(2) The flexural compressive unit-stress is less than the direct unit-stress; that is, S, is less than Then the combined unit-stress on the upper fibre is S, = S. S, (tensile); and that on the lower fibre is St = + (tensile).
The combined stress is represented by Fig. 43, d, and is all tensile.
Example. A steel bar 2 x 6 inches, and 12 feet long, is sub jected to end pulls of 45,000 pounds. It is supported at each end, and sustains, as a learn, a uniform load of 6,000 pounds. It is required to compute the combined unit-fibre stresses.
Evidently the dangerous section. is at the middle, k that is, M = 8 X 6,000 X 12 = 9,000 foot-pounds, or 9,000 X 12 = 108,000 inch-pounds. The bar being placed with the six-inch side vertical, c, = = 3 inches, and 1 I = X 2 X = 36 inches'. (See Art. 52.) 12 3 Hence S, = S2 = 108,0 00X 6,000 pounds per square inch.
Since A = 2 X 6 = 12 square inches, 45,000 So.- 1 = 3,750 pounds per square inch.
2 The greatest value of the combined compressive stress is S, = 9,000 3,750 = 5,250 pounds per square inch, and it occurs on the upper fibres of the middle section. The great est value of the combined tensile stress is S, = 9,000 + 3,750 = 12,750 pounds per square inch, and it occurs on the lowest fibres of the middle section.
Change the load in the preceding illustration to one of 6,000 pounds placed in the middle, and then solve.
pounds per square inch. = 21,750 " " " " 75. Flexure and Compression. Tmagine the arrowheads on P reversed; then Fig. 43,0, will represent a beam under com bined flexural and compressive stresses. The unit-stresses are computed as in the preceding article. The direct stress is a compression equal to P, and the unit-stress due to P is computed as in the article. Evidently the effect of P is to increase the compressive stress and decrease the tensile stress due to the flexure. In combining, we have two cases as before: (1) The flexural tensile unit-stress is greater than the direct unit-stress; that is, is greater than Then the com bined unit-stress on the lower fibre is = (tensile); and that on the upper fibre is = S, + (compressive).
The combined fibre stress is represented by Fig. 44, a, and is part tensile and part compressive.
(2) The flexural unit-stress on the lower fibre is less than the direct unit-stress; that is, is less than Then the com bined unit-stress on the lower fibre is = S, (compressive); and that on the upper fibre is = + (compressive).
The combined fibre stress is represented by Fig. 44, b, and is all compressive.
Example. A piece of timber 6 X 6 inches, and 10 feet long, is subjected to end pushes of 9,000 pounds. It is supported in a horizontal position at its ends, and sustains a middle load of 400 pounds. Compute the combined fibre stresses.
Evidently the dangerous section ie at the middle, and M = P/; that is, It occurs on the upper fibres of the middle section. The greatest value of the combined tensile stress is S, = 333 1 T 250 = 83 pounds per square inch.
It occurs on the lowest fibres of the middle section.
Change the load of the preceding illustration to a uniform load and solve.
Ans. 5 = 417 pounds per square inch.
= 83 " " " " (compression).
76. Combined Flexural and Direct Stress by fore Exact Formulas. The results in the preceding articles are only approxi mately correct. Imagine the beam represented in Fig. to be first loaded with the trans verse loads alone. They cause the beam to bend more or less, and produce certain flexural stresses at each section of the beam. Now, if end pulls are applied they tend to straighten the beam and hence diminish the flexural stresses. This effect of the end pulls was omitted in the discussion of Art. 74, and the results there given are therefore only approximate, the value of the greatest combined fibre unit-stress being too large. On the other hand, if the end forces are pushes, they in crease the bending, and therefore increase the flexural fibre stresses already caused by the transverse forces (see Fig. 45, b) . The results indicated in Art. 75 must therefore in this case also be regarded as only approximate, the value of the greatest unit fibre stress being too small.
For beams loaded in the middle or with a uniform load, the following formulas, which take into account the flexural effect of the end forces, may be used M denotes bending moment at the middle section of the beam; I denotes the moment of inertia of the middle section with respect to the neutral axis; S c, and have the same meanings as in Arts. 74 and 75, but refer always to the middle section; 1 denotes length of the beam; E is a number depending on the stiffness of the material, the average values of which are, for timber, 1,500,000; and for struc tural steel 30,000,000.* The plus sign is to be used when the end forces P are pulls, and the minus sign when they are pushes.
It must be remembered that S, and S, are flexural unit stresses. The combination of these and the direct unit-stress is made exactly as in articles 74 and 75.
Examples. 1. It is required to apply the formulas of this article to the example of article 74.
As explained in the example referred to, M = 108,000 inch pounds; 3 inches; and I = 36 Now, since l = 12 feet = 144 inches, 108,000 X = = = 8,284 pound 45,000 X 36+ 3.11 s 36f 10X 30,000,000 per square inch, as compared with 9,000 pounds per square inch, the result reached by the use of the approximate formula.
As before, = 3,750 pounds per square inch; hence = 8,284-3,750 = 4,534 pounds per square inch; and = 8,284 + 3,750 = 12,034 " " " " 2. It is required to apply the formulas of this article to the example of article 75.
As explained in that example, Ml = 12,000 inch-pounds; c, = e, = 3 inches, and I = 108 inches'.
Now, since 1 = 120 inches, 12,000 x 3 36,000 = S, S, = 362 pounds 9 000 x 8.64 108 ' 10 X 1,500,000