Slopes, General Considerations.—Table Showing the Usual Range of Slopes.—Table of Coefficients of Friction.—Notes on the Cohesion of Earth.—Form of a Slope.—Some Conditions Governing the Necessary Inclination.—Widening Earthworks within the Original Fences.

With regard to the inclination and the form of a slope, and the prevention of slips in earthwork, in any slip the surface must be affected; therefore, to determine the slope of permanent stability of any earth, whether in a cutting or an embankment, is of great importance. An unnecessarily flat slope is not only a monetary waste but may be a cause of instability, for it exposes a larger surface to deterioration by the weather. By what means is a correct decision to be attained? Experience has shown that certain earths under known conditions will repose at particular inclinations; however, to empirically assume that any earth will always stand under any circumstances is clearly imprudent and untenable. As a guide, a table of slopes for earths is valuable, but consideration of the consequences of movement, and the distinctive features of each case should govern a decision.

Should the earth be treacherous and require a flat slope, it may be advisable to reduce the quantity of excavation by the erection of a retaining wall, not only to effect a saving in expenditure, but also to permanently support the soil, and prevent instability.

An inspection of neighbouring quarries and the waste tips, sand or gravel pits, hills and cliffs, cuttings and embankments for roads, river banks, &c., will afford an indication of any local peculiarities caused by the climate or otherwise; and a comparison of the natural slopes assumed when the soil has to withstand vibration, and when not so disturbed, will enable a judgment to be formed of its deleterious effect upon the earth; but above all, an examination of the cuttings and embankments passing through or upon the same geological formation. Reliable evidence can be so obtained upon which the slope of permanent repose can be determined according to the dictates of science and experience, but the probable consequences of instability will demand a due regard to provision against contingent deteriorating influences which may almost destroy cohesion and render it necessary to rely solely upon frictional resistance, the remaining resisting power against movement of the earth, except in solid rock, which may stand with an overhanging or vertical face, whereas mud and quicksand may not be in a state of rest even when horizontal.

The following table of slopes for different earths has been carefully compiled to indicate the probable permanent slope, but it should not be separately considered from the other chapters in this book, as circumstances modify even a considerable range of inclinations; for instance, earths that will be stable at a certain slope in temperate climates will require a much flatter slope in tropical or very cold countries. Those named are more especially for artificially deposited or embanked earths subject to vibration, and, therefore, in cuttings the slope might be steeper, although not so in aluminous soils or those in which the particles become decomposed upon exposure. In Chapter II., under the head of each kind of earth, the approximate slopes of repose are more exactly named, as they would occupy too much space in a table. When not mentioned in Chapter II. the slope is given.

General Note.—The slopes must be flatter according to the amount of water in the same soil or to which it may be subject, the depth of a cutting and height of an embankment, and the presence of all other disturbing influences. Vide Chapter V. for the safe maximum depth of cuttings or height of embankments.

Note.—If an exposed coast, the rubble may require from 4 to 1 to 7 to 1 slope, depending upon its size, the currents, depth, and “fetch” of the sea, and solidity of the mass.

The usual slopes adopted for cuttings and embankments may be said to range from 1 to 1 for firm earth, having particles not seriously affected by water or weather, to 4 to 1, and the most frequent, 1 to 1 TO 1½ to 1 in cuttings and 1½ to 1 in embankments.

With respect to the chief organ of stability in earths other than rock, namely, the frictional resistance; friction during motion is generally considered to be less than the force necessary to overcome it when at rest, and undoubtedly this is the case when the surfaces are similar, and are smooth and hard and not easily impressed, as iron, granite, concrete, and metals generally; but when they are comparatively soft and incapable of resisting indentation at any pressure that they may have to bear, the difference between the coefficient of friction during motion and that at the commencement of motion or of repose will not be so marked, for other resistances may come into action not due solely to surface friction of the mass. A surface may become indented or roughened thus offering opposition to motion not existing at the commencement of movement, and particularly so in any earth of a mixed character possessing hard particles, such as boulders, or sand in clay. On the other hand, in the case of hard rock, solid clay, or other homogeneous earth, the difference between friction during motion and that of friction at rest may be reliably determined.

In soils of a granular or gritty nature small particles become detached during motion, and by pressure occupy or become wedged into any cavities upon the surfaces, and therefore offer resistance which is not alone due to friction of a mass upon a like mass. From this cause friction during motion may seemingly even become greater than during rest, but with material consisting of rounded particles that will not wedge, the friction upon a sliding surface may be lessened by reason of the grains revolving.

