AMERICAN SOCIETY OF CIVIL ENGINEERSINSTITUTED 1852TRANSACTIONSPaper No. 1192EXPERIMENTS ON RETAINING WALLS AND PRESSURES ON TUNNELS.By William Cain, M. Am. Soc. C. E.With Discussion by Messrs. J. R. Worcester, J. C. Meem, and William Cain.The most extended experiments relating to retaining walls are those pertaining to retaining walls proper and the more elaborate ones on small rotating retaining boards. The results referring to the former agree fairly well with a rational theory, especially when the walls are several feet in height; but with the latter, many discrepancies occur, for which, hitherto, no explanation has been offered.It will be the main object of this paper to show that the results of these experiments on small retaining boards can be harmonized with theory by including the influence of cohesion, which is neglected in deducing practical formulas. It will be found that the influence of cohesion is marked, because of the small size of the boards. This information should prove of value to future experimenters, for it will be shown that, as the height of the board or wall increases, the influence of cohesion becomes less and less, so that (for the usual dry sand filling) for heights, say, from 5 to 10 ft., it can be neglected altogether.The result of the investigation will then be to give to the practical constructor more confidence in the theory of the sliding prism, which serves as the basis of the methods to follow.Surcharged wall and pressure of granular materialFig. 1.As, in the course of this investigation, certain well-known constructions for ascertaining the pressure of any granular material against retaining walls will be needed, it is well to group them here. The various figures are supposed to represent sections at right angles to the inner faces of the walls with their backings of granular material. In the surcharged wall,Fig. 1, produce the inner face of the wall to meet the surface of the surcharge at.It is desired to find the thrust against the plane,,for 1 lin. ft. of the wall. Drawthrough,the foot of the wall, making the angle of repose,,of the earth with the horizontal and meeting the upper surface at.Since any possible prism of rupture, as,in tending to move downward, develops friction against both surfaces,and,the earth thrust on the wall will make an angle,,with the normal to,whereis the angle of friction of the earth on the wall. As the earth settles more than the wall, this friction will always be exerted. Again, as the wall, from its elasticity and that of the foundation, will tend to move over at the top on account of the earth thrust, the earth, with its frictional grip on the wall, will tend to prevent this, so that the friction is exerted downward in either case, and the direction of the earth thrust,,onis as given inFig. 1.However, if,a thin slice of earth will move with the wall, and the rubbing will be that of earth on earth, so thatin this case must be replaced by.This rule will apply in all cases that follow, without further remark, whereveris mentioned.Now draw,making the angle,,with,as shown; then drawparallel to,to the intersection,,withproduced. Froma parallel tois constructed, meetingat.Since theory gives the relation:,two constructions follow, by geometry, for locating the point,.By the first, a semicircle is described onas a diameter; at the point,,a perpendicular is erected to,meeting the semicircle in;thenis laid off equal to the chord,.By the second construction, a semicircle is described onas a diameter, a tangent to it,,fromis drawn, limited by the perpendicular radius, and finallyis laid off equal to.The point,,having been thus found by either construction, drawparallel toto the intersection,,with.is the plane of rupture. On laying off,and dropping the perpendicular,,fromon,the earth pressure,,onis given by,whereis the weight of a cubic unit of the earth; otherwise, the value ofis given bytimes the area of the shaded triangle,.If the dimensions are in feet, andis in pounds per cubic foot, the thrust,,will be given in pounds.InFigs. 2and 3, the retaining boards,,are vertical, andis drawn, making the angle,,with the vertical,.The upper surface of the earth is,and the constructions for locatingandare the same as forFig. 1.,in all the figures, represents the plane of rupture.[Footnote1]In all cases, the earth thrust found as above is supposed to make the angle,(as shown), with the normal to the inner wall surface.Retaining boards and pressure of granular meterialFig. 2.Retaining boards and pressure of granular meterialFig. 3.In the Rankine theory, pertaining, say, toFig. 2, the earth thrust on a vertical plane,,is always taken as acting parallel to the top slope. This is true for the pressure on a vertical plane in the interior of a mass of earth of indefinite extent, but it is not true generally for the pressure against a retaining wall. Thus, when,Fig. 2, is horizontal, Rankine’s thrust onwould be taken as horizontal, which entirely ignores the friction of the earth on the wall. The two theories agree whenandslopes at the angle of repose, in which case, asis parallel to,there is no intersection,.It is a limiting case in which, to compute the thrust,can be laid off from any point in,on drawingparallel to,etc. Asapproaches the natural slope, the point,,recedes indefinitely tothe right, and it is seen that the plane of rupture,,approaches indefinitely the line,,or the natural slope. This limiting case, on account of the excessive thrust corresponding, will be