TABLE 4......00056°36′0.111156°0.110557°0.1011057°0.0961½047°30′0.178149°0.176549°0.1651049°0.155InTable 4the results forandare practically the same, butforis 13% less than for.If the value,,for fresh earth slightly damp and lightly rammed, given by Leygue above, is even approximately correct, it is seen that, for such a filling, the effect of cohesion must be included to get results at all agreeable with experience or experiment.Recurring to the experimental retaining walls proper,Figs. 4to9, it is evident from the foregoing, that cohesion will affect the results inappreciably, except perhaps in the case ofFigs. 6and 7, where the height was about 0.6 ft. Assuming,it seems to be probable, from the results ofTable 4, that the thrust should be decreased in the ratio of 93:111. Effecting the construction for the new thrust, it is found that the point, I, falls within the base, 0.03 of its width forFig. 6(Trautwine’s wall), and 0.02 of its width forFig. 7(Curie’s wall).The theory advocated is thus practically sustained by all the experiments given above, either on retaining boards or retaining walls proper, when a coefficient of cohesion of aboutfor dry sand is used.The method of evaluating the thrust, given inFig. 11, is as valid when,or cohesion is neglected, as in the ordinary theory. The lines parallel to the thrust are now drawn directly from 1, 2, ..., to the intersection with,,..., and the greatest one is taken for the true thrust. Although the writer expressly disclaims any great accuracy in the values ofinTable 4, on account of the small scale of the drawings, nevertheless, the results by the construction forandor ½, were found to differ from computed values only 2, 3, 0, and 1% for the different cases, which should give confidence in the general conclusions, at least.The diagram,Fig. 11, with a slight modification, can be utilized to find the coefficient of cohesion,,at which the bank of earth will stand without a retaining board. Thus, let each line, as,representing the cohesive force acting along its proper plane, be extended to meet the corresponding;any such line measured to the scale of force and then divided by the length of the plane along which it acts, will give the cohesive force, in pounds per square foot, corresponding to no thrust on,for the particular plane considered. The greatest of these values is evidently the value offor which the filling will stand without a retaining board. The work can be much abbreviated by using a well-known principle, that the plane along which the unit cohesion is greatest (the plane of rupture) bisects the angle,,between the surface,,and the line of natural slope. Supposeto be this plane, then we have only to extendto meet,at 0, measure 10 to the scale of force, and divide by the length of,to the scale of distance, to find the coefficient desired. By either method it was found that a cohesive force of 7 lb. per sq. ft. was required to sustain a mass of earth with a vertical face,ft. high, whenwas horizontal.It was stated, in connection withEquations (1)and(2), referring to the thrust on a vertical wall of height,,with the earth surface sloping at the angle of repose, that this particular case would be discussed later. To show the influence of cohesion, the planes of rupture for such a wall, 2.4 ft. high, for various values of(in pounds per squarefoot), are given inFig. 13. The values of(forand)are as follows:Planes of rupture for a vertical wallFig. 13.The first value was found by computation, the others by construction. As is well known, the theoretical plane of rupture approaches indefinitely the natural slope asapproaches zero. For appreciable cohesion (and there is always some cohesion) the plane of rupture lies above the natural slope, with very materially decreasing normal components to the thrust asincreases. As the height of wall increases, the influence of cohesion diminishes. Thus, as shown above, for a wall 5 times 2.4 ft., or 12 ft. high, the weights of the prisms,,etc., are 25 times the former values, but the cohesive forces, which vary directly as,etc., are only 5 times the first values. Hence, if the former values ofare multiplied by 5, the new diagram of forces,Fig. 11, will be similar to the old one. Thus, for the wall 12 ft. high, the plane of rupture and the value of,for,correspond to the old values for,for,to the old values for.For fresh earth filling, slightly packed, it is possible that the values,,,may be reached, with a material reduction infrom the values given byEquations (1)and(2). As the height of the wall increases, say to 25 or 50 ft., the influence of cohesion, in diminishing the thrust, becomes very small, and it is better to ignore it altogether. In fact, as we know very little, and that imperfectly, of the coefficients of cohesion, it is perhaps safer, at present, to useEquations (1)and(2)in all cases. It is very evident, though, that for most cases in practice, the formulas give a very appreciable excess over the true thrust, and that the true plane of rupture never coincides with the natural slope.From all that precedes, it is seen that the results of experiments on small models in the past have proved to be very misleading, and that experiments on large models are desirable, and can alone give confidence. Leygue has made such experiments on retaining boards, from 1 to 2 m. (3.28 to 6.56 ft.) in height, simply to determine the surface of rupture. This is really the essential thing, for, as soon as the prism of rupture is known, the thrust is easily found. In a general way, the results agree with theory when the cohesion is neglected, though the curved surfaces of rupture were very irregular, particularly for the stone filling. The first two experiments were made with both dry and damp sand as a filling; the next six, with stones varying from 1.5 to 20 in. in diameter. In another series of five experiments, sand was used. In all the foregoing experiments, the surface of the material was horizontal. In three additional experiments, the walls were surcharged with sand as a filling. In one experiment, the wall was 6.56 ft. high and the surcharge was 3.28 ft.; in another experiment, the wall was 3.28 ft. high, and the sand, sloping from its top at the angle of repose, as in the former case, extended to 3.28 ft. above the wall, where the surface was horizontal.Applying the construction ofFig. 1, it was found that the plane of rupture passed, say, 2° above that given by experiment in the first case and about 3° below in the second. It will be evident from the construction ofFig. 