserted having rates 50% in excess of that of the long grades,
+without decreasing the capacity of the line. This statement, of course,
+is general in its bearings, each case being subject to its especial
+limitations, and subject to detailed calculations.
+
+It may be noted that the velocity of 60 ft. per sec., assumed at the
+foot of the grade, is probably higher than should be expected in
+practice; it insures, on the other hand, that quite enough has been
+allowed for momentum, and that the results are conservative.
+
+Arguments like the foregoing are always more or less treacherous; being
+based on statistics, they are naturally subject to material
+modifications in the presence of a larger array of data, therefore,
+material assistance in reaching practical conclusions can be given by
+the presentation of additional data.
+
+
+
+
+DISCUSSION
+
+
+C. D. PURDON, M. AM. SOC. C. E. (by letter).--Some years ago the writer,
+in making studies for grade revision, found that the tractive power of a
+locomotive up grade becomes less as the length of the grade increases,
+and in some unknown proportion. This was a practical confirmation of the
+saying of locomotive engineers, that the engine "got tired" on long
+grades. On a well-known Western railroad, with which the writer is
+familiar, experiments were made for the purpose of rating its
+locomotives. The locomotives were first divided into classes according
+to their tractive power, this being calculated by the usual rule, with
+factors of size of cylinders, boiler pressure, and diameter of drivers,
+also by taking one-fourth of the weight on the drivers, and using the
+lesser of the two results as the tractive power.
+
+Locomotives of different classes, and hauling known loads, were run over
+a freight division, the cars being weighed for the purpose; thus the
+maximum load which could be handled over a division, or different parts
+of a division, was ascertained, and this proportion of tonnage to
+tractive power was used in rating all classes.
+
+Of course, this method was not mathematically accurate, as the condition
+of track, the weather, and the personal equation of the locomotive
+engineers all had an effect, but, later, when correcting the rating by
+tests with dynamometers, it was found that the results were fairly
+practical.
+
+There were three hills where the rate of grade was the same as the rest
+of the division, but where the length was much in excess of other grades
+of the same rate.
+
+Designating these hills as _A_, _B_, and _C_, the lengths are,
+respectively, 2.44, 3.57, and 4.41 miles. There were no other grades of
+the same rate exceeding 1 mile.
+
+In one class of freight engines, 10-wheel Brooks, the weight of the
+engine was 197,900 lb.; tender, 132,800 lb.; weight on drivers, 142,600
+lb.; boiler pressure, 200 lb.; and tractive power of cylinders, 33,300
+lb.
+
+On Hill _A_ these engines are rated at 865 tons, as compared with 945 on
+other parts of the division. As the engine weighs 165 tons and the
+caboose 15 tons, 180 tons should be added, making the figures, 1,045 and
+1,125 tons. Thus the length of the grade, 2.44 miles, makes the tractive
+power on it 92% of that on shorter grades.
+
+On Hill _B_, the rating, adding 180 tons as above, is 1,160 and 1,230
+tons, respectively, giving 94% for 3.57 miles.
+
+On Hill _C_, the rating, with 180 tons added, is 1,130 and 1,230 tons,
+making 92% for 4.41 miles.
+
+Taking the same basis as the author, namely, 4.7 lb. per ton, rate of
+grade × 20, and weight on drivers, gives:
+
+Hill _A_,  18.078%, remainder of division, 19.462%
+Hill _B_,  20.068%,     "     "      "     21.279%
+Hill _C_,  19.549%,     "     "      "     21.279%
+
+It will be noted that the author uses the weight on the drivers as the
+criterion, but the tractive power is not directly as the weight on the
+drivers, some engines being over-cylindered, or under-cylindered; in the
+class of engines above mentioned the tractive power is 23.35% of the
+weight on the drivers.
+
+The writer made a study of several dynamometer tests on Hill _C_. There
+is a grade of the same rate, about 1 mile long, near this hill, and a
+station near its foot, but there is sufficient level grade between this
+station and the foot of the hill to get a good start.
+
+All the engines of the above class, loaded for Hill _C_, gained speed on
+the 1-mile grade, but began to fall below the theoretical speed at a
+point about 2-1/4 miles from the foot of the hill. This condition
+occurred when the trains stopped at the station and also when they
+passed it at a rate of some 16 or 18 miles per hour, the speed becoming
+less and less as the top of the hill was approached.
+
+The writer concludes that the author might stretch his opinion as to
+using heavier rates of grade on shorter hills than 10 miles, and indeed
+his diagram seems to intimate as much, and that, for economical
+operation, the maximum rate of grade should be reduced after a length of
+about 2 miles has been reached, and more and more in proportion to the
+length of the hill, in order that the same rating could be applied all
+over a division.
+
+This conclusion might be modified by local conditions, such as an
+important town where cars might be added to or taken from the train.
+
+While it does not seem practicable to the writer to calculate what the
+reduction of rate of grade should be, a consensus of results of
+operation on different lengths of grade might give sufficient data to
+reach some conclusion on the matter.
+
+The American Railway Engineering and Maintenance of Way Association has
+a Committee on "Railway Economics," which is studying such matters, but
+so far as the writer knows it has not given this question any
+consideration.
+
+The writer hopes that the author will follow up this subject, and that
+other members will join, as a full discussion will no doubt bring some
+results on a question which seems to be highly important.
+
+
+JOHN C. TRAUTWINE, JR., ASSOC. AM. SOC. C. E. (by letter).--In his
+collection of data, Mr. Randolph includes two ancient cases taken from
+the earliest editions (1872-1883) of Trautwine's "Civil Engineer's
+Pocket-Book," referring to performances on the Mahanoy and Broad
+Mountain Railroad (now the Frackville Branch of the Reading) and on the
+Pennsylvania Railroad, respectively.
+
+In the private notes of John C. Trautwine, Sr., these two cases are
+recorded as follows:
+
+     "On the Mahanoy & Broad Mtn. R. R., _tank_ Engines of 35 tons, _all
+     on 8 drivers_, draw 40 _empty_ coal cars weighing 100 tons, _up_ a
+     continuous grade of 175 ft. per mile for 3-1/2 miles; & around
+     curves of 450, 500, 600 ft. &c. rad., at 8 miles an hour. (1864)
+     This is equal to 77-14/100 tons for a 27-ton engine." (Vol. III, p.
+     176.)
+
+     "On the Penn Central 95 ft. grades for 9-3/4 miles, a 29-ton engine
+     all on 8 drivers takes 125 tons of freight and 112 tons of engine,
+     tender, & cars, in all 237 tons,[C] and a passenger engine takes up
+     3 cars at 24 miles an hour (large 8 wheels). When more than 3, an
+     auxiliary engine."
+
+It will be seen that Mr. Randolph is well within bounds in ascribing to
+the Mahanoy and Broad Mountain case (his No. 10) a date "certainly prior
+to 1882," the date being given, in the notes, as 1864; while another
+entry just below it, for the Pennsylvania Railroad case, is dated 1860.
+
+It also seems, as stated by Mr. Randolph, quite probable that the
+frictional resistance (6 lb. per 2,000 lb.) assumed by him in the
+calculation is far below the actual for this Case 10. The small, empty,
+four-wheel cars weighed only 4,400 lb. each. Furthermore, the "tons," in
+the Trautwine reports of these experiments, were tons of 2,240 lb. On
+the other hand, the maximum curvature was 12° 45' (not 14°, as given by
+the author), and the engine was a tank locomotive, whereas the author
+has credited it with a 25-ton tender.
+
+After making all corrections, it will be found that, in order to bring
+the point, for this Case 10, up to the author's curve, instead of his 6
+lb. per 2,000 lb., a frictional resistance of 66 lb. per 2,000 lb. would
+be required, a resistance just equal to the gravity resistance on the
+3.3% grade, making a total resistance of 132 lb. per 2,000 lb.
+
+While this 66 lb. per ton is very high, it is perhaps not too high for
+the known conditions, as above described. For modern rolling stock, Mr.
+A. K. Shurtleff gives the formula:[D]
+
+Frictional resistance, on tangent, }
+in pounds per 2,000 pounds         } = 1 + 90 ÷ C,
+
+where _C_ = weight of car and load, in tons of 2,000 lb. This would
+give, for 4,400-lb. (2.2-ton) cars, a frictional resistance of 42 lb.
+per 2,000 lb.; and, on the usual assumption of 0.8 lb. per 2,000 lb. for
+each degree of curvature, the 12.75° curves of this line would give 10
+lb. per ton additional, making a total of 52 lb. per 2,000 lb. over and
+above grade resistance, under modern conditions.
+
+In the 9th to 17th editions of Trautwine (1885-1900), these early
+accounts were superseded by numerous later instances, including some of
+those quoted by the author.
+
+In the 18th and 19th editions (1902-1909) are given data respecting
+performances on the Catawissa Branch of the Reading (Shamokin Division)
+in 1898-1901. These give the maximum and minimum loads hauled up a
+nearly continuous grade of 31.47 ft. per mile (0.59%) from Catawissa to
+Lofty (34.03 miles) by engines of different classes, with different
+helpers and without helpers.
+
+Table 2 (in which the writer follows the author in assuming frictional
+resistance at 4.7 lb. per 2,000 lb.) shows the cases giving the maximum
+and minimum values of the quantity represented by the ordinates in the
+author's diagram, namely, "Traction, in percentage of weight on
+drivers."
+
+It will be seen that the maximum percentage (16.1) is practically
+identical with that found by the author (16) for grade lengths exceeding
+17 miles.
+
+Near the middle of the 34-mile distance there is a stretch of 1.51
+miles, on which the average grade is only 5.93 ft. per mile (0.112%),
+and this stretch divides the remaining distance into two practically
+continuous grades, 19.39 and 13.13 miles long, respectively; but, as the
+same loads are hauled over these two portions by the same engines, the
+results are virtually identical, the maxima furnishing two more points
+closely coinciding with the author's diagram.
+
+
+TABLE 2.--TRACTIVE FORCE, CATAWISSA TO LOFTY.
+
+========================================================================
+Length of grade, in miles                           |    |    34.03
+                                                    |    |
+Grade {in feet per mile                             |    |    31.47
+      {percentage                                   |_A_ |     0.597
+                                                    |    |
+Resistances, in pounds per 2,000 lb.,               |    |
+    Gravity (=20 _A_) = 11.94. Friction = 4.70      |_B_ |    16.64
+                                                    |    |
+  Load:      | Cars.  | Locomotive.| Tender.        |    |
+  Maximum[E] | 1,561  |   44.60    |  25.25         |_C_ | 1,631
+  Minimum[F] | 1,031  |   60.50    |  34.50         |_C_ | 1,126
+                                                    |    |
+Traction (= _B_ _C_ ÷ 2,000 ) Maximum[E]            |_D_ |    13.60
+                              Minimum[F]            |_D_ |     9.38
+ Weight on Drivers:   | Locomotive.|  Helper.       |    |
+    Maximum[E]        |    21.60   |    63.00       |_E_ |    84.60
+    Minimum[F]        |    47.00   |    72.00       |_E_ |   119.00
+                                                    |    |
+Percentage ( = _D_ ÷ _E_ ).                         |    |
+  Maximum                                           |_F_ |    16.1
+  Minimum                                           |_F_ |     7.9
+========================================================================
+
+    FOOTNOTES:
+
+    [Footnote E: Giving maximum values of percentage, _F_.]
+
+    [Footnote F: Giving minimum values of percentage, _F_.]
+
+
+
+BEVERLY S. RANDOLPH, M. AM. SOC. C. E. (by letter).--The percentages
+given by Mr. Purdon would seem to indicate that the length of the grades
+did not affect the loads in the cases cited, but these percentages are
+so much below those shown in the table, for similar distances, as to
+indicate some special conditions which the writer has been unable to
+find in the text.
+
+The use of the percentage of weight on drivers which is utilized in
+traction as a measure of the efficiency of the locomotive, while,
+probably, not applicable to individual machines, is sound for the
+purposes of comparison of results to be obtained on various portions of
+a line as far as affected by conditions of grade and alignment. It has
+the advantage of disregarding questions of temperature, condition of
+track, character of fuel, etc., which, being the same on all portions of
+the line, naturally balance and do not affect the comparison. It is, of
+course, simply a method of expressing the final efficiency of the
+various parts of the locomotive, and, since it depends entirely on
+actual results already accomplished, leaves no room for difference of
+opinion or theoretical error.
