TABLE 7.,inpounds persquarefoot.,infeet.,in pounds per squarefoot.,in pounds per squarefoot.(1)(2)(3)(4)(5)(6)(7)1001530°7401 7904101 0001003030°1 7903 9001 0002 1601001545°8302 0102405701003045°2 0104 3705701 2504001530°086004804003030°8602 9604801 6404001545°096002704003045°9603 320270950The pressures given in Columns 5 and 7 are intended to apply to a long section of a tunnel, those in Columns 4 and 6 refer to a short section about the heading. The values forare intended to apply to what Mr. Meem styles “soft normally dry ground,” and it is hoped that he may approve the figures, as they are somewhere near his own. The coefficient,,refers to hard consolidated ground. Here the pressure is 0 at the working faces of the 15-ft. tunnel.Fig. 27shows the variation infor,,,for some of the larger values of,as obtained by the revised formula given in the foot-note. It will be observed that the above demonstration for finding the limiting value of,is perfectly independent of Janssen’s formula. In it the relation,,is only assumed to be true for this one value of,andneed be determined by experiment only for this value. The result is thus general, no matter howvaries for other values of.A glance at all the diagrams,[Footnote31]giving the experimental values ofandfor various depths, will show thatis far frombeing a constant for varying depths, though the assumption is found to lead to practical results, as obtained from Janssen’s formula. The experiments of Jamieson on 12 by 13½ by 67½-ft. wheat bins, and of Bovey on 12 by 14 by 44 ft. 10-in. bins, both of wood, indicate that the maximum pressure,,is realized, practically, for heights of about four diameters. Pleissner’s experiments on a wooden bin, 11.51 by 8.20 ft., show four and a half diameters, and Luft, for a concrete bin, 23 ft. in diameter, gives, say, three diameters, for the height corresponding tomaximum.Variation in vertical pressureFig. 27.These are wide variations, resulting from variations inand the coefficient of friction of the wheat on the walls of the bin. Asincreases, this ratio of height to diameter decreases. It would appear to be a serious objection to the use ofEquationsandif this ratio for maximumwas large, but it must be remembered that,for earth over tunnels, is not known. It is possibly larger than assumed. In any case,Equations (9), which were deduced independently of the modified Janssen formulas, appear to hold.The writer has read with much interest the very interesting “dry sand and wheat arching experiments,” referred to by Mr. Meem. It is seen from the above, that the writer believes in this arching of sand under certain conditions, for example, after some settlement. He does not see any reason for any arching in an unlimited mass of sand, levelat the top. The conjugate pressures here are vertical and horizontal; but, if a tunnel is bored through this mass, it tends to sink over the tunnel, and, only in consequence of that settlement, is a part of the weight of the sand directly over the tunnel transferred to the sides through the friction caused by the lateral thrust and the cohesion. Neither of these forces, both acting vertically upward, were in action, before the settlement. Mr. Meem gives the followingaccountof an interesting experiment:“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.”Considering the sand in the pipe alone, it affords a pretty illustration of the bin theory. Here,.Takeand,;alsolb. per cu. in. Therefore, makingin the formula forabove, we havelb. per sq. in. Hence the total pressure on the tissue paper islb., or say ¼ lb. Perhaps the paper can stand this. The pressure is reduced to 0.185 lb. on the paper if we include cohesion, takinglb. per sq. ft., as deduced from Leygue’s experiments on dry sand. This pressure would not be increased if the pipe, supposed to be vertical and filled with sand, was of great height, the weight of the additional sand being equal to the weight of the man