examined more carefully in the sequel.Iftheheight of the wall,,in feet, andtheweight of a cubic foot of earth, in pounds, then when,and the surface,Fig. 2, slopes at the angle of repose, the earth thrust, in pounds, is given by theequation:If, however,is not equal to,thenis directed at the angle,,to the normal to the wall, and the thrustis:The foregoing constructions, and the corresponding equations, are all derived from the theory of the sliding prism. The wedge,,Figs. 2and 3, is treated as an invariable solid, tending to slide down the two faces,and,at once, thus developing the full friction that can be exerted on these faces. In the case of actual rotation of the board,,it is found by experiment that each particle of earth in the prism,,moves parallel to,each layer parallel tomoving over the layer just beneath it.A similar motion is observed if the board,,is moved horizontally to the left. However, in the first case (of rotation) the particles atdo not move at all, whereas in the second (of sliding motion) the particles aboutmove, rubbing over the floor, which thus resists the motion by friction. A thrust, thus recorded by springs or other device, in the case where the wall moves horizontally, would give an undervaluation at the lower part ofand consequently the computed center of pressure onwould be too high. On that account, only the experiments on rotating boards will be considered in this paper.The theory of the sliding wedge, however, is justified, because no motion of either kind is actually supposed. The wedge,,is supposed to be just on the point of motion, it being in equilibrium under the action of its weight, the normal components of the reactions of the wall, and the plane,,and all the friction that can be exerted alongand.These forces remain the same, whatever incipient motionis supposed. The hypothesis of a plane surface of rupture, however, is not exactly realized, experiment showing that the earth breaks along a slightly curved surface convex to the moving mass. For the sake of simplicity, the theory neglects the cohesion acting, not only along,but possibly to a small extent along.This additional force will be included in certain investigations to be given later.These preliminary observations having been disposed of, the results of certain experiments on retaining walls at the limit of stability will now be given.Hope's wall of bricks laid in wet sandFig. 4.Baker's wall of pitch-pine blocks backed by macadam screeningsFig. 5.Trautwine's experimental wallFig. 6.Curie's wall of wood coated on the back by sandFig. 7.Figs. 4, 5, 6, and 7 refer to vertical rectangular walls backed by sand, except in the case of Fig. 5, where the filling was macadam screenings. The surface of the filling was horizontal in each case. To give briefly in detail the quantities pertaining to each wall, the following symbols will be used:andare positive when the resultant on the base strikes within the base, otherwise they are negative.Fig. 4represents Lieut. Hope’s wall of bricks laid in wet sand:,,,,.It was 20 ft. long, and was backed by earth level with its top.,.The overhang, at the moment of failure, was probably 4 in. Including this,.Fig. 5shows Baker’s wall of pitch-pine blocks, backed by macadam screenings, the level surface of which was 0.25 ft. below the top of the wall;,,,,,,the assumed angle of friction of timber on stone,,.Trautwine’s experimental wall is shown inFig. 6. Only the ratio of base to height, 0.35, was given by the author, but J. C. Trautwine, Jr., Assoc. Am. Soc. C. E., assures the writer that the walls were probably 6 in. in height, though certain notes refer to walls varying from about 4 to 9 in.,,,(assumed),,.The wall of Curie,Fig. 7, was of wood coated on the back by sand, so that.Also,ft.,ft.,,,.A retaining structure consisting of two boards hinged at the topFig. 8.These walls were all at the limit of stability, and the first two are of appreciable height, 10 ft. and 4 ft., respectively.The figures show that the theory, including the whole of the wall friction, agrees fairly well with experiment, but that the Rankine theory does not thus agree. In both theories, the thrust,,is supposed to act at one-third of the height from the base of the wall to the surface of the filling; but, in the Rankine theory, this thrust is assumed to act horizontally, whereas, in the other theory, it is supposed to act in a direction making the angle,,below the normal to the wall.On combining the thrusts with the weight of the wall, as usual, the resultant strikes the base produced, atin the first case (Rankine theory), but atin the second case.Figs. 4to 7 present a striking object lesson as to the inaccuracy of the Rankine method of treating experimental retaining walls.In the next experiments, however, referring to a retaining structure consisting of two boards, hinged at the top,Fig. 8, and backed by sand level at the top, the Rankine theory is applicable when the board,,is placed either at or below the plane of rupture, on the left of.The thrust onis then assumed to act horizontally, atabove,and is combined with the weight of the sand,,to find theresultant on the board. If the board is at the plane of rupture, this resultant will make the anglebelow the normal to;hence, if one assumes a less thrust on,especially if inclined downward, the new resultant onwill make an angle greater thanwith the normal to,which is inconsistent with stability.[Footnote2]The same reasoning applies whenlies below the plane of rupture.