DISCUSSION.J. R. Worcester, M. Am. Soc. C. E.(by letter).— In reading Professor Cain’s admirable paper relating to experiments on retaining walls, the writer has looked in vain for a word of caution as to the effect which time plays in modifying the condition of equilibrium within a mass of earth. The author evidently considers it necessary to allow (by using a factor of safety) for a possible lessening of the angle of friction on account of a change in the amount of moisture, and possible vibrations, but states that in a great majority of cases the greatest thrust will occur where the earth has been recently deposited. It would appear that he neglects a possibility, if not a probability, of a readjustment of the earth particles through the influence of time, by which the angle of friction is lessened if not wholly cancelled.The theory that cohesion in the earth and frictional resistance on the back of the retaining wall account for the experimental results seems indisputable, but such experiments must needs be carried through in a reasonable time, and, in that respect, at least, must needs differ from actual constructions which are intended to be permanent.It is well known that unbraced excavations can often be carried vertically to considerable depths in safety, but that not infrequently— as many have learned to their sorrow— such unbraced banks have subsequently caved in. The slides in the Culebra cut may be mentioned as a similar illustration. To be sure, the delayed motion of the earth (or rock) may be attributed to the effect of moisture, but that does not invalidate the argument, as one always has to reckon with water.It is also a matter of common knowledge that, in braced excavations, the pressure on the sheeting and bracing frequently increases in time to an extent enormously exceeding the original pressure. In many instances this has caused a failure long delayed.Another instance of the effect of time is found in many retaining walls, in which a very slow motion has occurred, though the walls appeared to be entirely stable when first built.A frictional or tangential force along the back of the wall may tend to prevent motion, but it is difficult to conceive of this tangential force being perpetually present on the back of a stationary wall after the back-filling has become settled and consolidated. In the interior of the mass of earth at rest the author admits that the reaction between particles is not along inclined lines, but rather that the lines of pressure are vertical and horizontal. The vertical lines of force are, of course, caused by gravity, and the horizontal lines by the tendency of particles to wedge in between those below and to spread them.If this conception of the forces within a mass of earth is reasonable, it would seem as if it might also be extended to the pressure against an immovable vertical wall. One must then consider whether it is possible for a horizontal pressure to cause the wall to move, without changing the conditions and introducing the inclined stresses. It would seem within the bounds of possibility that a very minute motion might be produced, and that this would be followed by a readjustment of stresses in the earth by which the forces would gradually resume the horizontal direction.The nature of the soil undoubtedly has much to do with this question. In some kinds of clay there appears to be a sort of viscosity, such as is frequently seen in pitch and other materials, or a tendency toward a slow flowing. No amount of pressure would cause a sudden motion, but time will effect a motion under a slight pressure or even the force of gravity alone. It appears that this condition is produced by the very minute particles, each moving individually into a position in which the surrounding forces balance. If one cuts a vertical face in such material one cannot force the exposed particles out of their position by crowding them from behind, but each in its turn will feel the pressure unbalanced and will slowly move out. This may not be true to the same extent with granular materials of large diameter, but a familiar instance is seen in fine wet sand. If a small excavation is dug in a wet beach sand, the banks will stand vertically at first, but, by watching closely, one may see the particles, beginning at the foot of the bank where there is most water, gradually moving out, overcoming the force of cohesion, and ever tending to seek a condition with a level surface. It seems quite likely that a similar tendency would exist in almost all soils, to greater or less degree, though perhaps it might be safely neglected in a mass of hard, irregularly-shaped fragments of stone which could interlock.The point which the writer wishes to make is that a word of caution should accompany this argument for the frictional and cohesive forces; that they cannot always be relied on; and that sometimes the Rankine theory may be better than the wedge theory in designing, even though it does not seem to fit the experimental results.Another warning may not be amiss, in considering the safe thickness to allow for retaining walls, and that is the effect of frost, where the surface of the ground is level and likely to retain moisture. The swelling force of freezing, under these circumstances, may be more than sufficient to overcome the beneficial effects of both cohesion and friction. Presumably this must be provided for in the “factor of safety,” and is in itself a justification for a very appreciable factor.It may be well to emphasize the fact that a large part of the author’s assumed factor of safety seems to be absorbed in keeping the resultant within the middle third of the base. The proportionsbetween width of wall and height, determined onpages 433and 434, are such as to keep this resultant just within the base. If, with these same proportions of wall, the factor were assumed so that the resultant were within the middle third, it would be found to be nearer 1¼ than 3. The author’s statement onpage 435, that he “does not advocate the middle-third limit method in design,” is not wholly clear, but the implication is that the resultant should be well within this limit. In this case, it seems as if the factor of safety would be wholly absorbed in thus locating the resultant, and would leave nothing for other elements of uncertainty.J. C. MEEM, M. Am. Soc. C. E.(by letter).—In the writer’s judgment the author has gone a step forward in developing the relation of cohesion and friction to walls and tunnels, but he has not given sufficient value to the larger consideration of what may be called “cohesive friction” induced by the lateral pressure of earth against retaining walls and other faces. This will be noted later in the discussion.The author states that “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.”[Footnote21]To a certain extent this is fully in accordance with the writer’s views, as noted in his papers on earth pressures,[Footnote22]and he feels justified in once more calling attention to the fact that, in his judgment, the only experiments which can definitely establish the value of earth pressures against walls and sheeted faces should be made on a large scale and against independently laid and independently braced horizontal sheeting. If these experiments are made in a homogeneous material, such as dry sand of a known angle of repose, it is believed that it will be conclusively and definitely shown that the pressures at a point above the middle plane will be greater than below it, and, further, that it will be proved that the pressure near the bottom, for example, of a 20-ft. trench, will not be perceptibly greater than that against a brace at the same distance from the bottom of a 40- or a 60-ft. trench.The writer agrees with the author that the theory of a sliding wedge is the best practical one for retaining walls, and if the face of the wedge be keyed tight, as stated by the author, or through the compacting of the material into a more solid mass, it will be seen that, with no break, the resultant