Vertical transverse section of a tunnelFig. 16.Fig. 16is a vertical transverse section of a tunnel,,and the earth,,extending over itfeet. If this tunnel has been driven by the use of a shield or poling boards, the ground will tend to settle over it, and part of the weight ofwill be sustained by cohesion and friction (resulting from the lateral thrust) exerted along the sides, vertically upward. The earth will probably arch itself, or form a series of domes superposed one upon the other, but the external forces acting on such domes will be the same as those acting on a corresponding horizontal lamina, and the theory, given in full in the Appendix, begins with the considerations pertaining to the equilibrium of such a lamina.If there was no settlement of the earth,,in relation to,then the vertical pressure per square foot onwould be(being the weight of a cubic foot of the earth in pounds), 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.Also, the lateral unit pressure,,at the level,,will be much less where settlement occurs. From the equations forand,given in the Appendix, the diagrams,Figs. 17and 18, have been constructed.In both diagrams, the weight of the earth was taken atlb. per cu. ft., and the cohesion of the earth atlb. per sq. ft. InFig. 17,and the curves forandwere laid off for a width of tunnel,,of 15 ft. and also for 30 ft. InFig. 18,,and curves are given forand,also forft. and 30 ft. for variousheights,.It will be perceived, in both figures, that when certain heights are attained, bothandcease to increase perceptibly, so that such values may be taken as corresponding toindefinitely large.Curves for vertical and lateral pressure, phi=45°Fig. 17.Curves for vertical and lateral pressure, phi=30°Fig. 18.A simple way of deriving these extreme values is given in the Appendix. The values of,andhave been taken here the same as those used by Mr. Meem, in framing his table of pressures,[Footnote13]which may be supposed to embody, in part, practical experience. The results found fromFigs. 17and 18 by the writer, for a depth of covering of several hundred feet, are uniformly much larger than those given by Mr. Meem. Are they too large for safety?In answeringthis 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 cohesion coefficients, 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. The results are submitted for discussion, in the hope that engineers will give their experience relative to the pressuresrealized in the timbering of tunnels, particularly through sand or earth not thoroughly consolidated.The value of,inFigs. 17and 18, is the average vertical unit pressure at the top of the tunnel. Experiments on grain bins lead to the inference that the pressure at the middle of the roof is greater than that at the sides, but no law of variation can be stated.The lateral unit pressure on the vertical sides of the tunnel lining at the top is given by the equation for,or by the corresponding diagram. The variation in this lateral pressure over the sides of the tunnel cannot be easily formulated, as so much of the weight of the earth, directly over the tunnel, has been transferred by a kind of arch action to the sides. Experience would better speak here.Table 5gives the values ofandforft. The figures in Columnsare taken from Mr. Meem’s table, previously referred to; those for Columnsare from the diagrams,Figs. 17and 18.In quoting Mr. Meem’s figures, the writer must not be understood as endorsing in any way his theory; but the results are of interest as embodying the conclusions of a practical engineer of large experience.TABLE 5.,in feet..,in pounds per squarefoot.,in pounds per squarefoot.....1545°1 4852 3004053001530°1 0352 1005406003045°3 2402 8004504003030°2 3252 600450750If the height,,of earth covering is 200 or 300 ft., the values given byFigs. 17and 18 are much larger than those given in Columns,which presumably represent Mr. Meem’s pressures for any height greater than 40 ft.In saturated earth, it has been customary, perhaps, to regard the earth as if it were gravel composed of solid spheres, like marbles, so that the water has free access in any direction. Thus, in the case of a retaining wall backed by such material, the water has full access practically to every part of the wall, and the wall is subjected to the full water pressure corresponding to its depth. It is likewise subjected to a thrust from the earth, corresponding toand,for the saturated material, but with a weight per cubic foot equal to that of the earth in air less the buoyant effect of the water. Thus,if a cubic foot of the porous earth, in air weighed 90 lb., and if the voids were 40%, then 1 cu. ft. of earth contains 