PROCEEDINGS

of the Tomsk Technological Institute

of Emperor Nicholas II,

Vol. 11. 1908. No. 3.

II.

F. K. Yasevich.

INVESTIGATION OF BALLASTS*).

Article 1; with an appendix of 3 plates of drawings.

1–115.

) The present article, in its original form, had already been printed before the publication of the present volume of the Proceedings, in books 2, 3, and 4 of the Journal of the Ministry of Ways of Communication* for 1908, where it was sent by the author as a completed note on the question under consideration. Here it is placed, in a somewhat altered form, as part of a complete monograph intended for publication.

List of Chapters and Paragraphs

Introduction.

§ 1. Importance of studying the established type of track superstructure.

§ 2. Practical necessity of formulas for stresses in rails.

§ 3. Imperfection of the formulas in use.

§ 4. Necessity of observations of track deformations.

I. The role of ballast as an element of the track superstructure.

§ 5. Ballast as the most unstable part of the superstructure.

§ 6. Definition of ballast and its principal properties.

§ 7. Magnitude of the elasticity of ballasts.

§ 8. Coefficient of yielding of the ballast layer \(C_0\).

II. Requirements imposed upon ballasts, and their fulfillment.

§ 9. Properties of ballast that affect the stability of the track, its strength, and the value of \(C_0\).

§ 10. Thickness of the ballast layer and its width.

§ 11. Drainage of the upper surface of the roadbed.

§ 12. Water permeability and uniformity in the size of ballast particles.

§ 13. Stability of ballast with respect to horizontal forces.

§ 14. Conditions for the strength of the ballast layer.

III. The question of ballast in connection with the properties of the other parts of the track.

§ 15. Determination of the vertical pressure of a locomotive axle upon a sleeper.

§ 16. Formulas for determining sleeper settlement and the introduction into them of the coefficient \(C_0\).

§ 17. Ordinary speeds of movement and experiments with electric locomotives.

§ 18. Determination of the influence of speed on the deflections and stresses of rails.

§ 19. N. Petrov’s method for determining dynamic deflections and stresses in rails.

§ 20. Application of N. Petrov’s formulas for determining vertical deformations of ballast.

§ 21. Observations of locomotive movements affecting the track.

IV. Experimental study of the physical and mechanical properties of ballast.

§ 22. Results of measurements of ballast sands of the Ryazan–Ural Railway.

§ 23. Laboratory experiments with ballasts of the Siberian Railway.

§ 24. The necessity of systematic observations at a special experimental station on the influence of static load on various parts of the superstructure.

V. Information on ballasts of foreign railways.

§ 25. Technical aspect of the question.

§ 26. Economic calculations of the advantage of using ballast of higher qualities.

VI. Characteristics of ballasts of Russian railways.

§ 27. Technical aspect of the question.

§ 28. Economic aspect of the question.

§ 29. Conclusion.

Introduction.

§ 1. The construction and principal dimensions of the railway track (the rails fastened to the sleepers, very rarely to longitudinal timbers; all this rests in a low layer of ballast, transmitting the pressure to the roadbed) have remained almost the same from the very beginning of railway construction. In different countries, under one or another set of traffic conditions, rails of various weights are used (for the standard gauge, from 18 to 38 pounds per running foot, i.e., from 24 to 52 kg per running metre), together with several other dimensions of sleeper (20–27 cm wide, 12–15 cm high, 2.5–2.7 m long), and ballast of various qualities (with a layer height of 10–35 cm beneath the underside of the sleeper), for which both the finest sand and a filling of very coarse pebble or crushed stone are used, sometimes artificially prepared. Yet the general composition of the type of permanent way remains everywhere one and the same, and proves to be fairly stable and inexpensive. This gives reason to think that the customary construction of the track is fully rational and, probably, for many years, if not forever, will remain without substantial changes. This thought, on the one hand, may inspire in technical science and in all who are interested in the safety of railway traffic and in its proper development—that is, in all humanity—a hope close to certainty, that the very practice of many hundreds of thousands of versts of railway track in all countries will gradually improve this basic type of track and will bring, in every part of the railway network, its arrangement and method of maintenance into full correspondence with the conditions and influences under which the track there is found. On the other hand, the circumstance that this same type is used everywhere and will be constructed and repaired for millions of rubles annually compels the most serious attention to be paid to a careful investigation of the role played by the separate parts of such a permanent way, and to clarifying what measures must be taken so that the condition of the track in each given place should fully correspond to those

Importance of studying the established type of permanent way.

by the conditions to which it is subjected and by those biological, so to speak, functions which it bears. Two subjects must constantly be studied by railway engineering—the construction of the permanent way and the construction of the locomotive. The proper arrangement and correspondence in the work of these two elements provide the foundation for the possible safety of movement. Technicians in the various fields of railway practice have to satisfy the demands of rational economy in the construction of various structures and in their operation, to strive toward the comprehensive satisfaction of the needs presented by the population for movement, both passenger and freight, and at the same time always to remember that railway affairs are constantly moving forward, developing, and that the growth of the demands placed upon railways will never cease—to carry, if possible safely, cheaply, and successfully, on the same road, ever greater quantities of freight and passengers at ever greater and greater speed.

The success and cheapness of transportation depend on the entire complex of the arrangements of the railway and the organization of its operation; the safety of movement, however, depends not solely, but of course above all, on the extent to which one may be certain that a locomotive set upon the existing track can safely travel along it. Concern for the safety of movement must be the first duty of railway engineers, since society entrusts to them the technical operation of railways and, consequently, the care of the lives of the passengers on all trains.

Abraham Lincoln, advocating the most rapid possible development of the network of American railways, said that for the country it was more important that the railway network be larger, on condition of cheaper construction, than that several tens of lives be saved at the cost of increasing the cost of construction while bringing the safety of movement to a minimum.¹ A judgment of this kind, even if it has some basis from the point of view of the state as a whole and in wartime, must in no case be laid at the foundation of the activity of engineers called upon to put into practice the principles developed by technical science and to perfect the entire railway mechanism, bearing great—

¹ Lincoln expressed the following thought about railways: “We are faced with a choice: to build solidly and little, or lightly, quickly, and much. In the first case we shall save each year a certain number of human lives; in the latter the whole Union will gain in its greatness and prosperity. I mourn the victims as a man, but as President of the Union I can only advise that they be made.” (See Railway Affairs for 1903, No. 1.)

…responsibility for those who suffer, having trusted this mechanism and relied upon the skill of the engineers and the authority of technical science.

§ 2. The very interesting and complex question of the deformations and stresses occurring, under the influence of the action of a locomotive, in all parts of the track is, for the most part, reduced by investigators to the finding of methods and formulas, very valuable for practice, for obtaining figures for the stress in those parts of the track for which, owing to the homogeneity of their material, a limiting norm of stresses can be established. Such parts are, above all, the rails. The study of the stress of the material of rails is placed first by persons who bear the general responsibility for the safety of traffic, since it appears indisputable to them that, by finding a correct method of determining the stress of the material in rails laid in a given place, they will have in their hands the possibility, if not of preventing, then of reducing to a minimum cases of rail fracture under passing loads. They think that, in order to eliminate the chief cause of railway wrecks, it is sufficient to see to the arrangement of the track in such a way that the material of the rails has a sufficient reserve of strength under the most unfavorable conditions of the action of the locomotive wheel upon the given rails. At the same time it is assumed that, of itself, there will be the same or a greater reserve in the remaining parts of the permanent way; which, being more remote from the point of application of the wheel forces, should not undergo significant changes of form.

Practical necessity of finding formulas for stresses in rails.

Perhaps not one of the technical questions with which engineering science is occupied interests railway administrators so much as precisely the question of the permissible limiting speeds of movement with a given permanent way. With the ever-increasing speed and intensity of traffic and the spread of the use of very heavy locomotives (with axle loads of up to 20 tons), most often, for deciding the question whether it is not extremely dangerous to leave the existing track without radical strengthening, there are no sufficient indications from practice, since the experience of other lines cannot be convincing as applied to the local conditions of track construction and operation. In such cases, attention turns to theory, of which it is assumed that, as all-embracing, it should give an answer for each given case.

The task of those investigating the indicated question is complicated in view of what has been said, since it is not sufficient to confine oneself to more or less

to a rigorous elucidation of the phenomenon and to base, upon a series of propositions which cannot be refuted, methods of calculation for the simplest cases of stresses acting in different parts; but it is still necessary to elucidate and indicate to what extent such calculations can, in different cases, encompass the phenomena in all their complexity; to explain to what extent the assumptions taken as the basis of the calculation correspond to reality in each place; and, in general, to make it possible for anyone using the proposed formulas to know within what limits he may trust them unconditionally, and to what extent they should serve merely to facilitate the complex process of forming for himself the most faithful view of the operation of the given track. Conclusions of the latter kind must be drawn on the basis of knowledge of the character of each of the separate parts of the track and of the results that can be derived from observations of the deformations of various existing tracks, often operating under exceptional conditions of the weight of the loads rolling over them, their speed, and the poor condition of the track itself.

The chiefs of railways are obliged, once or several times a year, when approving the schedule of trains on the separate sections of the railway, to prescribe for the guidance of the Traction Services the maximum speeds on separate parts and stretches, which must not be exceeded either when catching up time or under any other circumstances.

In order to assist in this matter with general indications, the Ministry of Ways of Communication proposes to be guided by methods for calculating the strength of rails that have become popular in technical science thanks to the works of Winkler, Zimmermann, Ast, Kholodetsky, and others, and recommends the use of one or another formula. Until 1899 no definite order had been issued on this question, and the administrations of various railways used different formulas, most often Winkler’s: \(M = 0.1888\,Pl\) (sometimes Zimmermann’s

\[ M = \frac{8 - \frac{k}{\mu} + 7}{16 \frac{k}{\mu} + 40}\,Pl \]

—for a static load, and Winkler’s

\[ M = \frac{0.1888\,Pl}{1 - \frac{0.1888\,Plv^{2}}{I\,v_{0}\,g}} \]

for a dynamic load). In 1899 the Engineering Council settled upon the formulas it considered the most correct, and they were placed, for the guidance of all chiefs and managers of railways, in the Technical Conditions for the Design of Main Lines approved by the Minister.

These formulas are as follows: 1) the Zimmermann formula

\[ R_s = M \frac{z}{J} = \frac{8 \frac{k}{\mu} + 7}{16 \frac{k}{\mu} + 40} \, Pl \cdot \frac{z}{J}, \quad \text{where} \quad \frac{k}{\mu} = \frac{12EJ}{0.89ab l^3 C} = \alpha, \]

which gives the static stress in the rail as a beam lying on four elastic supports (sleepers), and 2) the, so to speak, Zimmermann–Winkler formula

\[ R_d = \frac{8\alpha + 7}{16\alpha + 40} \cdot P'l \cdot \frac{z}{J}, \quad \text{where} \quad P' = P : \left(1 - \frac{8\alpha + 7}{16\alpha + 40} \cdot \frac{Plv^2}{EJg}\right), \]

where \(v\) is the speed in centimeters per second, and \(R_d\) is the stress in the rail at a given speed of motion. The limiting stress in the rail material was prescribed to be admitted for the first formula as \(14 \ \frac{\mathrm{kg}}{\mathrm{mm}^2}\), for the second \(20 \ \frac{\mathrm{kg}}{\mathrm{mm}^2}\) (and \(25 \ \frac{\mathrm{kg}}{\mathrm{mm}^2}\) for speeds above 60 versts per hour).

To make the task of establishing speeds on individual lines and their sections still easier, the Administration of Railways, having calculated by the indicated formulas and having introduced into the results further corrections and restrictions indicated by the practice of service of the superstructure (since the formulas mentioned sometimes lead to clearly erroneous figures), established for certain types of rails (\(32 \tfrac{1}{2}\), \(28 \tfrac{1}{2}\), \(24 \tfrac{1}{3}\), and \(22 \tfrac{1}{2}\) lb per running foot) those loads and the highest limiting speeds which must under no circumstances be exceeded (Order of the Minister of Communications of January 14, 1903, No. 14).

§ 3. With what care the results obtained from the formulas should be corrected.

However carefully the results obtained from the formulas may be corrected, still, in order to use these formulas, one must have a guarantee that they have at least something in common with the actual work of a railway track. Upon the closest examination it turns out that the indicated formulas, interesting in themselves, are completely unsuitable for the purpose for which they are intended, namely, to verify the general stability of a given track. The reason for this is that their derivation was made on the basis of a whole series of assumptions having little in common with reality and, in any case, not embracing all the circumstances of the work of the track.

1) Let us begin with the second formula.

The action of a wheel rolling along a rail may be regarded as divided, in the last resort, into three separate actions—the static vertical load, the dynamic vertical load, and an identical horizontal load. In the Zimmermann–Winkler formula, of all the additional actions that appear when axles are in motion (the following are usually counted: the galloping of the locomotive, i.e. the alternating overload of the front and rear axles, depending on its oscillation about the horizontal transverse axis; the influence, i.e. oscillation about the vertical axis, depending on the presence in the locomotive of parts with alternating motion; the overload of one or another wheel depending on irregularities and imperfect packing of the track; the successive increase and decrease of the wheel load, which is a consequence of the vertical component of the centrifugal force during the motion of cranks and counterweights of locomotive wheels; impacts from rail curvature, increasing with speed, etc.), only the centrifugal force is taken into account, according to Winkler’s assumption, arising from the fact that the wheel rolls between two sleepers not along a straight line, but along a concave curve. It turns out that, since the sleepers, when the wheel passes over them, sink only slightly more than when it passes over the rail span, the rolling stock therefore rolls almost in a plane—lowered, but parallel to the rail track in a state of rest¹; and in any case the versed sine of the arc along which the wheel rolls in the span between sleepers is expressed by a value of no more than \(1/4\) mm, which cannot produce any appreciable centrifugal force.

Encountering the reaction of the rail, the wheel, during the interval of time until it has reached the next sleeper from one sleeper, cannot have time to descend by the amount that would correspond to Winkler’s assumption. Thus calculations of the stress in the rail at one or another speed, when using the indicated dynamic formula, must lead to incorrect results. The mere fact that, in calculations by the formula giving the stress under static load, the limit of permissible stress in the steel of rails is established as

\[ 14 \frac{\mathrm{kg}}{\mathrm{mm}^{2}}, \]

whereas for the formula for checking rails under a moving load the limit

\[ 20 \frac{\mathrm{kg}}{\mathrm{mm}^{2}}, \]

is permitted, already indicates that some un-

…from these formulas, and perhaps both, do not give the actual stress in the material of the rail, and still less can they characterize the work of the entire track.

2) Under a static load on the rail, the stress may be taken as equal to a certain value obtained from the maximum bending moment (for a span equal to the distance between sleepers), divided by the section modulus of the rail profile and multiplied by a coefficient expressing the dependence on the transmission of the force to one or another number of elastic supports. This, in fact, is how Zimmermann reasons in deriving his static formula; but he permits himself to fix the indicated coefficient on the basis of consideration of an abstract, simplest case, whereas in reality the magnitude of this coefficient is very conjectural, since the degree of elastic settlement of the sleepers can scarcely be defined in the form of any constant figure and may be very varied for the same type of rails and even for the same composition of the superstructure of the track.

N. Petrov proposed in 1903 a new method for finding deflections and stresses at various speeds of motion, knowing (for example, from experiments, if not on the given track, then on another with the same principal dimensions of construction) the static deflections under the given load. Moreover, in those cases where no experimental results are available, for determining the static deflections he proposes using a formula which, in essence, differs in no way from Zimmermann’s static formula. N. Petrov stipulates that only observed static deflections can be reliable, and it may be said in advance that they will differ greatly from those obtained from the existing formulas, since the latter do not include all the circumstances that have an influence on the deformation of the track.

N. Petrov reduces the influence of dynamic load exclusively to the influence of 1) irregularities in the longitudinal profile of the rails, resulting from their wear or from improper rolling, and 2) flat spots on the tires, produced by braking. The results of computations by his formulas lead to conclusions with which one cannot agree, namely: 1) that an increase in speed of motion within the limits from 60 to 125 versts per hour causes almost no increase in the stresses of the rails or in the deflections of the sleepers; 2) that by increasing the rail profile one cannot achieve a reduction in its stress under the dynamic action of loads; and 3) that the sure means of reducing the stress in the rail to a high degree is improvement of the coefficient \(C\) of the ballast. It is evident that Petrov’s formulas are in no way better, if not…

of the older formulas of Winkler and Zimmermann, but the method of finding dynamic stresses by means of static methods is correct.

Necessity of observing deformations of the track. § 4. Proceeding from the idea that in such a complex question as the determination of stresses in the parts of the superstructure of the track it is necessary to rely not on more or less ingeniously constructed formulas derived from the theory of elasticity, but on observations of the deformations of the various parts of the track and, chiefly, of the deflections of the rail and changes in the position of the sleeper bed, one must conclude that, for judging any given track, and also for deriving known formulas that characterize the strength of various types of superstructure, the most important thing is to carry out measurements of the changes of form of the most diverse tracks under different conditions of traffic.

The experiments must consist in measuring both the vertical and the horizontal oscillations of the track.

Vertical oscillations have been observed by many foreign engineers (Weber, Flamache, Colard, Genty, Ast), and in Russia by Stetsevich and, in a more precise manner, by Vasyutinsky. The works of the latter, as well as those of Colard, are the most detailed. They also made certain observations of horizontal oscillations of the rail. Nevertheless, systematic measurements of deflections at various points of the span between the sleepers, and for different types of track and traffic, are very necessary, if only because Vasyutinsky observed the deflections of the rail and sleepers only in one place on the Warsaw–Vienna Railway. To measure the deformations of rails, and especially of the whole track, in the horizontal plane is far more difficult than in the vertical one; meanwhile it is a question whether these deformations are not chiefly what cause the derangement of the track. It is first necessary to ascertain the character and extent of the influence of the wobbling and lameness of each axle of the rolling stock, which, in all probability, are the cause both of horizontal deflections and of the twisting stress in the rails and their canting.

Since few such observations have been made, and correct conclusions have not been drawn from them, it is necessary to rely on the experience of existing railway tracks and on a careful study of every kind of movement that takes place both in the parts of the rolling stock and in the parts of the track when a train passes over it; in addition, statistics of incidents during movement must be taken into account, as well as the opinions of persons standing close to the given track as to the degree of its suitability for one or another kind of traffic, and the subjective opinions about it of experienced engineers versed in this matter.

When checking, in each particular case, the stability of the track by means of one or another of the formulas in use up to the present, it should be borne in mind that the result of the calculation can serve as only one of many considerations for one or another final evaluation of the track.

I. The Role of Ballast as an Element of the Track Superstructure.

§ 5. From the formulas in use one always expects an answer to the question of stresses throughout the track; moreover, in the best case they could give such answers with respect to individual parts of the track. If it is possible to speak of stresses acting throughout a track of a given type, then it would be most correct to seek those limits of wheel loads and speeds of motion at which the disturbance of the weakest parts of the given track by the passage of trains becomes so negligible that we can confidently say that, even under prolonged service, all parts of the track under these conditions experience stresses that do not exceed the limits permitted by the elasticity of those parts whose material possesses such properties, or the limits within which it is possible, by constant minor repair, to restore the proper mutual disposition of the parts. The study of the track would be a comparatively easy task if it were possible to regard it as a more or less homogeneous elastic body, or at least as a chain of bodies, each possessing a definite elasticity. Yet this is not the case. If we may consider the rails, the sleepers, and also the fully settled earthen roadbed as possessing quite definite elastic properties and as undergoing, during the passage of rolling stock, only vanishing deformations—that is, within the actual limits of forces and settlements—the same cannot be said of the ballast, which is an intermediate part, having almost no connection during the passage of a train either with the rails or with the ground, possessing no appreciable elasticity in small, uncemented masses, very heterogeneous in composition, and subject to the most varied influences: impacts of wheels, atmospheric agencies, accidents of laying and tamping, and so forth.

Therefore, if the task of rational design and maintenance of the superstructure is taken to be the assignment of its separate parts, in qualities and dimensions, in full conformity with one another, so that they constitute, as far as possible, a uniform whole without an excessive reserve of stability in one part while it is insufficient in another, it is necessary to pay the most serious attention to

the study of the properties of ballast and to find a means of assessing its conformity with the other parts of the permanent way. The insufficient strength of the roadbed manifests itself on long-built lines only in exceptional cases, in the form of local subsidences or heaving places, and in all such cases measures are taken to eliminate these defects or to provide reinforced protection for such places in the track; types or qualities of rails and sleepers that do not correspond to the dimensions of the traffic make themselves known directly by an obvious disorder of the track, by increased battering of the rails, by a short service life of the sleepers, etc.; whereas the nonconformity of the quality and profile of the ballast layer has a hidden effect in reducing the safety of traffic, in increasing the costs of track maintenance, and in more rapid wear of the parts both of the track and of the rolling stock.

At the conclusion of one of his articles dealing with the railway track (Die Schwelle und ihr Lager. Organ, 1898), Ast points out that, in comparison with the other elements of the permanent way, it is most difficult of all to strengthen the ballast so as to bring it into conformity with the increasing demands of traffic, since its resistance is limited by the properties of this material.

Under the conditions of Russian railway life this is especially applicable, since here, with the high cost of crushed stone and the scarcity of good coarse sands with a small admixture of clay, and with the impossibility, over long distances, of hauling ballast from quarries with material of good quality, local sands are used, often very unsatisfactory ones. In order to ascertain the possibility and procedure for strengthening the permanent way, it is necessary to study thoroughly the limits and properties of the resistance of the ballast layer for its various compositions.

Definition of ballast and its principal properties. § 6. Ballast is an intermediate layer that receives the pressure from the sleepers and transmits it to the roadbed; moreover, the ballast serves to unite all the sleepers—or, in other words, the whole permanent way—into one whole, preventing it from moving under the influence of horizontal forces, both lateral and longitudinal. Since the deflections of the sleepers are very unequal, depending on the position of their location beneath the axles of the train, and vary as the latter moves, it is of practical interest to know the average and limiting magnitudes of these deflections, as well as the limit beyond which the compression of a given ballast no longer remains elastic throughout its entire magnitude. In the theory of the permanent way developed by Winkler, Schwedler,

Leve and, finally, Zimmermann, have dispensed with a more detailed study of the action of the ballast layer by introducing into the formula a coefficient of ballast, or, more precisely, a coefficient of bedding, characterizing the greater or lesser compliance of everything lying below the sleepers. Before deciding how far a single number can characterize all the properties of ballast, on which the stability and strength of the track depend, it is necessary to examine more closely the significance of these properties. The ballast layer must protect the upper surface of the roadbed from damage and washout, and therefore it must: 1) not retain water within itself, i.e. consist of a water-permeable, easily drying, and therefore sufficiently dense and coarse material (gravel, crushed stone or coarse sand, slag, etc.), not subject to the action of frost or to being blown away by the wind; 2) have sufficient thickness for a possibly uniform distribution of the pressure from the train over the surface of the roadbed; and 3) itself be difficult to damage by impacts and abrasion.

Consisting of a large number of individual grains piled up to a considerable height, the ballast layer is, within certain comparatively small limits, elastic. Under considerable pressure the layer is compressed both by the compression of the individual grains and, chiefly, by the reduction of the dimensions of the voids between them. Since changes in the dimensions of these voids, whose dimensions are large in comparison with the grains, cannot be elastic, the elasticity of a loose body such as ballast must be attributed to the elasticity of the material of the grains themselves and to their coarseness in comparison with the voids; therefore the difference in the elastic properties of different ballasts can be explained only by the difference in the material of which they consist. The inelastic settlement of the ballast, corrected from time to time by tamping, depends on the shape and properties of the grains and on the size of the voids between them. Although, besides the ballast, the adjacent parts, such as the sleepers and the earthen roadbed itself, are also elastic, it is generally considered that their compressibility is much less than that of the ballast layer, and the latter is therefore often called the elastic foundation of the superstructure.

It is very interesting to determine to what extent elasticity is desirable in a railway track. To any significant degree it undoubtedly has an adverse effect on the expenses of operating the railway and on the very condition of the track. Indeed, the least resistance to motion is presented by a smooth, homogeneous, and as rigid as possible track, since all deflections of the rail surface contribute to the development of additional loads from moving loads, which lead to an increase in resistance.

of the train in motion and to disordering of the track in various directions. On the other hand, if it is impossible to attain an arrangement of all rail supports, i.e. sleepers, strictly on one level, given the inevitable deflections of the loaded rail in the intervals between sleepers and the presence of such weak places in the track as rail joints, an entirely rigid construction of the permanent way would be the cause of systematic blows experienced by the track and leading to wear both of its parts and of the rolling stock.

Magnitude of the elasticity of ballasts.

§ 7. Let us consider how the elastic properties of ballast are usually represented. Considering the ballast layer as consisting of a more or less homogeneous material, it is assumed that it possesses the quality inherent in all homogeneous bodies—elasticity; i.e., up to a certain limit of stresses (up to a certain pressure per unit area of the sleeper base), the sinking of a cross-tie into the ballast completely disappears after removal of the load and is in direct ratio to the pressure \(p\) per unit area of ballast. Winkler was the first to adopt that the deflection

\[ y=\alpha p,\quad \text{or}\quad p\ \frac{\mathrm{kg}}{\mathrm{cm}^{2}}=\frac{1}{\alpha}\,y=Cy\ \text{cm.}, \]

where \(C\) is a constant quantity depending on the properties of the ballast, or the coefficient of ballast. If \(y=1\), then \(C=p\), i.e. the coefficient of ballast expresses the pressure from the sleeper per sq. cm causing a settlement of 1 cm. Since the settlement of a sleeper depends not only on compression of the ballast, but also on the elastic properties of the roadbed (according to Wasiutyński’s experiments, the roadbed at a depth of 0.5 m under the rail settles by 1.2 mm, and at a depth of 1.5 m under the rail a settlement of 0.6 mm can be noted), it is incorrect to call the figure \(C\) derived from observations the coefficient of ballast; rather it should be called the coefficient of the bed (Bettungsziffer). A series of observations abroad gave results for the value of \(C\) from 2 to 9; on average, for gravel ballast \(C=3\), and for crushed stone or gravel on a layer laid dry \(C=8\)¹). In Russia, few observations have been made of deflections of the ballast layer (Stetsevich on the Tambov–Saratov and Baltic railways in 1890–94; Wasiutyński on the Warsaw–Vienna railway in 1897 and 1898). On average, for Russian railways the bed coefficient is taken as \(C=4\); general calculations of the permanent way are usually made under two assumptions: \(C=3\) and \(C=8\). Wasiutyński from

¹) See Stetsevich, On the Stability of the Permanent Way, p. 22.

of his experiments to conclude that the ballast coefficient proper, which he denoted by \(K\), for sand mixed with coarse gravel varies from 6.9 to 9.0, and for granite crushed stone from 4.6 to 6.5. The load under which a unit area of the roadbed can subside by 1 cm he denoted by \(N\) (the roadbed coefficient). Under a unit load, the ballast bearing area would subside, for an incompressible roadbed, by

\[ \frac{1}{K}; \]

and the roadbed bearing area will subside by

\[ \frac{1}{N}, \]

reduced in the ratio of the area of the roadbed that received the pressure from the sleeper to the area of the sleeper bed, i.e., it will subside by

\[ \frac{1}{nN}, \]

where \(n\) is approximately equal to

\[ \frac{a}{b} \]

(\(a\) being the distance between sleepers, \(b\) the width of their lower beds).

One may write

\[ \frac{1}{C}=\frac{1}{K}+\frac{1}{nN}. \]

This formula makes it possible, from the bedding coefficient \(C\) derived from experiments, to find, if necessary, the ballast coefficient. \(N\) is found if, on the same roadbed, the track is tested successively with two ballasts for which \(K\) is known. According to the tests carried out by Vasotyński, the settlement of the roadbed surface proved to be 3–4 times less than the settlement of the sleeper footings; i.e., \(N\) is about 5. The value of \(K\) in his experiments varied from 5.5 to 8.

Comparing the values of \(K\) for sand and crushed stone obtained from Vasotyński’s experiments, we see that for the former—which is undoubtedly worse not only in the sense of greater labor expenditure during constant maintenance, but also in the sense of track stability—the ballast coefficient is almost \(1 \frac{1}{2}\) times greater. In general, Vasotyński comes to the conclusion that the ballast coefficient varies within very narrow limits and that its magnitude often does not coincide with the general character of the principal qualities of the ballast as part of the permanent way, so that in general one cannot speak of improving the latter together with an increase in ballast. At the same time he points out that this coefficient changes not only depending on the qualities of the ballast, but also on the arrangement of the sleepers and the type of rails. On the same question we find an indication in L. F. Nikolai’s report to the Engineering Council in 1898 (Journal No. 153). It says there: “In the report on adapting the permanent way of the Moscow–Vindava Railway to the passage of heavy locomotives with an axle load of 14.5 tons, it was found that the value of the coefficient \(C\), which is extremely advantageous with respect to the work of the material in the rails, is affected by sufficient hardness of the roadbed soil, good quality of the ballast, and sufficient width of it.”

It is noteworthy that, according to older observations, the fluctuations in the value of \(C\) were much greater than among observers closer to us in time.

Thus, on the basis of Weber’s initial, very approximate determinations of the value of \(C\), it varied within the limits from 4 to 45; but Leve and other engineers adopted 12–16 in calculations, and already after Hentschel’s experiments these values were established as 3.5 and 8. Stetsevich obtained (not taking into account cases of exceptional yielding of the sleepers) almost the same values as Hentschel: \(3 \tfrac{1}{2}\), 5, and 9 kg/cm². Vasyutinsky, observing ballasts in one place and under good conditions, obtained the values \(5 \tfrac{1}{2}\), 8; adding 3 for poor grades of ballast gives the same figures as in the preceding case.

§ 8.

Coefficient of yielding of the ballast layer \(C_0\).

It is thought that the narrowing of the limits of fluctuation in the value of \(C\) is explained by the fact that, in more rationally arranged experiments, an attempt is made to explain as fully as possible the influence of the principal factors acting on the track; and later observers excluded many factors that seemed secondary but, in their complexity, have considerable significance. It does not follow from this, however, that the limits of variation of the ballast coefficient under its elastic oscillations and under normal conditions are greater than has recently been considered established (3–8). On the contrary, it seems undoubted that these limits, when considering only the elastic oscillations of the ballast, are considerably smaller; the absolute values of \(C\), perhaps, should be expressed not by the figures 3–8, but by 15–40. But the point is that elastic oscillations are only part of the deformations of the ballast layer under the influence of load; observing, however, the settlement of sleepers at various points of the track and under different conditions, we should probably obtain greater differences in the values, computed from these observations, of the coefficient of proportionality of total settlements to load. This coefficient, which we shall call \(\alpha_0 = \frac{1}{C_0}\), consists of two parts: \(\alpha = \frac{1}{C}\), corresponding to the elasticity of part of the settlement, and \(\alpha_1 = \frac{1}{C_1}\), corresponding to permanent deformation. In fact,

\[ C = \frac{C_0 C_1}{C_1 - C}. \]

We shall call \(C_0\) the coefficient of yielding of the ballast layer. Vasyutinsky, in his report on observations in 1898, says that on the Baltic Railway Stetsevich observed that some sleepers settled so little under the action of locomotive wheels that the sleeper coefficient was obtained as \(C = 45\). Vasyutinsky considers such a figure incredible.

Meanwhile, it seems risky to us, for the sake of making it possible to explain, to reject certain observations merely because they do not give figures confirming the accepted theory. It may be explained that the \(C_0\) observed in the case indicated is indeed close to 45, and, recognizing that here, just as in the other cases, \(C\) could not have been greater, we shall put 8–9; we conclude that the value of \(C_0\) came out so large because the sleepers were in such conditions that permanent deformations were almost absent, which does not happen in most cases.

Schubert, in his experimental boxes, observed that with dried ballast the sleepers sank under a pressure of 3.3 kg per 1 sq. cm, whereas before the embankment was watered the same sinking of the sleepers was observed under a pressure of 6.44 kg.\(^1\) This shows what influence one or another condition of the ballast layer and roadbed, accidental changes in them, etc., can have on the value of \(C_0\). Therefore, considering \(C\) constant for a given ballast, we must, from prolonged observations on a given section of track, establish the limits within which \(C_0\) may fluctuate on this section; and, in calculating the strength of the track, if we set ourselves the aim of guaranteeing, to a high degree, the safety of movement and do not count, at the given place, on careful inspection and maintenance of the track, we must introduce into the calculation the most unfavorable limit for \(C_0\).