11, omitting cohesion, that trial planes of rupture differing by 2 or 3° from the true one, give nearly the same thrust. Taking the average, these experiments on large models, tend, in a general way, to sustain the theory.In a paper by the late Sir Benjamin Baker, Hon. M. Am. Soc. C. E., “The Actual Lateral Pressure of Earthwork,”[Footnote8]two experiments by Lieut. Hope and one by Col. Michon, on counterforted walls, are given. Although such walls do not admit of precise computation, on account of the unknown weight of earth carried by the counterforts, through friction caused by the thrust of the earth in a direction perpendicular to the counterforts, still the computation was made, as the conclusions are interesting. Therefore, the first vertical wall of Lieut. Hope was examined, especially as Mr. Baker, using the Rankine theory, found, for this wall, the greatest divergence between the actual and the Rankine thrust, of any retaining wall examined.At the moment of failure, the wall was 12 ft. 10 in. high, the thickness of the panel was 18 in., and the counterforts were 10 ft. from center to center, projecting 27 in. from the wall, or 3 ft. 9 in. from the face, as inferred from the next example. As it is stated that the wall had the same volume as the 10-ft. wall previously examined in this paper (Fig. 4), the counterforts must have been 2 ft. thick. Assuming these dimensions, and using the values given;,or(say),weightof a cubic foot of earth, andweightof a cubic foot of masonry, we first computelb., the normal component of the earth thrust on a length of 1 ft. of wall. The normal thrust on the panel is thusand on the counterfort.The friction (acting vertically downward) caused by this thrust ison the panel andon the counterfort. The moment of these forces about the outer toe of the wall, totals 39 800 ft‑lb. The resisting moment of 10 ft. in length of combined panel and counterfort, about the outer toe, assuming the wall to be vertical, is 29 800 ft‑lb. If, to the latter, we add the moment of 17% of the weight of earth between the counterforts, supposed to be held up by the sides of the latter, the total moment exactly equals the first. However, at the moment of failure by overturning, the panels had bulged 4½ in. and the overhang at the top was 7½ in. Taking the moment of stability of the wall at 26 000 ft‑lb. (Mr. Baker’s figure), it is found that, for equilibrium, 24% of the weight of earth between the counterforts must be carried by them, When the earth was 8 ft. high, a heavy rain was recorded, so that, doubtless, some appreciable cohesion was exerted, though necessarily omitted in the computation.The experimental wall of Col. Michon was 40 ft. high, with very deep counterforts, only 5 ft. from center to center. The very heavy and wet filling between the counterforts, being treated as a part of the wall, a construction (made on the printed drawing) shows that the resultant of earth thrust and weight of wall passes through the outer toe. Doubtless the cohesion factor in this wall was large. In the paper mentioned, the details as to Gen. Burgoyne’s experimental walls are given. There were four of these walls, each 20 ft. long, 20 ft. high, and with a mean thickness of 3 ft. 4 in. Two of the walls were perfectly stable, as in fact theory indicates for all four walls if they were monolithic. The other two walls fell, one bursting outat 5 ft. 6 in. from the base, and the other (a vertical wall), breaking across, as it were, at about one-fourth of its height. As these walls consisted of rough granite blocks laid dry, it is highly probable that the breaks were due to sliding, owing to the imperfect construction; besides, “the filling was of loose earth filled in at random without ramming or other precautions during a very wet winter.”From a consideration of all the observations and experiments (some of them unintentional), Mr. Baker concludes that the theoretical thrust is often double the actual lateral pressure. He used the old theory, which neglects both cohesion and wall friction. If he had included them, the resulting theory would not have been so deficient “in the most vital elements existent in fact” as he charges against the “textbook” theory.However, the writer must be clearly understood as not recommending that cohesive forces be considered in designing a retaining wall backed by a granular material, such as fresh earth, sand, gravel, or ballast. It has been the main object of this paper to show that, although cohesive forces must be included in interpreting properly the results on small models and many retaining walls, yet, for walls more than 6 or 10 ft. in height, backed with dry fresh material, not consolidated, the cohesive forces can be practically neglected in design. Hence, experimenters are strongly advised to leave small models severely alone and confine their experiments to walls from 6 to 10 ft. high, backed by a truly granular material, such as dry sand, coal, grain, gravel, or ballast, where the cohesive forces will not affect the results materially. Further, it is evident that walls of brick in wet sand, or walls of granite blocks, etc., laid dry, are very imperfect walls. The overhang, just before falling, is large, and the base is often imperfect. For precise measurements, a light but strong timber wall on a firm foundation, seems to be best; and the triangular frame ofFig. 8seems to meet the required conditions very well, especially if the framing is an open one, with a retaining board only on one leg. The base thus becomes wider, and the overhang less, than with any rectangular wall.When the design of a wall to sustain the pressure of consolidated earth is in question, even if a perfect mathematical theory existed, it would still prove of little or no practical value, because the coefficients of friction and cohesion are unknown. The coefficient of friction at the surface can be easily found, but it is a difficult matter to find thecoefficient of cohesion, which doubtless varies greatly throughout the mass.Mr. W. Airy, in his discussion of Mr. Baker’s paper, states that he found the tensile strength of a block of ordinary brick clay to be 168 and of a certain shaley clay 800 lb. per sq. ft., the coefficients of friction for the two materials being 1.15 and 0.36, respectively. Cohesive resistance is more analogous to shear, but such figures indicate the wide variations to be expected, particularly in,the coefficient of cohesion. If this coefficient is to be guessed at, in order to substitute it in the supposed perfect formula, then it is plainly better to guess at the thickness of the wall in the first instance.As an illustration, consider the well-known equation:[Footnote9]which gives the height,,of vertical trench that will stand without any sheeting.In this equation,Thus, if,whence,the equation reduces toAs certain trenches with vertical sides have been observed to stand unsupported for heights of 15 or even 25 ft., the equation would seem to indicate that cohesive or shearing resistances of about 200 to 300 lb. per sq. ft. were required to cause equilibrium. If friction is not supposed to be exerted, thenand;and, for the same unsupported heights, the cohesion would be about doubled.Evidently, if cohesion, which (to judge from Mr. Airy’s experiments) may vary from one to several hundred pounds per square foot, has to be guessed at in order to determine,it is plainly better to guess atat once.The foregoing equation cannot be regarded as giving very accurate results, mainly because a plane surface of rupture is assumed, whereas, from both theory and observation, this surface is known to be very much curved; besides, the cohesion and friction along the ends of the break have been neglected. However, the hypothesis of a plane surface of rupture, the ends being supposed to be included, gives a greater value tothan the true one, whereas, neglecting the influence of the ends, it tends in the other direction; so that the equation may not err so greatly.Breaks in the sides of an unsupported trenchFig. 