+
+The writer has always considered an "under-cylindered" locomotive as a
+defective machine. All weight is a distinct debit, in the shape of wear
+and tear of track and running gear, resistance due to gravity on grades,
+interest on cost, etc. When this weight fails to earn a credit in the
+way of tractive efficiency, it should not be present.
+
+The statement relative to the performance of locomotives on "Hill _C_"
+is interesting, especially in that it appears to have been immaterial
+whether they made a dead start after stopping at the station or
+approached the foot of the hill at 16 to 18 miles per hour. The momentum
+would appear to be an insignificant factor.
+
+It is gratifying to note that Mr. Trautwine has been able to brace up
+the weak member of Table 1 so completely with his detailed data; also
+that his other results strengthen the conclusions reached in the
+paper.
+
+
+    FOOTNOTES:
+
+    [Footnote A: "The Economic Theory of Railway Location," 1887 edition,
+    p. 502.]
+
+    [Footnote B: _Transactions_, Am. Soc. C. E., Vol. L, p. 1.]
+
+    [Footnote C: "Nearly 200 tons _exclusive_ of eng. & ten." (Vol. III,
+    p. 176-1/10.)]
+
+    [Footnote D: American Railway Engineering and Maintenance of Way
+    Association, Bulletin 84, February, 1907, p. 99.]
+
+
+
+
+
+End of the Project Gutenberg EBook of Transactions of the American Society
+of Civil Engineers, Vol. LXX, Dec. 1910, by Beverly S.  Randolph
+
+*** END OF THIS PROJECT GUTENBERG EBOOK AMERICAN SOCIETY OF CIVIL ENGINEERS ***
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+<pre>
+
+The Project Gutenberg EBook of Transactions of the American Society of
+Civil Engineers, Vol. LXX, Dec. 1910, by Beverly S.  Randolph
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: Transactions of the American Society of Civil Engineers, Vol. LXX, Dec. 1910
+       Locomotive Performance On Grades Of Various Lengths, Paper No. 1172
+
+Author: Beverly S.  Randolph
+
+Release Date: July 3, 2006 [EBook #18747]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK AMERICAN SOCIETY OF CIVIL ENGINEERS ***
+
+
+
+
+Produced by Juliet Sutherland, Sigal Alon and the Online
+Distributed Proofreading Team at http://www.pgdp.net
+
+
+
+
+
+
+</pre>
+
+
+
+<p><span class='pagenum'><a name="Page_321" id="Page_321">[321]</a></span></p>
+
+
+
+
+<h2>AMERICAN SOCIETY OF CIVIL ENGINEERS</h2>
+
+<h3>INSTITUTED 1852</h3>
+
+<hr style="width: 45%;" />
+
+<h1>TRANSACTIONS</h1>
+
+<hr style="width: 45%;" />
+
+<h3>Paper No. 1172</h3>
+
+
+<h1>LOCOMOTIVE PERFORMANCE ON GRADES OF
+VARIOUS LENGTHS.</h1>
+
+<h2><span class="smcap">By Beverly S. Randolph, M. Am. Soc. C.&nbsp;E.</span></h2>
+
+<hr style="width: 45%;" />
+
+<h2><span class="smcap">With Discussion by Messrs. C.&nbsp;D. Purdon, John C. Trautwine, Jr.,
+and Beverly S. Randolph.</span>
+</h2>
+
+<hr style="width: 45%;" />
+
+
+
+<p>In the location of new railways and the improvement of lines
+already in operation, it is now well recognized that large economies
+can be effected by the careful study of train resistance due to grades
+and alignment, distributing this resistance so as to secure a minimum
+cost of operation with the means available for construction.</p>
+
+<p>While engaged in such studies some years ago, the attention of the
+writer was attracted by the fact that the usual method of calculating
+the traction of a locomotive&mdash;by assuming from 20 to 25% of the
+weight on the drivers&mdash;was subject to no small modification in practice.</p>
+
+<p>In order to obtain a working basis, for use in relation to this
+feature, he undertook the collection of data from the practical operation
+of various roads. Subsequent engagements in an entirely different
+direction caused this to be laid aside until the present time. The
+results are given in <a href="#table1">Table 1</a>, from which it will be seen that the
+percentage of driver weight utilized in draft is a function of the
+length as well as the rate of grade encountered in the practical operation
+of railways.</p>
+
+<p>In this table, performance will be found expressed as the percentage
+of the weight on the drivers which is utilized in draft. This
+is calculated on a basis of 6 lb. per ton of train resistance, for dates
+<span class='pagenum'><a name="Page_322" id="Page_322">[322]</a></span>prior to 1880, this being the amount given by the late A.&nbsp;M. Wellington,
+M. Am. Soc. C..&nbsp;E.,<a name="FNanchor_A_3" id="FNanchor_A_3"></a><a href="#Footnote_A_3" class="fnanchor">[A]</a> and 4.7 lb. per ton for those of 1908-10, as
+obtained by A.&nbsp;C. Dennis, M. Am. Soc. C.&nbsp;E.,<a name="FNanchor_B_4" id="FNanchor_B_4"></a><a href="#Footnote_B_4" class="fnanchor">[B]</a> assuming this difference
+to represent the advance in practice from 1880 to the present
+time. Most of the data have been obtained from the "Catalogue of
+the Baldwin Locomotive Works" for 1881, to which have been added
+some later figures from "Record No. 65" of the same establishment,
+and also some obtained by the writer directly from the roads concerned.
+Being taken thus at random, the results may be accepted as
+fairly representative of American practice.</p>
+
+<p>Attention should be directed to the fact that the performance of
+the 10-34&nbsp;E, Consolidation locomotive on the Lehigh Valley Railroad
+in 1871 is practically equal to that of the latest Mallet compounds on
+the Great Northern Railway. In other words, in the ratio between
+the ability to produce steam and the weight on the drivers there has
+been no change in the last forty years. This would indicate that the
+figures are not likely to be changed much as long as steam-driven
+locomotives are in use. What will obtain with the introduction of
+electric traction is "another story."</p>
+
+<p>These results have also been platted, and are presented in <a href="#fig1">Fig. 1</a>,
+with the lengths of grade as abscissas and the percentages of weight
+utilized as ordinates. The curve sketched to represent a general
+average will show the conditions at a glance. The results may at first
+sight seem irregular, but the agreement is really remarkable when the
+variety of sources is considered; that in many cases the "reputed" rate
+of grade is doubtless given without actual measurement; that the
+results also include momentum, the ability to utilize which depends
+on the conditions of grade, alignment, and operating practice which
+obtain about the foot of each grade; and that the same amount of
+energy due to momentum will carry a train farther on a light grade
+than on a heavy one.</p>
+
+<p>There are four items in <a href="#table1">Table 1</a> which vary materially from the
+general consensus. For Item 9, the authorities of the road particularly
+state that their loads are light, because, owing to the congested
+condition of their business, their trains must make fast time. Item 10
+represents very old practice, certainly prior to 1882, and is "second-hand."
+<span class='pagenum'><a name="Page_323" id="Page_323">[323]</a></span>The load consisted of empty coal cars, and the line was very
+tortuous, so that it is quite probable that the resistance assumed in
+the calculation is far below the actual. Items 15 and 17 are both
+high. To account for this, it is to be noted that this road has been
+recently completed, regardless of cost in the matter of both track and
+rolling stock, and doubtless represents the highest development of
+railroad practice. Its rolling stock is all new, and is probably in
+better condition to offer low resistance than it will ever be again, and
+there were no "foreign" cars in the trains considered. The train
+resistance, therefore, may be naturally assumed to be much less than
+that of roads hauling all classes of cars, many of which are barely
+good enough to pass inspection. As the grades are light in both cases,
+this feature of train resistance is larger than in items including
+heavier grades. Attention should be called to the fact that a line
+connecting the two points representing these items on <a href="#fig1">Fig. 1</a> would
+make only a small angle with the sketched curve, and would be
+practically parallel to a similar line connecting the points represented<span class='pagenum'><a name="Page_324" id="Page_324">[324]</a></span>
+by Items 13 and 16. There is, therefore, an agreement of ratios, which
+is all that needs consideration in this discussion.</p>
+
+<div class="figcenter" style="width: 629px;">
+<img id="fig1" src="images/fig1.png" width="629" height="459"
+alt="Fig. 1.&mdash;DIAGRAM SHOWING PERCENTAGE OF WEIGHT ON DRIVERS WHICH IS UTILIZED IN TRACTION ON GRADES OF VARIOUS LENGTHS"
+title="Fig. 1.&mdash;DIAGRAM SHOWING PERCENTAGE OF WEIGHT ON DRIVERS WHICH IS UTILIZED IN TRACTION ON GRADES OF VARIOUS LENGTHS" />
+<span class="caption">Fig. 1.&mdash;DIAGRAM SHOWING PERCENTAGE OF WEIGHT ON DRIVERS WHICH IS
+UTILIZED IN TRACTION ON GRADES OF VARIOUS LENGTHS</span>
+</div>
+
+<p>Wellington, in his monumental work on railway location, presents
+a table of this character. The percentages of weight on the drivers
+which is utilized in draft show the greatest irregularity. He does
+not give the length of the grades considered, so that it is impossible
+to say how far the introduction of this feature would have contributed
+to bring order out of the chaos. In his discussion of the table he
+admits the unsatisfactory character of the results, and finally decides
+on 25% as a rough average, "very approximately the safe operating
+load in regular service." He further states that a number of results,
+which he omits for want of space, exceeds 33 per cent. The highest
+shown in <a href="#table1">Table 1</a> will be found in Item 1 (0.06 mile, 0.066 grade),
+showing 33 per cent. There is no momentum effect here, as the grade
+is a short incline extending down to the river, and the start is necessarily
+a "dead" one. The reports of Item 3, which shows 31%, and
+Item 5, which shows 27%, state specifically that the locomotives will
+stop and start the loads given at any point on the grade.</p>
+
+<p>The results of a series of experiments reported by Mr. A.&nbsp;C. Dennis
+in his paper, "Virtual Grades for Freight Trains," previously referred
+to, indicate a utilization of somewhat more than 23%, decreasing
+with the speed.</p>
+
+<p>All this indicates that the general failure of locomotives to utilize
+more than from 16 to 18% on long grades, as shown by <a href="#table1">Table 1</a>, can
+only be due to the failure of the boilers to supply the necessary steam.
+While the higher percentage shown for the shorter grades may be
+ascribed largely to momentum present when the foot of the grade is
+reached, the energy due to stored heat is responsible for a large
+portion of it.</p>
+
+<p>When a locomotive has been standing still, or running with the
+steam consumption materially below the production, the pressure
+accumulates until it reaches the point at which the safety valve is
+"set." This means that the entire machine is heated to a temperature
+sufficient to maintain this pressure in the boiler. When the steam consumption
+begins to exceed the production, this temperature is reduced
+to a point where the consumption and production balance.</p>
+
+<p>The heat represented by this difference in temperature has passed
+into the steam used, thus adding to the energy supplied by the combustion<span class='pagenum'><a name="Page_325" id="Page_325">[325]</a></span>
+going on in the furnace. The engines, therefore, are able to
+do considerably more work during the time the pressure is falling
+than they can do after the fall has ceased.</p>
+
+<p>The curve in <a href="#fig1">Fig. 1</a> would indicate that the energy derived from
+the two sources just discussed is practically dissipated at 15 miles,
+though the position of the points representing Items 16, 18, 19, 20,
+and 21 would indicate that this takes place more frequently between
+10 and 12 miles. From this point onward the performance depends on
+the efficiency of the steam production, which does not appear to be
+able to utilize more than 16% of the weight on the drivers. The
+diagrams presented by Mr. Dennis in his paper on virtual grades,
+and by John A. Fulton, M. Am. Soc. C.&nbsp;E., in his discussion of that
+paper, indicate that similar results would be shown were they extended
+to include the distance named.</p>
+
+<p>From this it would appear that a locomotive is capable of hauling
+a larger train on grades less than 10 miles in length than on longer
+grades, and that, even when unexpectedly stopped, it is capable of
+starting again as soon as the steam pressure is sufficiently built up.
+Conversely, it should be practicable to use a higher rate of ascent on
+shorter grades on any given line without decreasing the load which
+can be hauled over it. In other words, what is known as the "ruling
+grade" is a function, strictly speaking, of the length as well as the
+rate of grade.</p>
+
+<p>In any discussions of the practicability of using a higher rate
+on the short grades, which the writer has seen, the most valid
+objection has appeared to be the danger of stalling and consequent
+delay. As far as momentum is relied on, this objection is valid.