[Footnote3]A surcharged wall of Curie's just at the limit of stabilityFig. 9.The retaining board, 1 m. square, was coated with sand, so thatfor damp sand. Hence, for a horizontal thrust on,the plane of rupture (which bisects the angle between the vertical and the natural slope) makes an angle ofwith the vertical. The board,,was set at this angle to the vertical, sand was filled in level with the top, and it was found that the structure was at the limit of stability whenm. In the meantime, however, the sand had dried out, so thatwas;hence, strictly, the construction ofFig. 1(for earth level with top of wall) applies; but, as the results can only differ inappreciably, the thrust on,acting horizontally, was computed forand combined with the weight of sand,,and the weight of structure, both acting through their centers of gravity, to find the resultant on the base,.It was found to cut it 0.11 of its width from the outer toe,;therefore.In the next experiment, the angle,,was 55°,m. and.Pursuing the same method, it is found that,or the resultant on the base passes practically through.The third experiment was on a smaller retaining board. Herem.,,,and.InFig. 9is shown a surcharged wall of Curie’s, just at the limit of stability, havingft.,ft. and the level upper surface of the surcharge being 4.26 ft. above the top of the wall. The surcharge extended over the wall at the angle,,corresponding to damp sand. Experiment gave.The wall was of brick in Portland cement. The ratio,.It was found, using theconstruction ofFig. 1, that taking the thrust,,as acting 1.24 ft. above the base, or at one-third of the height of the surface,,that;and further, that ifacts 1.303 ft. above the base, the resultant on the base passes exactly through the outer toe of the wall.Planes of rupture for a surcharged wallFig. 10.As the true position of the center of pressure on a surcharged wall has never been ascertained, as far as the writer knows, he has made a number of constructions, after the method illustrated inFig. 1, in order to find it.In place of making the construction for the special case above, it was thought that the results would be more generally useful if the natural slope was taken with a base of 3 and a rise of 2, and,therefore.The wall,,Fig. 10, was taken vertical and 20 ft. high. The surcharge sloped fromat the angleof repose to a point,,10 ft. above,from which point the surface of the earth was horizontal. The face of the wall,,was divided into twenty equal parts, 1 ft. each; and, by the construction ofFig. 1, the thrusts (inclined at the angle,,below the normal to the wall) were found for the successive heights of wall of 1, 2, 3, ... 20, ft., respectively, taking the weight of 1 cu. ft. of earth equal to unity. The successive planes of rupture are shown by the dotted lines inFig. 10. On the original scale (2 ft. to 1 in.), the upper plane of rupture (for a height of wall = 1 ft.) was found to pass slightly to the rightof.On subtracting successive thrusts, the thrusts on each foot of wall were obtained. These were plotted as horizontal ordinates at the center of each foot division of the wall, and the “peaks” were slightly rounded off, as shown on the figure. Since, with all care, mistakes amounting to 1% of the total thrusts can easily be made, it was proper to adjust the results in this manner to give the most probable unit pressures on the successive divisions of the wall. The centers of pressure, for heights of the wall varying from 5 to 20 ft., were easily obtained by taking moments about some convenient point; the results are given inTable 1.Callthe height of wall, measured fromdownward, and,the height of surcharge above the top of the wall; also, lettheratio of the distance from the foot of the wall considered to the center of pressure, to the height of the wall. The values of,for various ratios,,are given inTable 1.TABLE 1.0.3331.000.364......0.750.3642.000.3530.500.3641.500.356......1.250.3600.000.3331.110.362It is seen, asdiminishes, thatincreases, until for,the maximum value for,0.364, is attained and remains the same up to,after which it probably diminishes, because, for,.When some other flatter slope is given to,doubtless these values ofwill be altered, but, for the case supposed, they should prove serviceable in practice.Although the earth thrusts on successive portions ofare really inclined atbelow the normal to,they are laid off here at right angles to it, so that the area,,is equal to the total thrust on.If the unit pressures varied as the ordinates to the straight line,,as for a uniformly sloping earth surface, then, as is well known,.The area to the left ofgives the excess thrust which causesto exceed.Making use of the results of the table as approximately applicable in the foregoing example (Fig. 9), and taking the center of pressure onasabove the base, the resultant there is found to pass 0.02 outside of the base, therefore.This experiment on a surcharged wall, of the kind shown, is particularly valuable as being the only one of which any account has been given, as far as the writer knows.Recurring once more toFig. 