CHAPTER VII.

CALCULATED STRENGTH.

THE many plans prepared during the construction of the Kansas City Bridge involved a proportionate amount of mathematical calculation; much of this was of a simple and elementary character, devoid of general interest; but no account of the work would be complete which did not embrace a review of the stresses in the foundation works, the pressure upon the several foundations, and the strains in the superstructure, these being the points in which the computations were carried into the greatest detail, and which have the most important bearing on the general structure.

The foundation works embrace both the caissons, which were exposed to the pressure of the sand and water, and the upper works, which carried the suspended weight. The strains in the latter were of a simple character, and need not be enumerated here; those in the former arose from the pressure of the water, due both to the current and the depth, and the pressure of the sand, including also the friction caused by this pressure on the sides of the descending caisson. The effects of the current was computed, but, though important in determining the strength of the cables used in anchoring the water-deadener and placing the round caisson, it was too slight to influence the general results elsewhere.

WATER PRESSURE.

The greatest water pressure occurred when the caisson for Pier No. 1 was pumped out. The surface of the water was then 101.4, about a foot and a half above the ordinary low-water stage, and four feet and a half above the extreme low-water; the elevation of the rock was about 84, so that the pressure corresponded to a depth of very nearly 17.5 feet; this made the pressure on each horizontal foot of caisson 9,570 pounds, and the total pressure, estimating the

perimeter at 155 feet, 1,483,250 pounds, or a little less than 750 tons. The form of the caisson was such that the starlings braced themselves, and the only pressure which had to be carried by interior braces was that on the opposite long walls, the total strain on the braces being equal to the pressure on 55 feet, the length of one of these sides, or 483,300 pounds. This would have been carried by twenty-five braces, each eight inches square, with a strain scarcely exceeding 300 pounds on the square inch; but to avoid all possibility of accident, nearly double this strength of bracing was used. No other caisson was pumped out to nearly this depth; the round tub used at Pier No. 2, from its circular form, withstood the strains upon it without the aid of interior bracing.

SAND PRESSURE AND FRICTION ON SIDES OF CAISSONS.

The pressure.of the sand was considered the same as the thrust of a bank of earth, the particles of which have no mutual cohesion and computed by the formula :

in which P denotes the total pressure on each horizontal foot; w, the weight of a cubic foot of the earth or other material; h, the height of the bank in feet, and a, the angle which the natural slope of the material makes with a vertical line, being the complement of the angle of repose and determined by the relation :

Cot. a coefficient of friction of material on itself.

The application of this formula becomes somewhat complicated when the earth or sand is submerged. The action of the water is threefold: 1st, it gives buoyancy to the mass, thereby diminishing the weight; 2d, by acting as a lubricator on the surfaces in contact, it reduces the friction and increases the value of a ; 3d, the pressure due to its weight is added to the thrust of the bank. The two first of these are simple and easily provided for by making the proper changes in the values of w and a; the latter is of a more complicated nature, dependent largely on the character of the material. If the bank

* This formula is taken from Claudel, Aide Memoire, etc. 7ieme Edition, p. 1252. It is also found in a slightly modified form in Rankine's Civil Engineering, 4th Edition p. 322 (11.)

is formed of loose stones or coarse gravel, the pressures of sand and water remain distinct, each substance transmitting its own pressure, and the gravel alone producing friction; if the material is a water-tight clay, the weight of the water above is equivalent to that of any other load, increasing definitely the pressure of the clay and causing additional friction;* if, however, as is commonly the case, the material be something between these extremes, a fine sand, a silt, or mixture of the two, the water neither penetrates the whole with perfect freedom nor remains as a weight on the top of a substance which it does not enter; its action is therefore dependent on capillary attraction and matters which cannot be measured precisely; and while the total pressure would not differ materially from that in either of the two preceding cases, the portion of that pressure which is transmitted by the sand, and which alone produces friction, would be somewhat greater than in the case of the gravel and less than with the clay. This could be better guarded against by an empirical allowance than measured by exact computation; in estimating friction, accordingly, the calculations were made by the formula given above, but the value given to w was the full weight of the saturated sand, and not its submerged weight alone; this, undoubtedly, gave excessive results, but as no allowance was made for the portion of the pressure of the superposed water transmitted by the sand, this discrepancy was a little less than might at first be supposed.

