APPENDIX.In the experiments of Jamieson and Pleissner on the pressures in deep grain bins[Footnote15], the ratio,,of the lateral unit pressure,,on a vertical plane to the vertical unit pressure,,on a horizontal plane, was found by Pleissner to vary from 0.3 to 0.5 and by Jamieson to equal 0.6, for wheat in wooden bins of various sizes.This ratio,,increases somewhat with the depth of the grain, but the increase is slight after a depth of from 2½ to 3 times the width or diameter of the bin is reached.It is recognized that the proper value of,for a particular case, can only be determined properly by experiment, but it is interesting to note that, by the theory of earth pressure of an unlimited granular mass, level at the top, the ratio of the lateral to the vertical unit pressure, at any point in the mass, is,,and that this varies from 0.361 to 0.271, as,the angle of repose, varies from 28° to 35°, the values offor wheat, given in some of the experiments. Further, by reference to Jamieson’s experiments on a model bin of smooth steel, 1 ft. in diameter,[Footnote16]filled with sand, for which,,we find the experimental value ofto equalexactly for a height of sand of 2.5 ft., the value at 6 ft. and upward being 0.33.The theory of bin pressure is utterly different from the ordinary theory of earth pressure in an unlimited granular mass; but it is seen that the latter may be of some use in furnishing a value ofwhen experimental values are lacking, as in the case of various kinds of earth, both granular and more or less consolidated.An equation for,for an unlimited mass of earth, level at the top and having a coefficient of cohesion,,has been given by Scheffler,[Footnote17]and is asfollows:This reduces to the usual formula when.It is seen fromFig. 19, if we lay off at the depth,,,that the horizontal ordinates of the triangle,,measure the values of the first term of the right member ofEquation (1). The second term,,is constant, and is represented by the horizontal ordinates of the rectangle,.Thus the value ofat the depth,,is represented by.Diagram representing lateral pressureFig. 19.At,;but, above,the equation is inapplicable, for negative values of,corresponding to tension along,are inadmissible; hence, above,we must write.[Footnote18]InEquation (1), for,,thereforeSolving thisfor,For given values of,,and,having computed,we have, on subtractingEquation (2)fromEquation (1),which is true only foror.When,.If we put,the equation reduces to the ordinary form, and thus the center of pressure of the thrust on(or)acts atabove,and its amount is.and, putting,we have, fromEquation (4),Tunnel whose roof has settledFig. 20.Next consider the case of a tunnel of width,,and length,,which has been driven by shield or by use of timbering, so that an appreciable settlement of the roof occurs; then the weight of the earth vertically over the tunnel is partly carried by the adjacent walls of earth, by friction and cohesion, and it would seem that such walls can be supposed to take the place of vertical grain bin walls, and that the theory of bin pressures corresponding may be made to apply. The theory that will be developed, which includes the influence of cohesion, is simply a modification of that used in developing Janssen’s formula, as given by Mr. Ketchum,[Footnote19]and, for a ready comparison of results, his notation will be used.InFig. 20, letbe the distance from the roof of the tunnel to the surface of the earth. ComputefromEquation (3)and lay it off from the top down.The foregoing equation,,will be used as a semi-empirical formula to express the relation betweenand,but nowis no longer equal tobut at present is unknown. As before,.Properly,should be determined by experiments, but, from lack of such experiments, it will be computed from the formula above,Consider now the conditions of equilibrium of a horizontal slice of earth of depth,,the weight of which is.The top surface, at the depth,,is acted on by the force,,acting downward, and the bottom surface by the force,,acting upward. The total lateral force acting on the area,,isand this causes a frictional force of,acting upward. The cohesive or shearing resistance on the area,,acts upward, and its amount is.Placing the sum of the vertical forces acting on the slice equal to zero.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. Simplifying the above equation, and dividing by,Putting,and,on placing,,.It follows that,therefore.When,,thereforewhere,the Napierian base.Solving for,Substituting the values ofand,we have,in which it must be remembered that,The lateral thrust is nowgiven by,To get the pressures,and,at the top of the tunnel, replaceby.The weight of the upper stratum, of depth,is in part sustained by the cohesion of the sides, but asis generally small, this cohesive force can be neglected, as was done above.Equations (5)and(6)reduce to the ordinary bin formulas of Janssen, when,,and therefore,.The modification due to these terms is generally small, unlessis very large.For large values of,is small, and asincreases indefinitely,approaches as a limit the valueThis expression may be derived independently, and is of practical value when a very high surcharge is considered.Referring toFig. 20, it is evident that the maximum limit ofwould be realized if the weight of any horizontal lamina is entirelyheld up by the friction and cohesion of the sides; for then, for all lower slices,andremain the same.as given above.As seen, such a state is not exactly realized, but is practically true for great depths.For a long tunnel, the perimeter of the section,,can be taken as,whence,This value was used in all the computations.As a numerical illustration of the use ofEquations (5)and(6), suppose a tunnel,ft. wide, and, for the earth covering let;therefore,lb. per cu. ft., andlb. per sq. ft.We deduce,,ft., thereforelb.,.Equations (5)and(6)readily reduce now toThese formulas give the vertical and horizontal unit pressures at the top of the tunnel when,In computing the values ofandfor various depths of earth covering,,a short table of hyperbolic logarithms is a convenience. The curves given by the equations above are shown onFig. 17.An additional note with respect toEquation (1)may not be inappropriate. Scheffler, in deriving this equation, considered the conditions of equilibrium of an infinitesimal wedge of earth at the depth,.It was found that the horizontal pressure at the depth,,given byEquation (3), was zero, and it was assumed by the writer that there was no pressure on a vertical plane for a less depth. Thus, inFig. 21, there is no horizontal pressure on the plane,,where;consequently, the weight of the wedge,,is supported entirely by the normal reaction of the plane,,with the cohesion and friction acting along it.[Footnote20]To deduce,the total earth pressure on the vertical plane,,it is then admissible to treat the prism,,as the prism of rupture, the surface of rupture consisting of the plane,,making the anglewith the vertical and the plane,.Therefore,the weight of the prism,,is in equilibrium with,,and,whereis the normal reaction of,,andthetotal cohesion on.On balancing components parallel and perpendicular to the plane, and then following familiar methods, it can be shown that the true value ofcorresponds to,and that this value is,Deducing total earth pressure on a vertical planeFig. 21.The derivative of this, with respect to,gives the intensity,,at the depth,,exactly the same asEquation (1), and the subsequent deductions hold. Thus the fundamentalEquation (1), according to the interpretation given, is seen to correspond to a prism of rupture,,which is a little nearer the true one, having a curved surface of rupture, than the wedge,.The above refers to the pressure on a vertical plane of a mass of level-topped earth of indefinite extent; but suppose thatis the back of a retaining wall, and that a slight movement downward of the prism of rupture is imminent; then, if the earth along the plane,,can exert sufficient tension, the mass,,in descending, may drag down the wedge,,with it, so that the full friction and cohesion alongwill be added to that along.In other words, the prism of rupture must now be taken as the wedge,;hence, the value ofcorresponding is given by the equation above, on making,as this introduces, in the first equations for equilibrium, the fact that the prism of rupture is now the wedge,.It is only one step farther to find the greatest height at which the vertical face of an open trench will stand for given coefficients,and.On makingin the equation forabove when,we find, after reduction,a value which has been quoted elsewhere in this paper. It is double the value forgiven byEquation (3). The reason for this, though, is now evident; for the last equation follows as a consequence of assuming that the full cohesive and frictional resistances alongwere exerted; whereasEquation (1)ignores them.