REMARKS.

Remembering that the bending moment at any section is equal to the area of the shear diagram up to that section, the maximum bending moment will occur at the section where the shear is zero.

Let w, be the intensity of loading at any point distant 7 from the apex of the load diagram.

The maximum bending moment occurs at the section where the shear is zero, and is equal to the area of the shear curve; hence

For another method of arriving at this result, see p. 185.

The determination of these bending moments depends on the elastic properties of the beams, which are fully discussed in Chap. XI.

In all these cases the beam is shown built in at both ends. The beams are assumed to be free endwise, and guided so that the ends shall remain horizontal as the beam is bent. If they were rigidly held at both ends, the problem would be much more complex.

My

CHAPTER XI.

DEFLECTION OF BEAMS.

Beam bent to the Arc of a Circle.-Let an elastic beam be bent to the arc of a circle, the radius of the neutral axis being p. The length of the neutral axis will not alter by the bending. The distance of the skin from the neutral axis = y.

But we have (see p. 319) the following relation :

Central Deflection of a Beam bent to the Arc of a

Circle. From the figure we have

We shall shortly give another method for arriving at this result. General Statement regarding Deflection. — In speaking of the deflection of a cantilever or beam, we always mean the deflection measured from

a line drawn tangential to that part of the bent cantilever or beam which remains parallel to its unstrained position. The deflection & will be seen by referring to the figures shown.

The point at which the tangent touches the beam we shall

term the " tangent point." When dealing with beams, we shall find it convenient to speak of the deflection at the support, i.e. the height of the support above the tangent.

Free End

Free Bnd

FIG. 415.

Free End

FreeEnd

Free End

Deflection of a Cantilever. Let the upper diagram (Fig. 416) represent the distribution of bending moment acting on the cantilever, the dark line the bent cantilever, and the straight dotted line the unstrained position of the cantilever. Consider any very small portion yy, distant /, from the free end of the cantilever. We will suppose the length yy so small that the radius of curvature p, is the same at both points, y, y.. Let the angle subtending yy be 0, (circular measure); then the angle