pressures against the wall or face are virtually those, not only of a sliding, but of a solid wedge tending to slide on the plane of repose, the mass being compactly held together by cohesioninduced by lateral pressure. Although the general solidity of this mass is dependent on the stability of the wall or bracing, the pressures caused by the tendency of this solid wedge to slide are not affected materially by slight changes in form due to gradual settlement, which, in turn, may be caused by the normal yielding of the wall or shrinkage of the bracing, or by any small losses of material, as long as any of those or their sum is not sufficient to cause partial or final collapse. Assuming then that this theory of the solid wedge tending to slide is the true one, it is difficult to reconcile it with the results of experiments on revolving boards or revolving walls, except in so far as they relate to the more accurate determination of the coefficients of friction and cohesion.In connection with the experiments of Leygue, the writer wishes to refer to the dry sand and wheat arching experiments described in the final discussion of his paper, “The Bracing of Trenches and Tunnels, with Practical Formulas for Earth Pressures,”[Footnote23]and especially to the experiment made with dry sand in a 2-in. pipe, described in his paper on “Pressure, Resistance, and Stability of Earth.”[Footnote24]For clearer reference, the latter experiment is again described:A 2-in. pipe, 18 in. long, was filled with dry sand for a depth of 12 in., and a thin piece of tissue paper was pasted across the bottom. Then, with a wooden piston bearing on the sand, the latter would support the blow of a sledge hammer or the weight of a man without breaking the tissue paper.All these experiments seem to prove very clearly that pressure is distributed laterally, at least through perfectly dry material, and, with certain values known, it should not be difficult to compute the value of this thrust. For example, if, in the pipe experiment, the exact coefficient of friction of the granular particles against the inside of the pipe is determined to be 0.4, then it would take a pressure equal to the complement of this, or 0.6 of the weight, to hold the granular particles against the vertical side. Now, if the exact depth of sand is determined at which no further pressure (by reason of adding to this depth) is transmitted to the bottom, then, by finding the difference or loss of pressure at the bottom, it should be a simple matter to compute the amount of thrust developing sufficient friction over the given area to sustain the additional weight. This is what the writer, for want of a better term, has called “cohesive friction” induced by lateral pressure. While such experiments on a small scale give definite results with very dry sand or wheat, they will not be conclusive unless made on a very large scale in cases where dampness adds materially to the normal cohesion of small masses.As to the experiments on revolving boards or walls referred to bythe author, it would appear that, as noted, they eliminate consideration of the effect of pressure as increasing the values of the coefficient of friction and cohesion.The writer believes that it can be shown byFig. 22that the lateral pressure against sheeted faces or retaining walls is greater toward the top.Fig. 22is adapted from Fig. 5 in the writer’s paper, “Pressure, Resistance, and Stability of Earth.”Lateral pressure against sheeted facesFig. 22.It is supposed that a retaining wall has first been built along the face,,and that it has been back-filled with ordinary dry sand to the curved line;this is then covered with a heavy canvass or tarpaulin, and Rods 1, 2, 3, etc., are run through it in sufficient numbers to hold it in position when the area,,has been back-filled with dry sand. The tarpaulin is then keyed up tight by the rods bearing on washers of large area on the top plane,.If the sand in the area,,be then removed, the observer is asked to note whether, in his judgment, the resulting pressure is greater ator at.It must undoubtedly be true that if this experiment establishes the writer’s claim in the case of normally dry sand, it will tend to do so much more readily in the case of mixed sands and earths as ordinarily found in trenches. If this fact of the greater pressure toward the top be proved conclusively, as the writer believes it undoubtedly will be in time, it may account for some of the failures of retaining walls.It seems that in order to accord our views to the theory of greater pressure at the bottom, we must assume that any given area of the face of a wall is borne on by a prism of material reaching to the top of the bank, and that its pressure is measured by the weight of its cross-section by some multiple of its length. A few moments spent in examining the face of a tunnel drift, or sheeted trench in sharp sand or loam, will convince any one that this cannot be true. In effect, the face of a sheeted trench resists the pressure of what may be termed a series of vertical groined arches, the braces being at the various abutments and the sheeting supporting the more or less loose material between. In no other way can the fact be explained that individual sheeting planks may frequently be removed for a short time without danger, even when the bracing shows evidence of very heavy pressure; whereas the removal of even a single brace may cause collapse.A few years ago the writer was called to examine a 44-ft. trench which had collapsed. While it will probably never be known definitelyhow or where the failure first occurred, it may be of interest to note that within 50 ft. of the break and under conditions apparently similar to those which had existed there previously, the writer found intermediate sheeting planks near the bottom, behind which the pressure was not sufficient to force them out against the rangers, whereas no one could for a moment doubt that there was pressure against the braces or on the sheeting directly behind them. In order, then, that experiments on retaining walls or sheeted faces may be of value, the pressures must be measured against areas absolutely independent of each other; and the writer believes that this can best be done as stated heretofore.The authorstates:“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.”The writer thinks it is unfortunate that the author has not had an opportunity of visiting with him many trenches in which no break had occurred, and yet in which the bracing had to be strengthened continually near the top. This was especially true of a horizontally sheeted and braced shaft of large area. The writer believes that it is not out of place to express the hope that the author may still change his viewpoint, and may eventually regard only as phenomena well-braced trenches which in sand or gravel do not show evidences of heavier pressure near the top.Referring now to tunnel pressures, the authorstates:“If there was no settlement of the earth,,in relation to,* * * but, as most of the weight ofis carried by the sides, in case of sufficient settlement, the vertical unit pressure,,on,will be much less than.”The writer would criticize this view because it brings in the element of settlement as essential to its conclusions, and, therefore, is contrary to his belief that all earth is under stable conditions due to the lateral transmission of thrust due to weight, thereby causing the “cohesive friction” previously noted. This transmission of thrust results in what may be termed “arching stability,” which is unchanged when small areas of hard, dry ground are undermined. When normally dry soft ground is disturbed, however, there is a gradual settlement of tiers of strata, and as those above are, in turn, left without support, they settle on what may be termed a new centering and form themselves into new arches. If these strata are composed of sand and gravel, with little natural cohesion, the action of this settlement will be similar to that observed in an hour-glass and will “work” to thesurface in a short time. It should be noted, however, that as the voids below are filled and solidified by the pressure, the lateral thrust of this pressure causes the arching tendencies to be resumed again in each successive stratum, relieving the lower strata of the pressure or shock of fall from those above.The tables given by the author onFigs. 