[Footnote15:Given in detail in “The Design of Walls, Bins and Grain Elevators,” by Milo S. Ketchum, M. Am. Soc. C. E.]Return to text[Footnote16:Ketchum’s “Walls, Bins, and Grain Elevators,” Fig. 171.]Return to text[Footnote17:“Traité de Stabilité des Constructions,” p. 292; see also Remark at end of Appendix.]Return to text[Footnote18:Scheffler has not noted this fact, and consequently some of his deductions are open to objections. His theory, involving cohesion, is the only one the writer has seen.]Return to text[Footnote19:“The Design of Walls, Bins and Grain Elevators,” Chapter XVI.]Return to text[Footnote20:It may be well to remark here, that for cohesive earth, it has been proved, both theoretically and experimentally, that the surface of rupture is curved, and not a plane, as the theory assumes. However, assuming it to be a plane, and considering successive wedges of rupture of different heights, the bases of which lie on the same plane, it can be easily shown that certain of the upper wedges can be sustained by cohesion alone, and that the coefficient of cohesion required for stability varies from 0 at the surface to its maximum value at a certain depth,.Below this depth, friction in addition to cohesion is exerted, and stability is assured if we suppose the friction coefficient to increase from 0 atto its maximum value,,at some depth,.Below this depth, on the plane of rupture, the maximum values of both coefficients are exerted. Now, the ordinary wedge theory assumes, for simplicity, that these coefficients are constant all along the plane of rupture, which may be true at the instant of rupture, but not for a stable mass. It is possible, too, that rupture may be progressive, starting at the bottom.]Return to text
In the experiments of Jamieson and Pleissner on the pressures in deep grain bins[Footnote15], the ratio,,of the lateral unit pressure,,on a vertical plane to the vertical unit pressure,,on a horizontal plane, was found by Pleissner to vary from 0.3 to 0.5 and by Jamieson to equal 0.6, for wheat in wooden bins of various sizes.
This ratio,,increases somewhat with the depth of the grain, but the increase is slight after a depth of from 2½ to 3 times the width or diameter of the bin is reached.
It is recognized that the proper value of,for a particular case, can only be determined properly by experiment, but it is interesting to note that, by the theory of earth pressure of an unlimited granular mass, level at the top, the ratio of the lateral to the vertical unit pressure, at any point in the mass, is,,and that this varies from 0.361 to 0.271, as,the angle of repose, varies from 28° to 35°, the values offor wheat, given in some of the experiments. Further, by reference to Jamieson’s experiments on a model bin of smooth steel, 1 ft. in diameter,[Footnote16]filled with sand, for which,,we find the experimental value ofto equalexactly for a height of sand of 2.5 ft., the value at 6 ft. and upward being 0.33.
The theory of bin pressure is utterly different from the ordinary theory of earth pressure in an unlimited granular mass; but it is seen that the latter may be of some use in furnishing a value ofwhen experimental values are lacking, as in the case of various kinds of earth, both granular and more or less consolidated.
An equation for,for an unlimited mass of earth, level at the top and having a coefficient of cohesion,,has been given by Scheffler,[Footnote17]and is asfollows:
This reduces to the usual formula when.
It is seen fromFig. 19, if we lay off at the depth,,,that the horizontal ordinates of the triangle,,measure the values of the first term of the right member ofEquation (1). The second term,,is constant, and is represented by the horizontal ordinates of the rectangle,.Thus the value ofat the depth,,is represented by.
Diagram representing lateral pressureFig. 19.
Fig. 19.
At,;but, above,the equation is inapplicable, for negative values of,corresponding to tension along,are inadmissible; hence, above,we must write.[Footnote18]
InEquation (1), for,,therefore
Solving thisfor,
For given values of,,and,having computed,we have, on subtractingEquation (2)fromEquation (1),
which is true only foror.When,.
If we put,the equation reduces to the ordinary form, and thus the center of pressure of the thrust on(or)acts atabove,and its amount is.
and, putting,we have, fromEquation (4),
Tunnel whose roof has settledFig. 20.
Fig. 20.
Next consider the case of a tunnel of width,,and length,,which has been driven by shield or by use of timbering, so that an appreciable settlement of the roof occurs; then the weight of the earth vertically over the tunnel is partly carried by the adjacent walls of earth, by friction and cohesion, and it would seem that such walls can be supposed to take the place of vertical grain bin walls, and that the theory of bin pressures corresponding may be made to apply. The theory that will be developed, which includes the influence of cohesion, is simply a modification of that used in developing Janssen’s formula, as given by Mr. Ketchum,[Footnote19]and, for a ready comparison of results, his notation will be used.