+Within the limits of the load which can be handled by the steam, it
+has small value, as it is only a question of waiting a few minutes
+until the pressure can be built up to the point at which the load can be
+handled. As this need only be an occasional occurrence, it is not to
+be balanced against any material saving in cost of construction.</p>
+
+<p>The writer does not know of any experiments which will throw
+much light on the value of heat storage as separated from momentum,
+though the following discussion may prove suggestive:</p>
+
+<p>A train moving at a rate of 60 ft. per sec., and reaching the foot
+of a grade, will have acquired a "velocity head" of 56.7 ft., equivalent
+to stored energy of 56.7&nbsp;&times;&nbsp;2,000&nbsp;=&nbsp;113,400&nbsp;ft-lb. per ton. On a 0.002<span class='pagenum'><a name="Page_328" id="Page_328">[328]</a></span>
+grade (as in Item 15 of <a href="#table1">Table 1</a>) the resistance would be, gravity
+4&nbsp;lb.&nbsp;+&nbsp;train&nbsp;4.7&nbsp;lb.&nbsp;=&nbsp;8.7&nbsp;lb., against which the energy above given
+would carry the train through 113,400&nbsp;&divide;&nbsp;8.7&nbsp;=&nbsp;13,034&nbsp;ft., say, 2.5 miles,
+leaving 5 miles to be provided for by the steam production. Examining
+the items in the table having grades in excess of 10 miles, it will
+be noted that 16% is about all the weight on drivers which can be
+utilized by the current supply of steam. In Item 15 the energy
+derived from all sources is equivalent to 24.3%; hence the stored heat
+may be considered as responsible for an equivalent of 24.3%&nbsp;-&nbsp;16%&nbsp;=&nbsp;8.3%
+for a distance of 5 miles.</p>
+
+<h4 id="table1">TABLE 1.</h4>
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="Table 1">
+<tr class='topr'>
+  <td>Item No.</td>
+  <td>Length of grade, in miles.</td>
+  <td>Rate of grade.</td>
+  <td>Maximum curvature.</td>
+  <td>Compen<br />sation.</td>
+  <td>Gross weight of load, in tons.</td>
+  <td>Weight of tender, in tons.</td>
+  <td>Weight of locomotive, in tons.</td>
+  <td>Weight on drivers, in tons.</td>
+  <td>Percentage of weight on drivers utilized in draft.</td>
+  <td>Class.</td>
+  <td>Maker.</td>
+  <td>Railroad.</td>
+  <td>Reporting Officer.</td>
+  <td>Year.</td>
+  <td>Source of Data.</td>
+  <td>Remarks.</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'> 1</td>
+  <td class='tdright'> 0.06</td>
+  <td class='tdleft'>0.066</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'> 115</td>
+  <td class='tdright'>&nbsp;</td>
+  <td class='tdright'> 37.5</td>
+  <td class='tdrightnd'> 29</td>
+  <td class='tdright'>0.358</td>
+  <td class='tdcenter'> 8-28-1/3&nbsp;C</td>
+  <td class='tdcenter'>Baldwin.</td>
+  <td class='tdleft'>Morgan's Louisiana &amp; Texas</td>
+  <td class='tdcenter'>Newell Tilton, Asst. Supt.</td>
+  <td class='tdleft'>1880</td>
+  <td class='tdleft'>Baldwin Catalogue, 1881, p.&nbsp;134</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'> 2</td>
+  <td class='tdright'> 0.33</td>
+  <td class='tdleft'>0.0203</td>
+  <td class='tdleft'>25°20'</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'> 242</td>
+  <td class='tdright'>25</td>
+  <td class='tdrightnd'> 35</td>
+  <td class='tdrightnd'> 23</td>
+  <td class='tdright'>0.285</td>
+  <td class='tdcenter'> 8-28&nbsp;C</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Long Island</td>
+  <td class='tdcenter'>S. Spencer, Gen. Supt.</td>
+  <td class='tdleft'>1878</td>
+  <td class='tdleft'>Baldwin Catalogue, 1881, p.&nbsp;72</td>
+  <td class='tdleft'>10 miles per hour.</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'> 3</td>
+  <td class='tdrightld'>1.0</td>
+  <td class='tdleft'>0.06</td>
+  <td class='tdleft'>16°</td>
+  <td class='tdleft'>0.05</td>
+  <td class='tdright'> 192</td>
+  <td class='tdright'>22</td>
+  <td class='tdright'> 57.5</td>
+  <td class='tdrightnd'> 50</td>
+  <td class='tdright'>0.310</td>
+  <td class='tdcenter'>10-36&nbsp;E</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Atchison, Topeka &amp; Santa Fe</td>
+  <td class='tdcenter'>J.&nbsp;D. Burr, Asst. Engr.</td>
+  <td class='tdleft'>1879</td>
+  <td class='tdleft'>Baldwin Catalogue, 1881, p.&nbsp;115</td>
+  <td class='tdleft'>8 miles per hour. Stops and starts on grade.</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'> 4</td>
+  <td class='tdrightld'>1.3</td>
+  <td class='tdleft'>0.0127</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'> 600</td>
+  <td class='tdright'>16</td>
+  <td class='tdrightnd'> 40</td>
+  <td class='tdright'> 32.5</td>
+  <td class='tdright'>0.300</td>
+  <td class='tdcenter'>Mogul.</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Chillan &amp; Talcahuana</td>
+  <td class='tdcenter'>J.&nbsp;E. Martin, Local Supt.</td>
+  <td class='tdleft'>1879</td>
+  <td class='tdleft'>Baldwin Catalogue, 1881, p.&nbsp;100</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'> 5</td>
+  <td class='tdrightld'>1.4</td>
+  <td class='tdleft'>0.0128</td>
+  <td class='tdleftld'> 3°12'</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'> 750</td>
+  <td class='tdright'>15</td>
+  <td class='tdrightnd'> 51</td>
+  <td class='tdrightnd'> 44</td>
+  <td class='tdright'>0.270</td>
+  <td class='tdcenter'>10-34&nbsp;E</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Chicago, Burlington &amp; Quincy</td>
+  <td class='tdcenter'>H.&nbsp;B. Stone</td>
+  <td class='tdleft'>1880</td>
+  <td class='tdleft'>Baldwin Catalogue, 1881, p.&nbsp;116</td>
+  <td class='tdleft'>Stops and starts at any point on grade.</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'> 6</td>
+  <td class='tdrightld'>2.0</td>
+  <td class='tdleft'>0.01</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'>1,000</td>
+  <td class='tdright'>15</td>
+  <td class='tdrightnd'> 51</td>
+  <td class='tdrightnd'> 44</td>
+  <td class='tdright'>0.291</td>
+  <td class='tdcenter'>10-34&nbsp;E</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Chicago, Burlington &amp; Quincy</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>1880</td>
+  <td class='tdleft'>Baldwin Catalogue, 1881, p.&nbsp;116</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'> 7</td>
+  <td class='tdrightld'>2.2</td>
+  <td class='tdleft'>0.013</td>
+  <td class='tdleftld'> 3°</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'> 725</td>
+  <td class='tdright'>15</td>
+  <td class='tdrightnd'> 51</td>
+  <td class='tdrightnd'> 44</td>
+  <td class='tdright'>0.245</td>
+  <td class='tdcenter'>10-34&nbsp;E</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Chicago, Burlington &amp; Quincy</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>1880</td>
+  <td class='tdleft'>Baldwin Catalogue, 1881, p.&nbsp;116</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'> 8</td>
+  <td class='tdrightld'>2.5</td>
+  <td class='tdleft'>0.0144</td>
+  <td class='tdleftld'> 6°</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'> 400</td>
+  <td class='tdright'>27</td>
+  <td class='tdrightnd'> 42</td>
+  <td class='tdrightnd'> 32</td>
+  <td class='tdright'>0.237</td>
+  <td class='tdcenter'>10-32&nbsp;E</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>St. Louis &amp; San Francisco</td>
+  <td class='tdcenter'>C.&nbsp;W. Rogers, Gen. Mgr.</td>
+  <td class='tdleft'>1879</td>
+  <td class='tdleft'>Baldwin Catalogue, 1881, p.&nbsp;87</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'> 9</td>
+  <td class='tdrightld'>2.5</td>
+  <td class='tdleft'>0.004</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'>2,700</td>
+  <td class='tdright'>70</td>
+  <td class='tdright'> 96.7</td>
+  <td class='tdright'> 85.8</td>
+  <td class='tdright'>0.207</td>
+  <td class='tdcenter'> H&nbsp;6&nbsp;-&nbsp;A</td>
+  <td class='tdcenter'>Pa. R.R</td>
+  <td class='tdleft'>Cumberland Valley.</td>
+  <td class='tdcenter'>&nbsp;</td>
+  <td class='tdleft'>1910</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'>10</td>
+  <td class='tdrightld'>3.5</td>
+  <td class='tdleft'>0.033</td>
+  <td class='tdleft'>14°</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'> 100</td>
+  <td class='tdright'>25</td>
+  <td class='tdrightnd'> 35</td>
+  <td class='tdrightnd'> 35</td>
+  <td class='tdright'>0.160</td>
+  <td class='tdcenter'>&nbsp;</td>
+  <td class='tdcenter'>&nbsp;</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdcenter'>&nbsp;</td>
+  <td class='tdleft'>1910</td>
+  <td class='tdleft'>Trautwine's Pocket Book, Ed. 1882, p. 412</td>
+  <td class='tdleft'>Empty cars; many curves and reversions.</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'>11</td>
+  <td class='tdrightld'>3.6</td>
+  <td class='tdleft'>0.035</td>
+  <td class='tdleft'>10°</td>
+  <td class='tdleft'>0.05</td>
+  <td class='tdright'> 236</td>
+  <td class='tdright'>22</td>
+  <td class='tdrightnd'> 57.5</td>
+  <td class='tdrightnd'> 50</td>
+  <td class='tdright'>0.245</td>
+  <td class='tdcenter'>10-36&nbsp;E</td>
+  <td class='tdcenter'>Baldwin.</td>
+  <td class='tdleft'>Atchison, Topeka &amp; Santa Fe</td>
+  <td class='tdcenter'>J.&nbsp;D. Burr, Asst. Engr.</td>
+  <td class='tdleft'>1879</td>
+  <td class='tdleft'>Baldwin Catalogue, 1881, p.&nbsp;114</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'>12</td>
+  <td class='tdrightld'>4.0</td>
+  <td class='tdleft'>0.0085</td>
+  <td class='tdleftld'> 4°</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'>1,020</td>
+  <td class='tdright'>30</td>
+  <td class='tdrightnd'> 51</td>
+  <td class='tdrightnd'> 44</td>
+  <td class='tdright'>0.256</td>
+  <td class='tdcenter'>10-34&nbsp;E</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Missouri Pacific</td>
+  <td class='tdcenter'>John Hewitt, Supt. M.&nbsp;P.</td>
+  <td class='tdleft'>1880</td>
+  <td class='tdleft'>Baldwin Catalogue, 1881, p.&nbsp;112</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'>13</td>
+  <td class='tdrightld'>6.0</td>
+  <td class='tdleft'>0.0145</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'> 308</td>
+  <td class='tdright'>25</td>
+  <td class='tdrightnd'> 38</td>
+  <td class='tdrightnd'> 28</td>
+  <td class='tdright'>0.207</td>
+  <td class='tdcenter'>10-28&nbsp;D</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Western Maryland</td>
+  <td class='tdcenter'>D. Holtz, M. of Mach'y.</td>
+  <td class='tdleft'>1878</td>
+  <td class='tdleft'>Baldwin Catalogue, 1881, p.&nbsp;86</td>
+  <td class='tdleft'>12 miles per hour.</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'>14</td>
+  <td class='tdrightld'>6.0</td>
+  <td class='tdleft'>0.020</td>
+  <td class='tdleft'>10°</td>
+  <td class='tdleft'>0.05</td>
+  <td class='tdright'> 460</td>
+  <td class='tdright'>32</td>
+  <td class='tdrightnd'> 57.5</td>
+  <td class='tdrightnd'> 50</td>
+  <td class='tdright'>0.242</td>
+  <td class='tdcenter'>10-34&nbsp;E</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Atchison, Topeka &amp; Santa Fe</td>
+  <td class='tdcenter'>J.&nbsp;D. Burr, Asst. Engr.</td>
+  <td class='tdleft'>1879</td>
+  <td class='tdleft'>Baldwin Catalogue, 1881, p.&nbsp;114</td>
+  <td class='tdleft'>8 miles per hour.</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'>15</td>
+  <td class='tdrightld'>7.5</td>
+  <td class='tdleft'>0.002</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdleft'> C</td>
+  <td class='tdright'>6,152</td>
+  <td class='tdright'>86</td>
+  <td class='tdright'>134.5</td>
+  <td class='tdright'>109.5</td>