II. Requirements Imposed on Ballasts and Their Fulfillment.

§ 9. From the preceding it follows that a single value of the coefficient characterizing the elastic properties of ballast, i.e. the value of a single \(C\), does not give an exact idea of ballast as a constituent part of the track, since the stability, and especially the strength, of the latter depends on the condition of the ballast, which is in direct connection with such qualities of it as, for example, water permeability, uniformity of particles, etc., and also on the condition of the roadbed, which in turn depends to a significant degree on the qualities and thickness of the ballast layer. According to Schubert’s experiments it turned out that, with a thickness of the ballast layer equal to the distance between adjacent sleepers \(+0.20\) m, the roadbed undergoes no changes under the influence of train movement. Therefore the study of ballasts should not be limited to—

Properties of ballast affecting the stability of the track, its strength, and the value of \(C_0\).

\(^1\) Zheleznodorozhnoe delo, 1893, p. 152; Zeitschr. für Bauwes., 1891.

by examining the coefficient of its elastic settlements, but rather the most serious attention should be paid to those qualities which it is necessary to require of every ballast, and to the degree to which their possession characterizes the greater or lesser suitability of the given ballast. Therefore this degree must be expressed either by some exact verbal definitions, or by numerical coefficients. At the same time it is necessary to note that the stability of the ballast layer is very difficult to consider separately from the conditions of its strength, since the vertical movements of the sleepers, however small they may be and to whatever great degree the deformations may prove elastic, after some interval of time disturb the bedding and crush the ballast. The more stable the ballast layer, i.e. the smaller the residual deformation that takes place from the greatest forces of the rolling stock, the smoother the movement of trains and the lower the expenditure for track repairs. Owing to the frequent recurrence of ballast settlement, to the considerable horizontal forces arising in it during the movement of trains, and also owing to the friction of the undersides of the sleepers against the ballast, even the most stable ballast layer cannot fail to be disturbed. Moreover, there is a whole series of factors contributing to this disturbance, many of which depend on the transverse profile of the ballast layer.

We shall begin the consideration of the chief properties of ballast with the profile, i.e., strictly speaking, with the thickness of the ballast layer.

Quite apart from the fact that the thickness of the ballast layer is not in itself indifferent, even the value \(C\) must depend on it. To establish the coefficient of elasticity \((C)\), the deformation \((y)\) must be taken for a definite dimension of the body in the direction of the force; but usually this compression is measured for the whole ballast layer, it being assumed that within those limits in which the thickness of the ballast layer actually varies, its total compression depends almost not at all on this thickness. If calculations are made for a comparative assessment of the qualities of track of lines situated under approximately identical conditions and, in any case, for lines of the same type in a given country, it is entirely possible to disregard the influence of the height of the ballast layer on the magnitude of the bedding coefficient, since the law of distribution of ballast compression over the height of the layer is unknown, variations in the thickness of the ballast layer are not great, and, moreover, calculations of track stability are always made for the average conditions of some section of track, and not for a definite point; consequently, insignificant differences in the height of the ballast layer cannot be taken into account.

§ 10. In general, it is desirable that the height of the ballast layer be close to 35 cm beneath the bottom of the sleepers, because with such a size the pressure from the sleepers is distributed sufficiently uniformly over the roadbed. According to the Technical Conditions of the M. of R. for the design of main lines in Russia, a ballast depth of 53 cm (0.25 sazhen) is required under the foot of the rail, i.e. 38 cm (0.18 sazhen) under the bottom of the sleepers. According to Schubert’s observations, all deformation of the roadbed—for example, the appearance of so-called troughs—ceases even in the most northerly soils, if the thickness of the ballast layer under the sleeper is equal to the clear distance between sleepers plus 20 cm, i.e. about 80 cm; but usually 20–30 cm are taken. If the slope of pressure distribution in the ballast layer is considered equal to 1/2 of the base to the height, then, strictly speaking, twice the thickness would be required. In fact, on most Russian railways the thickness of the ballast is considerably less and amounts to about 17 cm under the bottom of the sleeper. On foreign railways the smallest thickness of ballast under sleepers is taken as 20 cm.

Very many observations, as well as experiments carried out by Schubert in trial boxes, indicate that the principal cause of the formation of slips and settlement of embankments built long ago is the trough-shaped depressions¹ that arise in the roadbed from the pressure of trains. Water is retained in these depressions, as a result of which the soil of the embankment becomes soaked, becomes unstable, and tends toward greater settlement. To raise a track that has settled too much, too large a quantity of ballast is required; acting like a wedge, it thrusts apart the sides of the embankment and, in the presence of moisture accumulated in the depressions, causes failures of the slopes. Of course, ballast troughs do not take the form of regular depressions, but they are conductors of water into the roadbed, and in this sense they may be compared with wedges.

XIV (in 1897), the Advisory Council of Engineers of the Railway Service recognized that, in order to ensure uniform transmission of pressure to the roadbed and, consequently, to protect it from the beginning of the formation of ballast troughs, the thickness of the ballast must be not less than 0.20 sazhen under the bottom of the sleepers under the best soil conditions; in addition, such a form must be given to the transverse section

of the roadbed, which would correspond to the best possible runoff of water seeping through the ballast. This provision is set forth in greater detail in the part of the present essay that concerns the actual condition of the ballast layer on Russian railways.

A sufficient height of ballast has another important significance—the prevention, in any case, even in very severe climates and with rapid transitions from warmth to cold, of the reduction of frost heaves.

The higher the quality of the ballast and the more favorable the soil conditions, the smaller the thickness of the ballast may be. Practice indicates that an excessive thickness of the ballast layer is of no benefit. The conditions for determining the thickness of the ballast, proceeding from the limiting pressure on the roadbed, are taken by L. F. Nikolai¹ as follows: that the pressure transmitted from the underside of the sleeper to the roadbed should not exceed 1.5 kg/cm² (0.6 poods/dm²), and in sandy ballast one half of the pressure-distribution angle is taken, while in ballast of fine crushed stone or coarse gravel—a single angle. Ast sets the limit of pressure on the roadbed at 2 kg per cm², but at the London Congress data were presented showing that on the majority of roads this pressure does not exceed 1.32 kg per cm². Of course, such calculations are conditional. Everything depends on what overload we agree to assume from dynamic action. For simplicity of calculation it is assumed that the pressure of the axle is taken up by a single sleeper.

For the instantaneous stability of the layer it is necessary that the ballast not be able to be squeezed out from under the sleepers. The deeper the sleepers lie in the ballast, the more stable the track, since the larger are the prisms of support at the sides of the sleepers that must be displaced in order for the sleepers to yield downward. As we see, a deeper bedding of the undersides of the sleepers is useful in the sense of counteracting not only horizontal but also vertical forces. Abroad, on lines with high traffic speeds, for the most part the sleepers are filled with ballast almost up to the height of the rail heads, limiting its top between the rails by a convex surface. The usual ballast profile in Russia satisfies the requirements of stability, owing to the comparatively low speed of trains.

In determining a sufficient thickness of the ballast layer one cannot be guided by theoretical calculations similar to the stability calculations made when designing foundations in sandy

ground. Indeed, let us try to make such a calculation and see that the required dimensions are obtained far from those which have been worked out by the practice of railway construction. Let us first seek at what depth from the upper surface of the ballast the soles of the sleepers must be located in order that their position may be stable, i.e., so that they cannot sink deeper into the sand, accompanied by the pushing out sideways of prisms of the sandy foundation. Considering the sleeper as a loaded footing, upon a unit of whose area there falls a pressure

\[ N=\frac{P}{bl}, \]

where by \(P\) we shall denote the pressure from the wheel of a locomotive, \(b\) the width of the sleeper, and \(l\) one half of it, we shall apply, for calculating the depth of the foundation \(h\), Rankine–Pauker’s formula:

\[ h \geq h' \tg^{4}\left(\frac{90-\varphi}{2}\right), \]

in which \(h'\) is the height of the column of sand replacing the loaded sleeper and equal to \(\frac{N}{\gamma}=\frac{P}{bl\gamma}\), where \(l\) is ordinarily equal to \(\frac{270}{2}\) meters, \(b=0.25\) m, and \(\gamma\) is the weight of a cubic unit of sand, about 1640 kg per cubic meter. Taking \(P=7000\) kg and \(\varphi=35^\circ\), we obtain \(h \geq 0.92\) m. For \(P=10000\) kg, \(h\) is about 1.32 m. Even agreeing that Rankine–Pauker’s formula gives a factor of stability twice as large as is needed, we have a necessary depth of the upper ballast of 0.46 m, i.e. 0.21 sazhen; whereas, if with the usual thickness of a sleeper, about 0.15 m, we filled the ballast up to the very head of the rail, we would still obtain a layer of upper ballast no greater than 0.28–0.30 m. Rational from the theoretical point of view, the thickness of the lower ballast, i.e. the depth from the sole of the sleeper to the surface of the roadbed, will be that at which between the sleepers there remain no gaps on the surface of the roadbed to which the pressure transmitted by the sleepers loaded by adjacent locomotive wheels does not extend. The depth \(k\), with tangent \(n\) of the angle of propagation of pressure, will be found as a function of the distance between the locomotive wheels (\(d=1.3\) m) and of the width of the sleeper sole (\(b=0.25\) m) from the formula \(d-b=2nk\), or \(1.05=2nk\), whence, putting \(n=0.7\), we have \(k=0.74\) m \(\infty\) 0.34 sazhen; and putting \(n=1\), we have \(k=0.52\) m \(\infty\) 0.24 sazhen. In this case the pressure per sq. cm of the sleeper sole, equal approximately to \(\frac{7000}{135\times25}=2.1\) kg, trans-

is imparted to the roadbed in the form of pressure \(\frac{7000}{(135+52)\cdot(25+104)} = 0.3\), i.e., 7 times less. When calculating for a wheel of 10 tons, with a distance of 1.5 m, \(k\) is considerably greater.

In accordance with such a great thickness of the ballast layer, the width of the roadbed for a single track would also have to be determined as not less than 4 sazhen.

In reality, however, people are satisfied with an upper ballast layer of 0.15–0.25 m and a lower layer of 0.20–0.30 m; that is, practice indicates that there is no need, in order to ensure the safety of traffic, to resort to disproportionate expenditures in striving to attain the figures indicated above.

This proves that the assumption that the pressure of the axle is always taken up by one sleeper is incorrect. This assumption is made in view of the fact that the dynamic action of the load is taken to be equal to 2–\(2\frac{1}{2}\) static loads. The calculation presented proves the incorrectness of the latter. Therefore, for determining the pressure on the surface of the roadbed and for determining the sufficient depth to which the sleepers are embedded in the ballast, one may quite well be satisfied with the values of the static load, with a small addition.

Since the condition of the track is the better the fuller the ballast layer, we propose to introduce the indicator of the thickness of this layer into the general figure characterizing ballast in the sense of its influence on the stability of the track. This figure is the greater, the less must be subtracted from the number characterizing ballast as a fully elastic body. If we agree to take this figure as \(C_0\), of which we spoke above, then let \(C_0\) be considered greater for that ballast layer whose thickness is greater. Having adopted this method also with respect to other properties of ballast—such as its water permeability, the slight susceptibility of the layer to damage by water and wind, the strength of the rocks composing it, the magnitude of the coefficients of friction and cohesion, etc.—and having, moreover, a method for determining the value of \(C_0\) by a system, so to speak, of points. Although it is very difficult to establish methods for assigning these points that are fully scientific and that exclude the subjective point of view, this can hardly serve as a great obstacle to, if not establishing, then at least testing such a method in so complex a question as the determination of a single figure characterizing ballast, which could be inserted into formulas intended for calculating, of course only approximately, the reserve of stability and strength of a given track.

It may be agreed to enter into the value of $C_0$, as one of the terms, a digit corresponding to the decimeters of thickness of the layer, counting from the bottom of the sleepers downward.

As regards the width of the ballast layer, in Russia it is usually taken, for roads of normal single-track gauge, as $1.45$ sazhens $\infty\infty$ $3.10$ m, abroad $3.30$–$3.60$ meters.

The greater the width of the layer, i.e. the more the ends of the sleepers are covered with ballast, the better the superstructure resists lateral forces tending to shift the track to the side. Since widening the ballast layer leads also to a widening of the roadbed, which is associated with large expenditures, they limit themselves to giving the ballast layer such a width that the ends of the sleepers on each side are covered by not less than 1 decimeter. It seems correct, in the event that this covering is increased, to add, for every decimeter of increase of the covering on one side of the sleeper, one point to the coefficient $C_0$. In exactly the same way, several points must be added to the coefficient $C_0$ if the length of the sleepers is increased by one or several decimeters against the normal length, which we shall take as $2.7$ m; and likewise $C_0$ must be reduced if the sleepers are shorter than normal. It is clear that the length of the sleeper and the corresponding width of the ballast layer very strongly affect its condition and the movement of particles in it.

§ 11. On all roads very serious attention must be paid to the drainage of the upper surface of the ballast by means of the appropriate shaping of the surface of the earthwork roadbed, or by means of drainage (for example, stone pitching) of the bottom of the ballast, and also to ensuring that the very surface of the roadbed is absolutely impermeable to water. The fulfillment of this places the ballast in considerably better conditions, and therefore, it would seem, in cases where the surface of the roadbed is in a well-arranged condition, one should add 1–3 to the coefficient $C_0$; for example, in the case of correct shaping (with slopes) add 1, if the top of the roadbed is drained—2, and if it is entirely impermeable—3. The XIV Congress of Engineers of the Ministry of Ways of Communication recommended, for better runoff of water over the surface of the roadbed, that in the transverse profile of the roadbed slopes be made with a gradient of 0.1–0.2 from the line of the bottom of the outer ends of the sleepers. In the case of application of such a form of roadbed, moreover when we are certain that there are no ballast troughs, that the correct designed convex form of the roadbed (with a crown) has been preserved under the ballast, that the berms around the ballast layer have not grown up, and that more or less rapid removal of water from the ballast ...

should be secured against slopes—it would be necessary to add from 1 to 5 to the value of $C_0$. For the most part it happens that, when the track is first laid quite without ballast, or on a very thin layer of it, the surface of the roadbed assumes a very irregular form; humps and pits are formed, and there is no mention of a crown, until ballast troughs appear, and only subsequent additions to the height of the ballast layer prevent derangement of the track and even collapses. In former times, on other lines, in view of inspection trips, in places where the thickness of the ballast layer was too small, it was the practice to make cuttings in the shoulders of the roadbed so that the ballast profile would appear complete. If the purpose of this manipulation was not to present a ballast layer insufficient in thickness as sufficient, then in itself this measure can only deserve approval. The Americans employ a different method for a more complete removal of water from the ballast layer. In those places where the ballast is of medium quality or outright poor (sometimes they use not even sand, but sandy earth), they give the outer surface of the ballast layer itself a convex form, so that the ends of the sleepers remain wholly or partly uncovered. This measure seems to us very rational and deserving of application in many places in Russia, where the fine sand used as ballast is most often not covered with coarser material. Such an external outline does not permit such a disturbance of the form of the ballast layer during rains, gives the water falling on the ballast an opportunity in large part to roll off its surface, and contributes to the speedier drying of the sleepers. Therefore it would seem that, in the event such a profile is adopted, it is proper to increase the value of the coefficient $C_0$ by 1–2.

Water permeability and uniformity in the coarseness of ballast particles.

§ 12. On that section of track which is considered with respect to its suitability for one or another kind of train, the ballast everywhere should be of one and the same quality; otherwise it will have to be assumed that its quality along the entire length of the section is the same as in the places with the poorest ballast.

One of the chief requirements imposed upon ballast is its water permeability, which depends on sufficient coarseness and uniformity of the grains and on the absence of any significant admixture of clay. The role of good ballast consists in not allowing the roadbed to become soaked, in not delaying the drying of the sleepers, and in itself being wet for as short a time as possible. It has already been indicated above that wet ballast has less cohesion,

consequently, it offers poorer resistance to lateral displacements, and even its vertical settlement is greater. It is very difficult for the track to become wet through, but if the ballast retains moisture over long periods, under its influence the track becomes less firm, and the resistance of the soil composing it is considerably reduced; consequently, the appearance of ballast pockets becomes more possible, as do uneven settlement of sleepers, the low strength of this settlement, etc. There are ballasts which, after drying, remain as loose as they were before being moistened, and there are those which, after drying, give a more or less dense or even friable mass. This is observed in ballasts containing much clay and, in general, fine particles. If the ballast is nonuniform in grain size, and, after its being washed through, similar lumps form in the spaces between its larger parts, this contributes to the retention of moisture in the bed. In general it has been established that water permeability is inversely proportional to the content of clay in the ballast, and also increases with the size and, chiefly, the uniformity of the grains composing it. If the admixture of clay in the ballast is very considerable (for example, 20% and more), then water is retained in it and the clay swells, and in winter heaves. From experiments carried out in 1906 in the mechanical laboratory of the Tomsk Technological Institute on samples from 14 ballast quarries¹, it appears that those ballasts are worse and less elastic which are less homogeneous in grain size. Considering that the principal component parts of any sandy ballast material are shingle, gravel, and sand, we recognize that for those ballasts which consist chiefly of shingle (or crushed stone) and in which the admixture of gravel and sand is less than 10%, to the value of the coefficient $C_0$ there should be added, say, 5, reducing this figure by 1 for each increase of the admixture by 10 percent. For ballasts consisting mainly of gravel (we count as gravel—beginning with a grain size of 5 cm and ending with 1 mm; and sand, from one mm to 0.2 mm), we add 1 to the value of $C_0$ in the case where the predominant element enters into the mixture in an amount of not less than 80%, and one more unit in the case where the average size of the sand, taken separately, is less than $1/2$ mm. With a considerable admixture of clay, $C_0$ should be reduced the more, the higher the clay content; for example, with a content up to 10%, by 1; up to

15% by 2, up to 17% by 3, with 19% by 4, with 20% by 5, etc. If the ballast is of fairly uniform coarseness and does not contain a large admixture of clay, the absolute magnitude of the coarseness itself does not play a large role, so that it does not seem necessary to increase the value of \(C_0\) with the average grain size. The diameter of the grain will be reflected indirectly in the indicated coefficient, i.e., water permeability, cohesion, and friction in the ballast are increased. If the ballast is of crushed stone or of one large pebble, then we have agreed to increase \(C_0\), but it must be noted that the coarseness of the crushed stone for ballast should not be too great—no more than \(2—2\frac{1}{2}''\). In the case of excessively coarse, angular crushed-stone ballast, the value of \(C_0\) should be reduced by 1 or 2. We shall dwell in more detail on the most appropriate coarseness of crushed stone in the chapter on ballasts on foreign railways.

Stability of ballast with respect to horizontal forces. § 13. Let us dwell on that aspect of the work of the ballast layer to which, ordinarily, little attention is paid—namely, on its resistance to lateral forces acting on the rails, and through them on the whole superstructure.

It is indisputable that a ballasted track becomes deranged less than one laid on the roadbed without ballast. The deformations of the ballast layer, and of the track in general, in the vertical direction are the result of the static action of loads. It seems to us that settlements increase very little with increasing speed of movement along the track of the same loads. With regard to deformations in the horizontal direction, the opposite is true: they increase appreciably with increasing speeds of movement. Therefore, in order to decide questions concerning the admission of certain or other speeds, it is very important to investigate the resistance of ballasts to horizontal forces.

The lateral resistance of an unloaded ballast layer is small. According to experiments carried out on French railways, in order to shift the track sideways by several centimeters, a force of 7 tons is required if the track is ballasted but not loaded by rolling stock. From the experiments carried out by Weber¹, it appears that with a lateral force of 1.5–2.5 tons the bond between sleeper and ballast is already disturbed. The width of the ballast layer has little significance for its stability. It is important that the ballast possess as much friction as possible; in this case, of course, more significant

dimensions of the sleepers, and also the strength of the material composing the ballast, and its resistance to atmospheric agencies. In order to displace a track loaded by rolling stock, a very great force is needed; it is estimated at approximately 20 tons, while the lateral action of a wheel upon the rails is estimated at \(1/3\) of its pressure, i.e., 2.5 tons¹. Consequently, the stability of the track in a direction perpendicular to its axis appears to be, as it were, assured. But the fact is that the lateral movement of a locomotive somewhat carries along even the parts of the track lying in front of its first axle. It is clear that the transition from a state in which the track can be displaced sideways by a force of only some 2 tons, to a state in which, being pressed down by the wheels, it appears very stable against displacement, cannot take place at once; and at this moment, within the limits of the ballast, rather considerable lateral forces already develop. Therefore all measures tending to increase the resistance of the track to lateral displacement are very useful. First of all, one should endeavor to obtain a ballast possessing great cohesion and friction against the surface of the sleepers; the benefit from increasing the width of the ballast layer (which entails a widening of the roadbed) is not justified by the expense. To increase lateral resistance, it may be useful to heap piles of ballast against the ends of the sleepers, but experiments are needed on this subject. It is very essential to increase the area over which the ends of the sleepers act upon the ballast. To this end, on some French railways transverse planks are used, fastened at the ends of the sleepers. The method of fastening is shown in Drawings 1 and 2. This measure increases the resistance of an unloaded track to lateral displacement by almost 40%. Instead of planks, stakes are sometimes driven in at the ends of the sleepers, preventing the sleepers from moving. On double track it is sufficient to connect, longitudinally, by pieces cut from old sleepers, the ends of the opposite cross-ties. The immobility of the track is also increased by connecting it in the direction of the track with the aid of plates or boards—sleepers lying on both sides of a joint, or a row of intermediate sleepers. The latter measure is more often used to counteract longitudinal displacement of the track, or so-called creep. Some see an advantage in arranging the track on longitudinal sleepers, as compared with a track on cross-ties, in that with longitudinal sleepers the possibility of displacement of the track both in the lateral direction and in the longitudinal direction is considerably—

…is considerably reduced, especially if the beds are connected at certain intervals by cross ties. Such a combination of both types of track-laying in the form of beds fastened together by ordinary sleepers is in great use on the railways of southern Austria.

Let us dwell somewhat longer on the question of the creeping of rails and sleepers, since its magnitude characterizes the resistance of the track to horizontal forces, and all measures against creeping generally reduce the disturbance of the track and, consequently, increase its stability and strength. The longitudinal creeping of the superstructure always begins with the fact that the rails, under the action of adhesion with the wheels of the rolling stock, braking, and impacts, slide over their bearing plates until the edges of the oval holes in the rail ends abut against the joint bolts connected with the fishplates, which with their aprons hook behind the joint sleepers. With further movement the joint sleepers are carried along, since the longitudinal force is capable of overcoming the resistance of the ballast surrounding the joint sleepers. Sometimes these sleepers do not shift, but only incline; this can be explained above all by insufficient tamping of them, or else by the particular qualities of the ballast: friction along the entire lower bed of the sleepers offers a very considerable resistance to longitudinal displacement, while this ballast is compressed comparatively easily. In winter, with frozen ballast, creeping is observed in very small dimensions; in spring, when the ballast thaws, and in summer, after heavy and prolonged rains, creeping appears especially sharply. Under the influence of even slight creeping, the gaps at the rail joints become very unequal, the joints deviate from the position of a rectangle, and during the passage of rolling stock strong oblique jolts are felt, causing considerable blows at the joints. The result of such blows is a comparatively rapid wear of the joint sleepers and of the sleepers adjacent to them, and a disturbance of the track. Creeping is stronger on double-track lines, on long gradients, at approaches to stations, where braking takes place, and so forth. The simplest measure for increasing the resistance to creeping is to involve in this resistance as large a number as possible of intermediate sleepers by means of special angle plates, clamped to the rail and fastened to the sleepers, or by arranging ties between the intermediate and the joint sleepers. The track is made as immovable as possible by driving stakes around the sleepers, which connect the superstructure directly with the roadbed. One of the very good measures counteracting creeping is the laying of joint sleepers on fine cobblestone crushed stone with…

tamping of them. The size of the broken stone should be from \(1/2\) to \(1\,1/2''\). If, in general, good tamping of the joint sleepers brings great benefit with respect to displacement of the track, then the benefit from using broken stone for tamping the sleepers is especially felt on sections where creep can manifest itself to a strong degree. Joint sleepers well tamped on broken stone resist the forces of creep so stubbornly that even under exceptional conditions (heavy braking) neither displacement by creep of sleepers tamped with broken stone, nor their turning, is observed. In general, the holding of joint sleepers, as being weaker with respect to their position, on broken-stone ballast, if the whole track is on sand ballast, must be recognized as very useful with respect to the stability and better preservation of the permanent way.

In view of what has been said, it is thought that, if special measures are taken against track creep, from 1 to 5 should be added to the value \(C_0\). In the majority of derailment cases, where no collision, fracture of part of the rolling stock, malicious act, broken rail, or defect in maintenance of the permanent way established by examination of the site of the accident has occurred, the cause is characterized by a general nonconformity of the speed developed by the train at that place with the degree of stability of the track. Most often, near the place of the accident, where the locomotive ran shortly before leaving the track, one observes successive widenings of the gauge between the rails, and these go in a wave-like manner now toward one line of rails, now toward the other, gradually increasing. This is explained by the sinuous motion of the locomotive, which, when the complete stability of the permanent way is disturbed, is capable of developing to such a degree that it displaces the track by considerable amounts. Probably the whole track is then in waviness, and at the places coinciding with the crests of waves lying in the horizontal plane, the pressure of the wheels causes the rails to turn outward. Does not the beginning of such a disturbance of the track, which ultimately leads to derailment, lie in the fact that the friction between the ballast and the sleepers becomes insufficient and the whole ballast is easily set in motion sideways? In view of this, the data of experiments on the resistance of the permanent way to lateral displacement are very interesting.

It may be established approximately that an unballasted track, i.e. if the sleepers are laid directly on the roadbed and in the absence of a train, begins to shift under a lateral force of \(1/2\) ton, and this displacement reaches considerable deflections (6–8 cm) under a force of 2 tons; if the sleepers are laid in ballast but are not covered with it, then the forces are respectively equal to 1 and 4 tons; and if the sleepers are covered from above and the ballast layer is wide, then the forces reach 2 and 8 tons.

If the track is loaded by rolling stock, the difference due to the type of track packing becomes very insignificant, and in general it may be said that, with forces of 2 tons, the connection between the sleepers and the ballast is already broken; but for a noticeable displacement of the track to the side (by 2 centimeters), a force of 12 tons must be applied. As regards the influence of the type of ballast, clean gravel does not always prove better, with respect to lateral displacement, than ordinary gravel with an admixture of clay. The advantage of sufficiently fine broken stone is beyond doubt; thus, if with the ordinary type of superstructure the force displacing an unloaded track is equal to 4 tons, then with broken-stone ballast a force of 8 tons is needed; with a loaded track, replacement of the ordinary ballast by broken stone has a somewhat smaller significance, thus increasing the resistance of the track from 12 to 16 tons.

For the absolute stability of the track, figures are important which show what force against lateral displacement the track can withstand when it is unloaded, because the lateral force of the locomotive is transmitted through the rails and the parts lying ahead of the moving locomotive, and also in view of the fact that, owing to the vertical rocking of the locomotive, an unloading of its axles occurs from time to time.

It is very difficult to evaluate each ballast in the sense of its resistance to lateral forces. The coefficients of adhesion and friction of loose bodies are determined with great difficulty and with little accuracy. The angle which characterizes these properties was discussed above. If there are experiments determining the resistance, in tons, to lateral displacement (for a noticeable amount—not less than 2 mm) of an unloaded ballasted track¹, then for every ton of resistance above, for example, 2, one should add one unit to the value \(C_0\). In the absence of such experiments, if the ballast consists of broken stone of medium size with sharp edges and corners, which bite into the lower bed of the sleeper, \(C_0\) should be increased by 2–3; for sandy, sharp-angled ballast add 1; for clayey ballast decrease by 1; if the sleepers are covered by ballast, add 1.

Conditions of strength of the ballast layer. § 14. We have hitherto had in view chiefly the stability of the ballast layer and the stability of the entire superstructure depending upon it; but, since the track works very often and, moreover, very intensively, besides questions of its stability, very important—

Investigation of Ballasts

... is the question of preserving for as long as possible the state that ensures this stability, i.e., the strength of the ballast layer.

The elasticity of the ballast layer is in general imperfect and varies within small limits. According to Engesser, the permissible compressive stress in the ballast from the standpoint of statics is \(p = 2\ \frac{\mathrm{kg}}{\mathrm{cm}^2}\), and the settlement of the layer should not exceed \(2 \tfrac{1}{2}\) mm. In actual practice the unit pressure from the sleeper is about \(3\ \mathrm{kg}/\mathrm{cm}^2\), and settlements of up to 4 mm are obtained. From this it is clear how important it is to place the ballast under conditions favorable to its strength; and, since the stability of the ballast layer is not assured at all points, it is necessary, by means of constant inspection and repair, to maintain the indicated conditions.

Since only well-tamped ballast offers sufficient resistance to the settlement of sleepers, it is necessary to see carefully that all sleepers are always tamped alike and that the tamping corresponds to the quality of the ballast, to the forces acting on the track, and also to the length of the sleepers. The fact is that a sleeper undergoes bending when a train passes, and its elastic line depends on its length; therefore, in order that the pressure from the rolling stock be distributed as uniformly as possible over the whole ballast layer, and so that the sleepers do not bend excessively, the sleeper should not be tamped with equal force along its length. Usually sleepers are tamped more weakly at the middle and at the very ends than under the rails; but the question of the rationality of this method has been little illuminated by strictly scientific experiments. Since the stress in the ballast often exceeds the limits of elasticity, and, with repeated action, residual deformations lead to disturbance of the tamping, current repair, consisting in restoring the state of the ballast under the sleepers imparted by the initial tamping, is the first condition of its strength. If the sleepers in the track are of such dimensions that the tamping under them holds for a long time; if no uneven settlements of sleepers or of their parts are observed, and the sleepers do not sag and do not spring, then one should add to the value of the coefficient \(C_0\) from 1 to 5.

The principal defects that make ballast of little strength are the following:

1) It can retain water; its fine particles turn into mud; when it freezes, the layer heaves; when it thaws, the tamping in it is disturbed. If the content of clayey impurities exceeds 15%, ballast should be considered of little suitability, as one that poorly lets water through and therefore dries poorly. With such ballast the surface of the roadbed may be washed away, and the resistance of the ballast layer to lateral displacement is considerably reduced. In order that a clayey roadbed not be so easily washed away, it is good, when the qualities of the ballast are poor, ...

to place beneath it a layer of clean sand, crushed stone, or slag. For better drainage of water from the roadbed and for greater strength of the ballast layer, its bottom is sometimes made of a row of stones. In cases where it is necessary to ensure that water cannot penetrate into the earthen roadbed, its surface—which must always be made convex—is covered with a layer of clay, which should be tamped or compacted by passes of a roller. Crushed-stone ballast retains water least of all and is almost not disturbed by the penetration of water; that is why it requires the least labor for renewing the tamping and the least material for replacing the parts carried away and worn out. The quality of the ballast also has an essential influence on the rapidity with which sleepers are destroyed by rotting. The most unfavorable in this respect is ballast which, by preventing the penetration of air, retains dampness. Such is fine sand. Gravel and crushed stone, if they are clean, dry out very quickly, and therefore the sleepers in them for the most part lie dry and are well preserved. If the admixture of clay in sand is very considerable, then water is retained in it, the clay swells, and air has almost no access; for this reason, in loamy and clayey sand, sleepers are also well preserved.

If the ballast consists of fine material or contains many fine particles, so that water passes through it rather slowly, it is very useful to give the profile such a form that a significant part of the water falling on the roadbed will, as far as possible, run off quickly over the outer surface of the ballast. In America a convex profile is very often used (Fig. 4). In this case the ends of the sleepers are partly open; the poor permeability to water of most Russian ballasts would require as wide a trial as possible of this measure. It has not yet been precisely established whether sleepers are preserved better or worse if their ends are not covered by a berm of sand; 2) depending on the shape and size of the particles, the hardness of the material, and other properties, ballast may be mobile and roll easily. The resistance of the track to horizontal displacement with such ballast is insignificant. This occurs with ballast of rounded form, sea or river sand. It is useful to add crushed stone or broken pebbles to such sand. Ordinary pit gravel, mixed with coarse-grained sand or small pebbles, gives, in the sense of immobility, good ballast, if it does not contain too much clay. Clean sand and gravel are much more mobile than crushed stone and broken rock, and a small admixture of clay, especially in gravel, proves useful. The comparative immobility of ballast may be characterized by the average

the size of its grains; but it would be more correct to continue the experiments for studying the magnitude of adhesion in the various kinds of ballast. In general, it may be said that coarser ballast is less mobile, just as it is known that fine sand is more easily beaten out from beneath the sleepers than coarse sand. 3) Fine ballast, unprotected on the outside, may be washed out, especially if water drainage is not provided, and may also be blown away by the wind. Russia is poor in good, coarse ballasts. The transport everywhere of high-quality ballast from the best quarries, scattered over considerable distances, could be organized, but would be too expensive; therefore, up to the present time, on the network of Russian railways sand is used as ballast, sometimes even of the most mediocre quality. Where possible, it is covered with a coarser material, for example shingle or a layer of crushed stone; sometimes such ballast is allowed to become somewhat overgrown with grass. Light watering of the ballast with petroleum or creosote residues is successfully used against blowing on American and French railways. If the ballast is too fine, then, besides the need to replenish it often, it presents a serious inconvenience by the large quantity of dust that it produces. Apart from the undesirability of dust for passengers and goods, it entails wear of the machinery, joints, rail heads, and all the rolling stock.