InFig. 20, letbe the distance from the roof of the tunnel to the surface of the earth. ComputefromEquation (3)and lay it off from the top down.
The foregoing equation,,will be used as a semi-empirical formula to express the relation betweenand,but nowis no longer equal tobut at present is unknown. As before,.
Properly,should be determined by experiments, but, from lack of such experiments, it will be computed from the formula above,
Consider now the conditions of equilibrium of a horizontal slice of earth of depth,,the weight of which is.
The top surface, at the depth,,is acted on by the force,,acting downward, and the bottom surface by the force,,acting upward. The total lateral force acting on the area,,isand this causes a frictional force of,acting upward. The cohesive or shearing resistance on the area,,acts upward, and its amount is.Placing the sum of the vertical forces acting on the slice equal to zero.
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. Simplifying the above equation, and dividing by,
Putting,and,
on placing,,.
It follows that,therefore.
When,,therefore
where,the Napierian base.
Solving for,
Substituting the values ofand,we have,
in which it must be remembered that,
The lateral thrust is nowgiven by,
To get the pressures,and,at the top of the tunnel, replaceby.
The weight of the upper stratum, of depth,is in part sustained by the cohesion of the sides, but asis generally small, this cohesive force can be neglected, as was done above.
Equations (5)and(6)reduce to the ordinary bin formulas of Janssen, when,,and therefore,.The modification due to these terms is generally small, unlessis very large.
For large values of,is small, and asincreases indefinitely,approaches as a limit the value
This expression may be derived independently, and is of practical value when a very high surcharge is considered.
Referring toFig. 20, it is evident that the maximum limit ofwould be realized if the weight of any horizontal lamina is entirelyheld up by the friction and cohesion of the sides; for then, for all lower slices,andremain the same.
as given above.
As seen, such a state is not exactly realized, but is practically true for great depths.
For a long tunnel, the perimeter of the section,,can be taken as,whence,
This value was used in all the computations.
As a numerical illustration of the use ofEquations (5)and(6), suppose a tunnel,ft. wide, and, for the earth covering let;therefore,lb. per cu. ft., andlb. per sq. ft.
We deduce,,ft., thereforelb.,.
Equations (5)and(6)readily reduce now to
These formulas give the vertical and horizontal unit pressures at the top of the tunnel when,
In computing the values ofandfor various depths of earth covering,,a short table of hyperbolic logarithms is a convenience. The curves given by the equations above are shown onFig. 17.
An additional note with respect toEquation (1)may not be inappropriate. Scheffler, in deriving this equation, considered the conditions of equilibrium of an infinitesimal wedge of earth at the depth,.It was found that the horizontal pressure at the depth,,given byEquation (3), was zero, and it was assumed by the writer that there was no pressure on a vertical plane for a less depth. Thus, inFig. 21, there is no horizontal pressure on the plane,,where;consequently, the weight of the wedge,,is supported entirely by the normal reaction of the plane,,with the cohesion and friction acting along it.[Footnote20]To deduce,the total earth pressure on the vertical plane,,it is then admissible to treat the prism,,as the prism of rupture, the surface of rupture consisting of the plane,,making the anglewith the vertical and the plane,.Therefore,the weight of the prism,,is in equilibrium with,,and,whereis the normal reaction of,,andthetotal cohesion on.
On balancing components parallel and perpendicular to the plane, and then following familiar methods, it can be shown that the true value ofcorresponds to,and that this value is,
Deducing total earth pressure on a vertical planeFig. 21.
Fig. 21.
The derivative of this, with respect to,gives the intensity,,at the depth,,exactly the same asEquation (1), and the subsequent deductions hold. Thus the fundamentalEquation (1), according to the interpretation given, is seen to correspond to a prism of rupture,,which is a little nearer the true one, having a curved surface of rupture, than the wedge,.
