BEAMS OF UNIFORM STRENGTH.

321

law of variation depends upon the mode of variation of the moment of flexure of the beam from point to point, and this depends on the

distribution of the load and of the supporting forces, in a way which has been exemplified in Articles 289 and 290. When the depth of the beam is made uniform, and the breadth varied, the vertical longitudinal section is rectangular, and the plan is of a figure depending on the mode of variation of the breadth. When the breadth of the beam is made uniform, and the depth varied, the plan is rectangular, and the vertical longitudinal section is of a figure depending on the mode of variation of the depth. The following table gives examples of the results of those principles :

The formula and figures for a constant depth are applicable to the breadths of the flanges of the L-shaped girders described in Article 298. In applying the principles of this Article, it is to be borne in mind, that the shearing force has not yet been taken into account; and that, consequently, the figures described in the above table require, at and near the places where they taper to edges, some additional material to enable them to withstand that force. In figs. 137 and 139, such additional material is shown, disposed in the form of projections or palms at the points of support, which serve both to resist the shearing force, and to give lateral steadiness to the beams.

300. Proof Deflection of Beams.-Reverting to fig. 130, it is evident that if a represents the proportionate elongation of the layer C C', whose distance from the neutral surface O O' is y, and if r be the radius of curvature of the neutral surface, we must have 1:1 +æ : :r:r+y;

and consequently, the radius of curvature is

and the curvature, which is the reciprocal of the radius of curvature, is expressed by the equation

Let

Р be the direct stress at the layer CC, and E the modulus

f elasticity of the material; then a =

Ρ

and consequently, the cur

E

the second value being deduced from the first by means of equation 4 of Article 293.

Suppose now that the beam is under its proof load, and let M denote the greatest moment of flexure arising from that load, I, the moment of inertia of the cross section at which that moment acts, and the distance from the neutral axis of that section to the layer where the limiting intensity ƒ of the stress is attained. Then the curvature will be,

The exact integration of this equation for slender springs, in certain cases, will be considered in a subsequent Article. For beams it is integrated approximately in the following manner :

Let the middle of the neutral axis of the section of greatest stress be taken as the origin of co-ordinates, and represented by A in figs.

141 and 142. For a beam supported at both ends and symmetrically loaded, A is in the middle of the beam (fig. 141). For a beam fixed at one end and projecting, A is at the fixed end (fig. 142). Let the beam be so fixed or supported that at this point its neutral surface shall be horizontal, and let a horizontal tangent, A X C, to that surface at that point be taken as the axis of abscissæ. Let A C, the horizontal distance from the origin to one end of the beam, be denoted by c, which, as in Articles 289 and 290, is the length of the projecting portion of a beam fixed at one end, and the half-span of a beam supported at both ends and symmetrically loaded. Let A X, the abscissa of any other point in the beam Let A B D be the curved form assumed by the neutral surface when the beam is bent, which form, in a beam supported at both ends, is concave upwards, as in fig. 141, and in a beam fixed at one end concave downwards, as in fig. 142. Let X Bv be the ordinate

= x.

of any point B in the curve A BD; being the difference of level between that point and the origin A. Let CD = v, be the greatest ordinate this is what is termed the deflection.

The inclination of the beam at any point B, is expressed by the equation

and the curvature, being the rate of variation of the inclination in a given length of the curve, is expressed by

But in cases which occur in practice, the curvature of the beam is so slight, that the arc i is sensibly equal to its tangent, the slope

d v

; and the elementary arc ds is sensibly equal to its horizontal dx projection dx; so that the following equations may be used without sensible error :—

Therefore, when the curvature at each point is given by equation 2, the slope and the ordinate are to be found by two successive integrations, as shown by the following equations :

dx;

I Mo

......(4.)

dx.

0 。 I Mo

The greatest slope i—that is, the slope at D-and the deflection or greatest ordinate v1, are found by performing the complete integrations between the limits x = 0 and x = c.

[Readers who are not familiar with the integral calculus are referred to Article 81 for explanations of the nature of the process of integration.]

MIo is a

I Mo

In both the integrals of the formulæ 4, the quantity numerical ratio depending on the mode of distribution of the loading and supporting forces, and the mode of variation of the section of the beam. Hence it is evident that we must have the complete

integrals

where m" and n" are two numerical factors depending on the distribution of the forces and the figure of the beam; so that the greatest slope and the deflection are given by the equations

n" ƒc..............
Eyo

(6.)

For beams of similar figures, and similarly loaded and supported, y is as the depth, and c as the length; hence, for such beams, the greatest slope under the proof load is directly as the length, and inversely as the depth; and the proof deflection is directly as the square of the length, and inversely as the depth.

M

The following table gives the values of the factors m" and n" for some of the more ordinary cases of beams of uniform section, in Μ Ιο which the ratio being simply equal to depends on the IM,' distribution of the load alone, and may be found by the aid of the tables of Articles 289 and 290.

Mo

M

I

For a beam of uniform strength and uniform depth, the quantity

is constant; hence in every such beam, in what manner soever it may be supported and loaded, the curvature is uniform, as in the case of Example I. of the above table. For a beam of uniform strength and uniform breadth, the quantity is constant;

Mh

I

and