4) The presence in the composition of the ballast of a large amount of lime and, in general, of soluble parts is undesirable, because it makes it less durable, and consequently less strong with respect to external forces and the action of frost and water; and, when iron sleepers are used, often quite unsuitable, because these sleepers require a strong and coarse ballast material; otherwise the latter is soon crushed and abraded, and the track becomes deranged. But even with wooden sleepers and with movement that is not especially heavy, the ballast is somewhat worn and requires replenishment and replacement of the part becoming unfit; this is less, the stronger the ballast material.

As we see from what has been set forth, the same qualities that ensure the stability of the ballast layer also contribute to its durability. Materials that are stronger, coarser, and possess greater friction of particles are also less subject to wear, washing out, and dispersion into dust. Consequently, it may be accepted that the percentage of annual replenishment of ballast in the track is the greater, the smaller the sum of the qualities ensuring the strength of the ballast. Taking as normal a loss of 5% of volume per year, one may stipulate, for those ballasts which require a larger quantity for annual repair, that the coefficient \(C_0\) be reduced by the number of percent above the indicated 5. Abroad they consider

the standard of annual ballast replacement is approximately 5%. In Russia the estimated standard for replenishing the ballast layer is also close to this figure (5–6 cubic sazhen per verst of track); in fact, on those roads where there is no upper crushed-stone ballast, 10–15% is used for replenishment (on the Siberian Railway up to 15 cubic sazhen per verst), since a large quantity of ballast is washed away by rains.

As we see, the material for ballast, apart from yielding a sufficient coefficient of the roadbed, must satisfy, to a greater or lesser degree, a great many conditions; therefore, in order to judge ballast correctly, it is necessary each time to study it comprehensively from the standpoint of its physical and mechanical properties. For general judgments it is accepted that the principal conditions of stability and strength are satisfied by the following materials for ballast (see the Technical Conditions for Track Laying in the Construction of the Siberian Railway):

a) quartz gravel with crushed stone from sandstone, syenite, granite (crushed stone from granite containing much feldspar, when comminuted, gives much clay, for which reason such crushed stone must not be used), limestone and basalt stones in a proportion of approximately 2:5;

b) grus;

c) clean non-clayey sand, sufficiently coarse, with grus or shingle.

The upper ballast must be of coarse, if possible pea-shaped, gravel or of crushed stone from \(3/8''\) to \(2''\) in diameter. The lower ballast (poured under the sleepers) consists of finer crushed stone and gravel, or else of coarse sand. Sometimes a layer of dry masonry is laid at the base of the ballast.

III. The Question of Ballast in Connection with the Properties of the Other Parts of the Track

Determination of the vertical pressure of a locomotive axle on a sleeper. § 15. The theory of stresses in the parts of the permanent way has been developed sufficiently coherently, but nevertheless it does not answer all questions relating to the dimensions and properties of these parts, especially of the ballast layer. This is explained, first, by the circumstance that the design of the rolling stock plays a major role in the magnitude of the stresses in the permanent way, and also that physical and atmospheric phenomena, which are difficult to take into account, enter into play.

Safety of movement on the one hand, and the smallest expenses for operation of the line on the other, are possible only under the condition of a sufficient reserve of resistance of the track in comparison with the produced ...

on it under the greatest pressure. Investigations are needed of the resistance of the sleepers and ballast, which successively transmit to the roadbed the pressures from moving loads; it is necessary to determine the role of each of these elements and the changes that can occur in them. Only a complete theory of the operation of all parts of the superstructure, guided by the results of numerous experiments and observations on the life of the track, can make it possible correctly to assess one or another measure adopted by railways in order to satisfy the requirements of their modern operation.

Since the conditions of stability of the ballast layer can be represented analytically—that is, introduced into formulas—only together with the conditions of stability and strength of the other parts of the superstructure, and principally the rails, then, having in view, in a more complete form than has been done hitherto, the introduction into formulas concerning the superstructure of the track of the dependence on the qualities of the ballast, we are compelled to dwell on the existing formulas for calculating the strength of rails and the magnitudes of the deflections of the superstructure under the action of wheels.

First of all it is necessary to investigate the vertical pressure produced by the leading wheel of a locomotive. The matter is that, during the motion of a locomotive, the pressure of the axles does not remain unchanged, but, partly owing to the relative motion of the parts of the locomotive with respect to one another, and partly from the action of the track, which changes its form under the influence of motion, there results now an increase, now a decrease of the axle pressure as compared with the static one. Zimmermann found that, for continuous beams lying on elastic supports, a given load produces at the point of its application the greatest bending moment in the case when, at this point, the influence of the neighboring load is not reflected. This takes place when the leading wheel of the locomotive produces full pressure, while the following wheel (as sometimes happens) is temporarily unloaded. The pressure of a wheel consists not only of its weight in the static state, but also of additional vertical forces arising: 1) from the oscillation of the springs during the general rocking of the locomotive while in motion, which is especially intensified by defects in the construction both of the track and of the locomotive; according to Brière’s experiments (very incomplete), it is assumed that the influence of spring oscillation alone is expressed by an increase of the pressure to 1.63 of the static value and by a decrease to 0.38; 2) from the rolling of the wheel not on a straight but on a bent rail; according to Winkler’s approximate calculations, based on Phillips’s experiments, the increase of pressure by the vertical component of the centrifugal force resulting from motion along a straight rail, for speeds of 36 kilometers ...

per hour 54, and 72—correspond to 5.12 and 22% (it was stated above that Winkler incorrectly attached such great importance to the deflection of the rail between the sleepers); 3) to the action of the wheel counterweights, arranged to reduce the rocking of the locomotive; this is estimated at approximately 50% of the static pressure; with somewhat unevenly worn tires it may be considerably greater. Since well-selected counterweights only balance, during the motion of a given locomotive at its average characteristic speed, the additional influences of the pins and cranks, it would seem that no great additional forces should be expected from their presence.

Thus the dynamic action of the wheel is customarily considered to be approximately \(2 \tfrac{1}{4}\) times greater than its static pressure¹. It is impossible to establish theoretically the coefficient characterizing the increase in pressure, and one has to be content with the indications of very incomplete experiments. This coefficient varies depending on the type of rolling stock and the manner in which it is maintained, on the weight and speed of trains, and on the condition of the track and the peculiarities of soil conditions. It would be very valuable to establish for each railway the value of the indicated coefficient for the greatest speed and for the various types of locomotives running on it.

When a wheel moves, the rails undergo a wave-like motion in the vertical plane, in which the lowest point constantly coincides with the point of application of the load; at this point the greatest bending moment is developed. The sleepers undergo vertical oscillation, i.e., they either sink below the level corresponding to the state of rest, or rise. The deflection of the rail between the sleepers, depending on the distribution of the load over the supports, differs from its deflection over the sleepers by a fraction of a millimeter; therefore, despite the vertical oscillation of the sleepers and of the transverse sections of the rails, the wheels must move along a line lying below the normal level of the rails, but almost parallel to that level. Such a phenomenon would take place with an unchanging wheel pressure, but in reality, over and above this pressure, considerable vertical components of the centrifugal forces developed by the counterweights are obtained. These vertical components are taken, as already indicated, as 50% and more of the static pressure of the wheels, and, moreover, depending on the position of the counterweights

when the wheel rotates, either increase or decrease the pressure of the wheels. The waves produced are the longer and smoother, the larger the diameter of the wheels fitted with counterweights; they are the steeper, the greater the speed, since with speed the vertical components of the forces produced by the counterweights increase. On the other hand, at a very high speed of motion, even an insufficiently rigid track loses part of the amplitude of its oscillations, since a certain time is needed for the rail to bend, while the same point of the rail, soon after it should have risen, must already begin to descend. To ensure greater safety of motion and the least possible wear of the rolling stock and of the parts of the track, it is necessary, on the one hand, to take measures to improve the construction of locomotives, so that their motion may be as smooth and correct as possible, and, on the other hand, to increase the rigidity of the track, since under this condition many additional forces cease to affect the successive changes in the pressure of the wheels.

The influence of counterweights on the running of a locomotive is illustrated by experiments carried out in 1893 at Purdue University (State of Indiana) with a real locomotive installed there for experimental purposes. Under the locomotive wheel, one after another, pieces of wire were brought (drawn by the rotation of the wheel along a roller replacing the rail), which were flattened into a strip, the maximum pressure of the counterweight intensifying the flattening. Even before the speed had reached 60 kilometers per hour, momentary lifts of the locomotive wheel above the roller began to be detected at each revolution, as was proved by the appearance of unflattened places on the wire, the moments of lifting coinciding with the upward motion of the counterweight. When the speed was increased to 90 miles per hour, the length of the part of the wire that retained a circular section corresponded approximately to \(55^\circ\) of the circumference of the wheel. The downward motion of the wheel took place much more rapidly than the upward motion, which is quite understandable, for in the latter case the wheel has to overcome, besides its own weight, also the elastic action of the spring, whereas when the wheel falls downward both of these contributed to the acceleration of the motion.

It should be regretted that the experiment described was made in a crude form and that the work of the wheel at the different moments of its rotation was not calculated. The result for a rigid rail with a large cross-section might have proved somewhat different from that for the wire that was subjected to the test. At the same time it should be noted that the influence on the track of all the wheels of a locomotive and of the springs resting on them, connected by the common frame and by the weight of the boiler, is to a considerable degree more uniform,

the effect of an individual wheel when the base on which it acts is insufficiently solid and rigid; a good arrangement of counterweights is of very great importance, and in considering the strength of the track one must always bear in mind that improvements in the construction of locomotives contribute to it to a high degree. Cases of damage to the track due to improperly balanced wheels occur often, but are not always sufficiently clarified. The Railroad Gazette, in which the above-mentioned experiments were described,¹ gives an example in which a freight, cold locomotive with its connecting rods removed, following with a train moving at a very high speed, cut up the track over a considerable distance, as a result of which an order was issued that in future locomotives with connecting rods removed should move at a speed of not more than 15 versts per hour. In many cases the dynamic action of the counterweights of locomotive wheels has been revealed quite clearly by vertical and horizontal distortions of the rails, recurring at equal intervals of distance which corresponded exactly to the circumference of the wheels.

One way or another, once we have agreed to regard the ratio of the dynamic action to the static action as known, we determine more accurately the stresses in the material of the rails which it undergoes when the locomotive wheels are placed in the most unfavorable manner with respect to the sleepers. We shall now see that even the calculation of the stress in only one part of the superstructure (the rails) under the static action of loads presents great difficulties.

Formulas for determining the settlement of sleepers and introduction into them of the coefficient \(C_b\). § 16. Let us suppose that the wheel is located above a sleeper and exerts upon it a pressure \(G\). The sleepers resist this pressure, but not enough to avoid settling altogether; consequently this resistance will be expressed by a quantity \(P'\) smaller than \(G\). The force \(G - P' = P\) produces bending of the rail, the deflection arrow \(y\) being equal to the difference between the amount of compression of the ballast under the indicated sleeper, \(y'\) (corresponding to the force \(P'\)), and the amount of compression \(y_0\) (Fig. 5) of the ballast under the neighboring sleepers (corresponding to the force \(P_0\) acting on the sleeper adjacent to the one under consideration). According to the general theory of bending of a freely supported beam,

\[ y' - y_0 = \frac{(G - P')(2a)^3}{48EJ}, \]

where \(a\) is the distance between sleepers. Denoting by \(B\) the quantity

\[ \frac{48EJ}{(2a)^3} = \frac{6EJ}{a^3}, \]

which represents nothing

—that is, as the force necessary for lowering the sleeper by one unit, we obtain

\[ y' - y_0 = \frac{G - P'}{B}. \tag{1} \]

The settlements \(y'\) and \(y_0\) depend on \(P'\), \(P_0\), and also on the dimensions of the bearing surface of the sleeper (\(b\) the width and \(l\) the half-length) and on the coefficient of the bed; namely,

\[ y'=\frac{P'}{Cbl} \quad \text{and} \quad y_0=\frac{P_0}{Cbl}. \]

The expression \(Cbl\) (the force necessary for pressing a half-sleeper into the ballast by 1 centimeter) is denoted by the single letter \(D\). It is preferable to introduce into this expression one more coefficient, depending on the elasticity and dimensions of the sleeper, since the sleeper itself is deformed¹, and then

\[ y'=\frac{P'}{Cmbl}=\frac{P'}{D} \quad \text{and} \quad y_0=\frac{P_0}{Cmbl}=\frac{P_0}{D}. \]

Consequently

\[ y' - y_0 = \frac{P' - P_0}{D}, \]

and since

\[ P' + 2P_0 = G \]

(when the pressure is transmitted to three sleepers), then

\[ y' - y_0 = \frac{G - 3P_0}{D}. \tag{2} \]

Comparing equations (1) and (2), we have

\[ \frac{G-P'}{B} = \frac{G-3P_0}{D}. \]

Hence, with the aid of \(P' + 2P_0 = G\), we find:

\[ P'=\frac{B+2D}{3B+2D}\,G;\qquad P_0=\frac{B}{3B+2D}\,G, \]

\[ y_0=\frac{P_0}{D} =\frac{B}{3B+2D}\cdot\frac{G}{D}, \qquad y'=\frac{P'}{D} =\frac{B+2D}{3B+2D}\cdot\frac{G}{D}. \]

Denoting \(\frac{B}{D}\) by the letter \(\alpha\), we obtain the expressions:

\[ P'=\frac{\alpha+2}{3\alpha+2}\,G,\qquad P_0=\frac{\alpha}{3\alpha+2}\,G; \tag{3} \]

it is also possible to write

\[ y'=\frac{\alpha+2}{3\alpha+2}\,G/D. \]

The values of the bending moments in the rail are the following:

\[ M_0=0;\qquad M'=P_0 a=\frac{B}{3B+2D}\,Ga, \]

or

\(M'=\dfrac{\alpha}{3\alpha+2}\cdot Ga\). There are similar formulae for other positions of the loads relative to the sleepers.

In reality the value of \(\alpha\) varies between 0.5 and 4; for these values the pressure \(P'\) is most often obtained as about \(0.5G\), but reaches as much as \(0.7G\).

The expression (3) indicated above for the maximum pressure on a sleeper, known as Schwedler’s formula and derived on the assumption that the pressure is transmitted to only 3 supports, can be obtained from the general expression of the so-called Clapeyron equation for the elastic line of a beam on many movable supports; moreover, one can trace how far it is more correct to assume the distribution of the load over the supports.

The general equation of the elastic line is

\[ \pm EJ\frac{d^2y}{dx^2}=M. \]

Considering two adjacent spans, each loaded with one load and a uniformly distributed load, we write, for each span separately up to the point of application of the concentrated load and beyond this point, expressions for the moments and integrate the equation twice (Fig. 6).

The constants obtained in the integration are found from comparison of the results obtained by integration for the common points of the considered parts of the beam and at the end of the ends; we obtain¹:

\[ 6EJ_0\left(\frac{y'_1-y'_0}{l_0}+\frac{y'_1-y'_0}{l_1}\right) = -\left[ M'_0l_0+2M_1\left(l_0+l_1\frac{J_0}{J_1}\right)+M'_2l_1\frac{J_0}{J_1} \right] +\frac{P_0a'_0(l_0^2-a'_0{}^2)}{l_0} + \frac{P_1a'_1(l_1^2-a'_1{}^2)}{l_1}\cdot\frac{J_0}{J_1} +\frac{q'_0l_0^3}{4} +\frac{q'_1l_1^3}{4}\cdot\frac{J_0}{J_1}. \]

Under the action of a system of loads in each span, the following is obtained analogously:

\[ 6EJ_0\left(\frac{y_1-y_0}{l_0}+\frac{y_1-y_2}{l_1}\right) = -\left[ M_0l_0+2M_1\left(l_0+l_1\frac{J_0}{J_1}\right)+M_2l_1\frac{J_0}{J_1} \right] +\sum \frac{P_0a_0(l_0^2-a_0^2)}{l_0} + \sum \frac{P_1a_1(l_1^2-a_1^2)}{l_1}\cdot\frac{J_0}{J_1} +\frac{q_0l_0^3}{4} +\frac{q_1l_1^3}{4}\cdot\frac{J_0}{J_1}. \]

The minus sign before the brackets may be omitted, only then it is necessary to regard as positive the resisting moments of the supports directed upward, and as negative the moments of the forces directed downward.

When the spans are equal and in each there acts only one concentrated load without a uniformly distributed load, i.e. \(l_0=l_1\), \(J_0=J_1\), and \(q_0=q_1=0\), then there will be:

\[ \frac{6EJ_0}{l_0}\cdot(2y_1-y_0-y_2) = (M_0+4M_1+M_2)l_0 + \frac{P_0a_0(l_0^2-a_0^2)}{l_0} + \frac{P_1(l_0^2-a_1^2)a_1}{l_0} \tag{4} \]

Using these equations for each pair of adjacent spans, we can find the relative settlements of the supports \(y_1-y_0\) and \(y_1-y_2\). Assuming the ballast to be elastic, we take the bearing resistances of the sleepers (or the pressures upon them) to be proportional to the indicated settlements. Denoting the resistance of a support by \(r\) and the coefficient of proportionality by \(k\), we have \(y=kr\), or

\[ y=\frac{1}{D}\,r, \]

where \(D\) is the load that must be applied to a sleeper of length \(a\) and width \(b\), at each of the two rails, in order that it sink into the ballast by 1 cm. If the bending of the sleeper itself is neglected, then

\[ D=\frac{Cab}{2}, \]

where \(C\) is the coefficient of the bed, expressing the magnitude of the load per unit area of ballast capable of compressing it by 1 cm.¹ From the use of the approximate formula for \(D\) instead of the exact one, the error will not exceed 5%; therefore, in practice it is possible to assume that the sleeper does not bend. In studying the influence of external forces on the permanent way, it is important to know the pressure on the sleepers (their bearing resistances \(r\)) and the maximum moments acting in the rail. It would seem that, knowing the greatest bearing resistances, we could determine the maximum moments by a simple calculation. But the point is that the greatest pressures on the sleepers will occur when the load is located over a sleeper, while the greatest moments in the rail will occur when the load is located at the middle between sleepers. Therefore, to find the former, it is necessary to assume that a load standing over a sleeper distributes its action among three, five, etc. sleepers; and to find the latter, it is necessary to assume the load at midspan and the number of sleepers as 4, 6, etc.

With a load over the middle one of three sleepers we have (Fig. 7) \(M_0=M_2=0\), and equation (4) gives

\[ \frac{6EJ_0}{l_0^2}(2y_1-2y_0)=+M_1. \]

Putting \(y_1-y_0=k(r_1-r_0)\) and, as here, \(M_1=v_0l_0\) and \(r_0=\dfrac{P-r_1}{2}\), we obtain

\[ \frac{6EJ_0}{l_0^3}\,k\left(r_1-\frac{P-r_1}{2}\right)=P-r_1. \]

If we put

\[ \frac{6EJ_0}{l_0^3}=\frac{1}{\mu}, \]

then

\[ \frac{k}{\mu}\left(r_1-\frac{P-r_1}{2}\right)=P-r_1, \]

whence

\[ r_1=P\cdot \frac{2+\dfrac{k}{\mu}}{2+3\dfrac{k}{\mu}} = \frac{2+\alpha}{2+3\alpha}\,P \quad \text{(Schwedler’s formula).} \tag{5} \]

Hoffmann analyzed the case in which the forces of three loads are applied to three sleepers through one intermediate sleeper. In this case the bearing resistance obtained is greater than by Schwedler’s formula, namely

\[ r_1=\frac{4\alpha+1}{8\alpha+1}\,P \quad \text{(Fig. 8).} \]

The notation

\[ \frac{1}{\mu}=\frac{6EJ_0}{l_0^3} \]

may be called by a single letter \(B\), which represents the force producing, at the middle of a freely supported beam of length \(2l_0\), a deflection arrow equal to unity. Indeed,

\[ f=1=\frac{8Bl^3}{48EJ_0}, \]

so that

\[ B=\frac{6EJ_0}{l_0^3}. \]

\[ D=\frac{Cb^4}{\eta s}\sqrt{\frac{4E'J'}{Cb}}, \]

where \(E'\) and \(J'\) refer to the sleeper, and \(\eta s\) is a number expressing the dependence between the length of the sleeper and the width of the track, and is given in special tables.

The ratio \(\dfrac{B}{D}=\dfrac{k}{\mu}\), which is denoted by the single letter \(\alpha\), for the types of rails and sleepers existing in Russia varies approximately from 0.3 to 3.

It must be considered whether, in all cases, one may make the assumption, in seeking the greatest pressure on a sleeper, that the load is distributed among three cross-ties. If we take 5 sleepers (Fig. 9), then the pressure will be:

\[ r_0=\frac{-3\alpha+\alpha^2}{7+34\alpha+5\alpha^2},\qquad r_1=\frac{11\alpha+\alpha^2}{7+34\alpha+5\alpha^2} \]

and the greatest

\[ r_2=\frac{7+18\alpha+\alpha^2}{7+34\alpha+5\alpha^2}. \tag{6} \]

From the first equality we see that \(r_0\), for values \(\alpha>3\), assumes positive values, i.e. the outermost cross-ties do not compress the ballast layer so long as \(\alpha<3\), and, consequently, up to this limit Shwedler’s formula is applicable; but when \(\alpha\) exceeds 3, for the greatest pressure of the cross-tie one must use formula (6).

In determining the stress in the rail from static pressure, Zimmermann’s formula is usually used:

\[ \max M=\frac{8\alpha+7}{16\alpha+40}Pl \]

It is obtained in the same way as the preceding one, if we assume 4 supports and the load in the middle. Formula (4) becomes

\[ \frac{6EJ_0}{l_0^2}(2y_1-y_0-y_1)=4M_1+M_2+\frac{3}{8}Pl_0. \]

From drawing 10 we have: \(M_1=r_0l_0;\quad M_2=\left(2r_0+r_1+\dfrac{P}{2}\right)l_0\). Substituting, instead of \((y_1-y_0)\), the equal quantity \(k(r_1-r_0)\), we obtain

\[ \frac{k}{\mu}(r_1-r_0)=6r_0+r_1+\frac{3}{8}P; \]

knowing that \(r_0+r_1=\dfrac{P}{2}\), we have

\[ r_0=\frac{4\alpha-3}{16\alpha+40}P; \]

\(r_0\) becomes positive for \(\alpha>3/4\); further,

\[ r_1=\frac{4\alpha+23}{16\alpha+40}P; \]

\[ \max M=\left(\frac{3}{2}r_0+\frac{1}{2}r_1\right)l_0 =\frac{8\alpha+7}{16\alpha+40}Pl_0 \tag{7} \]

For values \(\alpha<3/4\), one must take not 4, but 2 supports; then

\[ \max M=\frac{1}{4}Pl_0. \]

If 6 supports are taken, the corresponding \(r_0\) is obtained positive only for values \(\alpha>8.16\). Consequently, for values of \(\alpha\) intermediate from 0.75 to 8.16, Zimmermann’s formula must be applied (for the commonly encountered values of \(\alpha\), the quantity \(r\) is not greater than \(0.5P\)).

The introduced quantities for the supporting resistances and moments change somewhat if one takes into account the bending of the cross-ties, the influence of the weight of the rails and sleepers, the influence of nearby loads, and also the possibility of a gap existing under the sleeper as a result of poor tamping. On the magnitude of the resistance the first two—

circumstances affect it¹, reducing it by approximately 5%; others increase it by 5%, 20%.

Kholodetsky takes into account the influence of imperfect tamping of the sleepers as a result of accidental causes or of non-uniformity of the ballast, and assumes that the gap \(\eta\) between the sleepers and the ballast reaches 4 mm. The increment of the moment acting in the rail reaches \(\frac{1}{5}\eta Bl\), where \(B=\frac{6EJ_0}{l^3}\) is the load that must be applied to the rail at the middle of a double span (2\(l\)) in order to produce a deflection of 1 cm.

The influence of the bending of the sleepers is associated with an increase of both the bearing resistances and the bending moment by up to 5%.

The weight of the superstructure reduces the static acting moments by from 4 to 15%.

If one considers the influence not of a single load, but of a system of them, it turns out that the presence of neighboring loads reduces the bending moments by 3–14%. The bearing resistances, when the axles are close to one another, are obtained as 2–20% greater than if calculated by Schwedler’s formula,

\[ y=\frac{\frac{k}{\mu}+2}{3\frac{k}{\mu}+2}\cdot P. \]

At the same time it turns out that the maximum pressure is almost independent of \(\frac{k}{\mu}\) (at least within the limits of \(\frac{k}{\mu}\) from \(1/3\) to 2). It follows from this that, with an increase in the weight of the rail, one cannot expect a reduction of the bearing pressures of the rail on the sleepers; and on sections of track with poor ballast, where the tamping of the track is disturbed by the passage of four-axle locomotives, increasing the weight of the rail will not be a rational measure if it is intended to weaken the influence of bad ballast.

For such sections and locomotives, the strength of the track should be increased by other measures, namely: by reducing the distance between sleepers and by increasing the lower bed of the sleepers. Of these two measures preference must be given to the first, since with a reduction of the span not only are the maximum bearing resistances reduced, but also the stresses in the material due to bending of the rails. Of course, this latter consideration, as based on more or less conjectural formulae, is very much in need of confirmation by experiment.

The length of the fishplates has an insignificant influence on the increase of the moments acting on the fishplate; a somewhat greater influence on the reduction of the moments bending the fishplates is exerted by increasing the moment of inertia of the latter. The most substantial influence on the moments acting on the fishplates is exerted by the quality of the ballast, especially for heavy rails. Thus, when the ballast coefficient is reduced by half, the moments bending the fishplates increase by 22%. The moments increase with a reduction of the gap and with an increase of the joint span. With elastic action of the fishplates and absence of a gap between them and the rails, the pressure on the joint sleeper would be less than the intermediate ones; but under conditions closer to practice, the pressure

on the joint sleeper may be somewhat greater than \(P\). The presence of hidden irregularities may increase the moment still further. A rail section weighing, for example, \(24^{1/3}\) lb. is insufficient for the rails of this section to be joined by fishplates capable of resisting bending elastically; and in order to satisfy the latter condition the rail section would have to be increased to a degree all the greater the poorer the quality of the ballast. By improving the ballast layer under the joint sleepers and the sleepers adjacent to them, the stresses in the fishplates can be reduced.

From the foregoing it is evident how difficult it is to solve completely, by theoretical means, the question of the static effect of a load on rails; how closely all parts of the superstructure of the track are connected; and how much one has to imagine in making various simplifications and assumptions. All formulas for determining the effect of external forces on the track have been derived on the assumption that the forces act in the plane of longitudinal symmetry of the rail. In reality, the plane of action of the forces does not coincide with the plane of this symmetry, whereby the question of determining the stresses in the rails becomes more complicated and all formulas must be regarded as approximate (the stress from torsion may be taken approximately as \(5\ \mathrm{kg/mm^2}\), and that from horizontal forces approximately as \(2\ \mathrm{kg/mm^2}\)). At the same time, as we have seen, all reasoning concerning the stresses in the rails and the deflections of their supports is based on: 1) the theory of elastic bending of rails as multi-span beams, and 2) the hypothesis of elasticity of the ballast layer under compression.

The first would hold, at least at the very first instant of the action of the load on the rail, if all the sleepers were equally and well tamped; the second assumption does not correspond to reality, since complete recovery of every settlement of an individual sleeper by the elasticity of the ballast, especially with fine material, cannot be expected.

Regarding the settlement of a sleeper not as it would be if the ballast were ideally elastic and the sleepers ideally tamped and transmitted pressure quite uniformly over the entire area of the base, but desiring to know the greatest possible settlement of the sleepers, taking into account the existing properties of the ballast, we must use, for example, Zimmermann’s formula¹

\(y=\dfrac{16a^2+112a+11}{32a(2a+5)}\cdot\dfrac{P}{D}\),

introducing into it a sharper dependence on the quality of the ballast, the condition of the upper part of the roadbed, and the dimensions of the sleepers, since all this has a very considerable effect on the possible maximum deflections. Although this contradicts the preceding—

position concerning the elasticity of the ballast that was made the basis for deriving the indicated formula; but we shall be closer to the truth if, instead of the coefficient of bedding \(C\), which enters into the quantity \(a\), we introduce the coefficient \(C_0\), which characterizes the condition of the ballast much more closely than \(C\) does¹. If, in considering some particular track, we compute those settlements of the sleepers which, under unfavorable circumstances, may occur under the influence of the pressure of a locomotive wheel, this will give us material for judging the degree of suitability of the given track for the intended loads, and for comparing this track with others.

As regards the stress in the rails, it is not substantially changed by whether the sleepers, in addition to elastic deflections, also acquire permanent ones, depending on defects in the sleepers, the ballast layer, and the trackbed, since the acting moments, and consequently the stresses in the different sections of the rail, depend only on the difference in the settlements of sleepers situated at different distances from the wheel producing the pressure on the rails. Therefore it would seem correct to calculate the static stress in the material of the rails by following the indications of the theory of elastic supports. Consequently, using Zimmermann’s formula

\[ M=\frac{8a+7}{16a+40}\,Pl, \]

one should take as \(C\) the coefficient of bedding computed from observations on the assumption that the ballast is entirely elastic.

N. Petrov takes an entirely correct point of view when he says² that the deflections from the static action of loads at different points of the rail span should be taken from measurements in the track, and not from calculations; but, for lack of observations, he himself uses, in order to find the dynamic deflections and the stress in the rail, those calculations which are obtained from formulas analogous to Zimmermann’s and to those derived by Petrov on the assumption that the wheel pressure is distributed among four sleepers. We shall dwell longer on Petrov’s theory when considering those assumptions and formulas that are used to determine the dynamic action of loads on rails.

Ordinary speeds of motion and experiments with electric cars.

§ 17. Consideration of the influence on the track of a factor that undoubtedly plays a great role in the stability, and especially in the strength, of the track—namely the speed of motion—shall begin with an indication of the speeds at which trains of steam railways move, and with a description of experiments intended to clarify the possibility of applying high speeds.

Statistical data on the density and speed of traffic on the broad-gauge railways of Russia indicate that the average speed of passenger trains over the entire network is 27 versts per hour; moreover, on roads with heavy passenger traffic (not fewer than four pairs per day), i.e., on approximately 15% of the entire network, it is 32 versts per hour; on roads with medium passenger traffic (from 2 to 4 pairs per day), i.e., on approximately 65% of the network, it is 25 versts per hour; and on roads with light passenger traffic (fewer than 2 pairs per day), i.e., on approximately 20% of the network, it reaches 24 versts per hour. For the Finnish railways this speed is 31 versts per hour.

The greatest average speed of passenger trains is on the St. Petersburg–Warsaw line: 39½ versts per hour; next come: St. Petersburg–Moscow 38.3, Warsaw–Frontier, Warsaw–Aleksandrów and Warsaw–Brest 34.9 versts per hour, Vilna–Verzhbolovo 33.8, Moscow–Kursk 38.3, Moscow–Ryazan 32.4, Kiev–Odessa 32.3, Ryazan–Kozlov 32.2, etc.

The fastest train in the world is the “Atlantic-City Express,” running in regular service on the Philadelphia Reading line. According to the timetable it covers a distance of 89.3 kilometers in 49 minutes, which corresponds to an average speed of 109.3 kilometers per hour. On July 21, 1904, this train covered the indicated distance in 43 minutes, which corresponds to a speed of 124.8 kilometers per hour; and if the time for starting from rest, braking, and stopping is not counted, the average running speed is 135 kilometers per hour.

In trial runs on various railways of Europe and America, it has been possible, over short distances, even not on gradients but on horizontal sections, to reach a maximum speed of 144 kilometers per hour.

As regards the magnitude of the highest speed that can be safely developed for the track on individual sections of a line, it appears to depend on the arrangement of the permanent way, as well as on the design of the engines. Theoretical considerations and numerous observations indicate that rails, even when not fully fastened to the sleepers, are held in the proper position by the pressure of the wheels.

passing over it. The principal deformation is the settlement of the rails, depending on the quality of the ballast, and also on the degree of correctness of the tamping, which requires careful maintenance, a sufficient rail section, and again, as high a quality of ballast as possible.

