AMERICAN SOCIETY OF CIVIL ENGINEERS
INSTITUTED 1852

TRANSACTIONS

Paper No. 1192

EXPERIMENTS ON RETAINING WALLS AND PRESSURES ON TUNNELS.

By William Cain, M. Am. Soc. C. E.

With Discussion by Messrs. J. R. Worcester, J. C. Meem, and William Cain.

The most extended experiments relating to retaining walls are those pertaining to retaining walls proper and the more elaborate ones on small rotating retaining boards. The results referring to the former agree fairly well with a rational theory, especially when the walls are several feet in height; but with the latter, many discrepancies occur, for which, hitherto, no explanation has been offered.

It will be the main object of this paper to show that the results of these experiments on small retaining boards can be harmonized with theory by including the influence of cohesion, which is neglected in deducing practical formulas. It will be found that the influence of cohesion is marked, because of the small size of the boards. This information should prove of value to future experimenters, for it will be shown that, as the height of the board or wall increases, the influence of cohesion becomes less and less, so that (for the usual dry sand filling) for heights, say, from 5 to 10 ft., it can be neglected altogether.

The result of the investigation will then be to give to the practical constructor more confidence in the theory of the sliding prism, which serves as the basis of the methods to follow.

Fig. 1.

As, in the course of this investigation, certain well-known constructions for ascertaining the pressure of any granular material against retaining walls will be needed, it is well to group them here. The various figures are supposed to represent sections at right angles to the inner faces of the walls with their backings of granular material. In the surcharged wall, [Fig. 1], produce the inner face of the wall to meet the surface of the surcharge at

. It is desired to find the thrust against the plane,

, for 1 lin. ft. of the wall. Draw

through

, the foot of the wall, making the angle of repose,

, of the earth with the horizontal and meeting the upper surface at

. Since any possible prism of rupture, as

, in tending to move downward, develops friction against both surfaces,

and

, the earth thrust on the wall will make an angle,

, with the normal to

, where

is the angle of friction of the earth on the wall. As the earth settles more than the wall, this friction will always be exerted. Again, as the wall, from its elasticity and that of the foundation, will tend to move over at the top on account of the earth thrust, the earth, with its frictional grip on the wall, will tend to prevent this, so that the friction is exerted downward in either case, and the direction of the earth thrust,

, on

is as given in [Fig. 1].

However, if

, a thin slice of earth will move with the wall, and the rubbing will be that of earth on earth, so that

in this case must be replaced by

. This rule will apply in all cases that follow, without further remark, wherever

is mentioned.

Now draw

, making the angle,

, with

, as shown; then draw

parallel to

, to the intersection,

, with

produced. From

a parallel to

is constructed, meeting

Since theory gives the relation:

, two constructions follow, by geometry, for locating the point,

. By the first, a semicircle is described on

as a diameter; at the point,

, a perpendicular is erected to

, meeting the semicircle in

; then

is laid off equal to the chord,

. By the second construction, a semicircle is described on

as a diameter, a tangent to it,

, from

is drawn, limited by the perpendicular radius, and finally

is laid off equal to

The point,

, having been thus found by either construction, draw

parallel to

to the intersection,

, with

is the plane of rupture. On laying off

, and dropping the perpendicular,

, from

, the earth pressure,

, on

is given by

, where

is the weight of a cubic unit of the earth; otherwise, the value of

is given by

times the area of the shaded triangle,

. If the dimensions are in feet, and

is in pounds per cubic foot, the thrust,

, will be given in pounds.

In [Figs. 2] and 3, the retaining boards,

, are vertical, and

is drawn, making the angle,

, with the vertical,

. The upper surface of the earth is

, and the constructions for locating

and

are the same as for [Fig. 1].

, in all the figures, represents the plane of rupture.[Footnote 1] ] In all cases, the earth thrust found as above is supposed to make the angle,

(as shown), with the normal to the inner wall surface.

Fig. 2.

Fig. 3.

In the Rankine theory, pertaining, say, to [Fig. 2], the earth thrust on a vertical plane,

, is always taken as acting parallel to the top slope. This is true for the pressure on a vertical plane in the interior of a mass of earth of indefinite extent, but it is not true generally for the pressure against a retaining wall. Thus, when

, [Fig. 2], is horizontal, Rankine’s thrust on

would be taken as horizontal, which entirely ignores the friction of the earth on the wall. The two theories agree when

and

slopes at the angle of repose, in which case, as

is parallel to

, there is no intersection,

. It is a limiting case in which, to compute the thrust,

can be laid off from any point in

, on drawing

parallel to

, etc. As

approaches the natural slope, the point,

, recedes indefinitely to the right, and it is seen that the plane of rupture,

, approaches indefinitely the line,

, or the natural slope. This limiting case, on account of the excessive thrust corresponding, will be examined more carefully in the sequel.

the height of the wall,

, in feet, and

the weight of a cubic foot of earth, in pounds, then when

, and the surface

, [Fig. 2], slopes at the angle of repose, the earth thrust, in pounds, is given by the [equation]:

If, however,

is not equal to

, then

is directed at the angle,

, to the normal to the wall, and the thrust [is]:

The foregoing constructions, and the corresponding equations, are all derived from the theory of the sliding prism. The wedge,

, [Figs. 2] and 3, is treated as an invariable solid, tending to slide down the two faces,

and

, at once, thus developing the full friction that can be exerted on these faces. In the case of actual rotation of the board,

, it is found by experiment that each particle of earth in the prism,

, moves parallel to

, each layer parallel to

moving over the layer just beneath it.

A similar motion is observed if the board,

, is moved horizontally to the left. However, in the first case (of rotation) the particles at

do not move at all, whereas in the second (of sliding motion) the particles about

move, rubbing over the floor, which thus resists the motion by friction. A thrust, thus recorded by springs or other device, in the case where the wall moves horizontally, would give an undervaluation at the lower part of

and consequently the computed center of pressure on

would be too high. On that account, only the experiments on rotating boards will be considered in this paper.

The theory of the sliding wedge, however, is justified, because no motion of either kind is actually supposed. The wedge,

, is supposed to be just on the point of motion, it being in equilibrium under the action of its weight, the normal components of the reactions of the wall, and the plane,

, and all the friction that can be exerted along

and

. These forces remain the same, whatever incipient motion is supposed. The hypothesis of a plane surface of rupture, however, is not exactly realized, experiment showing that the earth breaks along a slightly curved surface convex to the moving mass. For the sake of simplicity, the theory neglects the cohesion acting, not only along

, but possibly to a small extent along

. This additional force will be included in certain investigations to be given later.

These preliminary observations having been disposed of, the results of certain experiments on retaining walls at the limit of stability will now be given.

Fig. 4.

Fig. 5.

Fig. 6.

Fig. 7.

Figs. 4, 5, 6, and 7 refer to vertical rectangular walls backed by sand, except in the case of Fig. 5, where the filling was macadam screenings. The surface of the filling was horizontal in each case. To give briefly in detail the quantities pertaining to each wall, the following symbols will be used: