The maximum moment at the center of a slab one foot wide equals in which W equals the total load on the width of one foot of the slab, and 1 equals the span of the slab, in feet; but by the prin ciples of Mechanics, this moment equals s Rig, in which /1 equals the modulus of transverse strength, in pounds per square foot; and h equals the thickness of the stone, in feet. Placing these iwo expressions equal to each other, and solving for we find: 242. Example. Assume that a culvert is covered with 6 feet of earth weighing 100 pounds per cubic foot. Assume a live load on top of the embankment equivalent to 500 pounds per square foot, in addition; or that the total load on the top of the slab is equivalent to 1,100 pounds per square foot of slab. Assume that the slab is to have a span (1) of 4 feet. Then the total load W on a section of the slab one foot wide, will be 1,100 X 4 = 4,400 pounds. Assume that the stone is sandstone, with an average ultimate modulus of 525 pounds per square inch (see Table XII), and that the safe value R is assumed to be 55 pounds per square inch, or 144 X 55 pounds per square foot. Substituting these values in the above equation for h, we find that h equals 1.29 feet, or 15.5 inches.
The above problem has been worked out on the basis of the live load which would be found on a railroad. For highways, this could be correspondingly decreased. It should-be noted that in the above formula the thickness of the stone it varies as the square root of the span; therefore, for a span of 3 feet (other things being the same as above), the thickness of the stone h equals 15.5 X 4 inches.
For a span of 2 feet, the thickness should be 15.5 X 1 4 11.0 inches.
Owing to the uncertainty of the true transverse strength of building stone, as has already been discussed in the design of Offsets for Footings (see sections 1S1-1S3), no precise calculation is possible; and therefore many box culverts arc made according to empirical rules, which dictate that the thickness shall be as follows: These values are slightly less than those computed above.
Although a good quality of granite, and especially of Bluestone flagging, will stand higher transverse stresses than those given above for sandstone, the rough rules just quoted are more often used, and are, of course, safer. When it is desired to test the safety of stone already cut into slabs of a given thickness, their strength may be computed from Equation S, using the values for transverse stresses as already given in Table XII.
243. Double Box Culverts. A box culvert with a stone top is generally limited by practice to a span of 4 feet, although it would, of course, be possible to obtain thicker stones which would safely carry the load over a considerably greater span. Therefore, when the required culvert area demands a greater width of opening than 4 feet, and when this type of culvert is to be used, the culvert may be made as illustrated in Fig. 74, by constructing an intermediate wall which supports the ends of the two sets of cover-stones forming the top. A section and elevation of a double box culvert of 3 feet span and a net height of 3 feet, is shown in Fig. 74. This figure also gives details of the wing walls and end walls. The double box culvert
illustrated in Fig. 75 has two spans, each of 4 feet. The stone used was a good quality of limestone. The cover-stones were made 15 inches thick.
244. End Walls. The ends of a culvert are usually expanded into end walls for the retention of the embankment. For the larger culverts, this may develop into two wing walls which act as retaining walls to prevent the embankment from falling over into the bed of the stream. An end wall is. especially necessary on the upstream end of the culvert, so as to avoid the danger that the stream will scour the i.)ank and work its way behind the culvert walls. The end wall is also carried up above the height of the top of the culvert, so as to guard still further against the washing of earth from the embank ment over the end of the culvert into the stream below. All of these details are illustrated in the figures shown.
Box eulverts arc sometimes constructed as dry masonry—that is, without the use of mortar. This should never be clone, except for very small culverts and when the stones are so large and regular that they form close, solid walls with comparatively small joints. A dry wall made up of irregular stones cannot withstand the thrusts which are usually exerted by the subsequent expansion of the earth embank ment above it.
245. Plain Concrete Culverts. Culverts may be made of plain concrete, either in the box form or of an arched type, and having very much the same general dimensions as those already given for stone box culverts. They have a great advantage over stone culverts in that they are essentially monoliths. If the side walls and top are formed in one single operation, the joint between the side walls and top becomes a source of additional strength, and the culverts are therefore much better than similar culverts made of stone. The formula developed above (Equation S) for the thickness of the con crete slab on top of a box culvert, may be used, together with the modulus of transverse strength as given for concrete in Table XII. This formula will apply, even though the slab for the cover of the culvert is laid after the side walls are built, and the slab is considered as merely resting on the side walls. If the side walls and top arc constructed in one operation so that the whole structure is actually a monolith, it may be considered that there is that much additional strength in the structure; but it would hardly be wise to reduce the thickness of the concrete slab by depending upon the continuity be tween the top and the sides.
246. Arch Culverts. Stone arches are frequently used for culverts in cases where the span is not great, and in which the design of the culvert (except for some small details regarding the wing walls) depends only on the span of the culvert. The design of some arch culverts used on the Atchison, Topeka & Santa Fe. Railroad (see Fig. 76, and also Fig. 74) is copied from a paper presented to the Amer ican Society of Civil Engineers by A. G. Allan, Asso. Mem. Am. Soc C. E. The span of these arches is 14 feet, and the thickness at the crown is 1S inches. A photograph of one of these arch culverts, which shows also many other details, is reproduced in Fig. 77.