If locomotives had no parts with reciprocating motion, the track would be less disturbed during movement, since the oscillation of the locomotive about the transverse axis (galloping), as well as the lateral rocking, would be much less. In this respect the results of experiments with electric locomotives are very interesting. During trial runs of such a motor, of the firm Siemens and Halske, in 1902,¹ it turned out that, when moving at a speed of 140–160 km per hour, the permanent way (of rails of 33.4 kg per linear meter on sand ballast) received considerable deformations; moreover, at speeds greater than 150 km per hour, short but strong galloping movements occurred in the track itself. From this it may be concluded that, even when ordinary locomotives move at considerable speeds, the alternating, sometimes smaller and sometimes larger, deformations of the track are only intensified owing to the presence of masses with reciprocating motion, while the principal cause of their occurrence lies in poor dressing of the rails. So long as the settlement of the rails with the sleepers is not especially great, the galloping motion is insignificant, the curvature of the rails does not attain large dimensions, and the motion is safe. Observations of rail settlement during these trial runs gave: at a speed of 80 km/h—\(1 \tfrac{1}{2}\)–2 mm; at 108 km/h, settlement of \(2\)–\(2 \tfrac{1}{2}\) mm; at 114 km/h, \(3 \tfrac{1}{2}\)–\(5 \tfrac{1}{2}\) mm; at 145 km/h—6–7 mm. In general, the indicated experiments convinced the participants that on a well-built and well-maintained railway track it is possible to run at twice the speed now being used. On the above-mentioned permanent way, for which the greatest permissible locomotive speed is considered to be up to 80 km per hour, it was possible safely to develop a speed of up to 120 km per hour, provided that the locomotive did not have such reciprocating motions as would strongly contribute to galloping oscillations. With excellent ballast and rails of 42 kg per linear meter, the track is capable of withstanding speeds up to 160 km per hour. Judging by the results of experiments with high-speed travel, it may be said that the ordinary railway track, even with light rails and ballast of not very high quality, is far stronger than is usually thought.

¹ Journal of the M. P. S., 1903.

At very high speeds in the experiments indicated above, almost all the curvatures of the rails and their deflections to the side proved to be not far from the joints, where the rails, consequently, were subjected to the greatest forces. Under the joints the sleepers were placed, as far as possible, closer to one another, and the ballasting was carried out very carefully, since settlement of the track was feared here. During the experiments the joints themselves did not suffer, whereas the more distant sleepers, placed on poorer ballast and more sparsely, yielded together with the ballast. This also explains the appearance of the aforementioned curvatures, which usually began near the third or fourth sleeper from the joint.

Determination of the influence of speed on the deflection and stresses of the rails.

§ 18. To determine the influence of the speed of motion of a train on the deformation of the superstructure, i.e. on the magnitude of the sinking of the sleepers into the ballast, and also on the stress in the rails, appears to be very difficult; for theory is able to elucidate this question only under the condition of a whole series of assumptions that place the investigation far from reality. Observations and inferences cannot embrace the entire aggregate of very complex factors and of the most varied conditions in which the track is found at its different points. It is simplest to write the dependence of the deflections and of the acting moments on the magnitude of the speed, if one takes into account the influence of the centrifugal force developed when the wheel moves along a rail bent between sleepers, and rejects the influence of other additional forces beyond the static action of the load transmitted by the wheel. Thus the formula for \(M_d\), proposed by the Engineering Council for calculating stresses in rails at different speeds, was composed:

\[ M_d=\frac{8\alpha+7}{16\alpha+40}\cdot Pl:\left[1-\frac{8\alpha+7}{16\alpha+40}\cdot\frac{Pl}{EJg}v^2\right]. \]

Substituting in place of \(M\) its value \(\dfrac{RJ}{z}\) and the expression for \(\alpha\), we shall find the stress in the rails during motion

\[ R_d= \frac{9\cdot\dfrac{12EJ}{abCl^3}+7} {18\cdot\dfrac{12EJ}{abCl^3}+40} \cdot \frac{l} {\dfrac{9\cdot\dfrac{12EJ}{abCl^3}+7} {18\cdot\dfrac{12EJ}{abCl^3}+40} +\dfrac{lv^2}{EJg}\cdot P} \cdot \frac{lz}{J}. \]

If one inserts here, for the different dimensions of the superstructure, the magnitude for \(v\) corresponding to the greatest speeds practiced by us, one obtains \(R_d\) of about \(20 \dfrac{\mathrm{kg}}{\mathrm{mm}^2}\).

Hence arose the requirement to determine the greatest permissible speed on the track for the given superstructure, on the basis of the preceding formula, under the condition of a limiting

\[ B_d = 20\ \frac{\mathrm{kg}}{\mathrm{mm}^2}. \]

One obtains

\[ V_{max} = \sqrt{ \frac{ 4000\left(\dfrac{108\,EJ}{abCl^3}-20\right) - \left(\dfrac{108\,EJ}{abCl^3}+7\right) \dfrac{zl^4}{E_gJ^2}\,P }{ 2000\left(\dfrac{108\,EJ}{abCl^3}+7\right) \dfrac{l}{EJg}\,P } }. \]

For speeds greater than 60 versts per hour, in view of the discrepancy between the results obtained and the indications of practice, it was deemed by the Engineering Council appropriate to substitute in the formula a limiting stress not of 20, but of 25 kg per mm².

This formula in no way corresponds to the truth and can hardly serve even as an empirical formula, the closeness to reality of which is achieved by the introduction of coefficients (in the present case by the limiting value \(R_d\)), determined on the basis of numerous experiments. Indeed, it has not been clarified even approximately what error we make by disregarding the influence of the raising or lowering of the axles, on which depends the oscillation of the springs transmitting the load to the wheels; the influence of the inertia of the rails and sleepers, oscillating during the motion of the train; the influence of the vertical and horizontal oscillations of the locomotive; the influence of irregularities of the tyres and rails, etc. Finally, the deflection of the rail between sleepers is by no means as great as is assumed by the indicated formula (Zimmermann–Winkler). Many believe¹ that during the motion of a train the centrifugal force does not manifest itself at all, since the wheel does not have time, while passing from sleeper to sleeper, to sink enough to bend the rail. Ast even assumes that the influence of the vertical component of the centrifugal force decreases with increasing train speed.

It was indicated above that the dynamic pressure on the rails is estimated at \(2\frac{1}{4}\) times. Thus, for an approximate calculation, one may find the deflection from the dynamic action if one computes the lowering of the sleeper under static action and multiplies the result by \(2\frac{1}{4}\).

In reality, the supporting resistances of the sleepers, as well as the moments acting in the rail as a beam, and consequently the deflection of the sleepers and the stress in the material of the rails under dynamic action of the whe—

...are¹ less than the static ones, taken \(2\frac{1}{4}\) times. This is explained by the fact that, when the body of the locomotive rotates about a transverse horizontal axis (when the locomotive pitches), and also when it rotates about an axis passing through the center of gravity of the locomotive and forming a certain angle with its longitudinal axis (when it rolls sideways), very unequal pressures of the individual wheels are obtained; and since the magnitude of the supporting resistances and of the acting moments is affected not only by the magnitude of the load located above the point for which they are determined, but also by the neighboring loads, then, even if one assumes that the first load increases \(2\frac{1}{4}\) times during motion, the supporting resistances and moments will not increase in the same ratio, because the other loads do not increase \(2\frac{1}{4}\) times, and some even decrease.

The true ratio of the dynamic to the static deflection of a sleeper is very difficult to derive from observations.

I shall mention here experiments carried out by me with the aim of determining what static load, approximately, could replace the effect of a train that had passed over a given place on the rail. Such experiments, of course, cannot give exact results, since at best, even if they are carried out as carefully as possible, one can determine only the force which, under static action, performs work equivalent to the action of the train on the rail. Under a passing train, a copper kopeck was placed on the flat part of the rail head. From the action of the wheels of the locomotive, tender, and cars (the train was a passenger train; locomotive axles, 4; tender axles, 4; car axles, about 80; place of test—near one of the stations of the St. Petersburg–Warsaw Railway; speed of the train about 40 versts per hour; the pressure of a locomotive wheel may be taken as 7 tons), the kopeck flattened and changed its dimensions: the initial thickness was 1.1 mm; it became 0.75 mm; the initial diameter was 21.5 mm; it became, on average, 23 mm. A similar kopeck of the same initial dimensions was placed under Amsler’s press, and it turned out that the same deformation occurred from a steady pressure on the kopeck of 35 tons. Wishing to determine to what extent repetition of the action affects the final deformation, a similar coin was subjected to a static load half as large, i.e. about 17 tons, applied and removed successively 10 times. After 10 loadings, the coin’s dimensions proved to be the same as those of the one that had been removed from the track after the passage of the train. Hence we conclude that the action of the passage of the wheels of a train

as many times as the number of its axles, can be replaced by the static action of one load, 5 times greater in magnitude than the pressure of a locomotive wheel. If it is assumed that the action of all the axles of the locomotive and tender is equal to five times the action of the heaviest axle of the locomotive, and the action of the wagon wheels is equated to the action of another five such wheels, we find that, if 10 such loads replacing the train were applied at a very low speed, then, in order to obtain the same results, it would be necessary under static action to increase all the loads by \(2^{1}/_{2}\) times.

Let us try to compare the limiting elastic stress of rail steel with that which is permitted when checking the strength of rails by a single static formula.

On the basis of theoretical investigations and the practice of rail service, technical regulations have been developed as to what permissible stress should be adopted in rail steel if the strength of rails is checked only by the formula for static bending:

\[ R=\frac{8\alpha+7}{16\alpha+40}\,Hl\cdot\frac{z}{J}\quad \text{(Zimmermann).} \]

For Russian rails, which are required to have a temporary resistance \(\geq 65 \frac{\mathrm{kg}}{\mathrm{mm}^{2}}\) (the elastic limit may be taken as about \(1/2\) of the temporary resistance, i.e., let us put it at \(32 \frac{\mathrm{kg}}{\mathrm{mm}^{2}}\)), the stress computed by Zimmermann’s formula is permitted, according to the rules of the Ministry of Ways of Communication, at no more than \(14 \frac{\mathrm{kg}}{\mathrm{mm}^{2}}\); that is, it is as though it were assumed that the dynamic action of loads can never exceed the static by more than \(\frac{32}{14}\), or \(2^{1}/_{3}\) times. Abroad, under the same conditions, only \(13 \frac{\mathrm{kg}}{\mathrm{mm}^{2}}\) is permitted. Perhaps the ratio between the proportional limit and the temporary resistance of steel is somewhat different there, but in any case, in view of the higher standards there for the strength of rail material, it must be concluded that abroad the maximum ratio between the dynamic and static effect of a load on the superstructure of the track is considered greater than \(2^{1}/_{3}\). It may be that the magnitude of the speed of train movement, which is greater abroad, plays a considerable role here. In any event, it would seem necessary to take, for average speeds, the ratio in question as not less than \(2^{1}/_{2}\). Then we may be sure that all accidental effects on the rails—impacts at the joints, lateral shifts, rotational

forces can act,—the sum of all influences will not raise the static action above the limit indicated by us.

The existence of rails, as it has turned out, is influenced not so much by the possession of a high resistance to rupture as, chiefly, by the stiffness both of the material and of the rail section. That is why, according to the most recent technical specifications for the supply of rails, acceptance is proposed to be carried out exclusively on the basis of the results of the impact test, while the testing of samples of rail steel in tension is made unnecessary. Sufficient stiffness of the rails must ensure that their material will not be strongly stressed; it may receive a stress close to \(2 \tfrac{1}{2}\) times the static stress, but the rails will bend little, the sum of the wheel loads will be distributed almost uniformly over the surface of the ballast and of the roadbed occupied by the rolling stock, and therefore there can be no question of the sleepers sinking into the ballast by an amount equal to \(2 \tfrac{1}{2}\) times the static deflections.

Method of N. Petrov for determining dynamic deflections and stresses in rails.

§ 19. It would seem that the stress in the material of the rails under the dynamic action of loads should be determined by the same formulas as the static stress, and that only the results should be multiplied by 2 at insignificant speeds, for example up to 50 versts per hour, and by \(2 \tfrac{1}{2}\) at higher speeds, admitting that limiting stress which corresponds to the elastic limit of the given material. The latter limit, if impact tests of the rails have been carried out, should be computed by the formula of the deflection arrow, using the deflection figures obtained in the test; the limit of proportionality obtained in this way will fully correspond to the hardness of the given material and to the stiffness of the section.

To determine the suitability of a given track for movement at one speed or another, it is not enough to ascertain that the rails have no chance of breaking; it is necessary to know whether excessive settlements of the sleepers and of the whole ballast will result. Before selecting the most suitable method for calculating the dynamic settlements of sleepers, let us dwell in more detail on Petrov’s work on the dynamic influences on rails.

In general, of course, an increase in the translational speed of a locomotive causes an increase in the settlements of sleepers and in the stresses in the ballast, if only because the greater the speed, the more strongly the influence of the inertia forces of counterweights, irregularities of the tire profile, potholes on the rails, and so forth is manifested. The most recent investigations, however, do not confirm the very widespread opinion that movement affects the superstructure of the track in proportion to the square of the translational speed.

N. Petrov¹ indicates that the growth of this influence proceeds not proportionally even to the speed, but considerably more weakly. From a comparison of the results of calculations by the formulas he proposed, N. Petrov concludes, among other things, that rails permitting travel at a speed of 100 versts per hour permit, with the same degree of safety, travel at a speed of about 125 versts per hour. He further indicates that an increase in the coefficient of the bed, or an increase in the number of sleepers, serves in the most essential way to reduce stresses in the rail, and that improvement of the track in this respect is incomparably more important and more useful than the introduction of rails of a heavier type.

Of course, all these conclusions, so long as we do not recognize the theory itself as correct, have little authority; but N. Petrov asserts that the results of observations do indeed fully confirm these conclusions.

The author dwells on the assumption that the static load of a wheel is distributed over four sleepers. From many experiments, among them those of engineer Vasotyinsky, one may be convinced that the load is transmitted through the rail to six sleepers, but the greatest share of the load is taken by the four middle ones, while no more than 0.1 of the whole load falls on the two outer ones.

N. Petrov finds by the usual method the deflection of four sleepers, which is proportional to the bearing resistances; moreover, instead of the coefficient \(C\) he uses the quantity \(K = 0.89 \dfrac{ab}{2} C\), indicating the load that must be applied to a half-sleeper in order to lower it by 1 cm.

Introducing the quantity \(A = \dfrac{EJ}{HL_3}\), he finds (drawing 14) the deflection of the sleepers when the load is in the middle span:

\[ Y_1 = \frac{ 288A^2 + 54A - (216A^2 + 186A + 7)\delta + (108A + 12)\delta^2 + (12A + 5)\delta^3 }{ 720A^2 + 336A + 15 }\cdot P, \]

\[ Y_2 = \frac{ 216A^2 + 192A + 15 - (72A^2 - 42A + 3)\delta - (144A + 27)\delta^2 - (36A + 15)\delta^3 }{ 720A^2 + 336A + 15 }\cdot P, \]

\[ Y_3 = \frac{ 144A^2 + 126A + 15 - (72A^2 - 42A + 3)\delta - (36A - 18)\delta^2 - (36A + 15)\delta^3 }{ 720A^2 + 336A + 15 }\cdot P, \]

\[ Y_4 = \frac{ 72A^2 - 36A + (216A^2 + 6A - 2)\delta + (72A - 3)\delta^2 + (12A + 5)\delta^3 }{ 720A^2 + 336A + 15 }\cdot P. \]

The ordinate of the elastic curve of any point distant from the middle sleeper by \(\delta L\) is found from the equation

\[ EJ \frac{d^2 x}{dx^2} = -Y_4(3L-\delta L)-Y_3(2L-\delta L). \]

After double integration and certain transformations:

\[ y=\frac{P}{6KA}\left\{\left[-6A-6+(6A+11)\delta-6\delta^2+\delta^3\right]Y_4+\left[12A-1+(6A+3)\delta-3\delta^2+\delta^3\right]Y_3\right\} \]

From considering the numerical results of these formulae it follows that, for large values of \(A\), a more significant part of the load is transmitted to the two outer supports; therefore it is advantageous for the rail to be stiffer, \(L\) smaller, and \(K\) smaller. The moment acting in the rail is greater the larger \(A\) is; therefore an increase in the moment of inertia of the rail, a decrease in the coefficient of the bed, and a decrease in the distance between sleepers entail an increase in the bending moment in the rail.

As for the deflections of the rail in the intervals between sleepers, they are very small; even in the worst case, i.e. for small values of \(A\) and \(C\), they reach only \(1/3\) mm.

From this it is seen that, in wishing to detect the influence of speed on the deflections of sleepers and the stresses in the rail, it is necessary to ascertain the value of the irregularities on the wheel tires and on the rails, since at a considerable speed of motion these irregularities have a very great effect on the work of the upper structure of the track, being, in absolute magnitude, often considerably greater than the deflections of the rails between sleepers.

Convexities and depressions along the length of the rails often reach up to \(1/3\) mm, and at the joints the difference in height of adjacent rails is as much as \(1/2\) mm and more. Depressions and eccentricity of wheels, however, often reach 5 mm. To find the influence of the dynamic load, N. Petrov composes the equation of the vertical motion of a wheel under the influence of the pressure of the spring from above and of the rail from below; since these pressures are not equal, an acceleration \(\frac{d^2 y}{dx^2}\) is obtained and, if we call the weight of the wheel \(q\), the weight of the parts lying above it \(qm\), and the resistance of the rail \(P\), then we obtain an equation of the form

\[ \frac{q}{g}\frac{d^2 y}{dt^2}=q(1+m)-P. \]

Calling the velocity of translational motion \(v=\frac{x}{t}\), we have

\[ \frac{dy}{dx}=\frac{dy}{dt}\cdot\frac{dt}{dx} \quad\text{and}\quad \frac{dt}{dx}=\frac{1}{v}, \quad\text{therefore}\quad \frac{dy}{dx}=\frac{1}{v}\cdot\frac{dy}{dt} \quad\text{and} \]

\[ \frac{d^2 y}{dx^2}=\frac{1}{v^2}\cdot\frac{d^2 y}{dt^2} \quad\text{and the equation of motion}\quad \frac{q}{g}\frac{d^2 y}{dt^2}=\frac{q}{g}\frac{d^2 y}{dt^2}v^2= \]

\[ =q(1+m)-P\ldots(7) \]

Having made the assumption that the pressure of the rolling stock is transmitted to 4 sleepers, and that we have to deal with elastic yielding of the supports, we obtain such a value for \(P\) at which it is very difficult to perform the integration of equation (7).

In view of this, N. Petrov proposed a method for obtaining the value \(y_i\) for the boundaries of sufficiently small intervals of the span between sleepers (for example, one divided into 20 parts), proceeding from the values \(h_i\), known in advance, corresponding to the static action of the load. Introducing here the influence of irregularities on the rails (a projection on the tire and its eccentricity may be equated to irregularities of the rail), let us suppose that the height of the projection at the given place is \(y'\), and the lowering of the rail itself, which does not affect the magnitude of the pressure of the spring, is \(\bar y\); then the whole deflection causing a change in the pressure of the spring is equal to \(y-\bar y-y'\); the load on the axle will change by

\[ mq\,\frac{y-\bar y-y'}{l}, \]

if \(l\) is the deflection arrow of the spring under the load \(mq\). The total pressure of the spring on the axle is

\[ qm-mq\cdot\frac{y-\bar y-y'}{l} = qm\cdot\frac{l-y-\bar y+y'}{l}. \]

The resistance of the rail under dynamic deflection by \(y\) we shall find from the magnitude of the deflection under \(h\) of the load on the rail \((1+m)q\). This resistance is equal to \((1+m)\,q\,\dfrac{y}{h}\).

The equation of motion will be

\[ \frac{q}{g}\cdot \frac{d^2(y-y')}{dt^2} = q+qm. \]

\[ \frac{l-y+y'+\bar y}{l} - (1+m)\frac{y}{h} \]

Integrating for the interval between \(y_{i-1}\) and \(y_i\), where the integrand may be replaced by the mean of the values of these functions, leads to the equations:

\[ y_i= \frac{ y_{i-1}+y'_{i}-y'_{i-1} +\dfrac{a}{v}\omega_{i-1} +g\dfrac{(1+m)}{4}\dfrac{a^2}{v^2} \left[ 2-\dfrac{m}{(1+m)} \left( y_{i-1}-2\bar y-y'_i-y'_{i-1}-\dfrac{y_{i-1}}{h_{i-1}} \right) \right] }{ 1+g\dfrac{(1+m)}{4}\cdot\dfrac{a^2}{v^2} \left\{ \dfrac{m}{m+1}\cdot\dfrac{1}{l}+\dfrac{1}{h_i} \right\} }. \]

\[ \omega_i= \omega_{i-1} +\frac{(1+m)}{2}\cdot\frac{a}{v} \left\{ 2-\frac{m}{(1+m)l} \left( y_i+y_{i-1}-2\bar y-y'_i-y'_{i-1} \right) -\frac{y_i}{h_i} -\frac{y_{i-1}}{h_{i-1}} \right\}. \]

With regard to the actual magnitude \(y'\), we note that it is the algebraic sum of the projection on the rail \(\eta\) and the projection beyond the regular surface of the body of revolution on the wheel \(\chi\); here \(\eta\) should be considered as reaching 2 mm, since, although along the very length of the rail,

differences in level along the horizontal of the rail head rarely exceed \(1/2\) mm, but at joints there are offsets of up to 2 mm.^1) With regard to \(\eta'\), it may be said that on many roads the wheels enter the depression only after the depressions have reached a depth of 5 mm.

From the tables compiled by N. Petrov on the basis of the formulae cited^2) and giving, for ten successive points in several spans between sleepers, \(h_i; y_i; \omega_i\) and

\[ \frac{M}{\frac{Pl}{4}}, \]

the following conclusions may be drawn:

The quantity \(A = \dfrac{EJ}{Kl^3}\), which characterizes the moment of inertia of the rail, the spacing of the sleepers, and the resistance of the bed, affects the deflection under the static action of a load in such a way that the deflection is the smaller, the larger \(A\) is; thus, for example, at \(A = 1/4\) (which corresponds approximately to a rail type of \(24\frac{1}{2}\) pounds per running foot with \(l = 70\) cm), the deflection from a load of 10 tons at the middle of the span between sleepers, with \(K = 10000\) (or \(c\) about 3), is 5.6 mm (and at the support 5.3); while at \(A = 2\) (corresponding to a rail of \(31\frac{1}{2}\) lb per running ft and \(l\) about 50 cm), the deflection at the middle of the span for the same \(K\) is 3.3 (and at the support greater, namely 3.5 mm). Far more than on \(A\), the lowering of the track depends on the magnitude of \(K\). If this quantity is made 20000 instead of 10000, the deflections are reduced by half; if it is made 30000, then by a factor of three, and so on. The deflections of the rail between sleepers are so insignificant, and smaller than the accidental irregularities of the rail, that the rail should be considered, when a train moves over it, as a straight line.

Under dynamic action the deflections increase only slightly in comparison with the static ones—at the very smallest \(A\), by only 12%.

As for the stress in the material of the rail, under static action it depends on \(A\); thus, at \(A = 1\) (and \(K = 10000\)),

\[ R = 18 \ \frac{\mathrm{kg}}{\mathrm{mm}^2}, \]

and at \(A = 2\) it is equal to \(12 \ \mathrm{kg}\) per \(\mathrm{mm}^2\). An increase in \(K\) likewise affects the reduction of stress in the rail; thus, doubling \(K\) reduces \(R\) by approximately 30%.

Under dynamic action the stress differs almost not at all from the static stress; that is, the result obtained is that, wishing to reduce the stress in the rail, we may resort to increasing its section,

^1) Stetsevich. On the Service of Steel Rails of the M. R. C., 1889.

^2) Taking into account the corrections introduced by I. Stetsevich (Railway Engineering, 1904).

but instead of this the coefficient of the bed may be increased. Thus, for example, the same reduction can be achieved by replacing a rail of the type 18 lb. per running foot with a rail of 24½ lb. per running foot, and by replacing a ballast with \(K = 10000\) by a ballast with \(K = 20000\).

Turning to the clarification of the influence that projections and flattened portions on the wheel tire may have on the rail, we see that the former, when their magnitude is up to 0.3 mm, have, in general, an insignificant effect on the stresses in the rails, increasing them by about 5%. In the case of a depression in the wheel or of a deviation of its rolling surface from the form of a body of revolution by 2 mm, the stress in the rail, at speeds of motion above 40 versts per hour, increases, for various values of \(A\), by almost 100%.

When \(K\) is doubled, the stress in the rail falls by about 30%. At low speeds (up to 40 versts per hour) the stresses differ only slightly from the static ones, while, as speeds increase beyond a certain limit, the increments of stress at different speeds differ only insignificantly among themselves. This leads to the conclusion that rails which permit motion at a speed of 100 versts per hour permit, with the same degree of safety, motion also at a speed of 125 versts per hour.

The absolute magnitudes of the stress in rails for wheels of an irregular kind show to what extent it is inadmissible to leave, in fast locomotives, wheels with depressions reaching as much as 2 mm in depth.

All these conclusions follow from the theory adopted by N. Petrov. If they are confirmed by practice, this may serve as an indication that this theory, although it does not encompass all the factors of the action on the track during the movement of loads over it, nevertheless, taking into account the most important of them, is a method suitable for deriving values of stresses in the rail and settlements of the track under trains that are more or less close to reality.

Therefore we shall try to examine to what extent the results of calculations by N. Petrov’s formulas do not contradict one another and the propositions verified by practice.

In order to make certain that the results obtained by Petrov, concerning deflections and stresses under the static action of a load, do not differ from Zimmermann’s conclusions, the following table presents the numerical magnitudes of the settlements of sleepers and of the stresses \(R_s\), according to the formulas of Zimmermann and Petrov, for the principal types of rails of Russian railways.

In addition, in the last column of the table is given the ratio between the stresses obtained by Petrov for a speed of 70 miles per hour and under the quiet action of the load.

The following types of rails have been taken into account:

Weight of rails per running foot, lb	Moment of inertia \(J\)	Section modulus \(W\)	Distance between sleepers \(l\)	Ballast coefficient \(C\)	\(\dfrac{k}{\mu}=\alpha=6A=\dfrac{12EJ}{0.89abCl^3}\)
\(32\tfrac{1}{2}\)	1476	219	89	2	4.76
\(32\tfrac{1}{2}\)	1476	219	89	4	2.38
\(32\tfrac{1}{2}\)	1476	219	89	8	1.19
\(32\tfrac{1}{2}\)	1476	219	80	4	3.28
\(32\tfrac{1}{2}\)	1476	219	74	2	8.30
\(32\tfrac{1}{2}\)	1476	219	74	4	4.15
\(32\tfrac{1}{2}\)	1476	219	74	8	2.08
\(32\tfrac{1}{2}\)	1476	219	69	4	5.11
\(28\tfrac{1}{2}\)	1223	180	89	2	3.94
\(28\tfrac{1}{2}\)	1223	180	89	4	1.97
\(28\tfrac{1}{2}\)	1223	180	89	8	0.99
\(28\tfrac{1}{2}\)	1223	180	80	4	2.72
\(28\tfrac{1}{2}\)	1223	180	74	2	6.86
\(28\tfrac{1}{2}\)	1223	180	74	4	3.43
\(28\tfrac{1}{2}\)	1223	180	74	8	1.72
\(28\tfrac{1}{2}\)	1223	180	69	4	4.32
\(24\tfrac{1}{3}\)	925	144	89	2	2.98
\(24\tfrac{1}{3}\)	925	144	89	4	1.49
\(24\tfrac{1}{3}\)	925	144	89	8	0.75
\(24\tfrac{1}{3}\)	925	144	80	4	2.06
\(24\tfrac{1}{3}\)	925	144	74	2	5.20
\(24\tfrac{1}{3}\)	925	144	74	4	2.60
\(24\tfrac{1}{3}\)	925	144	74	8	1.30
\(24\tfrac{1}{3}\)	925	144	69	4	3.20
\(24^{г}\)	736	119	80	2	3.28
\(24^{г}\)	736	119	80	4	1.64
\(24^{г}\)	736	119	80	8	0.82
\(24^{п}\) (St. Petersburg)	810	132	74	2	4.56
\(24^{п}\) (St. Petersburg)	810	132	74	4	2.28
\(24^{п}\) (St. Petersburg)	810	132	74	8	1.14
\(22\tfrac{1}{2}\)	707	118	80	4	1.57
\(22\tfrac{1}{2}\)	707	118	74	2	2.98
\(22\tfrac{1}{2}\)	707	118	74	4	1.99
\(22\tfrac{1}{2}\)	707	118	74	8	1.00
\(22\tfrac{1}{2}\)	707	118	69	4	2.45
\(21\tfrac{1}{3}\)	626	109	80	4	1.39
\(21\tfrac{1}{3}\)	626	109	74	4	1.76
18	469	87	74	2	2.64
18	469	87	74	4	1.32
18	469	87	74	8	0.66

\(R_s\) in kilograms per sq. cent.; \(P\), wheel pressure in kl.	\(y_s=\dfrac{2}{0{,}89abc}\dfrac{4\alpha+23}{16a+40}\)	\(y\), Petrov	\(R_s\) according to Petrov	\(R_{70}\) according to Petrov	Ratio \(\dfrac{R_{70}}{R_s}\)
\(0{,}158P.\)	\(0{,}068P/1000\)	\(0{,}078P/1000\)	\(0{,}158P.\)	\(0{,}355P.\)	\(2{,}2\)
\(0{,}136P.\)	\(0{,}039P/1000\)	\(0{,}046P/1000\)	\(0{,}131P.\)	\(0{,}350P.\)	\(2{,}6\)
\(0{,}114P.\)	\(0{,}022P/1000\)	\(0{,}027P/1000\)	\(0{,}112P.\)	\(0{,}289P.\)	\(2{,}6\)
\(0{,}132P.\)	\(0{,}037P/1000\)	\(0{,}043P/1000\)	\(0{,}131P.\)	\(0{,}323P.\)	\(2{,}5\)
\(0{,}144P.\)	\(0{,}062P/1000\)	\(0{,}071P/1000\)	\(0{,}143P.\)	\(0{,}298P.\)	\(2{,}1\)
\(0{,}128P.\)	\(0{,}035P/1000\)	\(0{,}041P/1000\)	\(0{,}126P.\)	\(0{,}303P.\)	\(2{,}4\)
\(0{,}109P.\)	\(0{,}020P/1000\)	\(0{,}024P/1000\)	\(0{,}107P.\)	\(0{,}261P.\)	\(2{,}4\)
\(0{,}129P.\)	\(0{,}034P/1000\)	\(0{,}039P/1000\)	\(0{,}125P.\)	\(0{,}287P.\)	\(2{,}2\)
\(0{,}185P.\)	\(0{,}071P/1000\)	\(0{,}082P/1000\)	\(0{,}182P.\)	\(0{,}371P.\)	\(2,\)
\(0{,}157P.\)	\(0{,}041P/1000\)	\(0{,}048P/1000\)	\(0{,}156P.\)	\(0{,}362P.\)	\(2{,}3\)
\(0{,}132P.\)	\(0{,}023P/1000\)	\(0{,}028P/1000\)	\(0{,}131P.\)	\(0{,}300P.\)	\(3,\)
\(0{,}153P.\)	\(0{,}039P/1000\)	\(0{,}045P/1000\)	\(0{,}152P.\)	\(0{,}336P.\)	\(2{,}2\)
\(0{,}170P.\)	\(0{,}064P/1000\)	\(0{,}074P/1000\)	\(0{,}169P.\)	\(0{,}314P.\)	\(1{,}9\)
\(0{,}149P.\)	\(0{,}037P/1000\)	\(0{,}042P/1000\)	\(0{,}147P.\)	\(0{,}319P.\)	\(2{,}2\)
\(0{,}126P.\)	\(0{,}021P/1000\)	\(0{,}025P/1000\)	\(0{,}126P.\)	\(0{,}271P.\)	\(2{,}2\)
\(0{,}145P.\)	\(0{,}035P/1000\)	\(0{,}040P/1000\)	\(0{,}144P.\)	\(0{,}305P.\)	\(2{,}1\)
\(0{,}217P.\)	\(0{,}076P/1000\)	\(0{,}088P/1000\)	\(0{,}218P.\)	\(0{,}390P.\)	\(1{,}8\)
\(0{,}183P.\)	\(0{,}043P/1000\)	\(0{,}050P/1000\)	\(0{,}184P.\)	\(0{,}379P.\)	\(2{,}1\)
\(0{,}155P.\)	\(0{,}024P/1000\)	\(0{,}030P/1000\)	\(0{,}155P.\)	\(0{,}320P.\)	\(2{,}1\)
\(0{,}179P.\)	\(0{,}041P/1000\)	\(0{,}048P/1000\)	\(0{,}176P.\)	\(0{,}361P.\)	\(2{,}1\)
\(0{,}203P.\)	\(0{,}067P/1000\)	\(0{,}078P/1000\)	\(0{,}203P.\)	\(0{,}373P.\)	\(1{,}7\)
\(0{,}175P.\)	\(0{,}039P/1000\)	\(0{,}045P/1000\)	\(0{,}173P.\)	\(0{,}341P.\)	\(2,\)
\(0{,}147P.\)	\(0{,}022P/1000\)	\(0{,}027P/1000\)	\(0{,}145P.\)	\(0{,}288P.\)	\(2,\)
\(0{,}171P.\)	\(0{,}037P/1000\)	\(0{,}043P/1000\)	\(0{,}170P.\)	\(0{,}323P.\)	\(1{,}9\)
\(0{,}242P.\)	\(0{,}074P/1000\)	\(0{,}086P/1000\)	\(0{,}239P.\)	\(0{,}385P.\)	\(1{,}6\)
\(0{,}204P.\)	\(0{,}042P/1000\)	\(0{,}050P/1000\)	\(0{,}203P.\)	\(0{,}374P.\)	\(1{,}8\)
\(0{,}172P.\)	\(0{,}023P/1000\)	\(0{,}029P/1000\)	\(0{,}171P.\)	\(0{,}311P.\)	\(1{,}8\)
\(0{,}216P.\)	\(0{,}069P/1000\)	\(0{,}080P/1000\)	\(0{,}216P.\)	\(0{,}360P.\)	\(1{,}7\)
\(0{,}184P.\)	\(0{,}040P/1000\)	\(0{,}047P/1000\)	\(0{,}182P.\)	\(0{,}353P.\)	\(1{,}9\)
\(0{,}155P.\)	\(0{,}022P/1000\)	\(0{,}027P/1000\)	\(0{,}154P.\)	\(0{,}295P.\)	\(1{,}9\)
\(0{,}204P.\)	\(0{,}043P/1000\)	\(0{,}050P/1000\)	\(0{,}203P.\)	\(0{,}378P.\)	\(1{,}9\)
\(0{,}221P.\)	\(0{,}076P/1000\)	\(0{,}088P/1000\)	\(0{,}221P.\)	\(0{,}390P.\)	\(1{,}8\)
\(0{,}200P.\)	\(0{,}041P/1000\)	\(0{,}048P/1000\)	\(0{,}198P.\)	\(0{,}362P.\)	\(1{,}8\)
\(0{,}168P.\)	\(0{,}023P/1000\)	\(0{,}028P/1000\)	\(0{,}166P.\)	\(0{,}300P.\)	\(1{,}8\)
\(0{,}196P.\)	\(0{,}039P/1000\)	\(0{,}046P/1000\)	\(0{,}194P.\)	\(0{,}344P.\)	\(1{,}8\)
\(0{,}213P.\)	\(0{,}044P/1000\)	\(0{,}053P/1000\)	\(0{,}213P.\)	\(0{,}384P.\)	\(1{,}8\)
\(0{,}209P.\)	\(0{,}042P/1000\)	\(0{,}049P/1000\)	\(0{,}209P.\)	\(0{,}360P.\)	\(1{,}8\)
\(0{,}291P.\)	\(0{,}077P/1000\)	\(0{,}090P/1000\)	\(0{,}289P.\)	\(0{,}402P.\)	\(1{,}4\)
\(0{,}243P.\)	\(0{,}044P/1000\)	\(0{,}054P/1000\)	\(0{,}245P.\)	\(0{,}387P.\)	\(1{,}6\)
\(0{,}207P.\)	\(0{,}024P/1000\)	\(0{,}030P/1000\)	\(0{,}208P.\)	\(0{,}327P.\)	\(1{,}6\)

All the figures in the table refer to a wheel pressure of 7 tons. Although a requirement is now being introduced in Russia to calculate the permanent way for a locomotive-wheel load of 10 tons, for general considerations we may continue to use the locomotive-wheel pressure of 7 tons that has hitherto been legalized.