14.In the discussion of the paper[Footnote10]by J. C. Meem, M. Am. Soc. C. E., E. G. Haines, M. Am. Soc. C. E., states that where breaks occur in the sides of an unsupported trench, the solid of rupture often approximates to a quarter sphere, surmounted by a half-cylinder of the same height, the radii of the sphere and cylinder being equal. InFig. 14, letrepresent the quarter-sphere,the half-cylinder, andthe face of the trench. According to the observations of Mr. Haines, when the part,,of the side of the trench is supported by sheeting and bracing, it sometimes happens that a part of the quarter-sphere,,breaks out, so that the semi-cylinder above would descend but for the bracing, the thrust of which, it is supposed, induces arch action in the earth.This is possible; but, if so, as the sheeting is not supposed to be carried to the bottom of the trench, there can be no vertical component in its reaction, and the thrust,,of the braces and sheeting, acting on,must be horizontal; further, the earth cannot act as a series of independent arches devoid of frictional resistance between them, but must act as a whole.Another way of explaining the phenomena is to suppose the horizontal thrust of the braces,,on the exposed face,,to causefriction at the back of the break of sufficient intensity to prevent the semi-cylinder from descending, just as a book can be held against a vertical wall by a horizontal push.To illustrate the principle, it will suffice to replace the semi-cylinder by the circumscribing parallelopiped,,and suppose it to be held up by the friction on the back face, with possibly cohesion acting on the three interior vertical faces. Thus, let,and;then the friction on the back face is,the cohesion on the three faces is,and the weight of earth,,equals.Hence, as friction and cohesion always act opposite to the incipient motion, or vertically upward in this case,Evidently, the value of,derived from this equation, gives an extreme upper limit, which is doubtless never attained, as there is nearly always some support from the earth which has not broken out below the level of.Where the sheeting and bracing are of sufficient size, are tightly keyed up, and extend to the bottom of the trench, or where the bank is supported by a retaining wall, the earth near the bottom cannot break out, and the equation is not valid.However, if, from any cause, such as insufficient sheeting, the break has taken place over even a part of,the mass,,above will tend to tip over at the top, giving the greatest pressure on the top braces. This appears to explain the phenomena observed by Mr. Meem and others in connection with some trenches.With regard to tunnel linings, as is well known, the vertical pressure on the top is generally small, the great mass of earth vertically over the tunnel being largely held up by the friction of the earth (caused by the earth thrust) on its vertical sides, exactly as in the case of tall bins, where most of the weight of the grain is held up by the sides of the bin, the theory being very similar in the two cases. In consolidated earth, cohesion assists very materially in this action.It might be inferred, from the facts of observation, that consolidated earth acts as a solid, though, of course, it differs from a solid in this: that its physical constants (cohesion, friction, etc.) vary enormously with the degree of moisture. It is likely that these constants alter with the depth, and likewise are subject to changes from shocks.It is a question too, whether, as is the case with loosely granular materials, friction acts (before rupture) at the same time with shear or cohesion in consolidated earth. From the interesting remarks[Footnote11]of Mansfield Merriman, M. Am. Soc. C. E., on internal friction, it seems probable that friction and shear exist at the same time in a solid; but, to reach sound conclusions, as he states, “further studies on internal friction and on internal molecular forces are absolutely necessary.”From the present state of our knowledge with respect to the theory and physical constants pertaining to consolidated earth, it would seem that experience must largely be the guide in dealing with it. The facts are supreme—the rational theory may come later.Similarly, for retaining walls backed by loosely aggregated, granular materials, the facts are supreme, and, on that account, they have been presented very fully in this paper; further, a theory has been found to interpret them properly. It is true that the fresh earth, from the time that it is deposited behind a retaining wall, begins to change to a consolidated earth, from the action of rains, the compression due to gravity, and the influence of those cohesive and chemical affinities which manufacture solid earths and clays out of loosely aggregated materials, and even cause the backing sometimes to shrink away from the wall intended to support it; but it is plain that the wall should be designed for the greatest thrust that can come on it at any time, and this, in the great majority of cases, will occur when the earth has been recently deposited.The cases which have been observed where the bank has shrunk away from the wall and afterward ruptured (after saturation, perhaps) are too few in number to warrant including in a general scheme of design, even supposing that a rational theory existed for such cases. A few remarks on the theory pertaining to the design of retaining walls may not be inappropriate. From the discussion of all the experiments referred to in this paper, the conclusion may be fairly drawn that the sliding wedge theory, involving wall friction, is a practical one for granular materials of any kind subjected to a static load. In practical design, however, vibration due to a moving load has to be allowed for; also the effect of heavy rains. Both these influences tend generally to lower the coefficients of friction and add to the weight of the filling. Mr. Baker says:“Granite blocks, which will start on nothing flatter than 1.4 to 1, will continue in motion on an incline of 2.2 to 1,[Footnote12]and, for similar reasons, earthwork will assume a flatter slope and exert a greater lateral pressure under vibration than when at rest.”Instances of slips in railway cuttings, caused by the vibration set up by passing trains, have been given by many engineers. The effect of vibration is most pronounced near the top of a retaining wall, and is evidently greater for a low wall than for a high one. All the influences cited can only be included under the factor of safety, and the writer recommends for walls from 10 to 20 ft. in height a factor of 3. This may be increased to 3.5 for walls 6 ft. high and decreased to 2.5 for walls 50 ft. high, or those with very high surcharges. In the application, the normal component of the earth thrust on the wall,,will alone be multiplied by the factor, the friction,,exerted downward along the back of the wall, being unchanged. This allows very materially for a decrease indue to rains and vibration, as well as for an increase in the thrust, due tobecoming less.Retaining wall subjected to earth thrustFig. 