This beam will be used if the depth of the beam is limited to 15 or 16 inches; but if it is not limited, it will be better to use an 18-inch 55-lb. beam. The table (C 70) shows opposite the 20-foot span a carrying capacity of 23.58 tons, and it should be noted that the weight is 5 pounds less than the 15-inch 60-lb. beam. Therefore the 18-inch beam is cheaper and should be used. This point is often overlooked by designers. After a beam is designed as closely as possible according to its carrying capacity, one should see if there is not a deeper and lighter, and therefore cheaper, beam which will carry as much or more weight.
If beam 4 is to be designed, it will be seen by (C 70) that an 18-inch 55-lb. carrying 23.58 tons (including its own weight) is too small. By (C 97) it is seen that there are also 18-inch beams weighing 60, 65, and 70 pounds. In the table (C 70), the column to the right of the 18-inch beam column shows that, for each pound above 55 pounds, the beam will carry 0.24 ton. The carrying capacities of the 18-inch beams are: The net capacity is the capacity after the weight of the beam has been taken from the total capacity. Since beam 4 requires the net capacity to be 23.63 tons, it is evident that an 18-in. 60-lb. I-beam satisfies this condition, and will be used, since by looking at the next deeper beam, the 20-inch., it is found that there is no 20-inch beam which is lighter and which gives 23.63 tons net or more.
By consulting the tables for channels and other shapes (C 73-80), it will be seen that the loads above are too heavy to allow any other shape but an I-beam to be used. Of course, for beam 4, two channels might be used, and each would have to have a net capacity of 23.63÷2 =11.82 tons, or three with net capacities each of 23.63÷3=7.88 tons. By (C 73), opposite the 20-foot span of the first column, it is seen that two 15-in. 33-lb. channels are just too light, and that three 12-in. 20.5-lb. channels only give 3x5.70=17.10 tons. By (C 101) it is seen that there are five 15-in. channels whose weights are 35, 40, 45, 50, and 55 pounds. By the column (C 73), to the right of the 15-in. channel column, it is noted that 0.20 ton is added for every pound weight of section. The 15-inch 35 and 40-pound channels have net carrying capacities as follows: The required net carrying capacity for one beam is 11.82 tons. Comparing this with the above computations, it is seen that two 40-pound 15-inch channels are sufficiently strong. They are, however, very costly, when compared with the one 60-lb. 18-inch I-beam, since the two channels weigh 80 pounds, whereas the single I-beam weighs 60 pounds. In general it may be said that when used as a simple beam, a single shape is more economical than two or more others.
Second Method of Design. A second method may be employed to determine the size of the required beam. This is by the tables (C 97-106). In this case a little more latitude is allowed in the unit-stresses employed. Either 16,000 or 12,500 pounds per square inch may be used as desired, in case of beams and channels; while for Z-bars or T-shapes, 16,000 or 12,000 may be used. In this case the beam is designed by the use of the coefficients in the columns headed C and C'.
Rule—For uniform loads, multiply the total load on the beam in pounds by the span in feet, and select the sized shape in the columns of the table which occur on the same line as the value under C or C', as the case may be, which is equal to, or larger than, and as close as possible to, the value obtained by multiplying.
For example, let it be required to design beam 3, which has a total load of 40,950 pounds, the allowable unit-stress being 16,000 pounds per square inch. The span is 20 feet. For this case, the required coefficient C = 40,950 X 20= 819,000. Now, looking in the column "C," we find 866,100 as nearest to and larger than 819, 000; and on the same line in columns 2 and 3, is found the 15-in. 60-lb. I-beam. This 866,100 is the total C for that beam, and from this must be subtracted the C due to the weight of the beam. The weight of the beam is 60X20=1,200 pounds, and its C is 1,200X 20=-24,000. This gives a net C of 866,100-24,000 = 842,100. This beam could be used; but, by further inspection, it is found that C for an 18-in. 55-pound I-beam is 943,000. As this last beam is stronger as shown by the larger C, and lighter than the 15-in. beam, it should be used. This is the same result ar rived at by the first method, and this method is much shorter.
Let it be required to design the same beam when the allowable unit-stress is 12,500 pounds per square inch. C' is obtained in the same man ner, and is equal to C—that is, 40,950 X 20 = 819,000. This time we must look in the 13th column, where C' is given (C 98). The value next above 819,000 is 853,000, which is for an 18-in. 70-lb. I-beam, as noted by following the same line over to the 2d and 3d columns. From this 853,000 must be subtracted the C' of the beam. The weight of the beam is 60 X20-1,200, and the C' is 1,200x20=24,000. The net C' for the 18-in. 70-pound beam is then 853,000-24,000 =829,000, which is larger than the required C' 819,000, and therefore shows it to be strong enough. It will be noted that a 20-in. 65-pound I-beam gives a C' of 974,700, which shows it to be stronger, and as it is also 5 pounds lighter than the 18-inch beam, it should be used unless the depth is limited to 18 inches for some reason.