Petrov’s merit consists in the fact that, rejecting the unsuitable hypothesis that the entire increment of stress in the rail during the motion of a load depends on the centrifugal force arising from the wheel’s travel along the bent rail, he made an attempt to determine the deflection and the stresses at different speeds, taking the dynamic stress to be equal to the static stress multiplied by the ratio of the deflections of a given point of the rail under the one and the other action of the load,

\[ R_d = R_s \cdot \frac{y_i}{h_i}. \]

From the tables given in his article it appears that \(y_i\) differs very little from \(h_i\), and consequently the stress under the dynamic action of the load only slightly exceeds the static stress. But the latter applies only to the ideal case in which the surface of the rail is perfectly smooth and the wheel in contact with it is bounded by a quite regular and ideally centered cylindrical or conical surface. If one introduces the influence of irregularities of the rail (which Petrov considers to be no more than 0.3 mm) and especially of hollows on the wheel or its eccentricity (which may be estimated as equivalent to an irregularity on the rail up to 2 mm high), one obtains stresses at considerable speeds of motion (approximately 60 versts per hour) that exceed the static stresses by a factor of two. At the same time it turns out that a further increase in speed (up to 125 versts per hour) increases the stress obtained only very little.

However valuable the attempt to illuminate, from a side different from what had been done earlier, the phenomena occurring in the permanent way during the rapid movement of a load over it, several remarks must be made concerning the results of Petrov’s theory.

1) In this theory no account is taken of the overload of the axle from the pitching and oscillation of the locomotive. The author assumes, for all speeds, a value for \(P\) not of 7.5 tons but of 10 tons¹, precisely with a view to the additional pressures; but it cannot be that these additional pressures do not grow together with the speed; some of these pressures, for example the vertical component of the centrifugal force of the counterweights, depend on the square of the speed of motion.

2) It is hardly possible, in the calculations, to base oneself on determining the irregularity of the tire corresponding to a protrusion on the rail, namely 2 mm, and to suppose that the unevenness of the rail will instantly cause a change in the position of the spring and a change in the pressure on the rail. In reality, the spring does not by itself support the whole body, which, even if the spring were so elastic and so rapidly transmitted vibrations as the author assumes, descends and rises only by the mean magnitude of the oscillation of all the springs and, in any case, increases the pressure on the rail not at the point where the irregular place of the wheel touches it.

3) Calculations made for the rails of Russian railways on the basis of the tables given by Petrov show that the stresses at considerable speeds will differ very little among rails of different types, and all of them are contained approximately within the limits

\[ 30—40\ \frac{\mathrm{kg}}{\mathrm{mm}^{2}}. \]

If these figures are not regarded as fictitious, useful only for a comparative evaluation of different types of permanent way, then the question arises: how can one calmly regard such large stresses in the material of the rails (exceeding the limit of proportionality)?

4) Let us recall the experiments carried out with electric motor cars.

The general conclusion of those who conducted these experiments is that, on a well-constructed and well-maintained rail track, movement at a speed twice as great as that accepted at the present time is possible. It must be supposed that a speed twice as great is meant on the condition that there be no parts with reciprocating motion in the rolling stock. This follows from the fact that, in the description of these experiments, it is said¹ that on a road where the maximum speed of 80 miles per hour is permitted, one may safely travel at a speed of 120 kilometers per hour, provided only that the locomotive does not contain parts with reciprocating motion, which strongly contribute to interweaving oscillations.

From this it may be concluded that the presence of the indicated parts has a strong influence on increasing the stress in the rails. Assuming that, at that limiting speed which, according to observations, may be permitted for the given rails, there acts in them a stress close to the elastic limit (otherwise buckling would not occur—

rails)—for example, \(0.4 \times 70 = 28 \ \dfrac{\mathrm{kg}}{\mathrm{mm}^2}\); we see that on those tracks where this stress is attained by ordinary locomotives at a speed of 100 km per hour (the permissible limiting speed being 80 km per hour), with electric locomotives, i.e. in the absence of parts with reciprocating motion, it will be reached only at \(120 + 20 = 140\) km per hour. Thus the presence of these parts is equivalent to an increase of speed by 40 km per hour.

One cannot ignore the influence of these parts, which is expressed chiefly in the appearance of horizontal forces acting on the rails and, without doubt, increasing together with the speed of the locomotive’s translational motion.

In view of this, Petrov’s theoretical conclusion that, within the limits from 60 to 120 km/h, the stress in the rails is almost independent of speed cannot be correct. If we regard a locomotive speed of 60 km/h as equivalent to an electric-locomotive speed of 100 km, and a locomotive speed of 120 as correspondingly 160 for an electric locomotive, then, on the basis of Petrov’s theory, we should obtain that a track which becomes deranged under the motion of an electric car at a speed of 160 km per hour would already be deranged by a locomotive at a speed of 100 km, which is not at all evident from the experiments. From the same experiments it follows that, if we wish to bring the speed of the electric locomotive from 120 to 160 km per hour, it is necessary to strengthen the upper structure by replacing rails weighing 33.4 kg per meter with rails of 42 kg/m, with the use of ballast of better quality. It follows that increasing the speed of the electric locomotive by 40 km/h requires increasing the weight of the rails by approximately 25%. The same increase in the speed of motion of a locomotive should require increasing the weight of the rail by approximately 50%. Proceeding from this consideration, we conclude that if, being satisfied with maximum speeds up to 60 km/h, we used rails of \(21 \tfrac{2}{3}\)—\(22 \tfrac{1}{2}\) pounds per linear foot, then, wishing to raise the limiting speed to 100 versts per hour, we must replace these rails with rails, for example, of \(32 \tfrac{1}{2}\) pounds per linear foot¹.

Application of N. Petrov’s formula for determining vertical deformations of ballast.

§ 20. If we agree to consider Petrov’s theory incorrect in the calculation of stress in a rail, if only because it takes into account only one of the factors of the effect of speed on the rail, we shall nevertheless consider his method of calculation suitable for deter—

...division against the sleepers and the ballast when the wheel moves at varying speed, many objections to his theory fall away, since impacts from worn tires contribute most of all to the disturbance of the ballast layer. It is indisputable that, with smooth, regular wheels, the rail material may perhaps often experience overstress during the motion of the train, but the ballast layer remains intact; whereas, with poor turning of the tires, the ballast layer is very severely disturbed. Many other secondary causes may also contribute to this disturbance, but they can be taken into account by means of certain practical coefficients. Indeed, the surface of the rails is humped and stepped at the joints (Stetsevich. Wear of Rails. Journal of the Ministry of Ways of Communication, 1889); considerable bends of the rails often result from extremely careless laying without ballast during construction of the road; damage and bends sometimes increase during the replacement of rails and sleepers; rotten, undersized sleepers are often found in the track, spaced from one another at greater distances than prescribed; the ballast is sometimes of insufficient quantity, of low quality, and very nonuniform, so that under one sleeper it works differently than under another, and so on. Taking all this into account, one need only establish, for each track, the maximum magnitude of vertical irregularities for which one should allow.

It remains to note that Petrov conducts all his reasoning on the assumption that the coefficient \(C\) (or, in his notation, \(K\)) varies within wide limits. He has in mind the coefficient of the bed, considering the ballast to be perfectly elastic. Of course the latter is incorrect. The value of \(C\) varies within very narrow limits; this has already been said above. Experiments with certain ballasts are described in the following chapter.

One may retain Petrov’s conclusion by admitting large fluctuations of the coefficient depending on the ballast, but then this coefficient must be regarded not as \(C\), but as \(C_0\), which in fact varies within considerable limits¹. This is precisely how it must be done if we are to apply Petrov’s method of calculation not to the computation of stresses in the rails, but to the calculation of ballast deformations in the vertical direction.

Let us examine in more detail to what extent an increase in the value of \(C_0\) may actually promote the stability of the track against dynamic forces acting upon it.

Since at very considerable speeds the rail does not have time to bend while the wheel is in the interval between two sleepers, in these cases motion on a properly constructed track must be smoother and quieter than at medium speeds; consequently, above a certain limit (it is close to 100 kil. per hour) the increase in the influence of the load, depending exclusively on speed, does not increase. Therefore, if the permanent way is designed for a speed of motion close to this limiting speed, then an increase in speed is not dangerous, and, even if in all cases the dynamic action is taken as 2.5 times greater than the static one, there is no need to fear for the track, although by the formulas stresses would be obtained that exceed the elastic limit of the materials of the permanent way. The magnitude of track oscillations under the passage of loads depends on the flexibility of the track, and also on the magnitude of the vertical component of the centrifugal forces of the counterweights, which depends on the speed of motion.

The significance of sufficient rigidity of the track, which can be characterized by a high coefficient of the bed, is very great, since, besides the derangement of the track and the greater cost of maintenance, strong lowering and raising of the locomotive reduce its tractive force, generate new dynamic forces acting on the track, and increase stresses in the parts of the locomotive. At very high speeds the oscillations become smaller; therefore here an increase in the coefficient \(C\) does not play the same role, but it is self-evident that at considerable speeds this coefficient must in any case be as large as possible.

The most effective and simplest measure for increasing the rigidity of the track is improvement of the ballast, since the settlement of the supports is proportional to the coefficient of the bed \((D = n\, Cb l)\). To achieve that this coefficient be equally high for considerable sections of the track is impossible, in view of the different properties of the roadbed, which influence the value of \(C\). Nevertheless it is necessary to strive toward this aim, using ballast of appropriate quality and giving the ballast layer sufficient thickness. With ballast for which \(C = 8\) (crushed stone on a good roadbed), the oscillations of the track are almost three times smaller than with \(C = 3\) (sand ballast on a weak roadbed). By using crushed stone as ballast, it is possible, on any roadbed, to bring the coefficient of the bed \(C\) up to 5. At the same time one must strive to increase \(C_0\).

The second measure for increasing the rigidity of the track is the use of sleepers with a larger base. For example, by introducing sleepers with a bed width not of 5 versh. (22 cm), but 6 versh. (27 cm), and making them somewhat longer, we shall raise the rigidity of the track by 20–30%.

The stiffness of the track, although to a lesser degree, is also affected by the quantity \(B=\frac{6EJ}{a^3}\), i.e., the force capable of bending the rail by 1 cm. Reducing the distance \(a\) between sleepers increases the quantity \(B\) in proportion to the cube of this distance. If it is desired to raise substantially the resistance of the track, this is easily achieved by bringing the sleepers closer together. In America the application of this measure goes so far that the value of \(a\) is brought down to 60 cm. In this case, proper tamping of the sleepers becomes inconvenient and is possible only when the width of their underside is small. By varying, on different sections of the track, the distance between the sleepers, one can give it uniform stiffness over considerable lengths; and this uniformity is very important for the safety of traffic. Increasing the weight of the rails also increases the resistance of the track, since \(B\) is proportional to the moment of inertia of the rail. But \(B\) is proportional to the first power of \(J\), and \(B\) itself has only an insignificant influence on increasing the stiffness of the track; consequently, increasing the weight of the rails, as a measure used exclusively for strengthening the resistance of the track, is not rational, all the more so because other measures give more noticeable results and are often associated with lower costs. “Even among railway engineers,” says Ast, “there is a widespread false opinion that strengthening the rails is the only means of increasing the strength of the track¹; they believe that by this means it is possible to eliminate, if not all, then the greater part of the shortcomings of the modern superstructure; at the Paris International Congress of 1889 the words were even uttered: Le joint robuste, c’est le rail lourd.” The erroneous opinion that the rails alone have essential importance in the whole aggregate of the structure is very old. As a consequence of this, rails have constantly been subjected to investigation and improvement; and even in our time, when some superstructure is being considered, the question arises chiefly about the weight of the rails used, whereas the weight of the rails by no means determines the merits of a given system of track superstructure. Efforts to improve the rail profile, apart from the undoubted importance of this factor, are also due to the fact that replacing rails presents no technical difficulties; that when the rails are strengthened, the entire system of track maintenance remains unchanged; and that there is confidence that the resistance of the track has been increased everywhere to the same degree. Moreover, with heavier rails, rail joints do not pre—

constitute such weak places in the track as are found with light rails. The greatest dynamic forces develop at the rail joints; therefore the ballast is destroyed and voids form chiefly under the sleepers adjacent to the joints, as a result of which these sleepers require more frequent tamping. The rail fastenings must be so arranged, and the distance between the joint sleepers so reduced, that the deflection of the rail is the same both at the joints and at other points. Modern systems of rail fastenings do not fully satisfy this requirement; therefore the study of improved joints must be connected with observations of the performance of the ballast under the joint sleepers.

The increasing demands of traffic can be satisfied not solely by strengthening the track (which is possible only within certain limits), but also, as an indispensable condition, by improving the design of the locomotive. Strictly speaking, the existing arrangement of the track should be regarded as the principal factor, and the arrangement of the rolling stock should be adapted to it, and not conversely; and, in the development of traffic, before undertaking the strengthening of the superstructure of the track over an entire railway, or over whole networks of them, measures should be taken to reduce the dynamic forces from the motion of the wheels, since then their pressure can be increased with greater safety. Such measures are: the possible reduction of the load on the leading wheels, which produce the greatest pressures on the track; with this, it becomes possible advantageously to increase the load on the second axle. It is desirable to increase the distance between the wheels and their diameter. Irregular circularity of the wheels (flats formed during braking, incorrect turning, etc.) has a very harmful effect on the track. Such defects of the wheels give rise to forces similar to hammer blows and greatly increase the static pressure. Brake axles should be loaded less heavily; it would be very desirable to introduce a braking system in which the shoes would not press on those parts of the wheels which come into contact with the rail. Locomotives with inside cylinders are much more stable than those with outside cylinders; therefore they should be especially recommended for fast trains. In locomotives with outside cylinders, when building them, the cylinders should, as far as possible, be brought closer to the center of gravity of the locomotive, which, generally speaking, should be as high as possible.

Provided that the rolling stock is of the existing type, the whole rail cannot be increased; sleepers and their laying are of more or less the same type; the stiffness of the track can be increased by applying the best possible ballast, by replacing the existing ballast with another, or by improving the balsast pathway by replacing the existing ballast with another or by improving ...

its quality, by one means or another (screening, washing, adding larger particles, etc.). The necessary conditions for ballast are: considerable hardness of the material, resistance to frost, and water permeability. It is preferable that it consist of particles of uniform size. Moreover, the ballast layer must be of sufficient thickness and must everywhere possess significant and, as far as possible, identical coefficients $C$ and $C_0$.

In the following chapter are set forth the results of an experimental investigation, as far as possible, of all the physical and mechanical properties, bearing upon the resistance of the track, of several specimens of ballast. The study of ballasts, of their properties, of what should be required of them, and also of methods of improving the quality of ballasts available on the railway network, must now become a pressing question in the operation of Russian railways, since the strengthening of rails carried out several years ago makes it necessary to reckon with the fact that an increase in carrying capacity now depends on improving the ballasts.

§ 21. Since it is possible to reason at all correctly about the action of a locomotive upon the track only after becoming thoroughly acquainted with the construction and running of locomotives, and chiefly with those additional oscillations which are inherent in this machine, and with the wear of tires, axle journals, coupling rods, etc., in May and June 1906 I undertook, on the Nikolaev Railway, a short practical course of familiarization with the construction and operation of locomotives.

Observations on the motion of a locomotive’s wheels traveling along the track.

I was interested chiefly in: 1) verifying whether each wheel of a locomotive moves along the rail following a sinuous line, as is stated in courses on railways and in courses on locomotives and in some special treatises¹; and 2) what categories of additional motions may be recognized as taking place, and to what degree, and which of them disturb the track most of all?

With respect to the first question, I came to the conviction that under ordinary conditions the wheel moves, so to speak, along a single taut thread running over the surface of the rail head, and the tire is in contact with the rail in general along one generatrix of a circle. It must—

but there is transverse pressure of the wheels’ tires on the right rail, then on the left; yet it is not so great as to overcome the friction of the wheel on the rail and to shift the tires over the surface of the rail to one side or the other. The sinuous motion of the tire along the rail—the wobble arising from an incorrect localization of what are, in reality, quite rhythmic lateral slidings of the whole locomotive frame along the axles within the existing clearances.

As to the second question, while agreeing that the commonly accepted kinds of supplementary action of a locomotive¹, appearing during its motion, may be observed, one cannot fail to point out that, in general, all these motions are more palpable for the boiler itself and for the whole frame, and are hardly reflected on the axles themselves taken separately. The majority of these motions are transmitted by the axles to the track in so confused and interfering a form that only sharp oscillations—for example, lowering and rising at rail joints, the effect when passing between straight and curved portions of the track, the intensified rocking of the locomotive when the actual speed does not correspond to that for which this locomotive was built and intended—noticeably influence the magnitude of the locomotive’s pressure on the track, which pressure in general is distributed over it rather uniformly. Below are given my notes on the aforementioned trips on locomotives, in the form in which I made them immediately upon returning from these trips.

NOTES ON TRIPS ON LOCOMOTIVES

on the 1st section of the Traction Service of the Nikolaev Railway.

16 May 1906.
I departed from St. Petersburg station at 1 o’clock in the afternoon on a freight locomotive of series Н, No. 1075, with freight train No. 991. The locomotive is of a heavy type with 4 coupled axles, compound, with two cylinders. Steam is supplied to the smaller cylinder, situated on the right-hand side, and then, after working, passes to the left side into the cylinder of larger diameter. The tender is large, on two swivelling two-axle bogies. The fuel is coal (Cardiff coal). The locomotive is intended for

Investigation of Ballasts

speeds of up to 48 versts per hour. They ran at a speed of approximately 35 versts per hour. Joy’s slide valves. Usually the steam admission is kept at 0.4–0.5. In general, engine drivers avoid running a reserve locomotive, i.e. without a train, since they receive a premium according to the number of axles hauled; in view of this they try to travel to the other end of the section in order to take trains in the return direction, by attaching themselves to another train with its locomotive, i.e. by double traction. For the passage of a single locomotive—without cars—only a little less fuel is consumed than with cars (some 30%). In terms of fuel consumption, heavy trains of large composition are generally advantageous. Such a train possesses a large force of inertia; by skillfully making use of it, one can cover a considerable part of the route with a very small expenditure of steam, especially if there are grades with short ascents, whereby such grades may be traversed by the same inertia and further travel continued by using the driving force of the large rolling stock. On the line there is one locomotive with direct action of steam. Although it hauls trains of smaller composition, it always burns somewhat more fuel than a Compound locomotive of the same arrangement; but in general the advantage of double expansion is little perceptible in practice, while the unevenness of running is greater.

On the Nikolaev Railway there are no locomotives with three or four cylinders. The engine driver says that on the Baltic Railway there is a four-cylinder locomotive, the work of one of the Russian factories. The performance of this locomotive is by no means praised. On the Moscow–Kursk Railway there is a similar locomotive, but of foreign construction; that one is very good. When there was a trial test of locomotives of different systems on the Nikolaev Railway, the mentioned locomotive broke the record, hauling, over the same grade, a considerably heavier train and at a higher speed than other locomotives could. When firing with coal, the rule is always to keep the locomotive under steam, since raising steam in an extinguished locomotive requires a large expenditure of fuel and much time.

When firing with petroleum, in general, during long stops they extinguish the firebox and only see to it that a small quantity of steam always remains, sufficient to put the injector into operation. If for some reason this is overlooked and all the steam cools off, they resort to steam admission from another locomotive and, in the extreme case, to raising a small quantity of steam with firewood. If the locomotive has cooled completely, then for this purpose a large quantity of firewood must already be spent. With coal firing, it is necessary to clean the ashpan at least 1–2 times a day. A train to Lyuban and back (2 × 77 = 150 versts) cannot be made without throwing out the ash twice. Fuel consumption with coal firing is regulated not only by shoveling in coal, but also by opening or closing the blower (on the run, the farthest of the firebox doors is opened).

On locomotive No. 1075 I traveled 50 versts to Tosna station. At this station we noticed that the bearing of one crank axle had heated up badly and its babbitt lining was beginning to run. They removed the cotter and took off the bearing (embracing the crank axle in the form of two half-rings with a vertical joint). It turned out that the outflow had only just begun. The shells were only slightly occupied; they knocked down the runs with a chisel and reassembled the crank.

Back from Tosna to Petersburg (50 versts) I rode on passenger locomotive series Н No. 42.

Or at a speed of 45 versts per hour. The Hausshälter apparatus installed on this locomotive is never checked by anyone. The pointer indicated 50 versts per hour, while the needle traced on the tape somewhat less than 45 versts.

The locomotive was a compound, two-cylinder, with Joy valve gear. The steam admission was kept at 0.4. The brakes, as on freight locomotive No. 1075, were Westinghouse. The 50 versts were covered, with one stop for transferring correspondence, in 65 minutes, i.e. about 50 versts per hour while under way.

The engineer says that this locomotive has not-quite-worn-in flats on all the wheels, from the time when it once became necessary to brake and reverse steam in view of an impending collision with another train. Since the flats on all the wheels simultaneously touch the rails, the running of the locomotive has therefore become considerably more restless than it had been before. In my impression, the running of this locomotive is very smooth, especially on tangents, and at an established speed of motion, apart from a slight ordinary pitching, no deviations from proper motion are perceptible.

As regards the motions of the locomotive themselves, in general I had imagined that the influence of the locomotive and the longitudinal pitching were considerably larger in absolute dimensions. Judging by eye, the amplitude of the oscillations of the upper part of the locomotive in the horizontal direction is no more, on average, than 5–6 mm, so that it does not appear possible to determine whether there is any noticeable sinusoidal motion of the tires along the rails. It seems that the wheels do not describe regular wavy lines along the rails, since in general all the secondary motions of the locomotive are frequent and not entirely rhythmic. It is very difficult strictly to subdivide these secondary motions into categories on the basis of impressions obtained during the journey on the locomotive, but one may accept the generally adopted division: into yawing, pitching, and rolling; moreover, in general, horizontal actions on the rails, as well as rolling, disturb the track more than all kinds of vertical pressures. The influence of counterweights, i.e. the successive overloading and unloading of the wheels due to the influence of the vertical components of the inertia forces both of the counterweights and of the crank pins, is manifested, apparently, only insignificantly and in any case not as some investigators suppose.

Jolts at poorly tamped places in the track are felt quite strongly, but in general in the form of a very smooth lowering and raising, and of an equally smooth change of pressure on the track—exactly like a boat on waves.

On the Nikolaev Railway the fully settled roadbed, the ballast, is not coarse, but of uniform grain, clean, and tamped very carefully.

The coarse type of rails distributes the pressure of the wheels over a larger number of sleepers.

The engineers point out that it is noticeable to them that riding on new rails is smoother and somewhat calmer, but in general they do not notice a fundamental improvement of the track from the replacement of the old 24⅓-pound rails. One engineer pointed out a defect of the new rails—that on them the locomotive often slips, which did not occur on the old ones. Whether this should be explained by a mismatch between the head profile of the rail of the new type and the profile of the tires, or by the quality of the material, or by a reduction of local deflections in the intervals between the sleepers, is difficult to say.

The driver asserts that wear of the tire around the whole circumference, being local, greatly increases the forces tending to turn the rails and in general to disturb the superstructure.

Bad ballast may be that which, by poorly allowing water to pass through, permits it to accumulate under the sleepers, which causes shocks that increase under the repeated action of the wheels. Ballast not covered with crushed stone directly receives the impacts and the considerable velocity of the water falling upon it, is washed out and blown away, and its tamping does not hold so long. Ballast that has not had time to settle, as, for example, on the new Vologda line, yields and is the cause of jumping and pitching of the locomotive. Settled ballast is not elastic; if it settles by an insignificant amount, it does so uniformly everywhere.

With double traction the track is disturbed considerably more than with single traction, since the track experiences a greater load and the oscillations of two large masses standing side by side.

On May 17. Departed from St. Petersburg station with a courier train at 9:45 in the morning, on locomotive series N No. 42. We traveled at a maximum speed of 72 versts per hour. The horizontal rocking of the locomotive is quite distinct, and its rhythm corresponds approximately to the running over 2--\(2 \frac{1}{2}\) sazhens, i.e. the circumference of the wheel, since the wheel diameter is about 6 feet. Up to a speed of approximately 40 versts per hour, no vertical oscillations are noticed, and the horizontal ones are insignificant. At higher speeds both increase. We reached Luban (77 versts). From there to Kolpino we rode on a freight locomotive, series Ob, No. 218. As regards slipping on the new rails, the driver of this train did not notice any difference between new and old ones, and explains slipping exclusively by moisture settling on them, and also by oil thrown out of the locomotive or by that with which the bolts are lubricated when the joints are bolted.

At times during the ride, increased rocking in the vertical direction was observed. It occurred on descents. It was noticeable that, simultaneously with the increase in vertical rocking, the horizontal rocking decreased. The switches are scarcely perceptible on the locomotive. From Kolpino we rode on a passenger locomotive, series N No. 43. Horizontal rocking is stronger on straight sections than on curves.

It may be supposed that the visible lateral rocking is the movement of the frame with the boiler about the axles, within the limits of the clearances between the axle journals and the frame; the axles themselves roll properly, and their wave-like motion, however closely one looks, is imperceptible.

The driver of the freight train observed that on new rails the friction of the axles against them is greater than on old rails, since with the removal of the rails he cannot, under the same conditions, do the speed he achieved on old rails. On new rails the tires wear out much less. Perhaps the reason for this is the softness of the material of these rails (32 1/2 pounds per running foot, made by the Bryansk Works).

In general, the drivers do not think that, on a solid, everywhere well-tamped track, there can be any significant irregularities in the running of the locomotive that strongly affect the track. Irregularities of motion must exist, since not everywhere can everything be equally well tamped. Swaying will inevitably occur if one

the end of one sleeper is beaten down less than the other; the wheel will not fail, on the next passage, to settle this place of the rail, and on the following passage to break up this place still more; since the rail springs, the depression may remain unnoticed for a long time.

The pressure of the Kreutzkopf on the slide-bars, now upward, now downward, cannot be great, since it is not observed that the slide-bars wear out to any significant degree.

On all the locomotives of the Nikolaev Railway, instead of two parallel guides there is one guide, along which the clip from the Kreutzkopf moves. There is the same kind of guide for the valve rod.

When a train passes over the sleepers, the movement of the ends of the sleepers can by no means be observed by eye. The packing is pressed into the sleeper; the clearance between the sleeper and the packing now disappears, now is restored; the sleeper is compressed and forced apart; finally, it bends, sinking somewhat into the ballast directly under the rails, while the ends and the middle remain immovably seated on the ballast; meanwhile, directly under the rails there is probably a gap between the bottom of the sleeper and the top of the ballast, of a millimeter, or even more.

On 18 May, from 12½ o’clock until 6 o’clock in the evening, I inspected the carriage workshop at St. Petersburg station, the locomotive workshop, the locomotive building (roundhouse), and the laying of tracks to the new passenger platforms. I examined in detail the construction of Pullman car bogies, several of which had been rolled out from under cars being repaired and were in a partly dismantled state. In the locomotive workshop I observed the turning of tires, and noted that the profile of a worn tire is a smooth curve; tires that I saw did not belong to the class of heavily worn ones: not more than 4–5 mm. At the place of greatest wear one does not observe a fairly wide flat area, which would appear if the rolling of the wheel along the rail took place along a tortuous line. As regards the depth of wear, the foreman in charge of this workshop explained to me that ordinarily one observes wear increasing regularly with the passage of time—after half a month or two months by 1 mm, after another month by 1½, after another month by 2, then 2½, and so on up to 4–5 mm. Greater wear is not permitted, and the wheel is sent for re-turning. But he had happened to observe wear of 7 mm and more, even up to 10. In such cases it turns out that, instead of increasing further, the wear begins to diminish and reaches 5 mm. It must be concluded that, with more prolonged service, the entire profile of the tire is worn down. Local potholes resulting from braking (in general they are not large, occupying an area no more than 5–8 cm in size and with a depth not exceeding 3–5 mm) are, with the passage of time, rolled out, which occurs from uniform wear along the whole rim of the wheel. The greatest wear of the tire is noticed at the places on the rim opposite the crank pins and near the counterweights. This must be explained by systematic impacts against the rail due to the centrifugal force of these masses. The lateral, horizontal motion of the locomotive causes the bearings to slide along the axle journals, since there are clearances between the axle-boxes and the wheel hubs. These clearances are made 3 mm for the middle axles and 5 mm for the end axles (for the middle axles they may be 1 mm, and for the end axles 7 mm). With such clearances the locomotive easily enters curves. If two axles are connected into one bogie on a pivot, then the clearances are made much smaller (1–3 mm). After a test [[unclear: continuation cut off at bottom of page]]

after several thousand versts the clearances increase; for example, at the middle ones they reach 5–6 mm, and at the end ones up to 8 mm. This increase results from wear of the side faces of the axleboxes, as well as of the side faces of the pedestals. In order to maintain the required distance between the edges of the axleboxes, with the jaws and the axle journals in an unchanged position, they resort to screwing copper wedges of the appropriate thickness to the side surfaces of the pedestals, or to welding on babbitt alloy slabs at these places. The increase of the clearances indicates, on the one hand, a constant lateral rocking, which causes pressure and friction of the axleboxes against the pedestals, and, on the other hand, that the boiler, rocking in the horizontal direction, and in general everything that rests on the bearings, rubs against the axle journal; moreover, since the journal is abundantly lubricated, this rubbing is easily explained. Thus all the lateral oscillatory motion of the locomotive, which, given the high location of the locomotive’s center of gravity, cannot be considerable (at the level of the bearings it may be no more than 5–6 mm), with enlarged clearances manifests itself in the form of rubbing of the axleboxes along the journals. If, however, the clearances are small, then the entire oscillation will not be expressed by this rubbing, and part of it (perhaps 2–3 mm) will also appear as a successive deflection, now to the right, now to the left, of the point of contact between the wheel and the rail. In such a case the geometric locus of the points of contact of the wheels and rails will probably be a wavy line. With an increase of the clearances between the axleboxes and the pedestals, an intensified rolling of the locomotive is observed; it is shaken in all directions, so that enginemen complain of the fatigue of being on the locomotive; the mechanism gets out of order, the rivets in the frame become loosened, etc.; a large increase of clearances may be dangerous for the integrity of the connecting rods, cranks, and piston rods. Developed clearances are also the causes of sharp oscillatory motion of the locomotive, since the impacts of the axlebox against the horns and pedestals are already large and are not absorbed by the mass of the locomotive, but produce jolts. It must be noted that in compound locomotives the even working of both sides is possible only with a steam cut-off of 0.5. In fact, enginemen more often allow themselves to run with a smaller cut-off, as a result of which the right side works more strongly. This leads to an increase of rocking and to disorder in the parts of the engine.