15.The effect is illustrated inFig. 15, where a retaining wall is supposed to be subjected to the earth thrust,,making an anglewith the normal to the face,,of the wall. The component ofnormal tois,the component acting downward alongis represented in magnitude and direction by,which equals.Suppose the factor of safety to be 3, thenis extended to,making;is drawn equal and parallel to;whencewill represent the thrust, which, combined with the weight of the wall, acting through its center of gravity, must pass through the outer toe of the wall.To see what thickness of a vertical rectangular wall corresponds to this factor of safety, 3, for,or a natural slope of 3 base to 2 rise, let it be assumed that the weights per cubic foot of earth and cut-stone masonry in mortar are in the ratio of 2:3; then, for level-topped earth, a computation shows that, for the factor, 3, the base of the wall must be.If the earth slopes indefinitely at the angle of repose from the top of the back of the wall, and a factor 2.5 is used, then the thickness will be.For brick masonry in mortar, the specific weight of which isof that of the filling, the foregoing thickness would be changed toand,respectively,being equal to the height of the wall.It must be noted especially, however, that if the original earth thrust, when combined as usual with the weight of wall, gives a resultant which passes outside of the middle third of the base of the wall as computed above, then the thickness must be increased, so that the resultant will at least pass through the outer middle-third limit. This ensures compression over the whole base and no opening of part of the joint under normal conditions. With regard to the thickness above of about one-third of the height, Mr. Baker states that hundreds of brick revetments have been built by the Royal Engineer officers, with a thickness of onlyfor a vertical wall. He advises, as the result of his own extensive experience, that the thickness be made one-third of the height for level-topped earth of average character, and that the wall be battered 1½ in. to the foot. He states, further, that, under no ordinary conditions of surcharge on heavy backing is it necessary to make the thickness of a retaining wall on a solid foundation more than one-half the height. The thicknesses computed above agree fairly well with those recommended by Mr. Baker, and it would seem that a table of thicknesses computed on the above basis should correspond to safe walls under ordinary conditions.It has been noted above thatEquation (1), corresponding to a slope of indefinite extent, probably gives too great a thrust; besides, there are no embankments with such a slope. An embankment from 100 to 150 ft. high, supported by a low wall, may approximate the conditions assumed, but, before it is finished, the earth has consolidated to such an extent that the actual thrust is doubtless much less than the computed one. The truth is that, in nearly all back-filling of ordinary earth, the cohesive and chemical affinities commence their work very soon after the filling is deposited, and consolidation is gradually effected; so that, as has been stated, the actual thrust is often much less than is estimated in the design of the wall, where cohesive forces are neglected. In many old walls, as has been observed, the consolidation has gone so far that the backing has shrunk away from the wall altogether. It would be hazardous, though,to allow for cohesion, in a wall backed by fresh earth, unless the surcharge was high and was a long time in building. Finally, it should be observed that the footing of a retaining wall should be wide, and should always be tilted at such an angle that sliding is impossible.A glance atFigs. 4, 5, and 6, will make it apparent that the Rankine and other theories differ in their results mainly because of the assumed difference of inclination of the earth thrust. In the design of walls, however, the method proposed (Fig. 15) will approximate in results those given by the Rankine theory, where, say, the earth thrust, whether inclined or not, is multiplied by the factor of safety. The writerdoes not advocate the middle-thirdlimit method in design, as it gives variable factors of safety for different types of walls. Besides, if the actual resultant on the base passes one-third of its width from the outer toe, there is no pressure at the inner toe, and the unit pressure at the outer toe is double the average. If vibration or other cause increases the thrust, the joint at the inner toe opens, and the pressure is concentrated too much near the outer toe. In the reinforced concrete wall, the earth thrust on a vertical plane through the inner toe is required. As this plane lies well within the earth mass, the thrust on it must be taken as acting parallel to the top slope, and its amount will be the same as that given by the Rankine theory.Although it is highly desirable to have more precise experiments on large models in order to draw sure conclusions, yet, as far as the experiments go—those which have been analyzed and discussed in this paper—the following conclusions may be stated:1.—When wall friction and cohesion are included, the sliding-wedge theory is a reliable one, when the filling is a loosely aggregated granular material, for any height of wall.2.—For experimental walls, from 6 to 10 ft. high, and greater, backed by sand or any granular material possessing little cohesion, the influence of cohesion can be neglected in the analysis. Hence, further experiments should be made only on walls at least 6 ft., and preferably 10 ft., high.3.—The many experiments that have been made on retaining boards less than 1 ft. high, have been analyzed by their authors on the supposition that cohesion could be neglected. This hypothesis is so far from the truth that the deductions are very misleading.4.—As it is difficult to ascertain accurately the coefficient of cohesion, and as it varies with the amount of moisture in the material, small models should be discarded altogether in future experiments, and attention should be confined to large ones. Such walls should be made as light, and with as wide a base, as possible. A triangular frame of wood on an unyielding foundation seems to meet the conditions for precise measurements.5.—The sliding-wedge theory, omitting cohesion but including wall friction, is a good practical one for the design of retaining walls backed by fresh earth, when a proper factor of safety is used.As the subject of pressures on the roof and sides of a tunnel lining has received much attention of late, the writer has concluded to extend this paper, so as to give a development of a theory, based on the grain-bin theory of Janssen, but modified to include the cohesive or shearing resistances of the earth in addition to the frictional resistances.
TABLE 4......00056°36′0.111156°0.110557°0.1011057°0.0961½047°30′0.178149°0.176549°0.1651049°0.155
TABLE 4.
