Because of the different assumptions made in each of the above cases, and also because of differences in the theories concerning the resistance of reinforced concrete (I 444), there is a considerable difference in the results obtained by different designers.
Let d = the thickness of the slab, in inches; E = the load of earth, in lb. per sq. ft.; H = the height of the embankment above the upper limit of the waterway, in feet; h = the clear height of the waterway, in feet; L = the live load, in lb. per sq. ft.; M = the maximum bending moment, in inch-pounds; S = the clear span of the culvert, in feet; w = the weight of a unit volume of the earth = 100 lb. per cu. ft.; W = the weight of a unit of volume of the concrete = 150 lb. per cu. ft.; Top of the Culvert. Considering the top slab as a beam fixed at the ends, the maximum bending moment occurs at the top side over the inside face of the side wall, and is equal to one twelfth of the total load multiplied by the span; or equation 1; in other words, it is sufficiently exact to omit the term containing d.
The amount of steel to be used is 1 to 1.5 per cent of soft steel, or 0.75 to 1.00 per cent of high carbon steel (1 463). The work ing stress in the steel may be assumed from 12,000 to 16,000 lb. per sq. in. for soft steel (1 470). The working stress in the concrete may be taken at 650 lb. per sq. in. (1 474) for the best concrete, provided the full load is not applied until the concrete has set for at least 30 days.
The thickness of the slab may then be determined by equation 11 or by equation 12, page 228, according to whether the amount of steel adopted is less or more than that given by equation 10, page 228. Assuming that the adopted steel ratio is less than that re
quired by equation 10, the thickness of the slab is given by equa tion 11, which is the b in equation 11 becoming 1 since only a portion of the cover one foot long is under consideration. M is the value of the moment from equation 1 above, f, is the adopted unit stress in the steel, p is the steel ratio, and j = 0.875 approximately.
The thickness of tht slab can readily be computed by equation 2. To the net thickness as thus computed should be added 1 to 2 inches for the proper embedment and protection of the steel.
Owing to the difficulty of really fixing the ends of a beam, it is quite common, in computing the stresses in reinforced concrete beams having nominally fixed ends, to assume that the maximum bending moment is more than that for a beam having ends abso lutely fixed. One method is to compute the moment on the assump tion that the beam has free ends, and then use eight tenths of the computed moment in making the design, which is equivalent to assuming that the maximum moment in the beam is 4 of the total load multiplied by the span, while the maximum moment in a beam having fixed ends is of the total load multiplied by the span. This method is frequently employed in the design of reinforced con crete box culverts, and is often specified, directly or indirectly, in the building ordinances of many cities as the method to be employed in computing the strength of a reinforced concrete beam having nominally fixed ends.
Bottom of the Culvert. The bottom slab is usually made the same as the top, since the load is substantially the same; and hence the floor requires no new computations.
Sides of the Culvert. The horizontal component of the earth pressure may be assumed as one third of the weight of the earth; or the horizontal pressure at the top of the side wall is } w H, and at the bottom +) w (H+ h), and the average is * w (2 H + h). The moment at the center of the side then is with sufficient accuracy Knowing the bending moment, the thickness of the side wall may be computed by equation 2, page 583. If the culvert is built mono lithic, then it is proper to take M in equation 2, page 583, as 0.8 of the value computed by equation 3 above.