0.6 cu. ft. of solids and the buoyant effect of the water is the weight of an equal volume of water orlb. Hence, the weight per cubic foot of earth in water islb.Similarly, for the pressures,and,at the top of a tunnel,must be replaced by 52.5, andandmust be found for the saturated material and these values substituted inEquations (5)and(6)of the Appendix. To these pressures must be added the corresponding water pressures for the full height of water, supposing it to have free communication everywhere, as in the case of the gravel filling. However, with sand, or earth with much fine material, the pores are more or less clogged up and there is perhaps intimate contact of a part of the earth with the roof of the tunnel, so that the water cannot get under it to produce a lifting effect, and if such intimate contact is found along any horizontal or vertical section, of the earth on either side of the section, it is plain that the buoyant effort of the water on a cubic foot of material will be much diminished.Mr. Meem deserves great credit, not only for calling attention to this, but especially for performing certain experiments to prove it.[Footnote14]The experiments were on sand, and only on a small scale, but the practical conclusion drawn from them is that the water pressure transmitted through sand having 40% voids is diminished about 40% in intensity. This occurs for a depth of only a few inches of sand, and presumably the diminution would be greater for sand several feet in depth. Of course, before definite values can be stated, experiments on a large scale should be made on every kind of material usually met; but, as a numerical illustration of the application, for the diminution mentioned—which is assumed to extend through the mass—it is seen that, in the examples of the retaining wall and also the tunnel, the weight per cubic foot of the earth in water must now be taken atlb. per cu. ft.This value replaces theinEquations (5)and(6)from which theandfor the top of the tunnel are found. To these values, add,for the water pressure, where the surface of the water extends a height,,above the top of the tunnel. Similarly, in the case of the retaining wall, add 0.4 of the full water thrust on the wall to that given by the earth, weighing only 75 lb. per cu. ft.As a numerical illustration, takeft.,,in airlb.,ft.,ft.; but we must now replaceby the weight in water, 75 lb., as found above. The values ofandare now found, byEquations (5)and(6)(Appendix), to be 1 917 and 246 lb. per sq. ft., respectively, for the saturated earth alone. To these values addlb. per sq. ft., water pressure, giving a total of 3 417 and 1 746 lb. per sq. ft., respectively, for the vertical and horizontal unit pressures at the top of the tunnel lining.In connection with this subject of underground pressures, it may not be inappropriate to make some concluding remarks on the maximum vertical pressures to which culverts may be subjected.LetFig. 16now represent a longitudinal vertical section along the axis of a road embankment, built over an arch culvert or box-drain,,the line,,passing through the summit of the arch or the top of the covering stone of the box-drain, and the lines,and,coinciding in part with the exterior sides of the abutments.There is a horizontal thrust of the earth on the medial plane,,acting at right angles to the plane of the paper, which tends to distribute the weight of the central portion partly toward the sides; but, ignoring this, it is seen that, if the earth everywhere settles uniformly, the maximum pressure per square unit at the top of the culvert is,and the total vertical pressure on the culvert is the weight of the earth vertically above it.If, however, the earth outside the abutment walls settles more than the walls (a case which may occur), then part of its weight, and that of the earth vertically above it, will be transferred, through friction and cohesion, along the planes,and,to the culvert, and thus the vertical pressure on the top of the culvert will be greater than in the first supposed case; but, if the reverse obtains, or if the culvert settles more than the earth outside the lines,and,or if the arch or covering stone descends in the middle relatively to the abutments, then part of the weight of the earth vertically over the culvert is transferred to the sides. For a comparatively rigid arch, the settlement is perhaps not enough to warrant us in making the maximum unit pressure less than.Exactly what settlement would warrant the use of the theory set forth in the Appendix it is impossible to say. If the unit pressure is taken as,we can rest assured that in most cases the real pressure is materially less.