InTable 4the results forandare practically the same, butforis 13% less than for.If the value,,for fresh earth slightly damp and lightly rammed, given by Leygue above, is even approximately correct, it is seen that, for such a filling, the effect of cohesion must be included to get results at all agreeable with experience or experiment.
Recurring to the experimental retaining walls proper,Figs. 4to9, it is evident from the foregoing, that cohesion will affect the results inappreciably, except perhaps in the case ofFigs. 6and 7, where the height was about 0.6 ft. Assuming,it seems to be probable, from the results ofTable 4, that the thrust should be decreased in the ratio of 93:111. Effecting the construction for the new thrust, it is found that the point, I, falls within the base, 0.03 of its width forFig. 6(Trautwine’s wall), and 0.02 of its width forFig. 7(Curie’s wall).
The theory advocated is thus practically sustained by all the experiments given above, either on retaining boards or retaining walls proper, when a coefficient of cohesion of aboutfor dry sand is used.
The method of evaluating the thrust, given inFig. 11, is as valid when,or cohesion is neglected, as in the ordinary theory. The lines parallel to the thrust are now drawn directly from 1, 2, ..., to the intersection with,,..., and the greatest one is taken for the true thrust. Although the writer expressly disclaims any great accuracy in the values ofinTable 4, on account of the small scale of the drawings, nevertheless, the results by the construction forandor ½, were found to differ from computed values only 2, 3, 0, and 1% for the different cases, which should give confidence in the general conclusions, at least.
The diagram,Fig. 11, with a slight modification, can be utilized to find the coefficient of cohesion,,at which the bank of earth will stand without a retaining board. Thus, let each line, as,representing the cohesive force acting along its proper plane, be extended to meet the corresponding;any such line measured to the scale of force and then divided by the length of the plane along which it acts, will give the cohesive force, in pounds per square foot, corresponding to no thrust on,for the particular plane considered. The greatest of these values is evidently the value offor which the filling will stand without a retaining board. The work can be much abbreviated by using a well-known principle, that the plane along which the unit cohesion is greatest (the plane of rupture) bisects the angle,,between the surface,,and the line of natural slope. Supposeto be this plane, then we have only to extendto meet,at 0, measure 10 to the scale of force, and divide by the length of,to the scale of distance, to find the coefficient desired. By either method it was found that a cohesive force of 7 lb. per sq. ft. was required to sustain a mass of earth with a vertical face,ft. high, whenwas horizontal.
It was stated, in connection withEquations (1)and(2), referring to the thrust on a vertical wall of height,,with the earth surface sloping at the angle of repose, that this particular case would be discussed later. To show the influence of cohesion, the planes of rupture for such a wall, 2.4 ft. high, for various values of(in pounds per squarefoot), are given inFig. 13. The values of(forand)are as follows:
Planes of rupture for a vertical wallFig. 13.
Fig. 13.
The first value was found by computation, the others by construction. As is well known, the theoretical plane of rupture approaches indefinitely the natural slope asapproaches zero. For appreciable cohesion (and there is always some cohesion) the plane of rupture lies above the natural slope, with very materially decreasing normal components to the thrust asincreases. As the height of wall increases, the influence of cohesion diminishes. Thus, as shown above, for a wall 5 times 2.4 ft., or 12 ft. high, the weights of the prisms,,etc., are 25 times the former values, but the cohesive forces, which vary directly as,etc., are only 5 times the first values. Hence, if the former values ofare multiplied by 5, the new diagram of forces,Fig. 11, will be similar to the old one. Thus, for the wall 12 ft. high, the plane of rupture and the value of,for,correspond to the old values for,for,to the old values for.For fresh earth filling, slightly packed, it is possible that the values,,,may be reached, with a material reduction infrom the values given byEquations (1)and(2). As the height of the wall increases, say to 25 or 50 ft., the influence of cohesion, in diminishing the thrust, becomes very small, and it is better to ignore it altogether. In fact, as we know very little, and that imperfectly, of the coefficients of cohesion, it is perhaps safer, at present, to useEquations (1)and(2)in all cases. It is very evident, though, that for most cases in practice, the formulas give a very appreciable excess over the true thrust, and that the true plane of rupture never coincides with the natural slope.
From all that precedes, it is seen that the results of experiments on small models in the past have proved to be very misleading, and that experiments on large models are desirable, and can alone give confidence. Leygue has made such experiments on retaining boards, from 1 to 2 m. (3.28 to 6.56 ft.) in height, simply to determine the surface of rupture. This is really the essential thing, for, as soon as the prism of rupture is known, the thrust is easily found. In a general way, the results agree with theory when the cohesion is neglected, though the curved surfaces of rupture were very irregular, particularly for the stone filling. The first two experiments were made with both dry and damp sand as a filling; the next six, with stones varying from 1.5 to 20 in. in diameter. In another series of five experiments, sand was used. In all the foregoing experiments, the surface of the material was horizontal. In three additional experiments, the walls were surcharged with sand as a filling. In one experiment, the wall was 6.56 ft. high and the surcharge was 3.28 ft.; in another experiment, the wall was 3.28 ft. high, and the sand, sloping from its top at the angle of repose, as in the former case, extended to 3.28 ft. above the wall, where the surface was horizontal.
Applying the construction ofFig. 1, it was found that the plane of rupture passed, say, 2° above that given by experiment in the first case and about 3° below in the second. It will be evident from the construction ofFig. 11, omitting cohesion, that trial planes of rupture differing by 2 or 3° from the true one, give nearly the same thrust. Taking the average, these experiments on large models, tend, in a general way, to sustain the theory.