10, it may be recalled that some authors have assumed the unit pressures onto vary as the ordinates to a trapezoid, so that the unit pressure atwas not zero (as it should be), but an amount assumed somewhat arbitrarily. In particular, Scheffler derived in this wayas an upper practical limit, and used it in making tables for use in practice.A remark must now be added (relative to all the experimental walls previously mentioned, except Trautwine’s), that the friction of the backing on the sides of the box in which the sand was contained has been uniformly neglected. Where the wall is long, this can have little influence, but where the length is not much greater than the height, as in the experiments, this side friction becomes appreciable.Darwin, as well as Leygue, endeavored to estimate the amount the full thrust (with no side friction) was reduced, by experimenting with sand behind a retaining board, or wall, enclosed in a box as usual, when a partition board was placed perpendicular to the wall and centrally in the mass, and comparing results with those found when the partition board was omitted. Leygue thus found, for walls having a length of twice the height, that the true or full thrust was diminished about 5% from the side friction, for level-topped earth, and as much as 15% for the surface sloping at the angle of repose.If this is true, then the experimental walls just considered would have to be thicker to withstand the actual thrust; or, to put it another way, for the given thickness, the theoretical thrust, including the side friction, would have to be made (as a rough average) about 5% less for the level-topped earth and (roughly) 15% less for the earth sloping at the angle of repose. From the figures it is seen that this will modify the results but slightly, not enough to alter the general conclusion that the theory advocated (including the wall friction) is practically sustained by the experiments, and that the Rankine theory is not thus sustained.Trautwine’s wall consisted of a central portion of uniform height, from which it tapered to the ends, the upper surface being at the angle of repose for the tapered ends. In this case no side friction was developed. The results agree in a general way with the others.In the many experiments on high grain bins, the enormous influence of the friction of the grain against the vertical walls or sides of the bin has been observed. In fact, the greater part of the weight of grain, even when running out, is sustained by the walls through this side friction. This furnishes another argument for including wall friction in retaining-wall design.In connection with this subject, it may be observed that many experiments, made to determine the actual lateral pressure of sand or its internal friction angle, are inconclusive, because an unknown part of the vertical pressure applied to the sand in the vertical cylinder or box was sustained by the sides of the cylinder or box. The ratio of lateral to vertical pressures, or the friction angle, cannot be precisely found until the proportion of the load sustained by the sides of the containing vessel has been ascertained experimentally. The writer is of the opinion that the best experiments to aid in the design of retaining walls are those relating to the rotation of retaining walls or boards. The few given herein are the best recorded, though some of them were on models which were too small. In fact, for the small models of Leygue and others, the effect of cohesion is so pronounced that some of the results are very misleading.As the experiments by Leygue[Footnote4]were very extensive, and evidently made with great care, they will be considered carefully in what follows.As preliminary to the discussion, however, it is well to give the essentials of Leygue’s experimental proof that cohesion and friction exist at the same time. A box without a bottom, about 4 in. square in cross-section and 4 in. high, was made into a little carriage by the addition of four wheels. The latter ran on the sides of a trough filled with sand which the bottom of the box nearly touched. The box was partly filled with sand, and the trough and box were then inclined at the angle at which motion of the box just began, the sand in the box resting on the sand in the trough, developing friction or cohesion or both, just before motion began. Only friction was exerted after motion began. The solution involves the theory of the inclined plane, but, to explain the principles of the method, it will suffice to suppose the trough and the sand in it to be horizontal, and that the bottomless box filled with sand is just on the point of moving, due to a horizontal force applied to it. The weight of the box and a part of the weight of the sand in it held up by the friction of the sides, is directly supported by the wheels resting on the sides of the trough; so that only a fraction of the weight,,of the sand in the box is supported directly by the sand in the trough. Call this amount.Then, for equilibrium, callingthe horizontal force, less the resistance of the carriage wheels, we have,[Footnote5]The value ofwas found by weighing: For the dry sand it varied from 0.79 to 0.65, for heights of the sand in the box varying from 1.2 to 3.5 in. For the damp sand and fresh earth (slightly moistened and slightly rammed) which can stand with a vertical face for the height of the box, the filling was loosened by many blows on the box, andwas taken equal to 1.Three suppositions were made: (1) that both cohesion and friction acted at the same time before motion; (2) that friction alone acted (); (3) that cohesion alone acted ().The results for various heights of sand in the box are given inTable 2.
AMERICAN SOCIETY OF CIVIL ENGINEERSINSTITUTED 1852
Paper No. 1192
By William Cain, M. Am. Soc. C. E.
With Discussion by Messrs. J. R. Worcester, J. C. Meem, and William Cain.