17and 18, show that the pressure per square foot on the roof of a 15-ft. tunnel at a depth of 150 ft., for instance, is approximately double that on the same tunnel at a 50-ft. depth. This assumption is not justified by any facts which have ever come under the writer’s notice or have been brought to his attention. The observation of any tunneling operations in soft normally dry ground, or the examination of existing structures, will convince any one that, after a depth of approximately twice the greatest diameter of opening has been reached, it is impossible to tell, by any difference in the pressure, the depth of the tunnel. While numberless instances could be cited to illustrate this fact, two which have come under the writer’s notice may be of interest.In the case of a 15-ft. tunnel passing the bed of an old underground stream, a considerable amount of ground was lost through the influx of sand which came in with the water. For several days after this the writer examined the surface directly overhead for evidences of settlement, and after some 4 or 5 days he found a hole some 12 ft. in diameter and 8 or 10 ft. deep, which was at once filled in. Had the mass of earth of this area and some 20 to 60 ft. high come down on the timbering suddenly, without any intervening “arch cushion,” it would undoubtedly have crushed it; and yet none of the night men had been conscious of even the slightest shock.In the other case, a heavy rain had caused a large pond to form over the heading of a 15-ft. tunnel, and before it could be drained away it broke through the unprotected face of the heading, virtually filling the whole tunnel with sand for some distance. This, however, had not caused the collapse of any of the bracing, and, before work could be resumed, it was necessary to re-excavate the material. This material was used to fill in the hole caused by the cave-in, and when operations were finally resumed, about 10 days later, the sectional shields, which had remained in position undisturbed, were started with less than the ordinary pressure, as indicated by the gauge on the hydraulic pump.The writer confidently believes that the assumption behind the reasoning by which the table inFig. 17was made is fallacious, and that the fallacy is found in the following quotations:“In answering this question, it must be remembered that, of the weight of earth directly over the tunnel, all has been transferred to the sides that it was possible to transfer, for the coefficients of friction and cohesion given. We know scarcely anything of the cohesioncoefficients, so that the value assumed,lb. per sq. ft., may not be near the truth. Certainly it must appear plain from this discussion that the values ofandmust be better known, for all kinds of earth, before reliable results can be attained.”* * * * * * * * *“In reality, an arch or dome of the earth should be considered in place of the horizontal stratum, but the result is the same, because the same vertical forces act in either case.”* * * * * * * * *“ReferringtoFig. 20, it is evident that the maximum limit ofwould be realized if the weight of any horizontal lamina is entirely held up by the friction and cohesion of the sides; * * *”* * * * * * * * *“As seen, such a state is not exactly realized, but is practically true for great depths.”Tunnel sectionFig. 23.Referring to the last quotation, the writer would go further and say that if the assumption is true that the spaces above a tunnel are considered as a series of horizontal layers dependent on the natural coefficient of friction and cohesion (not added to by pressure) to hold them up, that it would appear to be far preferable to calculate always on full pressure to the top than to assume that some of these strata may be sustained by what would appear to the writer to be largely chance conditions.It would appear that the author has considered cohesion and friction only as normally found in exposed faces, and as they would be developed between contiguous vertical columns of earth through which pressures were transmitted laterally; and, in tunnels as against vertical faces, he does not appear to have given sufficient weight to the essential factor that cohesion and friction, combined into what the writer has previously termed “cohesive friction,” are increased by the pressure in some definite relation to it.If, for example, on a tunnel section,,Fig. 23, a centering or core of sand,,is assumed, and over this a mass,,composed of magnetic particles which cause them to adhere to each other, it is not difficult to conceive that a thickness at the key,,would be reached where the core,,could be removed. The same result, approximately, may be reached by assuming that themass is sand or earth and is supported by the core,.As, however, in this case the lower part of the arch alongis composed of loose material, the support of some of it must be provided for in the area where the blending of the arch and the core is indeterminate. This supported area is arbitrarily assumed to be,or half,the point,,being determined as far as possible by experiment to be at the intersection of the vertical,,and the line,,bisecting the angle between that of repose and the vertical. These general deductions, exclusive of the exact determination of the location of the point,,appear to be borne out by all the experiments previously noted and others in the writer’s papers hereinbefore referred to, and in the author’s observations on grain bins, as noted in the followingquotationfrom his paper:“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 a retaining-wall design.”Not only is this “an argument for including wall friction,” but it seems to prove that this friction is increased relatively to the pressure, and that under stable conditions coherence is also induced by the pressure and friction.The writer is much gratified to find that the author concurs in the view that the area of water pressure is reduced in subaqueous tunnels and other submerged structures in sand or earth, and he concurs heartily with the author that experiments on a large scale, to determine the values of this reduction definitely with relation to the various materials, will be of the greatest value to the Profession.William Cain, M. Am. Soc. C. E.(by letter).—The writer is gratified by Mr. Worcester’s words of commendation. The walls or boards subjected to earth pressure were of various inclinations, and the surface slope of the earth was equally varied. A theory which stands the test of experiments in such variety seems to be pretty well established. If the various theories that have been proposed from time to time were subjected to this test, how many would survive? And yet no theory can claim to be a practical one unless it is found to agree fairly well with experiments. Mr. Worcester seems to think that the effect of time on retaining walls ought to be included. The effort was made to do this, by using a factor of safety and by multiplying only the normal component of the thrust on the wall by this factor, taken as 3 for ordinary cases; where the effect of frost is decided, the factor should be increased, and the back of the wall, for say 3 or 4 ft. down from the top, should be sloped forward to allowthe earth, in the expansion incident to freezing, to push its way up the inclined plane corresponding.If railway trains pass near a retaining wall, their weight should be replaced by an equal weight of earth, which is regarded as dead weight in computing the thrust. Vibration probably increases the thrust, and this increase moves the top of the wall over slightly, on account of the yielding of the earth foundation about the outer toe. On account of the imperfect elasticity of earth, this deformation may remain and increase in time, and thus lead to the ultimate failure of the wall. This lack of spring, or recovery, in the earth foundation, is probably the main cause of the increased leaning of walls with time. The remedy is to build a foundation course of masonry, projecting in front of the wall, of such width that the true resultant on its base shall pass through its center. The base, too, should be inclined, in order to prevent sliding. Of course, efficient drainage must be secured by the use of weep-holes and perhaps drains back of the wall.Mr. Worcester is of the opinion that the friction against the back of the wall is not a permanent feature, and suggests the Rankine formula as possibly a better one for design. For a surcharge of sufficient inclination, and especially when it slopes at the angle of repose, the Rankine thrust involves more friction at the back of the wall than the method illustrated inFig. 