This section is from the book "Cyclopedia Of Architecture, Carpentry, And Building", by James C. et al. Also available from Amazon: Cyclopedia Of Architecture, Carpentry And Building.
5. Figs. a, cases 3 and 4, page 55, represent simple beams on end supports, the first bearing a middle load P, and the other a uniform load W. Figs. b are the corresponding moment diagrams. Take P and "W equal to 1,000 pounds, and I equal to 10 feet, and satisfy yourself that the diagrams are correct.
45. Maximum Bending Moment. It is sometimes desirable to know the greatest or maximum value of the bending moment in a given case. This value can always be found with certainty by constructing the moment diagram, from which the maximum value of the bending moment is evident at a glance. But in any case, it can be most readily computed if one knows the section for which the bending moment is greatest. If the student will com-pare the corresponding shear and moment diagrams which have been constructed in foregoing articles (Figs. 13 and 17, 14 and 18, 15 and 19, 16 and 20), and those which he has drawn, he will see that - The maximum bending moment in a beam occurs where the shear changes sign.
By the help of the foregoing principle we can readily com-pute the maximum moment in a given case. We have only to construct the shear line, and observe from it where the shear changes sign; then compute the bending moment for that section. If a simple beam has one or more overhanging ends, then the shear changes sign more than once - twice if there is one overhanging end, and three times if two. In such cases we compute the bending moment for each section where the shear changes sign; the largest of the values of these bending moments is the maxi-mum for the beam.
The section of maximum bending moment in a cantilever fixed at one end (as when built into a wall) is always at the wall.
Thus, without reference to the moment diagrams, it is readily seen that, for a cantilever whose length is l, with an end load P, the maximum moment is Pl, " a uniform " W, " " " ½ Wl.
Also by the principle, it is seen that, for a beam whose length is I, on end supports, with a middle load P, the maximum moment is ¼ Pl, " uniform " W, " " " " 1/8 Wl.
46. Table of Maximum Shears, Moments, etc. Table B on page 55 shows the shear and moment diagrams for eight simple cases of beams. The first two cases are built-in cantilevers; the next four, simple beams on end supports; and the last two, restrained beams built in walls at each end. In each case I denotes the length.