+  <td class='tdright'>0.243</td>
+  <td class='tdcenter'>Mallet.</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Virginian Ry.</td>
+  <td class='tdcenter'>&nbsp;</td>
+  <td class='tdleft'>1910</td>
+  <td class='tdleft'><i>Engineering News</i>, Jan. 13, 1910.</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'>16</td>
+  <td class='tdright'> 9.75</td>
+  <td class='tdleft'>0.018</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'> 200</td>
+  <td class='tdright'>18</td>
+  <td class='tdrightnd'> 29</td>
+  <td class='tdrightnd'> 29</td>
+  <td class='tdright'>0.170</td>
+  <td class='tdcenter'>&nbsp;</td>
+  <td class='tdcenter'>&nbsp;</td>
+  <td class='tdleft'>Pennsylvania</td>
+  <td class='tdcenter'>&nbsp;</td>
+  <td class='tdleft'>1910</td>
+  <td class='tdleft'>Trautwine's Pocket Book, Ed. 1882, p. 412</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'>17</td>
+  <td class='tdrightld'>10.0</td>
+  <td class='tdleft'>0.006</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdleft'> C</td>
+  <td class='tdright'>6,173</td>
+  <td class='tdright'>86</td>
+  <td class='tdrightnd'>299</td>
+  <td class='tdrightnd'>265</td>
+  <td class='tdright'>0.203</td>
+  <td class='tdcenter'>Mallet.</td>
+  <td class='tdcenter'>Baldwin.</td>
+  <td class='tdleft'>Virginian Ry.</td>
+  <td class='tdcenter'>&nbsp;</td>
+  <td class='tdleft'>1910</td>
+  <td class='tdleft'><i>Engineering News</i>, Jan. 13, 1910.</td>
+  <td class='tdleft'>Road locomotive and helper.</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'>18</td>
+  <td class='tdrightld'>12.0</td>
+  <td class='tdleft'>0.018</td>
+  <td class='tdleft'>10°</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'> 280</td>
+  <td class='tdright'>30</td>
+  <td class='tdrightnd'> 51</td>
+  <td class='tdrightnd'> 44</td>
+  <td class='tdright'>0.160</td>
+  <td class='tdcenter'>10-34&nbsp;E</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Lehigh Valley, Wyoming Div.</td>
+  <td class='tdcenter'>A. Mitchell, Div. Supt.</td>
+  <td class='tdleft'>1871</td>
+  <td class='tdleft'>Baldwin Catalogue, 1881, p.&nbsp;112</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'>19</td>
+  <td class='tdrightld'>12.0</td>
+  <td class='tdleft'>0.022</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'> 850</td>
+  <td class='tdright'>74</td>
+  <td class='tdrightnd'>175</td>
+  <td class='tdrightnd'>156</td>
+  <td class='tdright'>0.166</td>
+  <td class='tdcenter'>D-D&nbsp;16</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Great Northern</td>
+  <td class='tdcenter'>Grafton Greenough.</td>
+  <td class='tdleft'>1908</td>
+  <td class='tdleft'>Baldwin Loco. Wks. Record, No.&nbsp;65, p.&nbsp;29</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'>20</td>
+  <td class='tdrightld'>13.0</td>
+  <td class='tdleft'>0.022</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'> 800</td>
+  <td class='tdright'>74</td>
+  <td class='tdrightnd'>177</td>
+  <td class='tdrightnd'>158</td>
+  <td class='tdright'>0.153</td>
+  <td class='tdcenter'>D-D&nbsp;1</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Great Northern</td>
+  <td class='tdcenter'>Grafton Greenough.</td>
+  <td class='tdleft'>1908</td>
+  <td class='tdleft'>Baldwin Loco. Wks. Record, No.&nbsp;65, p.&nbsp;29</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'>21</td>
+  <td class='tdrightld'>13.0</td>
+  <td class='tdleft'>0.022</td>
+  <td class='tdleft'>14°</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'> 415</td>
+  <td class='tdright'>50</td>
+  <td class='tdrightnd'> 91</td>
+  <td class='tdrightnd'> 83</td>
+  <td class='tdright'>0.154</td>
+  <td class='tdcenter'>Consol.</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Baltimore &amp; Ohio</td>
+  <td class='tdcenter'>F.&nbsp;E. Blaser, Div. Supt.</td>
+  <td class='tdleft'>1910</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdleft'>Very crooked line. Uncompensated.</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'>22</td>
+  <td class='tdrightld'>16.0</td>
+  <td class='tdleft'>0.0044</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'> 950</td>
+  <td class='tdright'>30</td>
+  <td class='tdrightnd'> 51</td>
+  <td class='tdrightnd'> 44</td>
+  <td class='tdright'>0.164</td>
+  <td class='tdcenter'>10-34&nbsp;E</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Central of N.&nbsp;J.</td>
+  <td class='tdcenter'>W.&nbsp;W. Stearns, Asst.Gen.Supt.</td>
+  <td class='tdleft'>1880</td>
+  <td class='tdleft'>Baldwin Catalogue, 1881, p.&nbsp;113</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class='midr'>
+  <td class='tdright'>23</td>
+  <td class='tdrightld'>20.0</td>
+  <td class='tdleft'>0.022</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'> 500</td>
+  <td class='tdright'>62</td>
+  <td class='tdrightnd'> 97.5</td>
+  <td class='tdrightnd'> 90</td>
+  <td class='tdright'>0.170</td>
+  <td class='tdcenter'>F&nbsp;8, Consol.</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Great Northern</td>
+  <td class='tdcenter'>Grafton Greenough.</td>
+  <td class='tdleft'>1908</td>
+  <td class='tdleft'>Baldwin Loco. Wks. Record, No.&nbsp;65, p.&nbsp;29</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class='botr'>
+  <td class='tdright'>24</td>
+  <td class='tdrightld'>20.0</td>
+  <td class='tdleft'>0.022</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdright'> 800</td>
+  <td class='tdright'>74</td>
+  <td class='tdrightnd'>177</td>
+  <td class='tdrightnd'>158</td>
+  <td class='tdright'>0.159</td>
+  <td class='tdcenter'>L-1, Mallet.</td>
+  <td class='tdcenter'> "</td>
+  <td class='tdleft'>Great Northern</td>
+  <td class='tdcenter'>Grafton Greenough.</td>
+  <td class='tdleft'>1906</td>
+  <td class='tdleft'>Baldwin Loco. Wks. Record, No.&nbsp;65, p.&nbsp;29</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+</table></div>
+
+
+<p>In proportioning grade resistance for any line, therefore, a locomotive
+may be counted on to utilize 24.3% of the weight on the drivers
+for a distance of 5 miles on a 0.002 grade without any assistance from
+momentum, and, in the event of an unexpected stop, should be able, as
+soon as a full head of steam is built up, to start the train and carry
+it over the grade. This is probably a maximum, considering the condition
+of the equipment of this Virginian Railway, as previously
+mentioned.</p>
+
+<p>Treating Item 14 in the same way, a distance of 2,310 ft. is
+accounted for by momentum, leaving, say, 5.5 miles for the steam, or
+the length of a 0.02 grade on which a locomotive may be loaded on a
+basis of tractive power equal to 24.2% of the weight on the drivers.</p>
+
+<p>From these figures it may be concluded that on lines having grades
+from 12 to 15 or more miles in length, grades of 3 to 5 miles in
+length may be inserted having rates 50% in excess of that of the long
+grades, without decreasing the capacity of the line. This statement,
+of course, is general in its bearings, each case being subject to its
+especial limitations, and subject to detailed calculations.</p>
+
+<p>It may be noted that the velocity of 60 ft. per sec., assumed at the
+foot of the grade, is probably higher than should be expected in
+practice; it insures, on the other hand, that quite enough has been
+allowed for momentum, and that the results are conservative.</p>
+
+<p>Arguments like the foregoing are always more or less treacherous;
+being based on statistics, they are naturally subject to material modifications
+in the presence of a larger array of data, therefore, material
+assistance in reaching practical conclusions can be given by the
+presentation of additional data.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_329" id="Page_329">[329]</a></span></p>
+
+<h2><a name="DISCUSSION" id="DISCUSSION"></a>DISCUSSION</h2>
+
+
+<p class="section"><span class="smcap">C.&nbsp;D. Purdon, M. Am. Soc. C.&nbsp;E.</span> (by letter).&mdash;Some years ago the
+writer, in making studies for grade revision, found that the tractive
+power of a locomotive up grade becomes less as the length of the
+grade increases, and in some unknown proportion. This was a practical
+confirmation of the saying of locomotive engineers, that the engine
+"got tired" on long grades. On a well-known Western railroad, with
+which the writer is familiar, experiments were made for the purpose
+of rating its locomotives. The locomotives were first divided into
+classes according to their tractive power, this being calculated by the
+usual rule, with factors of size of cylinders, boiler pressure, and diameter
+of drivers, also by taking one-fourth of the weight on the drivers,
+and using the lesser of the two results as the tractive power.</p>
+
+<p>Locomotives of different classes, and hauling known loads, were run
+over a freight division, the cars being weighed for the purpose; thus
+the maximum load which could be handled over a division, or different
+parts of a division, was ascertained, and this proportion of tonnage
+to tractive power was used in rating all classes.</p>
+
+<p>Of course, this method was not mathematically accurate, as the
+condition of track, the weather, and the personal equation of the
+locomotive engineers all had an effect, but, later, when correcting the
+rating by tests with dynamometers, it was found that the results were
+fairly practical.</p>
+
+<p>There were three hills where the rate of grade was the same as the
+rest of the division, but where the length was much in excess of other
+grades of the same rate.</p>
+
+<p>Designating these hills as <i>A</i>, <i>B</i>, and <i>C</i>, the lengths are, respectively,
+2.44, 3.57, and 4.41 miles. There were no other grades of the same
+rate exceeding 1 mile.</p>
+
+<p>In one class of freight engines, 10-wheel Brooks, the weight of the
+engine was 197,900 lb.; tender, 132,800 lb.; weight on drivers, 142,600
+lb.; boiler pressure, 200 lb.; and tractive power of cylinders, 33,300 lb.</p>
+
+<p>On Hill <i>A</i> these engines are rated at 865 tons, as compared with
+945 on other parts of the division. As the engine weighs 165 tons and
+the caboose 15 tons, 180 tons should be added, making the figures, 1,045
+and 1,125 tons. Thus the length of the grade, 2.44 miles, makes the
+tractive power on it 92% of that on shorter grades.</p>
+
+<p>On Hill <i>B</i>, the rating, adding 180 tons as above, is 1,160 and
+1,230 tons, respectively, giving 94% for 3.57 miles.</p>
+
+<p>On Hill <i>C</i>, the rating, with 180 tons added, is 1,130 and 1,230 tons,
+making 92% for 4.41 miles.</p>
+
+<p><span class='pagenum'><a name="Page_330" id="Page_330">[330]</a></span>
+Taking the same basis as the author, namely, 4.7 lb. per ton, rate
+of grade &times; 20, and weight on drivers, gives:</p>
+
+<div class="blockquot">
+<p>
+Hill <i>A</i>,&nbsp;&nbsp;18.078%, remainder of division, 19.462%<br />
+Hill <i>B</i>,&nbsp;&nbsp;20.068%, remainder of division, 21.279%<br />
+Hill <i>C</i>,&nbsp;&nbsp;19.549%, remainder of division, 21.279%<br />
+</p>
+</div>
+
+<p>It will be noted that the author uses the weight on the drivers as
+the criterion, but the tractive power is not directly as the weight on
+the drivers, some engines being over-cylindered, or under-cylindered;
+in the class of engines above mentioned the tractive power is 23.35%
+of the weight on the drivers.</p>
+
+<p>The writer made a study of several dynamometer tests on Hill <i>C</i>.
+There is a grade of the same rate, about 1 mile long, near this hill,
+and a station near its foot, but there is sufficient level grade between
+this station and the foot of the hill to get a good start.</p>
+
+<p>All the engines of the above class, loaded for Hill <i>C</i>, gained speed
+on the 1-mile grade, but began to fall below the theoretical speed at a
+point about 2-1/4 miles from the foot of the hill. This condition occurred
+when the trains stopped at the station and also when they passed it
+at a rate of some 16 or 18 miles per hour, the speed becoming less
+and less as the top of the hill was approached.</p>
+
+<p>The writer concludes that the author might stretch his opinion as
+to using heavier rates of grade on shorter hills than 10 miles, and
+indeed his diagram seems to intimate as much, and that, for economical
+operation, the maximum rate of grade should be reduced after a
+length of about 2 miles has been reached, and more and more in
+proportion to the length of the hill, in order that the same rating
+could be applied all over a division.</p>
+
+<p>This conclusion might be modified by local conditions, such as an
+important town where cars might be added to or taken from the
+train.</p>
+
+<p>While it does not seem practicable to the writer to calculate what
+the reduction of rate of grade should be, a consensus of results of
+operation on different lengths of grade might give sufficient data to
+reach some conclusion on the matter.</p>
+
+<p>The American Railway Engineering and Maintenance of Way
+Association has a Committee on "Railway Economics," which is studying
+such matters, but so far as the writer knows it has not given this
+question any consideration.</p>
+
+<p>The writer hopes that the author will follow up this subject, and
+that other members will join, as a full discussion will no doubt bring
+some results on a question which seems to be highly important.</p>
+
+
+<p class="section"><span class="smcap">John C. Trautwine, Jr., Assoc. Am. Soc</span>. C.&nbsp;E. (by letter).&mdash;In
+his collection of data, Mr. Randolph includes two ancient cases
+taken from the earliest editions (1872-1883) of Trautwine's "Civil<span class='pagenum'><a name="Page_331" id="Page_331">[331]</a></span>
+Engineer's Pocket-Book," referring to performances on the Mahanoy
+and Broad Mountain Railroad (now the Frackville Branch of the
+Reading) and on the Pennsylvania Railroad, respectively.</p>
+
+<p>In the private notes of John C. Trautwine, Sr., these two cases
+are recorded as follows:</p>
+
+<div class="blockquot"><p>"On the Mahanoy &amp; Broad Mtn. R.&nbsp;R., <i>tank</i> Engines of 35 tons, <i>all
+on 8 drivers</i>, draw 40 <i>empty</i> coal cars weighing 100 tons, <i>up</i> a continuous
+grade of 175 ft. per mile for 3-1/2 miles; &amp; around curves of
+450, 500, 600 ft. &amp;c. rad., at 8 miles an hour. (1864) This is equal to
+77-14/100 tons for a 27-ton engine." (Vol. III, p. 176.)</p>
+
+<p>"On the Penn Central 95 ft. grades for 9-3/4 miles, a 29-ton engine
+all on 8 drivers takes 125 tons of freight and 112 tons of engine, tender,
+&amp; cars, in all 237 tons,<a name="FNanchor_C_5" id="FNanchor_C_5"></a><a href="#Footnote_C_5" class="fnanchor">[C]</a> and a passenger engine takes up 3 cars at
+24 miles an hour (large 8 wheels). When more than 3, an auxiliary
+engine."</p></div>
+
+<p>It will be seen that Mr. Randolph is well within bounds in ascribing
+to the Mahanoy and Broad Mountain case (his No. 10) a date
+"certainly prior to 1882," the date being given, in the notes, as 1864;
+while another entry just below it, for the Pennsylvania Railroad case,
+is dated 1860.</p>
+
+<p>It also seems, as stated by Mr. Randolph, quite probable that the
+frictional resistance (6 lb. per 2,000 lb.) assumed by him in the calculation
+is far below the actual for this Case 10. The small, empty, four-wheel
+cars weighed only 4,400 lb. each. Furthermore, the "tons," in
+the Trautwine reports of these experiments, were tons of 2,240 lb.
+On the other hand, the maximum curvature was 12&deg; 45' (not 14&deg;, as
+given by the author), and the engine was a tank locomotive, whereas
+the author has credited it with a 25-ton tender.</p>
+
+<p>After making all corrections, it will be found that, in order to bring
+the point, for this Case 10, up to the author's curve, instead of his
+6 lb. per 2,000 lb., a frictional resistance of 66 lb. per 2,000 lb. would
+be required, a resistance just equal to the gravity resistance on the
+3.3% grade, making a total resistance of 132 lb. per 2,000 lb.</p>
+
+<p>While this 66 lb. per ton is very high, it is perhaps not too high
+for the known conditions, as above described. For modern rolling
+stock, Mr. A.&nbsp;K. Shurtleff gives the formula:<a name="FNanchor_D_6" id="FNanchor_D_6"></a><a href="#Footnote_D_6" class="fnanchor">[D]</a></p>
+
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="formatting formula">
+<tr>
+  <td class='tdleft'>Frictional resistance, on tangent,<br /> in pounds per 2,000 pounds</td>
+  <td valign='middle' class='tdleft' style="white-space: nowrap; font-size: 200%">}</td>
+  <td class='tdleft'>= 1 + 90 &divide; C,</td>
+</tr>
+</table>
+</div>
+
+<p>where <i>C</i> = weight of car and load, in tons of 2,000 lb. This would
+give, for 4,400-lb. (2.2-ton) cars, a frictional resistance of 42 lb. per
+2,000 lb.; and, on the usual assumption of 0.8 lb. per 2,000 lb. for each<span class='pagenum'><a name="Page_332" id="Page_332">[332]</a></span>
+degree of curvature, the 12.75&deg; curves of this line would give 10 lb.
+per ton additional, making a total of 52 lb. per 2,000 lb. over and above
+grade resistance, under modern conditions.</p>
+
+<p>In the 9th to 17th editions of Trautwine (1885-1900), these early
+accounts were superseded by numerous later instances, including some
+of those quoted by the author.</p>
+
+<p>In the 18th and 19th editions (1902-1909) are given data respecting
+performances on the Catawissa Branch of the Reading (Shamokin
+Division) in 1898-1901. These give the maximum and minimum loads
+hauled up a nearly continuous grade of 31.47 ft. per mile (0.59%) from
+Catawissa to Lofty (34.03 miles) by engines of different classes, with
+different helpers and without helpers.</p>
+
+<p><a href="#table2">Table 2</a> (in which the writer follows the author in assuming frictional
+resistance at 4.7 lb. per 2,000 lb.) shows the cases giving the
+maximum and minimum values of the quantity represented by the
+ordinates in the author's diagram, namely, "Traction, in percentage
+of weight on drivers."</p>
+
+<p>It will be seen that the maximum percentage (16.1) is practically
+identical with that found by the author (16) for grade lengths exceeding
+17 miles.</p>
+
+<p>Near the middle of the 34-mile distance there is a stretch of 1.51
+miles, on which the average grade is only 5.93 ft. per mile (0.112%),
+and this stretch divides the remaining distance into two practically
+continuous grades, 19.39 and 13.13 miles long, respectively; but, as the
+same loads are hauled over these two portions by the same engines, the
+results are virtually identical, the maxima furnishing two more points
+closely coinciding with the author's diagram.</p>
+
+
+<h4 id="table2">TABLE 2.&mdash;<span class="smcap">Tractive Force, Catawissa to Lofty</span>.</h4>
+
+
+<div class='center'>
+<table border="0" width="100%" cellpadding="0" cellspacing="0" summary="Tractive Force, Catawissa to Lofty">
+<tr class="topr2">
+  <td class='tdleft' colspan='2'>Length of grade, in miles</td>
+  <td class='tdleft'  style='width: 2.5em;'>&nbsp;</td>
+  <td class='tdrightld'  style='width: 5em;'> 34.03</td>
+</tr>
+<tr class="midr">
+  <td class='tdleft' rowspan='2'>Grade</td>
+  <td class='tdleft'>in feet per mile</td>
+  <td class='tdleft'>&nbsp;</td>
+  <td class='tdrightld'> 31.47</td>
+</tr>
+<tr class="midr">
+  <td class='tdleft'>percentage</td>
+  <td class='tdcenter'><i>A</i></td>
+  <td class='tdright'> 0.597</td>
+</tr>
+<tr class="midr">
+  <td class='tdleft' colspan='2'>Resistances, in pounds per 2,000 lb.,<br />
+  <span style="margin-left: 5.5em;">Gravity (=20 <i>A</i>) = 11.94. Friction = 4.70</span></td>
+  <td class='tdcenter'><i>B</i></td>
+  <td class='tdrightld'> 16.64</td>
+</tr>
+<tr class="midr">
+  <td colspan='2' rowspan='3'>
+    <table border="0" width="100%" cellpadding="4" cellspacing="0" summary="internal table 1 for Tractive Force, Catawissa to Lofty">
+    <tr class="midr">
+      <td class='tdcenter'><span style="margin-left: 2.5em;">Load:</span></td>
+      <td class='tdcenter'> Cars.</td>
+      <td class='tdcenter'> Locomotive.</td>
+      <td class='tdcenter'> Tender.</td>
+    </tr>
+    <tr class="midr">
+      <td class='tdcenter'><span style="margin-left: 2.5em;">Maximum<a name="FNanchor_E_1" id="FNanchor_E_1"></a><a href="#Footnote_E_1" class="fnanchor">[E]</a></span></td>
+      <td class='tdcenter'> 1,561</td>
+      <td class='tdcenter'> 44.60</td>
+      <td class='tdcenter'> 25.25</td>
+    </tr>
+    <tr class="botr2">
+      <td class='tdcenter'><span style="margin-left: 2.5em;">Minimum<a name="FNanchor_F_2" id="FNanchor_F_2"></a><a href="#Footnote_F_2" class="fnanchor">[F]</a></span></td>
+      <td class='tdcenter'> 1,031</td>
+      <td class='tdcenter'> 60.50</td>
+      <td class='tdcenter'> 34.50</td>
+    </tr>
+    </table>
+  </td>
+  <td class='tdcenter'>&nbsp;</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class="midr">
+  <td class='tdcenter'><i>C</i></td>
+  <td class='tdrightnd3'> 1,631</td>
+</tr>
+<tr class="midr">
+  <td class='tdcenter'><i>C</i></td>
+  <td class='tdrightnd3'> 1,126</td>
+</tr>
+<tr class="midr">
+  <td class='tdleft' rowspan='2'>Traction (= <i>B</i> <i>C</i> &divide; 2,000)</td>
+  <td class='tdleft'>Maximum<a href="#Footnote_E_1" class="fnanchor">[E]</a></td>
+  <td class='tdcenter'><i>D</i></td>
+  <td class='tdrightld'> 13.60</td>
+</tr>
+<tr class="midr">
+  <td class='tdleft'> Minimum<a href="#Footnote_F_2" class="fnanchor">[F]</a></td>
+  <td class='tdcenter'><i>D</i></td>
+  <td class='tdrightld'> 9.38</td>
+</tr>
+<tr class="midr">
+  <td colspan='2' rowspan='3'>
+    <table border="0" width="100%" cellpadding="4" cellspacing="0" summary="internal table 2 for Tractive Force, Catawissa to Lofty">
+    <tr class="midr">
+      <td class='tdcenter'><span style="margin-left: 2.5em;">Weight on Drivers:</span></td>
+      <td class='tdcenter'> Locomotive.</td>
+      <td class='tdcenter'> Helper.</td>
+    </tr>
+    <tr>
+      <td class='tdcenter'><span style="margin-left: 2.5em;">Maximum<a href="#Footnote_E_1" class="fnanchor">[E]</a></span></td>
+      <td class='tdcenter'> 21.60</td>
+      <td class='tdcenter'> 63.00</td>
+    </tr>
+    <tr>
+      <td class='tdcenter'><span style="margin-left: 2.5em;">Minimum<a href="#Footnote_F_2" class="fnanchor">[F]</a></span></td>
+      <td class='tdcenter'> 47.00</td>
+      <td class='tdcenter'> 72.00</td>
+    </tr>
+    </table>
+  </td>
+  <td class='tdcenter'>&nbsp;</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class="midr">
+  <td class='tdcenter'><i>E</i></td>
+  <td class='tdrightld'> 84.60</td>
+</tr>
+<tr class="midr">
+  <td class='tdcenter'><i>E</i></td>
+  <td class='tdrightld'> 119.00</td>
+</tr>
+<tr class="midr">
+  <td class='tdleft' colspan='2'>Percentage ( = <i>D</i> &divide;  <i>E</i> ).</td>
+  <td class='tdcenter'>&nbsp;</td>
+  <td class='tdleft'>&nbsp;</td>
+</tr>
+<tr class="midr">
+  <td class='tdleft' colspan='2'><span style="margin-left: 5.5em;">Maximum</span></td>
+  <td class='tdcenter'><i>F</i></td>
+  <td class='tdrightld2'> 16.1</td>
+</tr>
+<tr class="botr">
+  <td class='tdleft' colspan='2'><span style="margin-left: 5.5em;">Minimum</span></td>
+  <td class='tdcenter'><i>F</i></td>
+  <td class='tdrightld2'> 7.9</td>
+</tr>
+</table>
+</div>
+
+<div class="footnote"><p><a name="Footnote_E_1" id="Footnote_E_1"></a><a href="#FNanchor_E_1">
+<span class="label">[E]</span></a> Giving maximum values of percentage, <i>F</i>.</p></div>
+
+<div class="footnote"><p><a name="Footnote_F_2" id="Footnote_F_2"></a><a href="#FNanchor_F_2">
+<span class="label">[F]</span></a> Giving minimum values of percentage, <i>F</i>.</p></div>
+
+
+<p class="section"><span class='pagenum'><a name="Page_333" id="Page_333">[333]</a></span>
+<span class="smcap">Beverly S. Randolph, M. Am. Soc</span>. C.&nbsp;E. (by letter).&mdash;The percentages
+given by Mr. Purdon would seem to indicate that the length
+of the grades did not affect the loads in the cases cited, but these
+percentages are so much below those shown in the table, for similar
+distances, as to indicate some special conditions which the writer has
+been unable to find in the text.</p>
+
+<p>The use of the percentage of weight on drivers which is utilized
+in traction as a measure of the efficiency of the locomotive, while,
+probably, not applicable to individual machines, is sound for the purposes
+of comparison of results to be obtained on various portions of a
+line as far as affected by conditions of grade and alignment. It has the
+advantage of disregarding questions of temperature, condition of track,
+character of fuel, etc., which, being the same on all portions of the
+line, naturally balance and do not affect the comparison. It is, of
+course, simply a method of expressing the final efficiency of the various
+parts of the locomotive, and, since it depends entirely on actual results
+already accomplished, leaves no room for difference of opinion or
+theoretical error.</p>
+
+<p>The writer has always considered an "under-cylindered" locomotive
+as a defective machine. All weight is a distinct debit, in the shape of
+wear and tear of track and running gear, resistance due to gravity
+on grades, interest on cost, etc. When this weight fails to earn a
+credit in the way of tractive efficiency, it should not be present.</p>
+
+<p>The statement relative to the performance of locomotives on
+"Hill <i>C</i>" is interesting, especially in that it appears to have been
+immaterial whether they made a dead start after stopping at the
+station or approached the foot of the hill at 16 to 18 miles per hour.
+The momentum would appear to be an insignificant factor.</p>
+
+<p>It is gratifying to note that Mr. Trautwine has been able to brace
+up the weak member of <a href="#table1">Table 1</a> so completely with his detailed data;
+also that his other results strengthen the conclusions reached in the
+paper.</p>
+
+
+<div class="footnotes"><h3>FOOTNOTES:</h3>
+
+<div class="footnote"><p><a name="Footnote_A_3" id="Footnote_A_3"></a><a href="#FNanchor_A_3">
+<span class="label">[A]</span></a> "The Economic Theory of Railway Location," 1887 edition, p. 502.</p></div>
+
+<div class="footnote"><p><a name="Footnote_B_4" id="Footnote_B_4"></a><a href="#FNanchor_B_4">
+<span class="label">[B]</span></a> <i>Transactions</i>, Am. Soc. C.&nbsp;E., Vol. L, p.&nbsp;1.</p></div>
+
+<div class="footnote"><p><a name="Footnote_C_5" id="Footnote_C_5"></a><a href="#FNanchor_C_5">
+<span class="label">[C]</span></a> "Nearly 200 tons <i>exclusive</i> of eng. &amp; ten." (Vol. III, p. 176-1/10.)</p></div>
+
+<div class="footnote"><p><a name="Footnote_D_6" id="Footnote_D_6"></a><a href="#FNanchor_D_6">
+<span class="label">[D]</span></a> American Railway Engineering and Maintenance of Way Association, Bulletin 84,
+February, 1907, p. 99.</p></div>
+</div>
+
+
+
+
+
+
+
+<pre>
+
+
+
+
+
+End of the Project Gutenberg EBook of Transactions of the American Society
+of Civil Engineers, Vol. LXX, Dec. 1910, by Beverly S.  Randolph
+
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+The Project Gutenberg EBook of Transactions of the American Society of
+Civil Engineers, Vol. LXX, Dec. 1910, by Beverly S.  Randolph
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: Transactions of the American Society of Civil Engineers, Vol. LXX, Dec. 1910
+       Locomotive Performance On Grades Of Various Lengths, Paper No. 1172
+
+Author: Beverly S.  Randolph
+
+Release Date: July 3, 2006 [EBook #18747]
+
+Language: English
+
+Character set encoding: ASCII
+
+*** START OF THIS PROJECT GUTENBERG EBOOK AMERICAN SOCIETY OF CIVIL ENGINEERS ***
+
+
+
+
+Produced by Juliet Sutherland, Sigal Alon and the Online
+Distributed Proofreading Team at http://www.pgdp.net
+
+
+
+
+
+
+
+
+
+AMERICAN SOCIETY OF CIVIL ENGINEERS
+
+INSTITUTED 1852
+
+
+TRANSACTIONS
+
+Paper No. 1172
+
+
+LOCOMOTIVE PERFORMANCE ON GRADES OF
+VARIOUS LENGTHS.
+
+BY BEVERLY S. RANDOLPH, M. AM. SOC. C. E.
+
+WITH DISCUSSION BY MESSRS. C. D. PURDON, JOHN C. TRAUTWINE, JR.,
+AND BEVERLY S. RANDOLPH.
+
+
+
+
+In the location of new railways and the improvement of lines already in
+operation, it is now well recognized that large economies can be
+effected by the careful study of train resistance due to grades and
+alignment, distributing this resistance so as to secure a minimum cost
+of operation with the means available for construction.
+
+While engaged in such studies some years ago, the attention of the
+writer was attracted by the fact that the usual method of calculating
+the traction of a locomotive--by assuming from 20 to 25% of the weight
+on the drivers--was subject to no small modification in practice.
+
+In order to obtain a working basis, for use in relation to this feature,
+he undertook the collection of data from the practical operation of
+various roads. Subsequent engagements in an entirely different direction
+caused this to be laid aside until the present time. The results are
+given in Table 1, from which it will be seen that the percentage of
+driver weight utilized in draft is a function of the length as well as
+the rate of grade encountered in the practical operation of railways.
+
+In this table, performance will be found expressed as the percentage of
+the weight on the drivers which is utilized in draft. This is calculated
+on a basis of 6 lb. per ton of train resistance, for dates prior to
+1880, this being the amount given by the late A. M. Wellington, M. Am.
+Soc. C.. E.,[A] and 4.7 lb. per ton for those of 1908-10, as obtained by
+A. C. Dennis, M. Am. Soc. C. E.,[B] assuming this difference to
+represent the advance in practice from 1880 to the present time. Most of
+the data have been obtained from the "Catalogue of the Baldwin
+Locomotive Works" for 1881, to which have been added some later figures
+from "Record No. 65" of the same establishment, and also some obtained
+by the writer directly from the roads concerned. Being taken thus at
+random, the results may be accepted as fairly representative of American
+practice.
+
+Attention should be directed to the fact that the performance of the
+10-34 E, Consolidation locomotive on the Lehigh Valley Railroad in 1871
+is practically equal to that of the latest Mallet compounds on the Great
+Northern Railway. In other words, in the ratio between the ability to
+produce steam and the weight on the drivers there has been no change in
+the last forty years. This would indicate that the figures are not
+likely to be changed much as long as steam-driven locomotives are in
+use. What will obtain with the introduction of electric traction is
+"another story."
+
+These results have also been platted, and are presented in Fig. 1, with
+the lengths of grade as abscissas and the percentages of weight utilized
+as ordinates. The curve sketched to represent a general average will
+show the conditions at a glance. The results may at first sight seem
+irregular, but the agreement is really remarkable when the variety of
+sources is considered; that in many cases the "reputed" rate of grade is
+doubtless given without actual measurement; that the results also
+include momentum, the ability to utilize which depends on the conditions
+of grade, alignment, and operating practice which obtain about the foot
+of each grade; and that the same amount of energy due to momentum will
+carry a train farther on a light grade than on a heavy one.
+
+There are four items in Table 1 which vary materially from the general
+consensus. For Item 9, the authorities of the road particularly state
+that their loads are light, because, owing to the congested condition of
+their business, their trains must make fast time. Item 10 represents
+very old practice, certainly prior to 1882, and is "second-hand." The
+load consisted of empty coal cars, and the line was very tortuous, so
+that it is quite probable that the resistance assumed in the calculation
+is far below the actual. Items 15 and 17 are both high. To account for
+this, it is to be noted that this road has been recently completed,
+regardless of cost in the matter of both track and rolling stock, and
+doubtless represents the highest development of railroad practice. Its
+rolling stock is all new, and is probably in better condition to offer
+low resistance than it will ever be again, and there were no "foreign"
+cars in the trains considered. The train resistance, therefore, may be
+naturally assumed to be much less than that of roads hauling all classes
+of cars, many of which are barely good enough to pass inspection. As the
+grades are light in both cases, this feature of train resistance is
+larger than in items including heavier grades. Attention should be
+called to the fact that a line connecting the two points representing
+these items on Fig. 1 would make only a small angle with the sketched
+curve, and would be practically parallel to a similar line connecting
+the points represented by Items 13 and 16. There is, therefore, an
+agreement of ratios, which is all that needs consideration in this
+discussion.
+
+[Illustration: FIG. 1.--DIAGRAM SHOWING PERCENTAGE OF WEIGHT ON DRIVERS
+WHICH IS UTILIZED IN TRACTION ON GRADES OF VARIOUS LENGTHS]
+
+Wellington, in his monumental work on railway location, presents a table
+of this character. The percentages of weight on the drivers which is
+utilized in draft show the greatest irregularity. He does not give the
+length of the grades considered, so that it is impossible to say how far
+the introduction of this feature would have contributed to bring order
+out of the chaos. In his discussion of the table he admits the
+unsatisfactory character of the results, and finally decides on 25% as a
+rough average, "very approximately the safe operating load in regular
+service." He further states that a number of results, which he omits for
+want of space, exceeds 33 per cent. The highest shown in Table 1 will be
+found in Item 1 (0.06 mile, 0.066 grade), showing 33 per cent. There is
+no momentum effect here, as the grade is a short incline extending down
+to the river, and the start is necessarily a "dead" one. The reports of
+Item 3, which shows 31%, and Item 5, which shows 27%, state specifically
+that the locomotives will stop and start the loads given at any point on
+the grade.
+
+The results of a series of experiments reported by Mr. A. C. Dennis in
+his paper, "Virtual Grades for Freight Trains," previously referred to,
+indicate a utilization of somewhat more than 23%, decreasing with the
+speed.
+
+All this indicates that the general failure of locomotives to utilize
+more than from 16 to 18% on long grades, as shown by Table 1, can only
+be due to the failure of the boilers to supply the necessary steam.
+While the higher percentage shown for the shorter grades may be ascribed
+largely to momentum present when the foot of the grade is reached, the
+energy due to stored heat is responsible for a large portion of it.
+
+When a locomotive has been standing still, or running with the steam
+consumption materially below the production, the pressure accumulates
+until it reaches the point at which the safety valve is "set." This
+means that the entire machine is heated to a temperature sufficient to
+maintain this pressure in the boiler. When the steam consumption begins
+to exceed the production, this temperature is reduced to a point where
+the consumption and production balance.
+
+The heat represented by this difference in temperature has passed into
+the steam used, thus adding to the energy supplied by the combustion
+going on in the furnace. The engines, therefore, are able to do
+considerably more work during the time the pressure is falling than they
+can do after the fall has ceased.