I had occasion to examine, in locomotive sheds, the construction of a domed covering, the groaning turntable crane, a tender raised on lifting jacks with axles that had run hot. In the locomotive workshop I was present at the dismantling, lifting onto jacks, and rolling out of axles, with removal of the axleboxes, of a four-axle locomotive, series Н. I picked up two spikes, apparently long present in the track. One of them is strongly bent along its length inward toward the track—toward the head; in the other such a general bend is scarcely noticeable. In the first, beneath the head, on the face nearest the rail, within the limits of the bearing plate there is a dent 3–4 mm deep; the upper third of the length is generally thinned and is noticeably worn on the rear face. It was as though the upper part of the spike hole had become enlarged, and the spike strongly pressed upon and rubbed the bearing plate around it. The other spike, having a dent in the front face of only about 1 mm and a rear face worn in its upper part, is strongly worn in the same upper part along both side faces, as though the bearing plate had long been rubbing in the direction of the track, striking now one side face of the spike, now the other.

To consider the ballast brought in by work trains for laying tracks to the new passenger platforms. I found its quality to be very high. In appearance it is clean, to a considerable degree quartzose, i.e. hard, consisting of grains of small size, predominantly about \(1/4\) mm, with a very slight admixture of coarser gravel and with a not very large content of fine particles. There is probably no more than a few percent of clay in it. According to the track foreman’s report, this ballast lets water pass through excellently, does not become compacted, is always loose, and is sufficiently elastic in mass. In the track foreman’s opinion, ballast consisting of grains coarser than \(1/4\) mm is not in the least better than this, and is even worse, because it is not capable of giving a completely compact and stable bed. Bad ballast is that which contains any considerable quantity of clay. A test of the quality of ballast, besides washing, may be the drying of a lump of wetted ballast. If it remains friable, then it should be considered quite a satisfactory material.

Ballast with an admixture of very fine particles is not very friable; therefore any gap under the sleepers or near their side edge remains unfilled, and with repeated passages of wheels such a gap keeps increasing, making the position of the sleeper unstable. If, with such ballast, a gap has formed under the sleeper, then in the course of time many of the very finest particles are beaten into it, producing a cushion of dust; and this dust is even sometimes thrown out from the sides of the sleeper, and after rain, if the water is not carried away by the ballast layer quite satisfactorily, it is splashed out in the form of mud. In general, with bad ballast it is possible to have a track almost as strong as with good ballast, but more attention must be paid to track maintenance, and nearly twice as much labor will be spent on repairs. For the stability of the track, i.e. so that it may require less attention, it is far more important to have heavy rails, distributing the load over a considerable number of sleepers and bending very little. The dimensions of the sleepers are of incomparably lesser importance. The quality of ballast is highly valuable in the sense that it must possess satisfactory properties as regards permeability to water, resistance to washing out and to being blown away, uniform grain size in the absence of any considerable admixture of clay, friability, and a certain elasticity in mass. The circumstance that the ballast springs by 1–3 mm is very valuable. If it does not spring at all, as is observed in very fine and clayey ballasts, then after the given sleeper has been lowered, when it again rises by virtue of the springing of the rails, a gap remains under it, increasing with time. In the opinion of this track foreman, since the pressure of the sleeper spreads downward at a certain angle, the part that resists the pressure is not only the ballast under the sleepers but also between them; and although in a small volume the ballast is inelastic, in its entire mass it, provided proper uniform tamping is performed everywhere, causes a sleeper that has sunk by 2–4 mm to return to its former place.

In his opinion, the predominant importance for track maintenance is held by all kinds of horizontal movements of the locomotive; the additional vertical pressures, manifested in an increase of static pressure, are very little dangerous for the track. If only the running of the locomotive were calm, and there were no parts with alternating motion,

that along present-day tracks one could run locomotives not only twice, but even three times as heavy as now. Considerable speeds are more dangerous for the track than weight, because at high speeds every kind of oscillation and rocking reaches larger dimensions, and the effect of the locomotive, which only with careful maintenance of the track and fully timely tamping of weakened places is not very dangerous. As proof that lateral forces are great and dangerous, he points out that experience shows how risky it is to raise a rail above the sleepers (for example, in winter, by means of packings under the sleepers during heaving) by more than 10–12 mm, whereas stability, assuming the action of vertical forces only, is not diminished under such conditions.

Trip on 19 May to New Port.

I left at 12 h. 20 m.; arrived at New Port (14 versts) at 1 o’clock; from there, with a call at the St. Petersburg–Warsaw Railway (22 1/2 versts), returned at 2 h. 40 m.

I rode with a passenger locomotive, series N No. 219. On the return trip I stood at the front end of the locomotive. There it shakes less, although it seems to the engine driver that the rocking of the front end of the boiler is strong. Above the axles and along the middle length of the locomotive no increase in rocking is noticeable. I observed gaps at the axle journals. They are not entirely rhythmic; in magnitude not less than 5 mm.

On 27 May I rode along the Vologda Railway, only recently opened to traffic, as far as Nazi station (63 1/2 versts): there on locomotive series Ob. No. 1071, back on locomotive series Ob. No. 247. I did not notice a tangible difference in running on this road in comparison with the Nikolaev Railway. The track appears entirely firm and well constructed. The rails are 24-pound, with 6 bolts; the ballast is very fine, but of uniform composition and clean; the layer is thick; the sleepers are covered with ballast. In general I observed that the magnitude of rocking depends on the condition of the locomotive. One locomotive of the same type and over the same track runs considerably more quietly than another. Each locomotive seems to have certain speed limits at which the motion is almost entirely quiet, once it has become established. The locomotive is thrown most from side to side when entering curves and leaving curves.

Running down a grade, especially with the regulator closed, is the most unsteady. Whether this depends on the fact that the train then does not pull the locomotive so tightly, or that there is then little inertia to keep it from secondary motions, or, on the contrary, that it is large and all irregularities of the track make a greater impression on the locomotive. In calm, established motion the locomotive goes almost along a straight line or a regular curve in curves of considerable radius, and only from time to time rocks to the side because one of its axles at once settles onto some axle, whereby not all the wheels are on one level, which results from an incompletely straightened or tamped track.

Note:

On the Nikolaev Railway, on straight sections, the track is always kept with one side (the right) 2 mm higher than the left. This leads to a reduction of rocking and jolts.

When a train moves along a meridionally directed track from north to south, it passes onto an adjacent point of the wheel which has a greater linear velocity of rotation about the earth’s axis and, as it were, lags behind the track. Therefore, to avoid a lateral derailment, it is useful to raise the right rail. The same is true when moving from south to north. Here there is a complete analogy with Baer’s theory of the washing away of the right banks of rivers of meridional direction. I have not been able to learn whether the indicated elevation on the Nikolaev Railway was introduced already by the Austrians who built it on theoretical grounds, or whether its usefulness was later felt in practice.

IV. Physical and mechanical properties of ballast.

Results of measurements of ballast sands of the Ryazan–Ural Railway.

§ 22. Of sand used for railway ballast, two qualities are usually required: 1) that it be, as far as possible, coarse, and 2) that it contain as little admixture of clay as possible. For fine sand a certain admixture of clay is considered necessary to reduce the blowing away of the sand by the wind. Thus, in judging ballast, the first thing of importance is its coarseness. Most often, in the absence of more precise data, one has to characterize a given material by names: crushed stone, pebble, gravel, sand, the last being defined as coarse, medium, or fine. Not content with such a verbal subdivision of sands, their coarseness is sometimes determined by direct measurement, by sifting them through sieves. For the comparability of the results of sifting and for a more uniform characterization of different sands, it would seem necessary to distinguish, for example, three categories: 1) sand, 2) sand with an admixture of gravel, and 3) sand with an admixture of gravel and pebble; moreover, for each ballast material the average coarseness of the grains must be determined separately for each of its constituent parts. One may agree to call grains smaller than 1 mm sand, grains of size from 1 to 5 mm gravel, and larger grains pebble. To establish the average coarseness, we determine the percentage content of the sifted residues on sieves with different sizes of openings and, denoting by the letters \(d_1, d_2\) the mean diameters of the openings for each two successive sieves and by \(c_1, c_2, c_3\) the quantities of sand retained on them in percent, we find the average coarseness by the formula

\[ \frac{c_1 d_1 + c_2 d_2 + \cdots}{100}. \]

In the Laboratory of the Administration of the Ryazan–Ural Railway, ballasts from 54 quarries of this railway were measured. It turned out¹ that an admixture of gravel (in an amount from 3 to 50%) was found—

was found only in 7 samples, so that the remaining 47 quarries contain ballast consisting of sand alone, with grain size not exceeding 1 mm. The average coarseness of the sand for all 54 quarries ranges between 0.2 and 0.7 mm. At the same time, an admixture of gravel was found in sands with a coarseness of more than 0.4 mm; an admixture of clay (from 1 to 16%) was found in all sands, but in considerable quantity (8 to 16%) only in comparatively fine sands, i.e. with an average grain size up to 0.4 mm. Thus, judging from the measurements of the ballasts of the Ryazan–Ural Railway, it may generally be assumed, in cases where no test results are at hand, that the average coarseness of sand containing an admixture of gravel is always above 0.4 mm. Sands with a coarseness of 0.4 mm and above, both those containing gravel and those not containing it, rarely have a considerable (more than 8%) admixture of clay. No data are given in the article for judging the qualities of the sands listed in the table as material for ballasting. In the notes to the indicated table it is said that the following standards may be adopted: for the highest grades, an average coarseness of more than 0.5 mm; for the medium grades, a coarseness of 0.3–0.5 mm; and for the lower grades, a coarseness of less than 0.3 mm. An admixture of gravel improves the quality of the ballast. For coarser sands, as small a clay content as possible is desirable; in fine sand, some admixture of clay (not less than 1%) is sometimes desirable, since without it the sand is easily blown away by the wind.

§ 23. In the Laboratory of the Tomsk Technological Institute, samples of material from several ballast quarries of the Siberian Railway were investigated.

The grain size was measured by sifting through sieves with openings¹ of diameter 5 mm, 4, 3, 2½, 2, 1¾, 1¼, 0.9; 0.8; 0.7; 0.6; 0.5; 0.4 and 0.2. By the method indicated above, having the amount of residue on each of these sieves, the average coarseness of each of these sands was calculated.

For each of them, the clay content was determined by elutriation, as were the specific gravity, the weight of a liter in the loose state, the approximate angle of natural slope; and the degree of siftability and compactability of these sands was also characterized.

These data are collected in the following Table I. The quantities of residue on the various sieves are set out in Table II.

Table No. 1.

No. of ballast	Name of ballast	Grain size and composition of ballast	Clay content, in %	Specific gravity	Weight of 1 liter in loose condition	Angle of natural slope, degrees	Characteristic of appearance	Degree of drying and cohesion of the mass	Coefficients of ballast from experiments
1	Tomsk coarse (81 versts, Tomsk branch).	11% pebbles, avg. grain 10 mm; 18% gravel, avg. grain 2.3 mm; 71% sand, avg. grain 0.3 mm. Avg. grain 1.7 mm.	15	2.56	1310 gram.	33	Orange, of middling quality.	Dries well, strongly coheres.	\(C = 1.5\) \(C' = 2.5\).
2	Tomsk fine (81 versts, Tomsk branch).	1% gravel, avg. grain 3.2 mm; 99% sand, avg. grain 0.1 mm. Avg. grain 0.1 mm.	27	2.29	1028	36	Orange, of middling quality.	Dries well, very strongly coheres.	\(C = 2\) \(C' = 2\).
3	14th section, Siberian Railway, Byeloyarsk, 1859 versts (bank of the Chulym).	14% pebbles, avg. grain 7.4 mm; 18% gravel, avg. grain 2.5 mm; 68% sand, avg. grain 0.4 mm. Avg. grain 1.8 mm.	2	2.8	1627	28	Coarse gravel; is almost not washed out and not blown away.	Dries fairly well, coheres somewhat.	\(C = 1.5\) \(C' = 1.5\).
4	Yaya medium, 12th section of the Siberian Railway.	40% pebbles, avg. grain 10 mm; 28% gravel, avg. grain 2.3 mm; 32% sand, avg. grain 0.6 mm. Avg. grain 4.8 mm.	1	3.24	1826	30	River ballast of the best quality. Coarse gravel; is almost not washed out and not blown away.	Dries fairly well, remains loose.
5	Yaya fine, 12th section of the Siberian Railway.	8% pebbles, avg. grain 8 m.; 29% gravel, avg. grain 1.9 mm; 63% sand, avg. grain 0.5 mm. Avg. grain 1.5 mm.	1	2.59	1628	27	River ballast of the best quality. Coarse gravel; is almost not washed out and not blown away.	Dries poorly, remains loose.

No. of ballast	Name of ballast	Size and composition of the ballast	Clay content in %	Specific gravity	Weight of 1 liter in loose state	Natural angle of repose in degrees	Character of breakage	Degree of passability and cohesion	Ballast coefficient in experiments
6	Churilovsky fine, 9 versts, section 1.	Sand of medium size; 0.1 mm.	4	2.59	1333	23	Of the sand, stones, small and visible in about 10%.	Passes well, remains in a round pile after a good hour.	\(C = 2\) \(C' = 2.5\).
7	Churilovsky coarse, 9th verst, section 1.	Gravel 22%, avg. gr. 1.6 mm; sand 78%, avg. gr. 0.4 mm. Avg. gr. 0.7 mm.	2	2.69	1610	31	Of the sand, stones, small and visible in about 10%.	Passes well, rather strongly coalesces.	\(C = 2.5\) \(C' = 3\).
8	Churilovsky medium, 9 versts.	Gravel 27%, avg. gr. 1.6 mm; sand 73%, avg. gr. 0.4 mm. Avg. gr. 0.7 mm.	4	2.65	1610	28	Of the sand, stones, small and visible in about 10%.	Passes not very well, does not coalesce strongly.	\(C = 2.5\) \(C' = 3\).
9	9th section, No. 1 Krivotschekovsky, 1326 versts.	64% pebbles, avg. gr. 6.6 mm; 32% gravel, avg. gr. 2.9 mm; 4% sand, avg. gr. 0.4 mm. Avg. gr. 4.9 mm.	2	2.55	1550	32	Good small gravel with enclosing sand.	Passes well; remains loose.	\(C = 2.5\) \(C' = 3.5\).
10	9th section, No. 2 Krivotschekovsky, 1326 versts.	50% pebbles, avg. gr. 9.5 mm; 23% gravel, avg. gr. 2.3 mm; 27% sand, avg. gr. 0.5 mm. Avg. gr. 5.4 mm.	1	3.18	1763	30	Good small gravel with enclosing sand.	Passes fairly well; remains loose.	\(C = 2\) \(C' = 3\).

No. of ballast	Name of ballast	Coarseness and composition of ballast	Clay content, %%	Unit weight	Weight of 1 liter in the dry state	Angle of natural slope, in degrees	General characteristic	Degree of drying and cohesion of particles	Ballast coefficients from tests
11	Azei, 22nd sec. 2692 versts	42% pebbles, avg. size 12 mm, 4% gravel, avg. size 2.2 mm, 54% sand, avg. size 0.4 mm. Avg. size 5.2 mm	2	2.59	1808	27	River, well lying on the track, little washed out, and almost not blown away.	Dries very well; cohesion remains.	C = 2 C′ = 3
12	of the 5th section, No. 1 Omsk	3% gravel, avg. size 1.3 mm, 97% sand, avg. size 0.3 mm. Avg. size 0.3 mm	5	2.55	1510	30	From local deposits in the steppe. Fine, dusty, and blown out: 5–20%.	Dries poorly; remains rather loose.	C = 2 C′ = 2.5
13	of the 5th section, No. 2 Omsk	10% gravel, avg. size 1.3 mm, 90% sand, avg. size 0.3 mm. Avg. size 0.4 mm	12	2.55	1200	33	From local deposits in the steppe. Fine, dusty, and blown out: 5–20%.	Dries poorly; remains somewhat loose.	C = 1.5 C′ = 2
14	Zlobino, 2058 versts, 16th sec.	62% pebbles, avg. size 15 mm, 9% gravel, avg. size 2.3 mm, 28% sand, avg. size 0.3 mm. Avg. size 9.7 mm	3	2.69	1990	33	Ravine; well lies on the track, little washed out, and almost not blown away.	Dries well; remains cohesive.	C = 1.5 C′ = 2.5
15	Building sand Tomsk	Sand, avg. size 0.2 mm	10	2.5	1300	35	Ravine; rock of kaolin quality.	Dries well; cohesion remains.
16	Building crushed stone, from local stone	Medium size 30–60 mm							C = 1.5 C′ = 2

Table 2. Content in the ballast of particles of different coarseness (in grams).

Grain size	Ballast No. 1	Ballast No. 2	Ballast No. 3	Ballast No. 4	Ballast No. 5	Ballast No. 6	Ballast No. 7	Ballast No. 8	Ballast No. 9	Ballast No. 10	Ballast No. 11	Ballast No. 12	Ballast No. 13	Ballast No. 14	Ballast No. 15
from 9 to 10 mm	54	0	8—15.35	14—25.117	7—14.39	0	0	0	7—15.115	10—18.180	12—22.193	0	0	25—40.95	0
5—9	54	0	6—7.48	9—14.87	5—7.40	0	0	0	5—7.397	8—10.104	9—12.79	0	0	20—25.75	0
4—5	17	5	22	31	10				127	51	15			23
3—4	24	2	36	41	19	0	3	3	155	69	12			25
2½—3	16		17	23	12		4	5	45	24	4	2	3	7
2—2½	20	3	23	29	27	0	15	2	59	32	5		6	13
1¾—2	35		27	53	68		34	63	40	39	6	9	16	13
1¼—1¾	40	3	32	57	93		117	146	22	35	8	13	28	18
0.9—1¼	25		23	47	57	2	47	47	6	34	10	23	40	14
0.8—0.9	47	2	48	92	127	3	191	172		53	24	42	65	28	2
0.7—0.8	11	2	0	47	104	4	70	73	3	53	49	42		13	2
0.6—0.7	24	5	25	79	137	5	85	77	3	41	122	75	75	22	6
0.5—0.6	75	3	73	40	98	10	43	37	3	31	106	59	46	11	4
0.4—0.5	65	40	66	59	152	60	42	49	5	72	177	445	278	109	332
0—0.2	254	935	73	8	20	916	347	345	16	39	40	700	442	119	643

The testing of the ballasts just mentioned for crushing was carried out in a special box (drawing 13) with thick walls and provided in the middle part with two iron hoops to keep the sides of the box from spreading under the high pressures exerted on the sand poured into the box. The width of the free space inside the box was 23 cm, and the height of this space 19 cm. In order to study whether the thickness of the layer, and also the greater or lesser proximity of rigid walls to the sleeper, had a significant influence on the character of the transmission of pressure from the sleeper to the ballast, 2 boards were prepared for this box, which could be placed either one or both together on the bottom of the box, and two more, which could be set inside against its side walls in order to reduce the width of the opening. The length of the box (143 cm in the clear) made it possible to place in it, on a layer of sand, a model of a sleeper at \(1/4\) natural size (in comparison with type No. 1 pine sleepers supplied for main lines). To study the dependence of the ballast coefficient on the absolute magnitude of the sleeper bed, experiments were carried out in parallel also with a model of the same sleeper at \(1/4\) natural size, and also with an iron I-beam whose length and flange width were suitable for the latter model of a wooden sleeper (draw. 14).

The experiments were conducted in the following manner: on a layer of ballast of various sections (with a width from 16 to 23 cm and a height from 8 to 19 cm) a sleeper of \(1/2\) natural size was laid; on it, at the places corresponding to the rails, two thick iron pads were placed, and upon them the movable plate of Olsen’s machine pressed, which could develop and measure a force of up to 100 tons. It was necessary to use this machine because it was impossible to install on the base of any of the other machines available in the Laboratory so large a box. The box was installed in an unchanged position between the columns on the lower board of the machine, which was provided with brackets. The force was brought up to 4, and at most 10 kilograms per square centimeter of the sole of the sleeper or beam, which corresponded approximately to a unit pressure on the ballast 3–5 times greater than the greatest static pressure from the heaviest locomotives under the most unfavorable conditions. A preliminary pressure of 20–50 kg was applied to the sleeper so that, as far as possible, the entire sole of the sleeper would touch the ballast, and then the pointer of the lever apparatus, making it possible to measure the vertical distance between the movable plate of the machine and the upper face of the box, which stood immovably, was set at 0. By noting the lowering and raising of the movable plate when changing the pressure on the ballast, it was possible to determine the compression of the ballast layer and to conclude,

to what extent one or another ballast remains elastic. To investigate elasticity, the procedure was that an increase in the load, say by 100 or 200 kilograms, was carried out several times, each time removing the added load immediately after its addition.

As a general rule, it turned out that upon removal of a load that had just been added, or of the entire existing load, the upper board, and consequently also the upper surface of the ballast layer, rose very little, for the most part by 2 mm, and at pressures above \(0.5 \ \frac{kg}{cm^2}\) did not rise at all. The general conclusion obtained was that poorly packed ballasts are very little elastic, and, within the limits in which they possess this property, the magnitude of the elasticity remains almost the same for different ballasts. It was found that with an iron beam the elasticity of the same ballast is less than with a wooden one. This may be explained by the fact that the wooden sleeper itself is compressed, producing the impression that the ballast is being compressed, or by the fact that the unyielding iron beam transmits the pressure to the ballast more uniformly and therefore, without springing at all, penetrates into it more freely. In exactly the same way it may be observed that the springing of the ballast increases with the width of the sole of the sleeper. From the results of the observations, graphs were drawn for all the ballasts (Drawings 15 and 15′) of the dependence of the compression of the layer on the load per unit of surface of the ballast.

Within the limits from 1 to 3 \(\frac{kg}{cm^2}\), these lines of dependence may be considered fairly smooth, and the tangent of their angle of inclination to the horizon may be taken as the coefficient of the given ballast—\(C'\). Strictly speaking, when one speaks of the coefficient of ballast (coefficient of the bed), it has always hitherto been understood that the ballast is considered within the limits of its elasticity. The values of \(C'\) obtained from the graphs will not correspond to this condition, since the 14 kinds of ballast tested showed appreciable elasticity only up to pressures of 0.5, and at most 1 \(\frac{kg}{cm^2}\). Up to these limits all the lines of dependence run more gently than within the pressure limits of 1–3 \(\frac{kg}{cm^2}\), which are of greater practical interest, since they include the actual pressures on ballast from moving loads. We shall henceforth call the coefficient of each ballast, obtained from the graphs, up to the limit at which its elasticity disappears, \(C\), and further, \(C'\).

The predominant coefficient in the experiments was \(C = 2\), while the limits of variation for \(C'\) were from 1.5 to 3.5 (see Table 1).

Let us consider two extreme types, between which every ballast lies—crushed stone and fine sand. A crushed-stone layer, with sufficient width of the sleeper bearing surface, is somewhat elastic (for example, under a pressure of \(0.4 \frac{\text{kg}}{\text{cm}^2}\) the settlement was 4 mm; upon removal of the pressure the settlement became 3 mm). At pressures above \(1 \frac{\text{kg}}{\text{cm}^2}\), the elastic part of the deformation becomes considerably smaller. With a small width of the sleeper bearing surface, elastic deformation is not observed at all.

Fine sand is not elastic at all. Settlement of the sleeper after removal of the load is observed only by the amount of compression in the transverse direction of the sleeper itself. With coarse material, the resistance of the ballast, i.e. the coefficient \(C'\), depends on the width of the sleeper bed, but not to such a significant degree as is found for fine materials.

From the experiments it appeared that the greatest \(C'\) was given by ballast with an average grain size of 5 mm (ballast from Section 9, No. 1—\(C' = 3.5\)). Materials having a greater average coarseness but containing a larger admixture of fine sand, i.e. materials, so to speak, less uniform in grain size, gave smaller values for \(C'\). For gravel, as well as for coarse shingle, \(C'\) was found to be less than for coarse crushed stone.

The ballast of Section 14, despite its considerable coarseness (1.8 mm), gave comparatively worse results than fine sand. The composition of the ballast of Section 14 is as follows: shingle 14%, gravel 18%, and sand 68%. The ballast of Section 9, No. 1 gave better results than Section 9, No. 2, although the average coarseness of No. 2 is somewhat greater; but it turns out that in the first the admixture of sand is only 4%, whereas in the second it is 27%. The ballast of Section 5, No. 1 is somewhat finer than that of Section 5, No. 2, yet it showed greater resistance, since it contains less gravel. The Tomsk ballast, the coarser one, gave worse results than the finer one—in view of the homogeneity of the latter. I do not observe that an admixture of clay had an influence on \(C'\) in the case where it did not have any significant influence on homogeneity in the sense of coarseness.

If the sleeper bearing surface is reduced, \(C'\), generally speaking, falls rather considerably, but in ballasts of average quality a small fluctuation in the width of the bearing surface is not noticeably reflected in the resistance. With an increase in the thickness of the layer, \(C'\) increases, but very slowly.

Wetted ballast becomes still less elastic; its resistance is generally less than that of dry ballast. The compaction of sand has an influence on its resistance. This is especially noticeable for fine ...

sand. Repeated rolling and preliminary compaction of the ballast entails an increase in resistance to the same extent as for dry ballast.

§ 24. Experiments capable of yielding rigorously verified scientific propositions must be arranged almost as they are carried out in laboratories, i.e., by placing the materials for various separate tests under identical conditions and observing the influence of changes in some factors or others.

This was borne in mind by Schubert, who conducted his experiments with models (at 1/40 of full size) of railway track. Besides the incompleteness of such a study in so complex a matter as the deformation of railway track under the action of static and dynamic loads, the chief obstacle to determining the properties of ballast by such a method is that, in the models, the size of the ballast grains is incommensurable with the dimensions of the other parts of the track. Most of the experiments on which the study of the permanent way is now based have been carried out by moving locomotives over a point selected for observation on a track in service. These experiments have hitherto yielded few precise results and, not being fully systematic, have given only a general idea of the oscillations of the upper part of the roadbed and of the rails during the passage of rolling stock.

For the solution of many questions relating to the track, and chiefly to the ballast, experiments are necessary on a separate railway track provided for this purpose, or specially arranged in exact agreement and to the scale of an actual railway track. Loading the track with a moving locomotive is necessary for studying the character of the oscillations of the railway track; but to obtain the most important primary data—namely, on the deformations of the track under static loading—it is preferable to place a press with a manometer over the experimental track. This will make it possible to apply the force with greater accuracy at the designated points of the track, and, most importantly, to measure the magnitude of the load precisely and to vary it, which is impossible when a locomotive is used as the load. The installation of precise apparatus for measuring the resulting deformations is readily practicable only under the condition of such a separate track with a press. On this track one may trace the influence of the rail profile by carrying out the same experiments with rails of different weights; by changing the ballast, one may accurately determine the significance of its various properties for the stability of the track; finally, one may determine the limit of propagation within the embankment of any noticeable

The necessity of systematic observations at a special experimental station on the influence of static loading on various parts of the permanent way.

oscillations under blows against the rails, and also the degree of strength of the ballast layer under the influence of repeated loads. A diagram of such an experimental station, designed at the Office of Civil Engineering and Roads at the Tomsk Technological Institute, is presented in Drawing 16. Under a special shed an embankment is arranged, about 2 sazhens high, 2.60 sazhens wide at the top, with slopes of 1:1.5, and a platform for placing a hydraulic press with a manometer (the Collet dynamometer mentioned at the beginning of the present article, shown in Drawing 3, may also be used). The bottom of the platform is located at a height of about 2½ sazhens above ground level.

The press must be sufficiently movable that the force can be applied to any point over a distance of not less than 2 spans between the sleepers. The board to which its upper part is fastened slides in another similar cast-iron board, and they move in the transverse direction between guiding iron beams. In accordance with the load of 10 tons of a normal locomotive wheel now being introduced for track calculation, the press can be set with a force of up to 30 tons, and the corresponding parts of the frame and platform structure, which is assumed to be entirely wooden, on piles, are designed for this force.

Independently of these experiments with static loading, it is necessary to carry out on the track, according to a program developed on the basis of the results of the first experiments, a whole series of observations of the oscillation and deformation of the track during and after the passage over it of various loads moving at different speeds. This is intended to be done at several points of the Siberian Railway.

V. Information on Ballasts of Foreign Railways.

The technical aspect of the question. § 25. The question of ballast was considered specially for the first time at the railway congress in 1900. There are no separate treatises on this subject. It is presented in somewhat greater detail in several general works on railway engineering, and two reports were printed in the bulletins of the above-mentioned Congresses: one by Bauchal, containing information on ballasts mainly of European railways, and another by Feldpauch on the ballast of American railways. To these reports was attached, as an appendix, a note by Basyutinsky on his experiments of 1898 on the Warsaw–Vienna Railway.

Some information on ballast on foreign railways is found in the reports to the London Railway Congress by Ast.

and Hunt on the question of the normal type of track superstructure for railways on which high-speed trains run.

The thickness of the ballast beneath the bottom of the sleeper on various railways is as follows:

On Austrian railways—0.10—0.24—0.30 m.

On French railways—0.22—0.25—0.30 m, and under metal sleepers 0.50 m.

On Belgian railways—0.34 m.

On English railways—0.45 m.

In most cases the ballast consists of coarse gravel or crushed stone.

On the Dutch railways the upper part of the ballast consists of sand, and the lower part, 10—15 cm thick, of gravel.

On all foreign railways special attention is paid to draining the base of the ballast layer by means of drainage, and to ensuring that the ballast is absolutely permeable to water, while the base of the ballast layer, i.e. the surface of the roadbed, is impermeable to water. For his part, Ast believes that the thickness of the ballast layer should depend on the properties of the soil beneath the ballast. For maximum resistance, in his opinion, the thickness should be not less than 0.4 m, and under the bottom of the sleepers not less than 0.3 m. From Hunt’s report it follows that in England the ballast consists for the most part of two layers—the lower and the upper—the upper ballast serving as the bedding for the sleepers. The lower ballast is always coarser than the upper and consists of stone laid by hand, or of coarse crushed stone, and serves chiefly for drainage; while the upper ballast, finer, consists of gravel, slag, fine crushed stone, or ash. The thickness of the lower ballast layer is 12″, that of the upper 6—12″, with the sleeper either not covered by ballast at all, or covered only to a thickness of 1″—2″. Thus the thickness of the ballast up to the bottom of the rails is 17—22″; and if the thickness of the sleepers is taken as 5″, then the height of ballast beneath the bottom varies from 12 to 17″ (0.14 sazhen to 0.20 sazhen).

In America the lower ballast layer, consisting of stone or crushed stone, is 6—9″ thick; the upper, consisting of gravel or fine crushed stone, is 5—12″ thick, measured from the top of the sleepers. The total thickness of both layers varies from 13 to 21″, and, after subtracting the thickness of the sleepers, is 7—15″ (0.083—0.18 sazhen).

Let us also cite some information on the actual condition of ballast abroad and on the requirements imposed there upon it.

