I 105.600 Hence = = 6.6 inches'.
c 16,000 That is, an I-beam is needed whose section modulus is a little larger than 6.6, to provide strength for its own weight.
To select a size, we need a descriptive table of I-beams, such as is published in handbooks on structural steel.
Below is an abridged copy of such a table. (The last two columns con tain information for use later.) The figure illustrates a cross-section of an I-beam, and shows the axes referred to in the table.
It will be noticed that two sizes are given for each depth; these are the lightest and heaviest of each size that are made, but intermediate sizes can be secured. In column 5 we find 7.3 as the next larger section modulus than the one required (6.6); and this corresponds to a 124-pound 6-inch lbeam, which is probably the proper size. To ascertain whether the excess (7.3 — 6.6 = 0.70) in the section modulus is sufficient to provide for the weight of the beam, we might proceed as in example 1. In this case, however, the excess is quite large, and the beam selected is doubtless safe.
1. Determine the size of a wooden beam which can safely sustain a middle load of 2,000 pounds, if the beam rests on end supports 16 feet apart, and its working strength is 1,000 pounds per square.inch. Assume width 6 inches.
Ans. 6 X 10 inches.
2. What sized steel I-beam is needed to sustain safely a uniform load of 200,000 pounds, if it rests on end supports 10 feet apart, and its working strength is 16,000 pounds per square inch? Ans. 100-pound 24-inch.
3. What sized steel I-beam is needed to sustain safely the loading of Fig. 10, if its working strength is 16,000 pounds per square inch ? Ans. 14.75-pound 5-inch.
67. Laws of Strength of Beams. The strength of a beam is measured by the bending moment that it can safely withstand; or, since bending and resisting moments are equal, by its safe resist ing moment (SI e). Hence the safe strength of a beam varies (1) directly as the working fibre strength of its material, and (2) directly as the section modulus of its cross-section. For beams rectangular in cross-section (as wooden beams), the section modu lus is -bat, b and a denoting the breadth and altitude of the rectangle. Hence the strength of such beams varies also directly
as the breadth, and as the square of the depth. Thus, doubling the breadth of the section for a rectangular beam doubles the • strength, but doubling the depth quadruples the strength.
The safe load that a beam can sustain varies directly as its resisting moment, and depends on the way in which the load is distributed and how the beam is supported. Thus, in the first four and last two cases of the table on page 55, Therefore the safe load in all cases varies inversely with the length; and for the different cases the safe loads are as 1, 2, 4, 8, 8, and 12 respectively.
Example. What is the ratio of the strengths of a plank 2 X 10 inches when placed edgewise and when placed flatIvise on its supports ? .
When placed edgewise, the section modulus of the plank is X 2 X 102 = 33k, and when placed flatwise it is u X 10 X = 03; hence its strengths in the two positions are as 333 to q respectively, or as 5 to 1.
• What is the ratio of the safe loads for two beams of wood, one being 10 feet long, 3 X 12 inches in section, and having its load in the middle; and the other 8 feet long and 2 X 8 inches in section, with its load uniformly distributed.
Ans. As 135 to 100.
68. Modulus of Rupture. If a beam is loaded to destruction, and the value of the bending moment for the rupture stage is computed and substituted for 113 in the.formula SI c = M, then the value of S computed from the equation is the modulus of rupture for the material of the beam. Many experiments have been performed to ascertain the moduli of rupture for different materials and for different grades of the same material. The fol owing are fair values, all in pounds per square inch: • T Wrought iron and structural steels have no modulus of rup ture, as specimens of those materials will " bend double," but not break. The modulus of rupture of a material is used principally as a basis for determining its working strength. The factor of safety of a loaded beam, is computed by dividing the modulus of rupture of its material by the greatest unit-iIbre stress in the beam.