EMPIRICAL RULES FOR THICKNESS OF RETAINING WALLS. Below are three well-known empirical rules for the thickness of masonry retaining walls which are also applicable to walls of plain concrete. Notice that the first gives the lightest wall and the last the heaviest.
"Hundreds of revetments have been built by royal engineer officers in accordance with General Fanshawe's rule of some fifty years ago, which was to make the thickness of a rectangular brick wall, retaining ordinary material, 24 per cent of the height for a batter of ., 25 per cent for , 26 per cent for k, 27 per cent for
28 per cent for
30 per cent for
and 32 per cent for a vertical wall."* Baker's Bale. Sir Benjamin Baker, who had large ex perience in all kind of soils in building 9 miles of retaining walls of heights up to 45 ft. and with 34 miles of trenches of depths down to 54 ft. says:
"Experience has shown that a wall [to sustain earth having a level top surface], whose thickness is one fourth of its height, and which batters 1 or 2 inches per foot on the face, possesses sufficient stability when the backing and foundation are both favor able. This allows a factor of safety of about two to cover contin gencies. It has also been proved by experience that under no ordinary conditions of surcharge or heavy backing, is it necessary to make a retaining wall on a solid foundation more than double the above, or one half of the height in thickness. Within these limits the engineer must vary the strength according to the conditions affecting the particular case. Outside of these limits, the structure ceases to be a retaining wall in the ordinary acceptation of the term. As a result of his own experience, the writer [Sir Benjamin Baker] makes the thickness of retaining walls in ground of an average character equal to one third of the height from the top of the footings. The whole of the walls on the District railway [the Metropolitan District underground railways of London] were designed on this basis, and there has not been a single instance of settlement or overturning or sliding forward." Trantwine's Rule. Trautwine t recommends that "the thickness on the top of the footing course of a vertical or nearly vertical wall which is to sustain a backing of sand, gravel, or earth, level top surface, when the backing is deposited loosely (as when clumped from cars, carts, etc.), .for railroad practice, should not be less than the following: When the hacking is somewhat consolidated in horizontal layers, each of these thicknesses may be reduced; but no rule can be given for this. Since sand or gravel has no cohesion, the full dimensions as above should be used, even though the backing be deposited in layers. A mixture of sand, or earth with pebbles, paving stones, bowlders, etc., will exert a greater pressure against the wall than the materials ordinarily used for backing; and hence when such backing has to be used, the above thicknesses should be increased, say, about .1 to 1 part." Since the applied force is not known definitely, it is impossible to compute the factor of safety with any considerable accuracy.
Some designers consider a wall as safe against overturning if the theoretical factor of safety as computed by equation 12, page 467, is three or more; or if the theo retical center of pressure lies within the middle third of the base, i.e., if the approximate theoretical factor of safety as computed by
equation 13, page 468, is three or more. Not infrequently walls are built which by the ordinary theories are on the point of overturn ing, under the belief that the error in the theory provides a sufficient factor of safety; and such walls seem to stand satisfactorily.
There is but little danger of a stone- or brick masonry retaining wall's failing by sliding. For example, as suming the coefficient of friction to be 0.65 (Table 75, page 495), a wall 10 ft. high, having an average thickness of 25 per cent of the height, and weighing 150 lb. per cu. ft., will have a resistance to sliding due to friction alone = 0.25 X 10 X 150 X 0.65 = 2,437 lb. per lin. ft.; while according to Coulomb's theory (eq. 4, page 493) the horizontal thrust is only about 1,400 lb., or the factor of safety is nearly two. But there is certainly a vertical component of the earth pressure which is neglected in the above computation; and besides the effect of the cohesion of the mortar has been neglected. Further, it is claimed, with a considerable show of reason, that equation 4 gives a result twice or more too great. Hence the real factor of safety against sliding is probably considerably more than four.
Upon the showing of some such investigation as above, it has been customary to pay little or no attention to the factor of safety against sliding for stone or brick masonry walls; and now that retaining walls are usually built of concrete, there is still less need of considering the stability against sliding, unless perhaps upon the foundation, a subject which will be investigated presently (I 1025 28).
As a rule, it seems to be customary to assume that the only load upon the base of the wall is the weight of the masonry, and also to assume that the center of pressure is to be kept within the middle third of the base, and that consequently the maximum pressure is not more than twice the mean. Computed in this way, there is no likelihood that the masonry of an ordinary retaining wall will fail by crushing. For example, the base of a prismatic column consisting of 1 : 2 :6 portland cement concrete one month old would about be upon the point of failing by crushing if the column were 2,000 feet high (Table 31, page 197); and hence such concrete would be upon the point of crushing under a retaining wall 1,000 feet high, and a prismatic wall one tenth as high would have a factor of safety of ten, and a wall thicker at the bottom than at the top would have a greater factor of safety, which shows that with any ordinary retaining wall there is no probability of the masonry's failing by crushing.
On account of the showing of some such computations as the above, little or no attention is usually given in the design of a retaining wall to the factor of safety against crushing. Apparently, this has frequently led to the neglect of an adequate consideration of the maximum pressure on the soil under the foundation (I 1026-28).