15, where only one-third of the friction is used for ordinary cases; but, even granting, for the sake of argument, that at some time this friction is null and that subsequently rains and vibration cause an increased thrust, then the top of the wall moves over slightly, the earth will again get its frictional grip on the wall, so that this friction is always exerted when required for stability.It is, perhaps, customary to design a wall so that the resultant on its base shall pass one-third of the width of base from the outer toe. This procedure gives very different factors of safety, as hitherto defined, for different types of walls. The writer’s method aims to give equal security to all ordinary walls, by using a constant factor of safety.The writer is gratified that Mr. Meem has again recorded some of his valuable experiences, but regrets that he cannot regard some of his theories as convincing. With regard to the center of pressure on a retaining wall backed by fresh earth, Mr. Meem maintains that the intensity of pressure increases from the bottom upward, so that the center of pressure lies above the horizontal plane drawn at mid-height. This view has been shown to be untrue by experiment. Thus:(1) Leygue, in the experiments referred to onpage 420, found the value of the moment of the earth thrust about the inner toe,and also determined the plane of rupture. Using the corresponding wedge of rupture, the writer computed the normal component of the thrust. On dividing the moment by this, the distance of the center of pressure from the base was found to be, as an average, for all experiments on sand, 0.34 of the height, and for millet seed, 0.405 of the height.[Footnote25](2) The easily made experiment of Mr. Gifford[Footnote26]on the deflection of a cardboard retaining wall shows that the resultant thrust lies nearer the base than the top of the wall.(3) All the experiments discussed onpages 407–427 agree with the latter statement.(4) The resultant earth thrust on a wall must approach indefinitely water thrust asapproaches zero. The latter is known to act at one-third of the height above the base.(5) If a triangular wedge of rupture is assumed, it follows inevitably that the unit pressure increases with the depth, and that the resultant acts at one-third of the height above the base. This follows because the total pressure then varies with the square of the height, as in water pressure.(6) Let the contrary be assumed—that the pressure increases from the base upward—then, in a great depth of earth, the pressure at the top would be enormous enough to crush the hand if thrust in the earth. As everyone knows, the pressure on the hand is very slight, and this shows the absurdity of the hypothesis.It may be stated now that the proposed experiment, referring toFig. 22, would not prove Mr. Meem’s contention. For, if the earth belowwas removed, the thrust onwould have to be sufficient to prevent the whole mass,,from descending, which is far greater than,which balances only the thrust of the wedge of rupture, the inclined base of which passesthrough.A conclusive experiment could be made on a high retaining wall, backed by sand or grain (not in a bin, but unconfined except by the wall) after Jamieson’s manner in the case of grain bins, by inserting the rubber diaphragms, etc., at various points from the top down, and measuring the pressures.In respect to the distribution of pressure, the theory of the sheeted trench differs materially from that of the retaining wall. Much confusion has arisen from confounding them. On that account, and to meet many interesting points made by Mr. Meem, the writer will give a thorough discussion of retaining walls and sheeted or unsheeted trenches, backed by coherent earth.Level-topped earth ignoring vertical cohesionFig. 24.For an unlimited mass of level-topped earth, having both friction and cohesion, but ignoring the cohesion along the vertical plane,,Fig. 24, it was shown in the Appendix that the horizontal pressure on a vertical plane,,is that due to a certain prism,,and that its total amount, in pounds,is,whereis the weight in pounds of 1 cu. ft. of the earth,is the cohesion of the earth, in pounds per square foot,,The plane,,is found to bisect the angle between the natural slope and the vertical whenis horizontal, as shown byFig. 24.The value ofis given just belowEquation (4)in the Appendix. It acts atabove the base, since the distribution of stress onis linear.It follows, becauseEquation (7)is the usual one for the thrust of non-coherent earth of depth,that the total horizontal stress onof the earth endowed with cohesion, is exactly the same as that due to the same earth, but devoid of cohesion, having a free horizontal surface,,extended indefinitely in both directions, at a depth,as given byEquation (8), below the original free surface. The theory ignores any possible cohesion acting upward along,or any tension in the mass that may possibly drag down part of the wedge,,and thus decrease thethrust,.The formula is thus seen to give a thrust greater than the true one, or what may be called an upper limit. To realize the hypothesis more clearly, it may be said that if a vertical crack in the earth is assumed along,the resulting value ofwill be the same as that given byEquation (7). It is a fact of observation that sometimes earth which has been saturated and then dried out, cracks along one or more vertical planes. This indicates tension in the mass, which is overcome, however, at certain points (only) and thus vertical cracks appear.In the construction ofFig. 11, the full friction and cohesion which can be exerted on the length,(ofFig. 24), is supposed to be exerted. This construction then gives a lower limit to the thrust. As to which hypothesis will lead to the most probable value, it may be observed that the broken line of rupture,,Fig. 24, is nearer the true curved line of rupture (which is assigned both by theory and the facts of observation) than the straight line,;hence, the hypothesis leading toEquation (7)seems to be the more probable one.In the case of the open trench, supposeto be a vertical side of the trench; iflies below the level of,and a crack exists along,then, undoubtedly, the mass,,will move to the left, because there is an unbalanced force,,to cause the motion.The vertical height of an unsupported bank, where vertical cracks occur of depth,will then be that given byEquation (8), which is one-half the value usually given. This is generally an extreme lower value; for any earth endowed with much cohesion must be capable of exerting tension. If the tension exerted is sufficient to drag down(which can stand unsupported), that is, if the wedge,,acts as a whole, then the free unsupported height will be double that given byEquation (8). This is evidently an extreme upper limit, perhaps rarely attained; for vertical cracks have often been observed to precede a fall of earth into a vertical trench. For,,andft.,Equation (8)gives,whereas the usual equation giveslb. per sq. ft. The true value is possibly between these two extremes.