or to the pressure induced by the blow of the hammer. It seems natural, then, to infer that the pressures due to the blow or man, are sustained by the sides of the pipe, as in the case of the sand, though the conditions are not the same. In fact, in this case, the pressure on the paper is even less than before; for the blow, or the weight of the man causes the passive lateral thrust of the earth to be exerted, and this, for,is nine times the active thrust hitherto used, at least for an unlimited mass of earth. If this ratio is assumed to hold for the sand in the pipe, the value ofwill be changed to,and the total pressure on the paper will be onlyIt is hoped that experimenters may turn their attention to finding definite values of the coefficient of cohesion for all kinds of earth. From observations of unsupported trenches, it has been seen that values ofof from 100 to possibly 400 lb. per sq. ft., may beexpected. Résal states[Footnote32]that MM. Jacquinot and Frontard, in July and August, 1910, made some preliminary experiments on earth taken from a reservoir dam which was failing, and found for it aboutkg. per. sq. m., or say 409 lb. per sq. ft.; but,corresponding to.The latter result is startling. For findingandexperimentally, Résal suggests that a thin slice of earth be placed between two rough metallic plaques, pressed firmly together, and that the resistance to the relative displacement of the two plaques, for varying pressures, be recorded. By writing the relation between,,and the forces involved, for each experiment, values ofandcan be found by elimination. In conclusion, the writer believes that he has offered a satisfactory and comprehensive theory of earth pressure, for earth endowed with both cohesion and friction. The results are not on as satisfactory a basis for pressures on tunnels, but the formulas derived are submitted in the hope that engineers will subject them to the test of both experience and experiment.The writer returns sincere thanks to Messrs. Worcester and Meem for their helpful and stimulating discussion.[Footnote21:Page 426.]Return to text[Footnote22:“The Bracing of Tunnels and Trenches, with Practical Formulas for Earth Pressures,”Transactions, Am. Soc. C. E., Vol. LX, p. 1; and “Pressure, Resistance, and Stability of Earth,”Transactions, Am. Soc. C. E., Vol. LXX, p. 352.]Return to text[Footnote23:Transactions, Am. Soc. C. E., Vol. LX, p. 1.]Return to text[Footnote24:Transactions, Am. Soc. C. E., Vol. LXX, p. 352.]Return to text[Footnote25:See the writer’s “Retaining Walls,” sixth edition, p. 132.]Return to text[Footnote26:Transactions, Am. Soc. C. E., Vol. LX, p. 84.]Return to text[Footnote27:This is reviewed inEngineering News, January 19th, 1911.]Return to text[Footnote28:The formulas used will be found in the writer’s “Retaining Walls,” sixth edition, p. 96,et seq.]Return to text[Footnote29:This assumes that the values ofanddo not decrease for the greater depths.]Return to text[Footnote30:In the Appendix, the “semi-empirical” formula,,was assumed. It gives fairly correct values for small values of,but forlarge andlarge, it departs more from the truth than was at first surmised. Hence, by the reasoning above,will be made zero throughout, andEquations (5)and(6)of the Appendix will bechangedtoEquationsandwill not give accurate results forsmall, which is of no importance, as such values are never used; but they should give practically accurate results for the larger values of.]Return to text[Footnote31:Ketchum’s “The Design of Walls, Bins and Grain Elevators,” pp. 253–282.]Return to text[Footnote32:“Poussée des Terres,” Deuxième Partie, p. 327.]Return to text
TABLE 7.,inpounds persquarefoot.,infeet.,in pounds per squarefoot.,in pounds per squarefoot.(1)(2)(3)(4)(5)(6)(7)1001530°7401 7904101 0001003030°1 7903 9001 0002 1601001545°8302 0102405701003045°2 0104 3705701 2504001530°086004804003030°8602 9604801 6404001545°096002704003045°9603 320270950
TABLE 7.
