SAFE LOADS IN TONS OF 2,000 LBS. BY FORMULA 17,100 57

CORNER COLUMNS WITHOUT COVER PLATES.

11' COLUMN REDUCED FROM 14" COLUMN.

CHAPTER XII.

BENDING-MOMENTS.

THE bending-moment of a beam or truss represents the destructive energy of the load on the beam or truss at any point for which the bending-moment is computed.

The moment of a force around any given axis is the product of the force into the perpendicular distance between the line of action of the force and the axis, or the product of the force into its arm. In a beam the forces or loads are all vertical and the arms horizontal.

The bending-moment at any cross-section of a beam is the algebraic sum of the moments of the forces tending to turn the beam around the horizontal axis passing through the centre of gravity of the section.

EXAMPLE. Suppose we have a beam with one end securely fixed into a wall, and the other end projecting from it, as in Fig. 1. Let us now suppose we have a weight, which, if placed at the end of the beam, will cause it to break

Fig. 1.

W

breaks, as shown by the dotted lines, Fig. 1.

Now, it is evident that the destructive energy of the weight is greater, the farther the weight is removed from the wall-end of the beam, though the weight itself remains the same all the time. The reason for this is, that the moment of the weight tends to turn the beam about the point A, and thus produces a pull on the upper fibres of the beam, and compresses the lower fibres. As the weight is moved out on the beam, its moment becomes greater, and hence also the pull and compression on the fibres; and, when the

moment of the weight produces a greater tension or compression on the fibres than they are capable of resisting, they fail, and the beam breaks. Before the fibres break, however, they commence to stretch, and this allows the beam to bend: hence the name "bending-moment" has been given to the moment which causes a beam to bend, and perhaps ultimately to break.

There may, of course, be several loads on a beam, and each one having a different moment, tending to bend the beam; and it may also occur that some of the weights may tend to turn the beam in different directions: the algebraic sum of their moments (calling those tending to turn the beam to the right +, and the others —) would be the bending-moment of the beam.

Knowing the bending-moment of a beam, we have only to find the section of the beam that is capable of resisting it, as is shown in the general theory of beams, Chap. XIV.

To determine the bending-moments of beams mathematically, requires considerable training in mechanics and mathematics; but, as most beams may be placed under some one of the following cases, we shall give the bending-moment for these cases, and then show how the bending-moment for any other methods of loading may be easily obtained by a scale diagram.

CASE III.

Beam fixed at one end, loaded with both a concentrated and a distributed load.

Beam supported at both ends, loaded with concentrated load at centre.

Beam supported at both ends, loaded with a distributed load W.

Beam supported at both ends, loaded with concentrated load noi

al centre.

Bending-moment

= W x mx n

L
Fig.7

L

CASE VII.

Beam supported at both ends, loaded with two equal concen trated loads, equally distant from the centre.

Bending-moment

= W × m.

W

W

Fig. 8

From these examples it will be seen that all the quantities which enter into the bending-moment are the weight, the span, and the distance of point of application of concentrated load from each end.

The bending-moment for any case other than the above may easily be obtained by the graphic method, which will now be explained.

Graphic Method of Determining Bending

Moments.

The bending-moment of a beam supported at both ends, and loaded with one concentrated load, may be shown graphically, as follows:

Let W be the weight applied, as shown. Then, by rule under Case VI., the bending

with each end of the beam. If, then, we wished to find the bendingmoment at any other point of the beam, as at o, draw the vertical line y to BC; and its length, measured to the same scale as AB, will give the bending-moment at o.

Beam with two concentrated loads.

To draw the bending-moment for a beam with two concentrated loads, first draw the dotted lines ABD and ACD, giving the outline