The opinion established among Russian engineers regarding the quality of ballasts on the roads of Western Europe is that it is to them, chiefly, that foreign countries owe the possibility of safely permitting such high speeds of traffic and such considerable steadiness in the movement of trains. On the main lines abroad, indeed, the ballast is excellent and consists of material so coarse that it wears very little even when metal sleepers are used. For ballast there, besides natural materials, artificial ones are also employed. The former are used either after a certain processing (stone is broken into crushed stone, pebble and gravel are screened and sometimes washed), or directly in the form in which they are obtained from quarries (gravel, sand), if they are very clean and of uniform grain. Crushed stone for the upper structure of the railway track¹ is prepared from the same sizes and by the same methods as for highways, often at special plants, of which there are especially many in Belgium. The choice between hand breaking and machine breaking depends on local conditions. Stone crushers usually give the pieces of crushed stone an elongated form, but this defect can be eliminated by making the material pass through two mouths with jaws set at right angles to one another.

In America, movable stone crushers are used, mounted on railway platform cars. It is useful to have such movable machines in order to be able to break crushed stone in those places on the line where it turns out to have been delivered in excessively large pieces. The softer the rock and the less well it resists atmospheric influences, the larger the pieces must be. The best size is considered to be a measured end of 4–6 cm, but a ring of 8 cm is often used. All placers of sufficiently hard rocks are used directly as ballast—pebble, gravel, sand. For stones admixed with ordinary ballast, the maximum permissible higher size is taken to be approximately 10 cm; in the case of limestone rocks, and 6 cm for especially hard ones. The largest stones are sometimes broken; sometimes such stones go into the bedding under the ballast layer or into the drainage of the upper part of the roadbed.

Pebble, or individual small stones, make for ballast that is too mobile. The qualities of such material can be improved by breaking the larger parts. If there is a mixture of small stones and gravel, a certain admixture of sand does not lower the quality of the ballast. If there is much small stone in the mixture—

gravel, and little gravel; a considerable admixture of sand is desirable. If such a mixture is extracted from under water, it is worse for ballast than that from a dry quarry, since the water washes away all the finer particles and the mixture has too great a mobility. When the sand content is too high, it is useful to screen it out.

The clay contained in sand intended for ballast should be screened out or washed away. For washing, it is sometimes more convenient to flood the quarry.

One and the same machine can extract the sand, screen it, wash it, and remove the residue. An excavator with a rotating screen, used on the Paris–Lyon line, is described in Revue gén. de ch. d. f. for 1886. In the same journal for 1893, an apparatus for washing ballast for the Paris circular railway is described.

On many Austrian, Swiss, and French railways, they use exclusively grit, gravel, pebbles, crushed stone, or, in the extreme case, coarse-grained sand, since the richest subsoil deposits of conglomerates there provide almost everywhere an immense supply of excellent ballast; and this alone can explain the remarkable stability of the aforementioned roads.

Artificial ballasts are made either from materials specially manufactured for this purpose, or from the residues of factory production. In England and America, burnt clay has long been used. The burnt lumps are broken into pieces of approximately 3 inches.

The Chicago—Burlington and Quincy Railroad Company uses burnt-clay ballast on one branch, because good sand or gravel ballast cannot be obtained there; the artificial ballast, in practice, gives excellent results and is comparatively inexpensive. It has also been laid in considerable quantities on other roads in the United States.

To prepare this ballast, the ground is cleared for a length of from 50 to 150 fathoms, depending on the length of the ballast train, and a fire is made over this area. A layer of fine coal is laid on top, then a layer of clay 2–3 inches thick, again a layer of coal, on top again a layer of clay, and so on, in the proportion of 1 ton of coal to 6 cubic meters of ballast. After firing is completed, the heap has a height of up to 8 feet and a width of 20–30 feet; the firing continues for 4–5 months, during which a small gang of workers supervises the proper progress of the firing. One gang of workers burns an amount of ballast of 18–20 thousand cubic meters, and its cost, with loading onto platforms, comes to as much as 1½ dollars per 1 cubic meter. This ballast does not crumble and, after the first rain, ceases to give off dust (Railroad Gazette, 1886).

Slags (blast-furnace and locomotive) in a layer of 35 cm under the sleeper constitute an excellent ballast; the upper layer must often be changed. Locomotive slags are suitable for sidings; blast-furnace or glass slags are also good for main tracks.

Schubert, from his experiments, derived the dependence of the performance of sleepers of different types on the quality of the ballast (Zeitschr. d. Bauw. 1896 and 97, and Organ 1897).

He found that, owing to tamping and the passage of trains, a compacted part is formed under the sleeper in the ballast, the height and width of which depend on the distance between the sleepers and on the type of ballast. The more frequent the sleepers, the more firmly they sit and the less labor and material are required for repair.

Crushed stone of hard rocks gives results three times better in terms of expenditure of labor and six times better as regards repair material. Small stone of semi-flat form is better than a collection of large cubes.

Crushed stone has the advantage that its tamping is possible at any time, regardless of the weather; it requires greater effort, but lasts longer and the layer retains its elasticity longer; moreover, the layer holds with steeper slopes and, for the same height, occupies a smaller width. Sand must be covered with coarser material; its tamping is difficult in wet weather and in very dry weather. Tamping is carried out approximately 0.25 sazhen to each side of the rail.

If the middle were tamped, then, after disturbance of the tamping at the rails, the sleeper would begin to rock about the middle and might crack. With fine ballast, tamping is also extended to the ends of the sleepers, leaving the middle untamped. Usually the joint sleepers are tamped more strongly¹.

Very large pieces must be broken up, since the properties of the ballast depend on uniform composition, and the sleeper may dance when resting on large pieces.

Replacement of ballast (screening or washing out the earthy particles), with replenishment of the layer, should be carried out, as the practice of French railways has shown, every 15–20 years.

To increase the friction of the sleepers against the ballast and their resistance to lateral displacement, it is useful to widen the layer or to fill over the sleepers from above. Most often the so-called English profile of ballast is applied, i.e., almost without covering the sleepers, while the width of the ballast layer is only ins—

many times greater than the length of the sleepers. Under unfavorable conditions of the profile, strengthening the ballast layer increases the stability of the track, although this increase can be achieved by artificial means.

The annual wear of the ballast layer is from 2 to 10%.

In most notes and reports on ballast we encounter the following opinions:

The profile of the ballast layer should not be the same everywhere, but should depend on the quality of the ballast and on the strength and density of traffic. In drawing 18 are shown the profiles of the superstructure on certain foreign railways. The usual thickness of the ballast layer abroad is considered to be from 20 to 35 cm.

Taking the maximum pressure of one axle as \( \frac{6}{5} \) of its load, considering the pressure on one sleeper from the axle together with the corresponding part of the superstructure to be 20 t, one obtains, for unballasted track, a pressure from the sleeper of about \(3\ \frac{kg}{cm^2}\); with ballast (angle of distribution of pressure \(45^\circ\))—for the upper limit of thickness, 0.67; for the lower, \(1.16\ \frac{kg}{cm^2}\). If the ground poorly withstands the latter pressure, a detour around such a place must be made, or the soil or the subgrade must be replaced. Bauschall says that in general the greatest thickness of ballast should be taken as 35 cm (and even 30 cm). If it is necessary to reduce the pressure on the roadbed by means of a greater height of the ballast layer, then it is advantageous to do this only up to a ballast thickness limit of 45 cm; above this, the pressure on the roadbed must be reduced by another method. German engineers (Blum, on the basis of Schubert’s experiments) consider it useful to give the ballast a height equal to the distance between the sleepers in the clear \(+\ 20\) cm (for clayey roadbed).

Experiments have shown that although the pressure in the ballast does not spread along a straight line at an inclination of \(45^\circ\), but along a curve with convexity downward, nevertheless the sole of transmission of pressure does not differ from that calculated under the first assumption.

After a sufficient duration of service the track comes to a stable condition, so that it can remain without tamping for two years with 60 trains daily, i.e., it can become disarranged by 43,800 trains, or almost 1,314,000 axles.

According to experiments on French railways, the resistance of an unloaded track to lateral displacement over a considerable distance is 7 tons; that of a loaded track is three times as much. The use of side planks connecting the sleepers at the ends increases the lateral resistance to 10 t.

Most of all, the friction of the ballast against the lower bed of the sleepers resists the displacement of the track. With crushed-stone ballast, which cuts into the sleepers, the contact surface is considerably increased. Brière determined the actual lateral effect of a locomotive at 1.4 tons. In moderate cold, ballast protects the roadbed from frost heaves; moreover, on the French eastern railways it has been observed that the same thickness of ballast has different effects depending on its quality. As regards the preservation of sleepers, some consider covering them with ballast useful, others do not. Here much depends on the quality of the timber, climatic conditions, etc.

Covering dusty ballast with crushed stone is practiced especially in India and France. In northern Spain and southern France, ballast is allowed to become overgrown to a certain extent with grass so that the ballast will not produce dust; in America it is often watered with petroleum.

According to Bauchalf, the annual wear of ballast is close to 5%, which amounts, for a single track, to about \(1/12\) cubic meter per running meter, at a cost of 0.3 francs; that is, almost as much as the wear of the rails (reckoning their service life at 50 years) and the cost of sleepers per running meter of track.

In America, crushed-stone ballast is sometimes covered with finer material, which reduces the rattling of rolling stock. In width the ballast extends beyond the ends of the sleepers by 10–20 cm; on sharp curves these shoulders are widened on the outer side and sometimes special reinforcements are also applied.

Metal sleepers require better ballast, and tamping them is more difficult; for them a smaller volume of ballast is required, since their height is less.

With fine ballast, the passage of water through it is impeded; therefore they resort to improving the drainage of water on the outside, giving the upper surface of the ballast a slope from the very middle of the track, so that the ends of the sleepers are partly exposed. Instead of this American type, water drainage can be strengthened by transverse drains or by lining the whole roadbed with large stone.

In America, improvement of ballast in the sense of increasing thickness and raising quality grows simultaneously with the increase in the load and speed of trains. Here the desire to reduce dust and the cost of maintaining the track also plays a role. But to this day on many lines, where good ballast is not available nearby, sandy or clayey earth is used instead.

In America, devices for the rapid unloading of platforms with ballast are in greater use; they consist of a plow in the form of a triang—

... frame, moving successively over all the platforms by means of a rope and a shaft installed on the locomotive or on the platform nearest the locomotive.

In America the shortcomings of earth ballast have been recognized, and the chief one—the penetration of water into the roadbed—and a strengthened transition to good ballast has been noted. With ballast that is little permeable to water, the sleepers become loose and sink into the softened roadbed, and it is difficult to keep the track in proper condition.

It was thought that it was always good to place larger stones under crushed stone, but this proved incorrect, since earth accumulates between the large pieces and retains moisture.

On the Pennsylvania Railroad experiments were made to compare the suitability of ballast made of crushed stone of different sizes: 2, 4, and 6 cm. The most commonly used size, 6 cm, appears to be the worst, and alignment of the track with it is more difficult. The medium size proved to be the best.

Let us dwell further on the question of what qualities are required of ballast if iron sleepers are used.

Observations of the ballast layer on German railways show that, with a metal superstructure, far more attention must be paid to the quality of the ballast than with wooden sleepers.

Sections of track of considerable length, laid on wooden sleepers and on ballast of average quality, have their superstructure on a dry foundation; whereas with metal sleepers, owing to their greater elasticity, when trains pass there occurs a kind of pumping of moisture from the lower layers into the upper ones, from which ballast of average quality is transformed into a liquid mass, not providing a good foundation for the superstructure. Even with wooden sleepers the ballast may turn into mud, but this happens only when it has already become completely impermeable to water and therefore must be replaced by another. The causes of the above-mentioned phenomenon are as follows:

1) Metal sleepers bend more than wooden ones.

2) Owing to the firm connection of the rails with the metal sleepers, the latter follow all movements of the rails during bending; with wooden sleepers, however, there is a small amount of play between the rail foot and the heads of the spikes, and the sleeper itself is compressible.

3) The very shape of the metal sleeper promotes the formation of a void beneath it and, consequently, the pumping of moisture.

4) Wooden sleepers are laid much deeper in the ballast than metal ones.

Hence the conclusion that, with a metal superstructure, special measures must be taken to improve the ballast, which must possess the following qualities:

1) It must be capable, when tamped, of compacting under the sleepers into a dense mass, providing a reliable foundation for the sleeper, while at the same time not possessing the ability to scatter easily.

2) It must transmit uniform pressure to the roadbed, offer the greatest resistance to the sliding of the sleepers upon it, and readily allow water to pass through it.

The best ballast in this respect is broken stone, followed by gravel and sand. In Germany, broken-stone ballast is used, generally speaking, in small quantities, owing to its costliness in comparison with gravel and sand.

In the journal of the Special Conference under the Ministry of Ways of Communication of July 23, 1901, on the question of introducing iron sleepers on Russian railways, the conclusion of Engineer Renson is cited: that preference should be given to wooden sleepers over metal ones in the following cases:

1) on a poorly drained roadbed,
2) on a new embankment that has not yet settled,
3) where the soil of the roadbed is marshy, and
4) on ballast with low water permeability.

A metal sleeper, by its sharp edges and hard surface, destroys and grinds down the ballast very rapidly if it consists of soft rock.

Depending on the character of the rail fastenings to the iron sleepers, which do not permit displacement of the fastened parts, the track on iron sleepers must be more unyielding than on wooden sleepers; that is, the ballast must be absolutely good and the rails heavy.

Proceeding from the fact that in Russia the ballast is considerably worse than abroad, and the rails are lighter, the aforementioned Conference expressed doubt as to the expediency of switching to iron sleepers on Russian railways.

To conclude the technical side of the question of ballasts on foreign railways, we shall also cite information about one fairly complete method of laboratory testing of materials intended for ballast.

In Mittheilungen d. Königl. Technischen Versuchsanstalten for 1897 there is described a method used in that laboratory by Rudeloff for testing gravel and crushed stone intended for railway ballast. In testing, Rudeloff attempted, as far as possible, to place the ballast under the same conditions in which it works in the track, and to study it with respect to: 1) the blows it receives during tamping, 2) wear and destruction, depending on the movements of the sleepers during the passage of trains, 3) vertical pressure from the wheels of the locomotive, and 4) atmospheric agents.

To test the destructibility of ballast under tamping, a definite portion of it was poured into a box with thick walls and placed under a cement rammer, replacing in it the round rammer with a tip fashioned in the form of a tamper (dwg. 19). The two sides of the box rest on strong springs so that, after a blow, the ballast yields and the empty space is filled by another portion of it, just as occurs when tamping the track. After a known number of blows, the size of the material was determined by sieving, and the change in composition serves as a measure of the resistance of the given material.

In order to determine the effect on ballast of the repeated application of pressure from sleepers and of the friction of the latter, the same box, now without middle walls on springs, was filled with the material being tested, covered with a thick board, and blows were dealt upon it with a heavy rammer.

In addition, the material was tested in compression under high pressure (up to 100 tons) in a pipe with thick walls, by means of a piston fitted to it; the material was tested: 1) dry, 2) after being for not less than 2 hours under water, 3) frozen, and 4) moist, having been subjected to repeated freezing.

§ 26. In the literature, the undoubtedly prevailing opinion is that even a considerable difference in the cost of acquiring good ballast, as compared with poor ballast, should not force a departure from the general rule—to use ballast of the very highest qualities. But there are indications that in practice another rule finds application—that the most expedient ballast is the one at hand, i.e. always cheaper, provided, of course, that the principal requirements for ballast are met; this is adhered to even by some experienced engineers. Where high requirements are imposed, such as are made of the track abroad, where it is possible either to arrange washing of the material available on site for ballast or to bring the latter from distant places, it is there an unforgivable mistake to build a line and supply it with ballast of poor—

Economic calculations of the advantages of using ballast of higher qualities.

high-quality ones. In general, the rule followed there is to reduce the amount of labor for track maintenance by using ballast of higher quality. Engineer Gerdenstein writes (Zheleznodorozhnoe delo, 1886) that in France and Switzerland, on all sections where it was necessary to deal with ballast containing earthy admixtures, it was decided to get rid of the latter by means of thorough screening. The cost of ballasting, of course, increased considerably, but in return a perfectly dried roadbed was obtained. For cleaning and washing ballast, various kinds of rotating excavators with screens are in use. In Revue Gén. des chm. d. f. for 1886 there is an article by Pieron and Gasnier, in which they prove by figures that it is always more economical to use unpolluted ballast. We shall cite the figures used by the authors mentioned, while noting that they may differ considerably from the corresponding figures for Russia.

First of all, they compiled a table (given below) containing information on the expenditure of labor for track maintenance on many railway lines with ballast of various qualities. From this table it may be concluded that, with poor ballast, the maintenance of 1 kilometer of track requires, in round numbers, 200 workdays per year; with ballast of medium quality, 150; and with good natural ballast, or ballast cleaned of admixtures, 100 days. Consequently, the transition from poor ballast to good gives a 50% saving, or, reckoning a workday at 3.50 francs, a saving of 350 francs per kilometer per year, which corresponds to a capital of 7,000 francs.

Name of line	Length of observed section, km	Number of pairs of passenger trains	Number of pairs of freight trains	Weight of rail, kg per meter	Worker-days per year and kilometer: with poor ballast, double track	Worker-days per year and kilometer: with poor ballast, single track	Worker-days per year and kilometer: with ballast of medium quality, double track	Worker-days per year and kilometer: with ballast of medium quality, single track	Worker-days per year and kilometer: with new or cleaned ballast, double track	Worker-days per year and kilometer: with new or cleaned ballast, single track	Notes
Lines in two tracks	0.78	37	60	30	380	—	—	—	205	—	Maintenance work consisted of: dismantling the track, lifting, tamping, correcting cuts, correcting the ballast cushion, shifting switches and switches, track straightening, filling and leveling the ballast.
Lines in two tracks	7.8	32	57	—	—	—	276	—	—	—	Maintenance work consisted of: dismantling the track, lifting, tamping, correcting cuts, correcting the ballast cushion, shifting switches and switches, track straightening, filling and leveling the ballast.
Lines in two tracks	10.3	—	—	—	—	—	—	—	200	—	Maintenance work consisted of: dismantling the track, lifting, tamping, correcting cuts, correcting the ballast cushion, shifting switches and switches, track straightening, filling and leveling the ballast.
Lines in two tracks	—	—	—	—	—	210	—	—	—	—	Maintenance work consisted of: dismantling the track, lifting, tamping, correcting cuts, correcting the ballast cushion, shifting switches and switches, track straightening, filling and leveling the ballast.
Lines in two tracks	4.43	30	38	—	—	—	240	—	—	—	Maintenance work consisted of: dismantling the track, lifting, tamping, correcting cuts, correcting the ballast cushion, shifting switches and switches, track straightening, filling and leveling the ballast.
Lines in two tracks	11.69	32	25	—	372	—	—	—	210	—	Maintenance work consisted of: dismantling the track, lifting, tamping, correcting cuts, correcting the ballast cushion, shifting switches and switches, track straightening, filling and leveling the ballast.
Lines in two tracks	1.16	22	26	—	480	—	—	—	—	—	Maintenance work consisted of: dismantling the track, lifting, tamping, correcting cuts, correcting the ballast cushion, shifting switches and switches, track straightening, filling and leveling the ballast.
Lines in two tracks	4.3	—	—	—	—	—	290	—	—	—	Maintenance work consisted of: dismantling the track, lifting, tamping, correcting cuts, correcting the ballast cushion, shifting switches and switches, track straightening, filling and leveling the ballast.
On average . . . .	—	—	—	—	410	—	268	—	205	—	—

Investigation of Ballasts

Line name	Density of traffic, section km	Number of train pairs: passenger	Number of train pairs: freight	Rail weight, kg per meter	Working days per year and km: with poor ballast, double track	Working days per year and km: with poor ballast, single track	Working days per year and km: with ballast of average quality, double track	Working days per year and km: with ballast of average quality, single track	Working days per year and km: with new or cleaned ballast, double track	Working days per year and km: with new or cleaned ballast, single track	Note
Line No. 1, double track	40.09	8	11	—	—	—	—	155	—	—
Line No. 1, double track	22.81	12	16	—	—	—	—	145	—	—
Line No. 1, double track	5	10	10	—	—	—	—	150	—	88
Line No. 1, double track	On average . . .				—	—	—	150	—	88
Single-track line	12.10	4	2	—	—	—	—	164	—	—
Single-track line	19.39	5	12	—	—	—	—	—	—	123	Blast-furnace slag ballast.
Single-track line	15.7	5	3	—	—	—	—	—	—	97
Single-track line	13.7	8	9	37	—	—	—	—	—	92
Single-track line	43.25	8	3	—	—	—	—	—	—	95	Ballast from small slag.
Single-track line	5.4	4	3	—	—	—	—	139	—	72
Single-track line	8.61	6	1	30	—	—	—	132	—	—
Single-track line	On average . . .				—	—	—	145	—	96

Assuming that replacing poor ballast with good ballast gives a saving of 7,000 francs for each kilometer, we see that, apart from other advantages, from the purely commercial point of view it is proper to choose ballast of high quality if the price of a cubic meter of such ballast exceeds the price of poor ballast by no more than \(7000 : 1825 = 3.85\) francs. Here it is assumed that 1,825 cubic meters of ballast are needed for ballasting one kilometer of track. It remains to determine how much it may cost to replace one ballast with another. If the material at hand for ballast is a mixture of pebbles or coarse gravel with fine sand, an excellent result can be achieved by screening the mixture and washing out all the sand. This may cost 0.50–1 franc per cubic meter. If this method is inapplicable, the ballast must be brought from distant quarries. The authors estimate the delivery of a cubic meter of ballast over a distance of one kilometer at 0.03 franc (the cost of hauling a train of 20 cars, each of 5.5 cubic meters, with the return of the empty cars over a distance of 46 kilometers).

cost 163.40 francs; consequently, the cost per cubic meter and kilometer is \(163.4 : 46.110 = 0.03\) francs. Thus, for economic reasons, the choice may fall upon a quarry with poor ballast only in the event that good ballast is located farther away by \(3.85 : 0.03 = 128\) kilometers or more.

On an existing line with poor ballast, it remains to decide whether it is more advantageous to clean the ballast with the addition of the corresponding portion, or to replace all the ballast with new. According to the observations of the said authors, the former costs 6,570 francs per kilometer, and the latter 6,940 francs, i.e. less than the saving of 7,000 francs in maintenance work which results from the use of good ballast.

VI. Characterization of the Ballasts of Russian Railways

Technical aspect of the question. § 27. The condition of the ballast layer on Russian railways leaves much to be desired. There are official indications of this. For example, the report of the Department of Railways to the Minister of Ways of Communication, dated September 22, 1897, No. 1896, “On the improvement and replenishment of the ballast layer on state-owned and private railways,” begins with the words:

“Bearing in mind that on certain lines of the railway network the ballast is present in insufficient quantity or is of improper quality, by order of the Minister of Ways of Communication it was proposed to the Chiefs of state-owned railways and to the Managers of private railways to submit to the Department the following information:

1) on which versts the ballast layer must be replenished because of its insufficiency;

2) on which versts there is poor-quality ballast requiring replacement; and

3) on which versts the ballast layer must be covered with crushed stone.”

From the information delivered in response to this inquiry for certain railways, it is evident that on many lines the ballast is fine, sandy, in places clayey (e.g., the Polesye Railway); on other railways it is fine, silty, clayey sand (Baltic).

There are railways on which, in places, the height of the ballast layer is less than 0.15 sazhen.

On some railways the ballast is generally good (on the Vladikavkaz Railway, screened gravel of medium size, or a mixture of gravel with sand).

The predominant type of ballast in Russia is fine sand. Where this sand is not too fine and the admixture of clay in it is very small (Ivanogorod–Dombrovskaya line), the results obtained are satisfactory. On many lines the sand is too fine (Moscow–Kiev–Voronezh, certain lines of the Ryazan–Ural Railway, etc.). With heavier traffic on such lines it becomes necessary to cover the ballast with a layer of crushed stone. On the Rybinsk Railway part of the route is covered with small stone obtained when extracting ballast from quarries. Lines with intensified traffic are covered with a crushed-stone top ballast even in cases where the lower ballast consists of coarse sand (for example, the Nikolaev Railway, Moscow–Kursk).

On a very large number of lines a deficiency in the height of the ballast layer is felt, estimated at 10–20%.

The cost of ballast fluctuates within narrow limits, from 2 to 5 rubles. When replacing poor-quality ballast with new ballast, owing to the difficulty of this work without interrupting traffic, the cost per cubic sazhen reaches 10 rubles.

For the upper crushed-stone ballast, 30–50 cubic sazhens of crushed stone are required per verst of track, which, at its price of 25–30 rubles, amounts in round figures to 1000 rubles per verst of single track.

On Russian railways, in most cases, sandy materials are used for ballast, such as are at hand; and Russia, as is known, is rich partly in river sand, and chiefly in fine-grained ravine loose sand, easily shifting, blown out by winds, and still more easily washed out by rains and intermittent downpours. With frequent settlement of the ballast layer and untimely replenishment, the track is usually kept at the design level by means of an annually added layer of ballast.

The profile of the normal ballast layer for Russian railways is established in the technical conditions for designing main lines. The thickness of the ballast layer is specified on open line sections, counting the thickness from the foot of the rails along the rail lines: not less than 0.25 sazhen in dry cuttings or on embankments of sandy soil, when the height of such is not more than 1 sazhen. On station and siding tracks, and likewise between tracks, the thickness of the ballast layer may be reduced to 0.23 sazhen.

With ballast of pebbles or fine crushed stone, the indicated thickness of the ballast layer both on open line sections and at stations may be reduced by 0.05 sazhen. With clayey soils and generally viscous soils, [the text continues].

the thickness of the ballast layer under the rail tracks must be correspondingly increased. The width of the ballast layer at the level of the rail base must be not less than 1.45 sazhen, and the slope of the side slopes not less than one and a half to one.

All roads should in time pass over to such a profile, but at present there is a great variety of profiles, depending on the conditions of construction of the given line and partly on the conditions of operation, under the influence of which the original profiles of the ballast layer may have changed. Increasing the ballast profile on old embankments is in general very difficult, since widening, as well as raising the ballast layer, is associated with a reduction or complete disappearance of the berms on the upper edge of the roadbed. The latter entails serious inconveniences, since the ballast profile then poorly maintains its proper form and much material slips down the slope, sometimes filling the ditches and drains.

Recently the thickness of ballast on Russian railways has most often been set at 0.22 sazhen under the rail, with a layer volume per verst of 160 cubic sazhen. From many drawings it is evident that the width of the layer along the top on the main lines of Russia is generally sufficient: for single tracks it is about 3.1–3.4 m; there are some lines (for example, Ekaterinburg–Chelyabinsk) with a ballast width along the top of 2.76 m (1.29 s.). The height of the layer along the greater part is 0.40–0.47 m¹ and even (Nikolaev Railway) 0.75 m. On some lines the thickness under the rails is only 0.25 m (Ekaterinburg–Chelyabinsk), 0.37 (Vladikavkaz Railway), etc. Covering with crushed stone is practiced only exceptionally on lines with heavy traffic; the thickness of the crushed-stone layer is about 0.1–0.12 m, and the size of the crushed stone is 0.04–0.06 m.

In 1897, the Advisory Congress of Engineers of the Track Service resolved to consider it necessary to bring the thickness of ballast under the base of the sleeper up to 0.20 sazhen. In 1898 the Moscow–Vindava–Rybinsk Railway Society, in the draft Technical Conditions for construction of the Moscow–Vindava line, proposed a ballast thickness of 0.16 sazhen under the base of the rail. This question was then examined in the Engineering Council of the Ministry of Ways of Communication (the rapporteur was L. F. Nikolai). The Congress, arriving at the above-mentioned conclusion regarding a ballast thickness of 0.20 sazhen, had chiefly in view the protection of the roadbed, and especially of embankments, from ballast troughs, while recognizing that, in order to ensure a uniform transmission of pressure to the roadbed and consequently to protect it from ballast troughs—

—the thickness of the ballast must be not less than 0.20 sazhen under the best soil conditions.

At the same time, the Congress resolved: in order to ensure, as far as possible, good drainage from the roadbed of water seeping through the ballast, in view of the almost universally used method of ballasting the track after the preliminary opening of working or temporary traffic over an unballasted track, slopes should be made in the cross-section of the roadbed from the line of the underside of the outer ends of the sleepers, giving them gradients: in dry and sandy soils of 0.10, and in fat clayey soils—up to 0.20.

Thus the Congress establishes the thickness of the ballast layer only from the point of view of protecting the roadbed from the formation of ballast troughs.

There are also other factors on which the thickness of the ballast layer should depend, namely: speed and axle load, in connection with the type of rail and the spacing between sleepers.

Ast, in his report to the London Congress, concludes that, in order for the track to reach the limiting value of resistance (limite supérieure de la capacité de service), the thickness of the ballast layer must be not less than 0.4 meter, and beneath the underside of the sleepers not less than 0.3 m. In general, he considers that the thickness of the ballast layer should depend on the properties of the soil beneath the ballast.

L. F. Nikolai, in his report to the Engineering Council on the question of the normal thickness of ballast, indicates that the higher the quality of the ballast (the coefficient \(C\) itself depends on the sufficient firmness of the subgrade soil, the qualities of the ballast, and its sufficient width) and the more favorable the soil conditions, the smaller the thickness of the ballast may be. A sufficient height of ballast has yet another important significance, as was indicated in the resolution of the XIV Congress—to prevent the formation of frost heaves.

Stetsevich believes that the smallest thickness of the ballast layer beneath the underside of the sleepers should be 0.15 sazhen in cuttings, and in embankments up to 0.17 sazhen.

As conditions for determining the minimum thickness of ballast, L. F. Nikolai takes the following: the pressure transmitted from the underside of the sleeper to the roadbed must not exceed \(1.5 \frac{\text{kg}}{\text{cm}^{2}}\) (0.6 p. per sq. dm.); at the same time he takes, for sandy ballast, a half angle of distribution of pressure, and for ballast of small broken stone or coarse gravel—a single one. For the calculation it is assumed that the pressure of the axle is entirely received by one sleeper of normal dimensions. This method of calculation, independently of the type of rail, is justified by the fact that, judging

according to theoretical investigations, the pressure on the joint sleepers reaches up to \(1.09 P\), and, in addition, allowance must be made in the calculation for overload in motion, which overload is determined by a coefficient of 1.7.

Nikolai does not advise taking the rail section and the distance between sleepers into account in determining the necessary thickness of the ballast layer, using these elements only to reduce the stress in the rail and to give the track greater rigidity both in the vertical and in the horizontal directions.

On these data Nikolai determined the minimum thickness, for axle pressures of up to 12 tons, with firm ground—for sand ballast as 0.15 sazhen and for crushed-stone ballast as 0.10 sazhen.

These standards are accepted by him only for the roadbed in dry cuttings and on embankments not higher than 0.5 sazhen. For damp cuttings, and also for embankments higher than 0.5 sazhen, with the exception of embankments of sandy soil, he determines the minimum height of ballast as 0.17 and, respectively, 0.12 sazhen.

At the same time the presenter stipulates that the speed of movement is assumed to be up to 50 versts per hour, the width of the ballast layer at the top 1.45 sazhen, and that with an increase in axle load the thickness of the ballast layer must be correspondingly increased.

As for the question of whether, in the construction of lines, the standard height of the ballast layer should be attained at once, Engineer Stevich thinks that laying the ballast layer of the proper dimensions at the very beginning of construction of the line is of very substantial importance, both with respect to the strength of the track and its gradual strengthening, and chiefly in view of the difficulty, subsequently, during operation of the lines, of adding to the ballast layer¹. In view of this, it would be better, when opening temporary traffic, to leave the roadbed entirely without a ballast layer than to make one of insufficient dimensions. On the other hand, in his opinion, an excessive thickness of the ballast layer brings no substantial benefit, as was shown by the experience of the Riga–Orel and Riga–Dünaburg railways.

Let us cite some further information on the use in Russia of slags instead of sand.

Blast-furnace slags were used in places as ballast, but chiefly on access tracks; therefore there is little experience in this respect. In general it has become clear that:

1) the water permeability of slag is entirely satisfactory; a track laid on slag is washed out with more difficulty than one laid on fine sand;

2) In stability, a track laid on slag is not inferior to a track on fine ballast.

3) Although, by specific weight, coal slag is lighter than ballast, nevertheless, owing to the large size of the particles of which the slag consists, and its property of packing together better, it is blown away by the wind to a lesser degree in comparison with fine ballast.

As regards the use of hard-coal slag, it follows that for main lines it proves to be a weak material.

The 7th Deliberative Congress of Engineers of the Track Service of 1889, on the question of the Department of Railways concerning the possibility and expediency of using coal slag for ballasting the track, resolved: a) in view of the presence of sulphurous compounds in coal slag, the latter may exert a harmful influence on the iron parts of the permanent way; b) sleepers that have lain on coal slag become rotten comparatively quickly, and the wood takes on a bluish color; c) owing to the poor thermal conductivity of slag, frost heaves, when it is used instead of ballast, are diminished (Ryazan–Vyazma and Baltic railways).