The most extended experiments relating to retaining walls are those pertaining to retaining walls proper and the more elaborate ones on small rotating retaining boards. The results referring to the former agree fairly well with a rational theory, especially when the walls are several feet in height; but with the latter, many discrepancies occur, for which, hitherto, no explanation has been offered.
It will be the main object of this paper to show that the results of these experiments on small retaining boards can be harmonized with theory by including the influence of cohesion, which is neglected in deducing practical formulas. It will be found that the influence of cohesion is marked, because of the small size of the boards. This information should prove of value to future experimenters, for it will be shown that, as the height of the board or wall increases, the influence of cohesion becomes less and less, so that (for the usual dry sand filling) for heights, say, from 5 to 10 ft., it can be neglected altogether.
The result of the investigation will then be to give to the practical constructor more confidence in the theory of the sliding prism, which serves as the basis of the methods to follow.
Surcharged wall and pressure of granular materialFig. 1.
Fig. 1.
As, in the course of this investigation, certain well-known constructions for ascertaining the pressure of any granular material against retaining walls will be needed, it is well to group them here. The various figures are supposed to represent sections at right angles to the inner faces of the walls with their backings of granular material. In the surcharged wall,Fig. 1, produce the inner face of the wall to meet the surface of the surcharge at.It is desired to find the thrust against the plane,,for 1 lin. ft. of the wall. Drawthrough,the foot of the wall, making the angle of repose,,of the earth with the horizontal and meeting the upper surface at.Since any possible prism of rupture, as,in tending to move downward, develops friction against both surfaces,and,the earth thrust on the wall will make an angle,,with the normal to,whereis the angle of friction of the earth on the wall. As the earth settles more than the wall, this friction will always be exerted. Again, as the wall, from its elasticity and that of the foundation, will tend to move over at the top on account of the earth thrust, the earth, with its frictional grip on the wall, will tend to prevent this, so that the friction is exerted downward in either case, and the direction of the earth thrust,,onis as given inFig. 1.
However, if,a thin slice of earth will move with the wall, and the rubbing will be that of earth on earth, so thatin this case must be replaced by.This rule will apply in all cases that follow, without further remark, whereveris mentioned.
Now draw,making the angle,,with,as shown; then drawparallel to,to the intersection,,withproduced. Froma parallel tois constructed, meetingat.
Since theory gives the relation:,two constructions follow, by geometry, for locating the point,.By the first, a semicircle is described onas a diameter; at the point,,a perpendicular is erected to,meeting the semicircle in;thenis laid off equal to the chord,.By the second construction, a semicircle is described onas a diameter, a tangent to it,,fromis drawn, limited by the perpendicular radius, and finallyis laid off equal to.
The point,,having been thus found by either construction, drawparallel toto the intersection,,with.is the plane of rupture. On laying off,and dropping the perpendicular,,fromon,the earth pressure,,onis given by,whereis the weight of a cubic unit of the earth; otherwise, the value ofis given bytimes the area of the shaded triangle,.If the dimensions are in feet, andis in pounds per cubic foot, the thrust,,will be given in pounds.
InFigs. 2and 3, the retaining boards,,are vertical, andis drawn, making the angle,,with the vertical,.The upper surface of the earth is,and the constructions for locatingandare the same as forFig. 1.,in all the figures, represents the plane of rupture.[Footnote1]In all cases, the earth thrust found as above is supposed to make the angle,(as shown), with the normal to the inner wall surface.
Retaining boards and pressure of granular meterialFig. 2.Retaining boards and pressure of granular meterialFig. 3.
Retaining boards and pressure of granular meterialFig. 2.
Fig. 2.
Retaining boards and pressure of granular meterialFig. 3.
Fig. 3.
In the Rankine theory, pertaining, say, toFig. 2, the earth thrust on a vertical plane,,is always taken as acting parallel to the top slope. This is true for the pressure on a vertical plane in the interior of a mass of earth of indefinite extent, but it is not true generally for the pressure against a retaining wall. Thus, when,Fig. 2, is horizontal, Rankine’s thrust onwould be taken as horizontal, which entirely ignores the friction of the earth on the wall. The two theories agree whenandslopes at the angle of repose, in which case, asis parallel to,there is no intersection,.It is a limiting case in which, to compute the thrust,can be laid off from any point in,on drawingparallel to,etc. Asapproaches the natural slope, the point,,recedes indefinitely tothe right, and it is seen that the plane of rupture,,approaches indefinitely the line,,or the natural slope. This limiting case, on account of the excessive thrust corresponding, will be examined more carefully in the sequel.