The pressures given in Columns 5 and 7 are intended to apply to a long section of a tunnel, those in Columns 4 and 6 refer to a short section about the heading. The values forare intended to apply to what Mr. Meem styles “soft normally dry ground,” and it is hoped that he may approve the figures, as they are somewhere near his own. The coefficient,,refers to hard consolidated ground. Here the pressure is 0 at the working faces of the 15-ft. tunnel.Fig. 27shows the variation infor,,,for some of the larger values of,as obtained by the revised formula given in the foot-note. It will be observed that the above demonstration for finding the limiting value of,is perfectly independent of Janssen’s formula. In it the relation,,is only assumed to be true for this one value of,andneed be determined by experiment only for this value. The result is thus general, no matter howvaries for other values of.A glance at all the diagrams,[Footnote31]giving the experimental values ofandfor various depths, will show thatis far frombeing a constant for varying depths, though the assumption is found to lead to practical results, as obtained from Janssen’s formula. The experiments of Jamieson on 12 by 13½ by 67½-ft. wheat bins, and of Bovey on 12 by 14 by 44 ft. 10-in. bins, both of wood, indicate that the maximum pressure,,is realized, practically, for heights of about four diameters. Pleissner’s experiments on a wooden bin, 11.51 by 8.20 ft., show four and a half diameters, and Luft, for a concrete bin, 23 ft. in diameter, gives, say, three diameters, for the height corresponding tomaximum.
Variation in vertical pressureFig. 27.
Fig. 27.
These are wide variations, resulting from variations inand the coefficient of friction of the wheat on the walls of the bin. Asincreases, this ratio of height to diameter decreases. It would appear to be a serious objection to the use ofEquationsandif this ratio for maximumwas large, but it must be remembered that,for earth over tunnels, is not known. It is possibly larger than assumed. In any case,Equations (9), which were deduced independently of the modified Janssen formulas, appear to hold.
The writer has read with much interest the very interesting “dry sand and wheat arching experiments,” referred to by Mr. Meem. It is seen from the above, that the writer believes in this arching of sand under certain conditions, for example, after some settlement. He does not see any reason for any arching in an unlimited mass of sand, levelat the top. The conjugate pressures here are vertical and horizontal; but, if a tunnel is bored through this mass, it tends to sink over the tunnel, and, only in consequence of that settlement, is a part of the weight of the sand directly over the tunnel transferred to the sides through the friction caused by the lateral thrust and the cohesion. Neither of these forces, both acting vertically upward, were in action, before the settlement. Mr. Meem gives the followingaccountof an interesting experiment:
“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.”
Considering the sand in the pipe alone, it affords a pretty illustration of the bin theory. Here,.Takeand,;alsolb. per cu. in. Therefore, makingin the formula forabove, we havelb. per sq. in. Hence the total pressure on the tissue paper islb., or say ¼ lb. Perhaps the paper can stand this. The pressure is reduced to 0.185 lb. on the paper if we include cohesion, takinglb. per sq. ft., as deduced from Leygue’s experiments on dry sand. This pressure would not be increased if the pipe, supposed to be vertical and filled with sand, was of great height, the weight of the additional sand being equal to the weight of the man or to the pressure induced by the blow of the hammer. It seems natural, then, to infer that the pressures due to the blow or man, are sustained by the sides of the pipe, as in the case of the sand, though the conditions are not the same. In fact, in this case, the pressure on the paper is even less than before; for the blow, or the weight of the man causes the passive lateral thrust of the earth to be exerted, and this, for,is nine times the active thrust hitherto used, at least for an unlimited mass of earth. If this ratio is assumed to hold for the sand in the pipe, the value ofwill be changed to,and the total pressure on the paper will be only
It is hoped that experimenters may turn their attention to finding definite values of the coefficient of cohesion for all kinds of earth. From observations of unsupported trenches, it has been seen that values ofof from 100 to possibly 400 lb. per sq. ft., may beexpected. Résal states[Footnote32]that MM. Jacquinot and Frontard, in July and August, 1910, made some preliminary experiments on earth taken from a reservoir dam which was failing, and found for it aboutkg. per. sq. m., or say 409 lb. per sq. ft.; but,corresponding to.The latter result is startling. For findingandexperimentally, Résal suggests that a thin slice of earth be placed between two rough metallic plaques, pressed firmly together, and that the resistance to the relative displacement of the two plaques, for varying pressures, be recorded. By writing the relation between,,and the forces involved, for each experiment, values ofandcan be found by elimination. In conclusion, the writer believes that he has offered a satisfactory and comprehensive theory of earth pressure, for earth endowed with both cohesion and friction. The results are not on as satisfactory a basis for pressures on tunnels, but the formulas derived are submitted in the hope that engineers will subject them to the test of both experience and experiment.