+
+The curve in Fig. 1 would indicate that the energy derived from the two
+sources just discussed is practically dissipated at 15 miles, though the
+position of the points representing Items 16, 18, 19, 20, and 21 would
+indicate that this takes place more frequently between 10 and 12 miles.
+From this point onward the performance depends on the efficiency of the
+steam production, which does not appear to be able to utilize more than
+16% of the weight on the drivers. The diagrams presented by Mr. Dennis
+in his paper on virtual grades, and by John A. Fulton, M. Am. Soc.
+C. E., in his discussion of that paper, indicate that similar results
+would be shown were they extended to include the distance named.
+
+From this it would appear that a locomotive is capable of hauling a
+larger train on grades less than 10 miles in length than on longer
+grades, and that, even when unexpectedly stopped, it is capable of
+starting again as soon as the steam pressure is sufficiently built up.
+Conversely, it should be practicable to use a higher rate of ascent on
+shorter grades on any given line without decreasing the load which can
+be hauled over it. In other words, what is known as the "ruling grade"
+is a function, strictly speaking, of the length as well as the rate of
+grade.
+
+In any discussions of the practicability of using a higher rate on the
+short grades, which the writer has seen, the most valid objection has
+appeared to be the danger of stalling and consequent delay. As far as
+momentum is relied on, this objection is valid. Within the limits of the
+load which can be handled by the steam, it has small value, as it is
+only a question of waiting a few minutes until the pressure can be built
+up to the point at which the load can be handled. As this need only be
+an occasional occurrence, it is not to be balanced against any material
+saving in cost of construction.
+
+The writer does not know of any experiments which will throw much light
+on the value of heat storage as separated from momentum, though the
+following discussion may prove suggestive:
+
+A train moving at a rate of 60 ft. per sec., and reaching the foot of a
+grade, will have acquired a "velocity head" of 56.7 ft., equivalent to
+stored energy of 56.7 x 2,000 = 113,400 ft-lb. per ton. On a 0.002
+grade (as in Item 15 of Table 1) the resistance would be, gravity
+4 lb. + train 4.7 lb. = 8.7 lb., against which the energy above given
+would carry the train through 113,400 / 8.7 = 13,034 ft., say, 2.5
+miles, leaving 5 miles to be provided for by the steam production.
+Examining the items in the table having grades in excess of 10 miles, it
+will be noted that 16% is about all the weight on drivers which can be
+utilized by the current supply of steam. In Item 15 the energy derived
+from all sources is equivalent to 24.3%; hence the stored heat may be
+considered as responsible for an equivalent of 24.3% - 16% = 8.3% for a
+distance of 5 miles.
+
+
+TABLE 1.
+
+===========================================================================
+Item No.
+  |Length of grade, in miles.
+  |     |Rate of grade.
+  |     |      |Maximum curvature.
+  |     |      |      |Compensation.
+  |     |      |      |    |Gross weight of load, in tons.
+  |     |      |      |    |     |Weight of tender, in tons.
+  |     |      |      |    |     |  |Weight of locomotive, in tons.
+  |     |      |      |    |     |  |     |Weight on drivers, in tons.
+  |     |      |      |    |     |  |     |     |Percentage of weight on
+  |     |      |      |    |     |  |     |     |drivers utilized in draft.
+  |     |      |      |    |     |  |     |     |     |Class.
+  |     |      |      |    |     |  |     |     |     |
+  |     |      |      |    |     |  |     |     |     |
+--+-----+------+------+----+-----+--+-----+-----+-----+--------------------
+ 1| 0.06|0.066 |      |    |  115|  | 37.5| 29  |0.358| 8-28-1/3 C
+ 2| 0.33|0.0203|25 deg.20'|    |  242|25| 35  | 23  |0.285| 8-28     C
+ 3| 1.0 |0.06  |16 deg.   |0.05|  192|22| 57.5| 50  |0.310|10-36     E
+ 4| 1.3 |0.0127|      |    |  600|16| 40  | 32.5|0.300|Mogul.
+ 5| 1.4 |0.0128| 3 deg.12'|    |  750|15| 51  | 44  |0.270|10-34     E
+ 6| 2.0 |0.01  |      |    |1,000|15| 51  | 44  |0.291|10-34     E
+ 7| 2.2 |0.013 | 3 deg.   |    |  725|15| 51  | 44  |0.245|10-34     E
+ 8| 2.5 |0.0144| 6 deg.   |    |  400|27| 42  | 32  |0.237|10-32     E
+ 9| 2.5 |0.004 |      |    |2,700|70| 96.7| 85.8|0.207| H 6 - A
+10| 3.5 |0.033 |14 deg.   |    |  100|25| 35  | 35  |0.160|
+11| 3.6 |0.035 |10 deg.   |0.05|  236|22| 57.5| 50  |0.245|10-36     E
+12| 4.0 |0.0085| 4 deg.   |    |1,020|30| 51  | 44  |0.256|10-34     E
+13| 6.0 |0.0145|      |    |  308|25| 38  | 28  |0.207|10-28     D
+14| 6.0 |0.020 |10 deg.   |0.05|  460|32| 57.5| 50  |0.242|10-34     E
+15| 7.5 |0.002 |      |  C |6,152|86|134.5|109.5|0.243|Mallet.
+16| 9.75|0.018 |      |    |  200|18| 29  | 29  |0.170|
+17|10.0 |0.006 |      |  C |6,173|86|299  |265  |0.203|Mallet.
+18|12.0 |0.018 |10 deg.   |    |  280|30| 51  | 44  |0.160|10-34     E
+19|12.0 |0.022 |      |    |  850|74|175  |156  |0.166|D-D 16
+20|13.0 |0.022 |      |    |  800|74|177  |158  |0.153|D-D  1
+21|13.0 |0.022 |14 deg.   |    |  415|50| 91  | 83  |0.154|Consol.
+22|16.0 |0.0044|      |    |  950|30| 51  | 44  |0.164|10-34     E
+23|20.0 |0.022 |      |    |  500|62| 97.5| 90  |0.170|F 8, Consol.
+24|20.0 |0.022 |      |    |  800|74|177  |158  |0.159|L-1, Mallet.
+===========================================================================
+
+
+============================================================================
+  |Maker.  |Railroad.                   |Reporting Officer.           |Year.
+--+--------+----------------------------+-----------------------------+-----
+ 1|Baldwin.|Morgan's Louisiana & Texas  |Newell Tilton, Asst. Supt.   |1880
+ 2|   "    |Long Island                 |S. Spencer, Gen. Supt.       |1878
+ 3|   "    |Atchison, Topeka & Santa Fe |J. D. Burr,  Asst. Engr.     |1879
+ 4|   "    |Chillan & Talcahuana        |J. E. Martin, Local Supt.    |1879
+ 5|   "    |Chicago, Burlington & Quincy|H. B. Stone                  |1880
+ 6|   "    |Chicago, Burlington & Quincy|      "                      |1880
+ 7|   "    |Chicago, Burlington & Quincy|      "                      |1880
+ 8|   "    |St. Louis & San Francisco   |C. W. Rogers, Gen. Mgr.      |1879
+ 9|Pa. R.R |Cumberland Valley.          |                             |1910
+10|        |                            |                             |1910
+11|Baldwin.|Atchison, Topeka & Santa Fe |J. D. Burr,  Asst. Engr.     |1879
+12|   "    |Missouri Pacific            |John Hewitt, Supt. M. P.     |1880
+13|   "    |Western Maryland            |D. Holtz, M. of Mach'y.      |1878
+14|   "    |Atchison, Topeka & Santa Fe |J. D. Burr,  Asst. Engr.     |1879
+15|   "    |Virginian Ry.               |                             |1910
+16|        |Pennsylvania                |                             |1910
+17|Baldwin.|Virginian Ry.               |                             |1910
+18|   "    |Lehigh Valley, Wyoming Div. |A. Mitchell, Div. Supt.      |1871
+19|   "    |Great Northern              |Grafton Greenough.           |1908
+20|   "    |Great Northern              |Grafton Greenough.           |1908
+21|   "    |Baltimore & Ohio            |F. E. Blaser, Div. Supt.     |1910
+22|   "    |Central of N. J.            |W. W. Stearns, Asst.Gen.Supt.|1880
+23|   "    |Great Northern              |Grafton Greenough.           |1908
+24|   "    |Great Northern              |Grafton Greenough.           |1906
+============================================================================
+
+
+==============================================================================
+  |Source of Data.                          |Remarks.
+--+-----------------------------------------+---------------------------------
+ 1|Baldwin Catalogue, 1881, p. 134          |
+ 2|   "        "      1881, "   72          |10 miles per hour.
+ 3|   "        "      1881, "  115          | 8   "    "   "
+  |                                         |    Stops and starts on grade.
+ 4|   "        "      1881, "  100          |
+ 5|   "        "      1881, "  116          |Stops and starts at any point
+  |                                         |    on grade.
+ 6|   "        "      1881, "  116          |
+ 7|   "        "      1881, "  116          |
+ 8|   "        "      1881, "   87          |
+ 9|                                         |
+10|Trautwine's Pocket Book, Ed. 1882, p. 412|Empty cars; many curves and
+  |                                         |    reversions.
+11|Baldwin Catalogue, 1881, p. 114          |
+12|   "        "      1881, "  112          |
+13|   "        "      1881, "   86          |12 miles per hour.
+14|   "        "      1881, "  114          | 8   "    "   "
+15|_Engineering News_, Jan. 13, 1910.       |
+16|Trautwine's Pocket Book, Ed. 1882, p. 412|
+17|_Engineering News_, Jan. 13, 1910.       |Road locomotive and helper.
+18|Baldwin Catalogue, 1881, p. 112          |
+19|Baldwin Loco. Wks. Record, No. 65, p.  29|
+20|Baldwin Loco. Wks. Record, No. 65, p.  29|
+21|                                         |Very crooked line. Uncompensated.
+22|Baldwin Catalogue, 1881, p. 113          |
+23|Baldwin Loco. Wks. Record, No. 65, p.  29|
+24|Baldwin Loco. Wks. Record, No. 65, p.  29|
+==============================================================================
+
+In proportioning grade resistance for any line, therefore, a locomotive
+may be counted on to utilize 24.3% of the weight on the drivers for a
+distance of 5 miles on a 0.002 grade without any assistance from
+momentum, and, in the event of an unexpected stop, should be able, as
+soon as a full head of steam is built up, to start the train and carry
+it over the grade. This is probably a maximum, considering the condition
+of the equipment of this Virginian Railway, as previously mentioned.
+
+Treating Item 14 in the same way, a distance of 2,310 ft. is accounted
+for by momentum, leaving, say, 5.5 miles for the steam, or the length of
+a 0.02 grade on which a locomotive may be loaded on a basis of tractive
+power equal to 24.2% of the weight on the drivers.
+
+From these figures it may be concluded that on lines having grades from
+12 to 15 or more miles in length, grades of 3 to 5 miles in length may
+be inserted having rates 50% in excess of that of the long grades,
+without decreasing the capacity of the line. This statement, of course,
+is general in its bearings, each case being subject to its especial
+limitations, and subject to detailed calculations.
+
+It may be noted that the velocity of 60 ft. per sec., assumed at the
+foot of the grade, is probably higher than should be expected in
+practice; it insures, on the other hand, that quite enough has been
+allowed for momentum, and that the results are conservative.
+
+Arguments like the foregoing are always more or less treacherous; being
+based on statistics, they are naturally subject to material
+modifications in the presence of a larger array of data, therefore,
+material assistance in reaching practical conclusions can be given by
+the presentation of additional data.
+
+
+
+
+DISCUSSION
+
+
+C. D. PURDON, M. AM. SOC. C. E. (by letter).--Some years ago the writer,
+in making studies for grade revision, found that the tractive power of a
+locomotive up grade becomes less as the length of the grade increases,
+and in some unknown proportion. This was a practical confirmation of the
+saying of locomotive engineers, that the engine "got tired" on long
+grades. On a well-known Western railroad, with which the writer is
+familiar, experiments were made for the purpose of rating its
+locomotives. The locomotives were first divided into classes according
+to their tractive power, this being calculated by the usual rule, with
+factors of size of cylinders, boiler pressure, and diameter of drivers,
+also by taking one-fourth of the weight on the drivers, and using the
+lesser of the two results as the tractive power.
+
+Locomotives of different classes, and hauling known loads, were run over
+a freight division, the cars being weighed for the purpose; thus the
+maximum load which could be handled over a division, or different parts
+of a division, was ascertained, and this proportion of tonnage to
+tractive power was used in rating all classes.
