In order still further to simplify the equation of the section of the solid, let us remove the new origin of coordinates to the extremity of the breadth b of the middle cross section, by writing in (2), bx for x. The equation (2) then becomes

Hence it is obvious that the extremities of the variable breadth lie on a parabola which has its vertex at the extreme point of the middle breadth b.

The ground-sketch of the body is therefore a figure similar to a rhombus whose sides, however, are formed by four similar parabolas, every two of which coincide with their vertices at the extremities of the middle breadth b.

CHAPTER VIII.

THE DEFLECTION OF BEAMS.

59. WHEN a beam is subjected to a straining force of flexure, there is still another problem of which the solution is necessary that of knowing the change of form which the beam undergoes under the action of the straining force. To obtain this solution, we have only to express the moment of the resistance of the beam in terms of the radius of curvature of the curved or bent beam, and of the coefficient of elasticity -and this in fact we have already effected in general in Chapter IV. The deflection of the beam is estimated by the deflection of the neutral line of the beam which is defined in Art. 19.

60. PROP. To find the deflection of a beam having the same form and area of section throughout, supported at both ends and loaded in the middle.

We will suppose that the beam is horizontal in its original position, and the deflections so small, that the projection of the bent beam may be taken to find the moments of the external forces.

Let the horizontal line AB be the original position of the neutral line (Art. 19) of the

beam, and let this line be deflected by the weight W concentrated in the centre C, into the position AC'B. Then if M be the moment of the

Fig. 29.

straining weight W on any section P of the beam, I the moment

M

of inertia of this section about its neutral axis, p the radius of curvature of the corresponding neutral line, and E the modulus of elasticity, we have by Art. 22,

But if 7 be the length of the beam, and x, y, the coordinates of P in reference to rectangular coordinates originating at C as in the figure, then as in practice only very small flexions are admitted, we may denote the moment of flexure M about the section P (Art. 34) by

of which dy is equal to the tangent of the angle which a line

dx touching the neutral curve at P makes with AB. Wherefore as

this angle is supposed to be very small, the square of

be neglected. The equation (1) consequently becomes

dy

dx may

The radius of curvature in this case is taken negative, because the convex side of the neutral curve is presented downwards.

in (2) is constant, the integration of (2) readily

= 0,

without a constant in the second member, seeing that for x =

dy dx

is zero on account of the symmetry of the curve.

Inte

grating a second time, and remarking that when y = 0, x =

2

24 EI 2

an equation which is that of the neutral curve AC'B. The height or versed sine of arc (8), that is, CC', is obtained by taking x = 0, which gives

61. PROP. To find the deflection of the beam as in the last proposition when the weight W' (including its own weight) is uniformly distributed along the length of the beam.

Let w be the weight per unit of length uniformly distributed along the beam; then using the same notation and figure as in the last proposition, the moment of flexure about P arising from the weight lw W' is (Art. 35),

Integrating this equation twice, just as in the last proposition, we get for the curve of neutral fibres in this case the

In this equation write & for y when x = 0; then for the greatest height or versed sine of arc, we have

Cor. 1. The curve (1) deviates but very little from the parabola represented by the equation

for the difference y' — y has its maximum value corresponding to

= 11, the ratio of the difference of or

9

dinates of the two curves to the ordinate of the elastic curve.

Cor. 2. Equating the values of 8 in (4), Art. 60, and (2), Art. 61, we get

Hence the deflection produced by a weight uniformly distributed along the length of a beam resting on two supports at its

extremities, is equivalent to the deflection produced by

8

of

this weight condensed at the centre of the beam.

Cor. 3. It follows from Cor. 2 that when a beam is loaded

of the weight of the beam itself must be added