J. R. Worcester, M. Am. Soc. C. E.(by letter).— In reading Professor Cain’s admirable paper relating to experiments on retaining walls, the writer has looked in vain for a word of caution as to the effect which time plays in modifying the condition of equilibrium within a mass of earth. The author evidently considers it necessary to allow (by using a factor of safety) for a possible lessening of the angle of friction on account of a change in the amount of moisture, and possible vibrations, but states that in a great majority of cases the greatest thrust will occur where the earth has been recently deposited. It would appear that he neglects a possibility, if not a probability, of a readjustment of the earth particles through the influence of time, by which the angle of friction is lessened if not wholly cancelled.
The theory that cohesion in the earth and frictional resistance on the back of the retaining wall account for the experimental results seems indisputable, but such experiments must needs be carried through in a reasonable time, and, in that respect, at least, must needs differ from actual constructions which are intended to be permanent.
It is well known that unbraced excavations can often be carried vertically to considerable depths in safety, but that not infrequently— as many have learned to their sorrow— such unbraced banks have subsequently caved in. The slides in the Culebra cut may be mentioned as a similar illustration. To be sure, the delayed motion of the earth (or rock) may be attributed to the effect of moisture, but that does not invalidate the argument, as one always has to reckon with water.
It is also a matter of common knowledge that, in braced excavations, the pressure on the sheeting and bracing frequently increases in time to an extent enormously exceeding the original pressure. In many instances this has caused a failure long delayed.
Another instance of the effect of time is found in many retaining walls, in which a very slow motion has occurred, though the walls appeared to be entirely stable when first built.
A frictional or tangential force along the back of the wall may tend to prevent motion, but it is difficult to conceive of this tangential force being perpetually present on the back of a stationary wall after the back-filling has become settled and consolidated. In the interior of the mass of earth at rest the author admits that the reaction between particles is not along inclined lines, but rather that the lines of pressure are vertical and horizontal. The vertical lines of force are, of course, caused by gravity, and the horizontal lines by the tendency of particles to wedge in between those below and to spread them.
If this conception of the forces within a mass of earth is reasonable, it would seem as if it might also be extended to the pressure against an immovable vertical wall. One must then consider whether it is possible for a horizontal pressure to cause the wall to move, without changing the conditions and introducing the inclined stresses. It would seem within the bounds of possibility that a very minute motion might be produced, and that this would be followed by a readjustment of stresses in the earth by which the forces would gradually resume the horizontal direction.
The nature of the soil undoubtedly has much to do with this question. In some kinds of clay there appears to be a sort of viscosity, such as is frequently seen in pitch and other materials, or a tendency toward a slow flowing. No amount of pressure would cause a sudden motion, but time will effect a motion under a slight pressure or even the force of gravity alone. It appears that this condition is produced by the very minute particles, each moving individually into a position in which the surrounding forces balance. If one cuts a vertical face in such material one cannot force the exposed particles out of their position by crowding them from behind, but each in its turn will feel the pressure unbalanced and will slowly move out. This may not be true to the same extent with granular materials of large diameter, but a familiar instance is seen in fine wet sand. If a small excavation is dug in a wet beach sand, the banks will stand vertically at first, but, by watching closely, one may see the particles, beginning at the foot of the bank where there is most water, gradually moving out, overcoming the force of cohesion, and ever tending to seek a condition with a level surface. It seems quite likely that a similar tendency would exist in almost all soils, to greater or less degree, though perhaps it might be safely neglected in a mass of hard, irregularly-shaped fragments of stone which could interlock.
The point which the writer wishes to make is that a word of caution should accompany this argument for the frictional and cohesive forces; that they cannot always be relied on; and that sometimes the Rankine theory may be better than the wedge theory in designing, even though it does not seem to fit the experimental results.
Another warning may not be amiss, in considering the safe thickness to allow for retaining walls, and that is the effect of frost, where the surface of the ground is level and likely to retain moisture. The swelling force of freezing, under these circumstances, may be more than sufficient to overcome the beneficial effects of both cohesion and friction. Presumably this must be provided for in the “factor of safety,” and is in itself a justification for a very appreciable factor.
It may be well to emphasize the fact that a large part of the author’s assumed factor of safety seems to be absorbed in keeping the resultant within the middle third of the base. The proportionsbetween width of wall and height, determined onpages 433and 434, are such as to keep this resultant just within the base. If, with these same proportions of wall, the factor were assumed so that the resultant were within the middle third, it would be found to be nearer 1¼ than 3. The author’s statement onpage 435, that he “does not advocate the middle-third limit method in design,” is not wholly clear, but the implication is that the resultant should be well within this limit. In this case, it seems as if the factor of safety would be wholly absorbed in thus locating the resultant, and would leave nothing for other elements of uncertainty.
J. C. MEEM, M. Am. Soc. C. E.(by letter).—In the writer’s judgment the author has gone a step forward in developing the relation of cohesion and friction to walls and tunnels, but he has not given sufficient value to the larger consideration of what may be called “cohesive friction” induced by the lateral pressure of earth against retaining walls and other faces. This will be noted later in the discussion.
The author states that “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.”[Footnote21]To a certain extent this is fully in accordance with the writer’s views, as noted in his papers on earth pressures,[Footnote22]and he feels justified in once more calling attention to the fact that, in his judgment, the only experiments which can definitely establish the value of earth pressures against walls and sheeted faces should be made on a large scale and against independently laid and independently braced horizontal sheeting. If these experiments are made in a homogeneous material, such as dry sand of a known angle of repose, it is believed that it will be conclusively and definitely shown that the pressures at a point above the middle plane will be greater than below it, and, further, that it will be proved that the pressure near the bottom, for example, of a 20-ft. trench, will not be perceptibly greater than that against a brace at the same distance from the bottom of a 40- or a 60-ft. trench.
The writer agrees with the author that the theory of a sliding wedge is the best practical one for retaining walls, and if the face of the wedge be keyed tight, as stated by the author, or through the compacting of the material into a more solid mass, it will be seen that, with no break, the resultant pressures against the wall or face are virtually those, not only of a sliding, but of a solid wedge tending to slide on the plane of repose, the mass being compactly held together by cohesioninduced by lateral pressure. Although the general solidity of this mass is dependent on the stability of the wall or bracing, the pressures caused by the tendency of this solid wedge to slide are not affected materially by slight changes in form due to gradual settlement, which, in turn, may be caused by the normal yielding of the wall or shrinkage of the bracing, or by any small losses of material, as long as any of those or their sum is not sufficient to cause partial or final collapse. Assuming then that this theory of the solid wedge tending to slide is the true one, it is difficult to reconcile it with the results of experiments on revolving boards or revolving walls, except in so far as they relate to the more accurate determination of the coefficients of friction and cohesion.