Iftheheight of the wall,,in feet, andtheweight of a cubic foot of earth, in pounds, then when,and the surface,Fig. 2, slopes at the angle of repose, the earth thrust, in pounds, is given by theequation:
If, however,is not equal to,thenis directed at the angle,,to the normal to the wall, and the thrustis:
The foregoing constructions, and the corresponding equations, are all derived from the theory of the sliding prism. The wedge,,Figs. 2and 3, is treated as an invariable solid, tending to slide down the two faces,and,at once, thus developing the full friction that can be exerted on these faces. In the case of actual rotation of the board,,it is found by experiment that each particle of earth in the prism,,moves parallel to,each layer parallel tomoving over the layer just beneath it.
A similar motion is observed if the board,,is moved horizontally to the left. However, in the first case (of rotation) the particles atdo not move at all, whereas in the second (of sliding motion) the particles aboutmove, rubbing over the floor, which thus resists the motion by friction. A thrust, thus recorded by springs or other device, in the case where the wall moves horizontally, would give an undervaluation at the lower part ofand consequently the computed center of pressure onwould be too high. On that account, only the experiments on rotating boards will be considered in this paper.
The theory of the sliding wedge, however, is justified, because no motion of either kind is actually supposed. The wedge,,is supposed to be just on the point of motion, it being in equilibrium under the action of its weight, the normal components of the reactions of the wall, and the plane,,and all the friction that can be exerted alongand.These forces remain the same, whatever incipient motionis supposed. The hypothesis of a plane surface of rupture, however, is not exactly realized, experiment showing that the earth breaks along a slightly curved surface convex to the moving mass. For the sake of simplicity, the theory neglects the cohesion acting, not only along,but possibly to a small extent along.This additional force will be included in certain investigations to be given later.
These preliminary observations having been disposed of, the results of certain experiments on retaining walls at the limit of stability will now be given.
Hope's wall of bricks laid in wet sandFig. 4.Baker's wall of pitch-pine blocks backed by macadam screeningsFig. 5.Trautwine's experimental wallFig. 6.Curie's wall of wood coated on the back by sandFig. 7.
Hope's wall of bricks laid in wet sandFig. 4.
Fig. 4.
Baker's wall of pitch-pine blocks backed by macadam screeningsFig. 5.
Fig. 5.
Trautwine's experimental wallFig. 6.
Fig. 6.
Curie's wall of wood coated on the back by sandFig. 7.
Fig. 7.
Figs. 4, 5, 6, and 7 refer to vertical rectangular walls backed by sand, except in the case of Fig. 5, where the filling was macadam screenings. The surface of the filling was horizontal in each case. To give briefly in detail the quantities pertaining to each wall, the following symbols will be used:
andare positive when the resultant on the base strikes within the base, otherwise they are negative.
Fig. 4represents Lieut. Hope’s wall of bricks laid in wet sand:,,,,.It was 20 ft. long, and was backed by earth level with its top.,.The overhang, at the moment of failure, was probably 4 in. Including this,.
Fig. 5shows Baker’s wall of pitch-pine blocks, backed by macadam screenings, the level surface of which was 0.25 ft. below the top of the wall;,,,,,,the assumed angle of friction of timber on stone,,.
Trautwine’s experimental wall is shown inFig. 6. Only the ratio of base to height, 0.35, was given by the author, but J. C. Trautwine, Jr., Assoc. Am. Soc. C. E., assures the writer that the walls were probably 6 in. in height, though certain notes refer to walls varying from about 4 to 9 in.,,,(assumed),,.
The wall of Curie,Fig. 7, was of wood coated on the back by sand, so that.Also,ft.,ft.,,,.
A retaining structure consisting of two boards hinged at the topFig. 8.
Fig. 8.
These walls were all at the limit of stability, and the first two are of appreciable height, 10 ft. and 4 ft., respectively.
The figures show that the theory, including the whole of the wall friction, agrees fairly well with experiment, but that the Rankine theory does not thus agree. In both theories, the thrust,,is supposed to act at one-third of the height from the base of the wall to the surface of the filling; but, in the Rankine theory, this thrust is assumed to act horizontally, whereas, in the other theory, it is supposed to act in a direction making the angle,,below the normal to the wall.
On combining the thrusts with the weight of the wall, as usual, the resultant strikes the base produced, atin the first case (Rankine theory), but atin the second case.Figs. 4to 7 present a striking object lesson as to the inaccuracy of the Rankine method of treating experimental retaining walls.
