stress; m' h, the distance of the most severely-strained layer from the neutral axis; I, the moment of inertia of the greatest crosssection; m", n", m", n", numerical factors (see Table below). V. Greatest inclination under proof load;

VII. Greatest inclination under a given load, W;

..(5.)

.(6.)

..(7.)

IX. Given, the half-span, c, and the intended proof deflection, v1, of a proposed beam; to find the proper value of the greatest depth, ho; make

(taking " from the preceding table, and making m' ho, as before, denote the distance from the layer in which the stress is

neutral axis).

to the

X. To deduce the greatest stress in a given layer of a beam from the deflection found by experiment.

Let h be the depth of the beam at the section of greatest stress, and y the distance from the neutral axis of that section to that layer of the beam at which the greatest stress is required:

c, the half-span of a beam supported at both ends, or the length of the loaded part of a beam supported at one end; n", the factor for proof deflection, already explained;

E, the modulus of elasticity of the material;

v, the observed deflection;

then the intensity of the required stress is

XI. To find the deflection of an uniform beam produced by its own weight, or by an uniform load bearing a given proportion, 1+m, to the weight of the beam. Let be the heaviness of the

material of which the beam consists; 72 I ÷ S, the square of the radius of gyration of its cross-section; n", as before, the factor for

deflection under a given load; then, for a beam supported at both

A table of values of r2 will be given at p. 525. The application of this problem to shafts for transmitting power will be explained in the next Chapter.

440. Beam fixed at the Ends.-When a beam is not merely supported, but fixed in direction at its two ends, it bends into the form of a curve which has two points of inflection; being convex upwards at the points of support, and concave upwards in the middle. The following are the two most important cases; the cross-section of the beam being supposed uniform in both:

I. Load concentrated at middle of span. The bending moments at the points of support and at the middle of the span are equal and contrary, and each equal to half of the bending moment upon an equal and similarly loaded beam with ends merely supported; that is, M =

W c 4

II. Load uniformly distributed. The bending moment at the middle of the span is one-third, and the contrary bending moment at each point of support two-thirds, of what the bending moment in the middle of the span would be if the ends were merely supported. That is, the most severe bending moment is M

=

w c2 3

441. The Resilience of a Beam (A. M., 305) is the work performed in bending it to the proof deflection;-in other words, the energy of the greatest shock which the beam can bear without injury; such energy being expressed by the product of a weight into the height from which it must fall to produce the shock in

question. This, if the load is concentrated at or near one point, is the product of half the proof load into the proof deflection; that is to say, let P be the proof load; then the resilience is

Let W be the weight of a mass which is let fall upon the beam from the height 2. Then the whole height through which that mass falls, before the beam reaches its proof deflection, is z + v1; and the whole energy of the blow which it gives to the beam is W (+v1); which being equated to the resilience, gives the following equation:—

P v1.

2

..(2.)

an equation which enables any one of the four quantities, W, z, P, to be calculated when the other three are given.

If the load is distributed, the length of the beam is to be divided into a number of small elements, and half the proof load on each element multiplied by the distance through which that element is depressed. The integral of the products will be the resilience.

SECTION VI.-Of Resistance to Thrust or Pressure.

442. Resistance to Compression and Direct Crushing.-Resistance to longitudinal compression, when the proof stress is not exceeded, is sensibly equal to the resistance to stretching, and is expressed by the same modulus of elasticity, denoted by E (page 493). When that limit is exceeded, it becomes irregular. (See Article 420, page 493.)

The present Article has reference to direct and simple crushing only, and is limited to those cases in which the pillars, blocks, struts, or rods along which the thrust acts are not so long in proportion to their diameter as to have a sensible tendency to give way by bending sideways. Those cases comprehend

Stone and brick pillars and blocks of ordinary proportions; Pillars, rods, and struts of cast iron, in which the length is not more than five times the diameter, approximately;

Pillars, rods, and struts of wrought iron, in which the length is not more than ten times the diameter, approximately;

Pillars, rods, and struts of dry timber, in which the length is not more than about five times the diameter.

In such cases the Rules for the strength of ties (page 493) are approximately applicable, substituting thrust for tension, and using the proper modulus of resistance to direct crushing instead of the tenacity.

Blocks whose lengths are less than about once-and-a-half their diameters offer greater resistance to crushing than that given by the Rules; but in what proportion is uncertain.

The modulus of resistance to direct crushing, as the Tables show, often differs considerably from the tenacity. The nature and amount of those differences depend mainly on the modes in which the crushing takes place. These may be classed as follows:

I. Crushing by splitting (fig. 275) into a number of nearly prismatic fragments, separated by smooth surfaces whose general direction is nearly parallel to the direction of the load, is characteristic of very hard homogeneous substances, in which the resistance to direct crushing is greater than the tenacity; being in many examples about double.

IIIZ

Fig. 275.

Fig. 276.

B

Fig. 277.

Fig. 278.

II. Crushing by shearing or sliding of portions of the block along oblique surfaces of separation is characteristic of substances of a granular texture, like cast iron, and most kinds of stone and brick. Sometimes the sliding takes place at a single plane surface, like A B in fig. 276; sometimes two cones or pyramids are formed, like c, c, in fig. 277, which are forced towards each other, and split or drive outwards a number of wedges surrounding them, like w, w, in the same figure. In substances which are crushed by shearing, the resistance to crushing is always much greater than the tenacity; Sometimes the block splits into four wedges, as in fig. 278.

III. Crushing by bulging, or lateral swelling and spreading of the block which is crushed, is characteristic of ductile and tough materials, such as wrought iron. Owing to the gradual manner in which materials of this nature give way to a crushing load, it is difficult to determine their resistance to that load exactly. That resistance is in general less, and sometimes considerably less, than the tenacity. In wrought iron, the resistance to the direct crushing of pillars or struts of moderate length, as nearly as it can be 2 4

ascertained, is from to of the tenacity.

3 5

IV. Crushing by buckling or crippling is characteristic of fibrous substances, such as wood, under the action of a thrust along the fibres. It consists in a lateral bending and wrinkling of the fibres, sometimes accompanied by a splitting of them asunder.