In connection with the experiments of Leygue, the writer wishes to refer to the dry sand and wheat arching experiments described in the final discussion of his paper, “The Bracing of Trenches and Tunnels, with Practical Formulas for Earth Pressures,”[Footnote23]and especially to the experiment made with dry sand in a 2-in. pipe, described in his paper on “Pressure, Resistance, and Stability of Earth.”[Footnote24]For clearer reference, the latter experiment is again described:
A 2-in. pipe, 18 in. long, was filled with dry sand for a depth of 12 in., and a thin piece of tissue paper was pasted across the bottom. Then, with a wooden piston bearing on the sand, the latter would support the blow of a sledge hammer or the weight of a man without breaking the tissue paper.
All these experiments seem to prove very clearly that pressure is distributed laterally, at least through perfectly dry material, and, with certain values known, it should not be difficult to compute the value of this thrust. For example, if, in the pipe experiment, the exact coefficient of friction of the granular particles against the inside of the pipe is determined to be 0.4, then it would take a pressure equal to the complement of this, or 0.6 of the weight, to hold the granular particles against the vertical side. Now, if the exact depth of sand is determined at which no further pressure (by reason of adding to this depth) is transmitted to the bottom, then, by finding the difference or loss of pressure at the bottom, it should be a simple matter to compute the amount of thrust developing sufficient friction over the given area to sustain the additional weight. This is what the writer, for want of a better term, has called “cohesive friction” induced by lateral pressure. While such experiments on a small scale give definite results with very dry sand or wheat, they will not be conclusive unless made on a very large scale in cases where dampness adds materially to the normal cohesion of small masses.
As to the experiments on revolving boards or walls referred to bythe author, it would appear that, as noted, they eliminate consideration of the effect of pressure as increasing the values of the coefficient of friction and cohesion.
The writer believes that it can be shown byFig. 22that the lateral pressure against sheeted faces or retaining walls is greater toward the top.Fig. 22is adapted from Fig. 5 in the writer’s paper, “Pressure, Resistance, and Stability of Earth.”
Lateral pressure against sheeted facesFig. 22.
Fig. 22.
It is supposed that a retaining wall has first been built along the face,,and that it has been back-filled with ordinary dry sand to the curved line;this is then covered with a heavy canvass or tarpaulin, and Rods 1, 2, 3, etc., are run through it in sufficient numbers to hold it in position when the area,,has been back-filled with dry sand. The tarpaulin is then keyed up tight by the rods bearing on washers of large area on the top plane,.If the sand in the area,,be then removed, the observer is asked to note whether, in his judgment, the resulting pressure is greater ator at.It must undoubtedly be true that if this experiment establishes the writer’s claim in the case of normally dry sand, it will tend to do so much more readily in the case of mixed sands and earths as ordinarily found in trenches. If this fact of the greater pressure toward the top be proved conclusively, as the writer believes it undoubtedly will be in time, it may account for some of the failures of retaining walls.
It seems that in order to accord our views to the theory of greater pressure at the bottom, we must assume that any given area of the face of a wall is borne on by a prism of material reaching to the top of the bank, and that its pressure is measured by the weight of its cross-section by some multiple of its length. A few moments spent in examining the face of a tunnel drift, or sheeted trench in sharp sand or loam, will convince any one that this cannot be true. In effect, the face of a sheeted trench resists the pressure of what may be termed a series of vertical groined arches, the braces being at the various abutments and the sheeting supporting the more or less loose material between. In no other way can the fact be explained that individual sheeting planks may frequently be removed for a short time without danger, even when the bracing shows evidence of very heavy pressure; whereas the removal of even a single brace may cause collapse.
A few years ago the writer was called to examine a 44-ft. trench which had collapsed. While it will probably never be known definitelyhow or where the failure first occurred, it may be of interest to note that within 50 ft. of the break and under conditions apparently similar to those which had existed there previously, the writer found intermediate sheeting planks near the bottom, behind which the pressure was not sufficient to force them out against the rangers, whereas no one could for a moment doubt that there was pressure against the braces or on the sheeting directly behind them. In order, then, that experiments on retaining walls or sheeted faces may be of value, the pressures must be measured against areas absolutely independent of each other; and the writer believes that this can best be done as stated heretofore.
The authorstates:
“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.”
The writer thinks it is unfortunate that the author has not had an opportunity of visiting with him many trenches in which no break had occurred, and yet in which the bracing had to be strengthened continually near the top. This was especially true of a horizontally sheeted and braced shaft of large area. The writer believes that it is not out of place to express the hope that the author may still change his viewpoint, and may eventually regard only as phenomena well-braced trenches which in sand or gravel do not show evidences of heavier pressure near the top.
Referring now to tunnel pressures, the authorstates:
“If there was no settlement of the earth,,in relation to,* * * but, as most of the weight ofis carried by the sides, in case of sufficient settlement, the vertical unit pressure,,on,will be much less than.”
The writer would criticize this view because it brings in the element of settlement as essential to its conclusions, and, therefore, is contrary to his belief that all earth is under stable conditions due to the lateral transmission of thrust due to weight, thereby causing the “cohesive friction” previously noted. This transmission of thrust results in what may be termed “arching stability,” which is unchanged when small areas of hard, dry ground are undermined. When normally dry soft ground is disturbed, however, there is a gradual settlement of tiers of strata, and as those above are, in turn, left without support, they settle on what may be termed a new centering and form themselves into new arches. If these strata are composed of sand and gravel, with little natural cohesion, the action of this settlement will be similar to that observed in an hour-glass and will “work” to thesurface in a short time. It should be noted, however, that as the voids below are filled and solidified by the pressure, the lateral thrust of this pressure causes the arching tendencies to be resumed again in each successive stratum, relieving the lower strata of the pressure or shock of fall from those above.