In the next experiments, however, referring to a retaining structure consisting of two boards, hinged at the top,Fig. 8, and backed by sand level at the top, the Rankine theory is applicable when the board,,is placed either at or below the plane of rupture, on the left of.The thrust onis then assumed to act horizontally, atabove,and is combined with the weight of the sand,,to find theresultant on the board. If the board is at the plane of rupture, this resultant will make the anglebelow the normal to;hence, if one assumes a less thrust on,especially if inclined downward, the new resultant onwill make an angle greater thanwith the normal to,which is inconsistent with stability.[Footnote2]The same reasoning applies whenlies below the plane of rupture.[Footnote3]
A surcharged wall of Curie's just at the limit of stabilityFig. 9.
Fig. 9.
The retaining board, 1 m. square, was coated with sand, so thatfor damp sand. Hence, for a horizontal thrust on,the plane of rupture (which bisects the angle between the vertical and the natural slope) makes an angle ofwith the vertical. The board,,was set at this angle to the vertical, sand was filled in level with the top, and it was found that the structure was at the limit of stability whenm. In the meantime, however, the sand had dried out, so thatwas;hence, strictly, the construction ofFig. 1(for earth level with top of wall) applies; but, as the results can only differ inappreciably, the thrust on,acting horizontally, was computed forand combined with the weight of sand,,and the weight of structure, both acting through their centers of gravity, to find the resultant on the base,.It was found to cut it 0.11 of its width from the outer toe,;therefore.
In the next experiment, the angle,,was 55°,m. and.Pursuing the same method, it is found that,or the resultant on the base passes practically through.The third experiment was on a smaller retaining board. Herem.,,,and.
InFig. 9is shown a surcharged wall of Curie’s, just at the limit of stability, havingft.,ft. and the level upper surface of the surcharge being 4.26 ft. above the top of the wall. The surcharge extended over the wall at the angle,,corresponding to damp sand. Experiment gave.The wall was of brick in Portland cement. The ratio,.It was found, using theconstruction ofFig. 1, that taking the thrust,,as acting 1.24 ft. above the base, or at one-third of the height of the surface,,that;and further, that ifacts 1.303 ft. above the base, the resultant on the base passes exactly through the outer toe of the wall.
Planes of rupture for a surcharged wallFig. 10.
Fig. 10.
As the true position of the center of pressure on a surcharged wall has never been ascertained, as far as the writer knows, he has made a number of constructions, after the method illustrated inFig. 1, in order to find it.
In place of making the construction for the special case above, it was thought that the results would be more generally useful if the natural slope was taken with a base of 3 and a rise of 2, and,therefore.The wall,,Fig. 10, was taken vertical and 20 ft. high. The surcharge sloped fromat the angleof repose to a point,,10 ft. above,from which point the surface of the earth was horizontal. The face of the wall,,was divided into twenty equal parts, 1 ft. each; and, by the construction ofFig. 1, the thrusts (inclined at the angle,,below the normal to the wall) were found for the successive heights of wall of 1, 2, 3, ... 20, ft., respectively, taking the weight of 1 cu. ft. of earth equal to unity. The successive planes of rupture are shown by the dotted lines inFig. 10. On the original scale (2 ft. to 1 in.), the upper plane of rupture (for a height of wall = 1 ft.) was found to pass slightly to the rightof.
On subtracting successive thrusts, the thrusts on each foot of wall were obtained. These were plotted as horizontal ordinates at the center of each foot division of the wall, and the “peaks” were slightly rounded off, as shown on the figure. Since, with all care, mistakes amounting to 1% of the total thrusts can easily be made, it was proper to adjust the results in this manner to give the most probable unit pressures on the successive divisions of the wall. The centers of pressure, for heights of the wall varying from 5 to 20 ft., were easily obtained by taking moments about some convenient point; the results are given inTable 1.
Callthe height of wall, measured fromdownward, and,the height of surcharge above the top of the wall; also, lettheratio of the distance from the foot of the wall considered to the center of pressure, to the height of the wall. The values of,for various ratios,,are given inTable 1.
TABLE 1.0.3331.000.364......0.750.3642.000.3530.500.3641.500.356......1.250.3600.000.3331.110.362
TABLE 1.
It is seen, asdiminishes, thatincreases, until for,the maximum value for,0.364, is attained and remains the same up to,after which it probably diminishes, because, for,.
When some other flatter slope is given to,doubtless these values ofwill be altered, but, for the case supposed, they should prove serviceable in practice.
Although the earth thrusts on successive portions ofare really inclined atbelow the normal to,they are laid off here at right angles to it, so that the area,,is equal to the total thrust on.If the unit pressures varied as the ordinates to the straight line,,as for a uniformly sloping earth surface, then, as is well known,.The area to the left ofgives the excess thrust which causesto exceed.