The above refers to the pressure on a vertical plane of a mass of level-topped earth of indefinite extent; but suppose thatis the back of a retaining wall, and that a slight movement downward of the prism of rupture is imminent; then, if the earth along the plane,,can exert sufficient tension, the mass,,in descending, may drag down the wedge,,with it, so that the full friction and cohesion alongwill be added to that along.In other words, the prism of rupture must now be taken as the wedge,;hence, the value ofcorresponding is given by the equation above, on making,as this introduces, in the first equations for equilibrium, the fact that the prism of rupture is now the wedge,.
It is only one step farther to find the greatest height at which the vertical face of an open trench will stand for given coefficients,and.On makingin the equation forabove when,we find, after reduction,
a value which has been quoted elsewhere in this paper. It is double the value forgiven byEquation (3). The reason for this, though, is now evident; for the last equation follows as a consequence of assuming that the full cohesive and frictional resistances alongwere exerted; whereasEquation (1)ignores them.
[Footnote15:Given in detail in “The Design of Walls, Bins and Grain Elevators,” by Milo S. Ketchum, M. Am. Soc. C. E.]Return to text[Footnote16:Ketchum’s “Walls, Bins, and Grain Elevators,” Fig. 171.]Return to text[Footnote17:“Traité de Stabilité des Constructions,” p. 292; see also Remark at end of Appendix.]Return to text[Footnote18:Scheffler has not noted this fact, and consequently some of his deductions are open to objections. His theory, involving cohesion, is the only one the writer has seen.]Return to text[Footnote19:“The Design of Walls, Bins and Grain Elevators,” Chapter XVI.]Return to text[Footnote20:It may be well to remark here, that for cohesive earth, it has been proved, both theoretically and experimentally, that the surface of rupture is curved, and not a plane, as the theory assumes. However, assuming it to be a plane, and considering successive wedges of rupture of different heights, the bases of which lie on the same plane, it can be easily shown that certain of the upper wedges can be sustained by cohesion alone, and that the coefficient of cohesion required for stability varies from 0 at the surface to its maximum value at a certain depth,.Below this depth, friction in addition to cohesion is exerted, and stability is assured if we suppose the friction coefficient to increase from 0 atto its maximum value,,at some depth,.Below this depth, on the plane of rupture, the maximum values of both coefficients are exerted. Now, the ordinary wedge theory assumes, for simplicity, that these coefficients are constant all along the plane of rupture, which may be true at the instant of rupture, but not for a stable mass. It is possible, too, that rupture may be progressive, starting at the bottom.]Return to text
[Footnote15:Given in detail in “The Design of Walls, Bins and Grain Elevators,” by Milo S. Ketchum, M. Am. Soc. C. E.]Return to text
[Footnote16:Ketchum’s “Walls, Bins, and Grain Elevators,” Fig. 171.]Return to text
[Footnote17:“Traité de Stabilité des Constructions,” p. 292; see also Remark at end of Appendix.]Return to text
[Footnote18:Scheffler has not noted this fact, and consequently some of his deductions are open to objections. His theory, involving cohesion, is the only one the writer has seen.]Return to text
[Footnote19:“The Design of Walls, Bins and Grain Elevators,” Chapter XVI.]Return to text
[Footnote20:It may be well to remark here, that for cohesive earth, it has been proved, both theoretically and experimentally, that the surface of rupture is curved, and not a plane, as the theory assumes. However, assuming it to be a plane, and considering successive wedges of rupture of different heights, the bases of which lie on the same plane, it can be easily shown that certain of the upper wedges can be sustained by cohesion alone, and that the coefficient of cohesion required for stability varies from 0 at the surface to its maximum value at a certain depth,.Below this depth, friction in addition to cohesion is exerted, and stability is assured if we suppose the friction coefficient to increase from 0 atto its maximum value,,at some depth,.Below this depth, on the plane of rupture, the maximum values of both coefficients are exerted. Now, the ordinary wedge theory assumes, for simplicity, that these coefficients are constant all along the plane of rupture, which may be true at the instant of rupture, but not for a stable mass. It is possible, too, that rupture may be progressive, starting at the bottom.]Return to text