In deducing a slope of repose for earth, the lowest value of frictional resistance, whether during motion or at rest, should be taken, and always as if the surfaces were wet. The coefficients of friction, F, during motion usually range between 0·25 to 1·10, and the slope of repose, S to 1, is consequently found by the expression—

therefore, S would equal to 1
0·25 to 1
1·10 = 4 to 1 TO 0·91 to 1, say, 1 to 1.

This is the required inclination to prevent movement provided no pressure is exerted upon the surface, and not taking into consideration the disturbing and weakening effects of vibration and all other deteriorating influences, such as the variable degree of moisture of the soil, the irregularity of its character, the destruction of the continuity of the surface by trenches or drains in a slope, the effect of gravity to detach a mass, the process of excavation or deposition, and the expansion and contraction of soils of an argillaceous nature. When any earth becomes suddenly water-charged or deteriorated by any of the agencies previously and subsequently mentioned, movement may be expected, and it should be remembered that in the same soil the resisting powers to disintegration frequently vary, consequent upon inequality in the quantity of moisture, the roughness, evenness, smoothness, compactness, looseness, the degree of fineness of the earth, and also the manner in which strain is applied.

Friction upon a dry surface is almost invariably greater than that upon a wetted surface, and is so beyond all question upon any plane lubricated with an unguent. The disturbing and enfeebling effect of water may be judged from a careful analysis of many reliable experiments to ascertain the frictional resistance in the case of the same material in a dry and in a wet state on an unplaned surface of cast iron and on timber piles. It shows the following results in addition to those given in the table of coefficients of friction of earth upon earth in the next page.

The surface friction of masonry or brickwork upon dry clay is reduced by from 25 to 30 per cent. when the clay is wet.

The frictional resistance of an unplaned surface of cast iron upon wet sand is about 16 per cent. less than the resistance upon the same material when dry. In the case of timber piles, it is about 12 per cent. less, and about 40 per cent. less in sandy clay and gravelly clay soil.

In sandy gravel the difference in the resistances is small, being only from 5 to 10 per cent. less when the earth is wet.

Sand has about 20 per cent. more friction than sandy gravel, both materials being in a wet state; and below a depth of from 10 to 15 feet, the frictional resistance increases little in gravelly sand and gravelly soils.

In lieu of a sufficient example under similar conditions and circumstances, the most reliable method to ascertain the slope of repose of any earth is that of S, the slope of repose, to 1, = 1
F, any cohesion of the soil being disregarded and considered as a margin of stability liable to be much impaired; and, therefore, except in a mass, it is prudent to look upon it as non-existent in the whole of a slope; and when motion has commenced, as even a means of accelerating movement by causing lumps to become displaced instead of mere particles.

The following coefficients of friction during motion are here given in confirmation of the frictional resistance of an earth being an indicator of its slope of repose.

They have been tabulated from different authorities, are the average results of practical experiments, and have been compared with the inclination of slopes actually assumed under the ordinary conditions of work.

Friction is the chief cause of stability in granular soils and those readily affected by moisture which have for practical purposes no immutable cohesion. In few earths are both cohesion and friction of considerable and reliable value, one or the other quickly becoming impaired or destroyed. Movement is caused by such various means that each earth must be separately considered, and also the circumstances under which it is placed. The particles of the earth may be dissolved by water and become in a muddy state, or they may be considered insoluble as in clean sand and gravel, although in compact sand or gravel the cementing material may crack and weather. Provided it was certain any earth would always remain as originally formed, a condition which it is impossible to guarantee in work, the cohesion and frictional resistance being known, the correct slope could be mathematically determined, but as all earths are subject to varying deteriorating influences, such a deduction is only valuable as a guide for reasonable inference. Cohesion may be more quickly impaired by certain action than friction, and vice versÂ. Probably, of all soils, that to be most distrusted is one that expands and contracts, such as clay earth, which although possessing considerable cohesion may become upon drying a mere congregation of disconnected lumps ready to move upon the return of wet weather, thus its considerable power of cohesion may practically be one of the chief causes of a slip. On the contrary, clean sand, although devoid of cohesion, will not crack or have a greasy surface, but the particles may be washed away.

To ascertain that any earth is uniformly affected throughout the mass, and to prevent or provide against deteriorating influences is the chief aim. It is useless to declare any earth possesses considerable cohesion when the power can be quickly dissipated by ordinary atmospheric action and even become a cause of movement, and to rely for permanent stability upon such property. In ordinary earths, not rock, it will generally be found that cohesion is small or insignificant in soil having a coefficient of friction of some moment, and the reverse. In most earths friction, although it is affected in a greater degree by vibration, has to be relied upon, and not cohesion, as the latter is variable and may exist almost unimpaired in a lump, which nevertheless may become detached because of fissures. The coefficients of friction of different earths are also better known than the cohesion; but how easily even friction is impaired may be gathered from the sudden manner in which bridge cylinders will sink after having hung for days by surface friction, or been held by the transitory expansion of clay. Mr. Wilfrid Airy, B.A., in a series of experiments on the cohesion of earth, found that in strong brick loam it is about 168 lbs. per square foot, in compact clay and gravel it may reach 800 lbs. per square foot of section, and in clean sand it is practically nihil.

In rock the slope of repose depends whether the earth is unstratified or stratified; and if stratified, upon the dip of the strata and their resistance to the effects of the weather. A vertical face may be stable in unstratified or stratified rock, provided in the latter case it does not dip towards a cutting; or the requisite slope may range from ¼ to 1 to such an inclination that the cohesion and friction, which vary greatly, are sufficient to prevent movement. In sidelong ground should the rock dip parallel to the surface the hill slope may require to be flat, whereas the valley slope may stand vertically.

Although, perhaps, in many instances the slopes of cuttings and embankments have been arbitrarily fixed, it may be said, on the whole, no very serious interruptions to traffic have been caused from the sole want of sufficient initial flatness of a slope, as that will soon become known. In determining the inclination it is not the angle at which the earth will stand at the time of excavation or deposition, and for a few months after that is required to be ascertained, but that which will permanently suffice to prevent movement. It is well known that almost all freshly cut soil stands nearly vertically for a small depth for a few days in ordinary weather but then begins to crumble and finally break away.

What then will be the slope of permanent stability? This chiefly depends upon the degree of exposure, the effects of the weather, water, and vibration upon the soil, and the depth or height of a cutting or embankment.

Each earth requires to be duly considered; for instance, gravel and sand are pieces of rock, however small, and for earthwork purposes the particles may be regarded as insoluble in water; nevertheless in the case of sand, should it be charged with water, it may be necessary to treat it as a fluid, the same as mud, although water does not change the particles; however, slips in cuttings and embankments in sandy or gravelly soils are not usually caused by their becoming gradually saturated throughout their mass, but by a flow of water which creates water seams: the stability, therefore, is dependent upon equal percolation and drainage and protection of the surface; and the slope that should be given to a sand cutting is also governed by the quantity of water it will have to hold, and whether the sand is pure or loamy. The depth of a cutting in sand has a considerable influence upon the slope of stability, for frequently sand is in a dry state in the upper portion of a cutting, but beneath it is wet, and partakes more of a silty character, and therefore may stand at a steeper inclination in the upper part, but require a flat slope in the lower portion. The same conditions are found in all soils, and the sides of a cutting vary, one may be comparatively compact and free from water, the other in a wet state and disintegrated. They will not permanently repose at the same angle, the slope varying according to the degree of dryness, size, and uniformity of the particles. In all earths in which cohesion is liable to be quickly destroyed, a straight slope is the best as having an even surface, which prevents the formation of depressions and causes the water to drain away, and also offers the least surface to the weather. In all earth such as shingle, gravel, or sand, consisting of pieces of rock or rubble, or having round particles, it is important to remember that any surface disturbance may cause serious movement, particularly should it commence at the bottom of a slope, as then the revolving action may not cease until a flatter inclination is produced by material rolling from the top and not reposing until it nearly reaches the base. The quantity of moisture is the chief governing condition of the slope of repose in clay soils, and it should not be forgotten that this may vary considerably. Clay when only slightly moist may stand at a 1 to 1 or 1½ to 1 slope, but as it gradually becomes in a wet state will require a flatter inclination, and may not be at rest until the slope is at least 3 to 1, and it is not safe to rely upon a steeper slope than 3 to 1 in the case of almost any surface clay beds liable to become charged with water, and even 4 to 1, should the excavation be on the side of a clay hill and near houses, or the ground be loaded; but clay having some powers of cohesion which usually are greater as the clay is harder, is not so quickly disintegrated as in the case of more porous soil, and the form of the slope it assumes is not a straight line; that most usually approached by sand or gravel or a granular soil consisting of particles of the same character, although the lower part of the slope may be flattened consequent upon the erosion of the finer particles which crumble and become deposited at the base.

As a rule, the greater the cohesion of the soil the more curved is its natural slope, the greatest pressure being at the base where the inclination is flatter, and is steeper towards the top, as the ground may be held together by cohesion at a vertical face. The harder and looser the particles the straighter will be the slope, and if the ground gradually increases in firmness it will usually be nearly straight; but if the contrary condition exists, the natural slope will be flatter towards the base although nearly vertical for a few feet from the top.