Making use of the results of the table as approximately applicable in the foregoing example (Fig. 9), and taking the center of pressure onasabove the base, the resultant there is found to pass 0.02 outside of the base, therefore.This experiment on a surcharged wall, of the kind shown, is particularly valuable as being the only one of which any account has been given, as far as the writer knows.
Recurring once more toFig. 10, it may be recalled that some authors have assumed the unit pressures onto vary as the ordinates to a trapezoid, so that the unit pressure atwas not zero (as it should be), but an amount assumed somewhat arbitrarily. In particular, Scheffler derived in this wayas an upper practical limit, and used it in making tables for use in practice.
A remark must now be added (relative to all the experimental walls previously mentioned, except Trautwine’s), that the friction of the backing on the sides of the box in which the sand was contained has been uniformly neglected. Where the wall is long, this can have little influence, but where the length is not much greater than the height, as in the experiments, this side friction becomes appreciable.
Darwin, as well as Leygue, endeavored to estimate the amount the full thrust (with no side friction) was reduced, by experimenting with sand behind a retaining board, or wall, enclosed in a box as usual, when a partition board was placed perpendicular to the wall and centrally in the mass, and comparing results with those found when the partition board was omitted. Leygue thus found, for walls having a length of twice the height, that the true or full thrust was diminished about 5% from the side friction, for level-topped earth, and as much as 15% for the surface sloping at the angle of repose.If this is true, then the experimental walls just considered would have to be thicker to withstand the actual thrust; or, to put it another way, for the given thickness, the theoretical thrust, including the side friction, would have to be made (as a rough average) about 5% less for the level-topped earth and (roughly) 15% less for the earth sloping at the angle of repose. From the figures it is seen that this will modify the results but slightly, not enough to alter the general conclusion that the theory advocated (including the wall friction) is practically sustained by the experiments, and that the Rankine theory is not thus sustained.
Trautwine’s wall consisted of a central portion of uniform height, from which it tapered to the ends, the upper surface being at the angle of repose for the tapered ends. In this case no side friction was developed. The results agree in a general way with the others.
In the many experiments on high grain bins, the enormous influence of the friction of the grain against the vertical walls or sides of the bin has been observed. In fact, the greater part of the weight of grain, even when running out, is sustained by the walls through this side friction. This furnishes another argument for including wall friction in retaining-wall design.
In connection with this subject, it may be observed that many experiments, made to determine the actual lateral pressure of sand or its internal friction angle, are inconclusive, because an unknown part of the vertical pressure applied to the sand in the vertical cylinder or box was sustained by the sides of the cylinder or box. The ratio of lateral to vertical pressures, or the friction angle, cannot be precisely found until the proportion of the load sustained by the sides of the containing vessel has been ascertained experimentally. The writer is of the opinion that the best experiments to aid in the design of retaining walls are those relating to the rotation of retaining walls or boards. The few given herein are the best recorded, though some of them were on models which were too small. In fact, for the small models of Leygue and others, the effect of cohesion is so pronounced that some of the results are very misleading.
As the experiments by Leygue[Footnote4]were very extensive, and evidently made with great care, they will be considered carefully in what follows.
As preliminary to the discussion, however, it is well to give the essentials of Leygue’s experimental proof that cohesion and friction exist at the same time. A box without a bottom, about 4 in. square in cross-section and 4 in. high, was made into a little carriage by the addition of four wheels. The latter ran on the sides of a trough filled with sand which the bottom of the box nearly touched. The box was partly filled with sand, and the trough and box were then inclined at the angle at which motion of the box just began, the sand in the box resting on the sand in the trough, developing friction or cohesion or both, just before motion began. Only friction was exerted after motion began. The solution involves the theory of the inclined plane, but, to explain the principles of the method, it will suffice to suppose the trough and the sand in it to be horizontal, and that the bottomless box filled with sand is just on the point of moving, due to a horizontal force applied to it. The weight of the box and a part of the weight of the sand in it held up by the friction of the sides, is directly supported by the wheels resting on the sides of the trough; so that only a fraction of the weight,,of the sand in the box is supported directly by the sand in the trough. Call this amount.Then, for equilibrium, callingthe horizontal force, less the resistance of the carriage wheels, we have,[Footnote5]
The value ofwas found by weighing: For the dry sand it varied from 0.79 to 0.65, for heights of the sand in the box varying from 1.2 to 3.5 in. For the damp sand and fresh earth (slightly moistened and slightly rammed) which can stand with a vertical face for the height of the box, the filling was loosened by many blows on the box, andwas taken equal to 1.
Three suppositions were made: (1) that both cohesion and friction acted at the same time before motion; (2) that friction alone acted (); (3) that cohesion alone acted ().
The results for various heights of sand in the box are given inTable 2.