RETAINING WALLS 223. A retaining wall is a wall built to sustain the pressure of a vertical bank of earth. The stability of the wall is a comparatively simple matter when three quantities have been determined: (1) The intensity of the earth pressure; (2) The point of application of the resultant of the earth pressure; (3) The line of action of this pressure.
Unfortunately, earthy material is very variable in its action in these respects, depending on its condition. It is not only true that different grades of earthy material act quite differently in these respects, but it is also true that the same material will act differently under varying physical conditions, especially in regard to its saturation with water. On these accounts it is impracticable, even by experiment, to deter mine values which are reliable for all conditions.
It is also comparatively easy to formulate a theory regarding the pressure of earthwork which shall be based on certain theoretical assumptions. One of these as sumptions is that the so-called plane of 'rupture is a plane surface —or, in other words, that the line a b (Fig. 65) is a straight line. There is considerable evidence, and even theoretical grounds, for considering not only that the line a b is a curved line, but that the curve is variable, depending on the physical conditions. It is also assumed that the earthy ma terial acts virtually the same as a liquid with a density considerably greater than water; but there is ground for believing that even this assumption is not strictly warranted. Theoretically the prob lem is also very much complicated by the question of the earth pressure which may be produced by a surcharged wall. A surcharge is a bank of earth which is built above the height of the top of the retaining wall and sloping back from it. It certainly adds to the pressure on the earth immediately back of the wall itself and in creases the pressure on the wall.
224. Theoretical Formulze. In spite of the unreliability of theoretical formulie, for the reasons given above, certain formukc which are here quoted without demonstration are sometimes used for lack of better formulie as a guide in determining the thickness of a wall. For simplicity it is assumed that the rear face of the wall is vertical. The variation in the theory by attempting to allow for a slope of the rear face, merely complicates the theory; while the effect of such a variation from the vertical as is ever adopted is usually so small thatit is utterly swallowed up by the unavoidable uncertainties in the practical application of the theory. In Fig. 66,
Let E = Total pressure against rear face of wall on a unit-length of wall; w = Weight of a unit-volume of the earth; h = Height of wall; = The angle of repose with the horizontal—that is, the angle at :which that kind of earth will remain without further sliding.
Then, when the upper surface of the earth is horizontal, we have the equation: E w = tans (45°— ) (6) If the upper surface of the earth is surcharged with a bank of earth at a natural slope, or if the angle of slope of the surcharge = 0, then the equation becomes: An inspection of Equation 6 will show that the pressure E is greater for small values of 55. The angle of repose for various materials is not only variable for different grades of material, but is variable for the same grade of ma- ' E_• terial under various ' conditions of satu ration. A value of „ - 55 which is frequent- Fig. CAI. Pressure on Retaining Wall.
ly adopted is 30°.
This is considerably lower than the usual true value of 96 for dry material, and is usually a safe value of ˘) for any material (except quicksand) either wet or dry. The adoption of this value, there fore, generally means that the result is safe, and that the factor of safety is merely made somewhat larger.
225. Example. What will be the pressure per foot of length of the wall, for a wall IS feet high, the angle of repose for the earth being assumed at 30°? Solution. Here h = 18 feet, and = 30°. The weight of earth (w) is quoted as varying from 70 to 120 pounds per cubic foot according to the degree of and density of packing. When the earth is densely packed, its angle of repose is greater; therefore we arc safe in assuming a weight of 100 pounds per cubic foot for an angle (95) of 30°. Substituting these values in Equation 6, we have: (45° 100 X 324 2 =tan'' 30° X 16,200 5,400 pounds. 2 • Using this value of i = 30° gives us the simple relation that E = w or one-sixth of the unit-weight of the earth times the square of the height, for a wall without a surcharge.