As a proof how quickly clay becomes less stable and loses its cohesive power with the usual quantity of moisture in it, when first tipped it may assume a slope of 1 to 1 TO 1½ to 1, but upon exposure to the weather, which causes the lumps to waste away and the clay to swell from moisture and other agencies, the firmest clay in cuttings and embankments may be said to be unstable until a slope of at least 1½ to 1 is reached in moderate depths, and 2 to 1 TO 3 to 1 in high embankments or deep cuttings. In all earths the chief cause of movement of earth is water, and the main questions to be decided are so far as regards the inclination of the slope.

1. Should the surfaces of a cutting be drained and protected and be excavated to a comparatively steep slope; or,

2. Should they be left undrained and uncovered and be excavated to a flat slope. Provided a cutting can be readily drained and covered and there is no probability of any sudden or permanent increase of moisture, perhaps the first method is the more economical; but much depends upon the quantity of water held by the earth in its normal state, whether it is of the same character throughout, and the depth of a cutting. Should the beds be upheaved or intermixed, then a flat slope is necessary and no covering except a wall may make it stable at a steep slope, and, for instance, should clay be always in a semi-saturated condition, 3 to 1 is the least slope at which it will permanently stand, and it will usually require a more moderate inclination. A medium course to adopt is that of varying the inclination of the slopes, the steepest, of course, being at the top and the flattest towards the toe; this is in accordance with the laws of pressure and a mathematical investigation of the theoretically correct slope, which nearly corresponds with the actual slope a high embankment will assume when allowed to weather and settle: for by varying the inclination of the slope the latter becomes practically a curved line and approximates to that of the curve of equilibrium. In almost all slips the surface from which the fallen mass has become detached is curved, the upper part being concave and the lower slightly convex, the outline being caused from the upper portion falling, the lower receiving it and being pressed outwards; however, it may happen that the lower part of a slope has remained intact, and only the upper slipped and become deposited upon it.

The varying slope system has recently been adopted by Mr. Francis Fox, M. Inst. C.E., upon the Scarborough and Whitby Railway, where an embankment about 90 feet in height in treacherous clay had slopes of 1½ to 1 for the upper 30 feet in height, 2 to 1 for the middle 30 feet, and 3 to 1 for the bottom 30 feet. Formation width 28 feet. A calculation of the insistent weight per square foot, without a train, at stated heights gives the following results; taking the weight of the earth at 0·055 of a ton a cubic foot, or 1½ ton a cubic yard, and assuming the worst case, that of the earth for the width of the formation, viz., 28 feet, to act simply as a column 1 foot square and the load as not being distributed over the area of the entire base at any point.

At the base of the upper 30 feet, 1½ to 1 slopes, it would be about 1·65 ton per square foot.

At the base of the middle 30 feet, 2 to 1 slopes, it would be about 3·30 tons per square foot.

At the base of the lower 30 feet, 3 to 1 slopes, it would be about 4·95 tons per square foot.

If the weight of 1 foot lineal of the embankment is taken and considered as equally distributed over the whole area of the base at the 30 feet divisions, the strain per square foot would be as follows:

At the base of the upper 30 feet, 1½ to 1 slopes, about 1 ton per square foot.

At the base of the middle 30 feet, 2 to 1 slopes, about 1¾ ton per square foot.

At the base of the lower 30 feet, 3 to 1 slopes, about 2¼ tons per square foot.

Note.—The actual strain is probably approximate to a mean between the two values, and is nearer the latter than the former.

There is no reason why the varying slope system should not be adopted in embankments of clay or soil having considerable powers of cohesion, as the expense of trimming the slopes is very little more than making them to a straight line. To trim a slope to an elliptical, parabolic, or cycloidal curve would be a needless refinement requiring a template; moreover, the surface is better when it consists of straight lines, provided that at the junction of any two inclinations the point of meeting is sufficiently rounded to prevent a lodgment of water. In cuttings in non-weathering tenacious soil the upper part might be left at a steep slope for 5 or 6 feet from the surface of the ground.

It is evident that the depth of a cutting or the height of an embankment affects the stability of the slopes, but some soils are so weak that an embankment of even little height will not be at rest until the toe of the slope is supported. If the slopes were not pressed out the earth in the central portion of an embankment would stand at any weight less than that which would crush it when in its weakest condition; therefore, as the load increases downwards it is a logical deduction that the slope should be flatter as it approaches the base. In slips the form generally assumed in soils having considerable cohesive power nearly approaches that of a parabolic curve, which shows that a straight slope is not the correct one in tenacious soil, and theory confirms it.