[Footnote1:The writer refers to his “Retaining Walls,” Van Nostrand’s Science Series, No. 3, for the demonstrations pertaining to the above constructions, and to the derivations of formulas.]Return to text[Footnote2:A full discussion may be found in the writer’s “Retaining Walls.”]Return to text[Footnote3:The experiments pertaining toFigs. 7,8, and9are due to Curie. See Curie’s “Poussée des Terres” and “Trois Notes,” Gauthier-Villars, Paris. They are of especial interest in that they were undertaken to attempt to overthrow the theory advocated above.]Return to text[Footnote4:All the experiments of Leygue referred to in what follows may be found inAnnales des Ponts et Chaussées, November, 1885.]Return to text[Footnote5:We can suppose, here, the horizontal force to be the pull of a cord extending horizontally from the box and passing over a fixed pulley, and that at the free end of the cord a weight is applied. The friction of the pulley and carriage wheels could be found experimentally and allowed for, so that some fraction of this weight would equal.]Return to text[Footnote6:This method is an extension of that given by Professor H. T. Eddy in his treatment of earth thrust, in “Researches in Graphical Statics.”]Return to text[Footnote7:To attain the greatest accuracy, in constructions like that shown inFig. 11, the scale should be as large as possible; the arcs of circles, especially, must be drawn with a large radius, and the points,,,etc., determined with care. The angle,,can be computed and laid off by aid of a table of chords. The construction in this figure corresponds to a vertical height offt.,,.The value of the component,,perpendicular to,is now to be found, by drawing lines fromand,perpendicular and parallel to,to intersection, and measuring the component to scale. For,it is found thatis the plane of rupture. The line,,through the new,representing the thrust, is very small; but it can be easily magnified by laying off the polygon,,to a scale two or three times as large, and thus the thrust can be found as accurately as before.]Return to text[Footnote8:Minutes of Proceedings, Inst. C. E., Vol. LXV. p. 140: reprinted in Van Nostrand’s Science Series.]Return to text[Footnote9:In reference to this equation, see Appendix.]Return to text[Footnote10:“The Bracing of Trenches and Tunnels, With Practical Formulas for Earth Pressures.”Transactions, Am. Soc. C. E., Vol. LX. p. 1. A number of important facts brought out in this paper are of vital importance to constructors.]Return to text[Footnote11:“Mechanics of Materials,” Tenth Edition, p. 381.]Return to text[Footnote12:Perhaps this may be accounted for by supposing cohesion between the blocks at rest, which is destroyed by the motion, when only friction acts.]Return to text[Footnote13:Transactions, Am. Soc. C. E., Vol. LXX, p. 387.]Return to text[Footnote14:Transactions, Am. Soc. C. E., Vol. LXX, pp. 365–368.]Return to text
Vertical transverse section of a tunnelFig. 16.
Fig. 16.
Fig. 16is a vertical transverse section of a tunnel,,and the earth,,extending over itfeet. If this tunnel has been driven by the use of a shield or poling boards, the ground will tend to settle over it, and part of the weight ofwill be sustained by cohesion and friction (resulting from the lateral thrust) exerted along the sides, vertically upward. The earth will probably arch itself, or form a series of domes superposed one upon the other, but the external forces acting on such domes will be the same as those acting on a corresponding horizontal lamina, and the theory, given in full in the Appendix, begins with the considerations pertaining to the equilibrium of such a lamina.
If there was no settlement of the earth,,in relation to,then the vertical pressure per square foot onwould be(being the weight of a cubic foot of the earth in pounds), 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.Also, the lateral unit pressure,,at the level,,will be much less where settlement occurs. From the equations forand,given in the Appendix, the diagrams,Figs. 17and 18, have been constructed.
In both diagrams, the weight of the earth was taken atlb. per cu. ft., and the cohesion of the earth atlb. per sq. ft. InFig. 17,and the curves forandwere laid off for a width of tunnel,,of 15 ft. and also for 30 ft. InFig. 18,,and curves are given forand,also forft. and 30 ft. for variousheights,.
It will be perceived, in both figures, that when certain heights are attained, bothandcease to increase perceptibly, so that such values may be taken as corresponding toindefinitely large.
Curves for vertical and lateral pressure, phi=45°Fig. 17.Curves for vertical and lateral pressure, phi=30°Fig. 18.