The coefficient of friction of wet Kansas City sand upon itself was ascertained by experiment to be about .8; the least observed was .725, which corresponds very nearly to a = 54°; this, substituted in the above formula, gives

Substituting for w the immersed weight of the heaviest sand weighed, or 69.5 pounds:

And if w be made the full weight of a cubic foot of such sand saturated,

* In this case the actual and not the submerged weight of the clay must be used in computation; but the cohesion of the particles of clay is so great that this formula would give very excessive results.

Experiments indicated the coefficient of friction of dressed oak on sand to be .475, but in calculations it was generally assumed to be .5, which, substituted in equation (c.), gives for the friction corresponding to each horizontal foot of caisson;-calling the friction F

F= 4.51 h2;

(e.)

or, substituted in equation (d),

F= 8.56 h2.

(f.)

The average weight of saturated sand, however, did not exceed 125 pounds to the cubic foot, and the coefficients of friction adopted have been slightly excessive; the decimals may therefore be omitted, and the formula reduced to the convenient form :

The average friction in pounds on each superficial foot of caisson in contact with the sand may therefore be considered as eight times the average depth in feet of the cutting edge below the surface of the sand. This formula of course varies with the material, and in its present form is applicable only to the Missouri River.

The sand pressure on the caisson at Pier No. 5, when sunk twenty feet into the sand, that being the depth of sand immediately around it when the sinking was completed, computed by formula (c.), which would properly be used in this case, as the external water pressure, whether through sand or water, was balanced by an equal internal water pressure, was 3,608 pounds on each horizontal foot, or, estimating the perimeter as 155 feet, 559,240 pounds on the entire caisson; this was less than two-fifths of the water pressure on the caisson used at Pier No. 1, and was easily carried by internal braces. The sand pressure at No. 3 was never so great as this. In proportioning the inverted caisson for Pier No. 4, the timber-work was made strong enough to withstand the thrust of the sand, without the assistance of the beton. The formula by which the pressure should be computed in this case is :

in which h denotes the total depth of sand, and h' the depth above the top of the inverted caisson. Assuming h 40 and h' 28, this equation gives for the

=

=

pressure on each foot of perimeter 7,360 pounds, a weight which a timber wall at least fourteen inches thick and eleven feet wide would easily carry over the distances between the three cross-walls.

The planking of the caisson used for Piers 3 and 5 was not dressed, and the roughness of the timber increased the friction about one-quarter, changing formula (g.) to

The available weight was in each of these cases barely enough to overcome this friction, which accounts for the slow progress of the sinking and the interruptions caused by sand-slides. The greatest available weight of the caisson at Pier No. 3 was about 700,000 lbs., which is equivalent to the friction produced by 21.25 feet of sand, the perimeter of the caisson being 155 feet; though this was greater than the actual average depth of sand, the excess was too small for advantageous work. At Pier No. 5, on the 2d of July, 1868, the effective weight for each foot of perimeter of the caisson was 2183.5 pounds; the average depth of the surrounding sand was then 15.5, corresponding to a friction of 2402.5 pounds per horizontal foot; this deficiency was remedied by piling sand above the caisson, but the weight was always too small for good results. It may be noted, that a caisson whose weight is barely greater than that of the water it displaces, may be sunk by long-continued dredging; the amount of sand excavated will be many times the capacity of the caisson, but, as the external sand slides down and passes under the edge, it will slowly carry down the caisson.

The relation between weight and friction at Pier No. 4 is most plainly shown by the tables in Appendix E; the friction per square foot of rubbed surface, computed by formula (g.), is added to these tables for convenience in showing this relation. The advantage of having a sufficient excess of weight to cause the cutting edge to penetrate well into the sand, cannot be overestimated: it aids in feeding the excavators, reduces the amount of excavation, and precludes sand-slides.

PRESSURE ON FOUNDATIONS.

The pressure upon the foundations of the seven piers is given below. In these computations the masonry is assumed to weigh 155 pounds per cubic foot,