In the case of cuttings of considerable depth, apart from the question of the best form of slope, in order to lessen the velocity of the surface water and the extent of a slip, and cause supported weight upon the slopes, they can be divided by broad terraces or benchings, about 6 feet in width, and at vertical distances of 15 to 20 feet, upon which can be impermeable catchwater drains. It is important not to allow a flow in a straight line or nearly so, as the velocity of the water may erode the slope. If the nature of the earth and its resistance to erosion will allow, the benches or steps should be abrupt, in order to cause the greatest resistance and deviation from a direct discharge. The slopes of the catchwater drains may require to be covered in a flood district or one having a heavy rain or snowfall.—Vide Chapter IV.

In excavating a cutting in soil likely to slip, care should be taken that the surfaces are not strained by lumps being left upon them which are only retained by reason of the cohesion of the earth, as they will cause weak places less able to bear any pressure brought upon the slopes through the sides being deprived of continuity of support. Therefore, in such cases, the slopes should always be rough trimmed as quickly as possible after the gullet has been excavated. Also when the slope of a cutting is furrowed so that its surface consists of separate and unsupported masses of earth clinging to it, continuity of support is destroyed and the earth is more exposed to meteorological dissipation; and in non-cohesive soils such as sandy or gravelly earth, in which especially a straight and uncut surface is of importance, movement is incited. Therefore, when it is found necessary to insert open trenches in a slope, they should be at right angles to its foot, and the inserted material should be well packed so as to support the sides and to interfere as little as possible with the slope. Even in clay soils or any having considerable cohesion, trenches diagonally or transversely cut in the surface are generally inexpedient as disturbing and destroying the continuity of support of the surface and increasing its exposure to the action of the weather; and although they may be temporarily effectual as drains, such division of the inclined face cannot but induce a disunited condition which will at once be apparent should the trench become choked and it miscarry as a drain; and a temporary failure may so destroy the existing delicate equilibrium as to cause movement. Such trenches should be regarded with suspicion, as any stability caused by their draining or conducting away water may be effected at the cost of continuity of support, the deterioration of the resistance of the soil to weather, and the impairment of its frictional and cohesive properties. If placed upon a slope at right angles to the formation the preceding objections are removed.

It may become necessary to excavate or trim a slope to a steeper inclination than that which would otherwise be adopted and is considered to be its angle of repose, in order to widen a railway between the fences, or enlarge a station. Then, pre-eminently the question of the effects of a slip demand attention. The great majority of railway stations are located prior to the commencement of, or as the works progress, but additional accommodation which, for reasons of economy must proceed pari passu with the development of a district, is usually required some time after a railway has been opened for public traffic, and it may be imperatively necessary to confine the works within the boundaries of the land originally purchased. Consideration of the several chapters in this book will recall to the mind the chief points to be regarded. Fortunately, stations are seldom placed in deep cuttings or upon high embankments; but frequently when a station is opened, houses will be erected around it, thus causing any movement of the ground to be of serious moment and dangerous. Assuming the railway must be widened within the original fences, the position of the toe of the slope is fixed, and also that of the top of the slope. The questions then in the case of a cutting are principally:—

1. Will the slope be too precipitous for the earth to stand at one inclination?

2. Can it be made sufficiently steep for 5 or 6 feet from the top to obtain an inclination for the lower portion at which it will have permanent stability?

3. Can the earth be made to repose if the face is evenly protected under circumstances 1 and 2?

4. Is it necessary to erect a retaining wall to a distance a few feet from the original surface of the ground?

5. If a wall be necessary, what should be its lowest height consistent with the stability of the unsupported inclined earth above it?

Provided there is no appreciable superincumbent weight to be borne nearer than 10 feet from the top of the slope, and that the foundations of a building are at a considerable depth in the ground, and the surface and back drainage waters properly controlled, no retaining wall may be required, assuming the original inclination under ordinary conditions to be proved to be the permanent slope of stability; but should the distance of the face of a building from the toe of the slope after widening be insufficient to allow of the original slope being adopted, a retaining wall will be necessary, its height being chiefly governed by the proximity of any building, and the necessity of nearly maintaining the originally established steepest slope of repose in the case of any unsupported earth. The advantage of a sufficient cess in such a case is that it makes provision against deterioration of the surface, and causes an imaginary slope of the same inclination as the original slope to be contained within the space between the face of the building and that of the retaining wall at formation level. In sandy or loose soil if any buildings or wells near the site show signs of cracking, the excavation should at once be stopped to see what preventive measures are requisite, and pumping water out of a trench may be dangerous. Any retaining wall in such a position should be erected in short lengths, so that the earth and foundations are exposed to the weather as little as practicable.