Curves for vertical and lateral pressure, phi=45°Fig. 17.
Fig. 17.
Curves for vertical and lateral pressure, phi=30°Fig. 18.
Fig. 18.
A simple way of deriving these extreme values is given in the Appendix. The values of,andhave been taken here the same as those used by Mr. Meem, in framing his table of pressures,[Footnote13]which may be supposed to embody, in part, practical experience. The results found fromFigs. 17and 18 by the writer, for a depth of covering of several hundred feet, are uniformly much larger than those given by Mr. Meem. Are they too large for safety?In answeringthis 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 cohesion coefficients, 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. The results are submitted for discussion, in the hope that engineers will give their experience relative to the pressuresrealized in the timbering of tunnels, particularly through sand or earth not thoroughly consolidated.
The value of,inFigs. 17and 18, is the average vertical unit pressure at the top of the tunnel. Experiments on grain bins lead to the inference that the pressure at the middle of the roof is greater than that at the sides, but no law of variation can be stated.
The lateral unit pressure on the vertical sides of the tunnel lining at the top is given by the equation for,or by the corresponding diagram. The variation in this lateral pressure over the sides of the tunnel cannot be easily formulated, as so much of the weight of the earth, directly over the tunnel, has been transferred by a kind of arch action to the sides. Experience would better speak here.
Table 5gives the values ofandforft. The figures in Columnsare taken from Mr. Meem’s table, previously referred to; those for Columnsare from the diagrams,Figs. 17and 18.
In quoting Mr. Meem’s figures, the writer must not be understood as endorsing in any way his theory; but the results are of interest as embodying the conclusions of a practical engineer of large experience.
TABLE 5.,in feet..,in pounds per squarefoot.,in pounds per squarefoot.....1545°1 4852 3004053001530°1 0352 1005406003045°3 2402 8004504003030°2 3252 600450750
TABLE 5.
If the height,,of earth covering is 200 or 300 ft., the values given byFigs. 17and 18 are much larger than those given in Columns,which presumably represent Mr. Meem’s pressures for any height greater than 40 ft.
In saturated earth, it has been customary, perhaps, to regard the earth as if it were gravel composed of solid spheres, like marbles, so that the water has free access in any direction. Thus, in the case of a retaining wall backed by such material, the water has full access practically to every part of the wall, and the wall is subjected to the full water pressure corresponding to its depth. It is likewise subjected to a thrust from the earth, corresponding toand,for the saturated material, but with a weight per cubic foot equal to that of the earth in air less the buoyant effect of the water. Thus,if a cubic foot of the porous earth, in air weighed 90 lb., and if the voids were 40%, then 1 cu. ft. of earth contains 0.6 cu. ft. of solids and the buoyant effect of the water is the weight of an equal volume of water orlb. Hence, the weight per cubic foot of earth in water islb.
Similarly, for the pressures,and,at the top of a tunnel,must be replaced by 52.5, andandmust be found for the saturated material and these values substituted inEquations (5)and(6)of the Appendix. To these pressures must be added the corresponding water pressures for the full height of water, supposing it to have free communication everywhere, as in the case of the gravel filling. However, with sand, or earth with much fine material, the pores are more or less clogged up and there is perhaps intimate contact of a part of the earth with the roof of the tunnel, so that the water cannot get under it to produce a lifting effect, and if such intimate contact is found along any horizontal or vertical section, of the earth on either side of the section, it is plain that the buoyant effort of the water on a cubic foot of material will be much diminished.
Mr. Meem deserves great credit, not only for calling attention to this, but especially for performing certain experiments to prove it.[Footnote14]The experiments were on sand, and only on a small scale, but the practical conclusion drawn from them is that the water pressure transmitted through sand having 40% voids is diminished about 40% in intensity. This occurs for a depth of only a few inches of sand, and presumably the diminution would be greater for sand several feet in depth. Of course, before definite values can be stated, experiments on a large scale should be made on every kind of material usually met; but, as a numerical illustration of the application, for the diminution mentioned—which is assumed to extend through the mass—it is seen that, in the examples of the retaining wall and also the tunnel, the weight per cubic foot of the earth in water must now be taken atlb. per cu. ft.
This value replaces theinEquations (5)and(6)from which theandfor the top of the tunnel are found. To these values, add,for the water pressure, where the surface of the water extends a height,,above the top of the tunnel. Similarly, in the case of the retaining wall, add 0.4 of the full water thrust on the wall to that given by the earth, weighing only 75 lb. per cu. ft.
As a numerical illustration, takeft.,,in airlb.,ft.,ft.; but we must now replaceby the weight in water, 75 lb., as found above. The values ofandare now found, byEquations (5)and(6)(Appendix), to be 1 917 and 246 lb. per sq. ft., respectively, for the saturated earth alone. To these values addlb. per sq. ft., water pressure, giving a total of 3 417 and 1 746 lb. per sq. ft., respectively, for the vertical and horizontal unit pressures at the top of the tunnel lining.
In connection with this subject of underground pressures, it may not be inappropriate to make some concluding remarks on the maximum vertical pressures to which culverts may be subjected.
LetFig. 16now represent a longitudinal vertical section along the axis of a road embankment, built over an arch culvert or box-drain,,the line,,passing through the summit of the arch or the top of the covering stone of the box-drain, and the lines,and,coinciding in part with the exterior sides of the abutments.
There is a horizontal thrust of the earth on the medial plane,,acting at right angles to the plane of the paper, which tends to distribute the weight of the central portion partly toward the sides; but, ignoring this, it is seen that, if the earth everywhere settles uniformly, the maximum pressure per square unit at the top of the culvert is,and the total vertical pressure on the culvert is the weight of the earth vertically above it.
If, however, the earth outside the abutment walls settles more than the walls (a case which may occur), then part of its weight, and that of the earth vertically above it, will be transferred, through friction and cohesion, along the planes,and,to the culvert, and thus the vertical pressure on the top of the culvert will be greater than in the first supposed case; but, if the reverse obtains, or if the culvert settles more than the earth outside the lines,and,or if the arch or covering stone descends in the middle relatively to the abutments, then part of the weight of the earth vertically over the culvert is transferred to the sides. For a comparatively rigid arch, the settlement is perhaps not enough to warrant us in making the maximum unit pressure less than.Exactly what settlement would warrant the use of the theory set forth in the Appendix it is impossible to say. If the unit pressure is taken as,we can rest assured that in most cases the real pressure is materially less.
[Footnote1:The writer refers to his “Retaining Walls,” Van Nostrand’s Science Series, No. 3, for the demonstrations pertaining to the above constructions, and to the derivations of formulas.]Return to text[Footnote2:A full discussion may be found in the writer’s “Retaining Walls.”]Return to text[Footnote3:The experiments pertaining toFigs. 7,8, and9are due to Curie. See Curie’s “Poussée des Terres” and “Trois Notes,” Gauthier-Villars, Paris. They are of especial interest in that they were undertaken to attempt to overthrow the theory advocated above.]Return to text[Footnote4:All the experiments of Leygue referred to in what follows may be found inAnnales des Ponts et Chaussées, November, 1885.]Return to text[Footnote5:We can suppose, here, the horizontal force to be the pull of a cord extending horizontally from the box and passing over a fixed pulley, and that at the free end of the cord a weight is applied. The friction of the pulley and carriage wheels could be found experimentally and allowed for, so that some fraction of this weight would equal.]Return to text[Footnote6:This method is an extension of that given by Professor H. T. Eddy in his treatment of earth thrust, in “Researches in Graphical Statics.”]Return to text[Footnote7:To attain the greatest accuracy, in constructions like that shown inFig. 