The writer returns sincere thanks to Messrs. Worcester and Meem for their helpful and stimulating discussion.
[Footnote21:Page 426.]Return to text[Footnote22:“The Bracing of Tunnels and Trenches, with Practical Formulas for Earth Pressures,”Transactions, Am. Soc. C. E., Vol. LX, p. 1; and “Pressure, Resistance, and Stability of Earth,”Transactions, Am. Soc. C. E., Vol. LXX, p. 352.]Return to text[Footnote23:Transactions, Am. Soc. C. E., Vol. LX, p. 1.]Return to text[Footnote24:Transactions, Am. Soc. C. E., Vol. LXX, p. 352.]Return to text[Footnote25:See the writer’s “Retaining Walls,” sixth edition, p. 132.]Return to text[Footnote26:Transactions, Am. Soc. C. E., Vol. LX, p. 84.]Return to text[Footnote27:This is reviewed inEngineering News, January 19th, 1911.]Return to text[Footnote28:The formulas used will be found in the writer’s “Retaining Walls,” sixth edition, p. 96,et seq.]Return to text[Footnote29:This assumes that the values ofanddo not decrease for the greater depths.]Return to text[Footnote30:In the Appendix, the “semi-empirical” formula,,was assumed. It gives fairly correct values for small values of,but forlarge andlarge, it departs more from the truth than was at first surmised. Hence, by the reasoning above,will be made zero throughout, andEquations (5)and(6)of the Appendix will bechangedtoEquationsandwill not give accurate results forsmall, which is of no importance, as such values are never used; but they should give practically accurate results for the larger values of.]Return to text[Footnote31:Ketchum’s “The Design of Walls, Bins and Grain Elevators,” pp. 253–282.]Return to text[Footnote32:“Poussée des Terres,” Deuxième Partie, p. 327.]Return to text
[Footnote21:Page 426.]Return to text
[Footnote22:“The Bracing of Tunnels and Trenches, with Practical Formulas for Earth Pressures,”Transactions, Am. Soc. C. E., Vol. LX, p. 1; and “Pressure, Resistance, and Stability of Earth,”Transactions, Am. Soc. C. E., Vol. LXX, p. 352.]Return to text
[Footnote23:Transactions, Am. Soc. C. E., Vol. LX, p. 1.]Return to text
[Footnote24:Transactions, Am. Soc. C. E., Vol. LXX, p. 352.]Return to text
[Footnote25:See the writer’s “Retaining Walls,” sixth edition, p. 132.]Return to text
[Footnote26:Transactions, Am. Soc. C. E., Vol. LX, p. 84.]Return to text
[Footnote27:This is reviewed inEngineering News, January 19th, 1911.]Return to text
[Footnote28:The formulas used will be found in the writer’s “Retaining Walls,” sixth edition, p. 96,et seq.]Return to text
[Footnote29:This assumes that the values ofanddo not decrease for the greater depths.]Return to text
[Footnote30:In the Appendix, the “semi-empirical” formula,,was assumed. It gives fairly correct values for small values of,but forlarge andlarge, it departs more from the truth than was at first surmised. Hence, by the reasoning above,will be made zero throughout, andEquations (5)and(6)of the Appendix will bechangedto
Equationsandwill not give accurate results forsmall, which is of no importance, as such values are never used; but they should give practically accurate results for the larger values of.]Return to text
[Footnote31:Ketchum’s “The Design of Walls, Bins and Grain Elevators,” pp. 253–282.]Return to text
[Footnote32:“Poussée des Terres,” Deuxième Partie, p. 327.]Return to text