+
+Of course, this method was not mathematically accurate, as the condition
+of track, the weather, and the personal equation of the locomotive
+engineers all had an effect, but, later, when correcting the rating by
+tests with dynamometers, it was found that the results were fairly
+practical.
+
+There were three hills where the rate of grade was the same as the rest
+of the division, but where the length was much in excess of other grades
+of the same rate.
+
+Designating these hills as _A_, _B_, and _C_, the lengths are,
+respectively, 2.44, 3.57, and 4.41 miles. There were no other grades of
+the same rate exceeding 1 mile.
+
+In one class of freight engines, 10-wheel Brooks, the weight of the
+engine was 197,900 lb.; tender, 132,800 lb.; weight on drivers, 142,600
+lb.; boiler pressure, 200 lb.; and tractive power of cylinders, 33,300
+lb.
+
+On Hill _A_ these engines are rated at 865 tons, as compared with 945 on
+other parts of the division. As the engine weighs 165 tons and the
+caboose 15 tons, 180 tons should be added, making the figures, 1,045 and
+1,125 tons. Thus the length of the grade, 2.44 miles, makes the tractive
+power on it 92% of that on shorter grades.
+
+On Hill _B_, the rating, adding 180 tons as above, is 1,160 and 1,230
+tons, respectively, giving 94% for 3.57 miles.
+
+On Hill _C_, the rating, with 180 tons added, is 1,130 and 1,230 tons,
+making 92% for 4.41 miles.
+
+Taking the same basis as the author, namely, 4.7 lb. per ton, rate of
+grade x 20, and weight on drivers, gives:
+
+Hill _A_,  18.078%, remainder of division, 19.462%
+Hill _B_,  20.068%,     "     "      "     21.279%
+Hill _C_,  19.549%,     "     "      "     21.279%
+
+It will be noted that the author uses the weight on the drivers as the
+criterion, but the tractive power is not directly as the weight on the
+drivers, some engines being over-cylindered, or under-cylindered; in the
+class of engines above mentioned the tractive power is 23.35% of the
+weight on the drivers.
+
+The writer made a study of several dynamometer tests on Hill _C_. There
+is a grade of the same rate, about 1 mile long, near this hill, and a
+station near its foot, but there is sufficient level grade between this
+station and the foot of the hill to get a good start.
+
+All the engines of the above class, loaded for Hill _C_, gained speed on
+the 1-mile grade, but began to fall below the theoretical speed at a
+point about 2-1/4 miles from the foot of the hill. This condition
+occurred when the trains stopped at the station and also when they
+passed it at a rate of some 16 or 18 miles per hour, the speed becoming
+less and less as the top of the hill was approached.
+
+The writer concludes that the author might stretch his opinion as to
+using heavier rates of grade on shorter hills than 10 miles, and indeed
+his diagram seems to intimate as much, and that, for economical
+operation, the maximum rate of grade should be reduced after a length of
+about 2 miles has been reached, and more and more in proportion to the
+length of the hill, in order that the same rating could be applied all
+over a division.
+
+This conclusion might be modified by local conditions, such as an
+important town where cars might be added to or taken from the train.
+
+While it does not seem practicable to the writer to calculate what the
+reduction of rate of grade should be, a consensus of results of
+operation on different lengths of grade might give sufficient data to
+reach some conclusion on the matter.
+
+The American Railway Engineering and Maintenance of Way Association has
+a Committee on "Railway Economics," which is studying such matters, but
+so far as the writer knows it has not given this question any
+consideration.
+
+The writer hopes that the author will follow up this subject, and that
+other members will join, as a full discussion will no doubt bring some
+results on a question which seems to be highly important.
+
+
+JOHN C. TRAUTWINE, JR., ASSOC. AM. SOC. C. E. (by letter).--In his
+collection of data, Mr. Randolph includes two ancient cases taken from
+the earliest editions (1872-1883) of Trautwine's "Civil Engineer's
+Pocket-Book," referring to performances on the Mahanoy and Broad
+Mountain Railroad (now the Frackville Branch of the Reading) and on the
+Pennsylvania Railroad, respectively.
+
+In the private notes of John C. Trautwine, Sr., these two cases are
+recorded as follows:
+
+     "On the Mahanoy & Broad Mtn. R. R., _tank_ Engines of 35 tons, _all
+     on 8 drivers_, draw 40 _empty_ coal cars weighing 100 tons, _up_ a
+     continuous grade of 175 ft. per mile for 3-1/2 miles; & around
+     curves of 450, 500, 600 ft. &c. rad., at 8 miles an hour. (1864)
+     This is equal to 77-14/100 tons for a 27-ton engine." (Vol. III, p.
+     176.)
+
+     "On the Penn Central 95 ft. grades for 9-3/4 miles, a 29-ton engine
+     all on 8 drivers takes 125 tons of freight and 112 tons of engine,
+     tender, & cars, in all 237 tons,[C] and a passenger engine takes up
+     3 cars at 24 miles an hour (large 8 wheels). When more than 3, an
+     auxiliary engine."
+
+It will be seen that Mr. Randolph is well within bounds in ascribing to
+the Mahanoy and Broad Mountain case (his No. 10) a date "certainly prior
+to 1882," the date being given, in the notes, as 1864; while another
+entry just below it, for the Pennsylvania Railroad case, is dated 1860.
+
+It also seems, as stated by Mr. Randolph, quite probable that the
+frictional resistance (6 lb. per 2,000 lb.) assumed by him in the
+calculation is far below the actual for this Case 10. The small, empty,
+four-wheel cars weighed only 4,400 lb. each. Furthermore, the "tons," in
+the Trautwine reports of these experiments, were tons of 2,240 lb. On
+the other hand, the maximum curvature was 12 deg. 45' (not 14 deg., as given by
+the author), and the engine was a tank locomotive, whereas the author
+has credited it with a 25-ton tender.
+
+After making all corrections, it will be found that, in order to bring
+the point, for this Case 10, up to the author's curve, instead of his 6
+lb. per 2,000 lb., a frictional resistance of 66 lb. per 2,000 lb. would
+be required, a resistance just equal to the gravity resistance on the
+3.3% grade, making a total resistance of 132 lb. per 2,000 lb.
+
+While this 66 lb. per ton is very high, it is perhaps not too high for
+the known conditions, as above described. For modern rolling stock, Mr.
+A. K. Shurtleff gives the formula:[D]
+
+Frictional resistance, on tangent, }
+in pounds per 2,000 pounds         } = 1 + 90 / C,
+
+where _C_ = weight of car and load, in tons of 2,000 lb. This would
+give, for 4,400-lb. (2.2-ton) cars, a frictional resistance of 42 lb.
+per 2,000 lb.; and, on the usual assumption of 0.8 lb. per 2,000 lb. for
+each degree of curvature, the 12.75 deg. curves of this line would give 10
+lb. per ton additional, making a total of 52 lb. per 2,000 lb. over and
+above grade resistance, under modern conditions.
+
+In the 9th to 17th editions of Trautwine (1885-1900), these early
+accounts were superseded by numerous later instances, including some of
+those quoted by the author.
+
+In the 18th and 19th editions (1902-1909) are given data respecting
+performances on the Catawissa Branch of the Reading (Shamokin Division)
+in 1898-1901. These give the maximum and minimum loads hauled up a
+nearly continuous grade of 31.47 ft. per mile (0.59%) from Catawissa to
+Lofty (34.03 miles) by engines of different classes, with different
+helpers and without helpers.
+
+Table 2 (in which the writer follows the author in assuming frictional
+resistance at 4.7 lb. per 2,000 lb.) shows the cases giving the maximum
+and minimum values of the quantity represented by the ordinates in the
+author's diagram, namely, "Traction, in percentage of weight on
+drivers."
+
+It will be seen that the maximum percentage (16.1) is practically
+identical with that found by the author (16) for grade lengths exceeding
+17 miles.
+
+Near the middle of the 34-mile distance there is a stretch of 1.51
+miles, on which the average grade is only 5.93 ft. per mile (0.112%),
+and this stretch divides the remaining distance into two practically
+continuous grades, 19.39 and 13.13 miles long, respectively; but, as the
+same loads are hauled over these two portions by the same engines, the
+results are virtually identical, the maxima furnishing two more points
+closely coinciding with the author's diagram.
+
+
+TABLE 2.--TRACTIVE FORCE, CATAWISSA TO LOFTY.
+
+========================================================================
+Length of grade, in miles                           |    |    34.03
+                                                    |    |
+Grade {in feet per mile                             |    |    31.47
+      {percentage                                   |_A_ |     0.597
+                                                    |    |
+Resistances, in pounds per 2,000 lb.,               |    |
+    Gravity (=20 _A_) = 11.94. Friction = 4.70      |_B_ |    16.64
+                                                    |    |
+  Load:      | Cars.  | Locomotive.| Tender.        |    |
+  Maximum[E] | 1,561  |   44.60    |  25.25         |_C_ | 1,631
+  Minimum[F] | 1,031  |   60.50    |  34.50         |_C_ | 1,126
+                                                    |    |
+Traction (= _B_ _C_ / 2,000 ) Maximum[E]            |_D_ |    13.60
+                              Minimum[F]            |_D_ |     9.38
+ Weight on Drivers:   | Locomotive.|  Helper.       |    |
+    Maximum[E]        |    21.60   |    63.00       |_E_ |    84.60
+    Minimum[F]        |    47.00   |    72.00       |_E_ |   119.00
+                                                    |    |
+Percentage ( = _D_ / _E_ ).                         |    |
+  Maximum                                           |_F_ |    16.1
+  Minimum                                           |_F_ |     7.9
+========================================================================
+
+    FOOTNOTES:
+
+    [Footnote E: Giving maximum values of percentage, _F_.]
+
+    [Footnote F: Giving minimum values of percentage, _F_.]
+
+
+
+BEVERLY S. RANDOLPH, M. AM. SOC. C. E. (by letter).--The percentages
+given by Mr. Purdon would seem to indicate that the length of the grades
+did not affect the loads in the cases cited, but these percentages are
+so much below those shown in the table, for similar distances, as to
+indicate some special conditions which the writer has been unable to
+find in the text.
+
+The use of the percentage of weight on drivers which is utilized in
+traction as a measure of the efficiency of the locomotive, while,
+probably, not applicable to individual machines, is sound for the
+purposes of comparison of results to be obtained on various portions of
+a line as far as affected by conditions of grade and alignment. It has
+the advantage of disregarding questions of temperature, condition of
+track, character of fuel, etc., which, being the same on all portions of
+the line, naturally balance and do not affect the comparison. It is, of
+course, simply a method of expressing the final efficiency of the
+various parts of the locomotive, and, since it depends entirely on
+actual results already accomplished, leaves no room for difference of
+opinion or theoretical error.
+
+The writer has always considered an "under-cylindered" locomotive as a
+defective machine. All weight is a distinct debit, in the shape of wear
+and tear of track and running gear, resistance due to gravity on grades,
+interest on cost, etc. When this weight fails to earn a credit in the
+way of tractive efficiency, it should not be present.
+
+The statement relative to the performance of locomotives on "Hill _C_"
+is interesting, especially in that it appears to have been immaterial
+whether they made a dead start after stopping at the station or
+approached the foot of the hill at 16 to 18 miles per hour. The momentum
+would appear to be an insignificant factor.
+
+It is gratifying to note that Mr. Trautwine has been able to brace up
+the weak member of Table 1 so completely with his detailed data; also
+that his other results strengthen the conclusions reached in the
+paper.
+
+
+    FOOTNOTES:
+
+    [Footnote A: "The Economic Theory of Railway Location," 1887 edition,
+    p. 502.]
+
+    [Footnote B: _Transactions_, Am. Soc. C. E., Vol. L, p. 1.]
+
+    [Footnote C: "Nearly 200 tons _exclusive_ of eng. & ten." (Vol. III,
+    p. 176-1/10.)]
+
+    [Footnote D: American Railway Engineering and Maintenance of Way
+    Association, Bulletin 84, February, 1907, p. 99.]
+
+
+
+
+
+End of the Project Gutenberg EBook of Transactions of the American Society
+of Civil Engineers, Vol. LXX, Dec. 1910, by Beverly S.  Randolph
+
+*** END OF THIS PROJECT GUTENBERG EBOOK AMERICAN SOCIETY OF CIVIL ENGINEERS ***
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