§ 28. We shall cite some data on the cost of ballasting on Russian railways. Construction estimates and reports give an average cost of 1 cubic sazhen of ballast of 7–10 rubles. The average price of a cubic sazhen of ballast on the Siberian Railway, according to the report of the Mikhailovsky Commission, was determined at 7 r. 53 k. The full cost of a cubic sazhen of ballast on the Siberian Railway during operation is composed of the following component parts: 1) equipment of rolling stock for transporting ballast, leasing of quarries, administration (foremen, quarry supervisors, etc.), bonuses to agents of the Traction and Traffic Services, removal of turf, i.e. of the upper layers of earth covering the ballast at the site, construction of trenches and other auxiliary works—all together costs from 1 r. 20 k. on the western sections to 1 r. 60 k. on the eastern sections; 2) labor for loading the ballast and unloading it at the site costs from 2 r. (in the West) to 3 r. 15 k. (on the eastern sections); 3) transport, at an average distance from quarry to unloading site of 50–70 versts, costs on the western sections about 3 rubles, on the eastern 4 r. (a preferential tariff of 1/200 kopeck per pood-verst is applied; at the normal tariff the expense would be twice as great); here, certain expenses for repairs and amortization of rolling stock are not taken into account.

Fig. 22 shows the disposition of quarries on the line of the Siberian Railway. The greatest distance is between quar-

…sections: Omsk and Krivoshchekovo, namely 1324–746 versts = 578 versts, and the latter quarry serves 500 versts of track, which is a truly colossal figure. Over the entire indicated length there are no quarries near the road; there is one in the middle of the designated stretch, but it lies 40 versts from the road. It was used during construction of the line, but in operation it proved more advantageous to close it, because, owing to its poverty, it was impossible to organize haulage over a long distance and thereby appreciably shorten the carting distance from the main quarries; moreover, special equipment would have been required.

On the Siberian Railway, according to the annual estimates, provision is made for replenishing the ballast layer, which diminishes from wear, blowing away, and washout, at 5–6 cubic sazhens per verst; whereas in fact it is necessary to haul from the quarries for replenishment 14–20 cubic sazhens, i.e. instead of 4%—11% of the ballast present in the track. In addition, an annual appropriation is made for replenishing ballast after damage to it by rainstorms; for this purpose, 2000 cubic sazhens are allotted for the whole railway, i.e. somewhat less than a cubic sazhen per verst of track, whereas rainstorms wash out 4000–5000 cubic sazhens annually on the railway. Almost all of this expenditure for repairing damage to the ballast from rainstorms has to be made on the western sections of the railway, where the ballast is considerably worse. This circumstance clearly indicates the advantage of obtaining better ballast, even at a comparatively greater carting distance.

In very many quarries there is an abundant admixture of pebbles of various sizes in the sand. Often it is dumped onto the track together with the sand, but this should not be done; rather, all large pieces should be screened or picked out and then used for the upper ballast or for packing joint sleepers. In contracts for ballasting on the Siberian Railway we find prices for placing into prisms 1 cubic sazhen of pebbles (with screening from sand, without crushing) from 9 to 14 rubles, depending on the diameter of the stones; almost the same price is assigned for crushing and preparing these stones as broken stone. Loading into cars and unloading pebbles or broken stone costs 6 rubles, whereas the same work for sand costs 2 rubles.

It is very difficult to determine from contracts and reports the actual cost of ballasting with 1 cubic sazhen of ballast, since distribution is carried out by the railway using its own means, with the assignment of haulage expenses to the most varied sources of credit, and some expenses—for example, wear of rolling stock, etc.—are sometimes not taken into account at all. Therefore, to form a fully clear idea of what carting distance is the most advantageous, by and large…

of the greater part is impossible, such questions must most often be decided by eye and on the basis of general considerations, and also with regard to the convenience of the movement of work trains and of the change-over of locomotives.

The measurement of ballast for payment to the contractor is usually made according to the number of wagons; for each type of wagon there is a certain filling, which is determined on the basis of a trial weighing of a wagon loaded with ballast on wagon scales. Three percent of the volume is not subject to payment, as being assigned for unloading. Since the degree of loosening of ballast in passing from the compacted state is not the same, depending on the conditions of bedding of the beds, the quantity of admixtures, and even the degree of moisture (it fluctuates from 10 to 15%), and since, moreover, it is very difficult to see to the proper loading of all wagons, while it is always advantageous to the workers and to the contractor not to fill the wagons to the full measure, this method of measurement leads to misunderstandings and is always disadvantageous to the railway; it ought therefore to be abandoned, and acceptance should be made by measuring the volume selected from the ballast pit in the compacted body. It would be necessary each time to carry out a fairly detailed survey of the quarry by transverse profiles, but in return accuracy of accounting would be achieved. With the present method of accepting ballast, it is not necessary to determine what loosening one or another ballast will have; instead, the theoretical volume of ballast is spread at the indicated place, adding only 5–8% for the compaction of the layer in the embankment.

With the use in Russia of river ballast, and chiefly of fine-grained ravine sand, there is constant settlement of the track; and since the ballast layer is usually not replenished in due time, an average of 70–100 rubles per year is spent on Russian railways for continuous correction of track damage, i.e. for its general repair, for each verst of track.

On good gravelly, slightly clayey ballast, a track that has once been well tamped and corrected at the beginning of summer requires no work until late autumn, although, incidentally, the qualities of ballast that are costly for track stability—coarse grain and, at the same time, a high degree of cementation—greatly impede the replacement of sleepers, since ballast that has compacted into a dense mass does not yield to shovel and spade, and in order to remove a sleeper from it one has to break it up with picks or crowbars.

Assuming that, with a better quality of ballast, it would be possible to save on track repair and on replenishment of the ballast layer half of the above-indicated 100 rubles per verst, now required for the repair of one ballast layer, i.e. 50 rubles annually, one could reckon that,

of which 50 rubles constitute the annual interest, i.e. approximately 1000 rubles, to be spent on improving the quality of the ballast; this means that one may go so far as to nearly double the present average cost of the ballast, if only its quality be better. On the other hand, it is evident that, if we wished to bring better material for ballast from more distant localities, we would have to limit ourselves, for economic reasons, to twice, or at most three times, the distance now employed as the limit. Taking such a distance to be about 1000 versts, we see that in many cases it would be proper to organize, in certain districts of Russia, the preparation of good ballast at central points; but this cannot be done everywhere. For example, in Siberia, good ballast would have to be brought to many parts of the line from distances more considerable than 1000 versts; therefore here it would be necessary to develop a method for cleaning ballast and to try to apply, as far as possible, an appropriate profile both for the roadbed and for the ballast layer. All these questions ought to be resolved as thoroughly as possible, and for this a comprehensive investigation is necessary of the economic side of the question of preparing and transporting ballast on all Russian railways.

CONCLUSION.

§ 29. The most difficult task in the question of the permanent way is to determine the vertical and horizontal oscillations during the movement of loads along the track, and the increase in stresses from dynamic action.

The influence of the speed of movement on railways must, in essence, be the same as on ordinary roads. Thanks to the springs of the vehicles, instead of blows from the falling of wheels into depressions in the roadway, periodic increases of pressure upon it are obtained. On a bad road, as measurements on highways show, springs give about 33% saving in tractive force. The speed of movement plays no role, either in the sense of resistance or in the sense of the strength of the road, when the highway is in good condition; and even driving at a walking pace damages such a highway more than driving at a trot; if, however, the condition of the highway is poor, then the opposite is true.

Dupuit, from his experiments, concluded, among other things, that the tractive force of carts does not depend on speed.

It may be supposed that, with an increase in the stress in the carriageway, appearing during the movement of wheels over it, the resis-

...the development of tractive forces. If small irregularities in the shape of the wheels had a substantial influence on the stresses in the track structure, then this influence at high speeds would manifest itself in a considerable increase of these stresses, and consequently the tractive force of the cars could not fail to depend on their speed. A railway track, in any case, can be compared only with a well-maintained highway; therefore it may be assumed that an increase in the speed of motion, in itself, scarcely increases the vertical pressures on the rails. The increase of stress in the material of the rails from the action of speed, the deflections of the rails that appear at considerable speeds, and the disturbance of the track should be ascribed chiefly to horizontal and twisting forces arising in the rails during the motion of locomotives and cars, and also to the overload of individual wheels, which appears especially sharply during braking and when the speed of motion changes. The increase of stress from an irregular form of the tires, if one judges by analogy with ordinary roads, must be considerably less than the increase of it from the causes indicated above.

The circumstance that very many parts and conditions take part in the work of the superstructure of the track (rails, fastenings, sleepers, ballast, the soil of the roadbed, the more or less close contact of these parts, the nonuniformity of the ballast packing, local deflections and irregularities, the distribution of the load over individual wheels, the elasticity of the springs, and in general the construction of the rolling stock, etc.) indicates that to embrace all the phenomena occurring in the material of the track—especially when a train is moving over it—with any single formula, and to determine exactly the stresses in the individual parts, is inconceivable. But this does not exclude the possibility of finding such formulas as would encompass as many aspects of the phenomenon as possible and would provide grounds for judging the influence of the properties and dimensions of the various parts of the structure, and also, by introducing into these formulas correction coefficients established on the basis of numerous experiments performed, would make it possible to determine, more or less accurately, the actual stresses in the parts of the track and their relation to the allowable stresses. The bending of various beams, even when all possible special conditions under which the various parts of structures and machines work are taken into account, represents very complex phenomena; nevertheless, the theory of elasticity has developed into a coherent science, and the application of its formulas to the calculation of structures renders invaluable services to the engineer. The phenomena occurring in the parts of railway bridges are far more complex than those assumed in calculating bridges for a static load, and, as certain...

...by means of measuring instruments the deformations in the parts of trusses; the formulas are far from embracing the whole complex of phenomena, yet from all this it by no means follows that the present theory of bridges is merely self-deception, or that, as it is perfected, the theory will not illuminate to a sufficient degree the principal aspects of these complex phenomena.

Up to the present time the stresses in rails have been studied most of all, since this part of the track appears at first glance to be the most serious one, directly bearing the moving rolling stock; and since, owing to the closely studied elasticity of steel, it is known with very great accuracy what forces and deformations may be allowed in rails of one type or another. Considerations concerning the deformations of the remaining elements of the superstructure, for the most part, do not go beyond the realm of craft, since there is not a sufficient number of precise observations of these deformations, and it is very difficult to embrace them with a coherent theory. It seems most correct to determine the stresses in the separate parts of the track on the basis of the magnitudes of their changes of form under given conditions.

One cannot fail to note the significance which one or another arrangement of locomotive springs, and in general the whole construction of the rolling stock, has for the rail track. If, by the use of springs on an ordinary road, the necessary tractive force is considerably reduced and wear of the road proceeds much more slowly, then it may be stated with certainty that the strength of the railway track and its resistance to the dynamic action of the load depend to a high degree on the correspondence of the locomotive springs with the weight, the arrangement of the separate parts, and the speed of movement of the locomotive. Thus the solution of the question of track strength depends to a large extent on progress in the design of locomotives. Of serious importance here is the position of the locomotive’s center of gravity.

Raising the center of gravity entails a reduction in damage to the track and in wear of wheels, axles, axle-boxes, and frame. It makes the running of the locomotive smoother, owing to the better play of the springs during lateral oscillations and on curves.

When the wheels of a locomotive strike the rails with the flanges of the tires—whether under the action of centrifugal force while passing through a curve, or as a consequence of the sinuous motion of the locomotive—the force of the blow, other conditions being equal, is the greater the lower the center of gravity. The pressure of the locomotive on the outer rail under the action of centrifugal force or of the sinuous motion is inclined toward the horizontal the more strongly, the higher the center of gravity is, and the greater part of the pressure on the outer rail will be directed vertically. This reduces—

the danger of derailment is reduced, owing to the increase in the load on the outer wheels, while, on the other hand, the spreading action of the locomotive is diminished.

The influence on the stability of all parts of the track of irregularities in the circumference of locomotive wheels and of their improper centering may perhaps not be very great, but the influence of these irregularities on the stress in the rail is considerable. N. Petrov (Proceedings of the I.R.T.S., Nos. 7–8 for 1906) concludes that neglecting the vertical velocity acquired under the influence of the irregular shape of the wheels may lead to an incorrect determination of the ratio of dynamic pressure to static pressure by as much as 2.67 times. It is indisputable that, before a complete study of the form of the actual trajectory of the rolling of a load along the rails has been made, the question of stresses in the rails and of the stability of the track will not make serious progress.

In N. Petrov’s article “The State of the Question of Rail Deflection” (Railway Engineering, 1904, No. 5), the results are given of observations, mutually inconsistent, made by engineers who deserve equal confidence. Couard and Flamache indicate that the greatest deflections are obtained not when the most heavily loaded locomotive axles pass, but when tenders and cars with brakes pass, when, from the impacts produced by these wheels, deflections are obtained two and three times greater than when locomotive axles pass. Wasutinsky likewise says that the increase in the settlement of sleepers per ton of static load, caused by the greater dynamic action of tender wheels in comparison with locomotive wheels, ranges from 26 to 51%. According to Stetsevich’s observations, the maximum deflections occurred under the locomotive axles, and not a single case was found of a greater deflection under the wheels of the braking axles of a tender or of cars. Dudley’s experiments confirm the possibility, for braking wheels, of a considerable difference between the dynamic and the static pressure (up to 3.6 times).

In observing rail deflections, we are dealing with a very complex phenomenon; often, if differences in the qualities of the ballast are not taken into account, the results of the observations appear inexplicable. Stetsevich, dwelling on one of his tables, says that the track of the Baltic Railway with 18-pound rails is not only considerably more stable in the vertical direction than that of the Saratov Railway with 24-pound rails, but that the stress in the rails of this track (type 18 lb. per running fathom) is less than in the 24-pound rails laid on the Saratov Railway with equal spans between sleepers (the ballast coefficient on the first railway is given as 9, on the second—3.5).

The dynamic action of wheels in general, depending on their motion along a bending rail and on irregularities in the form of the rails and wheels, according to Flamache’s observations, at speeds from 70 to 100 kilometers per hour may exceed the static action by 100% of this latter value (Aste, report to the 4th Congress).

N. Petrov concludes that the influence of speed, in itself, is expressed by an increase of pressure by an insignificant amount, while the influence of unevenness on the rail and wheel flange is reflected in an increase of pressure by \(2^{1}/_{2}\) times.

According to Vasotyński’s observations on a rail of type IV (\(23.4\) lb per linear foot), with characteristic \(A\infty^{1}/_{4}\), the middle of the span between sleepers is pressed down more than the supports by \(0.005;\ 0.010;\ 0.025;\ 0.037;\ 0.058;\ 0.070\) and \(0.1\) mm,—on average \(0.044\), whereas according to Petrov’s calculations the difference should be \(0.024\) mm. With type V, with \(A\infty^{1}/_{3}\) (28.3 lb per linear foot), the lowering of the middle of the rail, when the wheel stands over this middle, is greater than the lowering of the rail over the sleeper when the wheel is over the sleeper; the observations gave this difference as \(0.015;\ 0.040;\ 0.060\); by calculation, however, it comes out as \(0.018\). For brake wheels the observations gave \(0\)—\(0.065\) instead of \(0.0145\), and \(0.005\)—\(0.14\) instead of \(0.0135\). Couard observed that sometimes, with an increase in the speed of motion, the magnitude of the sinking of the sleepers decreases.

Increases in the stability and strength of the track superstructure are usually achieved by: 1) increasing the number of sleepers under a rail length; 2) increasing the section of the rails; and 3) improving the quality of the ballast. Reducing the distance between sleepers to 50 cm was formerly practiced in countries where sleepers were very cheap, while rails were comparatively expensive, and where, owing to a shortage of ballast and of labor for track maintenance, closely laid sleepers were the only means of ensuring some degree of serviceability for tracks made of rails of light type; this arrangement is falling out of use. In Russia a spacing between sleepers of less than 70 centimeters was practiced only as a temporary measure (on the Nikolaev Railway—63.3 cm). Under normal conditions this spacing should be taken as 85—75 cm. The obvious advantages of heavy rails, as traffic becomes more intensive, compel railways increasingly to adopt larger rail sections. It must be noted that, under a static load, the strength of a beam on many elastic supports does not increase in proportion to the increase in the moment of resistance of its section. Indeed, a flexible beam transmits the load to a small number of supports, and predominantly to those supports nearest the place where the load is applied. With an increase in the stiffness of the beam this load is distributed more uniformly, and instead of

the arm of the acting moment is thereby increased, although the ballast load remains constant. As for the bearing pressures, they decrease as the stiffness of the rail increases. According to Wasiutyński’s observations, in passing from rails weighing 31.45 kg/m to rails of 38 kg/m, it was not found that the wheel load was transmitted to a more considerable number of sleepers, but the greatest settlement of the sleepers decreased by 18 to 39%.

The settlement of sleepers under load, even if under small loads it could be considered entirely elastic, in reality, after hundreds and thousands of loadings and unloadings, produces permanent deformations of their foundation, requiring tamping of the sleepers, raising of the settled track, and other corrections. The greater the pressure of the rail on the sleeper, the sooner its elastic settlement passes into permanent deformation, which, under the dynamic action of the wheel, produces very considerable additional stresses in the rail. Wasiutyński’s experiments show that the coefficient of the track bed is obtained, for the same ballast and substructure, as greater with an increase in the length of the sleepers or with a change to a stronger rail. This dependence has not yet been more closely investigated. The coefficient of the track bed for rails of 28.3 was found to be 70% greater than for rails of 23.4 lb/yd, i.e., it increased almost in the same ratio as the moment of inertia of the rail.

Wasiutyński holds the view that the gradual strengthening of the rail, and in general of the superstructure of main lines, has been caused almost exclusively by reasons of an economic, rather than a technical, nature—namely, by the cost of maintenance and repair of the track, which, with high train speeds and a weak, unstable superstructure, prove disproportionately expensive.

The insufficient technical development of questions connected with the calculation of the track superstructure is the reason why its strengthening has hitherto always been motivated by the necessity of reducing the stresses in the rails.

Abroad, the importance for the track of high qualities in the ballast layer has long been recognized, and, with proper clarification from the economic side of the question of cleaning natural ballasts or of transporting good grades of ballast material from great distances, this question has there been resolved so satisfactorily that, without adopting any other measures, railway tracks there withstand very heavy traffic excellently and provide a smooth ride. Improvement of the track must consist in increasing its overall stability and strength by means of a harmonious increase of these qualities in all its constituent parts. The strength of the track undoubtedly

depends most of all on the quality of the ballast; the expenses of track maintenance are connected with it to a considerable degree; finally, without a proper, comprehensive study of ballast and of its role in the permanent way, it is inconceivable not only to obtain a correct understanding of the action of moving loads on the track, but even to formulate correctly the methods for studying this question. This is why it seems necessary to us to study the properties of ballast experimentally at a specially arranged station, and then, after studying the deformations in the ballast layer and in the rails from static loads, to proceed to the study of these deformations in the track under the influence of moving locomotives and trains.

Summarizing everything presented in the present essay concerning the study of ballast as a part of the permanent way, the following propositions may be drawn:

I. In reducing the study of deformations and stresses in all parts of the track to the study of stresses in the rails under the influence of locomotives moving over them, formulas are usually derived which appear to be theoretically justified, but which, in view of the excessive simplification of the problem and the adoption of a whole series of unverified hypotheses, are incorrect; it is supposed that, by checking in each separate case the adequacy of the rail section, it becomes possible, if not to prevent, then to reduce to a minimum cases of rail fracture under passing loads and thereby to eliminate the chief cause of railway accidents.

II. Only the study of the deformations of all the component parts of the permanent way, separately and in combination, can lead to knowledge of its life and to the determination of those measures which are capable of increasing the safety of traffic. A reserve of strength in the rails does not yet mean that there is no smaller reserve of strength and stability in the other parts of the track; finally, a broken rail is only very rarely the cause of an accident.

III. The formulas by which stresses in rails are now calculated (and, by assumption, the stability of the permanent way) do not withstand strict criticism, especially the dynamic formula based on Winkler’s assumption.

It is preferable to take as a basis the method of N. Petrov—to compute the static stresses from deflections found by observation at various points of the rail span, and from the latter the dynamic stresses.

IV. For the possibility of the rational design of the aggregate of all parts of the permanent way, it is necessary, as comprehensively as possible—

it is more appropriate to study ballast as the most inelastic part and the least strong.

V. The elasticity of ballast is very insignificant and varies within very small limits, which is confirmed by more recent observations. Measuring the settlements of sleepers at different points of the track and under different conditions, we obtain larger differences in the figures of proportionality of total settlements to loads.

Therefore one should also study the coefficient of yielding of the ballast layer \(C_0\), characterizing the settlements of a given ballast both as elastic and as inelastic.

VI. Since the condition of the track is the better the fuller the ballast layer, \(C_0\) should be considered greater for that ballast layer whose thickness is greater. Taking the point method for assigning \(C_0\), to obtain \(C_0\) we add to \(C\) as many units as there are decimeters in the thickness of the lower ballast, i.e. from the roadbed to the sole of the sleepers.

Usually the half-width of the ballast layer is greater than the half-length of the sleeper by 10 cm; when this difference is increased by a decimeter, 1 should be added to the value of \(C_0\). Taking the normal length of a sleeper as 2.7 m, we add or subtract one unit for every decimeter of change in this length. If the surface of the roadbed is properly shaped, drained, or completely watertight, if we are certain that there are no ballast troughs, one may add another 1; and when shaping the top of the roadbed with slopes of 0.1–0.2, another unit.

In the case of using (with poor, fine ballast) an American convex profile of the ballast layer, one should add to the value of \(C_0\) from 1 to 5, depending on the correspondence of the profile with the qualities of the ballast material and on successful practical results from applying this measure at the given location.

VII. Laboratory experiments with ballasts from 14 quarries of the Siberian Railway showed that those ballasts are worse and less elastic which are less uniform in grain size. For ballasts consisting chiefly of pebble or crushed stone, and in which the admixture of gravel and sand is less than 10%, 5 should be added to the value of the coefficient \(C_0\), reducing this figure by one unit when the admixture is increased by 10%. For ballasts consisting predominantly of gravel (gravel—with grain diameter from 5 mm to 1 mm) and sand, we add a unit to \(C_0\) in the case where the predominant element of the two indicated enters into the mixture in an amount of not less than 30%, and another unit if the average coarseness of the sand, taken separately, is not less than \(1/2\) mm. With a clay content from 5 to 10% we reduce \(C_0\) by 1; with content up to 15%, by 2; up to 17%, by 3; at 19%, by 4; at 20%, by 5; and further,

subtracting 1 for each new percent of clay content. If the crushed-stone ballast consists of very coarse pieces, then the value of \(C_0\) should be reduced by 1–2.

If special measures have been taken against subsidence of the track (plates at the ends of the sleepers, tying several sleepers together, piles, especially careful tamping of the joint sleepers with fine gravel, etc.), 1–5 should be added to the value of \(C_0\).

If there are experiments determining the resistance of an unloaded ballasted track, \(C_0\) should be increased by the number of tons corresponding to a noticeable displacement (not less than 2 mm), minus 2. In the absence of such experiments, if the lower and upper ballast consist of medium-sized crushed stone with sharp edges and corners, \(C_0\) should be increased by 2–3; for sandy and shingle ballast, add 1; and for clayey, slippery ballast, reduce \(C_0\) by 1; if the sleepers are covered on top with ballast, add one unit to \(C_0\). If the sleepers in the track are of such dimensions that the tamping beneath them holds for a long time, if no uneven settlements of the sleepers or their parts are observed, and the sleepers do not bend and do not spring, then 1 to 5 should be added to the value of \(C_0\).

Taking as the normal annual loss 5% of the volume of the ballast layer, for those ballasts that require a larger quantity for annual repair, \(C_0\) should be reduced by the number of percent above the indicated 5.

VIII. Wishing to know the greatest possible settlement of the sleeper, taking into account the existing properties of the ballast, one should use, for example, Zimmermann’s formula

\[ y=\frac{16a^2+112a+11}{32a(2a+5)}\cdot\frac{P}{D}, \]

introducing into it \(C^0\) instead of \(C\). Since the stress in the rail is not essentially changed by whether the sleepers, beyond elastic deflections, also receive residual ones, then for calculating the strength of the rail itself, using Zimmermann’s formula

\[ M=\frac{8a+7}{16a+40}\,Pl, \]

one may omit \(C\) from it.

IX. From experiments with electric locomotives it may be concluded that, when locomotives move at a certain speed, the alternating, now smaller, now greater deformations in the track are only intensified by the presence of masses with variable motion; the chief cause of them lies in the impossibility of keeping the track ideally straightened.

X. The theory proposed by N. Petrov for calculating stresses in rails and deflections of the track under the dynamic action of a load should be regarded as not giving sufficiently correct results in calculating stresses in the material of the rails, but its method should be

to be used for determining the settlements of sleepers and ballast when loads move along the track at various speeds. In doing so, in his formula the coefficient \(C_0\) should be replaced by the coefficient \(C\), which he erroneously assumes to vary within considerable limits. For each track it would be necessary to establish the maximum magnitude of vertical irregularities, which should be introduced into such a calculation of settlements.

XI. For the solution of many questions concerning the track and, chiefly, the ballast, experiments are necessary with a railway track specially arranged for this purpose; moreover, above the track there should be placed a movable apparatus for producing static pressure on the track at any point of it, and, in addition, numerous observations of the track should be carried out under trains moving along it at various speeds.—

Plate of drawings

Fig. 1

(Detail drawing with dimensions.)

Visible dimensions and labels include: 0.50, 0.45, 0.35, 0.28, 0.10, 0.04.

Fig. 2

(Detail drawing with dimensions.)

Visible dimensions and labels include: 0.75, 0.70, 0.40, 0.35, 0.25.

Fig. 3

(Large longitudinal sectional drawing of a device mounted on timber and rail.)

Fig. 5

(Elevation of the same device between rails.)

Label: along AB.

Fig. 10

(Diagram of a loaded beam.)

Visible labels:
M₀ = 0, M₁, M₂, M₃ = 0, y₀, y₁, y₂, y₃, P.

Fig. 11

(Diagram of a loaded beam.)

Visible labels:
x, δx, y₁, y₂, y₃, P.

Fig. 12

(Curve/deflection diagram.)

Fig. 14

(Long rectangular member with dimensions.)

Visible dimensions include: 1.14 m.

Fig. 17

(Switch or rail-joint diagram.)

Visible labels include: switch, guard rail, wheel.

Fig. 20

(Track alignment diagram.)

Visible labels include: Khristiania, Kongsvinger, Elverum.

Fig. 1.
Dimensions visible: \(3'5''\), \(4'3''\), \(6'6''\), \(11'4''\).
Other inscription: [[unclear: handwritten note near the upper dimension line]].

Fig. 5.
Labels: \(P\), \(p_a\), \(p_b\), \(a\), \(b\), \(c\), \(r_2\).

Fig. 6.
Labels: \(x\), \(y\), \(p_a\), \(p_b\), \(p_c\), \(a\), \(b\), \(c\), \(d_a\), \(d_b\), \(d_c\).

Fig. 7.
Labels: \(r\), \(x\), \(y\), \(p_a\), \(p_b\), \(p_c\).

Fig. 8.
Labels: \(x\), \(y\), \(p_a\), \(p_b\), \(p_c\).

Fig. 9.
Labels: \(x\), \(y\), \(P\), \(p_1\), \(p_2\), \(p_3\), \(p_4\), \(p_5\), \(p_6\).

Fig. 13.
Dimensions visible: \(11.35\ \mathrm{m}\); other small dimensions [[unclear]].

Fig. 14.
Dimensions visible: \(15\), \(45\), \(150\), \(50\), \(85\), \(250\), \(35\), \(90\), \(95\).
Other labels: [[unclear: small handwritten annotations on tool drawings]].

Fig. 15.
Labels: \(a_c\).
Other small labels and dimensions: [[unclear]].

Fig. 16.
Large side view of an apparatus on a ballast slope.
Labels visible: [[unclear: handwritten label near the frame]], \(a_c\).

Fig. 17.
Two sectional views of a rail fastening or clamping device.
Label: “Fig. 17.”

Fig. 20.
Route/profile line with station or locality names visible:
Hammersmith; Fulham; Putney; [[unclear]]; [[unclear]]; [[unclear]]; Chelsea; Westminster; [[unclear]].

Ballast — Diagram No. 1 (9)

Crushed stone.

Passenger traffic.

Freight traffic.

Average line of passenger schedules [[unclear]].

[[unclear: graph label]] (6)

Crushed stone ½″–2″; layer depth 19; sleepers 23 cm.

Curve labels visible:

• average [[unclear]]
• [[unclear]]

Fine sand (15)

Layer depth 30; sleepers 23.

Curve labels visible:

• average [[unclear]]
• [[unclear]]

Fine sand (15)

Layer depth 16; sleepers 23.

Curve labels visible:

• [[unclear]]
• [[unclear]]

Fine sand (15)

Layer depth 12; sleepers 66.

Curve labels visible:

• [[unclear]]
• [[unclear]]

Crushed stone ½″–2″ (16)

Layer depth 30; sleepers 23.

Curve labels visible:

• [[unclear]]
• [[unclear]]

Ballast, slag (1)

Visible annotations:

• frozen ground
• frozen to a depth of 0.5 m
• at temperatures: 6, 8, 9.5
• ballast: crushed stone, 2.5 [[unclear]]

Ballast, slag (1)

Curve labels visible:

• [[unclear]]
• [[unclear]]

Ballast [[unclear]] (3)

Curve labels visible:

• line 100 cm from the rail [[unclear]]
• [[unclear]]

Ballast, sandy (2)

Visible annotations and curve labels:

• line [[unclear]] passenger [[unclear]]
• line [[unclear]] freight [[unclear]]

Ballast, sandy (2)

Curve labels visible:

• [[unclear]]
• [[unclear]]
• [[unclear]]

Ballast, sandy (2)

Visible annotations:

• line of the first track, passenger [[unclear]]
• line [[unclear]] freight [[unclear]]
• [[unclear]]

Fig. 15.

Curves of moments.

[[unclear: handwritten explanatory note about the straight lines indicating the position of the center of gravity and the axis of rotation of the ballast]]

• Ballast No. 1 (9).
• Ballast No. 5 and No. 1 (12).
• Ballast No. 5 and No. 2 (13).
• Ballast [[unclear: adjective]] (2).
• Wooden ballast (10).
• Spare ballast (14).
• Wire ballast (7).
• Metal ballast (8).
• [[unclear: adjective]] ballast (2).
• Canvas ballast (11).

Visible curve annotations include repeated ballast designations and weights, such as:

• “ballast 10 kl.”
• “ballast 9 kl.”
• “ballast 8 kl.”
• “ballast 7 kl.”
• “ballast 6 kl.”
• “ballast 5 kl.”
• “ballast 4 kl.”
• “ballast 3 kl.”
• “ballast 2 kl.”
• “ballast 1 kl.”
• “without ballast”
• “main ballast”
• “water ballast”
• “spare ballast”

Axes and scales are marked in centimeters and kilograms; several vertical scales show moment values in parentheses, for example \(1600\), \(3200\), \(4800\), \(6400\), \(8000\), \(9600\), \(11200\), \(12800\), \(14400\), and \(16000\).

Drawing 18

For the article by O. B. Davydov, Investigation of Ballasts.

Cross sections, drawing 6

Bridge — with wooden beams.

Bridge — with metal beams.

Bridge — in a ballast trough.

Bridge — [[unclear: with/without ballast]].

Bridge — with a solid ballast prism.

Bridge — according to the Klett system.

Bridge — [[unclear: cross-section label]].

Arrangement according to the M. North system

Arrangement according to Fig. 9: North

Bridge — single track.

Bridge — double track.

Arrangement according to Fig. 10: Middendorf

Cross section.

Arrangement according to Fig. 10: North

Fig. 5. Cross section of a bridge

Cross section of a bridge with gravel ballast.

Cross section of a bridge with a ballast prism of [[unclear: 5 m]].

Cross section of a bridge without ballast.

Bridge — single track.

Bridge — double track.

Bridge — [[unclear: with ballast trough]].

Bridge — with open-web flooring.

Metal bridges

Bridge — with wooden beams.

Bridge — with metal beams.

Bridge — with a solid ballast prism.

Arrangement according to Fig. 8: Vistin-Möller

Arrangement according to Fig. 8: J. Most

Arrangement according to the ideas of P. [[unclear]]

For the article by O. B. Davydov, Investigation of Ballasts.

1. Engineer Stoltzman is of a different opinion. In a report to the Assembly of Engineers of Ways of Communication (Proceedings for 1906 and 1907) he recommends, during construction, bringing the thickness of the ballast to 0.16 sazhen and, on main lines, to 0.20 sazhen. ↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩

2. Transactions of the Imperial Russian Technical Society, 1903. ↩