The tables given by the author onFigs. 17and 18, show that the pressure per square foot on the roof of a 15-ft. tunnel at a depth of 150 ft., for instance, is approximately double that on the same tunnel at a 50-ft. depth. This assumption is not justified by any facts which have ever come under the writer’s notice or have been brought to his attention. The observation of any tunneling operations in soft normally dry ground, or the examination of existing structures, will convince any one that, after a depth of approximately twice the greatest diameter of opening has been reached, it is impossible to tell, by any difference in the pressure, the depth of the tunnel. While numberless instances could be cited to illustrate this fact, two which have come under the writer’s notice may be of interest.
In the case of a 15-ft. tunnel passing the bed of an old underground stream, a considerable amount of ground was lost through the influx of sand which came in with the water. For several days after this the writer examined the surface directly overhead for evidences of settlement, and after some 4 or 5 days he found a hole some 12 ft. in diameter and 8 or 10 ft. deep, which was at once filled in. Had the mass of earth of this area and some 20 to 60 ft. high come down on the timbering suddenly, without any intervening “arch cushion,” it would undoubtedly have crushed it; and yet none of the night men had been conscious of even the slightest shock.
In the other case, a heavy rain had caused a large pond to form over the heading of a 15-ft. tunnel, and before it could be drained away it broke through the unprotected face of the heading, virtually filling the whole tunnel with sand for some distance. This, however, had not caused the collapse of any of the bracing, and, before work could be resumed, it was necessary to re-excavate the material. This material was used to fill in the hole caused by the cave-in, and when operations were finally resumed, about 10 days later, the sectional shields, which had remained in position undisturbed, were started with less than the ordinary pressure, as indicated by the gauge on the hydraulic pump.
The writer confidently believes that the assumption behind the reasoning by which the table inFig. 17was made is fallacious, and that the fallacy is found in the following quotations:
“In answering this question, it must be remembered that, of the weight of earth directly over the tunnel, all has been transferred to the sides that it was possible to transfer, for the coefficients of friction and cohesion given. We know scarcely anything of the cohesioncoefficients, so that the value assumed,lb. per sq. ft., may not be near the truth. Certainly it must appear plain from this discussion that the values ofandmust be better known, for all kinds of earth, before reliable results can be attained.”
* * * * * * * * *
“In reality, an arch or dome of the earth should be considered in place of the horizontal stratum, but the result is the same, because the same vertical forces act in either case.”
* * * * * * * * *
“ReferringtoFig. 20, it is evident that the maximum limit ofwould be realized if the weight of any horizontal lamina is entirely held up by the friction and cohesion of the sides; * * *”
* * * * * * * * *
“As seen, such a state is not exactly realized, but is practically true for great depths.”
Tunnel sectionFig. 23.
Fig. 23.
Referring to the last quotation, the writer would go further and say that if the assumption is true that the spaces above a tunnel are considered as a series of horizontal layers dependent on the natural coefficient of friction and cohesion (not added to by pressure) to hold them up, that it would appear to be far preferable to calculate always on full pressure to the top than to assume that some of these strata may be sustained by what would appear to the writer to be largely chance conditions.
It would appear that the author has considered cohesion and friction only as normally found in exposed faces, and as they would be developed between contiguous vertical columns of earth through which pressures were transmitted laterally; and, in tunnels as against vertical faces, he does not appear to have given sufficient weight to the essential factor that cohesion and friction, combined into what the writer has previously termed “cohesive friction,” are increased by the pressure in some definite relation to it.
If, for example, on a tunnel section,,Fig. 23, a centering or core of sand,,is assumed, and over this a mass,,composed of magnetic particles which cause them to adhere to each other, it is not difficult to conceive that a thickness at the key,,would be reached where the core,,could be removed. The same result, approximately, may be reached by assuming that themass is sand or earth and is supported by the core,.As, however, in this case the lower part of the arch alongis composed of loose material, the support of some of it must be provided for in the area where the blending of the arch and the core is indeterminate. This supported area is arbitrarily assumed to be,or half,the point,,being determined as far as possible by experiment to be at the intersection of the vertical,,and the line,,bisecting the angle between that of repose and the vertical. These general deductions, exclusive of the exact determination of the location of the point,,appear to be borne out by all the experiments previously noted and others in the writer’s papers hereinbefore referred to, and in the author’s observations on grain bins, as noted in the followingquotationfrom his paper:
“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 a retaining-wall design.”
Not only is this “an argument for including wall friction,” but it seems to prove that this friction is increased relatively to the pressure, and that under stable conditions coherence is also induced by the pressure and friction.
The writer is much gratified to find that the author concurs in the view that the area of water pressure is reduced in subaqueous tunnels and other submerged structures in sand or earth, and he concurs heartily with the author that experiments on a large scale, to determine the values of this reduction definitely with relation to the various materials, will be of the greatest value to the Profession.
William Cain, M. Am. Soc. C. E.(by letter).—The writer is gratified by Mr. Worcester’s words of commendation. The walls or boards subjected to earth pressure were of various inclinations, and the surface slope of the earth was equally varied. A theory which stands the test of experiments in such variety seems to be pretty well established. If the various theories that have been proposed from time to time were subjected to this test, how many would survive? And yet no theory can claim to be a practical one unless it is found to agree fairly well with experiments. Mr. Worcester seems to think that the effect of time on retaining walls ought to be included. The effort was made to do this, by using a factor of safety and by multiplying only the normal component of the thrust on the wall by this factor, taken as 3 for ordinary cases; where the effect of frost is decided, the factor should be increased, and the back of the wall, for say 3 or 4 ft. down from the top, should be sloped forward to allowthe earth, in the expansion incident to freezing, to push its way up the inclined plane corresponding.
If railway trains pass near a retaining wall, their weight should be replaced by an equal weight of earth, which is regarded as dead weight in computing the thrust. Vibration probably increases the thrust, and this increase moves the top of the wall over slightly, on account of the yielding of the earth foundation about the outer toe. On account of the imperfect elasticity of earth, this deformation may remain and increase in time, and thus lead to the ultimate failure of the wall. This lack of spring, or recovery, in the earth foundation, is probably the main cause of the increased leaning of walls with time. The remedy is to build a foundation course of masonry, projecting in front of the wall, of such width that the true resultant on its base shall pass through its center. The base, too, should be inclined, in order to prevent sliding. Of course, efficient drainage must be secured by the use of weep-holes and perhaps drains back of the wall.
