This section is from the book "Safe Building", by Louis De Coppet Berg. Also available from Amazon: Code Check: An Illustrated Guide to Building a Safe House.
The architect is sometimes called upon to build retaining-walls in connection with terraces, ornamental bridges, city reservoirs, or similar problems. Then, too, all cellar walls, where not adjoining other buildings, become retaining walls; hence the necessity to know how to ascertain their strength. Some writers distinguish between "face-walls" and "retaining-walls"; a face-wall being built in front of and against ground which has not been disturbed and is not likely to slide; a retaining-wall being a wall that has a filled-in backing. On this theory a face-wall would have a purely ornamental duty, and would receive no thrust, care being taken during excavation and building-operations not to allow damp or frost to get into the ground so as to prevent its rotting or losing its natural tenacity, and to drain off all surface or underground water. It seems to the writer, however, that the only walls that can safely be considered as "face-walls" are those built against rock, and that all walls built against other banks should be calculated as retaining-walls.
The cross-section of retaining-walls vary, according to circumstances, but the outside surface of wall is generally built with a "batter" (slope) towards the earth. The most economical wall is one where both the outside and back surfaces batter towards the earth. As one or both surfaces become nearly vertical the wall requires more material to do the same work, and the most extravagant design of all is where the back face batters away from the earth; of course, the outside exposed surface of wall must either batter towards the earth (A B in Figure 49) or be vertical, (A C); it cannot batter away from the ground, otherwise the wall would overhang (as shown at A D). Where the courses of masonry are built at right angles to the outside surface the wall will be stronger than where they are all horizontal.
Thus, for the same amount of material in a wall, and same height, Figure 50 will do the most work, or be the strongest retaining-wall. Figure 51 the next strongest, Figure 52 the next, Figure 53 next, next, and Figure 55 the weakest. In Figure 50 and Figure 52 the ioints are at right angles to the outside surface; in the other figures they are all horizontal. For reservoirs, however, the shapes of Figures 53 or 55 are often employed.
To calculate the resistance of a retaining-wall proceed as follows:
The central line or axis of the pressure O P or p of backing will be at one-third of the height of back surface, measured from the ground lines,1 that is at 0 in Figure 5G, where A 0 = 1/3, A B.
The direction of the pressure-line (except for reservoirs) is usually assumed to form an angle of 57° with the back surface of wall, or
L POB = 57°.
For water it is assumed normal, that is, at right angles to the back surface of wall.
If it is desired, however, to be very exact, erect 0 E perpendicular to back surface, and make angle E O P, or (L. x) = the angle of friction of the filling-in or backing. This angle can be found from Table X.
Height of Line of Pressure.
The amount (p) of the pressure P 0 is found from the following formulae:
If the backing is filled in higher that the wall,2 p=w.L2/2. sin2. (y-x)/sin2.y.sin.(y+x) (48)
If the backing is filled in only to the top level of wall, p=w.L2/2. sinx/sin(y+2x) (48)
Amount of pressure - General case.
Backing higher than Wall.
Backing level with Wall.
Where the earth in front of the outside surface of wall C D is not packed very solidly below the grade line and against the wall, the total height of wall N B (including part underground) should be taken, in place of A B (the height above grade line).
- The top slope of backing in this case should never form an angle with the horizon, greater than the friction angle.
Where p = the total amount of pressure, in pounds, per each run-ning foot in length of wall.
Where w= the weight, in pounds, per cubic foot of hacking.
Where L= the height of retaining wall above ground, in feet. See foot note 1, p. 98.
Where y = the angle formed by the back surface of wall with the horizon.
Where x = the angle of friction of the backing as per Table X.
Weight per cubic foot.
Angle of friction.
Very compact earth....................
Sharp broken stones...........
Dry rammed earth.................
Wet rammed earth.........
Even those who do not understand trigonometry can use the above formula?.
It will simply be necessary to add or subtract, etc., the numbers of degrees of the angles y and x, and then find from any table of natural sines, cosines, etc., the corresponding value for the amount of the new angle. The value, so found, can then be squared, multiplied, square root extracted, etc., same as any other arithmetical problem. Should the number of degrees of the new angle be more than 90°, subtract 90° from the angle and use the positive cosine of the. difference in place of the sine of whole, or the tangent of the difference in place of the co-tangent of the whole; in the latter case the value of the tangent will be a negative one, and should have the negative sign prefixed.