In a paper by the late Sir Benjamin Baker, Hon. M. Am. Soc. C. E., “The Actual Lateral Pressure of Earthwork,”[Footnote8]two experiments by Lieut. Hope and one by Col. Michon, on counterforted walls, are given. Although such walls do not admit of precise computation, on account of the unknown weight of earth carried by the counterforts, through friction caused by the thrust of the earth in a direction perpendicular to the counterforts, still the computation was made, as the conclusions are interesting. Therefore, the first vertical wall of Lieut. Hope was examined, especially as Mr. Baker, using the Rankine theory, found, for this wall, the greatest divergence between the actual and the Rankine thrust, of any retaining wall examined.
At the moment of failure, the wall was 12 ft. 10 in. high, the thickness of the panel was 18 in., and the counterforts were 10 ft. from center to center, projecting 27 in. from the wall, or 3 ft. 9 in. from the face, as inferred from the next example. As it is stated that the wall had the same volume as the 10-ft. wall previously examined in this paper (Fig. 4), the counterforts must have been 2 ft. thick. Assuming these dimensions, and using the values given;,or(say),weightof a cubic foot of earth, andweightof a cubic foot of masonry, we first computelb., the normal component of the earth thrust on a length of 1 ft. of wall. The normal thrust on the panel is thusand on the counterfort.The friction (acting vertically downward) caused by this thrust ison the panel andon the counterfort. The moment of these forces about the outer toe of the wall, totals 39 800 ft‑lb. The resisting moment of 10 ft. in length of combined panel and counterfort, about the outer toe, assuming the wall to be vertical, is 29 800 ft‑lb. If, to the latter, we add the moment of 17% of the weight of earth between the counterforts, supposed to be held up by the sides of the latter, the total moment exactly equals the first. However, at the moment of failure by overturning, the panels had bulged 4½ in. and the overhang at the top was 7½ in. Taking the moment of stability of the wall at 26 000 ft‑lb. (Mr. Baker’s figure), it is found that, for equilibrium, 24% of the weight of earth between the counterforts must be carried by them, When the earth was 8 ft. high, a heavy rain was recorded, so that, doubtless, some appreciable cohesion was exerted, though necessarily omitted in the computation.
The experimental wall of Col. Michon was 40 ft. high, with very deep counterforts, only 5 ft. from center to center. The very heavy and wet filling between the counterforts, being treated as a part of the wall, a construction (made on the printed drawing) shows that the resultant of earth thrust and weight of wall passes through the outer toe. Doubtless the cohesion factor in this wall was large. In the paper mentioned, the details as to Gen. Burgoyne’s experimental walls are given. There were four of these walls, each 20 ft. long, 20 ft. high, and with a mean thickness of 3 ft. 4 in. Two of the walls were perfectly stable, as in fact theory indicates for all four walls if they were monolithic. The other two walls fell, one bursting outat 5 ft. 6 in. from the base, and the other (a vertical wall), breaking across, as it were, at about one-fourth of its height. As these walls consisted of rough granite blocks laid dry, it is highly probable that the breaks were due to sliding, owing to the imperfect construction; besides, “the filling was of loose earth filled in at random without ramming or other precautions during a very wet winter.”
From a consideration of all the observations and experiments (some of them unintentional), Mr. Baker concludes that the theoretical thrust is often double the actual lateral pressure. He used the old theory, which neglects both cohesion and wall friction. If he had included them, the resulting theory would not have been so deficient “in the most vital elements existent in fact” as he charges against the “textbook” theory.
However, the writer must be clearly understood as not recommending that cohesive forces be considered in designing a retaining wall backed by a granular material, such as fresh earth, sand, gravel, or ballast. It has been the main object of this paper to show that, although cohesive forces must be included in interpreting properly the results on small models and many retaining walls, yet, for walls more than 6 or 10 ft. in height, backed with dry fresh material, not consolidated, the cohesive forces can be practically neglected in design. Hence, experimenters are strongly advised to leave small models severely alone and confine their experiments to walls from 6 to 10 ft. high, backed by a truly granular material, such as dry sand, coal, grain, gravel, or ballast, where the cohesive forces will not affect the results materially. Further, it is evident that walls of brick in wet sand, or walls of granite blocks, etc., laid dry, are very imperfect walls. The overhang, just before falling, is large, and the base is often imperfect. For precise measurements, a light but strong timber wall on a firm foundation, seems to be best; and the triangular frame ofFig. 8seems to meet the required conditions very well, especially if the framing is an open one, with a retaining board only on one leg. The base thus becomes wider, and the overhang less, than with any rectangular wall.
When the design of a wall to sustain the pressure of consolidated earth is in question, even if a perfect mathematical theory existed, it would still prove of little or no practical value, because the coefficients of friction and cohesion are unknown. The coefficient of friction at the surface can be easily found, but it is a difficult matter to find thecoefficient of cohesion, which doubtless varies greatly throughout the mass.
Mr. W. Airy, in his discussion of Mr. Baker’s paper, states that he found the tensile strength of a block of ordinary brick clay to be 168 and of a certain shaley clay 800 lb. per sq. ft., the coefficients of friction for the two materials being 1.15 and 0.36, respectively. Cohesive resistance is more analogous to shear, but such figures indicate the wide variations to be expected, particularly in,the coefficient of cohesion. If this coefficient is to be guessed at, in order to substitute it in the supposed perfect formula, then it is plainly better to guess at the thickness of the wall in the first instance.
As an illustration, consider the well-known equation:[Footnote9]
which gives the height,,of vertical trench that will stand without any sheeting.
In this equation,
Thus, if,whence,the equation reduces to
As certain trenches with vertical sides have been observed to stand unsupported for heights of 15 or even 25 ft., the equation would seem to indicate that cohesive or shearing resistances of about 200 to 300 lb. per sq. ft. were required to cause equilibrium. If friction is not supposed to be exerted, thenand;and, for the same unsupported heights, the cohesion would be about doubled.Evidently, if cohesion, which (to judge from Mr. Airy’s experiments) may vary from one to several hundred pounds per square foot, has to be guessed at in order to determine,it is plainly better to guess atat once.
The foregoing equation cannot be regarded as giving very accurate results, mainly because a plane surface of rupture is assumed, whereas, from both theory and observation, this surface is known to be very much curved; besides, the cohesion and friction along the ends of the break have been neglected. However, the hypothesis of a plane surface of rupture, the ends being supposed to be included, gives a greater value tothan the true one, whereas, neglecting the influence of the ends, it tends in the other direction; so that the equation may not err so greatly.
Breaks in the sides of an unsupported trenchFig. 14.
Fig. 14.
In the discussion of the paper[Footnote10]by J. C. Meem, M. Am. Soc. C. E., E. G. Haines, M. Am. Soc. C. E., states that where breaks occur in the sides of an unsupported trench, the solid of rupture often approximates to a quarter sphere, surmounted by a half-cylinder of the same height, the radii of the sphere and cylinder being equal. InFig. 14, letrepresent the quarter-sphere,the half-cylinder, andthe face of the trench. According to the observations of Mr. Haines, when the part,,of the side of the trench is supported by sheeting and bracing, it sometimes happens that a part of the quarter-sphere,,breaks out, so that the semi-cylinder above would descend but for the bracing, the thrust of which, it is supposed, induces arch action in the earth.
This is possible; but, if so, as the sheeting is not supposed to be carried to the bottom of the trench, there can be no vertical component in its reaction, and the thrust,,of the braces and sheeting, acting on,must be horizontal; further, the earth cannot act as a series of independent arches devoid of frictional resistance between them, but must act as a whole.
Another way of explaining the phenomena is to suppose the horizontal thrust of the braces,,on the exposed face,,to causefriction at the back of the break of sufficient intensity to prevent the semi-cylinder from descending, just as a book can be held against a vertical wall by a horizontal push.
To illustrate the principle, it will suffice to replace the semi-cylinder by the circumscribing parallelopiped,,and suppose it to be held up by the friction on the back face, with possibly cohesion acting on the three interior vertical faces. Thus, let,and;then the friction on the back face is,the cohesion on the three faces is,and the weight of earth,,equals.Hence, as friction and cohesion always act opposite to the incipient motion, or vertically upward in this case,
Evidently, the value of,derived from this equation, gives an extreme upper limit, which is doubtless never attained, as there is nearly always some support from the earth which has not broken out below the level of.
Where the sheeting and bracing are of sufficient size, are tightly keyed up, and extend to the bottom of the trench, or where the bank is supported by a retaining wall, the earth near the bottom cannot break out, and the equation is not valid.
However, if, from any cause, such as insufficient sheeting, the break has taken place over even a part of,the mass,,above will tend to tip over at the top, giving the greatest pressure on the top braces. This appears to explain the phenomena observed by Mr. Meem and others in connection with some trenches.
With regard to tunnel linings, as is well known, the vertical pressure on the top is generally small, the great mass of earth vertically over the tunnel being largely held up by the friction of the earth (caused by the earth thrust) on its vertical sides, exactly as in the case of tall bins, where most of the weight of the grain is held up by the sides of the bin, the theory being very similar in the two cases. In consolidated earth, cohesion assists very materially in this action.
It might be inferred, from the facts of observation, that consolidated earth acts as a solid, though, of course, it differs from a solid in this: that its physical constants (cohesion, friction, etc.) vary enormously with the degree of moisture. It is likely that these constants alter with the depth, and likewise are subject to changes from shocks.
It is a question too, whether, as is the case with loosely granular materials, friction acts (before rupture) at the same time with shear or cohesion in consolidated earth. From the interesting remarks[Footnote11]of Mansfield Merriman, M. Am. Soc. C. E., on internal friction, it seems probable that friction and shear exist at the same time in a solid; but, to reach sound conclusions, as he states, “further studies on internal friction and on internal molecular forces are absolutely necessary.”
From the present state of our knowledge with respect to the theory and physical constants pertaining to consolidated earth, it would seem that experience must largely be the guide in dealing with it. The facts are supreme—the rational theory may come later.
Similarly, for retaining walls backed by loosely aggregated, granular materials, the facts are supreme, and, on that account, they have been presented very fully in this paper; further, a theory has been found to interpret them properly. It is true that the fresh earth, from the time that it is deposited behind a retaining wall, begins to change to a consolidated earth, from the action of rains, the compression due to gravity, and the influence of those cohesive and chemical affinities which manufacture solid earths and clays out of loosely aggregated materials, and even cause the backing sometimes to shrink away from the wall intended to support it; but it is plain that the wall should be designed for the greatest thrust that can come on it at any time, and this, in the great majority of cases, will occur when the earth has been recently deposited.
The cases which have been observed where the bank has shrunk away from the wall and afterward ruptured (after saturation, perhaps) are too few in number to warrant including in a general scheme of design, even supposing that a rational theory existed for such cases. A few remarks on the theory pertaining to the design of retaining walls may not be inappropriate. From the discussion of all the experiments referred to in this paper, the conclusion may be fairly drawn that the sliding wedge theory, involving wall friction, is a practical one for granular materials of any kind subjected to a static load. In practical design, however, vibration due to a moving load has to be allowed for; also the effect of heavy rains. Both these influences tend generally to lower the coefficients of friction and add to the weight of the filling. Mr. Baker says:
“Granite blocks, which will start on nothing flatter than 1.4 to 1, will continue in motion on an incline of 2.2 to 1,[Footnote12]and, for similar reasons, earthwork will assume a flatter slope and exert a greater lateral pressure under vibration than when at rest.”
