(b) Divide this by the allowable unit-stress.
(c) The result is the required section modulus, S.
(d) Look in the column headed S (C 98-120), and, finding a value for S equal to it or next greater in value, pick out the size of the shape in the columns on the left hand side of the page.
Let it be required to design beam 3, Fig. 43, by this method, when the allowable stress is 16,000 pounds. According to the rule on page 175, the greatest moment is M.(40,950 X 20 X 12)÷8=1,228,500 pound inches. Then, by di viding this by 16,000, S is obtained. It is: S=(1,228,500)÷16,000 = 76.9.
By looking in the column headed "S" (C 98), the next highest value is 81.2, and this belongs to a 15-in. 60-pound beam. The S required for the beam must now be computed. The weight of the beam is 60 x20=1,200 pounds; the moment is (1,200 x20 x 12)÷-8=36,000 pound inches; and the S required is 36,000-i-16,000= 2.25. The net S for the beam in question is 81.20 —2.25=78.95, which, being greater than the 76.9 required, shows that the beam is strong enough; but, as before, it will be noticed that the 18-in. 55-pound beam, with S = 88.4, is lighter and stronger, and therefore cheaper. It will be used.
In case the allowable unit-stress was 10,000 pounds per square inch, as is likely to be the case in railroad bridges, the S required is 1,228, 500=10,000=122.85. This shows that a 20-in. 75-pound beam will probably do. This beam has a total S of 126.9, requires an S for its own weight of (75 X 20 X20 X 12)÷ (16,000 X 8)=2.81, thus giving a net S of 126.90-2.81=124.09, which is sufficient, as it is greater than the re quired S of 122.85.
Many engineers, in their specifications, allow that, in designing, the designer may neglect the weight of the beam if it is about one-tenth of the weight it carries. Under this condition, the weight of the beam in all the above cases would have been neglected.
As has been mentioned before, the beam is usually safe to resist shear when it will carry the load in bending. Let beam 4 of Fig. 43 be investigated to see if it is safe to resist shear.
The allowable unit-stress in shear will be taken as 10,000 pounds in this case. In any special case the specifications will give the stress al lowed. The total end reaction of beam 3 is 40,950÷2=20,475 pounds (see page 163). This is the greatest shear (see page 174). The num ber of square inches required to resist the shear is: 20,475+10,000=2.0475 sq. in.
On page 187 the beam has been designed and found to be a 15-in. 55-pound I-beam. In col umn 4 (C 97), the area of the beam is found to be 16.18 square inches, and this is amply safe in this respect. This great difference between the required and that given is not unusual in the de sign of beams, but is a usual fact. Beams large enough to carry the load so as to prevent break ing by bending, usually show a great excess when examined for shear. See page 212 for fur ther information on this point.
In some cases the moment is. so small that either an I-beam, a channel, or a Z-bar may be used. In such case, the three are designed, and the lightest (and therefore the cheapest) is taken.
For example, suppose the moment in a cer tain beam was 125,000 pound-inches, and the allowable unit-stress was 12,000 pounds per square inch. This would. require a section modulus of: S=125,000÷120,000=10.50.
By reference to column 11 (C 100), the near est section modulus is found to be 11.2, and this belongs to a 7-in. 17.5-pound I-beam. On (C 102), in column 11, is found a section modulus of 10.5, which belongs to a 9-in. 13-pound chan nel. In column 9 (C 104) is found the value 11.20, which is the nearest to the required (but must be greater); and this belongs to a by by 13/16-in. Z-bar, which weighs per linear foot. Further inspection shows that the by 3% by 21.0-pound Z-bar gives an S of 11.22; and since it weighs less, it is cheaper. It is now seen that either a 7-in. 17.5 lb. I-beam, or a 9-in. 13.25-lb. channel, or a in. 21.0-lb. Z-bar will be sufficiently strong to carry the load which produces the bending moment of 126,000 pound-inches. By looking at the weights per foot, it is seen that the channel is the best one to use, and hence the following is true: When the loads are vertical and the beam is also in a vertical position (that is, standing up on edge, with web or central portion perpendicu lar), and when one beam will be sufficient to carry the load, the channel is the most efficient section.