11, the scale should be as large as possible; the arcs of circles, especially, must be drawn with a large radius, and the points,,,etc., determined with care. The angle,,can be computed and laid off by aid of a table of chords. The construction in this figure corresponds to a vertical height offt.,,.The value of the component,,perpendicular to,is now to be found, by drawing lines fromand,perpendicular and parallel to,to intersection, and measuring the component to scale. For,it is found thatis the plane of rupture. The line,,through the new,representing the thrust, is very small; but it can be easily magnified by laying off the polygon,,to a scale two or three times as large, and thus the thrust can be found as accurately as before.]Return to text[Footnote8:Minutes of Proceedings, Inst. C. E., Vol. LXV. p. 140: reprinted in Van Nostrand’s Science Series.]Return to text[Footnote9:In reference to this equation, see Appendix.]Return to text[Footnote10:“The Bracing of Trenches and Tunnels, With Practical Formulas for Earth Pressures.”Transactions, Am. Soc. C. E., Vol. LX. p. 1. A number of important facts brought out in this paper are of vital importance to constructors.]Return to text[Footnote11:“Mechanics of Materials,” Tenth Edition, p. 381.]Return to text[Footnote12:Perhaps this may be accounted for by supposing cohesion between the blocks at rest, which is destroyed by the motion, when only friction acts.]Return to text[Footnote13:Transactions, Am. Soc. C. E., Vol. LXX, p. 387.]Return to text[Footnote14:Transactions, Am. Soc. C. E., Vol. LXX, pp. 365–368.]Return to text
[Footnote1:The writer refers to his “Retaining Walls,” Van Nostrand’s Science Series, No. 3, for the demonstrations pertaining to the above constructions, and to the derivations of formulas.]Return to text
[Footnote2:A full discussion may be found in the writer’s “Retaining Walls.”]Return to text
[Footnote3:The experiments pertaining toFigs. 7,8, and9are due to Curie. See Curie’s “Poussée des Terres” and “Trois Notes,” Gauthier-Villars, Paris. They are of especial interest in that they were undertaken to attempt to overthrow the theory advocated above.]Return to text
[Footnote4:All the experiments of Leygue referred to in what follows may be found inAnnales des Ponts et Chaussées, November, 1885.]Return to text
[Footnote5:We can suppose, here, the horizontal force to be the pull of a cord extending horizontally from the box and passing over a fixed pulley, and that at the free end of the cord a weight is applied. The friction of the pulley and carriage wheels could be found experimentally and allowed for, so that some fraction of this weight would equal.]Return to text
[Footnote6:This method is an extension of that given by Professor H. T. Eddy in his treatment of earth thrust, in “Researches in Graphical Statics.”]Return to text
[Footnote7:To attain the greatest accuracy, in constructions like that shown inFig. 11, the scale should be as large as possible; the arcs of circles, especially, must be drawn with a large radius, and the points,,,etc., determined with care. The angle,,can be computed and laid off by aid of a table of chords. The construction in this figure corresponds to a vertical height offt.,,.The value of the component,,perpendicular to,is now to be found, by drawing lines fromand,perpendicular and parallel to,to intersection, and measuring the component to scale. For,it is found thatis the plane of rupture. The line,,through the new,representing the thrust, is very small; but it can be easily magnified by laying off the polygon,,to a scale two or three times as large, and thus the thrust can be found as accurately as before.]Return to text
[Footnote8:Minutes of Proceedings, Inst. C. E., Vol. LXV. p. 140: reprinted in Van Nostrand’s Science Series.]Return to text
[Footnote9:In reference to this equation, see Appendix.]Return to text
[Footnote10:“The Bracing of Trenches and Tunnels, With Practical Formulas for Earth Pressures.”Transactions, Am. Soc. C. E., Vol. LX. p. 1. A number of important facts brought out in this paper are of vital importance to constructors.]Return to text
[Footnote11:“Mechanics of Materials,” Tenth Edition, p. 381.]Return to text
[Footnote12:Perhaps this may be accounted for by supposing cohesion between the blocks at rest, which is destroyed by the motion, when only friction acts.]Return to text
[Footnote13:Transactions, Am. Soc. C. E., Vol. LXX, p. 387.]Return to text
[Footnote14:Transactions, Am. Soc. C. E., Vol. LXX, pp. 365–368.]Return to text