Instances of slips in railway cuttings, caused by the vibration set up by passing trains, have been given by many engineers. The effect of vibration is most pronounced near the top of a retaining wall, and is evidently greater for a low wall than for a high one. All the influences cited can only be included under the factor of safety, and the writer recommends for walls from 10 to 20 ft. in height a factor of 3. This may be increased to 3.5 for walls 6 ft. high and decreased to 2.5 for walls 50 ft. high, or those with very high surcharges. In the application, the normal component of the earth thrust on the wall,,will alone be multiplied by the factor, the friction,,exerted downward along the back of the wall, being unchanged. This allows very materially for a decrease indue to rains and vibration, as well as for an increase in the thrust, due tobecoming less.
Retaining wall subjected to earth thrustFig. 15.
Fig. 15.
The effect is illustrated inFig. 15, where a retaining wall is supposed to be subjected to the earth thrust,,making an anglewith the normal to the face,,of the wall. The component ofnormal tois,the component acting downward alongis represented in magnitude and direction by,which equals.Suppose the factor of safety to be 3, thenis extended to,making;is drawn equal and parallel to;whencewill represent the thrust, which, combined with the weight of the wall, acting through its center of gravity, must pass through the outer toe of the wall.
To see what thickness of a vertical rectangular wall corresponds to this factor of safety, 3, for,or a natural slope of 3 base to 2 rise, let it be assumed that the weights per cubic foot of earth and cut-stone masonry in mortar are in the ratio of 2:3; then, for level-topped earth, a computation shows that, for the factor, 3, the base of the wall must be.If the earth slopes indefinitely at the angle of repose from the top of the back of the wall, and a factor 2.5 is used, then the thickness will be.
For brick masonry in mortar, the specific weight of which isof that of the filling, the foregoing thickness would be changed toand,respectively,being equal to the height of the wall.
It must be noted especially, however, that if the original earth thrust, when combined as usual with the weight of wall, gives a resultant which passes outside of the middle third of the base of the wall as computed above, then the thickness must be increased, so that the resultant will at least pass through the outer middle-third limit. This ensures compression over the whole base and no opening of part of the joint under normal conditions. With regard to the thickness above of about one-third of the height, Mr. Baker states that hundreds of brick revetments have been built by the Royal Engineer officers, with a thickness of onlyfor a vertical wall. He advises, as the result of his own extensive experience, that the thickness be made one-third of the height for level-topped earth of average character, and that the wall be battered 1½ in. to the foot. He states, further, that, under no ordinary conditions of surcharge on heavy backing is it necessary to make the thickness of a retaining wall on a solid foundation more than one-half the height. The thicknesses computed above agree fairly well with those recommended by Mr. Baker, and it would seem that a table of thicknesses computed on the above basis should correspond to safe walls under ordinary conditions.
It has been noted above thatEquation (1), corresponding to a slope of indefinite extent, probably gives too great a thrust; besides, there are no embankments with such a slope. An embankment from 100 to 150 ft. high, supported by a low wall, may approximate the conditions assumed, but, before it is finished, the earth has consolidated to such an extent that the actual thrust is doubtless much less than the computed one. The truth is that, in nearly all back-filling of ordinary earth, the cohesive and chemical affinities commence their work very soon after the filling is deposited, and consolidation is gradually effected; so that, as has been stated, the actual thrust is often much less than is estimated in the design of the wall, where cohesive forces are neglected. In many old walls, as has been observed, the consolidation has gone so far that the backing has shrunk away from the wall altogether. It would be hazardous, though,to allow for cohesion, in a wall backed by fresh earth, unless the surcharge was high and was a long time in building. Finally, it should be observed that the footing of a retaining wall should be wide, and should always be tilted at such an angle that sliding is impossible.
A glance atFigs. 4, 5, and 6, will make it apparent that the Rankine and other theories differ in their results mainly because of the assumed difference of inclination of the earth thrust. In the design of walls, however, the method proposed (Fig. 15) will approximate in results those given by the Rankine theory, where, say, the earth thrust, whether inclined or not, is multiplied by the factor of safety. The writerdoes not advocate the middle-thirdlimit method in design, as it gives variable factors of safety for different types of walls. Besides, if the actual resultant on the base passes one-third of its width from the outer toe, there is no pressure at the inner toe, and the unit pressure at the outer toe is double the average. If vibration or other cause increases the thrust, the joint at the inner toe opens, and the pressure is concentrated too much near the outer toe. In the reinforced concrete wall, the earth thrust on a vertical plane through the inner toe is required. As this plane lies well within the earth mass, the thrust on it must be taken as acting parallel to the top slope, and its amount will be the same as that given by the Rankine theory.
Although it is highly desirable to have more precise experiments on large models in order to draw sure conclusions, yet, as far as the experiments go—those which have been analyzed and discussed in this paper—the following conclusions may be stated:
1.—When wall friction and cohesion are included, the sliding-wedge theory is a reliable one, when the filling is a loosely aggregated granular material, for any height of wall.
2.—For experimental walls, from 6 to 10 ft. high, and greater, backed by sand or any granular material possessing little cohesion, the influence of cohesion can be neglected in the analysis. Hence, further experiments should be made only on walls at least 6 ft., and preferably 10 ft., high.
3.—The many experiments that have been made on retaining boards less than 1 ft. high, have been analyzed by their authors on the supposition that cohesion could be neglected. This hypothesis is so far from the truth that the deductions are very misleading.
4.—As it is difficult to ascertain accurately the coefficient of cohesion, and as it varies with the amount of moisture in the material, small models should be discarded altogether in future experiments, and attention should be confined to large ones. Such walls should be made as light, and with as wide a base, as possible. A triangular frame of wood on an unyielding foundation seems to meet the conditions for precise measurements.
5.—The sliding-wedge theory, omitting cohesion but including wall friction, is a good practical one for the design of retaining walls backed by fresh earth, when a proper factor of safety is used.
As the subject of pressures on the roof and sides of a tunnel lining has received much attention of late, the writer has concluded to extend this paper, so as to give a development of a theory, based on the grain-bin theory of Janssen, but modified to include the cohesive or shearing resistances of the earth in addition to the frictional resistances.