Mr. Worcester is of the opinion that the friction against the back of the wall is not a permanent feature, and suggests the Rankine formula as possibly a better one for design. For a surcharge of sufficient inclination, and especially when it slopes at the angle of repose, the Rankine thrust involves more friction at the back of the wall than the method illustrated inFig. 15, where only one-third of the friction is used for ordinary cases; but, even granting, for the sake of argument, that at some time this friction is null and that subsequently rains and vibration cause an increased thrust, then the top of the wall moves over slightly, the earth will again get its frictional grip on the wall, so that this friction is always exerted when required for stability.
It is, perhaps, customary to design a wall so that the resultant on its base shall pass one-third of the width of base from the outer toe. This procedure gives very different factors of safety, as hitherto defined, for different types of walls. The writer’s method aims to give equal security to all ordinary walls, by using a constant factor of safety.
The writer is gratified that Mr. Meem has again recorded some of his valuable experiences, but regrets that he cannot regard some of his theories as convincing. With regard to the center of pressure on a retaining wall backed by fresh earth, Mr. Meem maintains that the intensity of pressure increases from the bottom upward, so that the center of pressure lies above the horizontal plane drawn at mid-height. This view has been shown to be untrue by experiment. Thus:
(1) Leygue, in the experiments referred to onpage 420, found the value of the moment of the earth thrust about the inner toe,and also determined the plane of rupture. Using the corresponding wedge of rupture, the writer computed the normal component of the thrust. On dividing the moment by this, the distance of the center of pressure from the base was found to be, as an average, for all experiments on sand, 0.34 of the height, and for millet seed, 0.405 of the height.[Footnote25]
(2) The easily made experiment of Mr. Gifford[Footnote26]on the deflection of a cardboard retaining wall shows that the resultant thrust lies nearer the base than the top of the wall.
(3) All the experiments discussed onpages 407–427 agree with the latter statement.
(4) The resultant earth thrust on a wall must approach indefinitely water thrust asapproaches zero. The latter is known to act at one-third of the height above the base.
(5) If a triangular wedge of rupture is assumed, it follows inevitably that the unit pressure increases with the depth, and that the resultant acts at one-third of the height above the base. This follows because the total pressure then varies with the square of the height, as in water pressure.
(6) Let the contrary be assumed—that the pressure increases from the base upward—then, in a great depth of earth, the pressure at the top would be enormous enough to crush the hand if thrust in the earth. As everyone knows, the pressure on the hand is very slight, and this shows the absurdity of the hypothesis.
It may be stated now that the proposed experiment, referring toFig. 22, would not prove Mr. Meem’s contention. For, if the earth belowwas removed, the thrust onwould have to be sufficient to prevent the whole mass,,from descending, which is far greater than,which balances only the thrust of the wedge of rupture, the inclined base of which passesthrough.
A conclusive experiment could be made on a high retaining wall, backed by sand or grain (not in a bin, but unconfined except by the wall) after Jamieson’s manner in the case of grain bins, by inserting the rubber diaphragms, etc., at various points from the top down, and measuring the pressures.
In respect to the distribution of pressure, the theory of the sheeted trench differs materially from that of the retaining wall. Much confusion has arisen from confounding them. On that account, and to meet many interesting points made by Mr. Meem, the writer will give a thorough discussion of retaining walls and sheeted or unsheeted trenches, backed by coherent earth.
Level-topped earth ignoring vertical cohesionFig. 24.
Fig. 24.
For an unlimited mass of level-topped earth, having both friction and cohesion, but ignoring the cohesion along the vertical plane,,Fig. 24, it was shown in the Appendix that the horizontal pressure on a vertical plane,,is that due to a certain prism,,and that its total amount, in pounds,is,
whereis the weight in pounds of 1 cu. ft. of the earth,is the cohesion of the earth, in pounds per square foot,,
The plane,,is found to bisect the angle between the natural slope and the vertical whenis horizontal, as shown byFig. 24.
The value ofis given just belowEquation (4)in the Appendix. It acts atabove the base, since the distribution of stress onis linear.
It follows, becauseEquation (7)is the usual one for the thrust of non-coherent earth of depth,that the total horizontal stress onof the earth endowed with cohesion, is exactly the same as that due to the same earth, but devoid of cohesion, having a free horizontal surface,,extended indefinitely in both directions, at a depth,as given byEquation (8), below the original free surface. The theory ignores any possible cohesion acting upward along,or any tension in the mass that may possibly drag down part of the wedge,,and thus decrease thethrust,.
The formula is thus seen to give a thrust greater than the true one, or what may be called an upper limit. To realize the hypothesis more clearly, it may be said that if a vertical crack in the earth is assumed along,the resulting value ofwill be the same as that given byEquation (7). It is a fact of observation that sometimes earth which has been saturated and then dried out, cracks along one or more vertical planes. This indicates tension in the mass, which is overcome, however, at certain points (only) and thus vertical cracks appear.
In the construction ofFig. 11, the full friction and cohesion which can be exerted on the length,(ofFig. 24), is supposed to be exerted. This construction then gives a lower limit to the thrust. As to which hypothesis will lead to the most probable value, it may be observed that the broken line of rupture,,Fig. 24, is nearer the true curved line of rupture (which is assigned both by theory and the facts of observation) than the straight line,;hence, the hypothesis leading toEquation (7)seems to be the more probable one.
In the case of the open trench, supposeto be a vertical side of the trench; iflies below the level of,and a crack exists along,then, undoubtedly, the mass,,will move to the left, because there is an unbalanced force,,to cause the motion.
The vertical height of an unsupported bank, where vertical cracks occur of depth,will then be that given byEquation (8), which is one-half the value usually given. This is generally an extreme lower value; for any earth endowed with much cohesion must be capable of exerting tension. If the tension exerted is sufficient to drag down(which can stand unsupported), that is, if the wedge,,acts as a whole, then the free unsupported height will be double that given byEquation (8). This is evidently an extreme upper limit, perhaps rarely attained; for vertical cracks have often been observed to precede a fall of earth into a vertical trench. For,,andft.,Equation (8)gives,whereas the usual equation giveslb. per sq. ft. The true value is possibly between these two extremes.